Pecharsky V.K., Zavalij P.Y. Fundamentals of Powder Diffraction and Structural Characterization of Materials
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Chapter
7
Figure
7.47.
The model of the crystal structure of Gd5Si4 as determined by Rietveld
refinement. The slabs, which are infinite along
X
and
Z
and limited along
Y
to
-6
A,
are
shown as bonded Gd and Si atoms. The major difference between this structure and Gd5Ge4
(Figure
7.44)
is the presence of short Si-Si interslab bonds (which are connected by grey
lines) here, and the absence of short Ge-Ge bonds between the slabs in Gd5Ge4.
As a starting model, we will use the coordinates of atoms taken from
Table
6.49
together with the unit cell dimensions and all profile parameters
determined fiom Le Bail's full pattern decomposition. They are found in the
input file,
Ch7ExlZa.inp
for LHPM-Rietica. The powder diffraction data
are located in the file
Ch7Exl2-MoKa.dat
on the CD. As already
established in section
6.18.3
(see
Figure
6.43
in Chapter
6),
this model of the
crystal structure is also feasible but far from complete because the observed
and calculated intensities do not match well. Thus, the refinement strategy
will be similar to the previous example. The least squares fit here may
Crystal structure refinement
693
become complicated by the pseudosymmetry introduced when we derived
the coordinates of atoms assuming a monoclinic distortion of the
orthorhombic Gd5Ge4-type structure. The progress of Rietveld refinement is
illustrated in
Table
7.33.
Table
7.33. The progress of Rietveld refinement of the crystal structure of GdsSizGez using x-
ray powder diffraction data. Wavelengths used:
hKal
=
0.70932
A,
hKaz
=
0.71361
A.
Refined parameters R, R,D
RB
x2
Initial (profile parameters from Le Bail, model from 3x104 3x104 3x104 2x10'
Table
6.49,
overall
B
=
0.6
Scale factor
a
19.5 25.2 15.0 131.4
Scale and coordinates of all Gd atoms
a
9.19 11.9 6.85 29.4
Scale and coordinates of all Gd and SiIGe atomsa 6.51 8.53 4.46 15.2
Scale,
U,
V,
W,
q,
asymmetry,
a,
b,
c,
y,
sample 5.14 6.82 2.72 9.78
displacement, background, coordinates of all atoms,
overall displacement parameter, preferred orientation
along [001Ia
All, plus individual isotropic displacement parameters of 5.10 6.76 2.67 9.61
Gd atoms; SiIGe atoms displacements in overall
approximation; preferred orientation along [OOlIa
All as above plus occupancies of SiIGe sitesb 5.04 6.69 2.61 9.43
All parameters togetherb 5.03 6.69 2.59 9.41
a
Si and Ge atoms are distributed randomly and equally among four available
crystallographic sites.
Constrained to full occupancy, i.e.
gsi
+gee
=
100%.
The residuals steadily decrease when we include coordinates of Gd and
then SiIGe atoms into the least squares fit, followed by profile, lattice and
displacement parameters as seen in rows
2
through
6
in
Table
7.33. A
Fourier map, calculated at this point, reveals no additional atoms in the unit
cell. When compared to the two previously considered examples, an
additional degree of freedom in the
Gd5Si2Ge2 structure is the distribution of
the Si and Ge atoms among the corresponding lattice sites. Thus, as the next
step we will refine site occupancies constrained to full occupancy by Si/Ge
(i.e.
g~,
=
1
-
gsi).
Final refinement of all parameters converges to low
residuals, indicating that the model of the crystal structure is correct and
complete. The parameters of individual atoms in the crystal structure of
Gd5Si2Ge2 are listed in
Table
7.34 and found in the file
Ch7Exl2b.inp.
It is worth noting that the chemical composition of the material was not
restricted in any other way except to maintain the full occupancy of all Si/Ge
sites.' As follows from
Table
7.34, the chemical composition of the powder
is Gd5Si2.01(4,Ge1,99(4,, which is a nearly ideal match to the nominal
'
In some algorithms (e.g. GSAS), it is permissible to impose restrictions on chemical
composition, if the latter is known, by setting and fixing the total number of atoms of a
specific element in the unit cell during the refinement of population parameters.
694
Chapter
7
composition of the as-prepared sample. The observed and calculated powder
diffraction patterns are shown in
Figure
7.48.
