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CAPTURE OF $\mu$-MESONS BY NUCLEI AND THE STRUCTURE OF INTRANUCLEAR NUCLEON SHELLS
In a number of experiments carried out in 1947–1948,^1 a detailed investigation was made of the behavior of slow $\mu$-mesons of cosmic radiation entering absorbers with different atomic numbers and being stopped in them. These experiments, in principle, consisted of the following: a beam of positively or negatively charged $\mu$-mesons, selected by means of a magnetic field, entered an absorber $P$ (Fig. 1), after first passing through counters $M$. The lower row of counters $P$ recorded the electrons arising in the absorber $P$ in the decay of $\mu$-mesons. By measuring, for a large number of events, the time intervals between the entry of a $\mu$-meson into the absorber (operation of counter $M$) and the emergence from it of the decay particle (operation of counter $P$), it was possible to determine the mean lifetime $\tau_+$ or $\tau_-$ of mesons in absorbers with different $Z$. Such experiments showed that the mean lifetime $\tau_+$ of positively charged $\mu^+$-mesons does not depend on the atomic number of the absorber and is equal to $\tau_+ = 2.2\,\mu\text{sec}$.
Fig. 1.
This result was entirely understandable, since the slow $\mu^+$-mesons, repelled by the Coulomb field of nuclei, cannot approach them to distances at which nuclear forces begin to act, and diffuse in the substance until they decay. It follows from this that the mean lifetime $\tau_+ = 2.2$ microsec is equal to the mean lifetime of positively or negatively charged mesons in vacuum, where nuclear forces do not act on them. For negatively charged $\mu^-$-mesons an unexpected result was obtained. Measurements of the mean lifetime of these particles showed that $\tau_-$ is comparable with $\tau_+$, although smaller than it, and decreases rapidly with increasing $Z$. Before explaining the unexpectedness of this result, let us note that a $\mu^-$-meson, slowed down in an absorber, can disappear for two reasons: 1) it may decay and 2) it may be captured by a nucleus. The probability of decay is equal to $1/\tau_+$; the probability of capture we shall denote by $\Lambda$.
Then the total probability of disappearance of a $\mu^-$-meson in the absorber will be equal to:
\[ \frac{1}{\tau_-} = \Lambda + \frac{1}{\tau_+}, \tag{1} \]
where $\tau_-$ is the mean lifetime of a negatively charged meson, measured in experiments similar to that shown in Fig. 1. If $\mu$-mesons interacted strongly with nuclei, then the capture probability $\Lambda$, in agreement with any version of meson theory, would have to be immeasurably large in comparison with $\dfrac{1}{\tau_+}$, and upon absorption of $\mu^-$-mesons the decay electrons would not arise at all: all $\mu^-$-mesons, within an extremely short time ($<10^{-20}$ sec), would be absorbed by the nucleus, and such a lifetime, of course, could not be measured at all in the experiment shown in Fig. 1. From the fact that $\tau_-$, measured for substances with $Z<16$, proved comparable with $\tau_+$, there followed the conclusion—unexpected in 1947–1948—that the probability of capture of $\mu^-$-mesons by nuclei is small and comparable with the probability of decay $1/\tau_+$. This meant that slow $\mu$-mesons interact very weakly with nuclei. As is known, this conclusion was confirmed by the discovery of $\pi$-mesons. It turned out that it is precisely these particles which interact strongly with nuclei, while $\mu$-mesons are secondary with respect to $\pi$-mesons and arise in their decay.
