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

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492

Chapter

111,

(deg)a

FWHM

(deg)

111,

(deg)

FWHM

(deg)

5 1 36.788 0.1 11 182 57.339 0.131

229 37.673 0.111 3 5 57.698

34 38.157 0.111 41 57.965

27 39.268 0.152 325 58.526

136 40.215 0.152 116 59.021

456 41.035 0.152 296 59.156

3 14 42.378 0.152 124 59.367

40 44.182 0.126 148 60.186

63 44.630 0.126 78 61.212

123 44.877 0.126 15 61.610

123 45.631 0.126

Bragg angles are listed for the location of the CuKa, component in the doublet,

1 .54O593

Chapter

CRYSTAL STRUCTURE SOLUTION

6.1

Introduction

Assuming that both the crystal system, i.e. the "powder" Laue class, and

lattice parameters of a material have been established, the next step to be

undertaken is the solution of its crystal structure to find the distribution of

atoms in the unit cell. The problem is generally far from trivial and many

structure solutions in powder diffraction remain unique, yet all of them have

much in common because they connect reciprocal and direct spaces,

i.e. a

powder pattern and a distribution of the electron or nuclear density in a

crystal lattice. Although traveling a structure determination path in a powder

diffraction vehicle is nothing like cruising down an interstate in a latest

model Cadillac, knowing where to enter, how to proceed, and where and

when to exit is equally vital. Familiarity with crystallographic features of

related materials along with basic chemical and physical properties of the

material in question, such as probable oxidation and coordination states of

atoms together with the expected shortest interatomic distances is highly

desirable and, in some instances, may be required to complete the journey.

Below we will examine some practical applications of the theory of

kinematical diffraction to solving crystal structures from powder diffraction

data. When considering several rational examples in reciprocal space, we

shall implicitly assume that the crystal structure of each sample is unknown

and that it must be solved based solely on the information that can be

obtained directly from a powder diffraction experiment and from a few

other, quite basic properties of a polycrystalline material. The solution of a

number of crystal structures in direct space will be based on the previously

known structural data and supported by the results of powder diffraction

analysis, such as unit cell dimensions and symmetry.

494

Chapter

6.2

initio

methods of structure solution

The crystal structure solving process (Figure

6.1)

usually begins with

analyzing systematic absences to find the space group symmetry of a

material or at least to identify all probable space groups, if the corresponding

diffraction class combines more than one group (see sections

2.12.3

and

2.12.4).

The next step requires determining the content of the unit cell, i.e.

one must establish or estimate how many atoms of each kind, molecules or

groups of atoms are expected to be present in one unit cell.

some cases, it

is possible to narrow the space group selection based on this simple

information, especially when the multiplicities of the site positions are high

(e.g. in high symmetry groups) and a site with the lowest multiplicity still

places more atoms of a given type than necessary inside the unit cell.

Once the content of the unit cell has been established, a model of the

crystal structure should be created using either direct or reciprocal space

techniques, or a combination of both. Direct space approaches do not

mandate immediate use of the observed integrated intensities, while

reciprocal space methods are based on them.

Preliminary processing:

Indexing:

b,c,a,Wyj

,--------------

--------------

L-r

..----------,

,--i:.j~-i:--?

Figure

The flowchart illustrating crystal structure determination from powder diffraction

data. Preliminary processing and indexing are described in Chapters

and

respectively, and

have been assumed accomplished at an earlier stage. Structure completion and Rietveld

refinement are described in Chapter

Gravimetric density: Systematic absences:

unit cell content diffraction class

Full pattern decomposition:

h,,

ki,

I],

ybs,

l50b~l

Choose

Direct space approaches

Reciprocal space approaches

Crystal structure solution

495

6.2.1

Conventional reciprocal space techniques

In any of the reciprocal space methods, which are based exclusively on

the use of the observed structure factors, the powder diffraction pattern must

be deconvoluted and the integrated intensities of all, or as many as possible,

individual Bragg reflections determined with a maximum precision. Only

then, Patterson or direct phase angle determination techniques may be

employed to create a partial or compete structural model. Theoretical

background supporting these two methods was reviewed in section 2.14.

The Patterson technique is often called the heavy atom method because it

works best when a few atoms in the unit cell are markedly heavier than the

others and, therefore, have much stronger scattering ability in x-ray

diffraction. Squared structure amplitudes are employed as coefficients in the

Fourier transformation, and the resulting Patterson map yields a distribution

of the interatomic vectors in the unit cell. A structure model is then obtained

in direct space by analyzing the Patterson function. Direct phase

determination methods are based on generating probable phases that were

lost during the diffraction experiment. Observed structure amplitudes, which

have been normalized, are employed together with computed phases in the

Fourier transformation yielding the so-called E-maps, which contain direct

images of the electron density distribution, where maxima correspond to

positions of atoms in the unit cell.

Currently, Patterson and direct methods are the most frequently employed

"classical" structure solution approaches. The direct phase determination

methods are especially successful in solving structures from single crystal

data, but their use in powder diffraction increases progressively as the

quality of powder data improves, better deconvolution techniques are

developed and more precise individual structure factors become available.

6.2.2

Conventional direct space techniques

As an example of a direct space approach, consider a material, which is

an intermetallic compound. Many intermetallics form series of closely

related structures, the so-called isostructural compounds, where the

coordinates of atoms in the unit cell remain nearly identical and only the

distribution of different kinds of atoms varies among available

crystallographic sites. If this is the case, then the crystal structure may be

solved

via

a comparison with known structure types, by searching for

matching symmetry, lattice centering, unit cell contents including chemical

similarity of the components, and analogous relationships among the unit

cell dimensions. Another example is an organic compound with the well-

know configuration of its molecule.

some cases it may be relatively easy

Chapter

to model the packing of the identical molecules in the unit cell, and perhaps,

optimize their positions, orientations and, if required, conformations by

using energy minimization or other principles.

There are many ways to build a model of the crystal structure of a

polycrystalline material without first using the intensities of individual Bragg

reflections, which are hidden in powder diffraction due to partial or complete

overlapping. Most of the direct space approaches are, in effect,

trial-and-

error methods and they include some or all of the following components:

Obtaining a structural model from the analysis of potentially isostructural

compounds with partially or completely different chemical composition

but identical or similar stoichiometry. The search for isostructural

compounds may be conducted using a digitized powder pattern, unit cell

dimensions, stoichiometry,

and/or other suitable parameters.'

Obtaining a partial structural model from known similar or closely

related compounds, e.g. those with an identical framework or layers.

Similar structures are usually found from close relationships among all or

some unit cell dimensions. For example, layered intercalates with the

same type of the host layer should have two similar unit cell dimensions.

the case of simple structures or known building blocks, geometrical

modeling of a trial structure or several possible structures can be

performed (e.g. modeling of zeolites from polyhedra or building blocks

that are more complex). The resulting model may be further optimized or

immediately tested by using powder diffraction data.

Direct space modeling may be an especially powerful tool in the case of

intermetallic and related structures, many of which are derived from

close packing of incompressible spheres. Thus, when positions of large

atoms in the unit cell are known, the smaller atoms will likely occupy

voids of sufficient size.

Obviously, trial-and-error techniques require some, and often extensive,

chemical, crystallographic and physical knowledge about a specific class of

materials in addition to the availability of a structural database and some

experience in structural analysis.

6.2.3

Unconventional reciprocal and direct space strategies

No matter how advanced, every numerical data processing technique has

intrinsic limitations, especially when the complexity of the data increases or

when the required information is partially missing, as for example in single

crystal diffraction from macromolecules and proteins or in powder

Five examples of using the ICSD database in solving crystal structure form powder

diffraction data by structural analogy are discussed in: J.A. Kaduk, Use of the Inorganic

Crystal Structure Database as a problem solving tool, Acta Cryst.

B58,370

(2002).

Crystal structure solution 497

diffraction from conventional materials when the unit cell volume is large

and/or when the symmetry is low. Poor crystallinity and an excess of weakly

scattering elements in macromolecular compounds on one side and heavy

Bragg reflections overlaps on the other, affect both the quality and quantity

of the available diffraction data and, therefore, cause problems in both cases

that are similar in many ways. As a result, the conventional phase angle

determination methods may become, and often turn out to be ineffective.

Therefore, novel techniques potentially applicable to solving crystal

structures are under continuous testing and development. A recent collective

monograph on the structure determination from powder diffraction data

provides an excellent discussion of the problem and introduces different

approaches that may be used in its solution.'

this chapter, unconventional

structure solution methods are only briefly reviewed: most of them are still

controversial and do not always work well with different kinds of

compounds and data, although solutions of several complex structures have

been demonstrated. Summarized below are the genetic algorithm, maximum

entropy, maximum likelihood, and simulated annealing methods.

Genetic algorithm

is the direct space optimization method based on the

evolution principle, in which only the members that fit best into the

environment survive. The improved subsequent generation is obtained by

considering the current state of a complex system and events that are

equivalent to mating, mutation and natural selection.

powder diffraction,

the fit is defined as profile residual,

(see section

6.7).

