EXPERIMENTAL PROOF OF THE EXISTENCE OF EXCHANGE NUCLEAR FORCES

I

According to the authors of the papers under review \(^{1,2}\), one of the most significant experiments carried out on the 184-inch phasotron \(^{3}\) during its entire period of operation is the investigation of the scattering of neutrons by protons at energies of 90 and 40 MeV.

The study of the scattering of elementary particles (proton—neutron, proton—proton, neutron—deuteron, proton—deuteron) at energies from several hundred KeV to 15 MeV made it possible, with sufficient accuracy, to determine

...the binding energy of light nuclei. But at such low energies, when the de Broglie wavelength of the incident particles (in the center-of-mass system) considerably exceeds the range of action of the nuclear forces, it is impossible not only to distinguish exchange forces from non-exchange forces, but even the form of the dependence of the nuclear forces on distance does not affect the character of the scattering (Landau and Smorodinskii⁴). For example, Sleter measured the total cross section of \((n\text{-}p)\) scattering in the energy interval from 6 to 22 MeV and found that the experiments can be explained by six different theories.

[In the figure: “Ordinary”; “Exchange”; vertical axis \(\sigma(\theta)\cdot 10^{27}\,\text{cm}^2\); horizontal axis \(\cos\theta\), with angles \(0^\circ, 60^\circ, 90^\circ, 120^\circ, 180^\circ\).]

Fig. 1. Differential cross sections \(\sigma(\theta)\) for \((n\text{-}p)\) scattering at 100 MeV for ordinary and exchange forces, calculated for a potential well whose width is \(2.8\cdot 10^{-13}\,\text{cm}\). \(\theta\) is the neutron scattering angle in the center-of-mass system.

At energies up to 15 MeV the principal role is played by \(s\)-scattering, which is symmetric in the center-of-mass system, and indeed (with the exception of the erroneous experiment of Amaldi in 1942), numerous experiments in this energy interval have shown that in the center-of-mass system \((n\text{-}p)\) scattering is isotropic.

Exchange forces differ from non-exchange forces only when \(p\)-scattering begins to play a role (as well as higher odd moments). At 90 MeV the de Broglie wavelength of neutrons in the center-of-mass system is \(0.95\cdot 10^{-13}\,\text{cm}\), i.e., it is comparable with the radius of action of the nuclear forces \((2\cdot 10^{-13}\,\text{cm})\); therefore the energy 90 MeV is quite sufficient for clarifying the character of the interaction between protons and neutrons. The essential difference between exchange and non-exchange forces, which manifests itself in scattering, consists in the different character of the dependence of the differential cross section \(\sigma\) on the scattering angle \(\theta\). In the case of ordinary forces, when neutrons are scattered by protons, obviously the largest number of neutrons (in the center-of-mass system) must be scattered through small angles, i.e., the largest number of neutrons must fly forward (in the direction of the incident beam), and the largest number of protons—backward. In the laboratory system this means that the largest number of protons must be scattered at right angles to the incident neutron beam.

In the case of exchange forces everything occurs in the same way, except that in the process of scattering the neutron turns into a proton, and the proton into a neutron. Consequently, in the center-of-mass system the largest number of protons flies not backward but forward, and the largest number of neutrons flies backward. In the laboratory system, in this case, the largest number of protons flies in the direction of the incident beam. The character of the dependence of \(\sigma(\theta)\) on the neutron scattering angle \(\theta\) in the center-of-mass system for the two assumptions about the nuclear forces is given in Fig. 1. The experimental results obtained for \((n\text{-}p)\) scattering at 90 MeV (see Fig. 2) differ strongly from both curves of Fig. 1.

However, the presence of a maximum of \(\sigma(\theta)\) at \(\theta = 183^\circ\) indicates that the forces are (at least in part) exchange forces. Further interpretation of the results obtained is too uncertain, since so far too little is known...

few facts, and the form of \(\sigma(\theta)\) depends on many circumstances, in particular on the concrete form of the dependence of nuclear forces on distance.

