Nuclear Transformations under Bombardment by High-Energy Particles

V. I. Goldansky

In recent years a number of reports have appeared in print on the transformations of various nuclei under the action of neutrons (with energies up to 90 MeV), deuterons (up to 190 MeV), and α-particles (up to 380 MeV), observed in experiments carried out on the 184-inch phasotron at Berkeley. At the same time, several papers have appeared in print indicating possibilities for a theoretical interpretation of the distinctive character of nuclear transformations caused by fast particles. The present article gives a brief review of the experimental data and theoretical communications relating to nuclear transformations under bombardment by high-energy particles*). All the material may conveniently be divided into two parts—according to the two types of nuclear transformations characteristic of cases of bombardment by high-energy particles.

I. Processes of Deep Spallation

It is appropriate to call by this name the processes referred to in the original articles as “spallation.” The essence of these processes is that, from the original nucleus, when it is bombarded by fast particles, a large number of some simple particles are emitted (how many and precisely which ones, in most cases, has not yet been established), and a nucleus is formed that differs very strongly from the original one in mass number and in charge. A striking example of such deep spallation is the transformation of \( \mathrm{Cu}_{29}^{63} \) into \( \mathrm{P}_{15}^{32} \) under the action of 190 MeV deuterons or 380 MeV α-particles.

Deep-spallation processes, therefore, differ essentially from nuclear transformations at low bombardment energies—

) In doing so, we dwell in greater detail on papers published after the appearance of the sixth issue of the scientific abstract collection on foreign periodical literature (Theory of Elementary Particles. Experiments with Fast Particles. State Publishing House of Foreign Literature. Moscow, 1948). See also a number of reports in UFN 38–40*.

bombarding particles, when the compound nucleus emits no more than two light particles (i.e., neutrons, protons, deuterons, or α-particles) and the final nucleus is removed from the initial one by no more than two cells of the periodic system. At the same time, deep-spallation processes differ essentially from fission processes as well, for in the case of deep spallation only one “fragment” is formed—the final nucleus; the entire difference in mass number and atomic number between the initial and final nuclei is accounted for by the many light particles emitted by the initial nucleus.

The peculiar nature of deep-spallation processes is readily explained on the basis of Serber’s¹ ideas concerning nuclear reactions at high energies. At high energies of the bombarding particles, the compound-nucleus model loses its meaning. The time of collision between nucleons is large compared with the time of collision between the incident particle and a nucleon of the nucleus. Therefore the collision between the bombarding particle and the nucleus may be regarded simply as a collision of two nucleons. Hence, incidentally, it follows that there should not be a large difference between the action on a nucleus of a neutron or proton with energy \(E\), a deuteron with energy \(2E\), or an α-particle with energy \(4E\), since the magnitude \(E\) is much greater than the binding energy and the Coulomb barrier. Serber’s calculations showed that, at nucleon energies of the order of \(100\ \mathrm{Mev}\), its range in nuclear matter is approximately \(4\cdot 10^{-13}\ \mathrm{cm}\), and the energy loss in a single collision is close to \(25\ \mathrm{Mev}\). If the nucleon with which the incident particle has collided is near the edge of the nucleus, then it is knocked out of the nucleus with an energy of \(15\text{–}20\ \mathrm{Mev}\). If, however, it is near the center of the nucleus, then it dissipates the energy acquired in the collision with the flying particle in subsequent collisions with other nucleons, forming a “boiling nucleus” that emits several particles with energies of a few \(\mathrm{Mev}\).

In individual cases still higher values of the excitation energy are also possible (\(25\ \mathrm{Mev}\) is an average value). Then, obviously, the emitted particles will have higher energy, or a larger number of particles will be emitted—the larger, the greater the excitation energy. Since the transfer of large excitation energies is of low probability, the yield of products of the deep-spallation process should decrease with increasing differences in mass numbers and atomic numbers between the initial and final nuclei.

Deep-spallation processes have been studied in detail for the following targets: As, Cu, Sb, Rh, Pb, U.

Deep spallation under bombardment of uranium will be discussed below, together with the description of the fission of this element.

In the study of deep-spallation processes, many previously unknown isotopes were discovered. Such are, for example, \(\mathrm{Fe}^{52}\), \(\mathrm{Zn}^{62}\), \(\mathrm{Ge}^{66}\), \(\mathrm{Ge}^{67}\), \(\mathrm{Ge}^{68}\), \(\mathrm{As}^{71}\), \(\mathrm{Se}^{71}\), \(\mathrm{Se}^{72}\), \(\mathrm{Se}^{73}\), and others—in all

In investigating the products of bombardment of various nuclei by high-energy particles, about sixty new isotopes were discovered[^2]. Of primary interest is the comparison of the yield of various isotopes when a given target is bombarded. Figs. 1 and 2 show the dependence of the relative intensity of the yield of various isotopes on their atomic number and mass number upon bombardment for 190 MeV deuterons of copper[^3] and arsenic[^4]. The cross section of deep spallation processes can be estimated from the fact that

Fig. 1. Yield of various isotopes and isobars upon bombardment of copper by 190 MeV deuterons.

Fig. 2. Yield of various isotopes and isobars upon bombardment of arsenic by 190 MeV deuterons.

the yield cross section of \( \mathrm{Cu}^{61} \) upon bombardment of copper (constituting approximately 24% of the total copper yield) is of the order of \(10^{-25}\ \text{cm}^2\). As is seen from the figures, with increasing differences \(Z_{\text{initial}} - Z_{\text{prod}}\) or \(A_{\text{initial}} - A_{\text{prod}}\), the yield of the corresponding nuclei, in general, falls, although there are fairly sharp deviations from this regularity. These deviations may be significantly connected with the absence of identification of short-period isotopes (with \(T_{1/2}\) less than 10–15 min), whose activity after the procedures of chemical separation was already too small. The yield of isotopes with large \(T_{1/2}\) was also measured insufficiently accurately. In the case of copper, four such isotopes were noted (\(\mathrm{Zn}^{65}\), \(\mathrm{Mn}^{54}\), \(\mathrm{Ti}^{51}\), and \(\mathrm{Cl}^{34}\)), and their yield is not taken into account in Fig. 1.

