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Free induction NMR-signals in solids
Volume 19, number 5
PHYSICS LETTERS
FREE
INDUCTION
NMR-SIGNALS
15 November 1965
IN S O L I D S
R. HAUSSER and G. SIEGLE
I. Physihalisches Institut der Technischen Hochschule Stuttgart Received 12 October 1965
In c o n t r a s t to liquids, two 9 0 ° - p u l s e s of a r o t a t i n g m a g n e t i c field at the L a r m o r - r e s o n a n c e frequency produce in solids with one spin type I a m a x i m u m echo amplitude only when the r . f . - p h a s e difference of the p u l s e s is ~lr. The p r e v i o u s i n v e s t i g a t i o n of f o r m a t i o n and shape of this "solid echo" [1-6] are extended by the following t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s . A r b i t r a r y pulse lengths, the o f f - r e s o n a n c e b e h a v i o u r , and solid echoes f r o m t h r e e pulse s e q u e n c e s a r e d i s c u s s e d ; f u r t h e r m o r e the influence of a second n o n - r e s o n a n t type of s p i n s (unpaired e l e c t r o n s ) is d e s c r i b e d . 1. Two-pulse s e q u e n c e s . I n t e g r a t i o n of the equation of motion of the density m a t r i x for a r i g i d lattice [7] with s p i n s I y i e l d s the following equations when v e r y s h o r t p u l s e s o c c u r at t = 0 and t 1. a. No r . f . - p h a s e shift between the p u l s e s (X-X-sequence):
SXX = + C cos a 1 s i n a 2 s i n ( w o t - 6w(t-tl))F(t-tl) + -
C s i n a 1 sin(wot+Sw(t-tl))sin 6 w t l ( c o s 2 a2F(t)+ s i n 2 a2F(t-2tl)+K1) +
(1)
+ C s i n a I cos ~2 s i n ( W o t - 6 w ( t - t l ) ) c o s 5wtl(F(t ) +K2) . b. ~ r . f . - p h a s e shift between the p u l s e s (X-Y-sequence):
SXY = - C cos a I s i n ~ 2 c°S(Wot-w(t-tl))F(t-tl) + + C s i n a l sin(Wot" 6~°(t-tl)) cos 6wtl(cos 2 a 2 F ( t ) + s i n 2 a 2 F ( t - 2 t l ) +K1) +
(2)
- C s i n a I cos a 2 c o s ( w o t - 6co(t-t1)) s i n 6wtl(F(t) +K2) . (The notation follows that found in A b r a g a m ' s book [7], ~2~2
C
-
Ho
(2i+l)Nk T '
2
r(t) = wr(r
+ ~.
'
lxJ
t4 [HH ~HII, rx j j7~ 2
+
'")
,
K1,K 2 = e r r o r t e r m s which d e s c r i b e the v a r i a t i o n of the shape of the s i g n a l and its decay when t 1 is inc r e a s e d , 6w = deviation f r o m r e s o n a n c e , ~1, a2 = pulse lengths.) The t e r m s with F(t), F(t-tl) d e s c r i b e decays s t a r t i n g at t = 0 and t = t l , r e s p e c t i v e l y ; the F(t-2tl) t e r m r e p r e s e n t s the solid echo. Eqs. (1) and (25 differ f r o m one a n o t h e r only by t h e i r different dependence on 5¢o. They p r e d i c t periodic o s c i l l a tions of the echo amplitude with i n c r e a s i n g 6w (t 1 = const.) or t1(6¢o = const.; for e x p e r i m e n t a l c o n f i r m a t i o n see fig. 15, and also explain f u r t h e r e x p e r i m e n t s with v a r y i n g a l , a 2. Eqs. (1), (2) can be extended to include double pulse e x p e r i m e n t s in solids with s p i n s I and additional n o n - r e s o n a n t spins S, when the o p e r a t o r s d e s c r i b i n g the i n t e r a c t i o n s between the s p i n s [(HH) and s p i n s S(Hss), r e s p e c t i v e l y , c o m m u t e a p p r o x i m a t e l y with the i n t e r a c t i o n o p e r a t o r s p i n / - s p i n S (HIs = ~ CK~IKzSaz). This can K~ ot be fulfilled in s u b s t a n c e s with l = n u c l e a r spin, S = u n p a i r e d e l e c t r o n spin. In this case eqs. (1), (25 m a y be r e w r i t t e n by r e p l a c i n g
5¢o-~ ~ 6 W K = ~ C K K S a z , g K, ol
Sxx= SXy= C (vA4t_ tP+KiK) c o s K
356
s i n WO~
F(t-2tl)~RK(t-2tl) g
,
5wKtT cos 6wK(t-2tl)) ,
KI~K1K, (3) (t~1 = a 2 = 90 °) .
Volume 19, number 5
PHYSICS LETTERS
Fig. 1. Echo envelope produced by varying the pulse distance when 5co = 0. The envelope starts at ?. 10/~s. (Proton resonance at 28 Mc/s in hydrate crystals. )
15 November 1965
Fig. 2. Envelope of the stimulated echo after an X - X - Y - s e q u e n c e . J ,20/~s. a = rapid decay of the X - X - Y - e c h o ; b = rapid increase of the X-X-X-echo; c = slow decay of the echoes. (Proton resonance at 28 Mc/s in hydrate crystals.)
All odd functions of 5w K v a n i s h in taking the expectation value (<)). One of the m a i n , and e x p e r i m e n t a l l y verified, c o n c l u s i o n s f r o m (3) is that in g e n e r a l the o c c u r r e n c e of a solid echo is not r e s t r i c t e d to a s p e c i a l choice of the m u t u a l p h a s e s as in (1), (2) at 5w = 0. It also m a y d e s c r i b e the w e l l - k n o w n d i f f e r ence in the shapes of the echo and of the decay a f t e r a s i n g l e pulse [4, 5]. For the a p p r o x i m a t i o n made above, eq. (3) a g r e e s with the f o r m u l a given by Mansfield [6]. Adjusting for the d i s a p p e a r a n c e of the echo (or e c h o - m i n i m u m for s u b s t a n c e s with [ and S - s p i n s at low t i m e s t l ) after X - X s e q u e n c e s is an e x c e l l e n t method to find exact r e s o n a n c e ; an a c c u r a c y of A H / H o = 6 w / 7 [ H o ~ 5 × 10 -6 is attainable. 2. T h r e e - p u l s e s e q u e n c e s . C a l c u l a t i o n s for p u l s e s of any length a l , a2, a 3 o c c u r r i n g at t = 0, t l , t2, and having different m u t u a l p h a s e s , show, in a g r e e m e n t with our e x p e r i m e n t s , that in solids with equal s p i n s •echoes exist with m a x i m u m a m p l i t u d e s at t = 2t2, t 2 - t l , 2t~2-tl, 2 t 2 - 2 t l , as known f r o m e x p e r i m e n t s in liquids. But t h e r e i s no t h r e e - p u l s e sequence which produce all types of echoes s i m u l t a n e o u s l y when 5w = 0. C o r r e c t i o n t e r m s as in (1), (2) lead to s t r o n g a t t e n u a t i o n of the echoes through i n c r e a s e d time i n t e r v a l s between the p u l s e s . Only for the echo after an X - X - Y-sequence c o r r e s p o n d i n g to the s t i m u l a t e d echo in liquids does the c a l c u l a t i o n yield a n o t h e r behaviour. When the solution of the Bloch equation M z ( t 2 - t l ) = M o + ( M z ( t l ) - Mo) exp{- ( t 2 - t l ) / T 1 } i s valid and ot1 = ol2 = a 3 = 90 °, the s i g n a l is given as
S X X Y = - C cos W o ( t - t 2 ) [ ( 1 - e x p { - ( t 2 - t l ) / T 1 } ) F ( t - t 2 )
- exp{-(t2-tl)/T1}{f(t-t2-tl)+K}i
.
(4)
Upon i n c r e a s i n g t 2 - t 1 a r a p i d decay of the s t i m u l a t e d echo, c a u s e d by the c o r r e c t i o n t e r m s K, is obs e r v e d . Then a slow d e c r e a s e d e t e r m i n e d m a i n l y by T 1 follows. ( E x p e r i m e n t a l v e r i f i c a t i o n see fig. 2.) M e a s u r i n g T 1 by the envelope of the s t i m u l a t e d echo gives a p p r o x i m a t i v e v a l u e s of T1, for t h e K ' s depend on t 2 and d i s a p p e a r only when t 1 = 0 (i.e. 1 8 0 ° - 9 0 ° - s e q u e n c e s ) . An X - X - X - s e q u e n c e p r o d u c e s no echo for s m a l l v a l u e s t 2 - t l ; for g r e a t e r t 2 - t 1 a s t i m u l a t e d echo e x i s t s with the s a m e amplitude as after X-X-Y-sequences (~1 = a 2 = a 3 = 90°)" A detailed d e r i v a t i o n and d i s c u s s i o n of the f o r m u l a s given h e r e and e x t e n s i v e e x p e r i m e n t a l r e s u l t s a r e given in a publication u n d e r p r e p a r a t i o n . The a u t h o r s wish to e x p r e s s t h e i r c o r d i a l thanks to P r o f e s s o r H. O. K n e s e r for his support, to Dipl. Phys. F. Noack and Dippl. Phys. U. H~lberlen for useful d i s c u s s i o n s . The work was s p o n s o r e d in p a r t by the Deutsche F o r s c h u n g s g e m e i n s c h a f t .
References 1. E.L.Hahn, Phys. Rev. 80 (1950) 580. 357
Volume 19, number 5 2. 3. 4. 5. 6. 7.
PHYSICS LETTERS
15 November 1965
I.G. Powles and P.Mansfield, Physics Letters 2 (1962) 58. I.G.Powles and I.H.Strange, Proc.Phys.Soc.82 (1963) 6. G. Stegle, Diplomarbeit TH Stuttgart 1963. R. Hausser and G. Slegle, XII. Colloque Ampbre 1963. P.Mansfield, Phys.Rev. 137 (1965) A961. A.Abragam, Principles of nuclear magnetism, 1961. $$$$$
INTERPRETATION OF THE ANISOTROPY FROM A MONOCRYSTALLINE
OF
a PARTICLE SOURCE *
EMISSION
O. S. O E N Solid State Division, Oak Ridge National Labo~'atory, Oak Ridge, T e n n e s s e e , USA Received 1 October 1965
D o m e i j and B j o r k q v i s t [1] r e c e n t l y d i s c o v e r e d an a n i s o t r o p y in a p a r t i c l e e m i s s i o n f r o m a m o n o c r y s t a l l i n e s o u r c e . The a e m i s s i o n w a s m e a s u r e d n e a r <111) in a W s i n g l e c r y s t a l made a - a c t i v e by i n j e c t i n g 60 keV 222Rn ions. T h e i r data axe r e p r o duced in fig. 1. Two f e a t u r e s s e e m e s p e c i a l l y n o t e worthy: The p r o n o u n c e d dip having a h a l f - w i d t h of 1.7 ° with a m i n i m u m in the (111) d i r e c t i o n , and the m a x i m a n e a r ± 2° whose e x c e s s contains (ass u m i n g r o t a t i o n a l s y m m e t r y of the e m i s s i o n patt e r n) the d e f i c i t in the dip. A s i m i l a r phenomenon has been r e p o r t e d [2] f o r the (d, p) r e a c t i o n in a s i l i c o n c r y s t a l . The p u r p o s e of this l e t t e r is to i n t e r p r e t the f o r m e r r e s u l t s q u a n t i t a t i v e l y u s i n g a m o d e l which a s s u m e s the a - a c t i v e n u c l e i a r e s u b s t i t u t i o n a l l y l o c a t e d at l a t t i c e s i t e s . C o n s i d e r an a p a r t i c l e of e n e r g y E s t a r t i n g f r o m a l a t t i c e p o s i t i o n at a s m a l l angle ~ with r e s p e c t to a r o w of s t a t i o n a r y a t o m s of s p a c in g d. (d = 2.74 ,~ fo r W (111).) The a p a r t i c l e is s c a t tered by the neighbouring nucleus through an angle ~ such that it n o w m o v e s at an angle O, which for small angles b e c o m d s
