1 2 . I n te r a cti o n o f R a d i a ti o n w i th M a tte r
1 2 . 1 S c at t e r i n g T h e o r y
1 2 . 1 . 1 C r o s s S e c t i o n
1 2 . 1 . 2 T h e r m a l N e u t ro n S c a t t e r i n g
1 2 . 2 Em i s s i o n an d A b s o r p t i o n
1 2 . 2 . 2 A b s o r p t i o n
1 2 . 2 . 3 B l a c k b o d y R a d i a t i o n
1 2 . 3 W i g n e r - W e i s s k o p f T h e o r y
1 2 . 3 . 1 I n t e ra c t i o n o f a n a t o m w i t h a s i n g l e m o d e e . m . fi e l d
1 2 . 3 . 2 I n t e ra c t i o n w i t h m a n y m o d e s o f t h e e . m . fi e l d
1 2 . 4 S c at t e r i n g o f p h o t o n s b y at o m s
1 2 . 4 . 1 T h o m s o n S c a t t e r i n g b y F r e e E l e c t r o n s
1 2 . 4 . 2 R a y l e i g h S c a t t e ri n g o f X - r a y s
1 2 . 4 . 3 V i s i b l e Li g h t S c a t t e r i n g
1 2 . 4 . 4 P h o t o e l e c t r i c E ff e c t
1 2 . 1 S c a t t e r i n g Th e o r y
W e w a n t t o d e s c r ib e t h e in t e r a c t io n o f r a d ia t io n w it h m a t t e r a s a s c a t t e r in g p r o c e s s . S p e c ifi c a lly , w e a r e in t e r e s t e d in c a lc u la t in g t h e r a t e o f s c a t t e r in g ( a n d t h e n t h e c r o s s s e c t io n ) , w h ic h is n o t h in g e ls e t h a n t h e t r a n s it io n r a t e f r o m a n in it ia l s t a t e ( in it ia l s t a t e o f t h e m a t t e r + in c o m in g p a r t ic le ) a n d a fi n a l s t a t e ( fi n a l s t a t e o f t h e t a r g e t + o u t g o in g r a d ia t io n ) 3 9 .
T h is is a p r o b le m t h a t c a n b e s o lv e d b y T D P T . In s t e a d o f c o n s id e r in g a c o n s t a n t p e r t u r b a t io n a s d o n e t o d e r iv e F e r m i’s G o ld e n r u le , w e a n a ly z e t h e c a s e o f a s c a t t e r in g p o t e n t ia l, in it s m o s t g e n e r a l f o r m . W e d e s c r ib e a s c a t t e r in g
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t
F i g . 2 0 : M o d e l f o r s c a t t e ri n g : Le f t , p a r t i c l e t r a j e c t o r y , r i g h t t i m e d e p e n d e n c y o f t h e p o t e n t i a l .
e v e n t a s a p a r t ic le c o m in g c lo s e t o a t a r g e t o r a m e d iu m , in t e r a c t in g w it h it a n d t h e n b e in g d e fl e c t e d a w a y . T h u s , a s a f u n c t io n o f t im e , t h e in t e r a c t io n H a m ilt o n ia n V v a r ie s a s in t h e fi g u r e 2 0 .
3 9 A v e r y g o o d r e s o u r c e f o r s c a t t e r i n g t h e o r y i s C h e n , S . H . ; K o t l a rc h y k , M . , I n t e r a c t i o n s o f Ph o t o n s a n d N e u t r o n s w i t h M a t t e r , ( 2 0 0 7 ) , w h i c h w e f o l l o w c l o s e l y i n t h i s c h a p t e r .
W e w a n t t o c a lc u la t e t h e p r o b a b ilit y o f s c a t t e r in g f r o m a n in it ia l s t a t e t o a fi n a l s t a t e :
s c at t
I
I
P = | ( f | U ( t ) | i ) | 2 = | ( f | ( 1 1 − i I ∞ V ( t ′ ) d t ′ + . . . ) | i ) | 2
− ∞
N o t ic e t h a t w e c o n s id e r n e g a t iv e t im e s a s w e ll. T h is c o r r e s p o n d s t o t h e s o - c a lle d a d i a b a t i c s w i t c h i n g , s in c e t h e in t e r a c t io n is a s s u m e d t o b e t u r n e d o n s lo w ly f r o m t h e b e g in n in g o f t im e a n d t o g o d o w n t o z e r o a g a in f o r lo n g t im e s .
A . S c a tte r i n g a n d T r a n s i ti o n m a tr i c e s
In s c a t t e r in g p r o b le m s , t h e p r o p a g a t o r U I is u s u a lly c a lle d t h e s c a t t e r in g m a t r ix S . T o s im p lif y t h e c a lc u la t io n , w e c a n a s s u m e a g a in t h a t V is a c t u a lly t im e - in d e p e n d e n t . T h e n f r o m t h e fi r s t o r d e r T D P T w e o b t a in :
( 1 )
( f | S | i ) = − i V I ∞ e i ω f i t d t = − 2 π i δ ( ω
— ω ) V
f i
− ∞
f m 1
N o w c o n s id e r t h e s e c o n d o r d e r c o n t r ib u t io n :
f i f i
( f | S
| i ) = − ( f | V | m ) ( m | V | i ) d t 1 e
( 2 ) � Σ
m
� I ∞ i ω
t I t 1
d t 2 e
m i 2
− ∞ − ∞
i ω t
N o t ic e t h a t t h e la s t in t e g r a l is n o t w e ll d e fi n e d f o r t → − ∞ . T o s o lv e it , w e r e w r it e it a s
I t 1
i ( ω m i − i ǫ ) t 2 e i ω m i t + ǫ t � t 1
2 − − ∞
lim d t e = lim i
ǫ → 0 + − ∞ ǫ → 0 + ω m i − i ǫ
→ − ∞ →
N o w w h e n t a k in g t h e lim it t t h e e x p o n e n t ia l t e r m e ǫ t 0 ( t h u s g e t t in g r id o f t h e o s c illa t io n s ) . T h e n w e a r e le f t w it h o n ly
I t 1
d t 2 e
i ω m i t 2 = lim − i
e i ( ω m i − i ǫ ) t 1
a n d w e o b t a in ( s e t t in g n o w ǫ = 0 )
− ∞ ǫ → 0 + ω m i − i ǫ
( f | S
| i ) = i
V f m V m i d t 1
− ∞
( 2 ) Σ
m
I ∞ e i ( ω f i − i ǫ ) t 1
ω
m i
— i ǫ
Σ ( f | V | m ) ( m | V | i )
m
ω
i
— ω
m
= − 2 π i δ ( ω f − ω i )
L o o k in g a t t h e fi r s t a n d s e c o n d o r d e r o f t h e s c a t t e r in g m a t r ix , w e s t a r t s e e in g a p a t t e r n e m e r g e . W e c a n t h u s r e w r it e
( f | S | i ) = − 2 π i δ ( ω f − ω i ) ( f | T | i )
:
w h e r e T is c a lle d t h e t r a n s it i o n m a t r ix . It s e x p a n s io n is g iv e n b y :
ω i − ω m
m , n ( ω i − ω m ) ( ω i − ω n )
( f | T | i ) = ( f | V | i ) + Σ ( f | V | m ) ( m | V | i ) + Σ V f m V m n V n i + . . .
m
B . S c a tte r i n g Pr o b a b i l i t y
| ( | | ) |
W e c a n n o w t u r n t o c a lc u la t e t h e s c a t t e r in g p r o b a b ilit y : P S = f S i 2 . In o r d e r t o o b t a in t h e t o t a l s c a t t e r in g p r o b a b ilit y , w e w ill n e e d t o c o n s id e r a ll p o s s ib le fi n a l s t a t e s . W e f o u nd :
P s = 4 π 2 | ( f | T | i ) | 2 δ 2 ( ω f − ω i )
W e c a lc u la t e t h e s q u a r e o f t h e D ir a c f u n c t io n f r o m it s d e fi n it io n b a s e d o n t h e lim it o f t h e in t e g r a l:
2 π
2 π
t → ∞ π
δ 2 ( ω ) = 1 I ∞ d te i ω t δ ( ω ) = 1 I ∞ d tδ ( ω ) = lim
− ∞
− ∞
t
δ ( ω )
T h e n a lt h o u g h t h e p r o b a b ilit y is n o t s o w e ll d e fi n e d , s in c e it c o n t a in s a lim it :
t → ∞
P s = lim 4 π t | ( f | T | i ) | 2 δ ( ω f − ω i ) t h e r a t e o f s c a t t e r in g is w e ll d e fi n e d , s in c e it is W S = P S / ( 2 t ) :
W S = 2 π | ( f | T | i ) | 2 δ ( ω f − ω i )
T h is is t h e r a t e f o r o n e is o la t e d fi n a l s t a t e . If in s t e a d w e h a v e a c o n t in u u m o f fi n a l s t a t e s , w it h d e n s it y o f s t a t e s ρ ( ω f ) w e n e e d t o s u m o v e r a ll p o s s ib le fi n a l s t a t e s :
W S = 2 π I 2 π | ( f | T | i ) | 2 δ ( ω f − ω i ) ρ ( ω f ) d ω f = 2 π | ( f | T | i ) | 2 ρ ( ω i ) N o t ic e t h a t t o fi r s t o r d e r , t h is is e q u iv a le n t t o t h e F e r m i G o ld e n r u le .
1 2 . 1 . 1 C r o s s S e c t i o n
W e n o w u s e t h e t o o ls d e v e lo p e d in T D P T t o c a lc u la t e t h e s c a t t e r in g c r o s s s e c t io n . T h is is d e fi n e d a s t h e r a t e o f s c a t t e r in g d iv id e d b y t h e in c o m in g fl u x o f “ p a r t ic le s ” :
d Ω d E ∝
d 2 σ
W S ( Ω , E )
Φ i n c
W e c o n s id e r a p a r t ic le + m e d iu m s y s t e m , w h e r e t h e p a r t ic le is s o m e r a d ia t io n r e p r e s e n t e d b y a p la n e w a v e o f m o m e n t u m k k . In g e n e r a l, w e w ill h a v e t o d e fi n e a ls o o t h e r d e g r e e s o f f r e e d o m d e n o t e d b y t h e in d e x λ , e .g f o r p h o t o n s w e w ill h a v e t o d e fi n e t h e p o la r iz a t io n w h ile f o r p a r t ic le s ( e .g .e n e u t r o n s ) t h e s p in .
H H H → ± ∞
T h e u n p e r t u r b e d H a m ilt o n ia n is 0 = R + M ( r a d ia t io n a n d m e d iu m ) . W e a s s u m e t h a t f o r t t h e r a d ia t io n a n d m a t t e r s y s t e m s a r e in d e p e n d e n t , w it h ( e ig e n ) s t a t e s :
| i ) = | k i , m i ) , | f ) = | k f , m f )
w it h e n e r g ie s :
H R | k i ) = h ω i | k i ) , H R | k f ) = h ω f | k f ) , H M | m i ) = ǫ i | k i ) , H M | m f ) = ǫ f | m f )
a n d t o t a l e n e r g ie s : E i = h ω i + ǫ i a n d E f = h ω f + ǫ f .
d
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S c a tte r i n g R a te
T h e r a t e o f s c a t t e r in g is g iv e n b y t h e e x p r e s s io n f o u n d e a r lie r :
f i
h
f
W = 2 π | ( f | T | i ) | 2 δ ( E
i
— E )
A s u s u a l, w e w a n t t o r e p la c e , if p o s s ib le , t h e d e lt a - f u n c t io n w it h t h e fi n a l d e n s it y o f s t a t e s . H o w e v e r , o n ly t h e r a d ia t io n w ill b e le f t in a c o n t in u u m o f s t a t e s , w h ile t h e t a r g e t w ill b e le f t in o n e ( o f p o s s ib ly m a n y ) d e fi n it e s t a t e . T o d e s c r ib e t h is d is t in c t io n , w e s e p a r a t e t h e fi n a l s t a t e in t o t h e t w o s u b s y s t e m s .
( | | )
W e fi r s t d e fi n e t h e p a r t ia l p r o j e c t io n o n r a d ia t io n s t a t e s o n ly , T k f , k i = k f T k i . B y w r it in g t h e d e lt a f u n c t io n a s a n in t e g r a l w e h a v e :
2 π
f i
h
f
k f , k i
W = ( m | T
| m ) ( m | T †
| m ) 1 I ∞ e i ( ω f − ω i ) t e i ( ǫ f − ǫ i ) t / k
− ∞
i
i
k f , k i
f
2 π h
N o w , s in c e e − i H R t / k | m i ) = e − i ǫ i t / k | m i ) ( a n d s im ila r ly f o r | m f ) w e c a n r e w r it e
f k f , k i i f k f , k i i f k f , k i i
( m | T | m ) e i ( ǫ f − ǫ i ) t / k = ( m | e i H R t / k T e − i H R t / k | m ) = ( m | T ( t ) | m )
a n d o b t a in a n e w e x p r e s s io n f o r t h e r a t e a s a c o r r e la t io n o f “ t r a n s it io n ” e v e n t s :
f i
h 2
i
k f , k i
W = 1 I ∞ e i ( ω f − ω i ) t ( m | T †
( 0 ) | m
f
f
) ( m | T
k f , k i
i
( t ) | m )
− ∞
F i n a l d e n s i t y o f s ta te s
( L )
T h e fi n a l d e n s it y o f s t a t e s d e s c r ib e t h e a v a ila b le s t a t e s f o r t h e r a d ia t io n . A s w e a s s u m e d t h a t t h e r a d ia t io n is r e p r e s e n t e d b y p la n e w a v e s ( a n d a s s u m in g f o r c o n v e n ie n c e t h e y a r e c o n t a in e d in a c a v it y o f e d g e L ) , t h e fi n a l d e n s it y o f s t a t e s is
3
ρ ( k f ) d 3 k f = k 2 d k f d Ω
2 π f
W e c a n e x p r e s s t h is in t e r m s o f t h e e n e r g y , ρ ( k ) d 3 k = ρ ( E ) d E d Ω . F o r e x a m p le , f o r p h o t o n s , w h ic h h a v e k = E / h c
w e h a v e
ρ ( E ) = 2 = 2
k
( L ) 3 E 2 ( L ) 3 ω 2
2 π h 3 c 3 2 π h c 3
2 m
w h e r e t h e f a c t o r 2 t a k e s in t o a c c o u n t t h e p o s s ib le p o la r iz a t io n s . F o r n e u t r o n s ( o r o t h e r p a r t ic le s s u c h t h a t E = k 2 k 2 ) :
ρ ( E ) =
2 π h 2 =
( L ) 3 k ( L ) 3 √ 2 m E
2 π h 3
L
If t h e m a t e r ia l t a r g e t c a n b e le f t in m o r e t h a n o n e fi n a l s t a t e , w e s u m o v e r t h e s e fi n a l s t a t e s f . T h e n t h e a v e r a g e r a t e is g iv e n b y W S = f W f i ρ ( E ) d E d Ω ( a s s u m in g t h a t W f i d o e s n o t c h a n g e v e r y m u c h in d Ω a n d d E ) .
In c o m i n g F l u x
w e c a n e x p r e s s t h e t im e a s t = L / v , t h u s t h e fl u x is Φ = v 3
A t
T h e in c o m in g fl u x is g iv e n b y t h e n u m b e r o f s c a t t e r e r p e r u n it a r e a a n d u n it t im e , Φ = # . In t h e c a v it y c o n s id e r e d ,
L 3 . F o r p h o t o n s , t h is is s im p ly Φ = c / L , w h ile f o r m a s s iv e
m L 3
p a r t ic le s ( n e u t r o n s ) v = h k / m , y ie ld in g Φ = k k .
