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Birefringence effect in the Raman spectrum of a crystal which is not cut parallel to the principal axes—II. Application to a single crystal of LaBO3
Blrefrlngence effect inthe Ramm spectrum of a crystal whl&lsnotcutpadleltotbeprhcipalaxesII. Application to a single crystal of LaB03 A. RULMONT,J. P. FLMME, M. J. PoTnw Departement de Chimie G&n&aleet de Chimic Physique, Institut de Chimie, Sart Tii B-4OfSLiege 1, Belgium
par
B. M. WANKLYN Clarendon Laboratory, Oxford Univnsity, Parks Road, Oxford OX1 3PU, U.K. Abstract-The
Raman spectra of LaBOa have been studied on a single crystal the faces of which are not parallel to the principal axes. A symmetry assignment is proposed from the 90” scattered intensity as a function of the polarization direction of the incident light. The experimental measurements contirm the theorefcal predictions derived in a previous paper.
INTRODUCIION In a preceding paper [l], we have theoretically derived
relations giving the scattered Raman intensity as a function of the direction of polarization of the incident light; an orthorhombic crystal has been considered, the principal axis system of which, (oxyz) is rotated by an angle /I, with respect to the system of axes perpendicular to the faces (cx’ff), along the common z’= 2 axis. We have checked these theoretical predictions on a LaBOS monocrystal, grown and cut at the C&e&m Laboratc#yipOxibrd~ bExPEwMENT
AL SETTING
The spectra have been recorded on a PHO CODERG double monochromator usinn a He-Ne laser Q- 632.8 mn ; W =50 mW). The crystal is &led on a go&meter head. The direction of polarization of the incident tight can be rotated by a half-wave plate, whose orientation is calibrated in degrees; the rotation of this plate can induce slight displacements of the focus inside the sample, which imply small corrections of the excentric lens before the entrance slit of the monochromator.
B CpYsfAL DSCRIPTION 1. General description
The structure of the LaBOs cry&
crystal faces, has been checked by X-ray diffraction on each face separately. The indices found for the three faces are 022,200, 031, which allow us to deiine the lattice parameter corresponding to each direction (ti Fig. 1 in the previous paper). : 5,872 K y= bo : 8,257 A X’CO : 5,107 x z-a0
The axes n&ion uo, b,,, co are taken from ASTM file 12-762. There is no way to choose the x, y, z axes notationsonasymmetrybasis;thezaxishasbeenchosen as the preserved axis between the principal axis system (oxyz) and the axis system of the crystal faces (ofyz’); the /I angle between those two axis system is 30”. The refraction indices have been measured for the Na-line: Nd,y : 1,875 N,,,/Jx : 1,870 N,J/z : 1,795 The optical sign is negative and the angle 2V is around 20”.
which has been
prepered below 1488°C [Z] is isotypic with uagonitc; the space group is D$ with 4 molecular formulas in the
unit cell. The BO:- groups are distributed between the planes formed by the Lanthanum atoms, in layers which are parallel to the (100) plane. The boron atoms lie on two axes parallel to the a axis; the BO:- are turned by lso” with respect to each other, from one layer to the other [3,43. _
3. Factor group analysis The results of the factor group analysis can be summarized as follows:
l-,,, l--
=4A,+2Bl,+4B,+2B,,+2A,+3B,,
=1A,+2B1,+
1Bz,+2B3,
3BS”
lB,, +2Bz.+
I&,
ri,umd =4A,+2B1,+4B2,+2B,+2A,+4B,.
2. Single crystal characteristics The crystal is a parallelipiped (1x2~3 mm). The orientation of the principal axes with respect to the
+ l&,+
+2A,,+
+=,,+4& The correlation table for the BO:-
Table 1. 635
ions $. given in
636
A. RULMONT et al.
Fig. 1. Raman spectrum of LaB03 in an orientation corresponding to the tensor element a,,.
