LUMINESCENCE AND THE LAWS OF SPECTRAL TRANSFORMATION OF LIGHT

E. I. Adirovich

1. LUMINESCENCE AND OTHER TYPES OF RADIATION

The first step in a theory of luminescent processes must evidently be a precise definition of the very phenomenon of luminescence. This definition, however, has been achieved only quite recently, after almost four hundred years of experimental and theoretical study of luminescent phenomena*). Even in the most recent specialized literature one very often encounters inadequate, and sometimes simply erroneous, definitions of what luminescence is and how it differs from other kinds of radiation. We shall limit ourselves to two examples—references to Riehl’s monograph³, published in 1941, and to Kröger’s most recent monograph⁴, published in 1948. The shortcomings of Riehl’s definition are noted in the Russian translation of his book in a note by Academician S. I. Vavilov³ (see also⁵). Kröger’s monograph opens with the following words: “When a system absorbs energy in one form or another, this energy may be partially emitted again in the form of radiation. Such a phenomenon is called luminescence.” It is easy to see that, by virtue of the law of conservation and transformation of energy, Kröger’s definition includes all possible kinds of radiation of bodies, for example the glow of incandescent lamps, which has nothing in common with luminescence.

Definitions of luminescence in the books of Pringsheim and Fogel⁶, Maurice Curie⁷, and others, based, following Wiedemann⁸, on Kirchhoff’s law, express only a necessary but not sufficient condition for the observed radiation to be luminescence. A complete and rigorous definition of luminescence was given by S. I. Vavilov⁵, ⁹, ¹⁰, ¹¹, who supplemented Wiedemann’s definition with the criterion of the duration of afterglow.

*) For historical information on luminescence, see ¹,².

According to S. I. Vavilov’s definition, luminescence is an excess over the thermal radiation of a body in the case when this excess radiation has a finite duration considerably exceeding the period of light oscillations.

In its first part this definition separates the thermal radiation of bodies, obeying Kirchhoff’s law[^12],

\[ E_\lambda = A_\lambda E^b_\lambda = A_\lambda \frac{2\pi h c^2}{\lambda^5}\, \frac{1}{e^{\frac{hc}{kT\lambda}}-1}, \tag{1,1} \]

from the whole variety of nonequilibrium radiations, in the course of which

\[ E_\lambda > A_\lambda E^b_\lambda . \tag{1,2} \]

The second part of the definition distinguishes luminescence from all other kinds of nonequilibrium radiation (light scattering, bremsstrahlung, the Cherenkov effect, etc.), which are practically inertialess (\(\tau\) of the order of the period of light oscillations \(\sim 10^{-15}\) sec.).

The definition of luminescence by the criterion of duration is by no means merely a simple methodological device, but is deeply connected with the essence of the matter. Of the five characteristics of radiation—intensity, spectral composition, polarization, coherence properties, and afterglow duration—only the last distinguishes luminescence from other kinds of nonequilibrium radiation. The criterion of intensity unites the whole variety of nonequilibrium radiations, separating them from thermal radiation, which occurs according to Kirchhoff’s law. Changes in the spectrum and incoherence are observed not only in the phenomena of luminescence, but also in combination and Compton scattering. Conversely, in the luminescence of simple rarefied gases, as a rule, resonance lines are present, and when an atom is excited by a sharp line (\(\Delta \lambda \ll\) the natural width of the emission line), resonance luminescence occurs with preservation of the coherence properties[^13]. There is no universal difference in the polarization properties of luminescence radiation either. Only in the duration of the afterglow is a fundamental and essential distinction manifested. In the case of luminescence, the acts of absorption and emission are separated by intermediate processes and states (labile and metastable states of atoms and molecules, the internal photoelectric effect and the sticking of electrons in crystals, etc.). This accounts for the finite duration of luminescence, much exceeding the period of light oscillations.

In all other cases, the emission of nonequilibrium radiation arises and disappears practically instantaneously, following the application or cessation of the excitation (\(\tau < 10^{-14}\) sec.). All these kinds

nonequilibrium radiation were brought together by S. I. Vavilov under the term “forced radiation.” In the classical interpretation, luminescence, as radiation with afterglow, is analogous to proper oscillations, while all other types of nonequilibrium radiation, which have no afterglow, are similar to forced oscillations, disappearing simultaneously with the removal of the external perturbation of the system.

Owing to its equilibrium character, which makes possible the direct application of thermodynamic methods, temperature radiation proves phenomenologically to be the simplest kind of radiation. However, this is only phenomenologically so. Consideration of the elementary acts of which temperature radiation is composed shows that these are either forced processes (scattering, negative absorption, etc.) or luminescent processes (excitation by impacts and by black radiation, ionization, etc.).

Under a certain statistical combination of these and other processes, taking place under conditions of equilibrium of the body with black radiation, the temperature emission of the body occurs. Under other conditions the body emits, generally speaking, nonequilibrium radiation*). From what has been said it is clear that the elementary acts of which both temperature radiation and any nonequilibrium radiation of a body are composed are one and the same. The difference consists only in the statistics of the elementary acts, leading in some cases to fulfillment, and in others to violation, of Kirchhoff’s law.

Under conditions of temperature emission, the luminescent and forced parts of the radiation are experimentally inseparable owing to the inertial thermal excitation of the body. In order to separate

*) It is possible, however, for a body to emit temperature radiation also under nonequilibrium conditions. An example may be the emission of a tungsten spiral in incandescent lamps, as well as the emission of any macroscopic bodies whose temperature is maintained, by heat conduction, above the temperature of the surrounding medium. The nonequilibrium of the conditions here consists in the fact that no black radiation, isothermal with it, falls on the radiating body from outside. However, inside the body, except for the surface layer essential for the balance of the emitted energy, the radiation density corresponds to Kirchhoff’s law. This occurs because, for each elementary volume, all the rest of the mass of the body forms an isothermal cavity. Therefore the nonequilibrium of the external conditions does not disturb the temperature character of the radiation. Distortions may arise only when radiation hotter than the body itself falls on the body from outside. If absorption of this radiation leads to exchange of energy for heat, then in this case too the temperature radiation of the body is practically not disturbed. But if the conversion of radiant energy into other forms (thermal, chemical energy, etc.) does not play a dominant role, then, at sufficient excitation intensities, nonequilibrium emission in the absorbing layer becomes so considerable that it predominates in the emission spectrum over the temperature radiation of the entire remaining mass of the body.

instantaneous processes from processes lasting $\gg 10^{-15}$ sec., it is necessary to stop the excitation. That part of the glow which disappears at once is due to forced elementary processes, and the delayed glow to luminescent ones. However, it is impossible to stop the thermal excitation of a heated body, as a result of which an experimental separation of temperature emission into forced and luminescent components is not feasible. Therefore luminescence is called not the whole component of the glow caused by elementary processes extended in time, but only that part of it which is caused by nonisothermal excitation and is present in the emission in excess over temperature radiation.

Thus, luminescence and forced emission are phenomenologically more complex processes than temperature radiation, but microscopically simpler than it. Each of them represents a “pure” case, whereas equilibrium radiation is a complex combination of both luminescent and forced elementary processes. The inertia of the thermal excitation of a luminous body excludes the possibility of investigating the nature of elementary acts under conditions of temperature (equilibrium) glow. The path to the study of the elementary processes of absorption and emission of light in bodies opens under nonequilibrium conditions, when instantaneous superposition and instantaneous cessation of excitation are possible. Measurement of the time interval separating excitation and radiation makes it possible at once to decide the question of the physical nature of the glow process. Thus, for example, S. I. Vavilov^10,14,15 directly experimentally proved that the blue glow of glycerin and a number of other liquids, discovered in 1928 under ultraviolet excitation,^16 can be quenched by extraneous impurities. It was thereby proved that the elementary acts of glow have a finite duration ($\tau \gg 10^{-15}$ sec.), as a result of which the excited molecule has time to transfer the excitation energy to quencher molecules. And from this it followed directly that the blue glow, having an almost universal character, of very many “pure” liquids is not scattering, as was assumed at first,^16 but luminescence. It was established^14 that this luminescence is caused by the dissolution in the liquids under investigation of various organic residues carried in the air.

