Einstein - On the Theory of Radiation (1917)

8/12/2019 Einstein - On the Theory of Radiation (1917)

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Received March3 1917 1ON THE QUANTUM THEORYOF RADIATION

A EINSTEIN

The formal similarity between the chromatic distribution curve forthermal radiation and the Maxwell velocity-distribution law is toostriking to have remained hidden forlong.In fact, it was this similaritywhich led W. Wien, some time ago, to an extension of the radiationformula in his important theoretical paper, in which he derived hisdisplacement lawQ = v*j{vlT). (1)As is well known, he discovered the formula

= a . v exp (hvjkT), (2)which is still accepted as correct in the limit of large values ofv/T(Wien's radiation formula). Today we know th at no approach which isfounded on classical mechanics and electrodynamics can yield a usefulradiation formula. Rather, classical theory must of necessity lead toRayleigh's formula

koce vKT. (3)Next, Planck in his fundamental investigation based his radiationformula

e exp(hv/kT)- 1 K'on the assumption of discrete portions of energy, from which quantumtheory developed rapidly. It was then only natural that Wien's argument, which had led to eq. (2), should have become forgotten.Editor's note.This pap er was published as Ph ys. Zs. 18(1917) 121. I t w as firstprinted in Mitteilungen der Physikalischen Gesellschaft Zurich, No. 18, 1916.

63


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6 4 A . E I N S T E I N 1Not long ago I discovered a derivation of Planck's formula which

w as closely related toWien'soriginal argu m ent * an d which was basedon the fundamenta l assumption of quantum theory. This derivat iondisplays the relat ionship between Maxwell 's curve and the chromaticdistribution curve and deserves at tention not only because of i tssimplicity, but especially because it seems to throw some light on themechanism of emission and absorption of radiat ion by matter, aprocess which is stil l obscure to us. By postulating some hypotheseson the emission and absorption of radiation by molecules, whichsuggested themselves from quantum theory, I was able to show thatmolecules with a quantum-theoretical distribution of states in thermalequil ibrium, were in dynamical equil ibrium with the Planck radiat ion;in this way, Planck's formula (4) could be derived in an astonishinglysimple and general way. It was obtained from the condit ion that theinternal energy dis t r ibut ion of the molecules demanded by quantumtheory, should follow purely from an emission and absorption ofradiat ion.

But if these hypotheses on the interaction between radiat ion andmatter turn out to be jus t i f ied, they must produce ra ther more thanjust the correct stat ist ical distribution of the internal energy of themolecules: for there is also a momentum transfer associated with theemission and absorption of radiat ion; this produces, purely throughthe interaction between the radiat ion and the molecules, a certainvelocity distribution for the lat ter. This must evidently be identicalwith the velocity distribution of the molecules which is entirely dueto their collisions among themselves, i .e. i t must agree with theMaxwell distribution. It has to be required that the mean kineticenergy of a molecule (per degree of freedom) should be equal to \kTin a Planck radiat ion field of temperature T. This requirement shouldhold independently of the nature of the molecules under considerationand independently of the frequencies emitted or absorbed by them.We want to demonstrate in the present paper that this far-reachingreq uir em en t is in fact satisfied qu ite generally, th u s lendin g n ewsup po rt to our simple hy poth eses concerning th e elem entary processesof emission and absorption.

To obtain such a result however requires a certain extension of thehypo theses , whichhad beenup to now solely concerned w ith an exch ang e* Verh.d. Deutschen physikal .Gesellschaft 18 Nr. 13/14 (1916) 318. The arguments used in that paper are reproduced in the present discussion.


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ON TH E QU AN TU M TH EO RY OF RADIATION 65of energy. The question arises: does the molecule suffer an impulsewhen it emits or absorbs energy e? As an example, let us consider theemission of ra di at io n from t h e p o in t of view of classical electro dy nam ics.When a body emits an energy e, i t has a recoil (momentum) e/c,provided the whole of the radiat ion is emitted in the same direct ion.If , however, the emission process has spatial symmetry, such as in thecase of spherical waves, no recoil is produced at all . This secondpossibility is also of importance in the quantum theory of radiation.If a molecule abso rbs or em its energy e in th eform of radiation duringits t ransi t ion from one qua ntum -theore t ical ly possible s ta te to anoth er ,such an elementary process can be thought of as being par t ial ly orcompletely directional, or else symmetrical (non-directional). It willbecome apparent that we shall only then arr ive at a theory which isfree from contradictions, if we consider such elementary processes tobeperfectlydirect ion al; this embodies the ma in result of the sub sequen tdiscussion.1. F u n d a m e n t a l h y p o t h e s i s of q u a n t u m t h e o ry . C a n o n i ca l d i s -

tr ibut ion of s ta tesIn quantum theory a molecule of a given kind can only exist in a discrete set of states Z\, Zz, ... Zn, .. . , with (internal) energies e 1;s2, ...en, ..., apar t f rom i ts or ientat ion and translatory motion. I f suchmolecules belong to a gas at temperature T, the relative frequencyWn of such statesZn is given by the formula

Wn = pn exp ( - enjkT), (5)which corresponds to the canonical dis tr ibution of s tates in s tat is t icalmechanics. In this formula, k=RjN is the well-known Bo ltzman ncons tant , and p n is a nu m ber , indep end ent of T and characteristic forthe molecule and its nth quantum state , which can be cal led thestatistical 'weight ' of this state. Formula (5) can be derived fromB oltzm ann 's pr inciple, or from purely therm ody nam ical considerat ions.I t expresses the most extreme generalisation of Maxwell 's velocity-dis tr ibution law.

The la tes t fundamenta l developments in quantum theory are concerned with a theoret ical der ivation of the quantum-theoret ical lypossible states Zn and their weights p n . Fo r th e presen t basic invest igat ion, a detailed determ ination of th e qu an tum state s is not required.


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6 6 A . E I N S T E I N 12. H y p o th e se s o n th e r a d i a t i v e e x c h a n g e o f e n e r g yL etZn an dZm be twoquantum-theoretically possible states of the gasmolecule, whose energies are en a n d em, respectively, and satisfy theinequal i ty em>en. Le t us assume that the molecule is capable of atransi t ion from sta teZn in to stateZm with an absorption of radiationenergysm en', th at , similarly, th e tran sit ion from sta teZm to s ta te Znis possible, with emission of the same radia t ive energy. Let theradiat ion absorbed or em it ted b y th e molecule ha ve frequency vwhichis characteristic for the index combination (m , n) that we are considering.For the laws governing this t ransi t ion, we introduce a few hypotheses which are obtained by carrying over the known situationfor a Planck resonator in classical theory to the as yet unknown onein quantum theory .

(a) E mission o f radiation. According to Hertz, an oscillating Planckreson ator rad iates energy in th e well-known w ay, regardless of w heth eror not it is excited by an external field. Correspondingly, let us assumeth at a molecule m ay go from sta teZm to a s ta teZn and emit radia t ionenergy sm sn w ith frequency (i, w ithou t excitation from exte rnalcauses. Let the probabili ty dW for this to happen during the t imein te rva l dt, be

dW Al dt, A)where / I * is a constant characterising the index combinat ion underconsideration.

The statist ical law which we assumed, corresponds to that of aradioactive reaction, and the above elementary process correspondsto a reaction in which only y-rays are emit ted. I t need not be assumedhere that the t ime taken for this process is zero, but only that thistime should be negligible compared with the t imes which the moleculespends in sta tes Z\, etc.(b) Absorption of radiation. If a Planck resonator is located in aradiation field, the energy of the resonator is changed through thework done on the resonator by the electromagnetic field of theradiation; this work can be positive or negative, depending on thephases of the resonator and the oscillating field. We correspondinglyintroduce the fol lowing quantum-theoret ical hypothesis . Under theinfluence of a radiation density q of frequency v, a molecule can makea t ransi t ion from sta teZn to s ta teZm by absorbing radiat ion energy


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ON THE Q U AN T UM TH EO RY OF RADIATION 67ee, according to the probabil i ty law

&W = BJg d*. (B)We similarly assume that a transit ion Zm-+Zn, associated w ith al iberation of radiation energy emen, is possible under the influenceof the radiation field, and that it satisfies the probability law

dW = BIQ At. (B')B and * are constants. We shall give both processes the name'changes of state due to irradiation' .We now have to ask ourselves what is the momentum transfer tothe molecule for such changes of state. Let us first discuss the caseof absorption of radiation. If a radiation bundle in a given directiondoes work on a Planck resonator, the corresponding energy is removedfrom the radiation bundle. To this transfer of energy there also corresponds a momentum transfer f rom radiat ion bundle to resonator , bymomentum conservat ion. The resonator is thus acted upon by a forcein the beam direction of the radiation bundle. If the energy transferis negative, then the force acts on the resonator in the opposite direction. If the quantum hypothesis holds, we can obviously interpretthe process in the following way. If the incident radiation bundleproduces the t ransi t ionZn-^~Zm by absorption of radiation, a momentu m {emBn)lc is transferred to the molecule in the direction ofpropagat ion of the beam. For the absorpt ion process Zm->Zn, themomentum transfer has the same magni tude, but is in the opposi tedirection. Fo r the case wh ere th e molecule is acted u po n simu ltaneouslyby several radiat ion bundles , we assume that total energy emenassociated with an elementary process is removed from, or added to,a single such radia t ion bundle . Thus here , too , the momentum t ransferred to the molecule is (sm em)/c.For an energy transfer by emission of radiation in the case of aPlanck resonator , no momentum transfer to the resonator takes place,since emission occurs in the form of a spherical wave, according toclassical theory. As was remarked previously, a quantum theory freefrom contradictions can only be obtained if the emission process, justas absorption, is assumed to be directional. In that case, for eachelementary emission process Zm->Zn a m om entum of ma gni tude{emen)jc is transferred to the molecule. If the latter is isotropic, weshall ha ve to assu m e th a t all directions of emission are equ ally pro ba ble .


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68 A. EINS TE IN 1If the molecule is not isotropic, we arrive at the same statement if theorientat ion changes with t ime in accordance with the laws of chance.Moreover, such an assumption wil l a lso have to be made about thestatistical laws for absorption, (B) and (B'). Otherwise the constantsBan d B would ha ve to depe nd on th e direct ion, and this can b eavoided by making the assumption of isotropy or pseudo-isotropy(using time-averages).3 . Derivat ion o f the P lanck radiat ion lawWe now look for that part icular radiat ion densi ty Q , for wh ich th eexchange of energy between radiat ion and molecules in keeping withthe probabili ty laws (A), (B), and (B') does not disturb the moleculardistribution of states given by eq. (5). For this i t is necessary andsufficient that the number of elementary processes of type (B) takingplace per unit t ime should, on average, be equal to those of type (A)and (B') taken together. From this condit ion one obtains from (5),(A), (B), (B') the equation

p n exp (-enlkT)BQ = p m exp (~sm/kT)(Bls + A )for the elementary processes associated with the index combination(m, n).

If, in addition, Qtend s to infini ty w ith T, as will be assumed, therelat ion

pnB = pmBl (6)has to hold between the cons tants B and B. W e the n o btain fromour equat ion,

n ; D

exp [(em sn)lkT] 1as the condit ion for dynamical equil ibrium.

This expresses the temperature dependence of the radiat ion densi tyaccording to P lan ck 's law. F ro m W ien's displacem ent law (1) i t followsimmedia te ly tha t

An- ~ =


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ON THE Q UA NT U M TH EO RY OF RADIATION 69elec t rodynamic and mechanica l processes; for th e p resent , one ha s toconfine oneself to a treatment of the l imiting case of Rayleigh's lawfor high temperatures, for which the classical theory is valid in thel imit .

Eq. (9) is of course the second principal rule in Bohr 's theory ofspectra. Since i ts extension by Sommerfeld and Eps te in , th i s maywell be claimed to have become a safely established part of ourscience. I t also contains implici t ly the photochemical principle ofequivalence, as has been shown by me.

