The Theory of Heat Radiation (1914) — part 9 of 12

result. It is therefore useful to have a method which leads directly to the expression for the entropy of a system in the state of thermodynamic equilibrium, without requiring any considera tion of the state of thermodynamic equilibrium. This method is based on an important general property of the thermodynamic probability of a state of equilibrium.

We know that there exists between the entropy S and the ther modynamic probability W in any state whatever the general relation (164). In the state of thermodynamic equilibrium both quantities have maximum values; hence, if we denote the maxi mum values by a suitable index:

Sm = k \ogWm. (229)

It follows from the two equations that :

Wm *s^

— = e k W

Now, when the deviation from thermodynamic equilibrium is at

Cf Cf

all appreciable, — ^— — is certainly a very large number. Accord-

K

144

DIRECT CALCULATION OF THE ENTROPY 145

ingly Wm is not only large but of a very high order large, cam- pared with W, that is to say: The thermodynamic probability of the state of equilibrium is enormously large compared with the thermodynamic probability of all states which, in the course of time, change into the state of equilibrium.

This proposition leads to the possibility of calculating Wm with an accuracy quite sufficient for the determination of Sm, without the necessity of introducing the special condition of equilibrium. According to Sec. 123, et seq., Wm is equal to the number of all different complexions possible in the state of thermo dynamic equilibrium. This number is so enormously large com pared with the number of complexions of all states deviating from equilibrium that we commit no appreciable error if we think of the number of complexions of all states, which as time goes on change into the state of equilibrium, i.e., all states which are at all possible under the given external conditions, as being included in this number. The total number of all possible complexions may be calculated much more readily and directly than the number of complexions referring to the state of equilibrium only.

142. We shall now use the method just formulated to calculate the entropy, in the state of equilibrium, of the system of ideal linear oscillators considered in the last chapter, when the total energy E is given. The notation remains the same as above.

We put then Wm equal to the number of complexions of all states which are at all possible with the given energy E of the system. Then according to (219) we have the condition:

(230)

Whereas we have so far been dealing with the number of complex ions with given Nnj now the Nn are also to be varied in all ways consistent with the condition (230).

The total number of all complexions is obtained in a simple way by the following consideration. We write, according to (165), the condition (230) in the following form:

CO

E__N_ hv~ 2 10

146 ENTROPY AND PROBABILITY

or

: -!,-£-*

P is a given large positive number, which may, without restricting the generality, be taken as an integer.

According to Sec. 123 a complexion is a definite assignment of every individual oscillator to a definite region element 1, 2, 3, ..... of the state plane (/, ^). Hence we may charac terize a certain complexion by thinking of the N oscillators as being numbered from 1 to TV and, when an oscillator is assigned to the nth region element, writing down the number of the oscillator (n— 1) times. If in any complexion an oscillator is assigned to the first region element its number is not put down at all. Thus every complexion gives a certain row of figures, and vice versa to every row of figures there corresponds a certain com plexion. The position of the figures in the row is immaterial.

What makes this form of representation useful is the fact that according to (231) the number of figures in such a row is always equal to P. Hence we have "combinations with repetitions of N elements taken P at a time," whose total number is

N(N+l)(N+2) ..... _

12 3 ..... P (N-1)\P\ ( ?"

If for example we had N = 3 and P = 4 all possible complexions would be represented by the rows of figures:

1111 1133 2222

1112 1222 2223

1113 1223 2233

1122 1233 2333

1123 1333 3333

The first row denotes that complexion in which the first oscil lator lies in the 5th region element and the two others in the first. The number of complexions in this case is 15, in agreement with the formula.

143. For the entropy S of the system of oscillators which is

DIRECT CALCULATION OF THE ENTROPY 147

in the state of thermodynamic equilibrium we thus obtain from equation (229) since N and P are large numbers :

and by making use of Stirling's formula (171) l

P , iP \ P. P

If we now replace P by E from (231) we find for the entropy exactly the same value as given by (222) and thus we have demonstrated in a special case both the admissibility and the practical usefulness of the method employed.2

1 Compare footnote to page 124. See also page 218.

2 A complete mathematical discussion of the subject of this chapter has been given by H. A. Lorentz. Compare, e. g., Nature, 92, p. 305, Nov. 6, 1913. (Tr.)

