Use of the tables

Vacher, R.; Courtens, E.

doi:10.1107/97809553602060000641

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 2.4, p. 331

Section 2.4.5. Use of the tables

R. Vacher^a ^* and E. Courtens^a

^a Laboratoire des Verres, Université Montpellier 2, Case 069, Place Eugène Bataillon, 34095 Montpellier CEDEX, France
Correspondence e-mail: rene.vacher@ldv.univ-montp2.fr

2.4.5. Use of the tables

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The tables in this chapter give information on modes and scattering geometries that are in most common use in the study of hypersound in single crystals. Just as in the case of X-rays, Brillouin scattering is not sensitive to the presence or absence of a centre of symmetry (Friedel, 1913). Hence, the results are the same for all crystalline classes belonging to the same centric group, also called Laue class. The correspondence between the point groups and the Laue classes analysed here is shown in Table 2.4.5.1. The monoclinic and triclinic cases, being too cumbersome, will not be treated here.

Table 2.4.5.1 | top | pdf |
Definition of Laue classes

Crystal system	Laue class	Point groups
Cubic		$[432,\bar 43m,m\bar 3m]$
Cubic		$[23,\bar 3m]$
Hexagonal		$[622,6mm,\bar 62m,6/mm]$
Hexagonal		$[6,\bar 6,6/m]$
Tetragonal		$[422,4mm,\bar 42m,4/mm]$
Tetragonal		$[4,\bar 4,4/m]$
Trigonal		$[32,3m,\bar 3m]$
Trigonal		$[3,\bar 3]$
Orthorhombic	O

For tensor components $[c_{ijk\ell }]$ and $[p_{ijk\ell }]$ , the tables make use of the usual contracted notation for index pairs running from 1 to 6. However, as the tensor $[p'_{ijk\ell }]$ is not symmetric upon interchange of $[(k,\ell)]$ , it is necessary to distinguish the order $[(k,\ell)]$ and $[(\ell, k)]$ . This is accomplished with the following correspondence: $[\matrix{1,1 \to 1\hfill & 2,2 \to 2\hfill &3,3 \to 3\hfill\cr 1,2 \to 6\hfill & 2,3 \to 4\hfill & 3,1 \to 5\hfill\cr 2,1 \to \bar 6\hfill & 3,2 \to \bar 4\hfill & 1,3 \to \bar 5.\hfill}]$

Geometries for longitudinal modes (LA) are listed in Tables 2.4.5.2 to 2.4.5.8 . The first column gives the direction of the scattering vector $[\hat{\bf Q}]$ that is parallel to the displacement $[\hat{\bf u}]$ . The second column gives the elastic coefficient according to (2.4.2.6). In piezoelectric materials, effective elastic coefficients defined in (2.4.2.11) must be used in this column. The third column gives the direction of the light polarizations $[{\hat{\bf e}}]$ and $[{\hat{\bf e}'}]$ , and the last column gives the corresponding coupling coefficient $[\beta]$ [equation (2.5.5.11)]. In general, the strongest scattering intensity is obtained for polarized scattering ( $[{\hat{\bf e}} = {\hat{\bf e}'}]$ ), which is the only situation listed in the tables. In this case, the coupling to light ( $[\beta]$ ) is independent of the scattering angle $[\theta]$ , and thus the tables apply to any $[\theta]$ value.

Table 2.4.5.2 | top | pdf |
Cubic Laue classes $[C{_1}]$ and $[C{_2}]$ : longitudinal modes

