Violations of the Photochemical Reciprocity Law for Photographic Layers

A. L. Kartuzhanskii

1. Introduction

In photochemistry there is a known reciprocity law, or Bunsen–Roscoe law, according to which the photochemical action of a given radiation is determined by the product of its intensity and the time of its action, and does not depend on either of these quantities separately. In other words, intensity and time are reciprocal: a change in one of these quantities can always be compensated by an equal change in the other quantity in the opposite direction. Applied to a photographic layer, this means that if in one case a layer is exposed by a certain amount of illumination \(H\) at illuminance \(E_1\) and exposure time \(t_1\), and in another case by the same amount of illumination at illuminance \(E_2\) and exposure time \(t_2\), then the blackenings obtained in the two cases should be equal, since \(E_1 t_1 = E_2 t_2 = H\).

The reciprocity law was experimentally established for photochemical reactions in gases. It is valid for elementary photochemical processes, but in more complex cases it may be violated because of accompanying processes. This is precisely the situation in the case of the action of light on a photographic layer, which, as is known, is an aggregate of crystals of silver halide salts suspended in gelatin. As an experimental fact, peculiarities in the behavior of the photographic layer were known already to Bunsen and Roscoe themselves¹; while testing the limits of applicability of the law they had established, in 1862 they found certain deviations from reciprocity for silver chloride photographic layers, but regarded them as experimental error. Subsequent work established quite definitely the fact that the photographic layer does not obey the reciprocity law. The ever broader development of methods of photographic photometry, especially in astronomy

and spectroscopy, demanded a correct accounting of deviations from reciprocity. The number of works on this question grew continuously, but all the efforts of investigators were directed mainly toward establishing the regularities of the phenomenon. An explanation of the nature of this phenomenon appeared only comparatively recently, when the quantum-mechanical theory of the formation of the latent image had been developed and extensive factual material had accumulated.

Already in early investigations it was established that the reciprocity law can be violated in two cases. One case occurs when equal amounts of illumination are imparted to the layer in the form of a single exposure at different illuminances or, correspondingly, exposure times, and the blackenings obtained are not equal to one another. This phenomenon is usually called deviations from the reciprocity law, although it covers only the case of continuous illumination and does not include another possibility—intermittent illumination of the layer. Namely, if a given exposure is divided into a number of partial exposures, for example by dividing the exposure time, then the action of the sum of such partial exposures on the layer is not equivalent to the action of a single exposure, although in fact the illuminance and the illumination time remain the same. In the literature this phenomenon is commonly called the phenomenon of intermittent illumination.

The purpose of the present review is to consider both of the named phenomena—the experimental results, their explanations, and the conclusions that should be drawn from them for the practice of using photographic layers. Such a consideration, it seems to us, is of interest in connection with the ever wider use of photographic layers in various branches of science and technology. The variety of such applications of photography has naturally produced a variety of conditions under which layers are exposed; the changes in the properties of the layer connected with this are not always taken into account to the proper extent. We shall scarcely touch upon the history of the question, since, up to 1934, it is set forth with exhaustive completeness in the well-known monograph of K. V. Chibisov[^2], and later data may be found in the book by K. Mees[^3].

2. DEVIATIONS FROM THE RECIPROCITY LAW UNDER CONTINUOUS ILLUMINATION

2.1. Methods of expressing deviations from reciprocity

For the purposes of photographic photometry it has always seemed desirable to have an analytical expression for deviations from reciprocity. One of the first attempts in this direction was the work of the astrophysicist Schwarzschild[^4], who proposed that constancy of the blackening density \(D\) obtained on the layer would occur when the condition \(Et^p = \mathrm{const}\) is fulfilled, where \(p\) is some-

... a constant determined experimentally. It is evident that the exponent \(p\) must differ from unity, since for \(p=1\) the condition written expresses the Bunsen–Roscoe law. According to Schwarzschild and some of his contemporaries, for the case of low illuminances and long exposures, characteristic of the practice of astronomy, the exponent \(p\) turned out to be less than 1, on the average about 0.8. However, it was soon found that the exponent \(p\) does not remain constant when the illuminance is varied over wider ranges and, moreover, depends on a number of factors, in particular on the conditions of development. At that time a graphical method was proposed\(^5\) for expressing deviations from reciprocity, one that has fully retained its significance up to the present and is in fact the only generally accepted method for expressing the phenomenon of non-reciprocity. It consists in constructing a curve of equal blackenings, i.e. \(\lg Et=f(\lg E)\) or \(\lg Et=f(\lg t)\) for \(D=\mathrm{const}\); such a curve is commonly called an isopacity. By finding the equation of this curve, the problem of an analytical expression for deviations from reciprocity would be solved.

Fig. 1.

The properties of the isopacity should be considered in more detail. The general form of the isopacity \(\lg H=f(\lg E)\) or \(\lg H=f(\lg t)\), according to the data presently available, is shown in Fig. 1. The isopacity is a curve with a clearly pronounced minimum. Since the photosensitivity of a layer \(S\) is defined as the quantity inverse to the exposure \(H_D\) required to obtain a certain density of blackening \(D\), the isopacity thereby gives us the dependence of photosensitivity on illuminance or time ...

exposure. The minimum of the curve corresponds to the maximum light sensitivity of the layer, and the corresponding illuminance or exposure time is customarily called optimal. Times shorter than the optimal will hereafter be called short, and longer than the optimal—long; the same applies to illuminances. A portion of the curve parallel to the abscissa axis (a “plateau”) is observed only at very short exposure times (usually \(10^{-5}\) sec. and less); it indicates that at very short times the reciprocity law holds, i.e., equal values of \(H\) correspond to equal values of \(D\) for any exposure time. The existence of this portion of the isopaque was established recently\(^{6,7}\).

From the obvious equality \(\lg H=\lg E+\lg t\) it follows that either \(\lg E\) or \(\lg t\) may serve as the independent variable. A family of parallel straight lines bisecting the angle between the coordinate axes will represent lines of equal values of \(\lg t\) in the first case and equal values of \(\lg E\) in the second. Let us define the slope of the isopaque at an arbitrary point as \(\dfrac{d\lg H}{d\lg E}\) or \(\dfrac{d\lg H}{d\lg t}\). From the equation \(Et^p=\mathrm{const}\) we find that \(\dfrac{d\lg t}{d\lg E}=-\dfrac{1}{p}\), whence we immediately have:

\[ \frac{d\lg H}{d\lg E}=1-\frac{1}{p}, \qquad \frac{d\lg H}{d\lg t}=1-p. \tag{1} \]

Thus, at low illuminances and long exposure times (see Fig. 1), \(p<1\), as Schwarzschild found in his time, whereas at high illuminances and short exposure times \(p>1\). Excluding the plateau in the region of very short times, fulfillment of the reciprocity law (\(p=1\)) takes place only at the minimum of the isopaque; moreover, fulfillment of this law at a single point is devoid of physical meaning. Although the exponent \(p\) is not constant over wide ranges of variation of \(E\) or \(t\), it can always be regarded as constant over a small interval, i.e., the corresponding portion of the isopaque can be straightened; this procedure is known, in particular, in astronomy\(^{8}\). Only in this sense can one speak of a definite value of the exponent \(p\) in some interval.

If formally \(\lg E\) and \(\lg t\) are entirely equivalent as independent variables, in substance this is not quite so. Under the conditions of practical photography, a number of details of the object having different brightnesses are simultaneously projected onto different portions of the photographic layer; consequently, the layer is exposed by a set of illuminances for one exposure time that is the same for all portions of the layer. Therefore it is correct to compare with one another the characteristics of the layer (above all, light sensitivity) referred to definite exposure times; consequently, from this point of view the choice of \(\lg t\) as the independent variable is more

more expedient. Analogous conditions occur in spectroscopy, when a neutral step attenuator is placed before the slit of the spectrograph. In what follows (see Sections 2.2 and 2.3) we shall see that there are still a number of practical, as well as theoretical, considerations in favor of choosing \(\lg t\) as the independent variable.

The dependences \(\lg H=f(\lg E)\) and \(\lg H=f(\lg t)\) are not the only ones suitable for constructing isoopacs. Also known are isoopacs expressing the dependence \(\lg E=f(\lg t)\) for \(D=\mathrm{const}^{9}\). Such curves have not come into wide use, since they do not possess the visual clarity of isoopacs with ordinate \(\lg H\); moreover, their construction is more complicated. Among the merits of the curves \(\lg E=f(\lg t)\) is the possibility of directly finding the exponent \(p\) at any point: as has already been said,

\[ p=-\frac{d\lg E}{d\lg t}, \]

and therefore in the present case \(p\) is simply equal to the tangent of the angle of inclination of the curve, taken with the opposite sign.

Many attempts have been made to find an analytic expression for the isoopac. Thus, within narrow limits the isoopac is well described by the equation of a catenary, but over a wider interval of variation of the illumination systematic discrepancies with experiment were found,\(^{10}\) due to the fact that the real isoopac is not a symmetric curve. With approximately the same degree of accuracy the isoopac can be expressed by the equation of a hyperbola.\(^{5}\)

However, almost all the proposed equations have the common defect that they are of an approximating character—their derivation is not justified by any physical considerations. An equation proposed recently by P. V. Meiklyar has a fundamentally different character:\(^{11}\)

\[ \frac{H}{H_{0}}=\left(\frac{H_{\mathrm{p}}}{H_{0}}-1\right)\frac{1}{\sqrt{1+at}}+\frac{1}{2}\left(1+\sqrt{1+\frac{4A}{N_{0}^{2}}t}\right). \tag{2} \]

Here \(H_{0}\) is the optimal amount of illumination, \(H_{\mathrm{p}}\) the amount of illumination corresponding to the plateau of the isoopac (see Fig. 1), \(a\) is a quantity of order \(10^{4}\) (its physical meaning, as well as the meaning of the quantities \(A\) and \(N_{0}\), will be discussed in Section 2.3), \(A\) is a quantity of order 1, sharply dependent on temperature, and \(N_{0}\) is a quantity of order 10, dependent on the conditions of development. This equation describes the course of the isoopac well in general outline and gives fair agreement with experiment,\(^{11,12}\) as is demonstrated in Fig. 2, borrowed from Meiklyar’s work.\(^{11}\) This equation has a number of features, the most important of which, in our opinion, are the following:

a) It covers a very wide interval of exposure times, including the plateau, which none of the previously proposed equations...

was not described. At small exposure times \(\left(at \ll 1,\ \dfrac{4A}{N_0^2}t \ll 1\right)\) the first term has the predominant value, while the second term is equal to 1; at large times \((at \gg 1)\) the second term acquires the principal significance, and the first term goes to zero. At very small times \((at \ll 1)\) the first term becomes \(\left(\dfrac{H_{\mathrm p}}{H_0}-1\right)\); taking the second term into account, we obtain that \(H=H_{\mathrm p}\) (plateau).

