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On the Spectral Density of Radiation
A. A. Gershun
Insufficient penetration into the essence of concepts, when they are familiar and are perceived as elementary, gives rise to a particular danger that erroneous notions may arise. The customary ways of presenting the question of the distribution of radiation in a continuous spectrum may serve as an illustration of the validity of this proposition. Usually this distribution is characterized by a spectral curve. It is constructed so that the area bounded above by the spectral curve, on the right and left by two ordinates, and below by the abscissa axis, on which the spectral scale is laid off, is proportional to the radiant flux in the corresponding interval of the spectrum (or to a photometric quantity derived from it, for example to the energetic luminous intensity, measured by the flux per unit area). In this case the ordinates are regarded as a measure of the spectral intensity of the radiation.
Thus, for example, Fig. 1 shows, on a uniform wavelength scale, the distribution of energy in the spectrum of an absolutely black body at temperature \(T = 5000^\circ\mathrm{K}\). On the basis of the position of the maximum for the ordinates \(p_\lambda\) of this curve, one may, in the accepted form, express the general judgment that the greatest values of the spectral intensity of the radiation of an absolutely black body at the given temperature fall in the visible region of the spectrum (this region is marked on the scale of Fig. 1, the limits being conventionally taken as \(0.400\) and \(0.725\,\mu\)). Indeed, the wavelength \(\lambda_m\) corresponding to the maximum of the curve, determined by the relation \(\lambda_m T = \mathrm{const}\), is, at the given temperature, equal to \(\lambda_m = 0.579\,\mu\). This value is close to that which corresponds to the maximum of the spectral curve of the sensitivity of the eye, shown in the same figure by a dotted line.
However, one may with equal right assert that the maximum of the spectral curve of the distribution of radiation of a black body at the given temperature \(5000^\circ\mathrm{K}\) lies, say, in the infrared region of the spectrum! Indeed, one may, as is usually done in theoretical physics, take as the spectral coordinate not the wavelength but the frequency \(\nu\). Then, for the given
in this case one obtains the spectral curve shown in Fig. 2. Along the abscissa axis the frequencies are plotted in \(10^{14}\) hertz; along the ordinate axis—
Figure axis labels:
Vertical: Density \(p_\lambda\) of spectral distribution
Horizontal: Wavelength \(\lambda\) (in microns)
Fig. 1. Spectral curve of the energy distribution for an absolutely black body at \(T = 5000^\circ\mathrm{K}\) on a uniform wavelength scale.
Figure axis labels:
Vertical: Density \(p_\nu\) of spectral distribution
Horizontal: Frequency \(\nu\) (in \(10^{14}\) hertz)
Fig. 2. The same curve as in Fig. 1, but on a uniform frequency scale.
(The dotted line in both figures shows the spectral curve of the sensitivity of the eye.)
the values of the spectral intensity of radiation \(p_\nu\), on a uniform frequency scale, defined so that the area bounded by the curve
area, just as in the preceding case, gave the radiant flux. The maximum of this spectral curve falls at the frequency \(\nu_m = 2.94 \cdot 10^{14}\) hertz, to which there corresponds the wavelength \(1.019\,\mu\). Thus, for a spectral curve constructed on a uniform frequency scale, the maximum in the present case lies in the infrared region of the spectrum and is situated far from the maximum of the spectral sensitivity curve of the eye, also shown by a dotted line in the same figure.
The comparison carried out above contains nothing new and is based on the following elementary calculation, which is given in many textbooks. Let the radiant flux \(dP\) fall on the spectral interval bounded by wavelengths from \(\lambda\) to \(\lambda + d\lambda\), or by the corresponding frequencies \(\nu\) and \(\nu + d\nu\). The spectral curve of the distribution of radiation, in accordance with the definition given at the beginning of the article, is constructed as the curve of the dependence of \(p_\lambda\) on \(\lambda\), or of \(p_\nu\) on \(\nu\), where
\[ p_\lambda = \frac{dP}{|d\lambda|}; \qquad p_\nu = \frac{dP}{|d\nu|}. \]
Since
\[ \lambda \nu = c, \]
where \(c\) is the speed of light, it follows that
\[ \frac{|d\lambda|}{\lambda} = \frac{|d\nu|}{\nu}. \]
Hence follows the well-known relation *):
\[ p_\nu = \frac{1}{c}\lambda^2 p_\lambda . \]
Naturally, the positions of the maxima of the quantities \(p_\lambda\) and \(p_\nu\) do not coincide, since equal intervals of frequencies do not correspond to equal intervals of wavelengths or, in other words, the values of \(\lambda\) for which \(p_\lambda\) and \(\lambda^2 p_\lambda\) are maximal are different. The wavelength \(\lambda_m\), at which the maximum of the quantity \(p_\lambda\) occurs, and the frequency \(\nu_m\), at which the maximum of the quantity \(p_\nu\) occurs, do not correspond to one another \((\lambda_m \nu_m \ne c)\).
