Experimental Methods for Studying Fast Relaxation Processes

L. A. Tumerman

Among the various relaxation processes that take place over very small intervals of time (of the order of \(10^{-7}\)—\(10^{-9}\) sec.), the greatest attention of researchers has been attracted by the processes of decay of the spontaneous luminescence of excited atoms or molecules. In cases where the decay of luminescence follows the simple exponential law \(I = I_0 e^{-t/\tau}\) and where the observed time course of the phenomenon is not distorted by secondary factors—for example, by the “afterglow” of radiation, which can greatly increase the measured duration of luminescence, or by quenching, which can shorten this quantity—the measurement of \(\tau\) (the mean duration of luminescence) is of interest chiefly because it makes it possible to determine the most important atomic or molecular constants characterizing the radiating system from the point of view of one or another general conception of the mechanism of radiation. We now interpret the quantity \(\frac{1}{\tau}\) as the probability of a transition between the corresponding levels; in classical theory it made it possible to determine the “strength” of the corresponding virtual oscillator.

In those, more complicated, cases where we can establish deviations from the exponential law of decay, or where the actual course of this process is not determined solely by the a priori probability of the corresponding transitions, the study of the actual course of decay under definite conditions is of interest for investigating the kinetics of processes in which excited molecules or atoms participate. This includes the study of luminescence processes, photochemical reactions (in particular, the processes of their sensitization), the optics of gas discharge, and a number of others.

It is probably for this reason that methods for studying fast relaxation processes, which have been intensively developed in recent times and have reached a fairly high degree of perfection, developed mainly as methods for studying luminescence processes. However, both the theory of these methods and many structural features of the corresponding instruments (fluorometers) can, with appropriate modifications, be applied

are valuable also in the investigation of other processes that decay over very short intervals of time.

In accordance with this, in what follows we set forth in greatest detail the question of the methods and results of studying the decay of spontaneous luminescence. Only in the last paragraph shall we examine, by way of example, the question of applying these methods to the study of relaxation processes of artificial anisotropy, which determine the electro-optical Kerr effect and the Faraday effect. It is curious to note that the prototype of modern fluorometers was the apparatus of Abraham and Lemoine, intended for determining the inertia of the Kerr effect, and that in their modern development the methods of fluorometry have been most successfully applied precisely to the study of the relaxation of the Kerr effect.

1. METHODS FOR STUDYING THE DECAY OF LUMINESCENCE

(General survey)

Historically, the methods for studying the decay of spontaneous luminescence of excited atoms and molecules developed from the methods for investigating the considerably slower processes of phosphorescence decay. Any of these methods may be regarded as the development and complication of one or another construction of a phosphoroscope. However, as we shall see, when one passes to the investigation of processes occurring within a very short interval of time, of the order of \(10^{-8}\)—\(10^{-9}\) sec., not only do a number of technical difficulties arise, but a number of fundamental methodological features also appear with particular force—features that were unimportant or comparatively easily taken into account in the study of longer processes. All this leads to such a deep and radical reworking of the method that it is often difficult to discern in it the features of the simple phosphoroscope that is its prototype.

In those cases when the decay process of the luminescence is studied and its duration is large in comparison with the duration of the time interval required for measuring the intensity of the luminescence, the measurement scheme is extremely simple. The luminescence of the substance under investigation is excited for some interval of time; then the excitation is “switched off,” and the intensity of the luminescence is measured continuously or at definite intervals of time. It is clear, however, that this simple scheme can no longer be realized in those cases when the duration of the luminescence is comparable with the duration of the measuring process or of the process of “switching off” the excitation, which, strictly speaking, cannot be instantaneous. In these cases there arises the necessity of applying those specific phosphoroscopic methods of investigation whose originator should be considered E. Becquerel.

The principal essence of these methods consists in the fact that excitation is produced periodically—by brief “flashes”—and after each such flash the brightness of the glow is measured, corresponding to a definite interval \(\Delta t\) between the moments of excitation and observation. The repetition frequency of the excitation and observation cycles must obviously be sufficiently high so that the recording instrument—the observer’s eye or a photocell with the appropriate measuring device—can average its reactions to the periodically repeated action of the radiation under study; while the duration of each excitation flash and of the observation period must be small in comparison with the duration of the process under study. It is precisely for this reason that any reasonably perfect realization of such a simple scheme is possible only in the investigation of relatively long processes. In the cases that interest us, namely very short-lived glow processes, we are forced to take into account the finite duration both of the excitation interval and of the observation interval.

Attention to the necessity of taking this aspect of the matter into account was drawn by S. I. Vavilov^1, who showed that even in the investigation of a phosphor glow process lasting about \(10^{-3}\)—\(10^{-4}\) sec., the curves obtained directly on Becquerel’s phosphoroscope do not, generally speaking, coincide with the true decay curves of the glow and sometimes require rather complex and difficult processing and interpretation. In the investigation of still shorter-lived processes, these complications, of course, make themselves felt with even greater force.

The general idea of phosphoroscopic observation of the decay of a glow can be realized in extremely varied ways. But despite all the variety of constructions and methods proposed by different authors, they can all be reduced to two principal types.

In instruments of the first type, Becquerel’s idea is realized directly, and therefore it is natural to call all these instruments “Becquerel phosphoroscopes,” although in many modern instruments of this type the technical, constructive embodiment of the basic idea has gone very far beyond the simple instrument used by Becquerel himself. All these instruments are characterized by the presence of two “shutters” or “interrupters,” which periodically open for a short time the access of the exciting radiation to the luminescent substance and the access of the luminescence light to the photometer—visual or objective. In this case the excitation and observation cycles are repeated sufficiently often that the light is perceived by us without flicker, and the moments at which both shutters open are shifted by some interval \(\Delta t\). By varying the magnitude of this interval and taking into account the above-mentioned complications introduced by the finite duration of the excitation and observation periods, it is possible in many cases to follow the course of the decay directly in time, without resorting to a spatial sweep of the process.

In the simplest designs of phosphoroscopes of the Becquerel type, the interruption of both beams of light—the exciting and the emitted—is carried out by various mechanical devices, for example, rotating disks with holes. Thus, for example, the well-known two-disk Becquerel phosphoroscope, described in all textbooks, or the Vavilov and Levshin phosphoroscope shown in Fig. 1,[^2] is close in conception to Becquerel’s first single-disk model. Its construction is clear from the caption to the figure.

Fig. 1. Diagram of the single-disk phosphoroscope of Vavilov and Levshin.

Periodic interruption of the exciting beam of light from source \(Q\), as well as of the light emerging from phosphor \(F\), directed by lenses \(L_1\), \(L_2\) onto photometer \(O\), is produced by the rotating disk \(D\) with holes. The time interval \(\Delta t\) between the moment of excitation and the moment of observation is determined by the distance between the place on the disk (\(F\)) at which the phosphorescing object is imaged and the hole of the disk nearest to this place (Fig. b). Variation of this interval is accomplished by displacing the two diaphragms \(B_2\) and \(B_1\), rigidly connected with each other, and lens \(L_2\), which is done with the aid of micrometer screw \(S\).

In more recent installations, the mechanical interrupters are often replaced by corresponding electro-optical devices fed by an alternating voltage of the required frequency and form. Fig. 2 shows the scheme of such an apparatus, constructed by Britt[^3] for studying the decay of phosphors when excited by an electron beam. This apparatus operates as follows.

A rectangular voltage pulse with a definite and rather high repetition frequency is applied simultaneously to the control electrode of the cathode-ray tube and to the grid of the modulating tube of the generator feeding the piezoquartz. In this way, during a short interval of time the screen of the tube is excited by an electron beam, and at the same moment the quartz sends into the liquid a short train of high-frequency ultrasonic waves. Propagating in the liquid with a definite velocity, this train intersects the light beam emerging from the screen after a definite time interval \(\Delta t\) following the moment of excitation. The light diffracted by the ultrasonic wave acts on the photocell, which thus records the intensity of the radiation at a moment separated from the moment of excitation by the time \(\Delta t\). The magnitude of this interval \(\Delta t\) can be varied by moving the entire ultrasonic cell in the direction of propagation of the wave and thus changing the distance between the quartz and the light beam.

With all the technical complications and improvements introduced into this apparatus, one can readily discern in it the basic principles—

...general features characteristic of all phosphoroscopes of the Becquerel type, and its theory has no specific fundamental peculiarities*). The resolving power of Briggs’s apparatus is about \(10^{-5}\)—\(10^{-6}\) sec. The first points on the phosphorescence decay curve reliably measured by Briggs correspond to a time interval of about \(10^{-5}\) sec from the moment of excitation.

Fig. 2. Diagram of Briggs’s phosphoroscope.

In phosphoroscopes of another type, the direct study of the course of the process in time is replaced by the study of the brightness distribution in a certain stable spatial picture—a sweep of the process. This idea is most simply realized in phosphoroscopes of various designs in which a small excited region of the luminescent substance is carried, by one device or another—for example, by a rotating disk (Fig. 3)—into an unilluminated region, where it emits light; in its place, new portions of the substance are continuously introduced into the excitation region. With a proper choice of the speed of rotation of the disk, a gradually weakening band appears on it, the brightness of which at each point corresponds to the intensity of phosphorescence at a definite moment after excitation.

Fig. 3. Diagram of a phosphoroscope with a rotating disk.

Instruments of this type, with mechanical motion of the luminescent substance, are suitable, of course, only for the study of comparatively long processes (in practice, for processes with a duration

*) Briggs calls his apparatus an “ultrasonic fluorometer.” This name seems to us very unfortunate, since, on the one hand, in order to avoid confusion of concepts, it is desirable to reserve the name fluorometers for instruments possessing certain fundamental features, which will be indicated below, and, on the other hand, the instrument is in essence intended not for the study of fluorescence (spontaneous emission), but for the study of phosphorescence phenomena (recombination luminescence).

on the order of \(10^{-3}\)—\(10^{-4}\) sec.). To study processes of shorter duration, Wood\(^4\) proposed the following modification of this method (Fig. 4). A jet of distilled vapor, i.e., a stream of rapidly moving particles in a definite direction, is crossed at some point along its path by a narrow beam of exciting radiation perpendicular to the vapor jet. It is obvious that the luminescence along the jet will gradually weaken, and the study of the brightness of the luminescence at various places in the jet gives, as before, the possibility of reconstructing the time course of the decay of the luminescence and measuring its duration.

Fig. 4. Obtaining a spatial sweep of the luminescence of mercury vapor by Wood’s method. \(E\)—beam of exciting radiation.

Wood himself and later Rayleigh (the younger)\(^5\) applied this method to the study of the luminescence of mercury vapor, having a duration on the order of \(10^{-5}\) sec. (Fig. 5). By the same method, Koenig and Ellet\(^6\) determined the duration of luminescence of the cadmium line \(3261\,\text{Å}\) and found for it the value \(\tau = 2.5 \cdot 10^{-6}\) sec. Apparently, this value is close to the lower limit of the possibilities of the method, since in practice in these experiments it is difficult to obtain thermal velocities greater than \(10^5\) cm/sec.

In order that, by the same method, it should be possible to investigate still shorter luminescence processes, it is obviously necessary to obtain corpuscular beams with a considerably greater particle velocity. V. Wien studied for this purpose the distribution of luminescence intensity along a canal ray emitted from a discharge space with a comparatively high gas pressure into a space with a considerably greater rarefaction, where excitation processes do not occur and only the afterglow of previously excited particles takes place. The theory and technique of this method were investigated in very great detail and with care in a number of Wien’s works,\(^7,\*\) and the method was successfully applied by him and his collaborators to determine the duration of luminescence of lines of the Balmer series of hydrogen and a number of lines of other elements. Wien’s method makes it possible to measure luminescence durations on the order of \(10^{-8}\)—\(10^{-9}\) sec., but its field of application, like that of Wood’s method, is limited.

The group of instruments giving a spatial sweep of the time course of luminescence decay also includes the phosphoroscope constructed by S. I. Vavilov and V. L. Levshin\(^2\) for the study of luminescence of uranium salts. Here the luminescent substance remains stationary, but its luminescence is swept out by a rotating mirror into a long strip on frosted glass. Excitation is produced by a spark, closed by a discharger, which is set into

rotation by the same motor as the mirror. This ensures synchronization of the excitation and the sweep. With a single closure of the spark, we would see a luminous flash running across the screen, brightest at the moment the spark closes and gradually weakening as the mirror rotates. Since, however, this whole process is repeated 25 times per second, we see on the screen a stable, gradually weakening band, the distribution of brightness along which, as in the preceding methods, reproduces the time course of the luminescence. According to the authors’ estimate, the resolving power of this phosphoroscope can be brought to \(10^{-6}\) sec.

Fig. 5. Photograph of the luminescence of a jet of mercury vapor, obtained by Rayleigh by Wood’s method: \(a\)—photograph in visible light, \(b\)—at the wavelength 2537 Å. The vapor pressure is 10 mm. The jet is directed from top to bottom; the arrow marks the position of the exciting beam.

When we attempt to apply a phosphoroscope of any type to the study of processes with durations of the order of \(10^{-8}\)—\(10^{-9}\) sec., then, of course, the fundamental difficulties in interpreting the observed picture, discussed above, appear in full force. Because of this, not only the design, but also the very theory of these methods acquires an entirely specific character and requires special consideration.

It is easy to see that in this respect there is no fundamental difference between instruments of the two types discussed above. To the finite duration of the time interval during which observation is made in Becquerel-type phosphoroscopes there corresponds the finite width of that part of the luminous band or jet whose brightness is recorded by the photometer in instruments with spatial sweep; in exactly the same way, to the finite period of opening of the first shutter in the Becquerel phosphoroscope there corresponds, in Wood’s method, the finite width of the region over which the substance under study is subjected to the action of the exciting radiation. Thus all the fundamental difficulties that arise in the study of very short-lived processes apply equally to instruments of both types. However, the range of objects whose luminescence can be studied is considerably broader for phosphoroscopes

...of the Becquerel type, and it is precisely instruments representing a development of these phosphoroscopes that have been used in recent years with particular success for the study of the luminescence of solutions. For an instrument of this kind Vavilov proposed the name “fluorometer,” which has already become quite firmly established in the literature. The present article is devoted mainly to these instruments and methods.

2. THEORY OF FLUOROMETRIC MEASUREMENTS

The fundamental differences between fluorometric methods and ordinary phosphoroscopic methods are due to the fact that, when the duration of the process under study is of the order of \(10^{-8}\)—\(10^{-9}\) sec., we are no longer able to produce “flashes” of excitation whose duration would be shorter than the duration of the luminescence, and therefore cannot observe the luminescence in the interval between two such flashes, i.e., with the action of the exciting light on the luminescent substance excluded. In this case, obviously, in order for the method to be able to give unambiguous results, it is necessary that the elementary law of decay of the luminescence—that is, the law determining the course of the emission of each separate group of molecules or atoms excited at some definite instant of time—should depend neither on the presence in the medium of other excited centers nor on the intensity of the light acting on the substance while these molecules remain in the excited state.

These conditions above all restrict the range of application of the theory under consideration to processes of spontaneous—or, in general, monomolecular—luminescence. In processes of recombination luminescence, the probability of emission of each individual luminescence center depends on the total concentration of all electrons torn away from their centers and on their localization. Therefore the law of decay of luminescence of the recombination type, for example the luminescence of crystal phosphors, depends substantially on the duration and intensity of the preceding excitation and on the intensity of the light acting on the substance during the period of its emission (see, for example, the works of Antonov-Romanovskii\(^8\)). For this reason the theory of fluorometric measurements set forth below is inapplicable to cases of recombination luminescence, for which the corresponding theory would be considerably more complicated.

