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THE PASSAGE OF NEUTRONS THROUGH CRYSTALS AND POLYCRYSTALS.

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The p a s s a g e o f n e u t r o n s t h r o u g h c r y s . H25
t a l s and p o l y c r y s t a l s . . .
Hew Y o r k ,
1940.
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TH IS D IS S E R T A TIO N HAS BEEN M IC R O F IL M E D
E X A C T L Y AS R E C E IV E D ,
i.■.;.;.\RY
is. Y, Univ.
THE PASSAGE OF NEUTRONS THROUGH
CRYSTALS AND POLYCRYSTALS
By
M o r to n Hamermesh
A p r i l 2 4 f 1940
S u b m itte d in p a r t i a l f u l f i l l m e n t o f th e re q u ire m e n ts
f o r t h e d e g r e e o f D o c t o r o f P h i l o s o p h y a t New Y ork
U n iv e rsity *
TABLE OF CONTENTS
S e c tio n
In tro d u c tio n
I.
The G e n e r a l S c a t t e r i n g F o r m u l a e f o r Atoms
I I . S c a t t e r i n g by C r y s ta ls and A g g re g ates o f
C ry sta ls
I I I . D is c u s s io n o f S c a t t e r i n g E x p erim e n ts
IV . D is c u s s io n o f t h e S iz e o f M i c r o c r y s t a ls and
E s t i m a t e o f t h e S c a t t e r i n g A m p l i t u d e o f Fe
Iso to p e s
V. An I n t e r p r e t a t i o n o f t h e L a rm o r P r e c e s s i o n o f
th e S p in d u r in g P a s s a g e th ro u g h M a g h e tiz e d
M ed ia
V I. P o l a r i z a t i o n E x p erim en ts w ith F e rro m a g n e ts;
G e n e r a l P r i n c i p l e s a n d C h o ic e o f Form F a c t o r
V I I . E v a lu a tio n o f P o l a r i z a t i o n E x p erim e n ts
C o n tin u e d ; V e lo c ity D i s t r i b u t i o n o f th e
I n c i d e n t N e u tro n s
F o o tn o te s and R e fe re n c e s
INTRODUCTION °)
C o n tin u in g e a r l i e r i n v e s t i g a t i o n s on th e
t r a n s m i s s i o n o f n e u tr o n s th ro u g h m a c ro sc o p ic b o d i e s ,
an d i n p a r t i c u l a r on m a c ro sc o p ic and m ic ro s c o p ic
m a g n e tic e f f e c t s ,
we p r e s e n t i n t h i s p a p e r a d i s c u s s i o n
o f t h e I n f l u e n c e o f c r y s t a l s t r u c t u r e on s c a t t e r i n g a n d
p o l a r iz a tio n o f n e u tro n s .
I t h as b e e n t h e acc u sto m e d
p ro c e d u re to c o n s id e r th e n u c le a r c r o s s - s e c t i o n f o r th e
s c a t t e r i n g o f s lo w n e u t r o n s a s a s t r i c t l y
a d d itiv e p ro ­
p e r t y in d e p e n d e n t o f th e p h y s ic a l s t a t e o r th e c h e m ic a l
c o n s t i t u t i o n o f t h e s a m p le I n v e s t i g a t e d .
On t h i s b a s i s
t h e t o t a l c r o s s - s e c t i o n o f a c h e m i c a l com pound c o u l d ,
f o r e x a m p le , b e d e t e r m i n e d fro m a k n o w le d g e o f t h e c r o s s s e c tio n s o f th e c o n s titu e n t e le m e n ts.
S u c h a p r o c e d u r e i s v a l i d o n l y i f we a r e d e a l ­
in g w ith a su b s ta n c e o f s t r i c t l y
g as).
a m o rp h o u s s t r u c t u r e
(a
In a l l o th e r cases th e c r y s t a l l i n e s t r u c t u r e o f
t h e m a t e r i a l beco m es o f s i g n i f i c a n c e ;
I t w ill tu rn o u t
t o b e o f i m p o r t a n c e e v e n f o r s u b s t a n c e s co m p o sed o f s m a l l
m ic ro c ry s ta ls,
re su lts
to
and to c o m p le te ly change th e s c a t t e r i n g
f o r m a c ro sc o p ic c r y s t a l s .
th e s e r e s u l ts
is
in i ts
The t h e o r y l e a d i n g
e sse n tia ls
t h e sam e a s t h a t
f o r X - r a y s c a t t e r i n g , b u t show s g r e a t q u a n t i t a t i v e d i f ­
f e r e n c e s d u e t o p e c u l i a r i t i e s w h ic h a r e c h a r a c t e r i s t i c
o f th e i n t e r a c t i o n b e tw e e n n e u tr o n and a to m s .
I n d i s c u s s i n g a n y c r y s t a l p h en o m en o n i t i s
e s s e n t i a l to s e p a r a t e th e c o h e r e n t and I n c o h e r e n t s c a t t e r ­
i n g p r o c e s s e s w h ic h h a v e a l r e a d y p r e v i o u s l y b e e n f o u n d
t o p l a y so g r e a t a p a r t I n t h e t h e o r y o f X - r a y s c a t t e r i n g ;
b u t w h ile th e th e o ry o f X -ra y s c a t t e r i n g by a c h e m ic a l
e le m e n t had to ta k e in to a c c o u n t o n ly th o s e I n c o h e r e n t
p r o c e s s e s w h ic h a r e p r o d u c e d b y i n e l a s t i c
th e In c id e n t p a r t i c l e s
(stric tly
In e la stic
w ith th e l a t t i c e
sc a tte rin g ),
c o llis io n of
v ib ra tio n s
we h a v e i n t h e c a s e o f
n e u t r o n s two m ore c a u s e s f o r I n c o h e r e n t p r o c e s s e s .
are:
first,
These
th e e x is te n c e o f s p in — dependent fo rc e s
b e t w e e n t h e n e u t r o n a n d t h e a to m , w h ic h may b e o f a
n u c l e a r o r m a g n e tic ty p e ; and s e c o n d ly ,
th e d is o r d e r in
t h e l a t t i c e c a u s e d b y t h e random d i s t r i b u t i o n o f t h e
i s o t o p e s , w h ich I n g e n e r a l w i l l p ro d u c e s c a t t e r e d n e u t r o n
w aves,
t h e a m p litu d e o f w h ic h , a s w e l l a s th e p h a s e ,
w i l l show no c o r r e l a t i o n f o r t h e v a r i o u s I s o t o p e s .
The s e c o n d p a r t o f t h e p a p e r i s
co n cern ed w ith
th e q u a n ti t a t iv e e v a lu a tio n of p o l a r i z a t io n e x p e rim e n ts.
a)
a)
We h a v e p r e v i o u s l y p o i n t e d o u t t h a t t h e e x i s t i n g t h e o r i e s
of p o la riz a tio n e ffe c ts ,
w h ic h f a l l
th e c r y s t a l l i n e s t r u c tu r e as w e ll as
to ta k e I n t o a c c o u n t
th e v a rio u s ty p e s
o f s c a t t e r i n g , h a v e no f i e l d o f a p p l i c a t i o n w h a t s o e v e r ,
a n d t h a t i t was n e c e s s a r y a l s o
to c a l c u l a t e i n d e t a i l
th e e f f e c t o f la c k o f m a g n e tic s a t u r a t i o n i n a ferro m ag n e t
o n t h e i n t e n s i t y o f t h e s c a t t e r e d a n d t r a n s m i t t e d b eam .
We f i n d t h a t t h e p r e v i o u s
e stim a te s
of p o la riz a tio n e ffe c ts
a r e o f a h i g h e r o r d e r o f m a g n itu d e t h a n t h o s e o b t a i n e d by
th e rig o ro u s
th e o ry .
T hese c a l c u l a t i o n s
be used to g e th e r w ith p r e v io u s ly
o f c o u r s e h a v e to
o b ta in e d r e s u lts
on
m a g n e tic d e p o l a r i z a t i o n .
Among p r e v i o u s
n e u tro n s
in v e s tig a tio n s
th ro u g h c r y s t a l s ,
may b e m e n t i o n e d .
of th e passage of
H)
s'}
p a p e r s b y W ick a n d P o m e ra n c h u k
P o in tin g o u t th a t th e e la s tic
o f n e u t r o n s d o e s n o t o f f e r m uch I n t e r e s t ,
a n alo g o u s to
sin c e i t is
th e p ro b lem i n X -ra y s c a t t e r i n g ,
an e s s e n tia l d iffe re n c e a r is e s
W ick show s t h a t
as f a r as th e i n e l a s t i c
s c a t t e r i n g due to i n t e r a c t i o n w ith th e l a t t i c e
Is co n cern ed .
sc a tte rin g
T h is d i f f e r e n c e a r i s e s
v ib ra tio n s
from th e n o n r e l a t i v i s t i c
m o t i o n o f t h e n e u t r o n ( e n e r g y - momentum x V / 2 , n o t
momentum x V a s i n t h e X - r a y c a s e ) ,
th e i n e l a s t i c
sc a tte rin g .
w h ic h t e n d s t o re d u c e
S im ila rly ,
P om eranchuk d i s ­
c u s s e s m a i n l y t h e s l o w i n g down o f n e u t r o n s
f o r n e u t r o n s o f l o n g w ave l e n g t h .
in c ry s ta ls
The p h y s i c a l a s p e c t s
w h ic h a r e r e l e v a n t i n o u r p a p e r h a v e n o t b e e n t r e a t e d
by t h e s e a u t h o r s .
The a r r a n g e m e n t o f t h e p a p e r i s a s f o l l o w s :
A f te r g iv in g e x p re s s io n s f o r th e a m p litu d e s o f th e v a rio u s
t y p e s o f s c a t t e r i n g , we d e r i v e a n d c o l l e c t a n u m b er o f
fo rm u la e f o r th e s c a t t e r e d i n t e n s i t i e s p ro d u ced by b o d ie s
in d if f e r e n t c r y s ta llin e
sta te s.
The r e s u l t s a r e a p p l i e d
to e x p la in o b s e rv a tio n s on n o n - a d d itiv ity of c r o s s - s e c ti o n s ,
a n d t o d i s c u s s some I n f o r m a t i o n a b o u t t h e f i n e s t r u c t u r e
o f th e c r y s t a l s .
S e c t i o n 5 c o n t a i n s a d e r i v a t i o n w h ic h
show s t h a t t h e L a rm o r p r e c e s s i o n o f t h e n e u t r o n s p i n i n
th e I n t e r i o r o f a m a g n e tiz e d m a t e r i a l c an be lo o k e d upon
a s a p h en o m en o n o f d i s p e r s i o n .
The l a s t two s e c t i o n s a r e
d e v o te d to a q u a n t i t a t i v e e v a l u a t i o n o f n e u tro n p o l a r i ­
z a tio n ,
t a k i n g i n t o a c c o u n t a t o m i c fo rm f a c t o r s ,
c ry sta l
s t r u c t u r e , a n d wave l e n g t h d i s t r i b u t i o n o f t h e i n c i d e n t
b e am .
I n c o m p a r in g t h e t h e o r e t i c a l r e s u l t s w i t h v a r i o u s
l a r g e l y c o n t r a d i c t o r y o b s e r v a t i o n s we f i n d t h a t t h e o b ­
se rv e d p o l a r i z a t i o n e f f e c t s a r e th ro u g h o u t c o n s id e ra b ly
la r g e r th an th e t h e o r e t i c a l e s tim a te s .
5.
I,
The G e n e r a l S c a t t e r i n g F o r m u l a e f o r Atom s
B e fo re c o n s id e rin g th e e f f e c t o f th e c r y s t a l l i n e
s t r u c t u r e o f t h e s c a t t e r e r on t h e r e s u l t s o f t r a n s m i s s i o n
e x p e r i m e n t s we w i s h t o r e v i e w b r i e f l y
th e th e o ry o f th e
p a ssa g e o f n e u tro n s th ro u g h a m a t e r i a l c o n s is tin g o f
Independent s c a t t e r e r s .
We s h a l l b e d e a l i n g t h r o u g h o u t w i t h s l o w n e u t r o n s ,
s o t h a t o n l y t h e " s " w ave n e e d b y c o n s i d e r e d ;
a lso
in e la s tic
n u c le a r s c a t t e r i n g can b e e x c lu d e d s in c e th e en erg y o f
th e n e u tro n i s
sta te s .
l
to o s m a ll to e x c i t e
th e n u c le u s to h i g h e r
I f one ta k e s a c c o u n t o f th e d i f f e r e n t is o to p e s
p r e s e n t i n th e s c a t t e r i n g sa m p le ,
th e a m p litu d e o f th e
c o h e r e n t s c a t t e r i n g f ro m a s i n g l e n u c l e u s I s g i v e n b y :
c =
w h e r e ^bp |
iso to p e ,
7L, i
^
c
( 1. D
p
g iv e s th e r e l a t i v e abundance o f th e p 't h
Ip I s t h e n u c l e a r s p i n o f th e p ' t h
iso to p e ,
a n d a£ a n d a f a r e t h e a m p l i t u d e s o f t h e s c a t t e r e d w ave
f o r t o t a l a n g u l a r m om enta i p- £ a n d 1^41- r e s p e c t i v e l y *
(N e u tro n s p i n ^ )
The i n t e n s i t y o f t h e I n c o h e r e n t n u c l e a r
0
s c a tte r in g is p ro p o rtio n a l to :
6.