This table, written for the class $[C{_2}]$ , is also valid for the class $[C{_1}]$ with the additional relation $[p_{12} = p_{13}]$ . It can also be used for the spherical system where $[c_{44} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ , $[p_{44} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}= \hat{\bf u}]$	C	$[{\bf e}={\bf e}']$	$[\beta]$
	$[c_{11}]$		$[p_{13}^2 /c_{11}^{}]$
	$[c_{11}]$		$[p_{12}^2 /c_{11}^{}]$
$[(1,1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{11}^{} + c_{12}^{}) + c_{44}^{}]$		$[(p_{12}^{} + p_{13}^{})_{}^2 /4C]$
$[(1,1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{11}^{} + c_{12}^{}) + c_{44}^{}]$	$[(1, - 1,0)/\sqrt 2 ]$	$[(2p_{11}^{} + p_{12}^{} + p_{13}^{} - 4p_{44}^{})_{}^2 /16C]$
$[(1,1,1)/\sqrt 3 ]$	$[{\textstyle{1 \over 3}}(c_{11}^{} + 2c_{12}^{} + 4c_{44}^{})]$	$[(1,1, - 2)/\sqrt 6 ]$	$[(p_{11}^{} + p_{12}^{} + p_{13}^{} - 2p_{44}^{})_{}^2 /9C]$
$[(1,1,1)/\sqrt 3 ]$	$[{\textstyle{1 \over 3}}(c_{11}^{} + 2c_{12}^{} + 4c_{44}^{})]$	$[(1, - 1,0)/\sqrt 2 ]$	$[(p_{11}^{} + p_{12}^{} + p_{13}^{} - 2p_{44}^{})_{}^2 /9C]$

Table 2.4.5.3 | top | pdf |
Tetragonal $[T{_1}]$ and hexagonal $[H{_1}]$ Laue classes: longitudinal modes

This table, written for the class $[T{_1}]$ , is also valid for the class $[H{_1}]$ with the additional relations $[c_{66} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ ; $[p_{66} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}= \hat{\bf u}]$	C	$[{\bf e}={\bf e}']$	$[\beta]$
	$[c_{11}^{}]$		$[p_{12}^2 /c_{11}^{}]$
	$[c_{11}^{}]$		$[p_{31}^2 /c_{11}^{}]$
	$[c_{33}^{}]$		$[p_{13}^2 /c_{33}^{}]$
	$[c_{33}^{}]$		$[p_{13}^2 /c_{33}^{}]$
$[(1,1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{11}^{} + c_{12}^{}) + c_{66}^{}]$		$[p_{31}^2 /C]$
$[(1,1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{11}^{} + c_{12}^{}) + c_{66}^{}]$	$[(1, - 1,0)/\sqrt 2 ]$	$[(p_{11}^{} + p_{12}^{} - 2p_{66}^{})_{}^2 /4C]$

Table 2.4.5.4 | top | pdf |
Hexagonal Laue class $[H{_2}]$ : longitudinal modes

$[\hat{\bf Q}= \hat{\bf u}]$	C	$[{\bf e}={\bf e}']$	$[\beta]$
	$[c_{11}^{}]$		$[p_{12}^2 /c_{11}^{}]$
	$[c_{11}^{}]$		$[p_{31}^2 /c_{11}^{}]$
	$[c_{33}^{}]$		$[p_{13}^2 /c_{33}^{}]$
	$[c_{33}^{}]$		$[p_{13}^2 /c_{33}^{}]$
$[(1,1,0)/\sqrt 2 ]$	$[c_{11}^{}]$		$[p_{31}^2 /c_{11}^{}]$
$[(1,1,0)/\sqrt 2 ]$	$[c_{11}^{}]$	$[(1, - 1,0)/\sqrt 2 ]$	$[p_{12}^2 /c_{11}^{}]$

Table 2.4.5.5 | top | pdf |
Tetragonal Laue class $[T{_2}]$ : longitudinal modes

$[\hat{\bf Q}= \hat{\bf u}]$	C	$[{\bf e}={\bf e}']$	$[\beta]$
	$[c_{33}^{}]$		$[p_{13}^2 /c_{33}^{}]$
	$[c_{33}^{}]$		$[p_{13}^2 /c_{33}^{}]$

Table 2.4.5.6 | top | pdf |
Orthorhombic Laue class O: longitudinal modes

$[\hat{\bf Q}= \hat{\bf u}]$	C	$[{\bf e}={\bf e}']$	$[\beta]$
	$[c_{11}^{}]$		$[p_{21}^2 /c_{11}^{}]$
	$[c_{11}^{}]$		$[p_{31}^2 /c_{11}^{}]$
	$[c_{22}^{}]$		$[p_{32}^2 /c_{22}^{}]$
	$[c_{22}^{}]$		$[p_{12}^2 /c_{22}^{}]$
	$[c_{33}^{}]$		$[p_{13}^2 /c_{33}^{}]$
	$[c_{33}^{}]$		$[p_{23}^2 /c_{33}^{}]$

Table 2.4.5.7 | top | pdf |
Trigonal Laue class $[R{_1}]$ : longitudinal modes

$[\hat{\bf Q}= \hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
	$[c_{11}^{}]$			$[p_{12}^2 /c_{11}^{}]$
	$[c_{11}^{}]$			$[p_{31}^2 /c_{11}^{}]$
	$[c_{11}^{}]$			$[p_{41}^2 /c_{11}^{}]$
	$[c_{33}^{}]$			$[p_{13}^2 /c_{33}^{}]$
	$[c_{33}^{}]$			$[p_{13}^2 /c_{33}^{}]$

Table 2.4.5.8 | top | pdf |
Trigonal Laue class $[R{_2}]$ : longitudinal modes

$[\hat{\bf Q}= \hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
	$[c_{33}^{}]$			$[p_{13}^2 /c_{33}^{}]$
	$[c_{33}^{}]$			$[p_{13}^2 /c_{33}^{}]$

Tables 2.4.5.9 to 2.4.5.15 list the geometries usually used for the observation of TA modes in backscattering ( $[\theta=180^\circ]$ ). In this case, $[\hat{\bf u}]$ is always perpendicular to $[\hat{\bf Q}]$ (pure transverse modes), and $[{\hat{\bf e}'}]$ is not necessarily parallel to $[{\hat{\bf e}}]$ . Cases where pure TA modes with $[\hat{\bf u}]$ in the plane perpendicular to $[\hat{\bf Q}]$ are degenerate are indicated by the symbol D in the column for $[\hat{\bf u}]$ . For the Pockels tensor components, the notation is $[p_{\alpha \beta }]$ if the rotational term vanishes by symmetry, and it is $[p'_{\alpha \beta }]$ otherwise.

Table 2.4.5.9 | top | pdf |
Cubic Laue classes $[C{_1}]$ and $[C{_2}]$ : transverse modes, backscattering

This table, written for the class $[C{_2}]$ , is also valid for the class $[C{_1}]$ with the additional relation $[p_{12} = p_{13}]$ .

It can also be used for the spherical system where $[c_{44} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ , $[p_{44} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
$[(1,1,0)/\sqrt 2 ]$	$[(1, - 1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{11}^{} - c_{12}^{})]$			$[(p_{12}^{} - p_{13}^{})_{}^2 /2(c_{11}^{} - c_{12}^{})]$
$[(1,1,1)/\sqrt 3 ]$	D	$[{\textstyle{1 \over 3}}(c_{11}^{} - c_{12}^{} + c_{44}^{})_{}^{}]$	$[(1,1, - 2)/\sqrt 6 ]$	$[(1, - 1,0)/\sqrt 2 ]$	$[[{3(p_{12}^{} - p_{13}^{})_{}^2 + (p_{12}^{} + p_{13}^{} + 4p_{44}^{} - 2p_{11}^{})_{}^2 }]/72C]$

Table 2.4.5.10 | top | pdf |
Tetragonal $[T{_1}]$ and hexagonal $[H{_1}]$ Laue classes: transverse modes, backscattering

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
$[(0,1,1)/\sqrt 2 ]$		$[{\textstyle{1 \over 2}}(c_{44}^{} + c_{66}^{})]$		$[(0,1, - 1)/\sqrt 2 ]$	$[[(n_1^2 + n_3^2)^2 / 16n_1^4 n_3^4 C](n_1^2 p_{66}- n_3^2 p_{44}')_{}^2 ]$

Table 2.4.5.11 | top | pdf |
Hexagonal Laue class $[H{_2}]$ : transverse modes, backscattering

