Neut r on Scattering

C r oss Section

in 7 easy steps

1. Scattering P r obability (TDPT)

2. Adiabatic Switching of P otential

3. Scattering matrix (integral o v er time)

4. T ransition matrix (cor r elation of e v ents)

5. Density of states

6. Incoming flux

7. Thermal a v erage

1. Scattering P r obability

• P r obability of final scatter ed state , when e v olving under scattering interaction

P sca tt = | h f | U I ( t ) | i i | 2

• Time-dependent per turbation theor y (Dyson expansion)

2. Adiabatic Switching

S catt ering M edium

• Slo w s witching of potential

➞ time ∈ [- ∞ , ∞ ] P ar ticle

• V is a pp r o ximatel y constant

• P r opagator f or time t=- ∞ → t= ∞ is called the scattering matrix

| h f | U I ( t i = -1 , t f = 1 ) | i i | 2 = | h f | S | i i | 2

• S is expanded in series:

Z ∞

⟨ f | S (1) | i ⟩ = — iV fi e i u fi t dt = — 2 π i δ ( ω f — ω i ) V f i

—∞

h f | S (2) | i i = — h f |

X m

V | m i h m | V

! | i i

∞

dt 1 e i u fm t 1

—∞

t 1

dt 2 e i u mi t 2

—∞

• Be car eful with integration ( ɛ ...)

• First and second or der simplify to

h f | S (1) | i i = - 2 π i δ ( ω f - ω i ) h f | V | i i

h f | S (2) | i i = - 2 π i δ ( ω f

- ω i

) X h f | V | m i h m | V | i i

m ω i - ω m

• The scattering matrix is giv en b y the transition matrix

h f | S | i i = - 2 π i δ ( ω f - ω i ) h f | T | i i

• which has the f ollo wing expansion

h f | T | i i = h f | V | i i + X

m

h f | V | m i h m | V | i i

ω i - ω m

+ X

m,n

V fm V mn V ni

( ω i - ω m )( ω i - ω n )

+ .. .

• Scattering p r obability

P s = 4 π 2 | h f | T | i i | 2 δ 2 ( ω f - ω i ) = 2 π t | h f | T | i i | 2 δ ( ω f - ω i )

(not w ell defined because of t →∞ )

• Scattering rate

W S = 2 π | h f | T | i i | 2 δ ( ω f - ω i )

• T arget is left in one (of man y possible) state .

• Radiation is left in a contin uum state

• Separate the tw o subsystems

(no entanglement prior and after the

scattering e v ent)

and r e write the transition matrix

• T arget: | m k i , ✏ k

• Radiation: | k i , ω k

• Scattering rate W fi = 2 π | ⟨ f | T | i ⟩ | 2 δ ( E f — E i )

g o back to definition, using explicit states

Z ∞

W fi = | h m f , k f | T | m i , k i i | 2 e i ( u f + є f — u i — є i ) t

—∞

• W ork in Sch r odinger pict. f or radiation and Interaction pict. f or target:

h m f , k f | T I I | m i , k i i = h m f , k f | T S S | m i , k i i e i ( u f - u i ) t e i ( є f - є i ) t

= h m f , k f | T I t S r

( t ) | m i , k i i e i ( u f - u i ) t

= h m f | T k f

,k i

( t ) | m i i e i ( u f - u i ) t

• Scattering rate is then a cor r elation

1 Z 1 †

W fi = e i ( u f - u i ) t h m i | T

~ 2 -1

k f ,k i

( 0) | m f i h m f | T k f

,k i

( t ) | m i i

N O TE: Time e v olution of target onl y

(e .g. lattice n uclei vibration)

n y

n

n x

• # of states P k n ( E k ) ⇡ R d 3 n ( E )

• Plane w a v e in a cubic c a vity

3

k x = n x ! d 3 n = d 3 k L 2 ⇡

3

⇢ ( E ) dE d ⌦ = ⇢ ( k ) d 3 k = k 2 dk d ⌦

2 ⇡

• Photons, k = E / ~ c !

d k d E

= 1 / ~ c

✓ L ◆ 3 E 2 ✓ L ◆ 3 ω 2

2 π ~ 3 c 3 2 π ~ c 3

Massiv e par ticles, E = ~ 2 k 2

2 m

✓ L ◆ 3

k ✓ L ◆ 3 p 2 mE

6. Incoming Flux

• # scatter er per unit ar ea and time

< = # = v , since t = L /v , A = L 2 A t L 3

• Photons, < = c /L 3

• Massiv e par ticles, Φ =

~ k mL 3

7. Thermal A v erage

• A v erage o v er initial state of target

W S ( i ! ⌦ + d ⌦ , E + dE ) = ⇢ ( E ) X P i X W f i

i f

• Scattering C r oss Section

d 2 σ 1

d Λ dE = ~ 2

f

∞

dte i u fi t

—∞

† ( 0) T fi ( t )

