Systematic Absences

Learning Objectives

Section 3.6 derived the structure factor and showed that diffraction intensity is proportional to $|F(\mathbf{G})|^2$. That formula contains a small surprise: some reciprocal-lattice vectors $\mathbf{G}$, even though they satisfy Bragg's law geometrically, give $F(\mathbf{G}) = 0$. Their Bragg peaks are invisible. These "missing" reflections are called systematic absences, and they are not a bug — they are a fingerprint of the symmetry inside the cell.

By the end of this section you will be able to:

1. Explain, in one breath, why a reflection allowed by the Bravais lattice can still vanish — namely, that the basis atoms inside the cell add complex phasors that cancel.
2. Read off the centring rules for SC, BCC, FCC, and diamond by inspection, and recognise them as the visible consequence of basis symmetry in reciprocal space.
3. Identify systematic absences in a 2-D slice of reciprocal space and understand why iron's (100) peak is absent while its (110) peak is one of the brightest in the pattern.
4. Recognise glide planes and screw axes as the next layer of symmetry that produces its own absence patterns — one notch deeper than centring.
5. Compute $F(hkl)$ for any cubic basis with a 15-line Python script, and connect that calculation to what VASP writes to OUTCAR for structure-factor analysis.

One-line preview. Bragg's law tells you which directions the lattice can scatter into. The structure factor tells you which of those directions actually carry intensity once the basis atoms inside the cell add up their contributions. Systematic absences are the directions where that sum is exactly zero.

The Puzzle — Geometry Said Yes, Intensity Said No

Imagine you have a cube of α-iron, the BCC phase of iron, mounted on an X-ray diffractometer. You run a powder scan. The first six peaks come out at the angles $2\theta$ predicted by Bragg's law for the reflections (110), (200), (211), (220), (310), (222) — and only those reflections. The simple cubic peaks (100), (111), (210), (300) are not there. Not weak. Gone.

And yet, geometrically, those reciprocal-lattice vectors are perfectly legal. The Bravais lattice of BCC iron is a cubic lattice with $\mathbf{G}_{hkl} = h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3$ for every triple of integers — including (100). Bragg's law does not forbid (100). What is killing it?

This is the seed of the entire section. Systematic absences are cancellations inside the unit cell, written into the structure factor by the symmetry of the basis. They are not measurement noise. They are the signal — the part of the diffraction pattern that tells you whether your sample is BCC or simple cubic, FCC or BCC, NaCl or diamond, even when all four would give peaks at exactly the same Bragg angles.

Quick Recap — The Structure Factor

From Section 3.6, the amplitude scattered into the reflection labelled $\mathbf{G} = h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3$ is

where the sum runs over the $N_{\text{basis}}$ atoms of the basis, $(x_j, y_j, z_j)$ are their fractional coordinates inside the conventional cell, and $f_j$ is the atomic form factor — how strongly atom $j$ scatters. The Bragg intensity is

Two things to notice. First, the geometry of the Bravais lattice has already been spent in choosing $\mathbf{G}$; the structure factor only knows about what is inside one cell. Second, each basis atom contributes a unit phasor of magnitude $f_j$ at angle $\phi_j = 2\pi(h x_j + k y_j + l z_j)$. If those phasors happen to sum to zero, the reflection vanishes — no matter how loud Bragg's law is shouting.

Atoms as Complex Phasors

The structure-factor sum has the perfect mental picture: every basis atom is a unit arrow on the complex plane, pointing at an angle that depends on where the atom sits inside the cell, and which reflection we are asking about. Add the arrows tip-to-tail. Where the last arrow ends is $F(hkl)$.

If the path closes back to the origin, the reflection is forbidden.

If the path stops far from the origin, the reflection is bright.

That is the whole physics. Let's try it on the easiest case. For BCC there are two atoms per conventional cell, at fractional positions $(0,0,0)$ and $(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})$. Their phases for a reflection $(h,k,l)$ are $\phi_1 = 0$ and $\phi_2 = \pi(h+k+l)$. The sum collapses to a single neat formula:

The (100) peak of α-iron has $h+k+l=1$: forbidden. The (110) peak has $h+k+l=2$: allowed, $|F|^2=4f^2$. Mystery solved by a single sum of two complex numbers.

