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DECAY AND NUCLEAR INTERACTIONS OF STOPPED CHARGED MESONS
G. B. Zhdanov
I. INTRODUCTION
The great interest aroused over the last 10 years by studies of cosmic radiation is connected above all with the fact that it was precisely the primary component of this radiation arriving at the Earth that placed at the disposal of experimenters the most powerful natural tool for splitting atomic nuclei and elucidating their properties. This circumstance could be truly appreciated only after the first major successes of nuclear physics. Let us recall that the use of $\alpha$-particles as “natural” projectiles led physicists to the discovery of the neutron in 1932. The hypothesis of Ivanenko[^1], advanced soon thereafter, concerning the proton–neutron structure of the nucleus placed on the agenda the problem of nuclear forces, i.e., of such specific interactions between nuclear particles as must differ essentially from the well-known electromagnetic forces. In 1934 Tamm[^2], and simultaneously Ivanenko[^3], put forward the idea of describing these forces by means of the notions of emission and absorption by nuclear particles, nucleons, of certain other, lighter particles. It was natural to invoke for this purpose the particles then known—the electron and the neutrino—which made it possible, as it were, to connect the problem of nuclear forces directly with the process of $\beta$-decay of nuclei. Although this hypothesis failed because of enormous quantitative discrepancies with experiment, the very idea of such a solution of the problem was subsequently used with success by Yukawa[^4]. Yukawa (1935) proposed the existence of a new particle, described by generalized Maxwell equations and called by him the heavy photon. The mass of the heavy photon $\mu$ could be directly inferred from the radius of action of nuclear forces $r_0$ (with the aid of the relation $\dfrac{1}{r_0} = \dfrac{\mu c}{h}$) and had to be close to 200 electron masses. It was also assumed that $\beta$-decay is the result of the spontaneous decay of the heavy photon into an electron and a neutrino, and then inevitably
it followed conclusions also about other properties of the new particle—an integral charge of one sign or the other and an integral spin, and also about the lifetime (of the order of \(10^{-8}\) sec.) with respect to the process of spontaneous decay.
Already in the following year, 1936, in the cosmic-ray experiments of Anderson and Neddermeyer\(^5\), the first data appeared on the existence of charged particles with a similar mass—particles subsequently called mesotrons, or mesons\(^*\). As regards the spin of the new elementary particle, the situation has remained unclear to the present day; the only possibility so far available for studying the spin from the large secondary showers produced by mesons of high energy permits, as the analysis of Belen’kii\(^7\) has shown, zero and half-integral spin to be regarded as almost equally probable. On the other hand, the property of mesons—discovered after several years—of decaying in air, though with a somewhat longer lifetime than had been expected (of the order of several microseconds), appeared, it would seem, to be a weighty argument in favor of the identity of the heavy photon, or “nuclear meson,” with the meson of cosmic radiation. However, it was possible to verify that the newly discovered particles really have a direct relation to the processes of interaction of nucleons and \(\beta\)-decay of nuclei only after experimental facts had appeared which, first, characterize the degree of interaction of mesons with the nuclear particles of matter and, second, clarify the mechanism of meson decay. All these properties could be investigated with sufficient reliability only for stopped mesons, as will be seen from what follows.
Some information, however, was also obtained from experiments with the penetrating component of cosmic radiation. Let us list all these data, although they do not always agree sufficiently well with one another and not all are directly related to the properties of a definite type of mesons\(^ {**}\):
1) additional or “anomalous” absorption of mesons in air as compared with dense substances, as well as barometric and temperature effects for the hard component, interpreted as the result of spontaneous decay with a lifetime of \(1\)—\(3\) μsec.\(^8\);
2) additional scattering of mesons by the nuclei of matter as compared with the corresponding calculations for purely electromagnetic Rutherford scattering\(^9\);
* The data of 1936 showed only that the mass of the new particle has an intermediate value between the masses of the electron and the proton. The fact that this mass is close to \(200\,m_e\) was first established in the works of other authors\(^6\) in 1937.
** It should be noted here that the penetrating component by no means ought to be identified, as was done earlier, with sufficiently energetic mesons of a definite type. It includes, in particular, protons as well.
3) multiplication in matter of \(10\)—\(12\) penetrating particles, arising in the so-called “special showers” from lead, i.e. showers of an explosive character, consisting of electrons and penetrating \(\alpha\)-particles;
4) a whole series of properties of extensive atmospheric showers that do not fit within the framework of the cascade theory and lead to the hypothesis of a nuclear-cascade origin of these showers with the participation, possibly, of some nuclear-interacting mesons \(^{13—14}\);
5) a break in the power spectrum of the hard component at energies of about \(6\cdot 10^{10}\) eV, which can possibly be explained \(^{34}\) by the origin of ordinary mesons through the decay of short-lived mesons with strong nuclear interaction.
For elucidating the character of the nuclear interaction of mesons, all investigations of the phenomena of meson generation are also of great importance. In particular, the large probability of absorption, assumed in the usual theory of nuclear forces, of stopped mesons by nuclei leads also to the conclusion that the reverse process—direct generation of single mesons by the primary proton component of cosmic radiation—has large effective cross sections. In recent years a large body of factual material has been collected on the generation of various penetrating particles; however, only an insignificant part of all cases is known with certainty to have a direct relation to “ordinary” mesons with mass \(200\,m_e\) and lifetime \(2\) microsec. Here we shall confine ourselves only to a brief enumeration of the principal experimental data:
1) the investigations of Vernov and collaborators \(^{15}\), agreeing with the hypothesis of the generation of penetrating particles with lifetime \(2\) microsec by absorption of the primary proton component of cosmic rays;
2) numerous observations (at altitudes up to \(10\) km) of cases of generation of the above-mentioned “special showers,” also containing penetrating particles \(^{16—18}\);
3) the birth of slow mesons in “stars,” repeatedly observed both in the Wilson chamber \(^{19}\) and by the photographic-plate method \(^{20}\); it has also been established that, in general, a considerable fraction (20—30%) of slow mesons with lifetime \(2\) microsec at medium atmospheric altitudes is generated with energies not higher than 100 MeV \(^{21}\);
4) “artificial” generation of mesons by fast protons and \(\alpha\)-particles obtained with the aid of accelerators \(^{22,23}\).
Unfortunately, in investigations of the phenomena enumerated one could very often say of the observed mesons only that they were particles with a mass intermediate between the electron and the proton*). At the same time, the discoveries of groups of Soviet physicists under
*) The exception is constituted by experiments with accelerators, which have given (see below) in general the most precise of the known determinations of the masses of mesons of two types.
under the direction of Alikhanov and Alikhanian (1946–1947)^[24] showed that this characteristic alone is completely insufficient for an unambiguous identification of particles. By using the method of magnetic analysis in an apparatus of the type shown in Fig. 1a, the authors succeeded in observing a whole set of particles with masses ranging from 30 to 20,000 electron masses^[25–28] (see, for example, Fig. 1b), with both positive and negative charge. At the same time, some other properties of the newly discovered particles, called varitrons, were also studied.
In particular, the following were determined:
1) the total flux of these particles, amounting at an altitude of 3 km to a value of the order of 10% of the flux of the hard component;
2) the range, apparently due mainly to ionization losses and amounting to several hundred meters of air or, correspondingly, several centimeters of Pb for the greater part of the particles;
3) the ionizing power, which proved to be 2–3 times above normal, as was found in the work of Nikitin^[29] with proportional counters and is a consequence of the fairly large mass of the varitrons and their small range.
The question of the origin of varitrons still remains unresolved.
Another group of discoveries is connected, above all, with studies of stopped mesons by the method of thick-layer photographic plates. This method, proposed by Mysovskii^[30] and first applied to mesons by another Soviet physicist—A. P. Zhdanov^[31], has in recent years been improved and has now made it possible to register relativistic particles. In 1947–1948, Occhialini and C. F. Powell with collaborators established^[32] that those very slow mesons which stop in the photoemulsion can be conventionally divided into 4 groups: mesons with a mass of about \(300\,m_e\), decaying at the end of their range into other short-range mesons with a mass of about \(200\,m_e\); mesons with a mass of about \(300\,m_e\), ev—
Fig. 1a. Diagram of one variant of the “mass spectrograph” used at Alagez for investigating the masses of varitrons. Roman numerals designate the individual groups of counters (groups I, II and III together give the so-called “master pulse”).
...arising after the stopping of nuclear disintegrations, and, finally, the last group, consisting of mesons with masses on the average about \(200m_e\), whose track ends in the emulsion without producing secondary particles. As is known, all these mesons were given the names, respectively, \(\pi\)-, \(\mu\)-, \(\sigma\)-, and \(\rho\)-mesons. Further experiments showed that \(\mu\)-mesons decay with the emission of electrons, like the “ordinary” mesons of the hard component; therefore, in what follows, for brevity, we shall call the “ordinary” mesons \(\mu\)-mesons.
Fig. 1, b. Approximate distribution of particles according to their deflections in a magnetic field, obtained with the aid of the apparatus shown in Fig. 1, a. The mass values corresponding to various maxima on the curve of deflections of negative particles with ranges from 3.6 to 5.6 cm of lead are given. (The determination of mass was made from the maximum possible value of the range and the value of the momentum corresponding to the break on the left of the given peak.)
On the basis of the totality of all their data, Occhialini and Powell proposed the following conception\(^{20}\). \(\pi\)- and \(\sigma\)-mesons are identical particles, but with charges of different signs, generated by the neutron–proton component in processes of nuclear disintegration. All \(\mu\)- and \(\rho\)-mesons arise as a result of the decay of \(\pi\)- or \(\sigma\)-mesons and likewise can differ only in the signs of their charges.
Postponing for the time being a more detailed discussion of the properties of the indicated mesons, let us note here only one circumstance. The initial, very inaccurate determinations of the masses of \(\pi\)- and \(\mu\)-mesons, made by the method of counting developed grains along tracks, gave for the ratio ...
masses \(\dfrac{M_\pi}{M_\mu}\)—a value \(1.65\) and, in any case, substantially greater than \(1.4\). If one takes into account that the energy of the \(\mu\)-meson always proves to be close to \(4\ \mathrm{MeV}\), then one must conclude that in the decay only one more neutral particle is emitted; then the application of the laws of conservation of energy and momentum makes it possible to express the mass of this neutral particle \(\mu^0\), in fractions of the mass \(M_\mu\), through the ratio \(k=\dfrac{M_\pi}{M_\mu}\). For \(k=1.65\) and \(M_\mu=200m_e\) one obtained \(\mu^0=100m_e\); for \(k=1.4\), \(\mu^0=60m_e\), i.e., in any case the neutral particle could not be as light as the neutrino.
As will be seen from what follows, the last conclusion proved to be erroneous because of an excessively inaccurate determination of the masses. However, the hypothesis of the existence in cosmic radiation of a neutral meson or neutretto received new support from the analysis of quite different phenomena \(^{33,34}\). In particular, experiments on the investigation of the absorption, in the atmosphere and in dense substances, of the component producing nuclear disintegrations, performed in 1947–1948 by Gorbunov and Chuvilo \(^{33}\), showed that there exist neutral particles which, like charged mesons, undergo considerable additional absorption in air.
Apparently this is explained by their property of spontaneous decay, which, together with the absence of charge, is a characteristic feature of the neutretto.
The last group of experimental data \(^{22,23,35}\) on mesons of a new type, obtained in 1948 with the aid of accelerator methods, was, in essence, a further development of investigations of \(\pi\)-, \(\sigma\)-, \(\mu\)- and \(\rho\)-mesons. Using the same photographic plates as indicators, the investigators in this case had additional advantages: first, rich statistical material, making possible great accuracy in the determination of masses, and, second, the presence of a directed beam of artificial mesons in a strong magnetic field, which makes it possible to separate mesons of both signs. The new method of investigation very quickly made it possible to obtain the following two important results:
1) a more accurate determination of the masses \([M_\pi \approx M_\sigma \approx (285 \pm 5)m_e,\ M_\mu \approx (215 \pm 5)m_e,\ \dfrac{M_\pi}{M_\mu}=1.32 \pm 0.01]\) \(^{36}\) and of the mean lifetime \(^{37}\) of the decay \(\pi \to \mu\) \([\,\tau_\pi=(0.9^{+0.3}_{-0.2})\cdot10^{-8}\ \mathrm{sec.}\,]\);
2) strong evidence in favor of the original concept of Powell–Occhialini concerning the origin and nature of the indicated mesons.
However, it is still premature to regard this concept as proved for cosmic radiation. On the one hand, it is hardly reasonable to extend the conclusion about the two-stage process of generation of \(\pi\)- and \(\mu\)-mesons, made for protons with an energy of \(350\ \mathrm{MeV}\) at pro-
tons with energies of \(10^9\)—\(10^{10}\) eV, characteristic of the primary radiation. In addition, the properties of low-energy \(\pi\)- and \(\mu\)-mesons known to us are entirely insufficient for explaining the above-mentioned high-energy processes observed in extensive air showers and in the so-called “special showers,” and which apparently\({}^{14}\) constitute the processes of generation of the entire hard and soft component of cosmic rays.
II. DECAY OF STOPPED MESONS
For the study of the decay of stopped mesons, three principal methods are currently used: the method of delayed coincidences with counters, the Wilson chamber method with various control systems, and the photographic-plate method. Recently the hodoscope method (“magnetic analysis”) has also been used.
§ 1. Method of delayed coincidences with counters
The idea of the delayed-coincidence method appeared soon after the above-mentioned investigations of the “anomalous” absorption of mesons in air, which showed that the lifetime of the meson must be several microseconds, and it was known both from theoretical premises and from the analysis of photographs in the Wilson chamber (see below) that one of the decay products is an electron with a sufficiently large range.
Fig. 2. Schematic of one of the first installations\({}^{38}\) used for detecting the decay of stopped mesons.
To illustrate the delayed-coincidence method and its features, let us consider one of the first installations of this type\({}^{38}\) (Fig. 2). The idea of the experiment was to register the decay of mesons passing through counters \(I\) and stopping in the lead filter \(d\), by means of the charged particles which fly out of \(d\) with some delay and pass through the group of counters \(II\). The problem was solved with the aid of a special radio circuit for counting coincidences, which in one channel produced a pulse of width \(20\ \mu\text{sec}\), shifted by \(1.5\ \mu\text{sec}\), and in the second channel a narrow pulse not shifted in time. If all counters \(I\) are connected to the first channel, and counters \(II\) to the second, then the triggering
DECAY AND NUCLEAR INTERACTIONS OF MESONS
...of the numerator in the case of overlap of both pulses will mean that the discharge in counters \(II\) was delayed by a time from 1.5 to 21.5 microseconds relative to the discharge in counters \(I\). The experiments of Montgomery and collaborators\(^ {38}\) ended unsuccessfully, for the possible magnitude of the “useful” effect turned out to be approximately 10 times smaller than what should have been expected from an estimate of the number of mesons stopped in the lead filter \(d\). However, this failure excellently illustrates the main difficulties that must be overcome in this method. First of all, it should be taken into account that, notwithstanding the lead block \(A\) (Fig. 2), only a very small fraction (of the order of \(10^{-3}\)) of all particles passing through counters \(I\) and \(II\), from the filtered soft component of cosmic radiation, could correspond to the case of a meson “sticking” in filter \(d\) with subsequent emission of decay products in the required direction. Therefore, a considerable “background” from events of two types inevitably had to be superposed on the sought “useful” effect. First, there were accidental coincidences from the passage through \(I\) and \(II\) of two independent particles. Their number, with the resolving power of the coincidence circuit equal to \(10^{-5}\) sec., exceeded several times the “useful” effect from the decay of stopped mesons. At the same time, the differential method used by the authors for eliminating accidental coincidences by subtracting the effects observed with and without the lead filter \(d\) is not very accurate, since the influence of filter \(d\) on the flux of particles passing through counters \(II\) is not taken into account. Secondly, the “background” was formed at the expense of so-called “false” coincidences caused by delayed discharges in the counters themselves of group \(II\) when one and the same particle passed through counters \(I\) and \(II\) simultaneously. The probability of such a delayed discharge is especially large in counters containing electronegative gases, and in the present case the counters used contained precisely a mixture of argon (94%) with oxygen (6%). Of course, “false” delays are many times smaller for modern counters that do not contain electronegative gases; nevertheless, even up to the present time this effect determines the lower limit (of the order of several tenths of a microsecond) of the meson decay times investigated.
