AUTHOR Trigg, George L. TITLE Crucial Experiments in ...Photoelectric Effect, Magnetic Properties of Atoms, The Scattering of ... FILE COPY. Crucial Experiments in Quantum Physics

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AUTHOR Trigg, George L.TITLE Crucial Experiments in Quantum Physics.INSTITUTION Commission on Coll. Physics, College Park, Md.SPONS AGENCY National Science Foundation, Washington, D.C.PUB DATE 66NOTE 42p.; Monograph written for the Conference on the

New Instructional Materials in Physics (Universityof Washington, Seattle, 1965)

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EDRS Price MF-$0.25 HC-$2.20*College Science, *Experiments, InstructionalMaterials, *Physics, *Quantum Mechanics, ResourceMaterials, *Science History

ABSTRACTThe six experiments included in this monography are

titled Blackbody Radiation, Collision of Electrons with Atoms, ThePhotoelectric Effect, Magnetic Properties of Atoms, The Scattering ofX-Rays, and Diffraction of Electrons by a Crystal Lattice. Thediscussion provides historical background by giving description ofthe original experiments and events which contributed to them. Manyoriginal figures are given and quotes from original accounts arepresented. An attempt was made to present the essentials of themethod of each experiment and the difficulties of execution andinterpretation encountered. (PR)

FILE COPY

Crucial Experimentsin Quantum Physics

GEORGE L. TRIGG

Brookhaven National Laboratory

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GENERAL PREFACE

This monograph was written for the Conference on the New Instructional

Materials in Physics, held at the University of Washington in the sum-

mer of 1965. The general purpose of the conference was to create effec-

tive ways of presenting physics to college students who are not pre-

paring to become professional physicists. Such an audience might include

prospective secondary school physics teachers, prospective practitioners

of other sciences, and those who wish to learn physics as one compf.ment

of a liberal education.

At the Conference some 40 physicists and 12 filmmakers and design-

ers worked for periods ranging from four to nine weeks. Thc central

task, certainly the one in which most physicists participated, was the

writing of monographs.

Although there was no consensus eq a single approanh, many writers

felt that their presentations ought to put more than the customary

emphasis on physical insight and synthesis. moreover, the treatment was

to be "multi-level" --- that ts, each roncgraph would consist of sev-

eral sections arranged In increasing order of sophistication. Such

papers, it was hoped, could ue readily introduced into existing courses

or provide the basis for new kinds of courses.

Monographs were written In four content areas: Forces and Fields,

Quantum Mechanics, Thermal and Statistical Physics, and the Structure

and Properties of (latter. Topic selections and general outlines were

only loosely coordinated within each area in order to leave authors

free to invent :ew approaches. In point of fact, however, a number of

monographs do relate to others In comp)ementary ways, a result of their

authors' close, informal interaction.

Because of stringent time limitations, few of the monographs have

been completed, and none has been extensively rewritten. Indeed, most

wr.tters feel that they are barely more than clean first drafts. Yet,

because of the highly experimental nature of the undertaking, it is

essential that these manuscrits be made available for careful review

by other physicists and for trial use with students. Much effort,

therefore, has gone into publishing them in a readable format intended

to facilitate serious consideration.

So many people have contributed to the project that complete

acknowledgement is not possible. The National Science Foundation sup-

ported the Conference. The staff of the Commission on Clllege Physics,

led by E. Leonard Jossem, and that of the University of Washington

physics department, led by Ronald Geballe and Ernest M. Henley, car-

ried the heavy burden of organization. Walter C. Michels, Lyman G.

Parratt, and George M. Volkoff read and criticized manuscripts at a

critical stage in the writing. Judith Bregman, Edward Gerjuoy, Ernest

M. Henley, and Lawrence Wilets read manuscripts editorially. Martha

Ellis and Margery Lang did the technical editing; Ann Widditsch

supervised the initial typing and assembled the final drafts. James

Grunbaum designed the format and, assisted in Seattle by Roselyn Pape,

directeo the art preparation. Richard A. Mould has helped in all phases

of readying manuscripts for the printer. Finally, and crucially, Jay F.

Wilson, of the D. Van Nostrand Company, served as Managing Editor. For

the hard work and steadfast support of all these persons aLd many

others, I am deeply grateful.

Edward D. LambeChairman, Panel on theNew Instructional MaterialsCommission on College Physics

CRUCIAL EXPERIMENTS IN QUANTUM PHYSICS

PREFACE

This work serves a double function. On the one hand, it provides some

of the historical background of quantum theory for the student who has

already achieved some understanding of the thory; on the other hand,

it helps to make easier for the beginner the acceptance of some of the

strange-seeming concepts of quantum physics. In general, I have at-

tempted to present each experiment from the point of view of the aver-

age physicist of the time; I have made speciffo mention of the one con-

scious exception of this rule.

The list of experiments that are discussed is naturally somewhat

arbitrary. I have deliberately omitted some that pertained to utomism

but not to quantum theory, such as Milliken's work on the electronic

charge. I have also ignored some which, while their conclusions were

essential to later developconts in quantum theory, nevertheless wore

themselves primarily directed at questions of atomic structure, the ex-

periment of Geiger and Marsden is an outstanding example.'However, I

have tried to make the discussion of each experiment reasonably com-

plete, in the sense that the essentials o/ the method and the difficul-

ties both of execution and of interpretation are made clear.

I have benefited greatly from criticisms from Dr. Walter Michels,

Dr. Lawrence Wilets, and Mr. Roger Atlas.

George L. Trigg

CONTENTS

PREFACE

1 INTRODUCTION 1

2 BLACKBODY RADIATION 2

3 COLLISIONS OF ELECTRONS WITH ATOMS 10

4 THE PHOTOELECTRIC EFFECT 14

5 MAGNETIC PROPERTIES OF ATOMS 21

6 THE SCATTERIA'G OF X RAYS 25

7 DIFFRACTION OF ELECTRONS BY A CRYSTAL LATTICE 30

1 INTRODUCTION

It is natural human behavior to expectthat regularities which hold under afairly wide range of conditions willcontinue to hold under all conditions.Thus, for example, it would be futileto send a rocket toward Mars carryingequipment with which to take picturesof the surface of that planet unlesswe rere at least reasonably confidentthat the equipment, and the rocketitself, would behave near Mars in ac-cordance with the same laws we havediscovered on the surface of the earth.Similarly, we have faith that an auto-mobile designed five years ago in thelight of laws as they were known thenwill still operate properly tomorrow.

But confidence in such extensions,or extrapolations, is sometimes mis-placed. For example,-a famous law ofphysics, discovered in 1662 by Robert!oyle, states that the product of thevolume of a particular quantity of Gasand the pressure it exerts on thewalls of its container is constant aslong as the temperature of the as re-mains unchanged, any increase in oneof the two quantities being exactlycompensated by a decrease in the other.If the temperature is too low, how-ever, or the pressure too high, thelaw fells; the gas starts to condense,and there may be a substantial changeof volume with no compensating changeof pressure. The failure of an extra-polation is commonly linked with theexistenc of laws or phenomena not

1

envisioned in the original relation-ship. Thus the failure of Boyle's lawis related to the fact that gases cancondense to form liquids.

The branch of physics known asquantum theory has at its basis the re-sults of a rather small number of ex-periments, most of which revealedsuch failures of extrapolation. (In atleast one case the failure was of adifferent sort: The previous theorysimply gave no information.) They haveone feature in common, namely, thatthe additional concepts that must beconsidered in order to provide acoherent explanation are foreign toour ordinary experience. These novelconcepts are fundamental to quantumtheory, and it is their strangenessthat usually causes the greatest dif-ficulty in understanding the theory.

The present work consists of adescription and brief discussion ofseveral of these experiments as tueywere originally carried out, incaidingmany of the original figures and o:tenquoting from the original accounts.

It will be noted that all theseexperiments deal either with sizes farremoved from ordinary experience orwith subtle details of the radiationof light and ',eat. The key to an appre-ciation of then may be described,therefore, as a refuge' to be re-strained by "common sense" outside therealm where common sense has been ac-quired.

2 BLACKBODY

Historically, quantum physics origi-nated in an attempt to give a completedescription of the radiation from ablack body. In order to see how thiscame about, it is first necessary tounderstand what a black body is andwhat its properties are.

Whenever light strikes the sur-face of an object, two eff%lcis occur:Some of the light is reflected Ironthe surface, and some passes throughthe surface into Vie body of the ob-ject. The latter protton, in turn,undergoes at least one and possiblytwo further processes: Some for all)of it is absorbed; some may reach an-other surface of the object and passout, or be transmitted. We see an ob-ject, unless it is intrinsically lumi-nous, only because it reflects somelight into our eyes. Even the sky i6visible because of light scatteredtoward our eyes. The less light anobject reflects, the darker it appe-rsto be. If an object should absorb allthe light that fell on it, it wouldreflect none, and would appear per-fectly black.' An object of this sortis cailed an ideal black body. Nosuch body actually exists. Neverthe-less, it is quite possible - and use-ful! - to act as though it did, and todetermine many of the properties itwould have; this sort of idealizationis quite common in science.

