Relaxation Effects Associated with Magnetic Phase Transitions

21 Relaxation Effects Associated with Magnetic

Downloaded by UNIV OF MICHIGAN ANN ARBOR on May 18, 2017 | http://pubs.acs.org Publication Date: July 1, 1981 | doi: 10.1021/ba-1981-0194.ch021

Phase Transitions G . R . HOY Physics Department, Old D o m i n i o n University, Norfolk, VA 23508 M . R. C O R S O N Physics Department, B o w d o i n College, Brunswick, ME 04011

Mössbauer spectra of substances that magnetically order often show a central peak with prominent wings in the region of the ordering temperature. We show that such spectra can arise from the critical slowing down of the Mössbauer ions spin fluctuation rate. Using a dynamic model, detailed calculations are presented for Kramers (S = 1/2) and non-Kramers (S = 1) salts. The spin Hamiltonian parameters, the reduced magnetization, the form and strength of the hyperfine interaction, and the spin fluctuation rate can be determined. Our results showing the critical slowing down of spinfluctuationsin K FeO are presented. 2

4

We also give a qualitative discussion of how these microscopic results for the spin dynamics may fit into the framework of critical fluctuations.

T n t h e r e g i o n of m a g n e t i c phase transitions, so-called anomalous M o s s b a u e r spectra a r e often o b s e r v e d ( J ) , c o n s i s t i n g of s m a l l b r o a d e n e d peaks o n t h e w i n g s of a large c e n t r a l peak. S u c h spectra are n o t c h a r a c t e r i s t i c of s i n g l e - v a l u e d , static h y p e r f i n e fields. B e c a u s e of t h e difficulty i n t h e o r e t i c a l l y fitting s u c h spectra, these strange features are c o m m o n l y a t t r i b u t e d t o effects a r i s i n g f r o m s a m p l e t e m p e r a t u r e i n h o m o geneities, a d i s t r i b u t i o n of C u r i e o r N e e l temperatures i n t h e s a m p l e , o r c r i t i c a l s u p e r p a r a m a g n e t i s m . W h i l e these m a y b e t h e correct i n t e r p r e t a t i o n i n some i n d i v i d u a l cases, these anomalous spectra h a v e b e e n o b s e r v e d i n m a n y n o n m e t a l l i c systems i n v e s t i g a t e d b y m a n y different researchers. H e n c e , i t seems reasonable to search f o r a n u n d e r l y i n g

©

0065-2393/81 /0194-0463$05.00/0 1981 American Chemical Society

Stevens and Shenoy; Mössbauer Spectroscopy and Its Chemical Applications Advances in Chemistry; American Chemical Society: Washington, DC, 1981.

464

M O S S B A U E R S P E C T R O S C O P Y A N D ITS

p h y s i c a l reason f o r t h e i r o c c u r r e n c e .

C H E M I C A L APPLICATIONS

I n a d d i t i o n to p r o v i d i n g a q u a l i t a

t i v e i n t e r p r e t a t i o n of these spectra, w e also p r o p o s e to f o r m u l a t e the p r o b l e m i n s u c h a m a n n e r t h a t some q u a n t i t a t i v e progress c a n b e m a d e in understanding such phenomena. V e r y p o w e r f u l and general theoretical techniques

for calculating

M o s s b a u e r effect l i n e shapes i n the presence of r e l a x a t i o n n o w exist

(2-

B y p a r t i c u l a r i z i n g this g e n e r a l theory, i t is p o s s i b l e to f o r m u l a t e t h e

20).

p r o b l e m i n s u c h a w a y t h a t m a n y i n t e r e s t i n g p h y s i c a l parameters c a n b e


d e t e r m i n e d u s i n g M o s s b a u e r spectroscopy.

I n p a r t i c u l a r , one c a n observe

a n d also c l a r i f y the m e a n i n g of the " c r i t i c a l s l o w i n g d o w n " of a n

(21)

ion's s p i n

fluctuation

rate n e a r the c r i t i c a l t e m p e r a t u r e .

Furthermore, it

m a y be possible to b e g i n to c o n n e c t the results of these Mossbauer

measurements

with

microscopic

the u s u a l c r i t i c a l p h e n o m e n a

picture

c o n s i s t i n g of clusters of c o r r e l a t e d a t o m i c spins.

