Volume 34B. number 4 PHYSICS LETTERS 1 March 1971 LEVINSON 'S THEOREM AND S-WAVE NEUTRON-DEUTERON SCATTERING * I. H. SLOAN Department of Applied Mathematics. University of New South Wales. Kensington, N.S. W.. A~stralia Received 20 December 1970 S-wave phase shifts are calculated for a separable-potential neutron-deuteron model up to E n : 100 MeV, and are found to satisfy the modified Levinson's theorem proposed b.v Swan. The effect of the Pauli prin- ciple on Levinson's theorem is illustrated with a simple three-body model. Levinson's theorem [1], which states that 5(0) - 5(~) =n~, where 6(E) is the phase shift and n the number of bound states in the particu- lar partial wave, is essentially a theorem in potential scattering, and does not necessarily apply for more general interactions [2, 3]. In particular, Swan [2] pointed out fifteen years ago that Levinson's theorem must be modified for many cases of scattering by composite sys- tems, the modification proposed (but not rigor- ously proved) being that n should include any bound states excluded by the Pauli principle, as well as the actual physical bound states. For the particular case of n-d scattering, Swan's result is 5 (0) - 5(~o) = ~ for both the doublet and quartet S-wave phase shifts. (This discussion ignores non-central forces, so that L and S are both sup- posed to be good quantum numbers.) In the doublet case the ~ on the sight hand side correspond to the one physical bound state, the triton, whereas for the quartet case it corresponds to a bound state excluded by the Pauli principle. (The im- portant point here is that in the quartet case the spins of the two neutrons are necessarily paral- lel.) The separable-potential model of Aaron et al. [4] has proved to be at least qualitatively a suc- cessful model of n-d scattering, and it is there- fore a matter of interest to see if the S-wave shifts in this model satisfy Swan's prediction. This was not clear from the original calculations [4], in fact in that work the doublet and quartet phase shifts were both assumed to satisfy the un- modified Levinson's theorem. The tabulated quartet phase shifts in ref. [4] do indeed seem to * Research supported by the Australian Research Grants Committee. O0 180 150 "~ 120 9O 60 k (f-") 0.5 1.0 1-5 I ~ I I Bill ~1 III I L 4S 2S ~~ 1 5 10 50 100 3O 0 0 Neutron tab. energy (MeV) Fig. 1. Real parts of the S-wave n-d phase shifts belo~ 100 MeV. The arrow indicates the breakup threshold (3.339 MeV), above which the phase shifts are complex The horizontal scale is linear in the wave number k. satisfy 5 (0) - 6 (~) = 0, but on closer examination the table of phase shifts is seen to have a proba- ble discontinuity. To clarify the situation we have recalculated that S-wave phase shifts for the separable-poten- tial model of ref. [4], and extended the maximum energy from 14 to 100 MeV, using methods de- scribed elsewhere [5]. The results are shown in fig. 1. Above the breakup threshold (E n = 3.339 MeV) the phase shifts are of course complex, and in this region 5 is the real part of the phase shift. The results in fig. 1 clearly confirm Swan's prediction that 5(0) - 5(~) = 77 for both the doublet and quartet cases. In drawing fig. 1 we have assigned the value to the zero-energy phase shift in both cases, so that the doublet and quartet phase shifts both 243