Uniform Representation of the Characteristic Curves of Pure SilverBromide Emulsions, Unsensitized and Sensitizedt

LUDWIK SILBERSTEIN

Kodak Research Laboratories, Eastman Kodak Company, Rochester, New York

(Received June 8, 1942)

W HAT is usually called a characteristic curveor, in commemoration of Hurter and

Driffield, an H and D curve, is the plot of thephotographic density D, the common logarithm(Log) of the opacity produced by an exposure E,against Log E, the two variables, Log E and D,being treated as rectangular coordinates. Theopacity or the reciprocal of transparency is a

pure number and the exposure E is the radiantenergy thrown upon the photographic plate orfilm per unit area and is, in recent work, measuredand quoted in ergs/cm 2 .

The general form of a characteristic curve willbe familiar to most readers interested in pho-tography. The gradient or slope of the curve,

g = dD/d Log E

increases continuously (with increasing exposure)from zero up to a more or less distinct max-imum which is commonly called "contrast" or

"gamma," Y=gmax, and then decreases graduallyto nil, while the curve ascends steadily, tendingin normal conditions* asymptotically to a definite

extreme or limiting density, usually denoted byD.. This density is attained, of course, when all

silver halide grains present are blackened. Theo-retically, D,., corresponds to E = oo and thus toLog E= oo. In most cases, the slope g of the

curve varies in the neighborhood of gmax or Myvery slowly, almost imperceptibly, so that a moreor less extended stretch of the curve can scarcelybe distinguished from a straight line, and thisstretch is commonly designated by the practicalphotographer as the "straight-line portion" ofthe characteristic curve. (Within this stretch thedensity increases proportionally, or nearly so, toLog E.) Theoretically, however, assuming thatnot only D, but also its derivative g, are con-tinuous functions of the exposure, the rectilinear

t Communication No. 859, Eastman Kodak ResearchLaboratories.

* Apart, that is, from the so-called "reversal," wlhen thecurve reaches a maximum height and descends again withincreasing exposure.

stretch is infinitesimal, that is to say, the curva-ture vanishes only at a point, namely, where gattains its maximum, y, and this point has to beconceived as the inflection point of the curve. Thispoint, then, is characterized by dg/dE = 0. Its co-ordinates may be called the inflection-exposureand inflection-density and will, hereafter, be de-noted by E and D. In many cases, and notablyfor the unsensitized pure silver bromide emul-sions, the inflection point is placed very nearly athalf the height of the curve, D= Dm; the lowerpart of the curve, the so-called "toe," has thesame shape as its upper part or "shoulder," andthe inflection point is a true center of rotationalsymmetry. That is to say, a rigid rotation aboutthis point through two right angles brings thewhole curve into congruence with itself. In othercases, the characteristic curves show a pro-nounced asymmetry. In a certain class, however,of such cases, to be specified later, the non-symmetrical curve can be split into two perfectlysymmetrical ones. But even the symmetricalcurves differ considerably in shape from eachother in dependence on the peculiarities of theemulsions.

Since the early days of photographic theory, avariety of empirical mathematical formulae for

the density as function of the exposure has beenproposed and tried out. But, with a possibleexception of C. C. Lienau and R. A. Houstoun'sformula' and its modified form as retouched byJ. H. Webb,2 scarcely any of these formulae can

claim to represent adequately the observed curvesin their whole extent, from toe to shoulder.For the history of this subject the reader must bereferred to F. E. Ross's monograph on The Physicsof the Developed Photographic Image.'

The object of the present paper is to describea comparatively simple sensitometric formula,

1 C. C. Lienau, J. Frank. Inst. 218, 35-39 (1934); R. A.Houstoun, Phil. Mag. 21, 1113 (1936).

2 J. I-I. Webb, J. Opt. Soc. Am. 29, 314 et seq. (1939).3 Monograph No. 5 of the Eastman Kodak Company

(Van Nostrand, New York, 1924).

474

VOLUME 32J. . S. A.AUGUST, 1942

CURVES OF PURE SILVER BROMIDE EMULSION

originally derived in 1939 by an asymptotic proc-ess from the quantum theory of exposure, butnot hitherto published. This formula was sincetried out extensively by the writer and turnedout to represent very closely a great number ofcharacteristic curves observed on various emul-sions, under different development conditions.This fact of an extensive agreement with observa-tions may justify perhaps the epithet "uniform"applied to the formula, and to its superpositions,in the title of the paper. At the same time, how-ever, it must be said in advance that, in spite ofits quantic origin, the close agreement of theformula with experiment cannot be consideredas a decisive test in favor of the quantum theory,for reasons which will become clear hereafter. Infine, regardless of the manner of its derivation,the reader may take it, in the present connectionat least, for an empirical or semi-empiricalformula.

The reasoning which has led to the formulain question was as follows:

Let the exposure E amount to n quanta perunit area striking the plate haphazardly. Let abe the "size," i.e., the projected area, of a grainand e its coefficient of photographically efficientabsorption and, therefore y= Ean=cE (say), themean or the probable number of efficient quantaper grain. Then the probability that the grainshall absorb m such quanta is, rigorously,

Pm(fl) =prnM-e py-mPm(n)!

where =ea, q= -p. The size a of the grain issupposed here to be a pure number, namely, thefraction of that area which has been taken aboveas "unit area," and since this area will be ulti-mately assumed to contain many grains, a and,a fortiori, p= Ea is a small fraction of unity, whilen is a huge integer. The mean or the probablenumber of quanta absorbed by a grain is the sumof mPm(n) extended over all integer values of mand this is well known to be equal to np or y, asstated above.