The system,

represented by a crystal structure, is split into fragments; each "survives" or

"dies" depending on how it affects the fit.

addition to structure solution,2

this method can be, and has been applied to unit cell determination from

powder

dataq3 Another example is application of the genetic algorithm

method to solving protein structures by deconvoluting a Patterson f~nction.~

Maximum entropy

method is a powerful numerical technique, which is

based on Bayesian estimation theory and is often applied to derive the most

Structure determination from powder diffraction data. IUCr monographs on

crystallography 13.

David, K. Shankland, L.B. McCusker, and Ch. Baerlocher,

Eds., Oxford University Press, Oxford, New York (2002).

K.D.M. Harris, R.L. Johnston and B.M. Kariuki, The genetic algorithm: Foundations and

applications in structure solution from powder diffraction data, Acta

Cryst.

A54,

632

(1998); K. Shankland, B. David, and T. Csoka, Crystal structure determination from

powder diffraction data by the application of a genetic algorithm,

Kristallogr.

212,

550

(1997).

B.M. Kariuki, S.A. Belmonte, M.I. McMahon, R.L. Johnston, K.D.M. Harris, and R.J.

Nelmes, A new approach for indexing powder diffraction data based on whole-profile

fitting and global optimization using a genetic algorithm,

Synchrotron Rad.

87 (1999).

Chang and M. Lewis, Using genetic algorithms for solving heavy-atom sites, Acta

Cryst. D50,667 (1 994).

498

Chapter

probable values of missing data. For example, rolling a dice with six faces

numbered

through 6 gives equal probability to see each face up,

116,

assuming a completely random distribution.

this case, the average value of

multiple observations is (1+2+3+4+5+6)/6

3.5. The problem is in finding

the probability of events (faces up) when the distribution is not random and

only the average is

known,

e.g. when the average is 2.5. The maximum

entropy method results in the highest unbiased unique probability of events

(e.g. faces up in the case of a dice) using the entropy function, Cpiloppi. This

method, similar to the genetic algorithm approach, finds applications in

various fields of scientific analysis, especially in image processing and

reconstruction.'

powder diffraction, the maximum entropy technique can

be and is successfully employed both to restore the lost phase angles2 and to

determine the relative intensities of the overlapped

reflection^.^

Crystal

structure determination using the maximum entropy approach results in

solving the phase problem and therefore, it is a reciprocal space method.

Maximum likelihood

method, similar to maximum entropy, works in

reciprocal space and results in finding a raw model that has the best chance

to be improved by applying small steps to achieve full agreement between

the observed and calculated structure amplitudes: rather than the final

structure. Maximum likelihood and maximum entropy techniques are often

combined

Various energy minimization methods that work in direct space may be

employed in order to optimize positions, orientations and conformations of

molecules or structural fragments. The problem is the presence of multiple

local minima on the way to a global minimum of energy that requires an

initial structure model to be in the range of the global minimum, thus making

the search for the lowest energy exceedingly slow. Different approaches and

optimization functions may be employed to speed up the process. Typically,

the potential energy of a system is used in combination with some other

criteria. Thus, the

simulated annealing

method is used to explore local

minima quickly by generating multiple trial models using Monte Carlo or

Examples of image processing can be found at

http://www.maxent.co.uk~examples.htm.

C.J. Gilmore, Maximum entropy and Bayesian statistics in crystallography: a review of

practical applications, Acta Cryst.

A52, 561 (1996).

W. Dong and C.J. Gilmore, The

initio

solution of structures from powder diffraction

data: the use of maximum entropy and likelihood to determine the relative amplitudes of

overlapped reflections using the pseudophase concept, Acta Cryst.

A54,438 (1998).

A.J. Markvardsen, W.I.F. David, and

Shankland, A maximum-likelihood method for

global-optimization-based structure determination from powder diffraction data, Acta

Cryst.

A58,3 16

(2002).

G. Bricogne, A multisolution method of phase determination by combined maximization

of entropy and likelihood.

111. Extension to powder diffraction data, Acta Cryst.

A47, 803

(1 99

1).

Crystal structure solution

499

grid search methods. Simulated annealing resembles the physical process of

annealing by virtually heating a sample to a certain temperature, and it

includes several control parameters defining the search of the global

minimum that are analogous to the real annealing process. They are the

initial temperature, the rate of its decrease and the magnitude of random

atomic jumps. The initial temperature should be set high, somewhere near

the melting point; if it is too low, no changes occur or they occur very slowly

and when it is too high, the structure may "melt" and become amorphous.'

