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On the pressure dependence of the ferroelectric Curie temperatures of KH2PO4 and KD2PO4
Volume 26A. number 10
ON
THE
PHYSICS
PRESSURE DEPENDENCE TEMPERATURES OF
LETTERS
8 A p r i l 1968
OF THE FERROELECTRIC KH2PO 4 AND KD2PO 4
CURIE
R. BLINC and B. 7,EKS The University, Nuclear Institute J. Stefan, Ljubljana, Yugoslavia Received 26 February 1968
The dependence of the ferroelectric Curie temperatures (Tc) of KH2PO 4 and KD2PO 4 on hydrostatic p r e s sure has been evaluated for the protonic o r d e r - d i s o r d e r model in a four particle cluster approximation. The results agree rather well with the recent experimental data if the overlap of the protonic wave functions between the two sites in the H-bond is included, whereas the neglect of this overlap - as in the original Slater model - results in the wrong sign of the isotope effect in the TC p r e s s u r e coefficients.
U m e b a y a s h i et al. [1] h a v e r e c e n t l y shown that the f e r r o e l e c t r i c C u r i e t e m p e r a t u r e s (TC) of K H 2 P O 4 and K D 2 P O 4 d e c r e a s e w i t h i n c r e a s i n g h y d r o s t a t i c p r e s s u r e and S a m a r a [2] found a s i m i l a r d e c r e a s e of both T C and the C u r i e - W e i s s c o n s t a n t f o r a 90% d e u t e r a t e d K D 2 P O 4 c r y s t a l . T h e m o s t i n t e r e s t i n g r e s u l t of t h e s e s t u d i e s i s the d i s c o v e r y of a l a r g e i s o t o p e e f f e c t in t h e TC p r e s s u r e c o e f f i c i e n t s [1], w h i c h s e e m s to s u p p o r t the so-called proton tunneling model for ferroelect r i c i t y in c r y s t a l s of t h i s type. In o r d e r to s e e w h e t h e r t h i s i s r e a l l y the c a s e we d e c i d e d to e v a l u a t e the p r e s s u r e e f f e c t s f o r the " t u n n e l i n g " v e r s i o n [3] of t h e S l a t e r - T a k a g i S e n k o - U e h l i n g m o d e l as w e l l a s f o r the o r i g i n a l m o d e l [4] w h e r e the o v e r l a p of t h e p r o t o n i c w a v e f u n c t i o n s b e t w e e n the two s i t e s in the h y d r o g e n bond i s n e g l e c t e d . It can be shown [3,5] that the C u r i e t e m p e r a t u r e s a r e r e l a t i v e l y i n s e n s i t i v e to s m a l l c h a n g e s in the e n e r g y of a H 3 P O 4 o r a H 4 P O 4 d e f e c t , but d e p e n d s t r o n g l y on the S l a t e r s h o r t - r a n g e c o n f i g u r a t i o n e n e r g y E, the t u n n e l i n g m a t r i x e l e m e n t F and the l o n g - r a n g e d i p o l e - d i p o l e i n t e r a c t i o n e n e r g y y. T h u s : Tc(P) = T C [~(p), r ( P ) , ),(P)]. In e v a l u a t i n g t h e p r e s s u r e d e p e n d e n c e of t h e s e p a r a m e t e r s , we s h a l l a s s u m e that the p r i n c i p a l e f f e c t of p r e s s u r e is to r e d u c e the O - H - - O d i s t a n c e s s i n c e the h y d r o g e n bond i s the s o f t e s t bond in t h e c r y s t a l . A r e d u c t i o n of t h i s d i s t a n c e d e c r e a s e s t h e width 2~ of the p o t e n t i a l b a r r i e r , s e p a r a t i n g the two p r o t o n i c e q u i l i b r i u m s i t e s and s l i g h t l y increases the O-H distances. Neglecting this last e f f e c t one o b t a i n s
dTc
~(~Tc~
/~Tc\
1 da
w h e r e 77 = 2 q2 _ 1 and q2 = 2 r n E o { 2 / l ~ w i t h rn being the proton respectively deuteron mass, Eo the z e r o p o i n t e n e r g y and a = 1 - ~ 2~/a with t h e unit c e l l e d g e a = 7.45 A in c a s e of K H 2 P O 4 and a = 7.47 A in c a s e of K D 2 P O 4. In d e r i v i n g t h i s ex. p r e s s i o n E is t a k e n [3,5] to be p r o p o r t i o n a l to ~2, (as can be shown by a s i m p l e e l e c t r o s t a t i c c a l c u l a t i o n ) and F = Y o A, w h e r e A is a r e l a t i v e l y p r e s s u r e i n d e p e n d e n t r e n o r m a l i z a t i o n f a c t o r due to p r o t o n - l a t t i c e c o u p l o n g and w h e r e the " b a r g e " t u n n e l i n g m a t r i x e l e m e n t Fo is g i v e n in r e f . 6. The long-range interaction energy parameter c o n s i s t s of two p a r t s - a d i r e c t and a l a t t i c e m e d i a t e d p r o t o n - p r o t o n c o u p l i n g - both of w h i c h a r e p r o p e r t i o n a l to ~2 and y i e l d in the l i m i t [7] 2 { / a << 1 the s a m e p r e s s u r e d e p e n d e n c e . Ins e r t i n g the v a l u e s of the p a r a m e t e r s e, F, 7 u s e d in ref. 3 to d e s c r i b e the e q u i l i b r i u m d i e l e c t r i c p r o p e r t i e s of KI-I2PO 4 and K D 2 P O 4 - with L ~H = = 2.29 (as o b t a i n e d f r o m s t r u c t u r a l and s p e c t r o s c o p i c data) - and t a k i n g t h e v a l u e s of the d e r i v a t i v e s of TC f r o m the s a m e r e f e r e n c e and u s i n g d a / a dp = - 0.99 × 1 0 - 3 / k b a r a s o b t a i n e d [8] f r o m the m e a s u r e d e l e a s t i c s t i f f n e s s c o e f f i c i e n t s [9], one finds:
( d T c / d P ) H = [-1.81 - 2.07 - 0.082] d e g / k b a r = = - 4.70 d e g / k b a r
(2a)
( d T c / d P ) D = [-2.66 - 0 - 0.70] d e g / k b a r = = - 3.36 d e g / k b a r
468
7q! a
(2b)
Volume 26A, number 10
PHYSICS LETTERS
in s u r p r i s i n g l y good a g r e e m e n t with the e x p e r i m e n t a l data: ( d T c / d P ) H = - 4.52 d e g / k b a r and (dTC/dP)D = - 2.63 d e g / k b a r , a c c o r d i n g to ref. 1 and (dTc/dP) D ~ -3.9 d e g / k b a r a c c o r d i n g to ref. 2. If one, on the o t h e r hand, n e g l e c t s the t u n n e ling m a t r i x e l e m e n t F and fixed ~H and ED in the o r i g i n a l S l a t e r m o d el to account f o r the changes in the C u r i e t e m p e r a t u r e s on d e u t e r a t i o n (EH = = 60 c m -1, ED = 107 c m - 1 ) , one gets the wrong d i r e c t i o n of the i s o t op e e f f e c t s in the T C p r e s s u r e c o e f f i c i e n t s , s i n c e now (d Tc/dP) D > (d Tc/dP)H. T h i s s t a t e m e n t is s t i ll v a li d if one i n c l u d e s ~. The p r e s e n t r e s u l t s thus s e e m to show that the isotope e f f e c t s in the TC p r e s s u r e c o e f f i c i e n t s r e f l e c t the m a s s dependence of the p r o t o n i c o v e r lap. A m o r e d e t a i l e d account of this w o r k with the i nc l usi o n of the p r e s s u r e e f f e c t s in the C u r i e -
A FLIPPER-CHOPPER
FOR
8April 1968
W e i s s co n st an t s and the spontaneous p o l a r i z a tions will be published e l s e w h e r e .
