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

Подождите немного. Документ загружается.

572 Chapter

inset) was indexed employing

TREOR

and using 16 peaks below 20

40" in

the hexagonal crystal system with a

13.255,

5.265 8L,

801.1 8L3.

The

figure of merit,

FI6

335(0.0021, 23), is extremely high. IT0

indexing produces the same result but in a C-centered orthorhombic lattice

with aortho

ahex, cortho

chex, and bortho

d3ahex. Unit cell refinement using

150 reflections observed below 20

130" results in highly accurate unit cell

parameters (see Table

6.43

on page 574) and a sample displacement of

-0.1 12(3) mm for a 250 mm goniometer radius.' The analysis of systematic

absences

points

to the following possible space groups P63/mmc, P63m~,

20 30 40

60 70

90 100 110 120 130

Bragg angle,

(deg.)

Figure

6.31.

Powder diMaction pattern collected from a ground MII~(OH)~(VO~)~ powder

(screened through a 38 pm sieve) using Cu

radiation on a Scintag XDS2000

diMactometer in a step scan mode with a step 0.01 and counting time

seclstep in the

range of 7 520

70•‹ and 30 seclstep for 70

132' to improve counting statistics of

weak reflections observed at high Bragg angles. The high Bragg angle range has been scaled

to a constant counting time of

seclstep for further use of the data. The vertical bars indicate

calculated positions of the

Kal

components of all possible Bragg reflections. The inset shows

the scanning electron microscopy image of peculiar empty hexagonal-pipe particle

morphology in the as-received state. Diffraction data are located

the

files

Ch6Ex8-CuKa.xy

and

Ch6Ex8-CuKa.raw

on the CD.

The maximum absolute difference between the observed and calculated 20 was 0.005',

which is an exceptionally low value.

Crystal structure solution

573

Thermogravimetric analysis resulted in complex traces in both the

oxygen and nitrogen atmospheres with gradual

-2

and -4

weight losses,

respectively. The powder diffraction pattern of the thermal decomposition

product can be identified as a mixture of Mn2V207 and Mn203. Available

data only allow a qualitative assumption about the absence of organic or

water molecules, simultaneously pointing to the presence of a small amount

of hydroxyl groups because of the continuous weight loss.'

general it may

be assumed that this compound contains Mn cations, OH- groups and

individual or shared comer [vo413- tetrahedra, as in V2O7, V4012 or (VO3)n.

The latter conclusion is based on the color, since all other oxidation states or

coordinations of V would result in black, dark green or dark blue crystals.

This reasoning is provided here to show how various chemical and physical

information may be used when considering composition, predicting,

proposing, or solving the structure.

An identification attempt using the Powder Diffraction File failed as no

acceptable matches were found. Undoubtedly, such high quality of the

powder diffraction data should be sufficient to solve the structure from first

principles using either Patterson or direct methods. Yet, a structure solution

is not fully automated and therefore, the ICSD database was searched in the

following order:

All compounds containing oxygen and one or both of the metals, Mn and

resulted in 3413 entries.

All hexagonal and primitive trigonal systems were considered, thus

reducing the number of entries to 204.

Search for the unit cell volume in the range between 700 and 900 A3, i.e.

within -100

of the title compound, shortened the list to 16

compounds.

12 of them belong to a different diffraction class and two have different

cla

ratios, where

is much greater than

Two remaining entries belong to the P63mc space group symmetry and

have similar unit cell dimensions as shown in

Table

6.43.

Note that Mn6-x(OH)3(HP03)4 has a unit cell volume and dimensions

close to those of the title material.

fact, much closer than the Zn-

containing compound, and it is also present in the PDF file. Theoretically, it

may have been found by a powder pattern search-match. The search,

conducted among all inorganic compounds with a narrow (0.04') window

and 5 matching reflections, failed in this example because of the relatively

large discrepancies in the unit cell dimensions and, therefore, peak positions.

The Zn-containing structure may be easily modified to represent the

crystal structure of Mn-OH-V04 (the composition derived above) by

Compounds containing organic molecules or water of crystallization usually demonstrate

rapid weight loss, while hydroxyl groups are lost slowly over a broad temperature range.

Chapter

substituting Zn with Mn and

with

Therefore, it was chosen as the initial

model in the Rietveld refinement (see

Figure

6.32

and

Table

6.44).

