2π or not?

Single-crystal diffraction calculations rely on g-vectors, U-matrices for defining grain orientation, and B-matrices for conversion between the Cartesian grain system and reciprocal space.

In physics in particular, g-vectors G_hkl are defined as a perpendicular to the diffracting plane (hkl) with a norm of 2π/d_hkl.

In crystallography, authors often do not include the 2π factor in the calculation of G. In this case, the norm of G_hkl is 1/d_hkl or 2sin(θ)/λ.

Among multigrain XRD software:

• ImageD11 assumes ||G|| = 1/d,
• g-vector files gve generated by ImageD11 also assume ||G|| = 1/d,
• polyxsim, fitallb, or the xfab library use a 2π prefactor in the calculations of G and the B matrix,
• GrainSpotter multiplies g-vectors by 2π when reading in and uses a B matrix that includes the 2π internally.

For details on U and B matrices calculations, refer to

• fable_geometry_version_1.0.8.pdf,
• or the 1967 Busing and Levy publication: W. R. Busing and H. A. Levy, Angle calculations for 3- and 4-circle X-ray and neutron diffractometers, Acta Cryst. 22, 457-464 (1967) [doi: 10.1107/S0365110X67000970 ]