MESOATOMS

D. D. Ivanenko and G. E. Pustovalov

§ 1. INTRODUCTION

The supposition that negative mesons can revolve for some time around a nucleus before being captured by the nucleus or undergoing spontaneous decay, thus forming a mesoatom[^1], was experimentally confirmed several years ago. First of all it was shown that cosmic mesons, when stopped in heavy elements, give rise to $\gamma$-quanta with energies of the order of several MeV, which agrees in order of magnitude with the transition energy $2P \to 1S$ of a meson in heavy mesoatoms[^2]. The existence of $\pi$-mesoatoms ($\pi$-atoms) was definitively established by Camac and others[^3] in the study of the characteristic X-rays arising when mesons, produced with the aid of accelerators, are stopped. In 1952 $\mu$-mesoatoms ($\mu$-atoms) were discovered in a similar way by Fitch and Rainwater[^4], and their radiation was investigated throughout the entire periodic system up to $Z = 83$ in 1953.

In comparison with an ordinary electronic hydrogen-like atom, a mesoatom possesses a number of specific features: the proximity of the meson to the nucleus, the possibility of capture of the meson by the nucleus, the nuclear interaction of the meson with the nucleons of the nucleus, the simultaneous presence in the atom of particles of different kinds (electrons and a meson) in orbitals, different meson spins, etc. The study of mesoatoms is therefore of great interest and can provide much new information on the structure of nuclei and on particle interactions.

The existence of a mesoatomic stage when negative mesons are stopped in matter leaves a peculiar and very deep imprint on the processes of nuclear capture and decay of negative mesons. This circumstance has led to the appearance of a considerable number of works connected in one way or another with the study of the properties of mesoatoms. The phenomena accompanying the stopping of mesons in matter, the capture of mesons into an orbit by the Coulomb field of the nucleus, the formation of mesoatoms, radiative transitions and Auger transitions—called, in the case of mesoatoms, conversion transitions—and the capture of a meson from an orbit by the nucleus have been examined in detail in a number of works[^5–^8]*.

A $\mu$-meson, with its relatively long lifetime $\tau = 2.22 \cdot 10^{-6}\ \mathrm{sec}$ and small probability of absorption by nucleons in light mesoatoms, most often decays before it has time to be captured by the nucleus. Therefore the lifetime of such mesoatoms is determined in practice by the lifetime of the meson itself. However, for $\mu$-mesoatoms with $Z \gg 10$, nuclear capture begins to predominate, so that the lifetime of heavy $\mu$-mesoatoms is determined by the probability of capture by the nucleus and is of the order of $10^{-8}\ \mathrm{sec}$[^8]. On the other hand, even in the lightest mesoatoms a $\pi$-meson has a high probability of being captured by the nucleus before decay, as a result of which the levels of $\pi$-mesoatoms prove to have a considerable width. The time

* In addition to the literature on these questions cited in review[^7], see also works[^9–^20].

the lifetime of pionic hydrogen is only \(10^{-10}\) sec, whereas the intrinsic lifetime of the \(\pi\)-meson is \(2.6\cdot 10^{-8}\) sec.

In the first approximation, the values of the energies, orbital radii, and transition probabilities for mesoatoms are given by the formulas for ordinary electronic hydrogen-like atoms. The Bohr radius of the orbit is then \(\mu/m_e\) times smaller than the radius of the electronic orbit (\(m_e\) is the electron mass, \(\mu\) is the meson mass). Electrons rotating far from the nucleus do not exert any noticeable effect on a meson moving close to the nucleus. In heavy mesoatoms (for example, in \(\mu\)-mesolead), the meson spends approximately half of its lifetime, \(\sim 4\cdot 10^{-8}\) sec, inside the atomic nucleus, traversing during this time in nuclear matter, which has a density \(\sim 10^5\) t/mm\(^3\), i.e. \(\sim 10^{17}\) g/cm\(^2\), a distance of about 5 m. Therefore the shape and dimensions of the nucleus have a strong influence on the entire system of energy levels of the meson in the mesoatom. Consequently, a meson in a mesoatom is a much more effective means, compared with an electron in an ordinary atom, for investigating the properties of the nucleus: its dimensions, shape, the distribution over the volume of the nucleus of positive charge, the quadrupole electric moment of the nucleus, as well as forces of non-electromagnetic origin acting

Fig. 1. Influence of various properties of the nucleus and the \(\mu\)-meson on the magnitude of the transition energy \(2P\to 1S\) in \(\mu\)-mesoatoms of tantalum and lead\(^{21}\). \(Q\)—quadrupole electric moment of the nucleus; \(T\)—nuclear spin.

between the meson and the nucleons\(^{21}\). On the other hand, the study of mesoatoms can clarify certain information about the meson itself (for example, about its magnetic moment and mass). Fig. 1 shows how strongly the influence of various properties of the nucleus and meson affects the magnitude of the transition energy \(2P\to 1S\) in heavy \(\mu\)-atoms.

