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Crystal optics and raman intensities
Chemical physics 34 (1978) l-9 0 North-Holland Publishing Company
CRYSTAL OPTICS AND RAMAN INTENSITIES
RW. MUNN Department of Chemistry. LJMIST,Manchester M60 IQD. U.
T. LUTY Institute for Theoretical Chemistry, Catholic Ukrversity.Nijmegen. lXe Netherlands and institute of Organic and Physical Chemistry. Technical lEversi@, N-370 WrocZaw,Poland
A. MIERZEJEWSKI Institute of Organic and Physical Chemistry. Technical University, 50-370 Wroc&aw.Poland Received
29 May 1978
The theory of Raman scattering intensities is modified to take account of crystal optics. The radiation at a long distance from an oscillating electric dipole in a diclectic continuum is calculated for propagation along or perpendicular to a principal dielectric axis. The results are combined with results on the effect of transmission through the crystal surface to give an expression for the polarized Raman intensity in an orthgonal axis system with only one axis a principal dielectric axis. For 90” scattering in monoclinic crystals in the ate’ axis system, there may be up to three different intensity patterns for a given polarization, yielding ten distinct crystal settings. Sample calculations for naphthalene show that the intensity patterns for the same polarizations in distinct settings differ oniy slightly, and this will normally be the case.
. 1. Introduction In a previous paper [ I], we derived a new expression for the Raman scattering tensor in molecular crystals. Although the principles of Raman scattering were treated by Born and Huang [2], they emphasized that their derivation was strictly valid only for an isolated molecular system. For a crystal, the electric vector of the incoming light must be t&en as the macroscopic field Wide a region small compared with the wavelength of the light but large compared with atomic and molecular dimensions. This led us to calculate the scattering tensor from the high-frequency electric suscep-
In comparing these results with experiment, a difficulty arises. Our treatment [I] modities Born and Huang’s results for an isolated molecule to take account of the electrostatic interactions between the molecules in a crystal. However, the results for an isolated molecule must also be modified to take account of the optical properties of a crystal. Polarized light incident on a crystal is in general attenuated, refracted and depolarized on passing through the crystal surface to give a transmitted beam. Similar changes take place in the Raman scattered light on passing through the crystal surface to give an emergent beam. Furthermore, the calculation of the scattering within the crystal, viewed
tibility
as originating
of the crystal,
within
the point-dipole
rigid-
molecule approximation. Thls.appfoach yields the contribution to the intensity from both translational and rotational motions; in both cases it is proportional to the fourthpower of the local-field correction.
from
an oscillating
transition
dipole,
must take account of the anisotropic dielectric medium through which the radiation passes. For normal incidence on cuboidal crystals cut parallel to the principal axes of the dielectric tensor, with
2
.R. W.Mmm et al./Cr_wtoloptics and Ramanintensities
all directions of polarization parallel to these axes, one expects that the correct relative intensities should be obtained for different modes of vibration in a given crystal setting, even when the effects of crystal optics are ignored in the calculation. Cubic crystals are optically isotropic and so present no problem. For orthorhombic and axial crystals the principal axes are fixed by symmetry, and cuboids are usually readily cut parallel to these axes. Indeed, a misoriented cuboid may be detected by its depolarization of a polarized beam on transmission through the crystal [3] _However, for monoclinic crystals, only the b axis is a principal dielectric axis by symmetry, the other principal axes lying in the ac plane. Many important molecular crystals are monoclinic with an easy cleavage parallel to the ab plane (because of the molecular packing). It is then convenient to cut cuboids parallel to the crystal ah’ axes. For comparison with Raman intensity measurements on crystals cut like this, it becomes essential to include the effect of the crystal optics in calculating the theoretical intensity. In the present paper, we complete our modification of Born and Huang’s results by including these effects of crystal optics. We begin by deriving expressions for the electric field of the radiation emitted by an oscillating dipole in an anisotropic dielectric for waves propagating along a principal dielectric axis or in the plane of two principal axes. These suffice for the problems just mentioned and also for triclinic crystals provided one principal axis is located for cutting them. We then treat the changes due to transmission through the crystal surface and finally the emergent intensity_
vector, and W is a matrix which &the principal dielectric axes takes the form
W=
kf+bf
%k2
%k3
klk2
k$+b;.
k2k3
k2k3
k$+b;
i klk3
_
(2)
i
Here the kj are the Cartesian components Of k and - k2 z q&/c2 - k2, where the Ei are the principal components of the dielectric tensor. The inverse matrix W-l is written as bz zcf
w-1 = w/D,
(3)
where
and
(4)
D=detW=cfcic$-
[cfci(kf+k$
+c;c$(k;
lk$)+c;c;(k;+k;)]
+k2(c:k;
+c;k;
+c;k;).
(5)
Then the limiting electric field is given by E(r + L) = (hGio/q
c2r)
C, exp (ik, 0r) wm J, exp (iot) l
XC
2. Dipole radiation in an anisotropic continuum A general method for obtaining the asymptotic solution as r -, 03 for the waves due to a spatially localized source at the origin has been given by fighthill
141.
A modification of the method has been presented by Kogelnik and Motz [5], and app!ied to radiation in a plasma subject to a magnetic field (which has a complex uniaxial permittivity). The Fourier components of the electric field amplitude are given by Ek = iwpOW’l -Jk,
(1)
where Jk is the Fourier component of the current density source oscillating at frequency w, k is the wave-
m
IV,D,
IIK, I’/2
’
(6)
where km is a root ofD(k) = 0 such that V@ll+r (to give the correct direction) and such that (r. V&l)/ (W/W) is positive definite but fmite (the radiation condition, to give causally propagating waves). The quantities w,, J,,, andVkD, are evaluated fork = km. The quantity Km is the gaussian curvature K evaluated fork = km, where [4] K= c [
D;(DiiDkk-D$
R. W. Mum et aI./Ciystal optics and Ramarr irttet&ies
(q/c) being a permutation of (123) and the sums being taken over all three such permutations, withDi E ( VkD>i, Dii 3 (VkVk D>ii- WhenKm < 0, Cm = * i according as Vk Dm iI+r, and when K,n > 0, Cm = k 1 according as the surface D = 0 is convex to the direction of -I-VDm fork = km, i.e. according as [4] C Dii(Di” f 0;) - 2 cDJ~DiDk
3 0.
(8)
For a point dipolep exp(iwt) at the origin, the polarization is p exp(ior) 6 (r) and the current density is thus iwp exp(iwt)6 (r). Then J, = iwp/8n3, leading to
(c)
-k, =O , or
(15)
(d)
c~k~+C22k2’$C32k~+C~k?-=c32(C:+C~).
