NEW INSTRUMENTS AND METHODS OF MEASUREMENT

A NEW METHOD FOR STUDYING RELAXATION PROCESSES AND ITS APPLICATION TO THE STUDY OF SOME PHYSICAL PHENOMENA

N. A. Tolstoy and P. P. Feofilov

I. FOUNDATIONS OF THE METHOD

§ 1. Introduction

Every physical system that has been in an equilibrium state under the action of certain external conditions, and that passes, when the latter change, into a new equilibrium state, accomplishes this transition over some period of time. The study of transient processes (relaxation processes), along with equilibrium states, makes it possible to obtain much more complete information about the structure and nature of physical systems. Indeed, when a system is in equilibrium, the details of its mechanism turn out to be, to a considerable extent, “balanced” and hidden in a certain averaged picture. Conversely, if the mechanisms possess different kinetic properties, then during the transient process their “balance” may be disturbed, which in a number of cases may lead to a distinct revelation of the details of these mechanisms. It is therefore natural that the development of methods making it possible to study the instantaneous nonequilibrium states of a physical system in the course of a relaxation transition is of considerable interest for many areas of physical research.

For reasons connected with experimental technique, it is expedient to subdivide the diversity of physical relaxation phenomena into the following domains: a) the domain of phenomena occurring in times of \(10^{-7}\) sec and shorter; for brevity we shall call these phenomena very fast; b) the domain of phenomena occurring in times of \(10^{-1}\) sec and longer; we shall call these phenomena slow; c) the domain of phenomena occurring in the intermediate time interval—

time, i.e., between \(10^{-7}\) and \(10^{-1}\) sec.; we shall call these phenomena fast.

The method for investigating very fast processes is based chiefly on the study of the amplitude and phase relations between the high-frequency sinusoidal process exciting the system and the response process of change in the state of the system. It should be emphasized that, in cases where the mechanism of relaxing systems is at all complex, the method of investigation by means of exciting sinusoids leads to difficulties in the treatment of experimental results that are hardly surmountable.

The investigation of slow processes in most cases presents no methodological difficulties: it reduces either to visual recording of the readings of instruments indicating the transient states of the system at various moments of time, marked by a stopwatch, or to continuous recording of transient phenomena by means of recording instruments, loop oscillographs, etc.

The present article is devoted to questions of methodology for the investigation of fast relaxation processes\(^{1}\) (\(10^{-7}\)—\(10^{-1}\) sec.*). In this time interval very numerous and important physical phenomena take place. It is enough to point to phenomena occurring in semiconductors (photoconductivity, the barrier-layer photoeffect, unipolar conductivity, etc.), in crystal phosphors (photo-cathodo-, X-ray- and radioluminescence), in certain dielectrics (relaxation polarization), in colloidal solutions (phenomena of anisotropy induced by external fields, electrophoresis, etc.), in electrolytes (the Becquerel photoelectric effect), in solid and liquid solutions (long-duration fluorescence of certain compounds), in gases (gas-discharge processes accompanied by the formation of metastable atoms and ions), at surfaces (ion evaporation processes), in mechanically fast-relaxing systems (stress relaxation in polymers, etc.), and so on.

Considering the very diverse methods that have been used for studying the relaxation of such phenomena, one may note in them elements of methods used for the investigation of very fast and slow relaxation phenomena. Thus, for example, sinusoidal exciters were often used to excite a system relaxing in the indicated time interval. What was said above about the difficulty of deciphering the mechanism of the system from a periodic curve characterizing the response instantaneous states of the system in this case applies fully, of course, also to the field of the fast processes of interest to us. However, if in the case

*) In practice we have so far been able to cover the interval \(10^{-5}\)—\(10^{-1}\) sec. for light excitation and \(10^{-5}\)—\(10^{-1}\) sec. for electrical excitation. See also § 23.

very rapid relaxation processes, the problem of investigation is in itself so difficult and delicate that one can be reconciled to obtaining only very rough characteristics of the process (the mean relaxation time of the system); whereas in the case of rapid processes we have the right to aspire to a more detailed knowledge of the character of the relaxation process under study. Under a sinusoidal mode of excitation the external conditions, into equilibrium with which the system should come, change continuously; moreover, the character of the relaxation process by which the system responds to the changing external conditions changes continuously and cannot, by its very nature, serve as a clear characteristic of the relaxation properties of the system. In order to reveal these relaxation properties in their purest and most unconfused form, it is necessary to make the excitation conditions as simple as possible. It is perfectly clear that the simplest kind of excitation must be a transition from one constant external condition (excitation condition) to another constant condition, carried out instantaneously or, at least, so rapidly that during the transition time the change in the state of the system may be neglected. The use of such a kind of excitation, schematically represented in Fig. 1, a, is quite trivial for the case of slowly establishing systems. For rapid relaxation processes it is advisable to use periodic rectangular pulses (P-pulses) as the exciting factor (Fig. 1, b). Then, at a sufficiently high repetition frequency of the P-pulses, the relaxation pattern can be studied, with the use of an appropriate method, as a static one. This, of course, applies only to the case of reproducible processes, i.e., those which can be identically repeated an indefinite number of times. It is clear that such processes as explosions, most chemical reactions, etc., must in this sense be excluded from consideration.

Fig. 1.

Exciting P-pulses can be obtained by various methods, many of which are based on the use of the very convenient means available to modern experimental physics. This will be discussed in greater detail in the section devoted to particular methods.

The study of the relaxation process itself (both direct and inverse) excited by P-pulses is most expediently carried out with the aid of a cathode oscillograph, on whose screen the curve depicting the relaxation process, at a sufficiently high frequency of the P-pulses, excluding flicker, is presented to the eye as a static picture. The use of a cathode oscillograph presupposes, of course, the transformation of the relaxation process under study into an adequate electrical process. Such a transformation (necessary, of course, only when the process under study is not electrical in itself) can be effected chiefly by photoelectric methods. Methodological details will likewise be set forth in the section devoted to special methods. Having obtained on the screen of the oscillograph a curve depicting the course of the relaxation process under study in time, we approach our main task—the study of the law of the relaxation process and of the parameters of this law. Obtaining on the screen of the oscillograph the curve of the process under study in its “natural” form, i.e., in the coordinates “phenomenon—time,” is connected with the linearity of the sweep of the oscillograph’s electron beam. However, the curves depicting the rapid relaxation processes that interest us will in most cases have a form highly unfavorable for solving the main task posed. Indeed, when the curves are represented on a linear time scale, the main part of the curve, reflecting the greater part of the entire relaxation process, occupies a relatively very small portion of the time axis and is therefore presented in the form of an almost vertical branch. Conversely, the final stages of the curve, reflecting the asymptotic approach of the system to the stationary state—stages during which only an insignificant residual part of the overall process takes place—turn out to be undeservedly greatly stretched by the sweep (Fig. 2). In general, one may assert that a linear time scale is alien to the specific character of relaxation processes. It is far more expedient to use not a linear scale, but such a “functional” scale—

Fig. 2.

scale of time, in which at first “time” flows rapidly (the sweep is carried out at high speed), and then gradually slows its course (the sweep speed asymptotically tends to zero). In this case the steepness of the curve on the oscilloscope screen will vary within much smaller limits. It is obvious that, as the mechanism carrying out such a functional sweep, one may use, for example, some previously known relaxation process of electrical character, or a process that can be converted into an electrical one. The study of unknown relaxation processes in asymptotic time coordinates, created by previously known relaxation processes (or by processes imitating relaxation ones), has extremely great advantages and leads to two basic methods for solving the main problem indicated above, namely: 1) the method of straightening and 2) the method of partial times.

§ 2. The method of straightening

Let the form of the law of the relaxation process under study be known a priori up to one or several parameters, whose determination constitutes the main problem. Then for the sweep it is natural to use the same law, whose parameters can be varied arbitrarily in a continuous manner. It is obvious that, with a proper choice of the parameters of the sweep law (when they become equal to the corresponding parameters of the law of the process under study), the curve on the oscilloscope screen will degenerate into a straight line. Having measured the values of the parameters that give “straightening,” and substituting them into the formula expressing the law of the process, we completely solve the main problem. Let us note that the slope of the straight line on the screen, determined by the scales along both axes, plays no role in these measurements; only the very fact of straightening is important. Deviations of any line on the oscilloscope screen from straightness are visually easily recognized, which makes it possible to carry out measurements of this kind with sufficient accuracy. On the other hand, if in these measurements it turns out that for no values of the sweep parameters is it possible to obtain a straight line, then this indicates that the law of the process assumed to be known a priori in fact does not hold. It is clear that checking the applicability of the assumed law and establishing deviations from it are in many cases very important.

In an extremely large number of cases relaxation processes follow the simplest exponential law

\[ y = y_0\left(1 - e^{-t/\tau}\right) \quad \text{(for rise)} \tag{1} \]

or

\[ y = y_0 e^{-t/\tau} \quad \text{(for decay),} \tag{1′} \]

corresponding to the condition that the rate of approach to the equilibrium state \((-dy/dt)\) is proportional to the degree of removal of the system from this state \((y)\). This condition is satisfied by spontaneous and certain other processes that obey the equation

\[ \frac{dy}{dt} = - \frac{1}{\tau} y . \]

An exponential sweep

\[ x = x_0(1 - e^{-t/\tau}) \quad \text{(forward stroke),} \tag{2} \]

\[ x = x_0 e^{-t/\tau} \quad \text{(return stroke),} \tag{2′} \]

which rectifies such processes, can be produced by a very simple and convenient method, which we shall describe in some detail, since, as will be seen below, this type of sweep can be used successfully in the study not only of exponential processes but, in general, of any relaxation processes.

To obtain an exponential sweep we use the process of charging a capacitor \(C\) from a source of constant voltage \(V_0\) according to the circuit of Fig. 3, and the process of discharging this capacitor through a resistance \(R = R_1 + R_2\) after the voltage supply is suddenly interrupted. In order that the charging and discharging processes be repeated periodically, the circuit is fed by square-wave voltage pulses. These pulses are synchronized with the square-wave pulses that excite the relaxation system under study. If the system under study is excited by electrical square-wave pulses, it is naturally most advisable to use them also for supplying the sweep, provided that the power of the original square-wave pulses is sufficiently large so that the consuming devices do not affect one another and do not alter the pulse shape. (Sometimes, in order to avoid the latter, one has to use isolating tube circuits.) By connecting the capacitance \(C\) to the horizontal plates of the oscillograph through an appropriate amplifier, we obtain a sweep of the electron beam in the horizontal direction according to an exponential law; during the process of charging the capacitor the beam moves in one direction (for example, from left to right), and during discharge—in the opposite direction (from right to left).

Fig. 3. Circuit for obtaining an exponential sweep.

The abscissa of the electron beam follows the equations

\[ x=x_0\left(1-e^{-t/RC}\right)\quad \text{—(charging),} \tag{3} \]

\[ x=x_0 e^{-t/RC}\quad \text{—(discharging),} \tag{3'} \]

which, as is known, express the change of voltage across the capacitance \(C\) under the conditions of the circuit in Fig. 3. If the process under study is exponential and follows equations (1) and (1′), then, under the condition of the indicated synchronization of the sweep with the process, process (1) will be swept by process (3), and process (1′) by process (3′), and the trajectory

Fig. 4. Exponential sweep of an exponential process.

of the electron beam on the oscilloscope screen will be a loop similar to that shown in Fig. 4. It is not difficult to see that the branches forming the loop will be parabolas:

\[ 1-\frac{y}{y_0}=\left(1-\frac{x}{x_0}\right)^{RC/\tau} \tag{4} \]

and

\[ \frac{y}{y_0}=\left(\frac{x}{x_0}\right)^{RC/\tau}. \tag{4'} \]

When \(RC=\tau\), the parabolic branches of the loop degenerate into two straight-line segments merging with one another. Thus, in measuring the time constant \(\tau\) of exponential relaxation processes, we may use not only the criterion of straightening, but also the still more sensitive criterion of disappearance of the loop. The accuracy of determining \(\tau\), as experience shows, may under favorable conditions reach 1%. If the capacitance \(C\) is graduated in farads and the resistance \(R\) in ohms, then the value \(RC\) corresponding to straightening gives us \(\tau\) directly in seconds.

The variety of elements included in circuits (including nonlinear ones, for example complex electron tubes) available to radio engineering makes it possible to implement more complex functional sweeps as well, which could find application in the analysis of more complex relaxation curves. As an example, one may

indicate a hyperbolic sweep (Fig. 5), which straightens hyperbolas of the form

\[ y=\frac{y_0}{1+at}. \tag{5} \]

The resistance \(r\) periodically changes according to a linear law with time from 0 to \(r_2\) over the period \(t_0\), equal to the period of the exciting \(\Pi\)-pulse. It is obvious that the voltage across the resistance \(r_1\) in such a circuit will vary according to the law

\[ V=\frac{V_0}{1+\frac{r_2}{r_1}t}. \tag{6} \]

Changing \(r_1\), we can control the parameter characterizing the steepness of the hyperbola.

For obtaining a functional sweep that straightens a quadratic hyperbola of the form

\[ y=\frac{y_0}{(1+at)^2}, \tag{7} \]

one may use, for example, one of the following methods:

Fig. 5. Circuit for obtaining a hyperbolic sweep.

\[ \Lambda_1,\ \Lambda_2 \text{ — separating tubes} \]

\[ R_{11}+R_{12}=R_1,\qquad R_{21}+R_{22}=R_2 \]

Fig. 6. Circuit for obtaining a biexponential sweep.

1) electrical differentiation of the sweep curve (5), transforming it into a curve of type (7);

2) electrical integration of the curve being studied, of the form (7), transforming it into the curve (5);

3) squaring the sweep curve (5) with the aid of a mixer radio tube capable of multiplying the instantaneous values of the signals applied to its grids;

4) doubling the circuit shown in Fig. 5, in which, instead of the voltage \(V_0\), a voltage varying according to law (6) is applied.

Another example may be a biexponential sweep capable of straightening processes that are a superposition of two exponential processes:

\[ y=\frac{y_0}{A+B}\left(Ae^{-t/\tau_1}+Be^{-t/\tau_2}\right). \tag{8} \]

The circuit of such a sweep is, in essence, a doubled circuit of Fig. 3 and can be implemented as shown

on Fig. 6. The quantities $R_1C_1$ and $R_2C_2$ determine the time constants, respectively $\tau_1$ and $\tau_2$, while the ratio of the constants $A$ and $B$, which determines the relative weight of each of the exponentials, is given by the ratio of the resistances $r_1$ and $r_2$. For straightening with the aid of such a circuit it is, obviously, necessary to select three parameters.

In a number of cases it may prove expedient to use, as sweeps, not “electrotechnical processes,” but more complex physical relaxation processes whose parameters may vary under the influence of external conditions (temperature, excitation intensity, etc.), for example, processes of luminescence relaxation, photoconductivity, and the like. The use of “physical” sweeps may be advantageous in the study of relaxation processes in related systems.

§ 3. Method of partial times

Since, as experience shows, relaxation laws may have a very diverse character, the construction of complicated functional sweeps for each case is, generally speaking, a rather difficult matter and is not always practically justified. We shall show that, by changing the approach to the solution of the basic problem, one may, while restricting oneself to a simple exponential sweep, attain its complete solution by another, sufficiently simple, method. We shall distinguish two cases: a) the relaxation law is known up to the parameter being sought, and b) the relaxation law is completely unknown, and the solution of the basic problem consists in constructing an empirical curve in the form of a graph or table.

Fig. 7. Growth curve in an exponential sweep for various $c$. Falling curves are omitted.

a) “Half-life time.” Obviously, in order to find the unknown parameter $p$ of a relaxation law of known form $F(p,t)$ (we assume here that there is only one unknown parameter), it is sufficient to find a particular value of the function $F$ at some instant of time, or to find the time corresponding to some prescribed instantaneous value of the function $F$. From the particular value thus found we can always determine the sought parameter $p$, and this exhausts the problem.

Let us show how this problem can be solved experimentally with the aid of an exponential sweep. Place the ends of the investigated branch of the loop (i.e., the branch that corresponds

of the growth curve or of the decay curve of the process) at the diagonally opposite corners of the rectangle marked out on the scale grid covering the oscilloscope screen (Fig. 7). When the sweep parameter \(\tau=RC\) is varied, the shape of the curve will change, and the curve will intersect the vertical straight line \(AB\), which divides the rectangle in half, at different heights. Let us make the curve pass through the center of the rectangle, i.e., intersect the median line \(AB\) at the middle of its height. Then, if the vertical scale is linear and represents without distortion the instantaneous states of the relaxing system, or, what is the same thing, the instantaneous values of the function \(F(p,t)\) (which we have normalized to the height of the rectangle \(AB\)), then the instantaneous value of the function at this point is

\[ F(p,t_m)=\frac{F(p,t_0)}{2}\simeq \frac{F(p,\infty)}{2}, \tag{9} \]

where \(t_0\) is the duration of the exciting \(\Pi\)-pulse [we assume here that during this time the curve \(F(p,t)\) practically reaches its stationary value \(F(p,\infty)\)], and \(t_m\) is the time during which the curve grows to one half of its maximum value. It is clear that \(t_m\) is equal to the time during which the electron beam, moving from left to right according to an exponential law, traverses the first half of its path. This time is equal (in the case when the sweep beam during the time \(t_0\) practically reaches its maximum deflection, i.e., if \(t_0 \gg t_1\))

\[ t_m=\tau \ln 2. \tag{10} \]

Thus,

\[ F(p,\infty)=2F(p,\tau \ln 2). \tag{11} \]

From this equation we find the required parameter \(p\). For a decay process proceeding according to the law \(f(q,t)\), we obtain, in a completely analogous way, the equation

\[ f(q,0)=2f(q,\tau \ln 2), \tag{12} \]

from which the required parameter \(q\) is found.

The physical meaning of the time \(t_m\) is obvious: it is simply the “half-life” of the process under study. It is clear that in a number of cases the value of this characteristic of the process may be of interest even when the law of the process under study is completely unknown.

The foregoing refers to the simple limiting case in which, during the time \(t_0\), both the studied growth and decay processes and the sweep processes practically reach completion (which corresponds to complete stopping of the beam in the corners of the rectangle). The general case, taking into account incomplete relaxation, will be considered below.

b) Partial times. Let the form of the law characterizing the relaxation curve be completely unknown to us. Then, as we said above, the solution of the fundamental problem is the

construction of the curve under study in the form of a graph or a table. In other words, we must indicate a series of particular values of the ordinate of the curve corresponding to definite instants of time. It would seem that, for this purpose, it is sufficient to obtain the relaxation curve on a natural scale and, using the coordinate grid (either on the oscilloscope screen itself or after photographing the curve), to determine the values of the ordinate at various instants of time. With such a formulation of the problem we can, in essence, answer only the question of what the values of the ordinate of the curve are at various instants of time taken at equal intervals; moreover, the smallest practically possible magnitude of the latter is, generally speaking, insufficient for resolving in time the initial part of the curve (for, as we have already indicated above, sufficiently rapid relaxation curves are distinguished by a very steep initial decline) and is excessively small for the final parts of the curve. Thus, by taking as a basis a uniform time scale, we obtain an inconvenient and unnatural scale for studying the relaxation process.

Something quite different is obtained if, in this case as well, when it is necessary to determine an empirical curve “point by point,” we make use of an exponential sweep. Indeed, by varying at will the parameter of the sweeping exponential \(\tau\), we can, while leaving the sweep amplitude unchanged, increase at will the “resolving power” of the initial stages of the sweep, corresponding to the fastest initial stages of the process. The method of solving our problem then proves to be the following: we arrange for the curve of the process under study, in the exponential sweep, to go from one corner of a rectangle (defined by a transparent scale grid on the oscilloscope screen) to the other opposite corner, and, by varying the sweep parameter \(\tau\), we make the curve under study pass through various points of the vertical straight line \(AB\), which divides the rectangle in half (Fig. 7). Obviously, if the curve passes, for example, through a point \(a\), located at 70% of the total height of the median straight line \(AB\), this means that in the time

\[ t_m=\tau\ln 2 \]

the ordinate expressing the state of the system reaches 70% of its maximum value. Having compiled a table of values of the parameters \(\tau\) or of the quantities \(t_m\) (which we shall call partial times) corresponding to a set of specified changes in the ordinate of the curve under study, we may consider the stated problem solved. The values of the ordinates of our curve are most naturally chosen at equal intervals; if 5% is taken as the interval, then the measured curve can be constructed from 20 points, which, of course, is practically quite sufficient. It should be emphasized that, with this method of obtaining the relaxation process, we proceed from the subdivision of the physical process itself into equal stages,

that, obviously, is much more natural than the same subdivision of time.

c) Determination of the position of maxima and minima. As we shall see below, relaxation curves in some cases possess anomalous features; in particular, they have maxima and minima. If these extrema are reached at the very beginning of the process, then determining the time at which the extremum is reached presents the same difficulties, indicated above, that are generally encountered in the study of the initial stages of relaxation processes. The use of the method of partial times makes it possible in this case to determine the position of the extremum with great accuracy. In fact, by choosing the scanning $\tau$ so that the extremum falls on the median straight line (see, for example, Fig. 8), we directly determine, by formula (10), the partial time of the extremum.

§ 4. Segmental Regime

In practice we often encounter the case where, during the time of one P-pulse ($t_0$), the process under study or the exponential that scans it does not manage to reach a stationary value. (An increase of $t_0$, allowing a larger part of the curve under study to be covered, is limited by two circumstances: first, it is necessary to reproduce the relaxation process sufficiently often so that the curve on the oscilloscope screen does not flicker; second, in most cases an increase in the duration of the P-pulse is accompanied by an increase in the rise and fall time of the pulse, in other words, by a decrease in the steepness of its front, which leads to distortions in determining the initial stages of the process.) Let us consider the features introduced by operation under conditions of the absence of complete relaxation or, as we shall say, under conditions of the segmental regime.

Fig. 8. Determination of the partial time of an extremum by shifting the extremum onto the median line.

a) Segmentation of scanning exponentials. Let the electron beam be scanned along the $X$ axis according to the law

$$ x=x_0 e^{-t/\tau}. \tag{13} $$

At $t=0$ the coordinate of the beam is equal to $x_0$, and at $t=t_0$ $x=x_0 e^{-t_0/\tau}$; in other words, the beam traverses the distance $\Delta x=x_0(1-e^{-t_0/\tau})$. The beam traverses half of this distance $\Delta x/2$ in the time $t_m$. In this case its coordinate is equal to

$$ x_0-\frac{\Delta x}{2}=x_0 e^{-t_m/\tau}. $$

From this we obtain

\[ t_m=\tau \ln \frac{2}{1+e^{-t_0/\tau}} . \tag{14} \]

For the reverse course of the sweep, taking place according to the law
\(x=x_0(1-e^{-t/\tau})+x_0e^{-t_0/\tau}\), the same, of course, will hold.

Formula (14) is a generalization of formula (10). Obviously, for \(t_0 \gg \tau\), (14) turns into (10). Formula (14) usually has to be used in determining the final stages of relaxation curves in the method of partial times.

To calculate the correction factor in formula (14), it is necessary to know the quantity \(t_0\), i.e., the duration of the П-pulse. One may indicate another, more convenient, way of introducing the correction, based on determining the magnitude of the segmentation of an exponential sweep. Indeed, it is not difficult to show that if the sweep \(\tau\) is taken sufficiently large so that the sweep amplitude decreases by a factor of \(b\) (which is easily determined on the oscilloscope screen by switching the \(RC\)-circuit to an obviously fast, unsegmented sweep), then the quantity \(t_m\) is determined from the simple formula

\[ t_m=\tau \ln(1+b). \tag{15} \]

b) Segmentation of the curves under study. If during the time \(t_0\) the rising branch of the process being studied does not have time to reach saturation, and the decaying branch—complete relaxation, then, after establishment of the regime as a result of repeated П-shaped excitation, the rise will proceed from some constant value \(y_1\) to another constant value \(y_2\); correspondingly, the decay curve will begin from the value \(y_2\) and fall to the value \(y_1\). (We assume that the value corresponding to complete saturation is equal to \(y_0\), and the value corresponding to complete relaxation is equal to 0. See Fig. 9.)

Fig. 9. Segmentation of curves.

If the relaxation of the system is single-valued (see §5b), then the resulting “segments” are pieces of the curves that would be obtained in the regime of complete saturation and complete relaxation, and one may assume that the observed segment of the rising curve \(F(t)\) is

part of the complete curve, beginning at some point \((y = 0)\), removed from the point \(t = 0\) by a time interval \(t'\) (Fig. 9). In exactly the same way, the observed segment of the decay curve \(f(t)\) may be continued to the point \(y = y_0\), corresponding to complete saturation and removed from the point \(t_0\) by a time interval \(t''\). The condition for matching the growth and decay segments will obviously be the condition:

\[ F(t') = f(t'' + t_0), \]

\[ F(t' + t_0) = f(t''). \]

Solving this system of equations, we shall obtain the times \(t'\) and \(t''\) as functions of \(t_0\) and of the parameter \(a\), on which the functions \(F(t)\) and \(f(t)\) depend. The equations of the observed segments obviously have the form:

\[ y = F(a, t + t') \qquad \text{— growth} \]

and

\[ y = f(a, t + t'') \qquad \text{— decay,} \]

where \(t\) in the first equation is measured from zero, and in the second from \(t_0\). Having determined the “half-life time” of the segments, we can, from the equations

\[ 2F(a, t_m + t') = F(t') + F(t_0 + t') \tag{16} \]

and

\[ 2f(a, t_m + t'') = f(a, t'') + f(a, t_0 + t'') \]

find the required parameter \(a\).

This, generally speaking, rather cumbersome method of determining the complete curve from its segment is applicable, of course, only in the case when the character of the curves is known a priori with accuracy up to a parameter (it being assumed here that the relaxation is single-valued). It should also be borne in mind that solving the equations may prove very difficult.

In practice, however, in most cases it is much more convenient to make use of some experimental device that makes it possible to determine the position of the observed segment relative to the stationary states corresponding to the absence of excitation and to the exciting action of constant magnitude. For this it is sufficient, without disturbing the course of the process under investigation, to imitate for a short time the state of complete relaxation attained by the system after a greater or lesser time following cessation of the excitation. From this “zero” level we can then measure the positions of the ends of the observed segment and the position of the level corresponding to complete excitation (Fig. 10). In the case where the relaxation of the system is single-valued—which, as will be shown below, can be specially verified—we shall know exactly to what part of the complete,

of the relaxation curve corresponds to the observed segment. Some methods for the practical realization of devices simulating the state of complete relaxation will be described below.

Having determined the form of the segment under study and found its position relative to the levels of complete excitation and complete relaxation, we can extrapolate the relaxation law thus found to the entire relaxation curve only in the case where the relaxation is

Fig. 10. Marks of the zero level on segmented curves.

single-valued. Otherwise the segment of the decay curve cannot be continued to the level of complete saturation, and the segment of the growth curve cannot be continued to the level of complete relaxation. In this case the observed segment is not part of the complete relaxation curve, but represents an independent process determined by the specific form of the preceding (reverse) segment or, ultimately, by the segmentation regime, i.e. by the value \(t_0\) and the amplitude of the exciting pulse. By varying these quantities, we can make the segments under study begin at one and the same point, i.e. proceed from one and the same instantaneous state, and observe whether these segments will be identical or different. In doing so, we shall obviously be comparing the behavior of systems that have arrived at the given state with different prehistories. If the resulting segments remain identically the same, we are dealing with a system whose relaxation is single-valued; if we detect a difference in these segments in even one case, this will indicate the non-single-valuedness of the system’s relaxation (see § 5c).

In practice one often encounters the case in which one of the branches of the process (most often the growth) under all experimental conditions reaches a stationary value in time \(t_0\), while the other branch is segmented. It is not difficult to see that in this case the boundaries of the second branch of the curve are determined quite exactly: the curve goes “from the beginning” and breaks off at a certain distance from the end, which is determined with the aid of the device indicated above.

§ 5. Phenomenological characterization of a relaxation process

Very often, when beginning the study of the relaxation properties of a physical object, we do not have a clear idea of the mechanism governing the behavior of the object. Then, in studying the relaxation process, we set ourselves the task not of refining the details of a known mechanism and not of obtaining quantitative data concerning the latter, but of trying to determine the basic qualitative character of this mechanism. For this purpose it is essential to be able, from the very beginning, to indicate the main phenomenological features of the relaxation process by introducing certain characteristics of relaxation that do not depend on any hypotheses about the mechanism of the object being studied. Having studied the physical object from these points of view, we narrow the field for choosing hypotheses about the nature of the object. Let us indicate a number of such a priori characteristics of the relaxation process. Many of them, of course, are quite obvious.

a) Monotonicity. In some cases the growth curve or the decay curve may turn out to be nonmonotonic. Thus, for example (see § 9), the brightness of the yellow emission band of some crystal phosphors of the ZnS·Mn type reaches, during the ignition process, a value several times exceeding the stationary value of the brightness (Fig. 25, b).

Similar features are encountered in the process of the growth of the photo-emf of some photoelements and in the process of the afterglow of certain lines in the relaxation of neon emission (Fig. 36, b). These features clearly illustrate the considerations given at the very beginning of this article. Of course, in order to note these features, there is no need to use exponential sweep; however, if it is desired to determine the position of a maximum or minimum on the curve under study, then exponential sweep, as was indicated in § 3 c, makes it possible to do this in a very simple way.

Along with phenomena of nonmonotonicity in the course of the relaxation curve itself, one may note features consisting in a nonmonotonic change of the derivative. An example of such a process may be furnished by the decay of some crystal phosphors. This feature, like the preceding one, clearly indicates the presence of at least two different, simultaneously acting mechanisms underlying the system and governing its kinetics.

b) Symmetry. If the decay curve \(f(t)\) can be expressed through the growth curve \(F(t)\) by the formula

\[ f(t)=1-F(t) \]

(assuming that both functions are normalized to unity), then the loop on the oscilloscope screen will have the following property:

for any \(\tau\) of the development the branches of the loop will intersect the median straight line at equal distances from the center (Fig. 11). The fact that this condition is satisfied, in other words, that the symmetry criterion is satisfied, can be established extremely quickly without a detailed study of the curves. Verification of the symmetry of the curves under study may serve in a number of cases for choosing between hypothetical mechanisms of the process. Thus, for example, if a hypothetical mechanism presupposes the coexistence of several (perhaps very many) independent symmetric exponential processes (i.e., processes of the type \(a_n e^{-t/\tau_n}\) for decay and \(a_n(1-e^{-t/\tau_n})\) for rise), then the resultant decay and rise curves must be symmetric. On the contrary, the curves may be asymmetric in the case where they are based on a hypothetical mechanism presupposing the existence of a single, even relatively simple, process. Thus, for example, the brightness of luminescence in the process of the decay and flare-up of phosphorescence, following the simplest bimolecular mechanism of recombination without the participation of trapping levels, would have to change according to asymmetric laws:

Fig. 11.

\[ I \sim \frac{1}{(1+at)^2} \;—\; \text{decay}, \tag{17} \]

\[ I \sim \operatorname{th}^2(at) \;—\; \text{flare-up}. \tag{18} \]

Let us note that formula (18) leads to a curve with a nonmonotonically changing derivative (at first the steepness of the flare-up curve increases, and then decreases).

c) Uniqueness of relaxation. The form of the relaxation curve depends on the strength of the exciting factor or on the level of excitation of the system. All relaxation systems may be divided into 2 classes: 1) systems possessing unique relaxation, 2) systems possessing non-unique relaxation. The relaxation process in “unique” systems is uniquely determined (for the decay branch) by the instantaneous state of the system, beginning from which the process is considered by us; for the rise branch the process is uniquely determined by the instantaneous state of the system and by the strength of the exciting factor. The relaxation process in “non-unique” systems depends not only on the instantaneous state of the system, but also on the way in which the system arrived at this state, i.e., on the prehistory of the system. Thus, for example, a system, relax-

relaxing from the excitation level \(y_1\) (curve I in Fig. 12), behaves in the state \(y\) differently than the same system relaxing from the level \(y_2\) (curve II); the curve of its relaxation may have a derivative greater or smaller than the derivative of the curve corresponding to the first case. A single-valued system, however, will give, at different excitation levels, curves that merge with one another beginning at the point \(y\).

The single-valuedness of relaxation is connected with an important physical circumstance. The point is that the usually observed physical

Fig. 12.

relaxation process is a sequence of nonequilibrium macroscopic states, each of which is an average of a certain instantaneous microscopic picture. Each macroscopic state can be realized, in the simplest cases, by only one, but generally speaking by many, microscopic “distributions” (for example, the distribution of the relative number of electrons in different states or over different levels). Processes starting from different macroscopically identical microstates will, after some time, generally lead to microstates averaged into different macrostates. On the contrary, where micro- and macrostates are in one-to-one correspondence, processes starting from two identical macrostates will identically coincide at all subsequent moments of time. It follows that if we establish experimentally a non-single-valuedness of relaxation, then it is obvious that the system certainly possesses a sufficiently complex mechanism, in which “redistributions” of one kind or another take place. If, however, single-valuedness of relaxation follows from experiment, then we almost certainly are dealing with a system possessing a simple mechanism. In a number of cases the criterion of single-valuedness of relaxation makes it possible to decide the question of the applicability of one or another

theory. Thus, for example, the simple bimolecular theory of phosphorescence (without taking trapping levels into account) leads, as is not difficult to show, to an unambiguous relaxation*). But the process of decay of phosphors, as experience shows, is, generally speaking, a non-unique process, and therefore this circumstance alone is sufficient to conclude that such a theory cannot completely describe the real mechanism of crystal phosphors.

As a second example, let us consider a more general relaxation law, known in the theory of phosphorescence under the name of Becquerel’s law,

\[ y=\frac{y_0}{(1+at)^\alpha}. \tag{19} \]

Let us compare the “Becquerelian” relaxation processes that start from different saturation states \(y_1\) with the same process counted from the moment \(t_1\) when \(y\) in (19) has reached the value \(y_1\) (Fig. 12).

Then, putting \(t^*=t-t_1\), we have:

\[ y=\frac{y_0}{[1+a(t^*+t_1)]^\alpha} =\frac{y_0}{(1+at_1)^\alpha\left(1+\frac{a}{1+at_1}t^*\right)^\alpha}. \]

Denoting

\[ \frac{y_0}{(1+at_1)^\alpha}=y_1 \quad \text{and} \quad \frac{a}{1+at_1}=a_1, \]

we obtain

\[ y=\frac{y_1}{(1+a_1t^*)^\alpha}. \]

This means that shifting the origin of counting on the Becquerelian hyperbola leads to a hyperbola of the same degree, but with a different coefficient \(a\). Since \(a_1\) and \(y_1\) are connected by the relation

\[ \sqrt[\alpha]{\frac{y_1}{y_0}}=\frac{a_1}{a}, \tag{20} \]

we can formulate the condition of uniqueness of the Becquerelian process as follows: if, upon changing the initial level of the relaxation process, we obtain a hyperbola of the same degree \(\alpha\), and if the parameter \(a\) is proportional to the root of degree \(\alpha\) of the magnitude of the initial level \(y_1\), then such a process is unique.

From (20), in particular, it follows that for the simplest bimolecular phosphorescence (17), for which the initial intensity is proportional to the intensity of the exciting light \(E\), the parameter

*) This is connected with the fact that the state of the system is determined in this theory only by the number of electrons in the conduction band and by nothing more.

\(a\) must depend on the intensity according to the formula

\[ a=a_0\sqrt{E}. \tag{21} \]

d) Instantaneous relaxation time. After obtaining a table of partial times characterizing the relaxation curves, we have the possibility of reproducing the latter graphically on any scale. However, relaxation curves represented on a “natural” scale (i.e., in “phenomenon—time” coordinates) are for the most part very inexpressive. They are all more or less rapidly falling, usually monotonic, curves, the differences in the form of which are very difficult to characterize empirically, unless one succeeds in finding analytical formulas corresponding to the curves. Indeed, how is one to compare and describe, for example, the two curves shown in Fig. 13, one of which is an exponential and the other a hyperbola?

Fig. 13.

One may introduce special coordinates that give a clear empirical characteristic of relaxation and make it possible to indicate a certain quantitative measure of the relaxation process at each instant of time. Let us plot along the ordinate axis the reciprocal logarithmic derivative of the quantity under study, taken with the opposite sign, i.e., \(-\dfrac{y}{y'}\), denoting it by \(\Theta\), and along the abscissa axis the time \(t\). (The values of \(\Theta\) can be obtained graphically by differentiating the curve \(\ln y, t\).) Let us indicate the physical meaning of the quantity \(\Theta\). The logarithmic derivative of the curve under study shows the rate of decrease of the quantity under study referred to the degree of removal of this quantity from its equilibrium value (i.e., from zero). Thus \(-(\ln y)'\) is the specific rate of relaxation, and \(\Theta\), correspondingly, the specific slowness of relaxation. It is also not difficult to see that \(\Theta\) coincides with the relaxation time \(\tau\) of that exponential \(e^{-t/\tau}\) which at the given instant of time approximates the curve under study. Therefore the quantity \(\Theta\) may be called the instantaneous relaxation time—

…of this process. Obviously, if the curve under study is an exponential, then its instantaneous relaxation time

\[ \Theta=-\frac{1}{(\ln e^{-t/\tau})'}=\tau=\text{const.} \]

is constant and equal to \(\tau\). In this sense the exponential is indeed the simplest relaxation process. As experience shows, relaxation processes are very often observed that are characterized by a linear increase of the instantaneous relaxation time with time

\[ \Theta=A+Bt. \tag{22} \]

It is not difficult to show that the analytical law of such a relaxation is expressed by the formula

\[ y=\frac{y_0}{(1+at)^\alpha}. \tag{23} \]

Here \(a\) and \(\alpha\) are constants, with

\[ a=\frac{B}{A},\qquad \alpha=\frac{1}{B}. \]

Thus, in studying relaxation processes that follow Beckerel’s law (23), the constants \(a\) and \(\alpha\) can be determined from the slope of the straight line \(\Theta(t)\) and from the magnitude of the segment it cuts off on the ordinate axis (Fig. 14).

Fig. 14. Determination of the constants \(a\) and \(\alpha\) from the dependence \(\Theta(t)\).

In the general case, when the relaxation curve is neither an exponential nor a hyperbola, we can describe this curve by the time course of its instantaneous relaxation time. The appearance of sufficiently well-defined rectilinear horizontal or inclined sections on the graph \((\Theta,t)\) makes it possible to note the time intervals during which the relaxation is respectively exponential or hyperbolic. Examples of the use of the graph of instantaneous relaxation time will be given below.

d) A “very fast” process and a slow process. Experience shows that relaxation processes sometimes have an extremely fast initial stage. The rate of relaxation in this stage usually differs by several orders of magnitude from the rate accessible to our method. It is essential to be able to recognize such a process. Experimentally it is established quite simply: if, at the fastest exponential dis…

in the sweep carried out by the instrument there is a part of the curve that does not lend itself to “unfolding,” this means that the rate of the process lies outside the working time interval of the instrument. Moreover, although the “very fast” process and the “fast” process following it*) pass into one another practically smoothly (on the curve in the “natural” scale), with a rapid exponential sweep a characteristic break appears at the “junction” of the curves corresponding to the one and the other process.

A relaxation process not infrequently has, at the end, a stage of very slow change. Such a “tail” falls outside the time interval accessible to our method on the side of long times. The total change of the quantity under study in this final, slow stage can be determined, as was indicated above (§ 4 b), by creating an artificial “zero line.”

II. APPLICATIONS OF THE METHOD

A. LUMINESCENCE

§ 6. General remarks

The fruitfulness of studying establishment processes is especially clearly seen in the example of luminescence. The modern theory of phosphorescence has been built to a considerable degree on experimental results obtained in the study of luminescence-decay processes. In judgments about the mechanism of one or another luminescence process, the criterion of the law of decay is often decisive. However, a theory of crystal phosphors based chiefly on data concerning the remote stages of decay is only a first crude approximation, and for a detailed description of the variety of complex processes occurring during the excitation and afterglow of luminescent objects, a detailed study of the entire process is necessary, especially of its initial stages, where the “balance” of the system mechanism, mentioned at the beginning of this article, is disturbed most strongly**).

*) It is clear that although both processes usually proceed simultaneously from the very beginning, the “very fast” process manages to be completed even before the slower one has managed to bring the state of the system to any change. Therefore one may say (when there is a sharp difference in the rates of the processes) that first the very fast process proceeds from the beginning to the end, and only then does the second, fast, process begin.

**) Thus, for example, Antonov-Romanovskii and Krylova² speak of the transition from the nonequilibrium initial stage of recombination to the equilibrium subsequent stage.