Fig. 2.

b) It makes it possible to take into account the influence of the temperature of the layer and the development conditions on the isopac through the quantities \(A\), \(N_0\), \(a\), and \(H_{\mathrm p}\); as we shall see below (Section 2.2), both factors have little significance for the shape of the isopac.

c) Calculating the slope of the isopac at large values of \(t\), we find that

\[ \frac{d\lg H}{d\lg t} = \frac{1}{2} \left( 1-\frac{1}{\sqrt{1+\dfrac{4A}{N_0^2}t}} \right); \tag{2a} \]

\[ \left.\frac{d\lg H}{d\lg t}\right|_{t=\infty} = (1-p)_{\infty} = \frac{1}{2}, \]

i.e.,

\[ p \geq \frac{1}{2},\quad p_{\infty}=\frac{1}{2}. \]

This means that, for any large exposure times, an increase of the exposure, for example by a factor of \(n\) \((n>1)\), will necessarily produce an increase of the photosensitivity, and moreover by no less than a factor of \(n^{1/2}\). In a careful study of the experimental data of all authors who have worked in this field, we have in fact been unable to find a single case in which it would turn out that \(p<\dfrac{1}{2}\). At the same time, at sufficiently large exposure times one can see\(^{13}\) that in reality \(p\) tends to the limiting value \(\dfrac{1}{2}\) (see below, Fig. 5, borrowed from paper \(^{13}\)). Therefore the opinion sometimes expressed—that at very large exposure times a photographic layer loses the ability to accumulate light energy and that no increase in exposure is capable of increasing the photosensitivity of the layer—proves to be without foundation.

2.2. Regularities of deviations from reciprocity

Numerous studies have established that the form of the iso-opacity curve, and consequently the magnitude of deviations from reciprocity, depend substantially on a number of factors, including some that must necessarily be encountered in any application of a photographic layer. The most important of these are: the sensitivity of the layer and the conditions under which the photographic emulsion is prepared; the spectral composition of the incident radiation; the temperature of the layer during exposure; the conditions of development, in particular the development time; and the density of blackening for which the iso-opacity curve is constructed.

Let us attempt to give a summary of the general regularities associated with each of these factors.

The influence of the sensitivity of the layer on deviations from reciprocity has still not been clearly formulated by any of the authors who have dealt with this question. The reason for this lack of clarity is that one and the same value of the sensitivity of a layer can be achieved by very different procedures in the process of preparing the photographic emulsion; moreover, in their influence on the shape of the iso-opacity curve these procedures are sometimes opposed to one another. Given the complexity of the modern technological process for preparing an emulsion, it proves simply impossible to take into account the whole variety of this influence. K. V. Chibisov and co-workers came to this conclusion as early as 1933[^14] as a result of an extensive investigation; naturally, in the subsequent years the situation has become still more complicated.

On the average, according to the data of A. L. Kartuzhansky and P. V. Meiklyar[^7], the optimal exposure time is the greater, the less sensitive the layer: for the most highly sensitive materials (aerial-survey and cine-negative materials) it is \(1/30\)—\(1/100\) sec., whereas for low-sensitivity materials it reaches 1 sec. and more.

From this point of view the choice of exposure times in the GOST sensitometry system[^15] should be considered successful: \(1/20\) sec. for highly sensitive layers and \(\sim 10\) sec. for low-sensitivity layers. These exposure times either are optimal for the corresponding layers, or differ from the optimal ones not so substantially as to appreciably diminish the sensitivity of the layers.

The influence of certain individual procedures in preparing emulsions (if all other conditions are kept unchanged) has been established with sufficient definiteness. Thus, the introduction of gold salts into an emulsion during its preparation[^16] leads to a sharp decrease in deviations from reciprocity in the region of short exposure times, with a simultaneous increase in sensitivity.

... (Fig. 3). This method is very effective in preparing layers used with short exposures. Another method[^17] for reducing deviations from reciprocity in the region of short exposure times is to introduce into the emulsion small amounts of appropriate halide salts of divalent metals (lead, cadmium); however, in this case the photosensitivity does not increase. The chemical composition of the emulsion crystals (with respect to the halide) has a substantial influence on reciprocity failure. Thus, for bromosilver layers the plateau begins at exposure times of the order of \(10^{-5}\) sec.; as bromine is replaced by chlorine, i.e., for mixed chlorobromosilver layers, the plateau covers ever longer times, and for pure chlorosilver layers it extends almost to \(10^{-3}\) sec. These layers would be very convenient for work with short exposures; however, this is hindered by the relatively low overall sensitivity of such layers. The addition of iodide silver (not exceeding 5% in ordinary layers) has no noticeable effect on the plateau, but usually increases deviations from reciprocity in the region of short exposure times[^18].

Fig. 3.

Regarding the role of the spectral composition of the incident radiation in the shape of the isopac, it may now be regarded as reliably established that isopacs expressed in the form of the dependence \(\lg H = f(\lg t)\) for different wavelengths are arranged parallel to one another[^19]. This property of isopacs is essential for photographic photometry, since it eliminates the need to take reciprocity-failure deviations into account anew when the spectral composition of the radiation changes. In view of the importance of this fact, it has been checked repeatedly under various conditions[^7,^20]: for monochromatic radiations and for broad spectral intervals, in the region of the intrinsic sensitivity of silver halide and in the region of the sensitivity of sensitizing dyes; the slight departures from parallelism observed in these cases were of a random character. It was also shown that the introduction of a sensitizing dye into a layer can cause a considerable change in the magnitude of reciprocity-failure deviations at short exposure times[^18]; this change, however, proves to be the same for all wavelengths, and the parallelism of the isopacs is preserved both in the region of intrinsic sensitivity and in the region of sensitization.

It should be emphasized that the property of parallelism does not hold for isopacs expressed in the form \(\lg H = f(\lg E)\), and, as will become clear below, for quite definite physical reasons.

VIOLATIONS OF THE PHOTOCHEMICAL LAW OF RECIPROCITY

...quantities. This is one more argument in favor of choosing the parameter \(\lg t\) as the independent variable in constructing isoopacs. The parallelism of the isoopacs \(\lg H=f(\lg t)\), and the nonparallelism of the isoopacs \(\lg H=f(\lg E)\) for different wavelengths, has as its consequence a fact of substantial practical importance, empirically found by G. A. Tikhov\({}^{21}\): the spectral variation of the exponent \(p\) accurately follows the curve of the spectral sensitivity of the layer, with all its maxima and minima. If one keeps in mind that in Tikhov’s work the discussion concerns long exposure times, characteristic of the practice of astronomy, then the result obtained by him can easily be explained with the aid of Fig. 4. Here three parallel isoopacs are shown for wavelengths \(\lambda_1,\lambda_2\) and \(\lambda_3\) (let \(\lambda_1<\lambda_2<\lambda_3\)).

Fig. 4.

The sensitivity of the layer is maximal for that wavelength for which the isoopac is situated lowest—in our case for \(\lambda_1\), and minimal for the wavelength for which the isoopac is situated highest; consequently, the curve of spectral sensitivity in our example has a minimum at \(\lambda_2\). In Tikhov’s work the independent variable was \(\lg E\); therefore we must draw a section through our isoopacs by the line \(E=\mathrm{const}\) in the region \(p<1\). From the relation

\[ \frac{d\lg H}{d\lg t}=1-p \]

it follows that \(p\) is maximal where the slope of the isoopac at the point of intersection is minimal, and therefore \(p\) is the greater the lower the intersected isoopac lies, i.e., the higher the sensitivity. In our example, for the sensitivities \(S_{\lambda_1}>S_{\lambda_3}>S_{\lambda_2}\), and analogously for the exponent \(p_{\lambda_1}>p_{\lambda_3}>p_{\lambda_2}\), which expresses the regularity observed by Tikhov.

Proceeding from the fact of the parallelism of isoopacs at different wavelengths, one must regard as successful the choice of the illumination scale for the system of spectral sensitometry GOST\({}^{22}\), since the parallelism of the isoopacs should lead to constancy of the shape of the curve of the spectral sensitivity of the layer at any exposure time, if the curve is obtained on the illumination scale. This circumstance has indeed been confirmed experimentally.

The temperature of the layer during exposure has an extremely strong influence on the shape of the isoopac\({}^{13,23}\). For illustration we give a family of isoopacs borrowed from the work of Kartuzhanskii and Meiklar\({}^{13}\) (Fig. 5). Here one can clearly see a considerable temperature displacement of the minimum of the isoopac and of the boundary of the plateau; with decreasing temperature both of these points shift toward longer exposure times. At the temperature of liquid air \((-186^\circ\mathrm{C})\) the boundary

the plateau is shifted so much that the plateau covers the entire time interval investigated, and the reciprocity law is fulfilled at all exposure times. Another interesting feature of the family of isopacs presented here is that the isopacs corresponding to different temperatures intersect one another. If a number of vertical sections of this family are made, one obtains the dependence of the light-sensitivity of the layer on temperature at different exposure times. At short exposure times and high illuminances the sensitivity falls as the temperature is lowered, whereas in the region of long exposure times and low illuminances

Fig. 5.

the reverse dependence occurs. This also explains the contradictory data found in the literature concerning the temperature dependence of light-sensitivity. A third feature of this family is the decrease in deviations from reciprocity at long exposure times with decreasing temperature, with their simultaneous increase in the region of short times.

From the different dependence of sensitivity on temperature at different exposure times it follows that, under the conditions of aerial photography, it is necessary to take into account the possibility of a considerable decrease in the sensitivity of the layer in comparison with the sensitivity determined under room conditions, whereas in astrophotography various effects may be observed depending on climate and season; thus, outdoors, under winter conditions, an increase in sensitivity may be expected.

It is known \(^{24}\) that the fall of sensitivity with temperature is expressed in the region of sensitization much more sharply than in the region of intrinsic sensitivity, i.e., the spectral-sensitivity curve is deformed when the temperature of the layer changes. In addition \(^{13}\), the fall of sensitivity in the region of sensitization

the greater, the farther into the long-wavelength part of the spectrum the sensitization maximum lies. Such differences, however, are not accompanied by a violation of the parallelism of the iso-opacities for different wavelengths; the differences are only in the magnitude of the displacement of the entire iso-opacity as a whole parallel to the axis \(\lg H\), which leads to unequal temperature coefficients

\[ \frac{d\lg S_{\lambda}}{dT} \]

for different wavelengths \(^{23}\), not depending, however, on the exposure time.

The development conditions also affect the form of the iso-opacity to the strongest degree. We shall refer to Fig. 6, borrowed from the work of Kartuzhanskii and Meiklyar \(^{7}\), where a family of iso-opacities is shown, obtained for a highly sensitive negative film at different development times.

