Introduction to Solid State Physics for Materials Engineers. Emil Zolotoyabko
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Название: Introduction to Solid State Physics for Materials Engineers
Автор: Emil Zolotoyabko
Издательство: John Wiley & Sons Limited
Жанр: Физика
isbn: 9783527831593
isbn:
Figure 1.11 The presence of inversion center (C) in diamond structure (a) and its loss (X) in zinc-blende structure (b). Dissimilar atoms are indicated by different colors.
Combining local symmetry elements with translations creates novel elements of spatial symmetry – glide planes and screw axes. Therefore, spatial symmetry is a combination of local (point) symmetry and translational symmetry. As a result, 32 point groups + 14 Bravais lattices produce 230 space groups describing all possible variants of crystal symmetry, associated with charge distributions, i.e. related to geometrical points and polar vectors. Magnetic symmetry, linked to magnetic moments (axial vectors, see Section 1.2), will be discussed in Chapter 11.
1.2 Symmetry and Physical Properties in Crystals
Crystal symmetry imposes tight restrictions on its physical properties. Term “properties” relates to those that can be probed by regular (macroscopic) optical, mechanical, electrical, and other measurements, averaging over the actual atomic-scale periodicity of physical characteristics. Note that complete spatial symmetry of the crystal is revealed in diffraction measurements using quantum beams (X-rays, neutrons, electrons) with wavelengths comparable with translational periodicity. Note that crystal characteristics, even averaged over many translation periods, show anisotropy which is dictated by the crystal point group. Within this averaged approach, the symmetry constraints are formulated by means of the so-called Neumann's principle: the point group of the crystal is a sub-group of the group describing any of its physical properties. In simple words, the symmetry of physical property of the crystal cannot be lower than the symmetry of the crystal: it may be only equivalent or higher.
In practical terms, it means that if physical property is measured along certain direction within the crystal and then the atomic network is transformed according any symmetry element of its point group and measurement repeats, we expect to obtain the measurable effect of the same magnitude and sign as before. Any deviation will contradict particular crystalline symmetry and, thus, the Neumann's principle. Using mathematical language, physical properties are, generally, described by tensors of different rank, for which the transformation rules under local symmetry operations are well-known. Tensor rank defines the number of independent tensor indices, i, k, l, m,…, each of them being run between 1 and 3, if the 3D space is considered. In most cases, physical property is the response to external field applied to the crystal. Note that external fields are also described by tensors, which are called field tensors to distinguish them from crystal (material) tensors.
Figure 1.12 Illustration of the Biot–Savart law (Eq. (1.7)).
Tensors of zero rank are scalars. It means that they do not change at all under coordinate transformations related to symmetry operations. As an example of scalar characteristics, we can mention the mass density of a crystal. Tensor of rank one is a vector. It has one index i = 1,2,3, which enumerates vector projections on three mutually perpendicular coordinate axes within Cartesian (Descartes) coordinate system. It is easy to point out field vectors, for example, an applied electric field, ℰi, or electric displacement field, Di. As crystal vector, existing with no external fields, one can recall the vector of spontaneous polarization,
where r is the radius-vector connecting the element Δl and the observation point (see Figure 1.12). In turn, magnetic dipole moment, μd, is defined as an integral over the volume containing the current density distribution J:
(1.8)
Axial vectors are considered when analyzing magnetic symmetry and magnetic symmetry groups (Chapter 11).
Tensor of rank 2 has two independent indices i, k = 1, 2, 3. As a rule, it linearly connects two vectors, e.g. the vectors of the electric displacement field, Di, and external electric field, ℰk, i.e.
(1.10)