refinement parameters - penn state college of earth and...

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 sample displacement, sample transparency,

zero-shift errors

unit cell dimensions

preferred orientation, absorption, porosity,

extinction parameters

scale factor(s)

2

Peak shape parameters

What determines peak shape?

Instrumental

source image

flat specimen

axial divergence

specimen transparency

receiving slit

monochromator(s)

3

Peak shape parameters

What determines peak shape?

Spectral

inherent spectral width

most prominent effect - Kα1 Kα2 Kα3 Kα4 overlap

4

Peak shape parameters

What determines peak shape?

Specimen

mosaicity

crystallite size

microstrain

5

Peak shape parameters

Historical

Began with neutron diffraction

- peak shapes nearly Gaussian

6

Peak shape parameters

1st basic peak parameter - FWHM

Caglioti formula: H = (U tan2 θ + V tan θ + W)1/2

i.e., FWHM varies with θ or 2θ

7

Peak shape parameters

1st basic peak parameter - FWHM

Caglioti formula: H = (U tan2 θ + V tan θ + W)1/2

i.e., FWHM varies with θ or 2θ

8

Peak shape parameters

X-ray case more complicated

not usually Gaussian

not usually Lorentzian

usually need to mix the two somehow

9

Peak shape parameters

10

Peak shape parameters

4 most common profile fitting fcns

11

Peak shape parameters

4 most common profile fitting fcns

12

Peak shape parameters

4 most common profile fitting fcns

Γ(z) = ∫ tz-1 et dt0

13

Peak shape parameters

4 most common profile fitting fcns

14

Peak shape parameters

But peaks are usually asymmetric - even after α2 stripping!

15

Peak shape parameters

Finally - use NIST std 660 (LaB6) to determine broadening frominstrumental and spectral contributions & kept constant (U, V, W)

16

Scale factors

Constant factor which scales calculated intensities of all reflectionsin pattern up to the observed intensities

Determined by least squares refinement

17

Scale factors

Constant factor which scales calculated intensities of all reflectionsin pattern up to the observed intensities

Determined by least squares refinement

In GSAS, two types of scale factors:

histogram scale factor phase scale factor

18

Scale factors

Constant factor which scales calculated intensities of all reflectionsin pattern up to the observed intensities

Determined by least squares refinement

In GSAS, two types of scale factors:

histogram scale factor phase scale factor

At any point in pattern, calc'd I is then obtained as above – thencompared with obs'd I

19

Scale factor

When more than one phase, Sph calculated for each

w/o of each phase present automatically calc'd:

20

Lattice parameters

In general

1/d2 = 1/V2 (S11h2 + S22k2 + S33l2 + 2S12hk + 2S13hl + 2S23kl)

S11 = b2c2 sin2 α

S22 = a2c2 sin2 β

S33 = a2b2 sin2 γ

S12 = abc2 (cos α cos β − cos γ)

S13 = ab2c (cos γ cos α − cos β)

S23 = a2bc (cos β cos γ − cos α)

21

Lattice parameters

1/d2 = 1/V2 (S11h2 + S22k2 + S33l2 + 2S12hk + 2S13hl + 2S23kl)

GSAS: