Full Text
Physics
B. B. Dotsenko
On the Question of the Structure of the Proton
(Presented by Academician N. N. Bogolyubov, 28 XI 1957)
In the work of Chambers and Hofstadter \((^1)\), the electromagnetic structure of the proton was investigated experimentally. Of the six possible distribution functions for the density \(d(r)\) of charge in the proton, the authors consider the function
\[ d(r)=d_0 r \exp(-r), \tag{1} \]
to be the most preferable, where \(r\) is the distance to the center of the proton, expressed in the corresponding units.
A theoretical investigation of the question of the structure of the proton would be most consistent if this question were considered on the basis of a rigorous field theory. However, since such a consideration encounters serious difficulties, we shall solve this question approximately, proceeding from certain phenomenological premises.
In this connection we shall make use of the well-known representation of the proton as a core surrounded by a meson cloud \((^2,^3)\). Let this cloud be formed by one meson.
From the known relativistic equation for the \(\Psi\)-wave function of a particle of mass \(m\) and spin 0 moving in a field with potential \(V(r)\) (see, for example, \((^4)\)), we have:
\[ \Delta \Psi + \frac{1}{\hbar^2 c^2}\left[(E - V)^2 - (mc^2)^2\right]\Psi = 4\pi D, \tag{2} \]
where \(E = i\hbar \cdot \partial/\partial t\); \(D\) is the source density; \(\hbar\) and \(c\) are known constants. Assuming that the process of meson emission is completed and that the meson interacts with the core, we obtain:
\[ \Delta \Psi + \frac{1}{\hbar^2 c^2}\left[(E - V)^2 - (mc^2)^2\right]\Psi = 0. \tag{3} \]
To determine the interaction potential \(V\), we shall use I. E. Tamm’s hypothesis of the “dissociation” of a meson in the field of the core into a nucleon–antinucleon pair*. Then, as the interaction potential of such a pair with the core, one may take the usual Yukawa potential
\[ V(r)=-\frac{g_1 g_2}{r}\exp(-\varkappa r), \tag{4} \]
where \(r\) is the distance from the pointlike core to the center of the meson-pair**; \(g_1\) is the nuclear charge of the core; \(g_2\) is the “effective” charge of the meson-pair, \(\varkappa=\mu c/\hbar\); \(\mu\) is the mass of an ordinary \(\pi\)-meson.
* Obviously, as a result of such “dissociation,” the meson mass \(m\) increases in comparison with the mass of an ordinary \(\pi\)-meson. In the first approximation, for simplicity, we regard the core as fixed at the origin of coordinates.
** We note that the linear dimensions of such a pair \(\sim \hbar/2Mc\), where \(M\) is the nucleon mass, will be considerably smaller than the radius of the meson cloud \(\sim \hbar/\mu c\).
We seek the solution of equation (3) in the form
\[ \Psi(\mathbf r,t)=\exp\left(-i\frac{\varepsilon}{\hbar}t\right)\Phi(\mathbf r), \tag{5} \]
where \(t\) is time; \(\varepsilon\) is a constant with the dimension of energy, which for simplicity we take to be real. This means that we seek stationary solutions for our core–meson system, assuming that the violation of stationarity occurs, for example, in accordance with I. E. Tamm’s hypothesis, because of annihilation of the central core with an antinucleon arising when a pair is formed.
Introducing polar coordinates, we represent the spatial part of the function \(\Phi(\mathbf r)\) in the form
\[ \Phi(\mathbf r)=R(r)Y(\vartheta,\varphi). \tag{6} \]
By the usual procedure, from equation (3), taking account of (5) and (6), we obtain for the angular part \(Y(\vartheta,\varphi)\) of the function \(\Psi\) the known equation whose solutions are the spherical functions:
\[ Y_{lm}=\left(\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}\right)^{1/2}P_l^m(\cos\vartheta)e^{im\varphi} \tag{7} \]
(where \(l\) and \(m\) are integers: \(l=0,1,2,\ldots;\ m=-l,\ldots,-1,0,1,\ldots,l;\) \(P_l^m\) are the unnormalized associated Legendre functions of order \(m\)), and the equation for the radial part \(R(r)\):
\[ \frac{d^2R}{dr^2}+\frac{2}{r}\frac{dR}{dr}+ \left[ A+\frac{2\beta\varepsilon}{\hbar c}\frac{\exp(-\varkappa r)}{r} +\frac{\beta^2}{r^2}\exp(-2\varkappa r)-\frac{l(l+1)}{r^2} \right]R=0 \tag{8} \]
(where \(\beta=g_1g_2/\hbar c;\ A=(\varepsilon^2-m^2c^4)/\hbar^2c^2\)).