$[c_{66} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ ; $[p_{66} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}]$	C	$[{\bf e}']$	$[\beta]$
	$[c_{66}^{}]$		$[p_{16}^2 /c_{66}^{}]$
	$[c_{44}^{}]$	$[{\rm (0,0,1)}]$	$[p_{45}^2 /c_{44}^{}]$
$[(0,1,1)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{44}^{} + c_{66}^{})]$		$[p_{16}^2 /(c_{44}^{} + c_{66}^{})]$
$[(0,1,1)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{44}^{} + c_{66}^{})]$	$[(0,1, - 1)/\sqrt 2 ]$	$[[(n_1^2 + n_3^2)^2 /16n_1^4 n_3^4 C](n_1^2 p_{66}^{} - n_3^2 p_{44}')_{}^2 ]$

Table 2.4.5.12 | top | pdf |
Tetragonal Laue class $[T{_2}]$ : transverse modes, backscattering

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
		$[c_{44}^{}]$		$[{\rm (0,0,1)}]$	$[p_{45}^2 /c_{44}^{}]$
$[(1,1,0)/\sqrt 2 ]$		$[c_{44}^{}]$		$[(1, - 1,0)/\sqrt 2 ]$	$[p_{45}^2 /c_{44}^{}]$

Table 2.4.5.13 | top | pdf |
Orthorhombic Laue class O: transverse modes, backscattering

$[\hat{\bf Q}]$	C	$[{\bf e}']$	$[\beta]$
$[(1,1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{44}^{} + c_{55})]$	$[(1, - 1,0)/\sqrt 2 ]$	$[[(n_1^2 + n_2^2)^2 /16n_1^4 n_2^4 C](n_1^2 p_{55}' - n_2^2 p_{4\bar 4}')^2 ]$
$[(0,1,1)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{55}^{} + c_{66})]$	$[(0,1, - 1)/\sqrt 2 ]$	$[[(n_2^2 + n_3^2)^2 /16n_2^4 n_3^4 C](n_2^2 p_{66}' - n_3^2 p_{5\bar 5}')^2 ]$
$[(1,0,1)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{44}^{} + c_{66})]$	$[(- 1,0,1)/\sqrt 2 ]$	$[[(n_1^2 + n_3^2)^2 /16n_1^4 n_3^4 C](n_3^2 p_{44}' - n_1^2 p_{6\bar 6}')^2 ]$

Table 2.4.5.14 | top | pdf |
Trigonal Laue class $[R{_1}]$ : transverse modes, backscattering

$[c_{66} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ ; $[p_{66} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}']$	$[\beta]$
		$[c_{66}]$		$[p_{41}^2 /c_{66}^{}]$
	D	$[c_{44}]$		$[p_{14}^2 /c_{44}^{}]$
	D	$[c_{44}]$		$[p_{14}^2 /c_{44}^{}]$
$[(0,1,1)/\sqrt 2 ]$		$[{\textstyle{1 \over 2}}(c_{44}^{} + c_{66}) + c_{14}]$	$[(0,1, - 1)/\sqrt 2 ]$	$[[(n_1^2 + n_3^2)^2 /16n_1^4 n_3^4 C][{n_1^2 (p_{66}^{} + p_{14}^{}) - n_3^2 (p_{44}' + p_{41}^{})}]^{2}]$
$[(0,1, - 1)/\sqrt 2 ]$		$[{\textstyle{1 \over 2}}(c_{44}^{} + c_{66}) - c_{14}]$	$[(0,1,1)/\sqrt 2 ]$	$[[(n_1^2 + n_3^2)^2/16n_1^4 n_3^4 C][{n_1^2 (p_{66}^{} - p_{14}^{}) + n_3^2 (p_{41}^{} - p_{44}')}]^{2}]$

Table 2.4.5.15 | top | pdf |
Trigonal Laue class $[R{_2}]$ : transverse modes, backscattering

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
	D	$[c_{44}]$			$[(p_{14}^2 + p_{15}^2)/c_{44}^{}]$
	D	$[c_{44}]$			$[(p_{14}^2 + p_{15}^2)/c_{44}^{}]$

