Environmental Engineering Reference
In-Depth Information
temperature, but does not become soft, so Pr remains paramagnetic
down to very low temperatures. However, these calculations indicate
that R 0
0 . 92, so that the exchange is very close to the critical value
which would drive this incipient soft mode to zero energy. As we shall
discuss in Section 7.4.1, under these circumstances a variety of pertur-
bations may induce magnetic ordering.
A more elegant technique for obtaining such results is based on
a diagrammatic-expansion technique. The introduction of this method
requires a further development and refinement of the mathematical anal-
ysis of the Green functions, which falls outside the scope of this topic.
Nevertheless, we wish to discuss some essential problems connected with
the use of the technique for rare earth systems, so we will present it very
briefly and refer to the topics by Abrikosov et al. (1965), Doniach and
Sondheimer (1974), and Mahan (1990) for more detailed accounts.
Instead of the retarded Green function, introduced in eqn (3.3.12),
we consider the Green function defined as the τ -ordered ensemble aver-
age: G τ BA ( τ 1
.Here B ( τ ) is the equivalent of
the time-dependent operator in the Heisenberg picture, eqn (3.2.1), with
t replaced by
T τ B ( τ 1 ) A ( τ 2 )
τ 2 )
≡−
ihτ .The τ -ordering operator T τ orders subsequent oper-
ators in a sequence according to decreasing values of their τ -arguments,
i.e. T τ B ( τ 1 ) A ( τ 2 )= B ( τ 1 ) A ( τ 2 )if τ 1
τ 2 or A ( τ 2 ) B ( τ 1 ) otherwise. Re-
stricting ourselves to considering the Green function G τ BA ( τ ) only in the
interval 0
β ,where β =1 /k B T , we may represent it by a Fourier
series (corresponding to letting the function repeat itself with the period
β ):
τ
β
n
1
n = 2 πn
β
T τ B ( τ ) A
G τ BA ( τ )=
G τ BA ( n ) e −ihω n τ
=
;
.
(7 . 2 . 1 a )
n is an integer and the ω n are called the Matsubara frequencies .The
Fourier coecients are determined by
G τ BA ( n )= β
0
G τ BA ( τ ) e ihω n τ dτ.
(7 . 2 . 1 b )
The most important property of the τ -ordered Green function is that
it can be calculated by perturbation theory using the Feynman-Dyson
expansion. By dividing the Hamiltonian into two parts,
H 1 ,
and denoting the ensemble average with respect to the 'unperturbed'
Hamiltonian
H
H 0 +
=
H 0 by an index '0', it can be shown that
T τ U ( β, 0) B ( τ ) A (0)
0
G τ BA ( τ )=
,
(7 . 2 . 2 a )
U ( β, 0)
0
Search WWH ::




Custom Search