Measurements made during 1947–1948 for substances with small $Z$ ($4 \leq Z \leq 16$) showed that at $Z \sim 10$ the probability of capture of a $\mu^-$-meson is approximately equal to the probability of its decay. Therefore (see formula (1)) the mean lifetime of a negatively charged meson absorbed in a substance with $Z \sim 10$ is not equal to $2.2$ microsec, but is close to $1$ microsec. With increasing $Z$ the capture probability $\Lambda$ increases rapidly, proportionally to the fourth power of $Z$. Such a dependence of $\Lambda$ on $Z$, confirmed also by theoretical calculations[^2], can easily be qualitatively understood from the following considerations. Before being captured by the nucleus, the $\mu^-$-meson is usually located near the nucleus in its $K$-orbit. Therefore the capture probability is inversely proportional to the volume bounded by the $K$-orbit. This gives proportionality to $Z^3$, since the orbit radius is proportional to $1/Z$, and the volume to $1/Z^3$. In addition, the capture probability is proportional to the number of protons in the nucleus, $Z$, which can capture a $\mu^-$-meson located in the $K$-orbit, and thus one obtains $\Lambda \sim Z^4$. Such a sharp dependence of $\Lambda$ on $Z$ leads to the fact that for $\mu^-$-mesons, cap-
FROM CURRENT LITERATURE
stopped in sulfur (\(Z=16\)), the mean lifetime turns out to be approximately \(0.5\ \mu\mathrm{s}\), i.e., close to the limit that can in general be measured with Geiger counters. With a further increase in \(Z\), the mean lifetime \(\tau_-\) becomes so small that its measurement requires counters that make it possible to carry out measurements in the millimicrosecond time range. As is known, such requirements are met by scintillation counters made from certain organic substances, which possess a very short fluorescence afterglow time, of the order of \(10^{-8}\)—\(10^{-9}\) sec. The paper under review\(^3\) is devoted to measuring the probability of capture of slow \(\mu^-\)-mesons in substances with large \(Z\), carried out with the aid of scintillation counters. Until quite recently, the use of such counters for the study of cosmic radiation was limited by the impossibility of obtaining transparent crystals of scintillating substances with sufficiently large surface area and volume, as required for the study of rare cosmic-ray phenomena. For the experiments under consideration, liquid scintillation counters were used (a solution of terphenyl in toluene, \(3.5\ \mathrm{g}\) per liter), which have great transparency for their fluorescence radiation. The surface and volume of such counters can therefore be made sufficiently large, and since the afterglow time of these counters is close to \(3\cdot 10^{-9}\) sec, with their aid it becomes quite possible to measure mean lifetimes of the order of tens of millimicroseconds. Before considering the scheme of the experiments performed, let us recall that a \(\mu^-\)-meson stopped in a substance can be detected by two signs: by the appearance of the decay electron and by the neutrons arising when it is captured by a nucleus. Since \(\mu^-\)-mesons interact only weakly with nuclei, their penetration into the nucleus does not lead, as is known, to the formation of noticeable stars characteristic of the absorption of a \(\pi^-\)-meson by a nucleus. Most of the excitation energy (\(100\ \mathrm{MeV}\)) introduced into the nucleus by the \(\mu^-\)-meson is carried away by the neutrino, and only a small part of the energy, about \(15\ \mathrm{MeV}\), remains in the nucleus; in the case of heavy nuclei with a large Coulomb barrier this is insufficient for the ejection of a proton from the nucleus. Therefore the capture of a negatively charged \(\mu^-\)-meson by a nucleus leads in most cases to the emission of a neutron. The process of interaction between the \(\mu^-\)-meson that has entered the nucleus and an intranuclear proton is itself described by the scheme
\[ P+\mu^- \longrightarrow n+\nu, \tag{2} \]
where \(\nu\) is a neutrino, and \(n\) is a neutron.
Fig. 2.
It follows from what has been said that the capture of a \(\mu\)-meson by a nucleus can be detected by the emission of a neutron from the nucleus. It was precisely this method of detecting the absorption of \(\mu^-\)-mesons that was used in the paper under review. Its advantages over the method of registering decay electrons are obvious: at large \(Z\), when the capture probability \(\Lambda\) is considerably greater than the decay probability \(1/\tau_+\), most \(\mu^-\)-mesons are captured rather than decay, and therefore the measurement of \(\Lambda\) can be carried out with considerably greater statistical accuracy from the neutrons produced in capture than from the decay electrons. The scheme of the experiments performed is shown in Fig. 2. Here \(T\) is an absorber serving to stop slow \(\mu\)-mesons, and \(S_1\) and \(S_2\) are scintillation counters. Counter \(S_1\) had dimensions \(30\times 30\times 2.5\ \mathrm{cm}^3\), counter \(S_2\), \(30\times 7.5\times 7.5\ \mathrm{cm}^3\). Counter
\(S_1\) recorded the entry into the absorber of a slow meson. To register the particles that emerged from the absorber \(T\), a second scintillation counter \(S_2\) served. The radio circuit connected with this system of counters consisted of two parts. The first part of the circuit selected, with a sufficiently large resolving time, coincidences caused by the phenomenon under study: such coincidences were the coincidences of discharges in the Geiger counters \(A\) and in both scintillation counters \(S_1S_2\), not accompanied by a discharge in the anticoincidence counters (row \(X\)). Obviously, such cases corresponded to entry into the absorber of a slow meson and to the emission from it of a nonionizing particle, which did not cause the anticoincidence counters \(X\) to operate.
