SCATTERING AND ABSORPTION OF HIGH-ENERGY NUCLEONS

V. I. Goldanskii, A. L. Lyubimov, and B. V. Medvedev

CONTENTS

I. Production of high-energy neutrons. Their yield, angular and energy distribution . . . 532

II. Principal methods of registering high-energy nucleons . . . 550
    a) Reactions \( \mathrm{C}^{12} + X \to \mathrm{C}^{11} + X + n \) . . . 550
    b) Fission of nuclei of heavy elements . . . 553
    c) \(np\)-scattering with registration of recoil protons . . . 556
    d) Registration of high-energy charged particles based on the Cherenkov effect . . . 557

III. \(np\)-, \(pp\)-, \(nd\)- and \(pd\)-scattering . . . 558
    a) Review of experimental data . . . 558
    b) Theoretical works devoted to nucleon–nucleon scattering at high energies . . . 569

Cited literature—for Sections I–III . . . 583

IV. Total nuclear cross sections for high-energy neutrons

V. Elastic scattering of high-energy nucleons

VI. Inelastic collisions of high-energy neutrons

VII. Interaction of high-energy nucleons with nuclei (theoretical concepts)

VIII. Interaction with nuclei and absorption of nucleons that are components of cosmic rays

Cited literature—for Sections IV–VIII

This article gives a review of the literature data that have appeared in print over the last several years and are devoted to the scattering and absorption of nuclear particles artificially accelerated to energies of several hundred MeV, as well as of particles that are components of cosmic rays. These investigations are of fundamental importance for the study of the interaction of nucleons and, ultimately, for elucidating the nature of nuclear forces. Especially important from this point of view is the study of nucleon–nucleon scattering at high energies, which has already led to a whole series of interesting and unexpected results that have not yet received a sufficiently satisfactory theoretical explanation. Experiments with artificially accelerated high-energy particles for the first time make it possible to connect

data obtained under laboratory conditions with the results of cosmic-ray investigations are also included in this review. The main content of the review consists of questions concerning the production and registration of high-energy neutrons and the measurement of effective cross sections for the interaction of ultrafast nucleons with nuclei. The principal theoretical propositions proposed for explaining the experimental data are also presented.

1. PRODUCTION OF HIGH-ENERGY NEUTRONS, THEIR YIELD, ANGULAR AND ENERGY DISTRIBUTION

When various targets are bombarded by deuterons or protons accelerated to energies of the order of 100 MeV, high-energy neutrons are formed that are directed predominantly forward. In the case of bombardment by deuterons, the source of formation of such neutrons is chiefly the “stripping” reaction, in which one of the two nucleons constituting the deuteron, grazing the edge of the target nucleus, is captured, while the other flies past. In this way beams of protons and neutrons of equal intensity arise; however, the protons are bent by the magnetic field of the phasotron into the interior of the apparatus, while the neutrons continue to move forward, being distributed in a definite manner over angles near the direction of the tangent to the orbit of the deuterons at the point where the target is located. Theoretical consideration of the stripping reaction\(^{1,2}\) has led to definite assumptions concerning the stripping cross section, as well as the angular and energy distribution of the neutrons obtained.

The stripping cross section is

\[ \sigma_c=\frac{\pi}{2}RR_d, \]

where \(R\) is the radius of the target nucleus, and

\[ R_d=\frac{\hbar}{2\sqrt{M\varepsilon}}=2.1\cdot 10^{-13}\ \text{cm} \]

is the radius of the deuteron (\(\varepsilon=2.18\) MeV is the binding energy of the deuteron). For \(R=1.5\cdot 10^{-13}A^{1/3}\) cm, evidently,

\[ \sigma_c=5\cdot 10^{-26}A^{1/3}\ \text{cm}^2, \]

so that the stripping cross sections vary from the lightest to the heaviest targets rather weakly, by only a factor of 3.

Another source of formation of ultrafast neutrons when different nuclei are bombarded by high-energy deuterons is the reaction of splitting of deuterons by the Coulomb field of the nucleus. Theoretical consideration of this process has led to the conclusion that its cross section for deuterons with an energy of 200 MeV is \(^{2,3} 2\cdot 10^{-29}Z^2\ \text{cm}^2\), and for the heaviest nuclei about \(1.35\cdot 10^{-29}Z^2\ \text{cm}^2\). Thus, even for the heaviest nuclei, the cross section of this process does not exceed 35% of the cross section of the stripping reaction; for light nuclei the fraction of deuteron splitting by the Coulomb field of the nucleus in the formation of high-energy neutrons is altogether small.

The calculation of the angular and energy distribution of stripping neutrons was carried out\(^{1}\) under two limiting assumptions: \(R\ll R_d\) (transparent model of the nucleus) and \(R\gg R_d\) (opaque model of the nucleus). For

…estimates of the applicability of the results obtained, it is essential that both methods of calculation led to close results. The half-width of the neutron beam (i.e., the width between the points with intensities equal to one half of the maximum) is equal to \(1.533\,\theta_0\) in calculations for the transparent model and \(1.601\,\theta_0\) for the opaque model, where

\[ \theta_0=\sqrt{\frac{\varepsilon_d}{E_d}}\left[1-\frac{E_d}{8Mc^2}\right], \]

with \(\varepsilon_d=2.18\ \text{MeV}\) the deuteron binding energy, \(E_d\) its kinetic energy, and \(M\) the mass.

If one also takes into account the influence of Coulomb repulsion and multiple scattering of the nucleons in the nucleus (the first circumstance introduces a proportionality of the half-width to \(Z^2/R^2\), or \(Z^2/A^{2/3}\); the second circumstance, a proportionality to \(\rho/A\), where \(\rho\) is the target density), larger values of the beam half-width are obtained, and the heavier the target nucleus, the larger they are. At the same time, for lighter nuclei the Coulomb field has a noticeably stronger influence on the increase of the half-width than multiple scattering; in the case of heavy nuclei, however, these two effects are practically of equal strength (the action of the Coulomb field accounts for 90% in the case of Be and 60% in the case of U of the total increase of the half-width as compared with the simple calculation). Figure 1 gives the theoretical angular distribution of neutrons from the breakup reaction for Be and U targets. Figure 2 gives the dependence of the neutron-beam half-width on the atomic number of the target for the transparent and opaque nuclear models. At \(E_d=190\ \text{MeV}\) the half-width of the neutron beam changes from \(0.16\)–\(0.17\) radian (Be) to \(0.21\)–\(0.22\) radian (U). The calculation of the angular distribution of neutrons was also carried out for the case of deuteron breakup in the Coulomb field of the target nuclei. The results of such a calculation for a uranium target and 185-MeV

Fig. 1. Angular distribution of neutrons from the breakup reaction on Be and U targets for deuterons with energy \(190\ \text{MeV}\) (theory and experiment); dashed curves are for the transparent calculated model, solid curves for the opaque calculated model.

Fig. 2. Dependence of the half-width of the angular distribution of knock-on neutrons (at \(E_d = 190\) MeV) on the atomic number of the target (broken lines—the prediction of the theory, \(A\)—opaque model, \(B\)—transparent model; circles—experimental data).

Fig. 3. Theoretical angular distribution of neutrons from the stripping of deuterons with an energy of 185 MeV in the Coulomb field of uranium nuclei.

deuterons are given in Fig. 3. With such a mechanism of neutron formation, a sharp forward directionality of them is observed.

The energy distribution of the neutrons and protons produced in stripping must, according to the theory, be symmetric with respect to \(E_d/2\). In doing so, one must, of course, take into account that, in passing through the target, the deuterons lose part of their energy to ionization.

When \(200\)-MeV deuterons pass through a Be layer \(1.25\ \text{cm}\) thick, the deuteron energy falls to \(\sim 175\ \text{MeV}\), so that the mean energy of the neutrons thereby produced is \(90\)—\(95\ \text{MeV}\). Fig. 4 gives the energy distribution of stripping neutrons

Fig. 4. Theoretical spectrum of stripping neutrons at \(E_d = 190\ \text{MeV}\).

from \(190\)-MeV deuterons for the transparent and opaque models. In the first case the distribution function of neutrons with respect to energy has the form

\[ N(E)= \frac{(\varepsilon_d E_d)^{1/2}} {\left[\left(E-\frac{1}{2}E_d\right)^2+\varepsilon_d E_d\right]}, \]

in the second case

\[ N(E)= \frac{\varepsilon_d E_d} {\pi\left[\left(E-\frac{1}{2}E_d\right)^2+\varepsilon_d E_d\right]^{3/2}}. \]

The half-width of the beam in energy is obtained from this, equal to: for the transparent nuclear model—\(2\sqrt{\varepsilon_d E_d}\), for the opaque

model—\(1.533\sqrt{\varepsilon_d E_d}\), i.e. (for \(E_d=190\) MeV) 41 MeV and 31 MeV, respectively. Thus, the energy of the main part of the stripping neutrons is close to \(E_d/2\). Both formulas are, of course, inapplicable for \(E>E_d\).

When neutrons are formed as a result of the splitting of deuterons in the Coulomb field of nuclei, theory\(^3\) predicts an energy spectrum of another form—with two maxima, located at \(E_n=75\) MeV and 125 MeV, if the deuteron energy is 200 MeV. Fig. 5 shows such a theoretical spectrum for the splitting of deuterons with an energy of 185 MeV in the field of uranium nuclei.

Fig. 5. Theoretical spectrum of neutrons from the splitting of deuterons of energy 185 MeV in the Coulomb field of uranium nuclei.

Let us now turn to an experimental verification of the theoretical data set forth above.

In the experiment, the fluxes of neutrons\(^4\) and protons\(^5\) produced when a beryllium target 1.27 cm thick was bombarded by deuterons with an energy of 190 MeV were measured. To measure the neutron flux, a method was used based on counting recoil protons from a paraffin scatterer with the aid of a telescope of proportional counters. By this method it was established that, at the center of the neutron beam, the total flux of neutrons with energy greater than 66 MeV at a distance of 15.9 m from the target was \(10^8\) neutrons/cm\(^2\) sec.

Starting from the theoretical angular distribution and the indicated value of the flux, the authors\(^4\) determined the total yield of neutrons from the target to be \(2\cdot10^{11}\) neutrons/sec, which corresponds to the formation of approximately one neutron per 30 deuterons, for the deuteron current was about 1 μA.

To determine the proton yield, the charge of the protons admitted into a special lead chamber was integrated on a lead collector insulated from the walls of the chamber. In this way it was established that, for a deuteron current of 1 μA, the proton current was \(1/40\) μA, i.e. the yield of the stripping reaction is equal to 2.5%. The stripping-reaction cross section calculated from the indicated yield values (for a target thickness of 1.27 cm) proves to be 50–100% higher than that predicted by theory. Partly this discrepancy may be attributed to multiple passage of deuterons through the target.

Recently, relative yields of high-energy neutrons were measured for deuterons with an energy of 190 MeV on nuclei of seven elements from beryllium to uranium^6. The measurements were made at the center of the neutron beam by means of bismuth fission chambers. Calculations of the total neutron yield were carried out on the basis of the theoretical angular distribution for the stripping reaction. Table 1 gives the relative values of the intensity of the forward-directed neutron flux and of the total neutron yield according to^6. The neutron yield from a carbon target is taken as unity.

Table 1

Target	Be	C	Al	Cu	Ag	Pb	U
Relative yield of neutrons forward, at 0°	0.93	1.00	1.34	1.44	1.88	2.09	2.77
Total relative yield of high-energy neutrons . . .	0.93	1.00	1.37	1.58	2.19	3.65	3.86

Comparison of the neutron yield from different nuclei shows that for heavy nuclei the yield is approximately \(1/3\) greater than would follow if only the stripping reaction were taken into account. The experimental results agree satisfactorily with theory if the addition of the process of deuteron scattering in the Coulomb field of the nuclei is taken into account.

Along with the yield of neutrons and protons, experiments investigated the angular distribution of neutrons^7 and the energy distribution of protons^8 formed in the interaction of high-energy deuterons with various nuclei.

Carbon plates activated by the reactions \(C^{12}(n,2n)C^{11}\) and \(C^{12}(p,pn)C^{11}\) served as detectors of neutrons and protons. We shall dwell on these reactions in more detail later.

As targets in the study of the angular distribution of stripping neutrons, samples of Be, Al, Cu, Mo, Sn, Ta, Pb, and U of thickness 1.6 mm were used. The experimental results are presented together with the theoretical curves in Figs. 1 and 2.

The half-width of the neutron beam from different targets is approximately equal to \(\theta = 0.155 + 0.0006 Z\). The experimental data agree rather well with the theoretical ones; moreover, in the case of targets with small \(Z\), better agreement is observed for the “transparent model,” and in the case of large \(Z\), for the “opaque model.”

In experiments to study the spectrum of protons from the stripping reaction (a copper target 19 mm thick),\(^{8}\) the protons were turned by the magnetic field of the phasotron inside the chamber, thereby being separated by energy, and fell on stacks of carbon detectors; from the activation of these detectors, after subtraction of the neutron background, it was possible to determine the relative intensity of the proton beam at different energies. A diagram of these experiments is given in Fig. 6, and the data obtained are presented in Table II.

