Wigners interaction sphere

We now introduce a boundary surface over which the joining conditions can be applied. This is done in a manner quite analogous to the definition of the sphere of radius ro in MQDT (cf chapter 3). 

Take the origin of coordinates at the point where all the particles coalesce. We then define a sphere centred at the origin, whose radius is as small as possible, consistent with the requirement that tpE at the boundary has attained its well-separated form. Let a be the radius of this sphere. 

In nuclear physics, because of the nature of the short range interactions involved, this boundary has a definite physical identity. In atomic physics, there is no clear physical value for a, but one introduces the sphere for computational convenience (see, e.g., [371] for further discussion of this point), and it is generally advantageous to define as small a radius a as possible for the problem in hand. We call a the channel radius. 

For two energies E and E , neglecting any small differences in the particle interactions, the joining condition is achieved by applying Green s theorem on the sphere for kinetic energy terms. One obtains 

As discussed by Wigner [370], the left hand side of this equation does not vanish, because of the higher-order dependence of tp on E. 

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Derive the contribution Ues by making the Wigner-Seitz sphere approximation, in which inter-cell electrostatic interactions are neglected and the intra-cell potential energy is approximated by that of the Wigner-Seitz sphere. 

Liu (1961) noted that the wave functions of the 4f electrons on different rare earth (R) atoms in the solid state do not usually overlap. This is because the radius of the 4f shell is almost 0.35 A and the wave functions are therefore zero on the Wigner-Seitz sphere. There can therefore be no direct exchange, and exchange interactions between different R-magnetic moments must be mediated by the conduction electrons. Liu points out that there are two possible interaction types. In the first type the 4f magnetic moment on the R-atom polarizes the sp conduction bands of the compound via a direct s-f exchange interaction given by... 

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