ELASTIC SCATTERING OF NEUTRONS BY ATOMIC NUCLEI

T. A. Goloborodko

At the present stage of development reached by the science of the atomic nucleus, the task of systematically presenting the enormous experimental material and the numerous theoretical works scattered through the pages of scientific journals becomes an important one. Such a presentation is all the more necessary because nuclear physics has already become an academic discipline.

For a clear exposition of any science, it is necessary to systematize it, to divide it into sections united by a common idea. Nuclear physics still has no such firm systematization, and this makes its exposition difficult. It seems to us that the most natural division—at least at the present time—will be one based on the interaction of various particles with atomic nuclei ($\alpha$-particles, deuterons, protons, neutrons, and photons). Approximately such a division is carried out in the well-known monograph by Livingston and Bethe[^1].

Another serious difficulty for exposition is the incomplete reliability of the facts presented. There is no strict, generally recognized theory, and many conclusions drawn from experimental data may later prove to be erroneous. Such a situation, however, is inevitable.

The science of the nucleus is not a completed system, as, for example, the statistical theory of gases is. At this stage of development, the results of experiments possess the greatest reliability, provided that these experiments have been performed with the maximum possible accuracy. At present, the greatest attention must be paid to the systematic presentation of the most reliable experimental facts, grouping them, for ease of assimilation, according to some common features or connecting idea.

At present, it seems to us, the elastic scattering of neutrons by atomic nuclei can already be set apart as such a section of nuclear physics. In the aforementioned monograph, published in 1937, little space was allotted to it, since the experimental material

there was then very little. But in the ten-odd years that have passed since then, a large number of works have been done in this field, leading to very interesting and unexpected results.

1. SOME GENERAL CONSIDERATIONS AND PREMISES

As is well known, the many-body problem cannot be solved rigorously either in classical or in quantum mechanics. Up to now this problem has nevertheless been solved more or less successfully: in astronomy and in the atomic system—by means of perturbation theory, in the mechanics of gases—by means of the introduction of statistical laws. In the theory of the atomic nucleus these tried methods have proved of little use.

For the successful application of statistical laws it is necessary to deal with a system consisting of a very large number of particles. But the most complex nuclei contain only two hundred-odd particles. This number is certainly far too small for the successful application of the laws of statistical physics; and if the number of particles, as in light nuclei, is 10–20, then even their approximate application is completely impossible. If, moreover, one takes into account that, in applying statistical laws, we must from the very outset ignore all structure in the nuclear system, it will become obvious that with the aid of statistical theory we may hope only in some cases to obtain crudely approximate solutions.

But, on the other hand, the method of small perturbations is not applicable in full measure, since any particle, upon joining the nucleus, interacts with all nuclear particles with the same force with which they are bound to one another.

Thus a rather bleak picture is formed. For a successful solution of the nuclear problem it is necessary to invent some completely new methods of mathematical analysis, or, by constructing various models of the nucleus, subsequently to test them by experiments. It is possible that in the future such methods will be found, but for the time being, in their absence, the only path that remains is the application of known methods. They are in fact applied in the modern theory of the nucleus. On the basis of what has been said, one must always treat the conclusions of this theory with great caution, since they may be either entirely incorrect or only crudely approximate. Taking these difficulties into account, one may think that the empirical path will prove even more fruitful than conclusions of a theory based on doubtful premises.

After N. Bohr’s indications of the necessity, in any interaction of particles with the nucleus, of dealing with the many-body problem, Breit², Wigner³, Bethe and Placzek⁴, and other theorists created the “dispersion theory” of scattering, which nevertheless tries

to apply the perturbation method. This theory had little success. The application of the fundamental equation to various particular cases proved very difficult because of the already mentioned fundamental and mathematical difficulties.

Then Bohr\(^5\), Frenkel\(^6\), Weizkopf\(^7\), and others constructed a statistical theory of the nucleus, treated as an evaporating liquid drop. The transition of the initial nucleus, after the addition of an external particle, into a complex compound nucleus is conceived as heating of the drop; emission of a particle—as evaporation. This theory is mathematically simple, as a result of which it has been developed more than the dispersion theory; however, as has already been said, from the very outset it refuses to take account of any structure of the nucleus and is completely inapplicable to light nuclei. With the aid of this theory one can satisfactorily explain those phenomena that occur in the interaction of heavy nuclei with particles possessing high energy, i.e., when the internal structure of the nucleus plays a considerably smaller role than in interaction with particles possessing low energy\(*\).

It is not difficult to see that, for the case of elastic scattering of neutrons with low energy (approximately from \(0.03\) eV to \(3\) MeV), with which the numerous experiments described in this article were chiefly carried out, the application of the theory of the evaporating liquid drop is very doubtful. In this almost unique case, the dispersion theory has been applied with some success. We shall give here a brief presentation of its conclusions, in order then to compare them with the experimental data.

2. THEORETICAL INTERPRETATION OF ELASTIC NEUTRON SCATTERING

In accordance with N. Bohr’s idea\(**\), every nuclear process can be described in the following way. A particle \(P\) falls on a nucleus \(A\), which is in the state \(p\). A compound nucleus \(C\) is formed, possessing a certain number of excited energy levels \(W_r\). Then some particle \(Q\) is emitted, leaving the nucleus \(B\) in the state \(q\).

\(*\) The completely unsatisfactory state of contemporary nuclear theory is very vividly characterized by the words of one of the most prominent experts on this theory—Bethe. At the beginning of one of his latest articles he writes\(^8\): “At the present time we wish to attempt to develop a theory of the probability of particles sticking to a nucleus. For this purpose we shall consider the case of dense levels and try to develop the theory of the compound nucleus, neglecting all effects of individual levels. This theory will be related to the dispersion theory as the classical theory of a solid body, describing its phenomenological constants—elastic moduli, conductivities, and elasticities—relates to the quantum theory, which takes account of each (electronic and vibrational) quantum state of a crystal.”

\(**\) The two original papers of Bohr can be found in Uspekhi Fizicheskikh Nauk\(^9,10\). This theory is also set forth very fully in Bethe’s article\(^11\).

If the particles \(P\) and \(Q\) are identical, we have the case of scattering. The probability of any nuclear process is then calculated by the methods of perturbation theory. In view of the fact that a double transition always takes place, it is necessary to calculate the second-order perturbation. The basic formula will have the form*):

\[ \sigma = 4\pi^3 \bar{\lambda}^{\,2} \left| \sum_r \frac{ H^{A P p}_{C_r}\, H^{C_r}_{B Q q} }{ W_{A p}+W_P+E_P-W_{C_r}+\frac{1}{2} i\gamma_r } \right|^2 , \tag{1} \]

\(\sigma\) is the cross section of the given process, characterizing its probability. For elastic scattering, \(\sigma\) is defined as the ratio of the number of particles scattered into a given solid angle to the number of incident particles. If this angle is equal to \(4\pi\), then \(\sigma\) represents the total cross section for elastic scattering. \(\bar{\lambda}\) is the wavelength of the incident particle divided by \(2\pi\)

\[ \left(\bar{\lambda}=\frac{\hbar}{\sqrt{2ME_P}}\right), \]

\(H^{A P p}_{C_r}\) is the matrix element of the transition to the compound nucleus, and \(H^{C_r}_{B Q q}\) is the matrix element of the transition from the compound nucleus to the residual nucleus (\(H\) is the Hamiltonian of the interaction of the particle with the nucleus). \(W_{A p}\) is the potential energy of nucleus \(A\) in state \(p\); \(W_P\) and \(E_P\) are the potential and kinetic energy of the incident particle; \(W_{C_r}\) is the potential energy of the compound nucleus in state \(r\), and \(\gamma_r\) is the total effective width of any level \(r\) in the compound nucleus (it is the sum of the widths corresponding to the emission of particles of various kinds). For \(\gamma_Q^r\), corresponding to the emission of particle \(Q\), the relation holds

\[ \gamma_Q^r = 2\pi \left| H^{C_r}_{B Q q} \right|^2 . \tag{2} \]

When the energy of particle \(P\) is close to one of the resonance levels of the compound nucleus, the influence of all the other levels may be neglected, and the general formula (1) is transformed into the simpler one:

\[ \sigma_Q^P = \pi \bar{\lambda}^{\,2} \frac{\gamma_P^r \gamma_Q^r} {(E_P-E_r)^2+\frac{1}{4}\gamma_r^2} . \tag{3} \]

The application of this formula to concrete cases is well justified by the results of experiments.

In the case of elastic scattering of neutrons (or other particles) the general formula (1) will have the form

\[ \sigma = \pi \bar{\lambda}^{\,2} \left| \sum_r \frac{\gamma_p^r} {E_P-E_r+\frac{1}{2} i\gamma_r} \right|^2 . \tag{4} \]

*) The simplest derivation of this formula is given in the paper by Kapur and Peierls\(^{12}\). For our purpose—the comparison of experimental data with the theoretical ones in the case of elastic scattering—we considered it unnecessary to give its complete derivation.

Here, as in formula (3), \(W_{C_r}-W_{A_p}-W_P\) has been replaced by the resonance energy of the given level \(E_r\).

In applying these formulas to various specific cases (for example, resonance capture of neutrons), the “one-level formula” (3) is of the greatest importance. In calculating \(\sigma\) for elastic scattering, however, we shall nevertheless attempt, following Bethe \(^{11}\), to apply the general formula (4), since in this case we must take into account, in addition to the resonance effect, also scattering when the energy of the incident particle \(P\) is far from the energies of the resonance levels \(r\). In the course of these calculations there will become clear also those mathematical difficulties which do not permit the successful application of the general formula (1) to various specific cases.

The first difficulty arises in calculating the matrix element \(H_r^{Pp}\), defined by the relation

\[ H_r^{Pp}=\int \psi_{C_r}^{*}\,H\,\psi_{A_p}\psi_{P_p}\,d\tau . \tag{5} \]

Here \(\psi_{A_p}\), \(\psi_{P_p}\), and \(\psi_{C_r}\) are the wave functions of the initial nucleus, the incident particle, and the compound nucleus; \(\psi_{A_p}\) and \(\psi_{C_r}\) may be regarded as known, at least in principle; as for \(\psi_{P_p}\), here we have to make a choice among three possibilities:

1. \(\psi_{P_p}\) is the solution of the Schrödinger equation for one particle, with an attractive potential.
2. The same, with a repulsive potential.
3. \(\psi_{P_p}\) is a plane wave.

This ambiguity arises owing to the fact that, as has already been noted, we cannot solve rigorously the problem for the complete system: nucleus + particle. Of course, this ambiguity pertains only to our method of approximate solution of the problem, and not to the problem of the interaction itself. With an unfortunate choice of the zeroth-order potential and the zeroth-order wave function, we shall have to calculate a large number of approximations before arriving at the correct result.

It is quite obvious that the convergence in perturbation theory must differ greatly for these three particle potentials. If we choose an attractive potential, then (5) will have a value larger than for a plane wave, since in this case resonance can already occur for the wave function of one particle. A repulsive potential, on the contrary, will reduce (5) more than the zero potential of a plane wave, since with this potential \(\psi_{P_p}\) will decrease exponentially from the surface of the nucleus inward and will be smaller than the plane-wave function even at the very surface of the nucleus. From these considerations it follows that the best choice will be a repulsive potential. With this choice the basic requirement imposed on the zeroth-order function is fulfilled: that it

led to the higher approximation being small in comparison with the first approximation.

The choice of a repulsive potential can be justified logically by the fact that the incident particle must give up its energy to the nuclear particles immediately after contact with the surface of the nucleus. With a rapid loss of energy along the path of the particle into the nucleus, its wave function will decrease so quickly that we can no longer represent a free particle inside the nucleus. On the basis of these considerations, it seems quite logical to choose such a wave function of the incident particle as would, so to speak, act from inside outward within the nucleus.

Let us choose the magnitude of the potential \(V_0\) to be of the order of the nuclear interaction (about \(10\ \mathrm{MeV}\)), and its range equal to the radius of the nucleus. To investigate the influence of the higher approximation of perturbation theory, let us calculate the elastic scattering of particles with wavelength greater than the nuclear radius \(R\). The scattering may be divided into three parts.

1. Zero-order scattering, containing also the zero-order functions. Since the particle is slow*), the repulsive potential is practically impenetrable for it. The nucleus acts like a hard sphere of radius \(R\). Consequently, for zero-order scattering we shall have

\[ \sigma_0 = 4\pi R^2. \tag{6} \]

1. Second-order scattering, produced by energy levels close to the energy of this particle. This is resonance scattering. In what follows we shall denote the cross section of this scattering by \(\sigma_{\mathrm{res}}\).

2. Scattering produced by the spacings between the energy levels of the compound nucleus. This scattering depends only weakly on the energy of the particle incident on the nucleus, and may be included in the zero-order scattering.

