Reading a Powder Pattern: Position, Intensity, Width

Three messages hiding in one jagged curve

The last guide ground your material into millions of randomly oriented crystallites and collapsed their three-dimensional cloud of reflections onto a single axis: the diffractogram, a trace of counted intensity climbing and falling as the detector sweeps through the scattering angle 2 theta. It looks deceptively plain — a row of spikes on a gently drifting background. But do not read it as one signal. Each peak is really three measurements stacked in the same place, and this guide teaches you to pull them apart: WHERE the peak sits, HOW TALL it stands, and HOW WIDE it spreads.

There is one organizing idea to hold onto, and it is the oldest theme of this whole ladder: the lattice is not the crystal. A crystal is a lattice (where you stamp) plus a motif (the picture on the stamp). A powder pattern splits neatly along that same seam. Peak POSITIONS report the geometry of the lattice — the size and shape of the unit cell — because position is fixed by Bragg's law, which knows only about plane spacings. Peak INTENSITIES report the motif — which atoms sit where inside the cell — because intensity is set by the structure factor, the phased sum over the atoms. And peak WIDTHS report neither the lattice nor the motif but the imperfection: how small and how strained the crystallites are.

WHAT EACH FEATURE OF A POWDER PEAK ENCODES

  feature       you read off       it hands you        set by
  -----------   ---------------    -----------------   ----------------------
  POSITION      2 theta of peak    d-spacing -> cell   Bragg's law + geometry
                                   (lattice params)    (unit-cell size + shape)
  INTENSITY     height / area      the MOTIF           |F_hkl|^2 x p x Lp x DW
                                   (which atom where)  (structure factor + more)
  WIDTH (FWHM)  breadth of peak    crystallite size    Scherrer: size ~ 1/cos t
                                   + microstrain       strain broadens ~ tan t

  positions   -> WHERE the stamps sit   (the lattice)
  intensities -> WHAT is on each stamp  (the motif)
  widths      -> how small / how strained the crystal patches are

Position: from an angle to the unit cell

Start with the easiest reading. Each peak's angle is a direct measurement of an interplanar spacing, because Bragg's law lambda = 2 d sin theta rearranges to d = lambda / (2 sin theta). You know lambda — it is the clean Cu K-alpha of 1.54 angstrom that guide 1 taught you to carve out of the tube — so every peak angle converts straight to a d-spacing. Work a number: a plane family spaced d = 2 angstrom, lit by lambda = 1.54 angstrom, diffracts at theta = arcsin(1.54 / (2 times 2)) = arcsin(0.385) = 22.6 degrees, so the peak lands at 2 theta = 45.2 degrees on the trace. Read that angle backwards and the 2-angstrom spacing falls right out.

A single d-spacing is one number; the whole SET of them is the fingerprint of the cell. This is where indexing comes in — assigning Miller indices (hkl) to every peak, then solving for the lattice parameters. For a cubic crystal the algebra is friendly: the plane-spacing equation gives 1/d^2 = (h^2 + k^2 + l^2)/a^2, so combining it with Bragg's law makes sin^2 theta proportional to the integer s = h^2 + k^2 + l^2. The peaks therefore fall at sin^2 theta values in the ratios 1 : 2 : 3 : 4 : ... (or with certain integers missing, exactly the systematic absences of the last rung). Spot that integer sequence and you have both indexed the pattern and measured a to a fraction of a percent.

1. Read off the peak angles and convert each to sin^2 theta (for cubic; low-symmetry cells need a proper indexing program).
2. Divide every sin^2 theta by the smallest one — the ratios should come out near a set of integers s = h^2 + k^2 + l^2.
3. Match the integer sequence to a lattice type: 1,2,3,4,5,6,8,... is primitive P; 2,4,6,8,... (all even) is body-centered; 3,4,8,11,12,... is face-centered.
4. Assign (hkl) to each peak, then back-solve the lattice parameter a from a = d times sqrt(s) — averaging over all peaks (and favouring the high-angle ones, which are most sensitive to a).

Intensity: the motif written in peak heights

Positions gave you an empty box; intensities furnish it. A reflection's strength is proportional to the squared magnitude of the structure factor, |F_hkl|^2, where F_hkl = sum over the motif of f_j times exp(2 pi i (h x_j + k y_j + l z_j)). Unpack that in words: every atom in the cell scatters a wavelet whose strength is its atomic scattering factor f_j (recall from guide 1 that heavy atoms, with more electrons, scatter far harder than light ones), and whose phase is set by where the atom sits, (x_j, y_j, z_j), relative to the reflecting planes. Add those phased wavelets and you get F; square it and you get the peak height. So the RELATIVE tallness of the peaks is the coded message of the motif — move an atom and you re-shuffle the whole intensity profile, even though not a single peak changes angle.