§ 2. EXPERIMENTAL INVESTIGATION OF MESOATOMS

We have already said that mesoatoms can be obtained artificially with the aid of accelerators. A typical setup for investigating mesoatom spectra\(^{22}\) is shown in Fig. 2. Negative \(\pi\)-mesons, obtained by braking protons in a target inside the cyclotron, are focused by the magnetic field of the cyclotron itself and by special magnets and are directed through a system of counters into an absorber made of a substance containing the element whose mesoatoms it is desired to obtain. The beam also contains \(\mu\)-mesons, obtained

formed in the decay of \(\pi\)-mesons, and the number of \(\pi\)-mesons decreases while that of \(\mu\)-mesons increases as the distance from the cyclotron is increased. Consequently, with the aid of such an installation one can study both \(\pi\)- and \(\mu\)-mesoatoms.

The X-rays associated with the stopping of a meson in an absorber are recorded by a scintillation counter coupled to a photomultiplier. The pulses from the photomultiplier are detected by a multichannel amplitude analyzer, which is calibrated with the aid of some X-ray sources of known energy. In some cases the calibration was carried out by means of radiation from mesoatoms, for which the transition energies were known \(^{23}\).

For measuring the transition energies of certain mesoatoms, it proved possible to use critical-absorption techniques. In this method filters are used which are placed between the absorber and the photon counter. The transition energy is compared with the well-known sharp absorption edge of the \(K\)-series of X-rays of the filter material. By selecting filters with different \(Z\), one can establish the limits between which the transition energy lies. In particular, this method made it possible to refine the values of the masses of the \(\mu\)- and \(\pi\)-mesons \(^{24,25}\).

In the same installation it is possible to determine not only the magnitude of the transition energy, but also the quantum yield, i.e., the number of mesons whose stopping in the absorber was accompanied by the emission of a photon of a definite energy \(^{22,26—29}\). With the aid of the quantum yield one can find the ratio of the probability of capture of a meson by the nucleus from a given orbit to the probability of a radiative transition. This ratio, for example, for the probability of capture \(W_{2P}\) from the \(2P\) state and the probability \(W_{2P\to 1S}\) of the transition \(2P\to 1S\), must be proportional to \(Z^{2}\), since the probability of the radiative transition \(2P\to 1S\) is proportional to \(Z^{4}\), whereas the probability of nuclear capture is proportional to the number of protons in the nucleus \(Z\) and to the probability of finding the meson in the region occupied by the nucleus, i.e., to the quantity \(|\psi_{2P}(0)|^{2}\), proportional to \(Z^{5}\). Hence the probability of capture of a meson by the nucleus from the \(P\)-state is proportional to \(Z^{6}\). The experimental data \(^{22}\) agree with the dependence predicted theoretically by Marshak and Messier \(^{30}\) for this ratio on \(Z\):

\[ \frac{W_{2P}}{W_{2P\to 1S}}=0.2Z^{2}. \]

Fig. 2. Schematic of an installation for studying mesoatom spectra \(^{22}\).
a — general schematic: proton orbit with energy 240 MeV; Al target; cyclotron; shielding; meson channel; counter system; focusing magnets.
b — counter system: aluminum wedges; X-ray filter; meson beam; absorber in which mesoatoms are formed; photomultiplier.
\(1, 2, 3, 4, 6\) — scintillation counters registering mesons; \(5\) — counter (NaI crystal) registering photons arising in the transition of mesons in mesoatoms; counters \(1, 2, 3, 5\) are connected in coincidence, \(4, 6\) in anticoincidence. With the aid of aluminum wedges, one can ensure that the greater part of the mesons stops in the absorber.

Comparison of the experimental and theoretical values of the transition energies \(2P \to 1S\) in \(\mu\)-mesoatoms with values of \(Z\) from 13 to 83 led to an unexpected result for nuclear sizes \(^{4,31}\). It proved necessary, in the formula for nuclear radii \(R = R_0 A^{1/3}\), to take for the constant \(R_0\) the value \(1.2 \cdot 10^{-13}\ \text{cm}\) instead of the value \(1.4 \cdot 10^{-13}\ \text{cm}\), which had seemed to be well confirmed by many data. We shall not now dwell on the discussion concerning nuclear sizes, noting only that the distributions of protons and neutrons may not coincide, that the question concerns the radii of effective spheres in the case of nonspherical nuclei, and that for different effects the effective nuclear sizes may apparently differ somewhat.

§ 3. ACCOUNT OF THE NUCLEAR VOLUME

Owing to the smearing of the charge over the finite volume of the nucleus, the levels of mesoatoms will be shifted upward (weakening of the interaction) in comparison with those expected for point nuclei. This volume effect is well known in the theory of the isotopic shift of ordinary, i.e. electronic, atoms, where it is, however, far less significant than in the case of mesoatoms considered here.

The influence of the nuclear volume is greatest on \(S\)-levels, smaller on \(P\)-levels, still smaller on \(D\)-levels, etc. Because of this, in mesoatoms even the usual order of levels changes. For example, in \(\mu\)-mesosulfur the level \(2S\) lies above the levels \(2P_{1/2}\) and \(2P_{3/2}\), and the level \(3S\) lies above the levels \(3P_{1/2}\), \(3P_{3/2}\), \(3D_{3/2}\), and \(3D_{5/2}\). In \(\mu\)-mesolead, moreover, the levels \(3P_{1/2}\) and \(3P_{3/2}\) lie respectively above the levels \(3D_{3/2}\) and \(3D_{5/2}\) (Fig. 3).

