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On the divergent growth of molecular fluctuations in classical shear flow
Volume 33A. number 3
PHYSICS LETTERS
ZERO-FIELD
SPLITTING
OF
M n ++ IN L a 2 M g 3 ( N O 3 ) 1 2 "
19 October 1970
24H20
R. CHATTERJEE and D. Van ORMONDT
Department of Physics, The University of Calgary, Calgar~ 44, Alberta, Canada Received 16 September 1970
The zero-field splitting parameter D for Mn++ in La2Mg3(NO3)12" 24H20 ha~..l~en calculated from seven different mechanisms. The theoretical and experimental values agree f o r ' ~ e ' ~ , but not for the other.
A n d r i e s s e n et al. [1] c a l c u l a t e d the z e r o - f i e l d splitting p a r a m e t e r D of the s p i n - H a m i l t o n i a n t e r m
DIS2 -¼S(S+I) for the 6S ground state of Mn++ in lanthanum m a g n e s i u m n i t r a t e by taking into account the B l u m e - O r b a c h m e c h a n i s m and the P r y c e m e c h a n i s m . In this letter a calculation of the c o n t r i b u t i o n s of five m o r e p e r t u r b a t i o n m e c h a n i s m s , all based on the c r y s t a l l i n e e l e c t r i c field model, a r e given. In the double n i t r a t e c r y s t a l the m a n g a n e s e ions occupy two different t r i g o n a l l y d i s t o r t e d octahedral sites. The m a i n difference between the two s i t e s is in the degree of s y m m e t r y ; one site (I) p o s s e s s i n g i n v e r s i o n a l s y m m e t r y , while the other (II) does not. In both s i t e s the six n e a r e s t neighbours a r e water m o l e c u l e s . The coefficients A 0 and A 0 in the e x p r e s s i o n of the axial potential e n e r g y Hax = = A02r2Y2°( 0, 4~) + A°r4 Y°(O, ~), were calculated from lattice s u m s with the following charge d i s t r i bution: Mg: +2e, Ow: - e , H: +~e,1 La: 3e, N: +e, ON: - ~2e ( - e i s the e l e c t r o n charge). It was found that shifting the c h a r g e s of the c e n t r a l ions in the complexes [Mg(H20)6] ++ and [La(NO3)6]--- towards the ligands does not affect the lattice s u m s by m o r e than + 20%. The calculated p a r a m e t e r s a r e (site I)
A 0 (r2>= + 1834cm -1,
(siteI)
AO(r4)= - 3 9 2 c m -1 ,
(siteII)
A20 ( r 2 ) = -
(site II)
A 0 ( r 4 ) = + 136cm -1.
883cm -1 ,
The s p i n - o r b i t coupling p a r a m e t e r was chosen to be 300 cm-1. D is c a l c u l a t e d from the following seven perturbation mechanisms. (1) Wybourne relativistic mechanism [2]. This m e c h a n i s m was f i r s t taken into account by Wybourne in a second o r d e r p e r t u r b a t i o n calculation of the z e r o - f i e l d splitting of the 8S ground state of Gd +++. Van Heuvelen [3] d e r i v e d an e x p r e s s i o n for the case of Mn ++. F o r the double n i t r a t e this e x p r e s s i o n yields D(site I) = - 174 × 10-4 c m - 1 , D(site II) = + 88 x 10-4 cm-1. (2) Lulek mechanism [4]. Lulek d e r i v e d an e x p r e s s i o n for the s p i n - o r b i t coupling p e r t u r b e d by the c r y s t a l l i n e e l e c t r i c field in the form of i r r e d u c i b l e t e n s o r o p e r a t o r and calculated the z e r o - f i e l d s p l i t ting for a 4f7(8S) ion through a second o r d e r p e r t u r b a t i o n m e c h a n i s m . Following L u l e k ' s example, an e x p r e s s i o n for the p e r t u r b e d s p i n - o r b i t coupling ( ~ o ) was d e r i v e d independently in a s i m p l e o p e r a t o r form as given by 0
s.o
2
l+S++
The c o n t r i b u t i o n to D was obtained for 3d5(6S) ions by the method of the d e t e r m i n a n t s 3~/5 ~2 D = - 8~/-~ m 2 c 2
A0 [ E(4p)_E(6S)
147
Volume 33A. n u m b e r ',3
PttYS1CS
LETTERS
it) October 1970
w h e r e { i s t h e s p i n - o r b i t c o n s t a n t , a n d h, ,z a n d c h a v e t h e u s u a l m e a n i n g . R e s u l t s : i)(site I) = = - 2.9 x 10 - 4 c m - 1 , D ( s i t e II) = + 1.4 × 10 -4 c m -1, (3) Pryce mechanism [5]. T h e e x c i t e d c o n f i g u r a t i o n s 3 d 4 4 s a n d 3 d 4 4 d h a v e b e e n t a k e n i n t o a c c o u n t in t h i s s e c o n d o r d e r p e r t u r b a t i o n m e c h a n i s m . R e s u l t s : D ( s i t e I) = 19.0 × 10 -4 c m -1 , D ( s i t e II) = = -9.1 x 10-4 cm-1, (4) Blume-Orbach mechanis,~ [6]. T h e c o n t r i b u t i o n of A(~ i s m o r e i m p o r t a n t t h a n t h a t of A,~ in t h i s third order perturbation mechanism R e s u l t s : D ( s i t e I) = - 21.3 x 10 -4 e m - 1, D ( s i t e II) - + 7.0 × 10 -4 c m - 1 (5) Wvbourne mechanism [7], T h i s i s a t h i r d o r d e r p e r t u r b a t i o n m e c h a n i s m , f i r s t a p p l i e d by W y b o u r n e in t h e c a s e of l a n t h a n i d e s . It i s s c h e m a t i c a l l y r e p r e s e n t e d a s
'6SlHsoi4P 4p!Haxi 4D)~'4DjHspin_spini6S
.
R e s u l t s : D ( s i t e I) = - 3.1 x 10 -4 e r a - l , D ( s i t e II) = 1.5 × 1 0 - 4 c m - 1 . (6) Hutchison, Judd and Pope mechanism [8J. T h i s f o u r t h o r d e r m e c h a n i s m , represented as ~6Si H s o l 4 p ' < 4 p l H s o !
4D' 4Di
IIaxi 4 p ' 4 p i t t s o 1 6 8
which is schematically
,
h a s a l s o not b e e n a p p l i e d to M n ++ b e f o r e . R e s u l t s : D ( s i t e I) = - 10.1 × 10 -4 c m -1, l)(site II) = = 4.9 x 10 - 4 e m -1. (7) Walanabe mechanism [9]. T h i s i s a w e l l - k n o w n f o u r t h o r d e r p e r t u r b a t i o n m e c h a n i s m . R e s u l t s : D ( s i t e I ) = 1 . 9 x 1 0 - 4 c m -1, D ( s i t e II) < 1 x 1 0 - 4 c m - 1 . T h e c o m b i n e d c o n t r i b u t i o n s of t h e s e v e n m e c h a n i s m s a r e D ( s i t e I) = - 190.5 × 10 -4 c m -1 a n d D ( s t t e II) = +91.7 x 10-4era-1. A c c o r d i n g to t h e p r e s e n t r e s u l t s D i s a p p r o x i m a t e l y p r o p o r t i o n a l to A~, t h e ma~n c o n t r i b u t i o n o r i g inating from the relativistic mechanism. The contribution from Hutchison, Judd and Pope's mechanism i s t h e l a r g e s t of t h e t h r e e n e w ( f o r M n ++) m e c h a n i s m s i n t r o d u c e d in t h i s l e t t e r . T h e e x p e r i m e n t a l v a l u e s a r e [1] D ~ s i t e I ) = - 1 8 8 . 6 ± 0.4 × 1 0 - 4 c m -1 a n d D ( s i t e II) = +21.0 ~ 0 . 4 . 10 -4 c m -1 (at 296OK). F o r s i t e I t h e a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t i s s a t i s f a c t o r y . F o r s i t e II t h e m a g n i t u d e of t h e t h e o r e t i c a l v a l u e i s m u c h too h i g h , a l t h o u g h t h e s i g n i s c o r r e c t . No e x p l a n a t i o n f o r t h i s d i s c r e p a n c y h a s b e e n f o u n d y e t . H o w e v e r it w a s f o u n d e x p e r i m e n t a l l y t h a t a m e c h a n i s m w h i c h i s p r o p o r t i o n a l to A ° c a n n o t a l w a y s d o m i n a t e [10]: at 4.2OK t h e r a t i o of t h e q u a d r u p o l e c o u p l i n g c o n s t a n t s ( p r o p o r t i o n a l to A 0) of t h e m a n g a n e s e n u c l e i a t t h e two s i t e s in l a n t h a n u m m a g n e s i u m n i t r a t e i s 1.7, w h i l e t h e r a t i o of t h e D v a l u e s a t t h a t t e m p e r a t u r e i s 4.5. T h e t h e o r y m a y b e i m p r o v e d b y t a k i n g i n t o a c c o u n t odd o r d e r po~ t e n t i a l s t e r m s , w h i c h e x i s t a t s i t e II d u e to t h e a b s e n c e of i n v e r s i o n a l s y m m e t r y . T h e a u t h o r s w i s h to t h a n k P r o f e s s o r H. A. B u c k m a s t e r of t h i s D e p a r t m e n t f o r d i s c u s s i o n s a n d v a l u a b l e a d v i c e d u r i n g t h e p r o g r e s s of t h i s w o r k . One of t h e a u t h o r s , (D. V. O.) e x p r e s s e s h i s g r a t i t u d e to T h e U n i v e r s i t y of C a l g a r y f o r a w a r d i n g h i m a 1 9 6 9 - 7 0 p o s t - d o c t o r a l t e a c h i n g F e l l o w s h i p .
