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On the photoelastic constants for anisotropic stressed crystals
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
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On the photoelastic constants for anisotropic stressed crystals Daniele Rinaldi a,c , Fabrizio Daví b,c ,∗, Luigi Montalto a,c a b c
SIMAU, Universitá Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy DICEA, Universitá Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy ICRYS, Universitá Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy
ARTICLE
INFO
ABSTRACT
Keywords: Photoelasticity Photoelastic constant Anisotropic crystals Scintillating crystals
The need for high-quality and high-performance crystals for high-energy physics (e.g. scintillators) and biomedical applications require a good knowledge of their mechanical and optical properties, amongst the others. This is a mandatory step in many quality control process aimed at the improvement of the technological processes for crystals growth. Following the first analysis already done for Tetragonal crystals, in this paper we study the elasto-optic behavior of optically biaxial (Monoclinic and Orthorhombic groups) and uniaxial crystals (Trigonal and Hexagonal groups). We evaluate the photoelastic constants, the optic angle and the optic plane associated to various states of stress, in terms of the components of the Piezo-Optic tensor.
1. Introduction
where 𝑎 > 𝑏 are the semi-axes of the Cassini-like curves. Since such a parameter depends on the applied stress 𝐓 and we define a tensor of Photoelastic constants 𝐅𝜎 by the means of:
The study of the photoelastic behavior of a crystal by the means of pointwise conoscopic analysis allows for a not-destructive and "fast-toperform’’ technique which permits to detect the presence of internal stresses, which in turn are generally associated to the presence of defects and to quality degradation. As we showed in previous works, a reliable map of the stress in the crystal [1,2] can be obtained by the means of conoscopic techniques; moreover, since the link between the optical and mechanical properties of a crystal is the piezo-optic fourthorder tensor 𝛱, the same techniques can be used in order to estimate its components [3]. In [4] we presented, for the optically Uniaxial crystals of the Tetragonal group, a detailed study of the isochromate interference fringes (generated by the intersection of the crystal surfaces with the Bertin surfaces [5–8]) on the mechanical stresses, either applied or residual. In this paper, we wish to extend these results to anisotropic crystals of the Monocline, Orthorhombic, Trigonal and Hexagonal point groups [9–12]. We accordingly study Uniaxial and Biaxial crystals, leaving out only those of the Triclinic group or the optically isotropic crystals of the Cubic group or Isotropic materials like glass (vid. e.g. [13]). We limit our analysis to the fringes in the planes orthogonal to either the optic axis (for unstressed Uniaxial crystals) or to the optic axis acute bisectrix (for unstressed Biaxial ones) in such a way that, for low stress and small crystal thickness, these fringes are closed curves we called the Cassini-like curves [6,14]. It is possible therefore to define an experimentally measurable parameter, the Ellipticity Ratio 𝐶 [14]: 𝑎 𝐶 = − 1, 𝑏
𝐅𝜎 =
𝑑𝐶 | | , 𝑑𝐓 |𝐓=𝟎
in such a way that, to within higher-order terms in ‖𝐓‖, the ellipticity parameter 𝐶 is a linear function of the stress 𝐓: 𝐶(𝐓) = 𝐶(𝟎) + 𝐅𝜎 ⋅ 𝐓 + 𝑜(‖𝐓‖2 ). The Ellipticity Ratio 𝐶(𝐓) can be easily measured by the means of experimental photoelastic techniques and from a theoretical point of view, it can be evaluated for each of the symmetry group we analyze and for a general state of stress. We recall that a detailed knowledge of the dependence of 𝐶 on 𝐓 allows for either: • the design of a set of experiments aimed at the full knowledge of the piezo-optic tensor 𝛱 [3]; • the estimate of the residual stress within the crystal, generated for instance by the crystal growth process, provided we have detailed knowledge of 𝛱 [15,16]. The aim of this work, as in [4], is the evaluation of the ellipticity parameter 𝐶, the photoelastic constant tensor 𝐅𝜎 , the change in the optic angle and the rotation of the optic plane, in terms of the components of the piezo-optic tensor 𝛱 and of the dielectric impermeability tensor 𝐁0 in the unstressed state, under external (applied) or internal (residual) stress.
∗ Corresponding author at: DICEA, Universitá Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy. E-mail address: [email protected] (F. Daví).
https://doi.org/10.1016/j.nima.2019.162782 Received 5 August 2019; Accepted 16 September 2019 Available online 19 September 2019 0168-9002/© 2019 Elsevier B.V. All rights reserved.
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
Generally speaking we assume that the specimen and the crystal lattice are coherent in the sense that the specimen edges are parallel to the crystal lattice directions: however, for the Orthorhombic crystals we study also the case of non-coherent crystals and analyze the dependence of the results on the misalignment angle between specimen and crystal lattice. 2. Background: photoelastic crystals 2.1. Optical indicatrix and Bertin surfaces In this and in the following subsections, in order to make the paper self-contained, we shall recall some definitions and results which are presented elsewhere. The dielectric properties of an anisotropic crystal are described by the second-order symmetric and definite-positive dielectric impermeability tensor 𝐁, (vid. [9–11]) 𝐞=
1 𝐁𝐝 , 𝜖𝑜
(1)
where 𝐝 is the dielectric displacement vector, 𝐞 the electric field vector and 𝜖𝑜 the dielectric constant of vacuum. In Photoelastic crystals the dielectric impermeability is a linear function of the Cauchy stress tensor: (2)
𝐁(𝐓) = 𝐁𝑜 + 𝛱[𝐓] ,
where 𝛱 is the Maxwell piezo-optic tensor and 𝐁𝑜 is the dielectric impermeability in the unstressed state. By the positive-definiteness of 𝐁 we can define the Optical indicatrix or indices ellipsoid, i.e. is the locus of normalized constant dielectric energy [17] which summarizes at a glance all the information about the crystal optical anisotropy:
Fig. 1. The box represents the physical frame which may coincides with the specimen. The intrinsic crystal frame , related to the Bravais lattices and the principal frame 𝛴 associated to the eigenvectors {𝐮𝑘 }, 𝑘 = 1, 2, 3 of 𝐁 are also represented together with the optical indicatrix.
(3)
ordinary (multiplicity two) refraction indices and we define a crystal optically Positive (Negative) when 𝑛𝑒 > 𝑛𝑜 (𝑛𝑒 < 𝑛𝑜 ). For Biaxial crystals, if value of intermediate refractive index is closer to that of highest refractive index, the crystal is optically Negative, and if is closer to lowest refractive index then the crystal is optically Positive. A Biaxial crystal has two optic axes orthogonal to 𝐮𝜂 and lying in the (𝜉 , 𝜁 ) plane, the optic plane [5]; in this work we shall consider the 𝜁-axis as the acute bisector of the optic axes: we define 2𝜑 the acute angle between the optic axes which is given by [7] √ √ 𝐵𝜂 − 𝐵𝜁 𝐵𝜉 − 𝐵𝜂 cos 𝜑 = , sin 𝜑 = . (6) 𝐵𝜉 − 𝐵𝜁 𝐵𝜉 − 𝐵𝜁
𝐁𝐱 ⋅ 𝐱 = 1 ,
where 𝐱 is a typical point within the crystal. In the sequel, to evaluate the effects of applied and residual stresses on the crystal optical properties, we shall make use of different frames at 𝐱, namely: a A system relative to the crystallographic structure of the 14 Bravais lattices, the Intrinsic crystal frame ≡ [𝑎 , 𝑏 , 𝑐 ] where the coordinates are referred to the crystal lattice vectors {𝐚 , 𝐛 , 𝐜}; b An orthonormal frame, the Cartesian physical frame ≡ {𝑥 , 𝑦 , 𝑧} referred to the orthonormal base {𝐞1 , 𝐞2 , 𝐞3 } in which we shall define the components of 𝐁, 𝐓 and 𝛱 and which can be related to by the crystal symmetry. It represents the edges of a prismatic specimen, for instance;
An Uniaxial crystal has only an optic axis which coincides with a material symmetry axis: in this case 𝜑 = 0, the optic axis is directed as the eigenvector corresponding to min 𝐵𝑖 in positive (max 𝐵𝑖 negative) crystals and the optical indicatrix is an ellipsoid of revolution around the optic axis. The other surface which identifies the optical behavior of the material is the surface of equal phase difference or Bertin surface which can be obtained from the Fresnel equation [5,17]. The Bertin surfaces are fourth-order surfaces [6,7], whose mathematical aspects are studied in details into [8] (vid. Fig. 2-Left) and whose equation is
c The orthonormal Principal frame of 𝐁, 𝛴 ≡ {𝜉 , 𝜂 , 𝜁} where the coordinates are referred to the eigenvectors {𝐮1 , 𝐮2 , 𝐮3 } of 𝐁. We shall denote with 𝛴𝐁 and 𝛴𝐓 the principal frames for 𝐁(𝐓) and 𝐓 respectively; these frames and their reciprocal relations are presented in Fig. 1. In the general case the tensor 𝐁 is not diagonal in the frame : hence the optical indicatrix admits the following explicit representation in terms of the six independent components of 𝐁: 𝐵𝑥𝑥 𝑥2 + 𝐵𝑦𝑦 𝑦2 + 𝐵𝑧𝑧 𝑧2 + 2𝐵𝑥𝑦 𝑥𝑦 + 2𝐵𝑥𝑧 𝑥𝑧 + 2𝐵𝑦𝑧 𝑦𝑧 = 1 ;
𝜉 4 cos4 𝜑 + 𝜂 4 + 𝜁 4 sin4 𝜑 + 2𝜉 2 𝜂 2 cos2 𝜑 + 2𝜂 2 𝜁 2 sin2 𝜑
(4)
the parameter 𝐻, which has the dimension of a length, is defined as
however, in the principal frame 𝛴(4) reduces to: 2
2
2
𝐵𝜉 𝜉 + 𝐵𝜂 𝜂 + 𝐵𝜁 𝜁 = 1 ,
𝐵𝜉 ≥ 𝐵𝜂 ≥ 𝐵𝜁 ,
(7)
− 2𝜉 2 𝜁 2 sin2 𝜑 cos2 𝜑 − 𝐻 2 (𝜉 2 + 𝜂 2 + 𝜁 2 ) = 0 ;
𝐻= (5)
𝑁𝜆 , 𝑛𝑚𝑎𝑥 − 𝑛𝑚𝑖𝑛
(8)
where 𝑁 is the fringe order, 𝜆 is the wave length of the light source.
with the three refraction indices (𝑛𝜉 , 𝑛𝜂 , 𝑛𝜁 ) defined by 𝐵𝑖 = 𝑛−2 𝑖 , 𝑖 = 𝜉 , 𝜂 , 𝜁. According to the eigenvalue multiplicity a material can be optically Biaxial when all the eigenvalues are different, Uniaxial when two eigenvalues are equal and Isotropic whenever all the eigenvalues are equal. In Uniaxial crystals we say 𝑛𝑒 and 𝑛𝑜 the extraordinary and
2.2. Cassini-like curves The section of the Bertin surfaces with the crystal surface or a projection plane gives the analytical expression of the isochromate 2
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
By trivial calculations then we have: √ 𝑁(𝑢𝑜 ) 𝑑𝐶 | 1 𝐴(𝐾 , 𝜑𝑜 ) = = | 𝑑𝑢 |𝜎𝑖𝑗 =0 𝐷(𝑢𝑜 ) (1 − 𝑢𝑜 )2
(15)
1 𝑁,𝑢 (𝑢𝑜 )𝐷(𝑢𝑜 ) − 𝑁(𝑢𝑜 )𝐷,𝑢 (𝑢𝑜 ) , √ 1 − 𝑢𝑜 2𝐷(𝑢𝑜 ) 𝑁(𝑢𝑜 )𝐷(𝑢𝑜 )
+
where 𝑢𝑜 = sin2 𝜑𝑜 , being 𝜑𝑜 the optic angle in the unstressed state, and √ 𝑁(𝑢) = 1 + 2𝐾 2 (1 − 𝑢)𝑢 + 1 + 4𝐾 2 (1 − 𝑢) , √ 𝐷(𝑢) = 1 − 2𝐾 2 𝑢 + 1 + 4𝐾 2 (1 − 𝑢) ; we notice that (15) is a term which is independent on both the stress 𝐓 and the piezo-optic tensor 𝛱 and depends only on 𝜑𝑜 and on the adimensional ratio 𝐾 = 𝜁𝑜 ∕𝐻.1 For an unstressed uniaxial crystal, with 𝑢𝑜 = 0, the term (15) reduces to
Fig. 2. Left: the Bertin surfaces of a biaxial crystal for different orders 𝑁; in the uniaxial case they reduce to revolution surfaces around the 𝜁-axis. In the figure it is show also the optic plane, which contains the optic axes: the 𝜁 -axis is the bisector of the optic axes. Right: the Cassini-like curves representing the interference isochromate fringes as a section of the left figure with a plane 𝜁 = 𝜁𝑜 . Here 𝑎 > 𝑏 with 𝑎 = 𝑏 for uniaxial crystals.
𝐴(𝐾 , 0) = (1 +
2𝐾 2 ). √ 1 + 1 + 4𝐾 2
(16)
From (6) then we have: (𝐵𝜉 − 𝐵𝜂 ),𝑖𝑗 (𝐵𝜉 − 𝐵𝜁 ) − (𝐵𝜉 − 𝐵𝜁 ),𝑖𝑗 (𝐵𝜉 − 𝐵𝜂 ) 𝜕 𝐵𝜉 − 𝐵𝜂 𝜕𝑢 = = , 𝜕𝜎𝑖𝑗 𝜕𝜎𝑖𝑗 𝐵𝜉 − 𝐵𝜁 (𝐵𝜉 − 𝐵𝜁 )2
interference fringes, fourth-order plane curves which can be either closed or open and two-folded. The equations of the intersection curves between the Bertin surface and a generic plane are given into [8]: here we shall limit our analysis to planes which are orthogonal to the bisector of the optic axes in biaxial crystals or to the optic axis in uniaxial and hence with 𝜁 = 𝜁𝑜 . In both cases we have closed fourth-order curves 𝑓 = 𝑓 (𝜉 , 𝜂 , 𝜁𝑜 ) parameterized on 𝜁𝑜 that we call Cassini-like for their similarity with the Cassini’s curves [6]. Let {±𝑎 , ±𝑏} with 𝑎 > 𝑏 be the solutions of the equations 𝑓 (𝜉 , 0 , 𝜁𝑜 ) = 0 and 𝑓 (0 , 𝜂 , 𝜁𝑜 ) = 0, then it is possible to define the ellipticity ratio [14], an useful measurable parameter (Fig. 2-Right):
(17) with 𝑘 = 𝜉, 𝜂, 𝜁 , then (17), evaluated and since for 𝐓 = 𝟎 it is 𝐵𝑘 = 𝑛−2 𝑘 for 𝐓 = 𝟎, reduces to: −2 −2 𝜕(𝐵𝜉 − 𝐵𝜂 ) 𝑛𝜉 − 𝑛𝜂 𝜕(𝐵𝜉 − 𝐵𝜁 ) | 1 𝜕𝑢 | = . ( − )| | |𝐓=𝟎 𝜕𝜎𝑖𝑗 |𝐓=𝟎 𝑛−2 − 𝑛−2 𝜕𝜎𝑖𝑗 𝜕𝜎𝑖𝑗 𝑛−2 − 𝑛−2 𝜉 𝜁 𝜉 𝜁
From (13), (14), (15) and (18) then we have: 𝐹𝜎,𝑖𝑗 =
𝐹𝜎,𝑖𝑗 =
𝐚 ≡ sin 𝛽𝐞1 + cos 𝛽𝐞3 ,
𝐛 ≡ 𝐞2 ,
𝐜 ≡ 𝐞3 ,
(21)
In the frame the tensor 𝐁𝑜 for the Monoclinic group may be represented as (see e.g.[5]): ⎡ 𝐵𝑥𝑥 𝐁𝑜 ≡ ⎢ ⋅ ⎢ ⎣ ⋅
(12)
0 𝐵𝑦𝑦 ⋅
𝐵𝑥𝑧 0 𝐵𝑧𝑧
⎤ ⎥, ⎥ ⎦
(22)
and the principal frame 𝛴 is rotated about the monoclinic 𝑏-axis by an angle 𝛾𝑜 with: √ 𝐵 + 𝐵𝑧𝑧 𝐵 − 𝐵𝑧𝑧 2 𝑜 2 , 𝐵𝜉,𝜂 = 𝑥𝑥 ± ( 𝑥𝑥 ) + 𝐵𝑥𝑧 2 2
where the components of 𝐅𝜎 in the frame are given by: (13)
𝐵𝜁𝑜 = 𝐵𝑦𝑦 ,
Since by (11) we have that 𝐶 depends on 𝑢 = sin2 𝜑 and by (6)𝑢 depends on 𝜎𝑖𝑗 by means of (2), then we have: 𝜕𝐶 𝑑𝐶 𝜕𝑢 | = . | 𝜕𝜎𝑖𝑗 𝑑𝑢 𝜕𝜎𝑖𝑗 |𝜎𝑖𝑗 =0
(20)
Following [5,11,12], we choose the crystallographic frame with the 𝑏-axis directed as the 𝑦-axis and the 𝑐-axis directed as the 𝑧-axis, with the monoclinic 𝛽 angle comprised between 𝑎 and 𝑐. The relation between and the physical frame is (vid. Fig. 3).
For small stress the ellipticity parameter 𝐶 is
𝑖,𝑗 = 𝑥,𝑦,𝑧.
