Pecharsky V.K., Zavalij P.Y. Fundamentals of Powder Diffraction and Structural Characterization of Materials
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622
Chapter
7
Ch7ExOlh.inp,
is located on the
CD.
The resultant observed and calculated
powder diffraction patterns are shown in
Figure
7.7
and the final parameters
of individual atoms are listed in
Table
7.6.
Table
7.6. Structural parameters of the major phase (LaNi4.85Sno.ls), fully refined by the
Rietveld technique using powder diffraction data shown in
Figure
7.7. The refined unit cell
dimensions are:
a=
5.04479(3),
c
=
4.01303(3)
A.
The following anisotropic displacement
parameters are constrained by symmetry in all sites:
PI3
=
P23
=
0.
Atom Site
XYZ
ga
104h 104P22 1o4h3 1o4PI2
La I(a) 0 0 0 1 135(2) 135(2) 252(4) 68(1)
Nil 2(c) 113 213 0 1 194(2) 194(2) 130(4) 97(1)
Ni2 3(g) 112 0 112 0.944(2) 136(2) 104(3) 137(4) 52(2)
Sn 3(g) 112 0 112 0.056(2) 136(2) 104(3) 137(4) 52(2)
a
See the corresponding footnote in
Table
7.4.
10
20 30 40
50
60
Bragg
angle,
28
(deg.)
Figure
7.7. The observed and calculated powder diffraction patterns of LaNi485Sno.15 after the
completion of Rietveld refinement. Notations are identical to
Figure
7.2. The second set of
vertical bars indicates the calculated positions of the Kal components of Bragg peaks in the
impurity phase, which is a solid solution of Sn in Ni. The virtual absence of a Bragg peak at
20
z
9' and the presence of the same reflection at 20
E
20' when Cu Ka radiation was
employed (see
Figure
7.5) is the result of differences in the anomalous scattering (see also
Eqs. 2.101 and 2.108).
Crystal structure rejinement 623
Refinement using Cu
Ka
data
Refinement using
Mo
Ka
data
X
Figure
7.8.
Atomic displacement ellipsoids in the crystal structure of LaNi4,85Sno,15, shown at
99%
probability,
as
refined using
Cu
Ka
(left) and Mo
Ka
(right) powder diffraction data.
A
comparison of the results presented in Table 7.6 with those shown in
Table 7.4 indicates no discrepancies in the refined populations of Ni2 and Sn
atoms, which simultaneously occupy the 3(g) site, although a small
difference in the anisotropic displacement parameters of all atoms is evident.
Regardless of this difference, as seen in Figure 7.8, displacement ellipsoids
have nearly identical orientations and sizes. The preference should be given
to the results obtained using Mo
Ka
radiation because Bragg reflections at
higher sinelh were included into the determination of the individual
anisotropic displacement parameters.
On the other hand, the comparison of lattice parameters obtained using
Cu
Ka
(Table 7.4) and Mo
Ka
(Table 7.6) data indicates small but
statistically significant differences. Considering the fact that the longer
wavelength experiment includes reflections at higher Bragg angles, the
preference should be given to the unit cell dimensions obtained in the
refinement based on Cu
Ka
radiation data.
Given the observed small variations in the Rietveld refinement results, it
is much better to employ all available data while performing the combined
least squares fit of the model. This can be done using the majority of
available Rietveld refinement programs and is illustrated in the next section.
7.3.8
Combined refinement using different diffraction data
We will begin combined Rietveld refinement of the crystal structure of
LaNi4.ssSno.ls using profile parameters (background, zero shifts and peak
shape) determined for each set of diffraction data during independent
refinements. The starting unit cell dimensions have been chosen as
established from Cu
Ka
data and the starting population and individual
atomic displacement parameters are taken from the Mo
Ka
result. The
624
Chapter
7
corresponding input data file, Ch7ExOli.inp, is found on the CD. For
LHPM-Rietica, a single data file should contain both sets of powder
diffraction data, and these are found on the
CD
in the file
Ch7ExOl-Cu&MoKa.dat. The progression of Rietveld refinement is
illustrated in
Table
7.7. The initial residuals are shown in the first row in
Table
7.7, where expectedly, the figures of merit for the second set of data
(Mo
Ka)
are much worse than for the first (see
Table
7.3 and
Table
7.5).
