* in contracted (matrix) notation
Representative property tensors
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| 1. Species |
|
Two different symmetry-breaking phase transitions in a
crystalline material are said to belong to the same species G > F
if their high-symmetry phase belong to same macroscopic symmetry class G,
their low-symmetry phase belong to same macroscopic symmetry class
F and for a suitably chosen domain state they have the same
correspondence between the symmetry elements of symmetry groups G and F.
Species No.: serial number #... of a species from the list of all 212 species G … crystallographic point group or class expressing the symmetry of the high-symmetry (parent, prototypic) phase F … crystallographic point group of a representative domain state of the low-symmetry (descent, ferroic) phase or the associated oriented class. When necessary, the orientation of the symmetry elements in F with respect to G is indicated by extra subscripts |,/,_,+. Their meaning is given in Section 3. |
| 2. List of 32 crystallographic point group classes |
| triclinic | 1↑†, 1† |
| monoclinic | 2↑†, m↑†, 2/m† |
| orthorhombic | 222, mm2↑, mmm |
| tetragonal | 4↑†, 4†, 4/m†, 422, 4mm↑, 42m, 4/mmm |
| hexagonal |
3↑†,
3†,
32,
3m↑,
3m, 6↑†, 6†, 6/m†, 622, 6mm↑, 62m, 6/mmm |
| cubic | 23, m3, 432, 43m, m3m |
| 3. Orientation of two-fold axes and mirror plane operations of F with respect to G |
| Symmetry of point group G | Tetragonal and hexagonal | Cubic | ||||||
| Orientation of two-fold axes 2 and mirror planes m of subgroup F in rectangular coordinate system of point group G | [001] | <100> | <100> | <110> | ||||
| Notation | 2| | m| | 2– | m– | 2+ | m+ | 2\ | m\ |
| 4. Internal symmetry of the tabulated macroscopic tensors |
| Symbol | Matrix components | Tensor | Internal symmetry of a tensor | Physical property | |
| Symbol | Number | ||||
| c | c | 1 | Scalar | c | Heat capacity |
| ρ | ρ | 1 | Pseudoscalar | ε | Chirality |
| p | pi | 3 | Polar vector | V | Spontaneous polarization, pyroelectricity, electrocaloric effect |
| t | ti | 3 | Pseudovector | εV | Pyroaxiality |
| u ε | uμ εi,j | 6 | Tensor of the 2nd rank | [V2] | Strain*) Permittivity |
| g | gμ | 6 | Pseudotensor of the 2nd rank | ε[V2] | Symmetric optical activity*) |
| d r | diμ riμ | 18 | Tensor of the 3rd rank | V[V2] | Piezoelectricity*) Electrooptics*) |
| A | Aiμ | 18 | Pseudotensor of the 3rd rank | εV[V2] | Electrogyration*) |
| s | sμν | 21 | Tensor of the 4th rank | [[V2]2] | Elastic compliance*) |
| Q π | Qμ,ν πμ,ν | 36 | Tensor of the 4th rank | [V2]2 | Electrostriction*) Piezooptics*) |
i, j = 1,2,3; μ,ν = 1,2,...,6.
*) in contracted (matrix) notation
| 5. Layout of the tabulated arrays of the property-tensor matrices |
Internal symmetry of tabulated arrays
* in contracted (matrix) notation
Representative property tensors
| Polar property tensors (see details) | ||||||||||||||||||||||||||
| p | Pyroelectricity | |||||||||||||||||||||||||
| ε0Xe | Electric susceptibility | |||||||||||||||||||||||||
| β | Thermal expansion | |||||||||||||||||||||||||
| d | Piezoelectricity | |||||||||||||||||||||||||
| s | Elastic compliance | |||||||||||||||||||||||||
| Axial property tensors (see details) | ||||||||||||||||||||||||||
| ρ | Chirality | |||||||||||||||||||||||||
| t | Pyroaxiality | |||||||||||||||||||||||||
| Xax | Axial susceptibility | |||||||||||||||||||||||||
| g | Symmetric optical activity | |||||||||||||||||||||||||
| A | Electrogyration | |||||||||||||||||||||||||
| 6. Domain states | |
In general, phase transitions with broken macroscopic symmetry in
crystals result in several orientational states of the lower-symmetry
phase(principal macroscopic domain states). Sometimes, two or more distinct
principal macroscopic domain states can have common orientation of a
particular property tensor, for example same spontaneous strain.
Set of such domain states is then considered as a single ferroelastic domain state.
Similarly, ferroelectric domain state is constituted by all principal domain states
sharing the same spontaneous polarization vector.
The numbers of the domains states distinguishable by various tensor
properties is uniquely defined by the species describing the phase
transition associated with the domain structure of the material.
In the DOMAIN STATES
section of this website, we are listing the following quantities:
|
| 7. Fine classification of ferroelectric, ferroaxial and ferroelastic phase transtions |
| Classification of ferroic phase (Aizu) | |||
| fe (with respect to ferroelectricity) | ef | ● | faithful (full) ferroelectric, ne = n |
| ed | ◐ | degenerate (partial) ferroelectric, 1 < ne < n | |
| et | ○ | trivial ferroelectric, ne = 1 | |
| en | ○ | non-ferroelectric, ne = 0 | |
| fax (with respect to ferroaxiality) | axf | ● | faithful (full) ferroaxial, nax = n |
| axd | ◐ | degenerate (partial) ferroaxial, 1 < nax < n | |
| axt | ○ | trivial ferroaxial, nax = 1 | |
| axn | ○ | non-ferroaxial, nax = 0 | |
| fa (with respect to ferroelasticity) | af | ● | faithful (full) ferroelastic, na = n |
| ad | ◐ | degenerate (partial) ferroelastic, 1 < na < n | |
| at | ○ | trivial ferroelastic, na = 1 | |
| 8. Group-subgroup diagrams |
Cubic species - subgroups with a cubic summit:
Hexagonal species - subgroups with a hexagonal summit:
Tetragonal species - subgroups with a tetragonal summit:
Triclinic, monoclinic and orthorombic species - subgroups with triclinic, monoclinic and orthorombic summits: