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Dynamical aspects of the phase transition in KH2PO4-type ferroelectric crystals
Volume 26A, number 1
P HYS I C S L E T T E R S
Table i.
-H(T = 0) ; eV/molecule. Li + Na + K+ Rb +
Cs +
He-
2.56
2.45
2.43
2.26
2.22
Ne-
1.01
0.86
1.08
0.98
0.98
obtained: H e - , 1.514; one-, 1.834; A - , 2.264; K r - , 2.42A; X e ' , 2.62A, we can obtain the p r e dicted e n t h a l p i e s of f o r m a t i o n f r o m eqs. (1) and (2), if tabulated v a l u e s [6] of the o t h e r p a r a m e t e r s a r e used (except the s m a l l z e r o point "energy which we neglect) and the NaC1 s t r u c t u r e a s s u m e d . T he a r g i d e s , k r y p t i d e s and x e n i d e s a r e probably van d e r Waals (non-ionic) c r y s t a l s , s in c e t h e i r ionic binding i s p r e d i c t e d a s s m a l l o r n e g a ti v e , which is the r e a s o n f o r t h e i r e x c l u s i o n f r o m the table. The u n c e r t a i n t y in o u r p r e d i c t i o n s l a r g e l y r e s i d e s in the l a t t i c e p a r a m e t e r e s t i m a t e . It is h a r d to s ee how t h i s can be wrong by as much as ten p e r c e n t . F i n a l l y , one ought to s u g g e s t a r e a s o n why such compounds s e e m not yet to have been obs e r v e d **. It is c l e a r that the ionic potential c u r v e s f o r the r e a c t i n g s p e c i e s do not c r o s s the r e p u l s i v e p o r t i o n of the van d e r Waals atomic c u r v e s , ( s t r i c t l y , of c o u r s e , c u r v e s of identical
4 December 1967
s y m m e t r y must n e v e r c r o s s ) until r e l a t i v e l y s m a l l v a l u e s of i n t e r n u c l e a r d i s t a n c e a r e r e a c h e d , so that t h e r e may be a b a r r i e r a g a i n s t r e a c t i o n of the i s o l a t e d a t o m s . It t h e r e f o r e a p p e a r s that s y n t h e s i s at high p r e s s u r e m ay be n e c e s s a r y to y i el d the m a t e r i a l s we p r e d i c t . Such s a l t s should be a n t i - f e r r o m a g n e t i c at low t e m p e r a t u r e (with l o w e r l a t t i c e s y m m e t r y ) and should have band ed g es c o r r e s p o n d i n g to a t o m i c a l l y forbidden ( s - s ) t r a n s i t i o n s in the v i s i b l e .
References 1. M.P. Tosi, Solid state physics 16, (Academic Press, New York, London, 1964) p. 1. 2. B. L. Moisewitch, Advances in atomic and molecular physics 1, (Academic Press, New York, London, 1965) p.61. 3. W.H. Zachariasen, reported by C. Kittel, Introduction to solid state physics, Table 3.5, (Wiley, New York, 1956). 4. E.R. Dobbs and G.O. Jones, Repts. Prog. in Phys.20 (1957) 516. 5. F.S.Ham, Phys. Rev. 128 (1962)82, 2524. Alkali metal lattice parameters from Internation tables of X-ray crystallography, Vol. III, (Kynoch Press, Birmingham, 1962) 278. 6. American Institute of Physics Handbook, 2nd ed., 7-13; 7-139. (MacGraw-Hill, New York, 1963). 7. G. Thomas et L. Herman, Compt. Rend. 229 (1949) 1313.
DYNAMICAL ASPECTS OF THE PHASE IN KH2PO4-TYPE FERROELECTRIC
TRANSITION CRYSTALS
K. KOBAYASHI
Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan Received 28 October 1967
Dynamical aspects of the ferroelectric phase transition in KH2PO4-type crystals are elucidated on the basis of the coupling between the proton tunneling mode and the optical mode vibration of the [K-PO4] complexes along the c-axis.
