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(23400-2) $4.95 Paperbound unless otherwise indicated. Prices subject to change without notice. Available at your book dealer or write for free catalogues to Dept. 23, Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501. Please indicate Held of interest. Each year Dover publishes over 200 books on fine art, music, crafts and needlework, antiques, languages, literature, children's books, chess, cookery, nature, anthropology, science, mathematics, and other areas. Manufactured in the U.S.A. George Gamow THIRTY YEARS THAT SHOOK PHYSICS The Story of Quantum Theory “Dr. Gamow, physicist and gifted writer, has sketched an intriguing portrait of the scientists and dashing ideas that made the quantum revolution . . ." — Christian Science Monitor In 1900, German physicist Max Planck postulated that light, or radiant energy can exist only in the form of discrete packages or quanta. This profound insight, along with Einstein's equally momentous theories of relativity, completely revolutionized man's view of matter, energy, and the nature of physics itself. In this lucid layman's introduction to quantum theory, an eminent physicist and noted popularizer of science traces the development of quantum theory from the turn of the century to about 1980— from Planck's seminal concept (stdl developing) to anti-particles, mesons and Enrico Fermi's nuclear research. Gamow was not just a spectator at the theoretical breakthroughs which fundamentally altered our view of the universe, he was an active participant who made important contributions of his own. This “insider's" vantage point lends special validity to his careful, accessible explanations of Heisenberg's Uncertainty Principle, Niels Bohr's model of the atom, the pdot waves of Louis de Broglie and other path-breaking ideas. In addition, Gamow recounts a wealth of revealing personal anecdotes which give a warm human dimension to many giants of 20th-century physics. He ends the book with the Blegdamsvej Faust , a delightful play written in 1932 by Niels Bohr's students and colleagues to satirize die epochal developments that were revolution- izing physics. This celebrated play is available only in this volume. Written in a clear, lively style, and enhanced by 12 photographs (including candid shots of Rutherford, Bohr, Pauli, Heisenberg, Fermi and other notables), Thirty Years that Shook Physics offers both scientists and laymen a highly readable introduction to the brilliant conceptions that helped unlock many secrets of energy and matter and laid the groundwork for future discoveries. Unabridged Dover (1985) republication of the edition first published by Anchor Books, Doubleday & Co., New York, 1966. Numerous line drawings. 12 black-and- white photographic plates. Index. Notes on the text. 240pp. 5X * 89. Paperbound. ISBN 0-NAb-3Mfl c !5-X 90000 9 u.p $7.15 IN USA THIRTY YEARS THAT SHOOK PHYSICS The Story of Quantum Theory * GEORGE GAMOW Illustrations by the Author DOVER PUBLICATIONS, INC. NEW YORK Copyright © 1966 by R. Igor Gamow. All rights reserved under Pan American and Internationa) Copyright Conventions. Published in Canada by General Publishing Company, Ltd M 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd. This Dover edition, first published in 1985, is an unabridged and unaltered republication of the work first published by Doubleday & Co. Inc., New York, in 1966. Manufactured in the United States of America Dover Publications, Inc., 31 Hast 2nd Street, Mineola, N.Y. 11301 Library of Congress Cataloging in Publication Data Gamow, George, 1904-1968. Thirty years that shook physics. Reprint. Originally published: Garden City, N.Y. : Doubleday, 1966. Includes bibliographical references and index. 1. Quantum theory. 2. Physics — History. I. Title. QCI74.I2.G35 1985 530.l'2'09 85-6797 ISBN 0-486-24895-X TO THE FRIENDS OF MY YOUTH LIST OF PLATES Plate I Plate II Plate III Plate IV Plate V Plate VI Plate VII Plate VIII Plate IX Plate X Sir J. J. Thomson and Lord Rutherford Niels and Mrs. Bohr G. Gamow and W. Pauli facing page 4 facing page 32 facing page 64 W. Heisenberg and E. Fermi facing page 114 N. Bohr and A. Einstein, G. Gamow and L. Rosenfeld facing page 132 Professor Paul Ehrenfest facing page 146 Copenhagen Conference, 1930 facing page 152 Copenhagen Spring Conference, 1932 facing page 156 Solvay International Institute of Physics, Sixth Council of Physics, Brussels, 1930 facing page 214 Solvay International Institute of Physics, Seventh Council of Physics, Brussels, 1933 following Plate IX BIOGRAPHICAL PREFACE Thirty Years That Shook Physics displays Dr. Gamow’s artistic gift as well as his ability to expound science in the layman’s language. Dr. Gamow himself has ac- knowledged Sandro Botticelli as his master in portrai- ture, and for any art students who may happen upon this book it will make an interesting subsidiary exer- cise to find the Botticelli influence in the studies of Max Planck (page 6) and Niels Bohr (page 29). The fur- ther relation of Dr. Gamow’s style to the Pop Art of recent notoriety will be more readily apparent. The philosophically minded may find it significant that con- tinuity in the stream of art from the Italian Renaissance to the mid-twentieth century’s Madison Avenue should be expressed in the works of a mathematical physicist celebrated for his development of the Big Bang theory of cosmic creation. Even if they could write with comparable flavor and had equal mastery of the difficult science, few (if any) of today’s physicists could have produced a book like this one. It is a retrospective view of a crucial pe- riod of intellectual development, written by one who was there and who took active part. The great names you will find in the following pages were more than names to Dr. Gamow; these scientific giants, who re- made the universe in man’s mind, were his teachers, his friends, and his colleagues. Because of the cosmo- politan circumstances of his life, it was his great good luck to have been among those in attendance at many viil THIRTY YEARS THAT SHOOK PHYSICS of the momentous events of the thirty years that shook physics. Dr. Gamow was born on March 4, 1904, in Odessa, Russia. In early youth he turned to science and spent a year studying paleontology. This experience, he said later, equipped him “to tell a dinosaur from a cat by the shape of the little toes.’* He entered the University of Leningrad, from which he received a Ph.D. degree in 1928, and spent a year at the University of Gottin- gen, in Germany, on a traveling fellowship. In 1928- 29 he worked with Niels Bohr in Copenhagen and in 1929-30 with Ernest Rutherford at the Cavendish Laboratory, Cambridge, England. Dr. Gamow was twenty-four when he made his first major contribution to physical theory. Concurrently, but independently, he, on the one hand, and the Amer- ican physicist E. U. Condon and the British physicist R. W. Gurney, on the other, explained the emission of alpha-particles from radioactive atoms by applying to the process the then new methods of wave mechanics. Two years later, in 1930, he made the successful pre- diction that protons would be more useful than alpha- particles in die experiments popularly known as “atom- smashing," and in the same year he suggested the liquid drop model for the nuclei of heavy elements. In 1929 he collaborated with R. Atkinson and F. Houter- mans in formulating the theory that the sun's heat and light resulted from thermonuclear processes, and his theory of the origin of chemical elements through neu- tron capture dominated cosmological thinking at one period in the 1940s. He also has contributed to the fundamentals of biology, having proposed that the four nucleotides of the DNA molecule compose a code whose different combinations act as templates in the organization of the various amino acid molecules. Dr. Gamow's personal characteristics are almost as BIOGRAPHICAL PREFACE h formidable as his creative achievements. A giant, six feet three and well over 225 pounds, he is given to puckish humor, as readers of his Mr. Tompkins fan- tasies well know. When he and his student, R. Alpher, signed their names to the preliminary calculations of their paper. The Origin of Chemical Elements, in 1948, Gamow commented, “Something is missing," and, crediting Hans Bethe in absentia , made the signature “Alpher, Bethe and Gamow." He speaks six languages and is a frequent and popular lecturer with a heavily accented delivery that moved a friend to observe that the six languages were all different dialects of one lan- guage— 1 “Gamovian." Traces of Gamovian creep into his literary style from time to time, but the editor who would expunge them ruthlessly would be a pedant of the worst kind, insensitive to individual enrichment of the language. Dr. Gamow’s ability as a linguist, however accented, reflects the ground he has covered in his professional career. After his studies with Bohr and Rutherford, he returned to Russia as Master in Research at the Acad- emy of Sciences in Leningrad but left his native land for good in 1933. He lectured in Paris and London and at the University of Michigan summer school, then joined the faculty of George Washington University, Wash- ington, D.C., where he was professor of physics from 1934 to 1956. He became a United States citizen in 1940 and acted as a Navy, Army, Air Force and Atomic Energy Commission consultant during and after World War U. Since 1956 he has been on the faculty of the University of Colorado, Boulder. Dr. Gamow has written many technical papers and one technical book, Atomic Nucleus (Oxford Univer- sity Press, 1931, revised 1937 and 1949). His popular writing includes numerous Scientific American articles and the following books: X THIRTY YEARS THAT SHOOK PHYSICS Mr. Tompkins in Wonderland, Cambridge Univer- sity Press, 1939 Mr. Tompkins Explores the Atom, Cambridge Uni- versity Press, 1943 Mr. Tompkins Learns the Facts of Life, Cambridge University Press, 1953 Atomic Energy in Cosmic and Human Life, Cam- bridge University Press, 1945 The Birth and Death of the Sun, Viking Press, 1941 Biography of the Earth, Viking Press, 1943 One, Two, Three . . . Infinity, Viking Press, 1947 Creation of the Universe, Viking Press, 1952 Puzzle-Math (with M. Stem), Viking Press, 1958 The Moon, H. Schuman, 1953 Matter, Earth and Sky, Prentice-Hall, 1958 (2nd Edition, 1965) Physics: Foundation and Frontiers (with J. Cleve- land), Prentice-Hall, 1960 Atom and Its Nucleus, Prentice-Hall, 1961 Biography of Physics, Harper and Brothers, 1961 A Star Called the Sun, Viking Press, 1965 A Planet Called the Earth, Viking Press, 1965 He took up illustrating for the second Mr. Tompkins book when World War II interrupted communication between him and the English artist who had worked with him on the earlier book of the series. In 1956 he received the Kalinga Prize from UNESCO for his pop- ular interpretations of science for lay readers. Dr. Gamow was a member of the Academy of Sci- ence of the U.S.S.R. until, as he says, he was M fircd after leaving Russia." He is a member of the Royal Danish Academy of Sciences and the National Acad- emy of Sciences of the United States. John H. Durston PREFACE Two great revolutionary theories changed the face of physics in the early decades of the twentieth century: the Theory of Relativity and the Quantum Theory. The former was essentially the creation of one man, Albert Einstein, and came in two installments: the Spe- cial Theory of Relativity, published in 1905, and the General Theory of Relativity, published in 1915. Ein- stein’s Theory of Relativity called for radical changes in the classical Newtonian concept of space and time as two independent entities in the description of the physical world, and led to a unified four-dimensional world in which time is regarded as the fourth coordi- nate, though not quite equivalent to the three space coordinates. The Theory of Relativity introduced im- portant changes in the treatment of the motion of elec- trons in an atom, the motion of planets in the solar sys- tem, and the motion of stellar galaxies in the universe. The Quantum Theory, on the other hand, is the re- sult of the creative work of several great scientists start- ing with Max Planck, who was the first to introduce into physics the notion of a quantum of energy. The theory went through many evolutionary stages and gives us today a deep insight into the structure of atoms and atomic nuclei as well as that of bodies of the sizes familiar to our everyday experience. As of today Quan- tum Theory is not yet completed, especially in its rela- tion to the Theory of Relativity and the problem of elementary particles, being stalled (temporarily) by Xii THIRTY YEARS THAT SHOOK PHYSICS tremendous difficulties encountered on the way toward further development. It is the development of the Quantum Theory that this book will discuss. The author was first introduced to the idea of quanta and Bohr’s atomic model at the age of eighteen when he enrolled as a student in the University of Leningrad, and later, at the age of twenty- four, he had the good luck to become Bohr’s student in Copenhagen. During those memorable years at paa Blegdamsvef (the address of Bohr’s Institute) he had the opportunity of meeting many scientists who con- tributed to the early development of the Quantum Theory, and of taking part in their discussions. The account that follows is an outgrowth of those experi- ences, centered on the great and lovable figure of Niels Bohr. The author hopes that the new generation of physicists will find some interesting information in the pages that follow. January 1965 George Gamow CONTENTS BIOGRAPHICAL PREFACE vii PREFACE xi INTRODUCTION 1 I M. PLANCK AND LIGHT QUANTA 6 Statistical Mechanics and Thermal Radiation —Max Planck and the Quantum of Energy- Light Quanta and the Photoelectric Effect— The Compton Effect n N. BOHR AND QUANTUM ORBITS 29 Rutherford’s Theory of the Nuclear Atom- Quantizing a Mechanical System— Sommer- feld’s Elliptical Orbits— Bohr’s Institute m W. PAULI AND THE EXCLUSION PRINCIPLE 62 Quotas for Electron Levels— The Spinning Electron— Pauli and Nuclear Physics— The Neutrino IV L. DB BROGLIE AND PILOT WAVES 80 Schrttdinger’s Wave Equation— Applying Wave Mechanics V W. HEISENBERG AND THE UNCERTAINTY PRINCIPLE 98 Discarding Classical Linear Trajectories VI P. A. M. DIRAC AND ANTI-PARTICLES 118 Unifying Relativity and Quantum Theory— Anti-Particle Physics Xhr THIRTY YEARS THAT SHOOK PHYSICS VH B. FERMI AND PARTICLE TRANSFORMATIONS 139 The Forces Behind /J-Transforination— Using Fermi Interaction Laws— Fermi’s Research in Nuclear Reactions VIE H. YUKAWA AND MESONS 149 IX MEN AT WORK 154 APPENDIX BLEGDAMSVEJ FAUST 165 INDEX 219 INTRODUCTION The opening of the twentieth century heralded an un- precedented era of turnover and re-evaluation of the classical theory that had governed Physics since pre- Newtonian times. Speaking on December 14, 1900, at the meeting of the German Physical Society, Max Planck stated that paradoxes pestering the classical theory of the emission and absorption of light by material bodies could be removed if one assumed that radiant energy can exist only in the form of discrete packages. Planck called these packages light quanta. Five years later, Albert Einstein successfully applied the idea of light quanta to explain the empirical laws of photoelectric effect; that is, the emission of electrons from metallic surfaces irradiated by violet and ultraviolet light. Still later, Arthur Compton performed his classical experi- ment, which showed that the scattering of X-rays by free electrons followed the same law as the collision between two elastic spheres. Thus, within a few years the novel idea of quantization of radiant energy firmly established itself in both theoretical and experimental physics. In the year 1913, a Danish physicist, Niels Bohr, extended Planck’s idea of quantization of radiant en- ergy to the description of mechanical energy of elec- trons within an atom. Introducing specific “quantiza- tion rules” for the mechanical systems of atomic sizes, he achieved a logical interpretation of Ernest Ruther- ford’s planetary model of an atom, which rested on a 2 THIRTY YEARS THAT SHOOK PHYSICS solid experimental basis but on the other side stood in sharp contradiction to all the fundamental concepts of classical physics. Bohr calculated the energies of vari- ous discrete quantum states of atomic electrons and interpreted the emission of light as the ejection of a light quantum with energy equal to the energy differ- ence between the initial and final quantum states of an atomic electron. With his calculations he was able to explain in great detail the spectral lines of hydrogen and heavier elements, a problem which for decades had mystified the spectroscopists. Bohr’s first paper on the quantum theory of the atom led to cataclysmic de- velopments. Within a decade, due to the joint efforts of theoretical as well as experimental physicists of many lands, the optical, magnetic, and chemical properties of various atoms were understood in great detail. But as the years ran by, it became clearer and clearer that, successful as Bohr’s theory was, it was still not a final theory since it could not explain some things that were known about atoms. For example, it failed completely to describe the transition process of an electron from one quantum state to another, and there was no way of calculating the intensities of various lines in optical spectra. In 1925, a French physicist, Louis de Broglie, pub- lished a paper in which he gave a quite unexpected interpretation of Bohr quantum orbits. According to de Broglie, the motion of each electron is governed by some mysterious pilot waves, whose propagation velocity and length depend on the velocity of the elec- tron in question. Assuming that the length of these pilot waves is inversely proportional to the electron’s veloc- ity, de Broglie could show that various quantum orbits in Bohr's model of the hydrogen atom were those that could accommodate an integral number of pilot waves. Thus, the model of an atom began to look like some INTRODUCTION 3 kind of musical instrument with a basic tone (the in- nermost orbit with the lowest energy) and various overtones (outlying orbits with higher energy). One year after their publication, de Broglie’s ideas were extended and brought into more exact mathematical form by the Austrian physicist Erwin Schrddinger, whose theory became known as Wave Mechanics. While explaining all the atomic phenomena for which Bohr's theory already worked, wave mechanics also ex- plained those phenomena for which Bohr’s theory failed (such as the intensities of spectral lines, etc.), and in addition predicted some new phenomena (such as diffraction of an electron beam) which had not even been dreamed of, either in classical physics or in Planck-Bohr quantum theory. In fact, wave mechanics provided a complete and perfectly self-consistent the- ory of all atomic phenomena, and, as was shown in the late twenties, could explain also the phenomena of radioactive decay and artificial nuclear transforma- tions. Simultaneously with Schrbdinger’s paper on wave mechanics, there appeared a paper of a young Ger- man physicist, W. Heisenberg, who developed the treatment of quantum problems by using the so-called “non-commutative algebra,” a mathematical discipline in which a X b is not necessarily equal to b X a. The si- multaneous appearance of SchrSdinger’s and Heisen- berg’s papers in two different German magazines (Ann. der Phys. and Zeitsch. der Phys .) astonished the world of theoretical physics. These two papers looked as dif- ferent as they could be, but led to exactly the same results concerning atomic structure and spectra. And it took more than a year until it was found that the two theories were physically identical except for being expressed in two entirely different mathematical forms. It was as if America was discovered by Columbus, sail- 4 THIRTY YEARS THAT SHOOK PHYSICS tog westward across the Atlantic Ocean, and by some equally daring Japanese, sailing eastward across the Pacific Ocean. But there still remained one sharp thorn to the crown of the Quantum Theory, and it made itself felt pain- fully whenever one tried to quantize mechanical systems which, because of the very high velocities in- volved (close to the speed of light) required relativis- tic treatment. Many unsuccessful attempts had been made to unite the Theory of Relativity with the Theory of Quanta until finally, in 1929, a British physicist, P. A. M. Dirac, wrote his famous Relativistic Wave Equation . The solutions of this equation gave a per- fect description of the motion of atomic electrons at velocities close to that of light, and gave automatically, as an unexpected bonus, the explanation of their linear and angular mechanical momenta and magnetic mo- ments. Some formal difficulties connected with handling this equation led Dirac to suggest that along with or- dinary negatively charged electrons there must also exist positively charged anti-electrons . His prediction was brilliantly verified a few years later when anti- electrons were found to the cosmic rays. The theory of anti-particles was extended to elementary particles other than electrons, and today we have anti-protons, anti-neutrons, anti-mesons, etc. Thus, by 1930, only three decades after Planck’s momentous announcement, the Quantum Theory took the final shape with which we are now familiar. Very little theoretical progress was made to the decades that followed these breathtaking developments. On the other hand, these later years have been quite fruitful to the field of experimental studies, especially to the investi- gation of the numerous newly discovered elementary particles. We are still waiting for a breakthrough to the solid wall of difficulties which prevent us from un- INTRODUCTION 5 derstanding the very existence of elementary particles, their masses, charges, magnetic moments, and inter- actions. There is hardly any doubt that when such a breakthrough is achieved, it will involve concepts that will be as different from those of today as today’s con- cepts are different from those of classical physics. In the following chapters an attempt will be made to describe the growth of the Quantum Theory of energy and matter through the first thirty years of its turbulent development, stressing the conceptual differences be- tween “good old" classical physics and the new look physics has assumed in the twentieth century. CHAPTER I M. PLANCK AND LIGHT QUANTA The roots of Max Planck’s revolutionary statement that light can be emitted and absorbed only in the form of certain discrete energy packages goes back to much earlier studies of Ludwig Boltzmann, James Clerk Max- well, Josiah Willard Gibbs, and others on the statistical description of the thermal properties of material bod- ies. The Kinetic Theory of Heat considered heat to be the result of random motion of the numerous individual molecules of which all material bodies are formed. Since it would be impossible (and also purposeless) to follow the motion of each single individual mole- cule participating in thermal motion, the mathematical M. PLANCK AND LIGHT QUANTA 7 description of heat phenomena most necessarily use statistical method. Just as the government economist does not bother to know exactly how many acres are seeded by farmer John Doe or how many pigs he has, a physicist does not care about the position or velocity of a particular molecule of a gas which is formed by a very large number of individual molecules. All that counts here, and what is important for the economy of a country or the observed macroscopic behavior of a gas, are the averages taken over a large number of farmers or molecules. One of the basic laws of Statistical Mechanics, which is the study of the average values of physical properties for very large assemblies of individual particles in- volved in random motion, is the so-called Equiparti- tion Theorem, which can be derived mathematically from the Newtonian laws of Mechanics. It states that: The total energy contained in the assembly of a large number of individual particles exchanging energy among themselves through mutual collisions is shared equally (on the average ) by all the particles. If all par- ticles are identical, as for example in a pure gas such as oxygen or neon, all particles will have on the average equal velocities and equal kinetic energies. Writing E for the total energy available in the system, and N for the total number of particles, we can say that the aver- age energy per particle is E/N. If we have a collection of several kinds of particles, as in a mixture of two or more different gases, the more massive molecules will have die lesser velocities, so that their kinetic en- ergies (proportional to the mass and the square of the velocity) will be on the average the same as those of the lighter molecules. Consider, for example, a mixture of hydrogen and oxygen. Oxygen molecules, which are 16 times more 8 THIRTY YEARS THAT SHOOK PHYSICS massive than those of hydrogen, will have average ve- locity y/16 = 4 times smaller than the latter.f While the equipartition law governs the average dis- tribution of energy among the members of a large nru Fig. 1. Maxweir s distribution: the number of molecules having different velocities v is plotted against the velocities for three different temperatures, 100°, 400 and 1600°K. Since the number of molecules in the container remains constant, the areas under the three curves are the same. The average velocities of the molecules increase propor- tionally to the square root of the absolute temperature. t Since kinetic energy is the product of [mass] x [velocity] 2 , this product will remain the same if the mass increases by a factor 16 and velocity decreases by a factor 4. In fact, 4 s = 161 M. PLANCK AND LIGHT QUANTA 9 assembly of particles, the velocities and energies of in- dividual particles may deviate from the averages, a phenomenon known as statistical fluctuations. The fluc- tuations can also be treated mathematically, resulting in curves showing the relative number of particles hav- ing velocities greater or less than the average for any given temperature. These curves, first calculated by J. Clerk Maxwell and carrying his name, are shown in Fig. 1 for three different temperatures of the gas. The use of the statistical method in the study of thermal motion of molecules was very successful in explaining the thermal properties of material bodies, especially in the case of gases; in application to gases the theory is much simplified by the fact that gaseous molecules fly freely through space instead of being packed closely together as in liquids and solids. Statistical Mechanics and Thermal Radiation Toward the end of the nineteenth century Lord Ray- leigh and Sir James Jeans attempted to extend the sta- tistical method, so helpful in understanding thermal properties of material bodies, to the problems of thermal radiation. All heated material bodies emit elec- tromagnetic waves of different wavelengths. When the temperature is comparatively low— the boiling point of water, for example— the predominant wavelength of the emitted radiation is rather large. These waves do not affect the retina of our eye (that is, they are in- visible) but are absorbed by our skin, giving the sensa- tion of warmth, and one speaks therefore of heat or infrared radiation. When the temperature rises to about 600°C (characteristic of the heating units of an electric range) a faint red light is seen. At 2000°C (as in the filament of an electric bulb) a bright white light which contains all the wavelengths of the visible radio- 10 THIRTY YEARS THAT SHOOK PHYSICS tion spectrum from red to violet is emitted. At the still higher temperature of an electric arc, 4000°C, a con- siderable amount of invisible ultraviolet radiation is emitted, the intensity of which rapidly increases as the 3 Fig . 2. The observed distribution of radiation intensities for different frequencies v is plotted against the frequencies. Since the radiation energy content per unit volume increases as the fourth power of the absolute temperature T, the areas under the curves increase. The frequency correspond- ing to maximum intensity increases proportionally to the absolute temperature. M. PLANCK AND LIGHT QUANTA 11 temperature rises still higher. At each given tempera- ture there is one predominant vibration frequency for which the intensity is the highest, and as the tempera- ture rises this predominant frequency becomes higher and higher. The situation is represented graphically in Fig. 2, which gives the distribution of intensity in the spectra corresponding to three different temperatures. Comparing the curves in Figs. 1 and 2, we notice a remarkable qualitative similarity. While in the first case the increase of temperature moves the maximum of the curve to higher molecular velocities, in the sec- ond case the maximum moves to higher radiation fre- quencies. This similarity prompted Rayleigh and Jeans to apply to thermal radiation the same Equipartition Principle that had turned out to be so successful in the case of gas; that is, to assume that the total available energy of radiation is distributed equally among all possible vibration frequencies. This attempt led, how- ever, to catastrophic results! The trouble was that, in spite of all similarities between a gas formed by indi- vidual molecules and thermal radiation formed by elec- tromagnetic vibrations, there exists one drastic differ- ence: while the number of gas molecules in a given enclosure is always finite even though usually very large, the number of possible electromagnetic vibra- tions in the same enclosure is always infinite. To under- stand this statement, one must remember that the wave- motion pattern in a cubical enclosure, let us say, is formed by the superposition of various standing waves having their nodes on the walls of the enclosure. The situation can be visualized more easily in a sim- pler case of one-dimensional wave motion, as of a string fastened at its two ends. Since the ends of the string cannot move, the only possible vibrations are those shown in Fig. 3 and correspond in musical termi- nology to the fundamental tone and various overtones 12 THIRTY YEARS THAT SHOOK PHYSICS Fig. 3. The bade tone and higher overtones in the case of the one-dimensional continuum— for example, a violin string. of the vibrating string. There may be one half-wave on the entire length of the string, two half-waves, three half-waves, ten half-waves, ... a hundred, a thou- sand, a million, a billion ... any number of half- waves. The corresponding vibration frequencies of M. PLANCK AND LIGHT QUANTA 13 various overtones will be double, triple . . . tenfold, a hundredfold, a millionfold, a billionfold . . . etc., of the basic tone. In the case of standing waves within a three-dimen- sional container, such as a cube, the situation will be similar though somewhat more complicated, leading to unlimited numbers of different vibrations with shorter and shorter wavelengths and correspondingly higher and higher frequencies. Thus, if £ is the total amount of radiant energy available in the container, the Equi- partition Principle will lead to the conclusion that each individual vibration will be allotted E/, an infinitely small amount of energy! The paradoxicalness of this conclusion is evident, but we can point it even more sharply by the following discussion. Suppose we have a cubical container, known as “Jeans' cube,” the inner walls of which are made of ideal mirrors reflecting 100 per cent of the light falling on them. Of course, such mirrors do not exist and can- not be manufactured; even the best mirror absorbs a small fraction of the incident light. But we can use the notion of such ideal mirrors in theoretical discussions as the limiting case of very good mirrors. Such reason- ing, whereby one thinks what would be the result of an experiment in which ideal mirrors, frictionless sur- faces, weightless bars, etc., are employed, is known as a “thought experiment” ( Gedankenexperiment is the original term), and is often used in various branches of theoretical physics. If we make in a wall of Jeans* cube a small window and shine in some light, closing the ideal shutter after that operation, the light will stay in for an indefinite time, being reflected to and fro from the ideal mirror walls. When we open the shutter some- time later we will observe a flash of the escaping light. The situation here is identical in principle to pumping some gas into a closed container and letting it out again 14 THIRTY YEARS THAT SHOOK PHYSICS later. Hydrogen gas in a glass container can stay in- definitely, representing an ideal case. But hydrogen will not stay long in a container made of palladium metal, since hydrogen molecules are known to diffuse rather easily through this material. Nor can one use a glass container for keeping hydrofluoric acid, which reacts chemically with glass walls. Thus, Jeans* cube with the ideal mirror walls is after all not such a fantastic thing! There is, however, a difference between the gas and the radiation enclosed in a container. Since the mole- cules are not mathematical points but have certain finite diameters, they undergo numerous mutual col- lisions in which their energy can be exchanged. Thus, if we inject into a container some hot gas and some cool gas, mutual collisions between the molecules will rapidly slow down the fast ones and speed up the slow ones, resulting in even distribution of energy in ac- cordance to the Equipartition Principle. In the case of an ideal gas formed by point-molecules, which of course does not exist in nature, mutual collisions would be absent and the hot fraction of the gas would remain hot while the cool fraction would remain cool. The ex- change of energy between the molecules of an ideal gas can be stimulated, however, by introducing into the container one or several particles with finite though small diameters (Brownian particles). Colliding with them, fast point-sized molecules will communicate to them their energy, which will be communicated in turn to the other slower point-sized molecules. In the case of light waves the situation is different, since two light beams crossing each other’s path do not affect each other's propagation in any way.$ Thus, t To avoid an objection on the part of those readers who know much more than necessary for understanding this discussion, the author hastens to state that, according to modem quantum electrodynamics, some scattering of light by light roust be ex- M. PLANCK AND LIGHT QUANTA 15 to procure the exchange of energy between the stand- ing waves of different lengths, we must introduce into the container small bodies that can absorb and re-emit all possible wavelengths, thus permitting energy ex- change among all possible vibrations. Ordinary black bodies, such as charcoal, have this property, at least in the visible part of the spectrum, and we may imagine "ideal black bodies” which behave in the same way for all possible wavelengths. Placing into Jeans' cube a few particles of ideal coal dust, we will solve our energy-exchange problem. Now let us perform a thought experiment, injecting into an originally empty Jeans' cube a certain amount of radiation of a given wavelength— let us say some red light. Immediately after injection, the interior of the cube will contain only red standing waves extending from wall to wall, while all other modes of vibrations will be absent. It is as if one strikes on a grand piano one single key. If, as it is in practice, there is only very weak energy exchange among different strings of the instrument, the tone will continue to sound until all the energy communicated to the string will be dissi- pated by damping. If, however, there is a leak of en- ergy among the strings through the armature to which they are attached, other strings will begin to vibrate too until, according to the Equipartition Theorem, all 88 strings will have energy equal to 1/88 of the total en- ergy communicated. But if a piano is to represent a fairly good analogy of the Jeans' cube, it must have many more keys ex- tending beyond any limit to the right into the ultrasonic region (Fig. 4). Thus the energy communicated to one string in an audible region would travel to the right into the region of higher pitches and be lost in the pected because of virtual electron-pair formation. But Jeans and Planck did not know this. 16 THIRTY YEARS THAT SHOOK PHYSICS Fig. 4. A piano with an unlimited number of keys extending into the ultrasonic region all the way to infinite frequencies. The equipartition law would require all the energy supplied by a musician to one of the low-frequency keys to travel all the way into the ultrasonic region out of the audible range! infinitely far regions of the ultrasonic vibrations, and a piece of music played on such a piano would turn into a sharp shrill. Similarly the energy of red light injected into Jean r* cube would turn into blue, violet, ultraviolet, X-rays, y-rays, and so on without any limit. It would be foolhardy to sit in front of a fireplace since the red light coming from the friendly glowing cinders would quickly turn into dangerous high-frequency radiation of fission products! The runaway of energy into the high-pitch region does not represent any real danger to concert pianists, not only because the keyboard is limited on the right, but mostly because, as was mentioned before, the vi- bration of each string is damped too fast to permit a transfer of even a small part of energy to a neighboring string. In the case of radiant energy, however, the situ- M. PLANCK AND LIGHT QUANTA 17 atioo is much more serious, and, if the Equipartition Law should hold in that case, the open door of a boiler would be an excellent source of X- and y-rays. Clearly something must be wrong with the arguments of nine- teenth-century physics, and some drastic changes must be made to avoid the Ultraviolet Catastrophe, which is expected theoretically but never occurs in reality. Max Planck and the Quantum of Energy The problem of radiation-thermodynamics was solved by Max Planck, who was a 100 per cent classical physi- cist (for which he cannot be blamed). It was he who originated what is now known as modern physics. At the turn of the century, at the December 14, 1900 meeting of the German Physical Society, Planck pre- sented his ideas on the subject, which were so unusual and so grotesque that he himself could hardly believe them, even though they caused intense excitement in the audience and in the entire world of physics. Max Planck was bom in Kiel, in 1858, and later moved with his family to Munich. He attended Maxi- milian Gymnasium (high school) in Munich and, after graduation, entered the University of Munich, where he studied physics for three years. The following year he spent at the University of Berlin, where he came in contact with the great physicists of that time, Herman von Helmholtz, Gustav Kirchhoff, and Rudolph Clau- sius, and learned much about the theory of heat, tech- nically known as thermodynamics. Returning to Munich, he presented a doctoral thesis on the Second Law of Thermodynamics, receiving his Ph.D. degree in 1879, and then became an instructor at that university. Six years later he accepted the position of associate pro- fessor in Kiel. In 1889 he moved to the University of Berlin as an associate professor, becoming a full pro- 18 THIRTY YEARS THAT SHOOK PHYSICS fessor In 1892. The latter position was, at that time, the highest academic position in Germany, and Planck kept it until his retirement at the age of seventy. After retirement he continued his activities and delivered public speeches until his death at the age of almost ninety. Two of his last papers (A Scientific Autobi- ography and The Notion of Causality in Physics ) were published in 1947, the year he died. Planck was a typical German professor of his time, serious and probably pedantic, but not without a warm human feeling, which is evidenced in his cor- respondence with Arnold Sommeifeld who, following the work of Niels Bohr, was applying the Quantum Theory to the structure of the atom. Referring to the quantum as Planck's notion, Sommerfeld in a letter to him wrote: You cultivate the virgin soil, Where picking flowers was my only toil. and to this answered Planck: You picked flowers— well, so have /. Let them be, then, combined: Let us exchange our flowers fair, And in the brightest wreath them bind.§ For his scientific achievements Max Planck received many academic honors. He became a member of the Prussian Academy of Sciences in 1894, and was elected a foreign member of the Royal Society of Lon- don in 1926. Although he made no contribution to the science of astronomy, one of the newly discovered asteroids was called Planckiana in his honor. Throughout all his long life Max Planck was inter- ested almost exclusively in the problems of thermo- dynamics, and the many papers he published were im- 9 Scientific Autobiography by M. Planck. Translated by F. Gaynor. New York: Philosophical Library (1949). M. PLANCK AND LIGHT QUANTA 19 portant enough to earn him the honorable position of full professor in Berlin at the age of thirty-four. But the real outburst in his scientific work, the discovery of the quantum of energy, for which, in 1918, he was awarded the Nobel Prize, came rather late in life, at the age of forty-two. Forty-two years is not so late in the life of a man in the usual run of occupations or professions, but it usually happens that the most important work of a theoretical physicist is done at the age of about twenty-five, when he has had time to learn enough of the existing theories but while his mind is still agile enough to conceive new, bold revolutionary ideas. For example, Isaac Newton conceived the Law of Universal Gravity at the age of twenty-three; Albert Einstein created his Theory of Relativity at the age of twenty-six; and Niels Bohr published his Theory of the Atomic Structure at the age of twenty-seven. In his small way, the author of this book also published his most important work, on natural and artificial trans- formations of the atomic nucleus, when he was twenty- four. In his lecture Planck stated that according to his rather complicated calculations the paradoxical con- clusions obtained by Rayleigh and Jeans could be remedied and the danger of the Ultraviolet Catastro- phe avoided if one postulates that the energy of elec- tromagnetic waves ( including light waves) can exist only in the form of certain discrete packages, or quanta, the energy content of each package being directly pro- portional to the corresponding frequency. Theoretical considerations in the field of statistical physics are notoriously difficult, but by inspecting the graph in Fig. 5 one can get some notion of how Planck’s postulate “discourages" radiant energy from leaking into the limitless high-frequency region of the spectrum. In this graph the frequencies possible within a “one- 20 THIRTY YEARS THAT SHOOK PHYSICS ABSCISSA Fig. 5. If, according to Planck’s hypothesis, the energy cor- responding to each frequency v must be an integer of the quantity hv, the situation Is quite different from that shown in the previous diagram. For example, for v — 4 there are eight possible vibration states, whereas for v — 8 there are only four. This restriction reduces the number of possible vibrations at high frequencies and cancels Jeans’ paradox . dimensional” Jeans* cube are plotted on the abscissa axes and marked 1, 2, 3, 4, etc.; on the ordinate axes are plotted the vibration energies that can be allotted to each possible frequency. According to classical phys- ics any value of energy (that is, any point on the verti- cal lines drawn through 1, 2, 3, etc.) is permitted, the distribution resulting statistically in the Equipartition of Energy among all possible frequencies. On the other hand, Planck's postulate permits only a discrete set of energy values, equal to one, two, three, etc., energy packages corresponding to the given frequency. Since the energy contained in each package is assumed to be proportional to the frequency, we obtain the permitted energy values shown by large black dots in the diagram. M. PLANCK AND LIGHT QUANTA 21 The higher the frequency, the smaller is the number of possible energy values below any given limit, a fact which restricts the capacity of the high-frequency vi- brations to take up more additional energy. As a re- sult, the amount of energy that can be taken by high- frequency vibrations becomes finite in spite of their infinite number, and everything is dandy. It has been said that there are “lies, white lies, and statistics,** but in the case of Planck’s calculations the statistics turned out to be well-nigh true. He had ob- tained for energy distribution in thermal radiation spectrum a theoretical formula that stood in perfect agreement with the observation shown in Fig. 2. While the Rayleigh-Jeans formula shoots sky high, demanding an infinite amount of total energy, Planck’s formula comes down at high frequencies and its shape stands in perfect agreement with the observed curves. Planck’s assumption that the energy content of a radia- tion quantum is proportional to the frequency can be written as: E=hv where v (the Greek letter mi) is the frequency and h is a universal constant known as Planck’s Constant, or the quantum constant. In order to make Planck’s theo- retical curves agree with the observed ones, one has to ascribe to h a certain numerical value, which is found to be 6.77 X 10 -27 in the centimeter-gram-sec- ond unit system.H The numerical smallness of that value makes quan- tum theory of no importance for the large-scale phe- nomena which we encounter in everyday life, and it f The physical dimension of the quantum constant A is a prod- uct of energy and time, or /erg * sec/ in c.g.s. units, and is known in classical mechanics as action; action appears in many important considerations, such as the Hamilton’s Principle of Least Action. 22 THIRTY YEARS THAT SHOOK PHYSICS emerges only In the study of the processes occurring on the atomic scale. Light Quanta and the Photoelectric Effect Having let the spirit of quantum out of the bottle, Max Planck was himself scared to death of it and preferred to believe the packages of energy arise not from the properties of the light waves themselves but rather from the internal properties of atoms which can emit and absorb radiation only in certain discrete quantities. Radiation is like butter, which can be bought or re- turned to the grocery store only in quarter-pound pack- Fig. 6. Experimental Studies of the Photoelectric Effect. In (a) a primitive method for a demonstration of the photo- electric effect is illustrated. The ultraviolet radiation emitted by an electric arc ejects the electrons from a metal plate attached to an electroscope. The negatively charged leaves L, which have been repelling each other, lose charge and collapse. In ( 6 ) the modern method is shown. Ultraviolet M. PLANCK AND LIGHT QUANTA 23 ages, although the butter as such can exist in any de- sired amount (not less, though, than one molecule!). Only five years after the original Planck proposal, the light quantum was established as a physical entity exist- ing independently of the mechanism of its emission or absorption by atoms. This step was taken by Albert Einstein in an article published in 190S, the year of his first article on the Theory of Relativity. Einstein indicated that the existence of light quanta rushing freely through space represents a necessary condition for explaining empirical laws of the photoelectric ef- fect; that is, the emission of electrons from the metallic surfaces irradiated by violet or ultraviolet rays. radiation from an electric arc passes through a prism allowing only one selected frequency to fall on the plate. Turning the prism, one can select a monochromatic light and direct it to the plate. The energy of photoelectrons is measured by their ability to get through from plate to receiver, moving against the electric force produced by a potentiometer between plate and grid. 24 THIRTY YEARS THAT SHOOK PHYSICS An elementary arrangement for demonstrating pho- toelectric effect, shown In Fig. 6a, consists of a nega- tively charged ordinary electroscope with a clean metal plate P attached to it When a light from an electric arc A, which is rich in violet and ultraviolet rays, falls on the plate, one observes that the leaves L of the elec- troscope collapse as the electroscope discharges. That negative particles (electrons) are discharged from the metal plate was demonstrated repeatedly, by the Amer- ican physicist Robert Millikan (1868-1953) among others. If a glass plate, which absorbs ultraviolet radia- tion, is interposed between the arc and the metal plate, electrons are not given off, conclusive evidence that the action of the rays causes the emission. A more elaborate arrangement used for the detailed study of the laws of photoelectric effect is shown schematically in Fig. 6b. It consists of: 1. A quartz or fluoride prism (transparent for ultra- violet) and a slit permitting the selection of a mono- chromatic radiation of desired wavelength. 2. A set of rotating discs with triangular openings of various sizes, permitting a change in the intensity of radiation. 3. An evacuated container somewhat similar to the electron tubes used in radio sets. A variable electric potential is applied between the plate P, from which the photoelectrons are emitted, and the grid C. If the grid is charged negatively, and potential difference be- tween the grid and the plate is equal to or larger than the kinetic energy of photoelectrons expressed in elec- tron volts, no current will flow through the system. In the opposite case there will be a current, and its strength can be measured by the galvanometer GM. Using this arrangement, one can measure the number and the kinetic energy of electrons ejected by the inci- dent light of any given intensity and wavelength (or frequency). M. PLANCK AND LIGHT QUANTA 25 The study of photoelectric effect in different metals resulted in two simple laws: I. For light of a given frequency but varying inten- sity the energy of photoelectrons remains constant while their number increases in direct proportion to the intensity of light (Fig. 7a). II. For varying frequency of light no photoelectrons are emitted until that frequency exceeds a certain limit Vo, which is different for different metals. Beyond that frequency threshold the energy of photoelectrons increases linearly, being proportional to the difference between the frequency of the incident light and the critical frequency Vo of the metal (Fig. 7b). These well-established facts could not be explained on the basis of the classical theory of light; in some points they even contradicted it. Light is known to be short electromagnetic waves, and the increase of in- tensity of light must mean an increase of the oscillating electric and magnetic forces propagating through space. Since the electrons apparently are ejected from the metal by the action of electric force, their energy should increase with the increase of light intensity, in- stead of remaining constant as it does. Also, in the clas- sical electromagnetic theory of light, there was no rea- son to expect a linear dependence of the energy of photoelectrons on the frequency of the incident light. Using Planck’s idea of light quanta and assuming the reality of their existence as independent energy packages flying through space, Einstein was able to give a perfect explanation of both empirical laws of photoelectric effect. He visualized the elementary act of the photoelectric effect as the result of a collision between a single incident light quantum and one of the conductivity electrons carrying electric current in the metal. In this collision the light quantum vanishes, giv- ing its entire energy to the conductivity electron at the metallic surface. But, in order to cross the surface and Fig. 7. The Laws of Photoelectric Effect. In (a) the num- ber of electrons Is plotted as the function of the intensity of the incident monochromatic light. In ( b ) the energy of photoelectrons is shown as the function of the frequency of the incident monochromatic light for three different metals: A, B and C. M. PLANCK AND LIGHT QUANTA 27 to get into the free space, the electron must spend a certain amount of energy disengaging itself from the attraction of metallic ions. This energy, known by the somewhat misleading name of “work function,” is dif- ferent for different metals and is usually denoted by a symbol W. Thus, the kinetic energy K with which a photoelectron gets out of the metal is: K = h(v-v 0 )=hv-W where v 0 is the critical frequency of light below which the photoelectric effect does not occur. This picture explains at once the two laws derived from experi- ment. If the frequency of the incident light is kept con- stant, the energy content of each quantum remains the same, and the increase of light intensity results only in the corresponding increase of the number of light quanta. Thus more photoelectrons are ejected, each of them with the same energy as before. The formula giv- ing K as the function of v explains the empirical graphs shown in Fig. 7b, predicting that the slope of the line should be the same for all metals having a numerical value equal to h. This consequence of Einstein’s pic- ture of photoelectric effect stands in complete agree- ment with experiment and leaves no doubt of the real- ity of light quanta. The Compton Effect An important experiment proving the reality of light quanta was performed in 1923 by an American physi- cist, Arthur Compton, who wanted to study a collision of light quantum with an electron moving freely through space. The ideal situation would be to observe such collisions by sending a beam of light through an electron beam. Unfortunately, the number of electrons in even the most intense electron beams available is so small that one would have to wait for centuries for 28 THIRTY YEARS THAT SHOOK PHYSICS a single collision. Compton solved the difficulty by us- ing X-rays, the quanta of which carry very large amounts of energy because of the very high frequency involved. As compared with the energy carried by each X-ray quantum, the energy with which electrons are bound in the atoms of light elements can be disregarded and one can regard them (the electrons) as being un- bound and quite free. Considering a free collision be- tween light quantum and an electron in the same way as one considers a collision between two elastic balls, one would expect that the energy, and hence the fre- quency, of scattered X-rays would decrease with the increasing scattering angle. Compton’s experiments (Fig. 8) stood in complete agreement with this theo- retical prediction, and with the formula derived on the basis of conservation of energy and mechanical mo- mentum in the collision of two elastic spheres. This agreement gave additional confirmation of the exis- tence of light quanta. Fig. 8 . Compton scattering of X-rays. Notice that after the collision the wavelength of X-ray quantum Increases be- cause of loss of energy to the electron . N. BOHR AND QUANTUM ORBITS The discovery that light propagates through space and can be emitted or absorbed by matter only in the form of discrete energy packages (light quanta) whose en- ergy content is strictly defined by their vibration frequencies, had a profound influence on the contem- porary views concerning the structure of atoms them- selves. When, in 1897, J. J. Thomson proved by direct experiments that tiny negatively charged particles (electrons) can be extracted from atoms, leaving be- hind positively charged residues (ions), it became clear that atoms are not, as the Greek meaning of their name implies, indivisible constituent units of matter. 30 THIRTY YEARS THAT SHOOK PHYSICS but, quite to the contrary, rather complex systems formed by positively and negatively charged parts. Thomson visualized the atom as being formed by some positively charged substance distributed more or less uniformly through the entire body, with negatively charged electrons imbedded in it as are raisins inside a round loaf of raisin bread. The electrons are attracted to the center of the positive charge distribution and repelled by one another according to Coulomb’s law of electric interactions, and the normal state of the atom is attained when these two opposing sets of forces are in equilibrium. If an atom is disturbed (or, as physi- cists say, “excited”) by a collision with another atom or a passing free electron, its inner electrons (like strings in a grand piano) begin to vibrate around their equilibrium positions and emit a set of characteristic light frequencies, which should account for the ob- served line spectra. The atoms of different chemical elements possess different numbers of differently dis- tributed inner electrons with different characteristic frequencies, and thus differ in their observed optical spectra (Fig. 9). If Thomson’s model of an atom was accepted, it was possible, by the methods of classical mechanics, to calculate the equilibrium distri- bution of electrons within the body of an atom con- taining a given amount of inner electrons, and it was expected that the sets of calculated characteristic vi- bration frequencies would coincide with the observed line spectra of various elements. Thomson himself and his students carried out com- plicated computations to find the configurations of in- teratomic electrons for which the calculated vibration frequencies should coincide with the observed frequen- cies in the line spectra of various chemical elements. The results were disappointingly negative. The theoret- ically calculated spectra based on Thomson’s model Fig. 9. Compare the simplicity of the hydrogen spectrum , produced by the motion of only one electron, with the complexity and seeming lack of order caused by the two electrons in the spectrum of helium. Both spectra continue much farther into ultraviolet and infrared regions. 32 THIRTY YEARS THAT SHOOK PHYSICS looked not at all like the observed spectra of any of the chemical elements. It became more and more evi- dent that some revolutionary change should be made in Thomson’s classical model of the atom. This point was especially stressed by a young Danish physicist, Niels Bohr, who, after receiving his Ph.D. from the University of Copenhagen for a paper on the theory of the passage of charged particles through matter, ar- rived in 1911 at the Cavendish Laboratory of Cam- bridge University, in England, to join the group work- ing under its director, J. J. Thomson. Bohr argued that since light must no longer be treated as continuously propagating waves but instead be endowed with the mysterious additional properties of emission and ab- sorption as discrete energy packages of well-defined size, the classical Newtonian mechanics on which Thomson’s atomic model was based should be changed correspondingly. If the electromagnetic energy of light is “quantized”— that is, restricted to definite portions of one, two, three, or more light quanta (hv, 2hv, 3 hv, etc.)— isn’t it reasonable to assume that the mechanical energy of atomic electrons is quantized, too, that it can assume only a discrete set of values, the inter- mediate values being prohibited by some yet undiscov- ered law of nature? Indeed, it would be odd if atomic systems built according to the laws of classical New- tonian mechanics, as Thomson's atomic model was, should emit and absorb light in the form of Planck’s light quanta, which do not fit at all into the frame of classical physics! Rutherford’s Theory of the Nuclear Atom J. J. Thomson did not like these revolutionary ideas of the young Dane. A number of sharp arguments forced Bohr to decide to abandon Cambridge and Niels and Mrs. Bohr enjoying a motorcycle ride. (Photographed by the author ) N. BOHR AND QUANTUM ORBITS 33 spend the rest of his foreign fellowship at some place where his still vague ideas about the quantization of the electron’s motion in atoms would meet less opposi- tion. His choice was the University of Manchester, where the chair of physics was held by a New Zealand fanner’s son and former student of Thomson’s by the name of Ernest Rutherford, who later received the title of Sir Ernest and then Lord Rutherford of Nelson for his scientific discoveries. When Bohr arrived in Man- chester, Rutherford was in the midst of his epoch- making studies of the internal structure of atoms by shooting them with high-energy projectiles known as “a-partides" which were emitted by the then newly discovered radioactive elements. In earlier studies, mostly carried out at McGill University, in Canada, Rutherford had been able to prove that a-particles emitted by radioactive elements are nothing other than positively charged atoms of helium moving with tre- mendously high velocities never before encountered in physics. Emission of the a-partides from unstable heavy atoms of radioactive elements was often followed by the emission of electrons (/3-particles) and high-fre- quency electromagnetic radiation (y-rays) similar to ordinary X-rays but having much shorter wavelength. If one wants to crack something, one naturally chooses as a projectile a solid iron ball rather than a light ping- pong ball, and Rutherford figured that massive a-par- ticles would penetrate much more easily into the atomic interior than light /3-partides. The arrangement was quite simple (Fig. 10). A small amount of radioactive material, Ra for example, emitting a-particles was put at a pinhead and placed at a certain distance from a thin foil (F) made from the metal to be investigated. A thin beam of a-partides was formed after the ray passed a diaphragm (D). In passing through the foil, a-par- ticles collided with the constituent atoms and a fraction 34 THIRTY YEARS THAT SHOOK PHYSICS Fig. 10. Rutherford apparatus for studying angular de- pendence of a-scattering. of the particles were scattered in different directions on the opposite side of the foil. Falling on the fluores- cent screen (5) placed behind the foil, each a-particle produced a little spark (scintillation) at the point of impact. Observing these scintillations through a micro- scope (M), one could count the number of particles scattered at different angles from the original direction, just as in target shooting with firearms one can measure the deviation of the bullet holes from the bull’s-eye. In his experiments, Rutherford noticed that, whereas the majority of the particles passed through the foil almost without deflection, forming a luminous spot (bull’s- eye) opposite the diaphragm opening, some were scat- tered at a quite considerable angle. Somewhat different experimental arrangements have shown that in a few cases oc-particles are thrown almost backward toward the source. This observation stood in direct contradiction of what would be expected on the basis of Thomson's atomic model. By passing through the atom, the inci- N. BOHR AND QUANTUM ORBITS 33 dent a-particle can be deflected from its original track either by the electric attraction of interatomic electrons, or by the electric repulsion of the spread-out positive charge. Interaction with the electrons, which are al- most 10,000 times lighter than a-particles, certainly could not produce any noticeable deflection in a-par- ticle motion. On the other hand, the positively charged material in Thomson’s model was distributed too thinly through the entire body of the atom to be able to cause any appreciable deflection of the a-particles passing through it. Indeed, if we throw an iron ball at a piece of coal it will bounce off it at an odd angle, perhaps breaking the coal into several pieces. But, if we grind the same piece of coal into fine powder and throw the same ball through the resulting coal dust cloud, it will pass through without any deflection. The observed very large deflections in Rutherford’s scattering experiments definitely proved that the positive charge (associated with most of the mass) of the atom is not distributed all through its body, as in the previous example of coal dust cloud, but is concentrated, like the solid piece of coal, in a little hard nut— the nucleus. The experimen- tally observed dependence of the number of a-particles scattered at various directions on the angle of scat- tering stood in perfect agreement with the theoretical formula for the scattering of particles moving in a field of a repulsive central force whose strength is inversely proportional to the square of the distance. Thus was bom Rutherford’s atomic model. With its light negatively charged electrons moving through free space around a positively charged heavy nucleus in the center, it somewhat resembled the Solar System. Since Coulomb’s law of electric attraction is mathematically identical with Newton's law of gravity (both forces being inversely proportional to the square of the dis- tance), atomic electrons move around the nucleus 36 THIRTY YEARS THAT SHOOK PHYSICS along the circular or elliptic orbits, just as do planets around the Sun. But there is one great difference which lies in the fact that, whereas the Sun and the planets are elec- trically neutral, the atomic nucleus and electrons carry heavy electric charges. It is well known that the oscil- lating electric charges produce diverging electromag- netic waves. Rutherford's atomic model can be con- sidered a miniature broadcasting station operating on ultra-high frequency. Using the classical theory of elec- tromagnetic emission, one easily calculates that light waves emitted by electrons circling the atomic nucleus will take away into space all the electrons' energy within about one hundred-millionth of a second. Having lost all their energy, atomic electrons must fall into the nucleus and the atom cease to exist! Strictly speaking, similar losses of energy are ex- pected also in the case of planets of the Solar System. According to Einstein's General Theory of Relativity, the oscillation of gravitating masses also emits so- called "gravitational waves" which take away energy. But, because of the small value of Newton’s constant, planetary energy losses through gravitational emission are extremely small, and, since their formation some four or five billion years ago, the planets cannot have lost more than a few per cent of their original energy. Quantizing a Mechanical System But what to do about the atoms built according to Rutherford's model? Theoretically, as we have said, they cannot exist longer than one hundred-millionth of a second, but in reality they do exist for eternity. This was the question which confronted young Bohr upon his arrival in Manchester. Staggering contradictions of this kind between theo- N. BOHR AND QUANTUM ORBITS 37 retical expectations on the one side and observational facts or even common sense on the other are the main factors in the development of science. A. A. Michel- son’s failure to detect the motion of the Earth through a luminiferous ethert led Einstein to the formulation of the Theory of Relativity, which altered our com- mon sense notions about space and time and caused profound changes in classical physics. Similarly, the Ul- traviolet Catastrophe, discussed in the previous chap- ter, led Planck to a completely novel idea of light quanta. The theoretical impossibility of the experimentally proved Rutherford model of an atom resonated with Bohr’s hidden feeling that, if the electromagnetic en- ergy is quantized, mechanical energy must be quan- tized too, even though perhaps in a somewhat dif- ferent way. In fact, when an excited atom is emitting light quantum with the energy hv, its mechanical energy must decrease exactly by this amount. Since the atomic spectra consist of a series of discrete, sharply defined lines, the energy differences between various possible states of an atom must also have sharply defined val- ues, and so must the absolute energies of these states themselves. This leads to the idea that the atomic mechanism is somewhat similar to an automobile gear box. One can make it run in the 1st (bottom) gear, 2nd gear, 3rd gear, etc., but never in l%th gear or 3%th gear. . . . Let E it E a , E a , E a , etc., be the possible energy val- ues of different states of an atom arranged in increas- ing order (Fig. 11). An atom always has some internal energy, but when this energy has fallen to the lowest possible level, E u none of it is available for emission of a light quantum. This E\ level is the normal or t See Bernard Jaffe, Miehelson and the Speed of Light. Double- day, Science Study Series (1960). 