Chapter 7

X-Ray diffraction

Contents

Basic concepts and definitions Waves and X-rays Crystal structure Bragg’s law X-ray techniques

Basic concepts Crystal chemistry: study of relationship of internal

crystal structure to the physical and chemical properties of minerals

Crystals:• Lattice structure, with unit cell as basic building block• Atoms and inter-atomic distances: 0.1 – 0.5 nm• For investigation of crystals - waves with wavelengths in

this order Visual light: wavelength 400 – 700 nm X-Rays and neutron waves: 0.01 – 0.5 nm

Basic concepts

Quantum shells orbits wherein electrons move:• K, L, M, N, O, P, Q• No of electrons: 2N2 (N = 1,2,3,4,5,6 or 7)• M: 2 x 32 = 18

Sub-shells – dividing quantum shells• s, p, d, f• No of electrons: 4K – 2 (K = 1, 2, 3 or 4)• d: (4 x 3) – 2 = 10

Basic concepts Generation of X-rays

• Electrons emitted by source (heated W filament) • Accelerate electrons by using an electric field (control

strength of electric field to control speed of electrons)• Electrons collide with metal anode (Cu or Mo)• Very high-energy radiation emitted: X-rays*

How:• Accelerated electrons displace electrons of inner shell of

atom• Electron from higher shell fills electron ‘gap’ and release

excess energy as X-ray photons• Energy and wavelength of these photons correspond to

particular electronic transition of given atom of the metal anode

Generation of X-rays

Waves Amplitude Wavelength Path/phase difference

• Fig 7.4 Interference of waves

• Constructive• Destructive

Interference of waves

Bragg’s Law Based on the diffraction of X-rays or neutrons

from crystal surfaces at certain angles Basis of developing powerful tool for studying

crystals in the form of X-ray and neutron diffraction

How:• When X-rays (or neutrons) hit an atom the movement of

electrons (or spinning of nuclei) cause re-radiation of waves with the same frequency

Rayleigh scattering• Scattered waves interfere with each other constructively

or destructively to produce peaks at certain angles• Produces a diffraction pattern from which the analyses of

the crystal structure is done

Bragg’s law When X-rays are scattered from a crystal lattice, peaks of

scattered intensity are observed which correspond to the following conditions:

1. The angle of incidence = angle of scattering2. The path length difference is equal to an integer number of

wavelengths.

The condition for maximum intensity contained in Bragg's law above allow us to calculate details about the crystal structure, or if the crystal structure is known, to determine the wavelength of the x-rays incident upon the crystal.

Bragg’s equation: nλ = 2dsinθ• n: integer determined by the order given• λ: the wavelength of x-rays• d: spacing between atomic lattice planes• θ: angle between the incident ray and the scattering planes

Bragg’s law

Techniques: X-ray diffractometer

The x-rays are collimated into a strong X-ray absorber (usually lead)

Narrow resulting x-ray beam strike the crystal

Rotate crystal and detector to satisfy Braggs law for diffraction

Techniques: Diffractometer

Techniques: Diffractometry of powders

Randomly oriented small crystals or ‘crystallites’

Reflections scanned and record intensity as function of diffraction angle

The list of θ angles with different intensities are converted to d-spacings

Identify crystals by comparing with diffractions patterns of known minerals

Techniques: Diffraction pattern

Chapter 8

Physical properties of crystals

Definitions(Further self-study of this

chapter is optional)

Definitions

Thermal conductivity:• Transfer of heat through a mineral

through thermal vibrations• High in metals and minerals with

significant metallic bonding Thermal expansion

• Expansion of a crystal (increase in volume) with an increase in temperature

Definitions Piezoelectricity:

• The ability of a crystal to change its shape slightly (undergoes strain) when an electrical field is applied to it (also vice versa: applying stress can induce an electric field on these crystals)

Only possibly with some crystals with no centre of symmetry

Pyroelectricity• The ability of a prismatic crystal to develop opposite

electric charges on opposite ends when heated Common in trigonal tourmaline crystals

Magnetisism• The ability of a crystal to produce a magnetic moment

when a magnetic field is applied to it Only possible when crystal contains atoms or ions with unpaired

electrons• Strongest: Fe3+ and Mn2+ - has five unpaired 3d-electrons• Fe2+ - has four unpaired 3d-electrons

Chapter 3Q2

Chapter 3Q5

Chapter 4

Q8