Tables 2.4.5.16 to 2.4.5.22 list the common geometries used for the observation of TA modes in 90° scattering. In these tables, the polarization vector $[{\hat{\bf e}}]$ is always perpendicular to the scattering plane and $[{\hat{\bf e}'}]$ is always parallel to the incident wavevector of light q. Owing to birefringence, the scattering vector $[\hat{\bf Q}]$ does not exactly bisect $[{\bf q}]$ and $[{\bf q}']$ [equation (2.4.4.4)]. The tables are written for strict 90° scattering, $[{\bf q}\cdot {\bf q}' = 0]$ , and in the case of birefringence the values of $[{\bf q}^{(m)}]$ to be used are listed separately in Table 2.4.5.23. The latter assumes that the birefringences are not large, so that the values of $[{\bf q}^{(m)}]$ are given only to first order in the birefringence.

Table 2.4.5.16 | top | pdf |
Cubic Laue classes $[C{_1}]$ and $[C{_2}]$ : transverse modes, right-angle scattering

This table, written for the class $[C{_2}]$ , is also valid for the class $[C{_1}]$ with the additional relation $[p_{12} = p_{13}]$ .

It can also be used for the spherical system where $[c_{44} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ , $[p_{44} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
		$[c_{44}]$		$[(1, - 1,0)/\sqrt 2 ]$	$[p_{44}^2 /2c_{44}^{}]$
	D	$[c_{44}]$		$[(1,0,1)/\sqrt 2 ]$	$[p_{44}^2 /2c_{44}^{}]$
$[(1,1,0)/\sqrt 2 ]$		$[c_{44} ]$			$[p_{44}^2 /2c_{44}^{}]$
$[(1,1,0)/\sqrt 2 ]$	$[(- 1,1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{11}^{} - c_{12})]$			$[(p_{12}^{} - p_{13}^{})_{}^2 /4C]$
$[(1,1,0)/\sqrt 2 ]$	$[(- 1,1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{11}^{} - c_{12})]$	$[(1, - 1,0)/\sqrt 2 ]$	$[(1,1, - \sqrt 2)/2]$	$[(2p_{11}^{} - p_{12}^{} - p_{13}^{})_{}^2 /32C]$

Table 2.4.5.17 | top | pdf |
Tetragonal $[T{_1}]$ and hexagonal $[H{_1}]$ Laue classes: transverse modes, right-angle scattering

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
		$[c_{44} ]$		$[(q_1^{(1)},q_2^{(1)},0)]$	$[(q_1^{(1)}p_{4\bar 4}')^2 /c_{44}^{}]$
		$[c_{66}]$		$[(q_1^{(2)},0,q_3^{(2)})]$	$[\{[{(n_3 q_1^{(2)})^2 + (n_1 q_3^{(2)})^2 }]^{2}/n_3^4 c_{66}\}(q_1^{(2)}p_{66})^2 ]$
	D	$[c_{44} ]$		$[(q_1^{(5)},0,q_3^{(5)})]$	$[\{[{(n_3 q_1^{(5)})^2 + (n_1 q_3^{(5)})^2 }]^{2}/n_1^4 c_{44}\}(q_3^{(5)}p_{44}')^2 ]$
$[(1,1,0)/\sqrt 2 ]$		$[c_{44}]$		$[(q_1^{(7)},q_2^{(7)},0)]$	$[ [{(q_1^{(7)} + q_2^{(7)})p_{4\bar 4}' }]^{2}/2c_{44}]$
$[(1,1,0)/\sqrt 2 ]$	$[(1, - 1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{11} - c_{12})]$	$[(1, - 1,0)/\sqrt 2 ]$	$[(q_1^{(10)},q_1^{(10)},q_3^{(10)})]$	$[\{ [{2(n_3 q_1^{(10)})^2 + (n_1 q_3^{(10)})^2 }]^{2}/n_3^4 (c_{11} - c_{12})\}[{q_1^{(10)}(p_{11} - p_{12})}]^{2}]$
$[(0,1,1)/\sqrt 2 ]$		$[{\textstyle{1 \over 2}}(c_{44} + c_{66})]$			$[p_{66}^2 /(c_{44} + c_{66})]$

