rietx manual

Release 1.4.0. The manual is in two parts.

Part 1, Using rietx, is the task-ordered guide to the package and its public API: install it, run a fit, understand what the fit did, read the report it hands back, and drive it from a program. It assumes you know powder diffraction and not this package.

Part 2, Theory, is the equations behind that machinery, numbered and cross-referenced, with the conventions that decide whether a number transfers between Rietveld codes.

How this manual was written

Claude (Opus 5 and Fable 5) wrote this manual from the source code, its docstrings and the project’s design records, under the direction of Yue Wu, who wrote and maintains rietx.

Documentation written that way can go stale or describe something the code does not do, so this one is tested like code. Every rietx name and parameter path in Part 1 resolves against the installed package, every example runs in the test suite, and every threshold quoted in Part 2 is injected from the live package when the manual builds. Those guards catch names and numbers. They cannot catch a paragraph that explains the wrong thing, so if a page misleads you, please open an issue and say which page.

Who this is written for

This manual is written for a person to read, but parts of the package are built for a program: the FitReport and its three layers, capabilities(), the JSON every answer type dumps to, the streaming event ladder, and the diagnostic codes. They are documented here because a person has to understand them to trust, debug or extend what a machine does with them. FitReport also answers the question anyone looking at a plot has: where is this model wrong, and how much of that will the package stand behind?

Notes for agents are marked like this:

For agents

Read the agent skill first, then come back here for the object model. rietx skill --path prints it, and The agent skill is the same text rendered. The skill says what to do in what order, what to check before believing a number, and which measured findings should change what you do. This manual describes the surface and does not restate it.

How to read this manual

Believe a docstring over this manual. Every physics function in rietx cites its reference (author, year, journal) in its docstring, and the long derivations live in the module docstrings. This manual organises that material into numbered equations in Part 2 and into a task order in Part 1; it does not replace it. If the two disagree, please report it.

Both parts are guarded against drifting from the code, by different mechanisms. In Part 2 every threshold and fenced constant is injected from the live package when the manual builds, so a renamed constant breaks the build; each displayed equation carries a Source line naming the symbol it was transcribed from, and a test imports every one of them. In Part 1 every dotted name and every parameter dot-path resolves against the live package, every fenced example either runs or states why it cannot, and the walkthroughs are scripts from examples/ included verbatim and executed by the test suite.

Names follow Python’s convention: CapWords for classes, lower case for functions and modules, UPPER_CASE for module-level constants. So Refinement, RefinementResult and FitReport are classes, refine, read_pattern and capabilities are functions, rietx.viz.compare is a module, and PLAN_INFO is a constant.

A method or a field is written under the class that defines it, not under the variable you would hold it in. RefinementResult.plot is the plot method of a RefinementResult, and in your own code that line reads result.plot(...). Written this way the name resolves, so the test suite can check every name in Part 1 against the live package.

Parameter dot-paths are the other dotted thing here, and they are never capitalised. phases.0.cell.a and instrument.profile.w are data: addresses into the parameter table, not attributes of a class.

The two parts set a fit statistic differently, on purpose. Part 2 sets it as mathematics (\(R_{wp}\), \(\chi^2_{\mathrm{red}}\), \(\Delta d/d\)), because there it is a symbol in an equation, defined by one. Part 1 writes the same statistics as plain text (Rwp, χ², GoF), because that is the word on the GUI’s own header and in a console line, and Part 1 is about driving the package rather than deriving it. In either part a name in code font is the field and not the statistic: Statistics.rwp is where the number lives.

Part 1: Using rietx

The chapters run in the order a first session with the package runs: install it, get one fit to the end, learn what the objects hold and how their parameters are addressed, run a staged refinement and control it, read the numbers and the report, go back to any state the fit passed through, then the specialised jobs (indexing, series, quantitative phase analysis, exports, the CLI) and the API a program drives. The closing chapter is the stability promise.

