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

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502

Chapter

Nonetheless, even when the gravimetric density cannot be measured,

Eq.

6.4

still can be used, especially when the chemical composition of the material is

known precisely or when working with molecular compounds.

these

cases,

can be estimated from restrictions imposed by symmetry and from

approximately known range of densities for a specific class of compounds.

For example,

Eq.

6.4 is easily rearranged as

where n is the minimum number of molecules per unit cell, which is usually

the multiplicity of a general site position' and

is a variable integer, which

corresponds to the number of molecules with mass

in the asymmetric

(lln-th) part of the unit cell. For instance, in the space group P2Jm, n

and by varying z, the acceptable value of density

can be achieved.

Consider tetraphenylphosphonium (TPP) tetravanadate, which

crystallizes in the monoclinic crystal system in the space group

P2Jc. Its unit

cell volume is

6806.6(6) A3. The molecular weight of [(C6H5)4P]2V~01~

is 1058.55 a.m.u. The density range for this class of compounds (not the

easiest case but certainly a good example) may be estimated between 1.2 and

1.6

glcm3 given the presence of both the metal oxide core and large organic

molecule. The multiplicity of the general site position in this space group is

4. The V4Ol1 unit, however, can be centrosymmetric because it has four V

and ten

atoms plus an additional

atom that can be located in the center

of inversion, and there are two

TPP

molecules in the formula unit. Therefore,

a minimum multiplier for the number of molecules per unit cell (n) should be

set to 412

We now consider densities while varying z. When

1 and z

2 the densities are 0.5 16 and 1 .O33 g/cm3, respectively, which are too low.

When z

4, the density is 2.065 &m3, which is too high. Therefore,

the only possibility left and it results in the density of 1.549 g/cm3. The latter

value falls into the expected range, so the resulting total number of the

formula units in the unit cell

6. Considering that the multiplicity of

the general site is 4, there are 1.5 formula units in the asymmetric part of the

sinks.

high-density fluid is then slowly added until neutral buoyancy of the particle is

reached. The gravimetric density of the particle is then determined from the density of the

mixture of two fluids, provided their amounts are known. Obviously, the two fluids should

form an ideal solution, i.e. volume effects of mixing should be negligible.

In general, n in

Eq.

6.5

can be the multiplicity of a general or any special position or their

sum. The latter makes it complicated, but still keeps the solution integer with respect to

Note that symmetry of any special position and the molecule should agree; furthermore,

some special positions, i.e, those at the center of inversion where all three coordinates are

fixed, can be occupied by only one atom.

Crystal structure solution 503

unit cell. Thus, there should be

TPP ions in the general position, while the

vanadate molecules

(V40,,)

may occupy 1 general and 1 special, or

special

positions, where special positions are such that one of the

atoms are

located in the centers of inversion.

6.4

Pearson's classification

As noted in section 6.2, when the material of interest is an intermetallic

alloy, the solution of its crystal structure may be simplified because

intermetallics often form series of isostructural compounds.

contrast to

conventional inorganic and molecular compounds, stoichiometries of the

majority of intermetallic phases are not restricted by "normal" valence and

oxidation states of atoms and ions; therefore, crystal structures of metallic

alloy phases are conveniently coded using the classification suggested by

W.B. Pearson.' According to

Pearson, each type of the crystal structure is

assigned a specific code (symbol), which is constructed from three

components as follows:

The first position in a structure type symbol is occupied by a small letter

designating the crystal system of the material:

for cubic,

for

ietragonal,

for hexagonal, trigonal and rhombohedral, o for

orthorhombic,

for monoclinic, and a for triclinic (anorthic).

The second position in the symbol is occupied by a standard notation of

Bravais lattice. Thus, the first two elements in the Pearson's symbol are

letters and they classify all available alloy structures according to 14

Bravais lattices, as shown in Table

6.1.

The third (and last) position in Pearson's symbols is occupied by the total

number of atoms located in one unit cell of the compound.

For example, considering the crystal structure of copper, which has cubic

face-centered lattice (Figure

6.2)

and a total of 4 atoms in the unit cell, its

Pearson's symbol is cF4. On the other hand, if the material has Pearson's

symbol 0132, this means that its crystal structure is orthorhombic, and one

body-centered unit cell contains a total of 32 atoms.

Pearson's classification is insensitive to both chemical compositions and

stoichiometries of metallic alloys. It is quite useful because all known

intermetallic crystal structures are grouped according to their structural

symbols, which are quite simple. Thus, once the symmetry and the content

of the unit cell of a new alloy phase have been established, it only makes

sense to search for potentially isostructural compounds among those that

have identical Pearson's symbols.

W.B. Pearson, Handbook of lattice spacings and structures of metals, vol. 2, Pergamon

Press, New York (1967);

W.B.

