III Ideal Linear Oscillators

Chapter III
Ideal Linear Oscillators

135. The main problem of the theory of heat radiation is to determine the energy distribution in the normal spectrum of black radiation, or, what amounts to the same thing, to ﬁnd the function which has been left undetermined in the general expression of Wien’s displacement law (119), the function which connects the entropy of a certain radiation with its energy. The purpose of this chapter is to develop some preliminary theorems leading to this solution. Now since, as we have seen in Sec. 48, the normal energy distribution in a diathermanous medium cannot be established unless the medium exchanges radiation with an emitting and absorbing substance, it will be necessary for the treatment of this problem to consider more closely the processes which cause the creation and the destruction of heat rays, that is, the processes of emission and absorption. In view of the complexity of these processes and the diﬃculty of acquiring knowledge of any deﬁnite details regarding them, it would indeed be quite hopeless to expect to gain any certain results in this way, if it were not possible to use as a reliable guide in this obscure region the law of Kirchhoﬀ derived in Sec. 51. This law states that a vacuum completely enclosed by reﬂecting walls, in which any emitting and absorbing bodies are scattered in any arrangement whatever, assumes in the course of time the stationary state of black radiation, which is completely determined by one parameter only, namely, the temperature, and in particular does not depend on the number, the nature, and the arrangement of the material bodies present. Hence, for the investigation of the properties of the state of black radiation the nature of the bodies which are assumed to be in the vacuum is perfectly immaterial. In fact, it does not even matter whether such bodies really exist somewhere in nature, provided their existence and their properties are consistent with the laws of thermodynamics and electrodynamics. If, for any special arbitrary assumption regarding the nature and arrangement of emitting and absorbing systems, we can ﬁnd a state of radiation in the surrounding vacuum which is distinguished by absolute stability, this state can be no other than that of black radiation.

Since, according to this law, we are free to choose any system whatever, we now select from all possible emitting and absorbing systems the simplest conceivable one, namely, one consisting of a large number $N$ of similar stationary oscillators, each consisting of two poles, charged with equal quantities of electricity of opposite sign, which may move relatively to each other on a ﬁxed straight line, the axis of the oscillator.

It is true that it would be more general and in closer accord with the conditions in nature to assume the vibrations to be those of an oscillator consisting of two poles, each of which has three degrees of freedom of motion instead of one, i.e., to assume the vibrations as taking place in space instead of in a straight line only. Nevertheless we may, according to the fundamental principle stated above, restrict ourselves from the beginning to the treatment of one single component, without fear of any essential loss of generality of the conclusions we have in view.

It might, however, be questioned as a matter of principle, whether it is really permissible to think of the centers of mass of the oscillators as stationary, since, according to the kinetic theory of gases, all material particles which are contained in substances of ﬁnite temperature and free to move possess a certain ﬁnite mean kinetic energy of translatory motion. This objection, however, may also be removed by the consideration that the velocity is not ﬁxed by the kinetic energy alone. We need only think of an oscillator as being loaded, say at its positive pole, with a comparatively large inert mass, which is perfectly neutral electrodynamically, in order to decrease its velocity for a given kinetic energy below any preassigned value whatever. Of course this consideration remains valid also, if, as is now frequently done, all inertia is reduced to electrodynamic action. For this action is at any rate of a kind quite diﬀerent from the one to be considered in the following, and hence cannot inﬂuence it.

Let the state of such an oscillator be completely determined by its moment $f (t)$ , that is, by the product of the electric charge of the pole situated on the positive side of the axis and the pole distance, and by the derivative of $f$ with respect to the time or

\frac{d f (t)}{d t} = ḟ (t) .

(204)

Let the energy of the oscillator be of the following simple form:

U = \frac{1}{2} K f^{2} + \frac{1}{2} L ḟ^{2},

(205)

where $K$ and $L$ denote positive constants, which depend on the nature of the oscillator in some way that need not be discussed at this point.

If during its vibration an oscillator neither absorbed nor emitted any energy, its energy of vibration, $U$ , would remain constant, and we would have:

d U = K f d f + L ḟ d ḟ = 0,

(205a)

or, on account of (204),

K f (t) + L \ddot{f} (t) = 0 .

(206)

The general solution of this diﬀerential equation is found to be a purely periodical vibration:

f = C cos (2 π ν t - θ)

(207)

where $C$ and $θ$ denote the integration constants and $ν$ the number of vibrations per unit time:

ν = \frac{1}{2 π} \sqrt{\frac{K}{L}} .

(208)

136. If now the assumed system of oscillators is in a space traversed by heat rays, the energy of vibration, $U$ , of an oscillator will not in general remain constant, but will be always changing by absorption and emission of energy. Without, for the present, considering in detail the laws to which these processes are subject, let us consider any one arbitrarily given thermodynamic state of the oscillators and calculate its entropy, irrespective of the surrounding ﬁeld of radiation. In doing this we proceed entirely according to the principle advanced in the two preceding chapters, allowing, however, at every stage for the conditions caused by the peculiarities of the case in question.

The ﬁrst question is: What determines the thermodynamic state of the system considered? For this purpose, according to Sec. 124, the numbers $N_{1}$ , $N_{2}$ , $N_{3}, \dots$ of the oscillators, which lie in the region elements $1$ , $2$ , $3, \dots$ of the “state space” must be given. The state space of an oscillator contains those coordinates which determine the microscopic state of an oscillator. In the case in question these are only two in number, namely, the moment $f$ and the rate at which it varies, $ḟ$ , or instead of the latter the quantity

ψ = L ḟ,

(209)

which is of the dimensions of an impulse. The region element of the state plane is, according to the hypothesis of quanta (Sec. 126), the double integral

\iint d f d ψ = h .

(210)

The quantity $h$ is the same for all region elements. A priori, it might, however, depend also on the nature of the system considered, for example, on the frequency of the oscillators. The following simple consideration, however, leads to the assumption that $h$ is a universal constant. We know from the generalized displacement law of Wien (equation (119)) that in the universal function, which gives the entropy radiation as dependent on the energy radiation, there must appear a universal constant of the dimension $\frac{c^{3} u}{ν^{3}}$ and this is of the dimension of a quantity of action⁴⁷ ( $erg sec .$ ). Now, according to (210), the quantity $h$ has precisely this dimension, on which account we may denote it as “element of action” or “quantity element of action.” Hence, unless a second constant also enters, $h$ cannot depend on any other physical quantities.

137. The principal diﬀerence, compared with the calculations for an ideal gas in the preceding chapter, lies in the fact that we do not now assume the distribution densities $w_{1}$ , $w_{2}$ , $w_{3} \dots$ of the oscillators among the separate region elements to vary but little from region to region as was assumed in Sec. 129. Accordingly the $w$ ’s are not small, but ﬁnite proper fractions, and the summation over the region elements cannot be written as an integration.

In the ﬁrst place, as regards the shape of the region elements, the fact that in the case of undisturbed vibrations of an oscillator the phase is always changing, whereas the amplitude remains constant, leads to the conclusion that, for the macroscopic state of the oscillators, the amplitudes only, not the phases, must be considered, or in other words the region elements in the $f ψ$ plane are bounded by the curves $C = const .$ , that is, by ellipses, since from (207) and (209)

{(\frac{f}{C})}^{2} + {(\frac{ψ}{2 π ν L C})}^{2} = 1 .

(211)

The semi-axes of such an ellipse are:

a = C and b = 2 π ν L C .

(212)

Accordingly the region elements $1$ , $2$ , $3, \dots$ $n \dots$ are the concentric, similar, and similarly situated elliptic rings, which are determined by the increasing values of $C$ :

0, C_{1}, C_{2}, C_{3}, \dots C_{n - 1}, C_{n} \dots .

(213)

The $n$ th region element is that which is bounded by the ellipses $C = C_{n - 1}$ and $C = C_{n}$ . The ﬁrst region element is the full ellipse $C_{1}$ . All these rings have the same area $h$ , which is found by subtracting the area of the full ellipse $C_{n - 1}$ from that of the full ellipse $C_{n}$ ; hence

h = (a_{n} b_{n} - a_{n - 1} b_{n - 1}) π

or, according to (212),

h = (C_{n}^{2} - C_{n - 1}^{2}) 2 π^{2} ν L,

where

n = 1

2

3, \dots

From the additional fact that $C_{0} = 0$ , it follows that:

C_{n}^{2} = \frac{n h}{2 π^{2} ν L} .

