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Dielectric theory of strained molecular crystals: Elasto-optic coefficients of orthorhombic sulphur
Chemical Physics 39 (1979) 165-173 0 h’orth-Holland Publishing Company
DIELECTRIC ELASTO-OPTIC
THEORY
OF STRAINED
COEFFICIENTS
P.J. BOUNDS* and R.W.
Received 19 Dccembcr
MOLECULAR
CRYSTALS:
OF ORTHORHOMBIC
SULPHUR
MUNN
1978
The gcncral dielectric theory of molecular crystals is estended to calculate the strain derivatives of dielectric propertics, assuming n fised pohrizability relative to the molecular axes. The molecular rotation due to strain makes an importnnt contribution, for which an empirical prescription is given. To illustrate the theory, the elasto-optic coefficients for orthorhombic sulphur are calculated. They agee well with experiment when each sulphur atom is treated as 2 polarizable point. It is concluded that the polarizability can be treated 2s Iised, for esample in deducing changes of molecular orienta-
tion from birefringcnce measurements through phase transitions.
1. Introduction The t!teory of the dielectric properties of molecular crystals has recently been studied in some detail: representative methods are described in refs. [l-j] . Most of this work has been aimed at understanding the properties of fixed crystal structures rather than variation of such properties with strains in those structures. However, strain is a useful and important external constraint, arising not only from applied stresses but also from changes of temperature. A theory of the strain dependence of the dielectric properties of molecular crystals is therefore necessary to understand the pressure and temperature dependence of the dielectric constant and properties such as the elasto-optic coefficients_ In this paper we develop a treatment of the basic features of such strain-dependent phenomena, restricted for simplicity to effects linear in strain. The principles are exemplified by a calculation of the elasto-optic coefficients for orthorhombic sulphur. Because the present treatment aims to provide a microscopic interpretation of the macroscopic strain dependence of dielectric properties, we first consider * Present address: Division of Chemistry, Council of Canada,
Ottawa,
National Canada KlA OR6.
Research
macroscopic strain and its effects at a molecular level. We then derive the basic equations governing the elasto-optic coefficients_ Assuming that the molecular polarizability rotates rigidly with the molecule, we can express the strain derivatives of the polarizability in terms of the pdlarizability itself. The polarizability has to be fixed by ad hoc numerical assumptions or by additional experiments [ 1,2,4-71, but no further information is required to determine the strain derivatives. We then calculate the elasto-optic coeffcients for orthorhombic sulphur and compare the results with a complete set of values measured with a combination of static and dynamic methods [S] _
2. Strain A homogeneous
deformation
of an elastic continu-
um is described by a tensor D, which carries a vector x in the reference configuration into the vectorx’ in
the strained configuration: x’= D-x.
(1)
In general, this defomtation comprises not only a pure strain but also a bodily rotation which is of no significance in the absence of external constraints and so has to be eliminated. The separation.of strain and
166
P.J. Bounds, R.N. MmnfDielectric
rotation can be achieved in various ways [9], but to first order it is sufficient to take the strain as the symmetric part e of the displacement gradient tensor u = D- 1: e = f(u +Z),
(2)
where tfte tilde denotes the transpose. Higher-order effects are not considered in this paper. A crystal is of course not a continuum, but if all subfattices are at special positions only ftomogeneous deformations need be treated. If sublattices are at general positions, it becomes necessary to take account of further irzrerrz~lstrnifzs relative to the hoinogeneousfy deformed lattice. The specification of internal strain is not unique [IO] , but in molecular crystals it is usually convenient to specify the translations and rotations of molecules. in general, molecular deformations have also to be included, but in the present work the molecules are taken to be rigid. The problem wit11tfte internal strains is knowing what they are: they are fixed implicitly by tfte flomogeneous deformation or external strabl because at equilibrium they have to minimize an appropriate energy function [IO] , but knowledge of intermolecular poientiafs may be insufficient to calculate internal strains this way, and direct experimental measurements are seldom available. One may tfterefore have to resort to some empirical prescription for determining internal strains from externa1 strains. The main internal strain required is the molecular rotation_ The external strain rotates all vectors in the crystal (except those parallel to the principal axes of strain) even when there is no bodily rotation, and so could be used to fix the rotation of the molecular axes, except that the strain also distorts the axes so tftat they do not remain orthogonal. For anthracene this problem was treated [ 1 I] by assuming tflat on distortion one speciffed axis remained a molecular axis and with a second specified axis continued to define the plane of the two rotated axes. We use a similar assumption for orthorhombic suIphur, but derive its consequences algebraically rather than numerically. The results are therefore of genera1 appficability. The direction cosine matrix in the reference configuration is taken to have components a,, , where czis a crystal axis and A a molecular axis. The columns of a thus correspond to unit vectors parallel to tile
theory of srrained molecular crystals
molecular axes. After a deformation given by eq. (1) the new vectors form a new matrix a’ given by a’ = D-a.
(3)
Differentiation with respect to the displacement gradients u then gives aQh/all(rr
= &uoQTA
_
(4)
We denote the nlolecuIar axes by L. M and iv, and assume that the new L axis is parallel to the deformed L axis while the new N axis is perpendicular to the deformed L and M axes; this fixes tfte new M axis. The new direction cosine matrix is A. h’ow from our assumptions, (5)
A al. = akr. /ALI wflere h, is the stretcfl of the L axis, given by
(6) (summation is implied over repeated Greek subscripts). On differentiation we obtain, using eq. (4), d‘%,.WJ,
= ($6,(i
- ahL Q~,)Q&~,
(7)
which has to be evaluated in the reference configuration u + 0, which implies a’ + a and AL + 1, leaving aA,Liarlor
= (“,, - apLa,L)u~L -
(8)
For the new M axis we have A crx = Q>r&r
,
(9)
where ai2\, is given by the vector product of the deformed L and ill axes: a&
=f spy
’ Q;M’ Qpl.
(10)
with Eap_, the alternating tensor. An extension of the
algebra used for the L axis leads to the result for u + 0
By expressing aOL and apnr in terms of products of direction cosines and using the orthonormality property ‘CTL‘TL ’ aoManlT + ‘ohraihr = ‘UT*
we reduce eq. (11) to the form
(12)
P.J. Bomds. R. IV. Mmzrt/Dielectric theory of strained r~~obmchrcrystals
aA&,&)l& = -Q&
(13).
- ~&,\P,,,).
The new ,V axis is written as the vector product of the IV and L ases, leading to the u -+ 0 result
167
the numerical results for anthracene [I l] .
3. Methods of calculation
aA&l allo7 = EoPr kyL aA,,,/acf(rr f aplVa+.lallD71
-
3. I. Dielecrric rheoty
(14) The general expression for the electric susceptibility is 1121
Substitution f&m eqs. (8) and (13) and manipulaCons of the sort already described give eventually (15)
aA~Jal~Br =a,rh1a0A,Q7nr- a,L.aDhfaTL-
01)
x= F$&,
The rotation tensor is given by R = A-a-‘,
(16)
so that its dependence on strain is given by
is given by CY~/EOU, where ‘Ye is the effective polar-
aRL2hlaLiOT= @~aqh&)Q (summation
%J%z
(17) M over subscripts A imphed). The resuIt is
= 4&L%.
- $0ada7L
-
%TQd%~
where the sum runs over all subIattices k in the primitive unit cell. The reduced effective polarizability PI;
+ uL.N%
+aOAraTL(adakV - ati,,a,,)-
izability of the molecules on sublattice k and u is the volume of the primitive unit cell. The local-field tenSOTdk gives the IocaI electric field at sublattice k in terms of the macroscopic electric field, and can be obtained as [ 1,2,7]
(18) (23)
dk=.$kl&, Since Tr R = R, is invariant and equal to 3, it should be independent of strain, and eq. (18) confirms that = 0. The first-order rotation is also antiaR,,lau, symmetric, and eq. (18) duly yields a&_/au, = -aR,,/ht,,. kfence calcutating aR/i3u is less fotmidable than it might seem, since R has only three independent non-zero components. For calculations, eq.. (18) is conveniently written as aRaXlaIiC = ~,,~y,x - 8L1TYCTh + %JCKD- %oXC%r i- Z,J-&
- Z,) 3
(19)
where X& = sot apt,
Yap = awvnpIjTand 2, = f7dQp~ are readily calculable elements of matrices, X and Y being symmetric. The rotation tensor R as calculated includes the contribution of any bodily rotation arising from the antisymmetric part of u. However, because of its antisymmetry, this contribution is eliminated when eq. (19) is symmetrized to give the derivative of R with respect to e. The result is %J%T
= f [6,, ‘VT& f sQT‘Ubh- k&6,
+ (Z,+ ‘ZrO) (2,X - Z,)l1
- 1vm60, (20)
where W = X - Y. These algebraic results agree with
where the supermatrix A has submatrices n kk’ =1Q&L&J.
