MATHEMATICAL ANALYSIS OF A PHOTOGRAPHIC

CHARACTERISTIC CURVE

A. P. Ivanov and V. A. Loiko UDC 77.018

It is usual to determine by experiment the charac te r i s t i c blackening curve f rom which one de t e r - mines the numerica l values for the photographic constants of a material ; this does not allow one to examine in detail the effect of sequential variation in the different proper t ies of the mater ia l , since change in one pa rame te r over a wide range usually does not allow one to maintain the other pa rame te r s constant on ac - count of the emulsion preparat ion technique. As a resul t , one gets an incor rec t analysis of the trends. It is therefore more convenient to pe r fo rm a theoret ical analysis for the effects of numerous pa ramete r s on the exposure-blackening relationship.

The nature of the blackening is dependent on the scat ter ing of the radiation in the mater ia l , the fo r - mation of the latent image, and the development; the charac te r i s t i c curve has been calculated in [1] for two l imit ing situations, where the grain s ize was much g rea te r or much less than the wavelengths of the incident light.

Here we give a detailed analysis of the effects of the various factors on the charac te r i s t i c curve for coa r s e - g r a ined mater ia ls .

It is found [1] that the direct ional optical density is

r--I t n--I

D I I = - ~ - x ' - - i - - g ) . (1) b., (n @ 1)! (l@bl--b,, 'r)n+l+i-i (l~:-bl)n+l+i-i

i=0 i=0 g=0

Here

b~ -- 0"27k~C1 El, b,_ - 0.27k~,Ce Er C~ = (1 -" R) (1 - - R - - 4RL~_)) , hv[~ h~.[3 I - - R 2 (1 - - 4Lx)

C,= 2 ( 1 - - R 2) L , L : l / - ~ ' ~ ) 2 : 2 ( 1 - - A) Aq~, 1 - - R 2 (I - - 4Lr)

R - - 1 ' I - - A . / - / l _ A \ 2 - - 2 1 _ - j ~ - 3 - - x 1

| / + 8

We consider now in more detail the meaning of the pa r ame te r s appearing in (1).

The quantity e u represen t s the attenuation coefficient for the undeveloped film; it is related to the optical thickness 7 of the layer by T = eUh, where h is the geometr ica l thickness. The probabil i ty that a photon will survive in an e lementary volume is A = ~u/eu , where ~u is the scat ter ing pa rame te r for the undeveloped film.

In calculating the par t ic le size distribution, the volumes v are specified in te rms of the y-function

~n+l f (V) = ' V" e "-~',

n!

where n and fl a re pa r ame te r s of the function.

Transla ted f rom Zhurnal Prikladnoi Spektroskopii, Vol. 18, No. 2, pp. 300-304, February , 1973. Original ar t ic le submitted January 28, 1972.

�9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A eopy o f this article is available from the publisher for $15.00.

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-J -2,5 -2 -/5 / Lg~ Fig. 1. Cha rac t e r i s t i c curves for A = 1 (solid line) and 0.95 (broken).

It is found [I] that

The c h a r a c t e r i s t i c sca t t e r ing indicatrix for an e l emen ta ry volume of the undeveloped f i lm x(T), where y is sca t t e r ing angle, is the f i r s t coefficient in the expansion with r e spec t to L e g e n d r e polynomials :

l

xl --- @ S x(y) cosy d cosy. --1

The quantity e d is the attenuation index of the developed f i lm, when this has rece ived an infinite number of photons; this is dependent on the par t i c le d ispers ion , the development t ime , and the type of developer .

ed = ~l M n + 1

Here M is the number of undeveloped par t i c les in unit volume, while ~ is the coefficient of propor t ional i ty . If one t r a n s f e r s f rom the num ber of pa r t i c les in unit volume to the mass concentrat ion c, one can r e l a t e this to the absorpt ion capaci ty of the s i l ve r K:

d .5 ~ - - C ~ - K C ,

d

where d is the speci f ic g rav i ty of the developed par t i c le .

P a r a m e t e r r defines the min imum number of photons averaged over all g ra ins [2-4] that have to be absorbed in o rde r to make a grain develop; this is dependent on the energy of the absorbed photon, the pa r t i c l e s ize dis t r ibut ion, the dis tr ibut ion of the t r a p s over the g ra ins , and the t r ap depths.

If r = 1, i .e . , it is sufficient to ab s o rb one photon for a grain to develop, the fo rmula for the optical densi ty takes the s imple f o r m

Dll = ~ - ~ (n + 1) b~ (1 + b I .... b2~) n+l (I + ~)~+i " (2)

It is c l ea r f rom (1) that the a rgument in construct ing the cha r ac t e r i s t i c curve may not be the ex- posu re Et at the upper su r face of the l aye r (E is il lumination and t is t ime) but the genera l ized quantity B = (kb/hVfl)Et, where k b is the absorpt ion p a r a m e t e r of the undeveloped par t i c le , with hv the energy of the incident photon.

