MTENSOR - Tensor calculation for Magnetic Point Groups
Introduction
MTENSOR provides the symmetry-adapted form of tensor
properties for any magnetic point (or space) group. On the one hand, a
point or space group must be selected. On the other hand, a tensor must
be defined by the user or selected from the lists of known equilibrium,
optical, nonlinear optical susceptibility and transport tensors, gathered from
scientific literature. If a magnetic point or space group is defined and
a known tensor is selected from the lists the program will obtain the
required tensor from an internal database; otherwise, the tensor is
calculated live. Live calculation of tensors may take too much time and
even exceed the time limit, giving an empty result, if high-rank
tensors, and/or a lot of symmetry elements are introduced.
Working setting definition
Due to the fact that the intrinsic symmetry
properties associated to any of the tensors defined in MTENSOR are only
valid when the tensors are expressed in an orthogonal basis, the tensors
provided by MTENSOR are always expressed in an orthogonal basis. For
triclinic, monoclinic, trigonal or hexagonal groups theorthogonal setting will
be chosen following the conventions defined at Physical Properties of Crystals (Nye, 1957) Appendix B 282, and Standards on Piezoelectric Crystals (1949). These conventions establish that, for any group expressed in a non-orthogonal basis,
the orthogonal basis (a', b', c') required to express tensors can be obtained from the non-orthogonal basis
(a, b, c) according to the formula:
a' || ac' || c*b' || c' ⨯ a
This convention is followed when point/space
groups are expressed in a hexagonal setting, in a monoclinic setting
with the monoclinic axis along a basis vector, or a triclinic setting.
For any other setting, the program will work in the standard setting of
the point/space group provided.
Intrinsic symmetry: Jahn symbols
The symbols at the "Intrinsic symmetry" column (Jahn symbols) are combinations of the following characters:
V: Vector (polar and invariant under time-reversal 1'). The number of vectors is equal to the tensor rank. For
example, if the Jahn symbol is [V2][V2] the tensor rank is 4.
e: axial constant
a: time-reversal constant (inversion under 1')
[]: Denotes symmetric indexes. For example, if the Jahn symbol is [V2]V, then Tijk =Tjik
{}: Denotes antisymmetric indexes. For example, if the Jahn symbol is{V2}V, then Tijk =-Tjik
[]*: Tensor indices interchange under time-reversal operation. For example, if the Jahn symbol is [V2]*V,
then 1'Tijk =Tjik and, under the action of primed operations R', R' T ijk =R T jik .
{}*: Tensor indices interchange and the sign changes under time-reversal operation. For example, if {V2}*V,
then 1'Tijk =-Tjik and, under the action of primed operations R' , R' Tijk =-R Tjik.
*: The symmetry operations
including time reversal (primed operations) do not introduce any
restriction in the tensor but they connect it with another tensor
property. For example, if the Jahn symbol is V2*, then 1' Tij =Ttji, being Tt the tensor
corresponding to the other property. Thermoelectric Peltier effect and
Seebeck effect tensors are two examples for T and Tt.
Build your own tensor
This tool allows to calculate the symmetry-adapted
form of a matter tensor which neither itself nor another with the same
transformation properties is included in the list of known matter
tensors. For this purpose, the Jahn symbol of the tensor and the
magnetic point group should be provided.
Abbreviated notation for symmetric tensors
As a consequence of the nature of the magnitudes
related by the tensor, as well as the thermodynamic relations between
them (see Physical Properties of Crystals, Nye, 1957), a matter
tensor can present some intrinsic symmetries, i. e, it can be invariant
(symmetric) or inverted (antisymmetric) under the permutation of two or
more indexes. For example, the second-order magnetoelectric tensor αijk fulfills the relation:
αijk = αikj
because the tensor components remain invariant under a swap of Ej and Ek.
The elastic compliance tensor Sijkl, defined by the equation:
εij = Sijklσkl
fulfills the following relations:
Sijkl = Sjikl
Sijkl = Sijlk
Sijkl = Sklij
being the first two ones derived from the intrinsic symmetry of the tensors related by Sijkl:
εij = εji
σij = σji
and the third one derived from thermodynamic relations.
When the tensor is symmetric under the permutation
of two indices (in this case i and j, and k and l as well), the tensor
can be rewritten making the substitution ij -> u (also kl -> v in
this case) fulfilling:
u = i if (i = j)
u = 9 - (i + j) if (i ≠ j)
(and the same for the substitution kl -> v). The tensor is expressed now as Suv,
u = 1,...,6, v = 1,...,6). These new indices u and v, which must be
denoted as "ij" and "kl" respectively, can be symmetric as well; this is
the case for the elastic compliance tensor Sijkl. Although
it is customary to introduce factors of 1/2 (or even 1/4 in some cases)
in the relationship between some of the tensor coefficients expressed
in abbreviated and full-length notations, we will not adopt here this
convention. The correspondence between coefficients will be always taken
with unity factors. For example, our elastic compliance coefficients in
abbreviated notation will verify
S11=S1111, S16=S1112,
or
S45=S2313.
Similarly, the piezoelectric coefficients fulfill
d11=d111,
or
d36=d312.
This contrasts with the convention used in the book by J.F. Nye, Physical Properties of Crystals (1957).
In cases of higher rank, when more than two
indices can be interchanged, the abbreviated notation will be used for
the first two indices on the left and, if possible, the next two indices
on the right will be treated in the same way. For example, if the Jahn
symbol is [V3] the correspondence will be Tuk=Tijk,
u = 1,...,6, k = 1,2,3. If the symbol is [V4] double reduction is possible and the correspondence will be
Tuv=Tijkl, u = 1,...,6, v = 1,...,6.
MTENSOR