L u i z L e a l

O a k R i d g e N a t i o n a l L a b o r a t o r y

L e c t u r e s

P r e s e n t e d

a t t h e N u c l e a r

E n g i n e e r i n g

D e p a r t m e n t

o f t h e

M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y ( M I T )

Courtesy of Luiz Leal, Oak Ridge National Laboratory. Used with permission.

T h e r m a l N e u t r o n S c a t t e r i n g

K e r n e l D e v e l o p m e n t

T h e

p r o c e d u r e

c o n s i s t s

o f f i n d i n g t h e

a m p l i t u d e

o f t h e

s c a t t e r i n g w a v e f u n c t i o n

t h a t

l e a d s t o t h e d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n . I n d o i n g s o , a f e w t h i n g s w i l l b e a s s u m e d :

( 1 ) F i r s t B o r n a p p r o x i m a t i o n ;

( 2 ) F e r m i P s e u d o ‐ p o t e n t i a l ;

( 3 ) T i m e ‐ d e p e n d e n t S c h r ö d i n g e r e q u a t i o n

( T D S E )

The

f i r s t

t w o

a s s u m p t i o n s

h a v e

a l r e a d y

b e e n

di s c u s s e d . W h y a s s u m e t h e T D S E i s e x p l a i n e d a s

f o l l o w s : t h e e n e r g y d e p e n d e n c y ( e x p l i c i t l y ) i n

t h e

S c h r ö d i n g e r

e q u a t i o n

i m p l i e s

t h a t

t h e

e i g e n s t a t e s

o f t h e s c a t t e r i n g

s y s t e m a r e

k n o w n ,

i . e . ,

i n i t i a l s

a n d f i n a l

s t a t e s .

U s u a l l y

t h e y

a r e

n o t

k n o w n

a n d

e v e n i f t h e y

a r e

k n o w n

t h e r e

w i l l a

l a r g e

n u m b e r . I t

i s d e s i r e d

t o “ e l i m i n a t e ”

t h e

e x p l i c i t a p p e a r a n c e o f t h e e i g e n s t a t e s .

In s o d o i n g o n e u s e s

E  i  

 t

or

  2

 

2  

  V    i 

 2 m   t

L e o

V a n

H o v e

c h a m p i o n e d

t h i s

i d e a .

T h e

t i m e

d e p e n d e n c e

i s s u b s e q u e n t l y

u s e d

i n a

F o u r i e r

t r a n s f o r m a t i o n .

A s s u m i n g t h a t t h e i n t e r a c t i o n o c c u r r e d a t

t  t 0 . T h  e w a v e f u n c t i o n a t t h e d e t e c t o r a t t h e

p o s i t i o n

r f o r

t  t 0

i s a s o l u t i o n o f t h e e q u a t i o n

  2

    

  2 m 

( r , t )

 V ( r , t )   ( r , t )  i 

  t

F o r

t  t 0

b e f o r e t h e c o l l i s i o n h a s o c c u r r e d

t h e i n c i d e n t ‐ n e u t r o n w a v e f u n c t i o n i s

  

 inc

( r , t )

 e i ( k . r  w 0 t )

where



 k  neutron momentum

 w 0  neutron energy



T h e s o l u t i o n a t t h e d e t e c t o r a t ( r , t ) is

   

( r , t )   inc ( r , t )   scat ( r , t )

G i v e n t h a t f o r t h e i n c i d e n t w a v e  inc a w a y

f r o m t h e p o t e n t i a l r e g i o n

  2

   

  2

( r , t )

 i   

( r , t )  0

 2 m

 t 

inc

then

  2

     

  2

( r , t )

 i   

( r , t )  V ( r , t )  ( r , t )

 2 m

 t 

scat

A p p r o x i m a t i o n s :

( a ) B o r n a p p r o x i m a t i o n

I n t h e r i g h t ‐ h a n d s i d e o f t h e a b o v e e q u a t i o n :



 ( r , t )



  inc ( r , t )

( b ) F e r m i P s e u d o ‐ p o t e n t i a l



V n ( r , t ) 

where



2   2

m



n [ r



 R n ( t )]

R n  position of nucleus n at time t

a n  bound scattering related to the bound cross sec tion as

  4  a 2

( a 2 

 b ) 4 

H e n c e t h e e q u a t i o n t o b e s o l v e d i s

 2 2 mi   

    

  

  scat ( r , t )  4    a n  [( r

 R n ( t 0 ) ]   inc ( r , t )

   t   n 

N o t e t h a t

R n ( t 0 )

i s f o r r e a c t i o n s t h a t o c c u r r e d a t

the

t  t 0 .

M a k i n g

u s e

o f t h e

G r e e n ’ s

f u n c t i o n m e t h o d t o

 

s o l v e

t h e

e q u a t i o n

a b o v e .

