Mechanical and Optical Properties of Anisotropic Single-Crystal Prisms

Fabrizio Daví; Daniele Rinaldi

doi:10.1007/s10659-014-9511-4

Journal of Elasticity

August 2015, Volume 120, Issue 2, pp 197–224 | Cite as

Mechanical and Optical Properties of Anisotropic Single-Crystal Prisms

Authors
Authors and affiliations

Fabrizio DavíEmail author
Daniele Rinaldi

Article

First Online: 24 January 2015

Received: 12 May 2014

145 Downloads
5 Citations

Abstract

The great interest in scintillating crystals, is related to their applications in the high energy physics, biomedicine and security. For this reason a complete characterization and understanding of their structural, optical and mechanical properties at the macroscopic level is necessary. We must give a complete theoretical characterization of the mechanics and optics of bulk prismatic single-crystal bodies in order to design experiments. This work shall deal solely with the theoretical description within the framework of linear theories.

Keywords

Anisotropic materials Photoelasticity Stress analysis Scintillating crystals

Mathematics Subject Classification (2010)

74E10 74F99 74K10 78A10

This is a preview of subscription content, to check access.

Notes

Acknowledgements

This work, which is within the scope of the CERN R&D Experiment 18, Crystal Clear Collaboration (CCC), was supported solely with resources of the DICEA and SIMAU, Universitá Politecnica delle Marche, Ancona, Italy. We wish to thank Michel Lebeau, former CERN associate, for his continuous interest in our work. The authors wish to thanks the Reviewers for the very helpful comments.

Appendix

In this appendix we show how we obtained the linear relations (52) and (54) between the angles δ, φ and the applied stress T ₃₃.

We begin with (52): by using (46) into (47):

$$\tan\delta=\frac{B_{12}}{B_{1}-B_{11}}=\frac {2B_{12}}{B_{22}-B_{11}+\sqrt{(B_{22}-B_{11})^{2}+4B_{12}^{2}}}; $$

then, since the components of B depends on T ₃₃, by setting:

$$u(T_{33})=\frac{2B_{12}}{B_{22}-B_{11}+\sqrt {(B_{22}-B_{11})^{2}+4B_{12}^{2}}}, $$

we arrive at the non linear relation:

$$ \delta(T_{33})=\tan^{-1}\bigl(u(T_{33}) \bigr). $$

(85)

We take the Taylor expansion of (85) about T ₃₃=0:

$$\delta(T_{33})=\delta(0)+\delta'(0)T_{33}+o \bigl(T_{33}^{2}\bigr),\qquad(\cdot )'= \frac{d}{dT_{33}}, $$

where δ(0)=δ ₀, the value of the angle in the unstressed state and:

$$\delta'=\frac{1}{1+u^{2}}u'. $$

If we set:

$$v(T_{33})=2B_{12},\qquad w(T_{33})=B_{22}-B_{11} , $$

we can rewrite u as follows:

$$u=\frac{v}{w+\sqrt{v^{2}+w^{2}}}, $$

and accordingly:

$$u'=\biggl(\frac{v}{w+\sqrt{v^{2}+w^{2}}}\biggr)'= \frac{v'(w+\sqrt {v^{2}+w^{2}})-v\bigl(w'+{\frac{vv'+ww'}{\sqrt {v^{2}+w^{2}}}}\bigr)}{(w+\sqrt{v^{2}+w^{2}})^{2}}. $$

Since for T ₃₃=0 we have:

$$u=\tan\delta_{0},\qquad v=2B^{0}_{12},\qquad w=B^{0}_{22}-B^{0}_{11}, $$

and

$$v'=\mathbb {M}_{1233},\qquad w'=\mathbb {M}_{2233}- \mathbb {M}_{1133}, $$

then we have:

$$\frac{1}{1+u^{2}}(0)=\frac{1}{1+\tan^{2}\delta_{0}}=\frac{\cos^{2}\delta _{0}}{\sin^{2}\delta_{0}+\cos^{2}\delta_{0}}=\cos^{2} \delta_{0}, $$

and:

$$u'(0)=\frac{\mathbb {M}_{1233}(w+\sqrt{v^{2}+w^{2}})-v\bigl(\mathbb {M}_{2233}-\mathbb {M}_{1133}+{\frac{v\mathbb {M}_{1233}+w(\mathbb {M}_{2233}-\mathbb {M}_{1133})}{\sqrt{v^{2}+w^{2}}}}\bigr)}{(w+\sqrt{v^{2}+w^{2}})^{2}}. $$