𝐴(𝐾 , 0) 𝜕(𝐵𝜉 − 𝐵𝜂 ) | . | −2 |𝐓=𝟎 𝜕𝜎𝑖𝑗 𝑛−2 𝑜 − 𝑛𝑒
3.1. Monoclinic crystals
2.3. The photoelastic constants tensor
𝜕𝐶 | , | 𝜕𝜎𝑖𝑗 |𝜎𝑖𝑗 =0
(19)
3. Stress analysis in optically biaxial crystals
𝜕𝐶 | . (10) 𝐶(𝐓) = 𝐶(𝟎) + 𝐅𝜎 ⋅ 𝐓 + 𝑜(‖𝐓‖2 ) , 𝐅𝜎 = | 𝜕𝐓 |𝐓=𝟎 Relation (10) allows to evaluate the stress by means of theoretical considerations and experimental measurements, like in [1–3,6,14,16, 18], Finally, by means of (9), (7) and the definition of 𝑎 , 𝑏, we obtain an explicit expression of 𝐶 in terms of 𝑢 = sin2 𝜑 ∈ [0 , 1] (vid. [4]): √ √ √ 𝜁 1 + 2𝐾 2 (1 − 𝑢)𝑢 + 1 + 4𝐾 2 (1 − 𝑢) 1 √ √ 𝐶(𝑢) = − 1 , 𝐾 = 𝑜 . (11) √ 1−𝑢 𝐻 2 2 1 − 2𝐾 𝑢 + 1 + 4𝐾 (1 − 𝑢)
𝐹𝜎,𝑖𝑗 =
−2 −2 𝐴(𝐾 , sin 𝜑𝑜 ) 𝜕(𝐵𝜉 − 𝐵𝜂 ) 𝑛𝜉 − 𝑛𝜂 𝜕(𝐵𝜉 − 𝐵𝜁 ) | , − )| ( −2 −2 −2 |𝐓=𝟎 𝜕𝜎𝑖𝑗 𝜕𝜎𝑖𝑗 𝑛𝜉 − 𝑛𝜁 𝑛𝜉 − 𝑛−2 𝜁
which for uniaxial crystals with 𝑛𝜉 = 𝑛𝜂 = 𝑛𝑜 and 𝑛𝜁 = 𝑛𝑒 reduces to (vid. [4]):
𝑎 − 1 > 0, (9) 𝑏 in such a way that 𝐶 = 0 in unstressed uniaxial crystal. Clearly, by means of (7), (6) and (2) the ellipticity ratio may depend on the stress 𝐓 and we may define as in [4] a Photoelastic constants tensor 𝐅𝜎 such that:
𝐶=
𝐶(𝐓) = 𝐶(𝟎) + 𝐅𝜎 ⋅ 𝐓 ,
(18)
(23)
1 To arrive at these results we assumed 𝐻 = const. vid. e.g. the discussion in [8].
(14) 3
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
the principal frame 𝛴𝐁 for 𝐁(𝐓) is obtained by a rotation of 𝛾 about the monoclinic 𝑏-axis 𝐞2 : tan 𝛾 =
𝐵𝜉 − 𝐵𝑥𝑥 − 𝜋11 𝜎𝑥𝑥 𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥
(28)
,
whereas the optic angle changes by (6) from (24) into: √ (𝐵𝑥𝑥 − 𝐵𝑧𝑧 + (𝜋11 − 𝜋31 )𝜎𝑥𝑥 )2 + 4(𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥 )2 sin2 𝜑 = , 2(𝐵𝜉 − 𝐵𝑦𝑦 − 𝜋21 𝜎𝑥𝑥 )
and the optic axes are directed as (25) with (28) and (29) in place of 𝛾𝑜 and 𝜑𝑜 . The photoelastic constants tensor 𝐅𝜎 has only one non-zero component and accordingly ( 𝐴(𝐾 , sin 𝜑𝑜 ) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋31 − 𝜋11 ) + 4𝐵𝑥𝑧 𝜋51 𝐶(𝜎𝑥𝑥 ) = 𝐶(𝟎) + √ 𝑛−2 − 𝑛−2 2 (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 4𝐵𝑥𝑧 𝜉 𝜁
Fig. 3. The frames , and 𝛴.
tan 𝛾𝑜 =
𝐵𝜉𝑜 − 𝐵𝑥𝑥 𝐵𝑥𝑧
,
−
Crystals of the Monoclinic group accordingly are optically Biaxial and the optic angle is, provided 𝐵𝜉 > 𝐵𝜂 > 𝐵𝜁 , given by (6):2 √ 2 (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 4𝐵𝑥𝑧 , (24) sin2 𝜑𝑜 = √ 2 𝐵𝑥𝑥 + 𝐵𝑧𝑧 − 2𝐵𝑦𝑦 + (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 4𝐵𝑥𝑧
+
(25)
= ± sin 𝜑𝑜 (sin 𝛾𝑜 𝐞1 + cos 𝛾𝑜 𝐞3 ) + cos 𝜑𝑜 𝐞3 , being the direction spanned by 𝐮3 the acute bisectrix of the optic angle 𝜑𝑜 . For all the classes of the Monoclinic group the piezo-optic tensor 𝛱 has the following representation in the frame (vid. e.g.[11])3 : ⎡ ⎢ ⎢ [𝛱] ≡ ⎢ ⎢ ⎢ ⎢ ⎣
𝜋11 𝜋21 𝜋31 0 𝜋51 0
𝜋12 𝜋22 𝜋32 0 𝜋52 0
𝜋13 𝜋23 𝜋33 0 𝜋53 0
0 0 0 𝜋44 0 𝜋64
𝜋15 𝜋25 𝜋35 0 𝜋55 0
0 0 0 𝜋46 0 𝜋66
⎤ ⎥ ⎥ ⎥; ⎥ ⎥ ⎥ ⎦
(26)
⎡ 𝜎𝑥𝑥 𝐓≡⎢ ⋅ ⎢ ⎣ ⋅
As it is shown into [8], for the Monoclinic group any combination of stress make the optic plane to rotate about the monoclinic 𝑏-axis, aside for the shear stresses 𝜎𝑦𝑧 and 𝜎𝑥𝑦 which rotate the optic plane about a generic direction. In order to give an estimate for the components of 𝛱 we can load the specimen with an uniaxial load (either 𝜎𝑥𝑥 , 𝜎𝑦𝑦 or 𝜎𝑧𝑧 ) or with a biaxial load which induces the simple shear 𝜎𝑥𝑧 . We shall show here only the case for 𝜎𝑥𝑥 , the other cases leading to similar results. The eigenvalues for 𝐁(𝐓) are, from (2) and (26) 𝐵 + 𝐵𝑧𝑧 𝜋 + 𝜋31 𝐵𝜉,𝜂 = 𝑥𝑥 + ( 11 )𝜎𝑥𝑥 2 2 √ 𝐵 − 𝐵𝑧𝑧 𝜋 − 𝜋31 ± ( 𝑥𝑥 + ( 11 )𝜎𝑥𝑥 )2 + (𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥 )2 , 2 2 𝐵𝜁 = 𝐵𝑦𝑦 + 𝜋21 𝜎𝑥𝑥 ,
(30)
0 0 ⋅
𝜎𝑥𝑧 0 𝜎𝑧𝑧
⎤ ⎥; ⎥ ⎦
(31)
𝐵𝜉,𝜂 = 𝛼𝑜+ + 𝛼1+ 𝜎𝑥𝑥 + 𝛼2+ 𝜎𝑧𝑧 + 𝛼3+ 𝜎𝑥𝑧 ± 𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) (32) 𝐵𝜁 = 𝐵𝑦𝑦 + 𝜋21 𝜎𝑥𝑥 + 𝜋23 𝜎𝑧𝑧 + 𝜋25 𝜎𝑥𝑧 , where 𝑅=
√ (𝛼𝑜− + 𝛼1− 𝜎𝑥𝑥 + 𝛼2− 𝜎𝑧𝑧 + 𝛼3− 𝜎𝑥𝑧 )2 + (𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥 + 𝜋53 𝜎𝑧𝑧 + 𝜋55 𝜎𝑥𝑦 )2
and 2𝛼𝑜± = 𝐵𝑥𝑥 ± 𝐵𝑧𝑧 , 2𝛼1± = 𝜋11 ± 𝜋31 , 2𝛼2± = 𝜋13 ± 𝜋33 and 2𝛼3± = 𝜋15 ± 𝜋35 . The principal frame 𝛴𝐁 for 𝐁(𝐓) is rotated about 𝐞2 by the angle 𝛾:
(27)
For different eigenvalues ordering the relation (6) reads:
sin2 𝜑𝑜 =
) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋31 − 𝜋11 ) + 4𝐵𝑥𝑧 𝜋51 ) 𝜎𝑥𝑥 , √ 2 2 (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 4𝐵𝑥𝑧
and by (2) arrive at the principal values for 𝐁(𝐓)
tan 𝛾 = 2
𝑛−2 − 𝑛−2 𝜋 + 𝜋31 − 2𝜋21 𝜂 𝜉 ( 11 −2 2 𝑛−2 − 𝑛 𝜉 𝜁
with 𝜑𝑜 given by (24), the term 𝐶(𝟎) is given by (11) with 𝑢 = sin2 𝜑𝑜 and the principal refraction index (𝑛𝜉 , 𝑛𝜂 , 𝑛𝜁 ) are evaluated in the unstressed state. The results for 𝜎𝑧𝑧 and 𝜎𝑥𝑧 can be obtained from (30) by replacing 𝜋11 , 𝜋31 , 𝜋51 respectively with 𝜋13 , 𝜋33 , 𝜋53 , and 𝜋15 , 𝜋35 , 𝜋55 , whereas those for 𝜎𝑦𝑦 by replacing 𝜋11 , 𝜋31 , 𝜋51 , 𝜋21 with 𝜋21 , 𝜋32 , 𝜋52 , 𝜋22 . All together the twelve relations (30), (28) and (29), which are in general not independent, do not allow for the complete determination of the twenty independent components of 𝛱 in monoclinic crystals: they are however a starting point for the design of further experiments aimed to a complete characterization of such a piezo-optic tensor. For crystals such that we have a full knowledge of the components of 𝛱 the relations we obtained can be used to give an estimate for the residual stress within the specimen. We limit our analysis to specimen that we assume ‘‘thin’’ enough along the 𝑏-axis in order to assume that the stress components 𝜎𝑦𝑦 , 𝜎𝑥𝑦 and 𝜎𝑦𝑧 be negligible. Accordingly we may consider the plane stress 𝐓𝐞2 = 𝟎:
and the two optic axis are directed as: 𝐦1,2 = ± sin 𝜑𝑜 𝐮1 + cos 𝜑𝑜 𝐮3 ,
(29)
𝐵𝜉 − 𝐵𝑥𝑥 − 𝜋11 𝜎𝑥𝑥 − 𝜋13 𝜎𝑧𝑧 − 𝜋15 𝜎𝑥𝑧 𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥 + 𝜋53 𝜎𝑧𝑧 + 𝜋55 𝜎𝑥𝑧
;
(33)
the optic axis directions have the same expression as in (25) with 𝛾 as in (33) and 𝜑 given by:
𝛥𝐵𝑚𝑖𝑛 . 𝛥𝐵max
3 Here we used for the component of 𝛱 the Voigt notation with 1 = 𝑥𝑥, 2 = 𝑦𝑦, 3 = 𝑧𝑧, 4 = 𝑦𝑧, 5 = 𝑥𝑧 and 6 = 𝑥𝑦. We notice that the piezo-optic tensor is not symmetric in this representation: the relation between this notation (sometimes referred as the Voigt–Mandel notation) and the components in of a fourth-order tensor is discussed in [19].
sin2 𝜑 =
2𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) , 2𝛼𝑜 − 2𝐵𝑦𝑦 + (𝛼1− + 2𝜋21 )𝜎𝑥𝑥 + (𝛼2− + 2𝜋23 )𝜎𝑧𝑧 + (𝛼3− + 2𝜋25 𝜎𝑥𝑧 ) + 𝑅
(34) in place of 𝜑𝑜 and 𝛾𝑜 . 4
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
In the frame ≡ the tensor 𝐁𝑜 for the Orthorhombic group has the diagonal representation: ⎡ 𝐵𝑥𝑥 𝐁𝑜 ≡ ⎢ 0 ⎢ ⎣ 0
0 𝐵𝑦𝑦 0
0 0 𝐵𝑧𝑧
⎤ ⎥, ⎥ ⎦
(39)
and hence crystals of the Orthorhombic group are optically Biaxial; by taking into account that all the principal axes can be the bisectrix of the optic axis [5], then both the crystallographic and physical frames coincide with the principal frame 𝛴. Clearly, depending on the physical properties of a given crystal the −2 ordering of the eigencouples (𝐮1 , 𝑛−2 ), (𝐮2 , 𝑛−2 𝜂 ) and (𝐮3 , 𝑛𝜁 ) can be 𝜉 −2 and 𝐵 = 𝑛−2 . different from the ordering of 𝐵𝑥𝑥 = 𝑛−2 , 𝐵 = 𝑛 𝑦𝑦 𝑧𝑧 𝑥 𝑦 𝑧 Fig. 4. The frames and .
Remark 1. Both the BeAl2 O4 (Chrysoberyllium) and its variant Cr:BeAl2 O4 (Alexandrite) are Orthorhombic crystals with lattice parameters 𝑎 = 9.404 Å, 𝑏 = 5.476 Å, 𝑐 = 4.727 Å. The refraction indices are 𝑛𝑥 = 1.736(4), 𝑛𝑦 = 1.741(9) and 𝑛𝑧 = 1.734(5) at 𝜆 = 755 nm [21]. The components of the Dielectric Impermeability tensor are 𝐵𝑥𝑥 = 0.3317, 𝐵𝑦𝑦 = 0.3296, 𝐵𝑧𝑧 = 0.3324. Since 𝑛𝑦 > 𝑛𝑥 > 𝑛𝑧 , then 𝐵𝑧𝑧 > 𝐵𝑥𝑥 > 𝐵𝑦𝑦 . We remark that in the 𝛴 frame of Fig. 1 and Eq. (4) the ordering is 𝐵𝑦𝑦 = 𝐵𝜁 < 𝐵𝑥𝑥 = 𝐵𝜂 < 𝐵𝑧𝑧 < 𝐵𝜉 , by following the convention that the optic plane is spanned by the eigenvectors of 𝑛−2 and 𝑛−2 . The crystal 𝜉 𝜁 is positive with 𝜑 < 𝜋∕2 with respect to the 𝑦-axis. In fact from (6) the 𝑦-axis is the acute bisector and the optic plane is the plane (𝑦 , 𝑧), with, in an unstressed crystal 𝜑𝑜 = 30◦ . □
By (32) and (19) then we get the three components of the photoelastic constants tensor 𝐅𝜎 : 𝐹𝜎,𝑥𝑥 =
𝐴(𝐾 , sin 𝜑𝑜 ) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋11 − 𝜋31 ) + 2𝐵𝑥𝑧 𝜋51 ( √ 𝑛−2 − 𝑛−2 2 (𝐵 − 𝐵 )2 + 𝐵 2 𝜉 𝜁 𝑥𝑥
−
𝐹𝜎,𝑧𝑧 =
− 𝑛−2 𝑛−2 𝜂 𝜉 𝑛−2 − 𝑛−2 𝜉 𝜁
(𝛼1+
𝑧𝑧
𝑥𝑧
2𝛼𝑜− 𝛼1− + 𝐵𝑥𝑧 𝜋51 )) , √ 2 + 𝐵2 𝛼𝑜− 𝑥𝑧
𝐴(𝐾 , sin 𝜑𝑜 ) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋13 − 𝜋33 ) + 2𝐵𝑥𝑧 𝜋53 ( √ 𝑛−2 − 𝑛−2 2 (𝐵 − 𝐵 )2 + 𝐵 2 𝜉 𝜁 𝑥𝑥
𝑧𝑧
(35)
For all the classes of the Orthorhombic group the piezo-optic tensor 𝛱 has the following representation in the frame (vid. e.g.[12]):
𝑥𝑧
𝑛−2 − 𝑛−2 𝜂 𝜉
2𝛼 𝛼2− + 𝐵𝑥𝑧 𝜋53 − )) , (𝛼2+ 𝑜− √ −2 −2 𝑛𝜉 − 𝑛𝜁 2 + 𝐵2 𝛼𝑜− 𝑥𝑧 𝐹𝜎,𝑥𝑧 =
−
⎡ 𝜋11 ⎢ 𝜋21 ⎢ 𝜋 [𝛱] ≡ ⎢ 31 ⎢ 0 ⎢ 0 ⎢ ⎣ 0
𝐴(𝐾 , sin 𝜑𝑜 ) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋15 − 𝜋35 ) + 2𝐵𝑥𝑧 𝜋55 ( √ 𝑛−2 − 𝑛−2 2 2 (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 𝐵𝑥𝑧 𝜉 𝜁 𝑛−2 − 𝑛−2 𝜂 𝜉 𝑛−2 − 𝑛−2 𝜉 𝜁
(𝛼3+
2𝛼𝑜− 𝛼3− + 𝐵𝑥𝑧 𝜋55 )) . √ 2 + 𝐵2 𝛼𝑜− 𝑥𝑧
• The change in the ellipticity ratio:
⎡ 𝜎𝑥𝑥 𝐓≡⎢ 0 ⎢ ⎣ 0
𝛥𝐶 = 𝐶(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) − 𝐶(𝟎) = 𝐹𝜎,𝑥𝑥 𝜎𝑥𝑥 + 𝐹𝜎,𝑧𝑧 𝜎𝑧𝑧 + 𝐹𝜎,𝑥𝑧 𝜎𝑥𝑧 ; (36) with the three non zero components given by (35) • the change in the optic angle:4
0 0 0 𝜋44 0 0
0 0 0 0 𝜋55 0
0 0 0 0 0 𝜋66
⎤ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(40)
0 𝜎𝑦𝑦 0
0 0 𝜎𝑧𝑧
⎤ ⎥, ⎥ ⎦
(41)
we get: 𝐵𝜂 = 𝐵𝑥𝑥 + 𝜋11 𝜎𝑥𝑥 + 𝜋12 𝜎𝑦𝑦 + 𝜋13 𝜎𝑧𝑧 ,
(37)
(42)
𝐵𝜁 = 𝐵𝑦𝑦 + 𝜋21 𝜎𝑥𝑥 + 𝜋22 𝜎𝑦𝑦 + 𝜋23 𝜎𝑧𝑧 ,
where 𝜑(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) is given by (34) and 𝜑𝑜 by (24); • the rotation of the optic plane: 𝛥𝛾 = 𝛾(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) − 𝛾𝑜 ,
𝜋13 𝜋23 𝜋33 0 0 0
As it is shown into [8], for the Orthorhombic group a spherical stress 𝐓 = 𝜎𝑚 𝐈, any uniaxial stress and any uniform plane stress maintains the optic plane and changes only the amplitude of the optic angle 𝜑. Any shear stress instead rotate the optic plane about the direction orthogonal to the shear stress plane. Accordingly for any diagonal stress tensor
In order to give an estimate of the residual stress within the specimen, provided we have a complete knowledge of the components of 𝛱 and 𝐁𝑜 , we have the three independent relations:
𝛥𝜑 = 𝜑(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) − 𝜑𝑜 ,
𝜋12 𝜋22 𝜋32 0 0 0
𝐵𝜉 = 𝐵𝑧𝑧 + 𝜋31 𝜎𝑥𝑥 + 𝜋32 𝜎𝑦𝑦 + 𝜋33 𝜎𝑧𝑧 , with 𝛴 ≡ and with the optic angle which changes into:
(38)
where 𝛾 and 𝛾𝑜 are given respectively by (28) and (23)3 .
sin2 𝜑 =
𝐵𝑧𝑧 − 𝐵𝑥𝑥 + (𝜋31 − 𝜋11 )𝜎𝑥𝑥 + (𝜋32 − 𝜋12 )𝜎𝑦𝑦 + (𝜋33 − 𝜋13 )𝜎𝑧𝑧 𝐵𝑧𝑧 − 𝐵𝑦𝑦 + (𝜋31 − 𝜋21 )𝜎𝑥𝑥 + (𝜋32 − 𝜋22 )𝜎𝑦𝑦 + (𝜋33 − 𝜋23 )𝜎𝑧𝑧
3.2. Orthorhombic crystals
, (43)
since for small stress the same optic plane as in the unstressed case is maintained. By (42) and relation (19), then we have the following non-null components of 𝐅𝜎 :
3.2.1. Coherent prisms The crystallographic axis are orthogonal for Orthorhombic crystals; we say a prismatic crystal specimen coherent if the crystallographic and physical frames are parallel in which case we may assume ≡ with 𝑥 = 𝑎, 𝑦 = 𝑏 and 𝑧 = 𝑐 (vid. Fig. 4).
𝐹𝜎,𝑥𝑥 =
4 We notice that the optic angle 𝜑 can be also calculated independently as in [20].
𝐹𝜎,𝑦𝑦 = 5
𝐴(𝐾 , sin 𝜑𝑜 ) 𝑛−2 𝑧
− 𝑛−2 𝑦
𝐴(𝐾 , sin 𝜑𝑜 ) −2 𝑛−2 𝑧 − 𝑛𝑦
(𝜋31 − 𝜋11 − (𝜋32 − 𝜋12 −
−2 𝑛−2 𝑧 − 𝑛𝑥 −2 𝑛−2 𝑧 − 𝑛𝑦 −2 𝑛−2 𝑧 − 𝑛𝑥 −2 𝑛−2 𝑧 − 𝑛𝑦
(𝜋31 − 𝜋21 )) , (𝜋32 − 𝜋22 )) ,
(44)
D. Rinaldi, F. Daví and L. Montalto
𝐴(𝐾 , sin 𝜑𝑜 )
𝐹𝜎,𝑧𝑧 =
−2 𝑛−2 𝑧 − 𝑛𝑦
(𝜋33 − 𝜋13 −
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
𝑛−2 𝑧 𝑛−2 𝑧
− 𝑛−2 𝑥 (𝜋33 − 𝑛−2 𝑦
− 𝜋23 )) ;
with 𝜑𝑜 given by (43) with 𝜎𝑥𝑥 = 𝜎𝑦𝑦 = 𝜎𝑧𝑧 = 0. In the case of uniaxial stress in the plane normal to the optic bisectrix, (𝜎𝑥𝑥 , 𝜎𝑧𝑧 ) we may evaluate the stress provided the components of 𝛱 are known. If we consider instead shear stresses, for instance a shear stress 𝜎𝑥𝑦 , then we have: √ 𝐵𝑥𝑥 + 𝐵𝑦𝑦 𝐵 − 𝐵𝑦𝑦 2 𝜎 2 + ( 𝑥𝑥 𝐵𝜁 = − 𝜋66 )2 , 𝑥𝑦 2 2 √ 𝐵 − 𝐵𝑦𝑦 𝐵𝑦𝑦 + 𝐵𝑦𝑦 2 𝜎 2 + ( 𝑥𝑥 𝐵𝜂 = + 𝜋66 )2 , (45) 𝑥𝑦 2 2 𝐵𝜉 = 𝐵𝑧𝑧 , Fig. 5. Rotation of 𝜃 about the 𝑏-axis of the frame with respect to the frame. The 𝛴 frame (red) is rotated by 𝛾 with respect to and 𝜉. The prismatic crystal specimen is represented by the blue lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
and the principal frame 𝛴 is obtained by a rotation about 𝐞2 of an angle 𝛾: 2𝜋66 𝜎𝑥𝑦
tan 2𝛾 =
𝐵𝑦𝑦 − 𝐵𝑥𝑥
(46)
.
If we assume 𝐵𝜉 > 𝐵𝜂 > 𝐵𝜁 , then the optic axis are directed as:
𝐹𝜎,𝑥𝑧 = 0 ;
(47)
𝐦1,2 = ± sin 𝜑𝐮1 + cos 𝜑𝐮3 ,
unsurprisingly, the photoelastic constant associated with the shear stress 𝜎𝑥𝑦 is zero. Indeed in [8] it is show that in Orthorhombic crystal shear stresses make the optic plane to rotate without changing the optic angle and hence without modifying the Cassini-like curves up to a rigid motion of amplitude 𝛾.
where sin 𝜑 and cos 𝜑 are given by (6) and (45). With these relations we can evaluate some components of the piezooptic tensor 𝛱, provided the crystals is stress-free, by the means of a loading device which controls uniaxial stresses or the biaxial stress corresponding to the simple shear like 𝜎𝑥𝑦 . Moreover, by applying in turn shear stresses 𝜎𝑥𝑧 and 𝜎𝑦𝑧 in place of 𝜎𝑥𝑦 we may evaluate the components 𝜋55 and 𝜋44 . Conversely, if the components of 𝛱 are given, we can give an estimate of the residual stress within the specimen. To this regard, consider as an example a plane stress 𝐓𝐞2 = 𝟎 which could represents for instance a state of residual stress in a sample cut normally to the optical bisectrix 𝐞2 , like the Alexandrite of Remark 1; we assume that the stress 𝜎𝑦𝑦 , 𝜎𝑥𝑦 and 𝜎𝑦𝑧 have mean zero value over the volume: ⎡ 𝜎𝑥𝑥 𝐓≡⎢ ⋅ ⎢ ⎣ ⋅
0 0 ⋅
𝜎𝑥𝑧 0 𝜎𝑧𝑧
⎤ ⎥; ⎥ ⎦
In order to give an estimate of the residual stress within the specimen, provided we have a complete knowledge of the components of 𝛱 and 𝐁𝑜 , we have again the three independent relations (36), (37) and (38) where 𝐅𝜎 is given by (52), 𝜑(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 ) is given by (51) and the rotation of the optic plane is given by (50).
3.2.2. Non-coherent prisms A more typical experimental setting is when the prismatic specimen is not oriented as the crystallographic axes and hence an uniaxial stress in the frame leads to a more complex state of stress in the frame . Let assume as an example that the axis 𝑏 and 𝑦 are parallel and that the frame is rotated by an angle 𝜃 with respect to : we may assume that 𝜃 can be determined by XRD diffraction techniques, for instance (see Fig. 5).
(48)
by (2) we arrive at the principal values for 𝐁(𝐓) 1 (𝐵 + 𝐵𝑧𝑧 + 𝜋̂ 1+ 𝜎𝑥𝑥 + 𝜋̂ 3+ 𝜎𝑧𝑧 ± 𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 )) 2 𝑥𝑥 𝐵𝜁 = 𝐵𝑦𝑦 + 𝜋21 𝜎𝑥𝑥 + 𝜋23 𝜎𝑧𝑧 ,
𝐵𝜉,𝜂 =
If the specimen undergoes an uniaxial stress 𝜎𝑧𝑧 , for instance by means of a four-point bending test, then in the frame the stress tensor is represented by the following components:
(49)
here 𝜋̂ 1± = 𝜋31 ± 𝜋11 and 𝜋̂ 3± = 𝜋33 ± 𝜋13 and √ 2 𝜎2 𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) = (𝐵𝑧𝑧 − 𝐵𝑥𝑥 + 𝜋̂ 1− 𝜎𝑥𝑥 + 𝜋̂ 3− 𝜎𝑧𝑧 )2 + 4𝜋55 𝑥𝑧
𝐓≡
with the principal frame 𝛴 rotated about 𝐞2 by an angle 𝛾 measured from 𝐞1 : tan 𝛾 =
𝐵𝜉 − 𝐵𝑥𝑥 − 𝜋11 𝜎𝑥𝑥 − 𝜋13 𝜎𝑧𝑧 𝜋55 𝜎𝑥𝑧
If we assume 𝐵𝜉 > 𝐵𝜂 > 𝐵𝜁 , then the optic axis directions have the same expression as in (47) with the optic angle 𝜑 given by: sin2 𝜑 =
2𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) , 𝐵𝑥𝑥 + 𝐵𝑧𝑧 − 2𝐵𝑦𝑦 + 𝜋̃1+ 𝜎𝑥𝑥 + 𝜋̃3+ 𝜎𝑧𝑧 + 𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 )
𝐹𝜎,𝑧𝑧 =
𝐴(𝐾 , sin 𝜑𝑜 ) −2 𝑛−2 𝑧 − 𝑛𝑥
𝐴(𝐾 , sin 𝜑𝑜 ) −2 𝑛−2 𝑧 − 𝑛𝑥
(𝜋31 − 𝜋11 − (𝜋33 − 𝜋13 −
−2 𝑛−2 𝑥 − 𝑛𝑦 −2 𝑛−2 𝑧 − 𝑛𝑦 −2 𝑛−2 𝑥 − 𝑛𝑦 −2 𝑛−2 𝑧 − 𝑛𝑦
⎤ ⎥; ⎥ ⎦
(53)
𝐵𝑎𝑎 + 𝐵𝑐𝑐 1 + 𝜎𝑧𝑧 [𝜋1+ (1 − cos 2𝜃) + 𝜋3+ (1 + cos 2𝜃)] ± 𝑅(𝜎𝑧𝑧 ) 2 4 1 𝐵𝜁 = 𝐵𝑏𝑏 + 𝜎𝑧𝑧 (𝜋21 (1 − cos 2𝜃) + 𝜋23 (1 + cos 2𝜃)) , (54) 2
(51)
here 𝑅(𝜎𝑧𝑧 ) √ 𝜎 1 = (𝜎𝑧𝑧 𝜋55 sin 2𝜃)2 + (𝐵𝑐𝑐 − 𝐵𝑎𝑎 + 𝑧𝑧 [𝜋1− + 𝜋3− + (𝜋3− − 𝜋1− ) cos 2𝜃)]2 , 2 2
(𝜋31 − 𝜋21 )) , (𝜋33 − 𝜋23 )) ,
−𝜎𝑧𝑧 sin 2𝜃 0 𝜎𝑧𝑧 (1 + cos 2𝜃)
𝐵𝜉,𝜂 =
where 𝜋̃1+ = 𝜋31 + 𝜋11 − 2𝜋21 and 𝜋̃3+ = 𝜋33 + 𝜋31 − 2𝜋23 . By (49) and (19) then we get the components of the photoelastic constants tensor 𝐅𝜎 : 𝐹𝜎,𝑥𝑥 =
0 0 ⋅
From (2) with (53) we arrive at the following principal values (here the components of 𝛱 are evaluated in the frame , as well as those of 𝐁𝑜 ):
(50)
.
⎡ 𝜎 (1 − cos 2𝜃) 1 ⎢ 𝑧𝑧 ⋅ 2⎢ ⋅ ⎣
with 𝜋1± = 𝜋31 ± 𝜋11 and 𝜋3± = 𝜋33 ± 𝜋13 with (52)
tan 𝛾 = 6
𝜋55 𝜎𝑧𝑧 sin 2𝜃 . 𝐵𝜉 − 𝐵𝑎𝑎
(55)
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
Fig. 7. The relation between the frames and . The 𝑧-axis coincides with the crystallographic 𝑐-axis, the coordinates in the square brackets are referred to the lattice vectors of .
Fig. 6. The relation between the frames and .
radius. Accordingly, from an experimental point of view these stress are very difficult to detect. Any other state of stress makes the crystal optically biaxial and the optic plane rotates about a generic direction, depending on the applied stress: since it is not possible to obtain analytical closed-form solutions we need to search for approximate solution by following e.g. the approximation procedure proposed in [9]; this is beyond the scope of the present work and shall be dealt with into a forthcoming paper.
By (19) we obtain the following expression for the photoelastic constant: 𝐴(𝐾 , sin 𝜑𝑜 ) ((𝜋1− (1 − cos 2𝜃) + 𝜋3− (1 + cos 2𝜃)) (56) 𝐹𝜎,𝑧𝑧 = −2 2(𝑛−2 𝑎 − 𝑛𝑐 ) −2 𝑛−2 𝑎 − 𝑛𝑏
𝜋 (𝐵𝑎𝑎 + 𝐵𝑐𝑐 − (𝜋21 − 1− )(1 − cos 2𝜃) −2 2 𝑛−2 − 𝑛 𝑎 𝑐 𝜋3− − (𝜋23 − )(1 + cos 2𝜃))) . 2 −
(57)
4.1.2. Classes 3, 3̄ The piezooptic tensor 𝛱 for the crystallographic classes 3, 3̄ has the following representation in the frame :
4. Stress analysis in optically uniaxial crystals 4.1. Trigonal crystals
⎡ ⎢ ⎢ [𝛱] ≡ ⎢ ⎢ ⎢ ⎢ ⎣
For the crystals of the Trigonal group we follow [22] which, instead of the relations proposed into [11] or [12], gives the following relations between the frames and (Fig. 6) based only on the Trigonal angle 𝛼: √ 1 𝛼 𝛼 𝛼 1 (4 cos2 − 1)𝐞3 , 𝐚 = sin 𝐞1 − √ sin 𝐞2 + 2 2 3 2 3 √ 2 𝛼 1 𝛼 𝐛 = √ sin 𝐞2 + (4 cos2 − 1)𝐞3 , (58) 2 3 2 3 √ 𝛼 𝛼 1 𝛼 1 (4 cos2 − 1)𝐞3 , 𝐜 = − sin 𝐞1 − √ sin 𝐞2 + 2 2 3 2 3
0 𝐵𝑥𝑥 0
0 0 𝐵𝑧𝑧
⎤ ⎥, ⎥ ⎦
𝜋11 𝜋12 𝜋31 𝜋41 0 0
𝜋12 𝜋11 𝜋31 −𝜋41 0 0
𝜋13 𝜋13 𝜋33 0 0 0
0 0 0 0 𝜋44 𝜋14
0 0 0 0 2𝜋41 𝜋11 − 𝜋12
⎤ ⎥ ⎥ ⎥; ⎥ ⎥ ⎥ ⎦
𝜋14 −𝜋14 0 𝜋44 −𝜋45 𝜋25
−𝜋25 𝜋25 0 𝜋45 𝜋44 𝜋14
2𝜋62 −2𝜋62 0 2𝜋52 2𝜋41 𝜋11 − 𝜋12
⎤ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(61)
For the crystals of the Hexagonal group with 𝜑 < 𝜋∕2 with respect to the 𝑧-axis, we follow [12] and identify 𝐞1 with the crystallographic direction [1 , 1 , 0], to obtain the following relation between the frames and (Fig. 7): √ √ 3 3 1 1 𝐞2 , 𝐛 = 𝐞1 + 𝐞 , 𝐜 = 𝐞3 , (62) 𝐚 = 𝐞1 − 2 2 2 2 2
(59)
𝜋14 −𝜋14 0 𝜋44 0 0
𝜋13 𝜋13 𝜋33 0 0 0
4.2. Hexagonal crystals
̄ 32, 3𝑚 4.1.1. Classes 3𝑚, These are the higher-symmetry classes for which the piezooptic tensor 𝛱 admits the representation in : ⎡ ⎢ ⎢ [𝛱] ≡ ⎢ ⎢ ⎢ ⎢ ⎣
𝜋12 𝜋11 𝜋31 −𝜋41 𝜋52 𝜋62
As it is shown into [8], also for these Trigonal classes a spherical stress 𝐓 = 𝜎𝑚 𝐈, an uniaxial stress 𝜎𝑧𝑧 and an uniform plane stress 𝜎𝑥𝑥 = 𝜎𝑦𝑦 preserve the crystal optical symmetry and therefore the stressed crystal remains uniaxial. For any other state of stress the considerations made in the previous subsection still apply.
