ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report RESULTS OF INVESTIGATION OF LOW-INTENSITY RECIPROCITY LAW FAILURE Robert L. Martin Author Ernst Katz Project Supervisor Project 2158 U. S. AIR FORCE AIR RESEARCH AND DEVELOPMENT COMMAND CONTRACT NO. AF 18(600oo)-750 E.OD NO. 670-729-BR-1, PROJECT NO. R-355-40-10 August 1956

This report has also been submitted as a dissertation in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1956.

TABLE OF CONTENTS Page LIST OF TABLES vi LIST OF ILLUSTRATIONS viii ABSTRACT xi OBJECTIVE xiii CHAPTER I. INTRODUCTION 1 A. BRIEF DESCRIPTION OF PHOTOGRAPHIC EMULSIONS 1 1. Chemical Sensitization 2 2, Dye Sensitization 2 B. EFFECT OF LIGHT ON PHOTOGRAPHIC GRAINS AND DEFINITION OF PHOTOGRAPHIC LATENT IMAGE 3 C. CHEMICAL COMPOSITION OF PHOTOGRAPHIC LATENT IMAGE 3 D. SIZE AND DISTRIBUTION OF TEE PHOTOGRAPHIC LATENT IMAGE 4 E. MECHANISM OF LATENT-IMAGE FORMATION 5 CHAPTER II. MOTIVATION FOR EXPERIMENT OF LOW-INTENSITY RECIPROCITY LAW FAILURE VS GRAIN SIZE 11 A. SINGLE-TRAP MODEL 12 B. ORDER PRINCIPLE 16 C. MECHANISM OF LATENT-IMAGE FORMATION 20 CHAPTER III. APPARATUS USED FOR LOW-INTENSITY RECIPROCITY-LAWFAILURE VS GRAIN-SIZE EXPERIMENTS 30 A. GENERAL DESCRIPTION OF APPARATUS 30 1. Measurement of Density 30 2. Emulsions Used 31 3. Low-Intensity Reciprocity-Law-Failure Apparatus for Exposing Emulsions 32 B. SUMMARY OF DESCRIPTION OF APPARATUS 52 iii

TABLE OF CONTENTS (Continued) Page CHAPTER IV. EXPERIMENTAL PROCEDURE AND RESULTS 54 A. GENERAL PROCEDURE 54 B. RESULTS 55 1. Surface Development (exposed and stored at about 320C) 55 2. Internal Development (exposed and stored at about 320C) 59 3. Surface Development (exposed and stored at about 380C) 61 4. Internal Development (esposed and storedat about 38~C) 61 C. DETAILS OF GENERAL PROCEDURE 63 1. Preliminary Experiments 63 2. Storage of Plates and Exposure Program 64 3. General Finishing Procedure 67 4. Development Procedure 67 D. REPRESENTATION OF DATA 70 1. Correction for Nonuniformity over Surface of Plate 70 2. Plotting the Characteristic Curves 72 3. Constructing the Reciprocity-Law-Failure Curves 75 E. REMARKS CONCERNING VALIDITY OF EXPERIMENTS 79 1. Temperature and Humidity Control 79 2. Errors in Operation of Timing Mechanism 81 3. Errors in Intensity 81 4. Influence of Periodicity of Light-Source Intensity 83 5. Uncertainty in Wavelength 84 6. Variations Arising from Development Conditions 85 7. Errors in Measurement of the Developed Density 86 8. Change of Emulsion Characteristics with Time 86 9. Summary and Conclusion in Regard to Validity of Results 91 CHAPTER V. DISCUSSION OF CONTINUOUS AND TWO-TRAP MODELS AND RELATION TO EXPERIMENTS 92 A. TWO-TRAP MODEL 92 1. Derivation of the Reciprocity-Law-Failure Curves 92 2. General Properties of Reciprocity-Law-Failure Curves Derived from the Two-Trap Model 95 3. Analysis of Reciprocity Curves for f and r2 (a = 0) 100 B. EXTENSION TO N-TRAP MODEL 105 C. CONTINUOUS DISTRIBUTION OF TRAP DEPTHS 107 1. Some General Conclusions 108 2. Special Case s = 1/2 109 3. Special Case O < s < 1 109 iv

TABLE OF CONTENTS (Concluded) Page CHAPTER V. (Continued) D. SUMMARY AND INTERPRETATION OF RESULTS 115 1. Summary of Results 115 2. Interpretation in Terms of the Original Model with Exponential Trap-Depth Distribution 115 3. Interpretation in Terms of the Corrected Exponential Model 116 4. Interpretation in Terms of the Two-Trap-Depth Model 117 5. A Remark Concerning the Grain-Size Dependence 117 E. SUGGESTIONS FOR FURTHER WORK AND IMPROVEMENT OF TECHNIQUE 118 APPENDIX A. CALIBRATION OF NEUTRAL DENSITY STEP TABLETS (Graded Step Wedges) 120 APPENDIX B. INFLUENCE OF PERIODIC LIGHT INTENSITY ON LOW-INTENSITY RECIPROCITY BEHAVIOR 136 BIBLIOGRAPHY 139 v

LIST OF TABLES Table Page 3-I Grain Sizes for Pure AgBr Emulsions Derived from Precipitation Time 33 4-I Additional Information Related to Reciprocity-Law-Failure Experiments Used to Construct Figs. 4.1a and b 57 4-IIa Slope of Low-Intensity Reciprocity-Law-Failure Curves and Related Details for Pure Silver Bromide Emulsions T-4139 to T-4143 Using Internal Development (Exposure and Storage at About 33~C) 60 4-IIb Slope of Low-Intensity Reciprocity-Law-Failure Curves and Related Details for Pure Silver Bromide Emulsions T-4139 to T-4143 Using Internal Development (Exposure and Storage at About 38~C) 62 4-III Data Sheet for Experiment 67-1 71 4-IV Sidel Functions for Experiment 67-1 Shown with Corrected Densities from Which They Were Derived 73 4-V Rough Measurement of Absorption Coefficient a for Silver Bromide Emulsion 83 A-I Density Calibration of Step Tablet No. ST-49-2 as Provided by Eastman Kodak Company and Base-Two Density and Intensity Calculated Therefrom 125 A-II Intensity Ratios Between Steps for Column No. 6 of ST-49-3 as Measured by 931-A Photomultiplier Tube (April 1956) 126 A-III Results of Density Measurements of Column No. 6 of Step Tablets, Using 931-A Photomultiplier Tube 127 A-IV Variation of Transmission and Density Along the Sixth Step of ST-3 as Measured by 931-A Photomultiplier Tube 128 A-V Comparison of Intensity Transmitted Through the Eighth Step with That of Dye Filter of Density D2 = 7.30 +.05 129 vi

LIST OF TABLES (Concluded) Table Page A-VI Developed Densities of Type-33 Emulsion Resulting from Exposure of 32 Minutes Through ST-49-3 129 A-VII Log2 of Intensity Behind ST-1 (ST-49-1) 130 A-VIII Log2 of Intensity Behind ST-2 (ST-49-2) 131 A-IX Log2 of Intensity Behind ST-3 (ST-49-3) 132 vii

LIST OF ILLUSTRATIONS Figure Page 1.1 Typical reciprocity-law-failure curve showing qualitative influence of temperature. 9 3.1 Average grain size of pure AgBr emulsion as function of precipitation time (after Trivelli and Smith). 33 3.2a Reproduction of typical reciprocity-law-failure plate. 34 3.2b Schematic diagram of plate holder used in reciprocitylaw-failure experitsent 34 3.3a Inside view of plate-holder assembly. 36 3.3b Outside view of plate-holder assembly. 37 3.4 View of plate holder. 38 3.5a Slide-control mechanism. 39 3.5b View of plate-holder assembly and slide-control mechanism. 40 3.6 Wiring diagram for slide-control mechanism. 41 3.7 View of light-source shutter. 43 3.8 View of timing apparatus. 44 3.9 Wiring diagram for timing apparatus. 48 3.10 Cutaway drawing of water-cooled housing for light source. 49 3.11 Diagram of arrangement for reciprocity exposures. 50 4.la Low-intensity reciprocity-law-failure curves for pure AgBr emulsions T-4140, T-4141, and T-4142 (surface development at 32~ + 1~C). viii

LIST OF ILLUSTRATIONS (Continued) Figure Page 4.1b Low-intensity reciprocity-law-failure curves for pure AgBr emulsions T-4139, T-4141, and T-4143 (surface development at 32~ + 10C). 56 4.1c Relationship between a (the average cross-sectional area of grains in the emulsions T-4139 to T-4143) and the shift A log I of Figs. 4.la and b. 58 4.2 Slopes of low-intensity reciprocity curves for AgBr emulsions T-4139 to T-4143 (internal development). 59 4.3a Reciprocity-law-failure curves for T-4139 developed with different surface developers. 69 4.3b Reciprocity-law-failure curves for T-4140 developed with different surface developers. 69 4.4 Characteristic curves for emulsion T-4140 from experiments 67-1 and 70-3 (surface development). 74 4.5 Example of reciprocity-law-failure curves for T-4140 (surface development) constructed from composite curves shown in Fig. 4.4. 76 4.6 Characteristic curves for emulsion T-4140 from experiments 72-2 and 89-1 (internal development). 77 4.7 Example of reciprocity-law-failure curves for T-4140 constructed from composite curves of Fig. 4.6. 78 4.8a Characteristic curves for T-4141 emulsion exposed and stored at 380C (the amount of the dip in the higher curves is very unstable). 80 4.8b Reciprocity curves derived from characteristic curves of Fig. 4.8a. 80 L4.9 Effect of storage on reciprocity curves (surface development). 88 4.10a Possible distortion of reciprocity curves from change in sensitivity during exposure (regular order). 90 ix

LIST OF ILLUSTRATIONS (concluded) Figure Page 4.10b Possible distortion of reciprocity curves from change in sensitivity during exposure (reverse order). 90 5.la Reciprocity curves of log (E-Eo) vs log ~ (arbitrary units) for the two-trap model for difference between deep and shallow traps of rn.26 ev. 97 5.1b Reciprocity curves of log (E-Eo) vs log 5 (arbitrary units) for the two-trap model for difference between trap depths of ~r.10 ev. 97 5.2a Reciprocity curves of log (E-Eo) vs log t (arbitrary units) for the two-trap model for equal number of deep and shallow traps. 99 5.2b Reciprocity curves of log (E-Eo) vs log 5 (arbitrary units) for the two-trap model for about one deep to every 32 shallow traps. 99 5.3a Sample reciprocity curve of two-trap model with f < 1. 101 5.3b Sample reciprocity curve of two-trap model with f larger than in Fig. 5.3a. 101 ~5.4 Plot of log2(E-Eo) vs log2 q for continuous distribution (s = 1/2 and %0 = 28) and M = 224, 216, and 28 110 5.5 Plot of 2 s0/1+2-I, the integrand of Js(q,M) for s = 1/2 and s = 1/8, showing the range of integration when M = 216 (extended range for s = 1/8 curve shown in insert). 112 5.6 Typical experimental curve (of emulsion T-4140 from Fig. 4.la) compared to theoretical curves. 116 A-1 Step tablet ST-49-2 shown masked as used in reciprocitylaw-failure experiments. 133 A-2 Arrangement for measuring transmission of step tablets. 134 A-3 Characteristic curves of type-33 emulsion at different intensities. 135

ABSTRACT The physical properties of the photographic emulsion, the nature of the photographic latent image, the Mott and Gurney theory of latent-image formation, and a simple model used by Webb to calculate an effective electron trap depth are briefly described. This "single-trap-depth" model predicts a simple, isodense, low-intensity reciprocity-law-failure (rlf) curve whose slope rapidly approaches -1 with decreasing intensity of exposure. A quantitative theory of the mechanism of latent-image formation proposed by Katz (outlined in Chapter II) predicts noninteger low-intensity rlf slopes when an exponential distribution of trap depths is assumed, and therefore agrees better with experimental results. It is shown in Chapter V that for this model the limiting slope will alsobecome -1 for sufficiently low intensities (depending on the total number of traps per grain) but the slope for an extended intermediate section is still inversely proportional to the spread in the assumed exponential distribution. This dependence of rlf slope on spread of trap depths suggests that if their distribution varies with grain size, then a relationship between grain size and rlf slope might be expected. This provides the motivation for the experiments described in Chapters III and IV. A new type of apparatus for use with very low intensities and long times of exposure is described in detail. The principle advantages of this design were the compactness permitted by the use of slides instead of a sector wheel, the simultaneous exposure of three plates, and the flexibility possible with the automatic timing circuit which controlled the exposure program. Pure AgBr emulsions were used for the experiments of rlf vs grain size in order to avoid difficulties arising from the correlation between iodide content and grain size. Since the characteristics of these emulsions change more rapidly with time than those of ordinary commercial emulsions, a uniform program of storage before and after exposure was employed, with test experiments to demonstrate the validity of this procedure. The rlf slopes from the five pure AgBr emulsions, whose average grain size ranged from about.2 [2 to 5 p2, demonstrate a definite increase in slope with average grain size of emulsion used when developed with internal developer and a similar but less marked tendency with surface development. This effect was stronger when higher temperatures of storage and exposure were used. A systematic indication of more structure in the experimental curves than could be accommodated by either the "exponential-trap-depth" or "single-trap-depthl' model was observed. However, this was only slightly beyond the estimated experimental error. xi

ABSTRACT (Concluded) It is shown that a simple model assuming two discrete trap depths provides low-intensity rlf curves of considerable structure which are qualitatively similar to experimental curves. For example, they show an extended, almost straight section of noninteger slope. These curves are described in terms of the relative abundance and depths of deep and shallow traps in the grain. A method of obtaining these parameters by analysis of experimental curves is given. Suggestions for improvement of the experimental technique and possible additional experiments suggested by the "two-trap-depth" model are offered. xii

OBJECTIVE This investigation is part of a plan to check the consequences of theories of the photographic latent-image formation, with a view toward finding the number of atoms necessary to form a developable speck of latent-image silver. Specifically, the present study deals with the dependence of low-intensity reciprocity failure on grain size, other factors being unchanged. xiii

CHAPTER I INTRODUCTION The purpose of this work is to present the results of investigations which it is hoped will contribute to the understanding of the mechanism of formation of the photographic latent image. A. BRIEF DESCRIPTION OF PHOTOGRAPHIC EMULSIONS A typical photographic emulsion is composed of many (about 109 grains per cm2) microcrystals of silver halide (called photographic grains) suspended in gelatin and coated on a suitable supporting backing. The grains range from submicroscopic to several microns in diameter (for information about size frequency distribution see Chapter II of Reference 1). In the pure silver bromide emulsion used for the experimental part of this work the microcrystals appear in the form of regular triangular plates, with corners cut off, and some needle-shaped crystals. The thickness of the plates ranges from approximately.5 to.1 of their diameter. Since silver bromide is known to have cubic symmetry,2'3 the flat faces of the grains must be the (111) planes of the crystal. That the microcrystals in emulsions also have this same symmetry was verified by the presence of characteristic silver bromide rings in the x-ray patterns from emulsions. Most commercial emulsions (this discussion will be restricted to lightsensitive emulsions) are bromo-iodide emulsions. The addition of several percent of iodide increases the photographic sensitivity of the grain. It has long been known3 that the lattice distance increases slightly with in*The grains of most commercial emulsions, however, appear spherical.

2 creasing iodide content up to about forty-percent iodide, where the hexagonal iodide structure sets in. As early as 1923 Trivelli4 suggested that mechanical strains, which were detected by observing the microcrystals under polarized light, were instrumental in producing this increased sensitivity (recently Chateau and Pouradier6 have demonstrated that nucleation begins with high iodide concentration and, consequently, hexagonal symmetry followed by the main part of the microcrystal being cubic). Mitchell and 7,8 Hedges have recently given further evidence defining the role of crystalline imperfections in photographic sensitization. 1. Chemical Sensitization. -Emulsions also contain traces of substances called chemical sensitizers. These substances such as silver, sulfur, and gold (and more recently lead, thallium, rhodium and other heavy metals9) probably exist in the microcrystals at the interface between the gelatin and the silver bromide and/or regions of stress in the microcrystals.6'7 * 2. Dye Sensitization. —The spectral sensitivity of a silver halide emulsion is determined fundamentally by the spectral absorption of the silver halide microcrystals. Silver halides absorb negligibly for wavelengths in all except the violet end of the visible spectrum. It is often desirable to extend the photographic sensitivity to the red end of the visible spectrum. This is made possible by the following mechanism: certain dyes when adsorbed to the surface of the photographic grains can absorb light which the host grain cannot and will pass on the absorbed energy in a form which is usable in the photochemical process of making a grain developable. For simplicity, only emulsions which have not been dye sensitized will Photographic characteristics are also strongly dependent on the hydrogen ion concentration and silver ion concentration of the gelatin surrounding the microcrystals.

3 be considered in the following discussion. A discussion of dye sensitization is given by Mees. "0 B. EFFECT OF LIGHT ON PHOTOGRAPHIC GRAINS AIND DEFINITION OF PHOTOGRAPHIC LATENT IMAGE When light of suitable wavelength is incident upon a photographic emulsion, the individual grains act as "Yes-No" detectors of the radiation. Thus, if a grain absorbs a sufficient number of photons, it is changed so that subsequent treatment with a suitable reducing agent (called a developer) will cause it to be completely reduced (i.e., developed) to metallic silver, whereas grains that have not absorbed a sufficient number of photons will be unaffected by the developer.* The number of such developed grains per unit area as a function of position on the photographic plate determines the macroscopic developed negative image. (The grains unaffected by light and consequently not developed are washed out of the "negative." This is called fixing the emulsion.) A photographic latent image is said to have been formed in those grains which have been made developable by the absorption of light. C. CHEMICAL COMPOSITION OF PHOTOGRAPHIC LATENT IMAGE Although this work is concerned primarily with the mechanism of formation of the latent image, we will consider briefly some of the evidence for the generally accepted conclusion that the photographic latent image consists of minute specks of silver which have been photochemically deposited on the grain. For a historical review of the theories concerning the photographic latent image, see Mees, Bancroft, and Webb. Some unexposed grains act as if they were in an unstable state. In time these grains, although not exposed to light, become spontaneously developable. These are called fog grains.

1. Photolysis of bulk silver halides is known to produce metallic silver and free halogen. 2. Direct chemical analysis has demonstrated the presence of free metallic silver in heavily overexposed emulsions. Recently the presence of silver has been detected even for exposures in the upper part of a normal characteristic curve (see Mees, pp. 106 and 122). 3. Bullockl3 has compiled a long list of cases in which the grains of exposed emulsions (even for the exposures in the usual photographic range) behaved chemically as if there were minute deposits of silver upon them. D. SIZE AND DISTRIBUTION OF THE PHOTOGRAPHIC LATENT IMAGE Some of the evidence that the latent image involves only a few atoms of silver will now be cited. 1. Using measured values of the absolute energy required to produce developability in the most sensitive grains of a photographic emulsion, 12 Webb has estimated that from one to twenty quanta of light are involved; consequently, only a very few atoms of photolytically released silver are sufficient to render these grains developable. 2. Kornfeld and James 14 have used the Herschel effect to determine the number of infrared quanta which a grain must absorb in order to destroy its developability. They conclude that one quanta per grain will do this and consequently only a small number of silver atoms are involved in the latent image. Even 200 silver atoms (which is a generally accepted upper limit) constitute a relatively minute change in the grain since the average grain will contain about 109 ion pairs.* Microscopic examination of ex*Lattice spacing in AgBr is about 5.75 A; thus, in a grain of cross section 1 k2 and thickness.2 there will be (10-4)2 cm2 x 2 x 10-5 cm 4 ion pairs x ion pairs/grain. 1 grain (..75 x 10-) cmx

5 posed grains subjected to arrested development generally reveals several centers of development (this technique was used as early as 1917 by Hodgson,15 Svedberg,16 Toy,17 and ClarkL8 and later extended to electron microscopy by Harmm and Comerl9 and others). This is interpreted to mean that the latent image is often distributed throughout the grains and that the actual number of atoms per developable speck may be even smaller than the total number of silver atoms in a developable grain. 3. From the sequence-loop data of Webb, Katz20 has estimated that from 4 to 9 atoms of silver are required for a developable speck. 4. By treating very thin films of silver (evaporated onto glass) with developer, Reinders and DeVries21 concluded that clusters of as few as 4 atoms of silver are sufficient to initiate development. Berg22 later reevaluated this work and concluded that 3 or 4 were sufficient. This and other evidence indicates that the minimum number of silver atoms required for a latent image may be as small as 3, depending on the emulsion and/or developer used. E. MECHANISM OF LATENT-IMAGE FORMATION Mott and Gurney23 in 1938 proposed a theory of the formation of the photographic latent image based on the following properties of the silver halides (for a recent review of the properties of the silver halides, see Seitz ): 1. The silver halides show no electronic conductivity in the dark at room temperatures; they show photoconductivity when exposed to light of the same wavelength which produces the photographic latent image in corresponding silver halide emulsions. (Photoconductivity of photographic

emulsions has been measured recently and reported by West and its wavelength dependence found to be essentially the same as the spectral sensitivity of the emulsion.) 2. The photocurrent-vs-voltage curves indicate the presence of traps for the photoelectrons, other than the bromine atoms from which they were originally ejected. These traps function as local regions where the electron experiences a lower potential energy, and are due to impurities or crystalline imperfections. 3. Silver halides exhibit electrolytic conductivity, increasing strongly with temperature. This is attributed to the mobility of interstitial silver ions. From the quantum theory of solids the allowed energy states for the electrons in a crystal can in general be grouped into bands separated by gaps of energies forbidden to the electron. An insulator is then described as a substance for which the bands are completely filled up to a forbidden gap of energy > > kT, the higher bands being completely empty. Since all energy states are then occupied up to the gap, the Pauli exclusion principle will not allow any net motion of the electrons (any conductivity) except by jumping the gap into the empty conduction band. Electronic photoconductivity results if a photon absorbed by the crystal is able to supply enough energy to cause the transition of the electron from the filled to the conduction band. Electrons released to the conduction band are effectively free to move about under the influence of an electrical field applied to the crystal. The existence of impurities or crystalline imperfections may distort the periodic potential so that the electron experiences a lower potential in their vicinity and is therefore trapped (for a short time if the trap is shallow, i.e., the electron is weakly bound, or per