Interactive — Watch the Phasors Cancel

Time to feel it. The widget below draws every basis atom as a unit phasor on the complex plane. Pick a lattice (SC, BCC, FCC, Diamond), choose a reflection $(h,k,l)$, and watch the arrows add up tip-to-tail. The thick green or red arrow is the resultant $F$ from the origin to the end of the path.

Centring Rules — SC, BCC, FCC at a Glance

Repeat the phasor-sum exercise for each cubic Bravais lattice and the rules collapse into a small table. Memorise this — every working crystallographer can reproduce it from a back-of-envelope phasor argument.

Interactive — Allowed vs Forbidden Reflections

The next widget shows a slice of reciprocal space at fixed $l$: a 9×9 grid of integer reflections $(h,k,l)$. Bright cyan dots are allowed peaks (radius proportional to $|F|$); faint red crosses are systematically absent. Toggle between SC, BCC, FCC, Diamond and watch the forbidden zones snap into different patterns.

Three things to notice as you switch lattice. SC has no absences — every spot is bright. BCC kills exactly half of them on a checkerboard: $h+k+l$ odd disappears. FCC keeps only the "all-same-parity" reflections, leaving a much sparser pattern. Diamond starts from the FCC pattern and additionally extinguishes the even-only reflections with $h+k+l=4n+2$ — most famously, (200) and (222) vanish in silicon and diamond even though FCC admits them.

The reverse problem. Looking at a powder X-ray pattern, an experimentalist works backwards: which (h,k,l) peaks are present and which are missing? That pattern of presences and absences uniquely identifies the centring of the Bravais lattice. The widget above is, almost literally, what they are reading.

Why It Looks the Way It Does — Cells in 3D

The selection rules feel less mysterious once you can see, in real space, where the "extra" atom or atoms sit. Below: BCC on the left has one extra atom at the body centre; FCC on the right has three extra atoms on the face centres of one cell (six face atoms shown, each shared with a neighbour). Drag to rotate.

Compare these visually against the SC cell — just eight corner atoms and nothing in the middle. The BCC body-centre atom is what creates the $e^{i\pi(h+k+l)}$ phasor that cancels odd-sum reflections. The FCC face-centre atoms create three different phasors that cancel any reflection where $h, k, l$ have mixed parity. The symmetry of the basis is, quite literally, the reason for the absence pattern.

The Diamond Twist — When (222) Goes Missing

Diamond, silicon, and germanium all share the diamond-cubic structure: an FCC lattice with a two-atom basis, the second atom shifted by $(\tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4})$. That fractional quarter is what makes diamond stiff, semiconducting, and… diffractometrically peculiar. Its structure factor factorises cleanly:

The first factor enforces the FCC rule; the second factor adds an extra penalty for some of the reflections that FCC would otherwise allow. Walk through the cases:

Glide Planes and Screw Axes — Beyond Centring

Centring is the simplest source of systematic absences, but the space groups of Chapter 1 contain richer symmetries: glide planes (mirror + half-translation) and screw axes (rotation + half-translation). Each leaves its own footprint in reciprocal space.

The mechanism is identical to what we just saw. A glide plane forces atoms to come in pairs related by a half-translation; that half translation generates a phasor of $e^{i\pi}=-1$ for certain reflections and they cancel. A two-fold screw axis along $\mathbf{c}$ with translation $\tfrac{1}{2}\mathbf{c}$ kills all (00$l$) reflections with $l$ odd, in exactly the same way BCC kills the odd $h+k+l$.

The symmetry-to-absence dictionary the experimentalists use:

Computing F(hkl) from a VASP Cell

For a custom basis — say, a Mn-doped CdSe supercell where the symmetry-rule tables no longer apply directly — you compute $F(hkl)$ straight from the structure-factor sum. Fifteen lines of Python suffice.