_
(23)
the LkK being Lotentz-factor tensors. Given the Lkk,, calculated from the crystal structure [ 13) , the pk determine the d, and vice versa. Then given the strain derivatives of the Lkk [14J, the strain derivatives of the & determine those of the d, and hence by eq. (21) the strain derivative of x_ Ortborbombic sulphur crystallizes in the space group Fddd with four molecuIes in the primitive unit cell [IS] . The molecules are related in pairs by centres of symmetry, and the pairs ate related by twofold axes parallel to the crystallographic a and b axes. These relationships allow the same expressions to be used as for pyrene [ 121 : if molecules 1 and 2 are related by inversion they have identical effective polarliabilities and local fields. If molecule I is carried into molecule 3 by the twofold axis parallel to Q then the poIarizabilities can be written as $;I = Mt + M3*p,
(24)
#;t_=M;
(25)
Here
iM3*p-!
P.J. Bozczzds.R. IV_Mzzrz~z/Dielccrric rlzmr_sof straitzed zzzoleczdarcr_rsrats
168
M, ‘Ml1 +M12’
(26)
Mj = Ml3 +Mt4>
(27)
M; = M,, + M,,,
(25)
with Mali. equal to x -’ + LkK_ The matrix p decoupies the contributions of the two sets of sublattices and is defined as (W
P=Pj-d,.(Pl.dl)-‘-
Because sublattices 1 and 3 are crystallographically equivalent, Bj is obtained from Bt by orthogonal transformation with the matrix lJ describing the rotation about a. Eqs. (24) and (35) are then consistent only if
p-1 =.lJ*p*u.
(JO)
The form of p entailed by this equation has been given elsewhere [ 1,.S]; four components qre independent. Since polarizabilities are symmetric, the product M3 -p must be symmetric, and this leaves two components of p undetermined. Finally, in orthorhombic sulphur the molecules retain a twofold axis of symmetry parallel to c. This leaves only one component of p not determined by symmetry and allows Pf’ to be written as A(ltq2p p;’ = M, + (i(AB)“’ 10
q(AB)“z
0
B( I+(;‘)“’
0 )
0
c1
(31)
where 4 is the arbitrary parameter and A, B and Care the diagona1 elements of M . TJie parameter 4 can in irinciple be determined from Stark spectra [6] or Raman intensities [7] _ Ctherwise, the choice 4 = 0 is algebraically convenient but quite arbitrary [I 1,12]_ To obtain reasonable polarizabiliries from x it is usually necessary to take account of the molecular size, shape, and orientation. This is achieved by representing a molecuIe as a set of polarizablepoint submolecules [4,5] _The foregoing algebra can be retained if the Lorentz-factor tensors are calculated as averages over all pairs of submolecules on the two sublattices in question (eseluding contributions within the same molecule), which corresponds to averaging the local field over the submolecules. For orthorhombic sulphur we locate submolecules
at each sulphur atom. The resulting
polarizabilities are better than those calculated for a single point at the centre of each molecule, and differ little from those calculated for submolecules at the centre of each sulphur-sulphur bond in the molecule
[161. 3.2. E!asto-optic
coejjkietm
The fourth-rank elasto-optic tensor describes the strain dependence of the inverse relative permittivity tensor or indicatris: p = a&cljae.
(32)
Since L= ,1 + x, the elasto-optic tensor can be expressed in terms of 1 as p apP
= -0
+ &(ax,,moT)
(1 + XI;;,
63
this and similar strain derivatives the inner products relate not to the strain but to the quantity
where in
differentiated
with respect to strain.
The strain derivative of r is conveniently by writing eq. (21) in the form
evaluated
x= c (fWkX-” k.k’
(34)
where band d are supermatrices pk-Skpand dkli._ Here
having submatrices
dke = (1 - L-&“,
(35)
with the supermatrices 1 and L having submatrices 1Ske and LkK, so that together these equations yield
x= L_ cp-’ - L)&
(36)
The strain derivative of x is then ““+ aeor
[(p-t _ L)-t.($$) .. t
OT
. (p-1 _ L)-t
(37)
I kKY which can be manipulated to yield eventually ax=C; aear k
a$, .-.dk+%_-p_--
k aeor
a Lkr
k,a
d
d
aeo7
*Q.$. (38)
Results of this form have been obtained previously [71. Calculating
the elasto-optic coefficients requires
P.J. Bowuis. R. I%!dlutot/Dielectricrheorjfof strained tnolecrrlarcrJ’stals
. 169
the strain derivatives of the polarizability
and of the Lorentz-factor tensors, as eq. (35) shows, but not of the strain derivative of the local field tensor, as noted
proximation consonant with the weak intermolecular interactions in molecular crystals, and is probably less drastic than that used to determine the molecular ro-
above.
tation due to external strain. The effect of this rotation on the polarizability is obtained starting from tire polarizability in the strained crystal, R-wR-~. The result in the limit R + 1 is
The elasto-optic
coeflicients can also be calculated by espressing p in terms of one subIattice polarizability and p(cf. ref. [I I]), thus requiring a knowledge of the strain dependence of p. Although p itself is not uniquely defined, for a given p the &ruin derivative is determined by thdse of the polariznbilities and Lorentz-factor tensors, through eqs. (19) and (35). In practice, ap/ae is fixed by requiring ax/se to lwe the proper symmetry. This justities the neglect of the spurious non-zero ab and bc’ elements of x found in our birefringence calculations on anthracene and phenanthrene [ 1 l] ; these elements arise for one sublattice but are cancclled out by equal and opposite elements for the other sublattice, as required by sym-
where X is summed; the last result uses the symmetry of p and the antisymmetry of R. The direction cosine matrix for the molecular on sublattice 1 in orthorhombic sulphur is cos Q
metry. The present arguments show that our asump-
I
tion of e p independent of strain was inconsistent with the assumed strain derivatives of br; and Lx_~,_ If ap/ae is required, it is best calculated from eq. (29) and the contributions to eq. (38) for the various sublattices X-. Experimentally, ap/ae could be determined from the strain dependence of the extra messurements which deterntine p itself.
at = -sin@
3.3. Srraitzdepettdetm
of palarizability
The reduced effective polnrizability p depends on strain through the unit-cell volume u as well as through the molecular polarizability. Separating these effects, we obtain for a given sublattice
ablae,,, = -fisaT + ( 1/eov)adaeaT,
sin@
O\
COSQ
0
I
axes
(42j
, i
\o
I/
0
where Q = ir//2 + tan-‘(b/a) [ 17]_ We use this to evaluate the strain derivatives of R from eq. (20), taking the L axis as the molecular axis parallel to the crystal c axis and the M axis as the other axis in the mean plane of the free molecule (the plane perpendicular to the fourfold ssis). The results are sin 24
aRabIaem
1
cos29
= + ~0~2~5 -sin?@ 0
0
0 0 0
i )
(43)
(3%
since a In v/at=,, = IS,, [ 141. The strain dependence of oLarises from the molecular rotation induced by e and from changes in the polarizability itself in the molecular axes. It is not clear bow one should interpret changes in the ttzolecrdar polarizability with e-yrertzals!rain, and it is consistent with our assumption that the molecules are mechanically rigid to assume also that they are electronically rigid. Hence the polarizability referred to the nzoleatlar m-es is assmed to be itzdepetzdetrtof strain. This assumption has previously been used to calculate the temperature de-
pendence of birefringence in anthracene [ 1 I] and to calculate lattice Raman intensities [7]. It is an ap-
(44
I I
/O O O\
aR,,laem
=$ 0
0
1 ,
0
1
0
(45)
all referring to sublattice 1. For sublattice 3, @has to be replaced by -Q in eq. (42) and hence in eq. (43).
P.J. 6oumk.
170
3.4. Strait1depaldeme
R. IV. Mum fDie[ectric theory of srraiiied ?nolecdar crystak
ofLorem-firctor temors
The calculation of the strain derivatives oM,Jkk’) foi a homogeneous strain Ilas been = aLd(kk’)/ae,, describ$d elsewhere [ 14]_ This method would suffice if we treated each sulphur molecule as a single polarhable point at 3 special position in the unit cell. However, when the molecule is treated as eight submolecules, we have to sllow for’ the effect bf the rotation due to strain on the iorentz-factor tensors_ We have also to recognise that a homogeneous deformation of all atomic positions would change the size and shape of the molecule, which we have assumed to be rigid. The centre of mass of the molecule is not at a completely symmetry-determined position, having an arbitrary z component. There is therefore an additional possible internal strain in this direction superimposed on the homogeneous strain. We neglect this internal strain [ 1 I] , seeing no plausible way of prescribing it empirically. It could be determined espcrimentally or from 3n intermolecular potential. An external strain thus causes a homogeneous defortnation of the centres of mm and rigid rotations about the centres. We write the position of atom i on moIecule k in unit cell Ix r(lki) = r(K) t s(k) )
(46)
where f(lk) is the position of the ccntre of nisss. Af-
ter J hqnwgeneous deformtion D with its associated rotations Rk, the new atomic position is r’(Iki) = Do = Dv(lki)
+ Rk.s(ki) - D-s(k)
+ Rk-s(k).
(47)
(48)
By introducing r(K-i) in eq. (48) we show that the desired deformation ~911be treated as a hon~ogeneous defamation operating on a11atoms, combined with internal strains to return the tttonis to their original positions relative to the centre of mss and a rotation to reorient the molecules. The honlogeneous deformation cm be treated by the previous methods [ 14]_ The averaged Lorcntz-factor tensors are given by L(kk’) = s-’
C
[ L(ki. k’i’) - I(k. ii’)6kK] ,
(49)
i.i
where s iS the number of atoms per molecule. The L(kI: k’i’) are Lorentz-factor tensors caiculated in the usual way but with each atom ki taken as a sublattice.
The other term subtracts the contribution of the dipole interaction between atoms i and i’ on the some molecule. We denote the total derivatiye ofL&k’) with respect to strain ear by a script
?I,p,Jkk’)
to
distinguish it from the homogeneous strain derivative. The result is
op,m(kk’)= s-‘g (1c .(
~~.,i(ki. k’i’)
- $ [LMJki. k’i’)sT(ki. k’i’) + LapT(ki,k’i’)sO(ki,X-Ii’)] + [L&i.
x
k’i’) - Z~$k. ii’&,
aR#) ___ [ aem
1
aR$ (x-7 s,(C) - ----ss,(k’i’) ae wr
,
w-9
11
where the L ,&X.
k’i’) are qundrupole lattice sums and s(ki, k’i’) = s(h) - s(k’i’). The form of the second terns arises from the synumetrization of derivatives with respect to u to yield those with respect to e_ The homogeneous strain derivatives I!?@,,, are synimetric under interchange of c@and m [ 141 t but the other terms in eq. (50) lack this syn~metry and hence so do the total strain derivatives_ [ 131
For completeness,
we note that any internal dis-
placement w(k) of the centres of niass is added to eqs. (47) and (48). It contributes to the strain derivative in eq. (50) in the sane way as the rotation. but is independent of i and i’_The sunmation can then be performed
L&M’>
[
to yield the extra contribution
akfp) -$y 07
altry - ~ ae Oi-
(51)
1’
where L ap7(kk’) is the averaged quadrupole SNIHexcluding contributions
from within the same molecule.
4. Results Averaged Lorentz-factor tensors calculated from the structure given in ref. [ 151 are shown in table l_
These have been used to calculate the effective polarimbilities for a range of values of the parameter 4 in eq. (3 I), taking the static susceptibilities
3s x,
=
Lorentzrbctor tcnsor components Lop(M’), aucrsgud over the atoms for orthorhombic sulphur, referred to crystal wx. Components nut quoted arc zero by symmetry ~___ ~~
aa
11
---~-
0.462 0.676 0.348 0.074
12 i3 14
ab = ba
bb
cc
-0.178 0.643 0 0 -___
0.542 0.538 0.148 0.154
-0.004 -0.214 0.504 0.712
2.68, ,$,b = 2.8 1 and X,, = 3.66 [IS] *_ The component of the polarizability parallel to the crystal c axis is independent of 4, but the other components vary considerably. When the polarizabilities are transformed to the molecular axis system, it is found that the two components in the mean molecular plane (which are equal by symmetry in the free molecule) are closest for small values of 4 [ 161. We therefore take 4 = 0 for further calculations. The choice 4 = 0 makes the matrix pof eq. (29) equal to the unit matrix. The product ok - dk is then the same for all four sublattices k, and equal to $ x by eq. (21). Substitution of this result in eq. (38) yields 16 ax-’ --$-=t;~_$$, OT
(52) OT
.
’
LIT
which can be used directly in eq. (33) rewritten as Po~,or=(l+~-')~
!g(l
+f');;.
(53)
OT
Application of this equation is simplified by the fact that x is diagonal under orthorhombic symmetry, so that p is simply proportional to the corresponding strain derivative of 1-l. The symmetry relationships between the sublattices simplify the application of eq. (52) too, because the SUI:IS over k consist either of four equal terms or of terms which are equal and opposite in pairs. Numerical results for the strain derivatives are given in table 2, where B4wT = a(Pk&$aeoT, * Tbcse data, obtained by an immersion method, agree well wirh thoseofref.
[191.
Strain derivatives of the inverse polarizability and of the Lorentz-factor tensors, and calculated and esperimentsl e&to-optic coefficients for orthorhombic sulphur
Gllc.
apt.
0.50 0.40 0.31 0.34 0.41 0.38
0.32 f 0.31 c 0.27 t 0.27 f. 0.30? 0.31 k
0.10 0.41
0.20 k 0.0 1 0.23 + 0.01 0.37 2 0.01
0.11 0.04 0.17
0.14 + 0.04 0.02 + 0.03 0.12 f 0.04
______
_..__ ~~_~~ ~_~.------
aa, *a aa, bb
2.597 3.509 3.053 3.262 2.350 2.806 2.150 2.150
-1.159 0.513 0.611 0.770 -0.695 0.036
2.150
-0.537 -1.098 -0.75 I -1.312
MI,CC
bb,aa bb, bb bb, cc cc.aa cc. bb
cc, cc bc. bc ac, ac ab. ab --___
-0.328 -0.452 -0.025
I.475 0.760
.O.‘l
[S J 0.01 0.015 0.01 0.01
0.015 0.01-
and qM,or
=T
C!f@,J”“‘).
