- Email: [email protected]
Birefringence effect in the Raman spectrum of a crystal which is not cut parallel to the principal axes—I.
~oebmi~AActaVol.35k~62qlo633 Pqamon Press Ltd.. 1979. Pnnted m Great Britain
Birefringence effect in.the Raman spectrum of a crystal which is not cut parallel to the principal axes--I. A. RULMONTand J. P. FUMME Dtpartement de Chimie Gtnbale et de Chimie Pbysique, lnstitut de Chimie, Sat? Tilman par B-4000 Liege I, Belgium (Receiwd 29 March 1978) Ahstmct-The Raman intensity scattered by a single crystal is theoretically studied as a function of the polarization direction of the incident light. For some orientations of the crystal the effect of biiefringence changes the shape of the intensity curves; these cannot be deduced any more by the formulas generally accepted for the calculation of the scattered intensity. INIItODUCI-ION of the peaks &the Raman spectra of a crystal is to study the sample in an experimental system of axes which cointides with the principal axes of the crystal. In a 90” scattering experiment, the experimental system of axes is formed by the direction of the incident and scattered (ki, k,) wave vectors and by the perpendicular to these vectors. The non zero elements of the Raman tensor corresponding to a given representation are directly obtained from character tables. The assignment of the peaks can then be checked by two methods (i) The Raman spectrum can be studied as a function of the-rotation angle of the crystal along a suitable chosen axis. Ideally, the sample shape has to be cylindrical [ 11.We have previously reported an analysis of the different parameters in the interpretation of the intensity variations when the crystal shape is prismatic
The simplest way to make a symmetry assignment
unit cell point group is Dlk The crystal is supposed to be a parailelipiped. Three different reference systems have to be considered. The system of axes perpendicular to the crystal faces: Ox’y’z’;The priocipai system of axes: Oxyz; The experi~rnental systemofaxesOxY2. In order to avoid light refraction at the crystal faces, the experimental OXY2 system will always be parallel to the Oiji system: the incident and scattered wave vectors k, and k, will be perpendicular to the crystal faces. Only two of the faces of the crystal are supposed to be perpendicular to a principal axis which will be denoted by z. The orientation of the different axis systems is represented on Fig 1. The principal axis z is parallel to a Cs and a cristallographic axis; it is the rotation axis to convert the experimental axis system into the principal axis system. The angle between the x and i axes or the y and f axes is denoted B. The experimental illumination direction is chosen parallel to the z = z’ axis. The observation axis may be x’ or f and is therefore not parallel to a principal axis (x or y). The direction of polarization of the incident light makes a variable angle 4 with an arbitrady chosen reference axis in the experimental setting (X). The scattered light is always
PI. (ii) A convenient way to avoid the troubles due to light refraction at the crystal faces is to keep the faces always perpendicular to the k, and ki wave vectors and to rotate continuously the direction of polarization of the incident light, (which is the direction of the electric field vector associated with the electromagnetic radiation). In the latter method, the developments are diRerent according as the crystal faces are perpendicular to the principal axes or not. We intend to survey the general case of an &ho-rhombic crystal for which the faces axis system is rotated by an angle j? with respect to the principal axes system. It will be shown that the formulas usually accepted for the calculation of the Raman scattered light intensity have to be corrected. The experimental results obtained on a LaBOs crystal will confirm these theoretical predictions [4]. EXPERIMENTAL
CONDITIONS
As the theory developed hereafter will be applied to a LaBO, crystal of Aragonite structure, we consider that the S.AlA) 35/6-I
Fig. 1. Relative orientation of the different axes in the experimental setting 629
630
A. Rwt+rr
and J. P. FUME
analysed for a direction of polarization parallel or pqendicular to the reference axis. CALCTJLATION OF THE SCATI’ERED INTENSITIES
The intensity of the Raman scattered light in a given polarization direction is proportional to the square of the amplitude of the oscillating induced dipole moment component parallel to this direction. The crystal is studied in the experimental axis system, equivalent to the crystal faces axis system, but the scattering tensors are OX&Jknown from group theory in the principal axis system. As can be seen on Fig. 1, the latter system is obtained from the further by a rotation /I around the z axis; the tensor elements in the experimerital system can therefore be calculated by the formulas of Tables 1 and 2.
: a
xx’
azx
B3g:
Ozy
1. Bz# and BS, uibratioml modes The peaks corresponding to B2, and Boo modes can be observed when the P,, component of the scattered light is analysed (i.e. direction of polarization of the scattered light parallel to the Oz’ axis). By combining the equations (1) and (2) with the formulas of Table 2, the expression of P,, can be readily calculated as a function of the tensor elements ali, the angle j and the experimental variable angle 4:
sin C#~-811
(4)
The #-dependence of the R intensity is then given by: J,M)a (P,,J2 = Ef[aL cos2 (4 - j)+a& sin2 (4 -/I) +a&,, sin W-811
ayy’ OZL
According to this expression@), expected at an angle 4. given by :
Elg : Oxy 82g :
It can be seen from Tables 1 and 2, that equation (2) gives non-m P, components for the A, and B1, modes, and that equation (3) gives non-zero P, components for the B2, B3, modes. This feature must be taken into accuunt to make a symmetry assignment of the peaks appearing in the Raman spectra for various orientations of the sample [4].
P,~=~dcr,, cm M+B)+a,
Table 1. Tensor elements corrispending to each R active representation for an or&-rhombic crystal
A g
(3)
an extremum
(5) is
-k+L For the B2, modes (a., 10; see Table l), as well as for the B,, modes, the extremum will occur at q5,=/?. The second derivative of expression (5) gives at b=fl:
Table 2. Relations between the tensor elements in the crystal faces system and the tensor elements for the principle axis system We obtain the following results: for the B2# modes, IAb) will be maximum at 4=/?; for the B3, modes, ld4) will be minimum at 4 = fl. The periodicity of the 1X4) functions is 180”. 2. A, and B,, vibrational modes
With the crystal setting of Fig. 1, the amplitude of the incident electric field vector has the following components in the OYy’Y system : E,,=EO cos 4 E,.=E,sinq5
(1)
E,=O
a,, + ay, -a, -ayy cos 2/?-a,, sin 28
axfY.=-
2 + 2 The P,, expression (2) becomes: p,,=E,,
where Eo=lEil and -9@<4,(!9@‘. The components of the induced dipole moment are : P,. = a,.,.E,. + aijEy.
The peaks corresponding to the A, and B1, modes can be observed ifthe P,. component of the scattered light IS analysed. Classical calculations can be made in the same manner as above; some trigonometric transformations are invoked. The two first equations ofTable 2 are written as: a,--QY, aXSg. --sin 2p+a,, cos 28 (6) 2
(2)
~,,,
$,+CL,, 2
-axy
(7)
cos (28 -4)
sin(V-4)
1
(8)
Birefringence effect in the Raman spectrum of a crystal--I Using:
631
These classical conclusions are summarixed in the 6rst part of Table 3 (An = 0). THE BIBBFBINGENCE
The 4dependence
of the Raman intensity becomes:
rd~)a,P;5.=E~{[(a,cosB-a,sin8)2cos2(B-_)] + [(a, sin B-a,,, cos B)’ sin2 @ -4)]
+ [(axx cos fl -a_
sin /Ma,, sin fl-crxY cos B sin (V-d)])
(9)
An extremum is expected at an angle 4, given by: &J2U3 -4.) 2(a, cos fl -a, = (a, cos B -a,
sin /IMa, sin fi - azYcos /S)
sin j?)2-(a,,
sin fi -aw cos fi)2 (10)
for the Br, modes, a,=a,==O. An extremum is expected at i&-p; ii will be a minimum according to the second derivative of (9A which is: d2(Px,)2 = ZE$af, cm 2(28 - 43 d4’
(11)
for the A, modes (a-=0), the position and the type of the extremum depend on the relative value of a, and
EFFECT
The preceding conventional calculations are in fact, based on two hypothesis which need to be mentioned. (i) The two rays of light associated with the electric vector components E, and E, are travelling along the same path in the crystal. (ii) There is no phase shii between the two electric fields: they both travel at the same speed. For a biaxial crystal, the first condition is only verifkl if the incident light is propagating at normal incidence along a principal axis This is the reason why the z axis has been chosen hereabove as the illumination axis. Different cases of normal incidence on a biaxial crystal have been reviewed by BLQS [6]. The second condition is only fulfilled if the refractive indices of the two waves are equal. Whenever the light is not propagating -along an optical axis, these conditions will not be fulfilled in anisotropic crystals. In this section, we shall in fact consider the time variation of the propagation waves and introduce the phase shift between the two exciting electric gelds in the calculation of the scattered R intensity
1. Equations for the two propagating electric filds When the incident light is propagating along a axis z, the two waves in the crystal have the
principal
aYP
Table 3. Summary of the properdee of the curves I,(d) A.
