From Current Literature

On the Intranuclear Cascade Process

The development of techniques for accelerating charged particles makes it possible to investigate ever more deeply the processes occurring when particles interact with atomic nuclei. Of particular interest is the study of the interaction of high-energy nucleons with the nuclei of heavy elements, in which the development of an intranuclear cascade process is possible. Such a study was carried out1, 2 for protons with an energy of 375 MeV and neutrons with an energy of \(\sim 300\) MeV, obtained with the aid of a phasotron. Beams of these particles irradiated photographic plates, in the emulsion of which nuclear disintegrations occurred and were then analyzed. The tracks were counted, and it was experimentally confirmed that at least 80% of inelastic collisions in the emulsion occur with Ag and Br nuclei, whose mean mass number is \(\sim 100\). The contribution made by light nuclei (gelatin) could be estimated and subtracted from the total effect.

At bombarding-particle energies of 300–400 MeV, meson production can still be neglected, but at the same time one may expect a considerable development of the intranuclear cascade in such heavy nuclei as Ag and Br.

The experimental results1 were compared with theoretical calculations2 for a definite model of the interaction process, and a surprisingly good agreement between theory and experiment was found.

The main features of the working model underlying the calculation of the intranuclear cascade process were as follows. Heavy nuclei were described by means of the ordinary statistical Fermi gas model. In the ground state the ideal nucleon gas of the Fermi model is at zero temperature in a potential well of depth 31 MeV, equal to the sum of the maximum Fermi energy and the mean binding energy of the nucleons. The height of the Coulomb-field barrier was calculated to be 8 MeV for protons. The distribution of Fermi momenta was represented as a sphere with radius equal to the maximum momentum (\(P_{\max} = 22\) MeV). Since the effective radius of action of the nuclear forces for the scattering of nucleons of high energy by nucleons is small compared with the mean free path in nuclear matter, it was assumed that the incident nucleon interacts with the nucleon gas through a sequence of successive scattering acts on individual nucleons. In this connection, the asymptotic cross section for free scattering of nucleons by nucleons is used, though it is correspondingly reduced because of the Pauli principle, which excludes scattering into states with momenta lying inside the filled Fermi sphere. The nucleons of the nucleus on which scattering occurs (we shall conventionally call them recoil nucleons), in turn, act po-

in a similar manner, as a result of which an intranuclear nucleon cascade is formed, which continues to develop until the moving nucleons leave the nucleus, or until the energy of the nucleon becomes less than the height of the nuclear barrier (including the Coulomb field), and the nucleon is captured thermally by the nucleus. It is assumed that during the development of the intranuclear cascade process the nucleus remains in its ground state.

As a result of such a process, several fast nucleons are emitted, and the remaining excited nucleus evaporates in accordance with the thermodynamic theory of nuclear evaporation.

Thus, in this approximation the interaction is a cascade process of free scattering of nucleons by nucleons. The influence of the remaining nucleons appears only in the existence of a potential barrier, an initial distribution of Fermi impulses, and the Pauli principle, which forbids collisions corresponding to transition into a state already occupied by other nucleons. An estimate of the error introduced by such an approximation has shown that, at least for nucleons with energy greater than 50 MeV, the correction for simultaneous interaction with many nucleons is small.

As will be shown below, the experimental data testify to the applicability of such a simple and crude model for describing the complex process of interaction with heavy nuclei.

The calculation of the cascade process was carried out by a method widely used for describing schemes consisting of a number of successive processes ($A$, $B$, $C$, …), each of which is characterized by its own statistical distribution. In this method each statistical distribution is divided into equiprobable intervals, and then a calculation is made, beginning with a randomly chosen case from $A$, in sequence with a randomly chosen equiprobable case from $B$, then from $C$, and so on, until the final state is obtained. The statistical fluctuations in the distribution curve of the final state itself will then be the same for a certain number of cases in such a sampling calculation and the same number of experimentally observed cases of interaction.

The calculations are greatly simplified when considering the development of the cascade without taking into account the charge of the interacting nucleons. Such an approximation gives the correct general characteristics of all nucleons (protons and neutrons considered together). The ratio between the number of protons and neutrons was obtained in another way.

The possibility of replacing protons and neutrons by one general type of nucleons, characterized by an averaged cross section and density equal to the average density of nucleons in the nucleus, is due to the following circumstances. In Ag and Br nuclei there are approximately equal numbers of protons and neutrons. Nuclear forces apparently do not depend on charge; therefore one should expect that $\sigma_{n-n} \cong \sigma_{p-p}$. The effective cross sections for interactions of the type $n$-$p$ and $p$-$p$ vary with energy in approximately the same way and can be replaced by an averaged cross section shown in Fig. 1 by the dash-dotted line. In this figure the $n$-$p$ and $p$-$p$ cross sections are extrapolated to 400 MeV, and the $n$-$p$ cross section, actually anisotropic, is replaced by an equivalent isotropic cross section chosen so that it corresponds to the same energy transfer as in the actual cross section. Since it is known that the $p$-$p$ cross section is isotropic at all energies, the averaged cross section will also be isotropic. The mean free path ($\lambda$), as a function of the energy ($E$) of a nucleon moving inside the nucleus, has the form:

$$ \lambda(E)=\frac{1}{\rho \sigma(E)}. $$

where \(\rho\) is the density of nucleons in the nucleus, and \(\sigma(E)\) is the averaged cross section shown in Fig. 1.