References 1. H. Umebayashi, B.C. Frazer, G. Shirane and W. B. Daniels, Solid State Communications 5 (1967) 591. 2. G.A.Samara, Phys. Letters 25A (1967) 664. 3. R. Blinc and S. Svetina, Phys. Rev. 147 (1966) 430. 4. J.C.Slater, J.Chem. Phys. 9 (1941} 16: Y. Takagi, J. Phys. Soc. Japan 3 (1948} 271: M.E.Senko, Phys. Rev. 121 (1961) 1599; H. B. Silsbee, E.A. Uehling and V. H. Schmidt, Phys. Rev. 133 (1964) A165. 5. R. Blinc and B. ~ek~, Helv. Phys. Acta, to be published. 6. R. Blinc and D. Had~i, Mol. Phys. 1 (1958) 391. 7. G. E. Bacon and R. S. Pease, Proc. Roy. Soc. (London) A230 (1955} 359. 8. L.NovakoviS, private communication. 9. S. Hasussi~hl, Z. mr Kristal. 120 (1964) 401. B. Zwicker, Helv. Phys. Acta 19 (1946) 523.
POLARIZED
SLOW
NEUTRONS
O. STEINSVOLL and A. VIRJO *
Institutt for Atornenergi, Kjeller, Norway Received 1 March 1968
A conventional neutron spin resonance flipper for polarized slow neutrons has been rebuilt to allow pulsed (~flipper-chopper ~) operation. It is then possible to use time-of-flight technique to obtain the energy of the scattered neutrons. The construction and results of experimental tests are briefly discussed.
In a d i f f r a c t o m e t e r f o r p o l a r i z e d slow n e u t r o n s a ne ut r o n spin r e s o n a n c e flipping d e v i c e [1,2] is o r d i n a r i l y u s e d f o r r e v e r s i n g the d i r e c t i o n of the n e u t r o n p o l a r i z a t i o n v e c t o r with r e s p e c t to the m a g n e t i c g u i d e - f i e l d . When the p o l a r i z e d b e a m is r e f l e c t e d f r o m a c r y s t a l s a m p l e which has a s c a t t e r i n g c r o s s s e c t i o n ( e l a s t i c o r i n e l a s t i c ) depending on the p o l a r i z a t i o n d i r e c t i o n of the i n c o m i n g n e u t r o n b e a m , the i n t e n s it y in the s c a t t e r e d b e a m depends upon w h e t h e r the flipping d e v i c e is on o r off. It has b e e n s u g g e s t e d by Nathans [3] that this d e p e n d e n c e could be u t i l i z e d f o r m e a s u r i n g the e n e r g y s p e c t r u m of the p o l a r i z a t i o n dependent p a r t of the s c a t t e r e d n e u t r o n s f r o m the s a m p l e c r y s t a l by a t i m e - o f - f l i g h t p r o c e d u r e . F o r this p u r p o s e one should in s o m e way switch the f l i p p e r * Guest scientist on leave from Technical University of Helsinki, Otaniemi, Helsinki, Finland.
off o r on f o r sh o r t p e r i o d s of t i m e and r e p e a t t h i s switching at constant t i m e i n t e r v a l s . If the b u r s t s of r e v e r s e d n eu t r o n p o l a r i z a t i o n p r o d u c e d in this way have a p p r o p r i a t e p o l a r i t y , they a r e then c o n v e r t e d at the s a m p l e to b u r s t s of i n c r e a s ed n eu t r o n intensity that can be u s e d f o r t i m e o f - f l i g h t an al y si s. The m o s t obvious way to a c h i e v e this would be to switch the c u r r e n t in the f l i p p e r coil on and off by s o m e p u l si n g d ev i ce. " F l i p p e r - c h o p p e r s " b a s e d on this p r i n c i p l e have been s u c c e s s f u l l y t e s t e d in d i f f e r e n t l a b o r a t o r i e s [4-7]. F o r p r a c t i c a l r e a s o n s we have chosen, howe v e r , an o t h er method as a b a s i s f o r o u r f l i p p e r c h o p p e r c o n s t r u c t i o n , n a m e l y to change the m a i n (d. c.) m a g n e t i c f i el d a c r o s s the f l i p p e r coil at a constant r e p e t i t i o n r a t e . T h i s was effected by m e a n s of an e x t r a " c h o p p e r c o i l " wound around the f l i p p e r coil. When a sufficiently s t r o n g c u r 469
ON
THE
PHYSICS
PRESSURE DEPENDENCE TEMPERATURES OF
LETTERS
8 A p r i l 1968
OF THE FERROELECTRIC KH2PO 4 AND KD2PO 4
CURIE
R. BLINC and B. 7,EKS The University, Nuclear Institute J. Stefan, Ljubljana, Yugoslavia Received 26 February 1968
The dependence of the ferroelectric Curie temperatures (Tc) of KH2PO 4 and KD2PO 4 on hydrostatic p r e s sure has been evaluated for the protonic o r d e r - d i s o r d e r model in a four particle cluster approximation. The results agree rather well with the recent experimental data if the overlap of the protonic wave functions between the two sites in the H-bond is included, whereas the neglect of this overlap - as in the original Slater model - results in the wrong sign of the isotope effect in the TC p r e s s u r e coefficients.
