II Radiation at Thermodynamic Equilibrium. Kirchhoﬀ’s Law. Black Radiation

Chapter II
Radiation at Thermodynamic Equilibrium. Kirchhoﬀ’s Law. Black Radiation

24. We shall now apply the laws enunciated in the last chapter 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 surrounded 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 equilibrium, 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.⁷ In these cases every state corresponding to a maximum value of the entropy represents a state of thermodynamic equilibrium of the system. But only one of them, the one corresponding 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, disturbance 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).

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 consistent 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 impermeable 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 thermodynamic equilibrium the radiation of heat has everywhere and in all directions the same properties. Then $K_{ν}$ , the speciﬁc intensity of radiation of a plane polarized ray of frequency $ν$ (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 $d σ$ and diverging within a conical element $d Ω$ corresponds an exactly equal pencil of opposite 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 volume-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 scattering refers only to a change in direction of the energy radiated, not to a creation or destruction of it. We shall, therefore, calculate the energy emitted and absorbed in the time $d t$ by a volume-element $v$ .

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

d t v \cdot 8 π \int_{0}^{\infty} ε_{ν} d ν

where

ε_{ν}

, the coeﬃcient of emission of the medium, depends only on the frequency

ν

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 $d t$ is found by considering 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 $d σ$ and cuts out of the volume $v$ a part equal to $f s$ has the intensity (energy radiated per unit time)

d σ \cdot \frac{f}{r^{2}} \cdot K

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

2 d σ \frac{f}{r^{2}} \int_{0}^{\infty} K_{ν} d ν .

Hence the intensity of a monochromatic ray is:

2 d σ \frac{f}{r^{2}} K_{ν} d ν .

The amount of energy of this ray absorbed in the distance

s

in the time

d t

is, according to (4),

d t α_{ν} s 2 d σ \frac{f}{r^{2}} K_{ν} d ν .

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

d t 2 d σ \frac{f s}{r^{2}} \int_{0}^{\infty} α_{ν} K_{ν} d ν .

When this expression is summed up over all the diﬀerent cross-sections

f

of the conical elements starting at

d σ

and passing through

v

, it is evident that

\sum f s = v

, and when we sum up over all elements

d σ

of the spherical surface of radius

r

we have

\int \frac{d σ}{r^{2}} = 4 π .

Thus for the total radiant energy absorbed in the time

d

t by the volume-element

v

the following expression is found:

d t v 8 π \int_{0}^{\infty} α_{ν} K_{ν} d ν .

(25)

By equating the emitted and absorbed energy we obtain:

\int_{0}^{\infty} ε_{ν} d ν = \int_{0}^{\infty} α_{ν} K_{ν} d ν .

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 monochromatic radiation. This is readily seen from the following. The magnitudes of $ε_{ν}$ , $α_{ν}$ , and $K_{ν}$ 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 continuous 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 $K_{ν}$ for each separate frequency does not change with the time. We have therefore for each frequency the relation:

\begin{align} ε_{ν} & = α_{ν} K_{ν}, or & (26) \\ K_{ν} & = \frac{ε_{ν}}{α_{ν}}, & (27) \end{align}

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

27. Since $ε_{ν}$ and $α_{ν}$ depend only on the nature of the medium, the temperature, and the frequency $ν$ , 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 temperature. An exceptional case is when $α_{ν} = 0$ , that is, when the medium does not at all absorb the color in question. Since $K_{ν}$ cannot become inﬁnitely large, a ﬁrst consequence of this is that in that case $ε_{ν} = 0$ also, that is, a medium does not emit any color which it does not absorb. A second consequence is that if $ε_{ν}$ and $α_{ν}$ both vanish, equation (26) is satisﬁed by every value of $K_{ν}$ . 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 $\frac{ε_{ν}}{α_{ν}}$ can have, if numerator and denominator both vanish, there exists one particular one which depends in a deﬁnite way on the nature of the medium, the frequency $ν$ , and the temperature. This distinct value of the fraction is accordingly called the stable intensity of radiation $K_{ν}$ , in the medium, which at the temperature in question is diathermanous for rays of the frequency $ν$ .

Everything that has just been said of a medium which is diathermanous for a certain kind of rays holds true for an absolute 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 diathermancy 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 weakening 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 $d t$ 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,

d t v 8 π \int_{0}^{\infty} β_{ν} K_{ν} d ν .

(28)

The question as to what becomes of this energy is readily answered. On account of the isotropy of the medium, the energy scattered in $v$ and given by (28) is radiated uniformly in all directions just as in the case of the energy entering $v$ . Hence that part of the scattered energy received in $v$ which is radiated out in a cone of solid angle $d Ω$ is obtained by multiplying the last expression by $\frac{d Ω}{4 π}$ . This gives

2 d t v d Ω \int_{0}^{\infty} β_{ν} K_{ν} d ν,

and, for monochromatic plane polarized radiation,

d t v d Ω β_{ν} K_{ν} d ν .