The Trigonal group is optically Uniaxial and hence the tensor 𝐁𝑜 has diagonal representation in the frame , which coincides with the principal frame 𝛴: ⎡ 𝐵𝑥𝑥 𝐁𝑜 ≡ ⎢ 0 ⎢ ⎣ 0
𝜋11 𝜋12 𝜋31 𝜋41 −𝜋52 −𝜋62
The Hexagonal group is optically Uniaxial and hence the tensor 𝐁𝑜 has diagonal representation in the frame , which coincides with the principal frame 𝛴: ⎡ 𝐵𝑥𝑥 𝐁𝑜 ≡ ⎢ 0 ⎢ ⎣ 0
(60)
0 𝐵𝑥𝑥 0
0 0 𝐵𝑧𝑧
⎤ ⎥. ⎥ ⎦
(63)
Remark 2. An Hexagonal crystal is for instance the LaBr3 :Ce (Cerium doped Lanthanium Bromide) with a unit cell 𝑎 = 7.9648 Å, 𝑐 = 4.5119 Å. The refraction indices are 𝑛𝑜 = 2.04, and 𝑛𝑒 = 2.09 [24]. The components of the Dielectric Impermeability tensor are 𝐵𝑥𝑥 = 0.240, 𝐵𝑧𝑧 = 0.228. Since 𝑛𝑒 > 𝑛𝑜 , the crystal is optically positive. □
a complete characterization of these components was obtained, by the means of different techniques, in [23] for LiNbO3 (Lithium Niobate). In [8] we show that for these classes any spherical stress 𝐓 = 𝜎𝑚 𝐈, the uniaxial stress 𝜎𝑧𝑧 and the uniform plane stress 𝜎𝑥𝑥 = 𝜎𝑦𝑦 leave the crystal uniaxial, the circular interference fringe modifying only their 7
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
4.2.1. Classes 6, 6̄ and 6∕𝑚
When the crystal is acted by the plane stress 𝐓𝐞3 = 𝟎 with 𝜎𝑥𝑧 = 𝜎𝑦𝑧 = 𝜎𝑧𝑧 = 0 we have instead:
The piezooptic tensor 𝛱 for the crystallographic classes 6, 6̄ and 6∕𝑚 has the following representation in the frame [12]: ⎡ ⎢ ⎢ [𝛱] ≡ ⎢ ⎢ ⎢ ⎢ ⎣
𝜋11 𝜋12 𝜋31 0 0 −𝜋62
𝜋12 𝜋11 𝜋31 0 0 𝜋62
𝜋13 𝜋13 𝜋33 0 0 0
0 0 0 𝜋44 −𝜋45 0
0 0 0 𝜋45 𝜋44 0
2𝜋62 −2𝜋62 0 0 0 𝜋11 − 𝜋12
⎤ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
𝜋11 + 𝜋12 (𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) ± 𝑅(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 ) , 2 𝐵𝜁 = 𝐵𝑧𝑧 + 𝜋31 (𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) ,
𝐵𝜉,𝜂 = 𝐵𝑥𝑥 +
where
(64)
𝑅(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 ) =
(71)
√ 2 ) + 𝐾 (𝜎 − 𝜎 )𝜎 , 𝐾1 ((𝜎𝑥𝑥 − 𝜎𝑦𝑦 )2 + 4𝜎𝑥𝑦 2 𝑥𝑥 𝑦𝑦 𝑥𝑦
2 𝐾1 = (𝜋11 − 𝜋12 )2 + 4𝜋62 ,
(72)
𝐾2 = 2(𝜋11 − 𝜋12 )𝜋62 ,
As it is shown into [8], for these hexagonal classes a spherical stress 𝐓 = 𝜎𝑚 𝐈, an uniaxial stress 𝜎𝑧𝑧 and an uniform plane stress 𝜎𝑥𝑥 = 𝜎𝑦𝑦 preserve the crystal optical symmetry and therefore the stressed crystal remains uniaxial, the interference fringes in the planes orthogonal to the 𝑧-axis still being circumferences.
whereas the frame 𝛴 rotates about 𝐞3 by an angle 𝛾: tan 𝛾 =
𝐵𝜉 − 𝐵𝑥𝑥 2𝜋62 (𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) + (𝜋11 − 𝜋12 )𝜎𝑥𝑦
(73)
.
Any state of stress different from these changes the optical symmetry into a Biaxial one. In detail, as it was shown into [8], a simple uniaxial stress 𝜎𝑥𝑥 (or 𝜎𝑦𝑦 ) and the shear 𝜎𝑥𝑦 makes the crystal biaxial with the optic plane which rotates about the 𝑧-axis. In the case of the shear 𝜎𝑦𝑧 and 𝜎𝑥𝑧 the optic plane rotates to become orthogonal to the 𝑥- and 𝑦-axis respectively, with one of the optic axis directed as the 𝑧-axis.
Once again the crystal becomes biaxial and we may assume that (47) still holds with
Rather than analyze all these cases in detail, the straightforward analysis can be done as in the previous section, we shall limit to two cases: a ‘‘diagonal’’ stress tensor like (41) and the plane stress 𝐓𝐞3 = 𝟎. In the first case the results can be used to evaluate the components of 𝛱, whereas in the second case we can give an estimate for a residual stress.
When we use (20) to evaluate the photoelastic constant tensor components we notice that 𝐵𝜉 − 𝐵𝜂 is an homogeneous function of degree one of the principal stresses and its derivatives with respect to the cartesian stress components are homogeneous functions of degree zero: by a well-known result, the limit for 𝐓 → 𝟎 exists if and only if the derivatives are constant, say, they are independent on the stress components. Since this is not the case, we circumvent the problem by expressing the stress components in terms of the principal stresses (𝜎1 , 𝜎2 , 𝜎3 = 0) which are related to the cartesian components by the means of (cf. e.g. [25]): 𝜎 − 𝜎2 𝜎 + 𝜎2 + cos 2𝜓 1 , 𝜎𝑥𝑥 = 1 2 2 𝜎1 + 𝜎2 𝜎1 − 𝜎2 𝜎𝑦𝑦 = − cos 2𝜓 , (75) 2 2 𝜎1 − 𝜎2 𝜎𝑥𝑦 = sin 2𝜓 , 2 where 𝜓 is the rotation about the axis 𝐞3 of the principal stress frame 𝛴𝐓 :
sin2 𝜑 =
whereas the frame 𝛴 rotates about 𝐞3 ≡ 𝐮3 by an angle 𝛾: √ 2 𝜋11 − 𝜋12 ± (𝜋11 − 𝜋12 )2 + 4𝜋62 tan 𝛾 = 2𝜋62
(65)
(66)
we notice that the angle 𝛾 is independent on the stress. tan 2𝜓 =
The crystal becomes biaxial: provided we assume 𝐵𝜉 > 𝐵𝜂 > 𝐵𝜁 the optic axis are directed as in (47) with:
sin2 𝜑 =
√ 2 (𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) (𝜋11 − 𝜋12 )2 + 4𝜋62 2𝐵𝑥𝑥 − 𝐵𝑧𝑧 + (𝜋11 + 𝜋12 + 2𝜋31 )(𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) + (2𝜋13 + 𝜋33 )𝜎𝑧𝑧
.
√ 2 ; (𝜋11 − 𝜋12 )2 + 4𝜋62
𝜎𝑥𝑥 − 𝜎𝑦𝑦
(76)
.
(77)
(78)
𝐶(𝜎1 , 𝜎2 ) = 𝐹𝜎,1 (𝜎1 − 𝜎2 ) .
As in the previous section, in order to estimate the residual stress we have three relations, namely (73), (74) and (78), provided we know the components of 𝛱.
(68) ̄ 4.2.2. Classes 6𝑚2, 622, 6 mm and 6∕𝑚𝑚𝑚 For these higher-symmetry classes the piezooptic tensor 𝛱 admits the representation in :
these relations extend to hexagonal crystals the similar ones obtained into [4]. These relations are useful in order to obtain an estimate for the components of 𝛱 as it is shown into [3].
⎡ 𝜋11 ⎢ 𝜋12 ⎢ 𝜋 [𝛱] ≡ ⎢ 31 ⎢ 0 ⎢ 0 ⎢ ⎣ 0
Since for unstressed hexagonal crystals 𝐶(𝟎) = 0 and 𝜑𝑜 = 0 then the ellipticity factor is a linear function of 𝐅𝜎 whose non-null components are √ 𝜋 − 𝜋12 2 𝐴(𝐾 , 0) 2 , ( 11 ) + 4𝜋62 (69) 𝐹𝜎,𝑥𝑥 = −𝐹𝜎,𝑦𝑦 = −2 2 𝑛−2 𝑜 − 𝑛𝑒
𝜋12 𝜋11 𝜋31 0 0 0
𝜋13 𝜋13 𝜋33 0 0 0
0 0 0 𝜋44 0 0
0 0 0 0 𝜋44 0
0 0 0 0 0 𝜋11 − 𝜋12
⎤ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
(79)
and, provided we set 𝜋11 = 𝜋22 , 𝜋13 = 𝜋23 , 𝜋31 = 𝜋32 , 𝜋44 = 𝜋45 and 𝜋66 = 𝜋11 − 𝜋12 , all the results we obtained for Orthorhombic crystal still hold.
i.e.: 𝐶(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑧𝑧 ) = 𝐹𝜎,𝑥𝑥 (𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) .
.
and since 𝐶(𝟎) = 0 and 𝜑𝑜 = 0 then:
It is interesting to remark that from (65) we may obtain a simple relation between the dielectric permeability and the stress components:
𝑆(𝛱) =
2𝜎𝑥𝑦
With (75), then by (20) we arrive at √ 𝐴(𝐾 , 0) 1 𝐹𝜎,1 = −𝐹𝜎,2 = 𝐾1 + 𝐾2 sin 2𝜓 cos 2𝜓 , −2 2 𝑛−2 𝑜 − 𝑛𝑒
(67)
𝐵𝜉 − 𝐵𝜂 = 𝑆(𝛱)(𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) ,
2(𝐵𝑥𝑥 − 𝐵𝑧𝑧 ) + (𝜋11 + 𝜋12 − 2𝜋13 )(𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) + 2𝑅(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 )
(74)
In the first case we have that the eigenvalues of 𝐁(𝐓) are: 𝜋 + 𝜋12 (𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) + 𝜋13 𝜎𝑧𝑧 , 𝐵𝜉,𝜂 = 𝐵𝑥𝑥 + 11 2 √ 𝜋 − 𝜋12 2 2 (𝜎 − 𝜎 ) ± ( 11 ) + 𝜋62 𝑥𝑥 𝑦𝑦 2 𝐵𝜁 = 𝐵𝑧𝑧 + 𝜋31 (𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) + 𝜋33 𝜎𝑧𝑧 ,
2𝑅(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 )
(70) 8
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
5. Conclusions
[2] L. Montalto, N. Paone, L. Scalise, D. Rinaldi, A photoelastic measurement system for residual stress analysis in scintillating crystals by conoscopic imaging, Rev. Sci. Instrum. 86 (2015) 063102. [3] P.P. Natali, L. Montalto, F. Daví, P. Mengucci, A. Ciriaco, N. Paone, D. Rinaldi, Theoretical and experimental evaluation of piezo-optic parameters and photoelastic constant in tetragonal PWO, Appl. Opt. 57 (4) (2018) 730–737. [4] L. Montalto, F. Daví, D. Rinaldi, On the photoelastic constants and the Brewster law for stressed tetragonal crystals, Math. Methods Appl. Sci. 41 (8) (2018) 3103–3116. [5] E.E. Wahlstrom, Optical Crystallography, Wiley, New York, 1960. [6] D. Rinaldi, P. Pietroni, F. Daví, Isochromate fringes simulation by Cassini-like curves for photoelastic analysis of birifringent crystals, Nuclear Instrum. Methods Phys. Res. A 603 (2009) 294–300. [7] F. Daví, D. Rinaldi, Mechanical and optical properties of anisotropic single-crystal prisms, J. Elast. 120 (2) (2015) 197–224. [8] F. Daví, On the Bertin surfaces of photoelastic crystals, J. Opt. Soc. Am. A 32 (12) (2015) 2323–2337. [9] Y.I. Sirotin, M.P. Shaskolskaya, Fundamentals of Crystal Physics, Mir-Moscow, 1982. [10] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1985. [11] M.V. Perelomova, M.M. Tagieva, Problems in Crystal Physics with Solutions, MIR Publishers, Moscow, 1983, English trans.. [12] A. Authier (Ed.), International Tables for Crystallography. Volume D: Physical Properties of Crystals, Kluwer Academic Publ, Dordrecht, 2003. [13] H. Aben, C. Guillemet, Photoelasticity of Glass, Springer Verlag, 1993. [14] A. Ciriaco, F. Daví, M. Lebeau, G. Majni, N. Paone, P. Pietroni, D. Rinaldi, PWO Photo-elastic parameters calibration by laser-based polariscope, Nuclear Instrum. Methods Phys. Res. A 570 (2007) 55–60. [15] M. Lebeau, L. Gobbi, G. Majni, N. Paone, P. Pietroni, D. Rinaldi, Mapping residual stresses in PbWO4 crystals using photoelastic analysis, Nuclear Instrum. Methods Phys. Res. A 537 (2005) 207–210. [16] D. Rinaldi, A. Ciriaco, M. Lebeau, N. Paone, Quality control on pre-serial bridgman production of PbWO4 scintillating crystals by means of photoelasticity, Nuclear Instrum. Methods Phys. Res. A 615 (2010) 254–258. [17] M. Born, E. Wolf, Principles of Optics, VIIth Ed., Cambridge University Press, Cambridge, 1999. [18] L. Montalto, P.P. Natali, F. Daví, P. Mengucci, N. Paone, D. Rinaldi, Characterization of a defective PbWO4 crystal cut along the 𝑎 − 𝑐 crystallographic plane: structural assessmnet and a novel photoelastic stress analysis, J. Instrum. 12 (2017). [19] P. Helnwein, Some remarks on the compressed matrix representation of symmetric second-order and fourth-order tensors, Comput. Methods Appl. Mech. Eng. 190 (22–23) (2001) 2753–2770. [20] B.G. Mytsyk, Kost’ Ya. P., N.M. Demyanyshyn, V.M. Gaba, O.M. Sakharuk, Study of piezo-optic effect of calcium tungstate crystals by the conoscopic method, Opt. Mater. 39 (2015) 69–73. [21] P. Loiko, A. Major, Dispersive properties of alexandrite and beryllium hexaaluminate crystals, Opt. Mater. Express 6 (7) (2016) 263034. [22] M. Mehl, J.D. Hicks, C. Toher, O. Levy, R.M. Hanson, G.L.W. Hart, S. Curtarolo, The AFLOW library of crystallographic prototypes: part 1, Comput. Mat. Sci. 136 (2017) S1–S828. [23] O. Krupich, V. Savaryn, R. Vlokh, Precise determination of full matrix of piezooptic coefficients with a four-point bending technique: the example of lithium niobate crystals, Appl. Opt. 53 (10) (2014) B1–B6. [24] D.J. Singh, Structure and optical properties of high light output halide scintillators, Phys. Rev. B 82 (2010) 155145. [25] I.S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd edition 1956, McGraw Hill, 1946.
The results we presented allow either for an estimate the components of the piezo-optic tensor 𝛱 or an estimate of the average residual stress along the specimen thickness for crystals of four point groups, two optically Biaxial (Monoclinic and Orthorhombic) and two Uniaxial (Trigonal and Hexagonal). We limited our analysis to optical observations in the plane normal to the acute bisectrix of optic axes (for biaxial crystals) or normal to the optic axis for the uniaxial: this is because the interference fringes which can be observed by the means of conoscopy are closed ‘‘Cassini-like’’ curves and in such a case the ellipticity ratio 𝐶 is an useful parameter. By following the same approach of [4] and by making use of the results obtained in [8], we reduced our analysis to two cases which are interesting from an experimental point of view: a combination of three different uniaxial loads and a plane stress in the observation plane. For Orthorhombic crystals we also analyzed the possibility of a specimen which is not oriented along the crystallographic directions (the non-coherent case). In all cases explicit formulae for the optic angle 𝜑, the principal refraction indices and the rotation of the optic plane 𝛾 are given; we also provide for the explicit expressions of the components of the photoelastic constants tensor 𝐅𝜎 , which generalizes to a generic state of stress the notion of photoelastic constant 𝑓𝜎 previously introduced in [14] for uniaxial stress. In order to give an estimate for a residual stress when it is identified with a plane state of stress and hence represented by three stress components, provided the components of 𝛱 are given, we also obtained three independent relations which should allow to give a residual stress estimate. With these results at hand it should be possible to design an experimental set-up for the determination of the components of the tensor 𝛱 with a set of independent experiments or for a precise estimate of internal stress and a global quality evaluation of crystals. Acknowledgments The research leading to these results is within the scope of CERN R&D Experiment 18 ‘‘Crystal Clear Collaboration’’ and the PANDA Collaboration at GSI-Darmstadt. It has received funding from the Universitá Politecnica delle Marche, Italy, Progetto Strategico di Ateneo 2017: "Scintillating crystals: an interdisciplinary, applications-oriented approach aimed to the scientific knowledge and process control for application concerning life-quality improvement". References [1] L. Montalto, N. Paone, D. Rinaldi, L. Scalise, Inspection of birefringent media by photoelasticity: from diffuse light polariscope to laser conoscopic technique, Opt. Eng. 54 (8) (2015) 081210.