In the case of copper the yield of elements with odd \(Z\), as a rule, is higher than the yield of neighboring elements with even \(Z\).

When copper was bombarded with a beam of \(\alpha\)-particles with an energy of 380 MeV, the relative yield of the various isotopes, to within a factor of 2, remained the same as under deuteron bombardment, with the exception of the yield of \(Cl^{38}\), which increased by a factor of 6. The dependence of the yield of four products of the bombardment of copper by deuterons—Zn, Cu, Co, and Ni—on the deuteron energy was studied separately. It was found that the yields of \(Zn^{62}\) and \(Zn^{63}\) initially increase with increasing energy, reach a maximum at \(E_d \simeq 50\) MeV, and then begin to fall. At a deuteron energy of 140 MeV the zinc yield is approximately 5 times smaller than the maximum values. The dependence of the yields of Ni and Co on the deuteron energy is more complicated. For them, at small energies the yield increases, reaches a maximum, then falls, passes through a minimum, and finally increases again with energy at high deuteron energies. The cross-section values at deuteron energies approximately 5 times exceeding the threshold energies are substantially higher than would follow from the compound-nucleus theory. From these results the authors conclude that the formation of Zn, i.e. a transformation of the type \(Z \to Z + 1\), proceeds through an intermediate stage of deuteron capture (or of the proton from the deuteron) and formation of a compound nucleus, and not through proton–neutron exchange, whereas the formation of Ni and Co proceeds by the mechanism proposed by Serber, i.e. through transfer of part of the energy of the incident particle to some nucleon in the nucleus, followed by its redistribution.

Similar results were obtained by other authors \(^{6}\), who studied the dependence of the yield of Zn, Cu, Mn, Fe, and Co from copper, when the latter was bombarded with 190 MeV deuterons, on the deuteron energy. The deuterons were passed through a stack of copper plates, so that each of the plates corresponded to a definite deuteron energy. The yield of the various isotopes was compared with the yield of \(Na^{24}\) from thin aluminum foil at the same deuteron energy [the cross section of the reaction \(Al^{27}(d,\alpha p)Na^{24}\) for 190 MeV deuterons was found \(^{7}\) to be equal to \(0.048 \times 10^{-24}\ \text{cm}^{2}\)]. It was found that the yields of \(Mn^{52}\), \(Mn^{56}\), \(Fe^{51}\), and \(Co^{58}\) increased with energy, whereas the yields of \(Zn^{62}\), \(Zn^{63}\), and \(Cu^{64}\) were maximal at comparatively low energy and then decreased with increasing energy. The yields of Cu and Zn were of the same order as the yield of \(Na^{24}\); the yields of Fe and Co—elements more remote from the initial one—were approximately 10 times smaller.

When arsenic is bombarded with deuterons, among the elements formed as a result of deep spallation from \(Se_{34}\) to \(Ga_{31}\), i.e. those closest to the initial element, isotopes with a neutron deficiency noticeably predominate; among the lighter elements formed, isotopes with a deficiency and with an excess of neutrons are formed in approximately equal amounts. It was further noted that about 80% of the total yield of the deep-spallation process under

in bombarding a target with deuterons is the yield of isotopes whose mass numbers \(A_i > A_{\mathrm{As}} - 8 = 67\), which corresponds to reactions for which the required excitation energy of the original nucleus does not exceed 75 MeV.

The predominance noted in deuteron bombardment of the target (among the elements formed, close to the original one) of isotopes with a neutron deficiency was confirmed in the deuteron bombardment of antimony\(^8\) and rhodium\(^9\). In bombarding antimony (\(\mathrm{Sb}^{121}\) and \(\mathrm{Sb}^{123}\)) with 190 MeV deuterons, the formation of \(\mathrm{Te}^{118}\) \((d, 5n)\) and \(\mathrm{Te}^{119}\) \((d, 4n)\), as well as \(\mathrm{Sb}^{119}\) \((d; p, 3n)\) and other Sb isotopes whose mass numbers were not established (possibly these are isomers of already known antimony isotopes), was observed. In order to establish, from the two noted values of \(T_{1/2}\), what the half-lives of \(\mathrm{Te}^{118}\) and \(\mathrm{Te}^{119}\) are, the authors also bombarded antimony with 40 MeV deuterons, on the assumption that at this deuteron energy the yield of \(\mathrm{Te}^{119}\) relative to \(\mathrm{Te}^{118}\) would increase. Indeed, in this case the ratio of the activities of the two tellurium isotopes was found to be \(40:1\), and thus it was established that for \(\mathrm{Te}^{118}\), \(T_{1/2} = 6\) days, and for \(\mathrm{Te}^{119}\), \(T_{1/2} = 4.5\) days. In bombarding rhodium (\(\mathrm{Rh}^{103}\)) with 50 MeV deuterons, the formation of \(\mathrm{Pd}^{101}\) \((d, 4n)\) and \(\mathrm{Pd}^{100}\) \((d, 5n)\) was observed, and also, through the \(\beta\)-decay of palladium, the formation of \(\mathrm{Rh}^{101}\) and \(\mathrm{Rh}^{100}\). The yield cross sections of \(\mathrm{Pd}^{100}\), \(\mathrm{Pd}^{101}\), and \(\mathrm{Pd}^{103}\) were determined. It is interesting that the yield cross section increased with decreasing mass number of the palladium isotope. Thus, for the \((d, 2n)\)-reaction (\(\mathrm{Pd}^{103}\)) \(\sigma = 0.24 \cdot 10^{-26}\ \mathrm{cm}^2\), for the \((d, 4n)\)-reaction (\(\mathrm{Pd}^{101}\)) \(\sigma = 0.24 \cdot 10^{-24}\ \mathrm{cm}^2\), and, finally, for the \((d, 5n)\)-reaction (\(\mathrm{Pd}^{100}\)) \(\sigma = 1.28 \cdot 10^{-24}\ \mathrm{cm}^2\). From these results one may conclude that the transfer of very small excitation energies (as well as the transfer of very large energies) is a low-probability process. It is necessary, however, to point out certain contradictions between the results of deuteron bombardment of antimony and rhodium.

