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

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552

Chapter

Table

6.34. Coordinates of atoms in the unit cell of Nd5Si as determined from x-ray powder

diffraction data in the space group symmetry P4,2,2

(RF

27.1%).

Atom Site

Nd 1 8(b) 0.3714 0.0057 0.4561

Nd2 8(b) 0.1280 0.9903 0.8779

Nd3a 4(4 0.3103 0.3103 0

Si 1 8(b) 0.4220 0.2004 0.8095

Si2 8(b) 0.1778 0.2070 0.3108

The coordinates of Nd3 were modified from 0.8 103, 0.1897,3/4

(Table

6.33) to represent

the triplet in a standard notation, i.e.

0 (see

Table

6.30), by using the following

transformation:

112, 112-y, z-0.75.

Figure

6.22.

The model of the crystal structure of Nd5Si4 as determined from x-ray powder

difiaction data.

The value of the residual is higher than any of the RF's we have seen so

far. This is associated with the multiple overlapping Bragg peaks at high

angles, which becomes especially severe at 28

-70" (see

Figure

6.21).

a result, the full pattern decomposition produces individual intensities and

squared structure factors, which are affected by larger than usual errors. This

is easy to verify by eliminating high Bragg angle reflections: when only

reflections below 70" 28 are included in the computation, the corresponding

becomes 20.3%.

Crystal structure solution 553

The Pearson symbol of this crystal structure is tP36 and after consulting

Pearson's Handbook, it is easy to find that the crystal structure of Nd5Si4

belongs to the Zr5Si4-type.' Its model is shown in Figure

6.22.

The solution

of this crystal structure must be completed by Rietveld refinement and it will

be discussed in the next chapter.

6.13

Crystal structure

NiMn02(0H)'

This is an example of a quite simple inorganic compound, nickel

manganese oxide hydroxide NiMn02(OH), with relatively good but far from

perfect crystallinity. Due to a peculiar shape of the crystallites, which grow

as well-defined and elongated faceted needles (see inset in Figure

6.23), the

powder exhibits a tendency toward a complex preferred orientation. The

latter is always a complicating factor especially when a structure solution

from first principles should be ~ndertaken.~ This example also illustrates

how to detect and refine hydrogen atoms from powder data (see Chapter 7),

which may be a difficult and often impossible task to accomplish using x-

rays.

Powder diffraction data were collected using a specimen prepared

from a

thoroughly ground powder screened through a 38

pm sieve. First, a fast

experiment was conducted in the range 2

70" using a continuous

scan with a sampling step of 0.03" and a scan rate of 1 deglmin. Positions of

23 individual Bragg reflections at 29

60" were determined using a semi-

manual profile fitting. Indexing was performed using TREOR, which

resulted in an orthorhombic base-centered lattice with

possible

reflections,

F20

135 (0.003,47).

order to proceed with the structure solution, a high quality powder

diffraction pattern to 29,,,

110" was collected in a weekend experiment

(Figure 6.23, data files

Ch6ExO5-CuKa.xy

and

Ch6ExO5-CuKa.raw

are

located on the CD). The unit cell dimensions were refined using all 75

observed Bragg peaks resulting in a

2.8609(1) A,

14.6482(5) A, c

5.2703(2) A,

220.86(2) A3, and sample displacement of -0.123(6) mm

for a 250

rnrn

goniometer radius.

H.U.

Pfeifer and

Schubert, Crystal structure of Zr5Si4,

Metallk.

57,

884

(1966).

R. Chen, P.Y. Zavalij, MS. Whittingham,

J.E.

Greedan,

N.P.

Raju, and M. Bieringer, The

hydrothermal synthesis of the new manganese and vanadium oxides,

NiMn03H,

MAV307

and

MAo,75V4010~0.67H20

(MA

CH3NH3),

Mater. Chem.

(1999).

The

polycrystalline material was prepared by hydrothermally treating a mixture of

Li2C03,

N(CH3)4Mn04,

and

Ni(CH3C00)2

taken in a molar ratio

1.5:

200•‹C

for

days.

When preferred orientation effects are strong, the intensities of Bragg reflections become

biased by systematic errors.

correction is nearly impossible before the crystal structure

is solved and the preferred orientation refined using an acceptable model. These errors are

in addition to the errors introduced by deconvolution of the overlapped Bragg peaks.

