EBRUARY, 1939

The Effect of Gradual Light Absorption in Photographic Exposure*

LUDWIK SILBERSTEIN

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

(Received October 21, 1938)

IN our previous treatment of the subject all thesilver halide grains contained in the layer of

emulsion were assumed to be exposed to one andthe same light intensity. This is legitimate inthe case of a very thin coating, the so-calledsingle-layer coating, as applied in microphoto-graphic experiments. Under usual circumstances,however, when the thickness of the emulsionlayer is not negligible, the radiation is graduallyabsorbed in traversing the layer, so that thegrains placed at a greater depth receive a lesserexposure than those spread at or near the uppersurface of the layer. It has, therefore, seemedinteresting to investigate to what extent ourprevious formulae are modified by this factor.

The effect of gradual light weakening byabsorption has been treated by W. J. Albersheim.1

His investigation, however, is limited to theideal case of equal grains, and the resultingformula contains an integral which cannot beevaluated in a finite form and is thus notamenable to an easy quantitative discussion.The corresponding sensitometric curve, for singlequantum hits, is in Mr. Albersheim's ownopinion "unsatisfactory," though it is somewhatimproved by the assumption of double quantumhits. It may be mentioned also that Albersheim'selementary exposure formula does not imply atall the size of the grains.

Now, it has occurred to me that if instead ofequal grains a whole range of grain sizes isconsidered, especially of the exponential type offrequency distribution, the aforesaid integral canbe readily evaluated in finite form, leading to acompact aggregate of rational algebraic andlogarithmic terms. The reason for this reduci-bility is that an integration over all sizes simpli-fies (rationalizes) the expression which is theneasily integrated over the whole thickness of theemulsion layer. This remark will become plainfrom the actual derivation which follows.

* Communication No. 691 from the Eastman KodakResearch Laboratories.

1W. J. Albersheim, J. Soc. Mot. Pict. Eng., p. 417,October (1937).

The total thickness of the emulsion layerbeing b, let, in our previous notation, E= en bethe incident exposure (viz. at the upper surfaceX = 0), K the absorption coefficient of the emulsionand, therefore, if we assume Beer's simple law,

(1)

the exposure at a depth x. Let Na=f(a)da bethe number, per unit area of the base, of grainsof size (area) ranging from a to a+da, supposedto be uniformly distributed throughout thethickness of the emulsion. Then the number ofsuch grains contained within a layer of thicknessdx placed at a depth x will be Nadx/b, and thenumber of those affected by the exposure Ex, onthe single-quantum assumption,

Nadx( -e-aEx).

b

Thus, the total number of grains of this classmade developable within the whole layer,

b

ka =(Na! b) (1 -e-Ea)dx.

Instead of x introduce the integration variablez=e--. Then, by (1), Ex=Ez, dx= -dz/KZ, and

Na r d dzIJ (1 e-az)_.

bK eKb( z

Here e Kb = T is the transparency of the plate andbk= -log T, so that

Na " dzka = Na+- e-aE_

log TJT z

Thus, the density D=M~aka due to grains ofall sizes,**

D = MJaf(a)daM ro rb1 dz+--f af(a)f e-aE-*da,

log TJ a=O z=T Z

** We assume, of course, that the opaque emulsion withthe unaffected grains is removed in the process of fixingand thus does not contribute to the density D.

67

VOLUME 29J. . S. A.

E., = Ee-11x

LUDWIK SILBERSTEIN

where M = Log e=0.4343, or, since

Mf af(a)da = M- Na=Dm

is the limiting density,

M r rr dz]D = Dm+ I af(a)II eaE-zIda.log TOO L T ZJ

Now, as mentioned above, the bracketed in-tegral, relating to equal grains, is not reducibleto elementary functions.t But we can invert theorder of integrations with respect to z and a,writing

M 'dz 0 0D-D. = og -TfTJ af(a)e-aEzda, (2)

and then, for certain forms of the frequencyfunction f(a), the second integral becomes asimple algebraic function of z and the completeintegral can at once be evaluated.