38 THIRTY YEARS THAT SHOOK PHYSICS \ f( ObO i (obs) t \ + . ( exfect«a>) f ^S9 > 1 ^3*. . .. Fig. 11. Bohr's explanation of Rydbergs rule. ground state of the atom, in which it can exist for eter- nity. It is known as the zero point energy, and in the case of an oscillator is 1/2 hv. Suppose now that the atom is brought into an excited state with some higher energy E n . The energy can be achieved, for example, by subjecting the gas to a very high temperature, as in the atmosphere of the Sun, where die atoms are brought to excited states by violent thermal collisions among themselves. Another way to excite atoms is to pass a high-tension electric discharge through a glass tube filled with rarefied ga$4 The atoms are excited $ One must use rarefied gas in order to give the electrons suffi- ciently long time intervals between the collisions to regain energy lost in each collision by being accelerated in the applied electric field. Under normal atmospheric pressure gases do not conduct electricity, and once the tension becomes very high a sudden breakdown occurs in the form of a spark. N. BOHR AND QUANTUM ORBITS 39 by the impact of fast electrons rushing through the tube from the negative electrode (cathode) to the posi- tive one (anode). Such gadgets, in which the gas be- comes luminous on passage of the high-tension electric discharge, were originally known as Oeissler tubes af- ter their inventor, Heinrich Geissler. You see them ev- erywhere today in luminous street signs and other light- ing devices. When an atom is excited to the mth energy state E m , it can return to some lower energy state, E« (n < m) by liberating the balance of energy in the form of a light quantum. Thus we can write The two indices at v m , n indicate that this particular frequency in the spectrum corresponds to the transi- tion from the mth quantum state of motion to the nth quantum state. This picture of the light quanta emission resulting from the transition of the atom from a higher energy state to a lower one has a very interesting consequence. Suppose in the spectrum of some element we observe two lines corresponding to the transition from the 6th quantum state to the 4th and from the 4th to the 3rd (left side in Fig. 11). Then it is possible that the transi- tion can occur straight from the 6th state to the 3rd, and we find a line with a frequency V«.» = Ve.4 + V4.8 The situation shown on the right side of the same figure is the opposite one. From the fact that we observe the frequencies v St2 and v a ,s it follows that we may also observe the frequency Vg.2 — Vb,8 — V$,2 40 THIRTY YEARS THAT SHOOK PHYSICS The Swiss spectroscopist W. Ritz discovered this law of Addition and Subtraction when Niels Bohr was still a school boy. But in pre-quantum spectroscopy Ritz’s rule and other similar numerical regularities between observed frequencies were only puzzling riddles that could not be explained in a reasonable way. They be- came very helpful, however, for Niels Bohr, in his at- tempt to solve the problem of light emission and absorp- tion of atoms by introducing the idea of the discrete quantum states of atomic electrons. For his first studies Bohr selected the atom of hy- drogen, the lightest and presumably the most simply constructed atom, which was also known to possess a very simple spectrum. In 1885 a Swiss schoolteacher, J. J. Balmer, who was interested in regularities of atomic line spectra, discovered that the frequency of the visible part of hydrogen can be represented with great precision by a very simple formula. The frequen- cies of these lines, shown in Fig. 9 top (where they are plotted in terms of wavelength X = c/p), are given in the following table: ff a Vi = 4.569 X 10 M sec-*§ Hfi Vi— 6.168 x 10 14 sec -1 H y p 8 =6.908 x 10 14 sec -1 Ht v A = 7.310 X 10 M sec- 1 As the reader can himself verify, these figures can be obtained from the formula v m .n = 3.289 x 10 18 ^ — ~ 2 ^sec — 1 where m assumed the values: 3, 4, 5, 6.H For larger n * s § Sec -1 means “per second”; cm -1 means "per centimeter"; ap- ples $**i means "apples per dollar.” II The numerical coefficient in the above formula is usually denoted by R and known as the Rydberg constant, although it should be more properly called the Balmer constant N. BOHR AND QUANTUM ORBITS 41 the frequencies fall into the ultraviolet region, and the lines become more and more crowded, converging to the value 3.289 x 10 18 X \ = 6.225 X 10 14 sec" 1 In Bohr's picture of the relation between the emitted light quanta hv m , n and the energy states E m and E« (or levels) of the atom, the Balmer formula tells us that the mth line of the series is due to the transition from die mth state of an excited hydrogen atom to the second state (since 4 = 2 2 ). If instead of 4 ”2 a in Balmer's formula one substitutes and puts m = 2, 3, 4, etc., one obtains a sequence of lines which fall into the far ultraviolet region and were actually discovered by Theodore Lyman. If, on the other hand, one chooses for the first term in the Balmer !-J_ _L _1_ 9 *“ 32 ' or 16 4 2 * formula one obtains the light frequencies which fall into the far infrared region and were found by Frie- drich Paschen and Frederick Brackett, respectively. Thus the mechanical quantum states must look as is shown in Fig. 12, which also indicates the transi- tions resulting in the emission of Lyman, Balmer, Paschen, and Brackett series. Thus each line in the entire spectrum is character- ized by two indices, m and n, of the two quantum levels, between which the transition occurs (starting 42 THIRTY YEARS THAT SHOOK PHYSICS Fig. 12. Bohr's original model of the hydrogen atom. Ly- man series are in the ultraviolet, Balmer series in the visible part of the spectrum. Paschen and Brackett series are in Infrared. from with and finishing on nth). Since the energy of the light quantum is equal to the energy difference be- tween the state from which it started and the state at which it ended, the generalized Balmer formula should be rewritten as: ^)] or where the two quantities in the parentheses represent the energy levels E m and E n . The reason for writing these energies as negative quantities is that, conven- tionally, one ascribes zero energy to the state of the system when all its parts are at an infinite distance from one another. Thus, if the energy of the system is posi- tive, it will not hold together, and all its components will fly apart. In a stable system like that of the planets N. BOHR AND QUANTUM ORBITS 43 revolving around the Sun, or of electrons revolving around the atomic nucleus, the energy is negative, and it would require a supply of energy from the outside to take them apart To explain the values of energy of the different states of the hydrogen atom as given by the above for- mula, Bohr made two simplifying assumptions: 1st: That the atom of hydrogen, being the simplest atom of the entire periodic system of elements, con- tains only one electron. 2 nd: That different quantum states of the hydrogen atom correspond to the motion of that electron along the circular orbits with different radii. Making these assumptions, one should be able to find the quantum orbits of the electrons from the relation: Rh n 2 Consider the orbital motion of an electron in the nth excited state of the hydrogen atom, and write r» and v„ for the radius and the orbital velocity of the electron in the nth state. The electron’s mass is m„ and its charge —e, whereas the charge of the nucleus (proton in this case) is +e. The condition of circular motion of the electron is that the electrostatic attrac- ts tion force —-5 is balanced by the centrifugal force + OTV 2 r . From follows v = e V»*7 giving the velocity v of the electron necessary for its motion along the circle of the radius r. According to 44 THIRTY YEARS THAT SHOOK PHYSICS this equation of classical mechanics an electron can move along any circular orbit provided it has the neces- sary velocity. What is the quantum condition which selects only the orbits with the energies E n = — Rh/rfi? In the Quantum Theory of Radiation described in the previous section we stated that the vibration of a given frequency v can carry only the energy of one, two, three, or more light quanta so that E n — nhv (n = 1 , 2, 3, etc.). We can rewrite it in the form: meaning that the quantity E/v can be only an integer multiple of the quantum constant h. It may be men- tioned here that the physical dimension of h is: |action| = lenergyl |frequency| mass | ♦ [velocityl 2 |frequency| |mass| llengthl 2 |time| -1 • |time| 2 )mass| • Ggljy . [lengthl = |mass| * ]velocity| • |length|. The product of the mass of a particle by its velocity and by the distance it travels is a well-known quantity called the action , and plays an important part in classi- cal analytical mechanics. For example, “the principle of least action” formulated by a French mathematician, P. L. M. de Maupertuis in 1747, states that a particle subjected to mechanical forces will travel from point A to point B along a trajectory for which the “total action” from A to B will be either the smallest or the largest of all other possible trajectories between these two points. Planck’s Law of Light Quanta adds to Maupertius’s principle a supplementary condition that: N. BOHR AND QUANTUM ORBITS 45 The total action must always be an integer multiple of h. In the case of the closed circular trajectory of an electron around the nucleus, the quantum condition will demand that the product of the electron mass, its velocity, and the distance covered in one revolution must be an integer multiple of h. Thus, for the nth • v„ • 277 t„ = nh m, • g ■ ♦ 2 irr n = y/wi&n = lTTe\frn t \/r^~ nh r#== 4 JUm Bohr’s orbit: or We shall now calculate the total energy E* of an electron on the nth orbit, which is the sum of its kinetic energy K and potential energy U. Using the expression for velocity v = ef\Jm# given earlier, and remember- ing that the potential energy of two charges +e and —e located at the distance r apart is +eVr, we write: E n — K n + £/„ ss — + 2 mjr n £ r» - I£_£ - _i£ ” 2r, r„ 2 r* Substituting into that the expression for r m from the earlier formula, we obtain A 2 ’ n 2 which coincides with the empirical expression Rh n 2 46 THIRTY YEARS THAT SHOOK PHYSICS obtained from the Balmer formula if we put: „ _ 4w 2 e 4 /n, R ~ fp When Bohr substituted into this expression the nu- merical values e, m e , and h, he obtained R = 3.289 x 10 15 sec -1 , which is exactly its empirical value ob- tained by spectroscopic observation. Thus, the problem of quantization of a mechanical system was success- fully solved. Sommerfeld’s Elliptical Orbits The original Bohr paper on the hydrogen atom was followed soon by that of a German physicist, Arnold Sommerfeld, who extended Bohr’s ideas to the case of elliptical orbits. The motion of a particle in the field of central force is characterized in general by two (polar) coordinates, its distance r from the center of attraction and its positional angle (azimuth) in respect to the major axis of the ellipse as indicated in the figure (Fig. 13); r has the maximum value when = 0, decreases to its minimum value at = 7 r, and increases again to its maximum value at = 2ir. Thus, in contrast to Bohr’s circular orbits where r remains constant and only changes, the motion along Sommerfeld’s ellipti- cal orbits is characterized by two independent coordi- nates, r and . It follows that each quantized elliptical orbit must be characterized by two quantum numbers: azimuthal quantum number and the radial quantum number n T . Applying Bohr’s quantum conditions that the total mechanical actions for azimuthal and radial components of motion must be integer numbers n« and n r of h, Sommerfeld obtained for the energy of the quantized elliptical motion the formula: N. BOHR AND QUANTUM ORBITS 47 p Rfl Vr (n* + n r ) 2 This is exactly the same as Bohr’s original formula ex- cept that, instead of the square of an integer, the de- nominator is the square of a sum of two arbitrary integers which is, of course, an arbitrary integer itself. Putting Hr — 0 we get, as a special case, Bohr's circular orbits. If «r 9 ^ 0 we get elliptical orbits with different degrees of ellipticity. But the energies of all orbits cor- responding to the same sum h* + n r is exactly the Fig. 13. Circular and elliptical quantum orbits in the hy- drogen atom. The first circular orbit ( solid line) corre- sponds to the lowest energy of the electron. The next four orbits, one circular and three elliptical ( dashed lines) cor- respond to the same energy, which is higher than on the first orbit. The next nine orbits ( dotted lines), only four of which are shown in the figure, correspond to still higher energy ( the same for all nine). 48 THIRTY YEARS THAT SHOOK PHYSICS same in spite of their different shapes. The sum n# + n„ usually denoted simply by n, is known as the prin- cipal quantum number. It may be remarked here that relativistic treatment of the hydrogen atom gives a slightly different result because the mass of the particle according to Einstein's mechanics increases with its velocity, approaching the infinite value when the velocity approaches the velocity of light c. In fact, if m 0 is the "rest mass" of a particle (practically, its mass when it moves much more slowly than light), the mass at much higher velocity v is given by which tends to infinity when v approaches c. Since in elliptical motion the velocity varies for the different points of the trajectory (Kepler's Second Law), the mass of the electron varies too, and the calculations become more complicated. In this case, the energies of different orbits corresponding to the same principal quantum number become slightly different and a single level splits into several closely located components. Correspondingly a single spectral line, resulting from the transitions between two quantum levels character- ized by two principal quantum numbers m and n, splits into a number of components. This splitting, which can be observed only by using a spectral ana- lyzer with very high dispersive power, is known as the "fine structure” of spectral lines. The frequency differ- ences between the fine structure components depend on the so-called "fine structure constant” a given by 1 N. BOHR AND QUANTUM ORBITS 49 This quantity has no physical dimensions, being a pure number, and its smallness accounts for the closeness of fine structure components. If c were infinite, a would be zero and no fine structure at all would be observed. Another extension of Bohr's original theory came from realization that Sommerfeld’s elliptical orbit may not necessarily be in the same plane but may have dif- ferent orientation in space, making the atoms with many electrons moving along many different orbits look not like flat discs (as in the case of our Solar Sys- tem) but rather like three-dimensional bodies. Bohr's Institute Upon Bohr’s triumphant return to Denmark, the Royal Danish Academy of Science gave him financial support in order to build his own institute for atomic studies and grant fellowships to young theoretical phys- icists from all parts of the world who wanted to come to Copenhagen to work with him. Thus, there arose, at the street address Blegdamsvej 15ft, a building of the Universitetets Institut for Teoretisk Fysik, and next door to it a director's house for Bohr and his family. It may not be inappropriate to mention here that the Royal Danish Academy of Science draws its main fi- nancial support from the Carlsberg Brewery, which produces the best beer in the world. Many years ago the founder of the brewery willed the income from it to the Academy to be used for the development of sci- ence, and it was specified in his will that the palatial mansion which the old Carlsberg built for hims elf right in the middle of his brewery property was to be used as the residence of the most famous living Danish scientist When Bohr came to his fame and the previous occu- tt The Institute’s official address since has been changed to Bleg- damsvej 17. 30 THIRTY YEARS THAT SHOOK PHYSICS pant of the Carlsberg Mansion died in the early thirties, Bohr and his family moved into it. In Fig. 14 is given a sketch of a tie which was made for an anniversary of the well-known Danish biochemist Linderstr0m Lang, who for many years was the director of Carlsberg Brewery’s research laboratory, and shows a bottle of Fig. 14. Carlsberg Beer and Its consequences. N. BOHR AND QUANTUM ORBITS 51 Carlsberg beer. It is a symbol to everybody who worked on the Carlsberg fellowship in Bohr’s Institute. Bohr’s Institute quickly became the world center of quantum physics, and to paraphrase the old Romans, “all roads led to Blegdamsvej 17.” The Institute buzzed with young theoretical physicists and new ideas about atoms, atomic nuclei, and the quantum theory in gen- eral. The popularity of the institute was due both to the genius of its director and his kind, one might say fa- therly, heart. Whereas another genius of that era, Al- bert Einstein, though a very kind man too, never formed what is known as a “school” around him but worked usually with just a single assistant to talk to, Bohr fathered many scientific “children.” Almost every country in the world has physicists who proudly say: “I used to work with Bohr.” When Bohr once visited the University of Gottingen, he met a young German physicist Werner Heisenberg (see Chapter V), who, at the age of twenty-five, made important advances in the field of quantum mechanics. Bohr proposed that Heisenberg come to Copenhagen to work with him. The next day at the dinner given in Bohr’s honor at the University, two uniformed German policemen inter- rupted the meal and one of them, putting his hand on Bohr’s shoulder, announced, “You are arrested on the charge of kidnapping small children!” Of course, the “policemen” were actually two masquerading graduate students, and Bohr never went to jail; but Heisenberg went to Copenhagen! Many theoretical physicists from Europe and America went to Copenhagen for a year, two years, or more and then returned again and again in the later years: P. A. M. Dirac (see Chapter VI) and N. F. Mott (now the director of Cavendish Labo- ratory) from England; H. A. Kramers and H. Casimir from Holland; Wolfgang Pauli (Chapter III), Werner Heisenberg (Chapter V), and M. Delbriick (see Ap- 52 THIRTY YEARS THAT SHOOK PHYSICS pendix), and Carl von WeizsScker from Germany; L. Rosenfeld from Belgium; S. Rosseland from Nor- way; O. Klein from Sweden; G. Gamow and L. Landau from Russia; R. C. Tolman, J. C. Slater, and J. Robert Oppenheimer from the United States; Y. Nishina from Japan; etc. They went for a long stay, for a short visit, or just for the conferences which were held each spring. One of the most colorful visitors was Paul Ehrenfest, a professor at the University of Leyden. Ehrenfest was born in Vienna in 1880 and studied under Boltzmann, receiving his Ph.D. in 1904. In that year he married a Russian mathematician, Tatiana, and they moved to St. Petersburg (now known as Leningrad) and stayed there till 1912, when he was invited to the chair of physics at the University of Leyden. He remained there till his death in 1933. His works on statistical mechan- ics and the theory of adiabatic invariants are too ab- stract and complicated to be described in this volume, but he was an invaluable member of all scientific meet- ings because of his broad and deep knowledge of phys- ics and critical turn of mind which helped him to find holes (sometimes the wrong ones) in the newly pro- posed theory. He liked to call himself a "school- teacher,” and many of his students did very well in their future scientific careers. Once when 1$$ was traveling from Denmark to Eng- land through Holland, Ehrenfest invited me to stay in his home for a few days. He met me at the station, brought me to his home, and after showing me the guest room in which I was to sleep, said: "No smoking here. If you want to smoke go on the street.” At that time I smoked almost as much as today, so I got around It While relating his own reminiscences the author will depart from the academic convention of modesty and write in the first person. N. BOHR AND QUANTUM ORBITS 53 his regulation by puffing the cigarette smoke into the loading gates of a large Dutch stove in my room. He detested any smell except that of fresh air. One day his student Casimir (now the Scientific Director of Philips Radio Company) had an appointment with him in the afternoon. Before the meeting Cas (a shortening of Casimir, which means “cheese” in Dutch) went to a barbershop to have his hair cut and didn't notice until too late that the barber was rubbing lotion into his blond hair. He had to spend the two hours before his appointment with Ehrenfest walking the streets to let the smell of the lotion dissipate. And, of course, no- body would dare to tell Ehrenfest that Dutch Bols is better (or worse) than English gin! In an amateur play celebrated among scientists, the Blegdamsvej Faust (it is reproduced in English trans- lation in the Appendix to this book), Ehrenfest played the role of Faust whom Mephistophdes (Pauli) seduces by showing him a vision of Gretchen (the neutrino). The personality of Niels Bohr and the pleasures of life and work in his Institute are still warm in my mem- ories of the years from 1928 to his death, and I hope that a personal anecdote or two will give some of die flavor of that most remarkable man. After passing the comprehensive examination at the University of Leningrad in the spring of 1928, 1 man- aged to get permission from the Soviet Government to spend two months attending summer school at the Uni- versity of Gottingen. At that time the idea of “prole- tarian” and “capitalistic” sciences, which are supposed to be hostile to each other, had not developed in Soviet Russia, and the problem of going abroad was just a problem of getting permission to exchange so many Russian rubles into the equivalent amount of German reichsmarks. Presenting recommendations from several University professors, I managed to obtain a rather 54 THIRTY YEARS THAT SHOOK PHYSICS meager amount of German money and found myself on a ship sailing from Leningrad to German shores. Arrived at Gdttingen, I rented a typical student’s room and set myself to work. This was just two years after the discovery of wave mechanics (see Chapter IV), and everyone was busy extending Bohr’s original theory of atomic and molecu- lar structure into the new and more advanced field of wave mechanics. But I do not, and never did, like to work in crowded fields, and so decided to see whether something could be done about the structure of the atomic nucleus. At the time the nucleus was being studied experimentally, but no theory of its structure and properties had yet been attempted. During these two months in Gdttingen, I struck a gold mine, and was able to explain, on the basis of the wave mechan- ics, the spontaneous decay of radioactive nuclei as well as nuclear disintegration under the bombardment of particles shot at it from outside. As I found later, a very similar piece of work was done simultaneously by a British physicist, R. W. Gurney, in collaboration with an American physicist, E. U. Condon; in fact, our pa- pers on it were presented for publication almost on die same day. By the close of summer school in Gdttingen my money was coming to an end, and I had to leave for home. But on the way I decided to stop in Copenhagen to meet Professor N. Bohr, whose work I so admired. In Copenhagen I took the cheapest room in a shabby little hotel and went to Bohr’s Institute to see his secre- tary, Frpken (Miss) Schultz, about an appointment. (When I visited Copenhagen several years ago, just about a year before Bohr’s death, she was still on the job.) "Professor,” said she, “can see you this after- noon.” When I went into his study, I found a friendly, smiling, middle-aged man who asked me what my in- N. BOHR AND QUANTUM ORBITS 55 terests in physics were and what I was working on at the moment. So I told him about the work I had done in Gottingen on nuclear transformations, the manu- script of which had been sent in for publication but had not yet appeared. Bohr listened carefully and said: “Very interesting, very, very interesting indeed. How long are you going to stay here?” I explained that I had just enough money left for one more day. “But could you stay for a year,” asked Bohr, “if I arrange for you a Carlsberg fellowship of our Academy of Science?”§§ I gasped and finally managed to mumble, “Oh, yes, I could!” Then things moved quickly. Frpken Schultz obtained a very nice room for me in a pension run by Frpken Have, just a few blocks from the Insti- tute, which later became a “campground” for many young physicists coming to work with Bohr. The work in the Institute was very easy and simple: everybody could do whatever he wanted, and come to work and go home whenever he pleased. Another young fellow who came to stay in Frpken Have’s pension was Max DelbrUck from Germany. We both liked to sleep late in the morning, and Frpken Have devised a special method to get us up. She would come to my room and wake me up, “Dr. Gamow, you’d better get up. Dr. Delbriick has already had his breakfast and left for work!” Then she would carry out the same deception on the still sleeping DelbrUck: “Dr. DelbrUck, get up. Dr. Gamow has already left for work!” And then Max and I would collide in the bathroom. But still every- body made some progress in work, especially in the evening, which is the most inspiring time for theoretical physicists. This evening work in the Institute’s library was often interrupted by Bohr, who would say that he was very tired and would like to go to the movies. The S§ Of which I have now the honor to be a member. 56 THIRTY YEARS THAT SHOOK PHYSICS only movies he liked were wild Westerns (Hollywood style), and he always needed a couple of his students to go with him and explain the complicated plots in- volving friendly and hostile Indians, brave cowboys, and desperadoes, sheriffs, barmaidens, gold-diggers, and other characters of the Old West. But his theoreti- cal mind showed even in these movie expeditions. He developed a theory to explain why although the villain always draws first, the hero is faster and manages to kill him. This Bohr theory was based on psychology. Since the hero never shoots first, the villain has to de- cide when to draw, which impedes his action. The hero, on the other hand, acts according to a conditioned re- flex and grabs the gun automatically as soon as he sees the villain’s hand move. We disagreed with this theory, and the next day I went to a toy store and bought two guns in Western holders. We shot it out with Bohr, he playing the hero, and he "killed” all his students. Another remark of Bohr, inspired by Western mov- ies, pertained to the theory of probability. "I can be- lieve,” he said, “that a girl alone might be walking on a narrow trail somewhere in the Rockies and might lose her step, and, rolling down to the precipice, man- age to grab a tiny pine at the brink and so save herself from inevitable death. I can also imagine that, just at that time, a handsome cowboy might be riding the same trail, and, noticing the accident, tie his lasso to his horse’s saddle and descend into the precipice to save the girl. But it looks to me extremely improbable that at the same time a cameraman would be present to record this exciting event on film!” In his youth Niels Bohr was quite an athlete, and on the field of football (played in the Old World with a spherical ball kicked by the foot) was second only to his brother, the well-known mathematician Harald N. BOHR AND QUANTUM ORBITS 57 Bohr, who was the champion hall-keeper of the Copen- hagen-command. When during the Christmas vacation in 1930 I went with Bohr (who was at that time forty-five years old) to join a group of Norwegian scientists (Rosseland, Solberg, and “Old Man** Bjerknes) for skiing in the northern part of Norway beyond the polar circle, Bohr outskied all of us. One story I always like to tell or to write on the sub- ject of Bohr is about the evening in Copenhagen when Bohr, Fru (Mrs.) Bohr, the aforementioned Casimir, and I were returning from the farewell dinner given by Oscar Klein on the occasion of his election as a university professor in his native Sweden. At that late hour the streets of the city were empty (which cannot be said of Copenhagen streets today). On the way home we passed a bank building with walls of large cement blocks. At the corner of the building the crev- ices between the courses of blocks were deep enough to give a toehold to a good alpinist. Casimir, an expert climber, scrambled up almost to the third floor. When Cas came down, Bohr, inexperienced as he was, went up to match the deed. When he was hanging precari- ously on the second-floor level, and Fru Bohr, Casimir, and I were anxiously watching his progress, two Co- penhagen policemen approached from behind with their hands on their gun holsters. One of them looked up and told the other: “Oh, this is only Professor Bohr!’* and they went quietly off to hunt for more dan- gerous bank robbers. There is another amusing story illustrating Bohr’s whimsey. Above the front door of his country cottage in Tisvilde he nailed a horseshoe, which is proverbially instrumental in bringing luck. Seeing it, a visitor ex- claimed: “Being as great a scientist as you are, do you really believe that a horseshoe above the entrance to 58 THIRTY YEARS THAT SHOOK PHYSICS a home brings luck?” "No,” answered Bohr, "I cer- tainly do not believe in this superstition. But you know,” he added with a smile, "they say that it does bring luck even if you don't believe in it!” After the discovery of wave mechanics and Heisen- berg's formulation of the Uncertainty Principle, Bohr put all his energy into the semi-philosophical develop- ment of the Duality Point of View on micro-phenom- ena in physics, according to which every physical en- tity, be it a light quantum, an electron, or any other atomic particle, presents two sides of a medal. On one side it can be treated as a particle; on the other side as a wave. We will return in Chapter V to a more de- tailed discussion of this topic. Also working with his assistant, L. Rosenfeld, he extended the original un- certainty relation for a single particle to the case of the electromagnetic field, laying the foundation for a very complicated branch of die Quantum Theory known as quantum electrodynamics. In the later years, after the discovery of neutrons, Bohr became intensely interested in the then only par- tially developed theory of nuclear reactions. He showed that when a bombarding particle enters the nuclear interior it does not just kick out some of the nuclear particles as one billiard ball kicks the other, but re- mains there for a little while (about one ten-billionth of a second perhaps), distributing its impact energy among all other particles. Then this energy can be emitted in the form of y-ray quantum, or, being col- lected in some of the nuclear particles, kick them out Thus, for example, the first nuclear reaction studies by Rutherford should not be written as one used to write them: t N“ + *He« > 8 0 1T + xH 1 but rather as a three-step process: 59 N. BOHR AND QUANTUM ORBITS ,N M + 2 He 4 > sF 18 '* 1 > 9 O n + 1 H 1 In the symbols for the atomic nuclei of various chemi- cal elements the subscripts at the left are the atomic numbers of the elements; the superscripts at the right are the atomic weights of the isotopes under considera- tion. The intermediate short-lived product #F 18 (the excited nucleus of phoshorus isotope) is known as a “compound nucleus,” and the introduction of this no- tion considerably simplified the analysis of complex nuclear reactions. When I left Soviet Russia for good in 1933, I be- came a professor of physics at George Washington Uni- versity, in Washington, D.C., where, the next year, an old friend and former Bohr student. Dr. Edward Teller, joined me. Following the Copenhagen lead, the An- nual Conferences on Theoretical Physics were organ- ized under the auspices of George Washington Univer- sity and the Carnegie Institution of Washington, where Dr. Merle Tuve was carrying out important experimen- tal studies on nuclear physics. The Conference of 1939 had an especially good attendance with Niels Bohr (who was visiting the United States at that time) and Enrico Fermi (see Chapter VII) sitting in the first row. The first day of the conference passed quietly in dis- cussion of current problems, but the next day brought great excitement. Bohr came a little late that morning, carrying in his hand a radiogram from Dr. Lise Meitner in Stockholm (where she had emigrated from Nazi Germany). The message announced that her former collaborator. Professor Otto Hahn, and his co-workers in Berlin had discovered that barium and another ele- ment that turned out to be an isotope of krypton had appeared in a sample of uranium bombarded by neu- trons. She and her nephew, the theoretical physicist Otto Frisch, suggested that the experiment showed that 60 THIRTY YEARS THAT SHOOK PHYSICS a hard-bit uranium nucleus splits into two about equal parts. The reader can imagine the excitement of that day, and of the remaining days of the conference. That same night the experiment was repeated in Tuve’s labora- tory, and it was found that the fission of uranium by impact of one single neutron results in the emission of a few more new neutrons. The possibility of a branch- ing chain reaction and the large-scale liberation of nu- clear energy seemed open. With the newspaper report- ers politely shown from the conference room, the pros and cons of fission chain reaction were carefully weighed. Bohr and Fermi, armed with long pieces of chalk and standing in front of the blackboard, resem- bled two knights at a medieval tourney. Thus did nu- clear energy enter the world of man, leading to ura- nium fission bombs, nuclear reactors, and later to thermonuclear weapons! When World War II started, Bohr was in Copen- hagen, and he decided to “sit it out” through the Nazi occupation to give as much help as he could to his com- patriots. But one day he heard from the Danish under- ground that he was to be arrested by the Gestapo the next morning. That same night a Danish fisherman rowed him across the Sund to the Swedish shore, where he was picked up by a British Mosquito bomber. Mos- quito bombers were small, and the only place for Bohr was the vacant seat at the tail of the plane, usually occupied by a tail gunner. He could communicate with the cockpit only by the intercom. Somewhere over the North Sea the pilot wanted to ask how Bohr felt but couldn't get any response. Thoroughly alarmed, the pilot, on landing at the English airstrip, rushed to the tail of the plane and opened the door of the tail gun- ner's compartment. There was Bohr, quite safe and sleeping quietly! N. BOHR AND QUANTUM ORBITS 61 Coming from England to the United States, Bohr went directly to Los Alamos to further the work on the fission bomb. Due to the strict security regulations, he carried documents in the name of Nicholas Baker, and was known affectionately as Uncle Nick. There is a story that on one of his visits to Washington he met in the hotel elevator a young woman whom he had often seen in Copenhagen. She used to be the wife of a nu- clear physicist. Dr. von Halban, and often visited Co- penhagen with her husband. “Very glad to see you again, Professor Bohr,” she greeted him. “I am sorry,” said Bohr, “you must be mistaken. My name is Nicho- las Baker. But,” he added, trying to be polite without breaking the security regulations, “I do remember you. You are Mrs. von Halban.” “No,” she snapped, “I am Mrs. Placzek.” The point is that some time previously she had been divorced from her first husband and had married George Placzek, who in earlier years spent a considerable time working with Bohr. In the summer of 1960, when my wife and I were traveling in Europe, we went to Copenhagen to visit Bohr and his family. He was spending the summer in his country cottage in Tisvilde and invited us to be his guests for a few days. I found him just the same as he was when I saw him first in 1928, but, of course, much slower and less energetic. We had many interesting discussions about the difficulties in the present develop- ment of physics. It was therefore quite a shock when, some two years later, I heard on die radio that Niels Bohr had died. CHAPTER III W. PAULI AND THE EXCLUSION PRINCIPLE One of the most colorful visitors at Blegdamsvej was no doubt Wolfgang Pauli. Bom in Geffflany in 1900, he spent most of his life as a professor in Zurich, but would appear unexpectedly as a devil of inspiration wherever theoretical physics was cultivated. His reso- nating, somewhat sardonic laughter enlivened any con- ference when he appeared, no matter bow dull it was at the start. He always brought along new ideas, telling the audience about diem as be continuously walked to and fro along the lecture table, his corpulent body oscillating slightly. His mannerisms inspired someone W. PAULI AND THE EXCLUSION PRINCIPLE 63 to write a poem, of which I can recall only this frag- ment: When with colleagues he debates All his body oscillates. When a thesis he defends This vibration never ends. Dazzling theories he unveils. Bitten from his fingernails! Once, presumably on a doctor’s order, Pauli decided to lose weight and, as in everything else he attempted, succeeded very quickly. When, minus a number of pounds, he appeared again in Copenhagen, he was quite a different man: sad, humorless, and grunting instead of laughing. We all urged him to join us in delicious wiener schnitzel and good Carlsberg beer, and in less than a fortnight Pauli became again his old gay self. Politically, Pauli was anti-Nazi and would never raise his right hand in the “Heil Hitler” salute— except once. Lecturing at the University of Michigan, in Ann Arbor, he joined a gay boating party on the lake and, stepping out of the boat in the darkness, fell down, breaking his right arm at the shoulder. The arm was put into a cast with a support which held it up at a 45- degree angle. When he appeared next at his lecture he had the chalk in his left hand and was addressing students in the proper Nazi fashion. But he refused to be photographed until the cast was removed. Pauli started his scientific career very early and, at the age of twenty-one, wrote a book on the Theory of Relativity which (in the revised edition) still represents one of the best books on the subject He is famous in physics on three counts: 1. The Pauli Principle, which he preferred to call The Exclusion Principle. 2. The Pauli Neutrino, which he conceived of in 64 THIRTY YEARS THAT SHOOK PHYSICS the early twenties and which for three decades escaped experimental detection. 3. The Pauli Effect, a mysterious phenomenon which is not, and probably never will, be understood on a purely materialistic basis. It is well known that theoretical physicists cannot handle experimental equipment; it breaks whenever they touch it. Pauli was such a good theoretical physi- cist that something usually broke in the lab whenever he merely stepped across the threshold. A mysterious event that did not seem at first to be connected with Pauli’s presence once occurred in Professor J. Franck’s laboratory in Gfittingen. Early one afternoon, without apparent cause, a complicated apparatus for the study of atomic phenomena collapsed. Franck wrote hu- morously about this to Pauli at his Zurich address and, after some delay, received an answer in an envelope with a Danish stamp. Pauli wrote that he had gone to visit Bohr and at the time of the mishap in Franck’s laboratory his train was stopped for a few minutes at the Gottingen railroad station. You may believe this anecdote or not, but there are many other observations concerning the reality of the Pauli Effect! Quotas for Electron Levels The Pauli Principle, in contrast to the Pauli Effect, is much better established and pertains to the motion of electrons in atoms. In previous chapters we have de- scribed quantum orbits or, in more modern language, quantum vibration states in the Coulomb field of forces surrounding the atomic nucleus.! Since the hydrogen atom contains only one electron, this electron is free to occupy any possible energy state and, in the absence tSee the following chapter. Ir. Gamow and W. Pauli on Swiss lake steamer. ( Photographer unknown ) W. PAULI AND THE EXCLUSION PRINCIPLE 65 of excitation from outside, sits naturally in the state of lowest energy closest to the nucleus. If it is lifted to some state of higher energy by external force, it drops back to its original lowest state, emitting various lines of the hydrogen spectrum. But what happens in the atoms containing two, three, and more electrons? In Chapter II we derived two formulae for the hydrogen atom in its lowest state (n = 1). The radius of the orbit, or, more exactly, the average radius of the continuous function describing this state, is given by h 2 r 1 4w*e*m and the lowest energy by „ 4»r 2 e 4 m El jji- These formulae are obtained on the assumption that the electric force is equal to e*/r*. Suppose now that a single electron is revolving around the nucleus with the charge Ze where Z is its atomic number. In this case the force will be Ze*/r* instead of eVr 2 and in the foregoing formulae we must substitute Ze* for e* and Z*e 4 for e 4 . With the in- creasing atomic number Z the radii of the ground state will decrease as Z, and the absolute values of their energies will increase as Z*. If, instead of only one electron, we put in the Z electrons, and if they all crowd on the lowest level, the atoms forming the natu- ral system of elements will become smaller and smaller and more and more tightly bound. Of course, carrying out this argument, one must remember that electro- static repulsion between the electrons will tend to push them apart, but it can easily be shown that this repul- sion will not be strong enough to prevent the atoms of heavier elements from shrinking to a considerably 66 THIRTY YEARS THAT SHOOK PHYSICS smaller size. Thus the atomic volumes): would be ex- pected to decrease continuously, and rather rapidly, all the way from hydrogen to uranium as indicated by a broken line in Fig. ISa. The continuous line in the same figure representing experimental data does not look at all like that. It has only a very gentle slope, and is characterized mainly by its saw-shaped form, with the sharp peaks at the positions of the inert gases (He, Ne, A, Kr, Xe, etc.) which, as every chemist knows, are very reluctant to make compounds with other elements or among themselves. Also, if all the electrons of an atom were accumulated on the lowest energy level, the difficulty of extracting one electron from the atom would increase rapidly from the light to the heavy elements along the natural system (broken line in Fig. 15b). This again does not fit at all with the observed curve of the ionization potentials which char- acterizes that difficulty and is shown by a continuous line in the same figure. And, comparing the two curves, we notice that the maximum difficulty of extracting an atomic electron occurs at the same places where the atomic volumes have the smallest values. Thus, it looks as if the sequence of chemical elements can be visual- ized as a row of bodies with periodically varying sizes and resistance to the giving away of their electrons. The conclusion results that, with the addition of larger and larger numbers of electrons, the volumes occupied by various quantum states shrink, but that the number of states occupied by the electrons increases so that the total outside diameter of the atom remains approxi- mately constant. Therefore, there must exist some basic physical principle that prevents all the atomic electrons from crowding into the lowest quantum state; as soon t These can be calculated from the known atomic weights and densities of various elements, dividing the weight of 1 cm 8 of a given element by the weight of its atoms. W. PAULI AND THE EXCLUSION PRINCIPLE 67 4 . Fig. 15. The changes of atomic volumes and ionization potentials along the natural system of elements. Black cir- cles correspond to noble gases which, being saturated shells, have the highest binding. The black circles are alkali metals which begin to build new shells. as the “quota" for a given level is filled up, extra elec- trons must be accommodated on the other quantum states with higher energy. Pauli suggested that things can be settled satisfactorily if one permits only two electrons to occupy any given quantum state described by three quantum numbers: radial n r , azimuthal n*. 68 THIRTY YEARS THAT SHOOK PHYSICS and orientational «<>.§ In the original Bohr theory, in the course of which the Pauli Principle was first formu- lated, these three quantum numbers corresponded to the average diameters, eccentricities, and space orien- tations of the electron’s quantum orbits. In wave me- chanicsfl they represent the number of nodes in the complicated three-dimensional oscillatory motion of the ip functions. Using the Pauli Principle, Bohr and his co-workers (including, of course, Pauli himself) were able to construct the models of all atoms from hydrogen to uranium. Not only did they explain the periodic changes of atomic volumes and ionization potentials, but also they explained all other properties of atoms, their chemical affinities toward one another, their va- lencies, and other properties which many years earlier were summarized on a purely empirical basis and sys- tematized by the Russian chemist D. I. Mendeleev, in his periodic system of elements. All these developments fall outside the scope of this little book, the main pur- pose of which is to describe the revolutionary new ideas rather than their detailed consequences. The Spinning Electron The studies and interpretation of atomic spectra on the basis of Bohr’s theory, which relied on three quan- tum numbers (quite natural for three-dimensional space!) to describe the motion of atomic electrons, went happily ahead until, in the early twenties, three 9 We deviate here from conventional notations of quantum numbers for the sake of simplicity. In any branch of science the terminology becomes so cumbersome in the process of its prog- ress that it is very difficult to put it in a simple way for a reader who encounters all these complicated notions for the first time. 1 See the following chapter. W. PAULI AND THE EXCLUSION PRINCIPLE 69 quantum numbers suddenly turned out to be insuffi- cient. Studies of the Zeeman Effect (the splitting of spectral lines by strong magnetic fields) revealed that there are more components than three integers could account for, and to explain their existence a fourth quantum number was introduced. It was first called the “inner quantum number,” a name as good as any because nobody could account for the surplus splitting. Then, in 1925, two Dutch physicists, Samuel Goud- smit and George Uhlenbeck, made a bold proposal. This surplus line-splitting, they suggested, is not due to any additional quantum number describing the elec- tron’s orbit in the atom but to the electron itself. Ever since its discovery, the electron was regarded as a point characterized only by its mass and by its electric charge. Why cannot one think of it as a small electrically charged body rotating, like a top, about its axis? It would have a certain angular momentum, and a mag- netic moment, as any rotating charge has. A different orientation of the electron’s spin (as it was called) in respect to the plane of its orbit would account then for the additional components in line splitting. It was soon discovered that the proposal works and that, by ascribing to an electron a proper numerical value of spin (that is, angular mechanical momen- tum) and magnetic moment, one could explain all additional line components found by the experimental- ists. The magnetic moment of the spinning electron obligingly came out equal to the so-called Bohr’s mag- neton — that is, the minimum amount of magnetism which could be caused by its revolution around the nucleus. But then trouble appeared with the mechani- cal angular momentum of the spinning electron, which turned out to be only one-half of the regular angular momentum h/2n of atomic orbits. Many attempts were made to straighten out this dif- 70 THIRTY YEARS THAT SHOOK PHYSICS ficulty, and the problem was finally solved by P. A. M. Dirac, in a very unconventional way, only four years later (see Chapter VI). The reason why the introduc- tion of the spinning electron modified the Pauli Prin- ciple of atomic electron motion can be understood in the following way. As you will remember, this principle stated that only two electrons can occupy any given quantum orbit. Why two? After the discovery of the spinning electron, the original Pauli Principle was amended by the statement: “only two which possess opposite spin"; that is, are rotating in opposite direc- tions. The situation is illustrated graphically in Fig. 16 . Drawing (a) represents the old picture in which two point electrons, e t and e 2 , move along the same orbit. In drawing (b) we have the more recent presentation that two electrons can move along the same orbit only if one of them (e t ) rotates around its axis in the same direction as it revolves around the nucleus, while an- other (e s ) rotates in the opposite direction. It may be added that drawing (b) is not quite correct since the interaction between the magnetic moment of an elec- tron and the magnetic field within the atom in which it moves changes the orbit slightly so that we actually have two orbits with only one electron on each, (c) Thus the original Pauli Principle can be reformulated by permitting only one electron on each orbit if one takes into account the slight splitting of the original orbit. Pauli and Nuclear Physics We turn now to an entirely different field of Pauli’s activity in science: his contribution to the field of nu- clear physics. As everybody knows— or at least should know— radioactive elements emit three kinds of radia- ( 1 ) f) Flg . ,6. The according to (<0 tbe L y the same ° rb ^' ing the same l:# fnust have opp 4 n ^j according forces ^2,C£S - •* - " energy Jevel * 72 THIRTY YEARS THAT SHOOK PHYSICS tion: alpha (a), beta (/3), and gamma (y). The prin- cipal process of radioactive decay is the emission of a-particles, large chunks of unstable nuclei which were proved by Rutherford to be the nuclei of helium atoms. On the other hand, /3-particles are electrons which are sometimes emitted by nuclei following a-decay to re- store the balance between the charge and mass upset by ejection of an a-particle. Finally, y-rays are short electromagnetic waves resulting from the inner distur- bances caused by a- and /3-emission. For a given ra- dioactive element a-particles have exactly the same energy corresponding to the energy differences of the mother and daughter nuclei. The y-rays show com- plex sharp lines, much sharper, in fact, than the lines of optical spectra. All this activity indicates that atomic nuclei are quantized systems similar to atoms except that they are much smaller; since the nuclei are smaller their transformations, according to the quantum laws, in- volve much higher energies. But the physicists got a big surprise when James Chadwick discovered in 1914 that, in contrast to a-particles and y-rays emitted by radioactive nuclei, the /3-particles do not have well- defined energies. Quite to the contrary, their energy spectrum extends continuously from practically zero to very large values (Fig. 17). The possibility that this energy spread was due to some internal losses suffered by /3-particles in the process of escape from radioactive material was disproved definitely by careful experi- ments. Thus, one faced a situation in which the nuclear ledgers of income and expenditure of energy did not balance. Niels Bohr, who was first aroused by this ex- perimental finding, took the radical point of view that, if the experiments say so, the Law of Conservation of Energy really does not hold for /8-emission or (pre- sumably) for the /3-absorption processes. Indeed, this W. PAULI AND THE EXCLUSION PRINCIPLE 73 Fig. 17. Typical energy distribution curve of a typical /}• emitter . was the era when so many laws of classical physics were rejected under the impact of the newly developed Theory of Relativity and the Quantum Theory that no law of classical physics seemed to be unshakable. Bohr even tried to use this alleged non-conservation of energy in /3-decay processes to explain the seemingly eternal production of energy in stars. According to these little-known and never published views, stars con- tained in their interior large cores of nuclear matter having the same properties as ordinary atomic nuclei but being much larger (many kilometers rather than 10 -12 cm in diameter). These stellar cores, which were expected to be unstable, were emitting j3-particles of well-defined energy. They were surrounded by ordinary matter in a completely ionized state (plasma, we call it today) consisting of free high-energy electrons and ordinary bare nuclei. The energy of the electrons form- ing the base of these stellar envelopes was determined by the classical relation c=f*r where k is Boltzmann's constant and T the tempera- 74 THIRTY YEARS THAT SHOOK PHYSICS ture at the base of the envelope.tt On the other hand, the /3-particle emitted from the practically flat surface of the nuclear core always had the same energy de- termined by the inner properties of the nuclear fluid. Thus, there must have existed a dynamical equilibrium between the nuclear core and the surrounding ionized gas (plasma) similar to the equilibrium between the water and the saturated vapor above it. The number of /8-particles emitted by the radioactive core was equal to the number of free electrons from the envelope ab- sorbed by it But, whereas the energy of the absorbed free electrons from the envelope was determined by its temperature T, the energy of /3-particles emitted by the core was always the same, corresponding to a certain universal nuclear temperature, T 0 . Therefore, for T < T 0 there was a constant energy flow from the nuclear core into the envelope, and this flow, rising to the star’s surface, maintained its high temperature. By virtue of non-conservation of energy in /8-emission processes nothing changed in the nuclear core and the stars could shine eternally. Bohr spoke about this the- ory of his in a slightly critical fashion, but it looked as if he would not be greatly surprised if it were true. The Neutrino Pauli, who could not be called conservative in any sense of the word, was nevertheless strongly opposed to Bohr’s view. He preferred to assume that the bal- tt According to the mechanical theory of heat developed by Boltzmann and Maxwell in the middle of the last century, “Heat is nothing but the motion of molecules forming material bodies." They found that the energy of thermal motion (per molecule) is proportional to its absolute temperature— that is, the temperature counted from “absolute zero" at — 273 °C. The empirically determined coefficient of proportionality (or rather two-thirds of it) was called the Boltzmann constant. W. PAULI AND THE EXCLUSION PRINCIPLE 75 ance of energy violated by the continuity of /3-ray spectra was re-established by the emission of some other kinds of yet unknown particles which he called "neutrons.” The name of this “Pauli neutron” was later changed to "neutrino” after Chadwick's discovery of what today we call the neutron. Neutrinos were sup- posed to be particles carrying no electric charge and having no mass (or, at least, no mass to speak of). They were supposed to be emitted, being paired with /3-particles in such a way that the sum of their and the /3-particles' energies was always the same, which would of course re-establish the good old law of Con- servation of Energy. But, due to their zero-charge and zero-mass, they were practically undetectable, slipping between the fingers of the most skillful experimental- ists. Besides Bohr, another neutrinophobe was P. Eh- renfest, and heated verbal discussions and voluminous but never-published correspondence on the subject were exchanged among the three of them. As the years passed, more and more evidence was accumulated in favor of Pauli's neutrinos even though this evidence was circumstantial. It was not until 1955 that two Los Alamos physicists, F. Reines and C. Cowan, established beyond any doubt the existence of neutrinos by trapping them when they were escaping from the atomic piles of the Savannah River Atomic Energy Commission project. They found that the in- teraction between neutrinos and matter was so small that an iron shield several light years thick would be needed to reduce the intensity of the neutrino’s beam by one-half. Today neutrinos have a larger and larger place in the study of elementary particles and astro- physical phenomena; they may become the most im- portant elementary particles in physics. Like electrons, neutrinos were found to behave as little spinning tops, and their angular momenta are exactly equal to an 76 THIRTY YEARS THAT SHOOK PHYSICS O’* Totau SatTuvaTftA cJtWn/c sK»Uf Fig, 18, A comparison between (a) the saturation of electron shells in Bohr-Coster diagram of the sequence of atoms and (b) Mayer-J ensen’s diagram of the saturation of proton and neutron shells in the sequence of atomic nuclei. W. PAULI AND THE EXCLUSION PRINCIPLE 77 electron's. But since neutrinos carry no electric charge, their magnetic moment is equal to zero. It was later found experimentally that protons and neutrons also have the sam& spin as electrons and also obey the Pauli Principle. The latter fact is of great im- portance in the problem of internal structure of atomic nuclei, which are formed by an agglomeration of vari- ous numbers of protons and neutrons tightly bound to- gether by nuclear forces. As was first shown by G. 78 THIRTY YEARS THAT SHOOK PHYSICS Gamow in 1934, the natural sequence of atomic nuclei from hydrogen to uranium isotopes shows periodic changes in their various properties, similar to but much smaller than the changes of chemical properties of atoms in Mendeleev’s periodic system of elements. This periodicity indicated that atomic nuclei must have a shell structure similar to but probably more compli- cated than the shell structure of the atomic electron envelopes. The situation here is complicated by the fact that, whereas atomic envelopes are formed by only one kind of particle, namely electrons, the nuclei are formed by two kinds of particles, neutrons and protons, and that Pauli's Exclusion Principle applies to each kind separately. Thus, any given energy state charac- terized by three quantum numbers can accommodate two protons (with opposite spin) along with two neu- trons (also with opposite spin), and we actually have two systems of shells, one for protons and one for neu- trons, overlapping on one another. There is another difficulty. Because of the close packing of protons and neutrons in the nucleus, the calculations of energy levels become considerably more complicated. The problem was finally solved in 1949 by M. Goeppert Mayer and H. Jensen et al., who were able to prove that the neutron as well as the proton shells inside the nuclei have capacities of 2, 8, 14, 20, 28, 50, 82, and 126 particles each, as is shown schematically in Fig. 18. These numbers, known as "Magic Numbers,” per- mitted physicists to understand completely the observed periodicity in nuclear structure. Another important application of Pauli’s Principle can be found in the work of P. A. M. Dirac, who used it for the explanation of the stability of matter, as will be described in Chapter VI. On the basis of his theory, Dirac was led to the conclusion that to each "normal particle,” such as an electron, proton, neutron, and W. PAULI AND THE EXCLUSION PRINCIPLE 79 the hordes of other particles discovered during the last decade, there must exist an “anti-particle” with ex- actly the same physical properties but the opposite electric charge. This will be discussed in more detail in Chapters VI and VIII. To finish this present chapter, it is enough to say that it is just as difficult to find the branch of modern physics in which the Pauli Principle is not used as to find a man as gifted, amiable, and amming as Wolf- gang Pauli was. CHAPTER IV L. DE BROGLIE AND PILOT WAVES Louis Victor, Due de Broglie, bom in Dieppe in 1892, who became the Prince de Broglie on the death of an elder brother, had a rather unusual scientific career. As a student at the Sorbonne he decided to devote his life to medieval history, but the onset of World War I induced him to enlist in the French Army. Being an educated man, he got a position in one of the field radio-communication units, a novelty at that time, and turned his interest from Gothic cathedrals to electro- magnetic waves. In 1925 he presented a doctoral thesis which contained such revolutionary ideas concerning a modification of the Bohr original theory of atomic L. DE BROGLIE AND PILOT WAVES 81 structure that most physicists were rather skeptical; some wit, in fact, dubbed de Broglie’s theory "la Comidie Frangaise .** Having worked with radio waves during the war, and being a connoisseur of chamber music, de Broglie chose to look at an atom as some kind of musical in- strument which, depending on die way it is constructed, can emit a certain basic tone and a sequence of over- tones. Since by that time Bohr’s electronic orbits were fairly well established as characterizing different quan- tum states of an atom, he chose them as a basic pat- tern for his wave scheme. He imagined that each elec- tron moving along a given orbit is accompanied by some mysterious pilot waves (now known as de Broglie waves) spreading out all along the orbit The first quantum orbit carried only one wave, the second two waves, the third three, etc. Thus the length of the first wave must be equal to the length 2irri of the first quan- tum orbit the length of the second wave must be equal to one-half of the length of the second orbit i*2wr 2 , etc. In general, the nth quantum orbit carries n waves with the length £ 2 wr„ each. As we have seen in Chapter n, the radius of the nth orbit in Bohr’s atom is r "“ 4w® me* * From the equality of the centrifugal force due to the orbital motion, and the electrostatic attraction beween the charged particles, we obtain: mv B 2 _ e 2 r« or e 2 = mv n 2 r» 82 THIRTY YEARS THAT SHOOK PHYSICS Substituting this value of e 2 into the original formula, we get or 1 h 2 n 2 1 r " 4? r 2 m mv a 2 r n (2irr a ) 2 = h 2 n 2 mV Extracting the square root from both sides of this equa- tion we finally obtain: 2 irr n = it • h mv« Thus, if the length X of the wave accompanying an elec- tron is equal to Planck’s constant h divided by the me- chanical momentum mv of the particle, then and de Broglie could satisfy his desire to introduce waves of such a nature that 1, 2, 3, etc., of them would fit exactly into the 1st, 2nd, 3rd of Bohr’s quantum orbits (Fig. 19). The result given is mathematically equivalent to Bohr’s original quantum condition and brings in nothing physically new— nothing, that is, but an idea : the motion of the electrons along Bohr’s quan- tum orbits is accompanied by mysterious waves of the lengths determined by the mass and the velocity of the moving particles. If these waves represented some kind of physical reality, they should also accompany par- ticles moving freely through space, in which case their existence or non-existence could be checked by direct experiment. In fact, if the motion of electrons is al- ways guided by de Broglie waves, a beam of electrons under proper conditions should show diffraction phe- nomena similar to those characteristic of beams of light. L. DE BROGLIE AND PILOT WAVES 83 3 WAVES Fig. 19. De Broglie waves fitted to quantum orbits in Bohr’s atom model. Electron beams accelerated by electron tensions of sev- eral kilovolts (which are commonly used in laboratory experiments) should, according to de Broglie’s for- mula, be accompanied by pilot waves of about 10~ 8 cm wavelength, which is comparable to the wavelength of ordinary X-rays. This wavelength is too short to show a diffraction in ordinary optical gratings and should be studied with the technique of standard X-ray spectroscopy. In this method the incident beam is re- flected from the surface of a crystal, and the neighbor- ing crystalline layers, located about 10~ 8 cm apart, have the function of the more widely separated lines in op- tical diffraction gratings (Fig. 20). This experiment was carried out simultaneously and independently by 84 THIRTY YEARS THAT SHOOK PHYSICS Fig. 20. An incident wave, be it a short electromagnetic wave (X-ray) or a de Broglie wave associated with a beam of fast electrons, produces wavelets as it passes through the successive layers of a crystal lattice. Depending on the angle of incidence, dark and light interference fringes appear. ( P is the phase plane.) Sir George Thomson (son of Sir J. J. Thomson) in England, and G. Davisson and L. H. Germer in the United States, who used a crystal arrangement similar to that of Bragg and Bragg, but substituted for the beam of X-rays a beam of electrons moving at a given velocity. In the experiments a characteristic diffraction pattern appeared on the screen (or photographic plate) that was placed in the way of the reflected beam, and the diffraction bands widened or narrowed when the velocity of incident electrons was increased or de- creased. The measured wavelength coincided exactly L. DC BROGUE AND PILOT WAVES 85 in all cases with that given by the de Broglie formula. Thus the de Broglie waves became an indisputable physical reality, although nobody understood what they were. Later on a German physicist, Otto Stem, proved the existence of the diffraction phenomena in the case of atomic beams. Since atoms are thousands of times more massive than electrons, their de Broglie waves were expected to be correspondingly shorter for the same velocity. To make atomic de Broglie waves of a length comparable with the distances between the crystalline layers (about 10~ 8 cm). Stem decided to use the ther- mal motion of atoms, since he could regulate the veloc- ity simply by changing the temperature of the gas. The source consisted of a ceramic cylinder heated by an electric wire wound around it. At one end of the closed cylinder was a tiny hole through which the atoms es- caped at their thermal velocity into a much larger evac- uated container, and in their flight through space they hit a crystal placed in their way. The atoms reflected in different directions stuck to metal plates cooled by liquid air, and the number of atoms on the different plates was counted by a complicated method of chemi- cal microanalysis. Plotting the number of atoms scat- tered in different directions against the scattering angle, Stem obtained again a perfect diffraction pattern cor- responding exactly to the wavelength calculated from de Broglie's formula. And the bands became wider or thinner when the temperature of the cylinder was changed. When in the late twenties I was working at Cam- bridge University with Rutherford, I decided to spend Christmas vacation in Paris (where I had never been before) and wrote to de Broglie, saying that I would like very much to meet him and to discuss some prob- lems of the Quantum Theory. He answered that the 86 