(54)
Only terms which give non-zero net contribution are shown and these are the same for all k; there are also non-zero terms in which au. bb and cc are combined with ab orac (in either order), but these average to zero as explained above. The polarizability derivatives for the extensional strains are a11of similar size, because the first term in eq. (39) dominates and fl is not very anisotropic. The major effect is therefore the “dilution” of the polarizability density through the change in unit cell volume, leaving only minor contributions from molecular reorientation_ The Lorentzfactor tensor derivatives vary considerably in magnitude and may have either sign, but are mostly smaller than the polarizability derivatives. The experimental e&to-optic coefficients for orthorhornbic sulphur [8] are shown in table 2 with their quoted uncertainties. They were obtained by an ultrasonic diffraction method (supplemented by static measurements to determine the signs of the coefficients). Such dynamic methods produce a spatially-varying displacement rather than a pure strain, and there is then a contribution to the susceptibility change from the antisymmetric part of the displacement gradients u, corresponding to a local rotation [20]
172 The antisymmetric given by [20]
P.J_Bounds. R. IV_Mwm/Dielec~rictheory of strainedn~olenrlarcrwtafs
part of the elasto-optic tensor is
which in the principal dielectric axes reduces to (56) kit - E,-,‘) U&i = :(6,,spr - S&J (no summation implied)_ For orthorhombic sulphur we calculate’ahc,6, = -0.024, aac,ac = -0.029, and aab.ab = -0.005. These are no larger than the esperimental uncertainties shown in table 2 and will’therefore not be considered further_ The calculated elasto-optic coefficients are in reasonable agreement with the experimental ones. The experiments indicate few marked differences except that the coefficients involving shear strGns are significantiy smaller than the others. which are all about 0.3. The calculations show more vuriution between coefficients, but do indicate that those for shear strains are generally the smallest_ Most of the calculated (static) coefficients are larger than those measured (at ultrasonic frequencies). This suggests that there could be some systematic effect of frequency. At higher frequencies, fl and 1 are larger [2 I] and their inverses smaller_ If their strain derivatives behave similarly, one would expect the calculated coefficients to be larger than the measured ones, as observed_ The observed trend might also be partly attributable to the assumed form of the molecular rotation due to strain. More detailed examination shows that the calculations reproduce the observed orderings of the three coefficients for L$ = aa and cc and for UT= aa, while those for I@ = 66 and or = 66 are incorrect only becwse pbb 66 is quoted as less than ~aa,~~ and pbb,cc, whereas tileexperimental uncertainties permit the opposite ordering_ The near constancy of the coefticients for extensional strains can be attributed to the large and relatively isotropic polarizability of tbe S8 molecule. The calculations agree well with experinlent cnly when the molecule is treated as eight points. A singlepoint treatment is much simpler, and yields acceptable if somewhat poorer polarizabilities [ 161, but the calcdated elasto-optic coefticienrs bear little resemblance to the experimental ones except in sign and order of magnitude_ This is partly because of a
more anisotropic polariznbility, but-the major factor is greatly different Lorentz-factor tensor derivatives, lnany opposite in sign to those for the eight-point treatment_ It follows that calculation of the elastooptic coefficients is a more severe test of the model than calculation of the polarizability.
5. Discussion Previous calculations of elasto-optic coefficients have concentrated on cubic crystals, mostly ionic ones. It has generally been found difficult to explain even the signs of the coefficients {paamaa- paabb) and pab ab unless the ion polarizabilities are allowed to ch&ge with stnlin [X,23] ~though recent work finds most of the signs explicable with fixed polarizabilities [?LF]. A point-dipole treatment of the rare-gas solids assuming fixed polarizabilties [25] also show some qualitative disagreement with experiment. Improved agreement is obtained using a polarizability distribution, with overlap and exchange corrections deduced from pair-pohrizability calculations [3-61_ These corrections correspond to a strain-dependent polarizability, but it is difficult to see whether this or the spatial distribution is mainly responsible for the improvement. Corrections for a pair of atoms also seem inappropriate to the higher coordination in the solid. Against this background, the extent of agreement between our calculations and experiment is v&y satisfactory. The orderings of coefficients discussed above are the equivalents of the sign of (paa,w - paa&) when the symmetry is lowered from cubic to orthorhombic, and the three remaining coefficients are the equivalents of pab,ab _The detailed agreement with experimrnt.obtsined here depends firstly on the contribution from rotation of the molecular polarizability under strain. Though this is somewhat uncertain owing to rbe empirical assumptions about the rotation, it is a contribution which is absent in the cubic ionic and atomic solids; this may help to make the behaviour in molecular crystals easier to interpret. The other fwtor contributing to the agreement with experiment is the use of the eight-point treatment, as aheady discussed. It appears that taking account of the spatial distribution of polarizability in molecular crystals is necessary for good results [4,5] , and this
173
P.J. Boutrds, R. IV. blura~/Dielectric theory of straitled molecular crystals
may also be true in simpler crystals, as indicated by the calculations for the rare-gds solids [26]. It is also likely that a fixed spatially-distributed
polarizability
can explain the experimental results just as well as a strain-dependent point polarizability. (A local polarizability density is not strictly valid [27] , but for !inear dielectric behaviour may be adequate, especially for well-localized groups of electrons.) From the present work we conclude that the assumption of a fixed molecular polarizability is adequate to explain the strain dependence of the dielectric properties of molecular crystals, provided the molecules are realistically represented. We drew the same conclusion from our birefringence calculation for anthracene [ 1 I], but the agreement with experiment there was exaggerated because we omitted to allow for thermal expansion in converting optical path differences to birefringences. The weakness in these calculations is the need to assume a form for the molecular rotation due to strain. However, since it appears that the molecular polarizability can be taken as independent of strain, one can use experimental elasto-ootic coefficients or birefrinaence . changes to calculate the rotation. This approach has been applied to fluorene and. carbazole [28] , where the crystal structure allows only one degree of rotational
freedom.
A simplified
dielectric
theory
was
used, to avoid the arbitrary features of the general theory, but the present approach would also be applicable. Because birefringence measurements can be very sensitive,
with
calculations
of this sort
they
offer
valuable means of deducing subtle changes in tnolecular orientation at the many phase transitions which occur in molecular crystals.
a
Acknowledgement We have profited
from discussions
with Dr. D.A.
Dr. T. Luty, Professor J.W. Rohleder and Dr. W.J. Kusto. We thank the Science Research Council (UK) for a Research Studentship (P.J.B.). Dunmur,
References [I] P.G. Cummins, D.A. Dunmur and R.W. hlunn, Chem. Phys. Letters 22 (1973) 519; 36 (1975) 199. (21 F.P. Chen, D.hl. tI:nson and D. Fos, Chem. Phys. Letters 30 (1975) 337; 1. Chem. Phys. 63 (1975) 3878.
[3] L.hI. Mkenschicd and I. Vlieger,Physica 7.5(1974) 57; 79A (1975) 517; 8.5A(1976) 129. [4] T. Luty, Chem. Phys. Letters 44 (1976) 335. [5] P.I. Bounds and R-IV. hlunn,Chem. Phys. 24 (1977) 343. (61 D.A. Dunmur, W’.tI.Miller and R.W. hlunn, Chem. Phys. Letters 47 (1977) 592.
[7] T. Luty, A. Xlierzejetvski and R.W. hlunn, Chem. Phys. 29 (1978) 353. [8 S. Haussiihl and H.-J. Weber, Z. Arist. 132 (1970) 266. ,t:
R.W. blunn, J. Phys. CI Solid St. Phys. 11 (1978) L61. T.H.K. Barron, T.G. Gibbons and R.W. Munn, J. Phys.
C: Solid St. Phys. 4 (1971) 2805.
[I1 P.J. Bounds and R.W. Munn, Mol. Cryst. Liquid Cryst. [12
44 (1978) 301. A.K. Price, J.O. Wil!iams and R.W. blunn, Chem. Phys.
14 (1976) 413. [13 P.G. Cummins, D.A. Dunmur, R.W. Xlunn and R.J. Newham, Acta Cryst. A32 (1976) 847. [14) P-G. Cummins, D.A. Dunmur, R.W. hlunn
and R.J.
Newham, Acra Crysr. A32 (1976) 854. [15] S.C. Abraham%
Acta Cryst. 8 (1955) 661.
[Iti1 P.J. Bounds, Ph.D.Thesis, UhiIST, 1978.
[17] 1.Chen, Phys. Rev. B 2 (1970) 1060. [IS]
P.G. Cummins, Ph.D. Thesis, University of Sheffield, uli. 1974. 1191 W. Schmidt, Ann. Phys. (Leipzig) 11 (1903) 114. [20] D.F. Nelson and hl. Las, Phys. Rev. Letters 24 (1970) 379;Phys.Rev.B3(1971)2778; D.F. Nelson, P.D. Lzizay and hl. Las, Phys. Rev. B6 119721 3109. [21 J &doit~BGrnstein, Optical constants, 6th Ed., Vol. 2, Part 8 (Springer, Berlin, 1962). [22] H. hlueller, Phys. Rev. 47 (1935) 947. [23] K.G. Aggarwal and B. Szigeti, I. Phys. C: Solid St. Phys. 3 (1970) 1097. (241 J. Smit, 1. Appl. Phys. 49 (1978) 2935. [25] N.R. Werthamer, Phys. Rev. 185 (1969) 348; B6 (1972) 4075. 1261 D.W. Ostoby and V. Chandrasekharan, Phys. Rev. B16 (1977) 1706. [27] J.E. Sipe and J. van Kranendonk, hlol. Phys. 35 (1978) 1579; K.L. Clarke. P.A. Madden and A.D. Buckingham, Mol. Phys. 36 (1478) 301; D.W. Ostoby, J. Chem. Phys. 69 (1978) 1184. [28] J.W. Rohleder. Kristall. Technik 13 (1978) 517; W.J. Kusto and 1-W. Rohleder, hlol. Cryst. Liquid .Cryst., to be published.
