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surface,
∂ε k σ
∂k u
2
(2 π ) 3 h 2
σ
e 2 τ
1
|∇ k ε k σ |
σ uu
dS,
(5 . 7 . 63)
ε k σ = ε F
where dS is a surface element of the Fermi surface. Even without de-
tailed calculations, this expression shows that the conductivity may be
reduced substantially if the superzone gaps are able to eliminate sig-
nificant areas of the Fermi surface. Furthermore, the Fermi-velocity
factor puts different weight on the various regions of the Fermi surface
in the different components of the conductivity tensor. If k s is parallel
to the c -axis, as in the heavy rare earths, and if its length is close to
that of the Fermi wave-vector in the c -direction, only the cc -component
of the conductivity is appreciably affected by the superzone boundary.
For instance, an internal field of 2 kOe in the basal plane of Ho at 4
K, which eliminates the superzone energy gaps by inducing a transi-
tion from the cone to ferromagnetic ordering, increases the conductivity
along the c -axis by about 30%, while decreasing the b -axis component
by only about 1% (Mackintosh and Spanel 1964). As illustrated in Fig.
5.15, the anomalous increase in the resistivity in the helical phase of
Tb is eliminated by a magnetic field which is large enough to suppress
this structure, leaving only a weak maximum similar to that observed
in Gd, which has been ascribed to critical scattering of the conduction
electrons by magnetic fluctuations (de Gennes and Friedel 1958). This
anomalous increase is not observed in the basal plane and the resistivity
is little affected by a magnetic field (Hegland et al. 1963).
The theoretical calculations of the superzone effects within the free-
electron model give a semi-quantitative account of the experimental ob-
servations, with a small number of adjustable parameters. For example,
a superzone boundary normal to the c -axis, which intersects the Fermi
surface, gives a positive contribution to ζ cc ( T ) in (5.7.61) which is pro-
portional to δ/ε F , while ζ bb ( T ) decreases like ( δ/ε F ) 2 . Bearing in mind
the analogy between the real and free-electron Fermi surfaces mentioned
above, this corresponds well with the observations in, for example, Ho.
In addition, the model calculations suggest that the superzone gaps are
important for the value of the ordering wave-vector Q ,atwhichthe
exchange energy has its maximum (Elliott and Wedgwood 1964; Miwa
1965), by predicting a gradual reduction of the length of Q with the
increase of the size of the superzone gaps, which are proportional to
( q )is
somewhat dependent on the magnetization, because the nearly elastic
intra-band contributions to the exchange interaction depend on the den-
sity of states near the Fermi surface, as is also true in the ferromagnetic
case, according to (5.7.21).
J z
below the Neel temperature. Hence the exchange coupling
J
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