Abstract
The present review aims to systematize the main experimental data concerning the properties of $\pi$ -mesons and their interaction with nucleons and nuclei. The conclusions that can be drawn on the basis of these works are in most cases qualitative, since at present there is no sufficiently coherent theory of nuclear interactions. Therefore, we will often confine ourselves to reporting experimental results without their theoretical interpretation. The review does not consider processes of multiple production of $\pi$ -mesons, which begin to play a noticeable role at energies of the colliding particles above 1000 MeV. Part I considers the properties of $\pi$ -mesons and the regularities of their interaction with free nucleons. Part II is devoted to a review of works on the interaction of $\pi$ -mesons with nuclei.
Full Text
$\pi$-Mesons
(Review of Experimental Data)
L. M. Barkov and B. A. Nikol’skii
The purpose of the present review is to systematize the principal experimental data concerning the properties of $\pi$-mesons and their interaction with nucleons and nuclei. The conclusions that can be drawn on the basis of these studies are, in most cases, qualitative, since at present there exists no sufficiently complete theory of nuclear interactions. Therefore we shall often confine ourselves to reporting experimental results without their theoretical interpretation. The review does not consider processes of multiple production of $\pi$-mesons, which begin to play a noticeable role at energies of the colliding particles above 1000 MeV.
Part I considers the properties of $\pi$-mesons and the regularities of their interaction with free nucleons. Part II is devoted to a review of work on the interaction of $\pi$-mesons with nuclei.
PART I
INTERACTIONS OF $\pi$-MESONS WITH NUCLEONS
1. PROPERTIES OF $\pi$-MESONS
$\pi$-mesons are particles with a mass approximately equal to 270 electron masses, which interact strongly with nuclear matter. The existence of such particles was predicted by Yukawa, who introduced, to explain the nature of short-range nuclear forces, the so-called “heavy quanta”—mesons.
Three types of $\pi$-mesons are distinguished: positively charged, negatively charged, and neutral. The charge of electrically charged $\pi$-mesons is, in absolute value, equal to the charge of the electron.
$\pi$-mesons are unstable particles. In vacuum, $\pi^{+}$- and $\pi^{-}$-mesons decay into a $\mu$-meson and one neutral particle, apparently a neutrino. The energy of the $\mu$-meson produced in $\pi \to \mu$ decay is equal to 4.1 MeV$^{1}$. In studies of $\pi \to \mu$ decay in plates, in approximately 0.01% of cases so-called “short-range $\mu$-mesons”$^{2,3}$ were observed, whose track length was considerably smaller than the mean value of the ionization range of the $\mu$-meson in $\pi \to \mu$ decay. This phenomenon is satisfactorily explained if one assumes that here radiative decay of the $\pi$-meson takes place according to the scheme:
$$ \pi \to \mu + \nu + \gamma. \tag{1} $$
Table I gives the relative probability, calculated by Ioffe and Rudik$^{4}$, of radiative decay of the $\pi$-meson as a function of the ionization range of the $\mu$-meson.
According to Fry’s data[^3], for \(\dfrac{R}{R_0}=0.8\) the ratio
\[ \frac{w(\mu+\nu+\gamma)}{w(\mu+\nu)}=(3.3\pm1.3)10^{-4}, \]
which is in good agreement with calculated data. Fry et al.[^25]
Table I
Probability of radiative decay of the \(\pi\)-meson
| \(R/R_0\) | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 |
|---|---|---|---|---|---|---|---|
| \(\dfrac{w(\mu+\nu+\gamma)}{w(\mu+\nu)}\) | 4 | 2.8 | 1.9 | 1.3 | 0.88 | 0.52 | 0.24 |
\(R\)—length of the full ionization range of the \(\mu\)-meson,
\(R_0\)—range of the \(\mu\)-meson in the \(\pi\to\mu+\nu\) decay,
\(\dfrac{w(\mu+\nu+\gamma)}{w(\mu+\nu)}\)—relative probability of radiative decay of the \(\pi\)-meson as compared with the usual case, when the range of the \(\mu\)-meson \(\leq R\).
The decay of the \(\pi^+\) meson into a \(\mu\)-meson and an electron pair was also discovered:
\[ \pi\to\mu+\nu+2e . \]
The neutral \(\pi\)-meson decays into two \(\gamma\)-quanta: \(\pi^0\leftrightarrow\gamma+\gamma\). The decay of the \(\pi^0\)-meson with formation of an electron and positron was also observed, according to the scheme[^5]:
\[ \pi^0\to\gamma+e^+ + e^- . \tag{2} \]
The relative probability of the latter process is approximately \(1\%\).
Table II
Masses of \(\pi^+\)-, \(\pi^-\)-, and \(\pi^0\)-mesons
| Type of \(\pi\)-meson | Mass of the \(\pi\)-meson in electron masses | Reference |
|---|---|---|
| \(\pi^+\) | \(275.1\pm2.5\) | 6 |
| \(\pi^+\) | \(273.4\pm0.2\) | 7 |
| \(\pi^+\) | \(273.4\pm1.1\) | 8 |
| \(\pi^-\) | \(275.2\pm2.5\) | 9 |
| \(\pi^-\) | \(272.5\pm0.3\) | 7 |
| \(\pi^-\) | \(268.8\pm1.8\) | 10 |
| \(\pi^-\) | \(272.5\div273.3\) | 11 |
| \(m_{\pi^-}-m_{\pi^0}\) | \(10.6\pm2\) | 9 |
| \(m_{\pi^-}-m_{\pi^0}\) | \(8.8\pm0.6\) | 12 |
The masses of \(\pi\)-mesons have been determined by many authors. The results of the most accurate measurements are given in Table II.
As can be seen from Table II, the values of the masses of the \(\pi^+\)- and \(\pi^-\)-mesons are close to one another. The neutral meson is somewhat lighter than the charged ones:
\[ m_{\pi^-}-m_{\pi^0}\simeq \]
\[ \simeq 8.6 \text{ electron masses.} \]
There exists a point of view that all three types of \(\pi\)-mesons are different states of one and the same particle1, and the mass difference of the charged and neutral \(\pi\)-mesons is electromagnetic in character.
All three types of \(\pi\)-mesons have integral spin, i.e., obey Bose statistics. This follows, for example, from the fact that \(\pi\)-mesons are produced in nucleon–nucleon collisions2 in reactions of the type:
\[ p+p\rightleftarrows \pi^+ + d . \tag{3} \]
Since the total angular momentum of a system of two nucleons can take only integer values, the spin of the \(\pi\)-meson must be integral. At the same time the spin of the \(\pi^0\)-meson cannot be equal to unity, since a particle with spin \(S=1\) cannot decay into two \(\gamma\)-quanta*). It has been possible to determine the spin directly only for the case of the \(\pi^+\)-meson, by measuring the cross sections of the direct and inverse reactions (3). According to the principle of detailed balance, these cross sections are related by
\[ \left(\frac{d\sigma}{d\Omega}\right)_p = \frac{4}{3(2S+1)}\, \frac{p_p^2}{p_\pi^2} \left(\frac{d\sigma}{d\Omega}\right)_\pi , \tag{4} \]
where \(S\) is the spin of the \(\pi\)-meson, \(p_p\) and \(p_\pi\) are the momenta in the center-of-inertia system of the proton and the \(\pi\)-meson, respectively, \(\left(\frac{d\sigma}{d\Omega}\right)_p\) is the differential cross section of the process
\[ p+p\to \pi^+ + d, \]
and \(\left(\frac{d\sigma}{d\Omega}\right)_\pi\) is the differential cross section of the process
\[ \pi^+ + d\to p+p. \]
Of course, relation (4) retains its form also for the case when the total cross sections of reaction (3) are measured. From measurements of the differential and total cross sections of reaction (3) it was found that the spin of the \(\pi^+\)-meson is equal to zero\({}^{13}\). There are grounds for believing that the spins of the \(\pi^-\)- and \(\pi^0\)-mesons are likewise equal to zero. One argument in favor of such a conclusion is the fruitfulness of the hypothesis of isotopic spin\({}^{**}\), according to which all three types of \(\pi\)-mesons must be regarded as different states of one and the same particle.
Another important property of the \(\pi\)-meson is parity. The study of the reaction of capture of a slow \(\pi^-\)-meson by deuterium\({}^{9,14}\),
\[ \pi^- + d \to n+n, \tag{5} \]
made it possible to conclude that the \(\pi^-\)-meson is a pseudoscalar particle, i.e. possesses negative intrinsic parity. Indeed, if a \(\pi^-\)-meson is captured by deuterium in an \(S\)-state, then the total angular momentum of the system \((\pi^-+d)\) is equal to unity (under the assumption that the spin of the \(\pi^-\)-meson is zero), and the two neutrons formed in the reaction considered must have total angular momentum equal to unity. Since, further, the \(\Psi\)-function of the system of two neutrons must be antisymmetric, they can then be only in a \(P\)-state. A system of two nucleons in a \(P\)-state is described by an odd \(\Psi\)-function\({}^{***}\). Consequently, the system \((\pi^-+d)\) must also be described by an odd \(\Psi\)-function. Since the deuteron is a particle of even parity\({}^{****}\), reaction (5) is not forbidden only if the intrinsic parity of the \(\pi^-\)-meson is equal to \(-1\).
There is an indication that the \(\pi^0\)-meson is also a particle with odd intrinsic parity. In the works of Panofsky et al.\({}^{9}\) and Steinberger et al.\({}^{15}\) it was shown that the capture of a slow \(\pi^-\)-meson by deuterium with formation of a \(\pi^0\)-meson according to the reaction:
\[ \pi^- + d \to n+n+\pi^0 \tag{6} \]
*) L. D. Landau, DAN USSR 60, 207 (1948).
\({}^{**}\)) See the beginning of the following section.
\({}^{***}\)) The orbital parity of the \(\Psi\)-function of a system of particles is equal to \((-1)^L\), where \(L\) is the orbital angular momentum of the system. The intrinsic parity of the nucleons plays no role in this consideration, since the number of nucleons in reaction (5) does not change.
\({}^{****}\)) See, for example, A. Akhiezer and Pomeranchuk, Some Questions of Nuclear Theory, Gostekhizdat, 1951.
is not observed. In any case, according to the estimates of Steinberger et al.¹⁵, the probability of such a process is \(10^3\) times smaller than the probability of capture of a \(\pi^-\)-meson by reaction (5)*). The small probability of capture of a \(\pi^-\)-meson by reaction (6) can be explained if it is assumed that the parities of the \(\pi^-\)- and \(\pi^0\)-mesons coincide, since in this case reaction (6) is strongly forbidden on energetic grounds.
At present there are no direct experimental data on the determination of the parity of \(\pi^\pm\)-mesons; however, it is most probable that all three types of \(\pi\)-mesons are odd particles.
The results of measurements for determining the lifetime of \(\pi^+\)- and \(\pi^-\)-mesons
Table III
Values of the lifetime \((T)\) of \(\pi^+\)- and \(\pi^-\)-mesons relative to \(\pi \to \mu\) decay
| \(\pi^+\)-mesons | \(\pi^+\)-mesons | \(\pi^-\)-mesons | \(\pi^-\)-mesons |
|---|---|---|---|
| — in units of \(10^{-8}\) sec | reference | in units of \(10^{-8}\) sec | reference |
| \(2.8 \pm 0.6\) | 16 | \(2.54 \pm 0.11\) | 19 |
| \(2.85 \pm 0.25\) | 17 | \(2.62 \pm 0.12\) | 20 |
| \(2.55 \pm 0.19\) | 18 | \(2.65 \pm 0.12\) | 21 |
| \(\dfrac{T_{\pi^+}}{T_{\pi^-}} = 1.01 \pm 0.11\) | 17 | \(2.58 \pm 0.14\) | 22 |
| \(2.53 \pm 0.1\) | 23 | ||
| \(2.44 \pm 0.18\) | 18 |
relative to \(\pi \to \mu\)-decay, carried out in recent years, are given in Table III.
The \(\pi^0\)-meson decays into two \(\gamma\)-quanta:
\[ \pi^0 \to \gamma + \gamma \tag{7} \]
with lifetime \(T_{\pi^0} < 10^{-14}\) sec**).
As is seen from Table III, the lifetimes of \(\pi^+\)- and \(\pi^-\)-mesons relative to \(\pi \to \mu\)-decay are the same to within experimental errors. The \(\pi^0\)-meson apparently does not decay in a similar way, and it is unknown whether there exists a neutral analogue of the \(\mu^+\)- and \(\mu^-\)-mesons. Since one may imagine that the process of decay of the \(\pi^0\)-meson into two \(\gamma\)-quanta proceeds through an intermediate state with virtual formation of a proton and an antiproton according to the scheme***):
\[ \pi^0 \to p_1 + \widetilde{p}_1 \to p_2 + \gamma_1 + \widetilde{p}_1 \to \gamma_1 + \gamma_2, \]
the difference in the lifetimes of the charged and neutral \(\pi\)-meson may be connected with the additional possibility of radiative decay of the \(\pi^0\)-meson and is not proof of a difference between these particles.
*) The authors consider that the quoted value of the probability is an upper limit, since the cases they observed of formation of a \(\pi^0\)-meson by reaction (6) may be attributed to the presence in the deuterium of a small admixture of hydrogen.
**) According to Panofsky’s measurements²⁴, the most probable value of the lifetime of the \(\pi^0\)-meson is \(T_{\pi^0} = 5 \cdot 10^{-15}\) sec.
***) See, for example, E. Fermi, Elementary Particles, IL, 1951.
2. SCATTERING OF π-MESONS BY NUCLEONS
Phase analysis of the scattering of π-mesons by nucleons. Fermi used an analysis of the angular distribution of π-mesons in collisions with nucleons to test the validity of the hypothesis of charge invariance. According to this hypothesis, nucleons and π-mesons are assigned a new variable having the properties of spin—isotopic spin. It is assumed that the isotopic spin of nucleons is equal to \(1/2\), and the isotopic spin of π-mesons to 1. It is further assumed that the proton and the neutron represent one and the same particle and differ only in the values of the projections onto a certain axis of isotopic space, equal respectively to \(+1/2\) and \(-1/2\). In analogous fashion, the difference in the signs of the charges of π-mesons is equivalent to different values of the projections of isotopic spin. The projections of isotopic spin of \(\pi^+\)-, \(\pi^-\)-, and \(\pi^0\)-mesons are respectively \(+1\), \(-1\), and \(0\).
According to the hypothesis of charge invariance, the Hamiltonian of a closed system of nuclear-interacting particles is invariant with respect to rotations in isotopic space. In other words, one may regard the space of isotopic spin as isotropic. This leads to the following conclusions concerning processes associated with the interaction of nucleons and π-mesons:
a) The total isotopic spin of the system is conserved (conservation of the projection of isotopic spin is trivial, since this condition means conservation of the charge of the system).
b) The character of the interactions between particles can depend only on the value of the total isotopic spin and does not depend on its projection.
Of course, the hypothesis of isotopic spin is valid only approximately. Thus, for example, electromagnetic interactions of particles violate the principle of isotropy of isotopic space. However, in most of the processes of interaction of nuclear particles considered by us, electromagnetic interactions may be neglected. In the scattering reactions of π-mesons by nucleons considered here, the electromagnetic (Coulomb) interaction of the particles is sufficiently small that the problem of testing the hypothesis of isotopic invariance is, from this point of view, quite correct.
Let us consider the following processes of interaction of π-mesons with nucleons:
\[ \pi^+ + p \to \pi^+ + p, \tag{1} \]
\[ \pi^- + p \to \pi^- + p, \tag{2} \]
\[ \pi^- + p \to \pi^0 + n. \tag{3} \]
Reaction (3) is called “charge-exchange scattering.” From the point of view of the hypothesis of isotopic spin, the reactions
\[ \pi^- + n \to \pi^- + n, \tag{1'} \]
\[ \pi^+ + n \to \pi^+ + n, \tag{2'} \]
\[ \pi^+ + n \to \pi^0 + p \tag{3'} \]
differ from processes (1), (2), (3) only by the reversal of the sign of the projections of isotopic spin of all particles; and since isotopic space is assumed to be isotropic, the interactions of π-mesons with neutrons in reactions \((1')\), \((2')\), \((3')\) must be completely similar to the interactions with protons in reactions (1), (2), (3).
The independence of the character of the interaction of particles under simultaneous replacement, in the given reaction,
\[ \begin{array}{ccccc} p \to n &&& \pi^- \to \pi^+\\ n \to p & \text{and} && \pi^+ \to \pi^-\\ &&& \pi^0 \to \pi^0 \end{array} \]
is at present generally accepted and bears the name of the principle of charge symmetry.
According to the phase theory of scattering, the processes (1), (2), (3) can be completely described if the phases corresponding to all possible values of isotopic spin and of orbital and total angular momenta are specified. Since the isotopic spin of the meson–nucleon system can be either \(1/2\) or \(3/2\), and the possible orbital angular momentum at the considered \(\pi\)-meson energies does not exceed the value \(L=1^*)\), the processes of scattering of \(\pi\)-mesons by nucleons are described by six phases. We shall denote the phases of the \(S\)-waves corresponding to isotopic spin equal to \(3/2\) and \(1/2\) by \(a_3\) and \(a_1\), respectively. The phases of the \(P\)-waves will be denoted by \(a_{33}, a_{31}, a_{13}, a_{11}\), where the first subscript indicates twice the value of the isotopic spin, and the second subscript twice the value of the total angular momentum of the motion of the meson–nucleon system (see Table IV).
Table IV
Designations of the phases describing the scattering of mesons by nucleons
| State | \(T = 3/2\) | \(T = 1/2\) |
|---|---|---|
| \(S\) | \(a_3\) | \(a_1\) |
| \(P_{1/2}\) | \(a_{31}\) | \(a_{11}\) |
| \(P_{3/2}\) | \(a_{33}\) | \(a_{13}\) |
The phases are functions of the energy. Since meson scattering by nucleons is mainly determined by the \(S\)- and \(P\)-waves, the dependence of the cross sections of processes (1), (2), (3) on the angle will be described by a formula of the form
\[ \frac{d\sigma}{d\omega}=a+b\cdot \cos \vartheta+c\cdot \cos^2 \vartheta, \tag{4} \]
where \(\vartheta\) is the angle between the directions of the incident and scattered \(\pi\)-meson in the center-of-inertia system.
We shall denote the values of the coefficients \(a\), \(b\), and \(c\) in expression (4) for the reaction \(\pi^+ + p \to \pi^+ + p\) by \(a_+\), \(b_+\), \(c_+\). In an analogous way, the corresponding coefficients for the reaction \(\pi^- + p \to \pi^- + p\) will be denoted by \(a_-\), \(b_-\), \(c_-\), and for the reaction \(\pi^- + p \to \pi^0 + n\) by \(a_0\), \(b_0\), \(c_0\). Consequently, measurement of the angular distributions of mesons in scattering by nucleons for reactions (1), (2), and (3) makes it possible to determine nine constants: \(a\), \(b\), \(c\). If the hypothesis of charge invariance is valid, these nine constants must be expressible in terms of six phases (two \(S\)-phases and four \(P\)-phases). Below is given a table of the values of the coefficients \(a\), \(b\), and \(c\) for various \(\pi\)-meson energies.
We note that at \(\pi\)-meson energies above 200 MeV the description of meson–nucleon interactions with allowance for only \(S\)- and \(P\)-waves becomes not entirely satisfactory. As is seen from Fig. 1, the scattering cross section of \(\pi\)-mesons of energy 217 MeV is better described by a dependence of the form:
\[ \frac{d\sigma}{d\omega}=A+B\cdot Y_1(\vartheta)+C\cdot Y_2(\vartheta)+D\cdot Y_3(\vartheta), \]
\[ \underline{\hspace{2.5cm}} \]
\(^*)\) If one assumes that the interaction energy of a \(\pi\)-meson with a nucleon in the \(d\)-state is of the order \(mc^2\) (\(m\) is the mass of the \(\pi\)-meson), then the corresponding value of the \(d\)-phase at \(E_\pi = 130\) MeV is found to be of the order of fractions of a degree, whereas the \(S\)- and \(P\)-phases reach values of \(30^\circ\)–\(50^\circ\) (Fermi et al., Phys. Rev. 91, 155, 1953).