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Nuclear Inst. and Methods in Physics Research, A journal homepage: www.elsevier.com/locate/nima
On the photoelastic constants for anisotropic stressed crystals Daniele Rinaldi a,c , Fabrizio Daví b,c ,∗, Luigi Montalto a,c a b c
SIMAU, Universitá Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy DICEA, Universitá Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy ICRYS, Universitá Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy
ARTICLE
INFO
ABSTRACT
Keywords: Photoelasticity Photoelastic constant Anisotropic crystals Scintillating crystals
The need for high-quality and high-performance crystals for high-energy physics (e.g. scintillators) and biomedical applications require a good knowledge of their mechanical and optical properties, amongst the others. This is a mandatory step in many quality control process aimed at the improvement of the technological processes for crystals growth. Following the first analysis already done for Tetragonal crystals, in this paper we study the elasto-optic behavior of optically biaxial (Monoclinic and Orthorhombic groups) and uniaxial crystals (Trigonal and Hexagonal groups). We evaluate the photoelastic constants, the optic angle and the optic plane associated to various states of stress, in terms of the components of the Piezo-Optic tensor.
1. Introduction
where 𝑎 > 𝑏 are the semi-axes of the Cassini-like curves. Since such a parameter depends on the applied stress 𝐓 and we define a tensor of Photoelastic constants 𝐅𝜎 by the means of:
The study of the photoelastic behavior of a crystal by the means of pointwise conoscopic analysis allows for a not-destructive and "fast-toperform’’ technique which permits to detect the presence of internal stresses, which in turn are generally associated to the presence of defects and to quality degradation. As we showed in previous works, a reliable map of the stress in the crystal [1,2] can be obtained by the means of conoscopic techniques; moreover, since the link between the optical and mechanical properties of a crystal is the piezo-optic fourthorder tensor 𝛱, the same techniques can be used in order to estimate its components [3]. In [4] we presented, for the optically Uniaxial crystals of the Tetragonal group, a detailed study of the isochromate interference fringes (generated by the intersection of the crystal surfaces with the Bertin surfaces [5–8]) on the mechanical stresses, either applied or residual. In this paper, we wish to extend these results to anisotropic crystals of the Monocline, Orthorhombic, Trigonal and Hexagonal point groups [9–12]. We accordingly study Uniaxial and Biaxial crystals, leaving out only those of the Triclinic group or the optically isotropic crystals of the Cubic group or Isotropic materials like glass (vid. e.g. [13]). We limit our analysis to the fringes in the planes orthogonal to either the optic axis (for unstressed Uniaxial crystals) or to the optic axis acute bisectrix (for unstressed Biaxial ones) in such a way that, for low stress and small crystal thickness, these fringes are closed curves we called the Cassini-like curves [6,14]. It is possible therefore to define an experimentally measurable parameter, the Ellipticity Ratio 𝐶 [14]: 𝑎 𝐶 = − 1, 𝑏
𝐅𝜎 =
𝑑𝐶 | | , 𝑑𝐓 |𝐓=𝟎
in such a way that, to within higher-order terms in ‖𝐓‖, the ellipticity parameter 𝐶 is a linear function of the stress 𝐓: 𝐶(𝐓) = 𝐶(𝟎) + 𝐅𝜎 ⋅ 𝐓 + 𝑜(‖𝐓‖2 ). The Ellipticity Ratio 𝐶(𝐓) can be easily measured by the means of experimental photoelastic techniques and from a theoretical point of view, it can be evaluated for each of the symmetry group we analyze and for a general state of stress. We recall that a detailed knowledge of the dependence of 𝐶 on 𝐓 allows for either: • the design of a set of experiments aimed at the full knowledge of the piezo-optic tensor 𝛱 [3]; • the estimate of the residual stress within the crystal, generated for instance by the crystal growth process, provided we have detailed knowledge of 𝛱 [15,16]. The aim of this work, as in [4], is the evaluation of the ellipticity parameter 𝐶, the photoelastic constant tensor 𝐅𝜎 , the change in the optic angle and the rotation of the optic plane, in terms of the components of the piezo-optic tensor 𝛱 and of the dielectric impermeability tensor 𝐁0 in the unstressed state, under external (applied) or internal (residual) stress.
∗ Corresponding author at: DICEA, Universitá Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy. E-mail address: [email protected] (F. Daví).
https://doi.org/10.1016/j.nima.2019.162782 Received 5 August 2019; Accepted 16 September 2019 Available online 19 September 2019 0168-9002/© 2019 Elsevier B.V. All rights reserved.
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
Generally speaking we assume that the specimen and the crystal lattice are coherent in the sense that the specimen edges are parallel to the crystal lattice directions: however, for the Orthorhombic crystals we study also the case of non-coherent crystals and analyze the dependence of the results on the misalignment angle between specimen and crystal lattice. 2. Background: photoelastic crystals 2.1. Optical indicatrix and Bertin surfaces In this and in the following subsections, in order to make the paper self-contained, we shall recall some definitions and results which are presented elsewhere. The dielectric properties of an anisotropic crystal are described by the second-order symmetric and definite-positive dielectric impermeability tensor 𝐁, (vid. [9–11]) 𝐞=
1 𝐁𝐝 , 𝜖𝑜
(1)
where 𝐝 is the dielectric displacement vector, 𝐞 the electric field vector and 𝜖𝑜 the dielectric constant of vacuum. In Photoelastic crystals the dielectric impermeability is a linear function of the Cauchy stress tensor: (2)
𝐁(𝐓) = 𝐁𝑜 + 𝛱[𝐓] ,
where 𝛱 is the Maxwell piezo-optic tensor and 𝐁𝑜 is the dielectric impermeability in the unstressed state. By the positive-definiteness of 𝐁 we can define the Optical indicatrix or indices ellipsoid, i.e. is the locus of normalized constant dielectric energy [17] which summarizes at a glance all the information about the crystal optical anisotropy:
Fig. 1. The box represents the physical frame which may coincides with the specimen. The intrinsic crystal frame , related to the Bravais lattices and the principal frame 𝛴 associated to the eigenvectors {𝐮𝑘 }, 𝑘 = 1, 2, 3 of 𝐁 are also represented together with the optical indicatrix.
(3)
ordinary (multiplicity two) refraction indices and we define a crystal optically Positive (Negative) when 𝑛𝑒 > 𝑛𝑜 (𝑛𝑒 < 𝑛𝑜 ). For Biaxial crystals, if value of intermediate refractive index is closer to that of highest refractive index, the crystal is optically Negative, and if is closer to lowest refractive index then the crystal is optically Positive. A Biaxial crystal has two optic axes orthogonal to 𝐮𝜂 and lying in the (𝜉 , 𝜁 ) plane, the optic plane [5]; in this work we shall consider the 𝜁-axis as the acute bisector of the optic axes: we define 2𝜑 the acute angle between the optic axes which is given by [7] √ √ 𝐵𝜂 − 𝐵𝜁 𝐵𝜉 − 𝐵𝜂 cos 𝜑 = , sin 𝜑 = . (6) 𝐵𝜉 − 𝐵𝜁 𝐵𝜉 − 𝐵𝜁
𝐁𝐱 ⋅ 𝐱 = 1 ,
where 𝐱 is a typical point within the crystal. In the sequel, to evaluate the effects of applied and residual stresses on the crystal optical properties, we shall make use of different frames at 𝐱, namely: a A system relative to the crystallographic structure of the 14 Bravais lattices, the Intrinsic crystal frame ≡ [𝑎 , 𝑏 , 𝑐 ] where the coordinates are referred to the crystal lattice vectors {𝐚 , 𝐛 , 𝐜}; b An orthonormal frame, the Cartesian physical frame ≡ {𝑥 , 𝑦 , 𝑧} referred to the orthonormal base {𝐞1 , 𝐞2 , 𝐞3 } in which we shall define the components of 𝐁, 𝐓 and 𝛱 and which can be related to by the crystal symmetry. It represents the edges of a prismatic specimen, for instance;
An Uniaxial crystal has only an optic axis which coincides with a material symmetry axis: in this case 𝜑 = 0, the optic axis is directed as the eigenvector corresponding to min 𝐵𝑖 in positive (max 𝐵𝑖 negative) crystals and the optical indicatrix is an ellipsoid of revolution around the optic axis. The other surface which identifies the optical behavior of the material is the surface of equal phase difference or Bertin surface which can be obtained from the Fresnel equation [5,17]. The Bertin surfaces are fourth-order surfaces [6,7], whose mathematical aspects are studied in details into [8] (vid. Fig. 2-Left) and whose equation is
c The orthonormal Principal frame of 𝐁, 𝛴 ≡ {𝜉 , 𝜂 , 𝜁} where the coordinates are referred to the eigenvectors {𝐮1 , 𝐮2 , 𝐮3 } of 𝐁. We shall denote with 𝛴𝐁 and 𝛴𝐓 the principal frames for 𝐁(𝐓) and 𝐓 respectively; these frames and their reciprocal relations are presented in Fig. 1. In the general case the tensor 𝐁 is not diagonal in the frame : hence the optical indicatrix admits the following explicit representation in terms of the six independent components of 𝐁: 𝐵𝑥𝑥 𝑥2 + 𝐵𝑦𝑦 𝑦2 + 𝐵𝑧𝑧 𝑧2 + 2𝐵𝑥𝑦 𝑥𝑦 + 2𝐵𝑥𝑧 𝑥𝑧 + 2𝐵𝑦𝑧 𝑦𝑧 = 1 ;
𝜉 4 cos4 𝜑 + 𝜂 4 + 𝜁 4 sin4 𝜑 + 2𝜉 2 𝜂 2 cos2 𝜑 + 2𝜂 2 𝜁 2 sin2 𝜑
(4)
the parameter 𝐻, which has the dimension of a length, is defined as
however, in the principal frame 𝛴(4) reduces to: 2
2
2
𝐵𝜉 𝜉 + 𝐵𝜂 𝜂 + 𝐵𝜁 𝜁 = 1 ,
𝐵𝜉 ≥ 𝐵𝜂 ≥ 𝐵𝜁 ,
(7)
− 2𝜉 2 𝜁 2 sin2 𝜑 cos2 𝜑 − 𝐻 2 (𝜉 2 + 𝜂 2 + 𝜁 2 ) = 0 ;
𝐻= (5)
𝑁𝜆 , 𝑛𝑚𝑎𝑥 − 𝑛𝑚𝑖𝑛
(8)
where 𝑁 is the fringe order, 𝜆 is the wave length of the light source.
with the three refraction indices (𝑛𝜉 , 𝑛𝜂 , 𝑛𝜁 ) defined by 𝐵𝑖 = 𝑛−2 𝑖 , 𝑖 = 𝜉 , 𝜂 , 𝜁. According to the eigenvalue multiplicity a material can be optically Biaxial when all the eigenvalues are different, Uniaxial when two eigenvalues are equal and Isotropic whenever all the eigenvalues are equal. In Uniaxial crystals we say 𝑛𝑒 and 𝑛𝑜 the extraordinary and
2.2. Cassini-like curves The section of the Bertin surfaces with the crystal surface or a projection plane gives the analytical expression of the isochromate 2
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
By trivial calculations then we have: √ 𝑁(𝑢𝑜 ) 𝑑𝐶 | 1 𝐴(𝐾 , 𝜑𝑜 ) = = | 𝑑𝑢 |𝜎𝑖𝑗 =0 𝐷(𝑢𝑜 ) (1 − 𝑢𝑜 )2
(15)
1 𝑁,𝑢 (𝑢𝑜 )𝐷(𝑢𝑜 ) − 𝑁(𝑢𝑜 )𝐷,𝑢 (𝑢𝑜 ) , √ 1 − 𝑢𝑜 2𝐷(𝑢𝑜 ) 𝑁(𝑢𝑜 )𝐷(𝑢𝑜 )
+
where 𝑢𝑜 = sin2 𝜑𝑜 , being 𝜑𝑜 the optic angle in the unstressed state, and √ 𝑁(𝑢) = 1 + 2𝐾 2 (1 − 𝑢)𝑢 + 1 + 4𝐾 2 (1 − 𝑢) , √ 𝐷(𝑢) = 1 − 2𝐾 2 𝑢 + 1 + 4𝐾 2 (1 − 𝑢) ; we notice that (15) is a term which is independent on both the stress 𝐓 and the piezo-optic tensor 𝛱 and depends only on 𝜑𝑜 and on the adimensional ratio 𝐾 = 𝜁𝑜 ∕𝐻.1 For an unstressed uniaxial crystal, with 𝑢𝑜 = 0, the term (15) reduces to
Fig. 2. Left: the Bertin surfaces of a biaxial crystal for different orders 𝑁; in the uniaxial case they reduce to revolution surfaces around the 𝜁-axis. In the figure it is show also the optic plane, which contains the optic axes: the 𝜁 -axis is the bisector of the optic axes. Right: the Cassini-like curves representing the interference isochromate fringes as a section of the left figure with a plane 𝜁 = 𝜁𝑜 . Here 𝑎 > 𝑏 with 𝑎 = 𝑏 for uniaxial crystals.
𝐴(𝐾 , 0) = (1 +
2𝐾 2 ). √ 1 + 1 + 4𝐾 2
(16)
From (6) then we have: (𝐵𝜉 − 𝐵𝜂 ),𝑖𝑗 (𝐵𝜉 − 𝐵𝜁 ) − (𝐵𝜉 − 𝐵𝜁 ),𝑖𝑗 (𝐵𝜉 − 𝐵𝜂 ) 𝜕 𝐵𝜉 − 𝐵𝜂 𝜕𝑢 = = , 𝜕𝜎𝑖𝑗 𝜕𝜎𝑖𝑗 𝐵𝜉 − 𝐵𝜁 (𝐵𝜉 − 𝐵𝜁 )2
interference fringes, fourth-order plane curves which can be either closed or open and two-folded. The equations of the intersection curves between the Bertin surface and a generic plane are given into [8]: here we shall limit our analysis to planes which are orthogonal to the bisector of the optic axes in biaxial crystals or to the optic axis in uniaxial and hence with 𝜁 = 𝜁𝑜 . In both cases we have closed fourth-order curves 𝑓 = 𝑓 (𝜉 , 𝜂 , 𝜁𝑜 ) parameterized on 𝜁𝑜 that we call Cassini-like for their similarity with the Cassini’s curves [6]. Let {±𝑎 , ±𝑏} with 𝑎 > 𝑏 be the solutions of the equations 𝑓 (𝜉 , 0 , 𝜁𝑜 ) = 0 and 𝑓 (0 , 𝜂 , 𝜁𝑜 ) = 0, then it is possible to define the ellipticity ratio [14], an useful measurable parameter (Fig. 2-Right):
(17) with 𝑘 = 𝜉, 𝜂, 𝜁 , then (17), evaluated and since for 𝐓 = 𝟎 it is 𝐵𝑘 = 𝑛−2 𝑘 for 𝐓 = 𝟎, reduces to: −2 −2 𝜕(𝐵𝜉 − 𝐵𝜂 ) 𝑛𝜉 − 𝑛𝜂 𝜕(𝐵𝜉 − 𝐵𝜁 ) | 1 𝜕𝑢 | = . ( − )| | |𝐓=𝟎 𝜕𝜎𝑖𝑗 |𝐓=𝟎 𝑛−2 − 𝑛−2 𝜕𝜎𝑖𝑗 𝜕𝜎𝑖𝑗 𝑛−2 − 𝑛−2 𝜉 𝜁 𝜉 𝜁
From (13), (14), (15) and (18) then we have: 𝐹𝜎,𝑖𝑗 =
𝐹𝜎,𝑖𝑗 =
𝐚 ≡ sin 𝛽𝐞1 + cos 𝛽𝐞3 ,
𝐛 ≡ 𝐞2 ,
𝐜 ≡ 𝐞3 ,
(21)
In the frame the tensor 𝐁𝑜 for the Monoclinic group may be represented as (see e.g.[5]): ⎡ 𝐵𝑥𝑥 𝐁𝑜 ≡ ⎢ ⋅ ⎢ ⎣ ⋅
(12)
0 𝐵𝑦𝑦 ⋅
𝐵𝑥𝑧 0 𝐵𝑧𝑧
⎤ ⎥, ⎥ ⎦
(22)
and the principal frame 𝛴 is rotated about the monoclinic 𝑏-axis by an angle 𝛾𝑜 with: √ 𝐵 + 𝐵𝑧𝑧 𝐵 − 𝐵𝑧𝑧 2 𝑜 2 , 𝐵𝜉,𝜂 = 𝑥𝑥 ± ( 𝑥𝑥 ) + 𝐵𝑥𝑧 2 2
where the components of 𝐅𝜎 in the frame are given by: (13)
𝐵𝜁𝑜 = 𝐵𝑦𝑦 ,
Since by (11) we have that 𝐶 depends on 𝑢 = sin2 𝜑 and by (6)𝑢 depends on 𝜎𝑖𝑗 by means of (2), then we have: 𝜕𝐶 𝑑𝐶 𝜕𝑢 | = . | 𝜕𝜎𝑖𝑗 𝑑𝑢 𝜕𝜎𝑖𝑗 |𝜎𝑖𝑗 =0
(20)
Following [5,11,12], we choose the crystallographic frame with the 𝑏-axis directed as the 𝑦-axis and the 𝑐-axis directed as the 𝑧-axis, with the monoclinic 𝛽 angle comprised between 𝑎 and 𝑐. The relation between and the physical frame is (vid. Fig. 3).
For small stress the ellipticity parameter 𝐶 is
𝑖,𝑗 = 𝑥,𝑦,𝑧.
