The Theory of Heat Radiation (1914) — part 2 of 12

In addition to " black surfaces" the term "black body" is also used. According to G. Kirchhoff1 it denotes a body which has the property of allowing all incident rays to enter without surface reflection and not allowing them to leave again. Hence it is seen that a black body must satisfy three independent conditions. First, the body must have a black surface in order to allow the incident rays to enter" without reflection. Since, in general, the properties of a surface depend on both of the bodies which are in contact, this condition shows that the property of blackness as applied to a body depends not only on the nature of the body but also on that of the contiguous medium. A body which is black relatively to air need not be so relatively to glass, and vice versa. Second, the black body must have a certain minimum thickness depending on its absorbing power, in order to insure that the rays after passing into the body shall not be able to leave it again at a different point of the surface. The more ab sorbing a body is, the smaller the value of this minimum thick ness, while in the case of bodies with vanishingly small absorbing power only a layer of infinite thickness may be regarded as black. Third, the black body must have a vanishingly small coefficient of scattering (Sec. 8). Otherwise the rays received by it would be partly scattered in the interior and might leave again through the surface.2

11. All the distinctions and definitions mentioned in the two preceding paragraphs refer to rays of one definite color only. It might very well happen that, e.g., a surface which is rough for a certain kind of rays must be regarded as smooth for a different kind of rays. It is readily seen that, in general, a surface shows

1 O. Kirchhoff, Pogg. Ann., 109, p. 275, 1860. Gesammelte Abhandlungen, J. A. Earth, Leipzig, 1882, p. 573. In denning a black body Kirchhoff also assumes that the absorption of incident rays takes place in a layer "infinitely thin." We do not include this in our definition.

2 For this point see especially A. Schuster, Astrophysical Journal, 21, p. 1, 1905, who hae pointed out that an infinite layer of gas with a black surface need by no means be a black body.

GENERAL INTRODUCTION 11

decreasing degrees of roughness for increasing wave lengths Now, since smooth non-reflecting surfaces do not exist (Sec. 10), it follows that all approximately black surfaces which may be real ized in practice (lamp black, platinum black) show appreciable reflection for rays of sufficiently long wave lengths.

12. Absorption. — Heat rays are destroyed by " absorption." According to the principle of the conservation of energy the energy of heat radiation is thereby changed into other forms of energy (heat, chemical energy). Thus only material particles can absorb heat rays, not elements of surfaces, although some times for the sake of brevity the expression absorbing surfaces is used.

Whenever absorption takes place, the heat ray passing through the medium under consideration is weakened by a certain frac tion of its intensity for every element of path traversed. For a sufficiently small distance s this fraction is proportional to s, and may be written

a,s (4)

Here av is known as the " coefficient of absorption" of the me dium for a ray of frequency v. We assume this coefficient to be independent of the intensity; it will, however, depend in general in non-homogeneous and anisotropic media on the position of s and on the direction of propagation and polarization of the ray (example: tourmaline). We shall, however, consider only ho mogeneous isotropic substances, and shall therefore suppose that av has the same value at all points and in all directions in the medium, and depends on nothing but the frequency v, the tem perature T, and the nature of the medium.

Whenever av does not differ from zero except for a limited range of the spectrum, the medium shows "selective" absorption. For those colors for which av=0 and also the coefficient of scattering j8, = 0 the medium is described as perfectly "transparent" or "diathermanous." But the properties of selective absorption and of diathermancy may for a given medium vary widely with the temperature. In general we shall assume a mean value for «„. This implies that the absorption in a distance equal to a single wave length is very small, because the distance s, while small, contains many wave lengths (Sec. 2).

12 FUNDAMENTAL FACTS AND DEFINITIONS

13. The foregoing considerations regarding the emission, the propagation, and the absorption of heat rays suffice for a mathe matical treatment of the radiation phenomena. The calculation requires a knowledge of the value of the constants and the initial and boundary conditions, and yields a full account of the changes the radiation undergoes in a given time in one or more contiguous media of the kind stated, including the temperature changes caused by it. The actual calculation is usually very complicated. We shall, however, before entering upon the treatment of special cases discuss the general radiation phenomena from a different point of view, namely by fixing our attention not on a definite ray, but on a definite position in space.

