Pecharsky V.K., Zavalij P.Y. Fundamentals of Powder Diffraction and Structural Characterization of Materials

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662

Chapter

7.9

Rietveld refinement and completion of the

rn~~Mo,O~~

structure1

this section, we will illustrate the Rietveld refinement of a complex

molybdenum oxide-based layered structure containing 15 independent Mo

and

atoms with an intercalated organic methyl ammonium ion (one N, one

C and 6 H atoms) using conventional x-ray powder diffraction data. The

model of the Mo7OZ2 framework was obtained from the assumption that it is

nearly identical to that found in the T12M07022 compound, as identified by a

search of the ICSD database (section 6.15). Structure determination should

be completed by locating the methyl ammonium (CH3NH3f) ion from a

difference Fourier map and refining all relevant parameters. When located in

the unit cell, the N atom is clearly distinguished from carbon by much

shorter distances the former makes with oxygen, because of

N-H...O

hydrogen bonding.

The experimental powder diffraction pattern was collected from

to 100'

28 with a 0.02' step and a counting time varying from 30 seclstep at low

Bragg angles to 60 seclstep at high angles, scaled after the data collection to

a constant 30 seclstep counting time. The following parameters were used at

the beginning of this refinement:

The initial model of the structure (the molybdenum oxide layer) was

taken from

Table

6.42.

The default profile parameters were taken from the instrumental

parameter file Scintag.prm as described in section

7.6.

The sample shift parameter

4.49

for the sample displacement s

-0.098

mm, was obtained together with the unit cell dimensions during

lattice parameters refinement.

The space group is

C21c and the unit cell dimensions are

23.065,

5.5134,

19.561

122.93" as determined earlier.

The overall isotropic atomic displacement parameter was assumed at

Uiso=

0.015 A'.

The starting model with all necessary parameters is found on the CD in

the files Ch7Ex07a.e~~ and Ch7Ex07a.cif, and the experimental pattern is

in the file Ch7ExO7-CuKa.raw. Initially, only the phase scale factor and 6

coefficients of the shifted-Chebyshev polynomial to approximate the

background were refined, resulting in relatively high residuals, which are

listed in

Table

7.22.

The poor fit is mainly due to a mismatch between the

observed and calculated intensities as can be seen in

Figure

7.29.

P.Y.

Zavalij and MS. Whittingham, The crystal structure of layered methylammonium

molybdate (CH3NH3)2M07022 from X-ray powder data, Acta Cryst.,

C53,

1374

(1997)

Crystal structure refinement

663

ma,Mo,O,,,

37.6

R,,

46.8

Bragg angle,

(deg.)

II I

111

11 11111111 llllllllllllllll

lUlllllllllllUlllllllUillllllwMlII~IIIIB11

I""""'I'""""I""""'I"~""","'""'~,~~~~~l~~~"~"~I~"~

,,,,',.','''''

10 20 30 40 50 60 70 80 90

Bragg angle,

(deg.)

Figure 7.29. The observed and calculated powder diffraction patterns of ma2M07022 after the

initial Rietveld least squares minimization with only the scale factor and the background

refined. The difference

(YiobS

&ca'c)

is shown using the same scale as both the observed and

calculated data but the plot is truncated to fit within the range [-3000, 30001 for clarity. The

ordinate is reduced to -114 of the maximum intensity to better illustrate low intensity Bragg

peaks. The inset clarifies the range between 73 and 86' 20.

Table 7.22. The progress of Rietveld refinement of the crystal structure of

from

ray powder diffraction data.

Refined parameters RD RWD

Initial model from T12M07022 (Mo and

only) (CD: 93.3 93.5 98.8 1579

Ch7Ex07a.e~~

and

Ch7Ex07a.cif)

Scale and background (Figure 7.29) 37.6 46.8 38.2 396

Plus unit cell,

and preferred orientation (PO) along [I001 34.2 43.1 36.9 335

Plus coordinates,

Ui,,

and PO along [010] 19.4 25.6 15.2 120

Scale, background, PO,

asymmetry (a), and porosity a1, a2 13.7 18.8 12.0 64.3

Scale, background, PO,

coordinates,

Ui,,

(Figure 7.30); 12.9 17.9 10.6 58.5

(CD:

Ch7Ex07b.e~~

and

Ch7Ex07b.cif)

