Environmental Engineering Reference
In-Depth Information
3
LINEAR RESPONSE THEORY
This chapter is devoted to a concise presentation of linear response the-
ory, which provides a general framework for analysing the dynamical
properties of a condensed-matter system close to thermal equilibrium.
The dynamical processes may either be spontaneous fluctuations, or
due to external perturbations, and these two kinds of phenomena are
interrelated. Accounts of linear response theory may be found in many
topics, for example, des Cloizeaux (1968), Marshall and Lovesey (1971),
and Lovesey (1986), but because of its importance in our treatment of
magnetic excitations in rare earth systems and their detection by inelas-
tic neutron scattering, the theory is presented below in adequate detail
to form a basis for our later discussion.
We begin by considering the dynamical or
generalized susceptibility
,
which determines the response of the system to a perturbation which
varies in space and time. The
Kramers-Kronig relation
between the
real and imaginary parts of this susceptibility is deduced. We derive
the
Kubo formula
for the
response function
and, through its connection
to the dynamic
correlation function
, which determines the results of a
scattering experiment, the
fluctuation-dissipation theorem
,whichrelates
the spontaneous fluctuations of the system to its response to an external
perturbation. The energy absorption by the perturbed system is deduced
from the susceptibility. The
Green function
is defined and its equation of
motion established. The theory is illustrated through its application to
the simple Heisenberg ferromagnet. We finally consider the calculation
of the susceptibility in the
random-phase approximation
,whichisthe
method generally used for the quantitative description of the magnetic
excitations in the rare earth metals in this topic.
3.1 The generalized susceptibility
A response function for a macroscopic system relates the change of an
ensemble-averaged physical observable
B
(
t
)
to an external force
f
(
t
).
For example,
B
(
t
) could be the angular momentum of an ion, or the mag-
netization, and
f
(
t
) a time-dependent applied magnetic field. As indi-
cated by its name, the applicability of linear response theory is restricted
to the regime where
B
(
t
)
changes linearly with the force. Hence we
suppose that
f
(
t
) is suciently weak to ensure that the response is lin-
ear. We further assume that the system is in thermal equilibrium
before
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