Simulation of powder diffraction patterns

Simulation of powder diffraction patterns • generation of reflections • X-ray diffraction intensity • impact on quantitative phase analysis

“Selective reflection” For the often used Bragg-Brentano geometry – i.e. incident and diffracted beam are symmetrically aligned to the sample surface – this means:  Any detected signal results from lattice planes parallel to the sample surface.  From each crystal only a single lattice plane will generate detectable intensity. In powder diffraction only a small fraction of crystals (shown in blue) are oriented in a way fulfilling the diffraction conditions of reflection hkl. Using a “point” detector only a very small part of diffracted intensity will be detected.

 If crystals have special habit texture effects occur.

Crystal symmetry

reflection positions crystal system  metric ao=bo=co; a=b=g=90° are necessary but no sufficient conditions for an assumption that the given crystal is cubic.

multiplicity:

{hkl}Pg1  {hkl}Pg2

The fundamental sector depends on the point group symmetry. Attention: trigonal!

reflection intensities crystal structure The crystal structure is described by the entire atomic arrangement within the unit cell. simplification: reduction of the unit cell content to the asymmetric unit using the standard arrangement of symmetry elements in combination with lattice types (space group types and settings) given in the International Tables for Crystallography.

Symmetry versus crystal metric P m -3 m (221)

P 21/m

ao= 3.795Å

ao= 7.600Å bo= 7.600Å co= 7.600Å b = 90°

 = 4.131 g/cm³ ICSD 31865

Perovskite

(11)

 = 4.115 g/cm³ ICSD 29116

ICSD 31865

6 reflections

ICSD 29116

170 reflections

Reflection positions A special procedure generates in a list of reflections hkl. Using the lattice parameters their distance in reciprocal space and the resulting diffraction angles will be derived.

The hkl generation depends on the Laue group (cf. Friedel’s rule) or point group symmetry (anomalous dispersion). Using this approach no additional reflection will be calculated!

Friedel’s rule G. Friedel observed that the intensity distribution in diffraction patterns is centrosymmetric: (Comptes Rendus, Acad. Sci. (Paris) 157, 1533-1536 (1913)).

The Laue symmetry represents the point-group symmetry of the crystal extended by a center of symmetry (if not already present).

Friedel’s rule:

and are reflections on the same lattice plane , but from front and back side. In general is valid: . This is reasonable because the two sets of indices represent different sides of the same family of planes. Friedel’s rule is not valid under conditions of anomalous dispersion.

Reflection intensity The following equation enables the calculation of the powder diffraction intensity for any reflection hkl:

A PLG Mhkl Fhkl |Fhkl|

absorption correction polarization, Lorentz and geometrical factor plane multiplicity factor structure factor structure amplitude

Three different approaches Mhkl…plane multiplicity

1. single phase

(without corrections)

relative intensities, e.g. to compare with PDF lists

disadvantage: the absolute values do not match the common experience that metals show higher intensities than organic phases.

2. single phase

(with absorption)

reflects the scattering power for different phases or for the same phases considering different radiations

3. phase mixture

(„without“ absorption)

normalized intensity per unit volume (vol%); considering the x-ray density the phase fractions can be given in mass%

vol%

mass%

but.........How exact is the x-ray density?

X-ray density

x M rel...rel. mass of the unit cell Vuc...unit cell volume

• In quantitative Powder-XRD an incorrect x-ray •

density x has a remarkable influence on the derived phase fractions. Undefined crystal water or OH-groups in crystal structure prevents any consideration of their weight during calculation of x. example:

Ettringite

127 H-atoms are not given in the crystal structure description and are absent in the relative mass of the unit cell. After all for Mrel =2371 these are appr. 5%.

Ca6(Al(OH)6)2(SO4)3 + 25.7· H2O

• In such a case it seems to be better to distribute the H atoms arbitrarily in the unit cell, or to find another way to “correct” the density.

Structure factor

General equation

• The structure factor describes the electron density distribution in the unit cell.

(element, bonding type, degree of ionization etc.)

• In case of an incomplete occupation and/or existing substitutions by different atoms the average „effective“ number of electrons on each atomic position determines their scattering power.

ki

occupation of atomic position i

gj

substitution by atom j

fij

atomic scattering factor

HKL Laue indices for interferences of a lattice plane {hkl} x,y,z relative atomic coordinates Tij

temperature factor

• For isotypic structures the exponential term is identical (e.g. element lattices, semiconductors). • The so-called temperature factor describing vibrations of atoms has a distinct influence on the structure factor.

Structure types: examples For isotypic structures the exponential term is identically.

Primitive element lattice

fcc element lattice

• if H, K and L are all even or all odd: F=4f

All reflections are allowed. b PowderCell 2.0

a

c

b PowderCell 2.0

c

a

NaCl structure type

PowderCell 2.0

a

Ni3Al

Cu3Au

10%

33%

110

8%

27%

111

100%

100%

• if H, K, L are even: F=4(fNa+ fCl )

200

47%

47%

F=4(fNa- fCl )

210

4%

15%

211

3%

11%

220

27%

29%

• if H, K, L are mixed: F=0

b

HKL

F=0

100

• if H, K, L are odd: c

• all other:

Extinction laws & systematic absences Systematic absences occur as consequence of translational operations additionally to the defined translation lattice (basis vectors), i.e. lattice centerings, screw axes, and glide planes. One distinguishes between integral, serial, and zonal absences.

