Gyration and Permittivity of Ethylenediammonium Sulfate Crystals

Article abstract of DOI:10.1002/chir.22603

Ethylenediammonium sulfate (EDS) crystals were grown from aqueous solution and cleaved into thin (100-500 micron) plates. The 422 point group of EDS was confirmed by X-ray diffraction. The constitutive relations of EDS crystals were determined through generalized ellipsometry with an instrument that uses four photoelastic modulators (4PEM). The optical rotation at 500 nm, for example, was + 22.9°/mm along the optic axis and - 12.1°/mm perpendicular to the optic axis for the P4₁2₁2 crystals. Enantiomorphous twins frequently form across the (001) plane. Mirrored halves must be separated by cleavage in advance of optical measurements. Chirality 28:460-465, 2016.

Full text of DOI:10.1002/chir.22603

CHIRALITY 28:460–465 (2016)
Special Issue Article
Gyration and Permittivity of Ethylenediammonium Sulfate Crystals
SHANE NICHOLS, ALEXANDER MARTIN, JOSHUA CHOI, AND BART KAHR*
Department of Chemistry and Molecular Design Institute, New York University, New York City, New York, USA
ABSTRACT
Ethylenediammonium sulfate (EDS) crystals were grown from aqueous solution
and cleaved into thin (100–500 micron) plates. The 422 point group of EDS was confirmed by
X-ray diffraction. The constitutive relations of EDS crystals were determined through general-
ized ellipsometry with an instrument that uses four photoelastic modulators (4PEM). The optical
rotation at 500 nm, for example, was + 22.9°/mm along the optic axis and ꢀ 12.1°/mm perpen-
dicular to the optic axis for the P4 2 2 crystals. Enantiomorphous twins frequently form across
1
1
the (001) plane. Mirrored halves must be separated by cleavage in advance of optical measure-
ments. Chirality 28:460–465, 2016. © 2016 Wiley Periodicals, Inc.
KEY WORDS: gyration; optical activity; ellipsometry; polarimetry; chiral twinning; bianisotropic;
multiple reflections
In a review article at the start of this century, Kaminsky1
collected in a table all of the crystals for which the anisotropy
of circular birefringence had been measured over the
previous two centuries. The table lists only 28 substances,
Propagation of Light in an Optically Active Crystal
A complete description of light propagation in crystals
requires knowledge of the four tensorial optical functions of
the medium: the electric permittivity (ε), magnetic permeabil-
2
4 if excluding members of the large, isomorphous KDP
33
ity (μ), and the two magnetoelectric tensors (ρ and ρ ’).
(
KH PO ) family of crystals. The citations in this table are
2
4
They form the material constitutive relations that when
inserted into Maxwell’s curl equations give two first-order dif-
ferential equations for the electric and magnetic fields:
apportioned to decades as follows: 1960s (1), 1970s (1),
980s (3), 1990s (19). These four pieces of data tell a story.
1
Measuring the optical activity of anisotropic crystals has been
very difficult for a long time, but in the 1990s it looked like
experimental obstacles were finally broached with the intro-
duction of the HAUP (high accuracy universal polarimetry)
∇ꢁH ¼ ð∂=∂tÞðεE þ ρH Þ
ꢁE ¼ ð∂=∂tÞðμH þ ρ’EÞ
(1a)
(1b)
∇
2,3
technique; soon, it seemed, we might have well-defined
These equations are the basis for modeling electromagnetic
waves in crystals.34 Ab initio calculations of the constitutive rela-
gyration tensors for thousands of crystalline substances, just
4,5
35
as we have large collections of electric permittivity tensors.
tions are beyond reach for all but the simplest materials, and
Determining these quantities is important because it is
impossible to fully reckon chiroptical structure–property
relationships without embracing anisotropy. Average values
of bisignate quantities such as optical rotation (OR) or circu-
lar dichroism (CD) from measurements of solutions are not
easily correlated with structure. This problem has persisted
thus modeling of light/matter interactions hinges upon accurate
measurements of the tensor components.36 Our task is simpli-
fied as μ =I (the 3 × 3 identity tensor) for most materials at optical
T
frequencies, and reciprocity requires that ρ =ꢀ iα and ρ ’=iα ,
where T indicates the transpose and α is the gyration tensor.
Thus, our problem usually consists of measuring the compo-
nents of ε and α in the frequency range of interest.
6
since the discovery of optical rotation by Arago in 1811.
