Tensors for Physics

17.1 Time-Correlation Functions and Spectral Functions 355

SLor(ω)=π−^1

τ

1 +ω^2 τ^2

=π−^1

ν

ω^2 +ν^2

,ν=τ−^1. (17.10)

Theline widthis determined by the relaxation frequencyν=τ−^1.
The scattered intensity is proportional to

Iscat=e′μeνΔμν,λκe′λeκSLor(ω)=eμ′eνe′μeνSLor(ω)

=

1

2

(

1 +

1

3

(e′·e)^2

)

SLor(ω). (17.11)

The depolarized component, withe′·e=0, is^12 SLor(ω).
The time-correlation function and the spectral function are no longer isotropic, as
in (17.9) when external fields or an ordered structure render the system anisotropic.
An instructive example, as treated in [64], is considered next. Application of a mag-
netic field to a gas of rotating molecules causes a precessional motion of their rota-
tional angular momenta with the frequencyωB, cf. Sect.16.3.4. Ignoring the coupling
with the friction pressure tensor, the second of the (16.59) reduces to

∂

∂t

aμν−ωBHμν,μ′ν′aμ′ν′+νaμν= 0 ,

with the relaxation frequencyν=νa. With the help of the projection tensors intro-
duced in Chap. 14 in connection with the rotation of tensors, the solution of this
equation is written as

aμν(t)=Cμν,λκ(t)aλκ( 0 ), Cμν,λκ(t)=exp[−νt]

∑^2

m=− 2

exp[imωBt]Pμν,λκ(m).

(17.12)
Now the scattered intensity is proportional to

Iscat=e′μeνe′μeν

∑^2

m=− 2

WmSLor(ω+mωB),

e′μeνeμ′eνWm=

1

2

e′μeν

(

P(μν,λκm) +Pμν,λκ(−m)

)

eλ′eκ. (17.13)

According to the relations presented in Sect.14.5, the weight coefficientsWm, with
the property

∑ 2

m=− 2 Wm=1, are explicitly given by

W 0 = 3 (e·h)^2 (e′·h)^2 , W 1 =W− 1 =

1

2

[

(e·h)^2 +(e′·h)^2

]

− 2 (e·h)^2 (e′·h)^2 ,

W 2 =W− 2 =

1

2

−

1

2

W 0 −W 1. (17.14)