Fig. 6.

obtained for a highly sensitive negative film at different development times. The deformation of the iso-opacity with change in development time in Fig. 6 is quite typical for all layers and is also confirmed by other works \(^{25}\). Let us note first of all that a considerable decrease in the optimum exposure time is observed when the development time is increased; it is accompanied by a decrease in deviations from reciprocity in the region of short exposure times and by an increase in deviations in the region of long exposure times. The boundary of the plateau does not depend at all on the development time. It may also be established that the optimum amount of exposure decreases with development time according to a linear law

\[ \frac{1}{H_{0}}=\alpha t_{\mathrm{dev}}+\beta \qquad (\alpha,\ \beta\ \text{— constants}). \tag{3} \]

An increase in development time in short-time exposures brings a twofold benefit: it considerably increases the sensitivity (much more than in long exposures) and at the same time decreases the deviations from reciprocity at small exposure times. Therefore, in photographing or photometering short-lived phenomena, the longest possible development should be recommended.

Changing the development time alone by no means exhausts the whole possible variety of development conditions. It has been shown, however,^7 that for a given developer the form of the isopaque is uniquely determined by the value of the contrast coefficient—the so-called gamma (\(\gamma\))—regardless of the way in which this value of \(\gamma\) is attained (by the development temperature, the developer concentration, etc.). This rule holds even when passing to other developers that do not differ greatly in composition, above all in the developing agent,^7 for example, when passing from developer No. 1 (Chibisov’s)^15 to developer ID-19^26 or to developer No. 2 (A-12).^15 Therefore the regularities of Fig. 6 may be regarded as a sufficiently general expression of the change in the form of the isopaque that is caused by changes in development conditions in general. Although even a slight change in development conditions, especially at short development times, may substantially affect the magnitude of the deviations from reciprocity, for practical purposes it is sufficient to control the development conditions by the value of gamma; in this way a quite satisfactory reproducibility of the form of the isopaque is achieved.^7

Isopaques for different values of the blackening density \(D\), as numerous data in the literature show, differ considerably in their form. This property of isopaques is connected with the deformation of the characteristic curve of the layer when the exposure time is changed. As is known, the characteristic curve of a photographic layer is the curve expressing the dependence of the blackening density \(D\) on the logarithm of the quantity of exposure \(H = Et\), with a fixed value of one of the factors: \(E\) (the time scale) or \(t\) (the illumination scale). As an example we give in Fig. 7, borrowed from Ref. 7, a family of characteristic curves obtained on the illumination scale for a series of exposure-time values; for convenience the curves are divided into two separate groups.

Fig. 7.

Deformation of the characteristic curve does not occur in the exposure-time interval \(10^{-7}\)—\(10^{-5}\) sec, in which, as we know,

the reciprocity law is fulfilled (isopaque plateau). The coincidence of the curves over their entire extent shows that this law is observed for any densities of blackening. At exposure times greater than \(10^{-5}\) sec the curves do not coincide; deviations from the reciprocity law appear. In addition to a displacement of the curve as a whole, there is also a change in its shape, which causes the inequality of the deviations from reciprocity for different densities of blackening. First of all, the initial non-rectilinear portion of the characteristic curve is deformed; at very short exposure times this portion

Fig. 8.

of exposure is very extended, encompassing in our example a density interval up to \(D=1.1\) at \(t \leqslant 10^{-5}\) sec. The contraction of this portion occurs rather rapidly, and already at \(t=10^{-2}\) sec it reaches only \(D=0.55\). At the same time the contrast coefficient \(\gamma\)—the tangent of the angle of inclination of the rectilinear part of the characteristic curve—begins to increase. In our example \(\gamma\) begins to increase noticeably at \(t \gg 10^{-4}\) sec; for some other layers, according to the data of M. N. Alentsev\(^{27}\), \(\gamma\) is practically constant up to \(10^{-3}\) sec. The increase of \(\gamma\) continues up to times definitely exceeding the optimum exposure time; here \(\gamma\) passes through a maximum, after which it either changes very little (usually for highly sensitive layers, as in Fig. 7), or decreases monotonically (according to the data of G. S. Baranov\(^{28}\) for low-sensitivity layers), and sometimes quite significantly\(^{10}\).

Knowing the regularities of the deformation of the characteristic curve, we can immediately obtain the regularities of the change in the shape of the isopaque upon transition from one density of blackening to another. Indeed, the family of isopaques for different densities is nothing other than a series of horizontal sections of the family of characteristic curves; the curves of Fig. 7 correspond to the isopaques in Fig. 8. An essential feature of these isopaques

there is a clearly expressed decrease, first, in the optimal exposure time and, second, in the deviations from reciprocity in the region of short times, as one passes to ever smaller blackening densities. In the region of exposure times greater than the optimal, no substantial differences between the individual isopacs are observed, with the exception of the segment directly adjacent to the minimum. If the decrease of $\gamma$ at sufficiently long exposure times were expressed more sharply than in our example, then divergence of the isopacs in this region would have to be expected; in our case they are almost parallel. The boundary plateau, as is seen from Fig. 8, does not depend on the blackening density, i.e., deviations from reciprocity appear and disappear for all densities simultaneously.

Analogous regularities for the isopacs $\lg H=f(\lg E)$ may fail to exist. Thus, if the minima of all the isopacs in Fig. 8 were located on a straight line forming an angle of $45^\circ$ with the coordinate axes, i.e., on the line $\lg E=\mathrm{const}$, then this would mean that the position of the minimum (and, correspondingly, the optimal exposure) of the isopacs $\lg H=f(\lg E)$ does not depend on the density. In practice, families of isopacs are encountered for which the minimum is systematically displaced both to one side and to the other of the line $\lg E=\mathrm{const}$; sometimes an altogether irregular scatter of points is observed on both sides of this line. We can thus obtain entirely contradictory data on the nature of the deformation of an isopac as the blackening density changes. This also explains the indeterminacy of the results obtained in those works$^{10}$ in which the dependence $\lg H=f(\lg E)$ was used. Processing such results in the form $\lg H=f(\lg t)$ makes it possible to discern quite definite regularities, coinciding with those set forth above. Similar considerations can, with full justification, be extended also to the previously given family of isopacs for different development times (see Fig. 6). Therefore here too the literature data concerning the isopacs $\lg H=f(\lg E)^{10}$ do not give us definite regularities, and we are again convinced of the advantages of the dependence $\lg H=f(\lg t)$ over the dependence $\lg H=f(\lg E)$.

The insignificance of deviations from reciprocity for small blackening densities at short exposure times may create certain practical conveniences in cases where it is not required to have a distinct reproduction of the object photographed on the layer, and one can confine oneself merely to recording weak traces of its existence. Under such conditions there is a relative gain in sensitivity, and deviations from reciprocity may be neglected. Such are, for example, the conditions in photographing the glow of the screen of a cathode oscillograph$^{25}$.

Speaking of the deformation of the characteristic curve and of the deformation of the isopac associated with it, it is necessary to take into account that the coefficient

contrast \(\gamma\) at a given exposure time also depends on the wavelength of the acting radiation. On the other hand, isoopacity curves for a given blackening density and different wavelengths must be parallel to one another. It follows that the changes in \(\gamma\) with exposure time for any pair of wavelengths are not independent of one another, and that there must exist a certain relation between them. Indeed, it was shown\(^7\) that the contrast coefficients must satisfy the following “four-gamma relation”:

\[ \frac{1}{\gamma_{\lambda_1 t_1}}-\frac{1}{\gamma_{\lambda_2 t_1}} = \frac{1}{\gamma_{\lambda_1 t_2}}-\frac{1}{\gamma_{\lambda_2 t_2}} . \tag{4} \]

This relation connects the gammas for two wavelengths at any two arbitrary exposure times. If the dependence \(\gamma_\lambda=f(\lambda)\) is known for the exposure time \(t_1\), then it is sufficient to know \(\gamma\) for only one wavelength at the exposure time \(t_2\) in order, with the aid of relation (4), to obtain for this time the dependence \(\gamma_\lambda=f(\lambda)\) completely. Such a formula is useful for certain problems encountered in spectroscopic practice. On the other hand, in spectroscopic practice it is sometimes essential that the form of the curve \(\lg \gamma_\lambda=f(\lambda)\) be, as far as possible, independent of exposure time; this condition leads to the relation

\[ \frac{\gamma_{\lambda_1 t_1}}{\gamma_{\lambda_2 t_1}} = \frac{\gamma_{\lambda_1 t_2}}{\gamma_{\lambda_2 t_2}}, \tag{5} \]

which, generally speaking, is incompatible with (4), and hence also with the fact of the parallelism of isoopacity curves for different wavelengths. However, in practice\(^7\) this relation is fulfilled approximately with the same degree of accuracy as (4); it automatically reduces to (4) if \(\gamma_\lambda\) does not depend on the exposure time.

In principle, relation (4) can be obtained for any part of the characteristic curves and written in a more general form for gradients \(\left(g=\dfrac{dD}{d\lg H}\right)\), and not for gammas, which represent a special case of gradients; this is precisely what Wiltz and Webb\(^20\) did. If, however, this modified relation is used to construct a fourth characteristic curve from three known curves, such a construction gives an accuracy considerably lower than that required for constructing monochromatic curves in spectroscopy.

Among other factors affecting deviations from reciprocity, one should dwell on the additional exposure of the layer—the so-called method of double exposures\(^29\), the essence of which is as follows. If a photographic layer exposed by brief and intense illumination is then subjected to uniform additional exposure of low intensity and long duration, a significant enhancement of the action is observed.

of the first exposure, and the more so the shorter the first exposure was. Such a procedure leads to a considerable reduction in deviations from reciprocity in the region of exposure times smaller than the optimum (Fig. 9). In exactly the same way, if a layer that is to be exposed under weak and prolonged illumination is first subjected to uniform fogging of high intensity and short duration, then a considerable increase is observed in the effectiveness of the action of the second exposure—the greater, the longer the second exposure; in other words, a more or less complete disappearance of deviations from reciprocity in the region of exposure times greater than the optimum (Fig. 10).

Fig. 9.

Fig. 10.

Thus, double exposures, of which the first is short and the second long, invariably give a gain in sensitivity and reduce deviations from reciprocity to a minimum. It must be pointed out that the reverse order of fogging in double exposures is not effective under any conditions.

The method of double exposures finds practical application in the photographic recording both of short-lived processes[^25] and of weak luminescences, for example in spectroscopy. For the case of short-lived exposures the corresponding procedure has received the special name “latensification.” For the case of prolonged exposures this procedure falls under the more general concept of hypersensitization, which also includes certain chemical procedures leading to the same result, for example placing the photographic layer in an atmosphere of mercury vapor or ammonia[^30].

2.3. Physical causes of deviations from reciprocity

We have already said that deviations from reciprocity for any reactions are connected not with the primary photochemical act, but with secondary processes of a non-photochemical nature, the result of which may be, for example, the binding or destruction

of the initial substance or of the products of the photochemical reaction; and in solids there is still a number of specific phenomena. As a result of such interaction, the amount or the state of the photochemically formed substance changes, and this change depends on the conditions under which the photochemical reaction proceeds, in particular on its rate. In order to clarify what this interaction consists in in the case of the photolysis of emulsion crystals of a photographic layer, it is necessary briefly to set forth the mechanism of formation of the latent photographic image ^31,32.