PART IV

SYSTEM OF OSCILLATORS IN A STATION ARY FIELD OF RADIATION

CHAPTER I

THE ELEMENTARY DYNAMICAL LAW FOR THE

VIBRATIONS OF AN IDEAL OSCILLATOR.

HYPOTHESIS OF EMISSION OF QUANTA

144. All that precedes has been by way of preparation. Before taking the final step, which will lead to the law of distribution of energy in the spectrum of black radiation, let us briefly put together the essentials of the problem still to be solved. As we have already seen in Sec. 93, the whole problem amounts to the determination of the temperature corresponding to a mono chromatic radiation of given intensity. For among all conceiv able distributions of energy the normal one, that is, the one peculiar to black radiation, is characterized by the fact that in it the rays of all frequencies have the same temperature. But the temperature of a radiation cannot be determined unless it be brought into thermodynamic equilibrium with a system of mole cules or oscillators, the temperature of which is known from other sources. For if we did not consider any emitting and absorbing matter there would be no possibility of defining the entropy and temperature of the radiation, and the simple propagation of free radiation would be a reversible process, in which the entropy and temperature of the separate pencils would not undergo any change. (Compare below Sec. 166.)

Now we have deduced in the preceding section all the charac teristic properties of the thermodynamic equilibrium of a system of ideal oscillators. Hence, if we succeed in indicating a state of radiation which is in thermodynamic equilibrium with the system of oscillators, the temperature of the radiation can be no other *than that of the oscillators, and therewith the problem is solved.

145. Accordingly we now return to the considerations of Sec. 135 and assume a system of ideal linear oscillators in a stationary field of radiation. In order to make progress along the line proposed, it is necessary to know the elementary dynamical law,

151

152 A SYSTEM OF OSCILLATORS

according to which the mutual action between an oscillator and the incident radiation takes place, and it is moreover easy to see that this law cannot be the same as the one which the classical electro- dynamical theory postulates for the vibrations of a linear Hertzian oscillator. For, according to this law, all the oscillators, when placed in a stationary field of radiation, would, since their properties are exactly similar, assume the same energy of vibra tion, if we disregard certain irregular variations, which, however, will be smaller, the smaller we assume the damping constant of the oscillators, that is, the more pronounced their natural vibra tion is. This, however, is in direct contradiction to the definite discrete values of the distribution densities Wi, w^

ios, which we have found in Sec. 139 for the stationary

state of the system of oscillators. ' The latter allows us to conclude with certainty that in the dynamical law to be established the quantity element of action h must play a characteristic part. Of what nature this will be cannot be predicted a priori; this much, however, is certain, that the only type of dynamical law admis sible is one that will give for the stationary state of the oscillators exactly the distribution densities w calculated previously. It is in this problem that the question of the dynamical significance of the quantum of action h stands for the first time in the foreground, a question the answer to which was unnecessary for the calcula tions of the preceding sections, and this is the principal reason why in our treatment the preceding section was taken up first. 146. In establishing the dynamical law, it will be rational to proceed in such a way as to make the deviation from the laws of classical electrodynamics, which was recognized as necessary, as slight as possible. Hence, as regards the influence of the field of radiation on an oscillator, we follow that theory closely. If the oscillator vibrates under the influence of any external electro magnetic field whatever, its energy U will not in general remain constant, but the energy equation (205 a) must be extended to include the work which the external electromagnetic field does on the oscillator, and, if the axis of the electric doublet coincides with the z-axis, this work is expressed by the term Ezdf=Ezfdt. Here E2 denotes the z component of the external electric field- strength at the position of the oscillator, that is, that electric field-strength which would exist at the position of the oscillator,

THE ELEMENTARY DYNAMICAL LAW 153

if the latter were not there at all. The other components of the external field have no influence on the vibrations of the oscillator.