(214)

Thus the semi-axes of the bounding ellipses are in the ratio of the square roots of the integral numbers.

138. The thermodynamic state of the system of oscillators is ﬁxed by the fact that the values of the distribution densities $w_{1}$ , $w_{2}$ , $w_{3}, \dots$ of the oscillators among the separate region elements are given. Within a region element the distribution of the oscillators is according to the law of elemental chaos (Sec. 122), i.e., it is approximately uniform.

These data suﬃce for calculating the entropy $S$ as well as the energy $E$ of the system in the given state, the former quantity directly from (173), the latter by the aid of (205). It must be kept in mind in the calculation that, since the energy varies appreciably within a region element, the energy $E_{n}$ of all those oscillators which lie in the $n$ th region element is to be found by an integration. Then the whole energy $E$ of the system is:

E = E_{1} + E_{2} + \dots E_{n} + \dots .

(215)

$E_{n}$ may be calculated with the help of the law that within every region element the oscillators are uniformly distributed. If the $n$ th region element contains, all told, $N_{n}$ oscillators, there are per unit area $\frac{N_{n}}{h}$ oscillators and hence $\frac{N_{n}}{h} d f \cdot d ψ$ per element of area. Hence we have:

E_{n} = \frac{N_{n}}{h} \iint U d f d ψ .

In performing the integration, instead of

f

and

ψ

we take

C

and

φ

, as new variables, and since according to (211),

f = C cos φ ψ = 2 π ν L C sin φ

(216)

we get:

E_{n} = 2 π ν L \frac{N_{n}}{h} \iint U C d C d φ

to be integrated with respect to

φ

from

0

2 π

and with respect to

C

from

C_{n - 1}

C_{n}

. If we substitute from (205), (209) and (216)

U = \frac{1}{2} K C^{2},

(217)

we obtain by integration

E_{n} = \frac{π^{2}}{2} ν L K \frac{N_{n}}{h} (C_{n}^{4} - C_{n - 1}^{4})

and from (214) and (208):

E_{n} = N_{n} (n - \frac{1}{2}) h ν = N w_{n} (n - \frac{1}{2}) h ν,

(218)

that is, the mean energy of an oscillator in the $n$ th region element is $(n - \frac{1}{2}) h ν$ . This is exactly the arithmetic mean of the energies $(n - 1) h ν$ and $n h ν$ which correspond to the two ellipses $C = C_{n - 1}$ and $C = C_{n}$ bounding the region, as may be seen from (217), if the values of $C_{n - 1}$ and $C_{n}$ are therein substituted from (214).

The total energy $E$ is, according to (215),

E = N h ν \sum_{n = 1}^{n = \infty} (n - \frac{1}{2}) w_{n} .

(219)

139. Let us now consider the state of thermodynamic equilibrium of the oscillators. According to the second principle of thermodynamics, the entropy $S$ is in that case a maximum for a given energy $E$ . Hence we assume $E$ in (219) as given. Then from (179) we have for the state of equilibrium:

δ S = 0 = \sum_{1}^{\infty} (log w_{n} + 1) δ w_{n},

where according to (167) and (219)

\sum_{1}^{\infty} δ w_{n} = 0 and \sum_{1}^{\infty} (n - \frac{1}{2}) δ w_{n} = 0 .

From these relations we ﬁnd:

log w_{n} + β n + const . = 0

w_{n} = α γ^{n} .

(220)

The values of the constants $α$ and $γ$ follow from equations (167) and (219):

α = \frac{2 N h ν}{2 E - N h ν} γ = \frac{2 E - N h ν}{2 E + N h ν} .

(221)

Since $w_{n}$ is essentially positive it follows that equilibrium is not possible in the system of oscillators considered unless the total energy $E$ has a greater value than $\frac{N h ν}{2}$ , that is unless the mean energy of the oscillators is at least $\frac{h ν}{2}$ . This, according to (218), is the mean energy of the oscillators lying in the ﬁrst region element. In fact, in this extreme case all $N$ oscillators lie in the ﬁrst region element, the region of smallest energy; within this element they are arranged uniformly.