A v e r a g e o v e r i n i ti a l s ta te s
If t h e s c a t t e r e r is a t a fi n it e t e m p e r a t u r e T it w ill b e in a m ix e d s t a t e , t h u s w e n e e d t o s u m o v e r a ll p o s s ib le in it ia l s t a t e s :
ρ i =
e − β H M
Z
e − ǫ i /k b T
i
→ P i = L e − ǫ i /k b T
Σ Σ
W e c a n fi n a lly w r it e t h e t o t a l s c a t t e r in g r a t e a s :
W S ( i → Ω + d Ω , E + d E ) = ρ ( E ) P i W f i
i f
h 2
i
k f , k i
− ∞ − ∞
= ρ ( E ) Σ P i I ∞ e i ω f i t ( m | T †
f , i
( 0 ) | m
f
f
) ( m | T
( t ) | m ) d E d Ω = ρ ( E ) I ∞ e i ω f i t � T † ( 0 ) T
( t ) �
k f , k i
i
h 2
i f
f i
w h e r e ( ·) in d ic a t e s a n e n s e m b le a v e r a g e a t t h e g iv e n t e m p e r a t u r e .
1 2 . 1 . 2 T h e r m a l N e u t r o n S c a t t e r i n g
( L ) 3 m k f h k i ( m L 3 ) 2 k f
U s in g t h e s c a t t e r in g r a t e a b o v e a n d t h e in c o m in g fl u x a n d d e n s it y o f s t a t e e x p r e s s io n , w e c a n fi n d t h e c r o s s s e c t io n f o r t h e r m a l n e u t r o n s . F r o m
w e o b t a in
ρ ( E ) / Φ =
2 π h 2
/ m L 3
=
( 2 π h ) 3 k i
e f i
d 2 σ = h W = h ρ ( E ) 1 I ∞ i ω t
� T † ( 0 ) T f i ( t ) � =
1 ( m L 3 ) 2 k I ∞
e i ω f i t
� T † ( 0 ) T f i ( t ) �
f
d Ω d ω Φ Φ h 2 − ∞ i f 2 π 2 π h 2 k i − ∞ i f
| ) ( | ) −
I
N o w t h e e ig e n s t a t e s k i , f a r e p la n e w a v e s , r k = ψ k ( r ) = e i k · r / L 3 / 2 . T h e n , d e fi n in g Q = k i k f t h e t r a n s it io n m a t r ix e le m e n t is
T f i
( t ) = ( k f
| T ( t ) | k i ) = I
f
d 3 r ψ k
( r ) ∗ T ( r , t ) ψ k i
( r ) = 1 d 3 r e i Q · r T ( r , t ) L 3
a n d
L 3 L 3
T f i
( 0 ) † = 1
L 3
d 3 r e − i Q · r T ( r , 0 ) †
I
L 3
F e r m i P o te n ti a l
~
∼ ∼
T o fi r s t o r d e r , w e c a n a p p r o x im a t e T b y V , t h e n u c le a r p o t e n t ia l in t h e c e n t e r o f m a s s f r a m e ( o f t h e n e u t r o n + n u c le u s ) . Y o u m ig h t r e c a ll t h a t t h e n u c le a r p o t e n t ia l is a v e r y s t r o n g ( V 0 3 0 M e V ) a n d n a r r o w ( r 0 2 f m ) p o t e n t ia l. T h e s e c h a r a c t e r is t ic s s e e m t o p r e c lu d e a p e r t u r b a t iv e a p p r o a c h , s in c e t h e a s s u m p t io n o f a w e a k in t e r a c t io n ( c o m p a r e d t o t h e u n p e r t u r b e d s y s t e m e n e r g y ) is n o t s a t is fi e d . S t ill, t h e f a c t t h a t t h e p o t e n t ia l is n a r r o w m e a n s t h a t t h e in t e r a c t io n o n ly h a p p e n s f o r a v e r y s h o r t t im e . T h u s , if w e a v e r a g e o v e r t im e , w e e x p e c t a w e a k in t e r a c t io n . M o r e p r e c is e ly , t h e s c a t t e r in g in t e r a c t io n o n ly d e p e n d s o n t h e s o - c a lle d s c a t t e r i n g l e n g t h a , w h ic h is o n t h e o r d e r a V 0 r 0 . If w e k e e p a
~
≪
c o n s t a n t , d iff e r e n t c o m b in a t io n s o f V , r w ill g iv e t h e s a m e s c a t t e r in g b e h a v io r . W e c a n t h u s r e p la c e t h e s t r o n g n u c le a r p o t e n t ia l w it h a w e a k e r , p s e u d o - p o t e n t ia l V ˜ 0 , p r o v id e d t h is h a s a m u c h lo n g e r r a n g e r ˜ 0 , s u c h t h a t a V 0 r 0 = V ˜ 0 r ˜ 0 . W e c a n c h o o s e V ˜ 0 , r ˜ 0 s o t h a t t h e p o t e n t ia l is w e a k ( e V ) b u t t h e r a n g e is s t ill s h o r t c o m p a r e d t o t h e w a v e le n g t h o f t h e in c o m in g n e u t r o n , k r ˜ 0 1 . T h e n , it is p o s s ib le t o r e p la c e t h e p o t e n t ia l w it h a s im p le d e lt a - f u n c t io n a t t h e o r ig in .
2 π h 2
V ( r ) = a δ ( r ) µ
m n A
W e c a n a ls o d e fi n e t h e b o u n d s c a t t e r in g le n g t h , b = µ a ≈ A + 1 , w e r e m n is t h e n e u t r o n ’s m a s s a n d A t h e n u c le u s
m a s s n u m b e r . T h e n t h e p o t e n t ia l is
2 π h 2
V ( r ) = a δ ( r ) m n
N o t e t h a t b ( in t e r a c t io n le n g t h o r b o u n d s c a t t e r in g le n g t h ) is a f u n c t io n o f t h e p o t e n t ia l s t r e n g t h a n d r a n g e , w h ic h d e p e n d o n t h e is o t o p e f r o m w h ic h t h e n e u t r o n is s c a t t e r e d o ff .
T h e n t o fi r s t o r d e r t h e t r a n s it io n m a t r ix is T f i
p o s it io n r x ( t ) , w e h a v e :
= 2 π k 2 b , o r m o r e g e n e r a lly , if t h e r e a r e m a n y s c a t t e r e r s , e a c h a t a
m
n
T h e s c a t t e r in g c r o s s s e c t io n b e c o m e s
2 π h 2
Σ
m
T f i ( t ) = b x e
n x
i Q · r x ( t )
d 2 σ
1 k f I ∞
= e
f i
i ω t Σ
b x b y e
x
e
y
− i Q · r ( 0 ) i Q · r ( t )
d Ω d ω 2 π k i − ∞ x , y
N o t ic e t h a t s in c e t h e c o llis io n s a r e s p in - d e p e n d e n t , w e s h o u ld a v e r a g e o v e r is o t o p e s a n d s p in s t a t e s a n d r e p la c e b x b y
w it h b x b y .
S c a tte r i n g L e n g th s
N o t ic e t h a t b d o e s n o t d e p e n d e x p lic it ly o n p o s it io n , a lt h o u g h t h e p o s it io n d e t e r m in e s w h ic h is o t o p e / s p in w e s h o u ld
b ) δ x , y + b = b i + b c w h ic h d e fi n e s t h e c o h e r e n t s c a t t e r in g le n g t h b c = b a n d t h e in c o h e r e n t
c o n s id e r . W h a t is b x b y ? W e h a v e t w o c o n t r ib u t io n s . F o r x = y t h is is b 2 δ
x , y
, w h ile f o r x / / = /
2
2 2
y , it is b ( 1
— δ x , y
) . W e
x y
t h e n w r it e b b = ( b 2 — 2 2
i
b . If t h e r e a r e N s c a t t e r e r s , w e h a v e
x y
i
c
2
s c a t t e r in g le n g t h b 2 = b 2 —
L b b
= N ( b 2 + b 2 ) .
S tr u c tu r e F a c to r s
U s in g t h e s e d e fi n it io n , w e a r r iv e a t a s im p lifi e d e x p r e s s io n :
2
d σ = N k f � b 2 S ( Q , ω ) + b 2 S ( Q , ω ) �
d Ω d ω k i i S c
w h e r e w e u s e d t h e s e lf - d y n a m ic s t r u c t u r e f a c t o r
w h ic h s im p lifi e s t o
S ( Q , ω ) = 1 ∞ e i ω f i t 1 e − i Q · r x ( 0 ) e i Q · r x ( t )
I Σ
S
2 π − ∞ N x
2 π
S
S ( Q , ω ) = 1 I ∞ e i ω f i t D e − i Q · r ( 0 ) e i Q · r ( t ) E
− ∞
if a ll n u c le i a r e e q u iv a le n t ( s a m e is o t o p e ) , a n d t h e f u ll d y n a m ic s t r u c t u r e f a c t o r
2 π
N
S ( Q , ω ) = 1 I ∞ e i ω f i t 1 Σ e − i Q · r x ( 0 ) e i Q · r y ( t )
− ∞
x , y
T h e s t r u c t u r e f a c t o r s d e p e n d o n ly o n t h e m a t e r ia l p r o p e r t ie s . T h u s t h e y g iv e in f o r m a t io n a b o u t t h e m a t e r ia l w h e n o b t a in e d f r o m e x p e r im e n t s .
In te r m e d i a te S c a tte r i n g F u n c ti o n
F r o m t h e e x p r e s s io n s a b o v e f o r t h e s t r u c t u r e f a c t o r s , it is c le a r t h a t t h e y c a n b e o b t a in e d a s t h e F o u r ie r T r a n s f o r m ( w it h r e s p e c t t o t im e ) o f t h e q u a n t it ie s :
N
S
F ( Q , t ) = 1 * Σ e − i Q · r ( 0 ) e i Q · r x ( t ) +
a n d
x
N
F ( Q , t ) = 1 * Σ e − i Q · r x ( 0 ) e i Q · r y ( t ) +
x , y
T h e s e a r e c a lle d t h e in t e r m e d ia t e s c a t t e r in g f u n c t io n s . G o in g e v e n f u r t h e r , w e c a n w r it e e v e n t h e s e f u n c t io n a s a F o u r ie r T r a n s f o r m ( w it h r e s p e c t t o p o s it io n ) . F o r e x a m p le , f o r e q u iv a le n t t a r g e t s ( n o d is t r ib u t io n in is o t o p e n o r s p in ) , w e h a v e
F S ( Q , t ) = D e − i Q · r ( 0 ) e i Q · r ( t ) E
B y d e fi n in g a t h e p o s it io n o f a t e s t p a r t ic le , n ( R , t ) = δ ( R — r ( t ) ) , w e c a n c a lc u la t e t h e f o u r ie r t r a n s f o r m n ( Q , t ) :
n ( Q , t ) = I d 3 r e i Q · R n ( R , t ) = e i Q · r ( t )
T h e n w e h a v e F S ( Q , t ) = ( n ( Q , t ) n ( — Q , 0 ) ) . W e c a n a s w e ll d e fi n e t h e v a n - H o v e s p a c e - t im e s e lf c o r r e la t io n f u n c t io n ,
G s ( r , t ) = I d 3 r ′ ( n ( r ′ , 0 ) n ( r + r ′ , t ) )
w h ic h r e p r e s e n t s a c o r r e la t io n o f t h e t e s t p a r t ic le in s p a c e - t im e . T h e in t e r m e d ia t e s c a t t e r in g f u n c t io n is o b t a in e d f r o m G s a s
F S ( Q , t ) = I d 3 r e i Q · r G s ( r , t )
T h e s e fi n a l r e la t io n s h ip m a k e s it c le a r t h a t F S is t h e F o u r ie r t r a n s f o r m ( w it h r e s p e c t t o s p a c e ) o f t h e t im e - d e p e n d e n t c o r r e la t io n o f t h e t e s t p a r t ic le d e n s it y , n ( R , t ) , w h ic h o n ly d e p e n d s o n t h e t a r g e t c h a r a c t e r is t ic s .
E x a m p l e I: R e s ti n g , f r e e n u c l e u s
W e c o n s id e r t h e s c a t t e r in g f r o m o n e r e s t in g f r e e n u c le u s . W e n e e d o n ly c o n s id e r t h e s e lf d y n a m ic s f a c t o r a n d w e h a v e
b c = b = b :
d 2 σ k f 2
σ b k f
= b S ( Q , ω ) = S ( Q , ω )
d Ω d ω k i 2 π h k i
w h e r e w e in t r o d u c e d t h e b o u n d c r o s s s e c t io n σ b = 4 π b 2 ( w it h u n it s o f a n a r e a ) . S in c e t h e n u c le u s is f r e e , t h e in t e r m e d ia t e f u n c t io n is v e r y s im p le . F r o m
w e c a n u s e t h e B C H f o r m u la t o w r it e
F S ( Q , t ) = D e − i Q · r ( 0 ) e i Q · r ( t ) E
F S ( Q , t ) = e
2
D − i Q · [ r ( 0 ) − r ( t ) ] + 1 [ Q · r ( 0 ) , Q · r ( t ) ] E
T h e n w e w a n t t o c a lc u la t e [ r ( 0 ) , r ( t ) ] in o r d e r t o s im p lif y t h e p r o d u c t o f t h e t w o e x p o n e n t ia l. F o r a f r e e p a r t ic le ,
m
r ( t ) = r ( 0 ) + p t a n d [ r ( 0 ) , p ] = i h . T h e n w e h a v e
F S ( Q , t ) = e
2 m
= e e
D − i Q · [ r ( 0 ) − r ( t ) ] + i k Q 2 t E D − i Q · p /m E
2 m
+ i k Q 2 t
a n d f o r a n u c le u s a t r e s t ( p = 0 ) w e h a v e T h is g iv e s t h e s t r u c t u r e f a c t o r
a n d t h e c r o s s - s e c t io n
F ( Q , t ) = e i k t Q 2 / ( 2 m )
S
( )
h Q 2
S s ( Q , ω ) = δ ω — 2 m
=
d Ω d E 2 π h k i
δ ω —
2 m
S in c e Q = k f — k i , w e h a v e Q 2 = k 2 + k 2 — 2 k i k f c o s ϑ . A ls o , ω = E f — E i a n d k 2 = 2 m E a ≈ 2 A E a w h e r e w e
i f a
s u b s t it u t e d A f o r t h e m a s s o f t h e n u c le u s .
d E
W e c a n t h e n in t e g r a t e t h e c r o s s - s e c t io n o v e r t h e s o lid a n g le , t o fi n d d σ :
=
d E
δ ω —
2 m
2 π s in ϑ d ϑ =
4 E i
( A − 1 ) 2 / ( A + 1 ) 2 E i
δ ( x ) d x
d σ I π σ b k f ( h Q 2 ) A σ b I E i
D e fi n in g t h e f r e e - a t o m c r o s s s e c t io n σ f
w e h a v e
1 − 2
( )
σ f = 1 + A σ b
=
f
d σ { σ
( A + 1 ) 2 , f o r ( A − 1 f 2 E < E < E
4 A E A + 1
f
d E 0 , o t h e r w is e
T h is e x p r e s s io n f o r t h e c r o s s s e c t io n c a n a ls o b e o b t a in e d m o r e s im p ly f r o m a n e n e r g y c o n s e r v a t io n a r g u m e n t .
