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Method of refining powder diffraction data to find the crystalstructure, developed by Hugo M. RietveldSeminal papersH.M. Rietveld,
Acta Cryst.
H.M. Rietveld,
J. Appl. Cryst.
Overview of Rietveld refinement guidelinesL.B. McCusker, R.B. Von Dreele, D.E. Cox, D. Louer, P.Scardi,
J. Appl. Cryst.
Structure factors (form factors)
2.^
Multiplicity
3.^
Lorentz factor
LP factor
Polarization factor
Temperature factor or atomic displacement
6.^
Absorption
7.^
Preferred orientation
8.^
Extinction coefficients
fluorescentx-rays
heat
transmitted
electrons
beam
absorbing substance
incident beam
scattered x-rays (coherent & incoherent)
figure modified from N.F.M. Henry, H. Lipson, W.A.Wooster.
The Interpretation of X-ray Diffraction Photographs
, Macmillan: London, 1951.
amplitude of the wave scatteredby all the atoms in a unit cell
|^ F
hkl^
amplitude of the wave scatteredby one electron Fhkl
= ā f
exp [2n
Ļi (hu + kv + lw)]
It describes how the atom arrangement (u, v, w) affects thescattered beam.
fn
is the atomic scattering factor. The
intensity of a diffracted beam is proportional to |F |
Note:
exp(
Ļi) = - exp(
Ļi) = + exp(n
Ļi) = exp(-n
Ļi)
Fhkl
= ā f
exp [2n
Ļi (hu + kv + lw)]
A primitive cell with only one atomat the origin (0,0,0).
Fhkl
= f exp [
Ļi (h0 + k0 + l0)]
Fhkl
= f exp [
Ļi (0)]
Fhkl
= f
The intensity of the diffracted beam F
2 depends only on f
Cl^
Na
Fhkl
= f
eCl 2 Ļi(0)
eCl 2 Ļi(h/2+k/2)
+fCl
(^2) e Ļi(h/2+l/2)
eCl 2 Ļi(k/2+l/2)
+fNa
(^2) e Ļi(h/2+k/2+l/2)
Na^
2 Ļe i(l/2)
Na^
2 Ļe i(k/2)
Na^
2 Ļe i(h/2)
Fhkl
= f
{1 + eCl
Ļi(h+k)
Ļi(h+l)
Ļi(k+l)
+fNa
{e Ļi(h+k+l)
Ļil^
Ļik^
Ļih
Fhkl
= f
{1 + eCl
Ļi(h+k)
Ļi(h+l)
Ļi(k+l)
+fNa
Ļe i(h+k+l)
{1 + e
Ļi(-h-k)
Ļi(-h-l)
Ļi(-k-l)
Fhkl
= {1 + e
Ļi(h+k)
Ļi(h+l)
Ļi(k+l)
} { f
eNa Ļi(h+k+l)
recall: exp(n
Ļi) = exp(-n
Ļi)
Fhkl
= {1 + e
Ļi(h+k)
Ļi(h+l)
Ļi(k+l)
} { f
eNa Ļi(h+k+l)
If h+k+l is odd-odd-even or odd-even-even
{1 + e
Ļi(h+k)
Ļi(h+l)
Ļi(k+l)
If h+k+l is all even, { f
eNa Ļi(h+k+l)
} = f
Na
Fhkl
4 (f
) = 4 ( 18 eNa
-^ + 10 e -^ ) = 112 e
If h+k+l is all odd, { f
eNa Ļi(h+k+l)
} = f
Fhkl
4 (f
) = 4 ( 18 eNa
-^ - 10 e -^ ) = 32 e
Note: atomic scattering factor is the number of electrons in the ions
In a cubic structure (a = b = c){100} is (100), (010), (001), (-100), (0-10), (001) so M = 6{110} is (110), (-110), (1-10), (-1-10) so M = 12The {110} peak is expected to be TWICE as strong as the {100}
From cubic (a = b = c)
ā^
tetragonal (a = b ā c)
(100) M = 6
ā^
(100), (-100), (010), (0-10)
M = 4
(001), (00-1)
M = 2
(110) M = 12
ā^
(110)
M = 4
(101)
M = 8
(111) M = 8
ā^
(111)
M = 8
cubic tetragonal
(100)
(110)
(111)
Polarization factor = Ā½ (1 + cos
22 Īø)
LP factor = (1+cos
22 Īø
)/(sin
2 Īø^
cos
Īø)
Note 1. If a monochromator is used, then the polarizationfactor is Ā½ (1 + cos
22 Īø
cos
22 Īø
) whereM
ĪøM
is the Bragg
angle for the monochromator.Note 2. For neutron diffraction, polarization is a constant.
LP vs Īø figure from Cullity & Stock, 2001.
Thermal vibrations1. Unit cell expansion causes changes in the 2Īø positions.2. Decrease in the intensities of diffracted lines.3. Increase in the intensity of background scattering.
Figure from Cullity & Stock, 2001.
Idiffracted
incident
exp (-
Ī¼t)
Ī¼^ is the linear absorption coefficient, and
t^ is the thickness of
the sample.
-^
crystals are not randomly oriented
-^
examples: plate-like crystals or needle-shaped crystals March-Dollase functionP(Ī±) = (r
2 cos
2 Ī±+sin
2 Ī±/r)
-3/
P(Ī±)
pole distribution Ī±^
angle between hkl & PO vector r^
adjustable parameter
Spherical harmonic functionā¢ measurement of the pole densitydistributions of a number ofdiffraction planesā¢ more complex and thus morepowerful
