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Praise for Quantum
‘An exhaustive and brilliant account of decades of emotionally charged discovery and argument, friendship and rivalry spanning two world wars. The explanations of science and philosophical interpretation are pitched with an ideal clarity for the general reader [and] perhaps most interestingly, although the author is admirably even-handed, it is difficult not to think of Quantum, by the end, as a resounding rehabilitation of Albert Einstein.’
Steven Poole, Guardian
‘The reason this book is, in fact, so readable is because it contains vivid portraits of the scientists involved, and their contexts…This is about gob-smacking science at the far end of reason…Take it nice and easy and savour the experience of your mind being blown without recourse to hallucinogens.’
Nicholas Lezard, Paperback Choice of the Week, Guardian
‘Kumar is an accomplished writer who knows how to separate the excitement of the chase from the sometimes impenetrable mathematics. In Quantum he tells the story of the conflict between two of the most powerful intellects of their day: the hugely famous Einstein and the less well-known but just as brilliant Dane, Niels Bohr.’
Financial Times
‘Manjit Kumar’s Quantum is a super-collider of a book, shaking together an exotic cocktail of free-thinking physicists, tracing their chaotic interactions and seeing what God-particles and black holes fly up out of the maelstrom. He provides probably the most lucid and detailed intellectual history ever written of a body of theory that makes other scientific revolutions look limp-wristed by comparison.’
Independent
‘Quantum by Manjit Kumar is so well written that I now feel I’ve more or less got particle physics sussed. Quantum transcends genre – it is historical, scientific, biographical, philosophical.’
Readers’ Books of the Year, Guardian

‘Highly readable…A welcome addition to the popular history of twentiethcentury physics.’
Nature
‘An elegantly written and accessible guide to quantum physics, in which Kumar structures the narrative history around the clash between Einstein and Bohr, and the anxiety that quantum theory “disproved the existence of reality”.’
Scotland on Sunday
‘It would be a rare author who could fully address both the philosophical and the historical issues – an even rarer one who could make it all palatable and entertaining to a general audience. If Kumar scores less than full marks it is only because of the admirably ambitious scale of his book.’
Andrew Crumey, Daily Telegraph
‘Quantum: Einstein, Bohr and the Great Debate about the Nature of Reality by Manjit Kumar is one of the best guides yet to the central conundrums of modern physics.’
John Banville, Books of the Year, The Age , Australia
‘By combining personalities and physics – both of an intriguingly quirky nature– Kumar transforms the sub-atomic debate between Einstein, Niels Bohr and others in their respective circles into an absorbing and…comprehensible narrative.’
Independent
‘In this magisterial study of the issue, Manjit Kumar probes beyond the froth and arcane arguments and reveals what really lies behind the theory and ultimately what it means for the development of science…Here we have an erudite work that takes the debate into new territory.’
Good Book Guide
‘Quantum is a fascinating, powerful and brilliantly written book that shows one of the most important theories of modern science in the making and discusses its

implications for our ideas about the fundamental nature of the world and human knowledge, while presenting intimate and insightful portraits of people who made the science. Highly recommended.’
thebookbag.co.uk
‘This is the biography of an idea and as such reads much like a thriller.’ Ham & High
‘It is a revolution that, even if most people don’t fully realise yet, has changed the face of science – and of our understanding of the nature of reality – for ever. Beautifully written in a tour de force that covers the fierce debate about the foundations of reality that gripped the scientific community through the 20th century, this book also looks at the personal collision of thinking and belief between two of quantum theory’s great men, Albert Einstein and Niels Bohr… This is the world of Alice down the cosmic rabbit hole. Take a peek.’
Odyssey , South Africa
‘Kumar brings us through the detail of the various advances, confusions and mistakes, and what emerges clearly is a picture of how science works as a great international collective effort.’
Irish Times
‘The virtue of Kumar’s book is that it takes us deeper than many and in doing so gives an insight into what we don’t know.’
spiked-online.com
‘A dramatic, powerful and superbly written history.’ Publishing News
‘A fresh perspective on the debate.’ Press Association
‘The most important popular science book published this year.’

Bookseller
‘A quite marvellous book…Manjit Kumar does a great job of weaving together the science, the history and the human drama of it all, to create a book that, by the standards of most science books, can only be described as a page turner…It’s hard to recommend this book too highly.’
top10.supersoftcafe.com
‘A superbly written history of the 20th century’s most challenging scientific revolution ’.
Independent Bookseller’s Association Christmas Books Catalogue
‘Rich and intensively researched…this qualitative, narrative method is a great way to get your head around the most extraordinary and intellectually demanding theory ever devised. Kumar brings to life the wide spectrum of personalities involved in the development of the quantum theory, from the quiet and thoughtful Bohr, to the lively womanising Schrödinger…I had difficulty putting this book down.’
Astronomy Now
‘Accessible to the non-scientist…Quantum is a fine book in many ways.’ Socialism Today , April 2009

MANJIT KUMAR

QUANTUM
EINSTEIN, BOHR AND THE GREAT DEBATE ABOUT
THE NATURE OF REALITY
W. W. NORTON & COMPANY
New York • London

Copyright © 2008 by Manjit Kumar
First American Edition 2010
All rights reserved
For information about permission to reproduce selections from this book, write to Permissions, W. W. Norton & Company, Inc., 500 Fifth Avenue, New York, NY 10110
Library of Congress Cataloging-in-Publication Data
Kumar, Manjit. Quantum: Einstein, Bohr, and the great debate about the nature
of reality / Manjit Kumar.—1st American ed. p. cm.
Includes bibliographical references.
ISBN: 978-0-393-07829-9
1. Quantum theory—History—Popular works. I. Title.
QC173.98.K86 2009 530.12—dc22
2009051249
W. W. Norton & Company, Inc. 500 Fifth Avenue, New York, N.Y. 10110
www.wwnorton.com
W. W. Norton & Company Ltd. Castle House, 75/76 Wells Street, London W1T 3QT

For Lahmber Ram and Gurmit Kaur Pandora, Ravinder, and Jasvinder

CONTENTS
Prologue PART I: THE QUANTUM Chapter 1 The Reluctant Revolutionary Chapter 2 The Patent Slave Chapter 3 The Golden Dane Chapter 4 The Quantum Atom Chapter 5 When Einstein Met Bohr Chapter 6 The Prince of Duality PART II: BOY PHYSICS Chapter 7 Spin Doctors Chapter 8 The Quantum Magician Chapter 9 ‘A Late Erotic Outburst’ Chapter 10 Uncertainty in Copenhagen PART III: TITANS CLASH OVER REALITY Chapter 11 Solvay 1927 Chapter 12 Einstein Forgets Relativity Chapter 13 Quantum Reality PART IV: DOES GOD PLAY DICE? Chapter 14 For Whom Bell’s Theorem Tolls Chapter 15 The Quantum Demon
Timeline

Glossary Notes Bibliography Acknowledgements

Prologue
THE MEETING OF MINDS
Paul Ehrenfest was in tears. He had made his decision. Soon he would attend the week-long gathering where many of those responsible for the quantum revolution would try to understand the meaning of what they had wrought. There he would have to tell his old friend Albert Einstein that he had chosen to side with Niels Bohr. Ehrenfest, the 34-year-old Austrian professor of theoretical physics at Leiden University in Holland, was convinced that the atomic realm was as strange and ethereal as Bohr argued.1
In a note to Einstein as they sat around the conference table, Ehrenfest scribbled: ‘Don’t laugh! There is a special section in purgatory for professors of quantum theory, where they will be obliged to listen to lectures on classical physics ten hours every day.’2 ‘I laugh only at their naiveté,’ Einstein replied.3 ‘Who knows who would have the [last] laugh in a few years?’ For him it was no laughing matter, for at stake was the very nature of reality and the soul of physics.
The photograph of those gathered at the fifth Solvay conference on ‘Electrons and Photons’, held in Brussels from 24 to 29 October 1927, encapsulates the story of the most dramatic period in the history of physics. With seventeen of the 29 invited eventually earning a Nobel Prize, the conference was one of the most spectacular meetings of minds ever held.4 It marked the end of a golden age of physics, an era of scientific creativity unparalleled since the scientific revolution in the seventeenth century led by Galileo and Newton.
Paul Ehrenfest is standing, slightly hunched forward, in the back row, third from the left. There are nine seated in the front row. Eight men and one woman; six have Nobel Prizes in either physics or chemistry. The woman has two, one for physics awarded in 1903 and another for chemistry in 1911. Her name: Marie Curie. In the centre, the place of honour, sits another Nobel laureate, the most celebrated scientist since the age of Newton: Albert Einstein. Looking straight ahead, gripping the chair with his right hand, he seems ill at ease. Is it the winged collar and tie that are causing him discomfort, or what he has heard during the preceding week? At the end of the second row, on the right, is Niels Bohr, looking relaxed with a half-whimsical smile. It had been a good conference for him. Nevertheless, Bohr would be returning to Denmark disappointed that he had

failed to convince Einstein to adopt his ‘Copenhagen interpretation’ of what quantum mechanics revealed about the nature of reality.
Instead of yielding, Einstein had spent the week attempting to show that quantum mechanics was inconsistent, that Bohr’s Copenhagen interpretation was flawed. Einstein said years later that ‘this theory reminds me a little of the system of delusions of an exceedingly intelligent paranoic, concocted of incoherent elements of thoughts’.5
It was Max Planck, sitting on Marie Curie’s right, holding his hat and cigar, who discovered the quantum. In 1900 he was forced to accept that the energy of light and all other forms of electromagnetic radiation could only be emitted or absorbed by matter in bits, bundled up in various sizes. ‘quantum’ was the name Planck gave to an individual packet of energy, with ‘quanta’ being the plural. The quantum of energy was a radical break with the long-established idea that energy was emitted or absorbed continuously, like water flowing from a tap. In the everyday world of the macroscopic where the physics of Newton ruled supreme, water could drip from a tap, but energy was not exchanged in droplets of varying size. However, the atomic and subatomic level of reality was the domain of the quantum.
In time it was discovered that the energy of an electron inside an atom was ‘quantised’ it could possess only certain amounts of energy and not others. The same was true of other physical properties, as the microscopic realm was found to be lumpy and discontinuous and not some shrunken version of the large-scale world that humans inhabit, where physical properties vary smoothly and continuously, where going from A to C means passing through B. quantum physics, however, revealed that an electron in an atom can be in one place, and then, as if by magic, reappear in another without ever being anywhere in between, by emitting or absorbing a quantum of energy. This was a phenomenon beyond the ken of classical, non-quantum physics. It was as bizarre as an object mysteriously disappearing in London and an instant later suddenly reappearing in Paris, New York or Moscow.
By the early 1920s it had long been apparent that the advance of quantum physics on an ad hoc, piecemeal basis had left it without solid foundations or a logical structure. Out of this state of confusion and crisis emerged a bold new theory known as quantum mechanics. The picture of the atom as a tiny solar system with electrons orbiting a nucleus, still taught in schools today, was abandoned and replaced with an atom that was impossible to visualise. Then, in 1927, Werner Heisenberg made a discovery that was so at odds with common sense that even he, the German wunderkind of quantum mechanics, initially

struggled to grasp its significance. The uncertainty principle said that if you want to know the exact velocity of a particle, then you cannot know its exact location, and vice versa.
No one knew how to interpret the equations of quantum mechanics, what the theory was saying about the nature of reality at the quantum level. Questions about cause and effect, or whether the moon exists when no one is looking at it, had been the preserve of philosophers since the time of Plato and Aristotle, but after the emergence of quantum mechanics they were being discussed by the twentieth century’s greatest physicists.
With all the basic components of quantum physics in place, the fifth Solvay conference opened a new chapter in the story of the quantum. For the debate that the conference sparked between Einstein and Bohr raised issues that continue to preoccupy many eminent physicists and philosophers to this day: what is the nature of reality, and what kind of description of reality should be regarded as meaningful? ‘No more profound intellectual debate has ever been conducted’, claimed the scientist and novelist C.P. Snow. ‘It is a pity that the debate, because of its nature, can’t be common currency.’6
Of the two main protagonists, Einstein is a twentieth-century icon. He was once asked to stage his own three-week show at the London Palladium. Women fainted in his presence. Young girls mobbed him in Geneva. Today this sort of adulation is reserved for pop singers and movie stars. But in the aftermath of the First World War, Einstein became the first superstar of science when in 1919 the bending of light predicted by his theory of general relativity was confirmed. Little had changed when in January 1931, during a lecture tour of America, Einstein attended the premiere of Charlie Chaplin’s movie City Limits in Los Angeles. A large crowd cheered wildly when they saw Chaplin and Einstein. ‘They cheer me because they all understand me,’ Chaplin told Einstein, ‘and they cheer you because no one understands you.’7
Whereas the name Einstein is a byword for scientific genius, Niels Bohr was, and remains, less well known. Yet to his contemporaries he was every inch the scientific giant. In 1923 Max Born, who played a pivotal part in the development of quantum mechanics, wrote that Bohr’s ‘influence on theoretical and experimental research of our time is greater than that of any other physicist’.8 Forty years later, in 1963, Werner Heisenberg maintained that ‘Bohr’s influence on the physics and the physicists of our century was stronger than that of anyone else, even than that of Albert Einstein’.9
When Einstein and Bohr first met in Berlin in 1920, each found an intellectual sparring partner who would, without bitterness or rancour, push and prod the

other into refining and sharpening his thinking about the quantum. It is through them and some of those gathered at Solvay 1927 that we capture the pioneering years of quantum physics. ‘It was a heroic time’, recalled the American physicist Robert Oppenheimer, who was a student in the 1920s.10 ‘It was a period of patient work in the laboratory, of crucial experiments and daring action, of many false starts and many untenable conjectures. It was a time of earnest correspondence and hurried conferences, of debate, criticism and brilliant mathematical improvisation. For those who participated it was a time of creation.’ But for Oppenheimer, the father of the atom bomb: ‘There was terror as well as exaltation in their new insight.’
Without the quantum, the world we live in would be very different. Yet for most of the twentieth century, physicists accepted that quantum mechanics denied the existence of a reality beyond what was measured in their experiments. It was a state of affairs that led the American Nobel Prize-winning physicist Murray Gell-Mann to describe quantum mechanics as ‘that mysterious, confusing discipline which none of us really understands but which we know how to use’.11 And use it we have. Quantum mechanics drives and shapes the modern world by making possible everything from computers to washing machines, from mobile phones to nuclear weapons.
The story of the quantum begins at the end of the nineteenth century when, despite the recent discoveries of the electron, X-rays, and radioactivity, and the ongoing dispute about whether or not atoms existed, many physicists were confident that nothing major was left to uncover. ‘The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote’, said the American physicist Albert Michelson in 1899. ‘Our future discoveries,’ he argued, ‘must be looked for in the sixth place of decimals.’12 Many shared Michelson’s view of a physics of decimal places, believing that any unsolved problems represented little challenge to established physics and would sooner or later yield to time-honoured theories and principles.
James Clerk Maxwell, the nineteenth century’s greatest theoretical physicist, had warned as early as 1871 against such complacency: ‘This characteristic of modern experiments – that they consist principally of measurements – is so prominent, that the opinion seems to have got abroad that in a few years all the great physical constants will have been approximately estimated, and that the only occupation which will be left to men of science will be to carry on these measurements to another place of decimals.’13 Maxwell pointed out that the real reward for the ‘labour of careful measurement’ was not greater accuracy but the

‘discovery of new fields of research’ and ‘the development of new scientific ideas’.14 The discovery of the quantum was the result of just such a ‘labour of careful measurement’.
In the 1890s some of Germany’s leading physicists were obsessively pursuing a problem that had long vexed them: what was the relationship between the temperature, the range of colours, and the intensity of light emitted by a hot iron poker? It seemed a trivial problem compared to the mystery of X-rays and radioactivity that had physicists rushing to their laboratories and reaching for their notebooks. But for a nation forged only in 1871, the quest for the solution to the hot iron poker, or what became known as ‘the blackbody problem’, was intimately bound up with the need to give the German lighting industry a competitive edge against its British and American competitors. But try as they might, Germany’s finest physicists could not solve it. In 1896 they thought they had, only to find within a few short years that new experimental data proved that they had not. It was Max Planck who solved the blackbody problem, at a cost. The price was the quantum.

PART I

THE QUANTUM
‘Briefly summarized, what I did can be described as simply an act of desperation.’ —MAX PLANCK
‘It was as if the ground had been pulled out from under one, with no firm foundation to be seen anywhere, upon which one could have built.’ —ALBERT EINSTEIN
‘For those who are not shocked when they first come across quantum theory cannot possibly have understood it.’ —NIELS BOHR

Chapter 1

THE RELUCTANT REVOLUTIONARY
‘A new scientific truth does not triumph by convincing its opponents and
making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it’, wrote Max Planck towards the end of his long life.1 Bordering on cliché, it could easily have served as his own scientific obituary had he not as an ‘act of desperation’ abandoned ideas that he had long held dear.2 Wearing his dark suit, starched white shirt and black bow tie, Planck looked the archetypal late nineteenth-century Prussian civil servant but ‘for the penetrating eyes under the huge dome of his bald head’.3 In characteristic mandarin fashion he exercised extreme caution before committing himself on matters of science or anything else. ‘My maxim is always this,’ he once told a student, ‘consider every step carefully in advance, but then, if you believe you can take responsibility for it, let nothing stop you.’4 Planck was not a man to change his mind easily.
His manner and appearance had hardly changed when to students in the 1920s, as one recalled later, ‘it seemed inconceivable that this was the man who had ushered in the revolution’.5 The reluctant revolutionary could scarcely believe it himself. By his own admission he was ‘peacefully inclined’ and avoided ‘all doubtful adventures’.6 He confessed that he lacked ‘the capacity to react quickly to intellectual stimulation’.7 It often took him years to reconcile new ideas with his deep-rooted conservatism. Yet at the age of 42, it was Planck who unwittingly started the quantum revolution in December 1900 when he discovered the equation for the distribution of radiation emitted by a blackbody.
All objects, if hot enough, radiate a mixture of heat and light, with the intensity and colour changing with the temperature. The tip of an iron poker left in a fire will start to glow a faint dull red; as its temperature rises it becomes a cherry red, then a bright yellowish-orange, and finally a bluish-white. Once taken out of the fire the poker cools down, running through this spectrum of colours backwards until it is no longer hot enough to emit any visible light. Even then it still gives off an invisible glow of heat radiation. After a time this too stops as the poker continues to cool and finally becomes cold enough to touch.
It was the 23-year-old Isaac Newton who, in 1666, showed that a beam of white light was woven from different threads of coloured light and that passing it

through a prism simply unpicked the seven separate strands: red, orange, yellow, green, blue, indigo, and violet.8 Whether red and violet represented the limits of the light spectrum or just those of the human eye was answered in 1800. It was only then, with the advent of sufficiently sensitive and accurate mercury thermometers, that the astronomer William Herschel placed one in front of a spectrum of light and found that as he moved it across the bands of different colours from violet to red, the temperature rose. To his surprise it continued to rise when he accidentally left the thermometer up to an inch past the region of red light. Herschel had detected what was later called infrared radiation, light that was invisible to human eyes from the heat that it generated.9 In 1801, using the fact that silver nitrate darkens when exposed to light, Johann Ritter discovered invisible light at the other end of the spectrum beyond the violet: ultraviolet radiation.
The fact that all heated objects emit light of the same colour at the same temperature was well known to potters long before 1859, the year that Gustav Kirchhoff, a 34-year-old German physicist at Heidelberg University, started his theoretical investigations into the nature of this correlation. To help simplify his analysis, Kirchhoff developed the concept of a perfect absorber and emitter of radiation called a blackbody. His choice of name was apt. A body that was a perfect absorber would reflect no radiation and therefore appear black. However, as a perfect emitter its appearance would be anything but black if its temperature was high enough for it to radiate at wavelengths from the visible part of the spectrum.
Kirchhoff envisaged his imaginary blackbody as a simple hollow container with a tiny hole in one of its walls. Since any radiation, visible or invisible light, entering the container does so through the hole, it is actually the hole that mimics a perfect absorber and acts like a blackbody. Once inside, the radiation is reflected back and forth between the walls of the cavity until it is completely absorbed. Imagining the outside of his blackbody to be insulated, Kirchhoff knew that if heated, only the interior surface of the walls would emit radiation that filled the cavity.
At first the walls, just like a hot iron poker, glow a deep cherry-red even though they still radiate predominantly in the infrared. Then, as the temperature climbs ever higher, the walls glow a bluish-white as they radiate at wavelengths from across the spectrum from the far infrared to the ultraviolet. The hole acts as a perfect emitter since any radiation that escapes through it will be a sample of all the wavelengths present inside the cavity at that temperature.

