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On the ferroelectricity of KH2PO4 and KD2PO4 crystals
P h y s i c a X V , n o 11-12
D e c e m b e r 1949
ON THE FERROELECTRICITY O F KH2PO4 A N D K D 2 P 0 4 C R Y S T A L S by JEAN P I R E N N E University of Li6ge, Belgium, and Swiss Federal Polytechnicum, Ziirich "
Synopsis T h e f e r r o - e l e c t r i c p r o p e r t i e s of K H z P O 4 a n d K D 2 P O 4 c r y s t a l s e x h i b i t a m o s t s t r i k i n g , y e t u n e x p l a i n e d , i s o t o p i c e f f e c t : t h e C u r i e p o i n t , for ins t a n c e , is s h i f t e d b y as m u c h as 90 ° a n d o t h e r p h y s i c a l p r o p e r t i e s are surp r i s i n g l y d i f f e r e n t in b o t h c r y s t a l s . T h e a n a l y s i s of t h i s s i t u a t i o n leads t o a ,new t h e o r y in w h i c h t h e p r o t o n s a r e . m o v i n g in a c e r t a i n p o t e n t i a l well, t h e spontaneous polarisatoin arising from the electrostatic dipole-dipole intera c t i o n ; t h e i s o t o p i c e f f e c t h a s a s i m p l e q u a n t u m - m e c h a n i c a l origin. T h e case of a s q u a r e well p o t e n t i a l is s t u d i e d in d e t a i l as o f f e r i n g a s u i t a b l e d i s c u s s i o n base. I t a l r e a d y leads, in t h e i n n e r - f i e l d a p p r o x i m a t i o n , t o a q u a n t i t a t i v e a g r e e m e n t for t h e C u r i e p o i n t , s a t u r a t i o n p o l a r i s a t i o n , p a r a e l e c t r i c C u r i e c o n s t a n t a n d t o t a l e x t r a h e a t of b o t h c r y s t a l s . T h e w i d h of t h e well is f o u n d t o c o r r e s p o n d t o a s e p a r a t i o n of a b o u t 148 c m - I b e t w e e n t h e t w o f i r s t p r o t o n levels. T h e v a l u e s w h i c h h a v e t o be a s s i g n e d t o t h e L o r e n t z f a c t o r a n d t o t h e ionic c h a r a c t e r of t h e p r o t o n s are q u i t e r e a s o n a b l e a n d t h e c o n t r i b u t i o n of t h e g r o u n d t o t h e p o l a r i s a t i o n a p p e a r s t o be of t h e s a m e o r d e r of m a g n i t u d e as t h a t of t h e p r o t o n s . F u r t h e r e x p e r i m e n t s , a n d especially far i n f r a - r e d a b s o r p t i o n s p e c t r a , s e a m t o be e x t r e m e l y d e s i r a b l e in order to improve the model.