From what has been said it is clear why the meson lifetime \(\tau_0\), amounting to a value of the order of \(10^{-6}\) sec., is the most suitable for investigations by the method of delayed coincidences: for larger values of \(\tau_0\) there would be too many accidental coincidences, while for smaller values of \(\tau_0\) there would be too many “false” delays in the counters.
As for further investigations by the method of delayed coincidences, thanks to improvements it proved possible to make it, for a whole series of problems, not only the most reliable and accurate of all existing methods, but sometimes also completely irreplaceable. First of all, while remaining within the framework of the original idea of coinciding in time two pulses shifted with respect to one another, it proved possible...
significantly shortened, and both random and “false” coincidences can be taken into account more accurately. Let us consider, as an example, one of the recent works in this field, carried out by the present author together with Naumov[^39]. Here the ratio between the “useful” effect and random coincidences could have been made much more favorable by the following changes:
1) selection of the optimal interval of recorded delays ($\Delta t = 1—3$ $\mu$sec);
2) replacement of the lead filter $d$ by a graphite one, in which, as will be set forth below, not only positive but also negative mesons have time to decay;
3) replacement of the single groups of counters $I$ or $II$ by groups of double coincidences (Fig. 3) and the use of such a radio circuit as ensures the actual blocking of it by particles simultaneously passing through all groups of counters (by means of a sufficiently long “dead time” of the channels). Moreover, in itself
Fig. 3. Schematic of an apparatus for recording mesons stopped according to their decay[^43]. The apparatus records coincidences $C_1$ from counters $A$ and $B$, coincidences $C_2$ from counters $C$ and $D$, and then coincidences $C_2$ delayed with respect to $C_1$.
the determination and accounting of the number of random coincidences are made more precise and without the expenditure of any additional time on control experiments with removal of the filters. For this it is sufficient to record on two enumerators, in addition to coincidences of two normally shifted pulses from different branches of the apparatus $C_1$, $C_2$, also coincidences upon an additional shift of one of these pulses by a time of the order of 10 $\mu$sec. In the second case the total width of the pulses, which determines the resolving power of the circuit, and consequently also the number of random coincidences, remains unchanged, whereas the number of recorded decay events decreases many times because of the exponential law of this decay and practically drops out of the calculation.
For an accurate accounting of “false” coincidences one may use the counter configuration shown in Fig. 4, a. It turns out that, with the existing “dead time” of the radio-circuit channels, in this configuration delayed coincidences are caused almost exclusively by the effect of delay of the discharge in the central group of counters when one particle passes through all three groups.
Studies carried out with the aid of such an arrangement showed that the best results can be obtained with counters containing ethylene (mixed with argon) as the filling gas, although the alcohol counters usually used in the work of American authors can already give satisfactory results.
Finally, for a number of investigations using this method it is important to know the actual number of stopped mesons from the number of registered decay electrons, i.e. to determine the efficiency of the apparatus. For this purpose, a quantitative comparison is made between the observed number of delayed coincidences and the absorption curve of the hard component, determined in the same apparatus by the usual “telescope” method, i.e. by the method of non-delayed coincidences.
In order not merely to register the phenomenon of meson decay, but also to follow the law of this decay in time, the method of delayed coincidences was somewhat complicated so as to record, as accurately as possible (with an accuracy of the order of \(0.05\ \mu\mathrm{sec}\)), the time interval between the passage of the primary meson through one system of counters and the passage of the corresponding decay electron through another system of counters. This problem is solved either by photographing the corresponding pulses on one and the same oscilloscope sweep, together with photographing a calibration curve\(^{40}\), or by introducing a special radio-engineering device which, as a result of the superposition of the two indicated pulses, gives a certain resultant pulse depending (for example, in its amplitude) on the time separation between the two initial pulses. The latter method was first used in 1942\(^{41}\), and with its help it proved possible to obtain the most accurate of the existing decay curves in time for mesons in various substances (aluminum, brass, lead). The corresponding averaged curve (Fig. 5, b) agrees fairly well with the exponential
\[ f(t)=e^{-\frac{t}{\tau_0}}, \]
with the constant \(\tau_0=2.15\pm0.07\ \mu\mathrm{sec}\) for
Fig. 4, a). Scheme of a control experiment for determining the number of “false” coincidences of delayed type. Delays \(N\) (as fractions of the total particle flux through the apparatus, \(N_0\)) are recorded for channels III and IV of the radio circuit in relation to pulses in channels I and II. b). Dependence of the number of “false” coincidences on the minimum allowable delay time \(\Delta t\). Curves 1 and 2 refer to counters of different types.
time \(t\) from 1 to 10 μsec, and only in the initial portion is it distorted by the effect of discharge delays in the counters. The exponential character of the process, identical for all substances, testifies to the fact that what is actually observed is a process of spontaneous decay of mesons, in which the probability of the process depends neither on the time elapsed after the meson has stopped in the substance nor on the nature of this substance, and is a characteristic constant for the main part of the mesons making up the hard component of cosmic radiation.
Fig. 5, a). Diagram of the arrangement for investigating the law of decay of mesons stopped in matter during a time \(4t\). The arrangement recorded the delay of discharges \(\Delta t\) in the counters \(B\) relative to anticoincidences \((L, A_1, A_2, -MM)\).
b). One of the differential decay curves obtained by the authors.
Concerning the decay law obtained, several remarks should be made. First, the dependence of decay on time is always the same only for mesons at rest relative to the observer. As experiments on the “anomalous” absorption of moving mesons of more or less fixed energy have shown, for moving mesons the relativistic law of transformation of time \(t = t_0\sqrt{1-\beta^2}\) proves to be valid; and this means that, while the exponential character of the decay is preserved, the lifetime \(\tau\) changes as a function of the velocity of motion \(v=\beta c\) according to the law
\[ \tau=\frac{\tau_0}{\sqrt{1-\beta^2}}, \]
where \(\tau_0 = 2.15\) μsec. This fact is at present, although not very precise, nevertheless the most direct confirmation of the conclusions of the special theory of relativity. Secondly, the dependence of decay on time, generally speaking, does not depend on the substance only for positive mesons. As will be seen from what follows (Section III), negative mesons with a definite
with some probability “perish” also as a result of another, alternative process caused by interaction with nuclear particles, and this leads to the fact that in light substances their decay still plays a noticeable role, but the actual lifetime may be reduced many times. As a result, for a really existing mixture of positive and negative stopped mesons, the decay curve in time is no longer an exponential, although with some error it can be “fitted” by an exponential; thus, for example, Ticho^42, for decay in aluminum, obtained the value \(\tau_0 = 1.78\ \mu\mathrm{sec}\), which at one time prompted him to suggest a change of \(\tau_0\) with the altitude of the place of observation (the data of Nereson and Rossi^41, averaged for Al, brass, and Pb, referred to sea level, while Ticho’s data referred to an altitude of about \(4\ \mathrm{km}\)). As for a real dependence on the altitude of the place of observation, it could be connected with an admixture of mesons having a different, though not very different (otherwise they could not have been detected by this method), lifetime. To check this possibility, Rossi and collaborators^43 specially measured, at different altitudes up to 10 thousand meters, the ratios of the number of decay events for four time intervals:
\[ \Delta_1 t = 0.9—2.7\ \mu\mathrm{sec}; \]
\[ \Delta_2 t = 2.7—4.5\ \mu\mathrm{sec}; \]
\[ \Delta_3 t = 4.5—6.3\ \mu\mathrm{sec}; \]
\[ \Delta_4 t = 6.3—8.1\ \mu\mathrm{sec}. \]
The results of Rossi, presented in Fig. 6, show, although not with very good accuracy, that the lifetime apparently does not depend on altitude and, consequently, an admixture of mesons with somewhat different lifetimes appears improbable. (This conclusion, however, does not apply to possible decays with times \(\tau_0 \ll 10^{-7}\ \mathrm{sec}\) and \(\tau_0 \gg 10^{-4}\ \mathrm{sec}\).)
Fig. 6. Differential decay curves of stopped mesons at different altitudes.^43
The last question, which is of great importance for the whole technique of delayed coincidences, is how far the lifetime \(\tau_0\) is specific precisely to ordinary mesons with mass \(200\,m_e\), and whether, among other newly discovered types of mesons, there occur exactly the same lifetimes. Unfortunately, nothing definitive can yet be said on this point; some data obtained with the Wilson chamber and by the method of magnetic analysis will be given below.
In addition to studying the law of meson decay in time, the delayed-coincidence method can give, and has already given, some important information on the nature and properties of the decay products. In this field it is a very important supplement to investigations with a Wilson chamber, and not only from the point of view of auxiliary apparatus for controlling the chamber, but also from the point of view of obtaining a sufficient amount of statistical material.
As in experiments with a Wilson chamber, here above all the possibility is presented of investigating the charged decay products.
Fig. 7. Diagram of the apparatus for investigating meson decay by the delayed-coincidence method (Zhdanov and Khaydarov^44). Mesons stopping in filter d are registered by counters I, charged decay products by counters II and III.
For this purpose it was naturally necessary to take up the study of the absorption of the decay products in various substances. One of the arrangements intended for such investigations and belonging to the author of these lines^44 is shown in Fig. 7). In this arrangement the upper group of counters (I) registers mesons stopping subsequently in the filter d, while the decay electrons are registered in coincidences by two groups of counters (II and III) with filter D between them. Varying the thickness of filter D and its composition made it possible, first, to determine whether the charged decay products really undergo the large radiative losses expected for electrons with energies of tens of MeV, and, second, to check whether the absorption curve of these electrons could be reconciled with the assumption of a single energy for them, equal at the decay point to 50 MeV, as followed from generally accepted ideas. The corresponding experimental absorption curves in lead (curve 1) and graphite (curve 2*) are given in Fig. 8, the scales along the abscissa axis being reduced in both cases to thicknesses of material equivalent from the point of view of ionization losses.
As for the substantial role of non-ionization losses for the decay products, this circumstance is clearly illustrated even by a purely qualitative comparison of the two experimental absorption curves with one another. A quantitative comparison of curves 1 and 2 shows that the possible fraction of particles possessing only ionization losses (and, consequently, masses above 3–5
*) Investigations with a similar method have recently been carried out also by other authors^45,46, and the data obtained, referred to sea level, are in good agreement with the results set forth below for an altitude of about 4 km.
electronic masses), constitutes, in the total number of charged decay products, no more than 20–25%. This estimate makes it possible to conclude that, under the assumption that the mechanism of all decay processes with this lifetime is of one type, the decay products under the conditions of the given method are, at least for the most part, electrons, and not any light mesons.*)
Fig. 8. Absorption curves of charged meson-decay products in the \(D\) filter of the setup shown in Fig. 7.
\(1\) — experimental absorption curve in lead, \(2\) — experimental absorption curve in graphite, \(3\) — calculated absorption curve in graphite, \(E_0 = 50\) MeV, \(4\) — calculated absorption curve in graphite, \(E_0 = 25\) MeV.
Let us now turn to the answer to the second question—concerning the energy of the charged decay products; it turns out that, even in the case where all these are electrons with an energy of 50 MeV, the absorption curve would have a by no means step-like, but rather diffuse character, connected not only with the presence of a certain angular distribution of these electrons, but also with a whole series of processes that complicate the nature of their absorption in matter, even in such a light substance as graphite.
In order to calculate the absorption curves of electrons of a definite energy and, consequently, of a definite mean range, it is necessary, in addition to the angular distribution (taken to be naturally isotropic, since the decay of stopped particles is under consideration—
*) This method differs, in particular, from the thick-layer photographic-plate method in that it is not capable of detecting the presence of intermediate, rapidly decaying products of the primary decay, which have small ranges in matter.
account) five more factors leading to a “smearing” of the absorption curve. We shall confine ourselves here to listing these factors:
1) fluctuations of ranges, associated with Rutherford (multiple) scattering of the electrons;
2) scattering (mainly single) through large angles, carrying electrons outside the solid angle determined by the arrangement of counters II and III in Fig. 7;
3) fluctuations of ranges, associated with radiative losses;
4) the effect of conversion of photons of bremsstrahlung radiation into electrons;
5) annihilation of moving positrons, which constitute about half of all decay products.
The result of such a calculation is shown as curve (3) in Fig. 8, where, for comparison, the calculated curve (4) corresponding to absorption of decay electrons with initial energy 25 MeV is also given.
As is seen from comparison with the experimental curve (2), the latter can be satisfactorily explained only if it is assumed that the energy spectrum of the decay electrons is not limited to any one energy, but is smeared over the range from 45–55 MeV at least down to 15–25 MeV*). In this case a graphical calculation of the mean energy, over the spectrum, of the decay electrons from experimental curve 2, in comparison with curves 3 and 4, gives a value of 30–35 MeV.
The data obtained on the character of the spectrum of decay electrons cannot be reconciled with the previously accepted idea of meson decay (mesons, as is known[^47],[^74], having in the majority of cases a mass somewhat exceeding \(200\,m_e\)) into two particles, in particular an electron and a neutrino. On the other hand, this type of spectrum can easily be explained from the point of view of decay into three particles—an electron and two neutral ones. In particular, if these neutral particles are neutrinos, then, taking into account the laws of conservation of energy and momentum, it is easy to show that the electrons must have a maximum energy equal to \(\frac{1}{2}\mu c^2\), i.e. about 55 MeV. In analyzing the more complicated possibility consisting of decay into an electron, a neutrino and a neutral meson or neutretto with mass \(\mu^0\) \((\mu^\pm \to e^\pm + \nu + \mu^0)\), to determine the maximum electron energy in the spectrum \(E_{\max}\)
*) Strictly speaking, there should not have been a single energy \(E_0\) of the decay electrons already as a consequence of a peculiar “Doppler” effect for negative mesons moving near the atomic nucleus in the corresponding \(K\)-shells. However, as the corresponding calculations have shown[^48], the relative “smearing” of the electron energy due to this effect in decay into an electron and a neutrino has a magnitude of about \(\dfrac{Z}{137}E_0\), i.e. about 2 MeV in the case of graphite \((Z = 6)\).
one may also make use of the laws of conservation of energy and momentum, according to which
\[ E_{\max}=\frac{\mu c^{2}}{2}\left[1-\left(\frac{\mu^{0}}{\mu}\right)^{2}\right], \tag{1} \]
where \(\mu\) and \(\mu^{0}\) are, respectively, the masses of the initial meson and the neutretto. In this case the value \(E_{\max}=45\) MeV corresponds to a mass \(\mu^{0}\approx 70\), and thus, for the given method of determining \(E_{\max}\), all values of the neutretto mass from 0 to \(70m_e\) may be regarded as equally possible.
