NUCLEAR EMULSION TECHNIQUE*)

A. Beiser

CONTENTS

I. Properties of nuclear emulsions . . . . . . . . . . . . . . . . 378
I. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 378
I. 2. Characteristics of emulsions . . . . . . . . . . . . . . . . 378
I. 3. Temperature variation of sensitivity . . . . . . . . . . . . 383
I. 4. Regression of the latent image . . . . . . . . . . . . . . . 385

II. Formation of tracks and their analysis . . . . . . . . . . . . 388
II. 1. Formation of the latent image . . . . . . . . . . . . . . . 388
II. 2. Specific energy loss . . . . . . . . . . . . . . . . . . . . 390
II. 3. Range–energy relation . . . . . . . . . . . . . . . . . . . 394
II. 4. Identification of tracks . . . . . . . . . . . . . . . . . . 405
II. 5. Delta rays . . . . . . . . . . . . . . . . . . . . . . . . . 409
II. 6. Line of “thickening” . . . . . . . . . . . . . . . . . . . . 412
II. 7. Multiple scattering . . . . . . . . . . . . . . . . . . . . 414

III. Processing of emulsions . . . . . . . . . . . . . . . . . . . 417

III. 1. General considerations . . . . . . . . . . . . . . . . . . 417
III. 2. Temperature development . . . . . . . . . . . . . . . . . 423
III. 3. Diffusion of the developer . . . . . . . . . . . . . . . . 426
III. 4. Development by the two-bath method . . . . . . . . . . . 428
III. 5. Films . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
III. 6. Shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . 431

IV. Auxiliary techniques . . . . . . . . . . . . . . . . . . . . 434
IV. 1. Elimination of background . . . . . . . . . . . . . . . . . 434
IV. 2. Limiting the period of sensitivity . . . . . . . . . . . . 435
IV. 3. Detection of neutrons . . . . . . . . . . . . . . . . . . . 436
IV. 4. Gamma-ray spectra . . . . . . . . . . . . . . . . . . . . 440
IV. 5. Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 441
IV. 6. Deflection of particles by a magnetic field . . . . . . . . 445
IV. 7. Other methods of using plates . . . . . . . . . . . . . . . 448
IV. 8. Detection of tracks . . . . . . . . . . . . . . . . . . . . 450

Cited literature . . . . . . . . . . . . . . . . . . . . . . . . 453

) Rev. Mod. Phys. 24*, 237 (1952).

A. BEISER

I. PROPERTIES OF NUCLEAR EMULSIONS

I. 1. Introduction

A photoemulsion is a mixture of silver bromide dispersed in gelatin. Nuclear photoemulsions differ from ordinary ones by their very high concentration of silver. Ionizing particles, passing through such emulsions, act on the crystals of silver bromide in such a way that, after development, they form a series of black grains of colloidal silver, located along the trajectories of the particles. The more strongly a particle ionizes, the greater the number of such grains, and the greater the energy of the particle, the longer its track. There is a relation between these quantities which, under favorable conditions, makes it possible to identify the particle and determine its energy. In cases where measurements of the range and of the grain density are insufficient, a more complicated method is used for light particles, based on measurements of multiple scattering through small angles. In special cases, the deflection of charged particles by a strong magnetic field is used.

Photoemulsions were first used for recording particle tracks by Reinganum¹, who found that silver bromide grains encountered in the path of $\alpha$-particles become developable. This work was partly based on the results of an earlier investigation². Michl³ made the first quantitative estimates of $\alpha$-particle tracks. Proton tracks in an emulsion were first observed in 1925⁴. A review of the development of photodetection of nuclear particles has been given by Shapiro⁵.

The first photographic plates recorded only $\alpha$-particles. Later plates were made that recorded the tracks of low-energy protons⁶ and even more sensitive emulsions⁷.

During and after the war, great advances were made in the production of nuclear emulsions. In particular, in 1948 emulsions were obtained that were sensitive to all charged particles independently of their energy. The field of application of nuclear emulsions has expanded continuously, and at the present time they are an important research method in nuclear physics.

I. 2. Characteristics of emulsions

Physical properties. The composition of some dry nuclear emulsions, expressed as the number of grams of each element per 1 cm³ of dry emulsion, is given in Table I, and as the number of atoms per 1 cm³ of emulsion—in Table II. For Ilford G5 emulsion, the bromine and iodine contents are 1.496 and 0.026 g/cm³, respectively. The tables do not include an emulsion of type NTC, which has a much lower content of silver halide (65%) than other types

emulsion (81%). Gelatin is hygroscopic and, consequently, the composition of the emulsion will vary depending on the humidity of the air.

Table I

Composition of dry nuclear emulsions in g/cm³

Element	Ilford	Kodak	Eastman-Kodak
Silver	2.025	1.97	1.70
Bromine	1.465	1.44	1.22
Iodine	0.057	0.036	0.054
Carbon	0.30	0.27	0.34
Hydrogen	0.049	0.038	0.043
Oxygen	0.20	0.16	0.17
Sulfur	0.011	……	……
Nitrogen	0.073	0.080	0.11

The percentage content (by weight) of moisture in various emulsions at different relative humidities and at a temperature of 20°C is given in Table III.

Nuclear emulsions are usually applied to glass plates with a thickness of 1.25 to 1.40 mm. The thickness of the commonly used

Table II

Composition of dry nuclear emulsions in atoms per 1 cm³ (×10²²)

Element	Ilford	Kodak	Eastman-Kodak
Silver	1.17	1.14	0.99
Bromine	1.15	1.13	0.96
Iodine	0.03	0.02	0.027
Carbon	1.51	1.43	1.60
Hydrogen	2.93	2.39	2.64
Oxygen	0.75	0.65	0.68
Sulfur	0.02	……	……
Nitrogen	0.31	0.36	0.49

emulsions ranges from 25 to 600 μ. In special cases, both greater and smaller thicknesses can be obtained. The plates have sizes from 2.5 × 7.5 to 20 × 25 cm; moreover, small plates, when examined under a microscope, do not require additional cutting. Emulsions are also used in the form of

films. Such films are elastic and, moreover, permit relatively prolonged irradiation in vacuum. A rigid emulsion layer can be made by folding several films together,

Table III

Percentage moisture content (by weight) in nuclear emulsions at \(20^\circ\)C and various relative humidities

Relative humidity in %	Ilford	Kodak	Eastman-Kodak
0	1.41	…	…
30	2.06	1.3	…
50	2.65	2.6	2.2
60	2.95	…	…
70	3.7	3.5	4.0
85	5.17	…	…

which can then be separated and developed individually. To facilitate examination, after development the films are placed on glass plates.

The surfaces of nuclear emulsions are very sensitive to pressure\(^8\). Therefore, to protect against scratches, the surface of the emulsion is coated with a layer of gelatin. In Eastman-Kodak plates the thickness of this layer is \(0.5—1\mu\). The reduction of particle ranges in the emulsion owing to the presence of such a film can be allowed for\(^9\). Ilford plates also have a protective coating.

Table IV

Maximum recorded energy in MeV for various particles in Ilford nuclear emulsions

Particle	D1	E1	C2	B2	G5
Electron . . . . . .	…	…	0.03	0.07	all
\(\mu\)-meson . . . . . .	…	2	5.5	14.0	»
Proton . . . . . .	…	20	50	120	»
Deuteron . . . . . .	…	40	100	240	»
\(\alpha\)-particle . . . . . .	low	500	1500	all	»

Sensitivity. The sensitivity of an emulsion is conveniently expressed through the maximum energy of various particles that can be registered in it. In Tables IV, V, and VI are pre-

sensitivities of some emulsions to electrons, \(\mu\)-mesons, protons, deuterons, and \(\alpha\)-particles are given.

Table V

Maximum recorded energy in MeV for various particles
in Kodak nuclear emulsions

Particle	NT1a	NT2a	NT4
Electron	…	0.1	all
\(\mu\)-meson	2	20	»
Proton	20	200	»
Deuteron	40	400	»
\(\alpha\)-particle	500	all	»

Table VI

Maximum recorded energy in MeV for various particles
in Ilford nuclear emulsions

Particle	NTC	NTC3	NTA	NTB	NTB2	NTB3
Electron	…	…	…	0.03	0.2	0.4
\(\mu\)-meson	…	1	2	6	40	85
Proton	…	1.5	3	8	50	110
Deuteron	…	20	40	100	750	1500
\(\alpha\)-particle	low	100	200	800	all	all

Preparation of nuclear emulsions. The first emulsions for nuclear research were prepared by Mysovskii and Chizhov\(^{10}\), Blau and Wambacher\(^{11}\), and Zhdanov\(^{12}\).

Subsequently, recipes were developed for preparing high-sensitivity emulsions\(^{13–17}\).

Zhdanov\(^{12}\), on the basis of geometrical considerations, showed that the mean grain density \(\left\langle \dfrac{dn}{dx} \right\rangle\) of a track in an emulsion with silver-halide concentration \(c\) and density \(\rho\) is determined by the relation

\[ \left\langle \frac{dn}{dx} \right\rangle = \left(\frac{3}{2}\right) \left(\frac{cP}{\rho d}\right), \tag{I, 1} \]

where \(d\) is the mean grain diameter, and \(P\) is the probability that, for a given specific loss of energy, the intersection of the grain by a particle track makes it capable of being developed. It is evident from this equation that increasing \(c\) and decreasing \(d\) would give an emulsion with a high grain density. However, the quantities entering into equation (1.1) are not all independent, since \(P\) is approximately proportional to \(d^3\). Therefore, when \(d\) is decreased, \(P\) is reduced to an even greater degree and, consequently, in choosing these quantities one must arrive at a compromise. For the registration of weakly ionizing particles a high \(P\) is necessary, since otherwise such particles will not be recorded. Low-energy particles, on the contrary, ionize strongly, but form very short tracks. In such cases it is desirable to reduce \(d\), in order to be able to determine the range of these particles more accurately.

Dimmers’ method. The first emulsions prepared by Dimmers[^13] had relatively low sensitivity, so that, for example, for \(7 \div 8\) MeV protons the track density was \(\sim 50\) grains per 100 microns. These emulsions are prepared by the simultaneous addition of \(30\ \mathrm{cm}^3\) of a 60% solution of silver nitrate and \(30\ \mathrm{cm}^3\) of a 42% solution of potassium bromide to \(75\ \mathrm{cm}^3\) of a 6% gelatin solution. The process is carried out at a temperature from 40 to \(50^\circ\mathrm{C}\), and the silver and bromine solutions are added dropwise over a period of 30 minutes with continuous stirring. The emulsion is washed in cold water for several hours and then dried. The size of most of the resulting bromide grains is less than \(0.2\mu\) in diameter. When lower temperatures and weaker stirring are used, the grain size increases somewhat. Such emulsions are not very photosensitive and can be prepared under red or orange light.

As a result of further investigations[^15], Dimmers obtained an emulsion of better quality. First, \(25\ \mathrm{cm}^3\) of ethyl alcohol is added to a solution of \(4.5\ \mathrm{g}\) of gelatin in \(50\ \mathrm{cm}^3\) of water. The alcohol is introduced to prevent the formation of lumps and large grains. Then, under the conditions indicated above, the following solutions are added simultaneously: \(18.6\ \mathrm{g}\) of silver nitrate in \(30\ \mathrm{cm}^3\) of water and \(12.8\ \mathrm{g}\) of potassium bromide in \(30.5\ \mathrm{cm}^3\) of water. In this process a slight excess of bromide is desirable, which is achieved by introducing from \(0.5\) to \(1\ \mathrm{cm}^3\) of the bromide solution before adding the silver nitrate.

The resulting emulsion is poured into a flat tray and cooled until it solidifies; it is then washed for about eight hours in order to remove soluble potassium nitrate, melted, and applied to glass plates.

Helg and Ienni’s method. Helg and Ienni[^16], working according to Dimmers’ recipe, gave a more detailed method of preparing emulsions, which is as follows. To a solution of \(14\ \mathrm{g}\) KBr in \(23\ \mathrm{cm}^3\) of water, \(5\ \mathrm{cm}^3\) of a 10-percent solution is added.

CdBr₂(4H₂O) and 2 cm³ of a 10-percent KI solution. This solution and 30 cm³ of a 60-percent silver nitrate solution are added at a rate of ~1 cm³ per minute to the gelatin solution. To prepare the latter, 6.5 g of gelatin is first soaked for ~1 hour in 70 cm³ of water at 20°C. The gelatin is then melted by heating to 50°C with stirring. After the solutions have been mixed, the emulsion is left to ripen for 45 minutes at a temperature of 50°C, then poured into a porcelain dish and cooled on ice for six hours. The resulting gel is cut into pieces, placed in a loose mesh, and washed for ~16 hours to remove potassium nitrate. After this, two other solutions are prepared. The first is a solution of 2 g of chrome alum in 78 cm³ of water, to which 60 cm³ of ethyl alcohol, 42 cm³ of glycerin, and 0.75 cm³ of a 10-percent potassium bromide solution are added. The washed emulsion is melted at 35°C, and 9 cm³ of this solution and 5 cm³ of a 0.2-percent wetting agent are added to it. Then 1 cm³ of a 0.2-percent solution of acridine yellow (a sensitizing dye), which improves the characteristics of the emulsion, is added. The second solution (composed of a solution of 2 g of gelatin in 150 cm³ of water at 35°C, 5 cm³ of wetting agent, and 2.5 cm³ of a 2-percent chrome alum solution), after filtration, is used for preliminary coating of the glass. To cover an area of 2.5 × 7.5 cm, 1 cm³ of solution is required. After the substrate has hardened, it is coated with a layer of melted emulsion of the required thickness. Drying is preferably carried out in filtered air.

Yenni¹⁷ prepared electron-sensitive emulsions, but his method presupposes the use of specially treated gelatin containing various chemical sensitizers. In this method the sensitivity and grain size are greatly increased as a result of adding to the emulsion, during its ripening, a small amount of concentrated ammonia solution; after 5 minutes the ammonia is neutralized with citric acid. It was found that with this method of preparation the sensitivity of the emulsion increases threefold.

I. 3. Temperature Variation of Sensitivity

The sensitivity of nuclear emulsions varies significantly with the temperature during exposure. This variation has been investigated experimentally by several authors¹⁸–²⁰. The total number of developed grains was determined in the tracks of monoenergetic protons in emulsions irradiated at various temperatures. The results obtained in works¹⁹ and ²⁰ are given on

Fig. 1. In work \(^{18}\) the study was carried out only at low temperatures, and the results obtained agree with the curves shown.

Beiser \(^{21}\) found an empirical equation expressing the change in grain density with temperature during exposure in the form

\[ n = A \exp\left(-\frac{\varepsilon_1}{kT}\right) \left[1 - a \exp\left(-\frac{\varepsilon_2}{kT}\right)\right], \tag{I, 2} \]

where \(n\) is the grain density, \(T\) the temperature during exposure, \(A\) a function of the sensitivity of the emulsion and of the rate of energy loss of the particle forming the track, and \(a\), \(\varepsilon_1\), and \(\varepsilon_2\) are constants depending on the type of emulsion. The first exponential can be interpreted as being connected with the rate of collection of silver ions at sensitivity centers during exposure, and the second with the relative number of ions lost by the center during the same time as a result of thermal emission of electrons (compare Section II.1). The ions may move either into neighboring lattice sites free of \(\mathrm{Ag}^{+}\), or back into intermediate positions in the crystal.

Fig. 1. Change in the sensitivity of various nuclear emulsions as a function of temperature \(^{19,20}\).

Knowing the temperature dependence, one can determine the optimum exposure temperature corresponding to maximum sensitivity. For commonly used emulsions it proves to be about \(\sim 20^\circ\mathrm{C}\). If emulsions are used at other temperatures, then the decrease in sensitivity and the associated change in grain density in particle tracks can be taken into account with the aid of the curves given above. In addition, the decrease in sensitivity at low temperatures makes it possible to use thermal “shutters,” which make it possible to limit the time of sensitivity of the emulsion in experiments in which continuous registration of the phenomenon is undesirable \(^{18}\). Lord \(^{20}\) found that the sensitivity of the emulsion at \(-200^\circ\mathrm{C}\) is zero, and it is possible that irradiation at temperatures even somewhat exceeding this value will not give tracks distinguishable among the fog grains. Therefore storage of photographic plates at low temperatures may reduce the background caused by random radiation \(^{22}\).

I. 4. Regression of the latent image

The weakening of the latent image in nuclear emulsions in the interval of time between irradiation and development was first noted in Blau’s work^23^. It turns out that the density of grains in the track of a particle gradually decreases as a function of time and storage conditions. Below we shall consider the influence of various parameters on the course of regression; for this purpose we introduce the weakening coefficient \(F\), defined by the formula

\[ F=\frac{(N-N_0)}{N_0}, \tag{I, 3} \]

where \(N_0\) is the number of grains on a definite length of track when developed immediately after irradiation, and \(N\) is the number of grains when developed after storage for a time \(t\).

Influence of the surrounding medium. Experiments^24^ show that the regression of proton tracks is reduced by approximately 90% if the irradiated emulsion is kept in a vacuum before development. It follows from this that at least this part of the regression is caused by the action of the constituents of air. Some or all of the remaining part of the weakening may be explained^25^ by the thermal ejection of electrons from the centers of development of the latent image when these electrons acquire energy sufficient to re-enter the conduction band of the crystal. In this case the development center will lose silver ions and decrease in size. In some cases this decrease may be so considerable that the grain will not be developed.

Very great importance is attached to humidity. Albu and Faraggi^26^ investigated the change in regression as a function of the relative humidity of the atmosphere surrounding the emulsion (the regression rate was expressed by the time interval \(\theta\), at the end of which the number of grains was reduced by half). Their results are given in Fig. 2. A comparison of this curve with the curve obtained by Miso^27^ (Fig. 3) for the relation between the amount of water absorbed by the gelatin of the emulsion and the relative humidity convincingly shows that the rate of regression is an exponential function of the amount of moisture absorbed by the gelatin.

Fig. 2. Change of \(\theta\)—the time required for the grain density to decrease by \(1/2\)—as a function of relative humidity^26^.

Albu and Faraggi^26^ also investigated the influence of changes in the composition of the atmosphere surrounding the emulsion, at constant humid-

They found (Figs. 4 and 5) that the rate of regression in pure oxygen is twice as great as in air and approximately 10 times greater than in nitrogen. In all cases the regression proceeds more rapidly at the surface of the emulsion than in its depth.

Fig. 3. Relation between the amount of moisture absorbed by dry gelatin and the relative humidity.^27

Effect of temperature. The change of the rate of regression with temperature can be determined^28 by considering the effect of temperature on the rate of that chemical reaction (gas—solid) which is assumed, in the case of regression, to occur between some of the constituents of the surrounding atmosphere and the silver of the development centers in the exposed emulsion. This gives, for the rate of disappearance of development centers \(dN/dt\), the expression

\[ -\frac{dN}{dt}=Ce^{-k/T}, \tag{I, 4} \]

where \(T\) is the absolute temperature during storage, and \(C\) and \(k\) are constants. This means that regression, occurring under otherwise fixed conditions, will be an exponential function of \(1/T\), a result consistent with the experiments of Albouy and Faraggi.^26

Fig. 4. Rate of surface regression of the image in Ilford C2 nuclear emulsions kept in different gases at constant humidity.^26

Fig. 5. Rate of internal regression of the image in Ilford C2 nuclear emulsions kept in different gases at constant humidity.^26

Influence of storage time. To obtain the relationship between the magnitude of regression of the latent image and the storage time, it may be assumed that, other conditions being equal, the rate of regression is proportional to the number of development centers present[^28]. Since each of them, upon development, is capable of producing a visible grain, for the attenuation coefficient after integration we obtain:

\[ F=1-\exp(-ct). \tag{1,5} \]

The dependence of \(F\) on \(t\) is shown in Fig. 6 for different values of \(c\)—a constant depending on the type of emulsion and the storage conditions. The curves obtained agree well with the experimental data of Igoda and Kaplan[^29]. In some cases[^30] the regression rate is at first comparatively low, corresponding to a small initial value of \(c\), and then, after some time, increases rapidly, corresponding to a larger \(c\). It has been found[^31] that in most cases, during regression, the intervals between grains along the track increase according to an exponential law, which agrees with the attenuation coefficient \(F\) interpreted above.

Fig. 6. Change in the regression coefficient with storage time under different conditions[^28]. The experimental points are taken from the work of Igoda and Kaplan[^29]. The upper curve corresponds to storage in a saturated atmosphere, and the lower one to normal laboratory conditions.

Influence of the composition of the emulsion. Both \(pH\) and the grain size affect the sensitivity of the emulsion to regression[^26]. In general, the lower the \(pH\) and the finer the grains, the greater the rate of regression.

The introduction into the emulsion of various substances (for example, lithium, boron, or uranium compounds) having different \(pH\) values noticeably changes the regression rate. Since the sensitivity of a nuclear emulsion depends directly on the grain size (the larger the grains, the greater the sensitivity), plates such as G5, NT4, and NTB3 are less subject to regression than others[^32].

