Electron-strain interaction

Harley, Hayes, and Smith [18] had measured the zone-center vibron energy ha> 0) of the TmAs04 crystal under external magnetic held. In that crystal, like in TmV04, the dynamic coupling is not zero in the presence of the external magnetic held only. As it is shown on the Fig. 7 the electronic excitation is the soft mode at the absence of the electron-strain interaction only (go = 0, dashed line). However when the electron-strain constant is not zero (all other lines on the Fig. 7) the... 

The Hamiltonian of the crystal with possible ordering of the electric dipole moments [24] additionally to the traditional terms discussed above contains the energy of the polarized crystal, the electron-polarization interaction (similar to the electron-strain interaction), and the interaction with the external electric field. After... 

We can now extend the spin Hamiltonians by making combinations of T, with B, and/or S, and/or I, and since we are interested in the effect of strain on the g-value from the electronic Zeeman interaction (B S), the combination of interest here is T B S. 

The spin Hamiltonian encompassing electronic Zeeman plus strain interaction for a cubic S = 1/2 system is (Pake and Estle 1973 Equation 7-21) ... 

Note that the Zeeman interaction for a cubic system results in an isotropic g-value, but the combination with strain lowers the symmetry at least to axial (at least one of the 7 -, 0), and generally to rhombic. In other words, application of a general strain to a cubic system produces a symmetry identical to the one underlying a Zeeman interaction with three different g-values. In yet other words, a simple S = 1/2 system subject to a rhombic electronic Zeeman interaction only, can formally be described as a cubic system deformed by strain. 

Most of the experimental results on CJTE can be explained on the basis of molecular field theory. This is because the interaction between the electron strain and elastic strain is fairly long-range. Employing simple molecular field theory, expressions have been derived for the order parameter, transverse susceptibility, vibronic states, specific heat, and elastic constants. A detailed discussion of the theory and its applications may be found in the excellent review by Gehring Gehring (1975). In Fig. 4.23 various possible situations of different degrees of complexity that can arise in JT systems are presented. 

Fig. 8. BPR spectra of [3Fe-xS] clusters in oxidized hydrogenases, showing th influences of weak Ni-Fe-S electron-spin interactions, (a) Desulfovibrio desulfurican (strain Norway 4) hydrogenase, showing the spectrum of an isolated [3Fe-xS] cluster (b Chromatium vinosum hydrogenase the outer lines (Signal 2) correspond to interactio with Ni(lH) (c) Paracoccus denitrificans hydrogenase (d) Alcaligenes eutrophu membrane-bound hydrogenase. Spectra were recorded at approximately 20 K. Sample were provided by K. K. Rao, J. Serra, and K. Schneider.

Although the nature of the strain interaction may be different in crystals examined by EPR, electronic, and Raman spectroscopies, one would expect the remaining parameters to vary little. A given model must be able to account for all the experimental data. 

For analyzing noncovalent interactions, we typically evaluate V(r) on a three-dimensional surface of the molecule. For this purpose, we use the 0.001 au (electrons/bohr3) contour of its electronic density, as suggested by Bader et al. [51]. This surface encompasses at least 96% of the electronic charge of the molecule, and reflects its specific features, such as lone pairs, n electrons, strained bonds, etc., which is not true of surfaces created by overlapping atomic spheres. 

For the particular case of longitudinal optical modes, we found in Eq. (9-27) the electrostatic electron-phonon interaction, which turns out to be the dominant interaction with these modes in polar crystals. Interaction with transverse optical modes is much weaker. There is also an electrostatic interaction with acoustic modes -both longitudinal and transverse which may be calculated in terms of the polarization generated through the piezoelectric effect. (The piezoelectric electron phonon interaction was first treated by Meijer and Polder, 1953, and subsequently, it was treated more completely by Harrison, 1956). Clearly this interaction potential is proportional to the strain that is due to the vibration, and it also contains a factor of l/k obtained by using the Poisson equation to go from polarizations to potentials. The piezoelectric contribution to the coupling tends to be dominated by other contributions to the electron -phonon interaction in semiconductors at ordinary temperatures but, as we shall see, these other contribu-... 

We shall make use of the effective Hamiltonian formalism [14] that enables us to isolate effects of interest from irrelevant complications. We divide the electronic Hamiltonian into a strong part H° and a weak part H, and we shall suppose that H° is simple enough to be solved exactly. The Hamiltonian including the cubic field and interelectronic repulsion only is the usual choice for H in the case of the 3d group ions. Then H should include all other interactions (spin-orbit coupling, lower symmetry fields, electron-phonon interaction, external fields, strain etc). The most important assumption is that the perturbations, described by the H Hamiltonian (in particular the JT interaction) must be smaller relative to the initial splitting due to H°. In the case of the 3d metal ions the assumption is usually well justified. 

The physics of CITE looks very simple and clear. Because of the vibronic (electron-vibrational) interaction each Jahn-Teller (JT) molecule (center) is characterized with several energetically equivalent minima corresponding to a possible distortion of the initial (at the absence of the vibronic interaction) symmetry. In case of many JT centers in a crystal matrix an effective interaction caused by lattice strains around the centers takes place. This interaction breaks the equivalence of the minima. The preference of the specific distortions around each of the JT centers leads to the ordering of the local distortions - structural phase transitions. As each distortion is related to a specific electronic state (orbital) the JT structural transition is at the same time an ordering of orbitals. The last is a central question of the modern orbital physics. 

In this Hamiltonian k is the wave vector of the phonons, y is the phonon mode branch, go and Vmk are the electron-strain and the electron-phonon interaction constants. It is important to remind that as it was noted for the first time by Kanamori [3], the electron interaction with the homogeneous strain U should be considered separately from the electron-phonon interaction as that type of strain can not be represented by phonons. The introduction of the last ones depends upon the Born-Karman conditions that are changing at the structural phase transition. 

Equation (102) shows that MAQO can provide important information about the electronic parameters (extremal Fermi surface cross-sectional area, effective masses, electronic relaxation times) and about the electron-phonon interaction (strain derivatives of the cross-sectional area for different symmetry strains). With the help of this technique, combined with de Haas-van Alphen susceptibility measurements, one can put the deformation potential interaction and the temperature dependence of the elastic constants, discussed above in sect. 3.2, on a solid basis. In the following we discuss some compounds. 

At low temperatures, when only the ground state of the lanthanide ion in the crystal field is populated, the total magnetic moment of the ion is the sum of the induced (Van Vleck) moment and the intrinsic moment (the latter differs from zero only in the degenerate state). The contributions to the magnetostriction and the elastic constants due to changes in the intrinsic magnetic moment of the lanthanide ion with lattice strain can be written explicitly when considering the effective spin Hamiltonian. The latter contains a smaller number of independent parameters (constants of spin-phonon interaction) than the Hamiltonian of the electron-deformation interaction (18) and is more suitable in the description of experimental data. 

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