FROM CURRENT LITERATURE

QUASI-FREE SCATTERING OF NUCLEONS AND THE INTRANUCLEAR MOMENTUM DISTRIBUTION OF NUCLEONS

It is known that, when nuclei are bombarded by nucleons with energies of the order of hundreds of MeV, whose de Broglie wavelengths are already small in comparison with the dimensions of nucleons, the bombarding nucleons interact with individual nucleons of the target nucleus. If light nuclei are subjected to bombardment, with diameters close to the range of high-energy nucleons in nuclear matter, cases are very probable in which, after the very first collision between the bombarding nucleon and a nuclear nucleon, both nucleons participating in the collision leave the nucleus without further interaction with other nucleons. These cases correspond to quasi-free scattering of nucleons. The principal difference between quasi-free and free scattering of nucleons consists in the fact that, in free scattering, one of the nucleons is at rest in the laboratory coordinate system, whereas in quasi-free scattering the nucleons of the target nucleus possess definite momenta and energies. Therefore the angular and energy distributions of nucleons in free and quasi-free scattering prove to be somewhat different, and on the basis of this difference one can draw fairly definite conclusions about the intranuclear momentum distribution of nucleons. The presence of such a distribution manifests itself to one degree or another in almost all experiments on the bombardment of nuclei by high-energy nucleons or photons.

Various assumptions about the form of this distribution have been put forward, in particular, to explain the spectrum of deuterons knocked out by neutrons with an energy of 90 MeV from carbon nuclei, to describe the emission of high-energy protons in the nuclear photoeffect, to interpret the spectrum of \(\pi\)-mesons emitted at an angle of \(90^\circ\) to the primary beam when carbon nuclei are bombarded by photons with energies up to 330 MeV or by protons with an energy of 340 MeV. The intranuclear momentum distribution of nucleons determines, to a considerable extent, the angular distribution and spectrum of neutrons produced in the charge exchange of high-energy protons, and is also one of the reasons determining the applicability of the “bad geometry” method for determining the cross sections of inelastic collisions of high-energy neutrons.

What has been said is sufficient to show the broad application of the data on quasi-free scattering of high-energy nucleons.

During the last year five fairly detailed reports on such scattering have appeared in the literature. Four papers \(^{1–4}\) are devoted to quasi-free scattering of protons by deuterium, lithium, carbon, and oxygen nuclei. The fifth paper contains data concerning quasi-free

of neutron scattering by deuterium nuclei[^5]. Unfortunately, there are as yet no data on quasi-free \(nn\)-scattering. Meanwhile these data would in a definite sense be the most interesting in connection with the impossibility of observing free \(nn\)-scattering. The experimental technique of work on quasi-free scattering of nucleons was based either on recording, by means of coincidence circuits, the angular correlation of two nucleons (and comparing it with the analogous correlation in free scattering)[^1],[^2], or on measuring the energy distribution of protons or neutrons ejected at a definite angle to the primary proton or neutron beam[^5] (and comparing it with the corresponding distribution for free scattering).

In work[^1] quasi-free \(pp\)-scattering of protons with energy \(240\) MeV on deuterium nuclei was studied. The experiments were carried out with an internal, unextracted proton beam. Graphite and heavy paraffin were used as targets. From the activation of the copper foil due to the reaction \(\mathrm{C}^{12}(p,pn)\mathrm{C}^{11}\), the flux of protons incident on the target was determined. Coincidences of two protons flying out in directions the angle between which was close to \(90^\circ\) were recorded (at these energies, for free \(pp\)-scattering, the angles between the directions of the two protons, taking relativistic corrections into account, are somewhat less than \(90^\circ\)). Anthracene crystals located inside the chamber of the phasotron—one at the upper and the other at the lower pole of the magnet—served as counters.

Fig. 1.

