INTERACTION OF ELEMENTARY PARTICLES

L. A. Artsimovich, Leningrad

One of the principal tasks of the physics of the atomic nucleus is the study of the forces acting between elementary nuclear particles—neutrons and protons. The character of these forces ultimately determines the structure and properties of all atomic nuclei.

The stock of experimental data presently available is clearly insufficient for constructing an exhaustive picture of the interaction of elementary particles. Modern theories are in themselves powerless to give us a guiding idea for understanding these new types of interaction. Therefore the study of nuclear forces is still at the very earliest stage of its development. Nevertheless, even now we can extract from the experimental data a number of important indications concerning the character of nuclear forces. We shall try to analyze these indications, relying, as far as possible, only on firmly established facts, in order to separate solidly confirmed results from passing theoretical constructions.

Since there are two kinds of heavy elementary particles—neutrons and protons—there must be three types of elementary interactions: neutron—proton, proton—proton, and neutron—neutron. Of these three types of interaction, only the first two are at present accessible to direct experimental investigation; the interaction of two neutrons can now be judged only on the basis of indirect data.

Let us consider the interaction of a neutron and a proton. To clarify the character of this interaction we have at our disposal, first, data relating to the processes of collision of free neutrons with protons and, second, data on the structure and properties of atomic nuclei. Above all, one observation may be made. Since all nuclei consist of neutrons and protons, it follows that the forces between neutrons and protons, for certain states of motion of these particles, are forces of attraction. This assertion becomes quite obvious if one notes that the simplest nuclear system—the deuteron—consists only of one proton and one neutron.

Let us now turn to the analysis of data on collisions of neutrons with protons. The study of particle collisions is, as is well known, one of the most direct and fruitful methods for analyzing the forces acting between them. At present the following types of collisions of neutrons with protons are known:

1. Elastic scattering of neutrons by protons.

2. Capture of a neutron by a proton, accompanied by the emission of γ-rays. As a result a stable system is formed—the deuteron. This process is described by the simplest nuclear reaction:

$$ n^{1}+H^{1}\to H^{2}+h\nu . $$

1. Photodisintegration of the deuteron—a process inverse to the capture of a neutron by a proton. The nuclear reaction is:

$$ H^{2}+h\nu\to n^{1}+H^{1}. $$

All these processes are observed experimentally. Of these elementary processes, the elastic scattering of neutrons has been studied relatively best. In order to observe this phenomenon, a beam of neutrons is passed through a substance containing hydrogen (water, paraffin). As a result of an individual collision, the neutron deviates from its original direction of motion, while the proton, which before the collision was at rest, undergoes recoil and acquires kinetic energy (Fig. 1). Since the neutron and the proton have almost exactly the same mass, after the collision the angle between the directions of their motion must be equal to 90°. This conclusion follows from the laws of conservation of energy and momentum.

Fig. 1
1 — initial direction of motion of the neutron, 2 — direction of motion after collision with the hydrogen atom, 3 — recoil proton

It is obvious that measurements of the scattering of neutrons by protons may be based either on the registration of deflected neutrons or on the registration of recoil protons. Recoil protons are usually registered by means of a Wilson chamber, into which a substance containing hydrogen is placed. A parallel beam of neutrons is passed through the chamber. With the aid of the Wilson chamber one can obtain information above all about the character of the angular distribution of recoil protons (and consequently also of the scattered neutrons). This method is used, however, only in the case when the neutrons possess sufficiently large kinetic energy (of the order of several hundred kiloelectron-volts and higher). If the neutrons have a low velocity, the recoil protons will have a very small range, and they cannot be observed in the Wilson chamber.

Using another method, based on the registration of the scattered neutrons themselves, we do not encounter this inconvenience, since we have indicators for neutrons of the most varied energies (from 0.01 V to several million volts). Registration of neutrons is possible owing to the various nuclear reactions which they induce. The most commonly used indicators are those substances in which neutron bombardment leads to the creation of artificially radioactive elements. Thus, for example,

a silver plate can serve as an excellent indicator of slow neutrons. Neutrons, adhering to silver nuclei, lead to the formation of an unstable isotope, according to the reaction:

\[ \mathrm{Ag}^{109}+n^{1}\to \mathrm{Ag}^{110}. \]

The nuclei \(\mathrm{Ag}^{110}\) decay with the emission of electrons and turn into nuclei \(\mathrm{Cd}^{110}\). The number of electrons emitted by radioactive silver is proportional to the intensity of the neutron flux incident on the plate. Boron, lithium, rhodium, iodine, etc., are also convenient indicators of slow neutrons. Copper, aluminum, etc., can be used to register fast neutrons. In addition to the angular distribution, one can also measure the absolute value of the effective cross section for neutron scattering by protons. For this purpose it is necessary to measure the decrease in the intensity of a direct beam of neutrons that has passed through a layer of a substance containing hydrogen. If the neutron beam has a small divergence and is sufficiently monochromatic, and the neutron indicator is located at a sufficiently large distance from the scatterer, then the decrease in the intensity of the neutron flux is expressed by the following simple formula:

\[ I=I_{0}e^{-N\sigma x}. \]

Here \(I_{0}\) is the initial intensity of the neutron beam, \(I\) is the intensity of the beam that has passed through a layer of substance of thickness \(x\), \(N\) is the number of atoms in \(1\ \mathrm{cm}^{3}\), and \(\sigma\) is the effective scattering cross section. This formula is valid in such an interpretation, of course, only in the case where scattering by other atoms entering into the composition of the given substance can be neglected. If this cannot be done, then it is necessary to make special control measurements in order to determine what fraction of the effective cross section \(\sigma\) is due to collisions with protons. Such a check can usually be performed easily.

By the indicated methods the angular distribution was investigated and the effective scattering cross section was determined for a very wide range of neutron energies—from \(0.03\ \mathrm{eV}\) to several million electron-volts (angular distribution up to \(4\ \mathrm{MeV}\), cross-section value up to \(16\ \mathrm{MeV}\)). First of all, it was possible to find the law of the angular distribution of scattered neutrons, which can be formulated as follows: the number of neutrons scattered into a unit solid angle at an angle \(\vartheta\) to the initial direction of motion is proportional to \(\cos \vartheta\). This law assumes an even simpler form if one passes from the usual coordinate system, in which the proton is at rest before the collision, to the coordinate system in which the center of gravity of the two colliding particles is at rest. Such a coordinate system is better adapted for studying the regularities of scattering, since in it the asymmetry in the initial conditions for the neutron and the proton is eliminated. A simple recalculation shows that in this natural coordinate system the angular distribution of the scattered neutrons and recoil protons possesses spherical symmetry.

From this basic fact one can draw a quite definite conclusion about the character of the forces of interaction between the neutron and the proton. According to the principles of quantum mechanics, from the isotropy of the angular distribution of scattered neutrons it follows that the forces between the neutron and the proton have a very small range. The region within which these forces are concentrated must be smaller than the de Broglie wavelength of the incident neutron in order that the scattering be isotropic in space.

Experiment shows that even for neutrons having a very large energy and, consequently, a very small wavelength \((E_0 = 4—6\ \mathrm{mV},\ \lambda = 3—4 \cdot 10^{-13}\ \mathrm{cm})\), the scattering is isotropic. Therefore the range of action of the forces between the neutron and the proton must be less than \(4 \cdot 10^{-13}\ \mathrm{cm}\). The concept of “forces with a small range of action” has, of course, a qualitative rather than a quantitative character. It means that the potential well by means of which one may represent the energy of interaction of the neutron and the proton has the form of a deep well, outside of which the interaction energy is practically equal to zero.