Processes of deep spallation in bombardment with 190 MeV deuterons were also observed for lead\(^ {10}\). Along with fission of lead, the formation of various isotopes from bismuth to gold (\(Z = 83\text{--}79\)) was discovered. The most interesting result of these experiments was the discovery of new \(\alpha\)-radioactive isotopes of bismuth, of which, judging by their half-lives, there proved to be only three (a fourth \(\alpha\)-radioactive product, owing to its very short half-life—2 min—could not be chemically identified). Comparison of the yield of bismuth isotopes with different half-lives showed that, as the mass number decreases, the lifetime of the isotopes falls.

In all the works mentioned above, the yield of various products of deep-spallation processes from a definite given target was compared. There are also data on comparison

yields of definite products in the bombardment of various targets.

Hox11), 12) subjected ten different elements to bombardment by 90 MeV neutrons and compared the yield of various isotopes. Taking as unity the yield of \(C^{11}\) in the bombardment of carbon (the cross section of this reaction was found13 to be \(0.022 \cdot 10^{-24}\ \mathrm{cm}^2\)), he obtained relative yield values for 16 different reactions and, for 6 more reactions, upper limits of the yield. The data obtained by him are given in Table 1. Hox established an interesting dependence:

Table 1

Relative yield of certain isotopes in the bombardment of various elements by 90 MeV neutrons

*) Upper limits of the cross sections are given in parentheses.

if one compares the yield of a definite element from different nuclei that differ from one another in composition by 2 protons and 2 neutrons (i.e., by an \(\alpha\)-particle), then the logarithm of the yield falls linearly with increasing atomic number of the target. Thus, the yield of \(C^{11}\) from \(C^{12}\) and \(O^{16}\), \(F^{18}\) from \(F^{19}\), \(Na^{23}\), \(Al^{27}\), \(Mg^{24}\), and \(Si^{28}\), as well as \(Na^{24}\) from \(Mg^{24}\), \(Si^{28}\), \(S^{32}\), \(Al^{27}\), and \(P^{31}\), was investigated. The slope of the straight lines representing the dependence of the logarithm of the yield on the atomic number of the target proved in all cases (with the exception of the yield of \(F^{18}\) from \(Mg^{24}\) and \(Si^{28}\)) to be the same. The ratio of the cross sections of reactions in which a given element is formed from two nuclei differing in composition by an \(\alpha\)-particle proved to be approximately equal to three, in agreement with results obtained earlier by comparing the yields of \(C^{11}\) from carbon and oxygen14 when bombarded by protons (a difference in cross sections by a factor of 2.8), as well as

also by comparing the yields of \(N^{17}\) from different elements when they are bombarded with deuterons\(^{15}\), namely:

\[ \frac{\sigma_{F^{19}}}{\sigma_{Na^{23}}}=3.2;\qquad \frac{\sigma_{Na^{23}}}{\sigma_{Al^{27}}}=3.1;\qquad \frac{\sigma_{Al^{27}}}{\sigma_{P^{31}}}=3.6;\qquad \frac{\sigma_{P^{31}}}{\sigma_{Cl^{37}}}=3.0; \]

\[ \frac{\sigma_{Cl^{37}}}{\sigma_{K^{39}}}=3.1 \quad\text{and}\quad \frac{\sigma_{Mg^{24}}}{\sigma_{Si^{29}}}=3.6,\qquad \frac{\sigma_{Si^{28}}}{\sigma_{S^{32}}}=3.3. \]

In another work\(^{16}\), the yield of \(Mn^{52}_{25}\) and \(Mn^{56}_{25}\) was investigated in the bombardment of elements from Cr \((Z=24)\) to Sr \((Z=38)\). The results obtained are presented in Fig. 3. It is evident that the yield of Mn falls sharply as the atomic number of the target increases, beginning with Ni and higher. The data for targets whose nuclei differ by 2 protons and 2 neutrons, shown in Fig. 3, cannot be described by the above simple dependence of the logarithm of the yield on the target \(Z\). On the basis of comparisons of the yield of \(Mn^{52}\) (taken as unity) from manganese and \(Cu^{64}\) from copper—a reaction whose cross section was known to the authors\(^{16}\)—the cross section corresponding to the unit in Fig. 3 was determined, and proved to be equal to \(0.03\cdot 10^{-24}\ \text{cm}^{2}\). Since, of the two isomers of \(Mn^{52}\), the authors used in their measurements only the isomer with \(T_{1/2}=5.8\) days, so that the isomer with \(T_{1/2}=21\) min was not taken into account, the quoted value of the formation cross section of \(Mn^{52}\) is therefore underestimated.

Fig. 3. Yield of \(Mn^{52}\) and \(Mn^{56}\) in the bombardment of various elements with 190 MeV deuterons.

Additional information concerning deep-spallation processes can be extracted from data on the emission of secondary particles in the bombardment of various nuclei by 90 MeV neutrons, and also from investigations of star formation in the bombardment of nuclei by fast deuterons and \(\alpha\)-particles. Thus, from measurements of \(H\rho\) and the ranges of secondary particles it was found\(^{17}\) that, in the bombardment of carbon, copper, and lead by neutrons, a fairly large number of deuterons and protons with energies up to 90 MeV is formed; moreover, the maximum of the energy-distribution function for both deuterons and protons was found at 50–60 MeV. The cross section

the formation of deuterons with energies above 25 MeV in the bombardment of carbon by neutrons was found to be of the order of \(10^{-26}\ \text{cm}^2\); the cross section for proton emission was of the order of \(1/10\) of the total \(np\)-scattering cross section (equal at 90 MeV to \(8.3 \cdot 10^{-26}\ \text{cm}^2\))\(^{18}\).