(16)
These give four possible pairs of solutions, but for the combination (b)+(d) it is found that aD/ao = 0, which is inadmissible. On solving D(k) = 0 for the other three pairs, one finds that similarly tlD/aw = 0 for the combinations (a) f(d) and (b) + (c), leaving only (a) f (c), for which k, = 0 = kg and ki=cf
or
c$_
(17)
The radiation condition is satisfied for
E(r h -) 4 -(02eiwt/27rc0 c2r)
(9) The procedure used to calculate E is as follows. The direction r is specified, which specifies the direction of ?kD and permits two of the components of k to be expressed in terms of the third. The equation D(k) = 0 can then be sohed. For each solution km the signs of r- V,D and aD/aw are evaluated, where since D(k) = 0 sgn(aD/aw)=spn[~c:c5c~-k*(c:k:
3
,
+c$k~+c~k$]
(i)
k2 =-cl
,
(18)
(ii)
k? = -9
.
(1%
The curvature K is simplified because D, = 0 = 03, leaving K=(D11D33 -L&~D~,
(20)
where in general
-c; (c; + $1
+ 8cf k: ,
01)
wu and solutions not satisfying the radiation condition are discarded. For the remaining solutions the curvature K, is evaluated and thence the coefficient C,,, . Further manipulation gives E. The calculation of E as r 3 COalong a principal axis is relatively straightfonvard and serves as a check on more complicated solutions. With a view to propagation along the unique b axis in monoclinic crystals, we take r = (OJ, 0). Then we need D, =O=D3,
(11)
D13 =4k,k&
etc. Then for both cases (i) and (ii) it is found that K = Z/cz > 0. The convexity condition reduce; to C, = sgn(Drr f D,,), which yields Ci,= s$(c
sults yield
X (e-iclrp&l,
(12
(b)
cfkf +c;k;
(13) +c;k;
+c;k2
0,e-iC3~p3/c3) ,
(23)
where c1 = rzl w/c etc., with H1 a principal refractive
index. In the limit that all nI;become equal to unity,
etc. Then D1 = 0 implies either k, =O, or
(22)
E = (ozeiWrc2/4qrc2)
where in general
(a)
+c$),
=$(c5
with correspondingly for D3 = 0 either
+c;),(14)
this expression reduces to the standard result [6] for dipole radiation in vacuum. It can also be seen that radiation scattered parahel to a principal axis is a pure
transversevibration. A more substantial problem is presented by the cal-
4
R W. Mum et al./Crystal optics and Raman intensities
: -.. culation of E as i -+ m in the plane of two principal axes. A&in with a view to applications to monoclinic crystak, we take r = (r cos 6,0, r sin 0). Then we need Dz=Oand D,
sinO=D3 me1
(24)
From D2 = 0 we obtain two classes of solutions: (a) with k2 = 0, so that k is copIanar with r, and (b) with c:kT+c~k~+c32k~+c~k2=c~(c~fc32)
lead to ~$2 = -Q with all other elements zero. The constant Ci requires careful consideration. Comparison of c:P and Q using c: c; > ci cz shows that P and Q have the same sign and Ki > 0 if c; >c,2 oic~c~<~~c~.NowwithD,=O,dD,~= O=D2,,CiforKi>Oisgivenby Ci=sgn[D11D~fD22(01+D~)‘D33D:
(25)
- cf. eqs. (13) and (14). Of these solutions, those in class (b) are all inadmissible because they give aD/aw = 0. In class (a) there are four solutions (apart from ambiguities of sign) but two of those are independent of 0 and prove inadmissible under the radiation condition. The two remaining admissible solutions are
(35)
- 2DlD3D131 = sgn(Q + c;P) _
(36)
Since sgn Q = sgn P in this case, Ci = sgn Q or sgn P. The remaining case is specified by
c;c;
(37)
(i)
k =(-cz
cos 0, 0,-c2 sin 0))
(26)
which makes P < 0 and Q > 0 and thus Ki < 0. Then Ci is given by
(ii)
k = (-cf
cos e/c;, 0, -c:
(27)
Ci = i sgn(VkD*r)
sin e/c,),
where cf = c: sin% + cz c0s*e .
= i sgn(-29
(28)
These solutions satisfy k’ = c$ in case (i) and c+t
++;
=c;c:
(2%
in h:ase(ii), which are precisely the conditions for the two rays propagating in the plane of principai axes 1 and 3 [7], as might be expected. The difference from the usual crystal optics problem is that here k is not given directly but has to be determined from V,$Ill+r. With D, = 0 and D12 = 0 =Dz3 the curvatme becomes
(30)
Ki=p/Q, P=c&c;,
GJ*exp [i(cJt - c&3 E2 = 47rf+c’r
(39)
Q l/Z f9 -
0F
a*
[i(af - C2r)] 4q
c2 r
where -c;c$,
c: =cf c0fGe +c$ sin% _
(32) (33)
It is found that ki 0r = --car and
IVkDl=2c,lQl, while in wi the conditions k2 = 0 and bi = cz -k*
AQ=+
1,
=-I,
sgnP=sgnQ; sgnP=-s&Q.
In case (ii), the curvature is (34) =0
(40)
These results can be expressed in the single form
(31)
Q=c;c;
9
and for sgn P # sgn Q
E2 =
where
Q),
which is therefore -i. Collecting these results together and using sgn Q = Q/!Ql, we find for sgnP=sgn Q
eXp
In case (i), the curvature is evaluated as
(38)
Kii = c! P/c~~T:R 9 where
(42)
R. W. Munn et al.fCtysta1 optics and Raman intensities R=C,(C;-+Cf)+C;c;.
(45)
It is found that kii*r = -cLrand that IV@] = 2cxcz IRl/cz._ In wii all elements with any component parallel to the 2-axis are zero, leaving Wii=
x
(RcIc~/c~)
-sinecos e
side
0
0
0
0
0
cos*e
I
-sinecose
(46)
5
sults of this section in the form IE,I = (w2/4qrc2) Y$'pp,
(52)
where T denotes the direction of scattering r and the $8 are the components of 2 X 2 dimensionless scattering matrices with Q,fl# y_Taking the modulus of the component E, simplifies the Y (7) by removing factors of modulus unity which are ultimately eliminated in the expression for the Baman intensity. In the monoclinic crystal abc’ axes, Y fg) and Y cc’)are diag-
The curvature Kii’,O if ‘, an,d R $ave the same sign, i.e. ifczcI(cL +c,). In these cases Cii=-sgn(R
~~~C~p/C~),
that Cii = -S~ZZR or -sgn tiveifP>OandR
SO
(47) P. The
=~<~f~,“<~,(=~“~~)-=:=f.
curvature is nega-
(48)
when it is found that qi=iSgrlR=-i.
(49)
Together these results yield the components of E in the 1,Iplane: =
cd*
exp [i(ot - clr)] 47rfuC2Y
where nn w/c = cl c3/cL and n,, w[c = cl c3/cII for 0 taken as the angle between the a axis and the 1 axis in eqs. (28) and (33) - see also eq. (60) below. The elements of Y cc’)are obtained by interchanging 4 and c’ everywhere in the above expressions. The third matrix Y@) is diagonal in the principal axes but in the crystal axes takes the form Y(b) =(n,/n,n3)
(55)
(W nl sin28 frz3 co&% (~3 - pzl) sin 0 cos e\ where pL = p3 cos 0 - p1 sin 0 is the component of p perpendicular to r. By resolving these components of E along and perpendicular to r we find =.