Kirchhoff proved mathematically what potters had long observed in their kilns. Kirchhoff’s law said that the range and intensity of the radiation inside the cavity did not depend on the material that a real blackbody could be made of, or on its shape and size, but only on its temperature. Kirchhoff had ingeniously reduced the problem of the hot iron poker: what was the exact relationship between the range and intensity of the colours it emitted at a certain temperature to how much energy is radiated by a blackbody at that temperature? The task that Kirchhoff set himself and his colleagues became known as the blackbody problem: measure the spectral energy distribution of blackbody radiation, the amount of energy at each wavelength from the infrared to the ultraviolet, at a given temperature and derive a formula to reproduce the distribution at any temperature.
Unable to go further theoretically without experiments with a real blackbody to guide him, Kirchhoff nevertheless pointed physicists in the right direction. He told them that the distribution being independent of the material from which a blackbody was made meant that the formula should contain only two variables: the temperature of the blackbody and the wavelength of the emitted radiation. Since light was thought to be a wave, any particular colour and hue was distinguished from every other by its defining characteristic: its wavelength, the distance between two successive peaks or troughs of the wave. Inversely proportional to the wavelength is the frequency of the wave – the number of peaks, or troughs, that pass a fixed point in one second. The longer the wavelength, the lower the frequency and vice versa. But there was also a different but equivalent way of measuring the frequency of a wave: the number of times it jiggled up and down, ‘waved’, per second.10
Figure 1: The characteristics of a wave
The technical obstacles in constructing a real blackbody and the precision instruments needed to detect and measure the radiation ensured that no significant progress was made for almost 40 years. It was in the 1880s, when German companies tried to develop more efficient light bulbs and lamps than their

American and British rivals, that measuring the blackbody spectrum and finding Kirchhoff’s fabled equation became a priority.
The incandescent light bulb was the latest in a series of inventions, including the arc lamp, dynamo, electric motor, and telegraphy, fuelling the rapid expansion of the electrical industry. With each innovation the need for a globally agreed set of units and standards of electrical measurement became increasingly urgent.
Two hundred and fifty delegates from 22 countries gathered in Paris, in 1881, for the first International Conference for the Determination of Electrical Units. Although the volt, amp and other units were defined and named, no agreement was reached on a standard for luminosity and it began to hamper the development of the most energy-efficient means of producing artificial light. As a perfect emitter at any given temperature, a blackbody emits the maximum amount of heat, infrared radiation. The blackbody spectrum would serve as a benchmark in calibrating and producing a bulb that emitted as much light as possible while keeping the heat it generated to a minimum.
‘In the competition between nations, presently waged so actively, the country that first sets foot on new paths and first develops them into established branches of industry has a decisive upper hand’, wrote the industrialist and inventor of the electrical dynamo, Werner von Siemens.11 Determined to be first, in 1887 the German government founded the Physikalisch-Technische Reichsanstalt (PTR), the Imperial Institute of Physics and Technology. Located on the outskirts of Berlin in Charlottenburg, on land donated by Siemens, the PTR was conceived as an institute fit for an empire determined to challenge Britain and America. The construction of the entire complex lasted more than a decade, as the PTR became the best-equipped and most expensive research facility in the world. Its mission was to give Germany the edge in the appliance of science by developing standards and testing new products. Among its list of priorities was to devise an internationally recognised unit of luminosity. The need to make a better light bulb was the driving force behind the PTR blackbody research programme in the 1890s. It would lead to the accidental discovery of the quantum as Planck turned out to be the right man, in the right place, at the right time.
Max Karl Ernst Ludwig Planck was born in Kiel, then a part of Danish Holstein, on 23 April 1858 into a family devoted to the service of Church and State. Excellence in scholarship was almost his birthright. Both his paternal greatgrandfather and grandfather had been distinguished theologians, while his father became professor of constitutional law at Munich University. Venerating the laws

of God and Man, these duty-bound men of probity were also steadfast and patriotic. Max was to be no exception.
Planck attended the most renowned secondary school in Munich, the Maximilian Gymnasium. Always near the top of his class, but never first, he excelled through hard work and self-discipline. These were just the qualities demanded by an educational system with a curriculum founded on the retention of enormous quantities of factual knowledge through rote learning. A school report noted that ‘despite all his childishness’ Planck at ten already possessed ‘a very clear, logical mind’ and promised ‘to be something right’.12 By the time he was sixteen it was not Munich’s famous taverns but its opera houses and concert halls that attracted the young Planck. A talented pianist, he toyed with the idea of pursuing a career as a professional musician. Unsure, he sought advice and was bluntly told: ‘If you have to ask, you’d better study something else!’13
In October 1874, aged sixteen, Planck enrolled at Munich University and opted to study physics because of a burgeoning desire to understand the workings of nature. In contrast to the near-militaristic regime of the Gymnasiums, German universities allowed their students almost total freedom. With hardly any academic supervision and no fixed requirements, it was a system that enabled students to move from one university to another, taking courses as they pleased. Sooner or later those wishing to pursue an academic career took the courses by the pre-eminent professors at the most prestigious universities. After three years at Munich, where he was told ‘it is hardly worth entering physics anymore’ because there was nothing important left to discover, Planck moved to the leading university in the German-speaking world, Berlin.14
With the creation of a unified Germany in the wake of the Prussian-led victory over France in the war of 1870–71, Berlin became the capital of a mighty new European nation. Situated at the confluence of the Havel and the Spree rivers, French war reparations allowed its rapid redevelopment as it sought to make itself the equal of London and Paris. A population of 865,000 in 1871 swelled to nearly 2 million by 1900, making Berlin the third-largest city in Europe.15 Among the new arrivals were Jews fleeing persecution in Eastern Europe, especially the pogroms in Tsarist Russia. Inevitably the cost of housing and living soared, leaving many homeless and destitute. Manufacturers of cardboard boxes advertised ‘good and cheap boxes for habitation’ as shanty towns sprung up in parts of the city.16
Despite the bleak reality that many found on arriving in Berlin, Germany was entering a period of unprecedented industrial growth, technological progress, and economic prosperity. Driven largely by the abolition of internal tariffs after

unification and French war compensation, by the outbreak of the First World War Germany’s industrial output and economic power would be second only to the United States. By then it was producing over two-thirds of continental Europe’s steel, half its coal, and was generating more electricity than Britain, France and Italy combined. Even the recession and anxiety that affected Europe after the stock market crash of 1873 only slowed the pace of German development for a few years.
With unification came the desire to ensure that Berlin, the epitome of the new Reich, had a university second to none. Germany’s most renowned physicist, Herman von Helmholtz, was enticed from Heidelberg. A trained surgeon, Helmholtz was also a celebrated physiologist who had made fundamental contributions to understanding the workings of the human eye after his invention of the ophthalmoscope. The 50-year-old polymath knew his worth. Apart from a salary several times the norm, Helmholtz demanded a magnificent new physics institute. It was still being built in 1877 when Planck arrived in Berlin and began attending lectures in the university’s main building, a former palace on Unter den Linden opposite the Opera House.
As a teacher, Helmholtz was a severe disappointment. ‘It was obvious,’ Planck said later, ‘that Helmholtz never prepared his lectures properly.’17 Gustav Kirchhoff, who had also transferred from Heidelberg to become the professor of theoretical physics, was so well prepared that he delivered his lectures ‘like a memorized text, dry and monotonous’.18 Expecting to be inspired, Planck admitted ‘that the lectures of these men netted me no perceptible gain’.19 Seeking to quench his ‘thirst for advanced scientific knowledge’, he stumbled across the work of Rudolf Clausius, a 56-year-old German physicist at Bonn University.20
In stark contrast to the lacklustre teaching of his two esteemed professors, Planck was immediately enthralled by Clausius’ ‘lucid style and enlightening clarity of reasoning’.21 His enthusiasm for physics returned as he read Clausius’ papers on thermodynamics. Dealing with heat and its relationship to different forms of energy, the fundamentals of thermodynamics were at the time encapsulated in just two laws.22 The first was a rigorous formulation of the fact that energy, in whatever guise, possessed the special property of being conserved. Energy could neither be created nor destroyed but only converted from one form to another. An apple hanging from a tree possesses potential energy by virtue of its position in the earth’s gravitational field, its height above the ground. When it falls, the apple’s potential energy is converted into kinetic energy, the energy of motion.

Planck was a schoolboy when he first encountered the law of the conservation of energy. It struck him ‘like a revelation’ he said later, because it possessed ‘absolute, universal validity, independently from all human agency’.23 It was the moment he caught a glimpse of the eternal, and from then on he considered the search for absolute or fundamental laws of nature ‘as the most sublime scientific pursuit in life’.24 Now Planck was just as spellbound reading Clausius’ formulation of the second law of thermodynamics: ‘Heat will not pass spontaneously from a colder to a hotter body.’25 The later invention of the refrigerator illustrated what Clausius meant by ‘spontaneously’. A refrigerator needed to be plugged into an external supply of energy, in this case electrical, so that heat could be made to flow from a colder to a hotter body.
Planck understood that Clausius was not simply stating the obvious, but something of deep significance. Heat, the transfer of energy from A to B due to a temperature difference, explained such everyday occurrences as a hot cup of coffee getting cold and an ice cube in a glass of water melting. But left undisturbed, the reverse never happened. Why not? The law of conservation of energy did not forbid a cup of coffee from getting hotter and the surrounding air colder, or the glass of water becoming warmer and the ice cooler. It did not outlaw heat flowing from a cold to a hot body spontaneously. Yet something was preventing this from happening. Clausius discovered that something and called it entropy. It lay at the heart of why some processes occur in nature and others do not.
When a hot cup of coffee cools down, the surrounding air gets warmer as energy is dissipated and irretrievably lost, ensuring that the reverse cannot happen. If the conservation of energy was nature’s way of balancing the books in any possible physical transaction, then nature also demanded a price for every transaction that actually occurred. According to Clausius, entropy was the price for whether something happened or not. In any isolated system only those processes, transactions, in which entropy either stayed the same or increased were allowed. Any that led to a decrease of entropy were strictly forbidden.
Clausius defined entropy as the amount of heat in or out of a body or a system divided by the temperature at which it takes place. If a hot body at 500 degrees loses 1000 units of energy to a colder body at 250 degrees, then its entropy has decreased by –1000/500 = –2. The colder body at 250 degrees has gained 1000 units of energy, +1000/250, and its entropy has increased by 4. The overall entropy of the system, the hot and cold bodies combined, has increased by 2 units of energy per degree. All real, actual processes are irreversible because they result in an increase in entropy. It is nature’s way of stopping heat from passing spontaneously, of its own accord, from something cold to something hot. Only

ideal processes in which entropy remains unchanged can be reversed. They, however, never occur in practice, only in the mind of the physicist. The entropy of the universe tends towards a maximum.
Alongside energy, Planck believed that entropy was ‘the most important property of physical systems’.26 Returning to Munich University after his yearlong sojourn in Berlin, he devoted his doctoral thesis to an exploration of the concept of irreversibility. It would be his calling card. To his dismay, he ‘found no interest, let alone approval, even among the very physicists who were closely concerned with the topic’.27 Helmholtz did not read it; Kirchhoff did, but disagreed with it. Clausius, who had such a profound influence on him, did not even answer his letter. ‘The effect of my dissertation on the physicists of those days was nil’, Planck recalled with some bitterness even 70 years later. But driven by ‘an inner compulsion’, he was undeterred.28 Thermodynamics, particularly the second law, became the focus of Planck’s research as he began his academic career.29
German universities were state institutions. Extraordinary (assistant) and ordinary (full) professors were civil servants appointed and employed by the ministry of education. In 1880 Planck became a privatdozent, an unpaid lecturer, at Munich University. Employed neither by the state nor the university, he had simply gained the right to teach in exchange for fees paid by students attending his courses. Five years passed as he waited in vain for an appointment as an extraordinary professor. As a theorist uninterested in conducting experiments, Planck’s chances for promotion were slim, as theoretical physics was not yet a firmly established distinct discipline. Even in 1900 there were only sixteen professors of theoretical physics in Germany.
If his career was to progress, Planck knew that he had ‘to win, somehow, a reputation in the field of science’.30 His chance came when Göttingen University announced that the subject for its prestigious essay competition was ‘The Nature of Energy’. As he worked on his paper, in May 1885, ‘a message of deliverance’ arrived.31 Planck, aged 27, was offered an extraordinary professorship at the University of Kiel. He suspected it was his father’s friendship with Kiel’s head of physics that had led to the offer. Planck knew there were others, more established than he, who would have expected advancement. Nevertheless, he accepted and finished his entry for the Göttingen competition shortly after arriving in the city of his birth.
Even though only three papers were submitted in search of the prize, an astonishing two years passed before it was announced that there would be no winner. Planck was awarded second place and denied the prize by the judges

because of his support for Helmholtz in a scientific dispute with a member of the Göttingen faculty. The behaviour of the judges drew the attention of Helmholtz to Planck and his work. After a little more than three years at Kiel, in November 1888, Planck received an unexpected honour. He had not been first, or even second choice. But after others had turned it down, Planck, with Helmholtz’s backing, was asked to succeed Gustav Kirchhoff at Berlin University as professor of theoretical physics.
In the spring of 1889, the capital was not the city Planck had left eleven years earlier. The stench that always shocked visitors had disappeared as a new sewer system replaced the old open drains, and at night the main streets were lit by modern electric lamps. Helmholtz was no longer head of the university’s physics institute but running the PTR, the majestic new research facility three miles away. August Kundt, his successor, had played no part in Planck’s appointment, but welcomed him as ‘an excellent acquisition’ and ‘a splendid man’.32
In 1894 Helmholtz, aged 73, and Kundt, only 55, both died within months of each other. Planck, only two years after finally being promoted to the rank of ordinary professor, found himself as the senior physicist at Germany’s foremost university at just 36. He had no choice but to bear the weight of added responsibilities, including that of adviser on theoretical physics for Annalen der Physik. It was a position of immense influence that gave him the right of veto on all theoretical papers submitted to the premier German physics journal. Feeling the pressure of his newly elevated position and a deep sense of loss at the deaths of his two colleagues, Planck sought solace in his work.
As a leading member of Berlin’s close-knit community of physicists, he was well aware of the ongoing, industry-driven blackbody research programme of the PTR. Although thermodynamics was central to a theoretical analysis of the light and heat radiated by a blackbody, the lack of reliable experimental data had stopped Planck from trying to derive the exact form of Kirchhoff’s unknown equation. Then a breakthrough by an old friend at PTR meant that he could no longer avoid the blackbody problem.
In February 1893, 29-year-old Wilhelm Wien discovered a simple mathematical relationship that described the effect of a change in temperature on the distribution of blackbody radiation.33 Wien found that as the temperature of a blackbody increases, the wavelength at which it emits radiation with the greatest intensity becomes ever shorter.34 It was already known that the rise in temperature would result in an increase in the total amount of energy radiated, but Wien’s

‘displacement law’ revealed something very precise: the wavelength at which the maximum amount of radiation is emitted multiplied by the temperature of a blackbody is always a constant. If the temperature is doubled, then the ‘peak’ wavelength will be half the previous length.
Figure 2: Distribution of blackbody radiation which shows Wien’s displacement law
Wien’s discovery meant that once the numerical constant was calculated by measuring the peak wavelength – the wavelength that radiates most strongly, at a certain temperature – then the peak wavelength could be calculated for any temperature.35 It also explained the changing colours of a hot iron poker. Starting at low temperatures, the poker emits predominantly long-wavelength radiation from the infrared part of the spectrum. As the temperature increases, more energy is radiated in each region and the peak wavelength decreases. It is ‘displaced’ towards the shorter wavelengths. Consequently the colour of the emitted light changes from red to orange, then yellow and finally a bluish-white as the quantity of radiation from the ultraviolet end of the spectrum increases.
Wien had quickly established himself as a member of that endangered breed of physicist, one who was both an accomplished theorist and a skilled experimenter. He found the displacement law in his spare time and was forced to publish it as a

‘private communication’ without the imprimatur of the PTR. At the time he was working as an assistant in the PTR’s optics laboratory under the leadership of Otto Lummer. Wien’s day job was the practical work that was a prerequisite for an experimental investigation of blackbody radiation.
Their first task was to construct a better photometer, an instrument capable of comparing the intensity of light – the amount of energy in a given wavelength range – from different sources such as gas lamps and electric bulbs. It was the autumn of 1895 before Lummer and Wien devised a new and improved hollow blackbody capable of being heated to a uniform temperature.
While he and Lummer developed their new blackbody during the day, Wien continued to spend his evenings searching for Kirchhoff’s equation for distribution of blackbody radiation. In 1896, Wien found a formula that Friedrich Paschen, at the University of Hanover, quickly confirmed agreed with the data he had collected on the allocation of energy among the short wavelengths of blackbody radiation.
In June that year, the very month the ‘distribution law’ appeared in print, Wien left the PTR for an extraordinary professorship at the Technische Hochschule in Aachen. He would win the Nobel Prize for physics in 1911 for his work on blackbody radiation, but left Lummer to put his distribution law through a rigorous test. To do so required measurements over a greater range and at higher temperatures than ever before. Working with Ferdinand Kurlbaum and then Ernst Pringsheim, it took Lummer two long years of refinements and modifications but in 1898 he had a state-of-the-art electrically heated blackbody. Capable of reaching temperatures as high as 1500°C, it was the culmination of more than a decade of painstaking work at the PTR.
Plotting the intensity of radiation along the vertical axis of a graph against the wavelength of the radiation along the horizontal axis, Lummer and Pringsheim found that the intensity rose as the wavelength of radiation increased until it peaked and then began to drop. The spectral energy distribution of blackbody radiation was almost a bell-shaped curve, resembling a shark’s dorsal fin. The higher the temperature, the more pronounced the shape as the intensity of radiation emitted increased. Taking readings and plotting curves with the blackbody heated to different temperatures showed that the peak wavelength that radiated with maximum intensity was displaced towards the ultraviolet end of the spectrum with increasing temperature.
Lummer and Pringsheim reported their results at a meeting of the German Physical Society held in Berlin on 3 February 1899.36 Lummer told the assembled physicists, among them Planck, that their findings confirmed Wien’s

displacement law. However, the situation regarding the distribution law was unclear. Although the data was in broad agreement with Wien’s theoretical predictions, there were some discrepancies in the infrared region of the spectrum.37 In all likelihood these were due to experimental errors, but it was an issue, they argued, that could be settled only once ‘other experiments spread over a greater range of wavelengths and over a greater interval of temperature can be arranged’.38
Within three months Friedrich Paschen announced that his measurements, though conducted at a lower temperature than those of Lummer and Pringsheim, were in complete harmony with the predictions of Wien’s distribution law. Planck breathed a sigh of relief and read out Paschen’s paper at a session of the Prussian Academy of Sciences. Such a law appealed deeply to him. For Planck the theoretical quest for the spectral energy distribution of blackbody radiation was nothing less than the search for the absolute, and ‘since I had always regarded the search for the absolute as the loftiest goal of all scientific activity, I eagerly set to work’.39
Soon after Wien published his distribution law, in 1896, Planck set about trying to place the law on rock-solid foundations by deriving it from first principles. Three years later, in May 1899, he thought he had succeeded by using the power and authority of the second law of thermodynamics. Others agreed and started calling Wien’s law by a new name, Wien-Planck, despite the claims and counterclaims of the experimentalists. Planck remained confident enough to assert that ‘the limits of validity of this law, in case there are any at all, coincide with those of the second fundamental law of the theory of heat’.40 He advocated further testing of the distribution law as a matter of urgency, since for him it would be a simultaneous examination of the second law. He got his wish.
At the beginning of November 1899, after spending nine months extending the range of their measurements as they eliminated possible sources of experimental error, Lummer and Pringsheim reported that they had found ‘discrepancies of a systematic nature between theory and experiment’.41 Although in perfect agreement for short wavelengths, they discovered that Wien’s law consistently overestimated the intensity of radiation at long wavelengths. However, within weeks Paschen contradicted Lummer and Pringsheim. He presented another set of new data and claimed that the distribution law ‘appears to be a rigorously valid law of nature’.42
With most of the leading experts living and working in Berlin, the meetings of the German Physical Society held in the capital became the main forum for discussions concerning blackbody radiation and the status of Wien’s law. It was

the subject that again dominated the proceedings of the society at its fortnightly meeting on 2 February 1900 when Lummer and Pringsheim disclosed their latest measurements. They had found systematic discrepancies between their measurements and the predictions of Wien’s law in the infrared region that could not be the result of experimental error.
This breakdown of Wien’s law led to a scramble to find a replacement. But these makeshift alternatives proved unsatisfactory, prompting calls for further testing at even longer wavelengths to unequivocally establish the extent of any failure of Wien’s law. It did, after all, agree with the available data covering the shorter wavelengths, and all other experiments bar those of Lummer-Pringsheim had found in its favour.
As Planck was only too well aware, any theory is at the mercy of hard experimental facts, but he strongly believed that ‘a conflict between observation and theory can only be confirmed as valid beyond all doubt if the figures of various observers substantially agree with each other’.43 Nevertheless, the disagreement between the experimentalists forced him to reconsider the soundness of his ideas. In late September 1900, as he continued to review his derivation, the failure of Wien’s law in the deep infrared was confirmed.
The question was finally settled by Heinrich Rubens, a close friend of Planck’s, and Ferdinand Kurlbaum. Based at the Technische Hochschule on Berlinerstrasse, where at the age of 35 he had recently been promoted to ordinary professor, Rubens spent most of his time as a guest worker at the nearby PTR. It was there, with Kurlbaum, that he built a blackbody that allowed measurements of the uncharted territory deep within the infrared region of the spectrum. During the summer they tested Wien’s law between wavelengths of 0.03mm and 0.06mm at temperatures ranging from 200 to 1500°C. At these longer wavelengths, they found the difference between theory and observation was so marked that it could be evidence of only one thing, the breakdown of Wien’s law.
Rubens and Kurlbaum wanted to announce their results in a paper to the German Physical Society. The next meeting was on Friday, 5 October. With little time to write a paper, they decided to wait until the following meeting two weeks later. In the meantime, Rubens knew that Planck would be eager to hear the latest results.
It was among the elegant villas of bankers, lawyers, and other professors in the affluent suburb of Grunewald in west Berlin that Planck lived for 50 years in a large house with an enormous garden. On Sunday, 7 October, Rubens and his

wife came for lunch. Inevitably the talk between the two friends soon turned to physics and the blackbody problem. Rubens explained that his latest measurements left no room for doubt: Wien’s law failed at long wavelengths and high temperatures. Those measurements, Planck learnt, revealed that at such wavelengths the intensity of blackbody radiation was proportional to the temperature.
That evening Planck decided to have a go at constructing the formula that would reproduce the energy spectrum of blackbody radiation. He now had three crucial pieces of information to help him. First, Wien’s law accounted for the intensity of radiation at short wavelengths. Second, it failed in the infrared where Rubens and Kurlbaum had found that intensity was proportional to the temperature. Third, Wien’s displacement law was correct. Planck had to find a way to assemble these three pieces of the blackbody jigsaw together to build the formula. His years of hard-won experience were quickly put into practice as he set about manipulating the various mathematical symbols of the equations at his disposal.
After a few unsuccessful attempts, through a combination of inspired scientific guesswork and intuition, Planck had a formula. It looked promising. But was it Kirchhoff’s long-sought-after equation? Was it valid at any given temperature for the entire spectrum? Planck hurriedly penned a note to Rubens and went out in the middle of the night to post it. After a couple of days, Rubens arrived at Planck’s home with the answer. He had checked Planck’s formula against the data and found an almost perfect match.
On Friday, 19 October at the meeting of the German Physical Society, with Rubens and Planck sitting among the audience, it was Ferdinand Kurlbaum who made the formal announcement that Wien’s law was valid only at short wavelengths and failed at the longer wavelengths of the infrared. After Kurlbaum sat down, Planck rose to deliver a short ‘comment’ billed as ‘An Improvement of Wien’s Equation for the Spectrum’. He began by admitting that he had believed ‘Wien’s law must necessarily be true’, and had said so at a previous meeting.44 As he continued, it quickly became clear that Planck was not simply proposing ‘an improvement’, some minor tinkering with Wien’s law, but a completely new law of his own.
After speaking for less than ten minutes, Planck wrote his equation for the blackbody spectrum on the blackboard. Turning around to look at the familiar faces of his colleagues, he told them that this equation ‘as far as I can see at the moment, fits the observational data, published up to now’.45 As he sat down, Planck received polite nods of approval. The muted response was understandable.

After all, what Planck had just proposed was another ad hoc formula manufactured to explain the experimental results. There were others who had already put forward equations of their own in the hope of filling the void, should the suspected failure of Wien’s law at long wavelengths be confirmed.
The next day Rubens visited Planck to reassure him. ‘He came to tell me that after the conclusion of the meeting he had that very night checked my formula against the results of his measurements,’ Planck remembered, ‘and found satisfactory concordance at every point.’46 Less than a week later, Rubens and Kurlbaum announced that they had compared their measurements with the predictions of five different formulae and found Planck’s to be much more accurate than any of the others. Paschen too confirmed that Planck’s equation matched his data. Yet despite this rapid corroboration by the experimentalists of the superiority of his formula, Planck was troubled.
He had his formula, but what did it mean? What was the underlying physics? Without an answer, Planck knew that it would, at best, be just an ‘improvement’ on Wien’s law and have ‘merely the standing of a law disclosed by a lucky intuition’ that possessed no more ‘than a formal significance’.47 ‘For this reason, on the very first day when I formulated this law,’ Planck said later, ‘I began to devote myself to the task of investing it with true physical meaning.’48 He could achieve this only by deriving his equation step by step using the principles of physics. Planck knew his destination, but he had to find a way of getting there. He possessed a priceless guide, the equation itself. But what price was he prepared to pay for such a journey?
The next six weeks were, Planck recalled, ‘the most strenuous work of my life’, after which ‘the darkness lifted and an unexpected vista began to appear’.49 On 13 November he wrote to Wien: ‘My new formula is well satisfied; I now have also obtained a theory for it, which I shall present in four weeks at the Physical Society here [in Berlin].’50 Planck said nothing to Wien either of the intense intellectual struggle that had led to his theory or the theory itself. He had strived long and hard during those weeks to reconcile his equation with the two grand theories of nineteenth-century physics: thermodynamics and electromagnetism. He failed.
‘A theoretical interpretation therefore had to be found at any cost,’ he accepted, ‘no matter how high.’51 He ‘was ready to sacrifice every one of my previous convictions about physical laws’.52 Planck no longer cared what it cost him, as long as he could ‘bring about a positive result’.53 For such an emotionally restrained man, who only truly expressed himself freely at the piano, this was highly charged language. Pushed to the limit in the struggle to understand his new

formula, Planck was forced into ‘an act of desperation’ that led to the discovery of the quantum.54
As the walls of a blackbody are heated they emit infrared, visible, and ultraviolet radiation into the heart of the cavity. In his search for a theoretically consistent derivation of his law, Planck had to come up with a physical model that reproduced the spectral energy distribution of blackbody radiation. He had already been toying with an idea. It did not matter if the model failed to capture what was really going on; all Planck needed was a way of getting the right mix of frequencies, and therefore wavelengths, of the radiation present inside the cavity. He used the fact that this distribution depends only on the temperature of the blackbody and not on the material from which it is made to conjure up the simplest model he could.
‘Despite the great success that the atomic theory has so far enjoyed,’ Planck wrote in 1882, ‘ultimately it will have to be abandoned in favour of the assumption of continuous matter.’55 Eighteen years later, in the absence of indisputable proof of their existence, he still did not believe in atoms. Planck knew from the theory of electromagnetism that an electric charge oscillating at a certain frequency emits and absorbs radiation only of that frequency. He therefore chose to represent the walls of the blackbody as an enormous array of oscillators. Although each oscillator emits only a single frequency, collectively they emit the entire range of frequencies found within the blackbody.
A pendulum is an oscillator and its frequency is the number of swings per second, a single oscillation being one complete to and fro swing that returns the pendulum to its starting point. Another oscillator is a weight hanging from a spring. Its frequency is the number of times per second the weight bounces up and down after being pulled from its stationary position and released. The physics of such oscillations had long been understood and given the name ‘simple harmonic motion’ by the time Planck used oscillators, as he called them, in his theoretical model.
Planck envisaged his collection of oscillators as massless springs of varying stiffness, so as to reproduce the different frequencies, each with an electric charge attached. Heating the walls of the blackbody provided the energy needed to set the oscillators in motion. Whether an oscillator was active or not would depend only upon the temperature. If it were, then it would emit radiation into, and absorb radiation from, the cavity. In time, if the temperature is held constant, this dynamic give and take of radiation energy between the oscillators and the

radiation in the cavity comes into balance and a state of thermal equilibrium is achieved.
Since the spectral energy distribution of blackbody radiation represents how the total energy is shared among the different frequencies, Planck assumed that it was the number of oscillators at each given frequency that determined the allocation. After setting up his hypothetical model, he had to devise a way to share out the available energy among the oscillators. In the weeks following its announcement, Planck discovered the hard way that he could not derive his formula using physics that he had long accepted as dogma. In desperation he turned to the ideas of an Austrian physicist, Ludwig Boltzmann, who was the foremost advocate of the atom. On the road to his blackbody formula, Planck became a convert as he accepted that atoms were more than just a convenient fiction, after years of being openly ‘hostile to the atomic theory’.56
The son of a tax collector, Ludwig Boltzmann was short and stout with an impressive late nineteenth-century beard. Born in Vienna on 20 February 1844, he was, for a while, taught the piano by the composer Anton Bruckner. A better physicist than a pianist, Boltzmann obtained his doctorate from the University of Vienna in 1866. He quickly made his reputation with fundamental contributions to the kinetic theory of gases, so called because its proponents believed that gases were made up of atoms or molecules in a state of continual motion. Later, in 1884, Boltzmann provided the theoretical justification for the discovery by Josef Stefan, his former mentor, that the total energy radiated by a blackbody is proportional to the temperature raised to the fourth power, T4 or T×T×T×T. It meant that doubling the temperature of a blackbody increased the energy it radiated by a factor of sixteen.
Boltzmann was a renowned teacher and, although a theorist, a very capable experimentalist despite being severely shortsighted. Whenever a vacancy arose at one of Europe’s leading universities his name was usually on the list of potential candidates. It was only after he turned down the professorship at Berlin University left vacant by the death of Gustav Kirchhoff that a downgraded version was offered to Planck. By 1900 a much-travelled Boltzmann was at Leipzig University and universally acknowledged as one the great theoreticians. Yet there were many, like Planck, who found his approach to thermodynamics unacceptable.
Boltzmann believed that properties of gases, such as pressure, were the macroscopic manifestations of microscopic phenomena regulated by the laws of mechanics and probability. For those whose believed in atoms, the classical physics of Newton governed the movement of each gas molecule, but using

Newtonian laws of motion to determine that of each of the countless molecules of a gas was for all practical purposes impossible. It was the 28-year-old Scottish physicist James Clerk Maxwell who, in 1860, captured the motion of gas molecules without measuring the velocity of a single one. Using statistics and probability, Maxwell worked out the most likely distribution of velocities as the gas molecules underwent incessant collisions with each other and the walls of a container. The introduction of statistics and probability was bold and innovative; it allowed Maxwell to explain many of the observed properties of gases. Thirteen years younger, Boltzmann followed in Maxwell’s footsteps to help shore up the kinetic theory of gases. In the 1870s he went one step further and developed a statistical interpretation of the second law of thermodynamics by linking entropy with disorder.
According to what became known as Boltzmann’s principle, entropy is a measure of the probability of finding a system in a particular state. A wellshuffled pack of playing cards, for example, is a disordered system with high entropy. However, a brand-new deck with cards arranged according to suit and from ace to king is a highly ordered system with low entropy. For Boltzmann the second law of thermodynamics concerns the evolution of a system with a low probability, and therefore low entropy, into a state of higher probability and high entropy. The second law is not an absolute law. It is possible for a system to go from a disordered state to a more ordered one, just as a shuffled pack of cards may, if shuffled again, become ordered. However, the odds against that happening are so astronomical that it would require many times the age of the universe to pass for it to occur.
Planck believed that the second law of thermodynamics was absolute – entropy always increases. In Boltzmann’s statistical interpretation, entropy nearly always increases. There was a world of difference between these two views as far as Planck was concerned. For him to turn to Boltzmann was a renunciation of everything that he held dear as a physicist, but he had no choice in his quest to derive his blackbody formula. ‘Until then I had paid no attention to the relationship between entropy and probability, in which I had little interest since every probability law permits exceptions; and at that time I assumed that the second law of thermodynamics was valid without exceptions.’57

A state of maximum entropy, maximum disorder, is the most probable state for a system. For a blackbody that state is thermal equilibrium – just the situation that Planck faced as he tried to find the most probable distribution of energy among his oscillators. If there are 1000 oscillators in total and ten have a frequency , it is these oscillators that determine the intensity of radiation emitted at that frequency. While the frequency of any one of Planck’s electric oscillators is fixed, the amount of energy it emits and absorbs depends solely upon its amplitude, the size of its oscillation. A pendulum completing five swings in five seconds has a frequency of one oscillation per second. However, if it swings through a wide arc the pendulum has more energy than if it traces out a smaller one. The frequency remains unchanged because the length of the pendulum fixes it, but the extra energy allows it to travel faster through a wide arc. The pendulum therefore completes the same number of oscillations in the same time as an identical pendulum swinging through a narrower arc.
Applying Boltzmann’s techniques, Planck discovered that he could derive his formula for the distribution of blackbody radiation only if the oscillators absorbed and emitted packets of energy that were proportional to their frequency of oscillation. It was the ‘most essential point of the whole calculation’, said Planck, to consider the energy at each frequency as being composed of a number of equal, indivisible ‘energy elements’ that he later called quanta.58
Guided by his formula, Planck had been forced into slicing up energy (E) into hv-sized chunks, where v is the frequency of the oscillator and h is a constant. E=hv would become one of the most famous equations in the whole of science. If, for example, the frequency was 20 and h was 2, then each quantum of energy would have a magnitude of 20×2=40. If the total energy available at this frequency were 3600, then there would be 3600/40=90 quanta to be distributed among the ten oscillators of that frequency. Planck learnt from Boltzmann how to determine the most probable distribution of these quanta among the oscillators.
He found that his oscillators could only have energies: o, h , 2h , 3h , 4h … all the way up to nh , where n is a whole number. This corresponded to either absorbing or emitting a whole number of ‘energy elements’ or ‘quanta’ of size h . It was like a bank cashier able to receive and dispense money only in denominations of £1, £2, £5, £10, £20 and £50. Since Planck’s oscillators cannot have any other energy, the amplitude of their oscillations is constrained. The strange implications of this are manifest if scaled up to the everyday world of a spring with a weight attached.
If the weight oscillates with an amplitude of 1cm, then it has an energy of 1 (ignoring the units of measuring energy). If the weight is pulled down to 2cm and

allowed to oscillate, its frequency remains the same as before. However its energy, which is proportional to the square of the amplitude, is now 4. If the restriction on Planck’s oscillators applied to the weight, then between 1cm and 2cm it can oscillate only with amplitudes of 1.42cm and 1.73cm, because they have energies of 2 and 3.59 It cannot, for example, oscillate with an amplitude of 1.5cm because the associated energy would be 2.25. A quantum of energy is indivisible. An oscillator cannot receive a fraction of a quantum of energy; it must be all or nothing. This ran counter to the physics of the day. It placed no restrictions on the size of oscillation and therefore on how much energy an oscillator can emit or absorb in a single transaction – it could have any amount.
In his desperation Planck had discovered something so remarkable and unexpected that he failed to grasp its significance. It is not possible for his oscillators to absorb or emit energy continuously like water from a tap. Instead they can only gain and lose energy discontinuously, in small, indivisible units of E=h , where is the frequency with which the oscillator vibrates that exactly matches the frequency of the radiation it can absorb or emit.
The reason why large-scale oscillators are not seen to behave like Planck’s atomic-sized ones is because h is equal to 0.000000000000000000000000006626 erg seconds or 6.626 divided by one thousand trillion trillion. According to Planck’s formula, there could be no smaller step than h in the increase or decrease of energy, but the infinitesimal size of h makes quantum effects invisible in the world of the everyday when it comes to pendulums, children’s swings and vibrating weights.
Planck’s oscillators forced him to slice and dice radiation energy so as to feed them the correct bite-sized chunks of h . He did not believe that the energy of radiation was really chopped up into quanta. It was just the way his oscillators could receive and emit energy. The problem for Planck was that Boltzmann’s procedure for slicing energy required that at the end the slices be made ever thinner until mathematically their thickness was zero and they vanished, with the whole being restored. To reunite a sliced-up quantity in such a fashion was a mathematical technique at the very heart of calculus. Unfortunately for Planck, if he did the same his formula vanished too. He was stuck with quanta, but was unconcerned. He had his formula; the rest could be sorted out later.
‘Gentlemen!’ said Planck as he faced the members of the German Physical Society seated in the room at Berlin University’s Physics Institute. He could see Rubens, Lummer and Pringsheim among them as he began his lecture, ‘Zur

Theorie des Gesetzes der Energieverteilung im Normalspektrum’, On the Theory of the Energy Distribution Law of the Normal Spectrum. It was just after 5pm on Friday, 14 December 1900. ‘Several weeks ago I had the honour of directing your attention to a new equation that seemed suitable to me for expressing the law of the distribution of radiating energy over all areas of the normal spectrum.’60 Planck now presented the physics behind that new equation as he derived it.
At the end of the meeting his colleagues roundly congratulated him. Just as Planck regarded the introduction of the quantum, a packet of energy, as a ‘purely formal assumption’ to which he ‘really did not give much thought’, so did everyone else that day. What was important to them was that Planck had succeeded in providing a physical justification for the formula he had presented in October. To be sure, his idea of chopping up energy into quanta for the oscillators was rather strange, but it would be ironed out in time. All believed that it was nothing more than the usual theorist’s sleight of hand, a neat mathematical trick on the path to getting the right answer. It had no true physical significance. What continued to impress his colleagues was the accuracy of his new radiation law. Nobody really took much notice of the quantum of energy, including Planck himself.
Early one morning Planck left home with his seven-year-old son, Erwin. Father and son were headed to nearby Grunewald Forest. Walking there was a favourite pastime of Planck’s and he enjoyed taking his son along. Erwin later recalled that as the pair walked and talked, his father told him: ‘Today I have made a discovery as important as that of Newton.’61 When he recounted the tale years later, Erwin could not remember exactly when the walk took place. It was probably some time before the December lecture. Was it possible that Planck understood the full implications of the quantum after all? Or was he just trying to convey to his young son something of the importance of his new radiation law? Neither. He was simply expressing his joy at discovering not one but two new fundamental constants: k, which he called Boltzmann’s constant, and h, which he called the quantum of action but which physicists would call Planck’s constant. They were fixed and eternal, two of nature’s absolutes.62
Planck acknowledged his debt to Boltzmann. Having named k after the Austrian, a constant that he had discovered in his research leading up to the blackbody formula, Planck also nominated Boltzmann for the Nobel Prize in 1905 and 1906. By then it was too late. Boltzmann had long been plagued by ill health – asthma, migraines, poor eyesight and angina. Yet none of these were as debilitating as the bouts of severe manic depression he suffered. In September 1906, while on holiday in Duino near Trieste, he hanged himself. He was 62, and though some of his friends had long feared the worst, news of his death came as a

terrible shock. Boltzmann had felt increasingly isolated and unappreciated. It was untrue. He was among the most widely honoured and admired physicists of the age. But continuing disputes over the existence of atoms had left him vulnerable during periods of despair to believing that his life’s work was being undermined. Boltzmann had returned to Vienna University for the third and last time in 1902. Planck was asked to succeed him. Describing Boltzmann’s work as ‘one of the most beautiful triumphs of theoretical research’, Planck was tempted by the Viennese offer but declined.63 h was the axe that chopped up energy into quanta, and Planck had been the first to wield it. But what he quantised was the way his imaginary oscillators could receive and emit energy. Planck did not quantise, chop into h -sized chunks, energy itself. There is a difference between making a discovery and fully understanding it, especially in a time of transition. There was much that Planck did that was only implicit in his derivation, and not even clear to him. He never explicitly quantised individual oscillators, as he should have done, but only groups of them.
Part of the problem was that Planck thought he could get rid of the quantum. He only realised the far-reaching consequences of what he had done much later. His deep conservative instincts compelled him to try for the best part of a decade to incorporate the quantum into the existing framework of physics. He knew that some of his colleagues saw this as bordering on a tragedy. ‘But I feel differently about it’, Planck wrote.64 ‘I now know for a fact that the elementary quantum of action [h] played a far more significant part in physics that I had originally been inclined to suspect.’
Years after Planck’s death in 1947, at the age of 89, his former student and colleague James Franck recalled watching his hopeless struggle ‘to avoid quantum theory, [to see] whether he could not at least make the influence of quantum theory as little as it could possibly be’.65 It was clear to Franck that Planck ‘was a revolutionary against his own will’ who ‘finally came to the conclusion, “It doesn’t help. We have to live with quantum theory. And believe me, it will expand.”’66 It was a fitting epitaph for a reluctant revolutionary.
Physicists did have to learn to ‘live with’ the quantum. The first to do so was not one of Planck’s distinguished peers, but a young man living in Bern, Switzerland. He alone realised the radical nature of the quantum. He was not a professional physicist, but a junior civil servant whom Planck credited with the discovery that energy itself is quantised. His name was Albert Einstein.

Chapter 2

THE PATENT SLAVE
Bern, Switzerland, Friday, 17 March 1905. It was nearly eight o’clock in the
morning as the young man dressed in the unusual plaid suit hurried to work clutching an envelope. To a passer-by, Albert Einstein appeared to have forgotten that he was wearing a pair of worn-out green slippers with embroidered flowers.1 At the same time six days a week, he left his wife and baby son, Hans Albert, behind in their small two-room apartment in the middle of Bern’s picturesque Old Town quarter, and walked to the rather grand sandstone building ten minutes away. With its famous clock tower, the Zytloggeturm, and arcades lining both sides of the cobbled street, Kramgasse was one of the most beautiful streets in the Swiss capital. Lost in thought, Einstein hardly noticed his surroundings as he made his way to the administrative headquarters of the Federal Post and Telephone Service. Once inside he headed straight for the stairs and the third floor that housed the Federal Office of Intellectual Property, better known as the Swiss Patent Office. Here he and the dozen other technical experts, men in more sober dark suits, laboured at their desks for eight hours a day sorting out the barely viable from the fatally flawed.
Three days earlier, Einstein had celebrated his 26th birthday. He had been a ‘patent slave’, as he called it, for nearly three years.2 For him the job brought to an end ‘the annoying business of starving’.3 The work itself he enjoyed for its variety, the ‘many-sided thinking’ it encouraged and the relaxed atmosphere of the office. It was an environment Einstein later referred to as his ‘worldly monastery’. Although the post of technical expert, third class, was a humble one, it was well-paid and allowed him time enough to pursue his own research. Despite the watchful eye of his boss, the formidable Herr Haller, Einstein spent so much time between examining patents secretly doing his own calculations that his desk had become his ‘office for theoretical physics’.4
‘It was as if the ground had been pulled out from under one, with no firm foundation to be seen anywhere, upon which one could have built’, was how Einstein recalled feeling after reading Planck’s solution of the blackbody problem soon after it was published.5 What he sent in the envelope to the editor of Annalen der Physik, the world’s leading physics journal, on 17 March 1905 was even more radical than Planck’s original introduction of the quantum. Einstein knew that his proposal of a quantum theory of light was nothing short of heresy.

Two months later, in the middle of May, Einstein wrote to his friend Conrad Habicht promising to send four papers he hoped to see published before the year’s end. The first was the quantum paper. The second was his PhD dissertation in which he set out a new way to determine the sizes of atoms. The third offered an explanation of Brownian motion, the erratic dance of tiny particles, like grains of pollen, suspended in liquid. ‘The fourth paper,’ Einstein admitted, ‘is only a rough draft at this point and is an electrodynamics of moving bodies which employs a modification of the theory of space and time.’6 It is an extraordinary list. In the annals of science only one other scientist and one other year bears comparison with Einstein and his achievements in 1905: Isaac Newton in 1666, when the 23-year-old Englishman laid the foundations of calculus and the theory of gravity, and outlined his theory of light.
Einstein would become synonymous with the theory first sketched out in his fourth paper: relativity. Although it would change humanity’s very understanding of the nature of space and time, it was the extension of Planck’s quantum concept to light and radiation that he described as ‘very revolutionary’, not relativity.7 Einstein regarded relativity as simply a ‘modification’ of ideas already developed and established by Newton and others, whereas his concept of light-quanta was something totally new, entirely his own, and represented the greatest break with the physics of the past. Even for an amateur physicist it was sacrilegious.
For more than half a century it had been universally accepted that light was a wave phenomenon. In ‘On a Heuristic Point of View Concerning the Production and Transformation of Light’, Einstein put forward the idea that light was not made up of waves, but particle-like quanta. In his resolution of the blackbody problem Planck had reluctantly introduced the idea that energy was absorbed or emitted as quanta, in discrete lumps. However, he, like everyone else, believed that electromagnetic radiation itself was a continuous wave phenomenon, whatever the mechanism of how it exchanged energy when it interacted with matter. Einstein’s revolutionary ‘point of view’ was that light, indeed all electromagnetic radiation, was not wavelike at all but chopped up into little bits, light-quanta. For the next twenty years, virtually no one but he believed in his quantum of light.
From the beginning Einstein knew it would be an uphill struggle. He signalled as much by including ‘On a Heuristic Point of View’ in the title of his paper. ‘Heuristic’, as defined by The Shorter Oxford English Dictionary, means ‘serving to find out’. What he was offering physicists was a way to explain the unexplained when it came to light, not a fully worked-out theory derived from first principles. His paper was a signpost towards such a theory, but even that

proved too much for those unprepared to travel to a destination in the opposite direction to the long-established wave theory of light.
Received by the Annalen der Physik between 18 March and 30 June, Einstein’s four papers would transform physics in the years ahead. Remarkably, he also found the time and energy to write 21 book reviews for the journal during the course of the year. Almost as an afterthought, since he did not tell Habicht about it, he wrote a fifth paper. It contained the one equation that almost everyone would come to know, E=mc2. ‘A storm broke loose in my mind’, was how he described the surge of creativity that consumed him as he produced his breathtaking succession of papers during that glorious Bern spring and summer of 1905.8
Max Planck, the adviser on theoretical physics for the Annalen der Physik, was among the first to read ‘On the Electrodynamics of Moving Bodies’. Planck was immediately won over by what he, and not Einstein, later called the theory of relativity. As for the quantum of light, though he profoundly disagreed with it, Planck allowed Einstein’s paper to be published. As he did so he must have wondered about the identity of this physicist capable of the sublime and the ridiculous.
‘The people of Ulm are mathematicians’ was the unusual medieval motto of the city on the banks of the Danube in the south-western corner of Germany where Albert Einstein was born. It was an apt birthplace on 14 March 1879 for the man who would become the epitome of scientific genius. The back of his head was so large and distorted, his mother feared her newborn son was deformed. Later he took so long to speak that his parents worried he never would. Not long after the birth of his sister, and only sibling, Maja in November 1881, Einstein adopted the rather strange ritual of softly repeating every sentence he wanted to say until satisfied it was word-perfect before uttering it aloud. At seven, to the relief of his parents, Hermann and Pauline, he began to speak normally. By then the family had lived in Munich for six years, having moved so Hermann could open an electrical business in partnership with his younger brother Jakob.
In October 1885, with the last of the private Jewish schools in Munich closed for more than a decade, the six-year-old Einstein was sent to the nearest school. Not surprisingly in the heartland of German Catholicism, religious education formed an integral part of the curriculum, but the teachers, he recalled many years later, ‘were liberal and did not make any denominational distinctions’.9 However liberal and accommodating his teachers may have been, the anti-Semitism that

permeated German society was never buried too far beneath the surface, even in the schoolroom. Einstein never forgot the lesson in which his religious studies teacher told the class how the Jews had nailed Christ to the cross. ‘Among the children,’ Einstein recalled years later, ‘anti-Semitism was alive especially in elementary school.’10 Not surprisingly, he had few, if any, school friends. ‘I am truly a lone traveller and have never belonged to my country, my home, my friends, or even my immediate family, with my whole heart’, he wrote in 1930. He called himself an Einspänner, a one-horse cart.
As a schoolboy he preferred solitary pursuits and enjoyed nothing more than constructing ever-taller houses of cards. He had the patience and tenacity, even as a ten-year-old, to build them as high as fourteen storeys. These traits, already such a fundamental part of his make-up, would allow him to pursue his own scientific ideas when others might have given up. ‘God gave me the stubbornness of a mule,’ he said later, ‘and a fairly keen scent.’11 Though others disagreed, Einstein maintained he possessed no special talents, only a passionate curiosity. This quality that others had, however, coupled with his stubbornness, meant that he continued to seek the answer to almost childlike questions long after his peers were taught to stop even asking them. What would it be like to ride on a beam of light? It was trying to answer this question that set him on his decade-long path to the theory of relativity.
In 1888, aged nine, Einstein started at the Luitpold Gymnasium, and he later spoke bitterly of his days there. Whereas young Max Planck enjoyed and thrived under a strict, militaristic discipline focused on rote learning, Einstein did not. Despite resenting his teachers and their autocratic methods, he excelled academically even though the curriculum was orientated towards the humanities. He scored top marks in Latin and did well in Greek, even after being told by his teacher ‘that nothing would ever become of him’.12
The stifling emphasis on mechanical learning at school, and during music lessons with tutors at home, was in stark contrast to the nurturing influence of a penniless Polish medical student. Max Talmud was 21, and Albert ten, when every Thursday he began dining with the Einsteins as they adopted their own version of an old Jewish tradition of inviting a poor religious scholar to lunch on the Sabbath. Talmud quickly recognised the inquisitive young boy as a kindred spirit. Before long the two would spend hours discussing the books that Talmud had given him to read or had recommended. They began with books on popular science that brought to an end what Einstein called his ‘religious paradise of youth’.13

