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 agreement of the two states by a suitable coordination of their corresponding 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 $\left(f\psi \right)$ 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 $h\nu$. We have found exactly the same thing for the stationary state of the oscillators when they are exposed to uniform radiation, and the distribution density $w_n$ in the $n$th region element may be found from (270), if we remember that the $n$th region element contains the energies between $(n-1)h\nu$ and $nh\nu$. Hence:

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

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:

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 $p$,

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

and the space density of energy of uniform monochromatic unpolarized radiation of frequency $\nu$ is, from (159),

Since, among all the forms of radiation of differing constitutions, black radiation is distinguished by the fact that all monochromatic 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 $\nu$ but, as is usually done in experimental physics, to the wave length $\lambda$, by making use of (15) and (16) we obtain the expression

This is the specific intensity of a monochromatic plane polarized ray of the wave length $\lambda$ which is emitted from a black body at the temperature $T$ into a vacuum in a direction perpendicular to the surface. The corresponding space density of unpolarized radiation is obtained by multiplying $E_{\lambda }$ by $\frac{8\pi }{c}$.

Experimental tests have so far confirmed equation (276).52 According to the most recent measurements made in the Physikalisch-technische Reichsanstalt53 the value of the second radiation constant $c_2$ is approximately

More detailed information regarding the history of the equation 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 $\nu$,

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

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

158. For small values of $\lambda T$ (i.e., small compared with the constant $\frac{ch}{k}$) equation (276) becomes

an equation which expresses Wien’s54 law of energy distribution.

The specific intensity of radiation $K$ then becomes, according to (274),

and the space density of energy $u$ is, from (275),

159. On the other hand, for large values of $\lambda T$ (276) becomes

a relation which was established first by Lord Rayleigh55 and which we may, therefore, call “Rayleigh’s law of radiation.”

We then find for the specific intensity of radiation $K$ from (274)

and from (275) for the space density of monochromatic radiation we get

Rayleigh’s law of radiation is of very great theoretical interest, since it represents that distribution of energy which is obtained for radiation in statistical equilibrium with material molecules by means of the classical dynamics, and without introducing the hypothesis of quanta.56 This may also be seen from the fact that for a vanishingly small value of the quantity element of action, $h$, the general formula (276) degenerates into Rayleigh’s formula (283). See also below, Sec. 168 et seq.

160. For the total space density, $u$, of black radiation at any temperature $T$ we obtain, from (275),

or

and, integrating term by term,

where $\alpha$ is an abbreviation for

This relation expresses the Stefan-Boltzmann law (75) and it also tells us that the constant of this law is given by

161. For that wave length ${\lambda }_m$ the maximum of the intensity of radiation corresponds in the spectrum of black radiation, we find from (276)

On performing the differentiation and putting as an abbreviation

we get

The root of this transcendental equation is

and accordingly ${\lambda }_{m}T=\frac{ch}{\beta k}$, and this is a constant, as demanded by Wien’s displacement law. By comparison with (109) we find the meaning of the constant $b$, namely,

and, from (277),

while Lummer and Pringsheim found by measurements $0.294$ and Paschen $0.292.$

162. By means of the measured values57 of $a$ and $c_2$ the universal constants $h$ and $k$ may be readily calculated. For it follows from equations (277) and (288) that

Substituting the values of the constants $a$, $c_2$, $\alpha$, $c$, we get

163. To ascertain the full physical significance of the quantity element of action, $h$, much further research work will be required. On the other hand, the value obtained for $k$ enables us readily to state numerically in the C. G. S. system the general connection between the entropy $S$ and the thermodynamic probability $W$ as expressed by the universal equation (164). The general expression for the entropy of a physical system is

This equation may be regarded as the most general definition of entropy. Herein the thermodynamic probability $W$ is an integral number, which is completely defined by the macroscopic state of the system. Applying the result expressed in (293) to the kinetic theory of gases, we obtain from equation (194) for the ratio of the mass of a molecule to that of a mol,

that is to say, there are in one mol

molecules, where the mol of oxygen, $O_2$, is always assumed as $32\text{ gr}$. Hence, for example, the absolute mass of a hydrogen atom ($\frac{1}{2}{H}_2=1.008$) equals $1.62\times 10^{-24}\text{ gr}$. With these numerical values the number of molecules contained in $1{\text{ cm}}^3$ of an ideal gas at $0^{\circ }C.$ and $1$ atmosphere pressure becomes

The mean kinetic energy of translatory motion of a molecule at the absolute temperature $T=1$ is, in the absolute C. G. S. system, according to (200),

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

The elementary quantity of electricity or the free charge of a monovalent ion or electron is, in electrostatic units,

Since absolute accuracy is claimed for the formulæ here employed, the degree of approximation to which these numbers represent the corresponding physical constants depends only on the accuracy of the measurements of the two radiation constants $a$ and $c_2$.

