Von Laue condition for scattering and Ewald construction
Ewald came up with a geometrical construction to help visualize which Bragg planes are in the correct orientation to diffract.
In the Ewald sphere, we have two origins (which can make you uncomfortable until you realize that it is just a geometrical
construction that makes the mathematics of diffraction easy to picture). The origin of the crystal is at the center of the Ewald sphere, and the incoming X-rays are diffracted from that crystal. The origin of reciprocal space is at the point where the incoming X-ray beam
would exit the Ewald sphere. If we rotate the crystal, we rotate the Bragg planes and so we rotate reciprocal space in the same direction.
Since diffraction from a crystal is confined to points on the reciprocal lattice (corresponding to planes that can be specified by
integer indices), we can think of rotating the reciprocal lattice when we rotate the crystal. The following figure shows this
schematically, illustrating planes of points in the reciprocal lattice. The planes of points in the reciprocal lattice intersect the
Ewald sphere to give a circle of points in the diffracting condition. When the planes are aligned perpendicular to the X-ray beam, these
circles on the Ewald sphere will project onto circles of spots surrounding the direct beam position but, as we rotate the crystal (and
the reciprocal lattice) the circles on the Ewald sphere will be distorted and will project into what are called lines of spots.
Direct lattice and reciprocal lattice
A typical direct Bravis lattice vector in 2D can be written as : R=n1a
+n2b ,where a,b are the direct lattice primitive unit cell vectors.
The typical reciprocal lattice vector can be written as: G=kA+hB , where A,B are the reciprocal lattice primitive unit cell vectors.
The relation between the direct lattice primitive unit cell vectors and the reciprocal lattice primitive unit cell vector :
The relation between direct lattice vector R and reciprocal lattice vector G: eiGR=1
From the last relation we can see that R and G are vertical.
We describe Bragg planes with the direct lattice vector R and or with the reciprocal lattice vector G and the relation between
the indexes n,m,k and h is : n:m=1/k:1/h.
We can write Bragg law 2dsinθ=λm in terms of G and k (k=2π/λ):
Notice that if we take two G vectors with the same relation between k and h (for example-G1=(1,1),
G2=(2,2) ),the distance between two Bragg planes d is calculated according to the smallest G vector,
but the other G vector also satisfies the Bragg equation but for higher order n.
We define K as the vector of the incoming wave and K' as the vector of the outgoing wave and their relation with G as: G=K'-K or
The Ewald construction
When executing the function ewald(k), which can be downloaded in the download section ,
we can see selection of Bragg planes and representations of vectors G K and K' in the reciprocal lattice (lower
right window), Bragg planes and waves phases(upper left window), and the interference phase(upper right window).
Here are some examples of Bragg scattering from different planes:
in this movie we can see "scanning" of the points in the reciprocal lattice and the positive diffraction from different Bragg planes