The years at a Catholic school and instruction at home by a relative on Judaism had left their mark. Einstein, to the surprise of his secular parents, had developed what he described as ‘a deep religiosity’. He stopped eating pork, sang religious songs on the way to school, and accepted the biblical story of creation as an established fact. Then, as he devoured one book after another on science, came the realisation that much of the Bible could not be true. It unleashed what he called ‘a fanatic freethinking coupled with the impression that youth is intentionally being deceived by the State through lies; it was a crushing impression’.14 It sowed the seeds of a lifelong suspicion of every kind of authority. He came to view the loss of his ‘religious paradise’ as the first attempt to free himself from ‘the chains of the “merely personal”, from an existence which is dominated by wishes, hopes and primitive feelings’.15
As he lost faith in the teachings of one sacred book, he began to experience the wonder of his sacred little geometry book. He was still at primary school when his Uncle Jakob introduced him to the rudiments of algebra and began posing problems for him to solve. By the time Talmud gave him a book on Euclid’s geometry, Einstein was already well versed in mathematics not normally expected of a boy of twelve. Talmud was surprised at the speed with which Einstein worked through the book, proving the theorems and completing the exercises. Such was his zeal that during the summer vacation he mastered the mathematics to be taught the following year at school.
With a father and an uncle in the electrical industry, Einstein not only learnt about science through reading but was surrounded by the technology that its application could produce. It was his father who unwittingly introduced Einstein to the wonder and mystery of science. One day, as his son lay ill in bed with a fever, Hermann showed him a compass. The movement of the needle appeared so miraculous that the five-year-old trembled and grew cold at the thought that ‘Something deeply hidden had to be behind things.’16
The Einstein brothers’ electrical business initially prospered. They went from manufacturing electric devices to installing power and lighting networks. The future seemed bright as the Einsteins notched up one success after another, including the contract to provide the first electric lighting for Munich’s famous Oktoberfest.17 But in the end the brothers were simply outgunned by the likes of Siemens and AEG. There were many small electrical firms that prospered and survived in the shadow of these giants, but Jakob was over-ambitious and Hermann too indecisive for their company to be one of them. Beaten but not bowed, the brothers decided that Italy, where electrification was just beginning, was the place to start afresh. So in June 1894 the Einsteins relocated to Milan. All

except fifteen-year-old Albert who was left behind in the care of distant relatives to complete the three remaining years to graduation from the school he detested.
For the sake of his parents he pretended that everything was fine in Munich. However, he was increasingly troubled by the thought of compulsory military service. Under German law, if he remained in the country until his seventeenth birthday, Einstein would have no choice but to report for duty when the time came or be declared a deserter. Alone and depressed, he had to think of a way out, when suddenly the perfect opportunity arose.
Dr Degenhart, the teacher of Greek who thought Einstein would never amount to anything, was now also his form tutor. During a heated argument, Degenhart told Einstein he should leave the school. Requiring no further encouragement, he did just that after obtaining a medical certificate stating that he was suffering from exhaustion and required complete rest to recover. At the same time, Einstein secured a testimonial from his mathematics teacher that he had mastered the subject to a level required to graduate. It had taken him just six months to follow in the footsteps of his family and cross the Alps into Italy.
His parents tried to reason with him, but Einstein refused to go back to Munich. He had an alternative plan. He would stay in Milan and prepare for the entrance exams, the following October, of the Federal Polytechnikum in Zurich. Established in 1854, and renamed Eidgenossische Technische Hochschule (ETH) in 1911, the ‘Poly’ was not as prestigious as Germany’s leading universities. However, it did not require graduation from a gymnasium as a precondition for entry. To be accepted, he explained to his parents, he just needed to pass its entrance exams.
They soon discovered the second part of their son’s plan. He wanted to renounce his German nationality and thereby remove the possibility of ever being called up for military service by the Reich. Too young to do it himself, Einstein needed his father’s consent. Hermann duly gave it and formally applied to the authorities for his son’s release. It was January 1896 before they received official notification that Albert, at the cost of three marks, was no longer a German citizen. For the next five years he was legally stateless until he became a Swiss citizen. A renowned pacifist later in life, once he was granted his new nationality Einstein turned up for his Swiss army medical, on 13 March 1901, the day before his 22nd birthday. Fortunately, he was found unfit for military service because of sweaty flat feet and varicose veins.18 As a teenager back in Munich, it was not the thought of serving in the army that bothered him, but the prospect of donning a grey uniform on behalf of the militarism of the German Reich which he hated.

‘The happy months of my sojourn in Italy are my most beautiful memories’ was how Einstein, even after 50 years, recalled his new carefree existence.19 He helped his father and uncle with their electrical business and travelled here and there visiting friends and family. In the spring of 1895 the family moved to Pavia, just south of Milan, where the brothers opened a new factory that lasted little more than a year before it too closed. Although amid the upheaval he worked hard to prepare, Einstein failed the Poly entrance exams. Yet his mathematics and physics results were so impressive that the professor of physics invited him to attend his lectures. It was a tantalising offer, but for once Einstein took some sound advice. He had done so badly in languages, literature and history that the director of the Poly urged him to go back to school for another year and recommended one in Switzerland.
By the end of October Einstein was in Aarau, a town 30 miles west of Zurich. With its liberal ethos, the Aargau canton school provided a stimulating environment that enabled Einstein to thrive. The experience of boarding with the classics teacher and his family was to leave an indelible mark. Jost Winteler and his wife Pauline encouraged freethinking among their three daughters and four sons, and dinner each evening was always a lively and noisy affair. Before long the Wintelers became surrogate parents and he even referred to them as ‘Papa Winteler’ and ‘Mama Winteler’. Whatever the old Einstein said later about being a lone traveller, the young Einstein needed people who cared about him and he for them. Soon it was September 1896 and exam time. Einstein passed easily and headed to Zurich and the Federal Polytechnikum.20
‘A happy man is too satisfied with the present to dwell too much upon the future’, Einstein had written at the start of a short essay called ‘My Future Plans’, during his two-hour French exam. But an inclination for abstract thinking and the lack of practical sense had led him to decide on a future as a teacher of mathematics and physics.21 So it was that Einstein found himself, in October 1896, the youngest of eleven new students entering the Poly’s School for Specialised Teachers in the Mathematical and Science Subjects. He was one of the five seeking to qualify to teach maths and physics. The only woman among them was to be his future wife.
None of Albert’s friends could understand why he was attracted to Mileva Maric. A Hungarian Serb, she was four years older and a bout of childhood tuberculosis had left her with a slight limp. During the first year they sat through the five compulsory maths courses and mechanics – the single physics course offered. Although he had devoured his little sacred book of geometry in Munich, Einstein was no longer interested in mathematics for its own sake. Hermann

Minkowski, his maths professor at the Poly, recalled that Einstein had been a ‘lazy dog’. It was not apathy but a failure to grasp, as Einstein later confessed, ‘that the approach to a more profound knowledge of the basic principles of physics is tied up with the most intricate mathematical methods’.22 It was something he learnt the hard way in the years of research that followed. He regretted not having tried harder to get ‘a sound mathematical education’.23
Fortunately, Marcel Grossmann, one of the other three besides Einstein and Mileva enrolled on the course, was a better mathematician and more studious than either of them. It would be to Grossmann that Einstein later turned for help as he struggled with the mathematics needed to formulate the general theory of relativity. The two quickly became friends as they talked ‘about anything that might interest young people whose eyes were open’.24 Only a year older, Grossmann must have been an astute judge of character, for he was so impressed by his classmate that he took him home to meet his parents. ‘This Einstein,’ he told them, ‘will one day be a very great man.’25
It was only by using Grossmann’s excellent set of notes that he passed the intermediate exams in October 1898. In old age, Einstein could barely bring himself to contemplate what might have happened without Grossmann’s help after he began skipping lectures. It had all been so different at the beginning of Heinrich Weber’s physics course, when Einstein looked ‘forward from one of his lectures to the next’.26 Weber, who was in his mid-fifties, could make physics come alive for his students, and Einstein conceded that he lectured on thermodynamics with ‘great mastery’. But he became disenchanted because Weber did not teach Maxwell’s theory of electromagnetism or any of the latest developments. Soon Einstein’s independent streak and contemptuous manner began to alienate his professors. ‘You’re a smart boy’, Weber told him. ‘But you have one great fault: you do not let yourself be told anything.’27
When the final exams took place in July 1900 he came fourth out of five. Einstein felt coerced by the exams, and they had such a deterring effect upon him that afterwards he found ‘the consideration of any scientific problems distasteful to me for an entire year’.28 Mileva was last, and the only one to fail. It was a bitter blow for the couple who were now affectionately calling each other ‘Johonzel’ (Johnny) and ‘Doxerl’ (Dollie). Another soon followed.
A future as a schoolteacher no longer appealed to Einstein. Four years in Zurich had given rise to a new ambition. He wanted to be a physicist. The chances of getting a full-time job at a university were slim even for the best students. The first step was an assistant’s position with one of the professors at the Poly. None wanted him and Einstein began searching further afield. ‘Soon I

will have honoured all physicists from the North Sea to the Southern tip of Italy with my offer!’ he wrote to Mileva in April 1901 while visiting his parents.29
One of those honoured was Wilhelm Ostwald, a chemist at the Leipzig University. Einstein wrote to him twice; both letters went unanswered. It must have been distressing for his father to watch his son’s growing despair. Hermann, unknown to Albert then or later, took it upon himself to intervene. ‘Please forgive a father who is so bold as to turn to you, esteemed Herr professor, in the interest of his son’, he wrote to Ostwald.30 ‘All those in position to give a judgement in the matter, praise his talents; in any case, I can assure you that he is extraordinarily studious and diligent and clings with great love to his science.’31 The heartfelt plea went unanswered. Later Ostwald would be the first to nominate Einstein for the Nobel Prize.
Although anti-Semitism may have played a part, Einstein was convinced that it was Weber’s poor references that were behind his failure to secure an assistantship. As he grew increasingly despondent, a letter from Grossmann held out the possibility of a decent, well-paying job. Grossmann senior had learnt of Einstein’s desperate situation and wanted to help the young man whom his son held in such high regard. He strongly recommended Einstein for the next vacancy that arose to his friend Friedrich Haller, the director of the Swiss Patent Office in Bern. ‘When I found your letter yesterday,’ Einstein wrote to Marcel, ‘I was deeply moved by your devotion and compassion which did not let you forget your old luckless friend.’32 After five years of being stateless, Einstein had recently acquired Swiss citizenship and was certain it would help when applying for the job.
Maybe his luck had changed at last. He was offered and accepted a temporary teaching job at the school in Winterthur, a small town less than twenty miles from Zurich. The five or six classes Einstein taught each morning left him free to pursue physics in the afternoon. ‘I cannot tell you how happy I would feel in such a job’, he wrote to Papa Winteler shortly before his time in Winterthur ended. ‘I have completely given up my ambition to get a position at a university, since I see that even as it is, I have enough strength and desire left for scientific endeavour.’33 Soon that strength was put to the test when Mileva announced she was pregnant.
After failing the Poly exams a second time, Mileva returned to her parents in Hungary to await the arrival of the baby. Einstein took the news of the pregnancy in his stride. He had already entertained thoughts of becoming an insurance clerk and now vowed to find any job, no matter how humble, so that they could marry. When their daughter was born, Einstein was in Bern. He never saw Lieserl. What

happened to her, whether she was given up for adoption or died in infancy, remains a mystery.
In December 1901, Friedrich Haller wrote to Einstein asking him to apply for a vacancy at the Patent Office that was about to be advertised.34 The long search for a permanent job seemed at an end as Einstein sent off his application before Christmas. ‘All the time I rejoice in the fine prospects which are in store for us in the near future’, he wrote to Mileva. ‘Have I already told you how rich we will be in Bern?’35 Convinced that everything would be settled quickly, Einstein quit a year-long tutoring job at a private boarding school in Schaffhausen after only a few months.
Bern was home to some 60,000 people when Einstein arrived during the first week of February 1902. The medieval elegance of the Old Town quarter had changed little in the 500 years since it had been rebuilt following a fire that destroyed half the city. It was here that Einstein found a room on Gerechtigkeitgasse, not far from the city’s famous bear pit.36 Costing 23 francs a month, it was anything but the ‘large, beautiful room’ he described to Mileva.37 Not long after he unpacked his bags, Einstein went down to the local newspaper to place an advert offering his services as a private tutor of mathematics and physics. It appeared on Wednesday, 5 February and offered a free trial lesson. Within days it paid off. One of the students described his new tutor as ‘about five foot ten, broad-shouldered, slightly stooped, a pale brown skin, a sensuous mouth, black moustache, nose slightly aquiline, radiant brown eyes, a pleasant voice, speaking French correctly but with a slight accent’.38
A young Romanian Jew, Maurice Solovine, also came across the advert as he read his newspaper walking down the street. A philosophy student at Bern University, Solovine was also interested in physics. Frustrated that a lack of mathematics was preventing him from gaining a deeper understanding of physics, he immediately made his way to the address given in the newspaper. When Solovine rang the bell, Einstein had found a kindred spirit. The student and tutor talked for two hours. They shared many of the same interests and after spending another half hour chatting in the street, they agreed to meet the following day. When they did, all thoughts of a structured lesson were forgotten amid a shared enthusiasm for exploring ideas. ‘As a matter of fact, you don’t have to be tutored in physics’, Einstein told him on the third day.39 What Solovine liked about Einstein, as the two quickly became friends, was the care with which he outlined a topic or problem as lucidly as possible.

Before long, Solovine suggested that they read a particular book and then discuss it. Having done the same with Max Talmud in Munich as a schoolboy, Einstein thought it an excellent idea. Soon Conrad Habicht joined them. A friend from Einstein’s aborted stint teaching at the boarding school in Schaffhausen, Habicht had moved to Bern to complete a mathematics thesis at the university. United by their enthusiasm for studying and clarifying the problems of physics and philosophy for their own satisfaction, the three men started calling themselves the ‘Akademie Olympia’.
Even though Einstein came highly recommended by a friend, Haller had to make sure he was capable of doing the job. The ever-growing number of patent applications for all manner of electrical devices had made the hiring of a competent physicist to work alongside his engineers a necessity rather than a favour for a friend. Einstein impressed Haller sufficiently to be provisionally appointed a ‘Technical Expert, Third Class’ with a salary of 3,500 Swiss francs. At eight o’clock in the morning on 23 June 1902, Einstein reported for his first day as a ‘respectable Federal ink pisser’.40
‘As a physicist,’ Haller told Einstein, ‘you haven’t a clue about blueprints.’41 Until he could read and assess technical drawings, there would be no permanent contract. Haller took it upon himself to teach Einstein what he needed to know, including the art of expressing himself clearly, concisely, and correctly. Although he had never taken kindly to being instructed as a schoolboy or student, he knew that he needed to learn all he could from Haller, ‘a splendid character and a clever mind’.42 ‘One soon gets used to his rough manner’, Einstein wrote. ‘I hold him in very high regard.’43 As he proved his worth, Haller likewise came to respect his young protégé as a prized member of staff.
In October 1902, aged only 55, his father fell seriously ill. Einstein travelled to Italy to see him one last time. It was then, as he lay dying, that Hermann gave Albert his permission to marry Mileva – a prospect that he and Pauline had long opposed. With only Solovine and Habicht as witnesses, Einstein and Mileva married the following January in a civil ceremony at the Bern registrar’s office. ‘Marriage is,’ Einstein said later, ‘the unsuccessful attempt to make something lasting out of an incident.’44 But in 1903 he was just happy to have a wife that cooked, cleaned, and simply looked after him.45 Mileva had hoped for more.
The Patent Office took up 48 hours a week. From Monday to Saturday Einstein started at eight o’clock and worked until noon. Then it was lunch either at home or with a friend at a nearby café. He was back in the office from two until six. It left ‘eight hours for fooling around’ each day, and ‘then there’s also Sunday’, he told Habicht.46 It was September 1904 before Einstein’s ‘provisional’ position was

made permanent with a pay rise of 400 francs. By the spring of 1906 Haller was so impressed with Einstein’s ability to ‘tackle technically very difficult patent applications’ that he rated him as ‘one of the valued experts at the office’.47 He was promoted to technical expert, second class.
‘I will be grateful to Haller for as long as I live’, Einstein had written to Mileva soon after moving to Bern in the expectation that a job at the Patent Office would sooner or later be his.48 And he was. But it was only much later that he recognised the extent of the influence that Haller and the Patent Office exerted on him: ‘I might not have died, but I would have been intellectually stunted.’49 Haller demanded that every patent application be evaluated rigorously enough to withstand any legal challenge. ‘When you pick up an application, think that anything the inventor says is wrong,’ he advised Einstein, or else ‘you will follow the inventor’s way of thinking, and that will prejudice you. You have to remain critically vigilant.’50 Accidentally, Einstein had found a job that suited his temperament and honed his abilities. The critical vigilance he exercised in assessing an inventor’s hopes and dreams, often on the basis of unreliable drawings and inadequate technical specifications, Einstein brought to bear on the physics that occupied him. The ‘many-sided thinking’ his job entailed he described as a ‘veritable blessing’.51
‘He had the gift of seeing a meaning behind inconspicuous, well-known facts which had escaped everyone else’, recalled Einstein’s friend and fellow theoretical physicist Max Born. ‘It was this uncanny insight into the working of nature which distinguished him from all of us, not his mathematical skill.’52 Einstein knew that his mathematical intuition was not strong enough to differentiate what was really basic ‘from the rest of the more or less dispensable erudition’.53 But when it came to physics, his nose was second to none. Einstein said he ‘learned to scent out that which was able to lead to fundamentals and to turn aside from everything else, from the multitude of things which clutter up the mind and divert it from the essential’.54
His years at the Patent Office only heightened his sense of smell. As with the patents that inventors submitted, Einstein looked for subtle flaws and inconsistencies in the blueprints of the workings of nature put forward by physicists. When he found such a contradiction in a theory, Einstein probed it ceaselessly until it yielded a new insight resulting in its elimination or an alternative where none had existed before. His ‘heuristic’ principle that light behaved in certain instances as if it was made up of a stream of particles, lightquanta, was Einstein’s solution to a contradiction at the very heart of physics.

