The crystal, instead of
being seen as different sets of
parallel planes, may be represented by a normal drawn
perpendicular to each set of parallel planes from a common point as origin. The length of the normal is proportional to 1/dhkl .This length and direction of the normal is used to represent a set of parallel
planes.If a point is placed at the end of each such normal ,an array of points is generated. Each point
then represents a set of parallel equidistant lattice planes and hence, each point is represented by a set of Miller indices (hkl) of the crystal. This array of points is known as the reciprocal lattice. The reciprocal lattice
vector hkl has a direction same as the normal to the dhkl planes
and its magnitude is 1/dhkl
.
We see that the arrangements of the points
in the reciprocal lattice has the same symmetry as the
lattice points of the real
crystal.
Relation between reciprocal lattice and X-ray
diffraction: The concept of reciprocal lattice is very useful in X-ray crystallography. It
was Ewald who developed the relation between the diffracted X-ray beams. The concept of reciprocal lattice is particularly helpful in understanding diffraction of X-rays by crystal planes. Let
us rewrite the Bragg’s equation in reciprocal lattice as:
Here, we have tried to relate the magnitude
of the reciprocal lattice vector to diffraction angle and the wavelength of the X-ray. In order to see
the geometric consequence of this equation, let us imagine a sphere of radius 1/λ = AO as shown in figure:
Let AO also be the
direction of the X-ray beam incident on the crystal plane at C, the center of
the sphere. If θ is the Bragg’s angle, the reflected
beam will strike the sphere (shown as a circle here) at the point P making
an angle 2θ with the passing beam.
It should be noted that the angle between the incident beam and
AP is θ. Since
2 sinθhkl/λ = 1/dhkl,
we can see that
1/dhkl λ = OP.
Thus, the point P is the
reciprocal lattice point for the set of planes from which the X-ray beam is
reflected. Also, since the
Bragg’s diffraction conditions are satisfied, the diffracted beam will touch
the sphere (circle in the
picture) at point P. The same will be the case with other set of parallel planes except that they will
strike the sphere at some other point which are the reciprocal lattice points for the respective set
of planes. The three-dimensional sphere is called the sphere of reflection or Ewald
sphere.
