Griineisen electronic

Here k = is the compressibility, the averages are given by eq. (34) and y = -(din E /d In V) define electronic Griineisen parameters for a given CEF level. The close correspondence between )8 and the Schottky specific heat has been emphasized before (Ott et al. 1976, 1977, Liithi 1980a). The latter is described by a corresponding formula 1... 

We first begin with a short survey of the basic experimental evidence in sect. 4.1. The basic physical properties of the unstable-moment systems will be surveyed. In order to discuss their elastic and lattice dynamical properties that are of central interest here an outline of current theoretical concepts to describe unstable-moment systems, especially heavy-fermion compounds is given first (sect. 4.2). Because in heavy-electron compounds is much smaller than the CEF splittings they exhibit magnetoelastic anomalies as described in sect. 2. However, in addition, Griineisen parameter coupling to the heavy-electron bands leads to elastic effects below T. For a few typical Ce compounds this is studied within a microscopic model calculation in sect. 4.3. A more complete discussion... 

Microscopically, an important contribution to the Griineisen parameter coupling in HF Ce compounds comes from a strong volume dependence of the sf-hybridization strength V k) (Allen and Martin 1982, Razafimandimby et al. 1984). Therefore, sound waves of compressional character, e.g., longitudinal modes modulate the mixing term in the Hamiltonian of eq. (107) and consequently couple to electronic states. For smaU displacements one can write... 

The phenomenological Griineisen parameter electron-phonon coupling for the heavy-fermion compounds is very successful in describing the thermal expansion and temperature dependence of elastic constants. The parameters deduced from the different experiments agree with each other and they agree also with the parameters used in the microscopic description, sect. 4.3. 

On the basis of this expression, authors have determined the value ]r(T,F) = 0.57 for the electron Griineisen parameter, which is near to the theoretical value of 2/1. 

As stated previously, the Mattheissen formulation for resistivity holds only for temperatures greater than about 20% of the Debye temperature because all of the vibrational modes that can scatter electrons are not active at low temperatures. Griineisen used the Debye model to formulate an exact theory of the temperature dependence of resistivity for metals that extends the temperature dependence of resistivity to low temperatures. 

The Bloch-Griineisen formula is a special case of a more general expression. Within a variational solution of the Boltzmann equation we can, to a good approximation, write the resistivity Pei-ph that is limited by the scattering of conduction electrons by phonons as... 

Measurements of p T) for TiB2 by Williams et al. (9) from 4.2 to 300 K showed a resistivity minimum in the range 34-47 K. A normal resistivity term Pei.ph(r) could be extracted. It was well described by a Bloch-Griineisen expression varying as at low T and with Or = 720 K. The latter value seems reasonable, when compared with 0s see Table 1. The Kondo effect is caused by a spin-flip interaction between conduction electrons and localized magnetic moments of impurity atoms. Ni impurities are essential, but there are still several unsolved problems in the interpretation of the Kondo minimum in TiB2. 

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