7 haps permanently trapped if the trap is sufficiently deep). Using these ideas, Mott and Gurney postulated the following mechanism: 1. A photon absorbed by the silver halide crystal causes an electron to be liberated from one of the bromine ions into the conduction band of the crystal. 2. This photoelectron, which is now free to wander about in the crystal, is trapped by an imperfection, impurity, or silver speck (a latent image already started), remaining in the trap for a time dependent on the depth of the trap and the amount of thermal agitation. 3. If the photoelectron is trapped long enough for an interstitial silver ion to be attracted and to form a silver atom at the trap, then the next photoelectron may also be trapped at the same trap, forming an additional silver atom by attracting an additional silver ion. Thus, the process can continue to deposit silver at areas of the crystal which trap photoelectrons. 4. If the electron drifts off before the silver atom can be formed, or if it disintegrates again into a silver ion and a conduction electron, the electron may be trapped at another trap and start building a latent image, or it may be lost to the process of latent-image formation by recombination with a bromine atom. 5. Upon reaching a certain size (certain number of atoms of silver), the speck will become thermally stable and additional silver will be deposited in accordance with the Einstein photoequivalence law. This hypothesis explained many photographic effects such as the Herschel effect, reciprocity law failure (rlf), etc., and suggested experiments (for example, see Webb26). Its application to the reciprocity law failure will be described briefly. Since the early stages of formation of

8 the latent image are assumed to be thermally unstable, the process may become inefficient if the rate of quantum absorption becomes too slow. Thus, the photoelectron may be trapped and a silver ion attracted to it, beginning the latent-image formation. However, for very low intensities thermal agitation may cause its disintegration before another photoelectron can be captured at the same trap. Conversely, at high intensities the second photoelectron might follow the first so closely that the less mobile silver ion does not arrive in time to neutralize the space charge caused by the first photoelectron at the trap. Thus, the efficiency of photographic latent-image formation would be expected to decrease toward low intensities and also toward high intensities, with a certain optimum efficiency occurring at an optimum intensity between these extremities. Figure 1.1 is a representation of a typical experimental reciprocity curve where the logarithm of the exposure required to produce a constant photographic density is plotted as a function of the logarithm of the intensity during the exposure (an increased exposure required to produce a constant density implies a lower efficiency for producing the same photographic effect). The Mott and Gurney theory is in qualitative agreement with these experimental reciprocity curves. Furthermore, since the efficiency at low intensities decreases with increasing temperatures (thermal disintegration increases) while the efficiency at high intensities increases (silver ion mobility increases), the reciprocity curve for T2 > T1 would be expected to shift toward higher intensities. This is also in agreement with experimental results shown in the typical curve in Fig. 1.1. Although this theory has been very useful, it fails to explain some parts of the photographic process. The fate of the bromine atom produced by the ejection of the photoelectron is not explained. It is well known

9 LJU LCr - IIndicates Low x LOGI I(ARBITRARY UNITS) Intensity Region = INTENSITY Fig. 1.1. Typical reciprocity-law-failure curve showing qualitative influence of temperature. that the gelatin of the emulsion can function as a bromine acceptor, and that a bromine atom can effectively migrate (by hole exchange) from any point in the crystal and escape to the surface. It is not clear, however, why this nascent bromine does not immediately react with the latent-image silver (also situated largely at the surface in highly sensitized grains), thus destroying the photographic effect. Mitchell27 has suggested that photographic sensitizers act as acceptors of the photolytically released bromine* rather than as concentration specks which trap the photoelectrons. The latent image could then grow in the same way in a chemically active *CThe same idea was suggested in 197 by Hiclrman.28

10 region at the surface of the grain without interference from the bromine, as long as there was enough sensitizer to capture the bromine. With this description of the photolytic process in silver halides, Mitchell is able to explain most of the known effects in photographic emulsions. The quantitative description of the mechanism of latent-image formation which follows will be exclusively in terms of the more familiar Mott and Gurney theory, since only with respect to very few experiments has it been possible to distinguish experimentally between microscopic latentimage theories. It is hoped that this discussion will be useful in interpreting experimental results, and that it may in the future prove to be sufficiently general or adaptable so that it may apply to other microscopic theories.

CHAPTER II MOTIVATION FOR EXPERIMENT OF LOW-INTENSITY RECIPROCITY LAW FAILURE VS GRAIN SIZE The model for the mechanism of latent-image formation which assumes for simplicity that all electron traps are of the same depth has been used by Webb29'30 to relate properties of the reciprocity-law-failure curves to such quantities as the effective trap depth and activation energy for ionic conductivity (from high-intensity reciprocity failure) and to evaluate these quantities experimentally. However, this single-trap model inevitably predicts a low-intensity limiting slope of 0, -1, and -2 (i.e., a negative integer), thus disagreeing with experimental limiting slopes which lie between O and -1. It is the purpose of this chapter to show how if a continuous distribution of trap depths is assumed, an approximate solution yields noninteger limiting slopes which are in closer agreement with experimental results. It will then be shown that the low-intensity slope is related to the spread of the trap-depth distribution. Since the ratio of surface (deep) traps to internal (shallow) traps is probably dependent on grain size, a dependence of the low-intensity limiting slope of the reciprocity curve on grain size might be expected.* The similarity between the mechanism proposed for early stages of latentimage formation and phosphorescence in ZnS coupled with the results of Antanow-Romanowsky (see Mott and Gurney,31 p. 214) also suggests dependence of the reciprocity behavior on grain size. Antanow-Romanowsky found a dependence of the decay constant with grain size for ZnS phosphors. It should be noted, however, that Wallick's experiments,32 using ZnS samples separated according to size, showed the shape of the decay curves to be essentially independent of grain size, except that the decay rate was slightly greater in the first few seconds for the smaller grain-size samples. 11

12 This suggested relationship between low-intensity reciprocity slope and grain size provided the motivation for the experiments discussed in Chapter IV. The mathematical description of the process of latent-image formation suggested by Katz,33 showing the connection between trap-depth distribution and the reciprocity-law-failure curve, will be discussed below. It was also decided to investigate some refinements of this theory, which will be discussed in Chapter V. A. SINGLE-TRAP MODEL Before the review of the work of Katz, a brief discussion of the singletrap model (based on Webb's work 29) will be presented in order to show how integral slopes result. Let us assume, following Mott and Gurney, that photoelectrons are trapped at certain "sensitivity centers," subsequently attracting interstitial ions. The trapping of electrons followed by the attraction of silver ions proceeds at an inefficient rate at low intensity. This is because the newly formed atom may disintegrate by thermal ejection of an electron and departure of the silver ion. However, after the speck of silver reaches a certain critical size (say ns atoms) called the sub-latent image, it will then be thermally stable (i.e., the probability per unit time of disintegration of a speck of the size n < ns is much greater than for n > ns), and consequently each additional photoelectron will produce a silver atom.* The instability in the early stages of formation of the sub-latent image is manifest in the low-intensity reciprocity law failure; if the first silver atom were thermally stable, then the efficiency of forming silver would be independent of the rate of quantum absorption. *That the following atoms are added with quantum efficiency of unity is verified by the validity of the photoequivalence law.

13 Consequently, the exposure required to produce a certain density would be independent of the intensity of the light used for the exposure (rlf slope = 0). However, if ns > 1 (i.e., more than' one atom is required to form a stable sub-latent image), then the efficiency of the process, as measured by the number of quanta required to produce a given developed density, would decrease with decreasing intensity. The reciprocity-law-failure slope also would increase in some way with the number ns. In order to illustrate this, imagine the electron trapped within a potential barrier of depth U. As the electron is jostled about by the thermal motion of the lattice, it will have a certain probability of escape each time it hits the side of the barrier. This probability is given by the Boltzmann factor p = e-/kT, (2.1) where T is the absolute temperature and k the Boltzmann constant. If it vibrates in the trap with a frequency v per second, the probability of escape per second will be P v eU/kT. (2.2) Consequently, the average number of times the electron would escape during an exposure time t is The temperature dependence of v is assumed to be small compared to the effect in the exponent. ** In this section t is used to denote exposure time in order to reserve T to denote absolute temperature. In following sections the conventional notation T for exposure time and t for log T will be used. No confusion need occur, since the meaning of T will be clear from the context.

Pt = v e-U/kT t. (2.3) The average lifetime or critical time** T of an electron in a trap of depth U is then given by e U/kT (2 4) V Webb,29 applying the probability calculus formulae derived by Silberstein,34 obtained a useful expression for the probability P (of forming a *This formula was the basis of the method used by Webb to obtain U. Because of the dependence of P on the temperature, two rlf curves at the same density (i.e., the same P) but at different temperatures would be displaced with respect to each other. This means that at a higher tem. perature the time required to produce the same effect would be greater, the relation being log t2 = 1 - (2.3a) His measurements of the effective trap depth U for several emulsions gave U about.77 ev for the usual surface image and about.65 ev for the internal image. The probability that the speck will survive and grow to a full, stable sub-latent image is fundamentally dependent on the probability of producing photoelectrons within this time T (at low intensities the silver ion migration time is neglected). The dependence of this probability on the intensity of exposure as a function of the parameter ns can be estimated by assuming that the number of photons n occurring in a time T follows a Poisson distribution about the average number n, i.e., P(ns) e n Therefore, e-I s)ns I ns P(ns) n=.. IK << 1n n = KI where I = intensity, K = a constant (dependent on grain absorption, size, depth in emulsion, etc.). This shows that the probability of finding conditions favorable to the formation of a stable, sub-latent image decreases strongly with decreasing intensity I, and even more so with larger ns.

15 stable sub-latent image and subsequently a developable image) as a function of the intensity and the critical time T. His result was ks 1 - e-Ftns, (.5) where F (IT)ns'l I (2.6) Here t is time of exposure, ns is the number of photons which must occur in the critical time T in order to form a stable sub-latent image, and ks/N is the fraction of the total number of grains which become developable as a consequence of the exposure I x t. Thus, for constant ks/N this expresses the condition between I and t from which the reciprocity law failure at constant density can be derived. For ks/N to be constant, n5 -ln (IFt) ns I = constant Ftn5 (ns - 1) or (2.7) Insl E - (ns -l) ns Tns-1 Writing x = In I y = In E equation (2.7) becomes (ns - 1) In I + an E = Kn K (2.8) (ns - 1) x + y = In J zy =-(ns - ) *(2.9)

16 Notice that for ns = 1, dy/dx = 0, as was reasoned earlier. However, all observed reciprocity slopes are < 0; therefore, ns > 1. Nevertheless, one is forced to a weaker statement about the upper limit of ns. No slopes greater than one have been observed in normal results. Hence, it can be said only that ns is not necessarily greater than two. The fact that the low-intensity limiting slopes for many emulsions approach one is taken as an indication that ns = 2. Most reciprocity-lawiftailure slopes, however, lie between O and -1 and fit neither ns = 1 or ns = 2. One might hope to this point by assuming that some silver atoms had already been produced in the ripening process. Consequently, the effective number ns would be decreased slightly by this preexisting silver. This idea has been shown by Katz -(see last page of References 20, 335) to be incompatible with experimental results. Before showing how the assumption of continuous trap-depth distribution leads to noninteger slopes, the problem of relating single-grain theory to multigrain experiments will be considered briefly. B. ORDER PRINCIPLE Experimental investigations of the photographic behavior usually involve measurements of millions of grains (i.e., photographic density is a measure of the attenuation of light by the suspension of the developed grains). Theoretical investigations, however, are usually formulated in terms of properties of a single grain or an ensemble of identical grains..In an actual photographic emulsion the grains may vary considerably with regard to sensitivity, developability, and size (and therefore the "weight" of their contribution to the density). A further complication is the allor-nothing character of developability of a grain and the fact that there is no way to perform successive experiments on the same grain, since the

17 only known method of detecting the latent image in a grain is actually to develop that grain, thereby destroying the latent image as such. It is therefore not obvious how to associate the experimental measurements which result from an integration of a wide range of classes of grains with the theoretical results which concern a specific species of single grains. It is usually assumed that if the experimental results are expressed in terms of the exposure required to produce a constant density, they will be applicable to the theoretically predicted behavior of a single grain. 20,53 Katz has recently shown that it is indeed correct under certain limiting conditions to associate isodense measurements from photographic emulsions with single-grain theory and has suggested35 how to formulate the conditions of validity of this association. An elementary review of this consideration follows. Instead of an actual emulsion, in which the exposure required to produce a developable grain may vary greatly from grain to grain because of differences in degree of sensitization, grain size, absorption coefficient, depth in the emulsion, etc., consider first a simplified hypothetical emulsion composed of grains identical in every respect* except the grain size. The grains in such an emulsion will reach the state of developability in a definite order during the progress of an exposure if statistical fluctuations in the rate of absorption of quanta are ignored. The existence of such a definite order would mean that if several exposures were made (on a plate coated homogeneously with this emulsion) producing equal densities, these densities would in all cases be composed of *Note that if the grains were identical in every respect, the density-vsexposure curve would be a step function modified only by the statistical variation in the time between absorbed quanta.

equivalent grains. Thus, if the order in which these equivalent grains become developable were not disturbed (i.e., if the order principle were valid), the last grains to become developable would in all cases be equivalent, and the exposure which produced these densities would be exactly that required to produce the latent image (condition of developability) in these last grains. Suppose now that one of the exposure or development variables is varied, the exposure, however, being adjusted so that the final density will be the same. Then as long as the order principle is valid, the experimental results will still represent the relationship between that exposure or development variable and the exposure required (number of quanta absorbed by the grains) to produce a latent image in the equivalent last grains. For example, suppose N identical samples of this emulsion ('in each of which the grains differ only by their size) were subjected to N different exposures of intensity I, < I2 < I3... < IN and that the corresponding exposures required to produce a density Da were E1 > E2 > E3... > EN. Then the relationship between E and I represents the reciprocity behavior of the grain of that particular size* which became developable last at the density Da, and the corresponding results for a density Db > Da would represent the reciprocity behavior of another grain (in this case smaller) which became developable last at the density Db. That these conclusions refer only to the last grains becoming developable is a consequence of the all-or-nothing (step function) character of developability. The situation in an actual emulsion, however, is more complicated, Note, however, that when the characteristic curve, or the reciprocity curve of density vs intensity at constant exposure, is considered, then every point of the curve refers to a different-sized grain. Consequently, even for this simple emulsion the interpretation would be difficult.

19 since many factors will disturb the order principle applied to all the grains. For example, a group of grains which become developable after a certain time of exposure when situated at the surface of the emulsion would certainly require a longer time before becoming developable when situated deeper in the emulsion. Thus, an exposure causing the least sensitive grain at the surface to be just developable might also be just sufficient to make developable a more sensitive grain situated deeper in the emulsion. Consequently, the last developable grains might belong to two widely separated sensitivity classes, even when constant densities are compared. It is possible, however, to regard the total density as the sum of partial densities resulting from different grain classes each of which is homogeneous except for one grain variable. Hence, although the order principle is not applicable to the system of grains taken as a whole, it is consistent to assume its validity for each homogeneous class of grains contributing to the total developed density. For example, if the isodense reciprocity behavior of an actual emulsion is to be studied, the partial density corresponding to various grain sizes, depth of the grain in the emulsion, and so forth, can then be associated with a spread in intensity (calculable if the grain-size distribution and attenuation of intensity with depth in the emulsion are known). Using this approach, Katz was able to show that the deviation from the order principle caused by the inhomogeneity of grain size and light intensity* produces reciprocity curves which are practically identical with those which would be obtained if the light intensity were equal to the value at *These are factors which appear to cause the main deviation in the order principle, statistical fluctuation being accounted for in the discussion in Section C.

20 the average depth, and if the emulsion contained only grains of a certain average size. This conclusion refers to most common commercial emulsions. Thus, except for the knowledge that the actual curves are somewhat smoothed by these deviations in intensity, the deviation from the order principle need not disturb the association of the single-grain interpretation with the multigrain emulsion measurements, as long as the measurements are reduced to constant density. * C. MECHANISM OF LATENT-IMAGE FORMATION The probability of forming a latent image is fundamentally dependent on the probability that two photoelectrons are available simultaneously at the same trap and successively attract two interstitial silver ions, forming a stable diatomic silver speck. The assumption that a diatomic speck of silver is thermally stable is supported by the experiments of Webb,29 the theoretical work of Silberstein3 on the shape of the toe of the characteristic curves of photographic emulsions, and the observation of Katz (Section 10 of Reference 33) that the assumption of a triatomic stable speck would not predict correctly the experimental reciprocity bahavior. The probability that two successive photoelectrons will finally form a developable latent image in a grain is therefore dependent on: 1. The chance F(t) that a photoelectron released at t = 0 will still be alive at time t (i.e., it has not combined with a bromine atom) when the next photoelectron is released; 2. The chance c that the two successive photoelectrons which are"alive" will combine at the same trap to form a stable speck; and *This is essentially a review of the mathematical description of the mechanism of latent-image formation as presented by Katz.33

21 3. The probability P(T,G)dG that a photoelectron ejected at t = O will be followed by another at a time t between G and @ + dG, during an exposure of time T. F(t), called the survival function of the photoelectron, will be strictly dependent on the trapping mechanism of the grain. For example, the singletrap model will give F(t)- e-'t, while a suitable continuous distribution of traps will give F(t) (txo)-S. The derivation of these forms of F(t) will be given below. The chance c that two photoelectrons occurring simultaneously at the same trap will build a stable speck is dependent on properties of the microcrystal (for example, the mobility of the silver ions). It may indirectly depend on the trap-depth distribution but not through the interquantic time 9 [i.e., the dependence of c on trap-depth distribution is not dependent on the intensity of the absorbed light as is, for example, the P(G)]. Although no way of evaluating c is known, 0 < c < 1, by definition. The probability P(G)dG of finding an interquantic time between 9 and 9 + dG is purely a function of the statistics of light-absorption process and therefore dependent on the effective light intensity absorbed by the grain. The probability of forming at least one stable diatomic speck (and subsequently at least one developable speck causing the grain to be developable) can.be estimated very closely as follows.* The photoelectron released at t = 0 will have a chance F(G) of still being alive at t =, at which time the probability of the second photoelectron's being released is P(G)dQ. Thus the probability W of liberating For a slightly more rigorous derivation see Reference 20.

22 a second electron while the first is still "alive" is co W(1/8) = o F() P(,)dg 0 (2.10) Since the chance that two electrons combine to form a stable speck is c, the probability of building a stable speck per interquantic time is cW; thus, the probability of building a stable speck during an exposure of Ns interquantic times is NscW. It is then supposed that in order to construct a developable speck from a stable speck, an additional number of quanta No must be absorbed. Since No is assumed to be a constant for a particular grain, i.e., independent of the intensity of light, it is only the Ns = N-No interquantic times which enter into these probability calculations and determine the efficiency of production of developable specks as a function of the intensity. Consequently, the probability of forming a stable speck during an exposure of a total of N interquantic times is written as NscW = Q Then the probability P1 of forming at least one developable speck in the grain is P1 =1- e. Assuming the validity of the order principle, isodense exposures correspond to a constant probability that a developable speck will be produced in the type of grain made developable last. Consequently, the relationship between W(1/Q) or W(aI) and Ns for isodense exposure is W(aI) Ns = constant. (2.11) Note that c has been absorbed into the unknown constant and is of little interest.

23 This can be expressed in terms of the measurable exposure by noticing that the number of quanta during an exposure E = IT is N+1 = Ea when I is expressed as the number of quanta incident per second, a the effective area of the grain, and T the total time of exposure. Then Ns = N-No Ea-l-No and Equation 2.11 becomes W(aI) (E-Eo) = constant (2.11a) where Eo = (No+l)/a. In order to express the dependence of W upon I, both F(t) and P(O)dO are needed. If it is assumed that the a priori probability of absorption occurring in any small interval dG is constant, then for constant light intensity I = 1/aG, the probability of finding an interquantic time 9 is given by* P(G)dG = e- / dG/G (2.12) F(t) is calculated by considering an ensemble of identical grains having a trap-depth distribution m(U)dU [i.e., the fraction of traps having depths between U and U + dU is m(U)dU] and then evaluating the number of electrons alive in these grains at any time. Imagine that at t = 0 each grain absorbs a photon, and that the released photoelectron will be trapped (at random) in any of the traps in the grain. If the fraction n(U,t)dU of electrons found in traps of depth U were known at any time, t, after t = 0, then the F(t) could be calculated since 00 F(ty) = n(U,t,y)dU (2'13) The constant y is the probability per jump than an electron ejected from a trap will recombine with a hole rather than jump to an electron trap (probability a = l-y). This is valid only for exposures containing many interquantic times (see Section V of Reference 20).

24 The function n(U,t,y) is a solution of Urn dt n(U,t,y) = -x(U) n(U,t,y) - om(U) n(U',t,7) \(U')dU', (2.14) 0 U/kT with n(U,o,y) = m(U), where k(U) = ve as used in connection with the single-trap model and m(U)dU is the arbitrary relative trap-depth distribution with extremities Uo and Um. The net gain in n(Ut,y) is here expressed in terms of the rate of loss from the traps of depth U and the number gained from other traps of all depths. A general solution of (2.14) was not available; however, the main features of n(U,t,y) and, consequently, F(t,y) are seen from consideration of certain special cases. Case A. Discrete set of N trap depths N ri for U = Ui r 1 m(U) = 0 for U Ui i=l U1 < U2 < U3... < UN and UN - UN_1 > kT kT ".0025 ev The survival function for this case would be N N FA(t,y) =r aL e-ceat )-1 a=1e However, the discrete separation between UN, UN_1, > UN_2, etc. in the exponent of the exponent of the Nth and (N-l)th terms would cause the term containing UN to overwhelm the others rapidly, leaving FA(t,y) ~= C' e-N YCt (2.15)

25 where C and C' are constants of the order of unity. Case B. If a = 0 (7=1), Equation 2.14 can still be solved in complete generality for arbitrary m(U). The rigorous solution for F(t) is U FB(t,1) = m(U) e d(U)tt. (2.16) Case C. For a O a rigorous solution was not available. However, n(t) and hence F(t) can be developed in a power series in a: F(t,l-a) = FB(t,l) + G(t)c + H(t)a2 +... The G(t) and H(t) can be expressed in terms of Fg(t,l), and it was stated that the series appeared to converge rapidly. Further, if a plausible form of m(U) (see Equation 2.18 below) was assumed, it was possible to show that G(t);og ~ FB(t,l), g < 3, and H(t) << G(t). This would indicate that for a f O the general solution retains a functional dependence very similar to that of the FB(t,l). On the other hand, if a - 1 (r + 0), the probability that the electron will be lost to the photolytic process by recombination with its original hole is very small; thus, an equilibrium distribution among the various traps will be reached quickly. The distribution will gradually change, the shallow traps becoming less populated relative to the deeper traps as electrons are gradually lost by recombination with holes. Under these conditions the second term on the right of Equation 2.14 essentially loses its dependence on time, and an approximate solution was found by Katz to be

26 where 00 1 = JO m(U, m(U ) dU When he used a trial solution of the form F(t,y) = e-7Yt + yFl(t) + y2F2(t), Katz found that F1 0 O. Further, if the distribution of Equation 2.18 was assumed, the F2 term again became roughly proportional to FB(t,l). It was concluded that even though the integral differential Equation 2.14 was not solved in complete rigorous-form, the main characteristics of the behavior of the solutionare reflected in Cases A, B, and C. This analysis indicates that for single-trap depth or a discrete (energy separation of traps greater than kT) set of trap depths (O < y < 1) the survival function will be of the form FA(t,) = C' e-bt, (2.17) where b = Cyv e-UN/kT and C and C' 1. If, on the other hand, a continuous trap-depth distribution was assumed and 0 < y f 1, then it was argued that the time dependence of the survival function for continuous distribution was essentially the same as FB(t,l) (Equation 2.16). The remainder of this analysis will be restricted to a consideration of the effect of these two types of survival functions on the slope of the low-intensity reciprocity curves. Now in order to obtain an explicit time dependence for FC(t,l), it is necessary to postulate some plausible form of m(U). The form of m(U) used by Katz was suggested by the widely accepted idea that the sulfur impurities act as traps for the electron and that the effective depths of these shallow traps are increased by the presence of imperfections frozen, in during the ripening process. A simple assumption as to the form of the trap

27 adepth distribution thus formed is r(U) = 0 for U < Uo n(US) 1 e(Uo-U)/E for U > UO, (2.18) m(U) = 0 for U > UM where Uo is the ideal trap depth of an Ag2S speck and c is the average depth increase (or spread in the distribution) due to imperfections. Using Equations 2o16 and 2.18, the survival function becomes U= e (Uo-U)/c -% (U)t FB(t,l) e edU, U0 m where K(U) = v e-U /. Substituting q = Xot e(Uo-U)/kT, this reduces to 0 hot FB(tl) ( t)S ts em od e (2.19) ko -v e Uo/kT v, e,-UJm/kT to = _ e kTl ~nl% ~m J determinred- by 00 / m (J)dU - lInm M r,,lr~le M is the total nunber of electron traps in the grain (see footnote, p. i08). Then FP3(t'! = (xhot) FV(s) - s, s - e- d - s ot e d nmt << 1 ~(,t) (ot)sfor (2.20) hot ~ 1

28 Thus, the continuous distribution of trap depths leads to a survival function approximately* proportional to (xot)-s as compared to the discrete -bt set of trap depths which leads to a survival function proportional to e Direct substitution of survival functions (Equations 2.17 and 2.20) into (2.10)with(2.12) gives 00 a WA(I) = air e(b+aI)t dt b+aI (2.21) s d(aIt) WB(I) ~ (aI)S kos j (aI)-s t-s e-aIt d(I) (2.22) 00 = (ax)S Is s S e d- = Is(a%)s (l-s) To see the effect of these two forms of W on the low-intensity limiting slope of the reciprocity curve, consider W(I) (E-Eo) = constant as a description of the isodense reciprocity curve. ** Define x = log I y = log E then for the discrete case the isodense reciprocity curves are represented by aI baI (E-Eo) = constant - b+aI I(E-Eo) = a In Chapter V it will be shown, however, that even for this case the slope of the reciprocity curves will approach -l for low-enough intensities. A definition of x and y should really be in terms of dimensionless variables such as x = log I/ko and y = log E/Eo. However, convention is followed here since little difficulty results from this mis-notation; the quantity of interest being dy/dx.

29 where Eo = Eo + K. x + y + log ( - eYY) log (1 - e) log Thus for y >> Y, i.e., E >> Eo, dy/dx + -1. Similarly for continuous trap distribution, isodense curves are represented by IS(E-Eo) = constant = sx + y + log (1 - eYO-Y) = log. Then for y >> Yo, i.e., E >> Eo dy kT -- s - = dx E This analysis indicated that a discrete distribution of trap depths would give a limiting slope of -1 (which checks with Webb's results for the singletrap model), but that a continuous trap-depth distribution could result in reciprocity curves with noninteger limiting slopes. Furthermore, the analysis illustrated that these slopes would be inversely proportional to the spread in the trap-depth distribution.

CHAPTER III APPARATUS USED FOR LOW-INTENSITY RECIPROCITY-LAW-FAILURE VS GRAIN-SIZE EXPERIMENTS A. GENERAL DESCRIPTION OF APPARATUS 1. Measurement of Density. —A measurement of the photographic density affords a convenient means of evaluating how much the developed photographic emulsion has been affected by its exposure to light (that is, what fraction of the grains has been made developable). The density at a certain area of the developed emulsion is determined by allowing light of a constant intensity Io to shine on it and then measuring the intensity I (x,y) of the light transmitted through the developed emulsion at that area. The density is then defined by D(x,y) = log10 Ix For our experiments a Vincent-Sawyer type of microphotometer* was used to measure the density. In this instrument transmitted light is detected by a Weston photocell, the output of which is registered on a sensitive galvanometer. The usual linear scale was replaced by one recalibrated to read density directly. The efficiency of the photographic effect can also be measured by counting the percentage of grains affected by a particular exposure. Such a method involves a tedious program of grain counting, using a very good microscope. Since standard emulsions are coated much too thickly and Kindly made available through the cooperation of Professor R. A. Wolfe. 3o

31 densely to be practical for such observation, they would be extremely difficult to use without diluting and recoating them as single-grain-layer plates. 2. Emulsions Used. —An Eastman Kodak Type-33 4" x 10" spectroscopic plate was used for preliminary tests of the apparatus, reproducibility of exposure technique, development technique, etc. This plate was chosen because of its good uniformity, contrast, and sensitivity. For the experiments of low-intensity reciprocity law failure vs grain size, it was necessary to use emulsions with pure silver bromide grains, since in the usual bromoiodide emulsions the smaller grains contain a higher percentage of iodide than do the larger grains.37 Hence, use of a bromoiodide emulsion for study of properties of the grains of various sizes would inadvertently introduce the factor of changing iodide content. At first an attempt was made actually to separate fractions of various-sized grains from the same emulsions following essentially the method used by Renwick and Sease.37 This method was pursued to the point where it was evident that it was essentially feasible. However, it was put aside because it appeared that the advantage gained (i.e., greater certainty that samples of various grain sizes were from identical emulsion) was not sufficient to justify the amount of work required to obtain samples of various grain sizes in this way. See, however, remarks on grain-size dependence on page 116. Instead, the experiments were made by using the series of pure silver bromide emulsions,* manufactured as nearly as possible in the same way, except for the precipitation time of the formation of the silver bromide grains which occurs at the first step in the process of the manufacture of the emulsion. This resulted in a series of five emulsions supposedly Kindly supplied by Eastman Kodak Research Laboratories.

32 the same in every respect, except in grain size. In particular, the average grain size and "spread" in the distribution increased with the precipitation time. These emulsions were prepared according to the procedure described by Trivelli and Smith38'39 and were coated with a concentration of 1.5 mg/cm2 of AgBr to a thickness of 25pt. This provided a maximum density of about 3. The average grain sizes of these emulsions, T-4139 to T-4143, used for the experiments described below are shown in Table 3-I. They were obtained from the data of Trivelli and Smith39 by first plotting the average grain size as a function of the precipitation time for their pure silver bromide emulsions, then reading off the average grain size corresponding to the. precipitation times supplied with our emulsions (Fig. 3.1). A rough estimate was then made of the average grain size of emulsions T-4140 and T-4143 by counting the number of grains in several grain-size groups. These results can be seen in Table 3-I. 3. Low-Intensity Reciprocity-Law-Failure Apparatus for Exposing Emulsions. - a. General Description. The purpose of the apparatus to be described below was to provide a method of obtaining a specific array of exposures in which the rows had constant intensity and the columns constant time of exposure. The intensity decreases in powers of two (i.e., if I is the intensity at the front of the plate holder, the intensity along the nth row is I/2n). The times of exposure also advance in powers of two (i.e., the mth column has an exposure time of 2mT). This makes the exposure in the elements down a particular diagonal a constant (I/2n)(T2m) = IT 2m-n. Thus, if the reciprocity law were valid, the density down the diagonal would be constant. Figure 3.2a shows a plate which has been exposed in

E5 33j 0 bT4139 T4142 12 <2 T-4141 T4140 I 0 20 30 4 0 0 60 70 80 90 100 PRECIPITATION TIME IN MINUTES Fig, 3.1. Average grain size of pure AgBr emulsion as function of precipitation time (after Trivelli and Smith). TABLE- 3-T GRAIN SIZES FOR PURE AgBr EMULSIONS DERIVED FROM PRECIPITATION TIME Precipitation Estimated Average Results from Emulsion Coating Time Grain Size* Rough GrainNumber Number (minutes) (microns)2 Size Count T-4139 497528 1/2,2A.+ T-4140 497529 5,2 -5 ~ -3.3 T-4141 497530 25 1,5 -,3 T-4142 497531 45 2.6 +.4 T-4143 497532 85 4,7 +,6 6,3 ~+ 2,0 _ *From Trivelli and SmithA using Figz 3.1ope

Fig. 3.2a. Reproduction of typical reciprocity-law-failure plate. MONOCHROMATIC LIGHT SOURCE 8' FROM PHOTOGRAPHIC PLATE RACK AND PINION DRIVE I I/1,,','/'/, /-,,' //- ila ~~~~~~~ I6II / II EASTMAN KODAK NEUTRAL DENSITY STEP WEDGE 34 ~ ~ ~~4I PHOTOGRAPHIC PLATE Fig. 3oe2bo Schematic diagram of plate holder used in reciprocity-law-failure experiments.

55 the apparatus. Note the decrease in density up a diagonal, indicating the failure of the reciprocity law. The essential difference between this apparatus and that used by Jones and Huse and later by Webb lies in the method of obtaining the variations in the time of exposure. Because of space limitations the sector wheel, usually several feet in diameter, was replaced by a slide. The operation of the special plate holder which has the slide built into it is illustrated in Fig. 3.2b. This technique was permissible because the times of exposure were never less than about thirty seconds, so the corrections required due to the finite speed of the slide were easily determined by blank runs. Three such plate holders were placed adjacent to each other, which allowed many combinations of exposures of three plates to be made simultaneously. This was useful for trebling the output of the number of exposures, which inevitably requires a long time, or for including a standard plate to check the reproducibility of the operation of the apparatus, light source, etc. Figure 3.3a illustrates this arrangement as seen from the side toward the light source. Figure 3.3b illustrates the same unit from the outside (note the removable plate holders which are also separately illustrated in Fig. 3.4). This unit was constructed of aluminum and Dow metal. It was later blackened by anodizing. b. Slide-Control Mechanism. Above the plate-holder assembly the mechanism designed to move the slides (seen in Figs, 3.5a and 3.5b) consists essentially of a motor-driven reduction-gear system "G" connected to a rack and pinion which controls the opening and closing of the slides. The operation of the slide-control mechanism can be understood by a study of the wiring diagram in Fig. 3.6. The reversible motor is started by

Rack and pinion - drive for slides Photocell window Slides in position Removable to expose first plate 4 time columns holders Fig. 3.3a. Inside view of plate-holder assembly.

Arm which lowers standard source between light source and photocell Mechonism for moving slides synchronous motor Reversible motor for moving siiies Bulb of Recording thermometer Photoce ll ho us~i ng plate holders: I:, 5 b Outs:id-e view* of Ilate hoId.c aI sI'mby.y | i l 11112*0: >;; ^...-...... i le _W~i~ ISI 1111 _ _r1 o'i~i'i~i~i.......:':.-... Re m ov a

lii'B i~~~.'.i i'i' I::::::::::::::::::::::::::::::::::::::::j::jjj,:ij,:,:j,:,:,:::::::::::::::::-:::::::::::::::::::::::::::::::':':':':':':':':':':::::i'i'i'i'i'i'i'i'i'i'i'i'i::::::::::::::::::::::::::::::::::::::::::::'''''''''''''''''''''''''''''''''''''''':::':':':':':':':':':':':':':':':::::::::''''''''''''''''''''''''''':':' iiiiiiiiii::::::::::::::::::::::::::::::::::iii:::::::::::::: iiiiiiiiiiiiiiiiiiiiiiii::~:::~::::::::::::::::i'i'iii'iii'ii::::::::::::::::::::::::::::::::::::::::-:-: ~: i:i:i:i::::::::::: iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii ii;::::::::::: iiiiiiiiii:::::::::-: ii'iiiiiiiiiiiiiiiiiiiiiiiii::':':':'':':':'::::::::::::::::::::::::::::::::::: i'''i'''i'''::::::::::: i:i:i:i:::::::'i'i'i'i'i'i'iiiiiiii i:i:i:i:i:i:i:i:i iiiiiiiiiiiiii ":"':":':::::::::::::::i'i'i'i'i'i'i'i'i'i'i'i'i''''''''''' i:::::::::::::::::::::::::::::::::::::::::-:-:-:::-:~::-:~::::-:::-::-::-:::-:::'ii'ii:i.'iiiiii:iiiiii:iiiii.iliiii::::::::::::::::'::::::::::::::::::::::'::::'::'i'i::::~:::::: i:i ii i'i'i:: ~::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

39 Apollo~~~~~~~~~liF __ i~iii~i~i~i~iii~il~i~i~iii~iiiiir::i Fig. 3.5a. Slide-control mechanism.! I1_ - -k -l~~-r~,.,.~~~~~~~~~~~~ j~.~'-i.iiil~iii:,ii ~~ Fig e 3 e 5a. S~~~~~~~~~~~:iiiiiiiixii~'lide -control mechanisiiiiiiin

40 To external indicator lights i-Ar i -E~~~~~~~~~~~~~~~~~~~~~ -!~~~~~~~~~~4 Fig. 3. 5b. View of plate -holder assembly and slide-control mechanism.

Color Prong Channel Plug I Plug 2 A I Blue JA B 8 2 Orange 1KB C 3 Black 42 Ohms Quick stop H N L C D 4 Yellow mechanism D 4 Yellow E 5 White F 6 Red E 2-150,fd G 7 Brow Electrolytic in parallel H 8 Greein 9 Violet ~~RI ~ ~~~i~J 10 Red K I I White 3 L 12 Block M Small green N Red Motor * | In center cable 2 Prong on plug to releose coil of light source shutter I Meg. 14 Prong plug#2 to 15,0hms neon light position - indicator. 1N p2 3 4 5 6 7 e 9 j 0 I I 14 Prong___ —plug# I connected to timing device'-12 Strand cable toggle switch. a _ _-fc 7 b Normal position 0 8b closed c Id button out lc d open Microswitch Button in Ia b open action. ic ad closed Fig. 3.6. Wiring diagram for slide-control mechanism.

42 activation of relay R1 continuing to operate until the current in the coil of relay R1 is interrupted. This interruption occurs when any one of the microswitches Mi is depressed by the rotating arm N (Fig. 3.5a). When a microswitch is depressed, the relay can then receive the next starting pulse from the timing circuit. The slide can be set at any position independent of the timing circuit by use of the manual "override" switch S (Fig. 3.~5a). Figures 3.2b and 3.4 show that twelve columns can be exposed in succession to the light source as the slides advance to the extreme open position. The time program followed was: 1. The shutter for the light source (see Fig. 3.7) is opened manually. 2. An ordinary oven timer is set to start the timing mechanism at a convenient future time, which will be denoted as T = -2 min. The apparatus then proceeds automatically. 3. At T = 0 the motor receives a command from the timing apparatus (Fig. 3.8) and opens the slide to begin the exposure in the first column. (This process of opening requires about 9 sec.) The first microswitch, M1, is now depressed and consequently prepared to receive the next timing pulse. 4. At T = 210 min (= 1024 min = 17 hr 04 min) the pulse from the timing circuit causes the motor to advance the slides to begin the exposure in the second column. (This movement takes approximately 3 sec.) The second microswitch, M2, is now depressed and prepared to receive the third pulse. 5. At T = 29 + 210 (1536 min = 25 hr 36 min) the pulse from the timing circuit causes the motor to advance the slides to begin the exposure in the third column. (This movement takes approximately 3 sec.) The

43 Relay armature holding shutter open Smallapertur e for ii:::.: ii:I~~~~~~i;:iiiiX ~~~~~~~~~~~~~~rla ~B''.~~;~ ~~~,p: i~~:....~ —:i_:~::... - | _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:: Smal I aperture for very low intensity exposures Fig. 3.7. View of light-source shutter.

44 Rectifier for power supply of stepping switches /3 S2 S D K Extra timing motor and disk Fig~ 3.8. View of timing apparatus.

third microswitch, M3, is now depressed and prepared to receive the fourth pulse. These processes continue in an identical manner except that each time the interval between timing pulses is half the previous interval. 6. At T = 2n min the slide moves to the second from the final n=O microswitch, Mll, thus beginning the exposure of the next to the last column. 10 7. At T = E 2n + 20 min the slide moves to the final microswitch, n=O M12, beginning the exposure to the final column. 8. One minute later the final pulse depresses the relay (R2) which (a) causes the motor to run in the reverse direction until the slide is completely closed, and (b) simultaneously releases the relay which closes the shutter at the light source (Fig. 3-7). As a result of this program the columns received the total time of exposure given by the following table: To = 20 min T, = To + 20 min = (1 + 1) min = 21 min T2 = T1 + 21 min = (2 + 2) min = 4 min = 22 min Ts = T2 + 2 min = (4 + 4-) min = 8 min = 2 min Tn = =,_, + 2" —i n-= 2n T = Tn + 2n- min (2n-1+2n-1) min= 2 x 2n min = 2n m T11 =........................ = 211 min Note that this regular program is arranged so that the starting time of each column is different but all exposures end at exactly the same time (i.e., the instant the shutter at the light source closes). The time correction required to account for the movement of the slide is negligible except in the last column, where it may be about 35, i.e., 1.5 sec in 1 min, and here it can be corrected for in a simple way.

c. Neutral-Density Step Filter (step wedge). The variation in intensity to which the emulsion is exposed in the various rows is accomplished by the use of a so-called neutral-density step filter. This is an acetate film coated with a light-absorbing material. The concentration of this material is adjusted in steps for each row so that the light behind the filter is cut down in steps of powers of two (see Fig. A-I). Since the accuracy of the density of this step wedge was specified only to about 5%, it was desirable to check the value of the density at each point of the exposure matrix (see Tables A-VI, A-VII, and A-VIII for thetabulation of the corrected intensities behind the three step wedges used). The details of the method of calibration of the step wedge will be described in Appendix A. d. Timing Circuit. The timing circuit referred to above consists of a bank of three 25-position stepping switches, S1, S2, S3, with three banks of wiper contacts. The advance of these stepping switches is actuated by a primary timing pulse (see Fig. 3.8). This primary timing pulse usually occurs at a rate of two pulses per minute, but by replacing the synchronous motor, Ms (Fig. 3.8), or changing the number of notches on the disk D, it could -e varied from four per minute to 1/4 per minute. If the primary pulse occurs at the rate of two per minute, then the first stepping switch, S1, advances one revolution every 12.5 min. (The stepping switch has 25 positions.) The second stepping switch, S2, which is actuated only when the first is in the initial position (since its actuating coil sees the primary timing circuit only through the contacts of that step of the first stepping switch), will make one step every 25/2 = 12.5 min; hence, it requires 25 x 25/2 min to complete its cycle. The third stepping switch, S3, being similarly actuated by the second stepping switch, advances one position every 25 x 25/2 min.

47 Thus the third stepping switchwould complete a cycle in 25 x 25 x 25/2 min. This then gives a simple base 25 counting circuit. It is then only necessary to connect the microswitches to the contacts of the appropriate stepping switches in such a way as to start the motor at the prescribed times. The wiring for this is shown in Fig. 3.9. e. The Light Source. A commercial mercury vapor lamp (General Electric H-85 A-3) was used as a source of light. The radiation from this lamp was filtered through a Corning No. 5-74 filter so that the only radiation which was incident upon the emulsion was essentially theirLonochroma0 tic 4358 A Hg line. The lamp was maintained at a relatively stable temperature by the water-cooled housing shown in Fig. 3.10o Although the exact function of this unit is not understood, its use reduced temporal fluctuations in intensity from greater than 6% resulting from other techniques to less than 2%. For example, either blowing air on the outer bulb or putting a cooling jacket directly in contact with the outer bulb of the lamp resulted in eccentric behavior. Presumably, the arrangement which evolved is the most satisfactory because it allows the lamp to come to a rather high (about 300~C) equilibrium temperature which allows it to stabilize, andtlhecooling water simply carries off the excess heat energy radiated. The monochromatic light emerging from the filter is then diffused by passing through opal glass (ground glass does not diffuse the light uniformly). The intensity of the light source can be moderated by the diaphragm which reduces the effective area of the diffusing opal glass. A simple electromagnetically released shutter cuts off the light at the end of the experiment (see Fig. 3.7). This light-source housing and shutter system is mounted near the center of one end of a light-tight plywood box 8' x 2' x 2' which is painted

plug 5 M I0- 0- 0 O-, L- ~ o j ( go 0w P 0 0 0 o 0 0 0 0 a~~~~~~~0 0 0 0 0 0 _ __ I _ ~ 1 o0 0 A -28v DC (regulr pulse timer) Fig.9 Wiring diagra for t0ing apparatus 0 0o 0 ~ o o o 0 o o o o 03 oo oo i t ~/ is, r -C1 I LS2 I L__I _ 1 S. Uz.~e — Y Input plug X - 28v DC (constont) Telephone switch B + DC up gives zero reset A -28v DC(regulor pulse from primory timer) Fig. 3.9. Wiring diagram for timing apparatus,

49 Weston Photoelectric Cell I Neutral Density Filter Tubeartz1::~ ofHousing for Corning H85Tube of 5 -74) Filter Outer Glass Envelope /or Wa tuber Soapstone Retainer Auxiliary Opening to Fig 3.10. Cutaway drawing of water-cooled housing for ight Source

50 with a flat, black, nonturpentine-base* paint. The opposite end is occupied by the exposure mechanism. The inside of the box is diaphragmed as shown in Fig. 3.11 in order to eliminate effectively the low-angle reflection from the inside walls of the box. This arrangement gave a uniform illumination (within 2%) at the position of the plate holders. Photocell Housing Light Source Plate Holders Masonite Diaphrams To Prevent Low Angle Scattering Fig. 3.11K Diagram of arrangement for reciprocity exposures. f. Uniformity of Light Source. In order that the step wedge may perform its function properly, the light incidentuponit should be uniform over its entire surface. Rough checks with a photoelectric cell indicate that this condition was satisfied within about 2%. The fluctuations of the light intensity during the time of exposure present a greater problem. For best results the intensity should remain Turpentine is one of the many substances whose vapor will cause photographic emulsion' to fog. See E. R. Bullock, Reference 15.