The entropy $S$ of the system, which is in thermodynamic equilibrium, is found by combining (173) with (220) and (221)

\begin{align} S = & k N {(\frac{E}{N h ν} + \frac{1}{2}) log (\frac{E}{N h ν} + \frac{1}{2}) \\ - (\frac{E}{N h ν} - \frac{1}{2}) log (\frac{E}{N h ν} - \frac{1}{2})} . & (222) \end{align}

140. The connection between energy and entropy just obtained allows furthermore a certain conclusion as regards the temperature. For from the equation of the second principle of thermodynamics, $d S = \frac{d E}{T}$ and from diﬀerentiation of (222) with respect to $E$ it follows that

E = N \frac{h ν}{2} \frac{1 + e^{- \frac{h ν}{k T}}}{1 - e^{- \frac{h ν}{k T}}} = N h ν (\frac{1}{2} + \frac{1}{e^{\frac{h ν}{k T}} - 1}) .

(223)

Hence, for the zero point of the absolute temperature $E$ becomes, not $0$ , but $N \frac{h ν}{2}$ . This is the extreme case discussed in the preceding paragraph, which just allows thermodynamic equilibrium to exist. That the oscillators are said to perform vibrations even at the temperature zero, the mean energy of which is as large as $\frac{h ν}{2}$ and hence may become quite large for rapid vibrations, may at ﬁrst sight seem strange. It seems to me, however, that certain facts point to the existence, inside the atoms, of vibrations independent of the temperature and supplied with appreciable energy, which need only a small suitable excitation to become evident externally. For example, the velocity, sometimes very large, of secondary cathode rays produced by Roentgen rays, and that of electrons liberated by photoelectric eﬀect are independent of the temperature of the metal and of the intensity of the exciting radiation. Moreover the radioactive energies are also independent of the temperature. It is also well known that the close connection between the inertia of matter and its energy as postulated by the relativity principle leads to the assumption of very appreciable quantities of intra-atomic energy even at the zero of absolute temperature.

For the extreme case, $T = \infty$ , we ﬁnd from (223) that

E = N k T,

(224)

i.e., the energy is proportional to the temperature and independent of the size of the quantum of action, $h$ , and of the nature of the oscillators. It is of interest to compare this value of the energy of vibration $E$ of the system of oscillators, which holds at high temperatures, with the kinetic energy $L$ of the molecular motion of an ideal monatomic gas at the same temperature as calculated in (200). From the comparison it follows that

E = \frac{2}{3} L .

(225)

This simple relation is caused by the fact that for high temperatures the contents of the hypothesis of quanta coincide with those of the classical statistical mechanics. Then the absolute magnitude of the region element, $G$ or $h$ respectively, becomes physically unimportant (compare Sec. 125) and we have the simple law of equipartition of the energy among all variables in question (see below Sec. 169). The factor $\frac{2}{3}$ in equation (225) is due to the fact that the kinetic energy of a moving molecule depends on three variables ( $ξ$ , $η$ , $ζ$ ,) and the energy of a vibrating oscillator on only two ( $f$ , $ψ$ ).

The heat capacity of the system of oscillators in question is, from (223),

\frac{d E}{d T} = N k {(\frac{h ν}{k T})}^{2} \frac{e^{\frac{h ν}{k T}}}{{(e^{\frac{h ν}{k T}} - 1)}^{2}} .

(226)

It vanishes for $T = 0$ and becomes equal to $N k$ for $T = \infty$ . A. Einstein⁴⁸ has made an important application of this equation to the heat capacity of solid bodies, but a closer discussion of this would be beyond the scope of the investigations to be made in this book.

For the constants $α$ and $γ$ in the expression (220) for the distribution density $w$ we ﬁnd from (221):

α = e^{\frac{h ν}{k T}} - 1 γ = e^{- \frac{h ν}{k T}}

(227)

and ﬁnally for the entropy $S$ of our system as a function of temperature:

S = k N \{\frac{\frac{h ν}{k T}}{e^{\frac{h ν}{k T} - 1}} - log (1 - e^{- \frac{h ν}{k T}})\} .

(228)

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Chapter IIIIdeal Linear Oscillators

Chapter III
Ideal Linear Oscillators