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Strain optical constants of ADP and KDP
1. Phyr. Chum. &Ii&
Vol. 51. No. 8. pp. 915-919. 1990
0022.3697/90
13.00 + 0.00
0 19% Pergamon Press plc
Printed in Great Britain.
STRAIN
OPTICAL
CONSTANTS
OF ADP AND KDP
D. PRAVEENA and R. ETHIFUJ Department of Physics, University College of Science, Osmania University, Hyderabad-500007, India (Received
4 January
1989; accepted in revised firm
3 January 1990)
Abstract-A method to evaluate the strain optical constants of ammonium dihydrogen phosphate (ADP) and potassium dihydrogen phosphate (KDP), which belong to the tetragonal system of class 42m, is presented. The ions are treated as point dipoles which undergo displacement on application of a uniaxial stress. Considering (i) the anisotropies in the arrangements of the dipoles in the deformed crystal and (ii) the change in the density of the crystal, expressions for the change in the refractive indices for different directions of stress are developed and the corresponding strain optical constants (p,s) evaluated. Point dipoles, natural birefringence, artificial birefringence, electronic polarizability. strain optical constants, uniaxial stress, Poisson’s ratio and elastic compliances.
Keywordr:
INTRODUCTION Potassium dihydrogen phosphate (KDP) and ammonium dihydrogen phosphate (ADP) are among the first materials that were used and exploited for their non-linear optical and electro-optical properties. Their popularity still continues and many devices, such as electro-optic modulators and second harmonic generators, are developed using these crystals. This, naturally, stimulated theoretical studies and many models have been proposed as cited in [l]. Any study of the electro-optical effect in these types of crystal involves the strain optical effect via the converse piezo-electric effect [2]. A need was thus felt to understand the strain optical effect, which prompted us to develop an empirical model to explain the phenomena in these crystals. We used the “point dipole model” to establish the change in the refractive index induced by the alteration in the structural parameters due to a uniaxial stress and thus evaluate the various strain optical constants. This work can be viewed as an extension to (i) the mode1 suggested by Praveena et al. to evaluate the electronic polarizabilities of ions in KDP and ADP [3] and (ii) the model used by Kinase et al. to explain the correction to the dipole field due to the lattice deformation of a perovskite-type crystal [4]. Survey of the literature indicates that this work was undertaken for the first time in anisotropic crystals of the KDP type. THEORETICAL The general expression constants pijs is given by pij=
RELATIONS for
-2 an. --_1 rt:
hi
the
strain
optical
From the Pockels’ phenomenological theory of photoelasticity the number of independent nonvanishing strain optical constants for ADP and KDP are seven, viz. pII, p12,pIJ, PJ~, puv pu and Pa. The ions are treated as point dipoles and an ion at a point (x, y, z) shifts to a point (x’, y’, z’) on the application of a uniaxial stress. The new coordinates are given by x’=x(l
+A,)
Y’ = ~(1 + z’
=
A2)
z(l + A1),
(2)
where the As Q 1. The field at a lattice site (0, 0,O) due to a dipole of moment pXat (x’, y’, z’) with the electric vector of the light in the X-direction is given by E, = P~(~.X”~- R’2)/R’5 X’=x’*a Y’=f*Q Z’=z’*c R’2
=
X’2
+
y’2
+
Z”
(3)
and u and c are the lattice constants [5]. Neglecting the higher powers of the As, the above expression is written in terms of the underformed lattice site, as
&-=Ex+ccx
3X2(3 Y’ + 322 - 2x9 A I R7
3 Y2( Y2 + z* - 4X’)
(1)
+ cc,
where i is the direction parallel to the electric vector and j the direction parallel to the strain; i,j take values from 1 to 6.
+ & 915
R’
A2
3Z2( Y2 + 22 - 4X2) A R’ 3,
(4)
916
D.
PRAVEENA
and R. ETHIIW
Table 1. For the evaluation of p,,. pi2. p13.p3, and p33.Lattice sums Gs X tO-2’(cn03 Crystal ADP KDP
G-X,
G-X,
-0.0438 -0.053 1
0.0572 0.0614
Crystal
GmX,
GmX2
ADP KDP
-0.0427 -0.0544
0.0157 0.0226
Crystal ADP KDP
G-X, - 0.0439 - 0.0479
G-X2 0.0112 0.0146
Crystal ADP KDP
G,X, - 0.0428 -0.0505
G,Xr 0.0483 0.0573
G-X3
G,,,.,,Yr
-0.0347 -0.0341
0.0572 0.0614
G-Y2
G,. Yr
G-Z,
- 0.0437 -0.0531
-0.0347 -0.0341
-0.0134 - 0.0083
G,,,,,X3 G,,,,,Y, G,,Y2
G,Y3
G,Zt
0.0806 0.1095
0.0157 0.0226
-0.0427 -0.0544
0.0806 0.1095
0.0270 0.0318
G-A
G-Z3
-0.0139 -0.0083
0.0693 0.0681
GmZ2
G,Z3
0.0270 0.0318
-0.1612 -0.2189
Table 2. For the evaluation of pa. Lattice sums Gs x 10-24(cm)3 G-X,
G-Y,
0.0111 0.0168
0.0218 0.0263
G-Y,
G,,,,,,Yr
-0.0297 -0.0300
0.0183 0.0222
G,,,_X3
G,Yr
G,,,,,Yz
G, Y3
0.0482 0.0634
0.0215 0.0202
-0.0299 0.0289
-0.0781 -0.0903
where
0.0221 0.0216 G,$? 0.0212 0.0303
G-Z:
G-Z,
0.0185 0.0154
- 0.0294 -0.0390
GmZ2
GM23
-0.0782 -0.0863
0.0229 0.0269
The Gs are the lattice sums, and are a measure of the induced anisotropy by the uniaxial stress. The Gs in different directions were calculated considering a sphere of radius
x=x*a Y=y
G-Z,
*u
Z=z*c r = [(+)’
R2=X2+Y2+Z2 E,= /LJ~X*- R2>/R5 =~c,Dmx. Similar expressions were derived from E, and Ez. The first factor in eqn (4), viz. E, is due to natural birefringence. The Ds are lattice summations and are a measure of the anisotropy in the undeformed crystal; m represents the positive ion, n the negative ion. The Ds were evaluated in our earlier paper [3]. Equation (4) may be rewritten as
LX*
=C
GmX2
5x
3X2(3 YZ+ 322 - 2X2)
R'
WI
3Y2(YZ+z*-4X*)
LXJ=~
(6b)
R7
R'
and the values are given in Tables l-3. For the evaluation of the strain optical constants, (i) the change in the polarizability of the ions due to uniaxial deformation, and (ii) the change in the number of ions per unit volume were considered. The change in the refractive index due to (i) and (ii) was evaluated, from which the strain optical constants were calculated. (i) It is assumed as a first approximation that (a) the ions in the crystals have a spherical symmetry with respect to the electronic polarizability and that an effective value of a,,, and c(, can be attributed to the positive and negative ions, respectively, and (b) a, and z, are assumed independent of the strains induced in the crystal. The change in the polarizability is expressed by dame.. *,= a,[G,,,,,,X(Y, Z)a, + G,,,.,X(Y, Zkl
W
da,. xj- a,[G,X(Y,
Ub)
daxCY, .-)= da,,
3Z2( Y2 + 22 - 4X’)
'
(64
+ (+u)’ + (+)q”’
Z)a, + G,X(Y,
Z)a,l
2l+ danxCY, zj
(7c)
da, = da,, + dami
(7d)
Table 3. For the evaluation of pM. Lattice sums Gs x 10-24(Cm)3 Crystal ADP KDP
G-X, 0.0638 0.0655
G-X2 -0.0505 -0.0572
Crystal ADP KDP
GMX, 0.0022 0.0067
G,,,,X2 -0.0292 -0.0385
G-X, -0.0347 -0.0341 G,,,,,X, 0.0806 0.1094
G,,,,,,Y, -0.0505 - 0.0572
G, YZ 0.0638 0.0655
G, Yi - 0.0292 -0.0385
G,Yz 0.0022 0.0067
G,,,,,,Y3 - 0.0347 -0.0341 G,Y3
0.0806 0.1094
G,,,Jr -0.0134 -0.0083 G,Zt
0.0270 0.0318
G-Z2 -0.0134 -0.0083 GmZ2
0.0270 0.0318
G-Z3
0.0693 0.0682 Gmz3
-0.1612 -0.2189
Strain optical constants of APD and KDP
917
where N = number of ions ai = polarizability which yields
(i takes values from 1 to 6), where
per
volume
and
GmX(K Z) = GmX,(Y,,&)A, + GmX,(y,,