Table
7.7.
The progress of the combined Rietveld refinement of the crystal structure of
LaNi4.ssSno
Is
employing two sets of experimental data obtained using Cu
Ka
and Mo
Ka
radiations.
Refined parameters
Cu Mo
R, R,,
RB
R, Rw,
RB
Initial
11.78 14.45 11.45 523.6 612.5 589.1
Scale factors
"
6.97 10.08 4.40 22.44 27.02 8.12
Scale factors plus
a,
c
and zero shifts
8.00 11.38 4.60 7.55 11.39 1.40
As above plus peak shape, background
7.44 10.51 4.35 6.38 10.27 1.42
As above plus
Pv
(all) and population
7.29 10.39 4.13 6.43 10.30 1.53
parameter,
n,
in
3(g)
site
All plus an impurity phase
6.87 9.02 4.13 5.99 8.62 1.40
a
When multiple sets of data are used in a combined Rietveld refinement, the first set has a
fixed scale,
k
=
1,
but all other sets have their own scales in addition to a phase scale.
Thus, in our case when we have two sets of diffraction data and one crystalline phase, two
scale factors
(K
and
kM,
Ka)
have been refined independently (see
Eq.
7.10,
below).
When more than one set of experimental diffraction data is employed in
the combined Rietveld refinement, the minimized function (in the simplest
case of
Eq.
7.3) becomes
In
Eq.
7.10,
h
is the number of different sets of powder diffraction data,
ns
is the number of data points collected in the
sth
set, and k, is the scale
factor for the
sth
diffraction pattern, which appears because scattered
intensity is measured on a relative scale. Other notations are identical to
Eq. 7.3. Different scale factors, k, and
K,
are simple multipliers. Hence, they
strongly correlate, and usually are not refined simultaneously. Constraining
one of the scale factors (usually kl, for the first diffraction data set) at 1
enables successful refinement of the phase scale
(9
and scale factors of all
remaining sets of diffraction data (k2, k3,
. .
.,
kh). Equations 7.4, 7.6 and 7.7
are modified in the same way as
Eq.
7.3 for a combined Rietveld refinement.
Furthermore, it is often the case that x-ray and neutron, or conventional x-
ray and synchrotron data are used in combined refinements, therefore, the
Crystal structure refinement
625
corresponding groupings of Eqs. 7.3,
7.4, 7.6 and 7.7, modified as shown in
Eq. 7.10, are employed to express the minimized function.
Hence, the subsequent step is to refine both the phase scale factor and the
scale factor of the Mo
Ka
experiment. As shown in row two of
Table
7.7,
these two variables have a tremendous effect on the resulting figures of
merit. The residuals for Mo
Ka
data, however, remain far from the best and
a simple examination of the Rietveld plot
(Figure
7.9) indicates that this is
due to the inadequacy of the unit cell parameters determined from the Cu
Ka
experiment.
Thus, the next refinement step includes releasing unit cell dimensions
and zero shift parameters. The latter are usually different for different sets of
diffraction data. As seen in row three of
Table
7.7, overall the fit becomes
much better, although residuals for Cu
Ka
data are slightly increased. After
including all possible variables into the refinement (rows four and five in
Table
7.7) the fit improves for both sets of experimental data. The fully
refined model of the crystal structure of LaNi4,85Sno.15 without including the
impurity phase is
foundin the file
Ch7ExOlj.inp
on the CD.
Bragg angle,
20
(deg.)
10
20
30
40
50
60
Bragg angle,
28
(deg.)
Figure
7.9.
The observed (Mo Ka radiation) and calculated powder diffraction patterns of
LaNi,, ssSno,15 after combined refinement of scale factors only. The inset shows the expanded
view between 18.4 and
19.5"
of
20
to illustrate the inaccuracy of lattice parameters.
626
Chapter 7
As expected, adding the contribution from the impurity phase (again as
Le Bail's approximation) results in further reduction of the profile residuals,
see row six in Table 7.7. The final model of this crystal structure, as
determined using the combined Rietveld refinement in the two-phase
approximation, is found in the data file
Ch7ExOlk.inp
on the
CD.
Structural
parameters are listed in Table 7.8.
Table
7.8.
Structural parameters of the major phase (LaNi4&3no 15), fully refined by the
Rietveld technique using the combined powder diffraction data collected employing Cu
Ka
and Mo
Ka
radiations from the same specimen. The refined unit cell dimensions are:
a
=
5.04430(3),
c
=
4.01292(3)
A.
The following restrictions are imposed by symmetry:
PI3
=
P23
=
n
".
Atom Site
x
Y
z
g
lo4pll
~o~P~~
10~p~~ 104p12
La
l(a) 0
0 0 1 141(1) 141(1)
252(3) 71(1)
Nil
2(c) 113
213
0 1
205(2) 205(2) 151(4) 103(1)
Ni2
3(g)
112
0 112
0.956(2) 142(2) 117(3)
139(3) 59(1)
Sn
3(g) 112
0
112 0.044(2)
142(2)
117(3)
139(3) 59(1)
The resultant unit cell dimensions and structural parameters shown in
Table 7.8 are closer to those obtained in an independent Rietveld refinement
using Mo
Ka
data (see Table 7.6). Furthermore, the population of the 3(g)
site is slightly different from that obtained in both independent refinements.
The resulting chemical composition of the major phase is LaNi4.87(1)Sno.13(l),
which remains within two standard deviations from the alloy stoichiometry.
The Rietveld plots of both powder diffraction data sets are shown in
Figure 7.10 and Figure 7.11. Visual analysis of both figures indicates a good
fit, which was expected from the low residuals (Table 7.7). The model of the
crystal structure (Table 7.8) appears to be complete and makes both physical
(reasonable atomic displacement parameters) and chemical sense [no
overlapping atoms, the
3(g) sites are occupied simultaneously by atoms that
have close atomic volumes (Ni and Sn), the chemical composition of the
major phase established from x-ray data is nearly identical to the hown
composition of the alloy]. Therefore, the outcome of this crystal structure
determination may be accepted as satisfactory.
The presence of an impurity phase in this specimen may also be
accounted for by including its crystal structure into a "normal" Rietveld
refinement process rather than as a "Le Bail's" phase. For the sake of
illustration, this has been done and the set of fully refined parameters can be
found in the data file
Ch7ExOlm.inp.
It is unfeasible to refine the chemical
composition of the Ni-based impurity because of its low concentration in the
sample. Thus, only the scale factor, unit cell dimensions and overall atomic
displacement parameter have been refined for the impurity phase, while its
chemical composition was assumed as
Ni4.8sSno.ls. All residuals remain
practically identical to those shown in the last row of Table 7.7.
Crystal structure refinement
35
LaNi,,,,Sn,
,,,,
Cu
Kct
20 30 40 50 60 70 80 90 100 110 120
Bragg angle,
29
(deg.)
Figure
7.10.
The observed and calculated powder diffraction patterns of LaNi4.8sSno
(Cu Ka radiation) after the completion of the combined Rietveld refinement.
LaNi
,,,,
Sn,
,,,,
Mo
Kct
Rp
=
5.99%
kp
=
8.62%
R,
=
1.40%
x2
=
3.55
II I
I
IIIII IIIII II I111111
IIII
1111111111II11111IIII111111I1111111
I
I
I
I I I
II
I
10 20 30 40 50 60
Bragg angle,
29
(deg.)
Figure
7.11.
The observed and calculated powder diffraction patterns of LaNi4,8sSno,15
(Mo
Ka
radiation) after the completion of the combined Rietveld refinement.
62
8
Chapter
7
As noted before (Chapter 4), Rietveld refinement offers an opportunity to
determine the phase composition of the sample quantitatively. This
information can be found in the general output file after Rietveld refinement;
its extension is "out". The corresponding concentrations are as follows: the
major
LaNi4,s5Sno.15 phase makes 9734) and the impurity Nio.97Sno.03 phase
accounts for 2.5(3) per cent by weight of the alloy.