o
:
~/+
~
:
s/d + ~ (s),
t.6
>- t.4 ~" zb J ~-z 4.2 ,., _o t.0
~-
tY o_
~ o.8 ,-~ o.6 :~ t~ o z o.4 1"4
(1)
w h e r e s i s the i m p a c t p a r a m e t e r . Eq. (1) p r e d i c t s a m i n i m u m o r c r i t i c a l angle ** 0 = Oc, s i n c e the s c a t t e r i n g angle ¢p, in g e n e r a l , decreases with increasing s while the starting angle ~/increases. The differential scattering cross section per steradian is given by * Research sponsored by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation. ** It is interesting and perhaps instructive to note the close analogy to rainbow scattering first treated by Descartes in 1636. 358
4.8
0.2
0 -6
-4
-2
0
2
4 ,
EMISSION ANGLE (degPees from (H4>)
Fig. 1. Comparison of theory (curve a, static lattice; b, (~o2>)~ = 0.031 A; ((p2>)~ = 0.044 A) with experimental data points of Domeij and Bjorlelvist [1] for 5.49 MeV ot particles emitted from 222Rn ions embedded in a W single crystal.
PHYSICS LETTERS
FREE
INDUCTION
NMR-SIGNALS
15 November 1965
IN S O L I D S
R. HAUSSER and G. SIEGLE
I. Physihalisches Institut der Technischen Hochschule Stuttgart Received 12 October 1965
In c o n t r a s t to liquids, two 9 0 ° - p u l s e s of a r o t a t i n g m a g n e t i c field at the L a r m o r - r e s o n a n c e frequency produce in solids with one spin type I a m a x i m u m echo amplitude only when the r . f . - p h a s e difference of the p u l s e s is ~lr. The p r e v i o u s i n v e s t i g a t i o n of f o r m a t i o n and shape of this "solid echo" [1-6] are extended by the following t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s . A r b i t r a r y pulse lengths, the o f f - r e s o n a n c e b e h a v i o u r , and solid echoes f r o m t h r e e pulse s e q u e n c e s a r e d i s c u s s e d ; f u r t h e r m o r e the influence of a second n o n - r e s o n a n t type of s p i n s (unpaired e l e c t r o n s ) is d e s c r i b e d . 1. Two-pulse s e q u e n c e s . I n t e g r a t i o n of the equation of motion of the density m a t r i x for a r i g i d lattice [7] with s p i n s I y i e l d s the following equations when v e r y s h o r t p u l s e s o c c u r at t = 0 and t 1. a. No r . f . - p h a s e shift between the p u l s e s (X-X-sequence):
SXX = + C cos a 1 s i n a 2 s i n ( w o t - 6w(t-tl))F(t-tl) + -
C s i n a 1 sin(wot+Sw(t-tl))sin 6 w t l ( c o s 2 a2F(t)+ s i n 2 a2F(t-2tl)+K1) +
(1)
+ C s i n a I cos ~2 s i n ( W o t - 6 w ( t - t l ) ) c o s 5wtl(F(t ) +K2) . b. ~ r . f . - p h a s e shift between the p u l s e s (X-Y-sequence):
SXY = - C cos a I s i n ~ 2 c°S(Wot-w(t-tl))F(t-tl) + + C s i n a l sin(Wot" 6~°(t-tl)) cos 6wtl(cos 2 a 2 F ( t ) + s i n 2 a 2 F ( t - 2 t l ) +K1) +
(2)
- C s i n a I cos a 2 c o s ( w o t - 6co(t-t1)) s i n 6wtl(F(t) +K2) . (The notation follows that found in A b r a g a m ' s book [7], ~2~2
C
-
Ho
(2i+l)Nk T '
2
r(t) = wr(r
+ ~.
