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

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672

Chapter

fit, yet another 3% reduction has been achieved. All individual isotropic

displacement parameters are positively defined but

Ui,,

of Mn2,

and

were noticeably higher and increase continually, which points to possible

vacancies in these positions.'

The deficiency, if any, makes sense

fkom a structural point of view as

well, because the suspected defect sites are located along the tunnels, which

exist around the 6-fold and 3-fold axes, thus leaving the main framework

intact. At this point, the overall

U,,,

were assigned separately to all metal and

oxygen atoms and the occupancies of the three suspicious sites were refined

(Figure

7.34).

20 30 40 50 60 70 80 90 100 110 120 130

Bragg angle,

(deg.)

Figure

7.34.

The observed and calculated powder diffraction patterns of MII,(OH)~(VO~)~

after preferred orientation, individual atomic parameters and occupancies of Mn2, V2, and 06

were refined. The difference

(cobs

Ca")

is truncated to fit within the range [-2000,2000] for

clarity. The inset clarifies the range between 73 and 106" 20.

determined in the original Zn-based structure because their refinement is unfeasible using

the existing x-ray powder diffraction data due to the low scattering ability of a single-

electron hydrogen atom (also see section 7.7).

When sinO/h increases, both the atomic scattering functions and temperature factors

decrease exponentially (see sections 2.11.4 and 2.11.3, respectively). Thus, the

unreasonably high isotropic displacement parameters of selected atoms indicate that the

scattering ability of the respective sites is reduced. In the title compound, the only

reasonable explanation is that the suspected sites are partially occupied.

Crystal structure refinement

673

A difference Fourier map, calculated at this point, reveals an additional

small electron density maximum in the tetrahedral cavity next to the partially

occupied V2.' Thus, it is reasonable to assume that the V2 site splits into two

independent partially occupied positions with the coordinates, which

distribute V atoms in a random fashion in two adjacent tetrahedral positions

rather than being simply vanadium-deficient. We label these two sites as

V2a (corresponding to the former V2) and V2b (corresponding to the Fourier

peak). Refinement of this model slightly improves the fit. Subsequently,

additional profile parameters (Y,

Yo, and sample displacement) were included

in the refinement, followed by a typical procedure of refining the porosity in

the Suortti approximation with fixed atomic coordinates and Ui,,, and then

fixing the porosity parameters for the remainder of the refinement.

Obviously, the prohibitively short distance between V2a

and V2b

mandates that the following relationship holds:

Furthermore, the analysis of the values of the occupation factors refined for

V2a, V2b, and 06 point to the following relationship:

These two relationships can be easily programmed in GSAS and in the

majority of Rietveld software codes by using a constraint apparatus, which

was briefly discussed above (see section 7.3.3 and

Eq.

7.9).

Since the

constraints affect only the shifts that are determined during every least

squares refinement cycle but not the values of the related parameters, the

latter should be synchronized manually prior to imposing constraints. For

example, in our case when the computed shift for

gv2a

is 0.02, then the new

values of the constrained parameters (Eqs. 7.9, 7.11 and 7.12) will be

calculated as follows:

If before the beginning of the constrained refinement, gvza is not equal to

go6, e.g. they are 0.6 and 0.8, respectively, then after adding the shifts

according to Eq. 7.13, the corresponding values will become 0.62 and 0.82.

Thus, if needed, parameters constrained in this way should be matched

This peak is characterized by a reasonable tetrahedral configuration created by the oxygen

atoms, except that it is too close to the existing

V2.

Considering the deficiency of the

site, it is feasible that the structure contains two types of

[V04]

tetrahedra distributed

the

structure in a random fashion.

674

Chapter

manually: in this example, both

gv2a

and go6 should be set to identical values

and

gvza

and

gv2b

should sum up to unity.'