I n a d d itio n to t h i s p u re ly n u c le a r s c a t t e r i n g ,
t h e r e w i l l b e a n i n t e r a c t i o n b e t w e e n t h e m a g n e t i c moment
o f t h e n e u tr o n an d t h e m a g n e tic f i e l d due to t h e c u r r e n t
d i s t r i b u t i o n s o f th e e le c tr o n s
i n t h e a to m .
S in c e in
m ost c a s e s one d e a l s w ith m a t e r i a l s i n th e c r y s t a l l i n e
s ta te ,
th e o r b i t a l c u r r e n t s a r e e i t h e r a b s e n t (gro u n d
s t a t e S ) , o r a l m o s t c o m p l e t e l y q u e n c h e d by t h e c r y s t a l l i n e
fie ld ,
so t h a t t h e m a g n e t i c p r o p e r t i e s o f t h e a to m a r e
due a lm o st e n t i r e l y
to s p i n c u r r e n t s .
F or th e am p li-
t u d e o f t h e m a g n e t i c a l l y s c a t t e r e d wave we h a v e :
(1 .3 )
w h e r e Me =
m a ss o f t h e n e u t r o n ,
"fr = i t s
m a g n e t i c moment i n n u c l e a r
m ag n e to n s;
j ? - /?'
( T H e f
1
w h e r e k a n d k* a r e t h e i n i t i a l a n d f i n a l
p r o p a g a tio n v ftcto rs o f th e n e u tro n ;
s is
th e s p in o f th e n e u tro n in u n ite s of
2 -nr
X 5 i s t h e s p i n wave f u n c t i o n o f t h e i n c i d e n t
n e u tro n ,
F"^ i s t h e f o r m f a c t o r o f t h e s p i n d i s t r i b u t i o n
o f th e io n , an d
/i.
is
th e e p tn f u n c tio n d e s c r ib in g th e
o r i e n t a t i o n o f th e s p in S o f th e ground s t a t e o f th e
io n .
(T h e I n c i d e n t f l u x h a s b e e n n o r m a l i z e d t o 1 . )
In th e c a se o f ferro m ag n e ts a t s a t u r a t io n ,
th e s p in s of a l l th e io n s a r e r i g i d l y a lig n e d a lo n g
a d i r e c t i o n g i v e n b y t h e u n i t v e c t o r K. •
In t h i s c a se o n ly e l a s t i c
c o llis io n s
i n w h ic h
th e s p in s t a t e o f th e io n is unchanged a r e p o s s ib l e ,
a n d t h e e x p r e s s i o n f o r t h e m a g n e t i c a l l y s c a t t e r e d w ave
becom es:
( 1 .5 )
w h e re
( 1.
6)
(1 .7 )
T hen t h e t o t a l c o h e r e n t l y s c a t t e r e d wave
( i n c l u d i n g b o th m a g n e tic an d n u c l e a r s c a t t e r i n g ) b e c o m e s :^
I n a d d i t i o n to t h e above s c a t t e r i n g p r o c e s s e s ,
t h e r e w i l l b e p o s s i b l e c a p t u r e o f t h e n e u t r o n w h ic h w o u ld
l e a d t o a d e c r e a s e i n i n t e n s i t y o f t h e t r a n s m i t t e d beam*
The c r o s s - s e c t i o n
QT
f o r th is s t r a i g h t a b s o rp tio n w ill
o'
f o r s lo w n e u t r o n s f o l l o w t h e
law .
II.
S c a t t e r i n g by C r y s t a l s and A g g re g a te s o f C r y s t a l s
B e fo re p ro c e e d in g w ith c a l c u l a t i o n s ,
to f i x
we w i s h
th e n o m e n c la tu re to be u sed th ro u g h o u t.
The b a s i c u n i t i n t h e c r y s t a l I s t h e u n i t
c e l l w h ic h , by t h e r e p e t i t i o n o f a s e t o f t r a n s l a t i o n s ,
fills
o u t t h e w h o le s p a c e ,
fo rm in g a m i c r o c r y s t a l .
T he m i c r o c r y s t a l i s a c o m p o s i t e o f u n i t c e l l s
w h ic h a r e i n p e r f e c t a r r a n g e m e n t an d s c a t t e r c o h e r e n t l y .
A s i n g l e c r y s t a l i s com posed o f m i c r o c r y s t a l s
a l l o f w h ich a r e o r i e n t e d a p p r o x i m a t e ly t h e sam e, b u t
w hose r e l a t i v e p o s i t i o n s a r e f i x e d ' o n ly w i t h i n a d i s ­
ta n c e o f th e o rd e r of th e l a t t i c e
sin g le c r y s ta l.)
sp a c in g .
( M o s a ic
The s i n g l e c r y s t a l w i l l a c t m a c ro -
s c o p i c a l l y a s a hom ogeneous b o d y , b u t I n th e p r o p a g a t io n
o f d e B r o g l i e w aves o f t h e o r d e r o f t h e l a t t i c e
sp a c in g ,
th e se p a ra te m ic ro c ry s ta ls w ill n o t s c a t t e r c o h e re n tly .
To c a l c u l a t e
t h e s c a t t e r i n g fr o m t h e s i n g l e c r y s t a l ,
we m u s t f i r s t c a l c u l a t e
of its
th e s c a t t e r e d a m p litu d e f o r eac h
m i c r o c r y s t a l s a n d t h e n sum t h e I n t e n s i t i e s .
A p o l y c r y s t a l l i n e m a t e r i a l I s com posed o f
many s m a l l s i n g l e c r y s t a l s ,
a t random .
To c a l c u l a t e
or c ry s ta llite s ,
o rie n te d
t h e s c a t t e r i n g fro m a p o l y ­
c r y s t a l l i n e sa m p le , on e m u st a v e r a g e th e s c a t t e r i n g o f
its
su b u n its o v er a l l o r ie n ta tio n s of th e c r y s t a l l i t e s .
9.
By t h e p e r f e c t n e s s o f a c r y s t a l we mean t h e
l i n e a r d i m e n s i o n o f t h e m i c r o c r y s t a l s c o m p o s in g i t .
To c a l c u l a t e t h e t o t a l s c a t t e r i n g f r o m a p o l y ­
c r y s t a l l i n e s a m p l e , we m u s t i n t e g r a t e
th e u su a l d i f f r a c t i o n
i n t e g r a l o v e r a l l f i n a l d i r e c t i o n s o f th e n e u tro n , and
th en av erag e o v er a l l o r i e n t a t i o n s o f th e c o n s titu e n t
m ic ro c ry s ta ls.
We c o n s i d e r t h e n u c l e a r s c a t t e r i n g o f n e u t r o n s
o f wave l e n g t h A h y a m a t e r i a l h a v i n g s i m p l e c u b i c s t r u c t u r e
w ith l a t t i c e
d i s t a n c e a , an d o n ly one ty p e o f n u c le u s
in th e l a t t i c e .
The s c a t t e r i n g f r o m t h e c r y s t a l i n t o
/
7)
s o l i d a n g le
w ill be g iv e n by:
0“ is
=
C
th e c r o s s - s e c t i o n f o r c o h e re n t s c a t t e r i n g
(S ee s e c . I ,
1 .1 )
, a r e t h e n u m b e rs o f p l a n e s o f t h e
m ic r o c r y s ta l a lo n g each o f th e c o o r d in a te d i r e c t i o n s .
y
A
^ 4 ^ .8
au ^
Y
~
®
~
Y
=Kr-^~ A,
'
)
) A.
1C*
(c* * &- < * • & ) ’
{2<2)
10.
The z a x i s h a s b e e n t a k e n p e r p e n d i c u l a r t o
th e p a r t i c u l a r s e t o f p la n e s under c o n s id e ra tio n ; 3 a n d ^ 5
a r e th e p o l a r a n g le and azim u th f o r th e i n i t i a l d i r e c t i o n ,
and s i m i l a r l y f o r th e p rim e d q u a n t i t i e s Q an d p
•
L e t So be t h e p o l a r a n g le f o r B ragg r e f l e c t i o n
f ro m t h e p a r t i c u l a r s e t o f p l a n e s ,
o
w here j i j ^
c ^ e .
so t h a t
=
p + i *i- I j
(2 . 3 )
a n d (J,
l 3) a r e a t r i p l e
in te g e r s d e s c rib in g th e s e t of p la n e s .
th e i n i t i a l a zim u th i s u n n e c e s s a ry ,
a r r a n g e m e n t i s sy m m etric i n a z im u th .
g a n e ra lity ,
of
The a v e r a g e o v e r
s i n c e t h e w h o le
So w i t h o u t l o s s o f
we may t a k e
q
-
The d i f f r a c t i o n
.
o
l s A)
in te g r a l c o n s is ts of th re e f a c to r s ,
h a v i n g a s h a r p maximum a t
A i ~ m u ltip le o f ^
each
and d ro p p in g
f ro m t h e maximum v a l u e o f M L t o z e r o i f Ac c h a n g e s b y
a m o u n ts o f t h e o r d e r o f
; so th e sp re a d s
and
w i t h i n w h ic h t h e I n t e g r a n d i s a p p r e c i a b l e w i l l b e
S u b s t i t u t i n g f ro m ( 2 . 4 )
i n ( 2 .2 ) and t a k i n g term s o f
f i r s t o r d e r i n h f y A p & f we o b t a i n :
/{'■=
A
'-
A C '
( a & + A 3 ')
- Ay
*
8 0
a ***'
■&. +
Ca p * <
& ' ) ■ * - ©*
( 2 *5>
I n yA5 we o m i t a®-**®- { ~ ^ ~ r ) ,
a r g u m e n t of**'*-1'
s i n c e i t m e re ly s h i f t s
by a m u l t i p l e of"a .
T hen
J "
Let
(B ragg c o n d itio n * )
^
' t
=
^
oi
©.
x -
th e
e '^ d ^ r '
j Z ^(Ao)d. & eO aCar )
3 =
~^5
vj - -jjr^
; th e n
<*(^40
w h e re
rc<*.
/ \ , = X c*o 0.* x f
- ft A
-x
&
A ll th e v a ria b le s
th e lim its
y«a r e noW s e p a r a t e d .
of in te g r a tio n to i n f i n i t y
J
~*f
^x
o r u sin g
X
'
th is
T - ^
w here
H is
#
r
is
(
is
'
c*< 0 .
(2*6)
7 ^
t o b e summed o v e r a l l t r i p l e s IA|
(i)‘ £
T ir
X
t h e t o t a l num ber o f u n i t c e l l s *
B efo re c o n s id e rin g t h is
d e ta il,
we o b t a i n :
(2*3)
-
F in a lly
t® I *
f o r w h ic h
x.
\
E x te n d in g
fu n d a m e n ta l fo rm u la i n
we w i s h t o e x a m in e t h e c a s e w h e r e t h e i n i t i a l beam
s h a rp ly d e fin e d and i s
i n c i d e n t e x a c t l y a t th e B ragg
a n g le ,
and c a lc u la te th e t o t a l s c a t t e r i n g f o r t h i s f ix e d
o r i e n t a t i o n o f th e c r y s t a l .
sc a tte rin g
to be l a r g e .
I n t h i s c a s e we e x p e c t t h e
T h is i s o f I n t e r e s t i n e x a m in i n g
th e in flu e n c e o f secondary e x tin c tio n ,
fers
a te r r a w h ic h r e ­
t o t h e f a c t t h a t ' t h e i n c i d e n t b eam ,
i n g o in g th ro u g h
t h e s p e c i m e n , i s b e i n g d e p l e t e d d u e t o s c a t t e r i n g fro m
i n d i v i d u a l m i c r o c r y s t a l s , so t h a t l o w - l y in g m ic ro ­
c r y s t a l s a r e s h i e l d e d fro m t h e f u l l i n t e n s i t y o f t h e
beam b y m i c r o c r y s t a l s l y i n g a b o v e th e m .
T h is e f f e c t w i l l
b e n o t ic e a b l e I f th e s c a t t e r i n g o f th e g iv e n w a v e -le n g th
i n t h e sp e c im e n I s l a r g e , so to e s t im a t e I t s
im p o rta n c e
we c h o o s e t h i s m o s t f a v o r a b l e c a s e o f I n c i d e n c e a t t h e
B ragg a n g le .
A lso , f o r s i m p l i c i t y ,
we c o n s i d e r t h e
m ic ro c ry s ta ls to be p e rf e c tly a lig n e d , b u t n e v e rth e le s s
s c a t t e r in c o h e re n tly .
S lig h t v a ria tio n s
in o r ie n ta tio n
w o u ld p r o d u c e a b r o a d e n i n g o f t h e r a n g e o f r e f l e c t i o n ,
b u t have l i t t l e
H ere
e f f e c t on th e f i n a l r e s u l t .
A 0 =o
and
/ \ , = A e ' a n . 0. ^
A t
^
A, =
The t h r e e f a c t o r s h a v e m axim a
w id th s a r e g iv e n b y :
t
and t h e i r
13
A© ^ c
A,
cjrtbu
A .x \ , <XB' >
;'
.
ii_ W3
In c a lc u la tin g
te g ra tin g over
<2 - 1 0 >
th e t o t a l s c a t t e r i n g by I n ­
&<f , we s e e t h a t t h e r e i s a
and
c o m p e ti t io n b e tw e e n th e w id th s o f A, f and A 3 f and o n ly
one o f th e s e i s
e ffe c tiv e .
e x a c tl y a t th e B ragg a n g le ,
So f o r a beam i n c i d e n t
th e t o t a l s c a tte re d in te n s ity
s h o u ld b e p r o p o r t i o n a l t o h i? •
have s id e s
=
If
Jit
If
th en th e s c a t t e r i n g w i l l be
( 'A ) V «
V7C V
' ^
^
J.^c
th e s in g le c r y s t a l ( c r y s t a l l i t e )
th e r a t i o
of sc a tte re d
J ; .- = X
th e m ic ro c ry s ta ls
(&
A .
of
is £
(2 « u )
l& l
cm. o n a s i d e ,
t o i n i t i a l beam w i l l b e :
)4
- r
<S. 1 2)
-
F or F e, a rou g h e s tim a te o f th e e f f e c t can b e
m ad e : X l s ~ ^ c r a . ,
^
w h ile th e s iz e o f th e c r y s t a l l i t e s
is
2.x j o ~ 3 cm.