14. Let da be an arbitrarily chosen, infinitely small element of area in the interior of a medium through which radiation passes. At a given instant rays are passing through this element in many different directions. The energy radiated through it in an element of time dt in a definite direction is proportional to the area do-, the length of time dt and to the cosine of the angle 6 made by the normal of da with the direction of the radiation. If we make da sufficiently small, then, although this is only an approximation to the actual state of affairs, we can think of all points in da as being affected by the radiation in the same way. Then the energy radiated through da in a definite direction must be pro portional to the solid angle in which da intercepts that radiation and this solid angle is measured by da cos 6. It is readily seen that, when the direction of the element is varied relatively to the direction of the radiation, the energy radiated through it vanishes when

=!•

Now in general a pencil of rays is propagated from every point of the element da in all directions, but with different intensities in different directions, and any two pencils emanating from two points of the element are identical save for differences of higher order. A single one of these pencils coming from a single point does not represent a finite quantity of energy, because a finite amount of energy is radiated only through a finite area. This holds also for the passage of rays through a so-called focus. For

GENERAL INTRODUCTION 13

example, when sunlight passes through a converging lens and is concentrated in the focal plane of the lens, the solar rays do not converge to a single point, but each pencil of parallel rays forms a separate focus and all these foci together constitute a surface representing a small but finite image of the sun. A finite amount of energy does not pass through less than a finite portion of this surface.

15. Let us now consider quite generally the pencil, which is propagated from a point of the element da- as vertex in all direc-' tions of space and on both sides of do-. A certain direction may be specified by the angle 6 (between 0 and TT), as already used, and by an azimuth </> (between 0 and 2?r) . The intensity in this direction is the energy propagated in an infinitely thin cone lim ited by 6 and d+dd and 0 and <£+d<£. The solid angle of this cone is

d!2 = sin 0-d6-d<i>. (5)

Thus the energy radiated in time dt through the element of area do- in the direction of the cone dtt is:

dt do- cos ddttK = K sin 6 cos 0 dd d<t> do- dt. (6)

The finite quantity K we shall term the "specific intensity" or the "brightness," dtt the "solid angle" of the pencil emanating from a point of the element do- in the direction (0, 0). K is a positive function of position, time, and the angles 0 and <£. In general the specific intensities of radiation in different directions are entirely independent of one another. For example, on sub stituting TT — 0 f or 0 and TT + <£ f or </> in the function K, we obtain the specific intensity of radiation in the diametrically opposite direction, a quantity which in general is quite different from the preceding one.

For the total radiation through the element of area da toward one side, say the one on which 0 is an acute angle, we get, by integrating with respect to (/> from 0 to 2ir and with respect to

7T

0 from 0 to -

27T 2

I <*0 f

t/ o t/ o

ddK sin 0 cos 8 do- dt.

14 FUNDAMENTAL FACTS AND DEFINITIONS

Should the radiation be uniform in all directions and hence K be a constant, the total radiation on one side will be

TT K d<j dt. (7)

16. In speaking of the radiation in a definite direction (6, 0) one should always keep in mind that the energy radiated in a cone is not finite unless the angle of the cone is finite. No finite radiation of light or heat takes place in one definite direction only, or expressing it differently, in nature there is no such thing as absolutely parallel light or an absolutely plane wave front. From a pencil of rays called " parallel " a finite amount of energy of radiation can only be obtained if the rays or wave normals of the pencil diverge so as to form a finite though perhaps exceedingly narrow cone.