Part 1: Using rietx

• Installation
  • Requirements
  • Optional extras
  • The compiled kernels
  • Checking an install
  • Installing from source
  • Troubleshooting
  • Validation and accuracy claims
• A first refinement
  • If you have no data yet
  • The minimal call
  • refine or a Refinement session
  • Le Bail before Rietveld
  • Worked example: NAC on 11-BM
  • The RefinementResult object
  • Your own data
• Patterns, structures and instruments
  • Every refinable number is a Parameter
  • The pattern
  • The structure
  • The instrument
  • Calibrating an instrument once and reusing it
• The parameter table
  • A dot-path names one scalar
  • Reading the table
  • The four reasons a row is held
  • What the optimiser actually varies
  • Editing the table
  • Declared extra peaks
  • What a fit reports back
• How a refinement works
  • The parameter groups
  • Why the groups correlate
  • Constraining parameters to each other
  • Naming a variable of your own
  • Restraining a distance or an angle
  • Refinement plans
• Constraints and restraints
• Running a refinement
  • Two entry points, and where the settings live
  • Modes: what the intensities are allowed to do
  • Choosing a plan at run time
  • How hard each stage is converged
  • What a stage carries
  • Persisting a plan
  • Running one stage at a time
  • What each stage reports
  • Guards
  • Watching a run
  • Stopping a run
  • What the result records about the run
• Reading the numbers
  • Fit statistics
  • Structure agreement indices
  • The bonding geometry
  • The size and the strain, in physical units
  • How many observations there are
  • Which parameters the data could not separate
  • How finely the peaks were sampled
  • What the restraints did
  • What the statistics cannot tell you
  • What the absorption correction did
  • Diagnostics
  • Printing a result
  • Progress
• The fit report
  • Building a report
  • Layer 0: statistics independent of the model
  • Layer 1: attributing the misfit
  • Supporting evidence
  • Layer 2: suggested actions
  • Which parameter to free next
  • Reports at every stage
  • Comparing settings with rietx compare
• The refinement history
  • Switching the history on and off
  • What a node holds
  • Reading a tree
  • Naming a node
  • Going back, and forking
  • Recomputing a node
  • The tree’s identity
• Files and projects
  • Pattern files
  • Structures from CIF
  • Instrument profiles
  • Refinement files another program wrote
  • The .rex project directory
  • Run directories
  • The history log
  • Exports
• Indexing an unknown cell
  • Picking the peaks
  • Fitting peaks you name
  • Whether the list can be indexed
  • Running the search
  • The result object
  • The evidence view
  • From a candidate to a phase
  • The extinction symbol
  • Further reading
• Refining many patterns
  • A series is N refinements, chained
  • What comes back
  • The series fences
  • Telemetry, history and cancellation
  • Constraining a parameter across the series
  • A joint fit shares parameters instead
• How much of each phase
  • The table
  • What the fractions are fractions of
  • Microabsorption
  • Writing it out
• Exports and derived tables
  • The calculated pattern
  • Plotting the fit
  • The reflection list
  • Writing them out
  • Numbers with uncertainties
• The PowderLine recipe
  • Running one
  • Refusals
  • Unit conventions
  • The refinement plan
  • The four tables
  • The cross-engine fixtures
• The command line
  • rietx index: what is this cell?
  • rietx watch: a running refinement, live
  • rietx html: a saved result as a page
  • rietx compare: did that correction help?
  • rietx gui: the refinement GUI
  • rietx skill: the protocol, where your harness looks
• The GUI: a first fit
  • Install and start
  • Start with an example
  • What is on the screen
  • Run the fit
  • Read what happened
  • Then your own data
  • Where to go next
• The GUI panel by panel
  • The shape of the window
  • The header
  • The pattern
  • Parameters
  • Plan
  • Peaks
  • Model
  • Text
  • Series
  • Report
  • History
  • Build
• The GUI as a text document and a wire
  • The .rxt document
  • The keyboard
  • The command palette
  • The routes
• Calling rietx from a program
  • The Python API is the surface
  • The answer as JSON
  • Wrapping rietx in a tool call
  • capabilities()
  • Versioned contracts
  • Further reading: the agent skill
• The agent skill
  • Getting it
  • The protocol
  • Reference: 4/4b. The evidence behind the judging rules
  • Reference: 5. Read numbers, not pixels
  • Reference: 6. Abstention is a result. Do not convert it into a number
  • Reference: 7. The diagnostics are the channel for “your answer is wrong although Rwp is fine”
  • Reference: 7b-7f. Indexing: peak picking, the answer, the closed loop, the extinction screen
  • Reference: 7g. The project and recipe readers: what an imported file does not carry
  • Reference: 7h. The GSAS and GSAS-II readers and writers: what their files do not carry
  • Reference: 8. Twenty-four things that will surprise you, all measured
  • Reference: 9. The trajectory, and the history DAG as a search structure
  • Reference: 9b. Series: refine a ramp as a chain, and check it both ways
  • Reference: 9c. Many fits as one job: deciding an answer across fits
  • Reference: 9c. Many fits as one job: operating the run
  • Reference: 9d. A human is watching: run directories, and the window onto a long fit
  • Reference: The API index
• Compatibility
  • The preview position
  • The six version strings
  • The Python surface
  • Provisional by declaration
  • The JSON the package writes
  • The name and the formats are separate promises
  • What a default promises
• Glossary
  • Looking one up in code
  • Entry fields
  • Limitations
  • Parameters
  • Stage settings
  • Plan presets
  • Peak flags
  • Peak-list diagnostics
  • Peak origins
  • Reader options
  • Instrument preset fields
  • Indexing search settings

Part 2: Theory

Part 2 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 powder data. Fundamental-parameters profiles, neutron and time-of-flight data, and spherical-harmonics texture are not implemented today. They are planned for v2, behind seams the forward model already carries, and nothing in Part 2 describes them.