If not only n, but also the numbers m andn-m are great enough to warrant the applica-tion of Stirling's expression for the factorials,n! = (27rn) * nne-', the rigorous probability leads,

approximately, to

1 rd r (M-Y)2I exp[ d m

(2ry) 2 _ 2yX

as the expression for the probability that a grainshall absorb a to A quanta. This expression willbe familiar to most readers in connection with theusual statement of the well-known theorem ofBernoulli.4

Suppose, now, that a grain becomes develop-able whenever it absorbs at least r quanta (i.e.,r or more). Then the probability that the grainshall be made developable by an exposurey=aen=cE will be

1 ri '_ )

P(Y, r)= I exp M-y) dm(2 ry) ~ 2y

or, in terms of the error-function,

2 =

b(x) = f f e-u2du,1/7r o

P(y, r) =- ( °°) )-@i )

and since D is an odd function of its argumentand 4(oo)=1,

P(y, r)=-{1+@Q7)}.

If there are a great many equal grains spreadover our unit area, P(y, r) will give also thefraction of all these grains made developable bythe exposure y. And since the photographic den-sity D is proportional to the number of grainsmade developable and, in particular, Dm propor-tional to the total number of grains over unitarea,5 the resulting sensitometric formula is

D = D*P(y, r)=- 1 + ()| 1

It is this formula which will occupy our atten-tion in the present paper.

It replaces the more accurate quantum

Its simplified derivation is given in a note publishedin Phil. Mag. 26, 223 (1938). In the present case, np=yand q = -p .I in the radical.

5 Cf. J. Opt. Soc. Am. 28, 443 (1938).

475

LUDWIK SILBERSTEIN

formula

D f y2 y-

-=1-e-8 1+y+-+ .+ (A)Dm 2! (r-1)!

used in my previous publications.Rigorously, the Stirling expression for the fac-

torial r! and thus also the formula (1) would bevalid asymptotically, for r->oo, and may be ap-propriately referred to as the asymptotic quantumformula. Practically, however, the Stirling ex-pression is a fair approximation to r!, and thusalso (1) to (A), when r amounts only to severalunits.6 To what extent the asymptotic formula(1) approximates the correct quantic densitieseven for moderate values of r may best be seenfrom the following set of densities for y=r (in-flection point). By (A) we have, to three decimals,

r = 1 2 3 4 5 6 7D(r)/D,,=0.632 0.594 0.577 0.566 0.559 0.554 0.550,

while, by (1), D(r)/Dm=2. For r=6 or 7, thedifference amounts only to 10 percent, at theinflection point, and decreases gradually down tozero at higher or lower exposures. For values ofr, however, as low as 2 or 1 the formula (1) devi-ates from (A) considerably. In fine, for such lowvalues of the number r the asymptotic formula(1) can no longer be considered as an approxi-mate implication of the quantum theory of ex-posure, and still less so for fractional values, r < 1,when the formula (A) becomes meaningless.

Now, while none of the (A) formulae, as theystand, is adequate to cover satisfactorily any ofthe observed characteristic curves in their wholeextent, the asymptotic formula (1) has turnedout to represent very closely indeed a greatvariety of such curves, but always with surpris-ingly low values of the parameter r derivabledirectly from, and determined uniquely by, themain characteristics of the observed curve (Diand y; vide infra). In no case hitherto investi-gated, in fact, did r exceed 2.7 or 2.8, and in thegreat majority of cases was much less than that,descending to fractional values as low as 0.30 or0.25. This, then, is the reason why the closeagreement of the formula (1) with experimentalmeasurements cannot be invoked as a proof of

I E.g., for r= 12 the difference is about 0.007r! and evenfor r=6 amounts to 0.014r! only.

the quantum theory of exposure. The formula,moreover, has been derived above under theexplicit assumption that all grains are equal,while it has actually turned out to fit the curvesof emulsions with a wide range of grain sizes asclosely as the curves of almost uniform emulsions.The number r called for by a given emulsionwould thus have to be viewed as a kind of averageof the individual characteristics of the grainscomposing it.

We shall, accordingly, adopt here a purelyphenomenological attitude and, forgetting theoriginal meaning of r (as number of quanta re-quired for developability) consider it merely asa parameter of our formula to be determined fromsome directly measurable properties of the ob-served characteristic curve.

Before passing on to the applications of theproposed formula, it will be well to examine itsmain properties.

GENERAL PROPERTIES OF THEASYMPTOTIC FORMULA

The formula

D 1 y-r

D., 2 (2y)-

contains in all three parameters: the coefficient ofproportionality of our independent variable tothe exposure, c=y : E, the extreme density Di,and the number r. The parameter c affects onlythe position of the corresponding curve; itschange produces only a rigid shift of the wholecurve along the Log E axis. The parameter D.is a common factor of all ordinates, so that theshape of the curve is influenced by the parameterr only.

Needless to say, for y=0, =-1, D=0 andfor y-* o, =1 and D=Dm.

