Physics

S. S. Gershtein

Effective Cross Section for the “Stripping” of a $\mu$-Meson from a Proton to a Deuteron

(Presented by Academician L. D. Landau, 13 VIII 1957)

The experimentally observed dependence of the catalysis of the nuclear reaction $p + D = \mathrm{He}^3$ by $\mu$-mesons in hydrogen on the deuterium concentration is determined mainly by the process of stripping of the $\mu$-meson from the proton to the deuteron ($^1$). In the present work the effective cross section of this process is calculated.

The Hamiltonian of the $pD\mu$ system in mesoatomic units $(e=1;\ h=1;\ m_\mu=1)$ is

[
\hat H
=
-\frac{1}{2M_1}\Delta_{\mathbf R_1}
-\frac{1}{2M_2}\Delta_{\mathbf R_2}
-\frac{1}{2}\Delta_{\mathbf r}
-\frac{1}{r_1}
-\frac{1}{r_2}
+\frac{1}{R}
\tag{1}
]

($M_1$ and $M_2$ are the masses of the proton and deuteron; $R=|\mathbf R_1-\mathbf R_2|$ is the distance between the nuclei; $r_1=|\mathbf r-\mathbf R_1|$, $r_2=|\mathbf r-\mathbf R_2|$ are the distances of the meson to the proton and deuteron).

Assuming that the transition of the $\mu$-meson occurs between $K$-orbits, the wave function of the system may be written in the form

[
\Psi=A(R)\Sigma_g(\mathbf r,R)+B(R)\Sigma_u(\mathbf r,R);
\tag{2}
]

$\Sigma_g$, $\Sigma_u$ are the symmetric and antisymmetric wave functions of the $\mu$-meson in the field of two fixed nuclei (depending on $R$ as on a parameter)

[
\left(
-\frac{1}{2}\Delta_{\mathbf r}
-\frac{1}{r_1}
-\frac{1}{r_2}
+\frac{1}{R}
\right)\Sigma_{g,u}
=
E_{g,u}(R)\Sigma_{g,u}.
\tag{3}
]

Substituting (2) into (1) and taking account of (3), we obtain, multiplying respectively by $\Sigma_g$ and $\Sigma_u$ and integrating over the coordinates of the $\mu$-meson, the equations for $A(R)$ and $B(R)$:

[
-\frac{1}{2M}\Delta_R A
+
\left(E_g-\frac{1}{2M}K_{gg}-E\right)A
-\frac{1}{2M}K_{gu}B
-\frac{1}{M}S_{gu}\vec\nabla_R B
=0,
]

[
-\frac{1}{2M}\Delta_R B
+
\left(E_u-\frac{1}{2M}K_{uu}-E\right)B
-\frac{1}{2M}K_{ug}A
+\frac{1}{M}S_{gu}\vec\nabla_R A
=0,
\tag{4}
]

where

[
M=\frac{M_1M_2}{M_1+M_2};
]

[
\frac{1}{M}S_{gu}
=
\int
\Sigma_g
\left(
\frac{1}{M_1}\vec\nabla_{\mathbf R_1}
+
\frac{1}{M_2}\vec\nabla_{\mathbf R_2}
\right)
\Sigma_u\,(d\mathbf r),
\tag{5}
]

[
\frac{1}{M}K_{gg}
=
\int
\Sigma_g
\left(
\frac{1}{M_1}\Delta_{\mathbf R_1}
+
\frac{1}{M_2}\Delta_{\mathbf R_2}
\right)
\Sigma_g\,(d\mathbf r)
\quad \text{etc.}
]

For $R\to\infty$

[
\Sigma_{g,u}
=
\frac{1}{\sqrt{2}}\left(\psi(r_1)\pm\psi(r_2)\right);
\tag{6}
]

$\psi(r_1)$, $\psi(r_2)$ are the wave functions of the $\mu$-meson at the proton and deuteron, respectively.

