MESOATOMS

M. I. Podgoretskii

§ 1. INTRODUCTION

The stopping of negative mesons in matter is accompanied, as is known, by the formation of mesoatoms—systems analogous to hydrogen-like atoms, with the sole difference that the role of the electrons in mesoatoms is played by mesons. The formation of a mesoatom is due to the Coulomb interaction between the negative meson and the positive nucleus. Having fallen onto one of the mesoatomic levels, the meson can then pass to lower levels, emitting in most cases relatively hard photons. The decay of the meson or its capture by the nucleus occurs, as a rule, only after the meson has reached the lowest level, i.e. the \(K\)-shell of the mesoatom.

The presence of the mesoatomic stage leaves a distinctive and very deep imprint on the processes of nuclear capture and decay of negative mesons. This circumstance is the main reason that has led to the appearance of a fairly large number of works connected directly or indirectly with the study of the properties of mesoatoms. The study of mesoatoms is also of interest in itself, since in this field we encounter not only a repetition, on other scales, of the properties of the hydrogen atom, but also a number of essential differences. The discovery of new types of mesons makes the whole problem as a whole still more topical and in need of further detailed development.

The general considerations of the present review apply, as a rule, to negative mesons of any type. More detailed remarks, especially those connected with numerical estimates, apply only to \(\mu\)- and \(\pi\)-mesons, unless otherwise expressly stated.

§ 2. MECHANISM OF MESON CAPTURE BY THE COULOMB FIELD OF NUCLEI AND THE FORMATION OF MESOATOMS

The theory of this phenomenon was developed in \(^{1}\) and \(^{2}\) (see also \(^{3}\), p. 257) for the capture of a free electron by a proton, i.e. by a heavy Coulomb center of charge \(e\). The capture is assumed to be radiative, i.e. accompanied by the emission of a photon. The transition

to the case of meson capture that interests us is accomplished by a simple replacement of the electron mass \(m\) by the meson mass \(M\) and of the charge \(e\) by the nuclear charge \(Ze\)*).

If the kinetic energy of the meson \(E\) is much greater than the binding energy of the mesoatom

\[ W=\frac{MZ^{2}e^{4}}{2\hbar^{2}}\simeq 3000\cdot Z^{2}\ \text{eV}, \tag{1} \]

then capture is possible only into levels with zero orbital angular momentum \((l=0)\), and the effective capture cross section onto a shell with principal quantum number \(n\) is equal to

\[ \sigma_n=\frac{2^{7}\pi Z\hbar e^{2}\lambda^{5}}{3M^{2}c^{3}n^{3}}, \tag{2} \]

where \(\lambda\) is the meson wavelength, expressed in units of the mean radius of the mesoatomic \(K\)-shell,

\[ r=\frac{\hbar^{2}}{MZe^{2}}\simeq \frac{2\cdot 10^{-11}}{Z}\ \text{cm}. \tag{3} \]

The probability of capture rapidly increases as the meson velocity decreases, remaining, however, very small in absolute magnitude throughout the entire range of applicability of relation (2). Even at \(E\sim W\) (i.e., at \(\lambda\sim 1\)) \(\sigma\) is, in order of magnitude, equal to the nuclear geometrical cross section. On the other hand, the ionization range of a meson possessing energy \(E\sim W\) amounts to only a negligible part of the nuclear range. One may therefore conclude that the overwhelming majority of mesons are slowed down (without being captured), at least to such velocities that ionization already becomes impossible. The process of slowing down does not end there, since further energy losses may be caused by other processes (elastic collisions with atoms, etc.). Estimates given in a number of works (see, for example, \(^{4}\) and \(^{5}\)) show that in dense substances mesons have time to slow down to velocities considerably smaller than the velocities of orbital electrons, and only then are captured by the Coulomb field of nuclei. According to these estimates, the slowing-down time from the moment ionization ceases until the formation of a mesoatom is of order \(10^{-13}\) sec.**)

*) In the capture of mesons a conversion mechanism is also possible, involving the emission not of a photon, but of one of the electrons belonging to the atomic electron shell (see § 5). This circumstance practically does not affect the estimates made below.

**) A very crude experimental estimate of the slowing-down time follows from the fact that the decay of stopped negative \(\pi\)-mesons has never been observed. Since the lifetime of \(\pi\)-mesons is \(\sim 2\cdot 10^{-8}\) sec, it may be concluded that the slowing-down time is in any case less than \(10^{-10}\) sec.

MESOATOMS

The capture of mesons thus occurs at small kinetic energies, when \(E \ll W\). In papers \(^{1}\) and \(^{2}\) it is shown that under these conditions capture at levels with large values of the orbital angular momentum \(l\) becomes possible and very probable. Unfortunately, here the complete analogy with the problem of electron capture by a proton is already violated, since the electron shell may play an essential (and, for different atoms, substantially different) role, because the binding energy of electrons in atoms is of the same order as, or even greater than, the kinetic energy of the mesons.

