SOME NEW RESULTS IN MÖSSBAUER RELAXATION THEORY

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SOME NEW RESULTS IN MÖSSBAUER RELAXATION THEORY

F. Hartmann-Boutron

To cite this version:

F. Hartmann-Boutron. SOME NEW RESULTS IN MÖSSBAUER RELAXATION THEORY. Journal

de Physique Colloques, 1980, 41 (C1), pp.C1-223-C1-224. �10.1051/jphyscol:1980171�. �jpa-00219740�

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JOURNAL DE PHYSIQUE

Colloque C1, suppl6ment au

n O

1, Tome

41,

janvier 1980, page C1-223

SOME NEW RESULTS I N #ssBAUER RELAXATION THEORY

F. Hartmann-Boutron

Laboratoire de Spectrorne'trie Physique, B.P. 53 X, 38041 GRENOBLE-CEDEX (France).

A b s t r a c t . - W e d i s c u s s t h e v a l i d i t y o f t h e p e r t u r b a t i o n t r e a t m e n t s arid we d e r i v e some e x p r e s s i o n s for t h e Mejssbauer l i n e s h a p e b o t h w i t h o u t a n d w i t h t h e n o i s e a p p r o x i m a t i o n .

VALIDITY RANGE AND COMPARISON OF VARIOUS TREATMENTS n i n g o f a g i v e n Mossbauer l i n e become i d e n t i c a l a t The p r o b l e m o f r e l a x a t i o n e f f e c t s i n MBssbauer t h e t o p o f t h a t l i n e . I . e . , l i n e p r o f i l e s o b t a i n e d s p e c t r a h a s g i v e n r i s e t o s e v e r a l a p p r o a c h e s b a s e d w i t h b o t h m e t h o d s a r e v e r y s i m i l a r , a l t h o u g h c l i s on a l s t o c h a s t i c m o d e l s b l t h e r e l a x a t i o n e q u a t i o n

f o r t h e d e n s i t y m a t r i x c l t h e r e s o l v e n t method f o l - l o w i n g Fano. A l l t h r e e a p p r o a c h e s u s e L i o u v i l l e f o r - m a l i s m a n d l e a d t o r e l a x a t i o n s u p e r m a t r i c e s w h i c h e n t e r t h e MBssbauer l i n e s h a p e . Methods b l a n d c l h a v e b e e n compared i n g r e a t d e t a i l i n R e f

I

^{I )}^:^{w e}

summarize h e r e t h e c o n c l u s i o n s o f t h i s s t u d y . Both b l a n d c ) a r e p e r t u r b a t i o n m e t h o d s . I t c a n b e shown t h a t i f t h e " w h i t e n o i s e a p p r o x i m a t i o n " [WNAI i s v a l i d t h e y c a n d e s c r i b e t h e w h o l e r a n g e o f r e l a - x a t i o n e f f e c t s i n t h e Mossbauer s p e c t r a ( s m a l l , i n - t e r m e d i a t e o r l a r g e compared w i t h t h e d i s t a n c e b e t - ween l i n e s ) . On t h e c o n t r a r y i f t h e WNA i s n o t v a - l i d , t h e y a r e o n l y v a l i d f o r s m a l l r e l a x a t i o n e f - f e c t s [ a n d t h e n o n e c a n a p p l y t h e " s e c u l a r a p p r o x i - m a t i o n " ) . T h i s l a s t r e s u l t s o l v e s t h e a m b i g u i t i e s m e t i n p. 3 0 1 o f 121 ( n o t e ) , i n p. 307 o f 121 [ t o p o f f i r s t c o l u m n ) and i n p. 2117 o f 1 2 ) [ t h e s p u - r i o u s t h i r d o r d e r t e r m s a l w a y s c a n c e l o u t when t h e t h e o r y a p p l i e s ) .

Method b l l e a d s t o r e l a x a t i o n s u p e r m a t r i c e s R

r r

and S, w h i l e c ) l e a d s t o R ' ( o + i 7 ) a n d S'[w+i71.

