Wien’s Displacement Law

71. Though the manner in which the volume density $u$ and the specific intensity $K$ of black radiation depend on the temperature is determined by the Stefan-Boltzmann law, this law is of comparatively little use in finding the volume density $u_{\nu }$ corresponding to a definite frequency $\nu$, and the specific intensity of radiation $K_{\nu }$ of monochromatic radiation, which are related to each other by equation (24) and to $u$ and $K$ by equations (22) and (12). There remains as one of the principal problems of the theory of heat radiation the problem of determining the quantities $u_{\nu }$ and $K_{\nu }$ for black radiation in a vacuum and hence, according to (42), in any medium whatever, as functions of $\nu$ and $T$, or, in other words, to find the distribution of energy in the normal spectrum for any arbitrary temperature. An essential step in the solution of this problem is contained in the so-called “displacement law” stated by W. Wien,23 the importance of which lies in the fact that it reduces the functions $u_{\nu }$ and $K_{\nu }$ of the two arguments $\nu$ and $T$ to a function of a single argument.

The starting point of Wien’s displacement law is the following theorem. If the black radiation contained in a perfectly evacuated cavity with absolutely reflecting walls is compressed or expanded adiabatically and infinitely slowly, as described above in Sec. 68, the radiation always retains the character of black radiation, even without the presence of a carbon particle. Hence the process takes place in an absolute vacuum just as was calculated in Sec. 68 and the introduction, as a precaution, of a carbon particle is shown to be superfluous. But this is true only in this special case, not at all in the case described in Sec. 70.

The truth of the proposition stated may be shown as follows: Let the completely evacuated hollow cylinder, which is at the start filled with black radiation, be compressed adiabatically and infinitely slowly to a finite fraction of the original volume. If, now, the compression being completed, the radiation were no longer black, there would be no stable thermodynamic equilibrium of the radiation (Sec. 51). It would then be possible to produce a finite change at constant volume and constant total energy of radiation, namely, the change to the absolutely stable state of radiation, which would cause a finite increase of entropy. This change could be brought about by the introduction of a carbon particle, containing a negligible amount of heat as compared with the energy of radiation. This change, of course, refers only to the spectral density of radiation $u_{\nu }$, whereas the total density of energy $u$ remains constant. After this has been accomplished, we could, leaving the carbon particle in the space, allow the hollow cylinder to return adiabatically and infinitely slowly to its original volume and then remove the carbon particle. The system will then have passed through a cycle without any external changes remaining. For heat has been neither added nor removed, and the mechanical work done on compression has been regained on expansion, because the latter, like the radiation pressure, depends only on the total density $u$ of the energy of radiation, not on its spectral distribution. Therefore, according to the first principle of thermodynamics, the total energy of radiation is at the end just the same as at the beginning, and hence also the temperature of the black radiation is again the same. The carbon particle and its changes do not enter into the calculation, for its energy and entropy are vanishingly small compared with the corresponding quantities of the system. The process has therefore been reversed in all details; it may be repeated any number of times without any permanent change occurring in nature. This contradicts the assumption, made above, that a finite increase in entropy occurs; for such a finite increase, once having taken place, cannot in any way be completely reversed. Therefore no finite increase in entropy can have been produced by the introduction of the carbon particle in the space of radiation, but the radiation was, before the introduction and always, in the state of stable equilibrium.

72. In order to bring out more clearly the essential part of this important proof, let us point out an analogous and more or less obvious consideration. Let a cavity containing originally a vapor in a state of saturation be compressed adiabatically and infinitely slowly.

“Then on an arbitrary adiabatic compression the vapor remains always just in the state of saturation. For let us suppose that it becomes supersaturated on compression. After the compression to an appreciable fraction of the original volume has taken place, condensation of a finite amount of vapor and thereby a change into a more stable state, and hence a finite increase of entropy of the system, would be produced at constant volume and constant total energy by the introduction of a minute drop of liquid, which has no appreciable mass or heat capacity. After this has been done, the volume could again be increased adiabatically and infinitely slowly until again all liquid is evaporated and thereby the process completely reversed, which contradicts the assumed increase of entropy.”