It should also be borne in mind that, when considering the capture of mesons, the nucleus can no longer be regarded as a point Coulomb center. The correction associated with this circumstance must also depend on \(Z\), since, for example, for \(Z=1\) the radius of the mesoatomic \(K\)-shell is still a hundred times greater than the nuclear radius, whereas for \(Z \simeq 30\) the two radii already have approximately the same magnitude. It is therefore possible that the ratio between the probabilities of capture into different mesoatomic shells depends essentially on \(Z\).

Many experiments are arranged so that the slowing down of mesons takes place in a medium containing nuclei of various elements. There therefore arises the natural question of the relation between the probabilities of Coulomb capture of mesons by nuclei with one or another value of \(Z\). Here there may be two essentially different cases: capture in a chemical compound and capture in a mechanical mixture.

For chemical compounds the probabilities of Coulomb capture of mesons by the nuclei of each of the components are proportional to the corresponding atomic number \(Z\) (see \(^{5}\)). An exception is capture in compounds containing hydrogen isotopes, for example capture in paraffin \((\mathrm{CH}_2)\). The meson, together with the hydrogen nucleus that has captured it, forms a neutral system whose dimensions are approximately 250 times smaller than those of the hydrogen atom.

The system under consideration (mesoproton) diffuses inside the substance and, owing to the absence of an electron shell bound to it, can come very close to a carbon nucleus, which “intercepts” the meson. As a result of such a process, the meson initially captured by the proton ultimately ends up on one of the mesoatomic shells of carbon or of some other nucleus forming part of the chemical compound. The experimental existence of the “interception” effect has been established with great reliability (see \(^{6-12}\)*).

* Nuclear capture of stopped negative \(\pi\)-mesons in hydrogen and deuterium is accompanied, as is known, by the formation of \(\gamma\)-rays with energy of the order of 100 MeV, whereas capture by all heavier nuclei is associated with nuclear disintegrations and is not accompanied by \(\gamma\)-radiation of high energy. Capture in paraffin \((\mathrm{CH}_2)\) and LiH also does not lead to the formation of \(\gamma\)-rays, which testifies to the presence of “interception.”

Let us now consider the capture of mesons in mechanical mixtures. In practice, perhaps the only—but very important—example so far is capture in a photographic emulsion, which is a mechanical mixture of gelatin and AgBr grains, whose dimensions are of the order of tenths of a micron.

It has already been noted above that from the moment ionization ceases to the moment of capture no more than \(10^{-13}\) sec elapses. Even if one assumes that during all this time the meson moves rectilinearly with a velocity close to the orbital velocity of the outer electrons (\(10^8\ \mathrm{cm/sec}\)), even then it can be displaced by only \(10^{-5}\ \mathrm{cm}\), i.e. by less than the length of an AgBr crystal. Since in fact the velocity of the meson is decreasing all the time and the motion of the meson has a diffusive character (this is the main point), the true displacement must be many times smaller. An analogous (in any case, not contradictory) result can also be obtained from the diffusion estimates carried out in \(^{13}\).

It should therefore be considered that capture of a meson occurs precisely in that part of the mixture in which the given meson ceases to produce ionization. From this, in turn, it follows that the ratio of the capture probabilities is determined by the ratio of the specific ionization losses at small velocities in the different components of the mechanical mixture*). Such a conclusion is in approximate agreement with the results of experiments on the observation of decay electrons when negative \(\mu\)-mesons are stopped in photographic emulsion.

§ 3. PROPERTIES OF THE MESOATOM. NUCLEAR CAPTURE AND DECAY OF NEGATIVE MESONS

The structure of a mesoatom is determined mainly by the Coulomb interaction. The presence of a specific nuclear interaction in most cases has no appreciable effect. Indeed, the nuclear interaction of \(\pi\)-mesons with nucleons occurs, in the respect now of interest to us, in approximately the same way as the interaction of electric charges with the electromagnetic field: nucleons are able to absorb or emit \(\pi\)-mesons, but not to attract or repel them**). The possible energy levels of a mesoatom are therefore determined by the ordinary Coulomb field, and the presence of the nuclear interaction leads only to the fact that the meson may, with one probability or another, be captured by the nucleus. As a result there is a certain (sometimes very large) broadening of all levels, including the ground level of the mesoatom.

*) Similar considerations are given in \(^{14,15}\).

**) In the case of \(\mu\)-mesons the nuclear interaction is in general negligibly small in comparison with the Coulomb interaction (see, for example, \(^{16}\)).