Chapter

10 20 30 40 50 60 70 80 90 100 110

Bragg angle,

(deg.)

Figure

6.23. Powder diffraction pattern collected from the NiMnO,(OH) powder using Cu

radiation on a Scintag XDS2000 diffractometer. The experiment was carried out in a step scan

mode with a step

0.02'

and counting time 30 sec per step. The vertical bars indicate calculated

positions of the

Kal

components of all possible Bragg reflections. The inset shows the

scanning electron microscopy image of particle morphology in the as-received state.

Analysis of the systematic absences indicated three possible space

groups: Cmcm or one of its non-centrosymmetric subgroups Cmc2, or C2cm

(Ama2 in a standard setting). The pattern decomposition was carried out by

using a profile fitting procedure over manually selected small ranges of

Bragg angles.' For each group of peaks, present in the processed range, a

least squares profile fitting was conducted while refining both the positions

and full widths at half maximum of potentially resolvable Bragg reflections.

parabolic background, a mixing parameter of the pseudo-Voigt function,

and an asymmetry parameter were identical for all peaks within each

Semi-manual profile fitting was chosen over the full pattern decomposition to facilitate a

better control over the resultant integrated intensities extracted from groups of overlapped

Bragg peaks due to significant anisotropic peak broadening (see

Figure

6.24). Small fitting

ranges were chosen visually such that they contained one or more distinct Bragg

reflections clearly delimited by the background. Full pattern decomposition of this pattern

can be carried out using Le Bail or Pawley techniques. Both should converge to

4.2

R,.,,

5.5

and

4.1. We encourage the reader to undertake this effort and use

thus extracted intensities to solve the crystal structure as a self-exercise.

Crystal structure solution

555

selected range. The average

was

-2.5

%.I

order to maintain stability of

the non-linear least squares, the FWHM was constrained to be identical for

all peaks only in a few ranges at high Bragg angles. As can be seen from

Figure

6.24,

the distribution of full widths at half maximum is rather broad,

which is associated with the anisotropic peak broadening due to peculiar

shapes of the particles.

illustrating the solution of this crystal structure, we will use the

extracted intensities of 30 observed peaks at

70" with a total of 36

possible Bragg reflections. Only two pairs of reflections overlap nearly

completely in this range of Bragg angles and cannot be decisively re~olved.~

Four unobserved reflections were assigned some small intensity values, each

totaling about

of the intensity of the nearest observed peak, because the

presence of all, even near-zero-intensity, Bragg reflections increases the

chances of solving the structure using direct phase determination methods.

The absence of zero-intensity reflections is especially important when data

sets are small, as is the case here with only 36 possible reciprocal lattice

points.

V.V

I I

I I I I

10 20 30 40 50 60 70 80 90

Bragg angle,

(deg.)

Figure

6.24.

Full width at half maximum as a function of Bragg angle, observed in the

powder diffraction pattern of NiMn02(OH).

This value is considerably lower than

R,,

reachable during full pattern decomposition

because the extended background-only ranges are usually excluded from the semi-manual

profile fitting. Furthermore, an independent treatment of positions and full widths at half

maximum of Bragg peaks observed within the processed range enables a better fit between

the observed and calculated intensities.

The ability to see which Bragg peaks are resolvable with acceptable accuracy and those

which are not resolvable, is a benefit available in a semi-manual profile fitting.

556

Chapter

The indexed list of the individual structure factors

lFob,l

is found on the

CD in the file Ch6ExOS.hkl and the corresponding crystal data in the file

Ch6ExOS,ins, both are in the

SHELX

format. The intensities of all observed

Bragg reflections may also be employed in the solution of this crystal

structure, even with a greater success. Thus, the corresponding list is found

in the file Ch6ExO5-full.hk1, and we encourage the reader to use it as a

practical self-exercise.

Chemical composition of the crystals was established from microprobe

and thermogravimetric (TGA) analyses. According to microprobe data, the

ratio of Mn to Ni is 1:l. From TGA results, the amount of oxygen was

determined to be 1.5 times that of both metals, thus suggesting a 1:

stoichiometry, i.e. NiMn03. The amount of hydrogen cannot be accurately

established due to the complexity of the TGA trace, which has several

different weight losses in both oxygen and nitrogen atmospheres.