,
0 ~ - 7.3 j'-’0
cm.
.
(th e reaso n fo r th is
be seen In s e c tio n I I I . )
,
^
_ <f
X J O
U s in g t h e s e v a l u e s we o b t a i n :
3 L
X .
'
L A
*
/ ° ‘ -J
c h o ic e w i l l
So e v e n I n t h i s
fa v o ra b le case of r e f le c tio n
e x a c t l y a t t h e B ragg a n g le ,
th e seco n d ary e x tin c tio n w ill
be v e ry sm a ll f o r p o l y c r y s t a l l i n e I r o n .
sin g le c ry sta ls
O n ly f o r l a r g e
(Zi~J cm) w i l l i t b e i m p o r t a n t .
I n w hat
f o l l o w s we s h a l l d e a l c h i e f l y w i t h p o l y c r y s t a l l i n e s a m p l e s
f o r w h i c h we may n e g l e c t t h e e x t i n c t i o n .
E q u a tio n (2 .8 ) h as b een o b ta in e d f o r th e
s c a t t e r i n g o f n e u tro n s of a s i n g l e w a v e -le n g th by c r y s t a l s
o f s im p le c u b ic s t r u c t u r e .
In (2 .8 )
we h a v e n o t i n c l u d e d
th e e f f e c t o f th e te m p e ra tu re m o tio n o f th e n u c l e i in
th e c r y s t a l .
The t h e o r y o f t h i s
t)
b e e n g i v e n b y D eby e a n d W a l l e r .
v ib ra tio n
is
te m p e r a tu r e m o tio n h a s
T he e f f e c t o f t h e t h e r m a l
to d e crea se th e c o h e re n t s c a tte r in g o f a
c r y s t a l o f c u b i c sy m m etry b y t h e f a c t o r C
" * ]
w here B i s
w h ere:
— iv -
t 2 ' 13)
th e a n g le b e tw e e n f i n a l an d i n i t i a l d i r e c t i o n
o f t h e b e a m f '?»c<c- m a s s o f t h e a to m f o r m i n g t h e l a t t i c e ,
i
a?C>c)
ft i s t h e B o l t z m a n c o n s t a n t , - y i s t h e D eby e f u n c t i o n , a n d
X-
, w here © i s
t h e D eby e t e m p e r a t u r e a n d T
a b s o lu te te m p e ra tu re of th e c r y s t a l .
c o r r e s p o n d i n g t o i n d e x I Hi
th e
For th e r e f le c tio n
, we h a v e U i \ -
—
, so
1. 0-
t h e f a c t o r g i v i n g t h e d e c r e a s e i n c o h e r e n t s c a t t e r i n g fro m
- A M /1
t h e c r y s t a l becom es
C
(2 .1 4 .1 )
where
I t sh o u ld he n o ted t h a t th e te m p e ra tu re f a c t o r does n o t
depend on th e w a v e -le n g th ;
f o r a g i v e n D ebye r i n g
(d e fin ite
Ul ) I t I s t h e sam e f o r a l l w a v e - l e n g t h s .
re p re se n ts
th e e f f e c t o f th e z e r o - p o in t v i b r a t io n .
The ^
For Fe,
a n d t a k i n g T = 3 C>2~ >
e^fx)^-70i
A lso f o r F e ,
ta k in g
we g e t
^ ■
•L
x-
CL - Z ■2C A ° J
^
xv
ft™--
*/• ^
s o t h e t e m p e r a t u r e f a c t o r f o r i r o n b e co m es
e - .o t'ttje .i
( a .14. 5)
As t h e i n d e x j t i o f t h e D ebye r i n g i n c r e a s e s
sc a tte rin g
th e c o h e re n t
i s c u t down m o re a n d m o re ; f o r
f a c to r is a lre a d y
, th e
*73*
I n th e c a s e o f F e , a se c o n d c o r r e c t i o n m ust b e
m ade.
F e , a t room t e m p e r a t u r e , f o r m s a b o d y - c e n t e r e d
c u b ic l a t t i c e ,
w i t h two a to m s t o e a c h u n i t c e l l .
T h is
i n t r o d u c e s a c r y s t a l s t r u c t u r e f a c t o r w h ic h h a s t h e
fo llo w in g e f f e c t :
A ll r e f le c tio n s
f o r w h ic h -^T /c
i s odd d i s a p p e a r ;
I
a ll
fo r w h i c h ^ i s
even a r e q u a d ru p le d in I n t e n s i t y .
s h a l l r e f e r to t h e s e a s odd a n d ev en i& ! r e s p e c t i v e l y . )
(We
I n o u r s c a t t e r i n g fo rm u la f o r s im p le c u b ic
la ttic e
t h e f a c t o r N w h ic h g a v e t h e n u m b er o f u n i t c e l l s
re p re se n te d a lso
t h e t o t a l n um ber o f a to m s i n t h e c r y s t a l ,
w hereas f o r t h e b o d y - c e n te r e d c a s e ,
c e l l s = £ t h e num ber o f a t o m s .
t h e n u m b er o f u n i t
O ur f o r m u l a e w i l l b e
c o r r e c t f o r b o d y -c e n te re d l a t t i c e s
i f we u s e N t o d e ­
s i g n a t e t h e n u m b er o f a to m s i n t h e l a t t i c e a n d t a k e 2 / r ^j
Then t h e f i n a l s c a t t e r i n g fo rm u la becom es f o r s im p le
*
c u b ic l a t t i c e :
M fl.,.
'l. j
(2 .15 )
J /f;
ill
and f o r b o d y -c e n te re d l a t t i c e
*
(i <L i
Z\VC
\ ^j
( e . g . Fe)
-
*'
- */ %
V
(2.16)
i ^I
\Ie s e e t h a t t h e c o n t r i b u t i o n s t o t h e t o t a l c o h e r e n t
s c a t t e r i n g o f th e p o l y c r y s t a l , from d i f f e r e n t r i n g s
d e c r e a s e s w i t h i n c r e a s i n g /Al slu e t o e a c h o f t h e f a c t o r s
i n t h e summand.
A g iv e n v a lu e o f }i! c a n be r e a l i z e d i n
many d i f f e r e n t w a y s;
e .g .
can be g o tte n f o r
(0 ,1 ,1 );
we m u s t c o u n t a l l p e r m u t a t i o n s a n d c h a n g e s o f s i g n I n
f i n d i n g t h e m u l t i p l i c i t y ^ £|jof a t e r m i n t h e sum; f o r
S i n c e t h e r e h a v e n o t y e t b e e n made s i g n i ­
f i c a n t e x p e r i m e n t s w i t h m o n o c h r o m a tic beam s o f s l o w
n e u tr o n s i t i s n e c e s s a r y t o g e n e r a l i z e th e fo rm u la e
g i v e n f o r t h e c a s e o f r a t h e r w id e v e l o c i t y s p e c t r a *
We s h a l l i n t h i s s e c t i o n p r e s e n t som e c a l c u l a t i o n s made
f o r i l l u s t r a t i v e p u r p o s e s o n l y by u s i n g a M a x w e l l i a n d i s ­
trib u tio n .
In s e c tio n 6 and 7 c a lc u la tio n s w i l l be
p r e s e n te d u s in g e m p ir ic a lly d e te rm in e d d i s t r i b u t i o n
fu n c tio n s.
I n t h e c a s e o f a M a x w e ll d i s t r i b u t i o n ,
c a lc u la tio n s can be c a r r ie d o u t a n a ly tic a lly ;
th e
we so
f i n d f o r th e s c a t t e r e d i n t e n s i t y p ro d u ced by a bodyc e n te re d l a t t i c e s
2 -
/ A
r A
k f
7 ^
w h e re
(2 .1 7 .1 )
_
and
A
J Z ltk T
w here T i s
(2 .1 7 .2 )
th e te m p e ra tu re o f th e n e u tro n so u rc e .
So t h e e f f e c t i v e c o h e r e n t c r o s s - s e c t i o n <T o f
th e c r y s ta l,
n e u tro n s ,
a s d e t e r m i n e d b y u s i n g a beam o f t h e r m a l
w i l l be r e l a t e d to th e c o h e r e n t c r o s s - s e c t i o n
f o r i n d e p e n d e n t n u c l e i CT b y :
W
(T
(2 .1 8 ,
T hese s c a t t e r i n g fo rm u la e w i l l h e a p p l i e d to t h e e v a lu a ­
t i o n o f e x p e rim e n ta l r e s u l t s in s e c tio n I I I .
L e t us c o n s i d e r l i m i t i n g c a s e s f o r t h e v a r io u s
s c a t t e r i n g fo rm u la e .
la ttic e ;
(We c o n s i d e r o n l y s i m p l e c u b i c
s i m i l a r a rg u m e n ts h o ld f o r c r y s t a l s w ith b a s i s . )
In 2 .8 ,
by an i n t e g r a l ;
we c a n a p p r o x i m a t e t h e l a t t i c e
sum
t h i s g iv e s
The v a l i d i t y o f t h i s a p p r o x i m a t i o n b y i n t e g r a l i n c r e a s e s
w i t h i n c r e a s i n g nu m b er o f l a t t i c e p o i n t s ,
i.e .
d e c r e a s i n g 1.
So f o r s m a l l X , we f i n d t h e e f f e c t o f t h e c r y s t a l s t r u c t u r e
d is a p p e a rin g , as is
t o b e e x p e c t e d fr o m g e n e r a l r e a s o n i n g .
I n e q u a t i o n 2 . 1 5 , w h e re t h e e f f e c t o f t h e
te m p e ra tu re v ib r a tio n is
In clu d e d ,
g iv e s a q u ite d i f f e r e n t r e s u l t .
th is
lim itin g case
H e r e r e p l a c i n g t h e sum
b y a n i n t e g r a l R iv e s
so t h a t as
)
^
>
d
—» © ■
A g a in t h i s r e s u l t i s e a s i l y
in te rp re te d .
The i n c l u s i o n
o f t h e t e m p e r a t u r e e f f e c t m eans t h e i n t r o d u c t i o n o f a
f o rm -fa c to r f o r th e n u c le a r p o s itio n in th e c r y s t a l ;
f o r m - f a c to r ° , fo r
any
I n t h e e x p r e s s io n f o r s im p le c u b ic c r y s t a l s
c o rre sp o n d in g to 2 .1 8 ,
l e t u s c o n s i d e r t h e c a s e w h e re
The a p p r o x i m a t i o n b y I n t e g r a l g i v e s
ovf r' =
rr
— Z . vV - <*a- JT J V
, + K f t e & xk—) * *
^
tj -
j
by p a r t s
g i v e s 0> -= /
a-) cL v r
and i n t e g r a t i o n
■
AT
L et us c o n s id e r 2 .8 as A in c r e a s e s .
W ith
I n c r e a s i n g A , t h e n u m b e r o f t e r m s i n t h e sum d e c r e a s e s ,
a,
b u t th e i n t e n s i t y o f each r i n g
in c re a se s,
i n c r e a s e s a s /I .
As A
t h e r e w i l l b e d e f i n i t e p o i n t s a t w h ic h Debye
rin g s d isa p p e a r.
(T h e I
w ill d isa p p e a r a t A -
and a d is c o n tin u o u s drop in i n t e n s i t y o c c u rs ;
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),
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th e n
V
h e c a u se o f th e A f a c t o r ,
th e in te n s ity
t h e n e x t D ebye r i n g
is
cut o u t.
f i r s t D eb ye r i n g i s
l e f t and
If
in c re a se s u n til
, o n ly th e
z<-‘ > --
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Tc
th e c o h e re n t s c a t t e r i n g d isa p p e a rs
For
c o m p le te ly .
A p a r t fro m t h e c o h e r e n t s c a t t e r i n g we h a v e t o
d isc u ss
tw o t y p e s o f i n c o h e r e n t s c a t t e r i n g w h ic h f o r m a
b a c k g ro u n d b e tw e e n t h e Debye r i n g s .
in e la s tic
T here is
f i r s t th e
s c a t t e r i n g o r i g i n a t i n g fr o m p r o c e s s e s
in v o lv in g
e n e rg y t r a n s f e r b e tw e e n t h e n e u tr o n and t h e l a t t i c e .
T h is i n e l a s t i c s c a t t e r i n g i s ,
in th e c a se o f X -ra y s, g iv e n
by f o r m u l a e a n a l o g o u s t o :
<r(/ -
r
(2>19)
I n t h e c a s e o f n e u t r o n s c o n d i t i o n s a r e m o re d i f f i c u l t ,
m e n tio n ed a lr e a d y
in th e i n t r o d u c t i o n .
as
T he I n e l a s t i c
c r o s s - s e c t i o n w h ic h w o u ld a l r e a d y b e s m a l l f o r X - r a y s
i n c a s e t h e Debye t e m p e r a t u r e i s c o n s i d e r a b l y l a r g e r t h a n
t h e t e m p e r a t u r e o f o b s e r v a t i o n b e co m es o f e v e n l e s s
s i g n i f i c a n c e f o r th e s c a t t e r i n g o f slo w n e u t r o n s .