Of course, the raiiation that isabsorbed by a black body carries en-ergy, The internal energy of the bodywould be increased and the temperatureof the body would rise indefinitely ifthere were no rechanism by which thebody disposed of some energy. Themechanism is, simply, that the bodyradiates; in fact, not only a blackbody but any object left to itself in

'Actually, of course, we Mould not be able to seethe object at all, but only a chunk of space fromWhich f3 light reached us.

2

RADIATION

unchanging surroundings tends towarda state of equilibrium. When it hasreached the equilibrium state, itradiates as much energy per unit time- as much power - as it absorbs.2

The reason for interest in theproperties of A black body is somewhatabstruse. It begins with the fact thatany object gives off radiation, at arate which increases strongly with thetemperature of the body. The wave-lengths of the radiation range contin-uously over the spectrum, not only thevisible portion but also the ultra-violet, the infrared, and all otherparts; and the way in which the energyis distributed in wavelength alsochanges with the temperature of thebody, as well as being dependent onthe nature of the body. The measure-ment of this distribution is calledthe apectral emittance, designated byEA and d'fined as follows: The spec-tral emittance at wavelength A is theenergy associated with wavelengthsaround A radiated per unit time, perunit surface twee, per unit wavelength.Thus, EA dt dS dA is tie energy in thewavelength range A to A dA radiatedby a surface element of area dS in atime interval dt. Secondly, as hasbeen mentioned, any real object ab-sorbs a traction of the radiationstriking it. The fraction absorbed de-pends on the nature of the body and onthe wavelength of the incident light;let us denoto the fraction at wave-length A by Al,. Both AA end EA mayvary with temperature. There is arelationship between them, howeeer,deduced by O. R. Kirchhoff in intro-ducing the concept of a black body in

*it must be emphasised that this is not the sameas reflection. the characteristics of reflectedlight are delemined partly by the properties ofthe reflecting body but also partly by those ofthe incident light. Radiation, on tee other hand,Is not in any say affected by incident light.

BLACKBODY RADIATION 3

1860: The ratio between Ex and Ax foran arbitrary body at any given wave-length and temperature is equal to thespectral emittance of a black body atthat wavelength and temperature. Insymbols, if ex is the spectral emit-tance of a black body, the relation-ship is

Ex/AA = ex.

0. Lummer and E. Pringsheim, whose ex-perimental studies of blackbody radi-ation are to be described a littlelater, go on as follows3:

If one knows, therefore, the radia-tion of a black body as a functionof wavelength and temperature, thenone knows thereby the laws of radi-ation for all those bodies whoseabsorptive powe: is likewise givenas a function of wavelength andtemperature. Experimentally, thereverse process is likely to besimpler, by a study of the radia-tion of a body to explain the ab-sorption A with the help of theknowledge of e.

Evidertly, knowledge of ex wasdesireble, and by the end of thenineteenth century the problem had al-ready received considerable attention.Most of the efforts were empirical de-ductions from observations on realbodies. In 1896, F. Paschen, reportinghis own results of .his type in thejournal Annalen der Physik, cited somehalf-dozen earlier works; Paschen'eexpreeston, eA e Ck'sle-S/At, where c,C, and a are constants for a givenmaterial, was one of the simplest.The one empirical result that has re-tainti its validity was one given inthe Sittungsberichte der kOniglicheGesellschaft der Wissenschaft zu Wien,in 1879, by J. Stefan, that the totalemittance, the total power (regard-less of wavelength) radiated per unitarea, is proportional to the fourth

'Translated by G. L. T.

power of the absolute temperature ofthe radiating body.

Theoretical studies were not lack-ing. They were immensely simplifiedby the following property, also provedby Kirchhoff: If a region of space isbounded by '.aterial of whatever nature,provided only that it is not a mirror-like reflector, and these boundarywalls are maintained at a uniformtemperature T, the space will becomefilled with radiation identical inevery respect with that emitted by ablack body whose temperature is T. Themethod of proof is to show that other-wise it would be possible to bringabout a violation of the second law ofthermodynamics. As we shall see, thepossibility of experimental measure-ment of ex rests upon essentially thisproperty of a cavity.

In 1884, Ludwig Boltzmann pub-lished in Annalen der Physik two pa-prs which together provided a proofthat Stefan's empirical relationshipmust hold for a black body. (As a con-sequence, the relationship has becomeknovn as the Stefan-Boltzmann law.)His arguments involved a cavity whosewalls were not at a uniform tempera-ture, and which was subdivided bymovable pistons with shutters in them;he epplied the second law of thermo-dynamics to the transfer of energyfrom one wall to another by means ofvarious motions of the pistons and op-eration of the shutters, making use ofthe fact that the radiation exerts apressure on the pistons so that workis expended or absorbed in their no-tie. Nine years later, Willy Wienobtained two equally important results,which he published in the Sitzungsber-ichte der kOnigliche Preussische Aka-demie der Wissenschaften zu Berlin. Henoted that if a cavity were reducedin volune, the energy per unit volumein the cavity would be increased notonly by the confinement to a smallervolume but also by virtue of the workdone against radiation pre! 2. Thedensity can also be increasee by anincrease of temperature, and the secand law of thermodynamics relates the

4 CRUCIAL EXPERIMENTS IN QUANTUM PHYSICS

increases produced in the two ways.The relationship must hold not onlyover all, but also for the energy den-sity associated with every infinitesi-mal range of wavelengths. But themoving walls produce a change in wave-length of the reflected radiation, bythe Doppler effect; and so the tempera-ture change must also alter the wave-length distribution. The quantitativeconsequences of this were, first, thatthe wavelength A' at temperature T'that properly corresponds to the wave-length A at temperature T is given by

A'T' = AT; (2.1)

second, that the spectral emittancesfor corresponding wavelengths at dif-ferent temperatures are related by

ex/eA' = T5/T'5. (2.2)

In particular, if eA has a maximumvalue ex,,,, at some wavelength Am,then ex, Ma X satisfies the relation

eA max T'S = const.,

while Am satisfies the relation

AmT = const.

(2.2a)

(2.1a)

Equations (2.1) and (2.2) are known asWien's displacement laws. Together,although Wien did not note it, theyimply that the expression for eA mustbe of the general form

ex = X-5f(AT),

or, equivalently,

(2.3)

eA = T5F(AT),

where F(AT) = (AT)-51(XT). The spaci-fic form of the function f(XT) orF(XT) is not determined by these ar-guments; merely the fact that theydepend on A and T only through theproduct AT. This was the most thatcould be established on the basis ofclassical theory without the additionoz more or less questionable hypoth-eses.

Wien went on, in a later paper(1896) in Annalen der Physik, to ob-tain an explicit form for ex by makingassumptions about the process of radi-ation from a molecule. His result was

= CA- 5 e -e/AT (2.4)

in agreement with Eq. (2.3) and insupport of Paschen's empirical results;but some physicists regarded his as-sumptions as dubious.

One other theoretical attempt de-serves mention, although it actuallypostdated the experimental work. In1900, Lord Rayleigh approached theproblem in the following way: Theradiation in a cavity at equilibriummust consist of standing waves. It ispossible to calculate the number ofdifferent standing -wove modes of wave-lengths between A and A O. that canexist per unit volume of the cavity.According to classical statisticalconsiderations, each of these modesshould have the same average energy,namely an amount kT where T is theabsolute temperature and k is a uni-versal constant. These considerationslead to an expression for the energyper unit volume in the cavity per unitwavelength range, and thence to anequation for the spectral emittance,

eA = 8nckTA-4.

This equation agrees with Eq.(2.3). However, it has one serious de-fect: It yields an infinite value forthe total energy in the cavity.4 Theinfinite value is due to larger andlarger contributions from shorter andshorter wavelengths. Since short wave-lengths are associated with ultra-violet radiation, the divergence isoften referieo to as the "ultravioletcatastrophe.' In the light of modern

41t is interesting to note that there were pre-vious indications of the failure of the law ofequipartition of energy. Maxwell had noticed asearly as 1859 that the law was incapable of pro-viding an adequate explanation of the ratio ofspecific heats of gases.


knowledge, a more appropriate namemight be the "gamma-ray catastrophe.")

The experimental study was madepossible by a slight modification ofthe discussion that gave the equival-ence of radiation in a closed cavityand blackbody radiation. Suppose acavity of the type envisioned earlierhas a hole drilled through one wall.The hole must have a diameter largecompared with its length, but itsarea must be small compared with thatof the cavity walls. Under these cir-cumstances, any radiation that fallson tho hole is almost certain to passinto the cavity, there to be trappedby continual reflection back and forthby the walls, with some absorption ateach reflection, so that the hole isin that respect a good approximationto a black body. Moreover, the radia-tion coming out of the hole will be arepresentative sample of the radia-tion in the cavity, which has alreadybeen described as equivalent to black-body radiation.

The definitive experimental workwas carried out by Lummer and Pring-sheim in Charlottenburg, Germany. Thefirst step, reported in Annalen derPhysik in 1897, was to verify theStefan-Boltzmann relationship. Theprocedure is simple in principle -merely to measure the energy radiatedfrom a hole in a cavity maintained ata constant known temperature - but itrequired considerable care in its exe-cution.

