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22.903 Home Work Set No. 3 (Professor Sow-Hsin Chen) Spring Term 2005. Due May 2, 2005
1. There is a practically important case in light and x-ray scattering from isotropic fluids where the
coherent dynamic structure factor can be obtained rigorously. It is called the Rayleigh-Brillouin
scattering from fluids obeying linearized hydrodynamic equations. This is the case when radiation is
probing long wave length thermal density fluctuations in the fluids. Read J.P. Boon and S. Yip,
“Molecular Hydrodynamics”, McGraw Hill (1980), p.238-251, and derive the following results.
(A) The coherent intermediate scattering function for fluid of number density n is given by
F(k,t) ⎛ 1 ⎞ ⎡ 2 ⎛ κ ⎞ ⎤ 1
= ⎜1− ⎟ exp⎢−k ⎜
⎟t⎥ + exp −k 2Γt [cos(v skt ) + bk sin(v skt )]
S(k) ⎝ γ ⎠ ⎢⎣ ⎜⎝ nC p ⎟⎠ ⎥⎦ γ
(
)
(1)
where κ is the thermal conductivity, v s the adiabatic sound speed, η and ηv the shear and bulk
viscosities, and γ = Cp /C v , the ratio of specific heats at constant pressure and at constant volume,
⎛ κ ⎞ ⎛ η ⎞⎤
1⎡
4
Γ = ⎢(γ −1)⎜⎜
⎟⎟ + ⎜ L ⎟⎥ ; ηL = η+ ηv ,
2 ⎢⎣
3
⎝ nCp ⎠ ⎝ nm ⎠⎥⎦
⎛ γ ⎞1/2
⎛ ηL ⎞⎤
1 ⎡
b=
3Γ − ⎜ ⎟⎥ ; v s = ⎜
; χ T = isothermal compressibility.
⎟
⎢
⎝
⎠
nm ⎦
γv s ⎣
⎝ nmχ T ⎠
(B) The dynamic structure factor is
⎧
κ /nC p )k 2
(
S(k,ω) 1 ⎪⎛ 1 ⎞
= ⎨⎜1− ⎟
π ⎪⎝ γ ⎠ ω 2 + κ /nC k 2
S(k)
( p)
⎩
[
]
2
⎤⎫
⎡ 2
Γk 2 + bk (ω + v sk ) ⎥⎪
1 ⎢ Γk + bk (ω − v sk )
+
+
⎬
2γ ⎢ (ω − v k )2 + Γk 2 2 (ω + v k )2 + Γk 2 2 ⎥⎪
s
s
⎦⎭
⎣
( )
( )
Show from kinematic consideration that in order to measure Brillouin peaks of a fluid with adiabatic
sound speed vs, the incoming speed of neutrons must be larger than vs .
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2. Prove the following Sear’s expansion for the rotational intermediate scattering function of a water molecule:
∞
(
).
e−iQ⋅b(0) eiQ⋅b(t) = ∑ (2 +1)j2 (Qb) P bˆ (0) ⋅ bˆ (t)
=0
(2)
Where b is the vector from the center of mass of the molecule to the location of the hydrogen atom which is
scattering neutrons. Hint: start by expanding each exponential factor on the left-hand side of the above equation
by spherical harmonics and then use the addition theorem of the spherical harmonics to obtain the last factor in
this equation. This factor is called the rotational correlation function of the − th order.
3. In general, the self-intermediate scattering function or the intermediate scattering function of a tagged particle
in an inhomogeneous liquid (or confined liquid), is defined as:
Fs (Q,t) = ∫ ∫ e iQ⋅( r − r )G s (r ,t | r0 )p( r0 )d 3rd 3r0
0
where
G s ( r , t | r0 )
is the van-Hove space-time self-correlation function of the tagged particle and
(3)
p( r0 )
equilibrium distribution of the particle under the confinning potential V(r). Use this definition to show the
following four important properties.
(A) Conservation of probability:
Fs (Q = 0,t) = ∫ ∫ G s ( r ,t | r0 )p( r0 )d 3rd 3r0 = 1 .
(4)
(B) Initial condition:
G s ( r , t = 0 | r0 ) = δ( r − r0 )
(C)
leads to
Fs (Q, t = 0) = 1 .
(5))
Equilibrium distribution:
G s ( r , t = ∞ | r0 ) = p( r ) =
where Z is the normalization factor.
(D)Elastic Incoherent Structure Factor:
1 − V(r ) / k BT
e
Z
(6)
the
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Fs (Q,t = ∞) = EISF = ∫ p( r )e
iQ⋅ r 3
2
d r
(7)
In particular, use the last relation to calculate EISF for the following two cases.
Case One: When the tagged particle is confined inside a sphere of radius a,
⎡ 3j1 (Qa) ⎤2
EISF = ⎢
⎥ .
⎣ Qa ⎦
(8)
Case Two: When the tagged particle is confined in a harmonic potential well,
⎡ 1
⎤
ELSF = exp ⎢− Q 2 r 2 ⎥ ,
⎣ 3
⎦
where
r2
(9)
is the mean square vibrational amplitude of the particle in the well. Calculate this quantity in terms
of the parameters of the potential well.