Variational methods developed by Schwinger are applied to neutron-proton scattering at energies below 10 Mev. $S$-wave scattering alone is considered, and the tensor force is not taken into account. An expansion is obtained for the phase shift in powers of the energy. The coefficients can be evaluated explicity from the wave function. The first term of the series is related to Fermi's scattering length, the second term involves an "effective range." The third and higher terms turn out to be negligible.

The results are used to define an "intrinsic range" for a potential well of arbitrary shape. Thus a reasonable comparison of potential wells of different shapes is made possible. The relation between intrinsic range and effective range is discussed.

The experimental data on coherent and incoherent neutron-proton scattering are discussed in terms of a "shape-independent" approximation. The best value for the effective range in the triplet state is $r_t=(1.56\mathrm{}\pm 0.13)\times {10}^{-13\mathrm{}}$ cm. The effective range in the singlet state is not well determined by the present data.

The effect of higher, shape-dependent terms in the expansion of the phase shift is considered. These terms become more important as the well shape becomes more long tailed, but they are found to be negligible within experimental uncertainties for the four well-shapes considered here (square, Gaussian, exponential, and Yukawa).

The results for the scattering phase shifts can be extrapolated to negative energy to give an approximate algebraic equation for the energy of the bound state of the deuteron.

All the numerical results are shown in graphical form; interpolation formulas are provided where higher accuracy may be needed.

• Received 2 March 1949

DOI:https://doi.org/10.1103/PhysRev.76.18