As for the general form of the electron spectrum in decay according to the scheme \(\mu \to e+\nu+\mu^{0}\), it was first calculated, from quite natural assumptions, in the work of Horowitz and collaborators,\(^{49}\) who somewhat generalized the usual theory of \(\beta\)-decay. It turned out that the mean electron energy in the spectrum \(\overline{E}\) should lie within the limits \((0.6—0.7)E_{\max}\), while the “spread” of the spectrum, characterized by the quantity
\[ \Delta E=\sqrt{\left(\overline{E}\right)^{2}-\overline{\left(E^{2}\right)}}, \]
is about \(0.2E_{\max}\). Both properties of the spectrum agree well with the results cited above and with the data of Wilson’s chamber, set forth below.
In connection with the question of the decay mechanism, it was very important to check that the neutral particles participating in it are certainly not photons. This is also important for estimates of that energy which is transferred, in the decay of mesons, to the equilibrium soft component of cosmic rays. As is known,\(^{8}\) experimentally this latter question has still not been solved at all satisfactorily; although it has now become quite clear,\(^{50}\) that at least at medium and high altitudes this energy is clearly insufficient to explain the entire observed soft component, nevertheless a correct estimate of the share of the equilibrium component appears, in a number of cases, to be quite essential.
Fig. 9. Diagram of the apparatus for detecting delayed photons (together with electrons) in meson decay.\(^{51}\) Coincidences \((B, C)\), delayed with respect to coincidences \((A, B)\) by a time \(\Delta t=0.6—5.3\ \mu\text{sec}\), were registered.
Three works were devoted to attempts to detect photons emitted during the decay. In the first of them,\(^{51}\) by means of an apparatus schematically shown in Fig. 9, the frequency of two events was compared: on the one hand, the delayed (with respect to counters \(A\)) discharge of one of the groups of counters \(B\) or \(C\), and, on the other hand, the simultaneous discharge of counters \(B\) and \(C\), with delays within the same limits \((\Delta t=0.6—5.3\ \mu\text{sec})\). The measurements showed that events of the second type are registered approximately 65 times more rarely than events of the first type, and can be wholly explained by accidental coincidences, and not
with the emission of an electron and a photon (in opposite directions) in meson decay.
The next work[^52], in which the decay was investigated no longer in graphite, but in brass (i.e., only for positive mesons instead of mesons of both signs, as will be clear from what follows), led to the same negative result, but with greater completeness. The merits of these measurements (see the layout of the apparatus in Fig. 10) were: a wider interval of the delays studied (from 1.2 to 8.0 μsec), a larger and, moreover, quantitatively estimated probability of conversion in lead of photons recorded by the groups of counters \(D\) and \(E\), and, finally, better statistical accuracy of the measurements, connected with the fact that delayed photons could be recorded independently of electrons. As a result of these measurements—
Fig. 10. Layout of an apparatus for detecting delayed photons from stopped mesons[^52]. Anticoincidences \((D, E, C)\) and delayed coincidences in counters \((A, B)\) were recorded during a time \(\Delta t = 1.2\text{--}8.0\ \mu\text{sec}\).
Fig. 11. Layout of an apparatus with which the possibility of photon emission upon stopping mesons in an iron filter \((d)\) was investigated[^54]. The anticoincidences \((A, B, -C)\) played the role of the triggering master pulse; the remaining groups of counters were hodoscopic.
the authors were able to reject with confidence the possibility that in each act of meson decay (with a mean lifetime of 2 μsec) a photon with energy not less than 15 MeV is emitted. On the other hand, the indicated photons could appear not directly in the act of decay, but as a result of the secondary two-photon decay of neutral mesons participating in the process, with a lifetime not exceeding \(10^{-10}\) sec. Therefore the result presented also makes improbable the participation in the decay of a sufficiently heavy \((\mu^0 \gg 60\,m_e)\) neutral particle with integral spin. Indeed, theory predicts[^53] for such particles a very large probability of spontaneous two-photon decay, provided there is sufficiently strong interaction with nucleons obeying the Dirac equation.
Finally, a third work on the same subject, belonging to Piccioni \(^{54}\), also showed the absence of protons in the decay of positive mesons stopped in iron, independently of the lifetime for this process. For this purpose the apparatus (Fig. 11) was supplemented by a small hodoscope of five groups of counters \(H\), which made it possible to determine more reliably the direction of motion of the photon and thereby to exclude accidental coincidences. Again the negative result of the experiment allowed the author to conclude that photons are absent not only in decay, but also in processes of meson capture, as will be mentioned below in connection with the further discussion.
§ 2. The Wilson Chamber Method
In comparison with the method described above, investigations with the Wilson chamber suffer from one fundamental drawback: the working volume of the chamber usually does not allow the use, for studying mesons stopped in it, of any appreciable quantities of substance, and thereby strongly limits the possible number of observed decay events. Thus, for example, with average chamber dimensions (diameter about \(30\ \text{cm}\)) its working volume can be partitioned by plates with a total effective amount of substance of the order of \(100\ \text{g}\); at the same time, for apparatus with counters in the delayed-coincidence method, it is possible without difficulty to investigate meson decay on areas of the order of several thousand square centimeters and, consequently (with the same layer thicknesses as in the chamber), with effective quantities of substance up to \(10\ \text{kg}\). If one takes into account that at sea level about \(0.05\) meson per hour \(^{21}\) is stopped in each gram of substance, and that only a fraction of them gives tracks of decay electrons suitable for energy measurements, it becomes clear why the number of investigated decay events in Wilson-chamber plates until very recently was limited to one or two tens *). The number of photographs showing meson decay in the gas of a Wilson chamber is still smaller—it is measured in units, and the meson lifetime has been established in only a single case \(^{55}\). At the same time, in almost every work carried out by the method described above, decay events were recorded by the thousand, without which it would have been impossible to study sufficiently accurately the laws of decay in time for different substances and for mesons of different sign. A second, no longer fundamental, feature of most investigations with the Wilson chamber—a feature which may be regarded as both a drawback and an advantage, depending on the formulation of the problem—is that, when observing the entire track of the decay electron inside the chamber, it is impossible to record the time interval,
*) The only exception is a recent work \(^{65}\), in which 75 tracks of decay electrons were obtained.
during which the decay occurred. This means, on the one hand, the impossibility of reliably identifying the decaying meson with an “ordinary” meson (by its decay constant), and, in investigations with plates, simply a considerable probability of an accidental coincidence of two genetically unrelated events; on the other hand, the absence of fixed delay intervals of the electron relative to the primary meson makes it possible to hope for the discovery of decay processes of any mesons independently of their lifetimes, provided only that there are charged decay products with a sufficiently large range (for determining the energy and nature of the particles).
The last of the difficulties in the application of the Wilson chamber, though likewise not fundamental, is the difficulty of simultaneously determining the energy and mass of the decay products: since the maximum values of the ranges of decay electrons amount to about 20 g of light substance (see Fig. 8), it is, as a rule, impossible to combine curvature and range measurements, and one has to use some method or other of determining the particle energy, assuming in advance that all of them are electrons.
In spite of all this, investigations with the Wilson chamber also possess a number of indisputable advantages. Quite apart from the fact that they give a visual picture of each elementary act, it becomes possible to know not only the signs of the charges and the energy of the particles, primary and secondary, but also their masses, and moreover not on the average but in each individual case (although obtaining all the indicated quantities simultaneously is practically impossible).
In proceeding to a survey of the results of investigations of decay by the Wilson-chamber method, one should divide them into several groups according to the specific features of the technique and the problems posed.
The first in time was a series of results (beginning with the photograph published in 1938^56), relating to decay in the gas of the Wilson chamber. In this case, up to 1947 there was in general no doubt that the process was consistent with the conception of the decay of the meson into an electron and a neutrino. Two photographs^57,58, on which cases of meson decay in the gas were quite reliably recorded, testified that the meson can indeed decay with the emission of one fast charged particle; in this case the mass of the meson agreed with the usual value \(200 m_e\), known from other determinations of the mass^47, while the energy of the electron did not contradict the conception of the decay of this meson into an electron and a neutrino.
In later experiments of this kind it proved possible to observe a case of “anomalous” meson decay. In 1947, at an altitude of about 9 km, two photographs were obtained of the decay of a positive meson, apparently in the chamber gas (argon), with the emission of positrons with energies in the range 21–28 MeV^59,60. In this case the masses of the primary mesons were determined very inaccurately, but in one of the cases a mass value close to \(100 m_e\) was almost excluded. An analogous picture of decay
was observed in 1948,^61 but already at an altitude of about 4 km and with a positron energy of about 15 MeV.
More numerous cases of decay in plates placed in a Wilson chamber were observed by a number of authors; in some works^62,63 the electron energy was determined from an analysis of its scattering (calculation of the deflection angle in several thin plates), while in other experiments^64,65 it was determined from the radius of curvature of the track in a magnetic field. Only in one of the works^55 relating to the latter case,
Fig. 12. Diagram of the Wilson chamber used for the study of electron decay energy. Mesons stopped in the chamber and in plate \(d\) are registered by the “telescope” \((A, B)\); decay electrons (delayed by a time \(\Delta t = 0.7\text{–}4.7\ \mu\text{sec}\)) are registered by counters \(DC_1\) or \(DC_2\).
G. B. ZHDANOV
the Wilson chamber was controlled by delayed coincidences with an interval of possible delays \(\Delta t = 0.7—4.7\) μsec. The diagram of the corresponding apparatus is shown in Fig. 12, where counters \(A\) and \(B\) are intended for recording mesons stopping in an aluminum plate (6 mm Al) or, in general, inside the chamber, and counters \(DC_1\) and \(DC_2\) for recording decay electrons. It is interesting to note that, in contrast to other investigations, here almost all cases of decay agree with the earlier ideas of a single energy of the decay electrons, close to 45 MeV. The most complete of all investigations of decay in a Wilson chamber are, however, the results of the last work, carried out at sea level with the aid of a Wilson chamber freely falling in a magnetic field of 7250 gauss
Fig. 13. Diagram of a Wilson chamber used for the investigation of the spectrum of decay electrons \(^{65}\). The chamber was controlled by anticoincidences \((C_1, C_2,—C_3)\); \(d\)—plate (2 g/cm\(^2\) of graphite) intended for the absorption of mesons.
Fig. 14. Energy spectrum of decay electrons, obtained in the apparatus shown in Fig. 13. Along the ordinate axis are plotted the numbers of electrons falling in an energy interval of 10 MeV.
(see the arrangement diagram in Fig. 13) \(^{65}\). In this work it was possible to determine the electron energy for 75 cases of meson decay in the graphite plate, in the walls, or in the gas of the chamber. In individual cases the mass of the meson was also determined, and it proved to be in agreement with the value \(210—215\,m_e\), usually accepted at present as the most accurate. The spectrum of decay electrons obtained in the work is presented in Fig. 14. The authors also calculated the mean energy of the decay electrons, close to 34 MeV, and the mass of the meson \(\mu\) [equal to \((217 \pm 4)\,m_e\)] on the assumption that two neutrinos participate in the decay*).
*) If, on the contrary, one assumes the value of the meson mass \(\mu = (215 \pm 5\,m_e)\), then from the upper limit of the electron spectrum \(E_{\max} = 55\) MeV it follows that the maximum permissible mass of either of the neutral particles participating in the decay cannot exceed \(30\,m_e\) [see equation (1)].
Finally, the last group of experiments may conventionally include those in which the meson decay occurred outside the Wilson chamber. In one work, carried out with the apparatus shown in Fig. 15[^65], an entirely new problem was posed—to estimate the masses of those mesons which decay in the filter \(d\) in the time interval \(1\text{--}10\ \mu\mathrm{sec}\). In this case counters 1, 2, and 3, together with the Wilson chamber in a magnetic field, served to register mesons incident on \(d\) with a definite interval of ranges and with momentum values measured from the curvature of the tracks; counters 4 registered the decay electrons.
Fig. 15. Diagram of the Wilson chamber used to investigate the mass of the sign of mesons decaying in various substances (filter \(d\)) with a lifetime of about \(2\ \mu\mathrm{sec}\). The chamber was controlled by discharges in counters 4, delayed by a time \(\Delta t = 1\text{--}10\ \mu\mathrm{sec}\) relative to coincidences \((1, 2, 3)\).
Fig. 16. Momentum spectrum of mesons obtained in the apparatus shown in Fig. 15, for the case of decay of negative mesons in graphite. The graph indicates the intervals of permissible momentum values for mesons with masses \(100\), \(200\), and \(250\,m_e\).
The approximate distribution of mesons by momenta in one of the series of experiments is shown in Fig. 16, where the limits of momentum values permissible in this apparatus for particles with masses respectively \(100\), \(200\), and \(250\,m_e\) are also given. As is seen from Fig. 16, at sea level the greater part of at least those mesons which decay with a lifetime of \(2\ \mu\mathrm{sec}\) have masses close to \(200\,m_e\). Unfortunately, the large percentage (about 15%) of random and “false” delayed coincidences makes it impossible to answer the question to what extent the delayed coincidences may be caused by mesons with masses lying outside the above-indicated limits of \(100\) and \(250\,m_e\). In this connection one may also cite work[^67], in which, with the aid of a single Wilson chamber, the fact of decay was registered and the range of the meson measured, while the momentum was measured in an upper chamber with a magnetic field. It turned out that all five clearly expressed cases of decay of positive mesons in lead plates—
as the lower chamber gave mass values agreeing with the value \(200\)—\(220\,m_e\). Finally, in yet another paper\({}^{68}\), the arrangement of which is shown in Fig. 17, the Wilson chamber was controlled by mesons stopping in the graphite plate \(d\), and the energies of the decay electrons were determined from the number of aluminum plates traversed by them in the chamber (the symbols \(\oplus\) mark groups of counters connected in an anticoincidence circuit). The energies of three decay electrons were measured, equal respectively to 13, 18, and 50 MeV. The authors believe that in their apparatus, apparently, a considerable fraction of the electrons emitted from \(d\) in the direction of the chamber is “sifted out” only because their energy is less than 14 MeV and does not allow them to overcome the walls of all the instruments.
Fig. 17. Diagram of an apparatus with a Wilson chamber in which the energies of electrons from the decay of mesons in the graphite plate \(d\) were studied\({}^{68}\). The Wilson chamber was controlled by anticoincidences \((A, B, C, E, --X)\).
