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On the transverse electrical susceptibilities of KDP type crystals
Solid State Communications, Vol. 33, pp. 775—779. Pergamon Press Ltd. 1980. Printed in Great Britain. ON THE TRANSVERSE ELECTRICAL SUSCEPTIBILITIES OF KDP TYPE CRYSTALS B.K. Chaudhuri* Faculty of Science, Osaka University, Toyonaka, Osaka-560, Japan and S. Ganguli and D. Nath Department of Magnetism, Indian Association for the Cultivation of Science, Calcutta-7000 32, India (Received 21 September 1979; in revised form 20 October 1979 by H. Kawamura) Using statistical Green’s function technique exact expression for the transverse electrical susceptibility (Xa) for KDP type crystals has been derived with the total pseudo-spin lattice coupled mode Hamiltonian. Very good fittings of the experimental data of Xa for KH2PO4, RbH2PO4 and KH2AsO4 with the derived expression have been obtained with a single set of Blinc-de Geenes (BG) parameters being different for different salts. The temperature dependence of the E-mode character along the transverse a-axis has also been studied theoretically and compared with the recent experimental results available for KH2PO4 only. IT IS WELL KNOWN [1, 2] that the essential features of the ferroelectric behaviours in KDP (KH2PO4) type crystals obtained from different kinds of measurements along the polar axis (c-axis) can be explained with the pseudo-spin model [3,4]. Havlin eta!. [5] (hereafter referred to as HLU) showed that the informations of the Blinc-de Gennes (BG) parameters viz, the proton tunneling energy r (= 2~2)proton—proton interaction constant (J0) etc. could also be obtained from the knowledge of the experimental data of the transverse susceptibility (x0). But it is observed from their theoretical results that the values of the above mentioned parameters differ markedly from those obtained from Raman studies [6—8]. Recently Singh and Basu [9] also applied the theory of HLU to fit their experimental data of (Xa) for the RhH2PO4 crystal. Here also we find that the pseudo-spin parameters they obtained from the fitting of experimental (Xa T) data with the theory of HLU are different from those obtained from Raman spectroscopic studies [7, 10]. Further, Singh and Basu [9]also considered large variation ofJ0 with temperature which is not expected. It, therefore, appears to be interesting and also important to decide whether the BG parameters calculated from the transverse susceptibility data should actually be different from those obtained from measurements along c-axis. In this paper we have attempted to make an elabor. ate calculation using statistical Green’s function method On leave of absence from the Department of Magnetism, Indian Association for the Cultivation of Science, Calcutta, India. —
*
to find the exact expression for the transverse suscepti. bility and hence to calculate the BG parameters from fitting of the experimental (x0 T) data. Similar type of Green’s function technique has previously been used by us to calculate the static and the dynamic properties of KDP type crystals with success [11, 121. In the present calculations we have taken into account the total pseudo-spin lattice coupled Hamiltonian. We have considered the proton displacement along a-direction (the z direction) and hence the transverse electrical susceptibility is related to the Green’s function ((Sz(t)iSz(tl)>> and the longitudinal susceptibility is related to the Green’s function (
—
—
——~
.
.
.
calculations for the sake of simplicity. S?(a = x, y, z) 775
776
TRANSVERSE ELECTRICAL SUSCEPTIBILITIES OF KJTW TYPE CRYSTALS
Vol. 33, No.7
Table]. Pseudo-spin mode! (BG)parwn eters for KH2PO4, KH2AsO4 and RbH2PO4 Crystals
(cm’)
(cm’)
(cm312)
w0 (cm’)
j1~ x l0’~ (cgs)
80.00
440.90
340.00
25.60
153.00
1.80
2PO4 *{(a) (b) (c)
86.00 64.28
450.00 141.78
344.00
-.
153.00
(a) (b) (c)
26.30
281.80
21135
—
-
-
-
—
70.20
((a) RbH2PO4 ~(b) (d)
78.40 78.40 91.74
1) (cm
KH
KH2AsO4
(
13.90
3.90
—
483.60 473.00
21.09
362.70 357.00 127.19
—
115.00
1.60
—
—
4.05
25.43
125.00 126.00
—
1.23 5.51
—
(a), (b), (c) and (d) represent respectively the values of the parameters calculated by us [11, 12, 17J and used in this paper for calculations of Xa and w0, experimentally observed by (PS) [6—8,10], calculated by (HLU) [51 and (SB) [9].
*
ioo—
725
E
111111
-
I.)
U
7~_
~7Q
~
4,
TEMPERATURE (T K)
~ TEMPERATURE(T’K )
5~-
0,
~60-
RDP
--
~ SI ~20
200
-,
-
250
3cL
I
-
0
___ 70
~KIJA 130 TEMPERATURE(T’K
310
r ~) Fig. 2. Thermal variations of the pseudo-soft mode frequency for KDP and KDA calculated from our theory Fig. 1. Thermal variations of the dielectric constants of using (10). KDP, KDA and RDP. 0,., experimental points; continuous line, present theory; dotted line, theory of HLU. moment along the transverse axis and v0 is the volume are the ath component of the pseudo-spin variable S. of the unit cell with number of pseudo-spin n. The ~2is the tunneling frequency,J~1and V~are respecequation of motion with the above Green’s function tively the proton—proton and the proton—lattice may be written as [15] (in units of h = I and M = 1) coupling constants governing the interaction of proton 3i, 11 with optical of theM.K—PO4 w((Sf Sf> = ~ir ~L’ ~ 5Z1) + ((Sf,HISJ)). (3) reduced massvibrations of the system Q~andsystem Pg are with respectively the normal co-ordinates and the conjugate To linearize the complex Green’s functions like momenta. Wq are the bare harmonic phonon frequencies. ((S~S757)>, we use the decoupling [11, 12] The transverse susceptibility x 0 is calculated from the relation ((Sf57157)) (S7)((SrISf)~+(S~fX(SfS7)). (4) TEMPERA TURE(
-~-—
x
=
_~~~G2 z(w) VØ
where G~(w)is the Fourier transform of the Green’s function G~(t t’) = ((S~(t)JS~(t’))).pa is the dipole —
(2)
A All,Bthe = F, possible Q SxS~ Green’s and functions SZ) are calculated like ((AIB)) and(where decoupled using the relation (4). Then the required equations of motion from which G~(w)is obtained can be written in the matrix form as
Vol. 33, No. 7
TRANSVERSE ELECTRICAL SUSCEPTIBILITIES OF KDP TYPE CRYSTALS (5) 4~2 cli anh 2~22J~ tanh Xa = t VoT? 71 —a 0 —b(S~’) 01 for T< T~ where w —c _b(Sx) 0 2V~
=
where
[ a
=
_______
0
2ifl
0 [w 0
0
0
0]
0
0
0
—b
w
—
iw~
i
The energy spectrum obtained from the secular determinant Ml = 0 using (5a) may be expressed as
w
W~,2 =
i(Sx>
2ir
G~(w)j and G~(w)
=
~
—~—
00
j
forT>T~
~(SZWo,
b
lVq,
=
12 =
0
2V2 (Sb)
c
=
Vjq,
+i(2c2_1(Sz)Jo, (6) (Sf> (SZ),
iv’
‘~j~ ~“q =
Gq~(w)= 2&2(SX)
W2
—
{4c12
—
2~2(SX)Jq
—
2f1
=
—
77
—
tanh 3,~3i~
(8a)
.Jo(5Z> tanh ~ 2~
(8b)
Vo
l6l~2 (lOa)
(lfl(4&2
for T> T~
where 2—J L~= 4c~ 0~ltanh I3~2 2J 2 2~2 0 L~= 77 tanh~(3~. The transverse0dielectric constant ~a
(ila)
—
15
(llb) calculated from (12)
Equations (9a, b) are the most exact expressions for calculating theoretically the transverse electrical susceptibiities of KDP (KH 2PO4), KDA (KH2AsO4) RDP (RbH2PO4), and the other ofwKDP family. The model parameters (viz.members J~,JO,t~, in Table 1) have already been calculated 0inand our~2shown earlier papers [11, 12, 17] from fitting of the longitudinal dielectric properties and found to be agreed very well with the available Raman spectroscopic data [6—8, 10] (also shown in Table 1). Knowing J~and ~2from Table 1 we require only one parameter p~,to fit the exper(Ca T) curves of KDP, KDA and
imental [9,18]
—
RDP. values1.ofThe J.La very calculated for the bestbetween fitting are shownThe in Table good agreement the theoretical and the experimental (Ca T) curves —
where ~2 = [4~2 + 1(J 21and ~ = (kBfl’(kB is 0(s~)) the Boltzmann’s constant and T is the absolute ternperature). From (2) and (7) the expressions for the transverse susceptibilities above ahd below (when (S~)*0)T 0 can be written as —
_w~)2 +
forT
(9) using the general relation Ca = 1 + 4ITXa.
