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Elastic Tensors of Crystal Systems

Adapted from Nye Physical Properties of Crystals

Adapted from Weishaupt and Hochheimer, Phys. Rev. B, 57 (1998)

q vector is the measurement direction. Each angle gives one longitudinal and two transverse acoustic modes. The longitudinal modes are the first three on diagonals (c11, c22, c33). The transverse / shear are the last three on-diagonal components (c44, c55, c66).

q vector is the measurement direction. Each angle gives one longitudinal and two transverse acoustic modes. A cubic crystal has a=b=c so the longitudinal mode is the only on-diagonal component (c11). The transverse / shear is the last on-diagonal component (c44).

q vector is the measurement direction. Each angle gives one longitudinal and two transverse acoustic modes. A hexagonal crystal has a = b and c so the longitudinal modes measured in the two direction give the on diagonal components along each (c11, c33). It is possible to reassing c as 11 and a as 33 depending on choice of axis (note in fiber symmetry which is essentially hexagonal symmetry, this is the case. The transverse / shear is the last on-diagonal component (c44) or c66. Note C66 = 1/2(c11-c12). (c44).

q vector is the measurement direction. Each angle gives one longitudinaland two transverse acoustic modes. Fiber symmetry is like a hexagonal crystal with a = b and c so the longitudinal modes measured in the two direction give the on diagonal components along each (c33, c11). It is possible to reassing c as 11 and a as 33 depending on choice of axis. Personal preference is to put 11 down the axis of the fiber. The transverse / shear is the last on-diagonal component (c44) or c66. Note C66 = 1/2(c11-c12). (c44).

Connections Between Crystal Space Groups

 

 

 

Symmetry elements of crystal systems, Thurston (1974)
Crystal System Essential Symmetry
Triclinic None
Monoclinic One 2-fold axis or one plane
Orthorhombic Three orthogonal 2-fold axes or 2 planes intersecting via a 2-fold axis
Tetragonal Either a 4-fold rotation axis or a 4-fold rotation-inversion axis
Rhombohedral One 3-fold axis but no 6-fold rotation or 6-fold rotation-inversion axis
Hexagonal Either a 6-fold rotation axis or a 6-fold rotation-inversion axis
Cubic Four 3-fold rotation axes which also implies three 2-fold axes