Post on 05-Apr-2018
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Piezoelectricity
The piezoelectric effect is understood as the linear electromechanical
interaction between the mechanical and the electrical state in crystalline
materials with no inversion symmetry. The piezoelectric effect is a
reversible process in that materials exhibiting the direct piezoelectric
effect (the internal generation of electrical charge resulting from an
applied mechanical force) also exhibit the reverse piezoelectric effect(the internal generation of a mechanical force resulting from an applied
electrical field). -- Wikipedia
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Properties like piezoelectricity are only possible in crystalline
materials that have no center of symmetry. Otherwise a chargeseparation cannot be developed. So we must look to the symmetry
of the atoms in the unit cell to determine if an otherwise
macroscopic property exists.
To understand which crystal symmetries can support such an
effect, we need to look at the point group of the space group. We
are interested in the point group of the space group because
translational symmetry at the unit cell level does not affect the
existence of a property.
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Obtaining the the point group of a given space group is relatively
easy. We simply remove the space group centering operation
(P, I, F, A, B, C) and then convert any symmetry elementscontaining translations (screw axes and glides) to their non-
translational equivalents (rotations and mirrors).
Pbca ------------------> mmm
P42nm ------------------> 4mm
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One other point group can be eliminated because of its
symmetry elements. This is a cubic point group whosesymmetry elements prevent charge separation. It is 432.
This leaves a total 20 point groups that can sustain a property
like piezoelectricity these groups can be divided into two
classes, one of which are the polar point groups.
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Polar Point GroupsPoint groups having a unique axis that is not repeated in any
direction.
1 2 3 4 6
2 3 4 6m mm m mm mm
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The polar point groups can support spontaneous polarization
of charge without any mechanical stress due to a non-
vanishing electric dipole moment associated with the unit
cell. Materials in these systems also support pyroelectricity
(temperature driven charge distribution).
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The remaining 10 point groups also support the piezoelectric
effect but do not support a spontaneous polarization. Here the
stress applied to the material can be thought of as transforming
the point group from a non-polar one to a polar point group.
222 4 422 42 32
6 6 2 622 43 23
m
m m
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Phase Changes Can Affect Physical
Properties
3 4m m mm
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Birefrengence
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Light waves passing through a single crystal are affected differently
if they are aligned with a symmetry axis than if they are aligned
perpendicular to that direction. With initially unpolarized light, the
waves perpendicular to the symmetry axis will be refracted (bent)
differently from those aligned with the axis. This effect is known as
birefringence. If light is passed through a crystal along a symmetry
axis, it only sees one environment and the birefringence
disappears.
The effect is different for different crystal classes. So, symmetry
plays an important role in determining refractive index properties.
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Cubic
Because of the multiple 3-fold axes, cubic crystals do nothave a unique axis. The refractive index is isotropic and
birefringence cannot occur.
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Uniaxial
Tetragonal Hexagonal Trigonal
Principal symmetry axis (3, 4, 6) is the Optic Axis
Light entering off the Optic Axis sees two refractive indexes from:
Light vibrating parallel to optic axis
Light vibrating perpendicular to optic axis
Birefringence is observed.
Light entering parallel to the Optic axis sees one refractive index
Birefringence does not occur.
Isotropic
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Orthorhombic Monoclinic Triclinic
Crystals in these systems have three refractive indexes and two
principal optic axes along which light waves pass in an isotropic
manner. These materials are said to be bi-axial
Examine an elliptical solid to see why there are two principal
axes.
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J.F. Nye, "Physical Properties of Crystals" (1957)
Physical Property Dependencies
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Each property in the previous diagram is given a dimension. 0
represents a scalar, 1 a vector and in general n represents a n
rank tensor.
Look at pyroelectricity for example. The polarization P (a
vector) is produced by a temperature T (scalar). The
relationship is thus:
P pT
The pyroelectric effect, p, must also be a vector in thisequation. Symmetry helps determine the possible values in p.
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Triclinic
No symmetry restrictions: p = [ p1 p2 p3 ]
Monoclinic
Class 2: p = [ 0 p2 0 ] P vector along 2-fold
Class m: p = [ p1 0 p3 ] P vector in x-z plane
Orthorhombic
Class mm2: p = [ 0 0 p3 ] P vector along 2-fold
Class m: p = [ 0 0 0 ] No effect observed
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Tetragonal Hexagonal Trigonal
Polar classes: p = [ 0 0 p3 ]Other classes: p = [ 0 0 0 ]
Cubic
p = [ 0 0 0 ] No effect observed