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Crystal Diffraction. Lecture 16 G.J. Mankey The Bragg equation states that constructive interference occurs when the path length difference associated with reflections from adjacent crystal planes is an integral number of wavelengths:.

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crystal diffraction

Crystal Diffraction

Lecture 16

G.J. Mankey

bragg diffraction
The Bragg equation states that constructive interference occurs when the path length difference associated with reflections from adjacent crystal planes is an integral number of wavelengths:

This basic equation is the starting point for understanding crystal diffraction of x-rays, electrons, neutrons and any other particle which has a DeBroglie wavelength less than an interatomic spacing.

Bragg Diffraction





reciprocal space
A Bravais lattice is an infinite array of discrete points with an arrangement an orientation which appears exactly the same, from whichever of the points the array is viewed.

There are 14 Bravais lattices with primitive vectors a1, a2, and a3.

The set of all wave vectors k that yield plane waves with the periodicity of a given Bravais lattice is known as the reciprocal lattice.

The primitive vectors of the reciprocal lattice are found from:

Where cyclic permutations of i, j, and k generate the three primitive vector components.


Reciprocal Space






Ref: Ashcroft and Mermin, Solid State Physics (1976).

the ewald construction
The Ewald construction is a way to visualize the Laue condition that the change is wavevector is a vector of the reciprocal lattice -- K = k’ – k, where k = 2p/l is the magnitude of the incident wavevector.

A sphere of radius k is drawn such that its surface intersects a point in reciprocal space and it origin is at the tip of the incident wavevector.

Any points in reciprocal space which intersect the surface of the sphere indicate where diffraction peaks will be observed if the structure factor is nonzero.

In this example, only the origin intersects the sphere, so there will be no diffraction peaks.





The Ewald Construction
laue condition satisfied
Laue Condition Satisfied
  • In this case, the incident wavevector angle is adjusted such that two points lie on the surface of the sphere.
  • The diffraction angle is then half the angle between the incident and diffracted wavevectors.
  • Generally, only a few angles will yield the proper conditions for diffraction.






geometrical structure factor
There is a phase associated with diffraction of x-rays from individual atoms and the net ray scattered by a primitive cell is the sum of the individual rays:

So even if the Laue condition is met, the structure factor may be zero and no diffraction peak will be observed.

An example is the bcc lattice where the sum of the indices of the reciprocal lattice vectors must be even for the structure factor to be nonzero. If the sum of the indices is odd, then no peak will be observed.

For a polyatomic lattice this is replaced by the atomic form factor….

Geometrical Structure Factor
leed retarding field analyzer with integral electron gun
LEED: Retarding Field Analyzer with Integral Electron Gun
  • Monoenergetic electrons are accelerated toward the surface by the electron gun.
  • Diffracted electrons travel in the field-free region between the sample and Grid 1.
  • Girds 2 and 3 are biased to filter electrons which lose energy interacting with the surface.
  • The collector is positively biased and coated with phosphor to produce a light spot.
ewald sphere for leed
Ewald Sphere for LEED
  • In low energy electron diffraction, the electrons interact mainly with the surface atoms so the reciprocal lattice becomes regularly spaced rods perpendicular to the surface plane.
  • The Ewald construction gives the spot locations on the screen and hence the in-plane lattice vectors.
  • Intensity of the spots versus electron energy gives the d-spacing.




rheed refection high energy electron diffraction
RHEED: Refection High-Energy Electron Diffraction
  • Monoenergetic electrons at high energy >10 keV are incident at grazing incidence on a crystal surface.
  • The diffracted electrons are displayed on a phosphor screen located opposite the electron gun in the vacuum chamber.
  • Since high energy electrons are used, in-situ studies of crystal growth during deposition can be performed.
ewald construction for rheed
Ewald Construction for RHEED
  • In RHEED, High energy electrons are incident at grazing incidence.
  • The Ewald construction illustrated why straks are observed rather than spots--the reciprocal lattice rods intersect the Ewald sphere nearly tangentially.
rheed close up
RHEED Close-Up
  • For this example, with an in-plane spacing of 2.8 Å and a reciprocal lattice spacing of 2.26 Å-1, the first-order spot is found at an angle of 14º with respect to the surface plane.
  • The rod intersects nearly tangentially, so any broadening due to defects, phonons or energy losses will result in a streak.