164. Natural Units.—All the systems of units which have hitherto been employed, including the so-called absolute C. G. S. system, owe their origin to the coincidence of accidental circumstances, inasmuch as the choice of the units lying at the base of every system has been made, not according to general points of view which would necessarily retain their importance for all places and all times, but essentially with reference to the special needs of our terrestrial civilization.

Thus the units of length and time were derived from the present dimensions and motion of our planet, and the units of mass and temperature from the density and the most important temperature points of water, as being the liquid which plays the most important part on the surface of the earth, under a pressure which corresponds to the mean properties of the atmosphere surrounding us. It would be no less arbitrary if, let us say, the invariable wave length of Na-light were taken as unit of length. For, again, the particular choice of Na from among the many chemical elements could be justified only, perhaps, by its common occurrence on the earth, or by its double line, which is in the range of our vision, but is by no means the only one of its kind. Hence it is quite conceivable that at some other time, under changed external conditions, every one of the systems of units which have so far been adopted for use might lose, in part or wholly, its original natural significance.

In contrast with this it might be of interest to note that, with the aid of the two constants $h$ and $k$ which appear in the universal law of radiation, we have the means of establishing units of length, mass, time, and temperature, which are independent of special bodies or substances, which necessarily retain their significance for all times and for all environments, terrestrial and human or otherwise, and which may, therefore, be described as “natural units.”

The means of determining the four units of length, mass, time, and temperature, are given by the two constants $h$ and $k$ mentioned, together with the magnitude of the velocity of propagation of light in a vacuum, $c$, and that of the constant of gravitation, $f$. Referred to centimeter, gram, second, and degrees Centigrade, the numerical values of these four constants are as follows:

If we now choose the natural units so that in the new system of measurement each of the four preceding constants assumes the value $1$, we obtain, as unit of length, the quantity

as unit of mass

as unit of time

as unit of temperature

These quantities retain their natural significance as long as the law of gravitation and that of the propagation of light in a vacuum and the two principles of thermodynamics remain valid; they therefore must be found always the same, when measured by the most widely differing intelligences according to the most widely differing methods.

165. The relations between the intensity of radiation and the temperature expressed in Sec. 156 hold for radiation in a pure vacuum. If the radiation is in a medium of refractive index $n$, the way in which the intensity of radiation depends on the frequency and the temperature is given by the proposition of Sec. 39, namely, the product of the specific intensity of radiation $K_{\nu }$ and the square of the velocity of propagation of the radiation has the same value for all substances. The form of this universal function (42) follows directly from (274)

Now, since the refractive index $n$ is inversely proportional to the velocity of propagation, equation (274) is, in the case of a medium with the index of refraction $n$, replaced by the more general relation

and, similarly, in place of (275) we have the more general relation

These expressions hold, of course, also for the emission of a body which is black with respect to a medium with an index of refraction $n$.

166. We shall now use the laws of radiation we have obtained to calculate the temperature of a monochromatic unpolarized radiation of given intensity in the following case. Let the light pass normally through a small area (slit) and let it fall on an arbitrary system of diathermanous media separated by spherical surfaces, the centers of which lie on the same line, the axis of the system. Such radiation consists of homocentric pencils and hence forms behind every refracting surface a real or virtual image of the emitting surface, the image being likewise normal to the axis. To begin with, we assume the last as well as the first medium to be a pure vacuum. Then, for the determination of the temperature of the radiation according to equation (274), we need calculate only the specific intensity of radiation $K_{\nu }$ in the last medium, and this is given by the total intensity of the monochromatic radiation $I_{\nu }$, the size of the area of the image $F$, and the solid angle $\Omega$ of the cone of rays passing through a point of the image. For the specific intensity of radiation $K_{\nu }$ is, according to (13), determined by the fact that an amount

of energy of unpolarized light corresponding to the interval of frequencies from $\nu$ to $\nu +d\nu$ is, in the time $dt$, radiated in a normal direction through an element of area $d\sigma$ within the conical element $d\Omega$. If now $d\sigma$ denotes an element of the area of the surface image in the last medium, then the total monochromatic radiation falling on the image has the intensity