The relationships between the occupancies in this crystal structure have

both the chemical and physical sense. The V2 atom and the surrounding 4

oxygen atoms (three 04 and one 06) in a fully ordered structure create a

chain shown in

Figure

7.35a, where the occupied (grey) and empty (white)

tetrahedra are alternated. Vanadium atoms can also occupy pairs of corner-

sharing tetrahedra, thus forming a well-known V2O7 group (hatched in

Figure

7.35b). When a "mistake" occurs and a vanadium atom "jumps" to

the next empty site, the corner sharing pair of empty tetrahedra appears, in

which the shared corner must be vacated because the corresponding oxygen

atom is no longer bound to any vanadium atom. Therefore, vacancies on the

06 sites, constrained as shown in Eq. 7.12, should exist, as was confirmed

by the independent refinement of

go6.

The subsequent refinement included profile parameters X, Y, Xu, Y,, peak

asymmetry, sample displacement and transparency shift. Preferred

orientation was switched from the March-Dollase to the 8th-order spherical

harmonics expansion

variables total) and 12 coefficients of the shifted-

Chebyshev polynomial background approximation were employed. A

reasonably good fit, shown in

Figure

7.36, was achieved as a result.

V,O,

vacant site

Figure

7.35.

Chains of tetrahedra in the MQ(OH)~(VO~)~ structure. The fully ordered chain

(a) with the composition V04, where vanadium occupies every other tetrahedron (shaded) and

the remaining are vacant (white). The disordered chain

(b),

where some vanadium atoms

occupy pairs of neighboring tetrahedra, forming V2O7 groups (hatched), thus creating

vacancies on the 06 site and resulting in the chemical composition

(V04)1-x(V207)xiZ

Another example of synchronizing constrained parameters can be given considering the

coordinates of

V1 and 02

05 atoms in this structure. All of them are located in the 6(c)

sites, with the coordinates of the independent atom

xk.

Hence, when entering the

respective coordinate parameters it is necessary to ensure that

-x.

Crystal structure rejkement

675

Mn,(OH),(VO,),,

9.0

R,,=

11.7

10.7

90 95 100 105

Bragg angle,

(deg.)

I I I

I IIIIII

1111

1111l11111 IIIII

IIIIII

1111

1111111111111111

1111111111111111111lllllllllllllllllllllllllllll

111111111111111

11111

111111111111111111

20 30 40 50 60 70 80 90 100 110 120 130

Bragg angle, 20 (deg.)

Figure

7.36.

The final Rietveld plot of MII~(OH)~(VO~)~ data. The inset clarifies the details

between

and

106"

20.

The preferred orientation correction was accounted for in two ways

during the refinement. First, the March-Dollase approach with one texture

axis

[001] resulted in

1.247(2) and correction coefficients ranging from

0.52 to

1.39,

which gives the preferred orientation magnitude of 2.70.

Second, the sth-order spherical harmonics expansion, which corresponds in

this crystal system to six adjustable parameters (200,400, 600, 606, 800, and

806) was attempted with the March-Dollase preferred orientation correction

(i) left as is but fixed

(i.e. the spherical harmonics were in addition to the

March-Dollase model), or (ii) eliminated. Both ways result in practically an

identical result except for the magnitudes of the coefficients.

the second

case, the correction coefficients ranged from 0.61 to 1.54, which corresponds

to the preferred orientation magnitude of

2.52.

The fully refined model of the crystal (coordinates, population and

individual isotropic displacement parameters of atoms) and interatomic

distances in the crystal structure of MQ(OH)~(VO~)~ is found in

Table

7.25

and it is shown in

Figure

7.37

as the arrangement of the coordination

polyhedra of the Mn and

atoms.

676

Chapter

Table

7.25.

The final atomic parametersa9b and interatomic distances (in A) in the crystal

structure of

M~I~(OH)~(VO~)~ obtained from Rietveld refinement using GSAS. The fully

refined unit cell parameters are:

13.2292(1),

5.25467(6) A,

796.42(2) A3, the

space group is P63mc. Full set of crystallographic data is also found on the

in the files

Ch7ExOSc.exp and Ch7ExOSc.cif.

