Question

Prove that reciprocal lattice vector is perpendicular to plane

Answer 1

The reciprocal lattice defines a crystal in terms of vectors that are normal to a plane and whose lengths are the inverse of the interplanar spacing. To determine a normal to a plane in both crystal and cartesian co-ordinates (we use the latter (cartesian) form when plotting poles of planes on stereographic projections) we use the co-ordinate transformation matrix developed in the previous class.

We can now express the three unit cell vectors in cartesian co-ordinates using





and any crystal direction [uvw] as as the cartesian vector xyz





A vector in reciprocal space is defined as the vector perpendicular to a given plane whose length is the inverse of the spacing of that plane.

The 3 reciprocal lattice vectors can be found by using the fact that the cross product of any two vectors is a vector perpendicular to those two vectors. So if we consider the vectors  and , these define the plane (001) and hence the cross product  will be the normal to (001). If we normalize the vector by the volume of the unit cell then we will have defined the reciprocal lattice vector g001. 





Thus the reciprocal lattice vector in cartesian space of any plane (hkl) in crystal space is given by





The spacing of the crystal plane (hkl) is simply the inverse of the magnitude of the reciprocal lattice vector





While the angle between any two crystal planes (hkl) and (h’k’l’) is the angle between there normals i.e.,





We can easily convert back from a vector in cartesian space to one expressed in crystal co-ordinates by simply reversing the initial transformation of co-ordinates, i.e.,





hence the normal to the plane hkl, expressed with reference to the crystal co-ordinate system is






Let’s work through an example in which we determine the normal to the plane (110) in the monoclinic crystal b”-Mg2Si. (a=1.1534nm, b=0.405nm, c=0.683nm, b=106°).

First determine the transformation matrix M




Next, convert the three unit cell vectors into cartesian co-ordinates




Finally we can obtain the 3 reciprocal lattice vectors and hence the reciprocal lattice vector of (111). The interplanar spacing of the (111) plane is 0.327nm.