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It has been demonstrated in the previous section that the intrinsic symmetry of
the physical properties decreases the number of the independent tensor
components. Further reduction of the independent components of a physical
property tensor, and the zero value of certain components follow from the fact
that the crystal symmetry exerts some influence on the symmetry of the physical
properties. This fact is expressed by Neuman's principle formulated already in
the 19th century according to which *the symmetry elements of any physical
property of a crystal must include all the symmetry elements of the point group
of the crystal* :

(4.1) |

The well-known equations of transformation from an orthogonal *x _{1}*,

(4.2) |

(4.3) |

(4.4) |

(4.5) |

(4.6) |

(4.7) |

In some cases the tensor describing the physical properties is not polar, but axial (as for instance the tensor describing the optical activity or piezomagnetism). For axial (or pseudo) tensors the following transformation relation may be used as definition

(4.8) |

It is not difficult to find out whether the tensor representing any physical
property is polar or axial, since this can be easily decided by the eq. (2.1)
defining the physical property in question. If only one of the tensors
[*A*_{pqr...u}] and [*B*_{ijk...n}] in eq. (2.1) is axial (for instance magnetic
field is an axial tensor of rank one) also the property tensor
[*a*_{ijk...npqr...u}] as defined by eq. (2.1) will be axial, in every other
case the tensor is polar.

It should be remarked that if also the magnetic point groups are considered eq.
(4.6) and eq. (4.8) expressing the transformation properties of the tensor
components are valid only for conventional symmetry operations. If, however,
the conventional symmetry operations are combined with time-inversion which
actually happens in anti-symmetry operations (see [10-14]) the right sides of
eqs. (4.6) and (4.8) respectively should be multiplied with (-1) whenever eq.
(2.1) defining the physical properties contains the magnetic vector quantities
(magnetic field, magnetic induction, magnetization vector) odd times. Tensors
representing this type of properties are called *C*-tensors.^{1} For a more
detailed discussion of this problem the reader is referred to the
literature.^{1-3}

Considering the equations of transformation (4.6) and (4.8) and with regard to the above remark, the relationships between the components of the polar and axial tensors for a given crystal class can now be defined, since the invariance of a tensor with regard to any symmetry operation requires the relationship

(4.9) |

(4.10) |

(4.11) |

For every tensor component an equation of the type (4.10) and (4.11) respectively should be valid so that the tensor components must satisfy a system of these equations. Since this holds for every symmetry operation of a given crystal class, the number of the systems of equations between the tensor components will be equal to the number of the symmetry operations which may be performed in the given crystal class. However, in order to obtain every relationship among the components of a tensor representing any physical property in case of a given crystal class, it is not necessary to write down for every symmetry operation the system of equations of the type (4.10) and (4.11) respectively. It is well known from the group theory that for various crystal classes every symmetry operation may be deduced from a few basic symmetry operations. The application of the matrices corresponding to these basic operations (the generating matrices) are sufficient to obtain the effect due to the symmetry of a crystal class on the given tensor in question. Tables 2 and 3 summarize a series of generating matrices for every conventional crystal class.

These interrelations appear to be at first instance somewhat complicated, a
simple example, however, will help to obtain a better understanding. Let us
consider the form of the pyroelectric tensor in the crystal class 3 of the
trigonal system assuming that the *x _{3}* axis of the coordinate system is the
three-fold rotation axis. As one may see on consulting Tables 2 and 3 the
coordinate transformation related to the symmetry operation can be described
with the following matrix

(4.12) |

Taking into consideration the condition of invariance of the polar tensor as expressed in eq. (4.10) one obtains for the tensor components the following equations

(4.13) |

*p _{3}*=

Crystal system | Class symbol | Generating matrices | No. of symmetry elements | The choice of x, _{1}x, _{2}x crystal physical axes in relation
to the symmetry axes_{3} |
||

