intercepts → reciprocals → clear fractions → reduce → (hkl)
Module 04 · Crystallography for XRD
How to Find Miller Indices: Crystal Planes, Directions and XRD (hkl)
Three numbers in a bracket that tell you exactly which sheet of atoms a diffraction peak came from. This module gets you from zero to fluent — with a repeatable five-step method, twelve drawn planes, and the arithmetic that turns (hkl) into a 2θ value.
1, 1, ∞→
1, 1, 0→
(110)
Section 01What a Miller index actually is
A Miller index is a label for the orientation of a family of parallel atomic planes. It is not a length, not a position, and not a single sheet.
Atoms in a crystal sit on a repeating lattice. Any repeating arrangement automatically contains flat sheets of lattice points — and each sheet belongs to an infinite stack of identical, parallel, equally spaced sheets. The perpendicular gap between neighbouring sheets in one stack is the interplanar spacing, d, and it is the quantity an X-ray diffractometer actually measures.
Miller indices name the stack. The rule is:
h : k : l = 1/p : 1/q : 1/r
- p, q, r
- intercepts of the plane on the x, y, z axes, in units of the cell edges a, b, c (dimensionless)
- h, k, l
- the smallest set of integers in that ratio — the Miller indices (dimensionless)
- ( )
- brackets marking the trio as a plane, written with no commas: (110), not (1,1,0)
Two consequences follow immediately, and both are the reason the reciprocal is used at all:
| Property of the reciprocal rule | Why it matters |
|---|---|
| 1/∞ = 0 | A plane parallel to an axis never meets it. The reciprocal turns an unusable ∞ into a clean zero. Every zero in a Miller index means "parallel to that axis". |
| Intercepts in cell units, then reduced to integers | The same stack gets the same label in a 20 nm crystallite and a 2 cm single crystal. The index describes orientation only. |
| Large h, k, l ⇒ small d | Because h ∝ 1/p, high indices mean closely spaced, sparsely populated planes — weak, high-angle peaks. Low indices dominate a real pattern. |
The four brackets, and what each one means
These are confused more often than any other part of the subject. Learn all four now, together, and the confusion never starts.
| Notation | Object | Meaning | Example |
|---|---|---|---|
| (h k l) | One plane | A specific family of parallel planes with a specific orientation | (100) — cube face ⟂ x |
| {h k l} | Family of planes | All planes equivalent by the crystal's symmetry | {100} — all six cube faces |
| [u v w] | One direction | A specific line or vector through the lattice | [100] — the x-axis itself |
| ⟨u v w⟩ | Family of directions | All directions equivalent by symmetry | ⟨100⟩ — all six cube-edge directions |
Remember
Say it as separate digits. (110) is "one one zero", never "one hundred and ten". Double-digit indices are written with spaces, as in (11 0 0), to remove the ambiguity.
Section 02Setting up the axes on a unit cell
Everything in this module is measured on axes locked to the unit cell, not to the laboratory bench.
| Element | Definition | Practical rule |
|---|---|---|
| Origin O | A lattice point, taken as (0,0,0) | Free to place at any lattice point — and you will use that freedom when a plane passes through the origin. |
| x, y, z axes | Along the cell edges a, b, c | Perpendicular in cubic crystals, not in triclinic. Miller indices still work, because they never assume 90°. |
| Unit of length | 1 along x = one full cell edge a | Never use nanometres here. Intercepts are always fractions or multiples of the cell edge. |
| Negative side | Behind the origin, opposite to a, b, c | Written with a bar, e.g. (110), read "one, bar one, zero". |
Why cell units, not nanometres
If intercepts were in nanometres, silicon (a = 5.431 Å) and copper (a = 3.615 Å) would give different labels for the same geometric plane. In cell units, both give (111). The notation is deliberately blind to the size of the cell.