Such is the case especially for an exponentialsize-frequency distribution. In fact, if

f(a) = (N/a)e-afia

where N is the total number of grains over unitarea of the plate and a the average grain size,then

J'af(a)e-aEzdao

N co Nd=-I ae-a(Ez+fii)da =N

a -o (1 +aEz)2

Thus, by (2), and recalling that MNa=Dm,

D 1 1 dz

Dm log TOT z(1+aEz) 2

1 paE du=1+ 1

log TJET'U( +U) 2

du u 1Now, or = log + _

UJ u+u)2 l+u l+U

Thus, with the abbreviation y = E, the photo-

tj O(z/)dz is the so-called exponential integral,usually denoted by Ei(z).

graphic density formula ultimately becomes

D 1 I+Ty I 1

Dm log T lg +- 1+Ty(3)

Here T is the transparency of the unexposedmaterial, a proper fraction which is, generally,about 5 and sometimes even as low as 10,Thus the effect of the gradual light absorptionwill be considerable.

If we let T= 1-a tend to unity (perfecttransparency), then

1+Ty dylog -

1+y l+y1 1 dy

1+Ty +y (+y)2'

and since log T= -a, (3) reduces to

D y(2 +y)

Dm (l+y)2'(3.1)

identical with our previous formula based on theassumption of a unique exposure, equal for allgrains.

To see the effect of gradual absorption takethe case of T= 0.1. Then log T - 1/M and Eq.(3) becomes

D 1+y 1 1 =-Logl +MDm l+O.ly 1+O.ly +y

This gives the following set of relative densities.

IL IL 1 1 IY 32 16 8 4 2 1D/Dm 0.024 0.047 0.089 0.163 0.279 0.437

y 1.466 2 4 8 16 32D/Dm 0.536 0.615 0.776 0.892 0.957 0.986.

The corresponding curve, with

Log y Log (aE)

as abscissa and D/Dm as ordinate, is drawn inFig. 1, together with the curve (3.1) correspond-ing, caeteris aribus, to T= 1. The two curvesshow a marked difference.

Returning to formula (3), for any T let uswrite it briefly

Dm MNdD = F(y) = F(y).

log T log T

68

LIGHT ABSORPTION IN PHOTOGRAPHIC EXPOSURE

FiG. 1.

The gradient at any point of the D, Log ycurve is

Nag= yF'(y).

log T

The inflection point of the curve is determined by(dg/dy)= 0, i.e., by the equation

F'(y) +yF"(y) = 0.

Now, by (3),

T(2F+Ty) 2 +yF (y) = ~--~

(1+Ty)2 (l+y)2'

whence F", and the last equation becomes, aftersimple reductions, (1 + Ty)3 = T(1 +y)3 , or

1 +Ty- = T1, (4)

l+Y

a linear equation for = ad-, belonging to theinflection point. The corresponding density is,by (3),

1 1-Ti_=__ -. ~~~~~~(4.1)

D,, 3 (1+ TY) log T

Finally, substituting in the general expressionof the gradient g, we have gamma or the"contrast,"

Nalog Ty (y), (5)log T

where F(y) is as above.

Thus, e.g., for T= ,

y= E= 1.466, D/Dm=0.536,1(T= 0.1)

,y =0.259Na.)

For T= 1 we have, by formula (3.1),

5Y=2, y=(8/27)Na=0.296Na, (T=1)so that y is somewhat lowered by the gradualabsorption of light, while the inflection point isshifted from aE=' to 1.47, i.e., to an almostthreefold exposure, as shown in the diagram ofthe two curves.

We have, so far, adhered to the assumption ofsingle-quantum hits. The case of two or morelight quanta, as the minimum required for thedevelopability of a grain, can be treated withequal ease, leading again (for exponential fre-quency distribution) to rational algebraic in-tegrals. It may suffice to develop here in detailthe case of two quanta. At the same time,however, the general formula (2) can be written,for any distribution, in a more convenient form.

The exposure at a depth x being E = E e-kx, asbefore, let

D.(dx/b)

be the photographic density contributed by thelayer dx under this exposure. Since the distribu-tion of grains throughout the depth is supposedto be uniform, D itself stands for the densitywhich would be given by the whole coating if ithad received throughout a unique exposure E,i.e., in absence of absorption. Now, the totalactual density D is the sum or integral of thedensities contributed by the infinitesimal layers.Thus, if we write again e-KX=Z,

lD b l CT dZD- Ddx=- D,-boo bK, uZ

or, ultimately,

1 CT dzD= lo J D (Ez) (

where (Ez) is the density which would beobtained by giving to the whole emulsion a uniqueexposure Ez, in other words, as if the plate wereperfectly transparent. This equation holds, ofcourse, for any emulsion and independently of

69

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LUDWIK SILBERSTEIN

any theory of exposure, being based only on the

additivity of densities.Thus, if D is known as a function of exposure,

the density D, modified by the gradual absorp-tion, follows from it by the indicated integration,whether the D, E curve has been arrived attheoretically or is just an empirical curve.