In addition, at these energies relativism must be taken into account; it may give a correction of several tens of percent. Nevertheless, the authors carried out a large computational work to find the potential of nuclear forces. They first assumed that the potential energy of the forces acting between a proton and a neutron can be written in the following form:

\[ V(r)=g^2\,\frac{e^{-kr}}{r}\left(\frac{1+P}{2}\right), \tag{1} \]

where \(g^2\) has two values: for the triplet state (the spins of the proton and neutron are parallel) \(g^2=0.405\), and for the singlet state (the spins are antiparallel) \(g^2=0.280\). The constant \(g^2\) was chosen so as to satisfy the experimental data at low energies (the cross section for thermal neutrons and the binding energy of the deuteron). \(\frac{1}{k}=1.2\cdot 10^{-13}\ \text{cm}\) is equal to the Compton wavelength of a particle with mass \(326\,m_e\) (where \(m_e\) is the mass of the electron), and \(P\) is the exchange operator. From the formula given it is seen that 50% of the forces are regarded as exchange forces and 50% as ordinary ones. In Fig. 2, formula (1) corresponds to theoretical curve \(I\) for the differential cross section \(\sigma(\theta)^*\). The total cross section calculated by means of (1) is equal to \(\sigma_t=0.090\cdot 10^{-24}\ \text{cm}^2\), and is somewhat larger than the experimental value \(\sigma_t=0.076\cdot 10^{-24}\ \text{cm}^2\). However, expression (1) for centrally symmetric forces cannot explain the existence of the quadrupole moment of the deuteron; it is necessary to assume that the nuclear forces have a tensor character:

\[ V'(r)=g^2\left\{\frac{e^{-kr}}{r}+\gamma\left[\left(\frac{3(\sigma_1 r)(\sigma_2 r)}{r^2}-\sigma_1\sigma_2\right)\frac{e^{-kr}}{r}\right]\right\}\frac{1+P}{2}, \tag{2} \]

Fig. 2. Scattering cross section for 90 MeV. The solid curves were calculated theoretically for central forces (curve \(I\)) and tensor forces (curve \(II\)). The circles are values obtained experimentally.

where \(\sigma_1\) and \(\sigma_2\) are spin operators, and \(\gamma=0.16\). By means of (2) one can obtain the correct value of the electric quadrupole moment of the deuteron. The differential scattering cross section \(\sigma(\theta)\) for this case is shown in Fig. 2 (curve \(II\)). The total cross section in this case is still larger: \(\sigma_t=0.093\cdot 10^{-24}\ \text{cm}^2\). Besides the theoretical interpretation of the experiment reviewed here, a large number of theoretical calculations devoted to scattering of protons by neutrons is known at present \(^{5-10}\). However, the value of the indicated works is at present open to doubt, and formulas (1) and (2) should be regarded as empirical rather than theoretical.

\[ \text{*) Curves } I \text{ and } II \text{ were calculated in the nonrelativistic approximation.} \]

II

In view of the importance of the experiments, the measurement of \((n-p)\) scattering was carried out simultaneously by two methods by a large group of persons. The most accurate quantitative results were obtained in measurements using counters. Measurements with a Wilson chamber as a qualitative confirmation were also compared with the results obtained with counters. However, in the latter case the statistics are insufficient for a quantitative comparison of the two experiments.

The apparatus is shown schematically in Fig. 3. A beam of neutrons with an average energy of 90 MeV was obtained by bombarding a beryllium target with deuterons of energy 200 MeV. The neutron beam passed through an aperture in a 3-meter concrete shield. The background was collimated by means of two copper tubes 50 cm long and with an internal diameter from 1 to 7.5 cm. The intensity of the beam was measured by two methods (see Fig. 3):

Fig. 3. Arrangement of the apparatus for measuring \((n-p)\) scattering. Telescope I is used to measure \((n-p)\) scattering. Telescope II is used to measure the intensity of the neutron beam.

1) by the intensity of scattering of recoil protons (telescope II) at a constant angle of \(15^\circ\);

2) by the intensity of fission of bismuth under the action of neutrons.

