The lattice said WHERE; now, how bright?
The last three guides handed you a powerful gift and quietly withheld the other half. Bragg's law, the Laue conditions, and the Ewald construction all answer one and the same question — WHERE can a diffracted beam appear? Every one of them is pure geometry of the lattice: give me the unit cell, and I will tell you the exact angles at which reflections are allowed to flash out. But stand in front of a real diffraction pattern and something they never mention leaps out at you. The spots are not equally bright. Some blaze, some are barely there, and some that the geometry cheerfully permits are flatly missing. Where does that brightness come from, and where do the missing spots go? That is the whole business of this guide.
Recall the oldest split on this whole ladder: the lattice is a set of identical points, but the crystal is lattice plus motif — the actual atoms stamped onto every point. Bragg's law and the Ewald sphere see only the lattice, so they can only ever tell you the POSITIONS of reflections. The brightness is a message from the motif — from which atoms sit where inside the box. Make that idea quantitative and you get one clean statement that governs everything below: the measured diffracted intensity of a reflection is proportional to |F|^2, the squared magnitude of a single complex number called the structure factor. Positions come from the lattice; intensities come from the motif, packaged inside F.
Carry this headline through the whole rung: a powder pattern's peak POSITIONS give you the unit cell — the size and shape of the box — while the peak INTENSITIES give you the motif — what actually sits inside it. Two different questions, answered by two different halves of the same photograph. Confuse them and nothing about diffraction will ever quite click into place.
One atom's echo: the atomic scattering factor
Before we sum a whole motif, zoom all the way in to a single atom, because guide 1 already told us X-rays scatter off electrons, not off nuclei. One atom is a little cloud of Z electrons, and its scattering strength gets its own name — the atomic scattering factor f, also called the form factor. At the very lowest angle, where the incoming and outgoing beams are almost parallel, every electron in the cloud scatters in step, their wavelets add up perfectly, and the amplitude is simply Z, the atom's electron count (measured in units of the scattering from one free electron). So f(0) = Z: a lead atom (82 electrons) is a far louder scatterer than a carbon atom (6), which is exactly why heavy atoms dominate an X-ray pattern.
Now the crucial twist: f does not stay at Z — it falls off as the angle grows. The reason is beautifully physical. The electron cloud is not a point; it is a fuzzy ball roughly an angstrom across, comparable to the wavelength itself. At low angle the whole cloud scatters in unison, but as the scattering angle opens up, the wavelet from the near side of the cloud and the wavelet from the far side arrive increasingly out of step, so they partly cancel and the total amplitude sags. In practice f is plotted against sin(theta)/lambda: it starts at Z and decays smoothly. For carbon it slides from 6 down to roughly 2 by sin(theta)/lambda = 0.5 per angstrom. The lesson to bank: high-angle reflections are handed a weaker f before anything else even happens, so they tend to be faint.
Be honest about what f describes: it is the scattering of X-rays off the electron cloud, so it tracks the atomic number Z. Swap the probe and the whole picture shifts. Neutrons scatter off the tiny nucleus, which is essentially point-like, so their scattering length barely fades with angle and can make a light atom like hydrogen shout as loudly as a heavy one. Electrons scatter off the electrostatic potential far more strongly than X-rays, which is a gift for imaging thin samples but also drags in dynamical scattering, the reason TEM needs ultrathin foils. Same crystal, three different f's — pick your probe on purpose.
Summing the motif's wavelets: the structure factor
A real unit cell usually holds more than one atom, and here is the whole idea in one sentence: for a given reflection (hkl), every atom in the motif throws out its own scattered wavelet, and those wavelets add — but each arrives with its OWN phase, set by exactly where the atom sits in the cell. An atom parked deep between the reflecting planes lags one sitting on a plane; that lag is a phase shift. Do the geometry and the phase for an atom at fractional coordinates (x, y, z) comes out to precisely 2 pi times (h x + k y + l z). Add the wavelets, each of amplitude f_j and each turned by its own phase, and you have the structure factor: F_hkl = sum over j of f_j times exp[2 pi i (h x_j + k y_j + l z_j)]. It is nothing but the motif's own little Fourier sum, evaluated at the reflection (hkl).
- List the motif: write down every atom in the unit cell with its element (which fixes f_j) and its fractional coordinates (x_j, y_j, z_j).
- Pick the reflection you want, the integer triple (hkl).
- For each atom compute its phase, the angle 2 pi (h x_j + k y_j + l z_j), and read off its scattering factor f_j at that reflection's angle.
- Add the wavelets: F_hkl = sum of f_j times exp[2 pi i (h x_j + k y_j + l z_j)] — literally a sum of little arrows in the complex plane, each of length f_j turned by its phase.