Before applying the Eq. (6) to a particularform of D, we may derive from it some generalconclusions.

Write for the moment Ez = u. Then, sincedD/dE=zD'(u), we have from (6)

dD 1 TE D(TE)-D(E)= J 13(u)du= Eo

dE ElogTE log T

Now, the gradient at any point of the D, Log Ecurve is g = (E/M) (dD/dE). Thus,

D(TE) -D(E)a=~L

Log T(6.1)

If the D, E curve is given, say graphically, thegradient of the resultant D, Log E curve can

be read off directly by taking the difference of

pairs of ordinates D, at E and at TE. If, forexample, T=0.1, Log T= -1, and

g=D(E) -D(-oE).

Thus, if but a single point of the D, Log E curve

is known, the whole curve can be traced.

The inflection point of the D, Log E curve is

determined by dg/dE = 0, and thus by theequation

TP(T.2) = '(E2), (6.2)

and if the value of .E derived therefrom is sub-

stituted in (6.1), we have

11y = [D (TR) - D(E)] (6.3)

Log T

without ever evaluating the integral in (6).Let us now apply these general formulae to

the case of an exponential distribution f(a), asbefore, but under the assumption that, insteadof one, two quanta are required for the developa-

bility of a grain. When the gradual absorption

of radiation is disregarded, we have in thiscase2

_ y2(3+y)D=Dm (7

where y = aE, as before. Thus we have to substi-tute in (6)

y 2z2(3 +yz)D(Ez) =Dm (1+yz) 3 (7.1)

This gives

D rT z(3+yz)log T .-=y 2 I dz= y2{31+yI2},

D, (1 +Yz) I

where

T zdz 1 1 1 TY

Ji (1+yz)3 y2 2(1+u)2 l+u

T z2 dz 1 1+Ty

and 2 = ( =- log1+yz)3 Y3 1 +y

1 1 1 2+- ~ - --1.2y + (1+Ty)2 (1+y)2 y

Whence, after simple reductions, the requireddensity formula corresponding to the assumptionof two quanta,

D 1 1+Ty y Ty 1-= j log + - . (8)Dm log T l+y (l+y) 2 (1+Ty) 2

The first bracketed term is as in (3), for single-quantum hits, but the last two terms aremodified. This makes an essential difference.For, whereas, for a single quantum D is initiallyproportional to the first power of the exposure,we now have, from (8), the power series

D 3 8log T-=y 2 -(T 2 -1) -- (T3 )y+...* (8.1)

Dm 12 3

which starts with the square of the exposure,y =a2E2. Notice that log T<0, so that theterms of the series for D are intermittentlypositive and negative. Such, in fact, is also the

2 Cf. L. Silberstein and A. Trivelli, J. Opt. Soc. Am.28,441 (1938).

70

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LIGHT ABSORPTION IN PHOTOGRAPHIC EXPOSURE

series in absence of absorption, but the coeffi-cients of the terms are now numerically modified.

From (7) follows

= 6Dmy/(1 +y) 4,

so that the Eq. (6.2) for the inflection point of theD, Log E curve becomes

T 2y/(1 +Ty) 4=y/(I+y) 4,

whence the simple formula

y=aE= 1/V\T. (8.2)

For T= 1, y= 1, in agreement with our previousresult. With decreasing transparency the in-flection point is, of course, shifted forward, theinflection-exposure being proportional to thesquare root of the opacity. This value of gives at once, by (6.3) and (7), the formula forthe corresponding gamma,

-y (3+V\T)T-(1+3V/T) F(T)Dm Log T-(1+,T) 3 -= ~ , say.

Log T(8.3)

The ratio of -y and the limiting density is thusa function of the transparency T alone. For T= 1its value is 3/8M,* and for T= ' somewhatsmaller, namely, 0.308/M.

The formulae (8) to (8.3) hold under thetwo-quanta assumption. We might treat, withequal ease, the case of three or more quanta,which would give densities initially proportionalto the cube or higher powers of the exposure.But since none of the emulsions so far investi-gated shows an increase of density more rapidthan that of E2 , the two cases treated above willsuffice.