It should be especially noted those very few cases in which, in the Wilson chamber, it apparently proved possible to observe decay processes involving both neutral and charged mesons with masses clearly different from \(200\,m_e\). In one of these papers\({}^{69}\) the authors assert that in several cases they discovered the process of decay into an electron and a positron of a neutral meson with a mass from 13 to \(24\,m_e\), belonging to the penetrating component of cosmic radiation. In another paper\({}^{70}\), connected with the study of particles in penetrating showers, there was apparently observed one case of the decay of a neutral meson with a mass from 500 to \(1600\,m_e\) into positive and negative charged mesons, and one case of the decay of a positive meson with a mass of about \(1000\,m_e\) into a neutral and a lighter positive meson. As will be shown below, similar rare events have also been recorded by other methods of investigation.
§ 3. The Photographic-Plate Method
At the present time, after the development of highly sensitive photoemulsions\({}^{71}\) made it possible to register relativistic particles reliably, the photographic-plate method has become a very strong competitor of the Wilson chamber in the investigations of interest to us.
processes with stopped mesons. It is especially effective in those cases when, at the end of the meson track, secondary particles appear with short ranges of the order of hundreds of microns in emulsion, which corresponds to ranges measured in meters of gas in Wilson chambers at normal pressure. In these cases one can obtain an almost complete picture of the elementary act by measuring the masses and angles of emission of all charged particles. True, in contrast to the Wilson chamber, determination of the signs of the particle charges is, as a rule, excluded, since in order to obtain the necessary deflections, as distinct from scattering phenomena, magnetic fields that are too strong would be required, considerably exceeding the 15,000 gauss that are usually limited by the magnetic saturation of iron. However, this shortcoming, which is practically immaterial, for example, in experiments with “artificial” mesons, is compensated by the substantial advantage over the Wilson chamber—namely, an effective observation time many times greater. The latter is especially valuable for phenomena involving short-range particles, where almost all the advantages of the Wilson chamber are lost in the sense of using one or another control system.
In addition to the possibility of collecting a large statistical material, the advantage of continuous observation characteristic of photographic plates makes it possible almost to avoid the specific difficulties of the Wilson-chamber method associated with some uncertainty in knowing the absolute and relative effective observation time for particles with different ionization. This, in turn, creates the possibility of a reliable determination of the frequency of the phenomena studied and of the fluxes of stopping mesons, provided only that measures are taken72 to reduce sufficiently the influence of photoregression, i.e. the gradual weakening of the latent image of the tracks. It should be borne in mind that, in comparison with the Wilson chamber, the photographic emulsion offers substantially fewer possibilities in the choice of a desired material, homogeneous in chemical composition, in which the processes under study take place. Recently, however, there have also been new achievements in this direction through the use of layered photographic emulsions73.
Let us conclude the general remarks on the method with an assessment of the accuracy of measurements in experiments with photographic plates as compared with the Wilson chamber. Thus, for example, statistical processing of data on the scattering of mesons that have stopped in photographic emulsion makes it possible to obtain an accuracy of mass determination of about 10% (see below), whereas an investigation of meson masses in a chamber, analogous in frequency of cases, performed by Fretter, gives, according to Brode’s estimate74, an accuracy of about 2%. In determining the energies of individual decay electrons, both methods give, in general, the same accuracies (of the order of 10%), although the principles of energy determination are completely different: in the Wilson chamber one usually studies the curvature in a magnetic field or the range in matter (it is seldom possible in this case to combine both); in
case of photographic plates, the mean angle of Rutherford scattering is calculated.
Turning to a review of studies on meson decay carried out by the photographic-plate method, let us begin with one of the most recent works in time[^71], in which it proved possible to observe about 100 mesons that came to rest with the emission, at the end of their range, of a weakly ionizing relativistic particle. The authors consider that all these secondary particles (so-called $\eta$-particles) are electrons, while all the primary ones are ordinary $\mu$-mesons with a mass of $200\,m_e$, constituting the main part of the hard component. In support of their, as yet insufficiently substantiated, opinion they point out that in 9 of these 100 cases the observed $\mu$-mesons, in turn, were products of the decay of heavier stopped mesons ($\pi$-mesons), which will be discussed below. A picture typical of such a successive decay is shown in the microphotograph of Fig. 18. Analysis of the scattering of electrons along the track visible in the emulsion, often over a length of the order of a thousand microns or more, enabled the authors in 20 cases to determine the electron momenta at the point of their origin. Typical values of the electron momenta, in energy units, were, for example, $15 \pm 3$; $20 \pm 5$; $38 \pm 6$; $42 \pm 2$; $48 \pm 6$ MeV. These figures fully confirm the continuous character of the spectrum which must be ascribed to the decay electrons on the basis of data obtained by the methods of delayed coincidences and of the Wilson chamber. Moreover, the authors of the work hope, by means of further measurements with the photographic-plate method, to establish the shape of this continuous spectrum, which is of very great importance for judging the mechanism of the decay.
Let us now dwell in more detail on the decay of $\pi$-mesons, discovered[^75] and studied during the last two years precisely by the photographic-plate method. Already in the first investigations[^32] the authors drew attention to two characteristic features of such a process: in the transformation of a heavier stopped meson into a lighter one, the mass ratio (always measured with considerably greater accuracy and reliability than the mass of an individual particle) always remains the same, within the limits of the values $1.3$–$1.6$; moreover, the range of the secondary particle ($\mu$-meson) is always constant and close to $600\,\mu$, which corresponds to a $\mu$-meson energy of about 4 MeV. Further investigations[^76], with determination of the masses of individual particles from their scattering, gave the following mass values for the heavy ($M_\pi$) and light ($M_\mu$) meson: $M_\pi = (260 \pm 30)\,m_e$; $M_\mu = (205 \pm 20)\,m_e$. The values quoted were obtained from the analysis of only 20 cases, but, as will be shown below, they are in full agreement with other, more accurate data relating to “artificial” mesons. The fact that the energies of the $\mu$-mesons are always the same means, evidently, that the observed decay of the $\pi$-meson always occurs after the latter has lost practically all its kinetic energy and
Fig. 18. Microphotograph illustrating the sequential decay of a \(\pi\)- and \(\mu\)-meson in a photographic emulsion sensitive to relativistic particles (\(\eta\) is the decay electron).
besides the fact that among the decay products there is only one neutral particle, invisible in the photoemulsion. Unfortunately, for estimating the mass of the latter particle and, in general, for deciding the question of its nature (it could, in particular, be a photon), the accuracy of the data given above on the ratios of the masses \(M_\pi\) and \(M_\mu\) is quite insufficient.
The last characteristic of the decay that it has been possible to estimate from studies of cosmic radiation by the photographic-plate method is the lifetime of the \(\pi\)-meson. For this purpose, the character of the angular distribution was determined,^77 on the one hand, and the ratio between the number of heavy (\(\pi\)- and \(\sigma\)-) and light (\(\mu\)-) mesons at various distances from the photoemulsion to the dense substances surrounding the plates, on the other. It turned out that, when photographic plates were placed at a distance of about \(2\) m from the floor, the reverse flux (from below upward) of slow \(\rho\)-mesons stopping in the emulsion (i.e., mesons that do not give secondary charged particles at the end of their range) could be explained wholly or in large part by the decay of \(\pi\)- and \(\sigma\)-mesons coming (also from below upward) from the floor of the building; but in that case, from the exponential law of decay of heavy mesons in time (where, for slow particles, the dependence of the lifetime on velocity may be neglected), and also from the ratio in the fluxes of heavy and light mesons, one can determine the desired lifetime \(\tau_\pi\). It proved to be equal to \((6 \pm 3)\cdot 10^{-9}\) sec, with some underestimation if particles generated in the substance without the intermediate stage of a \(\pi\)-meson are present in the reverse flux of \(\rho\)-mesons.
In addition to the investigations carried out on mesons with masses \(200m_e\) and \(300m_e\) by the photographic-plate method, a number of individual cases of decay of mesons of larger masses have been found. Thus new direct confirmations have been obtained of the existence of a definite mass spectrum of mesons, established for the first time by Alikhanov and Alikhanyan by the method of magnetic analysis. First of all, one should point to the recently published work of Alikhanyan and collaborators,^78 in which, by the photographic-plate method, several cases of decay of barytrons of different masses (from \(750\) to \(7\)—\(9\) thousand \(m_e\)) with the emission of single particles of smaller mass were investigated.
New possibilities in this direction also appear in connection with the use of photoemulsions sensitive to relativistic particles. Of particular interest, in particular, is a recently published report of one case which apparently belongs to a class of phenomena completely unknown until now. The photograph shown in Fig. 19^71 is in question. In this photograph the authors paid special attention to the mass of the particle coming from the upper left-hand corner, which produced the upper “fork.” Measurement of the mass of this particle, carried out by comparison with the masses of known particles by the method of grain counting and checked by a second, less accurate method of scattering analysis, gave a mass value of \((1080 \pm 160)m_e\),
Fig. 19. A unique case of a phenomenon interpreted as the decay of a charged meson (particle \(k\)) with a mass of about \(1000\,m_e\) into three charged particles in a photographic emulsion sensitive to relativistic particles.⁷¹
whereby it could be guaranteed that values below \(700\,m_e\) and above \(1300\,m_e\) are practically excluded. This new particle, called the \(k\)-meson, could at the end of its path produce the ordinary “star,” observed in the photograph in the form of a triple “fork.” However, analysis of the ionizations and ranges of the particles emitted in this event showed that these cannot be either protons or electrons, but only mesons, one of which then causes nuclear disintegration (the lower “fork” in Fig. 19). In addition, by painstaking processing of the data on the measurement of angles (it was necessary to take into account the shrinkage of the photographic emulsion after development), the authors succeeded in constructing a spatial momentum diagram of the component parts of the upper “fork” and in showing, first of all, that all three tracks are coplanar, and consequently that in the process apparently neither neutral particles nor recoil nuclei take part. Finally, if one takes into account the rest energy of the \(k\)-particle and the initial energy of the particle that produced the second fork (apparently this is a \(\sigma\)-meson), then the obtained momentum diagram can be reconciled with the idea of the spontaneous decay of a stopped meson into three charged particles, each of which has a mass \(200\)—\(300\,m_e\).
The data indicated above, obtained photographically, we have brought together, with some additions, in Table I (see p. 506), without going into further details, since they will not help to form any sufficiently clear idea of the properties of this new group of particles. The small accuracy in the determination of the masses makes it impossible even to answer the question whether all the photographs refer to one and the same particles with mass about \(1000\,m_e\); it is clear only that their masses lie between the masses of the proton and of the \(\mu\)- and \(\pi\)-mesons investigated up to now. In the same table, for completeness, are also given the latest data on measurements of the masses of \(\mu\)- and \(\pi\)-mesons by the photographic-plate method, as well as separate results of investigations of heavy mesons independently of the phenomenon of their decay.
Let us consider, in conclusion, the main results of published investigations on the decay of stopped “artificial” mesons. To obtain these mesons, use was ordinarily made of a beam of \(\alpha\)-particles* accelerated in a powerful phasotron of diameter 184 inches (about \(4.5\,\mathrm{m}\)) to an energy of 350—380 MeV and directed onto a graphite target about \(1\,\mathrm{mm}\) thick. In one of the works\(^{35}\), special photographic plates with an emulsion-layer thickness of \(100\,\mu\) were arranged with respect to the beam in such a way (Fig. 20) that positive mesons emerging from the graphite target with energies from 2 to 5 MeV and deflected in the magnetic field of the phasotron would strike them (approximately at right angles to the surface). Combining the range of these mesons measured in the emulsion with the known radius
* Similar results could also be achieved with the aid of a proton beam accelerated to energies of 350 MeV\(^{29}\).
DECAY AND NUCLEAR INTERACTIONS OF MESONS
Table 1
Masses of stopped cosmic-ray mesons and their decay
(photographic-plate method)
| No. | Authors | Observation altitude above sea level | Number of observed cases | Method of mass determination | Most probable mass value (in electron masses) | Notes |
|---|---|---|---|---|---|---|
| 1 | Goldschmidt-Clermont et al.76 | $H_1=2800\ \text{m}$ $H_2=5500\ \text{m}$ |
20 | Processing of scattering data | $260\pm30$ ($\pi$-mesons) |
In all cases decay of a stopped $\pi$-meson occurred with emission of a $\mu$-meson |
| 2 | The same | $H_2=5500\ \text{m}$ | 20 | $205\pm20$ ($\mu$-mesons) |
Stopped mesons with masses from $750m_e$, belonging to at least 6 different types of particles, were observed; in 3 cases emission of a secondary charged particle was observed | |
| 3 | Alikhanian et al.78 | $H=3500\ \text{m}$ | Stopped mesons with masses from $750m_e$, belonging to at least 6 different types of particles, were observed; in 3 cases emission of a secondary charged particle was observed | |||
| 4 | Langer and Yagoda79 | $H=0$ | 3 | $400\text{–}600$ | Apparently, decay processes involving neutral mesons were observed | |
| 5 | Leprince-Ringuet80 | $H=4300\ \text{m}$ | 1 | Density of the developed grains | 700 | The meson caused nuclear disintegration |
| 6 | Lukirskii and Perfilov81 | $H=0$ | 2 | Energy release in nuclear disintegration | $400\text{–}700$ | The meson track was not observed directly because of the low sensitivity of the photoemulsion |
| 7 | Brown, Camerini et al.71 | $H=3500\ \text{m}$ | 1 | Grain density, scattering | $1080\pm160$ | Decay into 3 charged particles at the end of the range |
circumference described by mesons of a given sign, the investigators were able, with sufficiently high accuracy, to determine the masses of particles that were recorded by the photographic plates in a number of about 200. The observations [see also (32)] showed that all the mass values of particles falling into the photoemulsion are clearly distributed around two values—about 200 and about 300 \(m_e\), respectively, for \(\mu\)- and \(\pi\)-mesons. The latter could be distinguished by their characteristic property of giving, at the end of their range, a secondary, lighter meson (a \(\mu\)-meson) with a range of about \(600\,\mu\). (This corresponds to an energy of about 4 MeV for a \(\mu\)-meson mass of about \(200\,m_e\).) As for the \(\mu\)-mesons entering the emulsion already from the air, they could be wholly explained by the effect of partial decay of \(\pi\)-mesons in the air or in the substance of the target itself into these same \(\mu\)-mesons.
Fig. 20. Diagram of the arrangement of the target \(M\) and photographic plates \(\Phi\) in experiments on the investigation of “artificial” \(\pi\)-mesons. The mesons were generated under the action of a beam of \(\alpha\)-particles accelerated in the phasotron to an energy of 380 MeV.
