Systematic Absence Rule Checker
Apply the reflection condition rules for primitive, body-centered, face-centered, and C-centered lattices to check if an (hkl) reflection is allowed.
✅ What is a Systematic Absence Rule Checker?
A systematic absence (also called a systematic extinction) is a diffraction reflection that is always missing from a crystal's pattern, no matter what atoms occupy the structure, because the lattice's centering symmetry forces the structure factor F(hkl) to exactly zero for an entire class of (h,k,l) combinations. This calculator applies the shortcut reflection condition for each of the four common lattice centering types, primitive (P), body-centered (I), face-centered (F), and C-centered (C), to instantly tell you whether a candidate reflection is allowed or forbidden.
Crystallographers and materials scientists use these rules constantly when indexing a powder X-ray diffraction pattern: before assigning any observed peak to a specific set of Miller indices, they first rule out every (hkl) combination that the suspected lattice centering forbids. A student working through a BCC iron problem set immediately knows to skip (100) and (300) but expects to see (110) and (200). A researcher confirming an unknown sample's centering type checks which peaks are present or absent against each rule until only one centering type is consistent with the observed pattern.
A common point of confusion is mixing up systematic absences (a purely geometric consequence of lattice centering, true for every atom in the structure) with accidental or structural extinctions (specific cancellations that depend on the actual atomic positions and scattering factors within a more complex basis). The rules on this page cover only the geometric, centering-driven kind, the same rules that emerge directly from the structure factor sum for simple monatomic bases.
This calculator applies the four standard reflection conditions and also shows a reference table of the first twelve common low-index reflections classified for whichever lattice type you select, useful for crystallography coursework and for quick sanity checks while indexing a diffraction pattern.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — BCC Iron's Strongest Peak
Body-Centered (I): (h,k,l) = (1,1,0)
Example 2 — FCC's Forbidden (110) Peak
Face-Centered (F): (h,k,l) = (1,1,0)
Example 3 — A C-Centered Allowed Reflection
C-Centered (C): (h,k,l) = (2,2,0)
❓ Frequently Asked Questions
🔗 Related Calculators
What is a systematic absence in crystallography?
A systematic absence (or systematic extinction) is a diffraction reflection that is always missing from a crystal's pattern because of its lattice centering, regardless of what atoms occupy the structure. It follows directly from the structure factor F(hkl) evaluating to exactly zero for that class of reflections.
What is the reflection condition for a body-centered (I) lattice?
For a body-centered (I) lattice, a reflection (h,k,l) is allowed only when h+k+l is even. For example, (110) is allowed (1+1+0=2, even) but (100) is forbidden (1+0+0=1, odd).
What is the reflection condition for a face-centered (F) lattice?
For a face-centered (F) lattice, a reflection is allowed only when h, k, and l are all even or all odd (unmixed parity). For example, (111) and (200) are both allowed, but (110) and (211) are forbidden because they mix even and odd indices.
What is the reflection condition for a C-centered lattice?
For a C-centered lattice, a reflection is allowed only when h+k is even, the l index does not matter because C-centering only adds an extra lattice point on the ab-face. For example, (110) and (220) are allowed, but (100) and (211) are forbidden.
Does a primitive (P) lattice have any systematic absences?
No. A primitive (P) lattice has no centering condition, every (hkl) combination is an allowed reflection. Systematic absences arise specifically from the extra lattice points introduced by body-, face-, or side-centering.
Why can't Miller indices all be zero?
The combination (0,0,0) does not correspond to any real set of crystal planes, there is nothing to diffract from. This calculator blocks that input with a clear error since it has no physical meaning as a reflection.
How is this different from the Structure Factor F(hkl) Calculator?
The Structure Factor F(hkl) Calculator computes the exact complex sum over the basis atom positions and scattering factors, which is where these reflection condition rules come from in the first place. This calculator applies the resulting shortcut rules directly, without redoing the complex sum, for a fast allowed/forbidden check once you already know the lattice centering.
Are these rules the same for X-ray and neutron diffraction?
Yes. Systematic absences from lattice centering are a purely geometric consequence of the crystal lattice itself, independent of whether X-rays, neutrons, or electrons are used as the scattering probe. The same P, I, F, and C reflection conditions apply to all three techniques.
What does 'unmixed parity' mean for the F-centering rule?
Unmixed parity means h, k, and l are either all even numbers or all odd numbers, never a mix of the two. (2,0,0) is unmixed (all even), (1,1,1) is unmixed (all odd), but (2,1,0) is mixed (two even, one odd) and therefore forbidden.
Can I use this calculator to index an unknown powder XRD pattern?
This calculator helps you check candidate (hkl) assignments once you suspect a lattice centering, ruling out any reflection that violates the centering's condition. Full pattern indexing from scratch typically also needs the d-spacing formula for the crystal system, see the d-Spacing from Miller Indices Calculator for that step.
Why does BCC iron show a strong (110) peak but no (100) peak?
Iron's BCC structure is body-centered (I), so the reflection condition h+k+l even applies. (110) satisfies it (1+1+0=2, even) and is strongly allowed, while (100) violates it (1+0+0=1, odd) and is systematically forbidden, completely absent from the diffraction pattern no matter how long you measure.