The connection between the range of action of the forces and the angular distribution of the scattered particles follows from the basic principles of the quantum theory of particle collisions. A particle moving in the field of a fixed scattering center\({}^{1}\) may have various values of the angular momentum. In a central force field the angular momentum is quantized and can take only a series of discrete values corresponding to the azimuthal quantum numbers \(l = 0, 1, 2, 3,\ldots\) (\(S, P, D, F\)-states in optical terminology). The values of the angular momentum are respectively equal to:

\[ 0,\ \frac{h}{2\pi},\ 2\frac{h}{2\pi},\ 3\frac{h}{2\pi}, \ldots \]

[More precisely, to the azimuthal quantum number \(l\) there corresponds the angular momentum

\[ R = \frac{h}{2\pi}\sqrt{l(l+1)} . \]

] The greater the angular momentum, the farther, on the average, the scattered particle passes from the center. The presence of an angular momentum not equal to zero is equivalent to the existence of a fictitious potential barrier (the centrifugal-force barrier), which does not allow the particle to approach the scattering center.

If the range of action of the forces \(r_0\) is small in comparison with the wavelength of the incident particle \(\lambda\), then this additional potential barrier almost completely isolates the field of the scattering center. Consequently, a particle with a wavelength large in comparison with the range of action of the forces will be scattered only in the case when its angular momentum is equal to zero, i.e., when it is in an \(S\)-state with respect to the center of force.

\({}^{1}\) When we consider the collision of a neutron and a proton, using a “symmetrical” coordinate system, the role of the fixed scattering center is played by the center of gravity, since we may mentally make it a source of forces acting on each of the colliding particles.

In quantum mechanics, scattering by a center of forces is regarded as the diffraction of a plane wave by a spherical obstacle (if one uses the optical analogy, then it must be added that this obstacle is made of a transparent material with a variable refractive index). The plane wave corresponding to the incident particle can obviously be regarded as a superposition of a series of spherical waves corresponding to quantized values of the angular momentum with \(l=0, 1, 2, 3,\ldots\). On the basis of what was set forth above, we conclude that the influence of the scattering center for \(\lambda \gg r_0\) will be manifested only in the component with \(l=0\). Thus only the wave with \(l=0\) will undergo scattering, and only particles with zero angular momentum will be present in the scattered radiation. But zero angular momentum corresponds to spherical symmetry of the wave function (recall the spherical symmetry of the ground state of the hydrogen atom with \(l=0\)). Therefore the wave function of the scattered particles will also possess spherical symmetry, and the angular distribution will be characterized by complete isotropy in space. This result is entirely independent of any concrete ideas about the shape of the potential well; the details of its internal structure do not affect the character of the angular distribution of the scattered particles.

One should not think that such a connection between the size of the region scattering the wave and the angular distribution of scattering is due to specific quantum laws. It is a property of the diffraction problem under consideration. An analogous result is obtained if, for example, one studies the diffraction of sound waves by a rigid sphere of sufficiently small dimensions.

Let us now dwell on the properties of the simplest nuclear system—the deuteron, postponing for the time being the analysis of those data yielded by measurements of the magnitude of the effective cross section for the scattering of neutrons by protons. Experiments on the photodisintegration of the deuteron make it possible to find its binding energy; it is determined in the following way.

A beam of \(\gamma\)-rays of a prescribed wavelength (for example, \(\gamma\)-rays of ThC\('\) with energy \(2.62\) MeV) illuminates an ionization chamber or a Wilson chamber filled with heavy hydrogen. When the deuteron nucleus is split, the bond between the neutron and the proton is broken, and both particles fly apart in mutually opposite directions with the same kinetic energy, equal to

\[ E=\frac{h\nu-\varepsilon}{2}. \]

In this expression \(\varepsilon\) denotes the binding energy of the deuteron. The equality of the kinetic energies of the neutron and the proton follows from the fact that the total momentum of the system is zero both before and after the collision. The photon momentum may be neglected. The kinetic energy of the proton can be measured either by its range in the Wilson chamber, or by the ionization current pulse produced by it in the ionization chamber. This current pulse is proportional to the total number of ions formed by the proton, and consequently also to its

of kinetic energy. Having measured the kinetic energy of the proton, one can determine \(\varepsilon\) on the basis of the relation given above. According to Chadwick’s measurement it turns out to be equal to \(2.18\) MeV.

Thus, there are two basic facts on which one may rely in attempts to elucidate the nature of the forces of interaction between the neutron and the proton. First, the range of action of these forces must be less than \(4\cdot 10^{-13}\) cm. Second, the potential well must have such parameters that the energy of the ground level of the deuteron is equal to \(2.18\) MeV. These data, of course, are quite insufficient for determining the mathematical expression for the potential energy of interaction. However, by using them one can obtain qualitative information about the depth of the potential well. In order that, with a radius of the order of \(10^{-13}\) cm, a level with energy \(2.18\) MeV could fit in this potential well, it is necessary that its depth be of the order of several tens of MeV. This conclusion is not connected with any specific assumptions about the form of the potential well and follows from an analysis of the solutions of the problem of quantizing the motion of a particle in a field of forces with a small range of action. If one chooses some definite form of the dependence of the potential energy on the distance between the neutron and the proton, then one can establish a relation between the constants characterizing the potential well. Thus, for example, if we assume that the well has the form of a rectangular well (Fig. 2), then the relation between its depth and its radius can immediately be found. If, in addition, one of these constants is fixed, then the other is uniquely determined from the condition \(\varepsilon=2.18\) MeV. If, in particular, the radius \(r_0\) is taken equal to the classical electron radius, i.e. equal to \(2.8\cdot 10^{-13}\) cm, then the depth of the well turns out to be \(21.2\) MeV. For other values of \(r_0\) the following numbers are obtained:

Fig. 2. Potential well characterizing the interaction of a neutron with a proton

\[ r_0=1,\quad 2,\quad 3\ \ldots \times 10^{-13}\ \text{cm}, \]
\[ V_0=100,\quad 40,\quad 15\ \ldots\ \text{MeV}. \]

All these values are entirely equivalent, since each pair of quantities \(V_0\) and \(r_0\) has been chosen so as to satisfy the condition \(\varepsilon=2.18\) MeV. If one chooses some other mathematical expression for the potential energy, for example a Gaussian curve

\[ V=-V_0 e^{-\frac{r^2}{r_0^2}}, \]

then a somewhat different relation between \(V_0\) and \(r_0\) is obtained. There is almost unlimited scope in the choice of the form of the function \(V(r)\). The only condition to which

must be obeyed by this choice is that the function \(V(r)\) must tend to zero sufficiently rapidly as \(r\) increases beyond the known limits.

However, no matter how we choose the shape of the deuteron potential well, we cannot place in it even a single level, apart from the normal level with \(\varepsilon = 2.18\) MeV, corresponding to the lowest energy state with azimuthal quantum number \(l = 0\) (the fundamental \(S\)-state of the deuteron). Levels with \(l = 1, 2, 3, \ldots\), as well as higher levels with \(l = 0\), do not fit inside the potential well unless its radius is increased to \(r_0 > 5 \cdot 10^{-13}\,\text{cm}\), which is forbidden by the symmetry of the angular distribution of scattering. We arrive at the conclusion that the deuteron cannot have discrete excited states.

