On the Spectral Distribution of Radiation Power

M. M. Gurevich

1. The question of how the spectral distribution of power in the radiation of a source should be represented has been repeatedly discussed in recent years in specialized journals and has been resolved differently in different cases\(^{1,2,3}\). Even the legitimacy of posing the question of the position of the energy maximum in the radiation spectrum has been called into doubt.

At the present time\(^{4,5,6}\), it may be considered that the correct solution has been found, and since it diverges from what we have become accustomed to regard as elementary truths, dissemination of the new solution is urgently necessary.

One may speak of the distribution of power in the radiation spectrum of any source. The general foundations of the solution are most conveniently traced in the widely known laws of radiation of an absolutely black body, which alone will concern us below.

1. In practically all textbooks of optics, illuminating engineering, and related sciences, the spectral density of radiation \(p_\lambda\) is introduced, defined by the expression

\[ dP = p_\lambda\,d\lambda \; \text{watt}\cdot\text{cm}^{-2}, \tag{1} \]

where \(dP\) is the elementary power radiated from unit area in the spectral interval \(\lambda, \lambda + d\lambda\), while \(p_\lambda\) is expressed by the well-known Planck formula

\[ p_\lambda = \frac{c_1\lambda^{-5}}{\exp \dfrac{c_2}{\lambda T} - 1} \; \text{watt}\cdot\text{cm}^{-3}, \tag{2} \]

where \(c_1\) and \(c_2\) are constants, and \(T\) is the absolute temperature.

For the further exposition we shall need only the constant \(c_2\), which according to the latest data is taken to be equal to \(1.438\ \text{cm}\cdot\text{degree K}\).

By differentiating (2) with respect to the wavelength \(\lambda\), one readily obtains the equation

\[ \frac{x}{1-\exp(-x)}=5, \]

where

\[ x=\frac{c_2}{\lambda T}, \]

which leads to the value \(\frac{c_2}{\lambda T}=4.9651\), or to Wien’s displacement law in its usual form

\[ \lambda_m T=0.2896\ \text{cm}\cdot\text{degree K}, \tag{3} \]

where \(\lambda_m\) is the wavelength at which the spectral density \(p_\lambda\) has its maximum at the temperature \(T\) of an absolutely black body. It is considered that this wavelength indicates the place in the radiation spectrum where its maximum is located.

Fig. 1. Spectral distribution of the radiation of an absolutely black body with respect to a linear wavelength scale.

Next, by substituting (3) into (2), it is shown that the maximum of \(p_\lambda\) is proportional to the 5th power of the absolute temperature.

However, Planck’s formula can also be represented in another form if, instead of \(\lambda\), the wave number

\[ n=\frac{1}{\lambda}\ \text{cm}^{-1} \]

is taken as the independent variable. Then, by analogy with (1), it is assumed that the elementary power \(dP\), falling within the spectral interval \(n, n+dn\), can be written as

\[ dP=p_n\,dn\ \text{watt}\cdot\text{cm}^{-2}, \tag{4} \]

where the spectral density of radiation \(p_n\) must be expressed as

\[ p_n=\frac{c_1 n^3}{1-\exp\frac{c_2 n}{T}}\ \text{watt}\cdot\text{cm}^{-1}. \tag{5} \]

If \(p_n\) is differentiated with respect to \(n\) and the wave number \(n_m\) corresponding to the maximum of \(p_n\) is sought, it is not difficult to arrive at the equation

\[ \frac{x}{1-\exp(-x)}=3, \]

where

\[ x=\frac{c_2 n}{T}, \]

which leads to the solution:

\[ \frac{c_2 n}{T}=2.8214 \]

or

\[ \frac{T}{n_m}=T\lambda_m=\frac{c_2}{2.8214}=0.5097\ \text{cm}\cdot\text{degree K}. \tag{6} \]

The new value of \(\lambda_m\), differing from the former by a factor of 1.76, may with the same justification be regarded as the wavelength at which the maximum of the radiation occurs. In this case, substituting (6) into (5), we see that the maximum value of \(p_n\) is proportional only to the third power of the absolute temperature.

In Figs. 1 and 2 graphical representations are given of the functions \(p_\lambda\) and \(p_n\) with respect to linear scales of wavelengths and wave numbers. As follows also from (3) and (6), the greatest values of \(p_\lambda\) and \(p_n\) occur at different places in the spectrum. Thus, for example, for \(T=1800^\circ\mathrm{K}\) the maximum of \(p_\lambda\) occurs at the wavelength \(1.61\mu\), while the maximum of \(p_n\) occurs at the wavelength \(2.83\mu\). For \(T=5500^\circ\mathrm{K}\) (the sun), the maximum of \(p_\lambda\) lies in the middle of the visible spectrum at \(\lambda_m=0.527\mu\), while the maximum of \(p_n\) is in the infrared part at \(\lambda_m=0.927\mu\), etc.

Fig. 2. Spectral distribution of the radiation of an absolutely black body with respect to a linear scale of wave numbers.

One cannot pass over so substantial a difference, and the questions naturally arise: what, then, is correct, and if both are erroneous, then why, and what will the correct answer be?

1. If one looks closely at expressions (2) and (5), and also at Figs. 1 and 2, the first perplexity is to some extent dispelled when we notice that the spectral density \(p_\lambda\) is not

coincides with the spectral density \(p_n\). The first of them is the derivative of the radiation power (more precisely, of its surface density) with respect to wavelength, and the second—with respect to wave number. The dimensions of these two quantities are different, and therefore there is nothing surprising in the fact that they have different spectral distributions in one and the same radiation. At the same time, the dimensions of \(p_\lambda\) and \(p_n\) do not coincide with the dimension of the quantity whose distribution we would be interested in establishing, since what is at issue is the spectral distribution of the surface power density, having the dimension \([watt\cdot cm^{-2}]\). If we wish to preserve the condition by virtue of which the area bounded by the abscissa axis, the curve, and its two ordinates would be proportional to the surface power density of the radiation falling within the corresponding part of the spectrum, then it is obvious that a dimensionless quantity must be laid off along the abscissa axis.

Expressions (1) and (4) may be represented in the following form:

\[ dP=p_\lambda\,\lambda\,\frac{d\lambda}{\lambda}=p_n n\,\frac{dn}{n}. \tag{7} \]

If along the abscissa axis of the new graph (Fig. 3) one lays off \(\ln\lambda\), and if one takes into account that \(d(\ln\lambda)=\frac{d\lambda}{\lambda}=-\frac{dn}{n}=-d\ln n\), then equality (7) may be rewritten as:

\[ dP=\frac{c_1\lambda^{-4}}{\exp\frac{c_2}{\lambda T}-1}\,d(\ln\lambda) = -\frac{c_1 n^4}{1-\exp\frac{c_2 n}{T}}\,d(\ln n). \tag{8} \]

Then the quantity

\[ p_{\ln}=\frac{dP}{d(\ln\lambda)} = -\frac{dP}{d(\ln n)} = \frac{c_1\lambda^{-4}}{\exp\frac{c_2}{\lambda T}-1} = -\frac{c_1 n^4}{1-\exp\frac{c_2 n}{T}} \; watt\cdot cm^{-2} \tag{9} \]

will have all the properties which we required of it. Plotting it along the ordinate axis and the logarithm of the wavelength (or of the wave number) along the abscissa axis, we ensure proportionality between the area of the graph and the surface power density of the radiation in the corresponding part of the spectrum.

Let us see, however, where the maximum of the quantity \(p_{\ln}\) will now be located and what the magnitude of this maximum will be. Differentiating either of expressions (9), we arrive at the equation

\[ \frac{x}{1-\exp(-x)}=4, \]

where

\[ x=\frac{c_2}{\lambda T}=\frac{c_2 n}{T}, \]

from which it follows that \(c_2/\lambda T=c_2 n/T=3.9207\), and the quantities

\[ \frac{T}{n_m}=\lambda_m T=0.3668\ \text{cm}\cdot\text{degree K}. \tag{10} \]

Substituting the obtained values of \(\lambda_m\) or \(n_m\) into (9), we see that the maximum value of \(p_{1n}\) now increases in proportion to the fourth power of the absolute temperature.

From Fig. 3 it is seen that the largest values of \(p_{1n}\) occur at new positions. Thus, for \(T=1800^\circ\mathrm{K}\), \(\lambda_m=2.04\mu\). For \(T=5500^\circ\mathrm{K}\)

Fig. 3. Spectral distribution of the radiation of an absolutely black body with respect to a logarithmic scale of wavelengths or wave numbers.

(the Sun) \(\lambda_m=0.668\mu\), i.e. it coincides very well with the maximum of chlorophyll absorption.

1. Thus we have a third solution of the problem of the position of the maximum in the radiation spectrum of an absolutely black body. Dimensional considerations speak in favor of the last solution; however, for a final conclusion one would like to have more weighty evidence. This is all the more necessary because one can imagine an infinite set of different solutions connected with various functions of the wavelength that can be laid off along the axis of abscissas. Here there may even arise a general doubt as to the reality of such a concept as the maximum spectral density of radiation.