References [11 [21 [3] [4] [51 [6] [71 [~] [91 [10]
148
J . A n d r i e s s e n . G. De Jong and D. Van Ormondt, Phys. L e t t e r s 26A (1968) 617. B.G.Wybourne. J. Chem. Phys. 43 (1965) 4~506. A.Van Heuvelen. J. Chem. Phys. 46 (1967) 4903. T. Lulek. Phys. Stat. Sol. 39 (1970) K105. M . H . L . P r y c e . Phys, Rev. 80 (1950) 1107. R. R. Sharma. T. P, Das and R . O r b a e h . Phys. Rev. 149 (1966) 257. B.G.W.vbourne. Phys. Rev. 148 (1966)317. C.A.Hutchison. B.R. Judd and D.F. Pope. Proe. Phys. Soe. (I, ondon) B70 t1957) 51 ~. H. Watanabe, P r o g r . Theor. Phys. 18 11957} 405. R. de B e e r and D. wm Ormondt. Phys. L e t t e r s 27A (1968} i75.
PHYSICS LETTERS
ZERO-FIELD
SPLITTING
OF
M n ++ IN L a 2 M g 3 ( N O 3 ) 1 2 "
19 October 1970
24H20
R. CHATTERJEE and D. Van ORMONDT
Department of Physics, The University of Calgary, Calgar~ 44, Alberta, Canada Received 16 September 1970
The zero-field splitting parameter D for Mn++ in La2Mg3(NO3)12" 24H20 ha~..l~en calculated from seven different mechanisms. The theoretical and experimental values agree f o r ' ~ e ' ~ , but not for the other.
A n d r i e s s e n et al. [1] c a l c u l a t e d the z e r o - f i e l d splitting p a r a m e t e r D of the s p i n - H a m i l t o n i a n t e r m
DIS2 -¼S(S+I) for the 6S ground state of Mn++ in lanthanum m a g n e s i u m n i t r a t e by taking into account the B l u m e - O r b a c h m e c h a n i s m and the P r y c e m e c h a n i s m . In this letter a calculation of the c o n t r i b u t i o n s of five m o r e p e r t u r b a t i o n m e c h a n i s m s , all based on the c r y s t a l l i n e e l e c t r i c field model, a r e given. In the double n i t r a t e c r y s t a l the m a n g a n e s e ions occupy two different t r i g o n a l l y d i s t o r t e d octahedral sites. The m a i n difference between the two s i t e s is in the degree of s y m m e t r y ; one site (I) p o s s e s s i n g i n v e r s i o n a l s y m m e t r y , while the other (II) does not. In both s i t e s the six n e a r e s t neighbours a r e water m o l e c u l e s . The coefficients A 0 and A 0 in the e x p r e s s i o n of the axial potential e n e r g y Hax = = A02r2Y2°( 0, 4~) + A°r4 Y°(O, ~), were calculated from lattice s u m s with the following charge d i s t r i bution: Mg: +2e, Ow: - e , H: +~e,1 La: 3e, N: +e, ON: - ~2e ( - e i s the e l e c t r o n charge). It was found that shifting the c h a r g e s of the c e n t r a l ions in the complexes [Mg(H20)6] ++ and [La(NO3)6]--- towards the ligands does not affect the lattice s u m s by m o r e than + 20%. The calculated p a r a m e t e r s a r e (site I)
A 0 (r2>= + 1834cm -1,
(siteI)
AO(r4)= - 3 9 2 c m -1 ,
(siteII)
A20 ( r 2 ) = -
(site II)
A 0 ( r 4 ) = + 136cm -1.