Secondary deuterons and protons had a predominantly forward directionality; moreover, the directionality was more pronounced for deuterons and for particles of higher energy than for protons and for particles of lower energy.

Fig. 4. Energy distribution of secondary deuterons in the bombardment of carbon by 90 MeV neutrons.

Fig. 5. Energy distribution of secondary protons in the bombardment of carbon by 90 MeV neutrons.

The ratio of the number of deuterons and protons emitted in the direction of the primary neutron beam was, respectively: \(2/3\) for carbon, \(1/3\) for copper, and \(1/6\) for lead. Thus, for lighter nuclei a greater predominance of deuterons was observed. The results\(^{17}\) were in poor agreement with Serber’s theory, according to which one should expect a “peak” of protons in the forward direction with an energy of about 80 MeV. The comparatively low energy of the protons contradicts the data on the preferential forward directionality (experiment showed that the number of secondary particles per unit solid angle falls by a factor of two at an angle of \(12^\circ\) to the primary neutron beam), since the observed energy of the secondary particles could have been caused only by multiple scattering in the nucleus. A check of the determination of \(H\rho\) for the secondary protons by means of a Wilson chamber showed\(^{19}\) that some of the tracks were in fact not proton tracks, but deuteron tracks. Protons with energies up to 130 MeV and deuterons with energies of 23–70 MeV were observed. Later\(^{20}\), additional verification experiments were carried out. In a Wilson chamber in a magnetic field, by \(H\rho\) and ranges, three were identified

secondary particles—protons, deuterons, and tritons, produced when a carbon target was bombarded with \(90\) MeV neutrons. In Figs. 4 and 5 the spectra are given for secondary deuterons and protons emitted in two angular intervals. Among the 386 tracks investigated, 202 corresponded to protons, 162 to deuterons, and 22 to tritons. It was found that protons with energies \(32\)—\(62\) MeV are emitted isotropically, while the number of protons with energies \(62\)—\(98\) MeV emitted at an angle \(18^\circ \pm 3^\circ\) is half as large as in the direction of the primary beam. For deuterons, the angle corresponding to a twofold attenuation of the secondary flux was found to be \(18^\circ \pm 3^\circ\) for \(E_d = 30\)—\(58\) MeV and \(11^\circ \pm 2^\circ\) for \(E_d = 58\)—\(93\) MeV. The cross sections for the emission of secondary particles are given in Table II.

Table II

It was noted, moreover, that star formation occurs on carbon and oxygen; the authors\(^{20}\) consider that the corresponding cross section for bombardment of carbon by \(90\) MeV neutrons amounts to \(1/3\) of the total carbon cross section, i.e. about \((0.20 \pm 0.05)\cdot 10^{-24}\ \text{cm}^2\). Assuming that, among the particles formed in star disintegrations, \(12 \pm 3\%\) have sufficiently high energies (\(25\)—\(30\) MeV), the final value of the cross section for the knock-out from carbon of protons with energies exceeding \(32\) MeV and deuterons with energies exceeding \(25\) MeV is equal to \(\sigma = (24 \pm 12)\cdot 10^{-27}\ \text{cm}^2\).

The rather large cross sections for the knock-out of charged particles from light nuclei cause certain difficulties in experimental work devoted, for example, to the study of \(np\)-scattering by means of “telescopes” made of proportional counters operating in coincidence. These difficulties are connected with an increase in the load on each counter and, thereby, with an increase in the background of coincidences due to particles knocked out from the walls of the counters themselves. Apparently, precisely for this reason the apparatus for measurements of \(np\)-scattering\(^{21}\) was placed behind 10-foot concrete shielding, through which the neutron beam was collimated (and, in addition, copper

*) The presence of secondary particles with energies exceeding \(90\) MeV is explained by the presence in the bombarding beam of a noticeable number of neutrons with energies greater than the nominal one\(^{1}\).

diaphragms with diameters from 7.5 to 1 cm and a thickness of 50 cm). Under these conditions the magnitude of the random-coincidence background was small and was no longer associated with the knocking out of charged particles from the counter walls, since the coincidence background decreased by a factor of 1000 when one of the counter telescopes composing the telescope was moved from the telescope axis to a neighboring position.

With the aid of a Wilson chamber and thick-layer photographic emulsions, star formation associated with processes of deep spallation was observed. 1200 stars in a photographic emulsion were investigated when it was bombarded with deuterons of energies 35 MeV, 90 MeV, 130 MeV, and 190 MeV²². The rays of the stars could correspond to \(\alpha\)-particles, fragments, deuterons with energy less than 35 MeV, or protons with energy less than 17 MeV (desensitized plates were used, which did not record deuterons and protons of higher energies). Stars from \(\alpha\)-particles with energies up to 380 MeV were also investigated²³, and in some cases emulsions were used that permitted identification of fast protons and \(\alpha\)-particles up to the energy of the incident beam. It was found that at all energies of the bombarding deuterons and \(\alpha\)-particles the average number of rays in a star is approximately three. In bombardment with deuterons, the ratio of the number of rays in the direction of the primary beam to the number of rays in the opposite direction was approximately three; in the case of \(\alpha\)-particles, six. The results obtained are interpreted by the authors²⁴ in accordance with the ideas of a boiling nucleus excited by the fast bombarding particle. A theoretical calculation gave, for 190 MeV deuterons, an average number of rays of 4.4; if a correction is introduced for the insensitivity of the emulsion to fast protons and deuterons, this number is reduced to 3.3 for 190 MeV and to 1.1 for 35 MeV. The calculation was carried out for the heavy components of the emulsion—Ag and Br. The authors explain the excess of the experimentally observed average number of rays at low deuteron energies over the theoretically calculated value by star formation on the light components of the emulsion—carbon and oxygen, which disintegrate into 3 and 4 \(\alpha\)-particles. In favor of strong star formation on light nuclei, in the authors’ opinion, is also indicated by the preferential forward direction of the rays, caused, apparently, by the proper motion of the nuclei, for the process of nuclear evaporation is isotropic. Taking into account that at low energies neutrons, not recorded by the emulsion, evaporate preferentially, the authors determine the probability of formation of a star visible in the emulsion as \(0.1\sigma_0\) (where \(\sigma_0\) is the geometrical cross section) at \(E_d = 35\) MeV and as \(0.9\sigma_0\) at \(E_d = 190\) MeV.