Table 1. Correlation table BO:- ions f requcncy
iwlaccd
Site
Crystal
ion
rmgc -I (cm )
xz %h
939
(v,)
-
’ *
_
A’
-
D2h
L
::,
y
:;I
“i
606
C. SYMMETRY ASSIGNMENT OF THE RAMAN SPECl’RA
1. Direction of polarization of the incident and scattered light parallel to the crystal e&es The Raman spectra can be recorded for six different orientations corresponding to the tensor elements expressed in the axis system of the crystal faces: QxS, arSyS,aX.Y.,art,. The distribution of the non zero tensor elements as a function of the Raman active representations, have been already given [ 11.The interpretation of the Raman spectra in the various orientations must take into account the relative orientation of the two axis systems of the crystal. The spectrum corresponding to aZ,(=a,,ZS) contains uniquely vibrations of A,, representation. Unforttm-
ately, the intensity obtained in this orientation is very weak, as can be seen on Fig. 1. Figure 2 shows the spectra recorded for the four orientations aI,,,, aZSj, aXVYS, a,,,, when the crystal is illuminated along the z’ axis (which is also the principal axis z), and when the scattered light is observed along the f axis. These spectra are similar two by two. The exciting light (whose direction of polarization is no longer parallel to a principal axis), splits into two waves travelling along the z direction, with directions of polarization parallel to the principal axes (ox and oy); as the incident light along oz is perpendicular to a crystal face, there is no refraction effect. The observed spectra are thus related to couples of tensor elements; it is then possible to separate the peaks between two couples of representation: A,-BI, and BZB--Bss A complete assignment requires additional data.
637
Birefringence dfect in the Raman spectrum of a crystal-II
tions. According as either azY=O (I& modes) or a,=0 (Bs, modes), a maximum or a minimum is expected for an angle #,=fi=30”. This allows the symmetry assignment of the bands given in Table 2. At 943 cm-l, a weak peak could be assigned to a Bls vibration according to the correlation table; however some amount of leakage from the intense A, peak at this value must be taken into account too. (b) Vibrations belonging to the A,-BI1 representations (axy,a,, tensor elements). Typical intensity curves obtained for these representations are shown on Fig. 4. Some curves include well marked extrema whereas others are nearly independent of the angle 4. This can be explained from the equation giving the intensity for a A, modes as a function of the angle 4. Tbis equation (see equation (20) in El]), can be transformed so as to give the sum of a constant term and a function of 4. Ice Ei[a& cos2 /I ms2 (4-B)+
a& sin’ B sin’ (4 -8)] (1)
1
Fig. 2. Raman spectrum of LaBOs in several orientations. (Illumination axis parallel to 2).
= E$x& cos2 fi + Et[afi sin2 b -a& cos’ j] sin’ (4 - 8) (2) 1’
P’
2. Direction of polarization ojrhe scattered light parallel ~to the crystal edges: direction of polarization of the incident light rotated by an angle 4 around the z axis
(a)Vibmtions belonging to the B1,-Bs,
representa-
tion (q,, azYtensor elements.) Trpical curves represent-
ing the intensity variation as a function of the rotation angle of the direction of the incident light polarization, are shown on Fig. 3. As the peak are well separated, the intensities have been directly taken from the peakheights for these orientations. The accuracy was lower for the weak features at 280,315 and 327 cm-‘. Two types of behaviour have essentially been observed, in accordance with the theoretical predic-
i:..:....fi...*,...*
-90.
-SO*
0.
UT
SO’9
Fig. 4. Scattered intensity as a function of the angle (b for some A,-&, modes.
Fig. 3. Scattered intensity as a function of the angle 4 for some BZp-& modes.
A. RULMO~ et al.
638
Table 2. Raman frqucucies corresponding to aer and a,, Assignment
bahaviour
f rrquancy -I (cm )
at
b -
6
min.
113
B3g
IMX.
I 125 180
qir.
243
max.
280
mU.
315
max.
321
mu.
606
min.