Gd,Si,Ge,,
Mo
Ka
R,
=
5.03%
R,,
=
6.69%
R,
=
2.59%
X2
=9.41
Bragg angle,
20
(deg.)
Figure 7.48. The observed and calculated powder diffraction patterns of Gd5Si2Ge2 after
completion of Rietveld refinement. Compare this figure with Figure 6.43 in section 6.1 8.3.
A
total of 1532 independent Bragg reflections are possible in the examined range.
Table 7.34. Atomic parameters in the crystal structure of Gd5Si2Ge2 fully refined by Rietveld
technique using pseudo-Voigt peak shape function. The space group is P112,Ia. The unit cell
dimensions are: a
=
7.5814(3), b
=
14.8039(6),
c
=
7.7801(3)
A,
y
=
93.203(2)'.
Atom Site
x
Y
z
B
(A2) g
@)
Gd 1 4(e) 0.3258(4) 0.2485(2) 0.0045(3) 0.91(5) 100
Gd2
4(e) -0.0048(3)
0.0989(2) 0.1778(4) 0.88(6) 100
Gd3
4(e) 0.0180(3)
0.4023(2) 0.1799(4)
1.20(6) 100
Gd4
4(e) 0.3583(3)
0.8812(2) 0.1665(3)
1.05(5) 100
Gd5
4(e) 0.3286(4)
0.6226(2) 0.1768(3)
0.64(5) 100
Sil
a
4(e) 0.21 l(1)
0.2496(5) 0.3706(8)
0.8(2) 42(1)
Si2
a
4(e)
0.958(1) 0.251 l(6) 0.896(1)
0.3(2) 65(1)
Si3
a
4(e)
0.206(1) 0.9580(5) 0.470(1)
1.2(2) 44(1)
Si4
a
4(e)
0.156(1) 0.5413(5) 0.474(1)
1.5(2) 50(1)
a
Ge atoms occupy the same site with identical coordinates and displacement parameter.
The occupancy is
g~~=
100
-
gsi
%.
Population parameters were refined with the common
displacement parameter of all Si and Ge atoms to avoid potential correlations.
Crystal structure repnement
695
The refined model of the crystal structure of Gd5Si2Gez is shown in
Figure
7.49,
from which it is clear that similar to both Gd5Ge4 and Gd,Si4, it
is built from the same slabs. The Gd5Si2Gez structure is an intermediate
between the two parent compounds because here pairs of the slabs (B and
C)
are connected by short (Si,Ge)
-
(Si,Ge) bonds, while no similar bonds exist
between the next neighboring slabs
(A
to B and
D
to
C).
It is interesting to
note that these small but distinct differences between the three closely
related structures were established first from powder diffraction' and only
Figure 7.49. The model of the crystal structure of GdsSizGel as determined by Rietveld
refinement. The slabs, which are infinite along
X
and
Z
and limited along
Z
to
-6
A
are shown
as bonded Gd and Si atoms. This structure is intermediate between Gd5Ge4 (Figure 7.44) and
Gd5Si4 (Figure 7.47): only pairs of slabs,
B
and C, are connected with short (Si,Ge)
-
(Si,Ge)
bonds, while no such bonds exist between the pairs (A and
B
and C and
D).
'
V.K. Pecharsky and K.A. Gschneidner, Jr., Phase relationships and crystallography in the
pseudobinary system Gd5Si4-Gd5Ge4,
J.
Alloys Comp.
260,98
(1997).
696
Chapter
7
later were confirmed by a single crystal diffraction experiment.' Indeed, the
single crystal diffraction investigation was complicated by the inherent
twinning of the monoclinic phase, which has no effect on the powder
diffraction data.
The series of compounds existing in the pseudobinary
GdS(SixGel-x)4
system is exceptionally interesting because of a distinct role the crystal
structure plays in defining the magnetic properties of the alloys. As
discussed by Pecharsky and G~chneidner,~ all GdsSi4-type phases undergo a
second order paramagnetic to ferromagnetic transformation on cooling,
which occurs without changing the crystal structure. However, both
Gd5Si2Ge2- and SmSGe4-type phases order ferromagnetically simultaneously
with a crystallographic phase change, during which the slabs move with
respect to one another in a shear fashion, and all interslab bonds reappear as
shown schematically in
Figure
7.50.
Figure
7.50.
The nature of the crystallographic phase changes when the Gd5Ge4-type (left) or
the Gd5Si2Ge2-type structures (right) transform into the Gd5Si4-type structure. Short black
arrows indicate the directions and the numbers above the arrows the extent of the interslab
shear.
