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European Journal of Applied Sciences – Vol. 12, No. 1
Publication Date: February 25, 2024
DOI:10.14738/aivp.121.16257
Karppinen, M. (2024). Origin of Optical Activity in Cubic NaClO3 and NaBrO3 Crystals at 296K. European Journal of Applied
Sciences, Vol - 12(1). 174-184.
Services for Science and Education – United Kingdom
Origin of Optical Activity in Cubic NaClO3 and NaBrO3 Crystals at
296K
M. Karppinen
Institute of Chemistry, University of Uppsala
Box 531, Uppsala S-75121 Sweden
ABSTRACT
The optical activity is theoretically determined both in the dextro- and levorotatory
crystals of NaClO3 and NaBrO3. Both compounds have identical structures and they
crystallize in the space group P213 as ionic solids from the water solution at room
temperature. The net charges of Cl, Br and O atoms in the XO3
‒ ions are variables,
when the ratios of the electric moments of second rank in the two principal axis
directions and the ratios of the defined isomorphic and measured refractive indices
of the two crystals are iterated to a topological equivalence. The difference of the
net charges of Cl- and Br-atoms brings the opposite sense of optical rotation in these
compounds. The rotation is computed from the principal axis components of the
second electric moments and axial vectors derived from the point charge model.
The unit cells have four XO3
‒ ions of three-fold symmetries lying on the four
diagonals. The components of the dominant axial vectors of two ions, standing in
the right- and left-handed symmetries, are pointing towards each other in the
principal axis directions and contribute to the optical rotation. The left-handed
component against the propagation direction of the plane-polarized light changes
its handedness and rotation character making the compounds optically active.
Optical activity is observed in all three principal axis directions. NaBrO3 is dextro- and NaClO3 levorotatory in the right-handed coordinate axis system. The
compounds have opposite senses of optical rotation when crystallizing in the left- handed coordinate axis system, but the they are not enantiomorphs, because the
space group is not chiral.
INTRODUCTION
Optical rotation is a physical property, which also may occur in the cubic crystal classes 23 and
432. Crystals in cubic symmetry are free of disturbing double refraction. The optic axis is not
orientable and the optical rotation with the same sense and magnitude is possible to measure
in any principal axis direction. Dextro- and levorotatory crystals of NaClO3 have been reported
to grow up from a water solution (Marbach, 1856) [1], (Kondepudi, D. K., et al., 1990). [2]
Opposite optical rotations were observed in NaClO3 and NaBrO3 (Bijvoet et.al., 1960) [3] and
theoretical computations from the atomic polarizabilities were reported to lead to opposite
rotation senses in these crystals (Reijnhart, 1970) [4]. The experimentally observed optical
rotation in NaClO3 made both compounds as objects for early theory developments of optical
activity in the cubic crystals. Starting from Born’s (1922) [5] theory of oscillating dipoles
formed by the valence electrons to a more modern theory of atomic polarizabilities has met
some success. Further development of the atomic polarizabilities (Glazer and Stadnicka, 1986)
[6] has found out correlation both with structural and optical chirality associated with the
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Karppinen, M. (2024). Origin of Optical Activity in Cubic NaClO3 and NaBrO3 Crystals at 296K. European Journal of Applied Sciences, Vol - 12(1). 174-
184.
URL: http://dx.doi.org/10.14738/aivp.121.16257
atomic helical arrangements in several compounds. The atomic helices can be observed even in
NaClO3 and NaBrO3 crystals in some directions. However, these two crystals have up today
caused problems for all theoretical descriptions of optical activity because of the found opposite
sense of optical rotation though identical structures. [6] A cubic crystal in the class 23 has a
very specific symmetry. It has 4 three-fold axes in the direction of diagonals and 3 two-fold axes
in the principal axis directions all travelling through the midpoint of the unit cell. There are four
ClO3
‒ and BrO3
‒ ions with three-fold symmetries lying on the four diagonals in the unit cells of
NaClO3 and NaBrO3 crystals. Due to high symmetry some of the ions on the diagonals can have
opposite directions and may reorient and change the handedness of axial vectors with respect
to the principal axis directions of the chosen coordinate axis system in the space group P213.
This paper presents an alternative view of the origin of optical rotation in NaClO3 and NaBrO3
crystals relying on the distribution of valence electrons in a point charge model, which
generates a series of electric moments in the unit cell. Both compounds are ionic solids with the
integer charges on ions at room temperature. Na+ ion has a spherical noble gas core of electrons
around the nucleus, which makes it nonpolarizable and is ignored from the calculations. ClO3
‒
and BrO3
‒ are polar ions having valence electrons in the Cl-O and Br-O bonds, but these two
crystals are nonpolar due to symmetry. This study is focusing on the electric charges and
electric moments in the ClO3
‒ and BrO3
‒ ions and on the orientations and magnitudes of the
axial vectors of second rank with their screw rotational characters. According to this theoretical
approach to optical rotation the electric moments and axial vectors must have components with
different values in the principal axis directions though a cubic symmetry. They are possible to
obtain by iteration with the atomic net charges as variables within the Cl-O and Br-O atoms,
respectively. The iteration creates an identical asymmetry on the distributions of charge
densities in each ion in the unit cells of these two compounds. Cubic crystals are isotropic
materials. The isotropic indicatrix is not a quadric but a sphere, whose radius is the index of
refraction. A sphere around the cubic unit cell can always be drawn with eight corners of it on
the surface. The cubic crystals have also a measured specific refractive index with the same
magnitude in the principal axis direction. The magnitude of the isotropic refractive index is
reckoned in the direction of the diagonal and the ratio of the two refractive indices is used as a
reference, when the ratio of the magnitudes of electric moments in the principal axis directions
of NaBrO3 and an inverted ratio of them in NaClO3 are iterated to topological equivalences
(Encyclopedia Britannica).
STRUCTURAL BASIS
Figure 1: Reduced crystal structure of the unit cell of NaClO3 and NaBrO3
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Figure 2: Isotropic indicatrix with a diagonal andtrigonometric functions.
Accurate crystal structures of NaClO3 and NaBrO3 have been determined in the space group
P213 by X-ray diffraction and the models refined and corrected for anomalous dispersion.
(Abrahams and Bernstein, 1977). [7] A plot of the reduced crystal structure with four groups
of XO3 ions (X=Cl or Br) without Na ions is shown in Fig.1. (Farrugia, 1997) [8] Fig.2 is a
schematic presentation of the isotropic indicatrix round the unit cell with the trigonometric
functions, which relate the components of the second electric moment from one XO3 group on
the diagonal to three principal axis directions of the crystal. The Gr.2 and Gr.3 are pointing
upwards (in the positive c-axis direction) along the diagonals from 0,1,0 to 1,0,1 and from 1,0,0
to 0,1,1, respectively, while Gr.1 and Gr.4 point downwards from 1,1,1 to 0,0,0 and from 0,0,1
to 1,1,0, respectively, in the chosen constellation of the right-handed coordinate axis system.
Table 1: Refined atomic coordinates, lattice parameters and refractive indices for
NaClO3 and NaBrO3. [7] The refractive indices are computed from the dispersion
formulas for both crystals.[9]
x y z x y z
Na 0.06869 0.06869 0.06869 Na 0.0775 0.0775 0.0775
Cl 0.41822 0.41822 0.41822 Br 0.4067 0.4067 0.4067
O 0.30347 0.59293 0.50468 O 0.2882 0.5964 0.5085
a =6.57584 Å a = 6.70717 Å
n = 1.513553 (at 6328 Å) n = 1.612926 (at 6328 Å)
In the following calculations a light ray is considered to travers in the crystallographic a-axis
direction, when the electric moments of second rank are computed in the b- and c-axis
directions. The refined coordinates of Cl, respectively, Br atoms in Gr.1 (Table 1.), have same
values in the three principal axis directions. The values of second electric moments will be same
and produce equivalent values for the two axial vectors of second rank with the opposite
rotation senses. Hence the Gr.1 can’t contribute to optical rotation and therefore it is necessary
to make a similar analysis on three other groups in the unit cell. In the crystallographic
calculations the atomic coordinates must lie between 0 and 1 and the second electric moments
of the groups must have a common reference origin to be comparable with each other. The
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Karppinen, M. (2024). Origin of Optical Activity in Cubic NaClO3 and NaBrO3 Crystals at 296K. European Journal of Applied Sciences, Vol - 12(1). 174-
184.