Same as above with the region 20

15" excluded 10.5 14.5 11.8 31.8

Plus PO ratio and

7.2 9.4 7.1 13.5

sample shift,

and C1 located and added 5.4 7.4 5.9 8.3

All profile parameters

sample shift,

X,,

Y,,

with fixed 5.3 7.2 5.9 7.8

PO ratio (Figure 7.31 and Table 7.23);

(CD:

Ch7Ex07c.e~~

and

Ch7Ex07c.cif)

664 Chapter

It is highly likely that the intensity mismatches are caused by a relatively

crude initial structural model since the atomic coordinates were taken

directly from the T1-based structure, even without correcting for the lattice

distortion along

(compare the unit cell dimensions of both materials in

Table 6.41). Furthermore, a considerable preferred orientation could be

expected because of yet another distinctly layered structure (see Figure

6.30). Therefore, the subsequent refinement included unit cell dimensions,

grain size contribution to peak broadening

and preferred orientation (PO)

parallel to the [loo] direction, along which the M07022 layers are stacked.

Some improvement of the fit has been observed as a result. Unlike in the two

examples considered in sections 7.7 and 7.8, the preferred orientation is not

as strong here, possibly due to a less severe cleaving of the particles during

grinding.

A noticeable improvement, especially in the Bragg residual, occurs when

the coordinates of all atoms have been refined together with the isotropic

displacement parameters: individual for the Mo atoms and one parameter,

common for all oxygen atoms. A second preferred orientation parameter

along the

[OlO] axis was added as well, but its effect on the improvement of

the fit was quite small.

slightly negative overall isotropic displacement

parameter, U,,,

-0.002(4) A2, for the

atoms likely indicates a

contribution from the specimen porosity, which to a certain extent, also

incorporates other unaccounted systematic errors, e.g. absorption and beam

size exceeding the sample dimensions at low Bragg angles due to an

improper selection of the divergence slit aperture. Therefore, the porosity

effect was optimized in a subsequent refinement using two parameters

(a,

and a2) in the Suortti approximation. The latter refinement was carried out

after isotropic atomic displacement parameters were set to 0.015

for all

molybdenum and 0.020

for all oxygen atoms, and all atomic parameters

were kept fixed. The two porosity coefficients were refined to al

0.3 1 and

0.16 starting from the initial 0.40 and 0.40, respectively, after which

they were kept fixed through the end of the Rietveld refinement. Next, the

individual atomic parameters were released and re-refined. This substantially

improves the fit as shown in Figure 7.30.

Regardless of the considerable improvement, some mismatches between

the observed and computed intensities remain. At this point, we are still

missing one nitrogen and one carbon from the methyl ammonium ion, not

counting six hydrogen atoms, which will be nearly impossible to locate from

x-ray powder data (see section 7.7). Pinpointing the locations of carbon and

nitrogen in the unit cell and their inclusion into the model should indeed

improve the fit. However, another potentially deleterious effect on the

overall fit is the presence of a large intensity peak at low Bragg angle (28

go),

in front of which there is a much smaller and broader impurity peak.

Crystal structure refinement

665

Hence, similar to the previous example, the range below

15" was

excluded from further refinement. It is certainly worthwhile to note that this

exclusion eliminates -400 points (-9

from the profile, which contains

more than 4,600 data points total, but it leaves out only 3 (-0.3

of about

1000 possible Bragg reflections. As far as the structural model is of concern,

such truncation of the experimental data is indeed valid, and is often

employed in structure determination from powder diffraction. With this

modification, followed by several least squares minimization steps, profile

residuals decrease but

is slightly increased.

The subsequent Rietveld refinement of the preferred orientation ratio and

anisotropic peak broadening, further improves the fit resulting in

9.4

At this point, a difference Fourier map was computed and it produces

two peaks, which are notably stronger than the others. Their geometry is a

nearly ideal match with the two missing atoms

and C

of the methyl

ammonium ion. After including them into the model and completing the

Rietveld refinement, the final fi<(~i~ure

7.31)

is quite satisfactory-.

ma2Mo,02,,

=12.9%

&,=

17.9%

10.6

Bragg angle,

(deg.)

Figure

7.30.

The observed and calculated powder diffraction patterns of rn~~Mo~0,~ after

preferred orientation, individual atomic parameters of the Mo

and

atoms were refined

together with some profile parameters aid correction for porosity effects. The difference

(yrbS

Y,~"")

is shown using the same scale as both the observed and calculated data but the plot is

truncated to fit within the range [-3000,3000] for clarity. The inset clarifies the range between

73 and

86"

28.

