Generalized susceptibility function

Additional sophistication in calculational procedures led to further refinements to the band structure of the rare earth metals as reviewed by Liu (1978). The improved detail gave better results for the calculated generalized susceptibility functions (see. 

Fig. 6. Generalized susceptibility functions /(f) for Y, Dy, and Gd showing the pronounced maxima in Dy and Y corresponding to the occurrence of incommensurate magnetic order. Alloys of Gd (see e.g., Wenger 1986) or other rare earths with Y show incommensurate order similar to Dy. The ripples in the Gd function are ascribed to matrix element instaUlities. (After Liu et al. 1971.)...

The importance of the generalized susceptibility function xi ) in the theory of magnetism in metals was discussed by many authors (see the review by Herring, 1966). Crudely speaking, xiq) is the approximate response function of the band electrons to a sinusoidally modulated magnetic field. The true response function contains the dipole matrix elements, but if they are approximated as constant or a simple function of q, the simple form of xi ) in eq. (3.69) follows. 

Real Fermi surfaces are never perfectly planar. But the one-dimensional model is far less restrictive than it may appear to be. For example, if two sheets of Fermi surface of arbitrary shape may be brought into point-by-point coincidence by a translation vector Q, the generalized susceptibility function will have the logarithmic singularity at Q. Such two sheets of Fermi surface are said to nest , and the vector Q need not be oriented normal to the sheets. In practice even this less restrictive form of nesting is not realized because the... 

Fig. 3.65. The generalized susceptibility functions of one-, two-, and three-dimensional free electron models (Kasuya, 1%6).

Fig. 3.67. The generalized susceptibility functions for the perfect nesting model and the imperfect nesting model.

Fig. 3.68. The generalized susceptibility functions for Gd and Dy for wave vectors along the c-axis of the crystal (Evenson and Liu, 1%8).

Fig. 3.73. The effect of magnetic ordering on the generalized susceptibility function of the nearly free electron -model (Elliott and Wedgwood, 1964).

Equation 1.92 fulfills all the conditions imposed on a generalized susceptibility, that is [36], it is a complex function of frequency, where... 

In linear response theory the optical activity is obtained from the part of the generalized susceptibility involving the temporal correlations of the electric and magnetic polarization fields46,47. For a system such as a normal fluid, described by a statistical operator that is invariant under space and time translations, the appropriate retarded Green function is,... 

The retarded Green function is directly related to the generalized susceptibility... 

Linear response theory, applied to the particle velocity, considered as a dynamic variable of the isolated particle-plus-bath system, allows to express the mobility in terms of the equilibrium velocity correlation function. Since the mobility p(co) is simply the generalized susceptibility %vx(o ), one has the Kubo formula... 

However, from the point of view of linear response theory, the definitions (174) or (178) suffer from several drawbacks. Actually, the function X ( , tw) as defined by Eq. (174) is not the Fourier transform of the function X (, x), but a partial Fourier transform computed in the restricted time interval 0 < x < tw. As a consequence, it does not possess the same analyticity properties as the generalized susceptibility x( ) defined by Eq. (179). While the latter, extended to complex values of co, is analytic in the upper complex half-plane (Smoo > 0), the function Xi ( - tw) is analytic in the whole complex plane. As a very simple example, consider the exponentially decreasing response function... 

We saw earlier that a very simple form of the dispersion energy is obtained from frequency-dependent polarizabilities at the so-called uncoupled Hartree-Fock level. The sum over states appearing in second order RS perturbation theory is simply a sum over (occupied and virtual) orbitals. A first improvement of this simple model is obtained by including apparent correlation [140], i.e. by using frequency-dependent polarizabilities obtained from the TDCHF method [36,141]. This method was initially proposed in the context of the multipole expansion, but could be generalized [142-146] to charge density susceptibility functions (or polarization propagators), which avoids the use... 

Dzyaloshinski, Lifshitz, and.Pitaevski (5) have quantified van der Waals interactions using the methods of quantum field electrodynamics, characterizing the potentials by summation over dielectric susceptibility functions over all frequencies. This approach generally is referred to as the Lifshitz theory. The van der Waals force at small separations (less than 5 nm), at zero temperature (K), and between two materials (designated by subscripts 1 and 2) immersed in a third medium (designated by subscript 3) can be written... 

In turn J q) is described in terms of a bare s-f exchange function j f between conduction and 4f electrons and the generalized susceptibility Xst which represents the conduction electron response to the local exchange field and which is calculable from the band structure. The expression for J(q) for N spins is then... 

Fig. 12. Normalized exchange functions [J(0) - J( )]/(gj — 1) in the c-direction for the ferromagnetic phases of Er, Tb, Dy and Gd. Apart from differences in the generalized susceptibility, exchange and wave functions these curves would be identical for all rare earths. (After Lindgard 1978.)...

Bi = coefficients of operator equivalents in crystal field hamiltonian S = magnetoelastic coupling constant Bj(x) = Brillouin function B(q) = function entering spin-wave energy [see eq. (7.100)] e = (kT)- p = gmeHIkT c = c-axis lattice constant c = reciprical lattice constant C = basal plane elastic constant C] , C,-, = creation and annihilation operators for state m> on site 1 X = bulk susceptibility X, = atomic susceptibility x(d) - generalized wave number-dependent susceptibility function of conduction electrons... 

Peroxide Decomposers as Processing Stabiiizers. Alkyl and aryl phosphite esters (see Scheme 12) are effective melt stabilizers. They are often used in combination with hindered phenols (see Table 6). Phosphites generally function by a stoichiometric peroxidolytic mechanism (PD-S) Table 7 illustrates the benefits of using the commercial phosphites TNPP (AO 15) and Irgafos P-EPQ (AO 17, Table 3) for melt stabilization (95). The unique phosphite AO 37 and its phosphate transformation products AOs 38 and 39, (Table 7), which were shown (104,133-135) to operate by a catalytic mechanism (PD-C), are particularly effective at low concentration of the parent stabilizer molecule (see AO 37, Table 7) (95). Phosphites are, however, generally susceptible to hydrolysis. For example, hydrolysis of aryl phosphites leads to the formation of low molecular mass phenol and a... 

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