Formulation of the Problem I t is c o m m o n i n M o s s b a u e r spectroscopy to c o n s i d e r the M o s s b a u e r n u c l e u s to b e i n a n "effective m a g n e t i c field" p r o d u c e d b y t h e electrons i n the i o n . I n g e n e r a l it is n o t t r u e t h a t one c a n treat the p r o b l e m i n this way.

A m o r e g e n e r a l t y p e of i n t e r a c t i o n w o u l d b e a h y p e r f i n e t e r m

conventionally written f • A • tains o n l y a n A

zz

A SI, ZZ

Z

Z

If the h y p e r f i n e i n t e r a c t i o n tensor c o n

c o m p o n e n t , t h e n the h y p e r f i n e i n t e r a c t i o n c a n b e w r i t t e n

w h i c h c l e a r l y gives the effective m a g n e t i c field result.

T h e electrons t h a t interact w i t h the n u c l e u s are m o v i n g i n a c o m p l i c a t e d f a s h i o n . I n a d d i t i o n , t h e ions c a n i n t e r a c t w i t h e a c h o t h e r a n d / o r w i t h the l a t t i c e v i b r a t i o n s of the s o l i d . U n d e r c e r t a i n c o n d i t i o n s

these

d y n a m i c effects c a n be o b s e r v e d , a n d the correct i n t e r p r e t a t i o n of these effects contains r e m a r k a b l y d e t a i l e d i n f o r m a t i o n . B e c a u s e of the d i f f i c u l t y i n v i s u a l i z i n g the most g e n e r a l f o r m u l a t i o n of t h e t i m e - d e p e n d e n t

prob

l e m , i t is v e r y u s e f u l to g a i n some p h y s i c a l i n s i g h t b y c o n s i d e r i n g a less g e n e r a l case.

T o d o this, c o n s i d e r a

momentum, and total ionic spin S =

5 7

F e ion having no orbital angular 1/2.

I n the absence of a r e a l or

effective m a g n e t i c field, this K r a m e r s i o n w i l l h a v e a d o u b l y

degenerate

g r o u n d state i n d e p e n d e n t of the degree of a s y m m e t r y of the c r y s t a l l i n e environment

(Kramers

Theorem).

Assume,

for

simplicity, that

? • X • S h y p e r f i n e i n t e r a c t i o n c a n be r e p r e s e n t e d b y the effective netic field a p p r o x i m a t i o n . U n d e r these c o n d i t i o n s t h e

5 7

a n effective m a g n e t i c field e q u a l to — H or H w h e n S* = respectively.

F e n u c l e u s is i n 1/2 a n d — 1 / 2 ,

I f t h e s p i n system m a g n e t i c a l l y orders b e l o w some c r i t i c a l

t e m p e r a t u r e , the ground-state d e g e n e r a c y is r e m o v e d a n d the |S = Sz =

l/2>

the mag

and | l / 2 , — 1 / 2 >

I f the i o n undergoes

1/2,

eigenstates b e c o m e u n e q u a l l y p o p u l a t e d .

transitions b e t w e e n

these eigenstates

because of

some r e l a x a t i o n m e c h a n i s m i n the s o l i d , w e m u s t d e t e r m i n e the effective m a g n e t i c field at the n u c l e u s .


21.

H O Y A N D CORSON

Magnetic

Phase

465

Transitions

C o n s i d e r t h e cases of s l o w , fast, a n d i n t e r m e d i a t e , r e l a x a t i o n s h o w n i n F i g u r e 1. B y t h e t i m e - e n e r g y u n c e r t a i n t y p r i n c i p l e , t h e t i m e r e q u i r e d for a n u c l e u s t o " m e a s u r e " a n effective m a g n e t i c field is a p p r o x i m a t e l y t h e L a r m o r p e r i o d of the n u c l e u s i n t h a t field, w h i c h f o r

5 7

F e is t y p i c a l l y

10" s. I f t h e i o n i c s p i n fluctuation rate is s l o w , a n d a n i o n spends m a n y 9

L a r m o r p e r i o d s i n e a c h eigenstate, t h e n a M o s s b a u e r n u c l e u s experiences a m a g n e t i c field — i f or -\-H d e p e n d i n g o n t h e state o f t h e a t o m c o n t a i n i n g the n u c l e u s .

I n this case, t h e M o s s b a u e r s p e c t r u m is t h e B o l t z m a n n -


w e i g h t e d s u m of t w o spectra c h a r a c t e r i s t i c of these t w o fields. S i n c e — H a n d + H p r o d u c e t h e same M o s s b a u e r p a t t e r n , t h e r e s u l t i n g s p e c t r u m is as s h o w n i n the u p p e r c u r v e i n F i g u r e 1. If the ionic spin

fluctuation

r a t e is fast c o m p a r e d t o t h e L a r m o r

f r e q u e n c y , t h e n t h e n u c l e u s c a n n o t r e s p o n d t o the i n d i v i d u a l fields H and

— H, a n d a l l n u c l e i experience t h e same, s i n g l e - v a l u e d

effective

m a g n e t i c field w h i c h is t h e B o l t z m a n n - w e i g h t e d average of these t w o IONIC STATE S = 1/2

HYPERFINE EFFECTIVE MAGNETIC FIELD AT NUCLEUS

S,= 1/2

-H

S =-l/2 2

CASE I: SLOW RELAXATION (Relaxation frequency much lower than the nuclear Larmor frequency in the magnetic field H.) CASE 2= FAST RELAXATION (Relaxation frequency much higher than the nuclear Larmor frequency in the magnetic field H.)

CASE 3:

INTERMEDIATE RELAXATION

(Relaxation frequency comparable to the nuclear Larmor frequency in the magnetic field H.)

Figure 1. Calculated Mossbauer transmission spectra, including the effects of relaxation, for an S = V2 Kramers ion having eigenstates |S,M > of \V2, V2> and \V2, — V2>. In these spectra, the hyperfine interaction is assumed to be an effective magnetic field. S


466



values. F o r a system w i t h some m a g n e t i c o r d e r , a p o s s i b l e s p e c t r u m is s h o w n as the m i d d l e c u r v e i n F i g u r e 1. N o t e t h a t t h e o v e r a l l w i d t h of t h e fast r e l a x a t i o n s p e c t r u m is less t h a n t h a t of the s l o w r e l a x a t i o n spectrum. If the i o n i c s p i n fluctuation rate is c o m p a r a b l e to the n u c l e a r L a r m o r f r e q u e n c y , t h e n the effective m a g n e t i c field e x p e r i e n c e d b y t h e n u c l e u s is not w e l l defined. T h i s gives rise to a " r e l a x a t i o n - b r o a d e n e d " s p e c t r u m . T h e lowest c u r v e i n F i g u r e 1 shows s u c h a n e x a m p l e f o r a system h a v i n g


zero m a g n e t i c order. Next consider a ionic spin S =

5 7

F e ion w i t h no orbital angular momentum, total

1, a n d nondegenerate i o n i c eigenstates S* =

1,0, a n d — 1.

A g a i n assume the effective m a g n e t i c field a p p r o x i m a t i o n for t h e h y p e r f i n e i n t e r a c t i o n . T h i s effective m a g n e t i c field takes o n the values — H, 0, a n d +H

when S = z

1,0, a n d — 1 , r e s p e c t i v e l y . I n the p r e s e n c e of r e l a x a t i o n

effects, w e m u s t a g a i n d e t e r m i n e the r e s u l t i n g M o s s b a u e r spectra. F i g u r e 2 shows results s i m i l a r t o those i n F i g u r e 1 for t h i s n o n - K r a m e r s i o n . T h e r e are s e v e r a l i m p o r t a n t differences b e t w e e n these t w o sets of results. T h e i o n i c state |S = nucleus.

1, S — 0 > does n o t p r o d u c e a m a g n e t i c field at the z

T h u s , i n the s l o w r e l a x a t i o n l i m i t this state p r o d u c e s a single

p e a k at the c e n t e r of the p a t t e r n as s h o w n i n the u p p e r c u r v e i n F i g u r e 2. T h e t w o m i s s i n g peaks of the a c c o m p a n y i n g s i x - l i n e p a t t e r n f r o m states |1,1> a n d |1, — 1 > are h i d d e n b y the c e n t r a l peak. I n the fast r e l a x a t i o n l i m i t , as i n t h e m i d d l e c u r v e of F i g u r e 2, a l l n u c l e i e x p e r i e n c e the same s i n g l e - v a l u e d effective m a g n e t i c field w h i c h is the B o l t z m a n n - w e i g h t e d average of t h e three v a l u e s — H, 0, a n d + H . T h e m e a n i n g of this is t h a t t h e three fields — H, 0, a n d + H

h a v e lost

t h e i r i n d i v i d u a l i d e n t i t i e s . A s a c o n s e q u e n c e , the c e n t r a l p e a k i n t h e s l o w r e l a x a t i o n s p e c t r u m of F i g u r e 2 a t t r i b u t a b l e to the m a g n e t i c field of 0 is c o m p l e t e l y absent i n this fast r e l a x a t i o n l i m i t . A

more

complete

analysis a p p l i c a b l e to M o s s b a u e r e x p e r i m e n t a l

results i n c l u d i n g t i m e - d e p e n d e n t effects r e c e n t l y has b e e n f o r m u l a t e d i n a very useful a n d general w a y

(12),

a n d this m e t h o d is u s e d i n the

d e v e l o p m e n t p r e s e n t e d here. T h e details of the f o r m u l a t i o n c a n be f o u n d i n the o r i g i n a l reference, so w e w i l l o n l y g i v e a s u m m a r y . We

w i l l assume t h a t the

5 7

F e n u c l e u s is c o u p l e d

to its

atomic

electrons 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 , a n d t h a t i n the i o n i c system some r a n d o m process i n d u c e s transitions b e t w e e n t h e eigenstates of the i o n . T h i s " r e l a x a t i o n " process is a s s u m e d to b e s t a t i o n a r y M a r k o f f i a n , so t h a t i t is p o s s i b l e to o b t a i n c l o s e d - f o r m m a t h e m a t i c a l expressions f o r the r e s u l t i n g l i n e s h a p e u s i n g t h e s o - c a l l e d super o p e r a t o r f o r m a l i s m . essential result f o r fLis

the l i n e shape

as a f u n c t i o n of

emitted

The

energy

(12), FM—

£

6,(/*v|u(p)| iV) +

f

/


(1)

21.

HOY

A N D

Magnetic

CORSON

Phase

IONIC STATE S = I

HYPERFINE EFFECTIVE MAGNETIC FIELD AT NUCLEUS

•to

S = 0

s Downloaded by UNIV OF MICHIGAN ANN ARBOR on May 18, 2017 | http://pubs.acs.org Publication Date: July 1, 1981 | doi: 10.1021/ba-1981-0194.ch021

467

Transitions

= -I

CASE 1= SLOW RELAXATION (Relaxation frequency much lower than the nuclear Larmor frequency in the magnetic field H.) CASE 2

FAST RELAXATION

(Relaxation frequency much higher than the nuclear Larmor frequency in the magnetic field H.)

CASE 3= INTERMEDIATE RELAXATION (Relaxation frequency comparable to the nuclear Larmor frequency in the magnetic field H.)

Figure 2. Calculated Mossbauer transmission spectra, including the effects of relaxation, for an S = l non-Kramers ion having eigenstates |S,M > of |I,I>, \1,0> and 1>. In these spectra, the hyperfine interaction is assumed to be an effective magnetic field. S

where b

v

is t h e o c c u p a t i o n p r o b a b i l i t y of t h e i n i t i a l state |v>, p =

iT

— ifuo, A is the o p e r a t o r f o r e m i s s i o n ( o r a b s o r p t i o n ) of r a d i a t i o n , a n d u(p)