7.4
Rietveld refinement
of
CeRhGe31
This crystal structure was solved earlier (see sections 6.10 and 6.1 I), first
using x-ray and then using neutron powder diffraction data. The x-ray data
(Mo
Ka
radiation) were collected at room temperature, while the neutron
scattering experiment
(A
=
1.494
A)
was conducted at 200
K.
Hence,
combined Rietveld refinement is not feasible because of the differences in
the lattice and structural parameters of the alloy due to thermal
expansi~n,~
and we will use the two sets of data independently.
7.4.1
Refinement using x-ray diffraction data
Initial parameters for Rietveld refinement were assumed as follows:
background, peak shape, unit cell dimensions and zero shift as determined
from Le Bail's full pattern decomposition (see section
6.10), and the model
of the crystal structure from the
ab
initio
solution in the space group I4mm
Table
6.21.
Refinement of all free variables except the coordinates and displacement
parameters of individual atoms, beginning with the scale factor (rows one
through three in
Table
7.9), results in low residuals, basically confirming the
model of the crystal structure. When the coordinate parameters of atoms in
the unit cell were included in the refinement (row four,
Table
7.9), all
residuals improve, especially RB, which is lowered from 6.45 to 3.12%.
Similar to the previous example, the quality of the experimental data is quite
high, therefore, we easily refine individual isotropic and then individual
anisotropic displacement parameters of all atoms.
Finally, as may be established by a trial-and-error approach, a small
extinction correction further improves the agreement between the observed
'
0.
Zaharko,
V.K.
Pecharsky and
K.A.
Gschneidner, Jr., unpublished.
Displacement parameters of atoms are also expected to be different as the temperature of
the powder diffraction experiment varies. Furthermore, it is also feasible that atomic
positions may change due to generally anisotropic thermal expansion of crystal lattices.
These considerations are in addition to the most obvious cause (different lattice
parameters) preventing combined refinement using powder diffraction data collected at
different temperatures. In general, material may also be polymorphic but this is not the
case here, as was established in Chapter
6,
sections
6.10
and
6.1
1.
Crystal structure
refinement
629
and calculated intensities, as shown in the last row of
Table
7.9.
Attempts to
include preferred orientation assuming several possible preferred orientation
axes (such as [loo], [110], [OOl] and a few others) did not result in the
improvement of the Rietveld fit, thus indicating that in this experiment
preferred orientation effects are nonexistent within the accuracy of the
powder diffraction data.
The refined model of the crystal structure can be found in the file
Ch7Ex02a.inp
on the
CD
and the experimental data are located in the data
file
Ch7Ex02MoKa.dat.
The plot of the observed and calculated
intensities is sh& in
Figure
7.12
and the fully refined structural parameters
are listed in
Table
7.10.
Table
7.9.
The progress of Rietveld refinement of the crystal structure of CeRhGe3 using x-
ray powder diffraction data. Wavelengths used: hKal
=
0170932 A, hKaz
=
0.71361
A.
-
Refined parameters
R, Rw,
RB
xz
Initial (profile from Le Bail, model from
Table
6.24,
overall 1468 1557 1526 3x10'
B
=
0.5-A')
Scale factor 8.87 11.46 7.05 15.03
Scale, all profile,
a
and
c,
overall displacement 8.25 10.91 6.45 13.66
All above plus coordinates of individual atomsa 5.88 8.04 3.12 7.43
All above plus individual isotropic displacement parameters 5.78 7.92 3.07 7.22
All above plus individual anisotropic displacement 5.75 7.86 3.00 7.11
parametersb
All above plus extinctionC 5.42 7.54 2.14 6.54
"
In the space group I4mm the origin of coordinates along the Z-axis is not fixed by
symmetry. Therefore, the z-coordinate of one atom in the unit cell must be excluded from
the least squares at all times to avoid severe correlation problems. We choose the z-
coordinate of Ce atom (z
=
0.000) as the fixed coordinate parameter.
All atoms in this crystal structure are located in special site positions. This introduces
certain relationships between individual anisotropic displacement parameters:
PZ2
=
PI
for
the atoms in 2(a) sites (Ce, Rh and Gel);
PIz
=
PI3
=
PZ3
=
0 for all sites.