As is well known, the properties of KH2P Q and KD2P Q ferroelectric crystals exhibit a quite unusual isotopic effect, particularly striking for the C u r i e temperature T c, saturation polarisation Ps and extra specific heat c (see table I) 1). TABLE I Ps
Crystal
Tc
KH2PO4 KD.PO.t
123°K 213°K
Cb/cn# 5,25.10 -s 9,0.10 -6
4y~
½~
p2 s
cal.#hole 726 2120
RT c
Q = . f c dT
cal. mole
cal./mole
246 426
57,3 100
This surprising effect has not yet been explained. According to S 1 a t e r's theory 2), e = kT. In 2, e being the energy difference between dipoles pointing in c and a directions, but no reason is given why e should be nearly doubled when H is replaced b y D. On the - - 1019 - -
1020
JEAN PIRENNE
other h a n d , , takes account only of short range, chemical, correlation forces between nearest neighbours; it entirely neglects long range electrostatic dipole-dipole interaction energy - - W; at low temperature, when saturation is reached, W = (1)f p2, / being the L or e n t z factor; as / is presumably not very different from 4~/3, the neglected energy W turns out to be several times greater than ~. Moreover, the predicted transition is of the first order, whereas the experimental one appears to be of the second order ; it has been proposed to explain this b y a broadening effect due to internal stresses in the crystal, but this does not seen tenable, as the extra-specific heat curves are not at all symmetrical about T c. We propose therefore another model based on the following remarks !) Isotopic effects are purely quantum mechanical for any system involving only static interactions. 2) The observed isotopic effect is so important, that it is not presumably a secondary one, such as the result of a small isotopic lattice expansion 3). We regard it, on the contrary, as a primary effect of the quantisation of the protons motion. 3) The dipole-dipole interaction energy - - W is b y no means negligible, as has been pointed out above. However, the total extra heat required to depolarise the crystal, Q -~fc dT, is but a very small part of W. Therefore, the increase of - - W ,during the depolarisation process, must be compensated in some way, for instance b y a simultaneous decrease of the protons intrinsic energy (kinetic energy + potential energy between protons and neighbouring atoms). 4) The actual crystal polarisation may be conveniently divided into two parts: a) the proton polarisation Pp due to displacements of the protons, carrying with them a certain part of the electronic cloud and having therefore an effective charge 0 e < e; b) the polarisation P, of the "ground" refering to any other causes (electronic polarisation, ions displacements). T h e existence of P, requires a certain energy which also leads to a lowering of Q. 5) The question of whether dipole-dipole interaction can be responsible for spontaneous polarisation remains open, except for certain lattices where the polarised state is not the lowest one 4) 6). For other lattices, however, a strong support to this possibility is given b y 0 n s a g e r's theory, as we have showrr elsewhere 6). L u tt i n g e r and T i s z a's paper 5) is rather in favour of this possibility, as well. These considerations have lead us to test the following model as a
ON THE
FERROELECTRICITY
OF KH2PO 4 AND
KD2PO 4 CRYSTALS
1021-
base for discussions: the protons are moving in a certain potential well V (interaction between protons and neighbouring-atoms) and the spontaneous polarisation is entirely due to dipole-dipole interaction, Pg is supposed proportional to Pp. As a rough approximation a constant proton ionic character 0 is assumed. For a first investigation, the dipole-dipole interaction has been introduced by means of a molecular field/'Pp; then for a polarised state to be in equilibrium, V must be anharmonic. This has lead us to the study of the simple case where V is a square well potential in the direction of the spontaneous polarisation ; in perpendicular directions, a strong constant restoring force is assumed, so that transverse vibrations will not play any role at all *). Now, if H is replaced by D, every level is lowered by a factor 2 and becomes twice more polarisable; one can see from this that the polarisability, at any temperature, is greater with D; the saturation polarisation and C u r i e temperature are therefore higher, in agreement with experiment. Furthermore, the quantitative agreement is rather encouraging. Our model contains one main parameter, the energy E 0 of the first vibration level, and three secondary parameters,/', e and ~7 = Pp/P, which cannot reasonably be varied over very wide ranges. The following set has been chosen to get perfect agreement for both C u r i e temperatures and for the saturation polarisation of KH2PO 4 : E 0 = 0,683.10 -2 eV; /' == 5,97 ; ~ -= 0,27 ; ~ / = 0,206. We then get a rather good agreement with experiment for three other physical quantities: the saturation polarisation of KD2PO 4 (theoretical value: 9,85. l 0 - 6 Cb/cm2; experimental value: 9,0.10 -6 Cb/cm 2) and the C u r i e constants C of the paraelectric susceptibility law above the C u r i e point, ~ = no + C ( / T - - To), quoted in table II **). T A B L E II Crystal I(H2PO I K D~POI
C theoretical 274 243
C experimental 256,5 265
*) It h a s o f t e n b e e n a c c e p t e d , for i n s t a n c e in S 1 a t e r ' s t h e o r y , t h a t t h e p r o t o n s w e r e m o s t free a l o n g t h e h y d r o g e n b o n d s , w h i c h a r e a l m o s t n o r m a l to tile d i r e c t i o n of s p o n t a n e o u s p o l a r i s a t i o n . H o w e v e r , t h e n e u t r o n d i i f r a c t i o n e x p e r i m e n t s of G i b e r t a n d R o s s e 1 7) are r a t h e r in f a v o u r of t h e o p p o s i t e c o n c l u s i o n . A n y w a y , t h e r e a l d i r e c t i o n of e a s y m o u v e m e n t is n o t e s s e n t i a l , as l o n g as we use t h e m o l e c u l a r field a p p r o x i m a t i o n . **) T h e e f f e c t of t h e r m a l e x p a n s i o n has b e e n n e g l e c t e d as it w o u l d c h a n g e C b y less t h e n 3 % ; t h e s i t u a t i o n is t h e r e f o r e e n t i r e l y d i f f e r e n t front t h a t d i s e r i b e d b y v it n S a nt e n a n d J o n k e r ~) for BaTiO.~, w h e r e t h i s effect is p r e d o m i n a n t .