The probability that a collision inside the nucleus has not yet occurred at a distance \(x\) may evidently be written in the form

\[ g = e^{-\frac{x}{\lambda(E)}} . \]

A further simplification of the calculations is carried out by passing from three-dimensional geometry to two-dimensional geometry. Such a transition is possible owing to the fact that all final distributions must be symmetric with respect to the direction of the incident parallel beam of nucleons. The spherical nucleus is replaced by a two-dimensional circle, whose diameter is directed along the incident beam. At the same time, the three-dimensional character of the nucleus is taken into account by assigning to the segments of the circle, formed by chords parallel to the beam, a weight proportional to the areas of the corresponding rings in sections perpendicular to the beam.

Fig. 1.

In Fig. 2, a division of a semicircle with radius equal to the nuclear radius into 10 such segments is shown. For convenience, only half of the circle is divided, since the final distribution is symmetric with respect to the direction of the beam. If it is assumed that the same number of incident nucleons corresponds to each segment, then this introduces, in accordance with the true three-dimensional case, the Fermi sphere. The three-dimensional Fermi sphere is transformed into a circle of the same radius, and an element of area of this circle is assigned a weight corresponding to the element of three-dimensional volume obtained by rotation about the direction of the beam.

Consideration of the two-dimensional picture, among other things, makes it possible to count on the fact that it will be possible to compare the theoretical

of the angular distribution with the projection of the experimental angular distribution. Of course, the two angles do not correspond exactly to one another, but since both are characterized by similar two-dimensional properties, it is natural to expect that the basic properties of both angular distributions will also be the same.

The calculation of the intranuclear cascade process is carried out in the following order. It is assumed that each of 10 intervals of the momentum circle corresponds to an equal number of incident nucleons with an energy of 400 MeV.

Fig. 2.

When entering the nucleus, a nucleon acquires an energy of 31 MeV, corresponding to the nuclear barrier. For a nucleon with energy 431 MeV the quantity \(\lambda(E)\) is calculated. Then the quantity

\[ \frac{x}{\lambda(E)} \]

is divided into 1000 equally probable intervals, and one of them is chosen at random. Knowing \(\lambda(E)\), one obtains the value \(x\) and continues the trajectory of the incident nucleon into the depth of the nucleus by this amount. If in doing so the nucleon exits beyond the limits of the nucleus, this will correspond to the case of complete transparency of the nucleus. Then again, at random, one of 1000 equally probable intervals of Fermi momenta and one of 1000 intervals for the scattering angle are chosen. From these the angle between the scattered nucleons and their energy are determined. The same procedure is applied to secondary particles until they either leave the nucleus or are captured by the nucleus if their energy falls below 35 MeV (the barrier of the Coulomb field, averaged over protons and neutrons, is equal to 4 MeV).

Fig. 2 illustrates the order of the calculation. Here circles indicate those points at which collisions were supposed to occur but were prohibited by the Pauli principle.

The calculation was performed for the interaction of 90 nucleons with Ag and Br nuclei. Of these, 30 passed through the nucleus without interacting. Such a fraction, within the limits of error, corresponds to the theoretically calculated partial transparency of heavy Ag and Br nuclei. The effective cross section for interaction with all nuclei of the emulsion was calculated theoretically by the same method and was found to be \(\sigma = 0.6\,\sigma_{\text{geometr.}}\). This value com—

coincided with the experimental data obtained when emulsions were irradiated with protons of energy 375 MeV. The mean free path for the interaction of protons in the emulsion was determined in two ways: by counting the number of stars falling on a given area of the emulsion, and by counting the number of cases of inelastic interaction (formation of a star, sudden stopping, or inelastic scattering of a proton) found when tracing along individual proton tracks.

By the first method 404 stars were investigated; by the second, 34 cases of inelastic interaction found in the path of protons (with a total length of 1820 cm) in the emulsion. In stars the tracks were classified in accordance with Table I.

Table I

After introducing a correction for scanning in the first method of analysis of cases of stopping and inelastic scattering, as well as of stars with only gray tracks, both methods led to approximately the same value of the mean free path of the proton in the emulsion, corresponding to the cross section
\(\sigma = (0.56 \pm 0.11)\sigma_{\text{geom}}\) *). Thus the experiment well confirms the results of the calculation of the transparency of nuclei based on this mechanism of intranuclear interaction. We shall show that the same is confirmed by comparing the properties of stars formed by protons and by neutrons.