On the other hand, Weisskopf\(^9\) drew attention to the necessity of distinguishing between atomic states of the usual stable type and states excited by continuously incident light. The difference between these types of states is that for the first type the energy has a definite value, whereas for the second type the energy is indeterminate in the quantum sense. This circumstance makes it necessary to use the classical concept of the “lifetime” of excited atoms or mole-

of molecules, since this concept is associated with the concept of an energetically definite state and cannot be used in the ordinary way as a characteristic of states of the second type. In analyzing the theory of the fluorometer, Weisskopf pointed out the need for great caution in interpreting the results obtained. His calculations, however, apply to the case of fluorescence of atoms under strictly monochromatic excitation. As Weisskopf himself indicates (loc. cit., p. 110), fluorometric measurements give correct results if the excitation is produced by a broad range of frequencies near the resonance frequency. Moreover, apparently, Weisskopf’s considerations cannot be transferred to the case of fluorescence of complex molecules, which is not of a resonance character but, on the contrary, is characterized by broad absorption and emission bands shifted relative to one another in accordance with Stokes’ rule (cf. the work of Yablonsky[^10]). Therefore, in what follows this quantum-mechanical point of view is not taken into account, and we assume the possibility of conducting the discussion under the assumption that only states of the first type are present. With these reservations and limitations, the general theory of fluorometric measurements may be presented in the following form.

Let the law of variation of the intensity of the modulated exciting beam be expressed by a periodic function \(E(t)\) with period \(T\) or with cyclic modulation frequency \(\omega = \dfrac{2\pi}{T}\). The analogous function expressing the time course of the intensity of the luminescence excited by this light will be denoted by \(L(t)\), and the elementary law of decay by \(\Phi(t)\). In order not to introduce in all subsequent formulas inessential constant factors, we normalize the function \(\Phi(t)\) so that the equality

\[ \int_{0}^{\infty} \Phi(t)\,dt = 1. \tag{2.1} \]

holds.

Since, for practically realizable values of the excitation intensity, the number of molecules excited at each instant is proportional to the instantaneous value of the intensity of the exciting beam, the intensity of luminescence light observed at a certain instant \(t\), emitted by a group of molecules excited at a certain preceding instant \((t-\vartheta)\), is equal to \(E(t-\vartheta)\Phi(\vartheta)\,d\vartheta\). At each given instant, however, we observe luminescence emitted by various groups of molecules excited at all preceding instants of time. It follows from this that the functions \(E(t)\) and \(L(t)\) must be related to one another by the integral equation

\[ L(t)=\int_{0}^{\infty} E(t-\vartheta)\Phi(\vartheta)\,d\vartheta. \tag{2.2} \]

The kernel of this equation—the function \(E(t)\)—in all practically used devices is an even periodic function, whose expansion in a Fourier series has the form

\[ E(t)=\sum_{m=0}^{\infty} E_m \cos m\omega t . \tag{2.3} \]

Substituting this expansion into expression (2.2), we see that the Fourier-series expansion of the function \(L(t)\) has the form

\[ L(t)=\sum_{m=0}^{\infty} A_m E_m \cos(m\omega t-\varphi_m), \tag{2.4} \]

where

\[ \begin{aligned} A_m \cos \varphi_m &= \int_{0}^{\infty} \Phi(\vartheta)\cos m\omega \vartheta\, d\vartheta,\\ A_m \sin \varphi_m &= \int_{0}^{\infty} \Phi(\vartheta)\sin m\omega \vartheta\, d\vartheta. \end{aligned} \tag{2.5} \]

Thus, each harmonic component of the function \(L(t)\) is shifted in phase relative to the corresponding harmonic of the function \(E(t)\) by an amount \(\varphi_m\), determined by the relation

\[ \tg \varphi_m = \frac{\displaystyle \int_{0}^{\infty} \Phi(\vartheta)\sin m\omega \vartheta\, dt} {\displaystyle \int_{0}^{\infty} \Phi(\vartheta)\cos m\omega \vartheta\, dt}, \tag{2.6} \]

and the quantities \(A_m\), representing the ratios of the amplitudes of these components, are determined from the relations

\[ A_m^2 = \left[\int_{0}^{\infty} \Phi(\vartheta)\cos m\omega \vartheta\, d\vartheta\right]^2 + \left[\int_{0}^{\infty} \Phi(\vartheta)\sin m\omega \vartheta\, d\vartheta\right]^2 . \tag{2.7} \]

With the normalization condition we have adopted for the function \(\Phi(t)\), it is evident that \(A_0=1\), and since \(\Phi(t)\) is a decreasing function, then, generally speaking, the factors \(A_m\) are less than unity and decrease with the harmonic number. The deformation of the curve \(L(t)\) in comparison with the curve \(E(t)\) therefore consists in the fact that this curve is “flattened”: its maxima are lowered, and its minima are raised in comparison with the curve \(E(t)\).

In practice the modulation period of the exciting light, \(T\), is always much greater than the duration of the decay, so that for \(t \geq T\) one may take \(\Phi(t)=0\), and consequently in formulas (2.5) the upper limit of integration may be taken equal to \(T\).

Thus, having experimentally determined the values of the quantities \(A_m\) and \(\varphi_m\) for a sufficient number of harmonic components of the functions \(E(t)\) and

$L(t)$, we could determine the corresponding number of Fourier coefficients for a function which, in the interval $(0,T)$, i.e. practically over the entire duration of the luminescence process, coincides with the function $\Phi(t)$. In other words, we could reconstruct, with the required degree of accuracy, the course of the function $\Phi(t)$. In practice, however, such a way of determining the function $\Phi(t)$ is complicated and unreliable. All investigators who have applied the fluorometric method to the study of the luminescence-decay process have preferred to proceed in another way. On the basis of one or another general consideration, certain assumptions were made concerning the character of the function $\Phi(t)$, and the relations expected in this case between the functions $E(t)$ and $L(t)$ were computed. Comparison of the experimental data with the results of these computations makes it possible to determine the numerical values of the constants entering into the elementary decay law $\Phi(t)$ and, to a certain extent, to verify the correctness of the initial assumptions about the form of this function.

Suppose, for example, as is most often done, that the decay proceeds according to the simple exponential law

$$ \Phi(t)=\frac{1}{\tau}\cdot e^{-t/\tau}, \tag{2.8} $$

which corresponds to the most natural assumption that the probability of spontaneous emission by an excited particle does not change during the time it remains in the excited state. Such a decay law directly leads to the relations

$$ \operatorname{tg}\varphi_m=m\omega\tau,\qquad A_m=\frac{1}{1+m^2\omega^2\tau^2}=\cos\varphi_m. \tag{2.9} $$

Thus, in the case of exponential decay of luminescence, in order to determine the quantity $\tau$—the mean duration of luminescence or the mean “lifetime” of the excited state—it is sufficient to measure the phase shift between any two corresponding harmonic components of the functions $E(t)$ and $L(t)$, or the ratio of the amplitudes of these components.

In many cases one may, with practically sufficient accuracy, neglect all overtones of the fundamental frequency in the expansion of the function $E(t)$ in a Fourier series, i.e. one may assume that the modulation of the exciting light occurs according to the simple sinusoidal law

$$ E(t)=E_0+E_1\cos\omega t. \tag{2.10} $$

In this case, evidently,

$$ L(t)=E_0+A\cos(\omega t-\varphi), \tag{2.11} $$

where

$$ \operatorname{tg}\varphi=\omega\tau,\qquad A=\frac{1}{\sqrt{1+\omega^2\tau^2}}=\cos\varphi. \tag{2.12} $$

Thus, for purely sinusoidal modulation and exponential decay, the problem of determining the quantity \(\tau\) reduces simply to measuring the phase shift between the functions \(E(t)\) and \(L(t)\), or to determining the ratio of the amplitudes of their variable components.

Let us also note that the assumption we have made concerning the evenness of the function \(E(t)\) does not impair the generality of our conclusions. We would easily arrive at the same conclusions by writing the Fourier-series expansion for an arbitrary periodic function.

It is easy to see further how the relations derived make it possible to check the correctness of the assumption underlying them, namely that the decay of luminescence proceeds according to an exponential law. For this it is sufficient to measure the phase shift \(\varphi\) between the curves \(L(t)\) and \(E(t)\) at different modulation frequencies \(\omega_1,\ \omega_2,\ \omega_3,\ldots\) If the exponential law of decay really holds, then, calculating \(\tau\) from formulas (2.7) or (2.10), we shall obtain one and the same value of \(\tau\) for any pair of values \(\omega_1,\varphi_1;\ \omega_2,\varphi_2;\ldots\)

If the decay followed not an exponential law but some other law, the relations between the functions \(L(t)\) and \(E(t)\) would be different, and consequently the same mean duration of the process of decay of the luminescence would correspond in experiment to a different relation between \(\omega_i\) and \(\varphi_i\). For example, if all excited molecules emitted after one and the same time interval \(t_0\) following the instant of excitation, then, obviously, the mean duration of decay would be equal to \(t_0\); but in order to compute this quantity we would have to use the formula

\[ \omega t_0=\varphi \tag{2.13} \]

for sinusoidal modulation, or the formula

\[ m\omega t_0=\varphi_m \tag{2.14} \]

in the general case. As we see, in this case the phase shift between the corresponding harmonics of the functions \(E(t)\) and \(L(t)\) should be proportional to the modulation frequency \(\omega\) or to the number of the harmonic. All amplitude factors \(A_m\) would then be equal to unity, i.e. there would be no “flattening effect” of the curve \(L(t)\). It is not difficult to see that in this case the curve \(L(t)\) would be simply similar to \(E(t)\), but only shifted with respect to it along the time axis by the interval \(2\pi \dfrac{t_0}{T}\).

As an example illustrating what has been said about the possibility of checking the correctness of the initial assumptions concerning the type of the function \(\Phi(t)\), one may cite the results of fluorometric measurements of the quantity \(\tau\) for a solution of fluorescein in alcohol at room temperature. These measurements were carried out by Shimanovskii\(^{11}\) and later by us\(^{12}\) at various values of the modulation frequency, and in all cases

in some cases it was possible, with a sufficient degree of accuracy, to regard the modulation as sinusoidal. Table 1 gives the values of the cyclic modulation frequency $\omega$ and the corresponding experimentally obtained values of the phase shift $\varphi$, as well as the values of $\tau$ calculated from these values by formulas (2.10) and (2.11). As we see, the assumption of an exponential law of decay leads to good agreement of all measurement results, whereas formula (2.11) gives systematic discrepancies in the values of $\tau$ exceeding the possible random errors of measurement. This gives grounds for believing that the decay of dye solutions at room temperature does indeed occur according to an exponential law. The same kind of analysis of data obtained at low temperatures$^{12}$ shows that under these conditions, in a number of cases, very substantial changes in the form of this law are possible.

Table 1

Results of fluorometric measurements carried out at different values of the modulation frequency of the exciting light

(Solution of fluorescein in alcohol at room temperature)

Cyclic frequency $\omega$ (sec.$^{-1}$)	Observed values $\varphi$ (deg.)	$\tau \cdot 10^9$ sec. By formula (2.12)	$\tau \cdot 10^9$ sec. By formula (2.13)
1.017*)	29.6	4.9	5.4
1.407**)	37.6	4.65	5.5
1.984*)	47.1	4.1	5.4
2.387**)	52.0	3.8	5.35

It is of interest to note one more criterion for the exponential character of the decay. Since $\omega$ and $\tau$ are essentially positive quantities, under an exponential law of decay the quantity $\varphi$ can under no circumstances have values greater than $\frac{\pi}{2}$. The same is true for any law of decay that can be approximated in the form of the sum of some number of exponentials with positive coefficients. On the contrary, for the case in which there is a constant delay of emission for all molecules [formula (2.12)], and for other similar cases, the quantity $\varphi$ is in no way restricted and may have arbitrary values. For sufficiently large values

) Measurements by Shimanovsky$^{11}$.
*) Measurements by the author$^{12}$.

...frequencies \(\omega\), phase shifts \(\varphi\) exceeding \(\frac{\pi}{2}\) should be observed. Finally, certain conclusions about the character of the law of decay can also be drawn from a comparison of the quantities \(A_m\) and \(\varphi_m\).

Thus, although in essence fluorometric methods do not make it possible to reconstruct the law of luminescence decay directly from experimental data, analysis of the entire body of facts in many cases makes it possible to establish the functional character of this law; and then fluorometric methods make it possible, with fairly high accuracy, to determine the numerical values of the constants entering into this law.

As we have seen, the experimental problem is always reduced to the determination of the phase shift \(\varphi\) or of the amplitude factor \(A\), which characterizes the relation between the corresponding harmonics of the functions \(L(t)\) and \(E(t)\), or between these functions themselves. However, owing to the smallness of the period of these functions, direct study of them presents very considerable experimental difficulties, which it has been possible to overcome to a certain extent only in comparatively recent times. In older methods the functions \(L(t)\) and \(E(t)\) were replaced by another pair of functions, which preserve the same phase and amplitude relations but are more readily accessible to experimental observation. In essence, the difference between the numerous variants of the fluorometric technique proposed by various authors reduces to a difference in the methods of making this replacement.

Fig. 6. General schematic diagram of fluorometers with double modulation of the light beam.

3. FLUOROMETERS WITH DOUBLE MODULATION OF LIGHT BEAMS

The general scheme of installations of this type, in which their historical connection with the idea of Becquerel’s phosphoroscope is still clearly visible, is shown in Fig. 6. The exciting light from the source \(L\), after passing through the first modulating device \(M_1\), falls at \(F\) either on a scattering surface or on the fluorescent substance under investigation. The secondary beam of diffusely reflected light or, correspondingly, of fluorescence light is directed to the second modulating device \(M_2\), operating synchronously with the first, and its intensity after the second modulation is measured by one or another photometer \(Ph\).

If \(\theta\) is the time taken by the light to traverse the path \(M_1FM_2\), then, evidently, the measured mean values of the intensi-

…of the secondary beam in the case of scattered light and in the case of fluorescence are respectively equal to

\[ I(\theta)=\frac{1}{T}\int_{0}^{T} E(t-\theta)R(t)\,dt \tag{3.1} \]

and

\[ F(\theta)=\frac{1}{T}\int_{0}^{T} L(t-\theta)R(t)\,dt, \tag{3.2} \]

where \(R(t)\) is a function analogous to \(E(t)\) and expressing the course of the “transparency” of the second modulating device.

Substituting into the last formula expression (2.2) for the function \(L(t)\) and changing the order of integration, we obtain the following integral equation relating the functions \(F(\theta)\) and \(I(\theta)\):

\[ F(\theta)=\int_{0}^{\infty} I(t+\theta)\Phi(t)\,dt. \tag{3.3} \]

Except for the sign before \(\theta\), this equation is completely identical with equation (2.2), which relates the functions \(E(t)\) and \(L(t)\). Therefore the phase and amplitude relations between the corresponding harmonic components of the functions \(F(\theta)\) and \(I(\theta)\) are the same as those between the components of the functions \(L(t)\) and \(E(t)\), and we may with equal right use either pair of functions to study the law of decay of luminescence. The difference will be only in the sign of the phase shift \(\varphi\): whereas the curve \(L(t)\) lags behind the curve \(E(t)\), the curve \(F(\theta)\) leads the curve \(I(\theta)\). This is easily verified directly by substituting into expressions (3.1) and (3.2) the expansions of the functions \(E(t)\) and \(R(t)\) in a Fourier series and carrying out the corresponding integrations, taking into account the orthogonality of the system of functions \(\sin mx\), \(\cos mx\).

If the optical path length of the ray \(M_1FM_2\) is equal to \(2l\), then evidently

\[ \theta=\frac{2l}{c}, \]

and in the functions \(F(\theta)\) and \(I(\theta)\) we may replace the variable \(\theta\) by the variable \(l\), the period of the new functions \(F(l)\) and \(I(l)\) being equal to \(\lambda/2\), where \(\lambda=cT\) is the wavelength corresponding to the modulation period \(T\). Experimentally it is easy to obtain the functions \(F(l)\) and \(I(l)\) by changing the distance \(l\) between the modulating devices and the surface \(F\), and measuring the corresponding mean values of the light intensity after passage through the second modulating device. As we see, the experimental scheme here still fully corresponds to Becquerel’s phosphoroscope scheme, but the interpretation of the results obtained is entirely different from that with which we are dealing in the study of comparatively long processes.

High-frequency modulation of light was first used for measuring small time intervals by Abraham and Lemoine \(^{18}\)

in 1899. Their apparatus was directly intended for measuring the inertia of the electro-optical Kerr effect, i.e., for measuring the relaxation time of the artificial anisotropy produced by a field. However, the idea underlying it proved very fruitful and found application in many other cases as well, where it is necessary to measure very small intervals of time. These include, in particular, fluorometric measurements and modern methods for measuring the speed of light over short baselines.

The scheme of Abraham and Lemoine’s apparatus is shown in Fig. 7. The brightness of the spark \(S\), observed through the Kerr cell \(N_1, K, N_2\), depends on the phase shift between the ordinary and extraordinary components of the light beam passing through the Kerr condenser. By moving the mirrors \(M_1, M_2\), i.e., by increasing the interval between the moment of breakdown of the spark gap and the moment when the light passes through the condenser, Abraham and Lemoine observed a gradual weakening of the spark brightness and from these observations were able to establish that, for carbon disulfide, the relaxation time of the Kerr effect does not exceed \(10^{-8}\) sec.

Fig. 7. Diagram of Abraham and Lemoine’s apparatus for measuring the inertia of the Kerr effect.

In 1921, Wood\(^{14}\) was the first to apply Abraham and Lemoine’s apparatus to the study of the decay of photoluminescence. He illuminated crystals of platinum barium cyanide with a spark and observed them through a Kerr cell. Their green glow was still clearly visible, which would have been impossible if, between excitation and emission, there had existed a “dark pause” of duration \(10^{-7}\) sec or more.

Further improvement of the method for obtaining quantitative data on the duration of luminescence was carried out by Guttling\(^{15}\) in 1923. Placing a Wollaston birefringent prism between the Kerr condenser and the Nicol \(N_2\), Guttling measured the phase shift between the components of the elliptically polarized light in the condenser. He made measurements of this kind both for the light of the spark itself and for the light of the luminescence excited by this spark, and it turned out that, for one and the same optical path length, the phase shift for the exciting light was always greater than for the luminescence light. This is evidently due to an additional delay of the light in the luminescent substance. In order to obtain, with the exciting light, the same phase shift as with the fluorescence light, it was necessary in the first case to increase

the path length by a certain segment \(\Delta l\), i.e., to increase the time spent by the light in traversing this path by the amount \(\Delta t=\dfrac{\Delta l}{c}\). Assuming that all molecules excited at a certain instant emit after one and the same time interval \(\tau\) following excitation, Göttling supposed that this interval—the mean lifetime of the excited molecules, or the duration of the “dark pause”—is equal to \(\Delta t\).

Göttling’s assumption concerning the course of the decay of luminescence, however, is by no means compulsory or the only possible one. On the contrary, Wien’s direct observations of the decay of luminescence in excited atoms, as well as general ideas about the mechanism of emission, make it considerably more probable that the luminescence begins with its greatest intensity at the moment of excitation and decays according to an exponential law. Göttling’s method did not make it possible to choose between these hypotheses, and this served as the immediate reason for the profound reworking of it undertaken by Gaviola\({}^{16}\). It should be said in advance that Gaviola did not succeed in deciding the question of the functional character of the law of decay. The reasons for this are not of a fundamental but of a purely technical nature, and we shall not dwell on them here in greater detail. Nevertheless, Gaviola’s work was of exceptionally great importance for the entire subsequent development of the fluorometric method for studying luminescence-decay processes.

Fig. 8. Diagram of Gaviola’s fluorometer. The primary exciting beam of light is modulated by the Kerr cell \(N_1, K_1, N_2\); the secondary beam, emerging from the vessel with the fluorescing substance \(T\) or from the scattering surface \(S\), is modulated by the cell \(N_3, K_2, N_4\). The capacitors of both cells are supplied with voltage from one and the same tube generator of undamped oscillations.

The scheme of the last version of Gaviola’s fluorometer is shown in Fig. 8. It is easy to see that, from the fundamental point of view, it corresponds quite exactly to the general scheme of fluorometers with double modulation (Fig. 6), except for the circumstance that the quantity measured is not the intensity of the light after the second modulation, but the degree of ellipticity of its polarization, measured by the angle \(\delta\) through which the Nicol \(N_4\) must be turned in order to equalize the brightness of both fields of view produced by the Wollaston prism \(W\). This, however, has no fundamental significance, since from the measured angle of rotation of the Nicol \(\delta\), at which the fields of view obtained with the aid of the Wollaston prism are equalized, it is not difficult, in the cas-

...if necessary, to pass to the intensity of the corresponding beam. In practice such a need does not arise, since the positions of the maxima and minima of the curves \(\delta_I(l)\) and \(\delta_F(l)\), obtained respectively for the scattered light and the fluorescence light, coincide with the positions of the maxima and minima of the curves \(I(l)\) and \(F(l)\).

The modulation of both the primary and secondary light beams was carried out in Gaviola’s apparatus, as in earlier works, by Kerr cells; but an important methodological innovation was the fact that, instead of the previously used damped oscillations of Gaviola’s spark discharge, an alternating electric voltage of high frequency, obtained from a tube generator of undamped oscillations, was applied to the Kerr cells. Thus Gaviola for the first time achieved a purely periodic modulation of the light beams. To obtain a more advantageous characteristic of the Kerr cells, Gaviola applied to their plates, in addition to the alternating voltage, also a constant polarizing voltage approximately equal to the amplitude of the alternating one.

With the aid of this apparatus Gaviola measures the mean duration of luminescence \(\tau\) in the following way. By changing the distance between the first modulating device and the mirror \(S\), and by measuring each time the corresponding values of the quantity \(\delta\), he constructs from points the curve \(\delta_I(l)\), and then for the curve \(\delta_F(l)\) takes only one point, corresponding to some definite distance \(l = l_0\). Having determined on the curve \(\delta_I(l)\) that point \(l'\) for which \(\delta_I(l') = \delta_F(l_0)\), and denoting by \(\Delta l\) the difference \(l' - l_0\), Gaviola computes \(\tau\) from the formula \(\tau = \dfrac{2\Delta l}{c}\).

In this method of calculation there is, of course, a certain inaccuracy and inconsistency. Gaviola assumes that the decay of luminescence proceeds according to an exponential law, whereas the formula used by him would be strictly valid only in the case in which there were a constant “pause” between absorption and emission, as Göttling assumed. With an exponential law of decay, the curve \(\delta_F(l)\) is not only shifted relative to the curve \(\delta_I(l)\), but also deformed (the “flattening effect,” see above). Therefore the calculation of the phase shift from an arbitrary point is inaccurate and ambiguous. In addition, for this law of decay the calculation of \(\tau\) must, as we have seen, be made according to the formula \(\omega\tau = \tg \varphi\), and not according to the formula \(\omega\tau = \varphi\), which is equivalent to Gaviola’s formula.

Attention was drawn to these circumstances by Dushinsky\(^{18}\), who subjected the foundations of the theory of fluorometric measurements to a very thorough analysis. In his experimental investigation of the luminescence duration of the sodium \(D\)-lines he reproduced Gaviola’s apparatus, without making substantial changes in its construction, but considerably improved the method of calculating the duration of the excited state \(\tau\) from the experimental data obtained.

Hanfield\(^{19}\) also used Gaviola’s apparatus to investigate the luminescence of certain gases, including sodium vapor. The results he obtained are interesting, but he introduced nothing fundamentally new into the measurement technique.

The next important step forward in the development of fluorometric measurement techniques is connected with the work of Szymanowski\(^{11}\), published in 1935. From Fig. 9, which shows the schematic diagram of Szymanowski’s fluorometer, it is clear that this apparatus is essentially very similar to Gaviola’s. As there, the light is modulated by two Kerr cells with condensers \(K_1\) and \(K_2\); for different values of the distance \(l\), the ellipticity of the polarization of the light is measured after its passage through the second Kerr condenser \(K_2\). This measurement, however, is carried out not with a Wollaston prism and a Nicol, as Gaviola did, but with a Babinet–Soleil compensator \(K\) with a Szivessy half-wave plate, placed between the condenser \(K_2\) and the Nicol \(N_4\). Thus, in fact, what is measured is the additional phase shift \(\alpha\) between the components of the elliptically polarized beam that must be introduced in order to equalize the fields of view observed with a small telescope. This phase shift is evidently proportional to the displacement of the compensator \(\Delta k\).

Fig. 9. Schematic diagram of Szymanowski’s fluorometer.

A significant feature of Szymanowski’s method is that, taking into account Dushinsky’s considerations, he records the complete dependence of the quantity \(\Delta k\) (or \(\alpha\)) on the distance \(l\), not only for the scattered light (“main curve” in Fig. 10), but also for the fluorescence light (“fluorescence curve” in the same figure), and calculates the phase shift between the curves \(\alpha_l(l)\) and \(\alpha_F(l)\) from the displacement \(x\) of the minima of these curves, according to the formula

\[ \varphi = 4\pi \frac{x}{\lambda}. \tag{3.4} \]

By analogy with the known relation \(I \sim \sin^2 \frac{\varphi}{2}\), which connects the intensity of the light transmitted through a Kerr cell with the phase difference produced in this cell between the components of elliptically polarized light, Szymanowski assumes that the curves \(\sin^2 \frac{\alpha_l(l)}{2}\) and \(\sin^2 \frac{\alpha_F(l)}{2}\) are identical with the curves \(I(l)\) and \(F(l)\), respectively. This

incorrect. It is easy to show that, in the case where an alternating voltage is applied to the cell, such a simple transition from the setting of the compensator to the intensity that the cell would transmit without it, or conversely from the intensity to the displacement of the compensator, is no longer possible (see ^19). Nevertheless, the use of formula (3.4) to calculate the phase shift $\varphi$ of interest to us is quite legitimate, since the minima and maxima of the curves $a_I(t)$ and $I(t)$, on the one hand, and $a_F(t)$ and $F(t)$, on the other hand, coincide.

However, having thus determined the phase shift $\varphi$ considerably more accurately than had been done earlier, Shimanovskii then calculates the quantity $\tau$ (for exponential decay) not by the correct formula $\omega\tau=\operatorname{tg}\varphi$, but by Gaviola’s approximate formula: $\omega\tau=\varphi$, or

\[ \tau=\frac{2x}{c}. \]

For comparatively small values of $\varphi$ the error introduced thereby is small, but for large values it may already exceed the random errors of measurement (see Table I, p. 230).

Fig. 10. Phase shift between the components of light after passage through the second Kerr compensator ($K_2$) in Shimanovskii’s fluorometer, as a function of the distance between the modulating devices: dashed line—for scattered light (“main curve”); solid curve—for fluorescence light (“fluorescence curve”).

The most recent design of fluorometers with double modulation of the light beams is the apparatus built by the author jointly with Shimanovskii ^20 in 1937. Its most essential feature is the use of a new modulating device, replacing the Kerr cells used in all previous fluorometers. The need to find a new, more perfect device for high-frequency modulation of light was due to a number of inconveniences and shortcomings arising when Kerr cells are used for this purpose.

A fundamental drawback of using a Kerr cell as the modulating device in fluorometers is the circumstance that the excitation of luminescence is then always produced by linearly polarized light, and not by natural light; in exactly the same way, from the luminescence light the Nicol prism $N_3$ selects only one component corresponding to its orientation. As will be shown below (§ 7), under such conditions

under which the measured values of $\tau$ may differ from the true ones by a quite appreciable amount, and the measurement results depend on the mutual orientation of the planes of oscillation passed by the nicols $N_2$ and $N_3$.

Of the purely technical inconveniences connected with the use of the Kerr cell in these devices, the most important is the heating of nitrobenzene in the high-frequency field. The consequence of this is, on the one hand, a change in the Kerr constant, and, on the other, a change in the capacitance of the generator circuit and, consequently, a change in the modulation frequency. In Shimanovskii’s experiments, for example, the heating of the nitrobenzene proceeded at a rate of $4^\circ$ per minute, whereas even heating exceeding $0.5^\circ$ already noticeably distorted the results. Thus, the observer had only 7 seconds at his disposal for making the reading, after which it was necessary to remove the high voltage from the cell and wait until the nitrobenzene cooled to its former temperature. The troublesome nature of such a procedure is obvious.

To eliminate the indicated and other shortcomings of the Kerr cell, Dushinskii$^{18}$ attempted to construct a light modulator based on the dependence of the rotation of the plane of polarization in quartz on the magnitude of the voltage applied to the quartz. This attempt, however, did not give positive results because of the low intensity of the modulated light obtained in this way. Dushinskii’s idea was later developed somewhat by Broninghaus$^{21}$, but details of his work are lacking. Apparently, Broninghaus also did not succeed in realizing, on this principle, a modulating device that could have been used in a fluorometer.

The basis of our modulating device is the idea of using the so-called Debye–Sears effect, i.e. the phenomenon of diffraction of light by ultrasonic waves. The fundamental possibility of using this phenomenon to accomplish high-frequency modulation of light was pointed out as early as 1934 by Mandelstam, Landsberg and Papaleksi, and simultaneously by Carolus. Simultaneously with us, and independently of us, the same modulating device was used in his fluorometer by O. Maercks (see § 4).

The basic scheme of our modulating device is shown in Fig. 11. The luminous body of the source $L$ is imaged by a condenser $K$ on the horizontal entrance slit of the apparatus $S_1$, placed at the principal focus of the objective $O_1$. In the parallel beam of rays emerging from this objective there is placed a piezoquartz plate $Q$, to the electrodes of which a high-frequency voltage from a generator is supplied. Suitably arranged diaphragms block all the light passing by the quartz; the light that has passed through the quartz is focused by the objective $O_2$ on the exit slit of the apparatus $S_2$.

If the frequency of the voltage applied to the quartz coincides with the natural frequency of the mechanical oscillations of the quartz or with one of the harmonics of this frequency, then a standing ultrasonic...

wave of the corresponding frequency, at which diffraction of the light beam passing through the quartz takes place. In the plane of the slit \(S_2\), a clearly expressed diffraction pattern is then observed, containing spectra of several orders. Since the diffraction occurs on a standing ultrasonic wave, the light energy in this diffraction pattern is continuously redistributed among spectra of different orders. At the moments when the amplitude of the ultrasonic wave passes through zero, there is no diffraction, and all the energy of the light beam is concentrated in the central image of the slit—the spectrum of zero order. As the amplitude increases, the intensity of the diffraction pattern increases, and an ever more considerable part of the energy is transferred to spectra of higher orders at the expense of a corresponding decrease of the energy in the central image. If the frequency of the ultrasonic oscillations is \(\nu\), then this entire process of redistribution of energy is repeated periodically with frequency \(2\nu\). The slit \(S_2\) is adjusted so that it transmits only the central image (the zero-order spectrum) and stops all the light falling into spectra of higher orders. It is clear that in this case the intensity of the light passing through the slit changes periodically with frequency \(2\nu\). It would of course also be possible to arrange the slit \(S_2\) so that it transmitted not the central image of slit \(S_1\), but one of the diffraction spectra, or to replace slit \(S_2\) by a screen stopping the central image and transmitting all the diffracted light. Although in that case the depth of modulation reaches 100%, in practice isolating the central image proved more convenient.