N. A. TOLSTOY AND P. P. FEOFILOV

Among the numerous instruments intended for investigating the decay of luminescence, there are a number of instruments that cover the interval of fast processes of interest to us \((10^{-5}—10^{-1}\ \mathrm{sec})\). First of all, one should mention the widely known Becquerel phosphoroscope and its numerous modifications. Characteristic of instruments of this type is the determination of instantaneous values of the brightness of emission after some interval of time following the cessation of excitation by a light pulse, usually a rather short one. By varying the duration of the time interval between the moments of excitation and observation, one can obtain point by point the entire desired decay curve. It is essential that the measurement of brightness itself must in principle be carried out over some interval of time, during which the true (instantaneous) brightness changes somewhat; this leads to the obtaining of averaged data. Despite the extraordinary simplicity of the principle of the Becquerel phosphoroscope, its complete theory proves to be rather complicated. The theory of phosphoroscopic measurements, developed in the works of Wiedemann³ and Perrin and Delorme⁴, is applicable, as S. I. Vavilov⁵ showed, only to the case of exponential decay and in the general case leads to mathematical difficulties that have not yet been overcome. In addition to the fact that measurement with the aid of phosphoroscopes of the Becquerel type is extremely laborious and that carrying out any broad investigations of the influence of various physical factors on decay processes proves practically impossible, the Becquerel phosphoroscope and its modifications possess a number of substantial shortcomings. One may point to the difficulty of separating “instantaneous” (shorter than \(10^{-6}\ \mathrm{sec}\)) and longer processes, and to the difficulty of studying the processes of the rise of emission. Instruments based on obtaining a spatial sweep of the time course of the decay of emission (Wood’s phosphoroscope⁶, the spark phosphoroscope with the rotating mirror of S. I. Vavilov and V. L. Levshin⁷) are also not free from these shortcomings, which are inherent in all visual and photographic phosphoroscopes.

The development of experimental technique prompted attempts to apply photoelectric methods, with a cathode oscillograph as the recording instrument, to the study of the rise and decay of luminescence⁸˒⁹. Although in a number of cases photoelectric methods are inferior to visual ones with respect to sensitivity, even the simplest arrangement, in which the object under investigation is periodically excited by light pulses and the emission under study is received by a photoelectric device connected to a cathode oscillograph, makes the method objective and frees it from a number of the above-mentioned shortcomings of “classical” phosphoroscopes. However, in this form the method proves to be essentially qualitative, since, as was indicated above, measur-

tion of relaxation curves obtained on the screen of an oscillograph with a linear sweep in time is associated with considerable errors.

A good illustration of the unsuitability of sinusoidal excitation for the study of processes that proceed according to laws of any complexity is the attempt at an oscillographic study of the brightness of the luminescence of phosphors when they are excited by sinusoidally modulated light[^10]. Analysis of the phase and amplitude relations observed in this case between the exciting light and the luminescence led to serious mathematical difficulties and did not yield positive results.

§ 7. Oscillographic phosphoroscope with exponential sweep

The consistent application of the principles of the new method for studying relaxation processes, set forth in the first section of this article, made it possible to develop a method[^11] and to construct an instrument that makes it possible, with great rapidity and sufficient accuracy, to obtain curves of the rise and decay of luminescence under light excitation[^12]. The schematic circuit of the instrument is shown in Fig. 15. The disk \(D\), rotated by motor \(M\), is provided with projections that simultaneously close or open the access of light to the two slits \(\mathrm{Sh}_1\) and \(\mathrm{Sh}_2\). The duration of the light—

Fig. 15. Circuit of a phosphoroscope with exponential sweep.

in its impulse is equal to the duration of the dark interval. The relative steepness of the impulse front is determined by the ratio of the rise time of the impulse to its total duration or, what is the same, by the ratio of the slit width to the length of the projection on the disk, which in our apparatus was approximately \(1/500\)*). The mercury lamp SVD-250 (\(L_1\)), fed by direct current, illuminates, by means of the condenser \(K_1\), through the light filter \(\Phi_1\), the slit \(Sh_1\). The luminescence of the specimen under study \(L\) is received, through the light filter \(\Phi_2\), crossed with the light filter \(\Phi_1\), by the photomultiplier \(U_1\), whose load resistance \(r_1\) is connected to the input of the vertical plates of the cathode oscillograph \(KO\). If during the interval between the P-impulses (\(t_{01}\)) the specimen becomes practically completely extinguished (“complete-relaxation regime”), then the deflections of the beam on the oscillograph screen will be proportional to the instantaneous brightness of the luminescence. If, however, by the moment of the beginning of the next excitation the specimen still possesses some luminescence (“segmental regime”), then, in order to read off the intensity, a “zero line” must be introduced on the oscillograph screen (see § 4). In the present case, imitation of complete decay of the luminescence can be achieved by briefly and periodically closing the window of the multiplier \(U_1\) with the blades of an additional obturator \(O\). To determine the level of complete excitation, it is necessary, after stopping the disk \(D\) in such a position that access of the exciting light to the specimen is open, to measure the magnitude of the jumps of the beam on the screen when the multiplier window is closed and opened. With a linear sweep, a picture similar to that shown in Fig. 10, \(a\), will be obtained on the oscillograph screen; with an exponential sweep, one similar to that shown in Fig. 10, \(b\). To carry out the exponential sweep there is used, as was indicated in the first part of the article, the process of charging a capacitor by electrical P-impulses through an ohmic resistance. The incandescent lamp \(L_2\), fed by direct current, illuminates, by means of the condenser \(K_2\), the slit \(Sh_2\), and at the output resistance \(r_2\) of the multiplier \(U_2\) there arise electrical P-impulses synchronous with the light impulses exciting the luminescence. These electrical impulses charge one of the capacitances \(C\) through a resistance box \(R\), as a result of which at the input of the horizontal plates of the oscillograph there arises an alternating voltage varying according to the laws expressed by equations (3) and (3′). The value \(\tau\) (in seconds) in these equations will be equal to the product of the capacitance \(C\) (in farads) by the sum of the resistances \(r_2 + R\) (in ohms).

*) This ratio could be varied owing to a device allowing the slit width to be regulated. In those cases when the luminescence of the specimen proves weak, it is possible, by widening the slit, to illuminate it with light of greater intensity, at the cost of the steepness of the P-impulse front.

Having graduated the knobs controlling the capacitance and resistance banks, constructed according to the decade principle, we can directly read off the value of \(\tau\) in seconds. The time \(t_0\), which must be known when using the partial-times method, is determined by means of the stroboscope \(Cm\), rotating together with disk \(D\) and illuminated by lamp \(H\), or by means of a tachodynamo mounted on the axis of motor \(M\) and feeding a properly graduated voltmeter. A necessary condition for the proper operation of the instrument is the strict simultaneity of the opening and closing of slits \(Sh_1\) and \(Sh_2\). This is achieved by shifting one of the slits with a micrometer screw.

Fig. 16. \(a\)—the sweep lags, \(b\)—the sweep is phased, \(c\)—the sweep leads.

A rough criterion of simultaneity is the absence of vertical or horizontal jumps in the corners of the loop obtained on the oscilloscope screen (Fig. 16); the former simulate the presence of a “very fast” process, the latter—“dark” and “light” pauses.

The photomultipliers used to record the brightness of the glow are practically inertia-free (for the time interval of interest to us); however, the circuit by which they are connected to the oscilloscope has a certain time constant, whose magnitude is determined by the load resistance \(r_1\) and the capacitance of the connecting wires. As a result, the P-pulses turn out to be somewhat distorted (quite imperceptibly to the eye).

In order that these distortions be the same in the circuits of the first (measuring) and second (sweeping) multipliers, it is convenient to use the following procedure. By means of a half-silvered glass plate and a lens, the P-pulses of light emerging from some one slit are directed simultaneously into both multipliers (the second light source being extinguished).

Thus, in each of the multipliers there arise electrical P-pulses that are absolutely synchronous (by the method of their production). Let us apply some P-pulses to the vertical plates of the oscilloscope, the others to the horizontal plates, loading both multipli-

...to identical load resistances. Then, if the parasitic capacitances of the circuit in both multipliers are the same, we should obtain a straight line on the oscilloscope screen. Otherwise, by selecting the load resistances or additional small capacitances we shall bring the pattern on the screen to “straightening.” After such “electrical” adjustment we remove the glass plate and the lens and, allowing the light from each of the sources to pass through its own slit, move one of the slits along the circumference of the disk. At a certain position of this slit the loop on the screen will again close, which completes the optical adjustment.

The relaxation process recorded by the measuring multiplier is in fact distorted by the time constant of this multiplier. As a first approximation it is permissible to assume that this time constant is simply added to one or another partial time of the process. But the same distorting time constant also enters into the relaxation process generated by the second, sweeping multiplier. Therefore, if the distortions introduced by the two multipliers are selected, as indicated above, to be equal to one another, they drop out of the measurements. If, however, they must be made unequal (for example, if it is necessary to increase the sensitivity of the measuring multiplier by introducing a comparatively large load resistance), then the additional distortion arising as a result will have to be subtracted from the measurement data.

Special attention must be paid to the proper selection of the light filters through which excitation and observation are carried out. The requirements on the crossed character of the light filters must be extended beyond the visible part of the spectrum, since the photocathode of the multiplier may possess considerable sensitivity in the near infrared and ultraviolet regions (for example, an oxide-cesium cathode). Insufficiently complete crossing of the light filters leads to the fact that part of the exciting light enters the window of the multiplier and, appearing in the form of jumps on the oscilloscope screen, imitates the presence of instantaneous processes in the rise and decay of the luminescence. In order to make sure that the observed jumps are characteristic of the sample being studied and are not caused by insufficiently complete crossing of the light filters, the light filter through which excitation is performed should be moved behind the luminescent sample. If, with such a rearrangement of the filter, the jumps disappear, this indicates sufficiently complete absorption of the exciting light by the light filter standing in front of the multiplier window, and that the jumps are characteristic of the luminescence of the sample itself. In selecting light filters that eliminate the exciting light, it is necessary to keep in mind that very many glass and gelatin filters...

... themselves luminesce under the action of ultraviolet or short-wavelength visible light. The requirements for complete crossing may be substantially reduced if the multiplier is placed in the “shadow” of the exciting light, moving its window out of the exciting beam.

If, in order to increase the brightness of the glow, or for any other reasons, it is desirable to carry out excitation with the full light of the exciting source, or if the selection of crossed

Fig. 17. External view of the oscillographic phosphoroscope.

filters is impossible for one reason or another, then, to protect the multiplier from the exciting light, a second disk may be installed in front of the multiplier window, rotating on the same axis as the main disk of the instrument and having cutouts additional to the cutouts of the main disk.

In this case, of course, only the decay curve of the glow is accessible to study.

The general appearance of the instrument for studying photoluminescence, with a cathode oscilloscope connected to it (the third model of the oscillographic phosphoroscope), is shown in Fig. 17*).

* The instrument was made by mechanic L. A. Novozhilov, who took a large part in developing its design.

§ 8. Exponentially Decaying Luminophors

As was indicated in the first part of the article, the application of the new method proves most effective in the case of processes proceeding according to exponential laws:

\[ I = I_0(1 - e^{-t/\tau}) \quad \text{— build-up,} \]

\[ I = I_0 e^{-t/\tau} \quad \text{— decay.} \]

In this case, in order to determine the sought parameter of the law \((\tau)\), it is sufficient, having placed the specimen under investigation opposite slit III and rotating the knobs of the \(RC\)-circuit (“tau-meter”), to obtain straightening of the characteristic curve on the oscilloscope screen. The impossibility of straightening the curve on the screen immediately indicates the nonexponential nature of the process. The entire measurement process (in the case of exponential laws) takes several seconds; the accuracy

Fig. 18. Dependence of \(\tau\) on temperature (UKS—uranyl potassium sulfate).

of the measurements may reach approximately 1%. Such speed, as well as the possibility of following, from the picture on the oscilloscope screen, the most insignificant changes in the relaxation law, make it possible to undertake broad investigations into the influence of various physical factors on the luminescence of luminophors (temperature, intensity and spectral composition of the exciting light, spectral composition of the luminescence, external fields, etc.).

As an example one may cite the investigation of the luminescence of uranyl salts, most of which build up and decay according to strictly exponential laws\(^{13}\). Carrying out such an investigation with the aid of a Becquerel phosphoroscope would take an immeasurably longer time.

The study of the dependence of \(\tau\) on temperature showed that the decay rate sharply increases with increasing temperature,

(see Fig. 18, on which some of the obtained curves are presented). Along with the decrease of \(\tau\), an equally strong decrease in the luminescence yield is observed, which gives grounds to assume that the decrease of \(\tau\) with temperature is due to temperature quenching of luminescence, occurring according to a scheme of second-order collisions. Then the change in \(\tau\) should occur according to the law

\[ \frac{1}{\tau}=\frac{1}{\tau_0}+P, \tag{24} \]

where \(\tau_0\) is the intrinsic decay time, not distorted by the influence of quenching factors (the decay time at low temperature), and

Fig. 19. Dependence of the quenching probability on temperature.

\(P\) is the quenching probability—an increasing function of temperature. Processing of the data obtained showed (Fig. 19) that, to a first approximation, \(P\) increases with temperature according to the law

\[ P=Ae^{-u/kT}, \tag{25} \]

where \(u\) has the meaning of the energy which the quenching agent must possess in order to quench the excited center.

The values of \(\tau\) for crystals of uranyl potassium sulfate and for a solution of the same substance in sulfuric acid proved to be almost identical, despite the fact that the course of the curves of the temperature variation of \(\tau\) for these two states is completely different. This makes it possible to think that the molecular state of the medium (crystal, solution) affects only the probability of bringing quenching agents

agents to the excited centers, characterized by the quantity \(A\) in formula (25).

An illustration of the sensitivity of the method and of its peculiar possibilities may be the difference we found in the values of \(\tau\) for a crystalline plate of potassium uranyl sulfate when measured “by reflection” and “by transmission”\({}^{13}\). In some cases the magnitude of this difference may be very considerable (\(2.7 \cdot 10^{-4}\) and \(4.5 \cdot 10^{-4}\) sec). It can be shown that the cause of this difference is the reabsorption of the short-wavelength part of the luminescence while passing through strongly absorbing crystals (as a result of multiple reflections). It is very difficult to detect this phenomenon by means of ordinary phosphoroscopic methods, since there is always reason to suspect that the small differences observed are a consequence of experimental errors. Here, however, the difference in \(\tau\) is revealed immediately by the transformation of the straight line on the oscilloscope screen into a loop when the position of the specimen is changed.