𝐴(𝐾 , 0) 𝜕(𝐵𝜉 − 𝐵𝜂 ) | . | −2 |𝐓=𝟎 𝜕𝜎𝑖𝑗 𝑛−2 𝑜 − 𝑛𝑒
3.1. Monoclinic crystals
2.3. The photoelastic constants tensor
𝜕𝐶 | , | 𝜕𝜎𝑖𝑗 |𝜎𝑖𝑗 =0
(19)
3. Stress analysis in optically biaxial crystals
𝜕𝐶 | . (10) 𝐶(𝐓) = 𝐶(𝟎) + 𝐅𝜎 ⋅ 𝐓 + 𝑜(‖𝐓‖2 ) , 𝐅𝜎 = | 𝜕𝐓 |𝐓=𝟎 Relation (10) allows to evaluate the stress by means of theoretical considerations and experimental measurements, like in [1–3,6,14,16, 18], Finally, by means of (9), (7) and the definition of 𝑎 , 𝑏, we obtain an explicit expression of 𝐶 in terms of 𝑢 = sin2 𝜑 ∈ [0 , 1] (vid. [4]): √ √ √ 𝜁 1 + 2𝐾 2 (1 − 𝑢)𝑢 + 1 + 4𝐾 2 (1 − 𝑢) 1 √ √ 𝐶(𝑢) = − 1 , 𝐾 = 𝑜 . (11) √ 1−𝑢 𝐻 2 2 1 − 2𝐾 𝑢 + 1 + 4𝐾 (1 − 𝑢)
𝐹𝜎,𝑖𝑗 =
−2 −2 𝐴(𝐾 , sin 𝜑𝑜 ) 𝜕(𝐵𝜉 − 𝐵𝜂 ) 𝑛𝜉 − 𝑛𝜂 𝜕(𝐵𝜉 − 𝐵𝜁 ) | , − )| ( −2 −2 −2 |𝐓=𝟎 𝜕𝜎𝑖𝑗 𝜕𝜎𝑖𝑗 𝑛𝜉 − 𝑛𝜁 𝑛𝜉 − 𝑛−2 𝜁
which for uniaxial crystals with 𝑛𝜉 = 𝑛𝜂 = 𝑛𝑜 and 𝑛𝜁 = 𝑛𝑒 reduces to (vid. [4]):
𝑎 − 1 > 0, (9) 𝑏 in such a way that 𝐶 = 0 in unstressed uniaxial crystal. Clearly, by means of (7), (6) and (2) the ellipticity ratio may depend on the stress 𝐓 and we may define as in [4] a Photoelastic constants tensor 𝐅𝜎 such that:
𝐶=
𝐶(𝐓) = 𝐶(𝟎) + 𝐅𝜎 ⋅ 𝐓 ,
(18)
(23)
1 To arrive at these results we assumed 𝐻 = const. vid. e.g. the discussion in [8].
(14) 3
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
the principal frame 𝛴𝐁 for 𝐁(𝐓) is obtained by a rotation of 𝛾 about the monoclinic 𝑏-axis 𝐞2 : tan 𝛾 =
𝐵𝜉 − 𝐵𝑥𝑥 − 𝜋11 𝜎𝑥𝑥 𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥
(28)
,
whereas the optic angle changes by (6) from (24) into: √ (𝐵𝑥𝑥 − 𝐵𝑧𝑧 + (𝜋11 − 𝜋31 )𝜎𝑥𝑥 )2 + 4(𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥 )2 sin2 𝜑 = , 2(𝐵𝜉 − 𝐵𝑦𝑦 − 𝜋21 𝜎𝑥𝑥 )
and the optic axes are directed as (25) with (28) and (29) in place of 𝛾𝑜 and 𝜑𝑜 . The photoelastic constants tensor 𝐅𝜎 has only one non-zero component and accordingly ( 𝐴(𝐾 , sin 𝜑𝑜 ) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋31 − 𝜋11 ) + 4𝐵𝑥𝑧 𝜋51 𝐶(𝜎𝑥𝑥 ) = 𝐶(𝟎) + √ 𝑛−2 − 𝑛−2 2 (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 4𝐵𝑥𝑧 𝜉 𝜁
Fig. 3. The frames , and 𝛴.
tan 𝛾𝑜 =
𝐵𝜉𝑜 − 𝐵𝑥𝑥 𝐵𝑥𝑧
,
−
Crystals of the Monoclinic group accordingly are optically Biaxial and the optic angle is, provided 𝐵𝜉 > 𝐵𝜂 > 𝐵𝜁 , given by (6):2 √ 2 (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 4𝐵𝑥𝑧 , (24) sin2 𝜑𝑜 = √ 2 𝐵𝑥𝑥 + 𝐵𝑧𝑧 − 2𝐵𝑦𝑦 + (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 4𝐵𝑥𝑧
+
(25)
= ± sin 𝜑𝑜 (sin 𝛾𝑜 𝐞1 + cos 𝛾𝑜 𝐞3 ) + cos 𝜑𝑜 𝐞3 , being the direction spanned by 𝐮3 the acute bisectrix of the optic angle 𝜑𝑜 . For all the classes of the Monoclinic group the piezo-optic tensor 𝛱 has the following representation in the frame (vid. e.g.[11])3 : ⎡ ⎢ ⎢ [𝛱] ≡ ⎢ ⎢ ⎢ ⎢ ⎣
𝜋11 𝜋21 𝜋31 0 𝜋51 0
𝜋12 𝜋22 𝜋32 0 𝜋52 0
𝜋13 𝜋23 𝜋33 0 𝜋53 0
0 0 0 𝜋44 0 𝜋64
𝜋15 𝜋25 𝜋35 0 𝜋55 0
0 0 0 𝜋46 0 𝜋66
⎤ ⎥ ⎥ ⎥; ⎥ ⎥ ⎥ ⎦
(26)
⎡ 𝜎𝑥𝑥 𝐓≡⎢ ⋅ ⎢ ⎣ ⋅
As it is shown into [8], for the Monoclinic group any combination of stress make the optic plane to rotate about the monoclinic 𝑏-axis, aside for the shear stresses 𝜎𝑦𝑧 and 𝜎𝑥𝑦 which rotate the optic plane about a generic direction. In order to give an estimate for the components of 𝛱 we can load the specimen with an uniaxial load (either 𝜎𝑥𝑥 , 𝜎𝑦𝑦 or 𝜎𝑧𝑧 ) or with a biaxial load which induces the simple shear 𝜎𝑥𝑧 . We shall show here only the case for 𝜎𝑥𝑥 , the other cases leading to similar results. The eigenvalues for 𝐁(𝐓) are, from (2) and (26) 𝐵 + 𝐵𝑧𝑧 𝜋 + 𝜋31 𝐵𝜉,𝜂 = 𝑥𝑥 + ( 11 )𝜎𝑥𝑥 2 2 √ 𝐵 − 𝐵𝑧𝑧 𝜋 − 𝜋31 ± ( 𝑥𝑥 + ( 11 )𝜎𝑥𝑥 )2 + (𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥 )2 , 2 2 𝐵𝜁 = 𝐵𝑦𝑦 + 𝜋21 𝜎𝑥𝑥 ,
(30)
0 0 ⋅
𝜎𝑥𝑧 0 𝜎𝑧𝑧
⎤ ⎥; ⎥ ⎦
(31)
𝐵𝜉,𝜂 = 𝛼𝑜+ + 𝛼1+ 𝜎𝑥𝑥 + 𝛼2+ 𝜎𝑧𝑧 + 𝛼3+ 𝜎𝑥𝑧 ± 𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) (32) 𝐵𝜁 = 𝐵𝑦𝑦 + 𝜋21 𝜎𝑥𝑥 + 𝜋23 𝜎𝑧𝑧 + 𝜋25 𝜎𝑥𝑧 , where 𝑅=
√ (𝛼𝑜− + 𝛼1− 𝜎𝑥𝑥 + 𝛼2− 𝜎𝑧𝑧 + 𝛼3− 𝜎𝑥𝑧 )2 + (𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥 + 𝜋53 𝜎𝑧𝑧 + 𝜋55 𝜎𝑥𝑦 )2
and 2𝛼𝑜± = 𝐵𝑥𝑥 ± 𝐵𝑧𝑧 , 2𝛼1± = 𝜋11 ± 𝜋31 , 2𝛼2± = 𝜋13 ± 𝜋33 and 2𝛼3± = 𝜋15 ± 𝜋35 . The principal frame 𝛴𝐁 for 𝐁(𝐓) is rotated about 𝐞2 by the angle 𝛾:
(27)
For different eigenvalues ordering the relation (6) reads:
sin2 𝜑𝑜 =
) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋31 − 𝜋11 ) + 4𝐵𝑥𝑧 𝜋51 ) 𝜎𝑥𝑥 , √ 2 2 (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 4𝐵𝑥𝑧
and by (2) arrive at the principal values for 𝐁(𝐓)
tan 𝛾 = 2
𝑛−2 − 𝑛−2 𝜋 + 𝜋31 − 2𝜋21 𝜂 𝜉 ( 11 −2 2 𝑛−2 − 𝑛 𝜉 𝜁
with 𝜑𝑜 given by (24), the term 𝐶(𝟎) is given by (11) with 𝑢 = sin2 𝜑𝑜 and the principal refraction index (𝑛𝜉 , 𝑛𝜂 , 𝑛𝜁 ) are evaluated in the unstressed state. The results for 𝜎𝑧𝑧 and 𝜎𝑥𝑧 can be obtained from (30) by replacing 𝜋11 , 𝜋31 , 𝜋51 respectively with 𝜋13 , 𝜋33 , 𝜋53 , and 𝜋15 , 𝜋35 , 𝜋55 , whereas those for 𝜎𝑦𝑦 by replacing 𝜋11 , 𝜋31 , 𝜋51 , 𝜋21 with 𝜋21 , 𝜋32 , 𝜋52 , 𝜋22 . All together the twelve relations (30), (28) and (29), which are in general not independent, do not allow for the complete determination of the twenty independent components of 𝛱 in monoclinic crystals: they are however a starting point for the design of further experiments aimed to a complete characterization of such a piezo-optic tensor. For crystals such that we have a full knowledge of the components of 𝛱 the relations we obtained can be used to give an estimate for the residual stress within the specimen. We limit our analysis to specimen that we assume ‘‘thin’’ enough along the 𝑏-axis in order to assume that the stress components 𝜎𝑦𝑦 , 𝜎𝑥𝑦 and 𝜎𝑦𝑧 be negligible. Accordingly we may consider the plane stress 𝐓𝐞2 = 𝟎:
and the two optic axis are directed as: 𝐦1,2 = ± sin 𝜑𝑜 𝐮1 + cos 𝜑𝑜 𝐮3 ,
(29)
𝐵𝜉 − 𝐵𝑥𝑥 − 𝜋11 𝜎𝑥𝑥 − 𝜋13 𝜎𝑧𝑧 − 𝜋15 𝜎𝑥𝑧 𝐵𝑥𝑧 + 𝜋51 𝜎𝑥𝑥 + 𝜋53 𝜎𝑧𝑧 + 𝜋55 𝜎𝑥𝑧
;
(33)
the optic axis directions have the same expression as in (25) with 𝛾 as in (33) and 𝜑 given by:
𝛥𝐵𝑚𝑖𝑛 . 𝛥𝐵max
3 Here we used for the component of 𝛱 the Voigt notation with 1 = 𝑥𝑥, 2 = 𝑦𝑦, 3 = 𝑧𝑧, 4 = 𝑦𝑧, 5 = 𝑥𝑧 and 6 = 𝑥𝑦. We notice that the piezo-optic tensor is not symmetric in this representation: the relation between this notation (sometimes referred as the Voigt–Mandel notation) and the components in of a fourth-order tensor is discussed in [19].
sin2 𝜑 =
2𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) , 2𝛼𝑜 − 2𝐵𝑦𝑦 + (𝛼1− + 2𝜋21 )𝜎𝑥𝑥 + (𝛼2− + 2𝜋23 )𝜎𝑧𝑧 + (𝛼3− + 2𝜋25 𝜎𝑥𝑧 ) + 𝑅
(34) in place of 𝜑𝑜 and 𝛾𝑜 . 4
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
In the frame ≡ the tensor 𝐁𝑜 for the Orthorhombic group has the diagonal representation: ⎡ 𝐵𝑥𝑥 𝐁𝑜 ≡ ⎢ 0 ⎢ ⎣ 0
0 𝐵𝑦𝑦 0
0 0 𝐵𝑧𝑧
⎤ ⎥, ⎥ ⎦
(39)
and hence crystals of the Orthorhombic group are optically Biaxial; by taking into account that all the principal axes can be the bisectrix of the optic axis [5], then both the crystallographic and physical frames coincide with the principal frame 𝛴. Clearly, depending on the physical properties of a given crystal the −2 ordering of the eigencouples (𝐮1 , 𝑛−2 ), (𝐮2 , 𝑛−2 𝜂 ) and (𝐮3 , 𝑛𝜁 ) can be 𝜉 −2 and 𝐵 = 𝑛−2 . different from the ordering of 𝐵𝑥𝑥 = 𝑛−2 , 𝐵 = 𝑛 𝑦𝑦 𝑧𝑧 𝑥 𝑦 𝑧 Fig. 4. The frames and .
Remark 1. Both the BeAl2 O4 (Chrysoberyllium) and its variant Cr:BeAl2 O4 (Alexandrite) are Orthorhombic crystals with lattice parameters 𝑎 = 9.404 Å, 𝑏 = 5.476 Å, 𝑐 = 4.727 Å. The refraction indices are 𝑛𝑥 = 1.736(4), 𝑛𝑦 = 1.741(9) and 𝑛𝑧 = 1.734(5) at 𝜆 = 755 nm [21]. The components of the Dielectric Impermeability tensor are 𝐵𝑥𝑥 = 0.3317, 𝐵𝑦𝑦 = 0.3296, 𝐵𝑧𝑧 = 0.3324. Since 𝑛𝑦 > 𝑛𝑥 > 𝑛𝑧 , then 𝐵𝑧𝑧 > 𝐵𝑥𝑥 > 𝐵𝑦𝑦 . We remark that in the 𝛴 frame of Fig. 1 and Eq. (4) the ordering is 𝐵𝑦𝑦 = 𝐵𝜁 < 𝐵𝑥𝑥 = 𝐵𝜂 < 𝐵𝑧𝑧 < 𝐵𝜉 , by following the convention that the optic plane is spanned by the eigenvectors of 𝑛−2 and 𝑛−2 . The crystal 𝜉 𝜁 is positive with 𝜑 < 𝜋∕2 with respect to the 𝑦-axis. In fact from (6) the 𝑦-axis is the acute bisector and the optic plane is the plane (𝑦 , 𝑧), with, in an unstressed crystal 𝜑𝑜 = 30◦ . □
By (32) and (19) then we get the three components of the photoelastic constants tensor 𝐅𝜎 : 𝐹𝜎,𝑥𝑥 =
𝐴(𝐾 , sin 𝜑𝑜 ) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋11 − 𝜋31 ) + 2𝐵𝑥𝑧 𝜋51 ( √ 𝑛−2 − 𝑛−2 2 (𝐵 − 𝐵 )2 + 𝐵 2 𝜉 𝜁 𝑥𝑥
−
𝐹𝜎,𝑧𝑧 =
− 𝑛−2 𝑛−2 𝜂 𝜉 𝑛−2 − 𝑛−2 𝜉 𝜁
(𝛼1+
𝑧𝑧
𝑥𝑧
2𝛼𝑜− 𝛼1− + 𝐵𝑥𝑧 𝜋51 )) , √ 2 + 𝐵2 𝛼𝑜− 𝑥𝑧
𝐴(𝐾 , sin 𝜑𝑜 ) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋13 − 𝜋33 ) + 2𝐵𝑥𝑧 𝜋53 ( √ 𝑛−2 − 𝑛−2 2 (𝐵 − 𝐵 )2 + 𝐵 2 𝜉 𝜁 𝑥𝑥
𝑧𝑧
(35)
For all the classes of the Orthorhombic group the piezo-optic tensor 𝛱 has the following representation in the frame (vid. e.g.[12]):
𝑥𝑧
𝑛−2 − 𝑛−2 𝜂 𝜉
2𝛼 𝛼2− + 𝐵𝑥𝑧 𝜋53 − )) , (𝛼2+ 𝑜− √ −2 −2 𝑛𝜉 − 𝑛𝜁 2 + 𝐵2 𝛼𝑜− 𝑥𝑧 𝐹𝜎,𝑥𝑧 =
−
⎡ 𝜋11 ⎢ 𝜋21 ⎢ 𝜋 [𝛱] ≡ ⎢ 31 ⎢ 0 ⎢ 0 ⎢ ⎣ 0
𝐴(𝐾 , sin 𝜑𝑜 ) (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )(𝜋15 − 𝜋35 ) + 2𝐵𝑥𝑧 𝜋55 ( √ 𝑛−2 − 𝑛−2 2 2 (𝐵𝑥𝑥 − 𝐵𝑧𝑧 )2 + 𝐵𝑥𝑧 𝜉 𝜁 𝑛−2 − 𝑛−2 𝜂 𝜉 𝑛−2 − 𝑛−2 𝜉 𝜁
(𝛼3+
2𝛼𝑜− 𝛼3− + 𝐵𝑥𝑧 𝜋55 )) . √ 2 + 𝐵2 𝛼𝑜− 𝑥𝑧
• The change in the ellipticity ratio:
⎡ 𝜎𝑥𝑥 𝐓≡⎢ 0 ⎢ ⎣ 0
𝛥𝐶 = 𝐶(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) − 𝐶(𝟎) = 𝐹𝜎,𝑥𝑥 𝜎𝑥𝑥 + 𝐹𝜎,𝑧𝑧 𝜎𝑧𝑧 + 𝐹𝜎,𝑥𝑧 𝜎𝑥𝑧 ; (36) with the three non zero components given by (35) • the change in the optic angle:4
0 0 0 𝜋44 0 0
0 0 0 0 𝜋55 0
0 0 0 0 0 𝜋66
⎤ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(40)
0 𝜎𝑦𝑦 0
0 0 𝜎𝑧𝑧
⎤ ⎥, ⎥ ⎦
(41)
we get: 𝐵𝜂 = 𝐵𝑥𝑥 + 𝜋11 𝜎𝑥𝑥 + 𝜋12 𝜎𝑦𝑦 + 𝜋13 𝜎𝑧𝑧 ,
(37)
(42)
𝐵𝜁 = 𝐵𝑦𝑦 + 𝜋21 𝜎𝑥𝑥 + 𝜋22 𝜎𝑦𝑦 + 𝜋23 𝜎𝑧𝑧 ,
where 𝜑(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) is given by (34) and 𝜑𝑜 by (24); • the rotation of the optic plane: 𝛥𝛾 = 𝛾(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) − 𝛾𝑜 ,
𝜋13 𝜋23 𝜋33 0 0 0
As it is shown into [8], for the Orthorhombic group a spherical stress 𝐓 = 𝜎𝑚 𝐈, any uniaxial stress and any uniform plane stress maintains the optic plane and changes only the amplitude of the optic angle 𝜑. Any shear stress instead rotate the optic plane about the direction orthogonal to the shear stress plane. Accordingly for any diagonal stress tensor
In order to give an estimate of the residual stress within the specimen, provided we have a complete knowledge of the components of 𝛱 and 𝐁𝑜 , we have the three independent relations:
𝛥𝜑 = 𝜑(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) − 𝜑𝑜 ,
𝜋12 𝜋22 𝜋32 0 0 0
𝐵𝜉 = 𝐵𝑧𝑧 + 𝜋31 𝜎𝑥𝑥 + 𝜋32 𝜎𝑦𝑦 + 𝜋33 𝜎𝑧𝑧 , with 𝛴 ≡ and with the optic angle which changes into:
(38)
where 𝛾 and 𝛾𝑜 are given respectively by (28) and (23)3 .
sin2 𝜑 =
𝐵𝑧𝑧 − 𝐵𝑥𝑥 + (𝜋31 − 𝜋11 )𝜎𝑥𝑥 + (𝜋32 − 𝜋12 )𝜎𝑦𝑦 + (𝜋33 − 𝜋13 )𝜎𝑧𝑧 𝐵𝑧𝑧 − 𝐵𝑦𝑦 + (𝜋31 − 𝜋21 )𝜎𝑥𝑥 + (𝜋32 − 𝜋22 )𝜎𝑦𝑦 + (𝜋33 − 𝜋23 )𝜎𝑧𝑧
3.2. Orthorhombic crystals
, (43)
since for small stress the same optic plane as in the unstressed case is maintained. By (42) and relation (19), then we have the following non-null components of 𝐅𝜎 :
3.2.1. Coherent prisms The crystallographic axis are orthogonal for Orthorhombic crystals; we say a prismatic crystal specimen coherent if the crystallographic and physical frames are parallel in which case we may assume ≡ with 𝑥 = 𝑎, 𝑦 = 𝑏 and 𝑧 = 𝑐 (vid. Fig. 4).
𝐹𝜎,𝑥𝑥 =
4 We notice that the optic angle 𝜑 can be also calculated independently as in [20].
𝐹𝜎,𝑦𝑦 = 5
𝐴(𝐾 , sin 𝜑𝑜 ) 𝑛−2 𝑧
− 𝑛−2 𝑦
𝐴(𝐾 , sin 𝜑𝑜 ) −2 𝑛−2 𝑧 − 𝑛𝑦
(𝜋31 − 𝜋11 − (𝜋32 − 𝜋12 −
−2 𝑛−2 𝑧 − 𝑛𝑥 −2 𝑛−2 𝑧 − 𝑛𝑦 −2 𝑛−2 𝑧 − 𝑛𝑥 −2 𝑛−2 𝑧 − 𝑛𝑦
(𝜋31 − 𝜋21 )) , (𝜋32 − 𝜋22 )) ,
(44)
D. Rinaldi, F. Daví and L. Montalto
𝐴(𝐾 , sin 𝜑𝑜 )
𝐹𝜎,𝑧𝑧 =
−2 𝑛−2 𝑧 − 𝑛𝑦
(𝜋33 − 𝜋13 −
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
𝑛−2 𝑧 𝑛−2 𝑧
− 𝑛−2 𝑥 (𝜋33 − 𝑛−2 𝑦
− 𝜋23 )) ;
with 𝜑𝑜 given by (43) with 𝜎𝑥𝑥 = 𝜎𝑦𝑦 = 𝜎𝑧𝑧 = 0. In the case of uniaxial stress in the plane normal to the optic bisectrix, (𝜎𝑥𝑥 , 𝜎𝑧𝑧 ) we may evaluate the stress provided the components of 𝛱 are known. If we consider instead shear stresses, for instance a shear stress 𝜎𝑥𝑦 , then we have: √ 𝐵𝑥𝑥 + 𝐵𝑦𝑦 𝐵 − 𝐵𝑦𝑦 2 𝜎 2 + ( 𝑥𝑥 𝐵𝜁 = − 𝜋66 )2 , 𝑥𝑦 2 2 √ 𝐵 − 𝐵𝑦𝑦 𝐵𝑦𝑦 + 𝐵𝑦𝑦 2 𝜎 2 + ( 𝑥𝑥 𝐵𝜂 = + 𝜋66 )2 , (45) 𝑥𝑦 2 2 𝐵𝜉 = 𝐵𝑧𝑧 , Fig. 5. Rotation of 𝜃 about the 𝑏-axis of the frame with respect to the frame. The 𝛴 frame (red) is rotated by 𝛾 with respect to and 𝜉. The prismatic crystal specimen is represented by the blue lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
and the principal frame 𝛴 is obtained by a rotation about 𝐞2 of an angle 𝛾: 2𝜋66 𝜎𝑥𝑦
tan 2𝛾 =
𝐵𝑦𝑦 − 𝐵𝑥𝑥
(46)
.
If we assume 𝐵𝜉 > 𝐵𝜂 > 𝐵𝜁 , then the optic axis are directed as:
𝐹𝜎,𝑥𝑧 = 0 ;
(47)
𝐦1,2 = ± sin 𝜑𝐮1 + cos 𝜑𝐮3 ,
unsurprisingly, the photoelastic constant associated with the shear stress 𝜎𝑥𝑦 is zero. Indeed in [8] it is show that in Orthorhombic crystal shear stresses make the optic plane to rotate without changing the optic angle and hence without modifying the Cassini-like curves up to a rigid motion of amplitude 𝛾.
where sin 𝜑 and cos 𝜑 are given by (6) and (45). With these relations we can evaluate some components of the piezooptic tensor 𝛱, provided the crystals is stress-free, by the means of a loading device which controls uniaxial stresses or the biaxial stress corresponding to the simple shear like 𝜎𝑥𝑦 . Moreover, by applying in turn shear stresses 𝜎𝑥𝑧 and 𝜎𝑦𝑧 in place of 𝜎𝑥𝑦 we may evaluate the components 𝜋55 and 𝜋44 . Conversely, if the components of 𝛱 are given, we can give an estimate of the residual stress within the specimen. To this regard, consider as an example a plane stress 𝐓𝐞2 = 𝟎 which could represents for instance a state of residual stress in a sample cut normally to the optical bisectrix 𝐞2 , like the Alexandrite of Remark 1; we assume that the stress 𝜎𝑦𝑦 , 𝜎𝑥𝑦 and 𝜎𝑦𝑧 have mean zero value over the volume: ⎡ 𝜎𝑥𝑥 𝐓≡⎢ ⋅ ⎢ ⎣ ⋅
0 0 ⋅
𝜎𝑥𝑧 0 𝜎𝑧𝑧
⎤ ⎥; ⎥ ⎦
In order to give an estimate of the residual stress within the specimen, provided we have a complete knowledge of the components of 𝛱 and 𝐁𝑜 , we have again the three independent relations (36), (37) and (38) where 𝐅𝜎 is given by (52), 𝜑(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 ) is given by (51) and the rotation of the optic plane is given by (50).
3.2.2. Non-coherent prisms A more typical experimental setting is when the prismatic specimen is not oriented as the crystallographic axes and hence an uniaxial stress in the frame leads to a more complex state of stress in the frame . Let assume as an example that the axis 𝑏 and 𝑦 are parallel and that the frame is rotated by an angle 𝜃 with respect to : we may assume that 𝜃 can be determined by XRD diffraction techniques, for instance (see Fig. 5).
(48)
by (2) we arrive at the principal values for 𝐁(𝐓) 1 (𝐵 + 𝐵𝑧𝑧 + 𝜋̂ 1+ 𝜎𝑥𝑥 + 𝜋̂ 3+ 𝜎𝑧𝑧 ± 𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 )) 2 𝑥𝑥 𝐵𝜁 = 𝐵𝑦𝑦 + 𝜋21 𝜎𝑥𝑥 + 𝜋23 𝜎𝑧𝑧 ,
𝐵𝜉,𝜂 =
If the specimen undergoes an uniaxial stress 𝜎𝑧𝑧 , for instance by means of a four-point bending test, then in the frame the stress tensor is represented by the following components:
(49)
here 𝜋̂ 1± = 𝜋31 ± 𝜋11 and 𝜋̂ 3± = 𝜋33 ± 𝜋13 and √ 2 𝜎2 𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) = (𝐵𝑧𝑧 − 𝐵𝑥𝑥 + 𝜋̂ 1− 𝜎𝑥𝑥 + 𝜋̂ 3− 𝜎𝑧𝑧 )2 + 4𝜋55 𝑥𝑧
𝐓≡
with the principal frame 𝛴 rotated about 𝐞2 by an angle 𝛾 measured from 𝐞1 : tan 𝛾 =
𝐵𝜉 − 𝐵𝑥𝑥 − 𝜋11 𝜎𝑥𝑥 − 𝜋13 𝜎𝑧𝑧 𝜋55 𝜎𝑥𝑧
If we assume 𝐵𝜉 > 𝐵𝜂 > 𝐵𝜁 , then the optic axis directions have the same expression as in (47) with the optic angle 𝜑 given by: sin2 𝜑 =
2𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 ) , 𝐵𝑥𝑥 + 𝐵𝑧𝑧 − 2𝐵𝑦𝑦 + 𝜋̃1+ 𝜎𝑥𝑥 + 𝜋̃3+ 𝜎𝑧𝑧 + 𝑅(𝜎𝑥𝑥 , 𝜎𝑧𝑧 , 𝜎𝑥𝑧 )
𝐹𝜎,𝑧𝑧 =
𝐴(𝐾 , sin 𝜑𝑜 ) −2 𝑛−2 𝑧 − 𝑛𝑥
𝐴(𝐾 , sin 𝜑𝑜 ) −2 𝑛−2 𝑧 − 𝑛𝑥
(𝜋31 − 𝜋11 − (𝜋33 − 𝜋13 −
−2 𝑛−2 𝑥 − 𝑛𝑦 −2 𝑛−2 𝑧 − 𝑛𝑦 −2 𝑛−2 𝑥 − 𝑛𝑦 −2 𝑛−2 𝑧 − 𝑛𝑦
⎤ ⎥; ⎥ ⎦
(53)
𝐵𝑎𝑎 + 𝐵𝑐𝑐 1 + 𝜎𝑧𝑧 [𝜋1+ (1 − cos 2𝜃) + 𝜋3+ (1 + cos 2𝜃)] ± 𝑅(𝜎𝑧𝑧 ) 2 4 1 𝐵𝜁 = 𝐵𝑏𝑏 + 𝜎𝑧𝑧 (𝜋21 (1 − cos 2𝜃) + 𝜋23 (1 + cos 2𝜃)) , (54) 2
(51)
here 𝑅(𝜎𝑧𝑧 ) √ 𝜎 1 = (𝜎𝑧𝑧 𝜋55 sin 2𝜃)2 + (𝐵𝑐𝑐 − 𝐵𝑎𝑎 + 𝑧𝑧 [𝜋1− + 𝜋3− + (𝜋3− − 𝜋1− ) cos 2𝜃)]2 , 2 2
(𝜋31 − 𝜋21 )) , (𝜋33 − 𝜋23 )) ,
−𝜎𝑧𝑧 sin 2𝜃 0 𝜎𝑧𝑧 (1 + cos 2𝜃)
𝐵𝜉,𝜂 =
where 𝜋̃1+ = 𝜋31 + 𝜋11 − 2𝜋21 and 𝜋̃3+ = 𝜋33 + 𝜋31 − 2𝜋23 . By (49) and (19) then we get the components of the photoelastic constants tensor 𝐅𝜎 : 𝐹𝜎,𝑥𝑥 =
0 0 ⋅
From (2) with (53) we arrive at the following principal values (here the components of 𝛱 are evaluated in the frame , as well as those of 𝐁𝑜 ):
(50)
.
⎡ 𝜎 (1 − cos 2𝜃) 1 ⎢ 𝑧𝑧 ⋅ 2⎢ ⋅ ⎣
with 𝜋1± = 𝜋31 ± 𝜋11 and 𝜋3± = 𝜋33 ± 𝜋13 with (52)
tan 𝛾 = 6
𝜋55 𝜎𝑧𝑧 sin 2𝜃 . 𝐵𝜉 − 𝐵𝑎𝑎
(55)
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
Fig. 7. The relation between the frames and . The 𝑧-axis coincides with the crystallographic 𝑐-axis, the coordinates in the square brackets are referred to the lattice vectors of .
Fig. 6. The relation between the frames and .
radius. Accordingly, from an experimental point of view these stress are very difficult to detect. Any other state of stress makes the crystal optically biaxial and the optic plane rotates about a generic direction, depending on the applied stress: since it is not possible to obtain analytical closed-form solutions we need to search for approximate solution by following e.g. the approximation procedure proposed in [9]; this is beyond the scope of the present work and shall be dealt with into a forthcoming paper.
By (19) we obtain the following expression for the photoelastic constant: 𝐴(𝐾 , sin 𝜑𝑜 ) ((𝜋1− (1 − cos 2𝜃) + 𝜋3− (1 + cos 2𝜃)) (56) 𝐹𝜎,𝑧𝑧 = −2 2(𝑛−2 𝑎 − 𝑛𝑐 ) −2 𝑛−2 𝑎 − 𝑛𝑏
𝜋 (𝐵𝑎𝑎 + 𝐵𝑐𝑐 − (𝜋21 − 1− )(1 − cos 2𝜃) −2 2 𝑛−2 − 𝑛 𝑎 𝑐 𝜋3− − (𝜋23 − )(1 + cos 2𝜃))) . 2 −
(57)
4.1.2. Classes 3, 3̄ The piezooptic tensor 𝛱 for the crystallographic classes 3, 3̄ has the following representation in the frame :
4. Stress analysis in optically uniaxial crystals 4.1. Trigonal crystals
⎡ ⎢ ⎢ [𝛱] ≡ ⎢ ⎢ ⎢ ⎢ ⎣
For the crystals of the Trigonal group we follow [22] which, instead of the relations proposed into [11] or [12], gives the following relations between the frames and (Fig. 6) based only on the Trigonal angle 𝛼: √ 1 𝛼 𝛼 𝛼 1 (4 cos2 − 1)𝐞3 , 𝐚 = sin 𝐞1 − √ sin 𝐞2 + 2 2 3 2 3 √ 2 𝛼 1 𝛼 𝐛 = √ sin 𝐞2 + (4 cos2 − 1)𝐞3 , (58) 2 3 2 3 √ 𝛼 𝛼 1 𝛼 1 (4 cos2 − 1)𝐞3 , 𝐜 = − sin 𝐞1 − √ sin 𝐞2 + 2 2 3 2 3
0 𝐵𝑥𝑥 0
0 0 𝐵𝑧𝑧
⎤ ⎥, ⎥ ⎦
𝜋11 𝜋12 𝜋31 𝜋41 0 0
𝜋12 𝜋11 𝜋31 −𝜋41 0 0
𝜋13 𝜋13 𝜋33 0 0 0
0 0 0 0 𝜋44 𝜋14
0 0 0 0 2𝜋41 𝜋11 − 𝜋12
⎤ ⎥ ⎥ ⎥; ⎥ ⎥ ⎥ ⎦
𝜋14 −𝜋14 0 𝜋44 −𝜋45 𝜋25
−𝜋25 𝜋25 0 𝜋45 𝜋44 𝜋14
2𝜋62 −2𝜋62 0 2𝜋52 2𝜋41 𝜋11 − 𝜋12
⎤ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(61)
For the crystals of the Hexagonal group with 𝜑 < 𝜋∕2 with respect to the 𝑧-axis, we follow [12] and identify 𝐞1 with the crystallographic direction [1 , 1 , 0], to obtain the following relation between the frames and (Fig. 7): √ √ 3 3 1 1 𝐞2 , 𝐛 = 𝐞1 + 𝐞 , 𝐜 = 𝐞3 , (62) 𝐚 = 𝐞1 − 2 2 2 2 2
(59)
𝜋14 −𝜋14 0 𝜋44 0 0
𝜋13 𝜋13 𝜋33 0 0 0
4.2. Hexagonal crystals
̄ 32, 3𝑚 4.1.1. Classes 3𝑚, These are the higher-symmetry classes for which the piezooptic tensor 𝛱 admits the representation in : ⎡ ⎢ ⎢ [𝛱] ≡ ⎢ ⎢ ⎢ ⎢ ⎣
𝜋12 𝜋11 𝜋31 −𝜋41 𝜋52 𝜋62
As it is shown into [8], also for these Trigonal classes a spherical stress 𝐓 = 𝜎𝑚 𝐈, an uniaxial stress 𝜎𝑧𝑧 and an uniform plane stress 𝜎𝑥𝑥 = 𝜎𝑦𝑦 preserve the crystal optical symmetry and therefore the stressed crystal remains uniaxial. For any other state of stress the considerations made in the previous subsection still apply.
