VAVILOV’S LAW

B. I. Stepanov

On January 25, 1956, five years had passed since the untimely death of the great scientist, eminent statesman and public figure, President of the Academy of Sciences of the USSR, Academician Sergei Ivanovich Vavilov. The whole life of S. I. Vavilov was bound up with the development of Soviet science. He opened up new, unexplored paths in it, persistently revealing the laws of the surrounding world. S. I. Vavilov always devoted great attention to the application of the achievements of science in the national economy and culture of the country, using them for the benefit of mankind.

One of the most important sections of physical optics—the doctrine of luminescence—is forever associated with the name of S. I. Vavilov. To him belongs the formulation of the basic concepts of this science, including the scientific definition of the very concept of luminescence, the establishment of a whole series of fundamental regularities, as well as the development of the principal specific methods for investigating luminescence and the basic questions of the theory of the phenomenon. In the years when S. I. Vavilov began to study luminescence, completely erroneous views prevailed in the theory of luminescence. At present the doctrine of luminescence constitutes an entire branch of knowledge.

A central place in S. I. Vavilov’s works on luminescence is occupied by studies of the yield of luminescence. The present article is limited to a brief exposition of S. I. Vavilov’s works devoted mainly to the dependence of the yield on the wavelength of the exciting light, and to the current state of the question.

§ 1. VAVILOV’S WORKS

Measurement of fluorescence yield. In works of 1922 and 192412, S. I. Vavilov formulates the concept of fluorescence yield \(\Gamma\) as the ratio of the energy of the secondary radiation (the energy of fluorescence) to the energy of all absorbed radiation and carries out

the first reliable measurement of this quantity. The significance of the results obtained was exceptionally great. Before the work of S. I. Vavilov there existed an erroneous notion even as to the order of magnitude of the fluorescence yield. It was usually believed that luminescence is a secondary phenomenon accompanying the principal process—the conversion of absorbed energy into heat. Helmholtz[^3] believed that the efficiency coefficient of the fluorescence of an aqueous quinine solution was equal to 1/1200. Wiedemann[^4] estimated this quantity for Balmain’s luminous paint and obtained the value 1/22. From the theoretical ideas developed by Lorentz it likewise followed that \(\Gamma \ll 1\).

Absolute measurements of the fluorescence yield still present great difficulties and are most often carried out by comparison with an already studied standard. In the 1920s these difficulties were especially great. Nevertheless, the values obtained by S. I. Vavilov are distinguished by high accuracy and for a quarter of a century served as the sole initial material for all subsequent measurements. The value of the fluorescence yield of an aqueous fluorescein solution obtained by S. I. Vavilov is equal to 0.80. Thus it was proved that a considerable fraction of the absorbed energy is converted into radiation, and the possibility of using luminescence for technical purposes was substantiated. Twenty-five years later, Alentsev[^5] made new, more accurate measurements and obtained similar values.

The law of Vavilov. In 1922 S. I. Vavilov carried out the work “Dependence of the intensity of fluorescence on the wavelength of the exciting light”1. This raised an entire problem which even now is of primary importance. The question concerns the essence of those processes that occur in complex molecules during the absorption and subsequent emission of light. Between the act of absorption and the act of emission the molecule remains for some short time (for example, \(10^{-8}\) sec) in an excited electronic state. During this time a large number of the most varied processes can occur in the molecule: redistribution of vibrational energy within the molecule, partial conversion of electronic energy into vibrational energy of the nuclei, transfer of part of the energy to the surrounding medium, and sometimes complete transfer of the excitation energy to other similar molecules. In a number of cases, during this time a complex photochemical process of structural change or decomposition of the molecule takes place.

S. I. Vavilov outlined and subsequently carried out a broad program for the study of these processes, measuring the fluorescence yield and its duration, the polarization and the spectra of fluorescence. In doing so, various methods were used for varying the experimental conditions—changing the excitation conditions (the wavelength of the exciting light), the temperature and state of aggregation, and the properties of the surrounding medium.

(solvent), the addition of foreign substances to the solution, and a change in the concentration of the substance itself.

The study of the dependence of the fluorescence yield on the wavelength of the exciting light is of especially great importance. In Fig. 1 the curve \(\Gamma(\lambda_{\mathrm{exc}})\), obtained by S. I. Vavilov in 1927,[^6] is given; the results of this work have become classical. Along the ordinate axis is plotted the energy yield of luminescence of an aqueous solution of the ammonium salt of fluorescein.

Fig. 1. Dependence of the energy yield of fluorescence of a solution of fluorescein \(+\mathrm{NH}_4\mathrm{OH}\) in water on the wavelength of the exciting light (according to the data of S. I. Vavilov).

At first, from 254 to 410 \(m\mu\), there is a linear increase of the yield. Then, in the interval from 410 to 510 \(m\mu\), it remains almost constant and, finally, rapidly falls. Using the simple relation between the energy and quantum yields of fluorescence,

\[ \Gamma = \mathfrak{B}\,\frac{\nu_{\mathrm{em}}}{\nu_{\mathrm{exc}}}, \tag{1} \]

S. I. Vavilov came to the conclusion that in the region of the linear increase of the energy yield the quantum yield of fluorescence \(\mathfrak{B}\) does not depend on the frequency of the exciting light.

The results obtained by S. I. Vavilov for a fluorescein solution were confirmed in the study of the luminescence of solutions of other complex molecules.[^7–^9] They are also valid for many objects of a different nature. Recently Alentsev[^10] carried out systematic investigations of various luminescent substances in the liquid and solid state, with different degrees of complexity and with different mechanisms of luminescence. In all cases it was observed that the quantum yield of fluorescence does not depend on the frequency of the exciting light in the Stokes region of the excitation spectrum and begins to fall rapidly

in the region of excitation frequencies close to the maximum of the fluorescence band. This dependence has a universal character and has been given the name of Vavilov’s law.

Let us first consider Stokes excitation. The consequences of Vavilov’s law for this part of the excitation spectrum are extremely important. When light of different frequencies is absorbed, the molecule reaches different vibrational levels of the excited electronic state, possesses different stores of vibrational energy and, consequently, different properties. Nevertheless, both the position of the fluorescence spectrum and, most importantly, its quantum yield remain unchanged. From this one can draw only one conclusion: during the time \(\tau\) for which the molecules remain in the excited state, certain processes take place as a result of which, before the act of emission, the entire ensemble of emitting molecules finds itself in one and the same initial state. S. I. Vavilov connected this circumstance with the influence of the solvent, with the presence of a condensed phase. In paper \(^{11}\) he wrote: “The most lawful and correct point of view is that in which the center of interest in the luminescence of solutions is seen in the distinctive character of this type of photoluminescence, depending on the properties of the condensed solvent and on the dissolved molecule correspondingly altered and conditioned by it. The center of gravity of investigation in such an approach is concentrated on questions of the state of the liquid as a whole, its viscosity, pseudocrystalline structure, the slowness of Brownian displacements, the blurring of energy levels, electrolytic dissociation, etc. From this point of view the luminescence of solutions becomes a remarkable and scarcely replaceable means of studying the liquid state.” Thus, in the opinion of S. I. Vavilov, in studying the fluorescence of solutions or other condensed media we study the properties of the entire ensemble of emitting molecules as a whole. The properties of individual molecules are manifested, but in an averaged form, and because of this the properties of the luminescence of solutions are in many respects simpler than the properties of the fluorescence of vapors.

After the classical works devoted to the yield of fluorescence, S. I. Vavilov turns his attention to the laws of decay and the duration of luminescence, as one of the most important parameters characterizing luminescence. In a whole series of works \(^{12}\) he establishes the fundamental distinction between fluorescence and phosphorescence. In paper \(^{13}\), carried out jointly with Levshin, it is shown that the luminescence of uranyl salts decays according to the exponential law

\[ W = W_0 e^{-t/\tau}, \]

where \(W\) is the power of emission, and \(\tau\) is the mean duration of the excited state. There follows further a systematic development of direct and indirect methods for determining the duration of the excited state \(^{14}\). The latter was connected with a profound and comprehensive

studying the quenching of fluorescence by foreign substances and temperature quenching, as well as the polarization of luminescence^14,15. In works^16,11 an important experimental fact is established, one directly adjoining Vavilov’s law—the independence of the duration of the excited state of solutions from the wavelength of the exciting light. Subsequently this result was confirmed by many authors. However, in contrast to the spectrum of quantum yield, no changes are observed in the spectrum of the duration of the excited state upon transition into the anti-Stokes region of the excitation spectrum. This was established by Sveshnikov^17 for a fluorescein solution. An analogous result was obtained recently by Neporent and Borisevich^18 for the vapors of several phthalimide derivatives.

Quenching of fluorescence of the first and second kind. In two papers of 1936^19 S. I. Vavilov elucidates the mutual connection between the quantum yield of fluorescence and the duration of the excited state and establishes a fundamentally important distinction between two types of fluorescence quenching. The process of absorption of light may be accompanied by excitation of the electron shell of the molecule. It may happen, however, that electronic excitation does not occur and the energy of the incident light is spent on ionization or dissociation of the molecule or is distributed over vibrational-rotational degrees of freedom. In the three latter cases luminescence does not arise and the entire process will be perceived by the observer as quenching of luminescence. S. I. Vavilov called this type of quenching quenching of the first kind. It is characterized by great rapidity and is established in a time commensurate with the period of molecular vibration \((10^{-12}—10^{-14}\ \mathrm{sec})\). To quenching of luminescence of the second kind S. I. Vavilov assigned all processes leading to deactivation of the excited electronic state after excitation of the electron shell, for example, collisions of the second kind or chemical processes with excited molecules. At the present time, thanks to the works of Terenin and his school^20,21, it has been clarified that one of the most important processes of this kind is the conversion of electronic energy into the energy of vibrations of the nuclei, occurring without emission of light. All these processes depend on the duration of the excited state and on the properties of the medium (concentration of quenching substances, temperature, viscosity, etc.).

The kinetics of quenching of the second kind, evidently, does not depend on quenching of the first kind, since the latter is completed before the beginning of the second process. Denoting by \((1-s)\) the fraction of absorbed quanta expended on quenching of the first kind, S. I. Vavilov introduces the concept of the pure yield of quenching of the second kind

\[ \mathfrak{B}'=\frac{N_{\mathrm{em}}}{s\cdot N_{\mathrm{abs}}}. \tag{2} \]

Here \(N_{\mathrm{em}}\) and \(N_{\mathrm{abs}}\) are the number of acts of emission and absorption. The true

the quantum yield is equal to:

\[ \mathcal{B}=\frac{N_{\mathrm{em}}}{N_{\mathrm{abs}}}. \tag{3} \]

From (2) and (3) it follows that

\[ \mathcal{B}=s\mathcal{B}'. \tag{4} \]

In the presence of spontaneous radiation, formula (4) is made more specific. As a result of simple calculations, S. I. Vavilov obtains

\[ \frac{\mathcal{B}}{\tau}=\mathrm{const} \tag{5} \]

and

\[ \mathcal{B}=\frac{s\tau}{\tau_0}. \tag{6} \]

Here \(\tau_0\) is the duration of the excited electronic state in the absence of quenching of the second kind. Let us note that the derivation of (5) and (6) is valid only in the case when, under quenching, the probability of a transition with radiation does not change. This fact, specially noted by S. I. Vavilov, is sometimes forgotten, and formulas (5) and (6) are ascribed an excessively general significance.