By using again (46) and (47) we finally arrive at:

$$M=\delta'(0)=\frac{2\cos^{2}\delta_{0}}{B^{0}_{1}-B^{0}_{11}}\biggl(\mathbb {M}_{1233}+\tan \delta_{0}\frac{(\mathbb {M}_{2233}-\mathbb {M}_{1133})(B^{0}_{1}-B^{0}_{11})+4B^{0}_{12}\mathbb {M}_{1233}}{B^{0}_{1}-B^{0}_{2}}\biggr). $$

For (54) we start from (48):

$$ \varphi(T_{33})=\sin^{-1}\sqrt{ \frac{B_{2}-B_{3}}{B_{1}-B_{3}}}, $$

(86)

where the eigenvalues are given by:

$$\begin{aligned} B_{1} =&\frac{1}{2}\bigl(B^{0}_{11}+B^{0}_{22}+( \mathbb {M}_{1133}+\mathbb {M}_{2233})T_{33}\bigr) \\ &{}-\sqrt{\biggl(\frac{B^{0}_{11}-B^{0}_{22}+(\mathbb {M}_{1133}+\mathbb {M}_{2233})T_{33}}{2}\biggr)^{2}+ \bigl(B_{12}^{0}+\mathbb {M}_{1233}T_{33} \bigr)^{2}}, \\ B_{2} =&\frac{1}{2}\bigl(B^{0}_{11}+B^{0}_{22}+( \mathbb {M}_{1133}+\mathbb {M}_{2233})T_{33}\bigr) \\ &{}+\sqrt{\biggl(\frac{B^{0}_{11}-B^{0}_{22}+(\mathbb {M}_{1133}+\mathbb {M}_{2233})T_{33}}{2}\biggr)^{2}+ \bigl(B_{12}^{0}+\mathbb {M}_{1233}T_{33} \bigr)^{2}}, \\ B_{3} =&B^{0}_{33}+\mathbb {M}_{3333}T_{33} , \end{aligned}$$

and we can set:

$$\begin{aligned} u(T_{33}) =&\frac{1}{2}\bigl(B^{0}_{11}+B^{0}_{22}-2B_{33}+( \mathbb {M}_{1133}+\mathbb {M}_{2233}-2\mathbb {M}_{3333})T_{33} \bigr), \\ w(T_{33}) =&\biggl(\frac{B^{0}_{11}-B^{0}_{22}+(\mathbb {M}_{1133}+\mathbb {M}_{2233})T_{33}}{2}\biggr)^{2}+ \bigl(B_{12}^{0}+\mathbb {M}_{1233}T_{33} \bigr)^{2}, \end{aligned}$$

to write (86) as:

$$ \varphi(T_{33})=\sin^{-1}\sqrt{ \frac{u+\sqrt{w}}{u-\sqrt{w}}}. $$

(87)

By linearizing (87) about T ₃₃=0 we get

$$\varphi(T_{33})=\varphi(0)+\varphi'(0)T_{33}+o \bigl(T_{33}^{2}\bigr), $$

where φ(0)=φ ₀ the angle at the unstressed state and where by the chain rule:

$$ \varphi'(T_{33})=\sqrt{\frac{u+\sqrt{w}}{\sqrt{w}}} \frac{w'u-u'w}{2\sqrt{w}(u-\sqrt{w})^{2}}\sqrt{\frac{u-\sqrt{w}}{u+\sqrt {w}}}. $$

(88)