Two cavities were used, a copperone for temperatures up to 877°K andan iron one for temperatures from799°K to 1561°K. The copper cavity wasimmersed in a molten mixture of sodiumnitrate and potassium nitrate; thetemperature of the bath could be heldconstant to within one or two degreesfor as long as half an hour by con-trol of the supply of gas to the heat-ing flame. The iron cavity was heatedby means of a special double-walledoven shown in Fig. 2.1. The hot gasesfrom the flame passed aroun' the cav-ity inside the inner wall of the oven,then between the two walls, and then

Fig. 2.1 Diagram of the double-walledoven as used for heating the iron cavity.

into the chimney flue. Temperatures upto 755°K were measured by means ofmercury thermometers; higher tempera-tures, by a thermocouple.

The radiant power gas measured bymeans of a bolometer. In this device,radiation falls on one of two black-ened platinum wires and is absorbed,raising the temperature of the wireand therefore its electrical resist-ance. The increase in resistance ismeasured by comparison with the re-sistance of the other wire. Extensiveprecautions were taken tD ensure thatthe energy recorded came only from thecavity, and to correct for possiblevariations in the fraction of radia-tion absorbed in the air along thepath from cavity to sensing element.In fact, the only difficulty that wasnot almost entirely overcome was thatof achieving truly uniform tempera-tures in the iron cavity. The finalconclusion, based on observations overa range of temperature whose extremesdiffered by a factor of four, was thatthe Stefan-Boltzmann law is valid.

Lummer and Pringsheim then pro-ceeded to a study of how the emissivityvaries with wavelength at a giventemperature. The results were reportedin a series of three papers in theVerhandlungen der Deutschen physikali-schen Gesellschaft in 1899 and 1900.


Fig. 2.2 Comparison of Lummer and Prirg-sheim's experimental data (plain crossesand solid curves), with Wien's formula(circled crosses and dashed curves). Theshaded areas ',how the absorption by watervapor and carbon dioxide in the air.

Again tht basic concept was simple andthe basic procedure straightforward,and only the various precautions andcorrections were complicated. Variouscavities were used, at temperaturesfrom 85 to about 1800°K. The lowertemperatures were achieved by immer-sion in liquid air (85°K), boilingwater (373°K), and molten saltpeter(around 600°K, depending on exact com-position). Higher temperatures, up toabout 1800°K, were obtained by elec-trical heating. At such temperatures,by far the largest fraction of theradiation is in the infrared region of

the spectrum; the range of wavelengthsstudied was from about 1 micron toabout 18 microns (visible light coversthe range of wavelengths from about0.4 to about 0.7 micron). A substantialdifficulty in this spectral region isthat water vapor and carbon dioxide,both normally present in the atmos-phere, absorb strongly near certainwavelengths, especially around 1,8,2.7, and 4.5 microns. In the earliestwork, Lummer and Pringsheim merelyattempted to correct for this. Later,they enclosed the spectrometer andbolometer in a container in which theair was dried and chemically purgedof carbon dioxide, so that the neces-sary correction was greatly reduced.As in the work on the Stefan-Boltz-mann law, strict precautions weretaken to ensure that only the radia-tion of interest fell on the bolometer.

One method of presenting the re-sults is simply as a curve showing P-as a function of A for various temper-atures as in Fig. 2.2. From suchcurves, Lummer and Pringsheim deter-mined both the wavelength at which exwas maximum, and the maximum value,for testing Eqs. (2.1a) and (2.2a). Itwas simply a matter of seeing whetherthe appropriate combinations of fac-tors were indeed constants. Already intheir first report, they could makethis statements;

It can therefore be regarded asproven by this series of observa-tions that for the radiating bodyemployed the maximum energy in-creases with the fifth power of theabsolute temperature. Also theequation ART = A can be consideredproven, since the deviations of thevalues of A from the average valuelie within the observational errorspossible from the determination ofX,.

It is interesting to note thatLummer and Pringsheim repeated thesetests on each series of observations



they carried out, for, as they put it,"The fulfilment of these three laws[the third being Stefan's law] is theconditio sine qua non if one wishes todraw from the radiation measurementsany conclusion whatever about the formof the spectral equation (energycurve)." In fact, so firmly convincedwere they of the truth of this state-ment that they discarded one series ofobservations because the maximum valueof ex increased as T5" rather thanas T5.6

Figure 2.2 shows, together withone set of experimental plots of thesort just discussed, the curves repre-sented by Eq. (2.4) for the values ofT used. The agreement between theoryand experiment looks fair, but Lummerand Pringsheim were not satisfied, anddevised a means of making a more sen-sitive test. If one takes the logarithmof both sides of Eq. (2.4),7 the resultis

log ex = log (CA-5) (c/A1) log e,

which can be rewritten as

log eA = log (CA-5) (c log e/A)(1/T).

This has the form y = a + bx, wherey = log eA, a = log (CA-5),

b = (c log e/X), and x = 1/T. ThusWien's formula implies that whenlog ex for a fixed A is plotted against1/T, the resulting curve, called anisochromat, should be a straight line;the slope of the line is proportionala c, and the intercept on the log exaxis can be used to compute C. Thevalue of C might vary from one seriesof observations to another, but itshould be constant throughout any one

"They state that this may have been the result ofa poor adjustment which alloved the spectrometerto "look at" part of the cooler outer surface ofthe cavity as well as at the interior.?The reader is reminded that the logarithm ofa number y is the number a such that 10 = y;

that therefore 101°g Y y; that if two num-bers are equal, so are their logarithms; that

log (xy) log x + log y; and that log (xs)

= log [(10108 log (10' log x) s log x.

1.0

L_

Ni_

.

4.,...'

--'"" ---

1

-

..

'

I

1

3

an ION120.0007 0 I 0,0000 o. to o. 11 CO

Fig. 2.3 A set of isochromats from Lummerand Pringsheim's first report on black bodyradiation.

series; the values obtained for c, onthe other hand, should all be nearlythe same.

In their earliest report, Lummerand Pringsheim found that the iso-chromat6 seemed indeed to be straightlines (see Fig. 2.3), but gave valuesof C and c that varied with wavelength.At that stage, they were not suffi-ciently confident in their proceduresto regard the question as settled. Bythe time of the second report, theywere soundly enough convinced of theinvalidity of Eq. (2.4) that theylooked for, and reported finding,curvature in the isochromats. They say,however, "Nevertheless, before we passfinal judgment against the validity ofthe Wien-Planck equation [Planck hadsupported Wien's deduction on the basisof a different line of argument; seebelow], we consider it necessary toextend the studies over a larger tem-perature interval and wavelengthrange." Finally, in the third report,the evidence had become unquestionable;the curvature in the isochromats wasobvious (see Fig. 2.4 on next page).They firmly conclude that "Likewisebrought down to the ground therewithare all those extensive consequences



Fig. 2.4 A set of isochromats from Lummerand Pringsheim's third report.

that people have derived from the Wien-Planck equation."8

A discussion of Max Planck's in-troduction of the constant (customar-ily designated by h), that bears hisname, while not exactly part of theaccount of the experiment, is relevantand interesting. Planck had for sometime been interested in the problem ofblackbody radiation, being attractedto it by the "absolute" character ofthe distribution law - its independ-ence of the material of the cavitywalls. He made use of this independencein his work on the problem by takingas the walls an assemblage of harmonicoscillators. His approach was alwaysthrough the thermodynamics of theassemblage, with particular emphasison the thermodynamic quantity calledentropy. This quantity is a measureof the disorder in a system, and thesecond law of thermodynamics, withwhich Planck was thoroughly familiar,states that the natural tendency of asystem is to change in such a way thatits entropy increases. The equilibriumstate, consequently, is the state ofmaximum entropy; and for a cavity, theequilibrium state is characterized bythe cavity being filled with black-bcdy radiation. Planck's task, there-fore, was to calculate the entropy ofhis assemblage.

In his earlier work, Planck wasnot familial- with the "disorder" in-terpretation of entropy, and be con-sidered the entropy of an individualoscillator, which he sought to relate

to its energy U. Ho found that a basicquantity was the curvature R of thegraph of this relationship, and anerroneous assumption led him to con-clude that R must depend on U throughthe equation

R = a/U, (2.5)

where a is a positive quantity thatmight depend on frequency. The radia-tion law to which this led was justEq. (2.4), with a = X/c.

The correct radiation law, infact, was first obtained by a purelyempirical procedure. The experimentalresults showed that Eq. (2.5) neededmodification for large values of U.If we write the equation as1/R U/a, it can be seen that thesimplest alteration of the sort neededwould be to add a term in U2. Planckdid so, and obtained a radiation lawof the form

ex m X-5/(eb/XT_1) (2.6)

which proved to fit the experimentalresults extremely well.

It was only after he had come toa more complete understanding of en-tropy that Planck was able to justifyEq. (2.6). The properties of entropyshow that it is a measure of the prob-ability of the state involved. Theprobability, in turn, can be foundsimply by counting the number of dif-ferent microscopin arrangements - inthe present case, the number of waysof assigning energies to the individ-ual oscillators - compatible with thegiven over-all state, and by assumingthat each microscopic arrangement isequally probable. In order for thecounting process to be possible, how-ever, the energy cannot be a contin-uous variable but must be parceled outin multiples of a basic unit e, sothat U = 11E. When Planck put theseconcepts into his treatment, he foundthat the entropy S depended on U andE only through the combination U/E.Wien's displacement laws, on the otherhand, implied that S = f(U/P), where


y is the frequency of the oscillator.Consequently, one must have E. = hv:

The energy of an oscillator must be anintegral multiple of a basic unit pro-portional to the frequency.

The resulting radiation law was

8reheX A5(ech/kAT 1)'

where h is Planck's constant and k isanother universal constant, and c isthe speed of light; this reduces to

Wien's law foi small values of AT, andto Rayleigh's law for large values of

AT.The beginnings of quantum theory,

then, lay in an experiment whose re-sults could be understood only by theintroduction of an idea foreign toclassical theory: that in some sys-tems, energy is not infinitely subdi-visible, but is exchanged with therest of the universe only in discreteamounts, or quanta.

3 COLLISIONS OF ELECTRONS WITH ATOMS

In the early years of the twentiethcentury, the nature of the atom wasvery unclear. Obviously, the atom con-tained electrons, whose motions wererelated in some fashion to the fre-quencies of light emitted by the atom;but how the electrons and the remain-ing parts of the atom were fitted to-gether remained a mystery. Meanwhile,the atom itself provided an object ofstudy, and its properties were stud-ied with interest.

One property is the ionizationpotential: the amount of energy thatmust be supplied to knock one electronloose from the atomic structure. Oneway to supply the energy is by strik-ing the atom with an electron, and theenergy is measured in terms of theelectrical potential difference throughwhich the electron must be acceleratedto produce ionization. This potentialdifference is called the ionizationpotential.

J. Franck and G. Hertz, workingat the University of Berlin, had sup-posedly measured the ionization poten-tials of several substances that aregases at ordinary temperatures. Themethod was to produce electrons bymeans of a heated filament; acceler-ate them through a measured, variablepotential difference V maintained be-tween the filament and a grid; anddecelerate them again by a potentialdifference, which was ten volts greaterthan V, between the grid and a collec-tor plate. In the space between gridand collector, the electrons couldcollide with gas atoms. They could notunder any circumstances reach the col-lector. However (the argument ran),if V were greater than the ionizationpotential, the electrons would ionizesome atoms by collision; the electricfield would accelerate the positiveions toward the collector, which wouldthen register a current. The value ofthe ionization potential, then, was

10

the value of V for which current te-gan to flow to the collector. By thismethod, Franck and Hertz measured whatappeared to be ionization potentialsfor a half-dozen gages.

They hoped to correlate the ion-ization potentials with atomic radii,and for this purpose they wanted tomake measurements on metallic atoms.They felt that maintaining their ap-paratus at the higher temperaturesneeded to produce metal vapors wouldproduce errors - presumably, the cur-rents they worked with were so smallthat the decrease in the resistanceof the glass envelope, produced by theincrease in temperature, would giverise to stray currents large enoughto mask the desired effect. Accord-ingly, they devised a new form of ap-paratus. The grid N, instead of beingfairly close (5 mm) to the filament,was made about 4 cm away; he collec-tor plate G was placed only 1 or 2 mmfrom the grid, instead of some 2.5 cm.The potential between filament andgrid was variable and such as to ac-celerate the electrons, as before;the potential between grid and collec-tor was again decelerating, but smalland constant.

The principle of the method isbased on some assumptions which war-rant brief discussion. The first isthat when an electron and an atom un-dergo an elastic collision,9 the elec-tron loses only a negligible amountof energy. This assumption can bechecked merely by use of the laws ofconservation of energy and momentum,

'The reader is reminded that collisions areclassed as elastic or inelastic according to whatbecomes of the initial kinetic energy. If it re-mains as kinetic energy of the two collidingbodies, though perhaps shared differently, thecollision it; elastic. If some of it is absorbedso as to alter the inter4a1 state of one of the

bodies, than the collision is inelastic.

COLLISIONS OF ELECTRONS WITH ATOMS 11

and turns out to be completely valid.1°The second is that a collision betweenan electron and an atom will be elas-tic as long as the kinetic energy ofthe electron is less than the ioniza-tion energy. This will be discussedlater; but it seemed reasonable in thelight of the earlier experiments. Thethird assumption is that the proba-bility of a collision being inelastic,if the electron energy is large enoughfor that to be possible, is not verysmall compared to unity. This assump-tion, also, seemed to be borne out bythe earlier work.

The actual operation was well de-scribed by Franck and Hertz in thereport on their work, published in theVerhandlungen der Deutschen physikal-ischen Gesellschaft in 1914:

As long as the accelerating poten-tial is less than the decelerating,the current [to the collector] isnull. Then it will increase, untilthe accelerating potential has be-come equal to the ionization poten-tial. At this point, the electronswill undergo inelastic collisionsin the neighborhood of the grid andthereby ionize. Since they them-selves and the electrons set freeby ionization traverse only a verysmall potential until their passagethrough the grid, they pass throughit without an appreciable velocityand are incapable of running againstthe retarding field. The galvanome-ter current will, therefore, dropto zero as soon as the acceleratingpotlitial has become greater thanthe ionization potential. If one in-creases the accelerating potentialfurther, the point at which theelectrons undergo inelastic colli-sions moves inward from the grid.The electrons present after the in-elastic collisions, therefore, passon the way to the grid through a

1°If a body of mass M, initially at rest, isstruck by a body of mass m, and if the collisionis elastic, the incident body loses a fraction of4ts clergy which is at largest 4mM/(M + m)2.

potential which is equal to thedifference between the acceleratingpotential and the ionization pcten-tial. As soon as this differencehas become greater than the constantretarding potential between N andG, electrons can again go againstthe retarding field and the gal-vanometer current again grows.Since the number of rlectrons isincreased by the ionization, it ac-tually grows large, than the firsttime. However, as the acceleratingpotential becomes equal to twicethe ionization potential, the elec-trons undergo inelastic collisionsa second time in the neighborhoodof the grid. Since they therebylose all their energy, and thenewly produced electrons likewiuehave no appreciable velocity, nomore electrons can run against theretarding field. Thus, as soon asthe accelerating potential isgreater than twice the ionizationpotential, the galvanometer currentagain drops to zero. Since the samephenomenon repeats itself each timethe accelerating potential is equalto an integral multiple of the ion-ization potential, we should expecta curve having maxima of increasingsize, whose separation is equal tothe ionization potential."

Just such curves were obtained;an example is reproduced in Fig. 3.1(see next page). The maxima turned outto be quite sharp - the report gives afeeling that they were sharper thanthe authors had dared to expect - andFranck and Hertz imply a confidencethat their results are accurate to 0.1volt as compared with the 1-volt ac-curacy of the older method. The valueobtained for the ionization potentialof mercury was 4.9 volts. As a com-parison with the older method, theyremeasured the value for helium andfound that the two methods gave verysatisfactory agreement.

"Translated by G. L. T.


Fig. 3.1 Plot of collector current versusaccelerating potential, showing the equalspacing of the maxima. tNote: Several ef-fects combine to cause the spacing betweenmaxima to be slightly different from theposition of the first maximum.)

Such accuracy exceeded anythingpreviously obtained, and enabledFranck and Hertz to make a quantita-tive test of a theoretical proposalthat had been put forth several times:that the ionization energy should beequal to Planck's constant h timesthe frequency of one of the "propermotions" of the electrons. They feltit natural to choose a frequency thatwas very strongly absorbed by mercuryvapor, that corresponding to a wave-length of 2536 A. The potential indi-cated by the theory was 4.84 volts, inexcellent agreement with the measuredvalue.

At this stage, perhaps out of thevery excellence of the agreement,doubts began to arise. The possibilitypresented itself that the electronslost their energy not in ionizing themercury - after all, in this methodionization had not been directly ob-served - but in exciting radiation.They could not check this new possi-bility with the same apparatus, as thewavelength 2536 A lies well into theultraviolet, while the glass envelope

of their tube was ,.paque to ultravio-let light. Consequently, they built anew tube out of quartz, which is trans-parent to ultraviolet light. This tubewas much simpler than the other, in-volving merely a platinum filament toprovide electrons, a platinum grid to-ward which the electrons could be ac-celerated and on which they were col-lected, and a pool of mercury toprovide vapor. The whole bulb washeated to about 150°C by means of agas burner, and any radiation givenoff was analyzed by means of an ultra-violet spectrograph.

The results were somewhat sur-prising. When the potential betweenfilament and grid was less than 4.9volts, no radiation was emitted by themercury vapor.12 When the potentialwas greater than 4.9 volts, the mer-cury radiated, as expected; but itradiated only the wavelength 2536 A,despite the fact that many lines of themercury arc spectrum are more intensethan that at 2536 A and despite thefact that the wavelengths of many ofthese lines correspond to potentialsless than 4.9 volts. Franck and Hertzconcluded that in some collisions, theenergy that the electron had acquired,if large enough, was converted intoradiation; they remained convincedthat in other collisions at the sameenergy, the energy was used to ionizethe atom.

We should now abandon the pointof view of Franck and Hertz, and ex-amine these conclusions in the lightof later knowledge. In fact, if Franckand Hertz had been able to work withmercury or an alkali metal in both theearlier apparatus and the later one,they would have met a peculiar incon-sistency: For any of these substances,the two methods would have given dif-ferent results, although they gaveidentical results for helium and wouldhave done so for any of the substances

12The critical value in this arrangement actuallyturns out to be somewhat less than 4.9 volts be-cause the electrons already have some energywhen they are produced by the filament.

