Eight Lectures on Theoretical Physics/VI

by Max Planck

Sixth Lecture.

Heat Radiation. Statistical Theory.

Following the preparatory considerations of the last lecture we shall treat today the problem which we have come to recognize as one of the most important in the theory of heat radiation: the establishment of that universal function which governs the energy distribution in the normal spectrum. The means for the solution of this problem will be furnished us through the calculation of the entropy $S$ of a resonator placed in a vacuum filled with black radiation and thereby excited into stationary vibrations. Its energy $U$ is then connected with the corresponding specific intensity ${\mathfrak {K}}_{\nu }$ and its natural frequency $\nu$ in the radiation of the surrounding field through equation $(47)$:

${\begin{aligned}&(48){\color {White}.}\qquad &&{\mathfrak {K}}_{\nu }={\frac {\nu ^{2}}{c^{2}}}U.\end{aligned}}$

When $S$ is found as a function of $U$, the temperature $T$ of the resonator and that of the surrounding radiation will be given by:

${\begin{aligned}&(49){\color {White}.}\qquad &&{\frac {dS}{dU}}={\frac {1}{T}},\end{aligned}}$

and by elimination of $U$ from the last two equations, we then find the relationship among ${\mathfrak {K}}_{\nu }$, $T$ and $\nu$.

In order to find the entropy $S$ of the resonator we will utilize the general connection between entropy and probability, which we have extensively discussed in the previous lectures, and inquire then as to the existing probability that the vibrating resonator possesses the energy $U$. In accordance with what we have seen in connection with the elucidation of the second law through atomistic ideas, the second law is only applicable to a physical system when we consider the quantities which determine the state of the system as mean values of numerous disordered individual values, and the probability of a state is then equal to the number of the numerous, a priori equally probable, complexions which make possible the realization of the state. Accordingly, we have to consider the energy $U$ of a resonator placed in a stationary field of black radiation as a constant mean value of many disordered independent individual values, and this procedure agrees with the fact that every measurement of the intensity of heat radiation is extended over an enormous number of vibration periods. The entropy of a resonator is then to be calculated from the existing probability that the energy of the radiator possesses a definite mean value $U$ within a certain time interval.

In order to find this probability, we inquire next as to the existing probability that the resonator at any fixed time possesses a given energy, or in other words, that that point (the state point) which through its coordinates indicates the state of the resonator falls in a given “state domain.” At the conclusion of the third lecture we saw in general that this probability is simply measured through the magnitude of the corresponding state domain:

${\begin{aligned}&{\color {White}.(00)}\qquad &&\int d\varphi \cdot d\psi ,\end{aligned}}$

in case one employs as coordinates of state the general coordinate $\varphi$ and the corresponding momentum $\psi$. Now in general, the energy of the resonator, in accordance with $(40)$, is:

${\begin{aligned}&{\color {White}.(00)}\qquad &&U={\tfrac {1}{2}}Kf^{2}+{\tfrac {1}{2}}L{\dot {f}}^{2}.\end{aligned}}$

If we choose $f$ as the general coordinate $\varphi$ and put, therefore, $\varphi =f$, then the corresponding impulse $\psi$ is equal

${\begin{aligned}&{\color {White}.(00)}\qquad &&{\frac {\partial U}{\partial {\dot {f}}}}=L{\dot {f}},\end{aligned}}$

and the energy $U$ expressed as a function of $\varphi$ and $\psi$ is:

${\begin{aligned}&{\color {White}.(00)}\qquad &&U={\tfrac {1}{2}}K\varphi ^{2}+{\frac {1}{2}}{\frac {\psi ^{2}}{L}}.\end{aligned}}$

If now we desire to find the existing probability that the energy of a resonator shall lie between $U$ and $U+\Delta U$, we have to calculate the magnitude of that state domain in the $(\varphi ,\psi )$-plane which is bounded by the curves $U={\text{const.}}$ and $U+\Delta U={\text{const.}}$ These two curves are similar and similarly placed ellipses and the portion of surface bounded by them is equal to the difference of the areas of the two ellipses. The areas are respectively $U/\nu$ and $(U+\Delta U)/\nu$; consequently, the magnitude sought for the state domain is: $\Delta U/\nu .$ Let us now consider the whole state plane so divided into elementary portions by a large number of ellipses, such that the annular areas between consecutive ellipses are equal to each other; i. e., so that:

${\begin{aligned}&{\color {White}.(00)}\qquad &&{\frac {\Delta U}{\nu }}={\text{const}}=h.\end{aligned}}$