Scattered
A*
clar8ical
light
lalyaed
parallel
I
w.des
: An -
to ox’.
0
An
c
Bigm0de8 clarmical
0
: An -
I +c
llmyr
-$
An # 0
I
‘25
2
0
min.
B.
Scattered
light
> t:
lnalymd
5
: min.
parallel
to 0%‘.
-
alvaya
I z
0,
5
I min.
I
A. RU~MOWand J. P. FLAMME
632
following properties: The normal to the plane of the two waves obeys the Snell’s law and remains thus parallel to z. The two rays of light (direction of propagation of the energy) coincide with the wave normal. The electric field vector E remains then parallel to the electric displacement vector D. The direction of D associated with the two waves is given by the direction of the two axes of the ellipse obtained by cutting the optical indicatrix by a plane perpendicular to z. As z is a principal axis, the ellipse axes must also be principal axes. The two electric field vectors are therefore respectively parallel to one of the principal axes x or y, and can be expressed by the equations:
I,~A~=~~a~,~os~(~-~)~~~~~+af~sin~
(c$-/3)sinjI
-aXpYY cos (f#~ -/I) sin ($ -fl) sin 2/I cos 61 (18) This formula would give the result obtained by the conventional method [3] if8=0. Themain result of the introduction of the birefringence is that the scattered intensity depends now on the point considered inside the crystal through the parameter 1. The equations (18) canbewrittenas: A’(l)=&-C
cos 2~; &I
(19)
(The angle 4 is kept constant). A’( 1) is a periodical function of 1; the space period
1, is: 1
where n. and nr are the refraction indices, c is the light velocity and 1 is the length of the path of the light inside the crystal. w,, is the laser frequency, while oR will denote hereafter the scattered light frequency at which oscillates the induced dipole moment As 4 is the angle between the direction of polarisation of the incident light and the reference X=x’ axis : (13)
Eo,=.cJco8(r#l-fi) g=J?
For a He-Ne laser, I = 632.8 nm and 1p can easily be less than the thickness of the.cryst&along the illumination axis (e.g. a few mm) even if An is not very large. Experimentally the observed light is scattered by the whole crystal and the total intensity can be obtained by integrating (19) over the length e of the crystal:
c c
ftsxB-C
Jo
cos 2n +
2. A, vibrutional modes P,=a.J?cos(4-/?)sinwR P,=a$O
sin (4-p)
(
C
>
(14)
1%
sin oR t- (
Only the P,. component frequency oR is analysed
t-5
”
2nAn 1, I 0
cos 6 dB=B
(21)
The total scattered intensity reduces then to the B term : I,,= Ei[a& COS’(4-B)
cos2 @+a;,, sin2 (q5-p) sin2.j]
(22) The hnal effect of the birefringence is therefore to cancel the “sin 2(8-45)” - term in equation (9) giving the intensity variation for A, vibrational modes as a function of the angle 4. The shape of the intensity curve is drastically changed and the position of the extremum is now predicted always at an angle 4, = jI instead of being dependent of the values of a, and aY,.. 3. B,, vibrational modes
C >
of the scattered light at a
P,.=P,cos/3-P,sinfi
e
L =B-C-.-
siil @#J-/q
It is important to note that the two electric fields E, and Ey can induce only one unique dipole moment at each place in the crystal. This resultant moment is the vectorial sum of the moments induo%i by each of the two electric fields.
1 dl
A
(15)
The same treatment modes, giving
can be applied to the BI,
ltscc Eia&[sin2 (4 -/.I) cos2 /I+ cos’ (9 -8) sin’ /I] (23) An extremum is again predicted at an angle $,= /l instead of 28.
-&[a,
sin
(4
-8)
sin
fi]
sin
wR
(16)
This expression (16) can be written P,. = A sin (cuRt-6),
(17)
where 8=(2xvl/c)An=(2xl/~)An, An being the difference between the refractive indices. The intensity of the emitted light is proportional to the squared amplitude AZ [7].
