Miller Indices and X-Ray Diffraction (XRD)

Miller Indices

• Miller Indices are used to represent directions and planes in a crystal
• Miller Indices is a group of smallest integers representing a direction or plane

Steps to determine Miller Indices

• Obtain lengths of planar intercepts for each axis
• Take the reciprocal of these lengths
• Change the three numbers into the smallest set of integers using a common factor
• Enclose Miller Indices within parentheses, e.g., (012)

Tips for Miller Indices:

• Parallel planes have the same indices
• An index of 0 implies the plane is parallel to that axis

X-Ray Diffraction (XRD)

• XRD is a non-destructive technique for analyzing crystal structures
• XRD measures diffraction patterns to determine atomic arrangements

Applications of XRD:

• XRD is used to obtain structural information about crystalline solids
• XRD is useful in biochemistry to solve the 3D structures of complex bio-molecules
• Bridges physics, chemistry, and biology

Fields where X-ray diffraction is important:

• Solid-state physics
• Biophysics
• Medical physics
• Chemistry and Biochemistry

X-ray Nature

• X-rays are electromagnetic waves with a wavelength (λ) ≈ 0.01–10 nm
• X-rays have a smaller wavelength than visible light, higher energy, and are more penetrative

Historical milestones in X-ray discovery:

• 1895: Wilhelm Röntgen discovered X-rays
• 1912: Max von Laue demonstrated crystal diffraction
• 1913: Bragg formulated Bragg's Law

X-ray Production:

• X-rays are generated when high energy electrons strike an anode in a sealed vacuum

Key Component Materials

• Anodes are often made of copper, iron, or molybdenum

Formula

• λmin = (1240 / ΔV) nm

Bragg's Law

• William Henry Bragg and William Lawrence Bragg found a simple interpretation of von Laue's experiment
• They assumed that each crystal plane reflects radiation as a mirror and analyzed constructive and destructive interference
• 2d sin θ = nλ

Powder Diffractometers and Bragg-Brentano Geometry

• The incident angle (ω) is defined between the X-ray source and the sample
• The diffracted angle (2θ) is defined between the incident beam and the detector angle
• The incident angle ω is always one half of the detector angle 2θ

Types of Instruments

• In a θ:2θ instrument (fixed, Rigaku RU300), the tube is fixed, the sample rotates at θ °/min, and the detector rotates at 2θ °/min
• In a θ:θ instrument (e.g., PANalytical X’Pert Pro), the sample is fixed, and the tube rotates at a rate of -θ °/min, while the detector rotates at a rate of θ °/min

Single Crystal in a Diffractometer

• A single crystal specimen in a Bragg-Brentano diffractometer produces only one family of peaks in the diffraction pattern
• At 20.6 °2θ, Bragg's law is fulfilled for the (100) planes, producing a diffraction peak