Solid State Physics || Unit 01 || Crystal Diffraction & Reciprocal lattice ||The Von Laue Treatment ||Laue Diffraction equations 💥|| mynotes ||#physicsextreem

The Laue equations are derived from a simple static atomic model of a crystal structure. It describes effectively scattering of Xray from different atoms . The Bragg equation (2dSin@=n lambda) is derived as a direct consequence of the Laue equation. We consider the nature of the Xray diffraction pattern produced by identical scattering centre located at the lattice point of a space lattice.

We first look at the lattice points of a space lattice. Considering the scattering from any two lattice points P1,P2 in given figure, seperated by vector r.The unit incident wave normal is S0 and the unit scattered wave normal is S. We examine at a point 'a' long distance away ,the difference in phase of the radiation scattered by P1 and P2.

If P2A and P1B are the projections of vector r on the incident and scattered wave directions, the path difference between the two scattered wave is -----

where vector S=vector S0-vectorS

The vector S happens to represent the direction of the normal to a plane that reflects the incident direction into the scattering direction as shown in the following figure:---

If 2@ is the angle vector S makes with vector S0, then @ is the angle of incidence and from the figure we see that 

|S|=2.Sin@ as S and S0 are unit vectors.The phase fifference (phai) is equal to  (2.pai/lambda).path difference.

We have 

For this condition , the separate scattered amplitudes add up constructively and the intensity in the diffracted beam is maximum. If a,b,c are the primitive translation vectors , we have for the diffraction maxima:-------

Where h,k,l are integers . We have for direction cosine:---

This is Laue equation These eq have a direct geometrical interpretation .

The Laue equation state that in a diffraction direction the dcs are proportional to h/a, k/b,l/c.