X
3 i (tz3--nl)sinO
cOse
nl co~~e+r~~sin*e
I
appropriate to rotation by the angle 0 from the 1 axis to to the a axis.
w2 exp [i(wt - clr)] 47reuc2r
3. Effect of surfaces
(51) so that the scattered radiation remains purely transverse. When r is parallel to the 1 or 3 axes, eqs. (42) and (51) reduce correctly to forms analogous to eq. (23). For later use. it is convenient to summarize the re-
The effect of transmission through the crystal surfaces is most readily treated using the result that the two orthogonally polarized normal modes propagating in 2 given direction see the crystal 2s an isotropic medium with the refractive index appropriate to the given polarization [S]. Thus light propagating along the 2 principal axis is resolved into components polarized parallel to the 1 and 3 principal axes. Light propagating
6
R. IV.Muruzet al./Ctystal optics and Rarnan intensities
along the a or c’ axes is resolved into components polarized parallel to the 2 (b) and c’ or a axes. We denote the electric vector of the incident Iigbt by E” and that of the transmitted light by Et, so that Et is the macroscopic field required in the basic theory [ 11. We denote the electric vector of the scattered light by ES and that of the emergent light by Ee, so that Ee determines the observed scattered intensity_ The crystat is assumed to be in a medium of refractive index 1. All the light beams propagate at normal incidence to the crystal faces. The normal mode electric vectors are modified by a factor of 2/&--t 1) on transmission into the crystal and by 2rri/(rri + I) on emergence from the crystal, where ni is the appropriate crystal refractive index. Then we can define 2 X 2 transmittance matrices9(r) for each direction of propagation y such that Et=9(“1).@.
(56)
Referred to the crystal abc’ axes these matrices are g(a!
tl cos’0 f t3 sin%
g(b) = (tl -tj)
(tl -t$
sin 0 CDS0
3 sin 0 cos 0 fl sin% ft3 co&9 I (58)
gk’) =
(59)
2tZi/(rZi+ I), or equivalently with the ni replaced by i/f+.
4. Resultant intensities After passing through an analyser for (Ypolarization, the emergent beam has a power density described by a Poynting vector of magnitude S=E~CIEE~~ _
Here Ee is given in terms of ES by eq. (61), while ES is given by eq. (52). The dipole p is given by the Raman theory [l] in terms of Et and the quantityx e) = axlaQi, the derivative of the crystal electric susceptibility with respect to the normal coordinate for phonon mode j. Finally Et is related to E” by eq. (56). Raman intensities are commonly described solely in terms of the polarizations of the incident and emergent light. However, since the transmittance, scattering, and’ emergence matrices depend on the propagation direction as well as on the polarization direction, it is necessary to specify the intensities more precisely. We therefore use the conventional notation ~(4)s to denote light incident parallel to the y axis polarized parallel to the oraxis yielding emergent light parallel to the 6 axis polarized parallel to the P axis [ 131. Then the power scattered into unit solid angle, divided by the square of the amplitude of the electric vector of the incident light, can be written as [ 1,2] Irbm
where tj = 3/(tzj+ 1). These results are consistent with
those of ref. [9] when expressed in terms of the components of the dielectric tensor relative to the crystal axes [9-I 11, and can also be derived using the result
WI ‘4 = u~,*~/(‘T~c&I
+-rzz sin%)112 ,
(60)
for the refractive index appropriate to propagation at an angle 0 to the 1 axis in the IICplane with polarization perpendicular to this direction in the same plane. We can similarly define emergence matrices &(y) such that Ee =&(-i).ES _
(60
Referred to the crystal abc’ axes these matrices take the forms (57)~(59) with the tj replaced by ej =
(62)
=zi1(~(6).y(s).~ci).g(r))~~[2,
Zi = fiw; e. u2(tzif l)/87r2c3wi.
(63) (64)
Here ws is the frequency of the scattered light, u is the volume of the primitive unit cell, and jzi is the thermal average number of phonons in a mode of frequency Oj, so that eq. (63) gives the intensity for Stokes scattering. The result (63) differs from eq. (5) of [l] by the inclusion of the matrices & , Y, &rd 9 describing the effects of the crystal optics, and by rectifying the omission of a factor (ec u)2 _ Some general characteristics of the scattering intensity can be deduced from eq. (63) and the form of the various matrices in it. The following discussion assumes a 90” scattering geometry in a monoclinic crystal, with all matrices referred to the crystal ubc’ axes. When the propagation directions both lie in the QCplane, all the
7
R. W. Mum et al.JCrystal optics atid Raman intensities
optical matrices are diagonal and the intensity for (alp) polarization depends only on the C$ component of I(/). Thus for these propagation directions, the reiative intensities of modes in a given setting can be calculated from Zj and the x,&j). However, for incident light parallel to the b axis, 9is not diagonal and for emergent light parallel to the b axis, & and Y are not diagonal. The intensity then depends on a linear combination of two components of X(i), as noted previously [14]. In such cases even the relative intensities of modes in a given setting cannot be calculated from Zi and the x4(i) without also calculating the optical matrices. For 90” scattering there are twenty-Four different settings. (These can be derived from three directions of incidence with either of two polarizations yielding either of two directions of emergence with either of two directions of polarization_) These settings correspond to the restrictions y # CY, S # fl and y #S in the general setting ~(cY@&.For each of the three cases when the polarization does not change on scattering (CY = fl), there are two settings. For each of the six other cases (0~f P), there are three settings. The settings can be grouped in pairs differing only by reversal of the light path, i.e. ~(4) 6 and 6 (0~) y. For propagation in the IICplane, the relative intensities for these reversed settings depend on x,,(j) and I, and hence are equal because x(i) is symmetric. For propagation withy or 6 equal to b, the relative intensities for a reversed pair of settings appear different because one depends on E(b) 0Y(b) and the other on g@). However, it is readily verified in the principal axis system that &W oy(b) = n2 g(b) ,
(65)
and this result, together-with the symmetry of the optical matrices, makes the relative intensities equal for a reversed pair of settings. It is also found that for polarizations (+$I)with cy# fl where 01or fl is equal to b, the relative intensities are equal for pairs of settings_ differing only in the propagation direction in the IIC plane. Thesk equalities reduce to ten the number of independent settings giving different relative intensities, as summarized in table 1. As can be seen, there is a unique pattern of intensities only for the polarizations unchanged by scattering, (aa), (bb) and (c’c’); for (ab) and (bc’) polarizations there are two patterns, and for
Table 1 Crystal settings and relative intensities. All settings on the same line give the same relative intensities for different modes b (aa) c’ b (ab) c’ c’(ab) a b(ac’)a c’ (a&) b c’ (lx’) 0 a(bb)c’ a(bc’) b c’ (bc’) a a(c’c’) b
c’ (aa) b b (ab) a a(ba)c’ a @‘a) b b (da) c’ a (c’a) 12’ c’ (bb) a c’ (be’) b a (c’b) c’ b(c’c’) iz
a (ba) b
c’ (ba) b
b(c’b)c’
b (c’b) a
(ac’)
polarization there are three. It is therefore clear that the setting must be specified completely in reporting Raman intensity measurements not taken in the pripcipal axis system. Except for (no) and (c’c’) polarizations, it is always possible to choose a setting unaffected by the crystal optics. How far the intensity patterns differ between settings depends on the relative sizes of the components ofI@, on the optical anisotropy in the ac plane, and on the deviation of the crystal axes from the principal optical axes. To illustrate the variations, we have calculated the relative intensities for naphthalene in (ac’) and (ab) polarizations. We take the principal refractive indices as 121 = 1.525 and pz3= 1.945, with 0 = 23.4’ [15] _For the present purely illustrative purposes we use values of x(i) based on the free-molecule polarizability with no local-field correction, corresponding to the first line in table 6 of ref. [l] . The results are shown in table 2: the relative intensities in fact change little between settings. This is due to the small off-diagonal elements in $Icb), which is calculated to be
Table 2 Relative intensities of Raman scattering calculated lene in different settings wesp
for naphtlrz-
wexp Setting
Setting
cm A c’(ac’)a
c’(d)
51 7.0 74 ICI0 109 15.0 ---L----v
6.6 100 17.7
b
b(ac’)a
cm-’
c’(ab)a
b(ab)c’
7.3 100 14.4
46 71 12.5
0.9 6.9 100
1.4 9.2 100
8
R. W. Munnet al./Crystal optics and Raman intensities
so that there are only small admixtures of components Q&), where I$ does not correspond to the polarization; these components must therefore be particularly large to have any sizeable effect. The smallness of 9) can be traced to eq. (58) where its maximum value is f ItI - fa[ or ]rz3-~~j/(rz~ + l)(rza + 1). Even witha large birefringence n3 - rzl , as in naphthalene, the denominator reduces 9::) by an order of magnitude. It therefore appears that the relative intensities wifl change markedly between settings only in special cases of extreme anisotropy in x(i) and the refractive indices.