U m e b a y a s h i et al. [1] h a v e r e c e n t l y shown that the f e r r o e l e c t r i c C u r i e t e m p e r a t u r e s (TC) of K H 2 P O 4 and K D 2 P O 4 d e c r e a s e w i t h i n c r e a s i n g h y d r o s t a t i c p r e s s u r e and S a m a r a [2] found a s i m i l a r d e c r e a s e of both T C and the C u r i e - W e i s s c o n s t a n t f o r a 90% d e u t e r a t e d K D 2 P O 4 c r y s t a l . T h e m o s t i n t e r e s t i n g r e s u l t of t h e s e s t u d i e s i s the d i s c o v e r y of a l a r g e i s o t o p e e f f e c t in t h e TC p r e s s u r e c o e f f i c i e n t s [1], w h i c h s e e m s to s u p p o r t the so-called proton tunneling model for ferroelect r i c i t y in c r y s t a l s of t h i s type. In o r d e r to s e e w h e t h e r t h i s i s r e a l l y the c a s e we d e c i d e d to e v a l u a t e the p r e s s u r e e f f e c t s f o r the " t u n n e l i n g " v e r s i o n [3] of t h e S l a t e r - T a k a g i S e n k o - U e h l i n g m o d e l as w e l l a s f o r the o r i g i n a l m o d e l [4] w h e r e the o v e r l a p of t h e p r o t o n i c w a v e f u n c t i o n s b e t w e e n the two s i t e s in the h y d r o g e n bond i s n e g l e c t e d . It can be shown [3,5] that the C u r i e t e m p e r a t u r e s a r e r e l a t i v e l y i n s e n s i t i v e to s m a l l c h a n g e s in the e n e r g y of a H 3 P O 4 o r a H 4 P O 4 d e f e c t , but d e p e n d s t r o n g l y on the S l a t e r s h o r t - r a n g e c o n f i g u r a t i o n e n e r g y E, the t u n n e l i n g m a t r i x e l e m e n t F and the l o n g - r a n g e d i p o l e - d i p o l e i n t e r a c t i o n e n e r g y y. T h u s : Tc(P) = T C [~(p), r ( P ) , ),(P)]. In e v a l u a t i n g t h e p r e s s u r e d e p e n d e n c e of t h e s e p a r a m e t e r s , we s h a l l a s s u m e that the p r i n c i p a l e f f e c t of p r e s s u r e is to r e d u c e the O - H - - O d i s t a n c e s s i n c e the h y d r o g e n bond i s the s o f t e s t bond in t h e c r y s t a l . A r e d u c t i o n of t h i s d i s t a n c e d e c r e a s e s t h e width 2~ of the p o t e n t i a l b a r r i e r , s e p a r a t i n g the two p r o t o n i c e q u i l i b r i u m s i t e s and s l i g h t l y increases the O-H distances. Neglecting this last e f f e c t one o b t a i n s
dTc
~(~Tc~
/~Tc\
1 da
