1.7 Symmetry and every tensor

L1 Neumann's principle

Weis & Gaylord 1985 p. 198; point group 3m, IEEE 176 frame

A crystal property can have no less symmetry than the crystal. This is Neumann's principle, stated in 1833 P. For LiNbO₃ (point group 3m) it means every response must look the same after a 120° turn about c and after reflection in any of the three mirror planes. Most conceivable coefficients fail that test and are exactly zero; the survivors are tied together in fixed ratios.

Consequences you can see: the crystal has one optic axis, along c, so light travelling along c sees one index whatever its polarization; along any other direction it splits in two (02 Light). Heating or cooling puts charge only on the c faces. Compression along any axis never charges the x faces, which are perpendicular to a mirror plane P.

Neumann machine: piezoelectric d, 3 × 6 group 3m independent 4 / 18

property

generators

field

group

EC3C3²m yzm +60°m −60°

12345610000d15−2d222−d22d220d15003d31d31d33000

relationszero 10 tied 4

d16 = −2d22   d21 = −d22   d24 = d15
d32 = d31
zero: d11 d12 d13 d14 d23 d25 d26 d34
d35 d36

B = 1/n², ΔB_I = r_IjE_j derivedE = 0: group 3m, uniaxial

ΔB1 = 0
ΔB2 = 0
ΔB3 = 0
ΔB4 = 0
ΔB5 = 0
ΔB6 = 0

point at a component

d15 independent−d22 tied0 zeroc12 transpose (index symmetry)r11 allowed only under E
All operations off: every component independent. C3 on: most vanish, survivors pair up. Add one mirror: d keeps d15, d22, d31, d33. Then E along y: one mirror survives, the group reads m. Closure adds what the chosen elements imply (two mirrors give C3; C3 and one mirror give all three mirrors), drawn dashed. Under a field the matrix shows what crystal plus field allow; magnitudes are not computed. derived derived

Group3m

Independent 4/18

Property

Symmetry operations

All operations

Static field

EC3C3²m yzm +60°m −60°

point at a component

Relationszero 10 tied 4

d16 = −2d22   d21 = −d22   d24 = d15
d32 = d31
zero: d11 d12 d13 d14 d23 d25 d26 d34
d35 d36

Induced Δ(1/n²) derivedE = 0: group 3m, uniaxial

ΔB1 = 0
ΔB2 = 0
ΔB3 = 0
ΔB4 = 0
ΔB5 = 0
ΔB6 = 0

Switch every operation off: all components light. Turn on the threefold axis: most go dark and the survivors pair up. Add one mirror: d keeps d15, d22, d31, d33. Then put the field along y: one mirror line survives and the group reads m. Filled: independent; outlined with a link to its partner: tied; dot: zero; small dashed: transpose; gold: allowed only under the field. Elements implied by closure are drawn dashed. The lobes repeat once per element of the group. derived derived

L2 Axes and allowed forms

IRE 1949 / IEEE 176-1987 axes; Weis & Gaylord 1985 Eqs. (2)-(19)

Axes for tensors follow IRE 1949 / IEEE 176-1987, as used by Weis and Gaylord and by Volk and Wöhlecke.

Axis	Rule	Status
z	parallel to c, the threefold axis	P
x	along one of the equivalent a axes, perpendicular to a mirror plane; non-polar	P P
y	completes a right-handed set; lies in a mirror plane	P
+z sense	the c face that becomes negative on compression; equivalently positive on cooling; third test: cleavage terracing	P P-comp
+y sense	the y face that becomes negative on compression	P
IEEE 176-1987	+Z gives positive d₃₃, +Y positive d₂₂	P

Which face is +z

One z-cut plate, three tests; face A up as drawn

1 compress along c

          ||
     ====####====

  . . . . . . . . .     A
 +------------------+
 |        ^         |
 |        | Ps      |
 |        |         |
 +------------------+
  . . . . . . . . .     B

     ====####====
          ||

A 0 B 0 +z = ?

+z: the c face that becomes negative on compression P; +Z gives positive d₃₃ P

2 cool

  T steady    dT = 0

  . . . . . . . . .     A
 +------------------+
 |        ^         |
 |        | Ps      |
 |        |         |
 +------------------+
  . . . . . . . . .     B
[======================]
| n p n p n p n p n p  |
[======================]

A 0 B 0 +z = ?

+z: the c face that becomes positive on cooling; p₃ < 0 P

3 etch

|~~~~~~~~~~~~~~~~~~~~~~|
|                      |
|                      |
|  +----------------+  |  A
|  |       ^        |  |
|  |       | Ps     |  |
|  |       |        |  |
|  +----------------+  |  B
|                      |
|                      |
+----------------------+

A smooth B smooth +z = ?

The −c end of a domain etches much faster than the +c end P-comp

Compress, then cool, the same plate: face A turns negative, then positive, and both tests read +z = A. Compression pushes the cations toward their centred positions; cooling pulls them further out P. Turn the plate over and every sign follows the polarization. Readout: sign of each face, then the +z face. derived

Compress along c

A 0 B 0 +z = ?

The +z face is the c face that turns negative on compression P; +Z gives positive d₃₃ P

Cool on a Peltier stage

A 0 B 0 +z = ?