'
W.
Choe, V.K. Pecharsky, A.O. Pecharsky, K.A. Gschneidner, Jr., V.G. Young, Jr., and
G.J. Miller, Making and breaking covalent bonds across the magnetic transition in the
giant magnetocaloric material
Gd5(SizGez), Phys. Rev. Lett.
84,4617
(2000).
See V.K. Pecharsky and K.A. Gschneidner, Jr., Gd5(Si,Gel.J4: An extremum material,
Adv. Mater.
13,683
(2001), and references therein.
Crystal structure refinement 697
The coupled magnetic-crystallographic ordering, therefore, becomes a
first order transformation.
In
the ferromagnetic state only the Gd5Si4-type
crystal structure is stable. It is also important to note that the crystallographic
phase change as a function of temperature in this system was discovered first
using powder diffraction data' and two years later, it was confirmed from a
single crystal diffraction experiment by Choe et al. (see reference No.
1
in
the footnote on page 696).
7.13
Epilogue
In
the last three chapters of this book, we considered a number of
practical examples of the structure determination from powder diffraction
data. Their complexity ranged from quite simple structures, with no
coordinate degrees of freedom in the unit cell and only a few Bragg
reflections observed in the examined volume of reciprocal space
(e.g.
LaNi4.85Sn0.15) to rather complex inorganic and mixed inorganiclorganic
materials with numerous coordinate and population parameters and more
than a thousand possible Bragg reflections (e.g. ma2M07022 and Gd5Si2Ge2).
Successful determination of all these structures was enabled by both the high
quality of the experimental data and recent advancements in computational
crystallography. Only one of the considered examples (the anhydrous
monoclinic
FeP04) was based on relatively poor quality data, which was
unavoidable due to the physical state of the material. The problems
encountered in the latter case both during creating a model and Rietveld
refinement, may indeed serve as a highlight of the importance of data quality
in dependable structure determination from powder diffraction.
The fundamental simplicity of the powder diffraction experiment coupled
with the sophistication of the available numerical data processing techniques
and unsurpassed visual elegance, with which the results of the full profile
refinement are represented, have transformed the powder diffraction
technique into a modern and exceptionally important experimental tool in
structure determination. Without a doubt, the powder method is poised to
play a role in structure determinations with ever-increasing complexity.
A
recent example of the successful refinement of the sperm whale (Physeter
catodon) metmyoglobin structure from synchrotron powder diffraction data
is an excellent illustration of how far the technique has been already
ad~anced.~
'
L.
Morellon, P.A. Algarabel, M.R. Ibarra,
J.
Blasco,
B.
Garcia-Landa,
Z.
Arnold, and
F.
Albertini, Magnetic-field-induced structural phase transition in Gd5(Si1,8Ge2.2), Phys. Rev.
B
58,
R14721 (1998).
R.B. Von Dreele, Combined Rietveld and stereochemical restraint refinement of a protein
crystal structure,
J.
Appl.
Cryst.
32,
1084
(1999).
Chapter
7
Polycrystals have, and most certainly will continue to play a considerable
role in future advancements in natural sciences and engineering. Thus, it is
vitally important that structural details at the atomic resolution are known for
every material, physical, chemical, mechanical, and other properties of
which are under investigation. This will create a foundation for the future
understanding of critical structure-property relationships, which in turn, will
be an enabling step for a better understanding of both the nature and
properties of materials and judicious design of advanced materials and
devices, upon which we become so greatly dependent. We may only hope
that this book provides the necessary background and helps to develop
practical skills for many researchers, thus making them ready to take upon
a
challenging but a remarkably gratifying task
-
structure determination from
powder diffraction data.
Crystal structure refinement
7.14
Additional reading
The Rietveld method. WCr monographs on Crystallography
5,
R.A.
Young, Ed., International Union of Crystallography, Oxford University
Press, Oxford, New York (1993).
H.M. Rietveld, Line profiles of neutron powder-diffraction peaks for
structure refinement, Acta Cryst.
22,
15 1
(1
967)
H.M. Rietveld,
A
profile refinement method for nuclear and magnetic
structures, J. Appl. Cryst.
2,
65 (1969).
A.
K.
Cheetham, Structure determination from powder diffraction data:
an overview, in: Structure determination from powder diffraction data.
WCr monographs on Crystallography 13, W.I.F. David,
K.