What has happened since 2000 in the determination of the
chiroptical anisotropy of single crystals? Here is the update:
The tensor ε is preferably measured by generalized spec-
37
troscopic ellipsometry in reflection. Complications arising
from multiple reflections inside plane-parallel slabs are obviated
by ensuring that reflected light is taken only from a single
interface; any light that has propagated inside the crystal is
prevented from reaching the detector. The method is robust
and delivers ε to high precision over a broad spectral range.
Gyration is considerably more difficult to measure, as the
magnitude of the effect adds only a tiny contribution to the
refractive indices of natural crystals (synthetic “metamaterial”
7
–24
25–28
2
000s (18),
2010s (4).
The apparent building of
momentum from laboratories around the world has
dissipated. This is a consequence of the fact that HAUP is
challenging for reasons we discuss in the next section.
We have been keen to speed accurate chiroptical measure-
ments of anisotropic crystals. Here we describe such a
measurement for the determination of the gyration tensor of
ethylenediammonium sulfate (EDS), a chiral, uniaxial salt
3
that can be grown easily as spectacularly large (1 cm ) square
crystals. While the optical activity of EDS along the optic axis
th
29
*Correspondence to: Bart Kahr, Department of Chemistry and Molecular De-
sign Institute, New York University, 100 Washington Square East, Room 1001,
New York City, NY 10003. E-mail: bart.kahr@nyu.edu
Received for publication 29 November 2015; Accepted 29 February 2016
DOI: 10.1002/chir.22603
Published online 29 April 2016 in Wiley Online Library
(wileyonlinelibrary.com).
was measured in the 19 century, this crystal was not
widely appreciated until Cuccia et al. published its simple
3
0
synthesis in the Journal of Chemical Education. It is now
finding application in experiments related to asymmetric
autocatalysis.3
1,32
©
2016 Wiley Periodicals, Inc.

OPTICAL ROTATION OF EDS CRYSTALS
461
crystals with features on the order of the wavelength of visible
Ellipsometry
light can have substantially larger gyration). Moreover,
gyration has a vanishingly small contribution to the change
The complete 4 × 4 normalized Mueller matrix was measured using a
homebuilt ellipsometer with four photoelastic modulators described
3
8
42,46
in the polarization of light upon reflection ; therefore, α can
previously.
The 16 matrix elements were recovered simultaneously
only be recovered robustly by assaying its accumulated effect
without mechanically moving parts by digital demodulation of the
time-varying intensity signal. Measurements were made from 300 nm to
50 nm in steps of 2 nm at incident angles of 40°, 45°, and 50°. Measure-
ments at multiple azimuthal angles were unnecessary because the
tensors ε and α have axial symmetry for point group 422. The transmis-
sion Mueller matrix was measured in the same wavelength range at
normal incidence. A transmission measurement is not essential, but
serves as a direct measure of the α11 tensor component.
3
9,40
on light propagation over a considerable thickness
(on
7
the order of 100 μm).
In the thickness regime of 100 microns or more, the
multiple internal reflections inside the slab give rise to several
4
1
superimposed outgoing waves. For very thick crystals
measured at oblique incidence, the multiply reflected waves
can be spatially resolved and separately measured, but as
the HAUP method relies on transmission measurements at
normal incidence, the contribution of multiple reflections is
unavoidable, and has lead to artifactual oscillations in
Fitting Procedure
A model was constructed to fit the Mueller matrix data. The free
42
parameters were the dispersion relations for the components of ε and α,
a constant angle of incidence offset, and the crystal miscut. Starting from
an initial guess, the free parameters were refined using the Levenberg–
1
1,14
recovered values of α as a function of temperature.
More-
over, relying on normal incidence requires that crystals be
cut in multiple directions, the number of requisite directions
being equal to the number of unique components in α. This
requires first a mastery of the lapidary arts. Extensions of
HAUP to nonnormal incidence have not explicitly considered
the boundary conditions that must be treated to model
2
Marquardt algorithm to minimize the reduced χ error function. The stan-
dard deviation of each Mueller matrix element at each wavelength was
determined from three repeated measurements. For each data point,
the larger of either the standard deviation, or the value 0.0005 was used
2
for the error value in the calculation of the reduced χ figure of merit.
The threshold 0.0005 represents a reasonable estimation of systemic
error, whereas the measured standard deviation represents random error.
1
multiple reflections. Lastly, ε and the slab thicknesses must
be known independently for the HAUP approach.