Mechanism of regression. The explanation of the mechanism of regression proposed by Albouy and Faraggi[^26] is based on the experimental fact that approximately \(90\%\) of the effect may be associated with the influence of the atmosphere. They suggested that regression

is caused by the oxidation of the development centers by atmospheric oxygen in the presence of water, according to the reaction

\[ 2\mathrm{Ag}+ \mathrm{O}+ \mathrm{H}_2\mathrm{O}\to 2\mathrm{Ag}^{+}+2\mathrm{OH}^{-}. \tag{I, 6} \]

It is obvious that an excess of \(\mathrm{OH}^{-}\) ions in the emulsion (\(pH\) above 7) will hinder the reaction, whereas more acidic conditions accelerate it. The influence of humidity and of the oxygen concentration is confirmed by experiment.

Igoe \(^{33,34}\) proposed a hypothesis according to which the oxidation of the centers is caused by hydrogen peroxide formed in the emulsion in the immediate vicinity of the particle tracks. It is known \(^{35}\) that hydrogen peroxide is formed when ionizing radiation acts on water; however, a quantitative consideration shows the untenability of such an explanation.

Vineyard and Falla \(^{36}\) proposed that regression occurs because the silver atoms of the development centers can again form halide compounds with the ions of the bromide surrounding them \(^{37}\). Since this theory does not explain the different effects of the constituent parts of the atmosphere, it likewise cannot be accepted.

II. FORMATION OF TRACKS AND THEIR ANALYSIS

II.1. Formation of the latent image

The formation of the latent image (i.e., of the track of an ionizing particle capable of being developed) in a nuclear emulsion is essentially the same process as that occurring in ordinary light-sensitive emulsions. The important difference lies only in the mechanism of formation of ion pairs: in the first case the formation of ions occurs owing to the electrostatic interaction between the charged particle and the electrons of the atoms of the emulsion; in the second, as a result of the photoelectric emission of electrons caused by the incident photons. Gurney and Mott \(^{38}\) gave the first and most complete theory of the photographic process; its basic propositions, after certain modifications made by Mitchell \(^{39–41}\), are well substantiated.

According to the theory of Gurney and Mott, formation of the latent image occurs in two stages: one is characterized by the motion of electrons in silver bromide crystals, and the other by the motion of ions. When a particle passes through, some of the electrons belonging to bromine ions acquire energy sufficient for transition into unoccupied states of the conduction band of the crystal. These electrons then migrate freely in the crystal until they reach sites (sensitivity centers) characterized by the fact that their energy levels localized in some region are ...

are located below the levels of the conduction band. The sensitivity centers—groups of silver atoms or impurities—are usually situated on the surface of the crystal. The excess bromine slowly diffuses to the surface of the grain and here may be bound by gelatin. The sensitivity centers, now negatively charged, attract interstitial silver ions (Frenkel defects), which can move in the crystal lattice. These ions, combining with electrons, form silver atoms. Repetition of this process produces silver clusters of sufficient size to serve as development centers.

It should be expected that the formation of the latent image in nuclear emulsions is far less efficient, in terms of the use of the electrons produced during irradiation, than in photographic emulsions. In part, this inefficiency is explained by the short time taken by a particle to pass through a grain ^42^. For an $\alpha$-particle with energy $5$ MeV and a grain size of $0.3\,\mu$, this time is equal to $\sim 2 \cdot 10^{-14}$ sec. Since the migration of silver ions through the crystal is considerably slower than the migration of electrons, the sensitivity centers acquire a negative charge more rapidly than neutralization occurs, and the excess electrons will be repelled until a sufficient number of silver ions reaches the sensitivity center. During this time the free electrons may scatter and combine with silver ions at other points in the crystal and, consequently, the probability of forming a sensitivity center of a size sufficient for development is correspondingly reduced.

For weakly ionizing particles this effect is comparatively small. For example, it was found that the efficiency for electrons with energy $50$ keV in Kodak NT2a emulsions is comparable with the efficiency of light. This question is considered in Section II. 2.

Development of an irradiated emulsion consists in the deposition of additional silver atoms on the crystal of the development center in an amount sufficient to transform this crystal into a visible grain of colloidal silver. Photographic developers are weak reducing agents requiring the presence of silver clusters (“development centers”) as catalysts for their action. Therefore each silver bromide grain, during development, behaves individually. If there is a sufficiently large latent-image center in the silver bromide grain, the grain is completely reduced. Otherwise the grain will not be affected at all by development. The undeveloped grains are removed by fixing, usually in a hyposulfite solution, which facilitates the dissolution of silver bromide. The developed grains remain as inclusions in the gelatin.

II. 2. Specific Energy Loss

The density of the grains of a track formed by a particle in a nuclear emulsion is usually measured by the number of developed silver grains per unit length. Another quantity may also be used—the mean size of the gaps between grains—but the former method is usually preferred. The grain density depends on the degree of ionization produced in the halide grains and on their sensitivity, defined here as the number of electrons required for the grain to acquire the ability to be developed. If the sensitivity is regarded as constant, we find that the grain density is a function only of the specific energy loss of the ionizing particle. The loss of energy is caused by excitation of electrons and ionization of the atoms of the stopping substance in inelastic collisions.

Energy loss. The mean energy loss per unit length due to electronic collisions \(\left(-dE/dx\right)\) was obtained by Livingston and Bethe\(^{43}\) from quantum-mechanical considerations in the form

\[ -\frac{dE}{dx} = \frac{4\pi z^{2}e^{4}N}{mv^{2}} \left\{ Z\left[ \ln\left(\frac{2mv^{2}}{I}\right)-\ln(1-\beta^{2})-\beta^{2} \right] -C_{k} \right\}, \tag{II, 1} \]

where \(ze\) is the charge of the particle, \(v\) its velocity, \(N\) the number of atoms in \(1\ \mathrm{cm}^{3}\) of the stopping substance, \(Z\) its mean atomic number, \(I\) the ionization potential of the stopping substance, \(m\) the electron mass, \(\beta = v/c\); \(C_{k}\) is a correction term introduced in the case when \(v\) is comparable with the velocities of the electrons in the \(K\)-orbit, but large in comparison with the velocities of the electrons in other orbits. For particles with velocities less than \(\sim 5\cdot 10^{9}\ \mathrm{cm/sec}\), the terms of the relativistic correction containing \(\beta\) approximately cancel, as is seen from the expansion in a series of \(\ln(1-\beta^{2})\), and may be omitted. Hellwege\(^{44}\) found that for \(\alpha\)-particles with an energy of \(10\ \mathrm{MeV}\) in air, \(\ln\left(2mv^{2}/I\right)\) is equal to 5.1, whereas the sum of the two other terms in the square brackets amounts to only \(4.5\cdot 10^{-4}\). Further consideration of this equation was carried out by Wheeler and Ladenburg\(^{45}\).

If \(v\) is sufficiently large, i.e., if

\[ \frac{2mv^{2}}{I} > e, \]

where \(e\) is the base of natural logarithms, the specific energy loss \(-dE/dx\) will depend on \(v\) in accordance with the factor standing before the brackets in equation (II, 1). Thus, \(-dE/dx\) varies in this case inversely proportionally to \(v^{2}\), and as a result the grain density…

will increase in the direction of motion of the particle. At low energies (\(<1\) MeV for protons and \(<0.1\) MeV for \(\alpha\)-particles), when the specific energy loss begins to decrease, equation (II, 1) is not valid. This is connected with the fact that, in its derivation, the processes of capture and loss of electrons by the moving particles, which become appreciable at these velocities, were not taken into account. For example, \(\alpha\)-particles with an energy of \(0.8\) MeV have equal probabilities of carrying a single and a double charge[^46].

Fig. 7. Specific energy loss of various particles in air as a function of energy.

In Fig. 7 are shown curves (corresponding to equation (II, 1)) of the specific energy loss, expressed in MeV per 1 cm thickness of air, for various particles as a function of their energy. To obtain the energy loss in emulsion, these curves must be multiplied by the stopping power of the emulsion relative to air (see Section II. 3).

It is evident from the figure that all sufficiently energetic singly charged particles are characterized by an approximately equal—minimal—energy loss, increasing somewhat at higher energies. The minimal density of grains recorded in a nuclear emulsion of sufficient sensitivity will correspond to this quantity. Equation (II, 1) does not contain the mass of the particle, but only its charge and velocity. In Fig. 7, \(-dE/dx\) is expressed as a function of the particle energy. Since the particle energy is a function of the mass, the curves for different particles prove to be shifted relative to one another, although their form is identical. The minimal specific energy loss of a particle

corresponds to an energy of \(\sim 2m_0c^2\), where \(m_0\) is the rest mass of the particle. This energy is approximately equal to 1 \(Mэv\) for electrons, 200 \(Mэv\) for \(\mu\)-mesons, 300 \(Mэv\) for \(\pi\)-mesons, 2 \(Bэv\) for protons, 4 \(Bэv\) for deuterons, and 8 \(Bэv\) for \(\alpha\)-particles. Let us note that, since the minimum specific energy loss for \(\alpha\)-particles is four times greater than for singly charged particles, any track having a grain density less than four times the minimum grain density must have been formed by a singly charged particle. Tracks with a grain density exceeding four times the minimum density can, of course, be produced by any particle, and in this case other methods of analysis are necessary for determining its charge.

Grain density. Measuring grain density is a simple task if the grains are discrete and the fog is not too great. The first condition is satisfied if there are no more than 50 grains per \(100\ \mu\) of track length; above this limit individual grains are in most cases unresolved. The second condition determines the lower limit of the grain density at which the track can still be distinguished among the background grains. For plates with a relatively low background (less than four grains per 1 sq. micron), this limit is 20 grains per \(100\ \mu\). In Fig. 41 in Section IV.8, values are given for the minimum grain density at which the track is still distinguishable, as a function of the background density. In the case when several grains merge into one group, the number of grains \(n\) in such a group is determined more or less arbitrarily. Fowler and Perkins\(^{47}\) adopted \(n = 2.4l\), where \(l\) is the length of the group in microns. In this connection we note that the mean diameter of a developed grain is about \(0.3\)--\(0.4\ \mu\).

It may confidently be assumed that there is a direct proportionality between the specific energy loss of particles at different points of their track and the corresponding grain density. In Fig. 8 a graph is given of the dependence of the number of developed silver grains per \(100\ \mu\) of track length on the value of \(\frac{dE}{dx}\) for G5 plates\(^{47}\).

It is clear from the graph that for high values of \(\frac{dE}{dx}\) the proportionality breaks down, i.e. the sensitivity of the emulsion decreases when the specific energy loss exceeds a certain value. This is connected with the fact that sensitivity centers are not able to capture sufficiently rapidly the electrons formed in their immediate vicinity during the passage of a charged particle, and thereby to prevent the formation of a space charge with a high electron density. As a result of the recombination that occurs in this case, the effectiveness of the electrons in forming development centers decreases (Section II.1). If the specific losses

of energy less than a certain value, the efficiency of utilization of electrons (in the sense of the fraction of electrons participating in the formation of the latent image, out of the number of electrons originally formed) is almost constant and determines the linear part of the curve in Fig. 8.

At a density above 200 grains per 100 \(\mu\) saturation occurs, and it is no longer possible to estimate \(\dfrac{dE}{dx}\) for such tracks on the basis of grain counting, since with a continuous distribution of grains the proportionality between \(n\) and \(l\) is violated. However, in some cases other methods can be used for this purpose, such as, for example, the method of counting delta particles (Section II.5).

Fig. 8. Grain density in Ilford G5 emulsion as a function of specific energy loss\({}^{47}\).

According to the results of Debye and Hückel\({}^{48}\), the potential of the ionic cloud surrounding an individual ion in strong electrolytes is proportional to \(N^{1/3}\), where \(N\) is the number of ions present. Using the fact that \(N\) is proportional to \(\dfrac{dE}{dx}\), Blau\({}^{49}\) obtained the following expression for the grain density of singly charged particles:

\[ \frac{dn}{dx} = c \left\{ 1-\exp\left[ -b\left(\frac{dE}{dx}\right)^{-1/2} \right] \right\}, \tag{II, 2} \]

where \(b\) and \(c\) are experimentally determined constants depending on the composition of the emulsion and the developing method. The numerical value of \(b\) has not been definitively established, but apparently for Ilford C2 emulsion the value \(b=3\), obtained by Blau, is suitable, although in the region of low energies a somewhat smaller value fits better.

value, for example, 2.5. For \(c\) a value equal to 4 was obtained. The constant \(b\) is a measure of the efficiency with which the electrons released by the incident particle form the latent image, while \(c\) is the maximum possible density of grains in the emulsion, corresponding approximately to the number of halide grains per unit length of track in the undeveloped emulsion.

Van Rossum \(^{50}\) and Morand and van Rossum \(^{51}\) modified this equation and obtained

\[ \frac{dn}{dx} = c\left\{1-\exp\left[-bz\left(\frac{dE}{dx}\right)^{\frac12}-a^{\frac12}\right]\right\}. \tag{II, 3} \]

This equation agrees better with experiment. The constant \(a\) is here the minimum specific energy loss required for the development of emulsion grains under the given processing conditions. Fig. 9 gives a plot of the dependence of

\[ \ln\left[1-\left(\frac{1}{c}\right)\left(\frac{dn}{dx}\right)\right] \]

on

\[ \left(\frac{dE}{dx}\right)^{1/2} \]

for various \(b\) and \(c\). The graph was obtained for Ilford C2 plates loaded with boron and developed in ID19 developer, with \(x\) measured in microns and \(E\) in kev.

Fig. 9. Graphical expression of equation (II, 3), giving the relation between specific energy loss and grain density \(^{51}\).

The value \(a^{1/2}\) is obtained from the intersection of these curves with the abscissa axis and in this case is equal to \(1.2\) kev per micron. Both \(b\) and \(c\) increase with increasing development time. Morand and van Rossum \(^{51}\) found that \(b=0.314\pm0.02\) and \(c=1.8\pm0.1\) for two-hour development at \(5^\circ\)C in ID19 developer diluted \(1:3\), and that for four-hour development \(b=0.455\pm0.02\) and \(c=2.0\pm0.1\). In the presence of regression, \(a\) will increase, while \(b\) and \(c\) will decrease.

II.3. Range–energy relation

The loss of energy of a charged particle in matter occurs in discrete portions in random collisions with the electrons of the slowing-down substance. However, for a finite range this process may be regarded as continuous. A particle with initial energy \(E\), after passing through a distance \(R\) in the substance, expends

all its energy on the formation of ion pairs, whose number will depend on \(E\). Therefore one may expect that there is a definite dependence between the energy and the range of a given particle in a given stopping substance. Knowing this dependence, one can, by measuring particle ranges, determine their initial energy.

Fig. 10. Theoretical range–energy curve for protons in air up to an energy of 15 MeV\({}^{43}\).

Fig. 11. Theoretical range–energy curve for protons in air up to an energy of 250 MeV\({}^{52}\).

Theoretical dependence. Knowing the specific energy loss \(\dfrac{dE}{dx}\) and the initial energy \(E\) of the particle, the range can be calculated from the integral

\[ R=\int_{0}^{E}\frac{dE}{\dfrac{dE}{dx}}. \tag{II, 4} \]

Livingston and Bethe\({}^{43}\) considered this integral in detail for protons with energies up to 15 MeV in air, and Smith\({}^{52}\) extended the calculation to 10 BeV, using the equation

\[ R=R(15)+\int_{15}^{E}\frac{dE}{\dfrac{dE}{dx}}. \tag{II, 5} \]

The curves illustrating these theoretical results are given in Figs. 10 and 11. These calculations are based on the assumption that

energy is spent exclusively on ionization and excitation of the atoms of the slowing-down substance, which is approximately*) valid only up to energies of several hundred MeV. At higher energies Smith’s calculations are invalid, since meson production becomes appreciable. The curves in Figs. 10 and 11 may be used for nuclear emulsions by multiplying the range values by the stopping-power ratio of the emulsion to air (see below).

Combining equations (II, 1) and (II, 4), we obtain:

\[ R=\left(\frac{M}{z^{2}}\right) f(v), \tag{II, 6} \]

where \(M\) is the mass, \(z\) the charge, \(v\) the particle velocity, and \(f(v)\) is a function independent of both charge and mass. This equation makes it possible to obtain range–energy curves for any ionizing particle, if the curve is known for some particle with given mass and charge. Let \(M_a, z_a\) and \(M_b, z_b\) refer to two different particles \(a\) and \(b\) of the same velocity, passing through one and the same medium. Then

\[ R_b(v)=\left(\frac{z_a}{z_b}\right)^2\left(\frac{M_b}{M_a}\right)R_a(v). \tag{II, 7} \]

For the energies, obviously, the relation holds

\[ E_b=\left(\frac{M_b}{M_a}\right)E_a, \tag{II, 8} \]

and for the specific energy loss—the relation

\[ \left(\frac{dE}{dx}\right)_{b,v} = \left(\frac{z_b}{z_a}\right)^2 \left(\frac{dE}{dx}\right)_{a,v}. \tag{II, 9} \]

Using equation (II, 8) and carrying out some transformations, we obtain for particles of the same energy:

\[ R_b(E)= \left(\frac{z_a}{z_b}\right)^2 \left(\frac{M_b}{M_a}\right) R_a\left[\left(\frac{M_a}{M_b}\right)E\right], \tag{II, 10} \]

where \(R_a\left[\left(\frac{M_a}{M_b}\right)E\right]\) is the range of particle \(a\) with energy \(\left[\left(\frac{M_a}{M_b}\right)E\right]\), and

\[ \left(\frac{dE}{dx}\right)_{b,E} = \left(\frac{z_b}{z_a}\right)^2 \left(\frac{dE}{dx}\right)_a \left[\left(\frac{M_a}{M_b}\right)E\right], \tag{II, 11} \]

where \(\left(\frac{dE}{dx}\right)_a\left[\left(\frac{M_a}{M_b}\right)E\right]\) is the specific energy loss of particle \(a\) at energy \(\left[\left(\frac{M_a}{M_b}\right)E\right]\).

*) At high energies the particle also loses energy to nuclear disintegrations.

These equations are rigorously satisfied only under the condition \(z_a=z_b\), since then, for both particles, both the capture and the loss of electrons at low energies are the same. In this case all equations from (II, 7) to (II, 11) are exact, if for the reference particle the experimental values of the quantities entering into the formulas are used, irrespective of the fact that, in deriving equation (II, 1), the loss and capture of electrons were not considered. For particles with different \(z\), the calculated results differ from those observed experimentally. Thus, for example, for protons and \(\alpha\)-particles, Blackett and Lees\(^{53}\) found:

\[ R_{\text{proton}}= \left(\frac{M_p}{M_\alpha}\right) \left(\frac{z_\alpha}{z_p}\right)^2 R_\alpha-C; \tag{II, 12} \]

where \(C=0.20\ \mathrm{cm}\) of air. From Figs. 10 and 11 it is clear that this correction is appreciable only at very low energies. Since \(0.20\ \mathrm{cm}\) of air is roughly equivalent to \(1\ \mu\) of emulsion, and the scatter in ranges (see below) and experimental errors limit the accuracy of range determinations, this factor may be neglected when using equations (II, 7)—(II, 11).

Experimental dependence. The range–energy relation for \(\alpha\)-particles and protons has been investigated many times

Table VII

Dependence of range on energy for protons and \(\alpha\)-particles in Ilford B1 emulsion (measurements made up to an energy of \(13.0\ \mathrm{MeV}\) \(^{55}\) and then extrapolated \(^{56}\))

Energy (MeV)	Proton range (\(\mu\))	\(\alpha\)-particle range (\(\mu\))	Energy (MeV)	Proton range (\(\mu\))	\(\alpha\)-particle range (\(\mu\))
0,5	5,5	2,1	8,5	426,0	45,3
1,0	14,5	3,52	9,0	469,0	49,5
1,5	26,0	4,96	9,5	515,0	53,7
2,0	40,0	6,54	10,0	564,0	58,0
2,5	56,5	8,34	10,5	614,0	62,6
3,0	75,0	10,38	11,0	666,0	67,7
3,5	97,0	12,60	11,5	720,0	72,7
4,0	120,5	15,0	12,0	776,0	77,8
4,5	146,0	17,65	12,5	834,0	83,4
5,0	173,0	20,5	13,0	895,0	. .
5,5	202,0	23,6	15,0	1135	117
6,0	234,0	26,7	20,0	1870	201
6,5	269,0	30,0	25,0	2750	315
7,0	306,0	33,6	30,0	3760	464
7,5	345,0	37,5	35,0	4925	653
8,0	385,0	41,4

experimentally. The most complete data were obtained in the works of Lattes, Fowler, and Cuer^54,55 (for Ilford B1 emulsions) and of Bradner et al.^56 (for Ilford C2 emulsions). It turned out that no significant differences are observed in the stopping power of emulsions of the Ilford B1, B2, C2, C3, E1, and G5 types and Kodak NT2a.^56,57 Since the range–energy curves for Eastman-Kodak emulsions (Sec. I.3) are very close to the corresponding curves for other emulsions, and since the various emulsions differ only slightly in composition, it may be assumed that the results obtained in Refs. 54–55 are valid for all emulsions manufactured at the present time.