We p r o p o s e a new m o d e l f o r the f e r r o e l e c t r i c pha se t r a n s i t i o n of K H 2 P O 4 - t y p e c r y s t a l s . The p r o t o n tunneling mode is found to be s t r o n g l y coupled with the o p ti c a l mode of the [K-PO4] l a t t i c e v i b r a t i o n s along the c - a x i s v i a e l e c t r o static Coulomb i n t e r a c t i o n . The to ta l H a m i lt o n ia n of the s y s t e m is of the f o r m : H = H p + H L + H , L. Hp stands f o r the H a m i l t o n i a n of the p r o t o n ~ s tern which is e x p r e s s e d as: [1]
w h e r e QT is the tunneling f r e q u e n c y , J i ~ d e s ,d c r i b e s a d i p o l e - d i p o l e i n t e r a c t i o n b et w een p r o t o n s and Zi, Xi i s the z and x component of the I si n g spin a s s o c i a t e d with the i - t h B r a v a i s l a t t i c e s i t e of hydrogen bonds. H L d e s c r i b e s the l a t t i c e v i b r a t i o n , p a r t i c u l a r l y , the p o l a r mode v i b r a t i o n of [K-PO4] c o m p l e x e s along the c - a x i s . 55
Volume 26A, number 1
PHYSICS LETTERS
4 December 1967
(5)
%
5K
Hp, L r e p r e s e n t s the i n t e r a c t i o n between the p o l a r lattice v i b r a t i o n of [K-PO4] complexes and the p r o t o n - t u n n e l i n g motion and has the f o r m
= - ~q FqQqZ_q; Fq ~F= const. , for g ~ 0
B- site
A~ site
Hp, L
(3)
The two coupled modes of the s y s t e m a r e obtained within the f r a m e w o r k of a l i n e a r theory. ~+
=~
+
0
+
± ~(w2{k)_ ~2)2 + 16NF2122 I W
(4)
5K Tc
w~(k) d e s c r i b e s the p r o t o n - t u n n e l i n g collective P motion as obtained by de Genn~s, Tokunaga et al. [1] W= [(2~2T)2 + (J)2]~ and< So> is the t h e r m a l average of the Ising spin. F o r J > 0, eq. (4) a t t a i n s a m i n i m u m at k = 0 so that only the k - 0 mode i s c o n s i d e r e d , h e r e a f t e r , w_ tends to zero for T -~ Tc as W_ = B ( T - T c ) 2 , Tc : T o +
Fig. 1. Ferroelectric mode (cO_).
(5)
where To is the C u r i e t e m p e r a t u r e of the proton s y s t e m and has the form: kBT o = ~2T a r t a n h (4 ~2T/J) , by which the large isotope effect in Tc can be accounted for s a t i s f a c o r i l y [2]. w_ r e p r e s e n t s the proton tunneling "like" mode in which both s y s t e m s o s c i l l a t e in phase (fig. 1) and is r e s p o n s i b l e for the f e r r o e l e c t r i c phase t r a n s i tion. On the other hand, the w+ mode r e m a i n s a l m o s t unchanged for T ~ Tc. I.e. w2+ ~ 2 The i n c l u s i o n of the r e l a x a t i o n effect shows t~at in low frequency region, the d i e l e c t r i c d i s p e r s i o n is of the Debye type in a g r e e m e n t with H i l l I c h i k i ' s e x p e r i m e n t a l r e s u l t [3].
56
o
C
E(w) = ( T _ Tc) - i w T A
(6)
It can be concluded that the KH2PO4-type c r y s t a l s belong to the "mixed" type f e r r o e l e c t r i c s , that is to say, an o r d e r - d i s o r d e r a r r a n g e m e n t of p r o t o n s and an i n s t a b i l i t y of the lattice~vibration along c - a x i s o c c u r s i m u l t a n e o u s l y . Detailed d i s c u s s i o n will be published in the n e a r future.