COLLISIONS OF ELECTRONS WITH ATOMS 13

actually mPasured in the earlier appar-atus. The reason is that in no casewas there actually an ionization po-tential measured. The increases incollected current which Franck andHertz ascribed to positive ions had amuch different origin. The atoms ormolecules were being excited and wereemitting radiation, and the radiationwas producing photoelectrons from thecollector plate. The radiation emittedby mercury, and the radiation thatwould have been emitted by an alkalimetal, could not have produced thiseffect.13 In every case, the ioniza-tion potential is higher than the valuemeasured by Franck and Hertz in thiswork.

What, then do the measured valuesmean? They certainly are thresholds ofenergy beyond which something happensto the internal behavior of the atom.When the current to the collectordrops, it is a signal that the elec-trons have gained just enough energyto lose most of it in an inelasticcollision with an atom. When it dropsa second time, the electrons are gain-

"For a further discussion relating to this point,see Chapter 4.

ing enough energy to experience suchan encounter ':wise during their travelfrom filament to grid, and so on. Evi-dently there is a miniarm amount bywhich the internal energy of the atomcan be changed - and in that state-ment, forced on us by the experimentalfacts, is found the failure of extra-polation from experience on the ordi-nary scale. On the ordinary scale, itis perfectly acceptable for a dynamicalsystem to have characteristic frequen-cies, as an atom does; a pendulum or astretched string comes immediately tomind, and other more complicated ex-amples could be cited. But the energyof any such system can be changed byan arbitrary amount. Apparently, thesame is not true of an atomic system.It can exist only in certain stateswith certain, distinct energy values.Its state can change only from oneof these to another, and so its energycan change only by certain distinctamounts. This is the lesson of theFranck-Hertz experiment. And even asthe experiments weie being done, NielsBohr was taking it as one of the basicpostulates on which to build his the-ory of the atom, quite unaware thathis radical proposal was being testedand was passing the test.

4 T H E PHOTOELECTRIC LFFECT

We have seen in Chapter 2 that thequantum concept was originally intro-duced as an aspect of the behavior ofradiating oscillators rather than ofthe radiation itself. It was natural,however, to feel that such behaviormight impress itself at least par-tially on the radiation. The quantumconcept had existed only five yearswhen Albert Einstein seized upon thispossibility as an explanation for thepeculiar effects of radiation in thephotoelectric effect. It wa3 1914 be-fore a thorough, convincing test ofEinstein's ideas was reported.

The photoelectric effect was dis-covered late in the nineteenth cen-tury, and by 1914 had been studiedfairly extensively. The basic phenom-ena were known: A beam of light strik-ing the surface of a metal liberateselectrons from the metal, providedits frequency is greater than a criti-cal value dependent on the kind ofmetal. The electrons acquire in theprocess some kinetic energy, theamount of which increases with in-creasing frequency of the light ac-cording to a relationship whose formhad not been experimentally estab-lished with certainty in 1914. If theelectrons area collected at anotherelectrode and made to constitute acurrent, the magnitude of the currentis proportional to the intensity ofthe stimulating light. The entire pro-cess is virtually instantaneous.

In a sense, the very existence ofthe photoelectric effect can be re-garded as a failure of extrapolation,as there was nothing in classicaltheory that would have suggested sucha process. Even granted its existence,however, all but one of the propertiesjust listed are in conflict with whatwould be expected from classical the-ory. The dependence of photocurrenton light intensity is quite reasonable.But any "reasonable" assumption about

14

the mechanism would involve an inter-action of the electron with the elec-tric field of the light wave, and theintensity of an electric field wave isindependent of the frequency. The ap-pearance of the frequency as an essen-tial factor in tIA phenomenon, there-fore, is quite unexpected. The timedependence is an even more drasticallyshocking result. The most favorableassumption regarding the transfer ofenergy from the light to the electronis that some sort of resonance processtakes place; this assumption impliesthat the electron will absorb all theenergy incident on an area one wave-length square. Computations on thisbasis lead to the conclusion that fora beam of very low intensity, butstill sufficient for easy observationof the photoelectric effect, an elec-tron would take about 500 years to ac-cumulate energy equal to that observed.

Einstein's proposal of an alter-native theory had led R. A. Millikanto carry out an exhaustive experi-mental study at the Ryerson Laboratoryof the University of Chicago. His re-sults were first reported to a meetingof The American Physical Society inApril, 1914; and a detailed descrip-tion was published in The Physical Re-view in 1916. The status of the newtheory is well described in the follow-ing quotation from the introduction tothat paper:

It was in 1905 that Einsteinmade the first coupling of photoeffects and with [sic' any form ofquantum theory by bringing forwardthe bold, not to say the reckless,hypothesis of an electro-magneticlight corpuscle of energy hy, whichenergy was transferred upon absorp-tion to an electron. This hypothe-sis may well be called recklessfirst because an electromagneticdisturbance which remains localized

THE PHOTOELECTRIC EFFECT 15

in space seems a violation of thevery conception of an electromag-netic disturbance, and second be-cause it flies in the face of thethoroughly established facts of in-terference. The hypothesis was ap-parently made solely because itfurnished a ready explanation ofone of the most remarkable factsbrcIght to light by recent investi-gations, viz., that the energy withwhich an electron is thrown out ofa metal by ultraviolet light or x-rays is independent of the inten-sity of the light while it dependson its frequency. This fact aloneseems to demand. some modificationof classical theory or, at anyrate, it has not yet been inter-preted satisfactorily in terms ofclassical theory.

While this was the main if notthe only basis of Einstein's as-sumption, this assumption enabledhim at once to predict that themaximum energy of emission of cor-puscles under the influence of lightwould be governed by the equation

imv2 = V.e = by p, (I)

in which by is the energy absorbedby the electron from the light wave,which according to Planck containedjust the energy hy, p is the worknecessary to get the electron outof the metal and imv2 is the energywith which it leaves the surface,an energy evidently measured by theproduct of its charge e by theP.D. against which it is just ableto drive itself before being broughtto rest.

At the time at which it was madethis prediction was as bold as thehypothesis which suggested it, forat that time there were availableno experiments whatever for deter-mining anything about how P.D. var-ies with v, or whether the hypothe-tical h of equation (I) was anythingmore than a number of the same gen-eral magnitude as Planck's h. Nev-ertheless, the following resultsseem to show that at least five of

the experimentally verifiable rela-

tionships which are actually con-tained in equation (I) are rigor-ously correct. These relationshipsare embodied in the following as-sertions:

1. That there exists for each ex-citing frequency 4, above a cer-tain critical value, a definitelydeterminable maximum velocity ofemission of corpuscles.

2. That there is a linear relationbetween V and V.

3. That dV/dv or the slope of theVv line is numerically equal toh/e.

4. That at the crit'cal frequencyvo at which v = 0, p = hvo,i.e., that the intercept of theVv line on the v axis is theloweat frequency at which themetal in question can be photo-electrically active.

5. That the contact E.M.F.14 be-tween any two conductors isgiven by the equation

Contact E.M.F. = h/e(vo v;)

(Vo

No one of these points except thefirst had been tested even roughlywhen Einstein made his predictionand the correctness of this one hasrecently been vigorously denied byRamsauer. As regards the fourthElster and Geitel had indeed con-cluded as early as 1891, from astudy of the alkali metals, thatthe more electro-positive the metalthe smaller is the value of v atwhich it becomes photo-sensitive, aconclusion however ':rich later re-

'''When two different metals are electricallyconnected, either by direct contact cr throughan electrical circuit, a difference in electri-cal potential exists between them. This differ-ence is the contact potential difference or con-tact emf. It does not affect the flow of currentin the circuit, as the sum of all the contactemf's around a complete circuit vanishes. Itdoes, however, contribute to static effects.


searches on the non-alkaline metalsseemed for years to contradict.

During the ten years which haveelapsed since Einstein set up hisequation the fifth of the aboveassertions has never been testedat all, while the third and fourthhave never been subjected to care-ful experimental test under condi-tions which were even claimed topermit of an exact and definiteanswer, nor indeed can they be sosubjected without simultaneousmeasurements in vacuo of both con-tact potentials and photo-potentialsin the case of metals which aresensitive throughout a long rangeof observable frequencies. In mak-ing this statement I am not under-rating at all the exceptionallyfine work of Richardson and Compton,who in common with most other ob-servers interpreted their resultsin terms of Einstein's equation,but who saw the significance ofthat equation much more clearlythan most of their predecessors haddone. I am merely calling attentionto the fact that the slope mentionedin (3) and the intercept mentionedin (4) cannot possibly be deter-mined with any approach to certaintyunless the region of wave-lengthsopen to study is larger than it isin the case of any save the alkalimetals, and also, in the case of(4) unless simultaneous measure-ments are made in vacuo upon photo-potentials and contact E.M.F.'s.

As the last paragraph of the fore-going quotation implies, there hadbeen earlier attempts at verificationof Einstein's proposal. In particular,very careful studies had been reportedin 1912 by A. LI. Hughes from Cam-bridge, England, and, as mentioned, byG. W. Richardson and A. H. Comptonfrom Princeton. Nevertheless, a reviewof the subject published in 1913 con-cluded that the case was still open.

