Site-symmetry group

Table 8.3. IRs of the site symmetry point groups of the sublevels of np donor states in silicon split by a uniaxial stress, deduced from the IRs of the valley group. The site symmetry group under stress is indicated close to the stress orientation. The magnitudes of the splitting, independent of the value of m, are given in Table 8.2 (after [2])...

Table 8.5. Shift 8E with respect to zero-stress positions of the ls(T2) and ls(E) states of deep donors in silicon for compressive stresses along the [100] and [110] axes (units of EUT (S11 — Si2)/3). The components are labelled by the IRs of the appropriate site symmetry group. When crossing, components of the same symmetry may interact. This can occur with the Is (Ai) components of the Is (T2) and ls(E) for F// [110] as the former is deeper than the latter (after [14])...

The point qj of the orbit has a site-symmetry group Fj=RjFRj isomorphic to Fi. Thus, an orbit may be characterized by a site group Fi, (or any other from the set of groups Fj). The number of points in an orbit is equal to the index t=np/np. of the group Fj in F. 

If the elements Rj in (2.1) form a group P then the group F may be factorized in the form F = PFj. The group P is called the permutation symmetry group of an orbit with a site-symmetry group Fj (or orbital group). 

For this group there are 11 different Wyckoff positions denoted by letters from a to k. The number of crystallographic orbit points in the primitive unit cell (multiplicity) equals np/Uq where np = 16 is the order of the point group and Uq is the order of the site symmetry group Gg. The number of points in a Wyckoff position and their coordinates are given in the International Tables with respect to the conventional unit cell of the lattice (for the space group D j with a simple Bravais lattice, the... 

The difference between oriented site-symmetry groups of different Wyckoff positions is due to different orientations of the elements of the site-symmetry group G, with respect to the lattice. The difference arises when similar symmetry elements (reflections in planes and rotations about twofold axes of symmetry) occur in more than one class of elements of the point group F. Only eleven site groups [C2(2), Cs m), C2h 2./m), C 2 (2mm), CsyiZmm), 2(222), Ds(322), D2d(42m), D3d 32m), D hijnrnm), and >3 (62m)] can have different orientations with respect to the Bra-vais lattice. Oriented site- symmetry symbols show how the symmetry elements at a site are related to the symmetry elements of a space group. The site-symmetry... 

Diamond stmcture g. 2.5) is described by nonsymmorphic space group N227 with the face-centered cubic lattice the macroscopic cubic symmetry of this crystal appears as the direct product of the first carbon atom site symmetry group Td and inversion I at the center of C-C bond moving the first carbon to the equivalent second carbon atom in the primitive unit cell. 

Let the local functions Wj f r) = W<(r — q), (i = 1, 2,..., n ) be centered at point q of the direct lattice and span the space of the irrep / of the site-symmetry group Gq C G with matrices d > gq) and characters X H9q) 9q Gq) The nature of these functions depends on the physical problem under consideration. In the electron-band theory of crystals lF) (r — qA) are atomic functions of atom A. In phonon spectroscopy applications — qA) mean the components of atomic displacements of... 

In a cyclic model of a crystal with N primitive cells in the main region a band rep is an Np dimensional reducible rep of a space group. An induced rep is a particular case of a band rep as it satisfies both properties 1 and 2 with p = nqU/ n is the dimension of the site-symmetry group irrep for a point q belonging to the set of n, points in the unit cell). 

All simple induced reps may be generated by induction from the irreps of site-symmetry groups Ggi, of a relatively small number of q points forming the set Q in the Wigner-Seitz unit cell of the direct lattice. The set Q consists of... 

There is a formal analogy between simple induced reps and reps irreducible in the usual group-theoretical sense. However, this analogy is not complete. Indeed, the composite induced rep decomposition into simple ones is not always unique. This occurs whenever the site-symmetry group Gg is not the maximal isotropy one q Q). In this case, the group Gg is a subgroup of several maximal isotropy groups Gg-(q e Q). Consequently, the induced rep decomposition (3.70) will be different for different points. 

The first two columns of the table contain the labels of the induced reps in the q-basis (these labels number the rows of the table) the international symbols (Roman letters a, b, c and so on) of the Wyckoff positions (sites in direct space) and the Mulliken symbols of the irrep>s of the site-symmetry groups for these Wyckoff positions. For example, d a2u) and d(e ) are the labels of induced reps in q-basis for space... 

Finding the irreducible components of the site-symmetry group reps subduced by the irreps of the corresponding space group. 

Due to the considered symmetry difference of crystalline orbitals in MgO and Si crystals the nature of chemical bonding in these crystals is also different. Indeed, in ionic MgO crystal the splitting of the valence band allows the crystalline orbitals locahzed on an oxygen atom to be generated and transformed over aig and irreducible representations of the oxygen site-symmetry group O. In covalent Si crystal aU four sheets of the valence band have to be included in localization so that the locahzed orbitals found are centered at the middle of the Si — Si bond. 

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