5>)2 2&TZ(SX)Vg (7) 2 w~) (w The thermal averages (5X> and (5Z) can be calculated from our earlier work [22, 12] or from the calculations of Brout etaL [16] which have the form + ~(J0(S
=
w~) ± }1/2] xtanh~371
W~,2= ~~L22 +
0
and N is the number of hydrogen bonds involved. Solving (5) for the Green’s function G~(w)and putting (S~’>= 0 for obvious reason we have
(5X)
(10)
and
where ( ) denotes the thermal average of the enclosed operators, =
~[(L~+ w~)± {(L~_w~)2 + 8clV~tanh~l}L’u]
0
G~(w)
a
(9c) w
[_iSYl
G~(w)
=
(9b)
—
0
G~(w)]
O
(5a)
777
—
J~tanh j%’l)’
(9a)
thus obtained by using (9a, b) is observed from Fig. 1. This also indicates that the same parameters can be used to calculate the transverse and the longitudinal proper. ties of KDP type hydrogen bonded ferroelectric crystals with success. The values of (5Z) below T0 are obtained from 3). Thethe spontaneous polarization using the relation (similar to HLU) P4 = P0(Sz) (P0 = 2N~i0,N= 1022 cm effective massM was taken to be 4.6 x l0~gmwhich is of the same order of magnitude calculated by Kobayashi [19]. Further, to calculate x 0 from (9a, b) we also used the experimental values of vo given by Vain
778
TRANSVERSE ELECTRICAL SUSCEPTIBILITIES OF KDP TYPE CRYSTALS
eta!. [20]. The value of n was taken to be I indicating [21] the motion of all the ions in the unit cell. The values of dipole moments calculated by HLU are also very high compared to our calculated values shown in Table 1. HLU did not, however, give much importance to the values of dipole moments. The dipole moments calculated by us are also comparable with Pa = ez~ix= 1.2 x l0’8cgs (where e is the electronic charge and 2zXx is the distance of separation of the equilibrium positions in the hydrogen bond). We have already mentioned that the model parameters calculated by HLU do not agree with ours and those of PS as shown in Table 1 for comparison HLU accounted for the large difference between the calculated values of &2 with those of Fairall and Reese [22] (FR) to be due to the inadequacy of their (HLU) approximation and/or due to the lack of accurate experimental data. Our calculated value of 17 are very close to those of(PS) [7,8,10] and also comparable to the mean field result of(FR) [22]. This definitely indicates the validity of our calculated results. The BG parameters calculated by SB for RDP using the same model of HLU also deviate largely from our calculated values (Table I) and their [9] fitting of the experimental (Ca T) data is also not good as shown in Fig. I (represented by the dotted lines). Here we should also point out that SB not only considered large variation ofJo to get this fitting but also put [23] J~(effective value of pseudo-spin lattice coupling [19]) equal to J 0 (mean field value of pseudo-spin coupling with pure tunneling model [16]). In another paper Singh and Singh [23] used the (BG) parameters calculated by them earlier [9] using the HLU model to explain the temperature variation of the longitudinal velocity of RDP below T0 taking Jo/kB = .10*/kB 183 K (or .J0 = .J~’= 127 cm~).In our opinion this value of J~’is not correct and much lower than the actual value. —
It is found [3, 19] that the spin—lattice coupling has the effect of increasing the value of pure proton—proton coupling by the term Pj/w~(in the notation of PS, = J0 + G”). It is also observed [6—8,10] that for KDP type crystals .ig is of the order ofJo/0.75. If this is the case which is also supported by our calculations [11, 12], the (BG) parameters calculated by SB for RDP using the HLU model are not the appropriate values of the respective parameters and hence these values are not suitable for explaining the longitudinal velocity below T~.This is due to the fact that the behaviour of Ca below T,, is not well represented HLU model also pointed out by Pollina and Garland [24]. Our calculated model parameters shown in Table 1 are also used to calculate the pseudo-soft model frequency w1 (using 10) as a function of temperature. We call this mode pseudo-soft mode (PSM) since this mode
