TER VERKRIJGING VAN DEN GRAAD VAN DOCTOR IN DE WIS- EN NATUURKUNDE AAN DE RIJKSUNIVERSITEIT TE UTRECHT, OP GEZAG VAN DENnbsp;RECTOR MAGNIFICUS DR. H. R. KRUYT, HOOG-LEERAAR IN DE FACULTEIT DER WIS- EN NATUURKUNDE, VOLGENS BESLUIT VAN DEN SENAAT DERnbsp;UNIVERSITEIT TEGEN DE BEDENKINGEN VAN DEnbsp;FACULTEIT DER WIS- EN NATUURKUNDE TEnbsp;VERDEDIGEN OP DONDERDAG 10 JULI 1941, DESnbsp;NAMIDDAGS TE 4 UUR

ERNST KATZ

BIBLIOTHEEK DER RiJKSUNIV�RSITEITnbsp;UTRECHT

PROMOTOR: PROF. DR. M. G. J. MINNAERT.

DIT PROEFSCHRIFT WERD GROOTENDEELS ONDER LEIDING VAN WIJLEN PROF. DR. L. S. ORNSTEIN BEWERKT EN ONDERnbsp;DIE VAN DR. ]. M. W. MILATZ VOLTOOID.

The author is greatly indebted to the Independent Aid Co (New-York). especially to Miss M. D. Paschal, for grants during his study.

The ornament at the frontispiece was taken from a photomicrograph, published by J. H. Webb (Jl. of Appl, Phys. 11, p. 19, 1940). It represents silver-bromidenbsp;grains of a photographic emulsion.

At one time a drawing had to fulfil a twofold purpose: it represented a true image of an object and it showed an artisticnbsp;treatment. The artist and the �picturer� were identical to a largenbsp;degree and their work was an entity.

Between 1816 and 1839 Niepce and Daguerre invented a method of photography in the sense of producing a true and fixed image,nbsp;registered by the light itself. Photodecomposition of silver saltsnbsp;had been observed before, but these investigators utilized thisnbsp;process for the first time for this purpose technically on a largenbsp;scale t).

Innumerable �true imagesquot; devoid of any artistic quality have been produced since; on the other hand art produces picturesnbsp;deviating strongly from �reality�. �L�art pour I�artquot; appearednbsp;parallel with a trend towards �photography for true images�. Thisnbsp;is one of the many ways by which the principle of division ofnbsp;labour has divided mankind during the past century into twonbsp;species; the artist and the scientist. Will they ever reunite?

Though photography has rendered great services in all branches of science for the documentation, measurement, and understandingnbsp;of many phenomena, and though the photographic technique hasnbsp;been developed to a high level, no clear understanding has yetnbsp;been attained of the physical and physico-chemical processesnbsp;occurring in a photographic plate when a picture is taken. Of coursenbsp;certain groups of phenomena have been investigated carefully.nbsp;Theories have been advanced by several schools. But the large

number of theories for several essential parts of the photographic process proves that a general and satisfactory viewpoint has notnbsp;been reached yet.

Usually the difficulties arc not essential, but arise mainly from the complexity of the problems. The extensive researches, performed to elucidate the mechanism of the photographic process arenbsp;justified by the wide field of practical applications of the platenbsp;or film in science, industry and daily life. Also the plate, withnbsp;its fine grains of silverhalogenide, shows some photochemical ornbsp;photophysical properties � e.g. latent image formation � whichnbsp;could not be studied so easily with other objects. So their studynbsp;may well enrich photo-physics and -chemistry. Finally the photographic plate has several essential features in common with manynbsp;systems in radiobiology, which is due to the smallness of its photosensitive grains. It is more easily accessible to research though, andnbsp;may thus also contribute indirectly to progress in the latter field.

The physical and physico-chemical problems directly connected with photographic emulsions may roughly be divided into fournbsp;groups, concerning:

b. nbsp;nbsp;nbsp;effect of light or other agents (heat, other rays, pressure etc.)nbsp;upon the photographic material (latent image formation);

d. nbsp;nbsp;nbsp;quantitative description of the results (densitometry, etc.) andnbsp;their relation to the parameters of a, b and c (sensitometry, etc.).

This thesis deals mainly with a few aspects of latent image formation (cf. � 5). For the sake of orientation of the reader wenbsp;shall outline in the first chapter a number of fundamental conceptsnbsp;or problems and some work of other authors which has led us tonbsp;the investigations presented in this thesis.

The purpose of this section is, to resume a few basic facts and concepts. It is necessarily incomplete; for extensive information wenbsp;refer to handbooks.

The principle of preparation of photographic emulsions is; addition of silvernitrate to a moderately heated aqueous solutionnbsp;of gelatin and potassium halogenide (mostly bromide) in excess.nbsp;Linder suitable conditions a yellowish emulsion of very finenbsp;crystals or grains of silver halogenide is formed which solidifiesnbsp;after cooling to room temperature. Special after-treatmentsnbsp;(ripening processes) increase the average grain size of the emulsionnbsp;and its sensitivity to light. The role of gelatin is in the first placenbsp;stabilisation of the emulsion both in a mechanical and colloidalnbsp;sense. Also certain sulphuric impurities occurring at remarkablynbsp;low concentrations in gelatin are essential for good light sensitivitynbsp;of the emulsion 4).

A photographic plate consists of a glass plate covered at one side with a layer of ca. 20 [i of such an emulsion. It is oftennbsp;coated at the back side with a dark substance to avoid falsenbsp;reflections. Its sensitivity can be extended to colours of longernbsp;wave length than blue by adsorbing suitable substances to thenbsp;grains. In general these will sensitize the plate only for radiationsnbsp;which they are able to absorb, but an other mechanism is alsonbsp;known (shift of AgBr absorption, due to adsorbed ions).

The sensitive layer of the plate contains ca. 10� grains per cm^; their shape is mostly like a flat disk; they are oriented preferablynbsp;parallel to the plate. The average size of the grains usually liesnbsp;between 0.3 and 2 /x. The grain size distribution in one emulsionnbsp;has a rather flat maximum.

The absorption coefficient of the grains is very large for light of wave lengths below 4500 A. For unsensitized grains thenbsp;absorption for wave lengths above 4800 A is negligible. Innbsp;sensitized plates the adsorbed dye molecules absorb the light andnbsp;transmit some action to the grains (cf. � 4).

An ordinary photographic emulsion is susceptible to many influences. Among these are mainly light, containing radiation

of wave lengths below 4500 A (blue and violet); ultraviolet-, X-, y-, j8-, a-, atomic- and cosmic radiation, heat, mechanicalnbsp;pressure, etc.

The action of light has been studied most extensively. We shall not discuss other agents, though in some cases their action is morenbsp;easily understandable.

In order that light can act on a photographic plate, it is necessary that it is absorbed. In the beginning of this century the quanticnbsp;nature of all processes of emission and absorption of radiation wasnbsp;well established. Of course the photographic process should benbsp;understood in one line with all other absorption processes. Innbsp;1922 5) and again in 1931 6) Silberstein deemed it necessary tonbsp;introduce this idea into photographic literature.

It is rather amazing that at present authors are still found who do not yet take this into account'^) �).

If a sufficient quantity of light acts on a plate, it becomes darker, and black after long exposures. This phenomenon is callednbsp;direct blackening or print-out effect. The light, absorbed by thenbsp;grains, decomposes silverbromide (AgBr) into silver (Ag) andnbsp;bromine (Br). The Br is bound to gelatin, the Ag is responsiblenbsp;for the blackening.

It is generally known 3) from microscopic observations that the silver is deposited, concentrated or coagulated at discrete spotsnbsp;both at the surface and in the interior of the grains, for ordinarynbsp;plates as well as for sensitized ones 3), Also general evidencenbsp;indicates that such coagulation occurs only at places in the grainnbsp;where traces of Ag2S or Ag are present (due to the ripeningnbsp;process 3) 9)).

The effect of an exposure is measured either by determining chemically the amount A of Ag (or Br) formed per cm^, or bynbsp;determining the change in density d of the print-out effect. Thisnbsp;quantity d is defined as the logarithm (base 10) of the ratio of the

intensity Iq of light, incident on the plate for the measurement of d, and the intensity I transmitted by the plate: d = log (Iq/I).nbsp;The result is somewhat dependent on the colour of the measuringnbsp;light, on its angular distribution and on the aperture of thenbsp;measuring device. However these dependences are often of minornbsp;importance since they consist mainly of proportional changes ofnbsp;different densities.

The lower limit, down to which the first method has been applied, lies at exposures of about 10^^ quanta/cm^ absorbed or ca. 10^��nbsp;quanta per grain absorbed 3). The quantum efficiency (atoms Agnbsp;per quantum absorbed) is ca. 1 in these experiments.

The lower limit down to which the second method has been applied (Jurriens 1938 lo)) lies at exposures of about 10^*nbsp;absorbed quanta,/cm2 or ca. 400 quanta absorbed per grain.

The region of photographic exposures for practice lies between about 10^�� andnbsp;nbsp;nbsp;nbsp;absorbed quanta/cm^ or 1 to 100 or 1000

The results of measurements of the amount of silver A or the density d can be collected in an (A; I; t) or (d; I; t) surface,nbsp;(I = intensity of irradiation; t = time of exposure), or in sets ofnbsp;curves in plots of either A or d against one of the independentnbsp;variables I or t, with the other one as a parameter.

(A; t) lines for constant intensity of blue irradiation start linearly from the origin within the limits of error; for increasing time theynbsp;increase less rapidly 3) n).

(d;t) lines for constant I have a similar shape for blue irradiation according to Jurriens lo)^ but this author reports thatnbsp;upon intense red irradiation non sensitized plates yield curves withnbsp;a remarkable S-shape or threshold.

If a quantity of light, about 100 times smaller than would be required to produce a detectable print-out effect, acts on a plate,nbsp;something changes in the grains. A number of grains, dependentnbsp;on the conditions of exposure, becomes developable; i.e. certainnbsp;reducing substances (developers), which leave unexposed grains

unaltered, will be able to reduce these grains to Ag. Normally, if a grain is affected by a developer, it is reduced entirely and becomesnbsp;a black grain of silver.

After solving unreduced AgBr grains (fixing), washing and drying the plate we can measure its (developed) density, whichnbsp;we shall denote by D in contrast to the direct density d. Thenbsp;result is again somewhat dependent on the conditions of measurement.

If we adopt any standard method for development, fixing, density measurement etc., the developed density D is determinednbsp;by the state of the latent image. In this connection we shall introducenbsp;the term �developable density� of a latent image, this being thenbsp;density that would be obtained if this latent image should benbsp;developed etc. according to this standard method. The developablenbsp;density is not sufficient to characterize the latent image.

Many arguments point to the assumption that the latent image consists of a minute quantity of silver. In other words the latentnbsp;image and the print-out effect are essentially the same. We shallnbsp;adhere to this view, in agreement with the general trend innbsp;present-day literature.

Owing to the fact that no other method than development can show the presence of the latent image after normal exposures, allnbsp;evidence that it consists of Ag necessarily is of an indirect nature.nbsp;To obtain such indirect evidence two ways have mainly beennbsp;followed.

Firstly large AgBr crystals have been investigated, and their Ag-production was detectable after exposures well within the regionnbsp;of development of AgBr emulsions. Also a very small Ag contentnbsp;of unexposed AgBr emulsions was observed by Sheppard e.a. 4).

Secondly the behaviour of the latent image towards various chemical influences has been investigated. Extensive studies onnbsp;Ag-AgBr adsorbates by Reinders, L�ppo-Cramer e.a., provednbsp;that adsorbed Ag can behave just like the latent image does.

A peculiar feature of the latent image is its distribution through the grains. Microscopic observations of slightly developed grainsnbsp;show that the latent image is concentrated at certain centresnbsp;(sensitivity specks), just as in the case of print-out silver.nbsp;In fact, a grain is developable only if at least one speck at itsnbsp;surface has become sufficiently large (cf. also � 10.2). These

centres are, according to extensive researches of Sheppard, Trivelli, e.a., specks of Ag2S; also other irregularities in thenbsp;AgBr grain such as Ag atoms act as such to a lesser extent. A verynbsp;convincing argument for this view is that if AgBr emulsions arenbsp;made with gelatin containing a relatively small amount of sulphurnbsp;their sensitivity is very low, but is restored on addition of certainnbsp;sulphur compounds.

The mechanism of this concentration of silver has not been understood, till recently the theory of Gurney and Mott onnbsp;latent-image formation succeeded in explaining it satisfactorily (�4).

� 3. The developed density as a function of wave length, intensity and time of exposure.

In this section we continue the general remarks of � 2, pertaining more to the quantitative side of density production,

The transformation law (named thus by van KreveldIS)) expresses that equality of densities is independent of the methodnbsp;of development etc. This law is valid only in first approximation.nbsp;It is clear that it cannot be valid strictly. For instance: one developernbsp;reduces all grains in which n or more quanta have been �effective�,nbsp;and another one already acts on grains with (n�1) effectivenbsp;quanta. The grain size distributions for the two emulsions maynbsp;differ, so that equality in the former case will cause a slightnbsp;difference in the latter one.

For normally developed plates the density is in first approximation only a function of the energy, i.e, the product of intensity and timenbsp;of irradiation, called exposure (cf. � 3. 3). We shall first neglectnbsp;deviations from this �reciprocity-law�.

Considering a developed-density-log exposure curve of a general type, we distinguish three parts (fig. la).

in fig. lb against the exposure itself. It shows a typical S-shape, usually starting quadratically from the origin. The inflectionnbsp;tangent meets the abscissa at an exposure value which we shallnbsp;call threshold and the ordinate at a negatieve density value whichnbsp;we shall call density^defect. The middle part is more or less straight;nbsp;its slope is called contrast. This is the most important region fornbsp;photography. The produced straight part meets the log exposurenbsp;axis at an energy value called inertia. In the upper part we noticenbsp;saturation and then again a decrease of density with increasingnbsp;exposure (solarization, first order reversal) followed by severalnbsp;reversals of a higher orderly)

Schematical representation of density as a function of exposure, a. plotted against log exposure, withnbsp;nbsp;nbsp;nbsp;b. enlarged lower part I, plotted against

Within the mentioned approximation (reciprocity law) density-exposure curves for the different colours coincide in the first part if the exposure isnbsp;expressed in absorbed quanta per unit area. Consequently, in plots againstnbsp;incident energy the density-defect is independent of the colour.

The exact dependence of saturation density, contrast, inertia, threshold and density-defect on conditions of exposure and development is rather complicated.nbsp;We restrict ourselves to the following remarks:

The contrast is different in (D; log t) or (D; log I) plots. It also depends weakly on the value of the parameter (other variable): for parameter valuesnbsp;of the same order of magnitude the (D; log I) curves are practically parallel.

The same is true for (D; log t) curves. Also the contrast is a function of the conditions of development, especially of the time of development, but fornbsp;sufficiently long times the kind of developer has hardly any influence. Moreovernbsp;the contrast depends on the composition of the incident radiation (cf. Ch. V).nbsp;Usually it is higher for wave lengths for which the plate has been sensitized,nbsp;than for those absorbed directly in AgBr. This dependence and the fact thatnbsp;the shape of the curves in the lower part (I) somewhat depends on wavenbsp;length cause (D; log t) or (D; log I) curves for various colours to be innbsp;general distinctly non-parallel.

The reciprocity law states that the developed density D is only a function of the product of intensity I and time t, so only of thenbsp;energy E absorbed. It only holds with good approximation fornbsp;relatively small intervals of I and t; deviations are called reciprocitynbsp;law failure (r.l.f.).

a. log energy, required to produce a nbsp;nbsp;nbsp;b. density, produced by a given amount

given density Do, against log intensity nbsp;nbsp;nbsp;of energy Eo, against log intensity

The situation is usually shown in a r.l.f. diagram, in which log E = log I.t, required to produce a given density Dq is plottednbsp;against log I (fig. 2a) or against log t; in literature plots of thenbsp;density D, produced by a given energy Eq, as a function of log Inbsp;(fig. 2b) or log t are also found. The reciprocity law would innbsp;all cases yield a horizontal line, but the curves have a more or

less hyperbolic shape. Extensive investigations on questions concerning the r.l.f. were performed e.g. by Jones

The minimum or maximum in a plot with constant Dg or Eg corresponds to optimal efficiency: it is obtained for optimalnbsp;intensity Ig and optimal time tg. The optimal intensity is of thenbsp;order of magnitude of 100 quanta/sec. absorbed per grain.

almost similar shapes. This can be related to other properties of the (D; I; t) surface in the following way;

Often (D; log t) curves are practically parallel for large I-intervals, so the quantity y,, defined as

is independent of I for large regions of this variable, especially near and above the optimal intensity (cf. Jones e.a., l.c.). Since the vertical distancenbsp;between neighboring r.l.f. curves for constant density is inversely proportionalnbsp;to Yi , the independence ofnbsp;nbsp;nbsp;nbsp;of I is equivalent to parallelism of these

r.l.f. curves. Considering D as a function of log E and log I we have in one point (log E; log I):

In words: the ratio of the slopes of the r.l.f. curves in corresponding points of (D; log I) g and (log E; log I)p plots is � yi Hence, as far as y^ isnbsp;independent of I, we have, in connection with the mentioned parallelism, thatnbsp;the r.l.f. curves for D ~ constant and for E = constant are similar, i.e. relatednbsp;by an affine transformation in the ordinate direction.

The dependence of r.l.f. on wave length was investigated by Webb who found that (log E: logtjj^ lines are congruent fornbsp;different colours. They differ only by a translation in the log Enbsp;direction, the magnitude of which depends on the value of D andnbsp;on the wave length A (Webb's relation). The r.l.f. for one wavenbsp;length can be interpreted as an effect, connected with the meannbsp;time interval � between successive absorptions in the grains. Itnbsp;is highly probable, especially owing to the validity of Webb's

relation, that intensities of different wave lengths, which produce a given density D in the same time, (corresponding intensities withnbsp;respect to D) have the same average tjme interval � between successive absorptions per grain. Thus the influence of A. on the efficiencynbsp;with which a given density is formed, is reduced to the influencenbsp;of the average interval between successive absorptions in the grains.

Innumerable attempts have been made to describe the shape of the r.l.f. curves empirically.

If the relation between density, intensity and time of exposure is written in the form D = D(I.tP), p is only a slowlynbsp;changing function of I and D. For corresponding intensities pnbsp;does not depend on A, on account of Webb�s relation. The slowlynbsp;changing character of p(I, D) is apparent from the fact that thenbsp;slope in the (log E; log I)p plot is 1�p (fig. 2a), and constitutesnbsp;the basis for the Schwarzschild approximation with p = constant.nbsp;This is equivalent with a straight line in an r.l.f. diagram fornbsp;constant density, which shows the extent to which this approximation is applicable.

D=D

A quantitative theoretical discussion of the r.l.f. is one of the most fundamental problems of latent image formation. The decreasenbsp;of efficiency at each side of the optimal point has to be treatednbsp;separately. The decrease for lower intensities indicates that thenbsp;formation of a Ag atom is not merely a single-quantic process.nbsp;For higher intensities the curve exhibits the existence of processesnbsp;which are less effective if the mean interval between absorptionsnbsp;of successive quanta is too small; hence some factor limits the ratenbsp;of utilization of quanta.

field of observations more or less successfully. Then Gurney and MottI'^) and Mottos) presented a theory which had the greatnbsp;advantage of visualizing, from one standpoint, many importantnbsp;phenomena connected with latent-image formation and development,nbsp;and moreover linked these phenomena to electric properties ofnbsp;large crystals of AgBr, studied in recent times. Of course thenbsp;theory incorporates also many features of various previous theories.

2. nbsp;nbsp;nbsp;The latent image consists of silver, which is concentrated atnbsp;discrete sensitivity specks,

3. nbsp;nbsp;nbsp;All halogenides of alkali-like elements, e.g. of silver are semiconductors that show electronic photoconductivity and cationicnbsp;electrolytic conductivity in the crystalline state.

4. nbsp;nbsp;nbsp;The efficiency for the production of a measurable directnbsp;density is of the order of 1 absorbed quantum per atom Ag.

The last point indicates that practically all quanta, absorbed by the AgBr lead to the formation of a Ag atom. The first twonbsp;points indicate that this Ag atom is deposited at definite specks.nbsp;Since the absorption of quanta is not restricted to these specks,nbsp;a mechanism of transport for the energy has to be visualized,nbsp;between its absorption and the final Ag precipitation.