In choosing the positions of the counters, of course, some deflection of the scattered protons by the magnetic field was taken into account. For a given arrangement of the counters, the heavy-paraffin target was moved tangentially along the proton orbit or radially along the radius of the chamber, until a position corresponding to the maximum counting rate of coincidences was established. Owing to a certain finite angular resolution, this maximum counting rate corresponded not to a single point of the target position, but to a certain interval of tangential or radial displacements (the “tangential” and “radial plateau” on the graph of coincidence counting rate versus target position).

From the difference of the data for the paraffin \((\mathrm{CD}_2)\) and graphite \((\mathrm{C})\) targets, the effect of quasi-free \(pp\)-scattering on deuterium nuclei was computed. Such scattering was investigated at three selected scattering angles of one of the protons—\(90\), \(67\), and \(40^\circ\) in the center-of-mass system (c.m.s.). Changing these angles required some displacement of one of the two scintillation counters that recorded coincidences.

The measurement for an angle of \(90^\circ\) was compared, in order to absolutize the cross sections of quasi-free \(pp\)-scattering, with the corresponding data for free \(pp\)-scattering (for which

\[ \sigma_{240\,\mathrm{MeV}}(90^\circ)=4.97\cdot 10^{-27}\ \frac{\mathrm{cm}^2}{\mathrm{steradian}} \]

), and the measurements for angles \(67\) and \(40^\circ\) were compared with the measurements of elastic \(pd\)-scattering at \(54\) and \(31^\circ\) (in the c.m.s.) performed in the same work[^1]. Fig. 1 shows the differential cross sections of quasi-free \(pp\)-scattering on deuterium nuclei obtained in this way, and also the curve according to which the integral cross section was evaluated, equal to \((11\pm3)\times 10^{-27}\ \mathrm{cm}^2\). Starting from the geometrical conditions of the experiments and the limits of the tan-

tangential plate, it was possible to calculate the minimum and maximum scattering angles of two correlated protons. From the maximum scattering angle there was calculated the maximum energy transmitted to the forward-flying neutron in quasifree \(pp\)-scattering, and from the minimum scattering angle—the maximum energy of the neutron flying backward. From the extent of the radial plate, the maximum neutron energy was calculated for an angle of \(90^\circ\) (in the laboratory system). The results of these calculations are given in Table I. The angles of emission of the neutrons are indicated in the laboratory system, the scattering angles of the protons—in the c.m. system.

Table I

“Upper limits” of neutron energies in quasifree scattering \((240\ \text{MeV})\) on deuterium nuclei

It is difficult to give an exact definition of the term “upper limit” used in the table. But, in any case, the number of neutrons with this energy limit was 5–10 times smaller than the number of neutrons with energies corresponding to the peak of their energy distribution. It is obvious that the energy received by neutrons in quasifree \(pp\)-scattering is relatively small and does not exceed \(1.5\%\) of the energy of the initial protons.

The “upper limits” of the neutron energy increase with increasing proton scattering angle, and at a neutron emission angle of \(0^\circ\) are about twice as large as the values for \(180^\circ\). The value of this energy for \(90^\circ\) at a scattering angle of \(40^\circ\) is somewhat biased upward. But the accuracy of the experiments is not so great as to allow any special conclusions to be drawn from this fact.

Fig. 2.

Very detailed study was made of quasifree scattering of protons by deuterium nuclei in work 3, along with scattering by carbon and oxygen nuclei. The proton energy was \(340\ \text{MeV}\). In this work the extracted proton beam was directed onto various targets placed in a concrete building with walls \(170\ \text{cm}\) thick, protected from background. A specially calibrated argon ionization chamber served as the beam monitor. Protons scattered at a definite angle entered a magnetic field, where they were separated according to momentum. A diagram of the installation for recording the energy distribution of the scattered protons is shown in Fig. 2. The main recording was made in 35 channels, associated with the same number of Geiger tubes arranged in a row beyond the exit from the magnetic field. The geometrical position of each tube determined, for the given magnetic field, the interval of energies of the protons recorded by it. To reduce the background of accidental coincidences, the triggering amplifier-

