====== 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**<sub>hkl</sub> are defined as a perpendicular to the diffracting plane (hkl) with a norm of 2π/d<sub>hkl</sub>. 

In crystallography, authors often do not include the 2π factor in the calculation of **G**. In this case, the norm of **G**<sub>hkl</sub> is 1/d<sub>hkl</sub> or 2sin(θ)/λ.

Among multigrain XRD software:
  * [[software:imaged11|ImageD11]] assumes ||**G**|| = 1/d,
  * g-vector files [[fileformat:gve|gve]] generated by [[software:imaged11|ImageD11]] also assume ||**G**|| = 1/d,
  * [[software:polyxsim|polyxsim]], [[software:fitallb|fitallb]], or the [[software:xfab|xfab]] library use a 2π prefactor in the calculations of **G** and the **B** matrix,
  * [[software:grainspotter|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
  * {{ :processing: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: [[https://doi.org/10.1107/S0365110X67000970|10.1107/S0365110X67000970]] ]