THE REACTION $\pi^+ + d \rightleftarrows p + p$ AND THE SPIN OF $\pi$ MESONS

The study of the reactions of $\pi$ mesons with hydrogen and deuterium nuclei has led to a number of interesting conclusions, listed below: 1. The spin of $\pi$ mesons is integer. This conclusion was drawn on the basis of data on the formation of stars when $\pi$ mesons are captured by nuclei in photographic emulsions, and also from the condition of conservation of angular momentum in the reactions $p + p \to \pi^+ + d$ and $p + n \to \pi^+ + n$. 2. The neutral $\pi$ meson has spin equal to, or greater than, unity, for it decays into two $\gamma$ quanta, and such a decay is forbidden for a system with angular momentum $\hbar$. It should be noted that as early as 1948 L. D. Landau$^1$ pointed out that the existence of the reaction $\pi^0 \to 2\gamma$ implies that the spin of the $\pi^0$ meson is zero. 3. In view of the closeness of the masses and production cross sections of neutral and charged $\pi$ mesons, and also proceeding from the independence of nuclear forces from charge, it was natural to assume that the properties of $\pi^0$ and $\pi^\pm$ mesons are similar and that, in particular, the spin of charged $\pi$ mesons also cannot be equal to unity. 4. $\pi$ mesons are not scalar particles. This conclusion followed from data on the capture of $\pi^-$ mesons by deuterium nuclei, and also from experiments on the production of charged and neutral $\pi$ mesons by high-energy $\gamma$ quanta. Thus, as early as 1941 E. L. Feinberg$^2$ pointed out that if a $\pi^-$ meson is captured by a deuterium nucleus from an $S$ level, then the reaction $\pi^- + d \to n + n$ is forbidden for a scalar meson. On the basis of new experimental data, in a number of works by Soviet scientists conclusions were drawn concerning the pseudoscalar nature of charged and neutral $\pi$ mesons$^{3,4,5}$.

Fig. 1.

Until now, however, the spin of charged $\pi$ mesons had not been experimentally determined, and there was no convincing refutation of the fact that this spin is equal to or greater than two. Such an experimental determination of the spin of $\pi^+$ mesons was carried out this year$^{6,7}$ on the basis of studies of the differential angular cross sections of the reaction $\pi^+ + d \to p + p$. In Uspekhi Fizicheskikh Nauk it has already been reported$^8$ on works devoted to the inverse reaction $p + p \to \pi^+ + d$, and it was pointed out that the insufficiently accurate determination of the spectrum of the $\pi^+$ mesons formed near the maximum energy values prevented an unambiguous proof of the fact that this reaction proceeds precisely according to the scheme written above, and not according to the scheme: $p + p \to \pi^+ + p + n$. When, along with a $\pi^+$ meson, deuterons are formed, the mesons moving in the direction of the primary proton beam must have, in the laboratory system, an energy 4 Mev higher than when a proton and neutron are formed, and in the meson spectrum a sharp peak should be observed correspondingly at 70 or 66 Mev, while the upper boundary of the spectrum should be located at 74 or 70 Mev. The use of a magnetic analyzer with the best resolving properties, as well as careful monochromatization of the initial

From Current Literature

protons led to a refinement of the data on the formation of \(\pi^+\)-mesons in \(pp\)-interactions, and an image of the spectrum of \(\pi^+\)-mesons was obtained in Fig. 1, indicating that the reaction proceeds according to the scheme
\[ p+p\to \pi^+ + d^{9}. \]

The spin of the \(\pi^+\)-meson can be determined from data on the “detailed balance” of the direct and inverse reactions \(\pi^+ + d \rightleftarrows p+p\), since, according to the theory\({}^{10}\), the ratio of the differential angular cross sections of the direct and inverse reactions is equal to:

\[ \frac{\dfrac{d\sigma_{\pi^+ d}}{d\Omega}} {\dfrac{d\sigma_{pp}}{d\Omega}} = \frac{4}{3}\cdot \frac{p^2}{q^2(s+1)}, \]

where the differential angular cross sections and the momenta of the proton \((p)\) and meson \((q)\) are measured in the center-of-mass system, and \(s\) is the spin of the meson. The given relation is completely independent of meson-field theory and, in this lies the great advantage of determining the spin of the \(\pi^+\)-meson from this relation.

The differential angular cross sections of the reaction \(p+p\to \pi^+ + d\) were determined earlier\({}^{8,9}\). An experimental determination of the differential angular cross sections of the reaction \(\pi^+ + d \to p+p\) was carried out almost simultaneously, and by similar methods, in two papers\({}^{6,7}\).

Figure labels: meson beam, 60 or 75 MeV; magnet; counter 1; carbon absorber; counter 2; Al absorber; counter 3; counter 4; \(d\)-target; \(p\); \(p\).