Let us consider, for example, the question of the spectral curve of radiation of an absolutely black body determined by Planck’s formula. Introduce the auxiliary spectral coordinate:
\[ x = \frac{h}{kT}\cdot \nu = \frac{hc}{kT}\cdot \frac{1}{\lambda}, \]
where \(h\) is Planck’s constant, and \(k\) is Boltzmann’s constant.
*) It may also be written in the form
\[ \lambda p_\lambda = \nu p_\nu . \]
On the Spectral Density of Radiation
Then the expression for \(p_\nu\), if one operates with the radiant flux emitted from unit area, can be written in the form
\[ p_\nu=\frac{2\pi h}{c^2}\,\frac{\nu^3}{e^{\frac{h\nu}{kT}}-1} =\frac{2\pi k^3T^3}{c^2h^2}\,\frac{x^3}{e^x-1}. \]
The maximum of the quantity \(p_\nu\), as calculation shows, corresponds to the value \(x=2.821\) (the root of the equation \(\frac{xe^x}{e^x-1}=3\)).
The maximum of the quantity
\[ p_\lambda=2\pi hc^2\,\frac{\lambda^{-5}}{e^{\frac{hc}{kT\lambda}}-1} =\frac{2\pi k^5T^5}{c^3h^4}\,\frac{x^5}{e^x-1}, \]
as is known, corresponds to the value \(x=4.965\) (the root of the equation \(\frac{xe^x}{e^x-1}=5\)).
It follows from this that the maximum of the quantity \(p_\nu\) falls at the frequency \(\nu_m\), to which there corresponds a wavelength \(4.965/2.821=1.759\) times greater than the wavelength \(\lambda_m\) at which the maximum of \(p_\lambda\) occurs. Planck already pointed out the difference in the positions of the maxima of spectral curves on the wavelength and frequency scales. The fact that the maxima diverge so greatly that, when one of them lies in the visible region, the other is apparently in the infrared region of the spectrum, has not, apparently, attracted due attention up to now; yet this fact naturally leads to conclusions of a more general and broader character.
In essence, as Fabry noted, any quantity uniquely connected with the wavelength may be taken as the spectral coordinate.* It should be pointed out that, by the choice of the spectral scale, the maximum of the spectral curve of the energy distribution can be brought into any region of the spectrum
\[ \text{* } \]
- Thus, for example, one may construct the spectral curve of the distribution of radiation in a logarithmic scale of wavelengths or, what in the present case is the same thing, of frequencies, since \(|d\lg\lambda|=|d\lg\nu|\).
The ordinates of this curve will be the values of the quantity
\[ p_{\lg}=\frac{dP}{|d\lg\lambda|} =\frac{1}{\lg e}\lambda p_\lambda =\frac{1}{\lg e}\nu p_\nu. \]
In the case of absolutely black radiation,
\[ p_{\lg}=\frac{1}{\lg e}\,\frac{2\pi k^4T^4}{c^2h^3}\,\frac{x^4}{e^x-1}. \]
The maximum of the curve will correspond to the value \(x=3.921\). The maximum ordinate, with change of temperature, will increase proportionally to \(T^4\), i.e., in the same way as the total emitted flux, whereas the maximum value of \(p_\nu\) changes proportionally to \(T^3\), and \(p_\lambda\) proportionally to \(T^5\). This is also one of the advantages of this method of constructing the spectral curve.
electromagnetic waves, beginning, say, with X-rays and ending with radio waves.
This conclusion, as the author knows from many years of experience, is taken by many as a paradox. This is due to the fact that the spectral curve is commonly and widely misunderstood as a curve whose ordinates express the “intensity” of the individual monochromatic radiations entering into the composition of the given complex radiation. This is due to forgetting that strictly monochromatic radiation cannot be a carrier of energy and that the so-called spectral intensity of radiation is not the intensity of monochromatic radiation, but only an auxiliary quantity by which one must multiply an interval on the spectral scale in order to obtain the radiant flux. The position of the maximum of the spectral radiation curve in itself has no physical significance, in contrast to the spectral absorption curve or the spectral sensitivity curve of a receiver, whose ordinates do indeed correspond unambiguously to each monochromatic radiation, regardless of which spectral coordinate is used to characterize its position.