Extinction parameter 0.00001 was used as the initial approximation. Extinction (see Eq.
2.86) may correlate with phase scale factor, therefore, it may be necessary to keep the
scale factor fixed during the initial refinement of extinction.
Table
7.10.
Structural parameters of CeRhGe3, fully refined by the Rietveld technique
employing powder diffraction data collected from a ground powder using Mo Ka radiation.
The space group is I4rnm. The unit cell dimensions are:
a
=
4.39830(3),
c
=
10.03259(8) A.
Some of the anisotropic displacement parameters are fixed by site symmetry:
P12
=
PI3
=
P23
=
0. All sites are fully occupied.
Atom Site
x
Y
z
104x~ll 10~x8~~
~o~xP~~
Ce 2(a) 0 0 0.0000" 111(2) 111(2) 16(1)
Rh 2(a) 0 0 0.6589(1) 48(3) 48(3) 20(1)
Ge 1 2(a) 0 0 0.4209(1) 70(4) 70(4) 24(1)
Ge2 4(b) 112 0 0.2615(1) 1 60(5) 58(5) 26(1)
a
This coordinate was fixed to determine the origin of coordinates.
630
Chapter 7
10 20 30 40 50 60
Bragg angle,
20
(deg.)
Figure
7.12.
The observed and calculated powder diffraction patterns of CeRhGe3 after the
completion of Rietveld refinement. The data were collected from a ground CeRhGe3 powder
dusted on a flat sample holder using a rotating anode Rigaku TTRAX powder diffractometer
in a step scan mode with a step
A20
=
0.01'. All notations on the plot are identical to
Figure
7.2.
It is easy to verify by the calculation of the Fourier map that no additional
atoms are present in the unit cell of this material. All interatomic distances
are normal. The crystal structure is visualized in Figure 7.13, from which it
is easy to see that thermal ellipsoids of all atoms are reasonable.
When the final residuals obtained after the Rietveld refinement (Table
7.9) are compared with those obtained during Le Bail's full pattern
decomposition (Table
6.9), the differences are small, which serves as another
confirmation of the correctness of the structural model in the space group
I4mm.
In
this regard, it is useful to illustrate the Rietveld refinement of a
different model of the same crystal structure (see Table 6.15), which was
constructed earlier in the higher symmetry space group
-
I4lmmm. This
model was discarded based on high
RF,
and also based on the presence of
strong peaks on the Fourier map, which were too close to the atoms already
located in the unit cell.
Crystal structure refinement
Figure
7.13. The crystal structure of CeRhGe3 as determined from Rietveld refinement in the
space group I4mm (see
Table
7.10 and
Figure
7.12). Displacement ellipsoids are shown at
99% probability. See the footnote on page 634.
Beginning again with the background, peak shape and unit cell
parameters along with the zero shift, determined during Le Bail's full pattern
decomposition, we will attempt to perform Rietveld refinement of the crystal
structure model shown in
Table
7.1 1.
The corresponding data file,
Ch7ExOZb.inp, containing initial parameters for the Rietveld refinement
using LHPM-Rietica is found on the CD. The observed and calculated
intensities after the full profile least squares are plotted in
Figure
7.14.
Visual analysis of
Figure
7.14
immediately indicates that the agreement
between the observed and calculated intensity is poor. Furthermore, Rietveld
refinement of this model of the crystal structure, during which the
coordinates of atoms in the 4(e) site plus population of 4(d) and 4(e) sites
were optimized together with the overall isotropic displacement parameter,
results in the removal of
Rh
atoms from all sites: population of both sites by
Rh
becomes negative (see the data file Ch7Ex02c.inp on the CD).
Table
7.11. Coordinates of atoms in the unit cell of CeRhGe3 as determined from x-ray
powder diffraction data in the space group symmetry I4lmmm (earlier discarded as wrong, see
Table
6.15).
Atom
Site
x
Y
z
Ce
2(a) 0
0
0
0.25Rh+0.75Ge
4(d) 112
0
1 I4
0.25Rh+0.75Ge
4(e)
0 0
0.363