The same criterion of duration enabled S. I. Vavilov^15 to establish that the Cherenkov effect^17 does not belong to the class of luminescent phenomena, even before a detailed theory of this effect had been developed.

The Vavilov–Wiedemann definition of luminescence gives, as we see, the possibility of both a theoretical and a direct experimental delimitation of nonequilibrium radiation from tempera-

ture, and also luminescence from other kinds of nonequilibrium radiation. Let us dwell on its application to the problem of candoluminescence, which has clarified in many respects this extremely confused and obscure area of luminescent phenomena.

2. A FEW WORDS ON CANDOLUMINESCENCE

As early as 1864, in connection with the discussion and criticism of Stokes’ law, Tyndall[^19] and Aitken[^20] pointed out that bodies placed at the focus of a concave mirror concentrating upon them a beam of infrared rays emit visible light. Tyndall and Aitken saw in this phenomenon a violation of Stokes’ law concerning the obligatory increase of the wavelength of light in the process of luminescence. The phenomenon itself was called “calcescence” or “calorescence.” Without entering here into a discussion of the limits of validity of Stokes’ law, let us note only that Tyndall and Aitken’s example is not at all related to luminescence. “Calorescence” is nothing other than the thermal emission of bodies heated as a result of the absorption of infrared radiation. In the light of the modern Vavilov–Wiedemann definition, the delimitation of the luminescence of a body excited by some radiation from the thermal glow arising when the body is heated by this radiation presents no difficulty.

Considerably more complex and confused is the question of candoluminescence[^21],[^22], i.e., of the non-temperature glow of bodies in flames. The discussion that arose after Minchin’s report[^23] at the Oxford Conference on Luminescence in 1938 is indicative, as are the numerous contradictions in the statements of individual authors. Thus, for example, most authors, following Nichols, Howes, and Wilber[^22],[^24], see the cause of candoluminescence in oxidation–reduction processes occurring in flames. Such an interpretation of the phenomenon is in sharp contradiction with Tiede and Buscher’s assertion[^25] that the luminescence of zinc sulfide and willemite in a flame does not change if a thin quartz plate is placed between the sample and the flame. If this is so, then any chemical interaction of the sample with the flame is excluded. Some authors (see, for example,[^21]) emphasize the essential role of the activator in candoluminescing samples. Others (for example,[^24]) write that candoluminescence is also observed in unactivated samples incapable of luminescing under ordinary methods of excitation.

In principle, the possibility is not excluded of the glow of bodies during chemical processes in flames, as well as their excitation by ultraviolet radiation emitted in the reactions taking place in the flame. In each individual case, for each specific substance, the question of its capacity for candoluminescence must

be decided by direct experiment. In this connection, the only possible way to resolve this question is to apply the Vavilov–Wiedemann criterion. Other, seemingly obvious, indications that led to the assertion of candoluminescence were merely the cause of numerous scientific errors.

As one such indication it was pointed out that the brightness of certain bodies considerably exceeds the brightness of a black body in the same flame. Thus, Nichols, Howes, and Wilber[^32] assert that the intensity of the glow of niobium oxide in the blue region of the spectrum at \(560^\circ\text{C}\) exceeds by 85,000 times the corresponding intensity of black radiation (uranium oxide served as the black body). It is clear that, when \(E_\lambda(T) > E_\lambda^b(T)\), Kirchhoff’s law is certainly violated, and certainly \(E_\lambda(T) > A_\lambda E_\lambda^b(T)\) (since \(A_\lambda \leq 1\)); consequently, the corresponding glow of bodies in flames is not thermal. It was in precisely this way that attempts were made to explain, as candoluminescence, the bright glow of mantles in Auer (incandescent) burners*). Subsequently, however, it turned out[^27][^28] that the unusual brightness of Auer mantles is caused by the selectivity of their emissive power. Emitting in the infrared region of the spectrum considerably less energy than a black body, the Auer mantle assumes in the flame a considerably higher temperature than a black body under the same conditions. Therefore the observed brightnesses of the Auer mantle and of the black body placed side by side in the flame correspond to different temperatures. The temperature of the mantle \(T_1\) is higher than the temperature of the black body \(T_2\). And since in the visible part of the spectrum the absorptive power of the mantle is close to unity, it follows that \(E_\lambda(T_1) \simeq E_\lambda^b(T_1) > E_\lambda^b(T_2)\). Obviously, the same may be the reason for the increased brightness of any selective emitter in a flame, and the fact \(E_\lambda > E_\lambda^b\), as an indication of candoluminescence, must be regarded as unreliable unless equality of the temperatures of the investigated body and of the black body has been established by direct measurements.

Equally unreliable is the unusual color of the glow observed in some bodies (for example, blue) at comparatively low temperatures. This was vividly demonstrated in the experiments of V. M. Kudryavtseva and G. I. Sinyapkina[^29]. Heating in a muffle furnace a test tube with zinc oxide powder to \(800\text{—}900^\circ\text{C}\), V. M. Kudryavtseva and G. I. Sinyapkina observed a bright blue glow. Similar results were obtained with ZnS and with TiO\(_2\)[^30]. It is obvious that, outside contact with a flame, under conditions of heating by thermal conduction, the possibility of non-temperature radiation, the possibility of violation of Kirchhoff’s law, is certainly excluded. Indeed, if there existed some body with similar properties, then the exchange of radiation between it and a normal—

*) On the construction of incandescent burners, see [^26].

but by the emitting body would lead to a violation of the second law of thermodynamics, to the realization of a perpetual-motion machine of the second kind.

The cause of the luminescence-like blue radiation of ZnO, ZnS, and TiO₂, unusual to the eye, at 800–900° C (the temperature of orange-red incandescence) lies in the specific character of the absorption spectra of these substances. V. M. Kudryavtseva and G. I. Sinyakina showed (see also ³¹) that at these temperatures the absorptive capacity of ZnO, ZnS, and TiO₂ for \(\lambda \leqslant 460\,m\mu\) is close to unity and falls sharply on passing into the region of longer wavelengths. Consequently, in the ultraviolet and in the blue-cyan part of the spectrum these powders absorb (and hence thermally emit) like a black body, and farther on like a gray body. The corresponding theoretical curves of absorption \(A_\lambda\) and emission \(E_\lambda\) are shown in Fig. 1. The emission curves are constructed according to Kirchhoff’s formula (1, 1) for \(A_\lambda\) varying as shown at the top of Fig. 1. As a result, on the curve of thermal radiation of bodies having a continuous absorption band with a sharply breaking edge, an additional maximum appears (see the curve \(E_\lambda\) in Fig. 1). This is precisely the cause of the blue glow of ZnO, ZnS, and TiO₂ at 800–900° C. The experimental curves are shown in Fig. 2.

Fig. 1. Toward an explanation of the features of the temperature radiation of bodies with a sharp edge of the absorption band. The emissive power of such bodies \(E_\lambda = A_\lambda E^b_\lambda\) coincides in the short-wave part of the spectrum with the emissive power of a black body (curve \(E^b_\lambda\)), and at long wavelengths—with the emissive power of a gray body (curve \(E^g_\lambda\)). As a result, a maximum appears on the curve of temperature radiation.