Symbols and units

A quantity in Part 2 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; 2θ is the scattering angle and θ half of it
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: \(k = \sin\theta/\lambda\) and \(Q = 4\pi\sin\theta/\lambda\)
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.

Part 2 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 Glossary, which is generated from the package itself.

Part 2: Theory

• 1. The forward model
  • 1.1. Three intensity models
  • 1.2. The residual row layout
  • 1.3. Discreteness frozen per stage
• 2. Peak positions
  • 2.1. Lattice metric and Bragg’s law
  • 2.2. The wavelength–cell degeneracy
  • 2.3. Aberration shifts
  • 2.4. Wavelength scales
• 3. Peak profiles
  • 3.1. The instrument ⊕ sample width split
  • 3.2. The strain width bound
  • 3.3. The size coefficient bound
  • 3.4. Thompson-Cox-Hastings pseudo-Voigt
  • 3.5. Declared extra peaks
  • 3.6. The true Voigt, via the Faddeeva function
  • 3.7. Finger-Cox-Jephcoat axial divergence
• 4. Intensities
  • 4.1. The species scattering factor
  • 4.2. The structure factor
  • 4.3. Debye-Waller factors and ADP representations
  • 4.4. Anomalous scattering and the powder average
• 5. Intensity corrections
  • 5.1. Lorentz-polarisation
  • 5.2. Attenuation coefficients
  • 5.3. Capillary (cylindrical) absorption
  • 5.4. Flat-plate absorption
  • 5.5. Surface roughness
  • 5.6. Secondary extinction
  • 5.7. Preferred orientation (March-Dollase)
  • 5.8. Quantitative phase analysis and microabsorption
• 6. Microstructure
  • 6.1. Isotropic size and strain
  • 6.2. Reading a width as a size
  • 6.3. Reading a width as a strain
  • 6.4. Stephens anisotropic strain
  • 6.5. Symmetry and the allowed coefficients
  • 6.6. The positivity cone, the seed and the guard
• 7. Background
  • 7.1. Additive background models
  • 7.2. Explicit humps
  • 7.3. Model-free estimation
  • 7.4. Choosing the flexibility
  • 7.5. Background flexibility and bias
• 8. Estimation
  • 8.1. Objective and weights
  • 8.2. Agreement statistics
  • 8.3. Structure agreement indices
  • 8.4. The effective observation count
  • 8.5. Esds and the Bérar-Lelann inflation
  • 8.6. Esds of derived quantities
  • 8.7. Staged strategy and series
  • 8.8. Solvers
  • 8.9. Convergence
  • 8.10. The fp64 floor
  • 8.11. From fit to report
  • 8.12. Which parameter to free next
• 9. Parameterisation and constraints
  • 9.1. From tree to vector
  • 9.2. Site-symmetry degrees of freedom
  • 9.3. Magnetic moment constraints
  • 9.4. Soft restraints
  • 9.5. Weighting the restraints
• 10. Indexing
  • 10.1. The quadratic form
  • 10.2. The symmetry-allowed subspace
  • 10.3. What a peak list supports
  • 10.4. Volume from one line
  • 10.5. Figures of merit, in both directions
  • 10.6. Lattice ambiguity
• 11. Search engines
  • 11.1. Searching in the metric components
  • 11.2. Successive dichotomy
  • 11.3. Index-heuristic trial and error
  • 11.4. Zone indexing, assessed and declined
  • 11.5. The obliquity bound
  • 11.6. The reflection ceiling
• 12. Reading a paper against its own numbers
  • 12.1. Rouse b₂: validate against the integral
  • 12.2. Two Coelho papers vs. their own tables
  • 12.3. The FCJ corner at \(s = h\)
  • 12.4. µR, µt, and what \(\Delta R_{wp}\) judges

Citing rietx

If rietx contributed to published work, cite [Wu and Gaultois, 2026]. The repository carries the same record as CITATION.cff, which reference managers and GitHub’s “cite this repository” button read directly.

Bibliography

[AS64]

M. Abramowitz and I. A. Stegun, editors. Handbook of Mathematical Functions. Applied Mathematics Series 55. National Bureau of Standards, Washington, 1964.