The

second structure does not

look

promising, because it consists of Mn cations,

hydroxyl and

HP03

groups; the latter have a different geometry. It was not

tested at all because the first model results in a successful solution as will be

discussed in the next chapter, in section 7.10.

Table

6.43.

Unit cell dimensions of potentially closely related compounds identified as a

result of searching the ICSD database.

Compound

ICSD PDF Ref.

Zn7(OH)3(V04)3S04 12.8 l3O(6) 5.1425(2) 73 1.15 402-888

Mn6.,(OH)3(HP03)4

13.1957(6) 5.1 VO(3) 780.68 75-269 47-868

Title compound

13.2294(1)

5.25529(7)

796.54 Refined from profile fitting

Figure

6.32.

The model of the crystal structure of M~I~(OH)~(VO~)~ derived from

Zn7(OH)3(V04)3S04 assuming that Mn atoms substitute for Zn (large black spheres), and V

atoms occupy positions of both V (large dark-grey spheres) and

(large light-grey spheres).

Oxygen atoms are shown as medium size light-grey spheres and hydrogen atoms are depicted

using small black spheres.

K. Kato,

Kanke,

Oka, and

Zao, Crystal structure of zinc hydroxide vanadate(V)

Zn7(OH)3(S04)(V04)3, Z. Kristallogr.

213,

26 (1998).

M.P. Attfield, R.E. Morris, and A.K. Cheetham, Synthesis and structures of two

isostructural phosphites, Fell(HP03)8(OH)6 and Mnll(HP03)8(OH)6, Acta Cryst.

C50,

981

(1 994).

Crystal structure solution

575

Table

6.44.

Coordinates of Mn, V and

atoms in the unit cell of MII,(OH)~(VO~)~ as

assumed from the model of the crystal structure of Zn7(OH),(VO,),SO4. The space group

symmetry is PG3mc.

Atoma Site

Mnl (Znl) 12(d) 0.4266 0.0802 0

Mn2 (Zn2) 2(a) 0 0 0.8217

Vl (Vl) 6(c) 0.1513 -0.1513 0.0257

(Sl) 2(b) 113 213 0.7479

1 12(d) 0.0676 0.3460 0.8469

02 G(c) 0.8090 -0.8090 0.815

03 6(c) 0.5280 -0.5280 0.715

04 6(c) 0.3967 -0.3967 0.642

05 6(c) 0.9243 -0.9243 0.571

06 2(b) 113 213 0.024

Symbols in parentheses indicate the corresponding atoms in the parent

Zn7(OH)3(V04)3S04 structure.

6.17

Crystal structure

FePO4I

This example illustrates the derivation of a crystal structure based on a

suspected analogy with related compounds followed by geometry

optimization to enhance and improve the deduced structural model. Such a

complex approach in this case has been adopted because of poor crystallinity

of the material, which results in a low resolution of its powder diffraction

pattern (see

Figure

6.33), where the full widths at half maximum range from

0.25 to

0.55O.

Furthermore, the pattern is relatively complex, with as many

255

Bragg reflections possible for 28

37.5" when Mo

radiation is

employed.

The title compound was prepared by thermal decomposition of the

monoclinic dihydrate FeP04.2H20. The solid-state preparation reaction is

likely responsible for the poor crystallinity, and therefore, peak broadening2

The inadequate crystallinity of the material results in the insufficient

accuracy of both the peak positions and intensities. A serious lack of

resolution in this particular powder diffraction pattern, which occurs due to

the physical state of the powder, translates into considerable problems in

both the indexing and structure determination. When experimental data

collected using Cu

radiation were employed in the

ab initio

indexing,

IT0 and TREOR runs did not result in a reasonable solution.

Song,

P.Y.

Zavalij, M. Suzuki, and MS. Whittingham, New iron(II1) phosphate phases:

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

41,

5778 (2002).

This example is also an excellent illustration of a case where the physical state of a

material precludes single crystal diffraction analysis, and a powder diffraction experiment

becomes the only option for a solution of its crystal structure.

576

Chapter

FePO,:

Cu Ka and

Bragg angle,

(deg.)

Figure

6.33.

Powder diffraction patterns collected from a monoclinic FeP04 using Cu

radiation on a Scintag XDS2000 diffractometer (top, step scan, 0.02" step) and Mo

radiation on a rotating anode Rigaku TTRAX diffractometer (bottom, step scan, 0.01' step).

