EFG DETERMINATION IN MOHR SALT
[Fe(NH4)2(SO4)2·6H2O] APPLICATION OF THE
INTENSITY TENSOR TO THICK SINGLE
CRYSTALS
R. Zimmermann, R. Doerfler
To cite this version:
R. Zimmermann, R. Doerfler.
EFG DETERMINATION IN MOHR SALT
[Fe(NH4)2(SO4)2·6H2O] APPLICATION OF THE INTENSITY TENSOR TO THICK
SINGLE CRYSTALS. Journal de Physique Colloques, 1980, 41 (C1), pp.C1-107-C1-108.
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Colloque C1, supplkment au n " 1, Tome 41, janvier 1980, page C1-107
JOURNAL DE PHYSIQUE
EFG DETERMINATICN I N ElWR SALT [ F ~ ( N H ~ ) ~ ( s- o
APPLICATICN
~ o
OF M E INTENSITY TENSOR TO M I C K SINGLE
CRYSTALS
R. Zimmermann and R. Doerfler
PhysikaZisches Institub, 0-8520 ErZangen, W.-Germany.
1. I n t r o d u c t i o n
t e n s i t y t e n s o r . We find[5]
The e v a l u a t i o n of t h e angular dependence of
M6ssbauer l i n e i n t e n s i t i e s y i e l d s informat i o n on t h e e l e c t r i c f i e l d g r a d i e n t (EFG)
a t t h e nucleus. For 5 7 ~ quadrupolar
e
spect r a it i s r a t h e r s i m p l i f i e d by i n t r o d u c t i o n
of t h e i n t e n s i t y t e n s o r I ( p , q = x , y , z ) u , 2 1
P9
due t o t h e f a c t t h a t t h e e v a l u a t i o n of t h e
d a t a can be s e p a r a t e d i n t o two s t e p s . The
f i r s t s t e p i s c h a r a c t e r i z e d by f i t t i n g t h e
i n t e n s i t y t e n s o r t o t h e f r a c t i o n a l l i n e int e n s i t i e s I a s described by t h e formula
(no GKE)
(1
I=*
Ipqepeq
where ex,ey,ez a r e t h e d i r e c t i o n c o s i n e s of
t h e r r a y r e l a t i v e t o a c r y s t a l frame OxyZ.
The second s t e p i s t h e determination of t h e
p o s s i b l e EFG t e n s o r s . This s t e p i s s t r a i g h t
forward and r e a d i l y y i e l d s t h e manifold of
p o s s i b l e s o l u t i o n s f o r t h e EFG t e n s o r s (witho u t any f u r t h e r l e a s t squares analysis)b ,23
I n t h e p r e s e n t communication we want t o show
t h a t t h e s e p a r a t i o n of t h e e v a l u a t i o n i n t o
two s t e p s is s t i l l p o s s i b l e i n t h e case of
t h i c k absorbers.