Section 03Reading intercepts off a drawing
An intercept is the point where the plane crosses an axis, measured in cell units. There are only three cases, and you will meet all three constantly.
| Case | What you see | Intercept | Reciprocal |
|---|---|---|---|
| Cuts the axis | Plane crosses at a finite point | 1, ½, ⅓, 2 … | 1, 2, 3, ½ … |
| Parallel to the axis | Never meets it, however far extended | ∞ | 0 |
| Cuts on the negative side | Crosses behind the origin | −1, −½ … | −1, −2 … |
The one rule beginners break
If the plane passes through the origin, the intercept is 0 and 1/0 is undefined. Do not divide by zero. Move the origin to another lattice point — usually one cell along the nearest axis — and re-read the intercepts. The indices are unchanged, because a shift of one whole cell simply moves you to a parallel member of the same stack.
Section 04The five-step method
Use these five steps in this order, every time. The order is a genuine sequence — each step needs the output of the one before it.
-
Read the intercepts
Find where the plane crosses x, y and z, in cell units. Write ∞ for any axis it runs parallel to. If it passes through the origin, shift the origin first.
Why: three intercepts uniquely fix the orientation of a flat surface. This is the only geometric information the label will ever contain.
-
Write them as p, q, r
Keep fractions as fractions (½, ⅓). Do not convert to decimals — decimals hide the simple ratios you are about to exploit.
Why: Miller indices are a ratio, and ratios are read most reliably from exact fractions.
-
Take reciprocals: 1/p, 1/q, 1/r
Every ∞ becomes 0. Every ½ becomes 2. Every 1 stays 1.
Why: this removes infinity from the notation and makes the label proportional to plane density rather than plane position.
-
Clear the fractions
Multiply all three by the lowest common multiple of the denominators so all three become integers.
Why: indices must be integers — the law of rational indices — and that is what makes the notation finite and comparable between materials.
-
Reduce and enclose
Divide out any common factor, then write the trio in round brackets with no commas: (h k l). Negative values take a bar.
Why: reducing guarantees one unique label per orientation. Note the deliberate exception in XRD — see Section 12, where (200) is kept unreduced on purpose.
Worked once, slowly
A plane cuts x at ½, cuts y at 1, and runs parallel to z.
Steps 1–2: p, q, r = ½, 1, ∞ → Step 3: 2, 1, 0 → Step 4: already integers → Step 5: no common factor → (210).
Intercept → Index calculatorset the three intercepts
Reciprocals—
Clear fractions—
Miller index—
Reads as—
Section 05The plane atlas
Twelve planes, drawn in the same cube on the same axes. Learn to recognise these on sight; together they account for almost every peak in a routine powder pattern.
Read the atlas properly
Compare (100) and (200). They are drawn at the same angle — same orientation — but (200) sits halfway in. That is the point: (200) labels the stack whose spacing is half that of (100). Same direction, different d.
Section 06Worked examples
Each card shows the plane, the intercepts, the reciprocals, the clearing factor and the answer. Cover the right-hand side and try it before you read it.
Section 07Negative indices and zeros
When an intercept is negative
A plane can cut an axis on the far side of the origin. The intercept is then negative, its reciprocal is negative, and the index carries a bar above it.
| Intercepts (p, q, r) | Reciprocals | Index | Spoken |
|---|---|---|---|
| 1, −1, ∞ | 1, −1, 0 | (110) | one, bar one, zero |
| −1, 1, 1 | −1, 1, 1 | (111) | bar one, one, one |
| ½, −1, −1 | 2, −1, −1 | (211) | two, bar one, bar one |
| −½, ∞, 1 | −2, 0, 1 | (201) | bar two, zero, one |
A plane and its exact opposite are the same stack
(111) and (111) describe the same set of parallel planes with the same d-spacing — you have simply viewed the stack from the other side. In diffraction from a centrosymmetric crystal they are indistinguishable; this is Friedel's law.