THIRTY YEARS THAT SHOOK PHYSICS University would be closed but that he would be glad to see me in his home. He lived in a magnificent fam- ily mansion in the fashionable Parisian suburb Neuilly- sur-Seine. The door was opened by an impressive- looking butler. “Je veux voir Professeur de Broglie ” “Vous voulez dire , Monsieur le Due de Broglie ” retorted the butler. M O.K., le Due de Broglie ” said I, and was let into the house. De Broglie, wearing a silk house coat, met me in his sumptuously furnished study, and we started talking physics. He did not speak any English; my French was rather poor. But somehow, partly by using my broken French and partly by writing formulas on paper, I man- aged to convey to him what I wanted to say and to un- derstand his comments. Less than a year later, dc Broglie came to London to deliver a lecture at the Royal Society, and I was, of course, in the audience. He delivered a brilliant lecture, in perfect English, with only a slight French accent. Then I understood an- other of his principles: when foreigners come to France they must speak French. A number of years later when I was planning a trip to Europe and de Broglie asked me to deliver a special lecture in the institute of Henri Poincarl, of which he was a director, I decided to come well prepared. I planned to write the lecture down in my (still) poor French on board the liner crossing the Atlantic, have somebody in Paris correct the text, and use it as notes at the lecture. But, as everybody knows, all good reso- lutions collapse on an ocean voyage offering many dis- tractions, and I had to face the audience in the Sorbonne completely unprepared. The lecture went through somewhat stumblingly, but my French held, and every- body understood what I had to say. After the lecture L. DE BROGUE AND PILOT WAVES 87 I told de Broglie that I was sorry that I did not carry out my original plan of having the corrected French notes. "Mon Dieu!” he exclaimed, “it is lucky that you didn’t.” De Broglie told me about a lecture delivered by the noted British physicist R. H. Fowler. It is well known that since English is the best language in the world, the English are of the opinion that all foreigners should learn it, thus freeing themselves from the need to learn anyone else’s language. Since the lecture in the Sorbonne had to be in French, Fowler had prepared the com- plete English text of his lecture, and be sent it well in advance to de Broglie, who had personally translated it into French. Thus Fowler lectured in French, using the typewritten French text. De Broglie said that after the lecture a group of students came to him, "Mon- sieur le Professeur” they said, “we are greatly puzzled. We expected that Professor Fowler would lecture in English, and we all know enough English to be able to understand. But he did not speak English but some other language and we cannot figure out what language it was.” "Et parfois f* added de Broglie, “I had to tell them that Professor Fowler was lecturing in French!” SchrSdinger’s Wave Equation Creating the revolutionary idea that the motion of atomic particles is guided by some mysterious pilot waves, de Broglie was too slow to develop a strict mathematical theory of this phenomenon, and, in 1926, about a year after his publication, there appeared an article by an Austrian physicist, Erwin Schrodinger, who wrote a general equation for de Broglie waves and proved its validity for all kinds of electron motion. While de Broglie’s model of the atom resembled more an unusual stringed instrument, or rather a set of vi- 88 THIRTY YEARS THAT SHOOK PHYSICS brating concentrical metal rings of different diameters, Schrddinger’s model was a closer analogy to wind in* struments; in his atom the vibrations occur throughout the entire space surrounding the atomic nucleus. Consider a flat metal disc something like a cymbal fastened in the center (Fig. 21a). If one strikes it, it Fig. 21. Various vibration modes of. an elastic disc fastened in the center: ( a ) state of rest; ( b ) nodal point in the center; (c) one circular nodal line; (d) one radial nodal line; (e) two radial nodal lines; (/) three radial and two circular nodal lines. L. DE BROGLIE AND PILOT WAVES 89 will begin to vibrate with its rim moving periodically up and down (Fig. 21b). There exist also more com- plicated kinds of vibrations (overtones) like the pat- tern shown in Fig. 21c where the center of the plate and all the points located along the circle between the center and circumference (marked by heavy line in the figure) are at rest, so that, when the material bulges up within that circle the material outside the circle moves down, and vice versa. The motionless points and lines of a vibrating elastic surface are called the nodal points and lines; one can extend Fig. 21c by drawing higher overtones which correspond to two or more nodal circles around the central nodal point. Besides such “radial” vibrations there also exist the “azimuthal vibrations” in which the nodal lines are straight lines passing through the center as shown in Fig. 21d and e, where arrows indicate whether the membrane lifts or sinks in respect to the equilibrium horizontal position. Of course, die radial and azimuthal vibrations can exist simultaneously in a given mem- brane. The resulting complex state of motion should be described by two integers n r and giving the numbers of radial and azimuthal nodal lines. Next in complexity are the three-dimensional vi- brations such as, for example, the sound waves in the air filling a rigid metal sphere. In this case it becomes necessary to introduce die third kind of nodal lines and also the third integer n t giving their number. This kind of vibrations was studied in theoretical acoustics many years ago, and, in particular, Hermann von Helmholtz in the last century made detailed studies of the vibrations of air enclosed in rigid metal spheres (Helmholtz resonators). He drilled a little hole in the sphere, to let in sound from the outside, and used a siren which emitted a pure tone, the pitch of which could be changed continuously by changing the rota- 90 THIRTY YEARS THAT SHOOK PHYSICS tion speed of the siren's disc. When the frequency of the siren's sound coincided with one of the possible vi- brations of air in the sphere, resonance was observed. These experiments stood in perfect agreement with the mathematical solutions of the wave equation for sound, which is too complicated to be discussed in this book. The equation written by Schiddinger for de Broglie's waves is very similar to the well-known, wave equa- tions for the propagation of sound and light (that is, electromagnetic) waves, except that for a few years there remained the mystery of just what was vibrating. We will return to this question in the next chapter. When an electron moves around a proton in a hy- drogen atom the situation is somewhat similar to the vibration of gas within a rigid spherical enclosure. But whereas for Helmholtz vibrators there is a rigid wall preventing the gas from expanding beyond it, the atomic electron is subject to electric attraction of the central nucleus which slows down the motion when the electron travels farther and farther from the cen- ter, and stops it when it goes beyond the limit permitted by its kinetic energy. The situation in both cases is shown graphically in Fig. 22. In the figure on die left the "potential hole" (that is, the lowering of potential energy in die neighborhood of a certain point) resem- bles a cylindrical well; the figure on the right looks more like a funnel-shaped hole in the ground. The hori- zontal lines represent the quantized energy levels, the lowest of them corresponding to the lowest energy the particle can have. Comparing Fig. 22b with Fig. 12 of Chapter n, we find that the levels of the hydrogen atom calculated on the basis of Schrddinger's equation are identical with those obtained from Bohr's old theory of quantum orbits. But the physical aspect is quite different Instead of sharp circular and elliptic L* DE BROGLIE AND PILOT WAVES 9i Fig. 22. Quantum energy levels In a rectangular potential hole (a) and in a funnel-shaped potential hole (6). orbits along which point-shaped electrons run, we have now a full-bodied atom represented by multishaped vibrations of something which in the early years of wave mechanics was called, for lack of a better name, a ^-function (Greek letter psi) . It must be remarked here that the rectangular well potential distribution shown in Fig. 22a turned out to be very useful for a description of proton and neutron motion within the atomic nucleus, and later was used successfully by Maria Goeppert Mayer and, indepen- dently, by Hans Jensen for an understanding of the energy levels within the atomic nuclei and the origin of y-ray spectra of radioactive nuclear species. 92 THIRTY YEARS THAT SHOOK PHYSICS The frequencies of different ^-vibration modes do not correspond to the frequencies of the light wave emitted by the atom, but to the energy values of the different quantum states divided by h. Thus, the emis- sion of a spectral line necessitated the excitation of two vibration modes, say if/m and with the resulting composite frequency m ‘ n h ~ h ~ h which is the same as Bohr’s expression for the fre- quency of light quantum resulting from the transition of the atomic electron from the energy level E m to the lower energy level E n . Applying Wave Mechanics Apart from giving a more rational foundation to Bohr’s original idea of quantum orbits, and removing some discrepancies, wave mechanics could explain some phenomena well beyond the reach of the old Quantum Theory. As was mentioned in Chapter n, the author of the present book and, independently, a team consisting of Ronald Gurney and Edward Con- don successfully applied Schrddinger’s wave equation to the explanation of the emission of a-particles by radioactive elements, and their penetration into the nuclei of other lighter elements with the resulting trans- formation of elements. To understand this rather com- plicated phenomenon, we will compare an atomic nu- cleus to a fortress surrounded on all sides by high walls; in nuclear physics the analogy of the fortress walls is known as a potential barrier . Due to the fact that both the atomic nucleus and the a-particle carry a positive electric charge, there exists a strong repulsive Coulomb L. DB BROGLIE AND PILOT WAVES 93 forcef acting on the a-particle approaching a nucleus. Under the action of that force an a-particle shot at the nucleus may be stopped and thrown back before it comes into direct contact with the nucleus. On the other hand, a-particles that are inside the various nu- clei as constituent parts of them are prevented from escaping by very strong attractive nuclear forces (anal* ogous to the cohesion forces in ordinary liquids), but these nuclear forces act only when the particles are closely packed, being in direct contact with one an- other. The combination of these two forces forms a potential barrier preventing the inside particles from getting out and the outside particles from getting in, unless their kinetic energy is high enough to climb over the top of the potential barrier. Rutherford found experimentally that the a-particles emitted by various radioactive elements, such as ura- nium and radium, have much smaller kinetic energy than that needed to get out over the top of the barrier. It was also known that when a-particles are shot at the nuclei from outside with less kinetic energy than needed to reach the top of the potential barrier they often penetrate into the nuclei, producing artificial nuclear transformations. According to the basic princi- ples of classical mechanics, both phenomena were ab- solutely impossible, so that no spontaneous nuclear de- cay resulting in the emission of a-particles, and no artificial nuclear transformations under the influence of a-bombardment could possibly exist. And yet both were experimentally observed! If one looks on the situation from the point of view t During the early studies of electric phenomena, the French physicist Charles de Coulomb found that forces acting between charged particles are proportional to the product of their elec- tric charges and inversely proportional to the square of the dis- tance between them. This is known as Coulomb's Law. 94 THIRTY YEARS THAT SHOOK PHYSICS of wave mechanics, it appears quite different, since the motion of the particles is governed by de Broglie's pilot waves. To understand how wave mechanics ex* plains these classically impossible events, one should remember that wave mechanics stands in the same re- lation to the classical Newtonian mechanics as wave optics to the old geometrical optics. According to Snell’s Law, a light ray falling on a glass surface at a certain incidence angle i (Fig. 23a) is refracted at a smaller angle r, satisfying the condition sin i/sin r = n where n is the refractive index of glass. If we reverse the situation (Fig. 23b), and let a light ray propagat- ing through glass exit into the air, the angle of refrac- tion will be larger than that of incidence and we will have sin i/sin r — I/n. Thus a light ray falling on the interface between the glass and air at an angle of in- cidence greater than a certain critical value will not enter into the air at all but will be totally reflected back into the glass. According to the wave theory of light the situation is different. Light waves undergoing total internal reflection are not reflected from the mathe- matical boundaiy between the two substances, but penetrate into the second medium (in this case air) to the depth of several wavelengths X and then are thrown back into the original medium (Fig. 23c). Therefore, if we place another plate of glass a few wavelengths away (a few microns, in the case of visi- ble light), some amount of light coming into the air will reach the surface of that glass and continue to propagate in the original direction (Fig. 23d). The theory of this phenomenon can be found in the books on optics published a century ago and represents a standard demonstration in many university courses on optics. Similarly, de Broglie waves which guide the motion of a-partides and other atomic projectiles can pene- (s LA S 5 A»R £lA S? Fig. 23. Analogy Between Wave Mechanics and Wave Optics. In (a) we have the familiar picture of refraction of light entering from the rarer into the denser medium. In (6) we have the reverse case when the light entering from the denser into the rarer medium can be completely re * fleeted from the interface if the angle of incidence exceeds 96 THIRTY YEARS THAT SHOOK PHYSICS trate through the regions of space which are prohibited to particles by classical Newtonian mechanics, and a-particles, protons, etc., can cross the potential bar- riers whose height is greater than the energy of the incident particle. But the probability of penetration is of physical importance only for particles of atomic mass, and for barriers not more than 10~ 12 or 10“ 18 cm wide. Let us take, for example, a uranium nu- cleus which emits an a-particle after an interval of about 10 10 years. An a-particle imprisoned within the uranium potential barrier hits the barrier wall some 10 91 times per second, which means that the chance of escape after a simple hit is one out of 10 w X 3 • 10 7 X 10 81 as 3 • 10 88 hits (here 3 • 10 T is the number of seconds in a year). Similarly, the chances that an atomic projectile will enter the nucleus are very small for each individual hit, but may become considerable if a very large number of nuclear colli- sions are involved. It was shown in 1929 by Fritz Houtermans and Robert Atkinson that the nuclear collisions caused by intensive thermal motion, known as thermonuclear reactions, are responsible for the production of energy in the Sun and stars. Physicists are working hard now to produce the so-called “con- a certain critical value. According to the wave theory of light, the reflection takes place not on the mathematical surface separating the two media, but within a certain layer severed wavelengths thick. Thus, if the second layer of the denser medium is placed a few wavelengths beyond the first layer, a fraction of the incident light will not be totally reflected but will penetrate into the second dense layer propagating in the original direction. Similarly, according to wave mechanics, some particles can penetrate through the regions prohibited by classical mechanics, where the potential is higher than the original kinetic energy of the particles. L. DB BROGLIE AND PILOT WAVES 97 trolled thermonuclear reactions” which would supply us with cheap, inexhaustible, and harmless sources of nuclear energy. All this would have been impossible if Newton's classical mechanics had not been replaced by de Broglie-SchrOdinger wave mechanics. CHAPTER V W. HEISENBERG AND THE UNCERTAINTY PRINCIPLE Simultaneously with Schrodinger’s paper on wave me- chanics, which appeared in the Armalen der Physik, there appeared in another German magazine, the Physikalische Zeitschrijt, a paper by Werner Heisen- berg of Gottingen University dedicated to the same subject and leading to exactly the same results. But, to the great astonishment of the physicists who read these two papers, they started from entirely different physical assumptions, used entirely different mathe- matical methods, and seemed to have nothing to do with each other. As was described in the previous chapter, Schrd- W. HEISENBERG AND UNCERTAINTY PRINCIPLE 99 dinger visualized the motion of atomic electrons as being governed by a system of generalized three-dimen- sional de Broglie waves surrounding the atomic nu- cleus, whose shapes and vibration frequencies were de- termined by the field of electric and magnetic forces. Heisenberg, on the other hand, devised a more ab- stract model. He treated the atom as if it were com- posed of an infinite number of linear “virtual” vibra- tors with frequencies coinciding with all possible frequencies that the atom in question could emit. Thus, whereas in Schrodinger's picture the emission of a spectral line with the frequency v m , tt was considered a “cooperative result” of two vibration functions if/ m and ipnA in Heisenberg’s model the same spectral line was emitted by an individual vibrator which we may Call V m,n« In classical mechanics a linear vibrator is described by two numbers: its displacement from the position of equilibrium q and its velocity v, both quantities changing periodically in time. It is customary in ad- vanced mechanics to use, instead of the velocity v, the mechanical momentum t p, defined as the product of the particle’s mass by its velocity (p = my). For a given law of force acting on the vibrator, it will possess a well- defined frequency v. But the optical spectrum possesses t For simplicity's sake we use here only one instead of three quantum numbers for each vibration mode. t The notion of mechanical momentum, which Isaac Newton called amount of motion, was introduced in his book Mathe- matical Principles of Natural Philosophy and resulted from combination of tbe Second and Third Laws of Motion. If two particles, originally at rest, interact with one another, the acting forces Fjand F 2 are numerically equal and oppositely directed. On the other hand, the velocities gained during the period of interaction (Vj and v 2 ) are inversely proportional to the masses (m, and m 2 ) of tbe two particles. Thus the "amounts of motion” (or mechanical momenta as we call it today) are nu- merically equal and oppositely directed. This is the famous Law of the Conservation of Mechanical Momentum. 100 THIRTY YEARS THAT SHOOK PHYSICS a double manifold of frequencies that can be repre- sented by a table: I'll Vn vis I'M Pit I'M etc. V21 V22 *28 VSi v m Pat etc. v*l Vm i'as PS 4 i'as Vat etc. V41 Pit P48 Vii 1 '45 v*t etc. etc. etc. etc. etc. etc. etc. etc. Number arrays of this kind were known for a long time to mathematicians as matrices and were used suc- cessfully in the solution of various algebraic problems. Matrices can be finite if the indices m and n run from 1 to a given number, or infinite if both m and n run to infinity. In fact, there was developed a special branch of mathematics in which any given matrix (finite or infinite) can be represented by just one symbol printed in bold type. Thus a meant a matrix: a u &12 &1S a M Sis ai« an etc. &21 &22 &28 as4 a2s as* &27 etc. asi &83 ass a&4 ass ass ast etc. etc. etc. etc. etc. etc. etc. etc. etc. Like ordinary numbers, matrices can be added, sub- tracted, multiplied, and divided by one another. The rules of addition and subtraction are similar to those of ordinary numbers: one adds or subtracts, term by term. For example: W. HEISENBERG AND UNCERTAINTY PRINCIPLE 101 a±b= an =*= bn aw ± bi2 813 — bis etc. a 2 i b 2 i &22 ~ t)22 823 ± b 28 etc. asi — bsi &32 — b{2 833 — bas etc. etc. etc. etc. etc. From this definition it follows that the addition of matrices is subject to a commutative law; that is, a + b = b + a, just as3 + 7 — 7 + 3ora + b = b + a. But the laws of multiplication and division of matrices are more complicated. To obtain the term in the mth line and nth column of the product ab, one has to multiply term by term the entire sequence of terms in the mth line of a by the entire sequence of terms in the nth column of b and add all these products together. Schematically this rule can be represented by the fol- lowing diagrams where a circled dot of the product is given by the sum of products of dots placed in the squares: 8 = i-l 1st 2nd • t • —0-0-0- X 1 1 0 • • 1 1 (pea » e • O ' • 3rd • • • 1 0 < • « t e e e 1st 2nd 3rd Columns or: 102 THIRTY YEARS THAT SHOOK PHYSICS To become better acquainted with this procedure let us use numbers rather than letters for the elements of the matrix, and calculate the product of two ma- trices: 13 5 3 5 4 16 23 32 For 2 5 1 X 1 1 1 we get: 13 18 18 4 3 2 2 3 5 19 29 29 because 1X3 + 3X14-5x2 = 16; 1X5 + 3X1 + 5 x 3 = 23; etc. Let us now reverse the order of multiplicands, and calculate: 3 5 4 1 3 5 29 46 28 For 1 1 1 X 2 5 1 we get: 7 11 8 2 3 5 4 3 2 28 36 23 The result is quite different from the first case! The commutative law of multiplication, so common in arithmetic and ordinary algebra, does not hold in ma- trix calculus! That is why calculation with matrices is called non-commutative algebra. It must be remarked here that not all pairs of matrices necessarily give dif- ferent results when the order of multiplication is re- versed. If the result is the same, one says that these two matrices commute; if the result is different they do not commute. W. HEISENBERG AND UNCERTAINTY PRINCIPLE 103 The division of the matrices is defined in the same a 1 way as in ordinary algebra where j- = a • -j- and the value of -r (the reverse of that number) is subject to the condition b* ^ = 1. In non-commutative algebra ^ = a* where | satisfies the condition b*r = l and 1 = 1 0 0 0 0 etc. 0 1 0 0 0 etc. 0 0 1 0 0 etc. etc. etc. etc. etc. etc. etc. Heisenberg’s idea was that, since the frequencies of the spectral lines emitted by an atom represent an in- finite matrix: Vll Pl2 Pw Vu etc. V2l ?22 Vsa etc. vai V S 2 Vaa Va* etc. etc. etc. etc. etc. etc. The mechanical quantities, such as velocities, mo- menta, etc., should also be written in the form of ma- trices. Thus, the mechanical momenta and coordi- nates should be given by the matrices: 104 THIRTY YEARS THAT SHOOK PHYSICS and pu Pie Pis Pl4 etc. p 21 P 22 Pss P 24 etc. P21 p 22 Pss PS4 etc. etc. etc. etc. etc. eta 4 4ll Ql2 <]l3 9 * F/$. 26. Dirac 1 s picture of the energy level distribution of the particles with positive and negative mass. On the left (a) all negative energy levels are completely filled up, and only six ordinary electrons can exist on the normal positive levels. On the right (6) one of the electrons from a negative level Is lifted to a positive level, leaving behind it a "hole” which behaves as an ordinary positive electron with positive mass. If this extra electron from a positive level falls back into the hole (annihilation process of e and i the energy difference will be emitted as y-radiation. 128 THIRTY YEARS THAT SHOOK PHYSICS This equation can be obtained from the previous one by writing —mo instead of 4-m 0 , which means physically the introduction of negative mass. Thus relativistic mechanics permits in principle two separate sets of levels: those with rest energy +m 0 c 2 and higher, an- other with the Test energy — moc 8 and lower (Fig. 26). While the energy levels shown in the upper part of the diagram (£’>0) correspond to familiar types of motion of material particles (electron, proton, etc.), the energy levels in the lower part of the diagram (£<0) do not correspond to any physical reality. Par* tides having negative inertial mass do not correspond to anything observed in nature. Indeed, because of the negative value of their mass they would be accel- erated in the direction opposite to the force acting on them, and, in order to stop a moving particle of that kind, one should push it in the direction of its motion and not against it! Imagine two particles, let us say two elections, with numerically equal masses having, how- ever, opposite signs (+m and — m). According to the Coulomb law, they will be repelled by each other by electrostatic forces having the same numerical values, but acting in the opposite direction. If both particles had positive masses, this interaction would result in equal but oppositely directed accelerations (Fig. 27a), and they would fly away from each other with increas- ing velocities. If, however, one of the particles has a negative mass (Fig. 27b), it will be accelerated in the same direction as die other particle, and they will fly together, keeping a constant distance between them- selves and speeding up beyond any limit (n + i+v are other processes which are also subject to Fermi interaction laws. One is the absorption of the atomic electron by a nucleus which is unstable in respect to positive /3-decay. Instead of emitting a positive elec- tron and a neutrino, a nucleus may absorb a negative electron from its own electron shell, emitting a neutrino according to the formula: e (nucl.) a + e — > s _i(Nucl.) A + v + energy Since the atomic electron absorbed by the nucleus in such a process is one of the electrons from the K-shell (nearest to the nucleus), it is usually known as “K-cap- ture.” The simplest example of such a process is the unstable isotope of beryllium, Be 7 , which may trans- form either according to formulaft aLi 7 + i + v + energy or 4 Be 7 + (g)K-rten » sLi 7 + v + energy In the latter case, cloud chamber photographs show just a single track (that of aLi 7 ), and the situation is similar to an incident described by H. G. Wells in his 1 According to energy considerations, the first process occurs in the case of the free neutron, as well as in the case of neutrons bound inside the nucleus, whereas the second occurs only within the complex nuclei where the additional energy supply can be obtained from other nucleons. tt The lower index on the left gives the atomic number whereas the upper index on the right gives atomic weight. E. FERMI AND PARTICLE TRANSFORMATIONS 145 well-known story The Invisible Man, where a London constable was kicked in the pants from behind and, turning around, could not see anybody who could have kicked him. Observational studies of the K-capture processes showed that the frequency of their occur- rence agrees exactly with that predicted by Fermi’s theory. Another interesting process belonging to the same category is the H-H (Hydrogen-Hydrogen) reaction, first proposed by Charles Critchfield, which is respon- sible for the energy production in our Sun and other fainter stars4$ During the short interval of time while two colliding protons are in dose contact, one of them turns into a neutron through emission of a positive electron and a neutrino, forming the nudeus of deu- terium (heavy hydrogen) according to the equation: iP l + iP l — M^+^ + v+energy The probability of this process can be predicted exactly on the basis of Fermi’s theory. The last but not the least example of the Fermi in- teraction is the process by means of which F. Reines and C. Cowan directly proved the existence of neu- trinos. It is: i p' + v + energy > Reines and Cowan observed it in a chamber placed close to an “atomic pile,” at the Savannah River Atomic Energy Project The number of observed neu- trons and positive electrons formed simultaneously in the chamber subjected to extensive neutrino bombard- ment turned out to be exactly equal to that predicted by the Fermi theory. The interaction is so weak that, M In the case of brighter stars, Sirius for example, the main energy-producing reaction is the so-called carbon cyde, pro- posed independenUy by C. von Weizsacker and H. Bethe. 146 THIRTY YEARS THAT SHOOK PHYSICS in order to absorb one-half of the emitted neutrinos, one should use a liquid hydrogen shield several light years thick! Fermi’s theory of the processes involving neutrinos also applies to many cases of decay of new elementary particles discovered during more recent years, and one speaks today about the "Generalized Fermi Interaction." Fermi's Research in Nuclear Reactions Along with his theoretical studies, Fermi was involved in extensive experimental research on nuclear reactions in heavy elements bombarded by slow neutrons, and the formation of trans-uranium elements (z > 92), and for this work he received the Nobel Prize for 1938. Soon thereafter he came to the United States to live and was present at the 1939 conference at George Washington University at which Niels Bohr read a tele- gram from Use Meitner, a noted German physicist, (who by that time was living in Stockholm) containing very exciting news. She told him that her former col- laborators, Otto Hahn and Fritz Strassman, at Berlin University had found that a uranium nucleus hit by a neutron splits into two about equal parts, liberating vast amounts of energy. This announcement started a series of events which culminated, not too many years later, in nuclear bombs, nuclear power plants, etc., heralding the beginning of what one often calls the Atomic (Nuclear would be more correct) Age. Fermi took the leadership in the top-secret labora- tory at the University of Chicago, and on December 2, 1942 announced that on that afternoon the first chain reaction in uranium was achieved, thus initiating the first controlled release of nuclear energy by man. Since this book is devoted to the progress of the un- derstanding of the nature of things, and not to the prac- Professor Paul Ehrenfest explaining a difficult point to his audience. ( Photographed , probably, by Dr. S. Goudsmit) E. FERMI AND PARTICLE TRANSFORMATIONS 147 tical applications, we will omit any discussion of fission chain reaction, and finish this chapter by describing an interesting experiment carried out by Fermi in the newly invented fission reactor. For the first time it be- came possible to measure the mean life of a neutron, which ultimately decays into a proton, an electron, and a neutrino. The gadget used in that experiment is known as the “Fermi bottle,” although it was actually an evacuated spherical container somewhat resembling a Chianti bottle without a neck. As shown in Fig. 29, this sphere was placed inside an atomic pile and left - p _tR* c K e -tri^ck Fig. 29. Fermi’s bottle in a uranium pile, designed to measure the neutron’s mean life. 148 THIRTY YEARS THAT SHOOK PHYSICS there for a considerably long period of time while the pile was in operation. The fission neutrons crisscrossing the pile in large quantities in most cases entered and left the "Fermi bottle," passing without much difficulty through its walls. Once in a while, however, a neutron passing through the "bottle" would break up into a proton and an electron for which the walls of die "bot- tle" were impenetrable. Thus ordinary hydrogen gas gradually accumulated inside the "bottle" at a rate which depended on the chance that a neutron would break up while passing through. Measuring the amount of hydrogen accumulated in the "bottle" during a given period of time, one could easily estimate the neutron’s mean lifetime, which turned out to be about fourteen minutes. To learn more about Fermi’s activities along these lines the reader should turn to the book Atoms in the Family, written after his death by his wife, Laura. CHAPTER VIII H. YUKAWA AND MESONS The great success of Fermi’s theory of jB-decay raised the question whether or not it could also be applied to explain the attractive forces holding nucleons together. It was known at the time that the forces between two nucleons— be they two neutrons, a neutron and a pro- ton, or two protons— are identical except that in the last case one should add the ordinary Coulomb repulsion between the proton charges. Experiments have shown that in contrast to Coulomb forces, which decrease comparatively slowly with the distance (as 1 /r 8 ), nu- clear forces are more similar to the cohesive forces of classical physics. Just as two pieces of Scotch tape do 150 THIRTY YEARS THAT SHOOK PHYSICS not exert any force on each other, no matter how dose they are brought together, but stick tightly as soon as they come into contact, the forces between nudeons appear suddenly when they "touch each other," which occurs at the distance of about 10“ 18 cm. Once this happens, it takes about ten million electron volts of en- ergy to separate them again. Similar forces acting be- tween atoms are ascribed to the exchange of electrons between atomic shells as soon as they come into con- tact. The wave mechanical theory of these "exchange forces" was developed in 1927 by W. Heitler and F. London, who showed that the problem could be solved exactly in the simple case of two hydrogen atoms form- ing a diatomic molecule.t Heitler and London con- sidered two cases: (a) a hydrogen molecule ion, H 2 + , consisting of two protons and one electron; (b) a neu- tral hydrogen molecule, H 2 , formed by two protons and two electrons (Figs. 30a and 30b). Schrodinger’s wave equation for the electron’s mo- tion was solved exactly in both cases. The analytical result showed that there exists an equilibrium state of the lowest energy at certain distances R and R l be- tween the two nuclei. The computed energy of these equilibrium states turned out to be in perfect agree- ment with the measured dissociation energies of H 2 + and H 2 molecules. Thus, the notion of exchange forces between two identical atoms was firmly established in the field of quantum chemistry. It was only natural to assume that attractive forces between two nucleons could be understood on a simi- lar basis. When two nucleons were brought closely to- gether, an electron accompanied by a neutrino would jump to and fro between them, thus producing an at- tractive exchange force. It was a very good idea, but (alas!) it did not work. When, in 1934, D. Ivanienko t A diatomic molecule is one composed of two atoms. (a-) ( 4 ) Fig. 30. Exchange forces. ( a ) Heitler and London's theory of the forces which hold together two protons in an ionized and a neutral hydrogen molecule. ( b ) Three different pos- sibilities of explaining the nuclear forces by the exchange of pions iff, it, ft). 