1 Muon Spin Rotation: A n Exotic Probe of the Atomic Environment ARTHUR B. DENISON
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Department of Physics and Astronomy, University of W y o m i n g , Laramie, WY 82071 The muon, the
through
magnetic
The purpose mation
insulators
and
muonium.
components The
inforas a
or as in
combine
some
with
ferromagnetic
depending
on the nature
be
localized
or
of muonium
in solids
of the bound diffuse
over
an
metals
the
of the local field have been studied behavior
of
scale.
the type of
it may In
atomic
muon may remain
conductors,
semiconductors,
various
either
The
as in all metallic
to form
acts as a probe
on the
is to elucidate
be obtained.
electron
interpreted.
moment,
in a solid
of this review
that may
free particle
its magnetic
environment
is
electron,
which
a number
of
and varied may lattice
neighbors.
nphe
m u meson
(/JT),
or m u o n , is a sensitive a n d d e l i c a t e p r o b e
of
m a t t e r o n the a t o m i c scale. T h i s p a r t i c l e , w h i c h carries e i t h e r a p l u s or a m i n u s c h a r g e , m a y m i m i c the b e h a v i o r of t h e p r o t o n or e l e c t r o n i n matter a n d thus finds itself i n v o l v e d i n a v a r i e t y of p r o b e
situations
r a n g i n g f r o m c h e m i c a l reactions to static a n d d y n a m i c m a g n e t i c b e h a v i o r i n solids. D u e to the n o n c o n s e r v a t i o n of p a r i t y ( J ) , t h e w* - » /A* +
v decay
p r o d u c e s m u o n s w i t h t h e i r spins o p p o s i t e to t h e l i n e a r m o m e n t u m {IT is a p i m e s o n a n d v is n e u t r i n o ) . W i t h p r o p e r m o m e n t u m d e f i n i t i o n a p o l a r i z e d b e a m of m u o n s m a y be* o b t a i n e d .
T h e p o l a r i z e d m u o n s stop i n
m a t t e r a n d reflect t h e l o c a l m a g n e t i c e n v i r o n m e n t t h r o u g h t h e L a r m o r precession frequency.
T h e p o s i t i v e m u o n m a y r e m a i n free as
but i n
m a n y cases i t m a y c o m b i n e w i t h a n e l e c t r o n (e") t o f o r m m u o n i u m (/x e").
A t t h e e n d of t h e m u o n l i f e t i m e ( a b o u t 2.2 X
(lk - » e* + preferentially
v + in
+
—
10" s e c ) , the d e c a y 6
F) p r o d u c e s a p o s i t r o n ( e l e c t r o n ) , w h i c h is e m i t t e d the
direction
of
the
muon
spin.
The
f r e q u e n c y of the m u o n ( a n d therefore the l o c a l m a g n e t i c
precession
field)
0-8412-0472-1/80/33-186-003$05.75/1 ©
1980
American Chemical
Society
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
can be
4
SOLID STATE C H E M I S T R Y :
measured
with
positron
detectors
in a
A CONTEMPORARY
given
i m p o r t a n t q u a n t i t y is the d e p o l a r i z a t i o n t i m e .
solid
OVERVIEW
angle.
Another
T h i s characteristic time
reveals the d y n a m i c e n v i r o n m e n t of the m u o n t h r o u g h d e p h a s i n g of the spins o r i g i n a l l y i n p h a s e i n the b e a m . r e l a t e d to t h e T
2
T h i s d e p h a s i n g t i m e is closely
relaxation time often measured i n magnetic
resonance
e x p e r i m e n t s to o b t a i n i n f o r m a t i o n c o n c e r n i n g the i n t e r a t o m i c
motions
a n d correlations. I n t h e f o l l o w i n g text t h e m u o n s p i n r o t a t i o n (/*SR) m e t h o d w i l l b e d e s c r i b e d i n m o r e d e t a i l . T h e f o r m a t i o n a n d q u a n t u m d e s c r i p t i o n of m u o n i u m w i l l be considered.
A s u r v e y of the k i n d s of studies t h a t m a y
b e d o n e w i t h /xSR is g i v e n .
T h i s s u r v e y , a l t h o u g h not m e a n t to
exhaustive, s h o u l d g i v e the r e a d e r a g o o d i d e a o f w h a t is n o w
be
being
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d o n e i n the field. T h e field c u r r e n t l y is v e r y a c t i v e a n d n e w ideas a n d types of measurements are b e i n g g e n e r a t e d c o n s t a n t l y . S e v e r a l excellent r e v i e w articles c u r r e n t l y exist {2,3)
a n d are r e c o m m e n d e d to t h e r e a d e r
for b r o a d e r a n d m o r e d e t a i l e d i n f o r m a t i o n .
Method of
Measurement
F i g u r e 1 lists some of the i m p o r t a n t p r o p e r t i e s of the m u o n .
The
m u o n , a l e p t o n w i t h s p i n 1 / 2 , is p r o d u c e d f r o m p i o n d e c a y i n a n u c l e a r accelerator.
Several high-flux meson facilities (or meson factories)
exist
t o d a y . T h e p r i m a r y sites u s e d for /xSR w o r k are t h e L o s A l a m o s M e s o n Physics Facility ( L A M P F )
i n N e w M e x i c o , the T r i - U n i v e r s i t y
Meson
F a c i l i t y ( T R I U M F ) i n V a n c o u v e r , B r i t i s h C o l u m b i a , the S c h w e i z e r i s c h e s I n s t i t u t f u r N u k l e a r f o r s c h u n g ( S I N ) i n V i l l i g e n , S w i t z e r l a n d , a n d the R u s s i a n I n s t i t u t e for N u c l e a r Studies at D u b n a .
