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A Geometrical Representation

It turns out that the previous case, scattering by a one-dimensional chain, can be visualised by means of a geometrical figure known as the Ewald sphere (or sphere of reflection) plus the reciprocal lattice. As this idea crops up so often in crystallography texts, it is worth explaining briefly at this point. We saw in the previous figures that as the number of atom/molecule repeats (N) increases we get peaks of intensity, which are known as diffraction peaks. It is relatively easy to show that these peaks occur when a relationship between the chain (embodied by the repeat separation a) and the angle of scattering (embodied by S) is satisfied. This relationship is usually expressed in vector form and known as a von Laue condition, after von Laue who won the Nobel prize for his fundamental experiments on diffraction in 1912:

  a.S h, an integer
or in non-vector form:   
  a (2 sin θ/ λ) cos β h, an integer
where β, say, is the angle between the repeat separation a and the scattering S (β just happens to equal θ in the 1D-chain discussed above). It is not too difficult to show that all of this can be encapsulated by the following diagram:

2D Ewald projection

In this diagram the vector shown in cyan of length 1/λ represents the incident X-ray beam so. A circle is then drawn, of radius 1/λ, which represents all the potential scattering beams, of which two cases are illustrated. This circle is known as the Ewald sphere. Next a line (really a horizontal plane emerging from your screen) is drawn distant 1/a from the centre of the sphere. Where this plane intersects the sphere, the condition in the above equations is satisfied for h = 1 and diffraction can occur. Therefore diffraction will occur in the direction indicated by the vector shown in red. Similarly another plane can be drawn distant 2/a from the sphere centre and, for where this intersects the sphere, diffraction can again occur in the direction indicated by the vector shown in magenta, now with h = 2 (see above equations). Other cases can be drawn (e.g. h = -1, -2) though, as drawn in this particular case, the higher integer cases (3, 4 etc.) will occur outside the Ewald sphere and therefore not have an associated diffraction event. So we see that by drawing
  i. a sphere of radius 1/λ to represent the incident and potential diffracted X-rays,
  ii. a set of parallel planes 1/a apart to represent the repeated atoms/molecules a apart,
we predict when diffraction can occur. The former is known as the Ewald sphere and the latter is known as the reciprocal lattice (for this particular case). It is a construction often used and referred to in crystallography. Note that this construction tells you when diffraction will occur but does not tell you how intense the diffraction will be - you need to know what the repeating atom/molecule is to gauge that. One final point for this case is that this construction is really a 3D construction (3 dimensional; see below) of which the above diagram is a 2D-projection.

3D Ewald prijection
In three dimensions we have a sphere, and the reciprocal lattice of the 1D-chain is a set of parallel (2 dimensional) planes. The two intersect as curves and represent a continuous range of diffraction possibilities. As we proceed towards a three-dimensional crystal we will find that the diffraction possibilities decrease.

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© Copyright 1997-2006.  Birkbeck College, University of London.
Author(s): Paul Barnes
Simon Jacques
Martin Vickers