Table II

	57.2	64.1	70.5	78.7	87.0	95.4
Proton energy (MeV)	57.2	64.1	70.5	78.7	87.0	95.4
Relative intensity	0.212	0.282	0.595	0.751	0.955	1.00

	104	113	123	133	151	165	188
Proton energy (MeV)	104	113	123	133	151	165	188
Relative intensity	0.683	0.443	0.230	0.157	0.105	0.037	0.023

In this case, too, theory and experiment agree satisfactorily, but the experimental data are not accurate enough to make a choice between the transparent and opaque nuclear models.

When various nuclei are bombarded with high-energy protons, the source of the formation of superfast neutrons is the charge-exchange process, manifested in the fact that neutrons are emitted from the target with energies close to the energy of the protons and in directions close to the direction of the protons.

Let us first consider the essence of this process as applied to free np scattering. At low energies (up to 20–30 MeV), when the wavelength of the neutron or proton in the center-of-mass system \(\lambda \gg r_0\), where \(r_0\) is the radius of action of the nuclear forces (\(\sim 10^{-13}\) cm), the only collisions effective for scattering are those for which the orbital angular momentum \(l = 0\), since the impact parameter \(b = l\lambda\) must not exceed the value \(r_0\).

Therefore, at such energies, when \(\lambda \gg r_0\), only \(S\)-scattering neutron—proton is observed, isotropic in the center-of-mass system (c.m.s.). In this case, obviously, consideration of the charge-exchange process has no physical meaning, since at every given angle there are equal fluxes of neutrons and protons with the same energy.

However, at high energies, when the condition \(\lambda \gg r_0\) ceases to be fulfilled, terms with \(l \ne 0\) begin to play a role in the scattering; the scattering is no longer isotropic in the c.m.s. In this case, in scattering, for example, of neutrons on free protons, the angular distribution of both particles depends substantially

whether the forces acting between them are ordinary or exchange forces.

By ordinary forces here one also means those under the action of which the neutron and proton exchange spins (this effect appears only in the polarization in \(np\)-scattering), while by exchange forces one means those under the action of which an exchange of charges occurs between the proton and neutron, or an exchange of both charges and spins.

When ordinary forces act, the bombarding neutrons (or protons) must be scattered predominantly forward, while the recoil protons (or neutrons)—predominantly backward (in the center-of-mass system), at least in all cases in which the Born approximation is applicable (i.e., at sufficiently high energies for all potentials for which \(\lim_{r \to 0} rV(r)=0\)).

In the case of the action of exchange forces, the opposite picture should be observed—the scattered nucleons and the recoil nucleons, as it were, exchange angular distributions.

Fig. 6. Diagram of experiments for studying the spectrum of recoil protons.

If the forces are partly ordinary and partly exchange, then the angular distribution of both the scattered nucleons and the recoil nucleons must be characterized by two peaks—forward and backward, and these peaks may overlap to one degree or another. As experiments on \(np\)-scattering at high energies have shown, the angular distribution is almost symmetric with respect to \(90^\circ\) (in the c.m. system), i.e., the forces between the neutron and proton are half ordinary and half exchange. Differential

angular scattering cross sections at \(90^\circ\) are rather small, and therefore it may be assumed that the peaks of the recharged and unrecharged particles overlap only weakly, and that all protons directed at angles less than \(90^\circ\) (in the c.m. system), i.e., those which have received more than half the energy of the bombarding neutrons, are recharge protons. This is precisely how D. I. Blokhintsev defines nucleons of recharge origin\(^9\).

Let us now turn to the recharge of protons occurring not on free nucleons, but on nuclear nucleons.

The neutrons in a nucleus are not at rest, but possess a certain momentum distribution. Very roughly, this distribution is obtained from the Fermi-gas model, from which it follows that nucleons in a nucleus possess momenta corresponding to energies up to \(20\text{--}30\) MeV. It should be noted that in fact (see Section VIII) the “tail” of the momentum distribution extends considerably farther into the region of large momenta; however, for mean values this is apparently not very essential.

Therefore, in contrast to the case of collision with free nucleons, the nuclear recharge neutrons flying at a given angle will not have a definite energy. The half-width of the energy distribution for a fixed angle is estimated from the same considerations as for the stripping reaction, but now in place of the comparatively small deuteron binding energy there enters the considerably larger quantity of the order of the mean energy (\(20\text{--}30\) MeV) of a nucleon in the nucleus; i.e., the half-width should be of the order of \(\sqrt{E_{\text{nucl}}E_p}\), which for \(E_p = 350\) MeV is approximately \(100\) MeV.

The momentum distribution should lead to an additional broadening of the angular distribution of recharge neutrons by an amount of the order of \(\sqrt{\dfrac{\bar E_{\text{nucl}}}{E_p}}\), which for protons with an energy of \(350\) MeV is about \(15\text{--}18^\circ\).

The next phenomenon that arises in going over to bound neutrons consists in the fact that the effect of the Pauli principle, which forbids the proton after the collision to acquire a small momentum (since all states with small momenta in the nucleus are already occupied), leads to a strong suppression of forward emission of neutrons, which again contributes to the broadening of the angular distribution.

The third effect is that the proton may undergo not one, but several collisions in the nucleus. This leads to an additional broadening of the angular distribution of neutrons. There are grounds for believing that this effect plays a noticeable role even in light nuclei. In heavy nuclei its role should be very significant.

Thus, the angular and energy distribution of neutrons arising in the recharge of protons in nuclei may differ noticeably from the case of free \(np\)-scattering. The principal difference will consist in a noticeable broadening of the angular and energy

distributions; in particular, one may expect also the emission of neutrons at angles exceeding \(90^\circ\) (in the laboratory system), which is altogether impossible in free \(np\)-scattering.

The cross section for the production of high-energy neutrons by charge exchange must constitute a noticeable fraction of the total cross section for the inelastic interaction of high-energy nucleons with nuclei (see Section VI).

The way toward a theoretical determination of charge-exchange cross sections was indicated in the work of D. I. Blokhintsev\(^9\). In interesting papers by Soviet physicists I. Ya. Pomeranchuk, I. M. Shmushkevich\(^ {10}\), E. L. Feinberg and V. Ya. Fainberg\(^ {11}\), it was predicted that charge exchange should be accompanied by specific bremsstrahlung owing to the considerable acceleration or deceleration of the motion of the charges, and also by “magnetic” radiation due to a change in the direction of the magnetic moment of each particle and of the entire system as a whole during charge exchange, not accompanied by spin exchange. These authors calculated the probability of emission of such radiation, its spectrum, and angular distribution. Later the predictions of the Soviet physicists were confirmed by experiments.

In a large number of works the charge-exchange cross section of protons with energies from 110 to 385 MeV on various nuclei was determined, as well as the angular and energy distribution of the neutrons produced as a result of charge exchange.

The yield, angular distribution, and spectra of charge-exchange neutrons were studied with the aid of various high-energy neutron detectors (see Section II of the present review).

Figures 7 and 8 show the energy distribution of neutrons produced in the charge exchange of protons with energies of about 100–110 MeV on beryllium and carbon targets\(^{12,13}\), measured with coincidence telescopes that registered the protons knocked out by these neutrons from polyethylene targets. It is obvious that, in charge exchange on beryllium, a neutron spectrum is observed with a much more sharply expressed “peak” than in charge exchange on carbon. Moreover, the “peak” of the neutron spectrum for beryllium is located at an energy of 90–95 MeV, whereas the weakly expressed maximum in the case of carbon is at an energy of about 70 MeV. This difference is apparently connected with the presence of an “extra” neutron in the Be nucleus, which is manifested in the difference between the neutron binding energies in the nuclei Be\(^9\) (\(\sim 1.6\) MeV) and C\(^ {12}\) (18.7 MeV). The negative heat effect of charge exchange on beryllium—Be\(^9(pn)\)B\(^9\)—amounts to only about 1.85 MeV, whereas for carbon—C\(^ {12}(pn)\)N\(^ {12}\)—it is close to 18.9 MeV.

It must be noted, however, that the spectra given above are evidently strongly distorted because of the relatively broad energy distribution of the incident protons themselves after their passage through the target—owing to the radial oscillation of the orbit and the spread of ionization losses during multiple

Fig. 7. Spectrum of charge-exchange neutrons
(proton energy about \(110\) MeV, Be target).

Fig. 8. Spectrum of charge-exchange neutrons
(\(E_p \approx 100\) MeV, C target).

...of traversal of the target. According to[^13], the spread of the primary protons in energy after they have passed through a beryllium target is very similar to the spectrum of the resulting neutrons. The distortion of the spectrum of charge-exchange neutrons due to the difference in proton energies is less significant in the experiments described below, where the proton energy was higher.

The angular distribution of neutrons from charge exchange of protons with energy \(110\ \text{MeV}\) on Be, Al, Cu, and Pb proved to be rather close—with a half-width of \(54\text{--}59^\circ\) in the center-of-mass system[^14]. In the case of a carbon target, constancy of the charge-exchange-neutron flux was observed over a wide angular interval. A shortcoming of the work[^14] is the use, for registering charge-exchange neutrons, of a detector with a relatively low threshold (\(\sim 20\ \text{MeV}\))—the reaction \(\mathrm{C}^{13}(n,2n)\mathrm{C}^{11}\). Such a detector effectively registers not only charge-exchange neutrons, but also a noticeable fraction of the neutrons knocked out of the target nuclei in secondary processes; therefore the difference in the angular distribution of charge-exchange neutrons from different targets is smoothed out.

Recently, another report has appeared[^15] on the study, using a telescope of stilbene scintillation counters, of the angular distribution and spectrum of neutrons from charge exchange of \(110\ \text{MeV}\) protons on targets of D, Li, Be, C, Al, Cu, and Pb.

In the neutron spectrum for D, Li, and Be targets, a peak was observed in the high-energy region, apparently associated in part with the energy spread of the initial protons; in the case of the other four targets, however, the neutron distribution function in energy decreased smoothly with increasing energy, and there was no peak. For D, Li, and Be targets, the angular distribution was very close to the angular distribution for scattering of neutrons with energy \(90\ \text{MeV}\) by free protons. For the other targets the angular distribution deviated from the case of free \(np\)-scattering, and these deviations increased in the sequence Al, Cu, Pb, C.

The production of neutrons in the charge exchange of protons with initial energy of about \(170\ \text{MeV}\) on Be, C, Al, and U nuclei was studied in considerable detail[^16]. In this work the coincidence-telescope method was also used. Figures 9–12 show the spectra of charge-exchange neutrons on the four indicated targets for an angle of \(2.5^\circ\) to the axis of the neutron beam.

Table III (see p. 545) gives the values of the differential angular cross sections for the production of charge-exchange neutrons, calculated in[^16] on the basis of measurements of the neutron flux at angles \(2.5^\circ\) and \(5^\circ\) and measurements of the proton current, with a correction for possible multiple traversals of the target by protons. The mean proton energy given in Table III was obtained after introducing corrections for ionization energy losses.

Fig. 9. Spectrum of charge-exchange neutrons ($E_p \simeq 170$ MeV, Be target).

Fig. 10. Spectrum of charge-exchange neutrons ($E_p \simeq 170$ MeV, C target).

Fig. 11. Spectrum of charge-exchange neutrons ($E_p \simeq 170$ MeV, Al target).

Fig. 12. Spectrum of charge-exchange neutrons ($E_p \simeq 170$ MeV, U target).

Table III

Target	Be	C	Al	U
\(E_p\) average (MeV)	156	155	166	166
\(\sigma_{\text{charge exchange}}\times 10^{27}\ \text{cm}^2/\text{steradian}\) \((2.5^\circ)\)	\(91^{+45}_{-30}\)	\(47^{+23}_{-16}\)	\(122^{+61}_{-41}\)	\(1520^{+760}_{-507}\)
\(\sigma_{\text{charge exchange}}\times 10^{27}\ \text{cm}^2/\text{steradian}\) \((5^\circ)\)	\(99\pm19\)	\(72\pm15\)

The authors estimate the possible error of the experiments to be as large as a factor of 1.5, chiefly because of errors in determining the proton current and the effective thickness of the targets.

In a later work\({}^{25}\), carried out on the same phasotron at a proton energy of 170 MeV, the angular distribution of charge-exchange neutrons on a beryllium target was investigated. The neutrons were detected by counting recoil protons knocked out of a polyethylene target at an angle of \(12^\circ\) to the axis of the neutron beam.

The protons were registered with a coincidence telescope consisting of three scintillation counters; carbon filters were placed in front of the third counter, the thickness of which determined the energy threshold of the registered neutrons. In this way the angular distribution was studied for neutrons with energies greater than 50, 67, 83, 100, and 120 MeV.

As can be seen from Fig. 13, the angular distribution depends strongly on the registration threshold—for higher energies the angular distribution is more strongly directed forward. For the five indicated threshold values, the half-width of the angular distribution proved to be, respectively, \(67^\circ\), \(62^\circ\), \(56^\circ\), \(51^\circ\), and \(42^\circ\).