For given states of the initial nucleus and of the incident particle, the matrix elements \(\gamma^r_{Pp}\), on average, will not depend on the energy of the compound nucleus. This enables us to simplify the calculations by considering, along with slow particles, also fast particles. The sum over \(r\) in (4) will be produced mainly by the energy region with many levels, i.e. at high particle energies. Since for high energies \(\gamma\) drop out (the intervals between levels disappear), the main part of the sum will be produced by the energy region immediately bordering the region in which \(\gamma\) drop out. Let us denote the energy of the particle at which the decrease of \(\gamma\) begins,

*) The theoretical concept of a “slow particle” is based on comparing the wavelength of this particle with the mean nuclear radius. The region of “slow neutrons,” for example, under this definition may extend approximately up to \(1\ \mathrm{MeV}\). Experimentally, as is known, neutrons with energies approximately up to \(100\ \mathrm{eV}\) are called slow neutrons.

\(E_{\mathrm{cr}}\) (critical energy). Then in (4) we can replace the resonance denominator by the constant quantity \(E_{\mathrm{cr}}\) for all terms of the sum.

Substituting (5) into (4), we obtain:

\[ \frac{-\sum\limits_r \gamma'_{P_p}}{E_{\mathrm{cr}}} = -\frac{2\pi}{E_{\mathrm{cr}}} \sum_r \left| \int \psi_C^* H\psi_{A_p}\psi_P\,d\tau \right|^2 . \tag{7} \]

The volume element \(d\tau\) can be expressed as the product \(d\tau_A d\tau_P\), where \(d\tau_A\) is the volume element in the phase space of all particles of the initial nucleus, and \(d\tau_P\) that of the incident particle. The sum over \(r\) can be represented in the form

\[ \sum_r \left| \int \psi_C^* H\psi_{A_p}\psi_P\,d\tau \right|^2 = \int \left|H\psi_P\psi_P\right|^2 \,d\tau . \tag{8} \]

Integrating (8) with respect to \(d\tau_A\) and using the normalization with respect to \(\psi_{A_p}\), we can write:

\[ \int \left|H(r_A,r_P)\right|^2 \left|\psi_{A_p}(r_A)\right|^2 d\tau_A = U^2(r_P), \tag{9} \]

where \(U\) is a certain irregular function of the coordinates of the particle \(r_P\), having a magnitude of the order of the binding energy of particles in the nucleus. Introducing the value (9) and \(E_{\mathrm{cr}}\) into (4), we obtain:

\[ \sigma_{\mathrm{res}} = \frac{4\pi^3\hbar^2}{E_{\mathrm{cr}}^2} \left[ \int U^2\psi_P^2\,d\tau_P \right]^2 . \tag{10} \]

The wave function \(\psi_P\), normalized to unit energy, for the kinetic energy of the particle \(E_P\) and zero angular momentum, is a solution of the Schrödinger equation for one particle with potential \(V_0\). For \(r<R\) it will have the form:

\[ \psi_P = \frac{1}{\pi(2\hbar v)^{1/2}r} \left[ \frac{E}{V_0} \right]^{1/2} e^{k(r-R)}, \tag{11} \]

where \(k=(2MV_0)^{1/2}/\hbar\), and \(M\) is the mass of the particle (neutron). Substituting (11) into (10), with \(d\tau_P=4\pi r^2dr\), we obtain:

\[ \sigma_{\mathrm{res}} = \frac{\pi\hbar^2}{2M} \frac{(U^2)_{\mathrm{av}}^2}{E_{\mathrm{cr}}^2 V_0^2}. \tag{12} \]

Replacing \(U\), \(E_{\mathrm{cr}}\), and \(V_0\) by suitable numerical values, we can estimate \(\sigma_{\mathrm{res}}\). We have already taken \(V_0\sim 10\ \mathrm{MeV}\), and we set \(E_{\mathrm{cr}}\sim 8\ \mathrm{MeV}\) and \(U\sim 15\ \mathrm{MeV}\) (\(U\) must be greater than \(V_0\), since it contains \(V_0\) and the nuclear potential energy of binding of particles in the nucleus). Substituting these values into (12), we find

\[ 1.03\cdot0.79\cdot10^{-30} \frac{\mathrm{erg}^2\,\mathrm{cm}^2}{\mathrm{g}\cdot\mathrm{MeV}} = 1.03\cdot0.79\cdot0.63\cdot10^{-24}\ \mathrm{cm}^2 = 0.51\cdot10^{-24}\ \mathrm{cm}^2 . \tag{12a} \]

This quantity, as we shall see from the experimental works, is on average 10 times smaller than \(\sigma_0\). This indicates that the choice of a repulsive potential was made correctly. For a plane wave we would obtain\({}^{11}\)

\[ \sigma_{\mathrm{res}} = \frac{16\pi}{9\hbar^4}\, \frac{(U^2)_{\mathrm{av}}\rho M^2R^6}{E_{\mathrm{cr}}^2}. \tag{13} \]

Analogous calculations with several changed values of \(U\), \(V_0\), \(E_{\mathrm{cr}}\) lead to the value \(\sigma_{\mathrm{res}} \simeq 80\sigma_0\). For an attractive potential we obtain:

\[ \sigma_{\mathrm{res}} = \frac{\pi R^2}{2}\, \frac{(U^2)_{\mathrm{av}}}{E_{\mathrm{cr}}^2 V_0^{1/2} E_p^{\prime\prime\,1/2}} . \tag{14} \]

Since in this case the energy of the particle \(P\) enters the denominator, \(\sigma_{\mathrm{res}}\) increases rapidly as the neutron energy decreases.

The method set forth for solving the problem seems to be, to a considerable extent, an artificial adjustment for explaining the experimental data. That this artificiality is present is not denied by the author of this conclusion\({}^{*}\); however, apparently there is no better path. The repulsive potential is, of course, only a mathematical fiction introduced for the success of the calculation. Experiments on neutron scattering indicate that elastic scattering is in most cases most probable, whereas from the standpoint of the formation of a compound nucleus with preliminary distribution of the energy among all particles of the nucleus, and then its concentration on one particle, it follows that this process must have a negligible probability. To reconcile theory with experiment only two paths remain: 1) to exclude elastic scattering from the general scheme of the formation of an intermediate compound nucleus, and 2) while formally retaining the previous concepts, to create a suitable mathematical formulation. This latter path has been chosen in the present derivation. Further on, examples will be given of attempts to solve the problem by the first path.

3. SCATTERING OF NEUTRONS BY ATOMIC NUCLEI. EXPERIMENTAL DATA

A. Dependence \(\sigma = f(A)\)

The systematic study of the interaction of neutrons with atomic nuclei begins with the extensive work of Dunning and his collaborators on measuring the \(\sigma\) “absorption” of neutrons having thermal energy\({}^{13}\).

\({}^{*}\) “In our problem,” says Bethe, “we must use a certain clever trick in order to obtain rapid convergence, in spite of the large interaction. It is, of course, unclear whether our wave function is the most suitable for the final calculation, but in any case it will be more suitable than other functions”\({}^{11}\).

To obtain slow neutrons, a source of fast neutrons \((\mathrm{Rn}, \mathrm{Be})\) was placed at the center of a paraffin sphere, and the “absorber” was located directly in front of the ionization chamber. It is quite obvious that with such an experimental arrangement it is impossible to separate the effects of scattering and absorption. If, in some element, neutrons are strongly absorbed, so that the effect of scattering may be neglected, then the calculated absorption cross section will be correct; but in the converse case—strong scattering and weak absorption, or approximate equality of these processes—the value found for \(\sigma\) will not give us an idea of each process separately. Moreover, owing to the close position of the scatterer to the chamber, a large number of scattered neutrons will enter the chamber and systematically reduce the cross sections. This systematic error, made by Dunning, is all the more incomprehensible since the author already knew from earlier works, which he himself cites, that for most elements elastic scattering plays the principal role.

From the results given in Table I (2nd column) we see that for individual elements the measured \(\sigma\) are very large, and they must without doubt be attributed to absorption; but for most elements \(\sigma\) fluctuate around a certain average value, and which process they characterize is unknown. By careful comparison with later, more accurate data, it can be established that in those cases where absorption may be neglected, the found \(\sigma\) characterize elastic scattering; and to eliminate the above-mentioned error they must be increased by approximately 20%, as, for example, for Be, C, O, F, Mg. However, with considerable but not anomalously large absorption, it is not possible even approximately to separate absorption from scattering, although it is obvious that the latter predominates for most elements.

To separate these processes it was necessary to measure purely the elastic scattering of thermal neutrons, i.e. to measure not the number of neutrons passing through the scatterer, but the number of neutrons scattered to the sides. The most convenient method for this is the registration of neutrons scattered backward, at an angle \(\geq 90^\circ\) with respect to the direction of motion of the incident neutrons. It is quite obvious that in this case absorption will be completely excluded. Such measurements were carried out in a number of works by Mitchell et al.\(^{14}\). In these experiments the same neutron source \((\mathrm{Rn}, \mathrm{Be})\) was placed inside a paraffin parallelepiped at a depth of 6 cm. Directly on the surface of this paraffin block a silver detector was placed, and after irradiation by slow neutrons emerging from the paraffin, its activity was measured with a Geiger–Müller counter. After this, an analogous measurement was repeated with the scatterer under investigation placed directly on the detector. The areas of the scatterer and detector were equal

Table I

Elastic-scattering cross sections of slow neutrons by atomic nuclei

Element	Dunning et al.13	Mitchell and Murphy14	Goldhaber and Briggs18	Various investigators20–24
	\multicolumn{4}{c}{σ·10²⁴ cm²}
D	4,0		7,0	3,3
Li	45,0
Be	5,3		6,9	6,10
B	360
C	4,1	3,4	4,8 (4,8)*	4,9
N	11,3		8,2	∼ 3
O	3,3		4,2
F	2,5		4,1	3,7
Na	4,2		3,6	∼ 4,7
Mg	3,5	3,0	4,2	∼ 8,8
Al	1,5	1,0	1,6	1,5
Si	2,5		1,7	4,2
P	14,7		10,4 (13,6)	9,1
S	1,4	0,9	1,1	1,3
Cl	39		10
K	8,2		1,5
Ca	11,0		9,5
Ti	11,9		6,2
V	10		4,0
Cr	4,9	1,4	3,6
Mn	14,3	2,2	2,1
Fe	12,0	10,6	10,3 (13,6)	11,1
Co	35		∼5	3,7
Ni	15,4	18,0	12,4 (19,7)	16,1
Cu	7,5	8,3	8,6 (11,9)	8,3
Zn	4,7	3,7	5,2	4,2
Ga			<4
Ge	75
As	8,6		7,1	7,3
Se	19,8		12,7 (20,4)	12,0
Br	11,8		7
Sr	9,0		9,5
Y			14
Zr	16,7		16
Nb	14		5
Mo	7,1		6,7
Ru	12,5		5,9
Rh	115
Pd	10,0			3,0
Ag	55	6,3	(6,3)	9,5

*) The numbers placed in parentheses in the 4th column denote total σ, measured by Goldhaber and Briggs by the method of neutrons passing through a scatterer.

Continuation of Table 1

Element	Dunning et al. \(^{13}\)	Mitchell and Murphy \(^{14}\)	Goldhaber and Briggs \(^{15}\)	Various researchers \(^{20-24}\)
	\multicolumn{4}{c}{\(\sigma \cdot 10^{24}\ \mathrm{cm}^{2}\)}
Cd	3300	1,2		3,7
Sn	4,0	4,1	4,9	5,7
Sb	8,1		6,5	3,0
Te	8,2		5,3
J	94			2,7
Ba			8,2
La	80
Ce	25
Pr	25
Nd	220
Sm	4700
Eu	1000
Gd	30000
Ta	27		\(\sim 10\)
W	23		7,8	9,3
Re	89
Os	27		10,8
Ir	285
Pt	25		8,3 (21,5)
Au	88
Hg	380	4,7		12,0
Tl	11		14,2
Pb	8,6	7,7	12,9 (12,5)	9,6
Bi	8,2	10,2	8,9 (8,9)	8,7

By measuring the scattering of neutrons for different thicknesses of the scatterer, it was possible to construct a graph \(\sigma = f(\delta)\), where \(\delta\) is the thickness of the scatterer, and from this to find \(\sigma\). The linear portion of the curve indicated up to what thicknesses of the substance under investigation the scattering could be regarded as single scattering.

Analyzing this experimental setup, we note that it is not free from a certain systematic error. It is easy to see that under these conditions neutrons may be reflected many times from the scatterer and the paraffin before they are absorbed by the detector. This effect will systematically decrease \(\sigma\). From column 3 of Table 1, where the data of Mitchell et al. are given, one can see, on comparing them with the more accurate data of column 4, that this supposition is justified.

The problem of the elastic scattering of thermal neutrons was finally solved in 1937 by Goldhaber and Briggs \(^{15}\), who placed the source at a distance of 25 cm from the ionization chamber, and shielded the entire path of the neutron from the source to the scatterer and from the scatterer to the chamber with boron. The neutrons fell on the scatterer under

at an angle of 45°, and the flux of reflected neutrons was recorded by a boron chamber at the same angle. Thanks to this method the chamber was shielded from the direct flux of thermal neutrons, and the increase of ionization in it was produced only by neutrons elastically scattered by the given element. The results of these measurements are given in the 4th column of Table I.