Fig. 20. Dependence of the relaxation time of the luminescence of synthetic ruby on temperature.

Other exponentially decaying and flaring objects investigated by the new method were crystals of synthetic ruby. The red luminescence of these crystals is extremely favorable for the use of photomultipliers with an oxide-cesium photocathode. The law of decay is strictly exponential, with \(\tau\) of the order of \(10^{-3}\) sec. With increasing temperature \(\tau\) also decreases sharply (see Fig. 20). The quantity \(\tau\) does not depend on the wavelength of the exciting light, but is different for different lines of the luminescence spectrum. Measurements carried out with various light filters in front of the window of a multiplier with an oxide-cesium photocathode made it possible to verify the presence of intense radiation of ruby in the infrared region of the spectrum. The value of \(\tau\) for this radiation is approximately 2 times smaller than the value of \(\tau\) for the red doublet (\(\lambda \simeq 690\,\mathrm{m}\mu\)), which predominates in the visible radiation of ruby. The relative intensity of these infrared lines increases sharply with increasing chromium concentration in the specimen.

§ 9. Crystallophosphors

In the case of crystallophosphors, exponential processes are a relatively rare exception. Therefore, in investigating these luminophors one has to resort to the method of “partial times”; and since the durations of the afterglow of crystallophosphors are often rather large, the regime often proves to be segmented, and for determining the “zero line” and the saturation level one has to use an obturator.

The method, set forth in the first part of the article, for processing experimental data and based on the introduction of the concept of the “instantaneous relaxation time” \(\Theta\) (§ 5г), proves to be very useful in analyzing phosphorescence decay curves, since it makes it possible to judge the fulfillment of the hyperbolic law

Fig. 21. Decay curves in the coordinates \(\Theta, t\):
\(1\) — ZnS·Cu, \(2\) — ZnS·Ag, \(3\) — ZnS·Mn.

\[ I = \frac{I_0}{(1 + at)^\alpha}, \tag{23} \]

which is usually fulfilled in the late stages of the decay process. Investigation of the initial stages of the decay process of typical crystallophosphors based on zinc sulfide with copper or silver as activator showed\({}^{14}\) that, although the process as a whole does not obey Becquerel’s law (the course of the curve \(\Theta(t)\) in the very initial stages, i.e. approximately in the first millisecond or fractions of a millisecond, is nonlinear), it nevertheless rapidly develops into a “Becquerelian” process, corresponding to a linear growth of \(\Theta\) with time. Figure 21 shows the course of the curve \(\Theta(t)\) for

ZnS·Cu, ZnS·Ag, and ZnS·Mn. In the case of ZnS·Mn the graph \(\theta(t)\) is horizontal at first, which corresponds to the exponential onset of decay; subsequently the curve follows some other law.

Fig. 22 shows the course of the curve \(\theta(t)\) for ZnS·Cu at two different intensities of the exciting light \(E\). As can be seen, the exponent of the hyperbola \(\alpha\) on the “Becquerel” portions remains constant (the straight lines have an equal slope), while the value of \(a\) increases with increasing \(E\). In addition, it may be noted that the time during which the curve still does not follow Becquerel’s law decreases as the intensity of the exciting light increases. Fig. 23 shows that the dependences

\[ a \sim \sqrt{E} \tag{26} \]

[cf. formula (21)] and

\[ \alpha = \mathrm{const}. \tag{27} \]

hold. The same relations prove valid for ZnS·Cu phosphors with any concentrations of the activator.\(^{15}\)

Fig. 22. Course of the curve \(\Theta(t)\) at two different excitation intensities.

Investigating crystal phosphors, one can be convinced that the course of decay curves beginning from the same instantaneous luminescence intensities depends on the path by which these initial instantaneous values of the luminescence intensity were reached. The relations (26), (27) also lead to the same conclusion (cf. § 5c). Thus, the relaxation of crystal phosphors is, generally speaking, not single-valued.

A study of the rise of the luminescence of ZnS·Cu and ZnS·Ag phosphors showed that in the initial stages (approximately up to \(0.015\) sec) this process very often proceeds according to the exponential law \(I \sim E(1-e^{-t/\tau})\), or, in any case, according to a law very close to exponential. This can be judged both from data obtained by the method of “partial times” (Fig. 24), and directly, from the possibility of straightening the curves on the screen of an oscillograph. Thus, the process

rise proceeds in the initial stages in a substantially different way than this

Fig. 23.

follows from the simple theory of bimolecular recombination, which, as is known, gives the dependence (18)

\[ I \sim E\,\operatorname{th}^{2}(at). \]

The quantity \(\tau\) in the rise exponent depends strongly on the excitation intensity, decreasing as the latter increases. Experience has shown that in many cases the law

\[ \frac{1}{\tau}=\frac{1}{\tau_{0}}+k\sqrt{E}. \]

holds. Thus, the rise curve obeys the law

\[ I\sim E\left(1-e^{-t\left(\frac{1}{\tau_{0}}+k\sqrt{E}\right)}\right), \tag{28} \]

where \(k\) is a constant depending on temperature, \(E\) is the intensity of the exciting light, and \(1/\tau_{0}\) is a constant, often equal to zero.

Fig. 24. Rise curves:
\(1\)—ZnS·Cu, \(2\)—ZnS·Ag.

It is of some interest to note that the study of the rise of luminescence by the method described makes it possible, without any measurements, to investigate the dependence of the luminescence brightness on the intensity of the exciting light under instantaneous excitation.* Indeed, excitation

* As Antonov-Romanovskii has shown \(^{16}\), the quadratic dependence \(I_{\text{inst}}\sim E^{2}\) is characteristic of a bimolecular process, in contrast to the linear dependence \(I_{\text{inst}}\sim E\), characteristic of a monomolecular process.

can be regarded as instantaneous when, during the time of excitation, the build-up curve has time to reach a value small in comparison with the value which it would reach under continuously lasting excitation of the same strength. It is not difficult to understand that the initial part of the build-up curve caused by a \(\Pi\)-pulse of the exciting light is also the luminescence curve under the action of instantaneous excitation, for the phosphor, of course, “does not know” whether the excitation will break off or will continue during the time of the \(\Pi\)-pulse.

Fig. 25. Build-up of phosphor luminescence:
\(a\)—monotonic, \(b\)—with a “flash.”

Formula (28) shows that the brightness of phosphors under instantaneous excitation may obey neither a linear law \(I_{\mathrm{inst}}\sim E\), nor, at the same time, a simple quadratic law \(I_{\mathrm{inst}}\sim E^2\). Indeed, expanding formula (28) in a series for small \(t=t_1\), we have

\[ I_{\mathrm{inst}}\sim E\frac{t_1}{\tau}+kE^{3/2}t_1. \tag{29} \]

In many cases, the study of build-up processes can provide substantial information on the details of the mechanism of the process under investigation. To illustrate this, one may cite the case, studied by us, of the anomalous character of the build-up curves of luminescence of zinc sulfide phosphors activated by manganese\(^{17}\). The build-up of the luminescence of these phosphors, under definite physical conditions and at a definite concentration of activator, proceeds not along the usual monotonically increasing curve tending to a constant value (Fig. 25, \(a\)), but first reaches a sharp maximum, after which the luminescence brightness falls to a stationar-

of its value (Fig. 25, b). The brightness of the luminescence at the maximum may exceed the stationary brightness many times over. We have been able to observe “flashes” of fourfold brightness. Anomalies of the flare-up are observed only in the orange luminescence band of the phosphor, due to the presence of the activator—manganese; in the blue luminescence band, which is customarily attributed to excess zinc, the flare-up proceeds monotonically. The phenomenon is most sharply expressed in samples with a definite manganese concentration \((C = 10^{-3}\ \mathrm{g/g})\). The relative magnitude of the “flash” increases with increasing pulse duration \(t_0\).

Since for all \(t_0\) at which observations were made \((4 \cdot 10^{-1} > t_0 > 10^{-2})\), the flare-up reached the stationary value of the luminescence by the end of excitation, whereas the decay was practically complete only for sufficiently large \(t_0\) from the indicated interval, it may be thought that the magnitude of the effect is directly related to how fully the decay process preceding excitation has taken place.

To determine the position of the maximum on the flare-up curve, the method described in the first part of the article was applied, consisting in finding that parameter of the exponential sweep at which the maximum is located in the middle of the portion of the abscissa axis occupied by the curve under study on the oscilloscope screen (Fig. 8). As the results of these determinations showed, the maximum is reached several milliseconds after the beginning of excitation, and this value remains practically constant when the temperature and chopping frequency are changed, and decreases when the intensity of the exciting light is increased (Fig. 26).

Fig. 26. Dependence of \(I_{\max}/I_{\mathrm{st}}\) (curve 1) and \(t_{\max}\) (curve 2) on the intensity of the exciting light.

The totality of the characteristics of the phenomenon made it possible to put forward certain assumptions about the mechanism of the flare-up of the luminescence of the orange band of the phosphor ZnS·Mn. With a sufficiently long time \(t_0\) allowed for the decay process, the levels of the activator (Mn), transitions to which from the conduction band determine the yellow luminescence, have time to become depleted, and when the exciting light is switched on, transitions to these levels initially pro-

proceeds unhindered. With the passage of time, however, the activator levels are “clogged” by electrons, and further transitions from the conduction band to the activator levels occur only insofar as these levels have time to become freed. The stationary luminescence corresponding to equilibrium between the process of freeing the activator levels and the process of filling them with electrons from the conduction band is then weaker than the luminescence upon establishment of the equilibrium state, when the rate of filling of the activator levels can still exceed the rate at which they are freed. Although this proposed mechanism does not make it possible to explain completely the dependence on the activator concentration and on temperature, it agrees satisfactorily with the principal features of the phenomenon; one may think that it constitutes at least part of the complete, as yet unknown, mechanism explaining the phenomenon exhaustively. This example illustrates the idea expressed at the beginning of the article: that the investigation of relaxation processes, by making it possible to observe the mechanisms governing processes at moments when “balance” is disturbed, makes it possible to judge the properties of these mechanisms.

§ 10. Luminescence under cathode-ray, X-ray, and other types of excitation

The apparatus described is intended for the study of photoluminescence, i.e., luminescence under excitation by visible or ultraviolet rays. With equal success the method can also be extended to cases of other types of excitation. In Fig. 27 is shown the diagram of an installation constructed by us for the

Fig. 27. Diagram of an installation for studying cathodoluminescence.

study of the build-up and decay of luminescence under excitation by an electron beam (cathodoluminescence)*. In principle this apparatus is entirely analogous to the apparatus for studying photoluminescence. Excitation of the glow of specimen $L$ is produced by an electron beam obtained in the vacuum tube $T$. Here $K$ is the oxide cathode, $B$ is the Wehnelt cylinder, $\Phi$ is the focusing system, and $A_1$ and $A_2$ are anodes. For modulating the electron beam, a rectangular-pulse generator $\Gamma$, constructed according to the multivibrator circuit, is used. The rectangular pulses generated by it are fed to the Wehnelt cylinder and periodically cut off and lock the electron beam. (Good results are also obtained when the control voltage is applied to the deflecting plates. In this case the beam is instantaneously “removed” from the screen.) The excitation front can thereby be made very steep ($<10^{-6}$ sec). The glow of the specimen is received by a photoelectron multiplier $Y$, connected to a cathode oscilloscope. To realize exponential sweep, the electrical rectangular pulses generated in the second channel of the generator and fully synchronous with the rectangular pulses controlling the electron beam are fed through an $RC$ circuit (“taumeter”) to the horizontal plates of the oscilloscope. Determination of the “zero line” and of the level of stationary brightness of the glow, as in the case of photoluminescence, is carried out with the aid of a mechanical obturator. The procedure for studying the curves on the oscilloscope screen does not differ in any way from that described above.

It should be noted that, despite the extremely great variety of apparatuses proposed for investigating the kinetics of cathodoluminescence (see, for example, $^{18}$), all of them possess, to a greater or lesser degree, the shortcomings indicated above in the discussion of phosphoroscope designs. The apparatus described is to a considerable extent free of these shortcomings.

Without any fundamental changes, the method can also be applied to the study of the kinetics of luminescence excited by X-rays (roentgenoluminescence) or by radioactive radiations (radioluminescence). Modulation of the X-ray beam can be carried out either by a rotating disk opaque to rays of the given hardness, or by supplying the X-ray tube with high-voltage rectangular pulses. To interrupt a beam of radioactive radiation it is sufficient to take a metallic disk of appropriate thickness. Experiments carried out by us jointly with T. V. Timofeeva on applying the method to the radioluminescence of various crystalline phosphors showed that in some cases (ZnS·Cu, ZnS·Ag, and many others) the build-up and decay

* V. A. Arkhangelskaya and A. M. Bonch-Bruevich took part in the development of this apparatus.

radioluminescence proceeds in a substantially different way than under light excitation of the same phosphors: alongside a process with a duration of the order of \(10^{-3}\) sec., an intense, rapid process appears (\(\sim 10^{-5}\) sec.); in some cases (willemite, etc.) the laws prove to be practically the same as under light excitation.

The method may, of course, also be applied in more complex cases of combined excitation, when various excitation sources act on the specimen either simultaneously or successively, or when continuous excitation by one source is combined with intermittent excitation by another, etc.

It is expedient to apply the method also to the investigation of the effect of infrared rays on the excitation of a phosphor; moreover, if different time shifts between the exciting and “infrared” pulses are set and their relative durations are varied, the action of infrared rays can be studied at all stages of the processes of the rise and decay of luminescence.