D u e to the t y p e
i n t e r a c t i o n t h a t acts d u r i n g the p i o n d e c a y
the m u o n
(I),
of
magnetic
m o m e n t is d i r e c t e d i n the o p p o s i t e sense to t h e m u o n m o m e n t u m .
By
selecting muons w i t h i n a given m o m e n t u m range, one m a y obtain a highly polarized muon beam
(about 9 0 % ) .
T h e mass of t h e m u o n is
206.8 times the mass of the e l e c t r o n , w h i c h means i t is a b o u t o n e - n i n t h the mass of the p r o t o n . produced.
B o t h positive and negative muons
T h e l i f e t i m e of t h e m u o n is short ( a b o u t 2.2 X
may 10"
be sec),
6
so the e x p e r i m e n t s m u s t b e d e s i g n e d to o b t a i n t h e r e l e v a n t i n f o r m a t i o n w i t h i n just a f e w l i f e t i m e s . T h e m a g n e t i c m o m e n t is the k e y to the / A S R t e c h n i q u e , as the L a r m o r p r e c e s s i o n f r e q u e n c y
is m e a s u r e d to
obtain
the m a g n e t i c field at t h e site of t h e m u o n i n the s a m p l e u n d e r s t u d y . F o r t h e free m u o n t h e L a r m o r p r e c e s s i o n f r e q u e n c y
(/ ) L
is /
L
=
13.55
k H z / G X B ( G ) , w h e r e B is the m a g n e t i c field s t r e n g t h . T h e m e t h o d of d e t e c t i o n of the p r e c e s s i o n of t h e m u o n
magnetic
m o m e n t a g a i n relies o n the f a c t that p a r i t y is not c o n s e r v e d i n t h e w e a k d e c a y m o d e of the m u o n . T h e p o s i t i v e m u o n d e c a y s into a p o s i t r o n
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
(e ) +
1.
DENISON
Muon
Spin
7T
5
Rotation
fJL
+
+
+
SPIN OPPOSITE TO MOMENTUM IN DECAY
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SPIN
= I /2
MASS = 105.7 Mev = 206.8 m LIFETIME { T
M
e
) = 2.199 x I0" sec. 6
I
I et>
MAGNETIC MOMENT (/i^ ) » g^sJ
= 3.183^.
p
LARMOR FREQUENCY (f,) = 13.55 KHz /G xB Figure 1. a n d t w o n e u t r i n o s (n*
Properties
of the muon
e* + v -\- vn) in s u c h a w a y t h a t t h e p o s i t r o n e
is e m i t t e d i n a p r e f e r e n t i a l d i r e c t i o n w i t h respect t o t h e m u o n m a g n e t i c m o m e n t . I f y is the e n e r g y of d e c a y n o r m a l i z e d t o the m a x i m u m p o s s i b l e energy (52.8 M e V ) , t h e n = 2 j / { ( 3 -2y)
N(y,B)
2
+P(2y
-
l)cos®}
(1)
gives t h e d i s t r i b u t i o n o f positrons o f energy y ejected a t a n a n g l e © f r o m the m u o n m o m e n t , w h e r e P is t h e i n i t i a l p o l a r i z a t i o n . I f o n e integrates over t h e a l l o w e d energy spectrum, t h e total asymmetry f o r t h e decay positrons i s g i v e n b y
iV(0)-ATo[l + ^ P c o 8
0 j
(2)
T h i s p r e f e r e n t i a l f o r w a r d d i r e c t i o n o f d e c a y positrons w i t h respect t o t h e m u o n spin c a n be used to follow t h e m u o n precession w i t h positron detectors i n a g i v e n s o l i d angle w i t h respect to t h e s a m p l e target.
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
6
SOLID STATE C H E M I S T R Y : A C O N T E M P O R A R Y
F i g u r e 2 s h o w s a n i d e a l i z e d e x p e r i m e n t a l setup. of m u o n s is d i r e c t e d i n t o a s a m p l e target. essentially stop i n t h a t target.
OVERVIEW
A polarized beam
T h e muons slow d o w n
and
F o r most of t h e m a t e r i a l s s t u d i e d , t h e
s l o w i n g - d o w n process does n o t destroy the i n i t i a l m u o n p o l a r i z a t i o n . A s t h e m u o n enters the target, a c o u n t i n s c i n t i l l a t o r c o u n t e r S i is r e g i s t e r e d . If t h e m u o n stops i n the target, no c o u n t w i l l b e o b s e r v e d i n S . 2
d e n o t e a s t o p p e d m u o n as SiS* , w h e r e W is a n a n t i - S . 2
a c l o c k is started.