$I_{\nu }$ is of the dimensions of energy, since the product $d\nu dt$ is a mere number. The first integral is the whole area, $F$, of the image, the second is the solid angle, $\Omega$, of the cone of rays passing through a point of the surface of the image. Hence we get

and, by making use of (274), for the temperature of the radiation

If the diathermanous medium considered is not a vacuum but has an index of refraction $n$, (274) is replaced by the more general relation (300), and, instead of the last equation, we obtain

or, on substituting the numerical values of $c$, $h$, and $k$,

In this formula the natural logarithm is to be taken, and $I_{\nu }$ is to be expressed in ergs, $\nu$ in “reciprocal seconds,” i.e. , ${\left(\text{ seconds}\right)}^{-1}$, $F$ in square centimeters. In the case of visible rays the second term, $1$, in the denominator may usually be omitted.

The temperature thus calculated is retained by the radiation considered, so long as it is propagated without any disturbing influence in the diathermanous medium, however great the distance to which it is propagated or the space in which it spreads. For, while at larger distances an ever decreasing amount of energy is radiated through an element of area of given size, this is contained in a cone of rays starting from the element, the angle of the cone continually decreasing in such a way that the value of $K$ remains entirely unchanged. Hence the free expansion of radiation is a perfectly reversible process. (Compare above, Sec. 144.) It may actually be reversed by the aid of a suitable concave mirror or a converging lens.

Let us next consider the temperature of the radiation in the other media, which lie between the separate refracting or reflecting spherical surfaces. In every one of these media the radiation has a definite temperature, which is given by the last formula when referred to the real or virtual image formed by the radiation in that medium.

The frequency $\nu$ of the monochromatic radiation is, of course, the same in all media; moreover, according to the laws of geometrical optics, the product $n^{2}F\Omega$ is the same for all media. Hence, if, in addition, the total intensity of radiation $I_{\nu }$ remains constant on refraction (or reflection), $T$ also remains constant, or in other words: The temperature of a homocentric pencil is not changed by regular refraction or reflection, unless a loss in energy of radiation occurs. Any weakening, however, of the total intensity $I_{\nu }$ by a subdivision of the radiation, whether into two or into many different directions, as in the case of diffuse reflection, leads to a lowering of the temperature of the pencil. In fact, a certain loss of energy by refraction or reflection does occur, in general, on a refraction or reflection, and hence also a lowering of the temperature takes place. In these cases a fundamental difference appears, depending on whether the radiation is weakened merely by free expansion or by subdivision or absorption. In the first case the temperature remains constant, in the second it decreases.59

167. The laws of emission of a black body having been determined, it is possible to calculate, with the aid of Kirchhoff’s law (48), the emissive power $E$ of any body whatever, when its absorbing power $A$ or its reflecting power $1-A$ is known. In the case of metals this calculation becomes especially simple for long waves, since E. Hagen and H. Rubens60 have shown experimentally that the reflecting power and, in fact, the entire optical behavior of the metals in the spectral region mentioned is represented by the simple equations of Maxwell for an electromagnetic field with homogeneous conductors and hence depends only on the specific conductivity for steady electric currents. Accordingly, it is possible to express completely the emissive power of a metal for long waves by its electric conductivity combined with the formulæ for black radiation.61

168. There is, however, also a method, applicable to the case of long waves, for the direct theoretical determination of the electric conductivity and, with it, of the absorbing power, $A$, as well as the emissive power, $E$, of metals. This is based on the ideas of the electron theory, as they have been developed for the thermal and electrical processes in metals by E. Riecke62 and especially by P. Drude.63 According to these, all such processes are based on the rapid irregular motions of the negative electrons, which fly back and forth between the positively charged molecules of matter (here of the metal) and rebound on impact with them as well as with one another, like gas molecules when they strike a rigid obstacle or one another. The velocity of the heat motions of the material molecules may be neglected compared with that of the electrons, since in the stationary state the mean kinetic energy of motion of a material molecule is equal to that of an electron, and since the mass of a material molecule is more than a thousand times as large as that of an electron. Now, if there is an electric field in the interior of the metal, the oppositely charged particles are driven in opposite directions with average velocities depending on the mean free path, among other factors, and this explains the conductivity of the metal for the electric current. On the other hand, the emissive power of the metal for the radiant heat follows from the calculation of the impacts of the electrons. For, so long as an electron flies with constant speed in a constant direction, its kinetic energy remains constant and there is no radiation of energy; but, whenever it suffers by impact a change of its velocity components, a certain amount of energy, which may be calculated from electrodynamics and which may always be represented in the form of a Fourier’s series, is radiated into the surrounding space, just as we think of Roentgen rays as being caused by the impact on the anticathode of the electrons ejected from the cathode. From the standpoint of the hypothesis of quanta this calculation cannot, for the present, be carried out without ambiguity except under the assumption that, during the time of a partial vibration of the Fourier series, a large number of impacts of electrons occurs, i.e., for comparatively long waves, for then the fundamental law of impact does not essentially matter.