From the description of the method of this last work it can be seen that the authors took into account, as it were, all the systematic errors allowed by previous investigators, and their results should be regarded as correct. Such they were considered up to 1940, when works appeared indicating a dependence of the scattering cross sections of thermal neutrons on the aggregate state of the scatterer \(^{16-19}\). In the scattering of neutrons with such a large wavelength, interference effects from the various crystal lattices of the scatterers already begin to appear. This circumstance was not taken into account by Goldhaber and Briggs and may have led to considerable errors in the found \(\sigma\). In order to get rid of this influence of the structure of the scattering substance, the simplest course would be to repeat the work of Goldhaber and Briggs with neutrons of somewhat higher energy, for example \(1\)—\(3\) eV (group neutrons). Such a small change in neutron energy will not affect the measured values of \(\sigma\), while the influence of the aggregate state of the substance of the scatterers at such neutron energies is completely annulled.

These measurements, however, present considerable difficulties, owing to the fact that in the total flux of slow neutrons the fraction of group neutrons amounts to no more than \(2\)—\(3\%\). For the successful execution of all the measurements a prolonged investigation with powerful neutron sources would be required. Up to now no such systematic work has been carried out, but separate measurements have been made by various investigators \(^{20-24}\). In view of the fact that some of these measurements were performed by the method of neutrons transmitted through the scatterer, and others by the method of reflected neutrons, their results, given in the last column of Table I, are rather difficult to interpret; nevertheless the general conclusion will apparently be that the fundamental measurements of Goldhaber and Briggs deviate little from the truth, and only for certain elements, as for example for Si, may one suspect that the discrepancy is due to the reason sought.

The scattering of fast neutrons was also first studied by Dunning \(^{25}\). Using \((\mathrm{Rn}, \mathrm{Be})\) neutrons and placing the scatterer between the ionization chamber and the source, Dunning investigated 14 elements from H to Pb and found a monotonic increase of \(\sigma\) with increasing atomic number of the element. In the calculations he likewise did not take into account those neutrons which, after scattering, nevertheless enter the chamber. However, owing to the small radius of the chamber and the considerable distance to the scatterer, this error in the present case changes \(\sigma\) by no more than \(3\%\). In view of the fact that the gas ionization chamber

in these measurements was little sensitive to neutrons with energies below 0.5 MeV, which occur in the initial part of the neutron spectrum \((\mathrm{Rn}, \mathrm{Be})\); neutrons with energies greater than 3–5 MeV in this spectrum, as Dunning himself showed, are very few. The author believes that the values of \(\sigma\) found characterize the scattering of neutrons with energies from 3 to 5 MeV, which apparently predominate in the spectrum of neutrons produced in this reaction. The results of these measurements are given in Table II and in Fig. 1.

Fig. 1. Scattering of neutrons of different energies by atomic nuclei of various elements. Dependence \(\sigma = f(Z)\).

Legend in the figure:
\(\Delta\) — neutrons of thermal energy;
\(\times\) — \(E \sim 0.2\) MeV;
\(\square\) — \(E \sim 2.5\) MeV;
\(\circ\) — \(E \sim 3\text{–}5\) MeV;
\(\bullet\) — \(E \sim 90\) MeV.

The monotonic increase of \(\sigma\) is easy to explain, as Dunning did, by a simple increase in the volume of the scattering nucleus. Consequently, neutrons with energies of 3–5 MeV appear to be scattered according to the simple law (6). Resonance oscillations, if they exist, do not go beyond the statistical errors of the measurements.

For complete confidence in such a conclusion, which is of great importance for nuclear theory, more thorough checks with energetically homogeneous neutrons were, of course, necessary. After a detailed investigation by many experimenters of neutrons from the reaction \((\mathrm{D}, \mathrm{D})\), which showed their energetic homogeneity, \(^{26\text{–}30}\) the first such checks were carried out in 1939 in two studies. \(^{31\text{–}32}\) In both these works, as in all subsequent ones, the method of measurement remained unchanged: the scatterer was placed halfway between the neutron source and the detector (a gas chamber). Owing to the high power of the sources, the distance between the target irradiated by deuterons, the scatterer, and the detector could be significantly

considerably increase. In the first work the target of heavy ice, bombarded by deuterons at a discharge-tube potential of 300 KeV, was located at a distance of 40 cm from an ionization chamber of small dimensions; in the second, at a distance of 30 cm. All corrections for non-parallelism of the neutron beam, for the change in neutron energy within the thickness of the target, for fluctuations of the discharge-tube potential, and for the composition of the deuteron beam were carefully taken into account, and the statistical errors of the measurements did not exceed 5%. (In the preceding works they were 10–15%.) In the first work the neutron energy was \(2.4 \pm 0.12\) MeV; in the second, \(2.88 \pm 0.04\) MeV and \(2.46 \pm 0.04\) MeV.

The results of these works are given in Table II and in Fig. 1. With complete obviousness one observes the same irregular scatter from element to element as was found with thermal neutrons, but the amplitude of the oscillations is considerably smaller. Thus, the first checks showed that with decreasing neutron energy oscillations of \(\sigma\) appear, but the general tendency of monotonic increase is preserved. From these experimental results, therefore, a certain empirical law in the functional dependence \(\sigma=f(A)\) begins to take shape. To confirm this law it was extremely important to investigate, with the greatest possible completeness, the scattering of neutrons of intermediate energy, say from 0.1 to 0.5 MeV. These investigations were carried out chiefly by Soviet physicists\(^{33—8}\).

In the given energy interval the most suitable sources of homogeneous neutrons are the nuclear reactions: \((\mathrm{C}, \mathrm{D})\), \((\gamma\mathrm{ThC}^{\prime\prime}, \mathrm{D})\), \((\gamma\mathrm{ThC}^{\prime\prime}, \mathrm{Be})\), and \((\gamma\mathrm{RaC}, \mathrm{Be})\). The last three have a very low intensity in comparison with the first source; however, the photoneutrons obtained from them possess an important advantage. In them the energy spread produced by the different direction of emission with respect to the direction of the \(\gamma\)-quantum reaches on the average only 15%, whereas for neutrons obtained by bombarding various targets with deuterons in discharge tubes, the energy spread, produced by many causes, reaches 0.1 MeV and more.

Consequently, photoneutrons are the most suitable for such measurements*). The first check was made with neutrons from the reaction \((\gamma\mathrm{ThC}^{\prime\prime}, \mathrm{D})\). The binding energy of the deuteron, on the basis of many investigations\(^{39—43}\), is at present taken to be equal to 2.18 MeV. On the other hand, it has been established that in the \(\gamma\)-spectrum of \(\mathrm{ThC}^{\prime\prime}\) above 2.18 MeV there is only one line with energy 2.623 MeV (the presence of a line with energy \(\sim 3\) MeV in an amount of \(\sim 2—3\%\) is controversial\(^{44—46}\)). Hence it is easy to find that the photoneutrons which—

*) Owing to some strange misunderstanding, V. N. Kondrat’ev in his article\(^{47}\) asserts that photoneutrons have the greatest energy uncertainty. This assertion is completely erroneous.

which we shall in what follows call the second group, have an energy of 0.22 MeV. In the measurements the neutron source was a sphere of heavy water 5 cm in diameter, at the center of which RaTh was placed in an amount of approximately 100 mC. The detector was an artificially radioactive element (Dy, Rh, or Ag). For maximum activation it was placed at the center of a paraffin sphere 13 cm in diameter. The activity of the detector was measured on a Geiger–Müller counter with the scatterer placed between the source and the detector, and without it; from these data, using the general formula with a correction for the nonparallelism of the neutrons, the cross section was calculated. The results of the measurements are given in Table II and in Fig. 1.

The statistical errors of the measurements, owing to the weak source, are on the average 10–15%. From Fig. 1 we can see that the proposed empirical regularity is fully confirmed by these measurements. The general tendency toward a monotonic increase is preserved, and the magnitudes of the oscillations of \(\sigma\) occupy an intermediate position between the oscillations of \(\sigma\) obtained with thermal neutrons and with \((D,D)\) neutrons.

A second check with photoneutrons \((\gamma\mathrm{ThC}^{\prime\prime}, \mathrm{Be})\), carried out by a method analogous to that adopted in the preceding work, points to the same regularity. The energy of these photoneutrons was taken to be 0.4 MeV. The binding energy of the neutron in the beryllium nucleus, like the binding energy of the deuteron, is at present a firmly established quantity. On the basis of many determinations\(^{39–43}\) it is taken to be 1.63 MeV. Thus the photoneutrons of this reaction should have 0.88 MeV \(\left[(2.623 - 1.63)\frac{8}{9}\right]\); however, in a special paper by the author\(^{45}\) it was shown that this energy does not exceed 0.4 MeV. The decrease is apparently due to the existence in the beryllium nucleus of an excited level with an energy of the order of 0.45 MeV, still unknown at the present time. In what follows we shall call these photoneutrons the fourth group. The amplitudes of the oscillations of \(\sigma\), measured with these photoneutrons, as is seen from Fig. 1, are noticeably smaller than those observed in the scattering of photoneutrons with energy 0.22 MeV, but they nevertheless remain larger than with \((D,D)\) neutrons.

Checks were also made with photoneutrons \((\gamma\mathrm{RaC}, \mathrm{Be})\); however, in the present case, in studying the dependence \(\sigma = f(A)\), these measurements change the general picture little. Their significance will become clearer when we pass to the study of the functional dependence \(\sigma = f(E)\), where \(E\) is the energy of the scattered neutrons. The values found for \(\sigma\) are given in Table II.

With neutrons \((C,D)\), Amaldi et al.\(^{49}\) carried out a very extensive and accurate study. It will be discussed in detail in the presentation of the results on the scattering of neutrons by protons. In the work the general method of transmitted neutrons was used. The values of \(\sigma\) obtained are given in Table II and in Fig. 1 together with the photoneutron measurements.

Concluding the review of experimental works on the study of the dependence \(\sigma=f(A)\), we can draw several conclusions. First of all, in analyzing these data we notice a quite distinct “damping” of the oscillations of \(\sigma\), which have already been mentioned. The cross sections measured by Dunning, already within the limits of the statistical errors of the measurements, fit well on a straight line. In this case, however, it must be borne in mind that the neutrons (Rn, Be) are inhomogeneous, and it is quite possible to assume that the measured \(\sigma\)’s represent certain mean values. If it were possible to separate homogeneous groups of this neutron spectrum and measure the scattering \(\sigma\) of each group, then, probably, the same oscillations would be obtained as with neutrons (D, D).

It is easy to note that the number of values of \(\sigma\) measured with these neutrons and situated on one side or the other of Dunning’s straight line is approximately the same. Assuming that such a relation will also hold for \(\sigma\)’s measured with homogeneous neutrons with energy in the interval \(3\)—\(5\) MeV, and that the amplitudes of the oscillations will be of the same order, we come to the conclusion that the cross sections measured in scattering at any energy from 0 to \(\sim 5\) MeV will have a boundary indicated in Fig. 1 by the line \(AB\).

We do not know the true cause of the observed sharp oscillations of \(\sigma\), but their most probable cause is the resonance interaction of the neutron with the nucleus. Under this assumption it must be considered that in scattering characterized by cross sections lying, within the measurement errors, on the straight line \(AB\), resonance interaction is completely absent. This corresponds to the conclusions of theory, which indicate that at sufficiently high neutron energy the widths of the energy levels begin to overlap the intervals between them (see Section 2, and also the paper of Weisskopf et al.\(^{50}\)). Under these conditions the dependence of \(\sigma\) on the nuclear radius has the simple form: \(\sigma=\pi R^2\). Calculating from this \(R_1\) for the beginning of the boundary \(AB\) and \(R_2\) for its end, we obtain: \(R_1=4.7\cdot10^{-13}\ \mathrm{cm}\), \(R_2=12\cdot10^{-13}\ \mathrm{cm}\). These values agree well with nuclear radii calculated on the basis of other data.

Thus, the line \(AB\) indicates the geometrical boundary of nuclei. In reality this boundary probably must have the form of a curve, slightly convex in its middle part toward the abscissa axis, since the denser nuclei occupy the middle region.

Recently, \(\sigma\)’s for the scattering of neutrons with energy 90 MeV were measured. Unfortunately, these measurements end with \(\sigma_{\mathrm{Cu}}\). Of the heavy elements only \(\sigma_{\mathrm{Pb}}\) was measured. As is seen from Fig. 1, these \(\sigma\)’s in the region of light elements have values lying below the boundary \(AB\); for Cu and Zn they are somewhat higher. These oscillations can no longer be explained by resonance effects, but it is quite obvious that at such enormous energies neutrons can freely pass through the surface layer and penetrate into the depth of nuclei. The most suitable for determining the nuclear boundary would probably be systematic measurements of \(\sigma\) for neutron scattering with energy \(\sim 8\)—\(10\) MeV.