The results of the measurements^54 for protons and $\alpha$-particles up to an energy of 13 MeV (obtained from nuclear reactions) are given in Table VII.

Table VIII

Experimental values of the range–energy relation for protons in dry Ilford C2 emulsion^56

Energy in MeV	Range in $\mu$	Energy in MeV	Range in $\mu$
7.8	389	25.6	2849
16.4	1358	28.2	3369
17.6	1465	33.5	4597
22.3	2244	39.5	6123

For energies above 2 MeV the accuracy of the values given is $\pm 2\%$. Camerini and Lattes^58 extrapolated these results to 35 MeV with an assumed accuracy of $\pm 8\%$. The extrapolated values agree very well with Bradner’s experimental data for protons of energy 7.8–39.5 MeV from the Berkeley cyclotron (Table VIII); this also shows the reliability of the extrapolated values for $\alpha$-particles. For protons the accuracy of the measurements is at least 2%.

It is interesting to note the influence of atmospheric humidity on the magnitude of the range in emulsion. It was found that at a relative humidity of $\sim 80\%$ the range of protons with energy 17.6 MeV is 1497 $\mu$, and at humidities of 80 and 90% the range of 33.5-MeV protons is 4762 and 4936 $\mu$, respectively. This effect occurs owing to absorption of moisture by the gelatin of the emulsion (see below). In addition, in these experiments the range of 30-MeV protons in the glass of photographic plates was approximately determined. It turned out that the range in glass is larger by $18 \pm 4\%$ than in emulsion.

At high velocities the range–energy curves, plotted on a logarithmic scale, are close to straight lines (Fig. 12). Integrating the expression

\[ \frac{dE}{dx}=z^2 f_1(v)=z^2 f_2\left(\frac{E}{M}\right), \tag{II, 13} \]

we obtain:

\[ E=M f_3\left(z^2\frac{R}{M}\right). \tag{II, 14} \]

Using the linearity of the curves \(\ln R\) as a function of \(\ln E\), for \(f_3\) we find:

\[ f_3\left(z^2\frac{R}{M}\right)=K\left(z^2\frac{R}{M}\right)^n, \tag{II, 15} \]

where \(K\) is a constant, and the exponent \(n\) is constant over a comparatively wide region. In this region one can directly calculate the range–energy dependence for particles of any mass and charge without using equations (II, 7) and (II, 10). From equation (II, 14) we obtain:

\[ E=K z^{2n} M^{1-n} R^n . \tag{II, 16} \]

Lattes et al.\(^{59}\) found the values \(K=0.262\) and \(n=0.575\), if \(M\) is expressed in units of the proton mass, \(R\) in microns, and \(E\) in MeV.

Fig. 12. Range–energy dependence for various particles in nuclear emulsions.

Fig. 13. Relation between the energy of electrons and their range in emulsion\(^{61}\).

In Fig. 12 the curves for mesons obtained from equation (II, 16) are presented.

Determination of the range–energy dependence for electrons is greatly complicated by considerable scattering. Ross and Zajac\(^{60}\), using an electron spectrograph, and Gerz\(^{61}\), using an electron microscope and photoelectrons from \(\gamma\)-rays, determined this dependence for Kodak NT2a emulsion. Their results are summarized in Fig. 13, moreover

for energies above \(\sim 80\) kev (the maximum energy of electrons that can be recorded in NT2a emulsions) the data were obtained by extrapolation. It may be assumed that this curve is also valid for other electron-sensitive emulsions, taking into account errors in the measurement of electron ranges.

Fluctuations in particle ranges. We have assumed that the range of a particle of a given energy is perfectly constant and that all observed deviations are the result of experimental errors. However, evidently, owing to the discontinuous nature of the ionization process, fluctuations in the ranges of monoenergetic particles will occur. These fluctuations (scatter), amounting in air to only \(\sim 1\%\), become more appreciable in emulsion because of the finite dimensions and the relatively small number of grains forming a track. In addition, the halide grains are nonuniformly distributed in the gelatin and, consequently, there are regions with a grain concentration below the average concentration, which leads to additional uncertainty in the measurement of range. For ranges \(R_N\) of a beam of homogeneous particles, Rotblat\(^{62}\) defined the scatter \(\gamma\) in the form

\[ \gamma=\left(\frac{\pi}{2}\right)^{1/2} \left[ \frac{\Sigma_N (R_N-R_0)^2}{N} \right]^{1/2}, \tag{II, 17} \]

where \(R_0\) is the mean range, and \(N\) is the total number of tracks. The scatter may also be defined as the half-width \(\Delta R\) of the differential curve for an ordinate equal to half the maximum. For a Gaussian distribution

\[ \Delta R=\frac{2\gamma(\ln 2)^{1/2}}{\pi^{1/2}}=0.94\gamma . \tag{II, 18} \]

The scatter is usually expressed as a percentage relative to the range, i.e. as \(100\,\dfrac{\Delta R}{R}\), or relative to the energy,

\[ 100\frac{\Delta E}{E}=100\frac{n\Delta R}{R}, \]

where \(n\) is the exponent in equation (II, 15).

Fig. 14. Energy scatter in the ranges of protons and \(\alpha\)-particles in emulsion\(^{62}\) and protons in air\(^{43}\).

The scatter for different particles was determined experimentally\(^{55,63,64}\). In Fig. 14 are shown the results of determining the energy scatter of protons and \(\alpha\)-particles with energies \(0.4\)—\(8\) Mev\(^{62}\). For comparison, the same figure gives the corresponding curve for protons in air\(^{43}\).

Wet emulsions. When it is necessary to irradiate wet emulsions, as, for example, in the case of filling plates with deuterium (Section V.5), or when one wishes to introduce exact corrections for different moisture contents, the quantitative relations given above must be somewhat modified. The curves of the specific energy loss of protons in Ilford63 emulsions and in water (obtained from the range–energy curves for protons in oxygen and hydrogen66, 67) are shown in Fig. 15. The mean rate of energy loss in a wet emulsion can be determined from these curves by adding the corresponding values of

\[ \frac{dE}{dx} \]

for any given energy, multiplied by their relative weights. In this way, Kron and Shreder67 calculated the ranges of protons of various energies in Ilford emulsions containing different amounts of water (Table IX). These values can be recalculated for particles of different masses

Fig. 15. Specific energy loss of protons as a function of energy in Ilford emulsions and in water.

Table IX

Ranges of protons of various energies in emulsions with different water content. Ranges are given in microns

Energy in MeV	Relative volume of water in the emulsion	Relative volume of water in the emulsion	Relative volume of water in the emulsion	Relative volume of water in the emulsion	Relative volume of water in the emulsion	Relative volume of water in the emulsion	Relative volume of water in the emulsion	Relative volume of water in the emulsion
Energy in MeV	0.5	0.6	0.65	0.68	0.7	0.72	0.74	0.76
1	17.4	18.1	18.5	18.7	18.9	19.1	19.2	19.4
2	50.3	53	54.4	55.3	56	57	57	58
3	91.1	96	98	99.5	100	101	103	104
4	152	160	165	168	171	173	175	178
5	226	240	248	253	256	260	264	268
6	311	331	343	351	355	361	365	372
7	407	434	450	461	467	475	481	490
8	514	549	570	584	592	602	610	622
9	632	676	703	720	730	743	753	766
10	761	815	848	868	881	897	909	926
11	899	964	1003	1027	1043	1062	1076	1097
12	1046	1123	1168	1197	1216	1238	1256	1280
13	1204	1293	1346	1379	1402	1427	1448	1476
14	1379	1485	1542	1580	1606	1635	1660	1691

using the equations given above. In what follows another method will be described for determining the range–energy relation for wet emulsions, based on calculation of the stopping power.

In order, when determining the track length, to introduce a correction for swelling of the emulsion due to absorbed water, it is necessary to know the shrinkage coefficient of the emulsion \(S'\). Then the particle range is

\[ R=\left[Y^{2}+(S'Z)^{2}\right]^{1/2}, \tag{II, 19} \]

where \(Y\) is the projection of the track length onto the plane of the emulsion, and \(Z\) is the difference in depth of the ends of the track. The coefficient \(S'\) is equal to the ratio of the thickness of the emulsion during exposure to its thickness after development, and may be determined either by direct measurement or by calculation. In the latter case the equation

\[ S'=\frac{(ST_{0}+T_{w})}{T_{0}}, \tag{II, 20} \]

may be used, where \(S\) is the usual shrinkage factor (Section III.6), \(T_{0}\) is the thickness of the emulsion after development, and \(T_{w}\) is the thickness of the layer of added water. The thickness of the water layer, expressed in terms of mass, in a plate of size \(2.5\times 7.5\) cm is equal to

\[ T_{w}=0.517\,M_{\mathrm{H_2O}}\ \text{micron} \tag{II, 21} \]

or

\[ T_{w}=0.465\,M_{\mathrm{D_2O}}\ \text{micron}, \tag{II, 22} \]

where the masses of ordinary and heavy water are expressed in milligrams\({}^{67}\). The mass of water is determined by weighing the dry and wet emulsion. It should be noted that wet plates in air lose several milligrams of water per minute, although this amount can be reduced by using suitable vessels that preserve a saturated atmosphere.

Stopping power. The stopping power of nuclear emulsions, defined as the ratio of the ranges of a given particle in air at normal pressure to their ranges in the emulsion for specified energy intervals, is a very convenient quantity, since it makes it possible to convert immediately the values of the range–energy relation in air to the corresponding values in the emulsion.

The stopping power can be determined experimentally. However, one can also obtain an analytical expression for the stopping power for various homogeneous substances and use it, in particular, to determine the stopping power of wet emulsions and emulsions diluted with gelatin. The stopping power of an element with atomic number \(Z\) relative to air

is given by the relation

\[ s=\frac{B}{B_0}, \tag{II, 23} \]

where

\[ B=Z\ln\left(\frac{2mv^2}{I}\right) \tag{II, 24} \]

is the dimensionless quantity in the energy-loss formula (equation (II, 1)) for the atoms of the slowing substance, while \(B_0\) is the corresponding quantity for air. The stopping number \(B\) was estimated for certain elements by Livingston and Bethe \(^{43}\) for particles with velocities from \(1\cdot 10^9\) to \(5\cdot 10^9\) cm/sec, and Webb \(^{42}\), by interpolating from these data, obtained the stopping power of the atoms making up the emulsion. His results are given in Table X for \(\alpha\)-particles and protons with energies corresponding to the indicated velocities.

Table X.

Atomic stopping power for various velocities and particle energies \(^{42,43}\)

Velocity (in \(10^9\) cm/sec)	Energy in MeV \(\alpha\)-particles	Energy in MeV protons	Ag	Br	C	H	N	O	air
1.0	2.07	0.52	2.25	2.07	0.940	0.260	1.02	1.10	1.0
1.5	4.66	1.17	3.08	2.68	0.932	0.224	1.02	1.10	1.0
2.0	8.30	2.09	3.43	2.94	0.921	0.209	1.01	1.10	1.0
2.5	12.95	3.26	3.64	3.10	0.914	0.200	1.01	1.09	1.0
3.0	18.60	4.70	3.76	3.19	0.908	0.194	1.00	1.09	1.0
4.0	33.21	8.36	3.93	3.30	0.899	0.186	1.00	1.08	1.0
5.0	51.9	13.05	4.04	3.38	0.892	0.181	0.99	1.08	1.0

To calculate the stopping power of an emulsion from these data, one must use the method proposed by Kődz \(^{69}\) and Webb \(^{42}\). According to the definition of \(s\) for some specified substance, one may write:

\[ \frac{R_0}{R}=\left(\frac{N}{N_0}\right)s, \tag{II, 25} \]

where \(R_0\) is the range of the particle in air, \(R\) is its range in the substance, \(N_0\) is the effective number of atoms in \(1\ \text{cm}^3\) of air at normal pressure, calculated from the mean atomic weight, and \(N\) is the number of atoms of the slowing substance in \(1\ \text{cm}^3\). Since in practice the ratio

\[ \frac{\Delta R_0}{\Delta R} \]

of differential ranges for small

changes of energy, then a relation of the form will be used

\[ \frac{\Delta R_0}{\Delta R}=\left(\frac{N}{N_0}\right)s . \tag{II, 26} \]

Since

\[ N=\frac{kd}{A}, \tag{II, 27} \]

where \(d\) is the density of the substance, and \(A\) is the atomic weight of the elements composing it, then

\[ \frac{\Delta R_0}{\Delta R}=n\Sigma_i N_i\frac{s_i}{N_0}, \tag{II, 28} \]

where \(N_i\) is the number of atoms of the \(i\)-th kind with stopping power \(s_i\) in each molecule, and

\[ n=\frac{d}{\Sigma_i N_i A_i} \tag{II, 29} \]

is the number of molecules in \(1\ \mathrm{cm}^3\). Consequently,

\[ \frac{\Delta R_0}{\Delta R} = \left(A_0\frac{d}{d_0}\right) \left(\frac{\Sigma_i n_i s_i}{\Sigma_i N_i A_i}\right), \tag{II, 30} \]

or, in a form more convenient for calculation:

\[ \frac{\Delta R_0}{\Delta R} = \left(A_0\frac{d}{d_0}\right) \Sigma_i \frac{p_i s_i}{A_i}. \tag{II, 31} \]

The quantity

\[ p_i=\frac{N_i A_i}{\Sigma_j N_j A_j} \tag{II, 32} \]

is the weight fraction of each element in a compound substance.

Although nuclear emulsions are in fact a suspension of silver bromide crystals in gelatin and, consequently, are rather a mixture than a compound, equation (II, 31) may be applied provided that the existing inhomogeneity is negligible in comparison with the ranges of the particles. Substituting the values of \(s_i\) from Table X into equation (II, 31), one may, for the given emulsion, estimate \(\Delta R_0/\Delta R\) at various energies. To find the integral stopping power \(R_0/R\), it is necessary to divide the differential values of the range in air \(\Delta R_0\) for small energy intervals (one or two MeV) by the corresponding values of \(\Delta R_0/\Delta R\), and thus obtain the equivalent differential ranges \(\Delta R\) in the emulsion. Summation of these values of \(\Delta R\) gives the integral ranges in the emulsion \(R\). Division of \(R_0\), obtained—

summing the values of \(\Delta R_0\) over \(R\), gives the integral stopping power. Webb \(^{42}\) gives curves of \(\dfrac{R_0}{R}\) as a function of energy for an emulsion containing \(82\%\) silver bromide (Fig. 16). It may be expected that these curves are suitable for most of the emulsions produced. The experimental results of Lattes \(^{55}\) in the region below \(13\) MeV agree with these curves.

II. 4. Identification of tracks

In accordance with what has been said, on the basis of measurements of range and grain count it is possible to determine the mass and energy of particles. The methods described below considerably facilitate the identification of particles according to differences in the grain structure of the tracks. However, when electron-sensitive emulsions are used, most tracks have a discontinuous structure and other methods of identification are necessary. In particular, to determine the mass one may measure either the optical density or the scattering (Section II. 7) of the track. Tracks produced by particles with charge exceeding 2 may be analyzed by delta rays and by the length of the narrowing at the end of the track (Sections II. 5 and II. 6).

Fig. 16. Integral stopping power of nuclear emulsions as a function of particle energy \(^{42}\).

For the nuclei Li, Be, and B one may use the measurement of scattering in conjunction with ionization.

Tracks ending in the emulsion. The mass of particles whose tracks end in the emulsion is readily determined by grain counting. For singly charged particles, equation (II, 14) gives:

\[ E = M f\left(\frac{R}{M}\right). \tag{II, 33} \]

The total number of grains \(N\) in the track is undoubtedly a function of the initial energy of the particle \(E\) and, thus,

\[ N = M F\left(\frac{R}{M}\right), \tag{II, 34} \]

where \(F\) has one and the same form for all mass values.

Thus, for two particles \(a\) and \(b\)

\[ N_a=M_aF\left(\frac{R_a}{M_a}\right) \tag{II,35} \]

and

\[ N_b=M_bF\left(\frac{R_b}{M_b}\right). \tag{II,36} \]

If the values of \(F\left(\frac{R}{M}\right)\) in equations (II,35) and (II,36) are equal, then

\[ \frac{N_b}{N_a}=\frac{M_b}{M_a}=r \tag{II,37} \]

and

\[ \frac{R_b}{R_a}=\frac{M_b}{M_a}=r, \tag{II,38} \]

where \(r\) is the mass ratio. Consequently,

\[ \ln N_b-\ln N_a=\ln r \tag{II,39} \]

and simultaneously

\[ \ln R_b-\ln R_a=\ln r. \tag{II,40} \]

Figure 17 gives plots of the dependence of \(N\) on \(R\) on a logarithmic scale, taken from Lattes’ data \(^{69}\). From relations (II,39) and (II,40), it is evident that the points of equal values of \(F\left(\frac{R}{M}\right)\) for different curves are located at their intersections with a straight line drawn at an angle of \(45^\circ\). In this way one immediately obtains the value of \(r\). If, for some emulsion, graphs analogous to those shown in Fig. 17 are constructed, then on the basis of them one can identify singly charged particles.

Fig. 17. Variation of the total number of grains \(N\) as a function of the residual range \(R\) for tracks of various particles in Ilford C2 emulsions filled with boron \(^{69}\).

Since the curves for different particles are linear over the greater part of their length, we have

\[ F\left(\frac{R}{M}\right)=k'\left(\frac{R}{M}\right)^{n'} \tag{II,41} \]

and, consequently,

\[ N=k'M^{1-n'}R^{n'}, \tag{II,42} \]

as also in the case of the range–energy dependence. Here \(M\) is expressed in units of the proton mass, and for boron-filled Ilford C2 emulsions \(k'=76\) and \(n'=0.711\) \(^{39}\). For particles with charge \(z\) exceeding unity,

\[ N=k'z^{2n'}M^{1-n'}R^{n'}. \tag{II,43} \]

Another method of estimating the mass is based on the relation between the grain density \(\dfrac{dN}{dR}\) at any point of the track and the distance \(R\) from this point to the end of the track

\[ \frac{dN}{dR}=f'\left(\frac{R}{M}\right). \tag{II, 44} \]

Thus, at points with equal grain density, the ratio of the masses of particles \(M_a\) and \(M_b\) is determined by the equality

\[ \frac{M_b}{M_a}=\frac{R_b}{R_a}. \tag{II, 45} \]

Inaccuracies in the application of this method may arise in the presence of regression, if tracks formed at different times are compared. In addition, in the case of thick emulsions the degree of development may vary with depth, despite all precautions taken to reduce this effect to a minimum. However, under favorable conditions comparative mass measurements can be carried out with a good degree of accuracy. For example, this method was first applied to estimate the ratio of the masses of \(\pi\)- and \(\mu\)-mesons, and since in this case the tracks were formed at one and the same time and in the same region of the emulsion, the results obtained are sufficiently reliable.

Fig. 18. Specific energy loss of various particles in nuclear emulsions as a function of their residual range. \(10^4\ \mu\) of emulsion \(\cong 4.0\ \text{g}/\text{cm}^2\).

Tracks of particles ending in the emulsion. If a track does not end in the emulsion, the change in the grain density along it, when it is sufficiently large, makes it possible to determine the mass of the particle that produced this track. In Fig. 18 curves are plotted showing the relation between the residual range in the emulsion of various particles and their specific energy loss in \(\mathrm{MeV}/(\text{g}/\text{cm}^2)\)⁷⁰, ⁷¹. Each emulsion used must be calibrated in order to establish the corresponding relation between grain density and specific energy loss. To convert from experimental curves (such as those in Fig. 8) to the energy loss in \(\mathrm{MeV}/(\text{g}/\text{cm}^2)\), one should use the equality

\[ 10^4\ \mu\ \text{of emulsion}=4.0\ \text{g}/\text{cm}^2. \tag{II, 46} \]

For identification of a particle from a given portion of a track (assuming that the particle charge is equal to unity), the grain density is determined at two, preferably separated, points of the segment. The smallest density, corresponding to the greatest particle energy, is used to determine the residual ranges of the particles (by means of the curves in Fig. 18) for the given specific energy loss. The specific energy loss expected for these particles after passing through the second experimental point can be found by shifting downward along the ordinate, on the various curves, by the corresponding distance. The energy-loss value thus found, which corresponds to the greatest observed grain density, establishes the type of particle. The particle energy is then determined from the range–energy curves (Fig. 12) according to the expected residual range.

Fig. 18 may also be used for tracks ending in the emulsion. The grain density of the track at the point where the particle enters the emulsion, expressed in terms of energy loss, and the residual range are simply compared with these curves in order to identify the particle that produced the track.

