Finding the normal to a plane (hkl) in any crystal system using the reciprocal lattice construct

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.



The result is the vector, in cartesian space, that is normal to the plane (111) that would be used to plot the pole of (111) on a stereographic projection. To express this vector in terms of the crsyal co-ordinate system we simply reverse the initial transformation of co-ordinates by multiplying the vector xyx by the inverse of the matrix M.





Now, let’s see how to plot the poles of a given set of planes (hkl) on a stereographic projection for any crystal system......