Bond distances,

Atom Site

Mn 1 Mn2 V1 V2a

1.66 1 (6)3

05 6~ 0.9199(3) -0.9199(3) 0.586(2) 2.214(7),, 1.637(6)

2.302(8),

06 2b 113 213 0.118(3) 1.81(2)

~1~ 6c 0.433 -0.433 0.156

Ui,,

is 0.0145(2)

for metal atoms and 0.0140(8)

for

atoms.

The population parameters are as follows: gv2,

go,

0.819(4), gv2b

0.181(4)

gvza, g~,,2

0.873(4)

(1-2/3gv2b). All other sites are fully occupied.

This parameter was kept constant at all times to maintain a fixed origin of coordinates

along the Z-axis. Although GSAS enables automatic fixation of the origin of coordinates,

the refinement may become unstable, especially when the fit is far from the best.

Taken from the Zn-based compound and kept constant during the refinement.

addition to the already discussed vacancy relationships among V2a,

V2b and 06 atoms, the refined deficiency of the Mn2 site is nearly equal to

213 of the vacancies observed on the V2b site. Although the experimentally

established relationship is approximate, when obeyed exactly

(i.e. when g,,,

2/3gV2+,), the oxidation state of manganese in the Mn2 site is

3+,

while

it is 2+ for the Mnl site. All things considered, the following expression

describes the chemical composition of M~I~(OH)~(VO~)~:

This formula has been confirmed from single crystal diffraction data

(RF

2.2

%),

which give x

0.202(3).' The population of the Mn2 site is

0.792(4), slightly lower than expected 1

213x

0.865.

small difference is

acceptable because a tiny single crystal may not be representative of a large

polycrystalline sample, especially in the case of occupational disorder.

P. Zavalij,

Luta, and

M.S.

Whittingham, unpublished.

Crystal structure refinement

Figure

7.37.

The model of the crystal structure of

Mn7~y(OH)3(V04)4-,(V207)x12

shown along

the Z-axis. The Mnl sites (~n~') are shown as light spheres inside the dark grey [MnOs]

octahedra, the partially occupied Mn2 sites (~n~') are shown as light spheres in the light grey

[MnO,] octahedra, both the

and

sites are shown as dark spheres inside the light grey

[V04]

tetrahedra. Hydrogen atoms from the hydroxyl groups are not shown in this figure.

7.11

Rietveld refinement of the monoclinic FeP041

As established in the previous chapter (section 6.17), the anhydrous iron

phosphate has a relatively simple crystal structure, however, poor

crystallinity of the powder results in broad peaks where full widths at half

maximum vary from 0.2 to

0.5'

when Mo

radiation is employed.

Therefore, the resolution of the diffraction data is quite low, as was

illustrated in

Figure

6.33.

There are only

atoms in the asymmetric unit but

Rietveld refinement of the model is complicated by the inadequate quality of

the diffraction data. The model, derived from a suspected analogy with the

hydrated FeP04.2H20, cannot be completed based solely on the powder

diffraction data due to problems with the experiment. Thus, Rietveld

refinement considered in this section starts from the model improved by

Song, P.Y. Zavalij, M. Suzuki,

and

M.S. Whittingharn, New iron(II1) phosphate phases:

Crystal structure and electrochemical and magnetic properties, Inorg. Chem.

41,

5778

(2002).

678 Chapter

geometry optimization using quantum chemical minimization of

approximate coordinates of atoms, as was discussed in the previous chapter.

Furthermore, data quality precludes an unrestrained refinement of even the

optimized model, and similar to the example considered in section 7.8, soft

restraints should be imposed on the geometry of the

PO4 groups. We will,

therefore, take this opportunity to illustrate the role of soft restraint

weighting in Rietveld refinement.