[ACG+95]

A. Altomare, G. Cascarano, C. Giacovazzo, A. Guagliardi, A. G. G. Moliterni, M. C. Burla, and G. Polidori. On the number of statistically independent observations in a powder diffraction pattern. Journal of Applied Crystallography, 28:738–744, 1995. doi:10.1107/s0021889895005747.

[ACMR19]

A. Altomare, C. Cuocci, A. Moliterni, and R. Rizzi. Indexing a powder diffraction pattern. In International Tables for Crystallography Volume H: Powder Diffraction, chapter 3.4, pages 270–281. Wiley, 2019.

[And94]

Yu. G. Andreev. The use of the serial-correlations concept in the figure-of-merit function for powder diffraction profile fitting. Journal of Applied Crystallography, 27:288, 1994. doi:10.1107/s002188989300891x.

[Arm67]

B. H. Armstrong. Spectrum line profiles: the Voigt function. Journal of Quantitative Spectroscopy and Radiative Transfer, 7:61–88, 1967. doi:10.1016/0022-4073(67)90057-x.

[Aza55]

L. V. Azaroff. Polarization correction for crystal-monochromatized X-radiation. Acta Crystallographica, 8:701–704, 1955. doi:10.1107/s0365110x55002156.

[BPAC15]

S.-J. Baek, A. Park, Y.-J. Ahn, and J. Choo. Baseline correction using asymmetrically reweighted penalized least squares smoothing. Analyst, 140:250–257, 2015. doi:10.1039/c4an01061b.

[Bea67]

J. A. Bearden. X-ray wavelengths. Reviews of Modern Physics, 39:78–124, 1967. doi:10.1103/revmodphys.39.78.

[BLBSZ04]

J. Bergmann, A. Le Bail, R. Shirley, and V. Zlokazov. Renewed interest in powder diffraction data indexing. Zeitschrift für Kristallographie, 219:783–790, 2004. doi:10.1524/zkri.219.12.783.55873.

[BH88]

D. L. Bish and S. A. Howard. Quantitative phase analysis using the Rietveld method. Journal of Applied Crystallography, 21:86–91, 1988. doi:10.1107/s0021889887009415.

[BL91a]

A. Boultif and D. Louër. Indexing of powder diffraction patterns for low-symmetry lattices by the successive dichotomy method. Journal of Applied Crystallography, 24:987–993, 1991. doi:10.1107/s0021889891006441.

[BL04]

A. Boultif and D. Louër. Powder pattern indexing with the dichotomy method. Journal of Applied Crystallography, 37:724–731, 2004. doi:10.1107/s0021889804014876.

[Bri45]

G. W. Brindley. The effect of grain or particle size on X-ray reflections from mixed powders and alloys. Philosophical Magazine, 36:347–369, 1945. doi:10.1080/14786444508520918.

[BL91b]

J.-F. Bérar and P. Lelann. E.s.d.'s and estimated probable error obtained in Rietveld refinements with local correlations. Journal of Applied Crystallography, 24:1–5, 1991. doi:10.1107/s0021889890008391.

[CPR58]

G. Caglioti, A. Paoletti, and F. P. Ricci. Choice of collimators for a crystal spectrometer for neutron diffraction. Nuclear Instruments, 3:223–228, 1958. doi:10.1016/0369-643x(58)90029-x.

[Coe05]

A. A. Coelho. A bound constrained conjugate gradient solution method as applied to crystallographic refinement problems. Journal of Applied Crystallography, 38:455–461, 2005. doi:10.1107/s0021889805006096.

[Coe18a]

A. A. Coelho. Optimum Levenberg–Marquardt constant determination for nonlinear least-squares. Journal of Applied Crystallography, 51:428–435, 2018. doi:10.1107/s1600576718001784.

[Coe18b]

A. A. Coelho. TOPAS and TOPAS-Academic: an optimization program integrating computer algebra and crystallographic objects written in C++. Journal of Applied Crystallography, 51:210–218, 2018. doi:10.1107/s1600576718000183.

[CL70]

D. T. Cromer and D. Liberman. Relativistic calculation of anomalous scattering factors for X rays. Journal of Chemical Physics, 53:1891–1898, 1970. doi:10.1063/1.1674266.

[CL81]

D. T. Cromer and D. Liberman. Anomalous dispersion calculations near to and on the long-wavelength side of an absorption edge. Acta Crystallographica Section A, 37:267–268, 1981. doi:10.1107/s0567739481000600.

[Dav04]

W. I. F. David. Powder diffraction: least-squares and beyond. Journal of Research of the National Institute of Standards and Technology, 109:107–123, 2004. doi:10.6028/jres.109.008.