It is known that the energy levels of the stationary states of excited electrons in photoconductive crystals form quot;energy bands�.nbsp;If an electron is in such a state it may travel through the crystalnbsp;without requiring supply of energy, and is thus able to increasenbsp;the conductivity of the crystal. The energy band concerned is callednbsp;a conduction band. Light absorption is known to raise a Br-electronnbsp;of the lattice (fig. 3) from the ground state, in which it is boundnbsp;to its Br atom, into the conduction band, thus causing photoconductivity.

On the other hand the study of the electrolytic conductivity of crystals of the type of AgBr showed that this property has to benbsp;explained by assuming that the Ag ions in the lattice are boundnbsp;relatively losely to their equilibrium positions. The thermal agitationnbsp;will dislocate a silver ion relatively frequently from a lattice place

into an interlattice place (fig. 3), leaving a �hole� in the lattice. Eventually an equilibrium will be established between the formationnbsp;of the interlattice ions and their reunion with holes. The energynbsp;required to move such an ion to a neighboring interlattice positionnbsp;is relatively small so that electric fields will draw such ions tonbsp;the cathode.

The Gurney-Mott theory assumes for the latent image formation that the above mentioned photoelectron wanders through the crystalnbsp;until it becomes trapped at a certain sensitivity spot (potential hole)nbsp;which is to be expected at irregularities of the lattice, e.g. at thenbsp;edges or at places where Ag or Ag2S has been formed. Thenbsp;sensitivity speck is thereby negatively charged and neutralizes itselfnbsp;then by attracting a positive interlattice Ag ion. This Ag ion isnbsp;transformed into a Ag atom upon arrival at the speck. The latternbsp;then becomes ready to accept a following electron, etc,.

This explains the silver concentration and energy transport. The theory can explain many other phenomena qualitatively or

almost quantitatively. For instance; it requires that latent image formation is almost totally inhibited at liquid air temperaturesnbsp;because, ion diffusion is then practically absent. Experiments ofnbsp;Webb and Evans on latent-image formation at low temperaturesnbsp;were in very good qualitative agreement with theoretical expectationsnbsp;deduced from inhibition of ionic conduction.

Since a trapped electron must be neutralized before its successor can be effective, too high intensities will increase the possibilitynbsp;of the loss of an electron; qualitatively this explains the r.l.f.nbsp;for high intensities.

The authors assume that a sensitivity speck has a tendency (probability) to emit an electron into the crystal, due to thermalnbsp;agitation.

The remaining speck is positively charged on the average and will attract the electron so that it will preferently remain in thenbsp;neighborhood of the speck, which gives rise to a cloud likenbsp;probability distribution around the latter. As long as a speck isnbsp;positive on the average it cannot attract other interlattice Ag ionsnbsp;to produce a latent image. As soon as photoelectrons are producednbsp;in the crystal the dissociation is accordingly reduced to thenbsp;new equilibrium. If the number of photoelectrons (pressure ofnbsp;the electron gas) is sufficiently high, the speck will, on the average,nbsp;become negative and will start to grow by attracting Ag ions. Fornbsp;low light intensities the electron concentration will slowly if at allnbsp;reach this critical value, whereas for higher light intensities thenbsp;specks will sooner start to grow.

This should explain why low light intensities are less efficient in producing trapped electrons than higher ones (low intensity r.l.f.).

The action of developers is explained by taking into account that these reducing substances have a tendency to give off electrons.nbsp;These electrons are deposited on Ag specks produced by the actionnbsp;of the light at the surface of the grain, and are then neutralizednbsp;by Ag-ion diffusion until the entire grain is reduced in a similarnbsp;way as has been assumed for direct photolysis.

The sensitization by dyes is explained by assuming that a suitable adsorbed dye molecule can, after absorption of a quantum of light,nbsp;bring one of its own electrons into the conduction levels of thenbsp;AgBr grain. The electron is trapped and attracts Ag to form

Ag as discussed before; the positive dye ion is neutralized by the charge of a Br~ ion which becomes superfluous as soon as annbsp;Ag ion is reduced to Ag.

The Herschel effect of latent image erasure by infrared radiation is explained by remarking that these quanta are absorbednbsp;by Ag but not by AgBr. Their absorption in small Ag aggregatesnbsp;causes a photoelectric effect, i.e. sends away an electron into thenbsp;crystal which may become lost. The remaining positive Agnbsp;aggregate is neutralized by sending away an Ag ion into thenbsp;lattice: repetition of this process reduces the latent image to zero.

Finally we remark that the explanation of the WEiGERT-effect of photo adaptation, given by Cameron and Taylor 20) in 1934nbsp;as a result of investigations on Ag-AgCl systems runs practicallynbsp;along the same lines as the Gurney-Mott theory for thenbsp;Herschel effect.

Though its versatility renders the theory most valuable, much remains to be done before a quantitative explanation of thesenbsp;phenomena will be possible.

The work to be presented in this thesis consists of several more or less independent parts that are partly of an experimental andnbsp;partly of a theoretical nature.

The work of Jurriens 10) on the production of direct density, for exposures just above the region of ordinary exposures fornbsp;development, with the aid of sensitive absorption measurements,nbsp;had revealed a few points which we wished to investigate morenbsp;closely.

This author reports, that with blue light the {d;t)j curves sre straight lines for low exposures; for higher exposuresnbsp;they can be described by a (1 � exponential) saturation formulanbsp;quot;With the remarkably low saturation density of ca. 0.1. Thenbsp;reciprocity law failure decreases for low values of the exposurenbsp;so that it is probably absent for direct densities, corresponding to

latent images of exposures that would yield intermediate densities after development. It is of course still present for the developednbsp;densities themselves.

The (d:t)j curves for weakly absorbed radiations (e.g. red) show a marked S-shape, qualitatively resembling developed densitytime curves.

1. nbsp;nbsp;nbsp;to what extent the mentioned resemblance is inherent ornbsp;accidental.

2. nbsp;nbsp;nbsp;the causes for the S-shape of the (d; t) curves and its wavelength dependence.

In the course of the investigation a few related questions presented themselves.

The experimental arrangement and method is described in Ch. II. The measurements, results and discussions are to be found in Ch. III.

In the first place we shall discuss the Gurney-Mott theory somewhat more closely. In view of certain quantitative experimentalnbsp;results in literature we shall propose a modification of this theory.nbsp;(Ch. IV).

In the second place we shall discuss the effect of a mixture of colours on a photographic plate in relation to the effects of thenbsp;separate components (addition law). We shall see that thenbsp;empirical van Kreveld addition law^s) must be an approximation.nbsp;From the Gurney-Mott theory we deduce a theoretical additionnbsp;law which, however, appears to differ not much from van Kreveld�snbsp;law for many practical applications and may thus be considerednbsp;as its theoretical basis (Ch. V).

In the third place we shall discuss the �intermittency effect�. It is well known that the density produced by intermittentlynbsp;irradiated energy generally depends on the conditions of thenbsp;irradiation (frequency, dark to light ratio, average intensity etc.).nbsp;For high and for low frequencies of intermittency Webb linkednbsp;this effect to the reciprocity law failure. Moreover the discussion

of the experiments, given by W^ebb^i) and by Silberstein and Webb 22) showed that the photographic unit of action, that is thenbsp;region within which the absorption of a quantum can lead to thenbsp;production of the same final Ag speck, is of the order of magnitudenbsp;of one grain. We shall quantitatively extend the theory of thisnbsp;effect to the region of the intermediate frequencies, showing thatnbsp;Webb�s experimental results can be understood entirely in thisnbsp;region on principally the same basis as for high and low frequencies

Finally we shall give a comparative discussion of some problems and methods, common to photography and radiobiology (Ch. VII).

� 6, Apparatus.

We carried out our investigation with the apparatus of Jurriens, for the measurement of direct density, which we improved in severalnbsp;respects. We arrived after some trials at the arrangementnbsp;represented schematically in fig. 4. The light source LS, a 125 Vnbsp;500 W tungsten filament lamp, was connected with the laboratory�snbsp;constant tension 130 V batteries (constancy l^/oo)- It was placednbsp;in front of the lens L (f = 16 cm) at a distance of 20 cm at suchnbsp;an inclination that the illumination of the lens was as homogeneousnbsp;as possible. The light passed through a 2 cm water filter W and anbsp;set of filters FI (Schott amp; Gen. BG17, 3 mm and OG2, 2 mm,nbsp;transmitting about 10 % at 9000 A, 60 % at 8000 A, 100 %nbsp;between 7000 A and 5800 A and 0 % for wave lengths belownbsp;5500 A cf. fig, 5) and a shutter S. An image of the filament wasnbsp;formed at a distance of 70 cm on the lens L2 (f = 26 cm) whichnbsp;in turn focussed L via the mirror M and the diaphragm D2 on thenbsp;photographic plate P.

The intensity of this light could be varied with the aid of different diaphragms Dl.

The light, transmitted by the plate falls through a diaphragm D3 and a filter F4 on a Weston barrier layer cell Phi. In generalnbsp;F4 was an RG2 filter (2 mm), transmitting Agt; 6300 A. So thenbsp;�measuring� light and the �acting� light are obtained from thenbsp;same beam with different filters.

with some CaCl2, which improved its constancy. It was connected with a similar cell Ph2, and with a MoLL-galvanometer G. Thenbsp;second cell received its light from the same lamp LS through a rednbsp;filter F2 and some milk glass weakeners; its distance to LS couldnbsp;be adjusted accurately so that it almost compensated the action ofnbsp;the first cell.

Schematical diagram of the apparatus for the production and measurement of direct density. LS = light source; L = lens (f = 16 cm); W '= 2 cm waternbsp;filter; FI = Schott filter BG17 (3 mm) OG2 (2 mm); S = shutter;nbsp;L2 = lens (f = 26 cm); D1 = diaphragm for intensity control; M ~ mirror;nbsp;P3 = orange OG3 (2 mm) or red RGl (2 mm) filter; P = photographic plate;nbsp;D2 and D3 are diaphragms to avoid spurious light; Phi and Ph2 are Westonnbsp;barrier layer cells; F4 = red RG2 (2 mm) filter; G = galvanometer. Thenbsp;optical part of the figure is a projection on a vertical plane.

The deflection of the galvanometer could either be registered on a drum with photographic paper driven by a synchronous motor

or read off at a scale, both at ca. 4 meters distance. A suitable shunt made the deflection aperiodic.

A strip of 1.5 X 22 cm cut out of a photographic plate of 10 X 25 cm or 18 .X 24 cm lay on a carriage which could movenbsp;perpendicularly to the plane of drawing; by moving the carriagenbsp;we could expose various parts of the plate to the light, incidentnbsp;through D2. At the end of the carriage a vacuum thermoelementnbsp;was placed which was in the center of the light spot when thenbsp;carriage was at an end stop. It was connected with an othernbsp;MoLL-galvanometer, adapted to its small resistance, and served fornbsp;accurate determination of the intensity of the �actingquot; light atnbsp;the spot of the plate (�measuring spot�).

At the other end of the carriage a milkglas was fixed, adjoining to the photographic plate. It had been chosen so as to havenbsp;practically the same transmission of light to Phi as the plate.

If exposed strips were to be developed this was done with metol borax developer 23)^ during 6 min. at 18� C. The strip wasnbsp;developed in a large sort of test tube containing the developernbsp;which was placed into a thermostate.

The developed density of such strips was measured (after fixing, washing and drying) in the same arrangement as the directnbsp;densities. Incidentally we remark that the densities of such stripsnbsp;up to ca. 2, measured with our apparatus and with a Moll microphotometer, are proportional within � 3 %, their ratio being 1.40.

In the sequel we indicate the three types of acting light that were mostly used by yellow, orange and red. The meaning ofnbsp;these terms will be:

The intensity of the acting light, incident on P, will be expressed in relative units. In order to convert these units into absolute onesnbsp;we have determined one intensity with the aid of an absolutely

standardized thermopile too. In this way we obtained: I relative unit = 1.3 ,X 103 erg/cm2. sec.

The relative spectral intensities of the various types of acting light used were determined by comparing them through a doublenbsp;monochromator with the radiation of a standardized tungsten bandnbsp;filament lamp. The results are shown in fig. 5. We emphasize thatnbsp;the plates show such a steep gradient of absorption in thenbsp;Wave length region concerned that practically only the shortestnbsp;wave lengths present in the beam are active in density production.

After a nuinber of check experiments (see � 8) the following procedure for the production and measurement of (d; t) curvesnbsp;Was adopted as being most satisfactory.

1 � One hour previous' to the beginning of a measurement the ^ight source LS was switched on, the shutter S was open, the lightnbsp;Went through the diaphragm Dl, the filters of the right colournbsp;and through the milk glass. The current through LS was adjustednbsp;to 3.90 Amp. and checked from time to time. The filter F2 and

the position of the cell Ph2 were adjusted in such a way that small changes in intensity of LS caused no galvanometer deflection. Sincenbsp;the spectral transmission of F2 was not the same as that ofnbsp;W Fl F3 F4, the required point where compensation ofnbsp;intensity variations is achieved is not quite the same as the pointnbsp;where the entire deflection is compensated. An additional electricalnbsp;compensation (see fig. 4) brought the galvanometer image backnbsp;to the desired region of the scale or registration drum it necessary.

2. nbsp;nbsp;nbsp;The position of the galvanometer image was observed fromnbsp;time to time and when it had become very well constant (afternbsp;ca. 1 hour) its deflection on shutting S was measured, then thenbsp;carriage was moved to the other end, so that the thermoelementnbsp;received light after opening S again. The deflection of the othernbsp;galvanometer was a proportional measure of the intensity innbsp;arbitrary units.

3. nbsp;nbsp;nbsp;Meanwhile a plate strip was freed from backing, the shutter Snbsp;closed, the strip laid on the carriage and the carriage moved to thenbsp;milk glass end, whereupon S was opened again. Points 2 and 3 werenbsp;handled as quickly as possible, to obtain a stationary galvanometernbsp;deflection as soon as possible (see � 8, properties of the barriernbsp;layer cell).

4. nbsp;nbsp;nbsp;After about 10 min. the image was well stationary and wasnbsp;registered. The carriage was moved so as to expose a first spotnbsp;on the plate which was not registered; the transmission of the milknbsp;glass and the plate were practically equal, but for adaptation of thenbsp;cell after the small differences we waited one minute and movednbsp;then to the next spot, which was registered.

At the end of a period of about 30 min. the milk glass was again registered and then the same procedure of adaptation andnbsp;registration of a following spot was repeated, as many times asnbsp;the length of a strip allowed.

5. nbsp;nbsp;nbsp;At the end again milk glass, dark and thermoelement intensitynbsp;measurements as in 2 and 3 were performed.

6. nbsp;nbsp;nbsp;The registration drum rotated once in 2 h. 30 m. so that onenbsp;paper contained four deflection-time curves, each one enclosed

between two milk glass registrations, which were afterwards connected by a straight line (of small slope) in order to take intonbsp;account the drift of the entire device. The correctness of such anbsp;linear interpolation was checked by the continuity of successivenbsp;lines. An eventual error, introduced by this procedure was muchnbsp;smaller than the scattering of the individual curves; moreover itnbsp;was usually cancelled by the averaging of several observationsnbsp;(see � 9).

The registration was then converted into a density-time curve by correcting for this drift and taking into account the value ofnbsp;the entire deflection deduced from 2 and 5.

In a few cases the registration was replaced by scale readings; if the exposure had to be followed longer than ^/2 hour the latternbsp;procedure was always applied and the drift of the device wasnbsp;followed by intercepting the beam once in 10 minutes betweennbsp;P and Phi.

� 8. Experimental details.

For the sake of completeness we present here a list of experimental details, checks and difficulties connected with the measurements.

During the exposures the current of LS was checked from time to time. A few registrations for which the measurement of intensity with the thermoelementnbsp;at the beginning and at the end of a set of about 8 registrations differed morenbsp;than 2 % were discarded.

The vacuum thermoelement was mounted on the carriage in such a way that it could be placed in any part of the light spot. The image of L at Pnbsp;had a diameter of ca. 35 mm; D2 was placed so as to transmit the best 8 mm,nbsp;which did nowhere differ more than 1'% from the mean value. The entirenbsp;35 mm spot showed largest deviations of � 7 %, due to the inhomogeneousnbsp;emission of LS in different directions (inhomogeneous illumination of L). Thenbsp;Position and inclination of LS had been adjusted so as to reduce thesenbsp;inhomogenities as much as possible.

regularly as possible. This was checked with registration of flashes of known intervals.

The galvanometer house and circuit was grounded. All resistances etc. stood on grounded plates. The number of wire connectioiis was minimal. They werenbsp;insulated thermically with cotton-wool. Later the galvanometer was placed undernbsp;a glass box filled up with cotton-wool and containing a cup with CaCb to insurenbsp;still better freedom from thermal influences and humidity, which might causenbsp;thermo- or chemo-E.M.F.�s. The blank registration, without illumination of thenbsp;cells, was under these circumstances very satisfactory, showing fluctuationsnbsp;of ca. 0,3 mm.

Several Weston quot;photronic cellsquot; at our disposal were tested for constancy. The two best ones were used, after the experience of Jurriens in hermeticallynbsp;closed dry glass boxes containing CaCh. During their use their constancynbsp;decreased, so that they showed disagreable sensitivity fluctuations on thenbsp;registration of about 0.001 % later up to 0.01 % of the deflection with thenbsp;unexposed plates. This percentage was almost independent of the intensity. Thenbsp;fluctuations were rather annoying for our compensation measurements.

An a.c. photoelectric-cell amplifier after Milatz^^) would have been the other alternative for the measurements. After this author the device showsnbsp;an inconstancy of the same order of magnitude, (10��) for a light intensitynbsp;of 1 erg/sec.cm^. Preliminary experiments with higher intensities yielded anbsp;relative inconstancy of the same order of magnitude, the nature of which wenbsp;have not traced.

Though the use of the amplifier would have had the additional advantage of weaker intensities of the measuring light, we sticked to the Weston cells,nbsp;since they were at that time just as constant, and their use was less complicated.

Both compensating cells were connected parallelly with the galvanometer as indicated in fig. 4. This way yields larger deflections and better constancy ofnbsp;the cells than if they are connected in series^�). The time of indication of anbsp;cell depends on the intensity change it undergoes. It is largest for large changes.nbsp;It took sometimes many minutes before the deflection became constant. Thereforenbsp;we have always taken care to make intensity changes small (see � 7, 4). Fornbsp;small or slow changes the time of indication is about 5 seconds.

The long times of adaptation before every set of experiments were necessitated entirely by these cells. It may be mentioned that if the cells suffered largenbsp;intensity changes, they became constant sooner if they were at the same intensity level before and suffered the large changes only for a short time of the ordernbsp;of 10 minutes (see � 7, 5). The galvanometer deflection, after becoming constant,nbsp;is well proportional to the intensity, determined by the thermoelement. Howevernbsp;it shows a temperature dependence of -|- 1.5 %/�C. In our cellar room the

temperature varied during one day never more than 2�; the drift caused thereby Was small since it was for the greater part compensated by the other cell.

This diaphragm served to vary the intensity of the light without changing its relative spectral composition. In connection with the shape of the filamentnbsp;of LS, which consisted of six vertical wire spirals, the shape of Dl was anbsp;horizontal rectangular slit. With the aid of the vacuum thermoelement (cf. no. 2)nbsp;we found that its interposition did not decrease the homogenity. Round ornbsp;other diaphragms, wire gauze, etc. appeared to cause a large inhomogenity ofnbsp;the light spot on the plate.

In contrast to the method, employed by this author we preferred to let the plate stand still, so that both the measuring and the acting light wentnbsp;continuously through the same part of a plate, allowing a much more economicalnbsp;use of plates. On the other hand local irregularities of the plate sensitivitynbsp;became much more evident (see � 9).

Our measuring light was many times more intense than in Jurriens� case, so that it served also as acting light, and increased the sensitivity. The heatnbsp;production of our larger light source does not allow to place it immediatelynbsp;below the plate, as was done by Jurriens, but necessitates a large distancenbsp;between it and the plate and cell. In order to avoid convection near the platenbsp;and still have a large intensity we intercalated the system of lenses LI�L2 intonbsp;the beam. The convenience of working with the plate horizontally, combined withnbsp;the requirement for stability of the entire arrangement, led to the introductionnbsp;of the mirror M.

