Classical Mechanics (Kibble & Berkshire) · Problem 1.13

Velocity, acceleration and scalar angular velocity — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · vector-algebra-lame-equations

Problem

Lam\'e's equations The directions in which X-rays are strongly scattered by a crystal are determined from the solutions

\boldsymbol{x}

of Lam\'e's equations, namely

\boldsymbol{x} \cdot \boldsymbol{a} = L, \qquad \boldsymbol{x} \cdot \boldsymbol{b} = M, \qquad \boldsymbol{x} \cdot \boldsymbol{c} = N,

where

\{\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\}

are the basis vectors of the crystal lattice, and

L

M

N

are any integers. Show that the solutions of Lam\'e's equations are

\boldsymbol{x} = L\boldsymbol{a}^* + M\boldsymbol{b}^* + N\boldsymbol{c}^*,

where

\{\boldsymbol{a}^*, \boldsymbol{b}^*, \boldsymbol{c}^*\}

is the reciprocal basis to

\{\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\}