Vol. 33, No.7
does not vanish at T = T~but shows only a minima at T~(Fig. 2). The thermal variation of (PSM) as shown in Fig. 2 is identical to that of E-mode [13, 14] observed along the transverse a-axis of KDP type crystals. From Fig. 2 it is noticed that for KDP the minimum value of w1 at T~is about 78 cm1. This value is very close to the experimentally observed value [14] (~ 80 cm’ for KDP). We have also studied the (PSM) as a function of temperature for KDA (Fig. 2). Similar behaviour is also expected for RDP. It may finally be concluded from our theoretical work considering the total pseudo-spin—lattice coupled mode (PLCM) model that a single set of (BG) param. eters could be used to explain simultaneously the transverse and the longitudinal properties of KDP type crystals. Only the dipole moments are different along the two different directions which is obvious. Acknowledgement The authors are highly indebted to Prof. Y. Yamada of Osaka University for his valuable comments and suggestions. —
1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
REFERENCES P.G. De Gennes,Solid State Commun. 1, 132 (1963). M. Tokunaga & T. Matsubara,i~og.Theo. Phys. 35, 581 (1966). R.Blinc&B.Zeks,Adv.Phys. 21,693 (1972). M.E. Lines & A.M. Glass,Principles andApplications ofFerroelectrics and Related Materials, Clarendon Press, Oxford, (1977). S. Havlin, E. Litov & E.A. Uehling,Phys. Rev. B9, 1024 (1974). P.S. Peercy, Solid State Commun. 16,439 (1975); Phys. Rev. B12, 2725 (1975). P.S. Peercy & GA. Samara,Phys. Rev. B8, 2033 (1973). GA. Samara,Ferroelectrics 5,25 (1973). G.P. Singh & BK. Basu,Phys. Lett. 58A,39 (1976). P.S. Peercy,Phys. Rev. B9, 4868 (1974). S. Ganguli, D. Nath & B.K. Chaudhuri,Phys. Rev. (to be published). BK. Chaudhuri & M. Saha, Ferroelectrics 18, 213 (1978). S.Havlin E. Litov& H. Sompolinsky,Phys. Lett. 53.4,41(1975). K.E. Gauss, H. Happ & G. Rother,Phys. Status Solidi (b) 72,623 (1975). V .L. Bonch Bruevich & S.V. Tyablikov, The Green ~ Function Method in Statistical Mechanics, North Holland, (1962). R. Brout, K.A. Muller & H. Thomas, Solid State Commun. 4,507 (1966). BK. Chaudhuri,Ind. Pure & AppL Phys. 16, 831(1978). G. Busch,Helv.PhysActa. 11,269(1938). K.K. Kobayashi,J. Phys. Soc. Japan. 24,497 (1968). V.G.Vaks, N.E. Zein & BA. Strukov,Phys. S~tusSolidi 30,801 (1975). /.
Vol. 33, No.7 21. 22.
TRANSVERSE ELECTRICAL SUSCEPTIBILITIESOF KDP TYPE CRYSTALS
Ri. Elliott&A.P. Young,Ferroelectrics 7,23 (1974). C.W. Fairall&W. Reese,Phys. Rev. B6, 193 (1972).
23. 24.
779
G.P. Singh&S. Singh,J. Phys. C12, 995 (1979). R.J. Pollina & C.W. Garland,Phys. Rev. B12, 362 (1975).
*
to find the exact expression for the transverse suscepti. bility and hence to calculate the BG parameters from fitting of the experimental (x0 T) data. Similar type of Green’s function technique has previously been used by us to calculate the static and the dynamic properties of KDP type crystals with success [11, 121. In the present calculations we have taken into account the total pseudo-spin lattice coupled Hamiltonian. We have considered the proton displacement along a-direction (the z direction) and hence the transverse electrical susceptibility is related to the Green’s function ((Sz(t)iSz(tl)>> and the longitudinal susceptibility is related to the Green’s function (
—
—
——~
.
.
