LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 146

where the parameters, including the energy difference, usually depend
on ω . According to the general stability and causality requirements,
the poles of χ BA ( z )at z = z α α =( E α − i Γ α α must lie in the

lower half-plane, implying that Γ α α has to be positive (or zero). In the
case where

E α ) −

Γ α α ,the ω -dependence of these parameters
is unimportant, and the δ -function in (3.3.5) is effectively replaced by a
Lorentzian :

| E α − E α |

B A

α >< α |

<α | | | α>

χ BA ( ω )

Γ α α ( n α − n α )

( E α − E α − hω ) 2 +Γ α α

αα

(3 . 3 . 10)

hω Γ 0
( hω ) 2 +Γ 0

χ BA ( el ) , +

with a linewidth , or more precisely FWHM (full width at half maximum),
of 2Γ α α . In (3.3.10), we have added the quasi-elastic response due to a
pole at z = i Γ 0 , which replaces the one at z = 0. The corresponding

reactive part of the susceptibility is

−

B A

α >< α | <α | | | α>

χ BA ( ω )

( E α − E α − hω )( n α − n α )

( E α − E α − hω ) 2 +Γ α α

αα

Γ 0
( hω ) 2 +Γ 0

χ BA ( el ) . + (3 . 3 . 11)

The non-zero linewidth corresponds to an exponential decay of the oscil-
lations in the time dependence of, for instance, the correlation function:

e −iz α α t/h = e −i ( E α −E α ) t/h e − Γ α α t/h . S BA ( t ) ∼

The absorption observed in a resonance experiment is proportional
to χ AA ( ω ). A peak in the absorption spectrum is interpreted as an ele-
mentary or quasi-particle excitation ,orasa normal mode of the dynamic
variable A ,witha lifetime τ = h/ Γ α α .Apoleat z =

i Γ 0 is said to
represent a diffusive mode . Such a pole is of particular importance for
those transport coecients determined by the low-frequency or hydro-
dynamic properties of the system. Kubo (1957, 1966) gives a detailed
discussion of this subject. As we shall see later, the differential scatter-
ing cross-section of, for example, neutrons in the Born-approximation is
proportional to a correlation function, and hence to χ ( ω ). This implies
that the presence of elementary excitations in the system leads to peaks
in the intensity of scattered neutrons as a function of the energy transfer.
Finally, the dynamic correlation-functions are related directly to various
thermodynamic second-derivatives, such as the compressibility and the

−