Magnetic interactions between nuclei

Another mechanism of relaxation is associated with the magnetic interaction between nuclei and paramagnetic electrons (the so-called magnetic dipole interactions). This process is known as spin-spin relaxation time (T2). 

In these expressions the index i runs over electrons and a runs over nuclei. The Fermi contact term describes the magnetic interaction between the electron spin and nuclear spin magnetic moments when there is electron spin density at the nucleus. This condition is imposed by the presence of the Dirac delta function S(rai) in the expression. The dipole-dipole coupling term describes the classical interaction between the magnetic dipole moments associated with the electron and nuclear spins. It depends on the relative orientations of the two moments described in equation (7.145) and falls off as the inverse cube of the separations of the two dipoles. The cartesian form of the dipole-dipole interaction to some extent masks the simplicity of this term. Using the results of spherical tensor algebra from the previous chapter, we can bring this into the open as... 

A second perturbation of the nuclear levels is caused by the magnetic interaction between the nuclear magnetic moment and the surrounding electron spin density, creating an electric field gradient at the nucleus. This electric field gradient is a tensor that is described by the spatial orientations of its three principal axes Uxx,... 

Because of the narrow energy line width of the Fe57 y-radiation small relative displacements between source and absorber nuclear levels can be measured by the Mossbauer effect. The position of the nuclear energy levels is, influenced by electrostatic and magnetic interactions between the atomic electron shell and the Mossbauer nucleus. Three different types of interaction are of interest and will be described briefly. 

In Eq. (6) the index i designates the components x, y, z of the electron spin, and N is the sum of all the populated states AEn is the energy of the IVth electronic spin state relative to the lowest one. The nuclear. Hamiltonian, containing the magnetic hyperfine interaction between nucleus and paramagnetic valence shell, the nuclear Zeemann term ( —gNjMN o ), and the quadrupole interaction Hq, is given by... 

The other part of the hyperfme structure was suggested by Pauli [9] to be the result of the magnetic interaction between the nucleus and the electrons in the atom. If hfs is due to only a magnetic interaction, the difference between two neighbouring hyperfine levels follows an interval rule. However, accurate measurements of Eu, performed by Schuler and Schmidt [10], showed a deviation from this rule. Theoretical calculations performed by Casimir [11] showed that the experimental result was in agreement with an electric quadrupole interaction with the nucleus, thus establishing a quadrupole moment for Eu and 3Eu. 

Just from these simple considerations, we now know that the relaxation rate due to another nucleus interacting with the first is proportional to the square of the magnetic interaction between them regardless of the rate of the rotational tumbling (assuming, of course, that this is the only relaxation mechanism.)... 

H is an effective magnetic field which represents the magnetic interaction between the nucleus and the surrounding electrons. For a single electron it is given by... 

The major magnetic interaction between the electron and the nucleus. It is the major contribution to hyperfine coupfing and is maximized for S electrons because of their high spin density at the nucleus. 

The hyperfme parameters result from shifts in, or the removal of, the degeneracy of the nuclear energy levels s through the electric and magnetic interactions between the nucleus and its surrounding electronic environment. The expressions for the hyperfine parameters, the isomer shift, the quadrupole interaction, and the magnetic hyperfine field always contain two contributions, a nuclear contribution that is fixed for a given nuclide, and an electronic contribution that varies from compound to compound. 

The magnetic interaction between the nuclear magnetic dipole moment of iron and the magnetic hyperfme field generated at the nucleus by the surrounding electrons and magnetic dipoles is described by the Hamiltonian... 

This magnetic interaction, between the muon and the electron for the spherically symmetrical s state, sums to zero except where the electronic wavefunction overlaps the nucleus, hence the name contact interaction . Therefore the resulting hyperfine interaction constant. A, is a measure of the electron density at the nucleus. 

The last three terms of the spin Hamiltonian shape the Mossbauer spectrum because they describe the interaction of the nucleus with the atom, the solid and the external magnetic field. The term I - A - S describes the magnetic interaction between the nucleus and the atom. This acts through components of the atomic spin which are determined by the Boltzmann population of the spin Hamiltonian states shown in Figure 4.2. Thus the first three terms of the spin Hamiltonian act to determine values for the components of S that are used in the magnetic interaction that shapes the Mossbauer spectrum. The mechanisms involved in this nucleus-atom interaction I A S will be discussed in detail in the next section. The quadrupole interaction term represents the interaction of the nuclear quadrupole moment with the EFG produced by the atom and the lattice. The principal component of the EFG is V — d Vldz (V is the electric potential at the nucleus) and the asymmetry parameter =... 

MSssbauer Effect Source and absorber Mono-energetic y-rays 6-100 keV Mbssbauer spectrum (Doppler shifted (lines) 60 m 1 cm Interaction between nucleus and ils environment (local electric, magnetic fields bonds valency diffusion, etc.) 66... 

In the above I have attempted to show that the calculations, when considered impartially, suggest a limiting value of — 1.4500.4iZh, which represents an ionisation potential of about 24.37 V. Therefore the remaining difference which is not explained is about 1/10 V. Further corrections are possible, for instance for the motion of the nucleus, magnetic interactions between the electrons and relativistic mass changes. All of these appear to be on the order of 1/100 V or smaller, so that a precise calculation of these corrections in the present case is of only limited interest. 

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