DIELECTRIC ELASTO-OPTIC
THEORY
OF STRAINED
COEFFICIENTS
P.J. BOUNDS* and R.W.
Received 19 Dccembcr
MOLECULAR
CRYSTALS:
OF ORTHORHOMBIC
SULPHUR
MUNN
1978
The gcncral dielectric theory of molecular crystals is estended to calculate the strain derivatives of dielectric propertics, assuming n fised pohrizability relative to the molecular axes. The molecular rotation due to strain makes an importnnt contribution, for which an empirical prescription is given. To illustrate the theory, the elasto-optic coefficients for orthorhombic sulphur are calculated. They agee well with experiment when each sulphur atom is treated as 2 polarizable point. It is concluded that the polarizability can be treated 2s Iised, for esample in deducing changes of molecular orienta-
tion from birefringcnce measurements through phase transitions.
1. Introduction The t!teory of the dielectric properties of molecular crystals has recently been studied in some detail: representative methods are described in refs. [l-j] . Most of this work has been aimed at understanding the properties of fixed crystal structures rather than variation of such properties with strains in those structures. However, strain is a useful and important external constraint, arising not only from applied stresses but also from changes of temperature. A theory of the strain dependence of the dielectric properties of molecular crystals is therefore necessary to understand the pressure and temperature dependence of the dielectric constant and properties such as the elasto-optic coefficients_ In this paper we develop a treatment of the basic features of such strain-dependent phenomena, restricted for simplicity to effects linear in strain. The principles are exemplified by a calculation of the elasto-optic coefficients for orthorhombic sulphur. Because the present treatment aims to provide a microscopic interpretation of the macroscopic strain dependence of dielectric properties, we first consider * Present address: Division of Chemistry, Council of Canada,
Ottawa,
National Canada KlA OR6.
Research
macroscopic strain and its effects at a molecular level. We then derive the basic equations governing the elasto-optic coefficients_ Assuming that the molecular polarizability rotates rigidly with the molecule, we can express the strain derivatives of the polarizability in terms of the pdlarizability itself. The polarizability has to be fixed by ad hoc numerical assumptions or by additional experiments [ 1,2,4-71, but no further information is required to determine the strain derivatives. We then calculate the elasto-optic coeffcients for orthorhombic sulphur and compare the results with a complete set of values measured with a combination of static and dynamic methods [S] _
2. Strain A homogeneous
deformation
of an elastic continu-
um is described by a tensor D, which carries a vector x in the reference configuration into the vectorx’ in
the strained configuration: x’= D-x.
(1)
In general, this defomtation comprises not only a pure strain but also a bodily rotation which is of no significance in the absence of external constraints and so has to be eliminated. The separation.of strain and
166
P.J. Bounds, R.N. MmnfDielectric
rotation can be achieved in various ways [9], but to first order it is sufficient to take the strain as the symmetric part e of the displacement gradient tensor u = D- 1: e = f(u +Z),
(2)
where tfte tilde denotes the transpose. Higher-order effects are not considered in this paper. A crystal is of course not a continuum, but if all subfattices are at special positions only ftomogeneous deformations need be treated. If sublattices are at general positions, it becomes necessary to take account of further irzrerrz~lstrnifzs relative to the hoinogeneousfy deformed lattice. The specification of internal strain is not unique [IO] , but in molecular crystals it is usually convenient to specify the translations and rotations of molecules. in general, molecular deformations have also to be included, but in the present work the molecules are taken to be rigid. The problem wit11tfte internal strains is knowing what they are: they are fixed implicitly by tfte flomogeneous deformation or external strabl because at equilibrium they have to minimize an appropriate energy function [IO] , but knowledge of intermolecular poientiafs may be insufficient to calculate internal strains this way, and direct experimental measurements are seldom available. One may tfterefore have to resort to some empirical prescription for determining internal strains from externa1 strains. The main internal strain required is the molecular rotation_ The external strain rotates all vectors in the crystal (except those parallel to the principal axes of strain) even when there is no bodily rotation, and so could be used to fix the rotation of the molecular axes, except that the strain also distorts the axes so tftat they do not remain orthogonal. For anthracene this problem was treated [ 1 I] by assuming tflat on distortion one speciffed axis remained a molecular axis and with a second specified axis continued to define the plane of the two rotated axes. We use a similar assumption for orthorhombic suIphur, but derive its consequences algebraically rather than numerically. The results are therefore of genera1 appficability. The direction cosine matrix in the reference configuration is taken to have components a,, , where czis a crystal axis and A a molecular axis. The columns of a thus correspond to unit vectors parallel to tile
theory of srrained molecular crystals
molecular axes. After a deformation given by eq. (1) the new vectors form a new matrix a’ given by a’ = D-a.
(3)
Differentiation with respect to the displacement gradients u then gives aQh/all(rr
= &uoQTA
_
(4)
We denote the nlolecuIar axes by L. M and iv, and assume that the new L axis is parallel to the deformed L axis while the new N axis is perpendicular to the deformed L and M axes; this fixes tfte new M axis. The new direction cosine matrix is A. h’ow from our assumptions, (5)
A al. = akr. /ALI wflere h, is the stretcfl of the L axis, given by
(6) (summation is implied over repeated Greek subscripts). On differentiation we obtain, using eq. (4), d‘%,.WJ,
= ($6,(i
- ahL Q~,)Q&~,
(7)
which has to be evaluated in the reference configuration u + 0, which implies a’ + a and AL + 1, leaving aA,Liarlor
= (“,, - apLa,L)u~L -
(8)
For the new M axis we have A crx = Q>r&r
,
(9)
where ai2\, is given by the vector product of the deformed L and ill axes: a&
=f spy
’ Q;M’ Qpl.
(10)
with Eap_, the alternating tensor. An extension of the
algebra used for the L axis leads to the result for u + 0
By expressing aOL and apnr in terms of products of direction cosines and using the orthonormality property ‘CTL‘TL ’ aoManlT + ‘ohraihr = ‘UT*
we reduce eq. (11) to the form
(12)
P.J. Bomds. R. IV. Mmzrt/Dielectric theory of strained r~~obmchrcrystals
aA&,&)l& = -Q&
(13).
- ~&,\P,,,).
The new ,V axis is written as the vector product of the IV and L ases, leading to the u -+ 0 result
167
the numerical results for anthracene [I l] .
3. Methods of calculation
aA&l allo7 = EoPr kyL aA,,,/acf(rr f aplVa+.lallD71
-
3. I. Dielecrric rheoty
(14) The general expression for the electric susceptibility is 1121
Substitution f&m eqs. (8) and (13) and manipulaCons of the sort already described give eventually (15)
aA~Jal~Br =a,rh1a0A,Q7nr- a,L.aDhfaTL-
01)
x= F$&,
The rotation tensor is given by R = A-a-‘,
(16)
so that its dependence on strain is given by
is given by CY~/EOU, where ‘Ye is the effective polar-
aRL2hlaLiOT= @~aqh&)Q (summation
%J%z
(17) M over subscripts A imphed). The resuIt is
= 4&L%.