632
ma?z.
943
min.
I255
min.
1371
mU.
B26 B3*
B2g % 62R B28 B3g B2rJ B3g B3R
J
B2g
(ii) 4 vibrations, maximum or minimum at d,=jl =3@ ; can appear in the spectrum corresponding to a,; the ratio p depends on the relative values of the tensor elements a, and aW An assignment of the A,-&, vibrations is then possible; it has been summarized in Table 3. (c) Special features. From our data, it appears that the peaks at 114 and 127.. cm-t show accidental degeneracy. The arguments in favour of the degenaaey are the following These two peaks appear in all the spectra with a rather high intensity, despite of the fact that the leakage is rather low (see e.g. the peaks at 244 and 180 em-’ in Fig. 2). In the curve giving the intensity versustheangleQforthepeakat 114em-‘,amaximum
If the “sin2” term is small in this expression (2), the intensity variation as a function of the angle b, can be very small. The type of extremum is not sntiieient to make a definite assigmnent of the A,-&, modes (see Table 3 in Cl]), a distinction between the two types of modes can be obtained as follows. (i) Bi, vibrations. Minimum at +=jl-3@; do not appear in the .spectrum corresponding to a,,; the intensity ratio between the a,.,. and a,,,, spectra can be calculated from expression (1):
Table 3. Raman frqueucies corresponding to a,, a YPas, aI) frequency -I (El )
a=*
spcctm
IO r,y,/Iox,x,
behaviour at
0 -
(8)
(b)
Asrivnt
6 Cc)
I13
IUI.
0.43
127
tin.
1,86
A g
144
I crt.
(O,RO)
BIR AR
182
I c*t.
(1.75)
A
192
mu. I
212
wt.
222
min.
243
I est.
306
min.
318
I E.f.
0.58
(1,751 I .oo (0.75) 1.82 1.06
593-596
R
*R BIR A 8 A BIR Btg
Cd)
AR + 91g Residue
(631) +
BU.
1.08
1235
min.
I.55
1255
I ut.
1.00
943
Cd)
A
R
BIR A 8
(a) (+): observed in the a,, orientation. (b) I cst means that the peak intensity remains nearly constant as a function of the I$angle. (c) Intensity ratio of the peak in the two orientations: pzY, and a,,,,. The values in brackets are not very accurate (weak intensity). (d) The assignment of the peaks at 144 and 318 cm-’ (lA,+l&,) is deduced from the fact,that there are only nine A, and six Et, modes.
Birefringence dfect in the Raman spectrum of a crystal-11 occurs for 4= 30” when the scattered light direction of polarization is parallel to the i axis. If the scattered light corresponding to the B3, orientation were a leakage of the A. vibration, there would be no reason for the observed minimum when the emitted light has in fact a maximum of intensity. The top of the peak at about 595 cm-i, is slightly displaced from the orientation corresponding to the tensor elements a,,., to the orientation corresponding to a,,,,,. This displacement is of the order of 3 cm-l, to be compared with a full width at half maximum of 15 cm-i. Two neighbouring peaks are probably superposed and their relative intensity is varying from one orientation to the other. CONCLUSION The mman spectra of the single crystal of LaBOs have been completely assigned in spite of the unfavour-
639
able orientation of the crystal faces. The experimental data confirm the theoretical calculations and emphasize the need to take into account the birefringence effect in anisotropic_ crystals. _ Acknowledgments-One of us (J. P. F.) gratefully acknowledges the award of an ‘aspirant’ fellowship from the Belgian F.N.R.S. We are indebted to M. Vermeire for his technical help in the determination of the crystal orientation and to Professor G. MICE for the use of his curve resolver. Dr A. FIUNSOL~T has measured the refractive indexes.
