doi: 10.3934/jimo.2021183
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The C-eigenvalue of third order tensors and its application in crystals

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

2. 

Liquid Crystal Institute, Kent State University, Kent, OH 44242, USA

3. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

*Corresponding author: Yannan Chen

Received  January 2021 Revised  May 2021 Early access November 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China (11771405 and 12171168) and the Natural Science Foundation of Guangdong Province, China (2020A1515010489)

In crystallography, piezoelectric tensors of various crystals play a crucial role in piezoelectric effect and converse piezoelectric effect. Generally, a third order real tensor is called a piezoelectric-type tensor if it is partially symmetric with respect to its last two indices. The piezoelectric tensor is a piezoelectric-type tensor of dimension three. We introduce C-eigenvalues and C-eigenvectors for piezoelectric-type tensors. Here, "C'' names after Curie brothers, who first discovered the piezoelectric effect. We show that C-eigenvalues always exist, they are invariant under orthogonal transformations, and for a piezoelectric-type tensor, the largest C-eigenvalue and its C-eigenvectors form the best rank-one piezoelectric-type approximation of that tensor. This means that for the piezoelectric tensor, its largest C-eigenvalue determines the highest piezoelectric coupling constant. We further show that for the piezoelectric tensor, the largest C-eigenvalue corresponds to the electric displacement vector with the largest 2-norm in the piezoelectric effect under unit uniaxial stress, and the strain tensor with the largest 2-norm in the converse piezoelectric effect under unit electric field vector. Thus, C-eigenvalues and C-eigenvectors have concrete physical meanings in piezoelectric effect and converse piezoelectric effect. Finally, by numerical experiments, we report C-eigenvalues and associated C-eigenvectors for piezoelectric tensors corresponding to several piezoelectric crystals.

Citation: Yannan Chen, Antal Jákli, Liqun Qi. The C-eigenvalue of third order tensors and its application in crystals. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021183
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H. CheH. Chen and Y. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Appl. Math. Lett., 89 (2019), 41-49.  doi: 10.1016/j.aml.2018.09.014.

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Y. ChenL. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A: Mathematical and Theoretical, 51 (2018), 025206.  doi: 10.1088/1751-8121/aa98a8.

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J. Curie and P. Curie, Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées, Comptes Rendus (in French), 93 (1881), 1137-1140. 

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M. de JongW. ChenH. GeerlingsM. Asta and K. A. Persson, A database to enable discovery and design of piezoelectric materials, Scientific Data, 2 (2015), 150053.  doi: 10.1038/sdata.2015.53.

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G. Gaeta and E. G. Virga, Octupolar order in three dimensions, The European Physical Journal E, 39 (2016), 113.  doi: 10.1140/epje/i2016-16113-7.

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A. JákliT. Tóth-KatonaT. ScharfM. Schadt and A. Saupe, Piezolelectricity of a ferroelectric liquid crystal with a gltransition, Physical Review E, 66 (2002), 011701. 

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J. Jerphagnon, Invariants of the third-rank Cartesian tensor: Optical nonlinear susceptibilities, Physical Review B, 2 (1970), 1091.  doi: 10.1103/PhysRevB.2.1091.

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A. L. Kholkin, N. A. Pertsev and A. V. Goltsev, Piezolelectricity and crystal symmetry, In Piezoelectric and Acoustic Materials, (eds. A. Safari and E.K. Akdo gan), Springer, New York, (2008), 17–38.

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I. A. KulaginR. A. GaneevR. I. TugushevA. I. Ryasnyansky and T. Usmanov, Components of the third-order nonlinear susceptibility tensors in KDP, DKDP and LiNbO$_3$ nonlinear optical crystals, Quantum Electronics, 34 (2004), 657. 

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C. LiY. Liu and Y. Li, C-eigenvalues intervals for piezoelectric-type tensors, Appl. Math. Comput., 358 (2019), 244-250.  doi: 10.1016/j.amc.2019.04.036.

