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8.512 Theory of Solids II
Spring 2009
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8.512 Theory of Solids
Problem Set #1
1.)
(a)
Due: February 12, 2004
Prove the finite temperature version of the fluctuation dissipation theorem
χ ′′(q,ω ) =
and
1 − βω
(e −1)S(q,ω ) ,
2
S (q,ω ) = −2( n B (ω ) + 1) χ ′′(q,ω ) ,
€
r r
r
r
where S (q,ω ) = ∫ dx dt e−iq −x e iωx ρ( x,t ) ρ (0,
occupation
€ factor.
)T
−1
and n B (ω ) = (e βω −1) is the Bose
€
€
(b) Show that χ ′′(q,ω ) = −χ ′′(−q,−ω ) and S (−q,−ω ) = e− βω S (q,ω ) . In terms of the
scattering probability, show that this is consistent with detailed balance.
€
€
2.)
Neutron scattering by crystals.
We showed in class that the probability of neutron scattering with
r
r
r
2
momentum ki to k f is given by (2πb / M n ) S Q,ω where b is the scattering of
r r r
the nucleus Q = ki − k f and
( )
€
r
r
iωt
€ S Q,ω = ∫ dt e F Q,t
( )
€
( )
€
where
r r
r r
r
−iQ⋅ r (t )
F
Q,t =
∑
e j e iQ⋅ rl (0)
€
(
)
j,e
(1)
T
r
r
r r r
and rj (t) is the instantaneous nucleus position. Write rj = R j + u j , where R j
r
are the lattice sites,
€ and expand u j in terms of phonon modes
r
r
u j = ∑∑ λα
€
α
€
q
r
1
 i( qr ⋅ R j −ω q t )

aq e
+€
c.c.


2NMω q
€
€
(2)
r
where λα are the polarization vectors and α labels the transverse and
r r
longitudinal modes. Note that only Q ⋅ u j appear in Eq. (1). For simplicity,
r
assume the α modes are degenerate for each q so that we can always choose
r
r
r
€ one mode rwith λα parallel to Q . Henceforth we will drop the α label and λα
r
Qu j . Then
and treat Q ⋅ u j as scalar products
€
€
r r r
r
−iQ⋅ (
R j −R l )
€e
€ F
Q,t = ∑
€
F jl (t)
(
)
jl
€
where
€
F jl (t) = e
€
−iQu j (t ) iQul (0)
e
T
.
a) Show that
€
F jl (t) = e
−iQ ( u j (t )−ul (0))
e 2[
1
Qu j (t ),Qul (0)]
(3)
T
Furthermore, for harmonic oscillators, the first factor can be written as
€
e
−iQ ( u j (t )−ul (0))
T
=e
− 12 Q 2
( u j (t )−ul (0))
2
(4)
T
b) Using Eqs. (1–4) show that
€
F jl (t) = e
−2W
 Q2
1
exp
∑
(
2nq + 1) cosθ jl + isinθ jl
 2NM
q ω
q
(
where the Debye-Waller factor 2W is given by
€
2W =
(
and n q = 1 e
Q2
1
∑
(2nq + 1)
2NM q ω q
βω q
€
€
€
r
r r
−1 , θ jl = −ω q t + q ⋅ R j − Rl .
)
(
)

)

(5)
c) Expand the exp factor in Eq. (5) to lowest order and show that (V* is the
volume of reciprocal lattice unit cell)

r r
S (Q,ω ) = N V * e−2W 
∑δ Q − G

G

r r r
Q2
+
∑

( n q + 1)∑δ Q −
q − G δ (ω − ω q )
r 2N
M ω 
q
q
G
r r r

+ n q ∑ δ
Q + q − G
δ (ω + ω q )

G
(
)
(
(
)
)
d) Discuss the interpretation of various terms in Eq. (6) .
€
e) Even though we did not compute it explicitly, what experiment would
you propose to measure the polarization vector λα of a given mode at
energy ω q ?
€
€
(6)