- $0ada7L
-
%TQd%~
where the sum runs over all subIattices k in the primitive unit cell. The reduced effective polarizability PI;
+ uL.N%
+aOAraTL(adakV - ati,,a,,)-
izability of the molecules on sublattice k and u is the volume of the primitive unit cell. The local-field tenSOTdk gives the IocaI electric field at sublattice k in terms of the macroscopic electric field, and can be obtained as [ 1,2,7]
(18) (23)
dk=.$kl&, Since Tr R = R, is invariant and equal to 3, it should be independent of strain, and eq. (18) confirms that = 0. The first-order rotation is also antiaR,,lau, symmetric, and eq. (18) duly yields a&_/au, = -aR,,/ht,,. kfence calcutating aR/i3u is less fotmidable than it might seem, since R has only three independent non-zero components. For calculations, eq.. (18) is conveniently written as aRaXlaIiC = ~,,~y,x - 8L1TYCTh + %JCKD- %oXC%r i- Z,J-&
- Z,) 3
(19)
where X& = sot apt,
Yap = awvnpIjTand 2, = f7dQp~ are readily calculable elements of matrices, X and Y being symmetric. The rotation tensor R as calculated includes the contribution of any bodily rotation arising from the antisymmetric part of u. However, because of its antisymmetry, this contribution is eliminated when eq. (19) is symmetrized to give the derivative of R with respect to e. The result is %J%T
= f [6,, ‘VT& f sQT‘Ubh- k&6,
+ (Z,+ ‘ZrO) (2,X - Z,)l1
- 1vm60, (20)
where W = X - Y. These algebraic results agree with
where the supermatrix A has submatrices n kk’ =1Q&L&J.
_
(23)
the LkK being Lotentz-factor tensors. Given the Lkk,, calculated from the crystal structure [ 13) , the pk determine the d, and vice versa. Then given the strain derivatives of the Lkk [14J, the strain derivatives of the & determine those of the d, and hence by eq. (21) the strain derivative of x_ Ortborbombic sulphur crystallizes in the space group Fddd with four molecuIes in the primitive unit cell [IS] . The molecules are related in pairs by centres of symmetry, and the pairs ate related by twofold axes parallel to the crystallographic a and b axes. These relationships allow the same expressions to be used as for pyrene [ 121 : if molecules 1 and 2 are related by inversion they have identical effective polarliabilities and local fields. If molecule I is carried into molecule 3 by the twofold axis parallel to Q then the poIarizabilities can be written as $;I = Mt + M3*p,
(24)
#;t_=M;
(25)
Here
iM3*p-!
P.J. Bozczzds.R. IV_Mzzrz~z/Dielccrric rlzmr_sof straitzed zzzoleczdarcr_rsrats
168
M, ‘Ml1 +M12’
(26)
Mj = Ml3 +Mt4>
(27)
M; = M,, + M,,,
(25)
with Mali. equal to x -’ + LkK_ The matrix p decoupies the contributions of the two sets of sublattices and is defined as (W
P=Pj-d,.(Pl.dl)-‘-
Because sublattices 1 and 3 are crystallographically equivalent, Bj is obtained from Bt by orthogonal transformation with the matrix lJ describing the rotation about a. Eqs. (24) and (35) are then consistent only if
p-1 =.lJ*p*u.
(JO)
The form of p entailed by this equation has been given elsewhere [ 1,.S]; four components qre independent. Since polarizabilities are symmetric, the product M3 -p must be symmetric, and this leaves two components of p undetermined. Finally, in orthorhombic sulphur the molecules retain a twofold axis of symmetry parallel to c. This leaves only one component of p not determined by symmetry and allows Pf’ to be written as A(ltq2p p;’ = M, + (i(AB)“’ 10
q(AB)“z
0
B( I+(;‘)“’
0 )
0
c1
(31)
where 4 is the arbitrary parameter and A, B and Care the diagona1 elements of M . TJie parameter 4 can in irinciple be determined from Stark spectra [6] or Raman intensities [7] _ Ctherwise, the choice 4 = 0 is algebraically convenient but quite arbitrary [I 1,12]_ To obtain reasonable polarizabiliries from x it is usually necessary to take account of the molecular size, shape, and orientation. This is achieved by representing a molecuIe as a set of polarizablepoint submolecules [4,5] _The foregoing algebra can be retained if the Lorentz-factor tensors are calculated as averages over all pairs of submolecules on the two sublattices in question (eseluding contributions within the same molecule), which corresponds to averaging the local field over the submolecules. For orthorhombic sulphur we locate submolecules
at each sulphur atom. The resulting
polarizabilities are better than those calculated for a single point at the centre of each molecule, and differ little from those calculated for submolecules at the centre of each sulphur-sulphur bond in the molecule
[161. 3.2. E!asto-optic
coejjkietm
The fourth-rank elasto-optic tensor describes the strain dependence of the inverse relative permittivity tensor or indicatris: p = a&cljae.
(32)
Since L= ,1 + x, the elasto-optic tensor can be expressed in terms of 1 as p apP
= -0
+ &(ax,,moT)
(1 + XI;;,
63
this and similar strain derivatives the inner products relate not to the strain but to the quantity
where in
differentiated
with respect to strain.
The strain derivative of r is conveniently by writing eq. (21) in the form
evaluated
x= c (fWkX-” k.k’
(34)
where band d are supermatrices pk-Skpand dkli._ Here
having submatrices
dke = (1 - L-&“,
(35)
with the supermatrices 1 and L having submatrices 1Ske and LkK, so that together these equations yield
x= L_ cp-’ - L)&
(36)
The strain derivative of x is then ““+ aeor
[(p-t _ L)-t.($$) .. t
OT
. (p-1 _ L)-t
(37)
I kKY which can be manipulated to yield eventually ax=C; aear k
a$, .-.dk+%_-p_--
k aeor
a Lkr
k,a
d
d
aeo7
*Q.$. (38)
Results of this form have been obtained previously [71. Calculating
the elasto-optic coefficients requires
P.J. Bowuis. R. I%!dlutot/Dielectricrheorjfof strained tnolecrrlarcrJ’stals
. 169
the strain derivatives of the polarizability
and of the Lorentz-factor tensors, as eq. (35) shows, but not of the strain derivative of the local field tensor, as noted
proximation consonant with the weak intermolecular interactions in molecular crystals, and is probably less drastic than that used to determine the molecular ro-
above.
tation due to external strain. The effect of this rotation on the polarizability is obtained starting from tire polarizability in the strained crystal, R-wR-~. The result in the limit R + 1 is
The elasto-optic
coeflicients can also be calculated by espressing p in terms of one subIattice polarizability and p(cf. ref. [I I]), thus requiring a knowledge of the strain dependence of p. Although p itself is not uniquely defined, for a given p the &ruin derivative is determined by thdse of the polariznbilities and Lorentz-factor tensors, through eqs. (19) and (35). In practice, ap/ae is fixed by requiring ax/se to lwe the proper symmetry. This justities the neglect of the spurious non-zero ab and bc’ elements of x found in our birefringence calculations on anthracene and phenanthrene [ 1 l] ; these elements arise for one sublattice but are cancclled out by equal and opposite elements for the other sublattice, as required by sym-
where X is summed; the last result uses the symmetry of p and the antisymmetry of R. The direction cosine matrix for the molecular on sublattice 1 in orthorhombic sulphur is cos Q
metry. The present arguments show that our asump-
I
tion of e p independent of strain was inconsistent with the assumed strain derivatives of br; and Lx_~,_ If ap/ae is required, it is best calculated from eq. (29) and the contributions to eq. (38) for the various sublattices X-. Experimentally, ap/ae could be determined from the strain dependence of the extra messurements which deterntine p itself.
at = -sin@
3.3. Srraitzdepettdetm
of palarizability
The reduced effective polnrizability p depends on strain through the unit-cell volume u as well as through the molecular polarizability. Separating these effects, we obtain for a given sublattice
ablae,,, = -fisaT + ( 1/eov)adaeaT,
sin@
O\
COSQ
0
I
axes
(42j
, i
\o
I/
0
where Q = ir//2 + tan-‘(b/a) [ 17]_ We use this to evaluate the strain derivatives of R from eq. (20), taking the L axis as the molecular axis parallel to the crystal c axis and the M axis as the other axis in the mean plane of the free molecule (the plane perpendicular to the fourfold ssis). The results are sin 24
aRabIaem
1
cos29
= + ~0~2~5 -sin?@ 0
0
0 0 0
i )
(43)
(3%
since a In v/at=,, = IS,, [ 141. The strain dependence of oLarises from the molecular rotation induced by e and from changes in the polarizability itself in the molecular axes. It is not clear bow one should interpret changes in the ttzolecrdar polarizability with e-yrertzals!rain, and it is consistent with our assumption that the molecules are mechanically rigid to assume also that they are electronically rigid. Hence the polarizability referred to the nzoleatlar m-es is assmed to be itzdepetzdetrtof strain. This assumption has previously been used to calculate the temperature de-
pendence of birefringence in anthracene [ 1 I] and to calculate lattice Raman intensities [7]. It is an ap-
(44
I I
/O O O\
aR,,laem
=$ 0
0
1 ,
0
1
0
(45)
all referring to sublattice 1. For sublattice 3, @has to be replaced by -Q in eq. (42) and hence in eq. (43).