REFERENCES Specrrochim. Acta [II A. RaNom and J. P. Fm. 3SA, 629 (1979). J. Mat. Sci. La 8,lOSS (1973). VI B. WANKLYN, Handbuch der Anorganiscken Ckemie 28 B, 1310~‘s 386,&b Ed. (1961). 141G. K. abDULLAEv, G. G. DZNANN~OV and IL H. S. M-V, Aserb. Ajdzansky Kkim. Zh. 3, 117 (1976).
par
B. M. WANKLYN Clarendon Laboratory, Oxford Univnsity, Parks Road, Oxford OX1 3PU, U.K. Abstract-The
Raman spectra of LaBOa have been studied on a single crystal the faces of which are not parallel to the principal axes. A symmetry assignment is proposed from the 90” scattered intensity as a function of the polarization direction of the incident light. The experimental measurements contirm the theorefcal predictions derived in a previous paper.
INTRODUCIION In a preceding paper [l], we have theoretically derived
relations giving the scattered Raman intensity as a function of the direction of polarization of the incident light; an orthorhombic crystal has been considered, the principal axis system of which, (oxyz) is rotated by an angle /I, with respect to the system of axes perpendicular to the faces (cx’ff), along the common z’= 2 axis. We have checked these theoretical predictions on a LaBOS monocrystal, grown and cut at the C&e&m Laboratc#yipOxibrd~ bExPEwMENT
AL SETTING
The spectra have been recorded on a PHO CODERG double monochromator usinn a He-Ne laser Q- 632.8 mn ; W =50 mW). The crystal is &led on a go&meter head. The direction of polarization of the incident tight can be rotated by a half-wave plate, whose orientation is calibrated in degrees; the rotation of this plate can induce slight displacements of the focus inside the sample, which imply small corrections of the excentric lens before the entrance slit of the monochromator.
B CpYsfAL DSCRIPTION 1. General description
The structure of the LaBOs cry&
crystal faces, has been checked by X-ray diffraction on each face separately. The indices found for the three faces are 022,200, 031, which allow us to deiine the lattice parameter corresponding to each direction (ti Fig. 1 in the previous paper). : 5,872 K y= bo : 8,257 A X’CO : 5,107 x z-a0
The axes n&ion uo, b,,, co are taken from ASTM file 12-762. There is no way to choose the x, y, z axes notationsonasymmetrybasis;thezaxishasbeenchosen as the preserved axis between the principal axis system (oxyz) and the axis system of the crystal faces (ofyz’); the /I angle between those two axis system is 30”. The refraction indices have been measured for the Na-line: Nd,y : 1,875 N,,,/Jx : 1,870 N,J/z : 1,795 The optical sign is negative and the angle 2V is around 20”.
which has been
prepered below 1488°C [Z] is isotypic with uagonitc; the space group is D$ with 4 molecular formulas in the
unit cell. The BO:- groups are distributed between the planes formed by the Lanthanum atoms, in layers which are parallel to the (100) plane. The boron atoms lie on two axes parallel to the a axis; the BO:- are turned by lso” with respect to each other, from one layer to the other [3,43. _
3. Factor group analysis The results of the factor group analysis can be summarized as follows:
l-,,, l--
=4A,+2Bl,+4B,+2B,,+2A,+3B,,
=1A,+2B1,+
1Bz,+2B3,
3BS”
lB,, +2Bz.+
I&,
ri,umd =4A,+2B1,+4B2,+2B,+2A,+4B,.
2. Single crystal characteristics The crystal is a parallelipiped (1x2~3 mm). The orientation of the principal axes with respect to the
+ l&,+
+2A,,+
+=,,+4& The correlation table for the BO:-
Table 1. 635
ions $. given in
636
A. RULMONT et al.
Fig. 1. Raman spectrum of LaB03 in an orientation corresponding to the tensor element a,,.
