References#

Literature citations used throughout ad_hoc_diffractometer. Geometries, algorithms, and conventions are traced to their primary sources.

Diffractometer geometries#

Arndt & Willis (1966) : U.W. Arndt and B.T.M. Willis. Single Crystal Diffractometry. Cambridge University Press (1966); online edition 21 May 2010, ISBN 9780511735622. DOI: 10.1017/CBO9780511735622

Early monograph on the mechanical design and use of single-crystal diffractometers, including the chapter Design of diffractometers.

Busing & Levy (1967) : W.R. Busing and H.A. Levy. Angle calculations for 3- and 4-circle X-ray and neutron diffractometers. Acta Crystallographica 22, 457–464 (1967). DOI: 10.1107/S0365110X67000970

Foundational reference for the four-circle geometry, B matrix, U matrix, and UB matrix. Defines the orientation refinement least-squares procedure. Used by: fourcv — Eulerian Four-Circle (vertical), fourch — Eulerian Four-Circle (horizontal), kappa4cv — Kappa Four-Circle (vertical), kappa4ch — Kappa Four-Circle (horizontal).

Wyckoff (1985) : H.W. Wyckoff. Diffractometry. Methods in Enzymology 114, 330–386 (1985). DOI: 10.1016/0076-6879(85)14026-7

Figure 2(b) on p. 334 — the canonical Enraf-Nonius kappa diffractometer with explicit X/Y/Z coordinate axes and κ-rotation NEG/POS sense; the schematic cited by ITC Vol. C §2.2.6 as the reference for the kappa goniostat geometry. Used by: kappa4ch — Kappa Four-Circle (horizontal).

Bloch (1985) : J.M. Bloch. Angle and distance calculations for X-ray diffraction with the Z-axis geometry. Journal of Applied Crystallography 18, 33–36 (1985). DOI: 10.1107/S0021889885009858

Defines the Z-axis diffractometer geometry. Used by: zaxis — Z-Axis Four-Circle (Surface).

Vlieg et al. (1987) : E. Vlieg, A.E.M.J. Fischer, J.F. van der Veen, B.N. Dev, and G. Materlik. Surface X-ray diffraction: a study of relaxation in the Cu(110) system. Journal of Applied Crystallography 20, 330–337 (1987). DOI: 10.1107/S0021889887087266

Defines the five-circle geometry. Used by: fivec — Eulerian Five-Circle (Vlieg et al. 1987).

Lohmeier & Vlieg (1993) : M. Lohmeier and E. Vlieg. Angle calculations for a six-circle surface X-ray diffractometer. Journal of Applied Crystallography 26, 706–716 (1993). DOI: 10.1107/S0021889893006198

Defines the six-circle surface diffractometer geometry. Used by: sixc — Eulerian Six-Circle, Surface (Lohmeier & Vlieg 1993).

Evans-Lutterodt & Tang (1995) : K.W. Evans-Lutterodt and M.-T. Tang. Angle calculations for a ‘2+2’ surface X-ray diffractometer. Journal of Applied Crystallography 28, 318–326 (1995). DOI: 10.1107/S0021889895001063

Defines the S2D2 (2+2) diffractometer geometry. Used by: s2d2 — S2D2 General-Inclination Four-Circle.

Paciorek, Meyer & Chapuis (1999) : W.A. Paciorek, M. Meyer, and G. Chapuis. On the geometry of a modern imaging diffractometer. Acta Crystallographica A 55, 543–557 (1999). DOI: 10.1107/S0108767399000951

Mathematical description of a four-circle κ-goniometer (KM4CCD) with explicit coordinate frame (Fig. 1) and the κ-axis tilt parameters χ, α (Fig. 2). Cited by Sønsteby et al. (2013) as the basic mathematics of the six-axis κ instrument.

You (1999) : H. You. Angle calculations for a ‘4S+2D’ six-circle diffractometer. Journal of Applied Crystallography 32, 614–623 (1999). DOI: 10.1107/S0021889899001223

Defines the psic (4S+2D) six-circle geometry; axis sign conventions (mixed handedness); ψ angle definitions (eqs. 10–11). Used by: psic — Eulerian Six-Circle, 4S+2D (You 1999), kappa6c — Kappa Six-Circle.

ITC Vol. C §2.2.6 (2006) : International Tables for Crystallography, Volume C, Section 2.2.6, p. 36. Single-crystal X-ray techniques. DOI: 10.1107/97809553602060000577

Confirms the kappa 50° tilt convention; cites Wyckoff (1985, p. 334) for the schematic picture of the kappa goniostat. §2.2.6.2 documents the standard sign convention (right-handed for ω/χ/φ; left-handed for 2θ in Hamilton’s choice) — note that the presets shipped here follow Walko’s left-handed convention for ω/φ/2θ; see the Concepts page for the handedness discussion. Used by: kappa4cv — Kappa Four-Circle (vertical), kappa4ch — Kappa Four-Circle (horizontal), kappa6c — Kappa Six-Circle.

Thorkildsen, Larsen & Beukes (2006) : G. Thorkildsen, H.B. Larsen, and J.A. Beukes. Angle calculations for a three-circle goniostat. Journal of Applied Crystallography 39, 151–157 (2006). DOI: 10.1107/S0021889805041877

Vector formulation of the diffractometer angle-calculation problem, applicable to arbitrary goniostat designs (Eulerian, κ, and generalisations). Table 1 + equation (3) give the canonical κ-goniostat axes used by kappa4cv, kappa4ch, and kappa6c; §3 last paragraph explicitly notes the extension to additional rotation axes. Used by: kappa4cv — Kappa Four-Circle (vertical), kappa4ch — Kappa Four-Circle (horizontal), kappa6c — Kappa Six-Circle.

Sønsteby et al. (2013) : H.H. Sønsteby, D. Chernyshov, M. Getz, O. Nilsen, and H. Fjellvåg. On the application of a single-crystal κ-diffractometer and a CCD area detector for studies of thin films. Journal of Synchrotron Radiation 20, 644–647 (2013). DOI: 10.1107/S0909049513009102

Six-axis κ-diffractometer (KUMA6) at Swiss-Norwegian Beam Lines BM01A at the European Synchrotron Radiation Facility. Cites Paciorek (1999) for the basic mathematics and Thorkildsen et al. (2006) for the angular-calculation method used by the CrysAlis control software. The reference instrument for the kappa6c preset. Used by: kappa6c — Kappa Six-Circle.

Walko (2016) : D.A. Walko. Multicircle Diffractometry Methods. Reference Module in Materials Science and Materials Engineering, Elsevier (2016). DOI: 10.1016/B978-0-12-803581-8.01215-7

Comprehensive survey of diffractometer geometry designations (S/D system); kappa convention; zaxis, s2d2, fivec geometries. Figure 3 shows the kappa diffractometer in vertical-scattering layout (the kappa4cv reference). Used throughout the geometry factory descriptions.

Physical constants#

CODATA 2022 / 2019 SI : NIST CODATA 2022 recommended values. BIPM SI Brochure, 9th edition (2019).

\(hc = 12.398\,419\,843\,320\,026\,\text{keV·Å}\) — exact (h and c are defined constants since the 2019 SI redefinition).
\(h^2/(2m_n) = 81.804\,210\,235\,2\,\text{meV·Å}^2\) — from CODATA 2022 neutron mass \(m_n\).

See HC_KEV_ANGSTROM and NEUTRON_MEV_ANGSTROM2.

Numerical methods#

Nelder & Mead (1965) : J.A. Nelder and R. Mead. A simplex method for function minimization. The Computer Journal 7(4), 308–313 (1965). DOI: 10.1093/comjnl/7.4.308

Derivative-free simplex optimisation algorithm used by refine_lattice_simplex().

APS alignment session#

Walko (2020) : D.A. Walko, private communication (December 2020). Crystal alignment session at APS beamline 7-ID-C using a sapphire sample. Documented in Align a Four-Circle Diffractometer (fourcv).