[dW68]

P. M. de Wolff. A simplified criterion for the reliability of a powder pattern indexing. Journal of Applied Crystallography, 1:108–113, 1968. doi:10.1107/s002188986800508x.

[DK85]

R. D. Deslattes and E. G. Kessler. Experimental evaluation of inner-vacancy level energies for comparison with theory. In B. Crasemann, editor, Atomic Inner-Shell Physics, chapter 5, pages 181. Plenum Press, New York, 1985.

[DKI+03]

R. D. Deslattes, E. G. Kessler, P. Indelicato, L. de Billy, E. Lindroth, and J. Anton. X-ray transition energies: new approach to a comprehensive evaluation. Reviews of Modern Physics, 75:35–99, 2003. doi:10.1103/revmodphys.75.35.

[Dol86]

W. A. Dollase. Correction of intensities for preferred orientation in powder diffractometry: application of the March model. Journal of Applied Crystallography, 19:267–272, 1986. doi:10.1107/s0021889886089458.

[DWC99]

C. Dong, F. Wu, and H. Chen. Correction of zero shift in powder diffraction patterns using the reflection-pair method. Journal of Applied Crystallography, 32:850–853, 1999. doi:10.1107/s0021889899007396.

[DW50]

J. Durbin and G. S. Watson. Testing for serial correlation in least squares regression. I. Biometrika, 37:409–428, 1950. doi:10.2307/2332391.

[Eil03]

P. H. C. Eilers. A perfect smoother. Analytical Chemistry, 75:3631–3636, 2003. doi:10.1021/ac034173t.

[EM96]

P. H. C. Eilers and B. D. Marx. Flexible smoothing with B-splines and penalties. Statistical Science, 11:89–121, 1996. doi:10.1214/ss/1038425655.

[FCJ94]

L. W. Finger, D. E. Cox, and A. P. Jephcoat. A correction for powder diffraction peak asymmetry due to axial divergence. Journal of Applied Crystallography, 27:892–900, 1994. doi:10.1107/s0021889894004218.

[FT88]

R. X. Fischer and E. Tillmanns. The equivalent isotropic displacement factor. Acta Crystallographica Section C, 44:775–776, 1988. doi:10.1107/s0108270187012745.

[FW33]

R. Frisch and F. V. Waugh. Partial time regressions as compared with individual trends. Econometrica, 1:387–401, 1933. doi:10.2307/1907330.

[GKA02]

R. W. Grosse-Kunstleve and P. D. Adams. On the handling of atomic anisotropic displacement parameters. Journal of Applied Crystallography, 35:477–480, 2002. doi:10.1107/s0021889802008580.

[GKSA04]

R. W. Grosse-Kunstleve, N. K. Sauter, and P. D. Adams. Numerically stable algorithms for the computation of reduced unit cells. Acta Crystallographica Section A, 60:1–6, 2004. doi:10.1107/s010876730302186x.

[Hah02]

Th. Hahn, editor. International Tables for Crystallography, Volume A: Space-Group Symmetry. Kluwer Academic Publishers, Dordrecht, 2002.

[HJ39]

O. Halpern and M. H. Johnson. On the magnetic scattering of neutrons. Physical Review, 55:898–923, 1939. doi:10.1103/physrev.55.898.

[Ham65]

W. C. Hamilton. Significance tests on the crystallographic R factor. Acta Crystallographica, 18:502–510, 1965. doi:10.1107/s0365110x65001081.

[HTC84]

J. B. Hastings, W. Thomlinson, and D. E. Cox. Synchrotron X-ray powder diffraction. Journal of Applied Crystallography, 17:85–95, 1984. doi:10.1107/s0021889884011043.

[HS52]

M. R. Hestenes and E. Stiefel. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49:409–436, 1952. doi:10.6028/jres.049.044.

[Hig02]

N. J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia, 2nd edition, 2002.

[HF87]

R. J. Hill and H. D. Flack. The use of the Durbin–Watson d statistic in Rietveld analysis. Journal of Applied Crystallography, 20:356–361, 1987. doi:10.1107/s0021889887086485.

[HH87]

R. J. Hill and C. J. Howard. Quantitative phase analysis from neutron powder diffraction data using the Rietveld method. Journal of Applied Crystallography, 20:467–474, 1987. doi:10.1107/s0021889887086199.

[HS95]

J. H. Hubbell and S. M. Seltzer. Tables of X-ray mass attenuation coefficients and mass energy-absorption coefficients. Technical Report NISTIR 5632, National Institute of Standards and Technology, 1995.

[Hum82]

J. Humlíček. Optimized computation of the Voigt and complex probability functions. Journal of Quantitative Spectroscopy and Radiative Transfer, 27:437–444, 1982. doi:10.1016/0022-4073(82)90078-4.

[HFD+97]

G. Hölzer, M. Fritsch, M. Deutsch, J. Härtwig, and E. Förster. Kα₁,₂ and Kβ₁,₃ x-ray emission lines of the 3d transition metals. Physical Review A, 56:4554–4568, 1997. doi:10.1103/physreva.56.4554.

[KP90]

L. Kissel and R. H. Pratt. Corrections to tabulated anomalous-scattering factors. Acta Crystallographica Section A, 46:170–175, 1990. doi:10.1107/s0108767389010718.

[KA74]

H. P. Klug and L. E. Alexander. X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials. Wiley, New York, 2nd edition, 1974.

[KG76]

I. Křivý and B. Gruber. A unified algorithm for determining the reduced (Niggli) cell. Acta Crystallographica Section A, 32:297–298, 1976. doi:10.1107/s0567739476000636.

[LW78]

J. I. Langford and A. J. C. Wilson. Scherrer after sixty years: A survey and some new results in the determination of crystallite size. Journal of Applied Crystallography, 11:102–113, 1978. doi:10.1107/s0021889878012844.

[LVD04]

A. C. Larson and R. B. Von Dreele. General Structure Analysis System (GSAS). Technical Report LAUR 86-748, Los Alamos National Laboratory, 2004.

[LBDF88]

A. Le Bail, H. Duroy, and J. L. Fourquet. Ab-initio structure determination of LiSbWO₆ by X-ray powder diffraction. Materials Research Bulletin, 23:447–452, 1988. doi:10.1016/0025-5408(88)90019-0.

[LP82]

Y. Le Page. The derivation of the axes of the conventional unit cell from the dimensions of the Buerger-reduced cell. Journal of Applied Crystallography, 15:255–259, 1982. doi:10.1107/s0021889882011959.

[LL72]

D. Louër and M. Louër. Méthode d'essais et erreurs pour l'indexation automatique des diagrammes de poudre. Journal of Applied Crystallography, 5:271–275, 1972. doi:10.1107/s0021889872009483.

[Lov63]

M. C. Lovell. Seasonal adjustment of economic time series and multiple regression analysis. Journal of the American Statistical Association, 58:993–1010, 1963. doi:10.1080/01621459.1963.10480682.

[Mar32]

A. March. Mathematische Theorie der Regelung nach der Korngestalt bei affiner Deformation. Zeitschrift für Kristallographie, 81:285–297, 1932. doi:10.1524/zkri.1932.81.1.285.

[MVDC+99]

L. B. McCusker, R. B. Von Dreele, D. E. Cox, D. Louër, and P. Scardi. Rietveld refinement guidelines. Journal of Applied Crystallography, 32:36–50, 1999. doi:10.1107/s0021889898009856.

[MDGMH69]

W. H. McMaster, N. Kerr Del Grande, J. H. Mallett, and J. H. Hubbell. Compilation of X-ray cross sections. Technical Report UCRL-50174 Sec. II Rev. 1, Lawrence Radiation Laboratory, 1969.

[MS75]

A. D. Mighell and A. Santoro. Geometrical ambiguities in the indexing of powder patterns. Journal of Applied Crystallography, 8:372–374, 1975. doi:10.1107/s0021889875010801.

[NW06]

J. Nocedal and S. J. Wright. Numerical Optimization. Springer, New York, 2nd edition, 2006.

[OT13]

R. Oishi-Tomiyasu. Reversed de Wolff figure of merit and its application to powder indexing. Journal of Applied Crystallography, 46:1277–1282, 2013. doi:10.1107/s0021889813019833.

[OT14]

R. Oishi-Tomiyasu. Robust powder auto-indexing using many peaks. Journal of Applied Crystallography, 47:593–598, 2014. doi:10.1107/s1600576714000922.

[Paw81]

G. S. Pawley. Unit-cell refinement from powder diffraction scans. Journal of Applied Crystallography, 14:357–361, 1981. doi:10.1107/s0021889881009618.

[PMGT+15]

J. M. Perez-Mato, S. V. Gallego, E. S. Tasci, L. Elcoro, G. de la Flor, and M. I. Aroyo. Symmetry-based computational tools for magnetic crystallography. Annual Review of Materials Research, 45:217–248, 2015. doi:10.1146/annurev-matsci-070214-021008.

