Particle-Hole States in the Alpha Particle with Realistic Forces

We solve the Tamm-Dancoff equations, using the realistic separable potential of Tabakin, for the odd-parity excited states of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. A comparison is made with results obtained using the hard-core potentials of Brueckner, Gammel, and Thaler and of Hamada. The calculated spectrum is found to be strongly influenced by the $p$-state contributions, particularly by the effect of the tensor interaction. We predict that the $0^{-}$, $T=0\mathrm{}$ state lies close to the $2^{-}$, $T=0\mathrm{}$ state. The spin-orbit splitting between the $p_{\frac{1}{2}}$ and $p_{\frac{3}{2}}$ single-particle states is calculated and found in first order to depend only on the relative two-body spin-orbit interaction. Our estimate for the ratio of the probability of an $E1\mathrm{}$ transition from the upper $1^{-}$, $T=1\mathrm{}$ state to the ground state to that of an $E1\mathrm{}$ transition from the lower $1^{-}$, $T=1\mathrm{}$ state is calculated to be 1.6, compared with the experimental ratio which is ∼½. We also calculate the squared matrix elements and the total capture rate for muon capture in ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. The squared matrix elements are found to be equal within 10 percent, but the calculated total capture rate is smaller than the experimental rate. Our theoretical results tend to justify the use of the supermultiplet theory for the excited states of the $\alpha$ particle. We find that our seven equations for the energy splittings depend upon only four quantities. Consequently, we are able to obtain three relations, which give the two unobserved energy splittings and the ratio of the $E1\mathrm{}$ transition probabilities in terms of the four observed energy splittings. The two empirical results which can be compared with experiment are in excellent agreement.