The absolute cross sections for star formation by \(\alpha\)-particles of various energies are given below:

It is not known, however, on precisely which elements of the emulsion the obten-

...stars were counted, and therefore the figures given are averaged ones. The magnitudes of the cross sections evidently reach the order of geometrical ones. Star formation was also observed in a Wilson chamber with 90 MeV neutrons. Of the 499 stars investigated, 267 had two rays, 159 had three, 46 had four, 25 had five, and 2 had six. Hence the mean number of rays is \(2\frac{2}{3}\). If one takes into account, however, that in two-ray stars in 66% of cases rays with energies greater than could be recorded by nuclear emulsions were observed, then from the above figures it follows that, in observations with photographic plates, the mean number of rays would have been equal to three.

Conclusions. In bombarding various elements over a wide interval of mass numbers and atomic numbers with fast neutrons, deuterons, or \(\alpha\)-particles, processes of deep spallation were observed, connected with the evaporation from the bombarded nuclei of a considerable number of light particles (mainly protons, neutrons, and deuterons, and also, to a lesser degree, tritons and \(\alpha\)-particles). As a result, nuclei of elements are formed with mass numbers and atomic numbers significantly smaller than those of the original nuclei. The character of the processes of deep spallation depends little on the nature of the bombarding particles. The farther the final nucleus is from the initial one in mass number and atomic number, the more particles must “boil off” from the nucleus in the given process; therefore, in the general case, the less probable this process is, and the smaller is the yield of the corresponding final nucleus (exceptions are observed mainly for final nuclei comparatively close to the initial ones). The cross sections of deep-spallation processes are rather large and reach the order of \(10^{-25}\)—\(10^{-24}\ \mathrm{cm}^2\), i.e., values close to geometrical cross sections.

II. FISSION PROCESSES

Alongside the processes of deep spallation, the fission of heavy nuclei under bombardment by particles of high energy has been studied in considerable detail. At present, fission of ten heavy elements under the action of fast deuterons, neutrons, or \(\alpha\)-particles has been observed (besides uranium, thorium, and plutonium, also bismuth, lead, thallium, tantalum, mercury, gold, and platinum). A characteristic sign of fission by fast particles is the disappearance of the “double-humped” spectrum of the distribution of fission fragments by masses and energies, and the appearance instead of a single peak with a mass number corresponding approximately to half the mass number of the fissioning nucleus.

Physically this is apparently connected with the circumstance that, at large excitation energies of the fissioning nucleus, the “reaction coordinate” is blurred—the decay can occur not only along the line of the saddle of the energy relief, but also much higher

on both sides of this line. Thus, there are no grounds for obtaining a “two-humped” distribution curve of fragments by mass or energy, associated, as is sometimes assumed, with the necessity of passing through a saddle point. A characteristic example of a transitional spectrum from the case of fission by slow neutrons to the case of fission by fast particles is the mass spectrum of thorium fission fragments \(^{25}\) when bombarded by 32-MeV \(\alpha\)-particles, shown in Fig. 6, together with the mass spectrum of \( \mathrm{U}^{235} \) fission fragments under thermal neutrons.

Fig. 6. Mass spectrum of thorium fission fragments under 32-MeV \(\alpha\)-particles.

Recently a number of attempts have been made to find empirical regularities of fission under slow and fast particles.

Thus, for example, the suggestion was made \(^{26}\) that the mass spectrum of fragments of \( \mathrm{U}^{235} \) and \( \mathrm{Pu}^{239} \) is characterized by the fact that, for each heavy fragment, the length of the \(\beta\)-decay chain up to the formation of a stable nucleus is the same as that of the light fragment corresponding to the given heavy fragment. Proceeding from the fact of the excess of neutrons in the primary fission fragments of \( \mathrm{U}^{235} \), which are therefore inclined to \(\beta\)-decay, Kingdon \(^{27}\), using the quantity

\[ \frac{A - Z}{Z} \]

as a measure of neutron affinity, constructs the spectrum, shown in Fig. 7, of the distribution of fragments by atomic numbers, on the assumption that each pair of (light—heavy) fragments is characterized by the maximum value of the sum

\[ \frac{A_1 - Z_1}{Z_1} + \frac{A_2 - Z_2}{Z_2}, \]

where \(A_1 + A_2 = 236\), \(Z_1 + Z_2 = 92\). To explain the “single-humped” spectrum in fission

Fig. 7. Calculated spectrum of atomic numbers of \( \mathrm{U}^{235} \) fission fragments under slow neutrons.

on fast particles, it was suggested\(^ {28}\) that in this case, before fission occurs, there is no time for a redistribution of charges between the fragments, and therefore in each pair of (light–heavy) fragments

\[ \frac{A_1-Z_1}{Z_1}=\frac{A_2-Z_2}{Z_2}=\frac{A_0-Z_0}{Z_0}, \]

where the subscript 0 characterizes \(A\) and \(Z\) of the initial nucleus. The spectrum of fission fragments produced by fast particles has been studied in especially great detail for bismuth\(^ {28,29,30,31}\) and uranium\(^ {32,33,34}\). In an investigation of the ionization produced by fragments from the fission of bismuth\(^ {31}\) by 90-MeV neutrons, only a single peak was found in the energy distribution; the mean energy of the fragments was determined to be 71 MeV, and the most probable energy to be 74 MeV.