...and the counting system was effected by fourfold coincidences in proportional counters, three of which (“slit” counters) were located in the magnetic field, and the fourth was outside it, directly behind a row of Geiger tubes. The spectra of protons scattered at angles of 30 and 40° to the primary beam were investigated. The arrangement of the Geiger tubes made it possible to register scattered protons with energies from 90 to 350 MeV. The phasotron gave 66 pulses per second, each about 15–20 μsec in duration. After collimation the beam shape was rectangular, with a width of 3.5 cm and a height of 1.9 cm. The targets were larger than the beam, and in the horizontal plane had the form of a parallelogram. Targets of water, heavy water, graphite, and polyethylene were used; to determine the background, additional experiments were carried out without a target and with empty water containers. To simplify the difference counts underlying the data for hydrogen, deuterium, and oxygen, the dimensions of the various targets in the scattering direction were taken to be geometrically different, but corresponding to equal proton energy losses.

Fig. 3.

Since, along with protons, in each of the 35 channels of the magnetic spectrometer other charged particles with the same momenta could be registered, the authors discuss in detail the question of possible admixtures of deuterons, tritons, α-particles, and mesons in the beam of scattered protons. Analysis of the totality of the experimental data led the authors to the conclusion that, along with the protons, only deuterons could be registered, and even these only in negligible quantities. This conclusion was confirmed by direct experiments on pulse discrimination in different channels from two targets—D₂O and H₂O. In the channel corresponding to a proton energy of 315 MeV, one might have expected the appearance of deuterons elastically scattered at an angle of 40° with an energy of 177 MeV, as well as deuterons produced by capture of neutrons from oxygen nuclei by the bombarding protons. The amplitudes of pulses from such deuterons should have exceeded the amplitudes of proton pulses by a factor of 2.5. However, the deuterons could not be observed by pulse discrimination. For comparison of quasifree and free pp scattering, measurements were made of the spectra of protons scattered at angles of 30 and 40° (in the laboratory system) on hydrogen.

The half-width of the proton spectrum in the incident beam was 11 MeV. From this value, as well as from allowance for the resolving power of the spectrometer, one could calculate the expected energy distribution of the scattered protons. As is seen from Fig. 3, the calculations agree sufficiently well with the experiments on free pp scattering. Such agreement indicates a correct allowance for the resolving power of the spectrometer, which is determined by the width of the target, the width of the “slit” in the magnetic field of the first proportional counter, the width of the Geiger tubes, the energy losses of protons in the target, and scattering through small angles in the windows of the proportional counters and in air. The correctness of the rather cumbersome and laborious calculations of the resolving power...

FROM CURRENT LITERATURE

...of the spectrometer, confirmed by experiments on \(pp\)-scattering, ensures the reliability of the data on the quasi-free scattering of protons on deuterium, carbon, and oxygen nuclei. The results of experiments for such scattering are presented in Figs. 4 (\(\mathrm{H}, \mathrm{D}\), and \(\mathrm{C}\) for \(30^\circ\)), 5 (\(\mathrm{H}, \mathrm{D}\), and \(\mathrm{C}\) for \(40^\circ\)),

Fig. 4.

and 6 (\(\mathrm{C}\) and \(\mathrm{O}\) for \(40^\circ\)). Let us note that the relative scales along the ordinate in all the figures (including the two half-page figures, 3) are different. The interpretation of the data of work \(^{3}\), connected with the momentum distribution

Fig. 5.

of nucleons in nuclei, will be discussed below. For the moment we give in Table II the data of this work on the differential cross sections of quasi-free scattering of protons on deuterium—in comparison with the cross sections of \(pp\)- and \(pn\)-scattering.