The work at the University ofChicago had begun in 1905, presumablyimmediately upon publication of Ein-

stein's paper, and various aspects ofthe work had been reported in theyears from 1907 to 1912. Gradually,these earlier results, as Millikanputs it,

...revealed the necessity of ques-tioning the validity of all resultson photopotentials unless the ef-fects of surface films are elimi-nated either by rersoval of thefilms or by simultaneous measure-ment in vacuo of photopotentialsand contact E.M.F.'s or by bothprocedures at once. Accordingly Ihave initiated in 1910 on a some-what elaborate scale simultaneousmeasurements on photoeffects andcontact E.M.F.'s in vacuo on filmfree surfaces.

The investigation required a num-ber of more or less elaborate precau-tions, some of which will be discussedlater. The most bothersome, however,was the one mentioned in the last quo-tation, that of eliminating the effectsof surface films. The obvious solu-tion is to prepare the surfaces undersuch conditions that the films arepreventeJ from forming; and, despitethe formidable mechanical difficultiesinvolved, this was the course taken.

In all of this photoelectricwork at the Ryerson Laboratory thesame general method has been em-ployed, namely, the substances tobe studied or manipulated have beenplaced in the best obtainablevacuum on an electromagneticallycontrolled wheel and all the neededoperations have been performed bymovable electromagnets placed out-side. At new operations have beencalled for the tubes have by de-grees become more and more compli-cated until it has become not inap-propriate to describe the presentexperimental arrangement as amachine snop in vacuo. The opera-tions which are now needed in allthe tubes which are being used are:

1. The removal in vacuo of all sur-face films from all surfaces.

THE PHOTOELECTRIC EFFECT 17

Fig. 4.1 Diagram of one of the tubes used by Millikan inhis study of the photoelectric effect.

2. The measurement of the photocur-rents and photopotentials due tothese film-free surfaces.

3. The simultaneous measurement ofthe contact E.M.F.'s of thesurfaces.

A diagram of one of the tubesappears in Fig. 4.1. The three cylin-ders carried on the wheel W are castfrom the metals to be studied. Thewheel itself can be rotated by anelectromagnet, not shown, so as tobring any of the cylinders oppositeany of the other parts of the appara-tus. K is a rotating knife, which canbe moved back and forth along theaxis, and rotated, by the action ofthe electromagnet F on the armaturesX and 111. The metal to be studied isfirst brought opposite the knife,which is then advanced far enough tocut a thin shaving off the face of thecylinder; the cut is made by rotatingthe knife, which is then retractedagain, the shaving falling down belowthe wheel where it helps to removeany residual oxygen fror the bulb. Thefresh surface can then be turned op-posite either the plate S, for measure-ment of the contact emf, or the win-

dow 0 and electrodes 13 and C, formeasurement of photoelectric behavior.

Electrodes S and B were wade ofcopper, carefully treated so as tohave identical contact potentials. Themeasurement of the contact emf wasbased on the fact that tf there werea potential difference, of whateverorigin, between S and the cylinder, achange in the separation of S and thecylinder would cause some charge tomove through an electrometer connectedto them. Thus, when an external poten-tial was applied so as to give no mo-tion of charge, the external potentialmust be just canceling the contactpotential.

In studies of the photocurrentsthemselves, a beam of monochromaticlight entered through the window 0and struck the surface being studied.The photoelectrons were collected bythe double cylinder S and C, whichwere insulated from each other insidethe tube but, in the actual measure-ments, electrically connected outside.

One precaution is discussed inthe following quotation from the paper:

Since the aim was to test with theutmost possible accuracy the slope


of the line connecting frequencywith the maximum P.D. it was neces-sary first to know v with greatprecision and second to see that notrace of light of frequency greaterthan that being plotted's gotthrough the slit of the spectro-scope. To this end a . . . mercurylamp was used as a source . . . andonly such lines were chosen for useas had no companions anywhere onthe short wave-length side....Light filters to cut out strayshort wave-length light were alsoused.... Since the measurement wasto be made on the maximum P.D. andsince this increases with decreas-ing wave-length it was not of greatimportance that the source be ofgreat purity on the long wave-length side.

Millikan also considered and, asfar as possible, eliminated errorswhich had plagued other workers. Onewas "back leakage" of photoelectronsliberated from the collecting elec-trode by reflected light. If suchelectrons are present, the apparentcritical retarding potential will ac-tually be the potential at which the"forward" current just balances theback leakage. Richardson and Comptonhad been particularly conscious ofthis problem. Milliken avoided ft, forall but one of the wavelengths he used,by using a collecting electrode whoseown photoelectr.c threshold wavelengthwas shorter than that of the incidentlight. Another source of error was thefact that previous investigators hadused a very small range of wavelengths,none extending over a range equal tothe smaller limit, so that the workerswere forced to try to deduce the shapeof a curve from a very short portionof it. Millikan used a range nearlyfour times the lower limit. A third

"The reader should recall that, as sirted Inthe first paragraph of this chapter, It was al-ready known that an increase In frequency pro-duced an Increase In the energy of the cuittedelectrons.

source of error was light of shortwavelength which reached the sensitivesurface by diffuse reflection in themonochromator. This problem was elimi-nated, in cases where it was critical,by the use of light filters; usually,however, they were not necessary.

The resulting data were plottedas curves of photocurrent versus poten-tial difference for each of severalwavelengths. A set of such curves isshown in Fig. 4.2 (see next page). Theintercept of each curve with the hori-zontal axis gives the value of V forthat wavelength.

It will be seen...that the maximumpossible error in locating any ofthe intercepts is say two hundred-ths of a volt and that the totalrange of volts covered by the in-tercepts is more than 2.5. Eachpoint, therefore, of a potential-frequency curve should te locatedwith not more than a per ,.ent ofuncertainty. The frequencies are,of course, known with great preci-sion.

The values of the intercepts werethen plotted versus wavelength. Theplot obtained from the curves of Fig.4.2 is shown in Fig. 4.3 (see nextpage). "It will be seen that the firstresult is to strikingly confirm theconclusion...as to the correctness ofthe predicted linear relationship be-tween maximum P.D. and v, no pointmissing the line by more than abouta hundredth of a volt." If Eq. (4.1)is divided by e, the result is

V - (h/e)v - (h/e)vo,

a straight line whose slope is h/e.Thus, from the measured slope of theline in Fig. 4.3 and Millikan's ownearlier determination of e, he couldcompute h. The result was 6.56 x 10"7,in complete agreement with the valueoriginally computed by Planck from theconstants of the blackbody radiationlaws. Moreover, many determinations,

on different surfaces, gave the same

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therefore, could be obtained from the"raw" curve of Fig. 4.3 by shifting itupward by an amount equal to the con-tact emf. The resulting curve is showndashed in Fig. 4.3. It is this curvewhose intercept with the frequencyaxis gives vo.

The second method made use of thefact that if the intensity and fre-quency of the stimulating light arefixed, and the potential differenceused to accelerate the electrons to-ward the collector is varied, the col-lected current at first increases withincreasing voltage but eventuallyreaches a limiting value, the satura-tion current. For fixed frequency, thesaturation current is directly propor-tional to the intensity. The procedurewas to measure the intensity of theincident radiation, by means of athermopile," together with the cor-responding saturation photocurrent,for each of several wavelengths. Thesaturation photocurrent per unit in-tensity was then plotted against wave-length. The point at which the curvecrosses the wavelength axis is thecritical wavelength Ao, related to voby Aovo c. This method might be ex-pected to be less reliable than thefirst. Nothing was known in advanceas to the likely shape of the curve;and, in the examples shown in Milli-kan's paper, the curve was determinedonly by three or Pour points and thenextrapolated. Nevertheless, Millikanwas confident that he could locate theintercept to mithio 100 A; and cer-tainly the results seemed to bear himout, at least as Judged by agreementbetween the two methods.

I4A thermopile is a group of thernocoLples, con-nected electrically so as to act in conbinationand blackened so at to absorb incident radia-tion.

The fifth point mentioned inMillikan's introduction need not con-cern us further, except to mentionthat it was satisfactorily validated.

Millikan concludes his paper withfive pages of discussion of theoriesof photoemission. Most of this mate-rial has been made obsolete by laterdevelopments. Two portions, however,remain valid and warrant quotation:

Perhaps it is still too earlyto assert with absolute confidencethe general and exact validity ofthe Einstein equation. Nevertheless,it must be admitted that the pres-sent experiments constitute verymuch better Justification for suchan assertion than has heretoforebeen found, and if that equation beof general validity, then it mustcertainly be regarded as one of themost fundamental and far reachingof the equations of physics; for itmust govern the transformation ofall short-wave-length electromag-netic energy into heat energy. ...

The photoelectric effect then,however it is interpreted, if onlyit is correctly described by Ein-stein's equation, furnishes a proofwhich is quite independent of thefacts of black-body radiation ofthe correctness of the fundamentalassumption of the quantum theory,namely, the assumption of a discon-tinuous or explosive emission ofthe energy absorbed by the elec-tronic constituents of atoms fromether waves. It materializes, soto speak, the quantity "h" discov-ered by Planck through the studyof black body radiation and givesus a confidence inspired by noother type of phenomenon that theprimary physical conception under-lying Planck's work corresponds toreality.