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G. Lippmann, Principe de la conservation de l'électricité, Annales De Chimie Et De Physique, 24 (1881), 145-178. 

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D. R. Lovett, Tensor Properties of Crystals, 2$^nd$ Edition, Institute of Physics Publishing, Bristol, 1989.

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L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Computation, 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

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L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Advances in Mechanics and Mathematics, 39. Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.

[21]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974751.ch1.

[22]

L. QiY. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensors, J. Comput. Appl. Math., 221 (2008), 150-157.  doi: 10.1016/j.cam.2007.10.012.

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E. G. Virga, Octupolar order in two dimensions, The European Physical Journal E, 38 (2015), 63.  doi: 10.1140/epje/i2015-15063-x.

[24]

W. WangH. Chen and Y. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Appl. Math. Lett., 100 (2020), 106035.  doi: 10.1016/j.aml.2019.106035.

[25]

T. Zhang and G. H. Golub, Rank-one approximation to high order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.

[26]

W. N. ZouC. X. Tang and E. Pan, Symmetric types of the piezotensor and their identification, Proceedings of the Royal Society A, 469 (2013), 20120755. 

show all references

References:
[1]

D. Cartwright and B. Sturmfels, The number of eigenvalues of a tensor, Linear Algebra Appl., 438 (2013), 942-952.  doi: 10.1016/j.laa.2011.05.040.

[2]

H. CheH. Chen and Y. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Appl. Math. Lett., 89 (2019), 41-49.  doi: 10.1016/j.aml.2018.09.014.

[3]

Y. ChenL. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A: Mathematical and Theoretical, 51 (2018), 025206.  doi: 10.1088/1751-8121/aa98a8.

[4]

J. Curie and P. Curie, Développement, par pression, de l'électricité polaire dans les cristaux hémièdres à faces inclinées, Comptes Rendus (in French), 91 (1880), 294-295. 

[5]

J. Curie and P. Curie, Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées, Comptes Rendus (in French), 93 (1881), 1137-1140. 

[6]

M. de JongW. ChenH. GeerlingsM. Asta and K. A. Persson, A database to enable discovery and design of piezoelectric materials, Scientific Data, 2 (2015), 150053.  doi: 10.1038/sdata.2015.53.

[7]

G. Gaeta and E. G. Virga, Octupolar order in three dimensions, The European Physical Journal E, 39 (2016), 113.  doi: 10.1140/epje/i2016-16113-7.

[8]

S. Haussühl, Physical Properties of Crystals: An Introduction, Wiley-VCH Verlag, Weinheim, 2007.

[9]

A. Jákli, Electro-mechanical effects in liquid crystals, Liquid Crystals, 37 (2010), 825-837. 

[10]

A. JákliI. C. PintreJ. L. SerranoM. B. Ros and M. R. de la Fuente, Iezoelectric and electric-field-induced properties of a ferroelectric bent-core liquid crstal, Advanced Materials, 21 (2009), 3784-3788. 

[11]

A. JákliT. Tóth-KatonaT. ScharfM. Schadt and A. Saupe, Piezolelectricity of a ferroelectric liquid crystal with a gltransition, Physical Review E, 66 (2002), 011701. 

[12]

J. Jerphagnon, Invariants of the third-rank Cartesian tensor: Optical nonlinear susceptibilities, Physical Review B, 2 (1970), 1091.  doi: 10.1103/PhysRevB.2.1091.

[13]

A. L. Kholkin, N. A. Pertsev and A. V. Goltsev, Piezolelectricity and crystal symmetry, In Piezoelectric and Acoustic Materials, (eds. A. Safari and E.K. Akdo gan), Springer, New York, (2008), 17–38.

[14]

I. A. KulaginR. A. GaneevR. I. TugushevA. I. Ryasnyansky and T. Usmanov, Components of the third-order nonlinear susceptibility tensors in KDP, DKDP and LiNbO$_3$ nonlinear optical crystals, Quantum Electronics, 34 (2004), 657. 

[15]

C. LiY. Liu and Y. Li, C-eigenvalues intervals for piezoelectric-type tensors, Appl. Math. Comput., 358 (2019), 244-250.  doi: 10.1016/j.amc.2019.04.036.

[16]

G. Lippmann, Principe de la conservation de l'électricité, Annales De Chimie Et De Physique, 24 (1881), 145-178. 

[17]

D. R. Lovett, Tensor Properties of Crystals, 2$^nd$ Edition, Institute of Physics Publishing, Bristol, 1989.

[18] J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, 2$^nd$ Edition, Clarendon Press, Oxford, 1985.  doi: 10.1063/1.3060200.
[19]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Computation, 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[20]

L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Advances in Mechanics and Mathematics, 39. Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.

[21]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974751.ch1.

[22]

L. QiY. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensors, J. Comput. Appl. Math., 221 (2008), 150-157.  doi: 10.1016/j.cam.2007.10.012.

[23]

E. G. Virga, Octupolar order in two dimensions, The European Physical Journal E, 38 (2015), 63.  doi: 10.1140/epje/i2015-15063-x.

[24]

W. WangH. Chen and Y. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Appl. Math. Lett., 100 (2020), 106035.  doi: 10.1016/j.aml.2019.106035.

[25]

T. Zhang and G. H. Golub, Rank-one approximation to high order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.

[26]

W. N. ZouC. X. Tang and E. Pan, Symmetric types of the piezotensor and their identification, Proceedings of the Royal Society A, 469 (2013), 20120755. 