B. PHOTOEFFECT

§ 11. Methodology

Relaxation of photoelectric phenomena is the second, extensive and promising field of application of the method. The range of questions of interest in this connection includes: the photoeffect in resistive semiconductor photocells, in barrier-layer photocells, in gas-filled photocells, and the Becquerel photoelectric effect (in liquids). The relaxation of the phenomena listed lies mainly in the interval of times accessible to the method described. Relaxation processes in vacuum photocells are, for our method, “instantaneous” and lie outside the limits of its capabilities. Below some results will be presented relating to the relaxation of the resistive and barrier-layer photoeffect in semiconductors.

Whereas in the case of luminescence much attention has long been paid to relaxation phenomena, both methodologically and in principle, the relaxation of the photoeffect has been studied comparatively little and only sporadically. Apparently, the first attempt at a systematic study of such relaxation in semiconductors was the work of V. V. Balakov \(^{19}\), who used a P-pulse excitation regime. The method he developed is an electrical analogue of Becquerel’s optical method and therefore retains all the shortcomings of the latter.

Zhuze and Ryvkin \(^{20}\), as well as Lashkarev and co-workers \(^{21}\), studied the relaxation of photoconductivity with the aid of a cathode oscillograph. The methods they used gave, in general, pos—

...the possibility only of a qualitative judgment about the character of the relaxation. Together with D. B. Gurevich, we have developed a method for investigating the relaxation of the photoeffect^22, entirely analogous to the luminescence method described above. The layout of the apparatus is shown

Fig. 28. Layout of an apparatus for studying the photoeffect.

in Fig. 28, a. Here \(L_1\) is the light source (a 300-W motion-picture projection lamp, supplied with direct current), which excites the photoeffect in the specimen under study. The beam of exciting light is interrupted near slit \(S_1\) by disk \(D\), mounted on the shaft of motor \(M\). The rotational speed of the motor is determined with the aid of a stroboscope and a neon lamp. The photosensitivity or barrier-layer photoelement being investigated is illuminated through slit \(S_1\) by rectangular light pulses. The barrier-layer photoelement is connected directly to the input terminals of the oscillograph*), while the photosensitivity is connected in series with voltage source \(I.N.\) to load resistance \(z\), parallel to the terminals of the oscillograph. The choice of the magnitude of \(z\) is determined by the requirement that \(z\) be small in comparison with the resistance of the illuminated or darkened photosensitivity. Under this condition the voltage drop across \(z\) is proportional to the conductance of the specimen.

*) See § 2.

The light from the second incandescent lamp \(L_2\) (12 V, 15 W), powered by an accumulator, passes through the slit \(Sh_2\), located exactly on the same diameter as the slit \(Sh_1\), and falls on the photomultiplier \(F U\) (FEU-13), whose electrical P-pulses are used to produce an exponential sweep, in a manner and arrangement completely analogous to that described above.

Determination of the zero level. Since the processes of decay and growth of photoconductivity may, within the time of a single P-pulse \(t_0\), fail to reach completion, it is necessary to be able to refer segments of the observed curves to their asymptotic limits, i.e., to the level of dark conductivity and to the level of stationary photoconductivity. For this purpose we make use of the circumstance that, as experience shows, photoresistances (at least those of the semiconductor type) have no electrical inertia; in other words, the current passing through them follows the applied voltage without inertia. Then, choosing as the voltage source a generator of electrical rectangular pulses, we can imitate the absence of conductivity by the absence of voltage in the intervals between two P-pulses. The relaxation curve of photoconductivity on the oscilloscope screen will then, at a frequency of the electrical P-pulses sufficiently large in comparison with the frequency of the P-pulses of the exciting light, have the form of a dashed curve, beneath which, also dashed, a straight line will be traced corresponding to the absence of conductivity (Fig. 28, б). In order to determine the level of dark conductivity \(\sigma\) (from which, as from zero, it is customary to measure the photoconductivity \(\Delta\sigma\)), we switch off the exciting light and, having obtained the figure shown in Fig. 25, в, set on the oscilloscope screen the scale of the dark conductivity. Comparing the figure in Fig. 28, б with the figure in Fig. 28, в, we determine the zero level of the light (i.e., dark) conductivity. Having then obtained a figure similar to the latter, but with the light switched on and the motor disk stopped, we are able in an analogous way to determine the level of stationary photoconductivity.

In the case of barrier-type photocells the problem of determining the asymptotic limits is more complicated. We shall not dwell on this question here.

§ 12. Relaxation of photoconductivity in semiconductor photoresistances

The method described above was applied by D. B. Gurevich and ourselves to the study of a number of semiconductor photoresistances\(^{23,24}\). Se, Tl\(_2\)S, Bi\(_2\)S\(_3\), PbS′, CdS, MoS\(_2\), Cu\(_2\)O, and InSe were investigated at various illuminations, temperatures, and wavelengths of the exc—

exciting light. It turned out that in a number of cases the relaxation of photoconductivity follows an exponential law, whose time constant \(\tau\) is immediately determined by the straightening method.

However, much more often, especially in decay processes, nonexponential processes are encountered, which compel one to resort to the method of partial times. In this case, processing the results with the aid of the concept of the instantaneous relaxation time \(\Theta\) (see § 5g) made it possible to reveal new regularities. It turned out that in all cases where the dependence of the stationary photoconductivity \(\Delta\sigma_0\) on the illumination \(E\) is nonlinear (which corresponds, as experiment shows, to the condition \(\Delta\sigma_0 \gg \sigma\), where \(\sigma\) is the dark conductivity)\(^{25}\), the decay of photoconductivity corresponds to a linear growth of \(\Theta\) with time, i.e., proceeds according to the Becquerel law

\[ \Delta\sigma=\frac{\Delta\sigma_0}{(1+at)^2}. \tag{30} \]

Fig. 29 shows the course of \(\Theta(t)\) for a photoresistance sample made of \(\mathrm{Bi_2S_3}\) at various temperatures. As can be seen, the degree of hyperbola (30) \(\alpha\) increases with temperature. An analogous result holds for CdS (Fig. 30, a).

Fig. 29. Change in the slope of the straight lines \(\Theta(t)\) with temperature for photoresistances made of \(\mathrm{BiS_3}\).

The dependence of the degree \(\alpha\) and of the quantity \(a\) on the illumination is presented in Fig. 30, b and c, showing that \(\alpha\) does not depend on the illumination,

\[ \frac{d\alpha}{dE}=\text{const}\quad (\text{for }T=\text{const}), \tag{31} \]

whereas \(a\) is related to \(E\) by the simple dependence

\[ a \sim \sqrt{E} \qquad (T=\mathrm{const}). \tag{32} \]

These dependences are also typical of a number of other photoresistances.

It is very important to note that these dependences are completely analogous to those which occurred for the “Becquerel” portions of the decay curve of a typical ZnS·Cu crystallophosphor (see § 9, Fig. 21).

Fig. 30. Relaxation regularities in photoresistances made of CdS. Dependence of the degree of the hyperbola \(a\) on temperature (a) and illumination (b), dependence of the quantity \(a\) on illumination (c), and dependence of \(\frac{1}{\tau}\) on illumination (d).

Photoresistances exhibiting a nonlinear dependence of the stationary photoconductivity on illumination often reveal an exponential law of increase of the photoconductivity. This exponential law apparently has a nature quite different from that of the exponential law in linear photoresistances \((\Delta J_0 \sim E)\), for which the ignition exponent has \(\tau\) in the ши-

...within broad limits, independent of \(E\), whereas the exponent \(\tau\) of the first exponentials depends on the illumination \(E\) (within the investigated limits) according to the law

\[ \frac{1}{\tau}=\frac{1}{\tau_0}+k\sqrt{E}. \tag{33} \]

The quantity \(\dfrac{1}{\tau_0}\) is for the most part equal to 0. In addition, in linear photoresistances the decay of the photoconductivity also occurs according to an exponential law with the same \(\tau\), independent of \(E\). As an example, Fig. 30, \(g\), shows the course of \(\dfrac{1}{\tau}\) as a function of \(\sqrt{E}\) for CdS. We, together with D. B. Gurevich, have also investigated the dependences of all the above-mentioned parameters of photoconductivity relaxation on temperature for a number of substances. Details of these investigations may be found in special papers \(^{22,23,24,25}\).

Summarizing the results obtained in the investigation of photoresistances and comparing them with the results obtained in the investigation of luminescence, one can discover a remarkable parallelism in the laws of photoconductivity and luminescence, making it possible to classify photosensitive semiconductors in a manner completely analogous to the classification of luminescent objects. Tables I and II contain a summary of the principal stationary and kinetic laws of photoconductors and phosphors.

Examples of hyperbolic photoresistances are Se, InSe, Tl\(_2\)S, Bi\(_2\)S\(_3\), and CdS (at low temperatures). The exponential resistances include Cu\(_2\)O, and also CdS and Tl\(_2\)S at high temperatures, and MoS\(_2\).

From a comparison of the tables we see that substances conventionally called “hyperbolic photoresistances” may be compared with phosphors whose kinetics, according to accepted notions, corresponds to a bimolecular mechanism of recombination of light-liberated electrons with holes. On the other hand, “exponential photoresistances” may be compared with phosphors of the monomolecular type, although such a comparison is somewhat formal in character, in view of the fact that in monomolecular luminescence the electrons are not torn away from the “centers,” whereas in the “pseudomonomolecular” internal photoeffect they are, by the very meaning of the phenomenon, collectivized. The pseudomonomolecular character of the photoeffect is apparently connected with a sharp difference in the numbers of recombining partners. This is confirmed by the fact that exponential photoresistances possess, as Table I shows, a relatively greater dark conductivity. The same category includes the fact that at very low illuminations (i.e., at small photoconductivity \(\Delta\sigma\)) the dependence \(\Delta\sigma_0(E)\) becomes linear even for hyperbolic resistances. In

Table I

Basic regularities of photoresistances

Characteristic	Hyperbolic photoresistance	Exponential photoresistance
A. Stationary dependences	\(\Delta\sigma_0 \sim \sqrt{E}\) \(\sigma_0 < \Delta\sigma_0\)	\(\Delta\sigma_0 \sim E\) \(\sigma_0 \gg \Delta\sigma_0\)
B. Relaxation dependences: Decay	\(\Delta\sigma \sim \dfrac{\sqrt{E}}{(1+\alpha t)^a}\)	\(\Delta\sigma \sim E e^{-t/\tau}\)
Dependence on \(E\) at \(T=\mathrm{const}\)	\(a \sim \sqrt{E};\ \alpha=\mathrm{const}\)	\(\tau=\mathrm{const}\)
Dependence on \(T\) at \(E=\mathrm{const}\)	\(a\) increases with increasing \(T\)	\(\dfrac{1}{\tau}-\dfrac{1}{\tau_0}\sim e^{-u/kT}\)
Growth:	\(\Delta\sigma \sim \sqrt{E}\left(1-e^{-t/\tau}\right)\)	\(\Delta\sigma \sim E\left(1-e^{-t/\tau}\right)\)
Dependence on \(E\) at \(T=\mathrm{const}\)	\(\dfrac{1}{\tau}-\dfrac{1}{\tau_0}\sim \sqrt{E}\)	\(\tau=\mathrm{const}\)
Dependence on \(T\) at \(E=\mathrm{const}\)	\(\tau\) decreases with increasing \(T\)	\(\dfrac{1}{\tau}-\dfrac{1}{\tau_0}\sim e^{-u/kT}\)
General form of the relaxation dependence	\(\Delta\sigma=\sqrt{E}\, f_2(t\sqrt{E})\)	\(\Delta\sigma=E\varphi_2(t)\)

At these illuminations one should expect the expression of hyperbolic properties in exponential ones. An even more convincing example of such an expression may be furnished by the behavior of CdS upon an increase in temperature. At room temperatures this is a typical hyperbolic photoresistance. At temperatures beginning approximately at \(180^\circ\mathrm{C}\), the dark conductivity of CdS begins to increase sharply, and this increase is accompanied by the expression of hyperbolic properties in exponential ones: the dependence of \(\Delta\sigma_0\) on \(E\) becomes linear, the laws of growth and decay become exponential, with \(\tau\) equal and independent of \(E\), etc. Quite analogous behavior is also exhibited by \(\mathrm{Tl}_2\mathrm{S}\).

A comparison of Tables I and II shows that if one compares the photoconductivity of semiconductors with the light sum of phosphors with respect to both stationary,

Table II

Basic regularities of luminophors

Characteristic	Phosphorescence	Fluorescence
A. Stationary dependence	$I \sim E$	$I \sim E$
B. Relaxation dependences: Decay	$I \sim \dfrac{E}{(1+\alpha t)^2}$	$I \sim E e^{-t/\tau}$
Dependence on $E$ at $T=\mathrm{const}$	$\alpha \sim \sqrt{E};\ \alpha=\mathrm{const}$	$\tau=\mathrm{const}$
Dependence on $T$ at $E=\mathrm{const}$	$\alpha$ increases with increasing $T$	$\dfrac{1}{\tau}-\dfrac{1}{\tau_0}\sim e^{-u/kT}$
Rise:	$I \sim E\left(1-e^{-t/\tau}\right)$	$I \sim E\left(1-e^{-t/\tau}\right)$
Dependence on $E$ at $T=\mathrm{const}$	$\dfrac{1}{\tau}-\dfrac{1}{\tau_0}\sim \sqrt{E}$	$\tau=\mathrm{const}$
Dependence on $T$ at $E=\mathrm{const}$	$\tau$ decreases with increasing $T$	$\dfrac{1}{\tau}-\dfrac{1}{\tau_0}\sim e^{-u/kT}$
General form of the relaxation dependence for brightness	$I=E f_1(t\sqrt{E})$	$I=E\varphi_1(t)$
General form of the relaxation dependence for light sum	$L=\sqrt{E}\, f_2(t\sqrt{E})$	$L=E\varphi_2(t)$

also of the kinetic dependences, then complete identity is obtained. Such a comparison of properties established for heterogeneous objects (on the one hand, typical photoresistances; on the other, typical crystalline phosphors) finds its culmination in the study of the stationary and relaxation regularities of photoconductivity and phosphorescence in one and the same object. As D. B. Gurevich and the authors^26 showed, the laws governing the kinetics of the light sum of phosphorescence and the photoconductivity of CdS under excitation in a definite spectral region coincide not only in form, but also in the constants entering into these laws.

