Methods of Evaluating θR
Part of the The International Cryogenics Monograph Series book series (INCMS)
It has been mentioned in Section 3.7 that the Grüneisen—Bloch relation (equation 3.17) can often usefully aid an analysis of experimental resistivity—temperature data of not only the simpler monovalent metals but of many polyvalent and transition metals as well. This is in spite of the fact that the underlying theory makes use of a Debye vibrational spectrum for the lattice, assumes perfectly free conduction electrons, and ignores the role played by Umklapp processes. All the same, the Grüneisen—Bloch relation is not so precise as is superficially apparent, for the reason that, since the resistances of most metals are linear in temperature above θ R /4 (the temperature range which interests most people), it is only below this temperature that the relation receives a test that is at all exacting. A close scrutiny in fact reveals that the Grüneisen—Bloch equation never holds over the entire small-angle scattering region (between θ/4 and θ/20, say), even including the alkali metals Na and K, as shown by Dugdale and Gugan.139 Other equations have been proposed that also lead to ρ i ∝ T 5 at low temperatures and to ρ i ∝ T at high temperatures, but the superiority of the Grüneisen—Bloch equation resides in its relatively simple form, which facilitates comparison of its predictions in individual cases with experiment. Moreover, by inverting the computation, one may intercompare the behavior of metals by treating the experimental results as deviations from the Grüneisen—Bloch equation, which then takes on the role of providing the “standard form.” This is done by employing θ R , the characteristic temperature defined in Section 3.7, as a variable parameter and computing the value that it must possess at any temperature in order that the Grüneisen—Bloch equation may agree with experiment. Only if θ R turns out to be constant will the equation give perfect account of the experimental results. The extent to which it achieves this for Li, Na, and K may be judged by studying Fig. 31.139
KeywordsTable VIII Bloch Equation Umklapp Process Large Angle Scat Ideal Resistivity
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