Chapter 5
ADJUSTMENT OF STANDARD UNCERTAINTIES
5.1 Powder Diffraction Data (*.fos)
In powder diffraction, \(\sigma (\bm {h}_j )\)’s are estimated on the basis of the law of propagation of errors in combination with counting statistics: \begin {equation} \sigma ( \bm {h}_j ) = \frac {|F_\mathrm {o}(\bm {h}_j)|}{2} \left \{ \left [ \frac {1}{EI_\mathrm {o}(\bm {h}_j)} \right ]^2 \! + \left [ \frac {\sigma (s)}{s} \right ]^2 \right \}^{\! \frac {1}{2}} , \label {eq:sigma_Fo_1} \end {equation} where \(E\) is the factor to adjust \(\sigma ( \bm {h}_j )\), \(I_\mathrm {o}(\bm {h}_j)\) is the observed integrated intensity, \(s\) is the scale factor, and \(\sigma (s)\) is the standard uncertainty of \(s\) [6]. Both \(s\) and \(\sigma (s)\) results from Rietveld analysis or whole-pattern fitting with RIETAN-FP. \(E\) is selected in such a way that electron- or nuclear-density distribution which is physically and chemically reasonable results from MEM analysis. Convenient bash scripts, MPF_multi.command, for automatic MPF analyses enable us to change \(E\) as specified in *.prf (see 6.2.2).
In the case of angle-dispersive-type powder diffraction where the step width, \(\Delta 2 \theta \), is usually constant, \(E\) is approximately equal to \(1/\Delta 2 \theta \) (unit: rad\({}^{-1}\)), which is output as E(SCIO) in *.lst by RIETAN-FP1 [23]. On the other hand, \(E\) depends on the step width, \(\Delta t\), of the time-of-flight (TOF) in TOF neutron powder diffraction; a utility called Alchemy [24,25] can be used to convert files output by GSAS and Fullprof into *.fos or *.mem (see 5.2).
5.2 Single-Crystal Data (*.mem)
The estimation of \(\sigma ( \bm {h}_j )\) is difficult even in single-crystal X-ray or neutron diffraction; \(\sigma ( \bm {h}_j )\)’s are often underestimated or overestimated. With ERIS, \(\sigma ( \bm {h}_j )\)’s can be changed with the adjustment factor, \(E\), according to \begin {equation} \sigma ' \left ( \bm {h}_j \right ) =\frac {\sigma \left (\bm {h}_j\right )}{\sqrt {E}} \end {equation} without any conversion of \(\sigma \left (\bm {h}_j\right )\)’s in *.mem.
