ARTIFICIAL $\pi$-MESONS

A. B. Migdal and Ya. A. Smorodinskii

1. INTRODUCTION

In a large series of works, Alikhanov, Alikhanian, and their collaborators^1,2^ proved that, besides the particle with a mass of about $200\,m_e$ ($m_e$ is the electron mass), which bore the name meson, there exist charged particles with other masses. The new particles were called varitron(s) by the authors.*)

One of the varitrons, with a mass of about $300\,m_e$, was subsequently found by the photographic-plate method in cosmic rays by Lattes, Occhialini, and Powell^5^. The name $\pi$-meson was established for this particle. The previously known particle, with a mass of about $200\,m_e$, received the name $\mu$-meson. In the same work it was established that the $\pi$-meson decays into a $\mu$-meson and a neutral particle (possibly a neutrino; see below).

Somewhat later^6^ the same authors discovered that, along with decaying $\pi$-mesons, there exist particles, apparently of the same mass, whose tracks end not in a $\mu$-meson track but in nuclear disintegration—a “star” ($\sigma$-mesons). In addition, a large number of tracks corresponding to the same mass end without a $\mu$-meson and without a star ($\rho$-mesons).

It was established that $\pi$- and $\sigma$-mesons in fact differ from one another only in the sign of their charge (see ^3,4^). These particles are now denoted respectively as $\pi^+$- and $\pi^-$-mesons. As for the $\rho$-meson, it is a $\pi^-$-meson that produces nuclear disintegration not accompanied by the emission of charged particles.

The difference in the behavior of $\pi^+$- and $\pi^-$-mesons is entirely explained, as is known, by the difference in the sign of their charges. A $\pi^-$-meson, entering a condensed (i.e., non-gaseous) medium, is slowed down as a result of collisions with atoms and, in a time of the order of $10^{-12}$ sec,

*) A list of the literature on these works is given in the article by Nikitin and Weissenberg^3^. These same works are described in Zhdanov’s article^4^, which gives a detailed review of all works devoted to varitrons in cosmic rays.

its velocity becomes so small that it is captured into a closed orbit around one of the nuclei. This orbit is analogous to the \(K\)-orbit of an electron and differs only in that its radius, equal to

\[ a_{\pi}=\frac{\hbar^{2}}{Z m_{\pi} e^{2}}=\frac{0.19}{Z}\cdot 10^{-8}\ \text{cm}, \]

is \(\frac{m_{\pi}}{m_e}\) times smaller than the radius of the electron orbit (\(m_{\pi}\)—the mass of the \(\pi\)-meson).

In this formula \(Z\) is the nuclear charge, and the value of \(m_{\pi}\) is taken equal to \(276\,m_e\) (see below).

For the \(\pi^+\)-meson there is no bound state; therefore, after stopping it is not captured, but decays.

The subsequent fate of a \(\pi^-\)-meson captured into a \(K\)-orbit is determined by the ratio between the probability of its spontaneous decay and the probability of capture by one of the protons of the nucleus.

If the probability of capture of a \(\pi^-\)-meson by a nucleus is denoted by \(w_c Z^4\) (it is not difficult to see that this probability is proportional to \(|\psi(0)|^2 \sim \frac{1}{a_{\pi}^{3}}\sim Z^3\) and to the number of protons \(Z\)), and the probability of decay by \(w_d\), then the fraction of particles captured by the nucleus will be equal to

\[ \frac{w_c Z^4}{w_c Z^4+w_d} = \frac{1}{1+\frac{w_d}{w_c Z^4}}. \]

It is obvious that the ratio between the probabilities of decay and capture of negatively charged particles is a measure of the intensity of the interaction of these particles with nuclei and is one of the principal characteristics of the particle.

As is known, for \(Z>10\) the greater part of the \(\mu^-\)-mesons is captured by nuclei; for \(Z<10\) decay of \(\mu^-\)-mesons is observed. \(\pi^-\)-mesons, in contrast to \(\mu^-\)-mesons, do not decay in condensed matter, and their decay has been observed only in air. It follows from this that \(\pi^-\)-mesons interact much more strongly with nuclei.

From elementary considerations of detailed balance it follows that if \(\pi^-\)-mesons have a high probability of capture, then they must also have a high probability of formation in collisions of heavy particles. Such a process of artificial production of \(\pi\)-mesons was discovered in 1947 at the Berkeley cyclotron\({}^{7}\), which produced a beam of \(\alpha\)-particles with an energy of \(380\) MeV*).

These experiments marked the beginning of the second important stage in the study of varitrons; the first stage was the experiments of the group of Alikhanov and Alikhanyan.

*) See \({}^{8}\).

In general, we should expect that, under the action of particles of high energy, varitons of various masses should be formed. However, up to the present time it has been possible to observe only the production of \(\pi^+\)- and \(\pi^-\)-mesons (there is an indication of the production of a variton with a mass of about \(450\,m_e^{9}\)). This, apparently, must mean that only a few of the varitons interact sufficiently strongly with nuclei (at least among those whose production is energetically possible). The remaining varitons must have a very small production cross section and arise only as a result of the decay of other varitons.

The identity of artificial \(\pi^+\)- and \(\pi^-\)-mesons with those whose tracks were observed upon irradiation of photographic plates by cosmic rays was proved by measurements of their masses, and also by the fact that, like the tracks of “cosmic” \(\pi^+\)-mesons, the tracks of artificial \(\pi^+\)-mesons end in single tracks of \(\mu^+\)-mesons, while the tracks of \(\pi^-\)-mesons end in nuclear disintegrations, “stars”\(^{10}\) (see also a number of notes\(^{11\ 14}\)).

Precise measurements of the masses of \(\pi\)-mesons were made by Lattes, Gardner, and Bishop\(^{15—17}\), who measured the curvature of the path in a magnetic field and the range of the particles; their results are: \(m_{\pi^+}=285\pm 6\,m_e\). We shall also cite the result of van Rossum\(^{9*}\), giving for the mass the value

\[ m_{\pi}=(280\pm 15)\,m_e . \]

In the work of Lattes, Gardner, and Bishop the mass of the \(\mu^+\)-meson, which arises in the decay of the \(\pi^+\)-meson, was also determined; it was found to be \(m_\mu=216\,m_e\), which agrees with the results of measuring this quantity in experiments with cosmic rays.

In the work of Barkas, Bishop, Bradner, Gardner, and Smith\(^{18}\), new values of the masses are given, in agreement with the preceding ones:

\[ m_{\pi}=276\pm 6,\qquad m_{\mu}=210\pm 4. \]

From these data one can draw some conclusions about the mechanism of decay of the \(\pi^-\)-meson. It is well known that the laws of conservation of energy and momentum can be fulfilled only when the decay of a free particle proceeds into at least two particles. It is further known that the \(\pi^+\)-meson, in decaying, gives a \(\mu^+\)-meson whose track in the photographic emulsion always has one and the same magnitude—\(600\,\mu^{**}\), which corresponds to an energy of about \(4\) MeV. Such

*) The mass was determined both from the deflection in a magnetic field and by counting grains.

**) Five tracks measured in one of the papers\(^{10}\) had lengths respectively: \(625, 630, 612, 604, 560\,\mu\).

The relation between the energy of a particle and its range in photographic emulsion is given by the empirical formula\(^{42}\)

\[ E=0.251\,m^{0.419}R^{0.581}, \]

where \(E\) is the energy in MeV, \(m\) is the ratio of the particle mass to the proton mass, and \(R\) is the range in microns. (The formula is valid for Ilford B1, C2 plates and, apparently, with an error \(<2\%\), for Ilford B2, C3, and G5 plates.)

tracks can arise only in the case when the \(\pi^+\)-meson decays while having a velocity equal to zero, and the decay occurs into two particles: a \(\mu^+\)-meson and a neutral particle. If the decay occurred into 3 particles, then a spread in the energies of the \(\mu^+\)-mesons would be observed.

Denoting the mass of the neutral particle by \(m_\nu\), we write the law of conservation of energy in the form (we regard the \(\mu\)-meson as nonrelativistic)

\[ m_\pi c^2 = m_\mu c^2 + \frac{p^2}{2m_\mu} + \sqrt{m_\nu^2 c^4 + p^2 c^2}, \]

\[ \left(\frac{m_\nu}{m_\pi}\right)^2 = \left[\left(1-\frac{m_\mu}{m_\pi}\right)^2 + \left(1-\frac{E_\mu}{m_\pi c^2}\right)^2 -1\right], \]

where it has been taken into account that the momenta of the \(\mu^+\)-meson and of the neutral particle are equal to one another. Noting that

\[ \frac{p^2}{2m_\mu}=E_\mu=4\,\text{MeV}, \]

and substituting here the measured values of the masses of the \(\pi\)- and \(\mu\)-mesons (in doing so one may use the well-measured\(^{17}\) ratio \(\dfrac{m_\pi}{m_\mu}=1.32\pm0.01\)), we find that, within the accuracy of the experimental data, the neutral particle has a mass lying between zero and \(0.1\) of the mass of the \(\pi\)-meson. It is most natural to suppose that in the decay of the \(\pi\)-meson a \(\mu\)-meson and a neutrino are emitted (see below).

The lifetime of the artificial \(\pi^-\)-meson with respect to such a decay was measured by Richardson\(^{19}\). A beam of negative \(\pi\)-mesons, produced in the bombardment by \(\alpha\)-particles (with energy \(350\text{--}380\,\text{MeV}\)) of a carbon target situated inside a cyclotron, was directed into two spiral (more precisely, helical, i.e. bent along a screw line) channels of identical radius. A \(\pi\)-meson entering one of them traversed \(1/2\) of a circumference, while one entering the other traversed \(3/2\) of a circumference.

Photographic plates were placed at the exit ends of the channels. The number of tracks ending in the emulsion of the plate with a “star” served as a measure of the number of \(\pi\)-mesons that reached the plate. Therefore, knowing the ratio of the numbers of tracks in the two plates, it was possible to determine the relative number of \(\pi^-\)-mesons that decayed inside the channel. This number directly determines the lifetime of the mesons.

The energy of the mesons in these experiments was small (about \(10\,\text{MeV}\)), and relativistic effects were not substantial. It is curious to note-

considered that the adopted method of measuring the lifetime proves to be insensitive to the relativistic increase of this time with velocity.

This occurs because the time of motion of the meson in the channel increases with increasing velocity by the same factor—by \((1-\beta^2)^{-1/2}\)—as does the lifetime itself.

The lifetime of the \(\pi^{-}\)-meson was found to be

\[ \tau(\pi^-)=\left(1.1^{+0.31}_{-0.22}\right)\cdot 10^{-8}\ \text{sec.}, \]

and the half-life:

\[ \tau_{1/2}(\pi^-)=\left(7.7^{+2.1}_{-0.22}\right)\cdot 10^{-7}\ \text{sec.} \]

The lifetime of positive \(\pi\)-mesons was measured on the same cyclotron by Martinelli and Panofsky \(^{20}\).

In these experiments the \(\pi^{+}\)-mesons were obtained by bombarding a carbon target not with \(\alpha\)-particles, but with protons of energy 350 MeV, since the yield of mesons from such protons is 10 times greater than the yield from \(\alpha\)-particles of energy 380 MeV.

The mesons produced passed through channels \(1/2\), \(3/2\), and \(5/2\) circumference in length. Thus, in order to determine the lifetime it was possible to use not one ratio, as in Richardson’s experiments, but two. The \(\pi^{+}\)-mesons were identified by the track of the \(\mu^{+}\)-meson. The result of these experiments is as follows:

\[ \tau(\pi^+)=\left(1.97^{+0.14}_{-0.17}\right)\cdot 10^{-8}\ \text{sec.}; \]

\[ \tau_{1/2}(\pi^+)=\left(1.37^{+0.10}_{-0.12}\right)\cdot 10^{-8}\ \text{sec.} \]

These values differ noticeably from the corresponding values for \(\pi^{-}\)-mesons, and the discrepancy exceeds the experimental errors. Whether this discrepancy is real can be decided only by further experiments.