The binding energy of the deuteron characterizes the forces acting between the neutron and the proton with parallel spins. This follows from the experimental fact that the deuteron has nuclear spin equal to unity, i.e., angular momentum equal to \(\frac{h}{2\pi}\). Since the neutron and the proton, each separately, have spins equal to \(1/2\), it follows that in the deuteron the spins of both particles are parallel. The very fact that deuterons with spin equal to zero are absent indicates that the energy level for such a deuteron, in which the spins of the two particles are antiparallel, lies higher than the normal energy level of the deuteron with spin equal to unity. However, on the basis of this alone nothing can yet be said about how large the difference between the binding energies is for these two cases. It may be quite negligible and caused by secondary circumstances, such as magnetic splitting of the levels. Therefore, from data on the structure and binding energy of the deuteron, no information can be extracted about the interaction of a neutron and a proton with antiparallel spins. The situation is different in the study of neutron scattering by protons. In this phenomenon the dependence of the forces on the direction of the spins must show itself quite distinctly. In neutron scattering by protons, for every three cases of collision of particles with parallel spins there must be one case of collision with an antiparallel arrangement of the spins. This ratio is connected with the statistical weights of the two states. The antiparallel state gives a total spin equal to zero and, consequently, only one state in an external field. The parallel state gives a total spin equal to unity and therefore splits into three “substates” in an external field (spin projection: \(-1, 0, +1\)). The effective cross section \(\sigma\) measured experimentally is equal to:

\[ \sigma = \frac{3}{4}\sigma_p + \frac{1}{4}\sigma_a . \]

In this expression \(\sigma_p\) and \(\sigma_a\) denote the effective cross section for the cases of parallel and antiparallel spin arrangement. It turns out that for \(\lambda \gg r_0\) these cross sections can be calculated from the general theory of scattering without any new assumptions about the shape of the potential well. For an approximate calculation of each of them

it is necessary to know a single constant, namely, the binding energy. The calculation gives, without any other assumptions, the following formulas:

\[ \sigma_p=\frac{h^2}{\pi M}\frac{1}{|\varepsilon_p|+\frac{1}{2}E}, \]

\[ \sigma_a=\frac{h^2}{\pi M}\frac{1}{|\varepsilon_a|+\frac{1}{2}E}. \]

Here \(E\) is the energy of the neutron, \(M\) its mass, and \(\varepsilon_p\) and \(\varepsilon_a\) are, respectively, the binding energies for deuterons with parallel and antiparallel spins. Consequently,

\[ \sigma=\frac{h^2}{4\pi M} \left\{ \frac{3}{|\varepsilon_p|+\frac{1}{2}E} + \frac{1}{|\varepsilon_a|+\frac{1}{2}E} \right\}. \]

The value of \(\varepsilon_p\) is known. It is equal to \(2.18\ \mathrm{MeV}\). Measuring \(\sigma\) at various neutron energies, one can find the absolute magnitude \(\varepsilon_a\). Measurements at very low neutron energies, where \(\sigma\) is most sensitive to the difference between \(|\varepsilon_p|\) and \(|\varepsilon_a|\), gave:

\[ |\varepsilon_a|=105\,000\ \mathrm{eV}. \]

The sensitivity of \(\sigma\) to changes in \(|\varepsilon_a|\) is seen from the fact that if one assumes \(|\varepsilon_p|=|\varepsilon_a|\), then for very low energies the computed \(\sigma\) will be equal to \(2.4\cdot 10^{-24}\ \mathrm{cm}^2\), whereas the measured value of \(\sigma\) is equal to \(14.8\cdot 10^{-24}\ \mathrm{cm}^2\).

These measurements show that the energy level of the deuteron with zero spin lies much higher than that of the deuteron with spin equal to unity. Hence it follows that the forces between the neutron and the proton depend rather strongly on the orientation of the spins. The difference between the two types of interaction may be characterized by the difference in the depths of the potential well for the same range of action. For the interaction with parallel spins, \(\varepsilon=2.18\ \mathrm{MeV}\), and for a radius \(r_0=2.8\cdot 10^{-13}\ \mathrm{cm}\) the depth of a rectangular well is equal to \(21\ \mathrm{MeV}\). For the interaction with antiparallel spin orientation, the binding energy may be regarded as practically equal to zero, and for a range \(2.8\cdot 10^{-13}\ \mathrm{cm}\) the depth of the well turns out to be \(11.5\ \mathrm{MeV}\).

The value of \(|\varepsilon_a|\) was obtained on the basis of measurements of the scattering of very slow neutrons with energies of the order of \(1\ \mathrm{eV}\). Substituting this value into the formula for the effective scattering cross section, one can calculate \(\sigma\) for neutrons of any energy. The calculated and measured values of \(\sigma\) are given in Table 1. When comparing them with one another, it should be borne in mind that the calculation formula is approximate, while the experimental values were measured with an accuracy not exceeding \(10\text{–}15\%\). Only for small energies was \(\sigma\) measured with an accuracy of up to \(3\text{–}5\%\). Under these conditions one may state good agreement between theory and experiment, proving the correctness of the basic assumption that the potential well has the form of a deep well.

The scattering formula includes not the binding energy \(\varepsilon_a\) itself, but its absolute value. Therefore the question of the existence of a stationary state for the deuteron with antiparallel spins remains unresolved. If \(\varepsilon_a\) is positive, then such a stationary state can exist. If \(\varepsilon_a\) is negative, then the corresponding energy level is “virtual,” and a deuteron with antiparallel spins of the particles cannot be realized. The answer to the question of the sign of \(\varepsilon_a\) is given by measurements of neutron scattering in para- and ortho-hydrogen. The scattering process of very slow neutrons (\(E \simeq 0.02\ \mathrm{eV}\)) in para- and ortho-hydrogen must be different for the following reasons.

Table 1

1. In para-hydrogen, for very slow neutrons only elastic scattering can take place. At low temperature the molecules of para-hydrogen are at levels with rotational quantum number \(j=0\). In order to excite the molecule, i.e. to raise it to a higher energy level, the neutron must have sufficient energy. Therefore, for very slow neutrons, inelastic scattering accompanied by excitation of the rotational levels of the molecules is impossible. By contrast, for ortho-hydrogen molecules, which at low temperatures occupy levels with rotational quantum number \(j=1\), a transition under neutron impact from \(j=1\) to \(j=0\) is possible, with transformation of an ortho-hydrogen molecule into a para-hydrogen molecule. Such a transition is possible because the forces between the neutron and proton depend on the orientation of the spins.