When t h e WNA i s v a l i d R = R ' , S = S ' . When t h e WNA i s n o t v a l i d i t c a n be shown t h a t t h e m a t r i x e l e m e n t s o f R and R ' ( o r S a n d S ' l w h i c h c o n t r o l t h e b r o a d e -

s l i g h t l y b e t t e r .

EXPRESSIONS FOR THE MOSSBAUER LINESHAPE WHEN THE WNA I S NOT VALID

I/ L i n e s h a p e i n t h e i s o t r o p i c c a s e . I n R e f . ( 1

(

we u s e d t h e t e n s o r o p e r a t o r method i n o r d e r t o compute t h e M o s s b a u e r l i n e s h a p e f o r a n i o n w i t h a h y p e r f i n e

-+ + + +

c o u p l i n g A 1 . S i n t h e e x c i t e d s t a t e a n d A I .S i n g g

t h e g r o u n d s t s t e , when i t s e l e c t r o n i c s p i n S + i s s u b - m i t t e d t o a c o u p l i n g -gUBH.S w i t h a random f i e l d + +

Ff

whose f l u c t u a t i o n s a r e i s o t r o p i c a n d a r e c h a r a c t e r i - z e d by a c o r r e l a t i o n f u n c t i o n :

d t eiwt < H = [ o ) ~ = [ t ) > [ I ) [ t h i s i s a h i g h t e m p e r a t u r e c a l c u l a t i o n : kg T >>

A , Agl.

Here we w i l l n o t g i v e t h e g e n e r a l f o r m u l a s o b t a i - ned i n

I

(

b u t o n l y t h e i r a p p l i c a t i o n t o t h e Moss- b a u e r l i n e s h a p e o f [Yb 1 7 0 1 3 +

i n t h e e l e c t r o n i c dou- b l e t s t a t e Tg o r

r7

of c u b i c symmetry [ e f f e c t i v e e l e c t r o n i c s p i n S = 1 / 2 1 . ybl'O h a s a n e x c i t e d nu- c l e a r s t a t e I = 2 , whence two h y p e r f i n e s t a t e s F=5/2 w i t h E F =A a n d F=3/2 w i t h E =-3/2 A, a n d a g r o u n d F s t a t e I =O w i t h no h . f . s . L e t u s d e f i n e :

g

[ w h e r e I ( w l i s d e f i n e d a b o v e : Eq [ l ) J . T h e Moss- b a u e r l i n e s h a p e o f y b q 7 0 i s f o u n d t o b e ( p = r / 2 - i w ) :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980171

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-224 JOURNAL DE ^PHYSIQUE

A formula e q u i v a l e n t t o t h i s one has been d e r i v e d by Afanasev e t a 1 131. When t h e WNA i s v a l i d , Eq(21 reduces t o Eq(341 o f 141 w i t h X+Y = 1/TIS.

2/ Lineshape o f [Yb 17013+

i n u n i a x i a l symmetry. As- sume t h a t s t a t e I = 2 has a h y p e r f i n e s t r u c t u r e :

2 +

= A//IZSZ + a ( I Z - 2 1 w i t h S = l / Z , a n d t h a t S i s submitted t o a f l u c t u a t i n g f i e l d

:gl

⁼g//llBHZ(tl

+ +

X SZ + gLuBHl(tl .SL

.

m b e i n g t h e eigenvalue o f IZ, t h e MBssbauer l i n e s h a p e

3

( w l i s g i v e n by :

+ 2 2

Se C ^p⁺ i a [ m -2)/+ + 2~mL

2 2

m=-2

rp

⁺ia(m2-21/47 +

~ 2 '

⁺^Alp ^/4fi2 - (w?12

m I

r~

⁽³¹

w i t h WI =

-

^IL(cllw +i-)+- I,Iw-w-

2 1/2m -2 2 1/2,m + i ~ l

ILI

U) = (h:/n2~ l d t ewit < H + I ~ ) H - [ ~ I > / z 2

*w+1/2 ,m ⁼^fA//m/2 + afm - 2 )

When t h e WNA i s v a l i d = WL and t h i s e x p r e s s i o n reduces t o Eq(791 o f 15

I

WNA VALID : ANALOG FOR THE M.E. OF ABRAGAM-POUND COEFFICIENTS FOR PERTURBED ANGULAR CORRELATIONS

When r e l a x a t i o n i s v e r y f a s t t h e spectrum reduces t o a s i n g l e broadened l i n e

3

( w l a ~ e [I/(-iw+r/2+611.