2
2
We
W i t h t h i s event
I f t h e m u o n d e c a y s w h e n its m a g n e t i c
moment
p o i n t e d i n t h e d i r e c t i o n of S , t h e p o s i t r o n w i l l pass t h r o u g h S . 3
3
is
When
t h e p o s i t r o n is d e t e c t e d i n S , t h e c l o c k is s t o p p e d a n d the t o t a l event 3
is stored as a c o u n t i n t h e correct t i m e b i n of a m u l t i c h a n n e l a n a l y z e r . A s m a n y events are r e c o r d e d , a t i m e h i s t o g r a m is b u i l t u p , w h i c h s h o w s
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the t i m e - m o d u l a t e d d e c a y of the p r e c e s s i n g m u o n , as s h o w n i n F i g u r e 3. T h e s i g n a l c a n b e r e p r e s e n t e d as
N(t)
— No e - ' M l +A(t)
cosUt + -|
+
+
>||
l # » > — « | + ->||+ c | - + >n |#3> =
|
(5)
>||
| # 4 > = C | +
- > | | ~S\
-+>||
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with _
1
l~i_i_
T
x
i
2
r .
i
x
i
/
2
and x
_
(gefie -
1 H1
grfu)
E x p e r i m e n t a l l y o n e looks f o r t h e m u o n i u m p r e c e s s i o n
frequencies.
T h e o b s e r v e d f r e q u e n c i e s d e p e n d o n t h e m a g n i t u d e o f t h e a p p l i e d field a n d its d i r e c t i o n r e l a t i v e to t h e i n i t i a l m u o n p o l a r i z a t i o n . . M u o n i u m is f o r m e d w i t h t h e i n i t i a l p o l a r i z a t i o n d i r e c t i o n of t h e m u o n p r e s e r v e d so that t h e e l e c t r o n is c a p t u r e d i n a state either p a r a l l e l , | + , + > , o r a n t i p a r a l l e l , |+>"">> to t h e m u o n p r o j e c t i o n .
T h e general situation w i t h
respect to a r b i t r a r y m a g n e t i c field o r i e n t a t i o n is c o m p l i c a t e d , b u t o n e c a n e x a m i n e some s i m p l e cases. the states | + , + >
a
n
d
—
I n t h e case of a w e a k transverse
field,
> a r e n o t r e a l l y e i g e n states since t h e axis
of quantization (external magnetic
field)
is p e r p e n d i c u l a r t o t h e i n i t i a l
m u o n p o l a r i z a t i o n . N e v e r t h e l e s s , t h e s i t u a t i o n is n o t so c o m p l i c a t e d , since t h e c o u p l e d (
t r i p l e t ) s p i n state w i l l precess i n t h e m a g n e t i c
i n a direction dominated b y the electron moment. time histogram w i l l occur w i t h a frequency f
L
M u
field
T h e wiggles o n the
=
1/2(f
L
+
e
fjj ) 1
=
(1.4 M H z • G " ) . A t h i g h e r fields t h e correct c a l c u l a t i o n m u s t b e m a d e 1
(5)
to o b t a i n t h e frequencies c o r r e s p o n d i n g to t h e A m =
1 transitions
b e t w e e n t h e states s h o w n i n F i g u r e 4. T h e r e s u l t i n g h i s t o g r a m w i l l s h o w beats that m a y b e a n a l y z e d b y F o u r i e r transforms to o b t a i n t h e d e s i r e d frequencies. F o r m u o n i u m i n a l o n g i t u d i n a l field, t h e states | + , + > a n d |+>~~> are f o r m e d w i t h 5 0 % of t h e t o t a l m u o n i u m p o p u l a t i o n i n e a c h state. H o w e v e r , t h e state |+>~~> is n o t p u r e i n t h e sense t h a t i t is a s u p e r p o s i t i o n of states E a n d E , t h a t i s , 2
4
— > — s\E > 2
+ c|E >. Such a 4
m i x e d state w i l l oscillate i n t i m e b e t w e e n | + , — > a n d |—,+ > w i t h t h e
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
10
SOLID S T A T E C H E M I S T R Y : A C O N T E M P O R A R Y
frequency
a> 4. A t l o w e x t e r n a l m a g n e t i c
fields,
2
w4 =
OVERVIEW
wo a n d t h i s f r e
2
q u e n c y is too h i g h to resolve w i t h c u r r e n t a p p a r a t u s , so h a l f the a v a i l a b l e p o l a r i z a t i o n is lost.
A s the
that the p o l a r i z a t i o n of t h e JPmin, w h e r e P
min
=
(x
field
is i n c r e a s e d , h o w e v e r ,
|+,—>
— 1)/(x
2
2
one
can
state oscillates b e t w e e n
-\- 1),
with x =
w/w .
show
+1
a
R
d
T h e resulting
0
m u o n p o l a r i z a t i o n as a f u n c t i o n of m a g n e t i c field is t h e n g i v e n b y
P W
- l + (i)[aT5r]
(6)
Muonium in Insulating Solids: Depolarization Studies.
T h e behav
i o r of m u o n i u m i n a n e x t e r n a l m a g n e t i c field, d e s c r i b e d i n the p r e c e d i n g
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section, is for m u o n i u m i n a v a c u u m i n the absence of p e r t u r b i n g effects. D e v i a t i o n s f r o m s u c h b e h a v i o r for m u o n i u m i n solids p r o v i d e s i n f o r m a t i o n a b o u t the m a t e r i a l u n d e r s t u d y .
M u c h of
the current theory
of
d e p o l a r i z a t i o n , as w e l l as e a r l y m e a s u r e m e n t s , w a s d o n e b y t h e R u s s i a n group
( 5 , 6 , 7 , 8 ) . D e p o l a r i z a t i o n f r o m the a t o m i c
environment
comes
f r o m several causes, the most i m p o r t a n t of w h i c h is the s o - c a l l e d p r o p e r muonium mechanism (8).
