lithium niobate / Lab / Tensors

Tensors and frame conventions

Rotated tensor

IEEE Std 176-1987 crystal axes; wafer frame z = normal, x = propagation

Inputs

Tensor

Record

Cut

Euler α0.0°

Euler β0.0°

Euler γ0.0°

Frame

Record PKushibiki et al., IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 46 1315 (1999)

Rotated Voigt matrix, C/m^2derived

	1	2	3	4	5	6
1	0	0	0	0	3.6550	−2.4070
2	−2.4070	2.4070	0	3.6550	0	0
3	0.3280	0.3280	1.8940	0	0	0

180° z+z

180° about z flips c₁₄, s₁₄, p₁₄, p₄₁, e₂₂, d₂₂, r₂₂. Reversing +z flips every odd-rank component and keeps c, s, p, ε.

R, wafer axes (rows) in crystal coordinates

COMSOL order:

{0, -2.407000e+0, 3.280000e-1, 0, 2.407000e+0, 3.280000e-1, 0, 0, 1.894000e+0, 0, 3.655000e+0, 0, 3.655000e+0, 0, 0, -2.407000e+0, 0, 0}

JSON, with frame and rotation

{
 "tensor": "piezo-stress-e",
 "record": "piezo-stress-e/cln/kushibiki1999",
 "symbol": "e",
 "unit": "C/m^2",
 "source": "Kushibiki et al., IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 46 1315 (1999)",
 "doi": "10.1109/58.796136",
 "status": "P",
 "frame": {
  "convention": "IEEE Std 176-1987, 3m: x perpendicular to a mirror plane, z polar (+Z: positive d33)",
  "global": "z = wafer normal, x = propagation",
  "cut": "Z-cut, X-prop",
  "euler_zxz_deg": [
   0,
   0,
   0
  ],
  "rot180_about_z": false,
  "plus_z_flipped": false,
  "R_crystal_to_global": [
   [
    1,
    0,
    0
   ],
   [
    0,
    1,
    0
   ],
   [
    0,
    0,
    1
   ]
  ]
 },
 "voigt_order": "xx, yy, zz, yz, xz, xy; engineering shear strain",
 "voigt": [
  [
   0,
   0,
   0,
   0,
   3.655,
   -2.407
  ],
  [
   -2.407,
   2.407,
   0,
   3.655,
   0,
   0
  ],
  [
   0.328,
   0.328,
   1.894,
   0,
   0,
   0
  ]
 ]
}

Crystal axes in the wafer frame; Z (polar axis) in green

Rotated tensor: square area ∝ |value|, filled positive, hollow negative; green ring: sign flips under 180° about z, dashed ring: under +z reversal

Tensor and source

Status P

Cut

Euler α0.0°

Euler β0.0°

Euler γ0.0°

Frame sign

180° about z flips c₁₄, s₁₄, p₁₄, p₄₁, e₂₂, d₂₂, r₂₂. Reversing +z flips every odd-rank component and keeps c, s, p, ε.

COMSOL order:

{0, -2.407000e+0, 3.280000e-1, 0, 2.407000e+0, 3.280000e-1, 0, 0, 1.894000e+0, 0, 3.655000e+0, 0, 3.655000e+0, 0, 0, -2.407000e+0, 0, 0}

JSON with frame and rotation

{
 "tensor": "piezo-stress-e",
 "record": "piezo-stress-e/cln/kushibiki1999",
 "symbol": "e",
 "unit": "C/m^2",
 "source": "Kushibiki et al., IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 46 1315 (1999)",
 "doi": "10.1109/58.796136",
 "status": "P",
 "frame": {
  "convention": "IEEE Std 176-1987, 3m: x perpendicular to a mirror plane, z polar (+Z: positive d33)",
  "global": "z = wafer normal, x = propagation",
  "cut": "Z-cut, X-prop",
  "euler_zxz_deg": [
   0,
   0,
   0
  ],
  "rot180_about_z": false,
  "plus_z_flipped": false,
  "R_crystal_to_global": [
   [
    1,
    0,
    0
   ],
   [
    0,
    1,
    0
   ],
   [
    0,
    0,
    1
   ]
  ]
 },
 "voigt_order": "xx, yy, zz, yz, xz, xy; engineering shear strain",
 "voigt": [
  [
   0,
   0,
   0,
   0,
   3.655,
   -2.407
  ],
  [
   -2.407,
   2.407,
   0,
   3.655,
   0,
   0
  ],
  [
   0.328,
   0.328,
   1.894,
   0,
   0,
   0
  ]
 ]
}

Conventions

Voigt 1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = xz, 6 = xy

Each stored record is a 3m Voigt matrix in its crystal-physics frame. This page converts it to the full Cartesian tensor, applies T'_ij… = R_ia R_jb … T_ab…, and converts back. Shear strain is engineering strain, so s carries factors 2 and 4 and strain-charge d a factor 2 on shear columns; c, e, r, SHG d and p carry none.

Sources give the sign of the odd-rank and the 14-type components only together with a frame. A record whose frame is unstated can pair with another record in a different frame and give the wrong sign of c₁₄, e₂₂ or p₄₁; the frame-sign toggles show which entries are at stake.

Tensor	Voigt shape	COMSOL export	SI factor
Elastic stiffness c^E	6 x 6	Elasticity matrix cE (Voigt), 21 values	1e+9 → Pa
Compliance s^E	6 x 6	Compliance matrix sE (Voigt), 21 values	1e-12 → 1/Pa
Piezoelectric e	3 x 6	Coupling matrix eES, 18 values	1 → C/m^2
Piezoelectric d	3 x 6	Coupling matrix dET, 18 values (same order as eES)	1e-12 → C/N
Electro-optic r^S	6 x 3	6 x 3 row-major (no COMSOL material slot)	1e-12 → m/V
Electro-optic r^T	6 x 3	6 x 3 row-major (no COMSOL material slot)	1e-12 → m/V
SHG d	3 x 6	3 x 6 row-major	1e-12 → m/V
Photoelastic p	6 x 6	6 x 6 row-major (p is not symmetric: p14 differs from p41)	1 → 1
Permittivity ε^T/ε₀	3 x 3	3 x 3 column-major, 9 values	1 → 1
Permittivity ε^S/ε₀	3 x 3	3 x 3 column-major, 9 values	1 → 1

All records with sources and status: property tables; JSON at /api/v1/tensors.json.