# The Refraction Indices and Brewster Law in Stressed Isotropic Materials and Cubic Crystals

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**Bx**$\xb7$

**x**= 1, where

**B**is dielectric impermeability tensor, for an unstressed optically isotropic transparent material is a sphere [2], since the dielectric impermeability tensor is spherical:

**B**= n

_{o}

^{−2}

**I**,

**T**modifies the dielectric impermeability according to the Maxwell Equation:

**B**(

**T**) = n

_{o}

^{−2}

**I**+ Π[

**T**],

_{o}> 1 is the refraction index. Since the considered materials have a linear elastic behaviour, it makes sense to use the linear relations (2) which, from a mathematical point of view, holds true in the “small stress” regime. Such regime ranges, for brittle materials like crystals and glasses, over tens of MPa close to the ultimate tensile stress before the brittle fracture is reached and applies well to the cases we are treating. In the general case, besides some specific state of stress which leave the stressed crystal still optically isotropic and which were studied in details into [16], the tensor

**B**(

**T**) admits three eigencouples {(B

_{1},

**u**

_{1}), (B

_{2},

**u**

_{2}), (B

_{3},

**u**

_{3})} and it becomes either optically uniaxial or biaxial (The stress which leaves an optically isotropic material still optically isotropic is the spherical (idrostatic) stress of the form

**T**= σ

_{m}

**I**[16]).

**u**

_{1},

**u**

_{2},

**u**

_{3}} of the eigenvectors of

**B**(

**T**), hence it makes sense, for biaxial and uniaxial materials to define the Bertin Surfaces [19,20]:

_{min}and n

_{max}are the minimum and maximum refraction indices and 2$\phi $ is the angle between the optic axes, defined by:

_{1}> B

_{2}> B

_{3}and vice-versa with $\left|{B}_{1}-{B}_{2}\right|<\left|{B}_{1}-{B}_{3}\right|$ see ref [15,19,20].

_{1}= B

_{2}, it is $\mathsf{\rho}$ = 0 and (3) reduces to:

**F**

_{σ}such that:

**B**(

**T**) to a spherical tensor, in order to figure out what happens to the observed interference fringes. This means that for isotropic materials with n

_{max}= n

_{min}:

^{2}= ${\xi}^{2}+{\eta}^{2}$ reduces to:

^{4}− H

^{2}r

^{2}− d

^{2}H

^{2}= 0,

_{max}− n

_{min}$\to $0, thus disappearing progressively from the conoscopic field of observation.

_{max}− n

_{min}$\to $0, in fact the parameters a and b introduced before, diverge with a similar law of (11).

**B**(

**T**) and the principal refraction indices n

_{k}, k = 1, 2, 3 is:

**T**, then within the same hypotheses of (2) we may arrive at the linearized relation:

_{k}with respect to

**T**is not defined for

**T**=

**0**, thus making the linearization impossible. A way to remove this inconvenience was proposed into [32], where precisely such a problem was addressed and completely solved in a general manner: however for who is not accustomed to the very formal algebraic treatment given therein, here we shall recover the same results in a different and more conventional way.

**B**(

**T**) to the principal values σ

_{k}, k = 1, 2, 3 of

**T**, namely:

_{i}− n

_{j}= K

_{B}(σ

_{i}− σ

_{j}), i ≠ j, i,j = 1,2,3,

_{o}and Π; the question in this case is if it is possible to extend, and to which extent, the Equation (14) to other crystallographic symmetry group. The answer was obtained in a definitive manner into [33] and here we shall show how the same results can be easily recovered within our present approach.

**e**

_{1},

**e**

_{2},

**e**

_{3}} with coordinates (x,y,z) which is the reference coincident with the crystallographic axes, and Σ $\equiv $ {

**u**

_{1},

**u**

_{2},

**u**

_{3}} with coordinates (ξ,η,ζ), we already introduced, and which is the reference frame for the Optic Indicatrix and Bertin surfaces.

**e**

_{3}and accordingly we may observe only the effects of the plane stress

**Te**

_{3}=

**0**, since for the stress along the same direction, in absence of applied forces on the surfaces orthogonal to

**e**

_{3}, we may assume zero mean value: therefore they are not detectable by the means of a conoscopic analysis since it takes the mean values on the optic path along the same direction. Accordingly, the components of such a plane stress in the frame S are:

_{1}, σ

_{2}, 0) be the principal stress of Equation (15) and U the frame of the corresponding eigenvectors which is rotated with respect to S by an angle α given by:

## 3. Results

#### 3.1. Cubic Crystals

#### 3.1.1. Classes 23 and m3

**B**(

**T**), for a plane stress is given by (2) with piezo-optic tensor [13,14]

_{0}

^{−}

^{2}= B

_{0}:

**T**.

_{1}> B

_{2}but it is not possible to establish a-priori the complete ordering of the eigenvalues.

_{1}and n

_{2}blows up for

**T**=

**0**.

_{3}(we makes the same in the follow) ∆n = n

_{1}− n

_{2}, from (28):

_{1}and g

_{2}depend on the Π components π

_{13}, π

_{12}, π

_{11}and γ, meanwhile f

_{1}and f

_{2}depend also on the angles α.

#### 3.1.2. Classes $\overline{4}3m$, 432, m3m

#### 3.2. Isotropic Materials

**B**(

**T**) coincide with those of

**T**[32]: indeed:

_{1}− B

_{2}is a linear function of the principal stress difference:

_{21}(57) is proportional to the ∆B (56) in the same form of the Brewster law. Finally, the shear stress and the principal stress difference are valuable.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The frame S is referred to the crystallographic axes of the cubic crystal with axes (a,a,a); the frame Σ corresponds to the principal axes of the Optic Indicatrix. In absence of stress, since the optical indicatrix is a sphere we may assume S = Σ.

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**MDPI and ACS Style**

Rinaldi, D.; Natali, P.P.; Montalto, L.; Davì, F.
The Refraction Indices and Brewster Law in Stressed Isotropic Materials and Cubic Crystals. *Crystals* **2021**, *11*, 1104.
https://doi.org/10.3390/cryst11091104

**AMA Style**

Rinaldi D, Natali PP, Montalto L, Davì F.
The Refraction Indices and Brewster Law in Stressed Isotropic Materials and Cubic Crystals. *Crystals*. 2021; 11(9):1104.
https://doi.org/10.3390/cryst11091104

**Chicago/Turabian Style**

Rinaldi, Daniele, Pier Paolo Natali, Luigi Montalto, and Fabrizio Davì.
2021. "The Refraction Indices and Brewster Law in Stressed Isotropic Materials and Cubic Crystals" *Crystals* 11, no. 9: 1104.
https://doi.org/10.3390/cryst11091104