Muon Spin Rotation: An Exotic Probe of the Atomic Environment

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


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


(e ) +

1.

DENISON

Muon

Spin

7T

5

Rotation

fJL

+

+

+

SPIN OPPOSITE TO MOMENTUM IN DECAY


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.


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


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\

-+>||


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


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


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 .


Figure

1.

DENISON

Muon

Spin

Rotation

11

P l.O

0.8


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



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


2

1.

DENISON

Muon

Spin

Rotation

13

150

100 -

50


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


14


[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


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


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


moment.

16


A CONTEMPORARY

OVERVIEW


.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


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


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).


of


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



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


20

CHEMISTRY:

A CONTEMPORARY

OVERVIEW


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


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 ~" £

Muon Spin Rotation: An Exotic Probe of the Atomic Environment

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