Hence the complete energy equation reads:

Kfdf+Lfdf=Ezdf

or: Kf+Lf=E., (233)

and the energy absorbed by the oscillator during the time element eft is:

E,fdt (234)

147. While the oscillator is absorbing it must also be emitting? for otherwise a stationary state would be impossible. Now, since in the law of absorption just assumed the hypothesis of quanta has as yet found no room, it follows that it must come into play in some way or other in the emission of the oscillator, and this is provided for by the introduction of the hypothesis of emission of quanta. That is to say, we shall assume that the emission does not take place continuously, as does the absorption, but that it occurs only at certain definite times, suddenly, in pulses, and in particular we assume that an oscillator can emit energy only at the moment when its energy of vibration, U, is an integral mul tiple n of the quantum of energy, e = hv. Whether it then really emits or whether its energy of vibration increases further by absorption will be regarded as a matter of chance. This will not be regarded as implying that there is no causality for emission; but the processes which cause the emission will be assumed to be of such a concealed nature that for the present their laws cannot be obtained by any but statistical methods. Such an assumption is not at all foreign to physics; it is, e.g., made in the atomistic theory of chemical reactions and the disintegration theory of radioactive substances.

It will be assumed, however, that if emission does take place, the entire energy of vibration, U, is emitted, so that the vibration of the oscillator decreases to zero and then increases again by further absorption of radiant energy.

It now remains to fix the law which gives the probability that an oscillator will or will not emit at an instant when its energy has reached an integral multiple of e. For it is evident that the sta tistical state of equilibrium, established in the system of oscil-

154 A SYSTEM OF OSCILLATORS

lators by the assumed alternations of absorption and emission will depend on this law; and evidently the mean energy U of the oscillators will be larger, the larger the probability that in such a critical state no emission takes place. On the other hand, since the mean energy U will be larger, the larger the intensity of the field of radiation surrounding the oscillators, we shall state the law of emission as follows: The ratio of the probability that no emission takes place to the probability that emission does take place is proportional to the intensity I of the vibration which excites the oscillator and which was defined in equation (158). The value of the constant of proportionality we shall determine later on by the application of the theory to the special case in which the energy of vibration is very large. For in this case, as we know, the familiar formulae of the classical dynamics hold for any period of the oscillator whatever, since the quantity element of action h may then, without any appreciable error, be regarded as infinitely small.

These statements define completely the way in which the radiation processes considered take place, as time goes on, and the properties of the stationary state. We shall now, in the first place, consider in the second chapter the absorption, and, then, in the third chapter the emission and the stationary dis tribution of energy, and, lastly, in the fourth chapter we shall compare the stationary state of the system of oscillators thus found with the thermodynamic state of equilibrium which was derived directly from the hypothesis of quanta in the preceding part. If we find them to agree, the hypothesis of emission of quanta may be regarded as admissible.

It is true that we shall not thereby prove that this hypothesis represents the only possible or even the most adequate expression of the elementary dynamical law of the vibrations of the oscilla tors. On the contrary I think it very probable that it may be greatly improved as regards form and contents. There is, how ever, no method of testing its admissibility except by the investi gation of its consequences, and as long as no contradiction in itself or with experiment is discovered in it, and as long as no more adequate hypothesis can be advanced to replace it, it may justly claim a certain importance.

CHAPTER II ABSORBED ENERGY

148. Let us consider an oscillator which has just completed an emission and which has, accordingly, lost all its energy of vibra tion. If we reckon the time t from this instant then f or t = 0 we have/=0 and df/dt = Q, and the vibration takes place according to equation (233). Let us write E2 as in (149) in the form of a Fourier's series:

. 2irnt An cos -- +Bn sin ~ (235)

where T may be chosen very large, so that for all times t consid ered £<T. Since we assume the radiation to be stationary, the constant coefficients An and Bn depend on the ordinal num bers n in a wholly irregular way, according to the hypothesis of natural radiation (Sec. 117). The partial vibration with the ordinal number n has the frequency v, where

(236)

while for the frequency v0 of the natural period of the oscillator

Taking the initial condition into account, we now obtain as the solution of the differential equation (233) the expression

CO

/= ^/i [an(cos ut — cos o>0Q+6B(sin co£ -- sin co02)L (237)

i

where

A" 6> = Bi (238)

L(0)0 * — b)2) L(Uo — U)

155

156 A SYSTEM OF OSCILLATORS

This represents the vibration of the oscillator up to the instant when the next emission occurs.