E x a m p l e I I: S c a tte r i n g f r o m a c r y s ta l l a tti c e
→
W e c o n s id e r n o w t h e s c a t t e r in g o f n e u t r o n s f r o m a c r y s t a l. F o r s im p lic it y , w e w ill c o n s id e r a o n e - d im e n s io n a l c r y s t a l la t t ic e m o d e le d a s a 1 D q u a n t u m h a r m o n ic o s c illa t o r . T h e p o s it io n r x ( in 1 D ) o f a n u c le u s in t h e la t t ic e is t h e n t h e p o s it io n o f a n h a r m o n ic o s c illa t o r o f m a s s M a n d f r e q u e n c y ω 0 ,
w it h e v o lu t io n g iv e n b y t h e H a m ilt o n ia n
x = h
J
2 M ω 0
( a + a † )
p 2 M ω 2 1
H 0
= + 0 x 2 = h ω ( a † a + ) 2 M 2 2
If w e c o n s id e r n o v a r ia t io n o f is o t o p e a n d s p in f o r s im p lic it y , w e o n ly n e e d t h e s e lf - in t e r m e d ia t e s t r u c t u r e f u n c t io n is
1
F S ( Q , t ) = e − i Q · x ( 0 ) e i Q · x ( t ) = e − i Q · [ x ( 0 ) − x ( t ) ] e + 2 [ Q · x ( 0 ) , Q · x ( t ) ]
F ir s t r e m e m b e r t h a t
p ( 0 )
0
x ( t ) = x ( 0 ) c o s ( ω 0 t ) + M ω
s in ( ω 0 t )
f o r a n h a r m o n ic o s c illa t o r . T h e n [ x ( 0 ) , x ( t ) ] = [ x ( 0 ) , p ( 0 ) ] 1 s in ( ω t ) = i k s in ( ω t ) . A ls o w e h a v e
0
2 M ω 0
M ω 0
0 M ω 0 0
0
M ω 0
∆ x ( t ) = x ( t ) — x ( 0 ) = x ( 0 ) [1 — c o s ( ω t ) ] +
p ( 0 ) s in ( ω t ) = J h
( a e − i ω 0 t + a † e i ω 0 t )
W e w a n t t o e v a lu a t e ( e i Q ∆ x ( t ) ) . U s in g a g a in t h e B C H f o r m u la , w e h a v e
e
2 M ω 0
= e = e e e
i Q � k ( ae − i ω 0 t + a † e i ω 0 t )
α a − α ∗ a † − α ∗ a †
α a − | α | 2 [ a, a † ] / 2
2 M ω 0
w it h α = i Q V k e − i ω 0 t . S in c e [ a , a † ] = 1 , w e o n ly n e e d t o e v a lu a t e t h e e x p e c t a t io n v a lu e D e − α ∗ a † e α a E , b y
n ∗ m
e x p a n d in g in s e r ie s t h e e x p o n e n t ia ls :
e e =
D − α ∗ a †
a a
α a E Σ ( † m
n , m
n ) α ( — α )
n ! m !
2 n
O n ly t h e t e r m s w it h m = n s u r v iv e ( t h e o t h e r t e r m s a r e n o t d ia g o n a l in t h e n u m b e r b a s is )
D − α ∗ a †
α a E Σ (
† n n ) ( — | α | )
e e =
n
N o w ( a † n a n ) = n ! ( ( a † a ) n ) , t h u s w e fi n a lly h a v e
( a a )
( n !) 2
e
e
=
( a a )
D − α ∗ a † α a E Σ ( †
n
n ) ( — | α | )
n !
− | α | 2 ( a † a )
2 n
( )
Q 1 1 i Q
= e
T h is r e s u lt is a p a r t ic u la r c a s e o f t h e B lo c h id e n t it y , e A = e ( A 2 ) / 2 w h e r e A = α a + β a † is a n y c o m b in a t io n o f t h e c r e a t io n a n d a n n ih ila t io n o p e r a t o r s . F in a lly , w e o b t a in e d f o r t h e in t e r m e d ia t e f u n c t io n :
F S ( Q , t ) = e − i Q · x ( 0 ) e i Q · x ( t )
2 k k 2
= e − 2 M ω 0 ( ( n ˆ ) + 2 ) e + 2 M ω 0 s i n ( ω 0 t )
( )
W e c a n a ls o r e w r it e t h is u s in g t h e B lo c h id e n t it y U s in g t h e B lo c h id e n t it y , e A = e ( A 2 ) / 2 w h e r e A = α a + β a † is a n y c o m b in a t io n o f t h e c r e a t io n a n d a n n ih ila t io n o p e r a t o r s , w e c a n r e w r it e t h is a s
F S ( Q , t ) = e e =
e
e
2
= e e
2
D − i Q · x ( 0 ) i Q · x ( t ) E ( i Q ∆ x ) + 1 [ Q · x ( 0 ) , Q · x ( t ) ] − Q 2 ( ∆ x 2 ) / 2 + 1 [ Q · x ( 0 ) , Q · x ( t ) ]
N o w ,
( ∆ x 2 ) = ( x ( 0 ) 2 ) + ( x ( t ) 2 ) + 2 ( x ( 0 ) x ( t ) ) — ( [ x ( 0 ) , x ( t ) ] ) = 2 ( x 2 ) + 2 ( x ( 0 ) x ( t ) ) — ( [ x ( 0 ) , x ( t ) ] )
f r o m w h ic h w e o b t a in
S
F ( Q , t ) = e − Q 2 ( x 2 ) e Q 2 ( x ( 0 ) x ( t ) )
If t h e o s c illa t o r is in a n u m b e r s t a t e | n ) , w e h a v e
( ) ( ) 0
h h
x 2 = ( 2 n + 1 ) , x ( 0 ) x ( t ) = [2 n c o s ( ω t ) + e i ω 0 t ] 2 M ω 0 2 M ω 0
If w e c o n s id e r a n o s c illa t o r a t t h e r m a l e q u ilib r iu m , w e n e e d t o r e p la c e n w it h ( n ) t h . In t h e h ig h t e m p e r a t u r e lim it ,
( n ) ≫ 1 a n d w e c a n s im p lif y :
— k Q 2 ( n ) [ 1 − c o s ( ω t ) ] − Q 2 W / 2 Q 2 W ( t ) / 2
F S ( Q , t ) = e M ω 0 0 = e 0 e
w it h W = 2 ( n ) k a n d W ( t ) = W c o s ( ω t ) . T h is f o r m o f t h e in t e r m e d ia t e f u n c t io n is t h e s a m e e x p r e s s io n o n e w o u ld
0 M ω 0 0 0
2
o b t a in f r o m a c la s s ic a l t r e a t m e n t a n d t h e t e r m e − Q W 0 / 2 is c a lle d t h e D e b y e - W a lle r f a c t o r .
—
T h e in t e r m e d ia t e s t r u c t u r e f u n c t io n is t h u s a G a u s s ia n f u n c t io n , w it h a t im e - d e p e n d e n t w id t h , W 0 W ( t ) . If W 0 < 1 w e c a n m a k e a n e x p a n s io n o f t h e t im e - d e p e n d e n t t e r m :
S
1 + W 0 c o s ( ω 0 t ) + 2 W 0 c o s ( ω 0 t ) + . . .
F ( Q , t ) = e − Q 2 W 0 / 2 e Q 2 W 0 c o s ( ω 0 t ) / 2 ≈ e − Q 2 W 0 / 2
1 2 2
± —
T h e n t h e s t r u c t u r e f a c t o r , w h ic h is t h e F o u r ie r t r a n s f o r m o f F S w ill b e a s u m o f D ir a c f u n c t io n s a t f r e q u e n c ie s ω = n ω 0 c o r r e s p o n d in g t o t h e n - p h o n o n c o n t r ib u t io n t o t h e s c a t t e r in g . H e r e t h e t e r m s δ ( ω n ω 0 ) c o r r e s p o n d t o s c a t t e r in g e v e n t s w h e r e t h e e n e r g y h a s b e e n t r a n s f e r e d f r o m t h e n e u t r o n t o t h e o s c illa t o r , w h ile t e r m s δ ( ω + n ω 0 ) d e s c r ib e a t r a n s f e r o f e n e r g y f r o m t h e la t t ic e t o t h e n e u t r o n . T h e c o n s t a n t t e r m y ie ld s δ ( ω ) w h ic h d e s c r ib e s n o e n e r g y e x c h a n g e o r e la s t ic s c a t t e r in g ( z e r o - p h o n o n t e r m ) . N o t e t h a t t h e e x p a n s io n c o e ffi c ie n t , W 0 c a n b e e x p r e s s e d in t e r m s
0
o f t h e t e m p e r a t u r e , s in c e in t h e h ig h t e m p e r a t u r e lim it , ( n ) ≈ k b T , f r o m w h ic h W
= 2 k b T .
In t h e lo w t e m p e r a t u r e lim it , ( n ) → 0 . T h u s w e h a v e :
k ω 0
0 M ω 2
— k Q 2 { 2 ( n ) [ 1 − c o s ( ω t ) ] + 1 − e i ω 0 t } − Q 2 k Q 2 k Q 2 e i ω 0 t
F S ( Q , t ) = e
2 M ω 0
0 ≈ e
2 M ω 0 e 2 M ω 0
E x p a n d in g in s e r ie s t h e s e c o n d t e r m , w e h a v e
F S ( Q , t ) ≈ e − Q
2 M ω 0 1 + e i ω 0 t + e 2 i ω 0 t + . . .
2 M ω 0 2 2 M ω 0
2 k Q 2 "
h Q 2 1 ( h Q 2 ) 2 #
—
E v e n a t lo w t e m p e r a t u r e , t h e s t r u c t u r e f a c t o r ( t h e F o u r ie r t r a n s f o r m o f t h e e x p r e s s io n a b o v e ) is a s u m o f D ir a c f u n c t io n , a ls o c a lle d a p h o n o n e x p a n s io n . H o w e v e r in t h is c a s e o n ly t e r m s δ ( ω n ω 0 ) a p p e a r , s in c e e n e r g y c a n o n ly b e g iv e n f r o m t h e n e u t r o n t o t h e la t t ic e ( w h ic h is in it ia lly in it s g r o u n d s t a t e ) .
1 2 . 2 E m i s s i o n a n d Ab s o r p t i o n
A t o m s a n d m o le c u le s c a n a b s o r b p h o t o n s a n d m a k e a t r a n s it io n f r o m t h e ir g r o u n d s t a t e t o a n e x c it e d le v e l. F r o m t h e e x c it e d s t a t e , t h e y c a n e m it p h o t o n s ( e it h e r in t h e p r e s e n c e o r a b s e n c e o f a p r e e x is t in g e .m . fi le d ) a n d t r a n s it io n t o a lo w e r le v e l. U s in g T D P T a n d t h e q u a n t iz a t io n o f t h e fi e ld w e c a n c a lc u la t e t h e t r a n s it io n r a t e s .
1 2 . 2 . 1 E m i s s i o n
|e ,n>
|g ,n+1>
| � | �
F i g . 2 1 : M o d e l f o r e mi s s i o n : t h e a t o m ( m o l e c u l e ) m a k e s a t r a n s i t i o n f r o m t h e e x c i t e d l e v e l ( e ) t o t h e g r o u n d s t a t e ) g ) w h i l e t h e n u m b e r o f p h o t o n s i n t h e mo d e k , λ g o e s f r o m n t o n + 1 .
T h e r a t e o f e m is s io n is g iv e n s im p ly b y
2 π 2
W = | ( f | V | i ) | ρ ( E ) .
h f
W e s e p a r a t e t h e fi e ld a n d t h e a t o m ( o r m o le c u le ) le v e ls :
| i ) = | n k λ ) | e ) , | f ) = | n k λ + 1 ) | g )
A s w e a r e lo o k in g a t a t o m ic / o p t ic a l p r o c e s s e s t h e d ip o la r a p p r o x im a t io n is a d e q u a t e a n d t h e in t e r a c t io n is g iv e n b y :
V = — d k · E k = — e k r · E k . R e m e m b e r t h e e x p r e s s io n f o r t h e e le c t r ic fi e ld :
L 3
k λ
k λ
E k = Σ J 2 π h ω k ( a
e i k r + a †
e − i k r f k ǫ k λ
k , λ
· ≪
≪
h
2
k λ
T h e p o s it io n o f t h e e le c t r o n w h ic h m a k e s t h e t r a n s it io n c a n b e w r it t e n a s k r = R k + ρ k , w h e r e R k is t h e n u c le u s p o s it io n . S in c e t h e r e la t iv e p o s it io n o f t h e e le c t r o n w it h r e s p e c t t o t h e n u c le u s is ρ λ , w e c a n n e g le c t it a n d s u b s t it u t e r w it h R in t h e e x p o n e n t ia l ( ρ k k k 1 ) . T h is s im p lifi e s t h e c a lc u la t io n , s in c e R is n o t a n o p e r a t o r a c t in g o n t h e e le c t r o n s t a t e . T h e n , f r o m t h e r a t e :
w e o b t a in
W = 2 π | ( g | d k | e ) · ( n
+ 1 | E k | n k λ
) | ρ ( E f )
W =
ω k ′ , λ ′ ( n k λ + 1 | a k ′ λ ′ e + a k ′ λ ′ e
( 2 π e ) 2 Σ
L 3
k ′ , λ ′
( i k R † − i k R f
2
√
| n k λ ) ( g | k r · ǫ k ′ , λ ′ | e )
ρ ( E f )
S in c e w e a r e c r e a t i n g a p h o t o n , o n ly t e r m s ∝ a † s u r v iv e a n d s p e c ifi c a lly t h e t e r m w it h t h e c o r r e c t w a v e v e c t o r a n d p o la r iz a t io n : ( n k λ + 1 | a † k λ | n k λ ) = n k λ + 1 ( a ll o t h e r t e r m s a r e z e r o ) . T h e n w e h a v e :
( 2 π e ) 2 2
W = L 3 ω k , λ ( n k , λ + 1 ) |( g | k r · ǫ k , λ | e ) | ρ ( E f )
S in c e t h e a t o m is le f t in a s p e c ifi c fi n a l s t a t e , t h e d e n s it y o f s t a t e s is d e fi n e d b y t h e e .m . fi e ld :
ρ ( E f ) d E f = ρ ( h ω k ) h d ω k
A s ω k
= c k a n d ρ ( k ) d 3 k = L 3 k 2 d k d Ω = L 3 ω 2 d ω d Ω w e h a v e :
2 π 2 π c 3
( )
ρ ( E ) =
L 3 ω 2
2 π h c 3 d Ω
( | | )
W e d e fi n e t h e d ip o le t r a n s it io n m a t r ix e le m e n t f r o m t h e d ip o le o p e r a t o r d k = e k r , d g e = g d e . T h e r a t e o f e m is s io n is t h e n :
ω 3 2
2 π h c 3
W = k ( n k λ + 1 ) | k ǫ k λ · d k g e | d Ω
F r o m t h is e x p r e s s io n it e a s y t o s e e t h a t t h e r e a r e t w o c o n t r ib u t io n s t o e m is s io n : S p o n t a n e o u s e m is s io n :
ω 3 2
2 π h c 3
W = k | k ǫ k λ · d k g e | d Ω
w h ic h h a p p e n s e v e n in t h e v a c u u m e .m . a n d s t im u la t e d e m is s io n :
ω 3 2
2 π h c 3
W = k n k λ | k ǫ k λ · d k g e | d Ω
w h ic h h a p p e n s o n ly w h e n t h e r e a r e a lr e a d y n p h o t o n s o f t h e c o r r e c t m o d e .