4. Btp and BOgvibrational modes No birefringence. effect occurs for the Bzg and Bx, modes, taking into account the supposed crystal orientation. If the direction of polarization of the analysed light is parallel to the oz axis, the induced dipole moment is: P,, = P, = a&,
But for Bag vibrational
+ a&
(24)
modes, only the a:, tensor
Birefringence effect in the Raman spectrum of a crystal-I element is non zero, and for BSI modes, only the a,). tensor element is non zero. (E, is always zero as the
incident light is travelling along the z axis). Therefore only one ofthe two permitted waves creates in the crystal the oscillating induced dipole moment and no phase shift could occur: the results obtained by the usual method are still valid. Table 3 gives thus the expected variations of the band intensities for the different representations, taking into account this birefringenee effect. As will be shown in the following paper, the intensity variations in the Raman spectra of a LaBOJ crystal are in agreement
633
ponents. When it does not, the usual method is valid. In the classical formulas [3] used to calculate the R intensity, it is implicitely considered that the two exciting fields are in phase; this can lead to erroneous predictions, when the crystal faces are not all perpendicular to the principal axes. Acknowledgcmeuts-One of the authors (J. P. F.) gratefully acknowledges the award of a aspirant fellowship from the FNRS.
with these predictions. REFERENCES
.
CONCLUSlON
We have studied a particular type of crystal with an unusual orientation of the principal axis system. It has been shown that the correct Raman intensity variation for an anisotropic crystal includes a birefringence effect, if several conditions are fulfilled: the path of the two partial rays of light in the crystal must be the same; the refractive indices corresponding to each ray must be different (An#O); the oscillating dipole moment must depend on the value of each electric field com-
[ll P. DAWSON,M. M. IUttcntn~vu and G. R. WILKINKIN,
J. PhYS.a.240 (1971). 12) W. R~LMx&, Sjectrdchim. Acru 35A, 625 (1979). # R. Louno~. A& Phvs. 13.423 (1964). A. R-h, 1. P. Fti, M.-J. Porrma, B. WANKLYN,
Spectrochh.
Acta JSA, 635 (1979). T. bNKATAMwDU, =OrJ’ Of Groups and its Application to Physicnl Problem. Academic Press (1969). VI F. D. BLOSS,An Introduction to the Methods of Optical Crystallography, p. 163. Holt, Rinehart, Winston (1961). 1~ J. A. Ko~osmt~, Intro&tioa to the Theory of the Ramm Eflect. Reidel, Dordreeht, Holland (1972).
1~) S. B~UQAV~,
Birefringence effect in.the Raman spectrum of a crystal which is not cut parallel to the principal axes--I. A. RULMONTand J. P. FUMME Dtpartement de Chimie Gtnbale et de Chimie Pbysique, lnstitut de Chimie, Sat? Tilman par B-4000 Liege I, Belgium (Receiwd 29 March 1978) Ahstmct-The Raman intensity scattered by a single crystal is theoretically studied as a function of the polarization direction of the incident light. For some orientations of the crystal the effect of biiefringence changes the shape of the intensity curves; these cannot be deduced any more by the formulas generally accepted for the calculation of the scattered intensity. INIItODUCI-ION of the peaks &the Raman spectra of a crystal is to study the sample in an experimental system of axes which cointides with the principal axes of the crystal. In a 90” scattering experiment, the experimental system of axes is formed by the direction of the incident and scattered (ki, k,) wave vectors and by the perpendicular to these vectors. The non zero elements of the Raman tensor corresponding to a given representation are directly obtained from character tables. The assignment of the peaks can then be checked by two methods (i) The Raman spectrum can be studied as a function of the-rotation angle of the crystal along a suitable chosen axis. Ideally, the sample shape has to be cylindrical [ 11.We have previously reported an analysis of the different parameters in the interpretation of the intensity variations when the crystal shape is prismatic
The simplest way to make a symmetry assignment