For experiments with blue acting light (necessarily of much smaller intensity than red light) the mirror M could be replaced by a glass plate which reducednbsp;the red light and allowed the blue light from an auxiliary light source to reachnbsp;the plate from above, but this device has not yet been used up to the present.

In connection with the higher light intensities applied, the sensitivity of our arrangement was 3 to 10 times that of Jurriens, depending on circumstances.nbsp;Due to the mentioned proportionality of the cell current and its fluctuations,nbsp;this gain did not result in an increased accuracy. However, by using a smallernbsp;diaphragm D2, this state of affairs allowed a much more economical use ofnbsp;the plates.

We did not use the milk glass amplification, in order to reduce as much us possible complications arising from the �multiple scattering of the light,nbsp;especially for the interpretation of the higher densities. The cell Phi wasnbsp;placed below the plate in such a way as to cover a solid angle of about 1nbsp;steradian, which we deemed sufficient to measure a density, mainly due tonbsp;absorption of the Ag.

JuRRiENs� results with blue light showed a remarkable reproducibility (within �3%) notwithstanding great variations in coating thickness (�: 20 %) on various plates or on various spots of onenbsp;plate. This reproducibility was, according to this author, probablynbsp;due to the large absorption of the plate for blue light, which restrictsnbsp;the principal part of the density production to a thin superficialnbsp;layer of the emulsion.

A great disadvantage of the measurements with red light was a remarkable irreproducibility, due to the relatively small absorption,nbsp;combined with variations in coating thickness and probably alsonbsp;other factors (see below). This means that (d:t) curves can benbsp;obtained safely by this method, but (d; I) curves have to benbsp;considered with reserve, because in the former case only one spot

is illuminated, whereas in the latter one we have to use as many spots as intensities.

To illustrate this dispersion we represent in fig. 6 a number of (d;t) curves obtained with red light (RGl). The-curves a, b, cnbsp;were obtained at various spots of one plate, curve d belongs tonbsp;another plate of the same package. The figure is characteristic fornbsp;the dispersion of the results and for the general type of the curves.

The deviations between the curves are much larger than the dispersion of the points along any one of the curves. So the formernbsp;ones are due to variations of the plate rather than to inaccuraciesnbsp;of the method of measurement.

The deviations were not correlated with the extinction of the unexposed plate which consists mainly of scattering. We have alsonbsp;checked the dependence on temperature roughly by performing twonbsp;experiments, one after another on neighboring spots of one plate.nbsp;In the first experiment the density curve was taken with a f�hnnbsp;blowing air of room temperature towards the illuminated spot. Innbsp;the second experiment the f�hn blew air of 40� C. The latternbsp;curve showed a more pronounced S-shape and the sensitivitynbsp;increased about 20 %. Roughly this indicates that the temperaturenbsp;sensitivity for this intensity is of the order of 1 %/� C.

However the temperature deviations of the experiments represented in fig. 6 were less than 2� so that this cannotnbsp;account for the deviations observed between the curves.

Another factor which might possibly cause systematic errors is the humidity. We have performed similar experiments on daysnbsp;when our room was very dry and on other days when it wasnbsp;very humid, but could not observe any correlation herewith. Seenbsp;however for the effect of water � 16.

We have spent much time on checks for all sorts of possible systematic errors, but we have not succeeded to make the reportednbsp;dispersion disappear.

Since the shortest wave lengths that are present in our beam are the most active ones in the production of direct density, owing tonbsp;the rapid decrease of AgBr absorption with increasing wave length.nbsp;We remark tentatively that variations of the absorption spectra ofnbsp;the grains at various regions of the plate, which may eventuallynbsp;be due to adsorbed ions or ion-groups (see e.g. 3) p, 95) or a

sensitization by traces of some substance, which are not distributed homogeneously, may be the main cause for the observed lack ofnbsp;reproducibility.

We mention in this connection that red exposures taken with the purpose of detecting changes of the latent image betweennbsp;exposure and development, showed a similar unexplained largenbsp;scattering.

Finally we adopted the procedure to average always 7 to 9 curves obtained from one strip. But still these averages showednbsp;considerable variations from strip to strip (cut out of one plate),nbsp;much more than could be expected from the dispersion of the 7 ornbsp;9 individual curves of one strip. Of course the first and last stripnbsp;of a plate were discarded; they showed still larger deviations. Alsonbsp;the first and last place of each strip were never used to avoidnbsp;effects of inhomogenity near the edges of a plate.

Developed-density-exposure curves (cf. � 3, fig. 1) and direct-density-exposure curves especially for red light show a striking resemblance. The latter ones start with a more or less pronouncednbsp;S-shape (fig. 8), followed by a straight part, with a tendencynbsp;towards saturation for still higher densities (fig. 13 or Jurriens lo)nbsp;fig. 19). We wish to see to what extent this similarity is accidental.nbsp;This will be discussed in three parts, corresponding to the divisionnbsp;of the curves (fig. 1) into regions of low densities, of intermediatenbsp;ones and of saturation. Experiments to be described below shownbsp;that exposures to red light, producing equal direct and developednbsp;densities, differ much. We shall reverse the order in which wenbsp;shall consider these regions because we mainly investigated experimentally the region of low densities.

Saturation for developed density would be expected when all grains can be developed chemically and for print-out density whennbsp;3ll grains are entirely photolysed. For the latter case more exposurenbsp;Would be needed, but finally the same density would be reached,nbsp;since in both cases the total amount of AgBr contained in thenbsp;plate is reduced to Ag. But things are not so simple.

The solarization and high order reversals, of which up to eight have been observed i4)_ are complications which set in at densitiesnbsp;often far below the theoretical maximum. Several theories exist

We wish to thank Mr. K. H. J. Bokhove and Mr. M. Braak for experimental assistance.

about solarization 3). We believe that experiments have shown sufficiently clearly that it is a complication which is due to thenbsp;development. The number of developed grains, plotted against lognbsp;exposure, do not or hardly show this peculiarity fig. 118, p. 179;nbsp;neither does the print-out density-exposure relation show remarkable features in this region of exposures lo). Moreover severalnbsp;authors report that the effect disappears if development is prolongednbsp;appreciably 26) 27).

For high exposures the print-out density tends towards saturation without reversal (see e.g � 15 and 6) p, HI). The saturationnbsp;density is much lower than for developed densities. So we concludenbsp;that the saturation effects are due to different causes.

A related behaviour of print-out and developed density in this region can hardly be expected, if the entirely different ways arenbsp;considered in which the two types of density are produced.

For the direct effect all grains take part in producing density. In each grain the silver coagulates at the sensitivity spots whichnbsp;are distributed throughout its volume 6) p. HI. With increasingnbsp;exposure these specks grow; soon their number remains almostnbsp;constant (ca. 1000 cf. � 11).

In the case of the developed density the grains, on the surface of which at least one sufficiently large speck of Ag has beennbsp;produced by the exposure, will be developed entirely, and the othernbsp;grains are not developed at all. So with increasing exposure thenbsp;number of developable grains increases, the average size of thenbsp;developable grain on the contrary does not change much 3)nbsp;p. 233 and 267.

So if we should expose a plate of identical grains to light in such a way that all grains absorbed quanta simultaneously, thenbsp;direct density would gradually increase with exposure, but thenbsp;developable density would show the abrupt course of line 1 in fig. 7.

The reason why the developable-density-exposure relation has in reality a similar shape as the direct one, is that� the former isnbsp;a superposition of many such abrupt curves with scattered valuesnbsp;of the parameters that determine the value of the threshold in each

grain, which superposition yields a curve of type 2 fig. 7. Thenbsp;causes which are most likelynbsp;responsible for this effect are:

1. nbsp;nbsp;nbsp;variations in the number ofnbsp;quanta absorbed in each grain,nbsp;due to:

b. nbsp;nbsp;nbsp;non-homogenity of thenbsp;light intensity throughoutnbsp;the plate,

d. nbsp;nbsp;nbsp;possible variations of thenbsp;absorption coefficient fornbsp;different grains;

2. nbsp;nbsp;nbsp;variations in the number ofnbsp;absorbed quanta required tonbsp;produce developability, due to:

a. nbsp;nbsp;nbsp;variations in the effect ofnbsp;the ripening process, (e.g.nbsp;in the number of sensitivitynbsp;specks per grain and theirnbsp;location with respect tonbsp;the surface),

Several efforts to explain the entire shape of the density-exposure relation only on the basis of lb and Ic or 2a may be found in literature, and especially the treatment of the influencenbsp;of lb and Ic by Silberstein and Trivelli^s) indicates that thesenbsp;factors are important, though on the other hand Webb 29) showednbsp;convincingly that the influence of 2a may not be neglected. Mainlynbsp;these causes make a quantitative understanding of the developednbsp;density-exposure relation difficult. However they all lead to

the conclusion that an essential similarity between direct and developed density production can hardly be expected in this region.

If the thresholds for direct and developed density are mainly due to the same causes, they should lie at the same exposure. Jurriens�nbsp;direct density-exposure curves showed a slight indication of a non-

linear course near the origin for blue light, but quantitative comparison of the two thresholds was not possible from hisnbsp;measurements. However, for red light the direct threshold becomesnbsp;pronounced, so that an investigation of a possible analogy betweennbsp;the thresholds seemed promising.

thresholds for blue light. Though we succeeded to increase the sensitivity of the method of Jurriens for the measurement of directnbsp;density several times, an increase by a factor 100 could not benbsp;achieved. This would have been required to measure direct densitiesnbsp;in the reigon of exposures corresponding to developed densitiesnbsp;not far above the threshold for blue light. So we had to contentnbsp;ourselves with experiments on exposures with red light. Here bothnbsp;thresholds were well within the region of measurement.

The experiments were performed with the arrangement, described in Ch. II. We used Ilford Special Rapid plates (extra sensitivenbsp;H. amp; D. 400). Both the acting light and the measuring light werenbsp;�red RGl.� (see � 7).

On one strip a number of identical exposures were performed and the direct density as a function of time was recorded. Thenbsp;average result and its mean error is represented in fig. 8. The wellnbsp;pronounced S-shape in this direct-density-time curve has itsnbsp;threshold at an exposure of I = 27 (arb. units cf. Ch. 2) andnbsp;t = 2.3 min.

On each one of three similar strips exposures were made at the same intensity but with various times of exposure. The strips werenbsp;developed (see Ch. II), fixed, washed and dried. The concentrationnbsp;of the developer was different for these three strips namely: thenbsp;normal concentration = 1, 0.1, and 0.03.

The results indicate that for this intensity an exposure of 2.3 min. produces almost saturation density after development,nbsp;hence we may conclude that a common basis of the thresholdsnbsp;does not exist.

The direct threshold will be discussed somewhat more closely in � 17. The conclusion that the thresholds are not linked wasnbsp;probable a priori, owing to the well known experiments of Reindersnbsp;and Beukers, who showed that the developed threshold is probablynbsp;an effect of development. These authors proved that physicalnbsp;development of a glass plate on which minute quantities of silvernbsp;had been condensed, was possible only if the silver occurred innbsp;aggregates of several atoms. The transposition of this conclusion

to a photographic emulsion introduces some uncertainty which justified the attack of this question from an other side.

Additionally we exposed similar strips during shorternbsp;times and to lower intensities,nbsp;in order to see how much thenbsp;direct and developed thresholdsnbsp;differed. The lower intensitiesnbsp;were obtained both with thenbsp;aid of diaphragms (see Ch. II)nbsp;and with a neutral glass NG3nbsp;(Schott and Gen.) of ca.

The results confirm the well known fact that the threshold does hardly depend on I or t separately, but lies at an almost constantnbsp;exposure of iXt = ca. 55. Moreover they show that the developednbsp;inertia is about l/600th of the direct one. This figure may benbsp;dependent on circumstances, but shows the order of magnitude. Wenbsp;wished to see, whether there is an indication, that the two thresholdsnbsp;would come closer together if the degree of development approachesnbsp;zero. If this should be the case, it would indicate that thenbsp;difference in threshold is caused, at least partly, by the development.nbsp;In order to keep all other conditions constant we decreasednbsp;development by applying smaller concentrations of the developer,nbsp;each one during the same time of development. For this purposenbsp;we exposed each one of four strips to a given intensity duringnbsp;various times, so that the results would allow a determination of

the developed threshold. The strips were developed in concentrations 1, 0.33, 0.067 and 0.033, during 6 min.

The results confirmed the general expectation that the developed threshold is almost independent of the degree of development. Anbsp;slight tendency was observed in the sense of a decrease of thenbsp;threshold for lower concentrations of the developer.

We conclude herefrom that both thresholds are not merely quantitatively but also qualitatively different phenomena.

The fact that the difference between the thresholds remains, if development is extrapolated to concentration zero, is probably duenbsp;to the fixing process, the influence of which upon directly blackenednbsp;plates is to erase completely densities up to more than 0.05 as wasnbsp;shown by the following experiments. We �fixed� a number ofnbsp;strips with direct densities of 0.007 and 0.05 in fixing solutionsnbsp;of NaHSOs to which suitable amounts of NaOH were added. Thenbsp;pH of the solutions was determined with a Coleman pH-meternbsp;and varied between 5.7 and 9. In all cases the direct densitynbsp;decreased so much that not a trace of it could be seen nornbsp;measured.

For fully developed plates this effect is not present to a comparable extent, which is probably due to the much larger sizenbsp;of developed Ag grains as compared with the Ag specks of directnbsp;blackening.

� 11. On the relation between direct density and amount of silver (latent image).

An interpretation of the shape of direct-density-exposure curves in terms of photo-silver requires, that the relation between thenbsp;direct density d and the amount of silver A shall be known.

It is not our purpose to determine the relation between d and A in a strictly quantitative sense e.g. by entering into a detailednbsp;discussion of the absorption and sattering of Ag-AgBr-gelatinnbsp;systems such as the photographic plate; our problem is merely tonbsp;what extent these quantities are more or less proportional, so thatnbsp;an S-shaped (d;t) curve corresponds to a similar (A:t) curve.

Several arguments suggest that for not too high direct densities the two quantities are proportional. We mention the remarkably

simple relation, found by Jurriens, for blue light between direct density and exposure. Also we refer to Eggert and Noddack ^)nbsp;p. 133, who proved chemically that for higher print-out exposuresnbsp;each quantum yields one Ag atom approximately.

The same authors p. 145, performed experiments which proved that, though the developable latent image of a rather large exposurenbsp;is erased by a FeCls treatment which reoxidizes the Ag at thenbsp;surface of the grains, 99 % of the photolytic Ag remainednbsp;unchanged by this treatment. In other words only 1 % wasnbsp;destroyed by FeCls, or only 1 % was present at the surface ofnbsp;the grains. On the other hand experiments of Toy, Svedberg e.a.nbsp;with microscopic observations of partially developed grains,nbsp;indicated that the number of spots where the Ag is coagulatednbsp;or concentrated at the surface is of the order of 10. So the numbernbsp;of specks throughout the volume is ca. 1000. Jurriens appliednbsp;exposures in the region considered of the order of 10^�.�10^�nbsp;quanta/cm'2. After Eggert and Noddack 3) p. 100, the grainsnbsp;absorb 1/5 th of the light. Since the number of grains/cm^ is ca. 10�,nbsp;these exposures correspond to ca. 200�2.10^ quanta absorbed pernbsp;grain, which means on the average a few � ca. 200 atoms pernbsp;silver speck (linear dimensions up to ca. 6 atoms). This is stillnbsp;quite small compared with the wave length of light, so thatnbsp;linearity of amount of silver and absorption may be assumed.

On the other hand the straightness of Jurriens� curves shows that the absorption of the silver per atom is independent of the size of the silvernbsp;specks, even if it consists of a few atoms only.

We must be aware, however, of two indications which warn for too easy acceptance of this proportionality.

Firstly we know from experiments of Webb, performed in connection with investigations of reciprocity law failure for developed density, that for verynbsp;low exposures the efficiency of latent-image formation depends largely on thenbsp;intensity or rate at which a number of quanta is admitted. In this region ofnbsp;exposures, which cause direct densities far too low to be measurable at present,nbsp;simple proportionality of exposure and direct density can hardly be expected,nbsp;so that the arguments fail there.

In the second place the results of Jurriens show small but systematic deviations from proportionality, in the sense that his straight density-time curvesnbsp;pass above the origin (see � 12). Experiments and discussions to be presentednbsp;below will sustain the conclusion that both the density and the silver productionnbsp;deviate from proportionality with exposure in this region.

These remarks all tend to show that the amount of silver A and the direct density d are proportional or almost proportionalnbsp;in the region concerned, which is of importance as soon as wenbsp;have to discuss density time curves, e.g. produced by red radiation,nbsp;which show a marked S-shape. From what has been said it isnbsp;likely that such an S-shape is also present in the correspondingnbsp;(A; t) relation.

The S-shape is not caused by the colour of the measuring light: a plate was exposed by Jurriens to blue light,� which yieldednbsp;straight (d; t) lines, and to red light, which yielded S-shaped onesnbsp;but the density was always measured with the same red light.

We have performed similar check experiments by registering one (d; t) curve, with yellow acting and measuring light, one withnbsp;yellow acting and red measuring light, and one with red actingnbsp;and red measuring light (see Ch. II). The first and second curvenbsp;were similar; their difference amounted to a small constant percentage in the density direction. The second and third curve werenbsp;entirely different (for a closer description of the qualitativelynbsp;different shapes of (d; t) curves produced by yellow and red lightnbsp;see � 12). From these experiments it follows that, within thenbsp;wave-length region in which the shape of the (d; t) curves changesnbsp;considerably, its shape does not depend on the wave length of thenbsp;measuring light. In other words the S-shapes are not due to anbsp;special wave-length dependence of the absorption of the Ag specksnbsp;for the measuring light, but indicate S-shaped (A; t) relations.

For higher densities the (A; d) relation naturally becomes more complicated than simple proportionality but there is no reason to expect serious disturbancenbsp;of the analogous behaviour of (A; t) and (d; t) curves, until for still largernbsp;specks, the magnitude of which becomes comparable with the wave lengthnbsp;of light in AgBr, special resonance absorptions may become importantnbsp;(Cameron and Taylor 2'*)).

Fig. 10 represents a few (d: t) curves, for low densities. The curves �red�, quot;orange� and �yellow� were obtained following thenbsp;normal procedure described in Ch. II. The curve marked �blue�nbsp;is taken from the work of Jurriens for comparison.

We notice, that with increasing wave length the S-shape becomes more pronounced, at first in the upper part and then in the lowernbsp;part. Furthermore a remarkably straight part appears for highernbsp;exposures, also for those colours which show a pronounced S-shapenbsp;for low exposures. Its produced part passes above the origin.

These observations suggest that the straight part of the curves is extended towards the origin with decreasing wave length.nbsp;Indeed, upon closer consideration the �straight� lines, observed bynbsp;JuRRiENS for blue light, show distinct remnants of the S-shape,nbsp;which is present so distinctly for red light, in the neighborhoodnbsp;of the origin. We refer in this connection to fig. 11 (blue light),nbsp;and to table I of the thesis of this author, which indicate thatnbsp;straight lines fitting best the experimental points, systematicallynbsp;pass above the origin.

This makes doubtful the extrapolation of Jurriens' conclusion that the latent image is proportional to the exposure, into the regionnbsp;of ordinary exposures for development. On the other hand, itnbsp;suggests that for a study of the direct density in the latter region.

one has either to improve greatly the accuracy of the method, or one may study this matter for red irradiation, and see to what extentnbsp;the phenomena observed there can be expected to appear on anbsp;smaller scale for blue irradiation.

Since we have not succeeded sufficiently in the first way we have chosen the second one.

� 13. Experiments with various kinds of plates.

We have performed a few qualitative experiments with Ilford plates with orthochromatic and panachromatic sensitization. Thenbsp;results indicate that the S-shape is slightly connected with thenbsp;rate of absorption of the considered wave lengths; for the ortho-chromatic plates somewhat longer wave lengths were required tonbsp;produce an S-shape, than with ordinary �Ilford special rapid plates�,nbsp;and this effect was also present with panchromatic plates. However,nbsp;the differences appeared to be not so pronounced as would correspond to the shift of the limit of active wave lengths due tonbsp;the sensitisation.