Let us consider certain features in the structure of a mesoatom connected with the influence of the finite dimensions of the nucleus. It is known that even in the case of an ordinary electron shell, a correct understanding of the nature of the isotopic displacement of levels is possible only when the finite dimensions of the nucleus are taken into account. Such an allowance is all the more necessary in analyzing the properties of mesoatoms, whose dimensions are approximately \(250Z\) times smaller than the dimensions of the hydrogen atom. Under these conditions the nucleus must, generally speaking, be regarded not as a point Coulomb center, but as an aggregate of charges distributed continuously and, to a first approximation, uniformly inside a sphere of radius

\[ R \sim 1.4 \times 10^{-13} A^{1/3}\ \text{cm}. \]

For levels with large principal quantum numbers, especially at small atomic numbers \(Z\), the radii of the mesoatomic shells nevertheless turn out to be several tens of times larger than the nuclear radius. In these cases the influence of the finite dimensions of the nucleus should be regarded as a correction leading to a rather strongly expressed displacement of levels of the isotope-shift type. The magnitude of the displacement \(\Delta W\) can be simply calculated by applying perturbation theory. For a level with quantum numbers \(n\) and \(l\) the displacement is

\[ \Delta W_{nl} = \int \psi^*_{nl}(r)[\Delta U(r)]\psi_{nl}(r)\,d\tau, \tag{4} \]

where \(\psi_{nl}(r)\) is the unperturbed wave function of the state under consideration, and \(\Delta U(r)\) is the difference between the Coulomb potential of the point charge \(Ze\) and the potential of the distributed charges actually present.

Since \(\Delta U(r) \ne 0\) only inside the nucleus, the magnitude of the correction \(\Delta W_{nl}\) depends essentially on the behavior of the wave function \(\psi_{nl}(r)\) near \(r=0\). In particular, if \(l \ne 0\), then \(\psi_{nl}(0)=0\), and the first-approximation correction is close to zero, i.e. the displacement is practically absent. If, on the contrary, the orbital angular momentum \(l=0\), then the unperturbed wave function is finite at zero, which leads to a noticeable displacement of the levels. For example, the \(2S\) level turns out to lie above the \(2P\) level by an amount of the order of tens or hundreds of electron volts even in the case of small \(Z\).

Already for \(Z \simeq 15 \div 20\) the correction \(\Delta W\) becomes approximately equal to the quantity \(W\) itself, and formula (4) ceases to be valid. This is not surprising, since for \(Z \sim 30\) the radius of the mesoatomic \(K\)-shell turns out to be equal to the nuclear radius, and at the end of the periodic system the radius of the \(K\)-shell, calculated in accordance with (3), is several times smaller than the nuclear radius. It follows from this that in the case of large \(Z\) and small \(n\) the model of a point Coulomb center cannot be used even as a zero-order approximation.

A considerably closer approximation to the truth is provided by the modified Thomas model, according to which the negative meson

moves inside a uniformly charged positive sphere. The potential \(U(r)\) is in this case proportional to the square of the distance of the meson from the center of the sphere. The behavior of the meson should therefore be close to that of a three-dimensional symmetric oscillator; the energy levels, in particular, should be arranged almost equidistantly. It should, of course, be borne in mind that Thomson’s model is only a crude approximation to reality. Deviations from it are quite large even for the ground level and large \(Z\). A more detailed consideration of this question has been carried out in \(^{16}\).

Some information about the structure of the mesoatom can be obtained from comparison of experimental data on nuclear capture and decay of negative \(\mu\)-mesons stopping in various substances. Undisputed theoretical estimates (see §§ 4, 5) show that the times of mesoatomic transitions are exceedingly small in comparison with the “decay” lifetime of \(\mu\)-mesons \((\tau_p = 2.2\cdot 10^{-6}\ \mathrm{sec})\). Therefore, if a negative \(\mu\)-meson initially reaches even one of the outer shells, in the end it always finds itself on the \(K\)-shell of mesoatoms, after which it is either captured by the nucleus or decays, both processes proceeding independently of one another. It may therefore be asserted that the lifetime of the \(\mu\)-meson \(\tau\) satisfies the relation

\[ \frac{1}{\tau}=\frac{1}{\tau_p}+\frac{1}{\tau_{\mathrm{capt}}}, \]

and the fraction of mesons that have undergone decay is

\[ \alpha=\frac{\tau_{\mathrm{capt}}^{*)}}{\tau_p}. \]

Measurement of the lifetime \(\tau\), or of the fraction of decaying (or captured) mesons, therefore makes it possible to determine the probability of nuclear capture per unit time \((1/\tau_{\mathrm{capt}})\) as a function of the atomic number \(Z\), and then to compare it with the theoretically expected dependence, whose form is determined by the structure of the ground state of the mesoatom.