Fig. 2. Absorption (curve \(A_\lambda\)) and temperature emission (curve \(E_\lambda\)) of TiO₂ at 800° C (according to V. M. Kudryavtseva and G. I. Sinyakina).

In the coordinates \(\lg E_\lambda + 5\lg\lambda\) and \(\frac{1}{\lambda}\), the measurement results of V. M. Kudryavtseva and G. I. Sinyapkina fall on straight lines

\[ \lg E_\lambda + 5\lg\lambda = \lg A_\lambda + \lg C_1 - \frac{C_2}{T}\,\frac{1}{\lambda} \tag{2,1} \]

(Fig. 3), which signifies the exact fulfillment of Kirchhoff’s law*). Let us emphasize once again that under conditions of heating by thermal conduction this is a thermodynamically necessary result. Its violation would lead to a contradiction with the second law of thermodynamics.

Fig. 3. Dependence of \(\lg E_\lambda + 5\lg\lambda\) on \(\frac{1}{\lambda}\) for the temperature radiation of \(\mathrm{TiO_2}\) at \(800^\circ\mathrm{C}\). Theoretic straight lines depict Kirchhoff’s law (after V. M. Kudryavtseva and G. I. Sinyapkina).

The similarity of the glow of \(\mathrm{ZnO}\), \(\mathrm{ZnS}\), and \(\mathrm{TiO_2}\) under these conditions, where the possibility of luminescence is manifestly excluded, to the glow of these powders in flames made it possible to suppose that the latter glow is due to the same cause and is not luminescence. Continuing the investigations of V. M. Kudryavtseva and G. I. Sinyapkina, applying the Vavilov–Wiedemann criterion directly to the glow of \(\mathrm{ZnO}\) in various flames (the flames of a gasoline burner, illuminating gas, and a mixture of illuminating gas with hydrogen), as well as to the spectrum of zinc burning in oxygen, V. A. Sokolov \(^{33}\) established the exact fulfillment of Kirchhoff’s law in these cases as well. Thus it may be considered proven that \(\mathrm{ZnO}\) does not candoluminesce, and that the anomalous color of its temperature radiation is caused by the features of the absorption spectrum—black absorption at \(\lambda \leq 460\,\mathrm{m}\mu\) and gray absorption at larger wavelengths. The coincidence of the emission spectra when \(\mathrm{ZnO}\) is heated by thermal conduction, when \(\mathrm{ZnO}\) is present in flames, and when zinc powder burns in oxygen shows that the observed glow is not the direct result of chemical reactions, but is caused by zinc-oxide particles heated to incandescence in the process of combustion. V. A. Sokolov’s experimental results with \(\mathrm{CaO}\) and \(\mathrm{MgO}\) led to the same result.

\(\mathrm{ZnO}\), \(\mathrm{ZnS}\), \(\mathrm{TiO_2}\), \(\mathrm{CaO}\), and \(\mathrm{MgO}\) belong to those substances to which the ability to candoluminesce has hitherto been attributed \(^{21,22,23}\). The results of V. M. Kudryavtseva, G. I. Sinyapkina, and V. A. Sokolov compel one to take a critical attitude also toward the assertion—

* \((2,1)\) is equivalent to \((1,1)\) when \(h\nu \gg kT\). The condition \(h\nu \gg kT\) is obviously fulfilled for the entire visible spectrum.

...about the candoluminescence of other bodies. A new quantitative experimental study of the glow of these bodies in flames is necessary, in accordance with Vavilov–Wiedemann’s general definition of luminescence. The works of V. M. Kudryavtseva, G. I. Sinyapkina, and V. A. Sokolov testify to the fruitfulness of this path, which can and must, in every case, lead to a definite and unambiguous answer. Its significance, after many years of errors and misconceptions that have already become traditional, is especially great.

3. LAWS OF THE SPECTRAL TRANSFORMATION OF LIGHT

The laws of the spectral transformation of light—more precisely, the laws of the transformation of excitation energy into luminescence energy—are the only universal regularities of luminescent phenomena, valid for all luminophores in any aggregate states. It is precisely in this aspect that these laws have been considered in theory throughout the entire period of the scientific study of luminescence, beginning with the works of Stokes^18 and E. Becquerel^33. All practical applications of luminophores are based on the capacity to transform various electromagnetic and corpuscular radiations into visible light. In luminescent lamps^34 the conversion of invisible light into visible light is used directly for illumination. Luminescent screens^35, which make X-rays, radium rays, ultraviolet, infrared, and other rays accessible to visual observation, have opened to the eye the possibility of seeing through objects opaque to light, have made it possible to observe elementary nuclear processes accompanied by corpuscular and γ-radiation, and have extended the limits of microscopy*). Owing to this capacity, luminophores have found wide application in radiolocation and television, in roentgenology, in nuclear physics, in electron and ultraviolet microscopy, and in other fields of science and technology. The same capacity of luminescent bodies for the spectral transformation of light underlies luminescent methods of chemical and grade analysis^37.

The law of the spectral transformation of light was first formulated by Stokes^18 in the form of the assertion that the wavelength must necessarily increase

\[ \lambda_{\text{lum}} > \lambda_{\text{exc}} \tag{3.1} \]

in the process of luminescence. Stokes’ law, generally speaking confirmed by a large body of experimental material, is, however, often violated (see^38,39 and others). The classical considerations expressed concerning Stokes’ law are now of only historical interest. The physical content of the law

*) On a new branch of microscopy based on the phenomenon of luminescence, see the works of E. M. Brumberg^36.

Stokes was revealed by Einstein^40 on the basis of the simplest ideas of the quantum theory of light. A decrease in the frequency of light means, obviously, that in the act of luminescence only part of the energy absorbed by the phosphor is emitted \((h\nu_{\text{lum}} < h\nu_{\text{exc}})\). In other words,

Fig. 4. On the laws of Stokes and Lommel:
a—a phosphor obeying Stokes’ rule. When excited by light \(\lambda < \lambda'\), the entire luminescence band is emitted. When excited by \(\lambda > \lambda'\), only the longer-wavelength part of the luminescence band is emitted (shaded); b—a phosphor not obeying Stokes’ law: both \(\lambda < \lambda'\) and \(\lambda > \lambda'\) cause emission of the entire luminescence band. Upon excitation by light with wavelength \(\lambda > \lambda'\), the shaded part of the luminescence band belongs to anti-Stokes emission. Lommel’s rule is fulfilled in both cases.

the content of Stokes’ law consists in the fact that, in every elementary act of luminescence, part of the energy of the exciting radiation must be transformed into the internal energy of the luminescing body. Violations of Stokes’ law indicate the possibility of a reverse transition, of internal (thermal) energy

Fig. 5. Mirror symmetry of the absorption and luminescence bands of rhodamine 6G (after V. L. Levshin).

of the body into the energy of luminescence radiation. Thus, already in the first approach to Stokes’ law from the standpoint of quantum theory, its energetic meaning was disclosed behind the spectral facts.

The incorrectness of Stokes’ law as a general assertion, and its numerous experimental violations, led to another, expanded formulation of the law of spectral transformation of light, due to Lommel^41. If the original formulation of Stokes was contradicted by any fact of the emission of luminescence more short-wavelength than the exciting light*), then Lommel’s rule does not exclude such a possibility. According to it, the luminescence band as a whole, and also the maximum of intensity in it, must always be shifted toward the red side of the spectrum relative to the absorption band and its maximum (Fig. 4). In this case a partial—and sometimes considerable—overlap of the luminescence and absorption bands is possible. Unlike Stokes’ law, Lommel’s rule refers not to individual elementary acts, but to the radiation as a whole, composed of many elementary acts of absorption and emission of light. For solutions of dyes, V. L. Levshin^42 established a mirror symmetry of the absorption and luminescence bands (Fig. 5). As the experiments of N. Prilezhaeva^43 showed, Lommel’s rule also is sometimes violated and, consequently, is not general (Fig. 6).