The longer wave lengths were obtained with the aid of Schott-glasses RG2, 5, 8, 10 with cut-off wave lengths 6300, 6750, 7000, 7800 A respectively.

� 14. Experiments at various intensities of red and orange irradiation.

Notwithstanding the difficulties mentioned in � 8 we deemed a comparison of (d;t) curves at various intensities worth while.nbsp;The results show a large dispersion. We give in the figures,nbsp;presented in this section, averages of 7 or 9 individual observationsnbsp;taken immediately one after another on one strip of a plate. Thenbsp;error of this mean as determined by the deviations of the individualnbsp;observations amounts to about d D = 0.0007. For some reasonnbsp;which is so far not understood, observations on different stripsnbsp;of the same plate deviate much more than this amount (cf. � 9).

We have represented (d;t) curves without measuring points, because they were continuously registered; so the mean error innbsp;the curves is not shown in the figures.

Certain main lines can be deduced from these results though. In fig. 11 we present sets of (d; t) curves for various values of thenbsp;intensity I, of orange OG3 and red RGl radiation respectively.nbsp;The curves show an S'shape, but are not similar, neither bynbsp;multiplication in the d nor in the t direction. For high intensitiesnbsp;their difference becomes relatively small. In order to characterizenbsp;the curves we introduce four parameters namely: the steepest

gradient 7^; the gradient in the straight part after the S-shape is passed y; the density at which the inflection tangent reachesnbsp;the ordinate axis and the density at which the extension ofnbsp;the straight part reaches the ordinate axis S. We have tabulatednbsp;these quantities for all comparable data available in table I,

We conclude herefrom that both and S do not depend systematically on the intensity; probably their dependence on thenbsp;wave length is very weak in this region; y^ and y are more ornbsp;less proportional to each other and to I for low intensities, theynbsp;appear to approach saturation for higher I values.

The dependence of y and on I is illustrated in fig. 12. The fact that y, which is d' for higher densities than 0.02 or 0.03, isnbsp;only proportional to I for low values of the intensity, means thatnbsp;for higher intensities we have a definite reciprocity law failure.nbsp;Namely the reciprocity law would require d = d (I X t) or d�nbsp;= If(d). Another way to express the same facts is, that the

density time curves cannot be made congruent by multiplying their scales with a constant factor. The Schwarzschild exponent pnbsp;amounts for our intensity region to about 1.4. We have not checkednbsp;p for these intensities and developed density. General evidencenbsp;indicates that our intensity was below the optimum, so p lt; 1 fornbsp;the developed density.

� 15. Experiments on the course of (dj t) curves for higher densities.

The fact that Jurriens� exponential formula implies a remarkably low maximum of density for blue irradiation (ca. 0.1) and thatnbsp;preliminary observations on (d;t) curves in red light showed a

similar behaviour in this respect, made it seem worthnbsp;while to perform a few experiments in that direction. Thesenbsp;experiments were performednbsp;with Ilford Special Rapidnbsp;plates. The galvanometernbsp;deflections were not registerednbsp;but read off at the scale. Thenbsp;zero was frequently checkednbsp;without interrupting the exposure (� 7). Fig. 13 shows anbsp;curve taken with yellow lightnbsp;and two curves with red lightnbsp;of high and of low intensity.

A maximum density at some low value of d is not present,nbsp;but a low rate of densitynbsp;production sets in at the regionnbsp;of densities near d = 0.3. Thenbsp;first part of the curve can benbsp;represented as the difference

of a slowly ascending linear function and an exponential part. To represent these experiments, the time scale had to be reducednbsp;by a factor 25 as compared with previous figures so that not muchnbsp;can be seen of the S-shape near the origin.

We mentioned, that the humidity of the air is probably not responsible for the lack of reproducibility reported in � 8. Thesenbsp;experiments led us into a side line, worth to be reported briefly,nbsp;because the results are somewhat unexpected and not yetnbsp;understood.

In a first set of experiments we compared the (d; t) curves of plates that were soaked in water and those of ordinary platesnbsp;(Ilford Special Rapid, red light RGl). Of course the soaked platesnbsp;were still wet during the exposure, and the gelatin was still swollen.nbsp;The water changed the colour of the plates from yellow to greenish

In accordance herewith the (d;t) curves rise steeper, almost proportionally by a factor of about 2. The effect was independentnbsp;of the time during which the plate was soaked in the water, if thisnbsp;time was varied from 2 minutes to 25 minutes. Also after suchnbsp;plates had dried again, the green colour and the increasednbsp;sensitivity for direct density production by red remained unaltered.nbsp;Even after drying 4 days in a CaCl2 exsiccator the results werenbsp;the same, so that we have here an effect which is difficult tonbsp;reverse. Incidentally we remark that the density at which the rate ofnbsp;density production becomes small, as exposed in � 15, is not altered.

However, this water treatment caused a strong desensitization for developed density production by the red light. So the directnbsp;density and the developed one are influenced by water innbsp;opposite ways.

� 17. Discussion.

We are not able to give a definite interpretation of the experiments presented, but we wish to advance a few tentativenbsp;remarks from the standpoint of the Gurney-Mott picture.

In � 11 we made plausible that the S-shape occurs both in (d:t) and (A;t) curves.

The Herschel effect is known to occur in this wave-length region. This effect is present for latent images in the region suitednbsp;for development, so if the Ag specks are relatively small and thenbsp;direct density is below the region of our measurements. It has notnbsp;or hardly been observed for higher direct densities, far above thenbsp;region where the S-shape occurs 3) p. 322. The effect can benbsp;understood by the Gurney-Mott theory (see Ch. I and IV). Isnbsp;the occurrence of the Herschel effect perhaps related to that ofnbsp;the S-shape?

This question led us to consider the course of d�, the time derivative of d. Fig. 14 and 15 represent d' against t and dnbsp;respectively, pertaining to an exposure to orange light (OG3). Wenbsp;notice: d' rises from a very low value, probably almost zero,nbsp;proportionally with time, corresponding to a quadratic rise of d

the infrared, but we do not know of experiments concerning the effect of adsorbed Ag atoms on the absorption of AgBr. Whatever the precise cause may be, apparently thenbsp;sensitivity does increase with density.

If these considerations are correct we may conclude that the simple (d; t) curves for blue light, where we have to do neithernbsp;with a Herschel effect nor with a Becquerel effect, are the mostnbsp;general ones and the S-shapes for red are due to additionalnbsp;complications which are essentially connected with the Herschelnbsp;effect and with the Becquerel effect.

It is known 3) p, 93, that adsorption of OH-ions shifts the absorption limit of AgBr towards the infrared. It is plausible thatnbsp;the change in colour of the emulsion and its increasednbsp;sensitivity for direct blackening after the water treatment is due tonbsp;such an adsorption of OH-ions. The decreased sensitivity fornbsp;developed-density formation, reported in � 16 possibly means thatnbsp;the OH-ions are adsorbed mainly in the neighborhood of (superficial) sensitivity specks. Owing to their negative charge photoelectrons will then preferably settle at specks in the interior ofnbsp;the grain, thus becoming inaccessible for the developer.

We mentioned in Ch. I that the Gurney-Mott theory explains many phenomena connected with photography from one viewpoint.nbsp;This renders the theory most valuable. However, in view of certainnbsp;quantitative features we deem a slight modification necessary. Wenbsp;shall consider these features here in some detail.

In � 4 we briefly mentioned the explanation of this part of the r.l.f. in which it is assumed that a negatively charged speck needsnbsp;some time for its neutralization; subsequent electrons, that arrive toonbsp;soon, are repelled and thus a decrease in efficiency results.

We shall see that the time required for neutralization is of the order of 10�� sec. so that the intensities at which this type of r.l.f.nbsp;should become observable corresponds to rates of absorption,nbsp;comparable with one quantum per 10�� sec. per grain. In realitynbsp;these effects become markedly observable at intensities of the ordernbsp;of 1 quantum in 10�� sec., per grain.

This discrepancy indicates that, though the effect proposed by the authors may appear at much higher intensities, the observednbsp;deviations from reciprocity law at high intensities require an othernbsp;explanation.

An estimate of the average time of neutralization of captured electrons can be obtained in the following way.

Suppose the number of the interlattice silver ions per cm^ is N, their average spacing 1, with l^N'^-'l, the specific conductivity ofnbsp;AgBr is o- e.s.u. and the charge of the ion or electron � e e.s.u.nbsp;Then the current density J in a field of F e.s.u./cm, with ionsnbsp;moving at average speed v in the direction of the field, is:

The field, caused by an electron at distance 1, is F = e/l^, so that the time required to neutralize a trapped electron by a silvernbsp;ion is of the order:

/ l^Ne 1

at room temperature. The value o- = 0.5 X 10�^� Ohm~i cm^�^ ivas taken from Tubandt ¹) so).

Here Gurney and Mott assume that a sensitivity speck has a tendency to dissociate into an electron and a positive speck owingnbsp;to the thermal agitation. An arriving photoelectron has to overcomenbsp;this tendency, if the speck is to be charged negatively. At highernbsp;light intensities the higher outer electron vapour pressure reducesnbsp;the dissociation and succeeds in charging the speck negatively,nbsp;which is followed by attraction of interlattice silver ions etc. andnbsp;leads to the formation of the latent image, as described in � 4.nbsp;The maximal concentration of electrons that can be reached at

Additionally we mention that, according to measurements of Arzyby-SCHEW^^), the mobility of similar ions in similar lattices is of the order of magnitude of 1 cm^/sec.Volt at room temperatures. Assuming a similar valuenbsp;for the mobility of AgH~ ions in an AgBr lattice, this would correspond to anbsp;value for 1 of the order of 2.10�� cm or ca. 10� interlattice ions per grain.

a given light intensity depends on the factors determining the mean life time of the free electrons. The authors assume that an electronnbsp;may fall into a hole from one of the 6 adjacent halogen ions innbsp;a time of the order of 10�* sec., and that it will spend aboutnbsp;6.10�^^ of its life on these ions, the grain consisting of lO�ionnbsp;pairs. So the mean life time of an electron with respect to onenbsp;hole is of the order of 2.10* sec.; the life time is n times as smallnbsp;if n electrons (hence n holes) are present in the grain. This isnbsp;a characteristic time for the grain. If a critical electron concentrationnbsp;has to be established to start the process of latent image formation,nbsp;it follows that the intensity must exceed a given critical value ofnbsp;the order of at least 1 quantum per 2.10 * sec.. For emulsions withnbsp;fine grains (0.2 ft or 10 * ion pairs) the corresponding figure wouldnbsp;be 2 seconds.

Several points give rise to difficulties if this explanation is considered more closely.

In the first place it seems doubtful that a neutral Ag^S speck should have an appreciable tendency to emit electrons into thenbsp;conduction levels of AgBr, much larger than the AgBr itself, ifnbsp;it has a conduction band, lying somewhat below that of AgBrnbsp;(which is necessary to make it trap electrons at higher concentrations) since the dissociation electron has to be released from anbsp;ground level.

In the second place the estimate of the electron-life time would suggest that plates with fine grains cannot be made developablenbsp;by intensities of one quantum in 2 sec. or longer, whereas we knownbsp;that such intensities have an efficiency below optimal but by nonbsp;means negligible. Moreover it should be mentioned that if thenbsp;attraction between an electron and its hole is taken into accountnbsp;this discrepancy becomes still more pronounced.

Webb 32) advanced a somewhat different explanation for the r.l.f. at low intensities. This author performed experiments,nbsp;similar to those of Weinland^s), concerning the effect of a givennbsp;exposure which is partly admitted at the optimal intensity andnbsp;partly at a much lower one. It was found that the density ofnbsp;an entire low intensity exposure is almost equal to that of annbsp;exposure consisting of low intensity for the first half and of optimalnbsp;intensity for the second half of the energy; on the other hand the

density D2 produced by exposing entirely to the optimal intensity is almost equal to that of an exposure consisting of optimal intensitynbsp;for the first half and of low intensity for the second half of thenbsp;energy. In other words the efficiency of the intensity of the secondnbsp;half of the exposure is almost equal to that of the intensity ofnbsp;the first half. These results led this author to the hypothesis thatnbsp;a Ag speck is unstable against thermal agitation, as long as itnbsp;has not yet surpassed a critical size, which is about half of tnenbsp;size required for developability. The disintegration of such a smallnbsp;Ag speck is assumed to proceed in the same way as proposed bynbsp;Gurney and Mott for the Herschel effect, namely by the emissionnbsp;of an electron, followed by the expulsion of a Ag ion. The latternbsp;authors have pointed out that this type of neutralization is to benbsp;expected preferably for small Ag aggregates, consisting of a fewnbsp;atoms. According to Webb the instability of small Ag specks isnbsp;responsible for the lower efficiency of low intensities for buildingnbsp;developable Ag specks, i.e. the low-intensity r.l.f.

The essential difference of the latter type of explanation with respect to the former one is firstly that the ionization of the Ag2Snbsp;speck needs no longer to be assumed for the special purpose ofnbsp;explaining the r.l.f., and secondly that the ionization of Ag specksnbsp;is combined with the expulsion of Ag ions, an effect which isnbsp;known to exist from the Herschel effect.

3. The developability of an appreciable number of grains after absorption ofnbsp;about 10 quanta.

As has been discussed in Ch. Ill the sensitivity specks where electrons may stick are distributed throughout the volume of thenbsp;grain; at the surface we find about 1 %, so that grains whichnbsp;need only one quantum to be developable should be relatively rarenbsp;from this standpoint. Even grains that are developable by a fewnbsp;quanta should be very improbable, whereas we know that a largenbsp;fraction of the grains do belong to this group.

In fact, the probability to find 3 electrons at the same speck if 10 electrons are distributed at random over 1000 specks is ca. 10�^.

the large number of sensitivity specks in one grain it is possible to extend this picture so as to cover also these facts. The dissociation of sensitivity specks depends very strongly on the depth ofnbsp;their potential hole, so that if we assume a relatively small dispersionnbsp;in these depths, already a great preference has to be expected fornbsp;the deepest holes, which determine the maximum electron vapournbsp;pressure in the grain. It seems not implausible that holes at thenbsp;surface should be slightly deeper on the average than in the volume.nbsp;It was pointed out also by the mentioned authors that lack of roomnbsp;in the interior of the grain may be responsible for this. However,nbsp;in order to maintain the validity of the explanation for the lownbsp;intensity r.l.f. the deepest hole may on the other hand not benbsp;too deep.

In � 19 we shall show that another explanation, covering also the high intensity r.l.f., is more probable.

From the preceding section it has become plausible that some minor changes of the theory are necessary to make it applicable alsonbsp;to the cases mentioned, which caused some quantitative difficultiesnbsp;to the theory in its original form. These difficulties can be overcomenbsp;by pushing Webb�s idea one step further, and discussing whatnbsp;happens to electrons and ions that are produced by disintegrationnbsp;of unstable Ag aggregates.

Two factors are of importance in this connection: the probability of the emission of an electron from a Ag speck and the probabilitynbsp;that an emitted electron shall reach another sensitivity speck. Thenbsp;essential assumptions in our discussion are:

1. nbsp;nbsp;nbsp;the probability for the emission of an electron from a Agnbsp;aggregate, followed by the expulsion of Ag ion, decreasesnbsp;gradually as the aggregates become larger.

2. nbsp;nbsp;nbsp;the probability for an electron to reach an other potentialnbsp;hole (sensitivity speck) is almost unity.

These two circumstances cause an electron to travel hither and thither in te grain, being captured, neutralized, and released atnbsp;intervals, until it finally arrives at such a large Ag speck that itnbsp;is bound and neutralized stably, or unites with a positive hole and

is lost for latent-image production. In other words the mean life time for the electrons is considerably increased on one hand, bynbsp;the fact that they spend part of their life at the sensitivity specks;nbsp;on the other hand it is decreased, especially at low concentrationsnbsp;of electrons, due to the greater chance to be trapped at sensitivitynbsp;specks, so that an a priori estimate of its magnitude can hardlynbsp;be given. Moreover this picture ensures automatically that thenbsp;electrons are concentrated to those few specks which happen tonbsp;be the first ones to reach the stability size.

The fact that small Ag specks are unstable against thermal agitation as assumed by Webb (see above), that somewhat largernbsp;specks are stable, that still larger ones are developable (bindnbsp;electrons of the developer) and are destroyed by infrared quantanbsp;(ca. 1.5 e.V) as observed with the Herschel effect, indicatesnbsp;that the depth of a silver potential hole increases gradually withnbsp;its size, probably reaching asymptotically the value for large Agnbsp;bulks. Also the fact that the developability of a Ag speck dependsnbsp;on the reducing power (oxydo-reduction potential) of the developernbsp;points towards this conclusion.

The smallness of the probability of electron loss during one journey follows from several experiences. Firstly the quantumnbsp;efficiency for the production of a detectable amount of Ag isnbsp;almost 1, which indicates that then most electrons reach a sensitivitynbsp;speck. This experience suggests that also under other circumstancesnbsp;the capturing-probability per single journey of an electron will benbsp;negligible. Secondly the conservation of the amount of Ag uponnbsp;red irradiation of directly blackened plates in the experiments ofnbsp;Cameron and Taylor ^o) on the photoadaptation effect indicatenbsp;that this sort of electron transport occurs with very small losses.

We shall show to what extent the modified theory is able to describe experimental facts.

We have seen already that it automatically ensures Ag coagulation at a few specks which are situated preferably on thenbsp;surface of the grain, or in general at initial sensitivity specks withnbsp;deepest potential holes.

The low intensity reciprocity law failure is interpreted as the loss of the first electron if the second one arrives much laternbsp;than the mean life time of the first one, so that the chance tonbsp;reach the stability size with a given number of electrons becomesnbsp;smaller.

The high intensity reciprocity law failure is interpreted as an increased probability to find the electrons wandering simultaneouslynbsp;through the crystal, if they are excited at intervals that are muchnbsp;smaller than the mean life times of their predecessors. This statenbsp;of affairs will then favour the formation of more small Ag specksnbsp;above one large one.

Let us assume for example that a speck of two Ag atoms is stable and one of four atoms is developable. Then we consider four successive photoelectronsnbsp;of which the first and second one have formed a stable speck (1;2). If thenbsp;time between the third and fourth electron is shorter (higher intensities) thenbsp;chance that both will stabilize each other at a place (3; 4) different from (1:2)nbsp;is larger, which is a non developable configuration. If, however, the third electronnbsp;has enough time (optimal intensity), it will have an increased chance to formnbsp;(1;2;3), leading finally to the developable (1; 2; 3; 4) configuration.

It is probable in that case, that there is also a relatively larger chance for electron loss, since the latter must be expected to dependnbsp;quadratically on the electron concentration, namely on the productnbsp;of the concentrations of electrons and holes.

Both causes cooperate in decreasing the chances for the formation of developable specks. The optimal intensity (at room temperature:nbsp;interval between successive quanta ca. 10^� sec.) so correspondsnbsp;more or less to the mean life time of the first electrons. An additionalnbsp;feature of this modified view is that it automatically explains whynbsp;the region of optimal intensity is so short.

As soon as stable Ag specks are formed an electron will not have to wait for the next one in order to be bound stably, so itsnbsp;efficiency becomes independent of the intensity and equal to thatnbsp;of the optimal intensity; the low intensity r.l.f. for exposuresnbsp;following optimal intensity disappears, but we expect that the highnbsp;intensity r.l.f. remains for corresponding cases.

makes us expect that if the speed of warming up a film after exposures at low temperatures, is varied within wide limits, annbsp;optimal (rather low) speed should exist with respect to the obtainednbsp;developed density. For, at low temperatures the electrons arenbsp;trapped stably also in shallow potential holes, from which theynbsp;are released during the warming up period by the increasing thermalnbsp;agitation. If all of them are released almost simultaneously wenbsp;obtain an efficiency corresponding to very high light intensities,nbsp;on the other hand very slow warming up corresponds to low lightnbsp;intensities. In both cases the effect will be independent of thenbsp;intensity at which the electrons were produced originally. So thisnbsp;explains the absence of reciprocity law failure for low temperaturenbsp;exposures as observed by Webb. Moreover, if the electrons arenbsp;released and neutralized one by one, a much higher efficiencynbsp;in the production of developable density has to be expected.

In one case the sensitivity at liquid air temperature was found to be even greater than at moderately low temperatures (�78�)nbsp;for quite high intensities (cf. Webb l.c.), which is in qualitativenbsp;agreement with our explanation.

An additional feature of the proposed theory is, that it yields a basis for a qualitative explanation of the changes of developable density as a function ofnbsp;the time that elapses between exposure and development (Brush effect). A morenbsp;or less quantitative elaboration of these ideas lies beyond our present purpose.

In 1933 VAN Kreveld is) published the results of an investigation, concerning the developed density, produced by simultaneousnbsp;irradiation with various wave lengths on photographic emulsions.nbsp;Briefly this author found, that the photographic plate �addsnbsp;different radiations correctly�. For instance if an intensity I^ ofnbsp;colour ¹) a, admitted during time t produced the density D, and annbsp;intensity Ij, of colour b produced the same density D in the samenbsp;time t, then D was also produced in t by the mixture of both coloursnbsp;with components Yi Ig and Y2 Ij, or alg and (1 �a) Ij,.

This addition law has been confirmed for various types of emulsions, intensity ratios of components of the mixture, compositions of components, number of components, densities, timesnbsp;of exposure, conditions of development and finally for variousnbsp;methods of density measurement. More or less systematic errorsnbsp;of the order of 5 % occurred.

Linder certain extreme conditions, especially if D approached its saturation value, large systematic deviations of this simple lawnbsp;were established definitely.

Various attempts to generalize the addition law failed, (see � 28), except one, which is due to Webb. Besides extensive verificationsnbsp;of VAN Kreveld�s law for simultaneous colour addition, this authornbsp;presents experiments that indicate correct successive addition ofnbsp;corresponding intensities of various colours.

In this connection the word colour indicates a radiation of given relative spectral composition.

The remarkable simplicity of the addition law has lead VAN Kreveld to propose to use this law as the basis for thenbsp;standardization of colour sensitivity of photographic emulsions.

The aim of our discussion is to investigate to what extent the addition law can be justified theoretically and to what extent itnbsp;has to be considered as an approximation. For this purpose wenbsp;shall theoretically deduce a new addition law which can benbsp;interpreted more easily in terms of the underlying physicalnbsp;processes; from this law we shall be able to derive van Kreveld�snbsp;law under certain circumstances.

The formulation, which we shall use in dealing with the empirical law, is: Let be the energy required to produce a density D innbsp;time t by irradiation with the colour a, let Ej, be the same fornbsp;colour b, then a mixture E^,, composed of an energy ^E^ = X X Egnbsp;of the colour a and ]^Eg, = X Ej, of the colour b, will alsonbsp;cause D in t, provided that;

Because of the equality of t for Eg, Ej, and E^j the energies may be replaced by the corresponding intensities. The ratio ofnbsp;Eg and E(j depends in general on D; it does not depend on t onnbsp;account of Webb�s relation (see � 3 or 12).

Another way to state this law is: the efficiency with which the energy of a certain colour, irradiated at constant intensity,nbsp;contributes to the formation of a latent image of developablenbsp;density D (see Ch. I) depends only on D and t. It does not dependnbsp;explicitly on the blending ratio of simultaneously acting colours.

case that (D; t) curves for corresponding intensities of different colours were congruent. Then illumination with intensities ofnbsp;colour a and Ij, of colour b, that would cause equal densities fornbsp;one value of t, would do so for all values of t, in other words thenbsp;ratio Ig : Ij, had the meaning of relative colour sensitivity, whichnbsp;could then easily be interpreted as ratio of absorption of quanta.nbsp;Also in that case the relation of Webb would be almost self-evident,nbsp;since the processes in the grains are determined by the numbernbsp;of absorbed quanta; this number would be equal for all grainsnbsp;if the same density was produced in the same time and so thenbsp;reciprocity law failure should be equal also.

However in reality the situation is more complicated. The ratio Ig :nbsp;nbsp;nbsp;nbsp;depends on D, or as we, mentioned before, the contrast

y of an emulsion depends on the colour of the light, so that a constant colour sensitivity is not present. If the gradual productionnbsp;of density during exposure is considered for two colours separatelynbsp;and mixed (see fig. 16) it is hardly possible to predict on theoreticalnbsp;grounds how the ratio of the intensities should be chosen in relationnbsp;to the time of termination of the exposure.

This suggests that the empirical addition law is an approximation, be it a very good one. Especially the fact that the empirical addition law does not allow to establish a differential law sustainsnbsp;this idea, since one may well expect that the addition of the effectsnbsp;of the mixture components obeys some differential law. We willnbsp;start our discussion therefore from the differential point of view.

� 22. A new theoretical addition law.

We consider a plate which consists of grains of uniform size. Its sensitive layer is assumed to be so thin that, if a beam of lightnbsp;passes through the plate, practically no decrease of light intensitynbsp;results. If this plate is irradiated with a constant intensity I ofnbsp;colour a, both the developable density D and the rate of increasenbsp;of the developable density D' are determined by a, I and t; wenbsp;prefer to express D' as a function of the independent variablesnbsp;I and D, and of the colour a, which is possible by excluding fromnbsp;our discussion the region of solarization

Dg' = nbsp;nbsp;nbsp;D/(/;D).

If this plate is irradiated with an intensity I(t), varying with time, of colour a, the value of D' at each instant will in general not benbsp;given by (2), since the efficiency also depends somewhat on thenbsp;way in which the density D was produced. If I(t) can be expandednbsp;in a Taylor power series in t we may express this dependencenbsp;analytically in the form:

. . .).

Then also for a sufficiently slowly changing intensity (�adiabatic variation�) we may assume that the relation (2) remains valid.nbsp;In other words we assume that, within certain limits, the rate ofnbsp;density increase at a given density and intensity does not dependnbsp;sensitively on the way in which this density was produced.

So far we have considered the irradiation with one colour; let us now consider a mixture of two (or more) colours. It followsnbsp;then from most pictures about latent image formation, andnbsp;especially clearly from the Gurney-Mott theory that, as soonnbsp;as a grain has absorbed a quantum of either colour, some rathernbsp;involved process or chain of processes starts to proceed, ultimatelynbsp;leading to the formation of a Ag atom with an efficiency, in generalnbsp;dependent on the state of the grain and on the colour (energy)nbsp;of the quantum.

For simplicity we shall consider the Gurney-Mott picture here. The absorbed quanta yield a photoelectron which is subsequentlynbsp;neutralized by a Ag ion. The efficiency of the processes after thenbsp;absorption is in general determined by the number of Ag atoms,nbsp;the rate of their formation and the colour of the light.

The influence of the rate of formation of Ag atoms on the efficiency becomes evident in the reciprocity law failure (see Ch. Inbsp;and Ch. IV). Its influence may for example be pictured as thenbsp;distribution of Ag atoms over the sensitivity spots and the numbernbsp;of electrons that have not yet been neutralized stably, which maynbsp;in turn be characterized by D and D'.

The influence of the wave length is required by the noncongruence of (D; log t)j curves for different wave lengths, their slope being usually higher for wave lengths for which the platenbsp;has been sensitized. Various causes may be responsible for thisnbsp;fact. It is theoretically possible to ascribe it to a difference innbsp;number of grains that are sensitive to different colours (bothnbsp;contrast and maximum density dependent on colour, as assumednbsp;by VAN Kreveld). We deem this improbable since this would meannbsp;that usually more grains of a panchromatic emulsion are sensitivenbsp;to red than to blue. We agree with the mentioned author thatnbsp;also the grain size is not the selective factor. But besides thesenbsp;two possibilities we wish to point out that other factors may benbsp;responsible; for instance the efficiency of transfer of photoelectronsnbsp;from a sensitizer to the AgBr conduction levels may have lessnbsp;dispersion about its mean value than the corresponding efficiencynbsp;for the transfer of excited Br electrons to these levels.

Other theories which take into account the quantic nature of light absorption may correspond to other pictures, but will lead tonbsp;essentially the same conclusion, namely that the efficiency of anbsp;quantum after its absorption depends on the state of the grainnbsp;and so is practically determined by D, D' and the colour of thenbsp;acting light.

The differential addition law which may then be expected states: Let Ig be the intensity of the colour a required to produce D' atnbsp;a density D (which was produced with almost the same Ig justnbsp;before), correspondingly Ij, for the colour b, and correspondinglynbsp;for a mixture of a and b containing glm � ^nbsp;nbsp;nbsp;nbsp;colour a

The form of this addition law is almost the same as that of the empirical one, but the subscripts D,t are changed into D,D'. Asnbsp;was mentioned before, this law does not follow from van Kreveld�snbsp;law by differentiation. It is not possible to deduce a differentialnbsp;law from this empirical law at all, on account of the dependencenbsp;of 7 on A,. The theoretical law merely follows from the generalnbsp;picture of Gurney and Mott and others. The part of the lawnbsp;placed between parentheses, which requires that D was producednbsp;just before with almost the same intensity, may in most practicalnbsp;questions be omitted, since the influence of I' and higher derivatives,nbsp;or more generally the history of D, for which it stands, is usuallynbsp;very small.

� 23. The integrated differential addition law.

In order to be able to compare the differential law with VANnbsp;Kreveld�s law we will bring itnbsp;into an integral form.

We consider a (D, t) plot (see fig. 16), containing three linesnbsp;which connect the origin with thenbsp;point P(D, t), namely one fornbsp;the constant intensity I^p of thenbsp;colour a, one correspondingly fornbsp;Ij,p and one for the constant

lines will in general run rather closely together and the third linenbsp;will usually lie between the firstnbsp;and the second one.

Now the mixture line can also be described by type a radiationnbsp;of suitable intensity as a functionnbsp;of time Ia(t) and correspondinglynbsp;I]3(t). In practice both functionsnbsp;will closely fulfil the requirement

i,{t) nbsp;nbsp;nbsp;hit)

t\ nbsp;nbsp;nbsp;t\

The formal similarity with van Kreveld's law is obvious, but there is a distinct difference in the meaning of the coefficients ofnbsp;glm 3nd fjlni-

� 24. Conditions for agreement between both integral laws.

Though the discussion of � 23 has given rise to an addition law which is somewhat different from van Kreveld�s law, it isnbsp;not necessary that one of them excludes the other one.

We ask for the conditions that the two integral laws shall be valid simultaneously. There are three possibilities for an answernbsp;to this question.

In the first place it is obvious that the two laws will agree if Ia(t) is constant throughout time, thus identical with Igp, andnbsp;also I{,(t) = Ij,p. This would mean that the sets of density-timenbsp;curves for constant intensity for the colours a and b are congruent.nbsp;(An arbitrary relation between the parameters Ig and Ij, ofnbsp;congruent curves may still exist). We know that in practice thisnbsp;condition is not fulfilled; the situation is in general more likenbsp;that in fig. 16.

The second possibility for the simultaneous validity of both laws

The condition for this case is that:

for each colour separately, or at least, since only a narrow group of oqI paths is considered between Igp and Ij^p, correspondingnbsp;to different mixing ratios A : yu of a and b, the first variation ofnbsp;the integral (with fixed boundary points) shall be zero. If wenbsp;recall that 1/Ig(t) = Fg(D(t), D'{t)), variation calculus requiresnbsp;for first order independence of path:

We require this for the I^p curve along which F does not depend on t so that, omitting the subscripts a, we have:

Eliminating Dquot; from (6) and (7), we obtain a differential equation in Q and D' only, which can be solved without difficulties,nbsp;and leads to:

choosing D instead of t as independent variable. This transformation leads to a linear equation, with solution:

(10) with an arbitrary function (D).

Formula (10) is identical with the requirement that the (D; log t) jg _ curves in the neighborhood of I^p are parallelnbsp;in the log t direction.

D'=:X(I) W(D)¹)

Similarly for the b irradiation the requirement (10a) has to be satisfied, eventually with other functions .X and !F. Then bothnbsp;addition laws will be concordant.

In practice these requirements are fulfilled very well. (D; log t)[ curves are not parallel if large ratios of I are considered, but fornbsp;small ratios often even up to about 2, the curves are verynbsp;satisfactorily parallel. So our conclusion is that good term by termnbsp;agreement may be expected between both laws to the first order.

We may inquire whether also the second variation can be made to vanish, insuring a still better agreement. The condition therefore, as developed bynbsp;variation calculus, is:

d L d dL

dD dtdD

Both (lOa) and (11a) can only be fulfilled if (D;t)j curves are straight lines. This is a special case of the condition of p. 61 for strict agreementnbsp;between the two laws. It is generally not the case for photographic plates, sonbsp;that second order agreement can in general not be expected. Special formsnbsp;of (10a) may however cause such an agreement, since (11) is derived withoutnbsp;making any assumptions about .S�(I) or 9 (D).

D' = ^gt;(D)/lK(l//_C)

D' ^ (D) (1//-C)

In the third place we inquire about the conditions which cause agreement between both integral laws, allowing inequality ofnbsp;corresponding terms.

We proved that the path in the (D; t) plane for constant intensity will make the integral under discussion to be extremenbsp;(first variation zero). It can be shown easily (see below) that thisnbsp;extremum is a minimum. Hence the integrated differential law willnbsp;always yield somewhat larger terms than the empirical law, ifnbsp;second order effects are considered (see � 25), there being no chancenbsp;that the increase of one term will be compensated by the decreasenbsp;of the other one. So we conclude that in as much as the differentialnbsp;law is strictly valid and both laws do not agree term by term,nbsp;they will show no overall agreement.

However, it is possible that deviations from the differential law occur on account of the fact that when we describe the mixturenbsp;line with one colour the intensity changes arc not strictly adiabatic.nbsp;We wish to point out, that in such a case the integrated differentialnbsp;law remains valid, but the term by term agreement may be disturbed,nbsp;both terms undergoing changes that are equal in first approximationnbsp;and of opposite sign. If the mixture curve is described e.g. by thenbsp;colour a, the intensity l3(t) will decrease with time; for b it willnbsp;then increase with time. In connection with the experiments ofnbsp;Webb and of Weinland described in Ch. IV we may expect thatnbsp;the efficiency of these intensities will then be somewhat higher ornbsp;lower respectively than for the adiabatic case (intensities supposednbsp;below the optimum). The same will be true for Ig* and Ij,*, sonbsp;that the two terms of the integrated addition law change in opposite

directions. This situation can be formulated more precisely in the following way.

Suppose the (small) difference between l/Igp and l/Ig*. connected with the fact that the intensity changes are not strictlynbsp;adiabatic, is given by the parameter ju, of the mixing ratio.

Here we introduce the plausible first order approximation that the mixture curve divides the distance between the single-colournbsp;curves into the ratio jx ; X and that the effect of varying thenbsp;intensities shall be proportional to these distances with oppositenbsp;sign. Then the first-term difference is:

So we conclude that in first approximation deviations from the adiabatic changes of I do not influence the validity of and agreementnbsp;between both addition laws.

We believe that the considerations given in this section may give some theoretical foundation to the empirical van Kreveldnbsp;addition law and show why this law holds so astonishingly well.nbsp;The reasons are the parallelism of (D; log t)j curves for each colour

separately and the fact that, moreover, the (D;t)j curves for different colours run not widely differently.

Both the differential addition law and the empirical addition law have so far been considered for simplified conditions. Wenbsp;wish to apply them to real plates, in which not all grains receivenbsp;the same light intensity, due to the absorption of the incident light,nbsp;which causes an intensity gradient from superficial layers to deepernbsp;ones. Moreover, the effective intensities (rates of absorption ofnbsp;quanta per grain) in one layer differ from grain to grain, onnbsp;account of a large dispersion of the grain sizes. The constructionnbsp;of the VAN Kreveld law and of the new law, both in differentialnbsp;or integral form, is such, that any more complicated situation willnbsp;cause only second order deviations, as is also the case for eventual deviations between the vannbsp;Kreveld law and the integratednbsp;addition law. Indeed both laws,nbsp;according to their formulation,nbsp;refer to the case that the finalnbsp;density D, or the momentaneousnbsp;density D and the rate of densitynbsp;increase D' respectively, are equalnbsp;for each colour separately, whatever may be the light gradient,nbsp;grain size distribution, method ofnbsp;development of the plate, etc.. Sonbsp;D and D' will show maximumnbsp;(second order) deviations fornbsp;A. oo 0.5 (fig. 17). A generalnbsp;estimation of the order of magnitude of these deviations fornbsp;various cases is rather difficult.

The condition for first order agreement between the two laws was that /dt/I(t) should be extreme along a (D; t)j line. It is wellnbsp;satisfied, due to the parallelism of (D;t) curves for variousnbsp;intensities of one colour. The integral is a minimum, as may benbsp;seen by chosing part of the corresponding (D; t) path horizontallynbsp;or vertically. So the integral corresponding to the intensity Ig^)nbsp;describing the mixture line should be somewhat larger.

Assuming that the differential law is correct, we expect that VAN Kreveld�s law shows small systematic negative deviationsnbsp;(A /i lt; 1). These deviations should be negligible for densities,nbsp;where D is almost proportional to t, but will increase for highernbsp;densities, and also for densities in the neighborhood of thenbsp;threshold.

Van Kreveld published data for 27 plates measured at densities between 0.03 and 1.6. These show indeed a systematicnbsp;error of � 1.4 � 0.3 % which may partly be due to the causenbsp;mentioned (cf. also � 25. 2). For higher densities larger systematicnbsp;errors in the same sense occur. For a given time of exposure thenbsp;way in which the density reaches its saturation value as a functionnbsp;of the intensity may be different for different colours, especiallynbsp;so for sensitized plates, as is well known. In this saturation regionnbsp;the empirical law breaks down. The colour which tends soonernbsp;towards saturation will need extremely high intensities to producenbsp;the required density in the given time, so its term in the additionnbsp;law becomes negligible; however, in the mixture it is more efficient,nbsp;so that negative deviations result (A ;u, lt; 1; cf. van Kreveld l.c.nbsp;fig. 5). It is difficult at present to estimate what errors (if any)nbsp;the theoretical addition law will show in this region. Possibly innbsp;its differential form it may show some disagreement with experiments for very high densities, but at any rate the integrated formnbsp;will suffer herefrom much less, being not affected by this effectnbsp;for the largest part of the path of integration. The possibility tonbsp;apply this form of the law successfully in a larger region ofnbsp;densities may be considered as a feature of trustworthiness.

The rate of Ag production corresponds to the rate of absorption occuring in one grain. So the effective intensity is proportionalnbsp;to the grain area. The non-uniformity of the grains is from thisnbsp;standpoint similar to a non uniform illumination of equal grains,nbsp;so its effect will be of the same type as the effect of a light gradientnbsp;occuring in plates of finite thickness. A difference may be expectednbsp;with respect to the influence of wave length. The non-uniformitynbsp;owing to the grain is practically the same for all wave lengths, in asnbsp;much as the absorption spectra of the grains do not depend onnbsp;their sizes, but the gradient of light intensity owing to absorptionnbsp;will in general vary with wave length, which gives rise to anbsp;somewhat more complicated situation.

Rather special assumptions have to be made to extend the strict validity of the differential addition law to cases of non-uniformnbsp;illumination or grain size. The difficulty is mainly, that if thenbsp;intensities Ig, Ij, and l^j fulfil the addition law for the entire plate,nbsp;they will be adapted to some average depth and average grainsize.nbsp;But higher or deeper layers of the plate and smaller or larger grainsnbsp;will show distinct first order deviations in density production duenbsp;to Ig and Ij,. For the total density of the plate the higher densitynbsp;of higher layers and larger grains compensates the lower densitynbsp;of lower layers and smaller grains to the first order, but the secondnbsp;order effects that remain can hardly be estimated. Even their signnbsp;cannot be predicted on general grounds.

Van Kreveld has tried to extend his addition law to mixtures, the components of which are not of the same duration. Fornbsp;instance this author took for ; 10 seconds red, for Ij, � 50 secondsnbsp;violet and for the �mixture�: 10 seconds a combination of bothnbsp;immediately followed by 40 seconds of the violet component of

the mixture. From fig. 18 and � 23 it is clear that here large deviations must be expected fromnbsp;the standpoint of the differentialnbsp;law. Its integration for this casenbsp;yields the same as an ordinarynbsp;addition law for an exposure fromnbsp;O up to the point P (so alongnbsp;the common path of both components). Instead of putting I^p,

VAN Kreveld used and Ij^r. causing a decrease of the firstnbsp;term and an increase of thenbsp;second one. His intensities werenbsp;below the optimal one, so thatnbsp;the efficiency increases with increasing intensity. If we assumenbsp;linear (D; t)j curves, satisfyingnbsp;the relation D = I^'^^.t, which isnbsp;a reasonable approximation fornbsp;densities of the order of 0.3, andnbsp;if we assume adiabatic change of

intensity at P, it can be shown simply that the sign of the deviations of X p, from 1 is negative or positive if the intensitynbsp;is above or below the optimal one respectively. So one expectsnbsp;positive deviations in van Kreveld�s experiments, especially for lownbsp;densities. Indeed, large positive deviations of the order of 20 %nbsp;were found. By assuming Schwarzschild�s p = 0.75 the assumptions made above lead to A p. = 1.15, which is of the correctnbsp;order of magnitude.

Van Kreveld notices that his extended law would necessarily neglect any difference resulting from the sequence of the mixturenbsp;components: so e.g. 40 sec. violet followed by 10 sec. red violetnbsp;irradiation should yield the same density as 10 sec. red 4- violetnbsp;followed by 40 sec. of violet, whereas in practice such differencesnbsp;exist. Also here the integrated differential addition law correspondsnbsp;better to experimental evidence, since it does predict such differences.

Namely in the first example mentioned, integration yields the addition law, only to be applied between O and P; in the secondnbsp;case it should be applied betweennbsp;P' and R' (see fig. 19), so in bothnbsp;cases along the common path ofnbsp;both components, which maynbsp;involve quite different intensities.

A second extension of the empirical addition law (cf. vannbsp;KreveldIS) p. 69)^ namely thenbsp;time addition law, has beennbsp;recognized by this author himselfnbsp;to be in general incorrect. Thisnbsp;law would state: If yields Dnbsp;in tj and yields D in t2, thennbsp;both together yield D in ts withnbsp;l/tg = 1/ti l/t2. We fullynbsp;agree with his proof, that time-and intensity addition law excludenbsp;each other if the reciprocity lawnbsp;are not approximately optimal.

A third extension of the van Kreveld addition law has been given by Webb 34)_ for successive illumination with different colours.nbsp;This law states: Let the intensity 1^ of a colour a produce D in t,nbsp;the same Ij, for b, then an exposure of (t � x) to 1^ followed bynbsp;X to If, will also produce D.nbsp;nbsp;nbsp;nbsp;^

It is clear that this law can be expected to be an approximation which must show second order deviations, since it predicts thatnbsp;the sequence in which the colours are applied is not essential,nbsp;whereas it is known that reversal of the sequence causes small

changes, be it much smaller than in the first extension, on account of the average equivalence of 1,^ and Fig. 20 shows in annbsp;exaggerated way what the differential law makes us expect; Ij^nbsp;during Yl after Ig during Yl tnbsp;yields D A',\^ during Yi t afternbsp;Ijj during Yl t yields D�A¹).

The data published by Webb 34) allow an estimation of the difference A to be expected. In allnbsp;cases it falls within his experimental accuracy so that a checknbsp;cannot be obtained from thesenbsp;data.

Herefrom it follows that for many practical purposes Webb�snbsp;extension is a satisfactory approximation.

Summarizing we see that there are five approximations to describe colour addition of photographic plates.

1. nbsp;nbsp;nbsp;The (integrated) theoreticalnbsp;differential addition law (errorsnbsp;very small).

2. nbsp;nbsp;nbsp;Van Kreveld�s empirical law (errors small, except for largenbsp;densities).

3. nbsp;nbsp;nbsp;Webb�s law for successive addition (errors probably somewhat larger).

4. nbsp;nbsp;nbsp;Van Kreveld�s extended law for components of unequalnbsp;duration (errors large).

5. nbsp;nbsp;nbsp;The time addition law (errors very large except near optimalnbsp;intensity).

It is possible that the non-adiabatic change of effective intensity at the switch point causes deviations from what would be expected by simplenbsp;integration. If the .effect is analogous to the results of the experiments of Webbnbsp;and of Weinland mentioned in Ch. IV, a decrease of d would result.

The simplest case of an illumination varying with time is: illumination during a time t^ followed by darkness during a time t2,nbsp;and then succeeded by development etc.. Changes in developablenbsp;density (comp. � 3) as a function of t2 for given t^ appear in variousnbsp;papers; Brushes) observes an increase for 15 minutes, thereafternbsp;a decrease, Weinland^g) finds no effect, Mauz^t) observes anbsp;decrease, Hylan and Blair 3^) observe a decrease and give morenbsp;literature, A. A. Kruithof (priv. comm.) finds a decrease during thenbsp;first 15 min. followed by an increase, of several days, approaching tonbsp;an asymptotic value. We conclude from the discordance betweennbsp;these authors that the conditions of this effect are not yet clearnbsp;from an experimental point of view.

Another aspect of this question is reported by Long, Germann and Blair 39) who expose a paper to green light, so that development would yield a density of 1.3. Then they wait 40 minutesnbsp;and thereafter they continue the exposure. The second exposurenbsp;at first decreases the developable density. Several other observationsnbsp;may be found scattered in literature.

Also Webb�s 32) experiments on the efficiency of high or low intensities after applying part of the exposure at low or highnbsp;intensities respectively (see e.g. � 18, 2) should be mentioned innbsp;this connection.

But besides this more or less incidental research the question of density as a function of inconstant illumination has been attackednbsp;systematically only for the case of periodic intermittency. Up tonbsp;about 1925 many authors had found that intermittency caused annbsp;increase of density and just as many had found a decrease. Davis

and also Weinland (see MeidingerS) p. 307) clarified conditions appreciably, showing that the effect depends generally on thenbsp;intensity of illumination, sector dark to light ratio, and frequency,nbsp;in a rather complicated way. These authors recognized that thenbsp;sign of the effect was correlated with the reciprocity law failure.

In 1933 Webb 21) pushed the problem two steps forward. Firstly he showed that for frequencies, sufficiently above a critical frequency,nbsp;the density obtained was independent of the frequency and secondlynbsp;that it was equal to the density, obtained by a continuous exposurenbsp;with the same average intensity during the same time. He foundnbsp;that at the critical frequency approximately one quantum per flashnbsp;was absorbed per grain and interpreted his findings in a simplenbsp;way on the basis of the quantic nature of light absorption.nbsp;SiLBERSTEiN and Webb gave a more complete theory of thenbsp;phenomena observed. Zimmerman 8) found deviations for verynbsp;short flashes, but Webb^o), repeating Zimmerman�s experiments,nbsp;found his own theory comfirmed, which was a priori probablenbsp;already on general grounds.

We believe that the theory given by Silberstein and Webb is correct in its main lines but incomplete in a few points, whichnbsp;we endeavour to elaborate in � 28; also we wish to present a morenbsp;general method for the computation of the intermittency effect.

� 28. Discussion of the treatment of the intermittency effect by Silberstein and Webb.

In Webb�s experiments plates were exposed to the green Hg radiation in three ways namely: firstly to an intensity Iq, appliednbsp;continuously during time T^, secondly to an intensity I, appliednbsp;continuously during time qT^, and thirdly to an intensity I, appliednbsp;during time through an intermittency sector wheel of transmission q, rotating at a frequency f. The third type of exposures wasnbsp;performed at various frequencies (fig. 21). In all exposures of suchnbsp;a set the plate received the same amount of energy. Denoting the

Since the difference between Do(I,qt) and D (j^(Iq,t) is the reciprocity law failure, these relations link the intermittency effectnbsp;to the reciprocity law failure.

A plot of Df against log f yielded curves of the type of fig. 22, and the frequency at which the ascending part �reached� thenbsp;D ^ level was interpreted as the frequency at which on the averagenbsp;one quantum was absorbed per flash in one unit of action.

The basic idea of this interpretation is, that, �continuousquot; irradiation causes, in fact, a discrete number of absorption acts,nbsp;due to the quantic nature of the process of light absorption. Thenbsp;statistics of such absorptions will hardly differ from those fornbsp;�really� intermittent exposures of the same average intensity if thenbsp;frequency of intermittency is higher than or equal to the frequencynbsp;of quanta reception. The experiments roughly agree with thisnbsp;theory in as much as f^. and I are proportional.

The intensity of irradiation being known, the receptive area of a unit of action could thus be determined and appeared to be of

In SiLBERSTEiN and Webb�s paper both statistics are discussed quantitatively, and the authors point out that the �criticalnbsp;frequency�� cannot be defined sharply, due to the assymptoticnbsp;character of the curve. In its place they define a �fusion frequency�nbsp;which is the value of f where the curve deviates just observablynbsp;from its final value.

We believe that the treatment of the authors is qualitatively correct, but is quantitatively subject to the following remarks.

1. nbsp;nbsp;nbsp;For density production the theory should take into accountnbsp;that the statistics of Ag depositing and other links of the process,nbsp;are in general not equivalent to the statistics of absorption acts,nbsp;at least if the dispersion (root mean square deviation) in the timenbsp;that elapses between light absorption and silver production is notnbsp;negligible, compared with the time intervals between two absorptions in one unit or grain (interquantum time).

2. nbsp;nbsp;nbsp;No account is taken of the decrease of light intensity with thenbsp;depth of the emulsion. If this would be done by merely taking thenbsp;average intensity in the plate instead of the incident intensity, the

size of the units would be found about 1.4 times as large as given by the authors.

3. nbsp;nbsp;nbsp;The grain size distribution is not sufficiently taken intonbsp;account.

4. nbsp;nbsp;nbsp;An experimentally badly defined point is chosen to deducenbsp;the critical frequency from, namely the point at the high frequencynbsp;side of fig. 22 where the intermittency effect becomes �justnbsp;observable�. Moreover it would be desirable to use the whole curvenbsp;of density against log f, instead of one single point, for thenbsp;determination of a parameter like the effective receptive area.

5. nbsp;nbsp;nbsp;The quantity, which the authors choose as a (qualitative)nbsp;measure for the comparison of the intermittency effect of exposuresnbsp;of a given energy and a given average intensity, is the standardnbsp;deviation (root mean square deviation) of the intervals betweennbsp;successive absorptions. Their derivation of the formulae for thisnbsp;standard deviation can be replaced by a shorter and more generalnbsp;one. However we shall show that this quantity is not suited fornbsp;a well approximating description of the intermittency effect.

We shall give a theory satisfying the requirements of these remarks without making essentially new assumptions about thenbsp;mechanism of the intermittency effect. Also we shall discuss thenbsp;connection, that exists between the intermittency effect for thenbsp;entire frequency range and the reciprocity law failure, withoutnbsp;making assumptions about the details of the mechanisms of bothnbsp;effects.

The fact that Df approaches Dq as f 0 is self-evident. For low frequencies the difference Dj'�Dg will in general be comparablenbsp;with those mentioned in � 27. It is usually small compared withnbsp;the difference between very low and very high frequencies,nbsp;D �Dq. On the other hand the fact that the density obtainednbsp;with f -gt; CO approaches the effect of a continuous exposure withnbsp;with the same overall time and the same mean intensity is due to

the circumstance that the statistics of absorptions for both methods of exposure cannot be distinguished (see below). As for these twonbsp;points we quite agree with Webb, but the interpretation of thenbsp;critical frequency as corresponding to approximately one quantumnbsp;per flash per unit is subject to several restrictions.

In the first place it can be stated that the considerations of Webb are applicable without restrictions if the statistics of the processnbsp;of silver depositing are equal to those of quantum absorption. Thisnbsp;would be the case e.g. if no time would elapse between the absorption and the formation of silver. However, we know that somenbsp;time does elapse between the two processes. If this �relaxation�nbsp;time is constant, the two statistics are still equal. Generally thenbsp;relaxation time will scatter from absorption to absorption, aboutnbsp;a mean value. Its dispersion, measured e.g. by the root mean squarenbsp;deviation, will be of the same order of magnitude as its mean value.nbsp;In particular this is the case for processes, the occurrence of whichnbsp;has a fixed probability per unit time, independent of previousnbsp;occurrences. The Gurney�Mott theory suggests that the processes, which are responsible for this relaxation time, belong morenbsp;or less to this type (e.g. the chance for a wandering electron tonbsp;become stabilized is the same for every journey). In that case thenbsp;critical frequency, at which the statistics of Ag depositing becomesnbsp;markedly influenced, will only be proportional to the light intensity,nbsp;if the latter is sufficiently low. As soon as the mean interval betweennbsp;two successive absorptions (�interquantum time�) passes below thenbsp;relaxation time, the critical frequency will become independent ofnbsp;the intensity and will be determined by the relaxation time only.nbsp;Indeed, the fluctuations of the latter cause a decrease of the criticalnbsp;frequency because they make the statistics of Ag depositing morenbsp;irregular than those of quantum absorption.

Hence the reciprocal critical frequency (�critical time�) is equal to the largest characteristic time of the process (interquantum timenbsp;and relaxation time).

In the second place there may be several chain links in the processes, occurring between absorption and Ag precipitation,nbsp;dealing with unstable intermediate products. So the absorption ofnbsp;a number of quanta, selected by chance, does not lead to Agnbsp;production. Eventually this situation may be combined with a dark

reaction, effected by a system of limited capacity. It can be shown that in all these cases the observed critical time will be approximatelynbsp;equal to the largest one of the relaxation times or the interquantumnbsp;time. The purpose of these general remarks will become clear afternbsp;the discussion of the special examples given herebelow.

1. nbsp;nbsp;nbsp;Suppose we have an unstable intermediate product, which is formed bynbsp;one quatum and has to wait for the next one to be stabilized. For instance wenbsp;consider the electrons which have not yet been neutralized stably, and have tonbsp;wait for a following electron to form together a primary stable silver specknbsp;(cf. Ch. IV). In that case, for high light intensities and high frequencies, allnbsp;electrons will be stabilized in time, so that each electron leads to the productionnbsp;of a Ag atom. If the frequency is lowered, losses will become apparent as soonnbsp;as the dark intervals become comparable with the mean life time of the intermediate products (electrons), hence the critical time is of the order of thenbsp;mean life time of the unstable products. For low intensities the efficiency willnbsp;be much lower. It may readily be seen that the critical time will then be of thenbsp;order of the interquantum time.

2. nbsp;nbsp;nbsp;Suppose we have an unstable intermediate product in combination with

a dark reaction that is effected by a system of limited capacity. For instance we consider the wandering electrons that can be neutralized at a maximum rate,nbsp;which is determined by the number of sensitivity specks and the mobility ofnbsp;the Ag ions (cf. Ch. IV). In that case two relaxation times are to be considered:nbsp;firstly the mean life time of the intermediate products (electrons), with respectnbsp;to their instability, and secondly the relaxation time of the dark reactionnbsp;(neutralization by ion attraction), which may for instance be defined as itsnbsp;reciprocal maximum reaction velocity, or as the average time needed for thenbsp;neutralization of one electron. If the former is by far the smaller one, thenbsp;electrons which cannot be utilized immediately by the limiting reaction are lost.nbsp;For high light intensities, for which the interquantum time is small, comparednbsp;with the relaxationnbsp;nbsp;nbsp;nbsp;time of the limiting system,nbsp;nbsp;nbsp;nbsp;the latter sets the pace ofnbsp;nbsp;nbsp;nbsp;the

entire reaction, nbsp;nbsp;nbsp;andnbsp;nbsp;nbsp;nbsp;an intermittency effect willnbsp;nbsp;nbsp;nbsp;become observable only, ifnbsp;nbsp;nbsp;nbsp;the

dark times are nbsp;nbsp;nbsp;comparable with this relaxationnbsp;nbsp;nbsp;nbsp;time. On the other hand,nbsp;nbsp;nbsp;nbsp;for

low intensities nbsp;nbsp;nbsp;thenbsp;nbsp;nbsp;nbsp;incidence of quanta is thenbsp;nbsp;nbsp;nbsp;pace-setting factor, hencenbsp;nbsp;nbsp;nbsp;the

critical time will be of the order of the interquantum time. If, on the contrary, the life time of the intermediate products (electrons) is much longer than thenbsp;relaxation time of the dark process with the limited capacity (neutralization time),nbsp;it may be shown similarly that, in order to obtain an observable intermittencynbsp;effect, the dark times, hence the critical time, have to be comparable to thenbsp;larger one of this life time or the interquantum time.

In the third place it is almost self-evident that, if each absorption has a fixed probability to exert no influence on the process ofnbsp;silver formation, the effective intensity in connection with the

critical frequency has to be deduced from the interquantum time of the effective quanta only, as was already remarked by Webb ^i).nbsp;However, according to the above considerations concerning thenbsp;relaxation times, this remark applies only for those statistic lossesnbsp;which are independent of the interquantum time; losses connectednbsp;with reciprocity law failure are not to be discussed in the waynbsp;proposed by Webb, but according to the views developed here, innbsp;the way described for relaxation times. The probability for thenbsp;occurrence of independent statistic losses is small theoreticallynbsp;(cf. Gurney and Mott and Ch. IV) and experimentally, especiallynbsp;on account of the observed efficiency of ca. 1 quantum per atom Agnbsp;for directly blackening exposures.

In Ch. IV we have seen that the optimal intensity more or less corresponds to such a relaxation time. From the preceding discussionnbsp;it will be clear that, in order to deduce the receptive area of anbsp;unit from the critical frequency it is justified to proceed accordingnbsp;to Webb�s method, provided that the intensity is well below thenbsp;optimum, whereas for intensities above the optimum the receptivenbsp;area of a unit should be computed as if the intensity should havenbsp;been optimal.

The experimental results obtained by Webb indicate distinctly that the grain area computed from experiments with intensitiesnbsp;above the optimum is smaller than the analogous result for lownbsp;intensities (cf. table II). Moreover, this author published onenbsp;intermittency experiment performed at �75� ^i), with an intensitynbsp;far above the optimum, which yields an extremely small effectivenbsp;grain area (ca. 2 % of the value at room temperature). Obviouslynbsp;the result of this experiment corresponds to the relaxation timenbsp;belonging to the optimal intensity, and not to the interquantumnbsp;time of the intensity actually applied.

In order to be able to carry out the computations for the determination of the intermittency effect we shall consider the statistics of absorption of quanta in some detail. In the following sectionsnbsp;various questions, related with these statistics will occur. In ordernbsp;to avoid there interruption of the discussion by mathematical

derivations we shall collect the main part of the latter ones here.

Consider one AgBr grain with cross-section area a. Let the grain be irradiated with an intensity of I(t) quanta/cm^.sec., with I(t) a periodic function of time withnbsp;period T and frequency f = 1/T.

Our purpose is in the first place to find an expression for the distribution of the interquantum times. The probability W(t) that our grain shall absorb anbsp;quantum of light in the infinitesimal time interval between t and t dt is then:

W{t)dt = I{t)a* dt

in which the effective receptive area a* is smaller than or at most equal to 2a. If the grain would be optically opaque, a* would be equal to 2a; in most practicalnbsp;cases this condition will not be fulfilled, firstly on account of the rather largenbsp;reflection of the grains and secondly because their transmission is not zero.nbsp;However, within not too wide limits a* and a in one emulsion will practicallynbsp;be proportional (For a justification of the factor 2 see p. 93).

The chance N(ti,t2) that no absorption takes place during the time interval between ti and ta is the product of the chances for no absorption duringnbsp;all intervals dt between ti and ta.

After expanding the logarithm, neglecting terms of higher order than dt and replacing the summation by integration we arrive at:

The chance for an absOrbtion during the interval dti at ti, followed by an interval � of non-absorption and closed by an absorption during the intervalnbsp;dd 3t ti �is:

C(t,J)dt^d�=W{t,)N{t^,ti �) W{t^ (l)dt^dO

Denoting by Ti the total time of the exposure, the probability P(fl)d9 that an interval between two successive absorptions has a duration between 0 andnbsp;6 dS is given by:

P (�) = fd t, C {t,. �)IJ d 6 �W nbsp;nbsp;nbsp;C {t� 6).

taking oo instead of Ti as the upper limit of the integrals, since Ti, the total time of the experiment, will always be very large compared with the averagenbsp;spacing of absorptions and the periods of intermittency studied. With thisnbsp;approximation we can replace the denominator by;

Then ,we can replace the upper limit of both integrals by T, the period of I(t). Then the denominator becomes a*IT and so:

It is now possible in general to compute 0� and averages of other functions of �. We shall first compute �. On physical grounds we see that:

since a* I is the average number of quanta absorbed per second in one grain. Mathematically the same result is easily obtained by substituting (1) into:

J=Td(IJ.P{(l)

0^f)^T^IQKP,{f, Q)dQ.