Let us first consider the case of light nuclei, whose radius is much smaller than the radius of the mesoatomic \(K\)-shell. Capture of negative \(\mu\)-mesons occurs, as is known, according to the scheme \(\mu^-+p\to n+\) neutrino (see, for example, \(^{18}\)) and takes place only on protons. The probability of capture is therefore proportional to the number of protons in the nucleus, i.e., to the atomic number \(Z\). The probability of capture is, in addition, proportional to the square of the wave function of the \(\mu\)-meson near the nucleus, i.e., to the quantity \(|\psi_{10}(0)|^2\). Since the radius of the \(K\)-shell varies—

\[ \text{*) The question of the relation between the probabilities of decay and capture of stopped negative \(\mu\)-mesons is considered in great detail in the review article }^{17}. \]

varies with atomic number as \(1/Z\), then \(|\psi_{1c}(0)|^2 \sim Z^3\), whence it follows that the sought capture probability is

\[ \frac{1}{\tau_{\mathrm{cap}}} \sim Z \cdot Z^3 \sim Z^4{}^{*}). \tag{5} \]

What has been said is valid only in the limiting case of small \(Z\). In the opposite limiting case of very large \(Z\), the \(K\)-shell of the mesoatom is located entirely inside the nucleus. The mean radius, like all the other characteristics of the meson wave function, is then determined only by the density of the distribution of the positive charge of the nucleus. Since the latter is, in a first approximation, the same for all heavy nuclei, the capture probability also should not depend on \(Z\).

Summing up, one may say that the capture probability in the region of light nuclei should increase as \(Z^4\), after which the dependence becomes weaker and the corresponding curve ultimately reaches a “plateau.” It should, however, be stipulated that this last assertion is valid only in a limit that is practically not realized; in reality even the heaviest of the actually existing nuclei are not on the “plateau,” but in the transition region. Exact values of the probabilities of capture of negative \(\mu\)-mesons by various nuclei, calculated under the assumption of a uniform distribution of electric charge inside the nucleus, are given in \(^{16}\).

In the experiments carried out in this connection, either the decay of \(\mu\)-mesons (in the case of small \(Z\)) or nuclear capture (elements of the middle and end of the periodic system) is investigated. The measurements use delayed-coincidence circuits containing ordinary counters, if decay electrons are recorded, or scintillation counters, recording neutrons (and, perhaps, photons) accompanying the nuclear capture of \(\mu\)-mesons. The results indicate good quantitative agreement with the theory in the region of light and medium nuclei, up to copper (see \(^{20,21}\) and the reviews \(^{22}\) and \(^{17}\)). For heavy nuclei the experimentally measured capture probability turns out to be \(2\div 3\) times smaller than that calculated in \(^{16}\). This is not surprising, since the calculations are based on an extremely simplified model, which does not take into account even such phenomenological factors as the nonuniformity of the electric-charge distribution in the nucleus and the decrease in the ratio of the number of protons to the number of neutrons as the atomic number \(Z\) increases. The more exact calculations carried out in \(^{23}\) are in good agreement with experiment also in the region of heavy nuclei, and not only from the point of view of the general course of the phenomenon, but also in the details. The theory explains, for example,

\({}^{*}\) In the capture of negative \(\pi\)-mesons, in general, the same dependence on \(Z\) is expected (see, for example, \(^{19}\)).

a relatively very small (20–30%) difference in the probabilities of nuclear capture in Hg \((Z=80)\) and Pb \((Z=82)\), connected with the presence of nuclear shells (see \(^{21-23}\)). Meson capture appears here as a means for studying the structure of the nucleus.

It was shown above that the presence of the mesoatomic stage leads to the possibility of nuclear capture of negative mesons and determines the relation between the probabilities of capture and decay. As for the decay mechanism itself, in the main it remains the same as in the decay of positive mesons.

It should be emphasized, however, that some, sometimes quite important details may change substantially. The point is that the decaying meson is located on the \(K\)-shell of the mesoatom, i.e., it is not at rest but moves relative to the laboratory coordinate system. Consequently, owing to the Doppler effect, the kinematic characteristics of the decay change; for example, the lifetime of the meson and the energy of the decay products change \(^{24}\). Generally speaking, effects of this kind are small, at any rate for light nuclei, since the binding energies of various mesoatoms are usually small in comparison with the mesons’ own energies \((Mc^2)\) and with the kinetic energies of the decay particles. Nevertheless, in some cases they must be taken into account.

Let us consider, for example, the possible decay of a negative \(\tau\)-meson into three \(\pi\)-mesons*). The most characteristic feature of the decay of the \(\tau\)-meson, which was even taken as the basis for its identification, is the coplanarity of the tracks of the decay products, which is a consequence of the total momentum being equal to zero (decay of a stopped meson). This is certainly true if the decay of positive \(\tau\)-mesons is involved. But for negative \(\tau\)-mesons the total momentum is not equal to zero, and coplanarity may be violated. Elementary estimates, based on comparison of the magnitude of the momentum of the \(\tau\)-meson at the moment of decay and the momenta of the resulting \(\pi\)-mesons, show that even in the case of mesocarbon (the nucleus C \(+\) negative \(\tau\)-meson), the mean angle of noncoplanarity (i.e., the angle between the trajectory of one of the \(\pi\)-mesons and the plane formed by the trajectories of the other two \(\pi\)-mesons) is \(3\text{–}6^\circ\).

For mesonitrogen and mesooxygen the noncoplanarity is expressed still more strongly, while for mesobromine and mesosilver any traces of coplanarity disappear altogether**). The accuracy with which it can be

* ) At present it is not known whether such a process actually occurs, since in all cases of \(\tau\)-meson decay in photographic emulsion its sign could not be determined. On \(\tau\)-mesons see, for example, \(^{25}\).

