Theory#
Theory states each equation rietx computes, the paper it comes from, and the code symbol that runs it.
Every displayed equation carries a source line naming that symbol, and the build fails when the name stops importing.
The forward model turns a structure, an instrument and a source into a calculated pattern.
Peak positions places each reflection on the 2θ axis.
Peak profiles gives the shape and width of one peak.
Intensities gives the area under each peak.
Intensity corrections gives each factor that multiplies a reflection’s intensity.
Microstructure reads the width coefficients as a size and a strain.
Background gives the background models and the bias a flexible one causes.
Estimation states what the fit minimises and how each esd is computed.
Parameterisation and constraints builds the parameter vector and ties its entries.
Representation analysis derives the magnetic structures a propagation vector allows.
Superspace groups describes a crystal modulated along one direction.
Indexing says what a list of line positions can tell about a lattice.
Search engines finds a cell from that list.
Reading a paper against its own numbers shows how three misprinted equations were caught.
References lists the works these pages cite.
Conventions#
Theory states every convention by its physics rather than by its letter, because Rietveld codes disagree on the letters. They swap the size and strain terms X and Y (GSAS and FullProf), sign March-Dollase \(r\) differently, normalise Stephens \(S_{HKL}\) in three independent ways, and print either a transmission \(A\) or its reciprocal \(A^*\). Wherever a number could be transferred from the literature or from another code, the convention warning sits beside the equation. Match the θ-law, the limit and the sign of the effect, not the symbol.
Scope#
Constant-wavelength X-ray and neutron powder data. Fundamental-parameters profiles and time-of-flight data are not implemented today. They are planned for v2, behind seams the forward model already carries. The one exception is Time of flight: the calibration relation a time-of-flight bank is read with, which puts a flight time on a d-spacing. No time-of-flight pattern is refined. Spherical-harmonics texture, (5.17), assumes cylindrical sample symmetry: one histogram cannot see a pole figure.
Symbols and units#
A quantity in Theory is in the unit rietx stores it in, and that is one of these
unless the equation says otherwise. Anything else is written in brackets at the
right of the equation that introduces it ([rad] on a derivation done in
radians, [barn], [fm]), or stated in the sentence beside it.
quantity |
unit |
|---|---|
every angle |
degrees; |
a position on the pattern axis |
deg 2θ |
a peak width |
deg 2θ, as FWHM (never a standard deviation, never an integral breadth) |
a Gaussian width coefficient |
deg² 2θ, because it is a variance |
a length |
Å: cell edges, d-spacings, wavelengths, crystallite sizes |
a reciprocal length |
Å⁻¹: \(s = \sin\theta/\lambda = 1/(2d)\), the reciprocal lattice vector length \(1/d = 2s\), and the scattering-chapter \(Q = 4\pi\sin\theta/\lambda\) |
a reciprocal length squared |
Å⁻²: the indexing-chapter \(Q = 1/d^2\), which is the |
a displacement parameter |
Ų, with \(B_{\mathrm{iso}} = 8\pi^2 U_{\mathrm{iso}}\) |
an observed or calculated intensity |
counts |
a reflection intensity |
counts·deg 2θ, an area, because every profile here is normalised to unit area |
a linear attenuation coefficient |
cm⁻¹ |
a distance in the diffractometer |
mm: goniometer radius, specimen displacement, capillary offsets |
a magnetic moment |
\(\mu_B\) |
Ratios of two of these are dimensionless and are not marked: a transmission coefficient, a mixing fraction, a weight fraction, a scale, an occupancy, a Miller index, a multiplicity.
Theory names quantities by their physics. The unit, default and typical range
of a named parameter (what phases.0.cell.a or instrument.profile.w holds) is
in the parameter reference, which is generated
from the package itself.