Table 2.4.5.18 | top | pdf |
Hexagonal $[H{_2}]$ Laue class: transverse modes, right-angle scattering

$[c_{66} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ ; $[p_{66} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
		$[c_{44}]$		$[(q_1^{(1)},q_2^{(1)},0)]$	$[(q_1^{(1)}p_{4\bar 4}' + q_2^{(1)}p_{45}^{})^2 /c_{44}^{}]$
		$[c_{66}]$	$[(1,1,0)/\sqrt 2 ]$	$[(1, - 1,0)/\sqrt 2]$	$[p_{16}^2 /c_{66}]$
		$[c_{66} ]$		$[(q_1^{(2)},0,q_3^{(2)})]$	$[\{[{(n_3 q_1^{(2)})^2 + (n_1 q_3^{(2)})^2 }]^{2}/n_3^4 c_{66}\}(q_1^{(2)}p_{66})^2 ]$
	D	$[c_{44}]$		$[(q_1^{(5)},0,q_3^{(5)})]$	$[\{ [{(n_3 q_1^{(5)})^2 + (n_1 q_3^{(5)})^2 }]^{2}/n_1^4 c_{44}\}(q_3^{(5)})^2 (p_{44}^{'2} + p_{45}^2)]$
$[(0,1,1)/\sqrt 2 ]$		$[{\textstyle{1 \over 2}}(c_{44} + c_{66})]$			$[p_{16}^2 /(c_{44} + c_{66})]$
$[(0,1,1)/\sqrt 2 ]$		$[{\textstyle{1 \over 2}}(c_{44} + c_{66})]$			$[p_{66}^2 /(c_{44} + c_{66})]$

Table 2.4.5.19 | top | pdf |
Tetragonal $[T{_2}]$ Laue class: transverse modes, right-angle scattering

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}']$	$[\beta]$
		$[c_{44}]$	$[(q_1^{(1)},q_2^{(1)},0)]$	$[(q_1^{(1)}p_{4\bar 4}' + q_2^{(1)}p_{45})^2 /c_{44}^{}]$
		$[c_{44}]$	$[(q_1^{(2)},0,q_3^{(2)})]$	$[\{ [{(n_3 q_1^{(2)})^2 + (n_1 q_3^{(2)})^2 }]^{2}/n_1^4 c_{44}\}(q_3^{(2)}p_{45})^2 ]$
	D	$[c_{44}]$	$[(q_1^{(5)},0,q_3^{(5)})]$	$[\{ [{(n_3 q_1^{(5)})^2 + (n_1 q_3^{(5)})^2 }]^{2}/n_1^4 c_{44}\}(q_3^{(5)})^2 (p_{44}^{'2} + p_{45}^2)]$
$[(1,1,0)/\sqrt 2 ]$		$[c_{44}]$	$[(q_1^{(7)},q_2^{(7)},0)]$	$[[{(q_1^{(7)} + q_2^{(7)})p_{4\bar 4}' + (q_2^{(7)} - q_1^{(7)})p_{45}}]^{2}/2c_{44}]$

Table 2.4.5.20 | top | pdf |
Orthorhombic Laue class O: transverse modes, right-angle scattering