It should be noted that the lifetime of \(\pi\)-mesons agrees with the estimates made for \(\pi\)-mesons in cosmic rays \(^{21}\).

2. THRESHOLD OF THE REACTION OF \(\pi\)-MESON PRODUCTION

In the experiments described in the literature on the production of \(\pi\)-mesons, the energy of the \(\alpha\)-particles reached 380 MeV, the energy of protons and neutrons—345 MeV, and the energy of \(\gamma\)-quanta—335 MeV. At such energies of \(\alpha\)-particles and protons only one variton can be produced. The energy of the \(\gamma\)-quanta was sufficient for the production of two \(\pi\)-mesons. However, the existence of such a process has not been proven.

Although it seems beyond doubt that, at high energies, such processes must occur just like the processes of simultaneous production of more than two \(\pi\)-mesons, in this review we shall confine ourselves only to processes of single production of \(\pi\)-mesons.

Four types of reactions are possible which lead to the formation of mesons in collisions of nucleons, namely:

\[ \begin{aligned} n+p&\to n+n+\pi^+,\\ n+p&\to p+p+\pi^-,\\ p+p&\to p+n+\pi^+,\\ n+n&\to p+n+\pi^-, \end{aligned} \]

and also two reactions of formation of \(\pi\)-mesons by \(\gamma\)-quanta:

\[ \begin{aligned} \gamma+n&\to p+\pi^-,\\ \gamma+p&\to n+\pi^+. \end{aligned} \]

From elementary energy considerations it is not difficult to obtain the value of the minimum energy that the incident particle must have in order for the process of production of a \(\pi\)-meson to be possible, i.e., to find the reaction threshold.

Let the incident particle have mass \(m\). Denote the mass of the target nucleus (at rest) by \(M\), the mass of the particle produced by \(m_1\), and the mass of the remaining reaction products by \(M_1\).

The condition for the possibility of the reaction in the center-of-inertia system is written in the form

\[ E_0 \geq (M_1+m_1)c^2, \tag{1} \]

where \(E_0\) is the energy of the system (incident particle + target nucleus) in the initial state (in the center-of-inertia system). The equality sign corresponds to the case when all particles in the final state are at rest.

In order to express \(E_0\) in terms of the energy of the incident particle in the laboratory coordinate system, let us note that the expression \(E^2-c^2p^2\), where \(E\) and \(p\) are the energy and momentum, is a relativistic invariant, unchanged under transition from one coordinate system to another. Since \(p_0\)—the momentum in the center-of-inertia system—is equal to zero, we may write:

\[ \left[T+(m+M)c^2\right]^2-c^2p^2=E_0^2, \tag{2} \]

where \(T\) is the kinetic energy of the incident particle, and \(p\) is its momentum. Substituting into (2) the expression for \(c^2p^2\):

\[ c^2p^2=(T+mc^2)^2-m^2c^4=T^2+2mc^2T, \tag{3} \]

we obtain:

\[ E_0^2=2Mc^2T+(m+M)^2c^4. \tag{4} \]

From (4), with the aid of (1), we find the value of the minimum kinetic ...

energy \(T_{\min}\) of the absolute threshold:

\[ T_{\min}=\frac{c^{2}}{2M}\left[(M_{1}+m_{1})^{2}-(M+m)^{2}\right]. \tag{5} \]

We give a table of values of \(T_{\min}\) for the birth of a \(\pi\)-meson of mass \(276\,m_e\) for several possible reactions*).

Table 1

Threshold for several reactions leading to the formation of a \(\pi\)-meson

In calculating the threshold we assumed that the entire energy yield of the reaction goes into the formation of the \(\pi\)-meson. However, at high energies, when the incident particle passes almost freely through the nucleus, the collision in practice takes place only between two nucleons. In this case the actual energy release may turn out to be appreciably smaller than follows from the value of the threshold. This circumstance may lead to a discrepancy between the experimental threshold and that calculated from the masses of the nuclei participating in the reaction. We shall discuss it in Section 6.

*) The table is taken from Barkas’s note\(^{22}\). In that note the mass of the \(\pi\)-meson was taken to be \(286\,m_e\). We have correspondingly reduced all threshold values.

A. F. MIGDAL AND Ya. A. SMORODINSKY

3. PRODUCTION OF $\pi$-MESONS BY CHARGED PARTICLES*)

Artificial $\pi$-mesons were obtained at the Berkeley cyclotron by bombarding a target placed inside the cyclotron with a beam of protons or $\alpha$-particles. By changing the distance of the mounted target from the center of the magnetic field, it was possible to investigate the production of $\pi$-mesons at different energies of the bombarding particle. The particles produced were deflected by the magnetic field in the cyclotron itself and were recorded on photographic plates. The direction of deflection determined the sign of the charge of the produced $\pi$-meson. The photographic plates used in these experiments recorded mesons with energies in the range 2–10 MeV.

Jones and White^45 measured the relative magnitude of the production cross sections of $\pi^-$-mesons (with energy 2–10 MeV) by $\alpha$-particles of different energies. $\pi^-$-mesons were chosen because they are easier to identify by the “stars” at the ends of their tracks. The results of these authors are given in Table II**).

Table II

Production cross section of $\pi^-$-mesons with energy
2–10 MeV by $\alpha$-particles (carbon target)

Similar data for the production of $\pi^-$-mesons (energy 2–10 MeV) by protons in carbon were published in the same work and are given in Table III.

In these experiments the experimental conditions made it possible to record $\pi$-mesons whose direction of motion made an angle $<45^\circ$ with the direction of the incident beam.

*) Two cases of $\pi$-meson production by neutrons have been published. One of them was the production of a $\pi$-meson in a Wilson chamber^23, the other in a photographic emulsion^13. In both cases the neutrons were obtained when protons with energy 350 MeV passed through a thin target. Quantitative data on this process have not yet been published.

**) These data, like the data on $\alpha$-particles, are somewhat corrected in comparison with the preliminary communications^25,26.

The absolute value of the cross section was determined by Peterson (see \(^{45}\)), who measured the absolute number of \(\pi^{-}\)-mesons arising from

Table III

Cross section for the production of \(\pi^{-}\)-mesons with energy
\(2\text{–}10\) MeV by protons (carbon target)

\(\alpha\)-particles with energy \(390\) MeV at an angle of \(\pm 45^\circ\) to the direction of the beam of \(\alpha\)-particles and having energy \(2\text{–}5\) MeV. These measurements gave*):

\[ \sigma_{\mathrm C}=(3.0\pm0.8)\cdot10^{-32}\, \frac{\mathrm{cm}^{2}}{\mathrm{MeV}\cdot \mathrm{rad}\cdot \text{nucleus}} \quad (E_{\alpha}=390\ \mathrm{MeV}). \]

The cross section for meson production under the action of protons was measured in carbon \(^{29}\) and in lead \(^{30}\) by the method proposed by Richman and Wilcox \(^{28}\). In these works absorbers of Al and Cu were placed in front of the plates, and the cross section was measured for different values of the energy of the particle produced. The cross sections were measured both for \(\pi^{+}\)- and for \(\pi^{-}\)-mesons, the direction of motion of which made an angle of \(90^\circ\) with the direction of the primary beam.

The energy distribution of the mesons had a broad maximum near \(35\) MeV for carbon and near \(30\) MeV for lead. The maximum meson energy was \(140\) MeV (for C) and \(120\) MeV (for Pb).

The differential cross section for production of \(\pi\)-mesons (\(\pi^{+}\) and \(\pi^{-}\)), referred to one carbon atom, is equal to:

\[ d\sigma_{\mathrm C}(90^\circ)=(2.4\pm0.3)\cdot10^{-28}\, \frac{\mathrm{cm}^{2}}{\text{radian}\cdot \text{nucleus}} \quad (E_{p}=345\ \mathrm{MeV}). \]

*) The cross section given in \(^{15}\) was obtained using unconvincing theoretical assumptions about the \(\pi\)-meson spectrum, and therefore this estimate is of no interest.

The same cross section for lead

\[ d\sigma_{\mathrm{Pb}}(90)=(8.9\pm2.4)\cdot10^{-28}\, \frac{\mathrm{cm}^2}{\text{radian}\cdot\text{nucleus}} \quad (E_p=345\ \mathrm{MeV}). \]

The value of the cross section for \(\pi\)-mesons with the energy corresponding to the maximum of the distribution curve is:

\[ d\sigma_{\mathrm{C}}^{\max}(90^\circ)=(3.4\pm0.4)\cdot10^{-30}\, \frac{\mathrm{cm}^2}{\text{radian}\cdot\text{nucleus}} \left(\begin{array}{c} E_p=345\ \mathrm{MeV}\\ E_\pi\simeq 40\ \mathrm{MeV} \end{array}\right); \]

\[ d\sigma_{\mathrm{Pb}}^{\max}(90^\circ)=(2.1\pm0.4)\cdot10^{-29}\, \frac{\mathrm{cm}^2}{\text{radian}\cdot\text{nucleus}} \left(\begin{array}{c} E_p=345\ \mathrm{MeV}\\ E_\pi\simeq 30\ \mathrm{MeV} \end{array}\right). \]

Interesting results were obtained by measuring the relative number of \(\pi^+\)- and \(\pi^-\)-mesons.

For the case of production by \(\alpha\)-particles (380 MeV) of \(\pi\)-mesons with energies \(2\)–\(5\) MeV in carbon, this ratio (denoted by \(\pi^+/\pi^-\)) is approximately equal to \(1/5\) and falls sharply with increasing \(Z\) of the target. (The measurements were made with targets of Be, C, Al, Cu, In, Pb.)\(^{31}\)

For protons (\(E_p=345\) MeV), the ratio of the probabilities of production of \(\pi^+\)- and \(\pi^-\)-mesons for various elements is given in Table IV.

Table IV

Ratio of the probabilities of production by protons
(\(E_p=345\) MeV) of high-energy \(\pi^+\)- and \(\pi^-\)-mesons
(\(\pi^+/\pi^-\)) in various elements

The difference between the probabilities of formation of slow \(\pi^+\)- and \(\pi^-\)-mesons can be attributed to the Coulomb barrier, which impedes the escape of positive particles.

At high energies of \(\pi\)-mesons, the value of the ratio \(\dfrac{\pi^+}{\pi^-}\) is determined by the fact that \(\pi^-\)-mesons can be produced by protons only in collisions with neutrons, whereas \(\pi^+\)-mesons can be produced both in collisions with protons and in collisions with neutrons. Moreover, the cross section for formation of a \(\pi^+\)-meson in a collision of two protons is approximately twice as large as the cross section in a collision of a proton with a neutron (thus

as each of the protons can turn into a neutron). Then \(\pi^+/\pi^-\) should be equal to \(\dfrac{2Z+N}{N}=\dfrac{A+Z}{A-Z}\) (\(N\) is the number of neutrons in the nucleus). For heavy elements this quantity is approximately 2.5. In the case of meson production by neutrons, analogous reasoning leads to \(\dfrac{\pi^+}{\pi^-}=\dfrac{Z}{2N+Z}=\dfrac{Z}{A+Z}\) (about 0.3 for heavy elements).

4. PRODUCTION OF \(\pi\)-MESONS BY \(\gamma\)-QUANTA

Experiments on the production of \(\pi\)-mesons by \(\gamma\)-quanta are described in the work of McMillan, Peterson, and White \(^{33}\). These authors obtained \(\gamma\)-quanta by means of a narrow beam of electrons (from the Berkeley synchrotron) with an energy of \(335\ \text{MeV}\), incident on a platinum target (about \(1/2\ \text{mm}\) thick).

The \(\gamma\)-quanta fell on a graphite cylinder, around which (in section planes parallel to the generators) photographic plates were arranged. The production of both \(\pi^+\)- and \(\pi^-\)-mesons was observed. In this case the ratio \(\dfrac{\pi^\pm}{\pi^\mp}\) proved to be \(0.59 \pm 0.07\). The angular distribution of the mesons in these experiments corresponded to an isotropic distribution in angles*).

To determine the absolute value of the cross section it is necessary to know the number of quanta. Since in reality in the experiments described the quanta were not monochromatic, but were distributed in energy almost uniformly up to the upper limit of \(335\ \text{MeV}\) (the bremsstrahlung spectrum), the authors introduce a conventional number of quanta equal to the energy incident on the graphite cylinder divided by the limiting energy. Then the cross section, referred to such an effective quantum per 1 C atom, is equal to

\[ \sigma_C = 5 \cdot 10^{-28}\ \text{cm}^2 \left(E_\gamma^{\max} = 335\ \text{MeV}\right). \]

According to the error estimate made by the authors, the cross section found cannot differ from the true one by more than a factor of 2.

The distribution of \(\pi\)-mesons (of both signs) in energy, beginning with \(30\ \text{MeV}\), was also measured. The distribution has a maximum near \(35\ \text{MeV}\) and extends approximately to \(150\ \text{MeV}\)**). In this case

*) Since in a collision with a \(\gamma\)-quantum the velocity of the center-of-inertia system is comparatively small, this corresponds to an isotropic distribution in the center-of-inertia system.

**) The production of \(\pi\)-mesons was also observed at an electron energy (producing the \(\gamma\)-quanta) equal to \(200\ \text{MeV}\). In this case the \(\pi\)-meson distribution had a boundary near \(35\ \text{MeV}\), and the yield was considerably smaller. However, no quantitative results were obtained for this case.

the yield of \(\pi\)-mesons with an energy of about \(100\) MeV amounts to \(1/8\) of the maximum.

The magnitude of the ratio \(\dfrac{\pi^+}{\pi^-}\) is explained by the influence of the recoil of the nucleon. When a \(\pi^+\)-meson is produced, the proton with which the current is associated receives the momentum; when a \(\pi^-\)-meson is produced, the neutron with which the current is not associated receives the momentum.

Calculations give for this ratio\(^{34,45}\)

\[ \frac{\pi^+}{\pi^-} = \left[ 1-\frac{\varepsilon}{Mc^2} \left( 1-\frac{v}{c}\cos\theta \right) \right]^2, \]

where \(\varepsilon\) is the energy of the \(\pi\)-meson (including the rest energy), \(M\) is the mass of the nucleon, \(v\) is the velocity of the \(\pi\)-meson, and \(\theta\) is the direction of its motion (relative to the \(\gamma\)-quantum)*.

At \(\theta=90^\circ\) (the conditions of the experiments described), this ratio varies from \(1.55\) (40 MeV) to \(1.83\) (100 MeV), which agrees with the experimental data.

A report by Steinberger and Bishop\(^{41}\) has been published, in which they observed the production of \(\pi^+\)-mesons by photons in various substances. The photons had a bremsstrahlung spectrum with an upper limit of \(300\) MeV. The observations were made with scintillation counters, which recorded delayed coincidences between the \(\pi^+\)-meson and the electron formed as a result of the decay of the \(\mu^+\)-meson (which in turn had arisen in the decay of the \(\pi^+\)-meson). The quantitative results of these experiments have not been published. It is noted only that the cross section for the formation of \(\pi^+\)-mesons (at an angle of \(90^\circ\) in the laboratory system), referred to one proton, is smaller in the case of carbon than in the case of hydrogen (for which the cross section was determined from the difference of the cross sections in paraffin and in graphite).

5. EMISSION OF HIGH-ENERGY \(\gamma\)-QUANTA IN NUCLEON COLLISIONS\(^{35}\)

When various targets are bombarded with protons of energy \(340\) MeV, photons with energies up to \(200\) MeV arise, the yield of which falls by a factor of 100 when the proton energy drops to \(170\) MeV. Such a fall in the cross section cannot occur for ordinary bremsstrahlung. Moreover, these photons have a spectrum sharply different from the bremsstrahlung spectrum and therefore must be associated with some process having a threshold near \(170\) MeV.

The most natural assumption is that the proton produces a new particle with a mass of about

* This expression turns out to be somewhat exaggerated, if the \(\pi\)-meson has spin different from unity, since a very strong magnetic interaction is associated with such a meson.

300 MeV, which decays with a very short lifetime, emitting a photon. The value of the mass determined from the threshold agrees with the energy of the resulting quanta. Whether this particle is neutral or charged cannot be asserted unambiguously. The authors consider the particle neutral only on the grounds that at present they know of no charged particles that emit $\gamma$-quanta in their decay.

A study of the angular distribution of the $\gamma$-quanta showed that they are produced isotropically in directions in a coordinate system moving with velocity $0.32c$ relative to the laboratory system.

If the reaction occurred between the incident proton and a nucleon at rest, then the velocity of the center-of-inertia system would be, as is easy to calculate, $0.39c$.

In reality, the nucleons in the nucleus have different velocities. The greatest release of energy and, consequently, the greatest probability of the process corresponds to the case when the nucleons in the nucleus move toward the incident nucleon.

The value of the velocity $0.32c$ corresponds to the velocity of the center-of-inertia system of the incident proton (345 MeV) and a nucleon in the target nucleus moving toward the proton with an energy of about 10 MeV. This agrees well with the scale of the kinetic energy of intranuclear particles. The cross-section values are given in Table V.

Table V

Cross section for the production of photons by protons with energy 345 MeV
(carbon target)

A similar phenomenon has also been found in cosmic rays. Recently a photograph was published of a large star produced by an α-particle with an energy of \(10^6\)—\(10^8\) MeV. This star contains a large number of electron pairs, which were formed at some distance from the center of the star. From the angle of divergence of these pairs one could estimate the energy of the \(\gamma\)-quanta forming them.

If it is assumed that the \(\gamma\)-quanta are not formed directly in the star, then they should be regarded as a product of meson decay. From the distance from the center of the star to the place where the first pair appears, and from the energy of the \(\gamma\)-quanta, one obtains an estimate of the lifetime of such a meson:

\[ \tau_0 < 10^{-12}\ \text{sec}. \]

Thus, both works give a consistent picture of a new phenomenon. However, the question of its interpretation and, in particular, the basic question of whether we are dealing with a neutral meson or with a charged one decaying with the emission of a photon and a charged particle (see p. 152) still remains unresolved.

6. DISCUSSION OF THE EXPERIMENTAL DATA

The totality of the available data makes it possible to draw certain conclusions about the properties of \(\pi\)-mesons.

1. First of all let us consider the question of the spin of \(\pi\)- and \(\mu\)-mesons. From the data on the decay of the \(\mu\)-meson (cf. \(^{88}\)) it follows that the decay occurs with the emission of one electron and at least two neutral particles. It is natural to suppose that the \(\mu\)-meson decays into an electron and two neutrinos:

\[ \mu^{\pm} \to e^{\pm} + \nu + \nu . \]

In this case the spin of the \(\mu\)-meson should be half-integral. The number of \(\delta\)-electrons produced by \(\mu\)-mesons, measured from the intensity of ionization bursts in shielded chambers (showers produced by \(\delta\)-electrons), does not contradict spin \(1/2\) and, apparently, excludes the possibility of higher spin values.

The absence of stars when a \(\mu\)-meson is captured by a nucleus means that the rest energy of the \(\mu\)-meson is almost entirely carried away by a neutral particle. This is possible only if the neutral particle has a mass much smaller than the mass of the nucleon with which the \(\mu\)-meson interacts. Since capture of \(\mu^-\)-mesons is not accompanied by the emission of \(\gamma\)-quanta, it is natural to assume that capture of \(\mu^-\)-mesons proceeds according to the scheme:

\[ \mu^- + p \to n + \nu . \]

The production of \(\pi\)-mesons by nucleons and \(\gamma\)-quanta apparently occurs without the emission of any additional particles, according to the scheme

\[ p+n \to \pi^+ + n + n, \]

\[ \gamma+n \to \pi^- + p. \]

If, in the production of a \(\pi\)-meson, some other particle were emitted, then a noticeable part of the energy would be carried away, which does not correspond to experiment. From the scheme of \(\pi\)-meson production just given it follows that its spin is integral. A choice between spin values \(0,1,\ldots\) cannot at present be made.

The same conclusion can also be drawn from the mechanism of capture of a \(\pi\)-meson by nuclei.

Since, in the capture of a \(\pi^-\)-meson, a noticeable part of its rest energy goes into the formation of a star, it cannot be admitted that in this process some light particle is emitted, as happens in the case of capture of a \(\mu^-\)-meson. The most natural capture reactions are

\[ \pi^- + p + n \to n + n, \]

\[ \pi^- + p + p \to p + n. \]

The participation of the second nucleon in the reaction is necessary in order to satisfy the law of conservation of energy and momentum.

This view of the spins of \(\pi\)- and \(\mu\)-mesons is also consistent with the accepted mechanism of \(\pi\)-meson decay (see Section 1):

\[ \pi^\pm \to \mu^\pm + \nu. \]

1. The most natural mechanism of \(\pi\)-meson production would be taken to be a pair collision, i.e. the collision of an incident high-energy nucleon with one of the nucleons of the nucleus. In doing this one must take into account that the nucleons in the nucleus have a certain kinetic energy of internal motion, of the order of \(20\text{--}25\) MeV.

However, such a description gives an incorrect result at energies close to the reaction threshold and, in particular, gives an incorrect value of the threshold itself.

On the contrary, at high energies of the incident nucleon, satisfactory results are obtained.

Let us estimate, for example, what maximum energy a \(\pi\)-meson can have if the energy of the incident particle is specified.

In Section 3 it was indicated that, for proton energies of \(345\) MeV, the maximum energy of \(\pi\)-mesons is \(120\text{--}140\) MeV.

In a pair collision of a proton with energy \(345\) MeV and a nucleon in the nucleus with energy about \(25\) MeV, the kinetic energy of both particles in the center-of-inertia system (assuming that in the most favorable...

in the case when their velocities are directed toward one another) will be equal to*)
\(\frac{1}{2}(\sqrt{345}+\sqrt{25})^2 = 275\) MeV.

Of this energy, 140 MeV goes into the rest mass of the \(\pi\)-meson, while 135 MeV remains for its kinetic energy, which agrees with experiment**).

Such an estimate is not very sensitive to the choice of the value of the energy of internal motion. One may try to determine this value so as to obtain the experimental values of the maximum energy in the case of carbon (140 MeV) and lead (120 MeV). These values will be about 25 MeV for C and about 20 MeV for Pb. However, no profound meaning can be attached to these numbers; the difference may be explained by experimental errors and by the inaccuracy of the picture of a pair collision itself.

If we try to determine the internal energy for C from the experimental value of the threshold of the reaction of \(\pi\)-meson formation (170 MeV), we obtain a value of about 16 MeV***).

Further, in a pair collision one would expect that the cross section for \(\pi\)-meson formation on different elements should be, roughly speaking, proportional to the number of nucleons, i.e., proportional to \(A\). However, experiment shows that in going from C to Pb (\(A\) increases by a factor of 17) the cross section (at an angle of \(90^\circ\)) increases only by a factor of 4. A possible explanation of this fact is that the proton, in passing through the nucleus, loses energy with appreciable probability and ceases to be effective for the production of mesons. Then the cross section for heavy elements should grow as the geometrical cross section of the nucleus.

Very interesting are the experiments on the dependence of the cross section on the energy of the incident proton (Table II). In these experiments the energy interval of the \(\pi\)-mesons was the same for all proton energies (2–10 MeV).