Integration of the angular distribution led to the determination\({}^{25}\) of the total cross section for the production of neutrons with energies greater than 50 MeV, equal to \((99\pm40)\cdot 10^{-27}\ \text{cm}^2\). The differential cross section for an angle of \(4^\circ\) was found to be in satisfactory agreement with\({}^{16}\), equal to \((69\pm30)\cdot 10^{-27}\ \text{cm}^2/\text{steradian}\).

A brief report on the spectra of charge-exchange neutrons of protons with energy 245 MeV on Be, C, and Pb nuclei\({}^{17}\) is of interest because the spectra were investigated not only in the direction of the primary proton beam, but also at an angle of \(15^\circ\) to this direction.

Table IV gives the neutron energy values corresponding to the peak of the spectra, as well as the half-widths of the different spectra,

Table IV

	Be (0°)	C (0°)	Pb (0°)	Be (15°)	C (15°)
Peak position (MeV)	215	195	180	175	165
Spectrum half-width (MeV)	66	81	120	98	103

i.e., the energy interval between the points at which the value of the distribution function was half that at the top of the “peak.”

Figure labels: relative neutron intensity; laboratory angle. Registration threshold: 1 — 120 MeV; 2 — 100 MeV; 3 — 83 MeV; 4 — 67 MeV; 5 — 50 MeV.

Fig. 13. Angular distribution of charge-exchange neutrons
(\(E_p \approx 170\) MeV, detector thresholds from 50 to 120 MeV).

From the data of Table IV it is evident that the heavier the target nucleus and the larger the angle of emission of the neutrons relative to the primary proton beam, the lower the mean energy of the charge-exchange neutrons and the broader their energy distribution. The form of the neutron spectrum

(at an angle of \(0^\circ\)) from charge exchange of protons with an energy of about \(235\) MeV on beryllium, published quite recently \(^{26}\), is presented in Fig. 14.

In a number of works, the spectra, angular distribution, and yield of charge-exchange neutrons for protons with energies of about \(340\)—\(350\) MeV were investigated. Fig. 15 shows an approximate spectrum of such neutrons \(^{62}\). The angular distribution of neutrons in charge exchange on Be, Al, Cu, and U, measured with carbon detectors, proved to be close \(^{19}\). As is seen from Fig. 16, the half-width of the angular distribution for charge-exchange neutrons on the four indicated nuclei is \(54\)—\(59^\circ\).

Fig. 14. Spectrum of charge-exchange neutrons (\(E_p \simeq 235\) MeV, Be target).

For free scattering at a mean neutron energy of about \(260\) MeV, the half-width of the angular distribution is much smaller, only about \(10^\circ\) \(^{30}\). A calculation carried out by the authors \(^{19}\) under the assumption of a Fermi distribution of the momenta of nuclear nucleons, with an upper limit of their energy equal to \(30\) MeV, gives a half-width of the angular distribution of about \(36^\circ\). Additional broadening of the angular distribution of charge-exchange neutrons may be connected with secondary collisions of nucleons in the nucleus in which the charge exchange occurred, or with the inapplicability of the Fermi-gas model for finding the momentum distribution of nuclear nucleons.

Thus, the angular distribution of charge-exchange neutrons, measured for various nuclei with carbon detectors, proved to be completely identical for incident protons with energies of \(110\) MeV \(^{14}\) and \(340\) MeV \(^{19}\). This alone already indicates the insufficient clarity of the results of experiments with carbon detectors and the need for measurements of the angular distribution of charge-exchange neutrons using high-threshold detectors. A comparison of data obtained for different nuclei and with different detectors,

Fig. 15. Charge-exchange neutron spectrum
\((E_p \approx 350\ \text{MeV},\ \text{Be target})\).

Plot labels: \(N(E)\); \(E_n\), MeV; 280.

Fig. 16. Angular distribution of charge-exchange neutrons
\((E_p \approx 350\ \text{MeV},\ \text{different targets; detector—carbon})\).

Plot labels: intensity of neutrons in relative units; \(\psi_{\text{lab}}\); Al, Cu, U, Be.

can give a sufficiently clear picture of the momentum distribution of intranuclear nucleons.

A study of the angular distribution of neutrons produced in the charge exchange on beryllium of protons with an energy of 340 MeV, carried out with the aid of a bismuth fission chamber (threshold about 50–60 MeV), led to the results shown in Fig. 17[^21]. It is obvious that the angular distribution in the case of a bismuth detector is narrower than with a carbon detector. Having determined, with the aid of carbon detectors, the proton current and the absolute value of the neutron flux, the author[^31], integrating the data of the angular distribution, obtained the cross section for charge exchange of 340-MeV protons on beryllium, equal to \(10^{-25}\ \mathrm{cm}^2\) (\(\pm 50\%\)), i.e. amounting to more than 40% of the total cross section of beryllium for neutrons of energy 270 MeV[^32].

Fig. 17. Angular distribution of charge-exchange neutrons
(\(E_p \simeq 350\ \mathrm{MeV}\), Be target, detector—bismuth fission chamber).

According to a brief report on the charge exchange of protons with an energy of 385 MeV on beryllium, the peak of the spectrum of charge-exchange neutrons is located in this case at 310 MeV, and the half-width of the spectrum is about 100 MeV.

Summing up all that has been said, it should be noted that the data on neutron production in the charge-exchange process are considerably less definite than for neutron production in the stripping reaction.

Meanwhile, all the data concerning charge exchange are of undoubted interest not only for characterizing the neutrons obtained, but also for studying the internal momentum distribution of nucleons.

As follows from this section, the neutrons formed in the stripping reaction and especially in charge exchange are characterized by a broad energy distribution. In what follows, when speaking of the interaction with nuclei of neutrons of energy \(E_n\) MeV, we shall everywhere understand by this energy the position of the peak in the energy distribution of the neutrons.

II. PRINCIPAL METHODS OF DETECTING HIGH-ENERGY NUCLEONS

The literature describes methods of detecting high-energy nucleons based on the following four processes:

1. The reaction of knocking a neutron out of a carbon nucleus with the formation of the radioactive isotope \(C^{11}\).
2. Fission of nuclei of heavy elements (mainly bismuth nuclei) by high-energy nucleons.
3. \(np\)-scattering with subsequent detection of recoil protons.
4. Emission of visible light when high-energy protons move in media with a large refractive index at a velocity exceeding the velocity of light in these media—the Cherenkov effect.

Let us consider in outline all four methods of detecting high-energy nucleons.

a) Reactions \(C^{12}+X \to C^{11}+X+n\)

The knocking out of neutrons from a carbon nucleus can successfully be used for detecting high-energy nucleons for the following principal reasons: 1) the binding energy of a neutron in the \(C^{12}\) nucleus is relatively large and amounts to 18.6 MeV; therefore the threshold of the reaction in which a neutron is knocked out of a carbon nucleus is much higher than the threshold of other reactions of this type; 2) the \(\beta^+\)-active isotope \(C^{11}\) formed as a result of the reaction has a half-life convenient for observation (20.5 min.) and emits sufficiently penetrating positrons \((E_{\max} \simeq 1\ \text{MeV})\); 3) in the interaction of high-energy nucleons with the \(C^{12}\) nucleus, side radioactive products with half-lives at all close to that of \(C^{11}\) cannot be formed. To be convinced of this, it is enough to look through the list of radioactive isotopes of light nuclei; 4) as experiments have shown, the cross section for knocking neutrons out of the \(C^{12}\) nucleus changes little over a wide range of nucleon energies. In experiments at high energies four reactions of the type \(C^{12}(X,Xn)C^{11}\) were observed, namely, \(C^{12}(p,pn)C^{11}\), \(C^{12}(n,2n)C^{11}\), \(C^{12}(d,dn)C^{11}\), and \(C^{12}(\alpha,\alpha n)C^{11}\).

The method used for investigating the energy distribution of protons in the stripping reaction (Fig. 6) was described above. The same method was used\(^ {27}\) to study the excitation function of the reaction \(C^{12}(p,pn)C^{11}\)—the activity of \(C^{11}\) was determined as a function of the range of protons in carbon at three initial energies—65, 88, and 140 MeV. To determine accurately the end of the proton range, boric-acid plates were used, placed between carbon plates and registering protons down to the very lowest energies by the reaction \(B^{11}(pn)C^{11}\). After subtracting

Fig. 18. Excitation function of the reaction
\(C^{12}(p,pn)C^{11}\).

the neutron background, it was possible to determine the fraction of carbon activation by high-energy protons. In a later work\(^5\) the proton flux was determined and, thus, it was shown that the cross section of the reaction \(C^{12}(p,pn)C^{11}\) at \(E_p = 60\) MeV is approximately \((0.073 \pm 0.010)\cdot 10^{-24}\ \text{cm}^2\). The first investigations showed approximate constancy of the cross section of this reaction from 60 to 140 MeV. Later, the following values of the cross sections of the reaction \(C^{12}(p,pn)C^{11}\) at different energies, given in Table V, were used in the literature:

Table V

\(E_p\) (MeV)	75	100	105	146	240	340
\(\sigma \cdot 10^{24}\ \text{cm}^2\)	83	70	67	63	49	38
Literature references	28	29	28	51	30	21

Figure 18 shows the excitation function of the reaction \(C^{12}(p,pn)C^{11}\) on the basis of these data.

Theoretical attempts to calculate the reaction \( \mathrm{C}^{12}(p,pn)\mathrm{C}^{11} \) were made\(^{31}\) on the basis of considering the competition of four variants of the interaction of protons with the \(\mathrm{C}^{12}\) nucleus:

a) \(p + \mathrm{C}^{12} \to \mathrm{N}^{13*} \to \mathrm{C}^{12} + p + n,\)
b) \(p + \mathrm{C}^{12} \to p + \mathrm{C}^{12*} \to \mathrm{C}^{11} + n,\)
c) \(p + \mathrm{C}^{12} \to n + \mathrm{N}^{12*} \to \mathrm{C}^{11} + p\) and
d) \(p + \mathrm{C}^{12} \to \mathrm{C}^{11} + p + n\) (knockout).

The ratios of the cross sections for \(pp\)-, \(np\)-, and \(nn\)-interactions adopted for the calculations later proved to be incorrect; therefore it is not surprising that the absolute value of the calculated cross section turned out to be smaller than the experimental cross section (the discrepancy between calculation and experiment was especially large for the reaction \(\mathrm{C}^{12}(n,2n)\mathrm{C}^{11}\)).

Owing to the establishment of the cross sections of the reaction \(\mathrm{C}^{12}(p,pn)\mathrm{C}^{11}\), carbon detectors found rather broad application for determining the absolute magnitude and monitoring the proton current in phasotrons (for example, \(^{28,29,51}\)).

The cross section of the reaction \(\mathrm{C}^{12}(n,2n)\mathrm{C}^{11}\) was measured experimentally at neutron energy 90 MeV\(^{4}\). To determine the cross section of this reaction, it was necessary to measure, by another independent method, the number of neutrons. Such an independent method is the determination of the number of protons knocked out by neutrons from a hydrogen-containing target at some angle, if the angular distribution and the total \(np\)-scattering cross section are known; for finding the latter, the absolute magnitude of the neutron flux is immaterial.

It was by precisely such a comparison that the cross section of the reaction \(\mathrm{C}^{12}(n,2n)\mathrm{C}^{11}\) was determined\(^{4}\), and was found to be \((0.022 \pm 0.004)\cdot 10^{-24}\ \mathrm{cm}^{2}\). Later calculations\(^{21}\) of the cross section of this reaction for neutrons with energy 270 MeV led to the value \(0.017\cdot 10^{-24}\ \mathrm{cm}^{2}\).

To study the excitation functions of the reactions \(\mathrm{C}^{13}(d,dn)\mathrm{C}^{11}\) and \(\mathrm{C}^{12}(\alpha,\alpha n)\mathrm{C}^{11}\), deuterons and \(\alpha\)-particles were deflected\(^{32}\) by a pulsed electrostatic deflector onto a stack of carbon plates, each 6 mm thick at the beginning and 1.5 mm thick at the end of the range, where the cross section may vary more strongly with energy. Comparison of the activities of the various plates directly gave the dependence of activation on the residual range, i.e., on the energy. It was found\(^{32}\) that the cross sections of the indicated reactions increase rapidly at \(E_d\) about 160 MeV and \(E_\alpha\) about 250–300 MeV and thereafter remain approximately constant. At deuteron energy 190 MeV, the cross section of the reaction \(\mathrm{C}^{13}(d,dn)\mathrm{C}^{11}\) is\(^{6}\) \(65\cdot 10^{-27}\ \mathrm{cm}^{2}\).