By evaluating (88) for T ₃₃=0 we get the (54) with:

$$ K=\frac{(B^{0}_{11}-B^{0}_{22})(\mathbb {M}_{1133}-\mathbb {M}_{2233})+B^{0}_{12}\mathbb {M}_{1233}}{B^{0}_{1}+B^{0}_{2}-2B^{0}_{3}}. $$

(89)

A similar analysis can be done for the other two cases to obtain:

$$\begin{aligned} K =&\bigl(2B^{0}_{3}-B^{0}_{1}-B^{0}_{2} \bigr)\frac{((B^{0}_{11}-B^{0}_{22})(\mathbb {M}_{1133}-\mathbb {M}_{2233})+4B^{0}_{12}\mathbb {M}_{1233})}{(2B^{0}_{3}+B^{0}_{1}-3B^{0}_{2})(B^{0}_{1}-B^{0}_{2})} \\ &{}+\frac{2(B^{0}_{1}-B^{0}_{2})(\mathbb {M}_{1133}+\mathbb {M}_{2233}-2\mathbb {M}_{3333})}{ 2B^{0}_{3}+B^{0}_{1}-3B^{0}_{2}}, \end{aligned}$$

(90)

when B ₃>B ₁>B ₂ and

$$K=\hat{K}\sin^{4}\varphi_{0} $$

when B ₁>B ₃>B ₂, where $\hat{K}$ is given by (90).