ρ ( E )

th Φ

Neut r ons

• Using < inc and ⇢ ( E ) f or massiv e par ticles, the scattering c r oss section is:

d 2 σ 1 ✓ mL 3 ◆ 2 k Z 1 D E

Neut r ons

• Evaluate T f or neut r ons, with states

| k i,f i !

k ( r ) = e ik · r /L 3 / 2

• w e obtain T k f k i ( t, Q ) with Q = k f — k i

h k f | T ( t, r ) | k i i =

d 3 r ⇤ ( r ) T ( r, t )

3

k i ( r )

1

= L 3

d 3 r e iQ · r T ( r, t )

L 3

Neut r on T ransition Matrix

• W e still need to ta k e the expectation value with r espect to the target states,

T = 1 Z d 3 r e iQ · r h m | T ( r, t ) | m i

fi 3 f i L 3

• T is an expansion of the interaction potential, her e the n uclear potential

• anal yze at least first or de r ...

• Nuclear potential is v er y st r ong (V 0 ~30MeV)

• And shor t range (r 0 ~ 2fm)

• Not g ood f or per turbation theor y!

• F ermi a pp r o ximation

• What is impor tant is the p r oduct

a a V 0 r 3

(a = scattering length) if k r 0 ⌧ 1

• Replace n uclear potential with

w eak, long range pseudo-potential

neutr on wa v efunction

• Still, shor t range compar ed to w a v elength

• Delta-function potential!

2 π ~ 2

V ( r ) = a δ ( r ) µ

Scattering Length

• Fr ee scattering length a,

2 π ~ 2 2 π ~ 2

V ( r ) = a δ ( r ) ! b δ ( r )

µ m n

• bound scattering length b (include inf o about isotope and spin)

b = m n a ⇡ A + 1 a

µ A

(r educed mass: µ =

M m n )

M + m n

• T o first or de r , the transition matrix is just

the potential

T = 1 Z d 3 r e iQ · r h m | V ( r, t ) | m i

fi 3 f i L 3

• Using the n uclear potential f or a n ucleus at a position R, w e h a v e

1 Z

3 iQ · r

2 π ~ 2 2 π ~ 2

iQ · R ( t )

T fi = 3 d r e

L 3

b ( R ) δ ( R ) T ( r, t ) =

m n

b ( R ) e

m n

• T o first or de r , f or man y scatter at position r i

2 ⇡ ~ 2 X iQ · r

n

( t )

i

• The scattering c r oss section becomes

2 ∞ X D E

= e i u fi t b A b j e — iQ · r ` (0) e iQ · r j ( t )

d ⌦ d ω 2 π k i

— ∞ A ,j th

Scattering Lengths

• The bound scattering length depends on isotope and spin

• W e need to ta k e the a v erage b ` b j ! b ` b j

- j = ` , b ` b j = b 2

- j / = ` , b b = 2

` j b

• Finall y , b ` b j = b 2 δ j, ` + b 2 (1 — δ j, ` )

• Coher ent/Incoher ent scattering length:

2 2 2 2 2

Scattering lengths

• Cohe r ent scattering length

b c = b

• Cor r elations in scattering e v ents f r om the same target

(scale-length o v er which the incoming radiation is coher ent in a QM sense)

• Simple a v erage o v er isotopes and spins

Scattering Lengths

• Incohe r ent scattering length

b 2 = ( b 2

2

- b ) c j, `

• Cor r elation of scattering e v ents betw een

differ ent targets

• V ariance of the scattering length o v er spin states and isotopes

• A v eraging o v er the scattering lenght

D E

2 ∞ X

= e i u fi t — iQ · r (0) iQ · r ( t )

d ⌦ d ω 2 π k i

— ∞ A ,j th

w e obtain S S ( Q, ω )

2 ∞

= e i u fi t

X ( b 2 + b 2 )

D e — iQ · r ` (0) e iQ · r j ( t ) E

d ⌦ d ω 2 π k i —∞

A ,j

i c th

S ( Q, ω )