Chapter

Bragg angle,

(deg.)

I1 I

I I

II 111 11 1111111

llllllllllu~11111llllUllllllllllUllllllll1mwllllUmlII

I""""'I""""'I"""'"I'""""I""""'I'"~"~"I""""',""""'l"""'

10 20 30 40 50 60 70 80 90

Bragg angle,

(deg.)

Figure

7.31.

The final Rietveld plot of

data. Low Bragg angle range below

15"

was excluded from the final refinement because it contains an extremely strong peak

(nearly an order of magnitude stronger that all other peaks), which is susceptible to all kinds

of experimental errors and may strongly influence the least squares minimization.

Attempts to adjust other parameters, which can be potentially refined, do

not lead to a statistically significant improvement of the fit, or they have a

tendency to move the solution away from a global minimum,

andlor the free

variables out of an acceptable range. Certain combinations of simultaneously

refined variables cause the least squares to diverge, despite an already good

fit. For example, adding preferred orientation in a spherical harmonics

approximation gives a minuscule additional correction of 1.006 and,

therefore, does not improve the fit. When individual isotropic displacement

parameters were refined for light atoms

(0,

C, and

N),

they were chaotic:

from unexpectedly high to unphysical negative values, without lowering the

residuals. When anisotropic displacement parameters of the heavy Mo atoms

were refined, they too, were unphysical, resulting in the so-called open

ellipsoids, or represented abnormally strong anisotropy

tremendously

elongated or flattened ellipsoids. Thus, the fit shown in

Figure

7.31

may be

considered final and the resulting model of the crystal structure is found in

Table

7.23

and on the CD in the files

Ch7Ex07c.e~~

and

Ch7Ex07c.cif.

Crystal structure refinement

667

Table

7.23.

Final atomic parameters and interatomic distances (in A) of ma2M07022 obtained

from Rietveld refinement using GSAS. The refined unit cell parameters are: a

23.0707(3),

5.51522(7),

19.5669(2)

122.930(1)", V

2089.68(5)

A3,

space group C21c. All

crystallographic data can be also found on the

in the files

Ch7Ex07c.e~~

and

Ch7Ex07c.cif.

Atom

U~SO

Bond distances,

Mol Mo2 Mo3 Mo4

clb 0.265(1) 0.143(4) 0.196(2) 0.014(4)

Hydrogen bond distances for

are: 2.97(3)

to 05, 2.88(3)

to 02, 2.85(3)

to 010,

and 2.76(2)

to 01 1.

C1 distances: 1.46(3)

to N1,3.31(4)

to 02,3.25(3) A to 01, and 3.39(3) A to 02.

The correctness of the crystal structure is doubtless because the M070~~

layer is isotypical to both T1- and Cs-based compounds (see Table

6.41).

The

nearly identical layer also exists in the intercalate, containing a larger

organic molecule:

RMO~O~~.H~O, where

4,4'-bipyridinium (H-NC5H4-

C~H~N-H)".' The orientation of the methyl ammonium molecule (in other

words, the recognition of the nitrogen and carbon atoms) is also convincing

because the N atom is much closer to the

atoms than the

atom, which is

due to the formation of strong N-H.a.0 hydrogen bonds. The C-H...O

interactions are much weaker, as follows from the comparison of bond

lengths: 2.8 to 2.9

for N--.O and 3.3

or more for the C...O distances.

The

C-N

bond length, 1.46(3) A, is within the expected range (compare to

single crystal data for maV3O7, where the C-N bond length is 1.487(8)

A).

Stacking of the layers with the intercalated methyl ammonium ions forming

P.J.

Zapf,

R.C.

Haushalter, and

Zubieta. Crystal engineering of inorganiclorganic

composite solids: the structure-directing role of aromatic ammonium cations in the

synthesis of the "stepv-layered molybdenum oxide phase

[4,4'-H2bpy][Mo7022].H20,

Chem. Commun., 321 (1997)

668 Chapter

N-H...O

hydrogen bonds, which hold the layers together, is illustrated in

Figure

7.32.