** ) For a mesoproton, noncoplanarity practically cannot be observed. However, the formation inside an emulsion of a mesoproton is accompanied, as was noted above, by the “capture” of the meson by one of the heavier emulsion nuclei (C, N, O). Moreover, the formation of a mesoproton itself occurs very rarely, since the probability of Coulomb capture of a meson is proportional to \(Z\).

complanarity in the decay of the τ-meson in a photoemulsion has been established to be \(1 \div 2^\circ\). It may therefore be asserted that a significant fraction of the decay cases of negative τ-mesons (if such cases occur at all) do not satisfy the selection criterion adopted at present, and that the search for these cases presupposes abandoning the requirement of complanarity.

§ 4. RADIATIVE TRANSITIONS

The capture of a slow meson into one of the mesoatomic levels and the subsequent transitions to lower levels are accompanied by the emission of relatively hard photons. In the region where the nuclear dimensions are still small in comparison with the dimensions of the mesoatomic shells, the system of terms is analogous to the system of terms of hydrogen-like atoms, i.e.

\[ W_n=\frac{MZ^2 e^4}{2\hbar^2 n^2}, \tag{6} \]

where \(n\) is the principal quantum number. As is seen from (6), for \(Z\sim 10\) transitions between the lower levels are associated with the emission of photons whose energy amounts to hundreds of kiloelectronvolts, while for \(Z\sim 50\) the energy increases to several million electronvolts. Allowance for the finite dimensions of the nuclei does not change the order of magnitude of the energy of the photons under consideration.

The probabilities of radiative transitions are determined by the same expressions as the probabilities of the corresponding optical transitions in the hydrogen atom. For example, the lifetime for the transition from the \(2P\) level to the ground level \(1S\) is

\[ \tau=0.16\,\frac{m}{MZ^4}\cdot 10^{-8}\ \text{sec} \; *), \tag{7} \]

For transitions between higher levels the lifetime increases, remaining, however, in all cases negligibly small in comparison with the “decay” lifetime of all presently known kinds of charged mesons.

As for the selection rules, they are completely analogous to the corresponding optical rules for electric dipole transitions. In other words, only such transitions are possible for which

\[ \Delta l=\pm 1 \quad \text{and} \quad \Delta m=0,\ \pm 1, \]

where \(m\) is the magnetic quantum number **). As an example we shall indicate—

*) See \(^{26}\), p. 238.

**) We do not consider here the relativistic selection rules, since relativistic effects (fine structure of the mesoatomic levels, etc.) play no significant role in the cases of interest to us here. We note only that in this respect the behavior of π-mesons (particles with spin zero) should differ from the behavior of μ-mesons (half-integer spin). Very interesting considerations on the influence of vacuum polarization on the fine structure of mesoatoms are developed in \(^{27}\).

therefore, the radiative one-photon transition from the \(2S\) level is forbidden, since the angular momentum of the final \(1S\) state is also equal to zero.

Each meson captured by the Coulomb field of the nucleus sooner or later reaches the lower shells of the mesoatom (\(K\) or \(L\)). Therefore the stopping of a negative meson in matter with a large atomic number \(Z\) must be accompanied by the emission of one or two \(\gamma\)-quanta with energy \(W \sim 1 \div 5\) MeV, which can be detected through the formation of Compton electrons or electron–positron pairs. What has been said is confirmed by the results of experimental works \(^{28}\) and \(^{29}\). In both cases the apparatus consisted of a Wilson chamber containing thin plates of lead (or iron) intended for stopping cosmic-ray \(\mu\)-mesons. It was found that, for a total thickness of the lead plates of about \(1 \div 2\) cm, approximately \(30\%\) of the stops of negative \(\mu\)-mesons are accompanied by the formation of Compton electrons or pairs with energies of several megaelectronvolts. Such a result agrees in order of magnitude with the expected one.

A similar conclusion was also obtained in work \(^{30}\), in which registration of the \(\gamma\)-quanta accompanying the stopping of negative \(\mu\)-mesons in lead was carried out with the aid of a scintillation counter, and also in work \(^{31}\). It is possible, however, that some part of the observed \(\gamma\)-quanta is produced not in mesoatomic transitions, but through the de-excitation of nuclei excited in the nuclear capture of \(\mu\)-mesons.