What a zero really means
A zero is never "nothing". It is the fingerprint of a parallel axis.
| Index | Zeros in position | Geometry |
|---|---|---|
| (100) | k = 0, l = 0 | Parallel to both y and z; cuts only x. A cube face. |
| (110) | l = 0 | Parallel to z only; cuts x and y. A face-diagonal plane standing vertically. |
| (111) | none | Cuts all three axes. The slanted corner-cutting plane. |
| (011) | h = 0 | Parallel to x; cuts y and z. |
Section 08Reverse problems: draw the plane from the index
Run the five steps backwards. The whole reverse procedure is one line: invert each index to get the intercept, then join the three points.
| Given index | 1/h, 1/k, 1/l | Intercepts | How to draw it |
|---|---|---|---|
| (100) | 1, 1/0, 1/0 | 1, ∞, ∞ | Mark x = 1; draw a face parallel to y and z |
| (110) | 1, 1, 1/0 | 1, 1, ∞ | Join x = 1 to y = 1; extend the sheet vertically through z |
| (111) | 1, 1, 1 | 1, 1, 1 | Join x = 1, y = 1, z = 1 — a triangle cutting one corner off |
| (200) | ½, ∞, ∞ | ½, ∞, ∞ | Same as (100) but at x = ½ |
| (210) | ½, 1, ∞ | ½, 1, ∞ | Join x = ½ to y = 1; extend through z |
| (211) | ½, 1, 1 | ½, 1, 1 | Triangle through x = ½, y = 1, z = 1 |
| (221) | ½, ½, 1 | ½, ½, 1 | Triangle through x = ½, y = ½, z = 1 |
| (311) | ⅓, 1, 1 | ⅓, 1, 1 | Triangle through x = ⅓, y = 1, z = 1 |
| (110) | 1, −1, ∞ | 1, −1, ∞ | Shift the origin one cell along +y, then join x = 1 to y = −1 |
| (102) | 1, ∞, ½ | 1, ∞, ½ | Join x = 1 to z = ½; extend through y |
Drawing tip that saves marks
For fractional intercepts, draw the cube large and mark ½ and ⅓ on the axes before drawing the plane. Students who draw the plane first almost always place the triangle by eye and get the slope wrong.
Section 09Crystal directions [uvw]
A direction is a line, not a sheet. Its indices are found by a completely different procedure — no reciprocals are taken at any point.
Put the tail of the vector at the origin
Translate the line so it starts at (0,0,0). Directions are free vectors; sliding them changes nothing.
Why: only the vector's components matter, not where it sits.
Read the head coordinates in cell units
Write them as u′, v′, w′. Fractions are allowed at this stage.
Why: these three numbers already are the direction, in raw form.
Clear fractions, reduce, use square brackets
Multiply up to integers, divide out common factors, write [u v w]. Negatives take bars.
Why: a direction has no unique length, so only the smallest integer ratio is meaningful.
| Plane (h k l) | Direction [u v w] | |
|---|---|---|
| What it describes | Orientation of a sheet of atoms | A line through the lattice |
| Built from | Axis intercepts | Vector components |
| Reciprocals? | Yes | No — never |
| Brackets | ( ) single · { } family | [ ] single · ⟨ ⟩ family |
| Zero means | Parallel to that axis | No component along that axis |
| Physical use | Diffraction, cleavage, growth faces | Slip, growth axis, epitaxy, texture |
| Cubic-only bonus | In cubic crystals only, [h k l] is perpendicular to (h k l). This is false in every other crystal system. | |
Left: the (111) plane, a sheet cutting the corner. Right: the [111] direction, the body diagonal. In a cubic crystal the arrow is the normal to the sheet — which is exactly why the two notations are so easy to confuse.
Common mistake
Writing "the [111] plane" or "the (111) direction". Both are wrong and both cost marks in vivas. Round brackets are always planes; square brackets are always directions.