The experimental powder diffraction pattern was collected on a rotating

anode Rigaku

TTRAX powder diffractometer using monochromatized Mo

radiation from 5 to 50'

in a step scan mode with a 0.01' step and

counting time of 10 seclstep. The following parameters were employed at

the beginning of this refinement:

The initial structure model, derived and optimized in the previous chapter

(Table

6.47).

The default profile parameters from the instrumental parameter file

Rigaku.prm obtained from the refinement of the La& standard as

described in section 7.6.

The sample shift parameter

3.99 for the sample displacement s

-0.087 mm obtained together with the unit cell dimensions at a stage of

lattice parameters refinement.

The space group P2Jn and the unit cell dimensions

5.489, b

7.493,

8.055 A, and

95.81•‹, determined earlier.

The overall isotropic displacement parameter

U,,,

0.015 A2;

The initial model with all needed crystallographic parameters is found on

the

in the files Ch7Ex09a.e~~ and Ch7Ex09a.cif, and the experimental

data are in the file Ch7Ex09-MoKa.raw. Only the range from 5 to 37.6' 28

was used in all calculations.' The background was approximated manually

with a straight line (i.e. two coefficients of a shifted-Chebyshev polynomial

were employed in this approximation) since unambiguous automatic

background determination was impossible at the beginning of the refinement

due to heavily overlapped Bragg peaks. Only two data points (the first and

the last in the range) were used for background estimation. The initial

refinement of the phase scale factor resulted in

62.7% (Table

7.26)

and

a quite poor fit, which showed that all calculated Bragg peaks were too

narrow. Therefore, both peak broadening parameters,

and

were

manually increased by a factor of

10,

yielding an acceptable weighted profile

residual of 35.6% and resulting in a tolerable fit as shown in Figure

7.38.

This range of Bragg angles corresponds to a

20,,,

89'

when using Cu

radiation. The

use of Mo

radiation in this case was justified by a large goniometer radius

(285

rnm)

and therefore, potentially high resolution, and by the presence of the significant amount of

Fe in the material (iron strongly absorbs Cu

radiation, see

Table

2.3,

and creates a

substantial fluorescent background).

Crystal structure

refinement

679

Table 7.26.

The progress of Rietveld refinement of the crystal structure of anhydrous FeP04

using low resolution x-ray powder diffraction data.

Refined parameters RD

Rw~

Initial

(CD:

Ch7Ex09a.e~~

and

Ch7Ex09a.N)

75.2 77.2 98.6 1414.

Scale factor, linear background adjusted manually

57.4 62.7 67.4 933.

Peak broadening

and

multiplied by

then phase scale

29.4 35.6 32.6 301.

(Figure 7.38)

Plus two background coefficients, coordinates, PO4 group soft-

13.3 16.8 22.6 67.6

restrained with weight

Same but soft-restraints with weight increased to

11.1

13.8 13.8 46.2

Plus grain size broadening,

8.5 11.1 10.7 30.1

Plus unit cell dimensions,

[OlO],

overall

Ui,,

(Figure 7.40)

8.1 10.8 9.9 28.1

(CD:

Ch7Ex09b.e~~

and

Ch7ExO9b.cif)

Plus strain broadening,

then asymmetry and sample

7.3 9.7 6.2 22.7

displacement

Four coefficients of the background

6.5 8.6 4.1 17.9

peak asymmetry,

X,,

Y,;

(Figure 7.41, Table 7.27) (CD:

5.2 7.1 3.6 12.3

Ch7Ex09c.e~~

and

Ch7Ex09c.cif)

Bragg angle,

(deg.)

Figure 7.38.

The observed and calculated powder diffraction patterns of the anhydrous FeP04

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

background refined. The difference

(yiobS

&"lc)

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

-29" 20.

680

Chapter

Both the complexity and low resolution of the experimental data, coupled

with the possibility of far fi-om ideal coordinates of some or all atoms in the

optimized model of the crystal structure' present an interesting dilemma in

the selection of the next set of parameters for a subsequent Rietveld

refinement. Although it is obvious that profile parameters require further

improvement, it also appears that both the inadequacy of the initial fit and

low resolution of the data may not allow their unequivocal refinement. On

the other hand, atomic coordinates likely deviate significantly from their real

values, which is easily seen in

Figure

7.38

indicating significant

discrepancies between the observed and calculated intensities for many

Bragg reflections.