.
calculations for the sake of simplicity. S?(a = x, y, z) 775
776
TRANSVERSE ELECTRICAL SUSCEPTIBILITIES OF KJTW TYPE CRYSTALS
Vol. 33, No.7
Table]. Pseudo-spin mode! (BG)parwn eters for KH2PO4, KH2AsO4 and RbH2PO4 Crystals
(cm’)
(cm’)
(cm312)
w0 (cm’)
j1~ x l0’~ (cgs)
80.00
440.90
340.00
25.60
153.00
1.80
2PO4 *{(a) (b) (c)
86.00 64.28
450.00 141.78
344.00
-.
153.00
(a) (b) (c)
26.30
281.80
21135
—
-
-
-
—
70.20
((a) RbH2PO4 ~(b) (d)
78.40 78.40 91.74
1) (cm
KH
KH2AsO4
(
13.90
3.90
—
483.60 473.00
21.09
362.70 357.00 127.19
—
115.00
1.60
—
—
4.05
25.43
125.00 126.00
—
1.23 5.51
—
(a), (b), (c) and (d) represent respectively the values of the parameters calculated by us [11, 12, 17J and used in this paper for calculations of Xa and w0, experimentally observed by (PS) [6—8,10], calculated by (HLU) [51 and (SB) [9].
*
ioo—
725
E
111111
-
I.)
U
7~_
~7Q
~
4,
TEMPERATURE (T K)
~ TEMPERATURE(T’K )
5~-
0,
~60-
RDP
--
~ SI ~20
200
-,
-
250
3cL
I
-
0
___ 70
~KIJA 130 TEMPERATURE(T’K
310
r ~) Fig. 2. Thermal variations of the pseudo-soft mode frequency for KDP and KDA calculated from our theory Fig. 1. Thermal variations of the dielectric constants of using (10). KDP, KDA and RDP. 0,., experimental points; continuous line, present theory; dotted line, theory of HLU. moment along the transverse axis and v0 is the volume are the ath component of the pseudo-spin variable S. of the unit cell with number of pseudo-spin n. The ~2is the tunneling frequency,J~1and V~are respecequation of motion with the above Green’s function tively the proton—proton and the proton—lattice may be written as [15] (in units of h = I and M = 1) coupling constants governing the interaction of proton 3i, 11 with optical of theM.K—PO4 w((Sf Sf> = ~ir ~L’ ~ 5Z1) + ((Sf,HISJ)). (3) reduced massvibrations of the system Q~andsystem Pg are with respectively the normal co-ordinates and the conjugate To linearize the complex Green’s functions like momenta. Wq are the bare harmonic phonon frequencies. ((S~S757)>, we use the decoupling [11, 12] The transverse susceptibility x 0 is calculated from the relation ((Sf57157)) (S7)((SrISf)~+(S~fX(SfS7)). (4) TEMPERA TURE(
-~-—
x
=
_~~~G2 z(w) VØ
where G~(w)is the Fourier transform of the Green’s function G~(t t’) = ((S~(t)JS~(t’))).pa is the dipole —
(2)
A All,Bthe = F, possible Q SxS~ Green’s and functions SZ) are calculated like ((AIB)) and(where decoupled using the relation (4). Then the required equations of motion from which G~(w)is obtained can be written in the matrix form as
Vol. 33, No. 7
TRANSVERSE ELECTRICAL SUSCEPTIBILITIES OF KDP TYPE CRYSTALS (5) 4~2 cli anh 2~22J~ tanh Xa = t VoT? 71 —a 0 —b(S~’) 01 for T< T~ where w —c _b(Sx) 0 2V~
=
where
[ a
=
_______
0
2ifl
0 [w 0
0
0
0]
0
0
0
—b
w
—
iw~
i
The energy spectrum obtained from the secular determinant Ml = 0 using (5a) may be expressed as
w
W~,2 =
i(Sx>
2ir
G~(w)j and G~(w)
=
~
—~—
00
j
forT>T~
~(SZWo,
b
lVq,
=
12 =
0
2V2 (Sb)
c
=
Vjq,
+i(2c2_1(Sz)Jo, (6) (Sf> (SZ),
iv’
‘~j~ ~“q =
Gq~(w)= 2&2(SX)
W2
—
{4c12
—
2~2(SX)Jq
—
2f1
=
—
77
—
tanh 3,~3i~
(8a)
.Jo(5Z> tanh ~ 2~
(8b)
Vo
l6l~2 (lOa)
(lfl(4&2
for T> T~
where 2—J L~= 4c~ 0~ltanh I3~2 2J 2 2~2 0 L~= 77 tanh~(3~. The transverse0dielectric constant ~a
(ila)
—
15
(llb) calculated from (12)
Equations (9a, b) are the most exact expressions for calculating theoretically the transverse electrical susceptibiities of KDP (KH 2PO4), KDA (KH2AsO4) RDP (RbH2PO4), and the other ofwKDP family. The model parameters (viz.members J~,JO,t~, in Table 1) have already been calculated 0inand our~2shown earlier papers [11, 12, 17] from fitting of the longitudinal dielectric properties and found to be agreed very well with the available Raman spectroscopic data [6—8, 10] (also shown in Table 1). Knowing J~and ~2from Table 1 we require only one parameter p~,to fit the exper(Ca T) curves of KDP, KDA and
imental [9,18]
—
RDP. values1.ofThe J.La very calculated for the bestbetween fitting are shownThe in Table good agreement the theoretical and the experimental (Ca T) curves —
where ~2 = [4~2 + 1(J 21and ~ = (kBfl’(kB is 0(s~)) the Boltzmann’s constant and T is the absolute ternperature). From (2) and (7) the expressions for the transverse susceptibilities above ahd below (when (S~)*0)T 0 can be written as —
_w~)2 +
forT
(9) using the general relation Ca = 1 + 4ITXa.