The coefficients an and bn attain their largest values when co is nearly equal to co0. (The case co = co0 may be excluded by assuming at the outset that v0T is not an integer.)

149. Let us now calculate the total energy which is absorbed by the oscillator in the time from t = 0 to t = r, where

o)0 r is large. (239)

According to equation (234), it is given by the integral

(240)

the value of which may be obtained from the known expression for Ez (235) and from

00

•' =^j[an( — co sin co£+co0sin u0t)+bn(u cos cot — co cos «0OL (241) i

By multiplying out, substituting for an and bn their values from (238), and leaving off all terms resulting from the multiplication of two constants An and Bn, this gives for the absorbed energy the following value:

1 C ^^ \ An2

— I at > — cos co£( — co sin co£ + co0 sm co0 y-j-

L J <^-* | co02 — co2

o 1

~ry 2 * ~|

• sin co£(co cos co^ — co cos u0t) . (24 la)

C002— CO2 J

In this expression the integration with respect to t may be per formed term by term. Substituting the limits r and 0 it gives

1^ An2 [" shvW L ! co02-co2L 2

co0+co co0— co

ABSORBED ENERGY 157

In order to separate the terms of different order of magnitude, this expression is to be transformed in such a way that the difference co0 — co will appear in all terms of the sum. This gives

An2 C00— CO C00 . C00 — CO . COo + 3cO

• sin2coH — - sin - T- sin- — - — T L^co02-co2l2(co0+co) coo+co 2 2

i

C00 COo — CC

+- - sm 9T

C00 — CO A

co0 — co

r sm2 cor

CO C00— CO . C00 + 3cO CO C00— CO

• sin - —r • sm^— — r-{— - sm2 — - — r • co.+co 2 2 co0-co 2 ]

The summation with respect to the ordinal numbers n of the Fourier's series may now be performed. Since the fundamental period T of the series is extremely large, there corresponds to the difference of two consecutive ordinal numbers, An = l only a very small difference of the corresponding values of co, dco, namely, according to (236),

&n = l = jdv==1'd") (242)

2?r

and the summation with respect to n becomes an integration with respect to co.

The last summation with respect to An may be rearranged as the sum of three series, whose orders of magnitude we shall first compare. So long as only the order is under discussion we may disregard the variability of the An2 and need only compare the three integrals

I

sm2 cor

co0 co0 — co

sin - T ' sin

I C0(co0 + co)2(co0-co)^ 2 and

158 A SYSTEM OF OSCILLATORS

The evaluation of these integrals is greatly simplified by the fact that, according to (239), COOT and therefore also cor are large num bers, at least for all values of co which have to be considered. Hence it is possible to replace the expression sin2cor in the integral Ji by its mean value \ and thus we obtain:

Ji=r

4co0

It is readily seen that, on account of the last factor, we obtain

for the second integral.

In order finally to calculate the third integral J3 we shall lay off in the series of values of co on both sides of co0 an interval extending from coi(<co0) to co2(>co0) such that

C00— COi C02— C00

• and — - are small, (243)

C00 C00

and simultaneously

(co0 — COI)T and (co2 — COO)T are large. (244)

This can always be done, since co0r is large. If we now break up the integral «/3 into three parts, as follows:

it is seen that in the first and third partial integral the expression

COo — CO

sin2 --- r may, because of the condition (244), be replaced by its

^

mean value J. Then the two partial integrals become:

OJl OO

/Updu _ C _ Uodu __

2(co0+co)(co0-co)2 £ J 2(co0+co)(co0-co)2'

O 002

These are certainly smaller than the integrals :

CO

da) C da)

2U^o)~2 an

ABSORBED ENERGY 159

which have the values

1 - -£- and — (246)

2 co0(co0 — coi) 2(co2 — co0J

respectively. We must now consider the middle one of the three partial integrals:

W2

C00 C00 — CO

dco/ T • sin2 —T.