Table II

Cross sections of elastic scattering of fast neutrons by atomic nuclei, dependence $\sigma=f(A)$

Element	3–5 MeV	2.4 MeV	2.88 MeV	0.1–0.18 MeV	0.22 MeV	0.4 MeV	0.1 MeV	0.3 MeV	90 MeV
D			2.17						0.117
Li	1.6	2.10		2.0	2.3	2.5	0.9	1.0	0.314
Be				2.6	2.8	1.7	2.8	2.8	0.431
B		1.67	1.98	2.1	4.2	2.9	4.7	6.5
C	1.7	1.47	1.97	2.1	4.7	3.3	3.6	4.0	0.550
N	1.8	1.62	1.38	2.1				1.6	0.656
O		1.25	1.25	2.1	3.0	2.2		1.8	0.765
F		2.20		2.7	6.4	3.7		6.3
Na		2.24	2.37	3.4	3.2	3.3		3.5
Mg		1.85	2.25	3.3	3.5	3.7	6.5	8.5	1.03
Al	2.4	3.17	2.34	3.7	4.0	3.5	4.4	3.7	1.12
Si		2.90	2.77	3.2	7.2	4.4	3.8	2.3
P		2.18		4.4	2.4	1.8
S	2.7	2.40	3.12	2.6	2.3	2.5	1.1	1.0
Cl		2.80	3.42	2.7	3.8	4.6		3.5	1.38
K		4.18	3.13	3.8	7.4	4.3
Ca		3.83		4.9	4.1	5.2
Ti		2.08		4.4
Cr		3.27		3.5	5.7	4.2	2.5	3.3
Mn		4.18	3.82	4.9	7.2	1.6	5.2	5.8
Fe	3.0	3.10	3.15	3.7	3.6	3.1	3.4	2.8
Co		2.60		5.2	6.8	4.5		6.5
Ni		2.52		6.6	5.9	3.6	5.7	6.9
Cu	3.2	2.61	2.82	3.6	4.8	2.5	3.4	4.2	2.22
Zn	3.3	2.65	3.28	3.6	4.0	2.8	5.1	5.7	2.21
As				4.7	6.3	5.8	8.2	7.1
Se			4.05	4.5	5.8	5.0	6.9	8.3
Br				6.6	7.8	7.3
Sr				5.8
Mo			4.05		9.6	7.9	9.8	9.8
Ag				5.5	8.1	6.6		5.6
Cd				4.9	10.0	7.1	7.0	5.5
Sn	4.3		4.39	5.2	4.9	4.9		4.1
Sb				5.4	5.8	4.3	5.4	1.7
Te					5.7	5.3		5.1
I	4.6			6.6	4.7	7.8
Ba					7.1	5.1		8.1
W	5.3				9.4	6.3	7.2	10.4
Hg	5.8		5.34	6.8	7.8	6.4		9.7
Tl					7.0	7.2	11.3	10.0
Pb	5.7		6.74	7.2	5.7	4.8	8.8	9.0	4.53
Bi				7.7	8.4	5.4		8.1
Th				7.3
U				17.0					5.03

In Fig. 1, parallel to the line \(AB\), dashed lines are drawn characterizing the mean values of \(\sigma\) for neutrons of each energy with which measurements were made. On either side of these lines the number of values of \(\sigma\), measured with neutrons of the given energy, is approximately the same. The gradual lowering of these lines with increasing neutron energy undoubtedly indicates that the resonance oscillations regularly decrease as the energy of the scattered neutrons increases, and the values of \(\sigma\) approach \(AB\). It is curious to note that the line characterizing the mean values of \(\sigma\), measured with neutrons \((\mathrm{C}, \mathrm{D})\), coincides exactly with the line for photoneutrons \((\gamma \mathrm{ThC}'', \mathrm{Be})\), whose energy is \(0.4\ \mathrm{MeV}\). This gives reason to suspect that the energy of the neutrons \((\mathrm{C}, \mathrm{D})\) with which Amaldi et al. worked is greater than the value they indicated, \(0.1\text{--}0.18\ \mathrm{MeV}\).

The invariable increase of the oscillations of \(\sigma\) with decreasing energy of the scattered neutrons is completely at variance with the conclusions of scattering theory. As we saw in Section 2, the theory predicts for slow neutrons a monotonic increase of \(\sigma\), according to formula (6). This dependence should hold also for neutrons with energies up to \(1\ \mathrm{MeV}\). Resonance oscillations are allowed by the theory, but, as was shown in deriving \(\sigma_{\mathrm{res}}\), they can amount to no more than \(10\%\) of the total value of the cross section.

The observed empirical regularity of the increase of resonance interactions with decreasing energy of the scattered neutrons clearly contradicts the crude model of the nucleus as a structureless liquid drop. There is no doubt that it is closely connected with the as yet unknown structure of the atomic nucleus. Below we shall consider several such structural models; for now, in concluding our remarks, we may say that the efforts of many investigators who measured elastic-scattering cross sections have not been in vain, but have led to an interesting regularity which, in further investigations, will undoubtedly be fully elucidated.*)

B. Dependence \(\sigma = f(E)\)

This functional dependence was determined most clearly after the measurement of the scattering cross sections of photoneutrons \((\gamma \mathrm{RaC}, \mathrm{Be})\), which could be divided into two groups.

It is known that in the \(\gamma\)-spectrum of RaC above \(1.63\ \mathrm{MeV}\) there are 6 lines of different intensity (Table III). Using the considerable difference in energy of the strongest second line from the fifth and sixth lines, separation of the neutrons into the group produced by the second \(\gamma\)-line

*) Recently new phenomena of resonance scattering have been found \(^{51\text{--}53}\). In some elements having large capture cross sections (Ag, Rh, Co, Mn) the same large cross sections of resonance scattering were observed. It is possible that this phenomenon will prove to be closely connected with the regularity indicated.

Table III

Line No.	1	2	3	4	5	6
\(E\) MeV . . . .	1.69	1.75	1.82	2.09	2.20	2.42
Relative intensity	0.40	2.42	0.41	0.37	1.00	0.50

(group I), and the group produced by \(\gamma\)-lines 5 and 6, can be carried out by surrounding the detector (Ag, Rh, Dy, or a boron chamber) with layers of paraffin of different thicknesses. It was found experimentally\({}^{48}\) that with a paraffin sphere of diameter 6 cm the maximum activity of a detector placed at the center of this sphere, due to group-I neutrons, is already observed. With a sphere of diameter 10 cm, group-I neutrons are predominantly absorbed in the paraffin on their way to the detector, and activation is produced mainly by neutrons generated by the 5 and 6 \(\gamma\)-lines. It no longer appears possible to separate these two close groups in the same way. It is easy to find that the energy of group-I neutrons will be almost exactly equal to 0.1 MeV \(\left[(1.75-1.63)\frac{8}{9}=0.107\right]\). Calculation of the energies of the neutrons of the following, mixed group gives values of 0.51 and 0.71 MeV, but experimentally, by comparison with homogeneous photoneutrons \((\gamma\mathrm{ThC}^{\prime\prime},D)\), it was found that the mean energy of the neutrons of this group does not exceed 0.3 MeV. This decrease in energy, as also for neutrons \((\gamma\mathrm{ThC}^{\prime\prime},\mathrm{Be})\), is explained by the existence in \(\mathrm{Be}^{8}\) of an excited level with an energy close to 0.45 MeV.

Such a separation made it possible to measure \(\sigma\) with neutrons of four groups, corresponding to energies \(\sim 0.1, 0.2, 0.3,\) and \(0.4\) MeV, and for many elements to obtain four points on the curve \(\sigma=f(E)\). The experimental method did not differ in principle from that adopted in the earlier measurements. The complete results of the measurements are given in Table II, while typical curves for some elements are shown in Fig. 2.

In addition to these data with photoneutrons, less extensive investigations of the dependence \(\sigma=f(E)\) were carried out with \((D,D)\) neutrons by Aoki\({}^{55}\) and McMillan\({}^{56}\). The results for Si and Mg are shown in Fig. 2, and the complete data in Table IV. In these works the neutron energy \((D,D)\) was varied by changing the angle \(\varphi\) between the directions of the deuterons and the neutrons. In this way, as is seen from Table IV, one can investigate the energy interval 0.63 MeV. The energy spread, according to Aoki’s data, reached 120 KeV in his measurements; in McMillan’s it was considerably smaller, \(\sim 40\) KeV.

The study of the curves in Fig. 2 leads us to the conclusion that, in contrast to the irregular variation of \(\sigma\) as a function of \(A\), here we have quite regular variations, indicating the location of resonance levels in nuclei.

However, it immediately becomes obvious that these resonance processes in no way fit within the framework of existing nuclear theories. It is well known from the calculations of Bethe\(^5\) and other authors that the intervals \((D)\) between resonance levels and the widths of these levels \((\Gamma)\) in heavy nuclei are very small. The quantities \(D\) vary

Fig. 2. Dependence \(\sigma=f(E)\) in the scattering of neutrons by various elements.

according to an exponential law depending on the atomic weight of the element and the energy of the scattered neutrons. For heavy elements \(D\) does not exceed fractions of a volt, while \(\Gamma\) is of the order of \(0.001\) eV.

It is quite obvious that, in the scattering of neutrons differing in energy by \(0.1\) MeV and having an energy spread of about \(20\) KeV, one cannot expect resonance phenomena with such closely spaced levels. Under these conditions, the probability of hitting an individual level is equal to the probability of hitting, with a football, between the strokes of a diffraction grating, or of obtaining dispersion in this grating for radio waves several meters in length. In view of the overlap, by the energy spread of the scattered neutrons, of an enormous number of levels, we can expect only a monotonic decrease of the values of \(\sigma\) with increasing \(E\). On the curves of Fig. 2, however, we observe that the intervals of the oscillations and their amplitudes essentially do not differ in any way from the region of light nuclei, where energy gaps are possible

hardness \(D \simeq 150\ \mathrm{KeV}\) and higher, from the region of heavy nuclei, and the only explanation of this phenomenon can be the assumption that in all nuclei without exception there may exist resonance levels separated by intervals of hundreds of thousands of electron-volts.

Another characteristic feature of the observed phenomena, to which we have already more than once drawn attention, is the preservation of the general monotonicity of the increase of \(\sigma\) in the transition from light to heavier elements. The minima for heavy elements do not fall below a certain boundary, defined in Fig. 1 by the line \(AB\). At the end of the preceding section we suggested that this line is close to the boundary of the nuclei. If this assumption corresponds to reality, then it must now be admitted that in all elements the outer shells of atomic nuclei have approximately the same structure as light nuclei.

Up to now our conclusions have been only the most probable explanations of the discovered phenomena of “anomalous” scattering. We shall now introduce a hypothesis which, although it is somewhat less reliable than the assumptions stated above, nevertheless seems very probable. If it is considered possible, as many authors suppose, that light nuclei consist of \(\alpha\)-particles, then it must be admitted that the shells of heavy nuclei also consist of \(\alpha\)-particles. This hypothesis thus automatically divides a heavy nucleus into an inner part, a subnucleus, consisting chiefly of excess neutrons, and a shell consisting of \(\alpha\)-particles. When the nucleus as a whole is excited, the system of levels may be close to that predicted by statistical theories, but interactions of nuclei with neutrons (and other particles) are also possible in which the inner subnucleus takes almost no part in the process. From the point of view of the theory of the evaporating drop, such an interaction may be called local heating of the surface of the nucleus. Nor is it excluded that, in elastic scattering, a neutron interacts with an individual nuclear particle (\(\alpha\)-particle, proton), for some reason bound in the nucleus more weakly than the other particles.

As was already noted, in calculating \(\sigma\), all investigators assumed that the principal process in the passage of neutrons through the elements under study is purely elastic scattering without loss of energy*). For neutrons with energies up to \(0.5\ \mathrm{MeV}\) this assumption proves entirely justified. It has been verified experimentally\({}^{33}\). But in the scattering of neutrons \((D,D)\) with energy \(\sim 2.5\ \mathrm{MeV}\), as Nonaka\({}^{58}\) showed, hard \(\gamma\)-radiation is observed, emerging from the scatterer. The appearance of this radiation can be ascribed only to excitation of atomic nuclei, and not to absorption, since the measured \(\sigma\) values turned out to be the same in magnitude as those found by Aoki. In this case

*) The loss of energy in an elastic collision of neutrons with large masses of atomic nuclei may be neglected.

oscillations of \(\sigma\) are also observed, analogous to the oscillations found in the work of Aoki and McPhail.

The angular distribution in neutron \((D,D)\) scattering, which in the calculations of \(\sigma\) had been taken to be spherically symmetric in the laboratory coordinate system, also, at least for some elements, proved to be sharply asymmetric. As early as 1939 Aoki found that, for lead, forward scattering is almost 10 times greater than backward scattering. This was confirmed in the work of Kikuchi et al.^59; and, finally, recently Barshall and Ladenburg^60, using ring scatterers, were able to show with considerable accuracy the existence, for Al, Fe, Cu, Zn, and Pb, of both inelastic scattering and asymmetry in the distribution of neutrons; moreover, for each process separately the angular distribution proved to be different.