We thus obtain those portions $\Delta U$ of the energy which correspond to equal probabilities and which are therefore to be designated as the energy elements:

${\begin{aligned}&(50){\color {White}.}\qquad &&\epsilon =\Delta U=h\nu .\end{aligned}}$

If the determination of the elementary domains is effected in a manner quite similar to that employed in the kinetic gas theory, there exist, with respect to the relationships there found, very notable differences. In the first place, the state of the physical system considered here, the resonator, does not depend as there upon the coordinates and the velocities, but upon the energy only, and this circumstance necessitates that the entropy of a state depend, not upon the distribution of the state quantities $\varphi$ and $\psi$, but only upon the energy $U$. A further difference consists in this, that we have to do in the case of molecules with spacial mean values, but in the case of radiation with mean values as regards time. But this distinction may be disregarded when we reflect that the mean time value of the energy $U$ of a given resonator is obviously identical with the mean space value at a given instant of time of a great number $N$ of similar resonators distributed in the same stationary field of radiation. Of course these resonators must be placed sufficiently far apart in order not directly to influence one another. Then the total energy of all the resonators:

${\begin{aligned}&(51){\color {White}.}\qquad &&U_{N}=NU\end{aligned}}$

is quite irregularly distributed among all the individual resonators, and we have referred back the disorder as regards time to a disorder as regards space.

We are now concerned with the probability $W$ of the state determined by the energy $U_{N}$ of the $N$ resonators placed in the same stationary field of radiation; i. e., with the number of individual arrangements or complexions which correspond to the distribution of energy $U_{N}$ among the $N$ resonators. With this in view, we subdivide the given total energy $U_{N}$ into its elements $\epsilon$ so that:

${\begin{aligned}&(52){\color {White}.}\qquad &&U_{N}=P\epsilon .\end{aligned}}$

These $P$ energy elements are to be distributed in every possible manner among the $N$ resonators. Let us consider, then, the $N$ resonators to be numbered and the figures written beside one another in a series, and in such manner that the number of times each figure appears is equal to the number of energy elements which fall upon the corresponding resonator. Then we obtain through such a number series a representation of a fixed complexion, in which with each individual resonator there is associated a definite energy. For example, if there are $N=4$ resonators and $P=6$ energy elements present, then one of the possible complexions is represented by the number series

${\begin{aligned}&{\color {White}.(00)}\qquad &&1\quad 1\quad 3\quad 3\quad 3\quad 4\end{aligned}}$

which asserts that the first resonator contains two, the second $0$, the third $3$, and the fourth $1$ energy element. The totality of numbers in the series is $6$, equal to the number of the energy elements present. The arrangement of figures in the series is immaterial for any complexion, since the mere interchange of figures does not change the energy of a given resonator. The number of all the possible different complexions is therefore equal to the number of possible “combinations with repetition” of $4$ elements with $6$ classes:

${\begin{aligned}&{\color {White}.(00)}\qquad &&W={\frac {(4+6-1)!}{(4-1)!\;6!}}={\frac {9!}{3!\;6!}}=84,\end{aligned}}$

or, in our general case the probability sought is:

${\begin{aligned}&{\color {White}.(00)}\qquad &&W={\frac {(N+P-1)!}{(N-1)!\;P!}}.\end{aligned}}$

We obtain, therefore, for the entropy $S_{N}$ of the resonator system, in accordance with equation $(12)$, since $N$ and $P$ are large numbers,

${\begin{aligned}&{\color {White}.(00)}\qquad &&S_{N}=k\log {\frac {(N+P)!}{N!\;P!}}\end{aligned}}$

and with the aid of Sterling's formula $(16)$:

${\begin{aligned}&{\color {White}.(00)}\qquad &&S_{N}=k\{(N+P)\log(N+P)-N\log N-P\log P\}.\end{aligned}}$

If, in accordance with $(52)$, we now write $U_{N}/\epsilon$ for $P$, $NU$ for $U_{N}$ in accordance with $(51)$, and $h\nu$ for $\epsilon$, in accordance with $(50)$, we obtain, after an easy transformation, for the mean entropy of a single resonator:

${\begin{aligned}&{\color {White}.(00)}\qquad &&{\frac {S_{N}}{N}}=S=k\left\{\left(1+{\frac {U}{h\nu }}\right)\log \left(1+{\frac {U}{h\nu }}\right)-{\frac {U}{h\nu }}\log {\frac {U}{h\nu }}\right\}\end{aligned}}$

as the solution of the problem in hand.

