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Magnitude of a property in a given direction

We will often want to talk about the magnitude of a property in a particular direction. For example if we apply an electric field to graphite in the [143] direction and measure the current density in the same direction, we would like to be able to describe the result as a measurement of the conductivity of graphite in the [143] direction.

Formally, the applied electric field E is described by a vector, as is the resultant current j. However conductivity requires a second rank matter tensor, and this means that these two vectors (E and j) will not in general be parallel to each other, and we will only measure the component of j which is parallel to E. For practical reasons it is therefore sensible to define the conductivity in a particular direction as the component of j that is parallel to E divided by the magnitude of E, i.e. j||/E.

We can apply this definition to a general second rank matter tensor T. If a field q = q(l1 l2 l3) is applied then the response of the material will be p = Tq.

Along the vector q the magnitude of T is given by:

\[T = \frac{{{\bf{p}} \cdot {\bf{q}}}}{q} \times \frac{1}{q} = \frac{{{\bf{Tq}} \cdot {\bf{q}}}}{{{q^2}}} = \frac{{{T_{ij}}{q_i}{q_j}}}{{{q^2}}} = {T_{ij}}{l_i}{l_j}\]

This can be simply related to the representation surface, as we know that the surface is decribed by the equation Tij xi xj = 1, where xi = r li and r is the radius. This gives us r2 Tij li lj = r2 T = 1

Hence the radius of the surface and the magnitude of the property it describes in a given direction are related by: = 1 / r2 or = 1 / √T

The radius-normal property

The radius-normal property of a representation surface gives us a geometrical method of finding the effect of a second rank tensor for a given action, for example finding the current density for a given electric field.

The property states that for the tensor equation p = Tq, if we draw the representation surface of T and take q from the origin, the vector normal to the surface where q meets it, is parallel to the direction of p. The size of p is given by the previously explained 'magnitude in a given direction' formulae, i.e.

|p= |q| / r2

where r is the radius to the point on the representation surface.

The property will be demonstrated along the principal axes of T but it can be generalised to any basis.

Let vector q = q(l1 l2 l3) where q is the magnitude and l1, l2 and l3 are the direction cosines of q. The point Q is the point on the representation such that OQ is parallel to q, so that OQ = r(l1 l2 l3).

As the equation of the surface is T1 x12 + T2 x22 + T3 x32 = 1, the tangent plane at the point (a1 a2 a3) has equation T1 x1 a1 + T2 x2 a2 + T3 x3 a3 = 1.

Hence the normal is n = (T1 a1  T2 a2  T3 a3)  = r (T1 l1  T2 l2  T3 l3).

Now as q = q(l1 l2 l3) and we are working in the principal basis we find simply that p = q (T1 l1  T2 l2  T3 l3), showing us that p is parallel to n.

These properties of the representation surface give us a simple way of finding the magnitude of property in a certain direction.