Electric field at the nucleus

The moments discussed in this chapter are sometimes referred to as the outer moments of the distribution, in contrast to the inner moments for which the powers of r in the operator 6 in Eq. (6.5) are negative. The electric field at the nucleus and the field gradient at the nucleus are examples of inner moments, which will be discussed in chapter 8. 

Flere q,j is defined as the gradient of the electrical field at the nucleus undergoing NQR, due to the local electron density and the nearby nuclear charges ... 

Forced Electric Dipole Transitions. In more recent work, Judd (15) has given further attention to the problem of intensities. According to this work, under certain symmetry restricted circumstances, the Hamiltonian for the interaction of a lanthanide ion with its neighbors can contain spherical harmonics with fc = 1 if the electrons of the rare-earth ion produce an electric field at the nucleus that exactly cancels that... 

Electric Quadrupole Interaction (Quadrupole Splitting). This interaction exists when the electrons and/or the neighboring atoms produce an inhomogeneous electric field at the nucleus and when the nucleus possesses a quadrupole moment. The interaction between the nuclear electric quadrupole moment Q and the electric field gradient (EFG) is given by (15)... 

It should be pointed out that the derivatives of the Y, terms do not all vanish at the origin. There appears to be an electric field acting on the nucleus of the rare-earth or actinide ion - an impossibility if the ion is in equilibrium. The resolution of this paradox lies in the admixtures of d and g states into the f shell. The redistribution of electronic charge of the rare-earth or actinide ion produces a second electric field at the nucleus that exactly cancels the first. 

A nucleus possesses a nuelear quadrupole moment if it has a spin I greater than 1/2. In this case, its energy levels will be affected by an electric field gradient (EFG) at the nucleus, which leads to sphtting of the Mossbauer line. To detect this it is sufficient if at least one of the nuclear states involved in y-ray excitation possesses a quadrupole moment eQ, and that the electric field at the nucleus is inhomogeneous. This is usually the case if there is a non-cubic valence electron distribution or non-cubic lattice-site symmetry. 

Electric quadrupole interaction between the nuclear quadrupole moment and an inhomogeneous electric field at the nucleus. The observable Mossbauer parameter is the quadrupole splitting AEq . The information derived from the quadrupole splitting refers to oxidation state, spin state and site symmetry. 

Finally, the TDLDA provides perfect nuclear shielding, ite, effective electric field at the nucleus vanishes. 

Information concerning the symmetry of the electric field at the metal nucleus can be found from this latter parameter, AEq, which can also be measured directly by nuclear quadrupole resonance techniques. Additional information concerning the symmetry of the ligand around the metal can be deduced from x-ray, infrared, and nuclear magnetic resonance data. 

Et is the /-component of the electric field at the site of the nucleus and V is the electrostatic potential at this site. The Laplace equation gives the relation... 

Any electron i at a given distance r from a nucleus, besides feeling the nuclear attractive force, experiences repulsion from all the electrons that he inside and outside a spherical surface of radius r. Any of these repulsions affects the energy of electron i. However, as far as the electric field at the position of electron i is concerned, there is a difference arising from electrostatics, as follows ... 

The term M is the central part of the charge distribution of the nucleus. If we have two isotopes with different neutron number, the charge distribution will be slightly changed, thus changing the electric field inside the nucleus. For electrons with a nonzero wavefunction at the nucleus, this difference will cause a shift (field shift) in the energy levels between the Isotopes. The field shift can in an approximate way be described as... 

In nonspherical nuclei (/ > i) there exists a nonzero quadrupole moment for the nuclear charge density that contributes to the quadrupolar term (fourth term) in Eq. (1). In the expression for the electrostatic interaction energy for the nucleus in the potential produced by the electrons, the electric quadrupole moment of the nucleus is coupled to the gradient of the electric field at the nuclear site. This field gradient is, of course, due to the electrons. In diagonalized form one may write... 

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