Such a method of proof would be erroneous, because, by the process described, the change that originally took place is not at all completely reversed. For since the mechanical work expended on the compression of the supersaturated steam is not equal to the amount gained on expanding the saturated steam, there corresponds to a definite volume of the system when it is being compressed an amount of energy different from the one during expansion and therefore the volume at which all liquid is just vaporized cannot be equal to the original volume. The supposed analogy therefore breaks down and the statement made above in quotation marks is incorrect.

73. We shall now again suppose the reversible adiabatic process described in Sec. 68 to be carried out with the black radiation contained in the evacuated cavity with white walls and white bottom, by allowing the piston, which consists of absolutely reflecting metal, to move downward infinitely slowly, with the single difference that now there shall be no carbon particle in the cylinder. The process will, as we now know, take place exactly as there described, and, since no absorption or emission of radiation takes place, we can now give an account of the changes of color and intensity which the separate pencils of the system undergo. Such changes will of course occur only on reflection from the moving metallic reflector, not on reflection from the stationary walls and the stationary bottom of the cylinder.

If the reflecting piston moves down with the constant, infinitely small, velocity $v$, the monochromatic pencils striking it during the motion will suffer on reflection a change of color, intensity, and direction. Let us consider these different influences in order.24

74. To begin with, we consider the change of color which a monochromatic ray suffers by reflection from the reflector, which is

Fig. 5

moving with an infinitely small velocity. For this purpose we consider first the case of a ray which falls normally from below on the reflector and hence is reflected normally downward. Let the plane $A$ (Fig. 5) represent the position of the reflector at the time $t$, the plane $A\prime$ the position at the time $t+\delta t$, where the distance $AA\prime$ equals $v\delta t$, $v$ denoting the velocity of the reflector. Let us now suppose a stationary plane $B$ to be placed parallel to $A$ at a suitable distance and let us denote by $\lambda$ the wave length of the ray incident on the reflector and by $\lambda \prime$ the wave length of the ray reflected from it. Then at a time $t$ there are in the interval $AB$ in the vacuum containing the radiation $\frac{AB}{\lambda }$ waves of the incident and $\frac{AB}{\lambda \prime }$ waves of the reflected ray, as can be seen, e.g., by thinking of the electric field-strength as being drawn at the different points of each of the two rays at the time $t$ in the form of a sine curve. Reckoning both incident and reflected ray there are at the time $t$

waves in the interval between $A$ and $B$. Since this is a large number, it is immaterial whether the number is an integer or not.

Similarly at the time $t+\delta t$, when the reflector is at $A\prime$, there are

waves in the interval between $A\prime$ and $B$ all told.

The latter number will be smaller than the former, since in the shorter distance $A\prime B$ there is room for fewer waves of both kinds than in the longer distance $AB$. The remaining waves must have been expelled in the time $\delta t$ from the space between the stationary plane $B$ and the moving reflector, and this must have taken place through the plane $B$ downward; for in no other way could a wave disappear from the space considered.

Now $\nu \delta t$ waves pass in the time $\delta t$ through the stationary plane $B$ in an upward direction and $\nu \prime \delta t$ waves in a downward direction; hence we have for the difference

or, since

and

or, since $v$ is infinitely small compared with $c$,

75. When the radiation does not fall on the reflector normally but at an acute angle of incidence $\theta$, it is possible to pursue a very similar line of reasoning, with the difference that then $A$, the point of intersection of a definite ray $BA$ with the reflector at the time $t$, has not the same position on the reflector as the point of intersection, $A\prime$, of the same ray with the reflector at the time $t+\delta t$ (Fig. 6). The number of waves which lie in the interval $BA$ at the time $t$ is $\frac{BA}{\lambda }$. Similarly, at the time $t$ the number of waves in the interval $AC$ representing the distance of the point $A$ from a wave plane $CC\prime$, belonging to the reflected ray and stationary in the vacuum, is $\frac{AC}{\lambda \prime }$.