Heckerman and Perlman carried out a very detailed chemical investigation\(^ {28}\) of the fission fragments of bismuth bombarded by 190-MeV deuterons. In all, they chemically isolated 27 different elements, and measured the activity of 48 isotopes with atomic numbers from 20 to 63 and mass numbers from 45 to 149. Figure 8 gives the mass spectrum of the fragments that they obtained and, for comparison, the spectrum of fragments from \( \mathrm{U}^{235} \).

In the figure: 1 — Bi; 2 — \( \mathrm{U}^{235} \). Vertical axis: Yield (\( \sigma, \%\)). Horizontal axis: Mass number of fragments.

Fig. 8. Mass spectrum of fission fragments of bismuth bombarded by 190-MeV deuterons.

Work with bismuth fission fragments produced by fast particles involved a number of difficulties. Indeed, in the fission of uranium by slow neutrons the primary fragments give rise to a long chain of \(\beta\)-decays, and as one approaches the stable isobar (i.e., as one shifts to the right in the periodic system), the half-lives, as a rule, increase. It is therefore sufficient to investigate one or two isobars with long half-lives (for convenience of measurement) in order to determine the total yield of a fragment with a given mass number. A different picture is observed in fission by fast particles. From the relation written above,

\[ \frac{A-Z}{Z}=\mathrm{const}, \]

it is easy to calculate what the most probable value of \(Z\) is for a given \(A\), and it turns out that in a number of cases the most pro-

the formation of stable isotopes or short-lived electron- or positron-emitters transforming directly into a stable isotope was probable. In such cases the authors investigated isobars with larger or smaller numbers than the expected value of \(Z\). The yield in this case proved to be very small. Sometimes the calculated value of \(Z\) proved to be fractional, i.e., corresponding to the two most probable fragments. In these cases as well, the agreement between the expected and observed yields was violated.

Nevertheless, thanks to the investigation of a large number of different fragments, the authors obtained their mass spectrum, which, under the assumption

\[ \frac{A-Z}{Z} = \text{const} \]

corresponds to the value

\[ \frac{A-Z}{Z} \simeq 1.37. \]

The authors believe that the fission process occurs in three stages—the formation of a compound nucleus, the emission of neutrons by the compound nucleus, and the fission of the residual nucleus—and, specifically, adopt the following mechanism for the fission of bismuth:

\[ {}^{209}_{83}\mathrm{Bi}(d,12n){}^{199}_{84}\mathrm{Po}\to \text{fragments}. \]

It is obvious that, if this mechanism corresponds to reality, then such fission can occur only at sufficiently high excitation energies, sufficient for twelve neutrons to evaporate from the “boiling” nucleus. The authors cite data from unpublished calculations, according to which the fraction of such collisions of 190-MeV deuterons with heavy nuclei, in which an excitation energy of less than 100 MeV is transferred to the nucleus, amounts to only 25%. For the remaining 75% of collisions the distribution function of the excitation energy has a broad maximum around 150 MeV. The geometrical cross section of such collisions is approximately 2 barns. Meanwhile, the experimentally determined fission cross section is only 0.2 barn. The authors therefore believe that 90% of the geometrical cross section is accounted for by other nuclear reactions, including deep spallation processes, in which charged particles or a smaller number (but faster) neutrons are emitted from the nucleus without subsequent fission. The yield of products of deep spallation processes was not measured by the authors.

As will be seen from the subsequent discussion, in the fission of uranium by fast particles a different picture is observed—the fission cross section constitutes a substantial part of the geometrical cross section and is not inferior to the cross section of deep spallation processes. Proceeding from the liquid-drop model of the nucleus, the authors qualitatively explain the difference between the fission of bismuth and uranium by using the parameter \(\frac{Z^2}{A}\) as a criterion of a nucleus’s capacity for fission (with increasing \(\frac{Z^2}{A}\), for example within the limits of one atomic number, with a decrease in the number of neutrons the capacity for fission increases). If the compound nucleus \({}^{236}_{92}\mathrm{U}\)

\(\left(\dfrac{Z^2}{A}=35.5\right)\) is readily fissionable, then, according to this point of view, the compound nucleus \(\mathrm{Po}^{211}_{84}\) \(\left(\dfrac{Z^2}{A}=33.44\right)\) must, in order to acquire fissionability, first emit about 12 neutrons \(\left(\text{for } \mathrm{Po}^{199}_{84}\ \dfrac{Z^2}{A}=35.46\right)\).

In support of this point of view the authors cite comparative data from experiments on the fission of lead enriched in \(\mathrm{Pb}^{204}\) to 27.3% \(\left(\text{for this isotope } \dfrac{Z^2}{A}=32.96,\ \text{and for the compound nucleus } \mathrm{Bi}^{206}_{83},\ \dfrac{Z^2}{A}=33.44\right)\) and \(\mathrm{Pb}^{208}\) to 90.3% \(\left(\text{for this isotope } \dfrac{Z^2}{A}=32.33,\ \text{and for the compound nucleus } \mathrm{Bi}^{210}_{83},\ \dfrac{Z^2}{A}=32.80\right)\). Obviously, since \(\dfrac{Z^2}{A}\) for \(\mathrm{Pb}^{204}\) is greater than for \(\mathrm{Pb}^{208}\), fission of the first isotope requires the preliminary emission of a smaller number of neutrons, and therefore the fission cross section of \(\mathrm{Pb}^{204}\) should be larger. The difference between the fission cross sections of \(\mathrm{Pb}^{204}\) and \(\mathrm{Pb}^{208}\) should be smoothed out with increasing deuteron energy. Precisely this regularity was observed in the experiment (Fig. 9). It should be pointed out, however, that according to other data \(^{35}\) the fissionability of \(\mathrm{Pb}^{206}\) is 14 times smaller than that of \(\mathrm{Pb}^{207}\). The authors also investigated how the mass spectrum of bismuth fission fragments changes (from the yields of three fragments—\(\mathrm{Cu}^{67}\), \(\mathrm{Mo}^{99}\), and \(\mathrm{Ba}^{133}\)) in going from 190 MeV deuterons to 100 MeV deuterons and 380 MeV \(\alpha\)-particles, and established that the total yield increases with energy, while the mass spectrum broadens without a shift of the maximum. At \(A<110\), bismuth fission fragments on fast particles are mainly electron emitters, as are [[unclear: word continues on next page]]—

Fig. 9. Fission excitation function of lead with different isotopic compositions under 190 MeV deuterons.

the enormous majority of fission fragments on slow neutrons, at \(A = 110—125\)—mainly stable isotopes, and at \(A > 125\)—mainly positron emitters.

We shall not dwell here on the analysis of each of the fragments investigated or on the description of the chemical method of the work.

The mass spectrum of fragments from uranium fission by fast \((380\ \text{MeV})\) \(\alpha\)-particles, obtained by O’Connor and Seaborg \(^{33}\), is shown in Fig. 10.

Unlike bismuth, here the ascending branch in the region of large mass numbers, corresponding to processes of deep spallation, was also studied. It is evident from the figure that the dominant process is fission. The fission cross section of uranium by \(380\ \text{MeV}\) \(\alpha\)-particles was found to be close to 2 barns, i.e. close to the geometrical cross section and 10 times greater than the fission cross section of bismuth by \(190\ \text{MeV}\) deuterons. Fig. 10 gives the names of the ten elements studied by the authors. The latter, however, indicate that in the reaction a large number of \(\alpha\)-radioactive elements with \(A \geq 210\) were formed which are not indicated in the figure. The maximum of the mass spectrum of fragments lies at

\[ A < \frac{A_U + A_\alpha}{2} = 121, \]

namely at 115—118. On this basis the authors consider that before fission the compound nucleus emits several neutrons, similarly to what was discussed above for bismuth. Crosses in Fig. 10 denote later data \(^{34}\) on the yields of fragments of the platinum group (Ru, Pd, Os, Ir, Pt). A special additional investigation of these fragments was prompted by the circumstance that they are located in the most interesting parts of the mass spectrum: the light elements—near the maximum, the heavy ones—at the beginning of the branch transitional from fission processes to processes of deep spallation.

Fig. 10. Mass spectrum of fragments from uranium fission by \(380\ \text{MeV}\) \(\alpha\)-particles.

The insufficient definiteness of the investigations of uranium fission by fast particles is evident at least from the fact that one and the same author—Sborg—for a year before the publication of the work in which one broad maximum of the mass spectrum of fragments was noted,^33 still reported the presence of a “double-humped” spectrum,^32 although with a shallower depression than in fission by slow neutrons.

Let us now dwell briefly on the available data on the fission of other elements. For lead, thallium, platinum, and tantalum, the presence of a mass spectrum of fragments with one broad maximum was also established,^30 with different bombarding particles being used, namely: 380 MeV α-particles (Pb, Tl, Pt, Ta), 190 MeV deuterons (Pb, Tl), and 90 MeV neutrons (Pb). It turned out that the relative yield of light and heavy isotopes of one and the same element decreases as the energy of the bombarding particles is decreased. In this the authors see new confirmation of the fact that neutrons are emitted before fission—emission that is the stronger, the higher the energy of the bombarding particle.

Fig. 11. Excitation function of fission of various nuclei by fast neutrons (the unit corresponds to σ = 0.019 barn)

In addition, the dependence of the fission yield of six elements on the neutron energy^35 in the interval 25–90 MeV was investigated. The neutron energy was varied by moving, along the radius in a magnetic field, the lead target on which, by the “stripping” mechanism, neutrons were obtained from deuterons. Thorium was used as a monitor at all energies of the neutron beam—the pulses of fragment ionization chambers operating on thorium and on the element under investigation were compared. The results of the experiments are given in Fig. 11.

We shall not touch here on works devoted to fission and “explosions” of heavy nuclei under the action of cosmic radiation, although in individual cases it was established that the observed fissions occurred on fast neutrons, and even the mean number of secondary neutrons per fission event was determined (see, for example,^36).

In conclusion, let us briefly dwell on the theoretical work of Goldberger\(^{37}\), devoted to the interaction of fast neutrons with heavy nuclei.

The author proceeds from the picture of an initial individual nucleon–nucleon collision, in which a comparatively small fraction of the energy of the incident particle is transferred, and of the subsequent redistribution of energy inside the nucleus as a result of collisions of the excited nucleon with other nucleons of the given nucleus. On the basis of these ideas, the scattering of nucleons by heavy nuclei and the mean free path of nucleons in nuclear matter are calculated. The author then considers in detail the questions of energy transfer in collisions of neutrons with heavy nuclei, and derives the number and the energy distribution of the particles emitted by the nucleus in a collision with a fast neutron. The nucleus is treated as a mixture of two noninteracting Fermi gases—neutron and proton—where the depth of the common potential well is taken to be \(26\) MeV, and the depth of the highest of the filled states to be \(8\) MeV. The maximum Fermi energy of neutrons or protons is

\[ E_F=\frac{\hbar^2}{2M}\left(\frac{3\pi^2 N}{V}\right)^{2/3}, \]