S. I. Vavilov drew attention to the fact that equation (6) can serve as the basis for an experimental analysis of the nature of fluorescence quenching, since \(\mathcal{B}\), \(\tau\), and, to a certain extent, \(\tau_0\) are accessible to direct or indirect measurement. In many cases relation (6) makes it possible to decide whether quenching belongs to the first or the second kind; as a result, a number of possible interpretations of quenching are immediately eliminated (electrolytic dissociation, coagulation, chemical processes, salting out, etc.). If, when the conditions of excitation of luminescence are changed (temperature, wavelength of the exciting light, introduction of foreign substances, change in concentration, etc.), the values of \(\mathcal{B}\) and \(\tau\) change in parallel (their ratio remains unchanged), then the quenching of fluorescence belongs to the second kind, i.e., it is connected with a change in the properties of the molecule in the excited state. In the opposite case, fluorescence quenching should be assigned to quenching of the first kind.

S. I. Vavilov and his collaborators \(^{11,17,19,23,24,25,27,28}\) applied this method to the analysis of many specific types of fluorescence quenching and obtained very reliable results. Fluorescence quenching by foreign substances belongs to the second kind \(^{17}\). The same may be said of temperature quenching \(^{23}\). A decrease in the fluorescence yield with increasing concentration, associated with the association of molecules, is not accompanied by a change in \(\tau\) and belongs to quenching of the first kind. In this case an increase in concentration leads to the formation of nonfluorescing (or, otherwise, fluorescing) dimers \(^{26,27,28}\). The luminescent properties of the nonassociated molecules remain unchanged, but absorption in the dimers leads to a decrease in the yield.

Vavilov’s Law

Relation (6) makes it possible to draw important conclusions concerning the dependence of the quantum yield and the fluorescence lifetime of solutions on the wavelength of the exciting light. In the Stokes region of the excitation spectrum, \(\mathscr{B}\) and \(\tau\) are constant. Equation (6) is fully satisfied. It follows from this that the values of \(\mathscr{B}\) and \(\tau\) are completely determined by one and the same process of energy redistribution occurring in the excited state. In the anti-Stokes region of the excitation spectrum, \(\mathscr{B}\) decreases while \(\tau\) is preserved, and, consequently, relation (6) is not satisfied. S. I. Vavilov\(^{22}\) explained the decrease in yield by quenching of the first kind. He assumed that, under anti-Stokes excitation, strong inactive absorption takes place, not accompanied by excitation of the electron shell and subsequent fluorescence. In other words, upon absorption there occurs excitation of very high vibrational levels of the ground electronic state. This assumption by S. I. Vavilov, although supported by a number of authors\(^{18,25,26}\), long remained unclear. Indeed, in the ordinary literature on spectroscopy it is asserted that optical transitions within a single electronic state that are associated with a large change in vibrational energy are highly improbable. This is true, however, only for simple systems. In complex molecules and other complex systems, inactive absorption may be very considerable. In these systems the anharmonicity of the potential function is extremely large. It manifests itself in the redistribution of vibrational energy within the molecule, with the time of redistribution being comparable to the period of vibration. As the anharmonicity increases, the probability of transitions with large values of \(\Delta E_{\mathrm{vib}}\) increases. In extremely complex cases, for example in gray matter\(^{29}\), the probability of absorption of a quantum \(h\nu = \Delta E\) does not depend at all on the magnitude of the quantum.

An attempt at a thermodynamic justification of the decrease in the fluorescence yield under anti-Stokes excitation. The decrease in the quantum yield of fluorescence under anti-Stokes excitation must receive a correct explanation from the point of view of the processes occurring in individual molecules. Such an explanation, however, encounters serious difficulties, and a unified opinion has until now been absent. This led to attempts at a purely phenomenological explanation based on the application of the second law of thermodynamics. The first works in this direction were carried out by S. I. Vavilov\(^{30}\). Further studies belong to Adirovich\(^{31}\) and Landau\(^{32}\). In the Stokes region of the excitation spectrum, the quantum yield of solutions, when \(\nu_{\mathrm{exc}}\) is decreased, remains unchanged and therefore, according to (1), the energy yield continuously increases. If the quantum yield retained its value also under anti-Stokes excitation, then the energy yield could increase to any magnitude. Experiment shows that this does not occur, and after a certain frequency \(\nu'_{\mathrm{exc}}\), lying near the maximum of the band

emission, the value of the energy yield begins to fall. According to (1), this is associated with an even more rapid fall of the quantum yield. If, proceeding from thermodynamics, it were possible to prove that the energy yield cannot exceed unity, then this would constitute proof of the inevitability of a fall in the quantum yield and a kind of phenomenological explanation of the phenomenon under consideration.

Attempts to prove the relation

\[ \Gamma \leqslant 1 \tag{7} \]

were based mainly on the application of the method of ideal thermodynamic cycles. A number of such cycles were proposed by S. I. Vavilov, but the cycle proposed by Adirovich gained the widest currency. Adirovich reasoned as follows. Let there be two phosphors inside a vessel with mirror walls. Suppose that one of them transforms monochromatic light of frequency \(\nu\) into monochromatic light of frequency \(\nu'\) with yield \(\Gamma_1 > 1\), while the second performs the inverse transformation with yield \(\Gamma_2 = \frac{1}{\Gamma_1} < 1\). The phosphors are in contact with heat reservoirs whose temperatures are \(T_1\) and \(T_2\), with \(T_1 \ll T_2\). Each phosphor may, as desired, be opened or closed to radiation by a mirror shutter, whose displacement takes place perpendicular to the normal and is not accompanied by work against light pressure. It is simplest to imagine that the phosphors are applied from within to both ends of a cylindrical mirror cavity, as shown in Fig. 2.

Fig. 2.

Let both shutters 1 and 2 be closed and let monochromatic radiation of frequency \(\nu\) be in the vessel. Open the first phosphor. This will lead to the absorption of light of frequency \(\nu\) and the emission of light of frequency \(\nu'\), while the energy density of the radiation present in the cavity will increase by a factor of \(\Gamma_1\):

\[ u \to u\Gamma_1 > u. \tag{8} \]

Now lower the first shutter and carry out the same operation with the second phosphor. As a result, radiation of frequency \(\nu\) with density \(u\), which existed in the cavity before the experiment was performed, will be restored:

\[ \Gamma_1 u \to \Gamma_2 \Gamma_1 u \to u. \tag{9} \]

The sole result of the cycle carried out consists in the transfer of energy \((\Gamma_1 - 1)uV\) (\(V\) is the volume of the vessel) from the reservoir at temperature \(T_1\) to the reservoir at temperature \(T_2 > T_1\), which contradicts the second law of thermodynamics. Thus, in Adirovich’s opinion, it is proved that the energy yield of luminescence cannot exceed unity.

The method of thermodynamic cycles was subjected to serious criticism in a number of works by Pringsheim[^33]. The most serious objections have recently been presented by Antonov-Romanovskii, Stepanov, Fock, and Khapalyuk[^34]. In his reasoning Adirovich implicitly assumes that there exists a phosphor capable of converting light of frequency \(\nu'\) entirely into light of frequency \(\nu\), i.e., of absorbing only one frequency \(\nu'\), and emitting only one frequency \(\nu\). In fact, such a phosphor is thermodynamically impossible (even for \(\Gamma \leq 1\)). A substance at temperature \(T\), absorbing any frequency \(\nu\), emits it in accordance with Kirchhoff’s law. The same applies to the frequency \(\nu'\). Therefore, after the first shutter is opened, thermodynamic equilibrium of the radiation and the phosphor will be established in the cavity, and at least two frequencies, \(\nu\) and \(\nu'\), will be present, with densities determined by Planck’s formula at temperature \(T_1\). After the second operation is performed, the cavity will contain radiation of the same two frequencies, with a ratio of densities determined by the temperature \(T_2\). The initial state in the cavity (only one frequency \(\nu\)) will not be restored. The impossibility of the existence of a phosphor that entirely converts the frequency \(\nu'\) into the frequency \(\nu\) can also be proved by the following consideration. If such a phosphor actually existed, then, by placing it in an isothermal cavity with it, it would be possible to accumulate radiation of frequency \(\nu\), taking the system away from the state of equilibrium. In his reasoning Adirovich does not take into account the inevitable background of thermal radiation (at the frequencies \(\nu\) and \(\nu'\)), which leads to heat transfer from the hot body to the cold one and can compensate the reverse transfer caused by luminescence.

Analogous objections may also be raised when considering other thermodynamic cycles proposed in the literature. They all cannot serve as proof of relation (7).

Landau[^32] considered the question of the yield of luminescence from somewhat different points of view, taking into account the change in the entropy of radiation. His conclusions are valid only for the conversion of one monochromatic radiation into another monochromatic radiation, i.e., in the case when the absorption and emission spectra consist of narrow lines. Consideration of the problem in the case of absorption and emission of a continuous spectrum of arbitrary composition is still lacking. The author obtained an upper bound for the energy yield of luminescence. In practice it imposes almost no restrictions, since at sufficiently low radiation density the bound itself tends to infinity.

Disagreeing with S. I. Vavilov, Pringsheim[^33], [^35] suggested that the energy yield may exceed unity. He reasoned as follows. Suppose there exists a gas whose molecules, in the ground electronic state, possess several vibrational levels. If we illuminate it with light

with the energy of quanta corresponding to the distance between the excited vibrational level of the ground electronic state and the lowest vibrational level of the excited electronic state \((E_3 — E_1)\), then anti-Stokes lines will appear in the emission spectrum, corresponding to transitions from this level to lower vibrational levels of the ground state \(\left(\nu_{31}=\frac{E_3-E_1}{h}\right)\). The quantum yield in this model is equal to unity and, consequently, the energy yield is greater than unity.

This important discussion, of fundamental interest for the entire problem of anti-Stokes fluorescence, was not completed during S. I. Vavilov’s lifetime. He returned to it in his last work\({}^{36}\), published posthumously in the Collected Works. In this work S. I. Vavilov comes to the conclusion that in the majority of real cases the energy yield is indeed less than unity. However, in the case of rarefied vapors one may expect the appearance of anti-Stokes fluorescence with a yield somewhat exceeding unity. Analogous phenomena may be observed, in S. I. Vavilov’s opinion, under very brief excitation and the immediately following brief observation. Below, some new works in this direction will be described.

In recent years S. I. Vavilov repeatedly addressed the question of the mechanism of anti-Stokes fluorescence and drew a number of important conclusions. He decisively disputed Pringsheim’s point of view\({}^{35}\), which asserted that under Stokes and anti-Stokes excitation one and the same excited state arises, as a result of which the two methods of excitation are in principle no different from one another. S. I. Vavilov showed that under anti-Stokes excitation a part of the absorbed energy may be converted directly into heat, which accounts for the failure of relation (6). He objected just as decisively to Pringsheim’s opinion, which assumed that the anti-Stokes decrease in fluorescence yield is connected with the influence of the solvent. This point of view of S. I. Vavilov was finally confirmed in the work of Neporent and Borisevich\({}^{18}\), who established that rarefied vapors, like solutions, possess a decrease in yield under anti-Stokes excitation. S. I. Vavilov\({}^{30,37}\) also considered the temperature dependence of the yield under anti-Stokes excitation and showed, in particular, that at \(T=0\) the yield must be equal to zero, i.e., Stokes’ law must be obeyed.

Definition of luminescence. S. I. Vavilov is responsible for the exact scientific definition of the concept of luminescence\({}^{38}\). Not every glow is luminescence. Luminescence must be distinguished from simple thermal radiation, from Rayleigh scattering and reflection, from combination scattering, from bremsstrahlung radiation, and from Vavilov—Cherenkov radiation. All these types of glow occur-

Vavilov’s Law

are encountered in one combination or another, and methods must be devised for their experimental separation. The thermal radiation of bodies, i.e., the radiation of bodies in equilibrium with the surrounding medium, obeys Kirchhoff’s law. Any disturbance of equilibrium in a system caused by an external action leads to a change in the character of the glow. However, the effect of external excitation depends on the preceding state of the system, i.e., on the properties of the medium with which it was in thermodynamic equilibrium. Thus, for example, if the temperature of the medium is very high, then small external actions will not be able to disturb the equilibrium state, and the emission of the system will still have an equilibrium character[^39]. If, however, the temperature is low, then an external action leads to a significant change in the state of the system and to the appearance of luminescence. The total emission of a system includes, to some extent, ordinary thermal emission. This circumstance, first noted by Wiedemann[^40], was especially thoroughly substantiated by S. I. Vavilov. Thus, the characteristic feature of luminescence is the excess of the radiating capacity of a body over its radiating capacity at equilibrium.

In order to distinguish luminescence from other types of nonequilibrium emission, S. I. Vavilov carefully considered a whole series of possible criteria. It was sometimes assumed that, in contrast to scattering and other types of glow, luminescence is characterized by complete independence from the mode of excitation. S. I. Vavilov showed that this is incorrect. He also showed that luminescence cannot be distinguished by the criterion of incoherence of the glow. The most characteristic feature of luminescence should be considered the duration of the glow, i.e., the presence of emission after the action of the external exciting factor has ceased. Luminescence is always associated with a definite (even if small) duration of the stay of atoms and molecules in the excited state. During this time the most diverse processes may occur in the system. All other types of nonequilibrium emission are characterized by the fact that they arise or disappear practically instantaneously after the beginning or cessation of the action of an external cause (more precisely, within a time of the order of a light oscillation). Luminescence possesses inertia; it does not arise immediately and does not cease immediately. When afterglow is present, spontaneously occurring processes exist in the system; in the absence of afterglow, only forced processes do.

Thus, according to S. I. Vavilov’s definition, luminescence is an excess over the temperature emission of a body in the case when this excess has a finite duration exceeding the period of light oscillations. This definition had and has fundamental significance for the analysis of experimental facts. Measurement of the time interval separating excitation

and quenching, makes it possible to establish the nature of the glow. Using this method, S. I. Vavilov solved several very interesting problems.^38 In particular, he succeeded in correctly interpreting the essential nature of the new Vavilov—Cherenkov radiation discovered in 1933 and in proving that it does not belong to luminescence.^41

§ 2. QUANTUM YIELD OF LUMINESCENCE AND OF TOTAL QUENCHING WITH ACCOUNT TAKEN OF THE BACKGROUND OF THERMAL QUENCHING

According to S. I. Vavilov’s definition, the luminescence of any objects is always accompanied by a certain thermal quenching. The existence of this thermal background manifests itself to one degree or another in the study of any characteristics of luminescence, including the quantum yield. This is especially important at high temperatures or in the region of large wavelengths (the infrared region and radio waves). The usual expression for the yield

\[ \mathcal{B}=\frac{f}{f+d} \tag{10} \]

(\(f\) is the probability of a transition with emission of light, \(d\) is the probability of a radiationless transition), which does not take the thermal background into account, has certain, comparatively narrow limits of applicability. Since total quenching is the sum of two types of quenching, one must distinguish the quantum yield of luminescence and the quantum yield of total quenching. The expression for the quantum yield of total quenching was obtained quite recently by Stepanov,^39 and for the quantum yield of luminescence by Alentsev, Antonov-Romanovskii, Stepanov, and Fok.^42 The nonequilibrium process—luminescence—is here considered on the basis of the preceding thermodynamic equilibrium.

Fig. 3.

Quantum yield of quenching. Let us consider the simplest system of particles with two energy levels (Fig. 3). Suppose that before the illumination begins the system of particles is in thermodynamic equilibrium with the surrounding medium. The distribution of particles is determined by the usual Boltzmann formulas

\[ n_{1}^{\mathrm{equil}}=\frac{n}{1+e^{-\frac{h\nu}{kT}}},\qquad n_{2}^{\mathrm{equil}}=\frac{n e^{-\frac{h\nu}{kT}}}{1+e^{-\frac{h\nu}{kT}}}. \tag{11} \]

At equilibrium the principle of detailed balance is satisfied: the number of acts of quenching is equal to the number of acts of absorption of the equilibrium radiation \(u_{\nu}\), which always penetrates the given volume, and the number of nonoptical transitions \(2\to 1\) is equal to the number of nonoptical transitions \(1\to 2\).

VAVILOV’S LAW

The possibility of establishing thermodynamic equilibrium imposes quite definite relations on the constants characterizing the given system. These are, first of all, the relations between the Einstein coefficients:

\[ B_{21}=B_{12}=B,\qquad A_{21}=\frac{8\pi h\nu^3}{c^3}B_{21}=A, \tag{12} \]

as well as the relations between the probabilities of nonoptical transitions

\[ \frac{d_{12}}{d_{21}}=e^{-\frac{h\nu}{kT}}. \tag{13} \]

The ratio of the probabilities of nonoptical transitions depends on the temperature, since they characterize interaction with the surrounding medium. Thus, of the five constants \(B_{21}\), \(B_{12}\), \(A_{21}\), \(d_{21}\), and \(d_{12}\), only two are independent. The value of one of the Einstein coefficients is determined by the properties of the system itself; the value of the constant \(d_{21}=d\), in addition, by the character of the interaction with the medium.

Let us now assume that additional radiation from an external source falls on our system. As a result of absorption and subsequent fluorescence, thermal equilibrium will be disturbed: the number of particles on the first level will decrease, and the number of particles on the second level will increase. After some time a new dynamic (but not thermodynamic) equilibrium will be established. The numbers of particles \(n_1\) and \(n_2\) can be determined from the conditions

\[ dn_2=-dn_1=an_1dt+d_{12}n_1dt-fn_2dt-d_{21}n_2dt=0. \tag{14} \]

Here \(a=B(u_0+S)\) and \(f=A+B(u_0+S)\) are the probabilities of absorption and emission, if the radiation density in the surrounding volume is equal to \(u=u_0+S\), where \(u_0\) is the density of equilibrium radiation that existed before excitation, and \(S\) is the density of the external radiation disturbing the equilibrium. The probabilities of nonoptical transitions do not depend on the radiation density. Solving (14), we obtain:

\[ \left. \begin{aligned} n_2&=\frac{a+de^{-\frac{h\nu}{kT}}}{f+a+d\left(1+e^{-\frac{h\nu}{kT}}\right)}\,n \\ &=\frac{B(u_0+S)+de^{-\frac{h\nu}{kT}}}{A+2B(u_0+S)+d\left(1+e^{-\frac{h\nu}{kT}}\right)}\,n,\\[6pt] n_1&=\frac{f+d}{f+a+d\left(1+e^{-\frac{h\nu}{kT}}\right)}\,n \\ &=\frac{A+B(u_0+S)+d}{A+2B(u_0+S)+d\left(1+e^{-\frac{h\nu}{kT}}\right)}\,n. \end{aligned} \right\} \tag{15} \]

Using relations (12) and (13) and the values of \(u_0\) according to Planck’s formula, it can be shown that at very high temperatures formulas (15) pass over into the usual formulas (11), i.e., a weak external action does not disturb the thermodynamic equilibrium inside the system.

With the aid of (15) it is not difficult to calculate the absorption and release of energy through all channels. The total absorption of the incident radiation per second is equal to \(BSn_1 - BSn_2\) (stimulated emission is considered here as negative absorption). Only this difference is measured experimentally. Along with absorption of the incident radiation \(S\), there will also occur absorption of equilibrium radiation \(Bu_0(n_1 - n_2)\). The energy of spontaneous emission per second is equal to \(An_2\). The total heat release is equal to the difference \(d_{21}n_2 - d_{12}n_1\). It is not difficult to show that the total absorption of all light energy is equal to the sum of the energy of spontaneous emission and the released thermal energy.

Before the excitation was switched on (i.e., under thermal equilibrium) the heat balance was zero; the amount of heat released was equal to the amount of heat absorbed. After the excitation is switched on, the heat release \(d_{21}n_2\) predominates over its absorption \(d_{12}n_1\). Heat release occurs at the expense of the absorbed light energy. All the quantities under consideration depend not only on the constants of the substance itself \(A\), \(B\), \(d\) and on the density of the incident radiation \(S\), but also on the ratio \(\dfrac{h\nu}{kT}\). A change in temperature can lead to a strong change in the character of the processes taking place, especially at small \(\nu\). In the limit, as \(T \to \infty\) (\(n_2 \to n_1\)), the incident radiation ceases to be absorbed and no heat is released.

Let us call the quantum yield of emission the ratio of the number of transitions with emission of light to the total number of absorbed quanta (during the same time)

\[ \mathscr{B}_{\mathrm{em}}= \frac{An_2}{B(u_0+S)(n_1-n_2)} . \tag{16} \]

For a system with two levels, the quantum yield is equal to the energy yield. Substituting the values of \(n_1\) and \(n_2\) from (15), we obtain:

\[ \mathscr{B}_{\mathrm{em}}=\Gamma_{\mathrm{em}}= \frac{ 1+\dfrac{d\left(1-e^{-\frac{h\nu}{kT}}\right)} {A\left(1+\dfrac{S}{u_0}\right)} }{ 1+\dfrac{d}{A}\left(1-e^{-\frac{h\nu}{kT}}\right) }. \tag{17} \]

It follows from this that, for the system under consideration, the emission yield

is always less than unity. It is equal to unity only when \(S=0\) or \(T \to \infty\), and when \(\dfrac{d}{A} \ll 1\). The first two cases correspond to thermodynamic equilibrium (the number of emission events is equal to the number of absorption events). The third case corresponds to complete isolation from the medium, as a result of which nonradiative transitions are absent altogether. It also follows from (17) that the emission yield is always somewhat greater than \(\dfrac{A}{A+d}\). It is equal to \(\dfrac{A}{A+d}\) at very high density of the incident radiation \(S\) and, simultaneously, large values of \(\dfrac{h\nu}{kT}\). In both these cases the influence of the background thermal radiation may be neglected. It is precisely this case that we usually encounter when studying luminescence in the ultraviolet and visible regions of the spectrum.

The most characteristic consequence of (17) is the dependence of the emission yield on the intensity of the incident light. As \(S\) increases, the disturbance of thermal equilibrium becomes ever greater, the release of heat increases continuously, and the emission yield decreases from 1 to \(\dfrac{A}{A+d}\). The dependence of \(\mathscr{B}_{\text{em}}\) on \(S\) is especially strong when \(\dfrac{h\nu}{kT} < 1\). In this case the formulas obtained without taking account of the background thermal radiation have no meaning. In the limiting case \(\dfrac{h\nu}{kT} \ll 1\), i.e., in the case of very low frequencies and very high temperatures, \(\mathscr{B}_{\text{em}} = 1\), independently of the magnitude of the probability of nonradiative transitions and, consequently, no conversion into heat takes place.

The dependence of \(\mathscr{B}_{\text{em}}\) on \(S\) must also be taken into account when \(\dfrac{h\nu}{kT} \gg 1\), if the intensity of the incident radiation \(S\) is only slightly greater than the density of the equilibrium radiation \(u_0\). This circumstance must be taken into account, for example, when the flux of radiant energy from one light source passes through another light source, and also within a light source whose separate parts have different temperatures.

Quantum yield of luminescence. Expression (16) is of serious importance in calculations of the energy balance. However, under ordinary conditions it is not the quantum yield of emission that is measured, but the quantum yield of luminescence. Any receiver of radiant energy does not register the background of thermal emission corresponding to its temperature, since it is in equilibrium with it, but registers only deviations from this background. When there is no luminescent body, thermal radiation from the walls of the cavity in which it is located reaches the receiver. If between them one places any body of the same temperature, then it will block the path of part

quanta that fell on the receiver, but instead, according to Kirchhoff’s law, it will itself emit exactly the same number. The receiver does not register all these quanta. Thus, in determining the energy of luminescence, as well as the light energy absorbed by the luminescing body, we measure only the excess over the thermal radiation at the given temperature.

The quantum yield of luminescence must be determined in the following way:

\[ \mathscr{B}_{\text{lum}}=\Gamma_{\text{lum}} = \frac{\bigl[(A+Bu_0)n_2-Bu_0 n_1\bigr]h\nu} {BS(n_1-n_2)h\nu}. \tag{18} \]

In the denominator stands the power of the external radiation absorbed within the volume of luminescing substance under consideration. In the numerator of (18), \(An_2\) is the number of spontaneous transitions \(2\to1\) with emission of light per unit time, and \(Bu_0n_2\) is the number of transitions \(2\to1\) forced by the surrounding thermal radiation with density \(u_0\). This forced emission is long-lasting, since it continues even after the action of the excitation has ceased, until the number of molecules in the upper excited state falls to the value corresponding to thermodynamic equilibrium. At the same time, the forced emission \(BSn_2\) ceases immediately after the excitation is switched off. Moreover, for an isotropic substance the forced emission \(Bu_0n_2\) is also isotropic, whereas the direction of the forced emission \(BSn_2\) coincides with the direction of the incident wave. Thus, the term \((A+Bu_0)n_2\) in the numerator of (18) represents the total number of quanta emitted per unit time. The receiver registers only part of them, since absorption of thermal radiation \(Bu_0n_1\) occurs in the system. The power of luminescence is equal to the difference between the total emission power \((A+Bu_0)n_2h\nu\) and the power necessary to maintain the existing thermal radiation, \(Bu_0n_1h\nu\).

Substituting into (18) the values of \(n_2\) and \(n_1\) from (15), we obtain:

\[ \mathscr{B}_{\text{lum}}=\Gamma_{\text{lum}} = \frac{1} {1+\dfrac{d}{A}\left(1-e^{-\frac{h\nu}{kT}}\right)}. \tag{19} \]

From comparison of (19) and (17) it follows that the quantum yield of luminescence is smaller than the quantum yield of emission. They practically coincide for large values of \(\dfrac{h\nu}{kT}\) and not very large values of \(d\) (for electronic transitions at \(T\sim300—1000^\circ\)). In the infrared region the difference between \(\mathscr{B}_{\text{lum}}\) and \(\mathscr{B}_{\text{em}}\) may prove to be significant.

VAVILOV’S LAW

The quantum yield of luminescence (19), in contrast to the quantum yield of emission, does not depend on the density of the exciting radiation. For very large \(S\), expressions (19) and (17) coincide, i.e., the influence of the thermal background is also insignificant. For \(d=0\), i.e., in the absence of nonoptical transitions, \(\mathscr{B}_{\text{lum}}=1\). The quantity \(\mathscr{B}_{\text{lum}}\) is close to unity if the temperature is very high or the frequency very small. This still means that under such conditions the presence of external radiation does not disturb the equilibrium distribution over the energy levels, as a result of which the number of quenching acts is equal to the number of acts of nonoptical excitation, i.e., no heat is released.

Under the most frequently encountered conditions (electronic spectra and not high temperature), formula (19) becomes the usual formula (10)

\[ \mathscr{B}_{\text{lum}}=\mathscr{B}_{\text{em}}=\frac{A}{A+d}=\frac{f}{f+d}. \]

§ 3. ENERGY YIELD OF LUMINESCENCE OF A SYSTEM OF PARTICLES WITH THREE ENERGY LEVELS

The question of the energy yield of luminescence of a system of particles with three energy levels is of fundamental importance for the theory of S. I. Vavilov’s law. This question was considered by Antonov-Romanovskii, Stepanov, Fok, and Khapalyuk.³⁴ The model of a system of particles with three energy levels was originally proposed by Pringsheim³³ in order to prove the possibility of realizing a system with an energy yield exceeding unity. Although Pringsheim’s principal assertion was repeatedly disputed, the model itself was not considered, and it was not proved why it would not work. Pringsheim’s arguments were described in § 1.

Let us consider this model (Fig. 4) in considerably greater detail, taking into account radiationless transitions and the background of thermal radiation. It is altogether impossible not to take the thermal background into account in the present case, since the very appearance of particles at the initial level 2 is connected with the existence of thermal equilibrium with the medium. At \(T=0\), the number of particles \(n_2\), and consequently the magnitude of absorption of frequency \(\nu_{31}\), will be equal to zero. Introduce the following notation: \(\nu_{ji}\) is the frequency corresponding to the energy difference of the levels \(E_j-E_i\), \(c_{ij}\) is the total probability of transition between levels \(j\) and \(i\)

Fig. 4.

\[ \begin{aligned} c_{ij}&=B_{ij}u_{ij}+d_{ij} \qquad (i<j),\\ c_{ij}&=A_{ij}+B_{ij}u_{ij}+d_{ij} \qquad (i>j), \end{aligned} \tag{20} \]

\(d_{ij}\) is the probability of a nonoptical transition. All probabilities are calculated per unit time. The probability of an optical transition is equal to \(c_{ij}-d_{ij}\). Quantities without primes refer to the system in thermodynamic equilibrium, those with primes to the system under excitation with frequency \(\nu_{32}\). As for particles with two levels, when the excitation is switched on the probabilities of nonoptical transitions do not change. The probabilities of the optical transitions \(1\to 2\) and \(1\to 3\) also remain unchanged. Only the probabilities of the transitions \(2\to 3\) change substantially:

\[ c'_{23}=c_{23}+B_{23}S;\qquad c'_{32}=c_{32}+B_{32}S \tag{21} \]

\[ (S=S_{23},\ B_{23}=B_{32}). \]

The energy yield of luminescence of the system under consideration is equal to:

\[ \Gamma_{\mathrm{lum}}= \frac{\left[(c_{31}-d_{31})n_3-(c_{13}-d_{13})n_1\right]\nu_{31} +\left[(c_{32}-d_{32})n_3-(c_{23}-d_{23})n_2\right]\nu_{32}} {b_{23}(n_2-n_3)\nu_{32}S} + \]

\[ +\frac{\left[(c_{21}-d_{21})n_2-(c_{12}-d_{12})n_1\right]\nu_{21}} {B_{23}(n_2-n_3)\nu_{32}S}. \tag{22} \]

Expression (22) has been written by analogy with (18). Each square bracket of the numerator is multiplied by \(\nu_{ji}\) in order to pass from the number of quanta to the luminescence power at the given frequency. Proceeding from the principle of detailed equilibrium, it is not difficult to prove that

\[ \left. \begin{aligned} d_{ij}&=d_{ji}e^{-\frac{h\nu_{ji}}{kT}}\qquad (j>i),\\ c_{ij}&=c_{ji}e^{-\frac{h\nu_{ji}}{kT}}\qquad (j>i). \end{aligned} \right\} \tag{23} \]

The values \(n_1\), \(n_2'\), and \(n_3\) in the presence of excitation can be determined by solving the system of equations:

\[ \left. \begin{aligned} dn_1&=-(c_{12}+c_{13})n_1dt+c_{21}n_2dt+c_{31}n_3dt=0,\\ dn_2&=+c_{12}n_1dt-(c'_{23}+c_{21})n_2dt+c'_{32}n_3dt=0,\\ dn_3&=+c_{13}n_1dt+c'_{23}n_2dt-(c_{31}+c'_{32})n_3dt=0, \end{aligned} \right\} \tag{24} \]

\[ n_1+n_2+n_3=n. \]

Hence it follows that:

\[ \left. \begin{aligned} n_1&=n_1^{\mathrm{eq}}+\Delta n;\\ n_2&=n_2^{\mathrm{eq}}-(1+\alpha)\Delta n;\\ n_3&=n_3^{\mathrm{eq}}+\alpha\Delta n, \end{aligned} \right\} \tag{25} \]

where

\[ \Delta n = \frac{(c_{31}-c_{21}) \left(e^{-\frac{h\nu_{21}}{kT}}-e^{-\frac{h\nu_{31}}{kT}}\right)B_{23}S} {\left(1+e^{-\frac{h\nu_{21}}{kT}}+e^{-\frac{h\nu_{31}}{kT}}\right)} \times \]

\[ \times \frac{n} {c\left(1+e^{-\frac{h\nu_{21}}{kT}}+e^{-\frac{h\nu_{31}}{kT}}\right) +B_{23}S\,(c_{21}+2c_{12}+c_{31}+2c_{13})}, \]

\[ a=\frac{c_{13}+c_{21}+c_{12}}{c_{31}-c_{21}}; \qquad c=c_{21}c_{31}+c_{21}c_{32}+c_{31}c_{32}e^{-\frac{h\nu_{32}}{kT}}. \]

Substituting (25) into (22), we obtain an expression for the energy yield of luminescence

\[ \Gamma_{\mathrm{lum}}= 1- \frac{ d_{31}c_{21}\frac{\nu_{13}}{\nu_{32}} +d_{32}\left(c_{31}+c_{31}e^{-\frac{h\nu_{32}}{kT}}\right) -d_{21}c_{31}\frac{\nu_{21}}{\nu_{32}} } { c_{21}c_{32}+c_{21}c_{31}+c_{32}c_{31}e^{-\frac{h\nu_{32}}{kT}} }. \tag{26} \]

As in the case of a system of particles with two energy levels, the luminescence yield does not depend on the density of the incident radiation \(S\).