(29)

Here it must be carefully kept in mind that this uniformity of radiation in all directions holds only for all rays striking the element $v$ taken together; a single ray, even in an isotropic medium, is scattered in diﬀerent directions with diﬀerent intensities and diﬀerent directions of polarization. (See end of Sec. 8.)

It is thus found that, when thermodynamic equilibrium of radiation exists inside of the medium, the process of scattering produces, on the whole, no eﬀect. The radiation falling on a volume-element from all sides and scattered from it in all directions behaves exactly as if it had passed directly through the volume-element without the least modiﬁcation. Every ray loses by scattering just as much energy as it regains by the scattering of other rays.

Fig. 2.

29. We shall now consider from a diﬀerent point of view the radiation phenomena in the interior of a very extended homogeneous isotropic medium which is in thermodynamic equilibrium. That is to say, we shall conﬁne our attention, not to a deﬁnite volume-element, but to a deﬁnite pencil, and in fact to an elementary pencil (Sec. 21). Let this pencil be speciﬁed by the inﬁnitely small focal plane $d σ$ at the point $O$ (Fig. 2), perpendicular to the axis of the pencil, and by the solid angle $d Ω$ , and let the radiation take place toward the focal plane in the direction of the arrow. We shall consider exclusively rays which belong to this pencil.

The energy of monochromatic plane polarized radiation of the pencil considered passing in unit time through $d σ$ is represented, according to (11), since in this case $d t = 1$ , $θ = 0$ , by

d σ d Ω K_{ν} d ν .

(30)

The same value holds for any other cross-section of the pencil. For ﬁrst, $K_{ν} d ν$ has everywhere the same magnitude (Sec. 25), and second, the product of any right section of the pencil and the solid angle at which the focal plane $d σ$ is seen from this section has the constant value $d σ d Ω$ , since the magnitude of the cross-section increases with the distance from the vertex $O$ of the pencil in the proportion in which the solid angle decreases. Hence the radiation inside of the pencil takes place just as if the medium were perfectly diathermanous.

On the other hand, the radiation is continuously modiﬁed along its path by the eﬀect of emission, absorption, and scattering. We shall consider the magnitude of these eﬀects separately.

30. Let a certain volume-element of the pencil be bounded by two cross-sections at distances equal to $r_{0}$ (of arbitrary length) and $r_{0} + d r_{0}$ respectively from the vertex $O$ . The volume will be represented by $d r_{0} \cdot r_{0}^{2} d Ω$ . It emits in unit time toward the focal plane $d σ$ at $O$ a certain quantity $E$ of energy of monochromatic plane polarized radiation. $E$ may be obtained from (1) by putting

d t = 1, d τ = d r_{0} r_{0}^{2} d Ω, d Ω = \frac{d σ}{r_{0}^{2}}

and omitting the numerical factor

2

. We thus get

E = d r_{0} \cdot d Ω d σ ε_{ν} d ν .

(31)

Of the energy $E$ , however, only a fraction $E_{0}$ reaches $O$ , since in every inﬁnitesimal element of distance $s$ which it traverses before reaching $O$ the fraction $(α_{ν} + β_{ν}) s$ is lost by absorption and scattering. Let $E_{r}$ represent that part of $E$ which reaches a cross-section at a distance $r$ ( $< r_{0}$ ) from $O$ . Then for a small distance $s = d r$ we have

E_{r + d r} - E_{r} = E_{r} (α_{ν} + β_{ν}) d r,

or,

\frac{d E_{r}}{d r} = E_{r} (α_{ν} + β_{ν}),

and, by integration,

E_{r} = E e^{(α_{ν} + β_{ν}) (r - r_{0})}

since, for

r = r_{0}

E_{r} = E

is given by equation (31). From this, by putting

r = 0

, the energy emitted by the volume-element at

r_{0}

which reaches

O

is found to be

E_{0} = E e^{- (α_{ν} + β_{ν}) r_{0}} = d r_{0} d Ω d σ ε_{ν} e^{- (α_{ν} + β_{ν}) r_{0}} d ν .

(32)

All volume-elements of the pencils combined produce by their emission an amount of energy reaching $d σ$ equal to

d Ω d σ d ν ε_{ν} \int_{0}^{\infty} d r_{0} e^{- (α_{ν} + β_{ν}) r_{0}} = d Ω d σ \frac{ε_{ν}}{α_{ν} + β_{ν}} d ν .