Then a genera l f ami l i a r i t y with (1) shows that each of the constants may be desc r ibed in t e r m s of a r e s t r i c t e d number of genera l i zed p a r a m e t e r s , in spi te of the va r i e ty of p rope r t i e s of the photographic f i lm. As r e g a r d s pa r t i c l e s of d i a m e t e r much g r e a t e r than the wavelength, these p a r a m e t e r s a r e ed /eu ; A; xt; ~; r; n; kb/hVfl. The re fo re , to elucidate the physical p ic ture it is des i r ab le to analyze the effects of each such p a r a m e t e r on the fo rm of the curve .

We begin our analys is with the ra t io e d / e u. If the exposure tends to infinity, then (1) shows that D II ~ (ed /eu) r ; this means that 8 d / e u c h a r a c t e r i z e s the l imit ing blackening of the fi lm. It will be c lea r f rom this fo rmula that this ra t io appea r s on the r ight as a fac tor , so the optical densi ty on the f i lm at any instant is d i r ec t ly propor t iona l to e d / e u , and so the subsequent analysis is bes t p e r f o r m e d via the quantity D~ = Dl l / ( s which shows that the blackening at any instant of development is e x p r e s s e d in p r o p o r - tions of D ]l for B = ~.

We now examine the fo rm of the c h a r a c t e r i s t i c curve as affected by A and x I for r = 1, in which case we can use (2).

F igure 1 shows c h a r a c t e r i s t i c curves for n = 32, x 1 = 2.2, and X. H e r e and in Fig. 2 the absc i s s a is B* = 0.27 B. This range in photon surv iva l probabi l i ty covers a l a rge range of rea l emuls ions . F igure 1 shows that reduct ion in A causes a loss of con t ras t and sens i t iv i ty in the f i lm, which accords with the conclusions of [5]. We see f r o m (2) that these changes become m o r e apprec iab le at l a r g e optical th i ck - n e s s e s .

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Fig. 2. Charac te r i s t i c curves for: a, b) n = 32; a, c) r = 4; b, c) T = 4. The numbers on the curves are : at w; b) r; c) n.

We have calculated blackening curves for various indicatr ices that occur in photographic f i lms, which corresponded to x 1 f rom 1 to 2.85, and this showed that for A = 1 the effect on the contras t as x(y) increased was so slight that it could not be revealed in printed f igures.

The curves for different optical thicknesses (Fig. 2a) show that the contrasts and the maximum density increase with 7; these resul ts agree with experiment [5, 6]. In this and the subsequent cases the analysis has been pe r fo rmed for x I = 2.2 and A = 1.

Figure 2b shows charac te r i s t i c curves for various r; as r is reduced under otherwise constant con- ditions, the sensi t ivi ty on the fi lm increases , while the contras t is reduced.

Figure 2c shows the effects of f(v) on the blackening curve; the pa rame te r s n and fl charac te r iz ing f(v) have simple relat ionships to the volume corresponding to the peak in the distribution v 0 = n/fi and the half-width at the 0.6 level for f(v0), which is | = 2.48~n/fi . If we assume that fl is constant, then increase in n increases the half-width and the peak in the distribution. This variation in f(v) actually occurs during physical ripening of the emulsion, and the figure shows that increase in n (i.e., increase in v 0 and| ra ises the sensi t ivi ty of the film, while the cont ras t remains constant. This at f i rs t sight would appear to con- flict with experiment , but one must bear in mind that in the calculation we assume that the other p a r a m - e ters remained unaltered when f(v) varied, whereas many quantities are t rans formed during physical r ipening of an emulsion. This means that the contrast actually is reduced in the p rocess .

It must be borne in mind here that the comparison on the calculations with available experimental evidence is only qualitative; a quantitative compar ison is difficult because we do not have available suf- ficient information on the values of the optical pa ramete r s defining the charac te r i s t i c curve. However, the formulas given here show c lear ly the physical pattern of the phenomenon and enable one to elucidate the effects on the curve, such as the light sensi t ivi ty, the contras t coefficient, the useful photographic la t i - tude, and so on, while enabling one also to relate the sensi t ivi ty to the various pa rame te r s .

An analogous mathematical analysis of charac te r i s t i c curves can be per formed via the data of [1] for f ine-grained photographic mater ia ls .

1. 2. 3. 4. 5. 6.

L I T E R A T U R E C I T E D

A. P. Ivanov and V. A. Loiko, Vests[ Akad. Nauk BSSR, Ser. Fiz.-Mat. Navuk, No. 5, 113 (1971). S. P. Shuvalov, Zh. Fiz. Khim., 8, No. 3, 387 (19367. S. P. Shuvalov, Zh. Fiz. Khim., 8, No. 4, 514 (1936). S. P. Shuvalov, Zh. Fiz. Khim., 11, No. 3, 384 (1938). I. H. Webb, JOSA, 38, 1, 27 (1948). Yu. N. Gorokhovskii, Spectral Studies of the Photographic P roce s s [in Russian], Fizmatgiz , Moscow (1960).

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