&)A,
+ G,X,(y,,
%)A,
dn, = [(nf + 2)2/6n,]
and a, and a, from [3] are used. The as take different values depending on the environment in which they are placed as evident from [l 11. In ADP and KDP G_ = G, and G,, = G,, and, depending on the direction of the uniaxial stress, the As take different values. When a uniaxial stress is applied in a direction parallel to the X, axis, then @a) (9b) A, =
6h3c9
(94
where the as are the Poisson ratios. The Poission ratio [6] in an anisotropic substance is given by ahk
=
shk bkk
2
(10)
where h is the direction parallel to the lateral compression, k the direction parallel to the elongation, s, the elastic compliances [7,8] with h and k taking values from 1 to 6. Using eqn (10) the As for different cases are shown in Table 4. Using eqns (7a) and (7b) the change in the polarizability due to positive and negative ions are shown in Table 5 for different cases. (ii) Consider a cube of unit dimension. On the application of uniaxial stress, the change in the volume is do = (A, + A, + AJ) neglecting higher powers of the As. The change in the number of ions per volume is given by
(4~/3) C N da, + (4n/3) C ai dN
x
(8)
[ with i taking values from 1 to 6.
CALCULATIONS of pI19 p12, p13, p31 and ~33 The values of the As from Table 4, and the Gs from Table 1 were used in eqn (7) to evaluate the changes in the polarizabilities of the cations and anions due to the application of a uniaxial stress. The changes in the polarizabilities are shown in Table 5. The strain optical constants p,, , P,~, pI,, p3, and p,, were evaluated using eqns (12) and (I).
Et&&on
Evaluation of pM A rotation of 45” about the X-axis was done, the relation between the transformed coordinates with respect to the original coordinates being
[:;I=[:
_:k
$][j
I’=x m’ = W&)(Y
+ 2)
n’ = (l/J2)(z
+y).
(13)
Table 5. Change in polarizabilities of positive and negative ions due to uniaxial stress das x 1024km)-3 KDP
ADP
dN = (- N,,m’ du)/(Mv2),
(11)
where NA = Avogadro’s number, M = molecular weight, and m’/v = density. The change in the refractive index is obtained by differentiating the Lorentz-Lorenz equation of the form (n; - l)/(nf + 2) = (4n/3) c Nai,
dam eon (7a) PII P21 P3l P33 PI3 Pd.4 PM
0.27936 0.02143~ - 0.30076 0.7106~ -0.35536 -0.435lc -0.1736c
da, eqn (7b)
da, eon (7a)
da, eon (7b)
1.656.~ -2.9681a I.31236 -2.9273~ 1.46276 1.82784 - 5.57336
0.22716 -0.00087c -0.2364~ 0.7123~ -0.3562~ -0.3378c -0.1725~
1.6918r -2.2841~ 0.5921c - I.95756 0.97886 1.66896 -4.6984
Table 4. As for various piis KDP
ADP
1
2
3
PII PZI P31>
--L
-0.115c
0.19&
PII P2L P3I>
--t
P33 PI3 >
0.256~
0.2566
--t
P33 PI3 >
0.02823~
P44 PW
0 t
t --E
--t 0
1 (12)
I
Pd.4 P66
0 E
2
3
-0.0856
0.2823~
0.28236
--t
f --E
--f 0
D. PIUvszh’r,and R. Errua~
918
Table 6. Calculated values of piis are for I = 5890A [9, IO]p,s at I = 58908, ADP
Strain optical constants
Refractive indices used [9]
PII
n, = 1.5242
PZI= PI2 PI3 PI3 P3I Pu = Pss PM,
KDP
Experimental [91
n2 = 1.5242 n, = 1.5242 n, = 1.4789 n, = 1.4789 n4=n5= 1.5011 n6 = 1.5242
Calculated
0.319
0.2711
0.277 0.169 0.167 0.197 -0.058 -0.091
0.317 0.2033 0.2566 0.3208 0.0653 -0.2620
On the application of stress, the deformed lattice sites are given by x’ = I’(1 + A,)
y’=m’(l
+Ar)
z’=n’(l
+Ar).
