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Mechanism

Piezoelectric plate used to convert audio signal to sound waves

The nature of the piezoelectric effect is closely related to the occurrence of electric dipole

moments in solids. The latter may either be induced for ions on crystal lattice sites with

asymmetric charge surroundings (as in BaTiO 3 and PZTs) or may directly be carried by

molecular groups (as in cane sugar) . The dipole density or polarization (dimensionality [Cm/m 3]

) may easily be calculated for crystals by summing up the dipole moments per volume of the

crystallographic unit cell .[11] As every dipole is a vector, the dipole density P is a vector field.

Dipoles near each other tend to be aligned in regions called Weiss domains. The domains are

usually randomly oriented, but can be aligned using the process of poling (not the same as

magnetic poling) , a process by which a strong electric field is applied across the material, usually

at elevated temperatures. Not all piezoelectric materials can be poled .[12]

Of decisive importance for the piezoelectric effect is the change of polarization P whenapplying a mechanical stress. This might either be caused by a re-configuration of the dipole-

inducing surrounding or by re-orientation of molecular dipole moments under the influence of

the external stress. Piezoelectricity may then manifest in a variation of the polarization strength,

its direction or both, with the details depending on 1. the orientation of P within the crystal, 2.

crystal symmetry and 3. the applied mechanical stress. The change in P appears as a variation of

surface charge density upon the crystal faces, i.e. as a variation of the electrical field extending

between the faces caused by a change in dipole density in the bulk. For example, a 1 cm 3 cube of

quartz with 2 kN (500 lbf) of correctly applied force can produce a voltage of 12500 V.[13]

Piezoelectric materials also show the opposite effect, called converse piezoelectric

effect , where the application of an electrical field creates mechanical deformation in the crystal.

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Mathematical description

Piezoelectricity is the combined effect of the electrical behavior of the material:

Where D is the electric charge density displacement (electric displacement ), ε is permittivity and

E is electric field strength, and Hooke's Law:

Where S is strain, s is compliance and T is stress.

These may be combined into so-called coupled equations, of which the strain-charge form is:

In matrix form,

Where the matrix for the direct is piezoelectric effect and is the matrix for the converse

piezoelectric effect. The superscript E indicates a zero, or constant, electric field; the superscript

T indicates a zero, or constant, stress field; and the superscript t stands for transposition of a

matrix.

The strain-charge for a material of the 4mm (C4v) crystal class (such as a poled piezoelectric ceramic such as tetragonal PZT or BaTiO 3) as well as the 6mm crystal class may

also be written as (ANSI IEEE 176):

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Where the first equation represents the relationship for the converse piezoelectric effect and thelatter for the direct piezoelectric effect .[14]

Although the above equations are the most used form in literature, some comments about

the notation are necessary. Generally D and E are vectors, that is, Cartesian tensor of rank-1; and

permittivity ε is Cartesian tens or of rank 2. Strain and stress are, in principle, also rank-2 tensors.

But conventionally, because strain and stress are all symmetric tensors, the subscript of strain

and stress can be re- labeled in the following fashion: 11 → 1; 22 → 2; 33 → 3; 23 → 4; 13 → 5;

12 → 6. (Different convention may be used by different authors in literature. Say, some use 12

→ 4; 23 → 5; 31 → 6 instead.) That is why S and T appear to have the "vector form" of 6

components. Consequently, s appears to be a 6 by 6 matrix instead of rank-4 tensor. Such a re-

labeled notation is often called Voigt notation.

In total, there are 4 piezoelectric coefficients, , , , and defined as follows:

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Where the first set of 4 terms correspond to the direct piezoelectric effect and the second set of 4

terms corresponds to the converse piezoelectric effect .[15] A formalism has been worked out for

those piezoelectric crystals, for which the polarization is of the crystal-field induced type, that

allows for the calculation of piezoelectric coefficients from electrostatic lattice constants or

higher-order Madelung constants .[11]

Crystal classes

Any spatially separated charge will result in an electric field, and therefore an electric potential.

Shown here is a standard dielectric in a capacitor. In a piezoelectric device, mechanical stress,

instead of an externally applied voltage, causes the charge separation in the individual atoms ofthe material, .

Of the thirty-two crystal classes, twenty-one are non -centrosymmetric (not having a centre of

symmetry), and of these, twenty exhibit direct piezoelectricity (the 21st is the cubic class 432).

Ten of these represent the polar crystal classes, which show a spontaneous polarization without

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mechanical stress due to a non-vanishing electric dipole moment associated with their unit cell,

and which exhibit piezoelectricity. If the dipole moment can be reversed by the application of an

electric field, the material is said to be ferroelectric.

Polar crystal classes: 1, 2, m, mm2, 4, 4 mm, 3, 3m, 6, 6 mm. Piezoelectric crystal classes: 1, 2, m, 222, mm2, 4, 4, 422, 4 mm, 42m, 3, 32, 3m, 6, 6,

622, 6 mm, 62m, 23, 43m.

For polar crystals, for which P ≠ 0 holds without applying a mechanical load, the piezoelectric

effect manifests itself by changing the magnitude or the direction of P or both. For the non-polar,

but piezoelectric crystals, on the other hand, a polarization P different from zero is only elicited

by applying a mechanical load. For them the stress can be imagined to transform the material

from a non-polar crystal class ( P =0) to a polar one ,[11] having P ≠ 0.

Materials

Many materials, both natural and synthetic, exhibit piezoelectricity:

Naturally occurring crystals

Quartz

Berlinite (AlPO 4), a rare phosphate mineral that is structurally identical to quartz

Sucrose (table sugar)

Rochelle salt

Topaz

Tourmaline-group minerals

The action of piezoelectricity in Topaz can probably be attributed to ordering of the (F,OH)

in its lattice, which is otherwise centrosymmetric: Orthorhombic Bipyramidal (mmm). Topaz has

anomalous optical properties which are attributed to such ordering .[16]