w h e r e 77 = 2 q2 _ 1 and q2 = 2 r n E o { 2 / l ~ w i t h rn being the proton respectively deuteron mass, Eo the z e r o p o i n t e n e r g y and a = 1 - ~ 2~/a with t h e unit c e l l e d g e a = 7.45 A in c a s e of K H 2 P O 4 and a = 7.47 A in c a s e of K D 2 P O 4. In d e r i v i n g t h i s ex. p r e s s i o n E is t a k e n [3,5] to be p r o p o r t i o n a l to ~2, (as can be shown by a s i m p l e e l e c t r o s t a t i c c a l c u l a t i o n ) and F = Y o A, w h e r e A is a r e l a t i v e l y p r e s s u r e i n d e p e n d e n t r e n o r m a l i z a t i o n f a c t o r due to p r o t o n - l a t t i c e c o u p l o n g and w h e r e the " b a r g e " t u n n e l i n g m a t r i x e l e m e n t Fo is g i v e n in r e f . 6. The long-range interaction energy parameter c o n s i s t s of two p a r t s - a d i r e c t and a l a t t i c e m e d i a t e d p r o t o n - p r o t o n c o u p l i n g - both of w h i c h a r e p r o p e r t i o n a l to ~2 and y i e l d in the l i m i t [7] 2 { / a << 1 the s a m e p r e s s u r e d e p e n d e n c e . Ins e r t i n g the v a l u e s of the p a r a m e t e r s e, F, 7 u s e d in ref. 3 to d e s c r i b e the e q u i l i b r i u m d i e l e c t r i c p r o p e r t i e s of KI-I2PO 4 and K D 2 P O 4 - with L ~H = = 2.29 (as o b t a i n e d f r o m s t r u c t u r a l and s p e c t r o s c o p i c data) - and t a k i n g t h e v a l u e s of the d e r i v a t i v e s of TC f r o m the s a m e r e f e r e n c e and u s i n g d a / a dp = - 0.99 × 1 0 - 3 / k b a r a s o b t a i n e d [8] f r o m the m e a s u r e d e l e a s t i c s t i f f n e s s c o e f f i c i e n t s [9], one finds:
( d T c / d P ) H = [-1.81 - 2.07 - 0.082] d e g / k b a r = = - 4.70 d e g / k b a r
(2a)
( d T c / d P ) D = [-2.66 - 0 - 0.70] d e g / k b a r = = - 3.36 d e g / k b a r
468
7q! a
(2b)
Volume 26A, number 10
PHYSICS LETTERS
in s u r p r i s i n g l y good a g r e e m e n t with the e x p e r i m e n t a l data: ( d T c / d P ) H = - 4.52 d e g / k b a r and (dTC/dP)D = - 2.63 d e g / k b a r , a c c o r d i n g to ref. 1 and (dTc/dP) D ~ -3.9 d e g / k b a r a c c o r d i n g to ref. 2. If one, on the o t h e r hand, n e g l e c t s the t u n n e ling m a t r i x e l e m e n t F and fixed ~H and ED in the o r i g i n a l S l a t e r m o d el to account f o r the changes in the C u r i e t e m p e r a t u r e s on d e u t e r a t i o n (EH = = 60 c m -1, ED = 107 c m - 1 ) , one gets the wrong d i r e c t i o n of the i s o t op e e f f e c t s in the T C p r e s s u r e c o e f f i c i e n t s , s i n c e now (d Tc/dP) D > (d Tc/dP)H. T h i s s t a t e m e n t is s t i ll v a li d if one i n c l u d e s ~. The p r e s e n t r e s u l t s thus s e e m to show that the isotope e f f e c t s in the TC p r e s s u r e c o e f f i c i e n t s r e f l e c t the m a s s dependence of the p r o t o n i c o v e r lap. A m o r e d e t a i l e d account of this w o r k with the i nc l usi o n of the p r e s s u r e e f f e c t s in the C u r i e -
A FLIPPER-CHOPPER
FOR
8April 1968
W e i s s co n st an t s and the spontaneous p o l a r i z a tions will be published e l s e w h e r e .