Pearson, The crystal chemistry and physics of metals and

alloys, Wiley-Interscience, New York (1972).

Chapter

Table

6.1.

Pearson's symbols used to designate

types of Bravais lattices.

Crystal system Bravais lattice First two parts of Pearson's symbol

Cubic Primitive, P

Body-centered, I CI

Face-centered, F cF

Tetragonal Primitive, P tP

Body-centered, I tI

Hexagonal/Trigonal Primitive, P hP

Hexagonal/Rhombohedral Rhombohedral, R hR

Orthorhombic Primitive, P OP

Base-centered, C OC

Body-centered, I 01

Face-centered, F OF

Monoclinic Primitive, P mP

Base-centered, C mC

Triclinic (anorthic) Primitive, P UP

Tens of thousands of intermetallic phases have been systematized and

classified using Pearson's symbols. They are listed in a source commonly

known as Pearson's Handbook,' which is periodically updated and published

by ASM Internati~nal.~ The Handbook also provides detailed information

about the coordinates of atoms in unit cells of all known structure types of

metals, alloys and related phases, which makes it a valuable tool in the

structure solution of metallic materials.

6.5

Structure factors from powder diffraction data

When the

initio

solution of a crystal structure is attempted from

powder diffraction data, eventually integrated intensities and structure

factors of individual Bragg reflections will be required, see

Figure

6.1.

simple numerical integration (Eq. 2.64) is rarely applicable and nearly

always intensities can be determined only after decomposition (or

deconvolution) of partially overlapped Bragg reflections. Sometimes

decomposition is carried out peak by peak or group by group as was

described in Chapter 4, but more often individual observed structure factors

are determined by using the so-called full pattern decomposition techniques.

In addition to the determination of individual observed structure factors,

the full pattern deconvolution carries several supplementary functions:

First, is to verify the correctness of indexing, which is easily established

by visually comparing the observed and "calculated" patterns to ensure

that every observed peak has a matching calculated Bragg reflection.

Pearson's handbook of crystallographic data for intermetallic phases, P. Villars and L.D.

Calvert, Eds.,

2nd

edition, ASM International, Materials Park, OH

(1991).

ASM International on the Web:

http:Nwww.asm-intl.org/.

Crystal structure solution

505

Second, is to precisely determine the unit cell dimensions without

performing a semi-manual profile fitting (see Chapter 4).

Third, is to estimate the best figures of merit (see section 6.7), achievable

in a Rietveld refinement using the existing set of diffraction data.

The two related full pattern decomposition methods in common use today

were suggested by Pawley' and by Le Bail

aL2

Pawley's approach is based

on Eq. 2.48 and full pattern decomposition in the case of dual wavelength

data, when

Kal/Ka2

doublets are present, is performed by solving the

following system of equations using a least squares minimization:

The notations used in Eq.

6.6

are identical to Eq. 2.48. Individual

integrated intensities (Ik) are treated as free least squares parameters. Peak

shape function parameters are represented as described in section 2.9, and

Bragg peak positions, which affect the values of xk, are established by the

unit cell dimensions, see section 2.8. The background,

bi,

where

and

is the total number of measured data points, is modeled by one of the

several functions described in Chapter 4 (see Eqs.

4.1

to 4.6).

When peak shape functions and their parameters, including Bragg

reflection positions, are

known precisely and the background is modeled by

a polynomial function with

coefficients, the solution of Eq.

6.6

is trivial

because all equations are linear with respect to the unknowns (Bj, see Eq.

4.1, and Ik). It facilitates the use of a linear least squares algorithm described

in section 5.13.1. In practice, it is nearly always necessary to refine both

peak shape and lattice parameters in addition to

Bj and

to achieve a better

precision of the resultant integrated intensities. Thus, a non-linear least

squares minimization technique (see next section) is usually employed

during full pattern decomposition using Eq. 6.6.

G.S. Pawley, Unit-cell refinement from powder diffraction scans,

Appl.

Cryst.

357

(1981).

Le Bail, H. Duroy, and J.L. Fourquet,

initio

structure determination of LiSbW06 by

x-ray powder diffraction, Mat. Res. Bull.

23,

447

(1988). The method is also commonly

known as Le Bail extraction.

506

Chapter

Pawley's method works best when the complexity of a powder

diffraction pattern is relatively low (few hundreds or so Bragg peaks total).

When the number of independent Bragg peaks

(m)

exceeds several thousand,

the size of the least squares normal equation matrix increases proportionally

m2.

As a result, Pawley's algorithm may become unstable, especially if

there are multiple Bragg reflections with low intensity and/or when there is a

severe overlap. The latter is quite normal when the total number of

measurable Bragg peaks exceeds -1000 in the scanned range. In the case of

complex patterns, this algorithm has problems in keeping values of all

integrated intensities

non-negative, which is obvious from Eq. 6.6.