=

(p

— W

— iH )' . 0

x

1

T h e m a t r i x elements of A are e s s e n t i a l l y

C l e b s c h - G o r d a n coefficients t i m e s v e c t o r s p h e r i c a l h a r m o n i c s . T h e m a t r i x elements of u(p)

are c a l c u l a t e d b y i n v e r t i n g t h e s u p e r m a t r i x

d i m e n s i o n a l i t y is (2S + l ) ( 2 Z i + 1) (21 2

i o n i c s p i n a n d Ii a n d I

0

0

+ 1), w h e r e S is t h e

whose

(effective)

are the spins of the e x c i t e d a n d g r o u n d n u c l e a r

levels. T h e m a t r i x elements of t h e s u p e r H a m i l t o n i a n H

0

X

are g i v e n b y

(12), (2) T h e m a t r i x elements of the s u p e r r e l a x a t i o n m a t r i x W w i l l b e g i v e n later. I n o u r c a l c u l a t i o n w e d o n o t a c t u a l l y c o m p u t e the n e e d e d i n v e r s e f o r e v e r y v a l u e of

-

|

S(S

1) J

+

E(S/

+

-

S) 2

y

+

(4)

yaS

z

w h e r e D a n d E are t h e s p i n H a m i l t o n i a n p a r a m e t e r s , a is t h e

reduced

m a g n e t i z a t i o n of the i o n i c s p i n system i n t h e m e a n field a p p r o x i m a t i o n , a n d S is the s p i n of the i o n . T h u s H

0

(see E q u a t i o n 2 ) c a n b e w r i t t e n ,

Ho == H

ion

(5)

Hhyper

T h e r e l a x a t i o n process associated w i t h H(t)

is t a k e n to consist of

an

i n t e r a c t i o n t h a t causes the i o n to e v o l v e i n t i m e a c c o r d i n g to a s t a t i o n a r y M a r k o f f i a n c h a i n t h r o u g h its i o n i c e n e r g y levels Ei a n d its c o r r e s p o n d i n g eigenstates ^ .

E a n d ^ are d e t e r m i n e d b y o b t a i n i n g the eigenvalues a n d {

eigenvectors of the i o n i c H a m i l t o n i a n H

i o n

.

I n o u r case, b e c a u s e w e are

p r i m a r i l y i n t e r e s t e d i n the l o w - t e m p e r a t u r e r e g i o n , w e h a v e t a k e n t h e r e l a x a t i o n m e c h a n i s m to arise f r o m t h e " f l i p - f l o p " p a r t of t h e


dipole-

21.

Magnetic

H O Y A N D CORSON

dipole interaction.

Phase

469

Transitions

W e expect t h i s e n e r g y - c o n s e r v i n g

spin-spin inter

a c t i o n t o b e t h e d o m i n a n t m e c h a n i s m at l o w t e m p e r a t u r e s . completely correct for a n S =

( T h i s is n o t

1/2, d o u b l y degenerate system, w h i c h , i n

a d d i t i o n to t h e C l a u s e r - B l u m e t h e o r y is d i s c u s s e d b y D a t t a g u p t a ( 2 2 ) . A t h i g h t e m p e r a t u r e s , t h e l a r g e n u m b e r of p h o n o n s is e x p e c t e d t o cause s p i n - l a t t i c e r e l a x a t i o n to d o m i n a t e . ) A s s u m i n g t h a t t h e i o n i c system is i n i t i a l l y i n state fa, t h e t r a n s i t i o n


p r o b a b i l i t y f o r t h e i o n to flip f r o m state fa to state fa is g i v e n b y Qij =

o | < fa fa | S

+i

X

S.j + S.i S

+j

exp ( - ^ / f c D / E e x p

| fa fa > (6)

(-Et/kT)

i

w h e r e O is t h e r e l a x a t i o n rate p a r a m e t e r , S a n d S. a r e t h e u s u a l r a i s i n g +

and

l o w e r i n g operators,

respectively,

f a c t o r d e t e r m i n e s t h e p r o b a b i l i t y of undergo a spin

flip.

and the normalized Boltzmann finding

a n e i g h b o r w i t h w h i c h to

I t is i m p o r t a n t to r e a l i z e that at sufficiently l o w

temperatures i n a magnetically ordered sample, the transition probabilities for

flipping

c a n n o t o c c u r s i m p l y b e c a u s e t h e n u m b e r of n e i g h b o r s w i t h

w h i c h to e x c h a n g e s p i n has b e e n r e d u c e d

to zero f o r t h e u n f a v o r e d

spin orientation. An

i m p o r t a n t step of this c a l c u l a t i o n is t h e c o n s t r u c t i o n of t h e

s o - c a l l e d super o r L i o u v i l l e H a m i l t o n i a n .