'
lxJ
t4 [HH ~HII, rx j j7~ 2
+
'")
,
K1,K 2 = e r r o r t e r m s which d e s c r i b e the v a r i a t i o n of the shape of the s i g n a l and its decay when t 1 is inc r e a s e d , 6w = deviation f r o m r e s o n a n c e , ~1, a2 = pulse lengths.) The t e r m s with F(t), F(t-tl) d e s c r i b e decays s t a r t i n g at t = 0 and t = t l , r e s p e c t i v e l y ; the F(t-2tl) t e r m r e p r e s e n t s the solid echo. Eqs. (1) and (25 differ f r o m one a n o t h e r only by t h e i r different dependence on 5¢o. They p r e d i c t periodic o s c i l l a tions of the echo amplitude with i n c r e a s i n g 6w (t 1 = const.) or t1(6¢o = const.; for e x p e r i m e n t a l c o n f i r m a t i o n see fig. 15, and also explain f u r t h e r e x p e r i m e n t s with v a r y i n g a l , a 2. Eqs. (1), (2) can be extended to include double pulse e x p e r i m e n t s in solids with s p i n s I and additional n o n - r e s o n a n t spins S, when the o p e r a t o r s d e s c r i b i n g the i n t e r a c t i o n s between the s p i n s [(HH) and s p i n s S(Hss), r e s p e c t i v e l y , c o m m u t e a p p r o x i m a t e l y with the i n t e r a c t i o n o p e r a t o r s p i n / - s p i n S (HIs = ~ CK~IKzSaz). This can K~ ot be fulfilled in s u b s t a n c e s with l = n u c l e a r spin, S = u n p a i r e d e l e c t r o n spin. In this case eqs. (1), (25 m a y be r e w r i t t e n by r e p l a c i n g
5¢o-~ ~ 6 W K = ~ C K K S a z , g K, ol
Sxx= SXy= C (vA4t_ tP+KiK) c o s K
356
s i n WO~
F(t-2tl)~RK(t-2tl) g
,
5wKtT cos 6wK(t-2tl)) ,
KI~K1K, (3) (t~1 = a 2 = 90 °) .
Volume 19, number 5
PHYSICS LETTERS
Fig. 1. Echo envelope produced by varying the pulse distance when 5co = 0. The envelope starts at ?. 10/~s. (Proton resonance at 28 Mc/s in hydrate crystals. )
15 November 1965
Fig. 2. Envelope of the stimulated echo after an X - X - Y - s e q u e n c e . J ,20/~s. a = rapid decay of the X - X - Y - e c h o ; b = rapid increase of the X-X-X-echo; c = slow decay of the echoes. (Proton resonance at 28 Mc/s in hydrate crystals.)
