The Meson Theory of Nuclear Forces. Part II. Theory of the Deuteron

With the nuclear forces as derived in the first part of this paper the deuteron problem is integrated. In \textsection{}7, natural units are introduced, viz., the range of the nuclear forces $\frac{1}{\ensuremath{\kappa}}=\frac{\ensuremath{\hbar}}{\ensuremath{\mu}c}$ as the unit of length and the quantity ${E}_{0}=\frac{{\ensuremath{\mu}}^{2}{c}^{2}}{M}$ as the unit of energy. With the adopted value of the meson mass, $\ensuremath{\mu}=177$ electron masses, we have $\frac{1}{\ensuremath{\kappa}}={2.18}_{5}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm and ${E}_{0}=8.68$ Mev. The quantum states to be expected are discussed (\textsection{}8); each state is characterized by the total angular momentum $J$, the total spin $S=0 or 1$, and the parity. For some states, the orbital momentum $L$ is defined uniquely by these quantum numbers (in this case, $L=J$); for others, we have a linear combination of wave functions with two different values of $L$, viz., $J+1$ and $J\ensuremath{-}1$. The angular coordinates are eliminated from the Schr\"odinger equation (\textsection{}9) and the radial wave equations obtained. The order of the states is discussed qualitatively (\textsection{}10) and it is made plausible that the ground state is a combination of a $^{3}S$ and a $^{3}D_{1}$ state, both in the neutral and in the symmetrical theory. The next higher triplet state is probably $^{3}P_{1}$. Then the wave equation is solved numerically for the $^{1}S$ state (\textsection{}11), the position of this state being taken from experiments on the scattering of slow neutrons by protons. The method of the numerical solution is described, and a table constructed giving the interaction constant $a=\frac{(\frac{2M}{\ensuremath{\mu}}){f}^{2}}{\ensuremath{\hbar}c}$ as a function of the cut-off distance ${x}_{0}=\ensuremath{\kappa}{r}_{0}$; $a$ is found to depend only slightly on ${x}_{0}$. The next section (\textsection{}12) deals with the numerical integration for the triplet (ground) state.The following sections contain the results. In \textsection{}13, the results for the neutral theory are given. A cut-off distance of 0.32 to 0.40 of the range of the forces will fit the positions of singlet and triplet state; these figures are very reasonable from general considerations. The calculations are carried out with two alternative ways of cutting off, assuming the potential inside ${r}_{0}$ either to be zero, or to retain the value it had at ${r}_{0}$. It is found that all physically significant quantities are practically independent of the method of cutting off. The value of $\frac{{f}^{2}}{\ensuremath{\hbar}c}$ is 0.077 or 0.080, according to the method used. The wave function contains about 6.7 percent $^{3}D_{1}$ state, the rest being $^{3}S$. This means that the sum of the magnetic moments of proton and neutron should be about 0.04 nuclear magneton greater than the deuteron moment which is just reconcilable with the present experimental data (\textsection{}14). In \textsection{}15, the quadrupole moment of the deuteron is calculated. The values found are 2.70 and 2.61 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}27}$ ${\mathrm{cm}}^{2}$ using the two methods of cutting off; the sign is positive (cigar-shape). Both sign and magnitude are in good agreement with the experiments of Kellogg, Rabi, Ramsey and Zacharias as evaluated by Nordsieck ($Q=2.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}27}$). In \textsection{}16, the $^{3}P_{1}$ state is calculated; it is shown to be unstable and to have no appreciable influence on the scattering of neutrons of moderate energy.The symmetrical theory is discussed in \textsection{}17. The calculations for this theory give unacceptable results, particularly for the cut-off distance which must be chosen considerably larger than the range of the nuclear forces (${x}_{0}=1.3 \mathrm{to} 1.7$). The quadrupole moment comes out about ten times too large and of the wrong sign. These results are very regrettable since only the symmetrical theory gives a natural explanation of the $\ensuremath{\beta}$-decay and of the extra magnetic moments of neutron and proton. Therefore an alternative way of cutting off is tried in \textsection{}18, with even more unfavorable results. The theory of M\o{}ller and Rosenfeld is discussed (\textsection{}19) but it is not considered satisfactory because of its intrinsic complication. The question remains open whether the neutral theory is correct because it gives quantitative agreement for the deuteron, or the symmetrical theory which is qualitatively preferable but in violent quantitative disagreement both with the theory of the deuteron and with the $\ensuremath{\beta}$-decay.