Considerably more reliable and unambiguous results have been obtained in investigations, with the aid of scintillation counters, of the stopping of negative mesons in light substances (see \(^{32-34}\)). In work \(^{32}\) the radiative transition \(2P \to 1S\) was studied during the stopping of cosmic-ray \(\mu^{-}\)-mesons in carbon; moreover, not only was the very fact of emission of the corresponding photon registered, but its energy was also measured. The latter turned out to be approximately \(80\) keV, which agrees excellently with the theoretical prediction (\(77\) keV). The quantum yield, i.e. the number of photons of interest to us per one act of \(\mu\)-meson capture, is close to unity. Similar results were obtained in work \(^{33}\), devoted to the investigation of photons associated with the radiative transitions \(2P - 1S\) during the stopping of negative \(\pi\)-mesons in Be, C, and \(\mathrm{H_2O}\) (i.e. actually in oxygen). The essential difference is that the quantum yield proved to be considerably smaller, namely \(0.13 \pm 0.03\) for carbon and \(0.21 \pm 0.07\) for oxygen. The decrease in the value of the quantum yield means that, in contrast to the \(\mu\)-meson, a \(\pi\)-meson located at the \(2P\) level can undergo not only a radiative transition to the \(1S\) level, but also direct nuclear capture.

From a comparison of the experimental data with the known lifetime (7) for the radiative transition it follows that the life-

ni with respect to nuclear capture from the \(2P\) level is approximately \(3 \cdot 10^{-16}\) sec. in the case of oxygen and \(5.7 \cdot 10^{-16}\) sec. in the case of carbon. This agrees in order of magnitude with the theoretical estimates that can be obtained on the basis of a number of works (see, for example, \(^{19}\)).

Let us also note that, from the point of view of the simplest theory considered above (the nucleus—a uniformly charged sphere, etc.; see \(^{16}\)), one should expect that the quantum yield for carbon should be, contrary to the experimental data, greater than for oxygen. The authors\(^{33}\) attribute this discrepancy to peculiarities of the structure of the oxygen nucleus (closed nuclear shells). We have already encountered an analogous situation above when comparing the probabilities of nuclear capture of negative \(\mu\)-mesons in lead and mercury.

§ 5. CONVERSION TRANSITIONS\(^*\)

When negative \(\pi\)- and \(\mu\)-mesons are stopped in a photographic emulsion, in approximately \(20 \div 30\%\) of cases one observes tracks of slow electrons beginning at the point where the meson track ends (see \(^{15}\) and \(^{35—41}\)\(^ {**}\)). The energy of these electrons, determined rather roughly from their range in the emulsion, lies in the interval from \(10 \div 20\) kev to \(100 \div 150\) kev. Electrons with energies of several hundred kiloelectron-volts are also encountered occasionally. The sharply expressed lower boundary \((E \sim 15 \div 20\ \text{kev})\) is apparently due to purely instrumental causes. Indeed, at \(E \sim 10 \div 15\ \text{kev}\) the length of the electron track in a photographic emulsion is approximately \(2 \div 3\) microns, which makes reliable identification difficult.

At still lower energies identification is practically already impossible. The form of the energy spectrum (a rapid and monotonic decrease with increasing energy) is approximately the same both in the case of \(\mu^-\)-meson capture and in the capture of \(\pi^-\)-mesons. In some, rather rare, cases the formation of two or three, and sometimes even a larger number of slow electrons is observed, their tracks originating from one point.

The most accurate and detailed data (see Table I) were obtained in work \(^{41}\), in which 1000 stoppings of negative \(\mu\)-mesons in Ilford G-5 emulsion were investigated.

It is well known that Coulomb capture of a negative \(\mu\)-meson in gelatin is, as a rule (with probability \(\sim 90\%\)), accompanied by decay, whereas mesons captured in AgBr,

\(^*\) In principle, besides radiative and conversion transitions, some other types of transitions are also possible, occurring, however, with very small probability (see \(^{16}\)).

\(^ {**}\) In \(B^{42}\) one such case was observed when a negative \(\mu\)-meson was stopped in gas inside a Wilson chamber.

Table I

358 captures in gelatin

610 captures, occurring mainly in AgBr

practically never decay (see, for example, \(^{17}\)). The presence or absence of a relativistic decay electron is therefore a sign indicating, with a high degree of reliability, exactly where (in gelatin or in AgBr) the stopping of a negative μ-meson took place. The simultaneous appearance of tracks of a relativistic and a slow electron signifies the formation of a slow electron when the μ-meson is stopped inside the gelatin, and moreover in a stopping not accompanied by nuclear capture. It follows from Table I that the probability of such a process is approximately 5%. At the same time, the probability of formation of a slow electron when a \(\mu^{-}\)-meson is stopped in AgBr is equal to

\[ \frac{255}{610-\frac{0.1}{358}}=42\%, \]

i.e., many times greater.

The slow electrons under consideration apparently appear as a result of internal conversion associated with mesoatomic transitions. The phenomenon is entirely analogous in its nature to the well-known internal conversion accompanying radiative nuclear transitions.

Indeed, the radii of mesoatomic shells are approximately 250 times smaller than the radii of the corresponding electronic shells. Therefore the mesoatom formed after the capture of a negative meson plays, with respect to the electron shell, the role of a “nucleus” with atom-

number, one less than the atomic number of the original nucleus*).

If the meson is not in the \(K\)-shell of the mesoatom, then the “nucleus” proves to be in an excited state and passes to its ground state (i.e., to the ground state of the mesoatom) by means of one or several successive transitions, mainly radiative ones.