One honest complication: the raw peak you measure is NOT |F|^2 by itself. In a powder several geometric and physical factors scale each peak before it reaches you, and you must strip them off to recover the structure-factor amplitudes. The largest is the multiplicity factor p — in a powder all the symmetry-equivalent planes of a family diffract to the same angle and pile into one peak, so the 6 planes of the cube {100} family, the 8 of {111}, the 12 of {110} each boost their peak in proportion. On top of that ride the Lorentz-polarization factor (a smooth geometric weighting that rises at low and high angles), the Debye-Waller factor exp(-2M) that thermal jitter uses to dim the high-angle peaks, and absorption. Measured intensity is roughly |F_hkl|^2 times p times Lp times exp(-2M): read the motif only after dividing those out.

Put the two readings side by side and the division of labour is exact and beautiful. NaCl and KCl both crystallize in the rock-salt structure with the same kind of cell, so their peaks march in the same sequence — the positions barely distinguish them. But Na and K scatter differently, and Cl sits at a different fractional distance from each, so the relative peak HEIGHTS differ sharply. Position told you the box; intensity told you the furniture. That is why you can never solve a structure from positions alone: the cell is only the stage, and the intensities are the play.

Width: crystallite size and strain from the Scherrer equation

Now the third reading, and the most surprising, because it turns a defect of the sample into a measurement. A perfect, infinitely large crystal would give infinitely sharp peaks — mathematical spikes with zero width. Real crystallites are finite, and the smaller they are, the fewer parallel planes there are to enforce perfect Bragg cancellation just off the exact angle, so each peak spreads into a hump. That is peak broadening, and its size dependence is captured by the Scherrer equation: t = K lambda / (beta cos theta), where t is the crystallite size, beta is the extra peak width (in radians, as a full width at half maximum), theta is the Bragg angle, and K is a shape constant near 0.9. Small crystals, broad peaks; the width is an inverse ruler for size.

Run one through. Take the peak from before at 2 theta = 45.2 degrees, so theta = 22.6 degrees and cos theta = 0.923, with lambda = 0.154 nm. Suppose that, after you subtract the instrument's own broadening, the peak is beta = 0.3 degrees wide — which in radians is 0.3 times pi/180 = 0.00524 rad. Then t = 0.9 times 0.154 / (0.00524 times 0.923) = 0.1386 / 0.00484 = about 29 nm. A peak just three-tenths of a degree wider than the machine floor is telling you the crystallites are roughly 29 nanometres across. The subtraction of the instrument's own width is not optional: on a well-aligned diffractometer that floor is itself a tenth of a degree or so, and if you forget it you will report crystals far smaller than they are.

Now the honest limits, because this is a rule with several exceptions. First, the size you get is the CRYSTALLITE size — the coherently diffracting domain — not the metallurgical grain you would see in a micrograph; one grain can hold many subgrains, so the Scherrer number is usually the smaller one. Second, the method runs out of resolution above roughly 100 to 200 nm: bigger crystals give peaks so nearly sharp that the extra width vanishes into the instrument floor and cannot be measured. Third, and most important, size is not the only thing that broadens a peak — non-uniform microstrain does too, as a spread of slightly different d-spacings smears the angle. The two causes have different angular signatures (size broadening scales as 1/cos theta, strain broadening as tan theta), and separating them by watching how the width grows with angle — the Williamson-Hall analysis — is how you get an honest number for each. Finally, K near 0.9 depends on the crystal shape and on how you define the width, so treat a Scherrer size as good to a factor, not to three digits.

Phase identification, and the ceiling you hit

Put position and intensity together across the whole pattern and you have a fingerprint no two compounds share — a specific list of d-spacings each with a relative height. That is the engine of phase identification: measure a pattern, compare its peak list against a reference database (the ICDD Powder Diffraction File holds hundreds of thousands of entries), and read off which crystalline phases are present. It is genuinely like a police lineup — you are not solving the structure from scratch, you are matching an unknown against known suspects. This is the single most common thing a powder diffractometer does: is my sample the alpha or the beta polymorph, did the reaction go to completion, what fraction is still unreacted starting material.

Two honest cautions keep this from being magic. Matching relies on the relative intensities, but those are exactly what preferred orientation ruins: if plate-like or needle-like crystallites lie down non-randomly (texture), some peaks swell and others shrink, and the fingerprint stops matching the reference measured on a random powder. And identification only works for crystalline phases — an amorphous solid has short-range order but no long-range periodicity, so it gives a broad diffuse hump, not sharp peaks, and cannot be looked up. 'Amorphous' does not mean 'no order'; it means no lattice for Bragg's law to bite on.