152 THIRTY YEARS THAT SHOOK PHYSICS and I. Tamm computed the exchange force between two nucleons resulting from the Fermi interaction, they found that the expected binding energy was of the order of magnitude of 10~ 8 electron volts! No, this is not a misprint; one hundred millionth of one electron volt in- stead of ten million volts; just fifteen zeros too small! Apparently, Fermi's "weak" interaction could not be responsible for the strong binding of protons and neu- trons within the atomic nucleus. A year later (in 1935) a Japanese physicist, Hideki Yukawa, proposed a revolutionary idea to explain strong interactions between nucleons. If these interac- tions cannot be explained by the exchange forces aris- ing from the to and fro jumping of electron-neutrino pairs, there must exist an entirely new, and as yet un- detected, particle that does the jumping. To have the strength required by experimental evidence, that par- ticle must be about two hundred times heavier than an electron (or about ten times lighter than a proton). Also, its interaction with nucleons, characterized by Yukawa’s interaction constant y, must be about 10 14 times greater than Fermi's interaction constant g re- sponsible for /^-transformations, thus being comparable to the ordinary Coulomb interactions between electric charges. This hypothetical particle was known under many aliases: Yukon, Japanese electron, heavy elec- tron, mesotron, and finally meson. Two years later, after Yukawa's proposal, the particles with mass 207 times greater than that of electrons were found in cos- mic rays by C. Anderson and S. Neddermeyer at the California Institute of Technology, and they seemed to give a brilliant confirmation to the Yukawa hypothesis. But then came a temporary setback. Experiments car- ried out by M. Converi, E. Pancini, and O. Piccioni proved without any doubt that, although the new par- ticles had the mass of Yukawa’s hypothetical mesons, Typical Copenhagen Conference in 1930. First Row: Klein, Bohr, Heisenberg, Pauli, Gamow, Landau, Kramers. H. YUKAWA AND MESONS 153 their interaction with nucleons was 10 12 times less than was needed to explain the nuclear forces. It was not until 1947 that the British physicist C. F. Powell, by sending photographic plates into the upper atmosphere, found that the mesons observed at sea level (207 electron masses) were actually the decay products of heavier mesons (mass 264 electron masses) which were formed by cosmic rays at the upper fringes of the terrestrial atmosphere. Thus there are two kinds of mesons: the heavy and the light ones. The former are now known as w-mesons, or simply pions, while the latter go under the name of /t-mesons, or muons for short. The pions show very strong interactions with nucleons and there is hardly any doubt that they are the particles initially visualized by Yukawa as being responsible for nuclear forces. However, no exact the- ory of these processes (comparable, for example, with Dirac’s theory of anti-particles) has yet been de- veloped. CHAPTER IX MEN AT WORK The reader will have noticed that the chapters of this book have become shorter. This was not due to the author’s growing fatigue but rather to the fact that, after the glorious developments in its first thirty years, Quantum Theory ran into serious difficulties, and its progress was considerably slowed down. The last “com- pletely finished chapter" of this period was Dirac’s uni- fication of wave mechanics and Special Relativity, which resulted in the elegant theory of anti-particles. After they were found experimentally, anti-particles proved to behave exactly according to the theoretical predictions. MEN AT WORK 155 Fermi's theory of the processes involving the emis- sions and absorptions of electron-neutrino pairs be- comes a little vague when applied to more complicated processes such as, for example, the decay of a muon into one electron and two neutrinos. Also, the numeri- cal value of the Fermi constant g still cannot be derived from the values of the other fundamental constants of nature. (Similarly, the Rydberg constant R of old spectroscopy remained an empirical constant until Bohr published his theory of the hydrogen atom.) Similar difficulties exist in the case of Yukawa’s the- ory of strong interactions between elementary particles, and the numerical value of the constant y still remains unexplained. A huge number of new facts are continu- ously being discovered by experimental research, and a large number of empirical rules are formulated by in- troducing new notions such as ''parity," "strangeness," etc. On the whole, the situation today resembles in many respects that existing in optics and in chemistry toward the end of the last century when the regularities in the spectral series and chemical valency properties of different elements were well known empirically but not at all understood theoretically. Things changed abruptly for the better when the Quantum Theory of atomic structure was developed and threw bright light on all the painfully collected empirical facts. In the opinion of the author, the present stalemate in the the- ory of elementary particles will be broken up— maybe next year, maybe in the year a.d. 2000— by an entirely new idea which will differ from the present way of thinking just as much as the present way differs from the classical one. We have no crystal ball for predicting the future developments of theoretical physics, but, as a substitute for it, one could use a discipline known as "dimensional analysis." Everybody knows that all 156 THIRTY YEARS THAT SHOOK PHYSICS physical measurements are based on three basic units: Length (stadia, miles, leagues, meters, etc.) Time (years, days, milliseconds, etc.) Mass (stones, pounds, drachms, grams, etc.) Every physical quantity can be expressed through those three by the so-called “dimensional formulae.*’ For ex- ample, velocity ( V ) is the length (or distance) cov- ered per unit of time; density (p) is mass per unit of volume (i.e., length in the third power); energy (£) is the mass times the square of velocity; etc. One writes: |E| = \M\ L T 12 where the vertical bars indicate that this is not a nu- merical relation but a relation between the physical nature of the quantities involved. It does not matter here which particular units one uses, and one can write: |$| = |£| = |mark| = |franc| = | ruble | = etc. or |yard| = |foot| = | meter) = |arshin| = |light-year| = etc. Length, time, and mass (or less correctly, weight) have been selected in classical physics on some kind of an- thropomorphic basis, that is, on notions that we human beings encounter in everyday life. (“It is five miles away”; 'Til be back in an hour”; “Give me three pounds of ground round steak.”) But the selection of these particular units is really not necessary, and any three complex units, be they the strength of the electric current (Amp.), the power of an engine (H.P.), or the A typical Copenhagen Spring Conference (1932) at which the play Faust (a parody) was presented. In the first row: N. Bohr, P. A. M. Dirac, W. Heisenberg, P. Ehrenfest, ALDdbrudi, Lisa Meitner, with many other great brains ( guess them!) in the rows behind. (1 he author of this book missed the conference, being detained in the U.S.S.R.) MEN AT WORK 157 brightness of light (standard candle), can serve as basic units, provided they are independent of one an- other. However, in building a consistent theory of all physical phenomena, it is rational to select three fun- damental units, each of which governs a vast area of physics, and to express all other units through them. Which units should be the members of this trio? One of them doubtless should be the velocity of light in vacuum (c), which governs the entire field of electrodynamics and the Theory of Relativity. In fact, if one would assume that light propagates with infinite velocity (c = oo ), Einstein’s entire theory would reduce to the classical mechanics of Isaac Newton. Another member of the Universal Trio is, of course, the quantum constant (A), which governs all atomic phenomena. If one assumes A to be equal to zero, one returns again to Newtonian mechanics. The great merit of Dirac is that he succeeded in uniting Relativity and Quantum Theories and in his equations c and A occupy equally honorable positions. But what is the third universal constant needed to make the system of theoretical physics complete? One of the possible august candidates is, of course, New- ton’s gravitational constant. But closer considerations seem to indicate that this constant is not very suitable for collaborating with the other two to explain atomic and nuclear phenomena. Gravitational forces are very important in astronomy, explaining the motions of planets, stars, and galaxies. But in our human-size world gravitational attraction between material bodies is negligibly small, and one would be greatly surprised to see two apples placed a few inches apart on a table roll toward each other, driven by Newtonian attraction. Only extremely sensitive instruments permit us to mea- sure gravitational attraction between two normal-size 158 THIRTY YEARS THAT SHOOK PHYSICS bodies, f In the atomic and nuclear world die forces of gravity are quite insignificant; some 10 s0 times smaller than electric and magnetic forces! It was once suggested by Dirac that Newton's "constant of gravity" is not really a constant but a variable which decreases in in- verse proportion to the age of the Universe. And he may very well be right! So what then? Which universal constant will occupy the third seat? We can just as well begin with ancient Greek philosophers, who first conceived the idea of the atom: the smallest amount of matter. In his book The Analysis of Matter ,% Bertrand Russell writes: We might suppose, as Henri Poincard once sug- gested, and as Pythagoras apparently believed, that space and time are granular, not continuous— i.e. the distance between two particles may always be an inte- gral multiple of some unit, and so may the time be- tween two events. Continuity in the percept is no evi- dence of continuity in the physical process. In his book The Physical Principles of the Quantum Theory ,§ Werner Heisenberg wrote: Although it is perhaps possible in principle to diminish space and time intervals without limit by refinement of measuring instruments, nevertheless for the principal discussion of the concepts of the wave theory it is ad- vantageous to introduce finite values for values of space and time intervals involved in the measurements and only pass to the limit zero (Ax >0; Af— >0) for these intervals at the end of the calculations. It is pos- sible that future developments of the Quantum Theory will show that the limit, zero, for such intervals is ab- tSee Gravity by G. Gamow, published in 1962 in this same series. I New York: Dover Publications (1954), p. 235. {Chicago: University of Chicago Press (1930), p. 48. MEN AT WORK 159 s traction without physical meaning; however, for the present, there seems to be no reason for imposing any limitations. However, six years later Heisenberg changed his opin- ion about the thirteen words following "however" in his above-quoted sentence, and suggested that the "divergencies" occurring in various fields of the Quan- tum Theory might be eliminated through the introduc- tion of an elementary length of the order of magnitude of 10 -18 cm. What does the word “divergency” mean? In mathe- matics this term pertains to "infinite series," that is, to endless sequences of numbers that are to be added to- gether. For example, we can write: 1+2+3+4+5+ (and so on to infinity). Gearly, the result of the summation will be infinite. But what about: 1 + 1/2 + 1/3 + 1/4+ 1/5+ (and so on to infinity)? It can be shown that this summation also becomes in- finite, or diverges, as mathematicians say. On the other hand, the series: 1 + 1/1 + 1/2! + 1/3! + 1/4! + 1/5! + (and so on to infinity) (where n! means the product of all integers from 1 to n) converges, and is equal to 2.3026. . . . Similarly, the series 1 — 1/3! + 1/5! — 1/7! + (and so on to infinity) converges to the value 0. The results of calculations carried out in theoretical physics are often represented in the form of infinite series. If they converge, as they often do, we have a 160 THIRTY YEARS THAT SHOOK PHYSICS clear-cut answer and the definite numerical value for the physical quantity we are trying to calculate. But if the series diverges, the result makes no sense leading to the infinite value of the quantity under consideration. As an early example of such divergencies, let us con- sider a problem concerning the mass of an electron. If we visualize an electron as a tiny electrically charged sphere with the charge e = 4.80 X 10 -10 electrostatic units and the radius r<>, classical electrostatics tells us that the energy of the electric field surrounding it is 1 e 2 equal to •=• — . According to Einstein's Law of Equiv- 2* To alence of Mass and Energy, the mass of that field is 1 e 2 -t — t, . Since that mass should not exceed the ob- 2 served mass mo of an electron (= 0.9 x 10~ 27 gm), it follows that: 2^? 4Sm . or r »-W =2 ' 82 * 10 ' ,,cm However, if one assumes that the electron is a point charge (r 0 — 0), the mass of the electric field surround- ing it becomes infinite! On the other hand, there are many good theoretical reasons for assuming that the electron is a point charge. Similar contradictions began to arise in larger numbers in the course of the further development of particle physics, and one was always arriving at divergent (infinite) results unless one cut the infinite mathematical series arising from straight- forward calculations at a certain spot without any suf- ficient reason for doing so. Pauli humoristically called the work in this direction "Die A bschneidungsphysik” (Cut-off Physics). It was characteristic that the cut-off had always to be done at distances of the order of 10~ 18 cm. When in MEN AT WORK 161 later years the range of forces acting between the nucleons was measured experimentally with sufficient precision, it turned out to be 2.8 x 10~ 18 cm; that is, exactly the same as die so-called “classical radius of the electron," calculated theoretically on the assumption that its mass is entirely due to the electrostatic field surrounding it. It becomes more and more evident that there is a lower limit of distance, the elementary length X anticipated by Pythagoras, Henri PoincarS, Ber- trand Russel], Werner Heisenberg, and others, which is fundamental in physics. Just as no velocity can ex- ceed that of light c, no mechanical action can be smaller than elementary action h, no distance can be smaller than elementary length X, and no time interval can be shorter than elementary duration X/c. When we know how to introduce X (and X/c) into the basic equations of theoretical physics, we will be able to state proudly: “Now at last we understand how matter and energy work!” But, after the thirty fat years in the beginning of the present century, we are now dragging through the lean and infertile years, and looking for better luck in the years to come. In spite of all the efforts of the old-tim- ers like Pauli, Heisenberg, and others, and those of the younger generation like Feynman, Schwinger, Gell- Mann, and others, theoretical physics has made very little progress during the last three decades, as com- pared with the three previous decades. The situation may best be characterized by a letter which Pauli wrote to the author of this book about an attempt he made with Heisenberg to explain the masses of various ele- mentary particles which at that time were multiplying like rabbits. Here is an extract from the letter, the ma- jor part of which (omitted in this text) is devoted to the problems of fundamental biology: UNimtITY OF CAIIIORNM «itaifirt*( » »*••*•'* HMiiif 4 «•*»«*••*»• y •J’ &qmAi /&\ w 4 9 ** > ^ tfcCds+A tjf **fit ^ ** > ** J 4*4^**®., ^ c«^ w £4*X$tjfcJL Kf4. < 4 vdC *^XUC 4^^ wjCaL*td4*ci A 40tot «* • jui »' -4^^^ ^ AA * 44 y -A JmA fr foiaMifa OOJtC^. ***A& *~C£e£ 0 T****} ** **■* 4^* *« «»* —««» teis P&*p£%:tzzL' V&* 4 ^ mC&j* A* w*A* t ■*^*^' -? *+h./—X** £&. 'TituM,* Ajjl 4h4hZc < t£ Ae/acJb obt Fft. Letter from Poult MEN AT WORK 163 Seven years have passed since this letter was written, hundreds of articles have been published on the prob- lem of elementary particles, and still we are in darkness and uncertainty on the subject. Let us hope that in a decade or two, or, at least, just before the beginning of the twenty-first century, the present meager years of theoretical physics will come to an end in a burst of entirely new revolutionary ideas similar to those which heralded the beginning of the twentieth century. A U § T E I N C H1ST0R.IE manuscript after: J. W. von Goethe PRODUCED by: The Task Force of the I “Institute for Theoretical Physics Copenhagen Motto: Not to criticize . . . N. Bohr PROLOGUE Between Heaven and Hell PREFATORY REMARKS The early decades of the present century witnessed the heady development of the Quantum Theory of the atom, and during that era the roads of theoreticians of all nationalities led, not to Rome, but to Copen- hagen, the home city of Niels Bohr, who was die first to formulate the correct atomic model. It became cus- tomary at the end of each spring conference at Blegdamsvejt 15 (the then street address of Bohr’s In- stitute of Theoretical Physics) to produce a stunt per- taining to recent developments in physics. The 1932 conference, which coincided with the tenth anniversary of Bohr's Institute, followed closely on the British physicist Jfemes Chadwick's discovery of a new par- ticle having the same mass as a proton but deprived of any electric charge. Chadwick called it the neutron , the name which is now familiar to anybody interested in nuclear physics and in what is called, somewhat in- correctly, "atomic energy." But there was some mixup in terminology. A few years earlier Wolfgang Pauli used the same name for a hypothetical particle which had no mass and no charge and was, in his opinion, necessary to explain the violation of the Law of Conservation of Energy observed experimentally in the processes of radioactive Beta-decay. "Pauli's Neutron" was the subject of hot discussions among the physicists, but these discussions were exclusively oral or carried on by private corre- t Pronounced “Bll-dams-vl.” 168 THIRTY YEARS THAT SHOOK PHYSICS spondence, and the name was never “copyrighted” through appearance in any publication. Thus, when the discoveiy of Chadwick’s heavy neutron was an- nounced in bis 1932 paper in Nature, the name of Pauli’s weightless neutron had to be changed. Enrico Fermi proposed calling it the neutrino, which in Italian means a little neutron. In the following transla- tion die name of Pauli’s “neutron” in the original text is changed to the present name “neutrino,” the exis- tence of which had not at that time been demonstrated. Many physicists, especially Paul Ehrenfest, of Leiden, were very skeptical concerning Pauli’s hypothetical neutrino, and it was only in 1955 that its existence was indisputably proved by the experiments of Fred Reines and Cloyd Cowan, of the Los Alamos Scientific Lab- oratory. The pages that follow are the script of a play that was written and performed by several pupils of Bohr and staged at the spring meeting in 1932. (The author of this book was unable to participate in the play, the Soviet Russian Government having refused him a pass- port to attend the Copenhagen meeting.) The theme of this dramatic masterpiece has Pauli ( Mephistopheles ) trying to sell to the unbelieving Ehrenfest (Faust) the idea of the weightless neutrino (Gretchen). The Blegdamsvej Faust, rendered into English by Barbara Gamow, is reproduced in this book as an im- portant document pertaining to these turbulent years in the development of physics. The authors and the performers prefer to remain anonymous, except for J. W. von Goethe. Because of our failure to locate the original author or authors and the original artist, we are suggesting that the publisher deduct an appropriate moiety of the royalties which are to be paid, and hold this amount in escrow for a period of two or three years in the hope that the publication of the book may lead BLEGDAMSVEJ “FAUST” 169 to the discovery of the author and/or artist Failing that the sum of money could be presented to the In* stitute’s Library for the acquisition of new books. Thanks are due to Professor Max Delbriick for his, kind help in the interpretation of certain parts of the play. G.O. In the German text of this physics Faust, Goethe’s rhythms and rhyme schemes (see comparable passages in the original Faust) were followed closely but not exactly. Certain poetic license has been taken here too, with the result that this English version falls somewhere between the other two. Unfortunately, some of the lines from the original Faust, which were used verbatim in the German physics version, could not be used here. Also, there were a few puns In the German language for which it was nec- essary to attempt English substitutions. Some of the passages in prose In the German physics version have here been converted into verse and appear as speeches by the Master of Ceremonies. This was with the idea of making this Faust more playable on the stage. There is an amusing confusion of identity throughout: Gretchen is at times referred to as Gretchen and at other times as The Neutrino; Faust sometimes as Faust and at other times as Ehrenfest. But it all adds to the fun, and nobody’s the worse for it. And by the way, if this should be played on the stage, U would seem a good idea for the different minor characters (be they Human or Physical Concepts ) to wear signs indicating who they are: "Slater,” "Darwin,” "The Monopole,” "The False Sign," etc. Otherwise, the audience will be hopelessly muddled. B.G. WHOM THE CHARACTERS REPRESENT (Note: the Master of Ceremonies is played by Max Del- bruck, German physicist) ARCHANGEL EDDINQTON ARCHANGEL JEANS ARCHANGEL MILNE MBPHISTOPHELES THE LORD THB HEAVENLY HOSTS FAUST GRETCHBN OPPIB A TALL MAN MILLIKAN-ARIEL LANDAU (DAU) GAMOW SLATER A. Eddington, British astronomer J. Jeans, British astronomer E. A. Milne, British astronomer W. Pauli, German physicist Niels Bohr, Danish physicist "Extras” P. Ehrenfest, Dutch physicist The Neutrino R. Oppenheimer, American physicist R. C. Tolman, American physicist R. A. Millikan, American physicist L. Landau, Russian physicist G. Gamow, Russian physicist J. C. Slater, American physicist 172 THIRTY YEARS THAT SHOOK PHYSICS Dnuc P. A. M. Dirac, British physicist darwin C. Darwin, British physicist fowler R. H. Fowler, British physicist four gray wombn The Gauge Invariant, Fine Structure Constant, Neg- ative Energy, Singularity friendly photographer A friendly photographer waoner J. Chadwick, British physicist mystical chorus Everybody who can sing The Blegdamsvej Faust The THREE ARCHANGELS, THE LORD, THE HEAVENLY HOSTS, and MEPHISTOPHELES ARCHANGEL EDDINGTON As well we know, the Sun is fated In polytropic spheres to shine ; 1 Its journey, long predestinated, Confirms my theories down the line. Mail to Lemaitre’s promulgation 3 (Which none of us can understand)! As on the morning of Creation The brilliant Works are strange and grand. ARCHANGEL JEANS And ever speeding and rotating, The double stars shine forth in flight. The Giants' brightness alternating With the eclipse's total night 176 THIRTY YEARS THAT SHOOK PHYSICS Ideal fluids, hot and spuming, By fission turn to pear-shaped forms. 8 Mine are the theories that are winning! The atom cannot change the norms. ARCHANGEL MILNE The storms break loose in competition (The Monthly Notices as well!) 4 And bum with violent ambition Important tidings to foretell. At heat of 10 to 7th power The gas degenerates in flame. Permitting us our shining hour Of freest flight in Fermi's name. 8 THE THREE This vision fills us with elation (Though none of us can understand). As on the Day of Publication The brilliant Works are strange and grand. MEPHISTO (springing forward ) Since you, O Lord, yourself have now seen fit To visit us and learn how each behaves, BLEGDAMSVEJ “FAUST” 177 And since it seems you favor me a bit. Well— now you see me here ( turning to the audience ) among the slaves. On Stars and Worlds I've nothing for the jury. All that I know is how the folks complain. To me the theory's full of sound and fury. Yet here you are in ecstasy again, Approving views that shatter like a bubble, Sticking your nose in every kind of trouble. THE LORD But must you interrupt these revels Just to complain, you Prince of Devils? Does Modern Physics never strike you right? MEPHISTO No, Lord! I pity Physics only for its plight. And in my doleful days it pains and sorely grieves me. No wonder I complain— but who believes me? THE LORD You know this Ehrenfest? . . . MEPHISTO The Critic? 7 (A vision of the above appears ) 178 THIRTY YEARS THAT SHOOK PHYSICS THE LORD My knightl MEPHISTO Your knight, your slave and henchman. What's your bet? You still will lose, I warn you, if you let Me tempt this knight and lead him far astray. THE LORD Oh, this is really dreadful! Must I say . . . Jah, muss Ich sagen. . . . There is an essential 8 Failure of classic concepts— a morass. One side remark— but keep it confidential— Now what do you propose to do with Mass? MEPHISTO With Mass? Why, just forget it! BLEGDAMSVEJ “FAUST” 179 THE LORD But ... but this . . . Is very fn-ter-est-ing. Yet to try it . . . MEPHISTO Oh, Quatschl What rot you talk today! Be quiet! THE LORD But ... but ... but ... but .. . MEPHISTO That’s my hypothesis! THE LORD But Pauli, Pauli, Pauli, we practically agree. There’s no misunderstanding— that I guarantee. Naturlich, Ich bin einig. We might throw Mass away But Charge is something different— why Charge just has to stay! MEPHISTO What temperamental nonsense! Why not get rid of Charge? THE LORD I understand completely, but maa ]eg sparge, friend , 9 MEPHISTO Shut up! THE LORD But Pauli, surely you’ll hear me to die end? If Mass and Charge go packing, what have you, by and large? MEPHISTO Dear man, it’s elementary! You ask me what remains? Why bless me, The Neutrino! Wake up and use your brains! (Pause. Both pace to and fro ) 180 THIRTY YEARS THAT SHOOK PHYSICS THE LORD I say this not to criticize, but rather just to learn. . . , 10 But now I have to leave you. Farewell! I shall return! {He exits ) MEPHISTO From time to time it's pleasant to see the dear Old Man, I like to treat him nicely— as nicely as I can. He’s charming and he’s lordly, a shame to treat him foully— And fancy!— he’s so human he even speaks to Pauli! {He exits ) FIRST PART FAUST I have— alas— leaned Valence Chemistry, Theory of Groups, of the Electric Field, And Transformation Theory as revealed By Sophus Lie in eighteen-ninety-three. Yet here I stand, for all my lore, No wiser than I was before. M. A. I’m called, and Doctor. Up and down, Round and about, the pupils have been guided By this poor errin’ Faust and witless clown; They break their heads on Physics, just as 1 did. But still I’m better than the cranks, The Big Shots, monkeys, mountebanks. All doubts assail me; so does every scruple; And Pauli as the Devil himself 1 fear. I grab the eraser, like a frantic pupil, Before the magic X-ings disappear, 11 For what is written down on black, in white, Is apt to be acceptable and right. Du Ueber Gott! I still could do some teaching. I have no Guth nor Breit here at my side, 12 But I could use their aptitude for preaching To spread the tested gospel good and wide. 184 THIRTY YEARS THAT SHOOK PHYSICS Not even Hund nor hound could bear my lot , 18 So I'm The Critic, sad and misbegot. (mephisto bursts In like thunder, dressed as a traveling salesman ) Why all the noise? MEPHISTO I’m at your service, Sirl FAUST What do you take me for? A customer? MEPHISTO You used to be receptive and urbane. . . . These theories nowadays are wrong as rain; Therefore I want to show you something higher, For with it you can set the world on fire: ‘The Dance of the Golden Calf”— kaleidoscopic— The Radiation Theory is my topic. BLEGDAMSVEJ "FAUST” (Canon, sung by all) Bom-Heisenberg Heisenberg-Pauli Pauli-Jordan Jordan-Wigner Wigner-Weisskopf Weisskopf-Bom Born— Heisenberg 14 (etc.) (etc.) 185 MEPHISTO These are my own, Bone of my bone. Listen how, with spunk and spice, Precociously they give advice. Here the width of lines diverges In the wave-field’s vasty length. (The master of ceremonies protests by gesture ; mephisto repeats) Here the width of lines diverges In the wave-field’s loss of strength. FAUST Enough! You’ll not seduce me. I am cured. I'll never touch your reprints, rest assured. MEPHISTO I'm glad of that. (aside) (His argument has pith. The first old man that I can reason with!) (showing his wares) A Psi-Psi Stern? 1 * 186 THIRTY YEARS THAT SHOOK PHYSICS FAUST MEPHISTO A Psi-Psi Gerlach? FAUST MEPHISTO Electrodynamics? FAUST MEPHISTO By Heisenberg-Pauli? FAUST No sale! No sale! No sale! No sale! MEPHISTO With infinite self-energy? FAUST MEPHISTO Electrodynamics? No sale! FAUST No sale! BLEGDAMSVEJ “FAUST” 187 MEPHISTO By Dirac? FAUST No salel * & MEPHISTO With infinite self-energy? FAUST The same old story! MEPHISTO So I must show you something that’s uniquel FAUST You’ll not seduce me, softly though you speak. If ever to a theory I should say: “You are so beautiful!” and “Stay! Oh, stay!” Then you may chain me up and say goodbye— Then Til be glad to crawl away and die. MEPHISTO Beware alone of Reason and of Science, Man’s highest powers, unholy in alliance. You let yourself, through dazzling witchcraft, yield To all temptations of the Quantum field. Listen! As now the obstacles abate, You’ll know the fair Neutrino for your fatel 188 THIRTY YEARS THAT SHOOK PHYSICS GRETCHEN ( comes in and sings to faust. Melody: "Gretchen at the Spinning Wheel I" by Schubert ) My Mass is zero, My Charge is the same. You are my hero, Neutrino’s my name. I am your fate, And I’m your key. Closed is the gate For lack of me. Beta-rays throng 10 With me to pair. The N-spin’s wrong 17 If I’m not there. My Mass is zero, My Charge is the same. You are my hero. Neutrino’s my name. My psyche turns To you, my own. My poor heart yearns For you alone. BLEGDAMSVEJ “FAUST My lovesick soul Is yours to win. I can't control My trembling spin. My Mass is zero. My Charge is the same. You are my hero, Neutrino's my name. 189 (Exeunt omnes) MRS. ANN ARBOR’S SPEAKEASY 18 (otherwise known as Auerbach Keller) (American physicists sitting sadly at the Bar) MEPHISTO (springing forward behind the bar) Can no one laugh? Will no one drink? Pit teach you Physics in a wink. . . . (he winks exaggeratedly and knowingly at the physicists) Shame on you, sitting in a daze When as a rule you're all ablazel BLEODAMSVEJ “FAUST” 191 OPPIB {swallowing— Njum! Njuml — before speaking ) Your faultl You’ve brought no single word of cheer— No news, no X-ings. Bahl MEPHISTO {producing gretchbn) But both are herel {Lively applause and general tumult) A TALL MAN A shapely and appealing Signorina. . . . {to MEPHISTO) But tell me, have you been in Pasadena? MEPHISTO With Einstein, yes. He greets you in your harbor, This wunder - bar of Mrs. Annie Arbor. A TALL MAN Einstein! His curves! His fields! His whole arena! MEPHISTO {sings) 192 THIRTY years that shook physics A Monarch cherished dearly A Flea, just as a son, 1 ® And quite as much-or nearly— As Gra-vee-t&y-shee-un. The Monarch summoned Mayer , 20 Said Mayen "To be sure! I’ll make him tensors, Sire , 21 With junker curvature." '•m Attired as a dandy. The Flea was then displayed. Folks ate him up like candy So sweetly was he made. The Flea grew up, and later His Son was bom. The son 22 Kept challenging his pater But never got to run. BLECDAMSVEJ “FAUST” 193 A tS -0 Half-naked, fleas came pouring From Berlin’s joy and pride. Named by the unadoring: “Field Theories— Unified.” Now, Physicists, take warning, Observe this sober test. . . . When new fleas are a-borning Make sure they’re fully dressed! ALL Drunk though we are, we feel as fine As— hid— five hundred female swinel FAUST (known to be opposed to alcohol, steps forward and sings ) 194 THIRTY YEARS THAT SHOOK PHYSICS (tO MEPHISTO) Do you expect me to get well In all this chaos, din and hell? (to GRBTCHEN) You Skeleton, you Monster, here I stand. But do you recognize your lord and master? What holds me back? See here, I take your hand And shatter you! GRETCHEN Faust, Faust, I fear disaster! ( Exeunt omnes ) SECOND PART A Charming Region (faust sleeps, on a bed of roses. A plum tree grows, to the right. A terrific din announces the approach of the milukan-ariel) MILLIKAN-ARIEL (. from above ) Hear, oh hear the words of rubes (Wilson Chambers, Counting Tubes) I 2 * Thundering, for the spirit's ear, Comic Rays will now appear! The protons are creaking and chattering, Electrons are rolling and clattering. Light comes rushing— whither? whence? Heisenberg is really grumpy ; 24 Rossi, Hoffmann— both are jumpy . 