(14)
pu was calculated by replacing eqn (2) with eqn (14). The values of the Gs and of the corresponding As were taken from Tables 2 and 4, respectively, from which da and dN were calculated. These values were substituted in eqn (1).
Refractive indices used [IO]
Calculated
n, = 1.5094
Experimental I101 0.287
n2 = 1.5094 n, = 1.5094 n, = 1.4682 n3= 1.4682 n, = nJ = 1.4884 n, = 1.5094
0.282 0.174 0.122 0.241 -0.019 -0.064
0.2010 0.1963 0.0977 0.3298 0.0696 - 0.2482
(iii) that the induced anisotropy is attributed to the geometrical anisotropy brought about by the deformation due to the uniaxial stress and the change in number of ions per unit volume, the agreement between the calculated and the experimental values is good. However, there is considerable disparity between the calculated and experimental values in ph( and pM and this could possibly be due to changes in the local field brought about by the piezo-electric and electro-optic effect via the converse piezo-electric effect. The relation between the strain optical constants, piezo-electric and electro-optic constants is given by r$ = r$ + i p,d,,, j-l
Evaluation of psa A rotation of 45” about the Z-axis was done, the relation between the transformed coordinates with respect to the original coordinates being
171’ = (l/J2)0,
0.3812
where the d,s are piezo-electric constants, the ris are primary (true or clamped) electro-optic constants and the r;s are secondary (false or unclamped) electro-optic constants. For the ?2m class of ADP and KDP crystals, the piezo-electric and electro-optic constants are d,4 and &, and r4, and r63, respectively. Therefore,
- x)
n’=z.
(19
For the calculation of pss, eqn (15) was substituted in eqn (14) which in turn replaced eqn (2). The values of the Gs and of the corresponding As were taken from Tables 3 and 4, respectively, from which da and dN were calculated. The values were substituted in eqn (1). RESULTS AND DISCUSSION The calculated values of the strain optical constants and the experimentally reported values (9, IO] are shown in Table 6. In view of the approximations made in the model, namely: (i) that the ions in the crystals have spherical symmetry with respect to the electronic polarizability; (ii) that a, and a, are independent of the strains induced in the crystal; and
and
The deformation due to the above factors has not been incorporated in the present calculation and it is interesting to note that, considering the assumptions made in the present mode, there is a close agreement in the experimental and calculated values. Acknowledgemenf-The authors wish to thank Prof. K. G. Bansigir, former Head, Department of Physics, Jiwaji University, Gwalior, India, for useful discussions.
REFERENCES 1. Webcr H. J., 2. Ozolinsh M., 3. Praveena D., Ethiraj R., J.
Acta crysrallogr. A44, 320 (1988). Mater. Res. Bull. 17, 741 (1982). Siddiqui M. A. A., Kumar G. S. and Muter. Sci. Left. 8, 496 (1989).
Strain optical constants of APD and KDP 4. Kinase W., Uemura Y. and Kikuchi M., J. Phys. Chem. Solid. 30, 441 (1969). 5. Wyckoff R. W. G., Crystals Structures, II. Interscience, New York (1951). 6. Perelomova N. V. and Tagieva M. M.. Problems in Crystal Physics (Edited by M. P. Shaskol’skaya), p. 118. Mir Publishers, Moscow (1983). 7. Zwicker B., Helu. Phys. Acta 19, 523 (1946).
919
8. Price W. J. and Huntington H. B., J. Acousr. Sot. Am. 22, 32 (1950). 9. Narasimhamurty T. S., Veerabhadra Rao K. and Petterson H. E., J. Mater. Sci. 8, 577 (1973). 10. Veerabhadra Rao K. and Narasimhamurty T. S., J. Mater. Sci. 10, 1019 (1975). 11. Tessman J. R., Kahn A. H. and Shockley W., Phys. Rev. 92, 890 (1953).
