Bragg's Law Numericals: d-Spacing and Lattice Parameter
X-ray diffraction (XRD) is the standard way to measure the distance between planes of atoms in a crystal. In solid-state chemistry questions, the examiner gives you a wavelength and a diffraction angle and then asks for the plane spacing, the edge length of the unit cell, or even the density of the metal. All of these steps use one short relation, called Bragg's law, together with the cubic geometry you already know. This article shows the full chain of calculation so that no step feels like a jump.
The idea behind Bragg's law
When X-rays fall on a crystal, each plane of atoms reflects a small part of the beam. Two reflections from neighbouring planes travel different distances. The extra path is 2d sin θ, where d is the spacing between the planes and θ is the angle between the incident beam and the plane. The reflected waves strengthen each other only when this extra path is a whole number of wavelengths.
n = order of reflection (1, 2, 3 ...), λ = wavelength (nm or pm), d = interplanar spacing (same unit as λ), θ = glancing angle (degrees)
The detector usually records the angle 2θ, which is the angle between the incoming and the diffracted beam. Always halve it before using the formula. Since sin θ cannot be more than 1, the largest possible order is n = 2d/λ, rounded down.
From d-spacing to lattice parameter
For a cubic crystal with edge length a, the spacing of the planes with Miller indices (hkl) is given below. Rearranged, it gives the edge length directly.
a = edge length of the cube (nm), h, k, l = Miller indices (no unit)
For (100) the root is 1, for (110) it is √2 = 1.4142, for (111) it is √3 = 1.7321, and for (200) it is 2.
From lattice parameter to density
Z = atoms per unit cell (sc 1, bcc 2, fcc 4), M = molar mass (g/mol), NA = 6.02214 × 10²³ mol⁻¹, a = edge (cm), ρ = density (g/cm³)
Remember that 1 nm = 10⁻⁷ cm, so a in nm must be multiplied by 10⁻⁷ before cubing. This unit conversion is where many marks are lost.
Worked examples
Question: Cu Kα X-rays of wavelength 0.1541 nm give a first-order reflection at 2θ = 44.0° from a set of planes in a crystal. Find the interplanar spacing d.
Step 1: The detector angle is 2θ = 44.0°, so θ = 22.0°.
Step 2: sin 22.0° = 0.37461.
Step 3: For n = 1, d = nλ / (2 sin θ) = 0.1541 / (2 × 0.37461) = 0.1541 / 0.74922.
Step 4: 0.1541 / 0.74922 = 0.20568 nm.
Check by a second route: 2d sin θ = 2 × 0.20568 × 0.37461 = 0.15410 nm, which equals λ. Correct.
Answer: d = 0.2057 nm (about 205.7 pm)
Question: The reflection in the previous question is from the (111) planes of a face-centred cubic metal. Calculate the edge length a of the unit cell.
Step 1: For (111), h² + k² + l² = 1 + 1 + 1 = 3, so the root is √3 = 1.73205.
Step 2: a = d × √3 = 0.20568 × 1.73205.
Step 3: 0.2 × 1.73205 = 0.34641 and 0.00568 × 1.73205 = 0.00984, so a = 0.35625 nm.
Check by a second route, using the wavelength directly: a = λ√3 / (2 sin θ) = (0.1541 × 1.73205) / 0.74922 = 0.26691 / 0.74922 = 0.35625 nm. Both routes agree.
Answer: a = 0.3562 nm (356.2 pm)
Question: The metal above is fcc with a = 0.3562 nm and molar mass 58.69 g/mol. Calculate its density.
Step 1: For fcc, Z = 4.
Step 2: a = 0.3562 nm = 3.562 × 10⁻⁸ cm.
Step 3: a³ = (3.562)³ × 10⁻²⁴ cm³. First 3.562² = 12.6878, then 12.6878 × 3.562 = 45.194. So a³ = 4.5194 × 10⁻²³ cm³.
Step 4: Mass of the cell contents = Z × M = 4 × 58.69 = 234.76 g/mol.
Step 5: NA × a³ = 6.02214 × 10²³ × 4.5194 × 10⁻²³ = 27.2165 cm³/mol.
Step 6: ρ = 234.76 / 27.2165 = 8.626 g/cm³.
Check in SI units: a = 3.562 × 10⁻¹⁰ m, a³ = 4.5194 × 10⁻²⁹ m³, ρ = (4 × 0.05869) / (6.02214 × 10²³ × 4.5194 × 10⁻²⁹) = 0.23476 / 2.7217 × 10⁻⁵ = 8626 kg/m³ = 8.626 g/cm³. Same value.
Answer: ρ = 8.63 g/cm³
Question: For planes with d = 0.20568 nm and X-rays of λ = 0.1541 nm, what is the highest order of reflection possible, and at what 2θ does the second order appear?
Step 1: The largest value of sin θ is 1, so n(max) = 2d / λ = 2 × 0.20568 / 0.1541 = 0.41136 / 0.1541 = 2.67.
Step 2: Only whole numbers are allowed, so n = 1 and n = 2 are possible; n = 3 is not.
Step 3: For n = 2, sin θ = nλ / 2d = 2 × 0.1541 / 0.41136 = 0.3082 / 0.41136 = 0.74923.
Step 4: θ = sin⁻¹(0.74923) = 48.5°, so 2θ = 97.0°.
Check: sin θ(second order) must be twice sin θ(first order) = 2 × 0.37461 = 0.74922. Matches.
Answer: highest order n = 2; second-order peak at 2θ ≈ 97.0°
Exam details
- Source: practice problem written for this article (not a past-paper question)
- Relevant exams: IIT-JAM, GATE, CSIR-NET, CUET-PG, and university MSc entrance tests
- Topic: Bragg's law, interplanar spacing, cubic lattice parameter and density from XRD
- Difficulty: Moderate to advanced
Common mistakes
Using 2θ in place of θ. The instrument reports 2θ, but the formula needs θ. Halve the angle first, every time.
Mixing units. If λ is in pm and a is needed in cm, convert once at the start. For density, 1 nm = 10⁻⁷ cm and 1 pm = 10⁻¹⁰ cm.
Forgetting the order n. A peak at a larger angle may be a second-order reflection of the same planes, and it gives the same d only if you use n = 2.
Using the wrong Z. Simple cubic has 1 atom per cell, bcc has 2 and fcc has 4. Using 1 for every cell gives a density that is too small by a simple factor.
Setting the calculator to radians. Check that it is in degree mode before you press sin.
How exams test this topic
| Exam type | How the topic appears |
|---|---|
| IIT-JAM | Direct numerical on d or a from the angle; density of a cubic metal |
| GATE Chemistry | Bragg's law combined with fcc or bcc geometry and Miller indices |
| CSIR-NET | Conceptual and short numerical items on the highest order, systematic absences and plane spacing |
| CUET-PG | Single-step substitution in nλ = 2d sin θ |
| University MSc papers | Long answers linking XRD data to the unit cell and density |
A short checklist
Write the wavelength and its unit first. Halve 2θ. Find sin θ to five digits. Compute d. If the plane indices are given, multiply by the root of h² + k² + l². For density, convert a to cm, cube it, and use the correct Z. Finally, put the unit next to the answer. A good habit is to repeat the calculation backwards, as in the checks above, because one wrong power of ten ruins the whole answer.
Explore real crystal structures in the Crystal Lab, then test yourself on previous-year questions.
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