Nuclear Interaction of $\pi$-Mesons

When various nuclei are bombarded by protons and alpha particles accelerated to energies of 300–400 MeV, or by high-energy bremsstrahlung, a rather large yield of $\pi$-mesons is observed—a quantity sufficient for successful investigation of a number of properties of these particles.^1

Of considerable interest, in particular, is the problem of the nuclear interaction of $\pi$-mesons. Until recently only two simplest examples of such interaction had been studied—the reactions of negative $\pi^-$-mesons with hydrogen and deuterium nuclei, in which neutral $\pi^0$-mesons were discovered^2 and a number of interesting conclusions were drawn about the properties of $\pi^-$- and $\pi^0$-mesons.^3,4 During the past year, it has been

Several works have been carried out in which the interaction of \(\pi\)-mesons with complex nuclei was investigated. In these works it was found that, in addition to Coulomb scattering through small angles, elastic scattering of \(\pi\)-mesons by nuclei and inelastic collisions of \(\pi\)-mesons with nuclear nucleons are also observed. The cross sections of these processes were measured for \(\pi\)-mesons with various kinetic energies—in the interval from 30 to 110 MeV. Elastic scattering of \(\pi\)-mesons by nuclei has a diffraction character, the nucleus itself being regarded in this case as a black or semitransparent screen. Such scattering is associated with a relatively small loss of kinetic energy by the \(\pi\)-mesons and with deflection mainly through relatively small angles. Inelastic collisions of \(\pi\)-mesons are associated with the loss of the greater part of the kinetic energy or with the destruction of the \(\pi\)-mesons. In connection with such a nature of inelastic collisions, three kinds of these processes are distinguished in various works: 1. Inelastic scattering, in which the meson is not destroyed, but merely loses a large part of its kinetic energy, with the possible appearance of new secondary particles. 2. Stars, in the formation of which the meson is destroyed, but several charged secondary particles are produced, recorded directly in the experiment. 3. “Disappearance in flight,” when the meson is destroyed with the formation only of neutral secondary particles, not recorded in the experiment.

In most of the papers reviewed, the nuclear interaction of \(\pi\)-mesons was observed with the aid of thick-layer photoemulsions\(^{5,6,7,8}\) or a Wilson chamber\(^{9,10}\).

The difference between the methods used in individual works involving photoemulsions was reduced mainly to the fact that either a definite area of the photoemulsion was examined\(^{6,7}\), or a definite number of meson tracks along their entire length\(^{5,7,8}\). Both methods gave similar results\(^{7}\). Ilford G-5 emulsions, with a layer thickness of about 600\(^{5}\) or 400\(^{6,7,8}\) microns, were used. To determine the cross section of the nuclear interaction of \(\pi\)-mesons in the photoemulsion, the observed number of acts of such interaction was referred to the total length of the meson tracks examined or to the total length of tracks in the examined area of the emulsion. The range in the emulsion, calculated from the geometrical cross sections of the individual constituent nuclei, is estimated at 23–25 cm\(^{5,6}\). The kinetic energy of the \(\pi\)-mesons was usually specified with the aid of magnetic analysis of the meson beam. In determining

Table 1

corrections for the background of $\mu$-mesons, determined from the number of $\mu$–$e$ decays, and also for the background of electrons and protons. As a special check showed, the background effects were small and were estimated with sufficient accuracy.

Table I gives the principal data obtained in the first three works5, 6, 7 devoted to the nuclear interaction of negative $\pi$-mesons.

The values of the ranges of nuclear interaction of $\pi$-mesons in emulsion at meson energies of 30–38 and 30–50 MeV, obtained in5 and6, proved to be rather close. But a noticeable discrepancy between these two works is observed in the estimate of the fraction of different kinds of inelastic collisions in the total number of cases of nuclear interaction.

A detailed investigation of the nuclear interaction of $\pi$-mesons by means of photoemulsions was therefore additionally carried out over a wide interval of $\pi^-$-meson energies8. In8, all cases in which the loss of kinetic energy in scattering did not exceed 15% were classified as elastic scattering events. The geometrical cross sections of nuclei, on the basis of which the range was calculated,

\[ \lambda_{\text{geom}}=\frac{1}{\sum N_i\sigma_i}, \]

were taken to be

\[ \sigma_{\text{geom}}=\pi\hbar^2 \left[ \left( \frac{A^{1/3}}{\mu c} \right) +\frac{1}{p} \right]^2, \]

where $\mu=2.51\cdot 10^{-25}$, $z$ is the mass of the $\pi$-meson, and $p$ is the momentum of the meson. Thus, the finite size of the meson waves was taken into account in the formula for the cross sections. The principal data of8 are given in Table II.

