Optic Axis Rotation and Bertin Surface Deformation in Lead Tungstate (PWO) and Other Tetragonal Crystals by Stress and Misalignment of Crystallographic Cells: A Theoretical Study

Abstract

For tetragonal lead tungstate (PWO) and other tetragonal crystals, we study modifications of the Bertin surfaces induced by either the distortion of crystallographic cells, the applied plane stress, or cell misalignment with respect to the specimen faces. In both cases, the distortions of the Bertin surfaces result in the reshaping of the interference pattern observed by conoscopy. We provide, for different observation directions of the crystals, analytical relations that allow for the evaluation of the optic plane and the optical indicatrix rotation with or without stress. By the means of these relations, interference image reshaping allows us to detect, provided that some conditions hold, the crystallographic axes’ rotation. This work is a theoretical study aiming to evaluate the optic axes and crystallographic cell orientation by means of conoscopic observations.

1. Introduction

Nowadays, the structural characterization of crystals is crucial due to the increased requirements of the applications in which they are involved. Scintillators, in the form of bulky monocrystals, are widely used in a number of radiation detection apparatuses and in large experiments in the field of high-energy physics. With regard to this, studies have confirmed the crucial nature of the structural condition of the crystals in terms of lattice distortion, misalignment, and homogeneity over the crystal volume. In ref. [1], the effect of oriented lead tungstate crystals (PbWO₄, PWO, tetragonal crystal with a Scheelite/Stolzite structure; see Section 3 for more information) on the radiation length was evaluated, revealing a clear reduction in this property in the lead tungstate monocrystal [2,3,4]. The effect of the strong electric field (SF) due to the particular lattice direction led to a significant reduction in the radiation length; this was strictly related to the angle between the impinging radiation beam and the crystal lattice orientation, and the stronger effectiveness was confined to a few mrad. As detailed in [1], a high-energy particle, e.g., $e^{\pm }$, experiences the effect of a strong electric field when its trajectory is parallel or quasi-parallel to one of the lattice planes/axes. This strong field stimulates intense hard photon emission, as well as intense pair production acceleration in electromagnetic shower development. The angular range (that is, the misalignment between the lattice and the particle trajectory) at which this effect is significant is described by

$${\mathsf{\Theta }}_o={\displaystyle \frac{U_o}{m{c}^2}},$$

where $U_o$ is the axial potential well depth, m is the particle rest mass, and c is the speed of light.

Many other examples of the possible effects of the crystal lattice orientation and/or distortion can be found in the literature; however, there is a need to develop reliable methods for the characterization and monitoring of crystals’ structural conditions, which would complement the well-assessed analytical techniques but allow for more flexibility and a non-invasive approach.

Photoelasticity is a fast and reliable technique for the study and analysis of transparent media that, in recent decades, has been successfully used to investigate amorphous materials and crystals [5,6,7,8,9,10,11,12,13]. Thus far, it has proven useful mainly for the evaluation of the internal stress magnitude and its spatial distribution, especially in optically isotropic materials.

Recently, studies on the photoelastic constants and Brewster’s law have been extended to anisotropic crystals [14,15,16,17,18,19]; the main purpose of such theoretical analysis is the assessment of the internal stress via the elaboration of the interference fringe pattern obtained by conoscopy and the evaluation of photoelastic constants [11,12,13]. The theoretical elaboration behind this analysis is based on classical optics [20,21,22,23,24,25].

Cassini-like curves, which are the typical shapes of these interference patterns, are generated by the intersection of the three-dimensional Bertin surfaces [18,19,20,21,26] with the observation surface, generating fourth-order Cassini-like curves. From these curves, through conoscopic techniques and a dedicated algorithm, we can measure the desired photoelastic parameters [11,12,13,27]. It is worth noting that the Bertin surfaces are the iso-delay surfaces linked to the crystal’s optical indicatrix (OI) and therefore share the same frame, i.e., the principal frame of the stress-dependent dielectric impermeability symmetric tensor $\mathbf{B}=\mathbf{B}\left(\mathbf{T}\right)$ (here, $\mathbf{T}$ is the symmetric Cauchy stress tensor). Moreover, the eigenvalues of $\mathbf{B}$ determine the principal refraction indices [18,19,22,23,24]. Accordingly, the study of the tensor $\mathbf{B}$ highlights the properties of the Bertin surfaces and Cassini-like curves, and evaluating the relative changes in the photoelastic parameters allows for the description of the lattice cell distortion due to the applied stress and induced by other causes, such as defects, crystal growth inhomogeneity, and so on. In stress-free crystals, the eigenvector’s direction depends on the cell’s symmetry [23,24]; the stress deforms the lattice symmetry and, as a consequence, the material properties. This is an occurrence that is well known in XRD studies and magnetic or mechanical investigations of stressed materials, e.g., [28,29,30,31,32]. With regard to this, XRD measurements allow structural analysis and axis orientation measurements at the crystal surface; by means of a rocking curve (RC), they enable the detection of surface axis misorientation with respect to the crystal lattice. The RC allows for the detection of the angle between the direction of the crystalline plane, in the vicinity of the crystal surface, and the crystal sample surface. Moreover, the mosaicity of the crystal can be evaluated by estimating the spread of the RC and the lattice orientation at several locations across the sample. XRD analysis is a powerful and well-assessed technique, and it is very sensitive to misalignments of about a few mrad [1,33]. However, this technique has some limitations and constraints, which are related to the method itself and to the implementation device. Depending on the characteristics of the studied crystal, information comes from a small portion of the sample surface; typically, each XRD measurement carries information on a plane angle, whereas the crystallographic planes lie in a three-dimensional space. Therefore, multiple measurements are needed. To identify the direction of the plane in space, one has to obtain RCs in different directions, which is a further source of complexity and expense.

In recent decades, photoelastic techniques have been improved, extending their application from the detection of stress magnitudes for quality control [34,35,36] to the detection of crystallographic cell distortion and OI rotation. Photoelastic techniques are advantageous due to their easy implementation and non-invasiveness, which is important whenever a large number of samples must be quality-controlled. Photoelastic measurements can be taken both in planes normal to the optic axis (homeotropic alignment) and in planes parallel to the optic axis (plane alignment—for instance, plane $a-c$) [37]. It is important to note that photoelasticity gives mean information about the volume probed through the sample thickness [38], both for stress detection and axis rotation. In certain conditions, three-dimensional information can be obtained in a single shot.