References 1. P. G. de Gennes, Solid State Commun. 1 (1963) 132; M. Tokunaga, Progr. Theor. Phys. 36 (1966} 857. 2. R. Blinc, J. Phys. Chem. Solids 13 (1960} 204. 3. R.M. Hill and S. K. Ichiki, Phys. Rev. 130 (1962) 150.
P HYS I C S L E T T E R S
Table i.
-H(T = 0) ; eV/molecule. Li + Na + K+ Rb +
Cs +
He-
2.56
2.45
2.43
2.26
2.22
Ne-
1.01
0.86
1.08
0.98
0.98
obtained: H e - , 1.514; one-, 1.834; A - , 2.264; K r - , 2.42A; X e ' , 2.62A, we can obtain the p r e dicted e n t h a l p i e s of f o r m a t i o n f r o m eqs. (1) and (2), if tabulated v a l u e s [6] of the o t h e r p a r a m e t e r s a r e used (except the s m a l l z e r o point "energy which we neglect) and the NaC1 s t r u c t u r e a s s u m e d . T he a r g i d e s , k r y p t i d e s and x e n i d e s a r e probably van d e r Waals (non-ionic) c r y s t a l s , s in c e t h e i r ionic binding i s p r e d i c t e d a s s m a l l o r n e g a ti v e , which is the r e a s o n f o r t h e i r e x c l u s i o n f r o m the table. The u n c e r t a i n t y in o u r p r e d i c t i o n s l a r g e l y r e s i d e s in the l a t t i c e p a r a m e t e r e s t i m a t e . It is h a r d to s ee how t h i s can be wrong by as much as ten p e r c e n t . F i n a l l y , one ought to s u g g e s t a r e a s o n why such compounds s e e m not yet to have been obs e r v e d **. It is c l e a r that the ionic potential c u r v e s f o r the r e a c t i n g s p e c i e s do not c r o s s the r e p u l s i v e p o r t i o n of the van d e r Waals atomic c u r v e s , ( s t r i c t l y , of c o u r s e , c u r v e s of identical
4 December 1967
s y m m e t r y must n e v e r c r o s s ) until r e l a t i v e l y s m a l l v a l u e s of i n t e r n u c l e a r d i s t a n c e a r e r e a c h e d , so that t h e r e may be a b a r r i e r a g a i n s t r e a c t i o n of the i s o l a t e d a t o m s . It t h e r e f o r e a p p e a r s that s y n t h e s i s at high p r e s s u r e m ay be n e c e s s a r y to y i el d the m a t e r i a l s we p r e d i c t . Such s a l t s should be a n t i - f e r r o m a g n e t i c at low t e m p e r a t u r e (with l o w e r l a t t i c e s y m m e t r y ) and should have band ed g es c o r r e s p o n d i n g to a t o m i c a l l y forbidden ( s - s ) t r a n s i t i o n s in the v i s i b l e .
References 1. M.P. Tosi, Solid state physics 16, (Academic Press, New York, London, 1964) p. 1. 2. B. L. Moisewitch, Advances in atomic and molecular physics 1, (Academic Press, New York, London, 1965) p.61. 3. W.H. Zachariasen, reported by C. Kittel, Introduction to solid state physics, Table 3.5, (Wiley, New York, 1956). 4. E.R. Dobbs and G.O. Jones, Repts. Prog. in Phys.20 (1957) 516. 5. F.S.Ham, Phys. Rev. 128 (1962)82, 2524. Alkali metal lattice parameters from Internation tables of X-ray crystallography, Vol. III, (Kynoch Press, Birmingham, 1962) 278. 6. American Institute of Physics Handbook, 2nd ed., 7-13; 7-139. (MacGraw-Hill, New York, 1963). 7. G. Thomas et L. Herman, Compt. Rend. 229 (1949) 1313.
DYNAMICAL ASPECTS OF THE PHASE IN KH2PO4-TYPE FERROELECTRIC
TRANSITION CRYSTALS
K. KOBAYASHI
Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan Received 28 October 1967
Dynamical aspects of the ferroelectric phase transition in KH2PO4-type crystals are elucidated on the basis of the coupling between the proton tunneling mode and the optical mode vibration of the [K-PO4] complexes along the c-axis.