URL: http://dx.doi.org/10.14738/aivp.121.16257
atoms with equal initial charges of -0.25 in the three groups are transformed to the relative
coordinates with their centers of charge at the coordinate = 1/2 and they are listed in Tables 2.
and 3. Only the coordinates of Cl and Br atoms are written in Gr.3 and Gr.4.
Table 2: The refined and transformed atomic coordinates of the ClO3- ion in Gr.2, Gr.3
and Gr.4.
Gr.2 x y z Gr.2 x y z
Cl 0.58178 0.91822 0.08178 Cl 0.53647 0.46353 0.53647
O1 0.40761 1.00468 0.19653 O1 0.36230 0.54999 0.65122
O2 0.69653 1.09239 -0.00468 O2 0.65122 0.63770 0.45001
O3 0.49532 0.80347 -0.09239 O3 0.45001 0.34878 0.36230
Gr.3 Gr.3
Cl 0.91822 0.08178 0.58178 Cl 0.46353 0.53647 0.53647
Gr.4 Gr.4
Cl 0.08178 0.58178 0.91822 Cl 0.53647 0.53647 0.46353
Table 3: The refined and transformed atomic coordinates of the BrO3
-
ion in Gr.2, Gr.3
and Gr.4.
Gr.2 x y z Gr.2 x y z
Br 0.5933 0.9067 0.0933 Br 0.54325 0.45675 0.54325
O1 0.4036 1.0085 0.2118 O1 0.35355 0.55855 0.66175
O2 0.7118 1.0964 -0.0085 O2 0.66175 0.64645 0.44145
O3 0.4915 0.7882 -0.0964 O3 0.44145 0.33825 0.35355
Gr.3 Gr.3
Br 0.9067 0.0933 0.5938 Br 0.45675 0.54325 0.54325
Gr.4 Gr.4
Br 0.0933 0.5933 0.9067 Br 0.54325 0.54325 0.45675
The refined atomic coordinates on the left-hand columns are directly read from the structure
plots. The transformed coordinates of the oxygen atoms on the right-hand columns in Gr.3 and
Gr.4 have same values as in the Gr.2 in the both Table 2. and Table 3., due to cubic symmetry,
but not in same order. The coordinates in the right-hand columns are used in the calculations
of the second electric moments. The Gr.3. in both crystals can be excluded from the following
computations because of the same value on y- and z-coordinates. The groups Gr.2 and Gr.4 in
Table 2. and Table 3. can contribute to the optical rotation, when the plane polarized light
travels in the c-axis direction, which is selected as a temporary optic axis.
REFRACTIVE INDICES OF ISOTROPIC COMPOUNDS
In order to start the iteration of electric moments the refractive indices of the isotropic
indicatrix must be known in both crystals. The following equations are valid for these two
crystals. The Eg.1. gives the length of the radius vector r’ and the Eg.2. the magnitude of the
refractive index, n’, of the isotropic indicatrix according to Fig. 2, where α = 45°, β = sin-1(1/√3)°.
The other input values are measured lattice constants and refractive indices for both crystals
from Table 1.:
1/ r’ = (sin2α/a2 + cos2α/a2 + sin2β/a2) -1/2 Eg.1.
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1/ n’ = (sin2α/n2 + cos2α/n2 + sin2β/n2) -1/2 Eg.2.