Besides external applied loads, the stress in a crystal can be induced by various causes; for instance, the cells’ orientation can differ within the sample as a function of the position, since the crystal growth process generates inhomogeneous defects, dislocations, vacancies, etc. This stress warps and rotates the Bertin surface, as well as the OI and the optic axes, by an amount that is limited, and it can be modeled by classical linear photelasticity [39] since crystals are brittle materials with low ultimate tensile strength. Moreover, when we consider a crystal sample as cut from the boule, the crystallographic cell axes’ orientation does not match the facets of the sample, which is termed misalignment or miscut. From a theoretical point of view, we can distinguish these two contributions, which may have different magnitudes.

The use of crystals in scientific and technological applications requires detailed knowledge of crystal properties such as crystallographic cell distortion and the OI orientation. This type of study is directly associated with the production of increasingly high-performance materials. For instance, new calorimeters for high-energy physics are based on more highly performant scintillator crystals—an example is the OREO project, which proposes a small-angle calorimeter (SAC) based on the exploitation of perfectly oriented PWO crystals as a forward detection matrix. In fact, the alignment of the lattices along the whole crystal decreases the radiation length. Accordingly, highly homogeneous PWO crystals with c-axis misalignment limited to a few mrad (${10}^{-3}$ rad) are needed [1], and this is a new challenge in both the crystal growth and crystal analysis domains.

In this paper, we deal with the rotation of the principal directions of the tensor $\mathbf{B}$ with respect to macroscopic crystal sample facets. We investigate in detail the case of tetragonal crystals, with focused attention to PWO crystals. First, we consider the rotation of the principal frame with respect to the physical sample, followed by the rotation induced by internal stress: we evaluate the OI axes’ (and Bertin surfaces’) rotation in the case of small (linear) stress. Within the linear model, we estimate the application limits of the Sirotin approximation [23]; indeed, his approximation allows us to find analytical solutions to the generally complex eigencouple problem for $\mathbf{B}$. Here, we show that such an approximation is suitable when applied to PWO and other tetragonal crystals. We finally discuss the obtained results by comparing the two cases of domain rotation by misalignment and by stress.

2. Background

2.1. A Matter of Frames

When we deal with a crystal specimen for experimental purposes, we can identify different frames related to different physical and geometric properties of the specimen.

We call the first one the specimen physical frame $\Sigma \equiv \{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$, where ${\mathbf{e}}_i$ denotes orthonormal vectors whose coordinates are $(x_1,x_2,x_3)$, and they have one or more coordinate directions coincident with the edges of the specimen. This frame is important, from an experimental point of view, because most experiments are related to specimen edges or facets.

The second frame that we deal with is the crystallographic frame spanned by the crystallographic directions $[1,0,0]$, $[0,1,0]$ and $[0,0,1]$ of the crystallographic group. Since, in this paper, we deal solely with crystals of the tetragonal group, such a frame ${\Sigma }_c\equiv \{{\mathbf{c}}_1,{\mathbf{c}}_2,{\mathbf{c}}_3\}$ is still an orthonormal frame, whose coordinates we denote by $(z_1,z_2,z_3)$. Hence, for an unstressed tetragonal crystal, these directions are the principal directions for the dielectric impermeability symmetric tensor ${\mathbf{B}}_o$ and for the associated optical indicatrix and Bertin’s surfaces [19,20,21,22].

When the crystal is stressed, the dielectric impermeability tensor changes into a—still symmetric—tensor $\mathbf{B}\left(\mathbf{T}\right)$, which is given by the well-known Maxwell relation

$$\mathbf{B}\left(\mathbf{T}\right)={\mathbf{B}}_o+\mathsf{\Pi }\left[\mathbf{T}\right],$$

(1)

where $\mathbf{T}$ is the symmetric stress tensor, whose matrix in the frame ${\Sigma }_c$ is

$$\mathbf{T}=\left[\begin{array}{ccc}{\sigma }_{11}& {\sigma }_{12}& {\sigma }_{13}\\ ·& {\sigma }_{22}& {\sigma }_{23}\\ ·& ·& {\sigma }_{33}\end{array}\right],$$

(2)

and $\mathsf{\Pi }$ is the fourth-order piezo-optic tensor, whose components are denoted by ${\pi }_{AB}$, $A,B=1,\dots ,6$ according to the Voigt notation [22,24].

Therefore, we introduce a third frame ${\Sigma }^{\prime }\equiv \{{\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{u}}_3\}$, which is the orthonormal frame of the principal directions of $\mathbf{B}\left(\mathbf{T}\right)$, whose coordinates we denote by $(y_1,y_2,y_3)$. In this frame, we represent the optical indicatrix and Bertin’s surfaces for the stressed crystal (Figure 1).

For a hypothetical stress-free crystal, we may assume that the specimen is cut along the crystallographic direction (as in, e.g., [24]) and, therefore, the specimen and crystallographic frames coincide:

$$\Sigma \equiv {\Sigma }_c,x_i=z_i,i=1,2,3.$$

(3)

Unfortunately, this is not typically the case, since, even with the best devices, a small but still noticeable misalignment may remain. Therefore, the base vectors of the two frames are related by a rotation ${\mathbf{Q}}_1$:

$${\mathbf{c}}_i={\mathbf{Q}}_1{\mathbf{e}}_i,i=1,2,3.$$

(4)

Moreover, we cannot disregard the possibility that the crystal is stressed—for instance, by residual stress given by the growing, cutting and polishing processes. This means that, since there exists a rotation ${\mathbf{Q}}_2$ such that

$${\mathbf{u}}_i={\mathbf{Q}}_2{\mathbf{c}}_i,i=1,2,3,$$

(5)

The optical indicatrix and the Bertin’s surface that we observe are defined in the principal frame ${\Sigma }^{\prime }$, whose base vectors are rotated with respect to the frame $\Sigma$ (the frame that is most likely related to the experimental observation) by means of a rotation $\mathbf{Q}$ such that

$${\mathbf{u}}_i=\mathbf{Q}{\mathbf{e}}_i,i=1,2,3,\mathbf{Q}={\mathbf{Q}}_2{\mathbf{Q}}_1.$$

(6)

With this in mind, the main question that we try to answer in this paper is as follows:

If we are able (in some way) to measure $\mathbf{Q}$, how can we evaluate the contribution ${\mathbf{Q}}_1$, which is related to crystal misalignment, and the contribution ${\mathbf{Q}}_2$, which depends on the (possibly residual) stress?