S p o n ta n e o u s E m i s s i o n
k
θ
φ
|e ,0
d
ε 2
|g ,1 ε 1
F i g . 2 2 : G e o me t r y o f s p o n t a n e o u s e m i s s i o n
S in c e t h e p h o t o n s e m it t e d c a n h a v e a n y p o la r iz a t io n ǫ a n d a n y w a v e v e c t o r k k d ir e c t io n , w e h a v e t o s u m o v e r a ll p o s s ib ilit ie s . W e a s s u m e t h a t t h e d ip o le v e c t o r f o r m s a n a n g le ϑ w it h r e s p e c t t o t h e w a v e v e c t o r k . T h e n t h e t w o p o s s ib le p o la r iz a t io n v e c t o r s a r e p e r p e n d ic u la r t o k , a s in F ig . 2 2 . T h e r a t e is t h e s u m o f t h e r a t e s f o r e a c h p o la r iz a t io n W s p = W 1 + W 2 , e a c h p r o p o r t io n a l t o | d · ǫ k 1 , 2 | 2 ,
d · ǫ k , 1 = d s in ϑ c o s ϕ , d · ǫ k , 2 = d s in ϑ s in ϕ
W e t h u s o b t a in t h e t y p ic a l s in 2 ϑ a n g u la r d e p e n d e n c e o f d ip o la r r a d ia t io n ( a ls o s e e n f o r c la s s ic a l d ip o le s ) :
W s p = k | d g e |
ω 3
2 π h c 3
2 s in 2
ϑ d Ω
T h e t o t a l e m is s io n c o e ffi c ie n t , o r E in s t e in ’s e m is s io n c o e ffi c ie n t , is o b t a in e d b y in t e g r a t in g o v e r t h e s o lid a n g le :
2 2
A e = W d Ω =
k
| d g e | 2 π ( 1 — µ ) d µ =
k d 2
I ω 3 I 1 4 ω 3
Ω 2 π h c 3 − 1 3 h c 3
g e
G iv e n t h e r a t e , w e c a n a ls o c a lc u la t e t h e p o w e r e m it t e d , a s r a t e t im e s e n e r g y
P = h ω A
4 ω 4
= 2
k e k d g e
3
3 c
N o t ic e t h a t t h is is v e r y s im ila r t o t h e p o w e r e m it t e d b y a c l a s s i c a l o s c illa t in g d ip o le ( a s if t h e e .m . fi e ld w a s e m it t e d b y o r b it in g e le c t r o n s ) .
S ti m u l a te d E m i s s i o n
≫
s t s p
In t h e s t im u la t e d e m is s io n , W k λ = n k λ W k λ . O n ly p h o t o n s w it h t h e s a m e f r e q u e n c y ( k k ) a n d p o la r iz a t io n o f t h e o n e s a lr e a d y in t h e fi e ld c a n b e e m it t e d . T h e n , a s m o r e p h o t o n s in a p a r t ic u la r m o d e a r e e m it t e d , it b e c o m e s e v e n m o r e p r o b a b le t o p r o d u c e p h o t o n s in t h e s a m e m o d e : w e p r o d u c e a b e a m o f c o h e r e n t p h o t o n s ( i.e . a ll w it h t h e s a m e c h a r a c t e r is t ic s a n d p h a s e c o h e r e n t w it h e a c h o t h e r ) . If t h e a t o m s c a n b e k e p t in t h e e x c it e d ( e m it t in g ) le v e ls , w e o b t a in a L A S E R ( lig h t a m p lifi c a t io n b y s t im u la t e d e m is s io n o f r a d ia t io n ) . O f c o u r s e , u s u a lly it is m o r e p r o b a b le t o h a v e t h e p h o t o n s a b s o r b e d t h a n t o h a v e it c a u s e a s t im u la t e d e m is s io n , s in c e a t e q u ilib r iu m w e u s u a lly h a v e m a n y m o r e a t o m s in t h e g r o u n d s t a t e t h a n in t h e e x c it e d s t a t e , n g n e . A m e c h a n is m c a p a b le o f in v e r t in g t h e p o p u la t io n o f t h e a t o m ic s s t a t e s ( s u c h a s o p t ic a l p u m p in g ) is t h e n n e e d e d t o s u p p o r t a la s e r .
1 2 . 2 . 2 A b so r p t i o n
T h e r a t e o f a b s o r p t io n is o b t a in e d in a w a y v e r y s im ila r t o e m is s io n . T h e r e s u lt is
2 π k
2 ω 3 2
2 π h c 3
W = h | ( e | d | g ) · ( n k λ | E k | n k λ + 1 ) | ρ ( E f ) = k n k λ | k ǫ k λ · d k e g | d Ω
( a s ( n k λ | a k λ | n k λ + 1 ) = √ n k λ ) .
1 2 . 2 . 3 B l a c kb o d y R a d i a t i o n
W e c o n s id e r a c a v it y w it h r a d ia t io n in e q u ilib r iu m w it h it s w a ll. T h e n t h e p o la r iz a t io n a n d k k - v e c t o r o f t h e p h o t o n s is r a n d o m , a n d t o o b t a in t h e t o t a l a b s o r p t io n r a t e w e n e e d t o in t e g r a t e o v e r it , a s d o n e f o r t h e e m is s io n . W e o b t a in
W ab = W ab ( ϑ ) d Ω = n k
k d g e
I 4 ω 3 2
Ω 3 h c 3
f o r a g iv e n f r e q u e n c y ( a n d w a v e v e c t o r le n g t h ) . S im ila r ly , t h e t o t a l e m is s io n is o b t a in e d a s t h e s u m o f s p o n t a n e o u s a n d s t im u la t e d e m is s io n :
4 ω 3 2
h 3
W e = W s t + W s p = ( n k + 1 ) k d g e
3 c
In t h e s e e x p r e s s io n n k is t h e n u m b e r o f p h o t o n s in t h e m o d e k . S in c e w e a s s u m e d t o b e a t e q u ilib r iu m , n k d e p e n d s o n ly o n t h e e n e r g y d e n s it y a t t h e a s s o c ia t e d f r e q u e n c y ω k . T h e e n e r g y d e n s it y is g iv e n b y t h e e n e r g y p e r v o lu m e , w h e r e t h e e n e r g y is g iv e n b y t h e t o t a l n u m b e r o f p h o t o n s t im e s t h e ir e n e r g y , E = n k ρ ( ω k ) h ω k :
u ( ω k ) = h ω k ρ ( ω k ) n k / L 3
k
2 π c c 3 π 2
3
T h e n , f r o m t h e d e n s it y o f s t a t e s ρ ( ω ) = 2 L 3 ω 2 � d Ω =
L ω 2 , w e o b t a in
n k =
π 2 c 3
h ω 3 u ( ω k )
k
T h e r a t e s c a n t h e n b e w r it t e n in t e r m s o f t h e e n e r g y d e n s it y a n d o f E in s t e in ’s c o e ffi c ie n t s f o r a b s o r p t io n a n d e m is s io n :
4 π 2 2
B ab = 3 h 2 d
4 ω 3 2
→ W ab = B ab u ( ω k )
h 3
B e m = B ab , A e = k d g e
3 c
→ W e m = A e + B e m u ( ω k )
D e ta i l e d B a l a n c i n g
e m
ab
A t e q u ilib r iu m , w e n e e d t o h a v e t h e s a m e n u m b e r o f p h o t o n s a b s o r b e d a n d e m it t e d ( t o p r e s e r v e t h e ir t o t a l n u m b e r ) . T h e n N e W k = N g W k . U s in g E in s t e in ’s c o e ffi c ie n t , w e h a v e N e ( A + u B ) = N g B u w h ic h y ie ld s N e A = u B ( N g — N e ) .
T h is is t h e p r in c ip le o f d e t a ile d b a la n c in g .
W e c a n s o lv e f o r t h e e n e r g y d e n s it y : u = A / B . B u t f r o m t h e ir e x p lic it e x p r e s s io n s w e h a v e A / B = k ω 3 a n d f r o m
N g / N e − 1 π 2 c 3
t h e c o n d it io n t h a t a t o m s a r e in t h e r m a l e q u ilib r iu m , t h e ir p o p u la t io n r a t io is g iv e n b y N g = e − β E g
= e − β ( E g − E e ) =
N e e − β E e
e β k ω k ( s in c e h ω k is t h e e x a c t e n e r g y n e e d e d f o r t h e t r a n s it io n f r o m g r o u n d t o e x c it e d s t a t e ) . F in a lly , w e o b t a in t h e e n e r g y d e n s it y s p e c t r u m f o r t h e b la c k - b o d y :
h ω 3 / π 2 c 3
u ( ω k , T ) = e β k ω k — 1
1 2 . 3 W i g n e r - W e i s s k o p f Th e o r y
1 2 . 3 . 1 I n t e r a c t i o n o f a n a t o m w i t h a s i n g l e m o d e e . m . fi e l d
R e c a ll w h a t w e s t u d ie d in S e c t io n 1 0 .5 . W e c o n s id e r a g a in a t w o - le v e l s y s t e m ( a n a t o m ) in t e r a c t in g w it h a s in g le m o d e o f t h e e .m . fi e ld . T h e H a m ilt o n ia n s im p lifi e s t o H = H 0 + V , w it h
H = h ν a † a + h ω σ , V = h g ( σ a + σ
a † )
0 2 z + −
w h e r e g = 1 J ν 3 d · ǫ is t h e d ip o le o p e r a t o r .
2 k L
W e m o v e t o t h e in t e r a c t io n f r a m e d e fi n e d b y t h e H 0 H a m ilt o n ia n , U = e i H 0 t , t h e n H I = U V U † o r
+
I
H = h g e i ν t a † a e i ω σ z t / 2 ( σ
a + σ −
a † ) e − i ν t a † a e − i ω σ z t / 2 = h g � e i ( ω − ν ) t σ
a + e − i ( ω − ν ) t σ −
a † �
+
W e w ill u s e t h e n o t a t io n ∆ = ω — ν . W e w a n t n o w t o s t u d y t h e e v o lu t io n o f a p u r e s t a t e in t h e in t e r a c t io n f r a m e :
i h ψ ˙ = H I | ψ ) . W e c a n w r it e a g e n e r a l s t a t e a s | ψ ) = L n α n ( t ) | e , n ) + β n ( t ) | g , n ) . N o t ic e t h a t s in c e w e h a v e a T L S ,
σ + | e ) = 0 a n d σ − | g ) = 0 . T h e e v o lu t io n is t h e n g iv e n b y :
i h Σ α ˙ n | e , n ) + β ˙ n | g , n ) = h g Σ � α n σ − a † e − i ∆ t | e , n ) + β n σ + a e i ∆ t | g , n ) �
n n
n
= h g Σ � α n e − i ∆ t √ n + 1 | g , n + 1 ) + β n e i ∆ t √ n | e , n — 1 ) � W e t h e n p r o j e c t t h e s e e q u a t io n s o n ( e , n | a n d ( g , n | :
i h α ˙ n = h g β n + 1 ( t ) e i ∆ t √ n + 1
i h β ˙ n = h g α n − 1 ( t ) e − i ∆ t √ n
t o o b t a in a s e t o f e q u a t io n s :
i ∆ t √
{ α ˙ n = — i g β n + 1 e n + 1
β ˙ n + 1 = — i g α n e − i ∆ t √ n + 1
T h is is a c lo s e d s y s t e m o f d iff e r e n t ia l e q u a t io n s a n d w e c a n s o lv e f o r α n , β n + 1 .
F o r e x a m p le : w e c a n a s s u m e t h a t in it ia lly t h e a t o m is in t h e e x c it e d s t a t e | e ) a n d it d e c a y s t o t h e g r o u n d s t a t e | g )
( t h a t is , β n ( 0 ) = 0 , ∀ n ) . T h e n w e h a v e :
α n ( t ) = α n ( 0 ) e i ∆ t / 2 c o s ( ) — s in ( )
Ω n t i ∆ Ω n t
2 Ω n 2
β n ( t ) = — α n
( 0 ) e − i ∆ t / 2 2 i g √ n + 1 s in ( Ω n t )
Ω n 2
n
w it h Ω 2 = ∆ 2 + 4 g 2 ( n + 1 ) . If in it ia lly t h e r e is n o fi e ld ( i.e . t h e e .m . fi e ld is in t h e v a c u u m s t a t e ) t h e n α 0 ( 0 ) = 1 ,
w h ile α n ( 0 ) = 0 ∀ n = /
0 . T h e n t h e r e a r e o n ly t w o c o m p o n e n t s t h a t a r e d iff e r e n t t h a n z e r o :
α ( t ) = e i ∆ t / 2 " c o s ( Ω 0 t ) — J i ∆ s in ( Ω 0 t ) #
0 2
− i ∆ t / 2
1
∆ 2 + 4 g 2 2
β ( t ) = — e J
2 i g
∆ 2 + 4 g 2 2
s in ( Ω 0 t )
T h u s , e v e n in t h e a b s e n c e o f fi e ld , it is p o s s ib le t o m a k e t h e t r a n s it io n f r o m t h e g r o u n d t o t h e e x c it e d s t a t e ! In t h e s e m ic la s s ic a l c a s e ( w h e r e t h e fi e ld is t r e a t e d a s c la s s ic a l) w e w o u ld h a v e n o t r a n s it io n a t a ll. T h e o s c illa t io n s o b t a in e d in t h e q u a n t u m c a s e a r e c a lle d t h e v a c u u m R a b i o s c illa t io n s .
1 2 . 3 . 2 I n t e r a c t i o n w i t h m a n y m o d e s o f t h e e . m . fi e l d
In a n a ly z in g t h e in t e r a c t io n o f a n a t o m w it h a s i n g l e m o d e o f r a d ia t io n w e f o u n d t h a t t r a n s it io n s c a n o c c u r o n ly if e n e r g y is c o n s e r v e d . In t h e r e a l w o r ld h o w e v e r w e a r e a lw a y s c o n f r o n t e d w it h a fi n it e lin e w id t h o f a n y t r a n s it io n . In o r d e r t o fi n d t h e lin e w id t h w e n e e d t o lo o k a t a m u lt i- m o d e fi e ld .
