kappa4ch — Kappa Four-Circle (horizontal)

Four-circle kappa diffractometer, horizontal scattering plane. The chi circle is replaced by a kappa arm tilted at α = 50° from the vertical axis toward the longitudinal axis (along the incident beam), lying entirely in the vertical–longitudinal plane. Laboratory convention.

Walko (2016) designation: S3D1 (kappa)

Coordinate basis: Busing & Levy (BASIS_BL): transverse=+x, longitudinal=+y, vertical=+z.

Quick start

import ad_hoc_diffractometer as ahd

g = ahd.make_geometry("kappa4ch")
g.wavelength = 1.0  # Å
print(g.summary())

Demo geometry definition

This geometry is defined by the kappa4ch — Kappa Four-Circle (horizontal) factory function — see the source for the complete stage and mode configuration.

Stage layout

Static fallback (click to expand if the interactive figure above is blank)

Sample stages (base first):

Stage	Axis	Handedness	Parent
komega	vertical (−z in BL)	left-handed	base
kappa	(cos α)·vertical + (sin α)·longitudinal = (cos α)·ẑ + (sin α)·ŷ in BL (α = 50°)	right-handed	komega
kphi	vertical (−z in BL)	left-handed	kappa

Detector stages (base first):

Stage	Axis	Handedness	Parent
ttheta	vertical (−z in BL)	left-handed	base

How the kappa axis is defined

The kappa rotation axis is inclined by α from the omega axis, lying in the plane that contains both omega and the equivalent-Eulerian chi axis, and tilted from omega toward that chi direction (Walko 2016 §4.1; ITC Vol. C §2.2.6.2: “the κ axis is inclined at 50° to the ω axis”).

For kappa4ch the omega axis lies along the vertical line and the equivalent-Eulerian chi axis lies along the longitudinal direction (along the X-ray beam). The kappa arm therefore lies in the vertical–longitudinal plane, tilted by α from the vertical line toward +L:

\[ \hat{n}_{\kappa} \;=\; \cos\alpha \cdot \hat{V} \;+\; \sin\alpha \cdot \hat{L}. \]

In the BL basis (V=+ẑ, L=+ŷ) at α = 50° this is

\[ \hat{n}_{\kappa} \;=\; \cos 50° \cdot \hat{z} \;+\; \sin 50° \cdot \hat{y} \;=\; (0,\, 0.766,\, 0.643). \]

The transverse component is exactly zero: the kappa arm rises upward and toward the X-ray source/sample direction in the plane that contains the vertical omega axis and the beam.

Published reference figure. This convention matches Wyckoff (1985) Figure 2(b) on p. 334 — the canonical Enraf-Nonius kappa diffractometer in the horizontal-scattering laboratory layout. ITC Vol. C §2.2.6 cites this figure as the schematic for the kappa goniostat. Walko (2016) Figure 3 shows the same instrument rotated 90° onto the vertical scattering plane (the kappa4cv convention).

The omega axis sense (komega = −vertical = left-handed rotation about +V) follows Walko’s left-handed sign convention. ITC Vol. C §2.2.6.2 prefers a right-handed sign convention for omega/chi/phi; the two are equivalent up to motor-angle sign flips.

Virtual Eulerian angles omega, chi, phi are mapped to / from the real motors via the geometry-aware decomposition in eulerian_to_kappa_axes() and kappa_to_eulerian_axes().

Diffraction modes

Each mode is a ConstraintSet of 1 constraint (N − 3 = 1 for N = 4 DOF). Identical mode set to kappa4cv — Kappa Four-Circle (vertical). See Switch Diffraction Modes for usage and Work with Constraints and Diffraction Modes for changing constraint values at run time.

bisecting (default)

VirtualBisectConstraint: omega_virtual = ttheta / 2 enforced on the virtual Eulerian omega pseudoangle. Solved via the geometry-aware eulerian_to_kappa_axes() decomposition.

Computed	komega, kappa, kphi, ttheta
Constant during forward()	—

fixed_kphi

SampleConstraint: kphi held at declared value (default 0°) — real stage, no kappa inversion needed.

Computed	komega, kappa, ttheta
Constant during forward()	kphi

fixed_omega

Fix virtual Eulerian omega at declared value (default 0°) — see kappa4cv — Kappa Four-Circle (vertical) for details.

Computed	komega, kappa, kphi, ttheta
Constant during forward()	omega (virtual)

fixed_chi

Fix virtual Eulerian chi at declared value (default 90°).

Computed	komega, kappa, kphi, ttheta
Constant during forward()	chi (virtual)

fixed_phi

Fix virtual Eulerian phi at declared value (default 0°).

Computed	komega, kappa, kphi, ttheta
Constant during forward()	phi (virtual)

fixed_psi

ReferenceConstraint: azimuthal angle ψ validation filter. Set g.azimuthal_reference = (h, k, l) before calling forward(). Returns bisecting solutions only when the natural ψ for (h,k,l) matches the stored target. See Surface Geometry and the Reference Vector.

Extras (input)	n̂ (reference vector), ψ (target azimuth, degrees)
Extras (output)	psi (computed azimuth)

API reference

• kappa4ch — Kappa Four-Circle (horizontal)

• AdHocDiffractometer

• ConstraintSet

• BisectConstraint

• SampleConstraint

• ReferenceConstraint

• EwaldSphereViolation

• ConstraintViolation

References

• H.W. Wyckoff, Diffractometry, in Methods in Enzymology 114, 330–386 (1985), Figure 2(b) on p. 334 (canonical Enraf-Nonius kappa diffractometer). DOI: 10.1016/0076-6879(85)14026-7.

• D.A. Walko, Multicircle Diffractometry Methods, in Reference Module in Materials Science and Materials Engineering (Elsevier, 2016), §4.1, equation [16]. DOI: 10.1016/B978-0-12-803581-8.01215-7.

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

• International Tables for Crystallography, Vol. C, §2.2.6 (2006), p. 36 (α = 50° convention; cites Wyckoff 1985 for the schematic picture). DOI: 10.1107/97809553602060000577.