The Trigonal group is optically Uniaxial and hence the tensor 𝐁𝑜 has diagonal representation in the frame , which coincides with the principal frame 𝛴: ⎡ 𝐵𝑥𝑥 𝐁𝑜 ≡ ⎢ 0 ⎢ ⎣ 0
𝜋11 𝜋12 𝜋31 𝜋41 −𝜋52 −𝜋62
The Hexagonal group is optically Uniaxial and hence the tensor 𝐁𝑜 has diagonal representation in the frame , which coincides with the principal frame 𝛴: ⎡ 𝐵𝑥𝑥 𝐁𝑜 ≡ ⎢ 0 ⎢ ⎣ 0
(60)
0 𝐵𝑥𝑥 0
0 0 𝐵𝑧𝑧
⎤ ⎥. ⎥ ⎦
(63)
Remark 2. An Hexagonal crystal is for instance the LaBr3 :Ce (Cerium doped Lanthanium Bromide) with a unit cell 𝑎 = 7.9648 Å, 𝑐 = 4.5119 Å. The refraction indices are 𝑛𝑜 = 2.04, and 𝑛𝑒 = 2.09 [24]. The components of the Dielectric Impermeability tensor are 𝐵𝑥𝑥 = 0.240, 𝐵𝑧𝑧 = 0.228. Since 𝑛𝑒 > 𝑛𝑜 , the crystal is optically positive. □
a complete characterization of these components was obtained, by the means of different techniques, in [23] for LiNbO3 (Lithium Niobate). In [8] we show that for these classes any spherical stress 𝐓 = 𝜎𝑚 𝐈, the uniaxial stress 𝜎𝑧𝑧 and the uniform plane stress 𝜎𝑥𝑥 = 𝜎𝑦𝑦 leave the crystal uniaxial, the circular interference fringe modifying only their 7
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
4.2.1. Classes 6, 6̄ and 6∕𝑚
When the crystal is acted by the plane stress 𝐓𝐞3 = 𝟎 with 𝜎𝑥𝑧 = 𝜎𝑦𝑧 = 𝜎𝑧𝑧 = 0 we have instead:
The piezooptic tensor 𝛱 for the crystallographic classes 6, 6̄ and 6∕𝑚 has the following representation in the frame [12]: ⎡ ⎢ ⎢ [𝛱] ≡ ⎢ ⎢ ⎢ ⎢ ⎣
𝜋11 𝜋12 𝜋31 0 0 −𝜋62
𝜋12 𝜋11 𝜋31 0 0 𝜋62
𝜋13 𝜋13 𝜋33 0 0 0
0 0 0 𝜋44 −𝜋45 0
0 0 0 𝜋45 𝜋44 0
2𝜋62 −2𝜋62 0 0 0 𝜋11 − 𝜋12
⎤ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
𝜋11 + 𝜋12 (𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) ± 𝑅(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 ) , 2 𝐵𝜁 = 𝐵𝑧𝑧 + 𝜋31 (𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) ,
𝐵𝜉,𝜂 = 𝐵𝑥𝑥 +
where
(64)
𝑅(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 ) =
(71)
√ 2 ) + 𝐾 (𝜎 − 𝜎 )𝜎 , 𝐾1 ((𝜎𝑥𝑥 − 𝜎𝑦𝑦 )2 + 4𝜎𝑥𝑦 2 𝑥𝑥 𝑦𝑦 𝑥𝑦
2 𝐾1 = (𝜋11 − 𝜋12 )2 + 4𝜋62 ,
(72)
𝐾2 = 2(𝜋11 − 𝜋12 )𝜋62 ,
As it is shown into [8], for these hexagonal classes a spherical stress 𝐓 = 𝜎𝑚 𝐈, an uniaxial stress 𝜎𝑧𝑧 and an uniform plane stress 𝜎𝑥𝑥 = 𝜎𝑦𝑦 preserve the crystal optical symmetry and therefore the stressed crystal remains uniaxial, the interference fringes in the planes orthogonal to the 𝑧-axis still being circumferences.
whereas the frame 𝛴 rotates about 𝐞3 by an angle 𝛾: tan 𝛾 =
𝐵𝜉 − 𝐵𝑥𝑥 2𝜋62 (𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) + (𝜋11 − 𝜋12 )𝜎𝑥𝑦
(73)
.
Any state of stress different from these changes the optical symmetry into a Biaxial one. In detail, as it was shown into [8], a simple uniaxial stress 𝜎𝑥𝑥 (or 𝜎𝑦𝑦 ) and the shear 𝜎𝑥𝑦 makes the crystal biaxial with the optic plane which rotates about the 𝑧-axis. In the case of the shear 𝜎𝑦𝑧 and 𝜎𝑥𝑧 the optic plane rotates to become orthogonal to the 𝑥- and 𝑦-axis respectively, with one of the optic axis directed as the 𝑧-axis.
Once again the crystal becomes biaxial and we may assume that (47) still holds with
Rather than analyze all these cases in detail, the straightforward analysis can be done as in the previous section, we shall limit to two cases: a ‘‘diagonal’’ stress tensor like (41) and the plane stress 𝐓𝐞3 = 𝟎. In the first case the results can be used to evaluate the components of 𝛱, whereas in the second case we can give an estimate for a residual stress.
When we use (20) to evaluate the photoelastic constant tensor components we notice that 𝐵𝜉 − 𝐵𝜂 is an homogeneous function of degree one of the principal stresses and its derivatives with respect to the cartesian stress components are homogeneous functions of degree zero: by a well-known result, the limit for 𝐓 → 𝟎 exists if and only if the derivatives are constant, say, they are independent on the stress components. Since this is not the case, we circumvent the problem by expressing the stress components in terms of the principal stresses (𝜎1 , 𝜎2 , 𝜎3 = 0) which are related to the cartesian components by the means of (cf. e.g. [25]): 𝜎 − 𝜎2 𝜎 + 𝜎2 + cos 2𝜓 1 , 𝜎𝑥𝑥 = 1 2 2 𝜎1 + 𝜎2 𝜎1 − 𝜎2 𝜎𝑦𝑦 = − cos 2𝜓 , (75) 2 2 𝜎1 − 𝜎2 𝜎𝑥𝑦 = sin 2𝜓 , 2 where 𝜓 is the rotation about the axis 𝐞3 of the principal stress frame 𝛴𝐓 :
sin2 𝜑 =
whereas the frame 𝛴 rotates about 𝐞3 ≡ 𝐮3 by an angle 𝛾: √ 2 𝜋11 − 𝜋12 ± (𝜋11 − 𝜋12 )2 + 4𝜋62 tan 𝛾 = 2𝜋62
(65)
(66)
we notice that the angle 𝛾 is independent on the stress. tan 2𝜓 =
The crystal becomes biaxial: provided we assume 𝐵𝜉 > 𝐵𝜂 > 𝐵𝜁 the optic axis are directed as in (47) with:
sin2 𝜑 =
√ 2 (𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) (𝜋11 − 𝜋12 )2 + 4𝜋62 2𝐵𝑥𝑥 − 𝐵𝑧𝑧 + (𝜋11 + 𝜋12 + 2𝜋31 )(𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) + (2𝜋13 + 𝜋33 )𝜎𝑧𝑧
.
√ 2 ; (𝜋11 − 𝜋12 )2 + 4𝜋62
𝜎𝑥𝑥 − 𝜎𝑦𝑦
(76)
.
(77)
(78)
𝐶(𝜎1 , 𝜎2 ) = 𝐹𝜎,1 (𝜎1 − 𝜎2 ) .
As in the previous section, in order to estimate the residual stress we have three relations, namely (73), (74) and (78), provided we know the components of 𝛱.
(68) ̄ 4.2.2. Classes 6𝑚2, 622, 6 mm and 6∕𝑚𝑚𝑚 For these higher-symmetry classes the piezooptic tensor 𝛱 admits the representation in :
these relations extend to hexagonal crystals the similar ones obtained into [4]. These relations are useful in order to obtain an estimate for the components of 𝛱 as it is shown into [3].
⎡ 𝜋11 ⎢ 𝜋12 ⎢ 𝜋 [𝛱] ≡ ⎢ 31 ⎢ 0 ⎢ 0 ⎢ ⎣ 0
Since for unstressed hexagonal crystals 𝐶(𝟎) = 0 and 𝜑𝑜 = 0 then the ellipticity factor is a linear function of 𝐅𝜎 whose non-null components are √ 𝜋 − 𝜋12 2 𝐴(𝐾 , 0) 2 , ( 11 ) + 4𝜋62 (69) 𝐹𝜎,𝑥𝑥 = −𝐹𝜎,𝑦𝑦 = −2 2 𝑛−2 𝑜 − 𝑛𝑒
𝜋12 𝜋11 𝜋31 0 0 0
𝜋13 𝜋13 𝜋33 0 0 0
0 0 0 𝜋44 0 0
0 0 0 0 𝜋44 0
0 0 0 0 0 𝜋11 − 𝜋12
⎤ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦
(79)
and, provided we set 𝜋11 = 𝜋22 , 𝜋13 = 𝜋23 , 𝜋31 = 𝜋32 , 𝜋44 = 𝜋45 and 𝜋66 = 𝜋11 − 𝜋12 , all the results we obtained for Orthorhombic crystal still hold.
i.e.: 𝐶(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑧𝑧 ) = 𝐹𝜎,𝑥𝑥 (𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) .
.
and since 𝐶(𝟎) = 0 and 𝜑𝑜 = 0 then:
It is interesting to remark that from (65) we may obtain a simple relation between the dielectric permeability and the stress components:
𝑆(𝛱) =
2𝜎𝑥𝑦
With (75), then by (20) we arrive at √ 𝐴(𝐾 , 0) 1 𝐹𝜎,1 = −𝐹𝜎,2 = 𝐾1 + 𝐾2 sin 2𝜓 cos 2𝜓 , −2 2 𝑛−2 𝑜 − 𝑛𝑒
(67)
𝐵𝜉 − 𝐵𝜂 = 𝑆(𝛱)(𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) ,
2(𝐵𝑥𝑥 − 𝐵𝑧𝑧 ) + (𝜋11 + 𝜋12 − 2𝜋13 )(𝜎𝑥𝑥 − 𝜎𝑦𝑦 ) + 2𝑅(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 )
(74)
In the first case we have that the eigenvalues of 𝐁(𝐓) are: 𝜋 + 𝜋12 (𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) + 𝜋13 𝜎𝑧𝑧 , 𝐵𝜉,𝜂 = 𝐵𝑥𝑥 + 11 2 √ 𝜋 − 𝜋12 2 2 (𝜎 − 𝜎 ) ± ( 11 ) + 𝜋62 𝑥𝑥 𝑦𝑦 2 𝐵𝜁 = 𝐵𝑧𝑧 + 𝜋31 (𝜎𝑥𝑥 + 𝜎𝑦𝑦 ) + 𝜋33 𝜎𝑧𝑧 ,
2𝑅(𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑥𝑦 )
(70) 8
D. Rinaldi, F. Daví and L. Montalto
Nuclear Inst. and Methods in Physics Research, A 947 (2019) 162782
5. Conclusions
[2] L. Montalto, N. Paone, L. Scalise, D. Rinaldi, A photoelastic measurement system for residual stress analysis in scintillating crystals by conoscopic imaging, Rev. Sci. Instrum. 86 (2015) 063102. [3] P.P. Natali, L. Montalto, F. Daví, P. Mengucci, A. Ciriaco, N. Paone, D. Rinaldi, Theoretical and experimental evaluation of piezo-optic parameters and photoelastic constant in tetragonal PWO, Appl. Opt. 57 (4) (2018) 730–737. [4] L. Montalto, F. Daví, D. Rinaldi, On the photoelastic constants and the Brewster law for stressed tetragonal crystals, Math. Methods Appl. Sci. 41 (8) (2018) 3103–3116. [5] E.E. Wahlstrom, Optical Crystallography, Wiley, New York, 1960. [6] D. Rinaldi, P. Pietroni, F. Daví, Isochromate fringes simulation by Cassini-like curves for photoelastic analysis of birifringent crystals, Nuclear Instrum. Methods Phys. Res. A 603 (2009) 294–300. [7] F. Daví, D. Rinaldi, Mechanical and optical properties of anisotropic single-crystal prisms, J. Elast. 120 (2) (2015) 197–224. [8] F. Daví, On the Bertin surfaces of photoelastic crystals, J. Opt. Soc. Am. A 32 (12) (2015) 2323–2337. [9] Y.I. Sirotin, M.P. Shaskolskaya, Fundamentals of Crystal Physics, Mir-Moscow, 1982. [10] J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1985. [11] M.V. Perelomova, M.M. Tagieva, Problems in Crystal Physics with Solutions, MIR Publishers, Moscow, 1983, English trans.. [12] A. Authier (Ed.), International Tables for Crystallography. Volume D: Physical Properties of Crystals, Kluwer Academic Publ, Dordrecht, 2003. [13] H. Aben, C. Guillemet, Photoelasticity of Glass, Springer Verlag, 1993. [14] A. Ciriaco, F. Daví, M. Lebeau, G. Majni, N. Paone, P. Pietroni, D. Rinaldi, PWO Photo-elastic parameters calibration by laser-based polariscope, Nuclear Instrum. Methods Phys. Res. A 570 (2007) 55–60. [15] M. Lebeau, L. Gobbi, G. Majni, N. Paone, P. Pietroni, D. Rinaldi, Mapping residual stresses in PbWO4 crystals using photoelastic analysis, Nuclear Instrum. Methods Phys. Res. A 537 (2005) 207–210. [16] D. Rinaldi, A. Ciriaco, M. Lebeau, N. Paone, Quality control on pre-serial bridgman production of PbWO4 scintillating crystals by means of photoelasticity, Nuclear Instrum. Methods Phys. Res. A 615 (2010) 254–258. [17] M. Born, E. Wolf, Principles of Optics, VIIth Ed., Cambridge University Press, Cambridge, 1999. [18] L. Montalto, P.P. Natali, F. Daví, P. Mengucci, N. Paone, D. Rinaldi, Characterization of a defective PbWO4 crystal cut along the 𝑎 − 𝑐 crystallographic plane: structural assessmnet and a novel photoelastic stress analysis, J. Instrum. 12 (2017). [19] P. Helnwein, Some remarks on the compressed matrix representation of symmetric second-order and fourth-order tensors, Comput. Methods Appl. Mech. Eng. 190 (22–23) (2001) 2753–2770. [20] B.G. Mytsyk, Kost’ Ya. P., N.M. Demyanyshyn, V.M. Gaba, O.M. Sakharuk, Study of piezo-optic effect of calcium tungstate crystals by the conoscopic method, Opt. Mater. 39 (2015) 69–73. [21] P. Loiko, A. Major, Dispersive properties of alexandrite and beryllium hexaaluminate crystals, Opt. Mater. Express 6 (7) (2016) 263034. [22] M. Mehl, J.D. Hicks, C. Toher, O. Levy, R.M. Hanson, G.L.W. Hart, S. Curtarolo, The AFLOW library of crystallographic prototypes: part 1, Comput. Mat. Sci. 136 (2017) S1–S828. [23] O. Krupich, V. Savaryn, R. Vlokh, Precise determination of full matrix of piezooptic coefficients with a four-point bending technique: the example of lithium niobate crystals, Appl. Opt. 53 (10) (2014) B1–B6. [24] D.J. Singh, Structure and optical properties of high light output halide scintillators, Phys. Rev. B 82 (2010) 155145. [25] I.S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd edition 1956, McGraw Hill, 1946.
The results we presented allow either for an estimate the components of the piezo-optic tensor 𝛱 or an estimate of the average residual stress along the specimen thickness for crystals of four point groups, two optically Biaxial (Monoclinic and Orthorhombic) and two Uniaxial (Trigonal and Hexagonal). We limited our analysis to optical observations in the plane normal to the acute bisectrix of optic axes (for biaxial crystals) or normal to the optic axis for the uniaxial: this is because the interference fringes which can be observed by the means of conoscopy are closed ‘‘Cassini-like’’ curves and in such a case the ellipticity ratio 𝐶 is an useful parameter. By following the same approach of [4] and by making use of the results obtained in [8], we reduced our analysis to two cases which are interesting from an experimental point of view: a combination of three different uniaxial loads and a plane stress in the observation plane. For Orthorhombic crystals we also analyzed the possibility of a specimen which is not oriented along the crystallographic directions (the non-coherent case). In all cases explicit formulae for the optic angle 𝜑, the principal refraction indices and the rotation of the optic plane 𝛾 are given; we also provide for the explicit expressions of the components of the photoelastic constants tensor 𝐅𝜎 , which generalizes to a generic state of stress the notion of photoelastic constant 𝑓𝜎 previously introduced in [14] for uniaxial stress. In order to give an estimate for a residual stress when it is identified with a plane state of stress and hence represented by three stress components, provided the components of 𝛱 are given, we also obtained three independent relations which should allow to give a residual stress estimate. With these results at hand it should be possible to design an experimental set-up for the determination of the components of the tensor 𝛱 with a set of independent experiments or for a precise estimate of internal stress and a global quality evaluation of crystals. Acknowledgments The research leading to these results is within the scope of CERN R&D Experiment 18 ‘‘Crystal Clear Collaboration’’ and the PANDA Collaboration at GSI-Darmstadt. It has received funding from the Universitá Politecnica delle Marche, Italy, Progetto Strategico di Ateneo 2017: "Scintillating crystals: an interdisciplinary, applications-oriented approach aimed to the scientific knowledge and process control for application concerning life-quality improvement". References [1] L. Montalto, N. Paone, D. Rinaldi, L. Scalise, Inspection of birefringent media by photoelasticity: from diffuse light polariscope to laser conoscopic technique, Opt. Eng. 54 (8) (2015) 081210.
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