Slide rule with movable scale. The approximate linearity of the range–energy, range–grain-density, and residual-range–total-number-of-grains curves on a logarithmic scale was used by Beiser^72 to construct a simple slide rule for facilitating the evaluation of tracks. The scales of the rule are most conveniently divided as shown in Fig. 19. Calibration is carried out from experimental curves for each emulsion composition and development method. The scales of range \((R)\), energy \((E)\), grain density \((d)\), and number of grains \((n)\) are taken directly from the corresponding curves. The scale \(P\), which gives particle identification, is marked by choosing an arbitrary value of \(R\), finding the corresponding values of \(E\), \(d\), or \(n\) for different particles, and bringing these values into coincidence with \(R\). The type of particle is then marked opposite the corresponding arrow on the \(R\) scale. Arrow \(A\) is used for the dependence of \(R\) on \(E\), \(B\) for the dependence of \(R\) on \(n\), and \(C\) for \(R\) on \(d\).

Fig. 19. Movable slide rule for identifying tracks^72.

When the ruler is used, the type of particle is determined in the same way as from the curves. If the track ends in the emulsion, then the number of grains \(n\) at a distance \(R\) (in microns) from the end of the track is combined with \(R\), and the type of particle is found near the mark \(B\). As a check, or if the track does not end in the emulsion, one should measure the grain density \(d\) over a section \(100\,\mu\) long at two points of the track, possibly far apart. Then pointer \(C\) is set opposite the various particles on scale \(P\), and on scale \(R\) the ranges corresponding to the smallest grain density are read. The distance between the two experimental points is subtracted from the range values thus obtained; pointer \(C\) is again set to the various particles and, opposite the new ranges, the grain density is read. The density corresponding to the measured grain density at the second point determines the particle.

The energy of a particle stopped in the emulsion can be found from scales \(R\) and \(E\) by means of pointer \(A\). For tracks that do not end in the emulsion, \(E\) can be determined from the range corresponding to the smallest grain density.

II.5. Delta rays

Particles with charge greater than two, even at high energies, form tracks with an actually grainless structure. Such tracks have the appearance of a continuous silver thread (Fig. 20). Sufficiently

Fig. 20. Parts of the track left by an iron nucleus with energy \(62\) Bev in a stack of nuclear emulsions; found during work with cosmic rays at high altitude\(^{77}\).

energetic heavy ions are observed chiefly as a component of cosmic radiation at high altitudes\(^{73}\). The rate of energy loss of these particles is so great that they comparatively often produce secondary electrons with energies sufficient for registering their tracks in the emulsion. The number of such electrons (or delta rays) is a function of \(\frac{dE}{dx}\) and, together with measurement of ranges, can serve for determining the charge and energy of these particles.

Theoretical calculation of the number of delta rays. Following Bradte and Peters\(^{74}\), let us find the theoretical expression for

$n$—the number of delta rays per 1 cm of track length. Mott\({}^{75}\), for the number $dn$ of such rays with energy between $W$ and $W+dW$, obtained the expression

\[ dn=\frac{2\pi Nz^2e^4}{mv^2}\,\frac{dW}{W^2} \left[ 1-\frac{(1-\beta^2)}{2}\frac{W}{mc^2} + \frac{z\alpha\beta}{137} \left( \frac{1-\beta^2}{2\beta^2}\frac{W}{mc^2} \right)^{1/2} \left( 1-\frac{1-\beta^2}{2\beta^2}\frac{W}{mc^2} \right) \right], \tag{II, 47} \]

where $m$ is the electron mass, $z$ the charge of the incident particle, $v$ its velocity, and $N$ the electron density in the stopping substance. This expression is the cross section for elastic scattering of electrons by the Coulomb field of a nucleus of charge $z$, referred to a coordinate system in which the electron is initially at rest. As in the case of the expression for $\dfrac{dE}{dx}$, the mass does not enter into this equation.

Identification of delta rays is possible only when they are properly oriented for observation and have an energy lying within some definite interval. The upper energy limit $W_2$ depends on the sensitivity of the emulsion to electrons (see Section I. 2), and the lower limit $W_1$ on the criterion adopted for distinguishing short tracks of delta rays from background grains. Freier et al.\({}^{76}\), for example, adopted a minimum track length equal to $1.5\,\mu$, while Bradt\({}^{74}\) adopted a minimum number of grains equal to four. Both methods give a lower limit near 10 kev.

The maximum energy of delta rays for a given $\beta$ in the nonrelativistic approximation is equal to $2mc^2\beta^2$, where $m$ is the electron mass. The smallest value of $\beta$ that can lead to the formation of delta rays with maximum energy $W_2$ is therefore determined by the condition

\[ \beta \geqslant \frac{W_2}{2mc^2}. \tag{II, 48} \]

For $W_2=30$ kev, which corresponds to the maximum energy of electrons still recorded by Ilford C2 emulsion, this means that equation (II, 47) can be used for residual ranges of $\alpha$-particles in the emulsion exceeding $1200\,\mu$. For fully ionized carbon atoms the minimum residual range is close to $400\,\mu$, and for heavier atoms this range is still smaller.

For delta rays with energies between 10 and 30 kev the relativistic term in the square brackets gives a correction of less than 8% for values of $z$ up to $z=30$. Thus this term may be neglected without introducing errors exceeding the experimental ones. Integrating the remaining part of equation (II, 47), we obtain for $n$—

numbers of delta rays with energy in the interval from \(W_1\) to \(W_2\): the value

\[ n=\frac{2\pi Nz^{2}}{\beta^{2}}\left(\frac{e^{2}}{mc^{2}}\right)^{2} \left(\frac{mc^{2}}{W_{1}}-\frac{mc^{2}}{W_{2}}\right). \tag{II,49} \]

Irrespective of the criterion used in identification, experimentally only a certain fraction of this theoretical number of \(\delta\)-rays should be observed. Bradt and Peters\({}^{74}\) found that this experimentally observed fraction is \(16\%\) for \(\delta\)-rays formed in NTB emulsions by \(\alpha\)-particles with energies of 168 and 368 MeV (from a cyclotron), and \(10\%\) for \(\delta\)-rays formed in Ilford C2 emulsions by cosmic \(\alpha\)-particles with energies \(\sim 60\) MeV.

According to equation (II, 49), \(n\) should vary as \(z^{2}/\beta^{2}\). However, the dependence \(1/\beta^{2}\) is in fact not quite exact, as follows from Fig. 21\({}^{76}\). These curves give the number of delta rays with energy \(> W_1\) for values of \(W_1\) between 5 and 50 keV as a function of \(\beta\), relative to the corresponding value of \(n\) for \(\beta=1\) (i.e., \(v=c\)). If \(n\) varied as \(1/\beta^{2}\), then at \(\beta=0.35\) the ordinates of all these curves would be equal to \(1/(0.35)^2\), or 8.2, which is only approximately true. However, for \(W_1 \leqslant 10\) keV the deviation from the \(1/\beta^{2}\) dependence is in most cases practically negligible.

Experimental application. To determine the charge of the particle responsible for a track that has a definite number of \(\delta\)-particles at the residual range \(R\), it is necessary to know accurately the variation of \(n\) with \(R\) for different values of \(z\). Since

\[ \frac{n}{n_{\alpha}}=\frac{z^{2}\beta_{\alpha}^{2}}{z_{\alpha}^{2}\beta^{2}}, \tag{II,50} \]

Fig. 21. Ratio of the number of delta rays with energy greater than \(W_1\) to the corresponding number of them at \(v=c\), as a function of \(\beta\)\({}^{76}\).

where \(n_{\alpha}\) is the density of \(\delta\)-rays in the track of an \(\alpha\)-particle at the point corresponding to the residual range \(R_{\alpha}\) and velocity \(\beta_{\alpha}c\), while \(\beta^{2}\) is a known function of

\[ \frac{Rz^{2}}{M}\simeq \frac{Rz}{2} \quad (M\simeq 2z \text{ is the mass of the particle}), \]

determined by the expression

\[ \beta^{2}=\psi\left(\frac{Rz}{2}\right), \tag{II,51} \]

we obtain:

\[ n=\frac{z^{2}}{4}\frac{n_{\alpha}\psi'(R_{\alpha})}{\psi\left(\frac{Rz}{2}\right)}. \tag{II,52} \]

Figure 22 shows a plot of the dependence of \(\ln n\) on \(\ln R\) for different values of \(z\), calculated by Bradt and Peters \(^{74}\) with the aid of equation (II,52). For tracks ending in the emulsion, \(z\) is determined very accurately, since \(n\) can be measured at various values of \(R\) and, using the curves of Fig. 22, several independent determinations can be made for one and the same track.

Fig. 22. Change in the density of delta rays with range in emulsion for different values of \(z\) \(^{74}\).

When the particle does not come to rest in the emulsion, the value of \(z\) can be established from the change of \(n\) along the track. If \(n\) remains relatively constant over a track length of several \(\mathrm{g/cm^2}\), the curves of the dependence of \(n\) on \(R\) can be used to obtain upper and lower limits for \(z\). When \(n\) changes considerably, it is necessary to assume a value of \(z\) and then, by trial and error, successively obtain the true value. A less commonly used method for determining an approximate value of \(z\) consists in assuming that the track at the point of entry into the emulsion has minimum ionization, from which an upper limit of \(z\) is found. The lower limit of \(z\) is determined on the assumption that the length of the part of the track visible in the emulsion is equal to the actual residual range.

II.6. Length of “narrowing”

An ingenious method for determining the atomic number of multiply charged particles whose tracks end in the emulsion was proposed by Freier \(^{73,76}\). Such tracks at first widen and then

near the end of the particle’s range become narrower. An example of “narrowing” or “sharpening” of the track of a heavy nucleus is shown in Fig. 20, where various parts of the track left by an iron nucleus with an initial energy of 62 Bev in a nuclear-emulsion stack are shown[^77]. The “narrowing” of the track occurs as a result of electron capture by the initially “bare” nucleus when its energy becomes sufficiently small. At the same time the effective charge of the nucleus decreases and, consequently, so does the rate of energy loss. The length of the “narrowed” part of the track can serve for calculating an approximate value of the atomic number of the particle \(z\), if it is assumed that electron capture begins at a particle velocity equal to the velocity of the electrons in the \(K\)-orbit. Since the velocity of the \(K\)-electrons is proportional to \(z\), the length of the “narrowing” \(L\) should also be a function of \(z\). Under the assumption that the mass of the nucleus is \(2z\) times greater than the proton mass, for the energy at which the first electron is captured we obtain the value \(0.05 z^3\) Mev. Assuming that, at all energies below the indicated energy, the capture energy \(E\) of all subsequent electrons is equal to \(0.05 z z'^2\), where \(z'\) is the effective charge of the particle for each value of the energy, one can determine \(L\) by numerical integration of the expression

Fig. 23. Theoretical dependence between the length of track narrowing and the atomic number for heavy nuclei[^73,^76].

\[ L=\int_0^z\left(\frac{dx}{dz}\right)\,dz = \int_0^z \frac{\dfrac{dE}{dz}}{\dfrac{dE}{dx}}\,dz. \tag{II, 53} \]

The value of \(\dfrac{dE}{dz}\) can be obtained from the value of \(\dfrac{dE}{dx}\) and from the above dependence between \(E\) and \(z\), by multiplying the energy-loss curves for protons of the same velocity (Section II.2) by \(z'^2\). \(\dfrac{dE}{dx}\) for a proton can be found by differentiating the corresponding range–energy curve (Section II.3). Figure 23 gives the curve of the dependence between the atomic number and the length of the “sharpening,” obtained in this way[^73,^76]. In contrast to these results, which give \(L \simeq 0.5 z^2\), Hsiang and Morellet[^78] found experimentally that an equation of the form \(L=az^\alpha\) with \(a \simeq 1\) gives better agreement with their data. This question requires further investigation.

II. 7. Multiple Scattering

A charged particle moving in matter undergoes a large number of small deflections caused by elastic collisions with atomic nuclei. Bohr and Chudury\(^{79}\), studying the dependence of scattering on the mass and energy of the moving particle, proposed using scattering measurements to determine these quantities. Subsequent work by Perkins\(^{80}\), Occhialini and Powell\(^{81}\), and others showed the possibility of practical use of multiple scattering.

Theory. The initial theoretical treatment of scattering in a form suitable for comparison with experiment was carried out by Williams\(^{82,83}\). He estimated the mean angular deflection caused by the scattering of a particle with charge \(z\), momentum \(p\), and velocity \(v\), in traversing a distance \(x\) in a medium containing \(N\) atoms of atomic number \(Z\) per \(1\ \mathrm{cm}^3\), and found a Gaussian distribution about zero for this deflection. In measurements on photographic plates it is convenient to use the projections of the spatial scattering angles onto a plane. The mean value of this projection was found by Williams to be

\[ \langle \Phi \rangle = \frac{2ze^2 (Z^2Nx)^{1/2}}{pv} \left[ \ln\left( \frac{\Phi_{\max}^2}{\Phi_{\min}^2} \right) \right]^{1/2}, \tag{II, 54} \]

where \(\Phi_{\max}\) is the largest, and \(\Phi_{\min}\) the smallest, angles of deflection that can contribute to the observed scattering. In considering multiple scattering it is convenient to introduce the unit angle \(\delta\):

\[ \delta = \frac{2ze^2 (z^2Nx)^{1/2}}{pv}. \tag{II, 55} \]

An approximate value of \(\Phi_{\max}\) can be found by determining an angle \(\Phi_1\) such that, in traversing the distance \(x\), the particle undergoes on average one collision with scattering through an angle \(\Phi > \Phi_1\). This gives:

\[ \Phi_{\max} \simeq \Phi_1 = \left(\frac{\pi}{2}\right)^{1/4}\delta. \tag{II, 56} \]

Taking into account the screening caused by the electrons of the shells of the retarding atoms, Williams found that

\[ \Phi_{\min} = \frac{mcZ^{1/3}}{78.3p}, \tag{II, 57} \]

where \(m\) is the electron mass.

Equation (II, 54) can be rewritten in the form

\[ \langle \Phi \rangle = L\delta. \tag{II, 58} \]

In other calculations\(^{84-86}\), somewhat different expressions were obtained

for \(L\). However, in general \(\langle \Phi \rangle\) can be expressed in the form

\[ \langle \Phi \rangle = \frac{K z x^{1/2}}{pv}, \tag{II, 59} \]

where \(K\) is defined as the scattering constant and is equal to

\[ K = 2e^2 (Z^2 N)^{1/2} L. \tag{II, 60} \]

Although \(K\) depends mainly on the characteristics of the scattering medium and thus remains relatively constant for a given emulsion, it is of interest to estimate its variation with the particle velocity \(\beta c\) and the distance traversed \(x\). This estimate was carried out by Molière using the quantity

\[ \Omega_b = \frac{\pi \delta^2}{\Phi_{\min}^2}, \]

which is a measure of the mean number of collisions experienced by a particle over the distance \(x\). Fig. 24 shows the curve of the dependence of \(\Omega_b/x\) on \(\beta^2\) for singly charged particles (for Ilford G5 emulsions)\(^{87}\). Fig. 25 gives two curves characterizing the variation of \(K\) with \(\Omega_b\): \(a\)—theoretically calculated with allowance for all scattering angles \(\Phi\), and

Fig. 24. Value of \(\dfrac{\Omega_b}{x}\) as a function of \(\beta^2\) for singly charged particles in Ilford G5 emulsions\(^{87}\).

Fig. 25. Variation of the scattering constant \(K\) as a function of the parameter \(\Omega_b\): \(a\)—with allowance for all scattering angles \(\Phi\), and \(b\)—only with allowance for angles smaller than the fourth-power mean angle \(\langle \Phi \rangle\)\(^{87}\).

\(b\)—for the case in which all scattering angles exceeding the fourth-power mean have been excluded. The values of \(K\) given by curve \(b\),

approximately 10% smaller than the corresponding values of curve \(a\). These curves have been verified experimentally\(^{87}\).

Experiment. At present there exist several methods for measuring the multiple scattering of particles in nuclear emulsions. In one method the tracks are divided into equal parts (cells), usually \(100\,\mu\) long, and the angles \(\Phi_t\) between the tangents to the portions of the track in successive cells are determined directly\(^{88-90}\). In practice, what is measured here are the angles between tangents drawn visually to the track in each cell. A variant of this method was used by Lattimore in his work\(^{91}\). To reduce the experimental error associated with the difficulty of drawing the tangent visually, he used the angles \(\Phi_c\) between successive chords along the track. The relation between \(\Phi_c\) and \(\Phi_t\) is approximately as follows:

\[ \Phi_c = 0,816 = \Phi_t . \tag{II, 61} \]

Fowler used another method\(^{92}\). In his method, the coordinates of the track at the ends of the cells were measured, and the values obtained were used to determine the mean values of \(\Phi_t\) between successive chords. Goldschmidt and Scott\(^{93,94}\) proposed, for finding the scattering angle, using the difference between the actual length of the track in an interval and the length of the straight line joining the ends of the interval, i.e. the difference between the lengths of the track and the chord.

Theoretically, for the case in which \(\beta^2 \simeq 1\), \(K\) is found to be 24.45 if the cell size is \(100\,\mu\), \(\Phi\) is measured in degrees, and \(pv\) in \(M\mathrm{ev}\).

The experimental results of Gottstein et al.\(^{87}\) are given in Table XI. \(K_a\) is the scattering constant in whose determination no restriction was imposed that each angle considered be less than four times the mean angle, while \(K_b\) is the constant obtained with such a restriction. Apparently, the results of this study show that the use of \(K = 26.0\), as a rule, introduces no error exceeding 8%, which is less than the usual experimental uncertainty in measuring angles.

For tracks of charged particles, equation (II, 59), together with the experimental value \(\langle \Phi \rangle\), gives the particle energy \(E\). If the particle track ends in the emulsion and the residual range is \(R\), then comparison of the values \(E\) and \(R\) with the range–energy curves for different particles (Section II. 3) immediately gives the mass of the particle. For tracks not ending in the emulsion, measurement of \(\langle \Phi \rangle\) along the track can be used together with the range–energy curves to determine the particle mass by a method similar to that described in Section II. 4. Knowledge of \(\langle \Phi \rangle\) for portions of the track located at a known distance from one another.

gives the energy at these points; comparison with the range–energy curves again uniquely determines the mass corresponding to this change in energy. Relative mass values are easily obtained from the values of \(\langle \Phi \rangle\) corresponding to portions of tracks with equal grain density. Since at these points the particle velocities are equal, the ratio of the scattering angles is equal to the inverse ratio of the masses.

Table XI

Experimental values of \(K_a\) (scattering constant without restriction for \(\Phi\)) and \(K_b\) (scattering constant with the restriction \(\Phi > 4\langle\Phi\rangle\))\(^{87}\)

Type of particle	Energy in MeV	\(K_a\)	\(K_b\)
Positrons	105	\(26.7 \pm 0.6\)	\(26.2 \pm 0.6\)
Positrons	185	\(24.9 \pm 0.8\)	\(24.0 \pm 0.8\)
Protons	336	\(30.7 \pm 1.0\)	\(29.2 \pm 1.0\)
Protons and mesons	5—50	\(\ldots\)	\(26.1 \pm 0.7\)
Protons	9—35	\(\ldots\)	\(27.5 \pm 0.5\)

In measuring scattering, both subjective and instrumental errors may arise. It may be expected that false scattering (noise) will increase as the cell length decreases. Gottstein et al., using several microscopes, investigated “noise” for various cell lengths. For \(x = 50\,\mu\) the mean scattering angle caused by “noise” was about \(0.13^\circ\), for \(x = 100\,\mu\), \(\sim 0.09^\circ\), and for \(x = 200\,\mu\), \(\sim 0.055^\circ\), when measured on two different microscopes. A third microscope gave a smaller “noise,” equal to \(0.035^\circ\) for \(x = 200\,\mu\); however, the same rate of its increase with decreasing \(x\) was noted. In precise work, the angles of false scattering must be determined for the selected cell sizes and for each instrument separately. For this purpose one may use straight tracks of high-energy particles.

The obtained values of the “noise” are subtracted from the angles found in the investigation of the track.

III. PROCESSING OF EMULSIONS

III. 1. General considerations

When considering methods for processing emulsions, it is convenient to divide the latter into two categories: emulsions with a layer thickness less than \(100\,\mu\) and those with a thickness of \(100\,\mu\) and more. The former are usually called “thin” emulsions, and the latter “thick” ones. Processing

processing thin emulsions presents no particular difficulties compared with the processing of ordinary photographic films and plates, whereas in the case of thick emulsions the considerable time required for complete impregnation of the emulsion with solutions necessitates the use of a more complicated technique.

Development of thin emulsions. For thin emulsions two degrees of development are possible—“moderate” and “strong.” Moderate development is more suitable when it is intended to measure the grain density of comparatively dense tracks (for example, protons or $\alpha$-particles with energies of several $Mev$). For such measurements it is essential that the grains be discrete, and moderate

Table XII

Development processes for emulsions of thickness 10, 25, and 50 $\mu$, recommended by Eastman Kodak. Developer D19 of normal concentration; the entire process is carried out at $20^\circ C$

Process	Time
10 $\mu$, moderate development	2–4 min.
25–50 $\mu$, same	4 min.
25–50 $\mu$, strong development	20 min. without agitation, 10 min. vigorous agitation
Washing (running water)	10 min.
Fixing (fixing bath Kodak F5)	Twice the development time
Washing (running water)	1 hr.

development contributes to this. Another advantage of this method is a considerable reduction in fog density. However, in this case the sensitivity of the emulsion is not fully utilized, and therefore the tracks of more weakly ionizing particles may not be recorded. Strong development, on the other hand, gives full use of the sensitivity of the emulsion, although at the same time it leads to an increase in fog. In this case the tracks of strongly ionizing particles develop in the form of solid columns of silver grains, which makes grain counting impossible. In practice, the best method of development consists in carrying out a series of experiments in order to determine the development time that provides the best combination of track density and background.