These data do not strongly distort the regularities \(\sigma=f(A)\) and \(\sigma=f(E)\). The corrections that must be introduced into the values of \(\sigma\) found are small and, possibly, do not exist for all elements (apparently, chiefly for the heavy ones). Since the general monotonic increase of \(\sigma\) remains, this indicates to us that not only elastic, but also inelastic, scattering processes occur mainly—and perhaps exclusively—only at the surface of atomic nuclei.

As is well known, to illustrate his idea of the formation of a compound nucleus in every nuclear process, Niels Bohr proposed a vivid model in the form of a small cup filled with smooth balls. An extraneous ball, flying into this cluster of balls in the cup, gives up its kinetic energy, which is distributed equally among all the balls. Then one ball, having by chance received enough energy to lift it to the rim of the cup, flies out of it. This model, still cited in all expositions of the theory of the atomic nucleus, is apparently very far from the real processes that occur when an external particle is attached to a nucleus. In elastic and inelastic scattering, most likely no compound nucleus at all is formed in the sense that the attached particle remains in the nucleus for a very long time (of the order of \(10^{-13}\) sec.). It interacts with a small number of particles on the surface of the nucleus for a time, probably, of the order of \(10^{-21}\) sec. Only in amalgamation, leading to absorption with the formation of a radioactive or highly excited nucleus, is a transition of all the particles into an excited state possible; but the attached particle itself, probably, even in this case remains on the surface of the nucleus. Probably only at an enormous energy, of the order of 100 MeV, can the particle penetrate into the interior of the nucleus.

Graham^61 and Wilson^62 arrive at ideas analogous to those expressed here. The first author, citing the experimental facts on inelastic scattering set forth in this article, and some others, assumes that to explain them it is necessary to introduce the idea of “local excitation” or “local heating” of the surface

nucleus. Since the first interaction occurs with a small number of particles, the levels will be broad and separated by large intervals. The locally excited nuclei may subsequently either emit a neutron again, or form a compound nucleus of the usual type.

In the first case the escaping neutron will retain a large part of its initial energy, since it has not yet had time to be distributed among all the particles of the nucleus. This will be a case of inelastic scattering with a small loss of energy.

Thus, the author introduces a new concept of neutron scattering with a small loss of energy. Until now it had been assumed, as follows from the theory of the evaporating nucleus, that in inelastic scattering neutrons must lose up to 90% of their initial energy. Experimental confirmation of the hypothesis of a small loss of energy would be very valuable.

It is curious to note that Gragam expresses the firm supposition that in heavy nuclei there exist resonant vibrations of the same kind as were found in the region of the light elements. The results of the work with photoneutrons, set forth in this section, are apparently completely unknown to him.

Wilson, relying on his theoretical considerations, expressed as early as 1933,^(62) and on the data of various authors (in doing so he also does not cite the works with photoneutrons, apparently being completely unaware of them), asserts that in all nuclei there may exist intervals between levels of the order of 0.4 MeV. He then gives a calculation of such a model of the nucleus, which permits the existence of these intervals. He represents this model in the form of a thin flexible spherical layer (instead of a sphere), positively charged. He explains the formation of such a layer by the saturation of nuclear forces. If a nucleon cannot be strongly bound to more than 4 neighboring nucleons, then electrostatic forces must transform the sphere into a layer. This model of the nucleus, as we see, is close to those proposed by us, but in Wilson’s there remains no place for additional neutrons, and the radii of heavy nuclei calculated from this model prove to be too large: \(\sim 15 \cdot 10^{-13}\) cm.

The study of the functional dependence \(\sigma=f(E)\), as is seen from Table II, was carried out in greatest detail only in the energy region 0.1–0.4 MeV. In the region 2–3 MeV there are almost no measurements for heavy elements, and, moreover, after the works of Nonaka and Barshall the results of these measurements must be corrected for inelastic scattering and for the asymmetry of the angular distribution. The regions from 0 to 0.1 MeV and from 0.4 to 2 MeV long remained empty. Only quite recently have 2 papers appeared in the literature, extending the 0.1–0.4 MeV region somewhat in both directions.

In the work of Russell et al.^(63) photoneutrons obtained from artificially radioactive elements were used: Na, Sb, Ga, Mn, La.

Table IV

Dependence \(\sigma = f(E)\) according to data of various authors
(\(\sigma\) in units of \(10^{-24}\ \mathrm{cm}^2\))

Element	2.14	2.28	2.40	2.60	2.77	Notes, cited literature
D	2.27		2.20		2.11
C	1.58	1.53	1.64	1.67	1.95
O	1.22		1.16		1.25
Si	2.27	2.24	2.87	2.50	2.43
Sn	4.18	4.00	3.90	2.60	2.76
Pb	4.95	5.40	5.20	5.18	5.43
Bi	5.28	5.50	5.61	5.80	6.00	\(^{55}\ \Delta\sigma \sim 5\%\)

Element	2.34	2.41	2.49	2.57	2.65	2.80	Notes, cited literature
C	1.41	1.39	1.38	1.38	1.45	1.57
N	1.33	1.22	1.27	1.39	1.28	1.25
Na	2.74	2.69	2.69	2.63	2.50	2.38
Mg	2.19	1.94	1.76	2.14	2.54	2.34
Al	2.49	2.19	2.18	2.94	2.94	2.48	\(^{56}\ \Delta\sigma \sim 5\%\)

Element	0.35	0.72	0.97	2.0	4.0	5.5	Notes, cited literature
D	3.53	3.46	2.27	2.60	1.70	1.52
O	4.80	2.01	5.06	0.89	2.90	0.96	\(^{78}\ \Delta\sigma \sim 5\%\)

Element	10.6	12.9	16.5	18.0	19.6	21.1	Notes, cited literature
C	1.43	1.18	1.29	1.13	1.31	1.17	\(^{104}\ \Delta\sigma \sim 3\%\)

Continuation of Table IV

Element	0.024	0.13	0.14	0.22	0.62	0.83	Notes, cited literature
Be	5.0	4.3	4.3	4.2	3.3	3.1
B	5.5	4.9	4.7	4.3	—	2.3
C	4.6	4.3	4.3	4.1	3.3	2.9
O	3.6	3.5	4.1	3.0	3.5	4.9
F	3.5	5.7	—	6.9	4.6	4.1
Na	5.1	3.9	4.2	3.8	5.9	4.6
Mg	4.4	4.9	5.7	8.7	4.2	3.4
Al	0.8	5.3	3.2	3.2	4.1	3.5
P	3.8	3.1	3.1	3.2	3.1	3.5
S	1.0	4.2	4.5	2.9	—	2.2
K	1.2	1.6	1.9	1.7	2.3	2.7
Fe	2.2	4.1	3.9	3.3	—	2.7
Ni	23.0	6.4	4.2	5.8	3.7	3.5
Cu	8.0	6.2	5.9	5.3	—	3.8
Zn	9.9	6.6	5.4	5.1	4.7	4.3
Ag	8.0	8.1	8.1	7.8	7.3	7.2
Cd	6.9	7.6	7.3	7.3	—	7.1
Sn	5.9	6.4	6.4	6.3	6.8	6.7
Sb	6.3	6.9	6.4	6.1	6.7	6.4
I	7.0	6.6	6.5	6.1	6.8	6.7
W	13.8	10.0	9.4	8.0	7.8	7.7
Pb	11.4	10.6	10.6	8.6	6.1	5.8
Bi	12.1	10.2	9.7	8.0	—	5.9	77 Δσ ∼ 10%

With the aid of powerful neutron fluxes in plutonium-production installations (reactors), it was possible to obtain γ-ray sources equivalent in intensity to from 100 to 5000 mC of radium emanation. By irradiating D and Be with these γ-rays, the authors were able to obtain photoneutrons with energies from 0.024 to 0.83 MeV. Table V gives the complete data, obtained by various authors, on the determination of the γ-ray energies in these sources, as well as the energies of the neutrons obtained.

Russell et al. believe that, in the source itself, the neutrons decrease their energy as a result of collisions with the nuclei of the atoms of this substance. Therefore they give different data for thin and thick targets. It is difficult to judge how correct this calculation is, but apparently a decrease in the energy of all photoneutrons by 5000 eV is poorly justified, since most of the neutrons leave the target without undergoing scattering. The change in energy of some fraction of the neutrons due to this effect will only increase the energy spread, and not reduce the total energy of all neutrons by one and the same amount.

Measurements of σ were carried out by the usual method of transmitted neutrons. The detector was an ionization chamber filled with fluori-

Table V

Energy of photoneutrons and γ-rays

Reactions	$E_\gamma$ MeV $^{77}$	$E_n$ MeV $^{77}$	$E_n$ MeV, average $^{77}$	$E_n$ MeV, max. $^{77}$	$E_n$ MeV, average $^{105}$	$E_n$ MeV, max. $^{105}$
Na + Be	2.75	0.80 0.86	0.83	1.00	0.800	1.020
Na + D	2.72	0.200 0.245	0.22	0.27	0.220	0.320
Mn + Be	1.83	1.45 1.57	0.15	0.18	0.300 <0.150	0.375 <0.150
Mn + D	2.7	0.22	0.22	0.26
Ga + Be	1.99	0.27	0.27	0.32
Ga + D	2.50	0.13	0.13	0.16
Sb + Be	1.67	0.024	0.024	0.029	0.035	0.068
La + Be	2.47	0.66	0.62	0.75
La + D	2.50	0.135 0.126	0.13	0.16

with boron trifluoride (BF$_3$) and placed in a paraffin block measuring $7.5 \times 10 \times 15\ \text{cm}^3$. The chamber was cylindrical, with a diameter of $2.5\ \text{cm}$ and a length of $12.5\ \text{cm}$. It was located at a distance of $50\ \text{cm}$ from the source. Radioactive sources in the form of cylinders of diameter $1.25\ \text{cm}$ and length $5\ \text{cm}$ were placed in cylinders of D$_2$O weighing $34.5\ \text{g}$ and Be weighing $69.7\ \text{g}$. The measured $\sigma$ are given in Table IV, together with the data of Aoki and McPhail. Table IV gives the values of $\sigma$ obtained at the corresponding photoneutron energies. Some of them differ considerably from the $\sigma$ found by other investigators. The authors explain this difference by inaccuracies in determining the neutron energy. Such an explanation, apparently correct, indicates that significant changes in $\sigma$ can occur already in small energy intervals of the order of 10–15 KeV.

In the work of Barshall et al.$^{64}$, neutrons from the reaction (Li, $p$) were used. The neutron spectrum of this source has a maximum in the direction of the proton beam. By varying the proton energy and using a thin lithium target, equivalent in stopping power to 10 KeV, the authors investigated the region from 10 to 1000 KeV. They found in this region 10 resonance maxima for Al, shown in Fig. 2. The detector used was a proportional Geiger–Müller counter filled with BF$_3$. In subsequent work the authors hope in the same way to investigate heavier elements, where, according to theoretical predictions, they expect to find a monotonic decrease of $\sigma$ with increasing neutron energy, evidently not suspecting that analogous resonance phenomena have already been found for these elements.

T. A. GOLOBORODKO

4. SCATTERING OF NEUTRONS BY PROTONS

A. Theoretical premises

The theory of the deuteron, as is known, was created by Bethe and Peierls^65 on the basis of the single postulate of the small radius of action of nuclear forces*). The formula expressing the functional dependence \(\sigma = f(E)\) at first differed sharply from the experimental data and was corrected by introducing interaction forces depending only on the spin directions of the neutron and proton. After this correction the formula acquired its final form:

\[ \sigma=\frac{\pi\hbar^2}{M}\left\{\frac{3(1+\alpha_1 r_0)}{\varepsilon_t+E_0/2}+\frac{1-\alpha_2 r_0}{\varepsilon_s+E_0/2}\right\}, \tag{15} \]

where \(M\) denotes the reduced mass of the neutron–proton system, \(r_0\) the radius of action of the nuclear forces, \(E_0\) the energy of the scattered neutrons in the laboratory coordinate system (the proton is at rest), \(\varepsilon_t\) the binding energy of the ground level of the deuteron (2.18 MeV), \(\varepsilon_s\) the binding energy of the second (virtual) level (\(\leqslant 100\) KeV), and \(\alpha_1\) and \(\alpha_2\) are certain constants depending on \(M\), \(\varepsilon_t\), \(\varepsilon_s\), and \(\hbar\)**).

On the basis of the postulate of the small radius of action of nuclear forces (\(r_0 \leqslant 2\cdot 10^{-13}\) cm), the dependence (15) must be automatically satisfied in the scattering of neutrons of all energies up to \(\sim 15\) MeV. Thus, in the case of any sharp discrepancy between the experimental result and (15), there can be only two conclusions: 1) the experimental results are wrong and 2) the theory is wrong in its basic postulate.

Many experimental results, which will be set forth below, sharply contradict the Bethe–Peierls theory of the deuteron, and, in order to reconcile them, theorists introduced various modifications. We shall dwell briefly on two of them.

