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Fig. 5.15. The c -axis resistivity of Tb in the vicinity of T N = 230 K,
after Hegland et al. (1963). As the helical ordering develops, the magnetic
superzones cause a sharp increase in the resistivity, which disappears at
T C = 220 K. The superzones may also be eliminated by a magnetic field
in the b -direction, which suppresses the helical structure.
The agreement obtained between simple model calculations of the
variation of Q and that observed experimentally is surprisingly good,
to some extent fortuitously so. The band electrons are far from free-
electron-like in the rare earth metals, and the approximation in which
I ( n k ,n k ) is replaced by j ( k k +
) is rather crude. The effective free-
electron model, with j ( q ) proportional to a form factor 1+( Aq ) 2 1
where A
τ
0 . 2 ˚ Aand2 k F
2 . 8 A 1 , leads to a maximum in
J
( q )
0 . 3 A 1
at q
parallel to the c -axis, in the paramagnetic phase. In
N q J
1
this model,
( q ) is found to be an order of magnitude larger
than
J
( 0 ), and the same is the case with the interband contributions
(
= 0 ) to the exchange interaction, compared to the intra-band con-
tributions. However, various estimates indicate that all these terms are
of the same order of magnitude. Lindg ard et al. (1975) have made the
only existing ab initio calculation of
τ
( q ) in a rare earth metal, consid-
ering the simplest case of Gd, and they obtained a reasonable account
of the dependence on wave-vector, even though the magnitude differed
by as much as a factor of four from that determined experimentally.
Their calculations show that the exchange integral is dominated by the
J
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