## Guang S. He

Print publication date: 2014

Print ISBN-13: 9780198702764

Published to Oxford Scholarship Online: December 2014

DOI: 10.1093/acprof:oso/9780198702764.001.0001

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# (p.620) Appendix 7 Tensor Elements of the Nuclear Third-Order Susceptibility in the Born–Oppenheimer Approximation

Source:
Nonlinear Optics and Photonics
Publisher:
Oxford University Press

Since the tensor element $χijkl(3)nuc$ of the nuclear third-order susceptibility is invariant under the permutations (ij), (kl), and (ijkl), it can be denoted by a new compact symbol $Dll′$. The relations of the cartesian indices between these two symbols are:${ij,kl=xxyyzzyzxzxyl,l′=123456.$ The distributions of nonzero elements of Dll are the same as those of the elastic compliance constant tensors of crystals. The number of the independent nonzero elements is given in parentheses.

Crystal System

Crystal Class

Triclinic

${11¯$

$D11D12D13D14D15D16D12D22D23D24D25D26D13D23D33D34D35D36D14D24D34D44D45D46D15D25D35D45D55D56D16D26D36D46D56D66$

(21)

Monoclinic

${2m2/m$

$D11D12D130D150D12D22D230D250D13D23D330D350000D440D46D15D25D350D550000D460D66$

(13)

Orthorhombic

${222mm2mmm$

$D11D12D13000D12D22D23000D13D23D33000000D44000000D55000000D66$

(9)

Trigonal

${33¯$

$D11D12D13D14−D250D12D11D13−D14D250D13D13D33000D14−D140D4402D25−D25D2500D442D140002D252D142(D11−D12)$

(7)

${3m3¯m32$

$D11D12D13D1400D12D11D13−D1400D13D13D33000D14−D140D44000000D442D1400002D142(D11−D12)$

(6)

Tetragonal

${44¯4/m$

$D11D12D1300D16D12D11D1300−D16D13D13D33000000D44000000D440D16−D16000D66$

(7)

${4mm4224¯2m4/mmm$

$D11D12D13000D12D11D13000D13D13D33000000D44000000D44000000D66$

(6)

Hexagonal

${66¯6/m6226mm6/mmm6¯m2$

$D11D12D13000D12D11D13000D13D13D33000000D44000000D440000002(D11−D12)$

(5)

Cubic

${23m34324¯3mm3m$

$D11D12D12000D12D11D12000D12D12D11000000D44000000D44000000D44$

(3)

Isotropic

$[D11D12D12000D12D11D12000D12D12D110000002(D1100 −D12) 00002(D110 −D12) 000002(D11−D12)]$

(2)

See R. W. Hellwarth, Prog. Quantum Electron. 5, 1(1977); J. F. Nye, Physical Properties of Crystals (Oxford University Press, London, 1957).

(p.621)