Fig. 6. Absorption (I) and luminescence (II) bands of aniline vapor. The entire band of ultraviolet luminescence of aniline, with a maximum at 300 mμ, can be excited in the absorption band with a maximum at 370 mμ (according to N. Prilezhaeva).

The modern formulation of the law of spectral transformation of light was given by S. I. Vavilov^44, ^45, ^46, ^5, ^47, ^10 in the form of the following two propositions:

1. The energy yield of luminescence cannot exceed unity

\[ \rho \leqslant 1. \tag{3,2} \]

1. Under anti-Stokes excitation, i.e. when \(\nu_{\mathrm{exc}} < \bar{\nu}\), where \(\bar{\nu}\) is the mean value of the frequency in the radiation band, the energy yield of photoluminesc—

*) In the excitation of phosphors obeying Stokes’ law by a line belonging to the luminescence band, only its longer-wavelength part remains in the emission spectrum, while wavelengths smaller than \(\lambda_{\mathrm{exc}}\) disappear (see Fig. 4).

of the radiation must decrease with increasing difference of the frequencies \(\nu-\nu_{\text{exc}}\), and the more rapidly, the lower the temperature of the body.

To prove the first law, S. I. Vavilov proposed a method of ideal thermodynamic cycles, showing that the assumption of an energy yield of luminescence exceeding unity leads to a contradiction with the principles of thermodynamics.

Let us consider one of such cycles, proposed by E. I. Adirovich and cited, among other proofs, in S. I. Vavilov’s reply\(^{47}\) to P. Pringsheim’s objections\(^{48}\).

Inside a vessel with mirror walls there are two luminophores. Suppose that one of them transforms monochromatic light of frequency \(\nu\) into monochromatic light of frequency \(\nu'\) with yield \(\rho_1>1\), while the second performs the inverse spectral transformation with yield

\[ \rho_2=\frac{1}{\rho_1}<1. \]

The luminophores are in contact with heat reservoirs whose temperatures are respectively equal to \(T_1\) and \(T_2\), with \(T_1\ll T_2\). Each luminophore may, at will, be opened or closed to the radiation by a mirror shutter, whose displacement occurs perpendicular to the normal and therefore is not accompanied by work against light pressure. It is simplest to imagine that the luminophores are deposited from the inside on the bottoms of a cylindrical mirror cavity, as shown in Fig. 7.

Fig. 7. Toward the proof of the impossibility of \(\rho>1\).

Let both shutters be closed, and let there be monochromatic radiation of frequency \(\nu\) in the vessel. The energy of the radiation contained in the cavity is equal to \(\varepsilon\). Let us open the first luminophore. This will lead to absorption of frequency \(\nu\) and to emission of frequency \(\nu'\), and the energy of the radiation in the cavity will increase \(\rho_1\) times:

\[ \varepsilon\to \rho_1\varepsilon>\varepsilon. \tag{3,3} \]

Now let us lower the first shutter and carry out the same operation with the second luminophore. As a result, the radiation of frequency \(\nu\) with energy \(\varepsilon\), which was in the cavity before the experiment, will be restored in the cavity:

\[ \rho_1\varepsilon\to \rho_2\rho_1\varepsilon=\varepsilon. \tag{3,4} \]

The sole result of the cycle carried out consists in the transfer of energy \((\rho_1-1)\varepsilon\) from the reservoir with temperature \(T_1\) to the reservoir with temperature \(T_2\gg T_1\), which contradicts the second law of thermo-

...dynamics. Thus the impossibility of an energy yield of photoluminescence exceeding unity is proved.

An equivalent proof of the impossibility of \(\rho>1\) was given by L. D. Landau\(^{49}\). L. D. Landau showed that the entropy of monochromatic radiation is equal to zero, and therefore the change in entropy in the transformation of monochromatic light into monochromatic light of another frequency consists only of the change in entropy of the phosphor

\[ dS=\frac{dQ}{T}=\frac{I_a-I_l}{T}. \tag{3,5} \]

Here \(I_a\) is the intensity of the absorbed light, \(I_l\) is the intensity of the luminescence radiation. Since \(dS \geqslant 0\), it follows that

\[ \rho=\frac{I_l}{I_a}\leqslant 1. \tag{3,6} \]

Thus, Vavilov’s first law follows directly from the general principles of thermodynamics. It is essential that the values of the frequencies \(\nu\) and \(\nu'\) do not enter into its proof at all, i.e. the condition \(\rho\leqslant 1\) must be fulfilled for any ratio of the frequencies of the absorbed and emitted light.

Pure thermodynamics, as applied to luminescent phenomena, leads only to the limiting condition (3,2) for the energy yield at any frequencies of excitation and emission (this result is also generalized to the case of non-optical excitation of luminescence). Combining a thermodynamic consideration of the luminescence process as a whole with the quantum laws of the elementary acts of absorption and emission of light, S. I. Vavilov disclosed the spectral dependence of the luminescence yield. It has the simplest form for phosphors whose spectral composition of luminescence is, over wide limits, independent of the frequency of the exciting light (fluorescein, eosin, madder lake, etc.). If \(w(\nu',\nu)\) denotes the probability of emission of a quantum \(h\nu\) as a result of absorption of a quantum \(h\nu'\), then, by definition, the energy yield is equal to*)

\[ \rho=\frac{h\int \nu w(\nu',\nu)\,d\nu}{h\nu'}=\frac{\bar{\nu}}{\nu'}. \tag{3,7} \]

*) The mean value of the frequency in the emission spectrum (the “center of gravity” of the luminescence band) is determined as follows:

\[ \bar{\nu}=\frac{\int \nu w(\nu',\nu)\,d\nu}{\int w(\nu',\nu)\,d\nu}. \]

Consequently, in the simple form (3,7) the energy yield can be written only in the case when the quantum yield

\[ \rho_{\mathrm{kv}}=\int w(\nu',\nu)\,d\nu \]

is equal to unity. For \(\rho_{\mathrm{kv}}\ne 1\), the expression for the energy yield will differ from (3,7) by the constant factor \(\rho_{\mathrm{kv}}\), which does not disturb the linear dependence of \(\rho\) on \(\lambda_{\mathrm{exc}}\) for phosphors with a constant luminescence spectrum.

Here \(\bar{\nu}\) is the mean value of the frequency in the luminescence band. If, as is the case for the class of luminophores under consideration, \(w(\nu', \nu)=w(\nu)\)—does not depend on the frequency of the exciting light, then \(\bar{\nu}=\mathrm{const}\). In this case

\[ \rho \sim \frac{1}{\nu'} \sim \lambda_{\mathrm{exc}} \tag{3,8} \]

—the energy yield must be a linear function of the wavelength of the exciting light (section \(AB\) in Fig. 8).

In 1927 S. I. Vavilov\(^{50}\) was the first to carry out measurements of the luminescence yield over a broad range of wavelengths of the exciting light*) (from \(254\,m\mu\) to \(546\,m\mu\)). The measurements were carried out on fluorescein. In Fig. 9, reproducing the results of S. I. Vavilov, a linear section \(\rho(\lambda_{\mathrm{exc}})\) under short-wavelength excitation is clearly visible, confirming the validity of the initial theoretical ideas. Further, on passing to anti-Stokes excitation \((\nu' < \bar{\nu})\), a sharp drop in the luminescence yield is observed.