Fig. 3.
The second part of the circuit was intended for measuring the time intervals between the operation of counters \(S_1\) and \(S_2\) in the cases selected by the first part of the circuit. The time intervals were measured by means of a circuit known as a “chronotron.” The principle used in this circuit for measuring the time between the appearance of two pulses consists in the fact that the pulses from counters \(S_1\) and \(S_2\) are fed to the beginning and the end of a delay line, which is usually a high-frequency cable. If the pulses from \(S_1\) and \(S_2\) occurred simultaneously, they meet in the middle of the cable. In the remaining cases the addition of the pulses will occur at the point of the cable situated closer to the end into which the delayed pulse was introduced. By placing along the cable a sufficient number of nonlinear pulse receivers, sensitive to the summed pulses \(S_1 + S_2\) and insensitive to single pulses \(S_1\) or \(S_2\), one can carry out the measurement of the time \(\tau_{-}\) with an accuracy determined by the ratio of the time for propagation of the pulse along the cable to the number of pulse receivers placed along the cable. Without entering into the details of this method, which deserves special consideration, we shall give the final results of the measurements.
The authors obtained decay curves and measured the lifetime \(\tau_{-}\) of mesons in six elements: Fe, Cu, Sb, Hg, Pb, and Bi. The corresponding values of \(\tau_{-}\) are equal to \(163 \pm 27\), \(116 \pm 9\), \(99 \pm 11\), \(58 \pm 4\), \(76 \pm 4\), and \(68 \pm 5\) millimicroseconds. Knowing the value of \(\tau_{-}\), one can, from (1), calculate the capture probability \(\Lambda\). The corresponding values of \(\Lambda\) are shown in Fig. 3, on whose abscissa the nuclear charge of the element \(Z\) is plotted, and on the ordinate—the probability of capture \(\Lambda\). The curve corresponds to the theoretical dependence \(\Lambda = Z^4\). We see that, in the case of substances with small \(Z\) (Fe, Cu), the quantity \(\Lambda\) is in agreement with the predictions of the theory. Let us note that the values of \(\Lambda\) for elements with \(Z \leqslant 16\), measured in papers \(^{2}\), also lie well on this curve. In the case of elements with large \(Z\) there is observed a sharp divergence from the law \(\Lambda \sim Z^4\). The explanation of this divergence is the principal interest of the work under consideration. At the basis of the calculations \(^{2}\), which led to the law \(\Lambda \sim Z^4\), lay the simplest hypotheses about the structure of the nucleus: it was assumed that the nucleons in the nucleus form a Fermi gas, and that the number of protons in the nucleus is equal to the number of neutrons. This approximation proved sufficient for light nuclei. The observed, for large \(Z\), deviation from the law \(\Lambda \sim Z^4\) is explained by the fact that, in deriving this law, the complex character of the nucleus was not taken into account, whose nucleons form
as is known, closed shells. Calculations taking into account the excess of neutrons in heavy nuclei, and the influence of the presence of closed nucleon shells on the absorption of $\mu^-$-mesons, were carried out in work 4. These calculations showed that the capture probability in the case of heavy nuclei should indeed be considerably smaller than follows from the law $\Lambda \sim Z^4$. Considering Fig. 3, we see that $\Lambda$ for mercury ($Z=80$) is considerably larger than for lead ($Z=82$). This result, obtained with great statistical accuracy, seems incomprehensible, but it finds a full explanation from the standpoint of the theory of nucleon shells. Indeed, the shells in Hg have a less completed structure than the shells in Pb, and the larger value of $\Lambda$ for Hg reflects the fact that in the Hg nucleus there are additional, as compared with lead, final states for the neutron arising as a result of reaction (2). Thus, from the works considered it follows that the study of the probabilities of absorption of slow $\mu^-$-mesons by various nuclei is a new method for analyzing the structure of intranuclear nucleon shells. It is interesting to note that $\mu^-$-mesons, precisely because of their weak interaction with the nucleus, are a sufficiently “delicate” agent which may make it possible to obtain new data on intranuclear shells.
A. V.
CITED LITERATURE
- Conversi, Pancini, Piccioni, Phys. Rev., 71, 209 (1947); Ticho, Phys. Rev., 74, 1337 (1948).
- Wheeler, Rev. Mod. Phys., 21, 133 (1949).
- Keaffel, Harrison et al., Phys. Rev., 87, 942 (1952).
- Kennedy, Phys. Rev., 87, 953 (1952).