The primary photochemical act in ionic crystals of silver halide consists in the absorption of light quanta with the formation of free photoelectrons in the conduction band of the crystal—one for each absorbed quantum. Until recently it was believed ^33 that electrons are always liberated from halide ions. At present there are convincing data in favor of the view that the electron donors, at least in the region of the maximum of the intrinsic photosensitivity, are \(F\)-centers ^33, which should be regarded as electrons localized at vacant anion sites. However, from what follows it will become clear that, for the phenomenon of interest to us, the nature of the absorbing centers and of the electron donors plays no role. An electron that has entered the conduction band is able to migrate through the crystal and ultimately becomes fixed at some “trap”; this completes the primary electronic process. The possibility is also not excluded of recombination of the electron with atomic halogen or with a positive hole; in this case the subsequent stages leading to the formation of the latent image do not occur.

The role of a trap may be attributed to any disturbance of the periodicity of the lattice, inevitable in a real crystal; with each such disturbance there are associated certain local energy levels lying below the conduction band. In most cases the depth of such levels is small and comparable with the energy of thermal motion, as a result of which they are, as a rule, unable to retain electrons for the time required for the occurrence of the subsequent processes leading to the formation of the latent image. The principal role in trapping photoelectrons belongs to deep traps—centers of photosensitivity, specially created in the crystals in the process of manufacturing the photographic emulsion and consisting of particles of metallic silver located on the surface of the crystals ^34. Local levels can be of substantial importance in those cases where deep traps were not created during manufacture of the emulsion (low-sensitivity layers), or if the deep traps are already filled. Recently, however, P. V. Meiklyar ^35 found on larger crystals that the filling of shallow traps occurs first and only then of the deeper ones. The conclusions that he

makes of this fact, lead it to somewhat different views on the mechanism of formation of the latent image than those set forth here.

In emulsion crystals, at any temperatures different from absolute zero, there exists a certain number of lattice defects, whose occurrence is due to thermal motion; their number depends on temperature according to the law \(e^{-W/kT}\) (\(W\) is the activation energy for the formation of the corresponding defect). The existence of these defects explains the presence in silver halide crystals of dark (ionic) conductivity and its dependence on temperature. The defects may be both of the Frenkel type (ions that have left their normal positions at lattice sites and move through interstices), and of the Schottky type (vacant ionic sites without the presence of free interstitial ions in the lattice; they move by a relay mechanism). Until recently it was considered established that in silver halide the only kind of defects are cationic Frenkel defects, i.e. interstitial \(\mathrm{Ag}^+\) ions. From what follows, however, it will become clear that the particular kind of ionic defects is of no essential significance for the phenomenon of interest to us.

An electron trapped in the crystal attracts to itself one of the interstitial silver ions and, by neutralizing it, forms an atom of photolytic silver. Repeated repetition of such a process at one center leads to the formation of a latent-image center. The time required for neutralization can be calculated from data on the dark conductivity of silver halide salts. If \(\sigma\) is the specific conductivity and \(\varepsilon\) the dielectric constant (equal to 13.0 for \(\mathrm{AgBr}\) and 12.2 for \(\mathrm{AgCl}\)), then the neutralization time is \(t_{\mathrm{n}}=\frac{\varepsilon}{2\pi\sigma}^{32}\). Since for \(\mathrm{AgBr}\) \(\sigma\) is, at room temperature, about \(10^5\) absolute units, and for \(\mathrm{AgCl}\) about \(10^4\) absolute units,\(^{32}\) the values of \(t_{\mathrm{n}}\) are about \(10^{-5}\) and \(10^{-4}\) sec, respectively. The electronic process proceeds immeasurably faster than the ionic one, and its duration need not be taken into account. The relatively long duration of the ionic process is significant in connection with the fact that the simultaneous trapping of two electrons at a given place, as calculation shows,\(^{32}\) is impossible: electrostatic repulsive forces arise between them. Therefore the arrival of electrons at a given center of photosensitivity must not occur at a rate exceeding \(1/t_{\mathrm{n}}\), so as not to outstrip the arrival of ions. At a higher rate of arrival of electrons, some of them will have to be trapped at other centers, including small traps; this will lead to the dispersal of the photolytically formed silver over many centers, which slows their growth and causes the loss of some electrons altogether.

A latent-image center that imparts to the crystal the capacity for development cannot be just any photolytically formed silver particle. To impart developability, it is required that the particle possess some minimum size and be located in a surface part of the crystal accessible to the action of the developer. Therefore, if, owing to an excessively high rate of arrival of electrons, there occurs the simultaneous slow growth of several particles instead of the rapid growth of one particle, and also the growth of particles in the interior of the crystal instead of the growth of particles located on the surface, then this is an undesirable phenomenon. Its result is a decrease in the number of developable crystals, which entails a decrease in the macroscopically measurable density of blackening, i.e. a fall in the photosensitivity of the photographic layer. The rate of arrival of photoelectrons, in turn, is determined by the rate of absorption of quanta of the incident light. If the amount of exposure (the total number of quanta) is given, then the rate of absorption of quanta (the tempo) is determined exclusively by the illumination time of the crystal, or, correspondingly, by the illuminance. Consequently, there exists a direct connection between photosensitivity and exposure time, which is precisely the essence of the phenomenon of non-reciprocity.

In silver bromide crystals, at exposure times of less than \(10^{-5}\) sec., as follows from the foregoing, neutralization is completely unable to occur during the exposure, and the process of formation of the latent image is played out after the end of the exposure. Consequently, the exposure time, provided only that it does not exceed \(t_n\), has no significance for the course of the subsequent processes, and the sensitivity of the layer should not under these conditions depend on the exposure time; this is fully confirmed by experiment (Figs. 5, 6, 8). At exposure times greater than \(t_n\), the possibility appears for the formation of the latent image during the exposure, and several electrons can now be fixed at one and the same center (each time after neutralization of the preceding electron). Under these conditions the latent image is less dispersed, and the centers grow more rapidly, which leads to an increase in photosensitivity at exposure times greater than \(t_n\). A further increase in the exposure time should create ever more favorable conditions for the rapid growth of a small number of centers, and hence also for an increase in photosensitivity. However, photosensitivity passes through a maximum, to which the minimum of the isopaque corresponds, and then decreases. Here yet another physical process begins to make itself felt.

The residence time of an electron in one trap or another depends on thermal motion. If \(U\) is the depth of the trap, then the probability of thermal release of an electron from the trap is proportional—

by \(e^{-U/kT}\), and the time of residence of the electron in the trap is equal to \(1/(\nu e^{-U/kT})\), where \(\nu\) is the frequency of thermal vibrations. Under brief illumination, i.e. at a sufficient rate of arrival of electrons at traps, the probability of thermal liberation of an electron from a deep trap before the arrival of the next electron may be neglected, and only the instability of electrons in shallow traps need be taken into account. With a considerable increase in the illumination time, the time interval between the arrival of two successive electrons at the same center becomes comparable with, and then greater than,

\[ \frac{1}{\nu e^{-U/kT}} \]

even for deep traps. There appears a probability that an electron which had previously been fixed will leave the trap before the arrival of the next electron, and after it the silver ion attracted by it, which is bound to the trapping center by nothing else, will also depart. After the arrival of the next electron the process of formation of the latent image will have to begin anew, and it is possible that thermal dissipation will occur again, etc. Therefore, with increasing illumination time, the efficiency of use of photoelectrons for the formation of latent-image centers decreases, although the process of neutralization has time to occur in due course; as a result, the photosensitivity, after passing through a maximum, decreases monotonically with increasing illumination time.

The stability of a silver particle with respect to thermal destruction grows rapidly with an increase in the number of atoms in it, since the height of the potential barrier that the electron must overcome in leaving the particle increases. The thermal stability of a center is attained already at an early stage of its formation, and further growth proceeds without losses. Such centers, possessing stability but insufficient to impart developability to the crystal, are called subcenters of the latent image. At short illumination times they constitute a significant fraction of the total mass of the latent image, whereas at long times they are only an intermediate product, rapidly growing to the size of centers.

The explanation given for deviations from the reciprocity law leads to one very important consequence. If, at illumination times less than \(t_n(\sim 10^{-5}\) sec), neutralization occurs after the end of illumination, then all latent-image centers formed under these conditions consist of only one atom each. Such centers cannot impart developability to the crystals. We know, however, that under these conditions the layer loses photosensitivity only partially, i.e. some of the crystals nevertheless acquire developability. It is necessary to suppose that in the crystals, after illumination, processes occur which lead to a redistribution of the photolytically formed silver

between individual centers, as a result of which some centers are built up to the required sizes. Such processes do indeed exist, but they can be detected only under intermittent illumination of the layer (see Section 3.3).

The physical picture described is based on the existence of two processes—electronic and ionic—proceeding at incommensurable rates; the specific form of both processes is of no essential importance. Whether the donor of electrons is an \(F\)-center or a halide ion, whether the ion \(Ag^+\) will be attracted to the captured electron, as described here, or whether the ion \(Br^-\) will be repelled, as is envisaged in the schemes of P. D. Dankov\(^{36}\) and P. V. Meyklyar\(^{37}\), the decisive significance belongs to the noncoincidence in time of the acts of liberation of the photoelectron and its neutralization, i.e., to the two-stage character of the process of latent-image formation, as Meyklyar\(^{37}\) rightly pointed out. Experimental data on deviations from reciprocity, while very useful for many questions in the theory of latent-image formation, cannot in principle yield anything concerning the question of the character of the electronic and ionic processes.

Let us now give a brief explanation of the regularities of Section 2.2 on the basis of the concepts set forth. The parallelism of the iso-opacs \(\lg H = f(\lg t)\) for different wavelengths can easily be explained if it is borne in mind that the vertical shift between the iso-opacs characterizes the difference in absorption coefficients for the corresponding wavelengths, which does not depend on the time of exposure; therefore, at the corresponding points of the iso-opacs the number of absorbed (and not incident) quanta is the same. The rate at which photoelectrons enter the conduction band at a given exposure time thus proves to be the same for all wavelengths, and if subsequently all photoelectrons behave identically, irrespective of what quantum was absorbed (this assumption seems very probable), then the iso-opacs must be parallel. Conversely, at a given illumination the rate of liberation of photoelectrons depends on the wavelength, and the iso-opacs \(\lg H = f(\lg E)\) cannot be parallel for different wavelengths. Apparently, in general, any regularity in deviations from reciprocity must be clearly expressed precisely for iso-opacs \(\lg H = f(\lg t)\), since here the quantity determining the rate of absorption of quanta, and consequently the entire course of the physical processes responsible for the violation of the reciprocity law, has been chosen as the independent variable. The independent variable \(\lg E\) has no significance, and therefore here one cannot expect to obtain distinct regularities, as we had occasion to say above (see Section 2.2).

The influence of the temperature of the layer is manifested, according to the scheme set forth, in two ways. First, the probability of thermal ...

destruction of the center of the latent image at the beginning of its formation, characterized by the factor \(e^{-\frac{U}{kT}}\). As a result, as the temperature is lowered there occurs an increase in the optimum exposure time, a decrease in deviations at long exposure times, and an increase in photosensitivity at long exposure times. Secondly, the ionic conductivity of the crystal changes (it is proportional to the number of lattice defects) according to an analogous exponential law, which leads to a change in the rate of the neutralization process. The consequence of this is a shift of the plateau boundary in accordance with the dependence \(t_{\mathrm{n}}\sim \frac{1}{\sigma}\), an increase in deviations from reciprocity, and a fall in photosensitivity at short exposure times as the temperature is lowered. Recently, however, it was experimentally shown\(^{38}\) that in this case part of the fall in photosensitivity must be attributed to an electronic process, namely to the decrease in light absorption with temperature, which is confirmed by the change in the value of \(H_{\mathrm{p}}\) with temperature (see Fig. 5). If it is assumed that light is absorbed by \(F\)-centers, then on the basis of the dependence of \(H_{\mathrm{p}}\) on temperature one can determine the activation energy for the thermal formation of \(F\)-centers. Such a determination\(^{11}\) gives the result \((0.06\ \text{eV})\), which is in full agreement with the value of the activation energy determined by another method. The connection between the position of the plateau boundary and the ionic conductivity is also demonstrated by the displacement of this boundary parallel to the change in conductivity in going from bromosilver to chlorobromosilver and chlorosilver layers; there are also experiments\(^{39}\) in which the ionic conductivity was changed by means of pressure applied to the photographic layer during exposure.

The dependence of the shape of the isopaque on the conditions of development is due to the fact that the induction period of development (the time from immersion of the layer in the developer to the appearance of the first traces of blackening) depends substantially on the sizes of the centers of the latent image in the crystal. The larger the center, the shorter, generally speaking, the induction period; with sufficiently prolonged development one may expect developability to appear in crystals with very small centers—such as under other conditions would be regarded as subcenters. Since at long exposure times considerably larger centers are formed than at short exposure times, development for a long time has no special significance for the growth of photosensitivity at exposures of great duration, but it is of substantial significance at short exposures. Therefore the isopaques for different development times approach one another at long exposure times and diverge sharply at short ones. It must also be borne in mind that at short exposure times some of the centers are located in the depth

of the crystal depth. Such deep centers have no contact with the developer and do not lead to development of the crystal, regardless of their size. Even if the developer specially contains solvents of silver halide, in this case too contact between the developer and the silver particle does not arise immediately, and the development process is delayed.

The formation of subcenters also explains the essence of the double-exposure method. The first exposure—short and intense—creates subcenters, while the second—long and weak—builds them up. This eliminates both the reduced capacity for development characteristic of short exposures and the losses of photoelectrons in the initial stage of center formation characteristic of long exposures.

Let us consider how the physical essence of non-reciprocity is reflected quantitatively in Meiklar’s iso-opacity equation (2). Let us first turn to the second term, which describes the region of long illumination times on the basis of the picture set forth here. The factor \(A\) is the probability of thermal destruction of a center and has the form

\[ \nu \sum_{n=1}^{N_0} e^{-\frac{U_n}{kT}}, \]

where the summation over \(n\) (\(n\) is the number of atoms in the center) takes into account the probability of destruction of a center consisting of one, two, etc., atoms. Comparison with experiment shows that the first term plays the predominant role in the sum, i.e., a center of two atoms is already practically stable. The value of the thermal activation energy \(U_2\) for a center of two atoms, found from comparison with experiment, is about \(0.80\) eV, which is in full agreement with data obtained by another method\(^ {40}\). Thus,

\[ A=\nu e^{-\frac{U_1}{kT}}. \]

The quantity \(N_0\), appearing in the denominator of the coefficient

\[ \left(\frac{4A}{N_0^2}\right) \]

at \(t\), is the number of atoms in the center just attaining the size necessary for developability under optimal illumination conditions. Since, with increasing development time, ever smaller centers must acquire the ability to make crystals developable, \(N_0\) must decrease with development time. Bearing in mind that \(N_0\) is proportional to the optimal exposure \(H_0\), and taking into account that comparison with experiment leads for \(H_0\) to formula (3), we find that \(1/N_0\) is a linear function of development time. For a high-sensitivity negative layer, whose iso-opacities are presented in Fig. 6, \(N_0\) is equal to 18 for 2-minute development and decreases to 4 for 30-minute development. \(N_0\) does not depend on temperature.

For times shorter than the optimal time, the second term of equation (2) becomes unity, and the deviations from reciprocity are described by the first term, obtained on the basis of

several other assumptions than those set forth by us. Namely, it is assumed that in this case the photographically ineffective electrons are those which become trapped in shallow traps and determine the inertia of the photoconductivity in silver halide crystals. Leaving, under the action of thermal motion, the sites of temporary trapping, these electrons can recombine with positive holes or with halide atoms, and then they are in fact lost for the latent image. The decline of the photoconductivity after the end of exposure\({}^{35}\) occurs according to the law

\[ (1 + at)^{-\frac{1}{2}}, \]

whence the corresponding factor in the first term arose.

In order to recognize one or another explanation of deviations from reciprocity at short exposure times, the principal importance lies in establishing the duration of the process of latent-image formation up to its complete completion. If the decisive role is played by the quantity \(t_{\mathrm{H}}\), then the mechanism expounded here should be adopted; if such a role belongs to the quantity \(\tau = \frac{1}{a}\), then Meiklar’s explanation should be adopted. For all the curves presented here one can see that it is precisely the duration \(t_{\mathrm{H}}\) that plays a special role. Moreover, in order to explain the experimentally observed regularities in the change of the shape of the isoopac, it is necessary to assign to the relaxation parameter \(a\) a definite dependence on temperature, the chemical composition of the crystals, etc. Such data, if they exist, are only qualitative in character, and a refusal to recognize the most important role of the ionic process under conditions of short-time exposure appears premature. At the same time, relaxation processes must play an essential role in latent-image formation, especially under conditions of intermittent exposure; as will be shown in Section 3.3, it is precisely these processes that determine the regrouping of the latent image after the end of exposure, which occurs under very short-time exposure.

3. DEVIATIONS FROM THE RECIPROCITY LAW UNDER INTERMITTENT EXPOSURE

3.1. Methods of expressing the effect of intermittent exposure

In the case of continuous exposure, the conditions of exposure can be completely characterized by one variable—illuminance or exposure time. In the case of intermittent exposure, both the number of exposures and their duration, their distribution in time, etc., are significant. The number of variables increases, and it is necessary first of all to determine in what relation to one another

...they are with respect to one another. We shall confine ourselves to the case when all individual exposures are equal to one another in duration and illumination, and the dark pauses between them are also equal to one another; other conditions are not encountered in practice. We shall assume that the illumination is produced by Π-shaped light pulses, i.e. that the change of illumination from zero to \(E\) and conversely occurs instantaneously; this approximation is also justified in a number of cases. Let us denote:

\(t_0\) — the duration of an individual light pulse,

\(t'_0\) — the duration of an individual dark pause,

\(T_0\) — the duration of one complete period, i.e. the sum of a light pulse and a dark pause,

\(q\) — the ratio of the durations of the light pulse and the complete period (if the interruption is carried out by a rotating disk with slots, then this quantity is the transmission coefficient of the disk),

\(t\) — the total time of illumination (actual),

\(t'\) — the total time of dark pauses,

\(T\) — the total illumination time including the dark pauses,

\(n\) — the number of light pulses during the time \(T\),

\(f\) — the interruption frequency.

These nine quantities are connected with one another by six obvious relations following directly from the definitions of these quantities:

\[ \begin{gathered} T_0=t_0+t'_0,\qquad T=t+t',\\ t=nt_0,\qquad q=\frac{t_0}{T_0},\\ t'=(n-1)t'_0,\qquad f=\frac{n}{T}. \end{gathered} \tag{6} \]

For a large number of light pulses \((n\gg 1)\), the third of relations (6) takes the form \(t'\simeq nt'_0\); then the 4th, 5th, and 6th relations take the form:

\[ T\simeq nT_0,\qquad q\simeq \frac{t}{T},\qquad f\simeq \frac{1}{T_0}=\frac{q}{t_0}. \tag{6a} \]

Consequently, three independent variables remain, which may be chosen in any manner in accordance with the specific conditions of the experiment. This gives rise to a great variety of ways of expressing the results and makes their comparison with one another very difficult, which is what first of all becomes apparent when one becomes acquainted with the literature\(^2\).

It is necessary to note that almost all authors have tried in one way or another to compare the results obtained by them with the results of the action of continuous illumination equal in time or illumination, which led to the introduction of a number of further quantities. Thus, alongside \(\lg H\), as the ordinate some authors use...

\(\Delta D\) is the difference between the blackening densities produced by intermittent and continuous illumination, and \(\lg H\) plays the role of an independent variable; it should be noted that \(\Delta D\) quantitatively characterizes, to a greater extent, the properties of the layer and the developer than the illumination conditions. Attempts were also made to compare directly the deviations from the reciprocity law under intermittent and continuous illumination. In connection with this, the quantity

\[ E_{\mathrm{av}} = E \cdot \frac{t}{T} = qE \]

was introduced—the mean illumination, i.e., the illumination that would have had to act on the layer during the illumination time \(T\), provided that illumination took place during the dark pauses in the same way as during the light pulses. The physically real quantity

\[ q = \frac{t}{T} \]

turned out in a number of works to be replaced by the fictitious quantity

\[ \frac{E_{\mathrm{av}}}{E}; \]

the fictitious character of \(E_{\mathrm{av}}\) follows at least from the fact that the actual illumination can take only the values \(E\) and zero. All these attempts further complicated the already difficult question of how to express deviations from the reciprocity law under intermittent illumination.

The question of ways of expressing the phenomenon of intermittent illumination can be simplified and, at the same time, a sufficiently complete exposition can be given of the principal regularities of the phenomenon that are of theoretical or practical interest, if one makes use of the circumstance that the whole variety of practically encountered cases of intermittent illumination of a layer can be reduced to two principal types. The first type is the photographic recording of periodically repeated luminescence, connected either with the experimental impossibility of isolating a single luminescence because of its too high frequency, or with the low brightness of the luminescence, as a result of which multiple exposures of the layer are required in order to obtain noticeable blackening. The second type is photographing with the use of a moving exposure modulator, for example a rotating sector disk, often used in spectroscopic and spectrophotometric practice as a neutral attenuator. For the first problem it is necessary to compare the effect of \(n\) identical exposures with the effect of one such exposure, while for the second problem it is necessary to compare the effect of a total continuous exposure \(H\) with the effect of the sum of \(n\) exposures, each equal to

\[ \frac{H}{n}. \]

Accordingly, the choice of independent variables for each of these problems must be different.