51 constant within 4i/o during the entire period of exposure (often several days). Several attempts were made to maintain proper control of temperature and constant voltage for the light source. Sola transformers were ab.ndoned in favor of a Sorensen 1000-watt a-c voltage regulation supply. This was supplied by a 220-volt line stepped down to 110 volts, which gave a much steadier voltage than any of the 110 supply lines ~ Even this voltage did not, however, assure completely the constant intensity of the light source. At times a quick pulse from the 220 supply would turn off the source completely (quick changes are not moderated by the Sorensen which has a time constant of about 1/5 sec). A photocell was set adjacent to the exposure plates, and a Leeds and Northrup recorder kept a record of its evaluation of the intensity, However, since the sensitivity of the photocell a-nicd criaplifier circuit also fluctuated slightly in an unpredictable manner,* it was felt advisable to have this photocell periodically co~ are the mits::i-t'y of the light soulrce with that of a standard source.** Accord inglvy, a, shutt,+er containing the standard source was built so that thce po}"toelel could see thie light-source intensity for about 12 min, then llothliC for'bour 20 sec (in order to check the dark current), then the ml.i? arI s3ouarc e for 5 min o This made available a record of the intensity o t- ihnciden~t illtrml~nation compared to a standard illumination. Exposure of'th-e plates to tlle monochromatic light so rce was not interrupted by this procedure. Although the 81 x 2' x 2' box was quite light tight, it was even then enclosed in a small dark room built of masonite and 2" x 4" lumber, with *A model O20i system mnade by T'Photovolt Corporation" was used.'lhis standard source was a l/2-inch diameter ZnS-Cu Phosphor activated by t.-he radiation from tritium (fromTracer-Lab., Inc ). Since the half-life of t-ritium is about 125. years, this standard source decays at a rate of about 4f per year.

52 double doors to allow entry safely even though the outer room was not dark. Besides reducing the danger of stray light, the box makes it possible for the temperature and humidity to be controlled. g. Temperature and Humidity Regulation. The temperature in the dark room was regulated by the use of a fan blowing over a small electric heater. In the windstream a thermostat was placed so that it turned on the heater coils when the air temperature dropped below the desired temperature. The temperature of the emulsion was supposedly the same as the exposure mechanism in which it was held. This temperature was measured and recorded by a recording thermometer* whose bulb was placed on the aluminum base of the exposure mechanism. Recorded temperatures usually showed less than + 1~C variation during an experiment. Since it is preferable to keep the humidity constant also (the sensitivity of the emulsion is dependent on the relative humidity to which it is exposed), a commercial dehumidifying unit was installed in the box during the humid summer months. It was attached to a "hair" control unit which maintained a relative humidity at 35% + 4%. B. SUMMARY OF DESCRIPTION OF APPARATUS In brief, the essential characteristics of the apparatus used are these. The reciprocity-law apparatus, for use at low intensity and long exposure, is housed in a large, light-proof box which has temperature and humidity control. It is entered by a double-door system designed to exclude all light. The temperature for each experiment is recorded continuously during the experiment. The intensity of the monochromatic light source is maintained as constant as possible by an a-c voltage regulator supplying the *Made by Scientific Instrument Company of Detroit.

mercury vapor lamp. The intensity is also continually recorded and compared every 15 min to a source of constant intensity. The three 4" x 10" glass plates coated with emulsions are exposed in ten rows with constant intensity along each row, the intensity varying in powers of two steps from one row to the next. The columns on the plate are exposed for times advancing in powers of two up to 212 min. The net result is an ordered matrix of exposures of varying time and intensity, with constant exposure elements along the diagonals.

CHAPTER IV EXPERIMENTAL PROCEDURE ASD RESULTS In order to investigate the relationship between the grain size and the failure of the reciprocity law at low intensities, a series of five sulfur-sensitized pure silver bromide emulsions (with average grain size ranging from about.2 [t2 to 4.7 12) were exposed in the apparatus described in Chapter III. The general experimental procedure and results will be outlined in brief. Details will then be considered. A. GENERAL PROCEDURE The general procedure for the experiments was as follows: 1. All plates were kept in the refrigerator at 8~C until needed. The exposure was preceded by a prescribed prestorage period and followed by an additional prescribed post-storage period, all in the exposure room maintained at a constant temperature (for example, 32~C). 2. The developed densities D were read, corrected for variations in emulsion thickness, and converted to Sidel* values, S. The characteristic curves plotted from S vs the logarithm of the intensity are more linear and consequently easier to "smooth" than the corresponding densityvs-logarithm of intensity curves (see Kaiser 42). 3. From the Sidel values the family of characteristic curves of S vs log intensity of the exposure was plotted (the family parameter was the time of exposure). *If D = log10 (I iI), S = log10 (Io/I - 1); hence, S = log10 (10D - 1). 54

55 4. When the characteristic curves for several exposures using the same emulsions were sufficiently similar, they were averaged to obtain a composite characteristic curve. If they were not amenable to averaging, they were plotted separately (see page 72). 5. From the compositite characteristic curves (Sidel function vs log intensity at constant time) the reciprocity-law-failure curves were constructed for various constant values of density. B. RESULTS 1. Surface Development (exposed and stored at about 32~C). —The reciprocity law failure of emulsions exposed and stored at 320C and developed with surface developer are shown in Fig. 4.1a for emulsions T-4140, T-4141, and T-4142, and in Fig. 4,lb for emulsions T-4139, T-4141, and T-4143. Table 4-I shows additional data related to experiments from which these results were derived. The straight-line segments are not intended to indicate a functional relationship between intensity and exposure, but rather to identify the emulsion to which each point belongs. The uncertainty illustrated by the symbol 4 was estimated from the uncertainty in the intensity behind the step tablet (see Appendix A) and in reading the density and plotting the characteristic curves, and from the consistency between the characteristic curves which made up a composite characteristic curve. The symbol should actually be inclined thus, 8, since the time of the exposure is known to better than 1% and therefore the intensity is the major source of uncertainty. The projection along the exposure axis was used since the inclined symbol would make it difficult to distinguish between adjacent points. See p. 67.

56 Emulsion ALog I No. ~xrx/ - - - - - T-4140 +1.9.5 + 2 T-4141 0 1.5+.3 -— T-4142 -.9 2.6.4 -0 a:-: 03.4 (D 1 04) x LOG2(RELATIVE INTENSITY) -_ Fig. 4.la. Low-intensity reciprocity-law-failure curves for pure AgBr emulsions T-4140, T-4141, and T-4142 (surface development at 520 + 10C). Emulsion /- T- 4139 +3.2.2 r.I >' 2 9 o8 -— T -4143 -1.0 4.7+.6 x -LOG2 (RELATIVE INTENSITY) Fig. 4.1a. Low-intensity reciprocity-law-failure curves for pure AgBr 3mlIo 1'~1' ms 00. x _+.LON s= o (D=.3) -4l -12 -11 -I0 -9 -8 -7 -6'5 -4 -3 -2 - I X =LOG2 (RELATIVE INTENSITY)"-I~ Fig. 4.lb. Low-intensity reciprocity-law-failure curves for pure AgBr emulsions T-)-159, T-4141, and T-1v145 (surface development at 520 ~ 10C).

57 TABLE 4-I ADDITIONAL INFORMATION RELATED TO RECIPROCITY-LAW-FAILURE EXPERIMENTS USED TO CONSTRUCT FIGS. 4.1la AND b Calibration Calibrat ion Emulsion Experiment Package Temp, Background Strip Time, Number Number Number ~C Fog Density Density in T-4139 70-2 2 32.4.071 +.003.27 +.01 2 71-1 2 32.7.077 +.003.19 +.01 2 T-414o 67-1 2 32.2.118 +.005 1.00 +.1 2 70-3 2 32.7.095 +.005 1.00 +.1 2 T-4141 67-2 3 32.2.164 +.006 1.2 +.2 1 71-2 3 3207.13 +.01.9 +.1 1 T-4142 67-3 2 32.2.105 +.005.85 ~.15 1 68-1 2 31.2.100 +.01.85 +.1 1 68-2 3 31.2.114 +.005 1.00 +.2 1 71-3 3 32.7.112 +.005.53 +.15.5 These reciprocity-law-failure curves show the log2 of the exposure required to produce a density of.30 and 1.04 (Sidel of 0 and 1, respectively) plotted as a function of log2 of the intensity used in the exposure. In both figures the intensity and exposure units applied directly to the emulsion T-4141. * The reciprocity-law-failure curves for emulsion T-4140 would lie higher (i.e., require a higher intensity to produce the same density) than those of T-4141, when plotted with the same units of exposure and intensity. In order to facilitate the comparison of slopes, the curves for T-4140 were plotted as though they had been exposed to an incident intensity 21'9 (i.e., 3.74 times the incident intensity used to *For example, x 0 corresponds to the incident intensity Io (at the front of the plate holder), x = -1 corresponds to an intensity Io/2, x = -10 corresponds to Io x 2-10, etc. Io is about 23 x 108 photons/sec/cm2 (N"23 photons/sec/average grain) as measured by comparison to a similar light source calibrated by Uo S. Bureau of Standards.

58 expose T-4141). Similarly, the curves for T-4142 were plotted as though they had been exposed to an incident intensity 2-`9 Io (i.e.,.536 of that used for T-4141). The change in log I used to effect the shift of the curves is denoted by A log I in the legend of Figs 4.1la and b. A plot of A log2 I vs log2 of the average grain size (a) of the emulsions is shown in Fig. 4.1c. This indicates a linear relationship between the effective absorbed intensity and the average grain size of the emulsion. T-4143. 2 T-4142 T-4141 C~j 0 -4 -3 -2 -I 0 1 2 A LOG2 I Fig. 4.1c. Relationship between a (the average cross-sectional area of grains in the emulsions T-4139 to T-4143) and the shift A log I of Figs. 401a and b. The results in Fig. 401a show that within the experimental error these reciprocity-law-failure curves for the pure silver bromide emulsions of average grain sizes.5, 1.5, and 2.6 ~2 have essentially the same slope. The only systematic difference is possibly a slight tendency for the slope to decrease with decreasing grain size. The results shown in Fig. 4.lb represent the extremes of average grain sizes used. They should therefore show the greatest systematic difference

59 in slope vs grain size if a significant relationship could be inferred from these experiments. 2. Internal Development (exposed and stored at about 320C). -The reciprocity-law-failure curves of the pure silver bromide emulsions developed with internal image developer could in most cases be closely fitted to a straight line. Figure 4.2 shows the slopes of these reciprocitylaw-failure curves (for S = 1 and 0) plotted as a function of the log10 of average grain size of the emulsions used.. Table 4-IIa gives details related to experimental results shown in Fig. 4.2. It is clear that the emulsions of larger average grain size show larger reciprocity law failure when internal development is used. S:I.O(D:1.04)}T=330+ 2/ C S=O(D=.3) f i.2.3 Q. -.6 -.3.3.6 -J.2 T-413 / T-414T4141 T-4142 T-4143 LOGIO-.3 0 A3.6 LOGio AVERAGE GRAIN SIZE OF EMULSION Fig. 4.20 Slopes of low-intensity reciprocity curves for AgBr emulsions T-4139 to T-4143 (internal development). *See p. 68.

TABLE 4-IIa SLOPE OF LOW-INTENSITY RECIPROCITY-LAW-FAILURE CURVES AND RELATED DETAILS FOR PURE SILVER BROMIDE EMULSIONS T-4139 TO T-4143 USING INTERNAL DEVELOPMENT (Exposure and Storage at About 33~C) Calibration Emusicon Experiment Package Temp, Fog StSlope at S 0.0 Slope at S = 1.0 Weighed Average EmulnNumber Number ~C D y Density (D =.30) (D = 1.04) S = 0.0 S = 1.0 T-4139 88-1 3 35.0.145 +.01.27 +.02 *a = g 81-1 3 33.8.17 +.007.64 +.03.120 +.007.207 +.007.2 +.1 81-2 3 33.8.153 +~ 01.64 +.02 89-1 3 5.17.ol.51+.05 1 140 72-2 2 32.7.12.57+.07.225 ~.015.183.24 +.01 +.02.5 +.2 88-2 3 35.115 +.007.50.02.20 +.02.27 +.02.5 +.2 72-2 2 32.7.12 +.01.57 + 07.187 +.07.25 +.01 T-4141 74-3 3 30.25 +.01 1.05 +.06.27 +.02.29.02.251 1.5 +.3 89-2 3 35.145 +.02.54 +.05.24 +.03.22 +.04 +.02 +.025 T-4142 728-3 2 25.283 +.01 7564 + o6.27 ~.01.295 ~.01.277.251 72-3 2 32.5.27 ~. 01.64 +.06.277 a= -- +.007 +.015 2.6 ~.4 91-2 4 35.23 +.02.88 +.03.275 +.01.275 +.02 - +.07 73-3 3 30.5.22 +.01.60 +.03.31 +.02.31 +.10 T414 91-1 4 35 +.7.28 +.02.73 +.07.27 ~.04.26 +.02 T-4143 4.7 =.6 1 88A-3 4 34.25 ~.015.57 +.03.31 +.03.36 +.03 4.7 +.6.295.293 91-1 4 35 +.7.28 +.02.73+.07. +..26.o9 73-3 3 30.5.22 +.01.60 +.03 51 6 a is average grain size in (micron)2

3. Surface Development (exposed and stored at about 38GC). —The reciprocity behavior of the pure silver bromide emulsions stored and developed* at 380C was qualitatively the same as for those exposed and stored at the lower temperature, except that the slopes in general were slightly steeper. 4. Internal Development (exposed and stored at 38~C). —Results from the silver bromide emulsions stored and exposed at 380C and developed with internal developer are tabulated in Table 4-IIb. The slopes vs grain size may also be seen plotted in Fig. 4.2 for comparison with the results at 33~C. The characteristic and reciprocity-law-failure curves in Figs. 4.8a and b are given in order to exhibit a peculiarity which occurs at higher temperatures. This phenomenon was not studied further but indicates simpler results at lower temperatures. The above results definitely indicate an increase in reciprocity-lawfailure slope with average grain size of pure AgBr emulsions if internal development is used, and a similar but less marked tendency with surface development. A possible explanation of the paradoxical increase of slope with increasing density (smaller grain size) will be considered briefly on page 116. The consistent indication of more structure in the reciprocitylaw-failure curves than can be accommodated by present models will also be considered in Chapter V. Now, however, the procedure will be discussed in more detail to help evaluate the validity of the experiment. *These emulsions were developed with M-AA-1 (see p. 68). However, control plates with SW/2 showed no systematic difference within the experimental uncertainty. It should be pointed out that the emulsion T-4139 used here from the fourth package showed evidence of having been accidentally stored at room temperature; its surface reciprocity curve (at 32~) had slopes of about.05 and were almost straight (see Fig. 4.9). Thus, the slopes for internal development may also be too low.

TABLE 4-IIb SLOPE OF LOW-INTENSITY RECIPROCITY-LAW-FAILURE CURVES AND RELATED DETAILS FOR PURE SILVER BROMIDE EMULSIONS T-4139 TO T-4143 USING INTERNAL DEVELOPMENT (Exposure and Storage at About 38~C) Calibration Emulsion Experiment Package Temp, Fog Number Number Number C Density Strip Slope at S Nensity T-41359 103-3 4 38.0.175 +.01.69 + ol.10 +.01 100-2 4 38.0.11 +.005.36 +.01 95-1 4 38 +.5.08 +. 0oo.70 +.02 T-4140 99-2 4 38.0.10 +.005.70 +.005.16 +.01 101-2 4 38.5.11 +.01.65 +.01 T-4141 96-2 4 37.8.22 +.01.55~ 32 +.+0 102-2 4 38.o.16 +.005 75 +.o05o 99-3 4 38.o.21 +.01.72 +.02 T-4142 95-2 4 38 +.5.175 +.005.61 +.01.285 +.01 98-2 4 38.0.19 +.01.60 +.01 95-3 4 38 +.5.22 +.01.o59.02 T-4143 101-3 4 38.5.25 +.01.59 +.01.37 +.02 98-3 4 38.o.20 +.01.57 +.01

63 CO DETAILS OF GENERAL PROCEDURE 1. Preliminary Experiments.-The first apparatus used was essentially the same as the usual sector-wheel apparatus, except that slides were used instead of the wheel. The slide design occupied an area of about 18" x 24" compared to an area of about 72" x 72" for the sector wheel. Each slide had three slots which were one, two, and four units in length. The slides moved at the rate of 1 inch, 1/8 inch, 1/64 inch, and 1/512 inch per minute. This gave the desired exposure times increasing in powers of two. Because of its availability and uniformity of coating, type-33 emulsion on regular spectroscopic plate was used almost exclusively. The use of this primitive apparatus made it possible to discover several of the causes of poor reproducibility. Preliminary experiments indicated the necessity for modifying several factors: a. The intensity of the light source was dependent on the ambient temperature and the supposedly constant output voltage of the Sola transformers. These transformers regulated momentary voltage fluctuations, producing a fairly stable output voltage; however, if the average line voltage was high for several hours (as it was overnight), this output voltage would tend to shift to a significantly different value. b. The humidity could not be controlled. c. The slide design presented several serious problems associated with scattered light. d. Only one plate per several days could be produced for low-intensity work. e. The plate-holder design was inconvenient and introduced pressures on the plate. * *Pressure causes emulsions to become developable (Backstrom43)

64 The apparatus described in Chapter III was designed to correct these difficulties. The new apparatus produced such consistent results using type-33 emulsion that the program using the silver bromide emulsion of various average grain sizes was started. The correction of the more obvious factors influencing reproducibility, made possible by use of this new design, served to unmask more subtle factors. For example, insufficient attention to storage conditions was now found to be a major source of error. Systematic control experiments with various storage conditions led to the storage pattern which was finally adopted. 2. Storage of Plates and Exposure Program. — It has long been known that the "sensitivity" of photographic emulsions changes with time. One of the major difficulties in experiments using pure silver bromide emulsions is that chargt-s in these aspe cia emxut ions apparently occur about 100 time- faster than in commercial emulsions. Since the experiments required exposures of about two days, it was necessary either to devise some way to approximate the behavior of the emulsion while it was changing or to attempt to prestore it for a long enough period of time so that further change during the experiment could be neglected. The latter alternative was chosen. Thus, the majority of the plates used in the experiments reported here were subject to the following storage procedure. Upon receipt from Kodak Research Laboratories (transportation involved several days at unknown temperature conditions) the boxes of twelve 4" x 10" plates were stored in the refrigerator at about 8~C until needed. Exactly one week before the beginning of the experiment the boxes containing the three plates to be used where placed in the darkroom (maintained at a constant temperature, usually 32~ ~ 10C) in which the exposure was to

65 be made. The plates were allowed to warm up (about 1-1/2 hr) before opening the boxes, in order to prevent fogging which would be caused by condensation of moisture on cold plates. The plates to be used were then removed from the boxes in complete darkness, labeled with tungsten carbide pencil according to experiment number, and inserted carefully into empty labeled 4" x 10" x 1/2" double cardboard box containers.* A red "Series 1" safe light was used for a visual check only after the plates were safely in the light-tight containers or apparatus.** One week later the plate holders and the step tablets were carefully checked in the red light. Then in complete darkness each of the three plates was removed from the double cardboard containers and carefully inserted into the appropriate plate holder of the apparatus. Before the back piece was put on the plate holder and the plate holder inserted into the apparatus, a final check was made in darkness to be sure the plate was in its extreme left position (later to be slid about 1/4 in. to the right to expose the calibration columns). Next, the shutter at the light source was opened, the timing circuit set to zero, the slide-control mechanism placed in starting configuration, and the darkroom sealed. The record of the light intensity was then started and an ordinary clock timer was set in such a way as to start the timing circuit. The operation of the apparatus was then automatic*** until the completion of the timer program (usually) two days later. *These were empty containers in which 4" x 10" spectroscopic plates were originally packaged. **The pure silver bromide emulsions are insensitive to this light; however, bromoiodide emulsions occasionally used are slightly affected. ***See Chapter III, Section A-3.

66 Occasional checks were made to ascertain that the slides were moving according to schedule by observing the small neon indicators on the outside of the darkroom. These showed which microswitch the arm was depressing, hence indicating the position of the slides. Forty-five minutes after the mechanism completed the exposure (the shutters at the light source and the plate holders were then closed) the plate holders were taken out and the step tablets removed from between the emulsion and the front of the plate holders. The plates were then reinserted, but this time placed in the extreme right position ready to receive the calibration exposure (see Fig. 3-4). The three plate holders were then reinserted in their proper positions. The timing circuit was set to zero in preparation for the next experiment and then disconnected to eliminate interference with the calibration exposure. The slides were then opened by manually starting the motor (switch S of Fig. 3.5b). The calibration exposure was started by manually opening the light-source shutter. It was terminated by starting the motor in reverse, which instantly closed the lightsource shutter and then the slides. The time for this calibration exposure was me asured with a stop watch. The plates were then removed and replaced into their labeled cardboard containers, where they were stored* until they were developed 50 (+l) hr later. In the interim the next set of plates (having been stored two days later than the former) were run through the same cycle. This program produced nine plates per six-day week when the maximum exposure was 34 hr, 8 min (211 min = 2048 min). *It has been shown that more consistent results can be obtained by delaying development for a time equal to the duration of the longest exposure which occurred during the experiment.

