Environmental Engineering Reference
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can be neglected and, to leading order, =
E F
in the ω -derivative may
be replaced by ε k
,with k = k F ↑ . In the limit of zero temperature, the
free-electron model then gives
m m
N
q τ
=1+ 2 J
2 1
π
d ( )
|
j ( q +
τ
)
|
−∞
Im G m 2 ( q )
+∆+ ( h| q + τ | ) 2
2 m
1
× 2
µ 2 ,
h 2 k| q + τ |
m
1
subject to the conditions that k = k F ↑
and
| k q τ |
<k F ↓
.These
conditions imply that k F ↑
k F ↓
<
| q +
τ |
<k F ↑
+ k F ↓
, and that the lower
1ofthe µ -integral is replaced by ( h 2 q 2 +2 m ∆) / (2 h 2 k F ↑ | q +
bound
τ |
).
Because Im G m 2 ( q ) is odd in ω , the contribution due to the upper
bound in the µ -integral can be neglected (it is of the order F ).
Since
Im G m 2 ( q )
d ( )=Re G m 2 ( q , 0) =
A q
E q
1
π
,
−∞
the average mass-enhancement of the spin-up electrons at the Fermi
surface is
k F + k F
dq d q
4 π
m m
2 2 JA q
E q
=1+ N ( ε F )
2 k F ↑
q
|
j ( q )
|
,
(5 . 7 . 50)
k F ↓
k F −k F
and, by symmetry, m
/m is given by the same expression, except that
N ( ε F ) is replaced by
N ( ε F ). We note that the mass-enhancement only
depends on the static part of the susceptibility, i.e. G m 2 ( q , 0), and that
the magnitude of the mass-renormalization is intimately related to the
linewidth of the spin waves derived above in eqn (5.7.38). Utilizing this
connection, we can write the specific heat, in the zero-temperature limit,
k B T
N,
N
q
C = π 2
3
q
πE q
N ( ε F )+ 1
N ( ε F )+
(5 . 7 . 51)
where again the q -sum only extends over the primitive Brillouin zone.
With typical values of E q N
0 . 05, this expres-
sion predicts a doubling of the linear term in the heat capacity due to the
interaction between the conduction electrons and the spin waves, which
therefore has an appreciable effect on the effective mass of the electrons
near the Fermi surface. More detailed analyses (Nakajima 1967; Fulde
and Jensen 1983) show that the deformation of the electronic bands is
( ε F )
0 . 01 and 2Γ q /E q
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