Because of condition (243) wTe may write instead of this:

. n co0— co

dco

2(co0-co)2

Wl

and by introducing the variable of integration x, where

co — co0

X = T

and taking account of condition (244) for the limits of the integral, we get:

• 00

T C sin2 x dx T 4 J ~~x*~ =47r'

— CO

This expression is of a higher order of magnitude than the expres sions (246) and hence of still higher order than the partial inte grals (245) and the integrals Ji and J2 given above. Thus for our calculation only those values of co will contribute an appre ciable part which lie in the interval between coi and co2, and hence we may, because of (243), replace the separate coefficients Anz and Bn2 in the expression for the total absorbed energy by their mean values A02 and B02 in the neighborhood of co0 and thus, by taking account of (242), we shall finally obtain for the total value of the energy absorbed by the oscillator in the time r:

l--(A0*+B0*) T (247)

LJ O

If we now, as in (158), define I, the "intensity of the vibration

160 A SYSTEM OF OSCILLATORS

exciting the oscillator," by spectral resolution of the mean value of the square of the exciting field-strength Ez:

l,.d* (248)

we obtain from (235) and (242) :

i and by comparison with (248) :

Accordingly from (247) the energy absorbed in the time r be comes :

that is, in the time between two successive emissions, the energy U of the oscillator increases uniformly with the time, according to the law

?-i- (249)

dt 4L Hence the energy absorbed by all N oscillators in the time dt is:

N\

-~dt = Nadt. (250)

4L

CHAPTER III EMITTED ENERGY. STATIONARY STATE

150. Whereas the absorption of radiation by an oscillator takes place in a perfectly continuous way, so that the energy of the oscillator increases continuously and at a constant rate, for its emission we have, in accordance with Sec. 147, the following law: The oscillator emits in irregular intervals, subject to the laws of chance; it emits, however, only at a moment when its energy of vibration is just equal to an integral multiple n of the elementary quantum e = hv, and then it always emits its whole energy of vibration ne.

We may represent the whole process by the following figure in which the abscissae represent the time t and the ordinates the energy

, (p<e) (251)

FIG. 7.

of a definite oscillator under consideration. The oblique parallel lines indicate the continuous increase of energy at a constant rate.

dU dp

,;=:£=«> (252)

n

161

162 A SYSTEM OF OSCILLATORS

which is, according to (249), caused by absorption at a constant rate. Whenever this straight line intersects one of the parallels to the axis of abscissae U — e, U = 2e, ..... emission may possibly take place, in which case the curve drops down to zero at that point and immediately begins to rise again.

151. Let us now calculate the most important properties of the state of statistical equilibrium thus produced. Of the N oscillators situated in the field of radiation the number of those whose energy at the time t lies in the interval between U = ne-}-p and U--dU = ne--p--dp may be represented by

NRntpdp, (253)

where R depends in a definite way on the integer n and the quan tity p which varies continuously between 0 and e.

dp

After a time dt = — all the oscillators will have their energy in- a

creased by dp and hence they will all now lie outside of the energy interval considered. On the other hand, during the same time dt, all oscillators whose energy at the time t was between ne--p — dp and ne+p will have entered that interval. The number of all these oscillators is, according to the notation used above,

NRn, p_dpdp. (254)

Hence this expression gives the number of oscillators which are at the time t+dt in the interval considered.

Now, since we assume our system to be in a state of statistical equilibrium, the distribution of energy is independent of the time and hence the expressions (253) and (254) are equal, i.e.,

Rn, -d = Rn, =Rn> (255)

Thus Rn does not depend on p.

This consideration must, however, be modified for the special case in which p = 0. For, in that case, of the oscillators, N = Rn-\dp in number, whose energy at the time t was between

dp

ne and ne — dp, during the time dt = — some enter into the energy

a

interval (from U = ne to U+dU = ne--dp) considered; but all of them do not necessarily enter, for an oscillator may possibly emit all its energy on passing through the value U = ne. If the proba-

EMITTED ENERGY. STATIONARY STATE 163

bility that emission takes place be denoted by 17 ( < 1) the number of oscillators which pass through the critical value without emitting will be

NRn-i(l-tidp, (256)

and by equating (256) and (253) it follows that

Rn = Rn-i(]- —i|)j and hence, by successive reduction,

Rn = Ro(l-tin. (257)