Remark 1.

In an unstressed and aligned sample of a tetragonal crystal, we have $\Sigma \equiv {\Sigma }_c\equiv {\Sigma }^{\prime }$, with the tetragonal c-axis directed as ${\mathbf{e}}_3={\mathbf{c}}_3={\mathbf{u}}_3$.

Remark 2.

In previous papers on the same subject [18,19], we described these frames with different symbology, which we recall here for ease of reference:

• Σ: denoted by $\mathcal{S}\equiv \{\mathbf{i},\mathbf{j},\mathbf{k}\}$, coordinates $(x,y,z)$;
• ${\Sigma }_c$: denoted by $\mathcal{U}\equiv \{\mathbf{a},\mathbf{b},\mathbf{c}\}$, coordinates $[a,b,c]$;
• ${\Sigma }^{\prime }$: denoted by $\Sigma \equiv \{{\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{u}}_3\}$, coordinates $(\xi ,\eta ,\zeta )$.

2.2. Cassini-like Curves

If we observe the sample along a generic direction ${\mathbf{k}}^{★}$, then the Cassini-like curves are given by the intersection between the Bertin surface and the plane (Figure 2):

$$\pi \equiv \{\mathbf{x}\mid {\mathbf{x}}_o+\mathbf{x}·{\mathbf{k}}^{★}=0\}.$$

(7)

We assume that the plane of observation is parallel to one of the sample factes, namely ${\mathbf{x}}_o=d{\mathbf{e}}_3$ and ${\mathbf{k}}^{★}={\mathbf{e}}_3$, to obtain the equation for the plane $\pi$ of observation in the frame $\Sigma$:

$$x_3-d=0.$$

(8)

Let

$$\begin{array}{cc}\hfill & y_1^4{cos}^4\mathsf{\Phi }+y_2^4+y_3^4{sin}^4\mathsf{\Phi }+2{cos}^2\mathsf{\Phi }{y}_1^2{y}_2^2+2{sin}^2\mathsf{\Phi }{y}_2^2{y}_3^2-2{sin}^2\mathsf{\Phi }{cos}^2\mathsf{\Phi }{y}_1^2{y}_3^2\hfill \\ \hfill & -{\left(NH\right)}^2(y_1^2+y_2^2+y_3^2)=0,\hfill \end{array}$$

(9)

be the Bertin surface for a biaxial crystal surface $F(y_1,y_2,y_3)=0$ in the frame ${\Sigma }^{\prime }$, where the optic angle $\mathsf{\Phi }$ (defined as half the angle between the optic axes) is related to the eigenvalues of $\mathbf{B}$—for instance, with eigenvalues in the order $\{B_1\ge B_2\ge B_3\}$—by

$${cos}^2\mathsf{\Phi }={\displaystyle \frac{B_1-B_2}{B_1-B_3}},{sin}^2\mathsf{\Phi }={\displaystyle \frac{B_2-B_3}{B_1-B_3}},$$

(10)

and

$$H={\displaystyle \frac{\lambda }{n_e-n_o}}$$

(11)

where $\lambda$ is the wavelength of the laser used in the conoscopic device and with $N>0,N\in \mathbb{N}$ as the fringe order.

We remark that, to obtain the results for an uniaxial crystal, we simply set $\mathsf{\Phi }=0$ to get

$$y_1^4+y_2^4+2y_1^2{y}_2^2-{\left(NH\right)}^2(y_1^2+y_2^2+y_3^2)=0.$$

(12)

As was pointed out in [26], there is a wide range of stresses that leave a uniaxial crystal still uniaxial. We remark that Equations (9)–(11) represent the surfaces of equal delay from classical optics, whose sections are the Cassini-Like curves that model the interference fringes—see, e.g., [18,20,21,26].

For $\mathbf{Q}$, the rotation occurs between $\Sigma$ and ${\Sigma }^{\prime }$; then, if $Q_{ij}$ denotes the components of $\mathbf{Q}$ in $\Sigma$, we get

$$\begin{array}{cc}\hfill y_1& =Q_{1j}{x}_j=Q_{11}{x}_1+Q_{12}{x}_2+Q_{13}{x}_3,\hfill \\ \hfill y_2& =Q_{2j}{x}_j=Q_{21}{x}_1+Q_{22}{x}_2+Q_{23}{x}_3,\hfill \\ \hfill y_3& =Q_{3j}{x}_j=Q_{31}{x}_1+Q_{32}{x}_2+Q_{33}{x}_3,\hfill \end{array}$$

(13)

where, regardless of the representation that we choose (Euler angles, Rodriguez formula, direction cosine), the nine components $Q_{ij}$ depend on three parameters.

By using (8) in (13), we have

$$y_j(x_1,x_2)=Q_{j\alpha }{x}_{\alpha }+Q_{j3}d,\alpha =1,2,j=1,2,3.$$

(14)

Then, by using (14) in (9), we arrive at an equation for the fourth-order curves $\gamma$ in the plane $\pi$ in terms of the two coordinates $(x_1,x_2)$ in the plane, whose general form is

$$\begin{array}{cc}\hfill A_1{x}_1^4& +A_2{x}_2^4+2A_3{x}_1^2{x}_2^2+2A_4{x}_1^3{x}_2+2A_5{x}_1{x}_2^3\hfill \\ \hfill & +B_1{x}_1^3+B_2{x}_2^3+B_3{x}_1^2{x}_2+B_4{x}_1{x}_2^3+C_1{x}_1^2+C_2{x}_2^2+2C_3{x}_1{x}_2\hfill \\ \hfill & +D_1{x}_1+D_2{x}_2+E=0,\hfill \end{array}$$

(15)

where, of the fifteen parameters, those denoted by $A_i$ depend on $(\mathsf{\Phi },Q_{ij})$, and those denoted by $B_i$ depend on $(\mathsf{\Phi },Q_{ij},d)$, whereas $C_i,D_i,E$ depend on $(Q_{ij},\mathsf{\Phi },{\left(NH\right)}^2,d)$.