[PP66]

W. J. A. M. Peterse and J. H. Palm. The anisotropic temperature factor of atoms in special positions. Acta Crystallographica, 20:147–150, 1966. doi:10.1107/s0365110x66000318.

[Pet05]

V. K. Peterson. Lattice parameter measurement using Le Bail versus structural (Rietveld) refinement: A caution for complex, low symmetry systems. Powder Diffraction, 20:14–17, 2005. doi:10.1154/1.1810156.

[PHM93]

W. Pitschke, H. Hermann, and N. Mattern. The influence of surface roughness on diffracted X-ray intensities in Bragg–Brentano geometry and its effect on the structure determination by means of Rietveld analysis. Powder Diffraction, 8:74–83, 1993. doi:10.1017/s0885715600017875.

[Pol71]

E. Polak. Computational Methods in Optimization: A Unified Approach. Academic Press, New York, 1971.

[Pri04]

E. Prince, editor. International Tables for Crystallography, Volume C: Mathematical, Physical and Chemical Tables. Kluwer Academic Publishers, Dordrecht, 2004.

[Rao48]

C. R. Rao. Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Mathematical Proceedings of the Cambridge Philosophical Society, 44:50–57, 1948. doi:10.1017/s0305004100023987.

[Rie69]

H. M. Rietveld. A profile refinement method for nuclear and magnetic structures. Journal of Applied Crystallography, 2:65–71, 1969. doi:10.1107/s0021889869006558.

[RCYC70]

K. D. Rouse, M. J. Cooper, E. J. York, and A. Chakera. Absorption corrections for neutron diffraction. Acta Crystallographica Section A, 26:682–691, 1970. doi:10.1107/s0567739470001687.

[RCG+88]

C. G. Ryan, E. Clayton, W. L. Griffin, S. H. Sie, and D. R. Cousens. SNIP, a statistics-sensitive background treatment for the quantitative analysis of PIXE spectra in geoscience applications. Nuclear Instruments and Methods in Physics Research Section B, 34:396–402, 1988. doi:10.1016/0168-583x(88)90063-8.

[Sab85]

T. M. Sabine. Extinction in polycrystalline materials. Australian Journal of Physics, 38:507–518, 1985. doi:10.1071/ph850507.

[Sab88]

T. M. Sabine. A reconciliation of extinction theories. Acta Crystallographica Section A, 44:368–373, 1988. doi:10.1107/s0108767388000613.

[SVDJ88]

T. M. Sabine, R. B. Von Dreele, and J.-E. Jørgensen. Extinction in time-of-flight neutron powder diffractometry. Acta Crystallographica Section A, 44:374–379, 1988. doi:10.1107/s0108767388000625.

[Sch18]

P. Scherrer. Bestimmung der Größe und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1918:98–100, 1918.

[Sch78]

G. Schwarz. Estimating the dimension of a model. Annals of Statistics, 6:461–464, 1978. doi:10.1214/aos/1176344136.

[SAF+89]

D. Schwarzenbach, S. C. Abrahams, H. D. Flack, W. Gonschorek, Th. Hahn, K. Huml, R. E. Marsh, E. Prince, B. E. Robertson, J. S. Rollett, and A. J. C. Wilson. Statistical descriptors in crystallography. Acta Crystallographica Section A, 45:63–75, 1989. doi:10.1107/s0108767388009596.

[Sea92]

V. F. Sears. Neutron scattering lengths and cross sections. Neutron News, 3(3):26–37, 1992. doi:10.1080/10448639208218770.

[Shm01]

U. Shmueli, editor. International Tables for Crystallography, Volume B: Reciprocal Space. Kluwer Academic Publishers, Dordrecht, 2001.

[Smi77]

G. S. Smith. Estimating the unit-cell volume from one line in a powder diffraction pattern. Journal of Applied Crystallography, 10:252–255, 1977. doi:10.1107/s0021889877013478.

[SS79]

G. S. Smith and R. L. Snyder. F_N: a criterion for rating powder diffraction patterns and evaluating the reliability of powder-pattern indexing. Journal of Applied Crystallography, 12:60–65, 1979. doi:10.1107/s002188987901178x.

[Ste99]

P. W. Stephens. Phenomenological model of anisotropic peak broadening in powder diffraction. Journal of Applied Crystallography, 32:281–289, 1999. doi:10.1107/s0021889898006001.

[SE07]

G. W. Stinton and J. S. O. Evans. Parametric Rietveld refinement. Journal of Applied Crystallography, 40:87–95, 2007. doi:10.1107/s0021889806043275.

[SW44]

A. R. Stokes and A. J. C. Wilson. The diffraction of X rays by distorted crystal aggregates – I. Proceedings of the Physical Society, 56(3):174–181, 1944. doi:10.1088/0959-5309/56/3/303.