The most characteristic point of the curve isits midpoint,

y=r, D(r) = D.

7 The situation is, perhaps, modified by the frequentoccurrence of "clumps" of grains which are well knownto have the property that one only of the grains need- beeffectively exposed in order that all grains forming theclump shall be made developable. This would imply anapparent lowering of the r value conceived as an average.At best, the question at issue could for the present bequalified as an open question whose clinching, however,is not one of the purposes of this paper.

(1)

476

CURVES OF PURE SILVER BROMIDE EMULSION

We know beforehand that this will also be itsinflection point, for such was y = r for all thequantic curves (A) corresponding to r= 1, 2, 3,etc., of which (1) is but a limit. This will beverified presently as a differential property of (1)itself.

The midpoint, y = r, has also another remark-able property. In fact, consider any two pointsequidistant from it along the Log y axis or

Iy = mr and y =-r,

m

g = yD'(y)/M, where M= Log e, so that

M 1-g(y) = yu' exp (-u 2 ) .Dm Vdr

(2)

The inflection point y is determined by g'(y) = 0or, after removal of the common factor exp (-

2)(whose vanishing gives the obvious roots y =0and y= co), u'+y[u"-2uu'2]=0, and since

y+r2 (2y3) 2

I = _y+3r4 (2y5)

where m is any positive number whatever. Let ube the argument of the error function in (1).Then, for the first point,

(m-1)r /r -it=-_ = (m -1)1

(2mr) I 2m

and for the second point,

-m) -) -u.2m

H c (1/m-1)r(2rlm)' 2 (

Hence,

Dm DmD(y)=-{1+)(u) } and D(y*)=-{1-D(u)}

2 2or

D(y)+D(y*) =Dm.

This holds for every pair of points y and y* = r2 /y.Thus the curve is rotationally symmetrical and

the point y = r is the center of its symmetry. Arigid rotation of the curve about this pointthrough two right angles brings the whole curveinto congruence with itself, its shoulder and toebeing interchanged. In view of this property,this point will henceforth be called briefly thecenter of the curve and will be denoted by C.

Let us now consider the differential propertiesof the curve (1).

Writing for brevity u=(y-r)/(2y)i, recallingthe properties of the error-function, and denotingderivatives with respect to y by dashes, we have

D'(y) u'= -exp (-u 2 ).

Dm V/

The gradient of the curve is, by definition,8 We may mention that none of the curves (A) is strictly

symmetrical, but they tend to become so with ever in-creasing r.

this equation is reduced to

(y-r)[(y+r)2-y]=o,a cubic for y. One root is

y=r, (3)

as announced above, while the remaining tworoots are, for all values of r greater than ,

complex conjugate.'Thus (unless r<0.25, which is seldom the case)

the curve has a unique inflection point, =r.At the same time, Eq. (2) gives for y or g(r)

Mly/Dm = (r/27r) ,whence

(4)

a simple expression for the parameter r in termsof the directly observable attributes y, Dm of thecurve which is to be represented by the form-ula (1).

In view of the theorem proved a moment ago,only symmetrical curves (or nearly so) can becovered by that formula, which will henceforthbe written briefly

D=DmP(y, r).

A table of the function

P(y, r) =_ 1 +q(-) I

(la)

If r<4, these roots, y, y2 ='[1-2r4(1-4r)i], arenot only real, but both positive and give actual maxima,g(yi) =g(y2), of the slope curve. Since yIy2=r2, these twomaxima are placed symmetrically with respect to thecentral ordinate g(r) which now is a minimum (although,even for r 8, only 10 percent lower than either maximum).When r= 4, the two maxima coalesce with g(r) = -y whichthen is a "flat" maximum and which becomes more andmore "sharp" with increasing r.

477

My 2r = 27r

D.

LUDWIK SILBERSTEIN

which will be found sufficient for all actual casesis given at the end of the paper.

The process of adapting the formula to an ob-served curve is very simple and rapid. Thevalues of y and D,, being read off the given curve(the latter, Di, if need be, after a slight extrapo-lation of the highest measured densities), thevalue of the parameter r is readily determinedby (4). The position of the curve (la) is bestfixed by placing its center C(y=r) upon the ob-served curve, that is to say, simply at the ob-served height D= 'Dm. This gives the preciseinflection exposure E, say in ergs/cm 2 (whichotherwise could be read off but vaguely), andherewith also the last parameter c appearing iny=cE. In fact, c=r/E. By the way in which rhas been determined the two curves have now acommon inflection tangent at C and thus a con-tact of the third order. It suffices then to calcu-late D by (la) for a few. higher steps, D(2r),D(4r), etc., the fewer the greater r. The densitiesbelow the center are then codetermined by thesaid symmetry, viz., D(r/2) =D,,,-D(2r), D(r/4)=Dm-D(4r), and so on. As will be seen from the

P table, the extreme density is practically reachedin three such steps above the center for r = 1, andin five steps for r= 0.30 which is about the lowestr-value called for by any emulsion. And as manysteps below C lead, of course, almost to D=O.For r= 3, which is a little above the highestparameter ever encountered in practice, D(4r) isonly 4 percent below Dm and, correspondingly,D(r/4) = O.OO5Dm, which in view of the p.e. of thedensity measurements may be practically con-founded with zero.

The symmetry of a curve, with Log E asabscissa, is most directly expressed by using,instead of E or y=cE, as independent variable,

x = Log E-Log E= Log (E/E) = Log (y/y).

In terms of x, any symmetrical curve whateveris obviously expressed by

DmD=-l+f(x)}, (5)

2

where f(x) is some odd function of x, and thusf(0) = 0, and y = Df'(0), while f ( oo )=1.

To reduce our formula (1) to this form, noticethat the argument of the error-function 4 can be

written

u = (r/2) [ (cElr) - (rlcE) 1]=(r/2)E [(El-R) - (IE) ]

= (r/2)[lz O-o 10-x/2],or

xu = (2r) sinh

2M

Thus the formula becomes1 x \

D=-D sinh ), (lb)D=2 mi±Fb V2r,51ni

where r has still the value (4). Needless to say,@D has all the properties of f (x) stated above. Thegradient is now, simply, g(x) =D'(x), and g'(x)vanishes at x=0, the inflection point. Differ-entiating once more, one readily finds that thesign of g"(O) is that of 4-r. Hence, if yy=g(0) isto be a true maximum of the gradient, r cannotbe less than . If r<!, then g(O) is actually aminimum, bordered symmetrically by two max-ima. (See footnote 9.)

SUPERPOSITION OF TWO TERMS OF THEFORM (1). ASYMMETRICAL CURVES

As has already been mentioned, the character-istic curves of unsensitized, pure silver bromideemulsions are all (rotationally) symmetrical andcan be very closely represented by the formula(1) or (la), identical with (lb). When, however,any of these emulsions is sensitized by some spe-cial process or other, the symmetry of its curveis destroyed, and the curve can, therefore, nolonger be represented by a single term DmP(y, r).

Now, as has been remarked by Mr. A. P. H.Trivelli of these Laboratories (to whom I amobliged for all the experimental data used in thispaper), there are good reasons to suppose thatthe grains of an emulsion are differently affectedby the sensitizing process and that in this re-spect, the whole emulsion might, perhaps, beconsidered summarily as consisting of two groupsof grains or of two components, each behavingsensitometrically as the original emulsion, butendowed with different parameter values. If suchwere the case, the asymmetrical curves of thesensitized emulsion ought to be expressible bytwo terms of the form (la), that is to say, by

D=AP(yi, r,)+A2 P(y2 , r2),

478

CURVES OF PURE SILVER BROMIDE EMULSION

where Ai+A 2 =Dm, and yi=ciE. Such, in fact,has turned out to be the case for all samples sofar investigated. After some trials, moreover, itwas found that the best superposition of termsis what may be called a two-to-one mixture(A1 =2A2), so that the working formula became

D=Dm P(yi, r)+ P(y2, r2)}. (5)

It remained to determine the values of the fourparameters r, r2 and cl, c2 (implied in yl, Y2)

or, equivalently, the position, Log El, Log A2 ofthe centers of symmetry C and C2 of the twocomponent curves, so as to fit the observed asym-metrical curve. The limiting densities, A 1= Dmand A 2 = 3-D, of the component curves beingknown, the parameters ri, r2 are, in virtue of (4),reducible to the gammas of these curves,

3 MYi) 2r, = 27r- _

2 D

/3M(2 2r2= 2r , (4a)

Dm

be the point y = 1r, and sometimes the centerC1 itself, y=rl.) Thus also the position of theD, curve is fixed. Let it be drawn completely,with the aid of the P table. By construction, thelower part of this curve will follow closely thetoe of the observed curve, say up to Yi = r or ri,and beyond that will fall below the observedcurve, deviating from it gradually more andmore, the limit of the deviation being Dm. Wenow mark, for a number of points, the differenceDob-Di, as ordinate of these points. From thesmooth curve

Dobs-D,, Log E (B)

drawn through these points we read off its gammaand, in order to satisfy the requirement Dobs-D,=D2, we take this gamma for 72, which gives usthe parameter r2= 27r(3M'Y2 /Dm) 2 . The curve

D2= 3 DmP( 2, r2) (5.2)

but -yr, -y2 themselves are unknown and theirrelation to the resultant -y derivable from (5) isof a very intricate nature. Theoretically, the fourparameters, cl, c2 and r 1, r2 (or -y, 72), would bedetermined by four points of the observed curve.But the solving of the corresponding system offour equations containing, through P, the trans-cendental function b of the unknown arguments,would be an extremely laborious task.

To obviate these difficulties, the followingmethod has been adopted.

A preliminary investigation of several caseshaving shown that the toe of the observedcurve can be closely enough represented by thesingle term

(5.1)

the contrast (-yC) of this partial curve has beendetermined by identifying it with the gradientof the observed curve at a point well below itsmidpoint (Dm). In practice, one or two trialsturned out to lead to the best choice of -y. Fromthis value of -y' the parameter r is directlycomputed by the first of (4a). A few points,yl=ri, rl, ri, etc., of the curve (5.1) are thendetermined with the aid of the P table and oneof these points is placed on the observed curve,this point being so chosen that all the lowerpoints also fall upon or very closely to theobserved curve. (In most cases it turned out to

can now be computed, and its center C2 (withdensity -D,,i) being placed on the curve (B),the whole curve (5.2) can be drawn on the samechart in its proper position. This completes theprocess which is by no means laborious.