Morse et al.^67 introduced a special potential function of the form

\[ V=-2V_0 e^{\frac{2}{r_0}(r_1-r)}+V_0 e^{\frac{4}{r_0}(r_1-r)}. \tag{17} \]

) In view of the fact that an excellent exposition of this theory is available in the monograph by Bethe and Bacher, Nuclear Physics* (ONTI, 1938, Kharkov), we do not consider it necessary to present it here in full.

**) In 1943 an interesting article appeared by Smorodinsky^66, who, pointing out that in the form (15) the formula in fact contradicts the basic postulate of the theory, since \(\sigma\) turns out to depend on \(r_0\), to which different values may be assigned, modifies it into the following:

\[ \sigma=\frac{\pi\hbar^2}{M}\left\{\frac{3}{(A_1-B_1E_0)^2+E_0/2}+\frac{1}{(A_2+b_2E_0)^2+E_0/2}\right\}. \tag{16} \]

He determines the constants \(A\) and \(B\) from experimental data on neutron scattering. In this form formula (16) resembles the general dispersion formula (3). It seems to us that, in view of those sharp contradictions between theory and experiment which will be set forth below and which Smorodinsky completely ignores, a correct description of the dependence \(\sigma = f(E)\) cannot be expected from formula (16) either.

For \(r_1=0\) there will be attraction at all distances \(r\); for \(r_1\) different from 0, at small distances there will be repulsion, having a minimum at \(r_1=r\). The authors solve the basic Schrödinger equation for the deuteron with this function \(V\) and arrive at a dependence \(\sigma=f(E)\) different from (15). The results of their calculations of \(\sigma\) are shown in Fig. 3. For a neutron–proton interaction of the Majorana type\(^{68}\) there is no noticeable difference from (15), but, for exchange forces, maxima and minima and even discontinuities of the curve are observed for various \(r_1\).

Fig. 3. Scattering of neutrons by protons. Dependence \(\sigma=f(E)\) for the Morse potential.

Shar and Stein\(^{69}\) pointed out that (17) is equivalent to a narrow and deep potential well at the center with a broad and shallow rim. In their article they chiefly examine possible deviations from the symmetry of the angular distribution of neutrons when they are scattered by protons. In the Bethe–Peierls theory, as is easy to see, this distribution must be spherically symmetric in the center-of-mass coordinate system, owing to the fact that the wavelength of the scattered neutron is always considerably greater than the range of action of the nuclear forces. Under these conditions only the \(S\)-wave with angular momentum \(l=0\) is appreciably scattered. Scattering of waves with \(l\ne0\) begins to play a noticeable role only at neutron energies of the order of 15 MeV and higher.

An entirely different interaction mechanism is possible for forms of the potential well with a shallow rim. The phase \(\delta_l\) of a wave with \(l>0\) is mainly affected by this rim. The authors find that a rim with a depth from 0.5 to 1 MeV, extending to distances \(r=(8.4\text{—}11.2)\,10^{-13}\,\text{cm}\), is sufficient to explain the sharp asymmetry of the angular distribution of neutrons observed by Koori and Harkins (see below). In view of the fact that these data, as will be seen below, were confirmed with much greater accuracy in later investigations, theoretical concepts introducing the notion of a large radius of action of nuclear forces acquire important significance both in the theory of the deuteron and, apparently, in the general theory of atomic nuclei.

The theory of Bethe–Peierls, as well as the modern mesotron theories of the deuteron, which lead to forces with a small radius of action, are, in all probability, only a special case of a future, more general theory, which will take into account the experimentally discovered possibility of the action of nuclear forces at considerably greater distances.

B. Experimental Data

After the introduction of the second (singlet) level of the deuteron, much work was done by experimentalists to test formula (15). The very first experimental result of M. Goldhaber^70 sharply disagreed with this formula.

Fig. 4. Dependence \(\sigma=f(E)\) for the scattering of neutrons by protons. The theoretical curve, except for the region \(0.1\)—\(0.4\) MeV, was constructed from the data of Begh and Richman^108.

In the figure legend:

• \(\gamma\)-neutrons, \(E=0.024\)—\(0.83\) MeV
• Photoneutrons, \(E \approx 0.1\)—\(0.4\) MeV
• Neutrons \((C,D)\), \(E \approx 0.2\)—\(0.7\) MeV
• Neutrons \((Li,\alpha),(C,D),(D,D)\), \(E=0.35\)—\(6\) MeV
• Neutrons \((D,D)\), \(E=2.14\)—\(2.76\) MeV

Goldhaber worked with photoneutrons \((\gamma ThC'', D)\). A boron chamber surrounded by paraffin served as the detector. The measurements were made by the usual method of a transmitted neutron beam. A paraffin disk served as the scatterer. For the mean free path of neutrons in paraffin the value obtained was

\[ \lambda = 4.5 \pm 1.5\ \text{cm}, \]

which corresponds to \(\sigma = 3 \cdot 10^{-24}\ \text{cm}^2\), whereas the theoretical value, as is seen from Fig. 4, should be \(10 \cdot 10^{-24}\ \text{cm}^2\).

Such an enormous discrepancy is quite impossible to explain by any systematic experimental error, since the measurement method is extremely simple. It is curious to note that subsequently none of the experimenters who obtained contrary data tried to analyze critically the conditions of Goldhaber’s experiment. The author himself likewise later published neither experimental checks nor a critical analysis of his work.

In the second paper, Tuve et al.^71 used neutrons (C, D) and found the value \(\lambda = 2.7\) cm, which corresponds to the value \(\sigma = 4.7 \cdot 10^{-24}\ \mathrm{cm}^2\). Assuming that the energy of the neutrons with which they were working lay in the interval \(0.6\text{--}1.2\) MeV, the authors come to the conclusion that, within the limits of the statistical errors of the measurements, the result found does not contradict the theoretical value. However, as was subsequently shown by very accurate investigations^72, at a discharge-tube potential of \(0.9\) MeV, which was used in Tuve’s work, the energy of the neutrons (C, D) does not exceed \(0.35\) MeV. At such a neutron energy the found value of \(\sigma\) is almost half the theoretical one.

The third work, by Leipunsky et al.^73, was also performed with photoneutrons, but not from the reaction \((\gamma\mathrm{ThC}^{\prime\prime}, D)\), but from the reaction \((\gamma\mathrm{RaC}, \mathrm{Be})\). The neutrons obtained from this latter source, as we saw above, are energetically nonuniform. The detector was silver placed at the center of a water sphere 13 cm in diameter. The authors assumed that the neutrons had a mean energy of \(0.15\) MeV. Using a paraffin scatterer, they found the value \(\lambda = 1.5 \pm 0.6\) cm, which corresponded to the value \(\sigma = 8.5 \cdot 10^{-24}\ \mathrm{cm}^2\). Although this value is lower than the theoretical one, within the experimental error it agrees with it.

Comparing these two results, we can arrive at only two conclusions: 1) in Goldhaber’s measurements some systematic error was made which is difficult to find; 2) both results (and also the third—Tuve et al.) are correct, while the difference between them is explained by the different neutron energy, i.e., formula (15) is incorrect. All specialists apparently preferred the first explanation, since during three years no checks of these important discrepancies appeared.

In 1939 Amaldi et al.^49, in the work already cited above, together with measurements of \(\sigma\) for 38 different elements, carefully measured also \(s_{\mathrm H}\) and obtained the value \(3.3 \cdot 10^{-24}\ \mathrm{cm}^2\), in excellent agreement with Goldhaber’s result.

In view of the great importance of this question for the theory of the deuteron and for the entire theory of the atomic nucleus, and of the very strange subsequent consequences of this work, we shall analyze it in somewhat more detail.

Wishing apparently to bring the energy of neutrons (C, D) closer to the energy of photoneutrons, the authors reduced the discharge-tube potential to \(0.6\) MeV and used neutrons traveling at an angle \(\varphi = 90^\circ\) with respect to the direction of the deuterons. Basing themselves on data

Livingston and Bethe and their previous measurements of the neutron-yield curve, they believe that the neutron energy lay in the interval 0.1–0.18 MeV\(^{74}\)*).

The entire work was carried out very carefully. The tube regime was maintained within \(\sim 0.1\%\). Neutrons were detected with an ionization chamber filled with hydrogen at a pressure of 20 atm. The ionization current was measured with an Edelmann electrometer having a sensitivity of 200 divisions/volt. The neutron background did not exceed 1% of the total intensity. The distance between the chamber and the graphite target was 30 cm, the chamber diameter 3 cm, and the scatterer diameter 4 cm. The correction for nonparallelism of the neutrons amounted to no more than 3%. Taking into account the influence of two neutron groups of higher energy, which at a potential of 0.6 MeV amounted to 5.3 and 1.5 MeV, showed that they could introduce errors of no more than 5% and 2% into the value of \(\sigma\) found.

Analyzing the possible errors of the measurements, the authors arrive at the entirely correct conclusion that these could change the result by only a few percent. Only one cause could have led to a significant lowering of \(\sigma\)—an admixture of neutrons \((D,D)\) accumulating in the carbon target; however, as Bonner showed, these neutrons are always present only in negligible quantity. The authors, for their part, note that for this it would have been necessary to assume an enormous absorption of deuterons in a target heated in vacuum to red heat, which is completely improbable.

And so, after the carefully performed work and its detailed analysis, in which no reasonable cause could be found capable of distorting the result so strongly, the authors, several months later, printed a short note in which they reported that, under all the previous conditions, they found \(\sigma \simeq 7.5 \cdot 10^{-24}\ \text{cm}^2\).

It is usually considered that if the authors themselves, after checking earlier results, renounce them, then their latest result should be regarded as the more correct. This, of course, is most often the case, although there may also be exceptions to the general rule. We believe that in the present case there is precisely such an exception. The results of these two works show that not the first, but the second conclusion, given as the explanation of the contradictory data of Goldhaber and Leipunsky et al., is more consistent with reality. In this energy interval \(\sigma\) depends strongly on the neutron energy. In the present case the difference in the two obtained results is due, apparently, to the uncertainty in the neutron energy, which reached 100 KeV. In the first measurement, Amaldi et al. accidentally measured the scattering of neutrons very close in energy to 0.22 MeV (or to 0.4 MeV), and obtained a result coinciding with Goldhaber’s data—

*) One may think, as has already been shown in the analysis of Fig. 1, that the energy of these neutrons was closer to the value 0.4 MeV. At this neutron energy \(\sigma\) also has the value \(3 \cdot 10^{-24}\ \text{cm}^2\).

Elastic Scattering of Neutrons by Atomic Nuclei

Goldhaber, in the second measurement of the energy of the scattered neutrons, for some reason deviated from this value, and obtained a result strongly differing from the former one.

In order finally to clear up the confusion that had arisen in the cited works, it was necessary, first of all, to repeat M. Goldhaber’s measurements with the greatest possible accuracy. Such a check was carried out by the author of this article \(^{75}\). Table II gives \(\sigma_H\), found with an \(H_2O\) scatterer. The value

\[ \sigma = (5.0 \pm 1.0)\cdot 10^{-24}\ \text{cm}^2 \]

is in better agreement with Goldhaber’s value than with the theoretical one. However, this result is unreliable, since the measurements were made with a rather poor geometry of the experimental setup. After the separation of the photoneutrons \((\gamma\mathrm{RaC}, \mathrm{Be})\) into two groups, described in the preceding section, and the measurement of many cross sections with four groups of photoneutrons, measurements were also made of four values of \(\sigma\) for hydrogen. The execution of such work was naturally a check not only of Goldhaber’s result, but also of all the others in this region \(0.1\text{--}0.4\ \mathrm{MeV}\).

The main condition set in this work was the maximum attainable reduction of the solid angle from the scatterer to the detector. In the scattering of neutrons by heavy nuclei, the calculations may be carried out by assuming their spherically symmetric distribution, and the correction introduced for neutrons reaching the detector after their scattering is small. In scattering by protons, however, this correction increases greatly, since after scattering the neutrons go predominantly forward. Therefore, in this work the distance between the photoneutron source and the detector was brought up to the limiting possible value—\(40\ \text{cm}\). With an amount of RaTh \(\simeq 100\ \mathrm{mC}\), with which the work was carried out, in order to reduce statistical errors it was necessary to repeat many times the measurements with the scatterer and without it. Measurements made with paraffin scatterer thicknesses of \(0.5\ \text{cm}\) and \(1\ \text{cm}\) practically did not differ, and the final count led to the value

\[ \sigma = 3.0\cdot 10^{-24}\ \text{cm}^2, \]

in good agreement with the data of Goldhaber, Amaldi, and others, and with the previous result.

Similar measurements with neutrons of group I, as expected, coincided with the measurements of Leipunsky and others. A check with group III led almost to the same result as was obtained with group I. This indicated that the minimum on the Breit–Wigner curve, so long sought, is evidently very narrow, and that a deviation to one side or the other from it leads to values of \(\sigma\) very close to the theoretical ones. This circumstance confirms our conjecture as to the reason for the discrepancy between the two values of \(\sigma\) obtained by Amaldi and others.