The gravimetric density of the crystals was not measured but the content

of the unit cell may be established by using

Eq.

6.5 and the expectation that

the reasonable value of

should be between 4 and 5 g/cm3. The estimated

density assuming NiMn03 composition has a reasonable value of 4.86 g/cm3

when

The two closest numbers of formula units

3 or 5) are

impossible due to the restrictions imposed by symmetry: in a base-centered

lattice, sites with odd multiplicities are impossible. The next two closest

numbers

2 or 6) result in unrealistically low and high densities,

respectively. Thus, we assume that there are 4 Mn, 4 Ni and 12

atoms in

the unit cell.

Considering space group Cmcm first, the multiplicity of the general site

here is

16.

The group also includes several

and 4-fold sites, suggesting

that at least one special position with the multiplicity

is occupied by

oxygen. Both Mn and Ni atoms should occupy 4-fold sites. The remaining 8

atoms can occupy one position with the multiplicity 8 or two 4-fold sites.

The very short unit cell dimension,

2.86

8,,

and the presence of mirror

planes perpendicular to it further limits possible locations of atoms in this

unit cell: all atoms should be located in the mirror planes. Thus, only the

following sites 4(a): 0,0,0; 4(b): 0,1/2,0; 4(c): O,y,1/4; and 8(9: O,y,z may be

occupied. If

accepts a non-zero value, the distance between symmetrically

related atoms becomes -1.43

or shorter,' which is unrealistic. Another

possibility is to use the non-centrosymmetric groups. The space group C2cm

does not look promising because the absence of the mirror plane

perpendicular to

which is a very short unit cell edge, is highly unlikely as

Mirror planes are spaced at

112~.

Thus, every atom with the coordinates

x,y,z

has nearest

symmetrically equivalent atoms at

-x,y,z

and

1-x,y,z.

The pairs of atoms are separated by

2xa

and

(1-&)a,

respectively. The distances are at maximum when

114,

i.e. the

spacings are

a12

1.43

Crystal structure solution

discussed above. The non-centrosymmetric space group Cmc2, looks more

promising as it has a 4-fold special position in the mirror plane perpendicular

4(a) with coordinates

Oyz,

where all atoms can be located.

The structure was solved' using SHELXS-90 and partial least squares

refinement using SHELXL-97 programs.' The centrosymmetric space group

symmetry Cmcm was tested first, however, several attempts with varying

parameters produced no acceptable model. It may be difficult to recognize

the incorrect selection of space group symmetry based on a few failures to

find the model, especially when relatively low quality or truncated structure

factor data are employed

(e.g. those extracted from the powder pattern). If a

solution at certain conditions in the selected space group symmetry was not

found, this does not necessarily mean that it does not exist. Often, it may be

tricky to identify a true solution.

Taking into account that the number of formula units per cell

gives a preference to the space group Cmc2,, it was chosen for the next

attempt. At first, the direct phase angles determination using SHELXS-90

was attempted with all default parameters. The program automatically

assigns heavy atoms to the peaks from the E-map, and in this case, the first

three peaks were treated as Mn. Analysis of interatomic distances indicated,

however, that the second strongest peak cannot be a metal and therefore, this

solution was

aband~ned.~

The following step, which is usually recommended when working with

powder data and when the default parameters do not result in an acceptable

solution, is to decrease the minimum normalized structure amplitude (Efi,,)

employed in the generation of phases.

general, this reduction decreases

the probability of phase relationships (see Eqs. 2.144, 2.147, and 2.148) but

it increases the number of reflections included in the process.

our case,

decreasing E,, from the default 1.2 to Ed,

1.1, increases the number of

reflections from 11 to 14. The best solution, shown in

Table

6.35,

contains

the first two peaks that are suitable as metals and the next three can be

suitable as oxygen atoms.4 Peaks beginning from

and below are

unacceptable because they are too close to the already assigned peaks.

Ready to use reflection file

Ch6ExOS.hkl

and crystal data file

Ch6ExOS.ins

are found on

the

CD.

Both can be used as input files for SHELXS-90.