Einstein had long accepted that everything was composed of atoms and that these discrete, discontinuous bits of matter possessed energy. The energy of a gas, for example, was the sum total of the energies of the individual atoms of which it was made up. The situation was entirely different when it came to light. According to Maxwell’s theory of electromagnetism, or any wave theory, the energy of a light ray continuously spreads out over an ever-increasing volume like the waves radiating outwards from the point where a stone hits the surface of a pond. Einstein called it a ‘profound formal difference’ and it made him uneasy while stimulating his ‘many-sided thinking’.55 He realised that the dichotomy between the discontinuity of matter and the continuity of electromagnetic waves would dissolve if light was also discontinuous, made up of quanta.56
The quantum of light emerged out of Einstein’s review of Planck’s derivation of the blackbody radiation law. He accepted that Planck’s formula was correct, but his analysis revealed what Einstein had always suspected. Planck should have arrived at an entirely different formula. However, since he knew the equation he was looking for, Planck fashioned his derivation to get it. Einstein worked out exactly where Planck had gone astray. In his desperation to justify his equation that he knew to be in perfect agreement with experiments, Planck had failed to consistently apply the ideas and techniques he used or that were available to him. If he had done so, Einstein realised that Planck would have obtained an equation that did not agree with the data.
Lord Rayleigh had originally proposed this other formula in June 1900, but Planck had taken little, if any, notice of it. At the time he did not believe in the existence of atoms and therefore disapproved of Rayleigh’s use of the equipartition theorem. Atoms are free to move in only three ways: up and down, back and forth, and side to side. Called a ‘degree of freedom’, each is an independent way in which an atom can receive and store energy. In addition to these three kinds of ‘translational’ motion, a molecule made up of two or more atoms has three types of rotational motion about the imaginary axes joining the atoms, giving a total of six degrees of freedom. According to the equipartition theorem, the energy of a gas should be distributed equally among its molecules and then divided equally among the different ways in which a molecule can move.
Rayleigh employed the equipartition theorem to divide up the energy of blackbody radiation among the different wavelengths of radiation present inside a cavity. It had been a flawless application of the physics of Newton, Maxwell and Boltzmann. Aside from a numerical error that was later corrected by James Jeans, there was a problem with what became known as the Rayleigh-Jeans law. It predicted a build-up of an infinite amount of energy in the ultraviolet region of

the spectrum. It was a breakdown of classical physics that many years later, in 1911, was dubbed ‘the ultraviolet catastrophe’. Thankfully it did not actually happen, for a universe bathed in a sea of ultraviolet radiation would have made human life impossible.
Einstein had derived the Rayleigh-Jeans law on his own and knew that the distribution of blackbody radiation that it forecast contradicted the experimental data and led to the absurdity of an infinite energy in the ultraviolet. Given that the Rayleigh-Jeans law tallied with the behaviour of blackbody radiation only at long wavelengths (very low frequencies), Einstein’s point of departure was Wilhelm Wien’s earlier blackbody radiation law. It was the only safe choice, even though Wien’s law managed to replicate the behaviour of blackbody radiation only at short wavelengths (high frequencies) and failed at longer wavelengths (lower frequencies) of the infrared. Yet it had certain advantages that appealed to Einstein. He had no doubts about the soundness of its derivation, and it perfectly described at least a portion of the blackbody spectrum to which he would restrict his argument.
Einstein devised a simple but ingenious plan. A gas is just a collection of particles, and in thermodynamic equilibrium it is the properties of these particles that determine, for example, the pressure exerted by the gas at a given temperature. If there were similarities between the properties of blackbody radiation and the properties of a gas, then he could argue that electromagnetic radiation is itself particle-like. Einstein began his analysis with an imaginary blackbody that was empty. But unlike Planck, he filled it with gas particles and electrons. The atoms in the walls of the blackbody, however, contained other electrons. As the blackbody is heated, they oscillate with a broad range of frequencies resulting in the emission and absorption of radiation. Soon the interior of the blackbody is teeming with speeding gas particles and electrons, and the radiation emitted by the oscillating electrons. After a while, thermal equilibrium is reached when the cavity and everything inside it is at the same temperature T.
The first law of thermodynamics, that energy is conserved, can be translated to connect the entropy of a system to its energy, temperature and volume. It was now that Einstein used this law, Wien’s law and Boltzmann’s ideas to analyse how the entropy of blackbody radiation depended on the volume it occupied ‘without establishing any model for the emission or propagation of radiation’.57 What he found was a formula that looked exactly like one describing how the entropy of a gas, made up of atoms, is dependent on the volume it occupies. Blackbody radiation behaved as if it was made up of individual particle-like bits of energy.

Einstein had discovered the quantum of light without having to use either Planck’s blackbody radiation law or his method. In keeping Planck at arm’s length, Einstein wrote the formula slightly differently but it meant and encoded the same information as E=h , that energy is quantised, that it comes only in units of h . Whereas Planck had only quantised the emission and absorption of electromagnetic radiation so that his imaginary oscillators would produce the correct spectral distribution of blackbody radiation, Einstein had quantised electromagnetic radiation, and therefore light, itself. The energy of a quantum of yellow light was just Planck’s constant multiplied by the frequency of yellow light.
By showing that electromagnetic radiation sometimes behaves like the particles of a gas, Einstein knew that he had smuggled his light-quanta in through the back door, by analogy. To convince others of the ‘heuristic’ value of his new ‘point of view’ concerning the nature of light, he used it to explain a little-understood phenomenon.58
The German physicist Heinrich Hertz first observed the photoelectric effect in 1887 while in the middle of performing a series of experiments that demonstrated the existence of electromagnetic waves. By chance he noticed that the spark between two metal spheres became brighter when one of them was illuminated by ultraviolet light. After months of investigating the ‘completely new and very puzzling phenomenon’ he could offer no explanation, but believed, incorrectly, that it was confined to the use of ultraviolet light.59
‘Naturally, it would be nice if it were less puzzling,’ Hertz admitted, ‘however, there is some hope that when this puzzle is solved, more new facts will be clarified than if it were easy to solve.’60 It was a prophetic statement, but one that he never lived to see fulfilled. He died tragically young at the age of 36 in 1894.
It was Hertz’s former assistant, Philipp Lenard, who in 1902 deepened the mystery surrounding the photoelectric effect when he discovered that it also occurred in a vacuum when he placed two metal plates in a glass tube and removed the air. Connecting the wires from each plate to a battery, Lenard found that a current flowed when one of the plates was irradiated with ultraviolet light. The photoelectric effect was explained as the emission of electrons from the illuminated metal surface. Shining ultraviolet light onto the plate gave some electrons enough energy to escape from the metal and cross the gap to the other plate, thereby completing the circuit to produce a ‘photoelectric current’. However, Lenard also found facts that contradicted established physics. Enter Einstein and his quantum of light.

It was expected that increasing the intensity of a light beam, by making it brighter, would yield the same number of electrons from the metal surface, but with each having more energy. Lenard, however, found the exact opposite: a greater number of electrons were emitted with no change in their individual energy. Einstein’s quantum solution was simple and elegant: if light is made up of quanta, then increasing the intensity of the beam means that it is now made up of a greater number of quanta. When a more intense beam strikes the metal plate, the increase in the number of light-quanta leads to a corresponding increase in the number of electrons being emitted.
Lenard’s second curious discovery was that the energy of the emitted electrons was not governed by the intensity of the light beam, but by its frequency. Einstein had a ready answer. Since the energy of a light-quantum is proportional to the frequency of the light, a quantum of red light (low frequency) has less energy than one of blue light (high frequency). Changing the colour (frequency) of light does not alter the number of quanta in beams of the same intensity. So, no matter what the colour of light, the same number of electrons will be emitted since the same numbers of quanta strike the metal plate. However, since different frequencies of light are made up of quanta of different energies, the electrons that are emitted will have more or less energy depending on the light used. Ultraviolet light will yield electrons with a greater maximum kinetic energy than those emitted by quanta of red light.
There was another intriguing feature. For any particular metal there was a minimum or ‘threshold frequency’ below which no electrons were emitted at all, no matter how long or intensively the metal was illuminated. However, once this threshold was crossed, electrons were emitted no matter how dim the beam of light. Einstein’s quantum of light supplied the answer once again as he introduced a new concept, the work function.
Einstein envisaged the photoelectric effect as the result of an electron acquiring enough energy from a quantum of light to overcome the forces holding it within the metal surface and to escape. The work function, as Einstein labelled it, was the minimum energy an electron needed to escape from the surface, and it varied from metal to metal. If the frequency of light is too low, then the light-quanta will not possess enough energy to allow an electron to break the bonds that keep it bound within the metal.
Einstein encoded all this in a simple equation: the maximum kinetic energy of an electron emitted from a metal surface was equal to the energy of the lightquanta it absorbed minus the work function. Using this equation, Einstein predicted that a graph of the maximum kinetic energy of the electrons versus the

frequency of light used would be a straight line, beginning at the threshold frequency of the metal. The gradient of the line, irrespective of the metal used, would always be exactly equal to Planck’s constant, h.
Figure 3: The photoelectric effect – maximum kinetic energy of emitted electrons versus the frequency of light striking the metal
surface
‘I spent ten years of my life testing that 1905 equation of Einstein’s and contrary to all my expectations,’ complained the American experimental physicist Robert Millikan, ‘I was compelled to assert its unambiguous verification in spite of its unreasonableness, since it seemed to violate everything we knew about the interference of light.’61 Although Millikan won the 1923 Nobel Prize partly in recognition of this work, even in the face of his own data he balked at the underlying quantum hypothesis: ‘the physical theory upon which the equation is based is totally untenable.’62 From the very beginning, physicists at large had greeted Einstein’s light-quanta with similar disbelief and cynicism. A handful wondered if light-quanta existed at all or whether they were simply a useful fictional contrivance of practical value in calculations. At best some thought that light, and therefore all electromagnetic radiation, did not consist of quanta, but only behaved as such when exchanging energy with matter.63 Foremost among them was Planck.
When in 1913 he and three others nominated Einstein for membership of the Prussian Academy of Sciences, they concluded their testimonial by trying to excuse his light-quanta proposal: ‘In sum, it can be said that among the important problems, which are so abundant in modern physics, there is hardly one in which

Einstein did not take a position in a remarkable manner. That he might sometimes have overshot the target in his speculations, as for example in his light-quantum hypothesis, should not be counted against him too much. Because without taking a risk from time to time it is impossible, even in the most exact natural science, to introduce real innovations.’64
Two years later, Millikan’s painstaking experiments made it difficult to ignore the validity of Einstein’s photoelectric equation. By 1922 it was becoming almost impossible, as Einstein was belatedly awarded the 1921 Nobel Prize for physics explicitly for his photoelectric effect law, described by his formula, and not for his underlying explanation using light-quanta. No longer the unknown patent clerk in Bern, he was by then world-famous for his theories of relativity and widely acknowledged as the greatest scientist since Newton. Yet his quantum theory of light was just too radical for physicists to accept.
The stubborn opposition to Einstein’s idea of light-quanta rested on the overwhelming evidence in support of a wave theory of light. However, whether light was a particle or a wave had been hotly disputed before. During the eighteenth century and in the early years of the nineteenth, it was Isaac Newton’s particle theory that had triumphed. ‘My Design in this Book is not to explain the Properties of Light by Hypotheses,’ Newton wrote at the beginning of Opticks, published in 1704, ‘but to propose and prove them by Reason and Experiments.’65 Those first experiments were conducted in 1666, when he split light into the colours of the rainbow with a prism and wove them back together into white light using a second prism. Newton believed that rays of light were composed of particles or, as he called them, ‘corpuscles’, the ‘very small Bodies emitted from shining Substances’.66 With the particles of light travelling in straight lines, such a theory would, according to Newton, explain the everyday fact that while a person can be heard talking around a corner, they cannot be seen, since light cannot not bend around corners.
Newton was able to give a detailed mathematical account for a host of optical observations, including reflection and refraction – the bending of light as it passes from a less to a more dense medium. However, there were other properties of light that Newton could not explain. For example, when a beam of light hit a glass surface, part of it passed through and the rest was reflected. The question Newton had to address was why some particles of light were reflected and others not? To answer it, he was forced to adapt his theory. Light particles caused wavelike disturbances in the ether. These ‘Fits of easy Reflexion and easy Transmission’, as he called them, were the mechanism by which some of the

beam of light was transmitted through the glass and the remainder reflected.67 He linked the ‘bigness’ of these disturbances to colour. The biggest disturbances, those having the longest wavelength, in the terminology that came later, were responsible for producing red. The smallest, those having the shortest wavelength, produced violet.
The Dutch physicist Christiaan Huygens argued that there was no Newtonian particle of light. Thirteen years older than Newton, by 1678 Huygens had developed a wave theory of light that explained reflection and refraction. However, his book on the subject, Traité de la Lumière, was not published until 1690. Huygens believed that light was a wave travelling through the ether. It was akin to the ripples that fanned out across the still surface of a pond from a dropped stone. If light was really made up of particles, Huygens asked, then where was the evidence of collisions that should occur when two beams of light crossed each other? There was none, argued Huygens. Sound waves do not collide; ergo light must also be wavelike.
Although the theories of Newton and Huygens were able to explain reflection and refraction, each predicted different outcomes when it came to certain other optical phenomena. None could be tested with any degree of precision for decades. However, there was one prediction that could be observed. A beam of light made up of Newton’s particles travelling in straight lines should cast sharp shadows when striking objects, whereas Huygens’ waves, like water waves bending around an object they encounter, should result in shadows whose outline is slightly blurred. The Italian Jesuit and mathematician Father Francesco Grimaldi christened this bending of light around the edge of an object, or around the edges of an extremely narrow slit, diffraction. In a book published in 1665, two years after his death, he described how an opaque object placed in a narrow shaft of sunlight allowed to enter an otherwise darkened room through a very small hole in a window shutter, cast a shadow larger than expected if light consisted of particles travelling in straight lines. He also found that around the shadow were fringes of coloured light and fuzziness where there should have been a sharp, well-defined separation between light and dark.
Newton was well aware of Grimaldi’s discovery and later conducted his own experiments to investigate diffraction, which seemed more readily explicable in terms of Huygens’ wave theory. However, Newton argued that diffraction was the result of forces exerted on light particles and indicative of the nature of light itself. Given his pre-eminence, Newton’s particle theory of light, though in truth a strange hybrid of particle and wave, was accepted as the orthodoxy. It helped that Newton outlived Huygens, who died in 1695, by 32 years. ‘Nature and Nature’s Laws lay hid in Night; / God said, Let Newton be! And all was Light.’ Alexander

Pope’s famous epitaph bears witness to the awe in which Newton was held in his own day. In the years after his death in 1727, Newton’s authority was undiminished and his view on the nature of light barely questioned. At the dawn of the nineteenth century the English polymath, Thomas Young, did challenge it, and in time his work led to a revival of the wave theory of light.
Born in 1773, Young was the eldest of ten children. He was reading fluently by the age of two and had read the entire Bible twice by six. A master of more than a dozen languages, Young went on make important contributions towards the deciphering of Egyptian hieroglyphics. A trained physician, he could indulge his myriad intellectual pursuits after a bequest from an uncle left him financially secure. His interest in the nature of light led Young to examine the similarities and differences between light and sound, and ultimately to ‘one or two difficulties in the Newtonian system’.68 Convinced that light was a wave, he devised an experiment that was to prove the beginning of the end for Newton’s particle theory.
Young shone monochromatic light onto a screen with a single slit. From this slit a beam of light spread out to strike a second screen with two very narrow and parallel slits close together. Like a car’s headlights, these two slits acted as new sources of light, or as Young wrote, ‘as centres of divergence, from whence the light diffracted in every direction’.69 What Young found on another screen placed some distance behind the two slits was a central bright band surrounded on each side by a pattern of alternating dark and bright bands.
Figure 4: Young’s two-slits experiment. At far right, the resulting interference pattern on the screen is shown
To explain the appearance of these bright and dark ‘fringes’, Young used an analogy. Two stones are dropped simultaneously and close together into a still

lake. Each stone produces waves that spread out across the lake. As they do so, the ripples originating from one stone encounter those from the other. At each point where two wave troughs or two wave crests meet, they coalesce to produce a new single trough or crest. This was constructive interference. But where a trough meets a crest or vice versa, they cancel each other out, leaving the water undisturbed at that point – destructive interference.
In Young’s experiment, light waves originating from the two slits similarly interfere with each other before striking the screen. The bright fringes indicate constructive interference while the dark fringes are a product of destructive interference. Young recognised that only if light is a wave phenomenon could these results be explained. Newton’s particles would simply produce two bright images of the slits with nothing but darkness in between. An interference pattern of bright and dark fringes was simply impossible.
When he first put forward the idea of interference and reported his early results in 1801, Young was viciously attacked in print for challenging Newton. He tried to defend himself by writing a pamphlet in which he let everyone know his feelings about Newton: ‘But, much as I venerate the name of Newton, I am not therefore obliged to believe that he was infallible. I see, not with exultation, but with regret, that he was liable to err, and that his authority has, perhaps, sometimes even retarded the progress of science.’70 Only a single copy was sold.
It was a French civil engineer who followed Young in stepping out of Newton’s shadow. Augustin Fresnel, fifteen years his junior, independently rediscovered interference and much else of what Young, unknown to him, had already done. However, compared to the Englishman, Fresnel’s elegantly designed experiments were more extensive, with the presentation of results and accompanying mathematical analysis so impeccably thorough that the wave theory started to gain distinguished converts by the 1820s. Fresnel convinced them that the wave theory could better explain an array of optical phenomena than Newton’s particle theory. He also answered the long-standing objection to the wave theory: light cannot travel around corners. It does, he said. However, since light waves are millions of times smaller than sound waves, the bending of a beam of light from a straight path is very, very small and therefore extremely difficult to detect. A wave bends only around an obstacle not much longer than itself. Sound waves are very long and can easily move around most barriers they encounter.
One way to get opponents and sceptics to finally decide between the two rival theories was to find observations for which they predicted different results. Experiments conducted in France in 1850 revealed that the speed of light was slower in a dense medium such as glass or water than in the air. This was exactly

what the wave of light predicted, while Newton’s corpuscles failed to travel as fast as expected. But the question remained: if light was a wave, what were its properties? Enter James Clerk Maxwell and his theory of electromagnetism.
Born in 1831 in Edinburgh, Maxwell, the son of a Scottish landowner, was destined to become the greatest theoretical physicist of the nineteenth century. At the age of fifteen, he wrote his first published paper on a geometrical method for tracing ovals. In 1855 he won Cambridge University’s Adams Prize for showing that Saturn’s rings could not be solid, but had to be made of small, broken bits of matter. In 1860 he instigated the final phase of the development of the kinetic theory of gases, the properties of gases explained by maintaining that they consisted of particles in motion. But his greatest achievement was the theory of electromagnetism.
In 1819 the Danish physicist Hans Christian Oersted discovered that an electric current flowing through a wire deflected a compass needle. A year later the Frenchman François Arago found that a wire carrying an electric current acted as a magnet and could attract iron filings. Soon his compatriot André Marie Ampère demonstrated that two parallel wires were attracted towards one another if each had a current flowing through it in the same direction. However, they repelled each other if the currents flowed in the opposite directions. Intrigued by the fact that a flow of electricity could create magnetism, the great British experimentalist Michael Faraday decided to see if he could generate electricity using magnetism. He pushed a bar magnet in and out of a helix coil of wire and found an electric current being generated. The current ceased whenever the magnet was motionless within the coil.
Just as ice, water and steam are different manifestations of H O, Maxwell 2
showed in 1864 that electricity and magnetism were likewise different manifestations of the same underlying phenomenon – electromagnetism. He managed to encapsulate the disparate behaviour of electricity and magnetism into a set of four elegant mathematical equations. On seeing them, Ludwig Boltzmann immediately recognised the magnitude of Maxwell’s achievement and could only quote Goethe in admiration: ‘Was it a God that wrote these signs?’71 Using these equations, Maxwell was able to make the startling prediction that electromagnetic waves travelled at the speed of light through the ether. If he was right, then light was a form of electromagnetic radiation. But did electromagnetic waves actually exist? If so, did they really travel at the speed of light? Maxwell did not live long enough to see his prediction confirmed by experiment. Aged just 48, he died from cancer in November 1879, the year Einstein was born. Less than a decade later, in 1887, Heinrich Hertz provided the experimental corroboration that ensured

Maxwell’s unification of electricity, magnetism and light was the crowning achievement of nineteenth-century physics.
Hertz proclaimed in his paper outlining his investigations: ‘The experiments described appear to me, at any rate, eminently adapted to remove any doubt as to the identity of light, radiant heat, and electromagnetic wave motion. I believe that from now on we shall have greater confidence in making use of the advantages, which this identity enables us to derive both in the study of optics and electricity.’72 Ironically, it was during these very experiments that Hertz discovered the photoelectric effect that provided Einstein with evidence for a case of mistaken identity. His light-quanta challenged the wave theory of light that Hertz and everyone else thought was well and truly established. Light as a form of electromagnetic radiation had proved so successful that for physicists to even contemplate discarding it in favour of Einstein’s light-quanta was unthinkable. Many found light-quanta absurd. After all, the energy of a particular quantum of light was determined by the frequency of that light, but surely frequency was something associated with waves, not particle-like bits of energy travelling through space.
Einstein readily accepted that the wave theory of light had ‘proved itself superbly’ in explaining diffraction, interference, reflection and refraction, and that it would ‘probably never be replaced by another theory’.73 However, this success, he pointed out, rested on the vital fact that all these optical phenomena involved the behaviour of light over a period of time, and any particle-like properties would not be manifest. The situation was starkly different when it came to the virtually ‘instantaneous’ emission and absorption of light. This was the reason, Einstein suggested, why the wave theory faced ‘especially great difficulties’ explaining the photoelectric effect.74
A future Nobel laureate, but in 1906 a privatdozent at Berlin University, Max Laue wrote to Einstein that he was willing to accept that quanta may be involved during the emission and absorption of light. However, that was all. Light itself was not made up of quanta, warned Laue, but it is ‘when it is exchanging energy with matter that it behaves as if it consisted of them’.75 Few even conceded that much. Part of the problem lay with Einstein himself. In his original paper he did say that light ‘behaves’ as though it consisted of quanta. This was hardly a categorical endorsement of the quantum of light. This was because Einstein wanted something more than just a ‘heuristic point of view’: he craved a fullyfledged theory.
The photoelectric effect had proved to be a battlefield for the clash between the supposed continuity of light waves and the discontinuity of matter, atoms. But in