Let us introduce into the discussion the quantity \(\varepsilon\)—the energy in the center-of-mass system that remains with the nucleons after the collision (for an energy of the incident proton \(E_p\) and of the \(\pi\)-meson—about 5 MeV):

\[ \varepsilon = \frac{1}{2}\left(\sqrt{E_p}+\sqrt{16}\right)^2 - 145\ \text{MeV}. \]

(The internal energy is chosen so as to obtain the correct value of the threshold.)

) The energy is equal to the square of the relative velocity multiplied by one half of the reduced mass.
) Since experimentally the \(\pi\)-mesons were observed at an angle of \(90^\circ\) to the direction of the protons, their energy changes almost not at all in transforming to the laboratory coordinate system.
**) At the same time, the threshold agrees perfectly with the value given in Table I, in which it is assumed that the energy of the whole nucleus participates in the formation of the \(\pi\)-meson.

The experimental value of the cross section turns out to be proportional to \(\varepsilon^{3.5}\). Theoretically it appears natural that, at low nucleon velocities, the dependence of the production cross section on the nucleon energy would be determined only by the statistical factor. Then, in a pair collision, the cross section should be proportional to \(\varepsilon^{1/2}\), as follows from the magnitude of the statistical weight.

In the case of the formation of \(\pi\)-mesons by \(\alpha\)-particles, the cross section turns out to be approximately proportional to \(\varepsilon_1^{3.5}\), where \(\varepsilon_1\) is determined by the formula:

\[ \varepsilon_1=\frac{1}{2}\left(\sqrt{\frac{1}{4}E_{\alpha}+9}\right)^2-145\ \text{Mev}. \]

The expression for \(\varepsilon_1\) was chosen so that it would agree with the graphically extrapolated value of the threshold of the reaction of \(\pi\)-meson production by \(\alpha\)-particles (\(\sim 220\ \text{Mev}\))*.

In the production of \(\pi\)-mesons by \(\gamma\)-quanta, the meson energy reaches \(150\ \text{Mev}\) at \(E_{\gamma}=335\ \text{Mev}\) and \(35\ \text{Mev}\) at \(E_{\gamma}=200\ \text{Mev}\). These values agree well with the threshold value given in Table I (\(160\ \text{Mev}\)).

1. To elucidate the mechanism of \(\pi\)-meson formation, experiments on the capture of \(\pi\)-mesons by nuclei are of great interest. In such capture, the rest energy of the meson is converted into the kinetic energy of at least two nucleons, and in an appreciable fraction of cases protons with energies of about \(70\ \text{Mev}\) should be observed among the stars. In experiment, however, the number of fast protons is apparently considerably smaller than might have been expected.

Cheston and Goldfarb\({}^{39}\) investigated 317 cases of stopping of \(\pi\)-mesons. Of these, only 8 (\(\sim 2\%\)) were accompanied by the emission of protons with energies from 30 to \(70\ \text{Mev}\). In one case a proton with an energy of \(100\ \text{Mev}\) was found. In 30–40% of the cases no “stars” were found at all. Since in 65% of the cases of capture without “stars” tracks of recoil nuclei were observed, such cases may be interpreted as capture with emission of neutrons. The large relative number of “stars” consisting only of neutrons is explained by the fact that the emission of charged particles is hindered by the Coulomb barrier.

The small number of fast protons was noted earlier by Parkinson\({}^{40}\), who investigated stars arising from \(\pi\)-mesons in cosmic rays.

In this connection, experiments on the capture of \(\pi\)-mesons in hydrogen and deuterium are of interest.

*) We note that this quantity is appreciably higher than the value given in Table I (\(202\ \text{Mev}\)). The term 9 in the brackets is connected with the internal energy of the nucleons in the target nucleus and in the \(\alpha\)-particle.

A. B. MIGDAL AND Ya. A. SMORODINSKY

In capture in hydrogen the process cannot proceed in the usual way, since the hydrogen nucleus consists of only one proton. This process must be accompanied either by the emission of a \(\gamma\)-quantum*), or by the emission of a neutral particle. The experimental attempt to detect \(\gamma\)-quanta\(^{41}\) in the capture of \(\pi\)-mesons in LiH and CH\(_2\) gave, for the upper limit of the number of \(\gamma\)-quanta per capture, the values for CH\(_2\)—\(1/1000\) and for LiH—\(1/200\). This result apparently means that capture occurs with the emission of neutral particles.

Capture by a deuteron realizes in pure form the reaction considered above,
\[ \pi^- + p + n \to n + n. \]
These processes, together with the phenomenon of production, may prove essential for understanding the mechanism of interaction of \(\pi\)-mesons with nuclei.

CITED LITERATURE

1. A. Alikhanov, A. Alikhanian and S. Nikitin, Journ. of Phys. 9, 167 (1945).
2. A. Alikhanov, A. Alikhanian and A. Weissenberg, ZhETF 18, 301 (1947).
3. S. Nikitin and A. Weissenberg, introductory article to the book by Janossy “Cosmic Rays,” I. L., 1949.
4. G. B. Zhdanov, UFN, 39, 512 (1949).
5. C. M. G. Lattes, G. P. S. Occhialini a. K. F. Powell, Nature 160, 453 (1947).
6. C. M. G. Lattes, G. P. S. Occhialini a. K. F. Powell, Nature 160, 486 (1947).
7. E. Gardner a. C. M. G. Lattes, Science 107, 270 (1948).
8. K. M. Zaltses, G. P. S. Occhialini, K. F. Powell and F. K. Frank, UFN 34, 450 (1948).
9. L. van Rossum, Journ. de Phys. et Radium, VIII, 10, 402 (1949).
10. J. Burfening, E. Gardner a. C. M. G. Lattes, Phys. Rev. 75, 382 (1949).
11. F. L. Adelman a. S. B. Jones, Phys. Rev. 75, 1468A (1949).
12. G. P. S. Occhialini a. K. F. Powell, Nature 161, 551 (1948).
13. A. S. Bishop, J. Burfening, E. Gardner a. C. M. G. Lattes, Phys. Rev. 74, 1558 (1948).
14. W. Barkas, E. Gardner a. C. M. G. Lattes, Phys. Rev. 74, 1558 (1948).
15. C. M. G. Lattes, Phys. Rev. 75, 1468 (1949).
16. E. Gardner, Phys. Rev. 75, 1468 (1949).
17. A. S. Bishop, Phys. Rev. 75, 1468 (1949).
18. W. H. Barkas, A. S. Bishop, H. Bradner, F. M. Smith a. E. Gardner, Phys. Rev. 78, 86 (1950).

