3.2 Elastic constants and bulk waves

Sound in LN travels at 3.5 to 7.4 km/s depending on direction and polarization (3474 to 7357 m/s over all directions derived): 7.33 km/s for a compression wave along the polar axis, 3.59 km/s for a shear wave along the same axis (Kushibiki et al. 1999 Table II P). Because the crystal is piezoelectric, an acoustic wave carries an electric field with it, and that field makes the crystal stiffer for that wave: the compression wave along Z is 3.2% faster than it would be in a non-piezoelectric crystal with the same elastic constants (7328.6 vs 7102.1 m/s derived).

Switch the stiffening off and on: the Z-longitudinal velocity moves between 7102 and 7329 m/s and lands on the measured value, while the Y-propagating shear wave polarized along X does not move at all.

Bulk velocities, computed vs measured (m/s)

Elastic stiffness at constant E, 3m form: c₂₂ = c₁₁, c₂₃ = c₁₃, c₂₄ = −c₁₄, c₅₅ = c₄₄, c₅₆ = c₁₄, c₆₆ = (c₁₁ − c₁₂)/2; six independent constants (Weis & Gaylord 1985 Eq. 17 P).

Density: 4642.8 ± 0.2 kg/m³ (Kushibiki et al. 1999 Table IV P); 4636 (Ogi et al. 2002 P); 4647 CLN, 4635 near-SLN (Volk & Wöhlecke 2008 Table 1.1 P-comp). Temperature coefficients of c^E (Smith and Welsh): c₁₁ −1.74, c₁₂ −2.52, c₁₃ −1.59, c₁₄ −2.14, c₃₃ −1.53, c₄₄ −2.04 × 10⁻⁴/K (Weis & Gaylord 1985 Table 6 P-comp). LiTaO₃ for comparison: c₁₄ = −10.75 GPa, opposite in sign to LN in the same convention (Kushibiki et al. 1999 P); the other LiTaO₃ constants are not available here.

Christoffel equation with piezoelectric stiffening, for a plane wave with unit wave normal n (IEEE 176 Eqs. 46-50 P):

Three velocities with orthogonal polarizations per direction; their inverses over all directions form the slowness surface. The second term is the stiffening: the wave's longitudinal electric field (quasistatic, curl-free) adds stiffness proportional to the square of the effective piezoelectric constant. The velocity table in L1 compares the computed velocities with the measured ones; the agreement (< 0.6 m/s) is a consistency check, not an independent test, since Kushibiki et al. 1999 fitted its constants to these velocities. Along X the remaining shear branch is 4028.96 m/s derived. Along Y the X-polarized shear is purely elastic (unstiffened) and the other two waves are stiffened and couple longitudinal and transverse motion, as IEEE 176 Eq. 53 states for class 3m P.

Solver. Γ is real symmetric; a 3×3 Jacobi eigen-decomposition per direction gives velocities and polarizations. Group velocity and power-flow angle follow from the gradient of ρv² = u·Γ(n)·u with respect to n (Hellmann-Feynman; Γ is homogeneous of degree 2 in n, so v_g = ∇_n(uΓu)/(2ρv)); Jaeken & Cottenier 2016 is the algorithm reference S. At degenerate directions (shear along Z) the group velocity depends on the eigenvector picked and is not unique.

Rotation. Rotate full-rank tensors (c_ijkl, e_ijk, ε_ij) with a_ip; the Bond 6×6 matrices (Bond 1943) are equivalent and convenient for Voigt storage. IEEE 176 warns that compressed notation hides transformation properties (p. 8, after Table 2 P).

Unstiffened vs stiffened. Setting e = 0 gives the unstiffened velocities; stiffening vanishes for directions where e_mijn_mn_i = 0.

c₆₆ consistency. Kushibiki et al. 1999 lists c₆₆ = 72.09 while (c₁₁ − c₁₂)/2 = 72.095; computations enforce the symmetry relation.

Composition. Elastic and photoelastic properties change little with MgO doping (Andrushchak et al. 2009, abstract S). The dependence of SAW velocity on Li₂O content is not quantified in the sources held.