For \(d_{31}=d_{32}=d_{21}=0\), the energy yield of luminescence of such a model is equal to unity \((\Gamma_{\mathrm{lum}}=1)\). Such a result, generally speaking, is obvious, since for \(d_{31}=d_{32}=d_{21}=0\) light energy cannot be converted into heat.

Let us now suppose that the probabilities of nonradiative transitions are not equal to zero. In the numerator of the fraction (26) there is a negative term

\[ -d_{21}c_{31}\frac{\nu_{21}}{\nu_{32}}. \]

The denominator of this fraction is always positive, and, consequently, for sufficiently small \(d_{31}\) and \(d_{32}\) one may obtain \(\Gamma_{\mathrm{lum}}>1\). The values of the parameters \(d_{31}\), \(d_{32}\), and also \(d_{21}\), are determined by the properties of the given particular system, and it is quite possible that systems with small \(d_{31}\) and \(d_{32}\) exist in reality. Thus, the calculation presented shows that in some systems the energy yield may exceed unity.

An analogous calculation, carried out by Khapalyuk and Stepanov for a system with four levels, shows that the result obtained has general significance. For certain relations between the constants, the energy yield may exceed unity.

Systems possessing an energy yield exceeding unity are thermodynamically quite possible. In this case no violation of the second law of thermodynamics occurs. The whole process as a whole is still reduced to the transfer of energy from a light source with temperature \(T'\) to the entire system of particles together with the medium surrounding them, which are at temperature \(T<T'\). It is accompanied by peculiar internal transformations of energy in

to the system of particles and the surrounding medium. In the process of luminescence, the light energy emitted by the particles is greater than the energy absorbed by them. This energy is drawn from the vibrational energy (or thermal energy of another type) of the particles themselves and is transferred to the surrounding medium, which is at temperature \(T\) (and not to a light source at temperature \(T' > T\), which would be equivalent to a violation of the second law of thermodynamics). At the same time, the entire process is stationary in character and, consequently, the supply of vibrational energy is continuously replenished through interaction with the very same medium. What occurs are ordinary transformations of energy under the conditions of a nonequilibrium stationary process.

Pringsheim assumed that, for an output \(\Gamma > 1\) realized in his model, cooling of the system of particles occurs and the whole process is equivalent to the action of a refrigerator. This assumption is essentially correct, although it requires some refinements. In the present case the distribution of particles over energy levels differs greatly from the equilibrium distribution (when external excitation is switched on), and therefore the concept of temperature is in general inapplicable. Moreover, the average store of energy of all particles does not decrease but, as a rule, increases. Nevertheless, the temperature of the medium immediately surrounding the system of particles under study (for example, the walls of a glass cuvette) does indeed decrease. Around the system of particles a constant temperature gradient is formed, ensuring a continuous influx of heat and compensating the loss of energy through emission (with output \(\Gamma > 1\)). It follows from this that a substance with \(\Gamma > 1\) can be compared not with the refrigerator itself, but with the working substance (cooling agent) of a refrigerator.

Thus, the possibility of the existence of objects with an energy output greater than unity may be regarded as proven. Most likely, such objects will be certain atoms, and not molecules. Thereby a general thermodynamic justification for the fall of the output when \(\nu_{\mathrm{exc}}\) is decreased (within the anti-Stokes region) proves impossible. This does not, of course, deny the experimental fact itself—in all investigated complex systems the energy output is indeed less than unity.

§ 4. NEGATIVE LUMINESCENCE

In the cited work of Antonov-Romanovsky, Stepanov, Fok, and Kharalhok, a new important concept was introduced—negative luminescence. The meaning of this concept becomes clear in the analysis of formulas (25), which characterize the distribution of particles over energy levels when external excitation is switched on. According to (25), under the action of excitation the number of particles on levels 2 always decreases, and on levels 3 always increases (\(\alpha \Delta n\) is positive). The decrease

\(n_2 - n_2^{\mathrm{eq}}\)

is always greater (in absolute value) than the change

\(n_1 - n_1^{\mathrm{eq}}\).

This redistribution of particles over energy levels is directly and quite peculiarly manifested in the luminescence spectrum.

Let us calculate the emission power of our system for the frequencies \(\nu_{21}\), \(\nu_{31}\), and \(\nu_{32}\), as perceived by a receiver whose temperature coincides with the temperature of the medium. Before the excitation is switched on in the cavity where the system of particles and the radiation receiver are located, there is complete thermodynamic equilibrium, and the latter registers nothing. In this case all three quantities

\[ \left. \begin{aligned} W_{\text{lum}}(\nu_{31})&=\bigl[(A_{31}n_3+B_{31}u_{31}n_3)-B_{13}u_{31}n_1\bigr]h\nu_{31},\\ W_{\text{lum}}(\nu_{32})&=\bigl[(A_{32}n_3+B_{32}u_{32}n_3)-B_{23}u_{32}n_2\bigr]h\nu_{32},\\ W_{\text{lum}}(\nu_{21})&=\bigl[(A_{21}n_2+B_{21}u_{21}n_2)-B_{12}u_{21}n_1\bigr]h\nu_{21} \end{aligned} \right\} \tag{27} \]

are equal to zero. They represent the differences between the emission powers of the frequencies \(\nu_{31}\), \(\nu_{32}\), and \(\nu_{21}\) and the powers of absorption of thermal radiation. After the excitation is switched on and a stationary regime is established, the values \(n_i\) change \((n_i^{\mathrm{eq}}\to n_i')\), which will lead to a change in all three luminescence powers registered by the receiver. Substituting the values \(n_i'\) from (25) into (27), we obtain:

\[ \left. \begin{aligned} W_{\text{lum}}(\nu_{31})&=\\ &=Yh\nu_{31}(A_{31}+B_{31}u_{31})B_{32}Sc_{21}e^{-\frac{h\nu_{21}}{kT}} \left(1-e^{-\frac{h\nu_{32}}{kT}}\right),\\[6pt] W_{\text{lum}}(\nu_{32})&=\\ &=Yh\nu_{32}(A_{32}+B_{32}u_{32})B_{32}S(c_{13}+c_{12}) \left(1-e^{-\frac{h\nu_{32}}{kT}}\right),\\[6pt] W_{\text{lum}}(\nu_{21})&=\\ &=-Yh\nu_{21}(A_{21}+B_{21}u_{21})B_{32}Sc_{31}e^{-\frac{h\nu_{21}}{kT}} \left(1-e^{-\frac{h\nu_{32}}{kT}}\right), \end{aligned} \right\} \tag{28} \]

where

\[ Y=\frac{1}{c\left(1+e^{-\frac{h\nu_{21}}{kT}}+e^{-\frac{h\nu_{31}}{kT}}\right)+B_{32}S(c_{21}+2c_{12}+2c_{13}+c_{31})}. \]

It follows from this that the luminescence power at the frequencies \(\nu_{31}\) and \(\nu_{32}\) is positive. At the same time, the luminescence power at the frequency \(\nu_{21}\) is negative. In accordance with S. I. Vavilov’s definition, luminescence is the excess of total emission over thermal emission. In the cavity where the particle system under consideration is located, there exists a certain excess of quanta \(h\nu_{31}\) and \(h\nu_{32}\) in relation to

compared with the number of quanta present at thermal equilibrium (before excitation). At the same time, the number of quanta \(h\nu_{21}\) will be less than the number of quanta corresponding to equilibrium. For the frequencies \(\nu_{31}\) and \(\nu_{32}\) the receiver will register positive luminescence; for the frequency \(\nu_{21}\), negative luminescence.

Negative luminescence has not yet been observed experimentally. To measure it, one must work at high temperatures, use a powerful source of incident radiation, and employ a very sensitive receiver of radiant energy. If the receiver has the same temperature as the entire system, then it will give negative readings. This means that the number of quanta emitted by the receiver itself is greater than the number of quanta falling on the receiver from the luminescent substance.

It is not difficult to show that negative luminescence is a widespread, though usually barely noticeable, phenomenon. Any external action on any system leads to a disturbance of thermal equilibrium, to a change in the distribution over energy levels. If, before the external action, the emission of the system was characterized by ordinary thermal emission, then after the action it will be modified. For some frequencies the emission power will be greater than the equilibrium emission power; for others, less than it. In all the latter cases negative luminescence is manifested.

§ 5. DEPENDENCE OF THE QUANTUM YIELD OF FLUORESCENCE OF VAPORS ON THE WAVELENGTH OF THE EXCITING LIGHT

As was emphasized in § 1, Vavilov’s law was formulated only for condensed systems. In rarefied vapors, energy exchange with the medium is absent (during the time spent in the excited state). It is therefore no accident that investigations of the luminescence of vapors of complex molecules, carried out by Neporent \(^{21}\), made it possible to discover a new regularity: when \(\nu_{\mathrm{exc}}\) is lowered within the Stokes region, the quantum yield gradually increases. On passing into the region \(\nu_{\mathrm{exc}} < \nu'_{\mathrm{exc}}\), the quantum yield of vapors falls rapidly (just as in solutions). In Fig. 5, as an example, the curve \(\mathscr{B}(\nu_{\mathrm{exc}})\), obtained by Neporent and Borisevich \(^{18}\) for vapors of 3-aminophthalimide, is presented.

