Case Study: Coordinate Convention and UB Matrix#

This page analyzes the case study diffractometer to show how a basis vector assignment (the mapping of physical directions to Cartesian axes) leads to the B, U, and UB matrices used throughout ad_hoc_diffractometer.

The worked example below uses the You (1999) convention, but the procedure is identical for any right-handed orthogonal basis — only the numerical values of the axis vectors change. See Choose and Understand Basis Vectors for a tutorial on choosing a basis.

Reference: H. You, J. Appl. Cryst. 32, 614–623 (1999). DOI: 10.1107/S0021889899001223

Basis vector assignment#

Any right-handed orthogonal mapping of the three physical directions to Cartesian unit vectors is valid. The You (1999) convention assigns:

  • xHat: vertical (along the mu and nu rotation axes)

  • yHat: longitudinal (along the incoming beam direction)

  • zHat: transverse

This is a valid right-handed system: xHat × yHat = zHat. From You (1999) §2: “the x axis is defined along the vertical mu and nu axes and the y axis is defined along the incoming beam direction.”

Correspondence with You (1999)#

The You (1999) paper describes a 4S+2D six-circle geometry: four sample-orienting stages (mu, eta, chi, and phi) and two detector stages (nu, delta). This matches the case study equipment exactly.

Using R to denote a rotation matrix whose invariant axis is identified by a 1 on the diagonal:

  • R xHat: 1 at position [1,1] — rotation about the vertical axis

  • R yHat: 1 at position [2,2] — rotation about the longitudinal axis

  • R zHat: 1 at position [3,3] — rotation about the transverse axis

Right-handed rotation places +sin below the diagonal; left-handed rotation (equivalently, right-handed rotation about the negated axis) places +sin above the diagonal.

You angle

You matrix

Stage

Physical axis

Axis vector

mu

U

S2-1

vertical

+xHat

eta

X

S2-2

transverse

−zHat

chi

H

S2-3

longitudinal

+yHat

phi

M

S2-4

transverse

−zHat

nu

P

S1-1

vertical

+xHat

delta

D

S1-2

transverse

−zHat

Notes:

  • mu and nu share a colinear vertical axis (+xHat) with the same right-handed sense of rotation; the stages are mechanically independent.

  • eta, phi, and delta share the transverse axis (−zHat) with left-handed rotation.

  • chi is the only stage with a longitudinal axis (+yHat), right-handed.

Sign convention#

A left-handed rotation about an axis is equivalent to a right-handed rotation about the negated axis:

R_left-handed(+nHat, θ) = R_right-handed(−nHat, θ)
                         = R_right-handed(+nHat, −θ)

For stages where You (1999) uses a left-handed convention (eta, phi, delta), the signed axis vector is −zHat rather than +zHat. The physical rotation axes are the same; only the sign convention for positive rotation differs.

The B, U, and UB matrices#

The introduction of lattice constants a, b, c, α, β, γ and a fixed wavelength λ sets up the UB matrix formalism of Busing & Levy (1967).

B matrix#

The B matrix (Busing & Levy 1967, eq. 3) transforms Miller indices h = (h, k, l)ᵀ to the scattering vector in Cartesian crystal-frame coordinates:

Q_c = B h

B is constructed from the reciprocal lattice parameters derived from a, b, c, α, β, γ. B is not in general orthonormal. See Direct Lattice for the explicit construction.

U matrix#

The U matrix (Busing & Levy 1967, eq. 4) is the orthogonal matrix relating the phi-axis frame (attached to the innermost sample stage) to the crystal Cartesian frame:

h_phi = U Q_c = U B h

U corrects for the misalignment between the crystal axes and the diffractometer axes when all motor angles are zero.

To avoid the ambiguity noted by Walko (2016) — where both U and UB are sometimes called the “orientation matrix” — this package uses the following unambiguous names:

Symbol

Name

Meaning

B

B matrix

Maps Miller indices to crystal Cartesian coords; encodes a, b, c, α, β, γ

U

U matrix

Orthonormal; relates crystal Cartesian frame to the phi-axis frame

UB

UB matrix

Maps Miller indices directly to the phi-axis frame; determinable from reflections alone

UB as a practical entity#

Busing & Levy treat UB as a single practical entity (eqs. 29–31):

UB = Hc H⁻¹

where Hc and H are matrices of observed and indexed reflection vectors respectively. This allows UB to be determined even when lattice parameters are unknown.

Full diffraction equation#

The full diffraction equation (You 1999, eqs. 10–11) relates Miller indices h to the sample rotation matrices and the detector position:

h^M = M H X U_mu · UB · h

where U_mu, X, H, M are the motor rotation matrices for mu, eta, chi, and phi respectively, and h^M is the diffraction vector in the laboratory frame.

The detector position is determined by:

kf = k P D kf0

where D and P are the rotation matrices for delta and nu, k = 2π/λ is the wave number, and kf0 is the forward beam direction.

Note

You (1999) uses the symbol U for both the mu motor rotation matrix and the U (orientation) matrix. This package uses U_mu for the mu motor rotation to avoid ambiguity.

References#

  • W.R. Busing & H.A. Levy, Acta Cryst. 22, 457–464 (1967). DOI: 10.1107/S0365110X67000970

  • H. You, J. Appl. Cryst. 32, 614–623 (1999). DOI: 10.1107/S0021889899001223

  • D.A. Walko, Reference Module in Materials Science and Materials Engineering, Elsevier (2016).