1022
o N T H E F E R R O E L E C T R I C I T Y OF KH2PO 4 A N D KD2PO 4 C R Y S TA LS
Thus, six experimental constants are explained, within 9%, by our four parameters model. Furthermore, the secondary parameters values are quite reasonable, at least for /', not far from the L or e n t z value 4z~/3 = 4,18, and ~o, which does not differ very much from its value in water, ~ = 0,40. The ,/value 0,206, is rather low. It should be remembered however, that according to B u s h 9), proton polarisation and "ground" polarisation have the same order of magnitude. For the total extra heat Q, we have found the following (approximate) values: Qn = 25 cal/mole and QD = 90 cal/mole. Thus two more results are in agreement with experiment: 1°, QD > QH; 2 °, there is a nearly complete compensation between the variations of the dipole-dipole interaction and of the protons kinetic energy (this is probably the reason why the numerical agreement is less good for their difference Q). The above theory prodicts an absorption band of energy 3E o, i.e. in the region of 166 cm -1, for KH2PO 4, and 83 cm -1, for KD2PO 4. This is perhaps to be compared with the frequencies 195 cm -1 and 148 cm -1 observed by L a L a u 10) for these two crystals, in the infrared absorption bands. Most of these ideas were communicated at the Ziirich conference, July, 1948. Im indebted to Professors W. P a u l i and P. S c h e r r e r for stimulating interest to this work, partially carried out under the auspices of the Fonds National de la Recherche Scientifique (Brussels). Received J u l y 8th, 1949. REFERENCES 1) B. Z w i c k e r and P. S c h e r r e r , H e l v e t i c a physica Acta, 17. 346, 1944; ~r. B a n t l e , id. 15, 373, 1942; J. M e n d e l s s o h n and K. M e n d e l s s o h n , Nature 144,595, 1939. 2) J . C . S 1 a t e r, Journal of chemical Pllysics, 9, 16, 1941. 3) Considered by M a s o n, Phys. Rev. 72 , 854, 1947, as the cause of isotopic effi.ct of Roehelle salt. 4) J. H. v a n V l e e k , Journal of chemical Physics, 5, 320, 1937; J. A. S a u e l " and A. N. V. T e m p e r l e y , Proc. roy. Soc. 176, 203, 1940. 5) J . M . l . u t t i n g e r and L. l ' i s z a , Phys. Rev. 7 0 , 9 5 4 , 1946. 6) J. P i r e n n e, Helvetica physica Acta, ° 2 479, 1949. 7) A. G i b e r t andJ. Rossel, H e l v e t i c a p h y s i c a A c t a 14, 285, 1946. 8) J . H . v a n S a n t c n and G. H. J o n k e r, Nature, 159, 333, 1947. 9) G. B u s c h , Helv. phys. Acta, 11, 269, 1938. 10) C. L a L a u, Thesis, A m s t e r d a m 1947.