Section 10Families: {hkl} and ⟨uvw⟩
In a cubic crystal, (100), (010), (001) and their three negatives are geometrically different planes but physically identical — symmetry maps each onto the others. They share a d-spacing and diffract at the same angle. Collectively they are written {100}.
| Family | Members (cubic) | Multiplicity | h²+k²+l² | Note |
|---|---|---|---|---|
| {100} | Six cube faces | 6 | 1 | Largest d, lowest angle |
| {110} | Twelve face-diagonal planes | 12 | 2 | Close-packed plane in BCC |
| {111} | Eight corner-cutting planes | 8 | 3 | Close-packed plane in FCC |
| {200} | Six | 6 | 4 | Same orientation as {100}, half the spacing |
| {210} | Twenty-four | 24 | 5 | — |
| {211} | Twenty-four | 24 | 6 | Common slip plane in BCC |
| {220} | Twelve | 12 | 8 | — |
| {311} | Twenty-four | 24 | 11 | Prominent in FCC patterns |
Where this shows up in the lab
In a randomly oriented powder every member of a family contributes to the same peak — which is why a powder pattern has far fewer peaks than a crystal has planes. In a textured thin film that randomness is broken: one member dominates, its peak grows and the others shrink. Comparing measured intensity ratios against database ratios is the standard test for preferred orientation.
Section 11The seven errors that cost marks
Wrong
Intercepts 1, 1, ∞ → index (11∞)
Reciprocals were never taken.
Right
1, 1, ∞ → 1, 1, 0 → (110)
Reciprocal is step 3, not optional.
Wrong
Intercepts ½, 1, 1 → (½ 1 1)
Fractions left inside the bracket.
Right
½, 1, 1 → 2, 1, 1 → (211)
Indices are always integers.
Wrong
"The [110] plane in copper"
Right
"The (110) plane" or "the [110] direction"
Match the bracket to the object.
Wrong
Plane through the origin → intercept 0 → 1/0 = ∞ → index ∞
Right
Shift the origin one cell, re-read intercepts, then index. Never divide by zero.
Wrong
Negative intercept written as (1 −1 0) with a minus sign
Right
(110) — the bar goes above the index. Minus signs are acceptable only in plain text.
Wrong
Reducing an XRD peak label from (200) to (100)
Right
Diffraction labels stay unreduced. (200) means d(100)/2 — a real, distinct reflection.
Wrong
Assuming [hkl] ⟂ (hkl) in a hexagonal crystal such as ZnO
Right
That perpendicularity holds in cubic crystals only. Elsewhere use the reciprocal lattice.
Section 12Why every XRD peak carries an (hkl) label
Bragg's law converts a plane spacing into an angle. Miller indices convert a plane label into a spacing. Chain the two and the whole labelling system becomes obvious.
d(hkl) = a / √(h² + k² + l²) (cubic only)
- d(hkl)
- interplanar spacing of the (hkl) stack, in Å or nm
- a
- cubic lattice parameter, same unit as d
- h,k,l
- Miller indices of the reflecting planes, unreduced
λ = 2 d sin θ
- λ
- X-ray wavelength; Cu Kα = 1.5406 Å
- θ
- Bragg angle — half the 2θ value printed on your pattern
- 2θ
- the scan axis of the diffractometer, in degrees
So a peak label is a statement about geometry. When your software prints "(111) at 2θ = 43.30°" for copper, it is telling you: this intensity arrived because a stack of planes with spacing 2.087 Å satisfied Bragg's law at that angle, and that spacing corresponds to a = 3.615 Å with h²+k²+l² = 3.
Why some indices are missing
Not every plane produces a peak. Waves scattered by atoms inside the cell can cancel exactly — a systematic absence. The rules below come from the structure factor, and they are what let you identify a lattice type from peak positions alone.
| Lattice | Reflection present when | First allowed (hkl) | h²+k²+l² ratios |
|---|---|---|---|
| Simple cubic | All h, k, l | 100, 110, 111, 200, 210, 211 | 1:2:3:4:5:6 |
| Body-centred (BCC) | h + k + l = even | 110, 200, 211, 220, 310, 222 | 2:4:6:8:10:12 |
| Face-centred (FCC) | h, k, l all odd or all even | 111, 200, 220, 311, 222, 400 | 3:4:8:11:12:16 |
| Diamond cubic | All odd, or all even with h+k+l = 4n | 111, 220, 311, 400, 331, 422 | 3:8:11:16:19:24 |
Worked example — indexing a real cubic pattern
Four peaks measured with Cu Kα (λ = 1.5406 Å) at 2θ = 38.47°, 44.72°, 65.13°, 78.25°.