2. The second difference in the scattering of neutrons by para- and ortho-hydrogen molecules is connected with the difference in the character of the interference of neutron waves from two protons with parallel and antiparallel spins. In ortho-hydrogen both protons have the same spin direction and therefore give the same amplitude of the scattered waves. In para-hydrogen the spins of the protons are directed in opposite directions. Therefore, when a neutron undergoes scattering in a para-hydrogen molecule, it interacts strongly with one of the protons and interacts more weakly with the other. The amplitudes of the waves scattered by the two protons will be different. Therefore the waves scattered by the two protons will interfere with one another differently in the cases of para- and ortho-hydrogen. Moreover, the interference in a para-hydrogen molecule depends very strongly on the sign of \(\varepsilon_a\). If \(\varepsilon_a\) is positive (a stable level), then the waves scattered by the two protons of the para-hydrogen molecule will have the same phase. If, however, \(\varepsilon_a < 0\) (a virtual level), the waves scattered by the two protons will be shifted in phase by \(180^\circ\). In the subsequent

in this case destructive interference takes place, considerably reducing the probability of scattering of neutrons by the para-hydrogen molecule. The destructive interference must manifest itself the more strongly, the greater the neutron wavelength, i.e., the smaller its velocity. This is connected with the fact that at large wavelengths the phase shift arising from the geometrical path difference of waves scattered by the two protons tends to zero, and neutron waves scattered by the two protons in any direction will have opposite phases. Therefore, if the deuteron level with antiparallel spins is virtual, one should expect that the scattering intensity in ortho-hydrogen will considerably exceed the scattering intensity in para-hydrogen and, moreover, that scattering in para-hydrogen should rapidly decrease as the neutron velocity decreases. These considerations were first expressed by Teller and Schwinger, who also gave a quantitative theory of neutron scattering by ortho- and para-hydrogen molecules.

Experiments on the scattering of neutrons with thermal velocities, carried out by Stern and his collaborators, as well as by Dunning, Brickwedde, and others, showed that the scattering intensity in ortho-hydrogen is considerably greater than in para-hydrogen, and that the ratio of the effective scattering cross sections in the two modifications of hydrogen increases as the neutron energy decreases. According to Dunning, at a mean neutron energy equal to \(5\cdot 10^{-2}\,\mathrm{eV}\),

\[ \sigma_{\text{ortho}} = 56\cdot 10^{-24}\ \mathrm{cm}^{2}, \]
\[ \sigma_{\text{para}} = 29\cdot 10^{-24}\ \mathrm{cm}^{2}. \]

At a mean energy equal to \(2\cdot 10^{-2}\,\mathrm{eV}\),

\[ \sigma_{\text{ortho}} = 79\cdot 10^{-24}\ \mathrm{cm}^{2}, \]
\[ \sigma_{\text{para}} = 18\cdot 10^{-24}\ \mathrm{cm}^{2}. \]

It follows from this that the deuteron level with antiparallel spin orientation is virtual. Using spectroscopic terminology, we may express this result as follows: the only stationary state of the deuteron is the normal level \({}^{3}S\); the \({}^{1}S\) level is realized only in the continuous spectrum.

Such are the principal conclusions that can be drawn from experiments on neutron scattering in hydrogen and from data on the structure and binding energy of the deuteron. The chief deficiency of these results is the impossibility of establishing with any precision the shape of the potential well. Moreover, they characterize the interaction forces of the neutron and proton only for a definite state of motion of these particles, described by a symmetric wave function (\(S\)-state). From a number of facts to be discussed below (saturation of nuclear forces), it apparently follows that the interaction forces of the neutron and proton cannot be described by means of a simple potential function \(V(r)\). The form of this potential function,

probably must be different for the symmetric and antisymmetric states of the neutron–proton system. Therefore, the case of interaction that we have considered up to now is a very special one, and its study is quite insufficient for elucidating the complete picture of the interaction between the neutron and the proton.

In order to make the form of the potential well more definite and to clarify the character of the interaction for \(l\) not equal to zero, it is first of all necessary to proceed to the study of the scattering of neutrons with very high energy—of the order of \(20\ \mathrm{MeV}\) and above. For such neutrons \(\lambda\) is already not large in comparison with \(r_0\), and therefore the influence of the \(P\), \(D\), \(F\ldots\) waves should appear here. The isotropy of the angular distribution, which masks details of the structure of the potential well, must disappear at such energies. The simple formula for \(\sigma\), derived almost without any assumptions about the character of the interaction and, thanks to its internal noncommittal character, agreeing so well with experiment, must also lose its significance for neutrons of sufficiently high energies. Besides increasing the energy of the neutrons in scattering experiments, it is also necessary to increase the accuracy of measuring the magnitude of the effective scattering cross section and the form of the angular distribution. For this it is necessary, first of all, to create more powerful neutron fluxes. Both of these experimental tasks—increasing the neutron energy and increasing the intensity of neutron beams—are at the present time closely connected with progress in the construction of powerful cyclotrons. Finally, it should be noted that interesting results may be expected from a comparative study of the scattering of neutrons and protons by deuterons.

Other processes of interaction of neutrons with protons have been studied much less than elastic scattering.

The capture of neutrons by protons leads to the formation of deuterons and the emission of \(\gamma\)-rays. This process has the greater probability, the lower the energy of the neutrons. At small neutron energies the effective cross section varies approximately inversely proportionally to the square root of the energy. Methods for studying the capture of neutrons by protons are based either on recording the \(\gamma\)-rays that arise when neutrons pass through a hydrogen-containing substance, or on measuring the diffusion of slow neutrons in such substances. The latter method is worth dwelling on in somewhat more detail, since it is typical of that area of experimental nuclear physics in which the properties of slow neutrons are studied.

In order to obtain very slow neutrons, a beam of fast neutrons is made to pass through some hydrogen-containing substance. A neutron moving in the bulk of such a substance undergoes repeated elastic collisions with protons. It is easy to show that, on average, in one collision a neutron loses half of its kinetic energy. Therefore, after several tens of collisions the neutron energy becomes comparable with the energy of the thermal motion of the atoms of that substance,

in which the neutrons are slowed down. After this the neutron’s energy no longer decreases, since it is in thermal equilibrium with the atoms of the substance. If neutron capture did not occur, thermal neutrons could traverse very long paths in the substance and exist in a free state for an unlimited time. However, since a slow neutron has a rather high probability of sticking to a proton, its lifetime in a substance containing hydrogen will be limited. Its path length will likewise be limited. The path length of a slow neutron can be measured, and from it the effective capture cross section can be determined. For neutrons with a velocity of the order of \(3\cdot 10^{-2}V\), the effective capture cross section turns out to be \(4\cdot 10^{-25}\ \mathrm{cm}^2\). Less accurate measurements, based on the registration of \(\gamma\)-rays arising during capture, give for the effective capture cross section the value \(\sim 3\cdot 10^{-25}\ \mathrm{cm}^2\) (for the same neutron energy).

A priori two mechanisms are possible for neutron capture by protons: a) an ordinary dipole transition, b) a magnetic dipole transition.

In a simple dipole transition, a neutron with angular momentum equal to unity (a \(P\)-neutron) passes directly into the \(S\) state, i.e., to the ground level of the deuteron. In this, in accordance with the usual optical selection rules, \(l\) changes by one unit. However, such a transition must be very improbable, since neutrons in the \(P\)-state practically do not interact with the nuclear field of the proton (see above). Therefore one should expect that the effective cross section for capture occurring by means of a dipole transition must be very small. Moreover, it must decrease as the neutron energy decreases, since the neutron wavelength then increases and the role of the \(P\), \(D\), \(F\ldots\) states in the interaction process is reduced to zero.

However, owing to the dependence of the interaction forces on the orientation of the spins of the neutron and proton, there is possible not only an electric dipole transition of the neutron into a bound state, but also a transition accompanied by magnetic dipole radiation. The assumption that such a transition is possible is not a theoretical novelty, since magnetic dipole transitions are also encountered in ordinary spectroscopy. As is known, in the spectra of a number of atoms there are observed lines that do not obey the usual spectroscopic selection rules valid for transitions accompanied by electric dipole radiation. In many cases these lines owe their origin to magnetic dipole radiation (such, for example, is the origin of a number of lines in the spectrum of doubly ionized oxygen). From the standpoint of classical electrodynamics, an electric dipole transition is associated with oscillation of the electric dipole in the atom; a magnetic dipole transition, with oscillations of the magnetic dipole.