The +z face turns positive on cooling, since p₃ < 0 P

Etch

A smooth B smooth +z = ?

The −c end etches much faster than the +c end P-comp

Squeeze, then cool, the same plate: face A turns negative, then positive, and both tests name A as +z. Compression pushes the cations toward their centred positions; cooling pulls them further out P. Turn the plate over and the charges follow the arrow. Signs only. derived

Forms allowed in point group 3m

Voigt contraction 11→1, 22→2, 33→3, 23→4, 13→5, 12→6.

Property	Rank	Non-zero pattern	Independent	Computed	Status
Pyroelectric p	1	(0, 0, p₃)	1	1 / 3	P derived
Permittivity	2	diag(ε₁₁, ε₁₁, ε₃₃)	2	2 / 6	P derived
Any other symmetric rank-2 property (impermeability, thermal expansion, thermal conductivity)	2	diag(t₁₁, t₁₁, t₃₃)	2	2 / 6	derived
Piezoelectric d (3 × 6)	3	d₁₅ = d₂₄, d₁₆ = −2d₂₂, d₂₁ = −d₂₂, d₃₁ = d₃₂, d₃₃	4	4 / 18	P derived
Linear electro-optic r (6 × 3)	3	r₁₃ = r₂₃, r₃₃, r₂₂ = −r₁₂ = −r₆₁, r₄₂ = r₅₁	4	4 / 18	P derived
Elastic stiffness c (6 × 6)	4	c₁₁, c₁₂, c₁₃, c₁₄, c₃₃, c₄₄; c₂₄ = −c₁₄, c₅₆ = c₁₄, c₆₆ = (c₁₁ − c₁₂)/2	6	6 / 21	P derived
Elastic compliance s (6 × 6)	4	as c, but s₅₆ = 2s₁₄, s₆₆ = 2(s₁₁ − s₁₂)	6	6 / 21	P derived
Photoelastic p (6 × 6)	4	p₁₁, p₁₂, p₁₃, p₁₄, p₃₁, p₃₃, p₄₁, p₄₄; p₄₂ = −p₄₁, p₂₄ = −p₁₄, p₅₆ = p₄₁, p₆₅ = p₁₄, p₆₆ = (p₁₁ − p₁₂)/2	8	8 / 36	P derived

The factors 2 in d₁₆, s₅₆, s₆₆ come from the contraction (engineering shear strain), not from the crystal P derived. Neumann's principle must be applied to the full tensor, then contracted P.

Example of the result	Value	Status
Optic axis	c; negative uniaxial: n_o = 2.2866, n_e = 2.2028 at 632.8 nm, 24.5 °C, congruent	P-comp
Signs fixed by the axis rules	d₂₂ > 0 and d₃₃ > 0	P

L3 Signs and broken symmetry

non-polar x; residual sign freedom; Curie's principle under field

Why x is not polar. x is perpendicular to a mirror plane, so any charge displaced on one side is mirrored on the other; the x faces cannot charge under stress P. Practical orientation: in boules pulled along z, the mirror planes outcrop as three facets along the growth axis, 120° apart, which mark y P-comp.

Residual sign freedom. A 180° rotation about z keeps the axis rules for z but flips d₂₂; once +y is fixed by d₂₂ > 0 no freedom remains, so c₁₄ > 0 in any frame that obeys IEEE 176 or IRE 1949 derived. A 180° rotation about x flips z and therefore every odd-rank tensor (p, d, r, χ⁽²⁾) but not c or the photoelastic p derived.

Breaking the symmetry. A static field is a polar vector, so the crystal under field keeps only the elements of 3m that leave the field unchanged (Curie's principle) derived. E along z keeps all of 3m: the crystal stays uniaxial and n_o, n_e shift through r₁₃ and r₃₃. E along y, which lies in a mirror plane, keeps that one mirror: point symmetry m. E along x, which is perpendicular to a mirror plane, keeps no mirror (the mirror normal to x reverses E_x; the other two mirror normals sit at 60° and 120° to x): point symmetry 1. For either transverse direction the impermeability acquires terms through r₂₂ and r₅₁ and a z-cut plate becomes biaxial derived: E_x adds the xy term −r₂₂E_x, which turns the in-plane principal axes to 45°, and the xz term r₅₁E_x; E_y splits the xx and yy terms by ∓r₂₂E_y without turning them and adds the yz term r₅₁E_y derived. The Neumann machine shows this as the converse of Neumann's principle: lower the symmetry and new components appear.

Sources

3 cited in 1.7

Weis & Gaylord 1985R. S. Weis, T. K. Gaylord, "Lithium niobate: Summary of physical properties and crystal structure", Appl. Phys. A 37, 191-203 (1985). doi:10.1007/BF00614817
IEEE Std 176-1987IEEE Standard on Piezoelectricity, IEEE Std 176-1987 (1988); reprinted IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 717 (1996). doi:10.1109/IEEESTD.1988.79638
Volk & Wöhlecke 2008T. Volk, M. Woehlecke, Lithium Niobate: Defects, Photorefraction and Ferroelectric Switching, Springer Series in Materials Science 115 (Springer, Berlin Heidelberg, 2008). doi:10.1007/978-3-540-70766-0