If the points of the curve (5.2) lie close to thecurve (B), the superposition of the two compo-nent curves, D1+D2, will lie as closely to the ob-served curve. For the differences D2 - (DObs-D,)are manifestly identical with the deviations

A= (D1+D2)-Dob.,

which ultimately interest us.Some details of this general description of the

computation process will become more lucidfrom the concrete examples to which we nowturn.

I. SYMMETRICAL CURVES OF UNSENSITIZEDEMULSIONS

Since the process of adapting the single-termformula (1) to a symmetrical observed curve isstraightforward and simple enough, we shalldescribe it in some detail only for the first of ourconcrete examples. In the case of all furtherexamples it will suffice to give the final resultsonly and the corresponding graph or figure.The abscissa of each figure is Log E, with theexposure E, throughout at constant intensityand varying exposure time, expressed in ergs/cm2 ,unless otherwise stated in the diagrams. The

479

D, = 1 -D.P(yi, ri),

LUDWIIK SILBERSTEIN

D1.0 ohe

0.6

0.4

oz _I 73 0.73 173

LOG E (erq/m')

2.73

FIG. 1. N4X-7229, unfinished. 2' developmentin ferrous oxalate.

development time, in minutes, will be brieflydenoted by t.

1. Emulsion N4X-7229 ("unfinished"), t=2in Ferrous Oxalate

The observed curve (Fig. 1) shows, nearlyenough, the limiting density D= 0.87 and-y=0.8 0. Whence, by formula (4), to two deci-mals, r= 1.00. Thus formula (1) or (la) becomes

D=0.87P(y, 1.00).

This, with the aid of the P table, gives the follow-ing set of densities:

y/r= 1 2 4 8 16 11 12 1 1 -

D=0.43 5 0.66 0.81 0.86 0.87 0.21 0.06 0.01 0.00

The corresponding points are marked in Fig. 1by eyelets. The center C(y=1.00) being placedon the observed curve, which gives Log E = 1.05,the remaining points are all very close to theobserved curve. The greatest deviation A = D-Do b 9 amounts barely to - 0.02 low down in thetoe and to A =0.02 two steps above the center.These small deviations scarcely exceed the p.e.of the density measurements.

Equally close representations by D = D ,P(y, r)have been found for the same emulsion witht=3, 4, 6, 9 and 18 min. in ferrous oxalate, nay,with much the same values of the parameters rand Log E, the former oscillating between 0.88and 1.00 and the latter between 1.03 and 1.07.Of these five cases, it may suffice to quote oneonly.

3, 4. Emulsion S-124-1, t= 5 and 8 in DK-50

The formulae, are, respectively,

D=2.16P(y, 0.325) and D=2.64P(y, 0.305).

See Fig. 3, which shows a perfect agreement ofthe calculated points with both experimentalcurves.

5, 6. Emulsion S-124-1, t=5 and 8 in DK-50

These two curves are represented, respectively,by

D=1.76P(y, 0.31) and D=2.40P(y, 0.28)

very closely indeed, as will be seen from Fig. 4.

7, 8. Emulsion N4X-3131, t=6 and 8 in D-19

The solid curve drawn in Fig. 5 corresponds to

D = 1.60P(y, 1.10)

and represents closely enough both sets ofmeasured densities, marked by crosses for t=6and by eyelets for t=8.

9

Some eighty more experimental characteristiccurves have been similarly investigated incomparison with theoretical ones of the form (1).In order to save space in the columns of thisJournal, the results found may be representedsummarily (and not less instructively, perhaps)in a single diagram, namely, with D/Dm asordinate and the error function 'i itself ( a knownfunction of the exposure, containing c, r asparameters) as abscissa. All the particulartheoretical curves are then condensed in aunique straight line,

DID.n = -(1 (I ),

extending horizontally from ci = -1

01.0

2. Emulsion N4X-7229, t = 9 in Ferrous Oxalate

The observed curve (Fig. 2) is representedthroughout very closely by D= 1.02P(y, 0.985),with center C placed again at Log E= 1.05.The greatest deviation is A= -0.015 only.

0.2

.93 0.93 193LOG E (erq/cml)

(6)

up to

2.3

FIG. 2. N4X-7229, unfinished. 9' developmentin ferrous oxalate.

C

I Ic

l

480

CURVES OF PURE SILVER BROMIDE EMULSION

b = 1 (E = 0 to x ) and vertically from D/Dm = Oup to 1, and condensing, as it were, in its mid-point the inflection points of all those curves.The observed points (with their coordinatesderived from the original curvilinear diagrams)are represented by dots which fall more or lessclosely to this straight line. Such a diagram,summarizing a good number of results found fora variety of emulsions (viz., an x-ray emulsion,Cine Positive, and some ten more emulsions, atdifferent development conditions), is given inFig. 6 which ought to be self-explanatory. Itcontains nearly two hundred observed "points"or dots, not selected ones, of course. Thesepoints cluster throughout very closely about thestraight line (6). It will be seen that even at theupper part of the shoulder, tb=0.85 to 1.00,considerable deviations are rather sporadic only.This diagram seems to give a good cumulativeevidence in favor of the formula D =DmP(y r)as a faithful representative of the symmetricalvariety of the observed curves.

