<a id="references"></a>

# References

These are the works the Theory pages cite, by author and year.

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**[AS64]** M. Abramowitz and I. A. Stegun, editors. *Handbook of Mathematical Functions*. Applied Mathematics Series 55. National Bureau of Standards, Washington, 1964.

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**[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](https://doi.org/10.1107/s0021889895005747).

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**[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.

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**[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](https://doi.org/10.1107/s002188989300891x).

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**[ABS19a]** L. C. Andrews, H. J. Bernstein, and N. K. Sauter. A space for lattice representation and clustering. *Acta Crystallographica Section A*, 75:593–599, 2019. [doi:10.1107/s2053273319002729](https://doi.org/10.1107/s2053273319002729).

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**[ABS19b]** L. C. Andrews, H. J. Bernstein, and N. K. Sauter. Selling reduction versus Niggli reduction for crystallographic lattices. *Acta Crystallographica Section A*, 75:115–120, 2019. [doi:10.1107/s2053273318015413](https://doi.org/10.1107/s2053273318015413).

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**[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](https://doi.org/10.1016/0022-4073(67)90057-x).

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**[Aza55]** L. V. Azaroff. Polarization correction for crystal-monochromatized X-radiation. *Acta Crystallographica*, 8:701–704, 1955. [doi:10.1107/s0365110x55002156](https://doi.org/10.1107/s0365110x55002156).

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**[Bea67]** J. A. Bearden. X-ray wavelengths. *Reviews of Modern Physics*, 39:78–124, 1967. [doi:10.1103/revmodphys.39.78](https://doi.org/10.1103/revmodphys.39.78).

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**[Ber68]** E. F. Bertaut. Representation analysis of magnetic structures. *Acta Crystallographica Section A*, 24:217–231, 1968. [doi:10.1107/s0567739468000306](https://doi.org/10.1107/s0567739468000306).

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**[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](https://doi.org/10.1107/s0021889804014876).

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**[BC72]** C. J. Bradley and A. P. Cracknell. *The Mathematical Theory of Symmetry in Solids*. Clarendon Press, Oxford, 1972.

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**[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](https://doi.org/10.1080/14786444508520918).

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**[Bro04]** P. J. Brown. Magnetic form factors. In E. Prince, editor, *International Tables for Crystallography Volume C: Mathematical, Physical and Chemical Tables*, chapter 4.4.5. Kluwer Academic Publishers, Dordrecht, 2004.

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**[Bun82]** H. J. Bunge. Mathematical Aids. In *Texture Analysis in Materials Science*. Butterworths, London, 1982. [doi:10.1016/b978-0-408-10642-9.50019-2](https://doi.org/10.1016/b978-0-408-10642-9.50019-2).

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**[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](https://doi.org/10.1107/s0021889890008391).

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**[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](https://doi.org/10.1016/0369-643x(58)90029-x).

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**[CSTH06]** B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch. ISODISPLACE: a web-based tool for exploring structural distortions. *Journal of Applied Crystallography*, 39:607–614, 2006. [doi:10.1107/s0021889806014075](https://doi.org/10.1107/s0021889806014075).

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**[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](https://doi.org/10.1107/s0021889805006096).

<a id="id87"></a>

**[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](https://doi.org/10.1107/s1600576718001784).

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**[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](https://doi.org/10.1107/s1600576718000183).

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**[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](https://doi.org/10.1063/1.1674266).

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**[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](https://doi.org/10.1107/s0567739481000600).

<a id="id81"></a>

**[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](https://doi.org/10.6028/jres.109.008).

<a id="id100"></a>

**[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](https://doi.org/10.1107/s002188986800508x).

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**[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.

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**[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](https://doi.org/10.1103/revmodphys.75.35).

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**[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](https://doi.org/10.1107/s0021889886089458).

<a id="id102"></a>

**[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](https://doi.org/10.1107/s0021889899007396).

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**[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](https://doi.org/10.2307/2332391).