We will now introduce the temperature $T$ of the resonator, and will express through $T$ the energy $U$ of the resonator and also the intensity ${\mathfrak {K}}_{\nu }$ of the heat radiation related to it through a stationary state of energy exchange. For this purpose we utilize equation $(49)$ and obtain then for the energy of the resonator:

${\begin{aligned}&{\color {White}.(00)}\qquad &&U={\frac {h\nu }{e^{h\nu /kT}-1}}.\end{aligned}}$

It is to be observed that we have not here to do with a uniform distribution of energy (cf. fourth lecture) among the various resonators.

For the specific intensity of the monochromatic plane polarized ray of frequency $\nu$, we have, in accordance with $(48)$:

${\begin{aligned}&(53){\color {White}.}\qquad &&{\mathfrak {K}}_{\nu }={\frac {h\nu ^{3}}{c^{2}}}\cdot {\frac {1}{e^{h\nu /kT}-1}}.\end{aligned}}$

This expression furnishes for each temperature $T$ the energy distribution in the normal spectrum of a black body. A comparison with equation $(38)$ of the last lecture furnishes us then with the universal function:

${\begin{aligned}&{\color {White}.(00)}\qquad &&F(\nu ,T)={\frac {h\nu ^{3}}{e^{h\nu /kT}-1}}.\end{aligned}}$

If we refer the specific intensity of a monochromatic ray, not to the frequency $\nu$, but, as is commonly done in experimental physics, to the wave length $\lambda$, then, since between the absolute values of $d\nu$ and $d\lambda$ the relation exists:

${\begin{aligned}&{\color {White}.(00)}\qquad &&|d\nu |={\frac {c\cdot |d\lambda |}{\lambda ^{2}}},\end{aligned}}$

we obtain from

${\begin{aligned}&{\color {White}.(00)}\qquad &&E_{\lambda }|d\lambda |={\mathfrak {K}}_{\nu }|d\nu |,\end{aligned}}$

the relation:

${\begin{aligned}&(54){\color {White}.}\qquad &&E_{\lambda }={\frac {c^{2}h}{\lambda ^{5}}}\cdot {\frac {1}{e^{ch/k\lambda T}-1}}\end{aligned}}$

as the intensity of a monochromatic plane polarized ray of wave length $\lambda$ is emitted normally to the surface of a black body in a vacuum at temperature $T$. For small values of $\lambda T$ $(54)$ reduces to:

${\begin{aligned}&(55){\color {White}.}\qquad &&E_{\lambda }={\frac {c^{2}h}{\lambda ^{5}}}\cdot e^{-(ch/k\lambda T)},\end{aligned}}$

which expresses Wien's Displacement Law. For large values of $\lambda T$ on the other hand, there results from $(54)$:

${\begin{aligned}&(56){\color {White}.}\qquad &&E_{\lambda }={\frac {ckT}{\lambda ^{4}}},\end{aligned}}$

a relation first established by Lord Rayleigh and which we may here designate as the Rayleigh Law of Radiation.

From equation $(30)$, taking account of $(53)$, we obtain for the space density of black radiation in a vacuum:

${\begin{aligned}&{\color {White}.(00)}\qquad &&\epsilon ={\frac {48\pi h}{c^{3}}}\left({\frac {kT}{h}}\right)^{4}\cdot \alpha =aT^{4},\end{aligned}}$

wherein

${\begin{aligned}&{\color {White}.(00)}\qquad &&\alpha =1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots =1.0823.\end{aligned}}$

The Stefan-Boltzmann law is hereby expressed. In accordance with the measurements of Kurlbaum, we have the constant

${\begin{aligned}&{\color {White}.(00)}\qquad &&a={\frac {48\pi k^{4}}{c^{3}h^{3}}}\cdot \alpha =7.061\cdot 10^{-15}{\frac {\text{erg}}{{\text{cm}}^{3}{\text{deg}}^{4}}}.\end{aligned}}$

For that wave length $\lambda _{m}$ which corresponds in the spectrum of black radiation to the maximum intensity of radiation $E_{\lambda }$ we have from equation $(54)$:

${\begin{aligned}&{\color {White}.(00)}\qquad &&\left({\frac {dE_{\lambda }}{d\lambda }}\right)_{\lambda =\lambda _{m}}=0.\end{aligned}}$

Carrying out the differentiation, we get, after putting for brevity:

${\begin{aligned}&{\color {White}.(00)}\qquad &&{\frac {ch}{k\lambda _{m}T}}=\beta ,\quad e^{-\beta }+{\frac {\beta }{5}}-1=0.\end{aligned}}$