unit cell point group is Dlk The crystal is supposed to be a parailelipiped. Three different reference systems have to be considered. The system of axes perpendicular to the crystal faces: Ox’y’z’;The priocipai system of axes: Oxyz; The experi~rnental systemofaxesOxY2. In order to avoid light refraction at the crystal faces, the experimental OXY2 system will always be parallel to the Oiji system: the incident and scattered wave vectors k, and k, will be perpendicular to the crystal faces. Only two of the faces of the crystal are supposed to be perpendicular to a principal axis which will be denoted by z. The orientation of the different axis systems is represented on Fig 1. The principal axis z is parallel to a Cs and a cristallographic axis; it is the rotation axis to convert the experimental axis system into the principal axis system. The angle between the x and i axes or the y and f axes is denoted B. The experimental illumination direction is chosen parallel to the z = z’ axis. The observation axis may be x’ or f and is therefore not parallel to a principal axis (x or y). The direction of polarization of the incident light makes a variable angle 4 with an arbitrady chosen reference axis in the experimental setting (X). The scattered light is always
PI. (ii) A convenient way to avoid the troubles due to light refraction at the crystal faces is to keep the faces always perpendicular to the k, and ki wave vectors and to rotate continuously the direction of polarization of the incident light, (which is the direction of the electric field vector associated with the electromagnetic radiation). In the latter method, the developments are diRerent according as the crystal faces are perpendicular to the principal axes or not. We intend to survey the general case of an &ho-rhombic crystal for which the faces axis system is rotated by an angle j? with respect to the principal axes system. It will be shown that the formulas usually accepted for the calculation of the Raman scattered light intensity have to be corrected. The experimental results obtained on a LaBOs crystal will confirm these theoretical predictions [4]. EXPERIMENTAL
CONDITIONS
As the theory developed hereafter will be applied to a LaBO, crystal of Aragonite structure, we consider that the S.AlA) 35/6-I
Fig. 1. Relative orientation of the different axes in the experimental setting 629
630
A. Rwt+rr
and J. P. FUME
analysed for a direction of polarization parallel or pqendicular to the reference axis. CALCTJLATION OF THE SCATI’ERED INTENSITIES
The intensity of the Raman scattered light in a given polarization direction is proportional to the square of the amplitude of the oscillating induced dipole moment component parallel to this direction. The crystal is studied in the experimental axis system, equivalent to the crystal faces axis system, but the scattering tensors are OX&Jknown from group theory in the principal axis system. As can be seen on Fig. 1, the latter system is obtained from the further by a rotation /I around the z axis; the tensor elements in the experimerital system can therefore be calculated by the formulas of Tables 1 and 2.
: a
xx’
azx
B3g:
Ozy
1. Bz# and BS, uibratioml modes The peaks corresponding to B2, and Boo modes can be observed when the P,, component of the scattered light is analysed (i.e. direction of polarization of the scattered light parallel to the Oz’ axis). By combining the equations (1) and (2) with the formulas of Table 2, the expression of P,, can be readily calculated as a function of the tensor elements ali, the angle j and the experimental variable angle 4:
sin C#~-811
(4)
The #-dependence of the R intensity is then given by: J,M)a (P,,J2 = Ef[aL cos2 (4 - j)+a& sin2 (4 -/I) +a&,, sin W-811
ayy’ OZL
According to this expression@), expected at an angle 4. given by :
Elg : Oxy 82g :
It can be seen from Tables 1 and 2, that equation (2) gives non-m P, components for the A, and B1, modes, and that equation (3) gives non-zero P, components for the B2, B3, modes. This feature must be taken into accuunt to make a symmetry assignment of the peaks appearing in the Raman spectra for various orientations of the sample [4].
P,~=~dcr,, cm M+B)+a,
Table 1. Tensor elements corrispending to each R active representation for an or&-rhombic crystal
A g
(3)
an extremum
(5) is
-k+L For the B2, modes (a., 10; see Table l), as well as for the B,, modes, the extremum will occur at q5,=/?. The second derivative of expression (5) gives at b=fl:
Table 2. Relations between the tensor elements in the crystal faces system and the tensor elements for the principle axis system We obtain the following results: for the B2# modes, IAb) will be maximum at 4=/?; for the B3, modes, ld4) will be minimum at 4 = fl. The periodicity of the 1X4) functions is 180”. 2. A, and B,, vibrational modes
With the crystal setting of Fig. 1, the amplitude of the incident electric field vector has the following components in the OYy’Y system : E,,=EO cos 4 E,.=E,sinq5
(1)
E,=O
a,, + ay, -a, -ayy cos 2/?-a,, sin 28
axfY.=-
2 + 2 The P,, expression (2) becomes: p,,=E,,
where Eo=lEil and -9@<4,(!9@‘. The components of the induced dipole moment are : P,. = a,.,.E,. + aijEy.