It is worth noting that practically all non-traditional methods for solving

crystal structures have been initially developed for both powder and single

crystal diffraction data to manage intrinsic incompleteness or poor quality

that cannot be improved experimentally. Despite a variety of structure

solution approaches, traditional direct phase determination methods appear

to be the most common and successful when powder diffraction data are

adeq~ate.~ Patterson methods also work quite well but they require the

presence of a heavy atom and, perhaps, more extensive crystallographic

expertise. The non-traditional methods are generally employed when other

techniques fail and their use is somewhat restricted by both the complexity

and limited availability of computer codes.

6.2.4

Validation and completion of the model

Regardless of how the model was created it must be validated andlor

completed by computing the electron or nuclear (if neutron diffraction data

are available) density

distribution(s) in the unit cell to confirm the placement

of known atoms and locate missing atoms, if any (see sections 2.13 and

2.14). The observed structure factors and, therefore, intensities of individual

Bragg peaks are required at this stage of the structure solution process.

Fourier map calculations may be repeated several times using observed

structure amplitudes and calculated phase angles that have been refined by

including additional atoms found from previous distributions of electron

(nuclear) densities and by modifying the coordinates of the existing atoms to

match those of the corresponding peaks on the Fourier maps.

crystal structure solution does not end with the development of a

plausible model: after the model has been built completely,' multiple

structural and profile parameters should be refined to achieve the best

A.A.

Coelho, Whole-profile structure solution from powder diffraction data using

simulated annealing,

Appl. Cryst.

33,

899 (2000).

C. Giacovazzo, Direct methods and powder data: state of the art and perspectives, Acta

Cryst.

A52,33

996).

Especially when powder data are complex, it may be necessary to use Rietveld technique

to re-determine individual intensities before all structural details can be established using

Fourier or differential Fourier maps.

500

Chapter

possible agreement between the observed and calculated powder diffraction

patterns or, in other words, between the crystal structure and the observed

reciprocal space image. At the same time, the crystal structure should make

both chemical (e.g. oxidation states, charge balance, valence, coordination,

etc.) and physical

(e.g. interatomic distances, valence and torsion angles,

coordination polyhedra, etc.) sense. Rietveld refinement employing powder

diffraction data, which will be considered in the next chapter, is an important

step in both the validation and completion of the model.

6.3

The content

the unit cell

Consider

Figure

6.2,

which illustrates the crystal structure of elemental

copper. If the lattice parameters are known, so is the volume of its unit cell.

Furthermore, if we know the total number of atoms located in this or any

other unit cell, it is easy to calculate the gravimetric density of a material by

dividing the mass of all atoms located in one unit cell by its volume.

The inverse calculation is also possible: the content of the unit cell may

be determined from the known chemical composition, lattice parameters and

density of a material. Assume that the total mass of all atoms located in one

unit cell is

Also, assume that the unit cell volume is

The latter is

known from diffraction analysis as soon as lattice parameters have been

established (see

Eq.

5.41) Thus, provided the gravimetric density (p) of the

crystalline material has been measured, the mass of one unit cell can be

easily calculated:

Figure

6.2.

One unit cell in the crystal structure of elemental copper illustrating that a point

located in a corner contributes

118

of an atom, and a point located on a face contributes

112

an atom to the overall content of the unit cell. Similarly, a point located on an edge would

contribute

114

of an atom. The overall content of this unit cell is

8x118

112

Cu atoms.

Crystal structure solution

501

It is usually the case that the chemical composition of a material is

known as the stoichiometry of its molecule, or in general, as its formula unit.

Molecular mass or formula unit mass,

is given as

where

is the number of different atom types in a molecule or in a formula

unit;

is the total number of atoms of type j, and

is the molar mass of

atoms of type j. The number of molecules or formula units, Z, in the unit cell

is, therefore, found from

Equation 6.3 may be transformed into a more useful form, where the

mass of the formula unit is given in

a.m.u., the unit cell volume is in A3, and

the density is in

g/cm3:

For example, the experimentally measured density of copper is

8.92

g/cm3, the unit cell dimension of its cubic unit cell is

3.615 A, and

the molecular mass of a formula unit is 63.55 a.m.u., which is the molar

mass of copper, one atom per formula unit. Thus, Eq. 6.4 results in Z

3.99

4 atoms per unit cell. The same equation may be used to calculate the

density of a material when its crystal structure has been established. It is

worth noting that the computed value of the material's gravimetric density is

known as the x-ray density, and it is usually slightly higher than the

measured density because real materials always have some defects and

porosity that are not accounted in Eq. 6.4.

This method of calculating the contents of the unit cell requires

experimentally measured density,' which is not always available.

The two methods, most commonly used to measure gravimetric density, are pycnometric

and flotation. In a pycnometric technique, the volume of the material is determined from

the volume of

fluid displaced

the known mass of a material. In a flotation approach, a

small particle, which can be a single crystal, is placed in a low-density fluid, where it