In principle, several mesons may be present in an atom, so that one may speak of constructing an analogue of the periodic system of \(\mu\)-mesoelements. Obviously this will not apply to \(\pi\)-mesons in view of the bosonic character of \(\pi\)-mesons, which may be present in any number on one level. Since, owing to the finite volume of the nucleus, the order of levels is changed, the properties of the system of \(\mu\)-mesoelements will not coincide with the properties of the usual electronic Mendeleev system.

To find the eigenvalues of the energy and eigenfunctions of the meson in a mesoatom, we must therefore, having specified one or another most reasonable (according to nuclear-physics data) distribution of positive charge in the nucleus, solve the problem of the motion of the meson in the electric field of the nucleus. For \(\mu\)-mesons one usually solves the Dirac equation; for \(\pi\)-mesons—the relativistic scalar (or, coinciding with it in the present case, pseudoscalar) Klein–Gordon equation.

The simplest distribution is that of protons with a density constant inside the nucleus and equal to zero outside the nucleus. Then inside the nucleus there will be a Thomson oscillator potential, and outside it a Coulomb potential:

\[ \begin{aligned} V(r) &= \frac{eZ}{R}\left[\frac{3}{2}-\frac{1}{2}\left(\frac{r}{R}\right)^2\right]\quad (r<R),\\ V(r) &= \frac{eZ}{r}\quad (r>R). \end{aligned} \tag{1} \]

Finding the eigenfunctions and eigenvalues of the energy of the meson with a combined potential even of such a simple form is a rather difficult problem, one that has no exact solution.

To determine the change in the energy of the meson in light mesoatoms (up to \(Z \sim 10\)) due to the influence of the nuclear volume in the nonrelativistic approximation, one may use perturbation theory. As the perturbation one may take the difference between the Coulomb and Thomson potentials inside

…of the nucleus. Outside the nucleus the perturbation is equal to zero. Then

\[ \Delta E_{nl}=\int \psi_{nl}^{*}\,\Delta V\psi_{nl}\,d\tau =\int_{0}^{R}|R_{nl}(r)|^{2} \left\{\frac{Ze^{2}}{r}-\frac{Ze^{2}}{R}\left[\frac{3}{2}-\frac{1}{2}\left(\frac{r}{R}\right)^{2}\right]\right\}r^{2}\,dr . \tag{2} \]

Here \(R\) is the nuclear radius; \(R_{nl}(r)\) are the radial wave functions of the Kepler

Fig. 3. Energy levels of \(\mu\)-mesolead and \(\mu\)-mesoantimony, calculated:
\(1\)—without taking into account the volume of the nucleus, using the Dirac equation;
\(2\)—for a constant density of the nuclear charge distribution;
\(a\)—using the Dirac equation by the method of matching the wave functions at the boundary of the nucleus;
\(b\)—using the nonrelativistic Schrödinger equation by the variational method (for antimony, relativistic corrections of order \(\alpha^{2}Z^{2}\) are taken into account).
Near the levels their energies in MeV are written. The meson mass is \(207\,m_e\); the nuclear radius is \(1.2\cdot10^{-13}A^{1/3}\,\mathrm{cm}\) \({}^{44}\).

problem for the mesoatom. Making use of the values of the radial wave functions at \(r=0\), it is easy to obtain:

\[ \left. \begin{aligned} \frac{\Delta E_{n0}}{E_{n0}}&=-\frac{4}{5n}\left(\frac{ZR}{b}\right)^{2};\\ \frac{\Delta E_{n1}}{E_{n1}}&=-\frac{2(n^{2}-1)}{105\,n^{3}}\left(\frac{ZR}{b}\right)^{4},\\ \frac{\Delta E_{n2}}{E_{n2}}&=-\frac{2(n^{2}-4)(n^{2}-1)}{225\,n^{5}}\left(\frac{ZR}{b}\right)^{6}. \end{aligned} \right\} \tag{3} \]

Here \(b=\dfrac{\hbar^2}{\mu e^2}\) is the Bohr radius of the mesonic orbit, \(\mu\) is the reduced mass of the meson.

The influence of the size and shape of the nucleus is especially strong on the position of the energy levels of mesoatoms with large \(Z\). In this case the finite size of the nucleus cannot be regarded as a small perturbation, and the zeroth approximation cannot be taken to be the solution of the Kepler problem. Indeed, the radius of the Bohr orbit of a meson in a mesoatom already for \(Z\sim 30\) becomes comparable with the nuclear radius. For lead the radius of the Bohr orbit is 2.3 times smaller than the nuclear radius. In this case the influence of the nuclear volume becomes so considerable that the energy of the \(1S\) level turns out to be 2–2.5 times smaller than according to the formulas for point nuclei.

For heavy mesoatoms, as the unperturbed problem one may take the solution for an infinitely extended oscillator potential, and regard as the perturbation the difference between the Coulomb and oscillator potentials outside the nucleus[^32].

The same problem can be solved by the variational method[^8,^33], which, with a suitable choice of variational wave functions, gives quite good results also for mesoatoms with large \(Z\). The advantage of the variational method consists in the fact that the wave functions obtained in this way have a comparatively simple analytic expression and can be used in solving a whole series of other problems (see, for example, the following paragraph).

A more exact solution is obtained by the method of matching the wave functions at the boundary of the nucleus. In this method, at the nuclear boundary one requires continuity of the wave functions for the case of the Dirac equation and continuity of the wave function and of its first derivative for the case of the Klein–Gordon equation, i.e., for unnormalized wave functions, equality of the ratios:

\[ \left.\frac{F_i}{G_i}\right|_{r=R} = \left.\frac{F_e}{G_e}\right|_{r=R}, \qquad \left.\frac{R_i}{R_i'}\right|_{r=R} = \left.\frac{R_e}{R_e'}\right|_{r=R}. \tag{4} \]

Here \(F_i, G_i\) and \(F_e, G_e\) are the radial parts of the wave functions of the Dirac equation, and \(R_i\) and \(R_e\) are the radial parts of the wave function of the Klein–Gordon equation. The indices \(i\) and \(e\) refer to the solutions inside and outside the nucleus respectively; \(R\) is the nuclear radius. The energy enters the wave functions as a parameter, and therefore the relations (4) are equations for determining the eigenvalues of the energy.

The solutions of the Dirac and Klein–Gordon equations which vanish at infinity for the wave functions outside the nucleus can be expressed by means of Whittaker functions or else through combinations of degenerate hypergeometric functions. The solutions inside the nucleus may be sought in the form of generalized power series. The eigenvalues of the energy are found by numerical solution of equations (4). This method is suitable not only for a constant density of the distribution of protons in the nucleus, but also for any other spherically symmetric distribution of protons in a finite region of space, if it can be represented in the form of an expansion in powers of \(r\)1.

One can also pose the inverse problem: having experimental information on the energy levels of mesoatoms, it is required to select a suitable distribution of charge density in the nucleus. For this purpose the Dirac equation for the particular case of \(\mu\)-mesolead was solved by numerical integration on an electronic calculating machine for various forms of density distribution

charge in the nucleus[^37]. In this connection it turned out that, if an increase of the charge density toward the center of the nucleus leads to a shift of the levels \(1S\), \(2P_{1/2}\), and \(2P_{3/2}\) upward by approximately one and the same amount, then the level \(2S\) is shifted downward by an amount roughly four times smaller (Fig. 4). It is true that so far, for \(\mu\)-mesolead, only the energy of the transition \(2P \to 1S\) has been measured; therefore no conclusions can yet be drawn about the charge distribution.

Fig. 4. Various forms of the distribution of charge density in the nucleus and the corresponding energy levels of \(\mu\)-mesolead. The values of their energies in MeV are written beside the levels. (According to[^37]. Curves 1, 2, 3 belong to family I with values \(n = 0, 1, 3\), respectively; curve 4—to family II with \(n = 4\); curve 5—to family III with \(n = 8,\ s = 3\); 6—constant charge density. The energy of the transition \(2P_{3/2} \to 1S\) was chosen in all cases to be about \(6\) MeV, in agreement with experiment.)

Let us also note that the meson in a mesoatom in turn exerts some influence on the nucleus, causing a deformation (polarization) of the nucleus[^31],[^38]. In this process rotational levels of the nucleus may be excited, which leads to the appearance of a fine structure of the mesonic spectral lines[^39],[^40].

§ 4. VACUUM POLARIZATION IN MESOATOMS

An interesting and significant correction to the energy of mesoatomic levels has proved to be the allowance for vacuum polarization[^41],[^42]. Whereas the Lamb shift of the energy levels of the electron in hydrogen is due mainly to the correction to the field electromagnetic mass of the electron or to the influence of vacuum fluc-

…i.e., of zero-point photon oscillations, and only approximately \(1/25\) of the shift is due to vacuum polarization, i.e., the influence of fluctuations of vacuum electrons—positrons; in mesoatoms we have a different picture. Vacuum polarization of electrons—positrons changes the electrostatic potential of the nucleus in the field of which the meson moves, independently of the mass of the meson orbiting in the orbit, moreover at distances of the order of the Compton wavelength of the electron \(\sim 10^{-11}\) cm, i.e., larger than the dimensions of the nucleus but of the same order as the radii of mesonic orbits. At the same time the correction to the proper electromagnetic mass, inversely proportional to the square of the mass of the moving particle, will be considerably smaller in mesoatoms, owing to the larger mass of the meson, than in the case of the Lamb shift for the electron. The influence of vacuum polarization is noticeable at small \(Z\) and for levels with large orbital angular momenta, when the influence of the nuclear volume is small. This is clearly seen in Fig. 5.

Fig. 5. Dependence on \(Z\) of the magnitude of the splitting of the 2nd level in light mesoatoms due to various effects: 1 — relativistic splitting without taking into account the nuclear volume and vacuum polarization; 2 — the nuclear volume is taken into account; 3 — vacuum polarization is also taken into account. For zero the nonrelativistic energy level is taken. Meson mass \(207\,m_e\), nuclear radius \(1.2\cdot 10^{-13} A^{1/3}\) cm. Note that in mesoatoms lighter than boron, as a result of the influence of vacuum polarization, the normal order of the levels is restored (the level \(2P\) is above the level \(2S\)).\(^{44}\)

The shift of the energy level of a meson in a mesoatom, caused by vacuum polarization, is determined by the expression (see, for example,\(^{43}\))

\[ \Delta E_{nl}=-4\alpha e\int_{0}^{\infty} I_{nl}(k)\,\varphi_{0}(k)\times \]

\[ \times\left[ \frac{4k_{0}^{2}-2k^{2}}{3k^{2}} \left( 1-\frac{\sqrt{4k_{0}^{2}+k^{2}}}{k}\operatorname{Arsh}\frac{k}{2k_{0}} \right) +\frac{1}{9} \right]\,dk, \tag{5} \]

where

\[ I_{nl}(k)=\int_{0}^{\infty} r\,|R_{nl}(r)|^{2}\sin kr\,dr; \tag{6} \]