Fig. 11. Schematic diagram of a device for high-frequency modulation of light based on the Debye–Sears effect.

The piezoquartz plate used by us had dimensions \(3 \times 4\ \mathrm{cm}\) with a thickness of \(2\ \mathrm{cm}\). The frequency of the electromagnetic oscillations of the generator coincided, of course, not with the fundamental natural frequency of the mechanical oscillations of such a thick plate, but with one of its very high overtones.

In order that under these conditions sufficiently intense ultrasonic oscillations could be obtained, it was necessary to apply to the quartz plates a rather high voltage, of the order of several thousand volts. At a voltage frequency of 11–12 thousand kilohertz, which corresponds to a cyclic modulation frequency of about \(1.5\cdot 10^8\) Hz, the modulation was still very deep. According to an approximate estimate, the modulation depth under these conditions amounted to more than

Fig. 12. Schematic diagram of the Tumerpan and Shimanovsky fluorometer.

80%. In some experiments we succeeded in obtaining a sufficient modulation depth even at a considerably higher frequency (about \(5\cdot 10^8\) Hz).

The schematic diagram of the fluorometric apparatus described in the work of the author and Shimanovsky \(^{20}\) is shown in Fig. 12. In the right-hand part of this figure the modulating device is shown, the separate details of which are designated by the same letters as in Fig. 11. The exit slit of the modulator is located at the principal focus of the objective \(O_3\), which directs a parallel beam of modulated light onto the objective \(O_4\), which, after reflection from the mirror \(M\), gives an image of the slit \(S_2\) in the plane \(F\), where there is a scattering plate or a vessel with the fluorescent substance. The fluorescence light (or, correspondingly, the scattered light) is directed back by the objective \(O_5\), at whose principal focus the image of the slit \(S_2\) is located. This parallel beam is then turned by the prism \(P\) through \(90^\circ\), passes through the quartz in a direction perpendicular to the former one, and is then focused by the objective \(O_6\) on the slit \(S_3\) of the photometer \(Ph\), whi-

which, like slit \(S_2\), is adjusted so that only the zero-order spectrum of the diffraction pattern arising in its plane passes through it. In this case, of course, secondary modulation of the light is carried out, and thus in a single quartz plate \(Q\) both modulating devices \(M_1\) and \(M_2\) of the basic scheme shown in Fig. 6 are combined.

The parts of the apparatus enclosed in Fig. 12 by the dashed rectangle—the mirror \(M\), the vessel \(F\), and the objectives \(O_4\) and \(O_5\)—were placed on a carriage that could be moved parallel to the optical axis of the apparatus, changing the distance \(l\) within approximately 140 to 600 cm. In this way the curves \(I(t)\) and \(F(t)\) were recorded; from the displacement of their minima the phase shift between the curves \(E(t)\) and \(L(t)\) was calculated, and from this the value of \(\tau\) was determined.

Recently we have introduced a number of changes into this apparatus which are not of fundamental significance, but which greatly facilitate the work and increase the accuracy of the measurements. The most important of these changes is the replacement of the visual Gelhof–Fabry type photometer, which we used previously, by a secondary-electron multiplier. We have applied with great success in these measurements a secondary-electron multiplier with magnetic focusing of L. A. Kubetskii’s system, with an antimony–cesium cathode. The development of such multipliers was carried out in Kubetskii’s laboratory by S. M. Feinstein. As a light source we now use super-high-pressure mercury lamps (type SVDSh), the production of which has recently been established by the Moscow Electric-Lamp Plant. Finally, we now place on the movable carriage, which moves along the optical bench, only two mutually perpendicular vertical mirrors, which turn the parallel beam of modulated light incident on them through \(180^\circ\) and displace it somewhat parallel to itself. This beam then falls on the fixed objective \(O_4\) and is focused on the cuvette with the fluorescing vessel (or scattering surface), which are also fixed immovably. The image of slit \(S_2\) on the cuvette is in the focus of objective \(O_5\), which directs the parallel beam of secondary radiation onto the quartz. After passing through the quartz in a direction perpendicular to the former one, the light enters objective \(O_6\), which gives a diffraction pattern in the plane of the exit slit of the entire apparatus, and is then directed to the multiplier, whose current is measured by a mirror galvanometer with a sensitivity of about \(10^{-9}\) A per division and a very short period.

4. STROBOSCOPIC FLUOROMETERS

In 1938 O. Maercks\(^{22}\) proposed a new and very original design of fluorometer, based on the following simple and elegant idea. When we carry out stroboscopic observation of some periodic process, i.e. observe it under

of illumination modulated at a frequency equal to the frequency of the process observed, then, as is known, we see this process stopped in some position, which depends on the phase of the light modulation. Therefore, by carrying out such stroboscopic observation twice with the aid of modulated light beams shifted relative to one another in modulation phase, from a comparison of those positions in which we then see the observed process we can determine the phase shift between the modulated light beams employed.

In Meks’ construction the modulation of the light was effected by a device which in principle is completely identical with ours, except for the circumstance that in Meks’ case the system of standing ultrasonic waves on which diffraction takes place was produced in a liquid, whereas we used a system of waves arising in the quartz itself.

As a periodic process convenient for observation under stroboscopic illumination, Meks uses a traveling ultrasonic wave propagating in a liquid from an oscillating quartz, to which a voltage of the same frequency is applied as to the quartz used for modulating the light. As is known, with sufficient amplitude and depth of the ultrasonic field along the direction of the light ray, a standing ultrasonic wave can be directly observed under steady illumination in the form of a system of dark and light bands, the so-called “lines of convergence,” parallel to the front of the ultrasonic wave and perpendicular to the illuminating light beam. In the case of a traveling wave this pattern is continuously blurred by the motion of the wave and can be observed only under stroboscopic illumination. In this case, of course, the position of this system of bands is determined by the phase of the modulated light beam used for its illumination. Therefore, if such a system of bands is photographed once under illumination directly by the modulated light beam, and a second time under illumination by the fluorescence light excited by this beam, then we obtain two systems of bands displaced relative to one another. From this displacement it is easy to calculate the phase shift between the light beams and—in the case of an exponential law of fluorescence decay—the mean duration of emission \(\tau\).

In Meks’ apparatus the traveling wave is photographed in scattered light and in fluorescence light separately, which entails a number of inconveniences. This shortcoming is eliminated in the stroboscopic fluorometer constructed in our laboratory by M. D. Galanin. The scheme of this fluorometer, in principle identical with Meks’ fluorometer, is shown in Fig. 13. The part of the apparatus modulating the exciting light, enclosed between slits \(S_1\) and \(S_2\), is identical with the modulator described above on p. 238; the only difference is that here the slits \(S_1\) and \(S_2\), as well as the quartz plates \(Q_1\),

are arranged vertically. The modulated primary beam of light is focused at \(F\) on a cuvette with a fluorescing solution, on the inner side of the wall of which a spot of greenish oil paint had been applied, with a cutout in the form of an acute angle. The cuvette was positioned so that the middle part of the slit \(S_2\) was imaged by the lens \(O_3\) onto the surface of the fluorescing solution, while the upper and lower parts were imaged onto the surface of the paint. Owing to this, in photographs of the traveling wave propagating in the vessel \(T\), one can obtain at once three systems of fringes, the upper and lower of which are photographs of the wave in scattered light, and the middle one is the same photograph in fluorescence light. The peculiar shape of the vessel \(T\) was calculated so as to prevent the formation in the vessel of standing ultrasonic waves. The photographing was carried out with a camera with microscopic optics; moreover, as in Merks’s arrangement, the cylindrical lens \(Z\) compressed the image in the vertical direction, which made it possible to shorten the exposure substantially.

Fig. 13. Stroboscopic fluorometer according to Merks in the constructive design of M. D. Galanin.

Fig. 14. Microphotogram of photographs of a traveling wave in a liquid, taken under stroboscopic illumination by scattered light (curve \(a\)) and fluorescence light (curve \(b\)).

In Fig. 14 a microphotogram is reproduced of a photograph of a traveling wave, taken in scattered light and in fluorescence light. It is easy to see that the distribution of light intensity in the pattern recorded in these photographs corresponds to the curves \(I(l)\) and \(F(l)\). Indeed, if we denote by \(T(l,t)\) the “transparency”

of the ultrasonic field at the time \(t\) at a point located at a distance \(l\) from the quartz, then the instantaneous value of the intensity of the light field passing through the given point is equal to \(E(t)\cdot T(l,t)\), where \(E(t)\) is the intensity of the incident light. The mean value of this intensity observed by us is

\[ S_l(l)=\overline{E(t)\,T(l,t)} \tag{4.1} \]

for the pattern observed in scattered light, and

\[ S_F(l)=\overline{L(t)\,T(l,t)} \tag{4.2} \]

when photographed in fluorescence light.

Since the function \(T(l,t)\) is periodic with respect to \(l\) and with respect to \(t\), and in time it varies with the same period \(T\) as the modulated light, while in space it varies with the period \(\Lambda\), where \(\Lambda\) is one half of the wavelength of the ultrasonic wave in the liquid, it is quite obvious that the relation between the curves \(S_l(l)\) and \(S_F(l)\) is exactly the same as the relation between the curves \(I(t)\) and \(F(t)\) in fluorometers with twofold light modulation. The function \(T(l,t)\) replaces the function \(R(t)\), which determines the “transparency” of the second modulating device. Owing to the nonlinear distortions introduced by the blackening law of the photographic emulsion, we cannot assert the same with respect to the curves obtained by microphotometry of the photographs. But in any case the maxima and minima of the curves shown in Fig. 14 coincide with the maxima and minima of the curves \(S_l(l)\) and \(S_F(l)\), respectively.

Therefore the phase shift between the curves \(S_l(l)\) and \(S_F(l)\), or, equivalently, between the curves \(E(t)\) and \(L(t)\), can be calculated from the formula

\[ \varphi = 2\pi \frac{x}{\Lambda}. \]

Like all the preceding authors, Merx calculates \(\tau\) not by this formula, but from the relation

\[ \frac{\tau}{T}=\frac{x}{\Lambda}, \]

which is equivalent to the relation \(\varphi=\omega\tau\). As we have seen, in the case of exponential decay of fluorescence this formula is valid only in the approximation in which one may regard \(\varphi=\tg\varphi\).

The possibility of obtaining the curves \(S_l(l)\) and \(S_F(l)\) simultaneously, and the possibility of measuring the quantities \(x\) and \(\Lambda\) for several successive maxima and minima with subsequent averaging of the results, are undoubtedly major advantages of Merx’s fluorometer. It must be said, however, that in the practical performance of these measurements one has to encounter a large number of difficulties, the overcoming of which makes this method scarcely less laborious than ordinary fluorometric methods. This circumstance is also noted by Kirchhoff \(^{23}\), who reproduced the setup

Mercer, having introduced into it a number of technical improvements. One of the principal hindrances is the heating of the liquid as a result of absorption of ultrasonic waves. To combat this phenomenon Kirchhoff, for example, was forced to connect vessel \(T\) with a tank of several tens of liters capacity and to resort to continuous forced circulation of water between this tank and the vessel. A number of special measures and great care are also necessary in order to obtain undistorted results when microphotometering the photographs obtained.

5. FLUOROMETERS WITH MODULATION OF THE RECEIVER SENSITIVITY

From the general considerations concerning fluorometers with twofold modulation of the light beam, set forth in Section 3, it is clear that, in principle, the same curves \(I(l)\) and \(F(l)\) can be obtained if, instead of secondary modulation of the light beam, one causes it to act on some receiver whose sensitivity varies periodically with cyclic frequency \(\omega\), equal to the modulation frequency of the primary beam. Such a receiver may be, for example, a photocell to which an alternating voltage of frequency \(\omega\) is applied. By \(R(t)\) one must then, of course, understand the function determining the time course of the receiver sensitivity, and by \(I(l)\) and \(F(l)\)—the “response” of this receiver, respectively, to scattered modulated light or to fluorescence light.

Fig. 15. Diagram of Guttel’s apparatus for measuring the speed of light or the quantity \(\tau\).

Figure 15 shows the diagram of an apparatus, constructed on this principle, by A. Guttel\(^{24}\), which differs from the fluorometers described above only in that the second Kerr cell is replaced here by a photocell \(Ph\). An alternating high-frequency voltage from one and the same generator is applied to this photocell and to the first Kerr cell. The photocurrent is measured by galvanometer \(G\). The dependence of it on the length of the optical path of the beam is shown in Fig. 16.

Guttel’s apparatus was intended for measuring the speed of light. Therefore he recorded on it only the curve \(I(l)\). It is easy to see—

however, that by replacing mirror \(S\) with a fluorescent substance, one could by exactly the same method obtain the curve \(F(l)\), which would be shifted relative to the curve \(I(l)\) along the abscissa axis by some interval \(x\), from which the phase shift \(\varphi\) between the functions \(I(l)\) and \(F(l)\) is easily calculated.

Fig. 16. Dependence of the current in the Goettgen apparatus on the length of the path of the light beam.

In 1924 Webb\({}^{25}\) proposed a very original fluorometric method, with the aid of which he and subsequently a number of his collaborators (Slack\({}^{26}\), Rendall\({}^{27}\), Garrett\({}^{28}\)) measured the damping constant \(\tau\) for a number of mercury and cadmium lines. Fig. 17 shows a diagram of this apparatus in Rendall’s design, differing from Webb’s original design only in that here the source of modulated radiation and the receiver with modulated sensitivity are placed in separate vessels \(A\) and \(B\), whereas in Webb’s apparatus they were enclosed in one common vessel and separated from each other by a quartz plate.

Fig. 17. Diagram of Webb’s fluorometer with modulated receiver sensitivity.
(According to Rendall\({}^{27}\).)

The excitation of mercury atoms is produced in vessel \(A\) by impacts of electrons whose velocity varies periodically; moreover, during a certain part of the period their kinetic energy exceeds the excitation potential of the lines under study. To carry out this kind of excitation modulation, an equipotential incandescent cathode \(C\), grids \(g, g'\), and an anode \(O\) with a window transmitting the radiation are mounted in the vessel. All the electrodes have the form of cylinders. The grids \(g, g'\) have a constant positive potential relative to the cathode \(C\), close to the excitation potential of the lines under study; relative to the electrode \(O\) they are likewise at a certain positive potential. In addition, an alternating voltage is applied to the grids \(g, g'\) and to the cathode \(C\), the frequency of which can be varied within wide limits while maintaining its amplitude constant. The value of this amplitude is chosen so that it exceeds several times

difference between the excitation potential and the constant grid potential. In Rensdell’s experiments, for example, the constant potentials had the values \(U_{g,g'} = 0.0\ \mathrm{V}\), \(U_C = -7.5\ \mathrm{V}\), \(U_0 = -3.0\ \mathrm{V}\), while the amplitude of the alternating voltage \(U_{C-g,g'}\) was equal to \(2.5\ \mathrm{V}\), so that the resulting cathode–grid voltage oscillated between the values 5 and \(10\ \mathrm{V}\), i.e., during a considerable part of the positive half-period it exceeded the excitation potential, of the triplet \(6^3P_{0,1,2} - 7^3S_1\) studied by Rensdell (5461 Å, 4358 Å and 4047 Å), equal to \(7.7\ \mathrm{V}\). As the static characteristics obtained by various authors have shown, during this part of the period the excitation is approximately proportional to the difference between the grid potential and the excitation potential (the electron-excitation function in the interval investigated has a linear course), while during the remaining part of the period excitation is absent.