In the case where the transition energy \(W\) exceeds the binding energy \(I\) of the electronic shell, the transition can, as is known, take place by emission of one of the \(K\)-electrons, whose kinetic energy is

\[ E = W - I^{**}. \tag{8} \]

An analogous phenomenon may, of course, occur not only in transitions between internal levels of the mesoatom, but also in the very formation of the mesoatom, i.e., in the initial capture of a slow free meson by the Coulomb field of the nucleus. In the presence of several successive mesoatomic transitions, the formation of several slow electrons is also possible (see Table 1).

A systematic study of conversion electrons may prove useful for a more detailed clarification of the question of the structure of the mesoatom and of the mechanism of capture of negative mesons. It is necessary only to make sure that the appearance of slow electrons is indeed due to internal conversion in mesoatomic transitions, and not to any other causes.

Nuclear capture of negative mesons is accompanied by excitation of the corresponding nuclei, which can then de-excite by emitting \(\gamma\)-quanta. Consequently, the formation of conversion electrons is also possible. In practice, the process under consideration is unlikely to play a noticeable role in the formation of slow electrons. It is known, for example, that nuclear disintegrations in a photoemulsion, caused by nuclear-active cosmic particles of comparatively small energy, are very rarely accompanied by the appearance of electrons \(^{43}\). It is not excluded, it is true, that the excitation of the nucleus in this case has quite a different character than in meson capture. On the other hand, however, the excitation of nuclei in the capture of \(\pi\)- and \(\mu\)-mesons also proceeds in a completely different way, but this has no noticeable effect on the number and energy distribution of slow electrons.

*) The electronic shell in this case is rapidly “rearranged,” losing at least one electron. If the initial atomic number is equal to one, then after formation of the mesoatom the electronic shell is absent altogether.

**) Internal conversion with the participation of electrons of the \(L\)-shell or other higher shells is also not excluded (see below). However, conversion in the \(K\)-shell usually plays the main role, provided only that it is energetically possible.

It should therefore be concluded that the appearance of slow electrons is not connected, at least to any significant extent, with nuclear transitions.

A decisive argument, at least for Coulomb capture of mesons in light substances, is also the appearance of slow electrons simultaneously with relativistic decay electrons, i.e., in processes not connected at all with excitation of nuclei.

Let us consider one more possible cause for the appearance of slow electrons. It was already noted above that the formation of a mesoatom is accompanied by a rearrangement of the electron shell because of the change of the effective nuclear charge by one unit. If the change of charge takes place sufficiently rapidly (in comparison with the time of revolution of the electrons), then there occurs a kind of “shaking” of the electron shell, accompanied in some cases by ionization of the atom, i.e., by the appearance of slow electrons. In general outline the phenomenon is completely analogous to the “shaking” of the electron shell in $\beta$-decay, considered in works$^{44}$ and $^{45}$.

The energy of the free electrons formed is, in order of magnitude, equal to the binding energy of the electron shell from which the corresponding electron has been torn out. Therefore, in a photoemulsion one can observe only the result of “shaking” of the $K$-shells of Ag and Br (binding energy of order $10 \div 20$ keV). However, the probability of ionization of the $K$-shell

\[ p=\frac{0.64}{Z^2}, \]

i.e., is negligibly small in comparison with the probability of the actual appearance of slow electrons.

It follows from what has been said that the slow electrons observed when negative mesons are stopped in a photoemulsion should be associated with internal conversion due to mesoatomic transitions.

The theory of internal conversion has been considered by many authors; a detailed review is contained in $^{46}$. The ratio of the probability of conversion to the probability of photon emission $(\omega)$ depends, as is known, on the type of transition. For the case of electric dipole transitions that interests us,

\[ \omega = 8\pi\alpha\left(\frac{\alpha Z}{W}\right)^4 \frac{e^{-4n\,\operatorname{arcctg} n}}{1-e^{-2\pi n}}, \qquad n=\frac{\alpha Z}{\sqrt{2W-\alpha^2 Z^2}}, \tag{9} \]

where $\alpha=\dfrac{e^2}{\hbar c}\sim\dfrac{1}{137}$, and $W$ is the transition energy, expressed in units of $mc^2$*).

* If conversion occurs through mesoatomic transitions, then in (9) $Z-1$ should be substituted instead of $Z$, since the effective charge of the nucleus is reduced by one unit.

The probability of conversion falls rapidly as the transition energy increases, which qualitatively agrees well with the general form of the energy spectrum of the observed slow electrons*). The number of slow electrons also agrees in order of magnitude with that expected (see \(^{47}\)).

Of course, exact quantitative agreement cannot be expected, since one must also take into account the presently unknown relations between the probabilities of Coulomb capture of mesons onto different levels of the mesoatom for different photoemulsion nuclei. On the contrary, it has already been pointed out above that it is expedient to use experimental data on conversion electrons for a more detailed elucidation of the mechanism of Coulomb capture of mesons.