Fig. 8. Theoretical dependence of the energy yield of luminescence on \(\dfrac{\bar{\nu}}{\nu'}\). For \(\bar{\nu}=\mathrm{const}\), \(\dfrac{\bar{\nu}}{\nu'} \sim \lambda_{\mathrm{exc}}\) (after S. I. Vavilov).

Fig. 9. Energy yield of fluorescein luminescence as a function of the wavelength of the exciting light (after S. I. Vavilov).

Fulfillment of (3,8) means constancy of the quantum yield of luminescence for substances possessing an emission spectrum independent of the frequency of excitation. Following the measurements of S. I. Vavilov, the validity of (3,8) was confirmed by Harrison and Leighton\(^{52}\) on fluorescing petroleum oils, V. A. Fab-

*) The results of Nichols and Merritt\(^{51}\) available at that time, and the preliminary measurements of S. I. Vavilov\(^{44}\), concerned too narrow a spectral interval of excitation and did not permit a definite conclusion about linearity.

by Pringsheim53 on fluorescein, aesculin, and quinine bisulfate, by S. S. Solomin54 on seventeen fluorescing solutions, and also by other authors.

When the quantum yield is constant, expression (3.7) remains valid also in the case where the luminescence spectrum depends on the wavelength of the exciting light. In this case, however, \(\bar{\nu} \ne \mathrm{const.}\), but depends on \(\nu'\), and the simple linear dependence \(\rho \sim \lambda_{\mathrm{exc}}\) is violated.

For \(\nu'=\bar{\nu}\) (or \(\nu'=\rho_{\mathrm{q}}\bar{\nu}\), if \(\rho_{\mathrm{q}}\ne 1\)) the energy yield becomes equal to unity. A comparison of (3.7) with (3.2), i.e., a combination of the requirements of thermodynamics with the quantum conditions of absorption and emission of light, shows that under anti-Stokes excitation \((\nu'<\bar{\nu})\) formula (3.7), and consequently also the proposition that the quantum yield of luminescence is constant, cannot remain valid.

Formal thermodynamics, the result of which is inequality (3.2), does not allow one to draw any conclusions as to whether, for \(\nu'<\bar{\nu}\), the energy yield will decrease or will remain equal to its maximum possible value—unity. Invoking the ideas of statistical thermodynamics, S. I. Vavilov formulated the second law of the spectral transformation of light—the law of the decrease of \(\rho\) in the anti-Stokes region. Experimental confirmation of this law is provided by the descending part of the curve in Fig. 9 in the region of long-wavelength excitation, as well as by the results of measurements by all the other authors, which will be discussed below.

The basis of Vavilov’s second law consists of the following considerations. In every elementary act of anti-Stokes emission the energy yield

\[ \rho=\frac{\bar{\nu}}{\nu'} \]

exceeds unity. Consequently, every such elementary act is accompanied by a conversion of heat into luminescence radiation, i.e., it has a fluctuational character. Its probability is the smaller, the greater the energy difference—or, more precisely, the greater this difference in comparison with \(kT\). The rapid decrease in the probability of a fluctuation with increasing magnitude entails a fall in the energy yield of photoluminescence under anti-Stokes excitation. This fall cannot be compensated by a comparatively slow increase in the energy emitted in individual acts of light transformation.

In Fig. 8, to the right of point \(B\), the theoretical course of the dependence of \(\rho\) in the anti-Stokes region is shown, according to Vavilov’s second law. The fall must be the steeper, the larger the ratio

\[ \frac{h(\nu-\nu')}{kT}, \]

i.e., the lower the temperature. As \(T\to 0\), the fall becomes vertical. This means that, at sufficiently low temperatures, all wavelengths in the luminescence spectrum must be greater than the wavelength of the light exciting the luminophore. In other words, Stokes’ law in its original formulation on

necessary increase of the wavelength in luminescence follows from Vavilov’s second law as a limiting case for \(T \to 0\). Just as the constancy of the quantum yield of luminescence in the region where it does not contradict the requirement \(\rho \leqslant 1\), the fall of the energy yield under long-wavelength excitation has been widely confirmed experimentally on the most varied luminophors \(^{54,55,56}\), etc. Extensive investigations of the yield of luminescence of organic substances, uranium compounds, crystalline phosphors (\(\mathrm{ZnS}\), \(\mathrm{ZnCdS}\), \(\mathrm{Zn_2SiO_4}\)) and iodine vapors under short-wavelength and long-wavelength excitation were carried out by M. N. Alentsev \(^{57,58}\). In all cases in which anti-Stokes luminescence could be produced, a decrease of the yield toward increasing \(\lambda_{\text{exc}}\) was established (Figs. 10, 11, 12)*). In the most complex objects—crystallophosphors and, in part, solutions of uranium salts—a displacement was observed from \(\bar{\nu}\) toward the violet end of the spectrum at the beginning of the decrease of the energy yield of luminescence.

Fig. 10. Dependence of the luminescence yield on the wavelength of the exciting light: I—fluorescein in water (S. I. Vavilov). II—fluorescein in water (A. Yablonskii). III—fluorescein in water (M. N. Alentsev). IV—rhodamine B in glycerine (M. N. Alentsev).

Returning to the two laws of spectral transformation of light formulated at the beginning of the paragraph, we see that the first law is a necessary requirement imposed by thermodynamics on luminescence processes, independently of the relation between the excitation and emission spectra. In turn, the laws of quantum optics lead to a spectral dependence of the luminescence yield. Under Stokes excitation these laws allow one to expect proportionality of \(\rho\) and \(\lambda_{\text{exc}}\) in every case for luminophors with a luminescence spectrum independent of the excitation**).

*) In iodine vapor it proved possible to establish only the constancy of the quantum yield under Stokes excitation, since the absorption of long waves by iodine vapor is extremely small.

**) Experimentally this latter result, expressing the simple fact of constancy of the quantum yield, is fulfilled over wide limits if only the luminescence is not distorted by side processes connected with absorption of the excitation energy \(^{58,59}\).

Fig. 11. Energy yield of luminescence of a solution of uranyl sulfate in sulfuric acid (I), uranyl glass (II) and crystalline uranyl sulfate (III) as a function of the wavelength of the exciting light (after M. N. Alentsev).

Fig. 12. Energy yield of luminescence of several crystalline phosphors as a function of the wavelength of the exciting light:

\[ 1 — \mathrm{ZnSCuCo}\left(10^{-4}\,\frac{\mathrm{g\,Cu}}{\mathrm{g}}\right), \]

\[ 2 — \mathrm{ZnSCuCo}\left(10^{-5}\,\frac{\mathrm{g\,Cu}}{\mathrm{g}}\right), \]

\[ 3 — \mathrm{ZnSCdS}\ (25\%\,\mathrm{CdS}), \]

\[ 4 — \mathrm{ZnSCu}\left(10^{-4}\,\frac{\mathrm{g\,Cu}}{\mathrm{g}}\right) \]

(after M. N. Alentsev).

In the region of anti-Stokes excitation the constancy of the quantum yield of luminescence contradicts the requirements imposed by thermodynamics, and therefore is not fulfilled. The combination of quantum and thermodynamic propositions leads in this case to Vavilov’s second law, which expresses the necessity of a decrease in the energy yield of luminescence in the anti-Stokes region. Both laws are confirmed by all known experimental data*). The thermodynamic content of these laws, revealed by S. I. Vavilov behind the external spectral facts relating to photoluminescence, suggests that they remain valid also for all other methods of excitation of luminescence.