The mercury vapor pressure in vessel \(A\) could be regulated and controlled by changing the temperature of the liquid mercury in the side arm \(T\), placed in a water bath, and by superheating the vessel itself by about 80 degrees relative to this bath.

The receiving part \(B\) is a photoelectric cell with a cylindrical cathode \(P\), an anode \(W\), and a grid \(H\), to which the same alternating voltage is applied as to the grids \(g, g'\). In Rensdell’s apparatus the cathode was sodium, and the anode \(W\) was at a potential of \(6\ \mathrm{V}\) relative to the cathode. The current–voltage characteristics of this photoelectric cell show that the current reaches saturation at a grid potential of approximately \(0.5\ \mathrm{V}\), and falls to zero at a potential equal to \(-0.7\ \mathrm{V}\). Webb, who together with Messenger investigated the damping of the mercury resonance line 2534 Å, used a nickel cathode. In his photoelectric cell the current tended toward a definite saturation value even in the negative half-period of the potential at the anode.

All the authors mentioned measured the photocurrent by means of a quadrant electrometer. Rensdell compensated the dark current and the current caused by the illumination of the heated cathode by the current of an ionization chamber \(I\), which could be regulated by changing the thickness of a lead filter covering the opening of a metal box containing a radioactive preparation.

From what has been set forth it is clear that the measured value of the photocurrent depends on the time \(\theta\) spent by the light in traversing the path from source to receiver, on the modulation frequency \(\omega\), and on the damping constants of the luminescence, which enter into this function through the quantities \(A_m\) and \(\varphi_m\). In contrast to all the methods described earlier, in Webb’s method the quantity \(\theta\) was not varied; owing to the small distance between vessels \(A\) and \(B\) it may be regarded as equal to zero, but the modulation frequency \(\omega\) was varied. Thus, instead of the function \(F(\theta)\), here the function

\[ F(\omega) = E_0 R_0 + \sum_{m=1}^{\infty} \frac{E_m R_m A_m}{2} \cos(\Delta_m + \varphi_m), \tag{5.1} \]

having, for an exponential law of decay, the form

\[ F(\omega)=E_0R_0+\sum_{m=1}^{\infty}\frac{E_mR_m}{2\sqrt{1+m^2\omega^2\tau^2}}\cos\left(\Delta_m+\operatorname{arctg} m\omega\tau\right) \tag{5.2} \]

[\(\Delta_m\) is the phase shift between the corresponding harmonic components of the functions \(E(t)\) and \(R(t)\)].

Fig. 18 shows the general course of this function according to Rendell’s measurements for the mercury line 5461 Å. Along the ordinate axis are plotted the values of the quantity \(\dfrac{F_\omega}{F_0}\), i.e., the ratio of the photocurrent values at the given frequency \(\omega\) to its value at a frequency close to zero (60 cycles). Curves for \(\dfrac{F_\omega}{F_0}\) according to the measurements of Webb and other authors have the same form.

Fig. 18. Dependence of the ratio \(R=\dfrac{F_f}{F_0}\) on the modulation frequency \(f\), according to Rendell.

Assuming an exponential law of decay of the luminescence, Webb resorts to the following procedure for calculating \(\tau\) from the curve \(F(\omega)\). Approximating in a known way the functions \(E(t)\) and \(R(t)\) of his apparatus, he obtains for the ratio \(\dfrac{F(\omega)}{F_0}\) the analytical expression

\[ R(\omega)=\frac{F_\omega}{F_0}= \frac{1+\dfrac{\omega^2\tau^2}{2}(1-s)}{1+\omega^2\tau^2}, \tag{5.3} \]

where \(s\) is the ratio of the saturation currents in the positive and negative half-periods of the voltage.

It is obvious that for the values \(\omega=0\) and \(\omega=\infty\) the function \(R(\omega)\) has, respectively, the values: \((R)_0=\) and \((R)_\infty=\dfrac{1-s}{2}\). If now we denote by \((R)_{1/2}\) the quantity

\[ (R)_{1/2}=\frac{(R)_0+(R)_\infty}{2} =\frac{1}{2}\left(1+\frac{1-s}{2}\right) \]

and determine from the curve \(R(\omega)\) that value of the frequency \(\omega_c\) at which \(R(\omega_c)=(R)_{1/2}\), then it is easy to see that

\[ \omega_c\tau=1 \quad \text{or} \quad \tau=\frac{1}{\omega_c}. \tag{5.4} \]

It is not difficult to show within what limits and under what general assumptions about the nature of the functions \(E(t)\) and \(R(t)\) Webb’s method is valid. From formula (5.2) it follows that

\[ \left. \begin{aligned} (F)_0 &= E_0 R_0+\sum_{m=1}^{\infty}\frac{E_m R_m}{2}\cos\Delta_m,\\[4pt] (F)_\infty &= E_0 R_0,\\[4pt] (F)_{1/2} &= E_0 R_0+\frac{1}{2}\sum_{m=1}^{\infty}\frac{E_m R_m}{2}\cos\Delta_m . \end{aligned} \right\} \tag{5.5} \]

Thus, if \(\omega_c\) is that cyclic modulation frequency at which \(F(\omega_c)=(F)_{1/2}\), then in the general case the relation between \(\omega_c\) and \(\tau\) is given by the equation

\[ \frac{1}{2}\sum E_m R_m \cos\Delta_m = \sum E_m R_m \left[ \frac{\cos\Delta_m+m\omega_c\tau\sin\Delta_m}{1+m^2\omega_c^2\tau^2} \right]. \tag{5.6} \]

Restricting ourselves here to the first \(n\) terms of the expansion of the function \(F(\omega)\), we obtain an algebraic equation of degree \(2n\) with respect to \(\omega_c\tau\), giving \(2n\) values of this quantity. The method is therefore not single-valued. Only in the case when we put \(n=1\), i.e., restrict ourselves to the first harmonic component of the functions \(E(t)\) and \(R(t)\), and may take \(\Delta_1=0\), does Webb’s condition follow from formula (5.6): \(\omega_c\tau=1\).

Rendall obtained for the function \(R(\omega)\) the approximate expression

\[ R(\omega)=\frac{K^2+\dfrac{\omega^2}{2}}{K^2+\omega^2}, \tag{5.7} \]

where \(K=1/\tau\). Rendall estimates the magnitude of the terms thereby discarded in the right-hand side of the formula as \(10\%\). He determines the quantity \(\tau\) by choosing the value of \(K\) in formula (5.7) so that the curve calculated from it agrees as well as possible with the experimental one. It is easy to see that here too the accuracy of the method depends on the extent to which the higher harmonics of the functions \(E(t)\) and \(R(t)\) play a role.

In this connection it is interesting to note that it would be possible to eliminate completely the necessity of investigating the functions \(E(t)\) and \(R(t)\), and to make this method, in principle, completely exact, if, by means of a tuned circuit, we selected the first harmonic of the function \(F(\omega)\), i.e., if the measurement were made not with an electrometer, but with a resonant amplifier.

L. A. TUMERMAN

6. “PHASE” FLUOROMETERS

The essence of any fluorometric method, as we have seen, consists in measuring the phase or amplitude relations between the corresponding harmonic components of the “excitation function” \(E(t)\) and the “emission function” \(L(t)\). With an exponential law of decay, the problem reduces only to measuring the phase shift or the ratio of the amplitudes of these components, since the quantities \(A_m\) and \(\varphi_m\) are very simply related to one another \((A_m = \cos \varphi_m)\). All the other procedures—secondary modulation of the light beam, use of a variable optical path length of the ray, stroboscopic observation of one or another periodic process, or use of receivers with modulated sensitivity—are intended only to facilitate the measurement of the quantities \(A_m\) and \(\varphi_m\), but are not fundamentally necessary.

Fig. 19. Schematic diagram of a “phase” fluorometer.

The development of modern methods of radio-engineering measurements in the high-frequency range has in recent years made it possible to construct fluorometers in which the phase shift \(\varphi_m\) between the modulated light beams \(E(t)\) and \(L(t)\) is measured directly or, more precisely, between the corresponding photoelectric currents. We shall call such fluorometers “phase” fluorometers.

The schematic diagram of a fluorometer of this type, proposed by me \(^{12, 19}\), is shown in Fig. 19. The light emerging from the modulating device \(M\) is divided by the semitransparent plate \(P\) into two beams: one is directed immediately onto the photocell \(Ph_2\), the other excites the luminescence of the investigated substance \(F\), the glow of which acts on the photocell \(Ph_1\). The currents of both photocells, amplified by resonant amplifiers \(V_1\) and \(V_2\), which select the first harmonics of these currents, are fed to the plates of an oscillograph, and the phase shift between them is determined from the Lissajous figure obtained on the screen of the oscillograph. Instead of this, one may also, as shown in Fig. 20, introduce into one of the paths a calibrated phase-shifting device and compensate the phase shift to zero, or measure this phase shift by other, purely electrical methods.

The chief experimental difficulty that must be overcome in carrying out this plan lies in the necessity of obtaining a rather considerable amplification of weak high-frequency photocurrents. The problem is complicated by the fact that the sensitive amplifying device must operate in the immediate vicinity of

to a powerful generator whose frequency is exactly half the frequency to which the amplifiers are tuned. In the construction of a phase fluorometer of this type, carried out by A. A. Brandt and myself, antimony–cesium secondary-electron multipliers, kindly made for us by engineer M. I. Belyaev in the laboratory of the Moscow Electric-Lamp Plant, were used as the receiver. After the first high-frequency stage, the two photocurrents being compared were heterodyned by one and the same heterodyne, and the further amplification was carried out at the intermediate frequency. A block diagram of the entire apparatus is shown in Fig. 20. To reduce interference from the generator, the latter was built according to a push-pull circuit, which was carefully symmetrized in order to suppress as far as possible the second harmonic component in its radiation. The apparatus described was brought to preliminary tests, in which quite distinct Lissajous figures were obtained on the screen of the oscillograph, but we had no opportunity to test it directly in measurements of \(\tau\) for specific objects, since, with the outbreak of the war, the work had to be discontinued and the setup was dismantled.

Fig. 20. Block diagram of the “phase” fluorometer of A. A. Brandt and the author.

It seems to us nevertheless that, if brought to proper technical form, this fluorometer would have had substantial advantages in comparison with all the designs described earlier. Its principal advantage is that here the first harmonic components of the functions \(E(t)\) and \(L(t)\) are selected, so that all questions connected with the law of modulation of the light, effected with the aid of one or another device, disappear of themselves. In practice, what is essential is that the necessity is eliminated for the very painstaking and laborious point-by-point recording of the curves \(I(t)\) and \(F(t)\), or curves analogous to them. At the same time the whole apparatus becomes considerably less bulky and complex, since there is no need to create and precisely adjust an optical path several meters long. In this connection we are now working on restoring this setup.

7. PRINCIPAL RESULTS OF FLUOROMETRIC MEASUREMENTS

1) Study of the processes of quenching of the fluorescence of solutions of complex organic substances (dyes).

The study of the photoluminescence of complex organic molecules in solutions was the problem for the solution of which, first of all,

have been developed, and it is precisely to these objects that the largest number of results obtained so far pertains. The attention of investigators here was drawn above all by the possibility of comparing data on the lifetime of the excited state with data on changes in the yield and polarization of luminescence when such factors are varied as the concentration of the fluorescing substance itself or of a “quencher” added to the solution, the temperature of the solution, the nature and viscosity of the solvent, etc. All processes of this kind may, in the broad sense of the word, be called processes of fluorescence “quenching.” Elucidation of their mechanism is of extremely great importance for constructing a theory of the very phenomenon of fluorescence in these substances.

As S. I. Vavilov[^29] indicated, all quenching processes should expediently be divided into two groups: processes “of the first kind,” in which deactivation of the excited molecule without radiation occurs “instantaneously,” i.e., in a time considerably shorter than the normal lifetime of the excited state, and processes “of the second kind,” taking place during a time comparable with the time during which the molecule remains in the excited state. Processes of the first kind, evidently, do not change the mean duration of luminescence observed by us. Processes of the second kind, however, must shorten this duration, since in this case the probability that the molecule will be quenched is the greater the longer this molecule remains in the excited state. F. Perrin[^30] and later, in a more general form, S. I. Vavilov[^31] showed that in processes of the second kind the ratio of the fluorescence yield \(L\) to its duration \(\tau\) must retain a constant value, i.e., the condition must be satisfied

\[ \frac{L}{\tau} = \operatorname{const}\cdot K, \tag{7.1} \]

where \(K\) is the probability that the molecule, after excitation, has not undergone quenching of the first kind. Thus, comparison of data on the course of the quantities \(L\) and \(\tau\) makes it possible to take the first important step in elucidating the mechanism of the quenching process—to discriminate between processes that depend on time and those that do not.

The best studied case of quenching is quenching by foreign substances introduced into the solution—such as potassium iodide or aniline. According to Vavilov[^32], this phenomenon is explained by collisions of the second kind between excited molecules and quencher molecules. However, in such a simple form, where the whole process is interpreted entirely as a process of the second kind, the theory agrees poorly with the experimental data. It leads to the relation

\[ \frac{L_0}{L} = 1 + \operatorname{const}\cdot C_{\text{quench}} \tag{7.2} \]

(\(L_0\) is the fluorescence yield in the absence of quencher and \(C_{\text{qu}}\) is the concentration of the quencher), which is confirmed experimentally only for very small concentrations of the quencher. Therefore, in the refined theory proposed by Vavilov and Frank1, the idea is introduced that a molecule can undergo quenching of the first kind (“static” quenching) if, within the limits of a certain sphere of action, there is a molecule of the quenching substance. The probability that, within the sphere of action of the molecule, there will be no quencher molecule, i.e., that quenching of the first kind will not occur, is equal to \(e^{-\omega Nc}\), where \(c\) is the concentration of the fluorescent substance, \(N\) is the number of quenching molecules in \(1\ \text{cm}^3\) of the medium, and \(\omega\) is the difference between the radius of the “sphere of action” and the gas-kinetic radius of the fluorescent molecule. Under these assumptions, the dependence of the quantities \(L\) and \(\tau\) on the concentration of the quencher should be described by the formulas

\[ \frac{L_0}{L}=e^{\omega Nc}\left(1+\text{const.}\cdot C_{\text{qu}}\right), \tag{7.3} \]

\[ \frac{\tau_0}{\tau}=1+\text{const.}\cdot C_{\text{qu}}, \tag{7.4} \]

and the relation between \(L\) and \(\tau\) takes the form

\[ \frac{L_0}{L}=\frac{\tau_0}{\tau}\cdot e^{\omega Nc}. \tag{7.5} \]

Subsequently, Sveshnikov showed that the available experimental data on quenching can be explained quite satisfactorily even without the assumption of a “sphere of action” and a static quenching process, if one takes into account the second term of Smoluchowski’s diffusion formula, which Vavilov and Frank had neglected. Taking this term into account leads to the fact that the factor \(e^{\omega Nc}\) in formula (7.3) is replaced by some other factor, depending in a rather complicated way on the concentrations of the quencher and of the fluorescent substance, the diffusion coefficient, and the probability of quenching collisions. Both the Vavilov–Frank formula and the Sveshnikov formula describe the fluorescence-quenching data equally well; and, evidently, the question of whether a “sphere of action” and a process of static quenching actually exist can be resolved only by studying the change in the quantity \(\tau\) during the quenching process.

Such a study for a solution of uranyl in water was carried out by Shimanovsky2. His results are presented in Fig. 21 and in Table II. The theoretical curve for the yield \(\left(\frac{L_0}{L}\right)_{\text{theor}}\) was obtained from the curve for \(\tau_0/\tau\) by multiplying by the factor \(e^{\omega Nc}\), with \(\omega\) taken to be \(2.5\cdot 10^{-21}\), the value given by Vavilov and Frank. The experimental values for the ratio \(\frac{L_0}{L}\) were likewise taken from the work of these authors. The linear behavior of the ratio \(\frac{\tau_0}{\tau}\) and the good agreement

experimental and theoretical curves for \(\dfrac{L_0}{L}\) confirm the Vavilov and Franck formulas (7.3) and (7.4) and indicate that, at least in the present case, the quenching process is completely explained by the Vavilov–Franck theory, and that allowance for the second diffusion term, made by Sveshnikov, is of no essential significance. Unfortunately, however, we have data of this kind for only one solution, and therefore the question of the role of the various factors indicated cannot be considered clarified in a general form. The whole question still requires additional experimental investigation.

Fig. 21. Dependence of the yield \((L_0/L)\) and of the lifetime of the excited state \((\tau_0/\tau)\) on the concentration of the quencher \(C_{KI}\) for a solution of uranin in water. (According to Shimanovskii \(^{34}\).)

A more complicated and, perhaps, more interesting question is that of the nature of so-called concentration quenching, i.e. the decrease in the yield of luminescence as the concentration of the fluorescing substance increases.

Table II

Change in the lifetime of the excited state \(\tau\) and in the luminescence yield \(L\) as functions of the concentration of the quenching substance \((C_{KI})\) for a solution of uranin in water (according to Shimanovskii \(^{34}\))

Concentration \(C_{KI}\cdot 10^3\ \mathrm{g/cm^3}\)	\(\tau\cdot 10^9\ \mathrm{sec.}\)	\(\dfrac{\tau_0}{\tau}\)	\(\left(\dfrac{L_0}{L}\right)_{\mathrm{exp}}\)	\(\left(\dfrac{L_0}{L}\right)_{\mathrm{theor}}\)
0	4,47	1,0	1,0	1,0
4,15	3,8	1,18	1,41	1,22
8,3	3,07	1,46	1,79	1,58
16,6	2,07	2,16	2,85	2,52
41,5	1,0	4,47	5,9	6,55
83,0	0,47	9,5	18,2	20,3

The question of whether concentration-

to the quenching, a parallel decrease in the duration of the excited state, was posed already in the first at all reliable fluorometric work of Gaviola[^16]. Comparing the results of his measurements of the quantity $\tau$ for solutions of fluorescein in methyl alcohol and of rhodamine in glycerin at different values of the dye concentration with the corresponding data of Vavilov on the change in the luminescence yield of these solutions, Gaviola showed that the decrease in the yield and in $\tau$ begins at the same concentration values and that the course of the corresponding curves is in general similar, although an exact coincidence of these curves does not occur. These measurements, however, were not sufficiently accurate for the reality of these discrepancies to be regarded as beyond doubt, i.e., for us to be able to conclude with confidence that, in the case of concentration quenching, we have a combination of “static” processes (not depending on time) with processes taking place during the time the molecule remains in the excited state.

More complete and more accurate data on this question are given in the third of the above-cited[^11] papers of Shimanovskii, who investigated the dependence of $\tau$ on the concentration of the fluorescing substance for fluorescein and rhodamine dissolved in media of different chemical nature and viscosity: water, alcohols, and glycerin. Unfortunately, Shimanovskii did not carry out parallel measurements of the yield on the same solutions, limiting himself to comparison of his data with Vavilov’s data on the yield. Although one cannot be certain of the complete identity of the substances with which the different investigators made their measurements, nevertheless Shimanovskii’s results, reproduced in Figs. 22 and 23, make it possible to draw a number of very important conclusions.

The decrease in $\tau$ accompanying concentration quenching and, at the same time, the entirely certain absence of strict proportionality between the yield and $\tau$ show first of all that the process of concentration quenching is neither a pure first-order process nor a pure second-order process, but that static factors and time-dependent factors are combined in it. Further, attention is drawn to the fact that the linear dependence of the ratio $\dfrac{\tau_0}{\tau}$ on concentration, which, as we have seen, occurs in the case of quenching by foreign substances, is here rather an exception than a general rule. It occurs for solutions of the substances investigated in glycerin and is sharply violated in all other cases. Finally, special attention is deserved by the fact that, for solvents as different in viscosity as alcohols and glycerin, the course of quenching and of the change in viscosity, at least in the initial portion of the corresponding curves, is fairly close, whereas for aqueous solutions these curves go considerably more steeply. Both

These circumstances—especially the latter—clearly indicate that the process of concentration quenching cannot be regarded as a diffusion process analogous, in a certain sense of the word, to quenching by foreign substances.

Fig. 22. Dependence of the yield \((L_0/L)\) and of the duration of the excited state \((\tau_0/\tau)\) on the concentration of the fluorescent substance for fluorescein solutions:
a) in water, b) in ethyl alcohol, c) in isobutyl alcohol, d) in glycerin. (After Shimakovsky \(^{11}\).)

We cannot consider that quenching occurs as a result of collisions of the second kind between exci—

excited and unexcited molecules, and are forced to seek another mechanism of their interaction in order to explain the established facts.

A complete theory of the influence of concentration on the fluorescence of solutions, apparently fully adequate to the totality of the available experimental data, was constructed by S. I. Vavilov[^35]. Fundamental to this theory is the assumption of quantum-mechanical migration of energy, i.e. that if in a solution there is an excited molecule \(A\) and an unexcited molecule \(B\), then even at fairly considerable distances between these molecules there exists an appreciable probability of transfer of the excitation energy from molecule \(A\) to molecule \(B\), i.e. of a process of deactivation of the first molecule and excitation of the second. At greater distances between the molecules this transfer process does not lead to quenching and causes only depolarization of the fluorescence. However, as the concentration increases, the number of transfers grows whose result is quenching, i.e. the transition of the excitation energy into the thermal degrees of freedom of the excited molecule. This explains the fact that concentration quenching, as a rule, begins at higher concentration values than concentration depolarization. The process of quantum-mechanical energy migration takes place throughout the entire time during which the molecule remains in the excited state, i.e. it is a time-dependent process. However, along with this, in order to explain the absence of strict proportionality between \(L\) and \(\tau\), it is necessary to assume also the possibility of a static process of energy transfer, occurring in a time extremely small in comparison with \(\tau\), and taking place when the unexcited molecule is located within a certain “sphere of action” of the excited molecule.

Fig. 23. Dependence of the yield \((L_0/L)\) and the duration of luminescence \((\tau_0/\tau)\) for solutions of rhodamine \(B\) extra in glycerine and methyl alcohol. (After Shimanovskii[^11].)

The theory developed by Vavilov on these assumptions gives expressions for the functional dependence of the yield \(L\), the polarization \(p\), and the duration of luminescence \(\tau\) on the concentration of the fluorescing substance. The constants entering into the functions \(L(c)\), \(p(c)\), and \(\tau(c)\) can be determined unambiguously only in the case where we have experimental data on the variation of all three quantities \(L\), \(p\), and \(\tau\) with concentration.

Such a complete set of experimental data is at present available only for solutions of fluorescein and rhodamine \(B\) in glycerin; moreover, it is necessary to compare data from different investigators. Nevertheless, in these cases the theory leads to quite satisfactory agreement between the calculated and experimental quantities. The data given in Table III show, for example, how well, on the whole, Shimanovskii’s fluorometric measurements agree with the results of Vavilov’s theory.

Table III

Theoretical and experimental data on the change of \(\tau\) with increasing concentration of the fluorescing substance

Fluorescein in glycerin	Fluorescein in glycerin	Fluorescein in glycerin	Rhodamine \(B\) in glycerin	Rhodamine \(B\) in glycerin	Rhodamine \(B\) in glycerin
\(c \cdot 10^{-18}\)	\(\left(\dfrac{\tau}{\tau_0}\right)_{\mathrm{obs.}}\)	\(\left(\dfrac{\tau}{\tau_0}\right)_{\mathrm{calc.}}\)	\(c \cdot 10^{-18}\)	\(\left(\dfrac{\tau}{\tau_0}\right)_{\mathrm{obs.}}\)	\(\left(\dfrac{\tau}{\tau_0}\right)_{\mathrm{calc.}}\)
0.18	1.00	0.99	0.05	1.00	1.00
0.58	1.00	0.97	0.24	1.00	0.98
7.3	0.71	0.71	2.4	0.97	0.80
11.0	0.62	0.62	3.0	0.76	0.76
14.3	0.52	0.55	6.0	0.49	0.62
20.0	0.24	0.48	—	—	—

2) Change in the state of polarization of the radiation in the course of its decay

The partial polarization of the fluorescence of solutions, first discovered by Weigert3, can be understood and explained only on the basis of the idea that the properties of a molecule characterizing it as an absorbing and emitting system are determined by the structure of this molecule and are not connected with the exciting light. Since molecules oriented in different ways with respect to the electric vector of the exciting light have different probabilities of excitation, the initial distribution of the excited molecules is anisotropic; moreover, the character of this anisotropy is determined by the nature of the electric symmetry of that system of charges which plays the role of the elementary absorbing structure. If this initial anisotropy of the excited molecules remained unchanged, then the polarization characteristics of the radiation would be determined only by it and by the nature of the electric symmetry

elementary emitter. In those cases where the assumption of the invariance of the initial distribution of orientations of the excited molecules is justified, i.e., in those cases where we are dealing with so-called limiting polarization, the study of these characteristics, as S. I. Vavilov has shown \(^{37}\), makes it possible to draw a number of very valuable conclusions about the nature of elementary absorbers and emitters. It makes it possible to decide whether, in the process of absorption and emission, we should treat a given molecule as a dipole, a quadrupole, or some other system with a higher degree of symmetry.

However, in a number of cases the observed degree of polarization proves to be considerably less than the limiting one. The fluorescence of solutions of very low viscosity or very high concentration may, for example, be almost completely depolarized. In these cases, evidently, during the time the molecule remains in the excited state, processes take place in the medium which reduce the initial anisotropy of the orientations of the excited molecules. The study of such processes, leading to depolarization of the emission, is of great interest both for elucidating the mechanism of luminescence itself and for investigating the kinetics of processes occurring with the participation of excited molecules.

At present two processes are well studied which lead to the destruction of the initial anisotropy of the distribution of orientations of excited molecules in solution and to depolarization of the emission. These are, on the one hand, the rotation of excited molecules as a result of Brownian motion and, on the other hand, interaction between molecules of the dissolved fluorescent substance, manifested in a decrease of the observed degree of polarization as the concentration of the solution is increased. The first process is usually called rotational, the second—concentration depolarization.

In both cases the presence of depolarization must lead to the degree of polarization of the emission changing continuously over the course of its decay period \(^{38}\). It follows that, when fluorescence is excited by polarized light, the decay curve of the component of the emission polarized parallel to the electric vector of the exciting light \((J_{\parallel})\), and the decay curve of the component polarized in the perpendicular direction \((J_{\perp})\), must differ from one another and from the true law of fluorescence decay, i.e., from the law determining the decrease in the number of excited molecules. Accordingly, the mean values of the duration of luminescence of both components \((\tau_{\parallel}\) and \(\tau_{\perp})\) must be different and must differ from the true value \(\tau\).

For the case of rotational depolarization the corresponding calculations were carried out by Jablonski \(^{39}\), who showed that if fluorescence decays according to a simple exponential law

if \(J=J_0 e^{-t/\tau}\), then the laws of decay for the “parallel” and “perpendicular” components have the form

\[ \left. \begin{aligned} J_{\parallel} &= \frac{A}{3}\,[1+2p_0+2(1-p_0)e^{-\varphi t}]\,e^{-t/\tau},\\ J_{\perp} &= \frac{A}{3}\,[1+2p_0-(1-p_0)e^{-\varphi t}]\,e^{-t/\tau}, \end{aligned} \right\} \tag{7.6} \]

where

\[ \varphi=\frac{3(p-p_0)}{\tau(1-p)(1+2p_0)}, \tag{7.7} \]

and

\[ p=\frac{\displaystyle\int_0^\infty J_{\perp}\,dt} {\displaystyle\int_0^\infty J_{\parallel}\,dt} \]

is the depolarization coefficient of the radiation; \(p_0=-\dfrac{J_{\perp}}{J_{\parallel}}\) for \(t=0\).

In the presence of rotational depolarization, the relation between the limiting polarization \((p_0)\) and the polarization \(p\) observed under one or another set of conditions, for an arbitrary decay law \(\Phi(t)\), is expressed by the relation\(^4\)

\[ \frac{1}{p}-\frac{1}{3} = \left(\frac{1}{p_0}-\frac{1}{3}\right) \cdot \frac{1}{\displaystyle\int_0^\infty \Phi(t)\cdot e^{-\frac{RT}{V\eta}t}\,dt}, \tag{7.8} \]

which, for the exponential law of decay, becomes the well-known Perrin–Vavilov formula

\[ \frac{1}{p}-\frac{1}{3} = \left(\frac{1}{p_0}-\frac{1}{3}\right) \left(1+\frac{RT}{V\eta}\cdot\tau\right) \tag{7.9} \]

(here \(R\) is the gas constant, \(T\) the absolute temperature, \(V\) the molecular volume of the dissolved substance, and \(\eta\) the viscosity of the solvent). From this it is easily seen that the constant \(\varphi\) entering the Yablonskii formulas (7.6), (7.7) has the value

\[ \varphi=\frac{RT}{V\eta}. \tag{7.10} \]

If, using formulas (7.6), we calculate the mean values of the duration of luminescence for the components \(J_{\parallel}\) and \(J_{\perp}\), then we easily arrive at the relations

\[ \left. \begin{aligned} \frac{\tau_{\parallel}}{\tau} &= \frac{(1+2p_0)y^2+2(1-p_0)} {(1+2p_0)y^2+2(1-p_0)y},\\ \frac{\tau_{\perp}}{\tau} &= \frac{(1+2p_0)y^2-(1-p_0)} {(1+2p_0)y^2-(1-p_0)y}. \end{aligned} \right\} \tag{7.11} \]

where

\[ y = 1 + \varphi \cdot \tau . \]

In fluorometers of the Gaviola–Shimanovsky type, in which light modulation is effected by means of a Kerr cell, the excitation is always produced by polarized light, whose direction of vibration is determined by the orientation of the Nicol \(N_2\) (Fig. 9). At the same time, a component polarized in the plane of vibration of the Nicol \(N_3\) is always selected in the fluorescence light. It follows that, generally speaking, the results of measurements on these fluorometers depend on the mutual orientation of the Nicols \(N_2\) and \(N_3\). By crossing these Nicols we measure the value \(\tau_{\perp}\); by setting them parallel to one another, we find the value \(\tau_{\parallel}\). As Yablonsky showed, the true value of \(\tau\) can be obtained when the angle between the Nicols \(N_2\) and \(N_3\) is \(54^\circ 74'\). (In this, however, no account was taken of the necessity of introducing into the calculation results a correction due to the fact that the law of decay of the components \(J_{\parallel}\) and \(J_{\perp}\) is not exponential.)

