Full Text
Nucleon Sizes
Ya. Smorodinsky
One of the most interesting problems of modern physics is the elucidation of the properties of the nucleon. It is well known that neither the proton nor the neutron is described by the Dirac equation, although they have the same spin as the electron. This property of nucleons is usually associated with their interaction with \(\pi\)-mesons. Although there is still no more or less consistent meson theory, it is qualitatively clear that a “bare” nucleon must be surrounded by a meson cloud whose radius should be of the order of
\[ \frac{\hbar}{\mu c}=14\cdot 10^{-14}\ \text{cm}. \]
This means that at distances of this order of magnitude from the center of the nucleon, the properties of the latter must deviate from the properties of a point particle. The study of the physical properties of nucleons at distances \(<10^{-13}\) should therefore provide important information for understanding the nature of nucleons.
The most direct way for such a study is the investigation of the scattering of \(\gamma\)-quanta or electrons by protons.
The first results of the study of the scattering of \(190\ \text{MeV}\) electrons by protons were reported in 1954 by Hofstadter et al.\(^{1,2,3}\) These experiments showed that the field of the proton remains Coulomb-like down to distances of \(10^{-13}\ \text{cm}\), i.e., that the dimensions of the proton are smaller than \(10^{-13}\ \text{cm}\).*)
At the end of 1954, at the conference on high-energy particle physics,\(^{4}\) further experiments by Hofstadter were reported, as a result of which a more accurate value for the proton radius was obtained. Finally, in April 1955 a report appeared,\(^{11}\) in which the value of the mean-square radius of the proton was given as
\[ \langle R^2\rangle=(7.4\pm 2.4)\cdot 10^{-13}\ \text{cm}. \]
In these experiments the scattering of electrons with energies of 100, 188, and \(236\ \text{MeV}\) at angles \(>35^\circ\) was studied.
*) These same experiments gave for the mean-square radius of the deuteron the value \((1.5\pm 0.2)\cdot 10^{-13}\ \text{cm}\) and for the \(\alpha\)-particle \((1.4\pm 0.2)\cdot 10^{-13}\ \text{cm}\).
The obtained value of the radius is in agreement with the estimate obtained on the basis of data on the scattering of high-energy protons by protons (see 5).
Data on the elastic-scattering cross section and the total cross section (elastic scattering + meson production) in the energy interval \(700—1000\) MeV \(^{6,7}\) show that in this interval
\[ \sigma_{\text{elastic}}=\frac{1}{2}\sigma_{\text{total}} =22\cdot 10^{-27}\ \text{cm}^{2}(\pm 10\%). \]
The same conclusion is also reached in experiments on the scattering of \(\pi\)-mesons of energy \(1400\) MeV by protons. Their results indicate the existence of a region of very strong interaction (“black” \(^{12}\)) with radius \(R\simeq 5\cdot 10^{-14}\) cm.
Such a relation between the cross sections makes it possible to regard scattering in this interval as scattering by black spheres (considering the process of meson production as absorption of protons). As is known, the elastic-scattering cross section on a black sphere is equal to \(\pi r^{2}\). In the case of a collision of two protons, by \(r\) we must understand the sum of the radii of both particles \((2R)\). Using the data given above, we obtain:
\[ R\simeq 4\cdot 10^{-14}\ \text{cm}. \]
The same order of magnitude also follows from one more experiment. (True, in this case one can speak only of the absence of contradiction.) We have in mind the well-known Fermi–Yang experiments on the scattering of slow neutrons (see, for example, 6).
In these experiments the existence of an interaction between the electron and the neutron was shown. The magnitude of the interaction of the neutron with the electron is customarily characterized by the effective depth of a rectangular well having radius \(e^{2}/mc^{2}=2.80\cdot 10^{-13}\) cm. Such a model description is legitimate, since the scattering of slow neutrons is determined, as is known, only by the value
\[ \int H\,d^{3}x \]
—the integral of the interaction Hamiltonian over the whole volume.
Obviously, the effective depth \(V_{0}\) is related to this integral by the expression
\[ V_{0}=-\frac{3}{4\pi}\left(\frac{mc^{2}}{e^{3}}\right)^{3}\int H\,d^{3}x. \]
The experimental value (see 9)
\[ V_{0}=4200\pm 600\ \text{eV}. \]
The interaction of the (slow) neutron with the electron consists of two parts: the interaction of the magnetic moment with the electric field of the electron (Foldy)*) and the electrostatic interac-
*) It is interesting to note that the interaction of a slowly moving magnetic moment with an electrostatic field at first glance contradicts electrodynamics. In reality this interaction is
properties, connected with the fact that the meson cloud surrounding the neutron carries charge part of the time (the system as a whole, of course, remains neutral all the time).