The relative photographic density D/Dm isexpressed by (8) as a function of the variable yalone with the transparency T of the emulsionlayer or, practically, of the whole film,t asparameter, without any adjustable constants.The variable y has been defined by y aE,E= en, so that

y=aen,

where e is a purely numerical fraction, to beevaluated experimentally, and n the total

* Which is about 9/7 times the value corresponding tothe single-quantum assumption.

t If the opacity of the base is negligible.

number of light quanta, per unit area, pene-trating the layer of emulsion just at its upper-most surface. The number n is thus proportionalto the total exposure given to the film, sayexpressed in ergs/cm 2 . Let us henceforth denotethis exposure by E, dropping the previousmeaning of this symbol. If the exposure ismonochromatic, of wave-length , each light-quantum carries the energy hv=ch/X, where, inc.g.s. units, c=3.10 0, h=6.55-10- 27 (light ve-locity and Planck's constant). Hence, E=chn/Xand

y= (EdX/ch)E. (9)

With =4358A, as in the experiments to beconsidered presently, X/ch= 2.2 2. 101. The aver-age grain size a is a moderate fraction of asquared micron or 10-8 cm2 . Thus, if a standshenceforth for the average grain size expressedin A2, we may conveniently write

y = 2220EaE. (9.1)

Of course, if we plot the theoretical densitiesagainst Log y and the observed ones againstLog E, the passage from y to E amounts only toa rigid shift along the axis of abscissae, so thatfor a comparison of theory with experiment apreliminary knowledge of the numerical value ofthe coefficient in (9.1) is not needed. When,however, the theoretical and the observed sensi-tometric curves have been brought by such ashift into the best possible fit, formula (9.1)will enable us to evaluate the fraction e repre-senting the "intrinsic sensitivity" of the grainsof known size a.

Manifestly, before applying such rigid shifts,the scale of ordinates of the two curves, D orD/Dm, must be properly adjusted. Now, theobserved limiting density Dim even if availablein some cases, is not at all reliable for the purposeof such comparisons, owing to certain complica-tions** with regard to the covering power of theblackened grains at high densities D which, sofar, do not seem amenable to a mathematicaltreatment. Under these circumstances it is im-perative to limit the comparison to low andmoderate densities, not exceeding D =1, or so.Thus the factor D. has to be eliminated alto-gether, and this is best done by substituting for

** Especially for long developments.

71

LUDWIK SILBERSTEIN

y in (8.3) the observed gamma. Formula (8) willthen give D itself as function of y alone. Thisamounts to giving to the theoretical curve the(maximum) gradient of the observed one, and itremains then to be seen whether the "toes" andsome part at least of the so-called "straight line"portion of the two curves can be brought intoclose coincidence by a mere shift along the Log Eaxis. The "toe," moreover, and the steepness ofascent (gamma) are the most important charac-teristics of an emulsion or its sensitometriccurve.

To test the theoretical formulae developedabove, two pairs of films, No. 1340, 1341 andNo. 1342, 1343, were used. Each two werecoated with the same emulsion, but at differentconcentrations or with different numbers N ofgrains, per square micron of thein A2,

film, namely,

No. a N

1340. 0.44 0.31 21 44.31341. 0.44 0.31 10.5 -t2.21342. 0.72 0.53 8.5±t11343. 0.72 0.53 4.2±0.5

clash prohibitively with our assumed exponentialdistribution, f(a) = Ne-ala a. This is easily ac-complished. In fact, with this distribution func-tion we have

(a2) Av=- air-alada = a2f x2e-dx,ao T

i.e., (a2)A = 22. NW, 02 = (a2) A -d2 . Thus,

=a, (10)

a remarkably simple result. For an exponentialdistribution, standard deviation equal to averagesize. Neither pair of our films deviates from thisrelation very much, their ratios /a being abouti and 5/7. The sensitometric properties, more-over, are mainly influenced by a (apart from N),and only in a secondary way by . Thus, if thefoundations of the theory are correct, our for-mulae ought to cover approximately the facts.

Let us now turn to consider in detail the caseof the film No. 1340. (For the remaining films itwill then suffice to state concisely the results.)The set of densities measured on this film, with asix-minute development and after the subtrac-tion of the slight fog 0.04, starts as follows:

where a- is the standard deviation of grain sizes.Our formulae, corresponding to an exponentialsize distribution, contain of course only a, butthe values of are relevant, since they willenable us to find whether the size-frequencycharacteristics of these emulsions are at allapproximated by the assumed exponential dis-tribution function.