The correct explanation of the dependence \(\mathscr{B}(\nu_{\mathrm{exc}})\) as applied to Stokes excitation was given by Neporent \(^{21}\). He showed that the probability of radiationless conversion of electronic excitation energy into the vibrational energy of nuclei increases rapidly as the store of vibrational energy of the excited molecule \(E^{*}_{\mathrm{vib}}\) increases. On the basis of this supposition he succeeded in explaining many other important experimental facts as well. Neporent also supposed that the probabilities of optical transitions do not depend on \(E^{*}_{\mathrm{vib}}\). Conseq-

...this assumption is sometimes valid, but sometimes it may also be violated.

The probabilities of radiative transitions \(f(E^*_{\mathrm{vib}})\) and of radiationless transitions \(d(E^*_{\mathrm{vib}})\) are the basic characteristics of the excited state of a complex molecule. If they are known, then, knowing in addition the distribution of excited molecules over vibrational-energy reserves \(\rho^*(E^*_{\mathrm{vib}})\), one can calculate the quantum yield of fluorescence, the lifetime of the excited state, their dependence on temperature and on the wavelength of the exciting light, and also many other characteristics of luminescence. The transition probabilities \(f(E^*_{\mathrm{vib}})\) and \(d(E^*_{\mathrm{vib}})\) are determined by the internal properties of the molecule; the distribution function \(\rho^*(E^*_{\mathrm{vib}})\), in addition, depends on the excitation conditions and on the character of the surrounding medium. The study of the fluorescence yield, and also of the lifetime of the excited electronic state, which are very sensitive to various external influences, constitutes the most important means for determining \(f(E^*_{\mathrm{vib}})\) and \(d(E^*_{\mathrm{vib}})\) and for elucidating the nature of the processes occurring in the excited molecule. The fluorescence laws of solutions and of rarefied vapors differ strongly from one another. As S. I. Vavilov pointed out, solutions are characterized by a strong interaction between the excited molecules and the solvent. This interaction manifests itself in the transfer to the medium of the excess vibrational energy \(h\nu_{\mathrm{exc}} - h\nu_{\mathrm{el}}\), released in the process of absorption, and in the establishment of a stationary distribution over the vibrational levels of the excited molecules. An exact expression for the quantum yield of solutions with allowance for energy exchange was obtained in work \(^{43}\)*:

Fig. 5. Dependence of the quantum yield of fluorescence of 3-aminophthalimide vapor at \(t = 216^\circ\mathrm{C}\) on the frequency of the exciting light.

\[ \mathfrak{B} = s\,\frac{\bar f}{\bar f+\bar d} \simeq s\,\frac{f(\bar E^*_{\mathrm{vib}})}{f(\bar E^*_{\mathrm{vib}})+d(\bar E^*_{\mathrm{vib}})} . \tag{29} \]

* In formula (29) and in the following formulas the background of thermal radiation is not taken into account. When working in the visible region of the spectrum it is insignificant.

Here \(s\) is a multiplier taking into account quenching of the first kind, and

\[ \bar f=\int f\left(E_{\mathrm{vib}}^{*}\right)\rho^{*}\left(E_{\mathrm{vib}}^{*}\right)dE_{\mathrm{vib}}^{*}. \tag{30} \]

and

\[ \bar d=\int d\left(E_{\mathrm{vib}}^{*}\right)\rho^{*}\left(E_{\mathrm{vib}}^{*}\right)dE_{\mathrm{vib}}^{*} \tag{31} \]

are transition probabilities averaged over all values of \(E_{\mathrm{vib}}^{*}\) and characterizing the entire ensemble of molecules as a whole. Formula (29) is valid if, for all \(E_{\mathrm{vib}}^{*}\), the probability of energy exchange with the solution is greater than the probabilities \(f\left(E_{\mathrm{vib}}^{*}\right)\) and \(d\left(E_{\mathrm{vib}}^{*}\right)\). It follows from (29)—(31) that the exchange of vibrational energy with the solvent, if it is sufficiently large, leads to averaging of all the optical characteristics of the molecule and, above all, of the transition probabilities between energy levels. As a result of interaction with the solvent, the individual properties of separate molecules possessing different reserves of vibrational energy are completely lost (averaged), and the solution fluoresces as a single integral system.

In sufficiently strongly rarefied vapors, with which Neporent worked, collisions between molecules are sufficiently rare, as a result of which the excited molecule preserves its entire reserve of vibrational energy up to the transition to the lower electronic state. According to \(^{43}\), the quantum yield of vapors is equal to

\[ \mathcal{B} = s\int \frac{f\left(E_{\mathrm{vib}}^{*}\right)} {f\left(E_{\mathrm{vib}}^{*}\right)+d\left(E_{\mathrm{vib}}^{*}\right)} \rho^{*}\left(E_{\mathrm{vib}}^{*}\right)dE_{\mathrm{vib}}^{*} = \]

\[ = s\left[ \frac{f\left(E_{\mathrm{vib}}^{*}\right)} {f\left(E_{\mathrm{vib}}^{*}\right)+d\left(E_{\mathrm{vib}}^{*}\right)} \right]_{\mathrm{av}} \simeq s \frac{f\left(\bar E_{\mathrm{vib}}^{*}\right)} {f\left(\bar E_{\mathrm{vib}}^{*}\right)+d\left(\bar E_{\mathrm{vib}}^{*}\right)}, \tag{32} \]

where

\[ \bar E_{\mathrm{vib}}^{*}\;(\text{vapor}) = \bar E_{\mathrm{vib}}^{*}\;(\text{solution}) + h\nu_{\mathrm{exc}} - h\nu_{\mathrm{abs}}. \]

If

\[ \frac{\partial d\left(E_{\mathrm{vib}}^{*}\right)} {\partial E_{\mathrm{vib}}^{*}}>0, \tag{33} \]

then, with an increase of \(\nu_{\mathrm{exc}}\), \(\bar E_{\mathrm{vib}}^{*}\) increases and, consequently, \(d\) increases.

With the aid of formulas (29)—(33) it is also not difficult to explain the temperature quenching of fluorescence, i.e. the decrease of the yield as the temperature is raised. As the temperature increases, the distribution function of molecules over reserves of vibrational energy changes; the mean vibrational energy of the excited molecules always increases. If condition (33) is satisfied, then the increase of \(\bar E_{\mathrm{vib}}^{*}\) is accompanied by an increase of \(d\left(\bar E_{\mathrm{vib}}^{*}\right)\) and a decrease of the yi-

VAVILOV’S LAW

... yield. In a number of works by Levshin^26,44, Galanin^23, Sevchenko^24,45 and other authors it has been established that the quantum yield remains unchanged at low temperatures and falls rapidly at high temperatures. It follows from this that the probabilities of radiationless transitions begin to increase only at sufficiently large \(E^*_{\mathrm{col}}\). Temperature quenching is an intramolecular process; it is observed both in vapors and in solutions, and the influence of the solvent is of secondary importance.

Between the quantum yield of fluorescence and the lifetime of the excited state there exists a quite definite relation:

solutions

\[ \frac{\mathscr{B}}{\tau} = s \bar f = s \int f(E^*_{\mathrm{col}})\rho^*(E^*_{\mathrm{col}})\,dE^*_{\mathrm{col}} \cong s f(\bar E^*_{\mathrm{col}}), \tag{34} \]

vapors

\[ \frac{\mathscr{B}}{\tau} = s\left\{ \frac{f(E^*_{\mathrm{col}})} {\left[f(E^*_{\mathrm{col}})+d(E^*_{\mathrm{col}})\right]^2} \right\}_{\mathrm{av}} : \left\{ \left[ \frac{f(E^*_{\mathrm{col}})} {f(E^*_{\mathrm{col}})+d(E^*_{\mathrm{col}})} \right]^2 \right\}_{\mathrm{av}} . \tag{35} \]

These formulas were obtained by Stepanov^43. They are equivalent to formula (5), derived by S. I. Vavilov, but take into account the possible dependence of the probabilities of optical transitions on the store of vibrational energy of the excited molecule, as well as the difference between vapors and solutions. Under monochromatic excitation, i.e. when the maximum of the function \(\rho^*(E^*_{\mathrm{col}})\) is sufficiently sharp, expression (35) is close to (34).

Relation (34) shows that the ratio \(\mathscr{B}/\tau\) in solutions does not depend on the probabilities of radiationless transitions. When \(d\) is changed by introducing foreign quenching substances into the solution, the values of \(\mathscr{B}\) and \(\tau\) change in the same way and their ratio is preserved.

In vapors the ratio \(\mathscr{B}/\tau\) may depend on the probabilities of radiationless transitions. From (34) it follows that the ratio \(\mathscr{B}/\tau\) for solutions can change only when the form of the distribution \(\rho^*(E^*_{\mathrm{col}})\) changes, i.e. it depends only on temperature. Experiment shows that the ratio \(\mathscr{B}/\tau\) is unchanged over a wide temperature interval. For solutions of rhodamine B in alcohol and glycerin it is constant from \(-50^\circ\mathrm{C}\) to \(+50^\circ\mathrm{C}\) (Galanin^23). At higher temperatures the decrease of \(\tau\) is relatively slowed, and the ratio \(\mathscr{B}/\tau\) begins to diminish. This proves that at small \(E^*_{\mathrm{col}}\) the probability of the radiationless transition \(f\) does not depend on \(E^*_{\mathrm{col}}\). At higher \(E^*_{\mathrm{col}}\) the value of \(f\) begins to depend on \(E^*_{\mathrm{col}}\) (in the case of rhodamine B, to decrease).

§ 6. THE INFLUENCE OF FOREIGN GASES ON THE FLUORESCENCE OF VAPORS. TRANSITION TO SOLUTIONS

As we have seen above, the laws of fluorescence of vapors and solutions of complex molecules (above all, the dependence of the quantum yield on the frequency of the exciting light) differ markedly from one another. In the process of fluorescence of vapors, the individuality of the excited molecules, possessing one or another store of vibrational energy, is revealed. In the process of fluorescence of solutions, all the fluorescing molecules, together with the molecules of the solvent, manifest themselves as a single whole; the properties of the individual emitting centers are averaged.

For the purpose of studying the intermediate states between a rarefied gas and a solution, Neporent[^21] investigated the fluorescence of vapors of organic molecules with the addition of certain permanent gases (H₂, N₂, NH₃, C₂H₄, C₅H₁₂, etc.).*) Excitation was carried out at various frequencies in the Stokes region of the excitation spectrum. Figure 6 gives curves of the dependence of the fluorescence quantum yield of β-naphthylamine vapor on the pressure of the foreign gas—pentane. Along the abscissa are plotted the numbers \(Z\) of collisions of an excited β-naphthylamine molecule with molecules of the foreign gas (calculated per 1 second). The different curves correspond to different \(\lambda_{\text{exc}}\). It follows from the figure that, as the pressure of the foreign gases increases, the quantum yield increases. The enhancement of fluorescence is especially noticeable at small \(Z\). With a further increase in \(Z\), the growth of the yield slows down and, finally, ceases altogether.