The indicated method of determining masses gives a considerably higher accuracy than in investigations of cosmic radiation. As the available reports\(^{36}\) show, the mass usually obtained in this way for \(\pi\)-mesons is close to \(285\,m_e\), and for \(\mu\)-mesons close to \(215\,m_e\). Especially great accuracy was obtained for the ratio of the two masses, which turned out to be
\[ \frac{M_\pi}{M_\mu}=1.32\pm0.01. \]
Using the indicated value for the mass ratio, as well as the known energy of the \(\mu\)-meson at the moment of decay, and applying the laws of conservation of energy and momentum to the act of decay, one can easily calculate the mass \(\mu^0\) of the neutral particle which must be emitted in the decay together with the \(\mu\)-meson. For this it is necessary to solve a system of two simple equations, in which all energies and momenta are expressed in units corresponding to the rest energy of the electron:
\[ \left. \begin{aligned} \sqrt{1+\left(\frac{pc}{215}\right)^2} +\frac{\mu^0 c^2}{215} \sqrt{1+\left(\frac{pc}{\mu^0 c^2}\right)^2} &=1.32,\\ \sqrt{1+\left(\frac{pc}{215}\right)^2} &=\frac{223}{215}. \end{aligned} \right\} \tag{2} \]
(Here \(p\) is the momentum of the neutral particle, equal to the momentum of the \(\mu\)-meson.)
From the equations written down one obtains the following value for the mass of the neutral particle: \(\mu^0 = 15\,m_e\). In any case, if the possible inaccuracies of the measurements are taken into account, apparently \(\mu^0\) cannot exceed \(25\,m_e\). These figures agree with the latest data \({}^{71}\) for “cosmic” \(\pi\)-mesons, since the values for the mass ratios
\[ \frac{M_\pi}{M_\mu} = 1.33 \pm 0.05, \]
obtained in studies of cosmic radiation, do not contradict the value 1.32 used above. At the same time this means that the value \(100\,m_e\), cited at the beginning, for the mass of the hypothetical neutral meson participating in the decay \(\pi \to \mu + \mu^0\), is now clearly unsuitable, and it is quite possible that a very light particle of the neutrino type is involved here.
The coincidence of the last decay characteristic—the lifetime—also speaks in favor of the identity of “artificial” \(\pi\)-mesons with “cosmic” ones. The determination of the mean lifetime of “artificial” \(\sigma\)-mesons, carried out recently by Richardson \({}^{37}\) and described by us in the following section, gives the value
\[ \tau_\pi = \left(0.9^{+0.3}_{-0.2}\right)\cdot 10^{-8}\ \text{sec}. \]
This figure agrees well with the value
\[ \tau_\pi = (0.6 \pm 0.3)\times 10^{-8}\ \text{sec.}, \]
which was mentioned above for “cosmic” \(\pi\)-mesons.
§ 4. Method of magnetic analysis
Up to the present time, among the extensive experimental material \({}^{23—27,83}\) accumulated in the study of varitons by the hodoscope method placed in a magnetic field, information on decay is very limited.
First of all, in an arrangement of the type shown in Fig. 1, about 40 cases \({}^{83}\) were observed in which the absorption of a variton of one mass or another is accompanied by the emission from the same point of another particle, delayed by a time of about \(2\ \mu\text{sec}\). In this case a significant fraction of all primary particles had a mass of about \(200\,m_e\); however, in a number of cases decay was also observed for varitons with a mass sharply different from \(200\,m_e\) *).
In addition, the enrichment of the variton flux with lighter particles when a considerable thickness of lead (about \(10\ \text{cm}\)) is placed under the apparatus, absorbing all varitons arriving from the air, may also testify \({}^{26}\) to the decay character of the transformations of the stopped varitons. However, the possibility is not excluded that the indicated phenomenon is connected with a difference in the generation of varitons in lead and in air in their mass spectrum, and not only in effective cross sections.
*) Most interesting in this connection are the decays of particles with masses less than \(200\,m_e\), which cannot simply be reduced to “chain” decays involving \(\mu\)-mesons in the last act.
Finally, the presence of a sharp upper boundary of the momenta for particles belonging both to the soft and to the hard components is interpreted by the authors of studies \(^{26}\) from the point of view of the decay origin of the observed particles; here it is also necessary to assume that the decay of the initial particles with masses from 300 to \(9000\,m_e\) occurs only for stopped barytrons and only into two particles, one of which is assumed to be neutral.
Let us summarize all that has been set forth in this section with the following conclusions on the decay of stopped mesons:
-
At least two types of decay processes for stopped mesons are known: decay with emission of a lighter meson, in particular the decay of the \(\pi\)-meson with mean lifetime \(\tau_\pi\) about \(10^{-8}\) sec, and decay of the \(\mu\)-meson type with mean lifetime \(\tau_\mu = 2.15\cdot 10^{-6}\) sec and with emission of an electron.
-
In the first case the neutral product is either one neutral or a lighter meson, and therefore the spectrum of charged particles is represented practically by a single line.
-
In the second case the neutral products are apparently two neutrinos (or light mesons with mass not exceeding \(30\,m_e\)), while the spectrum of charged particles extends from 0 to a maximum energy value of about 55 MeV, or to its mean value of about 33 MeV.
-
Besides \(\pi\)-decay, there exist several other possible decay phenomena with transformation of a heavier charged meson into one or several lighter charged mesons and with the participation of some neutral mesons.
III. NUCLEAR CAPTURE OF STOPPED NEGATIVE MESONS
§ 5. Method of delayed coincidences
Soon after the idea was put forward of the identity of the mesons of cosmic radiation (with mass \(200\,m_e\)) and Yukawa’s hypothetical particles, several theoretical works \(^{84,85}\) appeared, devoted to the question of the interaction of stopped mesons with the atomic nuclei of the surrounding substance. The corresponding calculations, the results of which in general depended little on the choice of any possible variant of the usual*) theory of nuclear forces, showed that even in air, not to mention dense substances, the lifetime of negative mesons with respect to nuclear capture is considerably (approximately
*) By the “usual” theory of nuclear forces, here and below is meant any theory of exchange forces describing nuclear interactions of nucleons through processes of emission and absorption of single charged mesons.
DECAY AND NUCLEAR INTERACTIONS OF MESONS
by two orders of magnitude) shorter than the lifetime with respect to decay. As for positive mesons, owing to the Coulomb repulsive forces the probability of their capture by nuclei is always negligibly small in comparison with the probability of decay. The indicated results meant that, in any substances, only positive stopped mesons should undergo decay, whereas negative ones should always undergo capture by nuclei with the release of the entire rest energy of the meson (about 100 MeV), apparently through the process of nuclear disintegration. All the first investigations^{41,96} by the method of delayed coincidences seemed to have fully confirmed this point of view. Indeed, for the various substances investigated, from aluminum to lead, it was found that only about 40% of all stopped mesons give decay electrons. True, the accuracy of such estimates was always not very high (the probable error was approximately ±15%), both because of the difficulty of allowing for the solid angles covered by the corresponding counters (counters of group B in Fig. 5), and because of the difficulty of correctly allowing for the absorption of decay electrons in the material of the filters used (filter d in Fig. 5); moreover, the experiments mentioned in § 1 subsequently showed that the estimates of the energies and ranges of the decay electrons had been considerably overestimated. Nevertheless, the results obtained were in full agreement with the idea of the nuclear capture of all negative mesons (if one takes into account, as is usually done, that the ratio of the fluxes of mesons of both signs is
\[ \frac{N_+}{N_-} \simeq 1.2 \]
) and, it seemed, did not require further refinement.
Fig. 21. Diagram of the experiment in which the weak nuclear interaction of stopped negative mesons was first discovered^{87}. Anticoincidences (I, II,—IV) register mesons of a definite sign stopping in filter d; delayed discharges in counter group I·I correspond to decay electrons.
Therefore the results of a separate investigation of the behavior of mesons of different signs in two substances—iron and graphite—proved completely unexpected. In this work the apparatus shown in Fig. 21 was used. It differed from all previous apparatuses (see, for example, Fig. 5) in that the “telescope” I—II, intended for registering mesons stopping in filter d, included blocks of iron plates magnetized to saturation, and counters IV connected in an anticoincidence circuit; moreover, the experiments were carried out with periodic reversal of the direction of the magnetic field.
In each case mesons of one definite sign, possessing such a velocity as to be stopped in filter \(d\), had to deviate on their path through counters \(I\) and \(II\) and also pass through the side counters \(IV\). The effect \(N_d\) registered by the apparatus was thus referred to mesons of the opposite sign; in addition, a small “background” from mesons of the given sign was observed owing to the inefficiency of counters \(IV\) and to scattering. An experimental estimate of the magnitude of the “background” was the ratio of the decay effect, measured for negative mesons, to the effect from positive mesons.
Fig. 22. Diagram of the apparatus on which the small probability of capture of stopping mesons in light substances was confirmed\({}^{89}\). Counters of group \(III\) registered electrons from the decay of mesons of both signs in filter \(d\).
As a result of prolonged measurements it was established that, with an iron filter \(d\), the effect for negative mesons \(N_d^{-}\) amounts to about \(5\%\) of the effect for positive mesons \(N_d^{+}\), as should also have been expected for the indicated “background.” At the same time, in experiments with graphite it was found that
\[ \frac{N_d^{-}}{N_d^{+}} \simeq 75, \]
and this could not be explained otherwise than by the decay of negative mesons in graphite. In other words, for \(Z=6\) (graphite) the probability of nuclear capture proved, in contrast to the case \(Z=26\) (iron), to be considerably smaller than the probability of decay, in obvious disagreement with the predictions of theory. For a more exact judgment of the relative probabilities of decay and capture in graphite it was still necessary to verify whether the value
\[ \frac{1}{\varepsilon}=\frac{N_d^{+}}{N_d^{-}}\simeq 1.3 \]
corresponds to the real ratio of the number of surviving mesons of different signs. However, the only work devoted to this question\({}^{88}\) proceeded from the inverse problem: to estimate the fraction of negative mesons by comparing the decay effects in different substances, knowing the corresponding probabilities of meson capture and decay. The result obtained does not contradict the value \(\varepsilon=0.8\), although it is also very inaccurate. The fact of an exclusively weak interaction of mesons with atomic nuclei found its further confirmation and refinement in the work\({}^{89}\). In this work (see the diagram of the apparatus in Fig. 22) another, simpler, but less convincing method was used, which made it possible to improve the statistical accuracy of the measurements considerably. By registering the number of discharges in counters of group \(III\), delayed by a time \(\Delta t=1\text{--}6\ \mu\text{sec}\) relative to coincidences of discharges in groups \(I\)--\(II\), the authors were able to compare the number of decay electrons arriving—
DECAY AND NUCLEAR INTERACTIONS OF MESONS
…per meson stopped in filter \(d\), for different values of \(Z\) for the filter material. Here, if one specifies a definite value of the ratio of the numbers of stopped mesons of different sign
\[ \varepsilon=\frac{N^-}{N^+} \]
and denotes by \(\Lambda_d=\dfrac{1}{\tau_d}\) and \(\Lambda_c=\dfrac{1}{\tau_c}\) the probabilities of decay and capture of a (negative) meson, then for a “natural mixture” of mesons of both signs one can obtain the following dependence of the number of emitted decay electrons on time:
\[ N_d=N^+\left[1+\varepsilon\frac{\Lambda_d}{\Lambda_d+\Lambda_c}e^{-\Lambda_c t}\right]e^{-\Lambda_d t}. \tag{3} \]
It is assumed here that nuclear capture, like meson decay, proceeds according to an exponential law, as will be discussed below.
Suppose that it is possible to measure the numbers of decay electrons of each sign, \(N_d^+\) and \(N_d^-\), emitted by mesons over the entire time from \(t=0\) to \(t=\infty\) (and this is easy to determine in experiments with separation of mesons of different sign, in view of the exponential law for decay and capture). Then the sought quantity \(\Lambda_c\) for the given substance will be related to the experimental results by the simple relation\(^{40}\)
\[ \frac{1}{\varepsilon}\cdot\frac{N_d^-}{N_d^+} = \frac{\tau_0^-}{\tau_0^+} = \frac{\Lambda_d}{\Lambda_d+\Lambda_c}. \tag{4} \]
In work\(^{8}\) the finite time interval selected for recording decay electrons leads to a more complicated dependence between \(\Lambda_c(Z)\) and the total number of decay electrons \(N_d=N_d^+ + N_d^-\). However, here too, from the dependence \(N_d(Z)\), it is possible, first, to obtain unambiguously the dependence \(\Lambda_c(Z)\), and second, to determine such a value of the atomic number \(Z=Z_0\) at which the probabilities of decay \(\Lambda_d\) and capture \(\Lambda_c\) approximately coincide. For investigations in the apparatus shown in Fig. 22, the authors selected six substances, both simple and chemically complex, containing elements with atomic numbers from \(Z=4\) (Be) to \(Z=16\) (S). The results of these measurements (see Table II) can be represented in the form of two rows of figures characterizing the relative number of decay electrons \(N_d\) emitted by the given substance per unit of its stopping power (the latter approximately corresponds to the number of decay electrons per 1 stopped meson). In the first and second rows of figures, the effects obtained respectively for sulfur and SiC are taken as unity.
Analysis of the data in Table II shows, first, that the capture probability \(\Lambda_c\) at \(Z\leqslant 6\) is considerably smaller than the decay probability \(\Lambda_d\), is approximately equal to it at \(Z=10\), and finally \(\Lambda_c\gg\Lambda_d\) at \(Z>14\); second, the data of the first column make it possible directly to obtain the fraction of negative stopped mesons
\[ \varepsilon=\frac{N^-}{N^+}\simeq 0.7. \]
Table II
| Substance | Be | S | ||
|---|---|---|---|---|
| $N_d$ | 1.67 | 1.0 | ||
| Substance | C | NaOH | Al | SiC |
| $N_d$ | 1.7 | 1.4 | 1.0 | 1.0 |
The methods applied in both of the works described above could, under certain assumptions indicated above, yield all the quantities of interest to us in the present case, namely the function $\Lambda_c(Z)$ and the quantity $\varepsilon$. Nevertheless, in almost all subsequent works on the capture of stopped mesons, it was not simply a certain delay interval that was selected for registering the decay electrons; rather, the time of each delay was determined, with one or another accuracy, as was also done in the decay studies mentioned above (Section II). Indeed, it could happen that the experimentally observed significant difference in the decay of negative mesons for different substances is connected only with the fact that, when a meson enters a $K$ orbit near the nucleus, the character of the decay process changes (above all, accelerated decay with the same decay products could occur). On the other hand, one could not in advance reject with complete certainty yet another possibility: the capture of a meson that had reached the $K$ orbit could occur very rapidly, but the time required to enter this state for a meson that had lost its velocity as a result of ionization braking could be comparable with the decay lifetime $(2\cdot 10^{-6}\ \text{sec.})$ and, moreover, depend substantially on the atomic number $Z$.
As for the latter assumption, it proved possible to exclude it by corresponding calculations, carried out in several works$^{90,91}$, which followed very soon after the first experimental data on the “anomalously” low capture probability. It turned out that even in light gases (air) the time for entering the $K$ orbit is about $10^{-9}$ sec., and in solids it is considerably smaller.