5>)2 2&TZ(SX)Vg (7) 2 w~) (w The thermal averages (5X> and (5Z) can be calculated from our earlier work [22, 12] or from the calculations of Brout etaL [16] which have the form + ~(J0(S
=
w~) ± }1/2] xtanh~371
W~,2= ~~L22 +
0
and N is the number of hydrogen bonds involved. Solving (5) for the Green’s function G~(w)and putting (S~’>= 0 for obvious reason we have
(5X)
(10)
and
where ( ) denotes the thermal average of the enclosed operators, =
~[(L~+ w~)± {(L~_w~)2 + 8clV~tanh~l}L’u]
0
G~(w)
a
(9c) w
[_iSYl
G~(w)
=
(9b)
—
0
G~(w)]
O
(5a)
777
—
J~tanh j%’l)’
(9a)
thus obtained by using (9a, b) is observed from Fig. 1. This also indicates that the same parameters can be used to calculate the transverse and the longitudinal proper. ties of KDP type hydrogen bonded ferroelectric crystals with success. The values of (5Z) below T0 are obtained from 3). Thethe spontaneous polarization using the relation (similar to HLU) P4 = P0(Sz) (P0 = 2N~i0,N= 1022 cm effective massM was taken to be 4.6 x l0~gmwhich is of the same order of magnitude calculated by Kobayashi [19]. Further, to calculate x 0 from (9a, b) we also used the experimental values of vo given by Vain
778
TRANSVERSE ELECTRICAL SUSCEPTIBILITIES OF KDP TYPE CRYSTALS
eta!. [20]. The value of n was taken to be I indicating [21] the motion of all the ions in the unit cell. The values of dipole moments calculated by HLU are also very high compared to our calculated values shown in Table 1. HLU did not, however, give much importance to the values of dipole moments. The dipole moments calculated by us are also comparable with Pa = ez~ix= 1.2 x l0’8cgs (where e is the electronic charge and 2zXx is the distance of separation of the equilibrium positions in the hydrogen bond). We have already mentioned that the model parameters calculated by HLU do not agree with ours and those of PS as shown in Table 1 for comparison HLU accounted for the large difference between the calculated values of &2 with those of Fairall and Reese [22] (FR) to be due to the inadequacy of their (HLU) approximation and/or due to the lack of accurate experimental data. Our calculated value of 17 are very close to those of(PS) [7,8,10] and also comparable to the mean field result of(FR) [22]. This definitely indicates the validity of our calculated results. The BG parameters calculated by SB for RDP using the same model of HLU also deviate largely from our calculated values (Table I) and their [9] fitting of the experimental (Ca T) data is also not good as shown in Fig. I (represented by the dotted lines). Here we should also point out that SB not only considered large variation ofJo to get this fitting but also put [23] J~(effective value of pseudo-spin lattice coupling [19]) equal to J 0 (mean field value of pseudo-spin coupling with pure tunneling model [16]). In another paper Singh and Singh [23] used the (BG) parameters calculated by them earlier [9] using the HLU model to explain the temperature variation of the longitudinal velocity of RDP below T0 taking Jo/kB = .10*/kB 183 K (or .J0 = .J~’= 127 cm~).In our opinion this value of J~’is not correct and much lower than the actual value. —