The integral is evaluated by substituting the above values for the n�*' segment of Q; an arithmetical series of higher order results; its sum is determinednbsp;easily. So we obtain in a direct and more general way the result for: A'�{f) derivednbsp;by SiLBERSTEiN and Webb:

A^{f) = 0^ ((xcoth x/2 � 2) (1 � nbsp;nbsp;nbsp; 1)

We notice that for f-gt;0orx-gt;-oo; -y oa

- �2

and for f-gt;.ooorx-gt;-0: nbsp;nbsp;nbsp;,

the latter value being the same as for continuous irradiation with the same average intensity.

Examples of the probability distribution of Q = �/T a.nbsp;nbsp;nbsp;nbsp;for sector transmission q = andnbsp;nbsp;nbsp;nbsp;freq. =nbsp;nbsp;nbsp;nbsp;critical freq. (Q '= 1)

b....... q = nbsp;nbsp;nbsp;H ..nbsp;nbsp;nbsp;nbsp;^nbsp;nbsp;nbsp;nbsp;..nbsp;nbsp;nbsp;nbsp;.. (Q=l)nbsp;nbsp;nbsp;nbsp;_

c. � nbsp;nbsp;nbsp;�nbsp;nbsp;nbsp;nbsp;� q = M It � � twice the critical freq. (Qt=2).

Quite generally one can show foi (f), and for most other functions y of � e.g. log 6, that;

by considering the expression (1) for P (0) d (}. For f-gt;- oo we may replace the integrand factors by their mean values, since the length of the time intervalnbsp;of integration T approaches 0, and d � is considered as a small but finitenbsp;interval. The relation (1) is thus simplified to:

P{0)dl

which is equal to the S distribution for continuous exposure. This proves that, for sufficiently high frequencies of intermittency, intermittent and continuous 6nbsp;statistics are entirely indiscernible.

log Qif) = ldQ log Q.P^ if, Q) 0

This integral can be evaluated by substituting the values for Pi(f) given above for the n'l' segment of Q, followed for n = 0 by series expansion of Pi(f) aboutnbsp;Q = 8 = 0, whereas the other terms can be integrated. The resulting expressions have to be summed over n. They are suited for the computation of log 0(f)nbsp;especially for values of Q(f) of the order of magnitude of 1, since then thenbsp;series converges rapidly with n (frequencies near the critical frequency f^ which

lim log 0 {[) � log 6 � 0.250

For low frequencies (small Q) we use the following approximation for log 0(f). We assume that in the open parts of the sector the distribution of log Q isnbsp;practically undisturbed, relative to continuous illumination with intensity I (seenbsp;fig. 24). Denoting by (j' the mean interquantum time within one light period,nbsp;we have;

l//a* = F.q

and log Q' = log 0' � 0.250 =c log 6 log q �� 0,250.

Absorption of quanta during light periods of an intermittent exposure for low frequencies. During the light periods the Q distribution is almost equal to thatnbsp;corresponding to continuous irradiation with maximum intensity (comp. text).

The mean number of intervals in one light period qT is qT/0' � T/�, These intervals together with one large interval (1 � q)T yield

log � (�) = ((log � log q - 0,250) TjH log (1 - q) T)/{1 T/�) (8b)

log 0 (0) =: log � -)- log q � 0.250.

which is the same value as for continuous exposure to intensity I (8a). Finally we notice;

In Ch. V we deduced the relation connecting the action of colour mixtures with that of their components from general featuresnbsp;of the photographic process in a differential form, which couldnbsp;then be integrated, becoming comparable in this way withnbsp;experimental data.

This was achieved by considering the infinitesimal exposure elements of each colour and of the mixture, required to producenbsp;the same rate of density increase, starting from a given density.nbsp;In the present chapter we shall see, that intermittent light cannbsp;in a sense be considered as a �mixture� of different constantnbsp;intensities of the same colour. It is our purpose to show that herenbsp;too it is possible to deduce a differential relation, at least for lightnbsp;intensities below the optimal one (cf. � 29), by considering thenbsp;exposure elements of each intensity and of the intermittencynbsp;�mixture�, required to produce the same rate of density increase,nbsp;starting from a given density. This relation will then afternbsp;integration connect the intermittency curve (log exposure, requirednbsp;to reach a definite density with intermittent radiation of givennbsp;mean intensity, as a function of log frequency) and the r.l.f. curvenbsp;(log exposure, required to reach the same density with radiationnbsp;of constant intensity, as a function of the intensity at which thenbsp;light is admitted). The relation found will be compared with thenbsp;experimental results of Webb mentioned previously.

We shall first confine our discussion to a thin emulsion with homogeneous light intensity throughout, and with grains of equalnbsp;size, so with the same effective intensity (rate of absorption ofnbsp;quanta) for each grain. The elimination of these restrictions willnbsp;be effected thereafter; in contrast to the analogous case for colournbsp;addition this does not meet with essential difficulties here.

1. For intensities well below the optimum the influence of the n�2�^ quantum in one grain on the efficiency for density productionnbsp;of the n* one will be negligible, compared to that of the n�1='nbsp;quantum on the efficiency of the n��' one. In this connection wenbsp;shall introduce for the entire plate the hypothetical quantity:nbsp;differential efficiency R of an energy, admitted with interquantum

increase from D to D dD is assumed to be effected by equidistant absorptions in each grain at intervals 6, representing together annbsp;energy dE.

This assumption is plausible in connection with the finite mean life time of the electrons, excited by the absorption of a quantum.

It is well known that R(0,D) depends also weakly on the way in which the density D has been produced previously (historynbsp;of D); especially the intensity with which this production has takennbsp;place is a factor of influence, which means that the density is notnbsp;determined merely by the amount of latent-image silver. Howevernbsp;effects of this type become apparent only for light intensities abovenbsp;the optimum, according to the considerations of Ch. IV, whereasnbsp;our discussion is restricted to intensities below the optimumnbsp;(cf. � 29). Moreover, if a number of different ways are comparednbsp;to produce D with intensities, the time averages of which are equal,nbsp;the effect in question may be expected to be practically absent.

2. The differential efficiency of an additional exposure, immediately following the production of the density D, does notnbsp;depend on the way in which D has been produced (history of D),nbsp;in as much as the mean light intensity for the latter productionnbsp;lies well below the optimum and, moreover, is the same in the casesnbsp;to be compared.

The efficiency for the production of an increase in developable density from D to D aD, by quanta, admitted in such a waynbsp;that 0 shows a probability distribution P(0)d0, is then given by:

R{P{�);D)^fR((I.D).P(�)d6.

If we consider an exposure to intermittent light, the frequency of which is not too low¹), so that for a density increase AT) for

For low frequencies it seems possible to eliminate the effect of the larger changes of R during one flash by averaging over experiments with variousnbsp;phase angles of the initial point with respect to the intermittency cycle.

which R does not yet change much, P(G) is the same (several intermittencies during the production of AD), formula (9) can benbsp;integrated over the energy, from 0 up to the total energy admittednbsp;in the exposure in order to obtain a given density Dq. Thenbsp;same is true for a �continuous� exposure, with another P(0)nbsp;distribution (3).

3. We assume that R(6,D) can be written as the product of a function of 9 and a function of D. This assumption is equivalentnbsp;to parallelism of (D;logt)j curves in the log t direction ornbsp;parallelism of (log E; log I)^) curves in the log E direction. Innbsp;practice these parallelisms are realized in good approximation (compare the condition derived in Ch. V for the agreement between thenbsp;empirical and the theoretical addition law).

Then formula (9) leads after integration over the energies admitted to produce a given final density Dq, to:

The overall efficiencies for the entire production of Dq with a P(0) distribution or with equidistant intervals 0 respectively are:

R(iDo) = DJE(0.Do)

By multiplying both sides of (9a) by Dq we see that these quantities satisfy a relation of the same form as (9):

R(P(0):Do) = IR{Q,Do)P{0)di

This formula gives the connection between the efficiency of intermittent light and an �ideal� r.l.f. curve, i.e. a curve representingnbsp;the efficiencies of exposures with equal interquantum times without

This formula can be verified e.g. by differentiation of E(P(fl); Do) towards Do, taking into account that E('0,Do) X R(�.D) is independentnbsp;of

statistic fluctuations, both efficiencies still for plates with homogeneous light intensity and grains of equal size. This ideal curve is related to that which would be obtained experimentally withnbsp;such a plate by a transformation of averaging of neighboring points,nbsp;similar to that described in � 32 which takes into account thenbsp;statistic fluctuations that occur necessarily in real exposures. Bynbsp;the inverse transformation the experimental curve may be transformed into the �ideal� curve, which may then be compared withnbsp;the intermittency curves according to form. (9a), so that in principlenbsp;the problem is solved.

For practical purposes it is more convenient to compute the shape of the �intermittency curve� by which we mean a curve of

against log f, E(f) being the energy required to produce a given density Dq with intermittent exposures of frequency f, and withnbsp;a given average intensity. For the same reasons as mentioned for Dnbsp;in � 29 we have here: �(f)nbsp;nbsp;nbsp;nbsp;�(0) = 1 for f ^ 0 and �(f)

In order to proceed with the computation of �(f) the values of E, derived from (9a) have to be substituted into (10). For thenbsp;present we have followed an approximating method, in view of thenbsp;limited accuracy of the experimental data (see fig. 25). It appearsnbsp;to be quite satisfactory if a 0-region, covering a factor of e.g. 30nbsp;is considered. Within such a region the deviations of the r.l.f.nbsp;curves from straight lines are small. So we have used thenbsp;ScHWARZSCHiLD approximation:

Since the amplitude of the statistic fluctuations of log 9 for �continuous� radiation is independent of 0, as can be shown bynbsp;expressing P(9)d0 in terms of (log 0 � log 0) only, straight partsnbsp;of the �ideal� r.l.f. curve correspond to straight parts of the real

f} := 1/p � 1

e (�) = (log Hf) � log ^ (o�))/(log 0 (0) � log (Oo)).

Formula (12) is independent of the values of a and j3 in (11). This means that, in so far as this approximation is valid, thenbsp;�intermittency curves� for a given sector transmission q all havenbsp;the same shape. They would coincide if we should plot �(f) againstnbsp;log Q = log 0/T, since log 9(f) � log 0(oo) is a function of Qnbsp;only, as can be verified with the aid of form. (6), (8), etc. (� 30).nbsp;Substituting the results, obtained there, into (12) we find:

This formula has been used to compare theory and experiments, but before such a comparison will be carried out we shall firstnbsp;dismiss in � 32 the restrictions made, concerning the uniformity ofnbsp;light intensity and grain size.

A (more laborious) possibility to compute �(f) rigorously would be a graphical evaluation of the integral expression (9b).

Another suitable method to compute s (f) approximately, is to replace R(�.D�) bij a sum of two or more exponential functions of

If the ScHWARZSCHiLD approximation (11) is considered as part of a series expansion of log E in powers of (log � � log (j ), of which quadratic andnbsp;higher order terms have been omitted, it can be shown by rather laboriousnbsp;calculations, which we shall omit here, that this expansion converges rapidlynbsp;for functions describing the r.l.f., e.g. Webb�s function: log E = lognbsp;A�~� C�3�^(cf. Ch. I). If desired, closer approximations can be calculatednbsp;with the aid of this expansion.

We wish to emphasize that the evaluation of the integral (9b) cannot be carried out by series expansion of R(0',Do) about 0lt; of the form:

R (P(0): Do) = R (fl. Do) ! R'(�. D,)~A/1 ! = 0 ( Rquot; (W) A^/2l ... withnbsp;nbsp;nbsp;nbsp;A = 0 � 6.

It is tempting to break off this formula after the second order term; since Q is independent of f, this expansion seems to be a justification of the methodnbsp;of SiLBERSTEiN and Webb to characterize the intermittency effect by the quantity Closer consideration shows that the opposite is true, since the seriesnbsp;diverges, both for the Swarzschild approxsmation and for Webb�s approximation of the r.l.f. for alle values of f, even for f -gt;� oo, because;

for n -�-oo . This divergence is not essentially due to the fact that the formulae for P(fl) (1),(3),(6) have been derived with the approximation that the uppernbsp;limit of is oo instead of Ti. We have calculated the behaviour of /j�/n!nbsp;also in this case for f = oo and have found that the series remains divergentnbsp;for Ti gt; � , which condition is satisfied in all of Webb�s experiments wherenbsp;Ti was of the order of 100. �. For all f lt; oo it diverges then a fortiori, since:

j /!quot;(/�) I gt; I I �

Herefrom we conclude that the quantity is unsuited to describe the intermittency effect even approximately. This difficulty is avoided by using the rapidly convergent expansions of log E in log Q.

� 32. The effect of the non-uniformity of grain size and of light intensity.

We shall discuss now the effect of the non-uniformity of grain size and of light intensity throughout the plate, upon the shape of the �intermittency curve�nbsp;(e(f); log Q). Since log Q = log f � log I � log a*, and each of the twonbsp;last mentioned quantities has its own (assumed independent) dispersion, we havenbsp;to determine a weighed mean for �(f).

If the relative contribution to the total intermittency effect, by grains with log a lying between log a and log a dlog a and log I lying betweennbsp;log I and log I dlog I is

normalized so that integration over all values of log a and log I occurring yields 1, then the weighed mean to be considered is:

E{f)=lle(f). W.dlog l.dlog a.

The assumption of independent dispersion of the two quantities is equivalent with:

We assume that a and a* have similar distributions, and that the light intensity can be approximated by an exponential light decrease in the plate, causing a linear decrease of log I, the magnitude of which is directly connected withnbsp;the absorption of the plate.

averaged over log a* and log I, after Taylor, about the point (log a*, log I), chosen in such a way that

A log 1= A log a* = 0 �A log I. A log a*

In this connection log a* has to be defined as the value of log a* averaged with respect to its contribution to the density. So if N(a)da is the grain sizenbsp;frequency:

Similarly log I has to be defined as the average intensity, weighed with respect to its contribution to the intermittency effect, c.q. to its contribution tonbsp;the total density of the plate. We shall see below, that the influence of thenbsp;dispersion of I is negligible, compared with that of a.

We shall consider in this expansion of �(f) only terms up to the second order (the first order terms being zero on account of the definitions of the averagesnbsp;of log a* and log I, and the assumed independence of the dispersions of thenbsp;two quantities). It is then convenient to replace the expansion by the followingnbsp;form, which does not differ from it within the approximation mentioned:

The sum has to be extended over all (four) combinations of the and � signs. It yields an easy way to determine e (f) from e (f) if (d log I)^ andnbsp;{A log a*)^ are known. These quantities can be estimated in principle fromnbsp;absorption data and from grain size frequency data respectively (given by Webb).

We observe that the points near * (f) = H change hardly by the transformation (14) which smoothens the curves of �(f) against log Q by quot;pulling at the endsquot;. However points corresponding to a density which is just discerniblenbsp;from the value for Q -gt;� oo are shifted considerably (cf. fig. 25).

� (f) is again a function of (log f � loga* � log I) only so that it is now possible to compare the theoretical curves with the experimental data over theirnbsp;entire range.

� 334 Comparison of the theory with Wcbh�s experiments.

In fig. 25 we have compared the theory with the experimental data of Webb 21) 41), Y/e have plotted �(f) against log Q =

log f � log a * � log I. The thin curves have been computed according to form. (13) for the sector transmissions used. Theynbsp;have been corrected then for inhomogenity of light intensity andnbsp;grain size according to � 32, which yields the heavy lines. Thenbsp;experimental points of Webb were then plotted against log f �

log Q = log f '� log a* � log I. Each set of experimental points has been adjusted by a horizontal translation, so as to fit the curves best. Since thenbsp;other variables are known, log a* can be derived from these plots. Thin linesnbsp;for plates with homogeneous light intensity and grains of equal size. Heavynbsp;lines corrected for light gradient and grain size distribution occurring in thenbsp;real plates.

Solid symbols: Intensities above the optimum; open symbols; intensities below the optimum.

log ac � log I in the same figure, choosing c so as to make them fit the theoretical curves best. The adaptation of c meansnbsp;that each set of points can undergo a horizontal translation withoutnbsp;changing their relative positions.

We conclude that the agreement between the experimental points and the theoretical curves is quite good (within the experimentalnbsp;error). This agreement allows a relatively accurate estimation of c,nbsp;using all points of a set.

Though we have stated explicitely that our theory is only applicable to intensities below the optimum, we have seen, on thenbsp;other hand, in � 29 that intensities above the optimum may benbsp;expected in this connection to behave more or less as if the optimalnbsp;intensity would have been used. Therefore we have also comparednbsp;those data of Webb, which pertain to intensities above the optimum,nbsp;with the shape of the theoretical curve for low intensities. Thenbsp;agreement is better than would be expected.

In table II we collected the data, needed for a quantitative comparison of Webb�s results with the theory developed here andnbsp;with that of Silberstein and Webb.

as occurring in our theory are collected in col. 6 and 8. For log I an exponential light decrease in the plate and absorption of 50 % or 20 % (seenbsp;Meidinger p. 97 and col. 5) has been assumed. For log a Webb's grain-sizenbsp;frequency data have been used pertaining to the plates used for these experiments.nbsp;The dispersions of log a and log I about their mean values, needed for thenbsp;transformations of � 32, are collected in col. 7 and 9. The former ones arenbsp;deduced again (roughly) from Webb�s grain-size data, the latter ones againnbsp;from the same assumptions on the light decrease in the plate mentioned above.nbsp;The influence of the latter is small compared with that of the dispersion in grainnbsp;size, and both influences cause differences of the order of the experimental errornbsp;only, according to fig. 25.

The agreement between the experiments and the present theory is shown in col. 11; that with the theory of Silberstein and Webbnbsp;in col. 12, by comparing the grain-size data from microscopicnbsp;observations with those effective in intermittency experiments. Theirnbsp;ratio c may be ^ 2, since in the latter case the effective area ofnbsp;the entire grain surface (both sides) is found, the former casenbsp;referring to the cross section area.

The agreement with the present theory (coll. 11) is quite satisfactory for low intensities. Except for one case also fornbsp;intensities above the optimum the agreement is quite good especiallynbsp;if it is considered that the theory was not deduced for these cases:nbsp;according to � 29, c has been computed here as if the optimalnbsp;intensity had been active.

The agreement with Webb�s theory (col. 12) is decidedly less good, since values of c up to 12 occur for low light intensities;nbsp;this becomes even worse if it is taken into account that the meannbsp;light intensity was in most cases lower by a factor of about 1,4.nbsp;For light intensities above the optimum, where the intensity itselfnbsp;has been used, the disagreement shows up especially clearly innbsp;the case of the low temperature experiment, since experiments withnbsp;the same plate at room temperature with an intensity below thenbsp;optimum show that no deviating value of c should be expected.nbsp;The corresponding figure in col. 11 does not show such anbsp;discrepancy but fits in well with the other values of this column.

1. nbsp;nbsp;nbsp;Webb�s experiments are described better by the presentnbsp;theory than by that of Silberstein and Webb.

2. nbsp;nbsp;nbsp;For light intensities below the optimum the shape of thenbsp;jntermittency curves is understood quantitatively for all frequenciesnbsp;on the basis of the reciprocity law failure and needs no othernbsp;principles for its explanation.

3. nbsp;nbsp;nbsp;For light intensities above the optimum the same explanationnbsp;holds with good approximation, provided that the optimal intensitynbsp;is substituted when the calculation of the effective receptive grainnbsp;area is carried out.

� 34. Introduction.

The wide field of radiobiology is attacked nowadays along several lines. Between its problems and those of the field ofnbsp;photographic research a remarkable similarity appears to exist.nbsp;This is especially true for biochemical processes which only proceednbsp;under absorption of visible or near ultraviolet light and whichnbsp;have proved to be not merely photochemical reactions, but chainnbsp;processes, the occurrence of which is essentially bound to somenbsp;cellular structure. The effect of these structures is then oftennbsp;comparable to that of the grains in the photographic emulsion.

We will discuss in this chapter some problems and methods which are common to photography and to radiobiological processes.

We shall deal particularly with examples of methods which have been transposed from one field to the other one and we shallnbsp;indicate a few cases in which such a transposition may be expectednbsp;to yield new results. We shall chiefly consider the process ofnbsp;vision occurring in the human eye and the process of photochemicalnbsp;carbon dioxide assimilation in plant cells, for instance in Chlorella,nbsp;both processes having been studied extensively. Many remarks arenbsp;applicable to other processes as well.

Necessarily this chapter does not aim at completeness. It is restricted to a few, but typical examples which tend to connectnbsp;the fields in question.

� 35. Similar problems, methods and effects.

a photochemical effect. In photography the absorbing �pigment� is AgBr. For photochemical CO2 assimilation it is chlorophyll innbsp;green plants: similar compounds are used for this purpose in othernbsp;photosynthetic organisms; for vision the visual purple plays probablynbsp;an important role. The nature of the pigment(s) of vision is notnbsp;yet fully known. The effective absorption spectra of the pigmentnbsp;complexes involved in these processes are well established.

Especially for chlorophyll, bacteriochlorophyll and bacterioviridine it is known that in the living cell these pigments are bound to anbsp;protein complex. Also for visual purple this idea has beennbsp;advanced 42). This bond shifts the absorption spectrum of thenbsp;pigment towards longer wave lengths 42).

In photography the situation is somewhat different in so far as no (proper) protein is involved. However the adsorptive bond ofnbsp;smaller (colourless) molecules or ions (for instance OH-ions) tonbsp;AgBr is also known to shift the absorption spectrum to longernbsp;wave lengths (MeidingerS) p. 96, also Ch. III).

The sensitization of photochemical processes by the adsorption of suitable dyes to the original pigment complexes was discoverednbsp;in photography about 50 years ago by Vogel. For the assimilationnbsp;various authors have suggested that the action of chlorophyllnbsp;amounts to a sensitization of the photosynthetic process 43). Alsonbsp;it does not seem impossible that carotenoid substances, which arenbsp;practically always present at the side of chlorophyll, act asnbsp;sensitizers.

Metzner reports that phototaxis of colourless bacteria could be achieved with visible light after sensitization with eosine andnbsp;other dyes (cited after Wassink 44)). This is perhaps a specialnbsp;case of photodynamic sensitization, as studied extensively by

Neither the sensitization of assimilation nor that of vision to normally inactive wave lengths, by application of solutions ofnbsp;suitable dyes has been reached, as far as we know. On grounds ofnbsp;analogy this might be possible, as was remarked already bynbsp;Engelmann in 1883 46).

theoretical remarks about the differential addition law can easily be transformed, so that they are applicable to the correspondingnbsp;situations in assimilation and vision. We do not know of anynbsp;experimental checks in the former case, but the law is likely tonbsp;be valid.

For the eye the siuation is somewhat different, because the reaction product is not accumulated. Here the corresponding ratesnbsp;of the processes involved in vision have to be compared. Manynbsp;authors assume that the brightness impressions are additive. Thisnbsp;is not in agreement with the fact that brightness impressions cannbsp;not yet be measured by these authors, since they can only definenbsp;and establish equality of brightness impressions. So an additionnbsp;law ought to be formulated in such a way as to relate all factorsnbsp;to equal brightness impressions.

Although the conclusions, drawn from experiments on the brightness of colour mixtures were thus not always formulatednbsp;correctly, the addition law has essentially been verified long agonbsp;and has since been used as a basis for heterochromatic photometrynbsp;in the brightness region of constant relative colour sensitivity.nbsp;Generally it proved to be in good agreement with experimentalnbsp;evidence: however, when comparing saturated colours with theirnbsp;mixtures, positive deviations (A ja gt; 1, see Ch. V) werenbsp;established. For regions of lower brightness, where the relativenbsp;colour sensitivity depends on the brightness, the validity of thenbsp;law is not self-evident. In analogy to photography it was verifiednbsp;there by Bouma47)^ v/ho found excellent agreement, which constitutes a basis for his new four-dimensional colorimetric system.

The deviations in the behaviour of a photographic plate from reciprocity law are well known. For photosynthesis a rather largenbsp;region of intensities I and times t exists, in which the amount ofnbsp;CO2 assimilated is proportional to I .X t.

However, for higher intensities the well-known Bl.ackman saturation appears; for very low intensities theoretical considerationsnbsp;(cf. Wohl48)) indicate that the efficiency will decrease, since

long times of illumination initial changes and changes in,the cells occur, causing a reciprocity law failure.

In most other radiobiological problems the I X t law holds within a certain region of I and t, therebeyond deviations are observednbsp;(cf. Wassink44) ). In various cases these will have to be explainednbsp;probably in different ways.

Though the reciprocity law failure may be due to different causes, two features are of a quite general nature.

In the first place we mean the low intensity failure for more-quantic processes with unstable intermediate products. For photography SiLBERSTEiN and Trivelli 28) showed in general that for sufficiently low intensities the rate of density production isnbsp;proportional to I� if the reaction requires n effective quanta tonbsp;produce its minimum effect, and if the intermediate products, formednbsp;by less than n quanta, are not stable. (For most plates it was foundnbsp;that n = 2). Independently Wohl 48) reached the conclusion thatnbsp;the rate of CO,2 assimilation which requires probably 4 quanta fornbsp;the assimilation of one CO2 molecule, should be proportional to I^nbsp;for sufficiently low intensities. This principle is valid for all radiobiological reactions and yields a method for the determination ofnbsp;the number of quanta, required for such a process. This methodnbsp;has not sufficiently been utilized up to the present.

Secondly we refer to the Gurney�Mott explanation of the high intensity reciprocity law failure in photography i'^), and itsnbsp;dependence on temperature. The similarity of the curves of thenbsp;rate of Ag production against light intensity given there and curvesnbsp;of the rate of CO2 assimilation by C/i/ore//a 49) against lightnbsp;intensity is most striking. Also their temperature dependence isnbsp;quite similar (see fig. 26).

This type of saturation is explained commonly by the assumption of a chain of (at least) two partial processes. One of these processesnbsp;is assumed to be photosensitive but rather insensitive to temperaturenbsp;(in photography: electron transport to sensitivity specks), the othernbsp;one has the reverse properties (in photography: neutralization ofnbsp;trapped electrons by ion conduction) and moreover possesses anbsp;limited capacity, which is due for instance to a limited number of

100

reactive places. Such a system must have just these properties provided that intermediate products are not infinitely stable.

An important item in our discussion is the application of the idea of the unit of action in both fields. We shall discuss this matter-somewhat more in detail.

The idea of a unit of action implies that a number of pigment molecules is connected by some mechanism in such a way that,nbsp;if any molecule of a unit absorbs a quantum of light, this energynbsp;is furnished to one definite reactive molecule.

In photography it was known since long that silver is deposited at a few definite �concentration specks�� (Sheppard e.a.) independent of the place in the AgBr grain where the correspondingnbsp;quantum was absorbed.

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So for this process a unit appears to exist, of the order of magnitude of a grain.

From this example we see quite generally that in a unit of action the energy must be transported over a certain distance, to the placenbsp;of the final reaction. The development of more specified ideasnbsp;about the unit of action depends entirely on the possibilities tonbsp;visualize suitable mechanisms of energy transport.

For photography the Gurney�Mott theory proposes for this mechanism photoconduction of an electron from the place ofnbsp;absorption to some structural irregularity as expounded in Ch. I.nbsp;This is conceivable on the basis of general ideas, developed fromnbsp;the study of photoconduction and semiconduction in crystals.

A few years before the publication of this theory J. H. Webb 21) had developed an experimental method to determine the size ofnbsp;a photographic unit of action. This method is based on a studynbsp;of the intermittency effect, and has been discussed in detail innbsp;Ch. VI. The results of Webb�s experiments and their additionalnbsp;discussion in Ch. VI make plausible that the unit of photographicnbsp;action is of the order of magnitude of an entire AgBr grainnbsp;(ca. 109 molecules).

In the field of photosynthesis the idea of a unit of action of chlorophyll is about 10 years old. It was first introduced bynbsp;Emerson and Arnold so) to explain their observations regardingnbsp;saturation of CO2 assimilation by Chlorella in flashing light.nbsp;Gaffron and Wohl si) and Wohl 48) discussed these experimentsnbsp;more closely and collected all arguments that pointed towards thenbsp;existence of such a unit. Its magnitude was estimated at aboutnbsp;2000 molecules of chlorophyll. More recent work of Emerson,nbsp;Green and J. L. Webb S2) indicates that this number tends tonbsp;increase with the age of the cells up to almost ten times this value.

The problem of energy transport is here a very difficult one, since it is not known whether chlorophyll is present in the cellnbsp;in a state which is comparable with the crystalline one. In grananbsp;the chlorophyll is rather densely packed, so that processes similarnbsp;to photoconduction of electrons might not be impossible, especiallynbsp;since one knows from the work of Hanson ^3) and of Ketelaar 54)nbsp;that monomolecular films of chlorophyll on water tend to formnbsp;very regular tile-like structures. Whether the electrons are con-

102

ducted through a chlorophyll structure or through a protein complex is an open question.

Another argument for electron transport is the fact that in assimilation the light probably ultimately gives rise to a reducingnbsp;agent ^5).

We wish to point out that Webb�s method for the determination of the size of the unit is in principle applicable to the process innbsp;question. After the discussion of Ch. VI it is clear that the requirednbsp;intermittency experiments should not be performed in the regionnbsp;of light saturation, where the relaxation time of a limiting factornbsp;interferes, nor can any result be expected in the region where thenbsp;rate of assimilation is proportional to the intensity of irradiation.nbsp;Experiments of this type would only yield an independentnbsp;estimate of the size of the photosynthetic unit, if one succeeds tonbsp;perform them at low intensities where the rate of assimilation isnbsp;proportional to

Moreover experiments of the same type, performed at intensities a little below light saturation would yield an estimate of thenbsp;dispersion of the relaxation time (root mean square deviation fromnbsp;the average relaxation time) of the processes of energy transportnbsp;(Blackman reaction etc.). This question has so far not beennbsp;attacked at all.

In the study of vision Talbot�s law has since long been known. It states that the brightness impression of intermittent irradiationnbsp;of so high frequency that a continuous sensation results, is equalnbsp;to the brightness impression of �really� continuous irradiation withnbsp;the same average intensity. We now understand this law for verynbsp;high frequencies from the remarks of Webb for photography; it isnbsp;essentially connected with the quantic nature of light absorption.

In this case the critical frequency of Webb corresponds to the fusion frequency at which the sensation becomes continuous. Itnbsp;has been determined for a wide range of light intensitiesnbsp;(cf. S. Hecht e.a. se)). This frequency is not proportional to thenbsp;intensity in the region where measurements were performed, hutnbsp;it shows a tendency to become so for still lower intensities. Probablynbsp;relaxation times of secondary processes interfere; moreover it isnbsp;conceivable in this case that the absorption or the amount ofnbsp;pigment present, is dependent on the light intensity, which maynbsp;be an additional complication.

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It is likely, however, that for sufficiently low intensities both complications drop out, so that in this region Webb�s method shouldnbsp;enable one to determine the size of the unit of vision, if desirednbsp;for each eye-pigment separately, and for various regions of thenbsp;retina.

The resulting size of the unit may in principle exceed that of one rod or cone, because by this method the unit for the entirenbsp;process of vision is determined. But a priori dimensions smaller thannbsp;one rod or cone seem more probable.

Moreover, for the mentioned low intensity regions the theoretically lowest value of the just observable brightness incrementnbsp;must be expected to vary inversely proportionally with the root ofnbsp;the intensity, that is with the relative statistic variations in thenbsp;number of quanta absorbed per unit time in one unit of vision.

We terminate here the list of examples, which, though far from complete, intends to support the conviction that an interchange ofnbsp;ideas and methods between the fields of photography and ofnbsp;radiobiology will prove to be highly fruitful.

It is a pleasure to thank Mr. J. A. Smit for helpful discussions and valuable criticism during the preparation of this thesis.

SUMMARY.

This thesis contains several more or less independent parts, dealing with different aspects of latent-image formation, namelynbsp;with the formation and measurement of low direct densities, withnbsp;the Gurney-Mott theory, with the addition law for differentnbsp;colours, and with the relation between the reciprocity law failurenbsp;and the intermittency effect.

Experimental Part,

Accurate compensation measurements were performed in a way more or less similar to that followed by Jurriens, on the direct-density production of weakly absorbed wave lengths. The apparatusnbsp;used is described in Ch. II (fig. 4 and 5), the results of thenbsp;measurements are to be found in Ch. III. The mentioned radiationsnbsp;give rise to direct-density-time curves with a peculair S-shape fornbsp;densities near 0.01. This shape appeared to be connected in nonbsp;way with the similar shape of developed-density-time curves. Itnbsp;may be expected similarly in silver-time curves. We mainlynbsp;investigated the influences of wave length (fig. 10), of lightnbsp;intensity (fig. 11 and table I), and of water-addition on the shapenbsp;of these curves. The S-shape disappears gradually for decreasingnbsp;wave lengths. This sustains the view that it has to be considerednbsp;as the resultant of two components, one of which is connected withnbsp;the silver production, showing a pronounced BECQUEREL-effect,nbsp;whereas the other one is connected with a silver destruction,nbsp;probably related to the HERSCHEL-effect. The effect of water isnbsp;to sensitize the production of direct density and to desensitize thatnbsp;of developed density, in these wave length regions. This may benbsp;due to adsorption of OH� ions to the sensitivity specks.

Theoretical Part.

A quantitative discussion of the Gurney�Mott theory, especially with respect to both parts of the reciprocity law failure.

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on the basis of Tubandt�s conductivity measurements in AgBr, lead to a slight modification of the theory (Ch. IV).

On the basis of general views on latent-image formation the photographic action of a mixture of colours, as compared with thatnbsp;of the component colours separately, is discussed (Ch. V). Thisnbsp;leads to the formulation of a new theoretical differential additionnbsp;law, which is, .after integration, compared with the empiricalnbsp;VAN Kreveld addition law. The difference between the two lav/snbsp;appears to be negligible for many practical cases, so that the newnbsp;law may be considered as a theoretical justification of the empiricalnbsp;law, and at the same time the former shows why the empirical lawnbsp;fails under certain extreme conditions.

After a discussion of the treatment of the intermittency effect by SiLBERSTEiN and Webb we proceed to compute the effect, innbsp;a way, similar to that followed for the derivation of the additionnbsp;law, which leads again to a differential relation, connecting thenbsp;reciprocity law failure with the intermittency effect for allnbsp;frequencies, provided that the intensities used are well below thenbsp;optimum. After integration this relation is compared with Webb�snbsp;experimental results, with which it is in good agreement, yieldingnbsp;more reliable values of the effective receptive area of the unit ofnbsp;photographic action (Ch. VI).

Finally a brief survey is given of some problems which are common to photography and to radiobiology (Ch. VII).

LITERATURE.

23. nbsp;nbsp;nbsp;W. Reinders and M. C. F. Beukers Ber. VIII Int. Kong. wiss. u. angew.

107

31. nbsp;nbsp;nbsp;S. A. Arzybyschew......Phys. Zs. der Sowjetunion 11, 636, '37.

38. nbsp;nbsp;nbsp;M. C. Hylannbsp;nbsp;nbsp;nbsp;andnbsp;nbsp;nbsp;nbsp;J.nbsp;nbsp;nbsp;nbsp;M.nbsp;nbsp;nbsp;nbsp;Blairnbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;J. O. S. A.nbsp;nbsp;nbsp;nbsp;23, 353, '33 and 25,nbsp;nbsp;nbsp;nbsp;246, '35.

39. nbsp;nbsp;nbsp;C. C. Long,nbsp;nbsp;nbsp;nbsp;F.nbsp;nbsp;nbsp;nbsp;E.nbsp;nbsp;nbsp;nbsp;E.nbsp;nbsp;nbsp;nbsp;German n

44. nbsp;nbsp;nbsp;E.nbsp;nbsp;nbsp;nbsp;C. Wassink.......Openbare les, Utrecht 1940. nbsp;nbsp;nbsp;,

45. nbsp;nbsp;nbsp;H.nbsp;nbsp;nbsp;nbsp;Tappeiner........Ergebn. Physiol. 8, 698, nbsp;nbsp;nbsp;'09.

47. nbsp;nbsp;nbsp;P. J. Bouma.........Kon. Akad. v. Wet. Amst. nbsp;nbsp;nbsp;38,nbsp;nbsp;nbsp;nbsp;2, '35.

50. nbsp;nbsp;nbsp;R.nbsp;nbsp;nbsp;nbsp;Emerson andnbsp;nbsp;nbsp;nbsp;W.nbsp;nbsp;nbsp;nbsp;Arnoldnbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;Jl. of Gen. Physiol. 16,nbsp;nbsp;nbsp;nbsp;191,nbsp;nbsp;nbsp;nbsp;'33.

51. nbsp;nbsp;nbsp;H.nbsp;nbsp;nbsp;nbsp;Gaffron andnbsp;nbsp;nbsp;nbsp;K.nbsp;nbsp;nbsp;nbsp;Wohlnbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;Naturwiss. 24, 81, '36.

55. nbsp;nbsp;nbsp;E. C. Wassink and E. Katz .nbsp;nbsp;nbsp;nbsp;. Enzymologia 6, 145, '39.

STELLINGEN

Ten onrechte noemt P. J. Bouma die kleuren, die buiten de spectrale kegel liggen, niet re�el.

II.

Het instrument van H. B. G. Casimir en C. A. Crommelin ter demonstratie van interferentie figuren van ��n-assige kristallen innbsp;convergent licht kan constructief vereenvoudigd worden.

III.

Het bewijs, dat in sommige leerboeken voor de middelbare scholen van de �Reststelling� gegeven wordt, is fout.

IV.

Op soortgelijke wijze als Cesaro het sommatie begrip uitbreidt, kan men als volgt het integraal begrip uitbreiden.

(ngt;0)

Men bewijst gemakkelijk: Indien I(n) bestaat, dan bestaat ook I(n 1) en heeft dezelfde waarde. Het is dan zinvol 1(0) de waarde I(n) toenbsp;te kennen, ook al is F(x) niet integreerbaar in gewonen zin, en F(x)nbsp;Cesaro integreerbaar te noemen.

Op deze wijze komt men tot bevredigende uitkomsten voor oscillerend divergente integralen, zoals de integraal van 0 tot oo van sin x (= 1),nbsp;cos X (=0), X sin X (=0).

Het is mogelijk de verhouding van helderheids-indrukken, teweeggebracht door twee aan elkaar grenzende vlakjes van ongelijke helderheid, kwantitatief te schatten. Het gebruik van de aldus geschatte verhoudingen als grondslag voor een metriek der kleurenruimte is tenbsp;verkiezen boven dat van de juist onderscheidbare verschillen.

Voor experimenteel werk op het gebied der fotografische spectraal-photometrie is het gebruik van de op het intermittentie effect berustende roterende sector volgens Evans vaak te verkiezen boven het gebruiknbsp;van trapspleten of trapverzwakkers, vooral wanneer men met langenbsp;belichtingstijden te maken heeft.

Voor die kleurenfotografische platen, waarbij elementen met verschillende kleurgevoeligheid naast elkaar liggen, kan men betrekkelijk sterke afwijkingen van de somwet verwachten.

Meting van het absorptiespectrum van bacteri�n en andere micro-organismen die kleurstoffen bevatten vormt een snel en doeltreffend hulpmiddel bij de identificatie.

De rol, die een waterstof donator speelt bij de photosynthese van purperen zwavel-bacteri�n kan niet als secundaire reactie gezien wordennbsp;met (hypothetische) zuurstof, die uit de primaire photosynthetischenbsp;reacties zou vrijkomen,

De bepaling van het aantal neutronen, dat een bron per sec. uitzendt, met behulp van metingen in een watertank, volgens een methode waarbijnbsp;in elk punt het aantal thermische neutronen dat van beide kanten op eennbsp;(dunne) detector valt wordt bepaald, verdient de voorkeur boven denbsp;methode van Fermi,

r..

r.%-v

7quot;'

CONTRIBUTION TO THE UNDERSTANDING OF LATENT IMAGE FORMATIONnbsp;IN PHOTOGRAPHY

PROEFSCHRIFT