The Phenomenological Theory of Exchange Currents in Nuclei

Abstract

As was first pointed out by Siegert, the existence of exchange forces in nuclei implies the existence of accompanying exchange currents. Sachs has calculated an expression for these, by making the Hamiltonian containing exchange potentials gauge-invariant, and has applied it to the calculations of exchange magnetic moments in ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{3}$. The Hamiltonian obtained by Sachs is not the most general admissible one. More generally, the exchange current density is found to depend on a vector function whose irrotational part is completely determined by gauge-invariance but whose solenoidal part is arbitrary except for the requirements (following from conditions of translational invariance and symmetry in all nucleons on the Hamiltonian) that it be translationally invariant and antisymmetric under the exchange of the spin and space coordinates of each pair of nucleons. Making use of these conditions on the Hamiltonian, the explicit form of the dependence of the solenoidal part of the exchange current upon the spin and isotopic spin coordinates of the nucleons has been derived. In the resultant exchange moments, the irrotational part leads to the expression obtained by Sachs, while the solenoidal term contribution contains the spin operators of the nucleons in particular combinations, together with arbitrary functions of the nucleon separation. Villars' exchange moment expression, as obtained from meson theory, is included as a special case and hence the exchange contributions to the moments of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{3}$ are explicable on a phenomenological basis, contrary to the results obtained in Sachs' special case. The generality and significance of the results are discussed in relation to the various meson theories.