P H Y S I C A L R E V I E W V O L U M E 9 0 , N U M B E R 1 A P R I L 1 , 1 9 5 3

Grain Density in Nuclear Emulsions LAURIE M. BROWN*

Institute for Advanced Study, Princeton, New Jersey (Received January 2, 1953)

A simple model of a nuclear track emulsion is discussed in which it is assumed that a grain is made de­velopable only if a charged particle loses in the grain an energy greater than some threshold value rj. I t is estimated that a particle of minimum ionization loses, on the average, an energy of order of magnitude ?? in traversing the diameter of a grain and, therefore, that fluctuations in the energy loss are important. This has as a consequence that the shape of the curve of grain density versus energy and, in particular, the percentage relativistic increase of grain density may be sensitive to small variations of TJ.

IN recent years a number of observations have been reported concerning the magnitude of the relativistic

increase of grain density in nuclear track emulsions. While the experimental situation is far from clear, it would appear that previous attempts to reconcile these observations with the theory of ionization losses have not been entirely successful. It is the purpose of this note to suggest that the discrepancy may be due to the mechanism by which fast particles make grains de­velopable and to offer a simple model for such a mecha­nism; namely, an increased role is attributed to fluctu­ations in the energy loss.

The Bristol group1 in 1949 selected at random, from a single Kodak NT4 plate exposed under a 10 cm thickness of lead at 11 000 feet, 25 tracks longer than one mm and attributed to fast electrons and mesons. They found a sharp distribution of grain density with a mean of 22.5 grains/lOOju and a full width at half-maximum of 2 grains/100ju. Nonuniformity of develop­ment with depth gave variations along single tracks from 23.5 grains/100/x at the air surface of the emulsion to 21.5 at the glass. In addition, part of the width was ascribed to fading and to other instrumental errors. Their results, therefore, were consistent with no relativ­istic increase of grain density.

Subsequent observations made with Ilford G5 plates have confirmed these results in the extreme relativistic region. Thus Corson and Keck2 found that electrons of 10 Mev and 180 Mev gave the same grain density within a statistical error of db2 percent and Occhialini3

concluded that electrons of energies ranging from 7.5 Mev to 500 Mev gave the same grain density within a total experimental uncertainty of ±10 percent. How­ever, the most recent reports4"9 while reaffirming these

* National Science Foundation Post-Doctoral Fellow on leave-of-absence from Northwestern University for the year 1952-1953.

1 Brown, Camerini, Fowler, Muirhead, Powell, and Ritson, Nature 163, 47 (1949).

2 D. R. Corson and M. R. Keck, Phys. Rev. 79, 209 (1950). 3 G. P. S. Occhialini, Co mo Congress pNTuovo cimento Supple­

ment 6, 377 (1949)]. 4 E. Pickup and L. Voyvodic, Phys. Rev. 80, 89 (1950). 5 A. Morrish, Phil. Mag. 43, 533 (1952). 6 1 . B. MacDiarmid, Phys. Rev. 84, 851 (1951). 7 Dansyz, Lock, and Yekutieli, Nature 169, 364 (1952). 8 Daniel, Davies, Mulvey, and Pekeris, Phil. Mag. 43, 753

(1952). 9 M. M. Shapiro and B. Stiller, Phys. Rev. 87, 682 (1952).

conclusions (e.g., Morrish finds 0±0.25 percent for the increase from E/mc2= 20 to 1000), have also claimed to find an increase of from 5 percent to 10 percent in the range E/mc2 from 3.5 to 10 or 20. Table I summarizes their results expressed as (gPi—gmin)/gPi in percent, where gpi is the plateau value of grain density for E/mc2> 20 and gmm is the lowest observed value of the grain density.

On the theoretical side the Bethe-Bloch ionization loss theory has been corrected for polarization effects, important in condensed materials in the range of E/mc2

we are considering, by Fermi,10 Wick,11 Halpern and Hall,12 A. Bohr,13 and Huybrechts and Schonberg.14 The Bethe-Bloch theory has been found experimentally to hold15,16 in gases (as it should) for E/mc2 up to 100. The theories of the polarization effect predict a relativistic increase in ionization loss many times larger than the ob­served relative increase in grain density. A considerable advance in interpretation of this discrepancy has been made by Messel and Ritson17 who have pointed out that the average energy loss is not directly responsible for the grain density since secondary electrons of sufficiently high energy (6-rays) will leave the grain with most of their energy. They have proposed, there-

TABLE I. Summary of experimental information concerning the relativistic increase of grain density.

Observer

4 5 6 8a

9

E/m gmin /0

3.5 to 20 8dz2 6-12 to 20-1000 4.8±2

7 to 60 8±2.5 3.5 to 10 —8

> 3 12„3+4

Observed gpi (grains/100/x)

31.7 and 35.7 not given

36.6 various

21.2

a Including reference 7 and L. Voyvodic (to be published).

10 E. Fermi, Phys. Rev. 57, 485 (1940). 11 G. C. Wick, Nuovo cimento 1, 302 (1943). 12 O. Halpern and H. Hall, Phys. Rev. 73, 477 (1948). 13 A. Bohr, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd.

24, No. 19 (1948). 14 M. Huybrechts and M. Schonberg, Nuovo cimento IX, 764

(1952). 16 R. S. Carter and W. L. Whittemore, Phys. Rev. 87, 494

(1952). 16 J. E. Kuperian, Jr., and E. D. Palmatier, Phys. Rev. 85,

1043 (1952). 17 H. Messel and D. M. Ritson, Phil. Mag. 41, 1129 (1950).

95

96 L A U R I E M . B R O W N

fore, to replace the "maximum energy transferrin the theories of Fermi, etc., by a fixed quantity equal to the mean energy left in a grain by a 5-ray. This fixed energy has been taken quite reasonably as 5 kev, and the resulting "probable" energy loss is insensitive to this particular choice. With this modification, the ionization loss saturates for E/mc2 between 50 and 100 (depending on the particular theory used) at a value about 15 percent above the minimum value of the energy loss.

The effect, then, that we propose to discuss is a trough in the experimental grain density vs energy curve | to f as deep and 3 to 10 times as narrow as the corresponding trough in the theoretical curve of "probable" ionization loss vs energy. Since our purpose is to point out certain qualitative features of the process by which grains are made developable by fast particles, we introduce a simplified model of the photographic plate.

First, we assume that a grain will become developable, in a given plate under given chemical development, if, and only if, a minimum energy rj ev is released in the grain by the impinging particle. Secondly, we assume that the grains are spheres of AgBr of diameter a microns (/x) distributed at random. Since the specific volume of AgBr in the standard nuclear track plates is 0.455, the number of grains touched by a track is 0.455/f#=0.682/a per JU, §# being the mean chord. This should be equal to the observed maximum grain density gmax in the plate unless some of the grains are insensitive.

If we consider, say, a proton of velocity v with average rate of energy loss in AgBr=£(j3) ev/ju, where fi=v/c, then this leaves in the grain, on traversing a chord of length I microns, energy kl ev. Since the fraction of chords traversed of length greater than I is P(l) = l — l2/a2, the observed grain density should be

g = g m a x [ l - W / ^ 2 ) ] , (1)

if ka>rj and zero if ka<rj. This, however,neglects fluctuations in the energy loss which may safely be done, to first approximation, providing ka^>rj but not otherwise.

The question of the magnitude of rj thus becomes of primary importance for the interpretation of the rela­tion between energy loss and grain density in nuclear track plates. The minimum value kmin of h{E) is about 900 ev/fj, (the "probable" ionization loss of Messel and Ritson is somewhat less) so kmina is about 200 ev. Now it would appear that rj for a plate sensitive to minimum ionization is just of this order of magnitude. For example, Webb18 estimates that for an Eastman NTB plate sensitive to protons of energy up to 50 kev, the threshold sensitivity is about 1100 ev. But such a proton loses energy at a rate just five times kmi'n. Webb also notes that about forty light quanta (i.e.,

18 J. H. Webb, Phys. Rev. 74, 511 (1948).

about 100 ev) are required to make a grain developable and that grains are much less sensitive to ionization produced by fast particles than that produced by light because of the failure of the reciprocity law for short times. Furthermore, values of t\ obtained by fitting (1) to the curves for low energy protons agree with this estimate.

We see that the grain density produced by minimum ionization particles must be ascribed in large measure to fluctuations in the energy loss. This holds a fortiori if we consider the "probable" ionization loss rather than the mean as responsible for grain density.

We now consider fluctuations of a very special type; namely, we ask for the number of secondary electrons UE having an energy greater than E, produced in a collision with a fast particle. The Rutherford formula gives (using Z/A = 0.436 for AgBr):

nE=43.4//32E per micron of AgBr, (2)

E being measured in ev. If we put 0 = 1, E= 200 ev, we get n#—0.22/micron. Since the specific volume of AgBr is 0.455, this means that fluctuations of this type alone would account for almost half of the grains observed beyond the minimum of ionization. Note that this probability decreases about 6 percent from the ioni­zation minimum.

Of course, it is improper to extend the Rutherford formula to secondaries of energy so low as this in elements of medium atomic number, but N. Bohr19 has shown that it is proper to consider about half of the ionization loss as resulting from the "free" collisions, to which this formula is applicable while the remainder is the result of "resonance" effects obeying a more complicated distribution law for energy losses in single collisions, the latter depending on the ionization proba­bilities and energy level structure of AgBr. Detailed calculation is complicated by the fact that the mean ionization energy and 77 are of the same order of magni­tude.