4. ON THE THEORY OF THE SECOND LAW OF SPECTRAL TRANSFORMATION OF LIGHT

The basis of the theory of the spectral transformation of light under anti-Stokes excitation of luminescence is the idea, considered above, of S. I. Vavilov concerning the necessity of combining the principles of thermodynamics with the laws of quantum optics, with the discreteness of the processes of absorption and emission of energy. The quantitative theory was developed by L. D. Landau \(^{49}\) and by E. I. Adirovich \(^{61}\).

L. D. Landau considers a phosphor that is wholly under equilibrium conditions and emits thermal radiation. As was shown above, thermal radiation is composed of luminescent and forced elementary processes. If we understand conventionally by luminescence that part of the thermal radiation of a body which is due to processes with duration \(\tau \gg 10^{-15}\) sec., arising as a result of the absorption of black radiation, then it may be asserted that under such conditions the intensity of “luminescence” is less than the total intensity of the radiation emitted by the body and, a fortiori, less than the intensity of black radiation. Introducing the probability of spectral transformation of light \(w(\nu', \nu)\), this result may be written as:

\[ \operatorname{const.}\int w(\nu', \nu)\,\nu'^3 e^{-h\nu'/kT}\,d\nu' \ll \operatorname{const.}\,\nu^3 e^{-h\nu/kT}. \tag{4,1} \]

As is evident from (4,1), the probability \(w(\nu', \nu)\) is expressed not as the ratio of the numbers of quanta, but as the ratio of the energies of the radiation of frequency \(\nu\) emitted by the body and of the radiation of frequency \(\nu'\) incident upon the body. On the right in (4,1) stands the intensity of black radiation, and on the left—the intensity of “luminescence” excited by the incident

*) The only measurements that led to contradictory results belong to Tsenkovsky \(^{60}\). However, they were not confirmed in the verification experiments carried out in the same laboratory by Yablonsky \(^{55}\), and they are at variance with the results of experiments by all other authors. This permits one to assert that Tsenkovsky’s data are the result of experimental errors.

on a body by black radiation. The entire discussion refers to that part of the spectrum where \(h\nu \gg kT\), described by Wien’s formula\(^{12}\), which, as is known, takes into account the discrete quantum structure of the light field. Consequently, equation (4,1) is connected with the quantum and thermodynamic requirements imposed on radiation.

Replacing in (4,1) the integration over the entire spectrum of frequencies by integration over a small interval \(\Delta \nu'\) near some value of the excitation frequency \(\nu'\) only strengthens the inequality. Choosing \(\Delta \nu'\) sufficiently small, we obtain

\[ \Delta \nu'\, w(\nu', \nu)\,\nu'^3 e^{-h\nu'/kT} \ll \nu^3 e^{-h\nu/kT}, \tag{4,2} \]

whence there follows the necessary condition for \(w(\nu', \nu)\)

\[ w(\nu', \nu) \ll \left(\frac{\nu}{\nu'}\right)^3 \frac{e^{-\frac{h(\nu-\nu')}{kT}}}{\Delta \nu'} . \tag{4,3} \]

For what follows it is necessary to clarify the meaning of the quantity \(w(\nu', \nu)\). The whole discussion concerns the temperature emission of a body that is in equilibrium with black radiation. The energy belonging to the spectral region \(\nu', \nu' + d\nu'\) in the black radiation incident on the body is partly reflected and scattered by the body, partly converted into heat, and partly re-emitted after a certain excitation period of the body, which has a finite duration. This latter radiation is due to luminescent processes, and its spectral distribution is specified by the function \(w(\nu', \nu)\). If the intensity of the radiation incident on the body at the frequency \(\nu'\) is increased by an amount \(\Delta i(\nu')\), and it is assumed that the corresponding change in the state of the body may be neglected, then the emission intensity at the frequency \(\nu\) will increase by

\[ \Delta I(\nu) = w(\nu', \nu)\,\Delta i(\nu'). \tag{4,4} \]

\(\Delta I(\nu)\) is the excess over the temperature emission of the body at the frequency \(\nu\), having a duration significantly greater than the period of light oscillations. In other words, \(\Delta I(\nu)\) is the intensity of luminescence at the frequency \(\nu\), whereas \(\Delta i(\nu')\) is the exciting radiation incident on the body. Precisely because luminescence is due to the same processes as the long-duration component of the body’s temperature radiation, the quantity \(w(\nu', \nu)\), found from consideration of the behavior of the body under equilibrium conditions, also characterizes its luminescence so long, however, as the state of the body differs little from the equilibrium state. The quantitative meaning of the assertion that the nonequilibrium state differs little from the equilibrium state will be specified below. In those cases where this assertion is valid, it follows from (4,3) and (4,4) that the intensity of luminescence for a sufficiently large anti-Stokes

of the displacement \(\nu-\nu'\) must become arbitrarily small in comparison with the intensity of the exciting radiation.

It is not difficult to see that the entire region of linear luminescence, in which

\[ \frac{I}{i}=\mathrm{const.}, \tag{4,5} \]

corresponds to nonequilibrium states of the body that differ little from equilibrium. Indeed, the condition of linearity may be written as

\[ \frac{\Delta I}{\Delta i}=\frac{I_0}{i_0}=\mathrm{const.} \tag{4,6} \]

Understanding by \(I_0\) and \(i_0\), respectively, the equilibrium values of the intensity of the “luminescent” emission at frequency \(\nu\) and of the intensity of the black radiation of frequency \(\nu'\) incident on the body, we see that the constant in (4,6) is \(w(\nu',\nu)\). But from the condition of linearity (4,6) it follows that the same \(w(\nu',\nu)\) is equal to the ratio of the luminescence intensity \(\Delta I(\nu)\) to the excitation intensity \(\Delta i(\nu')\). Consequently, if the linearity condition is satisfied, the quantity \(w(\nu',\nu)\), determined by L. D. Landau from consideration of the equilibrium radiation of a body, also characterizes its luminescence. In this case linearity is not only a sufficient, but also a necessary condition for the applicability of L. D. Landau’s results. In the nonlinear region*).

\[ \frac{\Delta I}{\Delta i}\ne \frac{I_0}{i_0}=w(\nu',\nu), \tag{4,7} \]

and the probability \(w(\nu',\nu)\), determined from (4,3), no longer pertains to the transformation of light in the process of luminescence.

Alongside the linear luminescence of solutions of organic dyes, undistorted by saturation effects, the glow of crystalline phosphors often exhibits nonlinearity even

*) Let us denote by \(\widetilde{w}(\nu',\nu)\) the probability of transformation of \(h\nu'\) into \(h\nu\) at some, generally speaking nonequilibrium, excitation intensity \(i=i(\nu')\). By definition

\[ I=\widetilde{w}i, \]

whence it follows that

\[ dI=\widetilde{w}\,di+i\,d\widetilde{w}. \]

The increase in emission intensity is due to: 1) the transformation of \(di\) with the former probability \(\widetilde{w}\), and 2) the fact that, as a result of a change in the state of the body, the entire flux of energy \(i(\nu')\) incident on the body is transformed with a slightly changed probability. The condition of linearity and, at the same time, the condition of a “nearly equilibrium” state is, evidently,

\[ \left|\frac{d\widetilde{w}}{di}\right|\ll \left|\frac{\widetilde{w}}{i}\right|. \]

at low excitation intensities (see^62, 63, 64, etc.) (Fig. 13). Experience also teaches that the probability of the spectral transformation \(\nu' \to \nu\) in the process of luminescence depends, as a rule, not only on \(i(\nu')\), but also on the detailed spectral composition of the exciting radiation, on the energy absorbed by the body at other frequencies as well. A convincing example is the infrared quenching of luminescence^65, as well as experiments showing that simultaneous irradiation of phosphors by two frequencies leads not to an additive superposition of the results, but to nonlinear effects^66, which do not disappear even at arbitrarily low excitation intensities. This means that, under certain excitation conditions, the state of luminescent bodies may, generally speaking, differ so greatly from their state under conditions of thermodynamic equilibrium that neglecting this change of state is inadmissible. For the circle of phenomena of linear luminescence, L. D. Landau proved the necessity of a decrease of the energy yield of luminescence in the anti-Stokes case. However, for the general case of the luminescence of bodies in substantially nonequilibrium states, a proof of this law cannot be obtained from consideration of equilibrium emission in an isothermal cavity.