T h i s particular construction i :

i n t e r e s t i n g b e c a u s e the eigenvalues of t h e r e s u l t i n g super m a t r i x g i v e t h e a c t u a l energies o f a l l p o s s i b l e r a d i a t i o n s o f t h e system. I n a d d i t i o n , f r o m a t h e o r e t i c a l p o i n t of v i e w , t h e i n t r o d u c t i o n of this t y p e of results i n s i m p l i f y i n g t h e c o m m u t a t i o n

relations needed

operator

to solve t h e

t i m e - d e p e n d e n t p r o b l e m . T h e s u p e r H a m i l t o n i a n m a t r i x is d e f i n e d b y {fam fam \H *\fa'm 'fa'm ') g

e

0

hj,ilff$m ,m ' e

where m

g

e

g

—

—

e

g

f/

0

0

g

$fafa'$m ,m '

e

g

g

a n d m refer to t h e n u c l e a r g r o u n d a n d e x c i t e d states, r e s p e c e

tively. T h e r e l a x a t i o n super m a t r i x W is d e f i n e d as ( 1 2 ) , (fa m fa m \W\ fa' m ' fa' ra '). g

e

T o s i m p l i f y t h e n o t a t i o n l e t fa =

g

/x a n d fa =

e

v. T h e n , t h e m a t r i x

elements o f W a r e g i v e n b y ,

(7) where

is d e f i n e d i n E q u a t i o n 6.


470


W h e n c o n s t r u c t i n g the r e l a x a t i o n operator


s u p e r m a t r i x W , i t is

i m p o r t a n t to k n o w a r e l a t i o n s h i p i n v o l v i n g t h e d i a g o n a l elements, n a m e l y ,

(i|^|i)--Z(i|^|j), w h e r e this r e l a t i o n s h i p f o l l o w s f r o m the c o n c e p t of d e t a i l e d b a l a n c e .


Consideration of Various Cases I n this section w e present m o d e l c a l c u l a t i o n s s h o w i n g t h e o r e t i c a l 5 7

F e M o s s b a u e r spectra as the f o l l o w i n g p a r a m e t e r s are v a r i e d : the s p i n

H a m i l t o n i a n parameters D a n d E , t h e r e d u c e d m a g n e t i z a t i o n a, t h e f o r m a n d s t r e n g t h of the h y p e r f i n e i n t e r a c t i o n A ,A ,A ,(A =A , x

y

z

x

etc.), the

xx

i o n i c s p i n S, a n d r e l a x a t i o n r a t e p a r a m e t e r Q. W e c o n s i d e r three p o s s i b l e cases of the c r y s t a l l i n e field—the h i g h l y s y m m e t r i c a l one i n w h i c h D

=

E =

0,

0, the a x i a l l y s y m m e t r i c c r y s t a l l i n e field w h e r e D ^ O , a n d E =

a n d finally, the r h o m b i c c r y s t a l l i n e field w h e r e D ^

0 and E ^

0.

We

assume t h a t the i o n i c system f o l l o w s a W e i s s m a g n e t i z a t i o n c u r v e as i t m a g n e t i c a l l y orders w h e n the t e m p e r a t u r e is l o w e r e d b e l o w the t r a n s i t i o n t e m p e r a t u r e . W e i n c l u d e t w o forms f o r the h y p e r f i n e i n t e r a c t i o n , n a m e l y , the effective field case A

=

A

=

fine i n t e r a c t i o n A

=

A

=^= 0. W e also c o n s i d e r t w o cases f o r t h e

x

x

ionic spin—the S =

=

A

y

y

z

0 and A

z

^

0, a n d the i s o t r o p i c h y p e r

1 / 2 K r a m e r s system a n d the S =

A s i m i l a r c a l c u l a t i o n for S =

1 non-Kramers ion.

