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 & Management Optimization, doi: 10.3934/jimo.2021183
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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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar
[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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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