25 All this nonsense makes no sensei 198 THIRTY YEARS THAT SHOOK PHYSICS FAUST (awakening) Sweet rosy field— what soil am I caressing? And why familiar? Rosenfeld, they say , 26 To the gteengauge invariant gives a blessing . 27 This is his plum. (master of ceremonies appears ) (to the m.c.) What’s going on today? M.C. Walpurgis Nights: the Classical Poetical, And afterwards, the Quantum Theoretical. FAUST Excellent! I quite agree! THE CLASSICAL WALPURGIS NIGHT M.C. (makes a gesture of presentation ) The Classical— a potpourri! BLEGDAMSVEJ “FAUST” 1 99 FAUST (He leans forward, expecting. A long pause indi- cates that nothing is happening ) But nothing’s happening! M.C. Just wait and see! FAUST (He waits. Another long pause and again nothing happens) See here now, DelbrUck! . . . M.C. Faust, you must expect That with the Classical there's no effect Upon the audience. (dirac enters) DIRAC Correct! Correct! FAUST Why not skip this, and go to the Q.-T.? 200 THIRTY YEARS THAT SHOOK PHYSICS M.C. If we do that, I fail as an M.C., For first the Classical must duly dose. FAUST I have two different time-scales to propose For these Walpurgis Nights. As I’ve avowed, The First should go to limbo. DIRAC Not allowed! FAUST I then propose the Classical be moved Much farther back in time and place. M.C. Approved! THE QUANTUM THEORETICAL WALPURGIS NIGHT (At one side of the stage, to the back, the lord and i landau 38 appear, the latter bound and gagged) BLEGDAMSVEJ “FAUST” 201 THE LORD Keep quiet, Daul . . . Now, in effect. The only theory that's correct, Or to whose lure I can succumb Is LANDAU Um! Um-um! Um-urn! Um-um! THE LORD Don't interrupt this colloquyt Vll do the talking. Dau, you see, The only proper rule of thumb Is LANDAU Um! Um-um! Um-um! Um-um! (At the other side of the stage, to the back, appears the face of gamow, through bars ) 202 THIRTY YEARS THAT SHOOK PHYSICS GAMOW I cannot go to Blegdamsvej (Potential barrier too high!). This “conversation” is the hoak— The Lord, he really make the joke. Bounded and gaggled, mouse to toe, Dau can’t say “Nyetl” nor “Horosho!” M.C. (center stage ) Be careful! Achtungl Watch it! These Holes of P. Dirac 29 Can trip you in a second and flip you on your back! (He puts up a “Warning!” sign) THE MONOPOLE (steps forward and sings) Two Monopoles worshiped each other , 90 And all of their sentiments clicked. Still, neither could get to his brother, Dirac was so fearfully strict! (to the m.c.) But tell me— (Watch it! There’s a Hole!) Where is my darling Antipole? BLEGDAMSVEJ “FAUST” 203 M.C. (aside) (A Hole! My foot! More like a crater!) (to the MONOPOLB) Now just a minute— Here comes Slater. (slater steps forward with a bloody lance and THE GROUP DRAGON ) 31 M.C. (observing the characters running about on the stage ) Why do they run? Why does he roll? Who stabbed him with the bloody pole? Group Dragon, by this mortal blow We laid you low! Scaly with indices is he Who died of Anti-symmetry. 204 THIRTY YEARS THAT SHOOK PHYSICS Reduced to nothing, there he lies Stripped of his status and disguise. Group Dragon, by this mortal blow We laid you lowl {The false sign steps forward) FALSE SIGN 8 * All the theories expire or bring disappointment The Sign is forever the fly in your ointment. The reckoning’s splendid and everything’s fine— Nonetheless, at die zenith, in squeezes The Sign! (dirac and darwin are brought forward) M.C. Now here’s the revered One-dimensional Case; His name is Dirac— you remember his face. Three-dimensional Darwin is following next. {The false sign prances around dirac and pulls him to the side. But he has no access to darwin) Observe the False Sign; he’s annoyed and perplexed. This injures his pride. He can handle Dirac, But Darwin's a nut that he can’t seem to crack, For Darwin so far is like pie in the sky— He’s only a glint in a physicist’s eye. BLEGDAMSVBJ “FAUST” 205 (M.c. holds up a card that reads) THE EXCHANGE RELATION 88 PQ - QP = h/2iri Watch this! Darwin's turned himself into a P, (fowler arrives on the scene) And Fowler— he’s G— has arrived. As you'll see, They explain the Relation described on the card By leapfrogging madly all over the yard. (At each exchange flashes the sign “h/2iri.” With this goes a song ) : Thus exchanged are P and Q Time and time anew. Time and time anew. Still there ever hovers by: h/2iri, h/2iriJ 206 THIRTY YEARS THAT SHOOK PHYSICS They can never rest in peace Till they're gone as geese. Till they’re gone as geese. Still there ever hovers by: h/2m, h/2ml Attention! Attention! Their form is now altered (P and Q now suffer the painful metamorphosis into donkey-electrons and fall into one of Dirads Holes) To Donkey-electrons. Observe that they’ve fal- tered 84 And fallen, through carelessness (clumsy old chaps!), Into one of those Holes that are planted as traps. (The spin of the photon, dressed in Indian guise, slithers across the stage, accompanied by fugitive music) Attention again! Hoe's The Spin of the PhotorP* With some kind of Indian sari and coat on. (It’s dear that no modest, respectable Boron 86 Would traverse the platform without any do’es on!) BLEGDAMSVEJ “FAUST” 207 (dirac comes forward, followed by four gray WOMEN) THE FIRST The Gauge Invariant is my name. THE SECOND I'm of Fine Structure Constant fame. 87 THE THIRD Negative Energy— that’s me. 88 THE FOURTH {to THE THIRD) Just watch your grammar, Number Three! (to the others ) Sisters, into the reckoning You cannot and you may not spring. But in the end there I shall be, For I am Singularity Z 89 (the four stand to the side of the stage, to mingle in again later ) FAUST Four I saw come, one I saw go; And what they tried to say I do not know. The air is now so full of shades and spooks That we had best hang on to our perukes. DIRAC A strange bird croaks. It croaks of what? Bad luck! Our theories, gentlemen, have run amuck. To 1926 we must return; 40 Our work since then is only fit to burn. FAUST Then nothing should originate today? 208 THIRTY YEARS THAT SHOOK PHYSICS DIRAC (tO the FOURTH GRAY WOMAN) You, Singularity, just go away! THE FOURTH My place is here— and, if you please, don’t shoutl DIRAC Wench, through my magic I will get you out! THE FOURTH Am I not in Eigen fields? Does Radiation not contain me? My form to change forever yields, My power is such that none can chain me. Yet on the track, as on the waves, I stand among the frightened slaves. Always found, though never sought. Cursed before she's even caught. BLEGDAMSVEJ “FAUST” 209 DIRAC I don’t see your point! ( He exits, chased by singularity) m.c. (to dirac’s back ) You’ll see it soon— That woman’s going to chase you to the Moon! (to the audience ) Unless, of course, his long legs save the day. Three guesses! Will he make his getaway? (mephisto appears. Somebody knocks at the door. A friendly pbotographer looks questioningly in- side.) mephisto Come on, come on! Come in, come in! You baggy-trousered lout, you, With plate and film and click and din! 210 THIRTY YEARS THAT SHOOK PHYSICS ( pointing to faust) He shrivels up without you. FAUST ( Highly excited, he takes a pose far the press pho- tographer) To this fair moment let me say: “You are so beautiful— Oh, stayl" A trace of me will linger ’mongst the Great, Within the annals of The Fourth Estate. Anticipating fortune so benign, I now enjoy the moment that is mine! (He dies, and his body is carried out by the Press) MEPHISTO No pleasure was enough; no luck appeased him. The changing forms he wooed have never pleased him. The poor man clung to those who would evade him. All's over now. How did his knowledge aid him? M.C. (to the photographer's camera) Out, Light overpowering!— Magnesium-devouring, Thundercloud-showering, Ego-deflowering, Stinking One, Blinking One, Vex us no morel FINALE Apotheosis of the True Neutron WAGNER 41 {appears, as the personification of the ideal experi- mentalist, balancing a black ball on his finger, and says, with pride ) The Neutron has come to be. Loaded with Mass is he. Of Charge, forever free. Pauli, do you agree? 214 THIRTY YEARS THAT SHOOK PHYSICS MEPHISTO That which experiment has found— Though theory had no part in- is always reckoned more than sound To put your mind and heart in. Good luck, you heavyweight Ersatz— 48 We welcome you with pleasure! But passion ever spins our plots, And Gretchen is my treasure! MYSTICAL CHORUS Now a reality, Once but a vision. What classicality, Grace and precision! Hailed with cordiality, Honored in song, Eternal Neutrality Pulls us along! Solvay International Institute of Physics, Sixth Council of Physics, Brussels, October 20-26, 1930. First Row: Th . De Bonder, P. Zeeman, P. Weiss, A. Sommerfeld, Mme. Curie, P. Langevin, A. Einstein, O. Richardson, B. Cabrera, N. Bohr, W. J. De Haas. Second Row: E. Herzen, E. Henriot, J. Verschaffelt, Manneback, A. Cotton, J. Errera, O. Stern, A. Piccard, W. Gerlach , C. Darwin, P. A. M. Dirac, H. Bauer, P. Kapitza, L. Brillouin, H. A. Kramers, P. Debye, W. Pauli, J. Dor f man, J. H. Van Vleck, E. Fermi, W. Heisenberg. (Photographed by Benjamin Couprie) Solvay International Institute of Physics, Seventh Council of Physics, Brussels, October 22-29, 1933. First Row: E. Schrodinger, Mme. I. Joliot, N. Bohr, A. Joffe, Mme. Curie, O. W. Richardson, P. Langevin, Lord Rutherford, Th. De Donder, M. de Broglie, L. de Broglie, MUe. L. Meitner, J. Chadwick. Second Row:E. Henriot, F. Perain, F. Joliot, W. Heisenberg, H. A. Kramers, E. Stahel, E. Fermi, E. T. S. Walton, P. A. M. Dirac, P. Debye, N. F. Mott, B. Cabrera, G. Gamow, W. Bothe, P. Blackett, M. S. Rosenblum, J. Errera, Ed. Bauer, W. Pauli, J. E. Verschaffelt, M. Cosy ns (in back), E. Herzen, J. D. Cockroft, C. D. Ellis, R. Peierls, Aug. Piccard, E. O . Lawrence. L. Rosenfeld. (Photnoranhps) h - o Rpninmin CnunrioX NOTES ON THE TEXT PROLOGUE 1. Polytropic spheres are mathematical models of hot gaseous spheres representing stars. 2. Abbd Georges Lemattre, a Belgian astronomer who originated die Theory of the Expanding Universe. 3. . . . pear-shaped forms. Jeans* theory of the origin of double stara. 4. Monthly Notices of the Royal Astronomical Society, in which most British papers on theoretical astrophysics are published, 5. Fermi’s degenerated electron gas forms the interior of certain classes of stars (see Chapter VII). 6. A symbol characterizing the quantum theoretical un- certainty relation A g • Ap which will be encoun- tered later in the text. 7. The Critic. Professor Ehrenfest had a critical attitude toward many theoretical ideas, in particular toward Pauli’s hypothesis of the neutrino. 8. /eh, muss Ich sagen. . . . The accepted language in Bohr’s Institute was to a large extent German, be- cause of the many visitors from Central Europe. Bohr spoke it perfectly but often with Danishisms. One of his typical expressions was “muss Ich sagen” (“must I say”) which should have been, in correct German, "darf Ich sagen ” (“may I say”). The rea- son was that in Danish "darf* is "nun,” which is closer to the German "muss” or the English “must” 9. "nun jeg spfrge ” means “may I ask” in Danish. 10. “. . . not to criticize.” Another of Bohr’s typical ex- 216 THIRTY YEARS THAT SHOOK PHYSICS pressions which he always used when he did oot agree with someone’s statement. FIRST PART 11. X-ings. ("Ixerei” in German) is a word invented by Einstein and was often used about papers which con- tained too much complicated mathematics, (“X” is the unknown in school algebra) but little physical content. 12. E. Guth (in English his name means "good”) and G. Breit (in English, "wide”). 13. The German physicist, F. Hund (in English, "dog”) whose name was often used in the expression "work- ing like a dog.” 14. German physicists working on the Quantum Theory of Radiation. 15. Psi-Psi Stem (in English, "Psi-Psi Star— ifnft*) is an important quantity in quantum physics. Here refer- ring to Otto Stem and W. Gerlach, well-known ex- perimentalists. 16. Beta-rays. According to Pauli’s hypothesis, the neutrino is a particle which always accompanies the emission of a Beta-ray from the nucleus. 17. N-spin. According to the views of those days, the spin (axial rotation) of the nitrogen nucleus could not be explained without considering the spin of the hy- pothetical neutrino. 18. Referring to the University of Michigan at Ann Arbor, Michigan. 19. The Flea of the Monarch (Einstein) is the General Theory of Relativity. 20. Walter Mayer— pronounced "Myer” (rhymes with "Sire”!)— mathematician who assisted Einstein in the development of his theories. 21. Tensors are mathematical symbols used in the Theory of Curved Spaces. 22. The Son is the Unified Field Theory on which Einstein NOTES ON THE TEXT 217 worked for the last three decades of his life, but without much success. SECOND PART 23. Wilson Chambers, etc. Physical apparatuses used in the study of cosmic rays. 24. W. Heisenberg (see Chapter VI), who was at that period interested in the Theory of Cosmic Rays. 25. Bruno Rossi and G. Hoffmann, experimentalists study- ing cosmic rays. 26. Leon Rosenfeld, a Belgian theoretical physicist 27. “Gauge invariant,” a complicated notion in theoretical physics. In German it is called "Elche invariant .” By coincidence the German word “Eiche” also means “oak.” 28. Landau. See Niels Bohr and the Development of Physics, ed. by W. Pauli, New York, McGraw-Hill Book Co., Inc., 1955, page 70. 29. Dirac’s Holes (see Chapter VII for explanation). 30. Monopole (see Chapter VII for explanation). 31. The Group Theory, a complicated branch of mathe- matics, used in the Quantum Theory. 32. The False Sign refers to the putting of a + instead of a — , or vice versa, a mistake often made by absent- mindedness in mathematical calculations and lead- ing, of course, to wrong results. 33. The Exchange Relation. The basic postulate in the Heisenberg quantum mechanics. 34. Donkey-electrons. A jocular expression for an electron with negative mass (see Chapter VII). 35. A photon, or light quantum, can be considered as a rotating package of energy. 36. Re bosons (see Chapter IV). 37. Fine Structure Constant. The number 137, which is important in the theory of the atom. 38. Negative Energy. One of the mathematical difficulties appearing in the Quantum Theory. 218 THIRTY YEARS THAT SHOOK PHYSICS 39. Singularity. Another mathematical difficulty appear* ing in die Quantum Theory. 40. In the year 1926 wave mechanics was discovered. FINALE 41. Wagner. James Chadwick, a British physicist who dis- covered the neutron (heavy neutral particle) in the year this play was presented. 42. Ersatz. The neutron, with its large mass, cannot be considered as a substitute (“Ersatz") for the weight- less neutrino. INDEX Numbers appearing in italics refer to numbers of the figures in the text. Addition and subtractioni law of, 100 matrices. 101-4 Alpha-particles, 33-35, 72, 92-94, 96 Condon and Gurney, viii formula, Fowler, 140 motion of, 94-95 Rutherford, apparatus, 10 experiment, 93 scattering, 35, 140, 10 Analysis of Matter, Russell, 158 Anderson, Carl, 152 experiment, 133 Anti-particles, 4, 79, 118-38, 153, 154 experiment, 137-38 theory of, 4, 132-38, 154 Aristotle, 131 Atkinson, Robert, viii, 96 Atomic models, 1, 2, 68 Bohr, xii, xiv, 40, 60, 12, 19 de Broglie, 87 Heisenberg, 99 Pauli, 68 Rutherford, 1, 33-35, 36, 37 Schrodinger, 88, 21 Thomson, 30-35 Atomic Nucleus, Gamow, ix Atomic volumes, 66-68 changes of, 15 Atoms in the Family, L. Fermi, 148 Azimuthal, 46, 67 nodal lines, 89, 21e vibrations, 89, 21d Balmer, J. J., 40, 41, 12 formula, 40-43, 45-46, 9a Beta-decay, theory of, Fermi, 142-44, 149, 167 Beta-emission, theory of, Fermi, 141 Beta-transformation, mathe- matical theory of, Fermi, 141-42 Bethe, H„ 145 Bevatron, 136 Bitter, Francis, 119, 123 Bjerknes, 57 Bohr, Harald, 56-57 Bohr, Niels, vii, viii, ix, xii, 1, 2, 3, 18, 19, 29-59, 60, 61. 64, 68. 72, 73-74, 75. 77, 80, 81, 92, 114-16, 119, 132-33, 146, 11, 18, atomic model, xii, xiv, 2, 40, 68, 12. 19 Faust, play, 53, 165, 167, 168, 171 if Institute, xii, xiv, 49-59, 169 theories of, 19, 90, 118, 155 Boltzmann, Ludwig, 6, 52, 73, 74 Boltzmann’s constant, 73, 74 Bondi. H., 125 Brackett, Frederick, 41, 12 Bragg, W., and Bragg, W. L., Broglie, Louis Victor de, 2-3, 80-97 atomic model, 87 pilot waves, 2, 81-87, 94, 99, 113, 114, 19, 20 Brownian particles, 14 motion, 110 Carlsberg: Brewery, 49 research laboratory, 50 fellowships, 51, 55 mansion, 49-50 Cartesian coordinate system, 114 Casimir, H., 51, 53, 57 Chadwick, James, 167, 168, 17211 discovery of neutron, 75 220 THIRTY YEARS THAT SHOOK PHYSICS Chadwick (cant’d) experiments, 72, 141, 17 Chain reaction, first, 146 Classical mechanics. See New- tonian classical mechan- ics Clausius, Rudolph, 17 Cloud chamber, 112-13, 144 Compton, Arthur, experi- ments, 1, 27-28, 8 Compton Effect, 27-28 Condon, E. U., viil, 54, 92 Convert, M., experiment, 153 Cosmotron, 136 Coulomb, Charles de, 30, 35, 64, 92-93, 128, 132, 149, 152 Cowan, C., 75, 145, 168 Crime and Punishment, 121— 22 Critchfield, Charles. H-H re- action, 145 Darwin, C- 171 Davisson, G., 84 Delbriick, Max, 51, 55, 169 Derivatives: first, 123-25 second, 123-25 Diagram: Bohr-Costcr, 18a Mayer-Jensen, 18b Dimensional: analysis, 155-57 formulae, 156-57 Dirac, P. A. M., 4, 51, 70, 78- 79, 118, 120-38, 153, 157, 158, 172 ff, 28 experiments, 132 linear equation, 125-26 theory of anti-particles, 132-38, 153-54 Unification of Relativity and Quantum Theories, 157 Unification of Wave Me- chanics and Special Rela- tivity, 154 "Divergencies’', 159-60 Dostoevski, 121-22 Eddington, A., 171 Ehrentest, Paul, 52, 53, 75, 168, 170, 171 ff statistical mechanics, 52 theory of adiabatic invari- ants, 52 Einstein, Albert, xi, xiii, 1, 19, 23, 25, 48, 51, 106-7, 124-26, 140, 157 and Bohr, argument, 114- 16, 25 General Theory of Relativ- ity, 36, 130 Law of Conservation of Mass and Energy, 72, 75, 128, 135. 160, 167 photoelectric effect, two laws of, 25-27 formula, 26-27 relativistic mechanics, 126 Tlieory of Relativity, 19, 37, 106 Electrodynamics, 58, 157 Electromagnetic emission, the- ory of, classical, 36 Electromagnetic energy, quan- tized, 37 Electromagnetic waves, 9, 25, 80, 90, 110, 113, 123, 142 energy of, 19 short, 20 vibrations, 11 Electron levels, quotas for, 64-68, 15 Electron spin, 119, 125-26, 130. See also Spinning electron Energy packages. See Light quanta Equipartition law, 7, 8, 11, 13, 14. 15. 17, 20, 4 Euclid, 107, 110 Exchange forces, theory of, Heitler and London, 150-52, 30 computation of, Ivanienko and Tamm, 151-52 Exclusion Principle. See Pauli Principle Experiments: Anderson, 133 atom-smashing, viii Chadwick, 72 Compton, 1, 27-28, 8 Converi, 153 Cowan, 145, 168 Davisson, 83-84 INDEX 221 Experiments ( cont'd ) Dirac, 132 Fermi, 60, 146 Fermi bottle, 147-48. 29 Frisch, 59 Germer, 83-84 Hahn, 59-60 Heisenberg, 107-10, 24 Helmholtz, von, 89 Michelson, 106 Pancini, 153 piaiu), comparison with, 15- Piccioni, 153 Planck, 25-27 Reines, 145, 168 Rutherford, 33-35, 93 Stern, 85 string, 11-12, 3 Thomson, G., 83-84 Thomson, J. J., 29-32 Tuve laboratory, 60 Faust, 53, 165-218 Fermi, E., 139-49, 152, 168 Atoms in the Family, 148 constant g, 143, 152, 155 degenerated electron gas. study of, 140 experiment, nuclear energy, 59-60, 146 theory, of particle transfor- mation, formulation of, weak interaction, 152 Fermi’s bottle, experiment, 147-48, 29 Feynman, Richard, 161 Fine structure, 48-49 constant, 171 Fish, deep sea, 130-31 Fission reactor, 147 Formula, alpha-particle scat- _ tering, Rutherford, 140 & 87 - m - ms Frisch, Otto, 59 Gamow, Barbara, 168 Gamow, George, 19, 52-57. 171 if**’ Gravity, 123, 158 Kalinga prize, x list of books, x Pauli's letter to, 161, 31 Geissler, Heinrich, 39 Geissler tubes, 39 Gell-Mann, 161 General Theory of Relativity, xiii Generalized Fermi Interac- tion, 146 Germer, L. H., 84 Gibbs, Josiah Willard, 6 Gordon, W., wave equation. 125 Goudsmit, Samuel, 69, 119 Gurney, R. W„ viii, 54, 92 Hahn, Otto, 59, 146 experiment, 59-60 Hamilton's Principle of Least Action, 21 Heisenberg, Werner, 51, 98. 152, 159, 161, 31 atomic model, 99 experiment, 107-10, 24 matrix mechanics, 100-5 non-commutative algebra, 3 Physical Principles of the Quantum Theory, 158-59 quantum microscope, 24 uncertainty principle, 98- formulation of, 58 Heitler, W., 150, 30 Helium, atoms of, 33, 72 Helmholtz, Hermann von, 17. 89, 90 experiment, 89-90 H-H reaction, 145 Houtermans, Fritz, viii, 96 Hydrofluoric acid, 14 Hydrogen, 2, 7-8, 14, 68, 76, 145, 148, 150, 30 Hydrogen atom, 40-43, 48, 65. 90, 133, 145, 150, 12 ’ Dirac linear equation, 125- 26 orbits of, 29-61, 119, 13 theory of, Bohr, 155 Hyperons, 136 Interaction laws, Fermi, 144- 46, 152, 153 constant, Fermi, 143, 152, 1 55 222 THIRTY YEARS THAT SHOOK PHYSICS Interaction laws (corn’d) constant, Yukawa, 152 Ivanienko, D., 150 Jaffe, B., Mlchetson and the Speed of Light, 37 Jeans, Sir James, 9, 11, 15, 19, 171 formula of, 21 and piano analogy, 15-16 , 4 Jeans 1 cube, 13, 15, 16, 20, 115 Jensen, Hans, 76, 77, 91, 18b "Jingle bells,” method of, 112-13. See also Cloud chamber Kant, Immanuel, 106 Kapitza, Peter, 121-22 K-capture process, 114-45 Kepler’s Second Law, 48 Kinetic energy, 7-8 Kinetic Theory of Heat, 6 Kirchhoff, Gustav, 17 Klein, O., 52. 57 formula, 121 wave equation, 125 Kramers, H. A., 51 Landau, L., 52, 171 Light quanta (quantum), 6- 28. 29. 32. 37. 39, 41, 42, 44, 57, 110, 142 Linderstrpm Lang, 50 London, F., theory of ex- change forces, 150, 30 Lyman, Theodore, 41, 12 Magellan, 106 Magic numbers, 76 Magnets, Bitter, 119 Mathematical Aspects of Physics, Bitter, 123 Mathematical Principles of Natural Philosophy, Newton, 99 Matrix, matrices, 100-5 Maupertuis, P. L. M., 44 Maxwell, James C, 6, 74 curves, 9, 10, 21, 1 Mayer, M. Goeppert, 76, 77, 91, 18b Mechanical momentum, 4, 99, 103, 110-14 Meitner, Lise, 59, 146 Mendeleev, D. I., 68 Mendeleev’s periodic system of elements, 76 Mesons, 136, 149, 152-53 Michelson, A. A., 37 experiment, 106 Michelson and the Speed of Light, Jaffe, 37 Millikan, Robert A., 24, 171 Milne, E. A., 171 Minkowski, H., 124 Mott, N. F., 51 Muons, 153, 155 Neddermeyer, S-, 152 Negative mass, 128-30, 132 — 34, 26, 27b Neutrino, 74-76, 141-47, 150, 155, 168, 170, 171 ff anti-, 143 Newton, Sir Isaac, xiii, 99 constant of gravity, 36, 157-58 Law of Universal Gravity, 19, 35, 139 law of motion, 124 equation of, 124 Newtonian classical mechan- ics. 7, 21. 30, 32. 94, 96, 97, 99-106, 111, 113, 123, 124, 157, 23 equation of, 44, 105, 123, 125 Newtonian concept of space and time, xi Nishina, Y., 52, 121 formula, 121 Nobel Prize Dirac. 120 Fermi, 146 Planck, 19 Non-commutative algebra, 3, 102-4 Nuclear atoms, theory of the, Rutherford, 32-36 Nuclear physics, Pauli, 70-74 Oppenheimer, J. Robert, 52, 171 Orbits. 29-61, 119 circular. 36, 43, 44, 46, 47, 90-91, 13 of the electron, 81, 112 INDEX 223 Orbits ( corn’d ) elliptical, 36, 46-49, 90-91, energy of, 47 quantum, 2, 29 if, 43, 64, 68, 70,81-82,92, 112, 13, 19 Origin of Chemical Elements, The, Gamow, ix Pancini, E., experiment, 1S3 Particle: trajectory, 112 transformations, 139-48 See also anti-particles Particle transformation, the- ory of, Fermi, 140-48 Paschen, Friedrich, 41, 12 Pauli, Wolfgang, SI, S3, 62- 79, 160, 161, 167, 171 if atomic model, 68 Effect, 64, 140 letter to Gamow, 31 , 161 neutrino, 74-73, 168 Principle; 63-79, 130, 13, 16a amended, 70, 16b, 16c Second, 133 Theory of Relativity, 63 vacancy, 131 Photoelectric Effect, 22-27 experiments of, 6 laws of, 1, 24-27, 6b, 7 Einstein, 2S-27 Physical Principles of the Quantum Theory, Hei- senberg, 138-39 Piano, 30 ana Jeans' cube. 15-16, 4 Piccioni, O., experiment, 133 Pilot waves, de Broglie, 80-87, Pions, 133, 30 Planck, Max, vii, viii, xi, 1, 3, 4, 6—28, 32, 37 constant A, 21, 82, 143, 137 formula, 21 law of light quanta, 32, 44- Nobel Prize, 19 Notion of Causality In Physics, The, 18 photoelectric effect, 22-27 quantum theory, 19-21, 3 theoretical curves, 21 Scientific Autobiography, A, 18 Second Law of Thermody- namics, 17 theory of radiation-thermo- dynamics, 17 Planckiana, asteroid, 18 Planck's Constant or quantum constant, 21 Poincard, Henri, 138, 161 Potential barrier, 92-93, 96 Powell, C. F., 133 Principal quantum number, 48 Principle of least action, Mau- pertuis, 44 Pythagoras, 138, 161 Quantum orbits. See Orbits Radiant energy, 1, 13, 16-17, 19, 109, 113-16 light quanta, 1, 2 Radiation, 10, 11, 22, 72, 113 frequency of, 2 infrared, 9 pressure, effect of, 109 a uantum, 21 lermal, 9, 11 ultraviolet, 9 Raleigh, Lord, 9, 11, 19 formula of, 21 Heines, Fred, 73, 143, 168 Relativity and Common Sense, Bondi, 123 Ritz, W., 40 Rosenfeld, L., 32, 38 Rosseland, S., 32, 37 Royal Danish Academy of Science, 49 Russell, Bertrand, 161 The Analysis of Matter, 138 Rutherford, Lora Ernest, viii, ix, 33, 72, 83 alpha-particles, 33-33 experiment, 93 formula for scattering, 140 apparatus, 10 atomic model, 1, 33, 33-37 first studies of nuclear reac- tions, 38-39 theory of nuclear atom, 33- 36 224 THIRTY YEARS THAT SHOOK PHYSICS Rydberg constant R, 40, 153 Rydberg's rule, 11 Savannah River Atomic En- ergy commission project, 75 Schrddinger, Erwin, 97, 99, 114 atomic model, S8, 21 equations, 87-92, 105, 118, 123, 125, 150, 22 non-relativistic wave me- chanics, 124 equation of, 125 wave mechanics, theory of, 3 paper on, 98, 105 Schwinger, Julian, 161 Second and Third Laws of Motion, Newton, 99 Segr6, Emilio, 136 Slater, J. C„ 52, 171 Snell’s Law, 94, 23 Solberg, 57 Sommerfeld, Arnold, 18 elliptical orbits, 46-49 formula, 46-47 Special Theory of Relativity, xiii Spectral lines, 2, 3, 30, 40-41, 48, 69, 99, 105, 125, 142 Spectrum, spectra, 3, 30-32, 37, 48, 139 beta-ray, 75 energy of beta-particles, 72, 143, 17 helium, 9 hydrogen, 40, 9 Spinning electron, 68-70, 76, 16. See also Electron spin Statistical mechanics, bade law, 7, 9, 52 Stem, Otto, experiment, 85 Strassman, Fritz, 146 Study of degenerated electron gas, Fermi, 140 Tamm, L, 152 Teller, Edward, 59 Theodolite Tube, 107-10 Theory of Relativity, Einstein, xiii, 19, 37, 73, 106, 118, 119, 122, 124, 126, 157 Theory of Relativity, Pauli, 63 Thermal motion, 6-9 Thermodynamics, 17-18 Thermonuclear reactions, 96- 97 Thomson, Sir George, 84 Thomson, J. J., 29-32, 84 atomic model, 30-32, 34-35 experiments, 29-32 Thought experiment/Gedan- kenexperunent, 13-15 Telman, R. G, 52, 171 Trajectory, 44, 45, 107-12 experiment, 107-10, 24 linear, 105, 113 Tuve, Merle, experiment, 59- 60 Uhlenbeck, George, 69, 119 Ultraviolet Catastrophe, 17, 19, 37 Uncertainty principle, Heisen- berg, 98-117 Unification of Relativity and Quantum Theories, 4, 119, 122-31 Universal Gravity, Theory of, 139 Uranium, 93 atomic model, 68 first chain reaction, 146 isotopes, 76 nucleus, 96 Wave mechanics, viii, 3, 54, 58, 91-94, 97-99, 105 , 23 equation of, 4, 105, 123, 125 Weak interactions, 143, 152 WeizsMcker, Carl von, 52, 145 Wells, H. G., The Invisible Man, 144-45 Widened mathematical lines, 113 Work function, energy of, 27 Yukawa, Hideki, 149, 152-53 constant y, 152, 153, 155 mesons, 152-53 theory of strong interac- tions, 155 Zeeman Effect, 69 A CATALOG OF SELECTED DOVER BOOKS IN ALL FIELDS OF INTEREST A CATALOG OF SELECTED DOVER BOOKS IN ALL FIELDS OF INTEREST CONCERNING THE SPIRITUAL IN ART, Wassily Kandinsky. Pioneering work by father of abstract art Thoughts on color theory, nature of art. Analysis of earlier masters. 12 illustrations. 80pp. of text. 5X * 8H. 2341 1*8 Pa. $3.95 ANIMALS: 1,419 Copyright- Free Illustrations of Mammals, Birds, Fish, Insects, etc, Jim Harter (ed.). Clear wood engravings present, in extremely lifelike poses, over 1 ,000 species of animals. One of the most extensive pictorial sourcebooks of its kind. Captions. Index. 284pp. 9 * 12. 23766-4 Pa. $12 95 CELTIC ART: The Methods of Construction, George Bain. Simple geometric techniques for making Celtic interlacements, spirals, Kells-type initials, animals, humans, etc. Over 500 illustrations. 160pp. 9 x 12. (USO) 22923-8 Pa. $9.95 AN ATLAS OF ANATOMY FOR ARTISTS, Fritz Schider. Most thorough reference work on art anatomy in the world. Hundreds of illustrations, including selections from works by Vesalius, Leonardo, Goya, Ingres, Michelangelo, others. 593 illustrations. 192pp. 7K x 10*4. 20241-0 Pa. $9.95 CELTIC HAND STROKE-BY-STROKE (Irish Half-Uncial from "The Book of Kells'*): An Arthur Baker Calligraphy Manual, Arthur Baker. Complete guide to creating each letter of the alphabet in distinctive Celtic manner. Covers hand position, strokes, pens, inks, paper, more. Illustrated. 48pp. 814 x 11. 24336-2 Pa. $3.95 EASY ORIGAMI, John Montroll. Charming collection of 32 projects (hat, cup, pelican, piano, swan, many more) specially designed for the novice origami hobbyist. Clearly illustrated easy-to-follow instructions insure that even beginning papercrafters will achieve successful results. 48pp. 8Xx 1 1. 27298-2 Pa. $2.95 THE COMPLETE BOOK OF BIRDHOUSE CONSTRUCTION FOR WOOD- WORKERS, Scott D. Campbell. Detailed instructions, illustrations, tables. Also data on bird habitat and instinct patterns. Bibl iography. 3 tables. 63 illustrations in 15 figures. 48pp. 514 x 8& 24407-5 Pa. $1.95 BLOOMINGDALE'S ILLUSTRATED 1886 CATALOG: Fashions, Dry Goods and Housewares, Bloomingdale Brothers. Famed merchants' extremely rare catalog depicting about 1 ,700 products: clothing, housewares, firearms, dry goods, jewelry, more. Invaluable for dating, identifying vintage items. Also, copyright-free graphics for artists, designer^. Co-published with Henry Ford Museum fc Green- field Village. 160pp. 814 x 1 1. 25780-0 Pa. $9.95 HISTORIC COSTUME IN PICTURES. Braun & Schneider. Over 1,450 costumed figures in clearly detailed engravings — from dawn of civilization to end of 19th century. Captions. Many folk costumes. 256pp. 8% x l IK, 23150-X Pa. $1 1.95 CATALOG OF DOVER BOOKS STICKLEY CRAFTSMAN FURNITURE CATALOGS, Gustav Stickley and L, & J. G. Stickley. Beautiful, functional fumituie in two authentic catalogs from 1910. 594 illustrations, including 277 photos, show settles, rockers, armchairs, reclining chairs, bookcases, desks, tables. 183pp. 6H x 9H. 23838*5 Pa. $9.95 AMERICAN LOCOMOTIVES IN HISTORIC PHOTOGRAPHS: 1858 to 1949, Ron Ziel (ed.). A rare collection of 126 meticulously detailed official photographs, called "builder portraits," of American locomotives that majestically chronicle the rise of steam locomotive power in America. Introduction. Detailed captions, xi + 129pp. 9 x 12. 27393-8 Pa. $12.95 AMERICA'S LIGHTHOUSES: An Illustrated History, Francis Ross Holland, Jr. Delightfully written, profusely illustrated fact-filled survey of over 200 American lighthouses since 1716. History, anecdotes, technological advances, more. 240pp. 8x m. 25576-X Pa. $1 1.95 TOWARDS A NEW ARCHITECTURE, Le Corbusier. Pioneering manifesto by founder of "International School." Technical and aesthetic theories, views of industry, economics, relation of form to function, "mass-production split" and much more. Profusely illustrated. 320pp. 6H x 9U. (USO) 25023-7 Pa. $9.95 HOW THE OTHER HALF LIVES, Jacob Riis. Famous journalistic record, exposing poverty and degradation of New York slums around 1900, by major social reformer. 100 striking and influential photographs. 233pp. 10 * 7ft. 22012-5 Pla $10.95 FRUIT KEY AND TWIG KEY TO TREES AND SHRUBS, William M. Harlow. One of the handiest and most widely used identification aids. Fruit key covers 120 deciduous and evergreen species: twig key 160 deciduous species. Easily used. Over 300 photographs. 126pp. 5ft x 8ft. 20511-8 Pa. $3.95 COMMON BIRD SONGS, Dr. Donald J. Borror. Songs of 60 most common U.S. birds: robins, sparrows, cardinals, bluejays, finches, more*— arranged in order of increasing complexity. Up to 9 variations of songs of each species. Cassette and manual 99911-4 $8.95 ORCHIDS AS HOUSE PLANTS, Rebecca Tyson Northcn. Grow catlleyas and many other kinds of orchids — in a window, in a case, or under artificial light. 63 illustrations. 148pp. 5ft x 8Si 23261-1 Pa. $4.95 MONSTER MAZES, Dave Phillips. Masterful mazes at four levels of difficulty. Avoid deadly perils and evil creatures to find magical treasures. Solutions for all 32 exciting illustrated puzzles. 48pp. 11. 26005-4 Pa. $2.95 MOZART'S DON GIOVANNI (DOVER OPERA LIBRETTO SERIES). Wolf- gang Amadeus Mozart. Introduced and translated by Ellen H. Bleiler. Standard Italian libretto, with complete English translation. Convenient and thoroughly portable — an ideal companion for reading along with a recording or the per- formance itself. Introduction. List of characters. Plot summary. 1 21pp. 5ft x 8ft. 24944-1 Pa. $2.95 TECHNICAL MANUAL AND DICTIONARY OF CLASSICAL BALLET, Gail Grant. Defines, explains, comments on steps, movements, poses and concepts. 15-page pictorial section. Basic book for student, viewer. 127pp. 5ft * 8ft. 21843-0 Pa. $4.95 CATALOG OF DOVER BOOKS BRASS INSTRUMENTS: Their History and Development, Anthony Baines. Authoritative, updated survey of the evolution of trumpets, trombones, bugles, comets, French horns, tubas and other brass wind instruments. Over 140 illustra- tions and 48 music examples. Corrected and updated by author. New preface. Bibliography. 320pp. 554 * 8tt. 27574-4 Pa. 19.95 HOLLYWOOD GLAMOR PORTRAITS. John Kobal (ed.). 145 photos from 1926-49. Harlow, Gable, Bogart, Bacall; 94 stars in all. Full background on photographers, technical aspects. 1 60pp. 854 x II 54. 23352-9 Pa. $ 1 1 .95 MAX AND MORITZ, Wilhelm Busch. Great humor classic in both German and English. Also 10 other works: "Cat and Mouse," "Plisch and Plumm," etc. 216pp. 5Xx8& 20161-3 Pa. 3195 THE RAVEN AND OTHER FAVORITE POEMS, Edgar Allan Poe. Over 40 of the author’s most memorable poems: "The Bells," "Ulalume," "Israfel," "To Helen," "The Conqueror Worm," "Eldorado," "Annabel Lee," many more. Alphabetic lists of titles and first lines. 64pp. 5 Ms x 854. 26685-0 Pa. 31 00 SEVEN SCIENCE FICTION NOVELS, H. G. Wells. The standard collection of the great novels. Complete, unabridged. First Men in the Moon, Island of Dr. Moreau, War of the Worlds, Food of the Gods, Invisible Man, Time Machine, In the Days of the Comet. Total of 1,015pp. 554 x 8)4. (USO) 20264-X Clothbd. 