1905 there were still those who doubted the reality of atoms. On 11 May, less than two months after he finished his quantum paper, the Annalen der Physik received Einstein’s second paper of the year. It was his explanation of Brownian motion and it became a key piece of evidence in support of the existence of atoms.76
When in 1827 the Scottish botanist Robert Brown peered through a microscope at some pollen grains suspended in water, he saw that they were in a constant state of haphazard motion as if buffeted by some unseen force. It had already been noted by others that this erratic wiggling increased as the temperature of the water rose, and it was assumed that some sort of biological explanation lay behind the phenomenon. However, Brown discovered that when he used pollen grains that were up to twenty years old they moved in exactly the same way. Intrigued, he produced fine powders of all manner of inorganic substances, from glass to a piece of the Sphinx, and suspended each of them in water. He found the same zigzagging motion in each case and realised that it could not be animated by some vital force. Brown published his research in pamphlet entitled: A Brief Account of Microscopical Observations Made in the Months of June, July, and August 1827, on the Particles Contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies. Others offered plausible explanations of ‘Brownian motion’, but all were sooner or later found wanting. By the end of the nineteenth century, those who believed in the existence of atoms and molecules accepted that Brownian motion was the result of collisions with water molecules.
What Einstein recognised was that the Brownian motion of a pollen grain was not caused by a single collision with a water molecule, but was the product of a large number of such collisions. At each moment, the collective effect of these collisions was the random zigzagging of the pollen grain or suspended particle. Einstein suspected that the key to understanding this unpredictable motion lay in deviations, statistical fluctuations, from the expected ‘average’ behaviour of water molecules. Given their relative sizes, on average, many water molecules would strike an individual pollen grain simultaneously from different directions. Even on this scale, each collision would result in an infinitesimal push in one direction, but the overall effect of all of them would leave the pollen unmoved as they cancelled each other out. Einstein realised that Brownian motion was due to water molecules regularly deviating from their ‘normal’ behaviour as some of them got bunched up and struck the pollen together, sending it in particular direction.
Using this insight, Einstein succeeded in calculating the average horizontal distance a particle would travel as it zigzagged along in a given time. He

predicted that in water at 17°C, suspended particles with a diameter of onethousandth of a millimetre would move on average just six-thousandths of a millimetre in one minute. Einstein had come up with a formula that offered the possibility of working out the size of atoms armed only with a thermometer, microscope and stopwatch. Three years later, in 1908, Einstein’s predictions were confirmed in a delicate series of experiments conducted at the Sorbonne by Jean Perrin, for which he received the Nobel Prize in 1926.
With Planck championing the theory of relativity, and the analysis of Brownian motion recognised as a decisive breakthrough in favour of the atom, Einstein’s reputation grew despite the rejection of his quantum theory of light. He received letters often addressed to him at Bern University, as few knew he was a patent clerk. ‘I must tell you quite frankly that I was surprised to read that you must sit in an office for 8 hours a day,’ wrote Jakob Laub from Würzburg. ‘History is full of bad jokes.’77 It was March 1908 and Einstein agreed. After almost six years he no longer wanted to be a patent slave.
He applied for a job as a mathematics teacher at a school in Zurich, stating that he would be ready and willing to teach physics as well. With his application he enclosed a copy of his thesis that had earned him, at the third attempt, a doctorate from Zurich University in 1905 and laid the groundwork for the paper on Brownian motion. Hoping it would bolster his chances, he also sent all of his published papers. Despite his impressive scientific achievements, of the 21 applicants, Einstein did not even make the short list of three.
It was at the behest of Alfred Kleiner, the professor of experimental physics at Zurich University, that Einstein tried for a third time to become a privatdozent, an unpaid lecturer, at the University of Bern. The first application was rejected because at the time he did not have a PhD. In June 1907, he failed a second time because he did not submit a habilitationsschrift – a piece of unpublished research. Kleiner wanted Einstein to fill a soon-to-be-created extraordinary professorship in theoretical physics, and being a privatdozent was a necessary stepping-stone to such an appointment. So he produced a habilitationsschrift as demanded and was duly appointed a privatdozent in the spring of 1908.
Only three students attended his first lecture course on the theory of heat. All three were friends. They had to be, since Einstein had been allocated Tuesdays and Saturdays between seven and eight in the morning. University students had the choice of whether or not to attend courses offered by a privatdozent and none were willing to get up that early. As a lecturer, then and later, Einstein was often

under-prepared and made frequent mistakes. And when he did, he simply turned to the students and asked: ‘Who can tell me where I went wrong?’ or ‘Where have I made a mistake?’ If a student pointed out an error in his mathematics, Einstein would say, ‘I have often told you, my mathematics have never been up to much.’78
The ability to teach was a vital consideration for the job earmarked for Einstein. To ensure that he was up to the task, Kleiner organised to attend one of his lectures. Annoyed at ‘having-to-be-investigated’, he performed poorly.79 However, Kleiner gave him a second chance to impress and he did. ‘I was lucky’, Einstein wrote to his friend Jakob Laub. ‘Contrary to my habit, I lectured well on that occasion – and so it came to pass.’80 It was May 1909 and Einstein could finally boast that he was ‘an official member of the guild of whores’ as he accepted the Zurich post.81 Before moving to Switzerland with Mileva and fiveyear-old Hans Albert, Einstein travelled to Salzburg in September to give the keynote lecture to the cream of German physics at a conference of the Gesellschaft Deutscher Naturforscher und Ärtze. He went well prepared.
It was a singular honour to be asked to deliver such a lecture. It was one usually reserved for a distinguished elder statesman of physics, not someone who had just turned 30 and was about take up his first extraordinary professorship. So all eyes were on Einstein, but he seemed oblivious as he paced the podium and delivered what would turn out to be a celebrated lecture: ‘On the Development of Our Views Concerning the Nature and Constitution of Radiation’. He told the audience that ‘the next stage in the development of theoretical physics will bring us a theory of light that may be conceived of as a sort of fusion of the wave and of the emission theory of light’.82 It was not a hunch, but based on the result of an inspired thought experiment involving a mirror suspended inside a blackbody. He managed to derive an equation for the fluctuations of the energy and momentum of radiation that contained two very distinct parts. One corresponded to the wave theory of light, while the other had all the hallmarks of the radiation being composed of quanta. Both parts appeared to be indispensable, as did the two theories of light. It was the first prediction of what would later be called waveparticle duality – that light was both a particle and a wave.
Planck, who was chairing, was the first to speak after Einstein sat down. He thanked him for the lecture and then told everyone he disagreed. He reiterated his firmly held belief that quanta were necessary only in the exchange between matter and radiation. To believe as Einstein did that light was actually made up of quanta, Planck said, was ‘not yet necessary’. Only Johannes Stark stood up to support Einstein. Sadly, he, like Lenard, would later become a Nazi and the two of them would attack Einstein and his work as ‘Jewish Physics’.

Einstein left the Patent Office to devote more of his time to research. He was in for a rude awakening when he arrived in Zurich. The time he needed to prepare for the seven hours of lectures that he gave each week left him complaining that his ‘actual free time is less than in Bern’.83 The students were struck by the shabby appearance of their new professor, but Einstein quickly gained their respect and affection by his informal style as he encouraged them to interrupt if anything was unclear. Outside formal lectures, at least once a week he took his students along to the Café Terasse to chat and gossip until closing time. Before long he got used to his workload and turned his attention to using the quantum to solve a longstanding problem.
In 1819 two French scientists, Pierre Dulong and Alexis Petit, measured the specific heat capacity, the amount of energy needed to raise the temperature of a kilogram of a substance by one degree, for various metals from copper to gold. For the next 50 years no one who believed in atoms doubted their conclusion that ‘the atoms of all simple bodies have exactly the same heat capacity’.84 It therefore came as a great surprise when, in the 1870s, exceptions were discovered.
Imagining that the atoms of a substance oscillated when heated, Einstein adapted Planck’s approach as he tackled the specific heat anomalies. Atoms could not oscillate with just any frequency, but were ‘quantised’ – able to oscillate only with those frequencies that were multiples of a certain ‘fundamental’ frequency. Einstein came up with a new theory of how solids absorb heat. Atoms are permitted to absorb energy only in discrete amounts, quanta. However, as the temperature drops, the amount of energy the substance has decreases, until there is not enough available to provide each atom with the correct-sized quantum of energy. This results in less energy being taken up by the solid and leads to a decrease in specific heat.
For three years there was hardly a murmur of interest in what Einstein had done, despite the fact that he had shown how the quantisation of energy – how at the atomic level energy comes wrapped up in bite-sized chunks – resolved a problem in a completely new area of physics. It was Walter Nernst, an eminent physicist from Berlin, who made others sit up and take note as they discovered that he had been to see Einstein in Zurich. Soon it was clear why. Nernst had succeeded in accurately measuring the specific heats of solids at low temperatures and found the results to be in total agreement with Einstein’s predictions based on his quantum solution.

With each passing success his reputation soared ever higher, and Einstein was offered an ordinary professorship at the German University in Prague. It was an opportunity he could not refuse, even if it meant leaving Switzerland after fifteen years. Einstein, Mileva and their sons Hans Albert and Eduard, who was not yet one, moved to Prague in April 1911.
‘I no longer ask whether these quanta really exist’, Einstein wrote to his friend Michele Besso soon after taking up his new post. ‘Nor do I try to construct them any longer, for I now know that my brain cannot get through in this way.’ Instead, he told Besso, he would limit himself to trying to understand the consequences of the quantum.85 There were others who also wanted to try. Less than a month later, on 9 June, Einstein received a letter and an invitation from an unlikely correspondent. Ernst Solvay, a Belgian industrialist who had made a substantial fortune by revolutionising the manufacture of sodium carbonate, offered to pay 1,000 francs to cover his travel expenses if he agreed to attend a week-long ‘Scientific Congress’ to be held in Brussels later that year from 29 October to 4 November.86 He would be one of a select group of 22 physicists from across Europe brought together to discuss ‘current questions concerning the molecular and kinetic theories’. Planck, Rubens, Wien and Nernst would be attending. It was a summit meeting on the quantum.
Planck and Einstein were among the eight asked to prepare reports on a particular topic. To be written in French, German or English, they were to be sent out to the participants before the meeting and serve as the starting point for discussion during the planned sessions. Planck would discuss blackbody radiation theory, while Einstein had been assigned his quantum theory of specific heat. Although Einstein was accorded the honour of giving the final talk, a discussion of his quantum theory of light was not on the agenda.
‘I find the whole undertaking extremely attractive,’ Einstein wrote to Walter Nernst, ‘and there is little doubt in my mind that you are its heart and soul.’87 By 1910 Nernst believed that the time was ripe to get to grips with the quantum that he regarded as nothing more than a ‘rule with most curious, indeed grotesque properties’.88 He convinced Solvay to finance the conference and the Belgian spared no expense booking the plush Hotel Metropole as the venue. In its luxurious surroundings, with all their needs catered for, Einstein and his colleagues spent five days talking about the quantum. Whatever slim hopes he harboured for progress at what he called ‘the Witches’ Sabbath’, Einstein returned to Prague disappointed and complained of learning nothing that he did not know before.89

Nevertheless, he had enjoyed getting to know some of the other ‘witches’. Marie Curie, whom he found to be ‘unpretentious’, appreciated ‘the clearness of his mind, the shrewdness with which he marshalled his facts and the depth of his knowledge’.90 During the congress it was announced that she had been awarded the Nobel Prize for chemistry. She had become the first scientist to win two, having already won the physics prize in 1903. It was a tremendous achievement that was overshadowed by the scandal that broke around her during the congress. The French press had learned that she was having an affair with a married French physicist. Paul Langevin, a slender man with an elegant moustache, was a delegate at the conference and the papers were full of stories that the pair had eloped. Einstein, who had seen no signs of a special relationship between the two, dismissed the reports as rubbish. Despite her ‘sparkling intelligence’, he thought Curie was ‘not attractive enough to represent a danger to anyone’.91
Even though at times he appeared to waver under the strain, Einstein had been the first to learn to live with the quantum, and by doing so revealed a hidden element of the true nature of light. Another young theorist also learned to live with the quantum after he used it to resurrect a flawed and neglected model of the atom.

Chapter 3

THE GOLDEN DANE
Manchester, England, Wednesday, 19 June 1912. ‘Dear Harald, Perhaps I have
found out a little about the structure of atoms,’ Niels Bohr wrote to his younger brother.1 ‘Don’t talk about it to anybody,’ he warned, ‘for otherwise I couldn’t write to you so soon.’ Silence was essential for Bohr, as he hoped to do what every scientist dreams of: unveiling ‘a little bit of reality’. There was still work to be done and he was ‘eager to finish it in a hurry, and to do that I have taken off a couple of days from the laboratory (this is also a secret)’. It would take the 26year-old Dane much longer than he thought to turn his fledgling ideas into a trilogy of papers all entitled ‘On the Constitution of Atoms and Molecules’. The first, published in July 1913, was truly revolutionary, as Bohr introduced the quantum directly into the atom.
It was his mother Ellen’s 25th birthday when Niels Henrik David Bohr was born on 7 October 1885 in Copenhagen. She had returned to the comfort of her parents’ home for the birth of her second child. Across the wide cobbled street from Christianborg Castle, the seat of the Danish parliament, Ved Stranden 14 was one of the most magnificent residences in the city. A banker and politician, her father was one of the wealthiest men in Denmark. Although the Bohrs did not stay there long, it was to be the first of the grand and elegant homes in which Niels lived throughout his life.
Christian Bohr was the distinguished professor of physiology at Copenhagen University. He had discovered the role of carbon dioxide in the release of oxygen by haemoglobin, and together with his research on respiration it led to nominations for the Nobel Prize for physiology or medicine. From 1886 until his untimely death in 1911, at just 56, the family lived in a spacious apartment in the university’s Academy of Surgery.2 Situated in the city’s most fashionable street and a ten-minute walk from the local school, it was ideal for the Bohr children: Jenny, two years older than Niels, and Harald, eighteen months younger.3 With three maids and a nanny to look after them, they enjoyed a comfortable and privileged childhood far removed from the squalid and overcrowded conditions in which most of Copenhagen’s ever-increasing inhabitants lived.
His father’s academic position and his mother’s social standing ensured that many of Denmark’s leading scientists and scholars, writers and artists were

regular visitors to the Bohr home. Three such guests were, like Bohr senior, members of the Royal Danish Academy of Sciences and Letters: the physicist Christian Christiansen, the philosopher Harald Høffding and the linguist Vilhelm Thomsen. After the Academy’s weekly meeting, the discussion would continue at the home of one of the quartet. In their teens, whenever their father played host to his fellow Academicians, Niels and Harald were allowed to eavesdrop on the animated debates that took place. It was a rare opportunity to listen to the intellectual concerns of a group of such men as the mood of fin-de-siècle gripped Europe. They left on the boys, as Niels said later, ‘some of our earliest and deepest impressions’.4
Bohr the schoolboy excelled at mathematics and science, but had little aptitude for languages. ‘In those days,’ recalled a friend, ‘he was definitely not afraid to use his strength when it came to blows during the break between classes.’5 By the time he enrolled at Copenhagen University, then Denmark’s only university, to study physics in 1903, Einstein had spent more than a year at the Patent Office in Bern.6 When he received his Master’s degree in 1909, Einstein was extraordinary professor of theoretical physics at the University of Zurich and had received his first nomination for the Nobel Prize. Bohr had also distinguished himself, albeit on a far smaller stage. In 1907, aged 21, he won the Gold Medal of the Royal Danish Academy with a paper on the surface tension of water. It was the reason why his father, who had won the silver medal in 1885, often proudly proclaimed, ‘I’m silver but Niels is gold’.7
Bohr struck gold after his father persuaded him to abandon the laboratory for a place in the countryside to finish writing his award-winning paper. Although he submitted it just hours before the deadline, Bohr still found something to add, and handed in a postscript two days later. The need to rework any piece of writing until he was satisfied that it conveyed exactly what he wanted verged on an obsession. A year before he finished his doctoral thesis, Bohr admitted that he had already written ‘fourteen more or less divergent rough drafts’.8 Even the simple act of penning a letter became a protracted affair. One day Harald, seeing a letter lying on Niels’ desk, offered to post it, only to be told: ‘Oh no, that is just one of the first drafts for a rough copy.’9
All their lives, the brothers remained the closest of friends. Apart from mathematics and physics they shared a passion for sport, particularly football. Harald, the better player, won a silver medal at the 1908 Olympics as a member of the Danish football team that lost to England in the final. Also regarded by many to be intellectually more gifted, he gained a doctorate in mathematics a year before Niels received his in physics in May 1911. Their father, however, always maintained that his eldest son was ‘the special one in the family’.10

Dressed in white tie and tails as custom demanded, Bohr began the public defence of his doctoral thesis. It lasted just 90 minutes, the shortest on record. One of the two examiners was his father’s friend Christian Christiansen. He regretted that no Danish physicist ‘was well enough informed about the theory of metals to be able to judge a dissertation on the subject’.11 Nevertheless, Bohr was awarded his doctorate and sent copies of the thesis to men like Max Planck and Hendrik Lorentz. When no one replied he knew it had been a mistake to send it without first having it translated. Instead of German or French, which many leading physicists spoke fluently, Bohr decided on an English translation and managed to convince a friend to produce one.
Whereas his father had chosen Leipzig and his brother Göttingen, German universities being the traditional place for high-flying Danes to complete their education, Bohr chose Cambridge University. The intellectual home of Newton and Maxwell was for him ‘the centre of physics’.12 The translated thesis would be his calling card. He hoped that it would lead to a dialogue with Sir Joseph John Thomson, the man he described later as ‘the genius who showed the way for everybody’.13
After a lazy summer of sailing and hiking, Bohr arrived in England at the end of September 1911 on a one-year scholarship funded by Denmark’s famous Carlsberg brewery. ‘I found myself rejoicing this morning, when I stood outside a shop and by chance happened to read the address “Cambridge” over the door’, he wrote to his fiancée Margrethe Nørland.14 The letters of introduction and the Bohr name led to a warm welcome from the university’s physiologists who remembered his late father. They helped him find a small two-room flat on the edge of town and he was kept ‘very busy with arrangements, visits and dinner parties’.15 But for Bohr it was his meeting with Thomson, J.J. to his friends and students alike, which soon preyed on his mind.
A bookseller’s son from Manchester, Thomson had been elected the third head of the Cavendish Laboratory in 1884 within a week of his 28th birthday. He was an unlikely choice, after James Clerk Maxwell and Lord Rayleigh, to lead the prestigious experimental research facility, and not just because of his youth. ‘J.J. was very awkward with his fingers,’ one of his assistants later admitted, ‘and I found it necessary not to encourage him to handle the instruments.’16 Yet if the man who won the Nobel Prize for discovering the electron lacked a delicate touch, others testified to Thomson’s ‘intuitive ability to comprehend the inner working of intricate apparatus without the trouble of handling it’.17

The polite manner of the slightly dishevelled Thomson, the epitome of the absent-minded professor in his round-rimmed glasses, tweed jacket and winged collar, helped calm Bohr’s nerves when they first met. Eager to impress, he had walked into the professor’s office clutching his thesis and a book written by Thomson. Opening the book, Bohr pointed to an equation and said, ‘This is wrong.’18 Though not used to having his past mistakes paraded before him in such a forthright manner, J.J. promised to read Bohr’s thesis. Placing it on top of a stack of papers on his overcrowded desk, he invited the young Dane to dinner the following Sunday.
Initially delighted, as the weeks passed and the thesis remained unread, Bohr became increasingly anxious. ‘Thomson,’ he wrote to Harald, ‘has so far not been easy to deal with as I thought the first day.’19 Yet his admiration for the 55-yearold was undiminished: ‘He is an excellent man, incredibly clever and full of imagination (you should hear one of his elementary lectures) and extremely friendly; but he is so immensely busy with so many things, and he is so absorbed in his work that it is very difficult to get to talk to him.’20 Bohr knew that his poor English did not help. So with the aid of a dictionary he began reading The Pickwick Papers as he fought to overcome the language barrier.
Early in November, Bohr went to see a former student of his father’s who was now the professor of physiology at Manchester University. During the visit, Lorrain Smith introduced him to Ernest Rutherford, who had just returned from a physics conference in Brussels.21 The charismatic New Zealander, he recalled years later, ‘spoke with characteristic enthusiasm about the many new prospects in physical science’.22 After being regaled with a ‘vivid account of the discussions at the Solvay meeting’, Bohr left Manchester charmed and impressed by Rutherford – both the man and the physicist.23
On his first day, in May 1907, the new head of physics at Manchester University caused a stir as he searched for his new office. ‘Rutherford went up three stairs at a time, which was horrible to us, to see a Professor going up the stairs like that’, remembered a laboratory assistant.24 But within a few weeks the boundless energy and earthy no-nonsense approach of the 36-year-old had captivated his new colleagues. Rutherford was on his way to creating an exceptional research team whose success over the next decade or so would be unmatched. It was a group shaped as much by Rutherford’s personality as his inspired scientific judgement and ingenuity. He was not only its head, but also its heart.

Born on 30 August 1871 in a small, single-storey wooden house in Spring Grove on New Zealand’s South Island, Rutherford was the fourth of twelve children. His mother was a schoolteacher and his father ended up working in a flax mill. Given the harshness of life in the scattered rural community, James and Martha Rutherford did what they could to ensure that their children had a chance to go as far as talent and luck would carry them. For Ernest it meant a series of scholarships that took him to the other side of the world and Cambridge University.
When he arrived at the Cavendish to study under Thomson in October 1895, Rutherford was far from the exuberant and self-confident man he would become within a few years. The transformation began as he continued work started in New Zealand on the detection of ‘wireless’ waves, later called radio waves. In only a matter of months Rutherford developed a much-improved detector and toyed with the idea of making money from it. Just in time, he realised that exploiting research for financial gain in a scientific culture where patents were rare would harm the chances of a young man yet to make his reputation. As the Italian Guglielmo Marconi amassed a fortune that could have been his, Rutherford never regretted abandoning his detector to explore a discovery that had been front-page news around the world.
On 8 November 1895, Wilhelm Röntgen found that every time he passed a high-voltage electric current through an evacuated glass tube, some unknown radiation was causing a small paper screen coated with barium platinocyanide to glow. When Röntgen, the 50-year-old professor of physics at the University of Würzburg, was later asked what he had thought on discovering his mysterious new rays, he replied: ‘I did not think; I investigated.’25 For nearly six weeks, he did ‘the same experiment over and over again to make absolutely certain that the rays actually existed’.26 He confirmed that the tube was the source of the strange emanation causing the fluorescence.27
Röntgen asked his wife Bertha to place her hand on a photographic plate while he exposed it to ‘X-rays’, as he called the unknown radiation. After fifteen minutes Röntgen developed the plate. Bertha was frightened when she saw the outlines of her bones, her two rings and the dark shadows of her flesh. On 1 January 1896, Röntgen mailed copies of his paper, ‘A New Kind of Rays’, together with photographs of weights in a box and the bones in Bertha’s hand, to leading physicists in Germany and abroad. Within days, news of Röntgen’s discovery and his amazing photographs spread like wildfire. The world’s press latched on to the ghostly photograph revealing the bones in his wife’s hand. Within a year, 49 books and over a thousand scientific and semi-popular articles on X-rays would be published.28

Thomson had begun studying the sinister-sounding X-rays even before an English translation of Röntgen’s paper appeared in the weekly science journal Nature on 23 January. Engaged in investigating the conduction of electricity through gases, Thomson turned his attention to X-rays when he read that they turned a gas into a conductor. Quickly confirming the claim, he asked Rutherford to help measure the effects of passing X-rays through a gas. For Rutherford the work led to four published papers in the next two years that brought him international recognition. Thomson provided a brief note to the first, suggesting, correctly as it later proved, that X-rays, like light, were a form of electromagnetic radiation.
While Rutherford was busy conducting his experiments, in Paris the Frenchman Henri Becquerel was trying to discover whether phosphorescent substances, which glow in the dark, could also emit X-rays. Instead he found that uranium compounds emitted radiation whether they were phosphorescent or not. Becquerel’s announcement of his ‘uranic rays’ aroused little scientific curiosity and no newspapers clamoured to report his discovery. Only a handful of physicists were interested in Becquerel’s rays for, like their discoverer, most believed that only uranium compounds emitted them. However, Rutherford decided to investigate the effects of ‘uranic rays’ on the electrical conductivity of gases. It was a decision he later described as the most important of his life.
Testing the penetration of uranium radiation using wafer-thin layers of ‘Dutch metal’, a copper-zinc alloy, Rutherford found that the amount of radiation detected depended on the number of layers used. At a certain point, adding further layers had little effect in reducing the intensity of radiation, but then surprisingly it began to fall once again as more layers were added. After repeating the experiment with different materials and finding the same general pattern, Rutherford could offer only one explanation. Two types of radiation were being emitted, and he called them alpha and beta rays.
When the German physicist Gerhard Schmidt announced that thorium and its compounds also emitted radiation, Rutherford compared it with alpha and beta rays. He found the thorium radiation to be more powerful and concluded that ‘rays of a more penetrative kind were present’.29 These were later called gamma rays.30 It was Marie Curie who introduced the term ‘radioactivity’ to describe the emission of radiation and who labelled substances that emitted ‘Becquerel rays’ as ‘radioactive’. She believed that since radioactivity was not confined to uranium alone, it must be an atomic phenomenon. It set her on the path to discovering, with her husband Pierre, the radioactive elements radium and polonium.