This result has general significance for the theory of the solid state, for it shows that, despite the countless variety and complexity of the properties of real crystalline bodies,

in the latter can be uncovered with the aid of broadly conducted relaxation studies. Although, at the present state of theory, the laws themselves for the kinetics of bimolecular photoconductivity and phosphorescence cannot yet be derived a priori, owing to the unquestionable difficulty of this problem, it nevertheless becomes clear that these difficulties are common, if not identical, for both phenomena. Moreover, it is possible, by a detailed quantitative study of the relaxation of photoconductivity and phosphorescence on one and the same object, to establish the relation between the constants characterizing one process and the constants characterizing the other. In this case we shall obtain relations in which the unknown common features of both kinetics will have dropped out. Of particular interest would be studies carried out simultaneously with the study of the relaxation of the dielectric constant; the magnitude of the latter, as may be supposed, is connected, among other things, with the number of electrons trapped at local levels. (Concerning the method for studying the kinetics of the dielectric constant, see below, Sections B and E.)

Fig. 31. Decay of the photoconductivity of the photoresistance of indium selenide.

It should be pointed out that the use of the concept of an instantaneous relaxation time, which so well justifies itself in the study of fast processes on a cathode-ray oscillograph, proves no less useful in the study of slow relaxation processes carried out with the aid, for example, of a galvanometer and a stopwatch. Thus, for example, Fig. 31 shows the course of the decay of the photoconductivity of a photoresistance made of indium selenide, expressed in the coordinates \(\Theta(t)\). As can be seen, the Becquerel law revealed by this graph is satisfactorily fulfilled over a time interval from several seconds to several minutes. Interesting results, using the method described, were also obtained by P. V. Meyklyar in studying the kinetics of photoconductivity of silver-halide salts in time intervals characteristic both of fast and of slow processes\(^{27}\). The results obtained by him make it possible to hope that investigation of the kinetics of photoconductivity of these objects will prove very useful for the theory of photographic processes.

§ 13. Relaxation of the Photoeffect in Photovoltaic Cells

Although photovoltaic cells are very convenient objects for studying relaxation processes, the phenomena observed in them are of interest more from the technical than from the physical point of view. The main feature distinguishing them from photoresistances is the presence in them of electrical inertia, connected with the fact that, by its very structure, a photocell with a blocking layer contains electrical capacitances, sometimes quite considerable. The equivalent circuits of photocells usually given (see, for example, Fig. 32) take into account the existence

Fig. 32. Equivalent circuit of a photocell.

of the capacitance of the blocking layer. However, as we have established, such equivalent circuits are inadequate for describing the photocell as an electrical two-terminal network. The point is that the voltage in the electrical circuit (Fig. 32), charged through a resistance by an electrical Π-pulse, should vary according to an exponential law. Experiment, however, carried out in a manner similar to that described in Section B, shows that the voltage varies according to a more complicated, nonexponential law. This may be connected either with the fact that the capacitance and resistance of the photocell are not lumped, or with the fact that the equivalent circuit should in reality contain not one but two capacitances separated by a resistance. Further complications consist in the fact that the resistances in the equivalent circuit of the photocell depend both on the magnitude of the current flowing through the photocell and on its direction. This follows from the fact that the form of the relaxation curves changes substantially when the magnitude of the electrical Π-pulses is changed, and especially their direction. Thus, relaxation of the photo-emf induced in photocells by Π-pulses of light occurs under conditions in which the relaxation process of purely optical origin is, to an extremely great degree, masked by a complex electrical relaxation process. A change in any factor produces the most varied ...

changes in the relaxation process. As examples one may point to the following effects: 1) the electrical relaxation of a photocell changes substantially when the photocell is illuminated by steady light; 2) the light relaxation of a photocell changes substantially when a constant voltage is connected in series with the photocell, and this change itself depends sharply on the direction of the applied voltage. The latter circumstance is quite instructive: it would seem that the relaxation curves recorded by a cathode oscillograph cannot depend on the constant voltage connected to the input of the oscillograph in series with the source of the variable photo-emf under study, since the oscillograph does not register constant components of the voltage being studied, owing to the fact that a coupling capacitor is connected at the input of the oscillograph. This capacitor prevents the passage of direct current in the circuit.

However, owing to the presence of a source of constant voltage, a constant field is applied to the photocell in this circuit; as experience shows, this field changes the form of the relaxation curves and, consequently, affects the mechanism of the photocell. The essential point is that if the source of the additional voltage is now removed, the photocell still finds itself under the action of a superposed constant field created by the oscillograph capacitor, since this capacitor is charged precisely to such a value as compensates the constant component of the variable photo-emf produced by the photocell. It follows from this that if we simply connect the photocell to the input of a cathode oscillograph, we obtain not the pure relaxation of the photo-emf of the photocell, but the relaxation of the photo-emf of a photocell on which an additional field is superposed. In order to get rid of this additional field and work under simple conditions, it is necessary to connect a source of additional voltage, choosing its magnitude and direction so as exactly to compensate the field on the coupling capacitor. It is obvious that the magnitude of such an additional voltage must be different for different illuminations of the photocell, temperatures, frequencies of the Π-pulses of light, and so on.

From the foregoing it is easy to understand that the study of the relaxation of photocells leads to extremely varied data that are difficult to decipher. Among the most typical results it should be noted that the relaxation of the photoeffect usually depends only on the quantity of light falling on the photocell, but not on its illumination intensity (in other words, it is immaterial whether the given beam of light is concentrated at one point or spread over the surface of the photocell). This clearly indicates that the principal role in the relaxation is played by the electrical circuit of the photocell. Another typical result is that, in the transition

for small excitation intensities the relaxation curves become symmetric and turn into exponentials. Among the curious features of relaxation we note that, for a sulfur–silver photocell, at a sufficient excitation intensity and at certain repetition rates of the P-pulses, the rise curve of the photo-emf proves to be nonmonotonic, but reaches a noticeable maximum, then falling to the value corresponding to the stationary emf.

Thus, the relaxation of barrier-layer photocells constitutes a range of phenomena rather remote from the phenomena of photoconductivity relaxation. The decay curves of the photo-emf of barrier-layer photocells do not obey Becquerel’s law (30).

B. DIELECTRIC POLARIZATION

§ 14. Relaxation polarization

As is known, the emf on a capacitor charged by P-pulses of voltage through a resistance rises and falls according to an exponential law. This is true, however, only on the assumption that the polarizability of the dielectric from which the capacitor is made is a constant quantity, independent of both time and voltage. If one of these conditions is not fulfilled, then the emf on the capacitor may, generally speaking, rise and fall according to laws different from the exponential. As shown by G. I. Skanavi and A. I. Demeshina28, capacitors made of rutile TiO₂ with small additions of metals of the second group of Mendeleev’s table possess, along with an increased dielectric permittivity, peculiar dielectric losses at sonic and ultrasonic frequencies, dependent on temperature. This is connected, as shown in the cited work, with the existence of processes of “relaxation polarization,” caused by the displacement of ions in the loosened lattice of the dielectric. Such displacements of ions are accompanied by dissipation of energy and, while contributing to an increase in the polarizability of the substance, turn out to be rather slow. We, together with G. I. Skanavi and K. I. Lebedeva, undertook an investigation of the processes of charging and discharging capacitors made of the indicated dielectrics, using a method based on the principles set forth above29. The circuit of the apparatus used by us is shown in Fig. 33. The pulses produced in one of the channels of the rectangular-pulse generator charge the capacitor made of the dielectric under study, \(C_x\), through the ohmic resistance \(R_0\). To carry out a functional sweep, the synchronous pulses of the second channel are passed through a “taumeter.” In the very first experiments it was found that all capacitors made of rutile with small additions of metals of the second group charge and discharge along curves

which cannot be straightened on the screen of the oscillograph by using the usual “taumeter,” which is a capacitance charged through an ohmic resistance and makes it possible to carry out an exponential sweep in time. Although the method of partial

Fig. 33. Diagram of the setup for studying dielectric polarization.

times permits, in this case, the empirical curves to be obtained completely, it proved more expedient to construct (on the basis of certain ideas about the mechanism of the processes occurring in the dielectric) an equivalent circuit of the latter and, reproducing this circuit and varying its parameters, to achieve straightening of the curves on the screen. In the work of G. I. Skanavi and A. I. Demeshina it was established that in a number of cases the curve of dielectric losses has two maxima, which made it possible to assume the existence of two relaxation processes of different duration. Proceeding from this, one could attempt to ascribe to the dielectric the equivalent circuit shown in Fig. 34, consisting of the capacitance \(C_1\), characterizing the “elastic” polarizability of the dielectric, and the capacitances \(C_2\) and \(C_3\), connected in series with the resistances \(R_2\) and \(R_3\), and describing the relaxation parts of the polarization. Indeed, it turned out that whereas a simple “exponential taumeter” did not allow the straightening of the charging curves of the capacitors under investigation to be performed, a “taumeter” in which, through the ohmic resistance \(R\), a circuit assembled according to the diagram

Fig. 34.

Fig. 34, make it possible, with an appropriate choice of the parameters \(R, C_1, C_2, C_3, R_2, R_3\), to obtain satisfactory straightening. (We note that with a single relaxation circuit \(R_2C_2\) or \(R_3C_3\) it was not possible to obtain straightening.) Although the selection of such a number of parameters is rather cumbersome and, perhaps, not always sufficiently unambiguous, the use of the complicated “taumeter” is justified by the convenience of studying the influence of various physical factors on the parameters of the equivalent circuit, the calculation of which from other data would be very cumbersome. Thus, for example, temperature investigations of the indicated dielectrics showed that, when the temperature is lowered, the role of relaxation processes decreases. This circumstance is immediately apparent from the emerging possibility of straightening already with the aid of a simple “taumeter.” It is very interesting to note that an equivalent circuit with parameters chosen so that straightening takes place, when investigated by the method of ordinary harmonic analysis, gives the same frequency characteristics and loss characteristics as the substance under investigation.

G. GAS DISCHARGE

§ 15. Neon glow

As is known, the time of establishment and decay of the glow in a gas discharge may in some cases prove sufficiently long to be investigated by the oscillographic method described here. The inertia of the glow may be due, first, to processes related to the mechanism of the discharge itself (recombination of ions, processes in the plasma, etc.), and, second, to the duration of the stay of atoms and ions in the excited state. This duration may be sufficiently large if the excited atoms and ions have metastable levels or if the corresponding electronic transitions are forbidden.

The principle of the method for investigating the relaxation of the glow of a gas discharge remains the same as in all the cases considered above. Excitation of the gas discharge is carried out by rectangular electrical pulses applied to the electrodes of the discharge tube; the observed glow is detected by a photoelectric device connected to a cathode oscillograph, whose linear sweep is replaced by an exponential one. It goes without saying that the establishment time of the discharge itself must in this case be considerably less than the time during which the glow develops and decays. If, however, the delay of the glow is to a considerable extent a trivial consequence of the delay of the discharge,

then the latter may become the subject of a special investigation, which, in turn, can be carried out by the method described. To illustrate the applicability of the method to the study of gas-discharge luminescence, we may cite some results obtained by us in studying the luminescence of a discharge in neon. The discharge in a technical neon lamp was produced by a Π-shaped voltage obtained with the aid of a rectangular-pulse generator or of a device consisting of a rotating disk, which interrupted the light of an incandescent lamp, a photoelectron multiplier (FEU-13), and a power amplifier that also served to separate the channels through which the lamp and the exponential sweep were supplied. The arrangement is shown in Fig. 35. The luminescence of the discharge

Fig. 35. Schematic of the apparatus for studying neon luminescence.

was passed through a monochromator, which made it possible to isolate individual spectral lines, and was detected by a photoelectron multiplier FEU₂ connected to the vertical plates of an oscillograph. Through the second channel the Π-pulses were fed, via an \(RC\) circuit (“taumeter”), to the horizontal sweep of the oscillograph. The sensitivity of the apparatus made it possible to determine the times of ignition and decay of individual spectral lines of the discharge. These times proved to be different for different lines, being of the order of \(10^{-4}\) sec. Most of the spectral lines ignite and decay according to exponential laws, and the corresponding curves are straightened on the oscillograph screen. Having straightened the curve corresponding to some spectral line, and then rotating the drum of the monochromator, one can see how, as individual spectral lines pass through the exit slit of the monochromator, the straight line either turns into a loop or remains straight. In this way one can very simply and rapidly separate lines with a common value of \(\tau\), i.e., lines,

corresponding to transitions from one and the same electronic level.

In the ignition and extinction of some spectral lines of neon, curious “anomalies” are observed, among which the following may be mentioned:

1. A “dark pause” in ignition. The glow of some lines begins to ignite not immediately after voltage is applied to the tube, but after a certain dark interval, comparable in duration with \(\tau\) (Fig. 36, \(a\)). This cannot be

Fig. 36. Ignition and extinction of the glow of a neon lamp (in exponential sweep):
\(a\)—“dark pause” during ignition, \(b\)—“flash” during extinction.

connected with a delay in the onset of the discharge, since most spectral lines begin to ignite immediately after voltage is applied.

1. A “flash” in extinction. After the lamp is switched off, the glow of some lines does not begin to fade monotonically, as does the glow of most other spectral lines, but, on the contrary, first increases and only then fades (Fig. 36, \(b\)). The intensity of the “flash” may reach 20–30% of the glow intensity of the given line in the discharge.

D. ELECTRO-OPTICAL PHENOMENA IN COLLOIDS

§ 15. Method

The method presented for studying relaxation processes can also be successfully applied in the investigation of electro-optical phenomena in colloidal systems, where the development of the processes occurs in most cases over times falling within the interval of times characterizing rapid processes (\(10^{-1}\)—\(10^{-5}\) sec.). Optical phenomena observed when an electric field is applied to a colloidal system have repeatedly served as the subject of investigation*). They attracted attention, on the one hand,

*) See, for example, E. V. Shpolskii\({}^{30}\) (where a bibliography of the question is also given).

N. A. TOLSTOI AND P. P. FEOFILOV

are of interest because of their pronounced character in comparison with electro-optical phenomena in molecular liquids and gases, and, on the other hand, because of their appreciable inertial character, characterized by relaxation times lying in an interval comparatively accessible to observation. The study of these relaxation times promises to provide information on the microscopic kinetics of colloidal systems and on its dependence on the viscosity of the medium and on the size, shape, and electrical properties of the particles.

Electro-optical phenomena in colloidal systems are connected primarily with the orientation of anisotropic colloidal particles (polarization phenomena: electric double refraction and electric dichroism) and with the forced migration of charged particles (electrophoresis). The imperfection of the investigative methods that have been used often did not permit a sufficiently complete analysis of the observed phenomena and not infrequently led to a number of wholly erroneous conclusions, especially since the analysis is greatly complicated[^31] by the diversity of electro-optical phenomena in colloidal solutions.

The principal shortcoming of the methods that have been used, including oscillographic ones, was that they were based on the use of an alternating sinusoidal field. As we have repeatedly indicated, this type of excitation leads in somewhat complex cases to results that are practically not amenable not only to quantitative, but often even to qualitative interpretation.

It is much more expedient to use, in this case as well, rectangular voltage pulses applied to electrodes placed in the colloidal solution under investigation. The sudden application of an electric field to the colloidal system and its equally sudden removal make it possible to follow, from the development of the optical phenomena, the course of the forced orientation of the particles when a field of constant magnitude is applied, and the course of their disorientation when the field is switched off. These data, on the one hand, make it possible to draw conclusions about the forces acting on the particles and determined, in the final analysis, by the structure of the particles themselves (the anisotropy of their polarizability, a rigid dipole moment, and the anisotropy of their shape). On the other hand, by studying the relaxation of the phenomena after removal of the field, we can draw conclusions about rotational diffusion, determined by the viscosity of the medium and by the dimensions of the particles. Since the optical phenomena associated with the orientation of particles in a field are determined by the anisotropy of their optical properties, the study of these phenomena makes it possible to approach the investigation of the optical properties of individual colloidal particles.

To obtain rectangular electrical pulses, in principle the same circuit may be used as that which was employed for obtaining the pulses feeding the exponential sweeps in the circuits of installations for the investigation of luminescence and photo-

effect. The light pulses obtained by means of a rotating disk with cutouts are converted into electrical ones by means of a photoelectric device (for example, a photomultiplier) connected to a power amplifier. The amplified pulses are fed through a blocking capacitor of large capacitance, which ensures the absence of a constant voltage component,

Fig. 37. Pulse shape for studying electro-optical phenomena: a—pulses of the 1st kind, b—pulses of the 2nd kind.

Fig. 38. Disk for obtaining pulses of the 2nd kind.

to the Kerr cell (the constant component causes rapid coagulation of colloids and must be eliminated). In the study of electro-optical phenomena it is expedient to use pulses of two kinds: P-pulses of the 1st kind (Fig. 37, a), in which the constant voltage instantaneously changes sign to the opposite one, and P-pulses of the 2nd kind (Fig. 37, b), in which voltage pulses of opposite sign are separated by a zero interval equal in duration to the duration of the P-pulse. During this time interval the field on the cell is absent. To obtain P-pulses of the 2nd kind, one may use a disk with specially arranged cutouts (Fig. 38). The cutouts of the disk alternately close and open two slits illuminated by two independent incandescent lamps. Equality of the amplitudes of the positive and negative pulses can be achieved by changing the heating of the lamp and the potential applied to the grid of the power amplifier (operating in a nonlinear regime)*.

* P-pulses of the 1st kind can be obtained of considerably better quality if one uses a rectangular-voltage generator constructed according to the multivibrator scheme. In this case a much steeper front is obtained and, moreover, increasing the duration of the P-pulse has little effect on the steepness of the front.

\(\Pi\)-pulses of the 1st kind are the most reasonable means of acting on that part of the orientation phenomenon which is determined by the presence of a rigid dipole moment in the particle. With respect to that part of the orientation phenomenon which depends only on the magnitude, but not on the direction, of the field (for example, orientation as a result of anisotropy of the electric polarizability or as a result of electrophoresis), \(\Pi\)-pulses of the 1st kind are, as it were, a constant voltage. \(\Pi\)-pulses of the 2nd kind, containing intervals of complete absence of voltage, give the particles the possibility of relaxation in the absence of any external actions.

Fig. 39. Diagram of the apparatus for studying the time course of electro-optical phenomena.

The diagram of the apparatus constructed by us for studying the time course of electro-optical phenomena is shown in Fig. 39. Its principal elements are: an optical system with a Kerr cell, a source of \(\Pi\)-pulses constructed according to the scheme indicated above, a “taumeter,” a photoelement, an amplifier, and a cathode oscilloscope. Details concerning the construction of the apparatus may be found in the original article\(^{32}\).

§ 16. Electro-optical phenomena in hydrophobic colloids

The rationality of applying rectangular pulses for electro-optical investigations was revealed in the very first studies of colloidal solutions (bentonite, benzopurpurin, graphite, certain “liquid crystals,” etc.). Analysis of oscillo-

lographic curves obtained when changes were made in the optical part of the setup made it possible to establish with certainty that, contrary to the assertions of earlier investigators, the modulation of light when an alternating field is applied to a Kerr cell filled with a colloidal solution is caused not by electric birefringence, but by electric dichroism. Analysis of the shape of the modulation curves (Fig. 40) made it possible to draw definite conclusions about the character of the orientation of the particles in the field[^32]. The time required to reach the maxima and minima observed on the modulation curves and corresponding to the moment of attainment of the most perfect orientation (different from the equilibrium one!) can be determined by selecting the parameter of the exponential sweep for which the maximum appears in the middle of the curve on the oscilloscope screen (see above, Chapter 1, § 3c).

Fig. 40. Light modulation curves by a Kerr cell filled with a colloidal solution.

a — Electric light vector along the field

b — Electric light vector across the field

The use of type-II Π-pulses makes it possible, as was indicated, to study the relaxation of the induced anisotropy. This process proves to be rather slow (several seconds), and, since the investigation has to be carried out in a regime of strong segmentation, only the initial parts of the relaxation curves turn out to be accessible to measurement. The initial course of the curves can be approximated by an exponential with \(\tau\) of the order of \(10^{-1}\)—\(10^{-2}\) sec.

The application of Π-pulses to the Kerr cell made it possible to discover a new electro-optical effect, consisting in a monotonic increase and decrease of the scalar transparency of the medium near

electrodes of the cell. This phenomenon can be explained by the electrophoretic migration of particles in the near-electrode region. Investigation of the modulation of the light flux by the exponential-sweep method showed that it proceeds according to laws close to exponential, with $\tau$ of the order of $10^{-2}$—$10^{-3}$ sec.

E. OTHER POSSIBLE AREAS OF APPLICATION OF THE METHOD

The examples given by no means exhaust the areas of possible application of the method. It is applicable, in essence, to any periodically reproducible relaxation processes occurring in the indicated time interval, since modern experimental technique makes it possible to reduce practically any phenomena to electrical ones. Without attempting to cover all possible areas of application and without entering into the details of individual experiments, whether possible or already performed, we shall indicate some of them.

§ 17. Surface ionization of atoms and molecules

In the study of the surface ionization of metals, installations are used in which the heated surface of the metal under investigation is bombarded by a molecular beam. The ions torn from the surface are accelerated by a field and are detected in the form of an ion current. The inertia of the processes leads to a delay of the ion current relative to the molecular beam. Investigation of the curves of rise and fall of the ion current makes it possible to determine the “lifetime” of atoms adsorbed on the surface, and the heat of adsorption. Interruption of the molecular beam in these experiments is usually accomplished either by a rotating disk or by an oscillating shutter. The curves of variation of the ion current are recorded by a cathode oscillograph[^33]. Since these curves are exponential with $\tau$ lying in the interval accessible to the method, it is natural to apply the exponential-sweep method to the investigation of the processes, synchronizing the latter with the device that interrupts the molecular beam.

§ 18. Capacitance measurements

By charging the investigated capacitor $C_x$ with P-pulses of voltage through a known resistance $R_0$, one can determine the value of the capacitance $C_x$ very simply and quickly. The circuit for such a measurement coincides with the circuit described in § 9. Experience shows that if the source of P-pulses gives pulses with a sufficiently steep leading edge (for example, $2$–$5 \cdot 10^{-7}$ sec), then it is possible to measure capacitances of several centimeters, for example interelectrode capacitances in radio-

lamps. Such a “capacitometer” can be realized in the form of a simple attachment to a cathode-ray oscilloscope, consisting of a multivibrator generating square pulses of a specified frequency and an \(RC\)-circuit.

§ 19. Relaxation of the Actino-Dielectric Effect

It is known that many crystallophosphors change their dielectric constant under the action of light (the actino-dielectric effect). The process of establishment of such a light-induced increment of the dielectric constant is of considerable interest for the theory of phosphorescence.

Measurement of the relaxation of the dielectric constant can be carried out by means of the circuit shown in Fig. 41. In series with the specimen under study \(C_x\), enclosed in a transparent capacitor, a load capacitor of substantially larger capacitance \(C_0\) is connected to the vertical plates of the oscilloscope. A sufficiently high voltage high-frequency source is connected to the points \(AB\). This voltage is distributed between the capacitances \(C_x\) and \(C_0\) in proportion to their inverse values. When the magnitude of the capacitance \(C_x\), which occurs under the action of a square pulse of light, changes, the amplitude of the high-frequency oscillations on the load capacitance \(C_0\) will also change. The process of change of this amplitude will reflect the sought relaxation of the dielectric constant. By applying to the horizontal (sweep) plates an exponential sweep synchronized with the light pulses, we can again make use of our method. Here we shall see on the oscilloscope screen not a line (loop), but a continuous luminous band, in the form of which the high-frequency oscillations are displayed. However, the envelope of this band will be much brighter (at these points the beam stops), and we can study the form of the envelope in exactly the same way as the form of any other relaxation curve. It is essential, of course, only that the relaxation time be much greater than the period of the high-frequency oscillations. The appearance of the described pattern is presented in Fig. 42.

a — Linear sweep

b — Exponential sweep

Fig. 42.

§ 20. Establishment of Stationary Electrical Conductivity

Such processes as the establishment of an electric current in liquids in the presence of polarizing electrodes, or the establishment of a current in unipolar conducting systems (semiconductor rectifiers), may serve as an interesting subject of investigation. The methodology in this case is obvious, and we shall not dwell on it.

§ 21. Mechanical Relaxation

Apparently, it would be of interest to apply the method to the study of phenomena of mechanical relaxation, for example to the establishment and dissipation of stresses. As an object it would be expedient to choose transparent polymers and to record the phenomena by an optical method (by double refraction). The most difficult point of the methodology, apparently, is the creation of Π-pulses of the external force producing the stresses.

§ 22. Electrophysiological Phenomena

As is known, a whole series of electrophysiological processes can be recorded with the aid of a cathode oscilloscope and possesses an inertia lying within the interval accessible to our method. It is possible that an exact study of the character of electrical relaxation processes in living objects would provide the biologist with valuable information about their structure and properties.

§ 23. Possible Ways of Extending the Method

It is obvious that the limit of the time interval of applicability of the method described is connected not so much with fundamental as with technical difficulties, at any rate as far as short times are concerned. From all that has been set forth above it follows that we are faced here with three problems: 1) obtaining a Π-pulse with a sufficiently steep front, 2) reducing the inertia of the receiving device, 3) increasing the frequency band passed by the cathode oscilloscope.

Obtaining an electrical Π-pulse with a front having a steepness of the order of \(10^{-8}\) sec. and even less belongs to problems which, though not simple, are successfully solved by radio engineering. Π-pulses with a steepness of the order of \(10^{-7}\) sec., in general, can be obtained without difficulty with the aid of a simple multivibrator. The use of the latter allowed us to advance the study of those phenomena which are caused by electrical excitation (re-

dielectric relaxation, cathodoluminescence), down to times on the order of \(10^{-6}\) sec. The situation is more difficult with obtaining short \(\Pi\)-pulses of light. Mechanical obturation makes it possible, without special difficulty, to obtain a front steepness of \(10^{-5}\) sec, and with very great difficulty and loss of intensity, \(10^{-6}\) sec, which determines the present limit of our method for these phenomena. Some possibilities are offered by the use of raster-optics methods, although the greatest hopes apparently may be placed on the use of gas luminescence under the action of very short and powerful electrical pulses. The latter method should be extremely advantageous for a whole series of reasons (complete utilization of the light energy, absence of slits, etc.).

A reduction in the inertia of the receiving device (for non-electrical phenomena) and of the cathode oscilloscope is, on closer examination, a problem connected with the sensitivity or with the intensity of the phenomenon. The photoelectron multipliers used for recording optical phenomena are themselves inertial only in the region of about \(10^{-8}\) sec (the electron transit time in the multiplier). Everything else is a consequence of the connection circuit. With a sufficient reduction in sensitivity (small load resistance), the inertia of the photoelectron multiplier can be reduced to values not far from the indicated magnitude. The final link in which time distortions occur is the amplifier of the cathode oscilloscope. But if the Braun tube is used directly, then in this case too the inertia is reduced practically to values on the order of \(10^{-8}\) sec, of course with a loss of sensitivity. The use of very wide-band amplifiers makes it possible to find a compromise between sensitivity and lack of inertia.

From all that has been said, we see that, by using wide-band amplifiers or by achieving a high intensity of the processes under study, one may fully expect to extend the range of applicability of our method to \(10^{-7}\) sec, or even less. For photoluminescence phenomena, as already stated, the use of pulsed light sources would be especially advantageous.

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