The measurement with photoneutrons of group IV again led to a small value,

\[ \sigma = 3.2\cdot 10^{-24}\ \text{cm}^2. \]

This result, found for the first time for neutrons of such an energy, was confirmed in the work of Good and Goldhaber \(^{76}\), who found

\[ \sigma = 2.6\cdot 10^{-24}\ \text{cm}^2. \]

The results of these measurements are given in Table VI and in Fig. 4.

Table VI

Neutron scattering cross sections by protons

Neutron energy in MeV	\(\sigma \cdot 10^{24}\ \mathrm{cm}^2\)	Notes, references to the literature
0,024—0,029	18,2
	17,5
0,13—0,16	11,8
	12,0
(0,15—0,18)*)	11,4
	11,1
	10,9
0,22—0,27	10,0
	9,2
0,27—0,32	8,7
	9,1
0,62—0,75	5,6
	6,1
0,83—1,00	5,1
	4,9	\(\Delta\sigma \sim 10\%\) ^74
0,35	7,15
0,46	6,52
0,72	5,22
0,97	4,45
1,00	4,16
1,6	3,36
2,0	2,96
2,6	2,60
3,0	2,33
3,5	2,09
4,0	1,85
4,5	1,83
5,0	1,63
5,5	1,48
6,0	1,32	\(\Delta\sigma \sim 3\%\) ^75
0,1	9,0	(6,95) **)
0,2	3,0	(4,10)
0,3	8,5	(6,78) \(\Delta\sigma \sim 10\%\)
0,4	3,2	(3,74) ^75
0,1—0,18	3,3	\(\Delta\sigma \sim 5\%\) ^40
0,2	3,0	^70
0,15	\(\sim 8,0\)	\(\Delta\sigma \sim 30\%\) ^74
\(\sim 0,4\)	2,63	\(\Delta\sigma \sim 10\%\) ^76
0,35	4,7	^71
0,2	5,0	^36

*) The energy is doubtful, owing to the presence in the source of a second line (see Table V).
**) Theoretical values obtained under the assumption of the existence in the deuteron of a virtual \(P\)-level.

Continuation of Table VI

Neutron energy in MeV	\(\sigma \cdot 10^{24}\ \mathrm{cm}^2\)	Notes, literature references
2,14	2,78
2,27	2,70
2,43	2,51
2,60	2,51	\(\Delta\sigma \sim 5\%\)
2,76	2,40	55
2,88	2,32	32
2,4	2,40	31
3—5	1,6	57
25	0,39	97
6,5	1,40
9,3	0,92
10,6	0,78
12,8	0,83
14,8	0,61
16,5	0,66
18,1	0,55
19,6	0,52	\(\Delta\sigma \sim 3\%\)
21,1	0,41	104
90	0,033	103
4,1	1,73
12,5	0,69
13,5	0,69	98
15	0,61	90
14	0,70	101
15	0,66
1 eV	\(20 \pm 1,0\)	102

For approximately six years this test of formula (15) was the last one. In 1947 two papers appeared, one of which was carried out with photoneutrons\(^{77}\), obtained from the reactions \((\mathrm{Na}, \mathrm{Be}, \mathrm{D})\), \((\mathrm{Mn}, \mathrm{Be}, \mathrm{D})\), \((\mathrm{Ga}, \mathrm{Be}, \mathrm{D})\), and \((\mathrm{La}, \mathrm{Be}, \mathrm{D})\). The measurement methods used were analogous to those adopted in the already cited work of Russell et al. The measurements of \(\sigma\) with a paraffin scatterer, as is seen from Fig. 4, do not deviate noticeably from the theoretical curve.

In the second paper Bennett et al.\(^{78}\), in order to obtain suitable neutrons, used the reactions: \((\mathrm{Li}, p)\), \((\mathrm{C}, \mathrm{D})\), and \((\mathrm{D}, \mathrm{D})\). By bombarding a thin lithium target with protons of different energies it was possible to obtain neutrons with energies from 0,35 to 0,97 MeV. When the second reaction was used, neutrons with energies from 1 to 2 MeV were obtained, and with the third—from 2,6 to 6 MeV. The results of the measurements, carried out with an ionization chamber, are placed in Table VI and plotted in Fig. 4.

In analyzing all the data on measurements of \(\sigma_{\mathrm H}\), beginning with Goldhaber’s first work and ending with the two most recent ones, one can draw two

conclusion: 1) anomalous departures of points from the curve expressing the dependence (15) apparently exist only in the region of low energies, approximately up to 0.5 MeV; 2) the latest data with photoneutrons once again cast doubt on the reality of these anomalous departures.

Let us first examine the second conclusion. The energies of photoneutrons \((\mathrm{Na}, D)\) and \((\mathrm{ThC}'', D)\) are almost equal—0.27 and 0.22 MeV, respectively—or, taking into account the possibility of a decrease in the neutron energy in the source itself, as the authors do, 0.22 and 0.17 MeV. If in the past we had been dealing with only one work (for example, Goldhaber’s first work), we would hardly have had the audacity to assert that a difference of 5000 eV leads to such a sharp anomaly in the value of \(\sigma\). But, as we have seen from the detailed analysis of all the preceding works, this anomalous value of \(\sigma\) stubbornly appeared in four works, and in the last verification by the author of this article it was measured more than 10 times, with the mean value not differing by more than 20% from each individual measurement of \(\sigma\). It is quite obvious that such striking coincidences cannot be accidental. On the basis of Wettenberg’s latest data\(^{77}\) one can only conclude that the limits of the region in which this fall of \(\sigma\) occurs are very narrow, of the order of 10–15 KeV.

The sources \((\mathrm{Ga}, D)\) and \((\mathrm{Ga}, \mathrm{Be})\) give neutrons with energies of 0.16 and 0.32 MeV (or 0.13 and 0.27 MeV). Both the first and the second energies are close to the energies of groups I and II of the photoneutrons \((\mathrm{RaC}, \mathrm{Be})\). The values of \(\sigma\), as is evident from the corresponding tables and Fig. 4, agree well here, and we have nothing more to say about them. Unfortunately, among the photoneutrons obtained with artificially radioactive sources there are none that would be close to 0.4 MeV, i.e., to group IV of the photoneutrons \((\mathrm{ThC}'', \mathrm{Be})\); but in the following work we find neutron energies \((\mathrm{Li}, p)\) of 0.35 and 0.46 MeV, and two cross sections measured with these neutrons do not deviate from the theoretical curve. Thus, at this point, at a neutron energy very close to 0.4 MeV, apparently the same anomaly is repeated as at the energy 0.17 or \(\sim 0.2\) MeV.

Consequently, a detailed analysis of the results of all the works leads us to the conclusion that on the curve illustrating the dependence \(\sigma = f(E)\), two sharp minima are observed in the regions \(\sim 0.2\) and 0.4 MeV, and a maximum between them. There is as yet no theoretical explanation of this phenomenon in the literature; however, the author of this article received a private communication from M. Bendix, who, with the close participation of G. Beck, attempted to explain it by introducing a third level \(P\) (virtual). Four values of the cross sections calculated by him are given in Table VI.

As for the first conclusion, here one can only say that it is very probable; but to vouch with certainty that in further investigations in other regions analogous departures of \(\sigma\) will not appear is hardly possible.

We do not undertake at the present time to draw a final conclusion about the reality of the anomaly described. Many specialists adhere to the point of view that this anomaly does not exist. Only further, more precise investigations must give the final solution of this problem.

5. ANGULAR DISTRIBUTION OF NEUTRONS IN THEIR SCATTERING BY PROTONS

For checking the correctness of our ideas about the nature of nuclear forces and the law of their action, the most direct method is the study of the interaction of the neutron with the proton, as two elementary particles. This study consists, first, in investigating the functional dependence \(\sigma = f(E)\); second, in investigating the angular distribution of neutrons in their scattering by protons. The angular distribution is the most sensitive indicator of the details of the shape of the potential well, i.e. of the details of the interaction of the neutron with the proton. If nuclear forces can be represented in the form of a narrow and deep potential well, then the scattering of practically all neutrons will be spherically symmetric in the center-of-mass coordinate system.

Any deviation from a spherically symmetric distribution, recorded experimentally, will be a very important fact, since it can be explained only by assuming that the action of nuclear forces extends to distances considerably greater than \((1—2)\cdot 10^{-13}\) cm.

From this one can see what enormous importance, along with the study of the dependence (15), is acquired by the study of the angular distribution. It too was studied by many experimenters throughout the entire period of time that has passed since the discovery of the neutron; however, as will be seen from what follows, success until the very last time did not accompany the experimenters.

In one of the first works, Meitner and Philipp, with a neutron source (Rn, Be), measured in a Wilson chamber 100 recoil tracks of protons. They divided these tracks into 5 groups according to angular intervals and found a spherically symmetric distribution.

Auger and Monod-Herzen[^79], placing (Po, Be), just as Meitner and Philipp had done, outside the Wilson chamber, measured 180 tracks and divided them into two groups—“long” and “short.” For the first group of long tracks they obtained a spherically symmetric distribution; for the second, the maximum in the laboratory coordinate system was noticeably shifted toward larger angles. This indicates that the slower neutrons were scattered predominantly forward in the center-of-mass coordinate system.

F. Kory[^80] arrived at opposite results in two works. In this investigator’s experiments, (Po, Be) was surrounded by a thin layer

paraffin, from which the neutrons emerging from the source knocked out protons. This source was placed at the center of a Wilson chamber filled with hydrogen. All recoil tracks were divided into angular intervals of \(10^\circ\). Under these conditions, for a spherically symmetric distribution in the center-of-mass coordinate system, the maximum number of tracks should have been observed at \(45^\circ\). Kurie found a distinct maximum at \(25^\circ\). This meant that, in the center-of-mass coordinate system, the neutrons are scattered predominantly backward.

Dunning, likewise, in two papers\(^{81,82}\) directly studied the distribution of neutrons by means of ring scatterers made of paraffin and an ionization chamber connected to a proportional amplifier. The neutron source was the same reaction \((\mathrm{Rn}, \mathrm{Be})\). The scattering angle was varied by moving the scatterer between the source and the chamber. Unfortunately, Dunning did not measure the distribution in the interval \(0\)—\(45^\circ\), but since from \(45^\circ\) toward larger angles a gradual decrease of scattering is observed, it may be considered that his results indicate a symmetric distribution.

More extensive and precise measurements in a Wilson chamber were carried out by Harkins et al.\(^{83}\). As did Kurie, they placed a neutron source \((\mathrm{MsTh}, \mathrm{Be})\) at the center of the chamber, filled in one series of measurements \((A)\) with ethylene and in another \((B)\) with hydrogen. In all, 1000 tracks were measured, and their data, processed by Kurie’s method, led to the same result. On the basis of the energy yield curve obtained in one of the preceding works, the authors believe that the largest number of neutrons had an energy of about \(6\ \mathrm{MeV}\). In the chamber no tracks shorter than \(1\ \mathrm{cm}\) were observed, which corresponds to an energy of \(\sim 0.04\ \mathrm{MeV}\); however, the authors note—and this, as we shall see below, is extremely important—that from the lengths of the various tracks about \(25\%\) of all neutrons had energies lying in the interval \(0.1\)—\(1.0\ \mathrm{MeV}\).

All these works may be combined into one group. The characteristic feature here is the use of sources giving energetically inhomogeneous neutrons. Along with the initial part of the neutron spectrum, containing neutrons with energies on the average \(0.2\)—\(0.4\ \mathrm{MeV}\), there is a central part with energies of the order of \(3\)—\(5\ \mathrm{MeV}\) and the end of the spectrum, containing an insignificant number of neutrons of high energy. Neglecting this latter part, we see that in all the works, with the exception of Dunning’s, the angular distribution of neutrons from two regions, superposed on one another, was measured.

To the second group belong very precise investigations, in a Wilson chamber, of the angular distribution of energetically homogeneous neutrons \((D, D)\). The most extensive and accurate of these works is that of Dee and Gilbert\(^{84}\). A target of heavy aluminum hydroxide, \(1.2\ \mathrm{cm}\) in diameter, was placed at a distance of \(17\ \mathrm{cm}\) from

walls of a chamber 25 cm in diameter. The chamber was filled with a mixture of methane and argon at a pressure of 3.5 atm. Under this operating condition of the chamber the range of protons at an angle \(\varphi=0^\circ\) with respect to the neutron direction was 3.5 cm. The neutron energy was taken to be 2.4 MeV. In all, 2000 tracks were measured, and quite distinctly their maximum was recorded at \(\varphi=45^\circ\), i.e., the distribution was found to be spherically symmetric.

In the second work Bonner\({}^{85}\) used a target of \(\mathrm{P_2O_5 + D_2O}\), placed at a distance of 18.5 cm from the wall of a chamber 17 cm in diameter and 4.5 cm deep. He measured 1000 tracks. From this number he selected 303 tracks satisfying the conditions he had set: measurement of the length with an accuracy up to 1 mm and of the angle \(\varphi\) up to \(5^\circ\). The distribution of these selected tracks, as in the preceding work, satisfied a spherically symmetric distribution and, within the limits of the statistical errors of the measurements, did not differ from the distribution of all the tracks. The neutron energy was 2.6 MeV.

