Powder diffraction: identify an unknown metal

A pinch of metal powder, a beam of copper X-rays and one diffraction pattern. From the peak positions, work out the lattice type and the lattice parameter, then find the metal on a reference card.

Difficulty ●●○ ≈ 20 min

Aim Use ratios of

\sin^2\theta

to decide whether an unknown cubic metal is simple, body-centred or face-centred, calculate its lattice parameter

a

with Bragg’s law, and identify the metal.

Structure factor · multiplicity · Lorentz–polarisation

Powder pattern of an unknown metal

Unknown sample #

Anode

Computed powder pattern · λ = 154.06 pm

Peaks found by the instrument (below 1 % omitted). Look at the sin²θ ratios first: which column becomes whole numbers after multiplying by 1, 2 or 3?

#	2θ / °	I	sin²θ ratio	×2	×3
1	44.49	100	1.000	2.00	3.00
2	51.85	46	1.334	2.67	4.00
3	76.38	24	2.667	5.33	8.00
4	92.93	28	3.667	7.33	11.00
5	98.44	8	4.001	8.00	12.00

Which lattice do you think it is?

Reference card: room-temperature lattice parameters

Metal	Lattice	a / pm
α-Iron Fe	Body-centred cubic	286.65
Tungsten W	Body-centred cubic	316.52
Niobium Nb	Body-centred cubic	330.04
α-Polonium Po	Simple cubic	335.20
Nickel Ni	Face-centred cubic (ccp)	352.40
Copper Cu	Face-centred cubic (ccp)	361.49
Aluminium Al	Face-centred cubic (ccp)	404.95
Gold Au	Face-centred cubic (ccp)	407.82
Sodium Na	Body-centred cubic	429.06
Lead Pb	Face-centred cubic (ccp)	495.08

Which metal is it?

Procedure

Enter a sample number (1–99). The diffractometer scans 2θ = 10°–120° with Cu Kα₁ radiation (λ = 154.06 pm) and lists every peak its peak search finds.
Read the $\sin^2\theta$ ratios. Since $\sin^2\theta = \dfrac{\lambda^2}{4a^2}(h^2+k^2+l^2)$ , they are ratios of $h^2+k^2+l^2$ . Find the column that turns into whole numbers after multiplying by 1, 2 or 3.
Face-centred gives 3, 4, 8, 11, 12… (×3); body-centred gives 2, 4, 6, 8, 10… (×2); simple cubic gives 1, 2, 3, 4, 5, 6, 8… — with 7 missing.
Choose the lattice you have decided on. The table labels each peak $(hkl)$ and computes $a = d\sqrt{h^2+k^2+l^2}$ . If every peak gives the same $a$ , the indexing is self-consistent.
Compare your mean $a$ with the reference card, choose the metal and check. Repeat with a few more sample numbers.

What to notice

Simple and body-centred cubic give identical ratios for the first six lines (1:2:3:4:5:6); only the seventh line separates them — simple cubic skips 7, body-centred does not. With only six lines, decide by whether $a$ matches a real metal.
Pick the wrong lattice and the $a$ values drift from peak to peak — that drift is the test of an indexing.
High-angle peaks give the most precise $a$ : near θ = 90° a given reading error corresponds to the smallest error in $d$ .
Switch to Mo Kα and every peak moves to lower angle, but the indexing and $a$ stay the same — the crystal has not changed, only the ruler.

In a real lab

Real copper radiation contains the Kα₁/Kα₂ doublet, which splits high-angle peaks; software strips Kα₂ or a monochromator removes it.
Grind the sample fine (about 1–10 μm) and pack it flat with random orientation; coarse grains or preferred orientation badly distort the relative intensities.
A sample surface that sits off the goniometer axis shifts every peak; precise lattice parameters use an internal standard such as silicon powder.
X-rays are ionising radiation: diffractometers have interlocked enclosures that keep the beam off while the door is open — never bypass them.

All experiments

Procedure

What to notice

In a real lab

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