Structure Determination
Rietveld Refinement
Rietveld refinement of XRD data for
La(Zn½Ti½)O3
Hugo M. Rietvled, 1966
MSE 421/521 Structural Characterization
Structure Determination
Rietveld Refinement
Antiparallel La+3 displacements along [010]c = 0.1050 Å
Antiparallel La+3 displacements along [001]c = 0.1570 Å
°
In-phase tilting of octahedra about [100]c of 8.3
°
Antiphase tilting of octahedra about [010]c and [001]c of 2.4
Tilt system
a
c
b
La
Zn
Ti
O
a+b-b-
Atom
Wycoff
x
y
La
Zn
z
Uiso (Å2)
4e
0.250(2)
0.03308(9)
0.9933(2)
0.0168(2)
2c
0
0.5
0
0.0232(3)
Ti
2b
0
0
0.5
0.0160(4)
O1
4e
0.9605(3)
0.2059(3)
0.2234(3)
0.0155(4)
O2
4e
0.9593(3)
0.7211(3)
0.2971(3)
0.0200(5)
O3
4e
0.2415(3)
0.9837(1)
0.4224(2)
0.0144(3)
Space Group: P21/n
°
a = 7.8950 Å, b = 5.5964 Å, c = 5.5809 Å, β = 90.034
R. Ubic et al., Acta Crystallographica, B62 521 (2006).
MSE 421/521 Structural Characterization
Lattice Constant Determination
Nelson-Riley Function
2dsinθ = λ
Need to graph ameas vs some function for each hkl:
2 sin Plot a vs 2 cos Δ sin Δ 0
cos
then extrapolate to 0 (where 2θ = 180°)
∴ sin Δ cos Δ
∴
cos
Δ
cot Δ
Now since for a cube ,
we know that ∴
Δ
cot Δ
Lattice constant of Al
And cotθ 0 as 2θ 180° ∴ Δ 0
Note: Factor of ½ is irrelevant –
extrapolation doesn’t change
MSE 421/521 Structural Characterization
Strain Determination
Rietveld Refinement
2dsinθ = λ
∴d = (½λ)cscθ
∆d = (½λ)[-cscθcotθ]∆θ
1 cos Δ Δ
2 sin sin Δ ∴
sin 2
sin cos Δ Δ
sin tan Δ Δ
Δ2
#
tan 2tan 2tan ∴
MSE 421/521 Structural Characterization
Δ
#
4tan Includes shifts to both
right (compressive) and
left (tensile) so ∆d must
be divided by 2 and we
drop the “-”.
Can multiply by E to yield σ
Crystallite Size Determination
NOT grain size or particle size!!
2dsinθ = λ
t = xd
Multiply both sides by x
t = crystallite thickness
x = number of (hkl) relfecting planes
d = interplanar spacing of (hkl) planes
2% sin %
∴ 2& sin %
Now differentiate both sides w.r.t. θ and t:
2& cos Δ 2Δ& sin 0
•
•
•
only interested in |∆θ|, and since t cannot be negative, drop “-”
say ∆t = d (smallest possible increment)
dsinθ = λ/2
∴&
Δ& sin cos Δ
∴&
2cos Δ # cos Assumes
triangular peak
shape
&
0.9
# cos Assumes
Gaussian peak
shape
B = 2∆θ = FWHM [rad]
MSE 421/521 Structural Characterization
Effect of Crystallite Size
NOT grain size or particle size!!
MSE 421/521 Structural Characterization
Separating Size/Strain Effects
Williamson-Hall Plot
Gaussian: B2 = B2inst + B2s + B2ε
Lorentian: B = Binst + Bε + Bs
B2 = B2meas – B2inst
B = Bmeas - Binst
Determine experimentally
# #) #*
0.9
0.9
This is the equation of a line,
4+ tan ∴ # cos 4+ sin so plot Bcosθ vs 4sinθ for each peak
&
& cos ZnO nanoparticles
synthesized under
different processing
conditions
Prabhu et al., World J. Nano Sci. Eng., 4 21-28 (2014).
MSE 421/521 Structural Characterization
Mosaic Structure
of a single crystal
tiny (~1000 Å) blocks (subgrains), each slightly misoriented (< 1°)
•
If misorientation angle is dθ,
then diffraction of a parallel
monochromatic beam from a
"single" crystal will occur not
only at an angle of incidence θB
but at all angles between θB
and θB+dθ (peak broadening).
•
Strains increase intensity of
diffracted beam relative to that
theoretically calculated for an
ideally perfect crystal.
Very exaggerated
MSE 421/521 Structural Characterization