Vol. 51. No. 8. pp. 915-919. 1990
0022.3697/90
13.00 + 0.00
0 19% Pergamon Press plc
Printed in Great Britain.
STRAIN
OPTICAL
CONSTANTS
OF ADP AND KDP
D. PRAVEENA and R. ETHIFUJ Department of Physics, University College of Science, Osmania University, Hyderabad-500007, India (Received
4 January
1989; accepted in revised firm
3 January 1990)
Abstract-A method to evaluate the strain optical constants of ammonium dihydrogen phosphate (ADP) and potassium dihydrogen phosphate (KDP), which belong to the tetragonal system of class 42m, is presented. The ions are treated as point dipoles which undergo displacement on application of a uniaxial stress. Considering (i) the anisotropies in the arrangements of the dipoles in the deformed crystal and (ii) the change in the density of the crystal, expressions for the change in the refractive indices for different directions of stress are developed and the corresponding strain optical constants (p,s) evaluated. Point dipoles, natural birefringence, artificial birefringence, electronic polarizability. strain optical constants, uniaxial stress, Poisson’s ratio and elastic compliances.
Keywordr:
INTRODUCTION Potassium dihydrogen phosphate (KDP) and ammonium dihydrogen phosphate (ADP) are among the first materials that were used and exploited for their non-linear optical and electro-optical properties. Their popularity still continues and many devices, such as electro-optic modulators and second harmonic generators, are developed using these crystals. This, naturally, stimulated theoretical studies and many models have been proposed as cited in [l]. Any study of the electro-optical effect in these types of crystal involves the strain optical effect via the converse piezo-electric effect [2]. A need was thus felt to understand the strain optical effect, which prompted us to develop an empirical model to explain the phenomena in these crystals. We used the “point dipole model” to establish the change in the refractive index induced by the alteration in the structural parameters due to a uniaxial stress and thus evaluate the various strain optical constants. This work can be viewed as an extension to (i) the mode1 suggested by Praveena et al. to evaluate the electronic polarizabilities of ions in KDP and ADP [3] and (ii) the model used by Kinase et al. to explain the correction to the dipole field due to the lattice deformation of a perovskite-type crystal [4]. Survey of the literature indicates that this work was undertaken for the first time in anisotropic crystals of the KDP type. THEORETICAL The general expression constants pijs is given by pij=
RELATIONS for
-2 an. --_1 rt:
hi
the
strain
optical
From the Pockels’ phenomenological theory of photoelasticity the number of independent nonvanishing strain optical constants for ADP and KDP are seven, viz. pII, p12,pIJ, PJ~, puv pu and Pa. The ions are treated as point dipoles and an ion at a point (x, y, z) shifts to a point (x’, y’, z’) on the application of a uniaxial stress. The new coordinates are given by x’=x(l
+A,)
Y’ = ~(1 + z’
=
A2)
z(l + A1),
(2)
where the As Q 1. The field at a lattice site (0, 0,O) due to a dipole of moment pXat (x’, y’, z’) with the electric vector of the light in the X-direction is given by E, = P~(~.X”~- R’2)/R’5 X’=x’*a Y’=f*Q Z’=z’*c R’2
=
X’2
+
y’2
+
Z”
(3)
and u and c are the lattice constants [5]. Neglecting the higher powers of the As, the above expression is written in terms of the underformed lattice site, as
&-=Ex+ccx
3X2(3 Y’ + 322 - 2x9 A I R7
3 Y2( Y2 + z* - 4X’)
(1)
+ cc,
where i is the direction parallel to the electric vector and j the direction parallel to the strain; i,j take values from 1 to 6.
+ & 915
R’
A2
3Z2( Y2 + 22 - 4X2) A R’ 3,
(4)
916
D.
