kappa6c — Kappa Six-Circle

Six-circle kappa diffractometer with psic-style outer axes (mu, nu). The inner sample axes (komega, kappa, and kphi) replace the Eulerian chi circle. Transverse detector, vertical scattering plane.

Coordinate basis: You (1999) (BASIS_YOU): vertical=+x, longitudinal=+y, transverse=+z.

Quick start

import ad_hoc_diffractometer as ahd

g = ahd.make_geometry("kappa6c")
g.wavelength = 1.0  # Å
print(g.summary())

Demo geometry definition

This geometry is defined by the kappa6c — Kappa Six-Circle 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
mu	vertical (+x in You)	right-handed	base
komega	transverse (−z in You)	left-handed	mu
kappa	(cos α)·transverse + (sin α)·vertical = (cos α)·ẑ + (sin α)·x̂ in You (α = 50°)	right-handed	komega
kphi	transverse (−z in You)	left-handed	kappa

Detector stages (base first):

Stage	Axis	Handedness	Parent
nu	vertical (+x in You)	right-handed	base
delta	transverse (−z in You)	left-handed	nu

How the kappa axis is defined

kappa6c is structurally the kappa4cv sample stack mounted on top of the You (1999) mu and nu outer axes. The kappa rotation axis is therefore 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 kappa6c the omega axis lies along the transverse line and the equivalent-Eulerian chi axis lies along the vertical direction. The kappa arm therefore lies in the transverse– vertical plane, tilted by α from the transverse line toward +V:

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

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

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

The longitudinal component is exactly zero: at mu = 0 the kappa arm rises upward and outward in the vertical plane perpendicular to the incident beam, exactly as in kappa4cv.

Published references. The kappa-arm orientation matches Walko (2016) Figure 3 and the canonical κ-goniostat axes in Thorkildsen et al. (2006) Table 1 and equation (3). Sønsteby et al. (2013) describes the six-axis κ-diffractometer (“KUMA6”) at SNBL/ESRF BM01A whose angular calculations follow Thorkildsen (2006); §3 of Thorkildsen (2006) explicitly notes that “additional rotation axes in the goniostat design are easily incorporated in the formalism”. No single published figure depicts the kappa6c-on-mu-nu composite directly; this preset constructs it from the kappa4cv sample stack and the psic outer axes.

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().

Bisect pairs:

• Vertical: komega (transverse) ↔ delta (transverse) → komega = delta/2

• Horizontal: mu (vertical) ↔ nu (vertical) → mu = nu/2

Diffraction modes

Each mode is a ConstraintSet of 3 constraints (N − 3 = 3 for N = 6 DOF). See Switch Diffraction Modes for usage and Work with Constraints and Diffraction Modes for changing constraint values at run time.

bisecting_vertical (default)

VirtualBisectConstraint + SampleConstraint + DetectorConstraint: omega_virtual = delta / 2, mu = 0, nu = 0. The virtual-bisect condition is on the virtual Eulerian omega pseudoangle and is solved via the geometry-aware eulerian_to_kappa_axes() decomposition. Vertical scattering plane (psic-style).

Computed	komega, kappa, kphi, delta
Constant during forward()	mu = 0, nu = 0

fixed_kphi

SampleConstraint: kphi held at declared value (default 0°), mu = 0, nu = 0.

Computed	komega, kappa, delta
Constant during forward()	kphi, mu = 0, nu = 0

fixed_mu

SampleConstraint + BisectConstraint + DetectorConstraint: mu held at declared value (default 0°), komega = delta/2, nu = 0.

Computed	komega, kappa, kphi, delta
Constant during forward()	mu, nu = 0

fixed_nu

DetectorConstraint + BisectConstraint + SampleConstraint: nu held at declared value (default 0°), komega = delta/2, mu = 0. Analogous to psic fixed_nu.

Computed	komega, kappa, kphi, delta
Constant during forward()	nu, mu = 0

fixed_psi_vertical

Vertical bisecting with azimuthal angle ψ validation. Set g.azimuthal_reference = (h, k, l) before calling forward(). The solver returns bisecting solutions only when the natural ψ for the requested (h,k,l) matches the stored target. See Surface Geometry and the Reference Vector.

Computed	komega, kappa, kphi, delta
Constant during forward()	mu = 0, nu = 0
Extras (input)	n̂ (reference vector), ψ (target azimuth, degrees)
Extras (output)	psi (computed azimuth)

double_diffraction_vertical

Full 4D simultaneous solver in the vertical scattering plane: finds motor angles where both the primary (h₁,k₁,l₁) and secondary (h₂,k₂,l₂) reflections satisfy the Ewald sphere condition.

Computed	komega, kappa, kphi, delta
Constant during forward()	mu = 0, nu = 0
Extras (input)	h₂, k₂, l₂ (secondary reflection Miller indices)

zone_vertical

Zone mode (You 1999 §6, SPEC setmode 5). Q is confined to the plane spanned by two reciprocal-lattice vectors z0 and z1. Structurally identical to the psic zone_vertical mode, with the bisecting condition enforced on the virtual Eulerian omega pseudoangle (Walko 2016 eq. [16]) rather than the literal komega motor. Off-plane requests return an empty list with a warning.

g.modes['zone_vertical'].extras['z0'] = (1, 0, 0)
g.modes['zone_vertical'].extras['z1'] = (0, 1, 0)

Computed	komega, kappa, kphi, delta
Constant during forward()	mu = 0, nu = 0
Extras (input)	z0, z1 (Miller-index 3-tuples)
Extras (output)	in_plane_residual

bisecting_horizontal

BisectConstraint + SampleConstraint + DetectorConstraint: mu = nu/2, komega = 0, delta = 0. Horizontal scattering plane.