D e c e m b e r 1949
ON THE FERROELECTRICITY O F KH2PO4 A N D K D 2 P 0 4 C R Y S T A L S by JEAN P I R E N N E University of Li6ge, Belgium, and Swiss Federal Polytechnicum, Ziirich "
Synopsis T h e f e r r o - e l e c t r i c p r o p e r t i e s of K H z P O 4 a n d K D 2 P O 4 c r y s t a l s e x h i b i t a m o s t s t r i k i n g , y e t u n e x p l a i n e d , i s o t o p i c e f f e c t : t h e C u r i e p o i n t , for ins t a n c e , is s h i f t e d b y as m u c h as 90 ° a n d o t h e r p h y s i c a l p r o p e r t i e s are surp r i s i n g l y d i f f e r e n t in b o t h c r y s t a l s . T h e a n a l y s i s of t h i s s i t u a t i o n leads t o a ,new t h e o r y in w h i c h t h e p r o t o n s a r e . m o v i n g in a c e r t a i n p o t e n t i a l well, t h e spontaneous polarisatoin arising from the electrostatic dipole-dipole intera c t i o n ; t h e i s o t o p i c e f f e c t h a s a s i m p l e q u a n t u m - m e c h a n i c a l origin. T h e case of a s q u a r e well p o t e n t i a l is s t u d i e d in d e t a i l as o f f e r i n g a s u i t a b l e d i s c u s s i o n base. I t a l r e a d y leads, in t h e i n n e r - f i e l d a p p r o x i m a t i o n , t o a q u a n t i t a t i v e a g r e e m e n t for t h e C u r i e p o i n t , s a t u r a t i o n p o l a r i s a t i o n , p a r a e l e c t r i c C u r i e c o n s t a n t a n d t o t a l e x t r a h e a t of b o t h c r y s t a l s . T h e w i d h of t h e well is f o u n d t o c o r r e s p o n d t o a s e p a r a t i o n of a b o u t 148 c m - I b e t w e e n t h e t w o f i r s t p r o t o n levels. T h e v a l u e s w h i c h h a v e t o be a s s i g n e d t o t h e L o r e n t z f a c t o r a n d t o t h e ionic c h a r a c t e r of t h e p r o t o n s are q u i t e r e a s o n a b l e a n d t h e c o n t r i b u t i o n of t h e g r o u n d t o t h e p o l a r i s a t i o n a p p e a r s t o be of t h e s a m e o r d e r of m a g n i t u d e as t h a t of t h e p r o t o n s . F u r t h e r e x p e r i m e n t s , a n d especially far i n f r a - r e d a b s o r p t i o n s p e c t r a , s e a m t o be e x t r e m e l y d e s i r a b l e in order to improve the model.