The I n c o h e r e n t b a c k g r o u n d d u e t o t h e p r e s e n c e
o f iso to p e s c o n s titu te s
a p h e n o m e n o n w h ic h d o e s n o t o c c u e
w ith X -ray s t h a t i s n u m e r ic a lly o f g r e a t im p o rta n c e .
It
i s due to th e f a c t t h a t t h e s c a t t e r i n g a m p litu d e s o f v a r io u s
Iso to p e s c o n s titu tin g th e c r y s t a l have in g e n e ra l d i f f e r e n t
m ag n itu d e a s w e ll a s s i g n .
The s p h e r i c a l l y s y m m e t r i c a l
background i s produced w ith a s c a t t e r i n g c ro s s s e c t i o n
g i v e n b y l)
IJ
■ ' b ?! Ljy ?
• **
7\
<if +
***J
(2.20)
r
We s h a l l s e e l a t e r on t h a t i t p r o b a b l y a c c o u n t s f o r t h e
l a r g e r p a r t of in c o h e re n t s c a t t e r i n g o c c u rrin g in so
w e ll-a n a ly z e d a s u b s ta n c e as F e.
III.
D is c u s s io n o f S c a t t e r i n g E x p erim e n ts.
The d e l i b e r a t i o n s o f s e c t i o n 2 c o n t a i n t h e
t h e o r e t i c a l b a s is f o r th e e x p la n a tio n and e v a lu a tio n o f
s c a t t e r i n g e x p e rim e n ts.
In p a rtic u la r,
th ey g iv e a c c o u n t
o f c e r t a i n c h a r a c t e r i s t i c p h enom ena w h ic h w e r e o b s e r v e d
when n e u t r o n s w e r e s c a t t e r e d fr o m t h e sam e s u b s t a n c e s
p r e s e n t e i t h e r i n d i f f e r e n t c r y s t a l fo rm a o r I n d i f f e r e n t
to )
c h e m ic a l c o m b in a tio n s .
We s h a l l f i r s t d i s c u s s t h e o b s e r v a t i o n s made
w i t h F e , s i n c e t h e y w i l l t u r n o u t t o b e o f g r e a t im­
p o r t a n c e n o t o n l y f o r t h e e v a l u a t i o n o f s c a t t e r i n g ex­
p e rim e n ts, b u t a ls o f o r th e q u a n tita tiv e d e te rm in a tio n
of th e p o l a r i z a t i o n e f f e c t s
in fe rro m a g n e ts a s g iv e n
In s e c tio n 6.
E x p e r i m e n t s w e re p e r f o r m e d f i r s t o n a p o l y c r y e t a l l i n e s a m p le (
- 11 0 *
a s i n g l e c r y s t a l o f Fe
th erm al d i s t r i b u t i o n
The c a s e o i
fo llo w s:
* /ct
7 */
) a n d t h e n on
), u sin g a
o f n e u t r o n s a t room t e m p e r a t u r e .
t h e s i n g l e c r y s t a l may b e d e s c r i b e d a s
The r e s o l v i n g p o w e r o f t h e m i c r o c r y s t a l s ,
w h ic h
a r e a l l o r i e n t e d i n a p p r o x i m a t e l y t h e sam e d i r e c t i o n ,
l i m i t th e s c a t t e r i n g
w ill
to s m a ll r e g i o n s i n th e n e ig h b o rh o o d
o f t h o s e w a v e - l e n g t h s f o r w h ic h B r a g g r e f l e c t i o n o c c u r s .
T hese n e u tr o n s ,
la rg e ,
f o r w h ich t h e c o h e r e n t s c a t t e r i n g i s
r e p r e s e n t o n ly a s m a ll f r a c t i o n o f th e t o t a l
in c id e n t n e u tro n i n t e n s i t y ;
e n tire
so t h a t c o n s i d e r i n g t h e
i n c i d e n t n e u t r o n beam ,
th e e f f e c t i v e c o h e re n t
s c a t t e r i n g w ill be n e g lig e a b ly s m a ll.
Thus t h e d i f f e r e n c e b e t w e e n t h e r e s u l t s
t h e two c a s e s r e p r e s e n t s
in
j u s t th e c o h e re n t s c a t t e r i n g
o f t h e i n c i d e n t d i s t r i b u t i o n by t h e p o l y c r y s t a l l i n e
sa m p le .
se c tio n
T h is g iv e s f o r t h e e f f e c t i v e c o h e r e n t c r o s s O '= 5 * ' °
•
The r a t i o 5 ^
fo r iro n ,
has been c a lc u la te d n u m e ric a lly
w ith th e rm a l n e u tr o n s ,
tu re fa c to r.
U s in g t h i s
in c lu d in g th e tem p era­
c a lc u la te d r a tio
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,
a n d t h e e x p e r i m e n t a l v a l u e o f W h i t a k e r a n d B e y e r o- ■= •S'*"*
we f i n d
cr
-
C
-
, so t h e n u c l e a r a m p litu d e
’
7
We c a n s i m i l a r l y
u n d e r s t a n d on t h e b a s i s o f
o u r fo rm u la e th e s m a ll d e v ia ti o n s fo u n d by W h ita k e r an d
B e y e r when t h e c r o s s - s e c t i o n o f c e r t a i n m o l e c u l e s was
c o m p a re d w i t h t h e sum o f t h e c r o s s - s e c t i o n s
p o n e n ts .
o f t h e com­
No g e n e r a l p r e d i c t i o n s c a n b e m ade a s t o
sig n of th is
e ffe c t,
th e
b u t s in c e th e c r y s t a l c r o s s - s e c ti o n
f o r re a so n a b le v e lo c ity d i s t r ib u t i o n s
o f th e n e u tro n s
may d i f f e r b y a p p r o x i m a t e l y 2C$> f r o m t h e v a l u e f o r t h e
a m o rp h o u s s t a t e we c a n r e a d i l y
se e t h a t change o f l a t t i c e ,
of b a sis,
and of te m p e ra tu re f a c t o r can le a d to c ro s s -
s e c t i o n c h a n g e s o f t h e o r d e r o f m a g n i t u d e o f 10%.
One
m u s t a l s o k e e p i n m ind t h a t some o f t h e c o m p o n e n ts ( e . g .
C u .) may p o s s e s s r a t h e r l a r g e m i c r o c r y s t a l s a n d t h e r e f o r e
show a p a r t i a l e x t i n c t i o n e f f e c t w h ic h w i l l d i s a p p e a r
when Cu e n t e r s
i n t o t h e c h e m i c a l com pou nd.
S im ila rly ,
in th e c ase o f a llo y s
( e . g . p e rm a llo y
10)
i n v e s t i g a t i o n s by W h ita k e r and B e y e r, and N ix , B e y e r, and
•0
D u n n in g ) t h e p r e s e n c e o f a n o r d e r e d c r y s t a l l i n e s t r u c t u r e
s h o u l d p r o d u c e m a rk e d c h a n g e s i n t h e c o h e r e n t s c a t t e r i n g ,
and g iv e r i s e to la r g e d e v ia tio n s
fro m a d d i t i v i t y ;
changes
i n t h e p e r c e n t a g e c o m p o s i t i o n , o r t r e a t m e n t w h ic h b r e a k
up t h e o r d e r e d c r y s t a l l i n e s t r u c t u r e s h o u l d t h e n l e a d
b a c k t o t h e v a l u e g i v e n by a d d i t i v i t y .
On t h e b a s i s o f o u r p r e v i o u s d i s c u s s i o n ,
cu rren t b e lie f
n e u tro n s i s
th e
t h a t th e s c a t t e r i n g c r o s s - s e c t i o n f o r slo w
in d ep en d en t o f th e n e u tro n
lo n g e r be a c c e p te d .
e n e r g y c a n no
The r a t i o Sjj- h a s b e e n c a l c u l a t e d
f o r F e f o r m o n o c h r o m a tic beam s o f v a r y i n g w a v e - l e n g t h ,
u sin g e q u a tio n 2 .1 6 .
The r e s u l t s a r e p l o t t e d
As a l r e a d y m e n t i o n e d i n S e c t i o n 2 ,
i n G ra p h I .
in c r e a s in g w a v e -le n g th
w i l l mean a d e c r e a s e i n t h e n u m b er o f D e b y e - S c h e r r e r r i n g s ,
b u t t h e i n t e n s i t y o f e a c h r e m a i n i n g r i n g g o e s up a s
I r o n w i t h l a t t i c e d i s t a n c e 2 . 8 6 A0 i s b o d y - c e n t e r e d ,
A
.
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so t h a t th e lo w e s t o r d e r a p p e a rin g w i l l be ( 1 , 1 , 0 ) w ith
III r ( z
and H l & l ) » 12;
iro n d isap p ears f o r
t h e c o h e r e n t s c a t t e r i n g fro m
^
.
The l i m i t i n g v a l u e
f o r t h e c o h e r e n t s c a t t e r i n g a t room t e m p e r a t u r e a s
is,
fro m 2 , 1 8
-- - 3- - - Z
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,
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cjv/4
S e c t i o n IV .
D is c u s s io n o f th e S iz e of M i c r o c r y s ta ls and
E s t i m a t e o f t h e S c a t t e r i n g A m p litu d e o f F e I s o t o p e s
I t is
fo rm u la e ,
o f i n t e r e s t to s e e t h a t t h e p r e c e d in g
to g e th e r w ith th e o b s e rv a tio n s o f th e d i f f e r e n c e
i n c r o s s - s e c t i o n s f o r th e c r y s t a l and p o l y c r y s t a l l i n e
s ta te o f iro n ,
e n a b l e u s t o draw some c o n c l u s i o n s a b o u t
th e s i z e o f th e m ic r o c r y s ta ls and th e s c a t t e r i n g a m p li­
tu d e s o f t h e Fe i s o t o p e s .
F o r a rough d i s c u s s i o n ,
it
i s s u f f i c i e n t to
d e s c r i b e t h e w e a k e n i n g o f a beam p a s s i n g t h r o u g h a n
a g g re g a te o f c r y s t a l l i t e s
as fo llo w s:
b e r e d u c e d i n I n t e n s i t y by t h e f a c t o r C
The beam w i l l
w h e r e fJ- - h
'
p
-U 7C a ?
( S e e 2 . 1 2 ) when i t s t r i k e s a c r y s t a l l i t e a t t h e B r a g g
a n g le .
The p r o b a b i l i t y o f s u c h a n e v e n t i s
w h ic h r e p r e s e n t s
m ic ro c ry s ta l.
g iv e n by
,
th e r e s o l v i n g pow er o f th e p e r f e c t
T h i s p r o b a b i l i t y m u s t b e m u l t i p l i e d by
a f a c t o r w h ic h g i v e s t h e n u m b er o f p o s s i b l e s t r o n g r e ­
fle c tio n s
f o r o n e a n d t h e same w a v e - l e n g t h .
A lth o u g h
o n l y a r o u g h e s t i m a t e o f t h i s f a c t o r c a n b e m ad e , i t
su ffic ie n tly
to 15.
is
a c c u r a t e f o r o u r p u rp o se s to p u t i t e q u a l
The t o t a l w e a k e n in g o f a beam i s now d e t e r m i n e d
by th e v a r i o u s m u tu a lly e x c l u s i v e a l t e r n a t i v e s
beam s t r i k e 0 , 1 , 2 ,
t h a t th e
e tc . p ro p erly o rie n te d c r y s t a l l i t e s
w hen p a s s i n g t h r o u g h a t h i c k n e s s d o f t h e s u b s t a n c e .
The p r o b a b i l i t y o f n f a v o r a b l e e n c o u n t e r s i s g i v e n b y :
^ _
w here
^ y~
(P o isso n
/y%. i
D e n o tin g th e s i z e o f one c r y s t a l l i t e by Lf
^
fo rm u la ).
we o b t a i n f o r t h e i n t e n s i t y
2 : = X
tra n s m itte d :
Z
X
-
e ^
' - ^
(4a)
t-
T h is m u s t b e com pared w ith th e e x p e r i m e n t a l l y fo u n d a b -A /< fo t
-—
s o r p t i o n la w
-i-jC
__
w h e re
<T r e f e r s
o n ly to th e
c o h e re n t c ro s s - s e c tio n of th e p o ly c r y s ta l,
a lte r n a tiv e s a re p o ssib le :
tin c tio n ),
In th is
Now two
C ase 1)
case
(L arg e ex­
so
U s in g t h e e m p i r i c a l l y d e t e r m i n e d v a l u e CT= 5
3 ),
= A icr aL
(S ee s e c t i o n
■**.
we o b t a i n a s a n u p p e r l i m i t ,
T h is u p p e r l i m i t f o r t h e s i z e o f t h e p e r f e c t
m ic ro c ry s ta l is
its
of c o u rse h ig h ly e x a g g e ra te d .
I t ta k e s
maximum i n t h e c a s e o f l a r g e e x t i n c t i o n s i n c e t h e
w e a k e n i n g o f t h e beam by p a s s i n g t h r o u g h o n e c r y s t a l
g ra in is
th e n a lm o st c o m p le te .