Table XII gives the development instructions recommended by Eastman Kodak for thin emulsions of this firm, which can also be used for other photoemulsions of analogous composition. The formula of developer D19 is given in Table XIII.

Table XIII

Developer compositions

Amidol ($pH$ 7.2)	D19 b ($pH$ 10.0)	Azol ($pH$ 11.5)	ID19	Amidol-bisulfite ($pH$ 6.7)
(a) Amidol 3 g Sulfite 12 g (anhydrous) Distilled water 1 l (b) Amidol 4.5 g Sulfite 18 g (anhydrous) Potassium bromide (10% solution) 8 cm³ Distilled water 1 l	Metol 2.2 g Sulfite 72 g (anhydrous) Potassium bromide 8.8 g Distilled water to 2 l	Johnson’s “Azol” solution 16 cm³ Potassium bromide (1%) 88 cm³ Distilled water 384 cm³	Metol 4.5 g Sulfite 288 g (cryst.) Hydroquinone 17.5 g Soda 260 g (cryst.) Potassium bromide 8 g Distilled water to 2 l	Amidol 3.0 g Sulfite 6.7 g (anhydrous) Bisulfite 1.4 cm³ Distilled water 930 cm³

In this formula, metol and hydroquinone are in fact the active agents, while the other constituents are needed for various other purposes. Soda is used to regulate the $pH$ of the solution, and also to increase the rate of development and therefore is often called an accelerator. Sometimes the soda is replaced by other alkalis, such as caustic soda, borax, or sodium metaborate. In addition, the accelerator serves to soften the gelatin, which facilitates easier penetration of the solutions into the emulsion.

Since the presence of oxidation products of the developing agents has a harmful effect on the development process, a substance should be introduced that prevents the appearance of oxides. This function is performed by sulfite.

The fourth component, which is essentially included in all developers, potassium bromide, serves to weaken the action of the developing agent on halide grains not affected by irradiation.

At the same time, the effect of bromide on the development of the “exposed” grains is much smaller, which leads to preferential development of the latter. The introduction of potassium bromide thus considerably decreases the rate of fog formation, although it somewhat increases the development time. It may be expected that, with increasing development time, the fog will also increase. This is confirmed by the results of the work of Kotesa[^95]. Figure 26 gives the curve obtained in this work, showing the increase in the density of background grains with development time. These results were obtained under special conditions and, while illustrating the expected rate of increase of fog, are not, however, applicable as a quantitative characterization of the development process.

Fig. 26. Change in the mean density of fog grains with development time[^95].

Permissible illumination. Nuclear emulsions are not very sensitive to light and may be handled under a certain permissible illumination. Orange-red filters may be used for all emulsions except type NTA, for which a yellow filter is recommended. Plates should remain under such illumination until complete fixing, after which normal illumination may be used.

Stopping. After completion of the development stage, the action of the developer must be stopped immediately. In the case of very thin emulsions, immersion in running water is sufficient for this purpose. However, for emulsions with a thickness of 25 μ, 50 μ, and more, the usual method is to lower the pH below the value necessary for development by means of an acetic acid solution (0.5–1%). Another method, often used together with the one indicated (in temperature development), is rapid cooling. Since development is a chemical process and, consequently, its rate depends on temperature, such cooling leads to a rapid stopping of the action of the developer. When this method is used, cooling to a temperature of about 5° C is usually sufficient.

Surface precipitate. Owing to the high concentration of silver bromide in nuclear emulsions and owing to its partial solubility in the developer, during development of the emulsion a thin film of silver forms on its surface. This film may be so dense that it actually makes observation impossible and in any case interferes with accurate investigation

plates. This deposit can be removed in the stop bath with the aid of a wet chamois or simply with a finger. The swollen emulsion is easily deformed, and therefore removal of the film should be carried out carefully in order to avoid the appearance of distortions in the emulsion. The silver film may also be removed after development and complete drying of the emulsion with the aid of cotton fabric or a chamois moistened with alcohol. However, after the processing process is completed, the film becomes very strong and it is often difficult to remove it completely. Stiller^96 succeeded in wiping the plates before development by first soaking them in distilled water. A reduction in the surface deposit may be achieved^97 by development in an inert atmosphere or by using such a developer as amidol (see Table XIII), which usually gives almost no film.

Fixing. Fixing of the developed emulsion is carried out in a 30–40% solution (by weight) of sodium thiosulfate (hyposulfite) in distilled water. Ammonium thiosulfate also acts as a fixing agent, but its use leads to removal of part of the developed grains near the surface of the emulsion. To some extent this also applies to ammonium chloride (sal ammoniac), which can be used to reduce the fixing time. Nevertheless, Stiller et al.^98 use ammonium chloride at a concentration of 0.7% in their formula. In order to reduce staining, sodium bisulfite is introduced into the fixing bath at a concentration from 0.75%^98 to 3%.^99

During fixing, owing to the passage into solution of considerable quantities of soluble silver salts, it is necessary to use large volumes of fixer or to change the bath from time to time. The fixing time of the emulsion varies substantially depending on the thickness, temperature, and degree of agitation. For example, it is roughly proportional to the square of the thickness. An emulsion 400 μ thick becomes transparent after 18 hours, whereas transparency of an emulsion 1000 μ thick is attained only after ∼100 hours. The plate should be kept in the fixing solution for approximately 50% longer than the time required for complete removal of the residues of silver bromide from the emulsion. At higher temperatures (approximately up to 25° C) the fixing process is accelerated, but at the same time the danger of the appearance of a reticulation pattern increases (see below). When fixing thick emulsions it is recommended, before washing, gradually to reduce the concentration of hyposulfite by successively diluting the solution, in order to minimize distortions.

Agitation of solutions. Agitation of solutions during the processing of emulsions is practiced in photographic technique in order to reduce the processing time. In the stop bath and in the fixing solution agitation does no harm.^81 However, at the development stage agitation is undesirable, especially with

processing of thick emulsions, since it leads to a difference in the rate of development inside the emulsion and on its surface. In addition, vigorous stirring increases oxidation of the developer and, possibly, also surface deposition of silver and deformation of the emulsion.

There are two types of stirring—mechanical and gas. In mechanical stirring, the solution is set in motion either by means of a propeller and a small motor, or by rocking the cuvette with the solution and plates. When the plates are in a horizontal position, recommended for emulsions thicker than 100 μ, rocking produces a laminar flow that is very effective^81. Mechanical stirring is usually easy to carry out, but care must be taken that the motion is smooth and is not accompanied by turbulence. Another method, in which bubbles of an inert gas, usually nitrogen, are passed through the bath, was studied by Wilson and Vanzelos^100. The use of ordinary air is undesirable, since the grains of developed silver on the surface of the emulsion will then be oxidized. It goes without saying that the gas supplied must have the same temperature as the solution being stirred. Gas stirring gives a 50% reduction in fixing time. Of course, it is not necessary to know exactly the reduction in fixing time for any particular setup; the fixing time should be established from the clearing time, to which another \(1/2\) of this time is added.

Washing. After fixing, washing is required, usually for the same length of time as the fixing itself. As a rule, cold tap water is used, which is carefully passed through the vessel containing the fixed plates. The thiosulfate remaining in the emulsion must be

Table XIV

Composition of the hypo-indicator solution

Distilled water . .	180 cm³
Potassium permanganate . . . .	0.3 g
Caustic soda . . . . . . .	0.6 g
Distilled water . .	to 250 cm³

completely removed in order to avoid weakening of the developed image, since the sulfur of the thiosulfate actively combines with the silver of the image and forms silver sulfide. A simple indicator solution for checking the presence of thiosulfate in the wash water is given in Table XIV. Several drops of this solution, usually violet in color, added to water containing thiosulf-

phyt, produce, in less than a minute, an orange coloration, and at higher concentrations of hyposulfite—a yellow one.

Drying. Drying the emulsion after washing requires special care if deformation of the gelatin is undesirable. Since evaporation of water from the surface of the emulsion proceeds much faster than diffusion of water from within it, during drying, in order to eliminate stresses, it is necessary to maintain a humid atmosphere. The process of drying the emulsion depends on the laboratory conditions. Usually several days are required for drying at a relative humidity of 90%, and then several more days at lower humidities. Air movement over the surface of the emulsion should be avoided, since this increases distortions^99; to accelerate the process the temperature may be raised somewhat. Dilworth^101 noted that more rapid drying of the edges of the plate causes deformation of the emulsion. This can be avoided almost completely by surrounding the plate with a “guard ring” of other plates of the same kind; in this case much more uniform drying is obtained.

Storage of plates. When drying is completed, it is advisable to subject the plates to further treatment that prevents the emulsion from separating from the glass. This is especially necessary when working with thick emulsions, although it is not obligatory for thin ones \((< 200\,\mu)\), if before drying they were immersed in a bath that increases the plasticity of the emulsion. Coating the edges of the plates with shellac or lacquer is in most cases sufficient for Ilford emulsions, which adhere to glass better than Eastman emulsions. For the latter, and for plates of thickness \(600\,\mu\) and more, coating is required not only at the edges but over the entire surface of the emulsion. In the event that, despite all precautions, separation and cracking of the emulsion occur—as, for example, in the case of frequent or abrupt changes of temperature and humidity—a thin cover glass can be glued to the surface of the emulsion with glue dissolved in acid^102.

III. 2. Temperature development

Because some time is required for the developer solution to penetrate into the depth of thick emulsions, the halide grains located near the surface of the emulsion will be developed to a greater extent than those at the glass surface. In this case, different portions of tracks running at an angle to the surface of the emulsion would be developed unequally, which would make it impossible to measure the grain density and compare cases recorded at different depths of the emulsion. To overcome this difficulty, Dilworth, Occhialini, and Payne^103 proposed a method of “temperature development,” in which the emulsion is kept in the developer at a temperature significantly

lower than that required for its action, until the developer has completely permeated the emulsion. The developer then warms up, and development takes place. Development is stopped by rapidly lowering the temperature by immersing the plate in a cold stop bath.

Preliminary saturation of the emulsion. To accelerate the penetration of the developer into the emulsion, preliminary saturation of it with distilled water, with or without the addition of a wetting agent, is often used. In this case the gelatin swells and permits more rapid diffusion of the developer. The emulsion is then immersed in the developer, the temperature of which is usually maintained near \(5^\circ\text{C}\). At a higher temperature the rate of penetration of the developer into the emulsion is less than the rate of its action, while at a lower temperature the penetration time is excessively prolonged \(^{103–105}\).

Development. After the emulsion has been impregnated with cold developer, the temperature is raised so that the development process may begin. It is necessary here that fresh developer not enter the emulsion, since this would lead to uneven development. At present three methods are used: dilution of the developer, dry development, and mechanical protection of the surface of the emulsion. The first of these requires extremely rapid adjustment of the developer concentration in order to prevent diffusion of fresh developer into the emulsion and back again. In practice, a mixture is usually used consisting of one part of developer at the concentration at which the plates were saturated with cold developer and two parts of distilled water. This method is simple to carry out, but is unsuitable for precise work. In dry development \(^{99,100}\) the plates are removed from the bath with developer, the excess solution is removed from their surface with filter paper or another absorbent material, and they are then kept for the required time at the temperature chosen for development. The plates are placed glass side down on a heated surface, with good thermal contact. Dry development can be carried out at a temperature from 25 to \(30^\circ\text{C}\), which considerably reduces the time usually required at this stage (when developing in the solution itself, a temperature of \(\sim 20^\circ\text{C}\) is maintained). This method eliminates overdevelopment of the upper layers of the emulsion, but oxidation of the developer may occur on the surface of the emulsion, leading to underdevelopment. Such oxidation can be avoided if the heating of the plates is carried out in an atmosphere of inert gas. In the third method the surface of the emulsion is protected by glass plates coated with wax to prevent their sticking to the emulsion, or is covered with oil. In the latter case, before immersing the plate in the stop bath, the oil must be carefully removed.

In processing emulsions of great thickness (more than 1 mm), it is necessary to use additional methods of restraining the action of the developer until it has completely penetrated into the emulsion. Besides further lowering the temperature, which is permissible when nonfreezing solutions are used, one may also introduce more bromide into the developer; this slows the action of the developer, correspondingly increasing the development time. The same effect is produced by changing the pH: the development time increases as the acidity of the solution increases. Amidol developers are especially suitable for this, since they work in an acid medium¹⁰¹.

Some instructions for development in solution and by the dry method, with the corresponding formulas, are given in Tables XIII, XV, and XVI.

Table XV

Method of developing emulsions 100 and 200 μ thick, recommended by Eastman Kodak

Procedure	Temperature, °C	Time
Soaking with developer (D19, diluted 1:1)	5	30 min.
Development (with the addition of two parts water at 20° C)	20	30 min.
Acid stop bath (2%, mixing by bubbling nitrogen)	5	30 min.
Soaking with fixer solution (30% hyposulfite)	5	15 min.
Fixing (mixing by bubbling nitrogen)	20	5 min. longer than the clearing time
Washing (running water)	10	Minimum 1 hour

Reticulation. A reticular pattern is a consequence of distortion of the emulsion due to nonuniform swelling or shrinkage of the gelatin. In the development of nuclear emulsions, reticulation usually appears when plates are transferred from cold solutions into solutions of higher temperature, without holding them at an intermediate temperature. Reticulation is a visible image that arises as a result of the formation of folds on the surface of the emulsion²⁷ and the displacement of silver particles, which tend to concentrate on the crests of the gelatin folds. Being sharply pronounced, this phenomenon renders the plate unusable, but even

Table XVI

Method of developing emulsions of thickness 400 and 600 μ ^98

Procedure	Temperature, °C	Time
Preliminary saturation with water (distilled water)	Gradual decrease from room temperature to 5	100 min.
Saturation with developer (amidol (b))	5	100 min.
Dry development	23	20 min.
Dry cooling	23 → 8	5 min.
Acid stop-bath (1%)	5	100 min.
Removal of surface precipitate	—	—
Fixing: clearing	5	18 hours
Fixing: dilution of the solution	5	24 hours
Washing	5	24 hours
Plasticizing solution (10% glycerin solution)	5	30 min.
Drying (relative humidity 100% → 50%)	21	7 days

A moderate mesh worsens the accuracy of measuring tracks. A gradual increase in the temperature of the solution or the use of a sufficient number of intermediate baths makes it possible to avoid such distortion.

III. 3. Diffusion of the developer

The diffusion of developers into nuclear emulsions was studied in great detail by the Bristol group.^99 Experiments carried out with Ilford G5 emulsion of various thicknesses and with four different developers, with preliminary saturation with distilled water and without saturation, established a quantitative basis for the development process by the temperature method. To determine the diffusion time, the emulsions were exposed from the side of the glass support in such a way as to obtain an image only near the glass, in a layer of ∼3 μ. The emulsions were then developed, and the diffusion time was taken to be the time required to obtain an image.

The nominal and actual thicknesses of the emulsion before development are given in Table XVII. Preliminary saturation with distilled water in all cases was carried out for the course of

three hours, and in all cases the diffusion time of the developer decreased appreciably. The results obtained for the diffusion time of a number of developers (Azol, D19b, amidol (a), and amidol with bisulfite) at \(20^\circ\)C are given in Table XVIII for two experiments: with preliminary saturation with water and without such saturation.

Table XVII

Thickness of emulsions
(in microns)

Nominal	Actual
100	105
200	230
300	260
400	400
600	675
800	720
1000	1050

The ratio of the mean diffusion times of the developer in these two experiments proved to be equal to: for Azol—1.34, for D19b—1.70, for amidol—1.86, and for amidol with bisulfite—1.65. Measurements of the diffusion time for emulsions subjected to preliminary saturation with water at temperatures of 5, 10, and \(15^\circ\)C were carried out only with Azol, D19b, and amidol, since amidol and amidol with bisulfite at \(20^\circ\)C have practically the same diffusion time. The results are given in Table XIX, into which, for comparison, the results of measurements at \(20^\circ\)C are also included. On the basis of these experiments the following formula was obtained for the diffusion time

\[ T = kt^x, \tag{III, 1} \]

where \(T\) is the diffusion time, \(t\) is the thickness of the emulsion, \(x\) is a number \(\sim 1.4\), and \(k\) is a constant depending on the developer and the temperature.

Table XVIII

Diffusion time of various developers at \(20^\circ\)C with preliminary saturation of the emulsion with water and without it

Developer	105	230	260	400	675	720	1050
Azol, with preliminary saturation	4.5	17	22	38	96	96	210
Azol, without it	6	21	27	50	138	123	330
D19b, with preliminary saturation	3.5	9.5	14.5	21	50	64	140
D19b, without it	5	12	20	37	110	125	270
Amidol, with preliminary saturation	1.5	6	8.5	12	26	26	58
Amidol, without it	3.5	9.5	11.5	22	49	52	120
Amidol with bisulfite, with preliminary saturation	2	6.5	8	12	25	25	55
Amidol with bisulfite, without it	2	8.5	10	20.5	53	46	110

III. 4. Development by the two-bath method

In another method of development, which is also suitable for thick emulsions, two developer solutions are used.^106 The first solution consists of developing agents without alkali, as a result of which the developer, while diffusing into the emulsion, produces no action. The second bath, containing an excess of alkali, causes development. This method requires that the rate of penetration of the change in pH into the depth of the emulsion exceed the rate of diffusion of the developer itself—a condition which in fact is not fulfilled.^99 However, for emulsions of approximately \(400 \mu\), the two-bath method eliminates the possibility of the appearance of a network, which often forms during temperature development if appropriate precautions are not taken. Details of the two-bath method are given in Table XX.

Table XIX

Diffusion times of various developers in emulsions previously impregnated with distilled water at temperatures of 5, 10, 15, and 20 C

Developer	Temperature, °C	105	230	260	400	675	720	1050
Azol	5	11	45	50	100	270	300	300
Azol	10	8,5	27	34	70	185	208	210
Azol	15	8,5	25	30	63	145	150	210
Azol	20	4,5	17	22	38	96	95	210
D19b	5	6	21	27	58	165	172	270
D19b	10	5	16,5	19	37	108	115	210
D19b	15	4	12	17	27	72	68	220
D19b	20	3,5	9,5	14,5	21	50	64	140
Amidol	5	5	16	20	37	80	84	190
Amidol	10	3,5	11	13,5	22	51	53,5	110
Amidol	15	3	8	11,5	18	37,5	45	95
Amidol	20	1,5	6	8,5	12	26	26	58

Distortions arising during processing. In a number of cases in the use of nuclear emulsions it is important that there be no distortions of any kind; sometimes this is even more important than uniformity of development with depth. This is the case, for example, with magnetic deflection of particles in the air gap between two plates (Section IV. 6). The temperature method of developing thick emulsions, while giving comparatively uniform development with depth, nevertheless leads to large distortions.^107 This may be attri-

Table XX

Development by the two-bath method

Component / operation	Amount / conditions
Solution A
Metol	1.1 g
Sodium sulfite	24.0 g
Hydroquinone	4.4 g
Potassium bromide	2.0 g
Distilled water	to 2 l
Solution B
D19 developer	400 cm³
Distilled water	1600 cm³
Soda	16 g
Development procedure:
1. Preliminary saturation with distilled water	10 min
2. Solution A	30 min, with gentle agitation
3. Solution B	30 min, without agitation
4. 2% solution of acetic acid	15 min, with agitation
5. Fixing	Eastman F5 fixer, 6–8 hours at 74° F, with agitation
6. Washing in running water	2 hours

distorted by shocks experienced by the emulsions being tested as a result of temperature changes at different stages of processing. Although these shocks can be reduced to a minimum by careful heating, they cannot be eliminated completely. The two-bath method, because of the constancy of the temperature, gives much smaller distortions; however, it produces an increased density of background and surface grains. This makes it very difficult to determine accurately the points at which particle tracks enter the emulsion and, of course, in order to avoid distortions the emulsion must not be wiped to remove the surface deposit. Barbour \(^{107}\) found that the use of a more dilute developer at a lower-than-usual temperature reduces distortions to a minimum and at the same time does not lead to the appearance of too large a development gradient. For developing emulsions \(200\,\mu\) thick with D19 developer diluted in the ratio \(4:1\), at \(18^\circ\text{C}\) about 55 minutes were required; moreover, the diffusion time, it appears, occupies a smaller part of the total time for which the plate remains in the developer than in the case of a more concentrated solution at \(20^\circ\text{C}\).

