X-Ray Diffraction and Bragg's Law
Almost everything we know about the arrangement of atoms in solids comes from one equation with four symbols in it. Exam questions on X-ray diffraction are, without exception, arithmetic questions dressed in crystallographic language — convert an angle to a spacing, convert spacings to a lattice parameter, convert a lattice parameter to a density. This article works all three conversions through in full, and then indexes a real powder pattern.
Bragg's law
n = order of reflection (a whole number) · λ = X-ray wavelength · d = spacing between adjacent lattice planes · θ = the Bragg angle, measured between the incident beam and the plane, not the normal.
The reasoning behind it is short. X-rays scattered from two successive parallel planes travel path lengths that differ by 2d sin θ. When that difference is a whole number of wavelengths the waves arrive in phase and reinforce; at any other angle they cancel. So a crystal gives sharp spots or lines at a few discrete angles and darkness in between.
Three practical consequences follow immediately:
- The diffractometer reports 2θ, not θ. Halve it before you do anything else. This is the most common single-step error in the whole topic.
- sin θ cannot exceed 1, so λ ≤ 2d. With typical lattice spacings of a few ångström you need radiation of about 1 Å. Visible light at 500 nm = 5000 Å can never diffract from atomic planes, which is exactly why X-rays are used.
- The value of n is a convention. The n-th order reflection from the (hkl) planes is treated as the first-order reflection from planes labelled (nh, nk, nl); a second order (111) is indexed as (222). Setting n = 1 always and letting the indices carry the order is the standard practice.
Common laboratory wavelengths are worth knowing. Cu Kα1 is 1.5406 Å; because the Kα1 and Kα2 lines are usually not resolved, many papers quote the weighted Kα average of 1.5418 Å instead. Both are in use, so state which you are using. Mo Kα, about 0.7107 Å, is preferred for single-crystal work because the shorter wavelength reaches more reflections.
Plane spacings in a cubic crystal
Tetragonal: 1/d² = (h² + k²)/a² + l²/c²
Orthorhombic: 1/d² = h²/a² + k²/b² + l²/c²
Combining Bragg with the cubic case gives the working equation: sin²θ = (λ²/4a²)(h² + k² + l²)
That last line is the whole of powder indexing. Because λ and a are fixed for a given pattern, sin²θ is directly proportional to (h² + k² + l²). Take ratios of sin²θ and the constants cancel, leaving whole numbers that name the planes.
Which whole numbers appear depends on the lattice type, because some reflections cancel out systematically:
| Lattice | Condition for a reflection | Allowed h²+k²+l² | Ratio pattern |
|---|---|---|---|
| Simple cubic (P) | none | 1, 2, 3, 4, 5, 6, 8, 9 … | 1 : 2 : 3 : 4 : 5 : 6 (7 is missing — it is not a sum of three squares) |
| Body-centred (I) | h + k + l even | 2, 4, 6, 8, 10, 12 … | 1 : 2 : 3 : 4 after halving |
| Face-centred (F) | h, k, l all odd or all even | 3, 4, 8, 11, 12, 16, 19, 20 … | 3 : 4 : 8 : 11 … |
Worked example 1 — d-spacing from a single peak
A reflection appears at 2θ = 20.00° with Cu Kα1, λ = 1.5406 Å. Find d.
θ = 20.00 ÷ 2 = 10.00°
sin 10.00° = 0.173648
2 sin θ = 0.347296
d = λ ÷ (2 sin θ) = 1.5406 ÷ 0.347296 = 4.436 Å
Sanity check: 2d = 8.87 Å, comfortably larger than λ = 1.5406 Å, so the reflection is physically allowed. If a calculation ever returns 2d < λ, an arithmetic slip has occurred.
Worked example 2 — indexing a powder pattern
A cubic metal gives its first three lines at 2θ = 38.47°, 44.72° and 65.09° with Cu Kα1. Identify the lattice and find a.
Step 1 — halve, take sines, square them.
| 2θ | θ | sin θ | sin²θ |
|---|---|---|---|
| 38.47° | 19.235° | 0.32944 | 0.10853 |
| 44.72° | 22.360° | 0.38042 | 0.14472 |
| 65.09° | 32.545° | 0.53793 | 0.28937 |
Step 2 — divide by the smallest value.
0.10853 ÷ 0.10853 = 1.000
0.14472 ÷ 0.10853 = 1.333
0.28937 ÷ 0.10853 = 2.666
Step 3 — scale to whole numbers. Multiplying by 3 gives 3.00 : 4.00 : 8.00.
Compare with the table above: 3, 4, 8 is the face-centred cubic sequence, so the three lines are (111), (200) and (220).