(33)

31. If the scattering did not aﬀect the radiation, the total energy reaching $d σ$ would necessarily consist of the quantities of energy emitted by the diﬀerent volume-elements of the pencil, allowance being made, however, for the losses due to absorption on the way. For $β_{ν} = 0$ expressions (33) and (30) are identical, as may be seen by comparison with (27). Generally, however, (30) is larger than (33) because the energy reaching $d σ$ contains also some rays which were not at all emitted from elements inside of the pencil, but somewhere else, and have entered later on by scattering. In fact, the volume-elements of the pencil do not merely scatter outward the radiation which is being transmitted inside the pencil, but they also collect into the pencil rays coming from without. The radiation $E'$ thus collected by the volume-element at $r_{0}$ is found, by putting in (29),

d t = 1, ν = d r_{0} d Ω r_{0}^{2}, d Ω = \frac{d σ}{r_{0}^{2}},

to be

E' = d r_{0} d Ω d σ β_{ν} K_{ν} d ν .

This energy is to be added to the energy $E$ emitted by the volume-element, which we have calculated in (31). Thus for the total energy contributed to the pencil in the volume-element at $r_{0}$ we ﬁnd:

E + E' = d r_{0} d Ω d σ (ε_{ν} + β_{ν} K_{ν}) d ν .

The part of this reaching

O

is, similar to (32):

d r_{0} d Ω d σ (ε_{ν} + β_{ν} K_{ν}) d ν e^{- r_{0} (α_{ν} + β_{ν})} .

Making due allowance for emission and collection of scattered rays entering on the way, as well as for losses by absorption and scattering, all volume-elements of the pencil combined give for the energy ultimately reaching

d σ

d Ω d σ (ε_{ν} + β_{ν} K_{ν}) d ν \int_{0}^{\infty} d r_{0} e^{- r_{0} (α_{ν} + β_{ν})} = d Ω d σ \frac{ε_{ν} + β_{ν} K_{ν}}{α_{ν} + β_{ν}} d ν,

and this expression is really exactly equal to that given by (30), as may be seen by comparison with (26).

32. The laws just derived for the state of radiation of a homogeneous isotropic medium when it is in thermodynamic equilibrium hold, so far as we have seen, only for parts of the medium which lie very far away from the surface, because for such parts only may the radiation be considered, by symmetry, as independent of position and direction. A simple consideration, however, shows that the value of $K_{ν}$ , which was already calculated and given by (27), and which depends only on the temperature and the nature of the medium, gives the correct value of the intensity of radiation of the frequency considered for all directions up to points directly below the surface of the medium. For in the state of thermodynamic equilibrium every ray must have just the same intensity as the one travelling in an exactly opposite direction, since otherwise the radiation would cause a unidirectional transport of energy. Consider then any ray coming from the surface of the medium and directed inward; it must have the same intensity as the opposite ray, coming from the interior. A further immediate consequence of this is that the total state of radiation of the medium is the same on the surface as in the interior.

33. While the radiation that starts from a surface element and is directed toward the interior of the medium is in every respect equal to that emanating from an equally large parallel element of area in the interior, it nevertheless has a diﬀerent history. That is to say, since the surface of the medium was assumed to be impermeable to heat, it is produced only by reﬂection at the surface of radiation coming from the interior. So far as special details are concerned, this can happen in very diﬀerent ways, depending on whether the surface is assumed to be smooth, i.e., in this case reﬂecting, or rough, e.g., white (Sec. 10). In the ﬁrst case there corresponds to each pencil which strikes the surface another perfectly deﬁnite pencil, symmetrically situated and having the same intensity, while in the second case every incident pencil is broken up into an inﬁnite number of reﬂected pencils, each having a diﬀerent direction, intensity, and polarization. While this is the case, nevertheless the rays that strike a surface-element from all diﬀerent directions with the same intensity $K_{ν}$ also produce, all taken together, a uniform radiation of the same intensity $K_{ν}$ , directed toward the interior of the medium.

34. Hereafter there will not be the slightest diﬃculty in dispensing with the assumption made in Sec. 25 that the medium in question extends very far in all directions. For after thermodynamic equilibrium has been everywhere established in our medium, the equilibrium is, according to the results of the last paragraph, in no way disturbed, if we assume any number of rigid surfaces impermeable to heat and rough or smooth to be inserted in the medium. By means of these the whole system is divided into an arbitrary number of perfectly closed separate systems, each of which may be chosen as small as the general restrictions stated in Sec. 2 permit. It follows from this that the value of the speciﬁc intensity of radiation $K_{ν}$ given in (27) remains valid for the thermodynamic equilibrium of a substance enclosed in a space as small as we please and of any shape whatever.