All these films were exposed to light of wave-length 4358A, the exposure values being deter-mined in ergs/cm2. From the carefully measuredreflectance and transmission followed the T-values of these films (not blackened of course)for the same wave-length,

No. 1340.T 0.20

1341.0.371

1342.0.232

The diffuse densities obtained on these films fora set of exposures and after various times ofdevelopment, in a DK-50 developer at 650F,were measured in the usual way.

Before passing to compare them with ourformulae let us ascertain whether the size fre-quency characteristics of these emulsions do not

Log E'=.59D=0.05

2.89 1.19 1.49 1.790.19 0.47 0.89 1.40

and then increases proportionally to the in-crements of Log E'. Here E' stands for theincident exposure in ergs per cm2. Since thereflectance of this film is 35.5 percent, our last-defined E is, in this case,

E= 0.645E'.

(This conversion will be required only later, forthe evaluation of e.) The exposure ratio insuccessive steps is 2, as usual. The "straight-line"portion of this experimental sensitometric curveis practically reached at D = 0.80, so that, inaccordance with the preceding remarks, we maybe content with representing these five densities.

From the first two measurements we see thatwhen the exposure is doubled, the density is verynearly quadrupled, so that, initially, D E2.Now, we have seen that on the single-quantumhypothesis the power series for D starts with anE term and on the two-quanta assumption withan E2 term. Generally, as shown in a previous

72

LIGHT ABSORPTION IN PHOTOGRAPHIC EXPOSURE

paper,2 if at least r quanta are required for thedevelopability of each grain, the series startswith an Er term. In the present case, then, thesingle-quantum assumption is ruled out and weare driven to use the two-quanta formulae (8) to(8.3). It will be kept in mind, however, thatD c'3E2 (initially) indicates only the absence ofgrains made developable by a single quantumhit and the presence of a considerable percentageof grains requiring only two quanta, while othersmay require three or more quanta. For in thatcase, also, will the series start with an E2 term,being a superposition of series starting with E2 ,with E, and so on. Thus we might consider asuperposition of D formulae based on r =2, r =3,etc. But before resorting to such complicationswe shall first ascertain whether and to whatextent the observed densities can be covered bythe pure two-quantic formulae.

Now, in order to eliminate Din, as explainedabove, divide Eq. (8) by (8.3). This gives

D/-y = M/F(T) { as in (8) },

where F(T) is as in (8.3) and, for thisfilm, T=0.20. Thus F(T)=-0.545, and sinceM=0.4343, we have

D 1~~+y TY yl-0.797 log Y +- |ly 1+Ty (1+Ty)2 (1+y)2

Let us now substitute for y its observedvalue which is 1.44, as derived from a consider-able stretch (segment) of the "straight-line"portion of the experimental curve. In doing so,it must be kept in mind that, while our Dstands for the specular density, the observeddensities, and thus also gamma derived fromthem, are all diffuse densities which are, ofcourse, smaller than the specular ones. Generally,the relation between the specular and thediffuse densities, D and D', is expressed by thetheoretical and extensively tested formula'

I0 D=P- I10-D'+(1A lo1-OD"

where j3 and are constants, the latter being thefraction of the total scattered light flux containedin the normally emerging beam. By the definition

3L. Silberstein and C. Tuttle, "The Relation betweenthe Specular and the Diffuse Photographic Densities,"J. Opt. Soc. Am. 14, 365-373 (1927).

and the manner of derivation of our D thecoefficient v is nil, so that the formula is reducedto D= 3D'. Thus also y=3-y' and

D'/y' = D/y.

If, therefore, the observed gamma is simplysubstituted for our y, the last formula will givethe diffuse density, which will thus be directlycomparable with the observed densities.

Thus, substituting -y= 1.44, which amountsonly to giving to our theoretical curve the(maximum) slope of the observed sensitometriccurve, we have ultimately, for the diffuse densityas function of y - E,

l+y TY y 1D=1.148 log + -_

1+Ty (+Ty) 2 (+y) 2 '

T= , (No. 1340)

a formula containing no adjustable parameters.This gives, to three decimals, the following pairsof corresponding values:

Y= i 4 2 1 2 2 4 8D = 0.017 0.068 0.196 0.459 0.646 0.855 1.275 1.59.