It
is, a l s o o f i n t e r e s t
t o p o i n t o u t t h a t i n t h e c a s e o f m a r k e d e x t i n c t i o n no
p o l a r i z a t i o n e f f e c t c o u ld b e o b s e rv e d , s in c e i t
to
is
due
th e s l i g h t I n c r e a s e o r d e c r e a s e o f th e a to m ic c r o s s -
s e c t i o n p ro d u c e d by th e i n t e r f e r e n c e o f n u c l e a r an d
m a g n e tic s c a t t e r i n g .
so l a r g e a s
If
th e n u c le a r s c a t t e r i n g
to p ro d u c e e x t i n c t i o n ,
is a lre a d y
a s l i g h t change In
th e t o t a l a to m ic c r o s s - s e c t i o n can n o t b e o f Im p o rta n c e ,
A s i n g l e c r y s t a l a s d i s t i n g u i s h e d fro m a p o l y c r y s t a l l i n e
body h as co m p lete e x t i n c t i o n , and t h e r e f o r e can n o t g iv e
an o b se rv a b le p o la r iz a tio n e ffe c t*
C a se 2 ) :
T h is c o r r e s p o n d s t o t h e a b s e n c e o f
e x tin c tio n o r q u a n tita tiv e ly sp e a k in g ,
_ V
th e n u m e ra to r in C
-£ -/* * ■
I
•
S in c e
J
A
1 b now v e r y much s m a l l e r ,
t h e n u m b er o f l a y e r s o f t h e p e r f e c t m i c r o c r y s t a l m u s t b e
c o rre s p o n d in g ly red u c ed In o rd e r to keep th e v a lu e f o r
th e s c a t t e r i n g c o n s ta n t, and e q u al to th e o b se rv e d
-jjJ
/
I n th e l i m i t i n g c a s e e -/-/* - , th e e x p r e s s io n f o r th e
_u
a b s o r p tio n red u c es to
a s was t o b e e x p e c t e d ,
m ic ro c ry s ta l.
'
_ /s~q-A
>- £
, an d so becom es,
in d e p e n d e n t o f th e s i z e o f th e
T h is c a s e c o r r e s p o n d s t o t h e c o n d i t i o n
fo u n d i n n a t u r e , and le a d s
to a v a lu e o f
< r = / x / ' & ***•.
S i n c e t h i s c a s e i s p r e c i s e l y t h e o n e w h ic h we c o n s i d e r e d
in S e c tio n 2,
th e f a c t t h a t th e v a lu e s o f
cr
o b ta in e d
b y b o t h a r g u m e n t s a g r e e w o u ld i n d i c a t e t h a t o u r c h o i c e
o f som e o f t h e c o n s t a n t s u s e d i s
c o rre c t*
R e c e n t l y , M i l l e r a n d Du M o n d ^ a v e g i v e n
e s tim a te s of th e le n g th o f p e r f e c t l a t t i c e
A1 o f a p p r o x i m a t e l y 5 0 0 l a t t i c e d i s t a n c e s .
fin d
t h a t a f t e r d i s t o r t i o n and w o rk in g ,
r e d u c e d t o 150 t o 2 0 0 l a t t i c e d i s t a n c e s .
i n Ag a n d
They a l s o
th e le n g th Is
S im ila r be­
h a v i o r I n F e w o u ld l e a d t o n e g l l g e a b l e e x t i n c t i o n .
We c a n a l s o a t t e m p t t o e v a l u a t e a p p r o x i m a t e l y
t h e c o h e r e n t c r o s s - a e c t i o n o f t h e two I r o n i s o t o p e s
13)
w h ich a r e p r e s e n t i n t h e r a t i o o f 9 : 1 .
S in c e th e t o t a l
c ro ss- se c tio n of iro n is
(i.e .
f o u n d t o h e 12 n u c l e a r u n i t s
1 2 x 1 0 “ 2 ^ x cm*2 ) a n d t h e c a p t u r e c r o s s - s e c t i o n
to be a p p ro x im a te ly 3 .5 n . u . ,
8*5 u n i t s
we h a v e t o a c c o u n t f o r
as b e in g due to c o h e r e n t and in c o h e r e n t
sc a tte rin g .
Of t h i s ,
5 n . u . a r e fo u n d to b e c o h e re n t
s c a t t e r i n g , w h ile t h e i n c o h e r e n t c o n t r i b u t i o n o f 3*5 n . u .
is
due to
la ttic e ,
th ree cau ses:
in e la s tic
in c o h e re n t n u c le a r s c a tte r in g ,
due to is o to p e d is o r d e r .
sm a ll,
s c a t t e r i n g by t h e
and back g ro u n d
The i n e l a s t i c s c a t t e r i n g i s
w h ile th e in c o h e r e n t n u c le a r s c a t t e r i n g a ls o
p r o b a b l y d o e s n o t c o n t r i b u t e much t o t h e t o t a l b a c k 5*6
g ro u n d .
T he i s o t o p e F e
h a s p r o b a b l y no s p i n , b e i n g
a 4n n u c le u s , w h ile t h e i s o t o p e Fe
a n a n o m a lo u sly l a r g e c r o s s - s e c t i o n
w o u ld h a v e t o h a v e
to c o n t r i b u t e a
l a r g e p o r t i o n o f th e o b s e rv e d b a c k g ro u n d s i n c e i t fo rm s
o n ly 10^ o f th e e le m e n t.
L u m p in g t o g e t h e r t h e s e f i r s t
tw o c a u s e s , a n d a s c r i b i n g
t o th e m f o r p u r p o s e s o f a
ro u g h e s tim a te a c r o s s - s e c t i o n o f one n u c le a r u n i t ,
we o b t a i n a v a l u e o f 2 . 5 n u c l e a r u n i t s f o r t h e d i s o r d e r
c o n t r i b u t i o n as g iv e n by
Cp
is
w here
t h e c o n c e n t r a t i o n a n d -fp -
a m p litu d e o f th e p 't h
iso to p e ,
two i s o t o p e s r e d u c e s t o
x th e c o h e re n t
w h ich f o r t h e c a s e o f
~ -fj.')
(4 .2 )
A cco rd in g to S e c t i o n 3 , a c r y s t a l l i n e c r o s s s e c tio n of fiv e n u c le a r u n its
c o r r e s p o n d s t o a n a m o rp h o u s
c r o s s - s e c t i o n o f a p p ro x im a te ly se v en n u c le a r u n i t s .
We
can w ith th e s e d a ta a tte m p t to d e te rm in e th e s c a t t e r i n g
a m p l i t u d e a n d p h a s e f o r t h e two i s o t o p e s .
are:
( . 1 f i -+ •
1
A)
The e q u a t i o n s
- 7
T h e r e i s no r e s t r i c t i o n o f g e n e r a l i t y i n a s s u m i n g t h a t t h e
a m p litu d e o f th e a b u n d an t
sig n .
We t h e n o b t a i n
iso to p e s h a ll
have th e p o s i t i v e
t h e f o l l o w i n g two s e t s o f s o l u t i o n s
f o r th e a m p litu d e s :
e ith e r
a)
or
b)
f, = 3 - "
;
-fK - 7 4-*-
( = 2 .7 2-
b ) may b e e x c l u d e d a s b e i n g h i g h l y i m p r o b a b l y ;
i t w o u ld
le a d to an u n u s u a lly h ig h c r o s s - s e c t i o n o f 55 n .u , f o r
th e l e s s a b u n d a n t i s o t o p e Fe
•
We f i n a l l y
o b ta in th e
fo llo w in g e s tim a te f o r th e c o h e re n t c r o s s - s e c tio n s of
t h e two n u c l e i :
:
CT ^
10
F e** :
<T ^
4 .5
Fer t
I n a d d i t i o n to t h i s ,
^
i t sh o u ld be n o te d t h a t th e a m p li­
tu d e s have o p p o s ite s i g n s .
V. An I n t e r p r e t a t i o n o f t h e L a rm o r P r e c e s s i o n o f t h e S p i n
D u r i n g P a s s a g e t h r o u g h M a g n e t i z e d M e d ia .
We w i s h now t o a p p l y o u r r e s u l t s
to th e p r o ­
p a g a t i o n o f n e u t r o n s i n a f e r r o m a g n e t i c medium ( F e ) ,
m a g n e tiz e d to s a t u r a t i o n ,
w ith th e s p in s o f a l l io n s
r i g i d l y a lig n e d a lo n g th e d i r e c t i o n o f a u n i t v e c to r ^
•
The t o t a l s c a t t e r e d wave fro m a n i o n w i l l
b e g iv e n by 1 . 8 :
The p r e s e n c e o f t h e m a g n e t i c t e r m i n t r o d u c e s t h e p o s ­
s ib ility
o f changes in p o l a r i z a t i o n due to th e d i f f e r e n c e
i n s c a t t e r i n g o f t h e two s p i n s t a t e s
o f th e n e u tro n .
To
p u t th e p o s s i b i l i t y o f su ch r o t a t i o n o f th e p la n e o f
p o la riz a tio n c le a rly
In e v id e n c e ,
n e u t r o n beam o f w a v e - l e n g t h
sp a c in g Is le s s
th an iL .
t r a v e r s i n g a l a t t i c e w h o se
T hen no B ra g g maxima c a n o c c u r
and th e c o h e re n t s c a t t e r i n g
d ire c tio n .
l e t us c o n sid e r a
is c o n fin e d to th e fo rw a rd
We s h a l l r e s o l v e t h e s p i n a l o n g t h e d i r e c t i o n
o f th e m a g n e tx z a tio n ,
w h ic h i s
th e m ost c o n v e n ie n t r e ­
p re s e n ta tio n .
Suppose K a lo n g th e x a x i s ;
p o n e n ts
t h e n q h a s com­
T hen i n summing o v e r t h e
31
la ttic e ,
t h e c r o s s - t e r m s d r o p o u t b e c a u s e o f t h e sy m m etry
a n d o n l y t h e x c o m p o n e n t ( II t o K )
of th e c r y s t a l ,
re m a in s.
We c o n c l u d e t h a t no d e p o l a r i z a t i o n c a n b e p r o d u c e d by
th is p ro cess.
N e v e rth e le ss,
t h e two p o l a r i z a t i o n s t a t e s a r e
sc a tte re d d iffe re n tly .
We s h a l l t r e a t t h e p r o p a g a t i o n
o f each of th e p o l a r i z a t i o n s t a t e s
d is p e rs io n th e o ry ,
an y p o i n t ,
as in c l a s s i c a l
by a d d i n g t o t h e i n c i d e n t wave a t
t h e s c a t t e r e d w aves fro m t h e c e n t e r s
in th e
m edium , a n d d e t e r m i n e a n i n d e x o f r e f r a c t i o n .
y
x
For s im p lic ity ,
l e t t h e p l a n e OXZ c o n t a i n o n e s c a t t e r i n g
c e n te r p er u n it a re a .
m ust c a l c u l a t e
OP = a .
L e t K ib e a lo n g t h e z a x i s .
t h e t o t a l w ave a t t h e f i e l d p o i n t P , w h e re
K
-
£
=
ip,
o,
<)
cLx. dj —aj~
t
Y>
.‘Us**
/
V
I
We
JQ_
/ - chto
The m a g n e t i c a l l y s c a t t e r e d wave f r o m p l a n e OXY i s :
£ &
(w here
r
.-
f f e .‘— c f s j
J J
yz
6
^
tL^QdB d<f c^..
= 3. D * " f
*J
As we h a v e s e e n p r e v i o u s l y ,
re m a in s, g iv in g :
t|.t) £ 1
Q _ /CLOJt. c j
o n ly th e z com ponent
„
- ± 7:ir
o^e;
T he i n t e g r a l w i l l b e o s c i l l a t o r y a t
th e u p p er l i m i t ,
in a l l F r e s n e l d i f f r a c t i o n p ro b le m s, b u t j u s t as
o p tic a l case,
such co nv ergen ce d i f f i c u l t i e s
as
in th e
can be
e l i m i n a t e d by t h e i n t r o d u c t i o n o f a w e i g h t f a c t o r .
P a r tia l in te g ra tio n g iv e s :
±
-r s
T he s e c o n d t e r m =
n e g le c tin g th is
p a rt,
th is
d.
te rm .
sc a tte re d
p 00 ‘ 1
f ^ . Pu
; we p r o c e e d
j
I f we h a d i n c l u d e d t h e n u c l e a r
wave f r o m p l a n e OXY w o u ld b e :
f o r t h e tw o p o l a r i z a t i o n s t a t e s .
B u t t h e t o t a l w ave
a t p o i n t P w i l l b e t h e sum o f t h e i n c i d e n t wave a t P a n d
t h e s c a t t e r e d w aves fro m a l l p l a n e s b e l o w P .
o f t h e I n c i d e n t wave a t P i s
ta k e n a t z e ro ,
If
its
th e phase
phase a t
33.
OXY i s 6
.S o
f o r a th ic k n e s s d,
th e a m p litu d e o f
t h e t o t a l wave i s
jz : 0
^ £
e ,'
w here I i s
~ * S -^
t h e i n c i d e n t w a v e , a n d we h a v e a s s u m e d o n e
s c a t t e r i n g c e n t e r p e r u n i t v o lu m e .
T h is g i v e s f o r t h e
in d e x o f r e f r a c t i o n :
¥ r - 1( oC ±
+
\
r*-C
/
k.
and f o r th e r a t e of r o t a t i o n o f th e d i r e c ti o n o f p o l a r i ­
=
z a tio n :
) ( v e l o c i t y o f t h e beam )
llJ*L — 4
_
k
k h ~
From t h e m a c r o s c o p i c v i e w p o i n t ,
'
(3 . 1 )
th is ra te of ro ta tio n
s h o u l d b e g i v e n by t h e v e l o c i t y o f t h e L a rm o r p r e c e s s i o n
o f th e n e u tr o n In th e m a c ro sc o p ic f i e l d o f th e c r y s t a l .
—y
j
The m a g n e t i c m o m e n t / u n i t v o lu m e i s M —
^ ^
so
t h a t 5 .1 becom es:
./ _
&yL
Y*
M
< 5 - 2)
The t o t a l r o t a t i o n o f t h e p l a n e o f p o l a r i z a t i o n I s o f
c o u r s e o b t a i n e d by a d d i n g t o 5 * 2 t h e p r e c e s s i o n d u e t o
t h e o v e r a l l m a g n e t i c f i e l d H.
A f t e r a d d in g t h i s
term
w h i c h c a n n o t b e s p e c i f i e d w i t h o u t k n o w le d g e a b o u t a c c i ­
d e n t a l o u t s i d e f i e l d s a s w e l l a s t h e s h a p e o f t h e b o d y we
c a n s u m m a r iz e t h e r e s u l t s by s a y i n g t h a t i t
~ j-jfVs-M
is
th e v e c to r:
(5 . 3 )
w h ic h d e t e r m i n e s t h e t o t a l L a rm o r f r e q u e n c y o f p r e c e s s i o n .
O u r d e r i v a t i o n a l l o w s som e i n s i g h t I n t o
th e m anner and
t h e m e c h a n ism b y w h i c h t h e in h o m o g e n e o u s m a g n e t i c f i e l d
o f t h e i n d i v i d u a l a to m s a v e r a g e s o u t a n d l e a d s
to th e
w e ll-k n o w n r e s u l t f o r th e r o t a t i o n o f t h e p l a n e o f
p o la riz a tio n .
I t c a n b e show n e a s i l y
t h a t a n a n a lo g o u s
r e s u l t h o l d s w h en t h e d i r e c t i o n o f m a g n e t i z a t i o n i s no
l o n g e r assum ed to b e p a r a l l e l
to th e d i r e c t i o n o f i n ­
c i d e n c e b u t a l l o w e d t o m ake a n a r b i t r a r y
a n g le w ith i t .
S e c t i o n V I.
P o l a r i z a t i o n E x p erim e n ts w ith F e rro m a g n e ts ;
G e n e r a l P r i n c i p l e s a n d C h o i c e o f Form F a c t o r
P o l a r i z a t i o n e f f e c t s a c c o m p a n y in g t h e p a s s a g e
o f n e u tro n s th ro u g h fe rro m a g n e ts a r e due s o l e l y to th e
I n t e r f e r e n c e b e tw e e n n u c l e a r an d m a g n e tic s c a t t e r i n g .
S i n c e i t was p r o v e n t h a t t h e m a g n e t i c s c a t t e r i n g by
If)
f r e e e l e c t r o n i c s p i n s i s a l w a y s i n c o h e r e n t we h a v e t o
l i m i t t h e d i s c u s s i o n to t h e c a s e i n w h ich t h e e l e c t r o n i c
s p i n re m a in s u n changed d u r i n g t h e c o l l i s i o n p r o c e s s .
T h is i s
th e c a s e o n ly f o r f e r r o m a g n e ts , and n e u tro n s
of s u ffic ie n tly
sm a ll v e l o c i t y .
S im ila rly ,
o n ly th e
c o h e re n t p a r t o f th e p u re ly n u c le a r s c a t t e r i n g
re le v a n c e .
(i.e .
is of
N e ith e r th e In c o h e re n t n u c le a r s c a t t e r i n g
t r a n s i t i o n w ith change o f s p i n o f n u c le u s ) n o r
th e background due to is o to p e d i s o r d e r c o n t r i b u t e any­
t h i n g to th e m a g n e tic p o l a r i z a t i o n phenom enon; t h i s
is
t r u e f o r th e l a t t e r s i n c e t h e m a g n e tic s c a t t e r i n g
a m p l i t u d e I s t h e sam e f o r a l l
d i s a p p e a r s fr o m 2 2 c ? 'fp*- -
iso to p e s,
and t h e r e f o r e
We c a n t h e r e f o r e
l i m i t o u r s e l v e s to th e e x p e r i m e n t a l ly d e te rm in e d
15")
c o h e re n t n u c le a r c ro s s -s e c tio n *
From t h e p r e c e d i n g
r e m a r k s i t b e co m es a p p a r e n t t h a t t h e c o n s i d e r a t i o n o f
th e c r y s t a l l i n e s tr u c tu r e o f th e m a te r ia l is
d e c isiv e
i n any c a l c u l a t i o n o f p o l a r i z a t i o n e f f e c t s .
Not o n ly
does i t a llo w us to ta k e p r o p e r ly i n t o a c c o u n t th e
c o h e re n t and in c o h e re n t c o n tr ib u tio n s , b u t i t
is a ls o ,
as p r e v io u s ly rem ark ed , e s s e n t i a l i n th e I n t r o d u c tio n
o f t h e form f a c t o r f o r t h e m a g n e tic s c a t t e r i n g .
I t w o u ld b e p e r m i s s i b l e (C f* S e c t i o n 3 a n d
G ra p h I )
to a p p ro x im a te t h e c r y s t a l l i n e a g g r e g a t e by
a n a m o rp h o u s b o d y f o r s u f f i c i e n t l y
sm a ll w a v e -le n g th .
B ut s in c e th e d i s t r i b u t i o n o f th e m a g n e tic a lly a c tiv e
c u rre n ts
i n a n a to m e x t e n d s o v e r a d o m a in w h ic h i s
n o t s m a l l e v e n c o m p a re d w i t h t h e w a v e - l e n g t h o f t h e r m a l
n e u tro n s ,
t h e c a s e o f s m a l l w a v e - l e n g t h w o u ld b ecom e
u n i n t e r e s t i n g s i n c e t h e a t o m i c f o r m f a c t o r w o u ld a l m o s t
a n n i h i l a t e th e e f f e c t to b e e x p e c te d .
C h o o s in g o n t h e
o th e r hand s u f f i c i e n t l y la r g e w a v e -le n g th s,
we a r e no
lo n g e r a llo w e d to n e g l e c t t h e c r y s t a l s t r u c t u r e ,
as soon l n - S eo tlo n
a in o o ,
—i t may a f f o o t r t h o o o h o r c n t
o o a t t o r i n g b y a m o u n to o f t h o o r d e r o f--a 5 # »
s c a tte r in g d isa p p ea rs
C oh eren t
i n t h e c a s e o f F e f o r w ave­
le n g th s l a r g e r th a n 4 .0 4 A °.
T here does n o t t h e r e f o r e
e x i s t a r a n g e i n w h ic h t h e t h e o r y b a s e d on a n a m o rp h o u s
s t r u c t u r e Is even a p p ro x im a te ly c o r r e c t .
A q u a n tita tiv e e v a lu a tio n of p o la riz a tio n
phenom ena r e q u i r e s
t h e r e f o r e k n o w le d g e o f t h e w ave­
le n g th o f th e I n c id e n t n e u tro n s , as w e ll as o f th e
f o r m f a c t o r o f t h e s c a t t e r i n g a to m w h i c h i s a f u n c t i o n
o f w a v e -le n g th and s c a t t e r i n g a n g le .
se c tio n ,
assum e, f i r s t ,
I n c id e n t, and d is c u s s
fa c to r.
We s h a l l ,
in th is
t h a t m o n o ch ro m atic n e u tr o n s a r e
c h o ic e and i n f l u e n c e o f t h e form
S e c tio n 7 c o n ta in s
t h e m o re g e n e r a l t r e a t m e n t
o f a n i n c i d e n t sp e ctru m t o g e t h e r w ith a n u m e ric a l e v a lu a ­
tio n o f th e r e s u l t s .
P o la riz a tio n e ffe c ts
c an b e o b s e rv e d th ro u g h
e x p e r i m e n t s o n t h e s c a t t e r e d o r t r a n s m i t t e d b eam ; o n l y
e x p e rim e n ts o f t h e se c o n d ty p e h av e so f a r b e e n p e r ­
fo rm ed , f o r re a s o n s o f a v a i l a b l e I n t e n s i t y .
th e r e fo re l im i t o u rse lv e s
We s h a l l
to t h i s g ro u p o f phenom ena;
t h e t r e a t m e n t o f t h e s c a t t e r e d beam s c a n e a s i l y b e
c a r r i e d o u t w ith t h e a i d o f t h e fo rm u la e p r e s e n t e d h e r e
and in p re v io u s n o te s .
As t h e I n c i d e n t beam I s r e s o l v e d a l o n g t h e
—
^
d i r e c t i o n o f m a g n e t i z a t i o n d e n o t e d b y K. » t h e a d d i ­
t i o n a l s c a t t e r i n g p e r u n i t s o l i d a n g l e f o r t h e two s p i n
/6J
s t a t e s I s g i v e n by
± a C
D
^ i t ) ^ - ± ^ C
P ^
(6.X
J
T h is c a n b e ta k e n o v e r t o
th e c r y s t a l l i n e
s in c e b o th th e a m p litu d e s
C and D r e f e r to c o h e re n t p ro ­
c e sse s o n ly .
T he m a g n e t i z a t i o n i s
c a s e im m e d ia te ly
assum ed to b e p e r p e n d i c u l a r
to th e d i r e c t i o n o f in c id e n c e , a s i s
e x p e rim e n ts.
i n d e x JX|
In th e c r y s t a l l i n e
, th e av erag e of ^
#
of
Ul o n ly .
case,
fo r a rin g of
o v e r a z im u th g iv e s
)
The f o r m f a c t o r F w h ic h i s
th e case in a l l
-
*
(
c o n ta in e d in D i s
'
<6 -2 >
fu n c tio n
(From now o n we s h a l l w r i t e t h e f o r m
fa c to r se p a ra te ly :
D=
^
).
The f i n a l r e s u l t f o r t h e a d d i t i o n a l c r o s s s e c tio n p,
f o r th e c a s e o f a sim p le c u b ic c r y s t a l ,
in c lu d in g te m p e ra tu re e f f e c t ,
-jP
v
= CD
is:
2 1
t
(/ + u & ? )
>x~^
(6.3)
= HmCL- Q. . X- (
C
F 0 *J')O ^
^
lLi
a n d c o m p a rin g w ith
- A U r
^
7 f
?
( 2 - 15)
we o b t a i n :
f
w here
=
^
is
(«•*>
th e f r a c t i o n o f th e s c a t t e r e d n e u tro n s
' i n t h e r i n g o f i n d e x jjtl •
The sam e f o r m u l a w i l l a p p l y t o a n y o t h e r c a s e
(av erag e over v e lo c ity d i s tr ib u tio n ;
c ry s ta l.)
b o d y -c e n te re d
The " p ” i s a l w a y s g i v e n b y :
* 12- • P a v e r a g e o v e r
th e rin g s of
F(/* 0 ( ^ 4 ^ = )]
To f i n d t h e a d d i t i o n a l s c a t t e r i n g w f o r a c e n t i m e t e r
th ic k n e ss,
we m u l t i p l y b y t h e n u m b e r N o f a t o m s / c c .
( f o r i r o n N - 8 .4 8 x 102 2 );
w=Np.
Then t h e f r a c t i o n a l
in c r e a s e in tra n s m is s io n on m a g n e tiz a tio n i s :
g .-* ™
w h e r e CT c o n t a i n s
e - " T“
th e c ro s s -s e c tio n B
o f w e a k e n i n g o f t h e beam ( c a p t u r e ,
in g ,
(6 .5 )
f o r a l l o th e r causes
in c o h e re n t s c a t t e r ­
n u c le a r c o h e re n t s c a t t e r i n g ) , and x is
th ic k n e s s
tra v e rse d .
th e t o t a l
T h is g iv e s f o r th e f r a c t i o n a l
change in tra n s m is s io n
T h is r e s u l t a p p l i e s
(6 . 6 )
to b o th s i n g l e and d o u b le t r a n s ­
m is sio n e x p e rim e n ts.
W h ile t h e a m p l i tu d e C o f n u c l e a r s c a t t e r i n g
i s in d e p e n d e n t o f th e s c a t t e r i n g a n g le , D w i l l depend
on i t
th r o u g h t h e m a g n e tic fo rm f a c t o r .
m in a tio n o f f e r s a c e r t a i n
d iffic u lty
Its
d e te r­
due to o ur i n ­
c o m p le te k n o w le d g e o f t h e c o n s t i t u t i o n o f t h e m a g n e t i c a l l y
a c tiv e 3 -d s h e l l of F e.
F o r th e p u rp o se s o f a rou g h
17)
,'W
e s t i m a t e we h a v e p r e v i o u s l y uB ed S l a t e r s h y d r o g e n l c
f u n c t i o n f o r t h e wave f u n c t i o n o f t h e e l e c t r o n s i n
q u e stio n .
T h i s m e t h o d , w h i c h was a l s o a d o p t e d w i t h
•l )
e v e n s t e e p e r d e c r e a s e t o w a r d s t h e o u t s i d e b y Van V l e c k ,
d o e s n o t se em t o l e a d t o r e l i a b l e q u a n t i t a t i v e v a l u e s a s
s u c c e s s f u l l y as i t m ig h t h a v e f o r o t h e r a p p l i c a t i o n s
r e p o r te d by S l a t e r .