The peaks corresponding to the A, and B1, modes can be observed ifthe P,. component of the scattered light IS analysed. Classical calculations can be made in the same manner as above; some trigonometric transformations are invoked. The two first equations ofTable 2 are written as: a,--QY, aXSg. --sin 2p+a,, cos 28 (6) 2
(2)
~,,,
$,+CL,, 2
-axy
(7)
cos (28 -4)
sin(V-4)
1
(8)
Birefringence effect in the Raman spectrum of a crystal--I Using:
631
These classical conclusions are summarixed in the 6rst part of Table 3 (An = 0). THE BIBBFBINGENCE
The 4dependence
of the Raman intensity becomes:
rd~)a,P;5.=E~{[(a,cosB-a,sin8)2cos2(B-_)] + [(a, sin B-a,,, cos B)’ sin2 @ -4)]
+ [(axx cos fl -a_
sin /Ma,, sin fl-crxY cos B sin (V-d)])
(9)
An extremum is expected at an angle 4, given by: &J2U3 -4.) 2(a, cos fl -a, = (a, cos B -a,
sin /IMa, sin fi - azYcos /S)
sin j?)2-(a,,
sin fi -aw cos fi)2 (10)
for the Br, modes, a,=a,==O. An extremum is expected at i&-p; ii will be a minimum according to the second derivative of (9A which is: d2(Px,)2 = ZE$af, cm 2(28 - 43 d4’
(11)
for the A, modes (a-=0), the position and the type of the extremum depend on the relative value of a, and
EFFECT
The preceding conventional calculations are in fact, based on two hypothesis which need to be mentioned. (i) The two rays of light associated with the electric vector components E, and E, are travelling along the same path in the crystal. (ii) There is no phase shii between the two electric fields: they both travel at the same speed. For a biaxial crystal, the first condition is only verifkl if the incident light is propagating at normal incidence along a principal axis This is the reason why the z axis has been chosen hereabove as the illumination axis. Different cases of normal incidence on a biaxial crystal have been reviewed by BLQS [6]. The second condition is only fulfilled if the refractive indices of the two waves are equal. Whenever the light is not propagating -along an optical axis, these conditions will not be fulfilled in anisotropic crystals. In this section, we shall in fact consider the time variation of the propagation waves and introduce the phase shift between the two exciting electric gelds in the calculation of the scattered R intensity
1. Equations for the two propagating electric filds When the incident light is propagating along a axis z, the two waves in the crystal have the
principal
aYP
Table 3. Summary of the properdee of the curves I,(d) A.
Scattered
A*
clar8ical
light
lalyaed
parallel
I
w.des
: An -
to ox’.
0
An
c
Bigm0de8 clarmical
0
: An -
I +c
llmyr
-$
An # 0
I
‘25
2
0
min.
B.
Scattered
light
> t:
lnalymd
5
: min.
parallel
to 0%‘.
-
alvaya
I z
0,
5
I min.
I
A. RU~MOWand J. P. FLAMME
632
following properties: The normal to the plane of the two waves obeys the Snell’s law and remains thus parallel to z. The two rays of light (direction of propagation of the energy) coincide with the wave normal. The electric field vector E remains then parallel to the electric displacement vector D. The direction of D associated with the two waves is given by the direction of the two axes of the ellipse obtained by cutting the optical indicatrix by a plane perpendicular to z. As z is a principal axis, the ellipse axes must also be principal axes. The two electric field vectors are therefore respectively parallel to one of the principal axes x or y, and can be expressed by the equations:
I,~A~=~~a~,~os~(~-~)~~~~~+af~sin~
(c$-/3)sinjI
-aXpYY cos (f#~ -/I) sin ($ -fl) sin 2/I cos 61 (18) This formula would give the result obtained by the conventional method [3] if8=0. Themain result of the introduction of the birefringence is that the scattered intensity depends now on the point considered inside the crystal through the parameter 1. The equations (18) canbewrittenas: A’(l)=&-C
cos 2~; &I
(19)
(The angle 4 is kept constant). A’( 1) is a periodical function of 1; the space period
1, is: 1
where n. and nr are the refraction indices, c is the light velocity and 1 is the length of the path of the light inside the crystal. w,, is the laser frequency, while oR will denote hereafter the scattered light frequency at which oscillates the induced dipole moment As 4 is the angle between the direction of polarisation of the incident light and the reference X=x’ axis : (13)
Eo,=.cJco8(r#l-fi) g=J?