883cm -1 ,
The s p i n - o r b i t coupling p a r a m e t e r was chosen to be 300 cm-1. D is c a l c u l a t e d from the following seven perturbation mechanisms. (1) Wybourne relativistic mechanism [2]. This m e c h a n i s m was f i r s t taken into account by Wybourne in a second o r d e r p e r t u r b a t i o n calculation of the z e r o - f i e l d splitting of the 8S ground state of Gd +++. Van Heuvelen [3] d e r i v e d an e x p r e s s i o n for the case of Mn ++. F o r the double n i t r a t e this e x p r e s s i o n yields D(site I) = - 174 × 10-4 c m - 1 , D(site II) = + 88 x 10-4 cm-1. (2) Lulek mechanism [4]. Lulek d e r i v e d an e x p r e s s i o n for the s p i n - o r b i t coupling p e r t u r b e d by the c r y s t a l l i n e e l e c t r i c field in the form of i r r e d u c i b l e t e n s o r o p e r a t o r and calculated the z e r o - f i e l d s p l i t ting for a 4f7(8S) ion through a second o r d e r p e r t u r b a t i o n m e c h a n i s m . Following L u l e k ' s example, an e x p r e s s i o n for the p e r t u r b e d s p i n - o r b i t coupling ( ~ o ) was d e r i v e d independently in a s i m p l e o p e r a t o r form as given by 0
s.o
2
l+S++
The c o n t r i b u t i o n to D was obtained for 3d5(6S) ions by the method of the d e t e r m i n a n t s 3~/5 ~2 D = - 8~/-~ m 2 c 2
A0 [ E(4p)_E(6S)
147
Volume 33A. n u m b e r ',3
PttYS1CS
LETTERS
it) October 1970
w h e r e { i s t h e s p i n - o r b i t c o n s t a n t , a n d h, ,z a n d c h a v e t h e u s u a l m e a n i n g . R e s u l t s : i)(site I) = = - 2.9 x 10 - 4 c m - 1 , D ( s i t e II) = + 1.4 × 10 -4 c m -1, (3) Pryce mechanism [5]. T h e e x c i t e d c o n f i g u r a t i o n s 3 d 4 4 s a n d 3 d 4 4 d h a v e b e e n t a k e n i n t o a c c o u n t in t h i s s e c o n d o r d e r p e r t u r b a t i o n m e c h a n i s m . R e s u l t s : D ( s i t e I) = 19.0 × 10 -4 c m -1 , D ( s i t e II) = = -9.1 x 10-4 cm-1, (4) Blume-Orbach mechanis,~ [6]. T h e c o n t r i b u t i o n of A(~ i s m o r e i m p o r t a n t t h a n t h a t of A,~ in t h i s third order perturbation mechanism R e s u l t s : D ( s i t e I) = - 21.3 x 10 -4 e m - 1, D ( s i t e II) - + 7.0 × 10 -4 c m - 1 (5) Wvbourne mechanism [7], T h i s i s a t h i r d o r d e r p e r t u r b a t i o n m e c h a n i s m , f i r s t a p p l i e d by W y b o u r n e in t h e c a s e of l a n t h a n i d e s . It i s s c h e m a t i c a l l y r e p r e s e n t e d a s
'6SlHsoi4P 4p!Haxi 4D)~'4DjHspin_spini6S
.
R e s u l t s : D ( s i t e I) = - 3.1 x 10 -4 e r a - l , D ( s i t e II) = 1.5 × 1 0 - 4 c m - 1 . (6) Hutchison, Judd and Pope mechanism [8J. T h i s f o u r t h o r d e r m e c h a n i s m , represented as ~6Si H s o l 4 p ' < 4 p l H s o !
