A nucleus that has a spin quantum number has a non-spherical charge distribution. The magnitude of the charge deformation, , is given by
An asymmetric charge distribution around the nucleus causes an asymmetric electric field at the nucleus, characterised by a tensor quantity called the Electric Field Gradient (EFG) . The electric quadrupole interaction between these two quantities gives rise to a splitting in the nuclear energy levels. The interaction between nuclear moment and EFG is expressed by the Hamiltonian
There are two contributions to the EFG i) lattice contributions from charges on distant ions and ii) valence contributions due to incompletely filled electron shells. If a suitable coordinate system is chosen the EFG can be represented by three principal axes, , and . If an asymmetry parameter is defined using these axes as
The Hamiltonian for the quadrupole interaction can be rewritten as
The excited state of Fe has a spin . The EFG has no effect on the ground state but does remove degeneracy in the excited state, splitting it into two sub-states and where the states are higher in energy for positive . The energy eigenvalues for have exact solutions given by
The now non-degenerate excited states give rise to a doublet in the Mössbauer spectrum as illustrated in Figure 2.4. The separation between the lines, , is known as the quadrupole splitting and is given by
As the nuclear quadrupole moment is fixed the magnitude and sign of gives information about the sign of the EFG and magnitude of .
Dr John Bland, 15/03/2003