As is well known, the properties of KH2P Q and KD2P Q ferroelectric crystals exhibit a quite unusual isotopic effect, particularly striking for the C u r i e temperature T c, saturation polarisation Ps and extra specific heat c (see table I) 1). TABLE I Ps
Crystal
Tc
KH2PO4 KD.PO.t
123°K 213°K
Cb/cn# 5,25.10 -s 9,0.10 -6
4y~
½~
p2 s
cal.#hole 726 2120
RT c
Q = . f c dT
cal. mole
cal./mole
246 426
57,3 100
This surprising effect has not yet been explained. According to S 1 a t e r's theory 2), e = kT. In 2, e being the energy difference between dipoles pointing in c and a directions, but no reason is given why e should be nearly doubled when H is replaced b y D. On the - - 1019 - -
1020
JEAN PIRENNE
other h a n d , , takes account only of short range, chemical, correlation forces between nearest neighbours; it entirely neglects long range electrostatic dipole-dipole interaction energy - - W; at low temperature, when saturation is reached, W = (1)f p2, / being the L or e n t z factor; as / is presumably not very different from 4~/3, the neglected energy W turns out to be several times greater than ~. Moreover, the predicted transition is of the first order, whereas the experimental one appears to be of the second order ; it has been proposed to explain this b y a broadening effect due to internal stresses in the crystal, but this does not seen tenable, as the extra-specific heat curves are not at all symmetrical about T c. We propose therefore another model based on the following remarks !) Isotopic effects are purely quantum mechanical for any system involving only static interactions. 2) The observed isotopic effect is so important, that it is not presumably a secondary one, such as the result of a small isotopic lattice expansion 3). We regard it, on the contrary, as a primary effect of the quantisation of the protons motion. 3) The dipole-dipole interaction energy - - W is b y no means negligible, as has been pointed out above. However, the total extra heat required to depolarise the crystal, Q -~fc dT, is but a very small part of W. Therefore, the increase of - - W ,during the depolarisation process, must be compensated in some way, for instance b y a simultaneous decrease of the protons intrinsic energy (kinetic energy + potential energy between protons and neighbouring atoms). 4) The actual crystal polarisation may be conveniently divided into two parts: a) the proton polarisation Pp due to displacements of the protons, carrying with them a certain part of the electronic cloud and having therefore an effective charge 0 e < e; b) the polarisation P, of the "ground" refering to any other causes (electronic polarisation, ions displacements). T h e existence of P, requires a certain energy which also leads to a lowering of Q. 5) The question of whether dipole-dipole interaction can be responsible for spontaneous polarisation remains open, except for certain lattices where the polarised state is not the lowest one 4) 6). For other lattices, however, a strong support to this possibility is given b y 0 n s a g e r's theory, as we have showrr elsewhere 6). L u tt i n g e r and T i s z a's paper 5) is rather in favour of this possibility, as well. These considerations have lead us to test the following model as a
ON THE
FERROELECTRICITY
OF KH2PO 4 AND
KD2PO 4 CRYSTALS
1021-
base for discussions: the protons are moving in a certain potential well V (interaction between protons and neighbouring-atoms) and the spontaneous polarisation is entirely due to dipole-dipole interaction, Pg is supposed proportional to Pp. As a rough approximation a constant proton ionic character 0 is assumed. For a first investigation, the dipole-dipole interaction has been introduced by means of a molecular field/'Pp; then for a polarised state to be in equilibrium, V must be anharmonic. This has lead us to the study of the simple case where V is a square well potential in the direction of the spontaneous polarisation ; in perpendicular directions, a strong constant restoring force is assumed, so that transverse vibrations will not play any role at all *). Now, if H is replaced by D, every level is lowered by a factor 2 and becomes twice more polarisable; one can see from this that the polarisability, at any temperature, is greater with D; the saturation polarisation and C u r i e temperature are therefore higher, in agreement with experiment. Furthermore, the quantitative agreement is rather encouraging. Our model contains one main parameter, the energy E 0 of the first vibration level, and three secondary parameters,/', e and ~7 = Pp/P, which cannot reasonably be varied over very wide ranges. The following set has been chosen to get perfect agreement for both C u r i e temperatures and for the saturation polarisation of KH2PO 4 : E 0 = 0,683.10 -2 eV; /' == 5,97 ; ~ -= 0,27 ; ~ / = 0,206. We then get a rather good agreement with experiment for three other physical quantities: the saturation polarisation of KD2PO 4 (theoretical value: 9,85. l 0 - 6 Cb/cm2; experimental value: 9,0.10 -6 Cb/cm 2) and the C u r i e constants C of the paraelectric susceptibility law above the C u r i e point, ~ = no + C ( / T - - To), quoted in table II **). T A B L E II Crystal I(H2PO I K D~POI
C theoretical 274 243
C experimental 256,5 265
*) It h a s o f t e n b e e n a c c e p t e d , for i n s t a n c e in S 1 a t e r ' s t h e o r y , t h a t t h e p r o t o n s w e r e m o s t free a l o n g t h e h y d r o g e n b o n d s , w h i c h a r e a l m o s t n o r m a l to tile d i r e c t i o n of s p o n t a n e o u s p o l a r i s a t i o n . H o w e v e r , t h e n e u t r o n d i i f r a c t i o n e x p e r i m e n t s of G i b e r t a n d R o s s e 1 7) are r a t h e r in f a v o u r of t h e o p p o s i t e c o n c l u s i o n . A n y w a y , t h e r e a l d i r e c t i o n of e a s y m o u v e m e n t is n o t e s s e n t i a l , as l o n g as we use t h e m o l e c u l a r field a p p r o x i m a t i o n . **) T h e e f f e c t of t h e r m a l e x p a n s i o n has b e e n n e g l e c t e d as it w o u l d c h a n g e C b y less t h e n 3 % ; t h e s i t u a t i o n is t h e r e f o r e e n t i r e l y d i f f e r e n t front t h a t d i s e r i b e d b y v it n S a nt e n a n d J o n k e r ~) for BaTiO.~, w h e r e t h i s effect is p r e d o m i n a n t .
