Neutron Scattering
1
Cross
Section
in 7 easy steps
1. Scattering Probability (TDPT)
2. Adiabatic Switching of Potential
3. Scattering matrix (integral over time)
4. Transition matrix (correlation of events)
5. Density of states
6. Incoming flux
7. Thermal average
2
1. Scattering Probability
• Probability of final scattered state, when evolving under scattering interaction
Pscatt = | hf | UI (t) |ii |
2
• Time-dependent perturbation theory
(Dyson expansion)
3
2.Adiabatic Switching
• Slow switching of potential ➞ time ∈ [-∞,∞]
Particle
Scattering Medium
V
t
• V is approximately constant
4
3. Scattering Matrix
• Propagator for time t=-∞ → t= ∞ is called
the scattering matrix
| hf | UI (ti = -1, tf = 1) |ii |2 = | hf | S |ii |2
• S is expanded in series:
hf | S (1) |ii =
hf | S (2) |ii =
iVf i
hf |
Z
1
ei!f i t dt =
1
X
m
V |mi hm| V
5
2⇡i (!f
! Z
|ii
1
1
!i ) V f i
dt1 ei!f m t1
Z
t1
1
dt2 ei!mi t2
3. Scattering Matrix
• Be careful with integration (ɛ ...)
• First and second order simplify to
hf | S (1) |ii = -2⇡i6(!f - !i ) hf | V |ii
X hf | V |mi hm| V |ii
hf | S (2) |ii = -2⇡i8(!f - !i )
!i - !m
m
6
4.Transition Matrix
• The scattering matrix is given by the
transition matrix
hf | S |ii = -2⇡i8(!f - !i ) hf | T |ii
• which has the following expansion
X hf | V |mi hm| V |ii X
Vf m Vmn Vni
hf | T |ii = hf | V |ii +
+
+ ...
(!i - !m )(!i - !n )
!i - !m
m
m,n
7
4.Transition Matrix
• Scattering probability
Ps = 4⇡ 2 | hf | T |ii |2 5 2 (!f - !i ) = 2⇡t| hf | T |ii |2 5(!f - !i )
(not well defined because of t→∞)
• Scattering rate
2
WS = 2⇡| hf | T |ii | 5(!f - !i )
8
4.Transition Matrix
• Target is left in one (of many possible) state.
• Radiation is left in a continuum state
• Separate the two subsystems
(no entanglement prior and after the
scattering event)
and rewrite the transition matrix
9
4.Transition Matrix
• Target: |m i , ✏
|ki , !
Radiation:
•
• Scattering rate
k
k
k
2
Wf i = 2⇡| hf | T |ii | (Ef
go back to definition, using explicit states
Wf i = | hmf , kf | T |mi , ki i |
10
2
Z
1
1
e
i(!f +✏f
!i ✏i )t
Ei )
4.Transition Matrix
• Work in Schrodinger pict. for radiation and
Interaction pict. for target:
hmf , kf | TIt Ir |mi , ki i = hmf , kf | TSt Sr |mi , ki i ei(!f -!i )t ei(✏f -✏i )t
= hmf , kf | TIt Sr (t) |mi , ki i e
i(!f -!i )t
= hmf | Tkf ,ki (t) |mi i e
rate is then a correlation
• Scattering
Z
Wf i
1
= 2
~
1
-1
e
i(!f -!i )t
i(!f -!i )t
†
hmi | Tkf ,ki (0) |mf i hmf | Tkf ,ki (t) |mi i
NOTE:Time evolution of target only
(e.g. lattice nuclei vibration)
11
5. Density of states
•
• Plane wave in a cubic cavity
# of states
P
k n(Ek ) ⇡
2⇡
kx =
n x ! d3 n =
L
3
✓
⇢(E)dEd⌦ = ⇢(k)d k =
L
2⇡
✓
R
◆3
L
2⇡
12
d3 n(E)
ny
dn
n
d3 k
◆3
nx
k 2 dkd⌦
5. Density of states
•
dk
k = E/~c !
= 1/~c
Photons,
dE
✓ ◆3 2
✓ ◆3 2
E
!k
L
L
⇢(E) = 2
=2
3
3
2⇡
~c3
2⇡
~ c
• Massive particles,
⇢(E) =
✓
L
2⇡
◆3
~2 k 2
E=
2m
k
=
2
~
13
✓
L
2⇡
◆3 p
2mE
~3
6. Incoming Flux
• # scatterer per unit area and time
#
v
2
= 3 , since t = L/v, A = L
<=
At
L
• Photons,
< = c/L3
• Massive particles,
~k
=
mL3
14
L
7.Thermal Average
target
• Average over initial state of X
X
WS (i ! ⌦ + d⌦, E + dE) = ⇢(E)
Pi
i
f
†
Tif (0)Tf i (t)
E
Wf i
• Scattering Cross Section
2
d
1
= 2
d ⌦dE
~
Z
X
f
1
i!f i t
dte
1
15
D
⇢(E)
th
Neutrons
• Using
and ⇢(E) for massive particles,
the scattering cross section is:
2
<inc
1
d
=
d ⌦d!