Because of the symmetry of the interaction scheme under consideration, due to the approximate equality of the numbers of protons and neutrons in nuclei, and also because \(\sigma_{p-p} \simeq \sigma_{n-n}\), one should expect that a proton emitted by a nucleus as a result of bombardment of the latter by a fast neutron must be, in all respects, equivalent to a neutron emitted as a result of bombardment by a proton. On the other hand, a proton emitted from a star formed by a proton must be equivalent to a neutron knocked out of the nucleus by a fast neutron. Thus, the results of two experiments on irradiation of emulsions by neutrons and by protons can give information both about protons and about neutrons knocked out of the nucleus. It should also be expected that after the first collision the ratio between the number of fast protons and neutrons will be statistically shifted toward the particles whose charge coincides with the charge of the primary nucleon. An estimate shows that this ratio is expressed as \(3:1\); however, after 2–3 collisions it begins to approach 1. According to the calculation, the mean number of collisions in interaction with nuclei

*) The mean free path for interaction with the nuclei of emulsion D5 is 25 cm.

Ag and Br is equal to \(\sim 4.5\). It follows from this that approximately the same number of protons and neutrons must be emitted in the stars. Then, because of the indicated symmetry, one should expect the identity of stars produced by protons and neutrons of the same energy. This is in fact observed (Tables II and III). Table I compares stars produced by protons and neutrons according to the average number of tracks of different kinds (rows a and b), and Table III according to the number of fast charged particles.

Table I

Table III

Figure 3 shows the distribution of stars caused by protons according to the number of black tracks, and Fig. 4 shows the fraction of stars containing tracks of fast particles. The distribution in Fig. 3 cuts off at stars with 8 tracks. Evidently, this corresponds to the complete transfer of 400 MeV to an Ag or Br nucleus. If this energy is distributed uniformly among the particles with black tracks, then each of them corresponds on average to the transfer to the nucleus of an energy of \(\sim 50\) MeV. Such an interpretation is confirmed by the break in the distribution

stars with gray tracks in 6 rays (~300 MeV was contained in 6 black tracks, and ~100 MeV went into the formation of particles with a gray track). From Fig. 4 it is seen that, as the star magnitude increases, the fraction of

Fig. 3.

energy going increasingly into the formation of black tracks grows. In large stars, apparently, a complete redistribution of energy among them takes place.

The authors succeeded, by counting grains, in determining the energy of 26 particles with gray tracks and thus, in a number of stars, in estimating the total energy

Fig. 4.

carried away by all charged particles. It turned out that this energy is close to 200 MeV. This confirmed that, on average, the energy is distributed equally between protons and neutrons.

In Fig. 5 the distributions of stars from protons and neutrons containing tracks of fast particles are compared. Both curves are normalized to the same total number of stars. The curve for stars from protons is shifted by 1–2 tracks toward stars with a larger number of black tracks, which, apparently, is the result of the greater mean energy of the protons (375 MeV) compared with the neutrons (about 300 MeV). Indeed, in accordance with the above consideration, 1–2 tracks correspond to a difference of 50–100 MeV, and the energies of protons and neutrons differ by the same amount. The distribution of stars by the number of fast particles is a sensitive check of the mechanism of the intranuclear cascade process. Table IV shows agreement of experiment with theory in this case as well.

Fig. 5.

Table IV

A theoretical calculation makes it possible to predict correctly even the average number of black tracks, despite the fact that at low energies one might have doubted the applicability of the given interaction model and of the other approximations made in the calculation.

Fig. 6.

It turned out that, contrary to the widespread opinion that all black tracks belong to particles evaporated from the nucleus, in reality 20–30% of the black tracks belong to slow recoil nucleons. This conclusion is confirmed by analysis of the angular distribution of the tracks. We shall characterize this distribution by the angle between the direction of the beam and the projection of the track onto the plane parallel to this direction.

The angular distributions for stars formed by protons and neutrons are very close to one another for all types of tracks (Fig. 6). However

if the gray and gray-black tracks are sharply directed forward (the gray ones more strongly than the gray-black ones), then the distribution of black tracks may be regarded as the superposition of two distributions: isotropic and directed forward. The contribution of the isotropic distribution does not exceed 60–70%. However, if only the largest stars are selected, then the angular distribution of black tracks in them is already almost completely isotropic, owing to the development in them of powerful nucleon cascades in which the predominance of the original direction of the particles is lost because of the large number of collisions. We note that the abundance of black tracks of recoil nucleons means a much larger value of the mean free path in nuclear matter for these slow nucleons than follows from the cross section for free scattering. This indicates the operation of the Pauli principle, by virtue of which the path of these particles is increased.

As we see, the theory describes well the angular distribution of all kinds of tracks. Let us use it to calculate the mean number of black tracks in stars and compare this number with the experimental value. Within the limits of error, the two values agree (Table V).

Table V

Thus, the most important characteristics of nuclear interaction agree with the picture of an intranuclear cascade process, in which nucleons interact by means of collisions of the nucleon—nucleon type.

Л. Э.

CITED LITERATURE

1. G. Bernardini, E. T. Booth and S. J. Lindenbaum, Phys. Rev. 85, 826 (1952).
2. G. Bernardini, E. T. Booth, and S. J. Lindenbaum, Phys. Rev. 88, 1017 (1952).