We would like to suggest, therefore, (a) that the quantity of interest in comparing the grain density with the energy loss of high energy particles is neither the mean energy loss nor the "probable" energy loss, but rather the probability of a loss E> t\ in traversing a thickness such that &(#)<??, (b) that it is to be expected that the "free" collisions play a relatively larger role in this process than in the average or "prob­able" ionization loss processes, (c) that the shape of the grain density vs energy curve is very sensitive to ka/rj in plates responsive to minimum ionization so that one may expect larger or smaller relativistic increases correlated with the plateau grain density.

In particular, if (c) is true, then one should not average the curves corresponding to different values of gvh e v e n m the same plate, since they may have

19 N. Bohr, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 18, No. 8 (1948).

G R A I N D E N S I T Y I N N U C L E A R E M U L S I O N S 97

different shapes. In this connection it may be noted that in Fig. 2 of the article of Pickup and Voyvodic,4

grain density curves are given for two plates in which gpi had average values of 31.7 and 35.7 grains/lOOju. The two curves have been normalized to the same

1.

TH E difficulties encountered by attempts to cal­culate the partition function of a system con­

sisting of a large number of interacting particles for all values of temperature and volume are well known. I t has never been proved, for instance, that, in the limit of infinite number of particles at finite density, the partition function Q of a substance approaches the three analytic parts seemingly required for the explana­tion of condensation. I t seemed of interest, therefore, to re-examine the relation between the thermodynamic behavior and the density of energy levels p(E, V) (V=volume, E= energy) of a—macroscopically speak­ing—very small amount of substance, i.e., of a substance consisting of N interacting particles for which l<3CiV" «10 2 3 .

We have, for this purpose, considered a class of hypothetical, but not unreasonable, functions p(E, V)y

defined by certain regularity conditions. We have shown that a function of this class can yield fusion and con­densation (in the sense of a discontinuous change of the most probable energy and volume) and that the specific features of p(E, V) which lead to these transitions need not be "pathological." The case of fusion at fixed volume was included, although our description is essen­tially the same as the double tangent construction given by Seitz,1 in order to emphasize the distinction necessary for finite N between the thermodynamic quantities and the corresponding functionals of p(E, V) with which they are piecewise identical.

1 F. Seitz, Modern Theory of Solids (McGraw-Hill Book Com­pany, Inc., New York, 1940), p. 489. We do not assume the "actual curve" in Fig. 16 to be correct for N—• « .

plateau value, but one of them is considerably flatter than the other. I t is also of interest to note that Occhialini3 reports that for k((f) between 4&min and 2&min the grain density decreases only half as fast as the ionization loss.

In Sees. 3 and 4 we have examined the behavior of a substance under a volume independent force per unit area, realized, e.g., by a piston of finite weight with vacuum above the piston. While our treatment is similar to the double tangent construction of thermo­dynamics2 it leads to an unorthodox result. We find, directly from the consideration of the canonical en­semble, that features of p(E, V) which can be respon­sible for condensation, can nevertheless yield a smooth curve similar to the van der Waals curve for kTd \ogQ/dV. This is not a contradiction, since we can show that, if the function kTd \ogQ/dV has the shape of the van der Waals curve, it is not to be identified with the pressure for all values of the volume and that the results of the usual ad hoc re-interpretation of the van der Waals type curve can be justified, directly from the canonical ensemble, without the conventional ad hoc assumptions. We do not claim that N~x logQ does not —(in the limit of infinite N and finite density) approach a piecewise analytic function, but that condensation (in the limited sense defined above) can occur for values of N and V for which this limit has not yet been reached. In a later paper we will show that this type of condensation does not necessarily occur if a volume dependent force acts on the system, but that the par­tition function of a macroscopic system built up of a large number of systems of the type discussed here consists practically of the three analytic parts required for condensation in the usual sense.

To avoid considerations extraneous to our problem we will limit the discussion to a range of energies E^E0

2 J. C. Slater, Introduction to Chemical Physics (McGraw-Hill Book Company, Inc., New York, 1939), p. 179.

P H Y S I C A L R E V I E W V O L U M E 9 0 , N U M B E R 1 A P R I L 1 , 1 9 5 3

Remarks on the Theory of Fusion and Condensation

ARNOLD J. F. SIEGERT Department of Physics, Northwestern University, Evanston, Illinois

(Received December 18, 1952)

We have re-examined the relation between the thermodynamic functions and the energy level density p of a substance consisting of N interacting particles for a class of hypothetical but (in the case l^iV^lO23) not unreasonable functions p defined by certain regularity conditions. We have shown that the specific features of p which lead to condensation—in the sense of a discontinuous change of the most probable volume —under a volume independent force per unit area can yield partition functions which, in the condensation region, are smooth and of a shape similar to the van der Waals function. This is not a contradiction since for a substance under a volume independent force, the results of the conventional ad hoc re-interpretation of the van der Waals curve can be proved directly from the canonical ensemble, without the ad hoc assumptions.