We recently developed a new approach that seeks to
remedy the shortcomings in the current methods to recover
RESULTS AND DISCUSSION
Constitutive Relations of EDS
42
ε and α. From the reflection Mueller matrix of a single cut
measured at multiple angles of incidence, the crystal
thickness, the exact cut of the crystal, and the dispersion
relations for the components of ε and α can be fitted together.
Crucial to our approach is an exact treatment of boundary
conditions, and the multiple incoherent reflections inside
the slab. We applied this method to quartz, and the results
were in agreement with published dispersion relations.42
In the standard setting of the crystal (4-fold axis along z),
both ε and α are represented by diagonal matrices having
two unique components:
ε ¼ diagðε ; ε ; ε Þ
α ¼ diagðα11; α11; α33Þ:
(2a)
(2b)
11
11 33
As EDS has no absorption in the visible wavelength range,
the components of ε are suitably modeled by the real-valued
Sellmeier equation:
MATERIALS AND METHODS
2
B λ2
2i
Crystal Growth
B λ
1
i
ε ¼ ꢀ
ꢁ þ ꢀ
ꢁ þ D
(3)
ii
i
EDS was synthesized from freshly distilled ethylenediamine and
2
2
2
2
λ ꢀ C
λ ꢀ C
32
1i
2i
sulfuric acid according to a published procedure. Large single crystals
3
(
~0.5 cm ) of EDS were grown by slow evaporation from aqueous
3
0
where the first two terms are “oscillators” characterized by an
amplitude (B) and a location (C), and λ is the wavelength. The
description is phenomenological, as the oscillators are not
tied to structure, and only represent average contributions
to the dielectric; terms one and two account for electronic
and vibrational transitions in the material, respectively. The
extra term D accounts for very short wavelength contribu-
tions to the dielectric. There are in total five fitting parame-
ters for each component of ε.
solution. Crystals were cleaved normal to [001] to produce thin sections
of high optical quality.
X-ray Crystallography
EDS crystallizes in the tetragonal enantiomorphous space groups
P4 2 2 and P4 2 2. Enantiomorphs of EDS were easily distinguished from
1 1 3 1
one another by their optical rotation in a petrographic microscope.
Dextro- and levorotatory crystals were identified with absolute structures
by analyzing the anomalous dispersion of X-rays. The absolute structure
of the EDS enantiomorphs was determined from analysis of 246 select
Components of α were fit to the two-parameter dispersion
model:
4
3
Bijvoet pairs. The Hooft y parameter was –0.013(45) and was calculated
3
Biλ
4
4
45
using PLATON. The traditional Flack parameter was –0.03(12).
counterclockwise sense of optical rotation down [001], as
measured looking at the source, corresponds to space group P4 2.
αii ¼
(4)
ꢀ
ꢁ
2
2
2
A
λ ꢀ C
i
1 1
2
Conventionally, a positive OR is defined as a clockwise azimuthal rotation
when viewing the light source. However, it is well known that this
definition is inconsistent with a right-handed screw along the propagation
which is likewise phenomenological. Only one “oscillator”
positioned in the near UV was required to fit the experimental
data.
1
direction. All computations reported here are based on a right-handed
The dispersion relations in Eqs. 3 and 4, along with the
sample-dependent parameters were used to construct a
model of the ellipsometry data. Figure 1 compares the raw re-
flection data with the fitted values. In principle, the crystal
section used for analysis should have a surface normal
Chirality DOI 10.1002/chir
coordinate system where + z is the direction of light propagation. Thus,
for our purposes and for internal consistency with basic conventions for
vectors in physics, we treat a clockwise rotation when viewing the light
source as negative OR. As we aim to report other gyration tensors by this
method, we strive for consistent definitions.

4
62
NICHOLS ET AL.
Fig. 1. Comparison of the measured and fitted reflection Mueller matrix of EDS. Each spectral plot represents one element of the 4x4 Mueller matrix operator
measured at (red) 40°, (green) 45°, and (blue) 50° angles of incidence. The fitting of the data is given by the thin black lines. Matrix elements are normalized by
m00, therefore, the y axes are unitless.
exactly along [001], due to the high-quality cleavage of EDS,
but the miscut was refined and resulted in a 0.11° inclination
from [001], and an azimuthal angle between the instrument x-
axis and the small projection of the optical axis of 73.62°.
While the high-precision motorized goniometer of our device
ensures precise steps in the angle of incidence, an overall an-
gle of incidence offset originates from imperfections in sam-
ple mounting; the fitted offset was 0.25°. The crystal
thickness was fixed to the value 305 μm, as determined by op2-
tical and mechanical measurements. The overall reduced χ
2
was 11.77, and the R value was 0.9999. Fitted values for the
dispersion relations are summarized in Table 1 and presented
in Figure 2.
Modeling a Thick Crystalline Slab
Many theorems and devices in the field of optics are rooted
4
7
in the Jones 2×2 matrix polarization formalism, including
HAUP. The Jones formalism treats orthogonal components
of the light as 2×1 Jones vectors. The effect of an object that
changes the polarization of light can be described by a 2×2
Jones matrix that operates on the Jones vector. Usually, a
Jones matrix describes either an interface between two media
Fig. 2. Dispersion curves of EDS calculated from the fitting parameters in
Table 1 are plotted as black lines in the top and middle panels. Red line values
were recovered from the transmission Mueller matrix measurement
according to Eq. 10. The bottom panel shows the calculated OA according
TABLE 1. Fitted Dispersion Parameters of EDS
a
b
c
a
b
c
29
Elem.
Param.
Value
Elem.
Param.
Value
to Eq. 9. The green mark is an early measurement along the optic axis.
ε
11
B
C
B
C
D
B
C
11
11
21
21
1.0648
0.0103 μm
2.3712
92.3287 μm
1.2728
-0.0146
ε
33
B
C
B
C
D
3
B
C
13
13
23
23
1.0759
0.0102 μm
2.6485
94.8497 μm
1.2751
0.0301
(
an abrupt change in the polarization of light over zero
thickness), or a bulk region of space (a gradual change in
the polarization that accumulates during propagation).
How light propagates through a crystalline slab with
plane-parallel interfaces can be written in terms of the Jones
formalism, as shown in Figure 3. Eight unique matrices
describe the changes in polarization upon refraction, reflec-
tion, or propagation. We provided formulas for their
1
α
11
1
α
33
1
1
0.104 μm
3
0.100 μm
a
Tensor element.
Parameter in dispersion formulae.
b
c
42
Fitted value.
calculation elsewhere.
Chirality DOI 10.1002/chir

OPTICAL ROTATION OF EDS CRYSTALS
463
the same arithmetic in Eq. 5 results in the overall coherency
matrices for the outgoing fields. Mueller matrices are related
to the coherency matrix as:
2
3
1
0
0
1
i
0
0
1
M ¼ ACAꢀ ; A ¼ 4
1
6 1
ꢀ1 7
5:
(7)
0
0
1
0
0
ꢀi
M contains the same information as C but is more closely
related to measurement.
Fig. 3. Propagation of light in a crystalline slab can be broken into eight 2x2
Jones matrices describing refraction (T) or reflection (R) at interfaces, or prop-
agation (P). The total outgoing fields from the slab are composed of partial
waves that have each traversed the medium a different number of times.
Gyration and Optical Rotation
As light propagates through a medium with gyration, its
field components rotate about the propagation direction.
The amount of rotation is called the optical rotation (OR)
and is a more familiar measure than gyration. In an
anisotropic medium, OR can be a bisignate function of the
wave vector of the incident light. We define the OR tensor,
ϱ, as the object that takes the wave vector k, and returns
the amount of OR(ϱ) in direction of k as:
A portion of the light incident on the crystal is reflected
back into the incident medium; it undergoes a change in
0
1
polarization described by R in Figure 3. Light refracted
across the boundary undergoes a change in polarization
01
described by T , and the change in polarization by propaga-
+
tion to the other interface is given by P . Similarly, matrices
1
2
12
ꢀ
T
T
and R describe the effect of the far interface, while P
ϱ ¼ k ϱk:
(8)
k
describes propagation in the backward direction. Finally,
matrices T and R describe the effect of encountering the
first interface from the backward direction.