17. The specific intensity K of the whole energy radiated in a certain direction may be further divided into the intensities of the separate rays belonging to the different regions of the spec trum which travel independently of one another. Hence we consider the intensity of radiation within a certain range of fre quencies, say from v to v '. If the interval v'-v be taken suffi ciently small and be denoted by dv, the intensity of radiation within the interval is proportional to dv. Such radiation is called homogeneous or monochromatic.

A last characteristic property of a ray of definite direction, intensity, and color is its state of polarization. If we break up a ray, which is in any state of polarization whatsoever and which travels in a definite direction and has a definite frequency v, into two plane polarized components, the sum of the intensities of the components will be just equal to the intensity of the ray as a whole, independently of the direction of the two planes, provided the two planes of polarization, which otherwise may be taken at random, are at right angles to each other. If their posi tion be denoted by the azimuth ^ of one of the planes of vibration (plane of the electric vector), then the two components of the intensity may be written in the form

K.cosV+K/sinV and K.sin V + K/cos V (8)

Herein K is independent of }/. These expressions we shall call

GENERAL INTRODUCTION 15

the " components of the specific intensity of radiation of frequency v.11 The sum is independent of ^ and is always equal to the intensity of the whole ray Ky + K/. At the same time K,, and K/ represent respectively the largest and smallest values which

either of the components may have, namely, when [/ = 0 and ^ = 9'

Hence we call these values the " principal values of the intensi ties," or the "principal intensities," and the corresponding planes of vibration we call the "principal planes of vibration" of the ray. Of course both, in general, vary with the time. Thus we may write generally

I'

(9)

where the positive quantities K,, and K/, the two principal values of the specific intensity of the radiation (brightness) of fre quency v, depend not only on v but also on their position, the time, and on the angles 6 and 0. By substitution in (6) the energy radiated in the time dt through the element of area do- in the direc tion of the conical element dtt assumes the value

CO

dt do- cos 6 dQ I dv (K. + K/) (10)

I

and for monochromatic plane polarized radiation of brightness K,:

dt do- cos 0 dtt K, dv = dt do- sin 8 cos d dd d^ K, dv. (11) For unpolarized rays K, = K/, and hence

oo

K = 2 Cdv K,, (12)

= 2 Cdv K

and the energy of a monochromatic ray of frequency v will be: 2dt do- cos $ dti K, dv = 2dt do- sin 6 cos 0 dd d<j> K, dv.(13) When, moreover, the radiation is uniformly distributed in all directions, the total radiation through da toward one side may be found from (7) and (12); it is

27r da dt 1 Kvdv. (14)

I

16 FUNDAMENTAL FACTS AND DEFINITIONS

18. Since in nature K,, can never be infinitely large, K will not have a finite value unless Kv differs from zero over a finite range of frequencies. Hence there exists in nature no absolutely homogeneous or monochromatic radiation of light or heat. A finite amount of radiation contains always a finite although possi bly very narrow range of the spectrum. This implies a funda mental difference from the corresponding phenomena of acoustics, where a finite intensity of sound may correspond to a single definite frequency. This difference is, among other things, the cause of the fact that the second law of thermodynamics has an important bearing on light and heat rays, but not on sound waves. This will be further discussed later on.

19. From equation (9) it is seen that the quantity K,, the intensity of radiation of frequency v, and the quantity K, the intensity of radiation of the whole spectrum, are of different dimensions. Further it is to be noticed that, on subdividing the spectrum according to wave lengths X, instead of frequencies v, the intensity of radiation Ex.oi the wave lengths X correspond ing to the frequency v is not obtained simply by replacing v in the expression for K,, by the corresponding value of X deduced from

v = \ (15)

A

where q is the velocity of propagation. For if d\ and dv refer to the same interval of the spectrum, we have, not E^ = K,,, but Ex d\ = K, dv. By differentiating (15) and paying attention to the signs of corresponding values of d\ and dv the equation

qdX dv -'- *•

is obtained. Hence we get by substitution:

E, = «£. (16)

This relation shows among other things that in a certain spectrum the maxima of Ex and K,, lie at different points of the spectrum.