Hence, the following refinement step included a linear background and

coordinates of all atoms.2 In anticipation of considerable problems with the

least squares minimization and high probability of moving away from a

global minimum, soft restraints were employed to restrain the well-known

geometry of the phosphate group

P04.3 Its initial geometry, obtained as a

result of quantum chemical optimization, was nearly perfect: the

P-0

distances vary between 1.52 and 1.54 A, while the 0-P-0 angles were

between 107.8 and 110.2". The following restrains were imposed: the

P-0

distance of 1.53

0.01 A, and the 0-P-0 angles of 109.5

2.0'; the weight

was set to

The first five cycles of the refinement substantially improve the

fit, lowering R, by more than 20

down to 16.8

This reduction,

however, comes at the cost of worsening the PO4 geometry: the P-0

distances now range from 1.43 to 1.61

and the 0-P-0 angles vary from

103 to 117". The Fe-0 distances remain acceptable, and they range from

1.83 to 1.95

but one additional elongated Fe-0 bond of 2.27

emerges.

order to improve the geometry, the soft restraint weight factor was

increased to 10, and several subsequent least squares cycles were conducted.

The weighted residual further decreases and, most importantly, the geometry

of the

PO4 group recovers. The correctness of this adjustment is

demonstrated in

Figure

7.39, which illustrates relative shifts of all atoms as

functions of least squares cycle number. It is obvious, that setting the weight

to 4 does little

stabilize the convergence, while increasing the weight to 10

results in a rapid reduction of the magnitudes of atomic displacements over a

few refinement cycles.

It is worthy of reminding one that the quantum chemical optimization of the geometry has

been performed after Fe was substituted by

Al.

Positions (coordinates) of atoms in the unit cell are the strongest contributors into the

computed integrated intensities of Bragg reflections assuming that preferred orientation

effects are weak. For this powder, preferred orientation was expected (and later found) to

be minor due to small particle sizes and predominantly isotropic particle shapes.

thorough reader should be able to verify the correctness of this statement by attempting

Rietveld refinement without imposing soft restraints.

Crystal structure refinement 68 1

The subsequent refinement included unit cell dimensions, the grain size

peak broadening parameter

preferred orientation in the March-Dollase

approximation with the texture axis [OlO],' and the overall isotropic

displacement parameter

Ui:,,,,

in addition to the linear background, phase

scale and coordinates of individual atoms, which were still soft restrained

with the weight set to

10.' The preferred orientation correction in this

example is insignificant. The corresponding parameter (~0~0

0.986) results

in a range of correction factors varying between 0.98 and 1.04 and therefore,

it can be ignored. This refinement results in an obvious improvement of the

geometry of the crystal structure. The

P-0

distances now range from 1.48 to

1.58

and the

0-P-0

angles are from 102 to 115' (see the file

Ch7ExO9b.cif). The profile fit (Figure

7.40)

is also improved, with the

weighted residual lowering to 10.8%;

0123456

10 12

Least squares cycle

number

Figure

7.39.

Relative shifts of individual atoms (left-hand scale) and average shift

(A)

standard deviation

(o)

ratio during the first

cycles of the least squares refinement of the

coordinates of all atoms in the model of the crystal structure of the anhydrous monoclinic

FeP04. Both the P-0 distances and 0-P-0 angles were soft restrained with the weight of

during the first five cycles. The weight was set to

beginning from cycle No.

The first

five cycles indicate erratic shifts of P and

atoms. Beginning from the sixth cycle, the shifts

of all atoms steadily decrease and approach zero after cycle No.

12.

The axis was chosen after trying the three major crystallographic directions as potential

preferred orientation axes.

The same weight was maintained through the end of this Rietveld refinement.