67 3. General Finishing Procedure. —Development of the plates was done in a temperature-controlled (21~C) darkroom, using rocker agitation at a frequency of about 1 oscillation per 2 sec. Brush development was abandoned since control experiments showed that the rocker method produced equally good uniformity and was much simpler. After developing (which will be described below) and fixing** by rocker agitation, the plates were washed with running tap water. They were dried in a horizontal position at a slow rate (about three hours) and protected from drafts (following 44 Clark's suggestions for obtaining uniformity). About one day after the drying process the density of the plates was read using the VincentSawyer microphotometer. The average density for each element of the array of exposures was obtained by scanning the area of the element. 4. Development Procedure.-It has long been known that the internal latent image and the surface latent image display different properties. Therefore, more easily interpreted results might be expected with the use of either surface or internal development alone. Since most developers in use today develop to a variable and unknown extent both the surface and internal images, special developers were used. a. Surface Development. For development of the surface latent image only, the silver halide solvent (usually sodium sulfite) must be eliminated Kindly made available through the cooperation of Professor R. A. Wolfe. **Plates were fixed 5 min in Kodak X-Ray Rapid or F-6 Fixer. Controlled experiments showed no difference between plates fixed by either fixer. When plates were fixed and dried properly no change in density was observed among readings of the same plates made one hour, one day, one month, or one year after drying. Litera~ture on the subject of internal and surface development can be obtained from articles by Stevens, Marriage, and Berg415,6 and James, Vanselow, and Quirk.47

68 from the formulae. A simple developer, denoted by SW/2, using glycine as a developing agent was first used. Later this was replaced with a developer recently recommended by James, Vanselow, and Quirk.47 Bridging experiments were conducted which showed the relationship between the results using these two surface developers. This developer had two advantages over the SW/2; (1) it was more stable during a time of the order of the development time and consequently gave more reproducible results, and (2) by a slight modification it could be used as an internal developer. Following James, this primary developer will be designated by M-AA-lo* For surface development, the exposed plates (still in their labeled containers) were removed from storage at exposure temperature and taken to the darkroom (210C)o The SW/2 was then mixed for fifteen minutes in the rocking trays; 500 cc of fresh developer were used per plate developed. The development proceeded for eight minutes (fifteen minutes when M-AA-1 was used), followed by 1/2 minute in a 5% acetic acid stop bath. b. Internal Development. To develop the internal image the surface image was first, destroyed by a fivre-minute bath in a weak aqueous solution of potassium ferricyanide (3 grams/liter). The plates were then washed thoroughly for 2 minutes, 4 minutes, and 8 minutes in three successive baths of 500 cc of distilled water per plate. The plates were then de*Formula for this developer is the same as that used by Webb.29 The reciprocity results for T-4139 and T-4140 developed with M-AA-1 compared with results of the same emulsion developed with SW/2 are shown in Figs. 4.3a and b. M-AA-1 tends to develop to higher densities, yet produces less fog. Reciprocity behavior is only slightly different, displaying most difference in the very finest grained emulsions. *Since the emulsion volume is about (.03 cm x 4 x 10 in.2 x 6.45 cm2/in.2) X7~5 cm3, this would constitute a dilution of 500 x 500 x 500 in the ferricyanide solution, if equilibrium is attained in each bath.

69,, sOr A L[Plate# 74-1 (Experiment 74-1) M-AA-I (15min) X —-X developed with plate #74-2 / (Composite, see fig.4.1b)SW/2 ~~~~~~~4 - - Xt,>a/~~ ~ (8min) Cr ~ 0 x S-I w (D ~~~~~~~~-X -10 -8 -6 -4 2 Plate# 74- 2 (Experiment 74-2)M-AA-1(15min) x —-X developed with 7plate-F 74-1 4 (Composite, see fig.4.la) SW/2 (8 5/ ~ min.) /' n3 -I -10 -8 -6 -4 -2 0 LOG2 INTENSITY -so

70 veloped for fifteen minutes in 500 cc (per plate) of stock MA-A-1 developer to which three grams per liter of sodium thiosulfate (hypo) had been added. The hypo acted as a solvent of the silver bromide, thus laying bare the latent image under the surface of the silver bromide grain. Control experiments using 1.5 grams, 3 grams, and 4.5 grams of hypo per liter demonstrated that the resulting reciprocity behavior over this range of hypo concentration was essentially the same. After this development the procedure was the same as that used for the surface image. D. REPRESENTATION OF DATA The procedure used for converting the density readings into isodense reciprocity-law-failure curves will be illustrated with the data for a typical silver bromide emulsion (T-4140). Table 4-III is a copy of the data sheet of readings from the microphotometer. Density values were always read as density above average fog density on the plate. The readings of the calibration strips are shown in between the readings of the reciprocitylaw-failure exposure areas. The underscored numbers above the density readings are the "corrected" densities obtained in the following mannero 1. Correction for Nonuniformity over Surface of Plate.-It was assumed that variations in calibration density at different regions of the plate were due primarily to variations in the number of grains per unit area of the emulsion coating. The validation of this assumption will be considered in the next section. Therefore, the corrected density of the reciprocity-law-failure elements should be given by the relation Dc(t,x) Dr(t,x) cr(t,x) where

71 TABLE 4-III DATA SHEET FOR EXPERIMENT 67-1 Microphotometer Readings of Density (Compare with Fig. 3.2a) Log2 Time of Intensity Steps Exposure 0 1 2 3 4 5 6 7 8 9 10 Exposure 1.12 1.12 1.13 1.12 1.12 1.12 1.12 1.12.291.605 1.15 1.91 275 i 2.15 Extra readings due to 11.322.668 1.28 2.10 3 2.35 added filter. 1.09 1.09 1.09 1.08 1.07 1.07 1.06 1.09 (See Appendix A, Table A-VII).130.334.749 1.44 2.4 2.75 10.141.360.811 1.55 2.56 2.90 1.08 1.07 1.08 1.08 1.07 1.06 1.05 1.04 1.06 1.05.044.143.385.89 -7 2.8 9.047.152.413.960 1.80 3 3 l.o6 1.06 1.07 1.08 1.09 1.08 1.06 1.04.007.045.168.466 1.04 1.90 2.83 L 2.28 8.007.048.18o.503 1.1 2.05 3 2.30 1.06 1.06 1.07 1.08 1.08 1.08 1.063 1.05.007.0587.195.54 1.16 2.09 2.55 7.007.063.210 5.79 1.25 2.20 2.65 1.05 1.08 1.08 1.07 1.07 1.05 1.04 \ 1.03.018.o68.221.59 1.31 2.42 6.019.073.237.631 1.38 2.50 1.07 1.07 1.08 1.07 1.06 1.03 1.01 Calibration.020.0715.246.653 1.42 2.52 -- Corrected density strip.022.077.263.692 1.47 2.55 - Original density readings readings 1.09 1.08 1.07 1.06 1.04 1.02.990.015.075.274.715 1.50 2.6 4.016.081.289.740 1.52 2.55 1.08 1.06 1.05 1.03 1.01.980.962.009.081 297.782 1.56 2.67 3.009.084.301.782 1.52 2.55 1.04 1.03 1.00.992.962.950.007.088.329.827 1.61 2 Fog Density =.116 to.120.007.087.321.787 1.50 Zero set at.118 1.00.982.965.941.918.009.103.352.872.99 Calibration average.009.099.327.789 taken as 1.0.972.955.922.898.016 117.386 -.025.015.108.343

72 Dr(t,x) = the averaged density as read for the area of the element of the array at (tx), Dc(t,x) = the corrected density for the element of the array at (t,x), c = the average density of all calibration strips, and c(t,x) = the averaged density of the calibration strips in the region adjacent to the element (t,x) of the array. The corrected densities were then converted into corresponding Sidel values S by S = log10 (l0De 1) The purpose of using the quantity S rather than D is to obtain straighter characteristic curves. 2. Plotting the Characteristic Curves. -These Sidel values were then plotted as a function of the log2 intensity, using the corrected intensil ties (described in Appendix A). When the resulting points from more than one experiment were sufficiently close, they were plotted together. The family of smooth curves corresponding to the multiple set of points will be referred to as the composite characteristic curves. To illustrate this process, Table 4-IV shows Sidel values corresponding to the corrected densities of Table 4-III. These Sidel values from experiment 67-1 are shown as dots in Fig. 404; the circles represent results from experiment 70-3 which used the same emulsion under similar conditions. The results of the separate experiments match well, except for values of S < -.4~ This corresponds to a density of <.15, which is the region where variable background fog begins to disturb accurate density measurements. *A shift of about.07 log2 intensity units was made to give the best match of the most points. This small discrepancy is probably due to a difference in Io (denoted by xo in the legend) between the two experiments and/ or the different average coating thickness of the plates used.

73 TABLE 4-IV SIDEL FUNCTIONS FOR EXPERIMENT 67-1 SHOWN WITH CORRECTED DENSITIES FROM WHICH THEY WERE DERIVED (See Table 4-III) Log2 Time of Intensity Steps 0 1 2 3 4 5 6 7 8 9 10 Exposure.291.605 1.15 1.91 2.75 2.13 Extra readings due to 11 -.026.85 1.11 1.91 2.75 2.13 added filter. (See Appendix A, Table A-VII).130-.334.749 1.44 2.4 275 10 -.46.135.665 1.42 2.4 j 2.75.044.143.385.89 1.67 2.8 [ 9 -.97 -.41.155.83 1.67 2.8.007.045.168.466 1.04 1.90 2.83 2.28 8 -.96 -.33.285 1.00 1.90 2.83 2.28.007.0587.195.54 1.16 2.09 2.55 7 -.84 -.25.40 1.12 2.09 2.55.018.068.221.59 1.31 2.42 6 -.77 -.18.46 1.28 2.42.020.0715.246.653 1.42 2.52 5 -1.32 -.745 -.12.545 1.40 2.52.015.075.274.715 1.50 2.6 4 -.72 -.055.62 1.5 2.6.oo009.081.297.782 1.56 2.67 3 -.685 -.01.70 1.56 2.67.007.088.329.827 1.61 2 Fog Density =.116 to.120 -.65.05.75 1.61 Zero set at.118.009.103.352.872.99 Calibration average -.57.10.815 taken as 1.0.016.117.386 -.025 -1.45 -.51.155 *Densities below.02 (or S - 1.3) were ordinarily not plotted.

67-1 T4140 SW/2 developed with 67-2,3 t'l 1.00 0 0 t400 t3. ~ T=32 5+5 +500 0 t=7.00 t=6.00 t= 5.00 X0 =~~~~.I1~6 t tI.00 to 8.00 Fog density=.116-1.20 Calibration density = 1.00 ~. I 70-3 T4140 SW/2 developed with 70-2 21 ~~ T= 32.5 +_.5 Xo:.01 Fog density =.089-.097 Cal i bration density= 1.00 1+.1 -- Z —-i Sidel values when density not corrected LL6 o0t / // t= 2.00 t =.99 21 - 4 -12 -0 -8 -6 -4 - 2 LOG2 INTENSITY =x Fig. 4.4. Characteristic curves for emulsion T-4140 from exlperiments 67-1 and 70-3 (surface development)O

75 The similarity of the results from these two experiments constitutes the main justification of the method (described on p. 70) of making corrections for nonuniformity over the surface of the plate. To illustrate this, note that in the data sheet for experiment 67-1 (Table 4-III) the calibration strips indicate higher densities* (for the same exposure) at the low-intensity end of the plate. On the other hand, calibration strips for experiment 70-3 (data sheet not shown) show the opposite tendency, i.e., the higher densities of calibration strips occur at the high-intensity end of the plate. Such effects have been noted in many plates. Therefore, if the results of the two experiments match after the corrections have been made in the manner described, they will not match if the corrections are omitted. This is illustrated graphically in Fig. 4.4 where the Sidelvalues corresponding to densities before correction are plotted for t = 11 and t = 1 in the region near S = 0. 3. Constructing the Reciprocity-Law-Failure Curves. -Finally the reciprocity curves (shown in Fig. 4.5) are constructed from the composite characteristic curves simply by plotting the log2 exposure (t + x) required to produce a given Sidel value at various times of exposure. (Note again the points corresponding to experiments 70-3 and 67-1 when corrections of density were omitted. It is clear that 70-3 would display considerably more reciprocity failure than 67-1 if the correction were omitted.) Analogous to Fig. 4.4, Fig. 4.6 shows the composite curve obtained from two experiments using internal development instead of surface development. The reciprocity-law-failure curves obtained from Fig. 4.6 are shown in Fig. 4.7. Results of analogous experiments but at 38~C are seen These variations in plate sensitivity range from less than 2% for commercial type-33 plates to as large as 20% for pure silver bromide emulsions.

0~~~~~~~~~~~ N 9,~~~~~~~~~~~~~~~N 0 S=2 21 D0 0 1Nte ad nS= uvedntepit S=O ~~~~~~~~~~~~~~~~~~~ j~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 resulting when density correction was omitted -2 (for exp 70-3 and 67j1,respectively) \ q., -10 ~~~-8 -6 -4 -20 x=LOG2 INTENSITY Fig. 4i.5. Example of reciprocity-law-failure curves for TJ-414O (surface development) constructed from composite curves shown in Fig. 4 —4.k

72-2 T-4140 internal development with 72-3 xo=.Ol 0 ~~~~Xo,~~~~~~~~~~~='.~~001 o T= 32,70 c Fog density=.113-.127 tll t=l t=9 t=8 t= 7 Calibration density=.57 89-1 T-4140 internal development with 89-2 2 x0 =./01 — l.T =350c Fog density=.085-.091 t=5 0_0~0 0 t=. Calculation density 2.50 0 U: 0~~~~o o 00 0~~~~~io -2 -2 0 2 4 6 8 10 _I~ I LOG2 INTENSITY=x Fig 4.6L. Characteristic curves for emulsion T-4140 from experiments 72-2 and 89-1 (internal development).

9t' ~ S=2 9' +2 x LU 1 S=.225 =.015 C) 0 ax S=l (.9 0 ~U (' I >0S=.5 S=.:75+'.01 S=0 -2r~~~~P 9''4 o' 2 JO(~~~~~~~~~~~~~~~~ f" =f5 -13 — 10 -8 -6 -4 -2 x=LOG2 INTENSITY Fig. 4.7. Example of reciprocity-law-failure curves for T-4140 constructed from composite curves of Fig. 4.6,

79 in Figs. 4.8a and b. It will be noted that the discrepancy between the two experiments is greater than that for the corresponding experiments using surface development. This is characteristic of internal development. Irregularities in calibration density and fog density are more pronounced, and background fog density is considerably higher when internal development is used. E. REMARKS CONCERNING VALIDITY OF EXPERIMENTS A number of factors influencing these isodense reciprocity results will now be considered, since these curves are extended to intensities lower (and larger times of exposure) than normally used and since the results differ in some respects from usual reciprocity results. For example, the reciprocity curves of the experiments using surface development differ distinctly from the catenary, to which experimental points are usually fitted. 1. Temperature and Humidity Control — The sensitivity of photographic emulsions is known to be dependent on both temperature and humidity (see Webb48 and Mees,l pp. 127 and 253). Since long exposures must occur at different average times than short exposures, temperature or humidity fluctuations during the exposure can produce erroneous results. For example, suppose the temperature were higher during a short exposure than during a long exposure. The efficiency of the photographic process would then appear to be higher for the short exposure. This would make the low reciprocity failure appear greater than it actually is. Although the control of temperature and humidity during the experiments (during both storage and exposure) has already been described in Chapter III, it should be mentioned that sudden changes in weather often caused temperature and humidity fluctuations which were not adjusted by

8o 80 T=2 T=I T=O T=6 1=5 T=4 T=3 -2 I l l -10 -8 -6 -4 -2 0 LOG2 INTENSITY Fig. 4.8a. Characteristic curves for T-4141 emulsion exposed and stored at 380C (the amount of the dip in the higher curves is very unstable). 7 - I L I T ITX? 2 _2 X. S=-I. Fig. 4.8b. Reciprocity curves derived from characteristic curves of Fig. 4e.8a.

81 the control system used. The results from such experiments, however, were used only if their resulting characteristic curves were sufficiently similar to those from experiments run under normal conditions. Hence, errors due to temperature and humidity change during storage and exposure are effectively eliminated from the consideration of these results. 20 Errors in Operation of Timing Mechanism. —The starting time for each of the exposures of an experiment was usually observed by means of the external neon indicator lights. If for any reason the slide failed to move at the proper time (because of low line voltage, dirty relay contacts, etc.), the slide-driving motor could be started manually and the time of movement recorded. Except for such a failure, which was extremely rare, the movement of the slide could be predicted to within a second, even for exposures extending over a period of several days. Thus the probability of a hidden error in times of exposure was negligible and can be effectively eliminated from consideration. 3. Errors in Intensity.- a. Fluctuations of intensity of the light source during the time of the exposure were measured by comparing the light-source intensity with that of a standard source (see Chapter III). Control experiments were conducted to assure that the P radiation from the tritium activated standard source produced no increase in developable density. Although corrections could be made for temporal fluctuations in light intensity, this was seldom necessary since the intensity rarely varied by more than 1% (about.01 log2 units) during the experiments. Thus, errors due to temporal fluctuations in intensity are negligible. bo The spatial variations in light intensity over the area of the plate holders was checked by scanning the plate-holder area with a photos

82 multiplier tube. No variation greater than 2% was discovered. A test of the uniformity of illumination was also made by exposing Kodak types33 (which has uniform sensitivity over the plate) and Kodak SA-1 (which is high-contrast emulsion) plates held in position without the step, tablet. By systematic exposure of the plates with various orientations (including cutting the plate and interchanging the middle and the end), it was possible to separate the variations of illumination from variations of plate sensitivity. These tests demonstrated also that the illumination field was constant within 2% at the plate holders c. The resultant intensity through the step tablet in situ is treated in Appendix A. That this calibration is essentially correct is verified by how closely the characteristic curves of plates treated alike match when plotted together, using the resulting intensities from Tables A-VII, -VIII, and -IX in Appendix A (see IFigs. 4.4 and 4D6). The reason for occasional lower densities in the border elements of the intensity is not understood. d. The effective light inte nl ity varies with emulsion depth and grain size. Since lighlt is absorbed al- it penetrates into the emulsion, the grains at the bottom of the emuSonion layer will receive less radiation than those at the top. This variation was estimated* by measuring the amount of light transmintted through the thickness of a, representative sample of each emulsion to be used.`The results are shown in Table 4-Vo Note that the intensity is reduced by a factor of about 3 to 16 (i.e e, a log2 value of -1.6 to [4), The grains in an emulsion are subject to a variation in effective *The emulsion thickness varies as much as 50%, hence variations in intent sity with depth cannot be completely described unless the thickness is specified.

TABLE 4-V ROUGH MEASUREMENT OF ABSORPTION COEFFICIENT c~ FOR SILVER BROMIDE EMULSION Emulsion Transmission = Thickness t -Loge C in Units of Number I/Io in Microns Transmission 102/cm T-4139.06 +.005 30 + 3 2.8 ~.1 9.4 + 1 T-4140.08 +.oo005 20 +3 2.52 +.08 12.6 + 2 T-4141.15 +.01 27 +.3 1.90 +.08 7.0 1.18 +.o1 18 + 3 1.72.08 9.6 2 T-4142 (.10o.01o) (25 1.3) (2.3 +.1 (9.2 +2) T-4143.29 +.01 20 +.3 1.24 +.08 6.2 +1 If Io = light incident on emulsion sample, and I = light transmitted through emulsion sample, -ct then a is defined by I = Io e absorbed intensities due to variations in their cross-sectional area. (The thickness of grains of pure silver bromide emulsions are effectively constant over a large range.) Details of the size-frequency distribution curves for emulsions used here were not determined. However, the main features of the distribution can be found for this type of emulsion in the articles by Trivelli and Smith.38'39 Katz33 has shown that the effect of spread in intensity (such as that due to the depth of grain in the emulsion or the size of the grain) is to reduce the curvature of the reciprocity-law-failure curves (see Chapter II). 4. Influence of Periodicity of Light-Source Intensity.-The commercial a-c mercury arc lamp used as a light source for these experiments produced a periodic light intensity. Periodic intensity variations were observed with a cathode-ray oscilloscope. This showed that the intensity as a function of time can be described as 1(t) = T(1 +.74 cos o0t) where cu =

84 2i x 120 radians/sec and I is the average intensity. The model discussed in Chapter II assumed a continuous light source. Consequently, a set of exposures was made using a d-c supply to the lamp (since the lamp was designed for use with a-c supply and deteriorated rapidly with d-c, the polarity was changed several times during the exposure and the lamp voltage adjusted frequently to maintain constant intensity, within 2%). A type33 plate and two pure silver bromide plates were used in this experiment. The resulting curves were the same within experimental uncertainty as those obtained with the a-c source. A calculation for the effect of a periodic intensity on the reciprocitylaw-failure curves is given in Appendix B. For the low intensity, frequency, and the wave form of light used in these experiments, the effect on the grain is shown to be the same as that of constant intensity. 5. Uncertainty in Wavelength. — According to specifications supplied by the manufacturer, the Corning filter No. 5-74 (which is designed to pass 436 my radiation from a mercury arc source) transmits negligible radiation 0 O except from 4000 A to 4800 A. The only strong lines in this region of the mercury arc spectrum have wavelengths of 4339 A], 4347' A, and 4358 A, with relative strength of about 150, 200, and 3000, respectively (see M.I.T, wavelength table, Reference 49). From information supplied by the lamp manufacturers the 4358 A line appears to spread out (probably from pressure broadening), obscuring the other two lines. Therefore, the wavelength of 0 0 the radiation should be designated as 4350 ~ 10 A rather than 4358 A. The fine structure of this radiation was not investigated further. However, the source was observed by use of a hand spectrometer, and no radiation except the strong blue (435 my) line was detected. A change of 10 A at 435 my produces a shift in the reciprocity-law-failure curve (for the bromo

85 iodide emulsions used by Webb50,51) of about.04 log2 units along the lines of constant time. This shift is negligable in comparison with that caused by the difference in intensity between the bottom and top of the emulsion layer. 6. Variations Arising from Development Conditions.-Considerable care was taken to insure the use of fresh developer in order to reduce the number of unknown factors in the development process. For example, SW/2 was always mixed fifteen minutes before use, and MA-A-1 (which is quite stable) was kept at 8~C until one hour before use, then warmed to 21~C. For internal development the hypo was always added fifteen minutes before beginning the bleaching process. The procedure finally adopted (Section C-4 above) resulted in good reproducibility using SW/2 developer, excellent reproducibility using MA-A-l, and usually good reproducibility when i.nternal development was used. As was previously mentioned, the results of the internal development were insensitive to hypo concentration for the concentrations and development times used. The work of Professor Hautot and his collaborators at Liege (reviewed by Berg 52 indicates that results are more critically dependent on concentration of the silver solvent in the bleach bath than on the time of the treatment. Therefore, a standard stock solution made up in quantities sufficient for an entire program of experiments, and the last experiment in the group was checked against the first to show that no change had occurred due to deterioration of the silver solvents. Control experiments where variations of as much as 20% from standard development time were tried revealed no change in the shape or slope of

86 the isodense reciprocity curves for densities less than 2 in the T-4139 emulsion and for densities less than 1 in the T-4143 emulsion. 7. Errors in Measurement of the Developed Density. —The VincentSawyer microphotometer measures the specular rather than the diffuse density. The shape of the characteristic curves will therefore differ from those obtained by investigators who have measured diffuse density. However, if the order principle (see Chapter II) is valid, with grain size a principal factor determining the order, and if the density measured here is a monotonic function of the diffuse density, then the shape of the reciprocitylaw-failure curve obtained from characteristic curves using specular densities will not differ significantly from those obtained from measurement of diffuse density (except that the number chosen to designate the constant density of the curve will be too high). The unexposed areas of the plate were always masked to prevent the scattering of light into the detector from adjacent areas. This increased the validity of the reading of the higher densities (above D = 1) so that the densities could be read to within about 1% for D < 1, 2% for D < 2, and 4% for D < 3. Low densities become difficult to measure accurately when they approached the same order of magnitude as the variations in background fog density. If after "zeroing' the microphotometer on the average background fog the range of densities considered is limited from.05 to 1 for surface-developed plates and from.1 to 1 for internal-developed plates, the diffused density can be measured to within 1%. 8. Change of Emulsion Characteristics with Time o-It has long been *During the study of these effects it was noted that the slopes of the reciprocity curves changed less than 2% during change of as much as 50% in y (the slope of the characteristic curve of the emulsion).

87 recognized that characteristics of photographic emulsions change with time and that the change occurs more rapidly at elevated temperatures. Not only do the characteristics change before exposure, but also the density produced by a given exposure is dependent on the time between exposure and development. The post-exposure changes may vary from one emulsion to another. Following the procedure suggested by experienced workers in the field, the apparatus was designed so that all exposures ended at the same time, and development was delayed for a time approximately equal to the maximum exposure of the experiment. Preliminary experiments showed no reason to depart from this accepted procedure. However, it should be emphasized that the results quoted here apply specifically to the behavior of the latent image present in the photographic grain two days after the completion of the exposure. The pre-exposure changes are, in general, an increase in fog, an increase in sensitivity,* and a decrease in the reciprocity failure at low intensity. Since we were inot aware of any precedent concerning the treatment of the photographic emulsions before exposure (exzcept to keep them in cold storage until used), a brief preliminary investigation was undertaken to ascertain the procedure to be followed. It was discovered that the pure silver bromide emulsions change much more rapidly than most ordinary emulsions ** The procedures outlined in *Sensitivity here means the reciprocal of the log of the exposure required to produce a given density under normal exposure and development conditions. Fog level for these emulsions may increase from.07 to.12, sensitivity increase by one (log2 unit), and the reciprocity-law-failure slope decrease by several percent after one month at 21~C (controlled darkroom temperature).

88 Section C-2 were adopted as a compromise between a long prestorage time, which might introduce high background fog, and short prestorage time, which might leave the emulsion in a state such that it would continue to change during the exposure. A systematic study of changes in emulsion characteristics before exposure was not undertaken. However, when some of the preliminary experiments (where prestorage conditions were not maintained constant but had been recorded) were reconsidered, the following effects were noticed: a. The sensitivity of the T-4139 emulsion for low intensities was increased by as much as one log2 unit after storage for one week at the exposure temperature. This effect alone could cause considerable distortion of the reciprocity curves (see page 89). b. Prolonged prestorage inevitably reduced the slope and curvature of the reciprocity curves for T-4139 emulsions. Figure 4.9 shows the reciprocity curves for T-4139 (surface developed) prestored 49 days and poststored 14 days, all at 37~C, and is typical of the behavior of this emulsion under long storage conditions.'4 C) t=10 or 210 min S=l Uw 12 0 XI i2 Regular storage (Iweek.pre-storage, / —- 2 days post-storage) 49 days pre-storage, 15 days post-storage. Exp 87-1 l I I -10 -8 -6 -4 -2 0 x = LOG2 INTENSITY Fig. 4.9. Effect of storage on reciprocity curves (surface development).

89 c, These changes seemed to be most pronounced in the emulsion of smallest average grain size, T-4139 or T-4140. Figures 4.10a and b illustrate how a real reciprocity curve of -slope -.5 could be distorted by a change in sensitivy during the time of the exposure. For simplicity, the exposure required to produce the constant density was assumed to decrease linearly with time. The ideal reciprocity curves would then be displaced parallel along the lines of equal time as shown in the Fig. 4.10. The measured curve would then be found by selection of the points of various times of exposure from their ideal reciprocity curves. Figure 4.10a illustrates the shape of the measured curve with all exposures ending together (regular order), while Fig. 410b shows the shape resulting if all exposures began together (reverse order). In order to determine whether such a change in the emulsions did occur during the exposure, three plates were exposed simultaneously but in reverse order. Two T-4140 plates were exposed together and treated identically, except that the first plate was stored in the regular manner (for one week) while the second was prestored for only six hours. The third plate, T-4139, was also stored for only six hours before exposure. The resulting reciprocity curves from the first plate (regular storage but all exposures started together) could not be distinguished (within experimental uncertainty) from the curves corresponding to the same density from regular experiments (when all exposures ended together). The interpretation of this result was that no significant change in sensitivity of this emulsion occurred during the exposure. The curves from the second and third plates (corresponding to the same density) were, however, significantly steeper than those from the first. This was interpreted to mean that the emulsions had not been sufficiently prestored andwere still changing during the exposure.

9o t l0 or 210 min Ideal r I f curves associated with sensitivity of emulsion n=7 n=10 during the f=2" exposures n=6 w -2 (U) 0 L&J -4 -6 -10I -8 -6 -4 -2 x=LOG2 INTENSITY Fig. 4.lOa. Possible distortion of reciprocity curves from change in sensitivity during exposure (regular order). O; _ ml 10or~ / Ideolrf curves n=8 f=2" exposures -6 -10 -8 -6 -4 -2 Fig. 4e Ob, Pojblble distortion of reciprocity c-rv~s fror 0 -10 cangr exposure ese e

91 Similar experiments conducted at 380C indicated that little change in sensitivity during a two-day exposure occurred after one week of prestorage at that temperature. Separate tests were not done for internal development; the storage procedure was simply kept the same for the two types of development. 9. Summary and Conclusion in Regard to Validity of Results.-o-The foregoing consideration of individual factors which influenced the results of these experiments indicates that the main source of ambiguity results from significant changes in the sensitivity and reciprocity behavior of these pure silver bromide emulsions with timeat room temperature and above. However, evidence has been presented to show that the storage procedure adopted was adequate to assure sufficient stability during the experiment (at least with regard to the surface latent image). Thus, within the uncertainty quoted the results describe the reciprocity behavior of pure silver bromide emulsions (2 days after exposure) which have been cured for one week at the exposure temperature. It might also be noted that many combinations of plates and development were exposed simultaneously. For example, Plate No. Emulsion No. Development 1 T 4139 t Internal, developed together 2 T-4141 3 T-4139 Surface In this way any pecularity wth regard to operation of apparatus or storage procedure could in principle be discovered from similar irregularities in all three of the plates.

CHAPTER V DISCUSSION OF CONTINUOUS AND TWO-TRAP MODELS AND RELATION TO EXPERIMENTS Before discussing an interpretation of the experimental results given in the preceding chapter, some extension and refinement of the mathematical model outlined in Chapter II will.1 be considered for the case of two discrete trap depths. The shape of the low-intensity reciprocity curves can be predicted in terms of the trap depths, their relative abundance, and the recombination probability 7. The reciprocity curves of Ys* = log(E Eo) vs x = log aI (where a is the effective grain area) will be shown for several different choices of these parameters. Although experimental curves are given in terms of log E vs log I, the change to this form is trivial if Eo is known. Also, for decreasing intensities the c'r.ves become increasingly independent of Eo, The m-,thod used for the two-trap model is sufficiently simple so that also for 1d discrete trap depths- the form of W(I) can be written immediately, and although the resulting lo'g(EEEo) is not solved explicitly, some gene oraal conclusionrs,,;a:i be obtained simply. The case of a continiuous di.tribution m(U) wdill be dscussped further, and it will be shown tlhat when'the proper limit (instead of the approximations of Chapter II) are used, the limiting slope r CL t ot9ise wiL-l also be -1 after an extended region where the slope will be very nearly -kT/e. A. TWO-TRAP MODEL 1. Derivation of the Reciprocity-Law-Failure Curves.~-Suppose that The subscript s is used to suggest stable"' since E-Eo = Es is the exposure required to form a stable sublatent image~ 92

93 the traps for the electrons in a crystal can be separated into two classes, shallow traps of depth U1 and deep traps of depth U2, below the conduction band. The shallow traps might, for example, be due to mechanical imperfections and the deep traps due to added sulfur sensitizer. In order to apply this simplified description, it is assumed that AUi << U1 - U2 and AUi << zT (i = 1 or 2), where AUi is the spread of trap depths around some average Ui. The trapping mechanism would then be described in terms of the trap depths Ui, the fraction ri of number of traps of depth Ui and the recombination y = 1 - o (defined in Chapter II, p. 23). Under these conditions the integral differential equation which expresses ni(t,y) [where ni(t;y)dUi is the fraction of ensemble grains having electrons in traps of depths between Ui and Ui + dUi] in terms of t and 2, reduces to the following simultaneous linear differential equations: dn2 dtn (t,y) = -klnl(t,y) + or [xlnl(t,y) + k2n2(t,y)l dn2 (ty) = -%2n2(t,7) + Cr2[1nl (t,7) + %2n2(t,7)j,J where X1 = v e-Ui/kT as in Chapter II. The initial conditions are n,(O,) = rl (5.1a) n2(O,zY) = rs o Since a final solution in terms of F(t) = nl + n2 is much more easily obtained than a solution in terms of nj + n2 separately, it will be convenient to transform to new variables: u = nl + n2 = F(t) and V = n1 - n2

94 Then Equations 5.1 become d t+ % - arz + u( + 2) + 1 - arl(,l - k V = 0 [di + 2 - Cr2(X + 2) u +[ d k - 2 r2(X1 - 2) V = 0 o The general solution of this system for u is F(t) = u = a ekt + a2-k2t and, incidentally, for v is v = b ekt + b2 ek2t where kl and k2 are solutions of k2 - [1(l - rl) + 2(1 - ar2)] k + X1X2(1-) = 0. (5.4) Using W = J F(G) P(G)dG 0 with P(Q) = e/ d/ and writing = Ia = 1/=, W can immediately be expressed as W(rl) - a - + a2 a k Wh T+kl Tj+k2 (ai +a2 ) + (ajk2+a2k1)' + (k,+k2)h + kjk2 Thus, using (E-Eo)W(n,) = m, a constant, and omitting the log ['which simThe evaluation of kl+k2 and klk2 in terms of k1, k2, rl, r2, and ~ follows directly from Equation 5.4, while a2k1 + alk2 is found easily by use of the initial conditions: fx(O) = a1 + a2 from Equation 5.3 x(O) = n1(O) + n2(0) = r1 + r2 = 1 Qx(O) = alkl + a2k2 from Equation 5 3 (0) = il(O) + A2() = = klrl+arl[ir1+X2r2] - k2r2 + ar2[klrl+k2r2] then: alk2 + a2kl = k1 + k2 - (alkl + a2k2) = klr2 + kX2rl

95 ply constitutes a shift of the complete curves in the log(E-Eo) direction], ys(T;xi,riy) = log(E-Eo) = log(l + kl/n) + log(l + k2/r) (5.5) - log [1 + (rlk 2 + r2kl)/n] Ys(71Xi;riw) = log(E-Eo) = log r2+ [ (1-arl ) 1+(l- ar )2 l+l2 ( 1- a ) 12 + (rlk2 + r2%l)I where kl and k2 are the solutions of Equation 5.4, namely 2kl = (l- rlr) + a2(1 -ar2)+l[k l-ar) + A2(1-ar2)] - 4\lXa(l-) 2k2 =1l(l-ar1) + 2(1-ar2)- - [Xk(l1arl) + X2(l-ar2) ] - 4Xla2(1-a) Equation 5.5 is convenient for constructing the curves of ys vs x = log r = log I + log a, since it gives Ys as the sum (and difference) of three identically shaped curves, which will be called generating curves, differing only by their position along the log i axis. It should also be noted that ys is a function of TR and the five parameters rL, r2, k%, %2Y and a; however, there are only three independent parameters, r! or r2 (since r1 + r2 = 1) and %2/X% =f [since the curves in log r1 and log(E-Eo) are determined only to within additive constants], log a and log. 20 General Properties of Reciprocity-Law-Failure Curves Derived from the Two-Trap Model. —By differentiation of (5.5)or (5.5a), the general form of the slope of the reciprocity-law-failure curves is obtained. Thus, dys dys X - (X-z)l2 + 2Yn + ZY dx drj ~3 + (X+Z)n2 + (xz+Y)71 + ZY where X = (1 - arl) X1 + (1 - 2)X2 Y = r12( 1 - a ) Z = Ka + rr1X2+

96 The physical problem is concerned only with 9 > 0, then since X-Z = (l-c)(klrl + k2r2) > 0, it follows that dy/dx < 0 (equal applies at r = 0o, i.e., x = 0o, or a - 1). Furthermore, since X-Z < X+Z and 2Y < (XZ+Y), dys > dx - (equal applies at O = 0, iLe., x = - ). In the intermediate region where r^ [kla%(1'-a)]1/2 the slope depends on r and parametrically on r2, f, and a. The qualitative behavior of the reciprocity curves will now be considered for several special cases. The main characteristics of the reciprocity behavior can be obtained most simply for the case ac = 0, and consequently most of the discussion will be concerned with this case. If a = 0, Equation 5.5a reduces to ys(Sr2,f,o) = log (1 + to1) + log (1 + f-l) log [1 + (r2 + fr) ],(. where the change of variable from r to \ L is introduced to simplify the formalism. Here the influence of rr and f is easily illustrated, since the first two generating curves are independent of r2 (which would not be true for 0 < a < 1). a. Variations of r2 with f constant. Figures 5.1a and b show the effects of varying the relative number of deep traps. Figure 5.la illustrates the behavior for f = 2-15. At room temperatures this corresponds to a trap-depth difference U = -15 x ge 2 ev =.26 ev It is clear that the slope in the intermediate region decreases with the

97 16 r2 0 I-" 14 r2-= r =2-11 f f: 2- 16 _ r2 Variable 122 a 0 12,o - 20 w I 6 r= 420 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 z =LOG2 { Fig. 5.1ao Reciprocity curves of log (E-Eo) vs log (arbitrary units) for the two-trap model for difference between deep and shallow traps of No'26 ev. 4:~~'. r,,. r=O 12 - r2'2'- -.. ~ ~~ f: 2-6 % r2=2y r2 Variable 10 _r2'0 8 — r2 I W I~~~~~~~~r: w 6-J -204 -12 -0 -8 -6 -4 -2 0 2 2 z: LOG2 5 Fig~ 5.lbo Reciprocity curves of log (E-Eo) vs log ~ (arbitrary units) for the two-trap model for difference between trap depths of r4JlO ev.

98 addition of deep traps. This is in qualitative agreement with the observation that the slope of the low-intensity branch of the reciprocity curves decreases with increasing time of digestion, assuming that the U1 (shallow) traps are due to imperfections and the U2 (deep) traps are due to the added sulfur* (see Mees,l p. 204). Figure 5.lb illustrates the behavior for f = 2-6 which corresponds to AU-n.105 ev. This value was chosen since it corresponds to Webbts estimate of the difference between internal and surface traps (see footnote, p. 14), Note that for r2 = 2-2 there exists a nearly straight portion of the curve with a slope of about.3. b. Variation of f with r2 constant. Figures 5.2a and b show the effect of varying f at a fixed ratio of abundance of deep and shallow traps. In Fig. 5.2a (where the deep and shallow traps are present in equal abundance) there is a region associated with f N 2-5 where the curves tend to become straight for several log2 units. For smaller values of f the curves show two inflection points, and for increasingly smaller f a plateau develops at ys = 1, (log2 L/r2)o in Fig. 5.2b where the log2 ratio of the number of shallow to deep traps is 5, the plateau occurs at ys = 5. (Note that for f \ 2-5 the nearly straight section in the intermediate portion of the curve has a slope of about 7y ) c. a % 0 (qualitative remarks). If a > 0, the first two terms of Equation 5.5 are no longer independent of r2, and consequently the influence of f and r2 on the shape of the curves is more difficult to determine. In general, however, Xlr2 + ka2rl - kl - %E and 0 <_ k2 <I X2 as 1 _ o > 0. For a particular choice of r2 and f this means that the slope of the reciprocity curves will in general tend to decrease with increasing a until Some experiments suggested from this discussion will be mentioned at the end of the chapter.

99 12 f=l f 12~~~~= f Variable ~~~~~~iip 0=2~'0 f~' w 6 0 0 2-Io -,~ ~ 14 -'~r=LOG2'~lo ~ -,2 -6 o -8 -6 -22 20 ~~~~~~~~z —LOG?.2 Fig. 5.2a. Reciprocity curves of log (E-Eo) vs log 5 (arbitrarY units) for the twotraP modtel for equal number of deep and shallow traps w\'2i8 ~f Variable 12 = 2 10 10~ r2=1/32 -0 8 0 f=~1 U, -14 -l2- I0 -8 -6 -4 -2 00 -22 -20 -18 -16 -14 -12 -10 =LOG 8 - Fig. 5.2b. ReciprOCitY curves of log (E-E0) vs log i, (arbitrarY uis o the two..trap model for about one deep to every 52 shallOW traps.~

100 dys/dx a 0 when a = 1. This is physically reasonable since increasing a means that the probability of photoelectrons being lost by recombination decreases until none are lost when a = 1. Then the process becomes independent of the rate of quantum absorption (i.e., no reciprocity law failure). The analysis of the curves given below for a = O is not really a special case in the following sense. A particular curve determined by the parameters a = O, hX, X2, and r2 can also be described by another choice of a = a' > 0 with the proper choice of different 1n' and ha8 (and the same r2). This is possible since if ki(o = O, %1, a2, r2) with i = 1 or 2 are solutions of Equation 5.4, then ki'(1 > a' > O, Xi, X2, r2) can also be found which are solutions. This and the comparatively greater ease of describing the behavior are the justifications for restricting the following analysis to the case a = 0..o Analysis of Reciprocity Curves for f and r2 (a = O) —This analysis is given for the curves of ys = log2(E-Eo) vs log2, where 5 = ~/lThe values of f and r2 will be expressed in terms of measurable properties illustrated by sample curves. Figures 5.3a and b slow typ cai reciprocity curves in solid lines and three associated generating curtves in dashed lines. Curve I is the generating curve Yj(~- c) = log (1 + 1/la) where c is varied until "best fit" is obtained between Curve I a.-d the right branch of the reciprocity curve. Similarly Curve II is I"fitted" asymptotically at the extremes of both left and right branches by changing b in Y2(; b) = log (1 + 1/br). Curve III is shifted by changing g in Y3(5; bg) = log (1 + i/gb~) + log g until "best fit" is obtained for the left branch of the reciprocity curve. When Curves I, II, and III are drawn in this manner and a is assigned the value 1 [i.e.,

101 10 1 0 B= Log2 g - II 4 \ - 2 - z = Log2 t I/2 Logf -22 -20 -18 -16 -14 -12 -10 -8 6 -4 -2 0 2 z:LOG2; Fig. 5.3a. Sample reciprocity curve of two-trap model with f < 1. -8 LiW B (9 0 _J G z- Logao HI -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 z = LOG2C Fig. 5.3b. Sample reciprocity curve of two-trap model with f larger than in Fig. 5.3a.

102 for Curve I when log 0 = O, y5s(; 1) = 1], then G log2 g = - log r2 (r.6) B _ log b = log (r1 + r2/f) o (5-7) These determine, in principle, r2 and f in terms of the measurable quantities B and G since r2 2-G (5.6a) and 2-G f = 2B 1 + 2-G (57a) The following additional relationships are useful: The difference A2Ys(t) between the reciprocity curve and Curve II has a maximum separation at = f;/2 (5.8) The value of this maximum separation is b(l + f ( )c A 2Ys(o) = log (1 + bfl/2)2 (5s.9) and the slope of the two curves at Co is so = Ay(O) (l + r ( f/2 + / + r2f-1/2)-i - (1 + bf/2) (5.10) The usefulness of the above relationships is dependent on how critically the values of B, G, ZO, sO, and C are determined by the scheme described. For example, if f " r2, it is very difficult to decide where Curve III fits best, while if f ~ 1 and r2 ~ f, the fit of Curve II is much more difficult than that of Curve IIIL In this latter case ZO is quite ambig

103 uous, but so and C are more definite (see Figs. 5.1, 5.2, and 5-3). The fitting of Curve II is not difficult if sufficient lengths of the right and left branches are present. In order to see how closely and under what conditions Curves I and III can be fitted to the Ys(O) curve when drawn as described on page 100, consider the difference Yi(O) - Ys(O) A1y(O) Ys(S) - Y3(5) A3Y(), where Y1(%) - log (1 + t-) (i.e., Curve I) and ys3() = log [1 + r2(rl + r2/f)-l t-11 _ log r2 (i.e., Curve III) Using Equation 5.5a' for Ys(%) gives A1y(~) = log + b (511) + Similarly, A3y( ) = log (r2 + ) ( + bf) (5.12) For a possible method consider Ay(5) at the value of: for which the slope of a first approximation of the position of the generating curve is 1/2 (i.e., 1 = 1 for Curve I and 5 = r2/b for Curve III). To this position correspond first approximations for r2 and f. Using (5.11) and (5.12) gives Aly(l) = log 1 + r2 + r1f (5-13) 1 + f 1 + (2r1+l) f/r2 + r12(f/r2)2 (5.14) A3Y(r2/b) = log (5.14) 1 + f/r2 + rl(f/r2)2

104 The values so obtained for ALy and A3y permit placing the generating curves in a better position (second approximation). By repeating this process a series of usually rapidly converging successive approximations is obtained for the positions of the generating curves and the values of r2 and f associated with them. a. Example to illustrate this method applied to curves of Fig. 5.3a. Measurements of tne curves taken from a first "fit"' of the generating curves give G = 7 and B = 8. For simplicity the error in measuring will be omitted. Insertion into Equations 5.6a and 507a gives r2 = 27 and f a 225 (1.004) as a first approximation. Also Zo = -7~5 which with Equation 5.8 gives f = 2-15. As an additional check, inserting these values into Equations 5.9 and 5.10 gives C' 5.5 and.o ~o415 compared with values of C = 56 and do -.42 obtained by inspc tion of'the curves. With Equation 5.13 Aly(l) I.0'i and with Equation 5.14 A3y(ra2/b) tw& o0.73 which means that no further approximation seems warranted. b. Example using Fig.,5&3b. Here the first approximate T'fit" gave G and B values leading to r: a 22 and f X 2.59 Also Zo. -3 gives f 2-6 With these values a first approximation for Aly(l) and A3y(r2jb) yields.38 and. ollo A secolnd approximation based on these shifts gave the more aecuratp values G; c an d, B = 4 and has also been used to compute C = 1o15 and s,o = o33 whic.th compare well with the values C = 1.1ol and so = 033 obtained by inspection of the curves. The general conclusion from the above considerations of the two-trap model is that although the limiting slope ultimately approaches -1 at very low intensities, there is an extended intermediate region possible for which the curves bear a strong resemblane to expermental rela tie procity curves, With certain choices of relative abundance (r2) and relative prob

o105 ability of escape (f) for the deep and shallow traps, the curves of log(E-Eo) vs log I become nearly straight for an extended portion of this intermediate region. For higher values of f a plateau will tend to develop before the curve bends up to slope -1, thus giving two inflection points. (By introducing the influence of Eo, that is by plotting log E instead of (E-Eo), the curves with two inflection points can be somewhat straightened.) In this intermediate region the slope decreases with increasing number of deep traps, with increasing difference between trap depths and with increasing o. Thus, it is not necessary to assume a continuous trap-depth distribution in order to obtain noninteger V'limiting slopes" as required by experimento The two-trap model is sufficienrtly general to describe many experimental characteristics and the analMysis given above indicates a possible procedure for obtainingz the parameters f arnd r2 from experimental curves. It is interesting to compare the reciprocity curves obtained by Tutihusi,53 as a result of adding de-sensitizing dyes, with the curves of the twio-trap model when f << 1 and r2 >> f (se% for exrample, the curve for f = 2 of Fig. 5.2b). His explanation of the role of these dyes (which preslnzably were absorbed to the silver halide gra.ine-) i s qualitatively equivalent to assuming that additional shallow trap,- hEave been added to the silver halide grainas. The change in thle shape of the low-intensity branch of the reciprocity curves after the addition of these dyes is qualitatively that which would be predicted by the two-trap model, although the bends shown in his curves seem to be sharper than any shown here~ B, EXTENSION TO N-TRAP MODEL By arguments formally identical to those used for the two-trap model,

106 the solution for N traps of depths Ui and relative abundance ri is W(h) ai i=o Thus, N Ys = log(E-EO) = - logs ai +ki i=o where the ki are the N solutions of the Nth order secular determinant from N simultaneous differential equationsanalogous to Equation 5.1 and the /_ ai from initial conditions. In general the relation between ys and log n could be very complicated; however, it is easy to show that (just as for the two-trap model) the slope of the curve dys/dx approaches -1 and zero as r approaches 0 and, respectively (i.e., low- and high-intensity limits), and that it is bounded by these limiting values, since N Xai _ ki i=.l wdy5 1 ) i=l Therefore, j ai/ki lim dys 1 0 +O dx a/

107 and lim dys since the ki, ai, and v are all positive: 1 < dys < - dx - where the equalities hold only at r = 0 or Xo and never in the intermediate region. Thus, in the general case of an arbitrary, finite discrete distribution, even though an intermediate region of slope -1 < dy/dx < 0 might occur, for low enough intensities the slope will always approach -1. It would seem unlikely, then, that an extension to a quasi-continuous distribution with finite limits for the trap depth would produce a limiting slope between -1 and 0. * In the next section it will be shown that the limiting slope for the continuous distribution with finite limiting trap depths (discussed in Chapter II) is indeed -1 for sufficiently low intensities. C. CONTINUOUS DISTRIBUTION OF TRAP DEPTHS Here the solution is obtained for the special (although physically reasonable) distribution 0 for U < Uo m(U) = 1/E e-(Uo-U)/E for UO < U < Um (5.15) 0 for U >, *For a mathematically continuous distribution extending to infinite trap depths, considered as the limiting case of a finer and finer discrete distribution, it is possible that the limiting slope for lower and lower intensities is not -1. However, any physical grain will have a large but finite number of traps. The discreteness of its distribution implies that the limiting slope for very low intensities must be -1.

