Multiple Symmetry of Inversion-Solid State Physics-Presentation, Slides of Solid State Physics

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MULTIPLE SYMMETRY

OF INVERSION

The concept of inversion symmetry plays an

important role in solid state physics. It is one of

the factor determining degeneracy's of the energy

Band away from the high symmetry points of the

brillouin zone. In elemental semiconductors like

germanium, with inversion symmetric diamond

structures, the heavy and light hole band are

doubly spin degenerate. In compound

semiconductor each of these bands split by the

crystalline electric field of the non symmetric

zinc blendes lattice. Such symmetry

considerations are also expected to be of

importance for artificial semiconductor

superlattices.

The inversion operation occurs through a single point called the inversion center , i , located at the center of the molecule. (Note that the inversion center may or may not coincide with an atom in the molecule.) Each atom in the molecule is moved along a straight line through the inversion center to a point an equal distance from the inversion center. If the resulting configuration is indistinguishable from the original, we say there exists an inversion center in the molecule.

 Inversion: symmetry about a point.

 Center of inversion:

Every point on one side of the center is matched by

an equivalent point on the other side, located on

the same line through the center and at the

distance from the center. Symmetry about a point.

 Inversion Symmetry :

The diagram below illustrates the simplest of these inversion axes, namely that of inversion at a point, -1, where the symbol for the point of inversion is a small open circle:

In the previous figure, every point in the molecule (which happens to be an organic molecule called D - alanine) in the left-hand side of the figure, may be inverted through the inversion point so as to generate the molecule on the right-hand side ( L -alanine, the chemical name name being the same except for the labels D and L which emphasise the difference in handedness ). A point of inversion at the origin has the symmetry operator - x ,- y ,- z. Objects possessing this symmetry element are frequently described as centrosymmetric.

INVERSION SYMMETRY ( I ): EACH POINT IN THE OBJECT IS CONVERTED TO AN IDENTICAL POINT BY PROJECTING THROUGH A COMMON CENTER AND EXTENDING AN EQUAL DISTANCE BEYOND THIS CENTER. OBJECTS WITH I SYMMETRY ARE SAID TO BE CENTROSYMMETRIC. FIG.III.10.

Fig. III.10. (Left) A mirror symmetry operation. (Right) An inversion symmetry operation. docsity.com