Le Bail's approach is also based on Eq. 6.6 and it differs from the

Pawley's method in that the individual intensities remain unaltered during

each least squares cycle. They are extracted from a total observed intensity

of the pattern between the least squares cycles after subtraction of the

background. The extraction is performed using a decomposition approach

that has been employed in the Rietveld method since its development,' in

which intensity observed in every point of the powder pattern is divided

among different reflections proportionally to their calculated intensities:

cnlc

Yk,i

yi;!

pk$i

(Y?~'

where

pk%i

Eq.

6.7,

yk,Pb5s the pseudo-observed intensity of the

kth

reflection in the

ith

point,

pk,i

is the fractional contribution of the

reflection to the

ith

point,

yrb9s the observed total intensity in the

ith

point,

yk,ic"'c

is the calculated

intensity of the

kt"

reflection in the

ith

point. The main difference between

Rietveld and Le Bail decompositions is in the calculated intensities. The

former technique uses intensities computed from the model of the crystal

structure, while the latter approach uses intensity obtained from the previous

cycle during the decomposition. Initially, all "calculated" intensity values in

the Le Bail's method are set to arbitrary identical quantities, typically unity.

It is worth noting that in the Le Bail's decomposition, the number of free

least squares variables becomes independent of the number of Bragg

reflections and only background, peak shape and lattice parameters are

refined during each least squares cycle. A small inconvenience of Le Bail's

approach is that the unit cell dimensions should be known with a greater

precision than in Pawley's method. It also takes more least squares

H.M. Rietveld, Line profiles of neutron powder-diffraction peaks for structure refinement,

Acta Cryst.

22,

151 (1967);

H.M. Rietveld, A profile refinement method for nuclear and

magnetic structures,

Appl. Cryst.

65 (1969).

Crystal structure solution

507

refinement cycles to complete but the fit converges in much more complex

cases. The disadvantage of the approach suggested by Le Bail is the separate

handling of intensities for

Kal and Ka2 peaks during the decomposition

according to Eq. 6.7 and correction for the proper intensity ratio after the

decomposition, which increases the time required for the completion of the

fitting process. As far as suitable software is of concern, it is essential to

check the manual and find out how the presence of the

KallKa2 doublet is

handled because some computer codes do not address this issue and the

resulting intensities of Kal and Ka2 components may become unreasonable.

There is a variety of freely available software, which enables one to

deconvolute a powder diffraction pattern and determine either or all:

individual intensities, lattice and peak shape function parameters, and

observed structure factors of all possible Bragg reflections.

Freeware codes

include EXPO, Fullprof, GSAS, LHPM-Rietica, and others.'

addition to

free programs, nearly all manufacturers of commercial powder

diffractometers offer software for sale either as a package with the sale of the

equipment or as stand-alone

product^.^

6.6

Non-linear least squares

Both the full pattern decomposition and Rietveld refinement are based on

the non-linear least squares minimization of the differences between the

observed and calculated profiles. Therefore, the non-linear least squares

method is briefly considered here. Assume that we are looking for the best

solution of a system of

simultaneous equations with m unknown

parameters

m), where each equation is a non-linear function with

respect to the unknowns, xl, x2,

..,

x,.

a general form, this system of

equations can be represented as

The most extensive sources of various software links are found at the International Union

of Crystallography (www.iucr.org) and/or Collaborative Computational Project No.

(http://www.ccpl4.ac.uk)

Web sites.

Commercial manufacturers, which offer a variety of software products for processing

powder diffraction data are: Bruker

(http://www.bruker-axs.com/production/indexnn.htm),

Philips

(http://www-us.analytical.philips.com/products/xrd),

Rigaku/Molecular Structure

Corporation

(http://www.rigaku.com/xrd/index.jsp),

STOE

Cie, Gmbh

(http://www.stoe.com/products/index.htm)

and many others.

508

Chapter

Obviously, a linear least squares algorithm described in section 5.13.1 is

not directly applicable to find the best solution of Eq.

6.8.

some instances,

it may be possible to convert each equation in (6.8) into a linear form by

appropriate substitutions of variables and thus reduce the problem to a linear

case.

general, the least squares solution of Eq. 6.8 is obtained by

expanding the left hand side of every equation using Taylor's series and

truncating the expansion after the first partial derivatives of the respective

functions.' Hence, Eq. 6.8 may be converted into:

%cxP

,-.,

x,)

Ax,

+...+

(XP,...,~

EY, -fi(xP,...,x,)

iY;(xP ,...,x

Ax,

+...+

v2(x:7'*',xi)

Ax,

-f2

7...7xm)

(6.9)

<xP,...,xm)

Ax,

...+

vn<.P7...,xi)

Axm EYn

-fn&

,...,x:

)

as long as the corresponding derivatives exist and are finite. Equation 6.9 is

linear with respect to Ax1, AxZ,

...,

Ax,

and its solution is obtained by

applying conventional linear least squares technique as

The notations in Eq. 6.10 are as follows:

Taylor's series is an expansion of a real function about a point. In the case of a function of

one variable,f(x), the expansion about a point x

x0 is given as

where

(x-xO) and

is a remainder. Whenfi,, xi,

. . .

xk) is a function of

variables,

the expansion about a point (xI0, xzo,

xko) is obtained by substituting each derivative in

the equation above with the sum of partial derivatives multiplied, respectively, by

Ax1,

Axz,

. . .