• Using the dynamic structur e factors, w e can write the c r oss section as

d 2 σ

d ⌦ d ω

= N k f ⇥

k i

2 S s ( Q, ω ) + b 2 S ( Q, ω ) ⇤

• These functions enca psulate the target characteristics, or mor e pr ecisel y , the target

r esponse to a radiation of energ y and w a v e v ector Q ~

Structu r e Factors

• Self dynamic structur e factor (incoher ent)

1 Z ∞ * 1 X +

• Full dynamic structur e factor (coher ent)

1

S ( Q, ω ) = e i u fi t

* 1 X

e - iQ · r ` (0) e iQ · r j ( t ) +

2 π -1

N

A ,j

Intermediate Scattering Functions

• Self dynamic structur e factor

1 Z ∞

2 π —∞ * X +

F s ( Q, t ) =

• ​

1 e — iQ · r ` (0) e iQ · r ` ( t ) N

`

1 Z ∞

S ( Q, ω ) = e i u fi t F ( Q, t ) 2 π —∞

F ( Q, t ) = * 1 X e — iQ · r j (0) e iQ · r ` ( t ) +

` ,j

• These functions ar e the F ourier transf orm (wr t time) of the structur e factors

• The y contain inf ormation about the target and its time cor r elation.

• Examples:

- Lattice vibrations (phonons)

- Liquid/Gas diffusion

• P osition in F ( Q, t ) ⇠ D P ` ,j e — iQ · x j (0) e iQ · x ` ( t ) E

is the n uclear lattice position

• Model as 1D quantum harmonic oscillator

• position: x =

~

2 M u 0

( a + a † )

• Hamiltonian (phonons)

2

H = p 2 + M u x 2 = ~ ω ( a † a + 1 )

2 M 2 0 2

• Assumption: 1 D , 1 isotope , 1 spin state

➞ Self-intermediate structur e function:

F S ( Q, t ) = ⌦ e — iQ · x (0) e iQ · x ( t ) ↵

• Note: [ x ( 0) , x ( t )] = /

0 (but it ’ s a n umber)

• Use BCH f orm ula: e A e B

= e A + B e [ A,B ]

. . .

D — iQ · [ x (0) — x ( t )] + 1 [ Q · x (0) ,Q · x ( t )] E

• Simplify using (Bloch) f orm ula:

D e α a + β a † E

= e h ( α a + β a † ) 2 i

• w e get F S

• with

( Q, t ) = e - Q 2 h D x 2 i / 2 e + 1 [ Q · x (0) ,Q · x ( t ) ]

⌦ ∆ x 2 ↵ = 2 ⌦ x 2 ↵ + 2 h x ( 0) x ( t ) i - h [ x ( 0) , x ( t )] i

F S ( Q, t ) = e — Q 2 ( x 2 ⟩ e — Q 2 ( x (0) x ( t ) ⟩

• The cr ystal is usuall y in a thermal state .

• Calculate F(Q,t) f or a n umber state and then ta k e a thermal a v erage o v er

Boltzman distribution

h n | x 2 | n i =

h n | x ( 0) x ( t ) | n i =

~ (2 n + 1)

0

~ [2 n cos( ω 0 t ) + e i u 0 t ]

• Replace n ! h n i

• Lo w temperatur e h n i ⇡ 0

⌦ x 2 ↵ = ~

h x ( 0) x ( t ) i =

~ e i u 0 t

2 M u 0 2 M u 0

• Expand in series of

~ 2 Q 2

2 M

/ ( ~ ω 0

) = E

ki n

/E bind

and calculate the dynamic structur e factor

S S ( Q, ω ) = F ( F ( Q, t )

— Q 2

~ Q 2 h

~ Q 2

1 ⇣ ~ Q 2 ⌘ 2

δ ( ω — 2 ω

) + . . .

• Neut r on/q.h. o . energ y exchange

Zer o-phonon = no excitation one-phonon = 1 quantum of Energ y (elastic scattering)

— Q 2

~ Q 2 h

~ Q 2

1 ⇣ ~ Q 2 ⌘ 2

δ ( ω — 2 ω

) + . . .

tw o-phonons = 2 quantum of Energ y

n-p honons

High T emperatu r e

• W e find a “classical” r esult, wher e

F cl = F [ G cl ( x, t ) ]

and the space-time self cor r elation

(classical) function

G cl ( x, t ) dx is the

p r obability of finding the h. o . at x, at time t, if it was at the origin at time t=0.

F cl ( Q, T ) = e — ( k b T Q 2 /M u 2 )[1 — cos( u 0 t ) ]

z 0

MIT OpenCourseWare http://ocw.mit.edu

22.51 Quantum Theory of Radiation Interactions

Fall 2012

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