Now this method may evidently be used to derive the laws of black radiation in a new way, entirely independent of that previously employed. For if the emissive power, $E$, of the metal, thus calculated, is divided by the absorbing power, $A$, of the same metal, determined by means of its electric conductivity, then, according to Kirchhoff’s law (48), the result must be the emissive power of a black body, irrespective of the special substance used in the determination. In this manner H. A. Lorentz64 has, in a profound investigation, derived the law of radiation of a black body and has obtained a result the contents of which agree exactly with equation (283), and where also the constant $k$ is related to the gas constant $R$ by equation (193). It is true that this method of establishing the laws of radiation is, as already said, restricted to the range of long waves, but it affords a deeper and very important insight into the mechanism of the motions of the electrons and the radiation phenomena in metals caused by them. At the same time the point of view described above in Sec. 111, according to which the normal spectrum may be regarded as consisting of a large number of quite irregular processes as elements, is expressly confirmed.

169. A further interesting confirmation of the law of radiation of black bodies for long waves and of the connection of the radiation constant $k$ with the absolute mass of the material molecules was found by J. H. Jeans65 by a method previously used by Lord Rayleigh,66 which differs essentially from the one pursued here, in the fact that it entirely avoids making use of any special mutual action between matter (molecules, oscillators) and the ether and considers essentially only the processes in the vacuum through which the radiation passes. The starting point for this method of treatment is given by the following proposition of statistical mechanics. (Compare above, Sec. 140.) When irreversible processes take place in a system, which satisfies Hamilton’s equations of motion, and whose state is determined by a large number of independent variables and whose total energy is found by addition of different parts depending on the squares of the variables of state, they do so, on the average, in such a sense that the partial energies corresponding to the separate independent variables of state tend to equality, so that finally, on reaching statistical equilibrium, their mean values have become equal. From this proposition the stationary distribution of energy in such a system may be found, when the independent variables which determine the state are known.

Let us now imagine a perfect vacuum, cubical in form, of edge $l$, and with metallically reflecting sides. If we take the origin of coordinates at one corner of the cube and let the axes of coordinates coincide with the adjoining edges, an electromagnetic process which may occur in this cavity is represented by the following system of equations:

where $a$, $b$, $c$ represent any three positive integral numbers. The boundary conditions in these expressions are satisfied by the fact that for the six bounding surfaces $x=0$, $x=l$, $y=0$, $y=l$, $z=0$, $z=l$ the tangential components of the electric field-strength $E$ vanish. Maxwell’s equations of the field (52) are also satisfied, as may be seen on substitution, provided there exist certain conditions between the constants which may be stated in a single proposition as follows: Let $a$ be a certain positive constant, then there exist between the nine quantities written in the following square:

all the relations which are satisfied by the nine so-called “direction cosines” of two orthogonal right-handed coordinate systems, i.e., the cosines of the angles of any two axes of the systems.

Hence the sum of the squares of the terms of any horizontal or vertical row equals $1$, for example,

Moreover the sum of the products of corresponding terms in any two parallel rows is equal to zero, for example,