20. When the principal intensities K,, and K/ of all mono chromatic rays are given at all points of the medium and for all directions, the state of radiation is known in all respects and all

GENERAL INTRODUCTION 17

questions regarding it may be answered. We shall show this by one or two applications to special cases. Let us first find the amount of energy which is radiated through any element of area da toward any other element da'. The distance r between the two elements may be thought of as large compared with the linear dimensions of the elements da and daf but still so small that no appreciable amount of radiation is absorbed or scattered along it. This condition is, of course, superfluous for diather- manous media.

From any definite point of da rays pass to all points of da' . These rays form a cone whose vertex lies in da and whose solid angle is

da' cos (/, r) ^- —I"

where / denotes the normal of daf and the angle (v1 ', r) is to be taken as an acute angle. This value of d& is, neglecting small quantities of higher order, independent of the particular position of the vertex of the cone on da.

If we further denote the normal to da by v the angle 6 of (14) will be the angle (v, r) and hence from expression (6) the energy of radiation required is found to be :

ArAr'cos(r,r)-cos(/,r)

K- - — - — - at. (17)

For monochromatic plane polarized radiation of frequency v the energy will be, according to equation (11),

dada'cos(v,r)cos(i>',r)

K,, dv — - -dt. (18)

r2

The relative size of the two elements da and da' may have any value whatever. They may be assumed to be of the same or of a different order of magnitude, provided the condition remains satisfied that r is large compared with the linear dimensions of each of them. If we choose da small compared with da', the rays diverge from da to da', whereas they converge from da to da', if we choose da large compared with da'.

21. Since every point of da is the vertex of a cone spreading out toward da', the whole pencil of rays here considered, which is

18 FUNDAMENTAL FACTS AND DEFINITIONS

defined by do- and da', consists of a double infinity of point pencils or of a fourfold infinity of rays which must all be considered equally for the energy radiation. Similarly the pencil of rays may be thought of as consisting of the cones which, emanating from all points of do-, converge in one point of da' respectively as a vertex. If we now imagine the whole pencil of rays to be cut by a plane at any arbitrary distance from the elements da and da' and lying either between them or outside, then the cross-sections of any two point pencils on this plane will not be identical, not even approximately. In general they will partly overlap and partly lie outside of each other, the amount of over lapping being different for different intersecting planes. Hence it follows that there is no definite cross-section of the pencil of rays so far as the uniformity of radiation is concerned. If, how ever, the intersecting plane coincides with either da or da ', then the pencil has a definite cross-section. Thus these two planes show an exceptional property. We shall call them the two " focal planes" of the pencil.

In the special case already mentioned above, namely, when one of the two focal planes is infinitely small compared with the other, the whole pencil of rays shows the character of a point pencil inas much as its form is approximately that of a cone having its vertex in that focal plane which is small compared with the other. In that case the " cross-section " of the whole pencil at a definite point has a definite meaning. Such a pencil of rays, which is similar to a cone, we shall call an elementary pencil, and the small focal plane we shall call the first focal plane of the elemen tary pencil. The radiation may be either converging toward the first focal plane or diverging from the first focal plane. All the pencils of rays passing through a medium may be considered as consisting of such elementary pencils, and hence we may base our future considerations on elementary pencils only, which is a great convenience, owing to their simple nature.

As quantities necessary to define an elementary pencil with a given first focal plane da, we may choose not the second focal plane da' but the magnitude of that solid angle dtt under which da' is seen from da. On the other hand, in the case of an arbi trary pencil, that is, when the two focal planes are of the same order of magnitude, the second focal plane in general cannot be

GENERAL INTRODUCTION 19

replaced by the solid angle dfi without the pencil changing markedly in character. For if, instead of da' being given, the magnitude and direction of dti, to be taken as constant for all points of da, is given, then the rays emanating from dcr do not any longer form the original pencil, but rather an elementary pencil whose first focal plane is da and whose second focal plane lies at an infinite distance.