2⇡
✓
3
mL
2⇡~2
◆2
kf
ki
16
Z
1
-1
e
i!f i t
D
†
Tif (0)Tf i (t)
E
Neutrons
• Evaluate T for neutrons, with states
•
|ki,f i !
we obtain Tk
hkf | T (t, r) |ki i =
f ki
k (r)
(t, Q)
Z
L3
1
= 3
L
with
/L
3/2
Q = kf
ki
⇤
kf (r)T (r, t) ki (r)
d3 r
Z
=e
ik·r
3
d re
L3
17
iQ·r
T (r, t)
NeutronTransition Matrix
• We still need to take the expectation value
with respect to the target states,
Tf i
1
= 3
L
Z
L3
d3 r eiQ·r hmf | T (r, t) |mi i
18
Fermi Potential
• T is an expansion of the interaction
potential, here the nuclear potential
• analyze at least first order...
19
Fermi Potential
• Nuclear potential is very strong (V ~30MeV)
• And short range (r ~ 2fm)
• Not good for perturbation theory!
• Fermi approximation
• What is important is the product
0
0
a/
3
V0 r0
(a = scattering length) if
20
kr0 ⌧ 1
Fermi Potential
neutron wavefunction
• Replace nuclear
potential with
weak, long range
pseudo-potential
• Still, short range compared to wavelength
• Delta-function potential!
2⇡~2
V (r) =
a (r)
µ
21
Scattering Length
• Free scattering length a,
2⇡~2
2⇡~2
V (r) =
a (r) !
b (r)
µ
mn
• bound scattering length b (include info about
isotope and spin)
A+1
mn
b=
a⇡
a
µ
A
M mn
(reduced mass: µ = M + m
n
22
)
Transition Matrix
• To first order, the transition matrix is just
the potential
Tf i
1
= 3
L
Z
L3
d3 r eiQ·r hmf | V (r, t) |mi i
• Using the nuclear potential for a nucleus at
a position R, we have
Tf i
1
= 3
L
Z
2
2
2⇡~
2⇡~
3
iQ·r
iQ·R(t)
d re
b(R) (R)T (r, t) =
b(R)e
mn
mn
L3
23
Transition Matrix
• To first order, for many scatter at position r
i
2⇡~2 X iQ·ri (t)
Tf i (t) =
bi e
mn i
cross section becomes
• The scattering
Z
2
d
1 kf
=
d ⌦d!
2⇡ ki
1
1
ei!f i t
X
`,j
24
D
b` bj e
iQ·r` (0) iQ·rj (t)
e
E
th
Scattering Lengths
• The bound scattering length depends on
isotope and spin
• We need to take the average b b ! b b
- j = `, b b = b
j
6
=
`,
b
b
=
b
• Finally, b b = b + b (1 )
• Coherent/Incoherent scattering length:
` j
2
` j
2
` j
` j
b ` bj =
(b2
2
2
j,`
2
b )
j,`
2
j,`
25
+b =
2
bi
+
2
bc
` j
Scattering lengths
• Coherent scattering length
bc = b
• Correlations in scattering events from
the same target
(scale-length over which the incoming
radiation is coherent in a QM sense)
• Simple average over isotopes and spins
26
Scattering Lengths
• Incoherent scattering length
2
bi
=
(b2
2
- b )cj,`
• Correlation of scattering events between
different targets
• Variance of the scattering length over
spin states and isotopes
27
Cross-section
2
over
the
scattering
lenght
• Averaging
Z
D
X
1 k
1
d
f
=
d ⌦d!
2⇡ ki
e
i!f i t
1
b` bj e
d
1 kf
=
d ⌦d!
2⇡ ki
Z
e
th
`,j
we obtain
2
iQ·r` (0) iQ·rj (t)
E
SS (Q,! )
1
1
ei!f i t
X
D
(bi2 + bc2 ) e
`,j
iQ·r` (0) iQ·rj (t)
S(Q,! )
28
e
E
th
Cross-section
• Using the dynamic structure factors, we can
write the cross section as
⇤
kf ⇥ 2
d2
=N
bi Ss (Q,! ) + b2c S(Q,! )
d ⌦d!
ki
• These functions encapsulate the target
characteristics, or more precisely, the target
response to a radiation of energy !