PRAVEENA
and R. ETHIIW
Table 1. For the evaluation of p,,. pi2. p13.p3, and p33.Lattice sums Gs X tO-2’(cn03 Crystal ADP KDP
G-X,
G-X,
-0.0438 -0.053 1
0.0572 0.0614
Crystal
GmX,
GmX2
ADP KDP
-0.0427 -0.0544
0.0157 0.0226
Crystal ADP KDP
G-X, - 0.0439 - 0.0479
G-X2 0.0112 0.0146
Crystal ADP KDP
G,X, - 0.0428 -0.0505
G,Xr 0.0483 0.0573
G-X3
G,,,.,,Yr
-0.0347 -0.0341
0.0572 0.0614
G-Y2
G,. Yr
G-Z,
- 0.0437 -0.0531
-0.0347 -0.0341
-0.0134 - 0.0083
G,,,,,X3 G,,,,,Y, G,,Y2
G,Y3
G,Zt
0.0806 0.1095
0.0157 0.0226
-0.0427 -0.0544
0.0806 0.1095
0.0270 0.0318
G-A
G-Z3
-0.0139 -0.0083
0.0693 0.0681
GmZ2
G,Z3
0.0270 0.0318
-0.1612 -0.2189
Table 2. For the evaluation of pa. Lattice sums Gs x 10-24(cm)3 G-X,
G-Y,
0.0111 0.0168
0.0218 0.0263
G-Y,
G,,,,,,Yr
-0.0297 -0.0300
0.0183 0.0222
G,,,_X3
G,Yr
G,,,,,Yz
G, Y3
0.0482 0.0634
0.0215 0.0202
-0.0299 0.0289
-0.0781 -0.0903
where
0.0221 0.0216 G,$? 0.0212 0.0303
G-Z:
G-Z,
0.0185 0.0154
- 0.0294 -0.0390
GmZ2
GM23
-0.0782 -0.0863
0.0229 0.0269
The Gs are the lattice sums, and are a measure of the induced anisotropy by the uniaxial stress. The Gs in different directions were calculated considering a sphere of radius
x=x*a Y=y
G-Z,
*u
Z=z*c r = [(+)’
R2=X2+Y2+Z2 E,= /LJ~X*- R2>/R5 =~c,Dmx. Similar expressions were derived from E, and Ez. The first factor in eqn (4), viz. E, is due to natural birefringence. The Ds are lattice summations and are a measure of the anisotropy in the undeformed crystal; m represents the positive ion, n the negative ion. The Ds were evaluated in our earlier paper [3]. Equation (4) may be rewritten as
LX*
=C
GmX2
5x
3X2(3 YZ+ 322 - 2X2)
R'
WI
3Y2(YZ+z*-4X*)
LXJ=~
(6b)
R7
R'
and the values are given in Tables l-3. For the evaluation of the strain optical constants, (i) the change in the polarizability of the ions due to uniaxial deformation, and (ii) the change in the number of ions per unit volume were considered. The change in the refractive index due to (i) and (ii) was evaluated, from which the strain optical constants were calculated. (i) It is assumed as a first approximation that (a) the ions in the crystals have a spherical symmetry with respect to the electronic polarizability and that an effective value of a,,, and c(, can be attributed to the positive and negative ions, respectively, and (b) a, and z, are assumed independent of the strains induced in the crystal. The change in the polarizability is expressed by dame.. *,= a,[G,,,,,,X(Y, Z)a, + G,,,.,X(Y, Zkl
W
da,. xj- a,[G,X(Y,
Ub)
daxCY, .-)= da,,
3Z2( Y2 + 22 - 4X’)
'
(64
+ (+u)’ + (+)q”’
Z)a, + G,X(Y,
Z)a,l
2l+ danxCY, zj
(7c)
da, = da,, + dami
(7d)
Table 3. For the evaluation of pM. Lattice sums Gs x 10-24(Cm)3 Crystal ADP KDP
G-X, 0.0638 0.0655
G-X2 -0.0505 -0.0572
Crystal ADP KDP
GMX, 0.0022 0.0067
G,,,,X2 -0.0292 -0.0385
G-X, -0.0347 -0.0341 G,,,,,X, 0.0806 0.1094
G,,,,,,Y, -0.0505 - 0.0572
G, YZ 0.0638 0.0655
G, Yi - 0.0292 -0.0385
G,Yz 0.0022 0.0067
G,,,,,,Y3 - 0.0347 -0.0341 G,Y3
0.0806 0.1094
G,,,Jr -0.0134 -0.0083 G,Zt
0.0270 0.0318
G-Z2 -0.0134 -0.0083 GmZ2
0.0270 0.0318
G-Z3
0.0693 0.0682 Gmz3
-0.1612 -0.2189
Strain optical constants of APD and KDP
917
where N = number of ions ai = polarizability which yields
(i takes values from 1 to 6), where
per
volume
and
GmX(K Z) = GmX,(Y,,&)A, + GmX,(y,,