The effect recorded by the method of work \(^{3}\) is the total result of quasi-free \(pp\)- and \(pn\)-scattering. On the basis of the data of Table II the authors \(^{3}\) point to the closeness of the \(p(D)\)-scattering cross section to the sum of the cross sections of free \(pp\)- and \(pn\)-scattering. Exact agreement of the cross sections in the laboratory system is not to be expected, if only because, owing to the momentum distribution of nucleons in the deuteron, a given laboratory scattering angle may be observed in a whole interval of energies of the bombarding particles and of scattering angles in the c.m. system. The conclusion about the additivity of the cross section of quasi-elastic scattering of protons by deuterium nuclei is valid only if: 1) the interference effects in scattering by the given...

angles; 2) nucleons in the deuteron spend a large part of the time separated by a distance greater than the wavelength of the bombarding proton, so that they may be regarded as two independent scattering centers; 3) the intervals of the relative energies of the nucleons and of the scattering angles in the c.m.s., corresponding to the given laboratory scattering angle, are small in comparison with their mean values; 4) the cross sections of free \(pp\)-scattering near \(340\) MeV depend little on the energy; 5) the differential cross sections of free \(pp\)-scattering in the c.m.s. do not depend on the angles, so that in the laboratory system the differential cross sections are proportional to the cosine of the scattering angle.

Fig. 6.

Table II

Differential scattering cross sections in the laboratory system
(in units of \(10^{-27}\ \text{cm}^2/\text{steradian}\))

Most of these conditions (for example, 3, 4, 5) are apparently fulfilled. The cross section for quasi-free scattering of protons by carbon and oxygen nuclei was not determined, since protons were recorded only with energies greater than \(90\) MeV, which clearly corresponds to an underestimation of the cross sections in comparison with the recording of complete spectra. However, from extrapolation of the spectra to the region of low energies it was possible to conclude that the cross section is close to the values predicted by the model of scattering by free nucleons. As is seen from Fig. 6, the yield of protons at an angle of \(40^\circ\) from an oxygen nucleus is approximately \(16/12\) of the yield from a carbon nucleus—in agreement with the proportionality of the yield to the number of nucleons.

Quasi-free scattering of protons by carbon nuclei was also studied in work \(^{4}\) at a proton energy of \(240\) MeV. The energy distribution

protons in the interval 60–190 MeV, emitted at an angle of 90° to the primary beam, was studied in this work by means of thick photoemulsions. The results4 are presented in Fig. 7 together with three theoretical curves, which will be discussed below. For comparison of theory and experiment, the experimental values of the cross sections, shown in Fig. 7 by rectangles, have been enlarged in the drawing by a factor of 2.2. Such an increase

Fig. 7.

Fig. 8.

in the experimental cross sections when compared with theory is due to the fact that the cross section for quasi-free \(p(C)\)-scattering, calculated from the model of identical nucleon–nucleon collisions, gave the value \(0.378 \cdot 10^{-24}\ \text{cm}^2\), whereas the experimentally observed cross section for inelastic collisions of neutrons with energy 270 MeV with carbon nuclei is only \(0.144 \cdot 10^{-24}\ \text{cm}^2\). In the last of the papers reviewed, the quasi-free scattering of protons was studied, by means of a coincidence arrangement, in quasi-free \(pp\)-scattering on lithium nuclei2. The experiments were carried out with an extracted beam of protons with energy about 350 MeV. When two scintillation counters placed at angles whose sum was close to 90° were arranged symmetrically with respect to the beam direction, correlated coincidences of two protons were observed. The dependence of the coincidence counting rate—at a fixed position of one of the counters (\(\Phi = 30^\circ\) and \(\Phi = 45^\circ\))—on the angle between the counters was measured. This dependence is shown in Fig. 8, where the abscissa axis represents the quantity \(\Psi = \Phi + \theta - 90^\circ\), i.e. the deviation of the sum of the angles of the positions of the two counters from 90°. The triangles in Fig. 8 take into account the angular resolution of the detecting system and, in shape (but not in position on the abscissa axis), correspond to free \(pp\)-scattering.

The broadening of the angular dependence in the case of lithium, as compared with free scattering, is connected with the momentum distribution of the nucleons in the lithium nucleus. The differential cross section of quasi-free pp-scattering on Li at \(30^\circ\) turned out to be equal to

\[ (39 \pm 4)\cdot 10^{-27}\ \frac{\text{cm}^2}{\text{steradian}} \]

— almost exactly three times greater than the corresponding cross section for free scattering (see Table II).