For the functions (a(R)=(A+B)\dfrac{1}{\sqrt{2}};\ b(R)=(A-B)\dfrac{1}{\sqrt{2}}), describing the motion of the deuteron relative to mesohydrogen and of the proton relative to mesodeuterium, we have the equations

[
\begin{gathered}
-\frac{1}{2M}\Delta_R a+
\left[\frac{1}{2}(E_g+E_u)-\frac{1}{4M}(K_{gg}+K_{uu}+K_{gu}+K_{ug})-E\right]a+\
+\left{\frac{1}{2}(E_g-E_u)-\frac{1}{4M}(K_{gg}-K_{uu}+K_{ug}-K_{gu})\right}b
+\frac{1}{M}S\frac{db}{dR}=0,\
-\frac{1}{2M}\Delta_R b+
\left[\frac{1}{2}(E_g+E_u)-\frac{1}{4M}(K_{gg}+K_{uu}-K_{gu}-K_{ug})-E\right]b+\
+\left[\frac{1}{2}(E_g-E_u)-\frac{1}{4M}(K_{gg}-K_{uu}-K_{ug}+K_{gu})\right]a
+\frac{1}{M}S\frac{da}{dR}=0,\
S=(S_{gu})_R .
\end{gathered}
\tag{7}
]

It can be shown that, to within exponentially small terms and terms (\sim \dfrac{1}{R^4}) for (R\gg 1),

[
E_g+E_u \simeq -1,
]

[
K_{gg}+K_{uu}\simeq
2\int \psi(r_1)\Delta_{r_1}\psi(r_1)\,(dr)=-1
]

[
K_{gu}+K_{ug}\simeq
2\frac{M_2-M_1}{M_2+M_1}
\int \psi(r_1)\Delta_{r_1}\psi(r_1)\,(dr)
=-\frac{M_2-M_1}{M_2+M_1}.
]

The quantities (K_{gg}-K_{uu}), (K_{gu}-K_{ug}), (S) are exponentially small, and the corresponding terms in (7), containing (\dfrac{1}{M}), at distances (R) essential for the problem under consideration, may be neglected in comparison with (E_g-E_u).

In approximation (6), (E_g-E_u\simeq {}^{4}/_{3}Re^{-R}); a more accurate calculation (taking into account the distortion of the (\mu)-meson wave function near one nucleus by the action of the other) gives

[
E_g-E_u\simeq \frac{4}{e}Re^{-R}.
]

Assuming that the collision occurs at thermal velocities, in the relative motion of the nuclei it is sufficient to take into account only the (S)-wave. The equations for the radial functions (u=Ra(R)) and (v=Rb(R)) take the form

[
\begin{gathered}
\frac{d^2u}{dR^2}+2M\varepsilon u+M(E_u-E_g)v=0,\
\frac{d^2v}{dR^2}
+2M\left[\varepsilon+\frac{1}{2}\left(\frac{1}{M_1}-\frac{1}{M_2}\right)\right]v
+M(E_u-E_g)u=0 .
\end{gathered}
\tag{8}
]

(The energy of the mesoatom of hydrogen has been chosen as the zero of energy.) The term (\dfrac{1}{2M_1}-\dfrac{1}{2M_2}) takes into account the difference between the binding energies of the (\mu)-meson at the proton and the deuteron.

(M(E_u-E_g)), in the range of distances essential for the problem under consideration, can be approximated by the function (e^{-\alpha(R-R_1)}) with (R_1=3.38;\ \alpha=1-\dfrac{1}{R_1}\simeq 0.704). In this same region the energy (\varepsilon) may be neglected.

Equations (8) then take the form ((x=R-R_1)):

[
\frac{d^2u}{dx^2}+e^{-\alpha x}v=0,\qquad
\frac{d^2v}{dx^2}+k_0^2v+e^{-\alpha x}u=0;
\tag{9}
]

[
k_0^2=-\frac{M_2-M_1}{M_2+M_1}\simeq \frac{1}{3}.
]

As (x\to -\infty) ((R_0^2\ll e^{-\alpha x})), the solutions of equations (9) are

[
u+v=C_1\left[J_0\left(\frac{2}{\alpha}e^{-\alpha x/2}\right)+\delta N_0\left(\frac{2}{\alpha}e^{-\alpha x/2}\right)\right],
]

[
u-v=C_2K_0\left(\frac{2}{\alpha}e^{-\alpha x/2}\right),
\tag{10}
]

where (J_0(z), N_0(z)) are Bessel functions of the first and second kinds; (K_0(z)) is a Bessel function of imaginary argument; (C_1, C_2, \delta) are constants.