$[\hat{\bf Q}]$	C	$[{\bf e}']$	$[\beta]$
	$[c_{55}]$	$[(q_1^{(1)},q_2^{(1)},0)]$	$[\{ [{(n_2 q_1^{(1)})^2 + (n_1 q_2^{(1)})^2 }]^2 /n_2^4 c_{55}\}(q_1^{(1)}p_{55}')^2 ]$
	$[c_{66}]$	$[(q_1^{(2)},0,q_3^{(2)})]$	$[\{ [{(n_3 q_1^{(2)})^2 + (n_1 q_3^{(2)})^2 }]^2 /n_3^4 c_{66}\}(q_1^{(2)}p_{6\bar 6}')^2 ]$
	$[c_{66}]$	$[(0,q_2^{(3)},q_3^{(3)})]$	$[\{ [{(n_3 q_2^{(3)})^2 + (n_2 q_3^{(3)})^2 }]^2 /n_3^4 c_{66}\}(q_2^{(3)}p_{66}')^2 ]$
	$[c_{44}]$	$[(q_1^{(4)},q_2^{(4)},0)]$	$[\{ [{(n_2 q_1^{(4)})^2 + (n_1 q_2^{(4)})^2 }]^2 /n_1^4 c_{44}\}(q_2^{(4)}p_{4\bar 4}')^2 ]$
	$[c_{44}]$	$[(q_1^{(5)},0,q_3^{(5)})]$	$[\{ [{(n_3 q_1^{(5)})^2 + (n_1 q_3^{(5)})^2 }]^2 /n_1^4 c_{44}\}(q_3^{(5)}p_{44}')^2 ]$
	$[c_{55} ]$	$[(0,q_2^{(6)},q_3^{(6)})]$	$[\{ [{(n_3 q_2^{(6)})^2 + (n_2 q_3^{(6)})^2 }]^2 /n_2^4 c_{55}\}(q_3^{(6)}p_{5\bar 5}')^2 ]$
$[(1,1,0)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{44} + c_{55})]$	$[(q_1^{(7)},q_2^{(7)},0)]$	$[\{ [{(n_2 q_1^{(7)})^2 + (n_1 q_2^{(7)})^2 }]^2 /n_1^4 n_2^4 (c_{44} + c_{55})\}]$ $[\times(n_1^2 q_1^{(7)}p_{55}' + n_2^2 q_2^{(7)}p_{4\bar 4}')^2 ]$
$[(0,1,1)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{55} + c_{66})]$	$[(0,q_2^{(8)},q_3^{(8)})]$	$[\{ [{(n_3 q_2^{(8)})^2 + (n_2 q_3^{(8)})^2 }]^2 /n_2^4 n_3^4 (c_{55} + c_{66})\}]$ $[\times(n_2^2 q_2^{(8)}p_{66}' + n_3^2 q_3^{(8)}p_{5\bar 5}')^2 ]$
$[(1,0,1)/\sqrt 2 ]$	$[{\textstyle{1 \over 2}}(c_{44} + c_{66})]$	$[(q_1^{(9)},0,q_3^{(9)})]$	$[\{ [{(n_1 q_3^{(9)})^2 + (n_3 q_1^{(9)})^2 }]^2/n_1^4 n_3^4 (c_{44} + c_{66})\}]$ $[\times(n_3^2 q_3^{(9)}p_{44}' + n_1^2 q_1^{(9)}p_{6\bar 6}')^2 ]$

Table 2.4.5.21 | top | pdf |
Trigonal Laue class $[R{_1}]$ : transverse modes, right-angle scattering

$[c_{66} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ ; $[p_{66} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	$[{\bf e}']$	$[\beta]$
		$[c_{66} ]$	$[(0,q_2^{(3)},q_3^{(3)})]$	$[\{ [{(n_3 q_2^{(3)})^2 + (n_1 q_3^{(3)})^2 }]^2 /n_1^4 n_3^4 c_{66}\}(n_1^2 q_2^{(3)}p_{66} + n_3^2 q_3^{(3)}p_{41})^2 ]$
		$[c_{66}]$	$[(q_1^{(4)},q_2^{(4)},0)]$	$[(q_1^{(4)}p_{41})^2 /c_{66}]$
		$[c_{44} ]$		$[p_{14}^2 /c_{44}]$
	D	$[c_{44} ]$	$[(q_1^{(5)},0,q_3^{(5)})]$	$[\{ [{(n_3 q_1^{(5)})^2 + (n_1 q_3^{(5)})^2 }]^2 /n_1^4 n_3^4 c_{44}\} [{n_1^4 (q_1^{(5)}p_{14})^2 + n_3^4 (q_3^{(5)}p_{44}')^2 }]]$
$[(0,1,1)/\sqrt 2 ]$		$[{\textstyle{1 \over 2}}(c_{44}+ c_{66}) + c_{14}]$		$[(p_{66} + p_{14})^2 /2C]$
$[(0, - 1,1)/\sqrt 2 ]$		$[{\textstyle{1 \over 2}}(c_{44} + c_{66}) - c_{14}]$		$[(p_{66} - p_{14})^2 /2C]$

Table 2.4.5.22 | top | pdf |
Trigonal Laue class $[R{_2}]$ : transverse modes, right-angle scattering

$[c_{66} = {\textstyle{1 \over 2}}(c_{11} - c_{12})]$ ; $[p_{66} = {\textstyle{1 \over 2}}(p_{11} - p_{12})]$ .