It may be more expedient, instead of the position of the maximum of the spectral energy-distribution curve, to determine the median, i.e., to indicate the position of the middle monochromatic radiation, the ordinate for which divides the area bounded by the spectral curve and the axis of abscissae into equal left and right parts. It may be specified by the wavelength \(\lambda_{\mathrm{cp}}\), determined from the relation
\[ \int_0^{\lambda_{\mathrm{cp}}} p_\lambda\, d\lambda = \int_{\lambda_{\mathrm{cp}}}^{\infty} p_\lambda\, d\lambda \]
(the same energy falls on the longer wavelengths as on the shorter ones), or by the frequency \(\nu_{\mathrm{cp}}\), determined from the identical relation
\[ \int_{\nu_{\mathrm{cp}}}^{\infty} p_\nu\, d\nu = \int_0^{\nu_{\mathrm{cp}}} p_\nu\, d\nu . \]
The position in the spectrum of the middle radiation, unlike the position of the maximum, does not depend on the choice of the spectral scale (in particular, \(\lambda_{\mathrm{cp}}\nu_{\mathrm{cp}}=c\)).
In the case of an absolutely black body, the middle radiation corresponds to the value of the auxiliary quantity used above,
\[ x=\frac{h}{kT}\cdot \nu=\frac{hc}{kT}\cdot \frac{1}{\lambda}, \]
equal to \(3.503\). Hence
\[ \lambda_{\mathrm{cp}}=\frac{4.965}{3.503}\lambda_{\mathrm{m}}=1.417\lambda_{\mathrm{m}}, \]
since the wavelength \(\lambda_{\mathrm{m}}\) at which the maximum of the spectral curve on a uniform wavelength scale occurs corresponds to the value \(x=4.965\).
The displacement law may be stated on the basis of the relation
\[ \lambda_{\mathrm{cp}}T=\mathrm{const}. \]
ON THE SPECTRAL DENSITY OF RADIATION
Since this constant is equal to \(\dfrac{hc}{3.503\,k}\), then, taking \(C_2=\dfrac{hc}{k}=14380\ \mu\cdot\text{degree}\), we obtain:
\[ \lambda_{\mathrm{avg}}=\frac{4105}{T}, \]
where \(T\) is the temperature in degrees of the absolute scale, and \(\lambda_{\mathrm{avg}}\) is the mean wavelength in microns\(^*\).
Forgetting the peculiarities of the spectral distribution curve of radiation gives rise to erroneous notions. They arise already among secondary-school pupils, when they are told, and when textbooks written for them say, that “waves of different lengths carry different amounts of energy,” that “to each temperature there corresponds a definite wavelength, upon whose share falls the predominant role in the radiation,” and so on.
Still more undesirable consequences of disordered notions appear (and not so very rarely) in physics textbooks for higher schools, in special courses on optics. Here is one example of such errors. First an expression is found for the frequency \(\nu_m\) at which the intensity of radiation has its maximum value on a uniform frequency scale. After this, a transition is made to the displacement law by substituting \(\lambda_m=\dfrac{c}{\nu_m}\), where by \(\lambda_m\) the author now already means, as is shown by the numerical value he gives, that wavelength at which the maximum of the spectral radiation curve occurs on a uniform wavelength scale.
In lighting engineering and colorimetry, arguments about the uniformity or nonuniformity of the distribution of energy over the spectrum are carried on without the necessary qualifications. However, by an “equal-energy” or “equal-intensity” spectrum one may with equal justification agree to mean both such a distribution in which equal energies fall on equal intervals of wavelengths, and such a distribution in which equal energies fall on equal intervals of frequencies, and, strictly speaking, any arbitrary distribution of energy over the spectrum. Indeed, even a line spectrum is only a special case of a continuous spectrum, when the density of the energy distribution is very large near certain values of the wavelengths. By the choice of the spectral scale, the distribution curve can be transformed into any desired form.
\(^*\) The wavelength \(\lambda_{\mathrm{avg}}\) exceeds the value of the most customary quantity \(\lambda_m\) by almost one and a half times. It differs from the wavelength \(\lambda_m\) by less than one hundredth of one percent from the wavelength \(\lambda'\), which satisfies the condition that three quarters of the energy are concentrated in the region \(\lambda>\lambda'\), and one quarter in the region \(\lambda<\lambda'\). In the region of wavelengths shorter than \(\lambda_m\), a fraction of energy equal to \(0.25005\) is concentrated.
From the general propositions formulated above there also follows, in particular, the need for more rigorous formulations of Stokes’ law, which determines the position of the luminescence light spectrum relative to the absorption spectrum of the given substance. Lommel’s formulation, stating that “the emission spectrum as a whole and its maximum are always shifted, in comparison with the absorption spectrum and its maximum, toward longer waves,” cannot be recognized as sufficiently rigorous. The absorption spectrum does not depend on the choice of spectral scale, whereas the spectral curve of emission does.
One must also be very careful in drawing conclusions based on comparing the spectral distribution curve of radiation with the spectral sensitivity curve of a receiver. For such a comparison it is absolutely necessary to construct a new curve, multiplying the ordinates of the spectral curve of the energy distribution by the corresponding values of the spectral sensitivity of the receiver. Conclusions may be drawn on the basis of a judgment about the values of the quantity
\[ \frac{\int v_\lambda p_\lambda\, d\lambda}{\int p_\lambda\, d\lambda} = \frac{\int v_\nu p_\nu\, d\nu}{\int p_\nu\, d\nu}, \]
where \(v_\lambda = v_\nu\) is the spectral sensitivity of the receiver, and not on the basis of comparing the positions of the maxima of the radiation distribution curve and the receiver sensitivity curve. In scientific, textbook, and popular articles on optics, one often limits oneself to constructing, on a uniform wavelength scale, graphs like Fig. 1, and, solely on the basis of the good agreement between the positions of the maxima of the spectral distribution curve of natural radiation reaching the earth and the spectral sensitivity curve of the eye, draws the conclusion that the human eye is highly adapted to sunlight. It is enough to pass to another spectral scale and construct, for example, a graph of the type of Fig. 2, equivalent to the graph in Fig. 1, in order to be clearly convinced of the illegitimacy of such reasoning. A judgment about the adaptation of the eye to the spectrum of sunlight can be made only on the basis of the entire set of criteria, as was done in the book by Academician S. I. Vavilov The Eye and the Sun, which is an unsurpassed model of the scientist’s scientific and literary work.
In conclusion, it seems appropriate to dwell on a question of a terminological nature. In 1934, in the notes to the translation of Fabry’s well-known book[^1] on photometry, the author proposed the general concept of the spectral density of a photometric quantity as the ratio of the elementary value of the given quantity, produced by an infinitely narrow portion of the spectrum, to the corresponding interval on the spectral scale (an infinitely small interval of wavelengths or frequencies). This concept can
can be applied to any of the photometric quantities. Thus, for example, one may speak of the spectral density of radiant flux, the spectral density of luminous flux, the spectral density of energetic luminosity, the spectral density of illuminance, the spectral density of brightness, etc.
It is better to speak not of the spectral intensity of radiation or spectral luminosity, or of emissive power, but of the density of the spectral distribution or of the spectral density of radiant flux, the spectral density of energetic luminosity, the spectral density of energetic brightness, etc. In doing so, it must always be clearly indicated whether the spectral density is meant in the wavelength scale or in the frequency scale.
The use of the terms “density of the spectral distribution” or “spectral density” in the present case is all the more justified because the quantities under consideration, by the physical nature of the phenomenon they characterize, rest directly on the mathematical concepts of probability density and distribution density. Let us note that Academician A. N. Kolmogorov and other Soviet scientists have created, on the basis of probability theory, a rigorous mathematical theory of oscillations with a continuous spectrum ^2,^3. In this general theory it is precisely “spectral density” that is spoken of, with which the energy of oscillations is continuously distributed among different frequencies.
Thus, it is desirable to avoid the term “spectral intensity,” and one should use the concept of density of distribution over the spectrum and speak of the density of the spectral distribution or of the spectral density of the photometric quantity under consideration.
The word “density” will constantly remind us of those fundamental features of the spectral distribution curve of radiation to which the present note is devoted.
Cited Literature
- Sh. Fabry, General Introduction to Photometry, GTTI, 1934, p. 130.
- A. N. Kolmogorov, “Statistical Theory of Oscillations with a Continuous Spectrum.” General Collection of the Academy of Sciences of the USSR Dedicated to the Thirtieth Anniversary of the Great October Socialist Revolution. Reports. Publishing House of the Academy of Sciences of the USSR, 1948.
- A. N. Kolmogorov, “Statistical Theory of Oscillations with a Continuous Spectrum.” Jubilee Collection Dedicated to the Thirtieth Anniversary of the Great October Socialist Revolution, Part One. Publishing House of the Academy of Sciences of the USSR, 1947.