The peculiarity of this refinement is the relatively large unit cell with 17

non-hydrogen atoms in the asymmetric unit, resulting in 1035 possible

independent Bragg reflections and a total of 75 free least squares variables,

respectively. This example also illustrates that the location and the quality of

the determination of light

and

atoms, forming a small organic molecule

encapsulated between massive molybdenum oxide layers, is quite reliable,

provided sufficient quality powder diffraction data are available. The

preferred orientation in this case is less severe when compared to the two

previous examples:

zloo

0.690(3) and

zolo

1.091(4) with an approximate

1 :2 ratio, which corresponds to the correction range between 0.7 to 1.8, or to

the preferred orientation magnitude of about 2.6. The latter value is still

significant and it may be partially responsible for the unphysical anisotropic

displacement parameters of the Mo atoms.

Figure

7.32.

The crystal structure of

shown along the Y-axis in perspective view.

The Mo atoms are depicted as large dark spheres located inside translucent octahedra formed

by oxygen atoms (small light spheres). The

atoms are shown as small black spheres and

the

atoms as small grey spheres. The

N-[H]...O

hydrogen bonds are shown using thin lines

without hydrogen atoms. Unlike in the crystal structure of the

tmaV307

(see section

7.8),

the

coordinates of hydrogen atoms cannot be easily computed because methylammonium ion can

rotate around the

C-N

bond, although this rotation is restrained by the formation of

N-H...O

hydrogen bonds. Hydrogen atoms could not be located from a differential Fourier map

because of the presence of the large amount of strongly scattering atoms (Mo)

the material.

Crystal structure repnernent 669

7.10

Rietveld refinement

Mn7(OH)3(V04)41

Using this example and employing conventional x-ray powder data, we

will illustrate Rietveld refinement of a medium complexity inorganic

structure with 10 independent atoms occupying various sites in an acentric

hexagonal space group symmetry. Several sites are occupied partially and

therefore, we will learn how to perform a sensible refinement of the

chemical composition. The refinement begins from the isostructural and

fully ordered zinc-vanadate-sulfate crystal structure, as was established in

the previous chapter (section 6.16). During the least squares minimization,

partially vacant sites were identified and their populations were refined in

the first approximation independently, and then with reasonable restrictions

on chemical composition, until the complete convergence was achieved. The

correctness of the crystal structure determination from powder data was later

confirmed by single crystal diffraction work. This example also shows how

some of the constraints, realized in

GSAS,

can be invoked.

The idealized chemical composition of the title compound when all sites

are fully occupied is Mn7(OH)3(V04)4. The experimental data were collected

with a 0.01" step.

the range from 12 to 70"

the counting time was 15

seclstep, and it was increased to 30 seclstep from 70 to 132' 28.

this way,

it was possible to achieve better counting statistics at high Bragg angles,

where the scattered intensity is significantly reduced, while simultaneously

avoiding a potential for errors associated with, for example, continuously

varying or different divergence slit apertures. The high Bragg angle range

was scaled to a fixed counting time of

seclstep for all computations. The

following parameters were used at the beginning of this refinement:

The initial structural model was taken from Zn7(OH)3(S04)(V04)3, which

crystallizes in the same (P63mc) space group with

12.813 and

5.1425 8L, by replacing Zn with Mn and S with

as was assumed in the

previous chapter (Table 6.44).

The default profile parameters were taken from the instrumental

parameter data file

Scintag-prm,

as described above in section 7.6.

The sample shift parameter S,

5.13 for the sample displacement s

-0.1 12 mm, which was obtained together with the unit cell dimensions at

an earlier stage of the structure solution.

The space group is P63mc and the unit cell dimensions are a

13.229

and c

5.2553 8L, also determined earlier.

The overall isotropic displacement parameter

U,,,

0.0 15 8L2.

Zhang,

P.Y.

Zavalij, MS. Whittingham. Synthesis and characterization of a pipe-

structure manganese vanadium oxide by hydrothermal reaction.

Mater. Chem.,

137

(1999).

670

Chapter

The initial structural model with all other relevant parameters is found on

the

in the files Ch7ExOSa.exp and Ch7ExOSa.cif, and the experimental

data are in the file Ch7Ex08-CuKa.raw. Initially, only the phase scale

factor and six coefficients of a shifted-Chebyshev polynomial to

approximate the background were refined, which resulted in a reasonable fit

as shown in

Figure

7.33. The residuals, shown in

Table

7.24,

were quite low,

especially taking into account that the model of the crystal structure has been

adopted from a different compound, where the geometry is expected to be

somewhat different

(e.g. the

SO4

group is smaller than the

V04

group).

Combined with a potential for a preferred orientation in the specimen,' this

causes obvious but not severe mismatches between the observed and

calculated intensities (e.g. see the inset in

Figure

7.33).