II. ASYMMETRICAL CURVES OF SENSITIZEDEMULSIONS

We shall now consider a few applications of thetwo-term formula (5),

D= DmP(yi, r)+ DmP(y2, r2)-Di+D 2.

In order not only to show its applicability tonon-symmetrical observed curves, but also tobring out the effect of the process of sensitizing,it will be best to consider first an emulsion whoseunsensitized sample has already been investi-gated in Section I.

2.8

2.4

DV. 2.2

2.0

1.8

I.0

0.8

0.

0.4

0.2

5aO 2.80 0.00

FIG. 3. Emulsion S123-1. 5' and 8' development in DK-50.

02.8

a IN. DEV. 2.4

2.2

2.0

>s0 2.80 eoo1.8

I.'

i4

1.2

l.0

0.8

0.4

0.2

0 "o~~.8 0.00

FIG. 4. Emulsion S-124-1. 5' and 8' development in DK-50.

10. Emulsion N4X-7229 Sensitized, t=2in Ferrous Oxalate

This is the same emulsion as that treated in theexamples 1 and 2, above, but sensitized to formsilver sulfide specks.

Its observed curve is the upper solid curvedrawn in Fig. 7. Its pronounced asymmetry canbe seen at a glance. The limiting density, actuallyreached and maintained for a good stretch ofthe curve, is D=1.30. This is split, in nearaccordance with formula (5), into 0.87+0.43, sothat the two component curves become

Di=0.87P(y 1 , r) and D2=0.43P(y2, r2).

Having adopted for -y', after a little trial, thegradient at the lower part of the observed curve,(0.32-0.17): 0.30, i.e., y=0.50, we find, by(4a), r1= 0.39 and thus Di=0.87P(y 1 , 0.39),which gives the following set of densities D1 :

yi/rs =1 2 4 8 1632 I l I iDi =0.435 0.58 0.72 0.81 0.86 0.87 0.29 0.15 0.06 .01 .00

Dobs-Di =0.05 0.16 0.28 0.36 0.40 0.43 0.00 .00 .00 .00 .00

The D points are marked by dots in Fig. 7.One of these, D1 =0.29, being placed on theobserved curve, the dots below it fall all per-ceptibly on this curve, while the center C andthe successive dots (yi/r 1 = 1, 2, 4, etc.) fallgradually away from the observed curve, asshown by the values of Dob s-D 1 , given above.The placing of the D dots on the chart (or thecorrelation of the y and E scales) fixes, at thesame time, the position of the center C1, markedby an eyelet, giving for its abscissa,

Log E1 = 0.47.

481

LUDWIK SILBERSTEIN

D

1.4

086

0.4

0.2

'/8 r/4 '11 7.lz 11 r r ir 32r

FIG. 5. N4X-3131. Developed in D-19.

The differences DOb-Di are now plotted, in situ,against Log E, which gives the curve (so super-scribed) in the lower part of the chart. Thiscurve, which practically coincides with the axisup to about Log E = 0.17, is, unlike the resultantobserved curve, strikingly symmetrical andturns out to be closely expressible by a singleterm of the form (1). From its limiting densityand its gamma one readily finds r2 =0.925,always by (4a). This gives, for the requiredsecond component,

D2 =0.43P(y 2 , 0.925).

The points computed by this formula are markedby eyelets. The central point C2 (D2 =0.215)being placed on the (Dob-D,) curve, whichgives for its position

Log E2 = 0.89,

the remaining eyelets or D2 points array them-selves very closely along the (Db s- D,) curve.Manifestly, the deviations of the eyelets fromthis curve are identical with the deviations of thecorresponding values of the sum D+D2,marked by crosses, from the total observeddensities. The greatest of these deviations,A=D,+D 2 -Dobs, amounts barely to 0.02. Infine, the observed asymmetrical curve is veryclosely represented in its whole extent by

D=0.87P(y,, 0.39)+0.43P(y2, 0.925),

the sum of two symmetrical curves. The coeffi-cients ci, C2 = r,/E1, r 2 /22 implied in yi, Y2 are co-determined by the position of the centers C1, C2of these curves,

Log ,1= 0.47, Log E 2 = 0.89.

For the unsensitized emulsion, with exactlythe same development, we have found (Ex. 1) for

1.0 - .0

0.9 . 0.9

0.8 _ 0.8

0.6 / 0.7

0.6

05 0 , .S

OA

0.2

0.1

-1 -0.0 _0. -0.4 -0.% 0 0.2 0.4 0. 0.6 I

FIG. 6. Observed D/Dm plotted against F.

2.93 .93 0.93 1.93

FIG. 7. N4X-7229, sensitized. 2' developmentin ferrous oxalate.

the center of the single symmetrical curve

Log E= 1.05.

The reciprocal of the inflection exposure, atwhich half the limiting density is reached, maybe considered as a measure of the sensitivity of theemulsion, under given development conditions,s= 1/E. Similarly, for the two components ofthe sensitized emulsion, S,, S2= 1/B1, 1/E2. Thuswe have, in cm2/erg, for the unsensitized emul-sion, s=0.089, and for the primary'0 and thesecondary components of the sensitized emulsion,

Si = 0.34, S2 =0. 13

or s,=3.8s and S2= 1.45s. The process of sensi-tizing produces thus a kind of bipartition of theemulsion, increasing the sensitivity of its mainpart about four times and that of the remainderonly 1.5 times. The grains of the original emul-sion may well be considered as endowed withdifferent susceptibilities to the sensitizer depend-ent on some difference in their individual pecu-liarities (such as size) which is revealed only bythe sensitizing process. Whether the greater sus-ceptibility belongs, on an average, to the largergrains or not, has not, so far, been ascertained.