5. Discussion We have treated here aspects of crystal optics which affect the intensity of Raman scattering. The treatment has been confined to situations with normal incidence and emergence from cuboidal crystals along or perpendicular to a principal dielectric axis. Some of the results have been expressed in a form appropriate to monoclinic crystals using the abc’ crystal axes, but the results can be applied to other crystals with suitable reinterpretation The first part of the treatment solves the problem of dipole radiation in an anisotropic dielectric continuum, subject to the above restrictions on direction of propagation_ As might be expected, the results are a superposition of the normal modes for the appropriate direction of propagation, but for propagation away from the principal axes the amplitude is a complicated function of the refractive indices. The expressions may be of use in other applications. A rather general treatment of the radiation of electromagnetic fields in biaxial media has recently been given [17]. The second part of the treatment derives the corrections for the effect of the crystal surface on light passing through it. Multiple reflection and interference effects are neglected, and the crystal is assumed to be non-absorbing. In essence, the results are not new [8,9], but finding a convenient formulation in the literature is surprisingly difficult. The final part of the treatment combines the earlier parts to obtain an expression for the Raman intensity.
This is referred to a set of orthogonal axes, but transformation to describe less conventional geometries would be straightforward. As it stands, the expression can describe both 90” and 180° scattering, but its consequences are explored only. for 90” scattering in the monoclinic a&’ axis system. Of the twenty-four different crystal settings, ten are distinct_ This number e&ceeds the number of polarizations because the same polarization may give up to three patterns of relative intensities depending on different combinations of components of the susceptibility tensor derivative x(i) in different settings. However, except in extreme cases, the patterns of relative intensities for a given polarization will not vary much with setting, although the complete setting should nonetheless always be specified. It would be of interest to examine the variation of intensities with setting experimentally_ For example, the pattern of intensities for a given polarization is very similar for p-dichlorobenzene and p-dibromobenzene except for (bc’) polarization [ 161, and it may be that the difference is partly attributable to the use of the distinct settings c’(bc’)a and b(c’b)a for the two compounds. In favourable cases, changing the setting may help to locate a weak line or resolve it from another one by lending it intensity. Theoretical analysis of intensities is easiest when all light propagates in the ac piane, as no consideration of the optical matrices is required. This is possible for some polarizations, but not all. Realising the complications for b axis propagation, some workers [14] prudently avoided it, thereby obtaining only incomplete polarization results. The present quantitative analysis shows that such caution, though commendable, is not essential, because b axis propagation will normally introduce only a small amount of mixing of components of x (i). Although the present derivation of crystal optical effects has been applied to sideways and back scattering, the basic results could equally well be used for forward scattering. Allowance for absorbing crystals could also be included. Such modifications would allow the treatment to be applied to problems such as polarized infrared absorption spectra in transmission. The effect of the optical matrices on this problem has already been examined [8], though in terms slightly different from those used here.
R W. Munn et =I. fCtysta! opticsand Raman intensities References 1l] T. Luty, A. Mierzejewski and R-W. Munn, Chem. Phys. 29 (1978) 353. [2] M. Born and K. Huang, Dynamical theory of crystal lattices (Oxford Univ. Press, London, 1954), sections 19, 20 and 49. [3] J.W. Arthur and G.A. Mackenzie, J. Raman Spectry. 1 (197.5) 119. [4] ML Lighthill, Phil. Trans.‘Roy. Sot. London A252 (1960) 397. [S] H. Kogelnik and H. Motz, Electromagnetic theory and antennas, ed. EC. Jordan (Pergamon Press, New York, 1963) part 1, p. 477. [6] P. Lonain and D. Carson, Electromagnetic fields and waves, 2nd Ed. (Freeman, San Francisco, 1970) p. 602. 171 H.J. Juretschke, Crystal physics (Benjamin, Reading, Mass., 1974) ch. 9.4. [S] J. Herranz and J-M. Delgado, Spectrochim. Acta 31A (1975) 125.5.
9
[9] E.E. Koch, A. Otto and K.L. Kliewer, Chem. Phys. 3 (1974) 362. [lo] R.W. Munn, J.R. Nicholson, H.-P. Schwab and D.F. Williams, J. Chem. Phys. 48 (1973) 3828. 1111 P.J. Bounds and R.W. Munn, Mol. Cryst. Liquid Crysr. 44 (1978) 301. [12] E.A. Bahik, D.J. Gannon and J.A. Morrison, I. Chcm. Pbys. 58 (1973) 5639. [I31 T.C. Damen, S.P.S. Porto and B. Tell, Phys. Rev. 142 (1966) 570. [14] J.B. Bates, D.M_ Thomas, A. Bandy and E.R. Lippincott, Spectrochim. Acta 27A(1971) 637. [IS] A.N. Wiichell, Optical properties of organic compounds (Academic Press, New York, 1954). [I61 M. Ito, M. Suzuki and T. Yokoyama, Excitons, magoons and phonons in molecular crystals, ed. A.B. Zahlan (Cambridge Univ. Press, London, 1968) p- 1. [ 171 J.J. Stamnes and G.C. Sherman, J. Opt. Sot. Am. 68 (1978) 502.