Click any line in the panel below to see the values flowing through it. The script reproduces the SC, BCC, FCC, diamond rules from first principles and prints whether each reflection is allowed:

1import numpy as np
2
3# Fractional coordinates of the basis atoms in each
4# conventional cubic cell. (Bravais lattice + basis.)
5LATTICES = {
6    "SC":      [(0, 0, 0)],
7    "BCC":     [(0, 0, 0), (0.5, 0.5, 0.5)],
8    "FCC":     [(0, 0, 0), (0.5, 0.5, 0),
9                (0.5, 0, 0.5), (0, 0.5, 0.5)],
10    "Diamond": [(0, 0, 0),     (0.5, 0.5, 0),
11                (0.5, 0, 0.5), (0, 0.5, 0.5),
12                (0.25, 0.25, 0.25), (0.75, 0.75, 0.25),
13                (0.75, 0.25, 0.75), (0.25, 0.75, 0.75)],
14}
15
16def structure_factor(basis, hkl, f=1.0):
17    """F(hkl) = sum_j f_j * exp(2 pi i (h*x_j + k*y_j + l*z_j))"""
18    h, k, l = hkl
19    F = 0 + 0j
20    for (x, y, z) in basis:
21        phase = 2 * np.pi * (h * x + k * y + l * z)
22        F += f * np.exp(1j * phase)
23    return F
24
25reflections = [(1, 0, 0), (1, 1, 0), (1, 1, 1),
26               (2, 0, 0), (2, 2, 0), (2, 2, 2), (3, 1, 1)]
27
28for name, basis in LATTICES.items():
29    print(f"--- {name} ({len(basis)} atoms / cell) ---")
30    for hkl in reflections:
31        F = structure_factor(basis, hkl)
32        tag = "ALLOWED" if abs(F) > 1e-6 else "FORBIDDEN"
33        print(f"  F{hkl} = {F.real:+5.2f}{F.imag:+5.2f}i"
34              f"  |F|^2={abs(F)**2:6.2f}  [{tag}]")

Running the script prints (truncated):

1--- SC (1 atoms / cell) ---
2  F(1, 0, 0) = +1.00+0.00i  |F|^2=  1.00  [ALLOWED]
3  F(1, 1, 1) = +1.00+0.00i  |F|^2=  1.00  [ALLOWED]
4--- BCC (2 atoms / cell) ---
5  F(1, 0, 0) = +0.00+0.00i  |F|^2=  0.00  [FORBIDDEN]
6  F(1, 1, 0) = +2.00+0.00i  |F|^2=  4.00  [ALLOWED]
7  F(1, 1, 1) = +0.00+0.00i  |F|^2=  0.00  [FORBIDDEN]
8  F(2, 2, 2) = +2.00+0.00i  |F|^2=  4.00  [ALLOWED]
9--- FCC (4 atoms / cell) ---
10  F(1, 0, 0) = +0.00+0.00i  |F|^2=  0.00  [FORBIDDEN]
11  F(1, 1, 1) = +4.00+0.00i  |F|^2= 16.00  [ALLOWED]
12  F(2, 0, 0) = +4.00+0.00i  |F|^2= 16.00  [ALLOWED]
13--- Diamond (8 atoms / cell) ---
14  F(1, 1, 1) = +4.00-4.00i  |F|^2= 32.00  [ALLOWED]
15  F(2, 0, 0) = +0.00+0.00i  |F|^2=  0.00  [FORBIDDEN]
16  F(2, 2, 2) = +0.00+0.00i  |F|^2=  0.00  [FORBIDDEN]
17  F(3, 1, 1) = +4.00+4.00i  |F|^2= 32.00  [ALLOWED]

For a real VASP POSCAR, replace the hand-typed $(x, y, z)$ tuples with the fractional coordinates VASP wrote, and weight each phasor by the element-specific atomic form factor $f_j(\mathbf{G})$ (a known function of the reciprocal vector's magnitude — tabulated for every element in the International Tables, Vol. C). The forbidden / allowed verdict does not change for centring absences; only the magnitudes do.

Reverse-Reading a Powder Pattern

The most common practical use of systematic absences is identification. Imagine you have a powder pattern of an unknown cubic phase. The first seven Bragg peaks come out at $2\theta$ values that satisfy $d_{hkl} \propto 1/\sqrt{h^2+k^2+l^2}$ with sums-of-squares

versus a different sample where the sums are

These two sequences come from the same Bragg geometry but different centring. The first is BCC: the only allowed squared-sums are even (110, 200, 211, 220, 310, 222, 321 → 2, 4, 6, 8, 10, 12, 14). The second is FCC: only all-same-parity triplets, so 111, 220, 311, 400, 331, 422, 511 → 3, 8, 11, 16, 19, 24, 27. The list of sum-of-squares is the lattice fingerprint.