Bragg angles in the pattern collected using Cu

radiation have been converted to match Mo

radiation. The two patterns are shown with a substantial displacement along the intensity

axis for clarity. The vertical bars indicate calculated positions of Bragg reflections for the

locations of

Ka,

components. Experimental data, collected using Mo

radiation, are

located in the files

Ch6Ex09-MoKa.xy

and

Ch6Ex09-MoKa.raw

on the

CD.

A single crystal diffraction experiment conducted using a small and low

quality single crystal yielded only about 20 detectable Bragg reflections, all

at low angles. This truncated array of data was insufficient even for a

reliable automatic indexing, but after a visual inspection, a monoclinic or an

orthorhombic lattice with the unit cell dimensions

5.5,

7.5, and c

8.0

was clearly noticeable. A solution with similar unit cell dimensions

and

monoclinic angle around 95" was found by employing IT0 using Mo

data with 19 of 20 low Bragg angle peaks indexed. The unit cell

dimensions are

5.489(1),

7.493(1),

8.055(1)

95.81(1)".

The monoclinic symmetry of the lattice and the systematic absences (likely

those of the space group P2Jn) are the same as in the parent hydrate,

FeP04.2H20.

The identical symmetry and similar unit cell dimensions between the

hydrated and anhydrous iron phosphates are found in the two orthorhombic

Crystal structure solution

modifications (see

Table

6.45). Moreover, the latter have closely related

crystal structures, i.e. the same bonding in the FeP04 frameworks except for

the water of crystallization in the hydrated compound (see the reference

listed in footnote No.

on page

575

for more details). This fact can be used

to solve the crystal structure of the anhydrous monoclinic compound

assuming that FeP04 connectivity remains intact in the two monoclinic

modifications as well. As illustrated in

Figure

6.34, the coordinates of Fe, P

and four independent

atoms from the hydrated monoclinic compound

were incorporated as the initial model of the crystal structure of the

anhydrous phosphate (model A, listed in

Table

6.46). Water molecules,

present in the monoclinic FePO4.2H2O, were ignored in model A.

Table 6.45. The comparison of unit cell dimensions of the orthorhombic and monoclinic

modifications of hydrated and anhydrous iron phosphates.

Compound Space group a,

8,deg.

V,A3

FeP04.2H20 Pbca 9.867 10.097 8.705

867.3

FeP04 Pbca 9.171 9.456

8.675

752.4

FeP04,2H20

P2Jn

5.307 9.755 8.675 90.16 449.1

FeP04

P2Jn 5.489

7.493 8.055 95.81 329.7

Unfortunately, a straightforward Rietveld refinement of model A fails

because it is far from reality, in addition to the low resolution of powder

diffraction data.' Therefore, the initial model A must be improved before

attempting the Rietveld refinement. The improvement was achieved using

the following two approaches to geometry optimization:

The first optimization was conducted using DMo13' and

CASTEP3

routines that perform energy minimization-and geometry optimization

using density functional theory. These programs are included in Materials

St~dio.~ However, when the FeP04 model A was employed, this

optimization was unstable and did not converge in a reasonable number

Note the considerable contraction along the b-axis, Table 6.45, and the expected distortion

of the FeP04 framework upon dehydration, Figure 6.34a and Figure 6.34b.

DMol3 is a molecular optimization technique based on density functional theory quantum

mechanical approach. See

Delley, J. Chem. Phys. 92, 508 (1990),

Delley, J. Chem.

Phys. 94, 7245 (1991),

Delley, J. Phys. Chem. 100, 6107 (1996), and

Delley, J.

Chem. Phys. 113,7756 (2000).

CASTEP (Cambridge Sequential Total Energy Package) is the ab initio quantum

mechanical density functional theory approach enabling modeling of properties of solids.

See M. C. Payne, M. P. Teter, D. C. Allan,

A. Arias, and J. D. Joannopoulos, Rev. Mod.

Phys. 64, 1045 (1992),

Milman,

Winkler, J. A. White, C.

Pickard, M. C. Payne, E.

Akhmatskaya, and

H. Nobes, Int.

Quant. Chem.

77,

895 (2000), and M. D. Segall,

P. L.

Lindan,

Probert, C.