For the existence of bound solutions it is necessary that \(A<0\), i.e., that \(\varepsilon<mc^2\). Then as \(r\to\infty\) the function \(R(r)\) will behave like \(\exp(\pm\rho/2)\), where \(\rho=2r/r_0,\ r_0=(\sqrt{-A})^{-1}\). Obviously, we must take the minus sign in the exponent. Therefore we shall seek the solution of equation (8) in the form
\[ R(r)=\exp(-\rho/2)f(\rho). \tag{9} \]
The equation for \(f(\rho)\) has the form
\[ f''+\left(\frac{2}{\rho}-1\right)f' +\left\{ \frac{1}{\rho}\left(r_0Be^{-\alpha\rho}-1\right) +\frac{1}{\rho^2}\left[\beta^2e^{-2\alpha\rho}-l(l+1)\right] \right\}f=0, \tag{10} \]
where \(B=\beta\varepsilon/\hbar c,\ \alpha=\varkappa r_0/2\) (primes denote differentiation with respect to \(\rho\)). Expanding in this equation \(\exp(-\alpha\rho)\) and \(\exp(-2\alpha\rho)\) in known series, we seek, in accordance with (6), solutions of equation (10) in the form
\[ f(\rho)=\rho^j w(\rho),\qquad w(\rho)=\sum_{k=0}^{\infty}a_k\rho^k. \tag{11} \]
Proceeding in the usual way, we obtain:
\[ j=-\frac12\pm\sqrt{\left(l+\frac12\right)^2-\beta^2}; \tag{12} \]
\[ a_1=\left[\frac12+\frac{r_0(\varkappa\beta^2-B)}{2(j+1)}\right]a_0, \]
\[ a_k= \frac{1}{k(k+2j+1)} \left\{ a_{k-1}\left[k+j+r_0(\varkappa\beta^2-B)\right] + r_0^2\varkappa \sum_{\nu=0}^{k-2} (-1)^\nu a_{k-2-\nu} \frac{(r_0\varkappa)^\nu}{(\nu+1)!} \left( \frac{B}{2^{\nu+1}}-\frac{\beta^2\varkappa}{\nu+2} \right) \right\} \quad (k\geqslant 2). \tag{13} \]
Near \(r=0\), where one may restrict oneself to the initial terms of the expansions \(\exp(-\chi r)\) and \(\exp(-2\chi r)\), equation (8) can be reduced to a degenerate hypergeometric equation, and its solution will have a form analogous to (9), but \(\omega(\rho)\) here must be replaced by a degenerate hypergeometric function (of Pochtammer).
In the case of arbitrary values of \(\rho\), it is not difficult to see that, owing to the first term of formula (13), the \(k\)-th term of series (11), as \(k \to \infty\), will behave approximately as \((d/k!)\rho^k\) (\(d=\mathrm{const}\)), i.e., as \(\rho \to \infty\) the sum of this series will grow as \(\exp \rho\). Therefore it is necessary to terminate the increasing part of series (11) at some \(n\)-th term, taking
\[ n+j+r_0(\chi\beta^2-B)=0. \tag{14} \]
Then, instead of the increasing part of the series, we obtain a polynomial of degree \((n-1)\), while \(R(\rho)\) with the remaining (infinite) part of series (11) decreases rapidly as \(\rho \to \infty\).
From (14) it is not difficult to find \(\varepsilon\):
\[ \varepsilon = mc^2 \frac{ \beta^3(\mu/m)\pm (n+j)\{[\beta^2+(n+j)^2]-\beta^4(\mu/m)^2\}^{1/2} }{ \beta^2+(n+j)^2 }, \tag{15} \]
and then also \(r_0\) as a function of \(n\), \(j\), \(\beta\), and \(m\).
Thus, the wave function of a meson with mass \(m\), forming a cloud around the nucleus, has the form
\[ \Psi(\mathbf r,t) = \exp\left(-i\frac{\varepsilon}{\hbar}t\right) Y_l^m(\vartheta,\varphi)R(r), \tag{16} \]
where
\[ R(r)=e^{-\rho/2}\rho^j\sum_{k=0}^{\infty}a_k\rho^k. \tag{17} \]
From the normalization conditions
\[ a_0\simeq \left[ \frac{2(m^2c^4-\varepsilon^2)^{1/2}}{\hbar c} \right]^{j+1/2} [\Gamma(2j+3)]^{-1/2} \qquad (l=1,2). \]
Let us note that for \(\beta^2>0\), according to (12), \(j\) will be positive (i.e., \(R(r)\) will not have a pole at zero) only when \(l>1\). Thus the lowest state of the system under consideration will be the \(P\)-state.
The dependence of the meson-cloud density on distance agrees approximately with experiment for \(2j=1\). Hence \(\beta=g_1g_2/\hbar c\simeq 1.118\).
In conclusion I consider it my duty to express deep gratitude to Acad. N. N. Bogolyubov for valuable consultations.
Institute of Physics
Academy of Sciences of the Ukrainian SSR
Received
27 XI 1957
CITED LITERATURE
- E. E. Chambers, R. Hofstadter, Phys. Rev., 103, No. 5, 1454 (1956).
- E. Fermi, Lectures on \(\pi\)-Mesons and Nucleons, Moscow, 1956.
- E. Fermi, Elementary Particles, Moscow, 1952.
- H. Bethe, Lectures on the Theory of the Atomic Nucleus, Moscow, 1949.
- P. Gombás, The Problem of Many Particles in Quantum Mechanics, Moscow, 1952.
- V. I. Smirnov, A Course of Higher Mathematics, 2, Moscow, 1954, p. 139.