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is [
Nν/
(
hβ
)]
2
, and we finally obtain the following expression for the
resistivity, or rather its upper limit:
k
F
↑
+
k
F
↓
dq
d
Ω
q
4
π
∞
3
(4
k
F ↑
k
F ↓
)
2
j
u
ρ
uu
(
T
)
ρ
0
d
(
hω
)
k
F
↑
−k
F
↓
−∞
π
α
2
q
·
u
2
q
βhω
sinh
2
(
βhω/
2)
1
χ
αα
(
q
,ω
)
,
×|
j
(
q
)
|
(5
.
7
.
57
a
)
where
h
N
↑
(
ε
F
)+
N
↓
(
ε
F
)
j
u
,
3
2
V
N
πm
he
2
ε
F
m
ne
2
π
ρ
0
=
j
u
=
(5
.
7
.
57
b
)
n
=
νN/V
is the electron density, and
2
k
F
dq
d
Ω
q
2
q
·
u
2
q.
3
(2
k
F
)
4
j
u
=4
4
π
|
j
(
q
)
|
(5
.
7
.
57
c
)
0
For cubic symmetry,
ρ
uu
is independent of
u
and
q
·
u
2
can be replaced
by
q
2
/
3. In the high-temperature limit, we have
∞
sinh
2
(
βhω/
2)
α
1
π
βhω
χ
αα
(
q
,ω
)
d
(
hω
)
−∞
∞
βhω
α
β
α
1
π
4
χ
αα
(
q
,ω
)=
4
χ
αα
(
q
,
0) = 4
J
(
J
+1)
,
d
(
hω
)
−∞
1
recalling that
χ
αα
(
q
,
0) =
3
βJ
(
J
+ 1) in this limit. This result shows
that the magnetic resistivity saturates at temperatures which are so high
that the ions are uniformly distributed over the states in the ground-
state
J
-multiplet, since the condition
k
B
T
ε
F
is always satisfied:
ρ
uu
(
T
)
→
J
(
J
+1)
ρ
0
for
T
→∞
,
(5
.
7
.
58)
and
J
(
J
+1)
ρ
0
is called the saturation value of the
spin-disorder
re-
1)
2
, the spin-disorder re-
sistivity is proportional to the de Gennes factor, as observed (Legvold
1972). If the crystal-field splitting of the energy levels is neglected, this
factor also determines the relative magnitudes of the contributions of
magnetic rare earth-impurities to the resistivity of a non-magnetic host
(Kasuya 1959). However, in analysing the measurements of Mackintosh
and Smidt (1962) of the resistivity changes produced by small amounts
of heavy rare earths in Lu, Hessel Andersen (1979) found that such
crystal-field effects are indeed important at 4 K.
sistivity.
Since
ρ
0
contains the factor (
g
−
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