Fig. 6. Quantum yield of fluorescence of β-naphthylamine vapors as a function of the pressure of a foreign gas under excitation by different wavelengths.

The strengthening action of foreign gases increases with increasing absorbed quantum, i.e. with increasing store of vibrational energy of the excited molecules. At small \(\nu_{\text{exc}}\) (\(\lambda_{\text{exc}} = 3341\) Å), the quantum yield is in itself sufficiently large, and the strengthening action of pen-

*) We do not consider here the influence of quenching gases, for example oxygen. In this case other regularities appear, which lie outside the scope of the present article.

VAVILOV’S LAW

is insignificant; for large $\nu_{\text{exc}}$ the quantum yield at $Z=0$ is very small, and foreign gases increase it severalfold.

Enhancement of the fluorescence of vapors under the action of foreign gases is characteristic not only of $\beta$-naphthylamine, but also of complex molecules of phthalimide $^{18,46}$. It was observed even for the comparatively simple molecule of aniline $^{21}$.

An increase in the pressure of a foreign gas, leading to an increase in the quantum yield, is accompanied by a simultaneous increase in the lifetime of the excited state. This effect was observed by Kem and Rellefson $^{47}$.

Neporent correctly explained all the facts indicated, taking into account the dependence of the probability of radiationless transitions on the store of vibrational energy of excited molecules. The phenomenon under consideration is closely connected with all the other features of the fluorescence of vapors and solutions and makes it possible to interpret them from a unified point of view. The increase in quantum yield with increasing pressure of foreign gases is determined by the change in the distribution function of excited vapor molecules with respect to their stores of vibrational energy. When vapor molecules are excited by radiation with frequencies $\nu_{\text{exc}} > \nu_{\text{el}}$, they acquire an excess of vibrational energy equal to $h\nu_{\text{exc}} - h\nu_{\text{el}}$. The mean energy of all excited molecules increases by the same amount. In the absence of foreign gases the quantum yield is determined by formula (32). In the process of collisions with molecules of the foreign substance, the excited molecules will give up to them some part of their excess vibrational energy. As a result, their mean vibrational energy and, consequently, the probabilities of internal radiationless transitions decrease. If (33) is satisfied, then in accordance with (32) this will lead to an increase in the quantum yield and to an increase in the lifetime of the excited state.

During its lifetime an excited molecule may undergo a whole series of elastic collisions. The greater their number, i.e., the higher the pressure of the foreign gas, the greater the loss of vibrational energy and the greater the increase in the quantum yield. The action of foreign gases is especially effective at low pressures. A further increase in pressure gives an ever smaller effect. This result is natural. In collisions an excited molecule can lose only its excess vibrational energy. As soon as the mean vibrational energy of the excited molecule becomes equal to the mean vibrational energy of the unexcited molecule, it is completely stabilized and further collisions will not be effective.

Enhancement of fluorescence under the action of foreign substances is characteristic only of vapors and is not observed in solutions. In solutions the role of foreign substances is played by the solvent molecules, and as a result of the transfer of vibrational energy to the molecules

solvent, the form of the distribution function in the upper state is the same as in the lower state. From this point of view, the optical properties of vapors at high pressures of a foreign substance are equivalent to the optical properties of solutions.

The conclusion obtained is clearly confirmed by the data of Fig. 7, taken from the work of Borisevich^46. In the figure, on a single scale, the values of the fluorescence yield of vapors of 3,6-diaminophthalimide in the absence of foreign gases and with the addition of pentane at

Fig. 7. Quantum yield of vapors of 3,6-diaminophthalimide ($\times$—vapors at 265° C, $\bullet$—vapors with the addition of 450 mm Hg of pentane).

a pressure of 460 mm Hg are given. Along the abscissa axis $\nu_{\text{exc}}$ is plotted. The yield of solutions in the Stokes region of excitation does not depend on $\nu_{\text{exc}}$. In vapors there is a large horizontal portion of the curve $\mathcal{B}(\nu_{\text{exc}})$, but then, as $\nu_{\text{exc}}$ increases, the yield falls. The addition of pentane leads to a significant increase in the length of the horizontal portion. With a further increase in the pressure of pentane, the horizontal portion would become still longer and, in the limit, would give a horizontal line.

Thus, Vavilov’s law is obeyed fully not only for the fluorescence of solutions and other condensed media, but also for the fluorescence of vapors at sufficiently high pressures of a foreign gas. It is necessary only that the number of collisions during the excited state be sufficiently large and that continuous exchange of vibrational energy with the molecules of the foreign substance take place. In this way it is possible to determine the limits of applicability of Vavilov’s law.

In close connection with the phenomenon just described is another important phenomenon, recently discovered by Borisevich^46. He investigated the influence of foreign substances on the quantum yield of fluorescence of vapors of several phthalimides under excitation in a broad region of the spectrum, including the region $\nu_{\text{exc}} < \nu_{\text{el}}$. In the region $\nu_{\text{exc}} > \nu_{\text{el}}$, the usual phenomena of fluorescence enhancement are observed; in the region $\nu_{\text{exc}} < \nu_{\text{el}}$, a noticeable decrease in the quantum—

of the yield. Analysis of all the facts obtained led the author to the conclusion that the weakening of fluorescence arises as a result of the transfer of vibrational energy from molecules of a foreign gas to excited phthalimide molecules. If \(\nu_{\text{exc}} < \nu_{\text{el}}\), then in the process of excitation the molecules lose a certain amount of vibrational energy. In the process of collisions with molecules of a foreign gas this deficiency is compensated, \(\overline{E}^{*}_{\text{vib}}\) and \(d(\overline{E}^{*}_{\text{vib}})\) increase, which, according to (32), leads to a decrease of the quantum yield.

§ 7. INTRAMOLECULAR “MECHANISM” OF ANTISTOKES FLUORESCENCE

We have already indicated that the dependence \(\mathscr{B}(\nu_{\text{exc}})\) under antistokes excitation in vapors and solutions is practically the same. The influence of the solvent is of secondary importance and may be neglected; the process of conversion of energy into heat occurs in the fluorescing molecules themselves.

Experiment shows\(^{10,18}\) that the frequencies \(\nu'_{\text{exc}}\) corresponding to the onset of the drop in yield are always less than \(\nu_{\text{el}}\). It follows from this that not all molecules can participate in the absorption process, but only those which possess a sufficiently large reserve of vibrational energy. This fact determines the strong temperature dependence of antistokes fluorescence. At \(T = 0\) antistokes fluorescence is absent altogether. Conversely, the higher the temperature, the less specific the antistokes excitation, and the smaller the drop in yield (Vavilov\(^{20,37}\)). The experimental study of the influence of temperature, carried out by Tumerman\(^{25}\), agrees with these basic propositions.

In § 1 we already mentioned that the decrease of the yield when \(\nu_{\text{exc}}\) is decreased is not accompanied by a decrease in duration. This fact points to the specificity of quenching under antistokes excitation and is interpreted as evidence of first-order quenching.

In the work of Borisevich, briefly described in the preceding paragraph, it is shown that the addition of foreign gases at \(\nu_{\text{exc}} < \nu_{\text{el}}\), including also in the antistokes region at \(\nu_{\text{exc}} < \nu'_{\text{exc}}\), leads to a decrease of the quantum yield. This shows that, in collisions with molecules of a foreign gas, an increase in the reserve of vibrational energy occurs and, consequently, the reserve of vibrational energy of the excited molecule before the collision is relatively small, less than the average energy corresponding to the given temperature. Hence it follows unambiguously that antistokes quenching cannot be associated with a large value of nonradiative transitions \(d(\overline{E}^{*}_{\text{vib}})\), leading to deactivation of the electronically excited state.

The only explanation of the cause of the drop in quantum yield within the limits of the antistokes excitation region remains the initial-

...the initial assumption of S. I. Vavilov: under anti-Stokes excitation there occurs relatively strong absorption, not accompanied by excitation of the electron shell and subsequent fluorescence. Non-exciting absorption is connected with a transition to very high vibrational levels of the ground electronic state, with the energy of the absorbed light being directly converted into heat. In § 1 we have already considered certain objections to this point of view and have shown that they are not essential.

Proceeding from this basic idea, Levshin \(^{26}\) made an attempt to derive the function \(\mathscr{B}(\nu_{\text{exc}})\). A more exact expression, allowing direct experimental verification, has been obtained quite recently by Stepanov.

Let the system absorb external radiation \(\nu_{\text{exc}} > \nu_{\text{el}}\). It is most probable that the absorption is connected with excitation of the electron shell and a small increase of \(E_{\text{vib}}\) \((\Delta E_{\text{vib}} = h\nu_{\text{exc}} - h\nu_{\text{el}})\). Purely vibrational transitions not accompanied by excitation of the electron shell are considerably less probable (by several orders of magnitude). The total number of absorbed quanta is equal to

\[ N = N_{\text{el}} + N_{\text{vib}} = n_1 \int_{E_{\text{vib}}=0}^{\infty} \rho(E_{\text{vib}})\, a_{\text{el}}(E_{\text{vib}}, \nu_{\text{exc}})\, dE_{\text{vib}} + \]

\[ +\, n_1 \int_{E_{\text{vib}}=0}^{\infty} \rho(E_{\text{vib}})\, a_{\text{vib}}(E_{\text{vib}}, \nu_{\text{exc}})\, dE_{\text{vib}} . \tag{36} \]

Here \(n_1\) is the total number of molecules in the lower electronic state,

Fig. 8. Exciting (right) and non-exciting (left) absorption.

\(N_{\text{el}}\) is the number of absorption events accompanied by excitation of the electron shell, \(N_{\text{vib}}\) is the number of events of non-exciting absorption, and \(a_{\text{el}}\) and \(a_{\text{vib}}\) are the probabilities of exciting and non-exciting transitions. For \(\nu_{\text{exc}} > \nu_{\text{el}}\) the second term plays no role; the total amount of absorbed energy depends on the properties of the function \(a_{\text{el}}\), while the value of the yield is determined by quenching processes of the second kind occurring in the excited electronic state.

