EXAMPLE - Benzene Ring EXAMPLE - Benzene Ring From the atomic positions...
From the atomic positions...
...we can easily evaluate the
...we can easily evaluate the interatomic interatomic distancesdistances and from them we can calculate the powder
and from them we can calculate the powder diffraction response.
diffraction response.
dij
2θ
k
q 2θ k’
elastic scattering elastic scattering
Diffraction Diffraction ……
Powder diffraction:
many identical particles, in all possible 3D orientations, isotropically distributed!
We need the spherical average of the differential cross section.
Debye’s formula
Peter Debye,
Ann. Phys. 1915, 46,809.
II((qq)) II00((qq) - ) - ““constant termconstant term””
Case of fixed scattering lengths
Case of fixed scattering lengths (or if you factor the (or if you factor the ff’’s s andand TT’’s out):s out):
PDFPDF
The PDF The PDF
Sinc
Sinc Fourier transformFourier transform
PDFPDF DFADFA
Atomic structure
Interatomic
distance vectors
Interatomic
distance lengths and multiplicities
Powder diffraction pattern
PDF (from data)
PDF advantages:
PDF advantages:
- clearly
- clearly see some see some interatomic interatomic distancesdistances,, modeling can be strongly constrained
modeling can be strongly constrained DFA advantages:
DFA advantages:
- simpler experiments, no need for high q and - simpler experiments, no need for high q and
very clean data,
very clean data, very sensitive to very sensitive to size/shapesize/shape distributions
distributions, , complex mixtures are OKcomplex mixtures are OK