References 1. H. Umebayashi, B.C. Frazer, G. Shirane and W. B. Daniels, Solid State Communications 5 (1967) 591. 2. G.A.Samara, Phys. Letters 25A (1967) 664. 3. R. Blinc and S. Svetina, Phys. Rev. 147 (1966) 430. 4. J.C.Slater, J.Chem. Phys. 9 (1941} 16: Y. Takagi, J. Phys. Soc. Japan 3 (1948} 271: M.E.Senko, Phys. Rev. 121 (1961) 1599; H. B. Silsbee, E.A. Uehling and V. H. Schmidt, Phys. Rev. 133 (1964) A165. 5. R. Blinc and B. ~ek~, Helv. Phys. Acta, to be published. 6. R. Blinc and D. Had~i, Mol. Phys. 1 (1958) 391. 7. G. E. Bacon and R. S. Pease, Proc. Roy. Soc. (London) A230 (1955} 359. 8. L.NovakoviS, private communication. 9. S. Hasussi~hl, Z. mr Kristal. 120 (1964) 401. B. Zwicker, Helv. Phys. Acta 19 (1946) 523.
POLARIZED
SLOW
NEUTRONS
O. STEINSVOLL and A. VIRJO *
Institutt for Atornenergi, Kjeller, Norway Received 1 March 1968
A conventional neutron spin resonance flipper for polarized slow neutrons has been rebuilt to allow pulsed (~flipper-chopper ~) operation. It is then possible to use time-of-flight technique to obtain the energy of the scattered neutrons. The construction and results of experimental tests are briefly discussed.
In a d i f f r a c t o m e t e r f o r p o l a r i z e d slow n e u t r o n s a ne ut r o n spin r e s o n a n c e flipping d e v i c e [1,2] is o r d i n a r i l y u s e d f o r r e v e r s i n g the d i r e c t i o n of the n e u t r o n p o l a r i z a t i o n v e c t o r with r e s p e c t to the m a g n e t i c g u i d e - f i e l d . When the p o l a r i z e d b e a m is r e f l e c t e d f r o m a c r y s t a l s a m p l e which has a s c a t t e r i n g c r o s s s e c t i o n ( e l a s t i c o r i n e l a s t i c ) depending on the p o l a r i z a t i o n d i r e c t i o n of the i n c o m i n g n e u t r o n b e a m , the i n t e n s it y in the s c a t t e r e d b e a m depends upon w h e t h e r the flipping d e v i c e is on o r off. It has b e e n s u g g e s t e d by Nathans [3] that this d e p e n d e n c e could be u t i l i z e d f o r m e a s u r i n g the e n e r g y s p e c t r u m of the p o l a r i z a t i o n dependent p a r t of the s c a t t e r e d n e u t r o n s f r o m the s a m p l e c r y s t a l by a t i m e - o f - f l i g h t p r o c e d u r e . F o r this p u r p o s e one should in s o m e way switch the f l i p p e r * Guest scientist on leave from Technical University of Helsinki, Otaniemi, Helsinki, Finland.
off o r on f o r sh o r t p e r i o d s of t i m e and r e p e a t t h i s switching at constant t i m e i n t e r v a l s . If the b u r s t s of r e v e r s e d n eu t r o n p o l a r i z a t i o n p r o d u c e d in this way have a p p r o p r i a t e p o l a r i t y , they a r e then c o n v e r t e d at the s a m p l e to b u r s t s of i n c r e a s ed n eu t r o n intensity that can be u s e d f o r t i m e o f - f l i g h t an al y si s. The m o s t obvious way to a c h i e v e this would be to switch the c u r r e n t in the f l i p p e r coil on and off by s o m e p u l si n g d ev i ce. " F l i p p e r - c h o p p e r s " b a s e d on this p r i n c i p l e have been s u c c e s s f u l l y t e s t e d in d i f f e r e n t l a b o r a t o r i e s [4-7]. F o r p r a c t i c a l r e a s o n s we have chosen, howe v e r , an o t h er method as a b a s i s f o r o u r f l i p p e r c h o p p e r c o n s t r u c t i o n , n a m e l y to change the m a i n (d. c.) m a g n e t i c f i el d a c r o s s the f l i p p e r coil at a constant r e p e t i t i o n r a t e . T h i s was effected by m e a n s of an e x t r a " c h o p p e r c o i l " wound around the f l i p p e r coil. When a sufficiently s t r o n g c u r 469