Before going further, let us return once more to Figs. 1, 2, and 3 and see what their difference amounts to. In each of them

the area enclosed between the abscissa axis, the curve, and its two ordinates is proportional to the surface density of the radiation power in the corresponding portion of the spectrum. If the entire area encompassed by each of the curves is divided into a large number of vertical strips, then the ratio of their areas will represent a certain distribution of power over the spectrum, which will be determined only by the choice of the widths of these strips. There are, however, no grounds for changing the width of a strip chosen as the initial one in any region of the spectrum. With a constant strip width, the power distribution that they produce coincides with the curve bounding the entire area as a whole. Thus, the whole question of the distribution of power in the radiation spectrum may be reduced to another question: which spectral intervals should be regarded as equal? The function \(p_\lambda\) gives the spectral distribution of power in the case where intervals identical in the difference of wavelengths \(\Delta\lambda\) are regarded as equal. The function \(p_n\) gives another distribution, since in this case the intervals regarded as equal are those characterized by identical differences of wave numbers \(\Delta n\), which, naturally, is at variance with equality of \(\Delta\lambda\). The function \(p_{\ln}\) is characterized by equality of relative intervals \(\Delta\lambda/\lambda\) or \(\Delta n/n\), which differs both from equality of \(\Delta\lambda\) and from equality of \(\Delta n\). Each new function of wavelength, plotted along the abscissa axis, gives its own solution to the question of equality of spectral intervals and its own position of the radiation maximum.

1. The solution of the question posed should be sought along a path not connected with the necessity of comparing the radiation powers at different wavelengths. Let us consider the problem of the efficiency of the radiation of an absolutely black body incident on some selectively receptive surface. Let this be, for example, a light-sensitive receiver placed behind the slit of a monochromator, which admits to it only a narrow part of the entire radiation spectrum falling on the entrance slit of the instrument. As the temperature \(T\) of the absolutely black body is raised, the radiation power incident on the receiver will increase, but the efficiency, defined as the ratio of this power to the total power of the source, will be a function of \(T\), and moreover such a function that at some temperature \(T_m\) it will pass through a maximum value. Let the monochromator isolate some arbitrary but unchanging spectral band between \(\lambda\) and \(\lambda + d\lambda\). Its power

\[ dP = \frac{c_1 \lambda^{-5}}{\exp \frac{c_2}{\lambda T} - 1}\, d\lambda \ \text{watt}\cdot\text{cm}^{-2} \tag{11} \]

does not depend on the form in which we represent it. The coefficient

On the Spectral Distribution of Radiation Power

the efficiency can be written as follows:

\[ \eta=\frac{c_1\lambda^{-5}\,d\lambda}{\left[\exp\frac{c_2}{\lambda T}-1\right]\sigma T^4}. \tag{12} \]

Differentiating the last expression with respect to \(T\) and setting the derivative equal to zero, we obtain the former equation

\[ \frac{x}{1-\exp(-x)}=4, \]

where

\[ x=\frac{c_2}{\lambda T}, \]

which gives

\[ x=\frac{c_2}{\lambda T}=3.9207 \]

or

\[ \lambda T_m=0.3668\ \text{cm}\cdot\text{degree K}, \tag{10*} \]

where \(T_m\) is the temperature of an absolutely black body at which the efficiency reaches a maximum. In other words, the power falling in the selected band \(\lambda,\ \lambda+d\lambda\) will have the greatest value in the total radiation power of an absolutely black body when its temperature reaches \(T_m\) according to (10). But relation (10) differs from (10) only by a transposition of the symbol \(m\), which does not change the connection between \(\lambda\) and \(T\). Expression (10*) says that at temperature \(T_m\) the greatest relative radiation power falls in the spectral interval \(\lambda,\ \lambda+d\lambda\), i.e. that in this case the spectral radiation density has a maximum at the wavelength \(\lambda\).

If, instead of (12), we wrote the expression for the efficiency through \(p_n\) or through \(p_{\ln}\), we would arrive at the same results.

Thus, in finding the position of the maximum in the radiation spectrum of an absolutely black body by a method that does not require comparison of radiation powers in different parts of the spectrum, i.e. by a method that does not require resolving the question of what equal spectral intervals are, we arrive at a result coinciding with the one that appeared to us the most correct also from the standpoint of dimensions.

1. Therefore one must consider that: 1) the position of the maximum of radiation in the spectrum of an absolutely black body is determined by Wien’s law, in which, instead of the constant \(0.2886\ \text{cm}\cdot\text{degree K}\), the constant \(0.3668\ \text{cm}\cdot\text{degree K}\) should be substituted; 2) the radiation maximum increases in proportion to the fourth power of the absolute temperature, and not the fifth, as is usually assumed; 3) equal spectral intervals should be considered to be intervals characterized by equality of the ratios \(\Delta\lambda/\lambda\) (or \(\Delta n/n\)), instead of equality of the differences \(\Delta\lambda\), as is tacitly assumed in most cases,

while along the abscissa axis of the corresponding graphs one should plot \(\ln \lambda\) (or \(\ln n\)) on a linear scale, or \(\lambda\) (or \(n\)) on a logarithmic scale.

References

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