For the first problem, the independent variable must first of all be the duration of a single luminescence \(t_0\), as a characteristic of the recorded luminescence. In exactly the same way, an independent variable must be a quantity characterizing the total exposure, say the number of exposures \(n\). Finally, it is necessary to charac-

…to characterize the periodicity of exposure; from a number of quantities suitable for this purpose \((f, q, t_0' \text{, etc.})\), it is convenient to choose the quantity

\[ q=\frac{t_0}{T_0}. \]

For the second problem, the characteristic of the total exposure, usually \(T\), is chosen first of all as the independent variable (although in reality the amount of illumination cannot be written in the form \(E_{\mathrm{cp}}T\) instead of \(\sum Et_0\)). Next, it is necessary to characterize the overall attenuation of the luminous flux by the modulator, expressed by the quantity \(q\). Finally, a characteristic must be given of the subdivision of the total exposure; since the publication of Webb’s work\({}^{41}\), it has been customary to choose the quantity \(f\) for this purpose. Thus, practically the most rational choice is the systems of independent variables \((t_0, n, q)\) or \((T, f, q)\), and we shall limit our discussion to these systems.

The system \((t_0, n, q)\) was proposed by A. L. Kartuzhanskii and P. V. Meiklyar\({}^{42}\). These authors specified a definite value of \(t_0\) and constructed a family of curves \(\lg H=\varphi(\lg n)\) for \(D=\text{const}\) at various values of \(q\), with all the curves naturally coinciding at \(n=1\) (a single exposure of duration \(t_0\)). From relations (6) we have \(n=\dfrac{t}{t_0}\); with \(t_0=\text{const}\), the independent variable is in fact \(\lg t\), and the curves \(\lg H=\varphi(\lg n)\), by analogy with the case of continuous illumination, may be called isopaque curves. Since the point \(n=1\) serves as the transition from intermittent to continuous illumination, the proposed method of expression also makes it possible to compare directly the action of intermittent illumination with the actual illumination time \(nt_0=t\) and the action of continuous illumination with the same illumination time \(t\), and makes it possible to determine whether there are differences in deviations from reciprocity under intermittent and continuous illumination of the layer.

The system \((T, f, q)\) was applied most consistently by Webb\({}^{41}\). He constructed curves expressing the dependence \(\lg H=\varphi(\lg f)\) for \(D=\text{const}\) at specified values of \(T\) and \(q\). For comparison with continuous illumination, values of \(\lg H\) at illumination times \(T\) and \(t=qT\), or, respectively, illuminances \(qE\) and \(E\), were plotted on the same graphs; these values were compared with the limiting values of \(\lg H\) at very high and very low interruption frequencies.

3.2. Regularities of the phenomenon of intermittent illumination

Since the discovery of the phenomenon of intermittent illumination, a large number of works have accumulated whose aim was to establish whether, for this phenomenon, the same regularities exist as for deviations from reciprocity under continuous illumination\({}^{2}\).

Thus, immediately after the Schwarzschild relation had been established for continuous illumination \((Et^p=\mathrm{const})\), an attempt was made to find the same relation for intermittent illumination as well; it turned out that the exponent \(p\) for these two cases does not coincide in magnitude.

In a number of works the existence of quantitative differences between the two phenomena was confirmed; moreover, for a long time it was believed that intermittent illumination can give only a smaller (if not equal) blackening than continuous illumination corresponding to it in duration or illuminance. Only in 1926 was it possible to show \(^{43}\) that intermittent illumination can produce both a smaller and a greater photographic effect than continuous illumination, and that either result is determined by the level of illuminance. It was also shown \(^{43}\) that the form of the characteristic curve of one and the same layer under intermittent and continuous illumination is different, the differences depending on the density of the blackening and being maximal for the initial portion of the curve.

Another approach to the question of the relation between the action of intermittent and continuous illumination is the attempt by a number of authors to apply to the photographic layer Talbot’s law from physiological optics. At a sufficiently high frequency of flicker, the eye perceives the luminous flux from a flickering source as continuous, but weakened by as many times as the observation time (including the dark intervals) is greater than the true time of shining of the source. At a very low frequency of flicker this law is not fulfilled, and the eye perceives the true flux from the source. Although there are no grounds for regarding the eye and the photographic layer as identical in the sense of the perception of light, such an analogy was employed by many authors and received its fullest expression in Webb’s work \(^{41}\).

The conclusions of this work may be formulated in the form of the following three basic propositions:

1. At a sufficiently high frequency of interruption, above a certain critical frequency \((f_k)\), the photographic action of intermittent illumination is equivalent to the action of continuous illumination with mean illuminance \(E_{\mathrm{cp}}=qE\), i.e. of duration \(T\). In other words, the layer as it were redistributes the action of the light over the entire illumination time, including the pauses, and averages the illuminance according to Talbot’s law.

2. At a sufficiently low frequency of interruption, the photographic action of intermittent illumination is equivalent to the action of continuous illumination with the actual illuminance \(E\), i.e. of duration \(t\). In other words, the layer as it were “does not notice” the dark pauses, and simply sums all the individual exposures.

3. At a frequency of interruption below the critical one, but not very small, the action of intermittent illumination is intermediate

between the action of continuous illumination of duration \(T\) and the action of intermittent illumination of duration \(t\).

Schematically, these propositions are illustrated in Fig. 11. As can be seen, the sign of the effect (an increase or decrease in sensitivity at \(f < f_k\)) depends on the relative positions of the times \(T\) and \(t\) on the isopaque of continuous illumination. At low exposures \(\lg H_T > \lg H_t\), and a decrease in frequency leads to an increase in sensitivity; at high exposures the opposite occurs. This, according to Webb, expresses the relation between deviations from reciprocity under intermittent and under continuous illumination.

Fig. 11.

Webb’s point of view for a number of years enjoyed wide currency and is still set forth in most books on scientific photography and spectroscopy. In doing so it was completely overlooked that it represents the consequence of a purely formal analogy. True, subsequently a certain theoretical basis was adduced for it[^44]. However, when attempts were made to apply Webb’s conclusions in practice, various authors repeatedly encountered contradictions. As data accumulated, it became increasingly evident that these conclusions, generally speaking, are not confirmed experimentally[^45][^46][^47], except in individual special cases. Finally, it was shown[^42] that Webb’s own experimental data also do not correspond to the propositions stated above, and that Webb distorted his results in order to fit them into a prearranged scheme. On the other hand, at sufficiently high interruption frequencies the action of intermittent illumination, as it turned out[^42][^46], indeed does not depend on the frequency, and in this sense there exists a certain critical frequency obeying definite regularities.

As was established by the author, at frequencies below the critical one an increase in sensitivity is always observed, irrespective of

illumination and on the mutual position of the times \(T\) and \(t\) in the iso-opacity of continuous illumination (Fig. 12). The critical frequency does not depend on the conditions of development, on the density of blackening, on the choice of the photographic layer, i.e., on factors having decisive significance for deviations from reciprocity in continuous illumination. For the critical frequency the following two dependences are observed: 1) the critical frequency at a constant value of \(q\) varies proportionally to the square root of the illumination; 2) the critical frequency at constant illumination varies proportionally to the value \(q\); thus, the exposure duration

\[ (t_0)_k=\frac{q}{f_k}, \]

corresponding to the critical frequency, at a given illumination is the same for all values of \(q\).

Fig. 12.

Both of these dependences are of essential importance for explaining the mechanism of the phenomenon of intermittent illumination (see Section 3.3). At the same time, they show that, for a photographic layer, fundamentally different laws hold than for the eye, and an attempt to relate the results obtained to the data for continuous illumination is not successful.

Investigation of the dependence \(\lg H=\varphi(\lg f)\) is meaningful for obtaining regularities when expressing results in the system \((T, f, q)\), but not for establishing the relation between the action of intermittent and continuous illumination.

In light of what has already been set forth, the critical frequency no longer has the practical interest for spectroscopy that was previously attributed to it. This quantity serves only as the boundary of the region in which one may freely vary the number of revolutions of the modulating disk and, consequently, need not take special measures to maintain constancy of the speed of rotation. It is nevertheless useful to give some indicative values of \(f_k\). Thus, at \(t=2.5\cdot10^{-3}\) sec and \(q=1:3\) we have \(f_k\sim3500\ \mathrm{sec}^{-1}\), while at \(q=1:48\), \(f_k\sim200\ \mathrm{sec}^{-1}\). At \(t=1\) sec and \(q=1:3\) we have \(f_k\sim200\ \mathrm{sec}^{-1}\), while at \(q=1:48\)

\(f_k \sim 10\ \mathrm{sec}^{-1}\). Intermediate values can be obtained by using the dependences \(f_k \sim E^{1/2}\) and \(f_k \sim q\).

Let us proceed to the description of the regularities observed in the system \((t_0, n, q)\). As an example we shall give a series of data obtained for a highly sensitive negative film\({}^{42}\). In Fig. 13 we present

Fig. 13.

a family of iso-opacs of intermittent illumination for various values of \(q\) at \(t_0 = 0.96 \cdot 10^{-5}\ \mathrm{sec}\). The iso-opacs \(\lg H = \varphi(\lg n)\) are, in their general features, similar to the iso-opac of continuous illumination; in particular, there exists a certain optimum number of light pulses \(n_{\mathrm{opt}}\), at which the light sensitivity is maximal. This number \(n_{\mathrm{opt}}\) decreases, generally speaking, with decreasing \(q\), but at small values of \(q\) it becomes constant, despite the further decrease

Fig. 14.

Fig. 15.

of \(q\). The magnitude of the positive effect, characterized by the difference \(\lg H\big|_{n=1}^{n=n_{\mathrm{opt}}}\), is approximately equal to the analogous difference for continuous illumination \(\lg H\big|_{t=t_0}^{t=t_0 n_{\mathrm{opt}}}\); the latter, however, is not always fulfilled.

Passing to the iso-opacs for other, larger values of \(t_0\), for example \(2 \cdot 10^{-4}\ \mathrm{sec}\) (Fig. 14) and \(10^{-2}\ \mathrm{sec}\) (Fig. 15), we observe essentially the same regularities, with \(n_{\mathrm{opt}}\) decreasing all the time, while the iso-opacs corresponding to different values of \(q\),

come closer to one another. Finally, beginning with a definite value of \(t_0\) (in this series of isoopacs, \(C t_0 = 10^{-1}\) sec.), complete coincidence of all the isoopacs with one another sets in; at the same time the region of the positive effect disappears, i.e. \(\lg H\) has a minimum at \(n=1\), and for all values \(n>1\) increases monotonically, just as occurred in Figs. 13–15 for \(n>n_{\mathrm{opt}}\). The value \(t_0\) at which all the curves merge into one is precisely the optimum time of continuous illumination \(t_{\mathrm{opt}}\). Thus, for \(t_0>t_{\mathrm{opt}}\), summation of a series of exposures on the photographic layer leads only to a decrease in sensitivity. If the total exposure is evaluated

Fig. 16.

by the actual illumination time \(t = nt_0\), it turns out that continuous illumination of the same duration \(t\) has a greater photographic action than intermittent illumination. Only under these conditions, i.e. at small illuminances and large durations of the individual exposures (conditions characteristic of the experimental technique of the early period in the development of photography), is the action of intermittent illumination indeed always less than that of continuous illumination, as was believed before 1926.^{2,43} Under other conditions, for example in Fig. 13, the actions of intermittent and continuous illumination may stand in any relation to one another. Isoopacs of intermittent illumination in this sense give a certain generalization of the results obtained earlier.