22. Since the energy radiation is propagated in the medium with a finite velocity q, there must be in a finite space a finite amount of energy. We shall therefore speak of the "space density of radiation," meaning thereby the ratio of the total quantity of energy of radiation contained in a volume-element to the magni tude of the latter. Let us now calculate the space density of radiation u at any arbitrary point of the medium. When we consider an infinitely small element of volume v at the point in question, having any shape whatsoever, we must allow for all rays passing through the volume-element v. For this purpose we shall construct about any point 0 of v as center a sphere of radius r, r being large compared with the linear dimensions of v but still so small that no appreciable absorption or scattering of the radia tion takes place in the distance r (Fig. 1). Every ray which reaches v must then come from some point on the surface of the sphere. If, then, we at first consider only all the rays that come from the points of an infinitely small element of area da FlG

on the surface of the sphere, and

reach v, and then sum up for all elements of the spherical sur face, we shall have accounted for all rays and not taken any one more than once.

Let us then calculate first the amount of energy which is con tributed to the energy contained in v by the radiation sent from such an element da to v. We choose da so that its linear dimen sions are small compared with those of v and consider the cone of rays which, starting at a point of da, meets the volume v. This cone consists of an infinite number of conical elements with the

20 FUNDAMENTAL FACTS AND DEFINITIONS

common vertex at P, a point of da, each cutting out of the volume v a certain element of length, say s. The solid angle of such a

conical element is 2 where / denotes the area of cross-section

normal to the axis of the cone at a distance r from the vertex. The time required for the radiation to pass through the distance s is:

From expression (6) we may find the energy radiated through a certain element of a hence the energy is:

certain element of area. In the present case d& = — and 6 = 0;

-f -f

rdaJ~K = -S2-K da. (19)

This energy enters the conical element in v and spreads out into the volume fs. Summing up over all conical elements that start from da and enter v we have

Kda _Kdo- r2q r2q

This represents the entire energy of radiation contained in the volume v, so far as it is caused by radiation through the element da. In order to obtain the total energy of radiation contained in v we must integrate over all elements da contained in the sur face of the sphere. Denoting by dtt the solid angle — - of a

cone which has its center in 0 and intersects in da the surface of the sphere, we get for the whole energy:

• I K dQ.

'

The volume density of radiation required is found from this by dividing by v. It is

= - KdQ. lj

(20)

GENERAL INTRODUCTION 21

Since in this expression r has disappeared, we can think of K as the intensity of radiation at the point 0 itself. In integrating, it is to be noted that K in general depends on the direction (6, 0). For radiation that is uniform in all directions K is a constant and on integration we get:

..?

23. A meaning similar to that of the volume density of the total radiation u is attached to the volume density of radiation of a definite frequency uv. Summing up for all parts of the spec trum we get:

(22) Further by combining equations (9) and (20) we have:

•  (K.+  K/)  dfi,  (23) 
  

u,=— I i

ij

and finally for unpolarized radiation uniformly distributed in all directions:

STT K, u, = - (24)

CHAPTER II

RADIATION AT THERMODYNAMIC EQUILIBRIUM. KIRCHHOFF'S LAW. BLACK RADIATION

24. We shall now apply the laws enunciated in the last chap ter to the special case of thermodynamic equilibrium, and hence we begin our consideration by stating a certain consequence of the second principle of thermodynamics: A system of bodies of arbitrary nature, shape, and position which is at rest and is sur rounded by a rigid cover impermeable to heat will, no matter what its initial state may be, pass in the course of time into a permanent state, in which the temperature of all bodies of the system is the same. This is the state of thermodynamic equilib rium, in which the entropy of the system has the maximum value compatible with the total energy of the system as fixed by the initial conditions. This state being reached, no further increase in entropy is possible.