108 as was used in Chapter II (with a = 0). It will be shown that for intermediate intensities the reciprocity slope approaches -kT/e, but for extremely low intensities it will approach -1. The length of the intermediate nearly straight part of the curve will be shown to depend on the separation of the maximum and minimum trap depths. Using m(U) of Equation 5.15 gives Um (Uo-U)/e F(t, a = 0) d u-X(U)t du U,E where Um is defined by 00 m(U)du = M * JUm which with Equation 5.15 gives M = (%o/Xm)' (s = e/kT). Changing the variable by % = Xot eO,U)/k, substituting F into W = j d e reversing the order of integration and integrating with respect to t gives 1 s-l ~, T (+516) V + (l6 1l Some General Conclusions o,-That the low-intensity limiting slope must be -l (and the high-intensity limiting slope zero) can be shown directly from this form before integration, since *This simply states that for this quasi-continuous distribution of a large number M of traps the number of traps occurring at depths greater than Um is one compared to M- 1 for U < iUm For this reason it is sensible not to continue the distribution function beyond the point Um.

109 dy T dW _ ( w + eo e d dx W() dsh( + X0o) d Thus, lim dys -1 and lim dy 0+ dx T + c dx and dy > -1 (the equality holds for i = 0) 2. Special Case s = 1/2. —For s = 1/2 a simple solution can be obtained for W in closed form by putting = y2 into Equation 5.16 which then becomes W1/2(Ti) = ([/2 [arc tan A )21 arc tan Fig. 5)4 shows the reciprocity curves (of log2(E-Eo) vs log12 ) which result. Note that for an intermediate range the slope is very nearly -1/2 but that for sufficiently small T (depending on M) the slope changes to -1. This can also be seen directly from Equation 5l16a, by expansion for T << Xm < %o and for Am < T < Xo. 35 General Case 0 < s < l. -The reciprocity behavior for any s can be visualized after a transformation of the integration variable by e = % ko/0. The integral of Equation 5.16 then becomes* *This form can be integrated directly by use of a Laplace transform (see Reference 54, po 144), giving the result in terms of tabulated ~ functions (see Reference 55). The descriptive solution is, however, more useful here.

28 24-0 20 w 1 w 8-~~~~~~~~~~~~~~~~~~~~~~~ 43I-=2~5 01 44 40 36 32 28 24 20 16 12 8 4 0 4 8~~~~ U, 0~~~8= / n o=28 n 4,26 n 8

111 ~2 - Wsh(,M) = s A e-s% d =_ s(r1/Xo)s Js(,M),(5 16b) ~ I 1 + eb where I L= log q - log ko ~a = log - log m. In this form the integrand is a function of 0 and the single parameter sa. This permits a simple qualitative description of the reciprocity behavior and also provides a convenient form for numerical integration. In Fig. 5o5 the integrand is plotted for s = 1/2 and s = 1/8. If the range of integration spans the hump and is sufficiently large compared to the width of the hump, then the value of the integral will be only slightly dependent on T,, and consequently W will be very nearly proportional to i, giving an effective reciprocity slope of -s. On the other hand, for values of ~ such that the range of integration does not include the hump, the integral Js(rq,M) will become proportional to ~1 or,-s, depending on which side of the hump the range of integration occurs. The resulting form of W will then become proportional to r, or ~, producing effective reciprocity slopes -1 or 0, respectively (see, forl example, Fig. 5.5). It can therefore be seen that I - ~2 = log kX/ko (l/s)log M determines the length of the reciprocity curve for which the slope is nearly -s. The slope will change toward -1 when ~ is approximately'm. When ~ is approximately ko the slope will change to 0.* In this manner the important properties of the resulting reciprocity curves can be qualitatively described in terms of M, s, and ko. These same conclusions can be obtained from direct differentiation of J5(I,M) with respect to s e

.7 ~~~~~~~~~~~~~~~~~~~~~~~~~~I [ [ i [ i I III i.6 S~~l/ ~ 1.0.5.5 -.4 01~~~~~~~~~~~~~~~~~~~~~~~~~ I ID~1~~~~~~ -40 -20 0 20 40 60 80 00 t=:' 3 —_ _ __ _ _ CV~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~H 2 LOG 2 M-z.22 l/s LOG2 M -.1 FOR s: /8 1%.2/ s:1/2 0 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 I/s LOG MLOG2 =LOG2 //Xo FOR s=1/2 t2LOG21/Xm Fig. 55. Plot of 2in/l+2t 1o the integrandof Js(gM) for s = 1/2 and s = 1/8, showing the range of integration when M = 2 (extended range for s = 1/8 curve shown in insert).

113 4. Comparison with Measurable Quantities.a. Position of the bend toward slope -1. If an effective trap depth Ueff and an effective escape probability keff is defined by %eff = v e-Ueff/kT and Um m(U)kT-U du neff \ m m(U) du JUO then Xeff = s - i ls MI-/s O (5 - 17a);ef =1. -s F M 2 1 s M m. (5 17b):eff =n% -.The intensity at which the slope changes to -1 can be estimated relative to the optimum intensity by u.sing Equatilon 5e17b iff it is assumed that at optimum intensities Iopt+,medf 1/T - aIops, (that is, the average effective time needed to escape frorn traps ies the same order of magnitude as the interquantic times for optimal intensities). If the intensity for whi'ch the bend occurs is denoted by TB, then IB/ Iopt ~ xm/%opt. Thus, log IB - log Iopt log M + log (1/s-)' log M. Using M = 216 as a reasonable estimnate for the number of traps per average grain (see p. 1137 of Reference 33 ) and optimal exposure times of v 1/10 sec, the bend might be expected to occur at exposure times tB where log tB ~ (l+s) log M or tB, i04 sec. Preliminary experiments using pure

114 silver bromide emulsion of minimum sensitization (presumably having less surface traps) indicate that the bend occurs at higher intensities than for the more highly sensitized emulsions. This would be in qualitative agreement with the above remark. However, more work is needed to establish this result. b. Criticism of exponential distribution. Using Equation 5.17a, setting P = (l-s)/s and requiring Uo to be positive, it follows that P log M - log P < Ueff/kT. (5.18) Webb's 9 measurements of the effective trap depth yield Ueff =.77 ev. Depending on the model employed slight variations in this value may be expected, but no changes in order of magnitude. Consequently, Equation 5.18 sets a lower limit to the value of s for any given M. For example, M A 200 s Z.13 M l105 s >.27 M 1 108 s.3 ~ A special study would have to be made to ascertain whether or not these values contradict experiment, but from general evidence they seem to do so since slopes well below.2 are observed for emulsions whose grains are likely to have more than a few hundred impurity centers (traps). In addition, if increased digestion builds more traps, the present argument suggests an increase in s with increased digestion. Experimentally the reverse appears to be true. These two reasons throw doubt on the applicability of the exponential distribution. It is true that Equation 5.18 was derived assuming a = 0, and qualitative arguments can be made to show that nonzero values of C will relax somewhat the first point of criti

115 cism. It is noteworthy that this particular criticism is not applicable to the two-trap-depth model described in Section A. D. SIM~ARY AND INTERPRETATION OF RESULTS 1. Summary of Results. -No strong systematic dependence of the slope of the isodense reciprocity-law-failure curves on the average grain size for the pure silver bromide emulsions -was found from the experiments reported in Chapter IV when surface development was used. There is a tendency for the curves of lower density to display a slight increase in slope with the increasing average grain size of the emulsions used. The same emulsions developed with internal developer showed a definite increase in slope with average grain size. This increase in slope was also greater in the curves of lower density. 2. Interpretation in Terms of the Original Model with Exponential Trap-Depth Distribution. -The origirnal model using an exponential trapdepth distribution as formulated by Katz predicts a shape of the reciprocity curves which differs slightly but systematically from the experimental points, obtained by surface development. A slightly greater curvature for the experimental points is observed at the low-intensity end of the curves. The estimated experiLmental errors were sufficiently small to warrant consideration of this difference. This is illustrated in Fig. 5.6 where a set of experimental points can be compared to the theoretical curves from (E-Eo)IS = constant. The theoretical curve using s =.74 fits the lefthand points best, while the curve using s =.58 fits the right-hand points best. If s =.66, the curve falls within the range of error of all the experimental points. However, even in this case a slight but systematically recurring departure can be seen. This suggests that more structure

116 2r 0 0 ~J - Experimental points * Theoretical curves using Ls (E-Eo)=constant a).74 b).58 c).66 -2 I I I I I -9 -8 -7 -6 -5 -4 -3 -2 -I x = LOG2 INTENSITY Fig. 5.6. Typical experimental curve (of emulsion T-4140 from Fig. 4.1a) compared to theoretical curves. occurs in the experimental reciprocity curves than can be accommodated by the original exponential trap model. Another difficulty of this model is apparent from curves obtained with internal development, which have a systematic tendency to be convex. The model can only give concave curves. Another difficulty of this model was mentioned in Section C-4b. 3. Interpretation in Terms of the Corrected Exponential Model. —By introducing a slight correction to the original formulation of the exponential model it was shown that, for the case a = 0, the slope of nearly -s existed for an intermediate region of intensities and changed to -1 for extremely low intensities. This change in slope can, in principle, provide the added structure of the theoretical curves needed to fit the increased curvature at the low-intensity end of the experimental reciprocity curves. Qualitatively it was shown that the bend toward slope -1 may occur at intensities associated with the observed slightly excessive cur

117 vature for surface development. More work would be required to warrant quantitative statements with regard to this point. The points of criticism regarding the convexity of the reciprocity curves from internal development and concerning the minimum slope, mentioned in the previous section, remain valid. 4. Interpretation in Terms of the Two-Trap-Depth Model. —It was shown that noninteger "limiting" slopes in the low-intensity branch of the experimental reciprocity curves do not require a continuous trapdepth distribution and. that many experimental reciprocity curves could be matched by the appropriate choice of trap-depth ratios and relative abundance of trap with a model using two discrete trap depths. A possible method of analysis of curves of this type has been given for a = 0. Theoretical formulae are also given for a f 0. It was found that the results for a f O and trap depths U1 and Up can be re-expressed more simply in terms of the "'effective"' trap depths U21 and U21 with a = 0. Further work is required, however, to test the validity of the two-trap-depth model. 5. A Remark Concerning the Grain-Size Dependence. —The dependence of the reciprocity behavior on grain size as found in this investigation seems to be in conflict with the well-known fact that the average developable grain size of an emulsion decreases with increasing density. Indeed, our smal-11grained emulsions tended to have lesser slopes than the large-grained emulsions, while the slope at higher densities (smaller grains) of any given emulsion tended to be greater than that at lower densities. The solution to this paradox is probably due to the different conditions of digestion which are given to grains of given size, on the one hand when this size is the average size of the emulsion, and on the other hand when it is much smaller than the average size.

If it is assumed that digestion is carried for each emulsion to the point of optimizing the sensitivity of grains of average size, it can be shown that the difference in the dependence of the slope on grain size in the two cases mentioned above is not implausible. E. SUGGESTIONS FOR FURTHER WORK AND IMPROVEMENT OF TECHNIQUE 1. The changes of the sensitivity of pure silver bromide emulsions with time as a function of temperature, reported on page 64, suggest further study, both for improving the reproducibility of the reciprocity characteristics and for their own sake. 2. Extension of the reciprocity measurements to lower intensities might decide if the limiting slope is -1. 3. The use of an absorbing instead of a scattering filter is suggested in order to make the transmitted intensity independent of distance between filter and plate. 4. The use of two crossed step filters giving equal exposures along rows on the plate would produce results more efficiently. 5o An analysis of the N trap-depth model for N + Ao might provide a general solution of the integral differential equation (2.14). It might then be possible to determine characteristics of the trap-depth distribution from the "'fine structure" of the reciprocity-failure curve. 6. The results of the two-trap model coupled with the effect observed in the low-intensity branch of Tutihasi6s reciprocity curves suggest a possible technique for studying the properties of electron traps in a grain relative to those associated with an added desensitizer (shallow traps). The basic principle of such a technique would be to observe the change in structure of the intermediate section of the low-intensity reciprocity

119 curves as a consequence of adding various amounts of (supposedly shallow) additional traps. 7. It might also be fruitful to investigate the effect of temperature on the structure of such curves (both theoretically and experimentally). Since the deep traps dictate the behavior of the extremely low-intensity end while the effect of shallow traps is mostly seen in the first rise toward the "plateau," the effect of the temperature on each and, consequently, their absolute trap depth might result. 8. Information related to the density-vs-slope problem mentioned in Section D-5 might be obtained by continuing the technique of grain-size separation mentioned on page 31. Precoated fractions from the same emulsion but of different average grain size would be more likely to have the same digestion treatment.

APPENDIX A CALIBRATION OF NEUTRAL DENSITY STEP TABLETS (Graded Step Wedges) The step tablets, one of which is shown in Fig. A-i, were supplied* with a calibration of the densities as shown in Table A-I. Note that the area between the columns of the step tablet is masked. This reserves an area of the reciprocity-law-failure plate for calibration exposures. Also listed in Table A-I are the base-two densities 2Dn and resulting base-two intensities 2 n transmitted through the nth step (if Io is the incident intensity) calculated from the figure provided by the supplier. The accuracy of the calibration was quoted as + 5% of the density. For a density (base two) of 10, this corresponds to ~ 0~5 log2 units~ The interpretation of our results required a more accurate evaluation of the relative intensities transmitted through every element of the step tablet. Therefore, some of the details of the method of this evaluation have been included. The method used is essentially as follows: 1. The transmission of each step in a particular (the sixth) column of the step tablet was measured with a photomultiplier tube. The variations from a constant density along a step were incidentally explored, and a general tendency to exhibit higher transmissions at the end of the step was noted, 2. Type-33 plates were used in situ to measure the variations of the intensity in the other columns from that previously determined in the sixth *The three step tablets (ST-49-1, -2, and-3) were purchased from Eastman Kodak Company. 120

121 column. Then an array of relative intensity for all rows and columns was constructed. (See Tables A-VII, A-VIII, and A-IX ) 3. The ratio between the transmitted intensities of the steps was then checked, in situ, by exposure of a type-33 test plate to different known relative intensities. A comparison of the actual and the predicted displacements between the characteristic curves resulting from these known relative intensities provides a check on the measurements made in (1) above. Using monochromatic light (K = 4360 A) the ratio of rn of light transmitted through the nth step compared to that transmitted through the n-1th step was measured for the sixth column of the step tablet. The arrangement is shown schematically in Fig. A-2. By adjusting the dynode voltage of the 931-A photomultiplier tube, the photocurrent caused by the light without the filter present could be made to cause full-scale deflection (jo) of a sensitive galvanometer. Then, with the first step over tha aperture and the photocurrent measured (ji), the ratio of transmitted intensity to incident intensity would be ji/jo. The photocurrent jn through any step could be measured, but it is more accurate (since tne ratio jn/jo becomes rapidly very small) to measure successive Jn/jnli ratios. Tabulation of the resulting pbotocur:rents is seen in Table A-IIo At the bottom of thiL table are shown the densities calculated from these measurements. Errors quoted are estimanrt,es of error of reading the galvanometer and placement of the step wedge over the aperture. Table A-III shows the results for all three step tablets. Table A-IV shows variation in the log2 of the transmitted intensity x6,n and density of the sixth step of step tablet no. 3 (i.eo, variations of density along this step).

122 The above measurements establish the absolute diffuse density of each step in the sixth column for each of the three step tablets, and also demonstrate that the density along a step is not constant. Then the log2 In 6 Xn,6 is known from within + 1% for the highest intensity step to + 7% for the lowest intensity step. Since the cumulative error became rather high for the low-intensity (high-density) steps, it is of interest to compare the transmission of the eighth step with that of a dye filter whose transmission was known* to be 1/157 (i.e., it had a base-two density of 7.3). The intensity of light transmitted through it is compared with that of the eighth step of the step tablet in Table A-V. Note that the results check within the accuracy quoted. The variations of the density along each step were evaluated photographically by a series of exposures using type-33 emulsion with the step tablets located in the apparatus as they would be in the experiments. An example of the developed densities resulting from one such exposure of the 33 plate is given in Table A-VI. By use of previously measured values of intensity of the sixth column, a characteristic curve was established for the plate used. Using this curve, the variations in developed density at the various positions along a particular step could be converted into variations in density of the step (i.e., variations in transmitted intensity). In the example shown here the second, third, and fourth steps were the only ones accurate enough to be usable. Thus, many such overlapping exposures were needed to evaluate all the steps. *The density of this filter was measured by using the inverse-square law, with the photomultiplier tube as the detector.

123 Using the evaluation of the intensity of the light transmitted through the steps of the sixth column (measured by the photomultiplier tube with step tablet removed from its plate holder) in conjunction with the variation of transmitted intensity along each step (as measured photographically), the intensity behind each position of the step tablets used in the reciprocity-law-failure experiments were calculated and tabulated. Results of the complete calibration of all three step tablets are shown in Tables A-VII, A-VIII, and A-IX. Note that the variations in transmitted intensity for the sixth step of step tablet no. 3 are compatible with those measured by the photomultiplier tube (see Table A-III). In order to check in situ the validity of the measurements made by the photomultiplier tube, exposures with differing incident intensities upon step tablet no. 2 were given as follows (using type-33 emulsion for the test): Column 1 Blank Columns 2 and 3 Exposed 6 minutes with I, Columns 4 and 5 Exposed 12 minutes with I2 = 2-1'7 I1 Columns 8 and 9 Exposed 12 minutes with I1 Columns 10 and 11 Exposed 6 minutes with 12 Columns 6 and 7 Exposed 6 minutes with I1, but with plate holder mounted.95 feet from source instead of usual 4 feet. After exposure of each two columns the preceding ones were masked by black cardboard in the plane of the slide. The intensity ratio r2 1 = I2/I1 was evaluated by a series of measurements using the photovolt meter. The result was +log2r2, 1 = 1.68 +.08. Using the inverse-square law for estimating the intensity of light at a distance 4 feet from the aperture of about 1.65 cm2, the log2 ratio of intensity at.95 feet compared to I1 is 4.15 +.05. The resulting charac*Since the error caused by treating the extended source as a point source is about (1/30)2 <.5%, it was neglected.

124 teristic curves for exposures using these intensity ratios are shown in Fig. A-3 where the Sidel function is plotted as a function of intensity through the step tablet. The intensities used are those given in Table A-VIII where the logarithm of the incident intensity is arbitrarily taken as 10. Results from 12-minute exposures are similar and are shown in the dashed curves of Fig. A-3. These results together are intended to demonstrate that: 1. The evaluation of the logarithm of intensity xn for step tablet no. 2 as quoted in Table A-VIII is correct to within the error quoted. * 2. The deviation of the points from an almost straight line in the region -.5 < S < 1.5 indicates that the error in relative intensity from step to step ranges from about ~.02 log2 units for higher intensities to about +.05 log2 units for lower intensities. The intensity for the last step is uncertain to about.07 log2 units. *The separation of the curves is a measure of the intensity difference at which the several exposures were made. Note that they match (within the accuracy quoted) the actual intensity differences used for the exposures. This indicates that the calibration of the intensity ratios between the steps was correct.