Table V
Angular distributions of $\pi$-mesons scattered by protons in the center-of-mass system; the cross section depends on the angle according to the formula
\[ \frac{d\sigma}{d\Omega}=a+b\cos\vartheta+c\cos^2\vartheta \]
In units of $10^{-27}\ \text{cm}^2/\text{sterad}$
| Meson | $E_\pi$, MeV | $a$ | $b$ | $c$ | Reference |
|---|---|---|---|---|---|
| $\pi^+$ | 53 | $0.46\pm0.4$ | $-0.08\pm0.8$ | $3.6\pm1.5$ | 26 |
| $\pi^+$ | 78 | $1.9\pm0.3$ | $-1.7\pm0.4$ | $7.6\pm0.9$ | 26 |
| $\pi^+$ | 110 | $3.8\pm0.7$ | $-4.8\pm0.8$ | $12.1\pm1.5$ | 26 |
| $\pi^+$ | 135 | $3.9\pm2.3$ | $-7.1\pm2.8$ | $18.0\pm6.8$ | 26 |
| $\pi^+$ | 145 | $5.0\pm2.9$ | $-6.8\pm3.6$ | $25.0\pm10$ | 27 |
| $\pi^+$ | 150 | $7.7\pm0.3$ | $-2.3\pm0.4$ | $16.5\pm0.7$ | 64 |
| $\pi^+$ | 151 | $6.2\pm2.4$ | $-7.7\pm2.6$ | $19.0\pm5.0$ | 28 |
| $\pi^+$ | 151 | $7.7\pm1.1$ | $-3.3\pm1.4$ | $17.3\pm3.2$ | 65 |
| $\pi^+$ | 165 | $8.8\pm1.2$ | $-3.7\pm1.8$ | $21.1\pm3.4$ | 55 |
| $\pi^+$ | 170 | $8.9\pm0.3$ | $-1.2\pm0.5$ | $20.5\pm0.9$ | 64 |
| $\pi^+$ | 176 | $9.14\pm0.42$ | $-0.84\pm0.76$ | $20.2\pm1.6$ | 67 |
| $\pi^+$ | 188 | $10.0\pm3.5$ | $-4.1\pm2.0$ | $8.7\pm2.8$ | 28 |
| $\pi^+$ | 189 | $7.7\pm0.8$ | $3.1\pm1.3$ | $26.0\pm3.0$ | 68 |
| $\pi^+$ | 200 | $7.92\pm0.36$ | $1.04\pm0.55$ | $18.7\pm1.2$ | 69 |
| $\pi^+$ | 217 | $5.2\pm1.4$ | $7.5\pm2.6$ | $20.4\pm4.6$ | 69 |
| $\pi^+$ | 240 | $3.15\pm0.22$ | $4.58\pm0.37$ | $14.93\pm0.95$ | 67 |
| $\pi^+$ | 270 | $2.93\pm0.15$ | $4.73\pm0.50$ | $10.59\pm0.75$ | 67 |
| $\pi^+$ | 300 | $2.27$ | $2.41$ | $11.1$ | 70 |
| $\pi^+$ | 307 | $2.36\pm0.14$ | $5.2\pm0.38$ | $9.26\pm0.6$ | 67 |
| $\pi^+$ | 310 | $2.4\pm0.2$ | $4.9\pm0.4$ | $9.3\pm0.7$ | 71 |
| $\pi^-$ | 118 | $0.49\pm0.15$ | $0.16\pm0.21$ | $0.85\pm0.45$ | 29 |
| $\pi^-$ | 120 | $0.49\pm0.11$ | $0.34\pm0.16$ | $1.16\pm0.34$ | 26 |
| $\pi^-$ | 144 | $0.82\pm0.16$ | $0.73\pm0.24$ | $1.52\pm0.50$ | 26 |
| $\pi^-$ | 150 | $0.97\pm0.03$ | $0.44\pm0.05$ | $1.87\pm0.10$ | 64 |
| $\pi^-$ | 169 | $0.64\pm0.19$ | $0.47\pm0.3$ | $3.0\pm0.6$ | 30 |
| $\pi^-$ | 170 | $1.06\pm0.04$ | $0.42\pm0.07$ | $2.44\pm0.13$ | 64 |
| $\pi^-$ | 187 | $0.81\pm0.13$ | $0.35\pm0.20$ | $3.08\pm0.38$ | 34 |
| $\pi^-$ | 189 | $0.8\pm0.5$ | $0.36\pm0.2$ | $3.1\pm0.4$ | 68 |
| $\pi^-$ | 194 | $1.12\pm0.24$ | $0.05\pm0.40$ | $2.87\pm0.8$ | 30 |
| $\pi^-$ | 210 | $1.56\pm0.34$ | $0.5\pm0.47$ | $2.14\pm1.1$ | 30 |
| $\pi^0$ | 40 | $0.45\pm0.07$ | $-0.98\pm0.13$ | $0.54\pm0.21$ | 32 |
| $\pi^0$ | 65 | $0.89\pm0.03$ | $-1.38\pm0.09$ | $0.21\pm0.36$ | 33 |
| $\pi^0$ | 120 | $0.6\pm0.4$ | $-1.0\pm0.7$ | $3.2\pm1.7$ | 26 |
| $\pi^0$ | 144 | $1.05\pm0.5$ | $-1.9\pm0.5$ | $3.9\pm2.0$ | 26 |
| $\pi^0$ | 150 | $1.54\pm0.07$ | $-1.34\pm0.06$ | $3.63\pm0.20$ | 64 |
| $\pi^0$ | 165 | $1.9\pm0.8$ | $-1.1\pm0.7$ | $5.5\pm3.2$ | 55 |
| $\pi^0$ | 169 | $1.85\pm0.72$ | $-0.69\pm0.62$ | $4.25\pm2.3$ | 30 |
| $\pi^0$ | 170 | $1.96\pm0.08$ | $-0.84\pm0.08$ | $4.25\pm0.23$ | 64 |
| $\pi^0$ | 187 | $1.4\pm0.24$ | $-0.56\pm0.38$ | $5.63\pm0.88$ | 34 |
| $\pi^0$ | 189 | $1.9\pm0.5$ | $-0.3\pm0.6$ | $5.1\pm1.8$ | 68 |
| $\pi^0$ | 194 | $1.73\pm0.8$ | $-0.1\pm0.7$ | $5.9\pm2.6$ | 30 |
| $\pi^0$ | 210 | $0.84\pm0.7$ | $-1.94\pm0.73$ | $5.55\pm2.3$ | 30 |
| $\pi^0$ | 217 | $1.36\pm0.22$ | $-1.23\pm0.26$ | $4.82\pm0.76$ | 31 |
where \(Y_i(\vartheta)\) denotes the \(i\)-th Legendre polynomial, so that formula (4), which corresponds to \(D=0\), follows.
In view of the lower energies of the \(\pi\)-mesons, meson–nucleon interactions are analyzed below under the assumption that only \(S\)- and \(P\)-waves are significant.
The phases characterizing the scattering of \(\pi\)-mesons by nucleons are related to the coefficients \(a\), \(b\), and \(c\) in the following way\({}^{10}\):
\[ \left. \begin{aligned} a_+&=\frac{1}{4k^2}\left(|A_+|^2+|C_+|^2\right); & b_+&=\frac{1}{4k^2}\left(A_+B_+^*+A_+^*B_+\right);\\ c_+&=\frac{1}{4k^2}\left(|B_+|^2-|C_+|^2\right);\\[4pt] a_-&=\frac{1}{36k^2}\left(|A_-|^2+|C_-|^2\right); & b_-&=\frac{1}{36k^2}\left(A_-B_-^*+A_-^*B_-\right);\\ c_-&=\frac{1}{36k^2}\left(|B_-|^2-|C_-|^2\right);\\[4pt] a_0&=\frac{1}{18k^2}\left(|A_0|^2-|C_0|^2\right); & b_0&=\frac{1}{18k^2}\left(A_0B_0^*+A_0^*B_0\right);\\ c_0&=\frac{1}{36k^2}\left(|B_0|^2-|C_0|^2\right), \end{aligned} \right\} \tag{5} \]
where \(k\) is the wave number of the \(\pi\)-meson in the center-of-inertia system,
\[ \begin{aligned} A_+&=\exp(2i\alpha_3)-1,\\ B_+&=2\exp(2i\alpha_{33})+\exp(2i\alpha_{31})-3,\\ C_+&=\exp(2i\alpha_{33})-\exp(2i\alpha_{31}),\\ A_-&=A_++2\exp(2i\alpha_1)-2,\\ B_-&=B_++4\exp(2i\alpha_{13})+2\exp(2i\alpha_{11})-6,\\ C_-&=C_++2\exp(2i\alpha_{13})-2\exp(2i\alpha_{11}),\\ A_0&=\frac{3}{2}A_+-\frac{1}{2}A_-,\\ B_0&=\frac{3}{2}B_+-\frac{1}{2}B_-,\\ C_0&=\frac{3}{2}C_+-\frac{1}{2}C_- . \end{aligned} \]
Fig. 1. Differential cross section of the process \(\pi^-+p\to\pi^-+p\) at pion energy \(E_\pi=217\) MeV. The curves shown correspond to the expressions \(a+b\cos\vartheta+c\cos^2\vartheta\) and \(AY_0(\vartheta)+BY_1(\vartheta)+CY_2(\vartheta)+DY_3(\vartheta)\) with coefficients that best satisfy the experimental data. \(Y_i\) is the Legendre polynomial of \(i\)-th degree (M. Glicksman, Phys. Rev. 94, 1335 (1954)).
It would be possible to check the validity of the isospin hypothesis as follows: determine the phases, using for this purpose six equations out of the nine in system (5), and verify whether these phase values satisfy the three remaining equations. However, at the existing accuracy of determination of the coefficients \(a\), \(b\), and \(c\) (Table V), such a method proves ineffective. Usually a method of numerical selection of phases is used, whereby the best-found values of the coefficients \(a\), \(b\), and \(c\) are satisfied experimentally for all meson-scattering reactions. As a measure of the uncertainty in the determination of the phases one takes the sum:
\[ M=\sum_i\left(\frac{\Delta_i}{\varepsilon_i}\right)^2 . \tag{6} \]
where \(\varepsilon_i\) are the experimental errors in the determination of \(a\), \(b\), and \(c\), and \(\Delta_i\) is the deviation of the coefficients computed from the phases from those found experimentally.
The criterion for the validity of the charge-invariance hypothesis in the present case is how well all six phases describe the angular dependence of the cross section for \(\pi\)-meson scattering by nucleons. Fig. 2 shows the meson–nucleon interaction cross-section curves, calculated with the aid of the phase theory described above, as functions of the scattering angle in the center-of-mass system. The figure also gives the experimental values of the cross sections that were used for determining the phases. As can be seen from the figure, the experimental cross sections of processes (1), (2), (3) are described quite satisfactorily by the six phases, which testifies to the plausibility of the isotopic-spin hypothesis. However, with the existing accuracy of the experimental data, the phases cannot be determined uniquely. Thus, for example, the data given above on meson scattering of energy \(120\) MeV\({}^{26}\) are interpreted equally well by two groups of phases (the solutions of Fermi and Yang). Table VI gives these two solutions and the corresponding differential cross sections of processes (1) and (2) for three angles (\(45^\circ\), \(90^\circ\), and \(135^\circ\) in the laboratory coordinate system). The experimental values of the cross sections at these angles are also given.
Fig. 2. Differential cross sections of meson–nucleon interactions in the center-of-mass system at \(\pi\)-meson energy \(E_\pi = 120\) MeV. The smooth curves are dependences of the form \(a + b\cos\vartheta + c\cos^2\vartheta\), where the coefficients are calculated from the phase values found. (Fermi et al., Phys. Rev. 91, 155, 1953.)
From Table VI it is evident that an unambiguous choice between the first and second solutions can be made only with a very high experimental accuracy.
Let us note that, with the present accuracy of the experimental data, the Fermi and Yang solutions given above are not the only possible ones. In later works\({}^{33,36\text{--}38,73}\) it was shown that other solutions also exist; moreover, the experimental material presently available does not make it possible to conclude which solution is “correct.”
From equations (5) it is seen that simultaneous reversal of the signs of all phases does not change the values of the coefficients \(a\), \(b\), and \(c\). Therefore only the relative signs of the phases can be determined from these equations. The absolute sign of the phases describing \(\pi\)-meson scattering by nucleons can be determined if the interference effects of the Coulomb and nuclear interactions are taken into account. These effects can be substantial only at small scattering angles and low \(\pi\)-meson energies. Assuming that only the \(S\)- and \(P\)-phases are substantial at the \(\pi\)-meson energies under consideration, one may write the following expression for the differential cross section of reactions (1), (2), (3)\({}^{39}\), taking the Coulomb interaction into account:
\[ \frac{d\sigma}{d\omega} = \frac{1}{4k^2} \left\{ \left| -\frac{i\varepsilon\beta}{\sin^2 \dfrac{\vartheta}{2}} \exp\left(-i\varepsilon\beta \ln \sin^2 \frac{\vartheta}{2}\right) + P + Q\cos\vartheta \right|^2 + R^2\sin^2\vartheta \right\}. \tag{7} \]
Here \(\vartheta\) and \(k\) are the scattering angle and the wave number of the \(\pi\)-meson in the center-of-inertia system; \(\beta=\dfrac{me^2}{\hbar^2 k}\), where \(m\) is the reduced mass of the nucleon and the \(\pi\)-meson; \(\varepsilon\) is equal to \(+1\), \(-1\), and \(0\) for processes (1), (2), and (3), respectively; \(P, Q, R\) in the case of process (1) are expressed in terms of the phases in the following way:
\[ P=\exp(2i\alpha_3)-1;\quad R=\exp(2i\alpha_{33})-\exp(2i\alpha_{31}); \]
\[ Q=\frac{1+i\beta}{1-i\beta}\,[\exp(2i\alpha_{31})+2\exp(2i\alpha_{33})-3]. \]
In the case of processes (2) and (3), \(P, Q, R\) are expressed in terms of the phases more complicatedly.
The study of the interference of Coulomb and nuclear scattering may also give additional information as to which group of phases is the “correct” one. The existing experimental results are insufficient for a final solution of the question of the absolute sign of the phases at different \(\pi\)-meson energies.
Table VI
Cross sections of reactions (1) and (2), calculated by the phase theory of scattering for angles \(45^\circ\), \(90^\circ\), and \(135^\circ\) in the laboratory coordinate system at \(E_\pi=120\) MeV
\((\text{in units of } \times 10^{-27}\ \text{cm}^2/\text{sterad})\).
| Process | Experimental cross section (in units of \(\times 10^{-27}\ \text{cm}^2/\text{sterad}\)) | Calculated cross sections: first solution | Calculated cross sections: second solution |
|---|---|---|---|
| \(\pi^+ \to \pi^+\) | \(4,26\pm1,16\) \(5,75\pm1,16\) \(16,00\pm1,82\) |
\(4,15\) \(5,13\) \(15,15\) |
\(4,12\) \(5,14\) \(15,09\) |
| \(\pi^- \to \pi^-\) | \(1,06\pm0,14\) \(0,47\pm0,10\) \(0,97\pm0,18\) |
\(1,06\) \(0,48\) \(1,02\) |
\(1,06\) \(0,47\) \(1,02\) |
| Phases | Phases | Phases | Phases |
| First solution: | \(\alpha_3=-15,2^\circ;\) \(\alpha_{31}=3,9^\circ;\) |
\(\alpha_1=9,0^\circ;\) \(\alpha_{13}=1,8^\circ;\) |
\(\alpha_{33}=29,6^\circ;\) \(\alpha_{11}=-2,8^\circ;\) |
| Second solution: | \(\alpha_3=-15,4^\circ;\) \(\alpha_{31}=38,6^\circ;\) |
\(\alpha_1=9,0^\circ;\) \(\alpha_{13}=-1,4^\circ;\) |
\(\alpha_{33}=12,9^\circ;\) \(\alpha_{11}=3,8^\circ.\) |
Experimental data on the scattering of \(\pi\)-mesons of energy \(40\)–\(60\) MeV\({}^{33,40-44}\) agree best with the angular distributions calculated under the assumption that the phase \(\alpha_3<0\), the phase \(\alpha_1>0\), and the dominant \(P\)-phase is positive. In Yang’s solution this \(P\)-phase is \(\alpha_{31}\), while in Fermi’s solution it is \(\alpha_{33}\). It may be considered that Fermi’s solution is more reliable\({}^{33,37,38}\), and since a smooth change of the phases with increasing energy is more probable, it may be assumed that the phase \(\alpha_{33}\) remains positive throughout the entire energy interval of \(\pi\)-mesons under consideration. In what follows, when studying the dependence of the phases on energy, we shall consider only this solution.
Additional results on determining the sign of the phases and on determining the correctness of choosing one or another group of phases could be obtained if it were possible to measure the polarization of recoil protons in the scattering of \(\pi\)-mesons on hydrogen\(^{45}\). However, this method at present encounters insurmountable experimental difficulties.
Table VII gives the values of the phases that best interpret the angular dependences of meson–nucleon interactions at various \(\pi\)-meson energies. In accordance with what was said above, in Table VII
Table VII
Values of the phases describing the scattering of \(\pi\)-mesons on nucleons at various \(\pi\)-meson energies.
The \(\pi\)-meson energy is given in the laboratory system of coordinates. \(\eta=\dfrac{p}{mc}\) is the momentum of the \(\pi\)-meson in units of \(mc\) in the center-of-inertia system. The phases are given in degrees. For the notation of the phases see Table IV. The sign \(*\) indicates that the phases were calculated under the assumption that \(a_{11}=a_{13}=a_{31}=0\).
| \(E_\pi^{\mathrm{lab}}\), MeV | \(\eta_\pi^{\mathrm{c.i.}}\) | \(a_1\) | \(a_3\) | \(a_{13}\) | \(a_{33}\) | \(a_{11}\) | \(a_{31}\) | Reference |
|---|---|---|---|---|---|---|---|---|
| 22 | 0,49 | — 2,7 | 0,7 | 0 | 74 | |||
| 40 *) | 0,68 | 9,7 | — 2,6 | 5,7 | 40 | |||
| 40 | 0,68 | 8,3 | — 3,7 | 0,3 | 4,9 | —5,3 | — 0,7 | 32 |
| 40 | 0,68 | — 5,7 | 4,1 | 0,1 | 41 | |||
| 45 | 0,73 | — 5,7 | 4,4 | 2,4 | 42 | |||
| 53 | 0,80 | 0 | 9,0 | 2,0 | 26 | |||
| 58 | 0,84 | — 4,9 | 7,6 | — 1,8 | 43 | |||
| 65 | 0,90 | 10,9 | — 6,2 | —2,6 | 9,1 | 0,4 | — 1,9 | 33 |
| 78 | 0,99 | — 6,0 | 13,0 | — 3,0 | 26 | |||
| 120 | 1,25 | 9,0 | —15,0 | 2,0 | 30,0 | —3,0 | 4,0 | 26 |
| 135 | 1,33 | 10,0 | —14,0 | 2,0 | 38,0 | —5,0 | 5,0 | 26 |
| 150 | 1,44 | 9,0 | —10,0 | 2,0 | 51,5 | 2,0 | — 5,0 | 64 |
| 151 | 1,44 | —30,0 | 45,0 | 8,0 | 28 | |||
| 151 | 1,44 | —26,0 | 50,0 | 0 | 65 | |||
| 165 | 1,50 | 7,9 | —20,5 | 1,1 | 62,8 | —10,7 | 3,9 | 55 |
| 169 * | 1,52 | 10,0 | —15,0 | 65,0 | 31 | |||
| 170 | 1,52 | 10,0 | — 8,0 | 2,0 | 65,0 | 0 | — 8 | 64 |
| 187 * | 1,63 | 8,0 | 0 | 83,0 | 34 | |||
| 188 | 1,63 | —35,0 | 55,0 | 0 | 28 | |||
| 189 | 1,64 | —2,8 | —11,3 | —2,1 | 98,8 | —2,6 | —11,6 | 68 |
| 194 * | 1,65 | 14,0 | — 5,0 | 84,0 | 31 | |||
| 217 * | 1,75 | —4,0 | —23,0 | 100 | 31 | |||
| 217 | 1,75 | —22,5 | 114 | —11,6 | 69 | |||
| 240 | —14,0 | 114,4 | — 2,0 | 75 | ||||
| 260 | —12,0 | 116,0 | —11,0 | 72 | ||||
| 270 | —13,6 | 128,8 | — 4,3 | 75 | ||||
| 300 | —12,0 | 126,0 | —10,0 | 70 | ||||
| 307 | —13,0 | 133,7 | — 4,0 | 75 | ||||
| 310 | —16,8 | 129,6 | — 2,0 | 71 | ||||
| 400 | —22,0 | 158,0 | —22,0 | 72 |
only the Fermi solutions with phase \(a_{33}>0\) are given. At high energies the phases were calculated under the assumption that \(a_{11}=a_{13}=a_{31}=0\). It may be assumed that no large errors were thereby made in determining the phases, since according to the calculations of Fermi et al.\(^{26,33}\), the indicated phases at energies \(E_\pi=60—130\) MeV are small (see Table VII).
In the work of Hofmann et al.\(^{38}\) a detailed analysis was made of the various groups of phases satisfying the experimental results at a \(\pi\)-meson energy
120–200 MeV. The authors conclude that the most “probable” solution is one that, in general outline, agrees with the data of Table VII. This solution is presented in Table VIIa.
Table VIIa
Values of the phases at pion energies \(E_\pi = 120\text{–}217\) MeV, calculated by Hofmann \(^{38}\)
| \(E_\pi^{\mathrm{lab}}\), MeV | \(\alpha_1\) | \(\alpha_3\) | \(\alpha_{13}\) | \(\alpha_{33}\) | \(\alpha_{11}\) | \(\alpha_{31}\) |
|---|---|---|---|---|---|---|
| 120 | 8 | −12 | 2 | 30 | −4 | 6 |
| 144 | 14 | −13 | 3 | 46 | −5 | 5 |
| 169 | 7 | −4 | −1 | 64 | 7 | 3 |
| 194 | −14 | −13 | 0 | 90 | 5 | −16 |
| 217 | −4 | −20 | 7 | 107 | −7 | −14 |
In Fig. 3 are shown the dependences of the phases on the pion momentum in the center-of-inertia system according to the data of Table VII. As is seen from these figures, the phase \(\alpha_{33}\) is approximately proportional to \(\eta^3\). The phase \(\alpha_1\) does not depend on the energy in the pion energy interval 40–200 MeV (in the laboratory coordinate system).
At present there are no sufficiently complete experimental data on the scattering of pions of low energies (\(E_\pi < 40\) MeV in the laboratory coordinate system) by nucleons. Evidently, at sufficiently low pion energies we should expect the phases to depend on the pion momentum as
Fig. 3. Dependence of the phases \(\alpha_1\), \(\alpha_3\), and \(\alpha_{33}\) on the pion momentum \(p_\pi\) in the center-of-inertia system
\[
\eta_\pi=\frac{p_\pi}{m_\pi c}.
\]
\[ \alpha_l \sim \eta^{2l+1}, \]
where \(\eta\) is the pion momentum in the center-of-inertia system, and \(\alpha_l\) is the phase corresponding to pion scattering by nucleons with angular momentum \(l\).
From an analysis of the experimental data, Orear et al. \(^{46}\) found that the expressions
\[ \alpha_3=-0.11\,\eta,\quad \alpha_1=0.16\,\eta \quad \text{and} \quad \alpha_{33}=0.235\,\eta^3 \tag{8} \]
well interpret the available experimental data at low energies.
The phase values were also estimated in determining the energy levels of \(\pi\)-mesoatoms \(^{66}\) and proved to be approximately equal to the values \(a_1\) and \(a_3\) found by Orear. Of course, the indicated values of the coefficients cannot be regarded as final.
It is seen from Fig. 3 that the phase \(a_{33}\) becomes equal to \(\dfrac{\pi}{2}\) at a \(\pi\)-meson energy of \(190\text{–}200\) MeV. One might suppose that such a strong interaction indicates the existence of an excited state of the nucleon with spin and isotopic spin equal to \(3/2\). However, the idea that the process of meson scattering by nucleons proceeds through an intermediate state of an excited nucleon is not consistent with the comparatively weak dependence of the phase \(a_{33}\) on the energy in the region \(E_\pi=200\) MeV. This is best seen from the dependence of the total cross section of the process \((\pi^+ + p)\) on energy, determined mainly by the phase \(a_{33}\). As is seen from Fig. 4, \(\sigma_t(\pi^+ + p)\) has a maximum at \(E_\pi \approx 190\) MeV, with a “width” of the order of 100 MeV. If this maximum of the cross section corresponded to the fact that the reaction \((\pi^+ + p)\) proceeds through an intermediate state, then the lifetime of this state would be \(\sim 10^{-23}\) sec. Since the time during which a particle traverses a distance of the order of the range of nuclear forces is approximately of the same magnitude, the concept of an intermediate state in this case loses its meaning.
Fig. 4. Total cross sections of the process \(\pi^+ + p \to \pi^+ + p\). The dashed curve shows the maximum possible cross section for the interaction of \(\pi\)-mesons with protons in a state with total angular momentum equal to \(3/2\):
\[
\sigma_{\max}=8\pi\lambda^2.
\]
Dependence of the total cross sections of \(\pi\)-meson scattering by nucleons on energy. Measurements of the total cross sections of meson scattering by nucleons have been made with sufficient accuracy over a broad interval of \(\pi\)-meson energies to permit a substantial supplement to the information on the character of meson–nucleon interactions obtained from the analysis of the angular distributions of processes (1), (2), (3) (see the beginning of this section).
The total cross sections of the reactions
\[ \pi^+ + p \to \pi^+ + p, \tag{1} \]
\[ \pi^- + p \to \begin{cases} \pi^- + p,\\ \pi^0 + n. \end{cases} \tag{9} \]
were determined from the attenuation of a beam of \(\pi^+\)- and \(\pi^-\)-mesons after passage through an absorber of liquid hydrogen or some hydrogen-containing substance, and also by integrating the existing differential
cross sections. The total cross sections of the processes
\[ \pi^- + p \to \pi^- + p, \tag{2} \]
\[ \pi^- + p \to \pi^0 + n \tag{3} \]
were determined by integrating the corresponding differential cross sections.
The existing experimental data on the determination of the total cross sections of processes (1), (9), (2), and (3) are given in Tables VIII, IX, and X, and also in Figs. 4 and 5.
In Figs. 4 and 5 there are also shown the curves of the dependence of the maximum possible scattering cross sections of \(\pi^+\)- and \(\pi^-\)-mesons on protons in the state with isotopic spin \(3/2\) and total angular momentum equal to \(3/2\). These cross sections correspond to the phase \(\alpha_{33}=\pi/2\) and are equal to \(8\pi\lambda^2\) for the process \((\pi^+ + p)\) and \(8/3\pi\lambda^2\) for the process \((\pi^- + p)\), where \(\lambda\) is the wavelength of the \(\pi\)-meson divided by \(2\pi\).
Fig. 5. Integral cross section for scattering of \(\pi^-\)-mesons on nucleons. Curve \(a\) refers to the process \(\pi^-+p\to\pi^-+p\), curve \(b\) refers to the process \(\pi^-+p\to\pi^0+n\), and curve \(v\) refers to the total scattering cross section of \(\pi^-\)-mesons on protons, i.e., to the sum of processes \(a\) and \(b\). The dashed curve shows the maximum possible scattering cross section of \(\pi^-\)-mesons on protons in a state with angular momentum \(3/2\) and isotopic spin \(3/2\): \(\sigma_{\max}=\dfrac{8}{3}\pi\lambda^2\).
It should be noted that the total cross sections for the interaction of \(\pi\)-mesons with nucleons given in Tables VIII and IX, at energies \(E_\pi \gg 1000\) MeV, no longer characterize processes (1), (2), and (3), since at these energies multiple meson production becomes possible.3
From Figs. 4 and 5 it is seen that the total cross sections of the processes of \(\pi\)-meson scattering on nucleons reach their maximum value at a \(\pi\)-meson energy of \(\sim 190\) MeV in the laboratory coordinate system, which corresponds to an energy \(E_\pi=125\) MeV in the center-of-inertia system. The corresponding values of the total cross sections are \(\sim 190\cdot10^{-27}\ \text{cm}^2\) for the process \(\pi^+ + p \to \pi^+ + p\) and \(66\cdot10^{-27}\ \text{cm}^2\) for the process
\[ \pi^- + p \to \begin{cases} \pi^- + p,\\ \pi^0 + n. \end{cases} \]
With a further increase in the energy of the \(\pi\)-mesons, the total cross section of meson–nucleon interactions decreases; the maximum has a “width” of the order of 100 MeV in the center-of-inertia system. As was indicated earlier, the considerable width of the maximum is not consistent with the assumption that the scattering processes of \(\pi\)-mesons on nucleons proceed through an intermediate state of the nucleon–meson system.
In Fig. 6 are shown the dependences of the total cross sections for the interaction of \(\pi^+\)- and \(\pi^-\)-mesons with nucleons over the entire measured range of \(\pi\)-meson energies.
According to the principle of charge invariance, the total cross section for the interaction of \(\pi\)-mesons with nucleons in a state with isotopic spin \(1/2\) is expressed in terms of the interaction cross sections of \(\pi^+\)- and \(\pi^-\)-mesons with protons as follows:
\[ \sigma\left(T=\frac{1}{2}\right) = \frac{3}{2}\,\sigma(\pi^-+p) - \frac{1}{2}\,\sigma(\pi^++p). \tag{10} \]
π-Mesons
Table VIII
Total scattering cross sections of $\pi^+$ mesons on protons.
The sign * denotes the cross section of the process $(\pi^-+n)$, determined as $\sigma(\pi^-+\mathrm{D}_2\mathrm{O})-\sigma(\pi^-+\mathrm{H}_2\mathrm{O})$.
| $E_\pi$, MeV | $\sigma_{\mathrm{tot}}\times 10^{27}\ \mathrm{cm}^2$ | Method of determination | Reference |
|---|---|---|---|
| 33 | $6.4\pm2.0$ | by beam attenuation | 47 |
| 37 | $11.8\pm1.0$ | by beam attenuation | 48 |
| 40 | $10.9\pm3.0$ | integration of $\sigma(\vartheta)$ | 41 |
| 43 | $9.0\pm4.0$ | integration of $\sigma(\vartheta)$ | 49 |
| 44 | $9.8\pm1.5$ | by beam attenuation | 47 |
| 45 | $12.0\pm3.0$ | integration of $\sigma(\vartheta)$ | 42 |
| 53 | $20.0\pm4.0$ | integration of $\sigma(\vartheta)$ | 50 |
| 56 | $20\pm10$ | by beam attenuation | 51 |
| 56 | $17.6\pm2.2$ | by beam attenuation | 47 |
| 58 | $15.3\pm1.0$ | integration of $\sigma(\vartheta)$ | 43 |
| 65 | $20.4\pm2.0$ | integration of $\sigma(\vartheta)$ | 33 |
| 70 | $19.0\pm2.6$ | by beam attenuation | 47 |
| 75 | $41.0\pm15$ | integration of $\sigma(\vartheta)$ | 52 |
| 78 | $31\pm3$ | integration of $\sigma(\vartheta)$ | 26 |
| 79 | $48\pm10$ | by beam attenuation | 53 |
| 82 | $50\pm13$ | by beam attenuation | 51 |
| 109 | $80\pm10$ | by beam attenuation | 53 |
| 110 | $77\pm6$ | integration of $\sigma(\vartheta)$ | 26 |
| 115 | $95\pm15$ | by beam attenuation | 53 |
| 118 | $91\pm6$ | by beam attenuation | 51 |
| 127 | $125\pm15$ | by beam attenuation | 53 |
| 128 | $122\pm8$ | by beam attenuation | 59 |
| 133 | $135\pm15$ | by beam attenuation | 53 |
| 135 | $120$ | by beam attenuation | 54 |
| 135 | $126\pm4$ | by beam attenuation | 59 |
| 135 | $126\pm20$ | integration of $\sigma(\vartheta)$ | 26 |
| 136 | $152\pm14$ | by beam attenuation | 51 |
| 140 | $133\pm8$ | by beam attenuation | 79 |
| 142 | $150\pm8$ | by beam attenuation | 59 |
| 144 | $151\pm4$ | by beam attenuation | 79 |
| 145 | $169\pm23$ | integration of $\sigma(\vartheta)$ | 27 |
| 146 | $150\pm7$ | by beam attenuation | 56 |
| 150 | $165\pm5$ | by beam attenuation | 64 |
| 150 | $167\pm5$ | integration of $\sigma(\vartheta)$ | 64 |
| 151 | $151\pm19$ | integration of $\sigma(\vartheta)$ | 65 |
| 151 | $152\pm31$ | integration of $\sigma(\vartheta)$ | 28 |
| 152 | $175\pm6$ | by beam attenuation | 59 |
| 156 | $170\pm5$ | by beam attenuation | 59 |
| 157 | $162\pm7$ | by beam attenuation | 56 |
| 164 | $169\pm5$ | by beam attenuation | 79 |
| 165 | $188\pm5$ | by beam attenuation | 55 |
| 165 | $194\pm5$ | integration of $\sigma(\vartheta)$ | 55 |
Continuation of Table VIII
| \(E_{\pi}\), MeV | \(\sigma_{\mathrm{full}} \times 10^{27}\,\mathrm{cm}^{2}\) | Method of determination | Reference |
|---|---|---|---|
| 166 | \(179 \pm 7\) | by beam attenuation | 56 |
| 170 | \(195 \pm 6\) | by beam attenuation | 64 |
| 170 | \(202 \pm 6\) | integration of \(\sigma(\vartheta)\) | 64 |
| 171 | \(203 \pm 8\) | by beam attenuation | 59 |
| 171 | \(204 \pm 6\) | by beam attenuation | 56 |
| 173 | \(205 \pm 6\) | by beam attenuation | 56 |
| 174 | \(193 \pm 6\) | by beam attenuation | 79 |
| 176 | \(199 \pm 5\) | integration of \(\sigma(\vartheta)\) | 67 |
| 178 | \(\sim 200\) | by beam attenuation | 54 |
| 181 | \(187 \pm 7\) | by beam attenuation | 56 |
| 182 | \(166 \pm 20\) | by beam attenuation | 59 |
| 184 | \(196 \pm 6\) | by beam attenuation | 79 |
| 185 | \(188 \pm 8\) | by beam attenuation | 59 |
| 188 | \(159 \pm 34\) | by beam attenuation | 28 |
| 189 | \(182 \pm 7\) | by beam attenuation | 56 |
| 189 | \(194 \pm 5\) | integration of \(\sigma(\vartheta)\) | 68 |
| 169 | 171 | calculated from phases found in experiments on \((\pi^{-}+p)\)-scattering | 31 |
| 194 | 184 | calculated from phases found in experiments on \((\pi^{-}+p)\)-scattering | 31 |
| 217 | 174 | calculated from phases found in experiments on \((\pi^{-}+p)\)-scattering | 31 |
| 194 | \(200 \pm 6\) | by beam attenuation | 79 |
| 196 | \(202 \pm 14\) | by beam attenuation | 59 |
| 209 | \(179 \pm 6\) | by beam attenuation | 79 |
| 200 | \(178 \pm 4\) | integration of \(\sigma(\vartheta)\) | 67 |
| 210 | \(148 \pm 20\) | by beam attenuation | 56 |
| 214 | \(141 \pm 7\) | by beam attenuation | 56 |
| 217 | 151 | integration of \(\sigma(\vartheta)\) | 69 |
| 219 | \(156 \pm 7\) | by beam attenuation | 79 |
| 222 | \(148 \pm 7\) | by beam attenuation | 56 |
| 229 | \(132 \pm 7\) | by beam attenuation | 79 |
| 262 | \(111 \pm 7\) | by beam attenuation | 56 |
| 263 | \(107 \pm 7\) | by beam attenuation | 56 |
| 240 | \(127 \pm 4\) | integration of \(\sigma(\vartheta)\) | 67 |
| 270 | \(81 \pm 3\) | integration of \(\sigma(\vartheta)\) | 67 |
| 280 | \(88 \pm 11\) | by beam attenuation | 56 |
| 298 | \(75 \pm 5\) | by beam attenuation | 56 |
| 307 | \(68 \pm 2.4\) | integration of \(\sigma(\vartheta)\) | 67 |
| 310 | 69 | integration of \(\sigma(\vartheta)\) | 71 |
| 340 | \(48 \pm 9\) | by beam attenuation | 56 |
| 450 | \(27 \pm 6\) | by beam attenuation | 56 |
| 700 | \(17 \pm 6\) | by beam attenuation | 56 |
| 1000* | \(21 \pm 3\) | by beam attenuation | 57 |
| 1450* | \(29 \pm 3\) | by beam attenuation | 57 |
π-Mesons
Table IX
Total scattering cross sections of \(\pi^-\)-mesons on protons
| \(E_\pi\), MeV | \(\sigma_{\mathrm{tot}} \times 10^{27}\ \mathrm{cm}^2\) | Method of determination | Reference |
|---|---|---|---|
| 37 | \(13 \pm 7\) | by beam attenuation | 48 |
| 72 | \(15 \pm 8\) | by beam attenuation | 53 |
| 79 | \(20 \pm 8\) | by beam attenuation | 53 |
| 89 | \(21 \pm 8\) | by beam attenuation | 58 |
| 109 | \(31 \pm 9\) | by beam attenuation | 53 |
| 112 | \(31 \pm 9\) | by beam attenuation | 58 |
| 120 | \(33 \pm 3\) | integration of \(\sigma(\vartheta)\) | 26 |
| 127 | \(45 \pm 10\) | by beam attenuation | 53 |
| 133 | \(47 \pm 2.4\) | by beam attenuation | 59 |
| 135 | \(52 \pm 6\) | by beam attenuation | 58 |
| 140 | \(44 \pm 2.7\) | by beam attenuation | 80 |
| 144 | \(61 \pm 7\) | integration of \(\sigma(\vartheta)\) | 26 |
| 150 | \(55 \pm 2.0\) | by beam attenuation | 64 |
| 152 | \(61 \pm 3.0\) | by beam attenuation | 59 |
| 157 | \(63 \pm 2.4\) | by beam attenuation | 59 |
| 165 | \(67 \pm 1.5\) | by beam attenuation | 55 |
| 165 | \(65 \pm 2.2\) | integration of \(\sigma(\vartheta)\) | 55 |
| 169 | \(63 \pm 4\) | \(\sigma(\vartheta)\) | 30 |
| 170 | \(63 \pm 2\) | by beam attenuation | 64 |
| 176 | \(66 \pm 6\) | by beam attenuation | 58 |
| 179 | \(66 \pm 2.5\) | by beam attenuation | 59 |
| 184 | \(66 \pm 2.4\) | by beam attenuation | 80 |
| 187 | \(64 \pm 2\) | integration of \(\sigma(\vartheta)\) | 34 |
| 189 | \(68 \pm 3.4\) | \(\sigma(\vartheta)\) | 68 |
| 194 | \(74 \pm 5\) | \(\sigma(\vartheta)\) | 30 |
| 194 | \(65 \pm 2.5\) | by beam attenuation | 59 |
| 195 | \(63 \pm 2.5\) | by beam attenuation | 59 |
| 197 | \(72 \pm 2.5\) | by beam attenuation | 80 |
| 209 | \(57 \pm 2.9\) | by beam attenuation | 31 |
| 210 | \(64 \pm 5\) | integration of \(\sigma(\vartheta)\) | 30 |
| 215 | \(56 \pm 2.2\) | by beam attenuation | 59 |
| 216 | \(57 \pm 2.5\) | by beam attenuation | 80 |
| 217 | \(60 \pm 6\) | by beam attenuation | 58 |
| 217 | \(55 \pm 5\) | integration of \(\sigma(\vartheta)\) | 31 |
| 220 | \(52 \pm 2.3\) | by beam attenuation | 31 |
| 226 | \(53 \pm 2.0\) | by beam attenuation | 80 |
| 236 | \(46 \pm 2.4\) | by beam attenuation | 59 |
| 240 | \(43 \pm 2.3\) | by beam attenuation | 59 |
| 256 | \(38 \pm 1.9\) | by beam attenuation | 80 |
| 258 | \(38 \pm 3.4\) | by beam attenuation | 59 |
| 265 | \(44 \pm 6\) | by beam attenuation | 56 |
| 290 | \(34 \pm 1.1\) | by beam attenuation | 80 |
| 330 | \(24 \pm 5\) | integration of \(\sigma(\vartheta)\) | 78 |
| 335 | \(26 \pm 1.0\) | by beam attenuation | 80 |
| 340 | \(23 \pm 11\) | by beam attenuation | 56 |
| 363 | \(26 \pm 1.6\) | by beam attenuation | 80 |
| 393 | \(26 \pm 2.7\) | by beam attenuation | 80 |
| 450 | \(25 \pm 3\) | by beam attenuation | 56 |
| 470 | \(27 \pm 5\) | by beam attenuation | 60 |
| 510 | \(20 \pm 7\) | by beam attenuation | 56 |
| 600 | \(23 \pm 11\) | by beam attenuation | 56 |
| 700 | \(42 \pm 10\) | by beam attenuation | 56 |
| 840 | \(47 \pm 5\) | by beam attenuation | 60 |
| 1000 | \(49 \pm 4\) | by beam attenuation | 57 |
| 1370 | \(35 \pm 2.7\) | by beam attenuation | 77 |
| 1450 | \(34 \pm 3.5\) | by beam attenuation | 57 |
Table X
Total cross sections
| Reaction \(\pi^- + p \to \pi^0 + n\) | Reaction \(\pi^- + p \to \pi^0 + n\) | Reaction \(\pi^- + p \to \pi^0 + n\) | Reaction \(\pi^- + p \to \pi^- + p\) | Reaction \(\pi^- + p \to \pi^- + p\) | Reaction \(\pi^- + p \to \pi^- + p\) |
|---|---|---|---|---|---|
| \(E_\pi,\) MeV | \(\sigma_{\text{tot}}\times 10^{27}\ \text{cm}^2\) | Reference | \(E_\pi,\) MeV | \(\sigma_{\text{tot}}\times 10^{-27}\ \text{cm}^2\) | Reference |
| 20 | \(5.0\pm0.8\) | 61 | 26 | \(1.1\pm0.6\) | 46 |
| 30 | \(5.7\pm0.9\) | 61 | 65 | \(2.9\pm0.5\) | 33 |
| 34 | \(5.0\pm1.5\) | 62 | 118 | \(9.6\pm2.0\) | 29 |
| 40 | \(7.9\pm1.8\) | 32 | 120 | \(11.3\pm1.6\) | 26 |
| 42 | \(6.9\pm1.2\) | 61 | 144 | \(17.0\pm2.4\) | 26 |
| 65 | \(12.4\pm1.5\) | 33 | 150 | \(20.0\) | 64 |
| 120 | \(21.7\pm2.7\) | 26 | 165 | \(22.5\pm1.5\) | 55 |
| 144 | \(30.6\pm3.7\) | 26 | 169 | \(21.2\pm2.0\) | 30 |
| 150 | \(34.7\) | 64 | 170 | \(23.6\) | 64 |
| 165 | \(42.5\pm1.3\) | 55 | 187 | \(22.5\pm1.3\) | 34 |
| 169 | \(41.4\pm3.0\) | 30 | 189 | \(23.0\pm1.4\) | 68 |
| 170 | \(39.2\) | 64 | 194 | \(26.4\pm2.7\) | 30 |
| 187 | \(41.6\pm1.5\) | 34 | 210 | \(28.7\pm3.1\) | 30 |
| 189 | \(45.3\pm3.2\) | 68 | 217 | \(18.2\pm2.3\) | 31 |
| 194 | \(47.0\pm3.6\) | 30 | 330 | \(11.0\pm4.0\) | 84 |
| 210 | \(35.0\pm3.6\) | 30 | |||
| 217 | \(35.8\pm3.4\) | 31 | |||
| 295 | \(14.0\pm3.5\) | 76 | |||
| 330 | \(13.0\pm4\) | 84 |
Fig. 6. Total cross sections for the interaction of \(\pi^+\)- and \(\pi^-\)-mesons with protons. Cross sections of meson–nucleon interactions with isotopic spin \(T=3/2\) and \(T=1/2\), calculated by formulas (10) and (11).
The cross section for the interaction of mesons with nucleons in the state with isotopic spin \(3/2\) is determined by the cross section for the interaction of \(\pi^+\)-mesons with protons:
\[ \sigma\left(T=\frac{3}{2}\right)=\sigma(\pi^+ + p). \tag{11} \]
The cross sections of meson–nucleon interactions in states with isotopic spin \(3/2\) and \(1/2\), calculated from formulas (10) and (11), are also shown in Fig. 6.
It should be noted that the cross sections of meson–nucleon interactions with isotopic spin \(T=1/2\) and \(T=3/2\), calculated from relations (10) and (11), do not depend on the particular type of reaction and therefore are valid at any pion energies. As is seen from Fig. 6, at energies \(E_\pi=50\text{--}300\ \mathrm{Mev}\) the cross section \(\sigma\left(T=\dfrac{1}{2}\right)\) is small and the interaction cross section is determined mainly by \(\sigma\left(T=\dfrac{3}{2}\right)\). At higher energies, on the contrary, the cross section \(\sigma\left(T=\dfrac{1}{2}\right)\) predominates; it reaches its maximum value at \(E_\pi \simeq 1000\ \mathrm{Mev}\).
3. PRODUCTION OF \(\pi\)-MESONS IN NUCLEON–NUCLEON COLLISIONS
The production of \(\pi\)-mesons in nucleon–nucleon collisions becomes energetically possible if, in the center-of-inertia system of the two colliding nucleons, their total energy exceeds the rest mass of the \(\pi\)-meson.