In April 1898, as Curie’s first paper was published in Paris, Rutherford learned that there was a vacant professorship at McGill University in Montreal, Canada. Although acknowledged as a pioneer in the new field of radioactivity, Rutherford put his name forward with little expectation of being appointed, despite a glowing letter of recommendation from Thomson. ‘I have never had a student with more enthusiasm or ability for original research than Mr Rutherford,’ wrote Thomson, ‘and I am sure if elected, he would establish a distinguished school of physics at Montreal.’31 He concluded: ‘I should consider any institution fortunate that secured the services of Mr Rutherford as professor of physics.’ After a stormy voyage, Rutherford, just turned 27, arrived in Montreal at the end of September and stayed for the next nine years.
Even before he left England he knew that he was ‘expected to do a lot of original work and to form a research school to knock the shine out of the Yankees!’32 He did just that, beginning with the discovery that the radioactivity of thorium decreased by half in one minute and then by half again in the next. After three minutes it had fallen to an eighth of its original value.33 Rutherford called this exponential reduction of radioactivity the ‘half-life’, the time taken for the intensity of radiation emitted to fall by half. Each radioactive element had its own characteristic half-life. Then came the discovery that would earn him the professorship in Manchester and a Nobel Prize.
In October 1901, Rutherford and Frederick Soddy, a 25-year-old British chemist at Montreal, began a joint study of thorium and its radiation and were soon faced with the possibility that it could be turning into another element. Soddy recalled how he stood stunned at the thought and let slip, ‘this is transmutation’. ‘For Mike’s sake, Soddy, don’t call it transmutation’, warned Rutherford. ‘They’ll have our heads off as alchemists.’34
The pair were soon convinced that radioactivity was the transformation of one element into another through the emission of radiation. Their heretical theory was met with widespread scepticism but the experimental evidence quickly proved decisive. Their critics had to discard long-cherished beliefs in the immutability of matter. No longer an alchemist’s dream, but a scientific fact: all radioactive elements did spontaneously transform into other elements, the half-life measuring the time it took for half the atoms to do so.
‘Youthful, energetic, boisterous, he suggested anything but the scientist’, is how Chaim Weizmann, later the first president of Israel but then a chemist at Manchester University, remembered Rutherford. ‘He talked readily and vigorously on any subject under the sun, often without knowing anything about it. Going down to the refectory for lunch, I would hear the loud, friendly voice

rolling up the corridor.’35 Weizmann found Rutherford ‘devoid of any political knowledge or feelings, being entirely taken up with his epoch-making scientific work’.36 At the centre of that work lay his use of the alpha particle to probe the atom.
But what exactly was an alpha particle? It was a question that had long vexed Rutherford after he discovered that alpha rays were in fact particles with a positive charge that were deflected by strong magnetic fields. He believed that an alpha particle was a helium ion, a helium atom that had lost two electrons, but never said so publicly because the evidence was purely circumstantial. Now, almost ten years after discovering alpha rays, Rutherford hoped to find definitive proof of their true character. Beta rays had already been identified as fast-moving electrons. With the help of another young assistant, this time 25-year-old German Hans Geiger, Rutherford confirmed in the summer of 1908 what he had long suspected: an alpha particle was indeed a helium atom that had lost two electrons.
‘The scattering is the devil’, Rutherford had complained as he and Geiger tried to unmask the alpha particle.37 He had first noticed the effect two years earlier in Montreal when some alpha particles that had passed through a sheet of mica were slightly deflected from their straight-line trajectory, causing fuzziness on a photographic plate. Rutherford made a mental note to follow it up. Soon after arriving in Manchester, he had drawn up a list of potential research topics. Rutherford now asked Geiger to investigate one of those items – the scattering of alpha particles.
Together they devised a simple experiment that involved counting scintillations, tiny flashes of light produced by alpha particles when they strike a paper screen coated with zinc sulphide, after passing through a thin sheet of gold foil. Counting scintillations was an arduous task, with long hours spent in total darkness. Luckily, according to Rutherford, Geiger was ‘a demon at the work and could count at intervals for a whole night without disturbing his equanimity’.38 He found that alpha particles either passed straight through the gold foil or were deflected by one or two degrees. This was as expected. However, surprisingly, Geiger also reported finding a few alpha particles ‘deflected through quite an appreciable angle’.39
Before he could fully consider the implications, if any, of Geiger’s results, Rutherford was awarded the Nobel Prize for chemistry for discovering that radioactivity was the transformation of one element into another. For a man who regarded ‘all science as either physics or stamp collecting’, he appreciated the funny side of his own instant transmutation from physicist to chemist.40 After returning from Stockholm with his prize, Rutherford learnt to evaluate the

probabilities associated with different degrees of alpha particle scattering. His calculations revealed that there was a very small chance, almost zero, that an alpha particle passing through gold foil would undergo multiple scatterings resulting in an overall large-angle deflection.
It was while Rutherford was preoccupied with these calculations that Geiger spoke to him about assigning a project to Ernest Marsden, a promising undergraduate. ‘Why not,’ said Rutherford, ‘let him see if any alpha particles can be scattered through a large angle?’41 He was surprised when Marsden did. As the search continued at ever-larger angles, there should have been none of the tell-tale flashes of light that Marsden had seen, signalling alpha particles crashing into the zinc sulphide screen.
As Rutherford struggled to make sense of ‘the nature of the huge electric or magnetic forces which could turn aside or scatter a beam of alpha particles’, he asked Marsden to check if any were reflected backwards.42 Not expecting him to find anything, he was utterly astonished when Marsden discovered alpha particles bouncing off the gold foil. ‘It was,’ Rutherford said, ‘almost as incredible as if you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you.’43
Geiger and Marsden set about making comparative measurements using different metals. They found that gold scattered backwards almost twice as many alpha particles as silver and twenty times more than aluminium. Only one alpha particle in every 8,000 bounced off a sheet of platinum. When they published these and other results in June 1909, Geiger and Marsden simply recounted the experiments and stated the facts without further comment. A baffled Rutherford brooded for the next eighteen months as he tried to think his way through to an explanation.
The existence of atoms had been a matter of considerable scientific and philosophical debate throughout the nineteenth century, but by 1909 the reality of atoms had been established beyond any reasonable doubt. The critics of atomism were silenced by the sheer weight of evidence against them, two key pieces of which were Einstein’s explanation of Brownian motion and its confirmation, and Rutherford’s discovery of the radioactive transformation of elements. After decades of argument, in which many eminent physicists and chemists had denied its existence, the most favoured representation of the atom to emerge was the socalled ‘plum pudding’ model put forward by J.J. Thomson.
In 1903 Thomson suggested that the atom was a ball of massless, positive charge in which were embedded like plums in a pudding the negatively-charged electrons he had discovered six years earlier. The positive charge would neutralise

the repulsive forces between the electrons that would otherwise tear the atom apart.44 For any given element, Thomson envisaged these atomic electrons to be uniquely arranged in a set of concentric rings. He argued that it was the different number and distribution of electrons in gold and lead atoms, for example, which distinguished the metals from one another. Since all the mass of a Thomson atom was due to the electrons it contained, it meant there were thousands in even the lightest atoms.
Exactly one hundred years earlier, in 1803, the English chemist John Dalton first put forward the idea that atoms of every element were uniquely characterised by their weight. With no direct way of measuring atomic weights, Dalton determined their relative weights by examining the proportions in which different elements combined to form various compounds. First he needed a benchmark. Hydrogen being the lightest known element, Dalton assigned it an atomic weight of one. The atomic weights of all the other elements were then fixed relative to that of hydrogen.
Thomson knew his model was wrong after studying the results of experiments involving the scattering of X-rays and beta particles by atoms. He had overestimated the number of electrons. According to his new calculations, an atom could not have more electrons than prescribed by its atomic weight. The precise number of electrons in the atoms of the different elements was unknown, but this upper limit was quickly accepted as a first step in the right direction. The hydrogen atom with an atomic weight of one could have only one electron. However, the helium atom with an atomic weight of four could have two, three, or even four electrons, and so on for the other elements.
This drastic reduction in electron numbers revealed that most of the weight of an atom was due to the diffuse sphere of positive charge. Suddenly, what Thomson had originally invoked as nothing more than a necessary artifice to produce a stable, neutral atom took on a reality of its own. But even this new, improved model could not explain alpha particle scattering and failed to pin down the exact number of electrons in a particular atom.
Rutherford believed that alpha particles were scattered by an enormously strong electric field within the atom. But inside J.J.’s atom, with its positive charge evenly distributed throughout, there was no such intense electric field. Thomson’s atom simply could not send alpha particles hurtling backwards. In December 1910, Rutherford finally managed to ‘devise an atom much superior to J.J.’s’.45 ‘Now,’ he told Geiger, ‘I know what the atom looks like!’46 It was nothing like Thomson’s.

Rutherford’s atom consisted of a tiny positively-charged central core, the nucleus, which contained virtually all the atom’s mass. It was 100,000 times smaller than the atom, occupying only a minute volume, ‘like a fly in a cathedral’.47 Rutherford knew that electrons inside an atom could not be responsible for the large deflection of alpha particles, so to determine their exact configuration around the nucleus was unnecessary. His atom was no longer the ‘nice hard fellow, red or grey in colour, according to taste’ that he once, tonguein-cheek, said he had been brought up to believe in.48
Most alpha particles would pass straight through Rutherford’s atom in any ‘collision’, since they were too far from the tiny nucleus at its heart to suffer any deflection. Others would veer off course slightly as they encountered the electric field generated by the nucleus, resulting in a small deflection. The closer they passed to the nucleus, the stronger the effect of its electric field and the greater the deflection from their original path. But if an alpha particle approached the nucleus head-on, the repulsive force between the two would cause it to recoil straight back like a ball bouncing off a brick wall. As Geiger and Marsden had found, such direct hits were extremely rare. It was, Rutherford said, ‘like trying to shoot a gnat in the Albert Hall at night’.49
Rutherford’s model allowed him to make definite predictions, using a simple formula he had derived, about the fraction of scattered alpha particles to be found at any angle of deflection. He did not want to present his atomic model until it had been tested by a careful investigation of the angular distribution of scattered alpha particles. Geiger undertook the task and found alpha particle distribution to be in total agreement with Rutherford’s theoretical estimates.
On 7 March 1911, Rutherford announced his atomic model in a paper presented at a meeting of the Manchester Literary and Philosophical Society. Four days later, he received a letter from William Henry Bragg, the professor of physics at Leeds University, informing him that ‘about 5 or 6 years ago’ the Japanese physicist Hantaro Nagaoka had constructed an atom with ‘a big positive centre’.50 Unknown to Bragg, Nagaoka had visited Rutherford the previous summer as part of a grand tour of Europe’s leading physics laboratories. Less than two weeks after Bragg’s letter, Rutherford received one from Tokyo. Nagaoka wrote offering his gratitude ‘for the great kindness you showed me in Manchester’ and pointing out that in 1904 he had proposed a ‘Saturnian’ model of the atom.51 It consisted of a large heavy centre surrounded by rotating rings of electrons.52
‘You will notice that the structure assumed in my atom is somewhat similar to that suggested by you in your paper some years ago’, acknowledged Rutherford

in his reply. Though alike in some respects, there were significant differences between the two models. In Nagaoka’s the central body was positively-charged, heavy and occupied most of the flat pancake-like atom. Whereas Rutherford’s spherical model had an incredibly tiny positively-charged core that contained most of the mass, leaving the atom largely empty. However, both models were fatally flawed and few physicists gave them a second thought.
An atom with stationary electrons positioned around a positive nucleus would be unstable, because the electrons with their negative charge would be irresistibly pulled towards it. If they moved around the nucleus, like planets orbiting the sun, the atom would still collapse. Newton had shown long ago that any object moving in a circle undergoes acceleration. According to Maxwell’s theory of electromagnetism, if it is a charged particle, like an electron, it will continuously lose energy in the form of electromagnetic radiation as it accelerates. An orbiting electron would spiral into the nucleus within a thousandth of a billionth of a second. The very existence of the material world was compelling evidence against Rutherford’s nuclear atom.
He had long been aware of what appeared to be an intractable problem. ‘This necessary loss of energy from an accelerated electron,’ Rutherford wrote in his 1906 book Radioactive Transformations, ‘has been one of the greatest difficulties met with in endeavouring to deduce the constitution of a stable atom.’53 But in 1911 he chose to ignore the difficulty: ‘The question of the stability of the atom proposed need not be considered at this stage, for this will obviously depend upon the minute structure of the atom, and on the motion of the constituent charged part.’54
Geiger’s initial testing of Rutherford’s scattering formula had been quick and limited in scope. Marsden now joined him in spending most of the next year conducting a more thorough investigation. By July 1912 their results confirmed the scattering formula and the main conclusions of Rutherford’s theory.55 ‘The complete check,’ Marsden recalled years later, ‘was a laborious but exciting task.’56 In the process they also discovered that the charge of the nucleus, taking into account experimental error, was about half the atomic weight. With the exception of hydrogen, with an atomic weight of one, the number of electrons in all other atoms had to be approximately equal to half the atomic weight. It was now possible to nail down the number of electrons in a helium atom, for example, as two, where previously it could have been as many as four. However, this reduction in the number of electrons implied that Rutherford’s atom radiated energy even more strongly than had previously been suspected.

As Rutherford recounted tales from the first Solvay conference for Bohr’s benefit, he failed to mention that in Brussels neither he nor anyone else discussed his nuclear atom.
Back in Cambridge, the intellectual rapport that Bohr sought with Thomson never happened. Years later, Bohr identified one possible reason for the failure: ‘I had no great knowledge of English and therefore I did not know how to express myself. And I could say only that this is incorrect. And he was not interested in the accusation that it was not correct.’57 Infamous for neglecting papers and letters from students and colleagues alike, Thomson was also no longer actively engaged in electron physics.
Increasingly disenchanted, Bohr met Rutherford again at the Cavendish research students’ annual dinner. Held in early December, it was a rowdy, informal affair with toasts, songs and limericks following a ten-course meal. Once again struck by the personality of the man, Bohr seriously began thinking about swapping Cambridge and Thomson for Manchester and Rutherford. Later that month he went to Manchester and discussed the possibility with Rutherford. A young man separated from his fiancée, Bohr desperately wanted something tangible to show for their year apart. Telling Thomson that he wanted ‘to know something about radioactivity’, Bohr was granted permission to leave at the end of the new term.58 ‘The whole thing was very interesting in Cambridge,’ he admitted many years later, ‘but it was absolutely useless.’59
With only four months left in England, Bohr arrived in Manchester in the middle of March 1912 to begin a seven-week course in the experimental techniques of radioactive research. With no time to lose, Bohr spent his evenings working on the application of electron physics to provide a better understanding of the physical properties of metals. With Geiger and Marsden among the instructors, he successfully completed the course and was given a small research project by Rutherford.
‘Rutherford is a man whom one cannot be mistaken about,’ Bohr wrote to Harald, ‘he comes regularly to hear how things are going and talk about every little thing.’60 Unlike Thomson, who seemed to him unconcerned about the progress of his students, Rutherford was ‘really interested in the work of all people who are around him’. He had an uncanny ability to recognise scientific promise. Eleven of his students, along with several close collaborators, would win the Nobel Prize. As soon as Bohr arrived in Manchester, Rutherford wrote to a friend: ‘Bohr, a Dane, has pulled out of Cambridge and turned up here to get

some experience in radioactive work.’61 Yet there was nothing in what Bohr had done to date to suggest that he was any different from the other eager young men in his laboratory, except the fact that he was a theorist.
Rutherford held a generally low opinion of theorists and never lost an opportunity to air it. ‘They play games with their symbols,’ he once told a colleague, ‘but we turn out the real solid facts of Nature.’62 On another occasion when invited to deliver a lecture on the trends of modern physics, he replied: ‘I can’t give a paper on that. It would only take two minutes. All I could say would be that the theoretical physicists have got their tails up and it is time that we experimentalists pulled them down again!’63 Yet he had immediately liked the 26year-old Dane. ‘Bohr’s different’, he would say. ‘He’s a football player!’64
Late every afternoon, work in the laboratory stopped as the research students and staff gathered to chat over tea, cakes and slices of bread and butter. Rutherford would be there, sitting on a stool with plenty to say, whatever the subject. But most of the time the talk was simply of physics, particularly of the atom and radioactivity. Rutherford had succeeded in creating a culture where there was an almost tangible sense of discovery in the air, where ideas were openly exchanged and discussed in the spirit of co-operation, with no one afraid to speak – even a newcomer. At its centre was Rutherford, who Bohr knew was always prepared ‘to listen to every young man, when he felt he had any idea, however modest, on his mind’.65 The only thing Rutherford could not stand was ‘pompous talk’. Bohr loved to talk.
Unlike Einstein who spoke and wrote fluently, Bohr frequently paused as he struggled to find the right words to express himself, whether in Danish, English or German. When Bohr spoke, he was often only thinking aloud in search of clarity. It was during the tea breaks that he got to know the Hungarian Georg von Hevesy, who would win the 1943 Nobel Prize for chemistry for developing the technique of radioactive tracing that was to become a powerful diagnostic tool in medicine, with widespread applications in chemical and biological research.
Strangers in a strange country, speaking a language that both had yet to master, the pair formed an easy friendship that lasted a lifetime. ‘He knew how to be helpful to a foreigner’, Bohr said as he recalled how Hevesy, only a few months older, helped him ease into the life of the laboratory.66 It was during their conversations that Bohr first began to focus on the atom, as Hevesy explained that so many radioactive elements had been discovered that there was not enough room to accommodate them all in the periodic table. The very names given to these ‘radioelements’, spawned in the process of radioactive disintegration of one atom into another, captured the sense of uncertainty and confusion surrounding

their true place within the atomic realm: uranium-X, actinium-B, thorium-C. But there was, Hevesy told Bohr, a possible solution proposed by Rutherford’s former Montreal collaborator, Frederick Soddy.
In 1907 it was discovered that two elements produced during radioactive decay, thorium and radiothorium, were physically different but chemically identical. Every chemical test they were subjected to failed to tell them apart. During the next few years, other such sets of chemically inseparable elements were discovered. Soddy, now based at Glasgow University, suggested that the only difference between these new radioelements and those with which they shared ‘complete chemical identity’ was their atomic weight.67 They were like identical twins whose only distinguishing feature was a slight difference in weight.
Soddy proposed in 1910 that chemically inseparable radioelements, ‘isotopes’ as he later called them, were just different forms of the same element and should therefore share its slot in the periodic table.68 It was an idea at odds with the existing organisation of elements within the periodic table, which listed them in order of increasing atomic weight, with hydrogen first and uranium last. Yet the fact that radiothorium, radioactinium, ionium, and uranium-X were all chemically identical to thorium was strong evidence in favour of Soddy’s isotopes.69
Figure 5: The periodic table

Until his chats with Hevesy, Bohr had shown no interest in Rutherford’s atomic model. But he now had an idea: it was not enough to distinguish between the physical and chemical properties of an atom; one had to differentiate between nuclear and atomic phenomena. Ignoring the problem of its inevitable collapse, Bohr took Rutherford’s nuclear atom seriously as he tried to reconcile isotopes with the use of atomic weights to order the periodic table. ‘Everything,’ he said later, ‘then fell into line.’70
Bohr understood that it was the charge of the nucleus in Rutherford’s atom that fixed the number of electrons it contained. Since an atom was neutral, possessing no overall charge, he knew that the positive charge of the nucleus had to be balanced by the combined negative charge of all its electrons. Therefore the Rutherford model of the hydrogen atom must consist of a nuclear charge of plus one and a single electron with a charge of minus one. Helium with a nuclear charge of plus two must have two electrons. This increase in nuclear charge coupled to a corresponding number of electrons led all the way up to the then heaviest-known element, uranium, with a nuclear charge of 92.
For Bohr the conclusion was unmistakable: it was nuclear charge and not atomic weight that determined the position of an element within the periodic table. From here he took the short step to the concept of isotopes. It was Bohr, not Soddy, who recognised nuclear charge as being the fundamental property that tied together different radioelements that were chemically identical but physically different. The periodic table could accommodate all the radioelements; they just had to be housed according to nuclear charge.
At a stroke, Bohr was able to explain why Hevesy had been unable to separate lead and radium-D. If the electrons determined the chemical properties of an element, then any two with the same number and arrangement of electrons would be identical twins, chemically inseparable. Lead and radium-D had the same nuclear charge, 82, and therefore the same number of electrons, 82, resulting in ‘complete chemical identity’. Physically they were distinct because of their different nuclear masses: approximately 207 for lead and 210 for radium-D. Bohr had worked out that radium-D was an isotope of lead and as a result it was impossible to separate the two by any chemical means. Later, all isotopes were labelled with the name of the element of which they were an isotope and their atomic weight. Radium-D was lead-210.
Bohr had grasped the essential fact that radioactivity was a nuclear and not an atomic phenomenon. It allowed him to explain the process of radioactive disintegration in which one radioelement decayed into another with the emission of alpha, beta or gamma radiation as a nuclear event. Bohr realised that if

radioactivity originated in the nucleus, then a uranium nucleus with a charge of plus 92 transmuting into uranium-X by emitting an alpha particle lost two units of positive charge, leaving behind a nucleus with a charge of plus 90. This new nucleus could not hold on to all of the original 92 atomic electrons, quickly losing two to form a new neutral atom. Every new atom formed as the product of radioactive decay immediately either acquires or loses electrons so as regain its neutrality. Uranium-X with a positive nuclear charge of 90 is an isotope of thorium. They both ‘possessed the same nuclear charge and differed only in the mass and intrinsic structure of the nucleus’, explained Bohr.71 It was the reason why those who tried, failed to separate thorium, with an atomic weight of 232, and ‘uranium-X’, thorium-234.
His theory of what was happening at the nuclear level in radioactive disintegration implied, Bohr said later, ‘that by radioactive decay the element, quite independently of any change in its atomic weight, would shift its place in the periodic table by two steps down or one step up, corresponding to the decrease or increase in the nuclear charge accompanying the emission of alpha or beta rays, respectively’.72 Uranium decaying with the emission of an alpha particle into thorium-234 ended up two places further back in the periodic table.
Beta particles, being fast-moving electrons, have a negative charge of minus one. If a nucleus emits a beta particle, its positive charge increases by one – as if two particles, one positive and the other negative, that existed in harmony as a neutral pair had been ripped apart with the ejection of the electron, leaving behind its positive partner. The new atom produced by beta decay has a nuclear charge that is one greater than the disintegrating atom, moving it one place to the right in the periodic table.
When Bohr took his ideas to Rutherford he was warned about the danger of ‘extrapolating from comparatively meagre experimental evidence’.73 Surprised by this muted reception, he attempted to convince Rutherford ‘that it would be the final proof of his atom’.74 He failed. Part of the problem lay in Bohr’s inability to express his ideas clearly. Rutherford, preoccupied with writing a book, did not make the time to fully grasp the significance of what Bohr had done. Rutherford believed that although alpha particles were emitted from the nucleus, beta particles were just atomic electrons somehow ejected from a radioactive atom. Despite Bohr’s trying on five separate occasions to persuade him, Rutherford hesitated in following his logic all the way to its conclusion.75 Sensing that Rutherford was by now becoming ‘a bit impatient’ with him and his ideas, Bohr decided to let the matter rest.76 Others did not.