P.J. 6oumk.
170
3.4. Strait1depaldeme
R. IV. Mum fDie[ectric theory of srraiiied ?nolecdar crystak
ofLorem-firctor temors
The calculation of the strain derivatives oM,Jkk’) foi a homogeneous strain Ilas been = aLd(kk’)/ae,, describ$d elsewhere [ 14]_ This method would suffice if we treated each sulphur molecule as a single polarhable point at 3 special position in the unit cell. However, when the molecule is treated as eight submolecules, we have to sllow for’ the effect bf the rotation due to strain on the iorentz-factor tensors_ We have also to recognise that a homogeneous deformation of all atomic positions would change the size and shape of the molecule, which we have assumed to be rigid. The centre of mass of the molecule is not at a completely symmetry-determined position, having an arbitrary z component. There is therefore an additional possible internal strain in this direction superimposed on the homogeneous strain. We neglect this internal strain [ 1 I] , seeing no plausible way of prescribing it empirically. It could be determined espcrimentally or from 3n intermolecular potential. An external strain thus causes a homogeneous defortnation of the centres of mm and rigid rotations about the centres. We write the position of atom i on moIecule k in unit cell Ix r(lki) = r(K) t s(k) )
(46)
where f(lk) is the position of the ccntre of nisss. Af-
ter J hqnwgeneous deformtion D with its associated rotations Rk, the new atomic position is r’(Iki) = Do = Dv(lki)
+ Rk.s(ki) - D-s(k)
+ Rk-s(k).
(47)
(48)
By introducing r(K-i) in eq. (48) we show that the desired deformation ~911be treated as a hon~ogeneous defamation operating on a11atoms, combined with internal strains to return the tttonis to their original positions relative to the centre of mss and a rotation to reorient the molecules. The honlogeneous deformation cm be treated by the previous methods [ 14]_ The averaged Lorcntz-factor tensors are given by L(kk’) = s-’
C
[ L(ki. k’i’) - I(k. ii’)6kK] ,
(49)
i.i
where s iS the number of atoms per molecule. The L(kI: k’i’) are Lorentz-factor tensors caiculated in the usual way but with each atom ki taken as a sublattice.
The other term subtracts the contribution of the dipole interaction between atoms i and i’ on the some molecule. We denote the total derivatiye ofL&k’) with respect to strain ear by a script
?I,p,Jkk’)
to
distinguish it from the homogeneous strain derivative. The result is
op,m(kk’)= s-‘g (1c .(
~~.,i(ki. k’i’)
- $ [LMJki. k’i’)sT(ki. k’i’) + LapT(ki,k’i’)sO(ki,X-Ii’)] + [L&i.
x
k’i’) - Z~$k. ii’&,
aR#) ___ [ aem
1
aR$ (x-7 s,(C) - ----ss,(k’i’) ae wr
,
w-9
11
where the L ,&X.
k’i’) are qundrupole lattice sums and s(ki, k’i’) = s(h) - s(k’i’). The form of the second terns arises from the synumetrization of derivatives with respect to u to yield those with respect to e_ The homogeneous strain derivatives I!?@,,, are synimetric under interchange of c@and m [ 141 t but the other terms in eq. (50) lack this syn~metry and hence so do the total strain derivatives_ [ 131
For completeness,
we note that any internal dis-
placement w(k) of the centres of niass is added to eqs. (47) and (48). It contributes to the strain derivative in eq. (50) in the sane way as the rotation. but is independent of i and i’_The sunmation can then be performed
L&M’>
[
to yield the extra contribution
akfp) -$y 07
altry - ~ ae Oi-
(51)
1’
where L ap7(kk’) is the averaged quadrupole SNIHexcluding contributions
from within the same molecule.
4. Results Averaged Lorentz-factor tensors calculated from the structure given in ref. [ 151 are shown in table l_
These have been used to calculate the effective polarimbilities for a range of values of the parameter 4 in eq. (3 I), taking the static susceptibilities
3s x,
=
Lorentzrbctor tcnsor components Lop(M’), aucrsgud over the atoms for orthorhombic sulphur, referred to crystal wx. Components nut quoted arc zero by symmetry ~___ ~~
aa
11
---~-
0.462 0.676 0.348 0.074
12 i3 14
ab = ba
bb
cc
-0.178 0.643 0 0 -___
0.542 0.538 0.148 0.154
-0.004 -0.214 0.504 0.712
2.68, ,$,b = 2.8 1 and X,, = 3.66 [IS] *_ The component of the polarizability parallel to the crystal c axis is independent of 4, but the other components vary considerably. When the polarizabilities are transformed to the molecular axis system, it is found that the two components in the mean molecular plane (which are equal by symmetry in the free molecule) are closest for small values of 4 [ 161. We therefore take 4 = 0 for further calculations. The choice 4 = 0 makes the matrix pof eq. (29) equal to the unit matrix. The product ok - dk is then the same for all four sublattices k, and equal to $ x by eq. (21). Substitution of this result in eq. (38) yields 16 ax-’ --$-=t;~_$$, OT
(52) OT
.
’
LIT
which can be used directly in eq. (33) rewritten as Po~,or=(l+~-')~
!g(l
+f');;.
(53)
OT
Application of this equation is simplified by the fact that x is diagonal under orthorhombic symmetry, so that p is simply proportional to the corresponding strain derivative of 1-l. The symmetry relationships between the sublattices simplify the application of eq. (52) too, because the SUI:IS over k consist either of four equal terms or of terms which are equal and opposite in pairs. Numerical results for the strain derivatives are given in table 2, where B4wT = a(Pk&$aeoT, * Tbcse data, obtained by an immersion method, agree well wirh thoseofref.
[191.
Strain derivatives of the inverse polarizability and of the Lorentz-factor tensors, and calculated and esperimentsl e&to-optic coefficients for orthorhombic sulphur
Gllc.
apt.
0.50 0.40 0.31 0.34 0.41 0.38
0.32 f 0.31 c 0.27 t 0.27 f. 0.30? 0.31 k
0.10 0.41
0.20 k 0.0 1 0.23 + 0.01 0.37 2 0.01
0.11 0.04 0.17
0.14 + 0.04 0.02 + 0.03 0.12 f 0.04
______
_..__ ~~_~~ ~_~.------
aa, *a aa, bb
2.597 3.509 3.053 3.262 2.350 2.806 2.150 2.150
-1.159 0.513 0.611 0.770 -0.695 0.036
2.150
-0.537 -1.098 -0.75 I -1.312
MI,CC
bb,aa bb, bb bb, cc cc.aa cc. bb
cc, cc bc. bc ac, ac ab. ab --___
-0.328 -0.452 -0.025
I.475 0.760
.O.‘l
[S J 0.01 0.015 0.01 0.01
0.015 0.01-
and qM,or
=T
C!f@,J”“‘).
