Chapter 7 X-Ray diffraction

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

Basic concepts Crystal chemistry: study of relationship of internal crystal structure to the physical and chemical properties of minerals Crystal chemistry: study of relationship of internal crystal structure to the physical and chemical properties of minerals Crystals: Crystals: Lattice structure, with unit cell as basic building blockLattice structure, with unit cell as basic building block Atoms and inter-atomic distances: 0.1 – 0.5 nmAtoms and inter-atomic distances: 0.1 – 0.5 nm For investigation of crystals - waves with wavelengths in this orderFor investigation of crystals - waves with wavelengths in this order Visual light: wavelength 400 – 700 nm Visual light: wavelength 400 – 700 nm X-Rays and neutron waves: 0.01 – 0.5 nm X-Rays and neutron waves: 0.01 – 0.5 nm

Basic concepts Quantum shells orbits wherein electrons move: Quantum shells orbits wherein electrons move: K, L, M, N, O, P, QK, L, M, N, O, P, Q No of electrons: 2N 2 (N = 1,2,3,4,5,6 or 7)No of electrons: 2N 2 (N = 1,2,3,4,5,6 or 7) M: 2 x 3 2 = 18M: 2 x 3 2 = 18 Sub-shells – dividing quantum shells Sub-shells – dividing quantum shells s, p, d, fs, p, d, f No of electrons: 4K – 2 (K = 1, 2, 3 or 4)No of electrons: 4K – 2 (K = 1, 2, 3 or 4) d: (4 x 3) – 2 = 10d: (4 x 3) – 2 = 10

Basic concepts Generation of X-rays Generation of X-rays Electrons emitted by source (heated W filament)Electrons emitted by source (heated W filament) Accelerate electrons by using an electric field (control strength of electric field to control speed of electrons)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)Electrons collide with metal anode (Cu or Mo) Very high-energy radiation emitted: X-rays*Very high-energy radiation emitted: X-rays* How: How: Accelerated electrons displace electrons of inner shell of atomAccelerated electrons displace electrons of inner shell of atom Electron from higher shell fills electron ‘gap’ and release excess energy as X-ray photonsElectron 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 anodeEnergy and wavelength of these photons correspond to particular electronic transition of given atom of the metal anode

Generation of X-rays

Waves Amplitude Amplitude Wavelength Wavelength Path/phase difference Path/phase difference Fig 7.4Fig 7.4 Interference of waves Interference of waves ConstructiveConstructive DestructiveDestructive

Interference of waves

Bragg’s Law Based on the diffraction of X-rays or neutrons from crystal surfaces at certain angles 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 Basis of developing powerful tool for studying crystals in the form of X-ray and neutron diffraction How: 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 frequencyWhen 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 Rayleigh scattering Scattered waves interfere with each other constructively or destructively to produce peaks at certain anglesScattered 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 doneProduces 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: 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 scattering 2.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. 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 Bragg’s equation: nλ = 2dsinθ n: integer determined by the order givenn: integer determined by the order given λ: the wavelength of x-raysλ: the wavelength of x-rays d: spacing between atomic lattice planesd: spacing between atomic lattice planes θ: angle between the incident ray and the scattering planesθ: 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) The x-rays are collimated into a strong X-ray absorber (usually lead) Narrow resulting x-ray beam strike the crystal Narrow resulting x-ray beam strike the crystal Rotate crystal and detector to satisfy Braggs law for diffraction Rotate crystal and detector to satisfy Braggs law for diffraction

Techniques: Diffractometer

Techniques: Diffractometry of powders Randomly oriented small crystals or ‘crystallites’ Randomly oriented small crystals or ‘crystallites’ Reflections scanned and record intensity as function of diffraction angle Reflections scanned and record intensity as function of diffraction angle The list of θ angles with different intensities are converted to d-spacings The list of θ angles with different intensities are converted to d-spacings Identify crystals by comparing with diffractions patterns of known minerals 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: Thermal conductivity: Transfer of heat through a mineral through thermal vibrationsTransfer of heat through a mineral through thermal vibrations High in metals and minerals with significant metallic bondingHigh in metals and minerals with significant metallic bonding Thermal expansion Thermal expansion Expansion of a crystal (increase in volume) with an increase in temperatureExpansion of a crystal (increase in volume) with an increase in temperature

Definitions Piezoelectricity: 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)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 Only possibly with some crystals with no centre of symmetry Pyroelectricity Pyroelectricity The ability of a prismatic crystal to develop opposite electric charges on opposite ends when heatedThe ability of a prismatic crystal to develop opposite electric charges on opposite ends when heated Common in trigonal tourmaline crystals Common in trigonal tourmaline crystals Magnetisism Magnetisism The ability of a crystal to produce a magnetic moment when a magnetic field is applied to itThe 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 Only possible when crystal contains atoms or ions with unpaired electrons Strongest: Fe 3+ and Mn 2+ - has five unpaired 3d-electronsStrongest: Fe 3+ and Mn 2+ - has five unpaired 3d-electrons Fe 2+ - has four unpaired 3d-electronsFe 2+ - has four unpaired 3d-electrons

Chapter 3 Q2

Chapter 3 Q5

Chapter 4 Q8 Q8