The baric changes of the refractive properties of K 2 SO 4 crystals

The dependencies of the birefringence ∆ni of K2SO4 crystals in wide spectral (250–800 nm) ranges, as well as the effect of uniaxial mechanical stress on them have been studied. The values of ni for K2SO4 are found to be rather sensitive to the uniaxial stress. The spectral characteristics of the refractive indices ni of free and mechanically clamped K2SO4 crystals have been studied. The refractive indices turned out to increase under the effect of uniaxial stress. The variations in the refraction, electron polarizability and the position of the center of the effective ultraviolet absorption band have been calculated. It is shown that the positions of the isotropic points in these crystals can be controlled by thermal and spectral methods.


Introduction
Crystals K 2 SO 4 (PS) -are classic ferroelastics.The phase transition (PT) at 860 K from paraelastic to ferroelastic phase of space group symmetry D 16 2h (c 0 = 7.48 Å, b 0 = 10.07Å, a 0 = 5.76 Å, Z = 4) was revealed [1][2][3][4].In spite of a certain interest in PTs taking place in K 2 SO 4 crystals [5,6], their optical properties have not been studied well enough.There are practically no data in the literature concerning the effect of uniaxial stress on the optical indicatrix of these crystals.Although this stress does not generally change the crystal symmetry without affecting the quantities under investigation, they make it possible to selectively modify certain bond groups and certain structural entities of the crystal.
The study of the temperature and spectral characteristics of the refractive indices n i of the A 2 BX 4 [K 2 SO 4 , Rb 2 SO 4 , LiKSO 4 , and (NH 4 ) 2 BeF 4 ] crystals made it possible to establish the isotropic points, i.e. the intersection of the dispersion curves of the refractive indices n i at certain temperatures and wavelengths.The Rb 2 SO 4 crystals have three isotropic points.In the short wavelength range of the spectrum, the isotropic points slightly depend on temperature with , where, like in the long wavelength range, the shifts of the isotropic points are quite pronounced, especially along the Y -direction [7].The (NH 4 ) 2 BeF 4 crystals also have two isotropic points [8].In the LiKSO 4 crystals, a decrease of the temperature causes the shift of the point of the intersection of the n x and n z curves toward the long wavelength range of the spectrum [4].The K 2 SO 4 crystals have two isotropic points in the spectral range 250-850 nm as well as demonstrate the property of isospectrality [8].
The aim of this work is to study the birefringence ∆n i and the principal refractive indices n i of K 2 SO 4 crystals in a wide spectral (300-800 nm) range, as well as the effect of uniaxial mechanical stress applied along the main crystallophysical directions and the bisectors of the angles formed by these directions on the birefringence properties and on the behavior of the isotropic points.

Experiments
K 2 SO 4 crystals were grown by a slow evaporation of an aqueous solution.The grown crystals were of good optical quality and were of an orthrombic shape.The spectral dependencies of the ∆n i were measured using the interference technique.
To define the birefringence, a photographic recording of the interference pattern in the focal plane of a DFS-8 spectograph was used, which ensured a spatial resolution of extrema of various orders, excluded "smeared" patterns, and therefore allowed for their independent registration.A transmittance of the polarizing system, composed of crossed polarizers, with a specimen being installed between them in a diagonal position and normally to the incident light beam, is determined according to the relation where I 0 and I are intensities of the incident and transmitted light beams, respectively, λ is a wavelength, and n i − n j is the birefringence and d is a crystal thinkness in the direction of the incident light beam.The extremum positions satisfy the relation where k is an interference order.Under the variation of the temperature and due to the d(T)-and n i (T)-dependencies, the positions of interference extrema will shift, and the birefringence will be determined by the expression An uniaxial stress was developed making use of a special attachment to a nitrogen cryostat and an electric furnace, which allowed us to obtain a stress of about 200 bar.Assuming the mechanical stress, the birefringence is determined through the expression The temperature variations of ∆n and d were investigated independently for unstressed and stressed specimens.By varying either of the parameters (T or σ ), with another one being fixed, one can unambiguously determine the temperature or baric dependence of ∆n .
The principal refractive indices were determined using the immersion Obreimov method at room temperature.

Effect of the uniaxial stress on the birefringence
The dispersion of the ∆n is normal in the whole spectral interval of investigation.A comparison of ∆n with the corresponding values for Rb 2 SO 4 and LiKSO 4 isomorphous crystals showed that the cation substitution Li + → K + in the sublattice causes a birefringence growth by 0.008-0.010on the average, while the Li + → K + and Rb + → Li + substitutions cause an average decrease of ∆n i by 0.02 and 0.03, respectively.The results are primarily due to the variations of the n i , to the shifts of UV and IR absorption bands, as well as due to the changes of relevant additive refractions and electron polarizability.
The values of ∆n i for this crystal are sensitive to the uniaxial stress and change almost linearly with the stress as follows: ∆n z : δ(∆n z ) = +0.15• 10 −4 and −1.80 • 10 −4 (for σ x and σ y = 100 bar, respectively).In general, although σ m stress affects the ∆n i values, it does not substantially change their dispersion.
It was found that if the uniaxial stress is X-directed, then ∆n z increases and ∆n y decreases, for Y-directed, then ∆n x increases and ∆n z decreases (figure 1).Since the relations n x > n z > n y and ∆n x = n z − n y , ∆n y = n x − n z and ∆n z = n x − n y of K 2 SO 4 crystals were established, the increase of the anisotropy of optical indicatrix under σ m uniaxial stress action was observed.