When one of the nucleons was at rest before the collision, the energy threshold for production of a \(\pi\)-meson in the laboratory coordinate system is approximately \(290\ \mathrm{Mev}\). Consider the following reactions of pion production by nucleons:
\[ p+n \to \pi^0+n+p, \tag{1} \]
\[ p+p \to \pi^0+p+p, \tag{2} \]
\[ p+p \to \pi^+ + n+p, \tag{3} \]
\[ p+n \to \pi^- + p+p. \tag{4} \]
If we assume that under the simultaneous replacement in a given reaction
\[ \begin{gathered} n \to p \qquad \pi^- \to \pi^+,\\ p \to n \qquad \pi^+ \to \pi^-,\\ \pi^0 \to \pi^0 \end{gathered} \]
the character of the interaction does not change (the principle of charge symmetry), then the reactions (1)—(4) are the only independent processes of \(\pi\)-meson production in nucleon–nucleon collisions. The hypothesis of isotopic invariance imposes additional restrictions on the number of parameters describing \(\pi\)-meson production. According to this hypothesis, the total isotopic spin of the system of strongly interacting particles is conserved. Therefore, in the production of a \(\pi\)-meson in nucleon–nucleon collisions, three different states of the system containing two nucleons and the produced \(\pi\)-meson are possible.
Two of these states:
\[ \begin{aligned} T(N_1+N_2)&=1,\qquad T_\pi=1,\\ T(N_1+N_2)&=0,\qquad T_\pi=1 \end{aligned} \left\} \quad T(N_1+N_2+\pi)=1, \right. \]
correspond to the case in which the total isotopic spin of the system is equal to 1 \(\bigl(T(N_1+N_2+\pi)=1\bigr)\), and one—to the case in which the total isotopic spin of the system
is equal to 0:
\[ \begin{gathered} T(N_1+N_2)=1,\\ T_\pi=1,\\ T(N_1+N_2+\pi)=0. \end{gathered} \]
It follows from this that there are only three independent matrix elements corresponding to transitions with \(T=1\) and \(T=0\). We shall denote these matrix elements by \(M_1, M_2, M_3\):
\(M_1\) describes the production of a \(\pi\)-meson in the case when the total isotopic spin of the system is \(T=1\), and the state of the nucleons in the final state is a triplet in isotopic spin: \(T(N_1+N_2)=1\);
\(M_2\) describes the production of \(\pi\)-mesons in the case when \(T=0\);
\(M_3\) describes the production of a \(\pi\)-meson with \(T=1\), while the isotopic spin of the two nucleons in the final state is equal to zero: \(T(N_1+N_2)=0\).
The matrix element \(M_3\) also corresponds to the case of formation in the final state of a deuteron, whose isotopic spin is equal to zero.
It can be shown that the total cross sections of processes (1)—(4) are expressed only through the squares of the moduli of the matrix elements \(M_1, M_2, M_3\), which, generally speaking, are complex quantities. These expressions have the form:
\[ \begin{aligned} \sigma(np,\pi^0)&=\frac{1}{6}|M_2|^2+\frac{1}{2}|M_3|^2,\\ \sigma(pp,\pi^0)&=\frac{1}{2}|M_1|^2,\\ \sigma(pp,\pi^+)&=\frac{1}{2}|M_1|^2+|M_3|^2,\\ \sigma(np,\pi^+)&=\frac{1}{4}|M_1|^2+\frac{1}{6}|M_2|^2. \end{aligned} \tag{5} \]
Here \(\sigma(np,\pi^0)\) denotes the cross section of the reaction \(n+p\to n+p+\pi^0\), etc. Thus, an exact determination of the total cross sections of reactions (1)—(4) makes it possible to test the hypothesis of isotopic invariance.
The use of the general relations of quantum mechanics makes it possible, in the case of small energies, to predict the character of the angular and energy dependences of the cross sections for \(\pi\)-meson production in reactions (1)—(4). By “small energies” we understand in the present case energies at which
\[ \frac{R}{\lambda_N}\lesssim 1 \]
and
\[ \frac{R}{\lambda_\pi}\lesssim 1, \]
where \(\lambda_N\) is the wavelength of the relative motion of the nucleons in the final state, \(\lambda_\pi\) is the wavelength of the produced \(\pi\)-meson, and \(R\) is the radius of action of the nuclear forces.
In this case the \(\pi\)-mesons will be produced mainly in \(S\)- and \(P\)-states; the orbital angular momentum of the relative motion of the two nucleons in the final state also does not exceed the value \(L=1\). Hence the total angular momentum of the system cannot exceed \(J=3\). Using the conservation laws of parity and total angular momentum, as well as the generalized Pauli principle, one can determine the initial and final states of the system in \(\pi\)-meson production in nucleon–nucleon collisions at small energies\({}^{81}\) (see Table XI).
In Table XI the following notation is adopted:
The letters \(S, P, D, F\) describe the state of the system of two nucleons in the generally accepted notation. The letters \(s, p\) correspond to the angular momentum of the produced \(\pi\)-meson, equal respectively to 0 and 1. The subscript of the letters \(s\) and \(p\) denotes the total angular momentum of the system. In the second column
the table gives the total isotopic spin of the system \((T_\Sigma)\) and the isotopic spin of the nucleons in the final state \((T(N_1+N_2))\). Columns 5 and 6 of Table XI give the angular and energy dependences of the cross sections for the production of \(\pi\)-mesons in the center-of-mass system, corresponding to definite initial and final states of the system (\(\eta\) is the momentum of the \(\pi\)-meson in the center-of-mass system in units of \(m_\pi c\)).
Table XI
Initial and final states of the system of particles in \(\pi\)-meson production in nucleon-nucleon collisions for various values of the total isotopic spin
The explanation of the notation is given in the text.
| Matrix element | Isotopic spin | Initial state | Final state | Angular dependence in the center-of-mass system | Total cross section \(\times 10^{27}\ \mathrm{cm}^2\) |
|---|---|---|---|---|---|
| \(M_3\) | \(T_\Sigma=1\) \(T(N_1+N_2)=0\) |
\({}^{1}S_0\) none \({}^{1}D_2\) |
\({}^{3}S_1p_0\) \({}^{3}S_1p_1\) \({}^{3}S_1p_2\) |
isotropic — \(\dfrac{1}{3}+\cos^2\vartheta\) |
\(1.0\,\eta^3\) in deuteron formation \(1.5\,\eta^4\) in formation of \(n,p\) |
| \(M_3\) | \(T_\Sigma=1\) \(T(N_1+N_2)=0\) |
\({}^{3}P_1\) | \({}^{3}S_1s_1\) | isotropic | \(0.14\,\eta\) |
| \(M_2\) | \(T_\Sigma=0\) | \({}^{3}S_1\) or \({}^{3}D_1\) |
\({}^{1}S_0p_1\) | isotropic \(\dfrac{1}{3}+\cos^2\vartheta\) |
\(0.3\,\eta^4\) |
| \(M_2\) | \(T_\Sigma=0\) | none | \({}^{1}S_0s_0\) | — | — |
| \(M_1\) | \(T_\Sigma=1\) \(T(N_1+N_2)=1\) |
none | \({}^{1}S_0p_1\) | — | — |
| \(M_1\) | \(T_\Sigma=1\) \(T(N_1+N_2)=1\) |
\({}^{3}P_0\) | \({}^{1}S_0s_0\) | isotropic | \(\sim 0.01\,\eta^2\) |
| \(M_1\) | \(T_\Sigma=1\) \(T(N_1+N_2)=1\) |
\({}^{3}P_{0,1}\) \({}^{3}P_{0,1,2}\) or \({}^{3}F_2\) \({}^{3}P_{1,2}\) or \({}^{3}F_{2,3}\) |
\({}^{3}P_0p_1\) \({}^{3}P_1p_{0,1,2}\) \({}^{3}P_2p_{1,2,3}\) |
\(c+\cos^2\vartheta\) | \(0.2\,\eta^8\) |
| \(M_1\) | \(T_\Sigma=1\) \(T(N_1+N_2)=1\) |
\({}^{1}S_0\) none \({}^{1}D_2\) |
\({}^{3}P_0s_0\) \({}^{3}P_1s_1\) \({}^{3}P_2s_2\) |
isotropic — isotropic |
\(\sim \eta^6\) |
The numerical coefficients given in column 6, expressing the absolute values of the cross sections of the corresponding processes, were obtained from comparison with the experimental data \({}^{81}\).
Interference in the angular distributions of the produced \(\pi\)-mesons is possible only in the case when the states of the nucleons at the end of the reaction are identical and the incident nucleons are in the same spin state. Hence, in particular, it follows that the only possible interference between states differing in isotopic spin can occur in meson production by reaction (4). Only in this reaction can we expect an asymmetry of the angular distribution of the produced \(\pi\)-mesons with respect to the angle \(\vartheta=90^\circ\) in the center-of-mass system \({}^{81}\).
Of the pion-production reactions (1)—(4), the reaction \((p+p\to \pi^+)\) was studied in the greatest detail. As the very first experiments showed, the spectrum of \(\pi^+\)-mesons produced in \((p-p)\)-collisions has a sharp peak in the region of the maximum possible pion energies, which was explained by the strong interaction of the nucleons in the final state. A typical spectrum of \(\pi^+\)-mesons produced by protons with an energy of \(340\ \mathrm{MeV}\) is shown in Fig. 7. It is assumed that the pions forming the peak in the energy distribution are produced by the reaction with deuteron formation:
\[ p+p\to \pi^+ + d. \tag{3′} \]
This conclusion is confirmed by experiments on the simultaneous registration of the produced \(\pi^+\)-meson and deuteron by the coincidence method. With increasing energy of the incident nucleon, the production of a \(\pi^+\)-meson in \((p+p)\)-collisions with deuteron formation in the final state becomes ever less probable. Already at a proton energy \(E_p=440\ \mathrm{MeV}\), the production of \(\pi^+\)-mesons by the reaction \((p+p\to \pi^+)\) with deuteron formation is approximately equal to the cross section with the formation of unbound nucleons\(^ {89}\).
Fig. 7. Energy spectrum of \(\pi^+\)-mesons produced by protons with an energy of \(340\ \mathrm{MeV}\) at an angle of \(18^\circ\) with respect to the proton beam.
The angular distribution in the center-of-mass system of \(\pi^+\)-mesons produced by protons with energies in the interval \(320\)—\(500\ \mathrm{MeV}\) in reaction (3′) is well approximated by the dependence
\[ \frac{d\sigma}{d\omega}\sim a+\cos^2\vartheta, \]
where \(a=0.2—0.3\) (see Table XII). It should be noted that a similar angular distribution is also obtained in the study of the reaction inverse to (3′), i.e. \(\pi^+ + d \to p+p\). The corresponding data for this reaction are given in Table XIII. The results of various authors’ work on determining the total cross sections for \(\pi^+\)-meson production in proton-proton collisions are given in Table XIII and in Fig. 8.
Fig. 8. Total cross sections for \(\pi^+\)-meson production in \((p-p)\)-collisions at various energies.
Crawford and Stevenson\(^ {102}\) found that in the energy interval of incident protons \(E_p=310—338\ \mathrm{MeV}\) the differential production cross section can be approximated by the expression
\[ 4\pi\left(\frac{d\sigma}{d\omega}\right)_{\mathrm{c.m.}} = \left[a_1\eta+a_2\eta^3+a_3\eta^3\cos^2\vartheta\right]\cdot 10^{-27}\ \mathrm{cm}^2/\mathrm{sterad}, \]
Table XII
Total cross sections for the production of \(\pi^+\)-mesons in \((p-p)\) collisions
The proton energy is given in the laboratory coordinate system. The sign \((d)\) denotes works relating to the reaction \(p+p\to\pi^+ + d\); the sign \((np)\) denotes works relating to the reaction \(p+p\to\pi^+ + n+p\).
In the work marked with an asterisk, only the relative dependence of the cross section on energy was determined; it was then normalized to absolute values according to the data of Crawford \(^{83}\). “\(a\)” represents the isotropic part of the cross section in the center-of-mass system under the assumption that
\[ \frac{d\sigma}{d\omega}\sim a+\cos^2\vartheta . \]
\(E_{\pi}^{\mathrm{c.m.}}\) denotes the maximum possible energy of the produced \(\pi\)-meson in the center-of-mass system.
| \(E^{\mathrm{lab}},\ \mathrm{MeV}\) | \(E_{\pi}^{\mathrm{c.m.}},\ \mathrm{MeV}\) | \(\sigma_t \times 10^{28}\ \mathrm{cm}^2\) | \(a\) | Reference | Note |
|---|---|---|---|---|---|
| 311 | 10.1 | \(1.0\pm0.13\) | — | 81* | \(d\) |
| 315 | 11.7 | \(1.33\pm0.16\) | — | 81* | \(d\) |
| 321 | 14.3 | \(1.68\pm0.18\) | — | 81* | \(d\) |
| 324 | 15.5 | \(1.78\pm0.16\) | \(0.28\pm0.07\) | 83 | \(d\) |
| 327 | 17 | — | \(0.38\pm0.11\) | 84 | \(d\) |
| 330 | 18.1 | \(2.28\pm0.17\) | — | 81* | \(d\) |
| 332 | 19.0 | \(2.45\pm0.13\) | \(-0.32\pm0.05\) | 83 | \(d\) |
| 336 | 20.6 | \(2.64\pm0.19\) | — | 81* | \(d\) |
| 338 | 21.5 | \(2.69\pm0.26\) | \(0.29\pm0.08\) | 83 | \(d\) |
| 340 | 22.3 | — | \(0.33\pm0.04\) | 84 | \(d\) |
| 340 | 22.3 | \(2.8\pm1.0\) | — | 81 | |
| 341 | 23.0 | \(1.8\pm0.6\) | \(0.11\pm0.06\) | 85 | \(d\) |
| 342 | 23.5 | \(2.3\pm0.4\) | — | 82 | |
| 345 | 24.5 | \(3.5\pm0.9\) | — | 86 | |
| 365 | 33 | \(6.1\pm5\) | — | 86 | |
| 380 | 39.5 | \(10.6\pm2.6\) | — | 86 | |
| 381 | 40 | \(7.3\pm2.3\) | — | 87 | |
| 437 | 63 | \(13.5\pm1.3\) | \(0.2\pm0.02\) | 88 | \(d\) |
| 437 | 63 | \(47.5\pm2.4\) | — | 81 | |
| 440 | 63.7 | \(40.0\pm10\) | \(0.15\pm0.06\) | 89 | |
| 460 | 71.8 | \(15.4\pm1.6\) | \(0.24\pm0.03\) | 90 | \(d\) |
| 460 | 71.8 | \(18\pm7\) | — | 100 | \((np)\) |
| 460 | 71.8 | \(26\pm20\) | — | 101 | \((np)\) |
| 500 | 87 | \(41\pm20\) | — | 101 | \((np)\) |
| 540 | 102.5 | \(55\pm21\) | — | 101 | \((np)\) |
| 560 | 110 | \(50\pm9\) | — | 100 | \((np)\) |
| 580 | 117 | \(79\pm21\) | — | 101 | \((np)\) |
| 600 | 124.5 | \(84\pm21\) | — | 101 | \((np)\) |
| 620 | 131.7 | \(99\pm22\) | — | 101 | \((np)\) |
| 640 | 139 | \(107\pm22\) | — | 101 | \((np)\) |
| 660 | 145.8 | \(100\pm12\) | — | 100 | \((np)\) |
| 660 | 145.8 | \(119\pm23\) | — | 101 | \((np)\) |
| 660 | 145.8 | \(31\pm2\) | \(0.23\pm0.03\) | 91 | \(d\) |
Table XIII
Angular distribution in the center-of-mass system of the reaction \(\pi^+ + d \to p+p\) for various \(\pi^+\)-meson energies
The table also gives the proton energies in the laboratory coordinate system for the cross section of the inverse reaction \(p+p\to\pi^+ + d\).
| \(E_{\pi}^{\mathrm{c.m.}},\ \mathrm{MeV}\) | \(E_p^{\mathrm{lab}},\ \mathrm{MeV}\) | Angular distribution | \(\sigma_{\mathrm{tot}}\times 10^{28}\ \mathrm{cm}^2\) | Reference |
|---|---|---|---|---|
| 23 | 341 | — | \(2.84\pm0.5\) | 92 |
| 25 | 345 | \(0.22+\cos^2\vartheta\) | \(2.2\pm0.2\) | 93 |
| 40 | 381 | \(0.2+\cos^2\vartheta\) | \(6.6\pm0.6\) | 93 |
| 53 | 413 | \(0.18+\cos^2\vartheta\) | \(9.7\pm0.9\) | 93 |
| 76 | 470 | \((0.27\pm0.09)+\cos^2\vartheta\) | \(13.9\pm2.3\) | 81 |
| 94 | 515 | \((0.39\pm0.08)+\cos^2\vartheta\) | \(20.4\pm2.2\) | 81 |
where \(\eta\) and \(\vartheta\) are the momentum and the scattering angle of the \(\pi\)-meson in the center-of-inertia system;
\(a_1=0.138\pm0.015;\ a_2=0.200\pm0.078;\ a_3=2.44\pm0.17.\) The total meson-production cross section is expressed in the form
\[ \sigma_{\text{tot}}=(0.138\eta+1.01\eta^3)\cdot 10^{-27}\ \text{cm}^2, \]
\[ \eta=\frac{p_\pi}{m_\pi c}. \]
It follows from Fig. 8 that the curve of the dependence of the cross section of the process \(p+p\to\pi^+ + d\) on energy reaches its greatest value at a proton energy of \(600\text{--}640\ \text{MeV}\), which in the center-of-inertia system corresponds to a \(\pi\)-meson energy of \(120\text{--}140\ \text{MeV}\). With further increase of the energy of the incident protons, the \(\pi\)-meson production cross section shows a tendency to decrease, which indicates the possible existence of a maximum in the curve \(\sigma(pp\to\pi^+ d)=f(E_p)\) at incident-proton energies \(\sim 600\ \text{MeV}\). Such an energy dependence of the cross section of the process under consideration may be explained by the influence of the interaction of the \(\pi^+\)-meson with nucleons in a state with isotopic spin and total angular momentum \(3/2\), which, as follows from experiments on the scattering of \(\pi\)-mesons by nucleons, has a “resonance” at \(\pi\)-meson energies \(\sim 120\ \text{MeV}\) (in the center-of-inertia system). It is also seen from Fig. 8 that the cross section for the production of \(\pi^+\)-mesons with the formation in the final state of unbound nucleons increases monotonically with increasing energy of the incident particles up to \(E_p=660\ \text{MeV}\). The absence of a maximum in the cross section of this process \((pp+\pi^+ np)\) is apparently a consequence of the large number of possible final states in \(\pi\)-meson production, which mask the “resonance” phenomenon in the state \(T=3/2,\ J=3/2\). An analogous change of the cross section with energy is also observed in the production of \(\pi^0\)-mesons in \((p-p)\) and \((n-p)\) collisions (see below).
Fig. 9. Total cross sections for the production of \(\pi\)-mesons in nucleon-nucleon collisions.
As for the energy dependence of the \(\pi^+\)-meson production cross section in the reaction \((pp\to\pi^+ d)\) at low energies, as was indicated earlier, the expression
\[ \sigma=0.14\eta+1.0\eta^3, \]
where \(\eta\) is the momentum of the \(\pi\)-meson in units of \(m_\pi c\) in the center-of-inertia system, approximates the experimental results quite satisfactorily up to an incident-proton energy of \(\sim 500\ \text{MeV}\).
Reactions (1), (2), and (4) have been studied considerably less completely. The total cross sections and angular distributions for these processes, measured at certain values of the nucleon energy, are given in Table XIV; the total cross sections of these processes are shown in Fig. 9.
Let us note that the values of the total cross sections for the production of \(\pi^0\)-mesons given in Table XIV were found under the assumption of isotropy of the cross section
Table XIV
Total cross sections and angular distributions of reactions (1), (2), and (4)
Column “\(a\)” corresponds to the isotropic part of the cross section in the center-of-inertia system
\[
\frac{d\sigma}{dO}\sim a+\cos^2\vartheta .
\]
\(T_\pi\) is the maximum possible energy of the \(\pi\)-meson in the center-of-inertia system.
| Reaction | Energy of incident nucleon, MeV | \(T_\pi\), MeV | \(a\) | \(\sigma_{\text{tot}}\times 10^{28}\ \text{cm}^2\) | Reference |
|---|---|---|---|---|---|
| \(p+n\to\pi^0\) | 340 | 28 | — | 0,09 | 97 |
| \(p+n\to\pi^0+d\) | 392 | 50 | \(0,28^{+0,26}_{-0,14}\) | \(4,1\pm0,7\) | 94 |
| \(p+n\to\pi^0+d\) | 400 | 53 | \(0,20\pm0,06\) | — | 95 |
| \(p+n\to \begin{cases}\pi^0+d\\ \pi^0+n+p\end{cases}\) | \(E_{\text{eff}}=400^{*}\) | 53 | — | \(4,0\pm2,0\) | 103 |
| \(p+n\to \begin{cases}\pi^0+d\\ \pi^0+n+p\end{cases}\) | \(E_{\text{eff}}=590^{*}\) | 122 | — | \(59\pm15\) | 104 |
| \(p+n\to \begin{cases}\pi^0+d\\ \pi^0+n+p\end{cases}\) | 670 | 155 | — | \(78\pm16\) | 96 |
| \(p+p\to\pi^0+p+p\) | 341 | 28 | — | \(0,10\pm0,03\) | 97 |
| \(p+p\to\pi^0+p+p\) | 430 | 64 | — | \(4,5\pm1,5\) | 98 |
| \(p+p\to\pi^0+p+p\) | 460 | 72 | — | \(5,0\pm2,5\) | 105 |
| \(p+p\to\pi^0+p+p\) | 480 | 79 | — | \(4,4\pm3\) | 106 |
| \(p+p\to\pi^0+p+p\) | 480 | 79 | — | \(10,5\pm4,5\) | 107 |
| \(p+p\to\pi^0+p+p\) | 560 | 110 | — | \(12\pm3\) | 110 |
| \(p+p\to\pi^0+p+p\) | 650 | 137 | — | \(30\pm6\) | 108 |
| \(p+p\to\pi^0+p+p\) | 660 | 146 | — | \(34\pm4\) | 110 |
| \(p+p\to\pi^0+p+p\) | 670 | 155 | — | \(37\pm8\) | 96 |
| \(n+p\to\pi^-+p+p^{***}\) | 409 | 49 | \(A+B\cos\vartheta+C\cos\vartheta^{**}\) | \(1,6\pm0,4\) | 109 |
*) \(E_{\text{eff}}\) is the mean effective energy of the neutrons causing the production of \(\pi^0\)-mesons, calculated for a continuous energy spectrum of the neutron beam.