35. From the consideration of a system consisting of a single homogeneous isotropic substance we now pass on to the treatment of a system consisting of two diﬀerent homogeneous isotropic substances contiguous to each other, the system being, as before, enclosed by a rigid cover impermeable to heat. We consider the state of radiation when thermodynamic equilibrium exists, at ﬁrst, as before, with the assumption that the media are of considerable extent. Since the equilibrium is nowise disturbed, if we think of the surface separating the two media as being replaced for an instant by an area entirely impermeable to heat radiation, the laws of the last paragraphs must hold for each of the two substances separately. Let the speciﬁc intensity of radiation of frequency $ν$ polarized in an arbitrary plane be $K_{ν}$ in the ﬁrst substance (the upper one in Fig. 3), and $K'$ in the second, and, in general, let all quantities referring to the second substance be indicated by the addition of an accent. Both of the quantities $K_{ν}$ and $K'$ depend, according to equation (27), only on the temperature, the frequency $ν$ , and the nature of the two substances, and these values of the intensities of radiation hold up to very small distances from the bounding surface of the substances, and hence are entirely independent of the properties of this surface.

36. We shall now suppose, to begin with, that the bounding surface of the media is smooth (Sec. 9). Then every ray coming from the ﬁrst medium and falling on the bounding surface is divided into two rays, the reﬂected and the transmitted ray. The directions of these two rays vary with the angle of incidence and the color of the incident ray; the intensity also varies with its polarization. Let us denote by $ρ$ (coeﬃcient of reﬂection) the fraction of the energy reﬂected, then the fraction transmitted is $(1 - ρ)$ , $ρ$ depending on the angle of incidence, the frequency, and the polarization of the incident ray. Similar remarks apply to $ρ'$ the coeﬃcient of reﬂection of a ray coming from the second medium and falling on the bounding surface.

Now according to (11) we have for the monochromatic plane polarized radiation of frequency $ν$ , emitted in time $d t$ toward the ﬁrst medium (in the direction of the feathered arrow upper left

Fig. 3.

hand in Fig. 3), from an element $d σ$ of the bounding surface and contained in the conical element $d Ω$ ,

d t d σ cos θ d Ω K_{ν} d ν,

(34)

where

d Ω = sin θ d θ d φ .

(35)

This energy is supplied by the two rays which come from the ﬁrst and the second medium and are respectively reﬂected from or transmitted by the element $d σ$ in the corresponding direction (the unfeathered arrows). (Of the element $d σ$ only the one point $O$ is indicated.) The ﬁrst ray, according to the law of reﬂection, continues in the symmetrically situated conical element $d Ω$ , the second in the conical element

d Ω' = sin θ' d θ' d φ'

(36)

where, according to the law of refraction,

φ' = φ and \frac{sin θ}{sin θ'} = \frac{q}{q'} .

(37)

If we now assume the radiation (34) to be polarized either in the plane of incidence or at right angles thereto, the same will be true for the two radiations of which it consists, and the radiation coming from the ﬁrst medium and reﬂected from $d σ$ contributes the part

ρ d t d σ cos θ d Ω K_{ν} d ν

(38)

while the radiation coming from the second medium and transmitted through $d σ$ contributes the part

(1 - ρ') d t d σ cos θ' d Ω' K' d ν .

(39)

The quantities $d t$ , $d σ$ , $ν$ and $d ν$ are written without the accent, because they have the same values in both media.

By adding (38) and (39) and equating their sum to the expression (34) we ﬁnd

ρ cos θ d Ω K_{ν} + (1 - ρ') cos θ' d Ω' K' = cos θ d Ω K_{ν} .

Now from (37) we have

\frac{cos θ d θ}{q} = \frac{cos θ' d θ'}{q'}

and further by (35) and (36)

d Ω' cos θ' = \frac{d Ω cos θ q'^{2}}{q^{2}} .

Therefore we ﬁnd

ρ K_{ν} + (1 - ρ') \frac{q'^{2}}{q^{2}} K' = K

\frac{K_{ν}}{K'} \cdot \frac{q^{2}}{q'^{2}} = \frac{1 - ρ'}{1 - ρ} .

37. In the last equation the quantity on the left side is independent of the angle of incidence $θ$ and of the particular kind of polarization; hence the same must be true for the right side. Hence, whenever the value of this quantity is known for a single angle of incidence and any deﬁnite kind of polarization, this value will remain valid for all angles of incidence and all kinds of polarization. Now in the special case when the rays are polarized at right angles to the plane of incidence and strike the bounding surface at the angle of polarization, $ρ = 0$ , and $ρ' = 0$ . The expression on the right side of the last equation then becomes $1$ ; hence it must always be $1$ and we have the general relations:

ρ = ρ'

(40)

and

q^{2} K_{ν} = q'^{2} K' .

(41)

38. The ﬁrst of these two relations, which states that the coeﬃcient of reﬂection of the bounding surface is the same on both sides, is a special case of a general law of reciprocity ﬁrst stated by Helmholtz.⁸ According to this law the loss of intensity which a ray of deﬁnite color and polarization suﬀers on its way through any media by reﬂection, refraction, absorption, and scattering is exactly equal to the loss suﬀered by a ray of the same intensity, color, and polarization pursuing an exactly opposite path. An immediate consequence of this law is that the radiation striking the bounding surface of any two media is always transmitted as well as reﬂected equally on both sides, for every color, direction, and polarization.