Infinitesimal Rotation and Small Deviation

We assume that either the crystal is unstressed or that the effects of the stress are negligible, and we consider the case of an infinitesimal rotation. Let $(\theta ,\omega )$ be the angle and axis of rotation; then, from the Rodriguez formula with $sin\theta \approx \theta$, we get

$$\mathbf{Q}=\mathbf{I}+\theta \mathbf{W}+{o\left(\right|\theta |}^2),\mathbf{W}=\left[\begin{array}{ccc}0& W_{12}& W_{13}\\ -W_{12}& 0& W_{23}\\ -W_{13}& -W_{23}& 0\end{array}\right]=\left[\begin{array}{ccc}0& -{\omega }_3& {\omega }_2\\ {\omega }_3& 0& -{\omega }_1\\ -{\omega }_2& {\omega }_1& 0\end{array}\right].$$

(16)

Accordingly, in (13), we have

$$Q_{11}=Q_{22}=Q_{33}=1,Q_{21}=-Q_{12},Q_{31}=-Q_{13},Q_{32}=-Q_{23},$$

(17)

with

$$\theta =\sqrt{Q_{13}^2+Q_{23}^2+Q_{12}^2},W_{ij}={\theta }^{-1}{Q}_{ij},i\ne j;$$

(18)

the explicit expression for the Cassini-like curves $\gamma$ can then be obtained by means of (15).

As a further approximation step, we may assume that the rotation axis is close to the direction $x_3$ in the sense that

$${\omega }_1\approx ϵ,{\omega }_2\approx ϵ,{\omega }_3\approx 1,ϵ<<1;$$

(19)

then, as a consequence of these approximations, (15) is, within higher-order terms in $ϵ$, a function of $\theta$ and $ϵ$ that contains $\theta$ up to the fourth order.

It is important to consider that we implicitly assume that $\theta$ is a measurable small quantity, whereas ${\omega }_{1,2}$ are not. However, if we assume that

$$ϵ=sup\{\theta ,{\omega }_1,{\omega }_2\},$$

(20)

then (15) reduces to within $o\left({\theta }^2\right)$, i.e., to a simpler form that contains both $\theta$ and $ϵ$ up to the first order.

3. A Roadmap for Stress and Misalignment Effect Measurement in Lead Tungstate and Other Tetragonal Crystals

3.1. Stressed Lead Tungstate Crystals

PWO (lead tungstate, PbWO₄) is a body-centered tetragonal crystal with point group $4/m$ with $a=b=0.54619$ nm, $c=1.2049$ nm (ICDD card n. 19-708 and [40]), where c is the uniaxial optic axis (the values of both a and c can vary slightly depending on the production technique).

From an optical point of view, PWO crystals are uniaxial negative, with the extraordinary and ordinary refractive indices being, respectively, $n_e=2.1634$ and $n_o=2.234$, for visible radiation with $\lambda =632.8$ nm [41], which, by (11), give $H^2=80.39(\mathsf{\mu m}{)}^2$ in (9).

The tensor ${\mathbf{B}}_o$ for a crystal of the tetragonal group is represented, in the frame ${\Sigma }_c$, by the matrix

$${\mathbf{B}}_o=\left[\begin{array}{ccc}{S}_1& 0& 0\\ ·& S_1& 0\\ ·& ·& S_3\end{array}\right],$$

(21)

with $S_1=S_2=n_o^{-2}=0.200$ and $S_3=n_e^{-2}=0.213$, whereas the tabular representation for the piezo-optic tensor $\mathsf{\Pi }$ in the frame ${\Sigma }_c$ is, for the low-symmetry classes $4/m$, $\overline{4}$ and $4m$, the following [24]:

$$\mathsf{\Pi }=\left[\begin{array}{cccccc}{\pi }_{11}& {\pi }_{12}& {\pi }_{13}& 0& 0& {\pi }_{16}\\ {\pi }_{12}& {\pi }_{11}& {\pi }_{13}& 0& 0& -{\pi }_{16}\\ {\pi }_{31}& {\pi }_{31}& {\pi }_{33}& 0& 0& 0\\ 0& 0& 0& {\pi }_{44}& {\pi }_{45}& 0\\ 0& 0& 0& {\pi }_{45}& {\pi }_{44}& 0\\ {\pi }_{61}& -{\pi }_{61}& 0& 0& 0& {\pi }_{66}\end{array}\right].$$

(22)

The magnitude of the components of $\mathsf{\Pi }$ is about ${10}^{-12}$ (Pa)⁻¹ [17,42], whereas the ultimate tensile stress is $18.5÷31.7$ MPa [39]. Accordingly, we may assume that

$${\left(\mathsf{\Pi }\left[\mathbf{T}\right]\right)}_{ij}\approx {10}^{-5}÷{10}^{-6}.$$

(23)

Remark 3.

In this paper, we deal with PWO crystals whose class is $4/m$; however, the analytical results hold true also for the two classes $\overline{4}$ and $4m$, which share the same piezo-optic tensor (22). Furthermore, these results still hold for the other high-symmetry classes $4mm,\overline{4}2m,422$ and $4/mmm$ provided that we set

$${\pi }_{16}={\pi }_{61}={\pi }_{45}=0.$$

(24)

in (22).

Remark 4.

In this subsection, we assume that the crystal and the specimen are not misaligned, so that $\Sigma \equiv {\Sigma }_c$ and ${\mathbf{Q}}_1=\mathbf{I}$, ${\mathbf{Q}}_2=\mathbf{Q}$ in (4) and (6). Hence, our analysis is focused on stress estimation. We deal with two cases, namely when the conoscopic observation is either along the direction of the c-axis or along one of the a-axes. To summarize the theoretical content, a simulation and numerical example are given in Section 3.1.2.

3.1.1. Conoscopic Observations in the Direction of the c-Axis (Homeotropic Alignment)

In this case, we assume the following:

• The direction ${\mathbf{e}}_3\equiv {\mathbf{c}}_3$ coincides with the $c-$ axis.
• The mean stresses along the c-axis direction of the crystal specimen are zero, i.e.,

  $${\overline{\sigma }}_{i3}=0,i=1,2,3.$$

  (25)
• The stress is accordingly a plane stress in the plane orthogonal to the c-axis:

  $$\mathbf{T}=\left[\begin{array}{ccc}{\sigma }_{11}& {\sigma }_{12}& 0\\ ·& {\sigma }_{22}& 0\\ ·& ·& 0\end{array}\right].$$

  (26)
• The intersections of the unstressed Bertin surfaces with the plane $x_3=d$ are the circumferences parameterized on N,

  $$x_1^2+x_2^2=R^2,R^2\left(N\right)={\displaystyle \frac{{\left(NH\right)}^2}{2}}+\sqrt{{\left({\displaystyle \frac{{\left(NH\right)}^2}{2}}\right)}^2+{\left(NHd\right)}^2},$$

  (27)

  and hence the conoscopic observations show a pattern of circular fringes.

