Case Study: Describing a Diffractometer#

This case study presents the diffractometer geometry problem that started the ad_hoc_diffractometer project. The equipment described here is a six-circle diffractometer matching the You (1999) psic geometry; the analysis leads directly to the demo geometries in geometries and the AdHocDiffractometer class.

For a worked example showing that the choice of right-handed basis applied to this equipment is arbitrary — different basis assignments produce U/UB matrices that differ by a fixed rotation, but the physical motor-angle ↔ (h, k, l) conversion is invariant — see Case Study: Choice of Basis and the UB Matrix.

The equipment#

We have a piece of mechanical equipment consisting of rotary stages only (no translational stages). The rotational axes of all stages ideally coincide at a single point of intersection — the sample position. In practice, engineering tolerances cause this point to expand into a small 3-D volume known as the sphere of confusion.

The equipment rotates the sample in three-dimensional space and positions a detector to observe the scattered radiation. All stage angles are described at their zero-degree positions.

Reference frame#

The reference frame is defined by three mutually perpendicular directions. The positive sense of each is chosen to form a right-handed system:

  • vertical: opposite to the direction of gravitational acceleration (i.e., upward).

  • longitudinal: a chosen direction in the plane perpendicular to vertical, conventionally aligned with the nominal incident beam direction projected onto that plane, positive toward the equipment. This is a property of the instrument installation, not of the beam itself.

  • transverse: orthogonal to both vertical and longitudinal, with positive sense chosen so that the three directions form a right-handed system (vertical × longitudinal).

These directions are properties of the physical setup — they do not depend on how basis vectors are assigned to Cartesian axes. The mapping of vertical, longitudinal, and transverse to x, y, z (or any other symbol) is a separate choice made by the caller and encoded in a basis dict. Two common choices — You (1999) and Busing & Levy (1967) — are applied to this equipment in Case Study: Choice of Basis and the UB Matrix, where they are shown to produce U/UB matrices that differ by a fixed rotation without changing the underlying physics.

Equipment description#

The equipment consists of two independent stage stacks that share a common axis at their base.

Stack 1 — detector stack (2 stages)

  • Stage 1 (base)

    • Axis of rotation: vertical

    • Sign of rotation: right-handed (consistent with coordinate system)

  • Stage 2 (sits on stage 1)

    • Axis of rotation: transverse

    • Positive rotation: from longitudinal toward vertical

    • The detector is mounted on a long radial arm pointing at the sample

Stack 2 — sample stack (4 stages)

  • Stage 1 (base)

    • Axis of rotation: vertical, colinear with stack 1 stage 1

    • Sign of rotation: same as stack 1 stage 1

  • Stage 2 (sits on stage 1)

    • Axis of rotation: transverse

    • Sign of rotation: same as stack 1 stage 2

  • Stage 3 (sits on stage 2)

    • Axis of rotation: longitudinal

    • Sign of rotation: right-handed (consistent with coordinate system)

  • Stage 4 (sits on stage 3)

    • Axis of rotation: vertical

    • Sign of rotation: right-handed (consistent with coordinate system)

Questions#

  1. Assign basis vectors (xHat, yHat, zHat) to each axis of the reference frame. Describe the orientation of each stack and stage in terms of these basis vectors, including the sign of rotation. Describe the steps to compute the orientation matrix U.

  2. Is it possible to make different assignments of the basis vectors? What are the resulting stage orientation vectors? How does the U matrix differ?

Both questions are worked out in Case Study: Choice of Basis and the UB Matrix. The goal there is to show — using two different right-handed basis assignments applied to the same physical equipment described above — that the choice of basis is arbitrary: it changes the numerical values of the axis vectors and the U/UB matrices by a fixed rotation, but the physical conversion of motor angles ↔ (h, k, l) is invariant. The equipment also maps exactly to the six-circle psic geometry.