Calculator Inputs
Formula used
This calculator evaluates the complex structure factor for a chosen reciprocal‑lattice index (h,k,l):
Output values include Re(F), Im(F), |F|, phase = atan2(Im,Re), and intensity I = |F|². The simplified thermal term is a convenient modeling knob when detailed Debye‑Waller parameters are unavailable.
How to use this calculator
- Enter the reflection index h, k, and l for the plane of interest.
- Select an input mode: provide s directly, or provide θ and wavelength.
- Add one or more atoms using fractional coordinates within the unit cell.
- Set f to represent relative scattering strength; keep occupancy at 1 if unknown.
- Optional: enter B values to include a simple thermal reduction via exp(−B·s²).
- Press calculate to see Re, Im, |F|, phase, and intensity above the form.
- Use the export buttons to save results as CSV or PDF.
Example data table
Example inputs for a two‑atom basis (use as a quick test).
| h | k | l | θ (deg) | λ | Atom | x | y | z | f | occ | B |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0 | 1 | 15 | 1.5406 | A | 0 | 0 | 0 | 14 | 1 | 0.6 |
| 1 | 0 | 1 | 15 | 1.5406 | B | 0.5 | 0.5 | 0.5 | 8 | 1 | 0.8 |
Structure factor guide
1) Why the structure factor matters
Diffraction peak positions come from lattice geometry, but peak strength comes from the structure factor. It combines where atoms sit in the unit cell and how strongly they scatter. When contributions add in‑phase, the magnitude grows; when they cancel, reflections weaken or vanish.
2) Complex number output
The calculator returns Re(F) and Im(F), then computes |F| and phase. The phase is reported in degrees using atan2(Im,Re), which correctly handles all quadrants. Intensity is calculated as I = |F|², matching the common proportionality used for comparing reflection strengths.
3) Inputs tied to diffraction data
You enter h, k, l and a list of atoms with fractional coordinates (x,y,z). Fractions keep the model unitless and compatible with any cell size. The scattering factor f acts as a strength weight; for quick studies, heavier elements can be represented with larger f values.
4) Using s = sin(θ)/λ
Many datasets report angle and wavelength, while others work directly with s. This tool supports both. Because s increases with θ, high‑angle reflections are more sensitive to thermal motion and disorder. Keeping λ in a consistent unit ensures s remains physically meaningful.
5) Thermal motion with B
The thermal factor is modeled as DW = exp(−B·s²). If B is near zero, DW is close to 1 and amplitudes are unchanged. Larger B values reduce high‑s contributions first, mirroring how temperature and vibration soften sharp diffraction features in real measurements.
6) Reading the per‑atom table
Each row shows phase, DW, and the real/imaginary contributions. If two atoms have phases separated by about 180°, their contributions largely cancel. This table is useful for debugging symmetry, occupancy choices, and coordinate shifts before you run full refinements.
7) Practical use cases
Use the calculator to compare candidate basis models, test extinction conditions, or explore how moving an atom changes intensity. For example, shifting an atom by 0.5 along an axis flips its phase for odd indices, often turning reinforcement into cancellation.
8) Data quality tips and limits
Keep occupancy between 0 and 1 for physical sites, and avoid negative f values. The model uses a simplified isotropic thermal term and treats f as an input parameter, so it is best for analysis, education, and rapid comparisons rather than high‑precision crystallographic reporting.
FAQs
1) What do Re(F) and Im(F) represent?
They are the real and imaginary components of the summed complex phasors from all atoms. Together they define the magnitude and phase of the structure factor for the selected (h,k,l).
2) Why can intensity be zero even with atoms present?
Atoms can interfere destructively. If their phases cause near‑perfect cancellation, |F| becomes very small and the computed intensity |F|² approaches zero for that reflection.
3) Should x, y, z be in angstroms or fractions?
Use fractional coordinates between 0 and 1. Fractions reference the unit cell directly and avoid needing the cell length. This matches standard crystallographic conventions.
4) What is a good default for occupancy?
If you do not know site occupancy, start with 1. For mixed or partially filled sites, use values between 0 and 1 to scale that atom’s contribution proportionally.
5) How do I choose f values?
For quick comparisons, assign larger f to heavier or more strongly scattering atoms and smaller f to lighter atoms. If you have tabulated scattering factors, you can enter those directly.
6) What happens if I set B to zero?
DW becomes 1, so there is no thermal reduction and amplitudes depend only on occupancy and f. This is useful for idealized models or when thermal parameters are unavailable.
7) Why does phase matter if I only compare intensities?
Phase controls interference between atoms and between related reflections. Even when you focus on intensity, phase explains why certain reflections strengthen or disappear as coordinates change.