Hence there are, all told, at the time $t$ in the interval $BAC$

waves of the ray under consideration. We may further note that the angle of reflection $\theta \prime$ is not exactly equal to the angle

Fig. 6

of incidence, but is a little smaller as can be shown by a simple geometric consideration based on Huyghens’ principle. The difference of $\theta$ and $\theta \prime$, however, will be shown to be non-essential for our calculation. Moreover there are at the time $t+\delta t$, when the reflector passes through $A\prime$,

waves in the distance $BA\prime C\prime$. The latter number is smaller than the former and the difference must equal the total number of waves which are expelled in the time $\delta t$ from the space which is bounded by the stationary planes $BB\prime$ and $CC\prime$.

Now $\nu \delta t$ waves enter into the space through the plane $BB\prime$ in the time $\delta t$ and $\nu \prime \delta t$ waves leave the space through the plane $CC\prime$. Hence we have

but

Hence

This relation holds for any velocity $v$ of the moving reflector. Now, since in our case $v$ is infinitely small compared with $c$, we have the simpler expression

The difference between the two angles $\theta$ and $\theta \prime$ is in any case of the order of magnitude $\frac{v}{c}$; hence we may without appreciable error replace $\theta \prime$ by $\theta$, thereby obtaining the following expression for the frequency of the reflected ray for oblique incidence

76. From the foregoing it is seen that the frequency of all rays which strike the moving reflector are increased on reflection, when the reflector moves toward the radiation, and decreased, when the reflector moves in the direction of the incident rays ($v<0$). However, the total radiation of a definite frequency $\nu$ striking the moving reflector is by no means reflected as monochromatic radiation but the change in color on reflection depends also essentially on the angle of incidence $\theta$. Hence we may not speak of a certain spectral “displacement” of color except in the case of a single pencil of rays of definite direction, whereas in the case of the entire monochromatic radiation we must refer to a spectral “dispersion.” The change in color is the largest for normal incidence and vanishes entirely for grazing incidence.

77. Secondly, let us calculate the change in energy, which the moving reflector produces in the incident radiation, and let us consider from the outset the general case of oblique incidence. Let a monochromatic, infinitely thin, unpolarized pencil of rays, which falls on a surface element of the reflector at the angle of incidence $\theta$, transmit the energy $I\delta t$ to the reflector in the time $\delta t$. Then, ignoring vanishingly small quantities, the mechanical pressure of the pencil of rays normally to the reflector is, according to equation (64),

and to the same degree of approximation the work done from the outside on the incident radiation in the time $\delta t$ is

According to the principle of the conservation of energy this amount of work must reappear in the energy of the reflected radiation. Hence the reflected pencil has a larger intensity than the incident one. It produces, namely, in the time $\delta t$ the energy25

Hence we may summarize as follows: By the reflection of a monochromatic unpolarized pencil, incident at an angle $\theta$ on a reflector moving toward the radiation with the infinitely small velocity $v$, the radiant energy $I\delta t$, whose frequencies extend from $\nu$ to $\nu +d\nu$, is in the time $\delta t$ changed into the radiant energy $I\prime \delta t$ with the interval of frequency $(\nu \prime ,\nu \prime +d\nu \prime )$, where $I\prime$ is given by (85), $\nu \prime$ by (83), and accordingly $d\nu \prime$, the spectral breadth of the reflected pencil, by

A comparison of these values shows that

The absolute value of the radiant energy which has disappeared in this change is, from equation (13),

and hence the absolute value of the radiant energy which has been formed is, according to (85),

Strictly speaking these last two expressions would require an infinitely small correction, since the quantity $I$ from equation (88) represents the energy radiation on a stationary element of area $d\sigma$, while, in reality, the incident radiation is slightly increased by the motion of $d\sigma$ toward the incident pencil. The corresponding additional terms may, however, be omitted here without appreciable error, since the correction caused by them would consist merely of the addition to the energy change here calculated of a comparatively infinitesimal energy change of the same kind with an external work that is infinitesimal of the second order.