Step 4 — extract the lattice parameter from the first line, using h² + k² + l² = 3:
a² = λ²(h² + k² + l²) ÷ (4 sin²θ)
λ² = 1.5406 × 1.5406 = 2.37345 Ų
numerator = 2.37345 × 3 = 7.12034
denominator = 4 × 0.10853 = 0.43413
a² = 7.12034 ÷ 0.43413 = 16.401 Ų
a = √16.401 = 4.050 Å
Cross-check on the second line, (200), h² + k² + l² = 4:
a² = (2.37345 × 4) ÷ (4 × 0.14472) = 9.49380 ÷ 0.57888 = 16.400, so a = 4.050 Å. The two lines
agree to four figures, which is the real test that the indexing is right — a wrong assignment
gives lattice parameters that disagree.
Worked example 3 — density from the unit cell
Z = formula units per unit cell (1 for simple cubic, 2 for body-centred, 4 for face-centred), M = molar mass in g mol−1, a in cm, NA = 6.022 × 1023 mol−1.
Take the fcc metal above to be aluminium, M = 26.982 g mol−1, Z = 4, a = 4.050 Å = 4.050 × 10−8 cm.
a³ = (4.050)³ × 10−24 = 66.430 × 10−24 = 6.6430 × 10−23 cm³
Numerator: Z M = 4 × 26.982 = 107.928 g mol−1
Denominator: a³NA = (6.6430 × 10−23)(6.022 × 1023) = 40.004 cm³ mol−1
ρ = 107.928 ÷ 40.004 = 2.698 g cm−3
The measured density of aluminium is about 2.70 g cm−3. Agreement to three figures confirms both the indexing and the assumption Z = 4 — which is how this calculation is normally used: run in reverse, a measured density tells you how many formula units sit in the cell.
Worked example 4 — crystallite size from peak width
Bragg's law says where a peak appears. Its width carries separate information: very small crystallites produce broader peaks, because there are too few planes to cancel completely off the exact Bragg angle.
K ≈ 0.9 (a shape factor), β = full width at half maximum in radians, corrected for instrumental broadening, θ = Bragg angle.
The (111) peak at 2θ = 38.47° has a corrected FWHM of 0.50°. With λ = 1.5406 Å:
β = 0.50 × π ÷ 180 = 0.008727 rad
θ = 19.235°, cos θ = 0.94417
β cos θ = 0.008727 × 0.94417 = 0.0082400
D = (0.9 × 1.5406) ÷ 0.0082400 = 1.38654 ÷ 0.0082400 = 168 Å = 16.8 nm
Two honest limits, both examinable: the Scherrer equation is only reliable below roughly 100 nm, and it attributes all the broadening to size. Lattice strain broadens peaks too, so a Scherrer number is a lower bound on the crystallite size unless strain has been separated out.
What diffraction can and cannot tell you
Peak positions give the size and shape of the unit cell — that is Bragg's law. Peak intensities give the positions of the atoms inside the cell, through the structure factor, which is where systematic absences come from in the first place. A powder pattern is therefore excellent for identifying a known phase and measuring a lattice parameter, while a full single-crystal data set is needed to solve an unknown structure. Peak widths give crystallite size and strain. Three different features of the same pattern, three different kinds of answer.
Mistakes that cost marks
- Using 2θ in place of θ. Every diffractometer reports 2θ. Halving it is the first line of the working, always.
- Measuring θ from the normal to the plane. The Bragg angle is measured from the plane itself, unlike the angle of incidence in optics.
- Leaving β in degrees in the Scherrer equation. It must be in radians; forgetting the conversion inflates the answer by a factor of about 57.
- Mixing ångström with centimetres in the density formula. 1 Å = 10−8 cm, so a³ carries a factor of 10−24.
- Indexing before checking the ratios are consistent. Compute the lattice parameter from two or three different lines; if they disagree, the assignment is wrong.
- Forgetting that 7 never appears in a cubic h²+k²+l² series, because no three squares sum to 7. Its absence is not a systematic absence and does not indicate centring.
- Treating the Scherrer size as exact. It is an estimate with a shape factor of roughly 0.9 and it ignores strain.
Where this appears in the exam
| Exam | Typical demand |
|---|---|
| CSIR-NET Chemical Sciences | Indexing a cubic powder pattern, systematic absences, lattice parameter and density calculations, Scherrer estimates |
| GATE Chemistry | Bragg's law numericals, d-spacing formulas, unit-cell packing and coordination number |
| IIT-JAM / CUET-PG | Solid state chapter — unit cells, Z values, density from edge length |
| Class 12 CBSE | The solid state: cubic unit cells, packing efficiency, density calculation |
The physics is one line; the marks are in the arithmetic. Diffraction problems mix trigonometry, cubes of small numbers, ångström-to-centimetre conversions and Avogadro's number in a single calculation. The ABC Chemistry Calculator Suite keeps the scientific calculator, unit converter and scientific constants together in one page while you work through a pattern.
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