An applied stress $\mathbf{T}$, by the means of relation (1), warps these circumferences into closed Cassini-Like curves, the sections of the stressed (biaxial) Bertin surfaces (9) with planes $y_3=d$,

$$y_1^4{cos}^4\mathsf{\Phi }+y_2^4+2{cos}^2\mathsf{\Phi }{y}_1^2{y}_2^2+2d^2{sin}^2\mathsf{\Phi }(y_2^2-{cos}^2\mathsf{\Phi }{y}_1^2)-{\left(NH\right)}^2(y_1^2+y_2^2)+D=0,$$

(28)

where

$$D=d^2(d^2{sin}^4\mathsf{\Phi }-{\left(NH\right)}^2),$$

(29)

and rotates the frame $\Sigma \equiv {\Sigma }_c$ into a frame ${\Sigma }^{\prime }$ by rotation $\mathbf{Q}$.

As shown in [18], in the presence of the plane stress (26), when the piezo-optic tensor is given by (22), $\mathbf{Q}$ is a rotation of an angle $\gamma$ about ${\mathbf{e}}_3$, the biaxial optic axis lying in the optic plane spanned by ${\mathbf{u}}_3\equiv {\mathbf{e}}_3$ and ${\mathbf{u}}_1=cos\gamma {\mathbf{e}}_1+sin\gamma {\mathbf{e}}_2$. We remark that such a rotation is the mean effect of the applied stress on the probed volume in conoscopic observations. Since the Cassini-like curve (28) admits two axes of symmetry whose lengths are $2b\ge 2a$ (Figure 3), it becomes possible to define an ellipticity ratio $C=C({\sigma }_{11},{\sigma }_{22},{\sigma }_{12})$ [18],

$$C={\displaystyle \frac{b}{a}}-1=C_o\left(\left(\sqrt{{({\pi }_{11}-{\pi }_{12})}^2+4{\pi }_{16}^2}\right)({\sigma }_{11}-{\sigma }_{22})+\left(2\sqrt{{\pi }_{66}^2+{\pi }_{16}^2}\right){\sigma }_{12}\right),$$

(30)

where

$$C_o={\displaystyle \frac{1}{|S_1-S_3|}}(1+{\displaystyle \frac{2K^2}{1+\sqrt{1+4K^2}}}),K^2={\displaystyle \frac{d^2}{H^2}};$$

(31)

provided that we are able to measure the rotation $\gamma$ (Figure 3), since, in [18], we also obtained

$$tan\gamma ={\displaystyle \frac{-({\pi }_{11}-{\pi }_{12})({\sigma }_{11}-{\sigma }_{22})\pm A({\sigma }_{11},{\sigma }_{22},{\sigma }_{12})}{2\left(({\sigma }_{11}-{\sigma }_{22}){\pi }_{16}+{\sigma }_{12}{\pi }_{66}\right)}},$$

(32)

where

$$A({\sigma }_{11},{\sigma }_{22},{\sigma }_{12})=\sqrt{4{\left(({\sigma }_{11}-{\sigma }_{22}){\pi }_{61}+{\sigma }_{12}{\pi }_{66}\right)}^2+{\left(({\sigma }_{11}-{\sigma }_{22})({\pi }_{11}-{\pi }_{12})+2{\sigma }_{12}{\pi }_{16}\right)}^2}.$$

(33)

Then, in [17,18], an explicit evaluation of the stress by means of the ellipticity ratio C and the angle $\gamma$ was obtained. Indeed, it suffices to know ${\pi }_{16}$, ${\pi }_{61}$, ${\pi }_{66}$ and the difference ${\pi }_{11}-{\pi }_{12}$ to obtain from (31) and (32) the values of ${\sigma }_{12}$ and the difference ${\sigma }_{11}-{\sigma }_{22}$.

We note that, in the absence of stress, a rotation of the crystallographic frame is undetectable, since the CL curves are circles. For a detailed analysis of this and other cases, see, e.g., [18].

These results, which are valid for PWO [42] and for all lower-symmetry tetragonal classes $4/m,\overline{4}$ and $4m$, still hold for the higher-symmetry classes $4mm,\overline{4}2m,422$ and $4/mmm$ provided that we set ${\pi }_{16}={\pi }_{61}={\pi }_{45}=0$. We note that, as shown in [17], we may have a wide optic plane rotation with a magnitude of about $\pi /2$ rad due to the applied stress. In Figure 3, the experimental effects on the Cassini-like curve of the uniaxial load are represented; the theoretical and experimental processes are explained in [17].

3.1.2. Conoscopic Observations in the Direction Orthogonal to the a-c-Axes’ Plane Alignment

Here, we assume the following:

• The observation direction is ${\mathbf{e}}_1\equiv {\mathbf{c}}_1$, which we force to coincide with one of the a-axes.
• The mean stresses along the a-axis direction of the crystal specimen are zero, i.e.,

  $${\overline{\sigma }}_{i1}=0,i=1,2,3.$$

  (34)
• The stress is accordingly a plane stress in the plane orthogonal to one of the a-axes:

  $$\mathbf{T}=\left[\begin{array}{ccc}0& 0& 0\\ ·& {\sigma }_{22}& {\sigma }_{23}\\ ·& ·& {\sigma }_{33}\end{array}\right].$$

  (35)
• The intersections of the unstressed Bertin surfaces with the plane $x_1=d$ are

  $${(d^2+x_2^2)}^2-{\left(NH\right)}^2(d^2+x_2^2+x_3^2)=0.$$

  (36)

Conoscopic observations in the direction normal to the crystallographic plane $a-c$ require further calculation. We have already solved the problem of stress detection in [14,15]; here, we simply recall some of the fundamental points.

When we consider the plane stress (35) with the piezo-optic tensor (22), then, from (1), we have

$$\mathbf{B}\left[\mathbf{T}\right]=\left[\begin{array}{ccc}{S}_1+{\sigma }_{22}{\pi }_{12}+{\sigma }_{33}{\pi }_{13}& 0& -{\sigma }_{23}{\pi }_{45}\\ ·& S_1+{\sigma }_{22}{\pi }_{11}+{\sigma }_{33}{\pi }_{13}& {\sigma }_{23}{\pi }_{44}\\ ·& ·& S_3+{\sigma }_{22}{\pi }_{31}+{\sigma }_{33}{\pi }_{33}\end{array}\right]$$

(37)

and, by means of the Sirotin approximation [23], we obtain the following expressions for the eigenvalues of $\mathbf{B}\left(\mathbf{T}\right)$:

$$\begin{array}{cc}\hfill B_1& =S_1+{\pi }_{12}{\sigma }_{22}+{\pi }_{13}{\sigma }_{33},\hfill \\ \hfill B_2& =S_1+{\pi }_{11}{\sigma }_{22}+{\pi }_{13}{\sigma }_{33},\hfill \\ \hfill B_3& =S_3+{\pi }_{31}{\sigma }_{22}+{\pi }_{33}{\sigma }_{33},\hfill \end{array}$$

(38)

whose associated eigenvectors, still under the Sirotin approximation [23], are given by means of (16):

$$\begin{array}{cc}\hfill {\mathbf{u}}_1& ={\mathbf{e}}_1-W_{31}{\mathbf{e}}_3,W_{31}=-{\displaystyle \frac{{\sigma }_{23}{\pi }_{45}}{S_3-S_1}},\hfill \\ \hfill {\mathbf{u}}_2& ={\mathbf{e}}_2-W_{32}{\mathbf{e}}_3,W_{32}={\displaystyle \frac{{\sigma }_{23}{\pi }_{44}}{S_3-S_1}},\hfill \\ \hfill {\mathbf{u}}_3& =W_{31}{\mathbf{e}}_1+W_{32}{\mathbf{e}}_2+{\mathbf{e}}_3.\hfill \end{array}$$

(39)

Remark 5.

The Sirotin approximation, introduced in Section 20 of [23] and also discussed in detail in [43], is based on the following assumptions:

• The eigenvalues $(S_1,S_2,S_3)$ of ${\mathbf{B}}_o$ are greater than the components of $\mathsf{\Pi }\left[\mathbf{T}\right]$, namely

  $$inf\{S_1,S_2,S_3\}>>sup\{{\left(\mathsf{\Pi }\left[\mathbf{T}\right]\right)}_{ij},i,j=1,2,3\}.$$

  (40)
  In ref. [43], we show that such an assumption is equivalent to assuming that the rotation ${\mathbf{Q}}_2$ can be represented as in (16).
• The differences between the eigenvalues of ${\mathbf{B}}_o$ are greater than the components of $\mathsf{\Pi }\left[\mathbf{T}\right]$:

  $$\underset{h,k=1,2,3,h\ne k}{inf}\left\{\right|S_k-S_h\left|\right\}>>sup\{{\left(\mathsf{\Pi }\left[\mathbf{T}\right]\right)}_{ij},i,j=1,2,3\}.$$

  (41)

Provided that hypotheses (40) and (40) hold true, the eigencouples of $\mathbf{B}\left(\mathbf{T}\right)$ admit the representation (38), (39).

We further remark that Sirotin’s approximation introduces a further step of approximation with respect to the approximations described in Section Infinitesimal Rotation and Small Deviation, as was discussed in [43].

For PWO, since (23) holds, then we have (cf. [42])

$$\begin{array}{c}\hfill S_1=2·{10}^{-1}>>sup\{{\left(\mathsf{\Pi }\left[\mathbf{T}\right]\right)}_{ij},i,j=1,2,3\}\approx {10}^{-5}÷{10}^{-6},\\ \hfill S_3-S_1=1,3·{10}^{-2}>>sup\{{\left(\mathsf{\Pi }\left[\mathbf{T}\right]\right)}_{ij},i,j=1,2,3\}\approx {10}^{-5}÷{10}^{-6},\end{array}$$

(42)

and both of Sirotin’s hypotheses (40) and (41) are verified. Moreover, since

$${\mathbf{u}}_i·{\mathbf{u}}_j=\left\{\begin{array}{c}1+W_{i3}^2,i=j=1,2,\hfill \\ 1+W_{13}^2+W_{23}^2,i=j=3,\hfill \\ -W_{13}{W}_{23},i=1,j=2,\hfill \\ 0otherwise\hfill \end{array}\right.$$

(43)

then, by (42)₂ and within higher-order terms in Sirotin’s approximation, the principal frame (39) is an orthonormal one. For PWO, since we have [42] $W_{ij}\approx {10}^{-3}÷{10}^{-4}$, such an approximation is fully justified.

Remark 6.

In the case of higher-symmetry tetragonal classes $4mm,42m,422$ and $4/mmm$, since ${\pi }_{45}=0$, (39) reduces to a rotation around ${\mathbf{e}}_1$ of amplitude $\theta =W_{32}$. The results obtained here can be applied also to other crystals that share the same tetragonal scheelite/stolzite structure $AB{O}_4$, e.g., ${\mathit{CaWO}}_4$, ${\mathit{CaMoO}}_4$ and ${\mathit{BaWO}}_4$ crystals [42].

In the principal frame ${\Sigma }^{\prime }$ given by (39), the correspondence between the eigenvectors ${\mathbf{u}}_k$ that identify the new axes $y_k$ depends on the relative ordering of the eigenvalues $B_k$, $k=1,2,3$ [21]. However, provided that (42)₁ holds, it is reasonable to assume that, if $S_3$ is the greater eigenvalue of ${\mathbf{B}}_o$, then $B_3$ given by (38)₃ remains the greater eigenvalue of $\mathbf{B}\left(\mathbf{T}\right)$. Accordingly, we are left with only two possible cases, either $B_3>B_2>B_1$ or $B_3>B_1>B_2$ [14,15].

In the first case, the optic plane lies in the plane $(y_1,y_3)$; moreover, since, in the Sirotin’s approximation, the angle $\theta$ is given by

$$\theta =\sqrt{W_{31}^2+W_{32}^2},$$

(44)

then, by (39) and (42)₂, it yields $\theta \approx ·{10}^{-3}÷{10}^{-4}$ rad. Accordingly, the frames $\Sigma$ and ${\Sigma }^{\prime }$ are very close in PWO, and, as was evidenced in [14,15], their relative rotations can be neglected when calculating the internal stress. Hence, the optic plane can be assumed to lie in the plane $(x_1,x_3)$.

We note that the whole approximation procedure relies on the fact that the rotation remains an orthogonal tensor, i.e., ${\mathbf{Q}}^{-1}={\mathbf{Q}}^T$, also in the linear approximation (16), but within higher-order terms in $\theta$. Indeed, via the Neumann expansion of the inverse of the identity tensor plus a small term [44], we get

$${(\mathbf{I}+\theta \mathbf{W})}^{-1}=\mathbf{I}-\theta \mathbf{W}+{\theta }^2{\mathbf{W}}^2+\dots ={(\mathbf{I}+\theta \mathbf{W})}^T+o\left({\theta }^2\right),$$

(45)

and, accordingly, as we have already shown elsewhere [43], the Sirotin approximation holds for tetragonal crystals.

Since, by (16) and Sirotin’s approximation,

$$\begin{array}{cc}\hfill y_1& =x_1-W_{31}{x}_3,\hfill \\ \hfill y_2& =x_2-W_{32}{x}_3,\hfill \\ \hfill y_3& =x_3+W_{31}{x}_1+W_{32}{x}_2,\hfill \end{array}$$

(46)

then the intersection between the Bertin’s surfaces (9) in the frame ${\Sigma }^{\prime }$ and the observation plane $x_1=d$ is given by a relation that is the analog of (15) when Sirotin’s approximation holds and with $x_3$ in place of $x_1$:

$$\begin{array}{cc}\hfill & {(d-W_{31}{x}_3)}^4{cos}^4\mathsf{\Phi }+{(x_2-W_{32}{x}_3)}^4+{(x_3+W_{31}d+W_{32}{x}_2)}^4{sin}^4\mathsf{\Phi }\hfill \\ \hfill & +2{cos}^2\mathsf{\Phi }{(d-W_{31}{x}_3)}^2{(x_2-W_{32}{x}_3)}^2+2{sin}^2\mathsf{\Phi }{(x_2-W_{32}{x}_3)}^2{(x_3+W_{31}d+W_{32}{x}_2)}^2\hfill \\ \hfill & -2{sin}^2\mathsf{\Phi }{cos}^2\mathsf{\Phi }{(d-W_{31}{x}_3)}^2{(x_3+W_{31}d+W_{32}{x}_2)}^2\hfill \\ \hfill & -{\left(NH\right)}^2({(d-W_{31}{x}_3)}^2+{(x_2-W_{32}{x}_3)}^2+{(x_3+W_{31}d+W_{32}{x}_2)}^2)=0.\hfill \end{array}$$

(47)

The curves $F(x_2,x_3,N)=0$ described by Equation (47) depend on the choice of the set of parameters $\mathsf{\Phi },d,W_{31},W_{32}$ and $\mathbf{T}$. In Figure 4, we plot these curves (defined on the frame ${\Sigma }^{\prime }$) superimposed on the curves $F(y_2(x_2,x_3),y_3(x_2,x_3),N)=0$ for the crystal in the frame $\Sigma$. The superimposition of the two frames allows us to compare the small displacement between the two curves, caused by the same stress but observed in different planes (the observation plane $\pi$ and the ideal plane normal to $y_1$).

Remark 7.

The calculation of the R index, as given in [15], does not change; however, it is possible to reduce the measurement error—for instance, by choosing the fringe order appropriately. We wish to highlight that the present work concerns only the interference pattern obtained as a section of the Bertin surfaces [20,21] and does not take into account the effect of the lens system for image detection—a problem that will be addressed in a specific work.

If $B_3>B_1>B_2$, the dual case occurs, which means that the optic plane is rotated $\pi /2$ in $\Sigma$. Then, (46) becomes

$$\begin{array}{cc}\hfill y_1& =x_2-W_{32}{x}_3,\hfill \\ \hfill y_2& =x_1-W_{31}{x}_3,\hfill \\ \hfill y_3& =x_3+W_{31}{x}_1+W_{32}{x}_2,\hfill \end{array}$$

(48)

and (47) changes accordingly. This means that the Bertin’s surface is warped in a dual direction with respect to the sample frame $\Sigma$ in the example in Figure 4. The results are similar in both cases (when using the same set of parameters): the rotation is small but the calculations allow to evaluate the axes’ rotation if required. Furthermore, the results given in Figure 4 are corroborated by the model and the images explained and shown in [14,15].

3.2. Misalignment of Crystallographic Directions of Lead Tungstate and Other Tetragonal Crystals

Here, we assume that the crystal and the sample are misaligned with no applied or residual stress such that $\Sigma ={\Sigma }^{\prime }$ and ${\mathbf{Q}}_1=\mathbf{Q}$, ${\mathbf{Q}}_2=\mathbf{I}$ in (4), (5) and (6). Hence, the analysis given in Section 2.2 still applies. There are, however, several possible scenarios.

3.2.1. No Information About Both the Rotation Amplitude and the Axis of ${\mathbf{Q}}_1$

In this case, by using the Rodriguez formula for finite values of the amplitude $\theta$, we can write the components $Q_{ij}$ in Equation (15) in terms of the four unknown values $({\omega }_1,{\omega }_2,{\omega }_3,\theta )$ under the constraint ${\omega }_1^2+{\omega }_2^2+{\omega }_3^2=1$, where ${\omega }_k$$k=1,2,3$ are the components of the axis of rotation in the frame $\Sigma$:

$$\begin{array}{cc}\hfill Q_{11}& =1+(1-cos\theta )({\omega }_2^2+{\omega }_3^2),\hfill \\ \hfill Q_{22}& =1+(1-cos\theta )({\omega }_1^2+{\omega }_3^2),\hfill \\ \hfill Q_{33}& =1+(1-cos\theta )({\omega }_1^2+{\omega }_2^2),\hfill \\ \hfill Q_{12}& =-sin\theta {\omega }_3+(cos\theta -1){\omega }_1{\omega }_2,\hfill \\ \hfill Q_{21}& =sin\theta {\omega }_3+(cos\theta -1){\omega }_1{\omega }_2,\hfill \\ \hfill Q_{13}& =sin\theta {\omega }_2+(cos\theta -1){\omega }_1{\omega }_3,\hfill \\ \hfill Q_{31}& =-sin\theta {\omega }_2+(cos\theta -1){\omega }_1{\omega }_3,\hfill \\ \hfill Q_{23}& =-sin\theta {\omega }_1+(cos\theta -1){\omega }_2{\omega }_3,\hfill \\ \hfill Q_{32}& =sin\theta {\omega }_1+(cos\theta -1){\omega }_2{\omega }_3.\hfill \end{array}$$

(49)

By using a suitable numerical optimization procedure, we can arrive at an evaluation of both the amplitude $\theta$ and the axis of rotation.

3.2.2. No Information About the Axis of ${\mathbf{Q}}_1$ but the Rotation Amplitude Is Small

In this case, we may use the approximated Rodriguez Formula (16), which leads to the following components of ${\mathbf{Q}}_1$:

$$\begin{array}{cc}\hfill Q_{11}& =Q_{22}=Q_{33}=1,\hfill \\ \hfill Q_{12}& =-Q_{21}=-\theta {\omega }_3,\hfill \\ \hfill Q_{13}& =-Q_{31}=\theta {\omega }_2,\hfill \\ \hfill Q_{23}& =-Q_{32}=-\theta {\omega }_1,\hfill \end{array}$$

(50)

and the optimization procedure is significantly simplified.

3.2.3. Information About Both the Amplitude and the Axis of ${\mathbf{Q}}_1$

In the last case, since ${\mathbf{Q}}_1$ is completely known, then the analysis of the Cassini-like curves can be used to detect the presence of residual stress associated with a rotation ${\mathbf{Q}}_2\left(\mathbf{T}\right)$. By using, once again, an optimization procedure, we may evaluate the tensor $\mathbf{Q}={\mathbf{Q}}_1{\mathbf{Q}}_2$ and then the rotation ${\mathbf{Q}}_2={\mathbf{Q}}_1^T\mathbf{Q}$. Specifically, we can define a function of the stress $\mathbf{T}$ that provides a global measure of the stress by the means of the difference between the stress-induced rotation ${\mathbf{Q}}_2\left(\mathbf{T}\right)$ and the identity

$$f\left(\mathbf{T}\right)=\parallel {\mathbf{Q}}_2\left(\mathbf{T}\right)-\mathbf{I}\parallel .$$

(51)

Further analysis of the kind described in the previous subsection would allow for the better characterization of the residual stress.

A detailed analysis of these three cases will be the subject of future papers.

4. Discussion

Within the problem of crystallographic axis rotation and cell distortion (which remains a complex one), we can distinguish two cases that can affect each other: rotation due to stress and misalignment with respect to the specimen’s facets. The exact geometric solution of the OI axes’ rotation in the three-dimensional space can be quite complex and requires precise experimental conoscopic observations and a dedicated algorithm to evaluate, from the Cassini-like curves, the angular parameters. Moreover, we have to take into account the effects of the optical acquisition system and the need for a high-precision positioning system for the experimental layout. The assessment of problems related to the experimental phase is, however, beyond the scope of this work and will be discussed and addressed in future papers. From an experimental point of view, all theoretical assumptions are corroborated, at least in part, by experimental results and image acquisitions, as shown in [14,15,17]. However, we are aware that specific efforts are necessary in the experimental direction to apply the theoretical acquisitions and validate the conclusions.

Section 2.1 shows the general case of crystallographic cell rotation or distortion; there are several phenomena that induce cell distortion as a consequence. What is directly measured is the rotation of the OI frame, which is related to the crystallographic axes’ rotation and distortion.

In Section 3, we applied the method to the tetragonal and PWO cases (see Remark 4), taking into account the effects of plane stress. A general method for the solution is suggested in the case of linear photoelasticity, using the Sirotin approximation whenever it is required. We also show that the internal stress induces a small axis rotation of about ${10}^{-3}÷{10}^4$ rad. All the described methods allow us to measure the axes’ rotation via the analysis of the Cassini-like interference fringes obtained by laser conoscopy and conoscopic observations of the plane alignment.

In our previous papers, we showed the feasibility of the fitting of the Cassini-like curves, aimed at the evaluation of the elasto-optic parameters and internal stress [11,12,16]. In the present paper, we show that is also possible to obtain a fine analysis of the OI axes’ rotation linked to crystallographic cells’ misalignment. We remark that, since conoscopy is a volume-based technique that analyzes a whole sample section, we obtain the mean value of the OI axes’ rotation within the probed volume. This provides complementary information with respect to XRD, which instead analyzes only the sample surface. Moreover, the volume probed by conoscopy can be controlled, regulating the lens layout and allowing also for higher spatial sensitivity.

With respect to our other previous work conducted on this subject, here, we consider all possible sources of distortion of the interference fringes besides stress—either applied or residual. In particular, the effects of specimens cut with respect to the crystallographic direction have been fully described from a theoretical point of view.

5. Conclusions

Nowadays, crystals are required to have specific features for the exploitation of physical phenomena that have been not considered until now. The lattice orientation plays a crucial role, addressing the need for compact, fast and reliable detectors to be used in new, high-energy experiments. This theoretical work paves the way for the development of an experimental optical methodology to measure miscut and lattice misalignment in anisotropic crystals in a fast and non-contact manner so as to inspect the entire population of crystal samples. This work also identifies a limitation in distinguishing the misorientation of the lattice due to stress. A number of open questions arise. Some of them are related to the practical application of the approach—operatively, the fringe patterns are obtained by the illumination of a certain volume of the crystal, and the features of the conoscopic images are averaged to this volume; the evolution of the influence of each part of this volume on the final result is crucial in determining the sensitivity of the technique to variations in the probed volume. Moreover, this study focuses on tetragonal samples, particularly classes $4,\overline{4},4/m$, paying specific attention to PWO crystals. The results are, however, applicable also to the higher-symmetry tetragonal classes $4mm,\overline{4}2m,422,4/mmm$. We show that, in the crystallographic plane alignment $a-c$, the internal stress-induced axis rotation is small and, in most cases, negligible. We also evaluate the application limits of the Sirotin small-stress approximation. On the other hand, in the case of misalignment (also superimposed with small internal stress), we suggest a general technique for the calculation of the rotation angles. In both cases, the equation for the interference fringe curves, modeled as Cassini-like curves, has been determined.