Environmental Engineering Reference
In-Depth Information
described in Section 1.3, and reproduces a number of observed features.
In the calculations of Jensen (1982a), compared with the experimen-
tal results in Fig. 5.13, the spin-splitting of this surface gives rise to the
critical value
q
0
at which the linewidths abruptly increase. The finite life-
times below this cut-off are due to interband transitions between states
on sections of the Fermi surface with opposite spin, which intersect in
the primitive Brillouin zone after translation through a reciprocal-lattice
vector. These effects will also occur in calculations based on a realistic
band structure, whereas the behaviour at higher
q
is much more depen-
dent on the details of the energy bands.
5.7.2 The mass-enhancement of the conduction electrons
The processes in which the spin waves are scattered by the electron-
hole pair excitations of the conduction electrons, and which therefore
limit their lifetime, also have consequences for the conduction electrons.
The energies of the conduction electrons are changed, and hence also
their effective mass at the Fermi surface
m
∗
, as measured directly by
cyclotron resonance or the de Haas-van Alphen effect, or as determined
from the low-temperature heat capacity. In the zero-temperature limit,
the electronic part of the specific heat is
C
=
γT
=
m
∗
m
π
2
k
B
N
↑
(
ε
F
)
N,
(5
.
7
.
39)
γ
0
=
3
γ
0
T
;
ε
F
)+
N
↓
(
where
m
∗
=(
m
↑
+
m
↓
ε
F
instead of
ε
F
is meant to indicate that all the effects of the MF Hamil-
tonian, including the interband couplings in (5.7.7), are assumed to be
incorporated in
γ
0
or
m
.
In order to calculate
m
∗
, we shall utilize the Green functions of
the conduction electrons. Because these particles are fermions, it is
convenient to introduce an alternative type of Green function, in which
an anticommutator bracket replaces the commutator bracket occurring
in the definition (3.3.12), so that, for instance,
)
/
2 in the spin-polarized case. The use of
i
h
θ
(
t
t
)
c
k
↑
(
t
);
c
k
↑
(
t
)
t
)
c
k
↑
(
t
)
,c
k
↑
(
t
)
.
(5
.
7
.
40)
The Fourier transform obeys an equation of motion equivalent to eqn
(3
.
3
.
14
a
), except that the commutator on the
right
-hand side of this
equation is replaced by the anticommutator, or
hωG
↑
(
k
,ω
)
G
↑
(
k
,t
−
≡
+
=
−
−
{
}
];
c
k
↑
+
=
,c
k
↑
}
−
[
c
k
↑
,
H
{
c
k
↑
=1
.
(5
.
7
.
41)
is approximated by
H
s
, given by eqn (5.7.10), we obtain the non-
interacting value of the Green function
If
H
1
G
o
↑
G
↑
(
k
,ω
)
(
k
,ω
)=
(5
.
7
.
42)
hω
−
ε
k
↑
Search WWH ::
Custom Search