The Ewald Sphere of Reflection

Incident wave vector: K_I
Diffracted wave vector:K_D
K=K_D-K_I
|K_I|=|K_D|=1/λ=|K|
Angle between K_D and K_I is 2θ, sinθ=|K|/2|K_I| --> |K|=2sinθ/λ
When θ=θ_B,
|K_B|=2sinθ_B/λ, (1)
By pure geometry, we can know, nλ=2dsinθ_B, which is Bragg's Law.
If n=1,
2sinθ_B=λ/d , (2)
Combined (1) and (2), we have: |K_B|=1/d,
Define this vector, K_B, to be g so that:
K_B=g

The reciprocal lattice is a 3D array of points, each of which we will now associate with a reciprocal-lattice rod, or relrod for short. This geometry of relrod holds even when we tilt the specimen. The reason why we have relrods is the result of the shape of the specimen. At this stage, it is purely empirical construction to explain why we can still see the DP even when the Bragg condition is not exactly satisfied.

Construction of the Ewald Sphere:
1.Pick a point along the incident beam, this is your center of the the Ewald Sphere.
2.Draw a circle with a radius of 1/λ
3.Choose the intersection of the incident beam and your Ewald Sphere as the origin point to construct the reciprocal lattice.
When the sphere cuts through the reciprocal lattice point, the Bragg condition is satisfied. When it cuts through a rod, you still see a diffraction spot, even though the Bragg condition is not satisfied.
Ewald Sphere 2D