We want t o compute 6.

I/ Paramagnetic case. Assume a Nassbauer t r a n s i t i o n w i t h m u l t i p o l a r i t y L and a f l u c t u a t i n g H a m i l t o n i a n

+

A I . S l t 1 ( o r A

7

. 3 ( t ) 1 w i t h c o r r e l a t i o n t i m e T

g g 1 S

( e l e c t r o n i c r e l a x a t i o n t i m e ] . Tensor o p e r a t o r methods l e a d t o :

5 7 As a check, when I = 3/2, I ₌1/2 and L = 1 ( F e l

g

we r e c o v e r t h e B r a d f o r d M a r s h a l l r e s u l t 16

1

f o r b/R

= b TIS (see b above Eq(3.14) o f 161) :

On t h e o t h e r hand, when A = Ag, I = I we r e c o v e r g

Eq ( 4 ) above i s o f i n t e r e s t f o r i n t e r p r e t i n g t h e l i n e broadenings observed i n z e r t a i n Rare E a r t h NSssbauer experiments ( 5 . 0 . Ounlap. t o be published)

.

2/ Quadrupole case. Here one assumes t h a t w i t h r e s - p e c t t o an a x i s 0 2 ' :

= eQ V z e 2 , [3 I

I:,

- 1(1+11]/ 4 1 ( 2 1-11 f o r t h e e x c i t e d s t a t e (and an analogous e x p r e s s i o n f o r t h e ground s t a t e 1 and t h a t Oz' r e o r i e n t s r a n - domly w i t h a c o r r e l a t i o n t i m e rc. Here t o o one uses t e n s o r o p e r a t o r s . The f i n a l r e s u l t i s I R I

v2

a1 ² aI ² I + L + I

6 = ~ r c [ ~ + - ~ - 2 a 1 a I 21 + I ( - 1 )

.% g g

where t h e { ) i s a 6 j c o e f f i c i e n t and:

When aI = , I = I we r e c o v e r t h e Abragam Pound g

r e s u l t f o r t h e damping c o e f f i c i e n t o f t h e m u l t i p o l e o f o r d e r L i n PAC ( E q ( V I 1 481 o f 121 w i t h k = L1.

I n the case o f 5 7 ~ e , which has a quadrupole e f - f e c t o n l y i n t h e e x c i t e d s t a t e I = 3/2 :

When one takes account o f t h e d i f f e r e n c e i n t h e no- t a t i o n s and one r e p l a c e s r c by I / X , t h i s r e s u l t i s i n agreement w i t h t h e l i m i t i n g l i n e w i d t h 9 Q ~ / X which can be d e r i v e d from D a t t a g u p t a and Blume (171 E q ( 4 4 ) ) i n t h e f a s t r e l a x a t i o n l i m i t .

11

I

HARTMANN-BOUTRON F. J. Phys ( F r )

40

(19791 57 121 HARTMANN-BOUTRON F. Ann. Phys. ( F r l

9

⁽¹⁹⁷⁵¹

285

131 AFANASEV A.M.

et

Sov. Phys. JETP - 47 (19781 585. Phys. Rev. L e t t .

40

(19781 816

141 CHOPIN C.

et

J. Phys. C o l l .

37

(19761 C6-73 ( C o r f u l

151 GONZALEZ-JIMENEZ F. e t a1 Phys. Rev.

89

⁽¹⁹⁷⁴⁾

95. See e r r a t u m f o r Eq(791

:

Phys. Rev 2134 161 BRADFORD E.

et

Proc. Phys. Soc.

87

⁽¹⁹⁶⁶¹

831

17

1

DATTAGUPTA S.

et

Phys. Rev.

810

(19741 4540 t h e PAC r e s u l t f o r t h e damping c o e f f i c i e n t o f t h e

m u l t i p o l e o f o r d e r L (Eq(VI1 421 o f 121 w i t h k = L1.