&)A,
+ G,X,(y,,
%)A,
dn, = [(nf + 2)2/6n,]
and a, and a, from [3] are used. The as take different values depending on the environment in which they are placed as evident from [l 11. In ADP and KDP G_ = G, and G,, = G,, and, depending on the direction of the uniaxial stress, the As take different values. When a uniaxial stress is applied in a direction parallel to the X, axis, then @a) (9b) A, =
6h3c9
(94
where the as are the Poisson ratios. The Poission ratio [6] in an anisotropic substance is given by ahk
=
shk bkk
2
(10)
where h is the direction parallel to the lateral compression, k the direction parallel to the elongation, s, the elastic compliances [7,8] with h and k taking values from 1 to 6. Using eqn (10) the As for different cases are shown in Table 4. Using eqns (7a) and (7b) the change in the polarizability due to positive and negative ions are shown in Table 5 for different cases. (ii) Consider a cube of unit dimension. On the application of uniaxial stress, the change in the volume is do = (A, + A, + AJ) neglecting higher powers of the As. The change in the number of ions per volume is given by
(4~/3) C N da, + (4n/3) C ai dN
x
(8)
[ with i taking values from 1 to 6.
CALCULATIONS of pI19 p12, p13, p31 and ~33 The values of the As from Table 4, and the Gs from Table 1 were used in eqn (7) to evaluate the changes in the polarizabilities of the cations and anions due to the application of a uniaxial stress. The changes in the polarizabilities are shown in Table 5. The strain optical constants p,, , P,~, pI,, p3, and p,, were evaluated using eqns (12) and (I).
Et&&on
Evaluation of pM A rotation of 45” about the X-axis was done, the relation between the transformed coordinates with respect to the original coordinates being
[:;I=[:
_:k
$][j
I’=x m’ = W&)(Y
+ 2)
n’ = (l/J2)(z
+y).
(13)
Table 5. Change in polarizabilities of positive and negative ions due to uniaxial stress das x 1024km)-3 KDP
ADP
dN = (- N,,m’ du)/(Mv2),
(11)
where NA = Avogadro’s number, M = molecular weight, and m’/v = density. The change in the refractive index is obtained by differentiating the Lorentz-Lorenz equation of the form (n; - l)/(nf + 2) = (4n/3) c Nai,
dam eon (7a) PII P21 P3l P33 PI3 Pd.4 PM
0.27936 0.02143~ - 0.30076 0.7106~ -0.35536 -0.435lc -0.1736c
da, eqn (7b)
da, eon (7a)
da, eon (7b)
1.656.~ -2.9681a I.31236 -2.9273~ 1.46276 1.82784 - 5.57336
0.22716 -0.00087c -0.2364~ 0.7123~ -0.3562~ -0.3378c -0.1725~
1.6918r -2.2841~ 0.5921c - I.95756 0.97886 1.66896 -4.6984
Table 4. As for various piis KDP
ADP
1
2
3
PII PZI P31>
--L
-0.115c
0.19&
PII P2L P3I>
--t
P33 PI3 >
0.256~
0.2566
--t
P33 PI3 >
0.02823~
P44 PW
0 t
t --E
--t 0
1 (12)
I
Pd.4 P66
0 E
2
3
-0.0856
0.2823~
0.28236
--t
f --E
--f 0
D. PIUvszh’r,and R. Errua~
918
Table 6. Calculated values of piis are for I = 5890A [9, IO]p,s at I = 58908, ADP
Strain optical constants
Refractive indices used [9]
PII
n, = 1.5242
PZI= PI2 PI3 PI3 P3I Pu = Pss PM,
KDP
Experimental [91
n2 = 1.5242 n, = 1.5242 n, = 1.4789 n, = 1.4789 n4=n5= 1.5011 n6 = 1.5242
Calculated
0.319
0.2711
0.277 0.169 0.167 0.197 -0.058 -0.091
0.317 0.2033 0.2566 0.3208 0.0653 -0.2620
On the application of stress, the deformed lattice sites are given by x’ = I’(1 + A,)
y’=m’(l
+Ar)
z’=n’(l
+Ar).