To calculate R0 we repeat the above process for the special case when n = 0 and p = 0. In this case the energy interval in question extends from [7 = 0 to dU = dp. Into this interval enter in the

dp

time dt = — all the oscillators which perform an emission during a

this time, namely, those whose energy at the time t was between

e — dp and e, 2e — dp and 2e, 3e — dp and 3e

The numbers of these oscillators are respectively

NR0dp, NRidp, NR2dp,

hence their sum multiplied by r/ gives the desired number of emitting oscillators, namely,

Nr](R0+Rl+R2+ ) dp, (258)

and this number is equal to that of the oscillators in the energy interval between 0 and dp at the time t--dt, which is NR0dp. Hence it follows that

R0 = ri(Ro+Ri+R2+ ). (259)

Now, according to (253), the whole number of all the oscillators is obtained by integrating with respect to p from 0 to e, and summing up with respect to n from 0 to °° . Thus

N=N'^J Rn» dp = N 2 Rn€ (260)

n = 0 o

and

2Rn = -- (261)

€

Hence we get from (257) and (259)

K» = ~ (l->?)n. (262)

164 A SYSTEM OF OSCILLATORS

152. The total energy emitted in the time element dt = —

a

is found from (258) by considering that every emitting oscillator expends all its energy of vibration and is

..... )€

= N dp = Nadt.

It is therefore equal to the energy absorbed in the same time by all oscillators (250), as is necessary, since the state is one of statistical equilibrium.

Let us now consider the mean energy U of an oscillator. It is evidently given by the following relation, which is derived in the same way as (260) :

J

. (263)

From this it follows by means of (262), that

hv

Since rj <1, U lies between -— and oo . Indeed, it is immediately

ft

— hv

evident that U can never become less than — since the energy

&

of every oscillator, however small it may be, will assume the value e = hv within a time limit, which can be definitely stated.

153. The probability constant rj contained in the formulae for the stationary state is determined by the law of emission enun ciated in Sec. 147. According to this, the ratio of the probability that no emission takes place to the probability that emission does take place is proportional to the intensity I of the vibration exciting the oscillator, and hence

— = pl (265)

"n

where the constant of proportionality is to be determined in

EMITTED ENERGY. STATIONARY STATE 165

such a way that for very large energies of vibration the familiar formulae of classical dynamics shall hold.

Now, according to (264), rj becomes small for large values of U and for this special case the equations (264) and (265) give

and the energy emitted or absorbed respectively in the time dt by all N oscillators becomes, according to (250),

N\ NU

-- dt = ~T -r-dt. (266)

4L 4Lphv

On the other hand, H. Hertz has already calculated from Maxwell's theory the energy emitted by a linear oscillator vibrating periodically. For the energy emitted in the time of one-half of one vibration he gives the expression1

3X3

where X denotes half the wave length, and the product El (the C of our notation) denotes the amplitude of the moment / (Sec. 135) of the vibrations. This gives for the energy emitted in the time of a whole vibration

167T4C2

3X3

where X denotes the whole wave length, and for the energy emitted by N similar oscillators in the time dt

since X = — On introducing into this expression the energy U of v

an oscillator from (205), (207), and (208), namely

we have for the energy emitted by the system of oscillators

SrVtf

N -i* (267)

H. Hertz, Wied. Ann. 36, p. 12, 1889.

166 A SYSTEM OF OSCILLATORS

and by equating the expressions (266) and (267) we find for the factor of proportionality p

3c3

(268)

154. By the determination of p the question regarding the properties of the state of statistical equilibrium between the system of the oscillators and the vibration exciting them receives a general answer. For from (265) we get

1+pl

and further from (262)

Hence in the state of stationary equilibrium the number of oscillators whose energy lies between nhv and (n--l)hv is, from equation (253),

N J Rndp = NRnt = N^~^^ (270)

where n = Q, 1, 2, 3,

CHAPTER IV

THE LAW OF THE NORMAL DISTRIBUTION OF ENERGY. ELEMENTARY QUANTA OF MATTER AND ELECTRICITY

155. In the preceding chapter we have made ourselves familiar with all the details of a system of oscillators exposed to uniform radiation. We may now develop the idea put forth at the end of Sec. 144. That is to say, we may identify the stationary state of the oscillators just found with the state of maximum entropy of the system of oscillators which was derived directly from the hypothesis of quanta in the preceding part, and we may then equate the temperature of the radiation to the temperature of the oscillators. It is, in fact, possible to obtain perfect agree ment of the two states by a suitable coordination of their corre sponding quantities.