329.95 AMULETS AND SUPERSTITIONS. E. A. Wallis Budge. Comprehensive dis- course on origin, powers of amulets in many ancient cultures: Arab, Persian, Babylonian, Assyrian, Egyptian, Gnostic, Hebrew, Phoenician, Syriac, etc. Covets cross, swastika, crucifix, seals, rings, stones, etc. 584pp. 554 x 85L 23573-4 Pa. 312.95 RUSSIAN STORIES/PYGCKNE PACCKA3bl: A Dual-Language Book, edited by Gleb Struve. Twelve tales by such masters as Chekhov, Tolstoy, Dostoevsky. Pushkin, others. Excellent word-for-word English translations on facing pages, plus teaching and study aids, Russian/English vocabulary, biographical/critical introductions, more. 416pp. 554 * 8H. 26244*8 Pa. 38.95 PHILADELPHIA THEN AND NOW: 60 Sites Photographed in the Past and Present, Kenneth Finkel and Susan Oyama. Rare photographs of City Hall, Logan Square, Independence Hall, Betsy Ross House, other landmarks juxtaposed with contemporary views. Captures changing face of historic city. Introduction. Captions. 128pp. 854 * 11. 25790-8 Pa. 39.95 AIA ARCHITECTURAL GUIDE TO NASSAU AND SUFFOLK COUNTIES, LONG ISLAND, The American Institute of Architects, Long Island Chapter, and the Society for the Preservation of Long Island Antiquities. Comprehensive, well- researched and generously illustrated volume brings to life over three centuries of Long Island’s great architectural heritage. More than 240 photographs with authoritative, extensively detailed captions. 176pp. 854 x ) 1. 26946-9 Pa. 314.95 NORTH AMERICAN INDIAN LIFE: Customs and Traditions of 23 Tribes, Elsie Clews Parsons (ed.). 27 fictionalized essays by noted anthropologists examine religion, customs, government, additional facets of life among the Winnebago, Crow, Zuni, Eskimo, other tribes. 480pp. 654 x 954. 27377-6 Pa. 310.95 CATALOG OF DOVER BOOKS FRANK LLOYD WRIGHT S HOLLYHOCK HOUSE, Donald Hoffmann. Lav- ishly illustrated, carefully documented study of one of Wright's most controversial residential designs. Over 120 photographs, floor plans, elevations, etc. Detailed perceptive text by noted Wright scholar. Index. 128pp. 9K x 1014. 27133-1 Pa. 111.95 THE MALE AND FEMALE FIGURE IN MOTION: 60 Classic Photographic Sequences, Eadweard Muybridge. 60 true-action photographs of men and women walking, running, climbing, bending, turning, etc., reproduced from rare 19th- century masterpiece, vi + 121pp. 9 x 12. 24745-7 Pa. 510.95 1001 QUESTIONS ANSWERED ABOUT THE SEASHORE. N. J. Berrill and Jacquelyn Berrill. Queries answered about dolphins, sea snails, sponges, starfish, fishes, shore birds, many others. Covers appearance, breeding, growth, feeding, much more. 305pp. 514 x 8 k. 23366-9 Pa. $7.95 GUIDE TOOWL WATCHING IN NORTH AMERICA, Donalds. Heintzelman. Superb guide offers complete data and descriptions of 19 species: bam owl, screech owl, snowy owl, many more. Expert coverage of owl- watching equipment, conservation, migrations and invasions, etc. Guide to observing sites. 84 illustra- tions. xiii + 193pp. 5K x 8H. 27344-X Pa. $8.95 MEDICINAL AND OTHER USES OF NORTH AMERICAN PLANTS: A Historical Survey with Special Reference to the Eastern Indian Tribes, Charlotte Erichsen-Brown. Chronological historical citations document 500 years of usage of plants, trees, shrubs native to eastern Canada, northeastern U.S. Also complete identifying information. 343 illustrations. 544pp. 6H x 914. 25951-X Pa. 512.95 STORYBOOK MAZES, Dave Phillips. 23 stories and mazes on two-page spreads: Wizard of Oz, Treasure Island, Robin Hood, etc. Solutions. 64pp. 814 x | L 23628-5 Pa. 52.95 NEGRO FOLK MUSIC. U.S.A., Harold Courlander. Noted folklorist's scholarly yet readable analysis of rich and varied musical tradition. Includes authentic versions of over 40 folk songs. Valuable bibliography and discography, xi + 324pp. 5H x 8H. 27350-4 Pa. 57.95 MOVIE-STAR PORTRAITS OF THE FORTIES. John Kobal (ed.). 163 glamor, studio photos of 106 stars of the 1940$: Rita Hayworth, Ava Gardner, Marlon Brando, Clark Gable, many more. 176pp. 8H x UK. 23546-7 Pa. 51 1.95 BENCHLEY LOST AND FOUND, Robert Benchley. Finest humor from early 30s, about pet peeves, child psychologists, post office and others. Mostly unavailable elsewhere. 73 illustrations by Peter Arno and others. 183pp. 514 x 8H. 22410-4 Pa. |5.95 YEKL and THE IMPORTED BRIDEGROOM AND OTHER STORIES OF YIDDISH NEW YORK, Abraham Cahan. Film Hester Street based on Yekl (1896). Novel, other stories among first about Jewish immigrants on N.Y.’s East Side. 240pp. 514 x 814. 22427-9 Pa. 56.95 SELECTED POEMS, Walt Whitman. Generous sampling from Leaves of Grass Twenty-four poems include "I Hear America Singing," "Song of the Open Road," "I Sing the Body Electric," "When Lilacs Last in the Dooryard Bloom'd," "O Captainl My Captain!"— all reprinted from an authoritative edition. Lists of titles and first lines. 128pp. 5VU x 8'4. 26878-0 Pa. 5L00 CATALOG OF DOVER BOOKS THE BEST TALES OF HOFFMANN. E. T. A. Hoffmann. lOof Hoffmann’s most Important stories: “Nutcracker and the King of Mice/* "The Golden Flowerpot/ 1 etc. 458pp. 54 x 84. 21798-0 ft. $8.95 FROM FETISH TO COD IN ANCIENT EGYPT, E. A. Wallis Budge. Rich detailed survey of Egyptian conception of “God” and gods, magic, cult of animals, Osiris, more. Also, superb English translations of hymns and legends. 240 illustrations. 545pp. 54 x 84. 25803-3 Pa. $1 1.95 FRENCH STORIES/CONTES FRAN£AIS: A Dual-Language Book, Wallace Fowlie. Ten stories by French masters, Voltaire to Camus: “Micromegas" by Voltaire; "The Atheist's Mass” by Balzac; “Minuet” by de Maupassant; “The Guest” by Camus, six more. Excellent English translations on facing pages. Also French-English vocabulary list, exercises, more. 352pp. 54 x 84. 26443*2 Pa. $8 95 CHICAGO AT THE TURN OF THE CENTURY IN PHOTOGRAPHS: 122 Historic Views from the Collections of the Chicago Historical Society, Larry A. Viskochil. Rare large-format prints offer detailed views of City Hall, State Street, the Loop, Hull House, Union Station, many other landmarks, circa 1904-1913. Introduction. Captions. Maps. 144pp. 94 x 124. 24656-6 ft. $12.95 OLD BROOKLYN IN EARLY PHOTOGRAPHS, 1865-1929, William Lee Younger. Luna Park, Gravesend race track, construction of Grand Army Plaza, moving of Hotel Brighton, etc 157 previously unpublished photographs. 165pp. 84x114. 23587-4 Pa. *13.95 THE MYTHS OF THE NORTH AMERICAN INDIANS, Lewis Spence. Rich anthology of the myths and legends of the Algonquins, Iroquois, Pawnees and Sioux, prefaced by an extensive historical and ethnological commentary. 36 illustrations. 480pp. 54 x 84. 25967-6 Pa. *8.95 AN ENCYCLOPEDIA OF BATTLES: Accounts of Over 1,560 Battles from 1479 bc to the Present, David Eggenberger. Essential details of every major battle in recorded history from the first battle of Megiddo in 1 479 ac to Grenada in 1 984. List of Battle Maps. New Appendix covering the years 1967-1984. Index. 99 illustrations. 544pp. 64 x 94. 24913-1 Pa. $14.95 SAILING ALONE AROUND THE WORLD, Captain Joshua Slocum. First man to sail around the world, alone, in small boat. One of great feats of seamanship told in delightful manner. 67 illustrations* 294pp. 54 x 84. 20326-3 Pa. $5.95 ANARCHISM AND OTHER ESSAYS, Emma Goldman. Powerful, penetrating, prophetic essays on direct action, role of minorities, prison reform, puritan hypocrisy, violence, etc. 271pp. 54 x 84. 22484-8 ft. $5.95 MYTHS OF THE HINDUS AND BUDDHISTS. Ananda K. Coomaiaswatny and Sister Nivedita. Great stories of the epics; deeds of Krishna, Shiva, taken from puranas, Vedas, folk tales; etc. 32 illustrations. 400pp. 54 x 84. 21759-0 ft. $9.95 BEYOND PSYCHOLOGY, Otto Rank. Fear of death, desire of immortality, nature of sexuality, social organization, creativity, according to Rankian system. 291pp. 54 x 84. 20485-5 ft. $8.95 A THEOLOGICO-POLITICAL TREATISE, Benedict Spinoza. Also contains unfinished Political Treatise. Great classic on religious liberty, theory of govern- ment on common consent. R. Elwes translation. Total of 421pp. 54 x 84. 20249-6 ft. $8.95 CATALOG OF DOVER BOOKS MY BONDAGE AND MY FREEDOM, Frederick Douglass. Bom a slave, Douglass became outspoken force in antislavery movement. The best of Douglass' auto* biographies. Graphic description of slave life. 464pp. 5H x 8k. 22457*0 Pa. $8.95 FOLLOWING THE EQUATOR: A Journey Around the World, Mark Twain. Fascinating humorous account of 1897 voyage to Hawaii, Australia, India, New Zealand, etc. Ironic, bemused reports on peoples, customs, climate, flora and fauna, politics, much more. 197 illustrations. 720pp. 5H * 8H. 261 1S-1 Pa. $15.95 THE PEOPLE CALLED SHAKERS, Edward D. Andrews. Definitive study of Shakers: origins, beliefs, practices, dances, social organization, furniture and crafts, etc. 33 illustrations. 351pp. 5H x 8K. 21081-2 Pa. $8.95 THE MYTHS OF GREECE AND ROME, H. A. Guerber. A classic of mythology, generously illustrated, long prized for its simple, graphic, accurate retelling of the principal myths of Greece and Rome, and for its commentary on their origins and significance. With 64 illustrations by Michelangelo, Raphael, Titian, Rubens, Canova, Bernini and others. 480pp. 5H x 8H. 27584-1 Pa. $9.95 PSYCHOLOGY OF MUSIC, Carl E. Seashore. Classic work discusses music as a medium from psychological viewpoint. Clear treatment of physical acoustics, auditory apparatus, sound perception, development of musical skills, nature of musical feeling, host of other topics. 88 figures. 408pp. 5H x 8H. 21851-1 Pa. $9.95 THE PHILOSOPHY OF HISTORY. Georg W. Hegel. Great classic of Western thought develops concept that history is not chance but rational process, the evolution of freedom. 457pp. 5k * 8k. 201 12-0 Pa. $9.95 THE BOOK OF TEA, Kakuzo Okakura. Minor classic of the Orient: entertaining, charming explanation, interpretation of traditional Japanese culture in terms of tea ceremony* 94pp. 5H x 8k. 20070-1 Pa. $3.95 LIFE IN ANCIENT EGYPT, Adolf Erman. Fullest, most thorough, detailed older account with much not in more recent books, domestic life, religion, magic, medicine, commerce, much more. Many illustrations reproduce tomb paintings, carvings, hieroglyphs, etc. 597pp. 5k x 8k. 22632-8 Pa. $10.95 SUNDIALS, Their Theory and Construction, Albert Waugh. Far and away the best, most thorough coverage of ideas, mathematics concerned, types, construction, adjusting anywhere. Simple, nontechnical treatment allows even children to build several of these dials. Over 100 illustrations. 230pp. 5H x 8H. 22947-5 Pa. $7.95 DYNAMICS OF FLUIDS IN POROUS MEDIA. Jacob Bear. For advanced students of ground water hydrology, soil mechanics and physics, drainage and irrigation engineering, and more. 335 illustrations. Exercises, with answers. 784pp. 6Kx9U. 65675-6 Pa. $19.95 SONGS OF EXPERIENCE: Facsimile Reproduction with 26 Plates in Full Color, William Blake. 26 full-color plates from a rare 1826 edition. Includes "The Tyger," "London," "Holy Thursday," and other poems. Printed text of poems. 48pp. 5k x 7. 24636-1 Pa. $4.95 OLD-TIME VIGNETTES IN FULL COLOR, Carol Belanger Grafton fed.). Over 390 charming, often sentimental illustrations, selected from archives of Victorian graphics— pretty women posing, children playing, food, flowers, kittens and puppies, smiling cherubs, birds and butterflies, much more. All copyright-free. 48pp. 9k x 12K. 27269-9 to. $5.95 CATALOG OF DOVER BOOKS PERSPECTIVE FOR ARTISTS, Rex Vicat Cole. Depth, perspective of sky and sea, shadows, much more, not usually covered. 891 diagrams, 81 reproductions of drawings and paintings. 279pp. 5ft * 8tf. 22487*2 Pa. $6.95 DRAWING THE LIVING FIGURE, Joseph Sheppard. Innovative approach to artistic anatomy focuses on specifics of surface anatomy, rather than muscles and bones. Over 170 drawings of live models in front, back and side views, and in widely varying poses. Accompanying diagrams. 177 illustrations. Introduction. Index. 144pp. 8* x 1 IK. 26723-7 Pa. $8.95 GOTHIC AND OLD ENGLISH ALPHABETS: 100 Complete Fonts, Dan X. Solo. Add power, elegance to posters, signs, other graphics with 100 stunning copyright- free alphabets: Blackstone, Dolbey, Germania, 97 more— including many lower- case, numerals, punctuation marks. 104pp. 8K * 1 1. 24695-7 Pa. $8.95 HOW TO DO BEADWORK, Mary White. Fundamental book on craft from simple projects to five-bead chains and woven works. 106 illustrations. 142pp. 5ft * 8. 20697-1 Pa. $4.95 THE BOOK OF WOOD CARVING, Charles Marshall Sayers. Finest book for beginners discusses fundamentals and offers 34 designs. “Absolutely first rate . . well thought out and well executed.”— E. J. Tangerman. 1 18pp. 7ft * 10ft. 23654-4 Pa. $5.95 ILLUSTRATED CATALOG OF CIVIL WAR MILITARY GOODS: Union Army Weapons, Insignia, Uniform Accessories, and Other Equipment, Schuyler. Hartley, and Graham. Rare, profusely illustrated 1846 catalog includes Union Army uniform and dress regulations, arms and ammunition, coats, insignia, flags, swords, rifles, etc. 226 illustrations. 160pp. 9 * 12. 24939-5 P!a. $10.95 WOMEN'S FASHIONS OFTHE EARLY 1900s: An Unabridged Republication of “New York Fashions, 1909," National Cloak 8c Suit Co. Rare catalog of mail-order fashions documents women's and children's clothing styles shortly after the turn of the century. Captions offer full descriptions, prices. Invaluable resource for fashion, costume historians. Approximately 725 illustrations. 128pp. 8ft * 1 114. 27276-1 Pa. $11.95 THE 1912 AND 1915 GUSTAV STICKLEY FURNITURE CATALOGS. Gustav Stickley. With over 200 detailed illustrations and descriptions, these two catalogs are essential reading and reference materials and identification guides for Stickley furniture. Captions cite materials, dimensions and prices. 1 12pp. 6H * 9ft. 26676-1 Pa. $9.95 EARLY AMERICAN LOCOMOTIVES. John H. White, Jr. Finest locomotive engravings from early 19th century: historical (1804*74), main-line (after 1870), special, foreign, etc 147 plates. 142pp. 1 1ft x 8ft. 22772-3 Pa. $10.95 THE TALL SHIPS OF TODAY IN PHOTOGRAPHS, Frank O. Braynard. Lavishly illustrated tribute to nearly 100 majestic contemporary sailing vessels: Amerigo Vespucci, Clearwater, Constitution, Eagle, Mayflower, Sea Cloud, Victory, many more. Authoritative captions provide statistics, background on each ship. 190 black-and-white photographs and illustrations. Introduction. 128pp. 8ft x lift. 27163-3 Pb. $13.95 CATALOG OF DOVER BOOKS EARLY NINETEENTH-CENTURY CRAFTS AND TRADES, Peter Slockham (ed.). Extremely rare 1 807 volume describes to youngsters the crafts and trades of the day: brickmaker, weaver, dressmaker, bookbinder, ropemaker, saddler, many more. Quaint prose, charming illustrations for each craft. 20 black-and-white line illustrations. 1 92pp. 4X *6. 27293-1 Pa. $4.95 VICTORIAN FASHIONS AND COSTUMES FROM HARPER’S BAZAR, 1867- 1898, Stella Blum (ed.). Day costumes, evening wear, sports clothes, shoes, hats, other accessories in over 1,000 detailed engravings. 320pp. 9fc x ]2K. 22990-4 Pa. $13.95 GUSTAV STICKLEY, THE CRAFTSMAN. Mary Ann Smith. Superb study surveys broad scope of Stickley’s achievement, especially in architecture. Design philosophy, rise and fall of the Craftsman empire, descriptions and floor plans for many Draftsman houses, more. 86 black-and-white halftones. 31 line illustrations. Introduction. 208pp. 6H * 9K. 27210-9 Pa. $9.95 THE LONG ISLAND RAIL ROAD IN EARLY PHOTOGRAPHS, Ron Ziel. Over 220 rare photos, informative text document origin ( 1844) and development of rail service on Long Island. Vintage views of early trains, locomotives, stations, passengers, crews, much more. Captions. 8% x 1 IK. 26301-0 Pa. $13.95 THE BOOK OF OLD SHIPS: From Egyptian Galleys to Clipper Ships, Henry B. Culver. Superb, authoritative history of sailing vessels, with 80 magnificent line illustrations. Galley, bark, caravel, longship, whaler, many more. Detailed, informative text on each vessel by noted naval historian. Introduction. 256pp. 5K*8K. 27332-6 Pa. $6.95 TEN BOOKS ON ARCHITECTURE, Vitruvius. The most important book ever written on architecture. Early Roman aesthetics, technology, classical orders, site selection, all other aspects. Morgan translation. 331pp. 5K x 8H. 20645-9 Pa. $8.95 THE HUMAN FIGURE IN MOTION, Eadweard Muybridge. More than 4.500 stopped-action photos, in action series, showing undraped men, women, children jumping, lying down, throwing, sitting, wrestling, carrying, etc. 390pp. 7 fcx 109fi. 20204-6 Clothbd. $24.95 TREES OF THE EASTERN AND CENTRAL UNITED STATES AND CANADA, William M. Harlow. Best one-volume guide to 140 trees. Full descrip- tions, woodlore, range, etc. Over 600 illustrations. Handy size. 288pp. 4H * 6K. 20395-6 Pa. $5.95 SONGS OF WESTERN BIRDS, Dr. Donald J. Borror. Complete song and call repertoire of 60 western species, including flycatchers, juncoes, cactus wrens, many more — includes fully illustrated booklet. Cassette and manual 99913-0 $8.95 GROWING AND USING HERBS AND SPICES, Milo Miloradovich. Versatile handbook provides all the information needed for cultivation and use of all the herbs and spices available in North America. 4 illustrations. Index. Glossary. 236pp. 5K x 8& 25058-X Pa. $6.95 BIG BOOK OF MAZES AND LABYRINTHS, Walter Shepherd. 50 mazes and labyrinths in aH — classical, solid, ripple, and more — in one great volume. Perfect inexpensive puzzler for clever youngsters. Full solutions. 1 1 2pp. 8K x \ \, 22951-3 Pa. $4.95 CATALOG OF DOVER BOOKS PIANO TUNING. J. Cree Fischer. Clearest, best book for beginner, amateur. Simple repairs, raising dropped notes, tuning by easy method of flattened fifths. No previous skills needed. 4 illustrations. 201pp. 5K x 811. 23267*0 Pa. $5.95 A SOURCE BOOK IN THEATRICAL HISTORY. A. M. Nagler. Contemporary observers on acting, directing. make*up. costuming, stage props, machinery, scene design, from Ancient Greece to Chekhov. 61 1 pp. 5X x 811. 20515*0 Pa. $ 1 1.95 THE COMPLETE NONSENSE OF EDWARD LEAR, Edward Lear. All nonsense limericks, zany alphabets. Owl and Pussycat, songs, nonsense botany, etc., illustrated by Lear. Total of 320pp. 5H x 8 H. (USO) 20167-8 Pa. $6.95 VICTORIAN PARLOUR POETRY: An Annotated Anthology, Michael R. Turner. 1 17 gems by Longfellow, Tennyson. Browning, many lesser-known poets. "The Village Blacksmith," "Curfew Must Not Ring Tonight," "Only a Baby Small,*' dozens more, often difficult to find elsewhere. Index of poets, titles, first lines, xxiii + 325pp. 511 x 814. 27044-0 Pa. $8.95 DUBLINERS, James Joyce. Fifteen stories offer vivid, tightly focused observations of the lives of Dublin's poorer classes. At least one, "The Dead." is considered a masterpiece. Reprinted complete and unabridged from standard edition. 160pp. 5ttt x 814. 26870-5 Pa. $1.00 THE HAUNTED MONASTERY and THE CHINESE MAZE MURDERS. Robert van Gulik. Two full novels by van Gulik, set in 7th*century China, continue adventures of Judge Dee and his companions. An evil Taoist monastery, seemingly supernatural events; overgrown topiary maze hides strange crimes. 27 illustrations. 328pp. 511 x 8 H. 23502-5 Pa. $7.95 THE BOOK OF THE SACRED MAGIC OF ABRAMEUN THE MAGE, translated by S. MacGregor Mathers. Medieval manuscript of ceremonial magic. Basic document in Aleister Crowley, Golden Dawn groups. 268pp. 511 x 8 & 23211-5 Pa. $8.95 NEW RUSSIAN -ENGLISH AND ENGLISH-RUSSIAN DICTIONARY, M. A. O'Brien. This is a remarkably handy Russian dictionary, containing a surprising amount of information, including over 70,000 entries. 366pp. 4H x 6 H 20208*9 Pa. $9.95 HISTORIC HOMES OF THE AMERICAN PRESIDENTS, Second, Revised Edition, Irvin Haas. A traveler's guide to American Presidential homes, most open to the public, depicting and describing homes occupied by every American President from George Washington to George Bush. With visiting hours, admission charges, travel routes. 175 photographs. Index. 160pp. 814 x 11. 26751*2 Pa. $10.95 NEW YORK IN THE FORTIES, Andreas Feininger. 162 brilliant photographs by the well-known photographer, formerly with Life magazine. Commuters, shoppers. Times Square at night, much else from city at its peak. Captions by John von Hartz. 181pp. 914 x 10H. 23585-8 Pa. $12.95 INDIAN SIGN LANGUAGE, William Tomkins. Over 525 signs developed by Sioux and other tribes. Written instructions and diagrams. Also 290 pictographs. 1 1 1 pp. 6H x 914 . 22029-X Pa. $3.50 CATALOG OF DOVER BOOKS ANATOMY: A Complete Guide for Artists, Joseph Sheppard* A master of figure drawing shows artists how to render human anatomy convincingly. Over 460 illustrations. 224pp. 8K * 1 1*. 27279*6 Pa. $10.95 MEDIEVAL CALLIGRAPHY: Its History and Technique, Marc Drogin. Spirited history, comprehensive instruction manual covers 15 styles (ca. 4th century thru 15th). Excellent photographs; directions for duplicating medieval techniques with modern tools. 224pp. 8tt * 1 114. 26142*5 Pa. $1 1.95 DRIED FLOWERS: How to Prepare Them, Sarah Whitlock and Martha Rankin. Complete instructions on how to use silica gel, meal and borax, perlite aggregate, sand and borax, glycerine and water to create attractive permanent flower arrangements. 12 illustrations. 52pp. 554 * 8H. 21802*5 Pa. $1.00 EASY-TO-MAKE BIRD FEEDERS FOR WOODWORKERS, Scott D. Campbell. Detailed, simple*to*use guide for designing, constructing, caring for and using feeders. Text, illustrations for 12 classic and contemporary designs. 96pp. 554 x 8V4. 25847*5 Pa. $2.95 OLD-TIME CRAFTS AND TRADES, Peter Stockham. An 1807 book created to teach children about crafts and trades open to them as future careers. It describes in detailed, nontechnical terms 24 different occupations, among them coachmaker, gardener, hairdresser, lacemaker, shoemaker, wheelwright, copper-plate printer, milliner, trunkmaker, merchant and brewer. Finely detailed engravings illustrate each occupation. 192pp. 454 * 6. 27598-9 Pa. $4.95 THE HISTORY OF UNDERCLOTHES, C Willett Cunnington and Phyllis Cunnington. Fascinating, well-documented survey covering six centuries of English undergarments, enhanced with over 100 illustrations: 12th-century laced- up bodice, footed long drawers (1795), 19th-century bustles, 19th-century corsets for men, Victorian “bust improvers,'* much more. 272pp. 554 x 814. 27124-2 Pa. $9.95 ARTS AND CRAFTS FURNITURE: The Complete Brooks Catalog of 1912, Brooks Manufacturing Co. Photos and detailed descriptions of more than 150 now very collectible furniture designs from the Arts and Crafts movement depict davenports, settees, buffets, desks, tables, chairs, bedsteads, dressers and more, all built of solid, quarter-sawed oak. Invaluable for students and enthusiasts of antiques, Americana and the decorative arts. 80pp. 654 x 954. 27471-5 Pa. $7.95 HOW WE INVENTED THE AIRPLANE: An Illustrated History, Orville Wright. Fascinating firsthand account covers early experiments, construction of planes and motors, first flights, much more. Introduction and commentary by FredG Kelly. 76 photographs. 96pp. 814 x 1 1. 25662-6 Pa. $8.95 THE ARTS OF THE SAILOR: Knotting, Splicing and Ropework, Hervey Garrett Smith. Indispensable shipboard reference covers tools, basic knots and useful hitches; handsewing and canvas work, more. Over 100 illustrations. Delightful reading for sea lovers. 256pp. 5S * 8H. 26440-8 Pa. $7.95 FRANK LLOYD WRIGHT'S FALLINGWATER: The House and Its History, Second, Revised Edition, Donald Hoffmann. A total revision — both in text and illustrations— of the standard document on Fallingwater, the boldest, most personal architectural statement of Wright's mature years, updated with valuable new material from the recently opened Frank Lloyd Wright Archives. “Fasci- nating " — The New York Times. 116 illustrations. 128pp. 914 x |(ft. 27450*6 Pa. $10.95 CATALOG OF DOVER BOOKS PHOTOGRAPHIC SKETCHBOOK OF THE CIVIL WAR, Alexander Gardner. 100 photos taken on field during the Civil War. Famous shots of Manassas, Harper’s Ferry, Lincoln, Richmond, slave pens, etc. 244pp. I Ob x 8b. 22731-6 Pa. $9.95 FIVE ACRES AND INDEPENDENCE, Maurice G. Kains. Great back-to-the-land classic explains basics of self-sufficient farming. The one book to get. 95 illustrations. 397pp. 5b * 8H. 20974-1 Pa. $7.95 SONGS OF EASTERN BIRDS, Dr. Donald J. Borror. Songs and calls of 60 species most common to eastern U.S.: warblers, woodpeckers, flycatchers, thrushes, larks, many more in high-quality recording. Cassette and manual 99912-2 $8.95 A MODERN HERBAL, Margaret Grieve. Much the fullest, most exact, most useful compilation of herbal material. Gigantic alphabetical encyclopedia, from aconite to zedoary, gives botanical information, medical properties, folklore, economic uses, much else. Indispensable to serious reader. 161 illustrations. 888pp. 6H* 9b. 2-vol. set (USO) Vol. 1: 22798-7 Pa, $9.95 Vol. II: 22799-5 Pa. $9.95 HIDDEN TREASURE MAZE BOOK, Dave Phillips. Solve 34 challenging mazes accompanied by heroic tales of adventure. Evil dragons, people-eating plants, bloodthirsty giants, many more dangerous adversaries lurk at every twist and turn. 34 mazes, stories, solutions. 48pp. 8^4 * 1 1. 24566-7 Pa. $2.95 LETTERS OF W. A. MOZART, Wolfgang A. Mozart. Remarkable letters show bawdy wit, humor, imagination, musical insights, contemporary musical world: includes some letters from Leopold Mozart 276pp. 5K * 8H. 22859-2 Pa. $7.95 BASIC PRINCIPLES OF CLASSICAL BALLET, Agrippina Vaganova. Great Russian theoretician, teacher explains methods for teaching classical ballet. 1 18 illustrations. 175pp. 5b x 8H. 22036-2 Pa. $4.95 THE JUMPING FROG, Mark Twain. Revenge edition. Theoriginal story of The Celebrated Jumping Frog of Calaveras County, a hapless French translation, and Twain's hilarious * ’retranslation” from the French. 12 illustrations. 66pp. 22686-7 Pa. $3.95 BEST REMEMBERED POEMS, Martin Gardner (ed.). The 126 poems in this superb collection of 19th- and 20th-century British and American verse range from Shelley's ”To a Skylark” to the impassioned ’’Renascence” of Edna St. Vincent Millay and to Edward Lear’s whimsical ’’The Owl and the Pussycat. ” 224pp. 5b x 8H. 27165*XPa.$4.95 COMPLETE SONNETS, William Shakespeare. Over 150 exquisite poems deal with love, friendship, the tyranny of time, beauty’s evanescence, death and other themes in language of remarkable power, precision and beauty. Glossary of archaic terms. 80pp. 5Vfo x 8b. 26686-9 Pa. $1.00 BODIES IN A BOOKSHOP, R. T. Campbell. Challenging mystery of blackmail and murder with ingenious plot and superbly drawn characters. In the best tradition of British suspense fiction. 192pp. 5H x 8H. 24720-1 Pa. $5.95 CATALOG OF DOVER BOOKS THE WIT AND HUMOR OF OSCAR WILDE. Alvin Redman (ed.). More than 1 .000 ripostes, paradoxes, wisecracks: Work is the curse of the drinking classes; I can resist everything except temptation; etc. 258pp. 5H x 20602-5 Pa. $5.95 SHAKESPEARE LEXICON AND QUOTATION DICTIONARY. Alexander Schmidt. Full definitions, locations, shades of meaning in every word in plays and poems. More than 50,000 exact quotations. 1.485pp. 6H x 9K. 2-vol. set. Vol. 1: 22726-X Pa. $16.95 Vol. 2: 22727-8 Pa. $15.95 SELECTED POEMS, Emily Dickinson. Over 100 best-known, best-loved poems by one of America's foremost poets, reprinted from authoritative early editions. No comparable edition at this price. Index of first lines. 64pp. 5 5 /i6 x 8!L 26466-1 Pa. $1.00 CELEBRATED CASES OF JUDGE DEE (DEE GOONG AN), translated by Robert van Gulik. Authentic 18th-century Chinese detective novel; Dee and associates solve three interlocked cases. Led to van Gulik's own stories with same characters. Extensive introduction. 9 illustrations. 257pp. 5tt x 8tt. 2S5S7-5 Pa. $6.95 THE MALLEUS MALEFICARUM OF KRAMER AND SPRENGER, translated by Montague Summers. Full text of most important witchhunter's "bible," used by both Catholics and Protestants. 278pp. 6* x 10. 22802-9 Pa. $11.95 SPANISH STORIES/CUENTOS ESPAftOLES: A Dual-Language Book, Angel Flores (ed.). Unique format offers 13 great stories in Spanish by Cervantes, Borges, others. Faithful English translations on facing pages. 352pp. 5H x 8H. 25399-6 Pa. $8.95 THE CHICAGO WORLD'S FAIR OF 1893: A Photographic Record, Stanley Appelbaum (ed.). 128 rare photos show 200 buildings. Beaux- Arts architecture, Midway, original Ferris Wheel, Edison's kinetoscope, more. Architectural empha- sis; full text. 116pp. 8X x H. 23990-X Pa. $9.95 OLD QUEENS. N.Y., IN EARLY PHOTOGRAPHS, Vincent F. Seyfried and William Asadorian. Over 160 rare photographs of Maspeth, Jamaica, Jackson Heights, and other areas. Vintage views of DeWitt Clinton mansion, 1939 World's Fair and more. Captions. 192pp. 8% x 1 1. 26358-4 Pa. $12.95 CAPTURED BY THE INDIANS: 15 Firsthand Accounts. 1750-1870. Frederick Drimmer. Astounding true historical accounts of grisly torture, bloody conflicts, relentless pursuits, miraculous escapes and more, by people who lived to tell the tale. 384pp. 5H x 8H. 24901-8 Pa. $8.95 THE WORLD'S GREAT SPEECHES, Lewis Copeland and Lawrence W. Lamm (eds.). Vast collection of 278 speeches of Greeks to 1970. Powerful and effective models; unique look at history. 842pp. 5H x 8H. 20468-5 Pa. $14.95 THE BOOK OFTHE SWORD, Sir Richard F. Burton. Great Victorian scholar/ad- venturer's eloquent, erudite history of the "queen of weapons" — from prehistory to early Roman Empire. Evolution and development of early swords, variations (sabre, broadsword, cutlass, scimitar, etc.), much more. 336pp. 6Kx(N. 25434-8 Pa. $8.95 CATALOG OF DOVER BOOKS AUTOBIOGRAPHY: The Story of My Experiments with Truth, Mohandas K. Gandhi. Boyhood, legal studies, purification, the growth of the Satyagraha (nonviolent protest) movement. Critical, inspiring work of the man responsible for the freedom of India. 480pp. 5K x 8H. (USO) 24598*4 ft. 88.95 CELTIC MYTHS AND LEGENDS. T. W. Rolleston. Masterful retelling of Irish and Welsh stories and tales. Cuchulain, King Arthur, Deirdre, the Grail, many more. First paperback edition. 58 full-page illustrations. 512pp. 5K x 8& 26507-2 ft. 89.95 THE PRINCIPLES OF PSYCHOLOGY, William James. Famous long course complete, unabridged. Stream of thought, time perception, memory, experimental methods: great work decades ahead of its time. 94 figures. 1 ,89 1 pp. 5H x 6& 2-vol. set. Vol. 1: 20881*6 ft. 812.95 Vol. II: 20882*4 ft. 512.95 THE WORLD AS WILL AND REPRESENTATION, Arthur Schopenhauer. Definitive English translation of Schopenhauer’s life work, correcting more than 1 ,000 errors, omissions in earlier translations. Translated by E. F. J. Payne. Total o( 1,269pp. 5X x 8H. 2-vol. set Vol. 1: 21761*2 ft. 51 1.95 Vol. 2: 21762*0 ft. 51L95 MAGIC AND MYSTERY IN TIBET, Madame Alexandra David- Neel. Experiences among lamas, magicians, sages, sorcerers, Bonpa wizards. A true psychic discovery. 82 illustrations. 821 pp. 5H x 8!*. (USO) 22682*4 ft. 58.95 THE EGYPTIAN BOOK OF THE DEAD, E. A. Wallis Budge. Complete reproduction of Ani's papyrus, finest ever found. Full hieroglyphic text, interlinear transliteration, word-for-word translation, smooth translation. 538pp. 6H x 9^. 21866-X ft 59.95 MATHEMATICS FOR THE NONMATHEMATICIAN, Morris Kline. Detailed, college-level treatment of mathematics in cultural and historical context, with numerous exercises. Recommended Reading Lists. Tables. Numerous figures. 641pp. 5* x 8H. 24828-2 ft. 51 1.95 THEORY OF WING SECTIONS: Including a Summary of Airfoil Data, Ira H. Abbott and A. E. von Doenhoff. Concise compilation of subsonic aerodynamic characteristics of NACA wing sections, plus description of theory. 350pp. of tables. 693pp. 5H x 8H. 60586*8 ft. 514.95 THE RIME OF THE ANCIENT MARINER, Gustave Dor*, $. T. Coleridge. Dori’s finest work; 34 plates capture moods, subtleties of poem. Flawless full-size reproductions printed on facing pages with authoritative text of poem. "Beautiful. Simply beautiful." — Publisher's Weekly . 77pp. 9K x 12. 22805-1 ft. 56.95 NORTH AMERICAN INDIAN DESIGNS FOR ARTISTS AND CRAFTS- PEOPLE, Eva Wilson. Over 360 authentic copyright-free designs adapted from Navajo blankets, Hopi pottery, Sioux buffalo hides, more. Geometries, symbolic figures, plant and animal motifs, etc. 128pp. 8K x ll. (EUK) 25841-4 Pa. 57.95 SCULPTURE: Principles and Practice, Louis Slobodkin. Step-by-step approach to clay, plaster, metals, stone: classical and modern. 253 drawings, photos. 255pp 8»x u. 22960-2 ft. 510.95 CATALOG OF DOVER BOOKS THE INFLUENCE OF SEA POWER UPON HISTORY, 1660-1783, A. T. Mahan. Influential classic of naval history and tactics still used as text in war colleges. First paperback edition. 4 maps. 24 battle plans. 640pp. 5H * 8H. 25509-3 Pa. 312.95 THE STORY OF THE TITANIC AS TOLD BY ITS SURVIVORS. Jack Winocour (ed.). What it was really like. Panic, despair, shocking inefficiency, and a little heroism. More thrilling than any fictional account. 26 illustrations. 320pp. 5K*8H. 20610-6 Pa. 38.95 FAIRY AND FOLKTALES OFTHE IRISH PEASANTRY. William Butler Yeats (ed*). Treasury of 64 tales from the twilight world of Celtic myth and legend: '‘The Soul Cages/' “The Kildare Pooka/' “King O'Toole and his Goose," many more. Introduction and Notes by W. B. Yeats* S52pp. 5H * 8H. 26941-8 Pa. 38.95 BUDDHIST MAHAYANA TEXTS. E. B. Cowell and Others (eds.). Superb, accurate translations of basic documents in Mahayana Buddhism, highly important in history of religions. The Buddha-karita of Asvaghosha, Larger Sukha vat ivy uha, more. 448pp. 5H x 8*. , 25552-2 Pa. 39.95 ONE TWO THREE . . . INFINITY: Facts and Speculations of Science, George Gamow. Great physicist's fascinating, readable overview of contemporary science: number theory, relativity, fourth dimension, entropy, genes, atomic structure, much more. 128 illustrations. Index. 352pp. 5H * 8tt. 25664-2 Pa. 38.95 ENGINEERING IN HISTORY, Richard Shelton Kirby, et al. Broad, nontechnical survey of history's major technological advances: birth of Greek science, industrial revolution, electricity and applied science, 20th-century automation, much more. 181 illustrations. “. . . excellent . . /' — Isis. Bibliography, vii + 530pp. 5K * 8'4. 26412*2 Pa. 314.95 Prices subject to change without notice. Available at your book dealer or write for free catalog to Dept. GI, Dover Publications, Inc., 31 East 2nd St., Mineola, N.Y. 1 1501. Dover publishes more than 500 books each year on science, elementary and advanced mathematics, biology, music, art, literary history, social sciences and other areas.