Tensors ϱ and α are different because ϱ operates on the
wave vector k, while α operates on the field components E
and H. Fortunately, the relationship E × H = k requires that
E and H are always perpendicular to k. Therefore, to find
the OR in the direction of k, we average the gyration in the
plane perpendicular to k. Because we can always find a coor-
dinate system in which α and ϱ are diagonal, the relations:
10
10
These matrices can be used to compute the Jones matrix
relating the incident field to any wave having made a specific
number of passes through the medium (called “partial
waves”). For example, the Jones matrix for the first transmit-
1
2 + 01
ted partial wave is the matrix product T P T . The Jones
matrix for the total outgoing fields, for either transmission
ϱ11 ¼ πdðα22 þ α Þ=λ
33
(9a)
(9b)
(9c)
(
T) or reflection (R), are found by summing over the Jones
ϱ22 ¼ πdðα11 þ α33Þ=λ
ϱ33 ¼ πdðα11 þ α22Þ=λ
matrices for all transmission or reflection partial waves,
respectively. These infinite matrix summations have the
exact solutions
48
are sufficient to interconvert ϱ and α. The units of ϱ
constructed from Eq. 9 are in radians accumulated over a
thickness d.
1
2
þ
ꢀ1 01
T ¼ T P ½I ꢀ Qꢂ T
(5a)
(5b)
0
1
10
ꢀ
12
þ
ꢀ1 01
R ¼ R þ T P R P ½I ꢀ Qꢂ T
Evidently, the OR of EDS measured down the optic axis is
ϱ33 = 2πdα /λ, while perpendicular it is ϱ = πd(α11 + α )/λ.
1
0
ꢀ
12 +
11
11
33
where Q = R P R P , and I is the 2x2 identity matrix.
Regardless of whether or not multiple reflections are coher-
Because the Jones formalism treats the amplitudes of
waves rather than intensities, the summation implicit in
Eq. 5 is a coherent summation, the validity of which depends
on the coherence properties of the illumination. Lasers have
coherence lengths on the order of meters, therefore Eq. 5 is
valid for lasers, but light beams in spectroscopic devices that
use lamps have coherence lengths on the order of a few
microns. For crystals thick enough to recover α, Eq. 5 is inva-
lid. We must “switch off” the coherence between the partial
waves by adding together intensities rather than amplitudes.
Just as the product of a scalar amplitude with its complex
conjugate gives the intensity, the matrix Kronecker product
of a field vector with its complex conjugate gives its intensity,
or “coherency” vector. Likewise, Jones matrices are
converted to coherency matrices by:
ent, the normal incidence transmission Mueller matrix of an
[
001] section has the simple form:
2
3
1
0
0
0
0
0
0
6
cosð2ϱ Þ sinð2ϱ Þ 0 7
3
3
33
M ¼ 4
5
(10)
sinð2ϱ Þ cosð2ϱ Þ
0
1
33
33
0
0
and provides an independent measure of ϱ . An early
measurement along the optic axis at 589 nm agrees with
33
29
our spectroscopic measurement of ϱ33 (Fig. 1).
Twinning in EDS
We observed that roughly half of EDS crystals are
enantiomorphously twinned across (001), as in Figure 4.
Twin boundaries were consistently near the center of crys-
tals, suggesting that twinning occurs at the onset of growth.
Curiously, reflection from the twin boundary is noticeable to
the eye, yet the reflection coefficient for a perfect (001)
boundary between P4 2 2 and P4 2 2 crystals was calculated
2
3
ꢃ
ꢃ
ꢃ
ꢃ
J J
J J
J J
J J
1
1
11
11 12
12 11
12 12
ꢃ
ꢃ
ꢃ
ꢃ
ꢃ
6 J J
J J
J J
J J
7
11
21
11 22
12 21
12 22
C ¼ J⊗J ¼ 4
ꢃ
ꢃ
ꢃ
ꢃ 5
(6)
J J
J J
J J
J J
21
11
21 12
22 11
22 12
ꢃ
ꢃ
ꢃ
ꢃ
J J
J J
J J
J J
2
1
21
21 22
22 21
22 22
1
1
3 1
where J is any Jones matrix, ⊗ denotes the Kronecker prod-
uct, and * indicates the complex conjugate. Converting every
matrix in Eq. 5 to its coherency matrix and then carrying out
to be zero. We conclude that the boundary must contain
imperfections. Crystal sections free of twinning were used
for the aforementioned optical and X-ray characterization.
Chirality DOI 10.1002/chir

4
64
NICHOLS ET AL.
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1
2
2
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3
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ACKNOWLEDGMENTS
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BK and SMN are grateful for support from the US National
Science Foundation through award DMR1105000 and predoc-
toral fellowship support (DGE-12342536), respectively. The
authors are grateful for the assistance of Dr. Chunhua Hu at
the Department of Chemistry of New York University and
the Molecular Design Institute for the purchase of the single
crystal diffractometer.
2
2
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