Pickard, P. J. Hasnip, S. J. Clark, M. C. Payne, J.

Phys.: Cond. Matt. 14,2717 (2002).

Materials Studio suite of crystallographic programs, Accelrys Inc., San

Diego, CA,

http://www.accelrys.com/.

578

Chapter

of cycles. Therefore, an attempt was made to optimize model A, in which

~e~' is substituted with ~1~'. Both the latter and the former often form

similar or isostructural compounds because of the same oxidation states

and nearly identical radii. The optimization performed by

DM013

successfully converged in

cycles resulting in model B.' The initial

model A, intermediate model B

obtained after

optimization cycles,

and final model

are compared in

Figure

6.35.

It is easy to notice the

contraction of the octagonal ring where the water molecule was located

in the hydrate, which makes excellent chemical and physical sense.

Changes in the geometry of coordination polyhedra also make perfect

sense, where strongly distorted tetrahedra in model A (which are actually

parts of the octahedra in the hydrate) are optimized into almost ideal

tetrahedral configurations of oxygen around iron

atoms.2

Figure

6.34.

Analogy between the crystal structures of the orthorhombic hydrate FeP04.2H20

(a) and anhydrous orthorhombic FeP04 (b) motivates the use of the monoclinic hydrate

FeP04.2H20 structure (c) as the initial model of the monoclinic anhydrous FeP04 (d). The

crystal structures are shown as packing of the corresponding

[X04] polyhedra of iron (small

black spheres) and phosphorus (large white spheres) atoms without the corresponding oxygen

atoms from water molecules that complete coordination polyhedra of iron in the hydrates.

Oxygen atoms are located in the corners of the corresponding polyhedra.

The optimization run takes approximately 24 hours on a PC equipped with a single

GHz

processor.

It is worth noting that energy minimization was carried out without applying any

geometrical restriction. Nonetheless, quite reasonable geometry resulted.

Crystal structure solution

DMol3

cycles

DLS

DMol3

cycles

Figure

6.35.

Back-bone models of the crystal structure of the anhydrous monoclinic FeP04

projected along the X-axis: the initial model A derived from the monoclinic hydrate

FeP04.2H20 (a); model B1 after

optimization cycles as AlP04 using DMol3 (c); model B

after final optimization as

AIP04

(16 cycles) using DM013 (d); model

optimized using

DLS-76

(b).

parallel to the quantum-mechanical optimization, in which multiple

attempts took many days of computing and analyzing the results, a purely

geometrical optimization was attempted using the DLS-76 (Distance

Least Squares) program,' which is based on minimizing the differences

between the existing and desired distances that were set for Fe-0 and

P-

O to

1.88

and 1.53 A, respectively. Additionally, the

0-0

distances were

set to 3.07

and 2.50 A, respectively, for [Fe04] and [PO4] tetrahedra.

The process converges very quickly resulting in model

C,'

which is quite

similar to model

obtained from DMol3. We note, however, that the

latter was achieved without any restrictions imposed on the geometry of

the crystal structure. Therefore, if geometrical restrictions in the DLS

attempt are wrong or even somehow are far from correct, the algorithm

Ch. Baerlocher, A. Hepp, and

W.M.

Meier, DLS-76, a program for the simulation of

crystal structures by geometric refinement. Institute of Crystallography and Petrography,

ETH:

Zurich, Switzerland, (1 997).

The optimization run takes approximately

0.02

seconds on a PC equipped with a single

GHz processor.

Chapter

may not (and highly likely will not) converge to a reasonable model. The

final model

is compared with the initial model

and models

and

from the DM013 optimization in

Figure

6.35.

This model was not tested

further because model

was successful.

Model

obtained as a result of DM013 optimization, is shown in

Table

6.47

and it will be used as the initial approximation in Rietveld refinement

discussed in the next chapter, section 7.1 1.

Table

6.46.

Coordinates of Fe, P and

atoms in the unit cell of FePO,, model

as assumed

from the model of the crystal structure of the monoclinic FeP04.2H20.

Atom Site

Fe 4(e) 0.0914 0.6739 0.6916

P 4(e) -0.0870 0.3509 0.6839

1 4(e) -0.1 164 0.5068 0.6700

02 4(e) 0.1659 0.3221 0.7638

4(e) -0.0944 0.2814 0.5268

4(e) -0.3002 0.2937 0.783 1

Table

6.47.