In certain cases it may happen that, under the given conditions, the entropy can assume not only one but several maxima, of which one is the absolute one, the others having only a relative significance.1 In these cases every state corresponding to a max imum value of the entropy represents a state of thermodynamic equilibrium of the system. But only one of them, the one cor responding to the absolute maximum of entropy, represents the absolutely stable equilibrium. All the others are in a certain sense unstable, inasmuch as a suitable, however small, distur bance may produce in the system a permanent change in the equilibrium in the direction of the absolutely stable equilibrium. An example of this is offered by supersaturated steam enclosed in a rigid vessel or by any explosive substance. We shall also meet such unstable equilibria in the case of radiation phenomena (Sec. 52).

1 See, e.g., M. Planck, Vorlesungen uber Thermodynamik, Leipzig, Veit and Comp., 1911 (or English Translation, Longmans Green & Co.), Sees. 165 and 189, et seq.

22

RADIATION AT THERMODYNAMIC EQUILIBRIUM 23

25. We shall now, as in the previous chapter, assume that we are dealing with homogeneous isotropic media whose condition depends only on the temperature, and we shall inquire what laws the radiation phenomena in them must obey in order to be con sistent with the deduction from the second principle mentioned in the preceding section. The means of answering this inquiry is supplied by the investigation of the state of thermodynamic equilibrium of one or more of such media, this investigation to be conducted by applying the conceptions and laws established in the last chapter.

We shall begin with the simplest case, that of a single medium extending very far in all directions of space, and, like all systems we shall here consider, being surrounded by a rigid cover imper meable to heat. For the present we shall assume that the medium has finite coefficients of absorption, emission, and scattering.

Let us consider, first, points of the medium that are far away from the surface. At such points the influence of the surface is, of course, vanishingly small and from the homogeneity and the isotropy of the medium it will follow that in a state of thermody namic equilibrium the radiation of heat has everywhere and in all directions the same properties. Then K,,, the specific intensity of radiation of a plane polarized ray of frequency v (Sec. 17), must be independent of the azimuth of the plane of polarization as well as of position and direction of the ray. Hence to each pencil of rays starting at an element of area da and diverging within a conical element dti corresponds an exactly equal pencil of oppo site direction converging within the same conical element toward the element of area.

Now the condition of thermodynamic equilibrium requires that the temperature shall be everywhere the same and shall not vary in time. Therefore in any given arbitrary time just as much radiant heat must be absorbed as is emitted in each vol ume-element of the medium. For the heat of the body depends only on the heat radiation, since, on account of the uniformity in temperature, no conduction of heat takes place. This condition is not influenced by the phenomenon of scattering, because scat tering refers only to a change in direction of the energy radiated, not to a creation or destruction of it. We shall, therefore, cal-

24 FUNDAMENTAL FACTS AND DEFINITIONS

culate the energy emitted and absorbed in the time dt by a volume-element v.

According to equation (2) the energy emitted has the value

CO

• dt V'S-JT I €„ dv

V-STT I

Jo

where €„, the coefficient of emission of the medium, depends only on the frequency v and on the temperature in addition to the chemical nature of the medium.

26. For the calculation of the energy absorbed we shall employ the same reasoning as was illustrated by Fig. 1 (Sec. 22) and shall retain the notation there used. The radiant energy absorbed by the volume-element v in the time dt is found by con sidering the intensities of all the rays passing through the element v and taking that fraction of each of these rays which is absorbed in v. Now, according to (19), the conical element that starts from da and cuts out of the volume v a part equal to fs has the intensity (energy radiated per unit time)

da- ~2-K

or, according to (12), by considering the different parts of the spectrum separately:

2 da

Hence the intensity of a monochromatic ray is:

2 da * K, dv.

r2

The amount of energy of this ray absorbed in the distance s in the time dt is, according to (4),

dta,,s2da 9 K, dv. r2

Hence the absorbed part of the energy of this small cone of rays, as found by integrating over all frequencies, is:

RADIATION AT THERMODYNAMIC EQUILIBRIUM 25

When this expression is summed up over all the different cross- sections / of the conical elements starting at da and passing through v, it is evident that S/s = v, and when we sum up over all elements da of the spherical surface of radius r we have