From peak positions to lattice type: compute $\sin^2\theta$ for each peak, divide by the first one, look at the integer sequence. BCC gives 1, 2, 3, 4, 5, 6, 7. FCC gives the spaced-out 3, 4, 8, 11, 12 (modulo a common factor). Every undergraduate XRD lab discovers this trick on day one.

Common Pitfalls

1. Confusing the conventional cell with the primitive cell. BCC has 1 atom per primitive cell, but 2 per conventional cubic cell. The structure factor formula uses whichever cell you wrote your $\mathbf{b}_i$ in. Mixing them produces phantom or missing absences. The phasor visualiser above uses the conventional cubic basis; translate carefully if your software does something else.
2. Treating a weak peak as forbidden. Different atomic form factors mean a near-extinguished reflection can have small but nonzero intensity. NaCl's "forbidden" (111) is only forbidden when $f_{\text{Na}} = f_{\text{Cl}}$; in reality $f_{\text{Cl}} > f_{\text{Na}}$ by a few electrons, so (111) appears as a weak — but real — peak. Systematic absences from centring are exact; absences arising only from equal form factors are not.
3. Forgetting that the Diamond rule is multiplicative, not additive. Diamond is FCC times an extra factor. A reflection forbidden by FCC (e.g. (210)) stays forbidden in diamond — even though $h+k+l=3$ is odd, which would suggest "allowed" on the diamond-only rule. Always apply both factors, or use the formula $F_{\text{FCC}} \cdot [1 + e^{i\pi(h+k+l)/2}]$.
4. Numerical zero ≠ true zero. When you compute $F$ with floating-point arithmetic, $e^{i\pi}$ is $-1 + 1.22\!\times\!10^{-16}\,i$, not exactly $-1$. Use a tolerance — typically $|F| < 10^{-6}$ — instead of strict equality when classifying allowed vs forbidden.
5. Assuming absences disappear in supercells. A supercell's reciprocal lattice is denser than the primitive one's, so more reflections seem to appear. They are mostly the original-cell reflections re-indexed; the underlying selection rules still hold. When VASP prints structure factors for a 2×2×2 cell, divide the indices by 2 to recover the conventional-cell labelling before checking against the centring rules.

Summary

• A systematic absence is a reflection allowed by the Bravais lattice but extinguished by destructive interference among the basis atoms inside the cell — $F(hkl) = 0$.
• Each basis atom contributes a unit phasor at angle $2\pi(h x_j + k y_j + l z_j)$ on the complex plane. The sum is the structure factor; if the phasors close into a loop, the reflection vanishes.
• The cubic centring rules: SC allows everything, BCC requires $h+k+l$ even, FCC requires all $h, k, l$ the same parity, diamond adds the further requirement that $h+k+l \neq 4n+2$.
• Glide planes and screw axes generate their own absence rules — each symmetry operation that involves a half-translation will, somewhere in reciprocal space, force a phasor to $e^{i\pi}=-1$ and cancel a class of reflections. The pattern of absences is the experimental signature of the space group.
• The 15-line Python script implements the structure-factor sum directly. With a real basis (CdSe, Mn:CdSe) and element-specific $f_j(\mathbf{G})$, it gives the same numbers that $\texttt{vaspkit}$ or $\texttt{pymatgen}$ would print after a VASP run.
• Reading absences backwards from a powder XRD pattern is how experimentalists identify the centring of an unknown cubic phase. The BCC sequence $h^2+k^2+l^2 \in \{2,4,6,8,10,12,14\}$ and the FCC sequence $\{3,8,11,16,19,24,27\}$ are the two most recognised fingerprints in materials science.

Coming next: Section 3.8 — k-Point Grids and Convergence — where we leave diffraction behind and return to electronic-structure mode. The same Brillouin zone we have been navigating now becomes the integration domain for total energies; how finely we sample it (the $8\times8\times8$ in your KPOINTS file) controls every number VASP prints.