In addition to carbon, aluminum is used for registering high-energy protons (for example, \(^{6,33}\)), for the reaction \(\mathrm{Al}^{27}(p,3pn)\mathrm{Na}^{24}\), with formation of the isotope \(\mathrm{Na}^{24}\) with a half-life of 14.8 h and maximum \(\beta\)-spectrum energy 1.4 MeV, is characterized by a rather large cross section (\(10^{-26}\ \mathrm{cm}^{2}\)), constant at proton energies from 100 to 340 MeV\(^{33}\).

b) Fission of nuclei of heavy elements

The basic characteristics of the fission of heavy nuclei by high-energy particles have been examined in detail in review^34, and therefore we shall not dwell on them here.

Judging from all the experimental data cited in^34, such fission is not ordinary “barrier” fission, of the type of uranium fission by slow neutrons, but emissive fission, associated with the preliminary evaporation of a large number of neutrons. Emissive fission is characterized by high thresholds and a rapid increase of the fission cross sections with increasing energy of the fissioning particles above threshold, and also on going from lighter nuclei to heavier ones. A typical dependence of the cross sections of emissive fission on neutron energy up to 84 MeV and on the atomic number of the nucleus \((Z = 78—83)\) is shown in Fig. 19^35.

Fig. 19. Excitation functions for fission of various nuclei by neutrons. One unit on the ordinate axis corresponds to a cross section of \(1.9 \cdot 10^{-26}\ \text{cm}^2\).

The detector of fission by high-energy neutrons most frequently used is the ionization chamber, which records pulses from fragments of bismuth fission.

The design and method of operation of such chambers are described in detail in the literature^36, and we shall not dwell here on this question.

The excitation function for fission of bismuth by neutrons with energies up to 84 MeV is given in Fig. 19. At a neutron energy of 270 MeV the fission cross section of bismuth is approximately three times higher than at 90 MeV^6, while the ratio of the bismuth fission cross section to the cross section of the reaction \(C^{12}(n, 2n)C^{11}\) in the interval from 90 to 270 MeV increases uniformly with neutron energy by a factor of 3.56^34. Although the fission cross section of bismuth, unlike the cross section of the reaction \(C^{12}(n, 2n)C^{11}\), rather strongly var-

counting with energy, but in a number of cases, when it was necessary to have a higher-threshold detector, bismuth fission chambers were preferably used (for example, \(^{21,22}\)).

Fig. 20. Excitation functions for fission of uranium and thorium by deuterons.

Fission of nuclei lighter than bismuth can evidently serve as the basis for the operation of still higher-threshold detectors. Thus, the literature has reported the use of a gold fission chamber for recording high-energy neutrons \(^{37}\).

Fission of heavy nuclei can be used to register not only neutrons, but also charged particles of high energy. In particular, fission of the nuclei \( \mathrm{U}^{235}\), \( \mathrm{U}^{238}\), Th, Bi, and Au by \(\alpha\)-particles with energies up to 390 MeV, by deuterons up to 195 MeV, and by protons up to 340 MeV has been studied \(^{38}\). In these experiments differential ionization chambers were used, in which the constant

Fig. 21. Excitation functions for fission of bismuth and gold by deuterons.

ionization by the beam of charged particles in one half of the chamber compensated the ionization in the other half of the chamber, and in such

...under which conditions pulses from fission fragments were recorded in one half of the chamber. Part of the results obtained is presented in Figs. 20–23. From these figures it is seen that the fission cross section of the listed nuclei by protons is of the same order as the fission cross sections

Fig. 22. Excitation functions for fission of uranium and thorium by protons.

Fig. 23. Excitation functions for fission of bismuth and gold by protons.

by high-energy neutrons (for \(U^{238}\), at a neutron energy of \(90\) MeV, \({}^{30}\ \sigma_f = 1.4 \cdot 10^{-24}\ \text{cm}^2\), and for Th \(\sigma_f = 1 \cdot 10^{-24}\ \text{cm}^2\)). The fission cross sections of uranium and thorium by deuterons (and \(\alpha\)-particles) of high energies are still higher and exceed 50% of the geometric cross sections. The fission cross sections of bismuth and, especially, gold are much smaller and do not exceed \(0.5 \cdot 10^{-24}\ \text{cm}^2\) for bismuth and \(0.15 \cdot 10^{-24}\ \text{cm}^2\) for gold (\(\alpha\)-particles).

c) np scattering with registration of recoil protons.

A very broad application has been found for the method of observing high-energy neutrons based on the registration of recoil protons knocked out by neutrons from hydrogen-containing targets (for example, 18, 20, etc.). Paraffin, polyethylene, and polystyrene were used as materials for such targets. Comparison of the number of protons knocked out from such targets with the number of protons knocked out from a carbon target makes it possible to isolate the effect associated with np scattering.

The protons knocked out in np scattering at a definite angle to the neutron beam are registered by means of a “coincidence telescope.” In various works, the “coincidence telescope” contained from two to four proportional counters or scintillation counters (anthracene, stilbene) connected in a coincidence circuit. In this way, only protons traveling in a definite direction were registered. Filters of different thicknesses were placed in front of the last counter of the telescope. By varying the filter thickness, it was possible to cut off protons with energies less than various specified values \(E_p\). If the angle made by the axis of the coincidence telescope with the direction of the primary neutron beam is equal to \(\theta\), then it is evident that, for np scattering, the energy of the protons knocked out at this angle corresponds (so long as relativistic corrections are small) to the energy of the initial neutron

\[ E_n^0=\frac{E_p}{\cos^2\theta}. \]

From this follows an important advantage of the method of registering recoil protons—the possibility of setting and changing the neutron-registration threshold.

The “excitation function” of the recoil-proton detector is determined by the dependence of the differential cross section for np scattering at the angle \(\theta\) on the neutron energy. This dependence was obtained from the data of the coincidence-telescope method itself \(^{53,20}\): at different energies the angular distribution of np scattering was studied (see Section III), and the relative values of the differential angular cross sections were normalized to the total np-scattering cross section, determined by an independent method, from the attenuation of the neutron beam as a result of scattering in hydrogen-containing media under conditions of “good geometry” (see Section IV).

Thus, the method of registering recoil protons was used not only for relative, but also for absolute measurements of the neutron flux—in particular, for determining the cross section of the reaction \(C^{13}(n,2n)C^{11}\), as was already mentioned above.

The coincidence telescope was used not only as a detector of neutrons in np scattering, but also for determining the cross sections of pp scattering (see Section III) and the number of charged particles (protons, deuterons, tritons) knocked out by high-energy neutrons from various complex nuclei (see Section VI).

d) Registration of high-energy charged particles based on the Cherenkov effect

The most comprehensive method of registering high-energy particles is based on the phenomenon, discovered by the Soviet physicist P. A. Cherenkov\(^{40—43}\) in the laboratory of Acad. S. I. Vavilov, of radiation emitted when a charged particle moves in a medium with refractive index \(n\) at a speed exceeding the speed of light in that medium \(\left(\dfrac{c}{n}\right)\)—a phenomenon that has received the name of the Cherenkov effect. The theory of the Cherenkov effect was given by the Soviet physicists I. E. Tamm and I. M. Frank\(^{44,45}\) on the basis of classical electrodynamics. The quantum theory of the Cherenkov effect was constructed in the works of V. L. Ginzburg\(^{46}\) and A. A. Sokolov\(^{47}\). If a particle moves with speed \(v=\beta c\), then for \(\beta>\dfrac{1}{n}\) radiation occurs during such motion at an angle \(\theta\), for which \(\cos\theta=\dfrac{1}{n\beta}\). The greater the particle energy, the closer the angle \(\theta\) is to the limiting value for the given medium,

\[ \theta=\arccos \frac{1}{n}. \]

The Cherenkov effect was discovered using electrons as an example, but all its regularities remain valid for any charged particles with relativistic velocities. Experimentally, in particular, the Cherenkov effect has also been observed and used for registering protons with energy \(340\ \text{MeV}^{48}\), for which \(\beta=0.68\).

Since a detailed description of the method based on the Cherenkov effect has already recently been given in UFN\(^{52}\), we shall not dwell on it here. We note only that, in contrast to the detectors described above, a Cherenkov detector makes it possible to limit the energies of recoil protons from neutrons registered through them both from below and from above, i.e. it makes possible the registration of monoenergetic neutrons, the width of the energy interval being determined by the angular resolution of the system associated with the detector. In experiments with a coincidence telescope, such selection of a monoenergetic neutron “line” is possible only by supplementing the telescope with a counter, separated from the others by a filter of a definite specified thickness and connected in anticoincidence. Thanks to the use of a Cherenkov detector, it was possible, in particular, to measure for several nuclei the effective cross sections with respect to neutrons of energy \(406\ \text{MeV}\) and with a half-width of the energy distribution of only \(32\ \text{MeV}^{50}\). These neutrons were obtained from the charge exchange of protons with energy \(450\ \text{MeV}\), so that the peak of their total spectrum should have been located, as is evident from the sum of the data of Section 1, at a considerably lower energy than \(406\ \text{MeV}\), while the half-width of the total spectrum should have greatly exceeded \(32\ \text{MeV}\). Thus, the use of a Cherenkov detector made it possible to obtain much more definite results (with respect to neutron energy).

III. np-, pp-, nd- and pd-SCATTERING

Of special interest for theoretical nuclear physics are experiments devoted to the interaction of high-energy nucleons with the simplest nuclei—hydrogen and deuterium. The results of such experiments can give, and have already given, a number of new data on the character of the forces that determine the interaction of elementary nuclear particles—the proton and the neutron.

In addition, data on the interaction of high-energy neutrons and protons with hydrogen and deuterium nuclei have a direct application to the general picture of the interaction of high-energy nucleons with complex nuclei, since the first stage of the inelastic collision of a very fast nucleon can always be regarded as the collision of two free nucleons. Therefore, in the present section, along with a description of the technique and the main results of the experiments, we shall dwell on the most important theoretical premises and conclusions following from the experimental data.

a) Review of experimental data

For np-scattering in the high-energy region both total and differential angular cross sections were measured. A summary of all data on total np-scattering cross sections in the energy interval from 14 to 280 MeV is given in Table VII, in Section 4, together with the total cross sections for complex nuclei. The dependence of the total np-scattering cross sections on neutron energy is illustrated by curve 1 in Fig. 24.

The total np-scattering cross sections were determined from the difference effect of the removal of neutrons from the beam by graphite and polyethylene or paraffin scatterers. Experiments to determine total cross sections were carried out under conditions of the so-called “good” geometry (see Section VI), when, upon deflection even through the smallest angles—of the order of \(1^\circ\)—the neutrons can no longer reach the detector and be recorded.

For three values of the neutron energy—about 40, 90, and 260 MeV—not only total but also differential angular scattering cross sections were measured\(^{53,20,54,55}\), the absolute values of which were determined from the conditions of normalization to the total cross section:

\[ \sigma_t = 2\pi \int_0^\pi \sigma(\theta)\sin\theta\,d\theta, \]

where \(\theta\) is the scattering angle in the center-of-mass system (for the laboratory system in the case of np-scattering the upper limit of the integral is \(\pi/2\), with

\[ \sigma(\theta)=\frac{\sigma(\Phi)_{\mathrm{lab}}}{4\cos\Phi_{\mathrm{lab}}}. \]

For the determination of differential angular cross sections of np-scattering, three methods were used for recording recoil protons knocked out ...

...produced by neutrons—by means of a coincidence telescope consisting of three proportional counters \(^{53,20}\), in a Wilson chamber \(^{54}\), and by means of

Fig. 24. Dependence of the total cross sections of \(pr\)-, \(pp\)-, and \(nd\)-interactions on energy.

Visible labels in the figure: left scale; right scale; \(E\), MeV; \(\sigma \times 10^{27}\,\text{cm}^2\) for curves 1, 2; \(\sigma \times 10^{27}\,\text{cm}^2\) for curves 3, 4, 5.

Fig. 25. Schematic of the apparatus for determining differential cross sections of \(pr\)-scattering.

Visible labels in the diagram: beam; scatterer-monitor; telescope-monitor; scatterer; telescope-counter; filter-absorber; Bi-monitor; Al.

thick-layer photographic emulsions \(^{55}\). Coincidence telescopes were used most widely.

The scheme of this method is shown in Fig. 25. The apparatus depicted in this figure was used for counting protons scattered at

angles of \(0—60^\circ\) to the neutron beam. For larger angles a somewhat different setup was used, which we shall not describe here.

Neutrons with energies of 40 and 90 MeV were obtained from deuterons on a beryllium target 1.27 cm thick, and neutrons with energy 260 MeV—from protons with energy 350 MeV, after recharging on a target 5 cm thick.

Between the second and third counters of the main telescope there was placed an aluminum filter cutting off protons knocked out by neutrons with energy less than 66 MeV (for \(E_n=90\) MeV)\(^{55}\) or less than 200 MeV (for \(E_n=260\) MeV)\(^{30}\).

By placing absorbers (graphite and paraffin) between the monitors and the main scatterer, the authors determined the total cross sections for carbon and hydrogen, to the latter of which the differential angular cross sections were normalized. The results of measurements of differential cross sections at neutron energies of 40, 90\(^{53}\), and 260 MeV\(^{30}\) are presented in Fig. 26. The presence of a maximum in the forward scattering of protons indicates a considerable influence of exchange forces. A comparison of the data for different energies shows that, as the energy increases from 40 to 260 MeV, the differential scattering cross sections fall especially strongly at angles close to \(90^\circ\) in the c.m.s., while the cross sections for angles around 0 and \(180^\circ\) change more weakly.