78. As regards finally the changes in direction, which are imparted to the incident ray by reflection from the moving reflector, we need not calculate them at all at this stage. For if the motion of the reflector takes place sufficiently slowly, all irregularities in the direction of the radiation are at once equalized by further reflection from the walls of the vessel. We may, indeed, think of the whole process as being accomplished in a very large number of short intervals, in such a way that the piston, after it has moved a very small distance with very small velocity, is kept at rest for a while, namely, until all irregularities produced in the directions of the radiation have disappeared as the result of the reflection from the white walls of the hollow cylinder. If this procedure be carried on sufficiently long, the compression of the radiation may be continued to an arbitrarily small fraction of the original volume, and while this is being done, the radiation may be always regarded as uniform in all directions. This continuous process of equalization refers, of course, only to difference in the direction of the radiation; for changes in the color or intensity of the radiation of however small size, having once occurred, can evidently never be equalized by reflection from totally reflecting stationary walls but continue to exist forever.

79. With the aid of the theorems established we are now in a position to calculate the change of the density of radiation for every frequency for the case of infinitely slow adiabatic compression of the perfectly evacuated hollow cylinder, which is filled with uniform radiation. For this purpose we consider the radiation at the time $t$ in a definite infinitely small interval of frequencies, from $\nu$ to $\nu +d\nu$, and inquire into the change which the total energy of radiation contained in this definite constant interval suffers in the time $\delta t$.

At the time $t$ this radiant energy is, according to Sec. 23, $Vud\nu$; at the time $t+\delta t$ it is $\left(Vu+\delta \left(Vu\right)\right)d\nu$, hence the change to be calculated is

In this the density of monochromatic radiation $u$ is to be regarded as a function of the mutually independent variables $\nu$ and $t$, the differentials of which are distinguished by the symbols $d$ and $\delta$.

The change of the energy of monochromatic radiation is produced only by the reflection from the moving reflector, that is to say, firstly by certain rays, which at the time $t$ belong to the interval $(\nu ,d\nu )$, leaving this interval on account of the change in color suffered by reflection, and secondly by certain rays, which at the time $t$ do not belong to the interval $(\nu ,d\nu )$, coming into this interval on account of the change in color suffered on reflection. Let us calculate these influences in order. The calculation is greatly simplified by taking the width of this interval $d\nu$ so small that

a condition which can always be satisfied, since $d\nu$ and $v$ are mutually independent.

80. The rays which at the time $t$ belong to the interval $(\nu ,d\nu )$ and leave this interval in the time $\delta t$ on account of reflection from the moving reflector, are simply those rays which strike the moving reflector in the time $\delta t$. For the change in color which such a ray undergoes is, from (83) and (91), large compared with $d\nu$, the width of the whole interval. Hence we need only calculate the energy, which in the time $\delta t$ is transmitted to the reflector by the rays in the interval $(\nu ,d\nu )$.

For an elementary pencil, which falls on the element $d\sigma$ of the reflecting surface at the angle of incidence $\theta$, this energy is, according to (88) and (5),

Hence we obtain for the total monochromatic radiation, which falls on the whole surface $F$ of the reflector, by integration with respect to $\phi$ from $0$ to $2\pi$, with respect to $\theta$ from $0$ to $\frac{\pi }{2}$, and with respect to $d\sigma$ from $0$ to $F$,

Thus this radiant energy leaves, in the time $\delta t$, the interval of frequencies $(\nu ,d\nu )$ considered.

81. In calculating the radiant energy which enters the interval $(\nu ,d\nu )$ in the time $\delta t$ on account of reflection from the moving reflector, the rays falling on the reflector at different angles of incidence must be considered separately. Since in the case of a positive $v$, the frequency is increased by the reflection, the rays which must be considered have, at the time $t$, the frequency ${\nu }_1<\nu$. If we now consider at the time $t$ a monochromatic pencil of frequency $({\nu }_1,d{\nu }_1)$, falling on the reflector at an angle of incidence $\theta$, a necessary and sufficient condition for its entrance, by reflection, into the interval $(\nu ,d\nu )$ is

These relations are obtained by substituting ${\nu }_1$ and $\nu$ respectively in the equations (83) and (86) in place of the frequencies before and after reflection $\nu$ and $\nu \prime$.

The energy which this pencil carries into the interval $({\nu }_1,d\nu )$ in the time $\delta t$ is obtained from (89), likewise by substituting ${\nu }_1$ for $\nu$. It is

Now we have

where we shall assume $\frac{\partial K}{\partial \nu }$ to be finite.