The root of this transcendental equation is:

${\begin{aligned}&{\color {White}.(00)}\qquad &&\beta =4.9651;\end{aligned}}$

and $\lambda _{m}T=ch/k\beta =b$ is a constant (Wien's Displacement Law). In accordance with the measurements of O. Lummer and E. Pringsheim,

${\begin{aligned}&{\color {White}.(00)}\qquad &&b=0.294\ {\text{cm}}\cdot {\text{deg}}.\end{aligned}}$

From this there follow the numerical values

${\begin{aligned}&{\color {White}.(00)}\qquad &&k=1.346\cdot 10^{-16}{\frac {\text{erg}}{\text{deg}}},\quad {\text{and}}\quad h=6.548\cdot 10^{-27}{\text{erg}}\cdot {\text{sec}}.\end{aligned}}$

The value found for $k$ easily permits of the specification numerically, in the C.G.S. system, of the general connection between entropy and probability, as expressed through the universal equation $(12)$. Thus, quite in general, the entropy of a physical system is:

${\begin{aligned}&{\color {White}.(00)}\qquad &&S=1.346\cdot 10^{-16}{\overset {e}{\log }}W.\end{aligned}}$

In the application to the kinetic gas theory we obtain from equation $(24)$ for the ratio of the molecular mass to the mol mass:

${\begin{aligned}&{\color {White}.(00)}\qquad &&\omega ={\frac {k}{R}}=1.62\cdot 10^{-24},\end{aligned}}$

i. e., to one mol there corresponds $1/\omega =6.175\cdot 10^{23}$ molecules, where it is supposed that the mol of oxygen

${\begin{aligned}&{\color {White}.(00)}\qquad &&O_{2}=32{\text{g}}.\end{aligned}}$

Accordingly, the number of molecules contained in $1$ cu. cm. of an ideal gas at $0^{\circ }$ Cels. and at atmospheric pressure is:

${\begin{aligned}&{\color {White}.(00)}\qquad &&N=2.76\cdot 10^{19}.\end{aligned}}$

The mean kinetic energy of the progressive motion of a molecule at the absolute temperature $T=1$ in the absolute C.G.S. system, in accordance with $(27)$, is:

${\begin{aligned}&{\color {White}.(00)}\qquad &&L={\tfrac {3}{2}}k=2.02\cdot 10^{-16}.\end{aligned}}$

In general, the mean kinetic energy of progressive motion of a molecule is expressed by the product of this number and the absolute temperature $T$.

The elementary quantum of electricity, or the free electric charge of a monovalent ion or electron, in electrostatic measure is:

${\begin{aligned}&{\color {White}.(00)}\qquad &&e=\omega \cdot 9658\cdot 3\cdot 10^{10}=4.69\cdot 10^{-10}.\end{aligned}}$

This result stands in noteworthy agreement with the results of the latest direct measurements of the electric elementary quantum made by E. Rutherford and H. Geiger, and E. Regener.—

Even if the radiation formula $(54)$ here derived had shown itself as valid with respect to all previous tests, the theory would still require an extension as regards a certain point; for in it the physical meaning of the universal constant $h$ remains quite unexplained. All previous attempts to derive a radiation formula upon the basis of the known laws of electron theory, among which the theory of J. H. Jeans is to be considered as the most general and exact, have led to the conclusion that $h$ is infinitely small, so that, therefore, the radiation formula of Rayleigh possesses general validity, but, in my opinion, there can be no doubt that this formula loses its validity for short waves, and that the pains which Jeans has taken to place^[1] the blame for the contradiction between theory and experiment upon the latter are unwarranted.

Consequently, there remains only the one conclusion, that previous electron theories suffer from an essential incompleteness which demands a modification, but how deeply this modification should go into the structure of the theory is a question upon which views are still widely divergent. J. J. Thompson inclines to the most radical view, as do J. Larmor, A. Einstein, and with him I. Stark, who even believe that the propagation of electromagnetic waves in a pure vacuum does not occur precisely in accordance with the Maxwellian field equations, but in definite energy quanta $h\nu$. I am of the opinion, on the other hand, that at present it is not necessary to proceed in so revolutionary a manner, and that one may come successfully through by seeking the significance of the energy quantum $h\nu$ solely in the mutual actions with which the resonators influence one another.^[2] A definite decision with regard to these important questions can only be brought about as a result of further experience.

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1. In that the walls used in the measurements of hollow space radiations must be diathermanous for the shortest waves.
2. It is my intention to give a complete presentation of these relations in Volume 31 of the Annalen der Physik.