30/1/03: Structure and Scattering

A knowledge of the atomic structure of a material is a prerequisite to an understanding of its physical behavior. Tools such as scanning force, scanning tunneling and electron microscopy provide direct images of charge and particle densities - but the information is restricted to the surface of the material.
Most information about the bulk structure, especially at the Angstrom scale, is obtained by scattering of neutrons, electrons or photons (X-ray).

Waves diffract when they meet an obstacle; the interference pattern appearing at a plane remote from the diffracting object, caused by the constructive and destructive interference of wavefronts emerging from different parts of the obstacle, is characteristic of the diffracting obstacle. This principle is used to determine the structural arrangement of atoms in a material, where the diffracting object is the collection of atoms themselves. Diffraction is most effective when the wavelength of the incident radiation is comparable to the size of the diffracting object.

Usual materials have inter-particle spacing on the Angstrom scale - we must consider what energies the scattered particles must have to correspond to these wavelengths and what potentials these particles are scattered from.

For photons, energy E = h * c / lambda.
Visible light has lambda = 0.4 - 0.7 * 104 Angstrom
This corresponds to an energy of 1 eV and is suitable for probing structure on the scale of micron (e.g., laser light scattering studies of hard sphere colloidal dispersions).
The photons are scattered from variations in the dielectric constant or refractive index.

Probing structure at the Angstrom scale requires X-ray photons with energy  104 eV, which are scattered from variations in the dielectric constant due to variations in the electronic charge density. X-rays in this range can penetrate upto a mm. of matter and provide information about the "bulk".

Electrons have energy E = h2 / (2 * me *lambda2 )
[Rest mass of electron, me: 0.511 MeV/c2]
A wavelength of 1 Angstrom corresponds to an energy of ~ 100 eV
Scattered from electrostatic potential, which are often quite large - unless the smaple is thin (~ 1 micrometre) there will be problems with multiple scattering.

Neutrons have a larger rest mass than electrons [mn: 939.6 MeV/c2]
A wavelength of 1 Angstrom corresponds to an energy of 0.1 eV - this is about the order of thermal energy at room temperature. Also, the energies of typical excitations in condensed matter system are of the order of fraction of eV - so these excitations are best studied through "thermal" neutron scattering.
Scattered from nuclear forces.

Heavier charged particles (e.g., ions) are also used - but they interact too strongly with the electronic charge density. Hence, these are used more frequently to probe surface structure (to avoid problems with multiple scattering).

We will focus on elastic or quasi-elastic scattering in which changes in the energy of scattered particles can be neglected.

X-ray production:
Classical theory - accelerated electric charge radiates electromagnetic waves.
When rapidly moving electrons (in an accelerating electric field V) are suddenly brought to rest by collision with a target, electromagnetic radiation occurs.
This process of X-ray production is known as Bremsstrahlung ("braking radiation").
Such X-rays have a minimum wavelength lambdamin = (h * c) / (V * e)
In addition to the continuous spectrum generated by Bremsstrahlung, at high V we can see narrow spikes of X-ray radiation at wavelengths characteristic of the target material. These are due to electronic transitions within the inner shells of atoms. [The high energy of the incident electrons are able to disturb the inner shells of the atoms, where electrons are tightly bound.] If a K shell electron is knocked out, an electron from an outer shell (L or M shell) drops into this "hole" - giving up most of its energy in the form of X-rays (Kalpha or Kbeta lines, respectively, in the X-ray spectrum).

X-ray diffraction
von Laue in 1912 measured the wavelength of X-rays by diffraction from crystals. Subsequently, W H Bragg and W L Bragg used the technique of X-ray diffraction to determine the structure of crystals.

An atom in a constant electric field becomes polarized since its negatively charged electrons and positively charged nucleus experience forces in opposite directions - resulting in a distorted charge distribution equivalent to an electric dipole.
In the presence of an alternating electric field of an electromagnetic wave, the polarization changes back and forth with the same frequency as that of the electromagnetic wave - creation of an oscillating electric dipole (at the expense of some of the energy of the incoming wave).
The oscillating dipole radiates electromagnetic waves (of same frequency as the incident wave) and these secondary waves proceed in all directions - i.e., the secondary waves will have spherical fronts. A monochromatic beam of X-rays falling upon a crystal will be scattered in all directions within it. Owing to the regular arrangement of the atons, in certain directions the scattered waves will constructively interfere with one another, while in others they will destructively interfere. This is the princple behind Bragg's relation.

Atoms in a crystal may be thought of as defining families of parallel planes - with each family having a characteristic seperation (d) between its component planes (Bragg planes). Then, under the condition of specular reflection (i.e.,  angle of incidence = angle of reflection = theta), the path difference between two scattered rays should be equal to an integral multiple of the wavelength, i.e.,

2 * d * sin(theta) = n * lambda
where n = 1, 2, 3, ... is the order of the scattered beam. The scattered intensity (at angle theta) reflects a fluctuation or inhomogeneity of the system with periodicity (lambda / [2 * sin(theta)]).

However, the Bragg relation does not say anything about the intensity of the scattered beams. For this, we require a more rigorous treatment (making use of the Fermi's golden rule).
For details of this dereivation, see Chaikin & Lubensky, pp. 29-37.