Let us now turn to the results of works in which the time dependence of the decay was studied separately for negative and positive mesons in various light substances. A typical apparatus arrangement for these investigations, and the decay curves obtained for decay in NaF, are shown in Fig. 23$^{92}$. The numbers of decay electrons plotted on these graphs (on a logarithmic scale) for various delay times $\Delta t$ lie well on
straight lines; in this case, for positive mesons the slope of the straight line gives the lifetime \(\tau_0^+ \approx 2.1\ \mu\mathrm{sec}\), and for negative mesons \(\tau_0^- = 1.33 \pm 0.14\ \mu\mathrm{sec}\). In addition, extrapolation to \(\Delta t = 0\) of the corresponding integral decay curves (giving the number of decays with times greater than the given \(\Delta t\)) makes it possible to obtain the ratio of the total number of decayed mesons of the two signs, \(\dfrac{N_d^-}{N_d^+}\), and then, having determined, as indicated above, the values
\[ \Lambda_c + \Lambda_d = \frac{1}{\tau_0^-} \quad\text{and}\quad \Lambda_d = \frac{1}{\tau_0^+}, \]
and also having adopted the usual value
\[ \frac{N^-}{N^+} = \varepsilon = 0.8, \]
it was possible
Fig. 23. a) Diagram of the setup used to determine the mean lifetimes of mesons of different sign\(^{41}\); b) typical differential curves for the decay of these mesons in time (NaF as filter \(d\)).
to verify the validity of relation (4). Similar comparisons for a number of substances (see graphs 6 and 7 in Table III) led to the conclusion\(^{40}\) that, for negative stopped mesons, besides decay there really exists some alternative process, likewise with a probability independent of time; moreover this probability \(\Lambda_c\) depends very sharply on \(Z\), passing through the value \(\Lambda_c = \Lambda_d =\)
\(=4.65\cdot 10^{5}\ \mathrm{sec}^{-1}\) near \(Z=Z_0 \simeq 10\). Moreover, the results obtained mean that the charged particles possibly emitted as a result of capture cannot have ranges comparable with the ranges of the decay electrons.
In order to determine more accurately the form of the function \(\Lambda_c(Z)\), one should use all the available determinations of the quantity \(\tau_0^{-}\) that we have compiled in Table III. As Wheeler noted \(^{91}\), the dependence sought should be close to the function \(\Lambda_d\cdot\left(\dfrac{Z}{Z_0}\right)^4\), independently of the mechanism of capture; indeed, the factor \(Z^3\) is connected with the probability that a meson situated in its \(K\)-orbit of radius \(r_k\), inversely proportional to \(Z\), will be within the range of the nuclear forces, while the second factor \(Z\) is connected with the number of those protons which can absorb the given meson.
In treating the data of Table III it should be borne in mind that in experiments with chemically complex substances the probability of capture of a meson
Table III
Experimental determinations of the lifetimes of negative stopped mesons \(\tau_0^{-}\)
| Substance | \(Z_{\mathrm{eff}}\) | Literature references | Lifetime \(\tau_0^{-}\), in \(\mu\mathrm{sec}\) | Capture probability \(\Lambda_c=\dfrac{1}{\tau_0^{-}}-\Lambda_d\), in \(10^5\ \mathrm{sec}^{-1}\) | \(\dfrac{\tau_0^{-}}{\tau_0^{+}}\) \((\tau_0^{+}=2.15\ \mu\mathrm{sec})\) | \(\dfrac{1}{\varepsilon}\cdot\dfrac{N_d^{-}}{N_d^{+}}\) \((\varepsilon=0.8)^{40}\) |
|---|---|---|---|---|---|---|
| Be | 4 | 93 | \(\simeq 2.2\) | |||
| C | 6 | 94 | \(\leq 2.25\pm 0.2\) | \(0.25\) | \(1.0\ (-0.05)\) | |
| C | 6 | 95 | \(2.15\) | \(0.25\) | \(1.0\ (-0.05)\) | |
| H\(_2\)O | 7 | 40 | \(1.8\pm 0.15\) | \(0.9\pm 0.5\) | \(0.84\pm 0.07\) | \(0.83\pm 0.08^{40}\) |
| NaF | 10 | 92 | \(1.33\pm 0.14\) | \(2.9\pm 0.8\) | \(0.62\pm 0.07\) | \(0.60\pm 0.03^{40}\) |
| Mg | 12 | 40 | \(0.96\pm 0.06\) | \(5.6\pm 0.4\) | \(0.45\pm 0.02\) | \(0.56\pm 0.05^{40}\) |
| Mg | 12 | 96 | \(1.0\) | \(5.6\pm 0.4\) | \(0.45\pm 0.02\) | \(0.56\pm 0.05^{40}\) |
| Al | 13 | 40 | \(0.75\pm 0.07\) | \(9.0\pm 1.0\) | \(0.3\pm 0.03\) | \(0.40\pm 0.04^{40}\) |
| Al | 13 | 97 | \(0.7\pm 0.1\) | \(9.0\pm 1.0\) | \(0.3\pm 0.03\) | \(0.40\pm 0.04^{40}\) |
| S | 16 | 40 | \(0.54\pm 0.12\) | \(14\pm 5\) | \(0.25\pm 0.05\) | \(0.29\pm 0.04^{43}\) |
by the nucleus of each of the elements present is proportional to \(Z^{90}\); therefore the results entered in columns 4–6 of Table III will be referred to a certain effective value
\[ Z_{\mathrm{eff}}=\frac{\sum Z_i^{2}\dfrac{\tau_z^{-}}{\tau_z^{+}}}{\sum Z_i}, \]
approximately accounting for the share of each element in the total number of recorded decay electrons. Let us now use the data of column 5 of Table III to construct the function \(\Lambda_c(Z)\) on a logarithmic plot (Fig. 24); rectilinear graphical interpolation of the results according to the law
\[ \Lambda_c = \Lambda_d \cdot \left(\frac{Z}{Z_0}\right)^\alpha \]
makes it possible to determine the values of \(\alpha\) and \(Z_0\):
\[ \alpha = 4.1 \pm 0.5 \quad \text{and} \quad Z_0 = 11.5 \pm 0.5. \]
Fig. 24. Dependence of the probability \(\Lambda_c\) of nuclear capture of negative mesons on the atomic number of the substance \(Z\) (according to the data of Table III).
The method of separating mesons of different sign, despite all its clarity, suffers from certain shortcomings that make it difficult to achieve high measurement accuracy. Chief among these are the impossibility of completely excluding particles of one sign from the recorded effect, and the insufficient magnitude of the total effect, associated with the peculiarities of magnetic focusing. Therefore the subsequent experiments were supplemented by registration of individual delay times\({}^{98}\). In this case all delays within \(\Delta t\) from \(0.5\) to \(16.25\ \mu\text{sec}\) were taken into account, and all data were grouped into intervals of duration \(0.25\ \mu\text{sec}\) each.
The data for the point \(0.5\ \mu\text{sec}\) and for all points beginning with \(\Delta t = 9\ \mu\text{sec}\) and greater proved to be inaccurate: in the first case because of the large influence of delays of discharges in the counters, and in the second case because of the small magnitude of the “useful” effect compared with the “background” of accidental coincidences.
Differential decay curves were constructed for the seven investigated substances in the time interval from \(0.75\) to \(8.75\ \mu\text{sec}\). In all cases and for the entire time interval, the experimental points satisfactorily agree with straight lines (on a semilogarithmic scale) describing exponential decay with an average
by a lifetime of 2.15 μsec*). In addition, curves were obtained for the dependence on \(Z\) of the lifetime \(\tau^-\), or of the emission capacity of the substance unambiguously connected with it, \(N_d^{+}+N_d\). In doing so, allowance for corrections for the different stopping powers of different substances enabled the authors to use data for the extreme members of the series of substances investigated (Be and S) in order to separate the effects belonging to capture and decay (see also equation (4), which refers to the extrapolated quantities \(N_d^{+}\) and \(N_d\)). The curves obtained are, in general, in agreement with curves constructed on the basis of a dependence \(\Lambda_c(Z)\) of the type \(\left(\dfrac{Z}{10}\right)^4\), although instead of \(Z_0=10\) the value \(Z_0=7\) is somewhat better suited.
The principal conclusion to be drawn from the data listed above with regard to the nuclear capture of negative mesons is that, in comparison with theory, the interaction constant of these mesons with nuclear particles, in particular protons, turns out to be smaller by about 12 orders of magnitude\({}^{99}\) than the constant which is obtained by applying the same meson theory to the description of nuclear forces. As for the concrete mechanism of this interaction, one might suppose\({}^{99}\) that one of two possible nuclear reactions occurs:
\[ \left. \begin{aligned} p+\mu^- &\to n+h\nu,\\ X+\mu^- &\to n+Y. \end{aligned} \right\} \]
Here \(p\) is a proton, \(\mu\) a meson, \(n\) a neutron, \(h\nu\) a photon, \(X\) the original nucleus, and \(Y\) the final or some intermediate nucleus. In turn, the nucleus \(Y\) may emit some particles as a result of the large excitation energy transmitted to it by the nucleon which has directly absorbed the meson. If this excitation energy is sufficiently large, namely close to the total rest energy of the absorbed meson (i.e. 100 MeV), then “evaporation” of the nucleus with emission of protons and neutrons, as in ordinary “stars,” becomes quite possible; if the excitation is small, emission of a photon is more probable. Thus, among the products of meson capture one may expect the presence of:
1) photons with energies either of the same order, or, conversely, considerably smaller than 100 MeV;
2) neutrons with energies of at least several MeV;
3) protons with average energies usual for “stars,” 5–10 MeV.
As for protons, their very small ranges at the above-mentioned energies make it impossible to use such
* Agreement with the indicated exponential in the presence of mesons of both signs is explained by the fact that a substantial influence of capture on the decay curve studied occurs only for delays \(\Delta t<0.75\) μsec.
amounts of matter that would make it possible to obtain an effect noticeable against the background of random coincidences, even if proportional counters were used to reduce the background. On the other hand, the use of proportional counters instead of ordinary fast counters greatly hampers the separation of delayed coincidences with sufficient accuracy. In view of this, up to now all data allowing one to judge the possible emission of protons upon capture have been obtained by other methods (see § 6).
For the detection of photons, the method of delayed coincidences can be used with greater success, but in doing so one must bear in mind that, in view of the considerations set forth in § 1, only substances with atomic number \(Z \simeq Z_0\) will be accessible, since in other cases the necessary delay intervals will be too inconvenient.
It therefore seems most convenient to register nondelayed coincidences in an apparatus of the type shown in Fig. 11, taking special precautions to single out random coincidences, whose number increases substantially in comparison with the delayed-coincidence method. Such an experiment was carried out in the work mentioned above\(^{54}\), with the aim of detecting photons emitted at the moment when negative mesons stop in the iron filter \(d\) of the apparatus of Fig. 11. As in the case of positive mesons, the author could maintain that the stopping of a negative meson in heavy substances is not accompanied by the emission of photons of appreciable \((h\nu \geq 30\ \mathrm{MeV})\) energy.
Fig. 25. Diagram of an apparatus for investigating delayed neutrons\(^{100}\). Thermal neutrons are registered in counters \(N\), delayed by a time \(\Delta t = 4\text{–}81\ \mu\mathrm{sec}\) relative to the anticoincidences \((A, B, C)\).
The delayed-coincidence method offers the best possibilities for detecting neutrons. In this case, on the one hand, a very small “background” of random coincidences should be observed owing to the use of special neutron counters, while the useful effect remains of the same order as in the case of electrons, since the ranges of these particles in matter are comparable with one another; on the other hand, the time required for slowing the expected neutrons down to thermal velocities (about \(150\ \mu\mathrm{sec}\)) automatically determines the chosen interval of delays independently of the lifetime of the initial meson. These features of the phenomenon were used in two recent works that gave a substantially positive result. In one of them\(^{100}\) the apparatus shown in Fig. 25 was used. Registered (by anticoincidences \(A, B,—C\) in three groups of counters)
mesons stopping in a lead block of thickness \(d\) equal to 7 cm and delayed with respect to them by a time \(\Delta t\) from 4 to 84 μsec; neutrons entering counters \(N\), surrounded by paraffin as a moderator. The observed number of thermal neutrons (the differential effect associated with the removal of the cadmium screen surrounding counters \(N\) was measured) was \(0.34 \pm 0.04\) per hour, which exceeded the expected number of accidental coincidences by approximately a factor of 300. At the same time, the comparatively small value \((0.7\ \text{hour}^{-1})\) of another measured effect, associated not with stopped but with moving mesons (i.e. with coincidences \(AB\)), showed that the principal effect obtained was connected precisely with the capture of stopped mesons. No determination of the efficiency of the counters in this apparatus was made, but a rough estimate of it made it possible to say that, for each stopped meson, on the average about one neutron is emitted.
Another experiment\(^{101}\) differed from the preceding one chiefly in the separation of mesons of different sign by means of a “magnetic lens,” which made it possible to have an additional check on the correctness of the interpretation of the results. The effect recorded here turned out to be considerably smaller in absolute magnitude; it amounted to only four coincidences per 1200 negative mesons stopped in one of the lead blocks \(d\), in the complete absence of events in the case when positive mesons were recorded. An experimental determination of the efficiency of the apparatus for neutrons from an artificial source \((\mathrm{RaC}+\mathrm{Be})\) made it possible to give a more reliable estimate of the number of neutrons per capture process, and the result agreed with the preceding one. However, a repetition with the same method, but with better statistics of measurements not at sea level but at an altitude of about 4 km,\(^{102}\) led the authors to a higher estimate, namely 4–7 neutrons for each stopped meson, which already appears somewhat doubtful.
Unfortunately, all the experiments with delayed neutrons referred to the case of the stopping of mesons in lead, which does not exclude the possibility of interpreting them as due to nuclear fission. However, the results obtained are in satisfactory agreement also with other experiments\(^{103}\), in which, with the aid of an ionization chamber filled with \(\mathrm{BF}_3\), the total number of neutrons generated in different substances was determined at sea level by measuring their fluxes in a definite experimental configuration and determining the intensity of generation by solving the corresponding diffusion equation. The number of neutrons generated (from \(0.08\ \text{g}^{-1}\ \text{hour}^{-1}\) for graphite to \(0.22\ \text{g}^{-1}\ \text{hour}^{-1}\) for lead) proved to be of the same order as the number of stopping mesons, determined many times (see, for example,\(^{41}\)) by the usual method of delayed coincidences.
§ 6. The Wilson Chamber Method
Even at the time when the apparent identity of the “cosmic” meson with Yukawa’s nuclear particle did not arouse particular doubts, attention was drawn^84 to one circumstance that was incomprehensible from this point of view. Among the small number of sufficiently clear photographs in a Wilson chamber that recorded the ends of meson ranges in the chamber gas, in addition to cases of decay with the emission of a single electron, there were also several pictures showing that the stopping of a meson is not accompanied by the appearance of any charged particles. Among these pictures, two cases^6,104 should be especially noted, in which evidently negative mesons stopped in the gas; on the other hand, for positive mesons there apparently exist no reliable photographs in which the decay electron would be absent. In both of the indicated cases the Wilson chamber was filled with a gas sufficiently heavy (argon) for nuclear capture to be possible, in accordance with the results set forth in § 5. From the standpoint of the usual theory of nuclear forces, absorption of the meson should have led to excitation of the nucleus with an energy of the order of 100 MeV (the rest energy of the meson) and, consequently, to a clearly visible nuclear disintegration.