- Square the length: the intensity is proportional to |F_hkl|^2. If the arrows happen to cancel to zero, that reflection is systematically absent, however loudly the geometry says it should appear.
A worked example: why BCC hides (100)
Watch the machine make a reflection vanish. Take a body-centered cubic metal like alpha-iron: one kind of atom, so f is the same for both, sitting at just two spots in the cell, the corner (0, 0, 0) and the body centre (1/2, 1/2, 1/2). Feed those into the sum. F_hkl = f times [1 + exp(2 pi i times (h/2 + k/2 + l/2))] = f times [1 + exp(pi i (h+k+l))] = f times [1 + (-1)^(h+k+l)]. Now just read it. When h+k+l is EVEN, (-1)^(even) = +1 and F = 2f — a strong reflection. When h+k+l is ODD, (-1)^(odd) = -1 and F = 0 — nothing. Physically the wavelet from the body-centre atom arrives exactly half a wavelength behind the corner atom's for odd sums, and the two cancel dead. So (100), (111), (210) simply are not there, while (110), (200), (211) shine. These blanks are the systematic absences.
STRUCTURE FACTOR of a cubic ELEMENT (one atom type, scattering factor f)
BCC motif: (0,0,0) and (1/2,1/2,1/2)
F = f [ 1 + (-1)^(h+k+l) ]
h+k+l even -> F = 2f (reflection PRESENT)
h+k+l odd -> F = 0 (systematic ABSENCE)
FCC motif: (0,0,0),(1/2,1/2,0),(1/2,0,1/2),(0,1/2,1/2)
F = f [ 1 + e^(pi i(h+k)) + e^(pi i(h+l)) + e^(pi i(k+l)) ]
h,k,l ALL even OR ALL odd -> F = 4f (PRESENT)
h,k,l mixed parity -> F = 0 (ABSENCE)
first reflections simple-cubic BCC FCC
(hkl) h+k+l (all present)
(100) 1 mixed yes ABSENT ABSENT
(110) 2 mixed yes yes ABSENT
(111) 3 all-odd yes ABSENT yes
(200) 2 all-even yes yes yes
(210) 3 mixed yes ABSENT ABSENT
(211) 4 mixed yes yes ABSENT
(220) 4 all-even yes yes yes
Run the same sum for a face-centered cubic metal like copper — four atoms, at the corner and the three face centres — and a different rule drops out: F = 4f only when h, k, l are all even or all odd, and F = 0 for any mixed set. So copper shows (111) and (200) but kills (100) and (110). Notice the deep point: the corner atoms are identical in all three cases, yet simple-cubic, BCC and FCC each print a different pattern of surviving spots. The absences are the motif leaving its fingerprint on the reciprocal lattice. Reading that fingerprint backwards — from missing reflections to the centering, screw axes and glide planes that caused them — is precisely the trade of the very next guide.
Multiplicity, temperature, and reading the motif
Be honest: |F|^2 is the heart of the intensity, but it is not the whole measured peak. Two more effects deserve a place, and the first is multiplicity. In a powder — millions of tiny crystals pointing every which way — many symmetry-equivalent planes share the exact same spacing d and therefore pile into one and the same peak. In a cubic crystal the family {100} has 6 members, {110} has 12, and {111} has 8, so the (110) peak is fed by twelve equivalent planes at once. The powder peak height therefore scales as multiplicity times |F|^2 — and then still further factors (Lorentz-polarization, absorption) ride along on top. Quote |F|^2 as the core of the intensity, not the finished number.
The second effect is temperature. Atoms are never frozen; they jiggle about their sites, and a smeared-out atom is a slightly blurred scatterer, especially for the fine detail that high-angle reflections probe. The Debye-Waller factor captures this: it multiplies each intensity by exp(-2M), where 2M grows with (sin(theta)/lambda)^2 and with the mean-square vibration amplitude — so heat it up and the high-angle peaks are the first to fade. This is not damage and not disorder in the lattice; the atoms are still on their sites on average. It is just that a shivering target reflects a crisp high-angle wavelet a little less faithfully. Cool a crystal down and those far peaks brighten right back up.
Now the reward the whole guide was building toward: because F sums the atoms WITH their f's, the intensities read out the motif itself. The classic demonstration is NaCl against KCl. Both take the rock-salt structure, so both have the same allowed reflections. But in rock salt the all-odd reflections carry F = 4(f_cation - f_anion), a DIFFERENCE of scattering factors. In NaCl the cation and anion differ (Na+ has 10 electrons, Cl- has 18), so (111) is plainly there. In KCl, K+ and Cl- both carry 18 electrons — nearly identical clouds — so f_cation - f_anion collapses toward zero and (111) almost vanishes, making KCl mimic a simple cubic of half the cell edge. Same lattice, same positions; the intensities alone betrayed what atoms sat inside.