\(\alpha=\dfrac{e^2}{\hbar c}\) is the fine-structure constant; \(R_{nl}(r)\) are the radial wave functions of the meson; \(\varphi_0(k)\) is the Fourier component of the potential of the external field causing vacuum polarization, i.e., in our case, the potential of the nucleus; \(k_0=\dfrac{m_e c}{\hbar}\) (\(m_e\) is the electron mass). In the case of a point nucleus, when

\[ \varphi_0(k)=\frac{eZ}{2\pi^2 k^2} \tag{7} \]

and

\[ I_{10}(k)=k\,\frac{1}{(1+\zeta^2)^2}, \]

\[ I_{20}(k)=k\,\frac{1-12\zeta^2+32\zeta^4}{(1+4\zeta^2)^4}, \tag{8} \]

\[ I_{21}(k)=k\,\frac{1-4\zeta^2}{(1+4\zeta^2)^4}, \quad \text{etc.} \]

\[ \left(\zeta=\frac{bk}{2Z},\quad b=\frac{\hbar^2}{\mu e^2}\ \text{is the Bohr radius of the mesonic orbit}\right), \]

integral (5) can be evaluated by contour integration\({}^{44}\). This gives

\[ \Delta E_{nl}=-\frac{\alpha^3 Z^2\mu c^2}{3\pi n}\,K_{nl}(\varepsilon_n), \tag{9} \]

where

\[ \left. \begin{aligned} \varepsilon_n&=\frac{n m_e}{\mu\alpha Z},\\[4pt] K_{10}(\varepsilon)&=-\frac{11}{3}-4\varepsilon^2 +\pi\left(\frac{3}{2}\varepsilon+2\varepsilon^3\right) +(2-\varepsilon^2-4\varepsilon^4)\Phi(\varepsilon),\\[4pt] K_{20}(\varepsilon)&=-\frac{16}{3}-14\varepsilon^2 +\pi\left(\frac{3}{2}\varepsilon+7\varepsilon^3\right) +\frac{3}{4}(1-\varepsilon^2)^{-1}\\ &\quad+\frac{9}{4}(1-\varepsilon^2)^{-2} +\left[\frac{13}{4}+4\varepsilon^2-14\varepsilon^4 -\frac{9}{4}(1-\varepsilon^2)^{-2}\right]\Phi(\varepsilon),\\[4pt] K_{21}(\varepsilon)&=-\frac{14}{3}-10\varepsilon^2 +\pi\left(\frac{3}{2}\varepsilon+5\varepsilon^3\right) +\frac{5}{4}(1-\varepsilon^2)^{-1}\\ &\quad+\frac{3}{4}(1-\varepsilon^2)^{-2} +\left[\frac{11}{4}+2\varepsilon^2-10\varepsilon^4 -(1-\varepsilon^2)^{-1} -\frac{3}{4}(1-\varepsilon^2)^{-2}\right]\Phi(\varepsilon),\\[4pt] \Phi(\varepsilon)&=(1-\varepsilon^2)^{-\frac12} \ln\left\{\varepsilon^{-1}\left[1+(1-\varepsilon^2)^{\frac12}\right]\right\}. \end{aligned} \right\} \tag{10} \]

Analogous, although rather cumbersome, formulas can also be obtained for the shifts of higher levels. The results of calculations by these formulas are given in Figs. 6 and 7.

Let us consider the influence of the nuclear volume on vacuum polarization. It can be shown that taking into account the distribution of charge over the nuclear volume substantially diminishes the effect of vacuum polarization for atoms with large \(Z\) (see\({}^{45}\), and also\({}^{46}\)). If, as the source producing vacuum polarization, we take a nucleus with charge uniformly distributed throughout its volume, then the Fourier component of the nuclear potential (1) will be

\[ \varphi_0(k)=\frac{eZ}{2\pi^2 k^2}\,\frac{3}{R^2k^2} \left(\frac{\sin kR}{kR}-\cos kR\right). \tag{11} \]

Here \(R\) is the radius of the nucleus. To take into account the change in the meson wave functions, described

in order to determine the shift of levels in mesoatoms due to vacuum polarization, with account taken of only terms of order \(\alpha Z\). As shown in work \(^{47}\), taking account of higher-order terms for a point nucleus, even in the case of muonic uranium, leads to a shift that amounts to less than \(0.02\%\) of the level energy. Evidently, allowing for the volume of the nucleus can only reduce this figure.

For the sake of simplicity, using the nonrelativistic Schrödinger equation, under the influence of the finite nuclear volume, we take as trial wave functions for the \(1S\), \(2S\), and \(2P\) states, respectively,

\[ \left. \begin{aligned} R_{10}(r)&=2\left(q\frac{Z}{b}\right)^{3/2}\exp\left\{-q\frac{Z}{b}r\right\},\\ R_{20}(r)&=\frac{2\sqrt{3}\,s\left(s\frac{Z}{b}\right)^{3/2}}{\sqrt{s^{2}-qs+q^{2}}} \left[1-\frac{Z}{3b}(s+q)r\right]\exp\left\{-s\frac{Z}{b}r\right\},\\ R_{21}(r)&=\frac{2}{\sqrt{3}}\left(t\frac{Z}{b}\right)^{5/2}r\exp\left\{-t\frac{Z}{b}r\right\}, \end{aligned} \right\} \tag{12} \]

with variational parameters \(q\), \(s\), and \(t\), which, as usual, are determined from the condition of minimum energy, taking into account that the potential is defined by formulas (1). Then we have:

\[ \left. \begin{aligned} I_{10}(k)&=\frac{k}{(1+\zeta^{2}q^{-2})^{2}},\\ I_{20}(k)&=\frac{3s^{2}k}{s^{2}-sq+q^{2}}\times\\ &\quad\times\left[ \frac{1}{(1+\zeta^{2}s^{-2})^{2}} -\frac{s+q}{3s}\frac{3-\zeta^{2}s^{-2}}{(1+\zeta^{2}s^{-2})^{3}} +\frac{(s+q)^{2}}{3s^{2}}\frac{1-\zeta^{2}s^{-2}}{(1+\zeta^{2}s^{-2})^{4}} \right],\\ I_{21}(k)&=k\frac{1-\zeta^{2}t^{-2}}{(1+\zeta^{2}t^{-2})^{4}}. \end{aligned} \right\} \tag{13} \]

(We note that here \(q<1\), \(s,t<\frac{1}{2}\); for \(q=1\), \(s=t=\frac{1}{2}\), formulas (13) become (8).) To find the level shift, one must substitute (11) and (13) into (5). Figure 7 shows the values of the energy-level shift of \(\mu\)-mesoatoms, taking the nuclear volume into account, obtained by numerical integration. As an example, let us point out that for \(\mu\)-mesolead the polarization shift of the \(1S\) level without taking account of the nuclear volume is \(217\ \mathrm{keV}\), while with the volume taken into account it is only \(53\ \mathrm{keV}\); for the \(2S\) and \(2P\) levels the shifts are, respectively, \(37\) and \(33\ \mathrm{keV}\) without taking account of the nucleus, and \(17\) and \(28\ \mathrm{keV}\) with the volume taken into account.

Fig. 6. Dependence on \(Z\) of the magnitude of the shift of the energy levels of \(\pi\)-mesoatoms owing to vacuum polarization \(^{44}\).

The influence of vacuum polarization was in fact discovered experimentally. In order to obtain a value of the $\mu$-meson mass consistent with the data of other experiments, when comparing theory with the experimental values of the transition energies $2P \to 1S$ in $\mu$-mesocarbon, $3D \to 2P$ in $\mu$-mesophosphorus, and $4F \to 3D$ in $\mu$-mesosilicon, it was necessary to introduce a correction for vacuum polarization ^24.

Tables I and II give, for comparison, the relative changes in the transition energy $2P \to 1S$ in $\pi$- and $\mu$-mesoatoms with $Z \leqslant 10$, due to the principal

Labels in the figure: $\Delta E$ in keV; solid curve — “without taking account of nuclear volume”; dashed curve — “with taking account of nuclear volume”; mass of the $\mu$-meson $=207m_e$; nuclear radius $=1.4\cdot 10^{-13} A^{1/3}\,\text{cm}$; levels $1S$, $2S$, $2P$, $3S$, $3P$, $3D$; horizontal axis $Z$.

Fig. 7. Dependence on $Z$ of the magnitude of the shift of the energy levels of $\mu$-mesoatoms due to vacuum polarization ^44.

effects causing level shifts (apart from the shift of levels in $\pi$-mesoatoms due to the nuclear interaction of the $\pi$-meson with the nucleons of the nucleus). The uncorrected values of the transition energy were obtained by solving the nonrelativistic Schrödinger equation with a Coulomb potential. Relativistic corrections were found from the usual fine-structure formulas for the Dirac equation in the case of $\mu$-mesoatoms and the Klein–Gordon equation in the case of $\pi$-mesoatoms, with accuracy up to and including terms of order $\alpha^2 Z^2$. The finite nuclear volume was taken into account by perturbation theory (see formulas (3)), and vacuum polarization by formulas (10). For the meson masses and nuclear radii the following values were adopted: $m_\mu = 207m_e$, $m_\pi = 272.5m_e$, $R = 1.2\cdot 10^{-13} A^{1/3}\,\text{cm}$.

Table I

$\pi$-mesoatoms

Table II

$\mu$-mesoatoms

§ 5. INTERACTION OF THE π-MESON WITH THE NUCLEONS OF THE NUCLEUS

In π-mesic atoms the strong interaction of the meson with the nucleons making up the nucleus must manifest itself. The magnitude of the shift of the meson energy levels in a mesic atom, caused by this interaction, cannot be calculated directly, since a completed mesodynamics does not exist. However, it is possible to compare experiments on the scattering and absorption of π-mesons with the experimentally determined magnitude of the level shifts in π-atoms.