Fig. 2.

The schematic diagram of the apparatus\({}^{7}\) is shown in Fig. 2. A meson beam with an energy of 60–75 MeV\({}^{6}\), or 40 MeV\({}^{7}\), was produced by bombarding a carbon target\({}^{6}\) with protons of energy 380 or 240 MeV, or a beryllium\({}^{6}\) or aluminum target\({}^{7}\). In this process, mesons in a definite energy interval were selected by means of a magnetic analyzer. As special experiments\({}^{7}\) showed, the spread of the initial mesons in energy had little effect on the value of the differential angular cross section of the reaction, since this quantity depended only weakly on the energy of the \(\pi^+\)-mesons. The number of mesons was determined by means of two scintillating crystal counters.

(1 and 2 in Fig. 2). To reduce the energy of the mesons, a carbon absorber was used. The heavy-water target had a thickness of 2.5 g/cm\(^2\). In screening checks for background caused by the presence of an admixture of hydrogen, experiments were carried out with the heavy water replaced by ordinary water. The two protons formed in the reaction, flying apart at an angle of \(180^\circ\) in the center-of-mass system of \(\pi^{+}+d\), were registered by the coincidence count between two liquid scintillation counters\(^6\) or two NaJ crystal counters\(^7\). In front of one of the scintillation counters (in Fig. 2, in front of counter 4) an aluminum absorber was placed, whose thickness was sufficient to stop the scattered protons and deuterons, but at the same time transparent to the protons from the reaction \(\pi^{+}+d\).

Fig. 3.

The mean energy of the \(\pi^{+}\)-mesons interacting with the deuterium nuclei in the principal experiments was \(28^6\), or about 23 MeV\(^7\). This energy in the center-of-mass system corresponds to a proton energy of 340 MeV in the inverse reaction.

After introducing corrections for the geometrical efficiency of the counting system, for the admixture of \(\mu\)-mesons in the initial beam (about 10%), and for nuclear absorption of mesons and protons in the target (about 7%), the work\(^6\) yielded the results shown in Fig. 3.

In the same figure are indicated the differential angular cross sections for the reaction \(\pi^{+}+d \to p+p\), calculated from the experimental data for the inverse reaction, under two assumptions concerning the spin of the \(\pi^{+}\)-mesons (\(s=0\) and \(s=1\)). It is evident that the results of investigations of the direct and inverse reactions agree only when the spin of the \(\pi^{+}\)-meson is zero. Similar conclusions were also reached by the authors\(^7\), who calculated from their measurements at different angles the total cross section of the reaction \(\pi^{+}+d \to p+p\) and compared this cross section with the cross section obtained from a value of \(1.3\cdot10^{-28}\) cm\(^3\)/steradian for the inverse reaction at an angle of \(0^\circ\). A comparison of the cross sections obtained directly from experiment and by calculation from the data for the reaction \(p+p \to \pi^{+}+d\), under different assumptions about the angular distribution of the reaction products in the center-of-mass system and about the spin of the \(\pi^{+}\)-meson, is given in the table (see p. 124).

It is evident that, in the calculation of the total cross sections as well, the data for the direct and inverse reactions agree only when the spin of the \(\pi^{+}\)-meson is zero.

The authors\(^6\) and \(^7\) point out that if the spin of the \(\pi^{+}\)-meson is not equal to zero, then the data for the direct and inverse reactions can agree with the indicated

Calculated and measured cross sections of the reaction \(\pi^+ + d \to p + p\)

by the above method only in the case that both the \(\pi^+\)-mesons produced when bombarding beryllium or aluminum targets with protons and the \(\pi^+\)-mesons produced in the reaction \(p+p \to \pi^+ + d\) are fully polarized. However, such an explanation is extremely improbable. Therefore it should be considered that the spin of charged \(\pi\)-mesons, like the spin of neutral \(\pi\)-mesons, is equal to zero.

G. I.

REFERENCES

1. L. D. Landau, DAN 60, 207 (1948).
2. E. L. Feinberg, Journ. of Phys. (URSS) 5, 177 (1941).
3. A. M. Baldin and V. V. Mikhailov, ZhETF 20, 1057 (1950).
4. B. Ioffe, A. Rudik and I. Shmushkevich, DAN 77, 403 (1951).
5. V. B. Berestetskii and I. Ya. Pomeranchuk, DAN 77, 803 (1951).
6. R. Durbin, H. Loar, J. Steinberger, Phys. Rev. 83, 646 (1951).
7. D. Clark, A. Roberts, R. Wilson, Phys. Rev. 83, 649 (1951).
8. UFN 42, 571 (1950).
9. W. Cartwright, C. Richman et al., Phys. Rev. 81, 652 (1951).
10. R. Marshak, Phys. Rev. 82, 313 (1951).