All odd functions of 5w K v a n i s h in taking the expectation value (<)). One of the m a i n , and e x p e r i m e n t a l l y verified, c o n c l u s i o n s f r o m (3) is that in g e n e r a l the o c c u r r e n c e of a solid echo is not r e s t r i c t e d to a s p e c i a l choice of the m u t u a l p h a s e s as in (1), (2) at 5w = 0. It also m a y d e s c r i b e the w e l l - k n o w n d i f f e r ence in the shapes of the echo and of the decay a f t e r a s i n g l e pulse [4, 5]. For the a p p r o x i m a t i o n made above, eq. (3) a g r e e s with the f o r m u l a given by Mansfield [6]. Adjusting for the d i s a p p e a r a n c e of the echo (or e c h o - m i n i m u m for s u b s t a n c e s with [ and S - s p i n s at low t i m e s t l ) after X - X s e q u e n c e s is an e x c e l l e n t method to find exact r e s o n a n c e ; an a c c u r a c y of A H / H o = 6 w / 7 [ H o ~ 5 × 10 -6 is attainable. 2. T h r e e - p u l s e s e q u e n c e s . C a l c u l a t i o n s for p u l s e s of any length a l , a2, a 3 o c c u r r i n g at t = 0, t l , t2, and having different m u t u a l p h a s e s , show, in a g r e e m e n t with our e x p e r i m e n t s , that in solids with equal s p i n s •echoes exist with m a x i m u m a m p l i t u d e s at t = 2t2, t 2 - t l , 2t~2-tl, 2 t 2 - 2 t l , as known f r o m e x p e r i m e n t s in liquids. But t h e r e i s no t h r e e - p u l s e sequence which produce all types of echoes s i m u l t a n e o u s l y when 5w = 0. C o r r e c t i o n t e r m s as in (1), (2) lead to s t r o n g a t t e n u a t i o n of the echoes through i n c r e a s e d time i n t e r v a l s between the p u l s e s . Only for the echo after an X - X - Y-sequence c o r r e s p o n d i n g to the s t i m u l a t e d echo in liquids does the c a l c u l a t i o n yield a n o t h e r behaviour. When the solution of the Bloch equation M z ( t 2 - t l ) = M o + ( M z ( t l ) - Mo) exp{- ( t 2 - t l ) / T 1 } i s valid and ot1 = ol2 = a 3 = 90 °, the s i g n a l is given as
S X X Y = - C cos W o ( t - t 2 ) [ ( 1 - e x p { - ( t 2 - t l ) / T 1 } ) F ( t - t 2 )
- exp{-(t2-tl)/T1}{f(t-t2-tl)+K}i
.
(4)
Upon i n c r e a s i n g t 2 - t 1 a r a p i d decay of the s t i m u l a t e d echo, c a u s e d by the c o r r e c t i o n t e r m s K, is obs e r v e d . Then a slow d e c r e a s e d e t e r m i n e d m a i n l y by T 1 follows. ( E x p e r i m e n t a l v e r i f i c a t i o n see fig. 2.) M e a s u r i n g T 1 by the envelope of the s t i m u l a t e d echo gives a p p r o x i m a t i v e v a l u e s of T1, for t h e K ' s depend on t 2 and d i s a p p e a r only when t 1 = 0 (i.e. 1 8 0 ° - 9 0 ° - s e q u e n c e s ) . An X - X - X - s e q u e n c e p r o d u c e s no echo for s m a l l v a l u e s t 2 - t l ; for g r e a t e r t 2 - t 1 a s t i m u l a t e d echo e x i s t s with the s a m e amplitude as after X-X-Y-sequences (~1 = a 2 = a 3 = 90°)" A detailed d e r i v a t i o n and d i s c u s s i o n of the f o r m u l a s given h e r e and e x t e n s i v e e x p e r i m e n t a l r e s u l t s a r e given in a publication u n d e r p r e p a r a t i o n . The a u t h o r s wish to e x p r e s s t h e i r c o r d i a l thanks to P r o f e s s o r H. O. K n e s e r for his support, to Dipl. Phys. F. Noack and Dippl. Phys. U. H~lberlen for useful d i s c u s s i o n s . The work was s p o n s o r e d in p a r t by the Deutsche F o r s c h u n g s g e m e i n s c h a f t .
References 1. E.L.Hahn, Phys. Rev. 80 (1950) 580. 357
Volume 19, number 5 2. 3. 4. 5. 6. 7.