References

1.
Korzhik, M., Annekov, A., Gektin, A., Pedrini, C.: Inorganic Scintillators for Detector Systems: Physical Principles and Crystal Engineering. Particle Acceleration and Detection. Springer, Heidelberg (2006) Google Scholar
2.
Scheel, H.J., Fukuda, T.: Crystal Growth Tecnology. Wiley, Chichester (2003) Google Scholar
3.
Dhaanaraj, G., Byrappa, K., Prasad, V., Dudley, M. (eds.): Springer Handbook of Crystal Growth. Springer, Berlin (2010) Google Scholar
4.
Leitao, U.A., Righi, A., Bourson, P., Pimenta, M.A.: Optical study of LiKSO₄ crystals under uniaxial pressure. Phys. Rev. B 50(5), 2754–2759 (1994) ADSCrossRefGoogle Scholar
5.
Ishii, M., Harada, K., Kobayashi, M., Usuki, Y., Yazawa, T.: Mechanical properties of PWO scintillating crystals. Nucl. Instrum. Methods Phys. Res., Sect. A, Accel. Spectrom. Detect. Assoc. Equip. 376, 203–207 (1996) ADSCrossRefGoogle Scholar
6.
Mytsyk, B.G., Andrushchak, A.S., Kost’, Ya.P.: Static photoelasticity of gallium phosphide crystals. Crystallogr. Rep. 57(1), 124–130 (2011) ADSCrossRefGoogle Scholar
7.
Skab, I., Vasylkiv, Y., Savaryn, V., Vlokh, R.: Relations for optical indicatrix parameters at the crystal torsion. Ukr. J. Phys. Opt. 11(4), 193–240 (2010) CrossRefGoogle Scholar
8.
Vasylkiv, Y., Savaryn, V., Smaga, I., Skab, I., Vlokh, R.: On the determination of sign of the piezo-optic coefficients using torsion method. Appl. Opt. 50(17), 2512–2518 (2011) ADSCrossRefGoogle Scholar
9.
Krupych, O., Savaryn, V., Vlokh, R.: Precis determination of the full matrix piezo-optic coefficients with a four-point bending technique: the example of lithium niobate crystals. Appl. Opt. 53(10), B1–B7 (2014) CrossRefGoogle Scholar
10.
Lebeau, M., Gobbi, L., Majni, G., Pietroni, P., Paone, N., Rinaldi, D.: Mapping residual stresses in PbWO₄ crystals using photoelastic analysis. Nucl. Instrum. Methods Phys. Res., Sect. A, Accel. Spectrom. Detect. Assoc. Equip. 537, 154–158 (2005) ADSCrossRefGoogle Scholar
11.
Ciriaco, A., Daví, F., Lebeau, M., Majni, G., Paone, N., Pietroni, P., Rinaldi, D.: PWO photo-elastic parameters calibration by laser-based polariscope. Nucl. Instrum. Methods Phys. Res., Sect. A, Accel. Spectrom. Detect. Assoc. Equip. 570, 55–60 (2007) ADSCrossRefGoogle Scholar
12.
Rinaldi, D., Pietroni, P., Daví, F.: Isochromate fringes simulation by Cassini-like curves for photoelastic analysis of birefringent crystals. Nucl. Instrum. Methods Phys. Res., Sect. A, Accel. Spectrom. Detect. Assoc. Equip. 603, 294–300 (2009) ADSCrossRefGoogle Scholar
13.
Scalise, L., Rinaldi, D., Daví, F., Paone, N.: Measurement of ultimate tensile strenght and Young modulus in LYSO scintillating crystals. Nucl. Instrum. Methods Phys. Res., Sect. A, Accel. Spectrom. Detect. Assoc. Equip. 654, 122–126 (2011) ADSCrossRefGoogle Scholar
14.
Daví, F., Rinaldi, D.: Elastic moduli and optical properties of LYSO crystals. Theory and experiments. IEEE Ttrans. Nucl. Sci. 59(5), 2106–2111 (2012) ADSCrossRefGoogle Scholar
15.
Saint-Venant, A.-J.-C.B.: Mémoire sur la flexion des prismes. J. Math. Pures Appl., Ser. II 1, 89 (1856) Google Scholar
16.
Saint-Venant, A.-J.-C.B.: Mémoire sur la torsion des prismes. Mém. Savants étrang. 14, 233 (1856) Google Scholar
17.
Voigt, W.: Lehrbuch der Krystallphysik (mit Anschluss der Krystalloptik). Mathematischen Wissenschaften, vol. XXXIV. Teubner, Leipzig und Berlin (1910) Google Scholar
18.
Daví, F., Tiero, A.: On the Saint-Venant’s problem with Voigt hypotheses for anisotropic solids. J. Elast. 36, 183–199 (1994) CrossRefMATHGoogle Scholar
19.
Fichera, G.: Problemi Analitici Nuovi nella Fisica Matematica Classica. Quaderni GNFM, Gruppo Nazionale Fisica Matematica, Firenze (1985) Google Scholar
20.
Iesan, D.: Saint-Venant Problem. Lecture Notes in Mathematics, vol. 1279. Springer, New York (1985) Google Scholar
21.
Clebsch, A.: Theorie der Elastizität fester Korper. Teubner, Leipzig (1862) Google Scholar
22.
Gurtin, M.E.: The linear theory of elasticity. In: Handbook of Physics, vol. VIa/2. Springer, New York (1972) Google Scholar
23.
Nye, J.F.: Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press, London (1985) Google Scholar
24.
Authier, A. (ed.): International Tables for Crystallography. D—Physical Properties of Crystals. Kluwer, Dordrecht (2003) Google Scholar
25.
Born, M., Wolf, E.: Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) edn. Cambridge University Press, Cambridge (2003) Google Scholar
26.
Shelby, R.A., Smith, D.R., Wolf, E.: Experimental verification of a negative index of refraction. Science 292(5514), 77–79 (2001) ADSCrossRefGoogle Scholar
27.
Abbot, R.N. Jr.: Calculation of the orientation of the optical indicatrix in monoclinic and triclinic crystals: The point-dipole model. Am. Mineral. 78, 952–956 (1993) Google Scholar

Copyright information

Authors and Affiliations

Fabrizio Daví
- 1
Email author
Daniele Rinaldi
- 2
- 3

1.Dipartimento di Ingegneria Civile, Edile ed ArchitetturaUniversitá Politecnica delle MarcheAnconaItaly
2.Dipartimento SIMAUUniversitá Politecnica delle MarcheAnconaItaly
3.Istituto Nazionale Fisica Nucleare-INFN, sezione di PerugiaPerugiaItaly

Mechanical and Optical Properties of Anisotropic Single-Crystal Prisms

Abstract

Keywords

Mathematics Subject Classification (2010)

Notes

Acknowledgements

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article