T h e m u o n i u m e l e c t r o n is s t r o n g l y c o u p l e d to
the e n v i r o n m e n t a n d t h r o u g h s p i n i n t e r a c t i o n s relaxes r a p i d l y . T h e m u o n that is c o u p l e d
to the e l e c t r o n t h r o u g h the h y p e r f i n e i n t e r a c t i o n also
d e p o l a r i z e s r a p i d l y . A s a n e x t e r n a l l o n g i t u d i n a l field is a p p l i e d a b o v e the c r i t i c a l field, the spins are d e c o u p l e d restored. quartz.
a n d the m u o n i u m p o l a r i z a t i o n
F i g u r e 5 shows the r e s t o r a t i o n of p o l a r i z a t i o n of m u o n i u m i n T h e fit to the d a t a is excellent w h e n u s i n g E q u a t i o n 6 w h i c h
describes appears
the i d e a l case for
muonium
in a vacuum.
n e u t r a l to m u o n i u m
in many
respects.
E q u a t i o n 6 a n d the d e f i n i t i o n of x = fine constant ^
( w / w ) , to find t h e effective 0
(Si0 ) 2
using hyper
of the m u o n i u m i n a g i v e n substance b y fitting t h e d a t a
o b t a i n e d f r o m d e p o l a r i z a t i o n as a f u n c t i o n of m a g n e t i c as e x p e c t e d ,
Quartz
It is p o s s i b l e ,
the v a c u u m v a l u e of
the h y p e r f i n e
field.
coupling
I n quartz, constant
is
obtained. A m o d i f i c a t i o n of this process o c c u r s w h e n the e l e c t r o n w i t h the m u o n i u m is c h e m i c a l l y a c t i v e .
associated
I f the e l e c t r o n enters i n t o a
c h e m i c a l b o n d , the m u o n w i l l be f r e e for a t i m e before a t t a c h i n g to another electron.
T h e m u o n becomes r a p i d l y depolarized ( i n l o w
w h i l e i t is i n m u o n i u m b u t not so r a p i d l y as a free m u o n .
field)
Examining
this process i n d e t a i l ( 9 ) , one finds t h a t t h e d e p o l a r i z a t i o n f o l l o w s
p_,
M ( l + 2vr) 2 [ ( < o r ) ( l + vr + Z ) + ( l + 2 v r ) ] 2
2
2
2
U
)
w h e r e x has its u s u a l d e f i n i t i o n , v is the f l i p p i n g f r e q u e n c y of t h e e l e c t r o n i n m u o n i u m , a n d r is the l i f e t i m e of t h e m u o n i u m c o n f i g u r a t i o n .
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
Figure
1.
DENISON
Muon
Spin
Rotation
11
P l.O
0.8
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0.6
0
1000
3000
2000 H^Oe
Soviet Physics JETP
Figure 5. The residual polarization for muonium in quartz (Si0 ) as a function of magnetic field. The external magnetic field is perpendicular to the beam polarization (8). (Solid line predicted by Equation 6 in text.) 2
6 shows t h e close agreement w i t h t h e p r e d i c t i o n f o r m u o n i u m i n K C 1 as a f u n c t i o n o f m a g n e t i c field. I t w a s i m p o s s i b l e to o b t a i n v a n d T separately, a l t h o u g h t h e p r o d u c t vr w a s d e t e r m i n e d ( V T =
1.81 ±
0.10).
Feasible
b o u n d s o n t h e separate q u a n t i t i e s a r e d i s c u s s e d i n t h e o r i g i n a l p a p e r . A d d i t i o n a l d i s c u s s i o n of the m e c h a n i s m of d e p o l a r i z a t i o n of t h e m u o n also is g i v e n there, w h i c h p o i n t s u p t h e i m p o r t a n c e of i m p u r i t i e s i n t h e s a m p l e . I m p u r i t i e s s u c h as those f o u n d i n K C 1 g i v e rise to l o c a l m a g n e t i c fields, w h i c h d e p o l a r i z e t h e m u m e s o n . A g a i n , b y e x a m i n i n g t h e e x t e r n a l field
d e p e n d e n c e o n t h i s extra d e p o l a r i z a t i o n , o n e m a y estimate t h e
m a g n i t u d e of t h e l o c a l
field.
T h e s e authors r e p o r t a c a l c u l a t e d l o c a l
field o n t h e o r d e r of 50 G . M u o n i u m in Insulating
Solids: Precession Measurements.
Recall
that t h r o u g h t h e F o u r i e r analysis of a t i m e h i s t o g r a m t a k e n f r o m m u o n i u m p r e c e s s i o n i n a transverse field, t h e f r e q u e n c i e s c o r r e s p o n d i n g t o t h e A m =
1 transitions i n the B r e i t - R a b i d i a g r a m ( F i g u r e 4 ) m a y be obtained.
F i g u r e 7 shows t h e p o w e r s p e c t r u m r e s u l t i n g f r o m t h e F o u r i e r t r a n s f o r m of the h i s t o g r a m of m u o n i u m i n S i 0 t w o f r e q u e n c i e s so o b t a i n e d f o r S i 0 u r e d frequencies
2
2
(quartz) and i n p-doped Si. T h e are w i a n d w 4. F r o m these meas
at a k n o w n m a g n e t i c
2
field,
the m u o n i u m h y p e r f i n e c o u p l i n g constant.