The selection rules for magnetic dipole transitions are considerably less stringent than for electric dipole transitions. In particular, a transition is possible from one \(S\)-state to another \(S\)-state, i.e., a transition of the type \(l=0\to l=0\). It is easy to rea-

to show how a magnetic dipole transition can occur in the interaction of a neutron and a proton. If the neutron colliding with the proton has angular momentum equal to zero \((l=0)\), and the spins of both particles are oriented in opposite directions, then at the initial moment the whole system is in the \({}^{1}S\)-state. Since the forces between the neutron and the proton depend on the orientation of the spins, at the moment of collision the spin orientation may change from antiparallel to parallel. As a result the whole system may pass to the ground level of the deuteron \({}^{3}S\), with the emission of magnetic dipole radiation. Owing to the fact that in this case a neutron with \(l=0\) is captured, the interaction between the two particles will be sufficiently large (in contrast to the case when capture occurs at \(l=1\)). The probability of this process must increase rapidly as the neutron velocity decreases, since as the velocity decreases the collision time increases.

The assumption of such a mechanism for the capture of neutrons by protons was first put forward by Fermi. He also developed a quantitative theory of this phenomenon, which, in particular, leads to the conclusion that for slow neutrons the capture cross section must vary inversely proportionally to the velocity of the neutrons. Such a result is almost obvious, and, on the basis of the qualitative considerations given above, Fermi also gave formulas for calculating the absolute magnitude of the capture cross section.

The values of the neutron absorption coefficients in paraffin and water calculated from these formulas agreed with the measured values. However, this agreement between calculation and experiment should not be attributed very great significance, since the absorption coefficients were measured with an accuracy not exceeding \(30\text{--}40\%\).

The inverse process—the disintegration of the deuteron by \(\gamma\)-rays—has also been studied only very poorly. The chief difficulty in determining the magnitude of the effective cross section for this process lies in the fact that it is necessary to know the absolute intensity of the \(\gamma\)-rays (the number of photons) producing the disintegration. The determination of the absolute intensity of the \(\gamma\)-rays emitted by a given source is a very difficult problem, which at present has been solved more or less satisfactorily only for a limited number of cases. A second difficulty is connected with the fact that we have at our disposal very few sources giving monochromatic \(\gamma\)-radiation. Among natural radioactive sources, only the source ThC\('\) can be used for studying the photodisintegration of deuterons; it gives an intense \(\gamma\)-line with energy \(2.62\ \mathrm{MeV}\). Artificial sources of \(\gamma\)-rays, in particular various nuclear reactions accompanied by \(\gamma\)-radiation, are still very poorly studied and so far have not been able to serve for an exact quantitative determination of the effective cross section of the nuclear photoeffect. Therefore on the curve of the dependence of the cross section of the nuclear photoeffect on the photon energy only one point has now been determined: at \(h\nu=2.62\ \mathrm{MeV}\) the cross section is equal to \(\sim 5\cdot 10^{-28}\ \mathrm{cm}^{2}\).

The theory of this phenomenon is, to a certain extent, analogous to the theory that considers the photoionization of atoms. The electromagnetic field of the \(\gamma\)-quantum, interacting with the electric moment of the deuteron, causes the transition of the entire system from the stable state \({}^{3}S\) to the state \({}^{3}P\) in the continuous spectrum. Along with such processes, the interaction of the \(\gamma\)-quantum with the magnetic dipole moment of the deuteron is also possible, leading to a transition from the state \({}^{3}S\) to the state \({}^{1}S\). The total effective cross section is composed of the effective cross sections for the electric and magnetic transitions. At \(h\nu = 2.62\ \mathrm{MeV}\) the cross section should be equal to \(\sim 1\cdot 10^{-27}\). Thus, the theoretical value is twice as large as the experimental value, which, however, may be explained by experimental errors.

There is at present a certain contradiction between theory and experiment with respect to the angular distribution of the products of deuteron photodisintegration. As is known, in the ordinary atomic photoelectric effect the photoelectrons fly predominantly in the direction perpendicular to the direction of motion of the photon. The intensity of photoelectron emission per unit solid angle is proportional to \(\sin^{2}\vartheta\), where \(\vartheta\) is the angle with the direction of motion of the photon. The same should also be the case for the angular distribution of photoprotons and photoneutrons, if the splitting of the deuteron occurred only by interaction of the photon with the electric moment of the deuteron.

For magnetic dipole transitions a different picture is obtained. In this case the deuteron passes from the \({}^{3}S\) level to the \({}^{1}S\) level, for which spherical symmetry of the wave function is characteristic. Therefore the photoprotons and photoneutrons produced in this transition should possess an isotropic distribution in space. At \(h\nu = 2.62\ \mathrm{MeV}\) magnetic dipole transitions should constitute \(35\%\) of all photodisintegrations, and therefore the angular distribution of the products of deuteron decay should differ strongly from that which corresponds to a purely electric interaction. However, the experiments of Chadwick et al., who investigated the angular distribution of photoprotons with the aid of a Wilson chamber, give for the angular distribution a picture that corresponds completely to the pure case of photoelectric disintegration. It cannot, however, be fully certain that this discrepancy is not connected with the statistical play of small numbers characteristic of experiments with a Wilson chamber.

The data considered by us give, on the whole, the following picture:

1. The forces between the neutron and the proton manifest themselves only at distances of the order of \(2\cdot 10^{-13}\ \mathrm{cm}\) and less. At greater distances they are practically equal to zero. The potential well, by means of which the interaction energy can be represented, has the form of a deep well with an abrupt break. The mean depth of the potential well is of the order of several units multiplied by \(10^{7}\ \mathrm{MeV}\).

2. The interaction forces depend on the relative orientation of the spins of both particles. They are greater for parallel than for antiparallel orientation of the spins. The only stationary lev-

The deuteron has a level \({}^{3}S\). The deuteron has no discrete excited states. The level \({}^{1}S\), corresponding to the union of a proton with a neutron with antiparallel spins, is realized only in the continuous energy spectrum.

1. All the facts considered fit into the scheme of central forces.

Such a picture had taken shape by about 1938. However, if one takes into account the very latest experimental facts, one must state that in certain respects it cannot be maintained.

Last year Rabi and his collaborators, studying the behavior of a molecular beam of heavy hydrogen in a magnetic field, established that the observed splitting of the levels of the molecule \(\mathrm{H}_{2}^{2}\) can be explained only on the assumption that the deuteron is not an electrically completely symmetric system (i.e., does not have a spherical distribution of charge density). This conclusion of Rabi is based on the fact that the splitting of the levels of the molecule \(\mathrm{H}_{2}^{2}\), observed by the method of magnetic analysis, turns out to be more complex than it should have been if it were due exclusively to the previously known types of intramolecular interactions. It is therefore necessary to admit a new type of interaction between the elements of the structure of the molecule. The potential energy of this new type of interaction is such that it can be ascribed to intramolecular forces of the type that exist between a charge and a quadrupole. In the present case this means that this new interaction is caused by forces of attraction between the charge of one of the deuterons entering into the molecule \(\mathrm{H}_{2}^{2}\) and the quadrupole moment of the other deuteron.