(54)
Only terms which give non-zero net contribution are shown and these are the same for all k; there are also non-zero terms in which au. bb and cc are combined with ab orac (in either order), but these average to zero as explained above. The polarizability derivatives for the extensional strains are a11of similar size, because the first term in eq. (39) dominates and fl is not very anisotropic. The major effect is therefore the “dilution” of the polarizability density through the change in unit cell volume, leaving only minor contributions from molecular reorientation_ The Lorentzfactor tensor derivatives vary considerably in magnitude and may have either sign, but are mostly smaller than the polarizability derivatives. The experimental e&to-optic coefficients for orthorhornbic sulphur [8] are shown in table 2 with their quoted uncertainties. They were obtained by an ultrasonic diffraction method (supplemented by static measurements to determine the signs of the coefficients). Such dynamic methods produce a spatially-varying displacement rather than a pure strain, and there is then a contribution to the susceptibility change from the antisymmetric part of the displacement gradients u, corresponding to a local rotation [20]
172 The antisymmetric given by [20]
P.J_Bounds. R. IV_Mwm/Dielec~rictheory of strainedn~olenrlarcrwtafs
part of the elasto-optic tensor is
which in the principal dielectric axes reduces to (56) kit - E,-,‘) U&i = :(6,,spr - S&J (no summation implied)_ For orthorhombic sulphur we calculate’ahc,6, = -0.024, aac,ac = -0.029, and aab.ab = -0.005. These are no larger than the esperimental uncertainties shown in table 2 and will’therefore not be considered further_ The calculated elasto-optic coefficients are in reasonable agreement with the experimental ones. The experiments indicate few marked differences except that the coefficients involving shear strGns are significantiy smaller than the others. which are all about 0.3. The calculations show more vuriution between coefficients, but do indicate that those for shear strains are generally the smallest_ Most of the calculated (static) coefficients are larger than those measured (at ultrasonic frequencies). This suggests that there could be some systematic effect of frequency. At higher frequencies, fl and 1 are larger [2 I] and their inverses smaller_ If their strain derivatives behave similarly, one would expect the calculated coefficients to be larger than the measured ones, as observed_ The observed trend might also be partly attributable to the assumed form of the molecular rotation due to strain. More detailed examination shows that the calculations reproduce the observed orderings of the three coefficients for L$ = aa and cc and for UT= aa, while those for I@ = 66 and or = 66 are incorrect only becwse pbb 66 is quoted as less than ~aa,~~ and pbb,cc, whereas tileexperimental uncertainties permit the opposite ordering_ The near constancy of the coefticients for extensional strains can be attributed to the large and relatively isotropic polarizability of tbe S8 molecule. The calculations agree well with experinlent cnly when the molecule is treated as eight points. A singlepoint treatment is much simpler, and yields acceptable if somewhat poorer polarizabilities [ 161, but the calcdated elasto-optic coefticienrs bear little resemblance to the experimental ones except in sign and order of magnitude_ This is partly because of a
more anisotropic polariznbility, but-the major factor is greatly different Lorentz-factor tensor derivatives, lnany opposite in sign to those for the eight-point treatment_ It follows that calculation of the elastooptic coefficients is a more severe test of the model than calculation of the polarizability.
5. Discussion Previous calculations of elasto-optic coefficients have concentrated on cubic crystals, mostly ionic ones. It has generally been found difficult to explain even the signs of the coefficients {paamaa- paabb) and pab ab unless the ion polarizabilities are allowed to ch&ge with stnlin [X,23] ~though recent work finds most of the signs explicable with fixed polarizabilities [?LF]. A point-dipole treatment of the rare-gas solids assuming fixed polarizabilties [25] also show some qualitative disagreement with experiment. Improved agreement is obtained using a polarizability distribution, with overlap and exchange corrections deduced from pair-pohrizability calculations [3-61_ These corrections correspond to a strain-dependent polarizability, but it is difficult to see whether this or the spatial distribution is mainly responsible for the improvement. Corrections for a pair of atoms also seem inappropriate to the higher coordination in the solid. Against this background, the extent of agreement between our calculations and experiment is v&y satisfactory. The orderings of coefficients discussed above are the equivalents of the sign of (paa,w - paa&) when the symmetry is lowered from cubic to orthorhombic, and the three remaining coefficients are the equivalents of pab,ab _The detailed agreement with experimrnt.obtsined here depends firstly on the contribution from rotation of the molecular polarizability under strain. Though this is somewhat uncertain owing to rbe empirical assumptions about the rotation, it is a contribution which is absent in the cubic ionic and atomic solids; this may help to make the behaviour in molecular crystals easier to interpret. The other fwtor contributing to the agreement with experiment is the use of the eight-point treatment, as aheady discussed. It appears that taking account of the spatial distribution of polarizability in molecular crystals is necessary for good results [4,5] , and this
173
P.J. Boutrds, R. IV. blura~/Dielectric theory of straitled molecular crystals
may also be true in simpler crystals, as indicated by the calculations for the rare-gds solids [26]. It is also likely that a fixed spatially-distributed
polarizability
can explain the experimental results just as well as a strain-dependent point polarizability. (A local polarizability density is not strictly valid [27] , but for !inear dielectric behaviour may be adequate, especially for well-localized groups of electrons.) From the present work we conclude that the assumption of a fixed molecular polarizability is adequate to explain the strain dependence of the dielectric properties of molecular crystals, provided the molecules are realistically represented. We drew the same conclusion from our birefringence calculation for anthracene [ 1 I], but the agreement with experiment there was exaggerated because we omitted to allow for thermal expansion in converting optical path differences to birefringences. The weakness in these calculations is the need to assume a form for the molecular rotation due to strain. However, since it appears that the molecular polarizability can be taken as independent of strain, one can use experimental elasto-ootic coefficients or birefrinaence . changes to calculate the rotation. This approach has been applied to fluorene and. carbazole [28] , where the crystal structure allows only one degree of rotational
freedom.
A simplified
dielectric
theory
was
used, to avoid the arbitrary features of the general theory, but the present approach would also be applicable. Because birefringence measurements can be very sensitive,
with
calculations
of this sort
they
offer
valuable means of deducing subtle changes in tnolecular orientation at the many phase transitions which occur in molecular crystals.
a
Acknowledgement We have profited
from discussions
with Dr. D.A.
Dr. T. Luty, Professor J.W. Rohleder and Dr. W.J. Kusto. We thank the Science Research Council (UK) for a Research Studentship (P.J.B.). Dunmur,
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[3] L.hI. Mkenschicd and I. Vlieger,Physica 7.5(1974) 57; 79A (1975) 517; 8.5A(1976) 129. [4] T. Luty, Chem. Phys. Letters 44 (1976) 335. [5] P.I. Bounds and R-IV. hlunn,Chem. Phys. 24 (1977) 343. (61 D.A. Dunmur, W’.tI.Miller and R.W. hlunn, Chem. Phys. Letters 47 (1977) 592.
[7] T. Luty, A. Xlierzejetvski and R.W. hlunn, Chem. Phys. 29 (1978) 353. [8 S. Haussiihl and H.-J. Weber, Z. Arist. 132 (1970) 266. ,t:
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C: Solid St. Phys. 4 (1971) 2805.
[I1 P.J. Bounds and R.W. Munn, Mol. Cryst. Liquid Cryst. [12
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and R.J.
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P.G. Cummins, Ph.D. Thesis, University of Sheffield, uli. 1974. 1191 W. Schmidt, Ann. Phys. (Leipzig) 11 (1903) 114. [20] D.F. Nelson and hl. Las, Phys. Rev. Letters 24 (1970) 379;Phys.Rev.B3(1971)2778; D.F. Nelson, P.D. Lzizay and hl. Las, Phys. Rev. B6 119721 3109. [21 J &doit~BGrnstein, Optical constants, 6th Ed., Vol. 2, Part 8 (Springer, Berlin, 1962). [22] H. hlueller, Phys. Rev. 47 (1935) 947. [23] K.G. Aggarwal and B. Szigeti, I. Phys. C: Solid St. Phys. 3 (1970) 1097. (241 J. Smit, 1. Appl. Phys. 49 (1978) 2935. [25] N.R. Werthamer, Phys. Rev. 185 (1969) 348; B6 (1972) 4075. 1261 D.W. Ostoby and V. Chandrasekharan, Phys. Rev. B16 (1977) 1706. [27] J.E. Sipe and J. van Kranendonk, hlol. Phys. 35 (1978) 1579; K.L. Clarke. P.A. Madden and A.D. Buckingham, Mol. Phys. 36 (1478) 301; D.W. Ostoby, J. Chem. Phys. 69 (1978) 1184. [28] J.W. Rohleder. Kristall. Technik 13 (1978) 517; W.J. Kusto and 1-W. Rohleder, hlol. Cryst. Liquid .Cryst., to be published.