III.5. Films

The processing of films, i.e. nuclear emulsion without a glass backing, has certain special features owing to the appreciable \((\sim 25\%)\) lateral swelling of the films when immersed in solution.

which may subsequently lead to severe distortions. Moreover, the emulsion, once swelled, is extremely weak, and it must be handled carefully, avoiding its adhering to the walls of the vessel. Usually, during development, one end of the film is held with a stainless-steel clamp.

The method of processing 250 μ Eastman Kodak films is given in Table XXI. After the washing has been completely finished, the films are placed on glass plates somewhat larger in size than the films themselves. These plates should be coated with gelatin; either special plates supplied by the manufacturer may be used, or undeveloped but fixed and washed nuclear photographic plates. The films on such a glass backing are placed in a refrigerator

Table XXI

Development of 250 μ Eastman Kodak films

Procedure	Temperature, °C	Time
Saturation with developer (ID19b)	5	10 min.
Development (with addition of two parts of water at 20° C)	20	10 min.
Acid stop bath (2%, agitation)	5	10 min.
Fixing (30% hyposulfite solution)
saturation with solution	5	10 min.
fixing	20	5 min. longer than the development time
washing	15	Fixing time
Another method
Development (D19b) (agitation)	20	8 min.
Acid stop bath (2%, agitation)	20	5 min.
Fixing (30% hyposulfite solution, agitation)	20	Twice the development time
Washing	15	Fixing time

and are kept there until they harden. Drying is then carried out at 20° C.

With another method of processing^98 the films are mounted on the glass backing before development. The mounted films are then processed in exactly the same way as ordinary plates with the corresponding emulsion thickness. Of course, with this method the advantage of rapid development and fixing possible with films is lost, owing to penetration of the solutions from both sides, but the distortions are thereby considerably reduced.

III. 6. Shrinkage

The high concentration of silver bromide in nuclear emulsions leads to a considerable decrease in the thickness of the emulsion after fixing, during which the undeveloped silver is removed. If the ratio between the emulsion thicknesses before and after processing—the “shrinkage coefficient”—is known, then, in order to correct for this effect when determining the track length, one may use the formula

\[ R=\left[Y^{2}+(SZ)^{2}\right]^{1/2}, \tag{III, 2} \]

which expresses the initial (i.e., before processing) track length \(R\) in terms of \(S\), the shrinkage coefficient, and \(Y\) and \(Z\), the horizontal and vertical components of the measured track length after processing the emulsion. Deviations from this relation are observed for tracks of heavy particles going at large angles (\(>25^\circ\)) to the plane of the emulsion \(^{103,109}\). Such tracks exhibit a considerably smaller relative decrease in angle than do tracks going at smaller angles. This is illustrated in Fig. 27, which gives the dependence of the calculated track length on the angle for triton \(+\alpha\)-particle tracks from the disintegration of Li in C2 emulsion (Sec. IV.3). The difficulty of displacement of grains which, in the course of shrinkage, come into contact with one another and subsequently hinder the displacement of the surrounding gelatin is the most probable explanation of this effect. For the critical angle \(\theta_0\), after which deviations from equation (III, 2) appear, Rotblat and Tej \(^{109}\) give the equation

Fig. 27. Dependence of the calculated track length on its angle of inclination in the emulsion for triton \(+\alpha\)-particle tracks from lithium disintegration by slow neutrons \(^{109}\).

\[ \cos\theta_{0}= \left[ \frac{ S^{2}-\left\{\dfrac{R}{R-l}\right\}^{2} }{ S^{2}-1 } \right]^{1/2}, \tag{III, 3} \]

where \(R\) is the range, and \(l\) is the total length of the gaps between grains for tracks going parallel to the surface of the emulsion.

Shrinkage coefficient. In the ideal case the shrinkage coefficient is determined by the expression

\[ S=1+\frac{V_s}{V_g}, \tag{III, 4} \]

where \(V_s\) is the volume of the soluble substance, and \(V_g\) is the volume of the remaining gelatin. However, it is necessary to take into account the change in \(S\) as a function of humidity during irradiation of the emulsion and during its examination after processing, since the relative amount of moisture is different for the processed and unprocessed emulsion. Figure 28 gives the dependence of the moisture content on the relative humidity in processed \((a_0)\) and unprocessed \((a_e)\) emulsions in the form of the ratio between the volumes of water and gelatin. The empirical formula

\[ S=\frac{S_0+1.27a_e}{1+a_0} \tag{III, 5} \]

gives the shrinkage coefficient as a function of the water content in the emulsion, which can be determined with the aid of Fig. 28, knowing the relative humidity. The quantity \(S_0\) is the shrinkage coefficient for an absolutely dry emulsion and for Ilford emulsions; it is equal to \(\sim 2.22\).

Experimentally, \(S\) is determined either by direct measurement of the difference in depths between the upper and lower grains of the background by means of a fine calibrated microscope setting, or by Wineberger’s method[^110]. The latter, using equation (III, 2), measured the tracks of \(\alpha\)-particles from ThC′. Since \(R\) is constant for these tracks, the graph of the dependence of \(Y^2\) on \(Z^2\) for tracks going at different angles must give a straight line with slope equal to \(S^2\).

Fig. 28. Dependence of the moisture content in unprocessed and processed Ilford emulsions on the relative humidity. The first was calculated from the composition of the emulsion, the second determined experimentally[^109].

A very elegant and accurate method for determining \(S\) with the use of optical interference phenomena was proposed by Roudes[^111]. If an emulsion is applied to a glass substrate in the form of a wedge, then the angle

Fig. 29. Inclination of the emulsion wedge before and after development[^111].

Labels in Fig. 29: surface of the unprocessed emulsion; developed; glass substrate; \(A\), \(B\), \(C\), \(D\); \(\alpha\), \(\beta\).

\(\alpha\) between the surface of the emulsion and the surface of the glass can be determined very accurately by measuring the distance \(x\) between the interference fringes obtained with monochromatic light reflected from the photographic plate and from the plane-parallel plate of optical glass. After development, the angle of inclination and the distance between the fringes will change and will be equal, respectively, to \(\beta\) and \(x'\). From Fig. 29 it is clear that

\[ S=\frac{AC}{DC}=\frac{\operatorname{tg}\alpha}{\operatorname{tg}\beta}, \tag{III, 6} \]

and since

\[ \operatorname{tg}\alpha=\frac{\lambda}{2x} \tag{III, 7} \]

and

\[ \operatorname{tg}\beta=\frac{\lambda}{2x'}, \tag{III, 8} \]

where \(\lambda\) is the wavelength of light, \(S\) is determined directly from the equality

\[ S=\frac{x'}{x}. \tag{III, 9} \]

In practice, the determination of the distances \(x\) and \(x'\) was carried out simultaneously on a plate, one part of which was subjected to processing, while the other remained undeveloped; moreover, the line separating these two regions was perpendicular to the direction of inclination of the wedge. As a result of this investigation, for emulsions of types C2 and G5 at normal room humidity it was found that \(S=2.38\pm0.04\) and \(2.65\pm0.07\). These figures agree with the results of Reblat. Roulz also estimated the change of \(S\) with storage time in a saturated atmosphere; the results are given in Fig. 30. When working with cosmic radiation, measurement of shrinkage is possible with the aid of a well-aligned stack of plates irradiated in the upper layers of the atmosphere. Energetic heavy particles will penetrate through the entire stack, and their angle of incidence can be determined from their positions in successive plates. This angle and the one measured in the emulsion make it possible to determine \(S\)

Fig. 30. Increase in the shrinkage factor of Ilford G5 emulsion as a function of storage time at 100% relative humidity.

IV. AUXILIARY PROCEDURE

IV. 1. Removal of Background

In most cases it is necessary to remove background tracks, which are usually present in nuclear emulsions before their irradiation. Such tracks are most often caused by the decay of radioactive contamination, for example thorium, present in the emulsion and in the glass of the plate, although some background may be caused by cosmic radiation if the plates were transported by air.

Even brief storage near accelerators can lead to the formation of a considerable background due to recoil particles produced by fast neutrons, and, in the case of electron-sensitive emulsions, also due to γ-radiation.

The latent image is oxidized rather easily, and this property is used in various methods of background removal.

Perfilov \(^{112,113}\) and Powell \(^{63}\) proposed, for removal of background, a method of directly immersing plates in an oxidizing solution—chromic acid at concentrations up to 2%. This method is effective in removing background tracks, but at the same time the emulsion becomes insensitive to protons and lighter particles. However, the sensitivity of the emulsion after treatment with chromic acid nevertheless remains sufficient for recording \(\alpha\)-particles and fission fragments.

A more satisfactory method was proposed by Yagoda and Kaplan \(^{29}\). In this method hydrogen peroxide is used as the oxidizing agent, the emulsion being kept over a 3% solution at \(25^\circ\mathrm{C}\). Treatment for 3–4 hours is sufficient for 25–50-micron emulsions, whereas thicker emulsions require a considerably longer time. After removal of the background the plates must be dried carefully, since with very rapid drying the emulsion peels off. A disadvantage of this method is that prolonged action (\(>15\) hours) of hydrogen peroxide on the emulsion, necessary when treating thick layers, reduces its sensitivity. Moreover, this loss of sensitivity is nonuniform throughout the volume of the emulsion.

Later, Wiener and Yagoda \(^{114}\) proposed using water vapor as the oxidizing agent, as a means effective in removing background and at the same time giving a minimal loss of sensitivity. C2 emulsions 200 \(\mu\) thick were kept in an atmosphere of saturated water vapor at a temperature of \(35^\circ\mathrm{C}\) for 16 hours and then dried for one hour over anhydrous calcium chloride. Treatment in this manner does not affect the recording of low-energy protons and \(\alpha\)-particles,

however, one may expect some decrease in sensitivity for more weakly ionizing particles, such as fast protons and mesons. At present this method, apparently, is the best.

IV. 2. Limitation of the Time of Sensitivity

One of the main limitations on the use of nuclear emulsions is connected with their continuous sensitivity. This leads to ignorance of the circumstances (i.e., the time and place) accompanying the registration of an individual event. For example, in the study of cosmic radiation in the upper layers of the atmosphere with the aid of plates, the exact time and altitude at which a given event occurred cannot be determined, although, of course, differences in the frequency of occurrence of individual phenomena under different circumstances can be obtained on a statistical basis. Indeed, since an emulsion preserves all cases registered in it from the moment of manufacture until development, it is usually impossible by other, non-probabilistic methods to determine whether the events of interest to us occurred during the period of investigation. In view of these circumstances, the development of methods that make it possible to limit the time of sensitivity of nuclear emulsions is of great importance.

Thermal methods. Thermal methods for controlling the time of sensitivity, based on the change of emulsion sensitivity with temperature (Section I.4), have long been known\(^{18}\), but they have limited applicability. This is mainly due to the fact that the minimum sensitivity of an emulsion attainable under laboratory conditions with the use of liquid nitrogen is only \(1/3\) of the sensitivity at the optimum exposure temperature of \(20^\circ\text{C}\); more sensitive emulsions (Ilford G5, Kodak NT4, Eastman NTB3) at low temperatures reduce their sensitivity still less. It is clear from this that thermal methods cannot serve as an effective means of controlling the time of irradiation.

Method of reducing sensitivity. Another method\(^{115}\) is based on the dependence of the action of certain substances that reduce the sensitivity of an emulsion on the concentration of oxygen\(^{116}\). The dyes used for this purpose apparently act as catalysts for oxygen in its reaction with the silver bromide grains of the emulsion and are capable of preventing the formation of a latent image, but are insufficient for destroying an already existing latent image. These desensitizers (phenosafranine, yellow pinakryptol and green pinakryptol) probably compete with the sensitivity centers

grains in the capture of electrons emitted during irradiation. A preliminary investigation shows that this method can be used successfully to limit sensitivity; moreover, an emulsion desensitized in this way is exposed either at reduced air pressure or in an atmosphere of inert gas. However, for broad practical use of this method, further quantitative investigations are necessary.

IV. 3. Detection of Neutrons

Neutrons, being uncharged particles, are not registered directly by nuclear emulsions. However, they can be registered indirectly from recoil protons or from characteristic reactions occurring in the interaction of neutrons with elements specially introduced into the emulsion.

Another method using ordinary photographic emulsions was developed by Kalman^117. This method is based on the formation of strongly ionizing particles, such as tritons and $\alpha$-particles, in the interaction of slow neutrons with nuclei of $\mathrm{Li}^6$ or $\mathrm{B}^{10}$ (see below); these particles then activate suitable phosphors, the light from which is recorded by a photographic plate. The best arrangement for this purpose is an emulsion coated with a layer of phosphor, a thin film of aluminum foil to reflect most of the fluorescence light into the emulsion, and a layer of Li or B. In the case of fast neutrons the latter layer may be replaced by a layer of paraffin, in which recoil protons are formed that activate the phosphor.

Slow neutrons. For the detection of slow neutrons several reactions may be used, by introducing compounds of suitable elements into the emulsion (see Section IV. 5). The two reactions most often used are:

$$ \mathrm{Li}^6 + \mathrm{n}^1 \to \mathrm{He}^4 + \mathrm{H}^3 \tag{IV, 1} $$

and

$$ \mathrm{B}^{10} + \mathrm{n}^1 \to \mathrm{Li}^7 + \mathrm{He}^4 . \tag{IV, 2} $$

The disintegration of $\mathrm{Li}^6$ gives characteristic tracks about $\sim 40\,\mu$ long, produced by an $\alpha$-particle and a triton flying apart in opposite directions^15,118,119. These tracks are unusual in that the grain density in them increases in both directions from the point ($\sim 6\,\mu$ from one end) corresponding to the position of the intermediate nucleus $\mathrm{B}^{11*}$. The distribution of the ranges of particles from the decay in this reaction for Ilford C2 emulsion is shown in Fig. 31^119.

The tracks of $\alpha$-particles from the $\mathrm{B}^{10}$ reaction are less noticeable than those from the disintegration of $\mathrm{Li}^6$, and most have a length of about $4\,\mu$ in

corresponding to an energy of \(1.6\) MeV; the lithium nucleus, with an energy of \(0.9\) MeV, has a very short range owing to its comparatively large mass. However, the cross section of the reaction with boron is several times larger than the cross section of the reaction with lithium, and therefore it is more suitable for studying the intensity of slow neutrons. Fig. 32 gives the range distribution in C2 emulsion for

Fig. 31. Range distribution of \(\alpha\)-particles (the shorter tracks) and tritons from the \(Li^6\) fission reaction by slow neutrons in Ilford C2 emulsion[^119].

Fig. 32. Range distribution of \(\alpha\)-particles from \(B^{10}\) fission by slow neutrons in Ilford C2 emulsion[^119].

\(\alpha\)-particles from \(B^{10}\), the maximum corresponding to a range slightly less than \(4\,\mu\) being associated with the formation of a \(Li^7\) nucleus in an excited state.

It should be remembered that ordinary boron contains only about \(20\%\) of the isotope \(B^{10}\), the rest being \(B^{11}\), which does not react with slow neutrons. However, “enriched” boron can be obtained, containing up to \(\sim 96\%\) \(B^{10}\). Similarly, ordinary lithium consists of \(7.5\%\) \(Li^6\) and \(92.5\%\) \(Li^7\); higher concentrations of \(Li^6\) can also be obtained by enrichment. Filling with lithium borate[^34] can be used to combine the effects of both reactions. This will be discussed in greater detail below.

Sometimes the reaction

\[ N^{14} + n^1 \to C^{14} + H^1, \tag{IV,3} \]

is used; it gives proton tracks about \(7\,\mu\) long[^120]. The applicability of this method of detecting slow neutrons is limited by the small cross section of this process and by the low concentration of nitrogen in the gelatin of the emulsion. In investigations using this method, emulsions must be impregnated with nitrogen-rich compounds (for example, \(NaN_3\)[^121]). The introduction of uranium into the emulsion leads

toward the formation of fission-fragment tracks as a result of the capture of slow neutrons by a \(U^{235}\) nucleus.\(^{122}\) It is also possible to irradiate thin uranium foils placed on emulsion.\(^{123}\)

Fast neutrons. Fast neutrons can be detected from recoil protons arising in collisions of neutrons with the hydrogen atoms of the emulsion.\(^{64,119,124-129}\) The energy transferred by neutrons in collisions with heavier nuclei is insufficient for these nuclei to be able to form visible tracks. This is evident from the dependence of the recoil energy \(E\) on the initial neutron energy \(E_n\), the angle \(\theta\) between the recoil direction and the direction of motion of the incident neutron, and the mass number \(M\) of the recoil nucleus (here it is assumed that initially it is at rest):

\[ E=\left[\frac{4M}{(1+M)^2}\right](E_n\cos^2\theta). \tag{IV, 4} \]

The maximum value of \(E\) corresponds to \(\theta=0\) (head-on collision):

\[ E=\frac{4M}{(1+M)^2}E_n. \tag{IV, 5} \]

Complete transfer of energy, \(E=E_n\), is possible only for \(M=1\), i.e., for neutron–proton collisions. The possibility of neutron–neutron collisions is absent in this case. Thus, it turns out that the recoil energies for the heavier nuclei present in the emulsion will be much smaller than for hydrogen, and, consequently, almost all observed recoil tracks will be caused by the latter. The energies of recoil protons are related to the initial neutron energy by the relation

\[ E=E_n\cos^2\theta, \tag{IV, 6} \]

where the mean energy loss in each collision is equal to \(1/e\), or about \(0.37\).

When studying recoil tracks it should be remembered that neutrons may be scattered through small angles by heavy nuclei, in fact retaining their initial energy. Such neutrons will give recoil protons with energies greater than follows from the observed values of \(\theta\). If such protons are regarded as arising from collisions with unscattered neutrons, some error may be made in determining the neutron energy. Therefore, in studying the energy spectrum of neutrons it is advantageous to have the possibility of using a source of recoil protons situated outside the emulsion serving as the detector. In this case one can choose a suitable experimental geometry that permits an unambiguous determination of the energy. An example of the arrangement of an experiment using a thin polyethylene scatterer is given in Fig. 33.\(^{119}\) In addition, such a method greatly facilitates examination, since only the surface of the emulsion must be studied

Emulsions filled with boron may also serve for the detection of fast neutrons. In this case the reactions used are

\[ \mathrm{B}^{10} + n^1 \to \mathrm{He}^4 + \mathrm{He}^4 + \mathrm{H}^3 \tag{IV, 7} \]

and

\[ \mathrm{B}^{11} + n^1 \to \mathrm{Li}^8 + \mathrm{He}^4 . \tag{IV, 8} \]

The α-particles and triton from the reaction with \(\mathrm{B}^{10}\) form a three-pronged star, the total energy of whose particles is approximately equal to the neutron energy \(^{65}\). Cases with \(\mathrm{B}^{11}\) can be identified by the “lithium hammers” formed in the decay of \(\mathrm{Li}^8\) \(^{130}\). The energy of the initially emitted α-particle is 5.4 Mev, that of the \(\mathrm{Li}^8\) nucleus is 1.7 Mev, and the total energy of both α-particles formed as a result of the decay of \(\mathrm{Li}^8\) is 2.6 Mev. If electron-sensitive emulsions are used, it is also possible to observe the β-decay of \(\mathrm{Li}^8\) into \(\mathrm{Be}^8\) up to the formation of α-particles.

The possibility of applying the reaction

\[ \mathrm{Li}^6 + n^1 \to \mathrm{He}^4 + \mathrm{H}^3 \tag{IV, 9} \]

for the detection of both fast and slow neutrons has been investigated \(^{131,132}\). The main advantage of this method is that it does not require collimation of the neutron beam. The cross section for the disintegration of \(\mathrm{Li}^6\) by fast neutrons is only about \(0.1 \cdot 10^{-24}\ \mathrm{cm}^2\), i.e. 10 times smaller than the cross section for the formation of recoil protons, but filling emulsions with lithium having a higher percentage content of \(\mathrm{Li}^6\) than ordinary lithium partly compensates for this deficiency. Identification of the tracks of α-particles and tritons can be carried out by ordinary methods. In practice the sum of the lengths of both tracks and the angle \(\theta\) between them are measured. The energy of the incident neutron is then determined from curves analogous to those shown in Fig. 34, which give the neutron energy as a function of the total track length for different values of \(\theta\). For any given neutron energy there are two possible total ranges, corresponding to cases in which the greater part

Fig. 33. Scheme of an experiment for determining neutron energy spectra \(^{119}\).

Labels in the diagram: neutron source; polyethylene scatterer \(10.01\) mm thick; collimating slits; vacuum chamber; nuclear photographic plate \(200\ \mu\); \(10^\circ\).

energy is received either by the triton (the greater total range) or by the α-particle (the shorter range). The curves in Fig. 34, obtained for

Fig. 34. Neutron energy as a function of the total range in B-emulsion (in units of 1.25 μ) of the α-particle and triton formed in the disintegration of Li\(^6\), for various angles \(\theta^{131}\).

emulsions of the Ilford C2 and Eastman-Kodak NTA types, can, with the corresponding corrections, be used for emulsions of other stopping power.