L e t G ( r  r ' , t  t 0 ) b e t h e

G r e e n ’ s f u n c t i o n , h e n c e :

 2 2 mi     '  

   t   G ( r

 r , t

 t 0 ) 

4   ( r  r ' )  ( t  t 0 )

T h e s o l u t i o n f o r  scat ( r , t ) is

    

 scat ( r , t )    d  dt 0 G ( r  r ' , t  t 0 )  inc ( r ' , t 0 )  a n  [ r  R n ( t 0 )]

 t 0 n

w h e r e  i s t h e v o l u m e .  

S u b s t i t u t i n g t h e v a l u e s f o r

G ( r

 r ' , t  t 0 ) a n d

 inc

w e h a v e

 t 

 scat

( r , t )  i

 dt 0

( t  t 0

) 3 / 2  a

 d   [ r  R n

( t 0

)] 

  

 n

2  

exp  im

r  r '



  exp [ i ( k 0 . r

 w 0 t )]

  2  ( t  t 0 )  

T h e a b o v e e x p r e s s i o n

i s t h e

s o l u t i o n f o r t h e

s c a t t e r e d 

w a v e

 scat

a t t h e d e t e c t o r a t

t h e

p o s i t i o n r f o r

t h e t i m e

t  t 0

f o r a n i n  c i d e n t

n e u t r o n o f

e n e r g y

 w 0

a n d m o m e n t u m

 k 0 . I t i s

n o t

c l e a r h o w

t o o b t a i n t h e

s c a t t e r e d

a m p l i t u d e

f r o m

t h e

a b o v e

e q u a t i o n .

L e o n

V a n

H o v e

c a m e

u p w i t h a c l e a v e r t r a n s f o r m o f t h e t y p e

i d e a

o f u s i n g

a F o u r i e r

  

' iw ' t

scat ( r , t )   f ( r , w ) e

w '

and

 1

f ( r , w ' ) 

T



T

 dt e iw ' t 

0

scat



( r , t )

f ( r , w )

r e l a t e s t o t h e s c a t t e r e d w a v e

a m p l i t u d e .

A f t e r L O T S O F A L G E B R A

 1 T

  

f ( r , w ' ) 

 dt e iw ' t 0  a

 d  [ r

 R ( t ) ] e i  .

Tr ' 0 n

F o r

d e r i v a t i o n

o f t h e

a b o v e

e q u a t i o n

s e e :

T h e r m a l 49 ‐ 52

N e u t r o n S c a t t e r i n g b y

E n g e l s t a f f

p a g e s

T h e s c a t t e r e d n e u t r o n h a s

  

  k 0  k  momentum change

w '  w 0  w  energy change

T h e s c a t t e r i n g d i f f e r e n t i a l c r o s s s e c t i o n i s

d e f i n e d a s

d 2  T   2

 r ' 2

f ( r , w )

dEd  h  0

 0 and

 initial

and

final

neutron velocity

R e a d : T h e

E l e m e n t s

o f N e u t r o n

I n t e r a c t i o n

Theor y A n t h o n y F o d e r a r o p a g e 5 5 5

 2 1  

    

f ( r , w )

 T 2 r ' 2

 d  e  iw   a * a  dr "  dr e i  .( r  r " ) 

 m , n

   

 [ r "  R n (0 ) ]  [ r  R n (  )]

T h e b a r i n d i c a t e s t i m e a v e r a g i n g .

T h e d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n i s f i n a l l y o b t a i n e d a s

d 2 

1   

   



dEd  h 

 d  e  iw 

 a * a

 dr '  dr e i  . r 

0  m , n

    

 [ r "  R n (0 )  r ]  [ r  R n (  )]

T h e S p a c e ­ T i m e C o r r e l a t i o n F u n c t i o n ( V a n H o v e )

 1     

G ( r ,  )    dr '  [ r '  R n ( 0)  r ]  [ r  R n (  )]

m , n



G ( r ,  ) is not the Green ' s function ! !



I n t e r p r e t a t i o n o f t h e

G ( r ,  )

T w o p a r t s :



f o r

m  n

( d i a g o n a l t e r m s )

G s ( r ,  ) 

f o r m  n

( o f f ‐ d i a g o n a l t e r m s ) G d ( r ,  )

I n t e r p r e t a t i o n :



G s ( r ,  )  self

 correlation

function : a s e c o n d

I D E N T I C A L n u c l e u s i s p r e s e n t



G d ( r ,  )  distinct  correlation

function : a

s e c o n d D I S T I N C T n u c l e u s i s p r e s e n t

  

G ( r ,  )  G s ( r ,  )  G d ( r ,  )

w h e r e

 1 N

    

G s ( r ,  ) 

N

and

 

n  1

dr '  [ r '  R n ( 0)  r ]  [ r

 R n (  )]

 1 N

    

G d ( r ,  ) 

N

  dr '  [ r '  R n ( 0)  r ]  [ r

 R n (  )]