Shankland,
L.B. McCusker, and Ch. Baerlocher, Eds., Oxford University Press,
Oxford, New York (2002).
Armel Le Bail's Web site dedicated to structure determination from
powder data at
http://pcb4122.univ-lemans.fr/sdpd/index.html.
700
Chapter
7
7.15
Problems
Answers to all problems listed below are located in the file Chapter-7-
Problem-Solutions.pdf on the CD accompanying this book.'
1. Complete structure determination and perform Rietveld refinement of the
model of SrSi2, which you solved in problem 6, Chapter 6. The
experimental powder diffraction pattern is located on the CD in the file
Ch7PrOl-MoKa.dat.
2.
Complete the solution of the crystal structure and perform Rietveld
refinement of the model of LaNill,4Gel,6 from problem 7, Chapter 6. The
experimental powder diffraction pattern is located on the CD in the file
Ch7PrO2-CuKa.dat.
3. Complete the solution of the crystal structure and perform Rietveld
refinement of the model of Hf2Ni3Si4 from problem 8 in Chapter 6. The
experimental powder diffraction pattern is located on the CD in the file
Ch7Pr03-CuKa.dat.
4. Perform Rietveld refinement of the hexamethylenetetramine, C6H12N4,
using the model established in problem 5, Chapter 6. The experimental
powder diffraction pattern is located on the CD in the file
Ch7Pr04-CuKa.raw.
5. Complete the solution of the crystal structure and perform Rietveld
refinement of the model of vanadyl acetate, VO(CH3C00)2, using the
model established in problem
9,
Chapter 6. The experimental powder
diffraction pattern is located on the CD in the file
Ch7PrO5-CuKa.raw.
6. Complete the solution of the crystal structure and perform Rietveld
refinement of the model of manganese oxide, MnOz, which crystallizes in
the space group P42/mnm with
a
=
4.41,
c
=
2.88
A.
The gravimetric
density of the material is
p
=
5.10 g/cm3. Assume that manganese atoms
occupy the site 2(a): 0,0,0. The experimental powder diffraction pattern
is found on the CD in the file
Ch7Pr06-CuKa.raw.
7. Solve the crystal structure and perform Rietveld refinement of the model
of NaV205, which crystallizes in the space group Pmmn with
a
=
1 1.3 17,
b
=
3.61 1,
c
=
4.807
A.
It is known that V2O5 belongs to the same space
group symmetry with the unit cell dimensions
a
=
11.5 1,
b
=
3.564,
c
=
'
For problems
4
-
9
use instrumental parameters file
Scintag.prm
located on the
CD.
Crystal structure refinement
70 1
4.368
A.
The coordinates of atoms in V2O5 are: V in 4(f): 0.10,1/4,0.90;
01 in 4(f): 0.10,1/4,0.53; 02 in 4(f): -0.07,1/4,-0.00; 03 in 2(a):
1/4,1/4,-O.OO. The experimental powder diffraction pattern is found on
the CD in the file
Ch7Pr07-CuKa.raw.
8. Complete the solution of the crystal structure and perform Rietveld
refinement of the model of tungsten oxide peroxide hydrate,
W02(02)(H20), which crystallizes in the space group symmetry P2Jn
with
a
=
12.07,
b
=
3.865,
c
=
7.36 A,
f3
=
102.9". The location of W has
been found from a Patterson map and it has the coordinates x
=
0.680,
y
=
0.066, z
=
0.364. Note that W usually exhibits octahedral or square-
pyramidal coordination (with the peroxide group, 0-0, counted as one
ligand). The experimental powder diffraction pattern is found on the CD
in the file Ch7PrO8-CuKa.raw.
9. Locate the missing water molecule and perform Rietveld refinement of
zinc vanadate (Zn3(OH)2V207.2H20), which crystallizes in the space
group ~Tml with
a
=
6.05,
c
=
7.19
A
starting from the following model:
Zn in 3(e): 1/2,0,0;
V
in 2(c): 0,0, z,
z
=
0.25, 01 in 2(d): 2/3,1/3,z, z
=
0.88; 02 in 6(i): x,2xg,
x
=
0.15,
z
=
0.82; 03 in I(b): 0,0,1/2. The
experimental powder diffraction pattern is found on the CD in the file
Ch7Pr09-CuKa.raw. The data have been affected by a considerable
sample displacement error: -0.2 mm for a 250 mm goniometer radius.