[Suo72]

P. Suortti. Effects of porosity and surface roughness on the X-ray intensity reflected from a powder specimen. Journal of Applied Crystallography, 5:325–331, 1972. doi:10.1107/s0021889872009707.

[TM91]

J. C. Taylor and C. E. Matulis. Absorption contrast effects in the quantitative XRD analysis of powders by full multiphase profile refinement. Journal of Applied Crystallography, 24:14–17, 1991. doi:10.1107/s002188989000841x.

[TCH87]

P. Thompson, D. E. Cox, and J. B. Hastings. Rietveld refinement of Debye–Scherrer synchrotron X-ray data from Al₂O₃. Journal of Applied Crystallography, 20:79–83, 1987. doi:10.1107/s0021889887087090.

[Tob06]

B. H. Toby. R factors in Rietveld analysis: how good is good enough? Powder Diffraction, 21:67–70, 2006. doi:10.1154/1.2179804.

[Tob24]

B. H. Toby. A simple solution to the Rietveld refinement recipe problem. Journal of Applied Crystallography, 57:175–180, 2024. doi:10.1107/s1600576723011032.

[TVD13]

B. H. Toby and R. B. Von Dreele. GSAS-II: the genesis of a modern open-source all purpose crystallography software package. Journal of Applied Crystallography, 46:544–549, 2013. doi:10.1107/s0021889813003531.

[TST24]

A. Togo, K. Shinohara, and I. Tanaka. Spglib: a software library for crystal symmetry search. Science and Technology of Advanced Materials: Methods, 4:2384822, 2024. doi:10.1080/27660400.2024.2384822.

[TBB+96]

K. N. Trueblood, H.-B. Bürgi, H. Burzlaff, J. D. Dunitz, C. M. Gramaccioli, H. H. Schulz, U. Shmueli, and S. C. Abrahams. Atomic displacement parameter nomenclature. Acta Crystallographica Section A, 52:770–781, 1996. doi:10.1107/s0108767396005697.

[Vis69]

J. W. Visser. A fully automatic program for finding the unit cell from powder data. Journal of Applied Crystallography, 2:89–95, 1969. doi:10.1107/s0021889869006649.

[VD97]

R. B. Von Dreele. Quantitative texture analysis by Rietveld refinement. Journal of Applied Crystallography, 30:517–525, 1997. doi:10.1107/s0021889897005918.

[WK95]

D. Waasmaier and A. Kirfel. New analytical scattering-factor functions for free atoms and ions. Acta Crystallographica Section A, 51:416–431, 1995. doi:10.1107/s0108767394013292.

[Was63]

J. Waser. Least-squares refinement with subsidiary conditions. Acta Crystallographica, 16:1091–1094, 1963. doi:10.1107/s0365110x63002929.

[Wat94]

D. Watkin. The control of difficult refinements. Acta Crystallographica Section A, 50:411–437, 1994. doi:10.1107/s0108767393012784.

[Wei94]

J. A. C. Weideman. Computation of the complex error function. SIAM Journal on Numerical Analysis, 31:1497–1518, 1994. doi:10.1137/0731077.

[Wer64]

P.-E. Werner. Trial-and-error computer methods for the indexing of unknown powder patterns. Zeitschrift für Kristallographie, 120:375–387, 1964. doi:10.1524/zkri.1964.120.16.375.

[WEW85]

P.-E. Werner, L. Eriksson, and M. Westdahl. TREOR, a semi-exhaustive trial-and-error powder indexing program for all symmetries. Journal of Applied Crystallography, 18:367–370, 1985. doi:10.1107/s0021889885010512.

[WBWB74]

G. K. Wertheim, M. A. Butler, K. W. West, and D. N. E. Buchanan. Determination of the Gaussian and Lorentzian content of experimental line shapes. Review of Scientific Instruments, 45:1369–1371, 1974. doi:10.1063/1.1686503.

[Wil63]

A. J. C. Wilson. Mathematical Theory of X-ray Powder Diffractometry. Philips Technical Library, Eindhoven, 1963.

[WG26]

Yue Wu and Michael W. Gaultois. rietx: Python-API-first analysis and Rietveld refinement of powder diffraction data. 2026. URL: https://github.com/yue-here/rietx.

[ZA11]

M. R. Zaghloul and A. N. Ali. Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Transactions on Mathematical Software, 38(2):15, 2011. doi:10.1145/2049673.2049679.

[NIST05]

NIST. X-ray Transition Energies Database, Standard Reference Database 128. 2005. https://physics.nist.gov/PhysRefData/XrayTrans/.