The results will be: r’ = 5.694844 Å and n’ = 1,310775 => n/n’ = 1,154701 for NaClO3 and r’ =
5.808580 Å and n’ = 1,396835 => n/n’ = 1.154701 for NaBrO3.
The n/n’ ratio is also possible to obtain from the relation of the unit cell parameters:
• D/2 = √3·a/2 = r’ = 5.694844 Å and a/ r’ = 1.154701= n/n’ in NaClO3 and
• D/2 = √3·a/2 = r’ = 5.808580 Å and a/ r’ = 1.154700= n/n’ in NaBrO3,
where D= the length of the diagonal. The received ratio of the refractive indices (n/n’) is used
as a reference in the following iterations.
ITERATION OF NET ATOMIC CHARGES AND ELECTRIC MOMENTS
The following three equations are developed for multinuclear negative ions in ionic solids and
presented in the study of LiNaSO4. [10] They are used here to calculate the principal
components of the second electric moments of the ClO3
‒ and BrO3
‒ ions. Atomic charges are
variables in the iteration. The ionic charge of -1 is dealt between one halogen and three
symmetry related oxygen atoms.
Θ’xx= Σ (eixi
2 ‒ 2μx · xc + qX2) · cos β · sin α Eq.3.
Θ’yy = Σ (eiyi
2 ‒ 2μy · yc + qY2) · cos β · cos α Eq.4.
Θ’zz= Σ (eizi
2 ‒ 2μz · zc + qZ2) · sin β Eq.5.
where xi, yi and zi are the coordinates of atomic charges, ei; xc, yc and zc are the location
coordinates of the charge centers of the first electric moment components, μx, μy and μz; X, Y, Z
are the coordinates of the centers of second electric moment, where the total electric charge, q,
of the ionic system is placed. Due to symmetry X, Y and Z are placed at the coordinate = 1/2. The
equations are multiplied by the trigonometric functions to transform the components of the
second electric moment from the diagonal to the principal axis directions according to Fig. 2.
The values are computed from the input elementary charges and the atomic coordinates, which
are standing for ClO3
‒ and BrO3
‒ ions in Table 2. and Table 3., respectively.
When starting the iteration with the initial charges of ‒0.25 for all four atoms in the XO3 ions,
the components of second electric moments have equal values in the three principal axis
directions. The measured electron attachment enthalpies (= electron affinities) for Cl, Br and I
are, respectively, 349, 325 and 295 kJ/mol [11] and the iterated net atomic charge of ‒ 0.1106
of iodine in the LiIO3 crystal [12] support the supposition that the atomic charge of Cl is more
negative than the charge of Br in the XO3 ions. This assumption will lead to a negative and to a
positive sense of the optical rotation in NaClO3 and NaBrO3, respectively. The iterations are
adjusted to the closest four-decimal values of the final atomic charges. The iterated axial
components of the second electric moments are presented in Tables 4. and Table 5. for both
XO3 ions.
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Karppinen, M. (2024). Origin of Optical Activity in Cubic NaClO3 and NaBrO3 Crystals at 296K. European Journal of Applied Sciences, Vol - 12(1). 174-
184.
URL: http://dx.doi.org/10.14738/aivp.121.16257
Table 4: Iterated atomic charges and axial components of second electric moments are
basing on the x-, y- and z- coordinates of Gr.2 in Table 2.
Atom Charge Θ’xx = Θ’yy Θ’zz Θ’yy / Θ’zz n/n’
Cl -0.25
O -0.25 -0.006590141 -0.006590141 1
Cl -0.265
O -0.245 -0.006894202 -0.006051963 1.13916
Cl -0.2665
O -0.2445 -0.006924540 -0.005998078 1.15446 1.15470
Table 5: Iterated atomic charges and axial components of second electric moments are
derived from the x-, y- and z-coordinates of Gr.2 in Table 3.