The density corresponding to y =1 is practicallyidentical with the observed density 0.47 forLog E'=l.19, i.e., E'=0.155 ergs/cm2 . Thus, ifwe place the unit of our y scale at this (theincident) exposure, the correlation between thetheoretical and the observed values is as follows:

Dcaic = 0.07Dobs = 0.05

0.20 0.46 0.86 1.280.19 0.47 0.89 1.40.

The first four points are practically coincident, sothat the "toe" with a good part of the "straight"portion is represented very closely, as shown alsoin the diagram (Fig. 2). The agreement is closeenough even up to D =1.30. Beyond that theobserved densities exceed the theoretical onesconsiderably. These might be represented, per-haps, by amplifying and thus complicating ourformula, but for the present-and in view of theprevious remarks-this simple rendering of thelower and most characteristic part of the ob-served curve for film No. 1340 may suffice.

Since E' stands for the incident exposure andsince, for this film, E=0.645E', to y=1 corre-sponds E=0.645. 0.155, so that, by (9.1) and

73

LUDWIK SILBERSTEIN

The agreement up to, and a stretch beyond,D=1.14 is very close indeed, as shown also inFig. 3. Beyond 1.40 the observed densities are inexcess, as before. Turning to the evaluation of theintrinsic sensitivity , we have for this film, at thestated wave-length, the reflectance 34 percent, sothat E= 0.66E', and since y = 0.85 corresponds toE'=0.155, and =0.44, formula (9.1) gives

0.85 = 99.9e, whenceE= 0.0085. (No. 1341)

FIG. 2.

with the average grain size a= 0.44,

whencee = 0.0102.

This is slightly greater than the highessensitivity" hitherto determined,E= 0.0077 , for the experimental emulsi

Let us now consider the film No. 13lowest densities observed on this filmdevelopment are (with the same above)

Log E'= 2.59D =0.04

2.89 1.19 1.49 10.16 0.42 0.76 1

Here again D is initially proportionsthat the two-quanta formulae are t,Since, for this film, T=0.371, one= -0.357, and the observed gammawe derive from (8) and (8.3), in mucway as in the preceding case,

This is, to all purposes, the same as for the filmNo. 1340, the difference scarcely exceeding thelimits of error implied in the measurements oftheir transmission and reflectance and theuncertainty of the exposure values and of the

(No. 1340) adjustment of the two scales. Practically, thest "intrinsic calculated values are =0.010 and 0.09. Now,which was these two films were coated with batches of the.on No. 73.2 same emulsion, only at different concentrations.41. The six The satisfactory coincidence of their E-values orafter an 8' "intrinsic sensitivities" thus arrived at seems,

notation as therefore, to be one more confirmation of thesoundness of the foundations of the proposed

.79 0.09 theory.179 009 Let us now pass to the second pair of films.14 1.52. which were coated with a different emulsion

al to E2 , so (consisting of larger grains, etc.), again the sameo be used. for both films, but at different concentrations,finds F(T) and hence different Tvalues. The lowest densitiesbeing 1.27, measured on the film No. 1342, with an eight-h the same minute development, are, after the subtraction

of a slight fog (0.06), as in the preceding cases:

1+y TY yD= 1.543 log + Ty1+Ty (I+Ty)2 (1+y)2

T=0.371. (No. 1341)

This gives, for y=0.50, 0.75, 1,

D=0.223, 0.368, 0.502,

respectively. Whence, by graphical interpolation,D=0.42 for y=0.85, which would, then, corre-spond to Log E'=1.19. With this correlation ofthe exposure scales, y and E', the results are asfollows:

y=0 .2 13Dcai= 0.048Dobs= 0.04

0.4250.180.16

0.85 1.70 3.400.42 0.78 1.1440.42 0.76 1.14

6.801.371.52.

Log E'=2.59D=0.035

2.89 1.190.15 0.37.

The density D is, initially, again very nearlyproportional to the square of the exposure, thuscalling for an application of the two-quantaformulae. For this film, T=0.232, and therefore,F(T)=-0.503. The observed gamma is 1.12.Thus, proceeding exactly as in the first two cases,we have the density formula, converted in thisway (as before) into one for the diffuse densities,

1 +y TY y D=0.969 log + T i,1 +Ty (1+Ty)2 (l+y)2J

T=0.232. (No. 1342)

74

PHOTOGRAPHIC EXPOSURE

This happens to give for y = 1, D = 0.375, practi-cally identical with the observed density forLog E'= 1.19. This, then, fixes the adjustment ofthe y and E' scales, and the final results are:

Y= 1Dcaile =0.057

Dob, = 0.035

:12

0.160.15

1 2 4

0.375 0.69 1.010.37 0.69 1.06

8

1.241.39.