(I)

80 85 90 95 100 105

Bragg angle,

(deg.)

I I

I IIIIII

1111

Ill

1111111111

11111

I11l111

1111

1111111111111111

1111111111111111111lllllll

1111111111111111111lll II IIIIIIIIIIIII

11111

1111111111111111111

20 30 40 50 60 70 80 SO 100 110 120 130

Bragg angle,

(deg.)

Figure

7.33. The observed and calculated powder diffraction patterns of M~I~(OH)~(VO&

after the initial Rietveld least squares minimization with only the scale factor and the

background refined. The difference (Y:~'

Y?) is shown using the same scale as both the

observed and calculated data but the plot is truncated to fit within the range

[-2000,2000]

for

clarity. The inset clarifies the range between

and

106"

28.

It is not expected to be significant because of the three-dimensional metal oxide

framework (see

Figure

6.32), and not a layered structure as was always the case in the

three previous examples. Yet the obvious anisotropy of the crystallite shapes (see the inset

Figure

6.31) suggests that the possibility of a complex preferred orientation cannot be

completely excluded.

Crystal structure refinement

Table 7.24. The progress of Rietveld refinement of Mn7(0H)3(V04)4 from x-ray powder

diffraction data.

Refined parameters

RwD RB

Initial (CD:

Ch7ExOSa.exp

and

Ch7ExOSa.cif)

82.3 83.7 99.4 409

Phase scale and background (Figure 7.33) 20.8 27.0 31.5 42.6

Plus unit cell, grain size broadening

(3,

peak asymmetry

(a)

13.3 16.6 18.7 16.1

and preferred orientation (PO) along [OOl]

Plus coordinates and individual

Uiso

of all atoms 11.1 13.7 14.4 11.0

Overall

Uis0

(Mn and V), overall

Uiso

(0), population parameter 1 1.0 13.8 14.3 1 1.1

(g) for Mn2, V2 and 06, (Figure 7.34). CD:

Ch7ExOSb.exp

and

Ch7ExOSb.cif

Plus strain broadening,

and

Y,,

and sample displacement 10.7 13.4 13.4 10.5

Porosity, a, and a2, then coordinates of all atoms, overall

Uiso

10.7 13.3 13.7 10.4

(Mn, V) and overall

Uis0

(0); population parameters restricted as

g(V2a)

g(06)

g(V2b)

Plus

asymmetry, sample displacement,

X,,

Y,,

spherical 9.7 12.3 1 1.9 8.85

harmonics of 8th order (6 free variables), coordinates of all

atoms

Plus 12 coefficients of the background, transmission error 9.0 11.7 10.7 7.98

(Figure 7.36, Table 7.25);

CD:

Ch7ExOSc.exp

and

Ch7ExOSc.cif

The subsequent refinement included unit cell dimensions, grain size peak

broadening

(3,

peak asymmetry

(a)

and preferred orientation along [OOl].

The latter is a typical preferred orientation axis to try first in both the

hexagonal and tetragonal crystal systems. All parameters refined, result in

the noticeable improvement of the fit, lowering the weighted profile residual

by close to 10

from

to 16.6

Such a substantial change is, for the

most part, caused by adjustments in three parameters: Xfrom 0.023 to 0.044,

asymmetry

from 0.025 to 0.010, and preferred orientation along the [OOl]

axis, with the associated parameter changing from 1 to 1.25 (correction

factors vary between 0.5 and 1.4, which corresponds to the preferred

orientation magnitude of 2.8). The unit cell dimensions practically did not

change.

Refining coordinates of all atoms except Mn2 and

H1'

and individual

isotropic atomic displacement parameters results in little improvement of the

The only potentially free variable in the coordinate triplet of Mn2

(z)

was constrained to

0.8217 [this somewhat unusual value has been taken from the original paper on

Zn7(OH)3(S04)(V04)3] to maintain the fixed origin along the Z-axis. The origin of

coordinates is not fixed in the space group P63mc due to the absence of symmetry

elements (planes or axes), perpendicular to Z, or centers of inversion. Thus, the

coordinates (if none are fixed at a constant value) become severely correlated and the

whole structure may be shifted by any translation along the Z-axis.

similar situation has

already been discussed in the case of CeRhGe3 (see section 7.4), where there are no

symmetry elements fixing the origin of coordinates along the Z-axis in the space group

symmetry I4mm. The coordinates of the HI atom were constrained to be identical to those