1 Refinement parameters What are the parameters to be determined? atom positional parameters atom thermal motion parameters atom site occupancy parameters background function parameters peak shape parameters unit cell dimensions scale factor(s) sample displacement, sample transparency, zero-shift errors preferred orientation, absorption, porosity, extinction parameters

2 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 

3 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  axial divergence

4 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  axial divergence p 1 = –h 2 K 1 /3RR = diffractometer radius p 2 = –h 2 K 2 /3RK 1, K 2 = constants for collimator h = specimen width

5 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  flat sample

6 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  flat sample p 3 = –  2 /K 3  = beam divergence K 3 = constant

7 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  specimen transparency

8 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  specimen transparency p 4 = 1/2  eff R  eff = effective linear absorption coefficient

9 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 specimen displacement p 5 = –2s/R s = displacement

10 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 zero error

11 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 p 4, p 5, & p 6 strongly correlated when refined together

12 Peak shift parameters 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 p 4, p 5, & p 6 strongly correlated when refined together When instrument correctly aligned, generally need get only p 5

13 Peak shift parameters  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 In GSAS: where:

14 Preferred orientation In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles

15 Preferred orientation In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry

16 Preferred orientation In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry soso s S = s - s o

17 Preferred orientation In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry

18 Preferred orientation March-Dollase function (a la GSAS) plates needles

19 Preferred orientation March-Dollase function (a la GSAS) plates needles # symmetrically equivalent reflections multiplier in intensity equation

20 Preferred orientation March-Dollase function (a la GSAS) plates needles # symmetrically equivalent reflections multiplier in intensity equation

21 Preferred orientation March-Dollase function (a la GSAS) plates needles # symmetrically equivalent reflections multiplier in intensity equation preferred orientation parameter (refined)

22 Preferred orientation March-Dollase function (a la GSAS) plates needles # symmetrically equivalent reflections multiplier in intensity equation preferred orientation parameter (refined) angle betwn orientation axis & diffraction vector for hkl

23 Preferred orientation March-Dollase function (a la GSAS)

24 Preferred orientation Spherical harmonics (a la GSAS) hkl sample orientation

25 Preferred orientation Spherical harmonics (a la GSAS) hkl sample orientation harmonic coefficients harmonic functions