CRYSTAL OPTICS AND RAMAN INTENSITIES
RW. MUNN Department of Chemistry. LJMIST,Manchester M60 IQD. U.
T. LUTY Institute for Theoretical Chemistry, Catholic Ukrversity.Nijmegen. lXe Netherlands and institute of Organic and Physical Chemistry. Technical lEversi@, N-370 WrocZaw,Poland
A. MIERZEJEWSKI Institute of Organic and Physical Chemistry. Technical University, 50-370 Wroc&aw.Poland Received
29 May 1978
The theory of Raman scattering intensities is modified to take account of crystal optics. The radiation at a long distance from an oscillating electric dipole in a diclectic continuum is calculated for propagation along or perpendicular to a principal dielectric axis. The results are combined with results on the effect of transmission through the crystal surface to give an expression for the polarized Raman intensity in an orthgonal axis system with only one axis a principal dielectric axis. For 90” scattering in monoclinic crystals in the ate’ axis system, there may be up to three different intensity patterns for a given polarization, yielding ten distinct crystal settings. Sample calculations for naphthalene show that the intensity patterns for the same polarizations in distinct settings differ oniy slightly, and this will normally be the case.
. 1. Introduction In a previous paper [ I], we derived a new expression for the Raman scattering tensor in molecular crystals. Although the principles of Raman scattering were treated by Born and Huang [2], they emphasized that their derivation was strictly valid only for an isolated molecular system. For a crystal, the electric vector of the incoming light must be t&en as the macroscopic field Wide a region small compared with the wavelength of the light but large compared with atomic and molecular dimensions. This led us to calculate the scattering tensor from the high-frequency electric suscep-
In comparing these results with experiment, a difficulty arises. Our treatment [I] modities Born and Huang’s results for an isolated molecule to take account of the electrostatic interactions between the molecules in a crystal. However, the results for an isolated molecule must also be modified to take account of the optical properties of a crystal. Polarized light incident on a crystal is in general attenuated, refracted and depolarized on passing through the crystal surface to give a transmitted beam. Similar changes take place in the Raman scattered light on passing through the crystal surface to give an emergent beam. Furthermore, the calculation of the scattering within the crystal, viewed
tibility
as originating
of the crystal,
within
the point-dipole
rigid-
molecule approximation. Thls.appfoach yields the contribution to the intensity from both translational and rotational motions; in both cases it is proportional to the fourthpower of the local-field correction.
from
an oscillating
transition
dipole,
must take account of the anisotropic dielectric medium through which the radiation passes. For normal incidence on cuboidal crystals cut parallel to the principal axes of the dielectric tensor, with
2
.R. W.Mmm et al./Cr_wtoloptics and Ramanintensities
all directions of polarization parallel to these axes, one expects that the correct relative intensities should be obtained for different modes of vibration in a given crystal setting, even when the effects of crystal optics are ignored in the calculation. Cubic crystals are optically isotropic and so present no problem. For orthorhombic and axial crystals the principal axes are fixed by symmetry, and cuboids are usually readily cut parallel to these axes. Indeed, a misoriented cuboid may be detected by its depolarization of a polarized beam on transmission through the crystal [3] _However, for monoclinic crystals, only the b axis is a principal dielectric axis by symmetry, the other principal axes lying in the ac plane. Many important molecular crystals are monoclinic with an easy cleavage parallel to the ab plane (because of the molecular packing). It is then convenient to cut cuboids parallel to the crystal ah’ axes. For comparison with Raman intensity measurements on crystals cut like this, it becomes essential to include the effect of the crystal optics in calculating the theoretical intensity. In the present paper, we complete our modification of Born and Huang’s results by including these effects of crystal optics. We begin by deriving expressions for the electric field of the radiation emitted by an oscillating dipole in an anisotropic dielectric for waves propagating along a principal dielectric axis or in the plane of two principal axes. These suffice for the problems just mentioned and also for triclinic crystals provided one principal axis is located for cutting them. We then treat the changes due to transmission through the crystal surface and finally the emergent intensity_
vector, and W is a matrix which &the principal dielectric axes takes the form
W=
kf+bf
%k2
%k3
klk2
k$+b;.
k2k3
k2k3
k$+b;
i klk3
_
(2)
i
Here the kj are the Cartesian components Of k and - k2 z q&/c2 - k2, where the Ei are the principal components of the dielectric tensor. The inverse matrix W-l is written as bz zcf
w-1 = w/D,
(3)
where
and
(4)
D=detW=cfcic$-
[cfci(kf+k$
+c;c$(k;
lk$)+c;c;(k;+k;)]
+k2(c:k;
+c;k;
+c;k;).
(5)
Then the limiting electric field is given by E(r + L) = (hGio/q
c2r)
C, exp (ik, 0r) wm J, exp (iot) l
XC
2. Dipole radiation in an anisotropic continuum A general method for obtaining the asymptotic solution as r -, 03 for the waves due to a spatially localized source at the origin has been given by fighthill
141.
A modification of the method has been presented by Kogelnik and Motz [5], and app!ied to radiation in a plasma subject to a magnetic field (which has a complex uniaxial permittivity). The Fourier components of the electric field amplitude are given by Ek = iwpOW’l -Jk,
(1)
where Jk is the Fourier component of the current density source oscillating at frequency w, k is the wave-
m
IV,D,
IIK, I’/2
’
(6)
where km is a root ofD(k) = 0 such that V@ll+r (to give the correct direction) and such that (r. V&l)/ (W/W) is positive definite but fmite (the radiation condition, to give causally propagating waves). The quantities w,, J,,, andVkD, are evaluated fork = km. The quantity Km is the gaussian curvature K evaluated fork = km, where [4] K= c [
D;(DiiDkk-D$
R. W. Mum et aI./Ciystal optics and Ramarr irttet&ies
(q/c) being a permutation of (123) and the sums being taken over all three such permutations, withDi E ( VkD>i, Dii 3 (VkVk D>ii- WhenKm < 0, Cm = * i according as Vk Dm iI+r, and when K,n > 0, Cm = k 1 according as the surface D = 0 is convex to the direction of -I-VDm fork = km, i.e. according as [4] C Dii(Di” f 0;) - 2 cDJ~DiDk
3 0.
(8)
For a point dipolep exp(iwt) at the origin, the polarization is p exp(ior) 6 (r) and the current density is thus iwp exp(iwt)6 (r). Then J, = iwp/8n3, leading to
(c)
-k, =O , or
(15)
(d)
c~k~+C22k2’$C32k~+C~k?-=c32(C:+C~).
(16)
These give four possible pairs of solutions, but for the combination (b)+(d) it is found that aD/ao = 0, which is inadmissible. On solving D(k) = 0 for the other three pairs, one finds that similarly tlD/aw = 0 for the combinations (a) f(d) and (b) + (c), leaving only (a) f (c), for which k, = 0 = kg and ki=cf
or
c$_
(17)
The radiation condition is satisfied for
E(r h -) 4 -(02eiwt/27rc0 c2r)
(9) The procedure used to calculate E is as follows. The direction r is specified, which specifies the direction of ?kD and permits two of the components of k to be expressed in terms of the third. The equation D(k) = 0 can then be sohed. For each solution km the signs of r- V,D and aD/aw are evaluated, where since D(k) = 0 sgn(aD/aw)=spn[~c:c5c~-k*(c:k:
3
,
+c$k~+c~k$]
(i)
k2 =-cl
,
(18)
(ii)
k? = -9
.
(1%
The curvature K is simplified because D, = 0 = 03, leaving K=(D11D33 -L&~D~,
(20)
where in general
-c; (c; + $1
+ 8cf k: ,
01)
wu and solutions not satisfying the radiation condition are discarded. For the remaining solutions the curvature K, is evaluated and thence the coefficient C,,, . Further manipulation gives E. The calculation of E as r 3 COalong a principal axis is relatively straightfonvard and serves as a check on more complicated solutions. With a view to propagation along the unique b axis in monoclinic crystals, we take r = (OJ, 0). Then we need D, =O=D3,
(11)
D13 =4k,k&
etc. Then for both cases (i) and (ii) it is found that K = Z/cz > 0. The convexity condition reduce; to C, = sgn(Drr f D,,), which yields Ci,= s$(c
sults yield
X (e-iclrp&l,
(12
(b)
cfkf +c;k;
(13) +c;k;
+c;k2
0,e-iC3~p3/c3) ,
(23)
where c1 = rzl w/c etc., with H1 a principal refractive
index. In the limit that all nI;become equal to unity,
etc. Then D1 = 0 implies either k, =O, or
(22)
E = (ozeiWrc2/4qrc2)
where in general
(a)
+c$),
=$(c5
with correspondingly for D3 = 0 either
+c;),(14)
this expression reduces to the standard result [6] for dipole radiation in vacuum. It can also be seen that radiation scattered parahel to a principal axis is a pure
transversevibration. A more substantial problem is presented by the cal-
4
R W. Mum et al./Crystal optics and Raman intensities
: -.. culation of E as i -+ m in the plane of two principal axes. A&in with a view to applications to monoclinic crystak, we take r = (r cos 6,0, r sin 0). Then we need Dz=Oand D,
sinO=D3 me1
(24)
From D2 = 0 we obtain two classes of solutions: (a) with k2 = 0, so that k is copIanar with r, and (b) with c:kT+c~k~+c32k~+c~k2=c~(c~fc32)
lead to ~$2 = -Q with all other elements zero. The constant Ci requires careful consideration. Comparison of c:P and Q using c: c; > ci cz shows that P and Q have the same sign and Ki > 0 if c; >c,2 oic~c~<~~c~.NowwithD,=O,dD,~= O=D2,,CiforKi>Oisgivenby Ci=sgn[D11D~fD22(01+D~)‘D33D:
(25)
- cf. eqs. (13) and (14). Of these solutions, those in class (b) are all inadmissible because they give aD/aw = 0. In class (a) there are four solutions (apart from ambiguities of sign) but two of those are independent of 0 and prove inadmissible under the radiation condition. The two remaining admissible solutions are
(35)
- 2DlD3D131 = sgn(Q + c;P) _
(36)
Since sgn Q = sgn P in this case, Ci = sgn Q or sgn P. The remaining case is specified by
c;c;
(37)
(i)
k =(-cz
cos 0, 0,-c2 sin 0))
(26)
which makes P < 0 and Q > 0 and thus Ki < 0. Then Ci is given by
(ii)
k = (-cf
cos e/c;, 0, -c:
(27)
Ci = i sgn(VkD*r)
sin e/c,),
where cf = c: sin% + cz c0s*e .
= i sgn(-29
(28)
These solutions satisfy k’ = c$ in case (i) and c+t
++;
=c;c:
(2%
in h:ase(ii), which are precisely the conditions for the two rays propagating in the plane of principai axes 1 and 3 [7], as might be expected. The difference from the usual crystal optics problem is that here k is not given directly but has to be determined from V,$Ill+r. With D, = 0 and D12 = 0 =Dz3 the curvatme becomes
(30)
Ki=p/Q, P=c&c;,
GJ*exp [i(cJt - c&3 E2 = 47rf+c’r
(39)
Q l/Z f9 -
0F
a*
[i(af - C2r)] 4q
c2 r
where -c;c$,
c: =cf c0fGe +c$ sin% _
(32) (33)
It is found that ki 0r = --car and
IVkDl=2c,lQl, while in wi the conditions k2 = 0 and bi = cz -k*
AQ=+
1,
=-I,
sgnP=sgnQ; sgnP=-s&Q.
In case (ii), the curvature is (34) =0
(40)
These results can be expressed in the single form
(31)
Q=c;c;
9
and for sgn P # sgn Q
E2 =
where
Q),
which is therefore -i. Collecting these results together and using sgn Q = Q/!Ql, we find for sgnP=sgn Q
eXp
In case (i), the curvature is evaluated as
(38)
Kii = c! P/c~~T:R 9 where
(42)
R. W. Munn et al.fCtysta1 optics and Raman intensities R=C,(C;-+Cf)+C;c;.
(45)
It is found that kii*r = -cLrand that IV@] = 2cxcz IRl/cz._ In wii all elements with any component parallel to the 2-axis are zero, leaving Wii=
x
(RcIc~/c~)
-sinecos e
side
0
0
0
0
0
cos*e
I
-sinecose
(46)
5
sults of this section in the form IE,I = (w2/4qrc2) Y$'pp,
(52)
where T denotes the direction of scattering r and the $8 are the components of 2 X 2 dimensionless scattering matrices with Q,fl# y_Taking the modulus of the component E, simplifies the Y (7) by removing factors of modulus unity which are ultimately eliminated in the expression for the Baman intensity. In the monoclinic crystal abc’ axes, Y fg) and Y cc’)are diag-
The curvature Kii’,O if ‘, an,d R $ave the same sign, i.e. ifcz
~~~C~p/C~),
that Cii = -S~ZZR or -sgn tiveifP>OandR
SO
(47) P. The
=~<~f~,“<~,(=~“~~)-=:=f.
curvature is nega-
(48)
when it is found that qi=iSgrlR=-i.
(49)
Together these results yield the components of E in the 1,Iplane: =
cd*
exp [i(ot - clr)] 47rfuC2Y
where nn w/c = cl c3/cL and n,, w[c = cl c3/cII for 0 taken as the angle between the a axis and the 1 axis in eqs. (28) and (33) - see also eq. (60) below. The elements of Y cc’)are obtained by interchanging 4 and c’ everywhere in the above expressions. The third matrix Y@) is diagonal in the principal axes but in the crystal axes takes the form Y(b) =(n,/n,n3)
(55)
(W nl sin28 frz3 co&% (~3 - pzl) sin 0 cos e\ where pL = p3 cos 0 - p1 sin 0 is the component of p perpendicular to r. By resolving these components of E along and perpendicular to r we find =.
X
3 i (tz3--nl)sinO
cOse
nl co~~e+r~~sin*e
I
appropriate to rotation by the angle 0 from the 1 axis to to the a axis.
w2 exp [i(wt - clr)] 47reuc2r
3. Effect of surfaces
(51) so that the scattered radiation remains purely transverse. When r is parallel to the 1 or 3 axes, eqs. (42) and (51) reduce correctly to forms analogous to eq. (23). For later use. it is convenient to summarize the re-
The effect of transmission through the crystal surfaces is most readily treated using the result that the two orthogonally polarized normal modes propagating in 2 given direction see the crystal 2s an isotropic medium with the refractive index appropriate to the given polarization [S]. Thus light propagating along the 2 principal axis is resolved into components polarized parallel to the 1 and 3 principal axes. Light propagating
6
R. IV.Muruzet al./Ctystal optics and Rarnan intensities
along the a or c’ axes is resolved into components polarized parallel to the 2 (b) and c’ or a axes. We denote the electric vector of the incident Iigbt by E” and that of the transmitted light by Et, so that Et is the macroscopic field required in the basic theory [ 11. We denote the electric vector of the scattered light by ES and that of the emergent light by Ee, so that Ee determines the observed scattered intensity_ The crystat is assumed to be in a medium of refractive index 1. All the light beams propagate at normal incidence to the crystal faces. The normal mode electric vectors are modified by a factor of 2/&--t 1) on transmission into the crystal and by 2rri/(rri + I) on emergence from the crystal, where ni is the appropriate crystal refractive index. Then we can define 2 X 2 transmittance matrices9(r) for each direction of propagation y such that Et=9(“1).@.
(56)
Referred to the crystal abc’ axes these matrices are g(a!
tl cos’0 f t3 sin%
g(b) = (tl -tj)
(tl -t$
sin 0 CDS0
3 sin 0 cos 0 fl sin% ft3 co&9 I (58)
gk’) =
(59)
2tZi/(rZi+ I), or equivalently with the ni replaced by i/f+.
4. Resultant intensities After passing through an analyser for (Ypolarization, the emergent beam has a power density described by a Poynting vector of magnitude S=E~CIEE~~ _
Here Ee is given in terms of ES by eq. (61), while ES is given by eq. (52). The dipole p is given by the Raman theory [l] in terms of Et and the quantityx e) = axlaQi, the derivative of the crystal electric susceptibility with respect to the normal coordinate for phonon mode j. Finally Et is related to E” by eq. (56). Raman intensities are commonly described solely in terms of the polarizations of the incident and emergent light. However, since the transmittance, scattering, and’ emergence matrices depend on the propagation direction as well as on the polarization direction, it is necessary to specify the intensities more precisely. We therefore use the conventional notation ~(4)s to denote light incident parallel to the y axis polarized parallel to the oraxis yielding emergent light parallel to the 6 axis polarized parallel to the P axis [ 131. Then the power scattered into unit solid angle, divided by the square of the amplitude of the electric vector of the incident light, can be written as [ 1,2] Irbm
where tj = 3/(tzj+ 1). These results are consistent with
those of ref. [9] when expressed in terms of the components of the dielectric tensor relative to the crystal axes [9-I 11, and can also be derived using the result
WI ‘4 = u~,*~/(‘T~c&I
+-rzz sin%)112 ,
(60)
for the refractive index appropriate to propagation at an angle 0 to the 1 axis in the IICplane with polarization perpendicular to this direction in the same plane. We can similarly define emergence matrices &(y) such that Ee =&(-i).ES _
(60
Referred to the crystal abc’ axes these matrices take the forms (57)~(59) with the tj replaced by ej =
(62)
=zi1(~(6).y(s).~ci).g(r))~~[2,
Zi = fiw; e. u2(tzif l)/87r2c3wi.
(63) (64)
Here ws is the frequency of the scattered light, u is the volume of the primitive unit cell, and jzi is the thermal average number of phonons in a mode of frequency Oj, so that eq. (63) gives the intensity for Stokes scattering. The result (63) differs from eq. (5) of [l] by the inclusion of the matrices & , Y, &rd 9 describing the effects of the crystal optics, and by rectifying the omission of a factor (ec u)2 _ Some general characteristics of the scattering intensity can be deduced from eq. (63) and the form of the various matrices in it. The following discussion assumes a 90” scattering geometry in a monoclinic crystal, with all matrices referred to the crystal ubc’ axes. When the propagation directions both lie in the QCplane, all the
7
R. W. Mum et al.JCrystal optics atid Raman intensities
optical matrices are diagonal and the intensity for (alp) polarization depends only on the C$ component of I(/). Thus for these propagation directions, the reiative intensities of modes in a given setting can be calculated from Zj and the x,&j). However, for incident light parallel to the b axis, 9is not diagonal and for emergent light parallel to the b axis, & and Y are not diagonal. The intensity then depends on a linear combination of two components of X(i), as noted previously [14]. In such cases even the relative intensities of modes in a given setting cannot be calculated from Zi and the x4(i) without also calculating the optical matrices. For 90” scattering there are twenty-Four different settings. (These can be derived from three directions of incidence with either of two polarizations yielding either of two directions of emergence with either of two directions of polarization_) These settings correspond to the restrictions y # CY, S # fl and y #S in the general setting ~(cY@&.For each of the three cases when the polarization does not change on scattering (CY = fl), there are two settings. For each of the six other cases (0~f P), there are three settings. The settings can be grouped in pairs differing only by reversal of the light path, i.e. ~(4) 6 and 6 (0~) y. For propagation in the IICplane, the relative intensities for these reversed settings depend on x,,(j) and I, and hence are equal because x(i) is symmetric. For propagation withy or 6 equal to b, the relative intensities for a reversed pair of settings appear different because one depends on E(b) 0Y(b) and the other on g@). However, it is readily verified in the principal axis system that &W oy(b) = n2 g(b) ,
(65)
and this result, together-with the symmetry of the optical matrices, makes the relative intensities equal for a reversed pair of settings. It is also found that for polarizations (+$I)with cy# fl where 01or fl is equal to b, the relative intensities are equal for pairs of settings_ differing only in the propagation direction in the IIC plane. Thesk equalities reduce to ten the number of independent settings giving different relative intensities, as summarized in table 1. As can be seen, there is a unique pattern of intensities only for the polarizations unchanged by scattering, (aa), (bb) and (c’c’); for (ab) and (bc’) polarizations there are two patterns, and for
Table 1 Crystal settings and relative intensities. All settings on the same line give the same relative intensities for different modes b (aa) c’ b (ab) c’ c’(ab) a b(ac’)a c’ (a&) b c’ (lx’) 0 a(bb)c’ a(bc’) b c’ (bc’) a a(c’c’) b
c’ (aa) b b (ab) a a(ba)c’ a @‘a) b b (da) c’ a (c’a) 12’ c’ (bb) a c’ (be’) b a (c’b) c’ b(c’c’) iz
a (ba) b
c’ (ba) b
b(c’b)c’
b (c’b) a
(ac’)
polarization there are three. It is therefore clear that the setting must be specified completely in reporting Raman intensity measurements not taken in the pripcipal axis system. Except for (no) and (c’c’) polarizations, it is always possible to choose a setting unaffected by the crystal optics. How far the intensity patterns differ between settings depends on the relative sizes of the components ofI@, on the optical anisotropy in the ac plane, and on the deviation of the crystal axes from the principal optical axes. To illustrate the variations, we have calculated the relative intensities for naphthalene in (ac’) and (ab) polarizations. We take the principal refractive indices as 121 = 1.525 and pz3= 1.945, with 0 = 23.4’ [15] _For the present purely illustrative purposes we use values of x(i) based on the free-molecule polarizability with no local-field correction, corresponding to the first line in table 6 of ref. [l] . The results are shown in table 2: the relative intensities in fact change little between settings. This is due to the small off-diagonal elements in $Icb), which is calculated to be
Table 2 Relative intensities of Raman scattering calculated lene in different settings wesp
for naphtlrz-
wexp Setting
Setting
cm A c’(ac’)a
c’(d)
51 7.0 74 ICI0 109 15.0 ---L----v
6.6 100 17.7
b
b(ac’)a
cm-’
c’(ab)a
b(ab)c’
7.3 100 14.4
46 71 12.5
0.9 6.9 100
1.4 9.2 100
8
R. W. Munnet al./Crystal optics and Raman intensities
so that there are only small admixtures of components Q&), where I$ does not correspond to the polarization; these components must therefore be particularly large to have any sizeable effect. The smallness of 9) can be traced to eq. (58) where its maximum value is f ItI - fa[ or ]rz3-~~j/(rz~ + l)(rza + 1). Even witha large birefringence n3 - rzl , as in naphthalene, the denominator reduces 9::) by an order of magnitude. It therefore appears that the relative intensities wifl change markedly between settings only in special cases of extreme anisotropy in x(i) and the refractive indices.