An equally close representation by the two-

10 We may call it so since it makes up two-thirds of thewhole emulsion with regard to limiting density.

482

CURVES OF PURE SILVER

term formula has been obtained for the curvesof the same emulsion for five more develop-ment times, =3 to 18 min. in ferrous oxalate,as well as for 3 to 12 min. in DK-50. Of all thesecases, it will suffice to quote here only the follow-ing one, since it is directly comparable with thattreated in Ex. 2.

distinctly asymmetrical. The first is closelyrepresented by

D=1.05P(yi, 1.55)+0.53P(Y2, 1.55)

with centers placed at

Log E,=0.29 and Log E2=0.97

11. Same Emulsion, but with t = 9 inFerrous Oxalate

Proceeding exactly as in 10, one finds thesuperposition of two symmetrical curves

D= 1.06P(yi, 0.345)+0.53P(y2 , 1.69),

which agrees with the observed curve as closelyas in the last case. All details can be seen fromFig. 8, where all symbols and marks have thesame meaning as in Fig. 7.

The two centers of the symmetrical componentcurves are now placed at

Log El = 0.42 and Log E2 = 0.795,

while the unique center of the unsensitizedemulsion, with exactly the same development(Ex. 2) had the position

Log E= 1.05.

Thus the sensitivities are

s = 0.089; s1 = 0.38, s2=0.16; (t=9').

s is the same as, and S 1, S2 are but slightly greaterthan for 1= 2 min. The "bipartition" or the selec-tive effect of the sensitizer is thus, nearly enough,independent of the development time.

12. Emulsion N4X-7233, t=2 and 3 in DK-50

This is the same emulsion as in Ex. 1, butsensitized with active gelatin. Its two curves(Figs. 9 and 10 for t=2 and 3, respectively) are

0.93LOG E (N./et')

FIG. 8. N4X-7229, sensitized. 9' developmentin ferrous oxalate.

and Am= 0.035 as greatest deviation, and thesecond is equally well represented by

D= 1.19P(yi, 1.64) +0.59P(y2, 1.64)with

Log E1=0.25 and Log E2 = 0.91

and Am= it0.04. An interesting peculiarity of thiscase is that, to our approximation, r = r2 foreither development time, so that the two com-ponent curves are essentially the same, differingonly as to vertical scale and position. Thesensitivities of the components are

s1 =0.51, s2 =0.11 for t=2 min. in DK-50

ands = 0.56, s2 = 0.12 for t=3 min.

In fine, the sensitivity of the primary componentis about five times that of the secondary com-

D

I.'

OLW.

58

01, 0, . -IO

-0.55C D..

_ >~~~~l

0.13 .13

LOG E (erm-l)2.13

FIG. 9. N4X-7233, finished with active gelatin.2' development in DK-50.

1.13

LOG E (er9 /cm2

)

FIG. 10. N4X-7233, finished with active gelatin.3' development in DK-50.

483BROMIDE EMULSION

toc

LUDWIK SILBERSTEIN

LOG E

FIG. 11. Emulsion I. XX D, +D 2. D2 =0.76P(0.449E, 2.05).Di= 1.54P(O.158E, 0.50).

080

LOG (erqs/cm')

FIG. 12. XXX D,+D2 . C, C 2 , centers (inflectionpoints). Emulsion II. D=o1.13P(0.975E, 0.93). D2=0.57P(0.185E, 1.04).

ponent and the effect of the prolonged develop-ment on s1, S2 is almost negligible (although thetotal extreme densities for t = 2 and 3 differ fromeach other as much as 1.58 from 1.78). Thesymmetrical curves of the unsensitized emulsiondeveloped in DK-50 (not reproduced here) showthe inflection exposure Log E= 1.02 for theshorter, and 1.00 for the longer development andthus the sensitivity s=0.095 and 0.10, not differ-ing much from that found with ferrous oxalateas developer.

13. A Set of Seven Converted(Ammoniacal) Emulsions

These emulsions differ only as to their precipi-tation time (tp). Their investigation, throughthe two-term formula, has afforded the additionalinterest of ascertaining how the parameters ofthat formula, and especially the two sensitivitiesS1, S2 depend on the average grain size (a) of anemulsion which increases steadily with t,.The precipitation time affects not only a, butalso the standard deviation a and thus the shape

D

1.4

1.2

1.0

0.8

O.G

04

0.2

D2.0

1.8

1..

1.4

1.2

1.0

0

q6

0.4

0.2

D

to

.8

0.4

0.4

0.2

0

0.1

04

D , 1.20

0.80

0.40

2.80 t8o 0.00 1.80

LOG E (erqs/cmZ)

FIG. 13. XXX D1+D2 . Emulsion III.

0, 80

D,

2 f , -1 L

2.80 .8o 0.80 1.s0

LOG E (eiq^/cm)

FIG. 14. XXX D1 +D 2 . Emulsion IV.

8 b. 0.97

00.24

iso T.80 0.80 1.80

LOG E (eIqs/cm.)