Fig. 13. Nonlinear luminescence of zinc (\(I\)) and manganese (\(II\)) in a \( \mathrm{ZnS}\cdot 10^{-3}\ \mathrm{Mn}\)-phosphor under excitation by mercury lines \(312\,m\mu\) and \(366\,m\mu\) (after V. L. Levshin).

E. I. Adirovich^61 proposed another proof, based on the principle of detailed balance^67, 68 and leading, moreover, to the establishment of a relation between the probabilities of direct and inverse spectral transformation of light.

Let \(E_j\) be the energy possessed by a luminescent body (luminophore), maintained at a constant temperature \(T\), under conditions of stationary irradiation by light containing the frequencies \(\nu\) and \(\nu'\). The dependence of the state of the body on the temperature and irradiation conditions is expressed by the index \(j\).

Place this body (unexcited!) in a cavity with mirror walls and, by transferring heat to the body from an external reservoir, bring the total energy enclosed in the cavity to the value

\[ E = E_j + \varepsilon + h\nu', \tag{4,8} \]

where \(\varepsilon\) is the energy that black radiation of temperature \(T\) would possess when filling the cavity containing the luminophore;

After this we isolate the cavity from the surrounding world. In the system body + radiation contained in the cavity, an equilibrium state with temperature \(T' \ne T\) will be established.

Let us consider an ensemble consisting of a very large number \(N_0\) of such systems. After equilibrium has been established in it, at each instant of time all states compatible with the given total energy of the system \(E\) will be realized. The probability of finding the system in some state \(\alpha\) is equal to*)

\[ \omega(\alpha)=\frac{N(\alpha)}{N_0}=e^{\frac{S_\alpha}{k}}, \tag{4,9} \]

where \(S_\alpha\) is the entropy of the system corresponding to this state.

Let \(P(\alpha,\beta)\) denote the probability of transition of the system per unit time from state \(\alpha\) to state \(\beta\). Applying the principle of detailed balance to the ensemble, we obtain

\[ N(\alpha)P(\alpha,\beta)=N(\beta)P(\beta,\alpha) \tag{4,10} \]

or

\[ \frac{P(\alpha,\beta)}{P(\beta,\alpha)} = \frac{N(\beta)}{N(\alpha)} = \frac{\omega(\beta)}{\omega(\alpha)} . \tag{4,11} \]

According to (4,9),

\[ \frac{P(\alpha,\beta)}{P(\beta,\alpha)} = e^{\frac{S_\beta-S_\alpha}{k}} . \tag{4,12} \]

\(\alpha\) and \(\beta\) are any two states of the system, which may differ from the equilibrium state by an arbitrary amount. Considering them as fluctuations is only a device of reasoning, allowing one to show that for any two states of the system the ratio of the probabilities of the forward and reverse transitions is a universal function of the difference of the entropies of these states.

Let us apply this result to the problem of the spectral transformation of light.

Let \(\alpha\) denote such a state of the system in which 1) the body is in that state \(j\) in which it was in the process of luminescence considered above; 2) the radiation consists of black radiation of temperature \(T\) and one quantum \(h\nu'\). Let \(\beta\) denote such a state of the system in which 1) the body is in state \(k\), into which it can be transferred from state \(j\) as a result of absorption of the quantum \(h\nu'\) and emission of the quantum \(h\nu\)

*) \(\alpha\) denotes the set of parameters completely determining the state of the system. As such parameters one may choose the numbers \(n_j\), defined as follows. For radiation these are the numbers of quanta with frequencies \(\nu_j\). For a body they index all possible states, and the numbers \(n_j\) can take only two values—zero and one. In each state of the body all numbers \(n_j=0\), except one.

\((E_k-E_j=h(\nu'-\nu));\) 2) the radiation consists of black radiation of temperature \(T\) and one quantum \(h\nu\). It is obvious that both states \(\alpha\) and \(\beta\) belong to the number of possible states for the value \(E\) expressed by (4.8), and therefore will be present in the ensemble with definite probabilities. The probabilities of transitions between these two states are, respectively, the probabilities of the spectral transformations \(\nu'\to\nu\) and \(\nu\to\nu'\). Consequently,

\[ \frac{w(\nu',\nu)}{w(\nu,\nu')}= \frac{P(\alpha,\beta)}{P(\beta,\alpha)} = e^{\frac{S_\beta-S_\alpha}{k}} . \tag{4.13} \]

The difference of the entropies of the system is \(S_\beta-S_\alpha=\Delta S_1+\Delta S_2\), where \(\Delta S_1\) refers to the body, and \(\Delta S_2\) to the radiation. For \(\Delta S_1\), thermodynamics gives

\[ \Delta S_1 \geq \frac{\Delta Q}{T}. \tag{4.14} \]

Since the processes of absorption and emission of light are, in the general case, nonequilibrium processes, (4.14) must be written with an inequality sign. The exception, besides the trivial case of irradiation of the body by black radiation in equilibrium with it, is also the case when stationary conditions of luminescence are realized and the difference between the absorbed and emitted radiant energy is removed by thermal conduction. In this case the state of the body remains unchanged, and hence the sum of the entropy changes of the body caused by interaction with the radiation and by heat transfer is zero. This is valid for any amount of spectrally transformed radiant energy. Referred to one quantum of absorbed radiation, this condition is written as

\[ \Delta S_1-\frac{h\nu'-h\nu}{T}=0. \tag{4.15} \]

From (4.15) it follows that

\[ \Delta S_1=-\frac{h(\nu-\nu')}{T}. \tag{4.16} \]

Consequently,

\[ \frac{w(\nu',\nu)}{w(\nu,\nu')}= e^{-\frac{h(\nu-\nu')}{kT}}\, e^{\frac{\Delta S_{\mathrm{rad}}}{k}}, \tag{4.17} \]

If there is a transformation of monochromatic light into monochromatic light, i.e. if the absorption and emission lines are sufficiently narrow, then \(\Delta S_{\mathrm{rad}}=0\). This proposition, proved by L. D. Landau for the case when the entire spectrum of the radiation field consists of narrow lines, can by the same method\({}^{19}\) be proved also for the transformation of monochromatic light into monochromatic light occurring in a radiation field of any spectral composition.

In this case*)

\[ \frac{w(\nu',\nu)}{w(\nu,\nu')}=e^{-\frac{h(\nu-\nu')}{kT}} . \tag{4,18} \]

In contrast to (4,1), the probabilities \(w\) are expressed here as ratios of numbers of quanta, and not of energies. To pass to energy quantities, each of them must be multiplied by the ratio of the emitted quantum to the absorbed one, i.e. by the ratio of the corresponding frequencies. As a result we obtain**)

\[ \frac{\eta(\nu',\nu)}{\eta(\nu,\nu')}= \left(\frac{\nu}{\nu'}\right)^2 e^{-\frac{h(\nu-\nu')}{kT}} . \tag{4,19} \]

*) Strictly speaking, the probabilities \(w\) in the numerator and denominator of (4,18) refer to different states of the body, and they should have been written as \(w_\alpha(\nu',\nu)\) and \(w_\beta(\nu,\nu')\). However, the influence of the change of energy \(\Delta E=E_k-E_j=h(\nu'-\nu)\) on the macroscopic properties of the body may be neglected. The same result (4,18), but already directly for \(w_\alpha(\nu',\nu)\) and \(w_\alpha(\nu,\nu')\), can be obtained by a somewhat more complicated route. Let, instead of (4,18), \(E=E_j+\varepsilon+h\nu'+h\nu\). Denote by \(\beta\) and \(\gamma\) the states into which the system passes from the state \(\alpha\) after the transformation \(h\nu'\to h\nu\) and \(h\nu\to h\nu'\), respectively, takes place. To the state \(\beta\) there corresponds the energy of the body \(E'=E_j+h(\nu'-\nu)\) and the field energy \(\varepsilon+2h\nu\); in the state \(\gamma\), \(E''=E_j+h(\nu-\nu')\), and the field energy is \(\varepsilon+2h\nu'\). The same considerations lead us to

\[ \frac{w_\alpha(\nu',\nu)}{w_\beta(\nu,\nu')}=e^{\frac{S_\beta-S_\alpha}{k}}; \]

\[ \frac{w_\alpha(\nu,\nu')}{w_\gamma(\nu',\nu)}=e^{\frac{S_\gamma-S_\alpha}{k}}, \]

whence it follows that

\[ \frac{w_\alpha(\nu',\nu)}{w_\alpha(\nu,\nu')}= e^{\frac{S_\beta-S_\gamma}{k}} \frac{w_\beta(\nu,\nu')}{w_\gamma(\nu',\nu)} . \]

But \(P(\beta,\gamma)=[w_\beta(\nu,\nu')]^2\), since the transition \(\beta\to\gamma\) is effected as a result of the transformation of two quanta of frequency \(\nu\) into two quanta of frequency \(\nu'\). Similarly, \(P(\gamma,\beta)=[w_\gamma(\nu',\nu)]^2\).

According to (4,12)

\[ \frac{w_\beta(\nu,\nu')}{w_\gamma(\nu',\nu)} = \sqrt{\frac{P(\beta,\gamma)}{P(\gamma,\beta)}} = e^{\frac{S_\gamma-S_\beta}{2k}} . \]

Consequently,

\[ \frac{w_\alpha(\nu',\nu)}{w_\alpha(\nu,\nu')}= e^{\frac{S_\beta-S_\gamma}{2k}} = e^{-\frac{h(\nu-\nu')}{kT}} . \]

**) We retain here the notation of the original papers. \(\eta\) in formula (4,19) has the same meaning as \(w\) in formula (4,3), obtained by L. D. Landau [see \(^{49}\), formula (16)].

\(\eta\) for any two frequencies cannot exceed unity, since otherwise the second law of thermodynamics would be violated. Replacing the denominator in (4.19) by unity, we obtain

\[ \eta(\nu', \nu) \leq \left(\frac{\nu}{\nu'}\right)^2 e^{-\frac{h(\nu-\nu')}{kT}} . \tag{4.20} \]

This formula is analogous to formula (4.3), with the difference that it has been obtained not for an equilibrium state, but for any stationary state of the luminescing body.

The stationarity of the state, meaning that the body can remain in such a state for an arbitrarily long time when irradiated with light of a definite spectrum and intensity (in the general case, not in equilibrium with the body), is essential. Only for such states are (4.19) and (4.20) valid. For nonstationary states it is entirely possible that \(\eta(\nu', \nu)>1\), as occurs, for example, in the infrared readout of the stored light sum of a flash in a crystallophosphor. Only for stationary states is the transition from (4.13) to (4.17) possible in the course of the proof.

\(w(\nu', \nu)\) is the probability that the body, in the process of luminescence, converts a quantum \(h\nu'\) belonging to the radiation field into a quantum \(h\nu\). It is equal to

\[ w(\nu', \nu)=A_{\nu'}p_{\mathrm{kv}}(\nu', \nu), \tag{4.21} \]

where \(A_{\nu'}\) is the probability of absorption of a quantum \(h\nu'\), and \(p_{\mathrm{kv}}(\nu', \nu)\) is the quantum yield, equal to the conditional probability of emission of a quantum \(h\nu\) by a body that has absorbed a quantum \(h\nu'\). Correspondingly, in energy units,

\[ \eta(\nu', \nu)=A_{\nu'}\rho(\nu', \nu), \tag{4.22} \]

where \(\rho(\nu', \nu)\) is the energy yield of luminescence at frequency \(\nu\) upon excitation of the luminophore by frequency \(\nu'\).

From (4.19) and (4.20) we obtain

\[ \frac{\rho(\nu', \nu)}{\rho(\nu, \nu')}= \frac{A_{\nu}}{A_{\nu'}} \left(\frac{\nu}{\nu'}\right)^2 e^{-\frac{h(\nu-\nu')}{kT}} \tag{4.23} \]

and

\[ \rho(\nu', \nu)\leq \left(\frac{\nu}{\nu'}\right)^2 \frac{e^{-\frac{h(\nu-\nu')}{kT}}}{A_{\nu'}} . \tag{4.24} \]

Inequality (4.24) is a proof of Vavilov’s second law on the decrease of the energy yield of luminescence in the anti-Stokes region of the spectrum. Formulas (4.19) and (4.23) establish the relation between the probabilities of spectral transformation of light and the relation of the energy yields of luminescence for two spectral lines \(\nu\) and \(\nu'\).

Let us note that both probabilities in formula (4.19) refer to one and the same state of the body. The same is true for both outcomes \(\rho(\nu', \nu)\) and \(\rho(\nu, \nu')\) appearing in formula (4.23). An experimental verification of the obtained relations, connecting the characteristics of the direct and reverse spectral transformations of light, is of interest.

The proof carried out is valid for arbitrary nonequilibrium states of luminescent bodies. It applies, however, only to the transformation of monochromatic light into monochromatic light, i.e., to those cases in which both the absorption spectrum and the emission spectrum consist of narrow lines. The proofs given above of the first law of the spectral transformation of light are based on the same assumption. Consideration of the problem in the case of absorption and emission of a continuous spectrum of arbitrary composition is still lacking. An attempt undertaken in this direction by L. D. Landau\(^{49}\) relies, in addition to thermodynamics, also on the assumption of proportionality between \(I_{\mathrm{lum}}\) and \(i_{\mathrm{exc}}\), up to considerable luminescence intensities. In this case L. D. Landau obtains not \(\rho \leqslant 1\), but \(\rho \leqslant 1 + \dfrac{T}{T_{\mathrm{eff}}}\), where \(\dfrac{T}{T_{\mathrm{eff}}} \ll 1\). However, consideration by the same method of the luminescence of bodies for which the linearity \(I_{\mathrm{lum}} = f(i_{\mathrm{exc}})\) is violated at low brightnesses will lead to \(\dfrac{T}{T_{\mathrm{eff}}} > 1\). In principle, \(\dfrac{T}{T_{\mathrm{eff}}} \gg 1\) is also possible. This makes one think that the method of consideration leading to the inequality \(\rho \leqslant 1 + \dfrac{T}{T_{\mathrm{eff}}}\) does not give an exact (i.e., the least) upper bound for the possible values of \(\rho\). On the other hand, the necessity indicated by L. D. Landau of taking into account, along with the change in the entropy of the body, also the change in the entropy of the radiation restricts the region of applicability of all arguments relating to the transformation of monochromatic light into monochromatic light, since in this case \(\Delta S_{\mathrm{rad}} = 0\). The nearest and most urgent task consists in a theoretical consideration of the problem of the spectral transformation of nonmonochromatic light in the general case of broad luminescence bands.

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