4. A me th od for ca lcu la t ing the mo t ion o f m ole cu le s in tneradiat ion f ie ldWe now turn to a discussion of the motion of our molecules under theinfluence of radiation. For this we shall make use of a method whichis well known from the theory of Brownian motion, and which Iemployed on several occasions for numerical computations of motionsin a radiation field. To simplify the calculation we shall only considerthe case where the motions take place in just one direct ion, the X-direction of the coordinate system. Furthermore, we shall confineourselves to a calculation of the average value of the kinetic energyof the progressive motion, and we shal l thus not at tempt to prove thatsuch velocities v obey the Maxwell dis t r ibut ion law. The mass M ofthe molecule is assumed sufficiently large, so that higher powers ofvjc can be neglected in comparison with lower ones; we can thenapply the laws of ordinary mechanics to the molecule. Final ly, noreal loss of generality is introduced if we perform the calculations asi f the s tates with index m a n d n were the only possible states for themolecule.

T h e m o m e n t u m Mv of a molecule undergoes two different typesof change during the short t ime interval x. Although the radiat ion isequally consti tuted in al l directions, the molecule will nevertheless besubjected to a force originating from the radiation, which opposes them otion. Let th is be equal toRv, whereR is a constan t to be determ inedlater. This force would bring the molecule to rest, if it were not forthe i r regular i ty of the radiat ive interact ions which t ransmit a momentu m A of changing s ign and magni tude to the molecule during t ime r;such an unsystematic effect , as opposed to that previously ment ioned,will sustain some movement of the molecule. At the end of the short


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7 0 A . E I N S T E I N 1t ime in terval T, the momentum of the molecule will have the value

Mv Rvr + A.Since the velocity distribution is supposed to remain constant witht ime , the average of the absolute value of the above quanti ty must beequal to Mv; the mean values ofthe squares of these quanti t ies, takenover a long t ime interval or over a large number of molecules, musttherefore be equal:

(Mv Rvr + A)2 = (M).Since we were specifically concerned with the systematic effect of

v on the momentum of the molecule, we shall have to neglect theaverage value vA. Expanding the left-hand side of the equation, onetherefore obtains

A* = 2RMvh. (10)The mean square value vz, which the radiat ion of temperature Tproduces in our molecules by interacting with them, must be of the

same size as the mean square value v2 obtained from the gas laws fora gas molecule at tem pe ratu re T in th e kinetic theo ry of gases. Fo r th epresence of our molecules would otherwise disturb the thermal equil ibr ium between the thermal radiat ion and an arbi t rary gas held atthe same temperature. We must therefore have

\Mv* = \kT. (11)Eq. (10) thus becomes3/T = 2RkT. (12)

The investigation is now continued as follows. For a given radiation(Q (V)), A2 a n d R can be calculated, using our hypotheses on theinteraction between radiation and molecules. If the results are insertedin eq. (12), this equation must become an identi ty, if Qis expressed asa function of v and T, using Planck's equation (4).5. Calculat ion of RConsider a molecule of the kind discussed above, moving uniformlywith velocity v along the X-axis of the coordinate system K. We wishto f ind the average momentum which is transferred from the radiationfield to the molecule per unit t ime. In order to calculate i t , we have to


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1 O N T H E Q U A N T U M T H E O R Y O F R A D I A T I O N 7 1describe the radiation in a coordinate system K which is at restrelat ive to the molecule in question. For we had only formulated ourhypotheses on emission and absorption for the case of stat ionarymolecules. The transformation to the [coordinate] system K' has beencarried out in a number of places in the l i terature, part icularly accurately in Mosengeil 's Berlin dissertation. For completeness, however,I shall reproduce these simple arguments at this point .

Referred to K, the radiation is isotropic, i .e. the radiation offrequency range dv per unit volume, associated with a given infinitesimal solid angle dx relative to its direction of propagation, isgiven by

. * , (13)where Q dep end s only on th e frequency v, b u t no t on the direction.To this par t icular radiat ion there corresponds a par t icular radiat ionin K which is similarly characterised by a frequency range dv' and acertain solid angle dx'. The volume density of this part icular radiationis given by

d'e'( v',v')dv'-. (13')4n

This defines Q '. I t depends on the direction, which we shall define inthe usual way by means of the angle


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7 2 A . E I N S T E I N 1to the desired approximation,

v' = vll cos '= cosq> 1 cos2 ? ) , (16)c c

y , = y>. (17)With the same approximation, we have from (15)

v v l 1 H COSf j.Therefore, again to the same approximation,

Q(V)= gf^'H v' cosq>')or

Q(V)= e(v') + -J-(v' ) - v'cos?. . (18)OV c

Moreover , f rom (15) , (16) a n d (17) ,dv v- = 1 H cos


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ON TH E QU AN TU M TH EO RY OF RADIATION 73theory of the electromagnetic processes, the Doppler principle and theaberration law will at al l events remain preserved, and hence alsoeqs. (15)a n d (16).Fur th erm ore, the va l idi ty of the energy re la t ion (14)cer ta inly extends beyond wave theory; according to the theory ofrelat ivity, this transformation law also holds, e .g. , for the energyden sity of a mass mov ing with (almost) the velocity of l ight an d hav inginfinitesimally small rest density. Eq. (19) can therefore lay claim tobeing valid for any theory of radiation.

According to (B), the radiation associated with the solid angle dx'would give rise to B Q '(V', q>') dx'jAn elementary absorption processesof the type Zn^-Zm per second , if th e molecule were to be resto redto the s ta te Zn immediately after each such elementary process. Butin reali ty, the t ime for remaining in state Zn per second is equal toS-ipn exp(-SnlkT) from (5), wh ere th e abb reviation

S =pn exp(en/kT) + p m exp(sm/kT) (20)has been used. The number of such processes per second is thus really

^>Bexp(-a/ar)B^>') *For each such e lementary process a momentum [{emen)/c] cos -


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7 4 A . E I N S T E I N 1Here the effective frequency is again denoted by v (instead of v').But this expression represents the whole of the average momentum

transferred per unit t ime to a molecule moving with velocity v. For iti t clear that the elementary radiative emission processes, which takeplace in K' without interaction with the radiation field, have nopreferred direction, so that they cannot t r ansmi t any m om entum toth e m olecule, on a ve rage . T he final resu lt of ou r discussion is the reforeR = i k Q~ iv^)pB^[exV(-enlkT)][\ - exp(-hv/kT)]. (21)

6. C alcu latio n of I 5It is much simpler to calculate the effect of the irregularity of theelementary processes on the mechanical behaviour of the molecule,because the calculation can be based on a molecule at rest, to thedegree of approximation to which we had restr icted ourselves from theDeginning.

Consider an arbitrary event , causing a momentum transfer A to amolecule in the -X'-direction. This momentum can be assumed ofdifferent sign and magnitude in different cases. Nevertheless Aissupposed to satisfy a certain statistical law, such that i ts averagevalue vanishes. Now let fa, fa, . .. be the m om enta t ran sm it ted tothe molecule due to a number of mutually independent causes , sothat the to ta l momentum t ransfer A is given by

A = faThen, if the averages fa of the individual fa vanish,

A2 = S 4 (22)If the mean square values fa\ of the ind ividual m om en ta are allequal (A|=A2), and if I is the total number of events producing these

momenta , the re la t ionA* = lW (22a)

holds.According to our hypotheses , a momentum K={hvjc)coscp is t rans

ferred to the molecule for each absorption and emission process.Here , cp denotes the angle between the X-axis and a randomly chosen


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O NIHEQ UA N TU M TH EO RY OF RADIATION 75direction. One therefore obtains

F = J(Ai/cj*. (23)Since we assume that al l the elementary processes which occur canbe regard ed as m utu ally indepe nde nt events, w e are al lowed to use

(22a). T h e n I is the number of elementary events which occur intim e T. Th is is twice th e nu m be r of abso rption processes Zn-+Zmtak ing place in t ime T. W e therefore ha veI = PnB e x p ( -enlkT)QT (24)

and from (23), (24) and (22), = -^{) PnK e x p ( -sn/kT)Q . (25)

7. Conclus ionW e now ha ve to show tha t th e m om enta t ransferred f rom the radia t ionfield to the molecule according to our basic hypotheses, never disturbthe thermodynamic equil ibrium. For this , we need only insert thevalues for A \x and R determined by (25) and (21), after replacingin (21) the expression

(e-i -^-)[i-exP(-M^)]b y ghvf3kT, from (4). It is then seen immediately that our basicequat ion (12) is identically satisfied.