The comparison of isoopacs of intermittent and continuous illumination can be made quite visual. Take an isoopac of intermittent illumination, and for some value of \(n\) plot the values of \(\lg H\) for the corresponding values \(t = nt_0\) and

\[ T = t_0\left(1 + \frac{n-1}{q}\right) \]

under continuous illumination, using for the transition from continuous to intermittent illumination the point \(n=1\), where \(t = T = t_0\).

Such curves are plotted in Fig. 16 for isoopacs \(q=1:3\) and \(q=1:2130\), borrowed from Fig. 13. In neither of the two cases do the isoopacs of intermittent illumination coincide with the isoopacs of continuous illumination of duration \(T\) or \(t\), nor are they always intermediate between them, in complete contradiction to Webb.

The regularities of the system \((t_0, n, q)\) are significant for a number of applications of photography in technology. V. K. Prokof’ev\({}^{48}\) drew attention to the sharp difference in the characteristics of intermittent illumination in various light sources used in spectral analysis, for example in the arc and in the spark. Thus, for an alternating-current arc \(t_0 = 10^{-2}\) sec and \(q = 1:2\), whereas for a condensed spark \(t_0 = 2 \cdot 10^{-4}\) sec and \(q = 1:50\) (for clarity it is useful to return to Figs. 14 and 15). Therefore, when selecting analytical pairs of lines, one must not choose lines that differ in luminescence time, for example an arc line and a spark line. Strictly speaking, it is also necessary to take into account that in this case the brightness of the luminescence has a sinusoidal time form; it can be reduced to an equivalent Π-shaped form, but then the effective value of \(t_0\) changes. Moreover, a single flash of a condensed spark is in fact intermittent even within \(t_0 = 2 \cdot 10^{-4}\) sec, breaking up approximately into 20 pulses of \(10^{-5}\) sec each; therefore in this case the summation of pulses must be applied successively twice. To avoid complications, the practically only correct method of calibrating plates should be considered to be the obtaining of intensity marks from the same source and under the same conditions as the excitation of the lines under study; the regularities we have presented have, in this case, an evaluative significance. In general, owing to the complexity of the relationship between the action of intermittent and continuous illumination (cf. Fig. 16) and the obvious nonobservance of the rules following from Talbot’s law, it proves fundamentally incorrect to use continuous illumination for calibrating layers employed under conditions of intermittent illumination. Such a procedure, unfortunately, is still sometimes encountered in spectroscopic practice.

Another question for which these regularities are of substantial importance is the photometry of periodically operating pulsed light sources. Here, from the photographic point of view, the successful choice of the duration of the flash and of the dark pause between flashes is of essential importance, usually carried out by varying the electrical parameters of the supply circuit, if this is permissible under the operating conditions of the source. For such sources the luminescence time is of the order of magnitude from microseconds to milliseconds, and therefore, with a not too large number of summed flashes, a noticeable gain in sensitivity can be obtained. In this case one must strive to make the dark pause as

3 UFN, vol. LI, issue 2

less than the greater the number of pulses being summed. The regularities of Figs. 13 and 14 must necessarily be taken into account when choosing the operating regime of a pulsed light source.

As in continuous illumination, such factors as the temperature of the layer during exposure, the conditions of development, the choice of blackening density, etc., have a substantial influence on the results obtained. As for the spectral composition of the acting radiation, here, as in the case of continuous illumination, the form of the isoopaque does not depend on the wavelength; such a result appears quite natural, since the independent variable \(\lg n\) is in fact \(\lg t\).

Fig. 17.

The influence of temperature has been studied in detail only for the system \((t_0, n, q)\)^{13}. For the system \((T, f, q)\) it is known only that the critical frequency decreases with temperature^{23}. The temperature dependence of the isoopaques of intermittent illumination is illustrated by Fig. 17. Here we encounter some regularities known for continuous illumination; among them, first of all, is the distinct displacement of the minimum of the isoopaque, i.e. an increase of \(n_{\mathrm{opt}}\) with decreasing temperature. Also noteworthy is the considerable increase of the positive effect

\[ \left.\lg H\right|_{n=n_{\mathrm{opt}}}^{\,n=1} \]

with decreasing temperature; in the case of continuous illumination a similar dependence, although it exists (cf. Fig. 5), is weakly expressed.

In connection with the temperature regularities, it seems interesting to note the following. As we know, the isoopaques of intermittent illumination corresponding to different values of \(t_0\) do not coincide with one another (see, for example, Figs. 13 and 14). However, if the compared values of \(t_0\) are sufficiently small—less than a certain quantity \(\theta\), then, for equal values of \(q\), the isoopaques for these times coincide; if, however, at least one of the values of \(t_0\) exceeds \(\theta\), then the isoopaques of intermittent illumination do not coincide^{13}. The quantity \(\theta\), as it turns out, increases when the temperature is lowered. Thus, at a temperature of \(-20^\circ\) C the isoopaques coincide for values of \(t_0\) less than \(10^{-4}\) sec.; at a temperature of \(+20^\circ\) C the isoopaques coincide for values of \(t_0\) less than \(10^{-5}\) sec.; and at a temperature of \(+60^\circ\) the isoopaques with \(t_0 > 10^{-5}\) sec. differ among themselves in accordance with the previously established

dependence of \(n_{\mathrm{opt}}\) on \(t_0\). The dependence of \(\theta\) on temperature is of fundamental importance for elucidating the causes of the phenomenon of intermittent illumination.

The influence of development conditions has likewise been investigated only for the system \((t_0, n, q)^{42}\). The regularities obtained (Fig. 18) are fundamentally similar to the analogous regularities for continuous illumination (cf. Fig. 6).

The dependence on the density of blackening has been studied by many authors. Unfortunately, in most cases the presentation of the results was unsuccessful: the authors readily used the quantity \(\Delta D\) (see section 3.1), which depends too strongly on the properties of the layer and on the experimental conditions. As was to be expected, the characteristic curve under intermittent illumination coincides with the characteristic curve under continuous illumination neither on the illumination scale nor on the time scale \((\Delta D \ne 0)\), but is usually intermediate between them \(^{49}\).

Fig. 18.

All authors note a change in the shape of the characteristic curve in passing from continuous to intermittent illumination of the layer, especially for the initial portion of the curve. The magnitude and sign of \(\Delta D\), obtained in different works, do not coincide, since some authors used the time scale and others the illumination scale. When compared with the curve obtained on the time scale, \(\Delta D\) is negative at small illuminations \(^{43}\), and depends on \(q\) according to the equation \(\Delta D\left(\dfrac{q}{1-q}\right)^k=m\), where \(m\) and \(k\) are constants for the given layer \(^{49}\); at large illuminations \(\Delta D\) is positive, i.e. the action of intermittent illumination is greater than the action of continuous illumination equal to it. On the average, \(\Delta D\) is the smaller the more sensitive the layer \(^{49}\), although there are exceptions to this rule \(^{43}\).

Figure 19 gives, borrowed from work \(^{42}\), a family of characteristic curves for various values of \(n\) at fixed values \(t_0(=10^{-5}\ \mathrm{sec.})\) and \(q(=1:2000)\). As can be seen, the transition from \(n=1\) to \(n>1\) (from continuous illumination to intermittent) is accompanied by a rapid growth of \(\gamma\): at \(n\sim100\), \(\gamma\) under intermittent illumination is 25–35% greater than \(\gamma\) under continuous illumination with the same actual illumination time \(t\). At the same time the initial nonrectilinear portion of the curve is deformed. At small values of \(t_0\), when the initial portion under

under continuous illumination \((n=1)\) it is very extended; its deformation with increasing \(n\) occurs so rapidly that the curves intersect with their initial portions. As a result, on the isoopacs of intermittent illumination (Fig. 20) there may even appear an additional inflection, disturbing the monotonic course of the isoopacs; for large values of \(D\) this inflection is smoothed out. This feature distinguishes the isoopacs of intermittent illumination from the isoopacs of continuous illumination; otherwise (cf. Fig. 8) the regularities of the deformation of the isoopacs with change in the density of blackening are similar.

Proceeding from the regularities set forth in this section, it is interesting to estimate what error intermittent illumination introduces into the determination of the light sensitivity of photographic layers by the Hurter and Driffield sensitometry system, the system most widespread among us up to 1951 and even now not yet completely out of use. In this system intermittent illumination of the layer is provided on a time scale, carried out with the aid of a rotating disk with stepped cutouts. Consequently, for each field of the sensitogram, and hence also for each density of blackening, there are its own values of \(t_0\) and \(q\), different from those of neighboring fields; since the total exposure time \(T\) is not specified and is chosen depending on the sensitivity, \(n\) may also vary within wide limits.

Fig. 19.

Fig. 20.

Calculation of the ratio of sensitivities \(S_{\text{interr}} : S_{\text{noninterr}}\) by means of the stated regularities shows that under the most favorable conditions there is inevitably a discrepancy of \(\sim 20\%\) between the sensitivity according to H. and D., used in the exponometric formula, and the actual sensitivity of the layer under conditions of its practical use with continuous illumination; under unfavorable conditions this discrepancy may reach 2- to 3-fold. Direct measurements\({}^{28}\) have also repeatedly confirmed the existence of discrepancies of almost 2-fold. Such an error is completely

...is inadmissible, especially in the case of high-contrast materials. Therefore the rejection of the H. and D. system and its replacement by the GOST 2817-50 system is of great importance for bringing the conditions of layer testing closer to the conditions of their use.

3.3. Physical causes of the phenomenon of intermittent illumination

Unlike the reciprocity-failure phenomenon under continuous illumination, the theory of the phenomenon of intermittent illumination has been developed only slightly. This is apparently due to the fact that the literature was long dominated by the view that, in Webb’s work, the phenomenon of intermittent illumination had been reduced to deviations from reciprocity under continuous illumination and therefore did not require any special explanation. It is true that the introduction of Talbot’s law for the layer was so formal that subsequently an attempt was made to give a more convincing theoretical foundation to these works. Such a foundation was supposed to be, at first sight, the quite plausible assertion[^46] that, owing to the quantum nature of light, any illumination is essentially intermittent, and that the critical frequency is precisely the limiting frequency at which the differences between intermittent and continuous illumination disappear. This assumption denies any specificity whatsoever to intermittent illumination and asserts that the process of latent-image formation proceeds identically during illumination and during the dark pauses. The corresponding calculations did not yield quantitative agreement with experiment; to save the hypothesis as a whole, the rather implausible assumption was made that the emulsion crystal absorbs light not over its entire area, but only over a certain part of it, which in some cases[^23] amounts to less than 2% of the entire projection area of the crystal. The conditions for the formation of the latent image at frequencies below the critical one, where intermittent and continuous illumination should differ, were not analyzed at all.