(14)
pu was calculated by replacing eqn (2) with eqn (14). The values of the Gs and of the corresponding As were taken from Tables 2 and 4, respectively, from which da and dN were calculated. These values were substituted in eqn (1).
Refractive indices used [IO]
Calculated
n, = 1.5094
Experimental I101 0.287
n2 = 1.5094 n, = 1.5094 n, = 1.4682 n3= 1.4682 n, = nJ = 1.4884 n, = 1.5094
0.282 0.174 0.122 0.241 -0.019 -0.064
0.2010 0.1963 0.0977 0.3298 0.0696 - 0.2482
(iii) that the induced anisotropy is attributed to the geometrical anisotropy brought about by the deformation due to the uniaxial stress and the change in number of ions per unit volume, the agreement between the calculated and the experimental values is good. However, there is considerable disparity between the calculated and experimental values in ph( and pM and this could possibly be due to changes in the local field brought about by the piezo-electric and electro-optic effect via the converse piezo-electric effect. The relation between the strain optical constants, piezo-electric and electro-optic constants is given by r$ = r$ + i p,d,,, j-l
Evaluation of psa A rotation of 45” about the Z-axis was done, the relation between the transformed coordinates with respect to the original coordinates being
171’ = (l/J2)0,
0.3812
where the d,s are piezo-electric constants, the ris are primary (true or clamped) electro-optic constants and the r;s are secondary (false or unclamped) electro-optic constants. For the ?2m class of ADP and KDP crystals, the piezo-electric and electro-optic constants are d,4 and &, and r4, and r63, respectively. Therefore,
- x)
n’=z.
(19
For the calculation of pss, eqn (15) was substituted in eqn (14) which in turn replaced eqn (2). The values of the Gs and of the corresponding As were taken from Tables 3 and 4, respectively, from which da and dN were calculated. The values were substituted in eqn (1). RESULTS AND DISCUSSION The calculated values of the strain optical constants and the experimentally reported values (9, IO] are shown in Table 6. In view of the approximations made in the model, namely: (i) that the ions in the crystals have spherical symmetry with respect to the electronic polarizability; (ii) that a, and a, are independent of the strains induced in the crystal; and
and
The deformation due to the above factors has not been incorporated in the present calculation and it is interesting to note that, considering the assumptions made in the present mode, there is a close agreement in the experimental and calculated values. Acknowledgemenf-The authors wish to thank Prof. K. G. Bansigir, former Head, Department of Physics, Jiwaji University, Gwalior, India, for useful discussions.
REFERENCES 1. Webcr H. J., 2. Ozolinsh M., 3. Praveena D., Ethiraj R., J.
Acta crysrallogr. A44, 320 (1988). Mater. Res. Bull. 17, 741 (1982). Siddiqui M. A. A., Kumar G. S. and Muter. Sci. Left. 8, 496 (1989).
Strain optical constants of APD and KDP 4. Kinase W., Uemura Y. and Kikuchi M., J. Phys. Chem. Solid. 30, 441 (1969). 5. Wyckoff R. W. G., Crystals Structures, II. Interscience, New York (1951). 6. Perelomova N. V. and Tagieva M. M.. Problems in Crystal Physics (Edited by M. P. Shaskol’skaya), p. 118. Mir Publishers, Moscow (1983). 7. Zwicker B., Helu. Phys. Acta 19, 523 (1946).
919
8. Price W. J. and Huntington H. B., J. Acousr. Sot. Am. 22, 32 (1950). 9. Narasimhamurty T. S., Veerabhadra Rao K. and Petterson H. E., J. Mater. Sci. 8, 577 (1973). 10. Veerabhadra Rao K. and Narasimhamurty T. S., J. Mater. Sci. 10, 1019 (1975). 11. Tessman J. R., Kahn A. H. and Shockley W., Phys. Rev. 92, 890 (1953).