39. The second formula (41) establishes a relation between the intensities of radiation in the two media, for it states that, when thermodynamic equilibrium exists, the speciﬁc intensities of radiation of a certain frequency in the two media are in the inverse ratio of the squares of the velocities of propagation or in the direct ratio of the squares of the indices of refraction.⁹

By substituting for $K_{ν}$ its value from (27) we obtain the following theorem: The quantity

q^{2} K_{ν} = q^{2} \frac{ε_{ν}}{α_{ν}}

(42)

does not depend on the nature of the substance, and is, therefore, a universal function of the temperature $T$ and the frequency $ν$ alone.

The great importance of this law lies evidently in the fact that it states a property of radiation which is the same for all bodies in nature, and which need be known only for a single arbitrarily chosen body, in order to be stated quite generally for all bodies. We shall later on take advantage of the opportunity oﬀered by this statement in order actually to calculate this universal function (Sec. 165).

40. We now consider the other case, that in which the bounding surface of the two media is rough. This case is much more general than the one previously treated, inasmuch as the energy of a pencil directed from an element of the bounding surface into the ﬁrst medium is no longer supplied by two deﬁnite pencils, but by an arbitrary number, which come from both media and strike the surface. Here the actual conditions may be very complicated according to the peculiarities of the bounding surface, which moreover may vary in any way from one element to another. However, according to Sec. 35, the values of the speciﬁc intensities of radiation $K_{ν}$ and $K'$ remain always the same in all directions in both media, just as in the case of a smooth bounding surface. That this condition, necessary for thermodynamic equilibrium, is satisﬁed is readily seen from Helmholtz’s law of reciprocity, according to which, in the case of stationary radiation, for each ray striking the bounding surface and diﬀusely reﬂected from it on both sides, there is a corresponding ray at the same point, of the same intensity and opposite direction, produced by the inverse process at the same point on the bounding surface, namely by the gathering of diﬀusely incident rays into a deﬁnite direction, just as is the case in the interior of each of the two media.

41. We shall now further generalize the laws obtained. First, just as in Sec. 34, the assumption made above, namely, that the two media extend to a great distance, may be abandoned since we may introduce an arbitrary number of bounding surfaces without disturbing the thermodynamic equilibrium. Thereby we are placed in a position enabling us to pass at once to the case of any number of substances of any size and shape. For when a system consisting of an arbitrary number of contiguous substances is in the state of thermodynamic equilibrium, the equilibrium is in no way disturbed, if we assume one or more of the surfaces of contact to be wholly or partly impermeable to heat. Thereby we can always reduce the case of any number of substances to that of two substances in an enclosure impermeable to heat, and, therefore, the law may be stated quite generally, that, when any arbitrary system is in the state of thermodynamic equilibrium, the speciﬁc intensity of radiation $K_{ν}$ is determined in each separate substance by the universal function (42).

42. We shall now consider a system in a state of thermodynamic equilibrium, contained within an enclosure impermeable to heat and consisting of $n$ emitting and absorbing adjacent bodies of any size and shape whatever. As in Sec. 36, we again conﬁne our attention to a monochromatic plane polarized pencil which proceeds from an element $d σ$ of the bounding surface of the two media in the direction toward the ﬁrst medium (Fig. 3, feathered arrow) within the conical element $d Ω$ . Then, as in (34), the energy supplied by the pencil in unit time is

d σ cos θ d Ω K_{ν} d ν = I .

(43)

This energy of radiation $I$ consists of a part coming from the ﬁrst medium by regular or diﬀuse reﬂection at the bounding surface and of a second part coming through the bounding surface from the second medium. We shall, however, not stop at this mode of division, but shall further subdivide $I$ according to that one of the $n$ media from which the separate parts of the radiation $I$ have been emitted. This point of view is distinctly diﬀerent from the preceding, since, e.g., the rays transmitted from the second medium through the bounding surface into the pencil considered have not necessarily been emitted in the second medium, but may, according to circumstances, have traversed a long and very complicated path through diﬀerent media and may have undergone therein the eﬀect of refraction, reﬂection, scattering, and partial absorption any number of times. Similarly the rays of the pencil, which coming from the ﬁrst medium are reﬂected at $d σ$ , were not necessarily all emitted in the ﬁrst medium. It may even happen that a ray emitted from a certain medium, after passing on its way through other media, returns to the original one and is there either absorbed or emerges from this medium a second time.

We shall now, considering all these possibilities, denote that part of $I$ which has been emitted by volume-elements of the ﬁrst medium by $I_{1}$ no matter what paths the diﬀerent constituents have pursued, that which has been emitted by volume-elements of the second medium by $I_{2}$ , etc. Then since every part of $I$ must have been emitted by an element of some body, the following equation must hold,

I = I_{1} + I_{2} + I_{3} + \dots I_{n} .