Refractive indices
The variations in the refractive indices of the K 2 SO 4 crystals were studied through analyzing the effect of uniaxial σ m on ∆n i .
It has been established that the dispersion of the n i has no anomalies and is well described by the single-oscillator Sellmeier formula where B i is the parameter related to the strength, effective mass, and charge of UV-oscillators and λ 0i is the position of the effective center of the UV absorption band.Using the experimental dependencies n i (λ) and the well-known Lorenz-Lorentz formula, we calculated the effective parameters of the Sellmeier formula, electron polarizability α i , and specific refractivity R i of the crystals (table 1) (ρ and µ are the density and the molar mass of the crystal and N 0 is the Avogadro number).The comparison with the analogous characteristics of the isomorphic Rb 2 SO 4 crystal showed that the cation substitution K + → Rb + causes an increase of the n i by 0.004-0.007,the α i by (2 − 4) • 10 −25 cm 3 , and R i by 0.8-1.0cm 3 .Using the data on the spectral variations of ∆n i under the pressure σ m , we calculated the spectral dependencies of the piezooptical constants of the K 2 SO 4 crystals in the spectral ranges of 300-700 nm, respectively, by the formula where δn i is the induced change in birefringence for the light propagating along the i -axis, s im are the coefficients of elastic compliance.The second term in formula (7) describing the variations in the geometric dimensions of the crystal under pressure was ignored, because its contribution to the change of π im did not exceed 2-5%, i.e., it was comparable with the experimental accuracy.Using the well-known expressions for piezobirefringence of orthorhombic crystals (the traditional Pockels method [9]) we calculated the spectral characteristics of the absolute piezooptical constants of the K 2 SO 4 crystals by solving the systems of equations with nine unknowns π im where δ∆n m i is the change in birefringence along the i -direction under uniaxial stress applied along the m-axis, and n k and n m are the absolute values of the refractive indices of a free K 2 SO 4 crystal.
The analysis of the variations of the principal n i of K 2 SO 4 crystals was performed using the piezooptic constants and the formula where n oi (λ) are the spectral dependencies of the refractive indices for a mechanically free crystal.
Figure 2 shows the stress dependence of the principal n i for λ = 500 nm and various directions of the applied mechanical stresses at room temperature.The n i increase with stresses application.In the UV spectrum range, the n i are very sensitive to pressures, which is explained by a considerable dispersion of the absolute piezooptic constants.For the light propagating along the X-, Y-, and Z-axes, the dependencies n i (λ, σ) and formula (6) provide the calculation of the changes in α i and R i of the crystal (table 1).
It is established that the pressures up to σ m ∼ 200 bar raise the R i (on the average by ∼ 1E2 • 10 −25 cm 3 ) as well as the α i of the K 2 SO 4 crystal.
Probably, uniaxial pressures change the degree of ordering of the crystal and lead to a slight deformation and rotation of the framework formed by SO 2− 4 -tetrahedra and to displacements of K + ions, which can be seen from the changes in α i and R i .Using the dependencies n i (λ, σ) and formula (5), we also calculated the baric dependencies of the parameters of ultraviolet (λ 0i , B i ) oscillators (table 1).
It has been established that an increase of the n i (∂n/∂σ ∼ = 5 • 10 −6 bar −1 ) under pressure is caused by a decrease of the strength of the effective UV oscillator and by the displacement of the center of the effective UV absorption band to the long wavelength range (∂λ 0 /∂σ ≈ 0.5E2 • 10 −2 Å/bar ≈ 15 • 10 −5 eV/bar).
Indeed, upon differentiation of (5), we arrive at The baric displacement of the effective UV oscillator is very large.It showed that UV oscillator is situated in a high energy spectrum region.
Thus, studying the spectral dependencies of the refractive indices of free K 2 SO 4 crystals mechanically clamped by the stresses along the main crystallophysical directions, we established that the refractive indices n i increase because the applied pressure effects the electronic subsystem of the crystal.
We also calculated the variations in the electron polarizability, refraction and the parameters of the UV oscillators of free and mechanically clamped K 2 SO 4 crystals.It is seen that uniaxial pressures shift the center of the effective absorption band to the visible range of the spectrum, which seems to be associated with a decrease of the band gap.
It has been found that the birefringence is rather sensitive to the uniaxial stresses applied along the principal crystallophysical axes.Those stresses, applied along mutually perpendicular crystallophysical axes, σ i,m , result in the changes ∆n j different in signs and values (i, j, m = 1, 2, 3).