The congruence of the first five pairs of "points"is as close as before, and the curves (Fig. 4)continue to coincide up to about D=1.15,beyond which the calculated densities are indefect, as before. In fine, the toe only with a goodsegment of the "straight" portion of the observedsensitometric curve are very closely representedby the simple two-quantic formula. The reflect-ance of this film being 0.31, we have E=0.69E'.The average grain size is a = 0.72, and since y= 1corresponds to LogE'=.19 or E'=0.155, for-mula (9.1) gives, for this film, 1=171e, whence

e = 0.005 9. (No. 1342)

Lastly, the six lowest densities measured on thefilm No. 1343, with a five-min. development are,after the subtraction of 0.03 as fog,

Log E'=2.59 2.89 1.19 1.49 1.79 0.09D=0.03 0.12 0.28 0.51 0.73 0.96.

.0 . 0. .4 0.85 1.0 a.4O 6.80

FIG. 3.

Here we have again, initially, D m E2 , calling forthe two-quantic formulae. Since, in the presentcase, T=0.457, one finds F(T)=-0.286. Theobserved gamma is 0.73. Thus our two-quanticformula becomes

D=1.108log y+ Ty _ yl+Ty (1+Ty)2 (l+y)2

T=0.457, (No. 1343)

0,.4

.2

O.0

0.6

0.6

0.4

0.2

1 . L I 2 4 8

FIG. 4.

This gives the following set of "steps":

Y= I 2 1 2 4Dcac0.05 0.14 0.31 0.53 0.71

8

0.81.

Thus the first five observed densities are wellrepresented by correlating Log E'= 1.19 orE'=0.155 with little less than y=1, in roundfigures,

E'=0.16 with y=1.

The curves fit then closely enough up to aboutD =0.9 (cf. Fig. 5), beyond which the ob-served densities (which seem to tend to alimiting density 1.6) are in excess, very much asfor the first three films. The reflectance ofthis film is 0.30, so that y=l corresponds toE = 0.16- 0.7 = 0.112 erg/cm 2 . Since the averagegrain size is again a=0.72, formula (9.1) gives1 =2200-0. 7 2-0.11 2 e= 178E,

e = 0.0056. (No. 1343)

This is practically identical with the E valuefound for the preceding film, No. 1342, which wascoated with the same emulsion, but twice asdiluted. The coincidence of these numbers ob-tained from two very different observed sensi-tometric curves confirms again the intrinsicmeaning of the coefficient e (which otherwisemight be considered as a mere adjustableparameter of the theoretical formula) as ex-pressing the actual photoefficiency or "sensi-tivity" of a silver halide grain, in much the samesense of the word as when one speaks of theefficiency of a photoelectric cell. Moreover, aswas pointed out in previous papers, the photo-graphic and the photoelectric coefficients are ofmuch the same order.

a, 9 .49 T..9 0.09 = LOG, E'.P0SUE

LIGHT ABSORPTION IN 75

LUDWIK SILBERSTEIN

'j. iL I I Z

FIG. 5.

We now have, for the two pairs of films:

No.1340 . .0io)~1341 .. o00same emulsion,134 .0.00591343 . 0.0056 same emulsion.

The net result of the investigation is, so far,that the pure two-quantic* formula (8), with thegradual absorption of light taken into account

* That is to say, without the superposition of termscorresponding to three and more quanta as possibleminimum required for the developability of some grains.

(T<1), gives a close representation of the toeand the lower straight portion of the observedsensitometric curves of these films and thecorrelation of the exposure scales, y and E,yields for films coated with the same emulsionthe same value of the "intrinsic sensitivity" e.The total probable or average number of lightquanta per grain corresponding to an averagenumber of efficient quanta y =2 is n = 2/e, andthus about 200 and 330 in the case of the first andthe second pair of films, respectively, which isin good agreement with Messrs. Eggert andNoddack's older findings.

The preparation of further material and datafor testing the said formula and, in particular, theclaimed intrinsic nature of the coefficient e, isnow in progress in the Kodak Research Labora-tories. These data will enable us also to studythat coefficient as a function of the wave-length.

My thanks are due to the Emulsion ResearchLaboratory and to Dr. Webb and Mr. Lovelandof these laboratories for furnishing the experi-mental data utilized in this paper.

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