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**[Eil03]** P. H. C. Eilers. A perfect smoother. *Analytical Chemistry*, 75:3631–3636, 2003. [doi:10.1021/ac034173t](https://doi.org/10.1021/ac034173t).

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**[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](https://doi.org/10.1214/ss/1038425655).

<a id="id32"></a>

**[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](https://doi.org/10.1107/s0021889894004218).

<a id="id43"></a>

**[FT88]** R. X. Fischer and E. Tillmanns. The equivalent isotropic displacement factor. *Acta Crystallographica Section C*, 44:775–776, 1988. [doi:10.1107/s0108270187012745](https://doi.org/10.1107/s0108270187012745).

<a id="id117"></a>

**[FB81]** M. A. Fischler and R. C. Bolles. Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. *Communications of the ACM*, 24:381–395, 1981. [doi:10.1145/358669.358692](https://doi.org/10.1145/358669.358692).

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**[FS88]** H. D. Flack and D. Schwarzenbach. On the use of least-squares restraints for origin fixing in polar space groups. *Acta Crystallographica Section A*, 44:499–506, 1988. [doi:10.1107/s0108767388002697](https://doi.org/10.1107/s0108767388002697).

<a id="id139"></a>

**[FD79]** A. J. Freeman and J. P. Desclaux. Dirac-Fock studies of some electronic properties of rare-earth ions. *Journal of Magnetism and Magnetic Materials*, 12:11–21, 1979. [doi:10.1016/0304-8853(79)90328-7](https://doi.org/10.1016/0304-8853(79)90328-7).

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**[FW33]** R. Frisch and F. V. Waugh. Partial time regressions as compared with individual trends. *Econometrica*, 1:387–401, 1933. [doi:10.2307/1907330](https://doi.org/10.2307/1907330).

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**[GTDLF+12]** S. V. Gallego, E. S. Tasci, G. De La Flor, J. M. Perez-Mato, and M. I. Aroyo. Magnetic symmetry in the Bilbao Crystallographic Server: a computer program to provide systematic absences of magnetic neutron diffraction. *Journal of Applied Crystallography*, 45:1236–1247, 2012. [doi:10.1107/s0021889812042185](https://doi.org/10.1107/s0021889812042185).

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**[Gil56]** J. J. Gilvarry. The Lindemann and Grüneisen laws. *Physical Review*, 102:308–316, 1956. [doi:10.1103/physrev.102.308](https://doi.org/10.1103/physrev.102.308).

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**[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](https://doi.org/10.1107/s0021889802008580).

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**[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](https://doi.org/10.1107/s010876730302186x).

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**[Hah02]** Th. Hahn, editor. *International Tables for Crystallography, Volume A: Space-Group Symmetry*. Kluwer Academic Publishers, Dordrecht, 2002.

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**[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](https://doi.org/10.1103/physrev.55.898).

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**[Ham65]** W. C. Hamilton. Significance tests on the crystallographic R factor. *Acta Crystallographica*, 18:502–510, 1965. [doi:10.1107/s0365110x65001081](https://doi.org/10.1107/s0365110x65001081).

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**[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](https://doi.org/10.1107/s0021889884011043).

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**[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](https://doi.org/10.6028/jres.049.044).

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**[Hig02]** N. J. Higham. *Accuracy and Stability of Numerical Algorithms*. SIAM, Philadelphia, 2nd edition, 2002.

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**[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](https://doi.org/10.1107/s0021889887086485).

<a id="id59"></a>

**[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](https://doi.org/10.1107/s0021889887086199).

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**[Hof02]** D. W. M. Hofmann. Fast estimation of crystal densities. *Acta Crystallographica Section B*, 58:489–493, 2002. [doi:10.1107/s0108768101021814](https://doi.org/10.1107/s0108768101021814).

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**[HS98]** C. J. Howard and H. T. Stokes. Group-theoretical analysis of octahedral tilting in perovskites. *Acta Crystallographica Section B*, 54:782–789, 1998. [doi:10.1107/s0108768198004200](https://doi.org/10.1107/s0108768198004200).