Table 1.  Positive C-eigenvalues of a piezoelectric tensor of $ \alpha $-quartz
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 0.137536 1.0 0.0 0.0 0.0 0.997515 -0.0704604
2 0.137536 -0.5 0.866025 0.0 0.863873 0.498757 0.0704604
3 0.137536 -0.5 -0.866025 0.0 0.863873 -0.498757 -0.0704604
4 0.13685 -1.0 0.0 0.0 1.0 0.0 0.0
5 0.13685 0.5 0.866025 0.0 0.5 0.866025 0.0
6 0.13685 0.5 -0.866025 0.0 0.5 -0.866025 0.0
7 0.000686228 -1.0 0.0 0.0 0.0 0.0704604 0.997515
8 0.000686228 0.5 -0.866025 0.0 0.0610205 0.0352302 -0.997515
9 0.000686228 0.5 0.866025 0.0 0.0610205 -0.0352302 0.997515
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 0.137536 1.0 0.0 0.0 0.0 0.997515 -0.0704604
2 0.137536 -0.5 0.866025 0.0 0.863873 0.498757 0.0704604
3 0.137536 -0.5 -0.866025 0.0 0.863873 -0.498757 -0.0704604
4 0.13685 -1.0 0.0 0.0 1.0 0.0 0.0
5 0.13685 0.5 0.866025 0.0 0.5 0.866025 0.0
6 0.13685 0.5 -0.866025 0.0 0.5 -0.866025 0.0
7 0.000686228 -1.0 0.0 0.0 0.0 0.0704604 0.997515
8 0.000686228 0.5 -0.866025 0.0 0.0610205 0.0352302 -0.997515
9 0.000686228 0.5 0.866025 0.0 0.0610205 -0.0352302 0.997515
Table 2.  Positive C-eigenvalues of the piezoelectric tensor of Cr2AgBiO8
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 2.6258 0.872141 0.48317 0.0769254 0.589036 -0.394962 0.705012
2 2.6258 -0.872141 -0.48317 0.0769254 0.589036 -0.394962 -0.705012
3 2.6258 0.48317 -0.872141 -0.0769254 0.394962 0.589036 0.705012
4 2.6258 -0.48317 0.872141 -0.0769254 0.394962 0.589036 -0.705012
5 2.61806 0.961197 -0.275862 0.0 0.693742 0.136827 0.707107
6 2.61806 -0.961197 0.275862 0.0 0.693742 0.136827 -0.707107
7 2.61806 0.275862 0.961197 0.0 0.136827 -0.693742 0.707107
8 2.61806 -0.275862 -0.961197 0.0 0.136827 -0.693742 -0.707107
9 0.401605 0.0 0.0 1.0 0.830569 -0.556916 0.0
10 0.401605 0.0 0.0 -1.0 0.556916 0.830569 0.0
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 2.6258 0.872141 0.48317 0.0769254 0.589036 -0.394962 0.705012
2 2.6258 -0.872141 -0.48317 0.0769254 0.589036 -0.394962 -0.705012
3 2.6258 0.48317 -0.872141 -0.0769254 0.394962 0.589036 0.705012
4 2.6258 -0.48317 0.872141 -0.0769254 0.394962 0.589036 -0.705012
5 2.61806 0.961197 -0.275862 0.0 0.693742 0.136827 0.707107
6 2.61806 -0.961197 0.275862 0.0 0.693742 0.136827 -0.707107
7 2.61806 0.275862 0.961197 0.0 0.136827 -0.693742 0.707107
8 2.61806 -0.275862 -0.961197 0.0 0.136827 -0.693742 -0.707107
9 0.401605 0.0 0.0 1.0 0.830569 -0.556916 0.0
10 0.401605 0.0 0.0 -1.0 0.556916 0.830569 0.0
Table 3.  Positive C-eigenvalues of the piezoelectric tensor of RbTaO3
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 12.4234 0.804378 0.464408 -0.370541 0.695227 0.401389 -0.596277
2 12.4234 -0.804378 0.464408 -0.370541 0.695227 -0.401389 0.596277
3 12.4234 0.0 -0.928816 -0.370541 0.0 0.802779 0.596277
4 7.82245 0.677808 -0.391333 -0.622442 0.49743 -0.287191 -0.818587
5 7.82245 -0.677808 -0.391333 -0.622442 0.49743 0.287191 0.818587
6 7.82245 0.0 0.782666 -0.622442 0.0 0.574382 -0.818587
7 6.91463 0.677894 -0.391382 -0.622318 0.5 0.866025 0.0
8 6.91463 -0.677894 -0.391382 -0.622318 0.5 -0.866025 0.0
9 6.91463 0.0 0.782764 -0.622318 1.0 0.0 0.0
10 5.14766 0.0 0.0 -1.0 0.0 0.0 1.0
11 4.38052 0.0247105 0.0142666 -0.999593 0.826334 0.477084 0.299271
12 4.38052 -0.0247105 0.0142666 -0.999593 0.826334 -0.477084 -0.299271
13 4.38052 0.0 -0.0285332 -0.999593 0.0 0.954168 -0.299271