Other relations arise in the region \(\nu_{\mathrm{exc}} < \nu_{\mathrm{el}}\). In the act of exciting absorption not all molecules can participate, but only those for which \(E_{\mathrm{col}} > h\nu_{\mathrm{el}} - h\nu_{\mathrm{exc}}\) (Fig. 8). Thus,

\[ N_{\mathrm{el}} = n_1 \int_{E_{\mathrm{col}} = h\nu_{\mathrm{el}} - h\nu_{\mathrm{exc}}} \rho(E_{\mathrm{col}})\, a_{\mathrm{el}}(E_{\mathrm{col}}, \nu_{\mathrm{exc}})\, dE_{\mathrm{col}} = \]

\[ = n_1 a_{\mathrm{el}}(\nu_{\mathrm{exc}}). \tag{37} \]

Here \(a_{\mathrm{el}}(\nu_{\mathrm{exc}})\) is the probability of absorption of the frequency \(\nu_{\mathrm{exc}}\), averaged over all \(E_{\mathrm{col}}\), and measured experimentally. As \(\nu_{\mathrm{exc}}\) decreases (from the point \(\nu_{\mathrm{exc}}=\nu_{\mathrm{el}}\)), the value of \(a_{\mathrm{el}}(\nu_{\mathrm{exc}})\) falls rapidly (especially at low temperatures). At the same time, the probabilities of non-exciting absorption either do not change at all or change very slowly (\(a_{\mathrm{col}}\simeq \mathrm{const}\)). The corresponding transitions occur from all levels, starting with \(E_{\mathrm{col}}=0\), independently of \(\nu_{\mathrm{exc}}\) (see Fig. 8).

Figure 9 gives a graph of the function \(W_{\mathrm{abs}}(\nu)\)—the dependence of the absorption power on \(\nu_{\mathrm{exc}}\). The dashed line shows the power

Fig. 9. Dependence of the absorption power on frequency (the power of non-exciting absorption is shown by the dashed line).

of non-exciting absorption. It may be neglected for \(\nu_{\mathrm{exc}}>\nu_{\mathrm{el}}\) and in the region \(\nu_{\mathrm{exc}}<\nu_{\mathrm{el}}\) for small deviations from \(\nu_{\mathrm{el}}\). For large deviations from \(\nu_{\mathrm{el}}\), the power of non-exciting absorption becomes appreciable and then even predominant. As experiments carried out for other purposes and, unfortunately, not sufficiently systematically show, far from \(\nu_{\mathrm{el}}\) the absorption power (coinciding with the power of non-exciting absorption) indeed depends only very weakly on \(\nu_{\mathrm{abs}}\).

In the presence of non-exciting absorption, the quantum yield is equal to

\[ \mathcal{B} = \mathcal{B}'\, \frac{W_{\mathrm{el}}}{W_{\mathrm{col}}+W_{\mathrm{el}}} = \mathcal{B}'\, \frac{N_{\mathrm{el}}h\nu_{\mathrm{exc}}}{(N_{\mathrm{el}}+N_{\mathrm{col}})h\nu_{\mathrm{exc}}} = \mathcal{B}'\, \frac{N_{\mathrm{el}}}{N_{\mathrm{el}}+N_{\mathrm{col}}}. \tag{38} \]

Here \(\mathscr{B}'\) is the quantum yield for exciting absorption. In the region \(\nu_{\mathrm{exc}}>\nu_{\mathrm{el}}\), \(\mathscr{B}=\mathscr{B}'\); the magnitude of the yield is determined by the properties of the excited electronic state and, above all, by the store of vibrational energy \(\bar E_{\mathrm{vib}}\). Quenching of the first kind is practically absent. As \(\nu_{\mathrm{exc}}\) decreases, beginning from \(\nu_{\mathrm{el}}\), the ratio

\[ \frac{N_{\mathrm{el}}}{N_{\mathrm{el}}+N_{\mathrm{vib}}} \]

for some time remains equal to unity, despite the decrease in \(N_{\mathrm{el}}\). Subsequently \(N_{\mathrm{el}}\) and \(N_{\mathrm{vib}}\) become comparable, and the quantum yield \(\mathscr{B}\) becomes smaller than \(\mathscr{B}'\). The value of the frequency \(\nu'_{\mathrm{exc}}\) corresponding to the beginning of the anti-Stokes decline of the yield depends on the magnitude of \(N_{\mathrm{vib}}\) and on the rate of decrease of \(N_{\mathrm{el}}\). In different molecules it may assume very different values and is in no way connected with the maximum of the emission band. At the very edge of the absorption band the value of \(N_{\mathrm{el}}\) falls to zero, and the magnitude of the absorption is determined by the magnitude of \(N_{\mathrm{vib}}\). In this region of the excitation spectrum,

\[ \mathscr{B}(\nu_{\mathrm{exc}})=\mathscr{B}'\frac{N_{\mathrm{el}}}{N_{\mathrm{vib}}}. \tag{39} \]

Since \(N_{\mathrm{vib}}\) and \(\mathscr{B}'\) depend relatively little on \(\nu_{\mathrm{exc}}\), it follows that

\[ \mathscr{B}(\nu_{\mathrm{exc}})=C N_{\mathrm{el}}=C'a_{\mathrm{el}}(\nu_{\mathrm{exc}}). \tag{40} \]

It follows from (40) that the dependence of the quantum yield of anti-Stokes fluorescence on the frequency of the exciting light coincides with the course of decrease of the probability of absorption or, equivalently, of the absorption power. They have the same temperature dependence. An increase in \(T\) leads to an increase in the absorption power (for \(\nu_{\mathrm{exc}}<\nu_{\mathrm{el}}\)) and to a proportional increase in the quantum yield.

The existing experimental data do not permit an exact quantitative verification of formulas (38)—(40), but they agree with them qualitatively. The yield begins to fall precisely where the absorption power becomes especially small.

Formulas (38)—(40) have been obtained as applied to complex molecules. However, they are apparently applicable to any complex and condensed systems and, in particular, to crystal phosphors and uranium compounds. As Alentsev has shown\({}^{10}\), the decrease in the quantum yield of these objects begins long before the emission maximum, and precisely in that region where a sharp decrease in the absorption power occurs.

A systematic experimental investigation and verification of formulas (38)—(40) will make it possible to discover separate details of the phenomenon hitherto unknown, and to construct an even more exact theory of the process of anti-Stokes fluorescence.

Vavilov’s Law

Vavilov’s law was established in 1927 and since then has been the basis of the entire doctrine of luminescence. The further development of science has confirmed the complete validity of this law and has clarified its decisive significance in the interpretation of the complex processes taking place in a system after its excitation. The very content of Vavilov’s law has been continuously enriched, revealing its great importance and many-sided character.

References Cited

1. S. I. Vavilov, Works, vol. 1, article 8.
2. S. I. Vavilov, Works, vol. 1, articles 11, 14, 17.
3. H. Helmholtz, Hanb. d. physiol. Optik 2, 94 (1911).
4. E. Wiedemann, Wied. Ann. 37, 229 (1889).
5. M. N. Alentsev, ZhETF 21, 133 (1951).
6. S. I. Vavilov, Works, vol. 1, article 21.
7. S. S. Solomin, DAN SSSR 31, 741 (1941).
8. A. Jablonski, Acta Phys. Pol. 2, 97 (1933).
9. V. A. Fabrikant, Zeits. d. Sowjetunion 3, 567 (1933).
10. M. N. Alentsev, Proceedings of FIAN, No. 5 (1950); DAN SSSR 62, 607 (1948); 64, 479 (1949).
11. S. I. Vavilov, Works, vol. 2, article 21.
12. S. I. Vavilov, Works, vol. 1, articles 23, 25, 26, 37, 39, 44, 45.
13. S. I. Vavilov, Works, vol. 1, article 25.
14. S. I. Vavilov, Works, vol. 1, articles 26, 28, 33, 45, 50; vol. 2, article 6.
15. S. I. Vavilov, Works, vol. 1, articles 10, 13, 15, 20, 30, 44; vol. 2, articles 8, 18.
16. S. I. Vavilov, Works, vol. 1, article 16.
17. B. Ya. Sveshnikov, Proceedings of GOI 12, No. 108 (1938).
18. B. S. Neporent and N. A. Borisevich, DAN SSSR 447 (1954).
19. S. I. Vavilov, Works, vol. 1, articles 48 and 49.
20. A. N. Terenin, ZhFKh 43, 1 (1944); Acta Phys. Chim. URSS 18, 210 (1943); Izv. AN SSSR, ser. fiz. 9, 305 (1945); Photochemistry of Dyes, Publishing House of the Academy of Sciences of the USSR, 1947.
21. B. S. Neporent, UFN 43, 380 (1951); ZhFKh 13, 965 (1939); ZhFKh 21, 1111 (1947); ZhFKh 24, 1219 (1950).
22. S. I. Vavilov, Works, vol. 1, article 50.
23. M. D. Galanin, Proceedings of FIAN 5 (1950).
24. A. N. Sevchenko, Proceedings of GOI 15, 65 (1941); DAN SSSR 42, 349 (1944).
25. L. A. Tumerman, Proceedings of FIAN, issue 4 (1938).
26. V. L. Levshin, Luminescence of Liquid and Solid Substances, Gostekhizdat, 1951.
27. V. L. Levshin, ZhFKh 6, 1 (1935); DAN SSSR 1, 474 (1935).
28. V. L. Levshin, Acta Phys. Chim. USSR 2, 221 (1935).
29. B. I. Stepanov, ZhETF 28, 559 (1955).
30. S. I. Vavilov, Works, vol. 2, articles 23 and 24.
31. E. I. Adirovich, Some Questions of the Theory of Luminescence, Gostekhizdat, 1951.
32. L. D. Landau, Journ. of Phys. 10, 503 (1946).
33. R. Pringsheim, Zeits. f. Phys. 57, 739 (1929); Journ. of Phys. 10, 499 (1946).

1. V. V. Antonov-Romanovskii, B. I. Stepanov, M. V. Fok, A. P. Khapalyuk, DAN SSSR 105, 50 (1955).

2. P. Pringsheim, Fluorescence and Phosphorescence, IL, 1951.

3. S. I. Vavilov, Works, vol. 2, article 34.

4. S. I. Vavilov, Works, vol. 2, article 16.

5. S. I. Vavilov, Works, vol. 2, articles 20, 28, 29; introductory article to book 35.

6. B. I. Stepanov, DAN SSSR 49, 971 (1954).

7. E. Wiedemann, Wied. Ann. d. Phys. 34, 446 (1888); 37, 117 (1889).

8. S. I. Vavilov, Works, vol. 1, article 41.

9. M. N. Alentsev, V. V. Antonov-Romanovskii, B. I. Stepanov, and M. V. Fok, ZhETF 28, 253 (1955).

10. B. I. Stepanov, UFN 43, 380 (1951); Izv. AN BSSR, issue 5 (1954).

11. V. L. Levshin, ZhFKh 6, 991 (1935).

12. A. N. Sevchenko, Izv. AN SSSR, ser. fiz. 15, 613, 1951; Proceedings of the State Optical Institute, session in memory of S. I. Vavilov, Oborongiz, 1953.

13. N. A. Borisevich, DAN SSSR 99, 695 (1954).

14. H. S. Guttea and G. Rollefson, Journ. Am. Chem. Soc. 74, 28 (1954).