On the path of the neutron beam a target was placed, in one experiment made of polyethylene, in the other of graphite. The recoil protons from the target were measured by means of three Geiger counters (telescope I in Fig. 3). The target was always set parallel to the counters. Between the last two counters an absorber was placed, which stopped all protons with energy less than 66 MeV. This means that in the experiment only neutrons with energy greater than 66 MeV were taken into account.

All the apparatus was subjected to the careful checking usual in work with counter telescopes.

The apparatus shown in Fig. 3 was used only in measuring scattering at angles \(\Phi < 60^\circ\). Indeed, at large angles the effective thickness of the target increased greatly, and the presence of air and of the counter walls in the path of the scattered protons made the experiment very difficult. To avoid this, another apparatus was constructed and placed in a vacuum. In addition, the 3 counters of the telescope were assembled in one casing, with no intervening walls between the counters. In this case absorber \(A\) was placed before the entire telescope. With this apparatus measurements were made in the angular range \(\Phi\) from \(35^\circ\) to \(71.6^\circ\).

The ratios \(R_1\) and \(R_2\) of triple coincidences in telescope \(I\) to triple coincidences in telescope \(II\) (see Fig. 3) were measured for the scattering of neutrons by polyethylene (1) and graphite (2) targets. Similarly, the background \(R_3\) was measured and

\[ H=(R_1-R_3)-0.713(R_2-R_3) \]

was calculated, where 0.713 is the ratio of the number of carbon atoms per \(1\ \mathrm{cm}^2\) of the polyethylene and graphite targets.

The quantity

\[ \frac{H}{t}, \]

where \(t\) is the effective target thickness, is proportional to the differential scattering cross section \((n-p)\) \(\sigma(\Phi)\Delta\Omega\), where \(\Delta\Omega\) is the solid angle of the apparatus. To find the proportionality coefficient, the total scattering cross section was measured:

\[ \sigma_t=\int \sigma(\theta)\,d\omega =\int \sigma(\Phi)\,d\Omega =k\int \frac{H}{t}\frac{d\Omega}{\Delta\Omega}, \]

where \(\theta\) is the angle of neutron scattering in the center-of-mass system, \(\Phi\) is the angle of proton scattering in the laboratory system (see Fig. 3),

\[ \sigma(\theta)=\sigma(\Phi)\frac{d\cos\Phi}{d\cos\theta}; \qquad \theta=\pi-2\Phi, \]

and \(k\) is the required proportionality coefficient. It is true that the cross section \(\sigma(\Phi)\) at angles from \(71.6^\circ\) to \(90^\circ\) is unknown. This part of the \(\sigma(\Phi)\) curve is responsible for 15% of the total value of \(\sigma_t\). In specific calculations it was assumed that the true curve in the unknown angular region coincides with curve \(I\) in Fig. 2, which agrees with the data obtained in a Wilson chamber. The result of the experiment and calculations is shown in Fig. 2.

Similar experiments were carried out with neutrons having an average energy of 40 MeV. Neutrons of this energy were obtained by bombarding a beryllium target, placed at a smaller radius in the phasotron, with deuterons of energy 98 MeV. The accuracy of the result in this case was considerably lower. The form of the curve \(\sigma(\theta)\) at 40 MeV is qualitatively the same as at 90 MeV. However,

\[ \frac{\sigma(180^\circ)}{\sigma(90^\circ)}\simeq 1.5, \]

instead of 3.3 at 90 MeV.

CITED LITERATURE

1. J. Hadley et al., Phys. Rev. 75, 351 (1949).
2. K. Brueckner et al., Phys. Rev. 75, 555 (1949).
3. UFN 32, 396 (1947).
4. L. Landau and Ya. Smorodinsky, ZhETF 14, 269 (1944).
5. M. Camac and H. Bethe, Phys. Rev. 73, 191 (1948).
6. Y. Ashkin and T. Wu, Phys. Rev. 73, 972 (1948).
7. H. Massey et al. 73, 1403 (1948).
8. F. Rohrlich and J. Eisenstein, Phys. Rev. 75, 705 (1949).
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M. Rabinovich.