**) The coefficients \(A,B,C\), according to the data of work 109, turned out to be equal (in units of \(\times 10^{-27}\ \text{cm}^2/\text{sterad}\))
\[ \begin{aligned} A&=1,07\pm0,39,\\ B&=1,38\pm0,78,\\ C&=0,57\pm1,14. \end{aligned} \]
***) In the case of production of a \(\pi^+\)-meson in the reaction \(n+p\to\pi^++n+n\), the coefficient \(B\) has the opposite sign. Additional results on the determination of the cross section of the reaction \(p+p\to\pi^0+p+p\) are given in Fig. 10.
in the center-of-inertia system. However, the indicated quantities \(\sigma_{\text{tot}}\) do not change substantially if one assumes that
\[ \left(\frac{d\sigma}{dO}\right)_{\text{c.i.}}\sim\cos^2\vartheta, \]
where the angle \(\vartheta\) determines the direction of emission of the produced \(\pi^0\)-meson in the system
Table XIVa
| Reaction | \(E^{\pi}_{\mathrm{lab}},\ \mathrm{MeV}\) | Assumed form of the angular distribution of \(\pi^0\)-mesons | \(\sigma_{\mathrm{tot}}\), in units of \(10^{-28}\ \mathrm{cm}^2\) | Reference |
|---|---|---|---|---|
| \((pp,\pi^0)\) | 460 | \(\sim \cos^2\vartheta\) | \(4\pm2\) | 105 |
| \((pp,\pi^0)\) | 460 | isotropic | \(5\pm2.5\) | 105 |
| ” | 480 | \(\sim \cos^2\vartheta\) | \(6.2\pm2.8\) | 107 |
| ” | 480 | isotropic | \(10.5\pm4.5\) | 107 |
| ” | 650 | \(\sim \cos^2\vartheta\) | \(37\pm7\) | 108 |
| ” | 650 | isotropic | \(30\pm6\) | 108 |
| \((np,\pi^0)\) | 590 | \(\sim \cos^2\vartheta\) | \(63\pm17\) | 104 |
| \((np,\pi^0)\) | 590 | isotropic | \(59\pm15\) | 104 |
center-of-mass system. This is seen from Table XIVa, where the total cross sections for reactions of \(\pi^0\)-meson production in \((p-p)\) and \((n-p)\) collisions are given, calculated under various assumptions about the angular distribution of the produced \(\pi^0\)-mesons in the center-of-mass system.
In Figs. 9 and 10 it is seen that, as in the case of \(\pi^+\)-meson production in the reaction \((p+p\to \pi^+ + n + p)\), the cross section of reactions (1) and (2) increases monotonically with increasing energy and shows no maximum up to \(E_p=670\ \mathrm{MeV}\).
At low energies the cross sections of the processes \((pp,\pi^0)\) and \((np,\pi^0)\) satisfy the dependences
\[ \sigma(pp\to \pi^0)\sim \eta^8, \qquad \text{(see Fig. 10)} \]
\[ \sigma(np\to \pi^0 d)\sim \eta^3, \qquad \text{(see }{}^{94}\text{)} \]
\[ \sigma\left(np\to \begin{pmatrix} \pi^0 d\\ \pi^0 np \end{pmatrix}\right) \sim \eta^{3.3\pm0.5}, \qquad \text{(see }{}^{104}\text{)} \]
where \(\eta\) is the maximum possible momentum of the \(\pi\)-meson in the center-of-mass system. These relations correspond to the theoretical predictions summarized in Table XI.
Fig. 10. Cross sections for \(\pi^0\)-meson production in \((p-p)\)-collisions: \(\bullet\)—\(\left(\dfrac{d\sigma}{d\omega}\right)_{90^\circ}\) according to the data of work \(^{111}\), \(\times\)—\(\left(\dfrac{d\sigma}{d\omega}\right)\) according to the data of works \(^{97,98,96,106}\), \(\eta_{\max}\)—the maximum possible momentum of the produced \(\pi^0\)-meson in the center-of-mass system.
Let us examine reaction (1) in more detail. According to the hypothesis of isotopic invariance, the cross sections for production of \(\pi^+\)- and \(\pi^0\)-mesons with formation of a deuteron in the final state are described by one and the same matrix element. In accordance with the notation adopted by us, the cross sections of the indicated processes are written as follows:
\[ \sigma(p+p\to \pi^+ + d)=|M_3|^2, \tag{6} \]
\[ \sigma(n+p\to \pi^0 + d)=\frac{1}{2}|M_3|^2. \tag{7} \]
These expressions follow directly from equations (5), and, if the isospin hypothesis is valid, the production cross section of \(\pi^+\)-mesons (6) must be twice as large as the production cross section of \(\pi^0\)-mesons (7). According to Schluter’s measurements\(^{94}\), the cross section (7) at an incident-neutron energy of \(392\ \mathrm{MeV}\) is equal to \(0.41 \pm 0.07\ \mathrm{mbarn}\) and in the incident-neutron energy interval \(E_n = 340\text{--}450\ \mathrm{MeV}\) is approximated quite well by the dependence
\[ \sigma(n+p \to \pi^0+d)=(0.47\pm0.08)\eta^3\ \mathrm{mbarn}, \]
where \(\eta\) is the momentum of the \(\pi\)-meson in units of \(m_\pi c\) in the center-of-mass system. At the same time, as indicated above, the cross section (6) in this energy interval can be approximated by the expression \(\sigma(p+p\to\pi^+ + d)=0.14\eta+1.0\eta^3\). Thus, the predicted ratio of the cross sections (6) and (7) was indeed observed experimentally.
The angular distribution of \(\pi^0\)-mesons produced in the reaction \(n+p=\pi^0+d\) turns out to be symmetric with respect to the angle \(\vartheta=\dfrac{\pi}{2}\) in the center-of-mass system and is well approximated by the dependence\(^{94,95}\)
\[ \sigma(\vartheta)=a+\cos^2\vartheta. \]
According to Schluter’s data\(^{94}\), the coefficient \(a\) is equal to
\[ a=0.28^{+0.26}_{-0.14}. \]
This result also agrees with the predictions of the isospin-invariance hypothesis, according to which the angular distributions of \(\pi^+\)- and \(\pi^-\)-mesons produced in the reactions entering into (6) and (7) must be similar.
According to the measurements of Schluter\(^{94}\) and Hildebrand*) the production cross section of a \(\pi^0\)-meson in the reaction \((np,\pi^0)\) at \(E_n=400\ \mathrm{MeV}\), with formation of a deuteron, is approximately \(2\text{--}3\) times smaller than the cross section of this reaction with formation, in the final state, of unbound nucleons.
In the study of reaction (4), a substantial asymmetry of the angular distribution of the produced \(\pi^\pm\)-mesons with respect to the angle \(\vartheta=90^\circ\) in the center-of-mass system was discovered\(^{99,109}\). As stated above, this asymmetry is a consequence of interference of the final states of the produced \(\pi\)-mesons, described by the matrix elements \(M_1\) and \(M_2\) (see relations (5)).
4. PRODUCTION OF \(\pi\)-MESONS ON NUCLEONS UNDER THE ACTION OF \(\gamma\)-QUANTA
Production of \(\pi\)-mesons by photons is one of the simplest processes associated with the interaction of mesons with nucleons and, in the case of production on a proton, is described by the following reactions:
\[ \gamma+p\to\pi^+ + n, \tag{1} \]
\[ \gamma+p\to\pi^0+p. \tag{2} \]
The study of photoproduction of \(\pi\)-mesons on neutrons is considerably complicated by the fact that in this case it is necessary to take into account various nuclear effects, most of which at present can be accounted for only qualitatively. It should be noted, however, that the study of photoproduction of \(\pi\)-mesons on deuterium substantially supplements our information on the elementary processes (1) and (2). Here only those effects will be considered which are associated with the production of \(\pi\)-mesons on free nucleons; the results of experiments on photoproduction
*) Private communication to Rosenfeld\(^{81}\).
mesons on nuclei are given below. An intense beam of high-energy \(\gamma\)-quanta is obtained when electrons are decelerated in the Coulomb field of nuclei. Thus, in a platinum target of thickness \(\sim 0.5\) mm, about \(15\%\) of the energy of an electron beam of energy \(325\) MeV is converted into a directed beam of photons with an angular spread \(\sim 1^\circ\) \({}^{112}\). The energy spectrum of bremsstrahlung photons is shown in Fig. 11.
Fig. 11. Spectrum of bremsstrahlung \(\gamma\)-radiation obtained in the deceleration of electrons of energy
\(E_{\mathrm{el}} = 325\) MeV \({}^{112}\).
Fig. 12. Angular distribution of \(\pi^+\)-mesons produced under the action of bremsstrahlung \(\gamma\)-quanta with \(E_{\max}=320\) MeV.
In Figs. 12 and 13 are shown the angular and energy distributions of \(\pi^+\)-mesons produced on hydrogen under the action of \(\gamma\)-quanta of bremsstrahlung radiation with maximum energy \(E_\gamma^{\max}=320\) MeV \({}^{112,113}\). The cross sections shown in these figures are referred to the number of “effective \(\gamma\)-quanta”—\(Q\), which is defined as
\[ Q=\frac{1}{E_{\max}}\int_{0}^{E_{\max}} E_\gamma f(E_\gamma)\,dE_\gamma . \]
Here \(f(E_\gamma)\) is the spectrum of bremsstrahlung radiation with maximum energy \(E_{\max}\). The total cross section for production of \(\pi^+\)-mesons under the action of bremsstrahlung \(\gamma\)-radiation with maximum energy \(E^{\max}=320\) MeV was obtained by integrating the angular distribution given in Fig. 12 and was found to be equal to
\[ \sigma_t=(1.03\pm0.21)\times10^{-28}\ \mathrm{cm}^2/Q . \]
Fig. 13. Energy distribution of \(\pi^+\)-mesons produced under the action of bremsstrahlung \(\gamma\)-quanta with \(E_{\max}=320\) MeV at the angle
\(\vartheta_{\mathrm{lab}}=90^\circ\).
Since we do not have a monochromatic source of \(\gamma\)-quanta, in order to obtain the angular distribution of \(\pi\)-mesons produced by photons of a given energy, as well as the dependence of the meson photoproduction cross section on energy,
γ-quanta require the simultaneous measurement of both the energy and the direction of emission of the produced π-meson or recoil nucleon. The energy of the γ-quantum in meson production by reactions (1), (2) can also be determined if
Fig. 14. Energy dependence of the total cross section for production of a π⁺-meson in the reaction
\(\gamma + p \to \pi^+ + n\). The energy of the γ-quanta is indicated in the laboratory coordinate system. The energy of the π⁺-meson is indicated in the center-of-mass system.
Fig. 15. Energy dependence of the total cross section for production of a π⁰-meson in the reaction
\(\gamma + p \to \pi^0 + p\).
one registers the directions of emission of two particles: the meson and the recoil nucleon. However, this latter method has not been used, since it requires very good angular resolution. By the method indicated above, the cross section for production of π⁺- and π⁰-mesons as a function of the energy of the γ-quanta was measured in sufficient detail\(^{112,114—132}\). The curves of the dependence of the total photoproduction cross section for π⁺- and π⁰-mesons on the energy of the γ-quanta are shown in Figs. 14 and 15. As is seen from these figures, the photoproduction cross section for π⁺- and π⁰-mesons reaches its maximum value at γ-quantum energy \(E_\gamma = 310—320\) MeV, which corresponds to an energy of the produced π-meson of 110–115 MeV (in the center-of-mass system). With a further increase in the energy of the γ-quanta the cross section falls. The “width” of the maximum formed is approximately 100 MeV in the center-of-mass system.
Table XV
Dependence of the photoproduction cross section of π⁺-mesons on energy near the production threshold. The cross section is given in units of \(\times 10^{-29}\ \mathrm{cm^2/sterad}\), \(m_\pi\) and \(p_\pi\) are the mass and momentum of the π-meson in the center-of-mass system
| \(E_\gamma^{\mathrm{lab}}\), MeV | \(\left(\dfrac{d\sigma}{dO}\right)_{90^\circ}\) | \(\dfrac{d\sigma}{dO}\times \dfrac{m_\pi c}{p_\pi}\) |
|---|---|---|
| 165 | 0.48 | 1.35 |
| 175 | 0.64 | 1.31 |
| 185 | 0.69 | 1.16 |
At small γ-quantum energies the photoproduction cross section of π⁺-mesons is proportional to the momentum of the π-meson in the center-of-mass system\(^{118,125}\). The corresponding data are given in Table XV.
The dependence \(\sigma \sim p_\pi\), where \(p_\pi\) is the momentum of the π⁺-meson, together with the isotropy of the angular distribution of π⁺-mesons produced by γ-quanta with energy
$E_\gamma \lesssim 200$ MeV (see Fig. 16), indicates that the production of $\pi^+$ mesons near the production threshold occurs predominantly in the $S$ state.
The photoproduction cross section of $\pi^0$ mesons at low $\gamma$-quantum energies exhibits a much sharper dependence on energy in comparison with the photoproduction cross section of $\pi^+$ mesons. Mills and Koster$^{114}$ found that, for $\gamma$-quantum energies $E_\gamma = 170\text{--}240$ MeV,
\[ \sigma_{\mathrm{tot}}(\gamma + p \to \pi^0) \sim (E_\gamma - E_{\mathrm{thr}})^{2.2}, \]
where $E_{\mathrm{thr}}$ is the energy threshold for photoproduction of $\pi^0$ mesons. Such a dependence $\sigma = f(E)$ can be explained by the fact that, in the production of a $\pi^0$ meson, the contribution of the $S$ wave is small. The same conclusion can also be drawn from an analysis of the angular distribution of the produced $\pi^0$ mesons (see below). The angular distribution of $\pi$ mesons produced by $\gamma$ quanta was investigated over a wide
Fig. 16. Angular distributions of $\pi^+$ mesons produced by $\gamma$ quanta of various energies. The smooth curves $\sigma = f(\vartheta)$ at $E_\gamma = 200,\ 235,\ 265,\ 275$ MeV represent dependences of the form $\sigma(\vartheta) = A_+ + B_+ \cos \vartheta + C_+ \cos^2 \vartheta$, where $\vartheta$ is the emission angle of the $\pi$ meson in the center-of-mass system.
Fig. 17. Energy dependence of the coefficients $A_+$, $B_+$, and $C_+$, determining the angular distribution of $\pi^+$ mesons produced in the reaction $\gamma + p \to \pi^+ + n$.
range of $\gamma$-quantum energies*). As is seen from Fig. 16, the curves of the angular dependence of the photoproduction cross sections of $\pi^+$ mesons are asymmetric with respect to the angle $\vartheta = \pi/2$ in the center-of-mass system and are approximated rather well by the expression
\[ \sigma(\vartheta) = A_+ + B_+ \cos \vartheta + C_+ \cos^2 \vartheta, \tag{3} \]
written under the assumption that, at the $\gamma$-quantum energies under consideration, $\pi^+$ mesons are formed mainly in $S$ and $P$ states. The corresponding values of the coefficients $A_+$, $B_+$, and $C_+$ are given in Table XVI.
In accordance with the data of Table XVI, Fig. 17 gives the coefficients $A_+$, $B_+$, and $C_+$ as functions of the $\gamma$-quantum energy.
*) See $^{112,115,118,119,126\text{--}131,133}$.
Table XVI
Values of the coefficients \(A_+\), \(B_+\), and \(C_+\) in the relation
\(\sigma(\vartheta)=A_+ + B_+ \cos\vartheta + C_+ \cos^2\vartheta\), where \(\vartheta\) is the emission angle of the produced \(\pi^+\)-meson with respect to the direction of the \(\gamma\)-quantum beam in the center-of-inertia system
| \(E_\gamma^{\mathrm{lab}}\) | \multicolumn{3}{c|}{In units of \(10^{-30}\ \mathrm{cm}^2/\mathrm{sterad}\)} | \(\sigma_{\mathrm{tot}}\times 10^{28}\ \mathrm{cm}^2\) | Reference |
|---:|---:|---:|---:|---:|---:|
| | \(A_+\) | \(B_+\) | \(C_+\) | | |
| 200 | \(10,3 \pm 3,6\) | \(-(1,9 \pm 1,3)\) | \(-(1,8 \pm 2,9)\) | \(1,22 \pm 0,09\) | 119 |
| 230 | \(14,3 \pm 0,3\) | \(-(1,7 \pm 0,6)\) | \(-(6,2 \pm 1,0)\) | \(1,53 \pm 0,03\) | 128 |
| 235 | \(14,5 \pm 1,9\) | \(-(2,8 \pm 0,8)\) | \(-(4,5 \pm 1,6)\) | \(1,64 \pm 0,06\) | 119 |
| 255 | — | — | — | \(1,9 \pm 0,3\) | 112 |
| 260 | \(17,9 \pm 0,4\) | \(-(3,5 \pm 0,7)\) | \(-(8,6 \pm 1,2)\) | \(1,9 \pm 0,04\) | 128 |
| 275 | \(25,1 \pm 3,0\) | \(-(6,6 \pm 1,6)\) | \(-(14,6 \pm 1,6)\) | \(2,5 \pm 0,5\) | 126 |
| 290 | \(20,2 \pm 0,4\) | \(-(1,8 \pm 0,6)\) | \(-(7,7 \pm 1,1)\) | \(2,21 \pm 0,04\) | 128 |
| 320 | \(19,9 \pm 0,4\) | \(-(0,4 \pm 0,6)\) | \(-(7,8 \pm 1,0)\) | \(2,17 \pm 0,04\) | 128 |
| 350 | \(15,5 \pm 0,4\) | \(1,2 \pm 0,5\) | \(-(4,9 \pm 0,9)\) | \(1,74 \pm 0,03\) | 128 |
| 380 | \(11,6 \pm 0,3\) | \(1,7 \pm 0,4\) | \(-(3,9 \pm 0,7)\) | \(1,29 \pm 0,03\) | 128 |
| 410 | \(8,5 \pm 0,3\) | \(2,3 \pm 0,4\) | \(-(1,8 \pm 0,6)\) | \(0,99 \pm 0,03\) | 128 |
| 440 | \(6,2 \pm 0,3\) | \(2,9 \pm 0,4\) | \(-(0,4 \pm 0,6)\) | \(0,76 \pm 0,03\) | 128 |
| 470 | \(4,6 \pm 0,2\) | \(3,1 \pm 0,4\) | \(-(0,2 \pm 0,5)\) | \(0,57 \pm 0,02\) | 128 |
The angular distribution of \(\pi^0\)-mesons produced by \(\gamma\)-quanta is likewise well approximated by a dependence of the form
\(\sigma(\vartheta)=A_0+B_0\cos\vartheta+C_0\cos^2\vartheta\), where \(\vartheta\) is the emission angle of the \(\pi^0\)-meson in the center-of-inertia system. It is true that the experimental errors in this case are considerably larger than in determining the coefficients \(A_+\), \(B_+\), \(C_+\). Figure 18 gives the coefficients \(A_0\), \(B_0\), and \(C_0\) for various \(\gamma\)-quantum energies \(^{129-131}\). From Fig. 18 it is evident that the coefficient \(B_0\) is small and that the distribution \(\sigma(\vartheta)\), consequently, is symmetric with respect to the angle \(\vartheta=\dfrac{\pi}{2}\). The small magnitude of the coefficient \(B_0\) indicates that \(\pi^0\)-mesons are produced predominantly in the \(P\)-state. According to \(^{129}\), the total cross section for production of a \(\pi^0\)-meson in the \(S\)-state in the \(\gamma\)-quantum energy interval 200–300 MeV is
\[ \sigma_{\mathrm{tot}}(S)=(4\pm 3)\cdot 10^{-30}\ \mathrm{cm}^2, \]
i.e., approximately 3% of the value of the total photoproduction cross section of \(\pi^0\)-mesons at this energy.
Fig. 18. Energy dependence of the coefficients \(A_0\), \(B_0\), and \(C_0\), which determine the angular distribution of \(\pi^0\)-mesons produced by \(\gamma\)-quanta:
\[ \sigma(\vartheta)=A_0+B_0\cos\vartheta+C_0\cos^2\vartheta . \]
CONCLUSION
As is seen from Figs. 4, 5, 14, and 15, the energy dependence of the cross sections for photoproduction and scattering of \(\pi\)-mesons on nucleons exhibits many common features. The cross section of both processes reaches its largest value at a \(\pi\)-meson energy \(E_\pi=110\text{–}120\) MeV in the center-of-inertia system. The “width of the maximum” in the cross sections of these processes is also approximately the same. A detailed
analysis shows that the “resonance” in the cross sections for photoproduction and scattering of \(\pi\)-mesons on nucleons is determined by the interaction in a state with total angular momentum and isotopic spin \(3/2\). These features of the photoproduction and scattering cross sections may be regarded as an argument that both processes proceed through one and the same intermediate state of the excited nucleon. Such a view is also supported by the results of experiments on the production of \(\pi^+\)-mesons in \((p-p)\) collisions. As can be seen from Fig. 8, the cross section of this process reaches its greatest value at approximately the same \(\pi^+\)-meson energy in the center-of-mass system as do the processes of scattering and photoproduction of \(\pi\)-mesons. The absence of maxima in the energy dependence of the cross sections of the reactions \((p+p \to \pi^+ + n + p)\), \((p+p \to \pi^0 + p + p)\), \((n+p \to \pi^0 + n + p)\) can be explained by the large number of final states of the particles, which complicates the interpretation of the processes.
However, the greater “width of the maxima” in the cross sections of the scattering and photoproduction processes of \(\pi\)-mesons under consideration, and the associated short lifetime of the assumed excited state, contradict such an explanation. It is also contradicted by the fact that both the scattering process and the process of photoproduction of \(\pi\)-mesons on nucleons cannot be explained solely as the result of the decay of one definite excited state of the nucleon. This follows from the presence of an entire set of phases in \((\pi-p)\)-scattering and from the presence of \(S\) and \(P\) waves in the photoproduction of \(\pi\)-mesons.
Thus, it may be said that the existing experimental data indicate the presence of a deep internal connection between the processes of scattering and photoproduction of \(\pi\)-mesons on nucleons. At present, however, it is difficult to draw any definite conclusion regarding the validity of the hypothesis of the existence of an excited state of the nucleon responsible for the processes of scattering and production of \(\pi\)-mesons on nucleons.
