Self-consistent relaxation time approximation

The Self-Consistent Relaxation Time Approximation (SCRTA)... 

However, upgrading of the theory was not stopped. After the first attempt the MPK2 [247] and MPK3 [126] have arisen whose kernels are different. As can be seen from Table V, the kernel for MPK2 is actually defined as a solution of the transcendent equation. Another one, for MPK3, appears to be identical to that provided by MET. More recently a new theory named the self-consistent relaxation time approximation (SCRTA), was developed by Gopich and Szabo [180]. In the irreversible case the inverse relaxation time of SCRTA, kfc, is defined by the transcendent equation for kf shown in Table V. It is equivalent to the one expressed by the Laplace transformation of k(t) ck(kf) = 1 [177]. In spite of this complication, SCRTA is competitive in the concentration corrections with other theories. 

Gopich, I.V., Szabo, A. Kinetics of reversible diffusion influenced reactions The self-consistent relaxation time approximation. J. Ghem. Phys. 2002,117, 507. 

To make the discussion more quantitative, let us now consider the doping dependence of Rh(S,T) in terms of the f-f -/model using the saddle-point and relaxation time approximations, where FS and correlation effects are involved via the renormalized SB band (Eq. 33). As we have pointed out above, in our approach the SB quasi-particle band dispersion has to be determined in a self-consistent way at each doping level 5. This should be in contrast to the NZA SB mean-field approach to the t-f -/model of Chi and Nagi [61] where, in the 7 -> 0 limit, the calculation of transport properties is based on the simple replacement Sj — Sj = -2 (5[(cos x + cos ) + 2(r7r)cos cosAy] ofthe non-interacting band dispersion (Eq. 2). 

The relaxation of the structure in the KMC-DR method was done using an approach based on the density functional theory and linear combination of atomic orbitals implemented in the Siesta code [97]. The minimum basis set of localized numerical orbitals of Sankey type [98] was used for all atoms except silicon atoms near the interface, for which polarization functions were added to improve the description of the SiOx layer. The core electrons were replaced with norm-conserving Troullier-Martins pseudopotentials [99] (Zr atoms also include 4p electrons in the valence shell). Calculations were done in the local density approximation (LDA) of DFT. The grid in the real space for the calculation of matrix elements has an equivalent cutoff energy of 60 Ry. The standard diagonalization scheme with Pulay mixing was used to get a self-consistent solution. In the framework of the KMC-DR method, it is not necessary to perform an accurate optimization of the structure, since structure relaxation is performed many times. 

Pokrovskii VN, Volkov VS (1978b) The calculation of relaxation time and dynamical modulus of linear polymers in one-molecular approximation with self-consistency. (A new approach to the theory of viscoelasticity of linear polymers). Polym Sci USSR 20 3029-3037... 

It should also be noted that this section represents a novel application of irreversible thermodynamics to systems with long-range forces. The local field has been dealt with self-consistently. In the macroscopic theory of Section 13.7 local electroneutrality was imposed through Eq. (13.7.10), whereas in the fluctuation theory there is no constraint of electroneutrality. However because we applied Eq. (13.8.3b) we see that deviations from local electroneutrality decay on the time scale rf1. This is the ionic relaxation time. In Section 9.4 only an approximate theory was presented. 

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