Double cone of eigendirections in optically active ethylenediammonium selenate crystals

Circular birefringence (CB) is generally responsible for only a small perturbation to the state of light polarization in crystals that also exhibit linear birefringence (LB). As such, the magnetoelectric tensor of gyration, which gives rise to CB and optical activity, is less well determined than the electric permittivity tensor in optical models of the Mueller matrix. To visualize the effect of the magnetoelectric tensor on polarimetric measurements, reported here are experimental mappings of the Mueller matrix and of the CB in a new chiral crystal with accidental null LB at an accessible optical frequency. Single crystals of ethylenediammonium selenate (EDSeO₄) were synthesized and characterized by X-ray diffraction and Mueller matrix measurements in transmission and reflection. The crystals are isomorphous with the corresponding sulfate salt. They are tetragonal, space group P4₁₍₃₎2₁2. The constitutive relations of EDSeO₄ were recovered using a partial wave summation of incoherent reflections. The extraordinary and ordinary refractive indices cross at 364 nm (3.41 eV), a scenario commonly called the `isotropic point' or `iso-index point'. At this wavelength, the magnetoelectric tensor fully describes the polarization transformation, giving rise to a double cone of eigendirections.

Computing details

Program(s) used to refine structure: SHELXL2014/7 (Sheldrick, 2014).

(I)

Crystal data

C2H10N2O4Se	Z = 4
Mr = 205.08	F(000) = 408
Tetragonal, P43212	Dx = 1.991 Mg m^−3
a = 6.1352 (4) Å	Mo Kα radiation, λ = 0.71073 Å
c = 18.1726 (12) Å	µ = 5.44 mm^−1
V = 684.03 (10) Å^3	T = 300 K

Data collection

11625 measured reflections	θmax = 30.9°, θmin = 3.5°
1050 independent reflections	h = −8→8
961 reflections with I > 2σ(I)	k = −8→8
Rint = 0.024	l = −25→24

Refinement

Refinement on F^2	Hydrogen site location: inferred from neighbouring sites
Least-squares matrix: full	H-atom parameters constrained
R[F^2 > 2σ(F^2)] = 0.021	w = 1/[σ^2(Fo^2) + (0.031P)^2 + 0.0547P] where P = (Fo^2 + 2Fc^2)/3
wR(F^2) = 0.056	(Δ/σ)max < 0.001
S = 1.20	Δρmax = 0.30 e Å^−3
1050 reflections	Δρmin = −0.92 e Å^−3
42 parameters	Absolute structure: Flack x determined using 320 quotients [(I+)-(I-)]/[(I+)+(I-)] (Parsons, Flack and Wagner, Acta Cryst. B69 (2013) 249-259).
0 restraints	Absolute structure parameter: 0.024 (7)

Special details

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds involving l.s. planes.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Se01	0.24089 (3)	0.75911 (3)	0.7500	0.01878 (11)
O1	0.2729 (3)	1.0105 (2)	0.72059 (9)	0.0309 (4)
O2	0.2859 (3)	0.5895 (3)	0.68224 (10)	0.0361 (4)
C1	0.7477 (3)	0.2244 (3)	0.79131 (15)	0.0280 (5)
H1A	0.7878	0.0768	0.8048	0.034*
H1B	0.8461	0.3232	0.8165	0.034*
N1	0.5212 (3)	0.2677 (3)	0.81625 (10)	0.0260 (4)
H1C	0.5146	0.2568	0.8650	0.039*
H1D	0.4315	0.1708	0.7960	0.039*
H1E	0.4819	0.4014	0.8027	0.039*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Se01	0.01957 (12)	0.01957 (12)	0.01719 (17)	0.00108 (13)	−0.00098 (6)	−0.00098 (6)
O1	0.0401 (10)	0.0247 (8)	0.0280 (8)	−0.0062 (7)	−0.0057 (8)	0.0059 (7)
O2	0.0349 (10)	0.0404 (10)	0.0329 (9)	0.0048 (7)	0.0009 (8)	−0.0169 (8)
C1	0.0310 (12)	0.0292 (10)	0.0237 (12)	0.0058 (10)	−0.0032 (8)	0.0041 (8)
N1	0.0323 (10)	0.0259 (10)	0.0197 (8)	0.0015 (8)	0.0011 (7)	0.0015 (8)

Geometric parameters (Å, º)

Se01—O2^i	1.6357 (16)	C1—H1A	0.9700
Se01—O2	1.6357 (16)	C1—H1B	0.9700
Se01—O1^i	1.6439 (15)	N1—H1C	0.8900
Se01—O1	1.6439 (15)	N1—H1D	0.8900
C1—N1	1.486 (3)	N1—H1E	0.8900
C1—C1^i	1.521 (5)
O2^i—Se01—O2	110.61 (13)	N1—C1—H1B	109.0
O2^i—Se01—O1^i	109.39 (9)	C1^i—C1—H1B	109.0
O2—Se01—O1^i	109.09 (8)	H1A—C1—H1B	107.8
O2^i—Se01—O1	109.08 (8)	C1—N1—H1C	109.5
O2—Se01—O1	109.39 (9)	C1—N1—H1D	109.5
O1^i—Se01—O1	109.26 (12)	H1C—N1—H1D	109.5
N1—C1—C1^i	112.72 (16)	C1—N1—H1E	109.5
N1—C1—H1A	109.0	H1C—N1—H1E	109.5
C1^i—C1—H1A	109.0	H1D—N1—H1E	109.5

Symmetry code: (i) −y+1, −x+1, −z+3/2.