C o n s id e r t h e s a m e H a m ilt o n ia n a s u s e d in t h e p r e v io u s s e c t io n , b u t n o w w e t r e a t a fi e ld w it h m a n y m o d e s . T h e in t e r a c t io n H a m ilt o n ia n in t h e in t e r a c t io n f r a m e is g iv e n b y
k
V I = h Σ g k ∗ a k σ + e i ( ω − ν k ) t + g k a † σ − e − i ( ω − ν k ) t
k
Σ
W e c o n s id e r a c a s e s im ila r t o t h e o n e c o n s id e r a t t h e e n d o f t h e p r e v io u s s e c t io n , w h e r e in it ia lly t h e e .m . fi e ld is in t h e v a c u u m s t a t e a n d t h e a t o m ic t r a n s it io n c r e a t e s o n e p h o t o n . N o w , h o w e v e r , t h is p h o t o n c a n b e in o n e o f m a n y m o d e s . T h e s t a t e v e c t o r is t h e n :
| ψ ( t ) ) = α ( t ) | e , 0 ) + β k | g , 1 k )
k
( n o w t h e in d e x k in β k la b e l t h e m o d e a n d n o t t h e p h o t o n n u m b e r ) a n d t h e in it ia l c o n d it io n s a r e α ( 0 ) = 1 , β k ( 0 ) = 0 ,
{ L
∀ k . T h e s y s t e m o f e q u a t io n s f o r t h e c o e ffi c ie n t s a r e
α ˙ ( t ) = — i k g k ∗ e i ( ω − ν k ) t β k ( t )
β ˙ k ( t ) = — i g k e − i ( ω − ν k ) t α ( t )
˙
′
k
If w e c o n s id e r t h is t r a n s it io n a s a d e c a y p r o c e s s f r o m t h e e x c it e d t o t h e g r o u n d s t a t e , | α ( t ) | 2 g iv e s t h e d e c a y p r o b a b ilit y . T o s o lv e f o r α ( t ) w e fi r s t in t e g r a t e β :
α ˙ = — i g k e
k
k
— i g k e
Σ ∗ i ( ω − ν ) t ( I t
0
− i ( ω − ν ) t
′ ′ )
α ( t ) d t
W e c a n r e w r it e t h e e x p r e s s io n a s :
Σ
I
t
2
0
α ˙ = — | g k |
k
d t ′ e
− i ( ω − ν k ) ( t − t )
′
α ( t ′ )
L
A s s um pt i o n 1 )
� ρ ( k ) d 3 k , w it h t h e d e n s it y o f s t a t e s s e t b y ν
= c k a s u s u a l: ρ ( k ) d 3 k = 2 L 3 k 2 d k d ϕ s in ϑ d ϑ .
W e a s s u m e t h a t t h e m o d e s o f t h e e .m . f o r m a c o n t in u u m , s o t h a t w e c a n r e p la c e t h e s u m b y a n in t e g r a l k →
k
2 π
2
W e t h e n r e m e m b e r t h e e x p lic it f o r m o f t h e in t e r a c t io n c o u p lin g in t e r m s o f t h e d ip o le o p e r a t o r :
= | d |
a n d u s in g a g a in ν k = c k w e o b t a in
| g k |
ν k
4 h L 3 e g
2 s in 2 ϑ
4 | d e g | 2 I ∞
3 I t
′
′ − i ( ω − ν ) ( t − t ) ′
A s s um pt i o n 2 )
α ˙ = — ( 2 π ) 2 6 h c 3
ν k d ν k d t e
0 0
k α ( t )
≈
In o r d e r f o r t h e t r a n s it io n t o h a p p e n , w e s t ill n e e d ν k ω .
k
— ∞ ≈
T h is a llo w s t w o s im p lifi c a t io n s : i) w e c a n r e p la c e ν 3 w it h ω 3 in t h e in t e g r a l, a n d ii) w e c a n e x t e n d t h e lo w e r in t e g r a l lim it t o ( s in c e a n y w a y w e k n o w t h a t it w ill g iv e c o n t r ib u t io n s o n ly f o r ν k ω ) . B y f u r t h e r m o r e in v e r t in g t h e o r d e r o f t h e in t e g r a ls w e o b t a in
ν k d ν k
I ∞ 3
I t ′
I t ′
′ 3 I ∞
′
d ν k e
k
− i ( ω − ν ) ( t − t )
I t ′ ′ 3 ′ 3
d t · · · →
d t α ( t ) ω
=
d t α ( t ) ω 2 π δ ( t — t ) = 2 π α ( t ) ω
0 0 0 − ∞ 0
T h u s , t h e d iff e r e n t ia l e q u a t io n d e fi n in g t h e e v o lu t io n o f α ( t ) s im p lifi e s t o
1 d 2 ω 3 Γ
— —
α ˙ ( t ) = e g α ( t ) = α ( t ) 2 π 3 h c 3 2
H e r e w e d e fi n e d t h e r a t e o f s p o n t a n e o u s e m is s io n
e g
d 2 ω 3
Γ =
3 π h c 3
N o t ic e t h a t t h e d e c a y r a t e is r e la t e d t o E in s t e in ’s e m is s io n r a t e , a s Γ = A e / 4 π a s w e s h o u ld e x p e c t , s in c e it is r e la t e d t o t h e t o t a l e m is s io n ( a t a n y f r e q u e n c y ) f r o m t h e e x c it e d t o t h e g r o u n d s t a t e .
T h u s w e h a v e s im p ly α ( t ) = e − Γ t / 2 a n d t h e d e c a y p r o b a b ilit y P d = e − Γ t .
.0
.8
.6
.4
.2
5
10
15
20
1
0
0
0
0
F i g . 2 3 : Lo r e n t z i a n l i n e s h a p e , c e n t e r e d a t ω = 1 2 a n d w i t h a l i n e w i d t h Γ = 2
F r o m t h e e x p r e s s io n f o r α ( t ) w e c a n g o b a c k a n d c a lc u la t e a n e x p lic it f o r m f o r β k ( t ) :
—
I t ′
− i ( ω − ν ) t
− Γ t ′ / 2 1 — e − i ( ω − ν k ) t e − Γ t / 2
β k ( t ) = i d t g k e
′
0
k e = g k
( ν k — ω ) + i Γ / 2
Ω
T h e f r e q u e n c y s p e c t r u m o f t h e e m it t e d r a d ia t io n is g iv e n b y P ( ν k ) = ρ ( ν k ) L λ = 1 , 2 � d Ω | β k ( t ) | 2 in t h e lim it w h e r e
2
k
t → ∞ . 1 + e − Γ t ( 1 — 2 c o s [( ω — ν ) t ] Γ 2
P ( ν k ) ∝ lim ∼ 1 / + ( ω — ν k )
4
k
t → ∞ Γ 2 + ( ω — ν ) 2 4
T h u s t h e s p e c t r u m is a L o r e n z t ia n c e n t e r e d a r o u n d ω a n d w it h lin e w id t h Γ .
1 2 . 4 S c a t t e r i n g o f p h o t o n s b y a t o m s
In t h is s e c t io n w e w a n t t o s t u d y t h e s c a t t e r in g o f p h o t o n s b y e le c t r o n s ( e it h e r f r e e e le c t r o n s o r in a n a t o m ) . W e p r e v io u s ly s t u d ie d s im ila r p r o c e s s e s :
- S c a t t e r in g t h e o r y ( w it h a n e x a m p le f o r t h e r m a l n e u t r o n s )
- E m is s io n a n d a b s o r p t io n o f p h o t o n s ( in t h e d ip o le a p p r o x im a t io n )
N o t ic e t h a t t h e s e la s t p r o c e s s e s o n ly in v o lv e d a s in g le p h o t o n ( e it h e r a b s o r b e d o r e m it t e d ) . N o w w e w a n t t o s t u d y t h e s c a t t e r in g o f p h o t o n s , m e a n in g t h a t t h e r e w ill b e a n in c o m in g p h o t o n a n d a n o u t g o in g p h o t o n : t h is is a p r o c e s s t h a t in v o lv e s t w o p h o t o n s .
k ’ '
|A f
|A i
k
F i g . 2 4 : P h o t o n s c a t t e r i n g c a r t o o n
In o r d e r t o s t u d y a t o m - p h o t o n in t e r a c t io n w e n e e d o f c o u r s e t o s t a r t f r o m t h e q u a n t iz e d e .m . fi e ld :
p ˜ 2 1 1 ( e A ) 2 1
—
H = 2 m + h ω ( n + 2 ) = 2 m p c
W e c a n s e p a r a t e t h e in t e r a c t io n H a m ilt o n ia n a s :
+ h ω ( n + )
2
p 2 1 e e 2 2
H = H 0 + V = 2 m + h ω ( n + 2 ) + — 2 m c ( p · A + A · p ) + 2 m c 2 A
" H v V 0 J " v V V J
M o r e g e n e r a lly , if t h e r e a r e m a n y e le c t r o n s , t h e in t e r a c t io n H a m ilt o n ia n is g iv e n b y
V = — 2 m c [ p i · A ( r i ) + A ( r i ) · p i ] + 2 m c 2 A ( r i )
i
Σ e e 2 2
m c
W e a lr e a d y u s e d t h e fi r s t t e r m ( in t h e d ip o le a p p r o x im a t io n e p · A → d · E ) t o fi n d e m is s io n a n d a b s o r p t io n
p r o c e s s e s . A s s t a t e d , t h e s e p r o c e s s e s o n ly in v o lv e o n e p h o t o n . H o w d o w e o b t a in p r o c e s s e s t h a t in v o lv e t w o p h o t o n s ?
S in c e f r o m t h e t e r m p · A a n d in t h e fi r s t o r d e r p e r t u r b a t io n t h e o r y w e d o n o t g e t t h e m , w e w ill n e e d
i) e it h e r t e r m s ∝ A 2 , o r
ii) s e c o n d o r d e r p e r t u r b a t io n f o r t h e t e r m ∝ p · A .
k c
N o t ic e t h a t b o t h t h e s e c h o ic e s y ie ld t r a n s it io n s t h a t a r e ∝ α 2 = ( e 2 f 2 , t h a t is , t h a t a r e s e c o n d o r d e r in t h e fi n e
s t r u c t u r e c o n s t a n t .
k 1 2
T h u s w e w a n t t o c a lc u la t e s c a t t e r in g t r a n s it io n r a t e s g iv e n b y W = 2 π | K ( 2 ) + K ( 1 ) | 2 ρ ( E f ) , w h e r e
1
m c
i
• K ( 2 ) is t h e 2 n d o r d e r c o n t r ib u t io n f r o m V 1 = — e L p i · A i a n d
2 L 2
2
2
2 m c i
i
• K ( 1 ) is t h e 1 s t o r d e r c o n t r ib u t io n f r o m V = e A .
1
K ( 1 ) is in s t e a d z e r o , s in c e it o n ly c o n n e c t s s t a t e t h a t d iff e r b y o n e p h o t o n ( t h u s it ’s n o t a s c a t t e r in g p r o c e s s ) a n d w e n e g le c t h ig h e r o r d e r s t h a n t h e s e c o n d .
T h e in it ia l a n d fi n a l e ig e n s t a t e s a n d e ig e n v a lu e s a r e a s f o llo w ( w h e r e γ in d ic a t e t h e p h o t o n ) :
In it ia l
e − : | A i )
γ : | 1 k , λ , 0 k ′ , λ ′ )
t o t : | i )
F in a l
| A f )
| 0 k , λ , 1 k ′ , λ ′ )
| f )
In . E n e r g y F in . E n e r g y
ǫ i ǫ f
h ω k h ω k ′
E i E f
2
W e fi r s t e v a lu a t e K ( 1 ) f o r a s in g le e le c t r o n . W e r e c a ll t h e e x p r e s s io n f o r t h e v e c t o r p o t e n t ia l ( s e e S e c t io n 1 0 .3 ) :
A =
a k λ e + a k λ e
k ǫ k λ .
k Σ � 2 π h c 2 ( i k · r † − i k · r f
k , λ
L 3 ω
k
2
K ( 1 ) is p r o p o r t io n a l t o A 2 , b u t w e o n ly r e t a in t e r m s t h a t lin k t h e c o r r e c t m o d e s ( k , k ′ ) a n d t h a t a r e r e s p o n s ib le f o r t h e a n n ih ila t io n o f a p h o t o n in m o d e k a n d t h e c r e a t io n o f a p h o t o n o f m o d e k ′ . T h e s e a r e t e r m s ∝ a † k ′ a k . W e fi n d :
( 1 ) e 2
2 π k c 2
k
k
✭ k λ
k ′ λ ′ e
i ( k + k )
✭ k ✭ λ
k ′ λ ′ e
K 2 = ( f | V 2 | i ) = 2 m c 2 L 3 √ ω ω ′ k ǫ k λ · ǫ k k ′ λ ′ ✭
✭ ✭ ✭
k λ
k ′ λ ′
✭ k λ
k ′ λ ′ e
− k )
× ( f | a a †
e i ( k − k ′ ) · r + a † ✭ a ✭ ✭ ✭ − i ( ✭ k ✭ ′ · r + a † ✭ a ✭ †
✭ ✭ − ✭ ✭ ′ · r + a
a ✭ ✭ ✭ i ( k ✭ + k ′ ) · r | i )
| | —
W e n o w u s e t h e e q u a lit y ω k = c k a n d k k k k ′ = k q = p k / h ( t h e e le c t r o n r e c o il m o m e n t u m ) t o s im p lif y t h e e x p r e s s io n . T h u s w e o b t a in :
K
( 1 )
2
e 2
=
2 m c
2 π h c 2
L
k k
3 √
′ k ǫ k λ · ǫ k k ′ λ ′ ( A f | e
i � q · � r
| A i ) ( 0 k λ 1 k ′ λ ′ | a k λ a k † ′ λ ′ | 1 k λ 0 k ′ λ ′ ) ,
2
w h e r e t h e la s t in n e r p r o d u c t is j u s t e q u a l t o 1 . W e c a n n o w e x t e n d K ( 1 ) t o m a n y e le c t r o n s :
( 1 )
e 2 2 π h c
Σ i � q · � r
k k ′ i
K 2 = ( f | V 2 | i ) = 2 m L 3 √ k ǫ k λ · ǫ k k ′ λ ′ ( A f | e
i | A i ) .
T h is is t h e fi r s t c o n t r ib u t io n t o t h e s c a t t e r in g m a t r ix e le m e n t , fi r s t o r d e r in p e r t u r b a t io n t h e o r y f r o m t h e q u a d r a t ic t e r m in t h e fi e ld p o t e n t ia l.
1
W e n o w w a n t t o c a lc u la t e K ( 2 ) , t h e s e c o n d o r d e r c o n t r ib u t io n f r o m t h e lin e a r p a r t V 1 o f t h e p o t e n t ia l:
K = Σ
( 2 ) ( f | V 1 | h ) ( h | V 1 | i )
h
1 E i — E h
1
N o t e t h a t t h is t e r m d e s c r ib e s v ir t u a l t r a n s it io n s t o in t e r m e d ia t e s t a t e s s in c e f r o m fi r s t o r d e r t r a n s it io n s V 1 c a n o n ly c r e a t e o r a n n ih ila t e o n e p h o t o n a t a t im e . S o t h e r e a r e t w o p o s s ib le p r o c e s s e s t h a t c o n t r ib u t e t o K ( 2 ) ,
- fi r s t a b s o r p t io n o f o n e p h o t o n in t h e k λ m o d e f o llo w e d b y c r e a t io n o f o n e p h o t o n in t h e k ′ λ ′ m o d e : t h e in t e r m e d ia t e s t a t e is z e r o p h o t o n s in t h e s e t w o m o d e s .
- fi r s t c r e a t io n o f o n e p h o t o n in t h e k ′ λ ′ m o d e f o llo w e d b y a n n ih ila t io n o f t h e p h o t o n in m o d e k λ : t h e in t e r m e d ia t e s t a t e is o n e p h o t o n in e a c h m o d e .