According to Sec. 139, the " distribution density" w of the oscillators in the state of statistical equilibrium changes abruptly from one region element to another, while, according to Sec. 138, the distribution within a single region element is uniform. The region elements of the state plane (/^) are bounded by concentric similar and similarly situated ellipses which correspond to those values of the energy U of an oscillator which are integral multiples of hv. We have found exactly the same thing for the stationary state of the oscillators when they are exposed to uniform radia tion, and the distribution density wn in the nth region element may be found from (270), if we remember that the nth region element contains the energies between (n — l)hv and nh v. Hence :

.(pi)-' i/ P\ v

• •

.

• •

This is in perfect agreement with the previous value (220) of wn if we put

1 p\

a — — , and 7 =- p\

167

168 A SYSTEM OF OSCILLATORS

and each of these two equations leads, according to (221), to the following relation between the intensity of the exciting vibration I and the total energy E of the N oscillators :

pl' • (272)

156. If we finally introduce the temperature T from (223), we get from the last equation, by taking account of the value (268) of the factor of proportionality p,

2/^3 1

(273)

ikT-l

Moreover the specific intensity K of a monochromatic plane polarized ray of frequency v is, according to equation (160),

1

K =

2 hv (274)

G j rrt

and the space density of energy of uniform monochromatic unpo- larized radiation of frequency v is, from (159),

U = c3 ^T (275)

e kT —1

Since, among all the forms of radiation of differing constitutions, black radiation is distinguished by the fact that all monochro matic rays contained in it have the same temperature (Sec. 93) these equations also give the law of distribution of energy in the normal spectrum, i.e., in the emission spectrum of a body which is black with respect to the vacuum.

If we refer the specific intensity of a monochromatic ray not to the frequency v but, as is usually done in experimental physics, to the wave length X, by making use of (15) and (16) we obtain the expression

c2h 1 ci 1

X5^_i X5e^_i

This is the specific intensity of a monochromatic plane polarized ray of the wave length X which is emitted from a black body at the temperature T into a vacuum in a direction perpendicular to the

LAW OF NORMAL DISTRIBUTION OF ENERGY 169 surface. The corresponding space density of unpolarized radia

tion is obtained by multiplying Ex by — .

c

Experimental tests have so far confirmed equation (276). l According to the most recent measurements made in the Physi- kalisch-technische Reichsanstalt2 the value of the second radia tion constant C2 is approximately

ch c2 = — = 1.436 cm degree.

K

More detailed information regarding the history of the equa tion of radiation is to be found in the original papers and in the first edition of this book. At this point it may merely be added that equation (276) was not simply extrapolated from radiation measurements, but was originally found in a search after a connection between the entropy and the energy of an oscillator vibrating in a field, a connection which would be as simple as possible and consistent with known measurements.

157. The entropy of a ray is, of course, also determined by its temperature. In fact, by combining equations (138) and (274) we readily obtain as an expression for the entropy radiation L of a monochromatic plane polarized ray of the specific intensity of radiation K and the frequency v,

k^li c2K\ / c2K\ c2K, c2Kl L = — ( 1 +— ) log ( 1 +— • ) - — log — 278

c2 l \ hvz/ \ hv*/ hv3 hv* J

which is a more definite statement of equation (134) for Wien's displacement law.

Moreover it follows from (135), by taking account of (273), that the space density of the entropy s of uniform monochromatic unpolarized radiation as a function of the space density of energy u is

3 3 3

This is a more definite statement of equation (119).

1 See among others H . Rubens und F. Kurlbaum, Sitz. Ber. d. Akad. d. Wiss. zu Berlin vom 25. Okt., 1900, p. 929. Ann. d. Phys. 4, p. 649, 1901. F. Paschen, Ann. d. Phys. 4, p. 277, 1901. O. Lummer und E. Pringsheim, Ann. d. Phys. 6, p. 210, 1901. Tatigkeits- bericht der Phys.-Techn. Reichsanstalt vom J. 1911, Zeitschr. f. Instrumentenkunde, 1912, April, p. 134 ff.