The (pD\mu) system has a level with energy very close to (0) ((^{2})). As a result, a resonance occurs, and the constant (\delta) is negligibly small. The constants (C_1, C_2) are determined from the condition that the asymptotic form of the solution as (x\to+\infty) has the form

[
u\simeq x+\beta;\qquad v\simeq \gamma e^{ik_0x}.
]

The quantities (\gamma) and (\beta) are found by numerical integration of system (9) with the initial conditions (10)(^*).

For the solutions

[
u_1=v_1=J_0\left(\frac{2}{\alpha}e^{-\alpha x/2}\right)
]

and

[
u_2=-v_2=K_0\left(\frac{2}{\alpha}e^{-\alpha x/2}\right),
]

taken at (x=-2), the asymptotic form as (x\to+\infty), respectively, is

[
u_1\simeq a_1x+b_1,\qquad v_1\simeq c_1\cos(k_0x+\delta_1),
]

[
a_1=-0.489,\qquad b_1=1.257,\qquad c_1=1.06,\qquad \delta_1=258^\circ;
]

[
u_2\simeq a_2x+b_2,\qquad v_2\simeq c_2\cos(k_0x+\delta_2),
]

[
a_2=0.202,\qquad b_2=-0.281,\qquad c_2=0.254,\qquad \delta_2=29^\circ.
]

The effective cross section of “transfer” is

[
\sigma=4\pi\frac{k_0}{k}|\gamma|^2
=4\pi\frac{k_0}{k}\,
\frac{\sin^2(\delta_1-\delta_2)}
{\left(\frac{a_1}{c_1}\right)^2+\left(\frac{a_2}{c_2}\right)^2
-2\frac{a_1}{c_1}\frac{a_2}{c_2}\cos(\delta_1-\delta_2)}
\simeq 1.5\pi\frac{k_0}{k}.
]

(Between the numbers (a_1,a_2,b_1,b_2,\ldots) there is the relation

[
a_1b_2-b_1a_2=k_0c_1c_2\sin(\delta_1-\delta_2),
]

by virtue of which (\operatorname{Im}\beta=k_0|\gamma|^2).)

The value obtained agrees with that adopted to explain the experimental data in work ((^{1})),

[
\sigma=\pi\frac{k_0}{k}
]

(the best agreement is obtained for

[
\sigma=0.5\pi\frac{k_0}{k}
]

).

The value of the effective cross section given in ((^{3})), calculated in the Born approximation ((^{4})) (which is essentially the opposite limiting case), differs from ours by a factor of (1.5\cdot10^2). It must also be noted that in Jackson’s paper ((^{3})), because nonorthogonal wave functions were used in the calculations (Appendix D), the probability of mesomolecule formation is overestimated by three orders of magnitude in comparison with that calculated in ((^{2})). In this connection, in ((^{3})) an incorrect conclusion was drawn that the dependence of the catalysis of the pD reaction on the concentration (D) is determined by competition between the processes of “jumping” of the (\mu)-meson from the proton to the deuteron and the processes of formation of the mesomolecule (pp\mu). As shown in ((^{1})), the indicated dependence is determined by the probability of the “jump” of the (\mu)-meson to the deuteron during the lifetime of the (\mu)-meson with respect to the decay (\mu\to e+\nu+\tilde{\nu}).

In conclusion I express my deep gratitude to L. D. Landau and Ya. B. Zel’dovich for their interest in the work and valuable suggestions.

Received
1 VII 1957

CITED LITERATURE

1. Ya. B. Zel’dovich, A. D. Sakharov, ZhETF, 32, 947 (1957).
2. Ya. B. Zel’dovich, DAN, 95, 493 (1954).
3. J. D. Jackson, Phys. Rev., 106, 330 (1957).
4. J. D. Jackson, H. Schiff, Phys. Rev., 89, 359 (1953).

* I take this opportunity to express my gratitude to M. G. Neigauz and S. M. Lomnev for carrying out the numerical integration.