3.1 Piezoelectricity

Squeeze a lithium niobate crystal along certain directions and charge appears on its faces; apply a voltage and it changes shape. The effect needs a crystal without a center of symmetry; LN (point group 3m) qualifies. The response depends strongly on direction: pressing along the crystal Y axis gives a charge about 2.3 times larger than pressing along the polar Z axis (d₂₂ = 20.3 vs d₃₃ = 8.69 pC/N derived). One volt across a free crystal moves its faces by about 20 pm along Y.

Drag the press direction around the equator: the charge changes sign every 60°, the threefold axis seen directly. The largest response lies off every crystal axis, 29° below the basal plane.

            +Y
             |
 `-.         |         .-'
    `-.      |      .-'
       `-.   |   .-'
          `-.|.-'
 -X ········ ▲ ········ +X
          .-'|`-.
       .-'   |   `-.
    .-'      |      `-.
 .-'         |         `-.
             |

Constitutive equations, stress-charge form, full tensor and compressed (Voigt) notation P:

IEEE 176 states that this pair (independent variables S, E) is the only form that yields the field equations of the unbounded continuum; the other three forms are exact but used where boundaries make some variables approximately zero.

IEEE 176 Eqs. 37-42 P. Relations (Eq. 43): c^Es^E = 1, β^Sε^S = 1, c^D = c^E + e^th, ε^T = ε^S + d e^t, e = d c^E, g = β^Td, s^D = s^E − d^tg, β^T = β^S − g h^t, d = ε^Tg, h = g c^D.

3m form in the IEEE frame

IEEE 176 Table 8, class 3m; Weis & Gaylord 1985 Eq. 8 P. d has the same pattern except d₁₆ = −2d₂₂ (factor 2 from the strain convention).

LN values (congruent, SAW grade)

d₃₃ is the least certain coefficient: tabulated values span 6 to 16 pC/N, and the measured hydrostatic coefficient d_h = 6.31 pC/N implies d₃₃ of about 8.0 to 8.4 pC/N given d₃₁ of −0.85 to −1.04 pC/N, which rules out 12.5 to 16 (derived from the Graham 1976 and compiled d₃₁ values derived).

Longitudinal response along a unit vector n (polar angle θ from +Z, azimuth φ from +X), from the 3m d tensor:

d_L(+Y) = +d₂₂ and d_L(+Z) = +d₃₃. The sin 3φ term is the threefold axis made visible: in the XY plane the response alternates sign every 60°.

Coupling factors

IEEE 176 defines quasistatic material coupling factors from energy ratios, e.g. k₃₃² = d₃₃²/(s₃₃^Eε₃₃^T) = (s₃₃^E − s₃₃^D)/s₃₃^E, and thickness-mode factors k_t = e₃₃/(c₃₃^Dε₃₃^S)^1/2, k₁₅ = e₁₅/(c₅₅^Dε₁₁^S)^1/2 (Eqs. 124-125, Table 10 P). For the Kushibiki et al. 1999 set:

For a Y-cut plate no single thickness coupling factor exists for class 3m, because two stiffened waves couple at the electrodes (IEEE 176 Sec. 4.3, Eqs. 64-65 P).

Pick a matrix below; click a cell to see which full-tensor components it stands for. s₆₆ stands for four components, each one quarter of the Voigt value; c₆₆ for the same four with no factor.

Convention of 2s

Compressing symmetric index pairs into Voigt indices (1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = xz, 6 = xy) puts factors on some quantities and not others P:

Numerical checks: c₆₆ = (c₁₁ − c₁₂)/2 but s₆₆ = 2(s₁₁ − s₁₂); e₁₆ = −e₂₂ but d₁₆ = −2d₂₂; c₅₆ = c₁₄ but s₅₆ = 2s₁₄; p₆₆ = (p₁₁ − p₁₂)/2 (Weis & Gaylord 1985 Eqs. 16-17, 19 P). Software that stores tensors in Voigt form must state the convention. Full-rank tensors: /api/v1/tensors.json.

Sign frame

+Z is the end that becomes negative on compression (equivalently, positive on cooling); IEEE 1987, IRE 1949 (via Weis & Gaylord 1985 pp. 192-193) and IRE 1958 (via Volk & Wöhlecke 2008 p. 2) give the same +Z and +Y for LN (P P-comp P-comp). IEEE 176 states a 180° rotation about Z between its 1987 and 1949 axes for quartz (class 32) only. A 180° rotation about Z flips c₁₄, s₁₄, e₂₂, d₂₂, p₁₄, p₄₁ and r₂₂ together; once d₂₂ > 0 is imposed, no sign freedom remains, and c₁₄ > 0 follows derived. Flipping +Z (180° about X) flips every odd-rank tensor (e, d, r, χ⁽²⁾, pyroelectric p₃) but not c or p.

Clamped vs free

ε^T (free, below mechanical resonances) and ε^S (clamped, above them) differ by almost a factor 2 for ε₁₁ (83.3 vs 44.9 P), because d₁₅ and e₁₅ are large. The same mechanism makes r^T − r^S = p d large for r₂₂ and r₅₁ (Weis & Gaylord 1985 Eq. 13 P); see 2.2.

Isothermal vs adiabatic

IEEE 176 makes no notational distinction; adiabatic constants apply under rapidly varying conditions (IEEE 176 p. 7, text after Eq. 21 P).

Data spread

ε₃₃^S spans 23.7 (Yamada et al. 1967) to 29 (Savage 1966), both via Weis & Gaylord 1985 Table 1 P-comp, and ε₁₁^T 83.3 (Kushibiki et al. 1999 P) to 85.2 (Warner et al. 1967, via Weis & Gaylord 1985 P-comp). Values from different sets are shown as a spread, not blended.

Measurement

Kushibiki et al. 1999 obtained c^E, e and ε^S by fitting bulk-wave velocities measured on X-, Y- and Z-cut plates together with ε^T and density (Table II velocities, Tables IV-V constants P). Ogi et al. 2002 used resonant ultrasound spectroscopy at 30 °C P. Rotating these constants to other axes: rotate full-rank tensors (c_ijkl, e_ijk, ε_ij); the Bond 6×6 matrices (Bond 1943) are equivalent for Voigt storage, and IEEE 176 warns that compressed notation hides transformation properties.