(44)

43. The most adequate method of acquiring more detailed information as to the origin and the paths of the diﬀerent rays of which the radiations $I_{1}$ , $I_{2}$ , $I_{3}$ , $\dots I_{n}$ consist, is to pursue the opposite course and to inquire into the future fate of that pencil, which travels exactly in the opposite direction to the pencil $I$ and which therefore comes from the ﬁrst medium in the cone $d Ω$ and falls on the surface element $d σ$ of the second medium. For since every optical path may also be traversed in the opposite direction, we may obtain by this consideration all paths along which rays can pass into the pencil $I$ , however complicated they may otherwise be. Let $J$ represent the intensity of this inverse pencil, which is directed toward the bounding surface and is in the same state of polarization. Then, according to Sec. 40,

J = I .

(45)

At the bounding surface $d σ$ the rays of the pencil $J$ are partly reﬂected and partly transmitted regularly or diﬀusely, and thereafter, travelling in both media, are partly absorbed, partly scattered, partly again reﬂected or transmitted to diﬀerent media, etc., according to the conﬁguration of the system. But ﬁnally the whole pencil $J$ after splitting into many separate rays will be completely absorbed in the $n$ media. Let us denote that part of $J$ which is ﬁnally absorbed in the ﬁrst medium by $J_{1}$ , that which is ﬁnally absorbed in the second medium by $J_{2}$ , etc., then we shall have

J = J_{1} + J_{2} + J_{3} + \dots + J_{n} .

Now the volume-elements of the $n$ media, in which the absorption of the rays of the pencil $J$ takes place, are precisely the same as those in which takes place the emission of the rays constituting the pencil $I$ , the ﬁrst one considered above. For, according to Helmholtz’s law of reciprocity, no appreciable radiation of the pencil $J$ can enter a volume-element which contributes no appreciable radiation to the pencil $I$ and vice versa.

Let us further keep in mind that the absorption of each volume-element is, according to (42), proportional to its emission and that, according to Helmholtz’s law of reciprocity, the decrease which the energy of a ray suﬀers on any path is always equal to the decrease suﬀered by the energy of a ray pursuing the opposite path. It will then be clear that the volume-elements considered absorb the rays of the pencil $J$ in just the same ratio as they contribute by their emission to the energy of the opposite pencil $I$ . Since, moreover, the sum $I$ of the energies given oﬀ by emission by all volume-elements is equal to the sum $J$ of the energies absorbed by all elements, the quantity of energy absorbed by each separate volume-element from the pencil $J$ must be equal to the quantity of energy emitted by the same element into the pencil $I$ . In other words: the part of a pencil $I$ which has been emitted from a certain volume of any medium is equal to the part of the pencil $J = I$ oppositely directed, which is absorbed in the same volume.

Hence not only are the sums $I$ and $J$ equal, but their constituents are also separately equal or

J_{1} = I_{1}, J_{2} = I_{2}, \dots J_{n} = I_{n} .

(46)

44. Following G. Kirchhoﬀ ¹⁰ we call the quantity $I_{2}$ , i.e., the intensity of the pencil emitted from the second medium into the ﬁrst, the emissive power $E$ of the second medium, while we call the ratio of $J_{2}$ to $J$ , i.e., that fraction of a pencil incident on the second medium which is absorbed in this medium, the absorbing power $A$ of the second medium. Therefore

E = I_{2} (≦ I), A = \frac{J_{2}}{J} (≦ 1) .

(47)

The quantities $E$ and $A$ depend (a) on the nature of the two media, (b) on the temperature, the frequency $ν$ , and the direction and the polarization of the radiation considered, (c) on the nature of the bounding surface and on the magnitude of the surface element $d σ$ and that of the solid angle $d Ω$ , (d) on the geometrical extent and the shape of the total surface of the two media, (e) on the nature and form of all other bodies of the system. For a ray may pass from the ﬁrst into the second medium, be partly transmitted by the latter, and then, after reﬂection somewhere else, may return to the second medium and may be there entirely absorbed.

With these assumptions, according to equations (46), (45), and (43), Kirchhoﬀ’s law holds,

\frac{E}{A} = I = d σ cos θ d Ω K_{ν} d ν,

(48)

i.e., the ratio of the emissive power to the absorbing power of any body is independent of the nature of the body. For this ratio is equal to the intensity of the pencil passing through the ﬁrst medium, which, according to equation (27), does not depend on the second medium at all. The value of this ratio does, however, depend on the nature of the ﬁrst medium, inasmuch as, according to (42), it is not the quantity $K_{ν}$ but the quantity $q^{2} K_{ν}$ , which is a universal function of the temperature and frequency. The proof of this law given by G. Kirchhoﬀ l. c. was later greatly simpliﬁed by E. Pringsheim.¹¹

45. When in particular the second medium is a black body (Sec. 10) it absorbs all the incident radiation. Hence in that case $J_{2} = J$ , $A = 1$ , and $E = I$ , i.e., the emissive power of a black body is independent of its nature. Its emissive power is larger than that of any other body at the same temperature and, in fact, is just equal to the intensity of radiation in the contiguous medium.