1022
o N T H E F E R R O E L E C T R I C I T Y OF KH2PO 4 A N D KD2PO 4 C R Y S TA LS
Thus, six experimental constants are explained, within 9%, by our four parameters model. Furthermore, the secondary parameters values are quite reasonable, at least for /', not far from the L or e n t z value 4z~/3 = 4,18, and ~o, which does not differ very much from its value in water, ~ = 0,40. The ,/value 0,206, is rather low. It should be remembered however, that according to B u s h 9), proton polarisation and "ground" polarisation have the same order of magnitude. For the total extra heat Q, we have found the following (approximate) values: Qn = 25 cal/mole and QD = 90 cal/mole. Thus two more results are in agreement with experiment: 1°, QD > QH; 2 °, there is a nearly complete compensation between the variations of the dipole-dipole interaction and of the protons kinetic energy (this is probably the reason why the numerical agreement is less good for their difference Q). The above theory prodicts an absorption band of energy 3E o, i.e. in the region of 166 cm -1, for KH2PO 4, and 83 cm -1, for KD2PO 4. This is perhaps to be compared with the frequencies 195 cm -1 and 148 cm -1 observed by L a L a u 10) for these two crystals, in the infrared absorption bands. Most of these ideas were communicated at the Ziirich conference, July, 1948. Im indebted to Professors W. P a u l i and P. S c h e r r e r for stimulating interest to this work, partially carried out under the auspices of the Fonds National de la Recherche Scientifique (Brussels). Received J u l y 8th, 1949. REFERENCES 1) B. Z w i c k e r and P. S c h e r r e r , H e l v e t i c a physica Acta, 17. 346, 1944; ~r. B a n t l e , id. 15, 373, 1942; J. M e n d e l s s o h n and K. M e n d e l s s o h n , Nature 144,595, 1939. 2) J . C . S 1 a t e r, Journal of chemical Pllysics, 9, 16, 1941. 3) Considered by M a s o n, Phys. Rev. 72 , 854, 1947, as the cause of isotopic effi.ct of Roehelle salt. 4) J. H. v a n V l e e k , Journal of chemical Physics, 5, 320, 1937; J. A. S a u e l " and A. N. V. T e m p e r l e y , Proc. roy. Soc. 176, 203, 1940. 5) J . M . l . u t t i n g e r and L. l ' i s z a , Phys. Rev. 7 0 , 9 5 4 , 1946. 6) J. P i r e n n e, Helvetica physica Acta, ° 2 479, 1949. 7) A. G i b e r t andJ. Rossel, H e l v e t i c a p h y s i c a A c t a 14, 285, 1946. 8) J . H . v a n S a n t c n and G. H. J o n k e r, Nature, 159, 333, 1947. 9) G. B u s c h , Helv. phys. Acta, 11, 269, 1938. 10) C. L a L a u, Thesis, A m s t e r d a m 1947.