Atom Charge Θ’xx = Θ’yy Θ’zz Θ’zz / Θ’yy n/n’
Br -0.25
O -0.25 -0.00763804 -0.00763804 1
Br -0.235
O -0.255 -0.007267669 -0.008266485 1.13743
Br -0.2332
O -0.2556 -0.007223256 -0.008341930 1.15487 1.15470
The electric moments are derived from the atomic positional parameters in the unit cell. When
the unit cell contains four groups of XO3
‒ ions, the moments are valid in all four groups due to
symmetry. The necessary information needed to obtain, is associated with one group and the
calculations in the following will be limited to the Gr.2 in one quadrant ofthe unit cells ofNaClO3
and NaBrO3 crystals. Therefore, the obtained values of electric moments will be divided by 4
and the lattice constants by 2, when the magnitudes of the gyration tensor components are
reckoned. The Gr.2 is selected suitable for the calculations, because its axial components follow
the positive directions of the chosen right-handed coordinate axis system. The contributions to
the sense of optical rotation from the Gr.4 according to the information in Table 2. and Table 3.
are discussed down.
Right-Handed Coordinate Axis System
Axial Vectors of Second Rank and Their Handedness:
In the following presentations the crystallographic c-axis is selected to an optic axis. When a
plane polarized light travels in the positive a-axis direction of the cubic crystal, it interacts with
polar electric moments in the b- and c-axis directions of the crystal and creates, through vector
cross products, two axial vectors of second rank, which will be orthogonal to each other and to
the propagation direction of light. The axial vectors contain information of optical rotation with
opposite senses. The handedness of optical rotation is considered in the principal components
of the axial vectors of Gr.2 and Gr.4 FOR NACLO3 in the FIG. 3A AND FIG. 3B And for NaBrO3 in
the Fig. 3c and Fig. 3d, respectively, in the chosen right-handed coordinate axis systems.
The polar second electric moments, Θ’xx, Θ’yy and Θ’zz, are denoted by A, B, and C and the axial
vectors by B’ and C’ in the Gr.2 and Gr.4 for both NaClO3 and NaBrO3. ka is a wave vector of light.
The Figs.3a and 3c are drawn for Gr.2 according to the Fig. 1 The axial vectors B’= ka x C follows
the left-hand rule with an anti-clockwise rotation character. It dominates in NaBrO3, while C’=
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ka x B follows the right-hand rule with a clockwise rotation character, which dominates in
NaClO3.
The Figs. 3b and 3d are designed for Gr.4 in both compounds. They are standing with their left- handed coordinate axes, where the c-axis components point in a negative direction in the
chosen right-handed coordinate axis system. Due to the opposite handedness and reorientation
the axial vectors change their rotation senses, otherwise these compounds can’t be optically
active at all. The dominant clockwise rotation in B’= ka x C turns to anti-clockwise in NaBrO3
(Fig. 3d) and in C’= ka x B the dominant anti-clockwise rotation becomes clockwise in NaClO3
(Fig. 3b). The optical rotation is measurable and seen by an observer towards the light source
in the chosen c-axis direction as levorotation in NaClO3 and as dextrorotation in NaBrO3.
In the Fig. 4 the crystal structure is seen in the c-axis direction, where the Gr.3 and Gr.2 are
standing at the same positions as the Gr.2 and Gr.4 in Fig. 1. The atomic x- and y-coordinates in
Table 2. and Table 3. reveal that the principal components of axial vectors in the a- and b-axis
directions in the Gr.3 and Gr.2 are standing towards each other and in the Gr.1 and Gr.4 the
optical rotations are extinguished, when the a- and b-axes, respectively, are selected as optic
axes in both crystals. The situations will be similar as above, where the c-axis was acting as an
optic axis.
Summarized: Both crystals have two groups in their unit cells, where the components of
circularly polarized light waves can travel through the unit cell with the same rotational sense
and magnitude. The rotation is observable and measurable in all principal axis directions.
Gyration Tensor Components g11, g22 and g33 :
The gyration tensor is an axial tensor of second rank, the components of which are axial vectors
of second rank. [13] The components of the second electric moments in Table 4. and Table 5.
are converted to the principal gyration tensor components as follows:
In NaClO3: g11 = g22 = Θ’zz /4 · ( -1,602177 · 10 ‒19) · (a /2)
2 · 1016 Ccm2, g33 = (Θ’xx = Θ’yy) /4 · ( -
1,602177 · 10 ‒19) · (a /2) 2 · 1016 Ccm2. g33 is dominant.
In NaBrO3: g11= g22 = Θ’zz/4 · ( -1,602177 · 10‒19) · (a/2)
2·1016 Ccm2. g11 and g22 are dominants.
g33 = (Θ’xx = Θ’yy) /4· ( -1,602177 · 10 ‒19) · (a/2)
2 · 1016 Ccm2, where e = -1,602177 · 10 ‒19
Coulombs and a = lattice constant.
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Karppinen, M. (2024). Origin of Optical Activity in Cubic NaClO3 and NaBrO3 Crystals at 296K. European Journal of Applied Sciences, Vol - 12(1). 174-
184.
URL: http://dx.doi.org/10.14738/aivp.121.16257
The values of the two tensor components in both crystals are written in Table 6.
The ratios of the refractive indices n/n’, of the gyration tensor components, of the axial and
polar vectors of second rank and of the clockwise, right-hand (nR) and anticlockwise, left-hand
(nL) circularly polarized light waves, when they are transmitted in the axial vectors of the
crystals, can be expressed as follows and the rotations are seen by an observe against the light
source in the c-axis direction of the NaClO3 and NaBrO3 crystals:
In NaClO3: n/n’ = g33 / g22 = C’/ B’ = (ka x B) / (ka x C) = B / C = Θ’yy / Θ’zz = nR / nL=> Levorotation.
In NaBrO3: n/n’ = g22 / g33 = B’/ C’ = (ka x C) / (ka x B) = C / B = Θ’zz / Θ’yy = nL / nR =>
Dextrorotation.
Sense and Magnitude of Optical Rotation in NaClO3 and NaBrO3 Crystals:
The theory of optical rotation is basing on the Fresnel’s [14] genial assumption that the plane
polarized light is splitting up into the right- (nR) and left-hand (nL) circularly polarized waves,
when it enters the crystal in optic axis direction. The waves being in phase at the entrance
traverse with slightly different wavelengths so that one of the circularly polarized wave
components is ahead at the emergency and determines the sense of optical rotation in the
crystal. The magnitude and sense of the optical rotation per unit path d in both crystals are
given according to Eq. 6 [13]
ф =
d · π · ( nL − nR)
λ
radians Eq.6.
The refractive indices and the gyration tensor components are presented for a plate of crystal
with the thickness of 1 cm and the optical rotation with the thickness of 1 mm. The magnitude
and sense of the optical rotation in the Gr.2 in both crystals are expressed with λ = 6328 · 10‒8
cm according to Eq.7 and Eq. 8:
For NaClO3:
ф =
0.1 · π · (g22 − g33)
λ
rad. mm-1 Eq.7
=> ф = ‒ 0.01992 rad. mm-1 = ‒ 1.141 ° mm-1.
For NaBrO3
ф =
0.1 · π · (g22 − g33)
λ
rad. mm-1 Eq.8
=> ф = + 0.02502 rad. mm-1 = + 1.433 ° mm-1.
Table 6: Gyration tensor components g11, g22, and g33, magnitude and doubled sense of
optical rotation, Ф, in cubic NaClO3 and NaBrO3 crystals at 298K.
Crystal Space group g11=g22 · 10‒5
(Ccm2
) g33 · 10‒5
(Ccm2
) Ф (rad. mm‒1
) Ф (° mm‒1
)
NaClO3 P213 2.5972 2.9984 ‒2 · 0.0199 ‒2.28
NaBrO3 P213 3.7578 3.2539 + 2 · 0.0250 + 2.87
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The experimentally measured room temperature values are ‒2.44 and +1.65 °mm-1 with λ
=6328 Å. (Abrahams, Glass and Nassau,1977). [15] Theoretically computed values are ‒3.70
and +1.50 °mm-1 (Devarajan and Glazer,1986) [16] for optical rotation in NaClO3 and NaBrO3,
respectively.