Fig. 26. Summary of data on differential cross sections of \(np\)-scattering (solid curves—theoretical, from Ref. \(^{67}\)).

Similar results for a neutron energy of 90 MeV (also plotted in Fig. 28) were obtained also in experiments with a Wilson chamber\(^{54}\).

The method of thick-layer photoemulsions\(^{55}\) was used to check the data on \(np\)-scattering at a neutron energy of 90 MeV for angles \(36—180^\circ\) in the c.m.s.\(^{53}\) and to extend the investigated angular interval into the region of small angles. When coincidence telescopes are used, small neutron scattering angles, corresponding to large emission angles, i.e. to small recoil-proton energies, are difficult

observed. In the case of thick-layer photoemulsions, however, the threshold for registering protons corresponds, according to the author’s estimate[^55], to about \(1.5\) MeV, and therefore \(np\)-scattering could be observed starting from \(25^\circ\) in the c.m.s. The results of measurements with photoemulsions in the angular interval \(36\)—\(74^\circ\) coincided with the data of the coincidence-counter experiments.

As is seen from Fig. 32, the total cross sections of \(np\)-scattering decrease rather rapidly in the region from 14 to 156 MeV (by more than a factor of 16—from 770 to \(46 \cdot 10^{-27}\ \text{cm}^2\)), but change little in the region from 156 to 280 MeV—only by 30—40%, and at 260—280 MeV are practically constant. Such a dependence of the \(np\)-scattering cross sections contradicts the predictions of the theory, according to which throughout the energy interval of the order of 100—300 MeV and above (at least up to the region of the increase of the cross sections owing to the sharp increase of meson production) one should have observed

\[ \sigma_{np} \sim \frac{1}{E}. \]

These theoretical predictions, associated with ideas about the “transparency” of nuclei at high energies, also underlay calculations of the cross sections for the interaction of neutrons with complex nuclei.

An even more striking contradiction with theoretical predictions is observed for the cross sections of the \(pp\)-interaction.

Differential cross sections of \(pp\)-scattering were measured over a wide energy interval; here we shall consider results relating to the region from 30 to 340 MeV. Most of the experiments on \(pp\)-scattering at high energies were carried out using a proton beam extracted from the phasotron.

The general scheme of typical experiments[^56] is presented in Fig. 27. Protons on an orbit of radius 206 cm (\(E_p = 345\) MeV) entered, owing to vertical oscillations, two graphite blocks placed above and below the normal position of the orbit. As a result of multiple scattering in the graphite, some of these protons then entered a magnetic deflector, through which they emerged into a special evacuated exit channel about 2 meters long. In this channel there were collimating tubes with a diameter up to 2.5 cm, and a lithium absorber was also placed there in those cases when it was necessary to lower the proton energy (the absorber material was chosen with the aim of reducing multiple scattering to a minimum). On leaving the channel (through a window of \(70\ \text{mg}/\text{cm}^2\) aluminum), the protons were deflected by a special magnet into an evacuated tube about 7 meters long; after passing through it, they struck a target of polyethylene or liquid hydrogen.

The location of the target and of the counting system was surrounded on all sides by concrete shielding about 2 meters thick. In this way \(5 \cdot 10^6\)—\(5 \cdot 10^7\) protons/sec were delivered to the target in the form of a narrow and very nearly monochromatic beam about 1.25 cm in diameter. An ioniza-

...ionization chamber, preliminarily calibrated by comparison with a Faraday cylinder for high-energy protons (the collector was in the form of a lead block with a diameter and thickness of 15 cm and a magnetic field of 100 gauss for returning secondary electrons to the collector).

The protons were recorded by means of two stilbene scintillation counters (or three proportional counters), either in the form of a coincidence telescope for recording only one of the two protons participating in the collision, or connected in coincidence when placed at a definite angle for recording both protons, i.e., the scattering and recoil events. Owing to the identity of the scattered particles and recoil particles, total cross sections are obtained from the differential angular cross sections in the center-of-mass system by recalculating

$$ \sigma_t=\pi\int_0^\pi \sigma(\theta)\sin\theta\,d\theta, $$

where the factor before the integral is no longer \(2\pi\), as usual, but \(\pi\).

Fig. 27. General scheme of the experiments on the investigation of \(pp\)-scattering.

Visible labels in the figure: orbit of the deflected ion; magnetic deflector; exit window; slit; focusing magnet; evacuated tube; carbon scatterer; concrete; beam; duant; vacuum chamber; magnet pole; 306 cm; 575 m.

At low proton energy the sum of the angles of the two protons relative to the initial direction is always equal to \(90^\circ\), independently

from the magnitude of each of the angles. In addition, at low energies the conversion from laboratory differential cross sections to cross sections in the c.m. system in the case of equal masses is carried out by the simple formula
\(\sigma(\theta_{\mathrm{c.m.}})=\sigma(\Phi_{\mathrm{lab}})\dfrac{1}{4\cos\Phi}\), with \(\theta=2\Phi\). At energies of the order of 100 MeV and higher, relativistic corrections already become appreciable, and if the recoil angle of one proton is \(\varphi\), and the scattering angle of the other proton is \(\Phi\), then

\[ \tg(\varphi+\Phi)=\left(1+\frac{2Mc^{2}}{E}\right)\tg\Phi+\frac{2Mc^{2}}{E}\ctg\Phi \]

or, approximately,

\[ \varphi+\Phi=90^\circ-\frac{E}{4Mc^{3}}\sin 2\Phi, \]

i.e., the sum of the two angles is less than \(90^\circ\) and depends on the magnitude of the angles. The transition from laboratory-system angles to angles in the c.m. system is carried out by the formula

\[ \tg\frac{\theta}{2}=\sqrt{1+\frac{E}{2Mc^{2}}}\,\tg\Phi, \]

and the relation between the differential angular cross sections in the two systems has the form:

\[ \sigma(\theta)=\frac{\sigma(\Phi)}{4\cos\Phi}\, \frac{\left[1+\frac{E}{2Mc^{2}}\sin^{2}\Phi\right]^{2}} {1+\frac{E}{2Mc^{2}}}. \]

In carrying out experiments with polyethylene and graphite targets, the dependence of the coincidence counting rate on the angle between the directions from the target to the two counters was first investigated. At a proton energy of 345 MeV\(^{56}\), for a polyethylene target at \(\Phi=43^\circ\) a maximum of the coincidence counting rate was observed when the sum of the angles \(\varphi+\Phi=84—86^\circ\). Calculation for the indicated energy gives \(\varphi+\Phi=84.7^\circ\). The height of the maximum exceeded the background counting rate by more than a factor of 100.

For a carbon target over the entire angular interval \(80—90^\circ\) the coincidence counting rate is much smaller and, according to\(^{56}\), does not depend on the sum \(\varphi+\Phi\), which indicates the absence in this case of angular correlation between the two protons. It should be noted that, unfortunately, detailed data on background experiments with carbon targets are not given in works on \(pp\)-scattering, although the angular and energy distribution of protons knocked out of carbon, as well as the search for a possible partial correlation of angles in this case, are of independent interest for the study of the momentum distribution of nucleons in the \(C^{13}\) nucleus.

The sharp drop in the coincidence counting rate at \(\varphi+\Phi=90^\circ\) indicates the absence in the proton beam of admixtures of low-energy protons.

By the method described here—counting coincidences of two protons in experiments with a polyethylene target—the differential cross sections of pp scattering at angles \(36—89^\circ\) (in the c.m. system) were investigated in work \(^{56}\) at \(E_p = 345\) MeV, \(47—90^\circ\) at \(E_p = 250\) MeV, \(60—90^\circ\) at \(E_p = 164\) MeV, and \(63—89^\circ\) at \(E_p = 120\) MeV. Results were obtained by the same method for angles \(35—90^\circ\) at \(E_p = 146\) MeV \(^{51}\) and \(40—90^\circ\) at \(E_p = 105\) and \(75\) MeV \(^{38}\). At smaller angles the measurement of coincidences becomes difficult, for the energy of protons scattered through angles close to \(90^\circ\) (in the lab. system) is too small. Therefore, to extend the measurements to small angles (\(11.3—53.2^\circ\)) in the c.m. system at \(E_p = 345\) MeV \(^{56}\), it was necessary to record only one of the two protons, scattered through the smaller angle. Since, with such registration, the background from protons knocked out of carbon is already very large, instead of a polyethylene target in these experiments a liquid-hydrogen target was used, in the form of a steel tube with windows of steel \(100\) mg/cm\(^2\) thick, placed in an evacuated casing with similar windows.

At all energies from \(75\) to \(345\) MeV the differential angular cross sections for proton scattering in the c.m. system over a wide angular interval from \(15\) to \(90^\circ\) proved to be constant.

Other methods of investigating pp scattering were applied in \(^{57,58}\) for protons of energy \(240\) MeV. In these works the polyethylene and carbon targets were placed inside the phasotron chamber, and the experiments were carried out with the internal, non-extracted proton beam. In work \(^{57}\) the scattered protons were focused, depending on their energy, by the magnetic field of the phasotron onto various photographic plates located in a special block inside the chamber. This method was thus close to the method, described in Section I, for determining the spectrum of protons produced in the stripping reaction. The collimation system for the scattered protons deflected by the magnetic field was designed taking into account the relation between the emission angle and the proton energy in pp scattering. Since, when protons are knocked out of carbon, there is no such simple relation, protons from the carbon target were collimated more poorly, and this reduced the background. In \(^{58}\) coincidences were recorded between pulses from scattered protons and recoil protons in two scintillation counters located inside the phasotron chamber at an angle close to \(90^\circ\) (with relativistic correction), taking into account the deflecting action of the magnetic field.

Thus, differential angular cross sections were obtained for angles from \(8.7^\circ\) to \(71.9^\circ\) (photographic emulsions \(^{57}\)) and from \(27^\circ\) to \(90^\circ\) (counters \(^{58}\)) in the c.m. system, and at angles greater than \(13^\circ\) the scattering was isotropic.

It should be noted that the absolute values of the differential cross sections for pp scattering obtained in \(^{57}\) and confirmed more accur...

...detailed investigations^58, disagree with the data of work^56. The data of works^28 and ^51 agree rather with ^57 than with ^56. In Fig. 24 the data^56 on the cross sections are shown by curve 3-b, and the data of other works by curve 3-a. According to ^56, the absolute values of the differential cross sections at energies above 100 MeV are approximately 30% smaller than in other works. The reason for this discrepancy remains as yet unclear. What is essential, however, is not this discrepancy, but the fact that both from ^56 and from ^28,51,57,58 it follows that the decrease of the pp-scattering cross sections with increasing energy is insignificant.

Table VI

Proton energy in MeV	31,8	75	105	120	146	164	240	250	345
Literature references . .	59	28	28	56	51	56	58	56	56
Differential cross section (in \(10^{-27}\ \mathrm{cm}^2/\mathrm{steradian}\)) . . . . . . .	14,4	6,6	5,4	4,0	4,99	3,8	4,97	3,8	3,8
Total cross section (in \(10^{-27}\ \mathrm{cm}^2\)) . . . . . . .	88	42	34	25	31	24	31	24	24
Ratio of total cross sections \(\sigma_{pp}/\sigma_{np}\) . . . . .	0,3	0,4	0,5		0,67	0,5	0,85	0,66

Table VI gives the differential scattering cross sections in the c.m. system—mainly for angles of \(90^\circ\)—according to data from various works, as well as the total cross sections obtained by multiplying the differential cross sections by \(2\pi\), neglecting violations of the isotropy of scattering at small angles (in particular, without taking Coulomb scattering into account).

Figure 28 shows the results of the works cited above, and also gives the angular distribution of pp-scattering at an energy of 31.8 MeV^59, obtained with the aid of proportional counters (partly by a coincidence arrangement).

Thus, pp-scattering is characterized, first, by isotropy over a wide range of angles in the c.m. system and, second, by an almost exact constancy of the cross sections in the energy region from 120 to 345 MeV. In absolute magnitude the total pp-scattering cross sections (without Coulomb scattering) at energies of 30–250 MeV amount to from 30 to 85% (according to ^56, up to 66%) of the total np-scattering cross sections.

For the interaction of high-energy neutrons with deuterium, sufficiently accurate information is available only with respect to total cross sections.

Fig. 28. Summary of experimental data on the differential cross sections of \(pp\)-scattering.

Vertical axis: \(\sigma(\theta)\cdot 10^{27}\ \text{cm}^2/\text{steradian}\).
Horizontal axis: \(\theta^\circ\), C.M.

