X-Ray Diffraction Basics

Introduction

X-rays have wavelengths comparable to atomic spacings, so a periodic crystal can scatter them into strong diffraction peaks. Peak positions help determine plane spacings and cell dimensions, while peak intensities carry information about which atoms occupy positions. Bragg's relation gives the elementary geometry behind these observations.

Core explanation

For a family of parallel crystal planes separated by d, constructive reflection-like interference occurs when nλ=2d sinθ in the simple Bragg picture. Here λ is X-ray wavelength, n an integer order and θ the angle between the incident beam and the planes. Many diffractometers report 2θ, the angle between incident and diffracted beams; inserting the displayed 2θ value as θ would be a common factor-of-two angular error.

The path difference between waves scattered from successive planes is 2d sinθ. If that difference is an integer number of wavelengths, phases reinforce and a peak appears. This is a geometric explanation, not a claim that X-rays physically bounce from perfectly solid mirrored atomic sheets. Atoms throughout the crystal scatter waves, and periodicity organizes their interference.

If λ and θ are known, d=nλ/(2sinθ). For first-order n=1, λ=0.150 nm and θ=15°, d≈0.150/(2×0.2588)=0.290 nm. The answer uses the same length unit as λ. If the instrument reports 2θ=30°, first halve it to θ=15°.

A crystal has many plane families, so a powder diffraction pattern contains multiple peaks. Their positions can identify lattice spacings and phases. Peak intensities depend on atom types, positions, symmetry and experimental effects. Two structures can have similar peak positions yet different intensity patterns because their bases differ. Absence of a peak can reflect destructive interference, not necessarily absence of that geometric plane family.

Amorphous materials lack long-range periodicity and typically show broad scattering features rather than the sharp Bragg peaks of a well-ordered crystal. Real crystals have finite size, strain and defects, which can broaden peaks. Peak broadening is not proof of amorphous character by itself; context and modeling matter.

Diffraction can help estimate cell parameters, identify polymorphs and detect mixtures. It does not directly photograph bonds or prove one localized Lewis structure. A refined atomic arrangement is inferred by fitting scattering data to a structural model. Other methods may be needed to identify oxidation state or subtle disorder.

Step-by-step reasoning

1. Read X-ray wavelength and whether the instrument angle is θ or 2θ. 2. Select the diffraction order, usually n=1 in a simple exercise. 3. Rearrange nλ=2d sinθ to solve d. 4. Keep angle mode and wavelength units consistent. 5. Interpret peak positions separately from intensities and widths.

Visual explanation

Draw two parallel planes separated by d and two incoming rays at angle θ to the planes. Show the lower ray traveling an extra down-and-up path of 2d sinθ. Mark the detector angle between incoming and outgoing directions as 2θ.

Real-world analogy

Two synchronized ripples reinforce when one travels an extra whole number of wavelengths. Crystal diffraction peaks occur when repeated atomic layers make scattered X-ray paths line up in phase.

Real-world example

Powder X-ray diffraction can distinguish two polymorphs of the same compound because their periodic plane spacings and atomic arrangements produce different peak patterns despite identical chemical formula.

Why?

Why does amorphous glass lack a set of sharp crystal peaks? It lacks the long-range repeated spacings needed for scattering from many regions to reinforce at precise angles.

Common misconception

“The measured 2θ can be used directly as θ in Bragg's equation.” The equation uses the Bragg angle θ; a reported detector angle 2θ must be halved first.

Worked example

A peak appears at 2θ=40.0° using λ=0.154 nm, first order. Then θ=20.0°. Calculate d=0.154/[2sin20.0°]≈0.154/(0.684)=0.225 nm. Using 40° in the sine would give a much smaller wrong spacing. The calculation estimates one family of plane spacings, not the entire cell structure by itself.

Quick check

1. In Bragg's relation, what angle is used if a diffractometer reports 2θ=50°? Answer: θ=25°.

Exam focus

Write Bragg's equation with defined variables, halve 2θ when required and distinguish peak positions, intensities and broadening. Avoid describing diffraction as a direct photograph of atoms.

Advanced insight

Structure factors sum waves scattered by atoms in the unit-cell basis. Symmetry can make certain reflections systematically absent, which helps identify lattice centering and space-group possibilities.

Summary

Periodic crystal planes produce constructive X-ray interference when nλ=2d sinθ. Peak positions give spacings, while intensities and widths add structural information. Angle convention and model assumptions are essential.

Practice questions

1. Solve Bragg's equation for d. Answer: d=nλ/(2sinθ). 2. What feature typically distinguishes an amorphous pattern from a crystalline one? Answer: Broad scattering features rather than many sharp long-range-order peaks. 3. Do two identical formulas guarantee identical diffraction patterns? Answer: No. Different polymorph structures can have different plane spacings and intensities.