The magnetic interaction is described by the Hamiltonian
\[ H_m=-\mu_N\left(\frac{e\hbar}{2Mc}\right)^2\frac{1}{e}\operatorname{div}\mathbf{E}, \]
where \(\mu_N=-1.91\) nuclear magnetons, and \(M\) is the nucleon mass. Integrating over the volume, noting that \(\int \operatorname{div}\mathbf{E}\,d^3x=-4\pi e\) (\(\mathbf{E}\) is the field of the electron, with charge \(-e\)), and comparing with the formula written above, we obtain for the effective potential:
\[ V_{0m}=\frac{3}{4}|\mu_N|\left(\frac{m}{M}\right)^2 \left(\frac{\hbar c}{e^2}\right)^2 mc^2 . \]
Substituting the values of the constants, we find:
\[ V_{0m}=4080\ \text{eV}. \]
The electrostatic interaction is evidently determined by the Hamiltonian
\[ H_{\mathrm{el}}=-e\varphi \quad (\varphi\text{ is the potential of the meson cloud}). \]
Integrating over the volume and carrying out simple transformations (integration by parts), we obtain:
\[ \int H_{\mathrm{el}}d^3x = -4\pi e\int \varphi r^2dr = -\frac{4\pi e}{6}\int(\varphi r)''r^3dr . \]
But according to Poisson’s equation \((\varphi r)''=-4\pi \rho r\) (\(\rho\) is the charge density in the meson cloud, evidently negative). Denoting
\[ \int \rho d^3x=-\beta e;\qquad \int \rho r^4 d^3x=-\langle R^2\rangle \beta e \]
(\(\beta e\) is the effective charge of the meson cloud, \(\langle R^2\rangle\) is the mean square of its radius). The quantity \(\beta\) may be interpreted as the fraction of the time during which the meson cloud is charged.
Using these notations, we may write:
\[ V_{0\mathrm{el}}= \frac{\beta}{2}\, \frac{\langle R^2\rangle}{(e^2/mc^2)^2}\,mc^2 . \]
Thus, for the full effect we obtain:
\[ V_0=V_{0m}+V_{0\mathrm{el}} = \frac{3}{4}|\mu_N| \left(\frac{m}{M}\right)^2 \left(\frac{\hbar c}{e^2}\right)^2 \left[ 1+\frac{2\beta}{3|\mu_N|} \frac{\langle R^2\rangle}{(\hbar/Mc)^2} \right]mc^2 . \]
Turning to comparison with experiment, it must be noted that the accuracy of the experiment is low and it is impossible realistically to estimate the influence of the second
consequence of relativistic effects (in this way it was obtained by Foldy) and connected with the so-called “trembling” of the neutron in a region of dimensions \(\dfrac{\hbar}{\mu c}\)—a phenomenon characteristic of the Dirac equation.
(electrostatic) term, since the magnetic interaction alone already gives, within the errors, an explanation of the effect.
For an estimate, however, we may assume that the electrostatic interaction accounts for no more than 10% of the effect. Hence we obtain: $\beta \langle R^2\rangle \sim 0.3\left(\dfrac{\hbar}{Mc}\right)^2$.
Substituting $\langle R^2\rangle \sim (4\cdot 10^{-14})^2 \sim 3\left(\dfrac{\hbar}{Mc}\right)^2$, we obtain $\beta \sim 0.1$. This means that the neutron is approximately 10% of the time a proton surrounded by a negatively charged meson cloud.
Further refinement of the experiments will make it possible to check the validity of such estimates.
At present one can only conclude that the radius of the nucleon turns out to be appreciably smaller than the mean distance between nucleons in a nucleus, which is of order $\hbar/\mu c$. This conclusion is very important, since it explains the fact that nucleons in a nucleus to a considerable degree (as the shell theory shows) retain their properties (for example, the magnetic moment), which would be very strange if the sizes of the nucleon were equal to the distance between nucleons in nuclei. Thus the nucleus turns out to be comparatively “empty”: the nucleons occupy only about $1/40$ of its volume.
References
- I. A. McIntyre and R. Hofstadter, Phys. Rev. 96, 854 (1954).
- R. Hofstadter, R. McAllister and E. Wiener, Phys. Rev. 96, 854 (1954).
- R. Hofstadter, Bull. Amer. Phys. Soc. 29, No. 8, 29 (1954).
- Science News Letter 67, 117 (1955), February 19, 1955.
- Ya. Smorodinsky, Problems of Modern Physics, issue 7 (1954) (introductory article).
- A. M. Shapiro, C. P. Leavitt and F. F. Chew, Phys. Rev. 95, 663 (1954).
- L. W. Smith, A. W. Reynolds and G. Snow, Phys. Rev. 97, 1186 (1955).
- D. Hughes, Neutron Studies on Nuclear Reactors, IL, p. 197, 1954.
- B. D. Fried, Phys. Rev. 88, 1142 (1952).
- L. L. Foldy, Phys. Rev. 87, 693 (1952).
- R. Hofstadter and R. W. McAllister, Phys. Rev. 98, 217 (1955).
-
L. M. Eisberg, W. B. Fowler, R. M. Lea, W. D. Shephard, R. P. Shutt, A. M. Thorndike and W. L. Whittemore, Phys. Rev. 97, 797 (1955).
- *
Proof correction note
A report has recently been published that Serber and Rarita have also considered proton–proton scattering on the basis of the optical model (Serber and Rarita, Bull. Amer. Phis. Soc. 30, No. 3 HAZ (1955)).