where 1 %i s t h e t e n s o r of t h e c o f a c t o r s of
P9
1
t h e t r a c e l e s s i n t e n s i t y t e n s o r T =I --6
pq Pq 2 pq'
i . e . I $ ~ = Tyy T zz -v2yzl l&'~xzTyz-TzzTxy~ . *
Equations (1 ) ( 3 ) show t h a t t h e i n t e n s i t y
t e n s o r can be determined from t h e angular
dependenceof t h e absorption a r e a s . Usually
d i f f e r e n t absorbers w i l l be used f o r each
o r i e n t a t i o n . I n t h i s case t h e e f f e c t i v e
thickne,ss w i l l n o t be a simple f u n c t i o n of
t h e o r i e n t a t i o n . Therefore we might use t h e
following s e l f c o n s i s t e n t scheme f o r t h e d e termination of I
It i s based on t h e e x i pq'
s t e n c e of t h e i n v e r s e f u n c t i o n of S ( t I , a ) c3):
-
This expansioncanbeeasilydriventohigher
o r d e r . I f t h e p o l a r i s a t i o n were known t h e
dimensionless.area of t h e two quadrupolar
l i n e s , S1 und S2 would y i e l d t h e e f f e c t i v e
t h i c k n e s s and t h e f r a c t i o n a l i n t e n s i t y of
each l i n e
2. Thickness Correction and I n t e n s i t y T e n s o r
For a w e l l resolved absorption l i n e t h e d i mensionless alrea S (S=2A/ (fTlr) where A i s
t h e background c o r r e c t e d a r e a , f t h e r e c o i l l e s s f r a c t i o n of t h e source a n d f t h e natur a l l i n e width) i s given by [3]:
(2)
s=+ [ K ( ~ I ( I + ~ ) ) + K ( ~ I ~ I]I - ~ )
where t i s t h e e f f e c t i v e t h i c k n e s s of t h e
absorber, a i s t h e p o l a r i s a t i o n ( i - e . t h e
p o l a r i s a t i o n of t h e f - r a y , i f t h e absorber
were an e m i t t e r ) and K i s t h e s a t u r a t i o n
f u n c t i o n of t h e a r e a f o r a powder absorber
(a=O)[41. I n o r d e r t o c a l c u l a t e t h e a r e a S
we must express t h e p o l a r i s a t i o n by t h e in-
To s t a r t with we may assume t h a t t h e p o l a r i s a t i o n were zero. This y i e l d s t h e i n t e n s i t i e s I1 und T 2 , t h e angular dependence of
which determines t h e i n t e n s i t y t e n s o r v i a
equation ( 1 ) . With formula ( 3 ) we can c a l c u l a t e the polarisationof eachorientation
and i n s e r t them i n t o equ.(S) f o r a second
determination of t h e i n t e n s i t i e s Ik. We can
r e r u n t h i s scheme u n t i l s e l f consistency
i s achieved.
3. EFG Determination i n Mohr S a l t
Mohr s a l t , Fe (NH4) (SO4) 2' 6B20, has monoclin i c c r y s t a l symmetry. There a r e twoequival e n t l a t t i c e s i t e s per u n i t c e l l withsymme9
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980118
JOURNAL DE PHYSIQUE
C1-108
t r y r e l a t e d by r e f l e c t i o n i n ( 0 1 0 ) . Due t o
i s r e a s o n a b l e , i n view of t h e d - e l e c t r o n
the c r y s t a l symmetrytheintensity tensor
d e n s i t y of t h e g r o u n d s t a t e h a v i n g t h e shape
h a s one p r i n c i p a l a x i s a l o n g t h e twofold
of a d i s t o r t e d c i g a r r e . I t i s r a t h e r d i f f e -
a x i s b and i s hence c h a r a c t e r i z e d by i t s
r e n t from what has been found e a r l i e r [73 and
p r i n c i p a l components and one a n g l e d e s c r i -
confirms t h e n e c e s s i t y of t h i c k n e s s c o r r e c t i o n
bing t o rotationaroundtheb-axis. I n F i g . 1
. . .
1.00,
Fig.2: Asymmetry
. r
t h r e e quadrupole s p e c t r a a r e shown o u t o f a
parameter 9 a n d E u l e r
s e r i e s of 5. I t can be v i s u a l i z e d t h a t t h e
75
.
0.50
g
Fig.1: Quadrupolar spect r a of s i n g l e c r y s t a l s
100
l.l!lLl
t o\
1
V
0.25
0
The o r i e n t a t i o n of t h e
the
d,
c
axes system of t h e
5
0 2 0 LO 60 80
JI = orctan lVyz/VxZl
r of
l o c a l EFG r e l a t i v e
t o the principal
p--,J
180-y
of Mohr s a l t a t 300K.
a n g l e s a(, 8,
2-
0 w
$-ray i s a s i n d i c a t e d .