The results obtained by Rabi in these experiments forced him to ascribe to the deuteron in the normal state a quadrupole electric moment equal to \(2\cdot 10^{-27}e\) (\(e\) is the charge of the proton). From Rabi’s measurements, however, not only the magnitude but also the sign of the quadrupole moment is determined. It turns out that the quadrupole moment of the deuteron is positive, i.e., the electric charge is elongated along the spin axis. If we wished to represent the charge distribution in the deuteron by means of a highly simplified electrical model, then for this purpose one could use the model of a uniformly charged ellipsoid of revolution having an elongated shape (Fig. 3). The degree of asymmetry in this model is expressed by the ratio of the axes. From Rabi’s data for our model it follows that the major axis is \(3\)—\(4\%\) greater than the minor one. Thus the asymmetry has a quite appreciable magnitude.

Fig. 3

The discovery of the electrical asymmetry of the deuteron is of very great fundamental significance. Since the deuteron in the normal state lacks spherical symmetry, this means that the forces by which this simplest nuclear system is held together are not strictly central. The model of the interaction of the neutron with the proton, based on the pre—

...notions of central forces is destroyed and must be replaced by another model, in which the potential energy is a function not only of the distance between the two particles, but also depends on the angle between the radius vector and the direction of the spins.

Further, it follows from this that, since the spherical symmetry of the wave function disappears, one can no longer speak of a quantized value of the orbital angular momentum. In particular, the basic state \(^3S\), to which we assigned the quantum number \(l = 0\), must in fact be regarded as a certain quantum-mechanical mixture of simple states.

However, since the deviations from spherical symmetry are small, in practice they do not affect the deuteron model considered above, which may still be regarded as a good first approximation. The discovery of the quadrupole moment of the deuteron likewise does little to complicate our ideas about the mechanism of neutron scattering and capture. In any case, this discovery makes the task of constructing a unified theory of nuclear forces still more difficult.

For characterizing the interaction of a proton with a proton, at present only one group of facts can be used, relating to collisions of protons with one another. But an indisputable merit of these facts is the great accuracy of the experimental material, immeasurably exceeding the accuracy of the material obtained in all experiments on the study of collisions of neutrons with protons. This is quite natural, because we can control a beam of protons within very broad limits, whereas with a beam of neutrons we can do very little—even as regards its monochromatization.

Fig. 4. Diagram of an apparatus for investigating the scattering of protons by protons.
1 — diaphragms selecting the beam, 2 — ionization chamber for measuring the number of scattered protons, 3 — ionization chamber for measuring the intensity of the primary beam

The arrangement of all the basic experiments in which the scattering of protons by protons was studied is almost exactly the same. A parallel beam of monochromatic protons, accelerated in a high-voltage vacuum tube, passes through a scattering chamber filled with hydrogen at low pressure (Fig. 4). At a definite point of the beam there is aimed a receiver of the scattered protons (a multiplication counter or an ionization chamber connected to a linear amplifier). The receiver can rotate about an axis passing through this point of the beam and perpendicular to the beam. Therefore the angle at which the scattering is observed can be varied over wide limits. The scattering volume is selected by diaphragms, fixed...

connected with a receiver. The primary beam, having passed through the scattering chamber, enters an ionization chamber or a Faraday cylinder. By measuring the number of protons that have entered the receiver, as well as the number of protons in the primary beam, one can determine the probability of scattering for any specified angle. In doing this, of course, it is necessary to know the geometrical data of the apparatus and the pressure of the hydrogen. Experiments of this type were first carried out by Gerthsen in 1929; the scattering of comparatively slow protons with an energy of about 40 KeV was studied. In 1936–1939 a number of papers appeared by American investigators—Tuve and Hafstad, Herb and his co-workers—in which the scattering of protons with energies from 220 to 2392 KeV was investigated with great accuracy.

Before discussing the results of the experiments, it is necessary to clarify what scattering pattern should be expected if it is assumed that the nuclear forces are small and do not manifest themselves in proton collisions. In that case the interaction of the protons must reduce to the Coulomb repulsion of two point charges. The probability of scattering in this case is given by Mott’s formula, which represents a quantum-mechanical generalization of Rutherford’s classical formula as applied to the collision of two identical particles having spin equal to \(1/2\). Mott’s formula has the following form:

\[ f(\vartheta)=N\frac{e^4}{m^2 v^4} \left\{ \frac{1}{\sin^4\vartheta} + \frac{1}{\cos^4\vartheta} - \frac{1}{\sin^2\vartheta\cos^2\vartheta} \cos\left( \frac{2\pi e^2}{hv}\lg \operatorname{tg}^2\vartheta \right) \right\}\cdot 4\cos\vartheta . \]

Here \(\vartheta\) is the scattering angle, \(f(\vartheta)\) is the number of scattered particles per unit solid angle, \(v\) is the velocity of the proton, \(m\) is its mass, \(e=4.8\cdot 10^{-10}\) CGSE, and \(N\) is the number of hydrogen atoms per \(1\ \mathrm{cm}^2\) of the scattering layer. The presence of nuclear forces should modify the function \(f(\vartheta)\) and lead to deviations from Mott’s formula.

Fig. 5

Gerthsen, working with slow protons, obtained scattering results in his investigations that were in full agreement with Mott’s formula. Thus Mott’s formula received experimental confirmation for the first time. The absence of deviations from Mott’s formula in this case is quite understandable. Slow protons cannot, because of Coulomb repulsion, approach one another closely, and therefore nuclear forces are isolated from participation in the scattering.

However, the very first experiments of Tuve, Hafstad, and Heydenburg, carried out for protons with energies from 600 to 900 KeV, gave a sharp discrepancy with Mott’s formula. The nature of this discrepancy is most simply clarified with the aid of graphs. In Fig. 5, along the axis

on the abscissa is plotted the energy of the protons, and on the ordinate axis—the ratio of the observed scattering intensity to the value calculated by Mott’s formula. At small angles, for all energies the scattering is less than that calculated by Mott’s formula; at higher energies and larger angles it, on the contrary, considerably exceeds the theoretical value. Subsequently these measurements by Tuve and his collaborators were substantially refined by them and, in addition, extended into the region of lower energies—down to 220 KeV. It turned out that, when the energy is decreased to 450 KeV, the scattering drops sharply (at all angles) and becomes many times smaller than is required by Mott’s formula. At an energy equal to 450 KeV and \(\vartheta = 45^\circ\), it reaches \(1/30\) of the theoretical value. With a further decrease in energy, the measured values of the scattering probability tend toward the theoretical values. Thus, at a given angle, the ratio

\[ \frac{n_{\text{exp}}}{n_{\text{Mott}}} \]

with increasing energy first decreases, passes through a minimum, and then increases, passing through unity. This is manifested especially clearly at \(\vartheta = 45^\circ\) (Fig. 6).