The difference between \(\tau_{\parallel}\) and \(\tau_{\perp}\) disappears, and both of these quantities coincide with \(\tau\), only in two limiting cases: either when the quantity \(\varphi\) is so small that depolarization practically does not occur \((p=p_0,\ \rho=\rho_0)\), or when this quantity is so large that the radiation is completely depolarized \((p=0,\ \rho=1)\). The first of these cases can be realized in very viscous solutions, for example, in thoroughly dehydrated glycerin; the second—in solutions of very low viscosity, for example, in aqueous ones. The first experimental observations of the effects described belong to Shimanovsky\(^{42}\), who found a difference between \(\tau_{\parallel}\) and \(\tau_{\perp}\) for solutions of fluorescein in mixtures of glycerin with water. In an aqueous solution the values of \(\tau_{\parallel}\) and \(\tau_{\perp}\), according to Shimanovsky’s data, coincide; as the glycerin content is increased, the values of \(\tau_{\parallel}\) systematically fall, decreasing at \(\rho=0.8\) \((p \simeq 10\%)\) by about \(20\%\). The values of \(\tau_{\perp}\) remain constant. Shimanovsky explains this by the fact that the depolarization effect is here compensated by the difference between the true values of \(\tau\) for aqueous and glycerin solutions.

Shimanovsky’s results were checked and confirmed by Kessel\(^{43}\), whose principal experimental results are given in Fig. 24, where the experimental and theoretical curves are compared, representing the course of the ratios \(\dfrac{\tau_{\parallel}}{\tau}\) and \(\dfrac{\tau_{\perp}}{\tau}\) as functions of \(\rho\). The theoretical curves (\(C\) and \(D\)) were calculated by Yablonsky’s formulas (7.11).

As we see, on the whole these measurements are in satisfactory agreement with Yablonsky’s theory. Small but systematic discrepancies between experiment and theory can probably be explained by the fact that, in processing the measurements, a number of factors were not taken into account. Such a factor, shifting the experimental curve in the re-

side is, for example, secondary fluorescence, i.e., fluorescence caused by absorption of the short-wavelength part of the primary radiation. In addition, it should be borne in mind that in these measurements the values of \(\tau\) were calculated by the simplified formula \(\tau=\dfrac{2x}{c}\) \([x\) is the displacement of the minima of the curves \(I(l)\) and \(F(l)]\), which, as was shown, is not exact even in the case of a purely exponential law of decay, and still less in the case of the more complicated law expressed by formula (7.10).

Fig. 24. Ratios \(\dfrac{\tau_{\parallel}}{\tau}\) and \(\dfrac{\tau_{\perp}}{\tau}\) as functions of the depolarization coefficient of the radiation \(\beta\) (according to Kessel). Curves \(A\) and \(B\) are experimental; curves \(C\) and \(D\) are theoretical. Fluorescein solution in mixtures of water + glycerin.

For concentration depolarization, analogous calculations and measurements were recently carried out in our laboratory by M. D. Galanin4. The basis of these calculations was the general theory of the influence of concentration on the fluorescence phenomena of solutions, developed by S. I. Vavilov5 and proceeding from the idea of the possibility of migration of excitation energy in a solution, i.e., its transfer from one molecule to another by way of quantum-mechanical resonance. The probability of such transfer, referred to unit time and unit concentration and not accompanied by quenching, is denoted by Vavilov as \(\dfrac{1}{k_2}\). Calculations performed by Galanin showed that, under these assumptions, the components \(J_{\parallel}\) and \(J_{\perp}\) decay according to the laws

\[ \left. \begin{aligned} J_{\parallel} &= \frac{J_0}{s} \left( e^{-\frac{t}{\tau}} + \frac{4p_0}{3-p_0}\,e^{-\frac{t}{\tau'}} + \frac{4p_1}{3-p_1}\cdot\frac{c}{k_2}\cdot t e^{-\frac{t}{\tau'}} \right),\\ J_{\perp} &= \frac{J_0}{s} \left( e^{-\frac{t}{\tau}} - \frac{2p_0}{3-p_0}\,e^{-\frac{t}{\tau'}} - \frac{2p_1}{3-p_1}\cdot\frac{c}{k_2}\cdot t e^{-\frac{t}{\tau'}} \right), \end{aligned} \right\} \tag{7.12} \]

and the dependence of the ratios \(\dfrac{\tau_{\parallel}}{\tau}\) and \(\dfrac{\tau_{\perp}}{\tau}\) on the concentration of the solution must have the form

\[ \left. \begin{aligned} \frac{\tau_{\parallel}}{\tau} &= \frac{ 1+\dfrac{4p_0}{3-p_0}\left(\dfrac{\tau'}{\tau}\right)^2 +\dfrac{8p_1}{3-p_1}\cdot\dfrac{c}{k_2}\cdot\tau' \left(\dfrac{\tau'}{\tau}\right)^2 }{ 1+\dfrac{4p_0}{3-p_0}\cdot\dfrac{\tau'}{\tau} +\dfrac{4p_1}{3-p_1}\cdot\dfrac{c}{k_2}\cdot\dfrac{\tau'^2}{\tau} },\\[1ex] \frac{\tau_{\perp}}{\tau} &= \frac{ 1-\dfrac{2p_0}{3-p_0}\left(\dfrac{\tau'}{\tau}\right)^2 -\dfrac{4p_1}{3-p_1}\cdot\dfrac{c}{k_2}\cdot\tau' \left(\dfrac{\tau'}{\tau}\right)^2 }{ 1-\dfrac{2p_0}{3-p_0}\cdot\dfrac{\tau'}{\tau} -\dfrac{2p_1}{3-p_1}\cdot\dfrac{c}{k_2}\cdot\dfrac{\tau'^2}{\tau} }. \end{aligned} \right\} \tag{7.13} \]

Here \(\rho_0\) is the limiting polarization, \(\rho_1\) is the polarization of the radiation emitted by molecules excited by a single transfer of energy from the initially excited molecule, and

\[ \frac{1}{\tau_{\parallel}}=\frac{1}{\tau_{\perp}}+\frac{c}{k_2}. \]

These formulas were subjected to experimental verification. The measurements were carried out in solutions of very high viscosity (pure glycerin, sugar candies), in which the influence of rotational depolarization could be regarded as practically excluded. Since our fluorometer, with a modulating device based on the Debye–Sears effect, makes it possible to excite with unpolarized light, the true values of \(\tau\) could be determined directly; and, by introducing appropriately oriented polarizers into the exciting beam and into the fluorescence beam, the values of \(\tau_{\parallel}\) and \(\tau_{\perp}\) could also be measured.

Fig. 25. Ratios \(\dfrac{\tau_{\parallel}}{\tau}\) and \(\dfrac{\tau_{\perp}}{\tau}\) as functions of fluorescein concentration. (After Galanin.) The solid curves were calculated from formulas (7.13). Circles indicate experimental values for solutions in glycerin; triangles, for solutions in sugar.

For these same solutions, from measurements of concentration depolarization, the constant \(\dfrac{1}{k_2}\) was determined, with the aid of which, using formulas (7.13), one can calculate the theoretical course of the ratios \(\dfrac{\tau_{\parallel}}{\tau}\) and \(\dfrac{\tau_{\perp}}{\tau}\) as functions of the concentration of the solution. The experimental points, as can be seen from Fig. 25, lie on these curves quite satisfactorily.

Starting from the decay laws of the components \(J_{\parallel}\) and \(J_{\perp}\), expressed by formulas (7.12), one can further calculate the values of the degree of polarization of the radiation at different moments of the decay. Applying the general theory of fluorometric measurements to the decay of the components \(J_{\parallel}\) and \(J_{\perp}\), Galanin showed that the dependence, observed on the fluorometer, of the degree of polarization \(p\) on the length of the optical path \(l\) between the first and second modulations must have the form shown in Fig. 26 by the solid ...

curve. To check these relations, the photometer in the fluorometric setup was replaced by a Savart polariscope, and a compensating plate was placed in front of the slit \(S_3\). In this way it proved possible to measure directly the degree of polarization of the fluorescence at different values of \(t\). The values obtained, as is seen from Fig. 26, lie well on the theoretical curve. The value of Vavilov’s constant \(\dfrac{1}{k_2}\), necessary for constructing this curve, was determined for the given solution from independent measurements of concentration depolarization.

Fig. 26. Values of the degree of polarization of radiation measured on a fluorometer for a solution of fluorescein in glycerin as a function of the optical path difference between the first and second modulations. The solid curve is calculated; the circles are experimental points.

Thus Galanin’s measurements make it possible to trace almost perceptibly the changes in the state of polarization of the radiation in the process of its decay and, at the same time, give an entirely irrefutable proof of the correctness of the initial assumptions of Vavilov’s theory.

3) The influence of temperature on the duration and law of decay of luminescence of solutions.

Measuring fluorometrically the values of \(\tau\) for solutions of rhodamine \(B\), rhodulin orange, and fluorescein in glycerin with temperature variations from room temperature to \(+170^\circ\), Gaviola found a substantial difference in the behavior of these solutions. Whereas for the first two named substances the duration of the excited state decreased very significantly with increasing temperature (by a factor of 3–4), the values of \(\tau\) for fluorescein solutions remained unchanged within the limits of measurement accuracy. In the case of rhodamine it was also found that a change in concentration by a factor of 100 (from \(c = 2 \cdot 10^{-5}\)

up to \(c = 2 \cdot 10^{-3}\ \mathrm{g/cm^3}\)) was not reflected in this temperature effect in any way.

This difference between the behavior of different dyes became to a certain extent understandable after Levshin’s work,^46 which showed that in solutions of rhodamine there is temperature quenching, i.e. a decrease in the fluorescence yield with increasing temperature, whereas in solutions of fluorescein no such quenching is observed.

Fig. 27. Relative values of the yield (solid curves) and duration of luminescence (dashed curves) as functions of the temperature of rhodamine B solutions. Curves \(1\) and \(1'\) refer to the solution in glycerin; curves \(2\) and \(2'\) to the solution in isoamyl alcohol.

In full agreement with Gaviola’s results, Levshin found that the course of temperature quenching does not depend on the concentration of the fluorescing substance or on the nature of the solvent, and he suggested that temperature quenching is due to changes in the state of individual molecules, and not to interaction between molecules, i.e. that it is an intramolecular rather than an intermolecular process.

It is natural to suppose that the decrease in \(\tau\) observed by Gaviola is due to the fact that the process of temperature quenching is a process of the second kind, necessarily leading to a proportional shortening of the duration of luminescence. Apparently, in the main this explanation is correct, although simultaneous measurements made by me and M. D. Galanin of changes in the yield \((\rho)\) and duration of luminescence \((\tau)\) with increasing temperature do not fully confirm it from a strictly quantitative point of view. From the results of our meas—

measurements presented in Fig. 27 show that in the initial stage of quenching the changes in yield and in the duration of luminescence proceed in parallel, so that the ratio \(\rho/\tau\) remains constant; however, at later stages the yield falls appreciably more steeply than \(\tau\). The impression is created that at higher temperatures a process of the first order is manifested in quenching, one not accompanied by a change in \(\tau\). However, the whole question of the mechanism and nature of temperature quenching is still far from clarified and requires further investigation.

In 1936 Krem[^47], who at that time was working in Warsaw on Szymanowski’s fluorometer, published an investigation of the duration of luminescence of uranin solutions in the temperature interval from \(0\) to \(+70^\circ\). He found that in solutions of low viscosity (water, ethyl alcohol) and at low concentrations of the fluorescing substance, the duration of luminescence, which remains constant within the limits from \(+70^\circ\) to \(+30^\circ\), begins to increase rather noticeably with further lowering of the temperature, so that at \(0^\circ\) the values of \(\tau\) are approximately 20% greater than at room temperature. As the concentration of the dissolved substance is increased, and as the viscosity of the solvent is increased, this temperature effect gradually weakens and disappears.

An attempt to explain the effect observed by Krem on the basis of the assumption that in the medium there occurs some diffusion process of quenching by foreign substances, the effectiveness of which decreases when the temperature is lowered, leading to an increase in the yield and in \(\tau\), proved unsuccessful. If this explanation were correct, then, proceeding from the Vavilov–Frank theory of quenching, we should have expected the dependence of the ratio \(\tau_0/\tau\) on the quantity \(T/\eta\) to be linear. The comparison of these quantities made by Krem, however, gave a curve sharply differing from a straight line and tending toward saturation. Thus, Krem had to reject this explanation, and the question of the cause of the phenomenon he observed remained open.

The question of the influence of temperature on the duration and the law of decay of luminescence was subjected to a more detailed investigation in my work, where the values of \(\tau\) were measured for alcoholic solutions of fluorescein, uranin, eosin, rhodulin orange, and rhodamine in the temperature interval from room temperature down to \(-107^\circ\).

In Fig. 28, as an example, are given the curves \(F(l)\) obtained for a fluorescein solution at room temperature (curve 1), \(-65^\circ\) (curve 2), and \(-95^\circ\) (curve 3), as well as the curve \(I(l)\). As we see, the quantity \(x\)—the displacement of the minimum of the curve \(F(l)\) relative to the curve \(I(l)\)—increases rapidly and very considerably with decreasing temperature.

The same increase also occurs for all the other solutions except rhodamine (Table IV).

Table IV.

Values of the quantity \(x\) for different temperatures of fluorescent solutions

| Substance | \multicolumn? |
|---|---:|---:|---:|---:|---:|---:|
| | room temp. | \(-22^\circ\) | \(-53^\circ\) | \(-65^\circ\) | \(-95^\circ\) | \(-107^\circ\) |
| Fluorescein | 70 | 82 | — | 135 | 170 | — |
| Uranin | 75 | — | 130 | — | — | — |
| Eosin | 73 | — | — | 102 | — | 123 |
| Rhoduline-orange | 45 | 73 | — | — | 152 | — |
| Rhodamine-extra | 64 | — | — | — | 65 | — |

As can be seen from the data presented, the duration of luminescence of rhodamine does not depend on temperature, whereas for all the remaining substances it increases considerably as the temperature is lowered. However, a more careful analysis shows that here we are dealing not with a simple change in the numerical value of the constant \(\tau\) in the exponential law of decay, but with a substantial change in this law itself, which at low temperatures can no longer be regarded as not only purely exponential, but even monotonic.

This is indicated above all by the fact that in a number of cases we obtain for \(x\) values very close to \(\frac{\lambda}{8}\) or even exceeding this value. Applying formula (2.10), which, as we have seen, must certainly be applied in the case of an exponential law of decay, we would obtain enormous values of \(\tau\), and for \(x>\frac{\lambda}{8}\) we would obtain negative values. Even if we explain this latter, obviously absurd result by the unaccounted-for influence of higher har-

Fig. 28. Curves \(F(l)\) for solutions of fluorescein in alcohol at temperatures \(+20^\circ\) (curve 1), \(-65^\circ\) (curve 2), and \(-95^\circ\) (curve 3). Cyclic modulation frequency \(\omega = 1.41\cdot 10^8\).

of the modulation function, then nevertheless the values of \(x\) obtained would be so large that the effect of “flattening” of the curves \(F(l)\) should, for an exponential law of decay, have caused them to become straight lines. Meanwhile, as is seen from Fig. 28, no noticeable flattening of the curves \(F(l)\) occurs at all, despite their enormous shift along the abscissa axis.

The only possible explanation of these facts, in our opinion, is the assumption that the entire period during which the molecule remains in the excited state is divided into two parts: an interval of a “dark pause,” during which the probability of radiation is extremely small or equal to zero, and the following period of normal luminescence, during which the probability of radiation has the normal value \(p=\frac{1}{\tau}\). In other words, we must assume the existence of two essentially different types of excited state of the molecules, differing above all in the value of the probability of spontaneous radiation, and possibly in a number of other respects as well.

Somewhat idealizing the decay law to which such an assumption leads, we may therefore write it in the form

\[ \begin{cases} \Phi(t)=0 & \text{for } t<\theta,\\[6pt] \Phi(t)=\dfrac{1}{\tau}e^{-\frac{t-\theta}{\tau}} & \text{for } t>\theta, \end{cases} \tag{7.14} \]

where \(\theta\) is the duration of the dark pause. From this point of view we must consider that at a relatively high (room) temperature \(\theta=0\), and the decay law is practically exponential. Qualitatively, the elementary decay law must have the form shown in Fig. 29. As the temperature is lowered, the value of \(\theta\) increases, while \(\tau\) remains constant. It is not difficult to see that an increase of \(\theta\) in formulas (7.14) should not lead to a “flattening” of the curve \(F(l)\), but should cause only a parallel displacement of it by a segment \(x'\), related to \(\theta\) by

Fig. 29. Approximate form of the elementary decay law at low temperatures.

\[ \theta=\frac{2x'}{c}. \]

The most convincing proof of the correctness of the proposed explanation is given, it seems to me, by experiments with variation of the frequency

modulation. At the same time, if the decay law really has the form (7.14), then the displacement \(x\) at room temperature should change in accordance with the formula \(\operatorname{tg}\varphi=\omega\tau\); the change in the value of \(x\) upon lowering the temperature should not depend on the modulation frequency. The depth of the curves \(F(l)\) should also change only slightly, since the “flattening” of this curve is determined only by the factor

\[ \cos\varphi=\frac{1}{\sqrt{1+\omega^2\tau^2}}. \]

Experiment confirmed all these expectations. In Fig. 30 the curves \(F(l)\) are shown for a fluorescein solution at different temperatures, taken at the modulation frequency \(\omega=2.38\cdot 10^8\ \mathrm{sec}^{-1}\), and the corresponding curve \(I(l)\). As we see, at room temperature, instead of the former value \(x=-70\ \mathrm{cm}\), we obtain the value \(x=57\ \mathrm{cm}\), but substitution of each of these values into formula (2.10) gives one and the same value of \(\tau\). The changes in this quantity upon lowering the temperature, within the accuracy of the measurements, do not differ from those which we observed at the former, almost twice smaller modulation frequency. Thus, for example, if at a temperature of \(-70^\circ\) we previously had \(\Delta x=75\ \mathrm{cm}\), then from these curves we find \(\Delta x=83\ \mathrm{cm}\). It may also be noted that in these measurements as well the depth of the curves \(F(l)\) remains quite considerable and does not change when \(x\) increases. Let us note further that in every case it follows with certainty from these experiments that at low temperatures the decay law is not exponential and is not reducible to a sum of exponentials with positive coefficients. This follows from the fact that here \(\lambda/8=99\ \mathrm{cm}\), whereas for the temperature \(-70^\circ\) we have \(x=140\ \mathrm{cm}\). No reasonable correction can

Fig. 30. Curves \(F(l)\) for a solution of fluorescein in alcohol at the cyclic modulation frequency \(\omega=2.8\cdot 10^8\). \(1\)—curve \(I(l)\); \(2\)—at temperature \(+20^\circ\); \(3\)—at temperature \(-70^\circ\).

can here yield values of \(x\) smaller than \(\dfrac{\lambda}{8}\), while for an exponential law of decay values \(x>\dfrac{\lambda}{8}\) give absurd negative values of \(\tau\).

Adopting the law of decay (7.14), we can obtain, for all the solutions investigated, the dependence of the duration of the “dark pause” on temperature. As is seen from Fig. 31, over a fairly broad temperature interval this dependence is approximately linear.

Fig. 31. Dependence of the duration of the “dark pause” \(\theta\) on temperature: 1—fluorescein solution (circles) and uranine solution (triangles); 2—rhoduline-orange solution; 3—eosin solution. All solutions are alcoholic.

Further investigation revealed that the addition to the solution of substances capable of quenching fluorescence, but in concentrations considerably smaller than those required for noticeable quenching, leads to a sharp reduction in the duration of the dark pause (Fig. 32). We shall not dwell on this in greater detail, since, on the one hand, it lies outside the scope of the present review, and, on the other hand, the very mechanism of the dark pause—i.e., the physical essence of the distinction between the two types of excited states—and the reasons for the influence upon it are still wholly unclear, and we are compelled for the time being to refrain from proposing any hypotheses.

8. INVESTIGATION OF INERTIAL ELECTRO-OPTICAL KERR AND FARADAY EFFECTS

In all the preceding exposition we have considered chiefly the methods and results of studying the processes of decay of luminescence. It is not difficult to see, however, that the theory of fluorometric measurements developed above in principle—and, in many respects, the construction of fluorometers as well—can be extended to the case of investigating other rapidly occurring relaxation processes.

Of such processes, the Kerr effect in liquids has been studied in the greatest detail. Let us recall that the Abraham–Lemoine apparatus, from which modern fluorometers developed, was intended precisely for studying the inertia of the Kerr effect. The subsequent development and improvement of fluorometers, in turn, made it possible to determine with great completeness and accuracy the relaxation times of the anisotropy that gives rise to the Kerr effect.

Fig. 32. Dependence of the duration of the “dark pause” on the concentration of the quencher for solutions of uranin at a temperature of \(56^\circ\)C (curve 1) and eosin at a temperature of \(107^\circ\) (curve 2).

In the case of a dipolar liquid this anisotropy arises because the dipole moments of the molecules are oriented by the external electric field. In cases where the molecules of the liquid do not have a dipole moment but possess only anisotropy of polarizability, the Kerr effect is due to the orientation of dipoles induced by the field itself. In both cases, however, the inertia of this effect is determined by the relaxation time required for the dipoles oriented by the field to return again to a statistical distribution.

When the liquid under study is acted upon by an alternating electric field, the presence of a finite relaxation time of the anisotropy will manifest itself, on the one hand, in the appearance of a phase shift between the birefringence and the field strength, and, on the other hand, in a decrease in the magnitude of the Kerr effect as the period of variation of the field \(T\) approaches the relaxation time \(\tau\). It is easy to see that the relation between the functions describing the time course of the intensity of light modulated by the Kerr cell and the time course of the field strength (or of the applied voltage) will be the same as the relation between the excitation and emission functions \(E(t)\) and \(L(t)\) in the case of fluorescence: the corresponding harmonic components of these functions will be shifted in phase, and at the same time, generally speaking, the modulation depth will decrease as the frequency increases. As in the case of fluorescence, in most experimental works the determination of the relaxation time is preferred to be carried out—

to be carried out on the basis of measuring phase, rather than amplitude, relations between these functions.

Without dwelling on the earlier works of Hutton48, Beams and Lawrence49, and Ranzi50, which in concept are close to the Abrahama–Lemoine arrangement, and in their construction are very similar to fluorometers with a variable optical path length of the Gaviola type, we shall note only the most recent work in this direction, that of Hanle and Merks51. The arrangement of these authors is shown in Fig. 33. As is easy to see, it is extremely similar to the Merks fluorometer described above. Light, modulated by a Kerr cell consisting of a capacitor \(K_1\) and polaroids \(F_1\) and \(F_2\), is concentrated on the slit \(S\), and with its aid a stroboscopic observation (photographing) is made of a traveling ultrasonic wave sent into the liquid by quartz \(Q\).

Fig. 33. Arrangement of the Hanle and Merks apparatus for measuring the inertia of the Kerr effect.

The position of the bands recorded on the plate, as was found, is determined by the relation between the phase of the wave and the phase of the light modulation. Therefore, replacing one capacitor by another, filled with another liquid, and comparing the position of the bands recorded on the plate, we can determine the phase shift of interest to us between beams modulated by means of the Kerr effect in the two liquids being compared.

Thus, in essence, this method gives not the absolute value of the relaxation time, but the difference between the values of this quantity for the two liquids being compared. Usually the Kerr effect in the liquid under investigation is compared with the effect in CSr, for which the relaxation time, as is known from other data, is so small that it may be neglected. This is entirely analogous to the fact that in fluorometric measurements we compare the duration of luminescence with the duration of the scattering process, which may be considered practically instantaneous. In any case, the relaxation times of the anisotropy obtained by this method may be smaller than the true ones, but in no way larger than them.

The principal results obtained by Hanle and Merks are set forth in Table V. On general grounds we should expect that the relaxation time of the anisotropy will be directly proportional to the volume of the molecule and the viscosity of the medium, and inversely proportional to the absolute temperature. It must also depend to a certain extent on the shape of the molecule, but, generally speaking, it is difficult to expect a direct relation between the value of the relaxation time and the magnitude of the dipole moment of the molecule. As is seen from Table V, in the case of nitrobenzene in

in the small temperature interval investigated, the ratio \(\frac{\tau T}{\eta}\) remains constant within an accuracy of up to 10%. Also confirmed is the absence of a direct connection between the values of \(\tau\) and the magnitude of the dipole moment (see the data for \(\alpha\)-bromonaphthalene, \(o\)-dichlorobenzene, and dimethylaniline).

Table V

Value of the relaxation time \(\tau\), determined from the inertia of the Kerr effect
(according to Hanle and Merx)

Substance	Temperature	Kerr constant \(\beta \cdot 10^{7}\)	Viscosity \(\eta\)	Dipole moment \(\mu \cdot 10^{18}\)	Relaxation time \(\tau \cdot 10^{9}\) sec.	Error
Nitrobenzene	22	210	0.020	3.9	2.7	\(\pm 0.2\)
Nitrobenzene	31	195	0.017	3.9	2.3	\(\pm 0.2\)
Nitrobenzene	39	175	0.014	3.9	2.0	\(\pm 0.2\)
Nitrobenzene	47	160	0.012	3.9	1.4	\(\pm 0.2\)
Nitrobenzene	55	153	0.010	3.9	1.1	\(\pm 0.2\)
Acetophenone	42	67	*)	2.97	1.1	\(\pm 0.2\)
Octyl alcohol	40	\(-7.5\)	0.089	1.62	2.3	\(\pm 1\)
\(\alpha\)-Bromonaphthalene	35	10	—	1.5	2.3	\(\pm 1\)
\(o\)-Dichlorobenzene	32	42	—	2.2	0.7	\(\pm 0.7\)
Dimethylaniline	35	10	0.011	1.39	2.8	\(\pm 0.5\)
Methylnaphthalene	60	2.1	—	0	1.4	\(\pm 0.3\)
Carbon disulfide	24	3.2	0.0037	0	0	\(\pm 0.5\)
Carbon disulfide in paraffin oil	24	—	0.04	0	0	\(\pm 0.5\)
Carbon disulfide in mineral oil	24	—	0.02	0	0	\(\pm 0.5\)
Nitrobenzene in mineral oil	—	—	—	3.9	7	\(\pm 0.3\)

In addition to investigations of the inertia of the Kerr effect, the relaxation time of anisotropy can also be determined by other methods. For dipolar substances it can be calculated from the dependence of the dielectric constant and dielectric losses on the frequency of the field. Moreover, independently of the magnitude of the dipole moment, it can be determined from observations of birefringence under flow (the Maxwell effect) and from investigations of the intensity distribution in the wing of the Rayleigh scattering line. The results of determining this quantity by the latter method for a large number of liquids were recently published by I. L. Fabelinskii\({}^{52}\). Here attention is drawn to the fact that, whereas measurements of the inertia of the Kerr effect—both those performed by Hanle and Merx and those belonging to other authors—give for the relaxation time val-

*) \(\eta = 0.07\) at \(80^\circ\).

measurements is of the order of \(10^{-9}\) sec.; all the other methods give, for this quantity, values three to four orders of magnitude smaller: \(10^{-12}\)–\(10^{-13}\) sec. In Table VI, which is taken from the cited work of Fabedinskii, the values of \(\tau\) are compared for a number of substances for which this quantity was determined by at least two of the methods indicated. Consideration of these data leads to the unavoidable conclusion that the relaxation time which determines the inertia of the Kerr effect differs very substantially from the relaxation time that determines other effects. The causes and nature of this very interesting and puzzling phenomenon are still by no means clear.

Table VI

Comparison of the anisotropy relaxation times obtained by various methods

Substance	Kerr effect	Dispersion	Scattering	Maxwell effect	Viscosity \(\eta\)	Depol. scattering of light \(\Delta \cdot 100\)	Dipole moment \(\mu \cdot 10^{18}\)
Chloroform	3800	—	0.12	—	0.0057	10	1.05
Chlorobenzene	—	2.1	0.30	0.33	0.0080	57.5	1.55
Water	—	0.12	0.12	—	0.010	8.5	1.84
Acetone	—	0.23–1.4	0.11	—	0.0033	23.6	2.80
Methyl alcohol	—	0.7	0.14	—	0.0050	7	1.68
o-Dichlorobenzene	700	3.4	—	—	—	—	2.25
Nitrobenzene	2700	1.6	—	17	0.020	79	3.90
Benzene	—	—	0.24	0.24	0.0065	45	0
Xylene	—	—	0.11	0.47	0.0064	63	0.52

By essentially the same method as for the Kerr effect, Gans\(^{65}\) investigated the inertia of the Faraday effect. As can be seen from Fig. 34, Gans’s apparatus differs from the preceding one only in that the Kerr condensers \(K_1\) and \(K_2\) are replaced by coils, into which the substance under investigation is introduced.

One should distinguish three different kinds of Faraday effect—the rotation of the plane of polarization in a magnetic field: the diamagnetic Faraday effect, accompanying the Zeeman effect; the temperature-dependent paramagnetic effect, caused by the influence of the magnetic field on transition probabilities; and, finally, the paramagnetic effect that is independent of temperature and is associated with the establishment of orientations of electronic orbits or spins. The first two effects must—

must follow the field without inertia. In the case of a paramagnetic effect, however, the relaxation time must correspond to the time necessary for the “turning over” of the magnetic moments, i.e., it must be approximately equal to the time between two collisions of molecules. In a liquid this time must be of the order of \(10^{-11}\)—\(10^{-12}\) sec.

Fig. 34. Diagram of Ganle’s apparatus for investigating the inertia of the Faraday effect.

In accordance with this, Ganle carried out his investigations for three groups of substances. The diamagnetic Faraday effect he observed in carbon disulfide, water, ether, benzene, toluene, nitrobenzene, glycerin, carbon tetrachloride, and Faraday (lead) glass. As a typical representative of substances with a paramagnetic effect dependent on temperature, titanium tetrachloride was investigated. The temperature-independent paramagnetic effect was studied in concentrated solutions of cerium nitrate \([\mathrm{Ce}(\mathrm{NO}_3)_3\cdot 6\mathrm{H}_2\mathrm{O}]\), where it is due to the arrangement of the electron shells of the ion \(\mathrm{Ce}^{+++}\), and in solutions of \(\mathrm{FeCe}_3\cdot 6\mathrm{H}_2\mathrm{O}\) and \(\mathrm{K}_3\mathrm{Fe}(\mathrm{CN})_6\), in which the relaxation time is determined by the time of “turning over” of the spin. In accordance with what was to be expected theoretically, in none of these cases could any inertia of the effect exceeding \(0.5\cdot 10^{-9}\) sec be detected.

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