Table II

In parentheses in Table II is indicated the number of stars containing secondary protons with energies greater than 30 MeV. From a comparison of data on the nuclear interaction of \(\pi^{-}\)-mesons of different energies, the authors \(^{8}\) came to the conclusion that, as the meson energy increases, the fraction of inelastic collisions increases. This conclusion partly explains the discrepancy, mentioned above, between the results of works \(^{5}\) and \(^{6}\). Analyzing the magnitude of the energy carried away by secondary charged particles, the authors \(^{8}\) concluded that a large part of the energy is apparently carried away by secondary neutral particles.

The interaction of \(\pi^{-}\)- and \(\pi^{+}\)-mesons with carbon and aluminum nuclei was studied by means of a Wilson chamber \(^{9,10}\) partitioned by plates of the indicated materials of thickness \(0.32\) cm. In this, the formation of stars and scattering through small and large angles were observed. Scattering through small angles (up to \(20^\circ\)) was not taken into account, since such scattering is mainly a consequence of the Coulomb interaction associated with \(\pi\)–\(\mu\)-decay. Diffraction scattering was considered to be scattering through angles up to those corresponding to the first zero of the angular diffraction distribution (\(75^\circ\) for carbon and \(50^\circ\) for aluminum). Stars and scattering through angles exceeding those indicated above were classified as inelastic-collision events.

In 4187 events of passage through a carbon plate by \(\pi^{-}\)- and \(\pi^{+}\)-mesons (the total path in carbon was approximately 1340 cm) with a mean energy of about 45 MeV, 5 cases of diffraction scattering and 21 cases of inelastic collisions \(^{9}\), mainly stars, were observed.

From this the authors \(^{9}\) determined the cross section of inelastic collisions of \(\pi\)-mesons with carbon nuclei to be \((0.22 \pm 0.05)\cdot 10^{-24}\ \text{cm}^2\), with a geometrical cross section of \(0.37\cdot 10^{-24}\ \text{cm}^2\). Thus carbon nuclei are semitransparent to \(\pi\)-mesons of the indicated energy. In the case of a “black” nucleus, the number of diffraction-scattering events should have been, according to the calculations of the authors \(^{9}\), 26.

Analogous experiments for mesons with energy \(65 \pm 10\) MeV led to the observation of 44 inelastic collisions (mainly stars) and 13 events of diffraction scattering over a total path in carbon of 1980 cm \(^{10}\). According to the calculations of the authors \(^{10}\), for a “black” nucleus 57 inelastic collisions and 42 diffraction-scattering events should have been observed. Consequently, if \(\sigma_a\) is the cross section of inelastic collisions, \(\sigma_d\) the cross section of diffraction scattering, and \(\pi R^2\) the geometrical cross section of the carbon nucleus, then for \(\pi^{-}\)- and \(\pi^{+}\)-mesons of the indicated energy

\[ \frac{\sigma_a}{\pi R^2} \simeq 0.75 \]

and

\[ \frac{\sigma_d}{\pi R^2} \simeq 0.30. \]

When mesons with a mean energy of 45 MeV passed through an aluminum plate, over a path length of 424 cm, 12 inelastic collisions and 3 cases of diffraction scattering \(^{9}\) were observed. The geometrical cross section of aluminum nuclei is estimated at about \(0.60\cdot 10^{-24}\ \text{cm}^2\); the cross section of inelastic collisions from the indicated data is \((0.48 \pm 0.14)\cdot 10^{-24}\ \text{cm}^2\). The calculated number of diffraction-scattering events for a “black” aluminum nucleus is equal to 8. Thus, the discrepancies between the experimental data and the calculation for a “black” nucleus are smaller in the case of aluminum than in the case of carbon.

Along with the experiments considered, in which elementary events of the interaction of \(\pi^{-}\)- and \(\pi^{+}\)-mesons with complex nuclei were observed, very recently the interaction of \(\pi^{-}\)-mesons with various-

with nuclei was also investigated in macroscopic experiments\(^ {11}\), the scheme of which is shown in the figure. A collimated beam of \(\pi^-\)-mesons with an initial energy of \(100\) Mev was passed in these experiments through absorber plates made of various materials. The beam monitor consisted of two Stilbene scintillation counters placed in front of the absorber.

The detector of meson absorption was a liquid scintillation counter placed immediately behind the absorber. Since

the detector was placed flush against the absorber, Coulomb scattering of mesons through small angles did not lead to an attenuation of the meson beam, and the absorption was determined only by the nuclear interaction.