G. M. Sheldrick, Phase annealing in SHELXS-90: direct methods for larger structures,

Acta Cryst.

A46,

467 (1990); G. M. Sheldrick, SHELXL-97. University of Gottingen,

Germany, (1997). See the footnote on page

515

on how to obtain the programs.

Generally, a situation like that does not necessarily mean that the model of the crystal

structure cannot be completed using this solution. It may take longer and it may be harder

to make decisions about which peaks should be included, and what atom types should be

assigned to them.

The suitability of peaks as atoms has been judged based on the relative heights of the

peaks on the E-map and from the shortest interatomic distances. The distance Mn2

558

Chapter

Table

6.35.

Maxima localized from the E-map refined for the best solution obtained by

SHELXS-97 using 14 reflections with

Emin

employing intensity data at

70'.

Bond distances,

Peaka

Height Comment

~nl~ ~n2~

Mnl 0.0000 0.3018 0.4909 71 1.5

-0.5000 0.2276 0.3901 60.1

Mn is indistinguishable from Ni at this stage because the former contains 25 electrons,

while the latter has

electrons, i.e. both atoms have similar scattering factors.

This column lists the distances to the indicated atom in

The obtained model was refined by using SHELXL-97 to

which resulted in the coordinates listed in

Table

6.36. Two oxygen atoms,

02 and 03, converged into locations that are too close to Mnl and therefore,

were eliminated from the model. A difference Fourier map was calculated

using phase angles determined by Mnl, Mn2, and 01 and the first two

strongest maxima are at reasonable distances to both metal atoms.

This is a good place to comment on a nearly blind inclusion of 02 and

03 in the previous step: it was done without the proper analysis of the

geometry of the model,

i.e. bond angles and coordination polyhedra should

be always analyzed in addition to bond distances.' The insertion of

"incorrect" atoms could be more detrimental than the simple removal of

these atoms at a later point. Indeed, this may disable solving a structure as a

whole. Therefore, only those atoms, which are rational from chemical and

physical points of view, should be

includedadded to the model. If there is a

suspicion about a partially built model, a conventional Fourier map should

be computed and used to improve the coordinates of already known atoms

and verify their positioning in the unit cell (e.g. see section 6.10, where the

improper positioning of the

atom was easily detected from a Fourier map

calculated in the space group 14lmmm).

Two new oxygen atoms improve the geometry of the model and the

following refinement lowers

to 9.5%

(Table

6.37). Overall, this result

gives a great confidence in the correctness of the model. All peaks on the

(1.59

is quite short, but at this point in the structure solution it may be acceptable

considering a small number of reflections included into the computation of both the phases

and E-map.

Analysis of bond angles is usually unnecessary in the case of intermetallic structures,

where bonding is predominantly metallic. However, knowing bond angles is critical when

solving structures with principally covalent bonding.

Crystal structure solution

559

subsequent difference Fourier map (only two are listed in

Table

6.37)

are too

close to the already present atoms.

Table

6.36. Results of structure refinement (SHELXL-97) and the difference Fourier peaks

computed using all reflection data below 20

70' (RF

%).

Atom/ Uisd Bond distances,

Peak

Height Mnla Mn2a 02a 03a

Comment

Mnl 0.0000 0.3021 0.4777

~4 1.0000 0.2146 0.7047 2.8

This column lists the distances to the indicated atom in

Deleted.

Table

6.37.

Results of structure refinement (SHELXL-97) and the difference Fourier peaks

computed using all reflection data below 20

70' (RF

9.5

%).

Atom/

Uisd

Bond distances,

Com-

Peak

Height Mnla Mn2a Ola 02a ment

Mnl 0.0000

0.3047 0.4759

0.022

Mn2 0.5000

0.5004 0.7235

0.032

01 0.0000

0.4496

0.9497 0.000

2.01,2.16

02 0.5000 0.3920

0.4445 0.029 1.93 1.96,2.17

03 0.5000

0.2177 0.5605

0.007 1.97,2.22

Q1 -0.500 0.4302

0.9673 1.9 1.65 1.46

42 0.5000 0.4249

0.6171 1.5 1.24 1.03

This column lists the distances to the indicated atom in

Distances are unreasonably short.

