Section 2.2.15.2. EFG conversion formulas

The nuclear quadrupole interaction (NQI) represents the interaction of Q (the nuclear quadrupole moment) with the electric field gradient (EFG) created by the charges surrounding the nucleus, as described above. Here we briefly summarize the main ideas (following Petrilli et al., 1998) and provide conversions between experimental NQI splittings and electric field gradients.

Let us consider a nucleus in a state with nuclear spin quantum number with the corresponding nuclear quadrupole moment , where is the nuclear charge density around point and e is the proton's charge. The interaction of this Q with an electric field gradient tensor , splits the energy levels for different magnetic spin quantum numbers of the nucleus according to in first order of , where Q represents the largest component of the nuclear quadrupole moment tensor in the state characterized by . (Note that the quantum-mechanical expectation value of the charge distribution in an angular momentum eigenstate is cylindrical, which renders the expectation value of the remaining two components with half the value and opposite sign.) The conventional choice is . Hence, is the principal component (largest eigenvalue) of the electric field gradient tensor and the asymmetry parameter is defined by the remaining two eigenvalues through (2.2.15.3) shows that the electric quadrupole interaction splits the ()-fold degenerate energy levels of a nuclear state with spin quantum number I () into I doubly degenerate substates (and one singly degenerate state for integer I). Experiments determine the energy difference between the levels, which is called the quadrupole splitting. The remaining degeneracy can be lifted further using magnetic fields.

Next we illustrate these definitions for ⁵⁷Fe, which is the most common probe nucleus in Mössbauer spectroscopy measurements and thus deserves special attention. For this probe, the nuclear transition occurs between the excited state and ground state, with a 14.4 KeV radiation emission. The quadrupole splitting between the and the state can be obtained by exploiting the Doppler shift of the radiation of the vibrating sample. For systems in which the ⁵⁷Fe nucleus has a crystalline environment with axial symmetry (a threefold or fourfold rotation axis), the asymmetry parameter is zero and is given directly by As can never be greater than unity, the difference between the values of given by equation (2.2.15.5) and equation (2.2.15.6) cannot be more than about 15%. In the remainder of this section we simplify the expressions, as is often done, by assuming that . As Mössbauer experiments exploit the Doppler shift of the radiation, the splitting is expressed in terms of the velocity between sample and detector. The quadrupole splitting can be obtained from the velocity, which we denote here by , by where c = 2.9979245580 × 10⁸ m s⁻¹ is the speed of light and E_γ = 14.41 × 10³ eV is the energy of the emitted radiation of the ⁵⁷Fe nucleus.

Finally, we still need to know the nuclear quadrupole moment Q of the Fe nucleus itself. Despite its utmost importance, its value has been heavily debated. Recently, however, Dufek et al. (1995b) have determined the value Q = 0.16 b for ⁵⁷Fe (1 b = 10⁻²⁸ m²) by comparing for fifteen different compounds theoretical values, which were obtained using the linearized augmented plane wave (LAPW) method, with the measured quadrupole splitting at the Fe site.

Now we relate the electric field gradient to the Doppler velocity via In the special case of the ⁵⁷Fe nucleus, we obtain EFGs can also be obtained by techniques like NMR or NQR, where a convenient measure of the strength of the quadrupole interaction is expressed as a frequency , related to by The value can then be calculated from the frequency in MHz by where (h/e) = 4.1356692 × 10⁻¹⁵ V Hz⁻¹. The principal component is also often denoted as .

In the literature, two conflicting definitions of are in use. One is given by (2.2.15.10), and the other, defined as differs from the first by a factor of 2 and assumes the value . Finally, the definition of has been introduced here. In order to avoid confusion, we will refer here only to the definition given by (2.2.15.10). Furthermore, we also adopt the same sign convention for as Schwarz et al. (1990) because it has been found to be consistent with the majority of experimental results.