Frederick Soddy soon spotted the same ‘displacement laws’ as Bohr, but unlike the young Dane, he was able to publish his research without first having to seek approval of a superior. Nobody was surprised that Soddy was at the forefront of these breakthroughs. But no one could have guessed that an eccentric 42-year-old Dutch lawyer would introduce an idea of fundamental importance. In July 1911, in a short letter to the journal Nature, Antonius Johannes van den Broek speculated that the nuclear charge of a particular element is determined by its place in the periodic table, its atomic number, not its atomic weight. Inspired by Rutherford’s atomic model, van den Broek’s idea was based upon various assumptions that turned out to be wrong, such as nuclear charge being equal to half the atomic weight of the element. Rutherford was suitably annoyed that a lawyer should publish ‘a lot of guesses for fun without sufficient foundation ’.77
Having failed to gain any support, on 27 November 1913 in another letter to Nature, van den Broek dropped the assumption that the nuclear charge was equal to half the atomic weight. He did so after the publication of the extensive study by Geiger and Marsden into alpha particle scattering. A week later, Soddy wrote to Nature explaining that van den Broek’s idea made clear the meaning of the displacement laws. Then came an endorsement from Rutherford: ‘The original suggestion of van den Broek that the charge on the nucleus is equal to the atomic number and not to half the atomic weight seems to me very promising.’ He was writing in praise of van den Broek’s proposal a little more than eighteen months after advising Bohr against pursuing similar ideas.
Bohr never complained that he had missed out on being the first to publish the concept of atomic number, or those ideas that won Soddy the Nobel Prize for chemistry in 1921, due to Rutherford’s lack of enthusiasm.78 ‘The confidence in his judgement,’ Bohr fondly remembered, ‘and our admiration for his powerful personality was the basis for the inspiration felt by all in his laboratory, and made us all try our best to deserve the kind and untiring interest he took in the work of everyone.’79 In fact, Bohr continued to regard an approving word from Rutherford as ‘the greatest encouragement for which any of us could wish’.80 The reason why he could afford to be so generous, when others would have been left feeling disappointed and bitter, was what happened next.
After Rutherford dissuaded him from publishing his innovative ideas, by chance Bohr came across a recently published paper that grabbed his attention.81 It was the work of the only theoretical physicist on Rutherford’s staff, Charles Galton Darwin, the grandson of the great naturalist. The paper concerned the energy lost by alpha particles as they passed through matter rather than being scattered by

atomic nuclei. It was a problem that J.J. Thomson had originally investigated using his own atomic model, but which Darwin now re-examined on the basis of Rutherford’s atom.
Rutherford had developed his atomic model using the large-angle alpha particle scattering data gathered by Geiger and Marsden. He knew that atomic electrons could not be responsible for such large-angle scattering and so ignored them. In formulating his scattering law that predicted the fraction of scattered alpha particles to be found at any angle of deflection, Rutherford had treated the atom as a naked nucleus. Afterwards he simply placed the nucleus at the centre of the atom and surrounded it with electrons without saying anything about their possible arrangement. In his paper, Darwin adopted a similar approach when he ignored any influence that the nucleus may have exerted on the passing alpha particles and concentrated solely on the atomic electrons. He pointed out that the energy lost by an alpha particle as it passed through matter was due almost entirely to collisions between it and atomic electrons.
Darwin was unsure how electrons were arranged inside Rutherford’s atom. His best guess was that they were evenly distributed either throughout the atom’s volume or over its surface. His results depended only on the size of the nuclear charge and the atom’s radius. Darwin found that his values for various atomic radii were in disagreement with existing estimates. As he read this paper, Bohr quickly identified where Darwin had gone wrong. He had mistakenly treated the negatively-charged electrons as if they were free, instead of being bound to the positively-charged nucleus.
Bohr’s greatest asset was his ability to identify and exploit failures in existing theory. It was a skill that served him well throughout his career, as he started much of his own work from spotting errors and inconsistencies in that of others. On this occasion, Darwin’s mistake was Bohr’s point of departure. While Rutherford and Darwin had considered the nucleus and the atomic electrons separately, each ignoring the other component of the atom, Bohr realised that a theory that succeeded in explaining how alpha particles interacted with atomic electrons might reveal the true structure of the atom.82 Any lingering disappointment over Rutherford’s reaction to his earlier ideas was forgotten as he set about trying to rectify Darwin’s mistake.
Bohr abandoned his usual practice of drafting letters even to his brother. ‘I am not getting along badly at the moment,’ Bohr reassured Harald, ‘a couple of days ago I had a little idea with regard to understanding the absorption of alpha-rays (it happened in this way: a young mathematician here, C.G. Darwin (grandson of the real Darwin), has just published a theory about this problem, and I felt that it not

only wasn’t quite right mathematically (however, only slightly wrong) but very unsatisfactory in the basic conception, and I have worked out a little theory about it, which, even if it isn’t much, perhaps may throw some light on certain things connected with the structure of atoms). I am planning to publish a little paper about it very soon.’83 Not having to go to the laboratory ‘has been wonderfully convenient for working out my little theory’, he admitted.84
Until he had put flesh onto the bare bones of his emerging ideas, the only person in Manchester whom Bohr was willing to confide in was Rutherford. Though surprised by the direction the Dane had taken, Rutherford listened and this time encouraged him to continue. With his approval, Bohr stopped going to the laboratory. He was under pressure, since his time in Manchester was almost up. ‘I believe I have found out a few things; but it is certainly taking more time to work them out than I was foolish enough to believe at first’, he wrote to Harald on 17 July, a month after first sharing his secret. ‘I hope to have a little paper ready to show Rutherford before I leave, so I’m busy, so busy; but the unbelievable heat here in Manchester doesn’t exactly help my diligence. How I look forward to talking to you!’85 He wanted to tell his brother that he hoped to fix Rutherford’s flawed nuclear atom by turning it into the quantum atom.

Chapter 4

THE QUANTUM ATOM
Slagelse, Denmark, Thursday, 1 August 1912. The cobbled streets of the small,
picturesque town some 50 miles south-west of Copenhagen were decked out in flags. Yet it was not in the beautiful medieval church, but in the civic hall that Niels Bohr and Margrethe Nørland were married in a two-minute ceremony conducted by the chief of police. The mayor was away on holiday, Harald was best man, and only close family were present. Like his parents before him, Bohr did not want a religious ceremony. He had stopped believing in God as a teenager, when he had confessed to his father: ‘I cannot understand how I could be so taken in by all this; it means nothing whatsoever to me.’1 Had he lived, Christian Bohr would have approved when, a few months before the wedding, his son formally resigned from the Lutheran Church.
Originally intending to spend their honeymoon in Norway, the couple were forced to change their plans as Bohr failed to finish a paper on alpha particles in time. Instead the newlyweds travelled to Cambridge for a two-week stay during their month-long honeymoon.2 In between visits to old friends and showing Margrethe around Cambridge, Bohr completed his paper. It was a joint effort. Niels dictated, always struggling for the right word to make his meaning clear, while Margrethe corrected and improved his English. They worked so well together that for the next few years Margrethe effectively became his secretary.
Bohr disliked writing and avoided doing so whenever he could. He was able to complete his doctoral thesis only by dictating it to his mother. ‘You mustn’t help Niels so much, you must let him learn to write himself’, his father had urged, to no avail.3 When he did put pen to paper, Bohr wrote slowly and in an almost indecipherable scrawl. ‘First and foremost,’ recalled a colleague, ‘he found it difficult to think and write at the same time.’4 He needed to talk, to think aloud as he developed his ideas. He thought best while on the move, usually circling a table. Later, an assistant, or anyone he could find for the task, would sit with pen poised as he paced about dictating in one language or other. Rarely satisfied with the composition of a paper or lecture, Bohr would ‘rewrite’ it up to a dozen times. The end result of this excessive search for precision and clarity was often to lead the reader into a forest where it was difficult to see the wood for the trees.
With the manuscript finally completed and safely packed away, Niels and Margrethe boarded the train to Manchester. On meeting his bride, Ernest and Mary Rutherford knew that the young Dane had been lucky enough to find the

right woman. The marriage indeed proved to be a long and happy one that was strong enough to endure the death of two of their six sons. Rutherford was so taken with Margrethe that for once there was little talk of physics. But he made time to read Bohr’s paper and promised to send it to the Philosophical Magazine with his endorsement.5 Relieved and happy, a few days later the Bohrs travelled to Scotland to enjoy the remainder of their honeymoon.
Returning to Copenhagen at the beginning of September, they moved into a small house in the prosperous coastal suburb of Hellerup. In a country with only one university, physics posts rarely became vacant.6 Just before his wedding day, Bohr had accepted a job as a teaching assistant at the Lœreanstalt, the Technical College. Each morning, Bohr would cycle to his new office. ‘He would come into the yard, pushing his bicycle, faster than anybody else’, recalled a colleague later.7 ‘He was an incessant worker and seemed always to be in a hurry.’ The relaxed, pipe-smoking elder statesman of physics lay in the future.
Bohr also began teaching thermodynamics as a privatdozent at the university. Like Einstein, he found preparing a lecture course arduous. Nevertheless, at least one student appreciated the effort and thanked Bohr for ‘the clarity and conciseness’ with which he had ‘arranged the difficult material’ and ‘the good style’ with which it had been delivered.8 But teaching combined with his duties as an assistant left him precious little time to tackle the problems besetting Rutherford’s atom. Progress was painfully slow for a young man in a hurry. He had hoped that a report written for Rutherford while still in Manchester on his nascent ideas about atomic structure, later dubbed the ‘Rutherford Memorandum’, would serve as the basis of a paper ready for publication soon after his honeymoon.9 It was not to be.
‘You see,’ Bohr said 50 years later in one of the last interviews he gave, ‘I’m sorry because most of that was wrong.’10 However, he had identified the key problem: the instability of Rutherford’s atom. Maxwell’s theory of electromagnetism predicted that an electron circling the nucleus should continuously emit radiation. This incessant leaking of energy sends the electron spiralling into the nucleus as its orbit rapidly decays. Radiative instability was such a well known failing that Bohr did not even mention it in his Memorandum. What really concerned him was the mechanical instability that plagued Rutherford’s atom.
Beyond assuming that electrons revolved around the nucleus in the manner of planets around the sun, Rutherford had said nothing about their possible arrangement. A ring of negatively-charged electrons circling the nucleus was known to be unstable due to the repulsive forces the electrons exert on each other

because they have the same charge. Nor could the electrons be stationary; since opposite charges attract, the electrons would be dragged towards the positivelycharged core. It was a fact that Bohr recognised in the opening sentence of his memo: ‘In such an atom there can be no equilibrium [con]figuration without the motion of electrons.’11 The problems that the young Dane had to overcome were mounting up. The electrons could not form a ring, they could not be stationary, and they could not orbit the nucleus. Lastly, with a tiny, point-like nucleus at its heart, there was no way in Rutherford’s model to fix the radius of an atom.
Whereas others had interpreted these problems of instability as damning evidence against Rutherford’s nuclear atom, for Bohr they signalled the limitations of the underlying physics that predicted its demise. His identification of radioactivity as a ‘nuclear’ and not an ‘atomic’ phenomenon, his pioneering work on radioelements, what Soddy later called isotopes, and on nuclear charge convinced Bohr that Rutherford’s atom was indeed stable. Although it could not bear the weight of established physics, it did not suffer the predicted collapse. The question that Bohr had to answer was: why not?
Since the physics of Newton and Maxwell had been impeccably applied and forecast electrons crashing into the nucleus, Bohr accepted that the ‘question of stability must therefore be treated from a different point of view’.12 He understood that to save Rutherford’s atom would require a ‘radical change’, and he turned to the quantum discovered by a reluctant Planck and championed by Einstein.13 The fact that in the interaction between radiation and matter, energy was absorbed and emitted in packets of varying size rather than continuously, was something beyond the realm of time-honoured ‘classical’ physics. Even though like almost everyone else he did not believe in Einstein’s light-quanta, it was clear to Bohr that the atom ‘was in some way regulated by the quantum’.14 But in September 1912 he had no idea how.
All his life, Bohr loved to read detective stories. Like any good private eye, he looked for clues at the crime scene. The first were the predictions of instability. Certain that Rutherford’s atom was stable, Bohr came up with an idea that proved crucial to his ongoing investigation: the concept of stationary states. Planck had constructed his blackbody formula to explain the available experimental data. Only then did he attempt to derive his equation and in the process stumbled across the quantum. Bohr adopted a similar strategy. He would begin by rebuilding Rutherford’s atomic model so that electrons did not radiate energy as they orbited the nucleus. Only later would he try to justify what he had done.
Classical physics placed no restrictions on an electron’s orbit inside an atom. But Bohr did. Like an architect designing a building to the strict requirements of a

client, he restricted electrons to certain ‘special’ orbits in which they could not continuously emit radiation and spiral into the nucleus. It was a stroke of genius. Bohr believed that certain laws of physics were not valid in the atomic world and so he ‘quantised’ electron orbits. Just as Planck had quantised the absorption and emission of energy by his imaginary oscillators so as to derive his blackbody equation, Bohr abandoned the accepted notion that an electron could orbit an atomic nucleus at any given distance. An electron, he argued, could occupy only a few select orbits, the ‘stationary states’, out of all the possible orbits allowed by classical physics.
It was a condition that Bohr was perfectly entitled to impose as a theorist trying to piece together a viable working atomic model. It was a radical proposal, and for the moment all he had was an unconvincing circular argument that contradicted established physics – electrons occupied special orbits in which they did not radiate energy; electrons did not radiate energy because they occupied special orbits. Unless he could offer a real physical explanation for his stationary states, the permissible electron orbits, they would be dismissed as nothing more than theoretical scaffolding erected to hold up a discredited atomic structure.
‘I hope to be able to finish the paper in a few weeks,’ Bohr wrote to Rutherford at the beginning of November.15 Reading the letter and sensing Bohr’s mounting anxiety, Rutherford replied that there was no reason ‘to feel pressed to publish in a hurry’ since it was unlikely anyone else was working along the same lines.16 Bohr was unconvinced as the weeks passed without success. If others were not already actively engaged in trying to solve the mystery of the atom, then it was only a matter of time. Struggling to make headway, in December he asked for and was granted a few months’ sabbatical by Knudsen. Together with Margrethe, Bohr found a secluded cottage in the countryside as he set about searching for more atomic clues. Just before Christmas he found one in the work of John Nicholson. At first he feared the worst, but he soon realised that the Englishman was not the competitor he dreaded.
Bohr had met Nicholson during his abortive stay in Cambridge, and had not been overly impressed. Only a few years older at 31, Nicholson had since been appointed professor of mathematics at King’s College, University of London. He had also been busy building an atomic model of his own. He believed that the different elements were actually made up of various combinations of four ‘primary atoms’. Each of these ‘primary atoms’ consisted of a nucleus surrounded by a different number of electrons that formed a rotating ring. Though, as Rutherford said, Nicholson had made an ‘awful hash’ of the atom, Bohr had found his second clue. It was the physical explanation of the stationary states, the reason why electrons could occupy only certain orbits around the nucleus.

An object moving in a straight line has momentum. It is nothing more than the object’s mass times its velocity. An object moving in a circle possesses a property called ‘angular momentum’. An electron moving in a circular orbit has an angular momentum, labelled L, that is just the mass of the electron multiplied by its velocity multiplied by the radius of its orbit, or simply L=mvr. There were no limits in classical physics on the angular momentum of an electron or any other object moving in a circle.
When Bohr read Nicholson’s paper, he found his former Cambridge colleague arguing that the angular momentum of a ring of electrons could change only by multiples of h/2 , where h is Planck’s constant and (pi) is the well-known numerical constant from mathematics, 3.14….17 Nicholson showed that the angular momentum of a rotating electron ring could only be h/2 or 2(h/2 ) or 3(h/2 ) or 4(h/2 )…all the way to n(h/2 ) where n is an integer, a whole number. For Bohr it was the missing clue that underpinned his stationary states. Only those orbits were permitted in which the angular momentum of the electron was an integer n multiplied by h and then divided by 2 . Letting n=1, 2, 3 and so on generated the stationary states of the atom in which an electron did not emit radiation and could therefore orbit the nucleus indefinitely. All other orbits, the non-stationary states, were forbidden. Inside an atom, angular momentum was quantised. It could only have the values L=nh/2 and no others.
Just as a person on a ladder can stand only on its steps and nowhere in between, because electron orbits are quantised, so are the energies that an electron can possess inside an atom. For hydrogen, Bohr was able to use classical physics to calculate its single electron’s energy in each orbit. The set of allowed orbits and their associated electron energies are the quantum states of the atom, its energy levels E . The bottom rung of this atomic energy ladder is n=1, when the
n
electron is in the first orbit, the lowest-energy quantum state. Bohr’s model predicted that the lowest energy level, E , called the ‘ground state’, for the
1
hydrogen atom would be –13.6eV, where an electron volt (eV) is the unit of measurement adopted for energy on the atomic scale and the minus sign indicates that the electron is bound to the nucleus.18 If the electron occupies any other orbit but n=1, then the atom is said to be in an ‘excited state’. Later called the principal quantum number, n is always an integer, a whole number, which designates the series of stationary states that an electron can occupy and the corresponding set of energy levels, E , of the atom.
n
Bohr calculated the values of the energy levels of the hydrogen atom and found that the energy of each level was equal to the energy of the ground state divided by n2, (E /n2)eV. Thus, the energy value for n=2, the first excited state, is –13.6/4 =
1

–3.40eV. The radius of the first electron orbit, n=1, determines the size of the hydrogen atom in the ground state. From his model, Bohr calculated it as 5.3 nanometres (nm), where a nanometre is a billionth of a metre – in close agreement with the best experimental estimates of the day. He discovered that the radius of the other allowed orbits increased by a factor of n2: when n=1, the radius is r; when n=2, then the radius is 4r; when n=3, the radius is 9r and so on.
‘I hope very soon to be able to send you my paper on the atoms,’ Bohr wrote to Rutherford on 31 January 1913, ‘it has taken far more time than I had thought; I think, however, that I have made some progress in it in the latest time.’19 He had stabilised the nuclear atom by quantising the angular momentum of the orbiting electrons, and thereby explained why they could occupy only a certain number, the stationary states, of all possible orbits. Within days of writing to Rutherford, Bohr came across the third and final clue that allowed him to complete the construction of his quantum atomic model.
Figure 6: Some of the stationary states and the corresponding energy levels of the hydrogen atom (not drawn to scale)
Hans Hansen, a year younger and a friend of Bohr’s from their student days in Copenhagen, had just returned to the Danish capital after completing his studies in Göttingen. When they met, Bohr told him about his latest ideas on atomic structure. Having conducted research in Germany in spectroscopy, the study of the absorption and emission of radiation by atoms and molecules, Hansen asked Bohr if his work shed any light on the production of spectral lines. It had long been known that the appearance of a naked flame changed colour depending upon which metal was being vaporised: bright yellow with sodium, deep red with lithium, and violet with potassium. In the nineteenth century it had been discovered that each element produced a unique set of spectral lines, spikes in the

spectrum of light. The number, spacing and wavelengths of the spectral lines produced by the atoms of any given element are unique, a fingerprint of light that can be used to identify it.
Spectra appeared far too complicated, given the enormous variety of patterns displayed by the spectral lines of the different elements, for anyone to seriously believe that they could be the key to unlocking the inner workings of the atom. The beautiful array of colours on a butterfly’s wing were all very interesting, Bohr said later, ‘but nobody thought that one could get the basis of biology from the colouring of the wing of a butterfly’.20 There was obviously a link between an atom and its spectral lines, but at the beginning of February 1913 Bohr had no inkling what it could be. Hansen suggested that he take a look at Balmer’s formula for the spectral lines of hydrogen. As far as Bohr could remember, he had never heard of it. More likely he had simply forgotten it. Hanson outlined the formula and pointed out that no one knew why it worked.
Johann Balmer was a Swiss mathematics teacher at a girls’ school in Basel and a part-time lecturer at the local university. Knowing that he was interested in numerology, a colleague told Balmer about the four spectral lines of hydrogen after he had complained about having nothing interesting to do. Intrigued, he set out to find a mathematical relationship between the lines where none appeared to exist. The Swedish physicist, Anders Ångström, had in the 1850s measured the wavelengths of the four lines in the red, green, blue and violet regions of the visible spectrum of hydrogen with remarkable accuracy. Labelling them alpha, beta, gamma and delta respectively, he found their wavelengths to be: 656.210, 486.074, 434.01 and 410.12nm.21 In June 1884, as he approached 60, Balmer found a formula that reproduced the wavelengths ( ) of the four spectral lines: = b[m2/(m2–n2)] in which m and n are integers and b is a constant, a number determined by experiment as 364.56nm.
Balmer discovered that if n was fixed as 2 but m set equal to 3, 4, 5 or 6, then his formula gave an almost exact match for each of the four wavelengths in turn. For example, when n=2 and m=3 is plugged into the formula, it gives the wavelength of the red alpha line. However, Balmer did more than just generate the four known spectral lines of hydrogen, later named the Balmer series in his honour. He predicted the existence of a fifth line when n=2 but m=7, unaware that Ångström, whose work was published in Swedish, had already discovered and measured its wavelength. The two values, experimental and theoretical, were in near-perfect agreement.
Had Ångström lived (he died in 1874 aged 59), he would have been astounded by Balmer’s use of his formula to predict the existence of other series of spectral