Computed	mu, kappa, kphi, nu
Constant during forward()	komega = 0, delta = 0

fixed_delta

DetectorConstraint + BisectConstraint + SampleConstraint: delta held at declared value (default 0°), mu = nu/2, komega = 0. Horizontal plane with delta frozen.

Computed	mu, kappa, kphi, nu
Constant during forward()	delta, komega = 0

fixed_psi_horizontal

Horizontal bisecting with azimuthal angle ψ validation. Symmetric with fixed_psi_vertical in the horizontal plane. Set g.azimuthal_reference = (h, k, l) before calling forward().

Computed	mu, kappa, kphi, nu
Constant during forward()	komega = 0, delta = 0
Extras (input)	n̂ (reference vector), ψ (target azimuth, degrees)
Extras (output)	psi (computed azimuth)

double_diffraction_horizontal

Full 4D simultaneous solver in the horizontal scattering plane.

Computed	mu, kappa, kphi, nu
Constant during forward()	komega = 0, delta = 0
Extras (input)	h₂, k₂, l₂ (secondary reflection Miller indices)

zone_horizontal

Horizontal-plane analog of zone_vertical. Locks komega = 0, delta = 0; the bisecting condition mu = nu/2 together with the kappa motor pair solves any in-plane (h, k, l).

Computed	mu, kappa, kphi, nu
Constant during forward()	komega = 0, delta = 0
Extras (input)	z0, z1 (Miller-index 3-tuples)
Extras (output)	in_plane_residual

lifting_detector_mu

Out-of-plane mode: mu and komega frozen, nu and delta solved via the qaz constraint (tan(qaz) = tan(delta) / sin(nu), You 1999 eq. 18). qaz = 90° constrains the scattering to the vertical plane.

Computed	mu, nu, delta
Constant during forward()	mu = 0, komega = 0

lifting_detector_kphi

Out-of-plane mode: kphi and mu frozen, nu and delta solved via the qaz constraint (tan(qaz) = tan(delta) / sin(nu), You 1999 eq. 18). qaz = 90° constrains the scattering to the vertical plane.

Computed	kphi, nu, delta
Constant during forward()	kphi = 0, mu = 0

Mode cross-reference

Each kappa6c mode mapped to its equivalent psic mode (the kappa6c geometry shares the psic outer-axis ordering), the closest analog in SPEC’s kappa6c macros, and the Hkl/Soleil K6C hkl engine. Modes are grouped by scattering plane: vertical first, then horizontal, then the out-of-plane lifting-detector family.

mode	psic equivalent	SPEC kappa6c	Hkl/Soleil K6C
bisecting_vertical	bisecting_vertical	(2,0,5,0,0)	bissector_vertical
fixed_kphi	fixed_phi_vertical	(2,0,4,2,0)	constant_phi_vertical
fixed_mu	—	(2,0,5,2,0)	—
fixed_nu	—	(2,0,5,2,0)	—
fixed_psi_vertical	fixed_psi_vertical	(2,4,5,0,0)	psi_constant_vertical
double_diffraction_vertical	double_diffraction_vertical	—	double_diffraction_vertical
zone_vertical	zone_vertical	setmode 5	(TODO HklEngine "zone")
bisecting_horizontal	bisecting_horizontal	(1,0,6,0,0)	bissector_horizontal
fixed_delta	—	(1,0,6,2,0)	—
fixed_psi_horizontal	fixed_psi_horizontal	(1,4,6,0,0)	psi_constant_horizontal
double_diffraction_horizontal	double_diffraction_horizontal	—	double_diffraction_horizontal
zone_horizontal	zone_horizontal	setmode 5	(TODO HklEngine "zone")
lifting_detector_mu	lifting_detector_mu	(3,0,1,2,0)	lifting_detector_mu
lifting_detector_kphi	lifting_detector_phi	(3,0,4,2,0)	lifting_detector_kphi

The SPEC tuple is (g_mode1, g_mode2, g_mode3, g_mode4, g_mode5), where g_mode1 selects the scattering plane (1 = horizontal, 2 = vertical, 3 = qaz/lifting-detector), g_mode2 selects an optional reference-angle constraint (0 = none, 4 = ψ-fixed), and g_mode3–g_mode5 fix specific motor angles.

—: no documented analog exists in that package.

References: SPEC kappa6c macros; Hkl/Soleil K6C; Hkl source.

API reference

• kappa6c — Kappa Six-Circle

• AdHocDiffractometer

• ConstraintSet

• BisectConstraint

• SampleConstraint

• DetectorConstraint

• ReferenceConstraint

• EwaldSphereViolation

• ConstraintViolation

References

• H.H. Sønsteby, D. Chernyshov, M. Getz, O. Nilsen & H. Fjellvåg, On the application of a single-crystal κ-diffractometer and a CCD area detector for studies of thin films, J. Synchrotron Rad. 20, 644–647 (2013) (six-axis κ instrument; KUMA6 at SNBL/ESRF). DOI: 10.1107/S0909049513009102.

• G. Thorkildsen, H.B. Larsen & J.A. Beukes, Angle calculations for a three-circle goniostat, J. Appl. Cryst. 39, 151–157 (2006), Table 1, equation (3); §3 last paragraph (extension to additional rotation axes). DOI: 10.1107/S0021889805041877.

• D.A. Walko, Multicircle Diffractometry Methods, in Reference Module in Materials Science and Materials Engineering (Elsevier, 2016), §4.1, Figure 3, equation [16]. DOI: 10.1016/B978-0-12-803581-8.01215-7.

• H. You, Angle calculations for a 4S+2D six-circle diffractometer, J. Appl. Cryst. 32, 614–623 (1999) (psic outer axes and You coordinate basis). DOI: 10.1107/S0021889899001223.

• 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.