Axk

taken to the appropriate power and divided by the corresponding

n!.

Crystal structure solution 509

where

is a square matrix representing individual weights

(wi)

for each of

the available

data points, and

is the transpose of

The refined parameters are computed by using both the set of the original

0 0 0

,...,

which represents the initial approximation of the unknowns,

and the vector

Ax,

which has been obtained from least squares (Eq.

lo), as:

Ax,

...

Ax,

The standard deviations' for each refined parameter according to the least

squares method are calculated from

Recently, the term "standard uncertainty" (s.u.) is becoming more common than the

"standard deviation".

(ATWA);

f:Wi(yi)2

i=l

(3(xj)

...,

n-m

Chapter

(6.16)

where:

is the number of equations in Eq. 6.9,

is the number of unknown parameters in Eq. 6.9,

(A~wA),~' is the corresponding diagonal element of the inverse

normal equation matrix,

is the corresponding weight,

is the corresponding element of the vector

The major differences between the non-linear least squares technique and

the linear least squares method, described in section 5.13.1, are as follows:

The substitution of the original Eq. 6.8 with Eq. 6.9 requires the

knowledge of initial (i.e, approximate) values of parameters to be refined,

0 0 0

which are represented by the set

The least squares solution (Eq. 6.10) results in the shifts (vector

Ax,

Eq.

6.1 I), which shall be added to the corresponding initial parameters, as

shown in Eq. 6.15.

Because Eq. 6.9 is not exact, usually more than one cycle of a least

squares refinement is necessary to achieve a full convergence: during the

second and following least squares cycles, the new set of parameters as

obtained in the previous step from Eq. 6.15 is used as the initial

approximation. Thus, non-linear least squares refinement is an iterative

process, where the result of the next iteration depends on the result

obtained during the prior iteration.

Because of the iterative nature of non-linear least squares, convergence

may be difficult to achieve, especially when the initial approximation is

far from correct (Figure 6.3, left) or when the minimized function (see

Eqs. 5.33, 5.34 and 6.9) is poorly defined. The latter often occurs when

certain least squares parameters correlate.' Instead of converging, non-

linear least squares may diverge and become unstable, as illustrated in

Figure 6.3, right. Therefore, various numerical conditioning techniques

should be employed to improve both the convergence and stability of the

method. Their detailed consideration exceeds the scope of this book and

the reader is referred to a large amount of special literature covering this

subject.

Considering, for example, Eqs.

2.65

and

2.89,

when the phase scale

(K)

is refined together

with the population parameters of all atoms

(g'),

a complete correlation results: an increase

of the phase scale by a factor

will be completely offset by the reduction of all population

parameters by the factor

dk.

Crystal structure solution

Figure

6.3.

Two examples when the non-linear least squares technique may fail in finding the

best solution of Eq.

6.9:

left

the initial approximation

(xo)

is located near a false minimum;

right

the minimum is poorly defined. The arrows represent the possible outcomes of two

least squares cycles. In the case on the left the minimization ends in a false minimum. In the

case on the right the obtained shifts have correct signs but wrong magnitudes and instead of

converging (i.e. instead of all

becoming smaller), their absolute values continue to

increase. True solutions (i.e. global minima) are marked as

x,,,.

The inverse of the normal equation matrix, (ATwA)-', may be used to

evaluate the correlation coefficients

(pu)

among the pairs of free least

squares variables (xi and

xj):

The correlation coefficients vary from 0 to 1 (absolute) and when they are in

the range from -0.5 to 0.5, the associated parameters show little to no

correlation. When

lpUl

is unity, the corresponding variables (xi and

xj)

are

fully (100

correlated and one of them should be excluded from the

refinement. It is useful to check the matrix of the correlation coefficients

during the refinement, especially when the non-linear least squares process

appears unstable, which is usually detected as continuous worsening of one

or more numerical figures of merit (see section

6.7)

in addition to erratic

changes of the values of some free variables included into the refinement.

The analysis of correlation coefficients may help in identifying the non-

obvious and serious problems, and show which parameters strongly

correlate,

i.e. depend on each other, and therefore, cannot be successfully

refined together.