5. Discussion We have treated here aspects of crystal optics which affect the intensity of Raman scattering. The treatment has been confined to situations with normal incidence and emergence from cuboidal crystals along or perpendicular to a principal dielectric axis. Some of the results have been expressed in a form appropriate to monoclinic crystals using the abc’ crystal axes, but the results can be applied to other crystals with suitable reinterpretation The first part of the treatment solves the problem of dipole radiation in an anisotropic dielectric continuum, subject to the above restrictions on direction of propagation_ As might be expected, the results are a superposition of the normal modes for the appropriate direction of propagation, but for propagation away from the principal axes the amplitude is a complicated function of the refractive indices. The expressions may be of use in other applications. A rather general treatment of the radiation of electromagnetic fields in biaxial media has recently been given [17]. The second part of the treatment derives the corrections for the effect of the crystal surface on light passing through it. Multiple reflection and interference effects are neglected, and the crystal is assumed to be non-absorbing. In essence, the results are not new [8,9], but finding a convenient formulation in the literature is surprisingly difficult. The final part of the treatment combines the earlier parts to obtain an expression for the Raman intensity.
This is referred to a set of orthogonal axes, but transformation to describe less conventional geometries would be straightforward. As it stands, the expression can describe both 90” and 180° scattering, but its consequences are explored only. for 90” scattering in the monoclinic a&’ axis system. Of the twenty-four different crystal settings, ten are distinct_ This number e&ceeds the number of polarizations because the same polarization may give up to three patterns of relative intensities depending on different combinations of components of the susceptibility tensor derivative x(i) in different settings. However, except in extreme cases, the patterns of relative intensities for a given polarization will not vary much with setting, although the complete setting should nonetheless always be specified. It would be of interest to examine the variation of intensities with setting experimentally_ For example, the pattern of intensities for a given polarization is very similar for p-dichlorobenzene and p-dibromobenzene except for (bc’) polarization [ 161, and it may be that the difference is partly attributable to the use of the distinct settings c’(bc’)a and b(c’b)a for the two compounds. In favourable cases, changing the setting may help to locate a weak line or resolve it from another one by lending it intensity. Theoretical analysis of intensities is easiest when all light propagates in the ac piane, as no consideration of the optical matrices is required. This is possible for some polarizations, but not all. Realising the complications for b axis propagation, some workers [14] prudently avoided it, thereby obtaining only incomplete polarization results. The present quantitative analysis shows that such caution, though commendable, is not essential, because b axis propagation will normally introduce only a small amount of mixing of components of x (i). Although the present derivation of crystal optical effects has been applied to sideways and back scattering, the basic results could equally well be used for forward scattering. Allowance for absorbing crystals could also be included. Such modifications would allow the treatment to be applied to problems such as polarized infrared absorption spectra in transmission. The effect of the optical matrices on this problem has already been examined [8], though in terms slightly different from those used here.
R W. Munn et =I. fCtysta! opticsand Raman intensities References 1l] T. Luty, A. Mierzejewski and R-W. Munn, Chem. Phys. 29 (1978) 353. [2] M. Born and K. Huang, Dynamical theory of crystal lattices (Oxford Univ. Press, London, 1954), sections 19, 20 and 49. [3] J.W. Arthur and G.A. Mackenzie, J. Raman Spectry. 1 (197.5) 119. [4] ML Lighthill, Phil. Trans.‘Roy. Sot. London A252 (1960) 397. [S] H. Kogelnik and H. Motz, Electromagnetic theory and antennas, ed. EC. Jordan (Pergamon Press, New York, 1963) part 1, p. 477. [6] P. Lonain and D. Carson, Electromagnetic fields and waves, 2nd Ed. (Freeman, San Francisco, 1970) p. 602. 171 H.J. Juretschke, Crystal physics (Benjamin, Reading, Mass., 1974) ch. 9.4. [S] J. Herranz and J-M. Delgado, Spectrochim. Acta 31A (1975) 125.5.
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[9] E.E. Koch, A. Otto and K.L. Kliewer, Chem. Phys. 3 (1974) 362. [lo] R.W. Munn, J.R. Nicholson, H.-P. Schwab and D.F. Williams, J. Chem. Phys. 48 (1973) 3828. 1111 P.J. Bounds and R.W. Munn, Mol. Cryst. Liquid Crysr. 44 (1978) 301. [12] E.A. Bahik, D.J. Gannon and J.A. Morrison, I. Chcm. Pbys. 58 (1973) 5639. [I31 T.C. Damen, S.P.S. Porto and B. Tell, Phys. Rev. 142 (1966) 570. [14] J.B. Bates, D.M_ Thomas, A. Bandy and E.R. Lippincott, Spectrochim. Acta 27A(1971) 637. [IS] A.N. Wiichell, Optical properties of organic compounds (Academic Press, New York, 1954). [I61 M. Ito, M. Suzuki and T. Yokoyama, Excitons, magoons and phonons in molecular crystals, ed. A.B. Zahlan (Cambridge Univ. Press, London, 1968) p- 1. [ 171 J.J. Stamnes and G.C. Sherman, J. Opt. Sot. Am. 68 (1978) 502.