*) This process is the inverse of the process of formation of a \(\pi\)-meson by a \(\gamma\)-quantum. The lifetime of the meson is related to the formation cross section by the formula
\[ \frac{1}{\tau}=\sigma\cdot \frac{2c^{2}}{v}\cdot \frac{\mu}{\bar{\mu}}\cdot N, \]
where \(v\) is the velocity of the meson, \(\bar{\mu}\) is the reduced mass, and \(N\) is the density of nuclei. For capture from a \(K\)-orbit, \(N\) must be replaced by
\[ \psi^{2}(0)=\frac{1}{\pi}\cdot \frac{1}{a_{k}^{3}}=\frac{1}{\pi}\left(\frac{\bar{\mu}e^{2}}{\hbar}\right)^{3}. \]

1. J. R. Richardson, Phys. Rev. 74, 1720 (1948).
2. E. A. Martinelli and W. K. H. Panofsky, Phys. Rev. 77, 465 (1950).
3. U. Camerini, H. Muirhead, C. F. Powell and D. M. Ritoon, Nature 162, 433 (1948).
4. W. H. Barkas, Phys. Rev. 75, 1109 (1949).
5. W. Hartsongh, E. Hayward and W. H. Powell, Phys. Rev. 75, 905 (1949).
6. W. K. H. Panofsky and E. A. Martinelli, Rev. Sci. Instr. 20, 286 (1949).
7. S. B. Jones and R. S. White, Phys. Rev. 75, 1468 (1949).
8. S. B. Jones and R. S. White, Phys. Rev. 76, 588 (1949).
9. V. Peterson, Phys. Rev. 75, 1469A (1949).
10. H. Wilcox and C. Richman, Phys. Rev. 78, 85 (950).
11. C. Richman and H. Wilcox, Phys. Rev. 78, 85 (1950).
12. M. Weissbluth, Phys. Rev. 78, 86 (1950).
13. W. H. Barkas, Phys. Rev. 75, 1467A (1949).
14. H. Bradner and S. B. Jones, Phys. Rev. 78, 90 (1950).
15. E. M. McMillan, J. M. Peterson and R. S. White, Science 110, 579 (1949).
16. K. A. Brueckener and M. L. Goldberger, Phys. Rev. 76, 1725 (1949).
17. R. Bjorklund, W. E. Crandall, B. J. Mayer and U. F. York, Phys. Rev. 77, 213 (1950).
18. M. F. Kaplan, B. Peters and H. L. Bradt, Phys. Rev. 76, 1735 (1949).
19. U. L. Bradt, M. F. Kaplon and B. Peters, Helv. Phys. Acta 23, 24 (1950).
20. E. P. Hincks and B. Pontecorvo, Phys. Rev. 77, 102 (1950).
21. W. Cheston and L. Goldfarb, Phys. Rev. 78, 320A (1950).
22. D. H. Perkins, Phill. Mag. 40, 601 (1949).
23. W. K. H. Panofsky and H. F. York, Phys. Rev. 78, 89 (1950).
24. H. Bradner, F. M. Smith, W. H. Barkas and A. S. Bishop, Phys. Rev. 77, 462 (1950).
25. F. M. Smith, E. Gardner, H. Bradner, Phys. Rev. 77, 562 (1950).
26. Steinberger and A. S. Bishop, Bull. Amer. Phil. Soc. 25, No. 1, p. 11 (1950).
27. S. B. Jones and R. S. Cohite, Phys. Rev. 78, 12 (1950).
28. P. Benoist-Guetal, J. Prentki et J. Ratier, Comptes Rendus 230, 1146 (1950).
29. R. E. Marshak, Bull. Amer. Phys. Soc. 25, No. 3, p. 29 (1950).
30. A. J. Seriff, R. B. Leighton, C. Hsiao, E. W. Cowan and C. D. Anderson, Phys. Rev. 78, 290 (1950).

ADDENDUM AT PROOF CORRECTION TO § 5

1. Panofsky (reported in 47) found that when $\pi^-$ mesons are captured by protons, $\gamma$ quanta arise with an energy of about 140 MeV and with an energy equal to approximately one half of this value. The appearance of such photons can be explained by the fact that the $\pi^-$ meson is captured both with the emission of $\gamma$ quanta and with the emission of a new particle, which decays into two quanta (or into one quantum and another neutral particle). The width of the distribution of these quanta in energy is determined by the velocity of the decaying particle (assuming that in the system in which the particle is at rest the $\gamma$ quanta are distributed isotropically). According to the theory of effec-

of the Doppler effect, the energy of the \(\gamma\)-quanta in the laboratory system differs from their energy in the particle’s system by the factor \(\left(1-\dfrac{v}{c}\cos\theta\right)\), where \(v\) is the velocity of the particle, and \(\theta\) is the angle between the direction of \(v\) and the direction of emission of the \(\gamma\)-quanta.

Neglecting the recoil energy of the proton, one can determine the velocity of the particle from the relation

\[ \Delta c^2=\frac{mv^2}{2}, \]

where \(\Delta\) is the mass difference between the \(\tau\)-meson and the decaying particle.

Denoting the width of the energy distribution of the \(\gamma\)-quanta by \(D\), we obtain

\[ D=\sqrt{2\Delta\cdot mc^2}. \]

Using the experimental value

\[ D=35\pm 15\ \text{MeV}, \]

we find that

\[ \Delta=(4\pm 4)\ \text{MeV}. \]

1. Seriff, Leighton, Hsiao, Cowan, and Anderson\(^{48}\) reported the presence in cosmic rays of a neutral particle decaying, with a lifetime of about \(3\cdot 10^{-10}\) sec., into two charged particles.