IV. 4. Spectra of γ-rays

Photodisintegration of the deuteron provides a convenient and very accurate method for determining the energy spectrum of γ-rays by means of emulsions loaded with deuterium\(^{36, 67, 124, 133—139}\). The photon energy is determined by the sum of the energies of the proton and neutron produced in the reaction, and the binding energy of the deuteron. Since the energies of the proton and neutron are equal, knowledge of the proton energy alone is sufficient. The loading of emulsions with deuterium will be discussed in more detail in Sec. IV.5.

A more accurate determination of the energy is possible if the direction of the γ-ray beam is known, as, for example, in measuring the spectrum of a betatron. Kron and Schroeder\(^{67}\) gave, for the photon energy \(E\), the formula

\[ E = \frac{2E_p + W} {1 - \dfrac{2E_p}{m_1 c^2} + \left(\dfrac{4E_p}{m_1 c^2}\right)^{1/2}\cos\theta}, \tag{IV,10} \]

where \(E_p\) is the proton energy, determined from its range in the emulsion, and \(\theta\) is the angle between the directions of the proton track and the γ-ra-

rays, \(m_1\) is the mass of the deuteron and \(W=(m_2+m_3+m_1)c^2\), where \(m_2\) and \(m_3\) are the masses of the proton and the neutron. A series of curves of the dependence of \(E\) on \(\theta\) for constant \(E_p\) is shown in Fig. 35. From these curves one can determine \(E\), knowing the angle and energy of the photoproton.

There exist several other processes of nuclear photodisintegration which can be used to determine the energy of \(\gamma\)-rays, although in most cases they occur with greater absorption of energy and have a relatively small cross section. In particular, for this purpose the following two reactions have been proposed\({}^{34}\):

\[ \mathrm{Be}^9+\gamma \to 2\mathrm{He}^4+\mathrm{n}^1 \qquad (\mathrm{IV},\,11) \]

and

\[ \mathrm{C}^{12}+\gamma \to 3\mathrm{He}^4. \qquad (\mathrm{VI},\,12) \]

The fact that the indicated course of the first reaction is more probable than the emission of a photoneutron with formation of a residual \(\mathrm{Be}^8\) nucleus was shown by Gluckauf and Rennet\({}^{140}\). For the disintegration of \(\mathrm{C}^{12}\), \(7.16\) MeV is required, and as a result three-pronged stars of \(\alpha\)-particles are formed\({}^{141}\).

Fig. 35. Dependence of the kinetic energy of protons on the photoproton angle in deuteron photodisintegration\({}^{67}\).

IV. 5. Loading

The field of application of nuclear emulsions can be considerably broadened by introducing into them certain elements whose properties it is desirable to study. This method provides a convenient means of investigating the radioactivity of long-lived \(\alpha\)-emitters, for example samarium, and also makes it possible to detect neutrons by loading the emulsion with lithium or boron. Some questions connected with loading have been considered by Yagoda\({}^{34}\).

General considerations. Loading with a given substance is usually carried out by immersing the emulsion in the corresponding solution for a time depending on the concentration of the substance in the solution, the desired concentration of it in the emulsion, the thickness of the emulsion, and the temperature. This is followed by a very brief washing with water in order to remove residues of the solution from the surface of the emulsion and

then the plates are dried in an atmosphere free of dust. The exact amount of absorbed substance can be determined either by analysis of the loaded emulsion, or (knowing the concentration of the solution) by measuring the volume of solution absorbed by the emulsion.

Another method, which usually gives a less uniform distribution of the introduced substance, consists in applying a certain amount of solution to the surface of the emulsion and evaporating it to dryness. The advantage of this method is that the amount of introduced substance is known exactly and a minimum of solution is used. For example, for \(40\ \mathrm{cm}^2\) of emulsion surface, \(\sim 1\ \mathrm{cm}^3\) of solution is sufficient. In this case it is desirable to use an easily evaporating solvent, for example alcohol, so that evaporation proceeds more rapidly, and sometimes small amounts of a wetting agent for a more uniform coating.

Some substances, for example chromium and uranyl ions, tend to reduce the sensitivity of the emulsion into which they are introduced \(^{142,143}\). Heavy ions, for example lead and bismuth, also give a partial decrease in sensitivity \(^{144}\). If the properties of an element emitting \(\alpha\)-particles are being studied, then the absence of emulsion sensitivity to more weakly ionizing particles is not very important; however, when loading with substances that reduce sensitivity, an estimate of the degree of this reduction is necessary. Harmful effects, such as, for example, the indicated reduction in sensitivity, can be avoided by using the method of introducing insoluble substances into the emulsion (see below).

Loaded emulsions of industrial manufacture. Emulsions can be loaded with various substances during their manufacture. Thus, Ilford emulsions of types B2, C2, and E1 are loaded either with lithium or with boron, and C2 emulsions also with bismuth. Table XXII gives the composition of loaded Ilford emulsions in grams of each element per \(1\ \mathrm{cm}^3\). These figures are rela-

Table XXII

Composition of Ilford-type emulsions with loading in \(\mathrm{g}/\mathrm{cm}^3\)

Element	Li	B	Bi	Element	Li	B	Bi
Silver	1,84	1,77	1,39	Sulfur	0,038	0,010	0,002
Bromine	1,35	1,28	1,01	Nitrogen	0,083	0,064	0,062
Iodine	0,053	0,047	0,039	Lithium	0,016	...	...
Carbon	0,27	0,26	0,33	Sodium	...	0,025	0,08
Hydrogen	0,047	0,053	0,047	Boron	...	0,023	...
Oxygen	0,29	0,32	0,43	Bismuth	...	...	0,27

refer to 50% relative humidity and a temperature of \(20^\circ\text{C}\); for other, greatly differing humidity values these figures will change somewhat. There are also Eastman-Kodak emulsions of types NTA and NTB, loaded with lithium or boron, the approximate amount of the element per \(1\ \text{cm}^3\) being indicated for each pack of plates. The manufacture of emulsions loaded with beryllium has been discontinued because of its toxic properties.

Loading with deuterium. Deuterium can be introduced into emulsions by loading them with calcium nitrate \((\mathrm{Ca}(\mathrm{NO}_3)_2)\) containing heavy water of crystallization \(^{133}\). In this way a concentration of \(\mathrm{D}_2\mathrm{O}\) equal to 6% by weight can be achieved. Ilford supplies plates loaded with a stable deuterium compound. However, much higher concentrations can be achieved by immersing the emulsion directly in \(\mathrm{D}_2\mathrm{O}\) and exposing it in the wet state. In this case concentrations of \(\mathrm{D}_2\mathrm{O}\) from 30 to 80% by weight can be obtained. On absorbing about \(0.5\ \text{g}\) of \(\mathrm{D}_2\mathrm{O}\) per \(0.2\ \text{cm}^3\), the emulsion swells to approximately 3.5 times its initial thickness. The curves of specific energy loss and the range–energy curves for wet emulsions, of course, differ from the corresponding curves obtained under ordinary conditions. In Section II.3 various corrections for wet emulsions were discussed; these results can also be applied to the case of impregnation with \(\mathrm{D}_2\mathrm{O}\). Figures obtained on the basis of volume ratios are the same for \(\mathrm{H}_2\mathrm{O}\) and \(\mathrm{D}_2\mathrm{O}\); however, the greater weight of heavy water makes it necessary to introduce an appropriate conversion factor if weight ratios are used.

Loading with lithium borate. The various advantages of lithium and boron loading of emulsions for neutron detection can be combined by introducing lithium borate \((\mathrm{Li}_2\mathrm{B}_4\mathrm{O}_7)\) into the emulsion. Igoda \(^{34}\) described in detail a method for such loading.

Table XXIII

Composition of the bath for loading with lithium borate

Component	Amount
Distilled water (hot)	\(300\ \text{cm}^3\)
Boric acid (crystalline)	\(60\ \text{g}\)
Lithium carbonate	\(19\ \text{g}\)
Glycerin	\(20\ \text{cm}^3\)
Distilled water to	\(400\ \text{cm}^3\)

Table XXIII gives the composition of the solution used for loading with lithium borate. First the boric acid is dissolved and lithium carbonate is added. Then the solution is cooled, filtered, and diluted to \(400\ \text{cm}^3\). A lithium borate solution with a concentration of 10% is obtained. The introduction of glycerin is necessary only for emulsions thicker than \(\sim 50\ \mu\). During a 15-minute

during their stay in this solution, 30-micron plates absorb \(\sim 0.24\) mg of lithium borate per \(1\ \mathrm{cm}^2\).

Introduction of uranium. For the study of uranium fission caused by neutrons, the emulsion is impregnated with solutions of uranyl nitrate or uranyl acetate. Wigneron et al.\(^{145}\) impregnated a 40-micron Ilford C2 emulsion for 5 min with a 20% solution, then immersed it in ethyl alcohol and dried it in a stream of air. Green and Livesey\(^{146}\) used solutions of uranyl acetate acidified with weak acetic acid, and approximately determined the degree of loss of sensitivity at various concentrations. After impregnation with a 1% solution of uranyl acetate for a time varying from 1 hour for an emulsion \(20\ \mu\) thick to 12 hours for an emulsion \(100\ \mu\) thick, weak proton tracks, distinct \(\alpha\)-particle tracks, and very dense tracks of fission fragments were observed in Ilford C2 and B1 plates. Two-percent solutions made the plates insensitive to protons and reduced the grain density in \(\alpha\)-particle tracks. Four-percent solutions in fact almost completely excluded the possibility of recording \(\alpha\)-particles.

“Pies.” During impregnation, the introduced substance is distributed more or less uniformly throughout the volume of the emulsion. In contrast to this, in making “pies,” thin layers of various substances are introduced between layers of emulsion. Harding\(^{147}\), in studying cosmic radiation, used plates consisting of alternating layers of pure gelatin and emulsion (four 30-micron layers of emulsion were separated by three thin layers of gelatin). Hudson and Perkins\(^{148}\), also in the study of cosmic radiation, attempted to introduce a layer of lead phosphate, but the amount of lead that can be introduced in this way without decreasing the transparency of the emulsion is very small. Metallic foils placed between two plates were also used; after irradiation these were separated for processing and study\(^{149}\); however, this method has a limited field of application.

Filling with insoluble substances. Direct filling of an emulsion with insoluble substances is impossible. However, the method of preparing “pies,” proposed by Wigneron and Bogard\(^{150}\), makes it possible to introduce solid grains of a substance, included in a layer of gelatin, between two layers of emulsion. By this method, a suspension is first prepared from particles of suitable size in a solution of one part gelatin in 200 parts water, and several drops of this suspension are then spread over the surface of the emulsion. One drop is sufficient for \(\sim 10\ \mathrm{cm}^2\) of surface. After drying in a vacuum chamber, the plates are immersed in a water bath. Then the emulsion is removed from the second plate with a razor blade, and such a strip of emul-

is placed under water onto the first plate. Films may also be used for this purpose (Section III.5). The prepared “sandwich,” from which excess moisture has been removed, is heated at \(45^\circ\text{C}\) for 4 min and then dried under reduced pressure. Plates prepared in this way are very strong and do not require special handling or special processing.

This method can also be used for soluble substances and compounds which, when introduced into the emulsion in the form of solution ions, change the \(pH\) of the emulsion and, consequently, its sensitivity and development characteristics. It is especially suitable for the study of radioactive substances; the beginning of any track can be established unambiguously and, since the particles of the element under investigation are located between two layers of emulsion, tracks running in any direction can be studied. A similar method was used in some experiments \(^{14,151,152}\).

IV.6. Deflection of particles by a magnetic field

The momentum of a charged particle passing through the gas of a Wilson chamber is determined from the curvature of the track under the influence of a strong magnetic field. This method cannot be applied directly to nuclear emulsions, since the very short ranges of particles in the emulsion, together with scattering, require the presence of very high fields in order to produce a measurable curvature (possibly 100 or more times higher than when working with a Wilson chamber). However, if the deflection takes place in an air gap between two plates, then fields of acceptable strength are required to determine the momentum of particles crossing both plates and the gap between them \(^{107,153–157}\).

Experimental procedure. In carrying out the experiment, plates with a thickness of 100 or 200 \(\mu\) are rigidly fixed in a holder at a distance of several millimeters from one another (a distance of 3 mm is convenient) and are placed between the poles of a magnet. Thicker plates may give too great a distortion of the emulsion during development. The size of the plates must be greater than the size of the poles, so that during drying the useful area of the emulsion is not distorted. A margin of about 2.5 cm is sufficient.

When applying this method it is necessary to know precisely the relative position of the plates during irradiation. For this purpose it is convenient to use a collimated beam of X-rays, by means of which a series of correctly positioned marks is placed on the mounted plates. Barbour \(^{154}\), in his work, used as a template a lead plate 0.75 mm thick with holes 100 \(\mu\) in diameter, arranged at a distance of 0.5 cm from one another; Powell’s group \(^{153}\)

used a grid of fine lines (approximately \(20\mu\) wide), applied by means of a slit in a lead plate that was irradiated in various positions.

In permanent installations at sea level or at mountain altitudes one may use an electromagnet of such intensity as is necessary to obtain the required results. Fields up to 27,500 gauss have been used. In high-altitude experiments carried out with the aid of airplanes or balloons, low weight is necessary, and in this case weak permanent magnets have to be used. For work with balloons, Barbour \(^{107}\) constructed a 26-kilogram magnet with a soft-iron yoke and poles of area \(25.9\ \mathrm{cm}^2\), giving a field of \(13.3\) thousand gauss in a gap of \(6.25\ \mathrm{mm}\), with which he obtained satisfactory results.

Magnets for magnetrons are designed for several thousand gauss; they can be used if the field is increased by reducing the area of the pole pieces with the aid of soft iron. The increase of the field up to saturation is roughly proportional to the reduction of the area, and in this way fields above 10,000 gauss can be obtained in magnetron magnets.

Analysis of tracks. When the plates are examined, the positions of the various tracks are plotted on a large sheet of paper together with the marks from the X-rays. The angular orientations of the tracks in the plane of the emulsion and the angles of inclination \(\Phi\) into the depth must be known accurately, so that it may be determined which pair of tracks in two plates was caused by one and the same particle. To determine \(\Phi\), it is necessary to know the shrinkage coefficient. The selection of pairs of tracks must satisfy the following conditions:

1. The angles of inclination \(\Phi_1\) and \(\Phi_2\) (Fig. 36) must be equal or very close.

2. The angles of deflection \(\theta_1\) and \(\theta_2\) between the continuations of the tracks and the line joining the points of entry of the tracks into the air gap must be close, since the trajectory of the particle in the magnetic field is an arc of a circle.

3. The density of grains of the track near the emulsion surfaces must be the same. In the air gap the particle loses very little energy and, consequently, the rates of energy loss must be equal.

4. The distance \(d\) between the points of entry of the two tracks must be compatible with the angles of inclination \(\Phi_1\) and \(\Phi_2\) and with the distance \(\delta\) between the plates.

The radius of curvature \(\rho\) of the particle trajectory in the air gap can be obtained from the relation

\[ \rho=\frac{d}{2\sin\theta_1}=\frac{d}{2\sin\theta_2}. \tag{IV,13} \]

However, as Barbour\(^{107}\) indicated, for determining \(\rho\) the preferable relation is

\[ \rho=\frac{d}{2\sin\left(\frac{\alpha}{2}\right)}, \tag{IV,14} \]

since the total angular curvature \(\alpha\) depends exclusively on the difference between the measured angular orientations of the parts of the track. Unlike measurements of \(\theta\), errors in plotting angles and positions do not affect the value of \(\alpha\). The momentum \(p\) of the particle is found from the equality

Fig. 36. Top and side views of a “sandwich” irradiated in a magnetic field. The track of a deflected particle is shown.

\[ p=\frac{eH\rho}{c\cos\Phi}, \tag{IV,15} \]

where \(H\) is the field strength, \(c\) is the speed of light, and \(e\) is the electron charge.

It is possible to determine the mass \(M\) of the incident particle if it stops in the second emulsion and we assume that its charge is unity. Since

\[ R=Mf(v) \tag{IV,16} \]

and \(p=Mv\) in the nonrelativistic approximation, knowledge of \(R\) and \(p\) makes it possible to eliminate \(v\) and find \(M\). (For the range–energy dependence, see Sec. II.3.) Barbour\(^{107}\) calculated range–curvature curves in Ilford C2 emulsions for singly charged particles of several masses (Fig. 37); the curvature \(C\) is determined from the relation

\[ C=\frac{(10^3\cos\Phi)}{H\rho}. \tag{IV,17} \]

Substantial errors that may occur in these measurements, apart from errors in the measurements of angles caused by distortion of the emulsion (these errors can be reduced to a minimum—possibly to \(0.5^\circ\)), are connected only with the scattering of particles through small angles and with the determination of \(H\). In the case of scattering through large angles in the air gap, the direction of the particle may be changed so much that the two parts of the track will not correspond. Francinetti\(^{136}\) estimated the deviation that may be expected because of scattering in the gap, relative to the deviation caused by the magnetic field, and in all cases found that the uncertainty does not exceed 4%. Since the scattering is proportional to the square root of the air pressure\(^{138}\), for the altitudes at which balloon experiments are carried out this source of error is quite negligible. Of course, it is essential to know \(H\) accurately and, moreover, only such an area of the pole pieces can be used within which the field is homogeneous to an accuracy of at least a few percent.

Fig. 37. Theoretical range–curvature curves in Ilford C2 emulsion for singly charged particles of various masses\(^{107}\).

The possibility of attributing two parts of different tracks to the trajectory of one particle is greatly reduced if the density is less than 100 tracks per \(1\ \mathrm{cm}^2\). Francinetti calculated the relative number of such false coincidences among this number of tracks, isotropically distributed in two plates, and obtained for this quantity the value \(3.3 \cdot 10^{-3}\).

IV. 7. Other methods of using plates

Arrangement for the study of scattering. Much valuable information about nuclear forces and reactions can be obtained by determining the nature and angular distribution of particles produced or scattered when a substance is bombarded by a beam of collimated particles from an accelerator. In the case where the target is arranged in space so as to permit the investigation of particles emerging from it in any direction, the plates may be placed around the target, setting the plane of the emulsion along the direction of motion of the particles from the target or at certain suitable angles that allow the tracks to be identified. An arrangement of this kind was used, for example, by Talbot\(^{139}\) in studying the distribution of \(\alpha\)-particles from the reaction of the disintegration of \(\mathrm{Li}^7\) by protons. A simplified diagram of this arrangement is given

in Fig. 38. The known geometry of the arrangement of the plates during irradiation makes it possible to determine, from the direction of each track in the emulsion, the corresponding angle of its emission.

Another arrangement, with plates placed radially around the target, was used by Wilkinson^160. In his experiment the emulsion plane of the plates was also positioned along the motion of the particles, with each plate corresponding to a definite scattering angle. A modification of an arrangement of this type was adapted by Adair^161 for work with gas targets. In this case the orientation of the scattered particles relative to the incident beam was determined for each plate by means of slits (Fig. 39). This arrangement makes it possible simultaneously to irradiate 69 plates, placed at intervals of \(2.5^\circ\) over a range of \(160^\circ\) on each side of the beam. The appropriate choice of the angle \(\alpha\) is determined by the experimental conditions. Other arrangements were also constructed for the study of scattering^162–165.

Fig. 38. Simplified diagram of the arrangement of plates. The direction of motion of particles from the target is parallel to the plane of the emulsion.^159

Fig. 39. Diagram of the arrangement of slits in work with a gas target.^161

Stacks of plates. In studies of cosmic radiation, stacks of plates are commonly used in such a way that, under favorable conditions, tracks can be recorded in a whole series of consecutively arranged emulsion layers. A variant of this method was used by Bradt and Peters^70 in studying the heavy component of primary cosmic radiation. Although grain counting determines the specific energy loss more accurately than counting delta rays, the use of this method is disadvantageous at the maximum ratio of specific ionizations that can be estimated in plates of a given sensitivity. Thus, if some particle

produces a faintly distinguishable track; another, having an ionizing power approximately 15 times greater, gives a track too dense for grain counting. Bradt and Peters^70 circumvented this difficulty by using stacks of plates containing emulsions of high and low sensitivity. Eastman NTB3 and NTA plates were used, the former being processed in the usual way and the latter developed in D19 diluted 1:20. Figure 40 gives the grain densities corresponding to different specific energy losses for two emulsions, with the positions of tracks caused by various relativistic nuclei indicated. For this purpose special plates are made, consisting of alternating layers of Ilford G5 and G0 emulsions (lower sensitivity), with layer thicknesses of 400 μ for G5 and 200 μ for G0. Preliminary investigations show that in G0 plates the grain density, expressed as the number of grains per 100 μ of track length, is numerically equal to the energy loss in keV/μ up to a value of 60, at which saturation effects become significant.

Fig. 40. Variation of grain density as a function of the specific energy loss in Eastman Kodak NTB3 (curve I) and NTA (curve II) emulsions. The expected grain density for various relativistic nuclei is shown^70.