| 2θ (°) | θ (°) | sin²θ | Ratio | ×3 | (hkl) | d (Å) | a (Å) |
|---|---|---|---|---|---|---|---|
| 38.47 | 19.235 | 0.1085 | 1.000 | 3 | (111) | 2.338 | 4.050 |
| 44.72 | 22.360 | 0.1447 | 1.333 | 4 | (200) | 2.025 | 4.050 |
| 65.13 | 32.565 | 0.2898 | 2.670 | 8 | (220) | 1.432 | 4.050 |
| 78.25 | 39.125 | 0.3981 | 3.668 | 11 | (311) | 1.221 | 4.049 |
The method in four moves: compute sin²θ for each peak → divide all by the smallest → multiply by a small integer until the list is whole numbers → match that list against the ratio column above. If a is constant when you back-calculate it, the indexing is right.
d-spacing & 2θ calculatorcubic lattices
h²+k²+l²—
d-spacing—
2θ (Bragg)—
Reflection—
Copper powder pattern, Cu Kα. Every peak is one family of planes. Peaks move left when the lattice expands, broaden when crystallites shrink and change relative height when the sample is textured — but the (hkl) labels stay fixed by the lattice type.
Section 13Four case studies from real patterns
All 2θ values below are for Cu Kα, λ = 1.5406 Å.
Copper — FCC metal, a = 3.615 Å
| (hkl) | h²+k²+l² | d (Å) | 2θ (°) | I/I₀ | What it tells you |
|---|---|---|---|---|---|
| (111) | 3 | 2.087 | 43.30 | 100 | Close-packed plane; the slip plane of FCC metals |
| (200) | 4 | 1.808 | 50.43 | 46 | I(200)/I(111) is the standard texture check for Cu films |
| (220) | 8 | 1.278 | 74.13 | 20 | Used for strain analysis — higher angle, better resolution |
| (311) | 11 | 1.090 | 89.93 | 17 | Preferred peak for Scherrer fits: isolated and reasonably strong |
Silicon — diamond cubic, a = 5.4309 Å
| (hkl) | d (Å) | 2θ (°) | Note |
|---|---|---|---|
| (111) | 3.136 | 28.44 | Strongest peak; the standard alignment reflection in most labs |
| (200) | 2.715 | — | Forbidden. All even, but h+k+l = 2 is not a multiple of 4 |
| (220) | 1.920 | 47.30 | Allowed: all even, h+k+l = 4 |
| (311) | 1.638 | 56.12 | Allowed: all odd |
| (400) | 1.358 | 69.13 | Allowed: all even, h+k+l = 4 |
Sodium chloride — rock salt, a = 5.640 Å
| (hkl) | d (Å) | 2θ (°) | I/I₀ | Note |
|---|---|---|---|---|
| (111) | 3.256 | 27.37 | ~10 | Weak — Na⁺ and Cl⁻ scatter out of phase, so amplitudes subtract |
| (200) | 2.820 | 31.71 | 100 | Strongest — the two ions scatter in phase, so amplitudes add |
| (220) | 1.994 | 45.45 | 55 | Also an in-phase reflection |
| (222) | 1.628 | 56.47 | ~2 | Weak, same reason as (111) |
Zinc oxide — hexagonal wurtzite, a = 3.2495 Å, c = 5.2069 Å
Hexagonal crystals often use four indices, (h k i l), where i = −(h + k) is redundant but makes symmetry-equivalent planes look alike. (100) and (1010) are the same plane.
1/d² = (4/3)(h² + hk + k²)/a² + l²/c²
- a, c
- hexagonal lattice parameters, in Å
- h,k,l
- three-index Miller indices
| (hkl) | d (Å) | 2θ (°) | I/I₀ | Note |
|---|---|---|---|---|
| (100) | 2.814 | 31.77 | 57 | Prism plane |
| (002) | 2.604 | 34.42 | 44 | Basal plane ⟂ c. Dominates in c-axis-oriented films and nanorods |
| (101) | 2.476 | 36.25 | 100 | Strongest peak in random ZnO powder |
| (102) | 1.911 | 47.54 | 23 | — |
| (110) | 1.625 | 56.61 | 32 | — |
| (103) | 1.477 | 62.86 | 29 | — |
Real lab tip
Before trusting an indexing, back-calculate the lattice parameter from every peak. If a drifts systematically with angle you have a sample-height or zero-offset error, not a new phase. Constant a across all peaks is the cheapest quality check in the whole technique.