^o)
B e th e u s e d t h e w e l l - t a b u l a t e d d e n s i t y f u n c t i o n
o f H a r t r e e f o r t h e 3 - d s h e l l o f Cu+ , w h i c h was d e r i v e d
b y t h e m e th o d o f t h e s e l f - c o n s i s t e n t f i e l d .
t h is d e n s ity d i s t r ib u t i o n to
To a d a p t
th e c a se o f ir o n w ith a
s m a ll e r c e n t r a l c h a rg e , B e th e "ex pan ded" t h e e f f e c t i v e
r a d iu s o f ir o n w ith r e s p e c t to co p p er i n th e in v e r s e
ra tio
o f th e e f f e c t i v e c h a rg e s and c a l c u l a t e d t h e form
f a c t o r by n u m e ric a l i n t e g r a t i o n s .
I n t h e p r e s e n t p a p e r we a l s o u s e t h e H a r t r e e
d i s t r i b u t i o n w h ic h h a s so f a r p r o v e n t o b e o f h i g h e r
a c c u r a c y t h a n o u r dem an ds w o u ld r e q u i r e .
But in p la c e
o f th e r a t h e r rough n u m e ric a l i n t e g r a t i o n c a r r i e d o u t
b y B e t h e , we u s e t h e a n a l y t i c
e x p re ss io n f o r th e H a rtre e
a./J
d i s t r i b u t i o n a s adapted b y S l a t e r .
I t Is In te re stin g
to o b se rv e t h a t th e a c tu a l d e n s ity d i s t r i b u t i o n can
b e w e l l r e p r e s e n t e d b y t h e sum o f t h r e e h y d r o g e n i c
f u n c t i o n s o f w h ic h o n l y tw o a r e r e a l l y s i g n i f i c a n t .
The r e s u l t o f t h i s
a n a ly tic
c a l c u l a t i o n a g re e s w ith
B e t h e 's fo rm u la f o r s u f f i c i e n t l y
Q
s m a l l v a l u e s o f "^^3 ,
b u t o u r p r o b a b l y m o r e c o r r e c t m e th o d g i v e s a d e c i d e d l y
s h a r p e r d e c r e a s e o f t h e fo rm f a c t o r a * l a r g e a n g l e s .
F o r a Cu*" 3 - d e l e c t r o n ,
The u n i t o f l e n g t h I s
th e ^ r a d i a l d e n s ity =
th e Bohr r a d iu s a 0 .
b - 6 . 2 9 ; d - 2 . 6 5 ; g = 1 .2 8 .
S l a t e r g iv e s th e fo llo w in g d a ta :
In n er i n te r s e c t i o n =3*78;
i.e .
and O u te r i n t e r s e c t i o n - 1 .1 5 ;
i.e .
T h is g iv e s a = 1 7 . 975c
f =
.0 2 8 0 4 0
So we c a n w r i t e :
1----i r~~, a-7*r
=J
i
To c a l c u l a t e t h e f o r m f a c t o r ,
te
~^
■ A.
* J
A *
F w i l l c o n s i s t o f s i x te rm s:
c,
r ~ 2 1 £ w here ^
f
A*
i
2
3
4
5
6
>6
3 2 3 .1
3 5 .9 5
1 .0 0 8
1 .0
.0 5 6 1
.0 0 0 7 9
-1.3.?*- I
J
we m u s t e v a l u a t e :
J e r r ^ ^ d J L &
w here
j.ifA
p j.*
i n Bohr u n i t s ,
i s o f th e form :
1
' 1 2 .5 8
8 .9 4
i
:
7 .5 7
5 .3 0
3 .9 3 I
2 .5 6 I
!
(6 .7 )
42.
-
S- j ^
'
f-” -i+ ;- r ^ l
_
I- 9 ( & j + ( % f _
N o r m a l i z a t i o n i s o b t a i n e d b y r e q u i r i n g t h a t f o r p —* 0 ,
F —? 1 .
So we m u s t d i v i d e b y t h e n o r m a l i z a t i o n f a c t o r
c
• £ / A*t~
. When t h i s i s d o n e , we o b t a i n t h e
fo rm f a c t o r :
F--ZF
4.
( 6 .8 )
w h e re
/yn^_
(%:)]
f?
.276
.025
-298
.156
.038
£/*•
_____
2
3
4
5
6
•
j
F I== ^
I .227
/VIi
1
8.94
7.57
5.30
3.93
2 .5 6
■
The f o r m f a c t o r c o n s i s t s o f t h e s e s i x - t e r m s ,
s i m i l a r to th e s i n g l e - t e r m
and Johnson.
each
e x p re s s io n g iv e n by H a lp ern
G-raph I I g i v e s a c o m p a r i s o n b e t w e e n F a n d
t h e f o r m f a c t o r Pg a s g i v e n b y B e t h e .
T hey c o i n c i d e
f o r s m a l l v a l u e s o f p , b u t F f a l l s b e l o w *B a n d d e c r e a s e s
m ore r a p i d l y f o r l a r g e p .
F o r a D e b y e - S c h e r r e r r i n g o f i n d e x M lJ
We s e e t h a t F i s a f u n c t i o n o n l y o f M l •
xr7teK* l*L
For a p p lic a tio n
—
^
>o
►
i■
■'n«ua 'niifW
MX
A
iL
w
uM
jo.ft'i i\ u iU
^
-fc.
<J\
6^
oo
o
t o F e , we h a v e r e d u c e d t h e w a v e - l e n g t h t o 7 5 ^ t o t a k e
acco u n t of th e c o n tra c tio n o f th e c h arg e.
The v a l u e s
o f p f o r t h e f i r s t fe w r i n g s a r e I n d i c a t e d i n G r a p h I I .
S e c tio n V II.
E v a l u a t i o n of* P o l a r i z a t i o n E x p e r i m e n t s
C o n tin u e d ; V e lo c ity D i s t r i b u t i o n o f th e I n c i d e n t N e u tro n s
The s t r o n g d e p e n d e n c e o f t h e s c a t t e r i n g i n t o
r in g s o f d i f f e r e n t v a lu e s o f
Ui
on th e w a v e -le n g th , as
i t b e co m es a p p a r e n t fro m t h e f o r m u l a 2 . 1 6 a n d fro m t h e
d i s c u s s i o n a t t h e en d o f S e c t i o n 3» makes i t n e c e s s a r y
to r e f i n e th e c a l c u l a t i o n s o f S e c t i o n 2, a s f a r a s th e
v e lo c ity d is tr ib u tio n is
co n cern ed .
I t has been f r e ­
q u e n tl y o b s e r v e d t h a t th e M a x w e llia n d i s t r i b u t i o n u s e d
in p a ra g rap h 2 f o r th e p u rp o se o f i l l u s t r a t i o n can n o t
re a lly
c la im to b e a n a c c u r a t e r e p r e s e n t a t i o n o f th e
v e l o c i t i e s p r e s e n t i n t h e i n c i d e n t beam o f " t h e r m a l
n e u tro n s ."
N e ith e r a r e th e lo n g w a v e -le n g th s
v e lo c itie s )
re a lly p re se n t,
(sm a ll
sin c e th e in e la s tic
t r a n s i t i o n s a r e f a r t o o s l o w t o p r o d u c e them ( c f .
F o m e ra n c h u k j; n o r c a n i t b e e x p e c te d t h a t t h e c u rv e
fa lls
o f f a t h ig h v e l o c i t i e s
i n an e x p o n e n t ia l m anner,
s i n c e t h e w h o le s p e c t r u m s h o u l d e x t e n d e v e n b e y o n d
t h e Cd a b s o r p t i o n l i m i t .
A b s o r p t i o n e x p e r i m e n t s w i t h " c o o le d " n e u t r o n s
a l s o f a i l t o show r e a l a g r e e m e n t w i t h t h e -~pr
la w , when
t e m p e r a t u r e s o f t h e n e u t r o n s o u r c e a r e u s e d f o r com­
p a r iso n .
We h a v e t h e r e f o r e made u s e o f t h e v e l o c i t y
no
UUnmn* ^
c u r v e w h ic h h a s b e e n o b t a i n e d b y Segspfe a n d c o l l a b o r a t o r s
g iv in g th e fre q u e n c y o f n e u tro n s a s a f u n c tio n o f t h e i r
m e c h a n ic a lly d e te rm in e d sp e e d .
The r e s u l t i s c o n t a i n e d
i n G rap h 3 , w h e re a M a x w e ll d i s t r i b u t i o n h a s b e e n i n ­
c lu d e d f o r c o m p a riso n .
a r e n o t n o rm a liz e d ;
c o in c id e ,
s h a p e .)
(T h e d i s t r i b u t i o n s a s show n
t h e i r m axim a h a v e b e e n m ade t o
i n o r d e r t o show c l e a r l y
T h is c h o ic e b r i n g s w ith i t
th e d if f e r e n c e in
th e d isa d v a n ta g e
o f a n u m e ric a l i n t e g r a t i o n s i n c e no a n a l y t i c r e p r e ­
se n ta tio n of th is v e lo c ity
d istrib u tio n
is a v a ila b le .
The v e l o c i t y d i s t r i b u t i o n show s q u a l i t a t i v e l y a l l t h e
f e a tu r e s to be e x p e c te d .
A seco n d p o i n t to be ta k e n i n t o a c c o u n t i s
th e change i n th e v e l o c i t y d i s t r i b u t i o n w h ile th e
n e u tro n s p a ss th ro u g h i r o n .
T he c o h e r e n t s c a t t e r i n g
as w e ll as th e background a r e a p p ro x im a te ly v e l o c i t y
in d ep e n d en t;
th e c a p tu r e on th e o t h e r hand fo llo w s th e
l a w , a n d t h e r e f o r e re m o v e s t h e lo w v e l o c i t y n e u t r o n s
p re fe re n tia lly .
S in c e i t
is
th e lo n g w a v e -len g th ,
low
v e l o c i t y n e u tr o n s w h ich d e te r m in e t h e p o l a r i z a t i o n ,
th is c o r r e c tio n w i l l tu r n o u t to be o f s i g n i f i c a n c e .
In o rd e r t o d e te rm in e th e v e lo c ity d i s t r i b u t i o n
o f t h e n e u t r o n s a s a f u n c t i o n o f t h e t h i c k n e s s o f s a m p le
•J -N ‘V9VMXI ‘l i l O O *
*“
1 1 1 * 1 (7 0
N>
v>i
(j\
<
G--
ttfT
m
n
|-Tjpf»r H t
:±fi:
■+ >j | -pyf j -*tj -*• - -t-f4
JStJirti HS ttfi t
: tt$±~Hrt
tro
r
t r a v e r s e d , we h a v e p r o c e e d e d a s f o l l o w s :
The o b s e r v e d c a p t u r e c r o s s - s e c t i o n
of
F e f o r t h e r m a l n e u t r o n s i s ^ 3*5 x 1 0 ” 2^ cm^.
T h is
r e p r e s e n t s an a v e r a g e o f t h e c a p t u r e c r o s s - s e c t i o n C T =
‘T ^
o v e r th e th erm al d i s t r i b u t i o n .
-AY a
P V.
i.e .
or ^ 3 . 5 - x j o
=
iL r_ L — (7 .1 )
T
w h e r e ^ ( ir) i s t h e n e u t r o n d i s t r i b u t i o n I n d i c a t e d i n
G ra p h 3 a n d t a b u l a t e d i n T a b l e 1 .
T h is g i v e s , a f t e r
n u m e ric a l i n t e g r a t i o n ,
Cj; = 8 . 1 x 10 J<\
cr
=
(7 . 2 )
ir
>
The v e l o c i t y d i s t r i b u t i o n
> a fte r a
th ic k n e s s x h a s b een tr a v e r s e d i s g iv e n by:
(7 .3 )
U s in g e q u a t i o n 7 . 3 ,
th e v e lo c ity d i s t r i b u t i o n
has b een c a l c u l a t e d f o r th ic k n e s s e s o f 1, 2,
3 , a n d 4-
cm, a n d t h e n a n o r m a l i z a t i o n f a c t o r i n t r o d u c e d s o t h a t
I f * ™ * ’ 1 •
i s t a b u l a t e d i n T ab le 1 .
We m u s t a l s o
re p la c e th e c a lc u la tio n
jf
'<r
g i v e n i n S e c t i o n 3 o n t h e b a s i s o f e q u a t i o n 2 . 1 7 , by
a s i m il a r c a l c u l a t i o n u sin g th e c o r r e c t n e u tro n d i s ­
t r i b u t i o n ^D(v) .
From e q u a t i o n 2 . 1 6 we o b t a i n s
f)
9^ =
CT .
k
(4 Z
T ab le I
i V v) !
.8 8 '
. 3 8 ~~f
AJLC*
.9
1 .6 4
l.l
.8 2 6 ;i
.1 6 4
.0 7 1
! g 'M
.0 3 1
.4 1 6 !
