176
7 Quadrupole Contributions from Interface and Bulk
α D2
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|q sr (ω 2 )|mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|q sr (ω 2 )|nn|μ p |mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
+
g|μ p |nn|μ q |mm|q sr (ω 2 )|g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ q |nn|μ p |mm|q sr (ω 2 )|g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
,
(7.80)
α
Q
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|q sp (()|nn|μ r |mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ r |nn|q sp (()|mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
+
g|q sp (()|nn|μ q |mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ q |nn|q sp (()|mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
.
(7.81)
We notice that Eqs. (7.76) and (7.79), (7.80), (7.81) have analogous forms except
for the operators μ and q in the matrix elements.
In these equations, the first two terms in the square bracket which include (ω 1 −
ω mg + ii mg ) in the denominator are vibrationally non-resonant terms, while the
latter two terms including (ω 2 − ω mg + ii mg ) are vibrationally resonant terms. For
example, α D1 in Eq. (7.79) is decomposed as α D1 = α D1,res + α D1,nonres , where
α
D1,res
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
+
g|μ p |nn|q sq (ω 1 )|mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|q sq (ω 1 )|nn|μ p |mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
,
(7.82)
α
D1,nonres
pqrs
((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|μ r |mm|q sq (ω 1 )|g
(ω 1 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ r |nn|μ p |mm|q sq (ω 1 )|g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
.
(7.83)
α D2 and α Q in Eqs. (7.80) and (7.81) are decomposed into the vibrationally resonant
and nonresonant terms in the analogous way.
Expressions for whole system Finally, we note that Eqs. (7.76), (7.77), (7.78),
(7.79), (7.80), (7.81), (7.82), (7.83) include the damping parameters , since these
7 Quadrupole Contributions from Interface and Bulk
α D2
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|q sr (ω 2 )|mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|q sr (ω 2 )|nn|μ p |mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
+
g|μ p |nn|μ q |mm|q sr (ω 2 )|g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ q |nn|μ p |mm|q sr (ω 2 )|g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
,
(7.80)
α
Q
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|q sp (()|nn|μ r |mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ r |nn|q sp (()|mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
+
g|q sp (()|nn|μ q |mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ q |nn|q sp (()|mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
.
(7.81)
We notice that Eqs. (7.76) and (7.79), (7.80), (7.81) have analogous forms except
for the operators μ and q in the matrix elements.
In these equations, the first two terms in the square bracket which include (ω 1 −
ω mg + ii mg ) in the denominator are vibrationally non-resonant terms, while the
latter two terms including (ω 2 − ω mg + ii mg ) are vibrationally resonant terms. For
example, α D1 in Eq. (7.79) is decomposed as α D1 = α D1,res + α D1,nonres , where
α
D1,res
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
+
g|μ p |nn|q sq (ω 1 )|mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|q sq (ω 1 )|nn|μ p |mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
,
(7.82)
α
D1,nonres
pqrs
((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|μ r |mm|q sq (ω 1 )|g
(ω 1 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ r |nn|μ p |mm|q sq (ω 1 )|g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
.
(7.83)
α D2 and α Q in Eqs. (7.80) and (7.81) are decomposed into the vibrationally resonant
and nonresonant terms in the analogous way.
Expressions for whole system Finally, we note that Eqs. (7.76), (7.77), (7.78),
(7.79), (7.80), (7.81), (7.82), (7.83) include the damping parameters , since these