Section 14Quick reference card
Everything on one screen
Plane, forward
Plane, reverse
(hkl) → intercepts 1/h, 1/k, 1/l → join the points
Direction
tail at origin → head coordinates → clear & reduce → [uvw]. No reciprocals.
Zero
Parallel to that axis. Comes from an infinite intercept.
Bar
Negative intercept. (1̄10) reads "bar one, one, zero".
Through the origin?
Shift the origin one cell. Never take 1/0.
Cubic d-spacing
d = a / √(h²+k²+l²)
Bragg
λ = 2 d sinθ, with θ = 2θ ÷ 2
FCC allowed
h, k, l all odd or all even
BCC allowed
h + k + l even
Diamond allowed
All odd, or all even with h+k+l = 4n
Cubic only
[hkl] ⟂ (hkl). Not true in hexagonal, tetragonal or orthorhombic.
Decision path when you are stuck
| Situation | Do this |
|---|---|
| Plane passes through the origin | Move the origin one cell along an axis the plane does not contain, then re-read |
| Plane parallel to one axis | Intercept = ∞ → that index is 0 |
| Intercepts come out as 2, 3, ∞ | Reciprocals ½, ⅓, 0 → ×6 → (3 2 0) |
| You get a common factor, e.g. 2, 2, 4 | Reduce to (1 1 2) — unless labelling a diffraction peak, where it stays unreduced |
| You are asked for a direction, not a plane | Stop. Do not take reciprocals. Use head-minus-tail coordinates. |
Section 15Practice
A · Determine the Miller index (intercepts in cell units)
- 1, 1, 1
- 1, ∞, ∞
- ∞, 1, ∞
- 1, 1, ∞
- ½, ∞, ∞
- ½, 1, ∞
- ½, 1, 1
- ½, ½, 1
- ⅓, 1, 1
- 1, −1, ∞
- −1, 1, 1
- 2, 1, ∞
- 2, 3, ∞
- 1, ∞, ½
- ⅓, ½, 1
B · Reverse — state the intercepts, then sketch the plane
- (100)
- (010)
- (001)
- (110)
- (101)
- (011)
- (111)
- (200)
- (210)
- (211)
- (221)
- (311)
C · Multiple choice
- A zero in a Miller index means the plane is (a) at the origin (b) parallel to that axis (c) perpendicular to that axis (d) forbidden
- Reciprocals are taken when indexing (a) planes only (b) directions only (c) both (d) neither
- {111} in a cubic crystal contains how many planes? (a) 3 (b) 6 (c) 8 (d) 12
- For cubic crystals, d(200) equals (a) 2·d(100) (b) d(100)/2 (c) d(100) (d) d(100)/4
- In FCC, which is a forbidden reflection? (a) 111 (b) 200 (c) 210 (d) 220
- In silicon, (200) is absent because (a) h+k+l is odd (b) indices are mixed (c) all even but h+k+l ≠ 4n (d) d is too small
- [110] in a cubic crystal is (a) parallel to (110) (b) perpendicular to (110) (c) at 45° to (110) (d) undefined
- A plane cuts x at ⅓ and is parallel to y and z. Its index is (a) (300) (b) (100) (c) (013) (d) (3̄00)
- Which family has the largest d-spacing in a simple cubic crystal? (a) {111} (b) {110} (c) {100} (d) {211}
- Miller–Bravais indices (hkil) are used for (a) cubic (b) hexagonal (c) triclinic (d) all systems
- The sin²θ ratio 3:4:8:11 indicates (a) SC (b) BCC (c) FCC (d) diamond
- The redundant index i in (hkil) equals (a) h+k (b) −(h+k) (c) h−k (d) hk
- A strong (002) with weak (101) in ZnO indicates (a) impurity (b) amorphous content (c) c-axis texture (d) instrument error
- If a plane passes through the origin you should (a) index it as (000) (b) shift the origin (c) use ∞ (d) discard it
- The BCC condition for an allowed reflection is (a) h+k+l even (b) h+k+l odd (c) all odd (d) all even
D · Numerical
- Copper, a = 3.615 Å. Calculate d for (111), (200) and (220).