.2 1 0
.1 0 5
.4 7 6
.2 6 6
\o
K\
.
ir^
1 .5 2 1
.8 5 2
1 .5
3 .7 8
2 .2 8 6
! 1 .3 8 2
1 .7
4 .2 0
2 .6 9 4
1 .9
4 .4 2
2 .1
CO
1 .3 : 2 .7 2
.5 0 5
1 .7 2 8
1 .1 0 9
.7 1 1
2 .9 7 2
! 1 .9 9 8
1 .3 4 3
.9 0 3
4 .5 5
3 .1 7 5
2 .2 1 5
1 1 .5 4 5
I .0 7 8
2 .3
4 .6 2
3 .3 2 8
2 .3 9 7
1 .7 2 7
1 .2 4 4
2 .5
4 .6 3
3 .4 2 3
! 2 .5 3 1
1 .8 7 1
1 .3 8 3
2 .7
4 .5 8
3 .4 6 2
2 .6 1 6
1 .9 7 7
1 .4 9 4
2 .9
4 .4 8
3 .4 5 5
2 .6 6 3
2 .0 5 4
1 .5 8 4
3 .1
4 .3 0
3 .3 6 9
2 .6 3 9
:
2 .0 6 8
1 .6 2 0
3 .3
4 .1 2
3 .2 7 7
2 .6 0 6
i 2 .0 7 5
1 .6 4 8
3 .5
3 .9 6
; 3 .1 9 1
2 .5 7 1
: 2 .0 7 1
1 .6 6 9
3 .8 1 i 3 .1 0 7
2 .5 3 4
2 .0 6 6
1 .6 8 5
3 .9
3 .6 6 ! 3 .0 1 5
2 .4 8 3
2 .0 4 5
1 .6 8 6
4 .1
3 .5 2
2 .9 2 9
2 .4 3 6
! 2 .0 2 7
1 .6 8 6
4 .3
3 .3 9
2 .8 4 3
2 .3 8 4
i
1 .9 9 9
1 .6 7 7
4 .5
3 .2 5
2 .7 4 8
2 .3 2 2
1 .9 6 3
1 .6 6 0
4 .7
3 .1 2
2 .6 5 6
2 .2 6 1
1 1 .9 2 5
1 .6 3 9
4 .9
3 .0 0
2 .5 7 2
; 2 .2 0 5
: 1 .8 9 0
1 .6 2 0
5 .1
2 .8 7 ; 2 .4 7 5
2 .1 3 5
; i.8 4 i
1 .5 8 8
5 .3
2 .7 4 '
2 .3 7 7
2 .0 6 3
1 .7 8 9
1 .5 5 2
1 2 .4 0 8
9 .5 6 2
7 .2 8 4
....5 . 2 5 6
3 .7
H* ~
:
1 6 .6 3
I
w here j j y ) h a s b e e n assum ed t o b e n o r m a l i z e d .
The o r d e r
o f sum m ation and i n t e g r a t i o n may b e r e v e r s e d , g i v i n g
\
~
~TU
f * W
<7"5)
_ a.M«-
Prom e q u a t i o n 7 . 5 we s e e t h a t ^
c o n s i s t s o f a sum o f
t e r m s f o r d i f f e r e n t I Z l, e a c h i n v o l v i n g a n u m e r i c a l
fro m rr ~ I l i L
t o “«,
T h is h a s b e e n
__
aAT«c a r r i e d o u t , g i v i n g CT - . 7 5 , w h ic h c o r r e s n o n d s t o
<x
C = 7 .3 3 x lO -^ c m .
xv)
U s in g t ~ 1 . 9 3 ; S = 1 . 0 7 , g i v i n g D -- 5 .8 x 10“
in te g ra tio n of
it1-
n n . t-v
M = 8 .4 8 x 10^ ,
we h a v e c a l c u l a t e d w f o r m o n o c h r o m a tic
beam s o f v a r i o u s w a v e - l e n g t h s , u s i n g e q u a t i o n 6 . 4 a n d
th e r a t i o
g i v e n i n G ra p h 1 .
The v a l u e s o f w o b t a i n e d a r e show n i n G ra p h 4 .
The b r e a k s
i n t h e c u r v e o c c u r a t w a v e - l e n g t h s w h e re a
D eb y e-S cherrer rin g is c u t o u t,
as has a lre a d y been
d e s c r i b e d i n c o n n e c t i o n w i t h G ra p h 1 .
F o r a n i n c i d e n t beam o f t h e r m a l n e u t r o n s ,
e q u a t i o n 6 . 6 w i l l no l o n g e r a p p l y , b u t w i l l b e r e p l a c e d
by
( 7 *6 )
f o r t h e t r a n s m i s s i o n e f f e c t o f a sa m p le o f t h i c k n e s s
d ; we w r i t e w (x ) b e c a u s e t h e p o l a r i z a t i o n w i l l now v a r y
w ith d is ta n c e tr a v e r s e d ,
s in c e th e v e lo c ity d i s t r i b u t i o n
i s ch an g in g b e c a u s e o f p r e f e r e n t i a l a b s o r p t io n o f
■A 'H
ip-oa -nimoa
\
(
i 'V 'a
)
T a b le 2
LM
r
■
ur(a)
...
\ ■
r
i ^
-------
.2 8 1 .1 2 5
'.2 5 2
.1 1 2
.2 3 1 [ .1 0 3
.?...Y5T
vv
. 2 0 7 (.0 9 2
. 1 9 3 | *086
.0 2 2
.0 7 1
.0 2 0
1
.0 6 8
.0 1 9
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lo w -v e lo c ity n e u tro n s .
To e v a l u a t e
th e p o la r iz a tio n e f f e c t ,
c a l c u l a t e d w (x ) f o r 0 ,
1, 2,
we h a v e
3. 4 , u s i n g t h e v a l u e s o f
£ 1 *)t a b u l a t e d i n T a b l e 1 a n d u s i n g e q u a t i o n 6 , 4 .
The
f r a c t i o n fOtyLB h e r e g i v e n b y t h e IA) 1t h t e r m i n e q u a t i o n
7 . 5 , d i v i d e d b y t h e t o t a l v a l u e o f t h a t su m .
su lts
a r e sho w n i n T a b l e 2 .
I t sh o u ld be p o in te d o u t
t h a t th e p o la r iz a tio n f a c t o r 0 ^
w a v e -le n g th .
The r e ­
)
depends on
We h a v e t h r o u g h o u t u s e d a n a v e r a g e v a l u e
f o r A. - 1 . 5 3 A ° .
T h is 1b j u s t i f i e d
slo w ly v a ry in g f u n c t i o n o f
X
•
sin c e i t
A Ib o ,
is
a
as th e d i s ­
t r i b u t i o n c h a n g e s w h ile p a s s i n g th r o u g h t h e sa m p le ,
th e a v erag e w a v e -le n g th d e c re a s e s ,
so t h a t a t w o r s t,
we a r e o v e r - e s t i m a t i n g t h e p o l a r i z a t i o n e f f e c t .
We o b t a i n f r o m t h e d a t a o f T a b l e 2 f o r
th e tra n sm iss io n e f f e c t th e r e s u lts
sh ow n i n t h e
4 yj
f o l lo w in g t a b l e w here e x p e r im e n ta l r e s u l t s a r e l i s t e d
f o r c o m p a riso n :
■T h ic k n e s s
I % E ffe ct
i
| E x p e rim e n ta l
'
.8
1 .3
1 .9 5
4 .0
.1 4
.3 5
.7 5
2 .9
.7 6
1 .7 8
3»32
6 .0
As c a n b e s e e n f r o m t h e t a b l e ,
:
;
j
th e re e x is ts
a c o n s i d e r a b l e d i s c r e p a n c y b e tw e e n t h e o r e t i c a l a n d
o b s e r v e d valu eB f o r t h e t r a n s m i s s i o n .
We h a v e a l s o ,
i n o u r c o m p a r is o n , t o t a k e i n t o a c c o u n t t h e marked d e ­
p o l a r i z a t i o n e f f e c t due t o d e v i a t i o n from s a t u r a t i o n .
The d a t a g i v e n by t h e e x p e r i m e n t e r s d o e s n o t a l l o w us
t o drav; any v e r y d e f i n i t e c o n c l u s i o n a b o u t t h e s t a t e
o f s a t u r a t i o n o b t a i n e d , and we s h a l l t h e r e f o r e d e f e r
t h e more d e t a i l e d d i s c u s s i o n o f t h a t e f f e c t u n t i l l a t e r .
The p r e s e n c e o f t h e d e p o l a r i z a t i o n e f f e c t i s
c le a r ly
i n d i c a t e d by t h e marked d e v i a t i o n from t h e law :
P o la r iz a tio n e f f e c t »c (th ic k n e ssJ 2
We do n o t , a t t h e moment, s e e any p o s s i b i l i t y
o f r e c o n c il i n g th e o b serv ed d ata w ith th e p r e s e n t ly
a c c e p t e d v ie w s a b o u t t h e m a g n e t ic i n t e r a c t i o n b e t w e e n
n e u t r o n and i r o n a to m , and t h e c o n s t i t u t i o n o f t h e
Iron c r y s t a l s .
We i n t e n d t o r e t u r n to a f u l l e r d i s ­
c u s s io n o f th e q u e s tio n s In v o lv ed i n t h i s f i e l d
-
la t e r paper.
In a
I
50.
FOOTNOTES AND REFERENCES
0.
A p r e l i m i n a r y r e p o r t o f t h i s w ork a p p e a r e d i n t h e
a b stra c t:
H a l p e r n , H am erm esh, a n d J o h n s o n - P h y s .
R ev. 5 5 -H 2 5 A -1 9 3 9 .
1.
I.
H a lp e rn and Jo h n so n - P h y s. Rev.
51-9 9 2 -1 9 3 7
II.
"
"
"
- P h y s. Rev.
52-52-1937
III.
"
"
"
- P h y s. Rev.
5 5-898-1939
IV .
H a l p e r n an d H o l s t e i n - P h y s . R e v . 5 5 - 6 0 1 A - 1 9 3 9
V.
"
11
"
-
In p u b lic a tio n .
T h e s e a r e r e f e r r e d t o t h r o u g h o u t t h e p a p e r by t h e
c o r r e s p o n d i n g Roman n u m e r a l s .
2.
S e e V.
5*
B lo ch - P h y s. Rev. 50-259-1936
S ch w in g er - P h y s . Rev. 5 1 -5 4 4 -1 9 3 7
4.
Wick - P h y s . Z e i t s . 3 8 - 4 0 3 , 6 3 9 - 1 9 3 7
5.
P o m e ra n c h u k - P h y s . Z e i t s . S o w .-U n . 1 3 - 6 5 - 1 9 3 8
6.
See I I I .
7.
Laue - a r t .
" W e l l e n o p t i k " - E n z . d . M a th . W is s .
V -3-no. 24.
8.
D ebye - A nn. d e r P h y s . 4 3 - 4 9 - 1 9 1 4
W a lle r - Z e i t s . f . P h y sik 5 1 -2 1 3 -1 9 2 8 ; D is s .
i
U p p sa la -1925.
9.
The f i r s t two t e r m s a l o n e r e p r e s e n t t h e i n c o h e r e n t
s c a t t e r i n g due to i s o t o p e d i s o r d e r ;
th e l a s t term
i s due to sp in -d e p e n d e n c e o f n u c le a r f o r c e s .
(S e e I I I and R e f. 5 ) .
51
.
10.
W h ita k e r an d B e y e r - P hya. Rev. 55-1101-1939
11.
N i x , B e y e r a n d D u n n in g - P h y s . R e v . 5 7 - 5 6 6 A - 1 9 4 0
12.
M i l l e r a n d Du Mond - P h y s . R e v . 5 7 - 1 9 8 - 1 9 4 0
13.
A c tu a lly , Fe has fo u r s t a b le is o to p e s .
The c o n ­
c e n t r a t i o n o f Fe58 i s n e g l i g e a b l e ; F e ^ an d Fe57
h a v e b e e n lu m p e d t o g e t h e r a n d a r e r e f e r r e d t o
a s "Fe54".
14.
See I I and I I I .
1 5 - The b a c k - g r o u n d p r o d u c e d b y i n e l a s t i c
c o llisio n s
b e tw e e n t h e n e u tr o n and th e l a t t i c e v i b r a t i o n s
sh o u ld , s t r i c t l y
sp e ak in g ,
a ls o be c o n sid e re d .
We n e g l e c t i t i n t h e f o l l o w i n g d i s c u s s i o n b e c a u s e ,
a s show n by Wick ( 4 ) ,
i t w o u ld b e v e r y s m a l l e v e n
f o r X -ra y s , s t i l l s m a lle r f o r n e u tro n s and m o stly
p r e s e n t f o r l a r g e s c a t t e r i n g a n g l e s , w here i t s
16.
e ffe c t is
s t r o n g l y c u t down b y t h e f o r m - f a c t o r .
See I I I .
We h e r e h e g l e c t s m a l l e f f e c t s p r o p o r t i o n a l
t o D2 .
17.
See I I I .
18.
S l a t e r - P h y s. Rev. 3 6 -5 7 -1 9 3 0 .
1 9 . Van V l e c k - P h y s . R e v . 5 5 - 9 2 4 - 1 9 3 9 .
20.
H o ffm a n , L i v i n g s t o n , a n d B e t h e - P h y s . R e v .
51-214-1937.
21.
S l a t e r - P h y s. Rev. 4 2 -3 3 -1 9 3 2 .
52.
22.
S e e P o m e ra n c h u k - R e f . 5 .
23.
D u n n in g , P e g ra m , F i n k , M i t c h e l l , a n d S e g r e P h y s. Rev. 4 8 -7 0 4 -1 9 3 5
24.
A lv a re z and B lo ch - P h y s . R ev. 5 7 -1 1 1 -1 9 4 0 .
25.
P o w ers - P h y s . R e v . 5 4 - 8 2 7 - 1 9 3 8 .
26.
H a lp ern and Jo h n so n - P h y s. R ev. 5 7 -1 6 0 -1 9 4 0 .
hlBRART
N . Y . UHXtS
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