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2020 | OriginalPaper | Chapter

7. Third Rank Properties

Authors : Manuel Laso, Nieves Jimeno

Published in: Representation Surfaces for Physical Properties of Materials

Publisher: Springer International Publishing

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Abstract

Odd rank properties are, in Newnham’s words, null properties, meaning that they may vanish for certain point groups (like all centrosymmetric ones, see Sect. 3.​6). As a result, not all materials will display third rank properties. Also as a consequence of being of odd rank, the RS will consist of overlapping positive and negative lobes, as shown in Fig. 7.1.

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previous chapter Symmetric Second Rank Properties
next chapter Fourth Rank Properties
Footnotes
1
And all directed spheres of Chap. 5.
 
2
The RQ for \(\underline{\underline{B}}\) is the indicatrix for the refractive index [1].
 
3
No difficulty should arise from the use of r and \(\underline{\underline{\underline{r}}}\) in the same equation. Although the literal symbol is the same, their tensorial rank makes it impossible to confuse them.
 
4
How would you write this shear stress without resorting to a specific reference frame?.
 
5
For example, trigonal class 32.
 
6
Which is not a normal stress in the conventional axes to which \({{\,\mathrm{str}\,}}(\underset{{{{\backsim }}}}{d})\) is referred.
 
7
Check that (7.6) has three mirror planes at \(\dfrac{2\pi }{3}\) to each other; find their position (their \(\varphi \) values).
 
8
Try and identify the remaining geometric elements (axes, planes, etc.) of symmetry that leave (7.10) invariant. How many elements of the symmetry group (individual geometric transformations, each represented by its \(\smash {\underset{{{{\backsim }}}}{L}}\)) do these geometric elements of symmetry generate?
 
9
For rigorous treatments of tensor decomposition see [2‐5].
 
10
Because only one axis has to be oriented.
 
11
This is not the only possibility. Can you find another one?.
 
12
Can you find another two similar ways of constructing \(n_1n_2n_3\)?
 
13
You can think of \(\underline{P}\) as longitudinal, of \(\underset{{{{\backsim }}}}{d}\) as the upper half of a matrix like \(\underset{{{{\backsim }}}}{s}\) and then recall the block coupling (4.​35).
 
14
Check that for the example (7.17) the eigenvectors of: \(\underset{{{{\backsim }}}}{M}= \begin{bmatrix} 0&{}1&{}1\\ 1&{}0&{}1\\ 1&{}1&{}0 \end{bmatrix}\) form a basis in which \(d_{14}=d_{25}=d_{36}=0\). Is there anything special about this reference frame? What are the eigenvalues of \(\underset{{{{\backsim }}}}{M}\)?
 
Literature
1.
Born, M., Wolf, E.: Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Elsevier, Amsterdam (2013)
2.
Backus, G.: A geometrical picture of anisotropic elastic tensors. Rev. Geophys. 8(3), 633–671 (1970)CrossRef
3.
Hamermesh, M.: Group Theory and its Application to Physical Problems. Dover Publications Inc., New York (2003)
4.
Jerphagnon, J.: Invariants of the third-rank cartesian tensor: optical nonlinear susceptibilities. Phys. Rev. B 2(4), 1091 (1970)CrossRef
5.
Dinckal, C.: Orthonormal decomposition of third rank tensors and applications. In: Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering 2013, 3–5 July, 2013, London, UK, 139, vol. 144 (2013)
6.
Norris, A.N.: Quadratic invariants of elastic moduli. Q. J. Mech. Appl. Math. 60(3), 367–389 (2007)CrossRef
7.
Ahmad, F.: Invariants of a cartesian tensor of rank 3. Arch. Mech. 63(4), 383–392 (2011)
8.
9.
Qi, L.: Eigenvalues and invariants of tensors. J. Math. Anal. Appl. 325(2), 1363–1377 (2007)CrossRef
Metadata
Title
Third Rank Properties
Authors
Manuel Laso
Nieves Jimeno
Copyright Year
2020
Book
Representation Surfaces for Physical Properties of Materials

Print ISBN: 978-3-030-40869-5

Electronic ISBN: 978-3-030-40870-1

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-40870-1_7

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