Experiments on quasi-free scattering of high-energy neutrons by deuterium nuclei\(^5\), in which the spectrum of secondary protons was recorded, differed from all the proton experiments described above in that the incident neutrons had a broad energy distribution. These neutrons were obtained by the charge exchange of \(350\ \text{Mev}\) protons on a beryllium target. The peak of the neutron spectrum corresponded approximately to \(280\ \text{Mev}\), and the half-width was about \(100\ \text{Mev}\). In most of the measurements carried out in\(^5\), with the aid of a coincidence telescope consisting of three or four proportional counters or scintillation counters, the total number of protons knocked out of the target at various angles from 4 to \(58^\circ\) was determined. Graphite, paraffin, water, and heavy water were used as targets; from the difference paraffin—graphite free np-scattering was investigated, and from the difference heavy water—water, the effect of quasi-free scattering. With the aid of copper filters placed in front of the last counter in the telescope, a proton-registration threshold was set equal on average to \(200\cos^2\Phi\ \text{Mev}\) (where \(\Phi\) is the scattering angle). Here it must be taken into account that the thickness of the scatterers (\(\mathrm{H_2O}\) and \(\mathrm{D_2O}\)) used for measurements at different angles was different: for small scattering angles, when the proton energy is higher, thicker scatterers were used. Accordingly, the registration threshold varied more or less smoothly and changed from \((195\text{–}205)\cos^2\Phi\ \text{Mev}\) (\(4^\circ\)) to \((158\text{–}243)\cos^2\Phi\ \text{Mev}\) (\(58^\circ\)).

In two cases—at angles of 4 and \(22.5^\circ\) (in the laboratory system)—the energy distribution of protons from free and quasi-free np-scattering was studied in detail with the aid of a magnetic spectrometer. The spectrometer used in\(^5\) was very close to that described above in connection with the work\(^3\), and we shall therefore not dwell on it. In connection with the work\(^3\), the question of deuteron contamination in magnetic spectrometry of protons has already been considered. Additional measurements performed in\(^5\), and, in particular, comparison of the readings of the counter telescope and the spectrometer, showed that in spectrometer measurements in channels corresponding to proton energies less than \(225\ \text{Mev}\), the deuteron contamination may somewhat distort the results. Further data from work\(^5\) on the spectra of protons knocked out by neutrons in free (H) and quasi-free (D) scattering at angles of 4 and \(22.5^\circ\) are given in Fig. 9. Comparison of the ratios of the proton yield from \(n(D)\)- and \(n(H)\)-interactions at different angles shows, as is seen from Fig. 10, that this ratio is constant and equal to about 0.7.

From Fig. 9 the ratio of the proton yield from \(n(D)\) and \(n(H)\) is obtained as somewhat greater than 0.7, which is apparently connected with some deuteron contamination raising the points from \(n(D)\) at proton energies below \(225\ \text{Mev}\).

Analyzing their data, the authors\(^5\) point to the close similarity of the proton spectra in the case of free and quasi-free np-scattering. They explain the decrease of the proton yield in quasi-free scattering by 30% in comparison with free scattering by the following considerations: 1. At small angles the emission of a proton in an \(n(D)\)-interaction increases, owing to the action of the Pauli prohibition, for in this case there is a high probability that both neutrons remain in the \(S\)-state, which is possible only for antiparallel spins. 2. The differential cross sections of np-scattering, averaged over the motion of the proton in the deuteron, may differ from the cross sections for the case of a stationary proton. In addition, the proton energy

Fig. 9.

Fig. 10.

in the case of an n(D)-interaction, it may decrease because of the formation of a dineutron, which will lead to a shift of the proton spectrum toward lower energies and to an exit from the counter aperture during registration.

The authors⁵ indicate that the theoretical treatment of their data testifies to a high probability of spin reversal in exchange np collision, and to the fact that the odd forces in the triplet pp state are not repulsive.

Before considering the conclusions from the papers reviewed concerning the momentum distribution of nucleons, let us briefly dwell on some additional differences between the data for free and quasi-free scattering of nucleons.