In the third work Kruger et al.\({}^{86}\) used a cyclotron. The great power of the source made it possible to place the chamber at a distance of 41 cm from a heavy-ice target irradiated by deuterons with an energy of 1 MeV. The diameter of the chamber, filled in one series of measurements with hydrogen and in another with methane, was 13 cm and the depth 3.2 cm. From the distribution of 1163 tracks the authors found a spherically symmetric distribution.

In the fourth work, Lampson et al.\({}^{87}\) chose the photographic-emulsion method. At a neutron energy of 2.55 MeV they found that the maximum number of tracks was displaced to \(\varphi=25^\circ\). It may be thought that this result is due to some systematic error, which is easy to admit in this method of measurement.

With this work, published, like all the works on the angular distribution of \((\mathrm{D}, \mathrm{D})\) neutrons, in 1937, the verification of the laws of angular distribution in favor of a spherically symmetric one apparently comes to an end. Only in 1940 was a further paper by Barshall and Kanner\({}^{88}\) published, and in 1946 a paper by the author of the present article\({}^{89}\). We shall set out this last work after a critical analysis of the results of the works described, which was also the stimulus for undertaking it. As for the work of Barshall and Kanner, as they themselves indicate, their investigation was undertaken not to verify the angular distribution, which they regard as not open to doubt, but to test the method proposed by them for determining this distribution by means of an ionization chamber.

Table VII gives a complete summary of all the works by year. At first glance at this table it becomes clear that the most accurate measurements invariably lead to a spherically symmetric distribution and, consequently, for all neutron energies up to 15–20 MeV, this law of scattering must be regarded as valid.

Table VII

Angular distribution of neutrons in their scattering by protons

Researcher	Method of investigation	Neutron energy in MeV	Results of the investigation. Data are given in the center-of-mass system	Year of publication of the work
F. N. D. Curie	Wilson chamber	2.5—14.0	More neutrons go backward	1933
Auger and Monod-Herzen	Same	0.5—14.0	Spherically symmetrical	1934
Auger and Monod-Herzen	Same	<0.5	More forward	1934
Meitner and Philipp	” ”	0.2—14.0	Spherically symmetrical	1934
Dunning	Annular scatterer	0.5—14.0	Spherically symmetrical	1934
Harkins et al.	Wilson chamber	0.1—15.0	More backward	1936
Dee and Gilbert	Same	2.4	Spherically symmetrical	1937
Bonner	” ”	2.6	Same	1937
Kruger et al.	” ”	2.5	” ”	1937
Lamson et al.	Photographic emulsion	2.55	More backward	1937
Barshall and Kanner	Ionization chamber	2.5	Spherically symmetrical	1940
Amaldi et al.	Wilson chamber	14.0	More forward	1944
Goloborodko	Annular scatterer	0.2	Asymmetric distribution: maxima at angles of 45° and 135° and minima at angles of 90° and 180°	1947

And yet, with a more careful analysis, doubt arises as to the absolute reliability of such a conclusion. The fact that the sharpest deviations from symmetry are observed almost exclusively in the works of the first group, where inhomogeneous neutrons were used, can be explained, as indeed was done by all the experimenters, by those systematic errors which this inhomogeneity creates; however, one may think that the asymmetry is real. It may exist in the distribution of neutrons of low energy, of the order of 0.2—0.4 MeV, and disappear for neutrons of higher energy. Such an idea appears in the investigation of the results of Monod-Herzen, Curie, and Harkins. In these works the anomalous scattering is, as it were, produced by neutrons of low energy. It is superimposed on a powerful background of fast neutrons having sym—

metric distribution, and its share is so small that, in the total distribution, the deviations lie at the limit of the measurement errors.

Naturally, for an experimental verification of this assumption, the best method would be one by which it would be possible to weaken the influence of the fast neutrons and, having separated out the slower ones, to study their distribution.

Such a method, as we saw in studying the work on the measurement of \(\sigma\), was carried out by filtering neutrons with layers of paraffin of different thickness. There it was used rather successfully. In the case of studying the angular distribution, its application, as may be expected, will be still more effective, since the neutrons that must be separated differ in energy by several million electron-volts.

On the basis of these considerations, experiments were set up on the angular distribution of neutrons of the initial part of the spectrum (Re, Be).

In a special investigation it was found that this part consists of a homogeneous group with energy \(\sim 0.2\) MeV. For the measurements the method of ring scatterers was chosen. The detector was Rh, placed at the center of a paraffin sphere of diameter 5 cm. It is quite evident that, for fast neutrons, this paraffin sphere is only a scatterer; neutrons with energy of the order of 0.2 MeV, however, falling on it, are to a considerable extent slowed down to thermal velocities, or close to them, and activate the detector. To study scattering at a given angle, a separate ring scatterer of definite diameter was always used, placed midway between the source and the detector. Such a procedure ensured greater accuracy of measurement than moving the scatterer from the source to the detector, or conversely, since in this case the neutrons traverse equal paths before and after scattering.

Table VIII

Angular distribution of neutrons with energy \(\sim 0.2\) MeV in the laboratory coordinate system

Scattering angle \(\varphi\)	Ratio of intensities: \(\displaystyle \frac{I}{I_0}=\frac{\text{with scatterer}}{\text{without scatterer}}\)
\(25^\circ \pm 12^\circ\)	\(1.55 \pm 0.06\)
\(45^\circ \pm 12^\circ\)	\(1.05 \pm 0.05\)
\(68^\circ \pm 12^\circ\)	\(1.46 \pm 0.06\)
\(90^\circ \pm 13^\circ\)	\(1.11 \pm 0.05\)
\(113^\circ \pm 3^\circ\)	\(1.46 \pm 0.05\)
\(135^\circ \pm 13^\circ\)	\(1.02 \pm 0.03\)
\(155^\circ \pm 13^\circ\)	\(1.45 \pm 0.04\)

For measurements at \(\varphi = 25^\circ, 45^\circ\), and \(68^\circ\), the distance between the source and the detector was 20 cm; for measurement at larger angles it was reduced to 10 and 6 cm. Measurements of the angular distribution of neutrons in their scattering by carbon were carried out separately, and the values found were subtracted from the total effect with the paraffin scatterer. The scattering by carbon, as expected, was very small, \(\sim 10\%\) of the total effect. The results of these measurements are given in Table VIII, and also in Fig. 5, cor—

together with the data of Mano-Hertzen and Harkins. As is seen from this figure, our assumptions about the existence of scattering asymmetry in the region of low energies are fully justified. The results are in complete agreement with the data of these investigators, but the asymmetry, obscured in their measurements by a large number of fast neutrons, appears considerably more sharply in our measurements.

As is seen from Table VIII, regular oscillations of neutron scattering are unexpectedly observed also at angles \(\varphi > 90^\circ\). This effect disappears when neutrons with higher energy are scattered (which is achieved by increasing the dimensions of the paraffin sphere surrounding the detector) and reappears when neutrons of lower energy are scattered. This backscattering in the laboratory coordinate system is completely incompatible with the usual conception of the elastic interaction of two bodies of equal mass, and further investigations are necessary to confirm its reality.

Fig. 5. Angular distribution of neutrons in their scattering by protons. Points ● denote the distribution of neutrons with energy \(\sim 0.2\) MeV; ▲—with energy \(>0.2\) MeV; ■ denote the data of Mano-Hertzen; ×—the data of Corey and Harkins. Line \(T\) corresponds to a spherically symmetric distribution in the central coordinate system.

The general conclusion obtained from a critical analysis of all experiments on the angular distribution seems quite clear. In the energy region \(0.2\)—\(0.4\) MeV there exists a sharp asymmetry in the scattering of neutrons by protons. In the region \(2\)—\(3\) MeV the scattering does not deviate from spherically symmetric scattering in the central coordinate system. The law of action of nuclear forces is probably close to that proposed by Schar and Stein. The nuclear forces near the center are large and decrease rapidly with distance, but then this decrease becomes considerably smaller and the action of weak forces extends to distances probably \(10\)—\(20\) times greater than the width of the central well.

6. INTERACTION OF HIGH-ENERGY NEUTRONS WITH PROTONS

In the scattering of neutrons with energy of the order of 15 MeV and higher, whose wavelength is comparable with the width of the central potential well, deviations should be observed from spherical symmetry of the angular distribution and from the dependence \(\sigma = f(E)\), вы-

given by formula (15). The first measurements of \(\sigma\) with neutrons \((\mathrm{Li}, \mathrm{D})\), having an energy of about \(15\) MeV, carried out by Roberts et al.^90, led to the value \(\sigma = 0.61 \cdot 10^{-24}\ \mathrm{cm}^2\). Copper was used as the detector; its excitation threshold lies between \(12\) and \(13\) MeV. A Li target was bombarded with deuterons of energy \(0.9\) MeV. The neutrons were scattered by paraffin. The \(\sigma\) for carbon at this neutron energy proved to be \(1.13 \cdot 10^{-24}\ \mathrm{cm}^2\). The authors note that the value of \(\sigma\) found for hydrogen, at a deuteron binding energy of \(2.17\) MeV and a range of nuclear forces \(r_0 = 2.81 \cdot 10^{-13}\ \mathrm{cm}\), agrees well with the theoretical value.

Amaldi et al.^91, using the reactions \((\mathrm{Li}, \mathrm{D})\) and \((\mathrm{B}, \mathrm{D})\), measured the angular distribution of neutrons. They consider that under the conditions of their experiment the neutron energy of the first source was \(13.5\) MeV and of the second \(12.5\) MeV. Measuring the scattering intensity at the angles \(\varphi = \pi/2\) and \(\varphi = \pi\) in the center-of-mass coordinate system, they found, for

\[ E_n = 12.5\ \mathrm{MeV}, \]

the intensity ratio

\[ R = \frac{I(\pi)}{I\left(\frac{\pi}{2}\right)} = 0.71, \]

and for

\[ E_n = 13.5\ \mathrm{MeV}, \quad R = 0.52. \]

The study of the asymmetry of the angular distribution of neutrons and of the deviation of \(\sigma\) from the dependence (15) acquires very great importance in connection with the mesotron theory of the deuteron and, in general, of nuclear forces, developed by many theorists. As is known, in order to obtain the correct order of magnitude of the binding of nucleons in the nucleus, it is necessary to introduce a new particle with a mass of the order of \(200\) electron masses \((m)\). This particle, without charge, was given the name “neutretto.” The theory of nuclear forces using, for the exchange force between nucleons, only the neutretto was developed by Bethe^92. It received the name “neutral” theory. After the discovery of the mesotron in cosmic rays, theories were created that take into account exchange between nucleons of particles of three kinds with masses of the order of \(200\,m\). Two of them are positively and negatively charged, the third is neutral. These theories were called “symmetric.”

As Rarita and Schwinger^93 showed, the symmetric and neutral theories must lead to different values of \(R\). The symmetric theory, created by Møller and Rosenfeld^94, gives, for \(14\) MeV neutrons, the value \(R = 1.63\).^95 This value is in sharp contradiction with the data of Amaldi et al. Comparing their results with the theoretical ones, these authors come to the conclusion that they agree rather well with Ferretti’s data^96, obtained from Bethe’s neutral theory^92. Such a conclusion, excluding from the mechanism of interaction of nuclear particles the known observable charged mesotrons, naturally could not satisfy physicists and gave rise to new experimental and theoretical work.

Sher[^97], bombarding Li with 10 MeV deuterons, obtained neutrons with an energy of 25 MeV. He found the value \(\sigma=(0.39 \pm 0.03)\cdot 10^{-24}\,\text{cm}^2\). The theoretical value of \(\sigma\), calculated from the data of the symmetric theory by Rarita and Schwinger[^98] for this neutron energy, is \(0.395\cdot 10^{-24}\,\text{cm}^2\); from the neutral theory, \(\sigma=0.89\cdot 10^{-24}\,\text{cm}^2\). Thus, the value of \(\sigma\) found agrees better with the symmetric theory.

Ageno et al.[^98] repeated the measurements of Amaldi et al. with neutrons of three energies: \(E_1=4.1\) MeV, reaction \((\text{Be},\text{D})\); \(E_2=12.5\) MeV, reaction \((\text{B},\text{D})\), and \(E_3=13.5\) MeV, reaction \((\text{Li},\text{D})\). Carefully measuring the change in neutron intensity by means of a complex arrangement of three Geiger–Müller counters installed for coincidences, they found the corresponding values: \(\sigma_1=(1.73\pm0.06)\cdot10^{-24}\,\text{cm}^2\), \(\sigma_2=(0.69\pm0.11)\cdot10^{-24}\,\text{cm}^2\), and \(\sigma_3=(0.69\pm0.019)\cdot10^{-24}\,\text{cm}^2\). Analyzing the preceding theoretical and experimental data, they again come to the conclusion that their results agree better with the data of the neutral theory.

The latest theoretical investigations[^99,^100] indicate that, for a more complete clarification of the problem, further studies of neutron scattering with energies on the order of 100–200 MeV are necessary. Apparently, such measurements, quite feasible at the present time, will help to solve this interesting and tangled problem completely.

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