TABLE A-I DENSITY CALIBRATION OF STEP TABLET NO. ST-49-2 AS PROVIDED BY EASTMAN KODAK COMPANY AND BASE-TWO DENSITY AND INTENSITY CALCULATED THEREFROM Step No. 0 1 2 3 4 5 6 7 8 9 10 Diffuse density (base 10) (by Eastman Kodak Company).o0.34.63.92 1.22 1.51 1.80 2.08 2.37 2.66 2.95 (by Eastman Kodak Company).... Base2 density 0.17 1.13 2.09 3.05 4.05 5.02 5.98 6.91 7.87 8.84 9.81 og2(In/Io)* = Xn 9.83 8.87 7.91 6.95 5.95 4.98 4.02 3.09 2.13 1.16 0.19 Log2 (In/Io)* = Xn Our measurement 9.74 8.78 7.77 6.77 5.77 4.79 3.95 2.91 1.96 1.01.09 (Column 6) +.07 +.o6 +.05 +.05 +.04 +.03 +.03 +.02 +.02 +.01 +.01 Io = incident intensity arbitrarily assigned the value 210

TABLE A-II INTENSITY RATIOS BETWEEN STEPS FOR COLUIVI NO. 6 OF ST-49-3 AS MEASURED BY 931-A PHOTOMJLLTIPLIER TUBE (April 1956) Step No, 0 1 2 3 4 5 6 7 8 9 10 1.00*.82.432 1.00.500 1.00.510 1.00.505 All galvanometer readings +.005 1.00.500 1.00.510 1.00.518 1.00.50 1.00.50 1.00.50 1.00 Transmission.852.432.236 =1101.0556.0278.0142 00755.003568.00184.00092 d233 1.21 2.21 3.18 4.1- 5.17 6.14 7.09 8.0og 9. 10.09 BI~ase2 ~.ey +.03 ~0l ~,02 ~.05 o ~.;3o4 ~,o4 ~0.os ~.o6 ~.o6 ~.o0 *Photoeurrent from incident light through unrestricted aperture.

TABLE A-III RESULTS OF DENSITY \MEASUREME NTS OF COLUMN NO. 6 OF STEP TABLETS, USING 931-A PHOTOMULTIPLIER TUBE Step No. 0 1 2 3 4 6 7 8 9 10 Trial No. I 27 12 2.22 5.319 4,18 16 6.13 7.10 8.04 8.97 9.951 +o01 +,01 ~+02 ~,02 ~.05 ~.03 ~-.04 +.05 ~.05 ~.o-06 ~.07 ST-91 Trial No. II ~.25 1.23 2_19 516 4.13 512 6.19 7.18 8,16 9.00 9.94 ST-49-1l, Trial No. I I +,i+1.Oi tOi1.02 t02 o03 +.03 o,04.o5 +.o5 +.o6 07 Average of Trials No. I. 6 i 24 2.21 3517 4.16 o14 6,16 7,14 8i0o 8.98 9.95 +- 007 +e 02 + 0 2 + 02,Ol~ + ~.04 ~0,5 ~,.05 and II o.007 0 02 02.06 ST9-2 Trial No. I 26 1.25 2.23 5.22 4.22 5.20 6,15 7.09 8.04 8.99 9.92 +.o 1.o 1.0 1.21o~1o.o4 1.o5 1+.o5 1.o6 to +,o1!,01 1.02 %02 +,03 +,03 04 +,09 +.09 +.06 07 Trial o, I,25 1.22 2,25 5,2,5 4,17 5.15 6.15 7.12 8,11 9.12 10.12.25.21 2.21 3.8 4.17 5.17 6,14 7.09 8.09 9.09 10.09 ST-49-3 Trial No. I! 1011 +. 102 1.02 +,o3 +o0o +.o4 ~.o5 +.o0 1.o6 +-o7 Trials No, I Average of.24 1.21 2.22 5.20 4.17 5.16 6.14 -7.11 8.10o 9.11 10.11 Trials No. I 131~0.1~0.2~02~0.5~0.5 ~o and II + 0 i + 1 01 +.02 +~02 ~.02 +03 03 3 + 03 ~,03 + 4 a~nd JII........ -..

12 i3 TAB3LE A- IV VA.RIATION OF TIiA'{ISilSION AiD DENSITY ALO.,NriG Tags SIXTH STEP OF ST'5 AL MEASUIRED Y'931 -A P'HOTiIi.'LT, " LIER TUTBE Columnl No. mn Tr ansmission Log2 ive I6 Tr ransrni ss on 12.89 4.77 11.91.9-74 4,81 10 095 1. 5 4.87 9 195.994 4.84 8. 91 94 4.81 794 1.-. 00 4.86 6 o3,. i:)<0 4.85 5 1I4 1 a:" 4,86 4 4 97 i 4.87 9 52 L) c) 4 84 i; 90).196 2 4.79

129 TABLE A-V COMPARISON OF INTENSITY TRANSMITTED THROUGH THE EIGHTH STEP WITH THAT OF DYE FILTER OF DENSITY D2 = 7.30 +.05 Transmission Log2iffer- Computed Density ence of Trans- Density of Relative to of 8th Step Relative to mission Relative of 8th Step Step Tablet Dye Filter of Step Tablet (8 Step) Step Tablet 1.15.01.202 7. 10 +.05 7.14 t.04 No. 1 Step Tablet 1.19 +.01.250 7.05 +.05 7o11 +.04 No. 2 TABDLE A-VI DEVELOPED DLESITIES OF TYPE-533 EMVULSION RESULTING FROM EXPOSURE OF 32 MINUTES THROUGH ST-49-3 (last figure is uncertain) 0 2 4 6 8 12 -095.303.668 11.106.533 ~695 10.11a4.342.721 9.107.329.717 8. 102.312.705 7 o010 _027 108.320.711 1 25 1.87 2.50 2.9 6.010.024.100.319.720 1.24 1.87 2.50 2.9 5.o096 306.698 4,103.316 G710 3.0og6.318.712 ~2~. o98.320.690 1.o098. 308.699

150 TABLE A-VII LOG2 OF INTENSITY BEHIND ST-1 (ST-49-1) O 1 2 3 4 5 6 7 8 9 10 12 -.03.92 1.92 2.84 3.80 4.78 5.75 3.10 4.01 5.03 5.95 2 +.o8 +.06 +.o6 +.o05 +.05 +.04 +.04 +.03 +.04.5 +.O -.01.96 1.93 2.88 3.84 4.83 5.81 6.82 4.03 5.05 5.98 +.06 +.0o +.o5 +.04 +.o4 +.o3 +.03 +.02 +.05 +.05 +.05.00 0.97 1.93 2.88 3.86 4.83 5.81 6.81 7.78 5.08 6.oo00 +.0o6 +.05 +.o +.o4 +.o4 +.03 +.03 +.02.02 +.05 +.05.02.99 1.96 2.90 3.89 4.85 5.80 6.82 7.78 5.12 5.03 9 +.06 +.o05 +.o05 +.o4 +.o4 +.03 +.03 +.02 +.02 +.o05 +.05 8.02 1.01 1.97 2.91 3.89 4.85 5.82 6.84 7.78 8.75 s.o4 +.o6 +.05 +.o0 +.o4 +.04 +.03 +.03 +.02 +.02 +.01 +.05.o1.99 1.96 2.90 3.87 4.85 5.82 6.81 7.78 8.75 9.75 7 +.o6 +.os.05 +.04 +.o4 +.03 +.03 +.02 +.02 +.ol +.ol 6.05 1.03 1.98 2.90 3.87 4.84 5.82 6.81 7.78 8.75 9.73 +.o6 +.0 +. 05 +.o4 +.o4 +.03 +.03 +.02 +.02 +.O1 +.O1.03 1.03 1.96 2.91 3.86 4.83 5.80 6.81 7.78 8.75 9.74 5 +.o6 05 +o.o4 +. 04 +. 03 +.03. +. +.02 +.02 +. +. 4 ~05.99 1.97 2.90 3.89 4.85 5.80 6.80 7.79 8.75 9.73 +.o6 +.05 +.05 +.o4 +.o4 +.03 +.03 +.02 +.02 +.Ol +.Ol.o4 1.01 1.97 2.91 3.87 4.84 5.80 6.82 7.78 8.75 9.74 3+.0 + +.05 +.o5 +.o4 +.o4 +.03 +.03 +.02 +.02 +.Ol +.ol.o05.98 1.96 2.90 3.86 4.83 5.79 6.81 7.77 8.76 9.76 2+.o6 +. +.o4 +.o4 +.03 +.03 +.02 +.02 +.01 +.01 -.02.95 1.93 2.98 3.84 4.82 5.79 6.81 7.78 8.75 9.75 +.08 +.06 +.06 +.05 +.0o +.o4 +.o4 +.03 +.03 +.02 +.02 10 9 8 7 6 5 4 3 2 1 0 Step No.

131 TABLE A-VIII LOG2 OF INTENSITY BEHIND ST-2 (ST-49-2) O 1 2 3 4 5 6 7 8 9 10 -.12 01.91 1.89 2.85 3.83 4.75 5.75 6.75 7.75 8.76 9.73 12 +.08 +.07 +.o6 +.o6 +..o0 +.0o4 +.03 +.03 +.02 +.03.og09 1.01 1.93 2.91 3.87 4.83 5.79 6.77 7.78 8.77 9.73 +.07 +.06 +.o +.o +.04 +.o3 +.03.02.02 +.02 +. 03.11 1.04 1.96 2.91 3.87 4.81 5.78 6.75 7.79 8.77 9.73 10 ~+07 +.o6 +.0 +.o05 +.o4 +.03 +.03 +.02 +.02 +.ol +.03.09.98 1.91 2.87 3.86 4.82 5.80 6.76 7.79 8.78 9.73 +9.07 +.o6 +.os +.05 ~.o4 +.0 +.3. +..0 2 O +.03 8.05 97 1.96 2.87 3.86 4.81 5.78 6.77 7.78 8.78 9.72 +.07 +.o6 +.05 +.05 +.04 +.03 +.03 +.02 +.02 +.01 +.01.06.99 1.95 2.90 3.85 4.80 5.78 6.77 7.79 8.78 9.73 7 +.o7 +.o6 +.05 +.o5 +.o4 +.o3 +.o3 +.o2 +.02 +.ol +.ol 6.09 1.01 1.96 2.91 3.85 4.79 5.77 6.77 7.77 8.77 9.74 +.07 +o6 +.o05 +.o05 +.o04 +.03 +.03 +.02 +.02 +.O1 +.O1.01.95 1.93 2.92 3.86 4.79 5.77 6.76 7.77 8.78 9.74 5 +.07 +.06 +.05 +.05 +.04 +.03 +.o3 +.02 +.02 +.ol +.ol.02.91 1.93 2.92 3.85 4.78 5.78 6.78 7.80 8.77 9.74 +.07 +.o6 +.o05 +.o05 +.04 +.o3 +.03 +.02 +.02 +.Ol +.o01.08 1.00 1.94 2.93 3.85 4.79 5.78 6.79 7.79 8.78 9.74 3 +.07 +.o06 +.o05 +.05 +.04 +.03 +.o3 +.02 +.02 +.01 +.ol.oo00.95 1.90 2.88 3.86 4.80 5.78 6.78 7.78 8.78 9.74 2 +07 +.o6 +.05 +.05 +.o4 +.03 +.03 +.02 +.02 +.01 +.01.91 1.87 2.85 3.82 4.78 5.76 6.76 7.76 8.76 9.73 +.07 +.o6 +.o6 +.05 +.o4 +.o4 +.03 +.03 +.02 +.02 10 9 8 7 6 5 4 3 2 1 0 Step No.

132 TABLE A-IX LOG2 OF INTENSITY BEHIND ST-3 (ST-49-3) 0 1 2 3 4 5 6 7 8 9 10 12 -.17.80 1.84 2.84 3.78 4.77 5.79 6.75 7.73 8.75 9.75 +.o5 +.o4 +.o4 +04 +.o4 +.03 +.03 +.03 +.o02 +.02 +0.o3 ~07.89 1.92 2.86 3.85 4.83 5.81 6.78 7.76 8.77 9.75 +.04 +.03 + 0 3 +.0 3 +.3 02 +.02 ~.02 +.01 +.01 +.02 -.02.96 1.97 2.90 3.89 4.88 5.85 6.81 7.78 8.77 9.75 10 +.04 +.03.o03 +.03 +.o03 +.02 +.02.02 +.oi +.o0 +.02 -.10.87 1.89 2.88 3.88 4.86 5.84 6.77 7.77 8.77 9.75 9 +.o04 +.03 +.03 +.03 +0.o3 +.02 +.o0 +.02.ol +.ol +.02 8 -.10.88 1.90 2.86 3.85 4.82 5.83 6.79 7.76 8.76 9.75 +.o4 +.o3 +.o3 +.o3 +.03 +.02 +.02 +.02 +.ol +.oi +.02 -.07.91 1.92 2.86 3.85 4.86 5.84 6.79 7.77 8.77 4.75 7 +.o4 +.03 +~03 +.03 +.03 +.o02 +.o02 +. +.o +.ol +.ol -.12.88 1.89 2.89 3.87 4.85 5.83 6.78 7.77 8.78 9.75 +.o4.o03 +.o03 +.o3.o03.02 +.02 +.02 +.o0 +.o0 +.o0 -.16.88 1.87 2.81 3.83 4.85 5.83 6.79 7.77 8.77 9.74 +.o04 +0.o3 +.03 +.03 +.03 +.o0 +.o +.02 +.ol +.ol +.o01 -.17.86 1.92 2.87 3.87 4.86 5.82 6.79 7.77 8.77 9.75 +.o4.o03.o03.o03.o03 +.02 +.02 +.02 +.01 +.01 +.01 -.15.85 1.90 2.86 3.87 4.86 5.83 6.80 7.78 8.77 9.76 3 +04 ~03 ~-03 ~ +.03 +.03 02 +.02 +.02 +.01 +01 +.0 -.17.83 1.91 2.86 3.84 4.83 5.83 6.79 7.77 8.76 9.76 +.o04 1.03 ~.03 +.03 +.03 +.02 1.02 +.02.O01 +01 +o01 -.14.80 1.86 2.83 3.84 4.81 5.80 6.76 7.73 8.73 9.75 1.o5.o4 +.o4 +.o4 +.o4.o03.o03.o03 +.02 +.02 +.02 0 9 8 7 6 5 4 3 2 1 0 Step No.

........................~~~~~~~~~~~~~~~~~~~~~~~~~Lh...........~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~L~..............~ r ~- %:iiiiiiiiiiiiiiii~~~~~~~~~~~~~~~~~~~~~~~~~~~~......................~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Ii Fig. A-1. Step tablet ST-49-2 shown masked as used in reciprocity-law-failure expleriments.

134 x =4 358, Cross section of 8m..,m ~.4,step tablet,...,....'"-',!..-. ~':*'."!, ~'.3mm Hole _..__m/m HoleOpal glass'T Light scattered by opal glass Sensitive surface of 931-A photo tube Fig. A-2. Arrangement for measuring transmission of step tablets.

INo. 7 I INo.8&9'I- No.6 /No.6 / No.3 Shift predicted Shift observed N45 No. from intensity on graph N/ No. 2 ratio of incidence No. light No. Sidel value 10 / 4.2 1.5/ / a. 4.15 ~.05 4.1 1 4.15 -.5 J.2 1.70 1.5 1.70 b. 1.68~.08 1.6 8 z 0 i-I ~i 1.68 1.75 z / / w 0 /// a. Intensity variation by distance 2,>[ / / / / / ~~~~~~~f rom source / ~~~~~~~~/ / / Xb. Intensity variation by size of 4.15 6 Minute exposure --- 12 Minute exposure 0 1 2 3 4 5 6 7 8 9 10 LOG2 I - Fige A-3. Characteristic curves of type-33 emulsion at different intensities.

APPENDIX B INFLUENCE OF PERIODIC LIGHT INTENSITY ON LOW-INTENSITY RECIPROCITY BEHAVIOR The effect of a sinusoidal component of the intensity on the reciprocity behavior at low intensities, where the total exposure time contains many interquantic times, will be considered. It will be shown that for very low intensities and not too low frequencies the effect is negligible, while for high intensities the slope tends to the decrease toward -1. Katz56 has shown that for a periodic intensity, I(t) = I(t+T), P(G) is given by t+G T -a Jt I(v)dT a P(G) dG =IT dt I(t) I(t+G) e dG IT o Here, a is the effective area of the grain absorbing the quanta at an average rate given by I. This relationship was derived for exposures for which the total exposure time Te >> (where the bar over a quantity means the time average). If I = constant, then P(G)dG = aI e aI dG = 1/Ge "/"dG, which is the form used in Chapters IV and V. Using I(t) = I(1 + b cos wt), where b < 1 since I < 0, t+G aI T -aI dt (1+b cos T)dT P() = T J dt(l+b cos t)[l+b cost)[l+b cos (t+)1 e By letting a = w(t + G/2) and 1356

137 2b.1 iZ = sin 2 am 2 the P(G) can be integrated in terms of Bessel functions of order 0, 1, and 2. Using suitable recursion relations and expanding the resulting functions gives e /P(G) = 1 +b ( cos (oG - - sin wo@ + (1b)2 sin2 b4 2 Q Fl 1 1 2 wGl + (-z sin (cos WG + 1/2) - G- sin sG +sin + terms of order b/(l@). For normal grains < 120 quanta are required to produce developability; the lowest times of exposure used were 1 minute; thus > 60/120 sec. Since the frequency source used was 120 cycles/sec, o = 23~ x 120 rad/sec;.'. ao > 377. For lower intensities this number will be proportionally larger. Thus only the first two terms will be considered. Using F(t) = (G/T)-s where T is the characteristic time associated with the survival function the W becomes Ws 0 G/f\dG b2 1 -/G)-s W - (;/Tr) A e~ / ()9 - + 2 e - cos cu dG The first term is the same as that resulting from constant intensities, namely, We = (G/T)<S F(l-s). The integral in the second term can be evaluated by use of a Laplace transform (see p. 125 of Reference 54), giving (when terms of order (1/ cU)2 are neglected): r(l-s) s-1 [cos(1-s)x/2 + 14 sin(l-s)X/

138 Thus, neglecting the term in 1/X0 = (l-s) TSL )-S +2 ( s-l cos(l-s)it/2] The reciprocity behavior given by log(E-Eo) + log W = constant becomes then 2 Ys = log(E-E0) = - log [ + 2 cos(l-s)r/2i + constant For low intensities lim dys I+0 dx - s and for high intensities lim dys -1. I +- ~ dx The maximum rate of change of slope occurs at log(Ia) = log X + Llog 2/b2 - log cos(l-s)v/21] X log 1-s If Gax is again assumxed to be approximately 2 for the highest intensities used in the experiments then GMax/Ica ev 2/w - 2/377. The maximum change in slope due to this periodic source would occur then more than 7 log2 units beyond the high intensity end of the graphs shown. Furthermore, the value of Ys(G = 1/2) - ys (G = 1/2, b = 0) < log [1 + b2/(754)S-1 cos (l-s)r/2] ~ b2(754)1-s cos (1-s)5/2 <.03 (or.05 log2 units) for s = 1/2, b =.75. It is therefore not likely that the sinusoidal wave form (used in the experiments of Chapter IV) produced any noticeable effect.

BIBLIOGRAPHY 1. C. E. Kenneth Mees, The Theory of the Photographic Process, 2nd ed., Macmillan Co., 1954. 2. R. B. Wilsey, Phil. Mag., 42, 262 (1921). 3. R. B. Wilsey, J. Franklin Inst., 200, 739 (1925). 4. A. P. H. Trivelli, Rec. Trav. Chim., 42, 714 (1923). 5. A. P. H. Trivelli and S. E. Sheppard, The Silver Bromide Grain of Photographic Emulsions, Monogram No. 1 on Theory of Photography, Eastman Kodak Co., Rochester, N. Y., 1921. 6. H. Chateau and J. Pouradier, Science and Applications of Photography, Proceedings of the R.P.S. Centenary Conference, London, 1953, p. 26. 7. J. W. Mitchell and J. M. Hedges, Phil. Mag., 44, 223 (1953). 8. J. W. Mitchell and J. M. Hedges, Phil. Mag., 44, 357 (1953). 9. F. W. H. Mueller, Science and Applications of Photography, Proceedings of the R.P.S. Centenary Conference, London, 1953, P. 13. 10. C. E. Kenneth Mees, Science and Applications of Photography, Proceedings of the R.P.S. Centenary Conference, London, 1953, p. 26. 11. W. D. Bancroft, Trans. Faraday Soc., 19, 243 (1923). 12. J. H. Webb, J. Appl. Phys., 11, 18 (1940). 13. E. R. Bullock, Chemical Reactions of the Photographic Latent Image, Monogram No. 6, Eastman Kodak Co., Rochester, N. Y., 1927. 14. G. Kornfeld and T. H. James, J. Opt. Soc. Am., 33, 615 (1943). 15. M. B. Hodgson, J. Franklin Inst., 184, 705 (1917). 16. The Svedberg, Photo. J., 62, 310 (1922). 17. F. C. Toy, Photo. J., 61, 417 (1921). 139

140 BIBLIOGRAPHY (continued) 18. W. Clark, Photo. J., 66, 80 (1926). 19. F. A. Hamm and J. J. Comer, J. Appl. Phys., 24, 1495 (1953). 20. E. Katz, J. Chem. Phys., 18, 499 (1950). 21. W. Reinders and R. W. P. DeVries, Rec. Trav. Chim., 56, 985 (1937). 22. W. F. Berg, Phil. Mag. 3, 36, 337 (1948). 23. N. F. Mott and R. W. Gurney, Proc. Roy. Soc. (London), A164, 151 (1938). 24. F. Seitz, Rev. Mod. Phys., 23, 328 (1951). 25. W. West, Fundamental Mechanisms of Photographic Sensitivity, Butterworth, 1951, p. 99. 26. J. H. Webb, J. Opt. Soc. Am., 28, 249 (1938). 27. J. W. Mitchell, J. Photo. Sci., 1, 110 (1953). 28. K. C. D. Hickman, Photo. J., 67, 34 (1927). 29. J. H. Webb, J. Opt. Soc. Am., 40, 3 (1950). 30. J. H. Webb, J. Opt. Soc. Am., 32, 299 (1942). 31. N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals, 2nd ed., Oxford Univ. Press, 1948. 32. G. C. Wallick, Phys. Rev., 84, 375 (1951). 33. E. Katz, J. Chem. Phys., 17, 1132 (1949). 34. L. Silberstein, J. Opt. Soc. Am., 29, 432 (1939). 35. E. Katz, Bull. Am. Phys. Soc., 30, 33 (1955). 36. L. Silberstein, J. Opt. Soc. Am., 29, 67 (1939). 37. F. F. Renwick and V. B. Sease, Photo. J., 64, 360 (1924).

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