46. We shall now add, without further proof, another general law of reciprocity, which is closely connected with that stated at the end of Sec. 43 and which may be stated thus: When any emitting and absorbing bodies are in the state of thermodynamic equilibrium, the part of the energy of deﬁnite color emitted by a body $A$ , which is absorbed by another body $B$ , is equal to the part of the energy of the same color emitted by $B$ which is absorbed by $A$ . Since a quantity of energy emitted causes a decrease of the heat of the body, and a quantity of energy absorbed an increase of the heat of the body, it is evident that, when thermodynamic equilibrium exists, any two bodies or elements of bodies selected at random exchange by radiation equal amounts of heat with each other. Here, of course, care must be taken to distinguish between the radiation emitted and the total radiation which reaches one body from the other.

47. The law holding for the quantity (42) can be expressed in a diﬀerent form, by introducing, by means of (24), the volume density $u_{ν}$ of monochromatic radiation instead of the intensity of radiation $K_{ν}$ . We then obtain the law that, for radiation in a state of thermodynamic equilibrium, the quantity

u_{ν} q^{3}

(49)

is a function of the temperature $T$ and the frequency $ν$ , and is the same for all substances.¹² This law becomes clearer if we consider that the quantity

u_{ν} d ν \frac{q^{3}}{ν^{3}}

(50)

also is a universal function of $T$ , $ν$ , and $ν + d ν$ , and that the product $u_{ν} d ν$ is, according to (22), the volume density of the radiation whose frequency lies between $ν$ and $ν + d ν$ , while the quotient $\frac{q}{ν}$ represents the wave length of a ray of frequency $ν$ in the medium in question. The law then takes the following simple form: When any bodies whatever are in thermodynamic equilibrium, the energy of monochromatic radiation of a deﬁnite frequency, contained in a cubical element of side equal to the wave length, is the same for all bodies.

48. We shall ﬁnally take up the case of diathermanous (Sec. 12) media, which has so far not been considered. In Sec. 27 we saw that, in a medium which is diathermanous for a given color and is surrounded by an enclosure impermeable to heat, there can be thermodynamic equilibrium for any intensity of radiation of this color. There must, however, among all possible intensities of radiation be a deﬁnite one, corresponding to the absolute maximum of the total entropy of the system, which designates the absolutely stable equilibrium of radiation. In fact, in equation (27) the intensity of radiation $K_{ν}$ for $α_{ν} = 0$ and $ε_{ν} = 0$ assumes the value $\frac{0}{0}$ , and hence cannot be calculated from this equation. But we see also that this indeterminateness is removed by equation (41), which states that in the case of thermodynamic equilibrium the product $q^{2} K_{ν}$ has the same value for all substances. From this we ﬁnd immediately a deﬁnite value of $K_{ν}$ which is thereby distinguished from all other values. Furthermore the physical signiﬁcance of this value is immediately seen by considering the way in which that equation was obtained. It is that intensity of radiation which exists in a diathermanous medium, if it is in thermodynamic equilibrium when in contact with an arbitrary absorbing and emitting medium. The volume and the form of the second medium do not matter in the least, in particular the volume may be taken as small as we please. Hence we can formulate the following law: Although generally speaking thermodynamic equilibrium can exist in a diathermanous medium for any intensity of radiation whatever, nevertheless there exists in every diathermanous medium for a deﬁnite frequency at a deﬁnite temperature an intensity of radiation deﬁned by the universal function (42). This may be called the stable intensity, inasmuch as it will always be established, when the medium is exchanging stationary radiation with an arbitrary emitting and absorbing substance.

49. According to the law stated in Sec. 45, the intensity of a pencil, when a state of stable heat radiation exists in a diathermanous medium, is equal to the emissive power $E$ of a black body in contact with the medium. On this fact is based the possibility of measuring the emissive power of a black body, although absolutely black bodies do not exist in nature.¹³ A diathermanous cavity is enclosed by strongly emitting walls¹⁴ and the walls kept at a certain constant temperature $T$ . Then the radiation in the cavity, when thermodynamic equilibrium is established for every frequency $ν$ , assumes the intensity corresponding to the velocity of propagation $q$ in the diathermanous medium, according to the universal function (42). Then any element of area of the walls radiates into the cavity just as if the wall were a black body of temperature $T$ . The amount lacking in the intensity of the rays actually emitted by the walls as compared with the emission of a black body is supplied by rays which fall on the wall and are reﬂected there. Similarly every element of area of a wall receives the same radiation.