$[\hat{\bf Q}]$	$[\hat{\bf u}]$	C	Scattering plane	$[{\bf e}]$	$[{\bf e}']$	$[\beta]$
	D	$[c_{44}]$				$[p_{14}^2 /c_{44}]$
	D	$[c_{44}]$			$[(q_1^{(5)},0,q_3^{(5)})]$	$[\{ [{(n_3 q_1^{(5)})^2 + (n_1 q_3^{(5)})^2 }]^{2}/n_1^4 n_3^4 c_{44}\} [{n_1^4 (q_1^{(5)}p_{14})^2 + n_3^4 (q_3^{(5)}p_{44}')^2 }]]$

Table 2.4.5.23 | top | pdf |
Particular directions of incident light used in Tables 2.4.5.17 to 2.4.5.22

$[\varepsilon _1 = (n_2 + n_3 - 2n_1)/4n_1 ]$ , $[\varepsilon _2 = (n_1 + n_3 - 2n_2)/4n_2 ]$ , $[\varepsilon _3 = (n_1 + n_2 - 2n_3)/4n_3 ]$ .

Notation
$[{\bf q}^{(1)}]$	$[ - 2^{(-1/2)}(1 - \varepsilon _3)]$	$[2^{(-1/2)}(1 + \varepsilon _3)]$
$[{\bf q}^{(2)}]$	$[ - 2^{(-1/2)}(1 - \varepsilon _2)]$		$[2^{(-1/2)}(1 + \varepsilon _2)]$
$[{\bf q}^{(3)}]$		$[ - 2^{(-1/2)}(1 - \varepsilon _1)]$	$[2^{(-1/2)}(1 + \varepsilon _1)]$
$[{\bf q}^{(4)}]$	$[2^{(-1/2)}(1 + \varepsilon _3)]$	$[ - 2^{(-1/2)}(1 - \varepsilon _3)]$
$[{\bf q}^{(5)}]$	$[2^{(-1/2)}(1 + \varepsilon _2)]$		$[ - 2^{(-1/2)}(1 + \varepsilon _2)]$
$[{\bf q}^{(6)}]$		$[2^{(-1/2)}(1 + \varepsilon _1)]$	$[ - 2^{(-1/2)}(1 - \varepsilon _1)]$
$[{\bf q}^{(7)}]$	$[ - 2^{(-1/2)}(n_1 + n_3)(n_1^2 + n_3^2)^{(-1/2)}]$	$[2^{(-1/2)}(n_1 - n_3)(n_1^2 + n_3^2)^{(-1/2)}]$
$[{\bf q}^{(8)}]$		$[ - 2^{(-1/2)}(n_1 + n_2)(n_1^2 + n_2^2)^{(-1/2)}]$	$[2^{(-1/2)}(n_2 - n_1)(n_1^2 + n_2^2)^{(-1/2)}]$
$[{\bf q}^{(9)}]$	$[2^{(-1/2)}(n_3 - n_2)(n_2^2 + n_3^2)^{(-1/2)}]$		$[ - 2^{(-1/2)}(n_2 + n_3)(n_2^2 + n_3^2)^{(-1/2)}]$
$[{\bf q}^{(10)}]$	$[ - {\textstyle{1 \over 2}}(1 - \varepsilon _2)]$	$[ - {\textstyle{1 \over 2}}(1 - \varepsilon _2)]$	$[2^{(-1/2)}(1 + \varepsilon _2)]$

References

Friedel, G. (1913). Sur les symétries cristallines que peut révéler la diffraction des rayons Röntgen. C. R. Acad. Sci. Paris, 157, 1533–1536.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 2.4, p. 331