Legend: \(31.8\ \text{MeV}\) \((\bullet)\); \(75\ \text{MeV}\) \((\times)\); \(105\ \text{MeV}\) \((\circ)\); \(146\ \text{MeV}\) \((\bullet)\) and \(240\ \text{MeV}\) \((\nabla)\); \(120\ \text{MeV}\) \((\—\text{Po}^{56})\); \(164\ \text{MeV}\) \((+)\); \(250\ \text{MeV}\) \((\square)\); \(345\ \text{MeV}\) \((\text{Po}^{56})\); coincidences \((\times)\); one proton \((*)\).

These cross sections were determined from the difference effect of attenuation of the neutron beam by light and heavy water. Total cross sections for \(nd\)-interaction at neutron energies of \(14\)–\(280\ \text{MeV}\) are presented in Tab-

lice VII (Section IV) and are shown in Fig. 24 together with the np and pp cross sections. Similarly to the np-scattering cross sections, the total nd-interaction cross sections decrease strongly from 14 to 156 MeV (by more than a factor of 11) and change little from 156 to 280 MeV (by about 40%).

Since a significant part of the total nd-interaction cross section is elastic scattering, it would be incorrect to regard this cross section as the sum of the np- and nn-interaction cross sections. However, if the inelastic part of the nd cross section is separated out, then for this part the nonadditive effects will be weaker, and from the magnitude of the inelastic-collision cross section one can therefore obtain a certain idea of the nn-interaction cross section[^60]. At high energies the inelastic interaction of neutrons with deuterium reduces to the breakup of the deuteron into a proton and a neutron.

In a Wilson chamber filled with deuterium and D₂O vapor, with a magnetic field of 21,700 gauss, elastic and inelastic interactions of 90-MeV neutrons with deuterium nuclei were observed[^61]. The elastically scattered deuterons were directed mainly forward (approximately one third of them emerged in a narrow peak near 0°). At 30° a minimum of elastic scattering was observed, and at 80° a second maximum, corresponding to the direction of the recoil neutrons forward.

The protons from inelastic interaction were also directed mainly forward, but their angular distribution was broader: the intensity of proton scattering from 0 to 35° decreased only by a factor of 2. A small fraction of the protons (about 5%) was directed backward.

In a later work[^62], high-energy protons knocked out by neutrons of energy 270 MeV in their interaction with deuterium were observed. For comparison, at the same angles (4—58° in the laboratory system), protons from np-scattering were also recorded. The detector was a coincidence telescope with a filter corresponding, at an observation angle \(\theta\), to the threshold for proton registration

\[ E_p > 200 \cos^3 \theta \ \text{MeV}. \]

In addition, for two angles (4° and 22.5°) the proton spectra from nd-interaction were investigated. These spectra proved to be close to the spectra of protons from np-scattering at the same angles (the presence, at a given np-scattering angle, of a spectrum rather than a proton line is connected with the complicated energy distribution of the incident neutrons). But the yield of high-energy protons from nd-interaction was smaller than from np-scattering and amounted, at all angles, to about 70% of the latter.

Finally, let us dwell on experiments on pd-scattering at high energies. These experiments were carried out at deuteron energies of 190 MeV (dp-scattering, evidently analogous to pd-scattering at a proton energy of 95 MeV) and at proton energies of 146 and 240 MeV (pd-scattering).

The dp-scattering experiments were performed on a beam of 190-MeV deuterons extracted from the phasotron, by counting \(p—d\) coincidences on two stilbene counters and by comparing the data

for carbon and polyethylene targets[^63]. The range of angles over which it was possible to work with the coincidence arrangement was \(40—160^\circ\) in the c.m.s. The results of [^63] are given in Fig. 29. Later, measurements were made for angles \(15—40^\circ\) and \(160—170^\circ\), using a single counter[^64].

The elastic-scattering cross section, integrated over the angular interval investigated, was found to be \((29 \pm 3)\cdot 10^{-27}\ \mathrm{cm}^2\). The inelastic \(dp\)-interaction was studied separately at \(E_d = 190\ \mathrm{MeV}\)[^65].

Fig. 29. Sum of data on the differential cross sections of \(pd\)-scattering.

Such an interaction was observed from coincidences between two protons for cases in which both protons underwent a large change of momentum; it turned out that the angles between the directions of the time-coincident protons lay close to \(90^\circ\), i.e., the distribution was similar to that for free \(pp\)-scattering. The differential cross section for scattering of one proton through angles \(\Phi\) close to \(90^\circ\) relative to the other proton was approximately \(1/3\) of the corresponding cross section for free \(pp\)-scattering of protons with the same momenta.

Integration of the charged particles over angles gave, for the total \(dp\)-scattering cross section at \(190\ \mathrm{MeV}\)[^65] (which is analogous to \(pd\)-scattering at \(95\ \mathrm{MeV}\)), the value \((92 \pm 7)\cdot 10^{-27}\ \mathrm{cm}^2\)—without Coulomb scat-

cross section, within the experimental errors, coincides with the nd-scattering cross section at \(95\ \text{MeV}^{37}\), equal to \((104 \pm 6)\cdot 10^{-27}\ \text{cm}^2\).

Effects associated with interference or with a change in the scattering phase cannot show up when comparing data on pd- and nd-scattering. Therefore the agreement of the pd- and nd-scattering cross sections may serve as direct and unambiguous evidence for the equality of the pp- and nn-interaction cross sections. This important conclusion, however, was not drawn by the authors\(^{65}\). Of great interest is a refinement and comparison of the total pd- and nd-scattering cross sections over a wide energy interval.

In experiments on pd-scattering at \(E_p = 240\ \text{MeV}^{30,66}\), a method was used that was based on coincidences in the registration of scattered particles by scintillation counters placed inside the chamber of a phasotron. Coincidences were recorded between the scattering protons and the recoil deuterons for angles from \(20.5^\circ\) to \(100^\circ\) in the c.m. system. The protons bombarded a heavy-paraffin target. For a given position of the counters, displacement of the target by a small distance completely cut off the main count and, thus, made it possible to measure the background, consisting of random coincidences (this part of the background is proportional to the square of the proton current) and the effect from carbon in the target (this part of the background is proportional to the proton current). The proton current was determined from the yield of the reaction \(C^{12}(p,pn)C^{11}\).

For all measurements except angles of \(100^\circ\), inelastic pd-interaction occurred with the formation of a neutron and two protons; moreover, in a large number of cases a correlation of the angles between the two protons was observed as in pure pp-scattering, i.e., with transfer to the neutron of small energies and momenta (quasi-free pp-scattering).

The differential cross section of such inelastic pd-interaction was found\(^{66}\) to be, respectively, \(1.1\), \(1.9\), and \(3\cdot 10^{-27}\ \text{cm}^2/\text{steradian}\) at proton scattering angles of \(40^\circ\), \(67^\circ\), and \(90^\circ\) in the c.m. system. Integration over angles gave the total inelastic pd-interaction cross section at \(240\ \text{MeV}\), equal to \((11 \pm 3)\cdot 10^{-27}\ \text{cm}^2\).

The experimental results are shown in Fig. 29 together with data for 95 and \(146\ \text{MeV}^{85}\). It is evident that the differential cross sections of elastic pd-scattering decrease with increasing energy.

b) Theoretical studies devoted to nucleon–nucleon scattering at high energies

There are two possible ways of constructing a theory of nuclear forces. The first consists in reducing nuclear forces to the properties of the meson field; along this path practically valuable results have not yet been obtained. The second path consists in selecting such interaction potentials characterizing the nuclear forces as would lead to agreement with experiment.

Let us consider precisely this phenomenological route.

The simplest system in which nuclear forces act is a system of two nucleons. At energies below 300–400 MeV, which are at issue in the present review, such a system can apparently still be described nonrelativistically. In a system of two nucleons the total angular momentum \(j\) and the parity of the state are naturally conserved; in this case the latter reduces to the parity of the angular part of the wave function of the relative motion (all states with even orbital angular momenta are even, those with odd ones are odd). Imposing the usual requirement of invariance with respect to rotations and reflections, we arrive at the following general form of the Hamiltonian of an interaction that does not lead to velocity-dependent forces:

\[ H_{\mathrm{int}}(\mathbf r,\boldsymbol\sigma_1,\boldsymbol\sigma_2) = V_1(r)+V_2(r)(S^2-1)+V_3(r)S_{12}, \tag{1} \]

where \(\mathbf r\) is the relative radius vector, \(\boldsymbol\sigma_1/2\) and \(\boldsymbol\sigma_2/2\) are the spins of the nucleons,

\[ \mathbf S=\frac{1}{2}(\boldsymbol\sigma_1+\boldsymbol\sigma_2), \tag{2} \]

and

\[ S_{12}=3\,\frac{(\boldsymbol\sigma_1\mathbf r)(\boldsymbol\sigma_2\mathbf r)}{r^2} -\boldsymbol\sigma_1\boldsymbol\sigma_2 = 6\,\frac{(\mathbf S\mathbf r)^2}{r^2}-2S^2. \tag{3} \]

Expression (1) is symmetric with respect to the spins of the nucleons, that is, it does not change when \(\boldsymbol\sigma_1\) is interchanged with \(\boldsymbol\sigma_2\), or \(\boldsymbol\sigma_2\) with \(\boldsymbol\sigma_1\). Therefore the Hamiltonian commutes with the square \(S^2\) of the total spin, that is, the total spin of the system is conserved. Thus all states of the system split into singlet (spin \(=0\)) and triplet (spin \(=1\)) states.

The admission of velocity-dependent forces greatly enlarges the freedom in writing the Hamiltonian (see below).

The first two terms in (1) lead to central forces, the last to noncentral forces whose direction depends on the spins. The interaction described by this term is called tensorial. If it is absent, then in a two-nucleon system, along with parity, total angular momentum, and total spin, the orbital angular momentum is conserved; then the states can be classified with the aid of the usual spectroscopic symbols. For example, \({}^{3}F_3\) is an odd state with total angular momentum 3, spin 1, and orbital angular momentum 3.

In the presence of tensor forces the orbital angular momentum is not conserved; therefore all states with the same parity, spin, and total angular momentum merge into one, which is conventionally called a “mixture” of the initial states and denoted by the sum of the corresponding spectroscopic symbols.

Since the Hamiltonian contains terms depending on the spins, the interaction in singlet and triplet states can be

choose independently. Namely, in singlet states (1) reduces to

\[ {}^{1}H_{\mathrm{int}} = V_1(r) - V_2(r), \tag{4} \]

and in triplet states to

\[ {}^{3}H_{\mathrm{int}} = V_1(r) + V_2(r) + \left(6\frac{(\mathbf{S}\mathbf{r})^2}{r^2} - 4\right)V(r). \tag{5} \]

The neutron and the proton may be regarded as two states of one particle, differing in the values of an internal coordinate called the isotopic spin. By virtue of the Pauli principle the wave function is antisymmetric with respect to interchange of nucleons, which reduces to the simultaneous performance of (a) reflection at the origin of coordinates, (b) interchange of spins, and (c) interchange of isotopic spins. Since the behavior of the wave function with respect to (a) and (b) is determined by the parity and by the total spin, in a system of two nucleons all even singlet and odd triplet states are symmetric with respect to charge, while odd singlet and even triplet states are antisymmetric. Therefore in a system of two identical nucleons only even singlet and odd triplet states are realized (since the wave function must be symmetric in charge), whereas a system of two different nucleons may possess states of any parity and spin.

The forces between different pairs of nucleons could, of course, be related to one another in any way. A number of facts obtained from experiments at low energy speak in favor of charge symmetry of nuclear forces (which by no means signifies equality of the forces between any pairs of nucleons). The requirement of charge symmetry consists in the fact that the forces between two nucleons may depend only on whether the state is symmetric with respect to charge or not.

Thus, in a charge-symmetric theory one may choose the forces in states differing either in spin or in parity arbitrarily.

A table of possible states of two nucleons for small \(j\) is given on p. 572.

The eigenvalues of the operator \(\mathbf{S}^2 - 1\) are \(+1\) for triplet states symmetric with respect to spin and \(-1\) for antisymmetric singlet states. Since this operator acts only on spin variables, it follows that its action on the wave function reduces to the interchange of the spins \(\sigma_1\) and \(\sigma_2\), i.e.

\[ \mathbf{S}^2 - 1 = P_\sigma, \tag{6} \]

where \(P_\sigma\) is the spin-exchange operator.

In an analogous way one constructs the charge-exchange operator \(P_\tau\), with eigenvalues \(+1\) in states symmetric with respect to charge and \(-1\) in antisymmetric states. If it is present in the Hamiltonian, then at the moment of scattering the particles exchange charges.

The eigenvalues of the operator \(-P_{\sigma}P_{\tau}\) are \(+1\) in even states and \(-1\) in odd states. Taking into account the antisymmetry of the total wave function, it is easy to see that it is the coordinate-exchange operator, \(-P_{\sigma}P_{\tau}=P_x\).

Parity	\(j\)	0	1	2	3	4
Even	Singlet states	\({}^1S_0\)		\({}^1D_2\)
Even	Triplet states		\({}^3S_1+{}^3D_1\)	\({}^3D_2\)	\({}^3D_3+{}^3G_3\)
Odd	Singlet states		\({}^1P_1\)		\({}^1F_3\)
Odd	Triplet states	\({}^3P_0\)	\({}^3P_1\)	\({}^3P_2+{}^3F_2\)	\({}^3F_3\)	\({}^3F_4+{}^3H_4\)

States symmetric with respect to charge are enclosed in a frame.

According to the terminology now established, only forces whose action includes exchange either of charges or of coordinates are called exchange forces.

We can now write the following general form of the charge-symmetric Hamiltonian of the interaction of two nucleons:

\[ \begin{aligned} H_{\text{int}}={}&V_{11}(r)+V_{12}(r)P_{\tau}+V_{21}(r)P_{\sigma} +V_{22}(r)P_{\sigma}P_{\tau}+{}\\ &+V_{31}(r)S_{12}+V_{32}(r)S_{12}P_{\tau} =V_{11}(r)-V_{12}(r)P_{\sigma}P_x+{}\\ &+V_{21}(r)P_{\sigma}-V_{22}(r)P_x +V_{31}(r)S_{12}-V_{32}(r)S_{12}P_{\sigma}P_x . \end{aligned} \tag{7} \]

According to the general theory of scattering, the differential effective cross section for scattering by a central field, i.e. in the absence of tensor forces, is equal to

\[ d\sigma=\sigma(\vartheta)\,d\Omega=|f(\vartheta)|^2\,d\Omega, \tag{8} \]

where the scattering amplitude \(f(\vartheta)\) is related by

\[ f(\vartheta)=\frac{\bar{\lambda}}{2i}\sum_{l=0}^{\infty}(2l+1)\left[e^{2i\delta_l}-1\right]P_l(\cos\vartheta) \tag{9} \]

to the scattering phases \(\delta_l\), which are determined from the Schrödinger equation with the corresponding potential \(V(r)\). In formula (10), \(l\) is the degree of the Legendre polynomial, \(\vartheta\) is the scattering angle in the c.m. system, and \(\bar{\lambda}\) is the de Broglie wavelength for the relative motion.

Applying these formulas to nucleon scattering, one must first average over the spin directions; here the triplet states receive the statistical weight \(\frac{3}{4}\), and the singlet states \(\frac{1}{4}\) (the triplet and singlet states do not interfere). In the case of scattering of identical particles, the amplitude of the recoil particles must be added to the amplitude of scattering of the incident particles. We obtain:

for identical particles

\[ \begin{aligned} \sigma(\vartheta) &=\frac{1}{4}\left|{}^{1}f(\vartheta)+{}^{1}f(\pi-\vartheta)\right|^{2} +\frac{3}{4}\left|{}^{3}f(\vartheta)+{}^{3}f(\pi-\vartheta)\right|^{2} \\ &=\bar{\lambda}^{2}\sum_{l,l'\ \text{even}}(2l+1)(2l'+1)\sin{}^{1}\delta_l\sin{}^{1}\delta_{l'}\cos({}^{1}\delta_l-{}^{1}\delta_{l'})P_lP_{l'} \\ &\quad+3\bar{\lambda}^{2}\sum_{l,l'\ \text{odd}}(2l+1)(2l'+1)\sin{}^{3}\delta_l\sin{}^{3}\delta_{l'}\cos({}^{3}\delta_l-{}^{3}\delta_{l'})P_lP_{l'}^{*}); \end{aligned} \tag{10} \]

for different particles

\[ \begin{aligned} \sigma(\vartheta) &=\frac{1}{4}\left|{}^{1}f(\vartheta)\right|^{2} +\frac{3}{4}\left|{}^{3}f(\vartheta)\right|^{2} \\ &=\frac{\bar{\lambda}^{2}}{4}\sum_{l,l'}(2l+1)(2l'+1)\sin{}^{1}\delta_l\sin{}^{1}\delta_{l'}\cos({}^{1}\delta_l-{}^{1}\delta_{l'})P_lP_{l'} \\ &\quad+\frac{3\bar{\lambda}^{2}}{4}\sum_{l,l'}(2l+1)(2l'+1)\sin{}^{3}\delta_l\sin{}^{3}\delta_{l'}\cos({}^{3}\delta_l-{}^{3}\delta_{l'})P_lP_{l'}. \end{aligned} \tag{11} \]

To estimate the relative role of states with different \(l\), let us write the approximate relation

\[ L=\hbar\sqrt{l(l+1)}\simeq \hbar l=\frac{\hbar}{\bar{\lambda}}\rho, \tag{12} \]

where \(\rho\) is the impact parameter. It is clear that a noticeable role can

\[ \text{*) In formula (10), the Coulomb interaction present in the pp case has not been taken into account. However, at the energies of interest to us it will play a role only in the region of very small angles.} \]

only those \(l\) can play a role for which \(\rho \lesssim a\) (\(a\) is the range of action of nuclear forces), i.e. \(l \lesssim \dfrac{a}{\lambda}\). Therefore at low energies (less than 20 MeV) only spherically symmetric \(S\)-scattering occurs. Without dwelling on this case, which has been analyzed in detail in the excellent monograph by A. Akhiezer and I. Pomeranchuk,\({}^{2}\) we note only that from experiments at low energies it is impossible to determine the dependence of nuclear forces on distance and to obtain any information concerning the forces in odd states.

It would be natural to expect that, as the energy increased, \(P\)-waves, \(D\)-waves, etc., would begin to enter into the scattering. In the case of scattering of unlike nucleons (np), interference terms \(SP\), \(SD\), should then first appear, which would lead to the appearance in the angular dependence of terms with \(\cos \vartheta\) and \(\cos^{2}\vartheta\). Such terms add at \(\vartheta = 0\) and are subtracted at \(\vartheta = \pi\), i.e. they lead to a stretching of the total angular distribution forward. In the case of scattering of identical nucleons (pp and nn), because of the elimination of states antisymmetric with respect to charge (see the table on p. 572), only the interference \(S—{}^{1}D\) term \(\sim \cos^{3}\vartheta\) should appear, i.e. one should expect a stretching of the total angular distribution both forward and backward.

The experimental data did not justify these expectations.

Let us begin with np scattering. The symmetry of the experimental curves (see Fig. 26) of the angular distribution in the c.m.s. with respect to \(90^\circ\) unexpectedly indicates an almost complete (within the errors) disappearance of the odd Legendre polynomials from the differential cross sections. This indicates that the forces in odd states are zero (or very weak), which can be described theoretically by such a choice of potentials as leads to the appearance in (7) of the common factor \(\dfrac{1 + P_x}{2}\) (a theory with such an interaction is sometimes called an “even theory”).

The most detailed analysis of the experiments on np scattering was carried out in \({}^{67}\). The calculations for small \(l\) were performed numerically, and for large \(l\)—in the Born approximation; the computational errors did not exceed a few percent. Three types of radial dependence were tested: (a) a potential well, (b) an exponentially decreasing potential \(e^{-r/R}\), and (c) a Yukawa potential \(\left(\dfrac{R}{r}\right)e^{-(r/R)}\), with various combinations of constants and variants of exchange behavior, and agreement with the data at low energies was required of all variants.

As a result of the calculations it was found that assumption (a) is unsuitable, since it gives an incorrect dependence of the total cross section on energy; assumptions (b) (with \(R = 1.35 \cdot 10^{-13}\ \text{cm}\)) and (c) (with \(R = 0.75 \cdot 10^{-13}\ \text{cm}\)) can be used, with (c) giving

somewhat better angular distribution. In order that the angular distribution not be too flat near \(90^\circ\), it was necessary to introduce tensor forces, allowing at the same time a correct explanation of the quadrupole moment of the deuteron. The theoretical curves that agree best with the experiment are shown in Fig. 26.

Thus, although the experiments on \(np\)-scattering at high energies did lead to unforeseen consequences (the absence of forces in odd states), their interpretation within the framework of the existing theory was achieved comparatively easily. It should be noted, however, that the theory has in fact succeeded in explaining well only the angular distribution of \(np\)-scattering at a number of energies; as for the dependence of the cross section on energy, as is seen from Fig. 26, here the actual change of the cross section with energy proved to be appreciably smaller than the theoretical one.

The results of the \(pp\)-experiments turned out to be more unexpected. The most striking is the comparison of the three facts discovered here: (a) the cross section proved to be isotropic in the c.m. system for angles greater than \(15^\circ\); (b) in the energy region from 100 to 350 MeV the magnitude of the cross section changes hardly at all with energy; (c) the differential cross section in this region proved to be approximately \((4 \div 5)\cdot 10^{-27}\ \mathrm{cm}^2/\mathrm{steradian}\).

An angular-independent differential cross section can arise naturally in two cases. First, isotropic scattering is obtained when the range of the forces is much smaller than the de Broglie wavelength, when only the \(S\)-wave participates in the scattering. However, from (10) it follows that the maximum possible value of the differential \(S\)-cross section for scattering of identical nucleons is

\[ \left(\frac{d\sigma}{d\Omega}\right)_{S,\ \max} = \left(\frac{\lambda}{2\pi}\right)^2 \approx \frac{200}{E_{\mathrm{MeV}}}\, \frac{10^{-27}\ \mathrm{cm}^2}{\mathrm{steradian}} . \]

Therefore the assumption that in \(pp\)-scattering only the \(S\)-wave plays a role is in contradiction both with the observed magnitude of the cross section\(^*\) and with the experimentally found independence of the cross section from energy.

Secondly, an isotropic cross section arises naturally in scattering by an impenetrable sphere whose dimensions are considerably larger than the de Broglie wavelength, i.e. under conditions in which quantum effects cease to play a decisive role. In this case auto—

\(^*\) In this connection one should note the attempt\({}^{68}\) to circumvent this difficulty by ascribing to the nucleon a certain new degree of freedom, whose possible change during scattering would remove the exclusion, imposed by the Pauli principle, of a number of states from the \(pp\)-system. The authors\({}^{69}\) also discuss the question of what change in the mass of the nucleon, associated with the new degree of freedom, could be not in contradiction with experiment.

mathematically, the independence of the cross section from the energy is also achieved. However, for such a case actually to take place, the scattering sphere would have to be too large, and the cross section would turn out to be several hundred times larger than that observed.

Therefore the attempts made so far to explain the results of pp scattering have been directed toward interpreting the isotropy and energy independence as the result of the superposition of a number of effects, each of which depends both on the energy and on the angles.

On the other hand, if charge symmetry of the theory and the analysis of pp scattering carried out in \({}^{67}\) are retained, then there will remain no free parameters for describing pp scattering. Indeed, then (an even theory!) only the singlet states \({}^{1}S\), \({}^{1}D\), \({}^{1}G\), etc., can play a role in pp scattering. But successive even Legendre polynomials have different signs at \(90^\circ\); therefore, with identical signs of all phases, as is always the case for a monotonic potential, the interference terms will be negative at \(90^\circ\), which will lead to a cross section elongated forward and backward. The experiment clearly contradicts such a prediction.

If charge symmetry is abandoned, then one can freely dispose of the forces in the odd states, of which only \({}^{3}P\) is essential. However, central forces will then lead to a term \(\sim \cos^{2}\vartheta\) in the angular distribution (the \({}^{3}P\) state has no one with which to interfere!), i.e., the dip at \(90^\circ\) will only be intensified.

Three ways out of this difficulty have so far been proposed. First, it was proposed \({}^{70}\) to introduce strong tensor forces in the odd states, which is connected with abandoning charge symmetry. Because such forces are noncentral, (10) and (11) will no longer be satisfied; associated polynomials will enter the expression for the angular dependence along with Legendre polynomials, of which the principal role will be played by \((P_{1}^{1})^{2} \sim \sin^{2}\vartheta\), i.e., tensor forces can give the desired rise of the cross section at \(90^\circ\).

The magnitude of the tensor potential is determined from the value of the total cross section. To describe simultaneously the experiments at 32 MeV and at 340 MeV, it proved necessary to choose, for the tensor interaction, a strongly singular dependence

\[ \left(\frac{r_{1}}{r}\right)^{2} e^{-\frac{r}{r_{1}}}. \]

The choice of constants made in \({}^{70}\) was based on the preliminary value of the differential pp cross section, equal to \(5.5 \cdot 10^{-27}\ \text{cm}^{2}/\text{steradian}\) \({}^{8}\). The authors \({}^{66}\) give the results of a recalculation of the constants performed in order to achieve the best agreement with later experiments, and also compare the computed angular distribution with the experimental one (Fig. 30). It is seen from Fig. 30 that the theory agrees well with experiment for angles exceeding \(35^\circ\);

for smaller angles the theory greatly overestimates the width, apparently, of the peak of the angular distribution that exists at small angles.

It should be noted that the theory set forth operates, in essence, with two independent potentials: one, central, which gives the correct scattering at low energies, and another, tensor, which does not manifest itself at low energies and plays the main role at 350 MeV. By choosing the radial dependence of this second potential, the authors[^70] succeeded in obtaining satisfactory agreement also at the third point—at 32 MeV. However, the principal qualitative feature of the behavior of the cross section with energy, namely the rapid fall up to 100 MeV and constancy at energies above 100 MeV, does not follow from the theory, which leads to a smooth decrease of the cross section as the energy increases.