Fig. 6

The results of these measurements indicate with complete obviousness the presence of a strong nuclear interaction between two protons. At the same time, the nature of the deviations from Mott’s formula shows that attractive forces are manifested here. Qualitatively, under this assumption, the whole pattern of the change in

\[ \frac{n_{\text{exp}}}{n_{\text{Mott}}} \]

is immediately explained. At large distances only Coulomb repulsive forces act. Therefore small scattering angles and small energies correspond to

\[ \frac{n_{\text{exp}}}{n_{\text{Mott}}} = 1. \]

If, at a given scattering angle, the initial kinetic energy of the protons is increased, then the impact parameter will decrease. In order to be scattered through the same angle, the fast proton must come closer to the nucleus than the slow one. However, at small distances the attractive forces will begin to make themselves felt, and the potential energy of these forces will partially compensate the potential energy of the Coulomb field. The forces acting on the proton will weaken, and the scattering will decrease. With a further increase in the kinetic energy of the protons, we are dealing with collisions at very small distances, where the potential energy of the nuclear interaction exceeds the Coulomb repulsion many times over. Therefore, at sufficiently high energies, the scattering begins to increase rapidly.

Herb and his collaborators studied the scattering of still considerably faster protons, with an initial energy up to 2392 KeV. These experiments, carried out with exceptional care, rare in nuclear physics, led to results in complete agreement with Tuve’s results. When the proton energy is increased from 860 to 2392 KeV, the deviations from Mott’s formula increase sharply. The ratio \(\frac{n_{\text{exp}}}{n_{\text{Mott}}}\) reaches, at \(\vartheta = 45^\circ\) and \(E = 2392\) KeV, a value equal to 42.9. The data obtained by Tuve and Herb form a solid basis for constructing a model of the interaction of two protons. A quantitative interpretation of these data was given by Breit.

It turns out that the deviations from Mott’s formula observed in experiment can be explained if, as a model of the interaction, one chooses a narrow potential well with a radius of the order of \(2 \cdot 10^{-13}\) cm. By choosing a definite form for this potential well (Gaussian, rectangular), one can, just as in the case of the interaction of neutrons with protons, obtain, on the basis of the experimental scattering data, a relation between the constants of the well (for example, its depth and width). As to the concrete form of the potential function, in this case too we cannot obtain more detailed information, in view of the fact that the available experimental data on proton–proton scattering refer to such an energy region where \(\lambda \gg r_0\). However, by choosing a definite form of the function \(V(r)\) and specifying one of the parameters (for example, the range of the forces), one can determine another parameter (the depth of the potential well) and compare the resulting potential well with the potential well characterizing the neutron–proton interaction.

Calculations of the parameters of the potential function \(V(r)\), carried out on the basis of the data of Tuve and Herb, led to a remarkable result. It turned out that the interaction energy of two protons is almost exactly equal to the interaction energy of a proton with a neutron for antiparallel spins. If a rectangular form is adopted for the potential well, then, for a force range equal to \(2.8 \cdot 10^{-13}\) cm, the depth of the potential well characterizing the interaction of two protons is equal to 11.3 MeV\(^1\), whereas the depth of the potential well for a neutron and a proton with antiparallel spins at the same radius is equal to 11.5 MeV. In interpreting this result, one should first of all note that in the experiments of Tuve and Herb on proton–proton scattering only the forces acting between two protons with antiparallel spins are revealed. This follows from the fact that scattering in the nuclear field experiences only the \(S\)-wave with azimuthal quantum number \(l = 0\) (since at these proton energies \(\lambda \gg r_0\)).

\(^1\) The total potential energy for two protons will, obviously, be the result of superposing the Coulomb barrier on the potential well characterizing the specific nuclear interaction.

But for \(l=0\) the wave function of both protons is symmetric with respect to the coordinates and therefore must be antisymmetric with respect to the spins, since protons obey Fermi statistics. More simply, this can be expressed as follows: for \(l=0\) both protons are in the same state with respect to the coordinates and, consequently, by the Pauli principle they must have oppositely directed spins.

Therefore the following basic result can be formulated: the potential energy of the nuclear interaction for two protons with antiparallel spins is almost exactly equal to the potential energy of the interaction of a neutron and a proton with the same orientation of spins. In order to check whether this rule also extends to the case of parallel spin orientation, it is necessary to have data on the scattering of appreciably faster protons with energies of the order of \(10^7\) eV. Such data do not yet exist. Nevertheless, it seems extremely probable that in this case as well the interaction energy for two protons and for a proton with a neutron will be the same. Thus, apparently, a fundamental result is established: the specific nuclear forces do not depend on the charge of the particles. From considerations of symmetry this result is naturally to be extended also to the case of the interaction of two neutrons with each other, although we have no direct experimental data on this interaction.

It should be noted in passing that the results obtained make evident the absence of a stable system of two protons \((\mathrm{He}^2)\). In such a system, in the lowest energy state \((l=0)\), the protons, by the Pauli principle, must have oppositely directed spins. Under this condition their interaction energy is equal to the interaction energy of a neutron and a proton with antiparallel spins. In this case, as was already said above, no stable energy state of the system exists. Therefore \(\mathrm{He}^2\) also cannot exist. Coulomb repulsion evidently acts in the same direction and only increases the difficulty of binding two protons.

Up to now we have left aside that group of experimental data which concerns the structure of complex nuclei. Using these data, one can draw further conclusions about the character of nuclear forces. The most important are the data on the energetics of nuclei. The energy content of nuclei can be determined either on the basis of mass-spectroscopic measurements, since the mass of a nucleus \(M\) is directly related to its energy \(W\) by the relation \(W = Mc^2\), or from the energy balances of many-particle nuclear reactions. The second method gives especially good results for light nuclei, for which an enormous number of reactions are known and the energy balances have been measured. For heavy nuclei, one mainly has to use data from mass-spectroscopic analysis.

The principal energy characteristic of a nucleus is its binding energy, equal to the difference between the sum of the energies of the elementary particles into which the nucleus can be decomposed and the energy of the nucleus.

Herb and his collaborators studied the scattering of still considerably faster protons, with initial energy up to 2392 KeV. These experiments, carried out with an exceptional thoroughness rare in nuclear physics, led to results in complete agreement with Tuve’s results. As the proton energy is increased from 860 to 2392 KeV, the deviations from Mott’s formula increase sharply. The ratio

\[ \frac{n_{\text{exp}}}{n_{\text{Mott}}} \]

reaches, at \(\vartheta = 45^\circ\) and \(E = 2392\) KeV, a value equal to 42.9. The data obtained by Tuve and Herb form a solid basis for constructing a model of the interaction of two protons. A quantitative interpretation of these data was given by Breit.

It turns out that the experimentally observed deviations from Mott’s formula can be explained if, as the interaction model, one chooses a narrow potential well with radius of the order of \(2 \cdot 10^{-13}\) cm. Having chosen a definite form for this potential well (Gaussian, rectangular), one can, just as in the case of the interaction of neutrons with protons, obtain from the experimental scattering data the relation between the constants of the well (for example, its depth and width). In this case as well, we cannot obtain more detailed information about the specific form of the potential function, owing to the fact that the available experimental data on proton–proton scattering refer to such an energy region where \(\lambda \gg r_0\). However, having chosen a definite form of the function \(V(r)\) and specified one of the parameters (for example, the radius of action of the forces), one can determine the other parameter (the depth of the potential well) and compare the resulting potential well with the potential well characterizing the neutron–proton interaction.