Thorough readers who

try

to use the full array of diffraction data in order

to verify the solution of this simple crystal structure will easily find out how

important is the completeness of the data: the whole model may be obtained

from an E-map employing phase angles, generated by using

Efi,

1.1

The

refinement of atomic parameters, however, converges to

which is

much higher than

9.5

obtained for the data array truncated at

The full array of the individual structure factors is located on the CD in the file

Ch6ExOS-full.hk1.

Identical Bragg reflections have slightly different values of

l~~~,"~~l

when compared to

Ch6ExOS.hkl

because two files were created during two independent

profile-fitting attempts. Readers can use this file, or they can create their own list of the

observed structure factors by using Le Bail or

Pawley full pattern decomposition

approaches. In this example, the anisotropic peak broadening should be refined to achieve

good convergence between the observed and calculated powder diffraction profiles.

560

Chapter

70". This increase in RF is associated with the weaker and broader peaks and

with a substantial overlap at high Bragg angles, which disable accurate

determination of the individual integrated intensities.

similar situation was

observed in the intermetallic Nd5Si4, see section 6.12.

It is critical to realize that low RF alone, cannot be taken as a sufficient

evidence for the rationality of the structural model.' Far more important

measures of whether the model is correct, are its geometry (bond distances,

angles, coordination polyhedra) and electroneutrality of the crystal.

this

example, the structure consists of square pyramids for Mnl and octahedra

for Mn2, which share oxygen atoms and form a three-dimensional

framework as shown in

Figure

6.25.

Furthermore, this figure clearly

illustrates that there are no mirror planes perpendicular to

and confirms the

correctness of the space group Cmc2,. The resulting model appears

reasonable and complete except Mn and Ni atoms must be distinguished, and

locations of hydrogen atoms should be established. The latter may require a

different type of experimental data, while the former should be doable

during Rietveld refinement of the structural parameters using the existing x-

ray diffraction data, as will be discussed in Chapter 7, section 7.7.

Figure

6.25.

Polyhedral representation of the three-dimensional framework in the crystal

structure of NiMn02(OH) shown along the X-axis as the packing of square pyramids [MeOS]

and octahedra [Me06], where Me

Mn or Ni. Oxygen atoms are located in the corners of the

polyhedra around the metal atoms. The latter are shown as large dark spheres.

and profile residuals (Eqs. 6.18 to 6.22) can be easily lowered by increasing the number

of adjustable parameters in a completely unreasonable structural model. Thus, relevant

residuals should be employed to gauge the fit between the observed and experimental data

rather than as an exclusive measure of the rationality of the underlying crystal structure.

Crystal structure solution

561

6.14

Crystal structure

trn~V~0~~

This example illustrates an unconventional case of solving a crystal

structure that shows relatively broad and therefore, substantially overlapped

at high angles Bragg peaks in addition to a significant preferred orientation.

The latter occurs due to a distinct platelet-like shape of the crystallites. The

powder diffraction pattern

(Figure

6.26)

was collected from a

tetramethylammonium

(tma)

trivanadate powder

a black graphite-like

crystalline material

that was thoroughly ground and screened. The

Bragg angle,

(deg.)

Figure

6.26.

Powder diffraction pattern collected from the

tmaV307

powder using

radiation on a Scintag XDS2000 diffractometer. The experiment was carried out in a step scan

mode with a step 0.02' and counting time 60 sec per step. The vertical bars indicate calculated

positions of the

Ka,

components of all possible Bragg reflections. The inset shows the

scanning electron microscopy image of particle morphology in the as-received state.

Experimental data are available in the files

Ch6Ex06-CuKa.xy

and

Ch6Ex06-CuKa.raw

the

CD.

P.Y.

Zavalij,

Chirayil, and M.S. Whittingham, Layered tetramethylammonium

vanadium oxide

[N(CH&]V307

by x-ray Rietveld refinement, Acta Cryst.

C53,

879

(1997); tma

tetramethylammonium

[N(cH&]+.

The material in a form of a black

crystalline powder was prepared by hydrothermal treatment at

185T

of a mixture of

V2O5, tmaOH,

and

LiOH

taken in

:2:

molar ratio and acidified with

CH3COOH

6.5.