E x p lic it ly w e h a v e :
1
ǫ i — ǫ h + h ω k
K ( 2 ) = Σ ( A f | ( 0 k λ 1 k ′ λ ′ | V 1 | 0 k λ 0 k ′ λ ′ ) | A h ) ( A h | ( 0 k λ 0 k ′ λ ′ | V 1 | 1 k λ 0 k ′ λ ′ ) | A i )
h
ǫ i + h ω k — ( ǫ h + h ω k + h ω k ′ )
+ Σ ( A f | ( 0 k λ 1 k ′ λ ′ | V 1 | 1 k λ 1 k ′ λ ′ ) | A h ) ( A h | ( 1 k λ 1 k ′ λ ′ | V 1 | 1 k λ 0 k ′ λ ′ ) | A i )
h
1 2
1 2
N o t ic e t h a t K ( 2 ) h a s a n e x t r a f a c t o r ∝ ω k in t h e d e n o m in a t o r w it h r e s p e c t t o K ( 1 ) . T h u s a t h ig h e r e n e r g ie s o f t h e in c id e n t p h o t o n ( s u c h a s x - r a y s c a t t e r in g ) o n ly K ( 2 ) s u r v iv e s , w h ile a t lo w e r e n e r g ie s ( o p t ic a l r e g im e ) K ( 1 ) is m o r e
im p o r t a n t .
A . T y p e s o f S c a tte r i n g
D e p e n d in g o n t h e e n e r g y h ω o f t h e in c id e n t p h o t o n ( w it h r e s p e c t t o t h e io n iz a t io n e n e r g y E I o f t h e a t o m ) a n d o n t h e e la s t ic o r in e la s t ic c h a r a c t e r o f t h e s c a t t e r in g , t h e s c a t t e r in g p r o c e s s is d e s ig n a t e d w it h d iff e r e n t n a m e s .
≪ | — |
- R a y le ig h s c a t t e r in g ( L o w e n e r g y , E la s t ic ) : h ω E I , E h E I , E f = E I .
T h e fi n a l s t a t e h a s t h e s a m e e n e r g y a s t h e in it ia l o n e , E f = E i s in c e t h e s c a t t e r in g is e la s t ic . T h e s c a t t e r in g t h u s in v o lv e in t e r m e d ia t e v ir t u a l le v e ls , w it h e n e r g ie s E h . W e w ill fi n d a c r o s s s e c t io n σ ∝ ω 4 .
≪ /
- R a m a n s c a t t e r in g ( L o w e n e r g y , In e la s t ic ) : h ω E I , E f = E I .
U s u a lly t h e fi n a l s t a t e is a d iff e r e n t r o t o v ib r a t io n a l s t a t e o f t h e m o le c u le , s o t h e e n e r g y d iff e r e n c e b e t w e e n in it ia l a n d fi n a l s t a t e is s m a ll. If E f > E I t h e s c a t t e r in g p r o c e s s is c a lle d S t o k e s , o t h e r w is e if E f < E I t h e s c a t t e r in g p r o c e s s is c a lle d a n t i - S t o k e s .
≫
- T h o m s o n s c a t t e r in g ( H ig h e n e r g y , E la s t ic ) : h ω E I , E f = E I .
≪
T h is p r o c e s s is p r e d o m in a n t f o r , e .g ., s o f t x - r a y s c a t t e r in g . T h is t y p e o f s c a t t e r in g c a n b e in t e r p r e t e d in a s e m i c la s s ic a l w a y , in t h e lim it w h e r e t h e w a v e le n g t h λ is la r g e r t h a n t h e a t o m ic d im e n s io n s , λ a 0 . T h e c r o s s s e c t io n
is t h e n e q u iv a le n t t o w h a t o n e w o u ld o b t a in f o r a f r e e e le c t r o n , σ = 8 π r 2 w it h r 0 t h e e ff e c t iv e e le c t r o n r a d iu s .
3 0
≫ ≪
- C o m p t o n s c a t t e r in g ( H ig h e n e r g y , In e la s t ic ) : h ω E I , λ a 0 , E f = E I .
F o r v e r y h ig h e n e r g y , t h e w a v e le n g t h is s m a ll c o m p a r e d t o t h e a t o m ’s s iz e a n d t h e e n e r g y is m u c h la r g e r t h a n t h e e le c t r o n b in d in g e n e r g y , s o t h a t t h e fi n a l s t a t e o f t h e e le c t r o n is a n u n b o u n d s t a t e . T h u s t h is s c a t t e r in g is v e r y s im ila r t o C o m p t o n s c a t t e r in g ( in e la s t ic s c a t t e r in g ) b y a f r e e e le c t r o n .
N o t e t h a t f o r x - r a y s c a t t e r in g s t h e c la s s ifi c a t io n is s lig h t ly d iff e r e n t t h a n t h e o n e g iv e n a b o v e . T h e r e a r e t w o p r o c e s s e s t h a t c o m p e t e s w it h C o u lo m b s c a t t e r in g e v e n a t t h e x - r a y e n e r g ie s :
- E le c t r o n ic R a m a n s c a t t e r in g : a n in e la s t ic s c a t t e r in g p r o c e s s w h e r e t h e in it ia l a t o m ic s t a t e is t h e g r o u n d s t a t e a n d t h e fi n a l s t a t e a n e x c it e d , d is c r e t e e le c t r o n ic s t a t e .
- R a y le ig h s c a t t e r in g f o r x - r a y s : a n e la s t ic s c a t t e r in g p r o c e s s , w h e r e t h e fi n a l a t o m ic s t a t e is t h e s a m e a s t h e in it ia l s t a t e , s in c e t h e r e is n o a t o m e x c it a t io n .
In a d d it io n t o s c a t t e r in g p r o c e s s e s , o t h e r p r o c e s s e s in v o lv in g t h e in t e r a c t io n o f a p h o t o n w it h e le c t r o n s a r e p o s s ib le ( b e s id e s a b s o r p t io n a n d e m is s io n o f v is ib le lig h t t h a t w e a lr e a d y s t u d ie d ) . In o r d e r o f in c r e a s in g p h o t o n e n e r g y , t h e in t e r a c t io n o f m a t t e r w it h e .m . r a d ia t io n c a n b e c la s s ifi e d a s :
R a y le ig h / R a m a n P h o t o e le c t r ic T h o m s o n C o m p t o n P a ir S c a t t e r in g A b s o r p t io n S c a t t e r in g S c a t t e r in g P r o d u c t io n
∼ ∼ ∼ ∼ ≥
≥ ≫ ∼
h ω < E I h ω E I h ω E I h ω m e c 2 h ω > 2 m e c 2 e V k e V k e V M e V M e V V is ib le X - r a y s X - r a y s γ - r a y s h a r d γ - r a y s
B . S e m i - c l a s s i c a l d e s c r i p ti o n o f s c a tte r i n g
A c la s s ic a l p ic t u r e is e n o u g h t o g iv e s o m e s c a lin g f o r t h e s c a t t e r in g c r o s s s e c t io n . W e c o n s id e r t h e e ff e c t s o f t h e in t e r a c t io n o f t h e e .m . w a v e w it h a n o s c illa t in g d ip o le ( a s c r e a t e d b y a n a t o m ic e le c t r o n ) .
0
~ —
—
2
T h e e le c t r o n c a n b e s e e n a s b e in g a t t a c h e d t o t h e a t o m b y a ” s p r in g ” , a n d o s c illa t in g a r o u n d it s r e s t p o s it io n w it h f r e q u e n c y ω 0 . W h e n t h e e .m . is in c id e n t o n t h e e le c t r o n , it e x e r t s a n a d d it io n a l f o r c e . T h e f o r c e a c t in g o n t h e e le c t r o n is F = e E ( t ) , w it h E ( t ) = E 0 s in ( ω t ) t h e o s c illa t in g e le c t r ic fi e ld . T h is o s c illa t in g d r iv in g f o r c e is in a d d it io n t o t h e a t t r a c t io n o f t h e e le c t r o n t o t h e a t o m k x e , w h e r e k ( g iv e n b y t h e C o u lo m b in t e r a c t io n s t r e n g t h a n d r e la t e d t o t h e b in d in g e n e r g y E I ) is lin k e d t o t h e e le c t r o n ’s o s c illa t in g f r e q u e n c y b y ω 2 = k / m e . T h e e q u a t io n o f m o t io n f o r t h e e le c t r o n is t h e n e
m e x ¨ e = — k x e — e E ( t ) → x ¨ e + ω x e = — E ( t )
0 m e
W e s e e k a s o lu t io n o f t h e f o r m x e ( t ) = A s in ( ω t ) , t h e n w e h a v e t h e e q u a t io n
( — ω 2 + ω 2 ) A = — e E
1 e
0
→ A = E
0
0 m e 0
ω 2 — ω 2 m e
A n a c c e le r a t e d c h a r g e ( o r a n o s c illa t in g d ip o le ) r a d ia t e s , w it h a p o w e r
2 e 2 2
P = 3 c 3 a
w h e r e t h e a c c e la r a t io n a is h e r e a = — ω 2 A s in ( ω t ) , g iv in g a m e a n s q u a r e a c c e le r a t io n
( a 2 ) =
ω 2 e 2 1
0
)
(
E 0
T h e r a d ia t e d p o w e r is t h e n
c E 2
( )
P =
0
2
T h e r a d ia t io n in t e n s it y is g iv e n b y I 0 =
c E 2
0
~ ×
8 π
( r e c a ll t h a t t h e e .m . e n e r g y d e n s it y is g iv e n b y u =
1 E 2 a n d t h e
in t e n s it y , o r p o w e r p e r u n it a r e a , is t h e n I c u ) . T h e n w e c a n e x p r e s s t h e r a d ia t e d p o w e r a s c r o s s - s e c t io n r a d ia t io n
in t e n s it y :
P = σ I 0
T h is y ie ld s t h e c r o s s s e c t io n f o r t h e in t e r a c t io n o f e .m . r a d ia t io n w it h a t o m s :
8 π ( e 2
e 0
o r in S I u n it s :
σ = = 4 π r 2
8 π ( e 2 ) 2 ( ω 2 ) 2 2 ( ω 2 ) 2
e
3 4 π ǫ 0 m e c 2 ω 2 — ω 2 3 ω 2 — ω 2
w h e r e w e u s e d t h e c la s s ic a l e le c t r o n r a d iu s 4 0 :
w h ic h is a b o u t 2 . 8 f m ( 2 . 8 × 1 0 − 1 5 m ) .
1 2 . 4 . 1 T h o m so n S c a t t e r i n g b y F r e e E l e c t r o n s
0 0
≫ ≫
W e c o n s id e r fi r s t t h e T h o m s o n s c a t t e r in g , w h ic h is w e ll d e s c r ib e d b y t h e s c a t t e r in g b y f r e e e le c t r o n s . In t h is c a s e w e c o n s id e r t h u s o n e s in g le e le c t r o n . A ls o in g e n e r a l, t h e p h o t o n s h o u ld h a v e e n e r g y h ig h e n o u g h t h a t t h e e le c t r o n is s e e n a s f r e e e v e n if in r e a lit y it is p a r t o f a n a t o m ( t h u s t h e p h o t o n e n e r g y m u s t b e la r g e r t h a n t h e a t o m ’s io n iz a t io n e n e r g y , h ω E I o r in o t h e r t e r m s λ t h a n t h e a t o m ’s s iz e ) . N o t e t h a t in T h o m s o n s c a t t e r in g t h e fi n a l e le c t r o n is s t ill a b o u n d e le c t r o n ( e la s t ic s c a t t e r in g ) w h ile in C o m p t o n s c a t t e r in g t h e e le c t r o n is u n b o u n d ( in e la s t ic s c a t t e r in g ) . S t ill, s in c e t h e b in d in g e n e r g y is s m a ll c o m p a r e d t o t h e o t h e r e n e r g y a t p la y , t h e e le c t r o n c a n b e c o n s id e r e d a s a f r e e e le c t r o n , a n d m a n y o f t h e c h a r a c t e r is t ic s o f C o m p t o n s c a t t e r in g s t ill a p p ly .
In it ia l F in a l
e − : | A i ) | A f )
ϕ : | 1 k , λ , 0 k ′ , λ ′ ) | 0 k , λ , 1 k ′ , λ ′ )
t o t : | i ) | f )
E n :
p x :
p y :
In it ia l
m c 2 + h c k h k
0
F in a l
= h c k + p c + m c
′
J
2 2
2 4
= h k ′ c o s ϑ + p c o s ϕ
= h k ′ s in ϑ — p s in ϕ
e
4 0 T h e B o h r r a d i u s i s a d i ff e r e n t q u a n t i t y : r B r B ∼ 5 × 1 0 − 1 1 m
k 2
~ m e 2
w i t h s o m e c o n s t a n t s ( d e p e n d i n g o n t h e u n i t s c h o s e n ) t o g i v e a b o u t
T h e in it ia l a n d fi n a l s t a t e s , a s w e ll a s e n e r g ie s a n d m o m e n t u m a r e w r it t e n a b o v e . T h e y r e s u lt f r o m t h e c o n s e r v a t io n o f e n e r g y a n d m o m e n t u m f o r a r e la t iv is t ic e le c t r o n w h ic h is in it ia lly a t r e s t .
? Q u e s t i o n : W h a t i s t h e ra t i o k / k ′ ? W h a t i s ∆λ = λ ′ − λ ? ( T h i s i s t h e u s u a l C o mp t o n s c a t t e ri n g f o r m u l a ) .
F ro m c o n s e rv a t i o n o f e n e rg y a n d mo me n t u m a n d w i t h t h e g e o me t r y o f fi g u r e 2 5 , w e c a n c a l c u l a t e t h e e n e rg y o f t h e s c a t t e re d
p h o t o n .
γ
e
E γ + E e = E ′ + E ′ → h ω + m e c 2 = h ω ′ + √ | p | 2 c 2 + m 2 c 4
h → k = h → k ′ + p →
h k = h k ′ c o s ϑ + p c o s ϕ
{
→ h k ′ s i n ϑ = p s i n ϕ
k
c 2
[ ] √
F ro m t h e s e e q u a t i o n s w e fi n d p 2 = k ( ω ′ − ω ) h ( ω ′ − ω ) − 2 m c 2 a n d c o s ϕ = 1 − h 2 k ′ 2 s i n 2 ϑ / p 2 . S o l v i n g f o r t h e c h a n g e i n t h e w a v e l e n g t h λ = 2 π w e fi n d ( w i t h ω = k c ) :
o r f o r t h e f re q u e n c y :
2 π h
e
∆λ = m c ( 1 − c o s ϑ )
h ω ′ = h ω 1 +
h ω m e c 2
( 1 − c o s ϑ )
− 1
'
F i g . 2 5 : P h o t o n / E l e c t r o n c o l l i s i o n i n C o mp t o n a n d T h o m s o n s c a t t e r i n g .
1 2 2
A t t h e s e h ig h e n e r g ie s , K ( 2 ) ≪ K ( 1 ) t h u s w e c a n c o n s id e r o n ly t h e K ( 1 ) c o n t r ib u t io n , t h a t w e a lr e a d y c a lc u la t e d in
t h e p r e v io u s s e c t io n .
T o fi n d t h e s c a t t e r in g r a t e a n d c r o s s s e c t io n w e n e e d t h e de ns i t y o f s t a t e s :
( L )
3
ρ ( E f ) d E f = k ′ 2 d k ′ d Ω
2 π
w h e r e t h e fi n a l e n e r g y is E f
= h c k ′ + J p 2 c 2 + m 2 c 4 ≈ h c k ′ + p 2 ( n o n - r e la t iv is t ic a p p r o x im a t io n ) . T h u s w e n e e d t o
d k ′
c a lc u la t e d E f . N o t in g t h a t
e 2 m
e
p 2 / h 2 = | k — k ′ | 2 = k 2 + k ′ 2 — 2 k k ′ c o s ϑ
w e fi n d
m c k
d E f
h 2
= h c +
′ 2 k c o s ϑ ) = h c 1 + h k ( k ′ )
( 2 k
d k ′ 2 m
—
— c o s ϑ
= 1 + m c ( 1 — c o s ϑ )
S o lv in g t h e c o n s e r v a t io n o f e n e r g y a n d m o m e n t u m e q u a t io n s , w e fi n d
k ′ h k
k
− 1
m c k m c
S in c e h k ≪ m c , w e c a n t a k e o n ly t h e fi r s t o r d e r t e r m in 1 + k k ( k ′ — c o s ϑ f . T h is is g iv e n b y : 1 + k k ( 1 — c o s ϑ ) .
B u t t h is f a c t o r is j u s t e q u a l t o k / k ′ . T h u s w e fi n a lly h a v e :
= h c
k ′
→ ρ ( E f ) =
d Ω
2 π k c
d E f k ( L ) 3 k ′ 3
d k ′
h
k
k ’
ε’ 1
θ
ε 2
ε 1
φ
ψ
ε’ 2
F i g . 2 6 : W a v e v e c t o r s a n d p o l a r i z a t i o n s o f s c a t t e r i n g p h o t o n s . c o s γ = s i n ϑ c o s ψ
F in a lly , t o c a lc u la t e t h e c r o s s s e c t io n , w e r e c a ll t h e e x p r e s s io n f o r t h e in c o m in g fl u x o f p h o t o n s Φ = c / L 3 .
d σ W f i / d Ω
2 π ( 1 ) 2
ρ ( E f
) L 3
( e 2 ) 2 ( k ′ ) 2
d Ω = c / L 3
= h | K 2 |
=
d Ω c
2
m c 2 k
| ǫ k λ · ǫ k ′ λ ′ |
W it h t h e a n g le s d e fi n e d in F ig . 2 6 w e fi n d :
= r 2 k
d σ ( ω ′ ) 2
s in 2 γ
d Ω e ω k
— —
w h e r e ( s in γ ) 2 = 1 s in 2 ϑ c o s 2 ( ϕ ψ ) a n d r e is t h e c la s s ic a l e le c t r o n r a d iu s . T h e a v e r a g e d iff e r e n t ia l c r o s s s e c t io n ( a v e r a g e d o v e r t h e p o la r iz a t io n d ir e c t io n s ψ ) is t h e n g iv e n b y
= r 2 k
( 1 — s in 2 ϑ / 2 ) = r 2 k
( 1 + c o s 2 ϑ )
d Ω
e
ω k 2
e
ω k
� d σ � ( ω ′ ) 2 1 ( ω ′ ) 2
1 2 . 4 . 2 R a y l e i g h S c a t t e r i n g o f X - r a y s
1
R a y le ig h s c a t t e r in g u s u a lly d e s c r ib e s e la s t ic s c a t t e r in g b y lo w e n e r g y r a d ia t io n . It d e s c r ib e s f o r e x a m p le v is ib le lig h t s c a t t e r in g f r o m a t o m s : in t h a t c a s e , t h e p r e d o m in a n t c o n t r ib u t io n c o m e s f r o m t h e t e r m K ( 2 ) . R a y le ig h s c a t t e r in g a ls o d e s c r ib e s c o h e r e n t , e la s t ic s c a t t e r in g o f x - r a y s f r o m a t o m s ( e .g . in a c r y s t a l) a n d is a n im p o r t a n t p r o c e s s in x - r a y
d iff r a c t io n .
1 2 1
In t h e c a s e o f x - r a y s c a t t e r in g , t h e p h o t o n e n e r g y is la r g e r t h e n t h e e le c t r o n ic e x c it a t io n e n e r g y : h ω ≫ E b . T h e n w e h a v e , a s s t a t e d a b o v e , K ( 2 ) ≪ K ( 1 ) a n d w e c a n n e g le c t t h e K ( 2 ) c o n t r ib u t io n . A s w e a r e c o n s id e r in g n o w b o u n d
e le c t r o n s , t h e r e c o il is z e r o , a n d d E f = h c . T h e n t h e d e n s it y o f s t a t e s is s im p ly ρ ( E ) =
L 3 k ′ 2 d Ω .
T h e c r o s s s e c t io n is g iv e n b y
d k ′
f 2 π k c
2
=
d σ 2 π | K ( 1 ) | 2 ρ ( E f )
2 π c 2 r 2 ( 2 π h ) 2 1 ( L ) 3 k ′ 2
| ǫ k λ · ǫ k ′ λ ′ | 2 | ( A f | Σ e i � q · � r i | A i ) | 2
e
=
′
d Ω h c / L 3 d Ω h c / L 3 L 3 k k 2 π h c
i
= r e
2 ( ω k ′ )
ω
2 Σ i � q · � r i 2
| ǫ k λ · ǫ k ′ λ ′ | | ( A f | e
i
| A i ) |
C o n s id e r a n e la s t ic s c a t t e r in g p r o c e s s ( t h e in e la s t ic s c a t t e r in g is c a lle d R a m a n s c a t t e r in g f o r x - r a y s ) . If t h e in c o m in g x - r a y is u n p o la r iz e d , w e h a v e
=
e ( 1 + c o s 2 ϑ ) | ( g | e i � q · � r i | g ) | 2
d σ r 2 Σ
d Ω 2
i
→ | ( | L | ) |
W e d e fi n e f ( p ) = ( g | L i e i � q · � r i | g ) t h e a t o m ic f o r m f a c t o r .
L L �
1 ) N o t ic e t h a t f o r p 0 g i 1 g 2 = Z 2 ( t h e a t o m ic n u m b e r s q u a r e d ) . T h u s in g e n e r a l w e e x p e c t R a y le ig h s c a t t e r in g t o b e w e a k e r f o r lig h t e r e le m e n t s .
2 ) In g e n e r a l w e c a n r e w r it e t h e s u m a s a n in t e g r a l i e i � q · � r i → e i � q · � r ρ ˜ ( r ) d 3 r u s in g t h e c h a r g e d e n s it y ρ ˜ ( r ) =
I I
i δ ( r — r i ) . T h e n t h e a t o m ic f o r m f a c t o r t a k e s t h e f o r m :
f ( p ) = ( g | e i � q · � r ρ ˜ ( r ) d 3 r | g ) = e i � q · � r ρ ( r ) d 3 r
w it h ρ ( r ) = ( g | ρ ˜ ( r ) | g ) . T h e n t h e a t o m ic f o r m f a c t o r is t h e F o u r ie r t r a n s f o r m o f t h e c h a r g e d e n s it y .
S c a tte r i n g f r o m a c r y s ta l
→
In a c r y s t a l, w e c a n r e w r it e t h e e le c t r o n p o s it io n s w it h t h e s u b s t it u t io n r i R l + r li , w h e r e R l is t h e a t o m p o s it io n ( o r t h e n u c le u s p o s it io n o r t h e a t o m ic c e n t e r o f m a s s p o s it io n ) . T h e n w e n e e d t o s u m o v e r a ll a t o m s a n d a ll e le c t r o n s in t h e a t o m . T h e n w e h a v e t h e s t r u c t u r e f a c t o r :
l
G ( q ) = ( g | Σ e i � q · R � l e i � q · � r i l | g ) = Σ f ( q ) e i � q · R � l
In a c r y s t a l w e c a n r e w r it e t h e a t o m p o s it io n a s R lj = l 1 a 1 + l 2 a 2 + l 3 a 3 + r j . T h e n
w it h f l = ( g | L i e i � q · � r i l | g ) .
l, i l
" u n i v t V c e l l
J p o s i t " i o v n V J i n c e l l
G ( q ) = Σ f j ( q ) e i � q · � r j e i q ( l 1 a 1 + l 2 a 2 + l 3 a 3 ) = Σ F ( q ) e i q ( l 1 a 1 + l 2 a 2 + l 3 a 3 )
lj "
F v ( V q )
J l 1 , l 2 , l 3
F ( q ) is t h e f o r m f a c t o r f o r t h e u n it c e ll, w h ic h is t a b u la t e d f o r d iff e r e n t c r y s t a ls . T h e c r o s s s e c t io n c a n b e w r it t e n a s :
d σ r 2
s in 2 ( N 1 q a 1 / 2 ) s in 2 ( N 2 q a 2 / 2 ) s in 2 ( N 3 q a 3 / 2 )
| |
= e ( 1 + c o s 2 ϑ ) F ( q ) 2
d Ω 2 s in 2 ( q a 1 / 2 ) s in 2 ( q a 2 / 2 ) s in 2 ( q a 3 / 2 )
O n ly w h e n q a n = 2 π h t h e in t e r f e r e n c e s t e r m s d o n o t v a n is h : t h is is B r a g g ’s d iff r a c t io n la w .
1 2 . 4 . 3 V i s i b l e L i g h t S c a t t e r i n g
— ·
W h e n c o n s id e r in g v is ib le lig h t , t h e w a v e le n g t h is la r g e c o m p a r e d t o t h e a t o m ic s iz e . T h e n , in s t e a d o f u s in g t h e f u ll in t e r a c t io n V 1 + V 2 w e c a n s a f e ly s u b s t it u t e it w it h t h e e le c t r ic d ip o le H a m ilt o n ia n 4 1 , V = d k E k . T h is H a m ilt o n ia n d o e s n o t p r o d u c e a n y t w o - p h o t o n p r o c e s s t o fi r s t o r d e r , s o in t h is c a s e w e n e e d t o c o n s id e r t h e t e r m K ( 2 ) . T h is t e r m in v o lv e s v ir t u a l t r a n s it io n s . S in c e t h e d u r a t io n o f t h e s e t r a n s it io n s is v e r y s m a ll, w e d o n o t h a v e t o w o r r y a b o u t c o n s e r v a t io n o f e n e r g y . R e c a ll:
E i — E h
K ( 2 ) = Σ ( f | V | h ) ( h | V 1 | i ) ,
h
— · | ) | ) | ) | ) | ) | )
w h e r e V = d k E k . T h e in t e r m e d ia t e s t a t e s a r e e it h e r h = A h 0 k λ 0 k ′ λ ′ o r h = A h 1 k λ 1 k ′ λ ′ . It w o u ld b e o f c o u r s e p o s s ib le t o d e r iv e t h e s c a t t e r in g c r o s s s e c t io n f r o m t h e v e c t o r - p o t e n t ia l/ m o m e n t u m H a m ilt o n ia n , a n d in t h a t
c a s e b o t h t e r m s K ( 1 ) a n d K ( 2 ) s h o u ld b e in c lu d e d 4 2 .
2 1
T h e e le c t r ic fi e ld in t h e L o r e n t z g a u g e is
L 3
ℓ ξ
E = Σ J 2 π h ω ℓ ( a
ℓ ξ
e i ℓ · R + a † e − i ℓ · R f ǫ ℓ ξ ,
ℓ , ξ
a n d t h u s w e o b t a in f o r ( h | V 1 | i ) a n d ( f | V 1 | h ) :
′
- ( 0 k λ 1 k ′ λ ′ |
( a ℓ ξ e
i ℓ · R
+ a ℓ † ξ e − i ℓ · R f
| 0 k λ 0 k ′ λ ′ ) = e
− i k · R
δ ℓ , k ′
- ( 0 k λ 0 k ′ λ ′ | ( a ℓ ξ e i ℓ · R + a ℓ † ξ e − i ℓ · R f | 1 k λ 0 k ′ λ ′ ) = e i k · R δ ℓ , k
- ( 0 k λ 1 k ′ λ ′ | ( a ℓ ξ e i ℓ · R + a ℓ † ξ e − i ℓ · R f | 1 k λ 1 k ′ λ ′ ) = e i k · R δ ℓ , k
′
- ( 1 k λ 1 k ′ λ ′ |
( a ℓ ξ e
i ℓ · R
+ a ℓ † ξ e
− i ℓ · R f
| 1 k λ 0 k ′ λ ′ ) = e
− i k · R
δ ℓ , k ′
t h u s w e h a v e
K 1
=
( 2 )
2 π h √ i ( k − k ′ ) R Σ ( A f | d · ǫ k ′ | A h ) ( A h | d · ǫ k | A i )
L
3
h
ǫ i — ǫ h + h ω k
Σ ( A f | d · ǫ k | A h ) ( A h | d · ǫ k ′ | A i )
h
ǫ i + h ω k — ( ǫ h + h ω k + h ω k ′ )
ω k ω k ′ e +
T h e s c a t t e r in g c r o s s s e c t io n is g iv e n a s u s u a l b y d σ = W 3 . a n d t h e d e n s it y o f s t a t e ( a s s u m in g n o r e c o il) is
d Ω c /L
ρ ( E f ) =
d Ω .
2 π c
( L ) 3 k ′ 2
h
F in a lly t h e c r o s s s e c t io n is g iv e n b y :
) 2 (
) 3 ′ 2 3 Σ 2
2 π h
L k L
( d f h · ǫ k ′ ) ( d h i · ǫ k ) ( d f h · ǫ k ) ( d h i · ǫ k ′ )
(
d Ω = h L 3
ω k ω k ′ 2 π
h c c ǫ i
h
— ǫ h
+ h ω k
+
ǫ i — ǫ h
— h ω k ′
d σ = k k ′ 3
( d f h · ǫ k ′ ) ( d h i · ǫ k ) + ( d f h · ǫ k ) ( d h i · ǫ k ′ )
2
d Ω ǫ i
h
— ǫ h + h ω k ǫ i — ǫ h — h ω k ′
Σ
4 1 A u n i t a r y t r a n s f o r m a t i o n c h a n g e s t h e C o u l o m b - g a u g e H a mi l t o n i a n i n t o a n e x p a n s i o n i n t e r ms o f m u l t i p o l e s o f t h e e l e c t r o ma g n e t i c fi e l d s . F o r a t o mi c i n t e ra c t i o n s , o n l y t h e e l e c t r i c d i p o l e i s k e p t , w h i l e h i g h e r m u l t i p o l e s , s u c h a s m a g n e t i c d i p o l e a n d e l e c t r i c q u a d r u p o l e , c a n b e n e g l e c t e d . T h i s u n i t a r y t r a n s f o r ma t i o n i s d e s c r i b e , e . g . , i n C o h e n - T a n n o u d j i ’ s b o o k , A t o m- P h o t o n s I n t e r a c t i o n s
4 2 T h i s d e r i v a t i o n c a n b e f o u n d i n C h e n , S . H . ; K o t l a r c h y k , M . , I n t e r a c t i o n s o f Ph o t o n s a n d N e u t r o n s w i t h M a t t e r , ( 2 0 0 7 )
A . R a y l e i g h s c a tte r i n g
| ) | )
2
R a y le ig h s c a t t e r in g d e s c r ib e s e la s t ic s c a t t e r in g , f o r w h ic h ω k = ω k ′ s in c e A f = A i . T h e n w e c a n s im p lif y t h e c r o s s s e c t io n :
d σ Σ ( d · ǫ ) ( d · ǫ ) ( d · ǫ ) ( d · ǫ )
= k 4
i h k
h i k +
i h k
h i k
d Ω ǫ i
h
— ǫ h + h ω k ǫ i — ǫ h — h ω k
A t lo n g w a v e le n g t h s h ω k ≪ ǫ h — ǫ i , t h u s w e c a n n e g le c t ω k in t h e d e n o m in a t o r . T h e n
2
4
∝ ω k 2
i h
k
h i
k
d σ Σ ( d · ǫ ) ( d · ǫ )
d Ω
a n d s im p lif y in g w e o b t a in t h a t
h ǫ i — ǫ h
d σ 4
d Ω ∝ ω k
≪
T h is e x p r e s s io n c o u ld h a v e b e e n f o u n d f r o m t h e c la s s ic a l c r o s s s e c t io n w e p r e s e n t e d e a r lie r , in t h e s a m e lim it ω ω 0 . T h e R a y le ig h s c a t t e r in g h a s a v e r y s t r o n g d e p e n d e n c e o n t h e w a v e le n g t h o f t h e e .m . w a v e . T h is is w h a t g iv e s t h e b lu e c o lo r t o t h e s k y ( a n d t h e r e d c o lo r t o t h e s u n s e t s ) : m o r e s c a t t e r in g o c c u r s f r o m h ig h e r f r e q u e n c ie s p h o t o n s ( w it h s h o r t e r w a v e le n g t h , t o w a r d t h e b lu e c o lo r ) .
A s lig h t m o v e s t h r o u g h t h e a t m o s p h e r e , m o s t o f t h e lo n g e r w a v e le n g t h s p a s s s t r a ig h t t h r o u g h . L it t le o f t h e r e d , o r a n g e a n d y e llo w lig h t is a ff e c t e d b y t h e a ir . H o w e v e r , m u c h o f t h e s h o r t e r w a v e le n g t h lig h t is s c a t t e r e d in d iff e r e n t d ir e c t io n s a ll a r o u n d t h e s k y . W h ic h e v e r d ir e c t io n o n e lo o k s , s o m e o f t h is s c a t t e r e d b lu e lig h t r e a c h e s y o u . S in c e t h e b lu e lig h t is s e e n f r o m e v e r y w h e r e o v e r h e a d , t h e s k y lo o k s b lu e .C lo s e r t o t h e h o r iz o n , t h e s k y a p p e a r s m u c h p a le r in c o lo r , s in c e t h e s c a t t e r e d b lu e lig h t m u s t p a s s t h r o u g h m o r e a ir . S o m e o f it g e t s s c a t t e r e d a w a y a g a in in o t h e r d ir e c t io n s a n d t h e c o lo r o f t h e s k y n e a r t h e h o r iz o n a p p e a r s p a le r o r w h it e . A s t h e s u n b e g in s t o s e t , t h e lig h t m u s t t r a v e l f a r t h e r t h r o u g h t h e a t m o s p h e r e . M o r e o f t h e lig h t is r e fl e c t e d a n d s c a t t e r e d a n d t h e s u n a p p e a r s le s s b r ig h t . T h e c o lo r o f t h e s u n it s e lf a p p e a r s t o c h a n g e , fi r s t t o o r a n g e a n d t h e n t o r e d . T h is is b e c a u s e e v e n m o r e o f t h e s h o r t w a v e le n g t h b lu e s a n d g r e e n s a r e n o w s c a t t e r e d a n d o n ly t h e lo n g e r w a v e le n g t h s a r e le f t in t h e d ir e c t b e a m t h a t r e a c h e s t h e e y e s . F in a lly , c lo u d s a p p e a r w h it e , s in c e t h e w a t e r d r o p le t s t h a t m a k e u p t h e c lo u d a r e m u c h la r g e r t h a n t h e m o le c u le s o f t h e a ir a n d t h e s c a t t e r in g f r o m t h e m is a lm o s t in d e p e n d e n t o f w a v e le n g t h in t h e v is ib le r a n g e .
B . R e s o n a n t S c a tte r i n g
1
A n in t e r e s t in g c a s e a r is e s w h e n t h e in c id e n t p h o t o n e n e r g y m a t c h e s t h e d iff e r e n c e in e n e r g y b e t w e e n t h e a t o m ’s in it ia l s t a t e a n d o n e o f t h e in t e r m e d ia t e le v e ls . T h is p h e n o m e n o n c a n o c c u r b o t h f o r e la s t ic o r in e la s t ic s c a t t e r in g ( R a y le ig h o r R a m a n ) . A s s u m e t h a t h ω k = ǫ h — ǫ i f o r a p a r t ic u la r h in t h e s u m o v e r a ll p o s s ib le in t e r m e d ia t e le v e ls . T h e n , o n ly fi r s t t e r m im p o r t a n t in K ( 2 ) ( d e s c r ib in g fi r s t a b s o r p t io n a n d t h e n e m is s io n ) is im p o r t a n t . In o r d e r t o
k e e p t h is t e r m fi n it e , w e in t r o d u c e a fi n it e w id t h o f t h e le v e l, Γ . T h e c r o s s s e c t io n t h e n r e d u c e s t o :
2
d σ ( d f h · ǫ k ′ ) ( d h i · ǫ k ) " | ( d f h · ǫ k ′ ) ( d h i · ǫ k ) | #
d Ω ǫ h — ǫ i — h ω k — i h Γ / 2 k ω ≈ ǫ − ǫ
( ǫ h — ǫ i — h ω k ) 2 + h 2 Γ 2 / 4
= k k ′ 3 = k k ′ 3
k
h
i k ω k ≈ ǫ h − ǫ i
T h is c r o s s s e c t io n d e s c r ib e s R a m a n r e s o n a n c e a n d , f o r k = k ′ r e s o n a n c e fl u o r e s c e n c e .
1 2 . 4 . 4 P h o t o e l e c t r i c E ff e c t
In t h is s e c t io n w e w a n t t o u s e s c a t t e r in g t h e o r y o f a p h o t o n f r o m e le c t r o n ( s ) in a n a t o m t o e x p la in t h e p h o t o e le c t r ic e ff e c t . W e c o n s id e r t h e c a s e o f a n h y d r o g e n - lik e a t o m w it h a t o m ic n u m b e r Z a n d w e c a lc u la t e t h e d iff e r e n t ia l c r o s s s e c t io n
d σ = W f i
d ω Φ i n c
w h e r e W f i is t h e t r a n s it io n r a t e f o r t h e s c a t t e r in g e v e n t a n d Φ i n c is t h e in c o m in g p h o t o n fl u x . T h e in c o m in g p h o t o n fl u x c a n b e c a lc u la t e d b y a s s u m in g ( f o r c o n v e n ie n c e ) t h a t t h e s y s t e m is e n c lo s e d in a c a v it y o f v o lu m e V = L 3 ( s o t h a t t h e r e ’s o n ly o n e p h o t o n in t h a t v o lu m e ) . T h e in c o m in g fl u x o f p h o t o n s in t h e c a v it y is g iv e n b y t h e n u m b e r o f p h o t o n s p e r u n it a r e a a n d t im e :
# p h o to n s 1 c Φ = ti m e · A r e a = L / c L 2 = L 3
w h e r e t h e a r e a is L 2 a n d t h e t im e t o c r o s s t h e c a v it y is t = L / c . T h e t r a n s it io n r a t e W f i is g iv e n b y F e r m i’s G o ld e n R u le , a s s u m in g a n a t o m - p h o t o n in t e r a c t io n V a n d a d e n s it y o f fi n a l s t a t e ρ ( E f ) :
f i
h
f
W = 2 π | ( f | V | i ) | 2 ρ ( E )
H e r e t h e fi n a l d e n s it y o f s t a t e s ρ ( E f ) is e x p r e s s e d in t e r m s o f t h e m o m e n t u m p o f t h e s c a t t e r e d e le c t r o n a n d t h e s o lid a n g le d Ω w h e r e it is e j e c t e d . In d e e d , a s t h e p h o t o n is a b s o r b e d , t h e fi n a l d e n s it y o f s t a t e s is o n ly g iv e n b y t h e f r e e e le c t r o n , a g a in a s s u m e d t o b e e n c lo s e d in t h e v o lu m e V . T h e d e n s it y o f s t a t e s f o r t h e e le c t r o n is g iv e n b y t h e d e n s it y o f m o m e n t u m s t a t e s in t h e c a v it y L 3 a s s u m in g t h e e le c t r o n p r o p a g a t e s a s a p la n e w a v e :
ρ ( E f ) d E f = ρ ( p k ) d 3 p k =
p 2 d p d Ω
( L ) 3
2 π h
w it h t h e ( n o n - r e la t iv is t ic ) e n e r g y f o r t h e e le c t r o n g iv e n b y E f = p 2 / ( 2 m ) g iv in g d E f = p d p / m . F in a lly
ρ ( E f ) =
m p d Ω
( L ) 3
2 π h
W e n e x t w a n t t o c a lc u la t e t h e t r a n s it io n m a t r ix e le m e n t ( f | V | i ) , w h e r e V = — e A k · p k . T h e r e le v a n t s t a t e s a r e
t h e p h o t o n s t a t e s 1 � k λ ) a n d 0 � k λ )
m c
a n d t h e e le c t r o n m o m e n t u m e ig e n s t a t e s , w h ic h i n t h e p o s it io n r e p r e s e n t a t io n a r e
( | ) ( | )
ψ i ( k r ) = k r i e a n d ψ f ( k r ) = k r f e .
T h e m a t r ix e le m e n t b e t w e e n t h e r e le v a n t s t a t e s is t h e n :
V i f = — m c ( f e |
0 � k λ L 3 ω
� h , ξ
a � h ξ e + a � h ξ e ǫ � h , ξ · p k 1 � k λ
e ( Σ � 2 π h c 2 h
h
i � h · � r † − i � h · � r i )
= — Σ e J 2 π h
( f | ( ( 0
a 1 ) e i � h · � r + ( 0
a † 1
| i e )
) e − i � h · � r f ǫ
· p k | i )
� h , ξ
T h e o n ly s u r v iv in g t e r m is
m L 3 ω h e
� k λ
� h ξ
� k λ
� k λ � h ξ
� k λ
� h ξ e
i f
m L 3 ω k
V = — e J 2 π h
e
� k λ
( f | e i � k · � r ǫ
e
· k p | i )
T h e n t u r n in g t o t h e p o s it io n r e p r e s e n t a t io n o f | i e ) , | f e ) a n d o f t h e m o m e n t u m o p e r a t o r , w e c a n c a lc u la t e a n e x p lic it e x p r e s s io n . U s in g ψ i ( k r ) = ( k r | i e ) , ψ f ( k r ) = ( k r | f e ) a n d ǫ � k λ · p k = ǫ � k λ · ( — i h ∇ ) , w e h a v e :
( f | e i � k · � r ǫ
· p k | i ) = I
d 3 k r ψ ∗ ( k r ) e i � k · � r ǫ
· ( — i h ∇ ψ ( k r ) )
F in a lly
e � k λ e f V
� k λ i
m L 3 ω k
V
( f | V | i ) = — e J 2 π h I
d 3 k r ψ ∗ ( k r ) e i � k · � r ǫ
i
· ( — i h ∇ ψ ( k r ) )
f
� k λ
T h e fi n a l w a v e f u n c t io n ψ f is j u s t a p la n e w a v e w it h m o m e n t u m k q = p k / h ( in t h e v o lu m e L 3 ) . T h e in it ia l w a v e f u n c t io n is in s t e a d a b o u n d s t a t e . Y o u s h o u ld h a v e s e e n t h a t f o r a n h y d r o g e n - lik e a t o m t h e w a v e f u n c t io n is g iv e n b y
e − | � r | / a 2 2
ψ i ( k r ) = √ π a 3 , w h e r e a is t h e B o h r r a d iu s s c a le d b y t h e a t o m ic n u m b e r Z ( a = h / ( m e Z ) ) . R e p la c in g t h e e x p lic it e x p r e s s io n s f o r ψ i a n d ψ f in t h e p r e v io u s r e s u lt w e o b t a in :
m L 3 ω
1 I
( f | V | i ) = — e J 2 π h √
d 3 k r e
i ( � k − q � ) · � r
k ǫ � k λ · — i h ∇
( e − | � r | /a )
√
k L 3 V π a 3
V k λ
W e n o w d e fi n e ∆ k k = k k — q k a n d e v a lu a t e t h e in t e g r a l: � d 3 k r e i ∆ � k · � r k ǫ � · ∇ ψ i b y p a r t s :
I d 3
k r e
i ∆ � k · � r k ǫ · ∇ ψ = e
i ∆ � k · � r
ψ i | L 3 — i ∆ k k · k ǫ � k λ I
d 3 k r e
i ∆ � k · � r
ψ i ( k r )
� k λ i
V V
k λ
N o t ic e t h a t t h e w a v e f u n c t io n v a n is h e s a t t h e b o u n d a r ie s , s o t h e fi r s t t e r m is z e r o . A ls o , b y d e fi n in g ϑ t h e a n g le b e t w e e n ∆ k a n d r w e c a n r e w r it e t h e in t e g r a l a s :
k I 2 I π i ∆ k r c o s ( ϑ )
∆ k k · k ǫ � I
— i 2 π ∆ k · k ǫ � k λ d r r ψ i ( r ) e
0
s in ( ϑ ) d ϑ = — i
| ∆ k k |
d r ψ i ( r ) r s in ( ∆ k r )
J — — I
T o e v a lu a t e t h is la s t in t e g r a l, w e c a n e x t e n d t h e in t e r v a l o f in t e g r a t io n t o in fi n it y , u n d e r t h e a s s u m p t io n t h a t L ≫ a :
( f | V | i ) = —
e 2 π h ( i h ) ( i ∆ k k · k ǫ � k λ ) √ π a 3 ∞ e − r /a r s in ( ∆ k r ) d r m L 3 ω k | ∆ k k | 0
a n d u s e t h e e q u iv a le n c e � ∞ d r e − r /a r s in ( b r ) =
2 a 3 b
t o o b t a in :
� ·
0 ( 1 + a 2 b 2 ) 2
e 2 π h 2 h a 3 ∆ k k k ǫ a 3
— m L 3 ω k ( 1 + a 2 ∆ k 2 ) 2
· · — · — ·
N o t ic e t h a t ∆ k k k ǫ k = k k k ǫ k k q k ǫ k = q k k ǫ k s in c e k k a n d t h e p o la r iz a t io n a r e a lw a y s p e r p e n d ic u la r .
N o w c o n s id e r in g t h e d e n s it y o f s t a t e s a n d t h e in c o m in g fl u x o f p h o t o n s Φ i n c = c / L 3 w e o b t a in t h e s c a t t e r in g c r o s s s e c t io n :
d Ω = m c ω ( 1 + a 2 ∆ k 2 ) 4
d σ 3 2 e 2 a 3 q ( k q · k ǫ k ) 2
k
≫
W h e n t h e e n e r g y o f t h e in c o m in g p h o t o n is m u c h h ig h e r t h a n t h e e le c t r o n b in d in g e n e r g y , w e h a v e a ∆ k 1 . In t h is lim it , w e c a n r e w r it e t h e s c a t t e r in g c r o s s s e c t io n a s
d Ω = m c ω ( a 2 ∆ k 2 ) 4 ∝ a 8 ∝ a
d σ 3 2 e 2 a 3 q ( q k · k ǫ k ) 2 a 3 − 5
k
N o w t h e c o n s t a n t a is t h e B o h r r a d iu s s c a le d b y t h e a t o m ic n u m b e r Z
h 2
a = m e 2 Z
w e t h u s fi n d t h e w e ll- k n o w n Z 5 d e p e n d e n c e o f t h e p h o t o e le c t r ic e ff e c t c r o s s - s e c t io n .
MIT OpenCourseWare http://ocw.mit.edu
22.51 Quantum Theory of Radiation Interactions
Fall 201 2
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