Hence, neglecting small quantities of higher order,

Thus the energy required becomes

and, integrating this expression as above, with respect to $d\sigma$, $\phi$, and $\theta$, the total radiant energy which enters into the interval $(\nu ,d\nu )$ in the time $\delta t$ becomes

82. The difference of the two expressions (93) and (92) is equal to the whole change (90), hence

or, according to (24),

or, finally, since $Fv\delta t$ is equal to the decrease of the volume $V$,

whence it follows that

This equation gives the change of the energy density of any definite frequency $\nu$, which occurs on an infinitely slow adiabatic compression of the radiation. It holds, moreover, not only for black radiation, but also for radiation originally of a perfectly arbitrary distribution of energy, as is shown by the method of derivation.

Since the changes taking place in the state of the radiation in the time $\delta t$ are proportional to the infinitely small velocity $v$ and are reversed on changing the sign of the latter, this equation holds for any sign of $\delta V$; hence the process is reversible.

83. Before passing on to the general integration of equation (95) let us examine it in the manner which most easily suggests itself. According to the energy principle, the change in the radiant energy

occurring on adiabatic compression, must be equal to the external work done against the radiation pressure

Now from (94) the change in the total energy is found to be

or, by partial integration,

and this expression is, in fact, identical with (96), since the product $\nu u$ vanishes for $\nu =0$ as well as for $\nu =\infty$. The latter might at first seem doubtful; but it is easily seen that, if $\nu u$ for $\nu =\infty$ had a value different from zero, the integral of $u$ with respect to $\nu$ taken from $0$ to $\infty$ could not have a finite value, which, however, certainly is the case.

84. We have already emphasized (Sec. 79) that $u$ must be regarded as a function of two independent variables, of which we have taken as the first the frequency $\nu$ and as the second the time $t$. Since, now, in equation (95) the time $t$ does not explicitly appear, it is more appropriate to introduce the volume $V$, which depends only on $t$, as the second variable instead of $t$ itself. Then equation (95) may be written as a partial differential equation as follows:

From this equation, if, for a definite value of $V$, $u$ is known as a function of $\nu$, it may be calculated for all other values of $V$ as a function of $\nu$. The general integral of this differential equation, as may be readily seen by substitution, is

where $\phi$ denotes an arbitrary function of the single argument ${\nu }^{3}V$. Instead of this we may, on substituting ${\nu }^{3}V\phi \left({\nu }^{3}V\right)$ for $\phi \left({\nu }^{3}V\right)$, write

Either of the last two equations is the general expression of Wien’s displacement law.

If for a definitely given volume $V$ the spectral distribution of energy is known (i.e., $u$ as a function of $\nu$), it is possible to deduce therefrom the dependence of the function $\phi$ on its argument, and thence the distribution of energy for any other volume $V\prime$, into which the radiation filling the hollow cylinder may be brought by a reversible adiabatic process.

84-1. The characteristic feature of this new distribution of energy may be stated as follows: If we denote all quantities referring to the new state by the addition of an accent, we have the following equation in addition to (99)

Therefore, if we put

we shall also have

i.e., if we coordinate with every frequency $\nu$ in the original state that frequency $\nu \prime$ which is to $\nu$ in the inverse ratio of the cube roots of the respective volumes, the corresponding energy densities $u\prime$ and $u$ will be in the inverse ratio of the volumes.

The meaning of these relations will be more clearly seen, if we write

This is the number of the cubes of the wave lengths, which correspond to the frequency $\nu$ and are contained in the volume of the radiation. Moreover $ud\nu V=Ud\nu$ denotes the radiant energy lying between the frequencies $\nu$ and $\nu +d\nu$, which is contained in the volume $V$. Now since, according to (99a),

we have, taking account of (99b),

These results may be summarized thus: On an infinitely slow reversible adiabatic change in volume of radiation contained in a cavity and uniform in all directions, the frequencies change in such a way that the number of cubes of wave lengths of every frequency contained in the total volume remains unchanged, and the radiant energy of every infinitely small spectral interval changes in proportion to the frequency. These laws hold for any original distribution of energy whatever; hence, e.g., an originally monochromatic radiation remains monochromatic during the process described, its color changing in the way stated.

85. Returning now to the discussion of Sec. 73 we introduce the assumption that at first the spectral distribution of energy is the normal one, corresponding to black radiation. Then, according to the law there proven, the radiation retains this property without change during a reversible adiabatic change of volume and the laws derived in Sec. 68 hold for the process. The radiation then possesses in every state a definite temperature $T$, which depends on the volume $V$ according to the equation derived in that paragraph,

Hence we may now write equation (99) as follows:

or

Therefore, if for a single temperature the spectral distribution of black radiation, i.e., $u$ as a function of $\nu$, is known, the dependence of the function $\phi$ on its argument, and hence the spectral distribution for any other temperature, may be deduced therefrom.

If we also take into account the law proved in Sec. 47, that, for the black radiation of a definite temperature, the product $u{q}^3$ has for all media the same value, we may also write

where now the function $F$ no longer contains the velocity of propagation.

86. For the total radiation density in space of the black radiation in the vacuum we find

or, on introducing $\frac{T}{\nu }=x$ as the variable of integration instead of $\nu$,

If we let the absolute constant

the equation reduces to the form of the Stefan-Boltzmann law of radiation expressed in equation (75).

87. If we combine equation (100) with equation (99a) we obtain

Hence the laws derived at the end of Sec. 84-a assume the following form: On infinitely slow reversible adiabatic change in volume of black radiation contained in a cavity, the temperature $T$ varies in the inverse ratio of the cube root of the volume $V$, the frequencies $\nu$ vary in proportion to the temperature, and the radiant energy $Ud\nu$ of an infinitely small spectral interval varies in the same ratio. Hence the total radiant energy $U$ as the sum of the energies of all spectral intervals varies also in proportion to the temperature, a statement which agrees with the conclusion arrived at already at the end of Sec. 68, while the space density of radiation, $u=\frac{U}{V}$, varies in proportion to the fourth power of the temperature, in agreement with the Stefan-Boltzmann law.

88. Wien’s displacement law may also in the case of black radiation be stated for the specific intensity of radiation $K_{\nu }$ of a plane polarized monochromatic ray. In this form it reads according to (24)

If, as is usually done in experimental physics, the radiation intensity is referred to wave lengths $\lambda$ instead of frequencies $\nu$, according to (16), namely

equation (106) takes the following form:

This form of Wien’s displacement law has usually been the starting-point for an experimental test, the result of which has in all cases been a fairly accurate verification of the law.26

89. Since $E_{\lambda }$ vanishes for $\lambda =0$ as well as for $\lambda =\infty$, $E_{\lambda }$ must have a maximum with respect to $\lambda$, which is found from the equation

where $Ḟ$ denotes the differential coefficient of $F$ with respect to its argument. Or

This equation furnishes a definite value for the argument $\frac{\lambda T}{c}$, so that for the wave length ${\lambda }_m$ corresponding to the maximum of the radiation intensity $E_{\lambda }$ the relation holds

With increasing temperature the maximum of radiation is therefore displaced in the direction of the shorter wave lengths.

The numerical value of the constant $b$ as determined by Lummer and Pringsheim27 is

Paschen28 has found a slightly smaller value, about $0.292$.

We may emphasize again at this point that, according to Sec. 19, the maximum of $E_{\lambda }$ does not by any means occur at the same point in the spectrum as the maximum of $K_{\nu }$ and that hence the significance of the constant $b$ is essentially dependent on the fact that the intensity of monochromatic radiation is referred to wave lengths, not to frequencies.

90. The value also of the maximum of $E_{\lambda }$ is found from (107) by putting $\lambda ={\lambda }_m$. Allowing for (109) we obtain

i.e., the value of the maximum of radiation in the spectrum of the black radiation is proportional to the fifth power of the absolute temperature.

Should we measure the intensity of monochromatic radiation not by $E_{\lambda }$ but by $K_{\nu }$, we would obtain for the value of the radiation maximum a quite different law, namely,