5 MAGNETIC PROPERTIES OF ATOMS

Bohr's theory of atomic structure waspublished in a series of papers be-tween 1913 and 1915,17 and by the early1920's had been fairly thoroughly de-veloped by many workers, notablyArnold Sommerfeld. One of its funda-mental postulates was that certain dy-namical quantities relating to peri-odic motion could not take on arbitraryvalues, but only a set of discretevalues, integral multiples of Planck'sconstant h. This postulate accountednicely for the discrete energy statesdeduced frdm the experiment of Franckand Hertz described in Chapter 3 - notonly for their qualitative existence,but, in the case of the hydrogen atom,for their quantitative values as well.

An even more curious result ofthe application of this rule is thatif an atom whose complement of elec-trons has a nonzero angular momentumis in a magnetic field, the angularmomentum vector can make only certainwell-defined, discrete angles withthe direction of the field. The anglespermitted by the theory are those suchthat the compoent of the angular mo-mentum along the field is an inteuelamultiple (positive, negative, but notzero) of h/2; where h is Planck'sconstant." The reason that this re-sult was regarded as curious was justthe sort of extrapolation discussedin Chapter 1, for classically the valueof the component of a vector along anyspecified direction is restricted onlyby the condition that it cannot exceedthe magnitude of the vector.

"James Franck remarked in 1961 that he andHerta had not known of Bohr's work while theirown experiments (see Chapter 3) were going on,but that even if someone had told them it existedthey probably would rot have read it: The prob-lem of atomic structure was regarded as so arcanethat anyone clatwing to hare solved it was likelyto be a crackpot."The theory also maintained that the magnitudeof the angular momentum must be an integral mul-tiple of h/2s,

21

In a paper in the Zeitschrift furPhysik in 1921, Otto Stern made thefollowing statement: "Now, whether thequantum theoretical or the classicalconception is valid can be disting-uished by means of an experiment,completely simple in principle. Oneneeds only to study the deflectionthat a beam of atoms undergoes it asuitably inhomogeneous magneticfield."19 The theoretical basis forthe statement as follows: An elec-tron moving in one of the orbits ofBohr's theory, with an angular momen-tum L, has a magnetic moment i whichis proportional to L and along thesame line. (In an atom containingseveral electrons, both the angularmomenta and tt,q magnetic moments ofthe individual electrons add vector-tally, so that the total magneticmoment is related in this same way tothe total angular momentum.) ..When theatom is in a magnetic field B, the ac-tion of the field on the magnetic mo-ment produces a torque in a directionperpendicular to both ,u (and there-fore L) and B, and of magnitude pro-portional to B and to L, and depend-ent on the angle between B and L.The effect, according to classical dy-namics, is to cause L to precess aboutB - to trace out a cone whose axis isalong B. The frequency of precessionis proportional to B, the factor ofproportionality involving only theproperties of tne electron and naturalconstaats; for a field of 101 gauss,the period is ? x 10'1° second. Nowsuppose that the field is not uniform,but varies from point to point. Denoteby 013/0x the change in B per unit dis-placement in the x direction, and simi-larly for y and E. Then there will bea net force F on the atom, given by

l'Iraoslatet by G. L. T.


Fig. 5.1 Shape of magnet poles to give a_large value of 8B/az in the direction of B.

= Az 0g/ax + Ay 0E/ay + Az ai/az.

If this is averaged over a time longcompared with the period of preces-sion, the first two terms average tozero because Az and Ay vary sinusoid-ally, and there remains only

- A ,

The atom, therefore, is accelerated inthe direction of .913/3z.

Now imagine an electromagnetwhose poles are shaped as shown inFig. 5.1. Close to the knife-edge, thefield B is strongest and the directionof aB/8z is along B. An atom of massM traveling along parallel to theknife-edge and just above it will ex-perience an acceleration a - /m

(88/0z)/M, in a direction along013/11z and thus perpendicular to itsinitial motion. If it spends a time ttraversing the field, it will havebeen deflected from its original pathby an amount x la t2 iitz(A13/0z)t2/N.The time t is the speed v at which theatom travels divided into the lengthf of the magnet poles. The deflection,then, is

PA IIx M az v2'

(5.1)

Now consider an atom for whichL h/2v. Then according to quantumtheory, the component La of L alongB can only be +h/2t, and pa can beonly &p. An initial beam of atoms allhaving the same velocity would thenbe split into two by the action of the

field, and there would be no part ofthe beam undeflected.

The classical case is a bit morecomplicated. Let us write Eq. (5.1) as

orx CA z

x CA cos 0, (5.2)

where 0 is the angle between A and B.The classical concept is that 0 canhave all possible values, so thatcos 0 will range continuously from1 to +1. The initial beam is notsplit but spread, with the extent ofspread being equal to the amount bywhich the two components predicted byquantum mechanics would be separated.However, it is still possible that theclassical result would look scmewhatlike the quantum one, if the intensityalong the spread had maxima at theends and a minimum at the center. Wecan show that this is not the case.The intensity distribution is the num-ber of atoms per unit deflection; theprocedure, then, is to find the numberof atoms for which 0 is in a smallrange dO and the values of the deflec-tion correspoiAing to this same range,and take the ratio. It is shown in cal-culus that if x and 0 are related asin Eq. (5.2), then the range dx of xcorresponding to a very small rangedO of 0 is given by

dx = C sin 0 dO, (5.3)

which is then the denominator of ourratio. The numerator, at least as re-gards its dependence on 0, is obtainedby recognizing that the numbers fortwo different angles 0, and 0, are toeach other as the areas of two narrowbands around a sphere, one at colati-tude2° 0, and one at colatitude 02,hating equal widths. Figure 5.2 showssuch a band. Its area is the circum"ft ence of the band, 2;a, times the

"The colatitude of a point on a -There Is theangle that a radius to that point wakes sith theaxis of the sphere. It tg 90' minus the lati-tude.

MAGNETIC PROPERTIES OF ATOMS

Fig. 5.2 Band at colatitude 0 on a sphereof radius r.

width (r d0), provided dO is small.The radius a is equal to sin 0 timesthe radius r of the sphere, so thatthe area of the band is 211-2 sin 0 de.In the ratio of two such areas, allfactors cancel out except sin 01/sin 01.Thus, the number of atoms whose mag-netic moments make an angle with Bin the range dO around 0 is propor-tional to sin 0 dO. When this is di-vided by Eq. (5.3), the angular de-pendence cancels out. The result isthat according to classical theory,the initial spot is spread into aband of uniform intensity.

These calculations, however, haveassumed a beam involving a single vel-ocity. In actuality, the beam is ob-tained by vaporization and contains adistribution of velocities. This turnsout to improve matters. in the quan-tum case, each of the two spots isspread out somewhat, but the centralregion is still a minimum (Eq. (5.1)shows that if Ia cannot be zero, andOBiot # 0, a zero deflection can re-sult only from an infinite velocity.)In the classical case, on the otherhand, the single uniform streak i3 re-placed by a superposition of streakswhose lengths range from very smallto very large; the combination oro-duces a maximum of intensity at thecenter. It is evident that the distinc-tion is, as Stern maintained,clearcut.

23

Moreover, the experiment semssimple in principle. In practice, itis another matter. To begin with, theexperiment must be carried out invacuum so that the beam will not bedestroyed by scattering from gas mole-cules. This restricts the possiblelength f of the poles to a few centi-meters. It is possible `o make B ofthe order of 103 gauss, ,Ind aB /az ofthe order of 104 gauss pe. centimeter.The value of p is known. The combina-tion MO is twice the kinetic energyof the atom; and the kinetic energyis determined by the vapor temperature,which may be of the order if 1000°K.When such numbers are put into theformula, the deflection that can beexpected is found to be of the orderof 0.01 mm. The experiment was possi-ble, but it would evidently be a deli-cate one.

When Stern submitted the papercontaining the foregoing analysis, inlate August of 1921, he and a co-worker, Walther Gerlach, were alreadyoccupied with carrying out the experi-ment at Frankfurt am Main, Germany;and by the middle of November they hadpreliminary results - too preliminary,however, to permit a decision on themain question. They proceeded to makesome improvements on the apparatus (afootnote to the third paper commentsthat "It was possible for these to bew:orked out and tested by joint effortsduring the Christmas vacation""), andsubmitted firm conclusions on 1 March1922.

The experimental arrangement war-rants fairly detailed description. Thesubstance used was silver, which wasvaporized in an electrically heatedoven and escaped through a circularopening of area 1 mmt. At a distanceof 2.5 cm from the oven" was a dia-

/.11"Translated by G. L. T."This distance was increased grow the 1 cm usedin the preliminary 'work, so as to prevent theaperture from being plastered ovcr either byswollen silver splattered out of the oven or bytoo rapid incrustation by deposition Iron thebean.


Fig. 5.3 Splitting of a beam of silveratoms by an inhomogeneous field. Each scaledivision is I/20 mm.

phragm containing an approximatelycircular aperture of area 2 X 10-3 mm2,i.e., radius about 0.03 mm. Another3.3 cm beyond this was a second dia-phragm, whose opening was in the formof a slit of length 0.8 mm and width0.03 to 0.04 mm, oriented perpendicu-lar to the direction of B. Such tinydimensions were obviously necessaryto produce a beam which was not vastlylarger than the amount by which it wasdeflected. The slit was placed just atthe apex of one end of the knife-edgepole piece (see Fig. 5.1), and the setof openings was so adjusted that thebeam traveled parallel to the knife-edge. The magnet poles were 3.5 cmlong. The whole arrangement was housedin a casing whose walls were thickenough to prevent pressure or the mag-net poles from shifting the relative

positions. The "exposure times" wereeight hours; even then, the deposit ona glass pate at the far end of thepole pieces was too thin to be visible,and had to be darkened by precipita-tion of nascent silver.

The best exposure is reproducedin Fig. 5.3. Gerlach and Stern say:

Two other exposures yielded a re-sult identical in all essentialpoints, not, however, with thiscomplete symmetry. It must be saidhere that a reliable adjustment ofsuch small diaphragms by opticalmeans is very difficult, so thatthe achievement of such a completelysymmetric exposure as in Fig. 3 issurely in part the result of luck;misplacement of one diaphragm by afew hundredths of a millimeter isalready enough to make an exposure

The one characteristic common to allthree exposures was the clear separa-tion of the beam into two components.As the authors put it, "The splittingof the atomic beam in the magneticfield gives rise to two discrete beams.There are no undeflectc -d atoms detect-

able."22Reference to the preceding dis-

cussion shows that this result clearlyconfirms the quantum hypothesis as op-posed to the classical behavior. Theterminology of the time was used byGerlach and Stera: "We behold in theseexperimental results the direct ex-perimental proof of the quantizationof direction in a magnetic field.22

''Translated by G. L. T.

6 T H E SCATTERING OF X RAYS

X rays were identified as electromag-netic waves in 1912, by means of dif-fraction experiments suggested by vonLaue and carried out by Friedrich andKnipping. The behavior of electromag-netic waves was well understood onthe basis of Maxwell's theory. In par-ticular, it was an easy matter to en-visage a mechanism by which they werescattered, and it was a straightfor-ward procedure to compute the quanti-tative features of the scattering. Themechanism was that the varying electricfield of the wave set the electrons inthe scatterer into forced vibration;the electrons, in turn, because theywere being accelerated, emitted radia-tion. The scattered radiation was ofthe same frequency as the incident -it had to be, as it was just the fre-quency of vibration of the emittingelectrons. It had all the propertiesof the radiation emitted by an oscil-lating electric dipole: intensitysymmetrically distributed around theline of the electron's motion and, inany plane containing that line of mo-tion, varying as the square of thesine of the angle between the line ofmotion and the direction of propaga-tion; and polarization properties thatare of no concern in the present dis-cussion. Moreover, the fraction of theincident energy transferred into thescattered radiation should be inde-pendent of the frequency.

In the ten years following 1912,this theory met with steadily increas-ing difficulty. The firt discrepancywas that for x rays of very short wave-length, or for y rays, the scatteredintensity was greater in the forwarddirection (relative to the incidentradiation) than in the backward.This feature received, for a time, aquantitative explanation by ascribingto the electron a size comparable tothe x-ray wavelength and allowing forinterference between the rays scattered

25

by different parts of the electron. Asdata accumulated, however, it wasfound that the value that must be as-signed for the diazieter of the elec-tron varied with the wavelength of theincident radiation - obviously a mostunsatisfactory situation. In addition,a still more serious difficulty hadappeared. It as discovered that thescattered radiation was different infrequency from the incident. Arthur H.Compton, in a paper in The PhysicalReview in 1923, had this to say aboutthe situation:

Such a change in wave-length isdirectly counter to Thomson's the-ory of scattering, fcr this de-mands that the scattaring electrons,radiating as they do uecause oftheir forced vibrations when tra-versed by a primary X-ray, shallgive rise to radiation of exactlythe same frequency as that of theradiation falling upon them. Nordoes any modification of the theorysuch as the hypothesis of a largeelectron suggest a way out of thedifficulty. This failure makes itappear improbable that a satisfac-tory explanation of the scatteringof X-rays can be reached on thebasis of the classical electrody-namics.

Compton's idea was to apply tothe description of scattering the samequantum concept that had proved souseful in treating the photoelectriceffect (see Chapter 4). He expressedthe basic change in viewpoint, and itsconsequences, as follows:

According to the classical the-ory, each X-ray affects every elec-tron in the matter traversed, andthe scattering observed is that dueto the combined effects of all theelectrons. From the point of view


of the quantum theory, we may sup-pose that any particular quantum ofX-rays is not scattered by all theelectrons in the radiator, butspends all of its energy upon someparticular electron. This electronwill in turn scatter the ray insome definite direction, at an an-gle with the incident beam. Thisbeinding of the path of the quantumof radiation results in a change inits momentum.24 As a consequence,the scattering electron will recoilwith a momentum equal to the changein momentum of the X-ray. The en-ergy in the scattered ray will beequal to that in the incident rayminus the kinetic energy of therecoil of the scattering electron;and since the scattered ray must bea complete quantum, the frequencywill be reduced in the same ratioas is the energy. Thus on the quan-tum theory we should expect thewave-length of the scattered X-raysto be greater than that of the in-cident rays.

Having established this frame-work, Compton proceeded to build uponit a rather impressive structure.First, he derived the relationship

A A0 + (2h/mc) sin' le

between the incident wavelength A0,the scattered wavelength A, and thescattering angle 0; h is Planck's con-stant, m the mass of the electron,and c the sr'ed of light. The form ofthe relationship is reasonable: Thelarger the scattering angle, thegreater the momentum given to the elec-

''It is not cleat whether Compton's assumptionof momentum carried by a photon was his out orone that had been used before. The phraseologyof hts conclusions (see later) suggests that Ineither case, the idea was controversial. if itwas his own contribution, he apparently expectedhis readers to be able to deduce for themselvesthat the magnitude of the momentum carried by aphoton of energy E must be E'c, 'here c is thespeed of light; certainly, there sere at leasttwo methods of deduction available to them.

tron; therefore the greater its kine-tic energy, acquired from the incidentphoton, and the greater the reductionin photon frequency and the increasein wavelength. The details follow sim-ply from application of the laws ofconservation of energy and momentumto the scattering event, rememberingthat the electron must be treatedrelativistically. One other featureof the equation should be noted. Thefactor 2h/mc has the value 0.0484 A.It is the smallness of this quantitythat prevented the detection of theeffect prior to the discovery andstudy of x rays.

Compton then noted that when theshift was expressed in terms of fre-quency instead of wavelength, it hadthe same form as the expression givenby classical theory for the shift re-sulting from the scattering by an elec-tron moving in the direction of propa-gation, if the velocity of the elec-tron was properly related to theincident frequency. He said, "It isclear, therefore, that so far as theeffect on the wave-length is concerned,we may replace the recoiling electronby a scattering electron moving in thedirection of the incident beam" at anappropriate velocity.

The argument then ran as follows:

It seems obvious that sincethese two methods of calculationresult in the same change in wave-length, they must also result inthe same change in intensity of thescattered beam . . . . I have not,however, succeeded in showing rig-idly that if two methods of scat-tering result in the same relativewave-length at different angles,they will also result in the samerelative intensity at differentangles. Nevertheless, we shall as-sume that this proposition is true,and shall proceed to calculate therelative intensity of the scatteredbeam at different angles on thehypothesis that the scattering elec-trons are moving in the directionof the primary beam. . .

THE SCATTERING OF X RAYS 27

with a velocity given by the expres-sion he had previously noted. In theprocess, Compton obtained an expres-sion for the number of quanta scat-tered per unit time, so that he couldcompute the scattering absorption co-efficient, the fraction of energy ofthe incident beam removed per unit ofpath length. He could also determinewhat proportion of this energy reap-pears as scattered radiation and whatwas truly absorbed into recoil energyof the scattered electrons, therebydetermining the true scattering coef-ficient and the coefficient of trueabsorption due to scattering.

The details of the computationare rather intricate and of no particu-lar interest here. They led to expres-sions for several quantities that hadbeen or could be measured experiment-ally. The most obvious, and best known,is the change of wavelength. A discus-sion of this will be deferred tempo-rarily.

Next is the scattering absorptioncoefficient a. When a beam of x raysof single wavelength A is incident onone side of a slab of material ofthickness x, the intensity of theemergent beam is less than the inci-dent by a factor e-P2. The coefficientIL is called the linear absorption co-efficient, and is the sum of a termproportional to A3, due to the photo-

Fig. 6.1 Experimental values of the scat-tering absorption coefficient for carbon(crosses) compared with predictions basedon two different theories.

electric effect, and a second termwhich is just a. The linear absorptioncoefficient for carbon had just beenmeasured as a function of wavelength.From the data, Compton estimated thevalue of a. The resulting values, di-vided by the density p, are shownplotted as crosses in Fig. 6.1. Theupper, horizontal solid line is thevalue predicted by the classical the-ory, as calculated by Thomson; thelower solid curve is Compton's expres-sion. It can be seen that where thetwo theories give substantially dif-ferent results, the experimental val-ues clearly favor the quantum theory.25The circle shows a value for total

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AUTHOR Trigg, George L. TITLE Crucial Experiments in ...Photoelectric Effect, Magnetic Properties of Atoms, The Scattering of ... FILE COPY. Crucial Experiments in Quantum Physics