Table 1. Correlation table BO:- ions f requcncy
iwlaccd
Site
Crystal
ion
rmgc -I (cm )
xz %h
939
(v,)
-
’ *
_
A’
-
D2h
L
::,
y
:;I
“i
606
C. SYMMETRY ASSIGNMENT OF THE RAMAN SPECl’RA
1. Direction of polarization of the incident and scattered light parallel to the crystal e&es The Raman spectra can be recorded for six different orientations corresponding to the tensor elements expressed in the axis system of the crystal faces: QxS, arSyS,aX.Y.,art,. The distribution of the non zero tensor elements as a function of the Raman active representations, have been already given [ 11.The interpretation of the Raman spectra in the various orientations must take into account the relative orientation of the two axis systems of the crystal. The spectrum corresponding to aZ,(=a,,ZS) contains uniquely vibrations of A,, representation. Unforttm-
ately, the intensity obtained in this orientation is very weak, as can be seen on Fig. 1. Figure 2 shows the spectra recorded for the four orientations aI,,,, aZSj, aXVYS, a,,,, when the crystal is illuminated along the z’ axis (which is also the principal axis z), and when the scattered light is observed along the f axis. These spectra are similar two by two. The exciting light (whose direction of polarization is no longer parallel to a principal axis), splits into two waves travelling along the z direction, with directions of polarization parallel to the principal axes (ox and oy); as the incident light along oz is perpendicular to a crystal face, there is no refraction effect. The observed spectra are thus related to couples of tensor elements; it is then possible to separate the peaks between two couples of representation: A,-BI, and BZB--Bss A complete assignment requires additional data.
637
Birefringence dfect in the Raman spectrum of a crystal-II
tions. According as either azY=O (I& modes) or a,=0 (Bs, modes), a maximum or a minimum is expected for an angle #,=fi=30”. This allows the symmetry assignment of the bands given in Table 2. At 943 cm-l, a weak peak could be assigned to a Bls vibration according to the correlation table; however some amount of leakage from the intense A, peak at this value must be taken into account too. (b) Vibrations belonging to the A,-BI1 representations (axy,a,, tensor elements). Typical intensity curves obtained for these representations are shown on Fig. 4. Some curves include well marked extrema whereas others are nearly independent of the angle 4. This can be explained from the equation giving the intensity for a A, modes as a function of the angle 4. Tbis equation (see equation (20) in El]), can be transformed so as to give the sum of a constant term and a function of 4. Ice Ei[a& cos2 /I ms2 (4-B)+
a& sin’ B sin’ (4 -8)] (1)
1
Fig. 2. Raman spectrum of LaBOs in several orientations. (Illumination axis parallel to 2).
= E$x& cos2 fi + Et[afi sin2 b -a& cos’ j] sin’ (4 - 8) (2) 1’
P’
2. Direction of polarization ojrhe scattered light parallel ~to the crystal edges: direction of polarization of the incident light rotated by an angle 4 around the z axis
(a)Vibmtions belonging to the B1,-Bs,
representa-
tion (q,, azYtensor elements.) Trpical curves represent-
ing the intensity variation as a function of the rotation angle of the direction of the incident light polarization, are shown on Fig. 3. As the peak are well separated, the intensities have been directly taken from the peakheights for these orientations. The accuracy was lower for the weak features at 280,315 and 327 cm-‘. Two types of behaviour have essentially been observed, in accordance with the theoretical predic-
i:..:....fi...*,...*
-90.
-SO*
0.
UT
SO’9
Fig. 4. Scattered intensity as a function of the angle (b for some A,-&, modes.
Fig. 3. Scattered intensity as a function of the angle 4 for some BZp-& modes.
A. RULMO~ et al.
638
Table 2. Raman frqucucies corresponding to aer and a,, Assignment
bahaviour
f rrquancy -I (cm )
at
b -
6
min.
113
B3g
IMX.
I 125 180
qir.
243
max.
280
mU.
315
max.
321
mu.
606
min.
632
ma?z.
943
min.
I255
min.
1371
mU.
B26 B3*
B2g % 62R B28 B3g B2rJ B3g B3R
J
B2g
(ii) 4 vibrations, maximum or minimum at d,=jl =3@ ; can appear in the spectrum corresponding to a,; the ratio p depends on the relative values of the tensor elements a, and aW An assignment of the A,-&, vibrations is then possible; it has been summarized in Table 3. (c) Special features. From our data, it appears that the peaks at 114 and 127.. cm-t show accidental degeneracy. The arguments in favour of the degenaaey are the following These two peaks appear in all the spectra with a rather high intensity, despite of the fact that the leakage is rather low (see e.g. the peaks at 244 and 180 em-’ in Fig. 2). In the curve giving the intensity versustheangleQforthepeakat 114em-‘,amaximum
If the “sin2” term is small in this expression (2), the intensity variation as a function of the angle b, can be very small. The type of extremum is not sntiieient to make a definite assigmnent of the A,-&, modes (see Table 3 in Cl]), a distinction between the two types of modes can be obtained as follows. (i) Bi, vibrations. Minimum at +=jl-3@; do not appear in the .spectrum corresponding to a,,; the intensity ratio between the a,.,. and a,,,, spectra can be calculated from expression (1):
Table 3. Raman frqueucies corresponding to a,, a YPas, aI) frequency -I (El )
a=*
spcctm
IO r,y,/Iox,x,
behaviour at
0 -
(8)
(b)
Asrivnt
6 Cc)
I13
IUI.
0.43
127
tin.
1,86
A g
144
I crt.
(O,RO)
BIR AR
182
I c*t.
(1.75)
A
192
mu. I
212
wt.
222
min.
243
I est.
306
min.
318
I E.f.
0.58
(1,751 I .oo (0.75) 1.82 1.06
593-596
R
*R BIR A 8 A BIR Btg
Cd)
AR + 91g Residue
(631) +
BU.
1.08
1235
min.
I.55
1255
I ut.
1.00
943
Cd)
A
R
BIR A 8
(a) (+): observed in the a,, orientation. (b) I cst means that the peak intensity remains nearly constant as a function of the I$angle. (c) Intensity ratio of the peak in the two orientations: pzY, and a,,,,. The values in brackets are not very accurate (weak intensity). (d) The assignment of the peaks at 144 and 318 cm-’ (lA,+l&,) is deduced from the fact,that there are only nine A, and six Et, modes.
Birefringence dfect in the Raman spectrum of a crystal-11 occurs for 4= 30” when the scattered light direction of polarization is parallel to the i axis. If the scattered light corresponding to the B3, orientation were a leakage of the A. vibration, there would be no reason for the observed minimum when the emitted light has in fact a maximum of intensity. The top of the peak at about 595 cm-i, is slightly displaced from the orientation corresponding to the tensor elements a,,., to the orientation corresponding to a,,,,,. This displacement is of the order of 3 cm-l, to be compared with a full width at half maximum of 15 cm-i. Two neighbouring peaks are probably superposed and their relative intensity is varying from one orientation to the other. CONCLUSION The mman spectra of the single crystal of LaBOs have been completely assigned in spite of the unfavour-
639
able orientation of the crystal faces. The experimental data confirm the theoretical calculations and emphasize the need to take into account the birefringence effect in anisotropic_ crystals. _ Acknowledgments-One of us (J. P. F.) gratefully acknowledges the award of an ‘aspirant’ fellowship from the Belgian F.N.R.S. We are indebted to M. Vermeire for his technical help in the determination of the crystal orientation and to Professor G. MICE for the use of his curve resolver. Dr A. FIUNSOL~T has measured the refractive indexes.
REFERENCES Specrrochim. Acta [II A. RaNom and J. P. Fm. 3SA, 629 (1979). J. Mat. Sci. La 8,lOSS (1973). VI B. WANKLYN, Handbuch der Anorganiscken Ckemie 28 B, 1310~‘s 386,&b Ed. (1961). 141G. K. abDULLAEv, G. G. DZNANN~OV and IL H. S. M-V, Aserb. Ajdzansky Kkim. Zh. 3, 117 (1976).