D e f i n i n g

m  n  1

 a 2

 1  a * a 

N m , n

m n mn

 a  2 

1  a * a

m n

m  n

d 2 

dEd 

d 2 

 coh

dEd 

d 2 

 incoh

dEd 

A s s u m i n g

o n l y

c o h e r e n t

i n e l a s t i c s c a t t e r i n g

p r e s e n t , i . e . ,

d 2 

 a 2  a  2  a

d 2 

dEd 

 coh

dEd 

d 2 

a 2 

   



dEd 

h  0

  dr d 

e i (  . r  w '  ) G ( r ,  )

R e c a l l t h a t

a 2   b

4 

and





 0

d 2 

 b ˜ 



dEd 

4 

S (  , w ' )

w h e r e

S ˜ (  1

 i (   

 , w ' ) 

h

S ˜ ( 

 dr d  e

 . r  w '  ) G ( r ,  )

 , w ' )  Scattering Law



 , w ' ) can b e

d 2 

o b t a i n e d d i r e c t l y f r o m

m e a s u r e m e n t s o f

dEd 

P r o p e r t i e s o f



 , w ' )

D e p e n d s

o n l y o n t h e d y n a m i c s o f t h e

s c a t t e r c e n t e r

( b ) S u m r u l e r

˜   2

 w S (  , w ) dw  2 M

( c ) C o n d i t i o n d e r i v e d f r o m t h e d e t a i l e d

b a l a n c e

 w



S (  , w  )  e S (

  , w )

D e f i n i t i o n s :

  

S ˜ (  , w )  e 2 S (  ,  )

W h e r e

 

 2  2

and

  E  E 0

2 mAKT KT

S i n c e

  

2 2 2

  k 0  k  

 k 0  k  2 k 0 . k

 

cos 

 2  2

 

 2 ( k 2



 k 2  2 k

. k cos  )

2 mAKT

 

AKT

2 mAKT

Hence t h e d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n

b e c o m e s

d 2 

dEd 

  b

4  KT





e 2 S (  ,  )

E 0

Notations:

d 2  dEd  or

 s ( E 0  E ,  )

or

 s ( E 0  E ,  ) where

  cos 

S i m p l e E x a m p l e :

( 1 )

Scatter er i s a

s i n g l e

n u c l e u s

o f m a s s M .

O n l y c o h e r e n t s c a t t e r i n g i s a c c o u n t e d f o r ;

( 2 ) Scatter er i s f r e e a n d a t r e s t

T h i s

c o r r e s p o n d s t o

t h e

s i t u a t i o n

d e a l t

w i t h i n

t h e

t h e o r y

o f n e u t r o n

m o d e r a t i o n

w h e r e

t h e

c h e m i c a l r e g i o n e f f e c t s a r e n e g l i g i b l e .

 ( E

 E ,  )   b

E 

S ( , w )

s 0 4  E 

  cos 



 , w ' )

i s a d e l t a f u n c t i o n a s

˜  E 0  E  2 ( EE ) 

1 / 2

S (  , w )   ( E 0  E  )

A

  cos  and A  M

m

R e c a l l t h a t

 w  E 0  E

 2  2

2 M

A l s o ,

 E 0

 E  2 ( EE 0

A

) 1 / 2 

d   2  sin  d 

  cos 

d    sin  d  d   2  (  d  )

and

 s ( E 0  E )    s ( E 0  E ,  ) d 

4 

 s ( E 0  E )

 1

 2    s ( E 0  E ,  ) d 

 1

 

E  1 / 2  1

E  E  2( EE

) 1 / 2 

 ( E  E )  2  b     ( E  E  0 0 ) d 

4   E 0   1 A

C h a n g e o f v a r i a b l e s

E  E  2 ( EE

) 1 / 2 

x  E 0  E 

A

such that

2 ( EE

) 1 / 2

dx   0 d 

A

   1  x   E 0  E 

( E 1 / 2

 E 1 / 2 ) 2

A

 

E  1 / 2 A x 

 ( E  E )  2  b  

  ( x ) dx

s 0 4 

 E 0

 2( E 0

E ) 1 / 2



 s ( E 0

 E )   b A

4 E

0

R e c a l l t h a t

2

 A  1 

 b     free

 A 

 s ( E 0  E ) 

( A  1 ) 2

4 A

1

 free

0

 A  1  2 4 A

if      1    2



hence

A  1  ( A  1 )

 s ( E 0

 E ) 

1

( 1   ) E 0

 free

Energy range ?

A 2  2 A   1

E  ( A  1 ) 2 E 0

  1  E  E 0

   1  E  ( 1   ) E 0

 1

 ( 1   ) E



 s ( E 0  E )  





 free for ( 1   ) E 0  E

 E 0

 0 otherwise

( N u c l e a r R e a c t o r A n a l y s i s D u d e r s t a d t a n d

H a m i l t o n p a g e 4 4 )