3
one can readily calculate
A s mentioned previously, the
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
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12
SOLID S T A T E C H E M I S T R Y :
1000
A
CONTEMPORARY
OVERVIEW
3000
2000
B ( GAUSS ) Soviet Physics JETP
Figure 6. The residual polarization as a function of magnetic field in KCl. The effects of forming and reforming the muon atom in the chemical environment give rise to the curve above 200 G. Additional depolarization due to local magnetic fields is evident at low fields (9). (Solid line predicted by Equation 7 in text.) h y p e r f i n e constant is the same as that for m u o n i u m i n a v a c u u m . I n other w o r d s , the m u o n i u m i n S i 0
2
has f o u n d a spacious e n o u g h site to a c c o m
m o d a t e itself essentially u n p e r t u r b e d . f o u n d i n a n u m b e r of m a t e r i a l s
S u c h u n p e r t u r b e d m u o n i u m is
(2).
I n contrast to this v a c u u m l i k e b e h a v i o r are the results for t h e s e m i c o n d u c t o r s S i a n d G e . I n fact, t h e b e h a v i o r of m u o n i u m i n these m a t e r i a l s is c u r r e n t l y one of the most i n t e r e s t i n g p r o b l e m s i n /xSR. G u r e v i c h o b s e r v e d m u o n i u m i n G e t h a t s h o w e d the h y p e r f i n e f r e q u e n c y «
0.56 w ( v a c ) . B r e w e r et a l . (10) 0
p - d o p e d S i (see observed w
h y p
Figure 7).
(Si) «
(5)
o) ,(Ge) hyi
p u b l i s h e d t h e i r results i n 1973 for
F o r m u o n i u m i n the i n t e r s t i t i a l site t h e y
0.405 w ( v a c ) . T h e m u o n i u m b e h a v e s as t h o u g h i t 0
is s o m e w h a t s w o l l e n i n the site i n w h i c h i t finds itself.
Such muonium
i n w h i c h t h e m u o n a n d e l e c t r o n are essentially l o c a l i z e d , a l b e i t s w o l l e n , is c a l l e d d e e p - d o n o r m u o n i u m .
If one assumes t h a t t h e h y p e r f i n e i n t e r
a c t i o n f o l l o w s the contact i n t e r a c t i o n |^(0)|
2
oc ( 1 / r ) , w h e r e |^(0)| is 3
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
2
1.
DENISON
Muon
Spin
Rotation
13
150
100 -
50
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O 0.
150
15
100
10
50
50
150
100
FREQUENCY
( MHz ) Physical Review Letters
Figure 7. (a) The power spectrum from the time histogram for muonium in quartz (SiO ). The two frequencies near 130 MHz represent the transitions v and v in the Breit-Rabi diagram (10). (b) The power spectrum from muonium in Si. The upper frequencies come from a swollen muonium in Si and the lower two frequencies have their origin in a shallow donor muonium (10). B
12
3i
p r o p o r t i o n a l to t h e e l e c t r o n d e n s i t y at t h e m u o n site, t h e n t h e r a d i u s c a n b e c a l c u l a t e d to h a v e e x p a n d e d b y r o u g h l y 2 0 % . (11)
W a n g and Kittel
i n t e r p r e t e d t h e s w e l l i n g as d u e to s h i e l d i n g b y t h e v a l e n c e b a n d
electrons. observed
Coker, Lee, and Das h y p e r f i n e shift
by
(12)
using
h a v e a t t e m p t e d to p r e d i c t t h e a self-consistent,
H i i c k e l m o d e l w i t h a cluster of 31 atoms.
charge-extended
R e a s o n a b l e a g r e e m e n t is
f o u n d for b o t h the m o d e l s a b o v e for the d e e p - d o n o r m u o n i u m . A n o t h e r set of f r e q u e n c i e s has b e e n o b s e r v e d , h o w e v e r , t h a t are a n i s o t r o p i c w i t h respect to c r y s t a l o r i e n t a t i o n i n the e x t e r n a l field. T h e s e lines, t h e
s o - c a l l e d a n a m o l o u s lines, w e r e
difficult
to
interpret and
p r o d u c e d g a n d c/f- values that w e r e h a r d to e x p l a i n . T h e Swiss g r o u p
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
14
SOLID S T A T E C H E M I S T R Y :
[Patterson et a l . (13)]
A
CONTEMPORARY
OVERVIEW
f o u n d that b y r o t a t i n g the c r y s t a l a b o u t v a r i o u s
axes, t h e y c o u l d p r e d i c t f r e q u e n c i e s
quite well, using an
anisotropic
spin Hamiltonian. J/lIU* — c#\\i&
directions. T h i s descrip
t i o n , w h i c h is not u n c o m m o n i n e l e c t r o n s p i n resonance w o r k , a l l o w e d
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a m u c h s i m p l e r i n t e r p r e t a t i o n w i t h the reasonable p a r a m e t e r s | j^L
I=
92.1 ± 0.3 M H z
\jf
| =
17.1 ± 0.3 M H z
u
g* = 0
e
=
2.01 ± 2.2
0.01
±0.2
F u r t h e r w o r k is p r e s e n t l y b e i n g d o n e to e l u c i d a t e the n a t u r e of t h e site w i t h this s y m m e t r y .
Several possible models
are p u t f o r w a r d i n t h e
original paper. Positive Muons in Metal The muon
r e m a i n s as a free m u o n i n c o n d u c t o r s , w h e r e
the
m e t a l l i c electrons are c o r r e l a t e d a n d act to screen the /x c h a r g e r a t h e r +
t h a n c o n t r i b u t i n g a n e l e c t r o n t o m a k e the m u o n i u m a t o m . T w o g e n e r a l types of p r o b l e m s h a v e b e e n s t u d i e d t h a t h a v e p r o v e d m o s t e x c i t i n g : ( 1 ) t h e s t u d y of m u o n d i f f u s i o n i n metals a n d ( 2 ) t h e s t u d y of m a g n e t i c i n t e r a c t i o n s i n m a t e r i a l s . B o t h studies r e q u i r e k n o w l e d g e of t h e w h e r e abouts of t h e m u o n a n d therefore d e a l w i t h t h e p h y s i c s of i n t e r s t i t i a l sites, defects, a n d t r a p p i n g . Diffusion. muons
I n f o r m a t i o n c o n c e r n i n g t h e d i f f u s i o n a n d t r a p p i n g of
i n metals is o b t a i n e d
t h r o u g h the
depolarization time.
This
depolarization time depends on temperature a n d local e n v i r o n m e n t a l factors, s u c h as the m a g n e t i c s u r r o u n d i n g s a n d t h e m e c h a n i s m of t r a p p i n g / S e v e r a l metals h a v e b e e n e x a m i n e d . T h e results s h o w t h a t t h e b e h a v i o r of the m u o n is b y no means u n i f o r m for different m a t e r i a l s . I t is b e c o m i n g clear t h a t e v e n f o r a g i v e n m e t a l s a m p l e the e x p e r i m e n t a l results c a n b e q u i t e different, d e p e n d i n g o n the n u m b e r of i m p u r i t i e s i n the sample.
T h e results are sensitive to t h e m e t h o d
of
preparation and
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
1.
DENISON
Muon
Spin
15
Rotation
a n n e a l i n g , w h i c h m a y influence s t r u c t u r a l defects a n d dislocations. n e w results are a p p e a r i n g i n the l i t e r a t u r e (see,
Many
for e x a m p l e , 14r-22),
and
c o m m u n i c a t i o n w i t h v a r i o u s laboratories shows t h e l i v e l y a c t i v i t y i n this field, w i t h n e w a n d p u z z l i n g results a p p e a r i n g constantly.
Rather than a
c o m p l e t e s u r v e y of a l l the m a t e r i a l s e x a m i n e d to date, a d i s c u s s i o n of the g e n e r a l a p p r o a c h to u n d e r s t a n d i n g , the results w i t h some specific examples w i l l be g i v e n here. I n E q u a t i o n 3 the t e r m A(t)
carries the d e p o l a r i z a t i o n i n f o r m a t i o n .
R e c a l l that the d e p o l a r i z a t i o n t i m e is the c h a r a c t e r i s t i c d e p h a s i n g t i m e of the m u o n s d u e to d y n a m i c r e l a x a t i o n effects f r o m the s a m p l e itself. It has b e e n k n o w n for a l o n g t i m e i n the field of m a g n e t i c
resonance
r e l a x a t i o n that the r e l a x a t i o n times are d e t e r m i n e d by* the p o w e r spec Downloaded by 80.82.77.83 on May 18, 2017 | http://pubs.acs.org Publication Date: June 1, 1980 | doi: 10.1021/ba-1970-0186.ch001
t r u m of the t i m e - m o d u l a t e d l o c a l m a g n e t i c e n v i r o n m e n t . quantitatively predict
the r e l a x a t i o n times use
c h a r a c t e r i s t i c of the p e r i o d of the l o c a l m a g n e t i c success i n b o t h m a g n e t i c resonance o b t a i n e d b y u s i n g the f u n c t i o n A(t)
1 +
2
w h e r e T is t h e c h a r a c t e r i s t i c m a g n e t i c s
fluctuation
(9)
t/r]}
t i m e ( o f t e n the m u o n
represents a l a t t i c e s u m o v e r t h e n e i g h b o r i n g n u c l e a r
2
dipoles. T h i s f u n c t i o n reduces to a G a u s s i a n (Aoe~ n ) a
(T -»
Reasonable
(21)
2
0
h o p t i m e ) a n d o-
fluctuations.
a n d m u o n d e p o l a r i z a t i o n has b e e n
exp{-2crs r [ e x p ( - ^ / r ) -
— A
T h e o r i e s that
a correlation time T ,
for s l o w d i f f u s i o n
f o r fast d i f f u s i o n ( T -> 0 ) .
As
m e n t i o n e d , t h e experiments are d o n e as a f u n c t i o n of t e m p e r a t u r e
for
any
oo) a n d to a n e x p o n e n t i a l (e' ** )
ts
2
given
sample,
and
the
71
q u a n t i t y i n the
exponential
of
A(t)
is
measured. A s a n e x a m p l e of the results o b t a i n e d as a f u n c t i o n of t e m p e r a t u r e , the d a t a of G a u s t e r et a l . (14)
are s h o w n i n F i g u r e 8. T h e d a t a for C u
w e r e fit b y u s i n g the g e n e r a l expression ( E q u a t i o n 9 ) , w i t h a v a l u e f o r a
2
of 0.257 dz 0.003 s" . 1
T h i s v a l u e of a assumes t h e m u o n is t r a p p e d at 2
a n o c t a h e d r a l i n t e r s t i t i a l site. T h e results o n A l , o n the other h a n d , s h o w a s u r p r i s i n g l a c k of t r a p p i n g , e v e n at v e r y l o w t e m p e r a t u r e s .