Figure 4: Reduced view normal to (001) of the unit cell of NaClO3 and NaBrO3.
Figure 5: Reduced view normal to (010) of the left-handed structure of NaClO3 and NaBrO3.
Left-Handed Coordinate Axis System
When the x- and y-coordinates of oxygen atoms in both crystals are exchanged in Table 1., the
designed models follow the left-handed coordinate axis system. The Cl- and Br-atoms have the
same positions as in the Fig. 1, but the order of the groups has changed and differences in the
geometries of the oxygen atoms can be observed in the plot of NaClO3 in the Fig. 5. x-, y- and z- coordinates of Cl and Br atoms in the Gr.1 have same values as in Table 1. and can’t cause any
optical rotation. Gr.2 has the same values of the x-and z-coordinates (Table 2.) resulting in the
axial vectors with equal magnitudes, which extinguish each other with the same but opposite
rotation power. Gr.4 and Gr.3 can contribute to optical activity in the crystallographic c-axis
a
b
c
Gr.3
Gr.2
Gr.1
Gr.4
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Karppinen, M. (2024). Origin of Optical Activity in Cubic NaClO3 and NaBrO3 Crystals at 296K. European Journal of Applied Sciences, Vol - 12(1). 174-
184.
URL: http://dx.doi.org/10.14738/aivp.121.16257
direction as observed also in Gr.2 and Gr.4 in the Fig. 1 in the right-handed coordinate axis
system. But when compared to the Fig. 3a, the dominant axial vector C’ = kb x A in the Gr.4
follows left-hand rule with an anticlockwise rotation character in the left-handed coordinate
axis system, where the light ray travels in the b-axis direction. It is seen as a dextrorotation by
an observer towards the light source in the principal c-axis direction. This explains the
simultaneous existence of the single D- and L-crystals of NaClO3 in the water solution. The
plotting program draws always the structures in the right-handed coordinate axis system. So is
the case in the Fig. 5 also, but there can’t exist two crystal structures of NaClO3 with different O
atoms positions in the space group P213 in the same coordinate axis system and to make
difference the Fig. 5 is designed in the left-handed coordinate axis system.
DISCUSSION
Theoretical descriptions of optical activity are connected to the speed of light, when it is
transmitted through the optically active medium. The plane polarized light splits up into the
left- and right-handed circularly polarized light waves, which are passing the medium with
different velocities owing to the different wave lengths. The medium has some kind of left- and
right-handed character, which results in an observable optical rotation. The calculations in this
paper are open and transparent. The information needed for computations has been the
accurately measured parameters and the symmetry of the crystal structures. NaClO3 and
NaBrO3 are structurally similar and as chemical compounds they always have some kind of
electric charges associated with the atoms. In this analysis the point charges associated with Cl
and Br atoms have close to the same values, which implies that the magnitudes of optical
rotations can’t deviate much from each other between these two compounds with slightly
different values of the cell dimensions. No adjustments of any structural parameters have been
made in this study and the uncertainty in the final parameters of optical rotations is associated
with the last digit. The experimentally observed sense and the measured values of optical
rotations in NaClO3 and NaBrO3 are good. Electron densities can be studied around the atoms
by X-ray diffraction and the most reliable will be spherical models free of deformations, because
in the structural refinement analysis the atoms are designed to act as spherical scatterers. This
work will say that a knowledge of point charges can open new views in the interpretations and
explanations of the locked questions concerning the physical and chemical properties of
crystals.
ACKNOWLEDGEMENT
The author thanks Dr. L. J., Farrugia for his skilled worked plot-program ORTEP-3 for Windows.
University of Uppsala is acknowledged with pleasure.
References
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