In fact, the radiation $I$ starting from an element of area of a wall consists of the radiation $E$ emitted by the element of area and of the radiation reﬂected from the element of area from the incident radiation $I$ , i.e., the radiation which is not absorbed $(1 - A) I$ . We have, therefore, in agreement with Kirchhoﬀ’s law (48),

I = E + (1 - A) I .

If we now make a hole in one of the walls of a size $d σ$ , so small that the intensity of the radiation directed toward the hole is not changed thereby, then radiation passes through the hole to the exterior where we shall suppose there is the same diathermanous medium as within. This radiation has exactly the same properties as if $d σ$ were the surface of a black body, and this radiation may be measured for every color together with the temperature $T$ .

50. Thus far all the laws derived in the preceding sections for diathermanous media hold for a deﬁnite frequency, and it is to be kept in mind that a substance may be diathermanous for one color and adiathermanous for another. Hence the radiation of a medium completely enclosed by absolutely reﬂecting walls is, when thermodynamic equilibrium has been established for all colors for which the medium has a ﬁnite coeﬃcient of absorption, always the stable radiation corresponding to the temperature of the medium such as is represented by the emission of a black body. Hence this is brieﬂy called “black” radiation.¹⁵ On the other hand, the intensity of colors for which the medium is diathermanous is not necessarily the stable black radiation, unless the medium is in a state of stationary exchange of radiation with an absorbing substance.

There is but one medium that is diathermanous for all kinds of rays, namely, the absolute vacuum, which to be sure cannot be produced in nature except approximately. However, most gases, e.g., the air of the atmosphere, have, at least if they are not too dense, to a suﬃcient approximation the optical properties of a vacuum with respect to waves of not too short length. So far as this is the case the velocity of propagation $q$ may be taken as the same for all frequencies, namely,

c = 3 \times 1 0^{10} \frac{cm}{sec} .

(51)

51. Hence in a vacuum bounded by totally reﬂecting walls any state of radiation may persist. But as soon as an arbitrarily small quantity of matter is introduced into the vacuum, a stationary state of radiation is gradually established. In this the radiation of every color which is appreciably absorbed by the substance has the intensity $K_{ν}$ corresponding to the temperature of the substance and determined by the universal function (42) for $q = c$ , the intensity of radiation of the other colors remaining indeterminate. If the substance introduced is not diathermanous for any color, e.g., a piece of carbon however small, there exists at the stationary state of radiation in the whole vacuum for all colors the intensity $K_{ν}$ of black radiation corresponding to the temperature of the substance. The magnitude of $K_{ν}$ regarded as a function of $ν$ gives the spectral distribution of black radiation in a vacuum, or the so-called normal energy spectrum, which depends on nothing but the temperature. In the normal spectrum, since it is the spectrum of emission of a black body, the intensity of radiation of every color is the largest which a body can emit at that temperature at all.

52. It is therefore possible to change a perfectly arbitrary radiation, which exists at the start in the evacuated cavity with perfectly reﬂecting walls under consideration, into black radiation by the introduction of a minute particle of carbon. The characteristic feature of this process is that the heat of the carbon particle may be just as small as we please, compared with the energy of radiation contained in the cavity of arbitrary magnitude. Hence, according to the principle of the conservation of energy, the total energy of radiation remains essentially constant during the change that takes place, because the changes in the heat of the carbon particle may be entirely neglected, even if its changes in temperature should be ﬁnite. Herein the carbon particle exerts only a releasing (auslösend) action. Thereafter the intensities of the pencils of diﬀerent frequencies originally present and having diﬀerent frequencies, directions, and diﬀerent states of polarization change at the expense of one another, corresponding to the passage of the system from a less to a more stable state of radiation or from a state of smaller to a state of larger entropy. From a thermodynamic point of view this process is perfectly analogous, since the time necessary for the process is not essential, to the change produced by a minute spark in a quantity of oxy-hydrogen gas or by a small drop of liquid in a quantity of supersaturated vapor. In all these cases the magnitude of the disturbance is exceedingly small and cannot be compared with the magnitude of the energies undergoing the resultant changes, so that in applying the two principles of thermodynamics the cause of the disturbance of equilibrium, viz. the carbon particle, the spark, or the drop, need not be considered. It is always a case of a system passing from a more or less unstable into a more stable state, wherein, according to the ﬁrst principle of thermodynamics, the energy of the system remains constant, and, according to the second principle, the entropy of the system increases.

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Chapter IIRadiation at Thermodynamic Equilibrium. Kirchhoﬀ’s Law. Black Radiation

Chapter II
Radiation at Thermodynamic Equilibrium. Kirchhoﬀ’s Law. Black Radiation