i n t e n s i t y tensor a t
300K a s a f u n c t i o n
of t h e parameter?Y
which d e s c r i b e s t h e
10101
090
-
1
0
1
2
manifold of p o s s i b l e s o l u t i o n s i n monocli-
3
n i c c r y s t a l s . The Euler a n g l e s a r e d e f i n e d
vsloclty lmd5l
e f f e c t of o r i e n t a t i o n i s r a t h e r l a r g e . The
e v a l u a t i o n of t h e specera according t o sec-
a s i n [ll The a n g l e s B and d a r e i d e n t i c a l
w i t h t h e p o l a r a n g l e s a a n d Qof t h e p r i n c i p a l component
t i o n 2 y i e l d s t h e f o l l o w i n g p r i n c i p a l com-
v2.(loc'.
ponents of t h e macroscopic i n t e n s i t y t e n s o r
( z p a r a l l e l b-axis
of t h e u n i t c e l l ) at300K
= 0.273
I z z = 0.536
YY
The e r r o r i s about 0.004. Note, however,
t h a t I x x + I y y + I Z z = 3 / 2a c c o r d i n g t o t h e o r y [I>
The r o t a t i o n of t h e x-axis r e l a t i v e t o t h e
c - a x i s and v e r s u s t h e a - a x i s of t h e u n i t
Ixx = 0.691
I
c e l l i s 34.50+12hose d a t a c h a r a c t e r i z e t h e
a n g u l a r dependence of t h e i n t e n s i t i e s uniquely and should always be quoted a s a d i -
Fig.3:
r e c t r e s u l t of t h e experiment.
system of t h e i n t e n s i t y t e n s o r r e l a t i v e t o
For n o n o c l i n i c c r y s t a l s t h e e v a l u a t i o n of
t h e s l i g h t l y d i s t o r t e d water octahedron i n
~ r i e n t a ' t i o nof t h e p r i n c i p a l axes
t h e i n t e n s i t y t e n s o r y i e l d s a l i n e a r mani-
Mohr s a l t . One p o s s i b l e o r i e n t a t i o n of
f o l d of p o s s i b l e s o l u t i o n s t h e range of
v
which i s f o r t u n a t e l y very narrow f o r Mohr
s a l t . For a l l s o l u t i o n s t h e s i g n o f t h e qua-
~ ~ i s (shown.
~
References
[I]
i n Fig.2.
shown
r i v e d from X-ray a n a l y s i s [6].
s o r ( = macrpscopic EFG-tensor)
of t h e i n t e n s i t y t e n s o r . H e n c e l i t i s d i r e c t e d between t h e water molecules. T h i s r e s u l t
and Meth.
Munck and R.
Zimmermann, Miissbauer
Gruverman and C.W.
(1976) Eds.
S e i d e l , Plenum
P r e s s , New York, London
b]
Housley, R.W.
R.M.
Phys.Rev.
[4]
has been
p l o t t e d t o g e t h e r w i t h t h e water octahedron.
According t o Fig.2 t h e p r i n c i p a l component.
Ck>
V 2 s of t h e l o c a l EFG i s c l o s e t o t h e y-axis
E.
I.J.
I n Fig. 3 t h e
p r i n c i p a l axes system of t h e i n t e n s i t y ten-
Zimmermann, N u c l . f n s t .
R.
E f f e c t Methodology, Vol.10
c h a r a c t e r i z e d by a c e r t a i n o r i e n t a t i o n of
t h e s l i g h t d i s t o r t e d w a t e r o c t a h e d r o n a s de-
)
L2]
Each p o s s i b l e s o l u t i o n i s a l s o
t h e EFG ( c f . F i g . 2 ) which can be r e l a t e d t o
~
128 (1975) 537
d r u p o l e s p l i t t i n g i s n e g a t i v e . Theasymmetry
parametervariesbetween 0.6and0.8as
~
178
Grant and U.
Gonser,
(1969) 514
G.A.
Bykov and P.Z.
JETP
16
Hien, S o v i e t Phys.
(1963) 646
151 R. Zimmermann, t o be published
la
H.
A.M.
Montgomery, R.V.
Chastain, J. J . N a t t ,
Witkowksa, and E.C.
Acta c r y s t .
22
Lingafelter,
(1967) 775
171 R. I n g a l l s , K. Ono, and L. Chandler,
Phys.Rev.
172
(1968) 295