From Figs. 4 and 5 it is evident that the peaks of the proton spectra from quasi-free scattering are shifted toward lower energies in comparison with the peaks of free scattering, and this shift is the stronger the larger the scattering angle. In Fig. 8, the angles \(\Psi\) corresponding to the maximum counting rate of coincidences in quasi-free scattering are shifted to the left in comparison with free scattering, for which, for example, at \(\Phi = 45^\circ\) \(\Psi_{\text{max. count}} = -5.6^\circ\).

This shift is connected with three factors—the presence of a nuclear potential well, the binding energy of the nucleons in the nucleus, and the excitation energy of the residual nucleus.

Owing to the presence of the nuclear potential well, the energy of a nucleon scattered at an angle \(\Phi\) will no longer be \(E = E_0 \cos^2 \Phi\), but

\[ E' = (E_0 + V_0)\cos^2 \Phi - V_0 = E_0 \cos^2 \Phi - V_0 \sin^2 \Phi, \]

where \(V_0\) is the depth of the well. For \(V_0 = 30\) MeV the term \(V_0 \sin^2 \Phi\) is equal, for an angle of \(40^\circ\), to 12.4 MeV, and for \(30^\circ\), to 7.5 MeV. The shifts of the spectral peaks observed in the case of carbon in Figs. 4 and 5 are 12 MeV for \(30^\circ\) and 27 MeV for \(40^\circ\). An additional shift may be caused by the binding energy of the nucleons in the nucleus (in C¹² the binding energy of the proton is about 16 MeV, and that of the neutron about 18.6 MeV), as well as by the excitation energy of the residual nucleus. The latter, however, is small if both nucleons leave the nucleus without further collisions.

Let us finally consider the conclusions following from the papers reviewed concerning the momentum distribution of nucleons in nuclei. The authors of papers ², ³, ⁴ assumed various types of momentum distribution and checked to what extent the coincidence counting rates calculated from these distributions at different angles, or the energy spectra of the scattered protons, agree with experiment.

In paper ² the test of the momentum distribution was carried out very roughly. The starting relations used were:

\[ \mathbf{P} + \mathbf{p} = \mathbf{P}_1 + \mathbf{p}_1 \tag{1} \]

and

\[ E_P = E_{P_1} + E_{p_1} + E_{\text{nucleus}} + B, \tag{2} \]

where \(\mathbf{P}\) and \(\mathbf{p}\) are the momenta of the bombarding and nuclear nucleons inside the nucleus before and after the collision, \(E\) are kinetic energies, \(B\) is the sum of the binding energy of the proton knocked out of Li⁷ (10 MeV) and the excitation energy of the residual He⁶ nucleus (taken to be 5 MeV).

From (1) and (2) in the complementary case and for \(|\mathbf{p}| = |\mathbf{P}_{\text{nucleus}}|\) (which is obtained for \(E_{P_1}\) and \(E_{p_1} \gg V\), where \(V\) is the depth of the nuclear “well”) it follows that:

\[ 2p_1P_1 \cos(\theta + \Phi) = -2pP_1 \sin \psi = (1 + A^{-1})p^2 + 2Pp \cos \alpha + 2mB, \tag{3} \]

where \(\alpha\) is the angle between \(\mathbf{P}\) and \(\mathbf{p}\), \(\theta + \Phi\) is the angle between \(\mathbf{P}_1\) and \(\mathbf{p}_1\), \(m\) is the mass of the proton, and \(A\) is the mass number of the residual nucleus (\(A = 6\)).

Obviously, the probability of the encounter of two protons in the given plane in the angular interval from \(\alpha\) to \(\alpha+d\alpha\) is \(N_\alpha d\alpha=\dfrac{1}{2\pi}\,d\alpha\), which corresponds in the distribution in \(\psi\) to the quantity \(N_\psi d\psi\), whence

\[ N_\psi=\frac{1}{\pi}\cos\psi\left[\left(\frac{Pp}{P_1p_1}\right)^2-2\left\{\frac{mB}{P_1p_1}+\left(1+A^{-1}\right)\frac{p^2}{2P_1p_1}\right\}\sin\psi-\sin^2\psi\right]^{-1/2}, \tag{4} \]

where in deriving (4) a factor 2 has been introduced because of the identity of the particles, and the approximation

\[ 1\gg \left[ \frac{mB+\dfrac{1}{2}\left(1+A^{-1}\right)p^2}{Pp} \right]^2 . \]

If, finally, the momentum distribution of the protons is given in the form \(f(p)\), then the coincidence counting rate in the plane \(\mathbf{P}\mathbf{P}_1\) at the given angle \(\psi\) is proportional to

\[ \int_{0}^{p_{\max}} N_\psi(p)\,f(p)\,dS_p, \tag{5} \]

where the element of the momentum surface in the given plane is \(dS_p\sim p\,dp\).

Substituting into (5) the distribution for a Fermi gas, \(f(p)=\mathrm{const}\) for \(p_{\max}^2=2m\cdot20\,\mathrm{MeV}\), the authors\({}^{2}\) obtained a fairly satisfactory agreement of the calculated dependence of the coincidence counting rate on the angle \(\psi\) with experiment (see Fig. 8). However, besides the approximations already mentioned, they introduce one further crude assumption, namely that \(p_{P_1}=330\,\mathrm{MeV}\). For \(E_P=345\,\mathrm{MeV}\) and \(B=15\,\mathrm{MeV}\) such an assumption is valid only for \(E_{P_1}=E_{p_1}\) and \(E_{\text{nucleus}}=0\). Thus, proceeding from the fact that in the experiments\({}^{2}\) the registration angle of one of the protons was \(\Phi=45^\circ\), the authors adopted a relation characteristic of free scattering at \(45^\circ\). As a result, the agreement between calculation and experiment cannot be regarded as proof of the applicability of the Fermi-gas model, for the calculation is little sensitive to the form of \(f(p)\), being in general very crude.

In work\({}^{3}\), for comparison of experiment and theory, use was made of a formula given in\({}^{6}\) and directly relating the differential (in angles and energies) cross section for quasi-free scattering of nucleons to the momentum distribution:

\[ \frac{d^2\sigma}{dE\,d\Omega} = 2\int_{K}^{\infty} \frac{|a(p)|^2}{|\mathbf{p}_b-\mathbf{q}|} \sum_{i=1}^{n} \sigma_i \frac{m}{\hbar^2} p\,dp, \]

where \(N(p)=|a(p)|^2\); \(\mathbf{p}_b\) is the momentum of the bombarding particle, and \(\mathbf{q}\) that of the scattered nucleon; \(p\) is the momentum of a nuclear nucleon, \(K=|q-p_b\cos\Phi|/|\mathbf{p}_b-\mathbf{q}|\); \(\sigma_i\) is the differential cross section for scattering of the \(i\)-th nucleon from the nucleus, containing \(n\) nucleons, \(\Phi\) is the laboratory scattering angle, \(m\) is the mass of the nucleon. Agreement with experiment was tested for four types of momentum distribution. Two of them—for the case of deuterium—were obtained by expan-

by Fourier-transforming the wave function according to the relation

\[ a(p)=\frac{1}{(2\pi \hbar)^{3/2}}\int_{-\infty}^{\infty} e^{-i\frac{\mathbf{pr}}{\hbar}}\Psi(r)\,dV, \]

where \(N(p)=|a(p)|^2\). Such an expansion was applied both to the ordinary wave function of the form

\[ \Psi(r)=\frac{e^{-\alpha r}}{r}, \]

and to a function of the form

\[ \Psi(r)=\frac{e^{-\alpha r}}{r}+\frac{e^{-\beta r}}{r}, \]

which well approximates the numerical solution of the Schrödinger equation with a Yukawa potential. Here \((\alpha\hbar)^2=m\varepsilon\), where \(m\) is the mass of the nucleon, \(\varepsilon\) is the binding energy of the deuteron, and \(\beta/\alpha=7\).