As one of the possible reasons why a “star” was not observed in the chamber, Migdal and Pomeranchuk^84 considered the possibility of the meson leaving the illuminated field of the chamber in a nonionizing state. In this case, the solution of the diffusion equation led them to an estimate of the lengths of such “nonionizing ranges” of the order of 1 cm (in air at normal pressure).
However, the possibility of nuclear capture not accompanied by the emission of ionizing particles (in particular, protons) was fully confirmed by subsequent works, in which the stopping of mesons in thin plates partitioning the Wilson chamber was studied. Among these results, which are much more convincing than those mentioned above, both because of better statistics and because considerations of diffusion are inapplicable, it is sufficient to dwell on the last work of Chang^105. Although the absence of a magnetic field in the chamber did not allow the author to investigate the fate of certainly negative stopped mesons, nevertheless the large number of observed cases (about 80) makes it possible to regard the results obtained as sufficiently convincing *).
In Chang’s work the Wilson chamber was partitioned by a whole series (about 10) of thin plates (of Al, Fe, or Pb) and was controlled by a “telescope” \(BCD\) with three additional groups of counters \(A\), connected in an anticoincidence circuit and placed beneath the chamber and
*) Similar, although less extensive, material had also been obtained in an earlier work^63.
from the side of the “telescopic” counters (Fig. 26). In all variants of the experiments, plates were placed that were sufficiently thin (0.5 mm Pb, 0.7 mm Fe, 0.8 mm Al and, finally, 0.05 mm Al) so that protons from nuclear disintegrations produced by a meson in the plate could be detected with considerable probability; this applies especially to aluminum plates 0.05 mm thick, which corresponds to a range of protons with energy only 2.2 MeV (in the other cases the total thickness of the plate corresponded to a proton energy of up to 15 MeV).
Fig. 25. Diagram of the apparatus used to study mesons stopping in thin plates of a Wilson chamber. The chamber was controlled by anticoincidences (A, C, D, —A).
To characterize the possibilities of this method, let us point out that out of 60 photographs obtained on average in 20 hours of operation, only in one case was a meson stopped in a plate observed, although the mesons themselves constituted about 60% of all stopped particles (the fraction of electrons was reduced by means of a 12-centimeter block of lead placed above the “telescope”). In this identification of mesons was based on two features. First, the author distinguished them (though not always reliably) from electrons and protons by the character of the change in ionization density along the track crossing several plates (the minimum ionization density in the upper part of the track had to be 1 in the case of an electron, about 3 in the case of a meson, and about 8 in the case of a proton stopping at the center of the chamber); the character of scattering of the particle in the plates served as an additional check. Second, the frequency of the observed cases agreed well with the number of particles of the hard component absorbed, in ordinary measurements with the “telescope,” by a layer of material equivalent in mass to all the plates of the given Wilson chamber.
All the photographs obtained by him of mesons stopped in plates are divided by the author into the following four groups:
1) complete absence of any secondary charged particles;
2) emission of a secondary slow proton, observed in all in one case (and even that not a very reliable one), Fig. 27;
3) emission of energetic (with ranges not fitting in the chamber) secondary electrons from the end of the meson track (Fig. 28); in this case the latter evidently should be attributed to the positive meson;
4) appearance of electrons of low energy (apparently of tertiary origin) within 2–5 MeV near the end of the meson track,
...oriented, as a rule, almost in the direction of the end of the track of the stopped meson (Fig. 29).
The total number of cases of each type, referring to each kind of plate used, is given in Table IV, where the data in parentheses include not very reliable results (connected mainly with difficulties in identifying the stopped mesons).
A comparative analysis of the data presented, with account taken of a certain quantitative factor characterizing the probability of operation of the counter group switched to anti-coincidences from decay electrons (owing to which the corresponding number of cases proved to be underestimated), leads the author to the following conclusions concerning the fate of stopped negative μ-mesons.
Fig. 27. One of the rare cases of emission of a secondary slow proton, apparently connected with the stopped meson[^105].
Fig. 28. Decay of a stopped (apparently positive) meson in a thin iron plate inside a Wilson chamber[^105].
1) The energy brought in by the captured μ-meson is usually not directly carried away from the nucleus either by any charged particles or by photons (with energy above 20 MeV); the latter the author infers from the energies observed by him near the end of the meson track of electrons, converted mainly not by pair production, but by the Compton effect or photoelectric effect.
2) Capture is rather often accompanied by the emission of low-energy photons (up to 5 MeV), apparently connected both with the “de-excitation” of the nuclear excitation and with radiation of the meson during its preliminary transitions to ever lower energy levels near the nucleus (the binding energy of the meson on the K shell is about 9 MeV in lead).
Thus, from Chang’s experiments there follows, apparently without any special doubt, the conclusion that the energy released through capture of the μ-meson is carried away by some neutral particle (neutron, neutrino, or neutrino).
Quite different statistical data on the capture of mesons were obtained recently in the work of Vallis et al.,\(^{106}\) who, at an altitude of about 3800 m, used a high-pressure chamber (argon at a pressure of
Table IV
Frequency of various phenomena associated with a stopped meson (according to Chang\(^{105}\))
| Plate in which the meson stopped | Absence of secondary tracks | Presence of decay electron | Presence of secondary proton | Presence of tertiary electrons \((E=2\text{–}5\ \mathrm{MeV})\) |
|---|---|---|---|---|
| Al 0.05 mm | 3 (4) | 1 (2) | — | — |
| Al 0.8 mm | 6 (8) | 3 (5) | — | — |
| Fe 0.7 mm | 11 (14) | 7 (9) | — | 1 |
| Pb 0.5 mm | 17 (19) | 7 (10) | (1) | 1 (7) |
105 atm), surrounded by a thick layer (11 cm) of lead and controlled by means of coincidences of counters selecting the lines. In this work the identification of mesons was carried out by comparing the mean scattering angle with the range in the gas of the chamber, which made it possible
Fig. 29. Photograph in a Wilson chamber illustrating the phenomenon of emission of a secondary electron with an energy of 2–5 MeV, associated with a meson stopped in the plate.\(^{105}\)
in individual cases to distinguish mesons reliably from electrons and protons. The authors observed seven cases of emission of a heavy particle from the end of a meson track (Fig. 30), the mass estimate giving (from scattering and range) values of \(200\text{–}300\,m_e\). However, in another work\(^{107}\) with a high-pressure chamber, carried out at sea level, it was found that mesons stopped in the gas either
give decay electrons if they are positive, or else give no ionizing particles at all if they are negative (as in the preceding case, the chamber was in a magnetic field). It is obvious that the results obtained in \(^{106,107}\) can be reconciled with one another and with the preceding ones (Chang \(^{105}\)) only if it is assumed that in the composition of showers produced in lead, especially at great altitudes, there is a considerable number of charged particles capable, unlike \(\mu\)-mesons, of producing nuclear disintegrations upon capture with the emission of at least one proton.
Fig. 30. Photograph in a high-pressure Wilson chamber, apparently showing a nuclear disintegration caused by the capture in argon of a meson of mass \(200—300\,m_e\) \(^{106}\). \(AB\) is the track of the primary meson, \(BC\) is the track of the secondary proton.
The latter assertion seems very plausible if one takes into account other available data \(^{16}\) on the increased content of heavy particles in a chamber controlled by “special” showers; true, in the case mentioned the heavy particles apparently are more often generated by some neutral penetrating component present in “special” showers.
In conclusion, let us note that some data on the nature of the neutral particles emitted upon capture of a meson can be obtained by studying recoil nuclei in a Wilson chamber. Unfortunately, the data available on this question are so far very scanty: in only one paper \(^{109}\) is there mention of the observation of a recoil nucleus. The latter apparently indicates that the neutral particle mentioned was a neutron. However, other indications, in particular the absence of visible recoil nuclei in photographic plates, speak in favor of the rest energy of the \(\mu\)-meson being carried away by lighter neutral particles.
§ 7. Method of photographic plates
Among the numerous photographic data that make it possible to judge the fate of various negative “cosmic” mesons stopped in the photoemulsion, the two phenomena are the most reliable.
First, various authors have observed a very considerable number of cases of nuclear disintegrations formed at the end of a meson track (thus, for example, in one work alone\({}^{76}\) about 60 of them were observed). Moreover, in the overwhelming majority of those cases in which it was possible to estimate (usually from the mean scattering angle; see, for example,\({}^{109}\)) more or less accurately the mass of the primary particle (the \(\sigma\)-meson), it proved to be close to \(300\,m_e\) [in the most accurate work mentioned above\({}^{76}\), \(M_\sigma=(275\pm15)\,m_e\) was obtained].
Second, a significantly (approximately 10 times, for an altitude of 3 km\({}^{77}\)) larger number of meson tracks in the photoemulsion do not end in any strongly ionizing particles (the so-called \(\rho\)-mesons). At the same time, the prevalence of such cases makes it possible to assert that these are mesons with lifetime \(\tau_0=2\ \mu\mathrm{sec}\), forming the hard component; of them at least a considerable fraction belong to negative stopped mesons.
Measurements of the masses of \(\rho\)-mesons usually give an average value close to \(200\,m_e\) (see, for example,\({}^{76}\)). A more definite estimate of the possible fraction of those stopped negative \(\mu\)-mesons (with mass \(200\,m_e\)) which might be assigned to the category of \(\sigma\)-mesons was made in the work cited above\({}^{77}\) on the basis of the following considerations. On the one hand, a direct count of various cases shows that in the vertical flux (angular interval \(0\text{–}40^\circ\) with the vertical) the number of \(\sigma\)-mesons is about 6% of the number of \(\rho\)-mesons \(N_\rho\), while that of \(\pi\)-mesons is approximately 2% less. On the other hand, out of the total number \(N_\rho+N_\sigma\) the number of negative \(\mu\)-mesons \(N_{\mu^-}\) should be not less than 40% (as we indicated above, this number is not sufficiently reliable), and about half of this number, i.e. 20%, should, according to theoretical and experimental estimates, be captured by heavy Ag or Br nuclei. Assuming that in fact it should be \(N_\sigma=N_\pi^*\), the authors compare these 20% of \(N_\rho+N_\sigma\) with the experimentally obtained value \(N_\sigma-N_\pi\simeq2\%\), whence it follows that only 10% of the negative stopped \(\mu\)-mesons are capable of causing disintegrations. In any case, allowing for all possible inaccuracies and even under the most extreme assumption that all \(\sigma\)-mesons are negative \(\mu\)-mesons, the fraction of those negative \(\mu\)-mesons which, upon capture, cause a disintegration is found to be no more than 40%; and it is quite admissible that there are no such cases at all and, consequently, all the observed “stars” are associated with “heavy” \(\sigma\)-mesons \((M_\sigma=285\,m_e)\).
To what has been set forth one should also add a number of other features characterizing the nature and properties of the “cosmic” mesons that generate “stars” in photoemulsion.
* In other words, it is assumed that the number of negative and positive mesons with mass about \(300\,m_e\) is the same.
First of all it is necessary to note that $\sigma$-mesons are produced in the process of nuclear disintegrations with much greater probability than $\rho$-mesons. Thus, in the work richest in number of cases studied (about 10,000 “stars”)$^{20}$, among 20 mesons of secondary origin$^*)$ at least 17 were $\sigma$-mesons. It would seem that positive $\pi$-mesons, too, should have been emitted in the “stars” rather often; however, their actual absence can quite well be attributed to the large (in comparison with $\sigma$-mesons) initial energy associated with Coulomb repulsion.
Further, the use of highly sensitive photographic emulsions made it possible, in one of the recent works$^{17}$, to detect at the ends of the tracks of $\sigma$-mesons three cases of emission of low-energy electrons (of the order of 15 KeV). Interpreted as the result of the Auger effect in successive transitions of $\sigma$-mesons to their $K$-levels near emulsion nuclei, these cases confirm the initial assumptions, appearing in the calculations for determining the masses $M_\sigma$, that the disintegrations are caused precisely at the very end of the range of the $\sigma$-meson$^{**})$.
Finally, among the “nuclear-active” stopped mesons causing “stars,” mesons with masses exceeding $300\,m_e$ have also recently been found. Thus, for example, in Table I we cited the case of disintegration observed by Leprince-Ringuet$^{80}$, caused by a meson with mass not less than $700\,m_e$. Similar estimates of the masses (400—700 $m_e$) were also given by Lukirskii and Perfilov$^{81}$. These authors selected the so-called “momentumless” disintegrations, in which the sum of the momenta of all charged particles was equal to zero, and assumed that the sum of the momenta carried away by neutral particles in this case is also equal to zero, while their energy is equal to the total energy of the charged particles of the “star.” Then the total amount of energy released in the “star” served as a measure of the rest energy for the particle which caused the disintegration but was invisible because of the low sensitivity of the photographic emulsion. However, the method based on selecting “momentumless” disintegrations and calculating the energy released in them gives rise to objections of principle; indeed, the momentum falling in such a disintegration to the share of neutrons may turn out to be sufficiently large, and therefore the conclusion that such a “star” originated through the capture of a stopped particle appears doubtful.
The idea that mesons with mass about $300\,m_e$ are considerably more active in the sense of nuclear interactions is convincingly confirmed by the totality of published data
$^*)$ Phenomena of meson generation in nuclear disintegrations have been observed many times also in a Wilson chamber$^{110}$; however, in none of these cases can it be indicated exactly which mesons were produced.
$^{**})$ The fact that slow electrons are encountered in these observations still by no means indicates each time, apparently, the predominant role of radiation in the transitions of $\sigma$-mesons to one or another level near a nucleus.
about “artificial” mesons. As has already been mentioned, in one of the papers\(^{35,82}\) a comparison was made of the masses of “artificial” mesons (negative ones) that do and do not cause disintegrations at the end of their ranges. It turned out that the masses of the former are grouped around the value \(300\,m_e\), while the masses of the latter—around the value \(200\,m_e\). Recalling those first experiments which proved the “anomalously” weak nuclear interaction of \(\mu\)-mesons (the delayed-coincidence method), it is natural to compare the probability of spontaneous decay with the probability of capture also for “artificial” mesons. Above, mention has already been made of a determination\(^{37}\) of the lifetime \(\tau\) precisely for \(\sigma\)-mesons. For this purpose, plates registering disintegrations from mesons produced in the phasotron were placed at different depths in channels whose shape coincided with the trajectories of the \(\sigma\)-meson beam in the magnetic field of the phasotron; the decrease in the number of “stars” along the channel served as a measure of the lifetime of the \(\sigma\)-mesons in the beam. At the same time, neither in this nor in other experiments on the study of negative “artificial” mesons were there observed, apparently, cases of the decay \(\sigma \to \mu\). This means that, in contrast to ordinary \(\mu\)-mesons, for mesons with mass about \(300\,m_e\), even in light substances the lifetime emulsion for capture of the negative particle by the nucleus is considerably less than \(10^{-8}\) sec. This circumstance, together with the fact of rather large effective cross sections for generation in collisions of fast nucleons, permits one to hope\(^{111}\) that it is precisely the \(\pi\)- (and \(\sigma\)-) mesons that are the particles directly related to nuclear forces of the exchange type (although, as will be seen below, other possibilities cannot at present be excluded either).
Fig. 31. Distribution of nuclear disintegrations caused by stopped “cosmic” (A) and “artificial” (B) mesons according to the number of emitted charged particles in the “star”\(^{112}\).