Calculations of the parameters of the potential function \(V(r)\), made on the basis of the data of Tuve and Herb, led to a remarkable result. It turned out that the interaction energy of two protons is almost exactly equal to the interaction energy of a proton with a neutron when the spins are antiparallel. If one assumes a rectangular form for the potential well, then with a radius of action of the forces equal to \(2.8 \cdot 10^{-13}\) cm, the depth of the potential well characterizing the interaction of two protons is 11.3 MeV\(^{1}\), whereas the depth of the potential well for a neutron and a proton with antiparallel spins at the same radius is 11.5 MeV. In interpreting this result, it should first of all be noted that in Tuve and Herb’s experiments on proton–proton scattering only the forces acting between two protons with antiparallel spins are manifested. This follows from the fact that scattering in the nuclear field is experienced only by the \(S\)-wave with azimuthal quantum number \(l = 0\) (since at these proton energies \(\lambda \gg r_0\)).

\(^{1}\) The total potential energy for two protons will, obviously, be the result of superposing the Coulomb barrier on the potential well characterizing the specific nuclear interaction.

But for \(l=0\) the wave function of both protons is symmetric with respect to the coordinates and therefore must be antisymmetric with respect to the spins, since protons obey Fermi statistics. This can be expressed more simply as follows: for \(l=0\) both protons are in the same state with respect to the coordinates and, consequently, by the Pauli principle they must have oppositely directed spins.

Therefore the following fundamental result may be formulated: the potential energy of the nuclear interaction for two protons with antiparallel spins is almost exactly equal to the potential energy of the interaction of a neutron and a proton with the same orientation of spins. In order to verify whether this rule also extends to the case of parallel spin orientation, one would need data on the scattering of considerably faster protons with energy of the order of \(10^7\ \mathrm{eV}\). Such data do not yet exist. Nevertheless, it appears highly probable that in this case as well the interaction energy for two protons and for a proton with a neutron will be the same. Thus, apparently, a fundamental result is established: the specific nuclear forces do not depend on the charge of the particles. From considerations of symmetry, this result can naturally be extended also to the case of the interaction of two neutrons with one another, although we also have no direct experimental data on this interaction.

It should be noted in passing that the results obtained make evident the absence of a stable system of two protons \((\mathrm{He}^2)\). In such a system, in the lowest energy state \((l=0)\), the protons, by the Pauli principle, must have oppositely directed spins. Under this condition their interaction energy is equal to the interaction energy of a neutron and a proton with antiparallel spins. In this case, as was already said above, no stable energy state of the system exists. Therefore \(\mathrm{He}^2\) likewise cannot exist. The Coulomb repulsion acts, obviously, in the same direction and only increases the difficulty of combining two protons.

Up to now we have left aside that group of experimental data which relates to the structure of complex nuclei. Using these data, one can draw further conclusions about the character of nuclear forces. The most important are the data on the energetics of nuclei. The energy reserve of nuclei can be determined either on the basis of mass-spectroscopic measurements, since the mass of a nucleus \(M\) is directly related to its energy \(W\) by the relation \(W=Mc^2\), or from the energy balances of numerous nuclear reactions. The second method gives especially good results for light nuclei, for which an enormous number of reactions is known and energy balances have been measured. For heavy nuclei one must for the most part use data from mass-spectroscopic analysis.

The principal energetic characteristic of a nucleus is its binding energy, equal to the difference between the sum of the energies of the elementary particles into which the nucleus can be decomposed and the energy of the nucleus.

The analogy between these two kinds of interaction is, of course, purely formal, since the mechanism of nuclear and chemical interactions is completely different. However, in both cases, although for different reasons, the potential energy is a function not only of $r$, but also of the state of motion of the particles, i.e., of the symmetry or antisymmetry properties of the wave function.

It must be emphasized once again that a complete elucidation of the character of the interaction of elementary particles will be possible only after data are available to us on the behavior of particles in various quantum states. Everything that has been known to us up to now refers only to the $S$-state.

At present our basic information on the properties of nuclear forces can be reduced to the following conclusions:

1. The forces between heavy elementary particles have a small range of action. The energy of interaction amounts to several tens of millions of electron volts within the region of action of these forces and is practically equal to zero outside this region—at distances greater than $3 \cdot 10^{-13}\ \text{cm}$.

A concrete form of the interaction energy cannot yet be established. But the relation between the depth and the width of the potential well can be found for each concrete form of the potential function $V(r)$.

1. The forces between elementary particles do not depend on charge, i.e., they are identical for two protons, two neutrons, and a proton and a neutron.

2. The interaction forces depend on the relative arrangement of the spins of both particles. They are greater for parallel spins.

3. The interaction forces are not strictly central.

4. The interaction forces give saturation in the interaction of four partners with one another.

LITERATURE

I. Scattering of neutrons by protons

a) Investigation of the angular distribution

1. P. G. Kruger, W. E. Shoupp and F. W. Stallmann, Phys. Rev., 52, 678, 1937.
2. N. A. Dobrotin, Izv. Akad. nauk SSSR, ser. fiz., No. 1–2, 199, 1938.
3. P. I. Dee and C. W. Gilbert, Proc. Roy. Soc., A 163, 264, 1937.

b) Measurement of the effective scattering cross section

1. L. Simons, Phys. Rev., 55, 792, 1939.
2. W. H. Zinn, S. Seely and V. H. Cohen, Phys. Rev., 56, 260, 1939.
3. H. Aoki, Phys. Rev., 55, 795, 1939.
4. E. O. Salant, R. B. Roberts and P. Wang, Phys. Rev., 55, 984, 1939.

c) Investigation of scattering in para- and ortho-hydrogen

1. E. Teller, Phys. Rev., 49, 420, 1936.
2. J. Schwinger and E. Teller, Phys. Rev., 52, 286, 1937.

1. J. Halpern, I. Estermann, O. C. Simpson, O. Stern, Phys. Rev., 52, 142, 1937.
2. F. G. Brickwedde, J. R. Dunning, H. J. Hoge and J. H. Manley, Phys. Rev., 54, 266, 1938.
3. W. F. Libby and E. A. Long, Phys. Rev., 55, 339, 1939.

II. Disintegration of the deuteron by γ-rays

1. J. Chadwick and M. Goldhaber, Proc. Roy. Soc., A 151, 479, 1935.
2. J. Chadwick, N. Feather and E. Bretscher, Proc. Roy. Soc., A 163, 366, 1937.
3. H. v. Halban, Nature, 141, 644, 1938.
4. F. T. Rogers and M. M. Rogers, Phys. Rev., 55, 262, 1939.

III. Capture of neutrons by protons

1. R. Fleischmann, Z. Physik, 97, 242, 1935.
2. L. Arzimowitsch, I. Kurtschatow, G. Latyschew u. W. Chramow, Sow. Phys., 8, 472, 1935.
3. E. Fermi, Phys. Rev., 48, 570, 1935.
4. E. Amaldi and E. Fermi, Phys. Rev., 50, 899, 1936.

IV. Quadrupole electric moment of the deuteron

1. J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, J. R. Zacharias, Phys. Rev., 55, 318, 1939.
2. J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey and J. R. Zacharias, Phys. Rev., 57, 677, 1940.

V. Scattering of protons by protons

1. M. A. Tuve, N. P. Heidenburg and L. R. Hafstad, Phys. Rev., 50, 806, 1936.
2. L. R. Hafstad, N. P. Heidenburg and M. A. Tuve, Phys. Rev., 53, 239, 1938.
3. R. G. Herb, D. W. Kerst, D. B. Parkinson and G. J. Plain, Phys. Rev., 55, 998, 1939.
4. G. Breit, H. M. Thaxton and L. Eisenbud, Phys. Rev., 55, 1018, 1939.