For complex nuclei, a Gaussian distribution was tested,

\[ N(p)=e^{-\frac{p^2}{\chi^2\hbar^2}}, \]

with various values of \(\chi^2\hbar^2\), and, finally, the distribution proposed

Fig. 11.

by Chiu and Goldberger \(^{7}\) to explain the formation of fast deuterons when carbon is bombarded by neutrons with energy 90 MeV, namely:

\[ N(p)=\left|\frac{1}{p^2+\chi^2\hbar^2}\right|^2 \quad \text{for} \quad \frac{\chi^2\hbar^2}{2m}=18\ \text{MeV}. \]

All the distributions compared with experiment are presented in Fig. 11,

namely:

curve \(A\)—deuteron

\[ \Psi(r)=\frac{e^{-\alpha r}}{r}+\frac{e^{-\beta r}}{r}, \]

\[ N(p)=\left|\frac{1}{p^{2}+\alpha^{2}\hbar^{2}}-\frac{1}{p^{2}+\beta^{2}\hbar^{2}}\right|^{2}, \qquad \alpha^{2}\hbar^{2}=m\varepsilon,\quad \beta/\alpha=7, \]

curve \(B\)—deuteron

\[ \Psi(r)=\frac{e^{-\alpha r}}{r}, \]

\[ N(p)=\left|\frac{1}{p^{2}+\alpha^{2}\hbar^{2}}\right|^{2}, \qquad \alpha^{2}\hbar^{2}=m\varepsilon, \]

curves \(C, D, E\)—Gaussian distribution

\[ N(p)=e^{-\frac{p^{2}}{\chi^{2}\hbar^{2}}}, \]

where

\[ \frac{\chi^{2}\hbar^{2}}{2m}=12\ \text{MeV}\ (C),\quad 16\ \text{MeV}\ (D),\quad 20\ \text{MeV}\ (E), \]

curve \(F\)—distribution

\[ N(p)=\left|\frac{1}{p^{2}+\chi^{2}\hbar^{2}}\right|^{2}, \]

where

\[ \frac{\chi^{2}\hbar^{2}}{2m}=18\ \text{MeV}. \]

As is seen from Fig. 12, in the case of the deuteron the distributions \(A\) and \(B\) give such close results that it is difficult to give preference to either of them, especially since both distributions lead to agreement of the calculation with experiment. The divergence between the two variants affects only the tail of the distribution, for which there are no sufficiently accurate experimental data.

Fig. 12.

For complex nuclei (for example, carbon), good agreement with experiment is given by the Gaussian momentum distribution; moreover, as is seen from Fig. 12, the best agreement is achieved for

\[ \frac{\chi^{2}\hbar^{2}}{2m}=16\ \text{MeV}. \]

In Figs. 4 and 5 the theoretical curves for deuterium are plotted according to variant \(A\), and for carbon according to variant \(D\). The authors\(^3\) believe that any Gaussian distribution in the interval

\[ \frac{\varkappa^{2}\hbar^{2}}{2m}=14\text{--}19\ \text{MeV} \]

describes the experiments with carbon satisfactorily.

In work\(^4\) the experiment was compared with three theoretical curves. In Fig. 7, curve \(A\) corresponds to the degenerate Fermi-gas model \((E_{\max}=22\ \text{MeV})\), curve \(B\) to the Chew–Goldberger distribution, and curve \(C\) to the same distribution, but cut off at an energy of 72 MeV. Curve \(C\) is in best agreement with experiment. The calculation for the Fermi gas gives complete disagreement with experiment. Although in work\(^3\) the Fermi-gas model was not used for comparison with experiment, judging from agreement with experiment of calculations based on a Gaussian distribution, agreement with the Fermi-gas model is not to be expected there either. Therefore most of the data on the momentum distribution of nucleons in nuclei argue against the use of the Fermi-gas model and in favor of the Gaussian distribution.

To refine the form of the momentum distribution, further experiments are necessary—especially on determining the spectrum of the fastest secondary particles at different angles.

G. I.

CITED LITERATURE

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