For a He-Ne laser, I = 632.8 nm and 1p can easily be less than the thickness of the.cryst&along the illumination axis (e.g. a few mm) even if An is not very large. Experimentally the observed light is scattered by the whole crystal and the total intensity can be obtained by integrating (19) over the length e of the crystal:
c c
ftsxB-C
Jo
cos 2n +
2. A, vibrutional modes P,=a.J?cos(4-/?)sinwR P,=a$O
sin (4-p)
(
C
>
(14)
1%
sin oR t- (
Only the P,. component frequency oR is analysed
t-5
”
2nAn 1, I 0
cos 6 dB=B
(21)
The total scattered intensity reduces then to the B term : I,,= Ei[a& COS’(4-B)
cos2 @+a;,, sin2 (q5-p) sin2.j]
(22) The hnal effect of the birefringence is therefore to cancel the “sin 2(8-45)” - term in equation (9) giving the intensity variation for A, vibrational modes as a function of the angle 4. The shape of the intensity curve is drastically changed and the position of the extremum is now predicted always at an angle 4, = jI instead of being dependent of the values of a, and aY,.. 3. B,, vibrational modes
C >
of the scattered light at a
P,.=P,cos/3-P,sinfi
e
L =B-C-.-
siil @#J-/q
It is important to note that the two electric fields E, and Ey can induce only one unique dipole moment at each place in the crystal. This resultant moment is the vectorial sum of the moments induo%i by each of the two electric fields.
1 dl
A
(15)
The same treatment modes, giving
can be applied to the BI,
ltscc Eia&[sin2 (4 -/.I) cos2 /I+ cos’ (9 -8) sin’ /I] (23) An extremum is again predicted at an angle $,= /l instead of 28.
-&[a,
sin
(4
-8)
sin
fi]
sin
wR
(16)
This expression (16) can be written P,. = A sin (cuRt-6),
(17)
where 8=(2xvl/c)An=(2xl/~)An, An being the difference between the refractive indices. The intensity of the emitted light is proportional to the squared amplitude AZ [7].
4. Btp and BOgvibrational modes No birefringence. effect occurs for the Bzg and Bx, modes, taking into account the supposed crystal orientation. If the direction of polarization of the analysed light is parallel to the oz axis, the induced dipole moment is: P,, = P, = a&,
But for Bag vibrational
+ a&
(24)
modes, only the a:, tensor
Birefringence effect in the Raman spectrum of a crystal-I element is non zero, and for BSI modes, only the a,). tensor element is non zero. (E, is always zero as the
incident light is travelling along the z axis). Therefore only one ofthe two permitted waves creates in the crystal the oscillating induced dipole moment and no phase shift could occur: the results obtained by the usual method are still valid. Table 3 gives thus the expected variations of the band intensities for the different representations, taking into account this birefringenee effect. As will be shown in the following paper, the intensity variations in the Raman spectra of a LaBOJ crystal are in agreement
633
ponents. When it does not, the usual method is valid. In the classical formulas [3] used to calculate the R intensity, it is implicitely considered that the two exciting fields are in phase; this can lead to erroneous predictions, when the crystal faces are not all perpendicular to the principal axes. Acknowledgcmeuts-One of the authors (J. P. F.) gratefully acknowledges the award of a aspirant fellowship from the FNRS.
with these predictions. REFERENCES
.
CONCLUSlON
We have studied a particular type of crystal with an unusual orientation of the principal axis system. It has been shown that the correct Raman intensity variation for an anisotropic crystal includes a birefringence effect, if several conditions are fulfilled: the path of the two partial rays of light in the crystal must be the same; the refractive indices corresponding to each ray must be different (An#O); the oscillating dipole moment must depend on the value of each electric field com-
[ll P. DAWSON,M. M. IUttcntn~vu and G. R. WILKINKIN,
J. PhYS.a.240 (1971). 12) W. R~LMx&, Sjectrdchim. Acru 35A, 625 (1979). # R. Louno~. A& Phvs. 13.423 (1964). A. R-h, 1. P. Fti, M.-J. Porrma, B. WANKLYN,
Spectrochh.
Acta JSA, 635 (1979). T. bNKATAMwDU, =OrJ’ Of Groups and its Application to Physicnl Problem. Academic Press (1969). VI F. D. BLOSS,An Introduction to the Methods of Optical Crystallography, p. 163. Holt, Rinehart, Winston (1961). 1~ J. A. Ko~osmt~, Intro&tioa to the Theory of the Ramm Eflect. Reidel, Dordreeht, Holland (1972).
1~) S. B~UQAV~,