Environmental Engineering Reference
In-Depth Information
The above calculation has been performed for a Bravais lattice, but
the result (5.7.26) is readily generalized to a crystal with a basis of
p
ions, as the conduction electrons, in the approximation adopted, are
not affected by the presence of the basis. If the couplings between the
different sublattices are introduced in an equivalent manner to (5.1.1),
then
p
τ
,ω
)exp
i
·
τ
(5
.
7
.
29)
J
ss
(
q
,ω
)=
2
2
χ
+
−
|
j
(
q
+
τ
)
|
c
.
el
.
(
q
+
τ
ρ
ss
replaces (5
.
7
.
26
c
), where
ρ
ss
is the vector connecting the two sublattices
s
and
s
.
The interaction between the localized moments is effectuated via
virtual electron-hole pair-excitations of the conduction electrons. The
transmission of any time-dependent event may be disturbed in two ways;
either by the finite propagation-time of the pairs, or by a decay of the
pair states into unbound electron and hole excitations, the so-called
Stoner
excitations. The second effect produces by far the most impor-
tant correction to the instantaneous interaction, but we shall begin with
a discussion of the frequency-dependence of the real part of
J
(
q
,ω
), due
to the finite transmission time. Returning to the simple model leading
to (5.7.26), we find that the exchange coupling is proportional to the
susceptibility function
χ
+
−
c
.
el
.
(
q
,ω
), which for unpolarized free electrons
is the same as the
Lindhard function
(Lindhard 1954). If corrections of
the order
k
B
T/ε
F
are neglected, the real part at zero wave-vector is
N
k
Re
χ
+
−
c
.
el
.
(
0
,ω
)
=
f
k
↓
−
f
k
↑
1
hω
−
ε
k
↓
+
ε
k
↑
(5
.
7
.
30)
F
)
1+
hω
∆
.
N
k
f
k
↑
−
f
k
↓
1
=
=
N
(
∆
−
hω
From this result, we find immediately that the intra-band contribution
at zero frequency to
J
(
q
→
0
,
0) in eqn (5
.
7
.
26
a
)is2
j
2
(
0
)
F
), which
is the same as in (5.7.21). On the other hand, the interband contri-
butions differ in the two expressions, as the denominator in (5
.
7
.
26
a
)
involves the exchange splitting ∆, whereas that in (5.7.21) does not.
However, this difference can be neglected, as it is of the order (∆
/ε
F
)
2
times the intra-band contribution, which is beyond the order considered
in these calculations. In fact, since the starting Hamiltonian (5.7.6) is
invariant with respect to the choice of
z
-axis for the electronic spins
and the angular momenta, the spin-wave frequency must vanish when
q
→
0
and
A
=
B
= 0, according to the
Goldstone theorem
, which will
be discussed in the next chapter. Therefore
J
N
(
(
q
→
0
,
0) =
J
(
0
,
0),
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