4D' 4Di
IIaxi 4 p ' 4 p i t t s o 1 6 8
which is schematically
,
h a s a l s o not b e e n a p p l i e d to M n ++ b e f o r e . R e s u l t s : D ( s i t e I) = - 10.1 × 10 -4 c m -1, l)(site II) = = 4.9 x 10 - 4 e m -1. (7) Walanabe mechanism [9]. T h i s i s a w e l l - k n o w n f o u r t h o r d e r p e r t u r b a t i o n m e c h a n i s m . R e s u l t s : D ( s i t e I ) = 1 . 9 x 1 0 - 4 c m -1, D ( s i t e II) < 1 x 1 0 - 4 c m - 1 . T h e c o m b i n e d c o n t r i b u t i o n s of t h e s e v e n m e c h a n i s m s a r e D ( s i t e I) = - 190.5 × 10 -4 c m -1 a n d D ( s t t e II) = +91.7 x 10-4era-1. A c c o r d i n g to t h e p r e s e n t r e s u l t s D i s a p p r o x i m a t e l y p r o p o r t i o n a l to A~, t h e ma~n c o n t r i b u t i o n o r i g inating from the relativistic mechanism. The contribution from Hutchison, Judd and Pope's mechanism i s t h e l a r g e s t of t h e t h r e e n e w ( f o r M n ++) m e c h a n i s m s i n t r o d u c e d in t h i s l e t t e r . T h e e x p e r i m e n t a l v a l u e s a r e [1] D ~ s i t e I ) = - 1 8 8 . 6 ± 0.4 × 1 0 - 4 c m -1 a n d D ( s i t e II) = +21.0 ~ 0 . 4 . 10 -4 c m -1 (at 296OK). F o r s i t e I t h e a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t i s s a t i s f a c t o r y . F o r s i t e II t h e m a g n i t u d e of t h e t h e o r e t i c a l v a l u e i s m u c h too h i g h , a l t h o u g h t h e s i g n i s c o r r e c t . No e x p l a n a t i o n f o r t h i s d i s c r e p a n c y h a s b e e n f o u n d y e t . H o w e v e r it w a s f o u n d e x p e r i m e n t a l l y t h a t a m e c h a n i s m w h i c h i s p r o p o r t i o n a l to A ° c a n n o t a l w a y s d o m i n a t e [10]: at 4.2OK t h e r a t i o of t h e q u a d r u p o l e c o u p l i n g c o n s t a n t s ( p r o p o r t i o n a l to A 0) of t h e m a n g a n e s e n u c l e i a t t h e two s i t e s in l a n t h a n u m m a g n e s i u m n i t r a t e i s 1.7, w h i l e t h e r a t i o of t h e D v a l u e s a t t h a t t e m p e r a t u r e i s 4.5. T h e t h e o r y m a y b e i m p r o v e d b y t a k i n g i n t o a c c o u n t odd o r d e r po~ t e n t i a l s t e r m s , w h i c h e x i s t a t s i t e II d u e to t h e a b s e n c e of i n v e r s i o n a l s y m m e t r y . T h e a u t h o r s w i s h to t h a n k P r o f e s s o r H. A. B u c k m a s t e r of t h i s D e p a r t m e n t f o r d i s c u s s i o n s a n d v a l u a b l e a d v i c e d u r i n g t h e p r o g r e s s of t h i s w o r k . One of t h e a u t h o r s , (D. V. O.) e x p r e s s e s h i s g r a t i t u d e to T h e U n i v e r s i t y of C a l g a r y f o r a w a r d i n g h i m a 1 9 6 9 - 7 0 p o s t - d o c t o r a l t e a c h i n g F e l l o w s h i p .
References [11 [21 [3] [4] [51 [6] [71 [~] [91 [10]
148
J . A n d r i e s s e n . G. De Jong and D. Van Ormondt, Phys. L e t t e r s 26A (1968) 617. B.G.Wybourne. J. Chem. Phys. 43 (1965) 4~506. A.Van Heuvelen. J. Chem. Phys. 46 (1967) 4903. T. Lulek. Phys. Stat. Sol. 39 (1970) K105. M . H . L . P r y c e . Phys, Rev. 80 (1950) 1107. R. R. Sharma. T. P, Das and R . O r b a e h . Phys. Rev. 149 (1966) 257. B.G.W.vbourne. Phys. Rev. 148 (1966)317. C.A.Hutchison. B.R. Judd and D.F. Pope. Proe. Phys. Soe. (I, ondon) B70 t1957) 51 ~. H. Watanabe, P r o g r . Theor. Phys. 18 11957} 405. R. de B e e r and D. wm Ormondt. Phys. L e t t e r s 27A (1968} i75.