It is found [3, 19] that the spin—lattice coupling has the effect of increasing the value of pure proton—proton coupling by the term Pj/w~(in the notation of PS, = J0 + G”). It is also observed [6—8,10] that for KDP type crystals .ig is of the order ofJo/0.75. If this is the case which is also supported by our calculations [11, 12], the (BG) parameters calculated by SB for RDP using the HLU model are not the appropriate values of the respective parameters and hence these values are not suitable for explaining the longitudinal velocity below T~.This is due to the fact that the behaviour of Ca below T,, is not well represented HLU model also pointed out by Pollina and Garland [24]. Our calculated model parameters shown in Table 1 are also used to calculate the pseudo-soft model frequency w1 (using 10) as a function of temperature. We call this mode pseudo-soft mode (PSM) since this mode
Vol. 33, No.7
does not vanish at T = T~but shows only a minima at T~(Fig. 2). The thermal variation of (PSM) as shown in Fig. 2 is identical to that of E-mode [13, 14] observed along the transverse a-axis of KDP type crystals. From Fig. 2 it is noticed that for KDP the minimum value of w1 at T~is about 78 cm1. This value is very close to the experimentally observed value [14] (~ 80 cm’ for KDP). We have also studied the (PSM) as a function of temperature for KDA (Fig. 2). Similar behaviour is also expected for RDP. It may finally be concluded from our theoretical work considering the total pseudo-spin—lattice coupled mode (PLCM) model that a single set of (BG) param. eters could be used to explain simultaneously the transverse and the longitudinal properties of KDP type crystals. Only the dipole moments are different along the two different directions which is obvious. Acknowledgement The authors are highly indebted to Prof. Y. Yamada of Osaka University for his valuable comments and suggestions. —
1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
REFERENCES P.G. De Gennes,Solid State Commun. 1, 132 (1963). M. Tokunaga & T. Matsubara,i~og.Theo. Phys. 35, 581 (1966). R.Blinc&B.Zeks,Adv.Phys. 21,693 (1972). M.E. Lines & A.M. Glass,Principles andApplications ofFerroelectrics and Related Materials, Clarendon Press, Oxford, (1977). S. Havlin, E. Litov & E.A. Uehling,Phys. Rev. B9, 1024 (1974). P.S. Peercy, Solid State Commun. 16,439 (1975); Phys. Rev. B12, 2725 (1975). P.S. Peercy & GA. Samara,Phys. Rev. B8, 2033 (1973). GA. Samara,Ferroelectrics 5,25 (1973). G.P. Singh & BK. Basu,Phys. Lett. 58A,39 (1976). P.S. Peercy,Phys. Rev. B9, 4868 (1974). S. Ganguli, D. Nath & B.K. Chaudhuri,Phys. Rev. (to be published). BK. Chaudhuri & M. Saha, Ferroelectrics 18, 213 (1978). S.Havlin E. Litov& H. Sompolinsky,Phys. Lett. 53.4,41(1975). K.E. Gauss, H. Happ & G. Rother,Phys. Status Solidi (b) 72,623 (1975). V .L. Bonch Bruevich & S.V. Tyablikov, The Green ~ Function Method in Statistical Mechanics, North Holland, (1962). R. Brout, K.A. Muller & H. Thomas, Solid State Commun. 4,507 (1966). BK. Chaudhuri,Ind. Pure & AppL Phys. 16, 831(1978). G. Busch,Helv.PhysActa. 11,269(1938). K.K. Kobayashi,J. Phys. Soc. Japan. 24,497 (1968). V.G.Vaks, N.E. Zein & BA. Strukov,Phys. S~tusSolidi 30,801 (1975). /.
Vol. 33, No.7 21. 22.
TRANSVERSE ELECTRICAL SUSCEPTIBILITIESOF KDP TYPE CRYSTALS
Ri. Elliott&A.P. Young,Ferroelectrics 7,23 (1974). C.W. Fairall&W. Reese,Phys. Rev. B6, 193 (1972).
23. 24.
779
G.P. Singh&S. Singh,J. Phys. C12, 995 (1979). R.J. Pollina & C.W. Garland,Phys. Rev. B12, 362 (1975).