The motion of the meson in a mesic atom can be visualized as the totality of scatterings of the meson in the field of the nucleus. If, in the scattering of a meson by a nucleon, the energy transferred to the nucleon is neglected, then in the Born approximation the change in the scattering amplitude \(a\), as compared with the scattering amplitude in the Coulomb field of the nucleus (or the phase shift with zero angular momentum \(\delta_0\), i.e. in the \(S\)-state), as well as the magnitude of the energy-level shift for a bound state of the meson in first-order perturbation theory, will be proportional to the mean additional potential energy of interaction of the meson with the nucleon:

\[ a=-\frac{\delta_0}{k}=-\frac{\bar{\mu}}{2\pi\hbar^2}\int \Delta V\,d\tau;\qquad \Delta E=|\psi(0)|^2\int \Delta V\,d\tau. \tag{14} \]

Here \(k=\dfrac{p}{\hbar}\), \(\psi(0)=\dfrac{1}{\sqrt{\pi}}\left(\dfrac{Z}{b}\right)^{3/2}\) is the value of the meson wave function at zero (\(p\) is the meson momentum, \(b=\dfrac{\hbar}{\mu e^2}\) is the Bohr radius of the mesonic orbit, \(\bar{\mu}\) is the reduced mass of the meson). Hence, for the shift of levels in π-mesic hydrogen, using the relation of the theory of isotopic spin,

\[ a=\frac{2}{3}a(T=1/2)+\frac{1}{3}a(T=3/2), \tag{15} \]

it is easy to obtain:

\[ \frac{\Delta E}{E_{1S}}=-\frac{4}{b}a=-\frac{4}{b}\,\frac{2\delta_1+\delta_3}{3k}, \tag{16} \]

where \(\delta_1\) and \(\delta_3\) are the phase shifts characterizing the \(S\)-scattering of a π-meson by a nucleon due to the nuclear interaction in states of isotopic spin \(T=1/2\) and \(T=3/2\), \(E_{1S}=-\dfrac{e^2Z^2}{2b}\) is the energy of the unperturbed \(1S\) level. For a nucleus with \(Z\) protons and \(N\) neutrons we have:

\[ \frac{\Delta E}{E_{1S}}=-\frac{4Z}{bk} \left(\frac{2}{3}Z\delta_1+\frac{3N+Z}{3}\delta_3\right). \tag{17} \]

Here the additivity of the scattering effects on protons and neutrons has been assumed (in the scattering of π-mesons by neutrons only the state of isotopic spin \(T=3/2\) participates). Finally, for nuclei with an equal number of protons and neutrons, one obtains for the shift of the \(1S\) level a quadratic dependence on \(Z\) (see\({}^{48}\), and also\({}^{49}\)):

\[ \frac{\Delta E}{E_{1S}}=-\frac{8Z^2}{b}\,\frac{\delta_1+2\delta_3}{3k}. \tag{18} \]

Experimental observations\({}^{22,23,50,51}\) of the relative shifts of the \(1S\) level in various π-atoms up to fluorine not only confirmed the proportionality to \(Z^2\), but also gave good agreement with the results of the experiment on determining the phase shift in the scattering of π-mesons by nucleons. From (18)

taking into account the experimental data on the shift of the \(1S\) level (see Fig. 8), one obtains

\[ \frac{\delta_1+2\delta_3}{3\eta}=-1.1^\circ\pm0.2^\circ \left(\eta=\frac{k\hbar'}{\mu c}\right). \tag{19} \]

On the other hand, the phase analysis of scattering, carried out by Orear \(^{52}\), led to the relations \(\delta_1=+9.2^\circ\eta=0.16\eta,\ \delta_3=-6.3^\circ\eta=-0.11\eta\) (for the region \(20\text{–}60\) MeV), the substitution of which into formula (19) gives precisely \(-1.1^\circ\). This is consistent with the relation \((\delta_1-\delta_3)>0\). Taking these data into account, formula (18) assumes the form

\[ \frac{\Delta E}{E_{1S}}\simeq \frac{Z^2}{850}. \tag{20} \]

Fig. 8. Relative change in the magnitude of the \(2P\to1S\) transition energy in \(\pi\)-mesoatoms due to the nuclear interaction of the \(\pi\)-meson with the nucleons of the nucleus. Experimental results \(^{23,51}\): \(a\)—from the position of the peak in a multichannel amplitude analyzer; \(b\)—from the selective absorption of mesoatom radiation in substances with different \(Z\). Calculations for nuclei with \(Z=N\): 1—according to Deser—Goldberger—Baumann—Thirring \(^{48}\); 2—according to Brueckner \(^{53}\); \(x\)—calculations by formula (17):

\[ E_{2P}-E_{1S}\approx \frac{3}{4}E_{1S}. \]

It should also be noted that analogous calculations were carried out for the case when the nucleon, in the scattering of a \(\pi\)-meson in the intermediate state, acquires a large momentum. In the real case such an intermediate state corresponds to absorption of the meson by the nucleus with the formation of a star \((\pi^-+\text{nucleus}\to\text{star})\). Then, taking into account the experimental data on the probability of absorption of mesons by nuclei, for the shift of the \(1S\) level one obtains, for nuclei with \(Z=N\), according to Brueckner \(^{53}\):

\[ \frac{\Delta E_1}{E_{1S}}=\frac{Z^2}{985}+i\frac{Z^2}{2150}. \tag{21} \]

The imaginary part gives the width of the level connected with absorption of the meson from the \(K\)-orbit. The total shift is then determined by the sum of (20) and (21):

\[ \frac{\Delta E}{E_{1S}}=\frac{Z^2}{456}+i\frac{Z^2}{2150}. \tag{22} \]

Comparison of the experimental data on the directly observed shift of the \(1S\) level in mesoatoms with the formulas of Deser—Goldberger—Baumann—Thirring (20) and Brueckner (22) is shown in Fig. 8. The experiment gives good agreement with (20) and somewhat smaller values for the shift than (22), which Brueckner explains by the insufficient accuracy of measuring the phases of mesons in scattering with small energies, and also by the simplifications made in deriving these formulas.