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**[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.

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**[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](https://doi.org/10.1016/0022-4073(82)90078-4).

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**[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](https://doi.org/10.1103/physreva.56.4554).

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**[INO91]** Yu. A. Izyumov, V. E. Naish, and R. P. Ozerov. *Neutron Diffraction of Magnetic Materials*. Consultants Bureau, New York, 1991.

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**[Jar93]** M. Järvinen. Application of symmetrized harmonics expansion to correction of the preferred orientation effect. *Journal of Applied Crystallography*, 26:525–531, 1993. [doi:10.1107/s0021889893001219](https://doi.org/10.1107/s0021889893001219).

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**[KL72]** C. J. E. Kempster and H. Lipson. A rapid method of assessing the number of molecules in the unit cell of an organic crystal. *Acta Crystallographica Section B*, 28:3674, 1972. [doi:10.1107/s056774087200857x](https://doi.org/10.1107/s056774087200857x).

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**[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](https://doi.org/10.1107/s0108767389010718).

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**[KA74]** H. P. Klug and L. E. Alexander. *X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials*. Wiley, New York, 2nd edition, 1974.

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**[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](https://doi.org/10.1107/s0567739476000636).

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**[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](https://doi.org/10.1107/s0021889878012844).

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**[LVD04]** A. C. Larson and R. B. Von Dreele. General Structure Analysis System (GSAS). Technical Report LAUR 86-748, Los Alamos National Laboratory, 2004.

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**[LH74]** C. L. Lawson and R. J. Hanson. *Solving Least Squares Problems*. Prentice-Hall, Englewood Cliffs, NJ, 1974.

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**[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](https://doi.org/10.1016/0025-5408(88)90019-0).

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**[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](https://doi.org/10.1107/s0021889882011959).

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**[LM99]** C.-J. Lin and J. J. Moré. Newton's method for large bound-constrained optimization problems. *SIAM Journal on Optimization*, 9:1100–1127, 1999. [doi:10.1137/s1052623498345075](https://doi.org/10.1137/s1052623498345075).

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**[LF71]** E. J. Lisher and J. B. Forsyth. Analytic Approximations to Form Factors. *Acta Crystallographica Section A*, 27:545–549, 1971. [doi:10.1107/s0567739471001190](https://doi.org/10.1107/s0567739471001190).

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**[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](https://doi.org/10.1107/s0021889872009483).

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**[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](https://doi.org/10.1080/01621459.1963.10480682).

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**[MSCL01]** I. C. Madsen, N. V. Y. Scarlett, L. M. D. Cranswick, and T. Lwin. Outcomes of the International Union of Crystallography Commission on Powder Diffraction round robin on quantitative phase analysis: samples 1a to 1h. *Journal of Applied Crystallography*, 34:409–426, 2001. [doi:10.1107/s0021889801007476](https://doi.org/10.1107/s0021889801007476).

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**[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](https://doi.org/10.1524/zkri.1932.81.1.285).

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**[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](https://doi.org/10.1107/s0021889898009856).

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**[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.

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**[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](https://doi.org/10.1107/s0021889875010801).

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**[Mor78]** J. J. Moré. The Levenberg–Marquardt algorithm: implementation and theory. In G. A. Watson, editor, *Numerical Analysis*, volume 630 of Lecture Notes in Mathematics, pages 105–116. Springer, Berlin, 1978. [doi:10.1007/bfb0067700](https://doi.org/10.1007/bfb0067700).

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**[MT91]** J. J. Moré and G. Toraldo. On the solution of large quadratic programming problems with bound constraints. *SIAM Journal on Optimization*, 1:93–113, 1991. [doi:10.1137/0801008](https://doi.org/10.1137/0801008).

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**[NW06]** J. Nocedal and S. J. Wright. *Numerical Optimization*. Springer, New York, 2nd edition, 2006.

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