We have now completed the arguments which provide a s t rongsup po rt for th e hyp othese s sta ted in 2, concerning th e interact ionbetween matter and radiat ion by means of absorption and emissionprocesses, or in- or outgoing radiat ion. I was led to these hypothesesby m y endeavour to postula teforth e molecules, in th e simplest possiblemanner , a quantum-theoret ical behaviour that would be the analogueof the behav iour of a Plan ck reso nator in the classical theo ry. F rom th egeneral quantum assumption for mat ter , Bohr 's second postula te(eq. 9) as well as Planck's radiation formula followed in a natural way.

Most important , however, seems to me to be the result concerningth e m om entu m t ransfer to th e molecule due to th e absorpt ion a nd


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76 A. EINSTE IN 1emission of radiat ion. If one of our assumptions about the momentawere to be changed, a violat ion of eq. (12) would be produced; i tseems hardly possible to maintain agreement with this relat ion, imposed b y the theo ry of he at , oth er tha n on the basis of our a ssum ptions.The following statements can therefore be regarded as fairly certainlyproved.

If a radiat ion bundle has the effect that a molecule struck by i tabsorbs or emits a quanti ty of energy hv in the form of ra dia tion( ingoing radia t ion) , then a momentum hvjc is alw ays transferred tothe molecule. For an absorption of energy, this takes place in thedirection of propagation of the radiation bundle, for an emission inthe opposite direct ion. If the molecule is acted upon by severaldirectional radiation bundles, then it is always only a single one ofthese which part icipates in an elementary process of i rradiat ion; thisbundle a lone then determines the di rect ion of the momentum t ransferred to the molecule.

If the molecule undergoes a loss in energy of magnitude hv wi thoutexternal excitat ion, by emit t ing this energy in the form of radiat ion(outgoing radiat ion), then this process, too, is direct ional . Outgoingradiat ion in the form of spherical waves does not exist . During theelementary process of radiative loss, the molecule suffers a recoil ofma g n i t u d e hvjc in a direct ion w hich is only determ ined b y 'chan ce' ,according to the present state of the theory.

These propert ies of the elementary processes, imposed by eq. (12),make theformulation of a proper quantum theory of radiat ion appearalmost unavoidable. The weakness of the theory l ies on the one handin the fact that i t does not get us any closer to making the connectionwith wave theory; on the other , that i t leaves the durat ion and di rect ion of the elementary processes to 'chance' . Nevertheless I am fullyconfident that the approach chosen here is a rel iable one.

There is room for one further general remark. Almost all theoriesof thermal radiat ion are based on the study of the interact ion betweenradiation and molecules. But in general one restricts oneself to adiscussion of the energy exchange , wi thout t ak ing the momentumexchange into account. One feels easily justified in this, because thesmallness of the impulses transmitted by the radiat ion f ield impliesthat these can almost always be neglected in pract ice, when comparedwith other effects causing the motion. For a theoretical discussion,however, such small effects should be considered on a completely


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1 O N T H E Q U A N T U M T H E O R Y O F R A D I A T I O N 7 7equal footing with more conspicuous effects of a radiativeenergytransfer, since energy and momentum are linked in the closest possibleway. For this reason a theory can only be regarded as justified whenit is able to show that the impulses transmitted by the radiation fieldto matter lead to motions that are in accordance with the theoryof heat.

Related papersla A. Einstein, Vber einen die Erzeugung und Verwandlung des Lichtes betref-fendenheuristischenGesichtspunkt. Ann. d. Phys. 17 (1905) 132.lb A. Einstein,Zur Theorie der Lichterzeugungund Lichtabsorption.Ann. d. Ph ys .20 1906) 199.1c A. Einstein, Beitrdge zur Q udntentheorie. Verh. de r D . P hys ikal. Ges, 16(1914) 820.Id A. Einstein, Strahlungs-Emission und Absorption nach derQ uantentheorie.Verh. der D. Physikal. Ges. 18 (1916) 318.

Einstein - On the Theory of Radiation (1917)

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