Properties of reciprocal lattice

 RECIPROCAL LATTICE:-

  Many properties of crystalline substance are different in different directions. W.H.Miller explain the concept of miller indices to represent the planes and directions of a crystal. In real practice there are many set of planes with different orientations and spacings and it becomes difficult to visualise the slope of the planes. Since all parallel planes have a common normal and simillar indices , so the miller indices of a plane can not be uniquely defined. Therefore to understand many crystallographic problem we consider the normal to be set of planes rather than the planes themselves. The orientation of plane is understood by the direction of normal.

DEFINITION OF RECIPROCAL LATTICE:-

      The reciprocal lattice in Fourier space is an infinite periodic three dimensional array of points whose spacings are inversely proportional to the distance between planes in the direct lattice.

    The need for devising the concept of the reciprocal lattice was to tabulate the two important properties of the crystal plane :

• Slopes of the plane

• inter planer spacings

Time-crystals:

   Time crystals are systems where the role of space in ordinary Crystaline crystals is replaced by time. As in ordinary Crystals, a very large number of atoms, in principle an infinite number, display periodicity in space, similarly in time crystals they exibit periodicity in time . The importance of time crystals lies in the replacement of space by time.

    In ordinary Crystals, like NaCl, atoms are arranged in a repeating pattern and interaction between neighbour atoms keep the system rigid . A time crystal shows the same kind of rigid pattern as in regular crystal, but instead of being arranged in space, the pattern repeat in time. This repetition in time is small like earth's season which appear like clock work once a year.

CONSTRUCTION OF RECIPROCAL LATTICE:-

                 The concept of normals is used in the construction of reciprocal lattice.

Martin Julian buerger(1903-1986) an American crystallographer in 1948 was the first to construct the reciprocal lattice in his theoretical paper "crystallographic symmetry in reciprocal space".

     A reciprocal lattice from a direct lattice can be constructed as in the following steps: 

• Take any point as an origin in direct lattice.

• From this origin , draw normal to every set of planes in the direct lattice.

• Set the length of every normal drawn equal to the reciprocal of the interplaner spacing of the set of parallel plane( h,k,l ) it represents.

• place a point at the end of each normal.

       The assembly of these points represent a lattice arrays which is called as reciprocal lattice.

PROPERTIES OF RECIPROCAL LATTICE :

      Reciprocal lattice is an invariant geometrical object whose properties are fundamental in the theory of solids . The elementary geometrical properties are as follows-

• The reciprocal lattice is a lattice in Fourier space . Direct lattice is a lattice in ordinary space.

• The reciprocal of the reciprocal lattice is the direct lattice.

• The volume of a unit cell of the reciprocal lattice is inversely proportional to the volume of a unit cell of the direct lattice .

• Each point in a reciprocal lattice represent a set of parallel plane of the crystal lattice.

• Each vector of the reciprocal lattice is normal to a set of lattice plane of the direct lattice.

• when we rotate a crystal, both reciprocal lattice and direct lattice rotate.

• Unit cell of the reciprocal lattice need not to be parallelopiped. Actually , we do deal with the wigner- seitz cell of the reciprocal lattice.

This is known as Brillouin zone in k space.