PHYSICS LETTERS
15 November 1965
I.G. Powles and P.Mansfield, Physics Letters 2 (1962) 58. I.G.Powles and I.H.Strange, Proc.Phys.Soc.82 (1963) 6. G. Stegle, Diplomarbeit TH Stuttgart 1963. R. Hausser and G. Slegle, XII. Colloque Ampbre 1963. P.Mansfield, Phys.Rev. 137 (1965) A961. A.Abragam, Principles of nuclear magnetism, 1961. $$$$$
INTERPRETATION OF THE ANISOTROPY FROM A MONOCRYSTALLINE
OF
a PARTICLE SOURCE *
EMISSION
O. S. O E N Solid State Division, Oak Ridge National Labo~'atory, Oak Ridge, T e n n e s s e e , USA Received 1 October 1965
D o m e i j and B j o r k q v i s t [1] r e c e n t l y d i s c o v e r e d an a n i s o t r o p y in a p a r t i c l e e m i s s i o n f r o m a m o n o c r y s t a l l i n e s o u r c e . The a e m i s s i o n w a s m e a s u r e d n e a r <111) in a W s i n g l e c r y s t a l made a - a c t i v e by i n j e c t i n g 60 keV 222Rn ions. T h e i r data axe r e p r o duced in fig. 1. Two f e a t u r e s s e e m e s p e c i a l l y n o t e worthy: The p r o n o u n c e d dip having a h a l f - w i d t h of 1.7 ° with a m i n i m u m in the (111) d i r e c t i o n , and the m a x i m a n e a r ± 2° whose e x c e s s contains (ass u m i n g r o t a t i o n a l s y m m e t r y of the e m i s s i o n patt e r n) the d e f i c i t in the dip. A s i m i l a r phenomenon has been r e p o r t e d [2] f o r the (d, p) r e a c t i o n in a s i l i c o n c r y s t a l . The p u r p o s e of this l e t t e r is to i n t e r p r e t the f o r m e r r e s u l t s q u a n t i t a t i v e l y u s i n g a m o d e l which a s s u m e s the a - a c t i v e n u c l e i a r e s u b s t i t u t i o n a l l y l o c a t e d at l a t t i c e s i t e s . C o n s i d e r an a p a r t i c l e of e n e r g y E s t a r t i n g f r o m a l a t t i c e p o s i t i o n at a s m a l l angle ~ with r e s p e c t to a r o w of s t a t i o n a r y a t o m s of s p a c in g d. (d = 2.74 ,~ fo r W (111).) The a p a r t i c l e is s c a t tered by the neighbouring nucleus through an angle ~ such that it n o w m o v e s at an angle O, which for small angles b e c o m d s
o
:
~/+
~
:
s/d + ~ (s),
t.6
>- t.4 ~" zb J ~-z 4.2 ,., _o t.0
~-
tY o_
~ o.8 ,-~ o.6 :~ t~ o z o.4 1"4
(1)
w h e r e s i s the i m p a c t p a r a m e t e r . Eq. (1) p r e d i c t s a m i n i m u m o r c r i t i c a l angle ** 0 = Oc, s i n c e the s c a t t e r i n g angle ¢p, in g e n e r a l , decreases with increasing s while the starting angle ~/increases. The differential scattering cross section per steradian is given by * Research sponsored by the U.S. Atomic Energy Commission under contract with Union Carbide Corporation. ** It is interesting and perhaps instructive to note the close analogy to rainbow scattering first treated by Descartes in 1636. 358
4.8
0.2
0 -6
-4
-2
0
2
4 ,
EMISSION ANGLE (degPees from (H4>)
Fig. 1. Comparison of theory (curve a, static lattice; b, (~o2>)~ = 0.031 A; ((p2>)~ = 0.044 A) with experimental data points of Domeij and Bjorlelvist [1] for 5.49 MeV ot particles emitted from 222Rn ions embedded in a W single crystal.