The distance between the two meson-beam monitor counters was \(275\) cm. The maximum rate of coincidences in both counters reached, during the registration time of the second counter, about \(1.13\cdot10^{-8}\) sec, which corresponds to a rate of registered particles of about \(0.82\) sec. From this it could be concluded that there was no appreciable background of protons and electrons in the beam of initial mesons. The fraction of \(\mu\)-mesons, determined from the counting rate of \(\mu\)-\(e\) decay events, did not exceed \(5\%\) of the total number of \(\pi\)-mesons. The results of the experiments in the form of total cross sections for the interaction of \(\pi^-\)-mesons with various nuclei are given in Table III. The mean energy of the \(\pi^-\)-mesons, allowing for slowing down in the absorber, is estimated at \(85\) Mev. For comparison, Table III also gives the nuclear geometric cross sections \(\sigma_{\text{geom}}\), calculated from the formula

\[ \sigma_{\text{geom}}=\pi A^{\frac{2}{3}}\left(\frac{\hbar}{\mu c}\right)^2 \text{ cm}^2. \]

It is evident that for all nuclei, except hydrogen, the cross sections for \(\pi^-\)-mesons are close to the geometrical cross sections. For hydrogen, however, the cross section for \(\pi^-\)-mesons proved to be approximately 5 times smaller than the geometrical cross section.

In the case of hydrogen, 3 types of nuclear interaction of \(\pi^-\)-mesons are possible:

1. \(\pi^-+p\to\pi^-+p\) (scattering),
2. \(\pi^-+p\to\pi^0+n\) (exchange scattering), and
3. \(\pi^-+p\to n+h\nu\) (radiative absorption).

The latter reaction, as is known from the study of the reaction \(p+\hbar\nu \to n+\pi^+\), has a cross section of the order of \(10^{-28}\ \mathrm{cm}^2\), i.e., much smaller than for reactions 1 and 2, whose cross sections together amount to about \(10^{-26}\ \mathrm{cm}^2\).

Table III

Total cross sections for the interaction of \(\pi\)-mesons with energy 85 MeV with various nuclei

In the case of complex nuclei, to the probability of scattering there is added also the probability of absorption of \(\pi\)-mesons with the formation of stars, and this probability, as follows from \({}^4\), is the predominant one. As a result, the cross sections for the interaction of \(\pi\)-mesons with complex nuclei reach values of geometrical cross sections, despite the smallness of the cross section for the elementary meson–nucleon interaction. The smallness of the cross section for the interaction of \(\pi\)-mesons with protons also excludes such an interpretation of the interaction of mesons with complex nuclei according to which mesons are scattered many times inside the nucleus before being absorbed or re-emitted from the nucleus. Thus, definite difficulties arise in explaining the experimentally observed processes of inelastic scattering of mesons, in which a considerable loss of kinetic energy occurred \({}^{7,8}\).

Other difficulties in explaining the nuclear interaction of \(\pi\)-mesons were revealed in Bethe’s theoretical paper \({}^{12}\). To describe the nuclear scattering of \(\pi\)-mesons Bethe made use of the theory pertaining to the scattering of high-energy neutrons \({}^{13}\). According to this theory, the cross sections for inelastic collisions and for diffraction scattering are determined by two parameters—the coefficient of absorption in nuclear matter, proportional to the average cross section for the interaction of a nucleon with a nucleon (or, in the present case, a meson with a nucleon), and the coefficient of refraction of the neutron (meson) wave in the nucleus, associated with the depth of the nuclear potential well. Bethe determined the magnitude of this “nuclear well” potential \(V_0\) from the experimental values of the cross sections for diffraction scattering of mesons with energy \(45\) MeV \({}^{9,10}\): \(V_0 = 11 \pm 11\) MeV. If the potentials of a meson in a pion and a proton are equal, then the cross section for scattering of \(\pi\)-mesons by nucleons is \(\sigma_s = 4\pi a^2\) (in the case of isotropy of this scattering), where the quantity \(a\) is related to the potential \(V_0\) by the relation: \(a = -\dfrac{2}{3}\, r_0^3 \dfrac{\mu V_0}{\hbar^2}\), moreover

\(r_0 \simeq 1.5 \cdot 10^{-13}\) cm. Hence \(\sigma_s = 1 \div 5 \cdot 10^{-27}\ \text{cm}^2\), i.e., smaller than that obtained experimentally[^11]. In order to achieve agreement between the calculated and experimental values of the meson–nucleon scattering cross sections, one must assume that the scattering amplitudes of \(\pi\)-mesons by neutrons and protons are of opposite sign. Then, even for large amplitudes \(a_{\pi n}\) and \(a_{\pi p}\), the quantity \(V_0\) may be small. But if \(\pi\)-mesons are pseudoscalar particles, as has been convincingly shown, in particular, in the works of Soviet scientists[^3][^4][^14], then \(a_{\pi n}\) and \(a_{\pi p}\) must be of the same sign. Thus, the cross sections for the interaction of \(\pi\)-mesons with nucleons still remain to be refined.

G. I.

Cited Literature

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