where \(N\) is the number of neutrons or protons, \(V\) is the nuclear volume, and \(M\) is the nucleon mass. For simplicity the author assumes that in the nucleus there is only one kind of particle, defining \(N\) as \(N=N_p+\frac{1}{4}N_n\) (i.e., in essence, setting \(\sigma_{nn}=\frac{1}{4}\sigma_{np}\)). In this, the author starts from the fact that the interaction of neutrons with protons is determined half by exchange forces and half by ordinary forces. The radius of the Fermi sphere is then equal to

\[ P=\hbar\left(\frac{3\pi^2N}{V}\right)^{1/3}, \]

and the most probable change of momentum in a nucleon–nucleon collision is of order \(\frac{\hbar}{r_0}\), where \(r_0\) is the range of the nuclear forces. Since \(\frac{\hbar}{r_0P}\) is of order \(\frac{1}{2}\), an appreciable fraction of collisions is forbidden by the Pauli principle and, thus, the cross section decreases while the mean free path increases in comparison with the case when the exclusions are not taken into account. As a result of calculations, on which we shall not dwell here, the author obtains

\[ \frac{\bar{\sigma}}{N\sigma_T}=0.78 \]

(if the radius of the sphere of momenta is \(18\) MeV, and the well depth is \(26\) MeV). Taking \(\sigma_T=\sigma_{np\,90+26\,\mathrm{MeV}}=0.083\,\frac{90}{116}=0.064\) barn and \(N=113\) (for \(\mathrm{Pb}_{82}^{206}\)),

the author obtains the cross section of the intranuclear interaction (i.e., the geometrical cross section) \(\bar{\sigma}=5.63\) barns, whence, for \(R_{\mathrm{Pb}}=9.05\cdot10^{-13}\ \mathrm{cm}\), the mean free path in nuclear matter is found to be

\[ l=\frac{4}{3}\cdot\frac{\pi R^{3}}{\sigma}=5.52\cdot10^{-13}\ \mathrm{cm} \]

(in the case of neglecting the Pauli principle, \(l=4.3\cdot10^{-13}\ \mathrm{cm}\)). More exact calculations, taking into account the anisotropy of scattering, lead to a third value: \(l=6.20\cdot10^{-13}\ \mathrm{cm}\). On the basis of the obtained value of the mean free path the author calculates the energy and angular distribution of nucleons scattered by lead. In Fig. 12 the energy distribution is given at three different angles (in the laboratory system) for an initial neutron energy of \(90\ \mathrm{MeV}\). The break in the curves at the maximum appears because the prohibitions are taken into account. In Fig. 13 the angular distribution (in the laboratory system) of scattered neutrons is given, calculated only for one collision; in Fig. 14—the angular distribution with allowance for multiple scattering.

Fig. 12. Energy distribution of particles scattered by a heavy nucleus at different angles.

Fig. 13. Angular distribution of particles scattered by a heavy nucleus in one collision.

Fig. 14. Angular distribution of multiple scattering of particles by a heavy nucleus.

It is clear from Figs. 13 and 14 that the “peak” of the scattered nucleons is concentrated in a very small angle, and allowance for multiple

scattering changes the result little, since even in heavy nuclei cases of simple collisions strongly predominate. According to the author’s calculations, 15% of fast neutrons pass through the nucleus without collisions. In 85% of cases nucleon–nucleon collisions are observed in the nucleus, as a result of which the nucleus is excited and emission of particles from the nucleus may occur. Graphical calculations were carried out for the energy distribution of particles

Fig. 15. Distribution of residual heavy nuclei by excitation energy upon bombardment with 86.6 MeV neutrons.

Fig. 16. Energy distribution of particles emitted at all angles immediately after bombardment of a heavy nucleus with 86.6 MeV neutrons.

emitted from the nucleus at all angles immediately after the collision, and of the distribution of the excitation energy of the residual nucleus, the neutron energy being taken equal to 86.6 MeV (the depth of the potential well, the radius of the sphere of momenta, and the radius of the nucleus are indicated above). The results of these calculations are given in Table III and in Figs. 15 and 16.

Table III

The distribution of the excitation energy of the residual nucleus is presented in Fig. 15. The most probable value of the excitation energy corresponds to approximately 16 MeV; the mean value is $\simeq 42.5$ MeV. This energy is distributed among the nucleons, and the subsequent processes are described by the model of a “boiling” nucleus. Goldberger thus subdivides the particles emitted by the nucleus, when it is bombarded by fast neutrons, into those emitted “immediately” (within $10^{-22}$ sec.) after the collision and those evaporated by the “boiling” nucleus as a result of the distribution of the excitation energy. Experimentally, of course, such a subdivision is impossible; however, taking it into account may help in considering fission processes that require the prior emission of neutrons by the nucleus. The energy distribution of the particles emitted “immediately” is presented in Fig. 16. The maximum in the region of high energies corresponds, in the author’s opinion, to the large number of particles that have undergone only one collision in the nucleus.

Conclusions. When bombarded by fast neutrons, deuterons, and $\alpha$-particles, a rather large number of heavy nuclei undergo fission. The spectrum of fission fragments by mass and energy, in contrast to the case of fission by slow neutrons, has only one broad maximum, corresponding to a mass number somewhat smaller than the sum of the mass numbers of the initial nucleus and the bombarding particle. The latter circumstance is connected with the fact that before fission the heavy nucleus emits a fairly large number of neutrons (for example, twelve). The fission cross section of nuclei that also undergo fission by slow particles is close, at high energies of the bombarding particles, to the geometrical cross section and is not inferior to the cross section of deep spallation processes. In the case of nuclei that undergo fission only under the action of fast particles (the fission threshold of which is apparently connected with the necessity of neutron emission before fission), the fission cross section constitutes only a small part of the geometrical cross section and is inferior to the cross section of other nuclear processes, in particular deep spallation processes. In the latter case the fission cross section increases rapidly with energy, reflecting the dependence of the probability of multiple neutron emission on the energy of the bombarding particle.

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