20a Physical Properties

7/31/2019 20a Physical Properties

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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

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20a Physical Properties