Later experiments on the study of the \(K\)-series of the spectra of \(\pi\)-mesoatoms, performed with the aid of a proportional counter, which gives greater accuracy in this energy range (up to 100 keV) than scintillation counters, also showed agreement with the formulas of Deser et al., and not with Brueckner’s formulas (see Table III) \(^{54}\).

Thus, from the very fact of the existence of an additional upward shift of the \(1S\) levels after allowance for the volume of the nucleus, vacuum polarization, and relativistic corrections, there follows the fundamental conclusion that between nucleons and the meson there is strong repulsion at small distances of the order

\(\sim 4.2 \cdot 10^{-14}\) cm (approximately twice the Compton wavelength of the nucleon). Analogous forces of nuclear type, but leading to attraction, appear between an electron and a neutron in ordinary atoms and, apparently, manifest themselves rather noticeably in isotopic mixing (see, for example, in \(^{36}\), § 7, 2).

Let us note in this connection that the question of phase shifts in the scattering of mesons by nucleons has recently been examined in every respect, both experimentally and theoretically. Obviously, the elucidation of the interaction of pions with nucleons must in turn help to clarify the more complicated question of the interaction of nucleons through mesons, i.e., in particular, the question of nuclear forces and nucleon scattering. The values of the phases and their behavior with changing energy have been considerably refined recently and have confirmed the correctness of the hypothesis of charge independence of nuclear forces and the possibility of restriction to \(S\)- and \(P\)-scattering at moderate energies. As for theoretical conclusions, without going into details, nevertheless, in view of the special importance of this question, it should be noted that modern pseudoscalar mesodynamics has in the end succeeded, under certain simplifying assumptions, in satisfactorily explaining many of the basic regularities of the phase shift in \(S\)- and \(P\)-scattering of pions by nucleons in both states of isotopic spin \(T = \frac{1}{2}\) and \(T = \frac{3}{2}\), and, in particular, has led, in agreement with experiment, to the conclusion that the most significant is the phase shift \(\delta_{33}\) (\(P_{3/2}\)-wave, \(T = 3/2\)), which in the region near 200 MeV reaches \(90^\circ\), i.e., passes through a resonance. This resonance is also confirmed by analysis of meson photoproduction. Earlier, to explain the resonance, semiphenomenological models of an excited isobaric state of protons and others were put forward. In the region of small energies the experimental values of \(\delta_{33}\) grow as the cube of the momentum (\(q^3\)), and in the region roughly after 30 MeV even somewhat faster. These results are confirmed by analysis of pion scattering at energies up to 300 MeV \(^{54-56}\).

It should be noted that the principal phase \(\delta_{33}\) satisfies the Chew–Low relation obtained from the solution of the Low equation:

\[ \left(\frac{q^3}{\omega^*}\right)\operatorname{ctg}\delta_{33} = 8.05 - 3.8\omega^* \tag{23} \]

(\(\omega^*\) is the initial energy of the meson minus the rest energy of the proton). Hence, at small energies \(\delta_{33} = 0.235 q^3\), and the resonance value \(\delta_{33} = 90^\circ\) is reached at 192 MeV*). This important equation for the phases was established on the basis of conditions of causality and invariance, taking into account the presence of a meson cloud around the nucleon core. It is a step forward in comparison with the treatment of processes by perturbation theory and the Tamm–Dancoff method. Analysis of pion and \(\pi\)-mesoatom scattering makes it possible to determine the character of the specific nonelectromagnetic interaction of pions with nucleons (pseudoscalar, pseudovector coupling, the role of nonlinear terms in the coupling) and, in particular, to determine the magnitude of the interaction constant; the best value of the latter at the present day is: \(f^2/\hbar c = 0.08\), where \(f\) is the \(\pi\)-meson pseudovector “charge” of the nucleon.

In conclusion we emphasize that the successful theoretical analysis of meson scattering, carried out on the basis of very general relations, has been the principal achievement of mesodynamics in recent years and has convincingly shown that, contrary to various attempts to create pessimistic moods by declaring almost a “collapse” of mesodynamics, meson theory has not only not exhausted its possibilities but, on the contrary, is developing, “pulling itself up” to the level of electrodynamics. The questions of eliminating the deep difficulties of the present field theory,

*) On the theoretical analysis of the Low equation see \(^{57-65}\).

to a considerable degree common to electrodynamics and mesodynamics and connected above all with the divergence of the field contributions to masses, electric charges, and other coupling constants of particles, and all the more so with the failure of attempts to derive definite empirical values for these quantities, will, in all likelihood, be solved by a radical generalization of the theory, possibly in the direction of taking into account some minimal length.

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1. On the solution by the matching method with allowance for the finite volume of the nucleus, see works[^4,^34,^35], and also in[^36], §§ 7, 8. ↩