IV.8. Detection of tracks

In studying emulsions with a high background density it is desirable to be able to determine the probability of detecting tracks of low specific ionization. Although certain plates can register particles with energies corresponding to minimum ionization, identification of the tracks may be hampered by the presence of an excessively dense grain background. This situation is completely analogous to the dependence of the audibility of an individual sound on the intensity of background noise. Knowledge of the dependence of track visibility on the grain density of the track and of the background is therefore useful in assessing the applicability of specially developed plates for various purposes.

Straight tracks. Cotes^95 established that the distribution of individual grains, tracks, and background is essentially random, which reduces the problem of estimating the visibility of tracks in nuclear emulsions

to the more general question of estimating the visibility of a line formed by a row of black spots with randomly distributed intervals against a background of other randomly distributed spots. Berriman[^166], using this approach, prepared a series of diagrams of artificial background of different density and, by superimposing artificial tracks of different density, determined the visibility of the latter as a function of the background.

In preparing the background diagrams, tables of randomly selected numbers were used to localize the positions of individual spots on a sheet of ruled paper. At the marked points, round holes were punched with a die, and the sheets were then photographed against a black background. It was assumed that the background consisted of spherical grains of diameter 0.2; 0.3; 0.4; 0.5; 0.6, present in the ratio 1:4:6:4:1, respectively; the sizes and relative numbers of the holes were proportional to these numbers. In each subsequent diagram the number of holes of all series was increased by a factor of \(\sim 2\), and the density varied from an equivalent of \(2 \cdot 10^{-3}\) grains per sq. micron (at a 1500-fold transverse magnification) to \(500 \cdot 10^{-3}\) grains per sq. micron. In preparing the track diagrams, tracks of the maximum grain density were first constructed, and then, at randomly selected points along this track, grains were removed until the desired density was reached.

Two tracks of each density were simultaneously superimposed on the background diagrams, and the maximum fog density was determined at which: 1) both tracks were easily recognized, and 2) only one of the tracks was recognized. Condition 1) was defined by Berriman[^166] as good recognition of tracks, and condition 2), approximately corresponding to equal probability of missing and detecting a track, as satisfactory recognition. Figure 41 gives the results for good and satisfactory recognition of tracks of minimum density as a function of the corresponding background density (the graph is given on a logarithmic scale).

Fig. 41. Registration curves for tracks of minimum density as a function of background density[^166].

Very interesting is the interpretation of the two points of inflection on these curves. The first, corresponding to a track density of about 0.3 grains per micron, is the point at which “doublets” first appear—adjacent grains touching one another or

located very close together. Such doublets act to increase the visibility of the track. The second bend occurs at a background density of 0.1 grain per sq. micron; at this point the background becomes too dense to allow the tracks to be easily distinguished. Above this point the slope of the curves increases very rapidly.

The curves of Fig. 41 in fact overestimate the track densities practically necessary for their detection. When examining the emulsion under a microscope one can study the region near the field of view in which there is a doubtful track. Moreover, once the existence of a track has been established, for example in some region with a reduced fog or owing to a somewhat greater density of grains over a small portion of its length, finding the remaining part of the track is considerably facilitated. In the artificial diagrams only portions of tracks equivalent to a length of 50 μ were used. Focusing the microscope through the depth of the emulsion during examination helps in recognizing tracks which go even at small angles, at a higher background density than that which follows from Fig. 41. Consequently, while the course of these curves may be accepted as approximately correct, the numerical values can serve only for an approximate estimate of the visibility of tracks.

Detection of electron tracks. An experiment similar to that described above was carried out by Beiser \(^{167}\) for tracks of low-energy electrons. Such tracks, unlike those considered by Berriman \(^{166}\), undergo considerable scattering and are accordingly less easily visible. Moreover, in this case separate accumulations of fog grains may be mistaken for a track. The density of the artificial background was varied by a factor of 1.5 from \(1\cdot 10^{-3}\) to \(39\cdot 10^{-3}\) grains per sq. micron. To determine visibility, four reproductions of each background were used, and the images of electron tracks were superimposed on only three of them. For this purpose, 50-micron portions of tracks were reproduced.

Fig. 42. Curves of the relative probability of registration of electron and pseudoelectron tracks as a function of background density \(^{167}\).

In Fig. 42 the mean probability (in percent) is shown for the recognition of electron tracks as a function of background density; these results are based on a large number of individual trials. The selected criterion was clear visibility of the tracks;

in each case the observer had to determine whether there was a track or several tracks and their exact trajectories. The visibility curve shows the number of correct identifications referred to the number of tracks actually present. The other curve in Fig. 42, corresponding to the detection of pseudo-tracks, is the number of incorrect identifications divided by the actual number of tracks. It is seen from the figure that, with a background above approximately \(5 \cdot 10^{-3}\) grains per sq. micron, a certain fraction of tracks will be lost and, in addition, incorrect identification is possible. Of course, as follows from the considerations given above, the value of this density is somewhat smaller than the actual practical threshold.

The decrease in the slope of the visibility curve for background densities above \(10^{-2}\) grains per sq. micron is apparently connected with a more careful examination of dense backgrounds. The continuing increase in the recognition of pseudo-tracks also serves as an indication of the greater concentration of the observer, as a result of which the possibility of assuming the existence of a track increases. It was observed that persons whose work is at least partly connected with the extraction of signals (in the most general sense) from random noise consistently improved their recognition of tracks and made fewer incorrect identifications than persons who had less experience. Even experienced scanners were no better than, for example, people engaged in observing specific phenomena on an oscilloscope screen accompanied by background “noise.” Apparently, such a test is suitable for evaluating the capacity for scanning, especially in the investigation of cases such as those considered here.

REFERENCES

1. M. Reinganum, Physik. Zeits. 12, 1076 (1911).
2. S. Kinoshita, Proc. Roy. Soc. 83, 432 (1910).
3. M. Michl, Akad. Wiss., Wien 121, 1431 (1912).
4. M. Blau, Zeits. f. Physik 34, 285 (1925).
5. M. M. Shapiro, Rev. Modern Phys. 13, 58 (1941).
6. H. J. Taylor, Proc. Roy. Soc. 150, 382 (1935).
7. H. Wambacher, Zeits. wiss. Phot. 38, 38 (1939).
8. K. B. Mather, J. Opt. Soc. Am. 38, 1054 (1948).
9. P. B. Maurer and H. L. Reynolds, Phys. Rev. 73, 1131 (1948).
10. Mysovskii and Chizhov, Zeits. f. Physik 44, 408 (1927).
11. M. Blau and Wambacher, Monatsh. 61, 99 (1932).
12. Zhdanov, J. phys. et rad. 6, 233 (1935).
13. P. Demers, Phys. Rev. 70, 86 (1946).
14. P. Demers, Phys. Rev. 70, 974 (1946).
15. P. Demers, Can. J. Research 25, 223 (1947).
16. W. Hälg and L. Jenny, Helv. Phys. Acta 21, 131 (1948).
17. L. Jenny, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).

1. C. C. Dilworth, Cosmic Radiation, F. C. Frank and D. R. Rexworthy, ed. (Butterworths Scientific Publications, London, 1949).
2. E. M. Dolman, Rev. Sci. Instr. 21, 118 (1950).
3. J. J. Lord, Phys. Rev. 81, 901 (1951).
4. A. Beiser (unpublished).
5. M. Cosyns, C. C. Dilworth and G. P. S. Occhialini, Université Libre de Bruxelles Note n° 6 (1949).
6. M. Blau, Akad. Wiss. Wien 140, 623 (1931).
7. K. B. Mather, Phys. Rev. 76, 486 (1949).
8. A. Beiser, Phys. Rev. 80, 112 (1950).
9. G. Albouy et H. Faraggi, J. phys. et rad. 10, 105 (1949).
10. C. E. K. Mees, The Theory of the Photographic Process (Macmillan, New York, 1942).
11. A. Beiser, Phys. Rev. 81, 153 (1951).
12. H. Yagoda and N. Kaplan, Phys. Rev. 73, 634 (1948).
13. H. Wäffler and S. Younis, Helv. Phys. Acta 22, 414 (1949).
14. J. LaPalme and P. Demers, Phys. Rev. 72, 536 (1947).
15. R. Brawn, U. Camerini, P. H. Fowler, H. Muirhead, C. F. Powell and D. M. Ritson, Nature 163, 47, 82 (1949).
16. H. Yagoda and N. Kaplan, Phys. Rev. 71, 910 (1947).
17. H. Yagoda, Radioactive Measurements with Nuclear Emulsions (John Wiley and Sons, Inc., New York, 1949).
18. F. H. Krenz, Proceedings of the Conference on Nuclear Chemistry, Ottawa, 1947.
19. L. Winand and L. Falla, Bull. Soc. Sci. Liège, 18, 184 (1949).
20. A. Hautot, Sci. Ind. Phot (2) 19, 441 (1948).
21. R. W. Gurney and N. F. Mott, Proc. Roy. Soc. 164A, 151 (1938).
22. J. W. Mitchell, Sci. Ind. Phot. 19, 361 (1948).
23. J. W. Mitchell, Phil. Mag. 40, 249 (1949).
24. J. W. Mitchell, Phil. Mag. 40, 667 (1949).
25. J. H. Webb, Phys. Rev. 74, 511 (1948).
26. M. S. Livingston and Bethe, Rev. Modern Phys. 9, 245 (1937).
27. D. Halliday, Introductory Nuclear Physics (John Wiley and Sons, Inc., New York, 1950).
28. J. Wheeler and R. Ladenburg, Phys. Rev. 60, 754 (1941).
29. E. Rutherford, Phil. Mag. 47, 277 (1924).
30. P. H. Fowler and D. H. Perkins, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).
31. P. Debye and E. Hückel, Physik. Zeits. 24, 185 (1923).
32. M. Blau, Phys. Rev. 75, 279 (1949).
33. L. van Rossum, J. phys. et rad. 10, 402 (1949).
34. M. Morand and L. van Rossum, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).
35. J. H. Smith, Phys. Rev. 71, 32 (1946).
36. R. M. S. Blackett and D. S. Lees, Proc. Roy. Soc. 134, 658 (1932).
37. C. M. G. Lattes, P. H. Fowler and P. Cüer, Nature 159, 301 (1947).
38. C. M. G. Lattes, P. H. Fowler and P. Cüer, Proc. Phys. Soc. 59, 883 (1947).
39. H. Bradner, F. M. Smith, W. H. Barkas and A. S. Bishop, Phys. Rev. 77, 462 (1950).
40. J. Rotblat, Nature 165, 387 (1950).
41. U. Camerini and C. M. G. Lattes, Ilford Technical Data (Ilford Ltd., London, 1947).
42. C. M. G. Lattes, G. P. S. Occhialini and C. F. Powell, Proc. Phys. Soc. (London) 61, 173 (1948).

1. M. A. S. Ross and B. Zajac, Nature 162, 923 (1948).
2. R. H. Herz, Phys. Rev. 75, 478 (1949).
3. J. Rotblat, Progress in Nuclear Physics. J. O. R. Frisch, ed. (Butterworth. Springer Ltd., London, 1950).
4. C. F. Powell, G. P. S. Occhialini and D. L. Livesey, J. Sci. Instr. 23, 102 (1946).
5. N. Nereson and F. Reines, Rev. Sci. Instr. 21, 534 (1950).
6. C. M. G. Lattes and G. P. S. Occhialini, Nature 159, 331 (1947).
7. Aron, Hoffman and Williams, University of California Radiation Laboratory Report, No. 121 (1949).
8. V. E. Krohn and E. F. Schrader, Case Institute of Technology Nuclear Physics Laboratory Technical Report No. 8 (1951).
9. P. Cüer, Comptes Rendus 223, 1121 (1946).
10. C. M. G. Lattes, G. P. S. Occhialini and C. F. Powell, Nature 160, 486 (1947).
11. H. L. Bradt and B. Peters, Phys. Rev. 80, 943 (1950).
12. H. L. Bradt, M. F. Kaplon and B. Peters, Helv. Phys. Acta, 23, 24 (1950).
13. A. Beiser, Rev. Sci. Instr. 21, 933 (1950).
14. P. Freier, E. J. Lofgren, E. P. Ney, F. Oppenheimer, H. L. Bradt and B. Peters, Phys. Rev. 74, 213 (1948).
15. H. L. Bradt and B. Peters, Phys. Rev. 74, 1828 (1948).
16. N. F. Mott, Proc. Roy. Soc. 124, 425 (1929).
17. P. Freier, E. J. Lofgren, E. P. Ney and F. Oppenheimer, Phys. Rev. 74, 1818 (1948).
18. H. L. Bradt and B. Peters, Phys. Rev. 76, 156 (1949).
19. Hoang and Morellet, Comptes Rendus 231, 695 (1950).
20. D. M. Bose and B. Choudhuri, Nature 147, 240 (1941).
21. Perkins, Nature 159, 126 (1947).
22. C. F. Powell and G. P. S. Occhialini, Nuclear Physics in Photographs (Oxford University Press, London, 1947).
23. E. J. Williams, Proc. Roy. Soc. 169A, 531 (1939).
24. E. J. Williams, Phys. Rev. 58, 292 (1940).
25. G. Molière, Zeits. Naturforsch. 2a, 133 (1947).
26. G. Molière, Zeits. Naturforsch. 3a, 78 (1948).
27. H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949).
28. K. Gottstein, M. G. K. Menon, J. H. Mulvey, C. O’Ceallaigh and O. Rochat, Phil. Mag. 42, 708 (1951).
29. Y. Goldschmidt-Clermont, D. T. King, H. Muirhead and D. M. Ritson, Proc. Phys. Soc. 61, 183 (1948).
30. R. Davies, W. O. Lock and H. Muirhead, Phil. Mag. 40, 1250 (1949).
31. Y. Goldschmidt-Clermont, Nuovo Cimento 7, 33 (1950).
32. Lattimore, Nature 161, 518 (1948).
33. P. H. Fowler, Phil. Mag. 41, 169 (1950).
34. Goldsmit and Scott, Phys. Rev. 74, 1537 (1948).
35. Goldsmit and Scott, Phys. Rev. 76, 212 (1949).
36. A. C. Coates, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).
37. B. Stiller — private communication.
38. C. C. Dilworth, S. J. Goldsack, Y. Goldschmidt-Clermont and F. Levy, Phil. Mag. 41, 1032 (1950).
39. B. Stiller, M. M. Shapiro and F. W. O’Dell, Bull. Am. Phys. Soc. 26, No. 6, 16 (1951).
40. A. D. Dainton, A. R. Gottiker and W. O. Lock, Phil. Mag. 42, 396 (1951).

1. M. J. Wilson and S. Vaneslow, Phys. Rev. 75, 1144 (1949).

2. C. C. Dilworth, G. P. S. Occhialini and Vermaesen, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).

3. P. H. Stelson, Massachusetts Institute of Technology Laboratory for Nuclear Science and Engineering Technical Report, No. 47 (1950).

4. C. C. Dilworth, G. P. S. Occhialini and Payne, Nature 163, 102 (1948).

5. M. Mortier and L. Vermaesen, Centre de Physique Nucléaire de Bruxelles, Note No. 5 (1948).

6. E. Picciotto, Comptes Rendus 228, 173 (1949).

7. M. Blau and J. A. De Felice, Phys. Rev. 74, 1198 (1948).

8. J. Barbour, Phys. Rev. 78, 518 (1950).

9. J. Rotblat and C. T. Tai, Nature 164, 835 (1949).

10. J. Rotblat and C. T. Tai, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).

11. L. Vigneron, J. phys. et rad. 10, 305 (1949).

12. F. A. Roads, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).

13. N. A. Perfilov, DAN 42, 258 (1944).

14. N. A. Perfilov, DAN 43, 14 (1944).

15. M. Wiener and H. Yagoda, Rev. Sci. Instr. 21, 39 (1950).

16. A. Beiser, Rev. Sci. Instr. 21, 1025 (1950).

17. M. Blau and H. Wambacher, Nature 134, 538 (1934).

18. H. Kallmann, Research 1, 254 (1948).

19. W. E. Burcham and M. Goldhaber, Proc. Cambridge Phil. Soc. 32, 632 (1936).

20. J. C. Allred, A. N. Phillips, L. Rosen and F. K. Tallmadge, Rev. Sci. Instr. 21, 225 (1950).

21. J. S. Cheka, Phys. Rev. 74, 128 (1948).

22. P. Cüer, J. phys et rad. 8, 83 (1947).

23. K. Lark-Horovitz and W. A. Miller, Phys. Rev. 59, 941 (1941).

24. D. Froman, L. Rosen and B. Rossi, Bull. Am. Phys. Soc. 22, No. 3, 5 (1947).

25. C. F. Powell, Nature 145, 155 (1940).

26. C. F. Powell and Champion, Proc. Roy. Soc. 183A, 64 (1944).

27. C. F. Powell, Proc. Roy. Soc. 181A, 344 (1943).

28. W. M. Gibson and D. L. Livesey, Proc. Phys. Soc. 60, 523 (1948).

29. D. L. Livesey and D. H. Wilkinson, Proc. Roy. Soc. 195A, 123 (1948).

30. J. C. Grosskreutz, Phys. Rev. 76, 482 (1949).

31. E. Pickup, Phys. Rev. 74, 495 (1948).

32. G. R. Keepin and J. H. Roberts, Phys. Rev. 76, 154 (1949).

33. G. R. Keepin and J. H. Roberts, Rev. Sci. Instr. 21, 163 (1950).

34. W. M. Gibson, L. L. Green and D. L. Livesey, Nature 160, 534 (1947).

35. W. Bosley, J. D. Craggs and W. F. Nash, Nature 161, 1022 (1948).

36. G. Goldhaber, Phys. Rev. 74, 1725 (1948).

37. G. Goldhaber, Phys. Rev. 77, 753 (1950).

38. Wang and Wiener, Phys. Rev. 76, 1724 (1949).

39. C. H. Millar, A. G. W. Cameron and M. Glicksman, Can. J. Research 28A, 475 (1950).

40. P. V. C. Hough, Phys. Rev. 80, 1069 (1950).

41. E. Glückauf and F. A. Paneth, Proc. Roy. Soc. 165A, 229 (1938).

1. H. Hänni, V. L. Telegdi and W. Zünti, Helv. Phys. Acta 21, 203 (1948).

2. L. L. Green and D. L. Livesey, Nature 158, 272 (1946).

3. E. Broda, J. Sci. Instr. 24, 136 (1947).

4. E. Broda, Nature 158, 872 (1946).

5. T. San Tsiang, Ho Zah Wei, R. Chastel and L. Vigneron, J. phys. et rad. 8, 1, 26 (1947).

6. L. L. Green and D. L. Livesey, Proc. Cambridge Phil. Soc. 241A, 223 (1948).

7. J. B. Harding, Nature 163, 440 (1949).

8. P. E. Hodgson and D. H. Perkins, Nature 163, 439 (1949).

9. Y. Barbour and L. Green, Phys. Rev. 79, 406 (1950).

10. L. Vigneron and M. Bogardt, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).

11. Picciotto, Comptes Rendus 228, 2020 (1949).

12. Picciotto, Comptes Rendus 229, 117 (1949).

13. C. F. Powell and S. Rosenblum, Nature 161, 473 (1948).

14. Y. Barbour, Phys. Rev. 74, 507 (1948).

15. C. Franzinetti, Phil. Mag. 41, 86 (1950).

16. C. Franzinetti, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).

17. C. C. Dilworth, G. P. S. Occhialini and L. Vermaesen, Bulletin du Centre de Physique Nucléaire de L’Université Libre de Bruxelles No. 13a (1950).

18. H. A. Bethe, Phys. Rev. 70, 821 (1946).

19. F. L. Talbott, A. Busala and G. C. Weiffenbach, Catholic University of America, Technical Report No. 1 (1950).

20. T. R. Wilkins, J. Appl. Phys. 11, 35 (1940).

21. J. C. Allred, L. Rosen, F. K. Tallmadge and J. H. Williams, Rev. Sci. Instr. 22, 191 (1951).

22. J. Chadwick, A. N. May, T. G. Pickavance and C. F. Powell, Proc. Roy. Soc. 183A, 1 (1944).

23. S. Rubin, W. A. Fowler and C. C. Lauritsen, Phys. Rev. 71, 212 (1947).

24. A. N. May and C. F. Powell, Proc. Roy. Soc. 190A, 170 (1947).

25. L. Rosen, F. K. Tallmadge and J. H. Williams, Phys. Rev. 76, 1283 (1949).

26. R. W. Berriman, Fundamental Mechanisms of Photographic Sensitivity, J. W. Mitchell, ed. (Butterworths Scientific Publications, London, 1951).

27. A. Beiser, Rev. Sci. Instr. 23, 500 (1952).

28. Barbour, Rev. Sci. Instr. 20, 530 (1949).

29. Curtis and Osborne, Phys. Rev. 75, 1327 (1949).

30. Demers and Mathieu, Phys. Rev. 75, 1327 (1949).

31. Hough, Phys. Rev. 76, 163 (1949).

32. Singer, Office of Naval Research (L) Techn. Report No. 71 (1951).

33. Von Friesen, Nature 166, 686 (1950).

34. Feld, Massachusetts Institute of Technology Laboratory for Nuclear Science and Engineering Technical Report No. 8 (1948).