International | Schoenflies | |||||

Triclinic | 1 | C_{1} |
M_{0} |
1 | ||

S=_{2}C_{i} |
M_{1} |
2 | ||||

Monoclinic | 2 | C_{2} |
M_{2} |
2 | ||

m |
C_{1h}=C_{s} |
M_{3} |
2 | |||

2/m |
C_{2h} |
M, _{2}M_{3} |
4 | |||

Orthorhombic | 222 | V=D_{2} |
M, _{4}M_{2} |
4 | ||

mm2 |
C_{2v} |
M, _{5}M_{2} |
4 | |||

mmm |
V_{h}=D_{2h} |
M, _{5}M, _{6}M_{3} |
8 | |||

Tetragonal | 4 | C_{4} |
M_{7} |
4 | ||

S_{4} |
M_{8} |
4 | ||||

422 | D_{4} |
M, _{7}M_{4} |
8 | |||

4/m |
C_{4h} |
M, _{7}M_{3} |
8 | |||

4mm |
C_{4v} |
M, _{7}M_{5} |
8 | |||

V_{d}=D_{2d} |
M, _{8}M_{4} |
8 | ||||

4/mmm |
D_{4h} |
M, _{7}M, _{3}M_{5} |
16 | |||

Trigonal | 3 | C_{3} |
M_{9} |
3 | ||

S=_{6}C_{3i} |
M_{10} |
6 | ||||

32 | D_{3} |
M, _{9}M_{4} |
6 | |||

3m |
C_{3v} |
M, _{9}M_{5} |
6 | |||

D_{3d} |
M, _{10}M_{5} |
12 | ||||

Hexagonal | 6 | C_{6} |
M_{11} |
6 | ||

C_{3h} |
M_{12} |
6 | ||||

D_{3h} |
M, _{12}M_{5} |
12 | ||||

622 | D_{6} |
M, _{11}M_{4} |
12 | |||

6/m |
C_{6h} |
M, _{11}M_{3} |
12 | |||

6/mm |
C_{6v} |
M, _{11}M_{5} |
12 | |||

6/mmm |
D_{6h} |
M, _{11}M, _{5}M_{3} |
24 | |||

Cubic | 23 | T |
M, _{13}M_{2} |
12 | ||

m3 |
T_{h} |
M, _{14}M_{2} |
24 | |||

432 | O |
M, _{13}M_{7} |
24 | |||

T_{d} |
M, _{13}M_{8} |
24 | ||||

m3m |
O_{h} |
M, _{14}M_{7} |
48 | |||

identity | |||

inversion | fourfold inversion-rotation about x axis_{3} |
||

twofold rotation about x axis_{3} |
threefold rotation about x axis_{3} |
||

reflection in x plane_{1}x_{2} |
threefold inversion-rotation about x axis_{3} |
||

twofold rotation about x axis_{1} |
sixfold rotation about x axis_{3} |
||

reflection in x plane_{2}x_{3} |
sixfold inversion-rotation about x axis_{3} |
||

reflection in x plane_{1}x_{3} |
threefold rotation about [111] direction | ||

fourfold rotation about x axis_{3} |
threefold inversion-rotation about [111] direction | ||

The method used in this example may be applied in every case, though with higher rank tensors it may be in many cases rather tiresome.

Considerable time can be saved (with the exception of the trigonal and hexagonal
classes) by the direct inspection method worked out by Fumi,^{4}, which though
in principle not differing from the previous treatment leads to results in a
relatively short time. Fumi's method is based on the fact that in an orthogonal
coordinate system the polar tensor components transform in the same way as the
products of the corresponding coordinates (see eq. (4.6)). One must, however,
be careful not to commute the sequence of the factors, thus for instance instead
of the product *x _{1}x_{2}* one cannot write

Fumi's method may be studied by a simple example considering the form of a polar
second-rank tensor, e.g. the dielectric susceptibility tensor of the crystal
class 2 of the monoclinic system. Assuming that the two-fold symmetry axis
coincides with the *x _{2}* coordinate axis the symmetry operator will transform
the coordinates in the following way

(4.14) |

(4.15) |

(4.16) |

(4.17) |

(4.18) |

(4.19) |

The forms of tensors or matrices composed of tensor components for the various crystal classes can be found properly tabulated in the special literature, see for example [1, 2, 5-7, 16, 17].

Finally it should be observed that in some cases simple geometric considerations
enable the determination of the independent components of the tensors
representing the physical properties. It is easy to see that no pyroelectric
effect can exist in a crystal possessing a centre of symmetry. This means that
for these crystals every component of the pyroelectric tensor is zero, *p* = [0,
0, 0], because in these crystals if the vector of polarization were pointed in a
given direction the vector should appear also in the opposite direction as a
result of Neumann's principle, consequently its value can only be zero.

**Copyright © 1984, 1998 International Union of
Crystallography**