Since Webb’s principal experimental results proved to be refuted, the need also disappeared for those theoretical considerations by which these results were explained. The experimental data clearly indicate the impossibility of reducing the action of intermittent illumination to the action of continuous illumination, and the physical picture of latent-image formation makes it possible to discern the reason for such a difference in the existence of processes occurring after the end of illumination, during the dark pauses. Such processes are the already considered processes of neutralization, which determine the growth of the latent image, and of thermal destruction of the centers of the latent image, as well as the aforementioned process of regrouping of the latent image.

the latter, as will now be shown, is identical with the process of relaxation of photoconductivity in crystals of silver halide salts^35.

As has already been said, under the action of light in AgBr and AgCl crystals not only deep traps but also shallow traps are filled with photoelectrons (moreover, apparently, the shallow ones are filled before the deep ones). After illumination ceases, the photocurrent in the crystals does not disappear at once, since electrons are released from the shallow traps owing to thermal motion, and the released electrons again appear in the conduction band before finally becoming fixed at a deep trap or recombining. This is what constitutes the process of photoconductivity relaxation. The decrease of the photocurrent after the end of illumination takes place according to the hyperbolic law \((1+at)^{-\alpha}\), where \(\alpha\) is close to \(0.5\) over a rather wide range of illuminations, while the parameter \(a\) is of the order of \(10^3\)—\(10^4\ \mathrm{sec}^{-1}\) and varies proportionally to the square root of the illumination, as was to be expected for a photoresistance with a bimolecular character of the decay of photoconductivity^50. The quantity

\[ \tau = \frac{1}{a} \]

may serve as a measure of the duration of the relaxation process.

The mechanism of the relaxation process may be directly applied to the explanation of the process of rearrangement of the latent image after the termination of a short-time illumination; the necessity for the existence of such a process is connected with the fact that some silver centers, initially consisting of a small number of atoms and not possessing developability, acquire the capacity for development after the illumination has ended and, consequently, grow (see section 2.3). In short-time illumination, as has already been said, the fixing of electrons, which is the beginning of the process of latent-image formation, occurs simultaneously at many points of the crystals, including such points as are shallow traps. After the illumination has ended, not all fixed electrons behave in the same way; those of them which became fixed at shallow traps are easily released from them and may, at least in part, fall into deep traps, supplementing the centers that arose there during the illumination. If we take Meiklejohn’s point of view (see above), then probably all electrons are initially fixed at shallow traps and only afterward pass into deep ones. In one way or another, the duration of such a process \(\tau\) must be decisive for the completion of the process of latent-image formation under conditions of intermittent illumination of high frequency.

Although in considering the relaxation of the photocurrent the discussion concerned only the fixing and transfer of electrons, while here neutralizing ions, possibly connected with the fixed electrons, are also involved, this

...does not introduce fundamental changes. The motion of an ion after the electron as the latter leaves the place where it was fixed (provided that the electron does not recombine with a halide atom or hole) takes place in a time of the order of \(10^{-5}\) sec., which is much less than \(\tau(>10^{-4}\) sec.), and does not limit the rate of the relaxation process. The exception is those electrons which, together with the ions that neutralized them, become finally trapped in shallow traps and initiate the growth of additional, smaller centers of the latent image.

The process of regrouping of the latent image described here has so far been confirmed experimentally only by indirect data relating to continuous illumination. Under intermittent illumination, one can also point to a direct, photographically observable confirmation of its existence. Such is the dependence \(\lg H=\varphi(\lg f)\) and the associated dependences \(f_k \sim \sqrt{E}\) at \(q=\mathrm{const}\) and \(f_k \sim q\) at \(E=\mathrm{const}\). From the relation \((t_0)_k=\dfrac{q}{f_k}=\mathrm{const}\) (for \(E=\mathrm{const}\)) it follows that the critical duration of the light pulse is a quantity characterizing a certain process determined only by the level of illumination. From the dependence \(f_k \sim \sqrt{E}\) it follows that \((t_0)_k \sim \dfrac{1}{\sqrt{E}}\), i.e., the duration of this process depends on the illumination according to the same law as \(\tau\); the order of magnitude of \((t_0)_k\) and \(\tau\) is also the same. All this makes it possible to identify \((t_0)_k\) with the duration of the relaxation process and to represent the dependence \(\lg H=\varphi(\lg f)\) in the following way.

During one light pulse \(t_0\), a certain number of photoelectrons were liberated in the emulsion crystal, filled shallow traps, and only after a time \((t_0)_k\) became finally fixed. If the interruption frequency is such that the duration of an individual pulse \(t_0\) is less than \((t_0)_k\) (i.e., \(f>f_k\)), then the relaxation process occurs chiefly during the dark intervals; consequently, the photographic action should not depend on \(t_0\), but may depend on the duration of the dark interval. By reducing the frequency, we make \(t\) equal to or greater than \((t_0)_k\); in this case the relaxation process is mostly completed during illumination, and the efficiency of utilization of the photoelectrons increases, just as it did in the case of continuous illumination when the illumination time passed through the value \(t_{\mathrm{n}}\). Therefore the photosensitivity has the smallest, and moreover constant, value for all frequencies when \(f>f_k\), while for \(f<f_k\) it increases (see Fig. 12).

If at high frequencies the relaxation process takes place chiefly during the dark intervals, then it may turn out that the duration of the dark interval will also be insufficient

for its completion. Therefore one should expect a dependence of the limiting value \((\lg H)_k\) for \(f \gg f_k\) on the duration of the dark pause, of course for a given level of illumination. Indeed, at high illuminations \((\lg H)_k\) decreases with decreasing \(q\), i.e., the photosensitivity increases with increasing dark pause. As the illumination decreases this dependence gradually becomes the opposite—\((\lg H)_k\) increases with decreasing \(q\): here the gradually increasing probability of thermal destruction of the latent image during the dark pauses begins to make itself felt.

The explanation of the isopachs of intermittent illumination also requires taking the relaxation process into account. It may be assumed that the optimal conditions for the formation of latent-image centers under intermittent illumination are those under which, during the dark pause, the regrouping process has time to occur, i.e., the transfer of electrons from shallow traps to deep ones, but thermal destruction of the centers formed does not have time to occur. Increasing the number of light pulses from \(n=1\) leads to a decrease in the number of photoelectrons produced during the action of each pulse, and the process of forming stable centers proceeds more completely. As the number of light pulses increases, however, the number of electrons released during one pulse becomes ever smaller, and the centers formed also have the possibility of being partially destroyed before the beginning of the next pulse. Equilibrium of these two processes corresponds to the optimal number of pulses; therefore the decrease of \(n_{\mathrm{opt}}\) with increasing duration of the dark pause, i.e., with decreasing \(q\), becomes understandable. True, at very small values of \(q\), \(n_{\mathrm{opt}}\) ceases to change (Figs. 13 and 14), but this is connected with the fact that, at such long pauses, thermal destruction begins to have an effect even in the region \(n < n_{\mathrm{opt}}\). It is precisely for this reason that the isopach \(q = 1:500\,000\) (Fig. 13) lies entirely higher than the isopachs \(q = 1:2130\), \(1:698\), and even \(1:48\), although all the other isopachs mutually intersect.

The idea of the dual role of the dark pause also makes it possible to explain why, with increasing temperature, \(n_{\mathrm{opt}}\) decreases, and also why the positive effect

\[ \lg H \left|_{n=1}^{\,n=n_{\mathrm{opt}}}\right. \]

decreases (see Fig. 17)—here the probability of thermal destruction increases according to the already known law

\[ e^{-\frac{U}{kT}}. \]

In an analogous way one explains the decrease of \(n_{\mathrm{opt}}\) with increasing \(t_0\): since the growth of the latent image occurs to an ever greater extent during illumination, the positive role of the dark pauses diminishes, while the probability of destruction of the latent image increases. When \(t_0\) becomes equal to the optimal time of continuous illumination, the dark pause has only a negative significance, promoting destruction; therefore

VIOLATIONS OF THE PHOTOCHEMICAL LAW OF RECIPROCITY

precisely at \(t_0 \gg t_{\mathrm{opt}}\) the positive effect disappears completely, and \(n_{\mathrm{opt}}=1\). The negative role of dark pauses under these conditions has as its consequence a more rapid fall of the light sensitivity with increasing number of pulses \(n\) than is observed for continuous illumination of the same actual duration \(t=nt_0\).

The explanation given for the dependence \(\lg H=\varphi(\lg n)\) shows that the cause of the decrease in light sensitivity at \(n<n_{\mathrm{opt}}\) is the formation of a large number of small centers, whereas at \(n \gg n_{\mathrm{opt}}\) it is the thermal destruction of the latent image in the initial stage of its formation. These same causes account for the decrease in light sensitivity at illumination times smaller and larger than the optimum in the case of continuous illumination. There is nothing surprising in the fact that the regularities of the change in the form of the isopaque as a function of a number of factors are in principle the same in both cases (cf. Figs. 17 and 5, 18 and 6, 20 and 8).

The inadequacy of explaining the isopaque of intermittent illumination solely by the ionic process, without the participation of the relaxation process, can be demonstrated by the following example. For the isopaques of Fig. 13, \(t_0=0.96\cdot 10^{-5}\) sec., and, consequently, neutralization each time ends almost immediately after the action of the light pulse has ceased. Already at \(q=1:3\) the duration of the dark pause is more than sufficient for completion of neutralization; however, the isopaque continues to deform. In this case \(n_{\mathrm{opt}}\) continues to decrease and reaches a constant value only at \(q\approx 0.01\) (more precisely, between \(q=1:48\) and \(q=1:698\)), which corresponds to a dark-pause duration of \(\sim 10^{-3}\) sec., approximately equal to \(\tau\).

At the same time, some regularities of the phenomenon of intermittent illumination are definitely connected precisely with the ionic process. These include, for example, the fact noted in Section 3.2 that the isopaques of intermittent illumination coincide for two different values of \(t_0\), if both these values are less than a certain definite quantity \(\theta\), which decreases with increasing temperature. The values \(\theta\)—\(10^{-4}\) sec. at \(-20^\circ\)C, \(10^{-5}\) sec. at \(+20^\circ\)C, and \(10^{-6}\) sec. at \(+60^\circ\)C—correspond precisely to the quantity \(t_{\mathrm{n}}=\dfrac{\varepsilon}{2t\sigma}\), but are much smaller than \(\tau=\dfrac{1}{a}\). This means that the isopaques coincide only for those values of \(t_0\) for which neutralization occurs exclusively during the dark pauses. If, however, one of the values of \(t_0\) exceeds \(t_{\mathrm{n}}\) (but may nevertheless be considerably smaller than \(\tau\)), then the corresponding isopaque undergoes deformation caused exclusively by the ionic process.

We have considered the last example in order to emphasize that, while taking into account the very important role of the relaxation process for the formation of the latent image, especially under conditions of intermittent

illumination, does not exclude, but, on the contrary, supplements the physical picture that was given earlier. The study of intermittent-illumination phenomena thus makes it possible to refine our ideas about the mechanism of formation of the latent image, relating it to processes that had hitherto been known only for coarser, non-emulsion crystals of silver halide.

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