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 12.4234 0.804378 0.464408 -0.370541 0.695227 0.401389 -0.596277
2 12.4234 -0.804378 0.464408 -0.370541 0.695227 -0.401389 0.596277
3 12.4234 0.0 -0.928816 -0.370541 0.0 0.802779 0.596277
4 7.82245 0.677808 -0.391333 -0.622442 0.49743 -0.287191 -0.818587
5 7.82245 -0.677808 -0.391333 -0.622442 0.49743 0.287191 0.818587
6 7.82245 0.0 0.782666 -0.622442 0.0 0.574382 -0.818587
7 6.91463 0.677894 -0.391382 -0.622318 0.5 0.866025 0.0
8 6.91463 -0.677894 -0.391382 -0.622318 0.5 -0.866025 0.0
9 6.91463 0.0 0.782764 -0.622318 1.0 0.0 0.0
10 5.14766 0.0 0.0 -1.0 0.0 0.0 1.0
11 4.38052 0.0247105 0.0142666 -0.999593 0.826334 0.477084 0.299271
12 4.38052 -0.0247105 0.0142666 -0.999593 0.826334 -0.477084 -0.299271
13 4.38052 0.0 -0.0285332 -0.999593 0.0 0.954168 -0.299271
Table 4.  Positive C-eigenvalues of the piezoelectric tensor of NaBiS2
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 11.6674 0.762919 0.0 -0.646494 0.693139 0.0 -0.720804
2 11.6674 -0.762919 0.0 -0.646494 0.693139 0.0 0.720804
3 7.93831 0.0 0.0 -1.0 0.0 0.0 1.0
4 7.11526 0.0 0.0 -1.0 1.0 0.0 0.0
5 0.6222 0.0 0.0 -1.0 0.0 1.0 0.0
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 11.6674 0.762919 0.0 -0.646494 0.693139 0.0 -0.720804
2 11.6674 -0.762919 0.0 -0.646494 0.693139 0.0 0.720804
3 7.93831 0.0 0.0 -1.0 0.0 0.0 1.0
4 7.11526 0.0 0.0 -1.0 1.0 0.0 0.0
5 0.6222 0.0 0.0 -1.0 0.0 1.0 0.0
Table 5.  Positive C-eigenvalues of the piezoelectric tensor of LiBiB2O5
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 7.73762 0.302351 0.0148322 0.953081 0.234203 0.707114 0.667187
2 7.73762 -0.302351 0.0148322 -0.953081 0.234203 -0.707114 0.667187
3 0.499616 0.902379 0.320695 -0.287865 0.675213 -0.698513 -0.236998
4 0.499616 -0.902379 0.320695 0.287865 0.675213 0.698513 -0.236998
5 0.205796 0.0 1.0 0.0 0.780252 0.0 -0.625465
6 0.12562 0.0 1.0 0.0 0.0 1.0 0.0
7 0.0913135 0.0 1.0 0.0 0.625465 0.0 0.780252
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 7.73762 0.302351 0.0148322 0.953081 0.234203 0.707114 0.667187
2 7.73762 -0.302351 0.0148322 -0.953081 0.234203 -0.707114 0.667187
3 0.499616 0.902379 0.320695 -0.287865 0.675213 -0.698513 -0.236998
4 0.499616 -0.902379 0.320695 0.287865 0.675213 0.698513 -0.236998
5 0.205796 0.0 1.0 0.0 0.780252 0.0 -0.625465
6 0.12562 0.0 1.0 0.0 0.0 1.0 0.0
7 0.0913135 0.0 1.0 0.0 0.625465 0.0 0.780252
Table 6.  Positive C-eigenvalues of the piezoelectric tensor of KBi2F7
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 13.5021 0.970501 0.209737 0.118907 0.972258 0.0506481 0.228363
2 4.46957 0.981961 0.189047 -0.00361752 0.22771 -0.414908 -0.880908
3 0.544863 0.759805 -0.368785 0.535439 0.0616756 0.870474 -0.488334
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 13.5021 0.970501 0.209737 0.118907 0.972258 0.0506481 0.228363
2 4.46957 0.981961 0.189047 -0.00361752 0.22771 -0.414908 -0.880908
3 0.544863 0.759805 -0.368785 0.535439 0.0616756 0.870474 -0.488334
Table 7.  Positive C-eigenvalues of the piezoelectric tensor of BaNiO3
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 27.4628 0.0 0.0 1.0 0.0 0.0 1.0
2 6.89822 0.0 0.0 1.0 $ y_1 $ $ \pm\sqrt{1-y_2^2} $ 0.0
No. $ \lambda $ $ {\bf {x}}^ \top $ $ {\bf {y}}^ \top $
1 27.4628 0.0 0.0 1.0 0.0 0.0 1.0
2 6.89822 0.0 0.0 1.0 $ y_1 $ $ \pm\sqrt{1-y_2^2} $ 0.0
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