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5
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84926805316
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In silicon, ``low'' density is roughly below 3 times 1019 electrons/cm3_ and ``high'' temperature is above about 77 K.
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6
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84926848910
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As noted in Ref. 1, macroscopically, τe is unique only to within an arbitrary tensor which in indicial notation takes the form M[ki]j,k where the bracketed indices indicate antisymmetry. Interestingly, the microscopic derivation from quantum mechanics appears to select this tensor to be zero when τe is written as in (2.3a), i.e., (2.6) is of the same form as (3.11) only in this case.
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7
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84926826621
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As is well known, the lowest-order density dependence is that for which the gas pressure is linear in density (ideal gas). Similarly, for the density-gradient dependence the lowest-order form is obtained when the double-pressure (Ref. 1) is proportional to the density gradient.
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8
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84926805315
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It should be noted that the macroscopic equations remain valid (given satisfaction of continuum assumptions) no matter how the gradient dependence arises and therefore can have wider applicability. An extreme example is that macroscopic equations similar to those presented in Sec. II apply also to completely classical situations of kinetic theory in which gradient effects result from two-body correlations. In that context the equations are known as Burnett equations, see S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, 3rd ed. (Cambridge, New York, 1970), Chap. 15.
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11
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84926870744
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Solid State Theory (Dover, Toronto 1980).
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Harrison, W.1
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13
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84926848909
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It may be noted that this demonstration can also be reached directly from (3.9), if we identify -U/q with the electric potential cphi and use the fact that, when the system is in macroscopic equilibrium, cphi = - cphiestar apart from an additive constant. Equation (3.9) then becomes a relationship between cphie and ρe and by a reversion of series an expression of the same form as (2.7a) is obtained. The derivation given in the text is preferable in that it requires no a priori identification of -U/q with cphi.
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14
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84926848908
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In contrast, the lowest-order terms of density-gradient expansions used previously in classical theory (Ref. 8) and in density functional theory [e.g., see J. Callaway and N. H. March, in Solid State Physics, edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, New York, 1984), Vol. 38, p. 186] accurately approximate the full expansions over very narrow ranges only and therefore are of limited usefulness.
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