- Using Cu Kα (1.5406 Å), calculate 2θ for the copper (220) reflection.
- A cubic sample gives its first FCC peak at 2θ = 38.47°. Find d and then a.
- An unknown cubic pattern gives sin²θ ratios 1 : 2 : 3 : 4 : 5 : 6. Which lattice type is it?
- Silicon (111) is measured at 2θ = 28.44°. Back-calculate a and compare with 5.4309 Å.
- For a cubic crystal with a = 4.00 Å, which is the highest-index reflection observable with Cu Kα? (Hint: d must be at least λ/2.)
- ZnO, a = 3.2495 Å, c = 5.2069 Å. Calculate d(002) and its 2θ.
- A plane has intercepts 2a, 3b, 6c in an orthorhombic cell. Find its Miller index.
E · Short answer and viva
- Why are reciprocals used instead of the intercepts themselves? Give two reasons.
- State the difference between (100) and {100} in one sentence each.
- Why is (200) not reduced to (100) when labelling a diffraction peak?
- Explain why NaCl has a weak (111) but a very strong (200).
- Why do low-index planes give the strongest diffraction peaks?
- Is [111] perpendicular to (111) in ZnO? Justify your answer.
- What physically changes in a crystal when a peak shifts to lower 2θ?
- How would you identify preferred orientation using index labels and intensities?
- Why does a powder pattern show fewer peaks than the number of plane families present?
- Give one mechanical and one optical property that depends on a specific plane family.
Answer key — objective sections
A · 1 (111) · 2 (100) · 3 (010) · 4 (110) · 5 (200) · 6 (210) · 7 (211) · 8 (221) · 9 (311) · 10 (11̄0) · 11 (1̄11) · 12 (120) · 13 (320) · 14 (102) · 15 (326)
B · Intercepts are 1/h, 1/k, 1/l: (100)→1,∞,∞ · (010)→∞,1,∞ · (001)→∞,∞,1 · (110)→1,1,∞ · (101)→1,∞,1 · (011)→∞,1,1 · (111)→1,1,1 · (200)→½,∞,∞ · (210)→½,1,∞ · (211)→½,1,1 · (221)→½,½,1 · (311)→⅓,1,1
C · 1 b · 2 a · 3 c · 4 b · 5 c · 6 c · 7 b · 8 a · 9 c · 10 b · 11 c · 12 b · 13 c · 14 b · 15 a
D · 1 · d(111) = 2.087 Å, d(200) = 1.808 Å, d(220) = 1.278 Å. 2 · 2θ = 74.13°. 3 · d = 2.338 Å, a = 4.050 Å (aluminium). 4 · simple cubic. 5 · d = 3.136 Å, a = 5.431 Å. 6 · d ≥ 0.7703 Å ⇒ h²+k²+l² ≤ 26.9, so the highest observable is h²+k²+l² = 26, i.e. (431)/(510). 7 · d(002) = 2.604 Å, 2θ = 34.42°. 8 · reciprocals ½, ⅓, ⅙ → ×6 → (321).
Companion tools
The two calculators on this page, a Bragg/d-spacing solver, a Scherrer crystallite-size calculator and a printable blank-cube practice sheet are all available on AdvanceMaterialsLab.com. Use them to check your hand calculations — not to replace them. In a viva you get a pen, not a browser.
Module 4 · Mastering Miller Indices — from the Complete Course on X-Ray Diffraction Analysis, an original educational resource of AdvanceMaterialsLab.com. All figures generated from first principles; all values computed for Cu Kα, λ = 1.5406 Å.
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