FIG. 15. XXX D 1+D2 . Emulsion V.

2.00000LOG E (eCqs/o^)

FIG. 16. XXX D1 +D2 . Emulsion VI.

.- D -0C1I- ~~~~~~~~~~, 0.4I- , 02131

200 00 0.00

LOG E (Crq*/c.,)

FIG. 17. Emulsion VII.

of the size-frequency distribution curve. Both,a and a, will be quoted presently, but in thepresent connection we shall be concerned onlywith the average grain size a.

The emulsions will, for brevity, be referred toas emulsions I,I ItoVII in ascendingorder of their

484

1.00 200

CURVES OF PURE SILVER BROMIDE EMULSION

precipitation times. The corresponding values of for space economy) will suffice, perhaps, to showa and a- (in 42), for the seven emulsions are: that the asymptotic formula D=DmP(y, r) and

Em. I II III IV V VI VII its superposition, D=AiP(y 1, ri)+A 2 P(y2 , r2),a=0.34 0.52 0.75 0.99 1.56 2.07 2.89 even with the particular ratio A 1 : A2 =2: 1,o=0.18 0.28 0.44 0.64 1.17 1.72 2.72 may claim to be uniformly close, and easily

The characteristic curves of these seven handled, representatives of the characteristicemulsions, each for a 2-min. development in curves of (respectively) non-sensitized andDK-50, are the upper solid curves, marked Dobs, sensitized emulsions, so far, that is, pure silverin Figs. 11 to 17, corresponding to emulsions I bromide emulsions.to VII. The calculated sums Di+D2 are marked The somewhat intricate complications pro-by crosses. The complete expressions found for duced by the addition of iodide are now underD1 and D2 are inscribed in the figures; they con- investigation and their discussion must betain the values of the proportionality coefficients relegated to a future opportunity.c1, c2 explicitly. Thus, e.g., for the emulsion , My best thanks are due to Mr. A. P. H.

Di = 1 54P(0. 158E, 050), Trivelli for the experimental material utilizedD2=0 .76P(0.449E, 2.05), in this paper and for some valuable suggestions.

C1 and C2 are the centers of symmetry of thecomponent curves.

It will be seen from Figs. 11 to 17 that allthe observed curves are represented throughoutvery closely by D1 +D2 .

From the position of the two centers C1 and C2,the sensitivities si and s2 are derived directly, asexplained before. Their values are:

Em. Ia = 0.34si = 0.31S2 = 0.22

II III IV V VI VII0.52 0.75 0.99 1.56 2.07 2.891.07 2.63 2.51 4.17 5.89 7.590.18 0.41 0.32 0.50 0.46 0.68

While s2 shows comparatively small and erraticvariations, the sensitivity s1 of the primary com-ponent of the emulsion is (apart from a slightupward jerk at a= 0.75) very nearly a linearfunction of the average grain size of the wholeemulsion, steadily ascending with this grain size.

An equally close representation by the two-term formula has been obtained for the curvesof the same seven emulsions found experimen-tally at a few other development times in DK-50extending up to 8 min. and at six developmenttimes in ferrous oxalate ranging from 1 to 18min. The behavior of the two sensitivities s1, s2,derived from the corresponding parameter values,has turned out to be very much the same asabove. The peculiarities of these curves (and oftheir representation) being quite similar to thoseshown in Figs. 11 to 17, these curves need not bereproduced here.

The numerous examples given in this paper(with many others worked out, but omitted here

APPENDIX

The following table of the function

P(y, r) = _(2_y) i) }

will be found convenient and sufficient for most or allpractical applications:

TABLE I. P(y, r)

\yr\ r 2r 4r 8r 16r 32r 64r

0.25 0.500 .638 773 892 970 997 1.0000.30 0.500 .651 794 912 980 999 1.0000.35 0.500 .662 813 928 987 999 1.0000.40 0.500 .673 829 941 991 1.000 1.0000.45 0.500 .682 843 952 994 1.000 1.0000.50 0.500 .692 856 960 996 1.000 1.0000.55 0.500 .700 867 967 997 1.000 1.0000.60 0.500 .708 877 973 998 1.000 1.0000.65 0.500 .716 887 977 999 1.000 1.0000.70 0.500 .723 895 981 999 1.000 1.0000.75 0.500 .730 903 984 999 1.000 1.0000.80 0.500 .736 910 987 1.000 1.000 1.0000.85 0.500 .743 917 989 1.000 1.000 1.0000.90 0.500 .749 923 991 1.000 1.000 1.0000.95 0.500 .755 928 992 1.000 1.000 1.000

1.00 0.500 .760 933 993 1.000 1.000 1.0001.10 0.500 .771 942 995 1.0001.20 0.500 .781 950 997 1.0001.30 0.500 .790 956 998 1.0001.40 0.500 .799 962 998 1.0001.50 0.500 .807 967 999 1.0001.60 0.500 .815 971 999 1.0001.70 0.500 .822 975 999 1.0001.80 0.500 .829 978 9995 1.0001.90 0.500 .835 981 1.000 1.000

2.00 0.500 .841 983 1.000 1.0002.20 0.500 .853 987 1.0002.40 0.500 .863 990 1.0002.60 0.500 .873 992 1.0002.80 0.500 .882 994 1.0003.00 0.500 .890 995 1.000

485