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pinned to the Fermi surface, and occurs within a narrow interval with a
width corresponding to the spin-wave energies. This implies that, even
if the electronic energies
ε
k
σ
appearing in the magnon Green-functions
were replaced with
E
k
σ
, due to higher-order processes, this modification
would not be of much importance. The total electronic heat capacity
is
C
e
=
k
σ
E
k
σ
df
k
σ
/dT
, when the imaginary part of the self-energy is
neglected. The extra contribution due to the coupling to the spin waves
is linear only at the lowest temperatures (
k
B
T<
0
.
05
E
q
), after which it
increases more rapidly than linearly to its maximum at
k
B
T
0
.
15
E
q
.
Above
k
B
T
0
.
3
E
q
, this contribution becomes negative and finally dies
out when
k
B
T
E
q
. This variation with temperature was described
by Fulde and Jensen (1983), and has been discussed in the context of
the phonon interaction by Grimvall (1981). The bosons (magnons and
phonons) do not contribute directly to the linear term in the heat capac-
ity, which is thus a characteristic phenomenon of the Fermi gas. How-
ever, the departure from the linear variation when
k
B
T>
0
.
05
E
q
may
be influenced by the spin-wave contribution
≈
∞
e
−βhω
C
m
=
q
1
π
d
dT
1
2Γ
q
(
hω
)
3
(
hω
)
2
− E
q
(
T
)
2
+
2Γ
q
hω
2
d
(
hω
)
1
−
−∞
2
5
y
2
+
3
k
B
T
q
dT
n
q
+
2
+
π
2
E
q
(
T
)
d
2Γ
q
πE
q
4
7
y
4
,
+
···
y
=
βE
q
/
2
π
q
(5
.
7
.
52)
to first order in Γ
q
/E
q
. The first term is the RPA spin-wave contribu-
tion (5.3.3) derived before, which dominates strongly at elevated tem-
peratures. However, in the low-temperature limit, the second term is
of the same order of magnitude as the non-linear corrections to eqn
(5.7.51). For comparison, the last term in this equation is multiplied
by the factor
1+3
/
(5
y
2
)+5
/
(7
y
4
)+
···
when the higher-order tem-
perature effects are included. The additional contribution due to the
non-zero linewidth of the bosons is normally not considered in the lit-
erature. It may be added to the pure electronic contribution derived by
Fulde and Jensen (1983), by replacing
yL
(
y
)with2
yL
(
y
)+
L
(
y
)intheir
eqn (17
a
). The mass-enhancement effect increases proportionally to the
inverse of
E
q
(Γ
q
∝
A
q
). On the other hand, the interval in which the
linear variation occurs is diminished correspondingly, requiring a more
careful consideration of the higher-order modifications.
In the metals, the itinerant electrons also interact with the phonons,
and this leads to an entirely equivalent enhancement of their mass. This
effect has been calculated for the whole rare earth series by Skriver and
Mertig (1990), who find an increase of the band mass due to coupling to
the phonons of typically about 35% for the heavy elements. Assuming
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