Question

Derivation of braggs law using laue equation

Answer 1

Let

{\displaystyle \mathbf {a} \,,\mathbf {b} \,,\mathbf {c} }

be the primitive vectors of the crystal lattice

{\displaystyle L}

, whose atoms are located at the points

{\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} }

that are integer linear combinations of the primitive vectors.

Let

{\displaystyle \mathbf {k} _{\mathrm {in} }}

be the wavevector of the incoming (incident) beam, and let

{\displaystyle \mathbf {k} _{\mathrm {out} }}

be the wavevector of the outgoing (diffracted) beam. Then the vector

{\displaystyle \mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {\Delta k} }

is called the scattering vector (also called transferred wavevector) and measures the change between the two wavevectors.

The three conditions that the scattering vector

{\displaystyle \mathbf {\Delta k} }

must satisfy, called the Laue equations, are the following: the numbers

{\displaystyle h,k,l}

determined by the equations

{\displaystyle \mathbf {a} \cdot \mathbf {\Delta k} =2\pi h}

{\displaystyle \mathbf {b} \cdot \mathbf {\Delta k} =2\pi k}

{\displaystyle \mathbf {c} \cdot \mathbf {\Delta k} =2\pi l}

must be integer numbers. Each choice of the integers

{\displaystyle (h,k,l)}

, called Miller indices, determines a scattering vector

{\displaystyle \mathbf {\Delta k} }

. Hence there are infinitely many scattering vectors that satisfy the Laue equations. They form a lattice