We p r o p o s e a new m o d e l f o r the f e r r o e l e c t r i c pha se t r a n s i t i o n of K H 2 P O 4 - t y p e c r y s t a l s . The p r o t o n tunneling mode is found to be s t r o n g l y coupled with the o p ti c a l mode of the [K-PO4] l a t t i c e v i b r a t i o n s along the c - a x i s v i a e l e c t r o static Coulomb i n t e r a c t i o n . The to ta l H a m i lt o n ia n of the s y s t e m is of the f o r m : H = H p + H L + H , L. Hp stands f o r the H a m i l t o n i a n of the p r o t o n ~ s tern which is e x p r e s s e d as: [1]
w h e r e QT is the tunneling f r e q u e n c y , J i ~ d e s ,d c r i b e s a d i p o l e - d i p o l e i n t e r a c t i o n b et w een p r o t o n s and Zi, Xi i s the z and x component of the I si n g spin a s s o c i a t e d with the i - t h B r a v a i s l a t t i c e s i t e of hydrogen bonds. H L d e s c r i b e s the l a t t i c e v i b r a t i o n , p a r t i c u l a r l y , the p o l a r mode v i b r a t i o n of [K-PO4] c o m p l e x e s along the c - a x i s . 55
Volume 26A, number 1
PHYSICS LETTERS
4 December 1967
(5)
%
5K
Hp, L r e p r e s e n t s the i n t e r a c t i o n between the p o l a r lattice v i b r a t i o n of [K-PO4] complexes and the p r o t o n - t u n n e l i n g motion and has the f o r m
= - ~q FqQqZ_q; Fq ~F= const. , for g ~ 0
B- site
A~ site
Hp, L
(3)
The two coupled modes of the s y s t e m a r e obtained within the f r a m e w o r k of a l i n e a r theory. ~+
=~
+
0
+
± ~(w2{k)_ ~2)2 + 16NF2122
(4)
5K Tc
w~(k) d e s c r i b e s the p r o t o n - t u n n e l i n g collective P motion as obtained by de Genn~s, Tokunaga et al. [1] W= [(2~2T)2 + (J
Fig. 1. Ferroelectric mode (cO_).
(5)
where To is the C u r i e t e m p e r a t u r e of the proton s y s t e m and has the form: kBT o = ~2T a r t a n h (4 ~2T/J) , by which the large isotope effect in Tc can be accounted for s a t i s f a c o r i l y [2]. w_ r e p r e s e n t s the proton tunneling "like" mode in which both s y s t e m s o s c i l l a t e in phase (fig. 1) and is r e s p o n s i b l e for the f e r r o e l e c t r i c phase t r a n s i tion. On the other hand, the w+ mode r e m a i n s a l m o s t unchanged for T ~ Tc. I.e. w2+ ~ 2 The i n c l u s i o n of the r e l a x a t i o n effect shows t~at in low frequency region, the d i e l e c t r i c d i s p e r s i o n is of the Debye type in a g r e e m e n t with H i l l I c h i k i ' s e x p e r i m e n t a l r e s u l t [3].
56
o
C
E(w) = ( T _ Tc) - i w T A
(6)
It can be concluded that the KH2PO4-type c r y s t a l s belong to the "mixed" type f e r r o e l e c t r i c s , that is to say, an o r d e r - d i s o r d e r a r r a n g e m e n t of p r o t o n s and an i n s t a b i l i t y of the lattice~vibration along c - a x i s o c c u r s i m u l t a n e o u s l y . Detailed d i s c u s s i o n will be published in the n e a r future.
References 1. P. G. de Gennes, Solid State Commun. 1 (1963) 132; M. Tokunaga, Progr. Theor. Phys. 36 (1966} 857. 2. R. Blinc, J. Phys. Chem. Solids 13 (1960} 204. 3. R.M. Hill and S. K. Ichiki, Phys. Rev. 130 (1962) 150.