Cda = J r* =

Thus for the total radiant energy absorbed in the time dt by the volume-element v the following expression is found:

00

f «,K,

Jo

dtvSir I av K, dv. (25)

Jo

By equating the emitted and absorbed energy we obtain:

oo

= I OL K,, dv.

f *, ^ = r

•Jo tJ o

A similar relation may be obtained for the separate parts of the spectrum. For the energy emitted and the energy absorbed in the state of thermodynamic equilibrium are equal, not only for the entire radiation of the whole spectrum, but also for each monochro matic radiation. This is readily seen from the following. The magnitudes of ev, «„, and Ky are independent of position. Hence, if for any single color the absorbed were not equal to the emitted energy, there would be everywhere in the whole medium a con tinuous increase or decrease of the energy radiation of that particular color at the expense of the other colors. This would be contradictory to the condition that K,, for each separate frequency does not change with the time. We have therefore for each frequency the relation:

e, = a, K,, or (26)

K,= — ' (27)

av

i.e. : in the interior of a medium in a state of thermodynamic equi librium the specific intensity of radiation of a certain frequency is equal to the coefficient of emission divided by the coefficient of absorp tion of the medium for this frequency.

26 FUNDAMENTAL FACTS AND DEFINITIONS

27. Since €„ and av depend only on the nature of the medium, the temperature, and the frequency v, the intensity of radiation of a definite color in the state of thermodynamic equilibrium is completely defined by the nature of the medium and the tempera ture. An exceptional case is when <*„ = (), that is, when the medium does not at all absorb the color in question. Since K, cannot become infinitely large, a first consequence of this is that in that case e, = 0 also, that is, a medium does not emit any color which it does not absorb. A second consequence is that if €v and OLV both vanish, equation (26) is satisfied by every value of K,,. In a medium which is diathermanous for a certain color thermodynamic equilibrium can exist for any intensity of radiation whatever of that color.

This supplies an immediate illustration of the cases spoken of before (Sec. 24), where, for a given value of the total energy of a system enclosed by a rigid cover impermeable to heat, several states of equilibrium can exist, corresponding to several relative maxima of the entropy. That is to say, since the intensity of radiation of the particular color in the state of thermodynamic equilibrium is quite independent of the temperature of a medium which is diathermanous for this color, the given total energy may be arbitrarily distributed between radiation of that color and the heat of the body, without making thermodynamic equilibrium impossible. Among all these distributions there is one particular one, corresponding to the absolute maximum of entropy, which represents absolutely stable equilibrium. This one, unlike all the others, which are in a certain sense unstable, has the property of not being appreciably affected by a small disturbance. Indeed we shall see later (Sec. 48) that among the infinite number of

values, which the quotient — can have, if numerator and denom inator both vanish, 'there exists one particular one which depends in a definite way on the nature of the medium, the frequency v, and the temperature. This distinct value of the fraction is accordingly called the stable intensity of radiation K,, in the me dium, which at the temperature in question is diathermanous for rays of the frequency v.

Everything that has just been said of a medium which is dia thermanous for a certain kind of rays holds true for an absolute

RADIATION AT THERMODYNAMIC EQUILIBRIUM 27

vacuum, which is a medium diathermanous for rays of all kinds, the only difference being that one cannot speak of the heat and the temperature of an absolute vacuum in any definite sense.

For the present we again shall put the special case of diather mancy aside and assume that all the media considered have a finite coefficient of absorption.

28. Let us now consider briefly the phenomenon of scattering at thermodynamic equilibrium. Every ray meeting the volume- element v suffers there, apart from absorption, a certain weaken ing of its intensity because a certain fraction of its energy is diverted in different directions. The value of the total energy of scattered radiation received and diverted, in the time dt by the volume-element v in all directions, may be calculated from expression (3) in exactly the same way as the value of the absorbed energy was calculated in Sec. 26. Hence we get an expression similar to (25), namely,

CO

ft K, dv. (28)