However, it should be borne in mind that even upon capture of \(\sigma\)-mesons by nuclei, evidently not all the rest energy \(\mu c^2\) is released in the form of kinetic energy of the products of nuclear disintegration. This is indicated, for example, by a recently performed\(^{112}\) study of the distribution according to the number of particles in “stars” caused in one case by mesons of cosmic radiation, and in another case by artificially generated negative mesons. In both cases rather similar pictures are obtained (Fig. 31), which testify to the release of energy in the “stars” considerably
less than the value $\mu_\pi c^2 = 150\ \mathrm{MeV}$. Unfortunately, the absence of similar data for experiments with high-sensitivity plates does not allow the indicated estimate of the energy release in the “star” to be considered sufficiently reliable.
Summarizing the experimental material presented in this section, one may assert the following:
1) the probability of interaction of $\mu$-mesons with nuclei has proved to be many orders of magnitude smaller than what followed from the existing theory of nuclear forces;
2) among the products of $\mu$-meson capture, neutrons are in any case observed considerably more often than protons, which, together with the fact of emission of photons of low energy, indicates a comparatively small excitation of the nucleus upon meson capture;
3) for $\pi$-mesons the probability of nuclear capture has proved to be many times greater; moreover, for the same mesons both the excitation of the nucleus upon capture and the effective cross section for the inverse process—the production of these particles by fast nucleons—turned out to be many times greater;
4) there are indications of the existence of nuclear-active mesons of other masses, different from $200$ or $300\,m_e$.
IV. POSSIBLE THEORETICAL IDEAS ON THE NATURE AND PROPERTIES OF $\mu$- AND $\pi$-MESONS
The experimental data set forth above on the decay and capture of stopped mesons have led to fundamental changes in theoretical ideas not only for phenomena connected with cosmic radiation, but also, in part, for the problem of nuclear forces. Below an attempt will be made at a brief review of the various qualitative and quantitative constructions that have appeared recently, which make it possible to give a satisfactory, though as yet by no means unambiguous, explanation of one or another of the processes that interest us here. At the same time it will become evident that some new experimental data are still required both for removing the indicated ambiguity and for a decisive quantitative test of one or another of the possible hypotheses.
§ 8. Decay of the $\mu$-meson
The totality of experimental data on the decay products of the $\mu$-meson (i.e., a particle with mass about $200\,m_e$ and lifetime about $2\ \mu\mathrm{sec}$), and in particular on the spectrum of the decay electrons, compels one first of all to consider the possibility of decay not into two (as was formerly thought), but into three particles. In this case, both the more complex character of the decay and the absence of a direct quantitative connection of this phenomenon with nuclear $\beta$-decay lead to the following uncertainties in the initial assumptions of possible
theory: 1) the spin of the initial meson and of the products of its decay is not known (with the exception of the electron); 2) the mass of the neutral decay products is not known, while the mass of the initial meson is known insufficiently accurately.
The data set forth in Section III compel one to suppose that a half-integer spin of the μ-meson is more probable.* As for the nature of the remaining decay products, then, apart from the general consideration concerning the half-integer spin of all elementary particles known up to now, some argument in favor of the half-integer spin of \(11^{3}\) will become clear in the analysis of the results obtained. Most fully, all the possibilities presented for a theory of decay with the participation of particles of half-integer spin have been considered quantitatively in work 114. In this work the case is considered of the transformation of a charged meson either into a neutral one (the process \(\mu^{\pm}\to\mu^{0}+e^{\pm}+\nu\)), or into one of the light particles (the process \(\mu^{\pm}\to e^{\pm}+2\nu\)). The cause of the indicated quantum transition is formally an interaction with the field of light particles (\(e\) or \(\nu\)). As in Fermi’s theory of β-decay, in which the decay is treated as a process of the type \(n+\nu\to e^{-}+p\), here one can calculate the electron spectrum, i.e. the probability \(dW_e\) that an electron with momentum \(p_e\) will be emitted. This probability is expressed by the usual formulae of perturbation theory through the square of the matrix element of the interaction energy \(H\). As in the theory of β-decay, the interaction energy can be expressed through one of the five simplest possible relativistically invariant combinations of Dirac matrices. Thus, for example, in the vector variant of the theory
\[ H=g(1-a_H a_L)(\tau_H\tau_L+\tau_H^{*}\tau_L^{*})\sigma(X_H-X_L), \tag{5} \]
where the index \(H\) refers to the principal particle (meson), and \(L\) to the light particles (electron, neutrino); the operators \(\tau\) are connected with the transition of a heavy or light particle into a particle of another kind (for example, the transformation \(\mu\to\mu^{0}\)); \(X\) are coordinates.
The interaction constant of the “principal” particle with the field of light particles, \(g\), can be determined by computing the total probability of the process (by integration over the electron spectrum) and comparing this probability with the value
\[ \frac{1}{\tau_0}\simeq 5\cdot 10^{-5}\ \mathrm{sec}^{-1}. \]
In almost all variants of the theory the value of the constant \(g\) (\(\sim 10^{-49}\ \mathrm{erg}\cdot\mathrm{cm}^{3}\)) is obtained, in order of magnitude, as coinciding with the analogous constant obtained for the process of spontaneous decay of a free neutron (or β-decay of light nuclei). If substantial significance is attached to this circumstance (for the present it appears to be no more than a coincidence), then it is possible not only to justify the choice of the initial assumptions about the half-integer spins of all particles, but also to choose among the various variants of the given theory. The point is that, in contrast to β-decay, in the present case the various variants of the theory give both
* More will be said about this below.
various forms of the electron spectrum as well as different values (for a given \(g\)) for the total probability of the process.
As for the form of the electron spectrum, so far all variants of the theory are almost equally not in contradiction with experiment, giving
Fig. 32. Forms of the momentum spectra of electrons from decay according to the calculations of Uehler and Tiomno\(^{114}\).
a) Curves for the decay \(\mu \to e+\nu+\mu^{0}\) in the vector variant of the theory for mass values \(\mu = 220\,m_e\) and \(\mu^{0}=0 \div 30\,m_e\).
b) Spectra for the process \(\mu \to e+2\nu\) (for \(\mu = 210\,m_e\)) according to the various variants of the theory (\(1\)—vector and pseudovector, \(2\)—tensor, \(3\)—scalar and pseudoscalar).
spectra spread from 0 to the value \(p_{\max}\), determined by the masses of the heavy particles \(\mu\), \(\mu^{0}\) participating in the process\(^*\), and shifted from \(p=\tfrac{1}{2}p_{\max}\) toward larger momenta (Fig. 32, a and b).
§ 9. Nuclear capture of the \(\mu\)-meson
At present, to explain the weak interaction of \(\mu\)-mesons with nucleons, possibilities of three types are envisaged: either to continue to associate the theory of nuclear forces precisely with \(\mu\)-mesons,
\(^*\) As was indicated, the maximum permissible value of the mass \(\mu^{0}\) is \(30\,m_e\).
modify at the root the initial conceptions of this theory, or to regard the interaction under consideration as a second-order process, possible only through the mediation of other nuclear-active mesons, in particular $\pi$-mesons, or, finally, to consider capture as altogether independent of any nuclear forces.
To the first group of possible conceptions belongs, first of all, the attempt to construct a theory of nuclear forces using the idea of space quantization proposed by Ambartsumian and Ivanenko115; in such a theory one can explain, under certain additional assumptions, why for light nuclei, which have a small radius, the process of capture of a nuclear-active meson proceeds with a very small probability116.
The possibility connected with the paired-meson character of the exchange interaction of nucleons has been investigated in greater detail. Calculations of this kind of interaction with the participation of neutral ($\mu^0$) and charged ($\mu$) mesons with half-integer spins are given in the paper by Markov and collaborators117; moreover, in the latter case it is possible to explain the experimental behavior of the curve of the probability of capture of a $\mu$-meson, if one assumes that the mass difference $\mu-\mu^0$ lies within the limits $10—25\,m_e$. As was shown, however, by the analysis of Weiskopf118 and the experiment of Neher119, who compared the probability of capture in different isotopes of boron, the conceptions indicated above cannot be applied directly, since experiment does not reveal the required dependence of the process on the magnitude of the mass defects of the final products of the reaction. A further development of the same idea, but already independently of nuclear forces, belongs to Wheeler120. In his paper (jointly with Tiomno) he also supposes that the $\mu$-meson and the neutral meson $\mu^0$ associated with it in processes of pair exchange interaction possess spins $\dfrac{1}{2}$ and, consequently, obey the Dirac equations. In this case it is possible to calculate the probability of capture of a $\mu$-meson with the aid of the analogy already mentioned above with Fermi’s theory of $\beta$-decay (more precisely, $K$-capture).
In the present case, in comparison with decay, there appears an additional difficulty connected with the proper allowance for the wave functions of the nucleons in the nucleus. However, in the analysis of three completely different nuclear models it turns out that, apparently, in all cases the excitation function of the final nucleus as a function of the energy of this excitation falls off strongly already at energies of the order of $15\,\mathrm{MeV}$ for any mass of the neutral meson $\mu^0$. This distribution of excitations agrees well with experiment (see the summary at the end of Section III). As for the total probability of capture, it can naturally be brought into agreement with experiment by an arbitrary choice of the interaction constant $g$. In doing so, the authors draw attention to the coincidence of the value obtained for $g$ (in order of magnitude) with that which had already been obtained earlier for interactions with light particles of nucleons
...and mesons separately. However, much more important here is the correct (from the experimental point of view) conclusion about the weak excitation of the final nucleus owing to the emission of a neutral meson, which carries away the greater part of the rest energy of the \(\mu\)-meson. It is precisely this circumstance that is the strongest argument in favor of the half-integral value of the spin of the \(\mu\)-meson used in the above-mentioned decay hypothesis.
Hypotheses of the second type now appear more natural, since experiment has demonstrated both the stronger nuclear interaction of \(\pi\)-mesons and the possibility of mutual transformation of the \(\pi\)- and \(\mu\)-meson (to be sure, only a process of the type \(\pi \to \mu + \mu^{0}\) is directly known; however, from general considerations one can assert the existence of the inverse process and calculate its probability). In this case it is possible to relate quantitatively the probability of capture of a \(\mu\)-meson, as a two-stage process, to the known probabilities of the two indicated simplest processes; independently of the particular variant of the theory, agreement with experiment (in order of magnitude) can be obtained[^121].
Let us consider here the only quantitative theory of this type available so far, connected with the so-called two-meson hypothesis of Bethe and Marshak[^111]. Bethe and Marshak assume that the \(\pi\)-mesons investigated by the photographic-plate method are precisely those nuclear particles which, together with other possible particles, are responsible for the interaction of nucleons in stable nuclei. The unavoidable approximate Yukawa relation for nuclear particles
\[ r_{0} \sim \frac{\hbar}{\mu c} \]
(\(r_{0}\) is the radius of action of the nuclear forces, \(\mu\) is the mass of the meson) in the present case, as special calculations show[^122], is in any case satisfied no worse than for particles with mass \(200m_{e}\). If one abandons in advance the attempt to reduce to the same \(\pi\)-mesons phenomena connected with the \(\beta\)-decay of nuclei, and also neglects the possibility of the existence of charged mesons of other masses, connected in one way or another with the appearance of \(\mu\)-mesons, then it is possible to explain, without any substantial contradictions, the totality of all available data on \(\pi\)- and \(\mu\)-mesons.
Indeed, as experiments on the production of “artificial” \(\pi\)-mesons show, the weak nuclear interaction of \(\mu\)-mesons may lead to the necessity of attributing the hard component of cosmic rays observed experimentally entirely to the decay of \(\pi\)-mesons generated in the upper layers of the atmosphere by the primary proton component. In this case, owing to the considerable difference in the lifetimes of \(\pi\)- and \(\mu\)-mesons, all existing observations of the decay of particles of the hard component will give the time \(\tau_{0} \approx 2\) microsec. This also applies to the data of Vernov and collaborators[^5], which make it possible to distinguish the altitude variation of the hard component almost up to the very boundary of the atmosphere from the absorption curve of primary pro-
tons and the curve of generation of \(\mu\)-mesons, and to calculations\(^{123}\) of the spectrum of the hard component at different altitudes. If one assumes that fast \(\pi\)-mesons have a considerable (though smaller than geometrical) effective cross section for nuclear interactions, then, as noted above, one succeeds in satisfactorily explaining\(^{34}\), in addition, such phenomena as the break in the spectrum of the hard component in the region of higher energies. The most important thing that the two-meson hypothesis now provides is an explanation of the fact of the weak nuclear interaction of \(\mu\)-mesons at a very large, close to geometrical, effective cross section for absorption of the primary proton component. In this case the decay character of the intermediate process of generation of mesons of the other type is probably the only possible way to remove the indicated difficulty\(^{124}\), since the possibility connected with a multiple character of the process of generation of \(\mu\)-mesons\(^{125}\), in its turn, apparently requires\(^{126}\) that the meson field have a large constant of interaction with nucleons.
However, despite all its merits, the two-meson hypothesis at present also gives rise to serious difficulties in the understanding of a number of phenomena of cosmic radiation. First of all, from the indicated point of view it is impossible to explain all processes connected with the generation and decay of varitrons. Also remaining aside are the phenomena, noted above (Section III), of nuclear interactions of mesons with masses different from \(200\) or \(300\,m_e\).
In this connection let us note that recently several attempts\(^{127,128}\) have been made to construct theories describing mesons with a definite mass spectrum; however, the difficulties encountered along this path do not yet allow the indicated concepts to be applied to the phenomena of cosmic radiation.
In conclusion we shall return once more to the problem of determining the spin of mesons of various types, so important for the construction of the corresponding theories. In this direction quite definite results may be given precisely by experiments concerning the capture of stopped mesons by nuclei. In addition to the considerations presented above, connected with the magnitude of nuclear excitation upon capture and compelling one to assume\(^{36}\) half-integer spin for the \(\mu\)-meson and integer spin for the \(\pi\)-meson, possible experiments with meson capture in hydrogen acquire great interest. Apart from questions connected with the specific behavior of such systems as mesohydrogen\(^{130}\), the indicated processes should help in studying the nature of the products of capture and in determining the spin of the meson with a far greater degree of definiteness than in the case of capture by more complex nuclei and, in general, in any other experiments. At the same time the practical feasibility of such problems is excellently illustrated by the investigations, mentioned above\(^{106,107}\), with high-pressure Wilson chambers. As for experiments on the investigation of the decay products of these or other mesons, they may possibly make it possible in the near future to solve
several other problem: knowing the spins of all charged particles participating in the process, to determine the spin of the neutral meson (if it turns out that the “ordinary” neutrino is insufficient to explain all decay processes). Of great importance \(^{131}\) is the study of the decay products also for resolving the question of the equilibrium soft component. At the same time, difficulties in correctly determining its intensity experimentally \(^{8}\) have already led to the fact that precisely knowledge of the nature of the decay makes it possible to draw definite conclusions about the total energy and flux of the equilibrium electron-photon component, and not the other way around.
In conclusion, the author thanks S. N. Vernov and M. A. Markov for their critical review of the manuscript and a number of valuable comments.
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