Let us compare, in this connection, the formation of conversion electrons in atoms of gelatin**) and in \(AgBr\). It follows from (6) that for small \(Z\) (gelatin) only a few mesoatomic transitions have energies in the interval \((2 \div 10)\cdot 10^4\) ev, whereas for large \(Z\) (\(AgBr\)) there are considerably more such transitions. In addition, at a fixed transition energy the probability of conversion increases very rapidly with increasing \(Z\) (see (9)). One may therefore expect that the main part of the conversion electrons is formed when mesons are captured by the Coulomb field of \(Ag\) and \(Br\) nuclei.

This conclusion is in full agreement with the data presented in Table I, according to which the probability of conversion upon capture of a \(\mu\)-meson in \(AgBr\) is approximately 10 times greater than upon capture in gelatin.

What has been said of course also applies to the Coulomb capture of negative \(\pi\)-mesons. In a carefully performed work \(^{39}\) it was shown that, in the Coulomb capture of \(\pi^{-}\)-mesons, the observed conversion electrons are associated with the formation of one-prong, but not many-prong, “stars.” One may therefore conclude that many-prong “stars” arise mainly as a result of nuclear absorption of \(\pi^{-}\)-mesons in gelatin and that, on the contrary, nuclear capture in \(AgBr\) leads chiefly to the formation of one-prong and, apparently, radiationless “stars.” This conclusion is also confirmed by other, co-

*) In accordance with (8), the energy spectrum of the conversion electrons should be linear. However, in practice the linear structure cannot be detected because of the proximity of the lines to one another (the complex composition of the photoemulsion) and the poor resolving power of the adopted method of measuring electron energies (by range).

**) We note, incidentally, that from the fact of the simultaneous appearance of a slow electron and a relativistic decay electron in the Coulomb capture of a \(\mu^{-}\)-meson in gelatin it necessarily follows that \(\mu^{-}\)-mesons manage to decay before nuclear absorption, i.e. they interact only very weakly with nuclei. This important proposition was established in its time from quite different considerations (see, for example, \(^{17}\)).

with completely independent considerations (see, for example, ^38 and ^48, and also ^41).

Thus, conversion electrons are formed mainly upon the capture of mesons in silver and bromine, possessing in this case an energy of several tens of kiloelectronvolts. Such an energy corresponds to mesoatomic transitions between states with rather large principal quantum numbers, \(n=5 \div 10\), from which it follows that the initial capture of mesons occurs chiefly into high levels of the mesoatom.

In this connection one may also expect that, when heavier negative mesons are stopped, conversion electrons should be observed more often than when \(\mu\)- and \(\pi\)-mesons are stopped, since the greater the mass of the meson, the larger the number of mesoatomic transitions falling in the energy region of interest to us \((10 \div 100\ \text{keV})\).

At first sight it seems that, for small \(Z\), the assumption of meson capture into levels with large quantum numbers is in contradiction with the experimental results, for in the subsequent radiative transitions a fairly considerable fraction of the mesons should have found themselves on the \(2S\) level (see ^26, p. 238), which, as is known, is metastable, since the one-photon radiative transition \(2S—1S\) is forbidden. Therefore the transitions \(2S \to 1S\) should have occurred by emission of conversion electrons of rather high energy, which, as was shown above, are in fact observed only rarely.

True, we have already seen above (see § 3) that the influence of the finite dimensions of the nucleus leads to an “isotopic shift” of the \(2S\) level, which turns out to be situated above the \(2P\) level, so that the radiative transition \(2S \to 2P\) becomes possible, with the subsequent radiative transition \(2P \to 1S\). However, for light nuclei \((Z<10)\) the probability of the radiative transition \(2S \to 2P\) proves to be negligibly small in comparison with the probability of the conversion transition \(2S \to 1S\) (see ^16).

The indicated difficulty was considered and resolved in work ^49. The difference of the energy terms \(2S\) and \(2P\), calculated in accordance with (4), even for the smallest \(Z\), exceeds the ionization potential of the outer electron shells of the atom.

Under these conditions the conversion transition \(2S \to 2P\) becomes possible (with conversion on the outer electron shell), accompanied by the subsequent radiative transition \(2P \to 1S\). It also turns out that the probability of the conversion transition \(2S \to 2P\) is many times greater than the probability of the direct conversion transition \(2S \to 1S\). At the same time, the conversion electrons arising in the transition \(2S \to 2P\) possess very small energy (tens of eV) and cannot be observed in a nuclear emulsion.

In conclusion, it should be noted that in the capture of both \(\pi\)- and \(\mu\)-mesons, peculiar condensations consisting of several grains, formed at the very end of the meson track (“blobs”; see, for example, \(^{35}\) and \(^{39}\)), are observed rather often. The nature of these condensations has not yet been studied in detail. There is no doubt, however, that in many cases they are connected with the appearance of very slow conversion electrons, and in other cases with recoil nuclei arising in the nuclear capture of mesons.

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