~
and wavevector Q
29
Structure Factors
• Self dynamic structure factor (incoherent)
1
SS (Q,! ) =
2⇡
Z
1
e
i!f i t
1
*
1 X
e
N
iQ·r` (0) iQ·r` (t)
e
`
• Full dynamic structure factor (coherent)
1
S(Q,! ) =
2⇡
Z
1
-1
e
i!f i t
*
1 X -iQ·r` (0) iQ·rj (t)
e
e
N
30
`,j
+
+
Intermediate Scattering Functions
• Self dynamicZ structure factor
1
SS (Q,! ) =
2⇡
1
ei!f i t Fs (Q, t)
*
1
1 X
Fs (Q, t) =
e
N
iQ·r` (0) iQ·r` (t)
e
structure factor
• Full dynamic
Z
`
1
S(Q,! ) =
2⇡
+
1
ei!f i t F (Q, t)
1
*
1 X
F (Q, t) =
e
N
`,j
31
iQ·rj (0) iQ·r` (t)
e
+
• These functions are the Fourier transform
(wrt time) of the structure factors
• They contain information about the target
and its time correlation.
• Examples:
- Lattice vibrations (phonons)
- Liquid/Gas diffusion
32
Crystal Lattice
•
Position in F (Q, t) ⇠
DP
iQ·xj (0) iQ·x` (t)
`,j e
e
is the nuclear lattice position
harmonic oscillator
• Model as 1D quantum
q
x
=
(a
+
a
)
position:
•
• Hamiltonian (phonons)
~
2M !0
H=
2
p
2M
+
M !02 2
2 x
33
†
= ~!0 (a a +
†
1
2)
E
Crystal Lattice
• Assumption: 1D, 1 isotope, 1 spin state
➞ Self-intermediate structure function:
⌦
FS (Q, t) = e
iQ·x(0)
• Note: [x(0), x(t)] =6 0
• Use BCHDformula: e e
e
(but it’s a number)
A B
FS (Q, T ) = e
↵
iQ·x(t)
= eA+B e[A,B] . . . E
1
iQ·[x(0) x(t)] + 2 [Q·x(0),Q·x(t)]
34
e
formula:
• SimplifyD using (Bloch)
E
e
• we get
• ⌦with↵
x2
↵a+ a†
(↵a+
h
=e
FS (Q, t) = e
a † )2 i
-Q2 hDx2 i/2 + 12 [Q·x(0),Q·x(t)]
e
⌦ 2↵
= 2 x + 2 hx(0)x(t)i - h[x(0), x(t)]i
FS (Q, t) = e
Q2 hx2 i
35
e
Q2 hx(0)x(t)i
• The crystal is usually in a thermal state.
• Calculate F(Q,t) for a number state and
then take a thermal average over
Boltzman distribution
hn| x2 |ni =
hn| x(0)x(t) |ni =
• Replace
~
(2n
2M !0
+ 1)
~
2M !0 [2n cos(!0 t)
n ! hni
36
+ ei!0 t ]
Phonon Expansion
• ⌦Low↵ temperature
x2 =
•
~
2M !0
hni ⇡ 0
hx(0)x(t)i =
~
i!0 t
e
2M !0
~ 2 Q2
2M /(~!0 )
= Ekin /Ebind
Expand in series of
and calculate the dynamic structure factor
SS (Q,! ) = F(F (Q, t)
h
2 ~Q2
Q 2M !
0
(!) +
SS (Q,! ) ⇡ e
1
2
⇣
~Q2
2M !0
37
⌘2
~Q2
2M !0
(!
(!
!0 )+
2!0 ) + . . .
Phonon Expansion
• Neutron/q.h.o. energy exchange
Zero-phonon = no excitation
(elastic scattering)
SS (Q,! ) ⇡ e
2 ~Q2
Q 2M !
0
1
2
h
⇣
one-phonon = 1 quantum of Energy
(!) +
~Q
2M !0
2
⌘2
~Q2
2M !0
(!
(!
!0 )+
2!0 ) + . . .
two-phonons = 2 quantum of Energy
n-phonons
38
High Temperature
• We find a “classical” result, where
FScl = F[Gscl (x, t)]
and the space-time self correlation
cl
(classical) function Gs (x, t)dx is the
probability of finding the h.o. at x, at time t,
if it was at the origin at time t=0.
cl
Fz (Q, T )
=e
(kb T Q2 /M! 02 )[1 cos(!0 t)]
39
MIT OpenCourseWare
http://ocw.mit.edu
22.51 Quantum Theory of Radiation Interactions
Fall 2012
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