Moreover there are relations of the following form:

and hence

If the integral numbers $a$, $b$, $c$ are given, then the frequency $\nu$ is immediately determined by means of (306). Then among the six quantities $e_1$, $e_2$, $e_3$, $h_1$, $h_2$, $h_3$, only two may be chosen arbitrarily, the others then being uniquely determined by them by linear homogeneous relations. If, for example, we assume $e_1$ and $e_2$ arbitrarily, $e_3$ follows from (307) and the values of $h_1$, $h_2$, $h_3$ are then found by relations of the form (308). Between the quantities with accent $e\prime$, $e\prime$, $e\prime$, $h\prime$, $h\prime$, $h\prime$ there exist exactly the same relations as between those without accent, of which they are entirely independent. Hence two also of them, say $h\prime$ and $h\prime$, may be chosen arbitrarily so that in the equations given above for given values of $a$, $b$, $c$ four constants remain undetermined. If we now form, for all values of $a$ $b$ $c$ whatever, expressions of the type (305) and add the corresponding field components, we again obtain a solution for Maxwell’s equations of the field and the boundary conditions, which, however, is now so general that it is capable of representing any electromagnetic process possible in the hollow cube considered. For it is always possible to dispose of the constants $e_1$, $e_2$, $h\prime$, $h\prime$ which have remained undetermined in the separate particular solutions in such a way that the process may be adapted to any initial state ($t=0$) whatever.

If now, as we have assumed so far, the cavity is entirely void of matter, the process of radiation with a given initial state is uniquely determined in all its details. It consists of a set of stationary vibrations, every one of which is represented by one of the particular solutions considered, and which take place entirely independent of one another. Hence in this case there can be no question of irreversibility and hence also none of any tendency to equality of the partial energies corresponding to the separate partial vibrations. As soon, however, as we assume the presence in the cavity of only the slightest trace of matter which can influence the electrodynamic vibrations, e.g., a few gas molecules, which emit or absorb radiation, the process becomes chaotic and a passage from less to more probable states will take place, though perhaps slowly. Without considering any further details of the electromagnetic constitution of the molecules, we may from the law of statistical mechanics quoted above draw the conclusion that, among all possible processes, that one in which the energy is distributed uniformly among all the independent variables of the state has the stationary character.

From this let us determine these independent variables. In the first place there are the velocity components of the gas molecules. In the stationary state to every one of the three mutually independent velocity components of a molecule there corresponds on the average the energy $\frac{1}{3}\bar{L}$ where $\bar{L}$ represents the mean energy of a molecule and is given by (200). Hence the partial energy, which on the average corresponds to any one of the independent variables of the electromagnetic system, is just as large.

Now, according to the above discussion, the electromagnetic state of the whole cavity for every stationary vibration corresponding to any one system of values of the numbers $a$ $b$ $c$ is determined, at any instant, by four mutually independent quantities. Hence for the radiation processes the number of independent variables of state is four times as large as the number of the possible systems of values of the positive integers $a$, $b$, $c$.

We shall now calculate the number of the possible systems of values $a$, $b$, $c$, which correspond to the vibrations within a certain small range of the spectrum, say between the frequencies $\nu$ and $\nu +d\nu$. According to (306), these systems of values satisfy the inequalities

where not only $\frac{2l\nu }{c}$ but also $\frac{2ld\nu }{c}$ is to be thought of as a large number. If we now represent every system of values of $a$, $b$, $c$ graphically by a point, taking $a$, $b$, $c$ as coordinates in an orthogonal coordinate system, the points thus obtained occupy one octant of the space of infinite extent, and condition (309) is equivalent to requiring that the distance of any one of these points from the origin of the coordinates shall lie between $\frac{2l\nu }{c}$ and $\frac{2l(\nu +d\nu )}{c}$. Hence the required number is equal to the number of points which lie between the two spherical surface-octants corresponding to the radii $\frac{2l\nu }{c}$ and $\frac{2l(\nu +d\nu )}{c}$. Now since to every point there corresponds a cube of volume $1$ and vice versa, that number is simply equal to the space between the two spheres mentioned, and hence equal to

and the number of the independent variables of state is four times as large or

Since, moreover, the partial energy $\frac{\bar{L}}{3}$ corresponds on the average to every independent variable of state in the state of equilibrium, the total energy falling in the interval from $\nu$ to $\nu +d\nu$ becomes

Since the volume of the cavity is $l^3$, this gives for the space density of the energy of frequency $\nu$

and, by substitution of the value of $\bar{L}=\frac{L}{N}$ from,

which is in perfect agreement with Rayleigh’s formula (285).

If the law of the equipartition of energy held true in all cases, Rayleigh’s law of radiation would, in consequence, hold for all wave lengths and temperatures. But since this possibility is excluded by the measurements at hand, the only possible conclusion is that the law of the equipartition of energy and, with it, the system of Hamilton’s equations of motion does not possess the general importance attributed to it in classical dynamics. Therein lies the strongest proof of the necessity of a fundamental modification of the latter.