This lack
of t r a p p i n g i n A l c a n be p a r t i a l l y r a t i o n a l i z e d i f one assumes t h a t t h e concentration
of
t r a p p i n g sites is l o w ,
so
that a m u o n
cannot
find
vacancies i n its l i f e t i m e or, a l t e r n a t i v e l y , that s u c h sites are n o t d e e p traps, so that d e t r a p p i n g occurs q u i c k l y . T h e e x p l a n a t i o n f o r this l a r g e difference b e t w e e n these t w o m a t e r i a l s w i t h s i m i l a r f a c e - c e n t e r e d
cubic
( F C C ) lattices is n o t c o m p l e t e at this t i m e . T h e basic m o d e l of diffusion g i v e n is f u r t h e r s u b s t a n t i a t e d i n C u b y a b e a u t i f u l set of experiments d o n e b y C a m a n i et a l . (22) p r e t e d b y H a r t m a n n (23).
as i n t e r
T h e c o p p e r n u c l e u s , i n a d d i t i o n to possessing
a m a g n e t i c d i p o l e m o m e n t , has a r e a s o n a b l y l a r g e q u a d r u p o l e
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
moment.
16
SOLID S T A T E C H E M I S T R Y :
A CONTEMPORARY
OVERVIEW
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.30
800
T
1000
1200
1400
Solid State Communications
Figure 8.
The depolarization rate of the diffusing /A in (a) Cu and (b) Al as a function of temperature (14) +
I n t h e case of m u o n d i f f u s i o n i n C u the m u o n distorts the site at w h i c h it sits. S u c h a d i s t o r t i o n p r o d u c e s a n electric field g r a d i e n t different f r o m zero, w h i c h w i l l i n t e r a c t w i t h the n e i g h b o r i n g n u c l e a r q u a d r u p o l e
mo
ments.
the
T h e o r i e n t a t i o n of the n e i g h b o r i n g n u c l e i , a n d therefore
resulting dipole sum a , 2
d e p e n d s o n the i n t e r a c t i o n of t h e
quadrupole
m o m e n t w i t h the i n d u c e d e l e c t r i c field g r a d i e n t a n d t h e i n t e r a c t i o n of the nuclear magnetic dipole moment w i t h any magnetic
field
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
present.
1.
DENISON
Muon
Spin
17
Rotation
T h e d e p o l a r i z a t i o n of the m u o n spins w a s m e a s u r e d as a f u n c t i o n of a n applied external magnetic
field.
T h e results so o b t a i n e d , u s i n g a single
c r y s t a l of C u o r i e n t e d i n a k n o w n w a y i n the e x t e r n a l m a g n e t i c are s h o w n i n F i g u r e 9. T h e s e results, w h i c h d e p e n d o n the
field,
competition
b e t w e e n the n u c l e a r o r i e n t a t i o n d u e to the e l e c t r i c field g r a d i e n t a n d t h e e x t e r n a l m a g n e t i c field, agree extremely w e l l w i t h the theory. I n a d d i t i o n , the a m o u n t measured.
of d i s t o r t i o n d u e to the presence of
the m u o n has
been
F i g u r e 10 shows the results of d e p o l a r i z a t i o n m e a s u r e d as a
f u n c t i o n of external field a n d as a f u n c t i o n of c r y s t a l o r i e n t a t i o n . A best fit to the d a t a occurs for a n a s s u m e d
expansion
of the l a t t i c e site
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about 5 % .
t (/x sec. ) Figure 9. Measured asymmetry N(t) in Cu(lll) at 80 K as a function of magnetic field strength and direction: (%), 4800 G; (O), 3500 G (22).
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
of
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Ol 40
I
I
100
1000
. B
e x t
1
(GAUSS)
Physical Review Letters
Figure 10. Depolarization rate of the fi in a single crystal of Cu as a function of magnetic field strength and direction: ( ), the predicted values for undistorted site; ( ), the predicted values for a dilation of the site by 5%. Cu(100): (9), 80 K; (O), 20 K. Cu(110): (+), 80 K; (O), 20 K. Cu(lll): (A), 80 K; (£s) 20 K (22). +
9
R e c e n t results o n n i o b i u m a n d v a n a d i u m (17,18,19,20)
show more
c o m p l i c a t e d b e h a v i o r of the d e p o l a r i z a t i o n e x p o n e n t i a l as a f u n c t i o n of t e m p e r a t u r e . F i g u r e 11 illustrates the g e n e r a l b e h a v i o r of the d e p o l a r i z a tion function.
T h e feature of c o n c e r n
is t h e d i p c o r r e s p o n d i n g
decrease i n d e p o l a r i z a t i o n rate at p a r t i c u l a r temperatures. are v e r y d e p e n d e n t o n t h e p u r i t y of the samples
used.
Such The
to
a
dips
current
i n t e r p r e t a t i o n c a n b e q u a l i t a t i v e l y m e n t i o n e d here, a l t h o u g h t h e details of the t h e o r y are s t i l l b e i n g w o r k e d out (15,17,19),
a n d i t is c l e a r t h a t
m o r e d a t a are r e q u i r e d . I n t h e h i g h - t e m p e r a t u r e r e g i o n ( D )
multiphonon
processes o c c u r that essentially k e e p t h e m u o n u n t r a p p e d , r e s u l t i n g i n a n a v e r a g i n g of t h e l o c a l m a g n e t i c e n v i r o n m e n t , g i v i n g s m a l l d e p o l a r i z a t i o n . A s the t e m p e r a t u r e goes d o w n the m u o n
finds
( C ) , traps d o b e c o m e effective,
itself stationary i n the v i c i n i t y of
d i p o l e s , w h i c h q u i c k l y d e p h a s e the m u o n spins. d i p ( B ) t w o m o d e l s are c u r r e n t l y u s e d .
nuclear
so t h a t magnetic
I n t h e r e g i o n of
the
F o r t h i s effect i n N i , G r e b i n n i k
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
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1.
DENISON
Muon
Spin
Rotation
19
temperature Figure 11.
General behavior of the depolarization rate of muons in the body-centered cubic metals Nb and V
et a l . ( 1 5 ) h a v e p r o p o s e d a k i n d of q u a n t u m t u n n e l i n g as a m e c h a n i s m f o r d i f f u s i o n , w h i c h a c t u a l l y increases as t h e t e m p e r a t u r e decreases. m o d e l treats t h e r e g i o n a t v e r y l o w t e m p e r a t u r e ( A ) as b e i n g i n a l l o w i n g t h e m u o n to diffuse r e a d i l y a n d find traps. theory
This
effective
T h e opposing
( 1 9 ) argues t h a t i n r e g i o n B t h e m u o n d i f f u s i o n i n a c l a s s i c a l
sense has s l o w e d to t h e p o i n t w h e r e i t is difficult t o find a d e p o l a r i z i n g t r a p a n d y e t is fast e n o u g h that m o t i o n a l a v e r a g i n g , as m e n t i o n e d a b o v e , is s t i l l effective.
I n this l a t t e r m o d e l at v e r y l o w t e m p e r a t u r e s , t h e m u o n
is s l o w e d t o t h e p o i n t t h a t t h e m o t i o n a l a v e r a g i n g d i s a p p e a r s . T h e d i f f u s i o n e x p e r i m e n t s d o relate to some f u n d a m e n t a l questions a b o u t t h e b e h a v i o r of a m u o n i n m a t t e r . Is t h e m u o n l o c a l i z e d o r is i t s p r e a d o u t i n t h e q u a n t u m m e c h a n i c a l sense? H o w does t h i s c h a r a c t e r c h a n g e as a f u n c t i o n of t e m p e r a t u r e ? field,
T h e m u o n a n d its r e s u l t i n g strain
a p o l a r o n , m a y m o v e together t h r o u g h a c r y s t a l b u t a g a i n c h a n g e
c h a r a c t e r as a f u n c t i o n of t e m p e r a t u r e .
T h e f u t u r e s h o u l d b r i n g some
answers t o these questions. Magnetism.
T h e m u o n has b e e n u s e d as a p r o b e i n a v a r i e t y of m a g
n e t i c m a t e r i a l s . W o r k has b e e n d o n e i n t h e m e t a l ferromagnets as w e l l as in both
metallic a n d nonmetallic antiferromagnets.
Several
different
aspects of m a g n e t i z a t i o n a n d t h e o r i g i n of m a g n e t i s m h a v e b e e n e x a m i n e d l o o k i n g a t b o t h static a n d d y n a m i c effects.
Perhaps the simplest a n d
cleanest t y p e of m e a s u r e m e n t is t o m o n i t o r t h e c h a n g e i n t h e l o c a l m a g netization above a n d below the critical temperature. Figures 1 2 ( a ) a n d
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
20
CHEMISTRY:
A CONTEMPORARY
OVERVIEW
Downloaded by 80.82.77.83 on May 18, 2017 | http://pubs.acs.org Publication Date: June 1, 1980 | doi: 10.1021/ba-1970-0186.ch001
SOLID S T A T E
12(b)
s h o w the d r a m a t i c c h a n g e i n l o c a l m a g n e t i z a t i o n i n N i i n t h e
p a r a m a g n e t i c a n d f e r r o m a g n e t i c state. F i g u r e 1 2 ( c )
shows the g e n e r a l
g r o w t h of m a g n e t i z a t i o n a n d t h e closeness t o t h e f a m i l i a r
Brillouin
function. Measurements have been made i n ferromagnetic N i , F e , a n d C o a n d i n G d [see
R e v i e w b y A . S c h e n c k (24)]
as w e l l as a n t i f e r r o m a g n e t i c D y
Holt et al.; Solid State Chemistry: A Contemporary Overview Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
1.
DENISON
Muon
Spin
ol Downloaded by 80.82.77.83 on May 18, 2017 | http://pubs.acs.org Publication Date: June 1, 1980 | doi: 10.1021/ba-1970-0186.ch001
21
Rotation
0
I
I
200
I
I
400
I
TEMPERATURE(K°)
J
600 Physics Letters
Figure 12c. The local magnetic field at the stopped muon site as a function of the applied magnetic field: (A), (%), poly crystal, (Q), single crystal, a n d H o ( D y has b o t h a n a n t i f e r r o m a g n e t i c a n d f e r r o m a g n e t i c p h a s e as the t e m p e r a t u r e is l o w e r e d ) .
T h e s e studies b e c o m e q u i c k l y n o n t r i v i a l
w h e n the o r i g i n of the o b s e r v e d field is q u e s t i o n e d i n d e t a i l . T h e i n t e r n a l ( s i t e ) field is g i v e n g e n e r a l l y as -Btot = B ext ~" £