3 / 2 has b e e n c o n s i d e r e d p r e v i o u s l y

(23).

H o w e v e r , i n that case, the o f f - d i a g o n a l elements of the e l e c t r o n n u c l e a r magnetic hyperfine interaction were neglected. I n o u r c a l c u l a t i o n s a l l the p a r a m e t e r s D , E , A , x

A, y

A

Zy

a n d O are

expressed i n e n e r g y u n i t s s u c h that the n u m b e r e i g h t corresponds to a t y p i c a l i r o n M o s s b a u e r e x p e r i m e n t a l l i n e w i d t h of 0.30 m m / s .

F o r energy

c o n v e r s i o n purposes note t h a t one of o u r u n i t s e q u a l s : 3.75 X 10"

2

1.8 X 1 0

1

S =

9

e V , 2.09 X 10" K , 2.89 X 1 0 "

1/2

5

K r a m e r s System.

21

mm/s,

e r g , or 1.46 X 10" c m " .

F o r this case t h e

5

spin Hamiltonian

p a r a m e t e r s D a n d E h a v e n o effect, as c a n be seen f r o m the f o r m of H I n F i g u r e 3 w e c o n s i d e r the K r a m e r s i o n system w i t h S =

1/2.

C T

.

T h e first

c o l u m n shows t h e c a l c u l a t e d M o s s b a u e r spectra w h e n the i o n i c system f o l l o w s a W e i s s m o l e c u l a r field m a g n e t i z a t i o n c u r v e as the t e m p e r a t u r e is l o w e r e d , a n d t h e i o n i c r e l a x a t i o n rate O is h e l d constant at a f a i r l y l a r g e v a l u e ( h e r e l a r g e means c o m p a r e d to the v a l u e of the h y p e r f i n e i n t e r a c t i o n c o u p l i n g constant as d i s c u s s e d i n the p r e v i o u s s e c t i o n ) .

The

s e c o n d c o l u m n shows the spectra f o r t h e case of a constant s m a l l v a l u e of r e d u c e d m a g n e t i z a t i o n o-, b u t w i t h t h e r e l a x a t i o n rate O d e c r e a s i n g several orders of m a g n i t u d e . T h e t h i r d c o l u m n is s i m i l a r t o the

by

second

b u t the system is a s s u m e d to b e a b o v e its m a g n e t i c t r a n s i t i o n t e m p e r a t u r e


21.

SPIN


Magnetic

H O Y A N D CORSON

1/2

Phase

AXG=AYG=0

471

Transitions

AZG=-80

Figure 3. Calculated Mossbauer spectra, including the effects of relaxation, for an S = V2 Kramers ion. The hyperfine interaction is assumed to be an effective magnetic field. a n d t h u s has a zero v a l u e of r e d u c e d m a g n e t i z a t i o n o\ T h e d a s h e d - a r r o w p a t h c o n n e c t i n g t h e S p e c t r a a t o b t o c a n d t h e n d o w n C o l u m n 1 shows a p o s s i b l e sequence of spectra i f t h e r e is a " c r i t i c a l s l o w i n g d o w n " of t h e ions spin

fluctuation

rate i n t h e t r a n s i t i o n r e g i o n as t h e t e m p e r a t u r e is

lowered. F i g u r e 4 shows t h e same c a l c u l a t i o n s as i n F i g u r e 3 except t h a t i n F i g u r e 4, t h e h y p e r f i n e i n t e r a c t i o n is t a k e n to be i s o t r o p i c . t w o i m p o r t a n t n e w features s h o w n i n this

figure.

T h e r e are

First, notice the t w o

u p p e r c u r v e s i n C o l u m n 1. T h e p a t t e r n a c t u a l l y n a r r o w s as t h e t e m p e r a t u r e is l o w e r e d b e l o w t h e t r a n s i t i o n t e m p e r a t u r e . T h e reason f o r this is e x p l a i n e d b y t h e second

i m p o r t a n t feature, seen i n C o l u m n 3, w h i c h

shows t h e spectra r e s u l t i n g f r o m t h e i s o t r o p i c h y p e r f i n e i n t e r a c t i o n w h e n

Relaxation Effects Associated with Magnetic Phase Transitions

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