The considered reactions of \(\pi\)-meson scattering on nucleons and meson production in nucleon-nucleon collisions testify to the validity of the hypothesis of isotopic invariance as applied to such processes.
PART II
INTERACTIONS OF \(\pi\)-MESONS WITH NUCLEI
In studying the processes of interaction of \(\pi\)-mesons with nuclei, as well as in studying the production of \(\pi\)-mesons on nuclei, it should be borne in mind that \(\pi\)-mesons, interacting strongly with nuclear matter, are a powerful means of investigating the nucleus. However, in studying reactions in which a nucleus participates, only particular questions concerning elementary interactions can be solved. This is connected with the fact that information about the nucleus is very incomplete, and therefore conclusions about elementary processes drawn from experiments with nuclei depend on various assumptions concerning nuclei.
As will be shown below, all the results of experiments on the interaction and production of \(\pi\)-mesons on nuclei can be explained within the framework of the assumption that the forces of interaction of mesons with nucleons bound in a nucleus do not differ substantially from the forces of interaction of mesons with free nucleons, and that the interaction of \(\pi\)-mesons with a nucleus occurs as a result of the interaction of mesons with individual nucleons moving in the nucleus. With the existing accuracy and completeness of the experimental data this does not mean that there is no difference in the interaction of \(\pi\)-mesons with free nucleons and with nucleons bound in the nucleus; however, one may think that such a difference is small.
5. INTERACTION OF π-MESONS WITH NUCLEI
In the interaction of π-mesons with nuclei the following processes may be observed: elastic and inelastic scattering of π-mesons, scattering of π-mesons with charge exchange, and, finally, absorption of π-mesons in the nucleus. Experiments on the interaction of mesons with photographic-emulsion nuclei can serve to obtain a general picture of the interaction of π-mesons with nuclei.
Table XVII
Mean free paths of π-mesons in photographic emulsion for various types of interaction
The mean free path corresponding to the geometrical cross section for interaction with photographic-emulsion nuclei is equal to 27 cm.
| Meson energy, MeV | Inelastic scattering | “Disappearing in flight” | “Stars” | Total nuclear interaction | Literature |
|---|---|---|---|---|---|
| 20—35, $\pi^-$ | 225 | 180 | 82 | 45 | 134 |
| 30—50, $\pi^-$ | 318 | 239 | 39 | 31 | 135 |
| 60—90, $\pi^-$ | 123 | 246 | 31 | 22.8 | 135 |
| 100—110, $\pi^-$ | 82 | 145 | 34 | 20.5 | 135 |
| 190—230, $\pi^-$ | 94 | 282 | 55 | 30.5 | 136 |
| 40, $\pi^+$ | — | — | 69 | 69 | 137 |
| 35—50, $\pi^+$ | 1560 | — | 62 | 55 | 138 |
| 62, $\pi^+$ | 135 | — | 40 | 31 | 139 |
| 70—80, $\pi^+$ | 250 | — | 36 | 31 | 138 |
In Table XVII a summary is given of experimental data for mean free paths for various types of interaction of π-mesons with nuclei. It is seen from the table that the principal role in nuclear interaction is played by the absorption of π-mesons by the nucleus (“stars”). “Disappearing in flight,” which may be interpreted mainly as the result of absorption of π-mesons with the emission from the nucleus of neutrons only, must also be attributed to the absorption of π-mesons. Indeed, disappearing in flight is not observed in the interaction of $\pi^+$-mesons, since in this case the emission from the nucleus of protons rather than neutrons is more probable. Approximately one quarter of the nuclear interaction is inelastic scattering of π-mesons.
It is also seen from the table that there is a weak dependence of the interaction cross section of π-mesons with nuclei on energy; moreover, at energies of 50—200 MeV the photographic-emulsion nuclei are practically opaque to π-mesons. In addition, a difference is observed in the interaction of π-mesons of different signs. The difference in the interaction of $\pi^+$- and $\pi^-$-mesons with nuclei can be explained by the different action of the Coulomb field of the nuclei on $\pi^+$- and $\pi^-$-mesons. The action of the Coulomb field may manifest itself, first, in a different change in the density of the beam of incident $\pi^+$- and $\pi^-$-mesons, owing to the attraction of $\pi^+$ and repulsion of $\pi^-$-mesons by the Coulomb field of the nucleus, and, second, in a different change in the kinetic energy of $\pi^+$- and $\pi^-$-mesons when approaching the nucleus by the magnitude of the Coulomb potential in the nucleus. The action of the first factor can be estimated from simple quasiclassical considerations. Taking this factor into account gives for the nuclear-interaction cross section the multiplier $(1 \pm V_{\mathrm{coul}}/E_\pi)^{-1}$.
In Fig. 19 the dependence
\[ \frac{\sigma_{\text{nucl}}}{\sigma_{\text{geom}}\left(1 \pm V/E_\pi\right)} \]
on the pion energy is plotted; from this it is seen that, as expected, the curves for \(\pi^+\)- and \(\pi^-\)-mesons are displaced relative to one another by twice the Coulomb potential (the average Coulomb potential for photoemulsion nuclei is approximately \(8\) MeV). Thus the action of the Coulomb field completely explains the experimentally observed difference in the interaction cross section of \(\pi^+\)- and \(\pi^-\)-mesons with nuclei.
Fig. 19. Dependence of
\[ \frac{\sigma_{\text{nucl}}}{\sigma_{\text{geom}}\left(1 \pm \dfrac{V}{E_\pi}\right)} \]
on the pion energy for photoemulsion nuclei. For \(\pi^+\)-mesons \(\left(1+\dfrac{V}{E_\pi}\right)\) is taken, for \(\pi^-\)-mesons \(\left(1-\dfrac{V}{E_\pi}\right)\); \(V\) is the average Coulomb potential of the photoemulsion nuclei, \(E_\pi\) is the kinetic energy of the \(\pi\)-mesons.
Owing to the strong interaction of \(\pi\)-mesons with nuclear matter, the cross section of the nuclear interaction of \(\pi\)-mesons with heavy nuclei is equal to the geometrical one over very wide intervals of pion energies\(^{145,160,161}\). The experimentally observed decrease in the interaction cross section of 50 and 37 MeV \(\pi^+\)-mesons with lead should be explained not by the transparency of nuclei for \(\pi^+\)-mesons of this energy, but by the action of the Coulomb field of the nucleus, which was discussed above. In Fig. 20 data are presented that characterize the transparency of light nuclei for \(\pi\)-mesons with energies from 20 to 350 MeV. As is seen from the figure, even light nuclei are not transparent for \(\pi\)-mesons with energies of 100–125 MeV; the curve of the nuclear-interaction cross section has a broad maximum in the energy region corresponding to the maximum of the interaction cross section of \(\pi\)-mesons with free nucleons. The increase in the width of the maximum in the interaction cross section of \(\pi\)-mesons for nuclei is evidently connected with the presence of internal motion of nucleons in the nucleus. The total interaction cross sections of \(\pi\)-mesons with nuclei (nuclear + elastic) are approximately twice as large as the pure nuclear interaction cross section of \(\pi\)-mesons\(^{165}\). In connection with the foregoing it is clear that experiments on the scattering of \(\pi\)-mesons by light nuclei can most likely answer the question of the character of the interaction of \(\pi\)-mesons with nucleons bound in the nucleus. Experiments on the scattering of \(\pi\)-mesons by deuterium\(^{53,150,166}\) have shown that the cross section \(\sigma(\pi + d)\) is almost equal to the sum of the interaction cross sections of \(\pi^+\)- and \(\pi^-\)-mesons with a free proton. This circumstance indicates that the interaction of mesons with the nucleus can be interpreted as the result of independent interaction of \(\pi\)-mesons with neutrons and protons bound in the nucleus. Such an interpretation of the interaction of \(\pi\)-mesons with nuclei is also confirmed by studies devoted to the investigation of the final products of reactions in absorption and inelastic scattering of \(\pi\)-mesons by nuclei\(^{140,164}\).
Elastic scattering. Elastic scattering of \(\pi\)-mesons by nuclei can be described phenomenologically by treating the nucleus as a medium with a complex potential \(U = V + iW\) and solving the problem of scattering of \(\pi\)-mesons by such an obstacle. The real part of the complex potential represents the average potential energy of the interaction of \(\pi\)-mesons in the nucleus; the imaginary part of the potential \(iW\) determines the absorption and inelastic scattering of \(\pi\)-mesons as they pass through the nucleus. Usually the interaction potential of \(\pi\)-mesons
nucleus is taken in the form of a rectangular well. If the scattering problem is solved with the aid of optical formulas for an absorbing medium, then this solution is called the “optical model.” The problem of neutron scattering by nuclei was first solved in this way,^155 and then elastic scattering of \(\pi\)-mesons was considered.^156 The condition for the applicability of the optical model is that the wavelength of the incident particle be small in comparison with the dimensions of the obstacle. In particular, the optical model does not take into account the reflection of the incident \(\pi\)-meson wave from the boundary of the nucleus.
Another method of solving the problem of \(\pi\)-meson scattering consists in solving the Schrödinger equation with the potential in the nucleus \(U = V + i\sigma\) and finding such parameters \(V\) and \(\sigma\) as best satisfy the experimental data. In this model the reflection of the \(\pi\)-meson wave from the boundary of the nucleus is taken into account, and it makes it possible to take into account the interference of the Coulomb and nuclear interactions. Table XVIII gives the values of the mean interaction potential of \(\pi\)-mesons in the nucleus, \(V\), obtained by both methods.
Fig. 20. Dependence of \(\dfrac{\sigma_{\text{nucl}}}{\sigma_{\text{geom}}}\) on the energy of \(\pi\)-mesons.
The values of the potential \(V\) presented in the table were determined with an error amounting to approximately \(50\%\) of the measured quantity. The sign of the potential was determined from the observed interference of the Coulomb and nuclear interactions. The sign of the potential turned out to be negative; this shows that the interaction of \(\pi\)-mesons with nuclei at these \(\pi\)-meson energies is attractive. In processing the experiment the parameter \(\sigma\) was also determined; however, a more illustrative characteristic is the mean free path for absorption and inelastic scattering of \(\pi\)-mesons in nuclear matter, which is easily calculated from \(\sigma\). Such values of the mean free path are given in Fig. 21, where it is seen that this length rapidly decreases with increasing \(\pi\)-meson energy and at energies \(\sim 100\) MeV amounts to only \(2 \cdot 10^{-13}\) cm. It is natural to relate the parameters of the optical model to the parameters characterizing the scattering of \(\pi\)-mesons by nucleons, assuming the validity of the picture of pair meson–nucleon collisions in the nucleus. Such a relation was obtained^143 when neglecting the motion of the nucleons and their interaction inside the nucleus,
\[ k^2 = k_0^2 + 4\pi N f(0^\circ). \]
Here \(k_0\) is the wave vector of the \(\pi\)-meson outside the nucleus, \(k\) is the wave vector of the \(\pi\)-meson in the nucleus, \(f(0^\circ)\) is the amplitude for scattering of \(\pi\)-mesons by a nucleon through an angle \(0^\circ\), a quantity which, generally speaking, is complex, \(N\) is the number of nucleons per unit volume.
Table XVIII
Values of \(V\) obtained from experiments on elastic scattering of \(\pi\)-mesons by nuclei
| \(\pi\)-meson energy, MeV | Sign of \(\pi\)-meson | Nucleus | \(V\) | Method of determination | References |
|---|---|---|---|---|---|
| 33 | \(+\) | C, Al, Cu | 16 | Optical model | 143 |
| 46 | \(\pm\) | C, Al, Cu | 25 | » | 143 |
| 48 | \(+\) | C | 15 | » | 144 |
| 60 | \(\pm\) | He | 20 | » | 140 |
| 62 | \(-\) | Photoemulsion nuclei | 20 | From the solution of the Schrödinger equation | 157 |
| 62 | \(+\) | C | \(-18\) | » | 44 |
| 68 | \(+\) | C, Al, Cu | 19 | Optical model | 143 |
| 80 | \(+\) | Al | \(-20\) | From the solution of the Schrödinger equation | 168 |
| 80 | \(-\) | Al | \(-30\) | From the solution of the Schrödinger equation | 168 |
| 105 | \(+\) | He | 18 | Optical model | 140 |
| 125 | \(-\) | C | 30 | From the solution of the Schrödinger equation | 145 |
volume. Since the values of the scattering amplitude are known from experiments on the interaction of \(\pi\)-mesons with free nucleons, the expression written above can be used to obtain the value of the vector \(k\), and consequently also \(V\) and \(\sigma\). In this way reasonable values of the parameters \(V\) and \(\sigma^{143}\) are obtained.
Another approach to explaining the angular distribution of elastically scattered \(\pi\)-mesons consists in replacing the problem of scattering of \(\pi\)-mesons by nuclei with the problem of scattering of \(\pi\)-mesons by individual nucleons bound in the nucleus.
Calculations\(^{169}\) were carried out of the angular distribution of elastic scattering of \(78\) MeV \(\pi^\pm\)-mesons by lithium, under the assumption that the interaction of a \(\pi\)-meson with the nucleus occurs through the interaction of the \(\pi\)-meson with individual nucleons of the nucleus, and that the forces of interaction of \(\pi\)-mesons with the nucleons of the nucleus do not differ from the forces of interaction of \(\pi\)-mesons with free nucleons. The results of the calculation are in qualitative good agreement with the experimental results.
In the plot: ordinate \(\lambda \cdot 10^{13}\) cm; abscissa \(E_\pi\), MeV. Filled points: optical model \((140,142–144,168)\). Open circles: from the solution of the Schrödinger equation \((145,168)\).
Fig. 21. Dependence of the mean free path of \(\pi\)-mesons in nuclear matter on the \(\pi\)-meson energy.
Charge-exchange cross section. In the scattering of \(\pi\)-mesons by protons, the charge-exchange cross section is relatively large, amounting to approximately \(2/3\) of the total cross sec-
[[unclear: beginning of word]] of the interaction of \(\pi\)-mesons with a proton. Therefore one should expect that, also in the interaction of \(\pi\)-mesons with nuclei, charge exchange of \(\pi\)-mesons occurs with high probability. The results of measurements of the charge-exchange cross section for \(\pi\)-mesons on nuclei are given in Table XIX. It is seen from the table that
Table XIX
Charge-exchange cross section
| Meson sign | \(E_\pi,\ \mathrm{MeV}\) | Nucleus | \(\sigma_{\mathrm{exch}}\times 10^{27},\ \mathrm{cm}^2\) | Literature |
|---|---|---|---|---|
| \(\pi^-\) | 34 | H | \(5.0 \pm 1.5\) | 62 |
| \(\pi^\pm\) | 34 | D | \(1.7 \pm 0.6\) | 62 |
| \(\pi^\pm\) | 34 | C, O | \(\lesssim 5\) | 62 |
| \(\pi^+\) | 44 | Be, O | \(\sim 1\) | 151 |
| \(\pi^+\) | 50 | Pb | \(37 \pm 26\) | 163 |
| \(\pi^\pm\) | 60 | He | \(22 \pm 7\) | 140 |
| \(\pi^-\) | 105 | He | \(64 \pm 13\) | 140 |
| \(\pi^-\) | 120 | H | \(21.7 \pm 2.7\) | 26 |
| \(\pi^-\) | 120 | D | \(14 \pm 2\) | 166 |
| \(\pi^-\) | 125 | C | \(20^{+20}_{-10}\) | 145 |
| \(\pi^-\) | 125 | Pb | \(100^{+80}_{-40}\) | 145 |
the charge-exchange cross section on nuclei depends strongly on the energy of the \(\pi\)-mesons, and the charge-exchange cross section at low energies (\(\sim 40\ \mathrm{MeV}\)) is close to zero. The sharp decrease of the charge-exchange cross section at low energies can be explained with the aid of the Pauli principle: in charge exchange, for example, of a \(\pi^-\)-meson, a neutron is produced instead of a proton, the lower energy states for which are occupied by other neutrons of the nucleus. Therefore charge exchange of a \(\pi^-\)-meson is possible only when a comparatively large energy is transferred to a nucleon in the nucleus.
Inelastic scattering. In Fig. 22 is shown the ratio of the inelastic-scattering cross section to the total cross section for nuclear interactions of \(\pi\)-mesons for photoemulsion nuclei as a function of the energy of \(\pi^\pm\)-mesons; it is seen that the inelastic-scattering cross section tends to zero at \(\pi\)-meson energies of the order of \(20\)–\(30\ \mathrm{MeV}\). Such behavior of the inelastic-scattering cross section of \(\pi\)-mesons resembles the behavior of the charge-exchange cross section and can likewise be explained on the basis of the Pauli principle, which forbids small energy transfers in the interaction of \(\pi\)-mesons with individual nucleons of the nucleus. Consequently, the principal process in the nuclear interaction of \(\pi\)-mesons of low energies with a nucleus
Fig. 22. Dependence of \(\dfrac{\sigma_{\mathrm{inel}}}{\sigma_{\mathrm{nuclear}}}\) on the energy of \(\pi\)-mesons for photoemulsion nuclei.
is the absorption of $\pi$-mesons. In Fig. 22 it is seen that the inelastic-scattering cross section for $\pi^+$-mesons goes to zero at high energies, whereas the inelastic-scattering cross section of $\pi^-$-mesons does not. This difference can be explained by the fact that, on approaching the nucleus, $\pi^+$-mesons are slowed down, while $\pi^-$-mesons are accelerated by the Coulomb field.
Let us consider the angular and energy distributions of inelastically scattered $\pi$-mesons. Fig. 23 gives the angular distribution, and Fig. 24 the energy distribution of the inelastically scattered mesons. In Fig. 24 the arrow indicates the energy that a $\pi$-meson scattered on a free nucleon would have with the maximum possible energy transfer. A characteristic feature of inelastic scattering is the large energy loss and the strong dependence of the cross section on angle, resembling the angular dependence of the cross section for scattering of $\pi$-mesons on the proton. The experimental results on inelastic scattering are consid—
Fig. 23. Angular distribution of inelastically scattered $\pi$-mesons on carbon and lead $^{145}$.
Fig. 24. Energy distribution of inelastically scattered $\pi$-mesons.
ered on the basis of the picture of meson–nucleon collisions in the nucleus. By the method of random trials a calculation was made of the inelastic scattering of $62$ MeV $\pi^+$-mesons after the first collision of a meson with a nucleon in the nucleus $^{139}$. In the calculation it is assumed that: 1) inelastic scattering in the nucleus occurs as a result of meson–nucleon collisions; 2) the distribution of nucleons by momenta in the nucleus corresponds to the Fermi-gas model; 3) the cross section for the interaction of $\pi^+$-mesons with nucleons in the nucleus does not differ from the corresponding cross sections for the interaction of $\pi$-mesons with free nucleons; 4) the Pauli principle forbids
collisions with small energy transfer; 5) the Coulomb field reduces the kinetic energy of \(\pi^+\)-mesons as they approach the nucleus and, in addition, for \(\pi^+\)-mesons in the nucleus there exists a potential well of depth \(15\) MeV. The results of the calculation are shown in Figs. 25 and 26, and comparison with Figs. 23 and 24 shows qualitative agreement with experiment. A more detailed calculation for the case of the interaction of \(160\) MeV \(\pi^-\)-mesons with photoemulsion nuclei, taking into account the possibility of two or more collisions of \(\pi\)-mesons with nucleons of the nucleus, shows that the model of pair meson–nucleon collisions in the nucleus gives results that agree well with experimental data on the angular and energy distribution of inelastically scattered \(160\) MeV \(\pi^-\)-mesons on nuclei \(^{167}\). It turns out that the number of \(\pi\)-mesons which have emerged from the nucleus after 2, 3, etc. collisions with nuclear nucleons is approximately 30% of all inelastically scattered \(\pi\)-mesons, while the mean interaction potential of \(160\) MeV \(\pi^-\)-mesons with photoemulsion nuclei is \(-(24 \pm 5)\) MeV. Such a value of the mean interaction potential for a \(\pi^-\)-meson in the nucleus approximately corresponds to the data obtained from the optical model. Thus, the results of this calculation agree well with the experimental results. Analysis of “stars” observed in photographic plates in the inelastic scattering of \(\pi\)-mesons also confirms that inelastic scattering is the result of a small number of pair meson–nucleon collisions in the nucleus \(^{153}\).
Fig. 25. Angular distribution of inelastically scattered \(\pi^+\)-mesons, obtained by calculation by the Monte Carlo method (energy of the \(\pi^+\)-mesons \(62\) MeV).
Fig. 26. Energy distribution of inelastically scattered \(\pi^+\)-mesons, obtained by calculation by the Monte Carlo method (energy of the \(\pi^+\)-mesons \(62\) MeV).
Absorption of \(\pi\)-mesons. Experiments performed with photographic plates and with a Wilson chamber make it possible to analyze the “stars” that arise when \(\pi\)-mesons are absorbed by nuclei. From analysis of the “stars” one can conclude that in more than 60% of cases the \(\pi\)-meson is absorbed by two nucleons—a proton and a neutron \(^{44,138,142}\). However, cases were observed that are difficult to explain by absorption of a \(\pi\)-meson by two nucleons: cases in which there was complete transfer of energy, including the energy associated with the rest mass of the \(\pi\)-meson, to one nucleon.
The probability of capture of a \(\pi^+\)-meson in a nucleus can be expressed in terms of the probability of capture of a \(\pi^+\)-meson in deuterium by the formula \(^{154}\)
\[ \frac{1}{z}\,\sigma(\pi^+ + A \to \text{star}) = \Gamma \sigma(\pi^+ + d \to 2p), \]
where \(\Gamma\) takes into account, first, the probability of quasideuteron formations in
nucleus and, secondly, factors connected with the motion of nucleons in the nucleus, and prohibitions for certain final states of nucleons in the nucleus. Table XX gives the results of determining \(\Gamma\).
Table XX
Values of \(\Gamma\) obtained by different authors
| \(E_\pi\), MeV | Meson sign | \(\Gamma\) | Reference |
|---|---|---|---|
| 20 | \(+\) | 5.6 | 142 |
| 39 | \(+\) | 3.2 | 142 |
| 50 | \(+\) | 10 | 138 |
| 62 | \(+\) | 3.3 | 44 |
6. PHOTOPRODUCTION OF \(\pi\)-MESONS ON NUCLEI
Of all reactions of meson production on nuclei, photoproduction is of especially great interest because of the relative simplicity of the interpretation of the experimental data. First, here there is an interaction of the well-studied electromagnetic field with nucleons and, secondly, in photoproduction of \(\pi\)-mesons a simple system of particles arises in the final state.
All experiments on the photoproduction of \(\pi\)-mesons on nuclei have been performed with nonmonochromatic \(\gamma\)-quanta produced when electrons are decelerated in matter with large \(Z\) (see Fig. 11). Since in most experiments spectra of \(\gamma\)-quanta with maximum energy greater than or of the order of 300 MeV were used, and the threshold for producing \(\pi\)-mesons by \(\gamma\)-quanta only slightly exceeds the energy corresponding to the rest mass of the meson, a rather broad spectrum of \(\gamma\)-ray energies effectively takes part in photoproduction. Unlike experiments on the photoproduction of mesons on free nucleons, in studying the photoproduction of \(\pi\)-mesons on nuclei it is impossible to isolate cases of production by \(\gamma\)-quanta of a definite energy. Exceptions are experiments on the “elastic” photoproduction of mesons, when in the final state only two particles are formed, or cases when all reaction products in the final state are registered. (By elastic photoproduction we shall mean the production of \(\pi\)-mesons without excitation or breakup of the nucleus.)
Photoproduction of \(\pi\)-mesons on deuterium. The production of \(\pi\)-mesons by \(\gamma\)-quanta on deuterium is the most reliable source of information concerning the production of \(\pi\)-mesons on the neutron. An important characteristic is the magnitude of the ratio of the photoproduction cross sections of \(\pi^+\)- and \(\pi^-\)-mesons in the reactions \(\gamma + D \to \pi^+ + n + n\) and \(\gamma + D \to \pi^- + p + p\). Experimental data for the magnitude of the ratio \(\sigma_{\pi^-}/\sigma_{\pi^+}\) are given in Table XXI.
Table XXI
Magnitude of the ratio \(\sigma_{\pi^-}/\sigma_{\pi^+}\) in photoproduction of \(\pi\)-mesons on deuterium
| \(E_\gamma\) max., MeV | \(E_\pi\), MeV | \(\theta^0_{\text{lab. frame}}\) | \(\sigma_{\pi^-}/\sigma_{\pi^+}\) | Reference |
|---|---|---|---|---|
| 330 | 65–145 | 26 | \(0.90 \pm 0.23\) | 171 |
| 330 | 40–105 | 90 | \(0.5 \pm 0.5\) | 171 |
| 310 | 65 | 135 | \(1.19 \pm 0.12\) | 172 |
| 318 | 70 | 45 | \(0.96 \pm 0.10\) | 173 |
| 318 | 70 | 90 | \(1.09 \pm 0.12\) | 173 |
| 318 | 70 | 135 | \(1.21 \pm 0.17\) | 173 |
| 310 | 46 | 180 | \(0.85 \pm 0.26\) | 115 |
| 310 | 34 | 90 | \(1.06 \pm 0.25\) | 116 |
| 310 | 54 | 90 | \(1.49 \pm 0.3\) | 116 |
| 310 | 84 | 90 | \(1.3 \pm 0.35\) | 116 |
It is seen from the table that, within the errors of the experiment, the ratio $\sigma_{\pi^-}/\sigma_{\pi^+}$ does not depend on the energy or on the emission angle of the $\pi$-meson. The weighted mean value of the ratio $\sigma_{\pi^-}/\sigma_{\pi^+}$, which can be obtained from the table, is found to be 1.1. This result shows that the cross section for photoproduction of $\pi$-mesons by $\gamma$-quanta on a nucleon depends only weakly on the charge of the nucleon.
A comparison of experiments on photoproduction of $\pi$-mesons on deuterium and hydrogen makes it possible to find the cross section for production of $\pi$-mesons by $\gamma$-quanta on a neutron. The experimentally measured value of the ratio $(\sigma_{\pi^0})_D/(\sigma_{\pi^0})_H$ was found to be 1.8 over a broad interval of $\gamma$-quantum energies[^133_189]. This result indicates that the cross sections for production of $\pi^0$-mesons by $\gamma$-quanta on a proton and on a neutron are also equal to one another. Of special interest are experiments on “elastic” production of $\pi^0$-mesons on deuterium. Theoretical calculations have shown[^175_176] that, in elastic production of $\pi^0$-mesons on deuterium near the production threshold, strong interference effects should be observed, and the magnitude of the elastic-production cross section depends strongly on the relative signs of the interaction constants of the neutron $g_n$ and proton $g_p$ with the meson field1. The cross section for elastic production in the production of $\pi$-mesons by $\gamma$-quanta on deuterium near the production threshold is, in magnitude, comparable with the “inelastic” production cross section in the case $g_n=-g_p$, and approximately 30 times smaller in the case $g_n=g_p$. Experiments show[^178_191_192] that the magnitude of the cross section and the angular distributions of $\pi^0$-mesons in elastic photoproduction of $\pi^0$-mesons on deuterium correspond to the calculated data for the case $g_n=-g_p$. This result confirms the validity of the hypothesis of isotopic invariance, on the basis of which precisely such a relation is obtained between the interaction constants of the proton and neutron with the meson field[^170].
The relatively large magnitude of the elastic photoproduction cross section of $\pi^0$-mesons near threshold can be explained by the small momentum of the incident $\gamma$-quantum, insufficient to break up the system of nucleons bound in the nucleus. Effects associated with the small magnitude of the momentum of the incident $\gamma$-quantum appear especially strongly in photoproduction of the $\pi^0$-meson on helium, owing to the large binding energy of helium. In works[^179_193_194] it was found that in this case production of a $\pi^0$-meson in elastic collisions occurs with high probability, up to $\gamma$-quantum energies of the order of 300 MeV; moreover, in the energy interval $E_\gamma=140-200$ MeV the photoproduction cross section is predominantly elastic. The absolute magnitude of the elastic cross section for photoproduction of $\pi^0$-mesons on helium, as in the case of production on deuterium, is found to be sufficiently large, which agrees with the assumption of positive interference, i.e., with the assumption that the signs of the interaction constants in photoproduction of $\pi^0$-mesons on a proton and on a neutron are different.
In studying the production of $\pi^0$-mesons by $\gamma$-quanta on deuterium, one can clarify the question of the magnitude of the spin interaction of $\gamma$-quanta with nucleons. Indeed, if in the production of $\pi$-mesons on deuterium by the reaction
\[ \gamma + D \to \pi^- + p + p \]
both nucleons have small relative energies in the final state, this means that they are in an $S$-state and therefore their spins are antiparallel. On the other hand, it is known that the spins of the neutron and proton in deuterium are parallel. Therefore, if in the production of $\pi^-$-mesons a large number of cases is observed in which the relative energy of the two protons
is small, then this means that in photoproduction of \(\pi^-\)-mesons spin flip occurs, i.e., the spin part of the interaction is large. Experiments performed with photographic plates filled with deuterium and irradiated in a beam of \(\gamma\)-quanta with maximum energy \(250\ \mathrm{MeV}\) show\({}^{186}\) that in 40% of the cases of meson production by the reaction \(\gamma + D \to \pi^- + p + p\), the protons of the final state have a relative kinetic energy less than \(12\ \mathrm{MeV}\). Comparison of the experimental results with a theoretical calculation carried out in the impulse approximation*) taking into account the interaction of the particles in the final state\({}^{187}\) shows that the experimental data agree well with theoretical calculations under the assumption of a purely spin interaction in the photoproduction of \(\pi^-\)-mesons on the neutron. It should be noted that comparison of the calculated results with experiment, as well as analysis of the reaction products in the photoproduction of \(\pi^-\)-mesons on deuterium, gives convincing evidence for the validity of the one-nucleon model in the photoproduction of \(\pi\)-mesons. The character of the interaction of \(\gamma\)-quanta with the proton in the photoproduction of \(\pi^+\)-mesons on deuterium was studied in a number of works\({}^{171-173,115,116,188}\). Comparison of the results of calculations\({}^{174}\) with experimental data also indicates a strong spin dependence of the interaction of \(\gamma\)-quanta with protons\({}^{188}\).
Photoproduction of \(\pi\)-mesons on complex nuclei. The total cross section for the interaction of high-energy \(\gamma\)-quanta with nuclear matter is relatively small, of the order of \(10^{-28}\ \mathrm{cm}^2\) per nucleon. Therefore even the heaviest nucleus is transparent to the \(\gamma\)-quanta participating in the photoproduction of \(\pi\)-mesons. Owing to this circumstance, the results of experiments on photoproduction of \(\pi\)-mesons on nuclei are easier to interpret than the results of experiments on production of \(\pi\)-mesons by nucleons, in the interpretation of which the opacity of the nucleus for nucleons must be taken into account.
Fig. 27. Yield of \(\pi^0\)-mesons at an angle of \(90^\circ\) to the \(\gamma\)-quantum beam as a function of \(E_{\gamma\max}\). The curves are the result of calculating the yield of \(\pi^0\)-mesons in photoproduction on carbon under various assumptions concerning the dependence \(\sigma_{\pi^0}=k(E_\gamma-E_{\gamma0})^n\) (\(E_{\gamma0}\) is the photoproduction threshold).
In contrast to experiments on photoproduction of \(\pi\)-mesons on hydrogen, in experiments on photoproduction of \(\pi\)-mesons on nuclei there are no direct measurements of the dependence of the photoproduction cross section on the energy of the \(\gamma\)-quantum. This is connected with the fact that the \(\gamma\)-quantum beams are not monochromatic, and the presence of internal motion of nucleons in the nucleus excludes the possibility of an unambiguous determination of the \(\gamma\)-quantum energy from the emission angle and energy of the \(\pi\)-meson, as is done in experiments on hydrogen.
If it is assumed that the photoproduction cross section of \(\pi\)-mesons on nuclei depends on the \(\gamma\)-quantum energy as \(\sigma=k(E_\gamma-E_{\gamma0})^n\), where \(k\) is a constant, \(E_{\gamma0}\) is the threshold photoproduction energy, and \(E_\gamma\) is the \(\gamma\)-quantum energy, then, averaging over the spectrum of \(\gamma\)-quantum energies from \(E_{\gamma0}\) to \(E_{\gamma\max}\), one can find for different values of \(n\) the expected yield of \(\pi\)-mesons as a function of \(E_{\gamma\max}\).
In Fig. 27 the experimental data are compared with the results of a calculation of the yield of photomesons as a function of \(E_{\gamma\max}\) for different values of \(n\)\({}^{133}\). From Fig. 27 it is seen that up to \(300\ \mathrm{MeV}\) the dependence of the \(\pi\)-meson production cross section on carbon on the energy of the \(\gamma\)-quanta is best approximated—
*) In calculations in the impulse approximation using one or another model, the motion of nucleons in the nucleus is taken into account, but the interaction of the reaction products in the final state is not taken into account.
is described by the curve \(\sigma = k(E_\gamma - E_0)\). The cross section for photoproduction of \(\pi\)-mesons on a nucleus is thus a weaker function of energy than the cross section for photoproduction of \(\pi\)-mesons on a proton. This difference in the energy dependences of the cross sections is connected with the presence of the internal motion of nucleons in the nucleus.
Let us consider the dependence of the cross section for photoproduction of \(\pi\)-mesons on atomic weight. Figs. 28 and 29 give experimental results \(^{172,180}\), which show that the cross section for photoproduction of \(\pi\)-mesons on nuclei at \(\gamma\)-quantum energies \(\sim 320\) MeV depends on the atomic weight as \(A^{2/3}\). Other studies \(^{181,182,195,196}\) also confirm this dependence.
Fig. 28. Yield of \(\pi^0\)-mesons at an angle of \(45^\circ\) to the beam of \(\gamma\)-quanta with \(E_{\gamma\max}=325\) MeV as a function of \(A\). (According to data of \(^{180}\).)
Fig. 29. Yield of \(\pi^+\)- and \(\pi^-\)-mesons \((E_\pi=65\) MeV) at \(90^\circ\) to the beam of \(\gamma\)-quanta with \(E_{\gamma\max}=310\) MeV as a function of \(A\). (According to data of \(^{172}\).)
Since the cross section for the interaction of \(\pi\)-mesons with nuclear matter is large and the mean free path is comparable with the distance between nucleons in the nucleus, it is natural to explain the observed dependence of the photoproduction cross section as the result of reabsorption of \(\pi\)-mesons. A quantitative estimate of this effect \(^{183}\) shows that, for reasonable values of the meson mean free path, it is possible, at least for heavy nuclei, to explain the dependence \(\sigma \sim A^{2/3}\) by meson reabsorption.
Experimental results on photoproduction of \(\pi\)-mesons on light nuclei are presented in Fig. 30 and in Table XXII.
Table XXII
Cross sections for photoproduction of \(\pi^+\)-mesons on light elements
| Element | \(d\sigma/d\Omega\), \(10^{30}\ \text{cm}^2/\text{sterad}\) per proton \(Q\), \(45^\circ\) | \(d\sigma/d\Omega\), \(10^{30}\ \text{cm}^2/\text{sterad}\) per proton \(Q\), \(90^\circ\) | \(d\sigma/d\Omega\), \(10^{30}\ \text{cm}^2/\text{sterad}\) per proton \(Q\), \(135^\circ\) | \(Q_{\text{total}}\times 10^{30}\ \text{cm}^2\) per proton | Reference |
|---|---|---|---|---|---|
| H | \(9.2 \pm 0.9\) | \(9.3 \pm 0.6\) | \(6.0 \pm 0.4\) | \(103 \pm 3\) | 182 |
| D | \(6.7 \pm 0.7\) | \(7.1 \pm 0.5\) | \(4.6 \pm 0.3\) | \(77 \pm 3\) | 182 |
| He | \(3.1 \pm 0.3\) | \(4.7 \pm 0.4\) | \(3.8 \pm 0.3\) | \(48 \pm 2\) | 182 |
| C | \(2.6 \pm 0.4\) | \(3.5 \pm 0.3\) | \(2.5 \pm 0.3\) | \(38 \pm 3\) | 184 |
As is seen from Table XXII, the cross section for photoproduction of \(\pi\)-mesons on a proton decreases approximately as \(A^{-1/3}\). In \(^{112}\) it is shown that the decrease of the cross section
of photoproduction of \(\pi\)-mesons (per proton) for very light nuclei can be explained mainly by the Pauli principle and by the small reabsorption of the produced \(\pi\)-mesons.
The change in the shape of the spectra with increasing \(A\) can also be explained by the Pauli principle. The prohibitions due to the Pauli principle must manifest themselves especially strongly for complex nuclei and mesons having a large kinetic energy. This is seen in Fig. 30.
Fig. 30. Energy spectra of \(\pi^+\)-mesons emitted at angles \(45^\circ\), \(90^\circ\), and \(135^\circ\), in the photoproduction of \(\pi\)-mesons on light elements by \(\gamma\)-quanta with maximum \(\gamma\)-quantum energy
\[
E_{\gamma \max}=318\ \text{MeV}.
\]
Very interesting are experiments on the photoproduction of \(\pi\)-mesons in which the ratio \(\sigma_{\pi^-}/\sigma_{\pi^+}\) was measured for different elements \({}^{172,197}\). Figure 31 presents the ratio of the photoproduction cross sections of \(\pi\)-mesons of different signs for various nuclei \({}^{172}\). There, for comparison, the difference in masses is also shown for nuclei whose charge is one unit smaller and one unit larger than the charge of the target nucleus. These isobars correspond to the nuclei that are formed in the photoproduction of \(\pi^+\)- and \(\pi^-\)-mesons. From comparison of the curves it follows that the energy gain due to the difference in masses of the final nuclei in the photoproduction of \(\pi^+\)- and \(\pi^-\)-mesons has a substantial effect on the production of mesons of different signs. It is clear that this effect must be especially strong in the case of the production of \(\pi\)-mesons by \(\gamma\)-quanta of low energies. Table XXIII gives data confirming this.
Fig. 31. The ratio \(\sigma_{\pi^-}/\sigma_{\pi^+}\) in the photoproduction of \(\pi\)-mesons having energy \(65 \pm 15\) MeV and recorded at an angle \(135^\circ\) to the beam of \(\gamma\)-quanta with maximum energy
\[
E_{\gamma \max}=310\ \text{MeV}.
\]
Also shown is the difference in masses of neighboring, in \(Z\), isobaric target nuclei
\[
(\Delta = M_{Z-1}-M_{Z+1}).
\]
Table XXIII
Ratio \(\sigma_{\pi^-}/\sigma_{\pi^+}\) in photoproduction of \(\pi\)-mesons on beryllium
| \(E_{\gamma \max},\ \mathrm{MeV}\) | \(E_\pi,\ \mathrm{MeV}\) | \(\theta^\circ_{\text{lab. c. k.}}\) | \(\sigma_{\pi^-}/\sigma_{\pi^+}\) | Reference |
|---|---|---|---|---|
| 310 | 50 | 135 | \(2.12 \pm 0.06\) | 172 |
| 310 | 65 | 135 | \(2.27 \pm 0.07\) | 172 |
| 256 | 50 | 135 | \(2.56 \pm 0.16\) | 172 |
| 256 | 65 | 135 | \(3.7 \pm 0.3\) | 172 |
| 310 | 54 | 90 | \(1.65 \pm 0.05\) | 185 |
| 270 | 54 | 90 | \(2.0 \pm 0.1\) | 185 |
| 250 | 54 | 90 | \(2.4 \pm 0.3\) | 185 |
| 225 | 54 | 90 | \(3.3 \pm 0.5\) | 185 |
Conclusion. All experimental results on the photoproduction of \(\pi\)-mesons on nuclei can be explained on the assumption that the production of mesons by \(\gamma\)-quanta occurs on individual nucleons of the nucleus with cross sections corresponding to production on free nucleons. To explain the regularities observed experimentally, it proves sufficient to take into account the internal motion of the nucleons in the nucleus, the absorption of the mesons produced inside the nucleus, and the Pauli exclusion principle. Owing to the relatively small magnitude of the momentum of \(\gamma\)-quanta in the photoproduction of \(\pi\)-mesons near the production threshold, effects associated with the structure of the nucleus appear sharply. At high \(\gamma\)-quantum energies these effects are not observed. In particular, at \(\gamma\)-quantum energies \(\sim 500\ \mathrm{MeV}\), the ratio \(\sigma_{\pi^-}/\sigma_{\pi^+}\) no longer depends on the structure of the target nuclei, but is determined exclusively by the ratio of the number of neutrons and protons in the nucleus \({}^{197}\).
7. PRODUCTION OF \(\pi\)-MESONS BY NUCLEONS ON NUCLEI
The production of \(\pi\)-mesons by nucleons on nuclei appears to be a process difficult to interpret, comprising such complicated phenomena as the passage of the nucleons that cause the production of \(\pi\)-mesons through the nucleus and the interaction of the produced \(\pi\)-mesons with the nucleus. Almost all experimental results on the production of \(\pi\)-mesons by nucleons on nuclei can be explained on the assumption that \(\pi\)-meson production occurs in individual nucleon-nucleon collisions undergone by the nucleon as it passes through the nucleus. The influence of the nucleus is then manifested in the following: 1) the presence of intranuclear motion of nucleons in the nucleus “smears out” the energy and angular distribution of the produced mesons; 2) the strong interaction of \(\pi\)-mesons with nuclear matter leads to the fact that a considerable fraction of the mesons produced in the inner parts of the nucleus is absorbed; 3) the nucleon causing meson production rapidly loses its energy in passing through the nucleus, and therefore the production of \(\pi\)-mesons occurs nonuniformly over the volume of the nucleus. Thus, because of the complexity of the processes occurring in the nucleus during \(\pi\)-meson production, it is sometimes difficult to give an unambiguous interpretation of the observed regularities.
Dependence of the \(\pi\)-meson production cross section on the energy of the nucleon. Figure 32 shows the dependence of the cross section for production of \(\pi\)-mesons on carbon on the proton energy near the production threshold; the cross section depends strongly on the proton energy, and the threshold for \(\pi\)-meson production in the collision of a nucleon with a nucleus is substantially lower than the threshold for \(\pi\)-meson production in the collision of two nucleons, which is approximately \(290\ \mathrm{MeV}\). The lowering of the threshold
production may be caused, first, by the presence of internal motion of nucleons in the nucleus and, second, by the possibility of producing a meson on the nucleus as a whole or on a group of nucleons of the nucleus. The absolute threshold for $\pi$-meson production
Fig. 32. Dependence of the cross section for $\pi$-meson production in the reaction $p + C \to \pi$ on the proton energy.
Fig. 33. Cross section for the production of $\pi^+$- and $\pi^-$-mesons on carbon at an angle of $90^\circ$ to the beam of protons with energy 340 MeV.
Fig. 34. Cross section for the production of $\pi^+$- and $\pi^-$-mesons on lead at an angle of $90^\circ$ to the beam of protons with energy 340 MeV.
in the reaction $N + A \to \pi + A'$ corresponds to the case in which the nucleus $A'$ is formed essentially in the unexcited ground state, and the kinetic energy of the $\pi$-meson relative to the nucleus is zero. However, since the wavelength of the incident
nucleon is small in comparison with the dimensions of nuclei and is equal to \(\sim 2.5\cdot 10^{-14}\) cm for a nucleon with an energy of 340 MeV, the production of \(\pi\)-mesons on the nucleus as a whole or on any large group of nucleons in the nucleus is improbable. Therefore, the production of \(\pi\)-mesons on a nucleus is usually interpreted as the result of nucleon–nucleon collisions in the nucleus. The small value of the threshold is explained by the fact that the nucleons in the nucleus are in motion. Indeed, if a nucleon in the nucleus moves with an energy of 20 MeV toward the incident nucleon, the threshold for \(\pi\)-meson production exceeds 140 MeV only slightly.
Energy spectrum of \(\pi\)-mesons. Another feature of the production of \(\pi\)-mesons on nuclei is the form of the energy spectrum of the mesons produced. In Figs. 33 and 34 are shown the energy spectra of \(\pi^+\)- and \(\pi^-\)-mesons produced on carbon and lead nuclei by protons with an energy of 340 MeV[^201]. Similar spectra have also been obtained in other works[^200,^87,^86,^203,^219]. The experimental results have been analyzed under the assumption that the production of \(\pi^+\)-mesons occurs in individual nucleon–nucleon collisions in the nucleus[^87,^86,^204]. In the calculations it was assumed that the mesons are produced by the reaction \(p+p\to\pi^+ + d\) and that the produced \(\pi^+\)-mesons do not interact with the nucleons in the nucleus before leaving the nucleus. Indeed, from experiments on the production of \(\pi^+\)-mesons in \(p-p\) collisions it is known that at such energies production of a \(\pi^+\)-meson with formation of a deuteron is most probable (see Fig. 8). The second assumption is approximately valid for light nuclei of the carbon type. In calculating the spectrum of \(\pi^+\)-mesons it is necessary to specify the momentum distribution of nucleons in the nucleus. Calculations were carried out for three distributions[^87]:
- Degenerate Fermi gas
\[ w(\mathbf{p})\,d\mathbf{p}=\frac{3}{4\pi p_0^3}\,d\mathbf{p}\quad \text{for } p\leqslant p_0; \]
\[ w(\mathbf{p})\,d\mathbf{p}=0\quad \text{for } p>p_0, \]
where \(p\) is the maximum momentum of a nucleon in the nucleus.
- Gaussian distribution
\[ w(\mathbf{p})\,d\mathbf{p}=\frac{1}{\left(\pi p_0^2\right)^{3/2}}\,e^{-p^2/p_0^2}\,d\mathbf{p}. \]
- Chew–Goldberger model
\[ w(\mathbf{p})\,d\mathbf{p}=\frac{\beta(\beta+\alpha)^2}{\pi^3(\alpha^2+\beta^2)(\beta^2+p^2)}\,d\mathbf{p}. \]
Fig. 35. Comparison of the results of theoretical calculations of the spectrum of produced \(\pi^+\)-mesons for three nuclear models with experimental data on the production of \(\pi^+\)-mesons by protons on carbon. The parameters used in the calculation were chosen so as best to satisfy the experimental data.
A comparison of the calculation results for various nuclear models with the experimental data is given in Fig. 35. From Fig. 35 it is seen that, with a suitable choice of parameters of the distribution of nucleons in the nucleus with respect to energies, all the models qualitatively explain the observed energy spectrum of \(\pi\)-mesons. The parameters of the distribution of nucleons in the nucleus with respect to energies used in the calculation coincide with the parameters found in the analysis of the inelastic scattering of fast nucleons by light nuclei[^220]. This testifies to the validity of the assumption that the production of \(\pi\)-mesons on nuclei occurs in nucleon–nucleon collisions. However, it is impossible to draw any conclusion as to the validity of one or another nuclear model because of the crudity of the consideration.
The total cross section for the production of a \(\pi\)-meson on a proton in the nucleus is greater than the total production cross section found in \(p-p\) collisions at the same energy of the incident protons. Thus, the cross section for meson production on carbon by protons
with an energy of \(340\) MeV is equal to \(^{206}(7.6 \pm 0.7)\cdot 10^{-27}\ \mathrm{cm}^2\) per nucleus, or \(1.2\cdot 10^{-27}\ \mathrm{cm}^2\) per proton in the carbon nucleus*), whereas the cross section for the production of \(\pi^+\)-mesons in the collision of two protons is only \(0.3\cdot 10^{-27}\ \mathrm{cm}^2\). Such an increase in the production cross section in a nucleus can, however, be explained by the internal motion of nucleons in the nucleus and by the strong dependence of the production cross section on the relative energy of the colliding nucleons.
From the curves shown in Figs. 33 and 34 it is seen that the spectra of \(\pi^+\)- and \(\pi^-\)-mesons produced on carbon and lead differ somewhat. A comparison of the energy spectra of mesons produced on carbon and lead shows that, in the case of \(\pi\)-meson production on lead, the relative probability of producing mesons of high energy is noticeably smaller. This may be connected with the high probability of absorption of energetic mesons in the heavy nucleus and with the fact that the energy of the incident nucleon is strongly reduced as a result of collisions with the nuclear nucleons. The latter circumstance evidently manifests itself more strongly in the case of meson production on a heavy nucleus.
A comparison of the spectra of \(\pi^+\)- and \(\pi^-\)-mesons produced on lead shows that the maxima in the spectra are shifted relative to one another. One of the reasons explaining this shift may be the action of the Coulomb field of the nucleus, which decreases the energy of \(\pi^-\)-mesons and increases the energy of \(\pi^+\)-mesons. A comparison of the spectra of \(\pi^+\)- and \(\pi^-\)-mesons produced on carbon, where the action of the Coulomb field is small, shows that the character of these spectra is also different. From the figure it is seen, in particular, that in the production of \(\pi^+\)-mesons the latter are formed with greater energy than \(\pi^-\)-mesons. This circumstance may be connected both with the Pauli principle and with the fact that in the reaction \(p+n\to\pi^-+p+p\), \(\pi^-\)-mesons are produced with lower kinetic energy than \(\pi^+\)-mesons in the reactions \(p+p\to\pi^+ + p+n\) and \(p+p\to\pi^+ + d\). The spectra of \(\pi^+\)- and \(\pi^-\)-mesons produced on beryllium and carbon by protons with an energy of \(660\) MeV have an analogous character \(^{223,224}\). At an incident-proton energy \(E_p=660\) MeV, the cross section for \(\pi^+\)-meson production per proton proves to be approximately three times smaller than the production cross section on a free proton. In the production of \(\pi\)-mesons by protons of lower energies (\(E_p=340\) MeV), however, the production cross section on a nuclear nucleon exceeds the production cross section on a free nucleon. The increase in the \(\pi\)-meson production cross section in the interaction of \(340\) MeV nucleons with the nucleons of a nucleus, as compared with the production cross section on a free particle, may be explained by the motion of nucleons in the nucleus. At proton energies of \(\sim 660\) MeV, the cross section of the reaction \(p+p\to\pi^+\) reaches its maximum value, and the influence of the nucleus is reduced merely to the absorption of the produced \(\pi\)-mesons.
Dependence of the \(\pi\)-meson production cross section on \(A\). The interpretation of experiments on the production of \(\pi\)-mesons by nucleons on nuclei is more complicated than the interpretation of photoproduction. In analyzing meson production by nucleons, in addition to absorption of the produced \(\pi\)-mesons in the nucleus, it is also necessary to take into account the fact that the nucleus is not transparent to the nucleons causing the production. Figs. 36 and 37 give the dependences of the \(\pi\)-meson production cross section on the atomic weight \(A\). From the figures it is seen that the production cross section of \(\pi\)-mesons, at least for not very light elements, follows the law \(A^{2/3}\). This conclusion is confirmed also in other works \(^{208,209,226,16}\). The observed deviations from the law \(A^{2/3}\) may be explained by the action of various
*) The contribution of the reaction \(p+n\to\pi^+ + n+n\) in the nucleus may be neglected, since, by charge symmetry, the cross section of this reaction is equal to the cross section of the reaction \(p+n\to\pi^-+p+p\), while the cross section of the latter reaction is several times smaller than the cross section of the reaction \(p+p\to\pi^+ + d\) (or \(p+n\)). This follows from the large value of the ratio \(\sigma_{\pi^+}/\sigma_{\pi^-}\) in meson production by protons with an energy of \(340\) MeV on carbon (see Table XXIV).
nuclear factors, which manifest themselves especially under specific experimental conditions. In \(^{210}\) the yield of low-energy mesons was measured. In this case deviations from the dependence \(\sigma \sim A^{2/3}\) can be explained by the influence of the Coulomb field of the nucleus, which deforms the spectrum of the produced low-energy mesons. The dependence of the production cross section of \(\pi^+\)-mesons on \(A\) should here be weaker than follows from the law \(\sigma \sim A^{2/3}\); this was indeed observed. In fact, the Coulomb field of the nucleus accelerates the produced \(\pi^+\)-mesons, as a result of which the number of low-energy \(\pi^+\)-mesons produced on heavy nuclei decreases sharply. In \(^{211}\) the production of \(\pi^+\)-mesons was studied
Fig. 36. Dependence of the production cross section of \(\pi^+\)- and \(\pi^-\)-mesons at an angle of \(90^\circ\) to a beam of protons with energy \(381\) MeV as a function of \(A^{87}\).
Fig. 37. Dependence of the production cross sections of \(\pi^0\)-mesons by protons with energy \(340\) MeV on the number of nucleons in the nucleus \(^{207}\).
at small angles \((\theta = 0^\circ \div 4^\circ)\) with respect to a beam of protons with energy \(340\) MeV. The obtained dependence of the production cross section on atomic weight is well explained on the assumption of a large interaction cross section of the incident protons and of the produced \(\pi\)-mesons with the nucleons of the nucleus. In the calculations presented, agreement was achieved when, as the indicated interaction cross sections, the cross sections obtained from experiments on the absorption and scattering of mesons and nucleons by nuclei were adopted. The same effect was observed also in the production of \(\pi^0\)-mesons by protons with energy \(670\) MeV \(^{16}\). Finally, deviations from the dependence \(\sigma \sim A^{2/3}\), obtained in the production of \(\pi^0\)-mesons on nuclei near the production threshold \((E_p = 240\) MeV), can be explained by the influence of the specific structure of the nuclei. In this case the probability of meson production increases in those cases in which, as a result of the reaction, a strongly bound nucleus is formed.
Ratio of the production cross sections of mesons of different signs. In experiments on meson production by nucleons on nuclei, a large difference was found in the magnitudes of the production cross sections of \(\pi^+\)- and \(\pi^-\)-mesons by protons. It was found that the production cross section of \(\pi^+\)-mesons by protons on light nuclei is approximately an order of magnitude larger than the production cross section of \(\pi^-\)-mesons. This difference
in the production cross sections of $\pi$-mesons of different signs could have been expected on the basis of the meson-production cross sections in $p-p$ and $p-n$ collisions for “free” nucleons (see Figs. 8 and 9).
An analogous effect is also observed in the production of $\pi$-mesons by neutrons \(^{214,215,219,223}\). In this case, in accordance with the principle of charge symmetry, the production cross section of $\pi^-$-mesons exceeds the production cross section of $\pi^+$-mesons. As is seen from Table XXIV, the ratio of the total production cross sections of $\pi^+$- and $\pi^-$-mesons by protons on carbon, $\sigma_{\pi+}/\sigma_{\pi-}=14\pm4$, agrees, within the limits of measurement errors, with the inverse ratio $\sigma_{\pi-}/\sigma_{\pi+}$ found in experiments on the production of $\pi$-mesons by neutrons on light nuclei.
Table XXIV
Ratio $\sigma_{\pi+}/\sigma_{\pi-}$ in meson production by protons and neutrons
* denotes the ratio $\sigma_{\pi-}/\sigma_{\pi+}$ for the case of $\pi$-meson production by neutrons.
| Nucleus | Incident particle | $E_N$, MeV | $\theta_{\mathrm{lab.\ c.m.}}^\circ$ | $E_\pi$, MeV | $\sigma_{\pi+}/\sigma_{\pi-}$ | Reference |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| D | $p$ | 340 | 0 | 30 | $5.8 \pm 2.5$ | 221 |
| D | $p$ | 340 | 0 | 60 | $22.1 \pm 3.5$ | 221 |
| D | $p$ | 340 | 0 | 88 | $36.5 \pm 6.2$ | 213 |
| D | $p$ | 340 | 90 | Average over the $\pi$-meson spectrum | $\sim 12$ | 212 |
| D | $p$ | 381 | 90 | Same | $\sim 25$ | 87 |
| He | $n$ | $\sim 300$ | 0—180 | » | $16 \pm 8$* | 219 |
| He | $p$ | 340 | 0 | 30 | $3.4 \pm 2.2$ | 221 |
| He | $p$ | 340 | 0 | 60 | $10.8 \pm 2.1$ | 221 |
| He | $p$ | 340 | 0 | 120 | $35.1 \pm 18.3$ | 221 |
| Be | $p$ | 340 | 0 | 30 | $4.3 \pm 1.2$ | 221 |
| Be | $p$ | 340 | 0 | 60 | $9.2 \pm 1.2$ | 221 |
| Be | $p$ | 340 | 0 | 120 | $12.7 \pm 1.8$ | 221 |
| Be | $p$ | 660 | 24 | Average over the $\pi$-meson spectrum | $5.3 \pm 0.6$ | 224 |
| C | $n$ | $\sim 300$ | 90 | 50—65 | $\sim 14$* | 214 |
| C | $p$ | 340 | 0 | 30 | $4.5 \pm 1.6$ | 221 |
| C | $p$ | 340 | 0 | 60 | $10.9 \pm 1.6$ | 221 |
| C | $p$ | 340 | 0 | 120 | $14.5 \pm 1.9$ | 221 |
| C | $p$ | 340 | 0 | Average over the $\pi$-meson spectrum | $29.5 \pm 1.2$ | 206 |
| C | $p$ | 340 | 90 | Same | $5.1 \pm 1$ | 201 |
| C | $p$ | 340 | 90 | » | $7.8 \pm 0.04$ | 206 |
| C | $p$ | 340 | 180 | » | $9.3 \pm 2.3$ | 159 |
| C | $p$ | 340 | 0—180 | » | $14.4 \pm 4$ | 206 |
| C | $p$ | 345 | 0—180 | » | $10 \pm 3$ | 86 |
| C | $p$ | 365 | 0—180 | » | $10 \pm 3$ | 86 |
| C | $p$ | 380 | 0—180 | » | $10 \pm 3$ | 86 |
| C | $p$ | 381 | 0—180 | » | $12.6 \pm 3.4$ | 87 |
| C | $n$ | $\sim 600$ | 90 | » | $5.4 \pm 1.1$* | 223 |
| C | $p$ | 660 | 24 | » | $7.0 \pm 0.8$ | 224 |
| C | $p$ | 660 | 90 | » | $5.0 \pm 0.7$ | 223 |
| O | $n$ | $\sim 300$ | 0—180 | » | $16 \pm 3.7$* | 215, 219 |
| Al | $p$ | 345 | 90 | » | $2.8 \pm 1.5$ | 203 |
| Cu | $p$ | 345 | 90 | » | $3.3 \pm 1.5$ | 203 |
| Cu | $p$ | 381 | 90 | » | $4.4^{+5}_{-2}$ | 87 |
| Pb | $p$ | 340 | 0 | 30 | $0.83 \pm 0.6$ | 221 |
| Pb | $p$ | 340 | 0 | 60 | $1.7 \pm 1.0$ | 221 |
| Pb | $p$ | 340 | 0 | 120 | $6.7 \pm 3.2$ | 221 |
| Pb | $p$ | 340 | 90 | Average over the $\pi$-meson spectrum | $1.5^{+4}_{-0.4}$ | 201 |
| Pb | $p$ | 381 | 90 | Same | $4.3^{+4}_{-2}$ | 87 |
From Table XXIV it is seen that there is a tendency for the ratio \(\sigma_{\pi+}/\sigma_{\pi-}\) to increase with increasing \(\pi\)-meson energy, which is strongly manifested for light nuclei. This indicates that \(\pi\)-mesons produced in \(n—p\) collisions inside the nucleus carry away substantially less kinetic energy than \(\pi\)-mesons produced predominantly in \(p—p\) collisions inside nuclei. An increase in the ratio \(\sigma_{\pi+}/\sigma_{\pi-}\) with increasing energy of the produced \(\pi\)-mesons is observed to the same extent for nuclei of different atomic weight (D, He, C). It follows from this that this effect cannot be explained by the Pauli principle, and that it is based on the regularities of \(\pi\)-meson production in the reactions \(p + p \to \pi^+\) and \(p + n \to \pi^-\). An increase in the ratio \(\sigma_{\pi+}/\sigma_{\pi-}\) with increasing \(\pi\)-meson energy was observed in experiments on meson production by 660 MeV protons \(^{224}\).
From Table XXIV a decrease in the ratio \(\sigma_{\pi+}/\sigma_{\pi-}\) is seen on passing to heavier nuclei. This may be explained, first, by the excess number of neutrons in heavy nuclei and, second, by the high probability of charge exchange of protons as they pass through the nucleus. The latter cause is apparently the main one, since the neutron excess even in such a heavy nucleus as lead is only \(\sim 50\%\), whereas the ratio \(\sigma_{\pi+}/\sigma_{\pi-}\) changes by several times.
Effects associated with the action of the Coulomb field of nuclei. The action of the Coulomb field of nuclei, deforming the spectra of \(\pi^+\)- and \(\pi^-\)-mesons, was discovered in a number of works. When comparing the spectra of slow \(\pi^+\)- and \(\pi^-\)-mesons produced on photoemulsion nuclei by protons and neutrons, respectively \(^{222}\), it was noted that these spectra, which by virtue of charge symmetry should be identical, turned out to be shifted relative to one another in energy by an amount equal to twice the mean Coulomb potential for photoemulsion nuclei. The action of the Coulomb field, deforming the spectrum of charged particles emitted from nuclei, was well known from the theory of \(\beta\)-decay. In the theory of \(\beta\)-decay, formulas are derived that make it possible to take into account the deformation of the spectra of electrons and positrons emitted from nuclei. However, these formulas, valid for particles with wavelength \(\lambda \gg R\) (\(R\) is the radius of the nucleus), cannot be applied to the case of meson production, since the wavelength of a meson even of very small energy is comparable with nuclear dimensions. Calculations with an exact estimate of the action of the Coulomb field for the case of meson production are complicated and require knowledge of the charge distribution in the nucleus. The simplest, approximately correct point of view is that according to which the action of the Coulomb field leads simply to a shift of the spectra of \(\pi^+\)- and \(\pi^-\)-mesons by the value of the Coulomb potential in the nucleus. Fig. 38 schematically shows the potential in a nucleus for mesons of different signs. Obviously, the indicated point of view is equivalent to the assertion that at the moment of production in the nucleus the mesons do not “feel” the presence of the Coulomb field, and after leaving the nucleus are simply accelerated by it. The different character of the action of the Coulomb field of nuclei on the produced \(\pi^+\)- and \(\pi^-\)-mesons is strongly manifested in the value of the ratio \(\sigma_{\pi+}/\sigma_{\pi-}\) when
Fig. 38. Potential energy for \(\pi\)-mesons of different signs in a nucleus as a function of distance from the center of the nucleus.
decrease of the meson energy\(^{209,217,218}\). Typical results of measurements of the ratio \(\sigma_{\pi^+}/\sigma_{\pi^-}\) for light and heavy nuclei are given in Table XXV.
Table XXV
Ratio \(\sigma_{\pi^+}/\sigma_{\pi^-}\) in the irradiation of carbon and uranium by protons with an energy of 340 MeV
\((\theta = 90^\circ)^{217}\)
| Nucleus | \(E_\pi\) | 12.5 MeV | 27 MeV | 37 MeV |
|---|---|---|---|---|
| C | 5.4 | 6 | 9.0 | |
| U | 0.2 | 0.9 | 1.2 |
It is seen from the table that for the uranium nucleus \((Z = 92)\) the ratio \(\sigma_{\pi^+}/\sigma_{\pi^-}\) for small \(\pi\)-meson energies is much less than unity, which can be explained by the action of the Coulomb field of the nucleus. As was already noted in considering the dependence of the cross section for \(\pi\)-meson production on \(A\), for mesons of low energy a deviation from the law \(\sigma \sim A^{2/3}\) was observed. It is interesting to note that introducing a correction for the action of the Coulomb field by taking into account the shift of the spectra of low-energy \(\pi\)-mesons by the magnitude of the Coulomb potential leads to the result that the production cross section for a fixed energy of a meson of either sign proves to be proportional to \(A^{2/3}\) up to the heaviest nuclei\(^{202}\).
Mechanism of \(\pi\)-meson production on nuclei. As is evident from the foregoing, meson production on nuclei may be interpreted as the result of meson production on individual nucleons in the nucleus. Thus, the energy spectra of the produced mesons can be obtained from the known cross sections for meson production on free nucleons, taking into account the internal motion of the nuclear nucleons. Under the same assumptions, but allowing for the absorption of the produced mesons, one can satisfactorily explain the magnitudes themselves of the production cross sections of both \(\pi^+\) and \(\pi^-\)-mesons on nuclei. These circumstances testify in favor of the picture of meson production in pair nucleon–nucleon collisions inside the nucleus. We shall present some results that also testify in favor of the assumption that mesons are produced in the nucleus in nucleon–nucleon collisions.
Fig. 39. Angular distribution of \(\pi^-\)-mesons produced by neutrons with energy \(\sim 300\) MeV on helium.
Figure 39 gives the angular distribution, in the laboratory coordinate system, of \(\pi^-\)-mesons produced in the reaction \(n + \mathrm{He} \to \pi^-\)-neutrons by neutrons with energy 300 MeV. It was shown\(^{219}\) that this spectrum, transformed to the center-of-mass system of two colliding nucleons, takes the form \(a + b\cos^2 \theta_{\mathrm{c.m.}}\), characteristic of the angular distribution of \(\pi\)-meson-
mesons produced in the reaction \(p + p \to d + \pi^+\). This shows that the experimentally measured angular distribution of \(\pi\)-mesons produced on nuclei can also be explained on the assumption that meson production occurs in nucleon–nucleon collisions in the nucleus. Convincing evidence that meson production occurs in pair nucleon–nucleon collisions is also provided by an analysis of \(\pi\)-meson production on very light nuclei. In a study carried out with a Wilson diffusion chamber, where the products of the reaction \(n + \mathrm{He} \to \pi^-\) \((E_n \sim 300\ \mathrm{MeV})\) were investigated, it was shown that even for meson production on such a strongly bound nucleus as He, the contribution of “elastic” meson production in the reaction \(n + \mathrm{He} \to \pi^- + p + \mathrm{He}\) amounts to less than 10% of the value of the \(\pi^-\)-meson production cross section. Since the mean neutron energy in these experiments was only \(\sim 300\ \mathrm{MeV}\), one may assert that at higher energies of the incident nucleons \(\pi\)-meson production in non-pair collisions practically does not occur. Moreover, even the small percentage of “elastic” meson production can be interpreted as the result of the strong interaction of nucleons in the final state, while the production of \(\pi\)-mesons itself proceeds through the pair interaction of the incident nucleon with a nuclear one\(^{219,225}\).
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