Coordinates of Fe, P and

atoms in the unit cell of the anhydrous FePO,, model

obtained from DMol3 ~ptimization.~

Atom Site

04 4iej 0.3450 0.3548 0.3720

order to directly compare models A and

the atomic coordinates of the former should

be transformed as:

xtA

0.5,

yIA

y,,

zIA

0.5 because atomic coordinates

undergo several transformations during the optimization process.

6.18

Empirical methods of solving crystal structures

addition to reciprocal and direct space techniques considered in the

previous sections, a large variety of approaches may be employed to create a

model of the crystal structure in direct space. One of these, i.e. the

geometrical method, has been implicitly employed in section 6.9, where the

location of a single La atom in the unit cell was established from a simple

analysis of the unit cell dimensions and from the availability of low

multiplicity sites in the space group symmetry P61mmm. Here we consider

more complex example, i.e. the solution of several crystal structures

occurring in the series of Gd5(Si,Gel-,)4 alloys.' These examples illustrate

the power of the powder diffraction method in detecting subtle details of the

V.K. Pecharsky and K.A. Gschneidner, Jr., Phase relationships and crystallography in the

pseudobinary system Gd5Si4-Gd5Ge4, J. Alloys Comp. 260,98 (1997).

Crystal structure solution

581

atomic distribution in the unit cell in addition to highlighting how structural

information is an enabling step in establishing critical structure

properties

corre1ations.l It is worth noting that nearly always, empirical techniques

require extensive literature searches to find out as much as possible about the

crystal structures of closely related materials.'

According to Smith, Tharp and

John~on,~ both the silicide and germanide

of Gd at

5:4

stoichiometries belong to the same type of crystal structure; the

distributions of atoms in their unit cells are essentially identical to the

orthorhombic Sm5Ge4-type str~cture.~ Furthermore, as reported by

Holtzberg, Gambino and

McGuire,' extended solid solutions based on both

binary compounds exist in the GdS(SixGel.,)4 system in addition to the

formation of an intermediate phase with an unknown crystal structure near

the Gd5Si2Ge2 stoichiometry. The coordinates of atoms in the unit cell of

SmSGe4 are listed in

Table

6.48.

Powder diffraction patterns collected from three different samples, which

belong to three different phase regions in the Gd5(SixGel-J4 system are

shown in

Figure

6.36.

Both the similarities and differences are noteworthy:

the patterns have distinct clusters of Bragg peaks in the regions -10

-18"

and -21

-26", however, a conspicuous variation in peak

intensities from one pattern to another is also observed.

This simple visual analysis of powder diffraction patterns is usually a

good indicator that there are detectable changes in the atomic structures of

the materials in question but the overall structural motif remains closely

related. Based on this conclusion and assuming that at least one of the

materials belongs to the

SmSGe4 type, it should be possible to establish

details of atomic distributions in these three lattices, provided the quality of

diffraction data is ~ufficient.~

V.K. Pecharsky and K.A. Gschneidner, Jr., Gd5(SixGel.x)4: An extremum material, Adv.

Mater.

13,

683 (2001).

Examples considered in this section are also similar to the case of hydrated and anhydrous

FeP04 discussed in the previous section. The major difference is in the better crystallinity

of the Gd5(SixGel.,), materials and in the resulting higher quality of powder diffraction

data, which facilitate a straightforward Rietveld refinement of individual atomic and

profile parameters without the need for a preliminary quantum mechanical

and/or

geometrical optimizations.

G.S. Smith, A.G. Tharp, and

Johnson, Rare earth

germanium and -silicon compounds

at 5:4 and

5:3

compositions, Acta Cryst. 22, 940 (1967).

G.S. Smith,

Johnson, and A.G. Tharp, Crystal structure of Sm5Ge4, Acta Cryst. 22, 269

(1967).

F. Holtzberg, R.J. Gambino, and T.R. McGuire, New ferromagnetic

5:4

compounds in the

rare earth silicon and germanium systems, J. Phys. Chem. Solids 28,2283 (1967).

Holtzberg

al.,

J. Phys. Chem. Solids 28, 2283 (1967), also noted the differences in the

powder diffraction patterns of Gd5Ge4, Gd5Si2Ge2, and Gd5Si4. However, considering the