Fig. 30. Comparison of the theoretical calculation of \(pp\)-scattering with strong tensor interaction with experimental data. In the calculations the following potentials were used: for singlet states,

\[ {}^{1}V=(-25.3\,\text{MeV})\,\frac{1+P_x}{2}\quad \text{for } r<r_1 \]

and

\[ V=0 \quad \text{for } r>r_1; \]

for triplet states:

\[ {}^{3}V=(-25.3\,\text{MeV})\,\frac{1+P_x}{2}\,\frac{r_2}{r}e^{-r/r_2} + \]

\[ +(-48.3\,\text{MeV})\,\frac{1+P_x}{2}\,\frac{r_2}{r}e^{-r/r_2}S_{12} + \]

\[ +(-15.25\,\text{MeV})\,\frac{1+P_x}{2}\left(\frac{r_3}{r}\right)^2 e^{-r/r_3}S_{12}; \]

\[ r_1=2.615\cdot 10^{-13}\,\text{cm};\quad r_2=1.35\cdot 10^{-13}\,\text{cm}; \]

\[ r_3=1.6\cdot 10^{-13}\,\text{cm}. \]

The second possibility, which was proposed[^71][^72] to compensate for the dip at \(90^\circ\), does not require abandoning the charge symmetry of nuclear forces and consists in allowing a nonmonotonic central potential, namely, that in singlet states the forces of attraction are replaced at small distances by forces of repulsion.

The introduction of a nonmonotonic potential leads to the result that the scattering phases which are positive at low energies change sign as the energy increases; moreover the \(S\)-phase changes sign first, while for higher phases the change of sign occurs at considerably higher energies. In the region where the \(S\)- and \(D\)-phaseshiqizo

have different signs, the interference \(SD\)-term leads not to a decrease but to an increase of the cross section near \(90^\circ\).

No repulsive forces can be introduced into the triplet interaction because of the smallness of the range \((1.7\cdot 10^{-13}\ \text{cm})\); therefore it is taken from \(^{67}\). To achieve the actual isotropy of the pp cross section at \(350\ \text{MeV}\), in addition to introducing repulsive forces, it is necessary to strengthen somewhat the tensor interaction.

Fig. 31. Curves of the dependence of the differential pp cross section at an angle of \(90^\circ\) on the energy, calculated in the theory with repulsive forces, for different radii of the repulsive core. Points—experiment \(^{56}\).

The only free parameter in the theory turns out to be the radius of the central repulsive core in singlet states. If its value is fixed, then the remaining parameters of the even singlet potential are determined from pp and np scattering at low energies. The parameters of the even triplet potential are determined by the binding energy of the deuteron and by np scattering at low energies. Finally, np scattering at high energies determines the almost complete absence of forces in odd states, while the fraction of tensor forces is found from obtaining the correct value of the cross section at \(90^\circ\) at \(350\ \text{MeV}\).

The arbitrariness in the choice of the radius of the repulsive core can be used to obtain the correct dependence of the differential cross section of pp scattering at an angle of \(90^\circ\) on the energy. Curves of the dependence of \(\sigma(90^\circ)\) on the energy for different radii of the repulsive core are shown in Fig. 31.

The best agreement with experiment was given by the interaction potentials: singlet

\[ V= \begin{cases} \infty, & r<r_0,\\[4pt] V_{0s}e^{-\frac{r-r_0}{r_s}}\dfrac{1+P_x}{2}, & r>r_0, \end{cases} \tag{13} \]

and triplet

\[ V=\left\{0.5+0.5P_x+\left(0.3+0.7P_x\right)\gamma S_{12}\right\}V_{0t}e^{-\frac{r}{r_t}} . \tag{14} \]

with the parameters

\[ \begin{gathered} r_0=0.6\cdot 10^{-13}\ \mathrm{cm};\quad r_s=0.40\cdot 10^{-13}\ \mathrm{cm};\\ r_t=0.75\cdot 10^{-13}\ \mathrm{cm},\quad \gamma=1.84;\\ V_{0s}=375\ \mathrm{MeV};\quad V_{0t}=69\ \mathrm{MeV}. \end{gathered} \tag{15} \]

Figure 32 gives a comparison with experiment of the effective cross section for pp scattering calculated in this way (345 MeV), and again satisfactory agreement is obtained for angles greater than \(35^\circ\).

The introduction of repulsive forces and a change in the tensor forces naturally lead to a certain worsening of the agreement of the angular distribution of pp scattering with experiment (Fig. 33; 260 MeV); however, the greatest difficulty is the increase of the total pp cross section, which in \(^{67}\) came out larger than the experimental value.

Fig. 32. Comparison of the theoretical calculation of pp scattering with repulsive forces with experimental data.

Finally, a third possibility for obtaining, for pp scattering, an angular distribution close to isotropic, which also does not require abandoning charge symmetry, consists in introducing forces that depend strongly on velocity, namely, in adding to the Hamiltonian spin-orbit terms \(\sim LS\), i.e., terms depending on the mutual orientation of the orbital \(L\) and spin \(S\) angular momenta \(^{73,74}\). The spin-orbit interaction also leads to the appearance, in the expression for the angular distribution of triplet \(P\)-scattering, of a term proportional to \(\sin^3\theta\). In order to obtain a satisfactory value of the total pp cross section at 350 MeV, the radial dependence of such an interaction again has to be chosen to be strongly singular. Unfortunately, calculations of the spin-orbit interaction were carried out only in the Born approximation, which is obviously inapplicable in this case, and were not brought to a serious comparison with experiment; therefore, so far only quali-

...substantial results, and the question of the possibility of a quantitative explanation of the experiment along this path remains open.

All three paths proposed for explaining, within the framework of the existing phenomenological theory, the peculiarities of nucleon–nucleon scattering at high energies discovered in recent years have failed to explain the totality of the experimental facts. All of them have led only to the derivation of several essentially semi-empirical formulas, suitable for explaining some new results and at times directly contradicting others. It is noteworthy that all these three paths share the common feature that each of them required the introduction of forces of a strongly singular character, i.e., forces varying with distance more strongly than Coulomb forces. Apparently, only this singular dependence on distance is as yet a sufficiently established property of nuclear forces, and it is very probable that more detailed information about nuclear forces will not be obtained until nonstatic and relativistic effects have been understood to a degree sufficient for a quantitative comparison of calculations with experiments.

Fig. 33. Comparison of the theoretical calculation of \(np\)-scattering (computed with the same potentials as the \(pp\)-scattering in Fig. 32) with experimental data.

Such an understanding is unlikely to be achieved on the basis of a phenomenological description; substantial progress in this direction will evidently be possible only on the basis of the concepts of the meson theory of nuclear forces, whose state for the time being leaves much to be desired. One may think that the existing concepts will require a fundamental restructuring here, and that only radically new physical ideas will be able to point the way out of the situation that has arisen. Thus, it is possible that an explanation of the features of nucleon interaction at high energies can be achieved at the price of abandoning the notion, now accepted in quantum field theory, of the localizability of elementary particles.\(^{86}\)

The attempts made so far to take relativistic corrections to nucleon scattering into account are hardly deserving of serious atten-

SCATTERING AND ABSORPTION OF HIGH-ENERGY NUCLEONS

... If they continue the phenomenological point of view[^75], then so many parameters appear in the theory that it becomes possible to obtain almost any results; it is essential, however, that a noticeable difference from nonrelativistic calculations, even for the highest energies so far attained, can be obtained only at the cost of introducing strong relativistic corrections (20–30%) even in such a “slightly relativistic” problem as the theory of the deuteron. If, on the other hand, they proceed from some meson model of nuclear forces[^76], then the results, although characterized by greater definiteness, turn out to be very far from experiment.

The only source for obtaining information on the interaction of free neutrons is experiments on the scattering of neutrons by neutrons bound in nuclei. Naturally, the most suitable for this is the scattering of neutrons by deuterons; for this reason this problem has been subjected to careful theoretical consideration[^77–^81].

The problem of neutron scattering by the deuteron is a three-body problem; not only the solution, but even the formulation of such a problem in general form is not yet possible because of the insufficiency of our information on the interaction of nucleons. However, a number of features of neutron scattering by deuterons at high energies make it possible to simplify the problem considerably and to change its formulation in such a way that it is possible not only to carry the calculations through to the end, but also in some cases to restrict the initial assumptions about the nuclear interaction to the information obtained directly from experiment.

First of all, from the nuclear point of view the deuteron is a very “extended” system: the mean distance between the nucleons composing it is several times larger than the range of nuclear forces. Therefore one may consider separately the scattering of the incident neutron by each of the nucleons of the deuteron. However, because of the coherence of the scattering, it is not cross sections that must be added, but amplitudes. Furthermore, the collision time is much less than the “characteristic nuclear time,” in the present case the period of revolution of the deuteron. Owing to this circumstance, the wave function of the deuteron does not have time to change during the collision; one may say that during the collision the action of the nuclear forces between the nucleons of the deuteron does not have time to manifest itself. Therefore the bound nucleons may be replaced by free ones, whose wave packets are constructed so as to give the correct momentum distribution. In other words, the role of the interaction of the nucleons in the deuteron is limited to the formation of their momentum distribution, while during the collision with the incident neutron this interaction is, as it were, switched off. These simplifications split the problem of the scattering of a fast neutron by the deuteron into two independent stages: the calculation of the scattering amplitudes on each of the nucleons of the deuteron, which are regarded as being in states with definite momenta, and subse-

corresponding summation of these amplitudes over the two nucleons and over all momenta present in the deuteron wave function.

Thus, it is possible to express the total cross section for scattering of neutrons by deuterons through the sum of the \(np\)- and \(nn\)-scattering cross sections and an interference term. Unfortunately, the value of the latter depends not only on the \(np\)- and \(nn\)-scattering cross sections, but also on scattering amplitudes, which are not determined experimentally and require some assumptions concerning nuclear forces. Therefore, from an analysis of the total \(nd\)-cross section alone it proves impossible to draw any conclusions about the magnitude of the \(nn\)-cross section.

By virtue of a number of considerations one may assert that the interference term is due mainly to elastic scattering. This prompted a calculation of the inelastic part of the \(nd\)-cross section^77. In this case the interference term is eliminated, and it becomes possible to express the total \(nn\)-cross section through: the experimentally measured \(np\)- and \(nd\)-cross sections; a factor expressing the small reduction of the \(np\)-cross section for a bound proton owing to the Pauli principle (it can be estimated from the angular distribution of fast protons arising in deuteron disintegration); and, finally, the probability that after the collision of one of the deuteron nucleons with the fast neutron, the former again forms, together with its old partner, a deuteron (calculated^77 from the wave function of the deuteron ground state). As a result, for the total \(nn\)-cross section at \(90\) MeV the value \((35 \div 40)\cdot 10^{-27}\ \mathrm{cm}^2\) was obtained, in good agreement with the value \(36\cdot 10^{-27}\ \mathrm{cm}^2\) of the total \(pp\)-cross section at the corresponding energy. A confirmation of the equality of the forces between two neutrons and two protons is also the fact, mentioned above, of the closeness of the total \(nd\)- and \(pd\)-cross sections at \(95\) MeV.

Besides finding the forces between two neutrons, the study of the scattering of fast nucleons by deuterons also makes it possible to obtain important information concerning the spin dependence of nuclear forces, since the deuteron is a system with a fixed relative orientation of the spins of the nucleons composing it. As I. Ya. Pomeranchuk showed^78,79, for this purpose it is especially convenient to consider comparatively the exchange collisions of nucleons with deuterons and with free nucleons, in which the greater part of the momentum is carried away by one particle. Unfortunately, the results obtained have not yet been subjected to numerical comparison with experiment.

Finally, let us touch upon one subtle effect in \(np\)-scattering. Owing to the presence in the \(np\)-interaction of noncentral tensor forces, the protons and neutrons produced in \(np\)-scattering and flying at an angle to the primary beam must be partially polarized. This polarization can be detected if one directs a beam of neutrons, produced by charge exchange on various targets, onto a hydrogen target and studies the dependence of the differential effective cross section of secondary scattering on the azimuthal angle.

coal (taking as the axis the direction of the beam of primarily scattered, i.e., charge-exchange, neutrons). Calculations were made\(^ {83}\) of the azimuthal asymmetry expected on the basis of the interpretation of \(p\)-\(p\) scattering carried out in\(^ {67}\), and the corresponding experiments were performed\(^ {84}\). It turned out that the polarization effect corresponding to that expected occurs if the first scattering of the protons takes place on a deuterium target, i.e., on the neutrons of the deuteron, and is absent if the first scattering takes place on a lithium target, i.e., on lithium neutrons. Thus, substantially different results were obtained for “free” neutrons (deuteron) and bound neutrons (lithium). The accuracy of the experiment, however, was not great enough for the result obtained to be considered a decisive confirmation of the proton-neutron interaction chosen in\(^ {67}\).

(To be continued in the next issue)

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