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Temperature Dependence of Magnetic Characteristics

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Dynamic Spin-Fluctuation Theory of Metallic Magnetism

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Abstract

In this chapter we study spin-correlation effects in metals at finite temperatures. We start with qualitative estimates of the correlation effects on the magnitude and relaxation time of a single-site spin.

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Notes

  1. 1.

    This is simply the sum of squares of electronic spins in different orbital states, since in these conditions the mean value of the scalar product of spins of different electrons is zero. The same argument applies in the derivation of (13.11).

  2. 2.

    These results are valid only in the case of non-interacting electrons.

  3. 3.

    If detailed calculations of the generalized susceptibility χ(q, ω) were available, the Fourier-transform of correlator (13.21) could be obtained by only triple (rather than sixfold) integration over the Brillouin zone. Indeed, by the fluctuation-dissipation theorem (2.56), we have

    $$\displaystyle{ F(\omega ) \propto \coth (\beta \hslash \omega /2)\int \mathop{\mathrm{Im}}\chi (\mathbf{q},\omega )\,\mathrm{d}\mathbf{q}. }$$

    However, calculations of χ(q, ω) are usually given only in a few symmetrical directions of q (see, e.g. [7]), and so this formula cannot be used.

  4. 4.

    The function F(ω) is even.

  5. 5.

    Since the calculations with νkσ-dependent matrix element and with constant matrix element yield similar results, for brevity, we give numerical results only for the former case.

  6. 6.

    A similar expression can be obtained from the formula given in the footnote 3 using the Kramers-Kronig relation for ℏω ≪ k BT.

  7. 7.

    Use of the constant matrix element approximation is the same as setting the form-factor (3.28) to unity.

  8. 8.

    In the DSFT we use g 2μ B 2∕2 units. If we come back to g 2μ B 2 units, one can see that

    $$\displaystyle{\chi = \frac{\chi ^{0}} {1 - 2u\chi ^{0}},}$$

    and the Stoner constant I must be compared with u (see (4.32)).

  9. 9.

    Compare the results obtained in the DNA with those obtained in the SLA, where the intersite correlations are not taken into account at all.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    Nikolai B. Melnikov

  2. Institute of Metal Physics, Russian Academy of Sciences, Ekaterinburg, Russia

    Boris I. Reser

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Melnikov, N.B., Reser, B.I. (2018). Temperature Dependence of Magnetic Characteristics. In: Dynamic Spin-Fluctuation Theory of Metallic Magnetism. Springer, Cham. https://doi.org/10.1007/978-3-319-92974-3_13

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