Additionally, absent reflections can also be caused by atomic scattering vectors that cancel each other out. They are not easily predictable and are therefore called accidental absences (e.g. KCl or Si, cf. exercise).

Pb

The b glide plane systematically generates additional atomic positions shifted by [0,1/2,0]. However, there is no lattice point (primitive lattice) so that also the given red positions belong to one and the same lattice point. Since they form something like an additional lattice plane between the origin and the (010), the first reflection is called 020.

X-ray atomic scattering factor fij

 The form factor is related to the scattering density distribution in an atom.

 Tables for all atoms and some selected ions as function of sinq/l are given in the International Tables for Crystallography (IT).

 Nowadays: use approximation of Cromer & Mann (ajq, bjq and cj are listed IT (Vol. C)) The curvature is given by a 9 parameter approximation based on quantum-mechanical calculations.

 The parameters are not given for all ions. When in doubt please select the atoms. Neutrons are scattered by the core and not the electron shell. Since nucleus are very small fN does not shows any angular dependence. Also there are no differences between ions and atoms!

Atomic scattering factor fij atomic scattering factor for l=1.5418Å

• form factor fij is equivalent to the atomic number at low angles, but it drops rapidly at high sinq/l • The scattering power for atoms and ions are slightly different.

• sinq/l is proportional to 1/dHKL! Therefore the scattering power can be related to the lattice plane distances, and for a fixed l also presented in 2q scale. • Up to 50° in 2q, i.e. mostly for low indexed reflections, differences between ions may influence the reflection intensities.

Ions or atoms? Ions and atoms in fayalite original: Fe2+, Si4+, O2-

Instead of the ions Fe2+, Si4+ and O2- in FeSi2O4 in the structure file

• Fe2+ has been substituted by Fe (red curve)

Intensity deviation in %

• after that Si4+ has been substituted by Si (yellow curve)

• finally O2- has been replaced by O (blue curve).

(Iion-IFayalite)/Imax · 100%

There is really a distinct effect visible but because of the different signs for the single reflections virtually no influences on the quantitative phase analysis must be expected. However, such influence cannot be excluded (structure dependent!).

O, O- or O2-?

Ca6 (Al (O H)6)2 (S O4)3 + 25.7· H2 O

The oxygen in ettringite is generally described as O2-. However, the oxygen is placed in different coordination:

• in crystal water, • in OH- ions, • in sulfate tetrahedrons. Is there any difference? Is it correct to use O2-, or should one better adapt the ionization? H2O: with H-pos.: without H-pos.:

2H++O2O

OH-: with H-pos.: without H-pos.:

H++O2O-

SO4:

O2-

Influence of atomic positions assumption: identical lattice parameters

Forsterite

Fayalite

D(x,y,z) 0.

0.

0.28

1/4

0.4292

0.0975

1/4

Mg2+ 4d Si4+ 4d

0.9915 0.4262

0.2774 0.094

1/4 1/4

-0.0059 0.0026 0.003 0.0035

0. 0.

4c

0.7687

0.0928

1/4

O2-

4c

0.2076

0.4529

1/4

O2O2-

4d 4d

0.7657 0.2215

0.0913 0.4474

1/4 1/4

0.003 0.0015 -0.0139 0.0055

0. 0.

O2-

8d

0.2884

0.1637

0.0383

O2-

8e

0.2777

0.1628

0.0331

0.

Fe2+ 4c

0.9853

Si4+ 4c O2-

0.

0.

Mg2+ 4a

0.

0.

0.

0.

Fe2+ 4a

0.0107

0.0009

0.0052

• Practically for both structures all atomic positions are identical • Does this difference will have an influence on the powder pattern?

Structure data of bad quality influence the quantitative phase analysis remarkably! 2+  replace Mg by Fe2+

in forsterite

Substitutions and occupations example: Forsterite Mg2SiO4

&

Fayalite Fe2SiO4

• powder patterns show differences both in reflex positions and in intensity distribution • therefore: lattice parameters and atomic positions are equalized, and only Fe2+ substitutes Mg2+

Mg2SiO4

Fe2SiO4

Atomic Substitution/Occupation Assumptions:

SOF…usually used for site occupation factor here: Substitution and Occupation Factor

1. no lattice parameter deviations 2. atomic positions are identical (in first approximation OK) Note also the variation in absolute intensity!

100% 40% 50% 60% 70% 80% 90% 30% 20% 10% 0% Fe

Temperature factor Tij When an atom thermally vibrates, it effectively appears bigger to the diffracting x-rays. This gives rise to an atomic scattering form factor which decreases faster at higher q.

Influence of Tij

Problems: • often not exactly known • cannot be seriously refined using powder diffraction data • sometimes units are confusing • for quantitative analysis reliable approximations are required

Bij Debye-Waller factor: B = 8p²U = 2p²u²  80U u vibrational amplitude (in relative coordinates) Urs anisotropic temperature factors brs anisotropic temperature factors:

b11=2p² U11 a*

Relative and absolute Tij influences absolute intensity IB

Intensity

Fayalite

• Relatively, the impact of Bij increases at higher Bragg angles.

Intensity deviation in %

Relative deviation IB/IICSD

Relative deviation IB/Max(IICSD) Intensity deviation in %

• During comparison of calculated diffractograms the influence of Bij influence becomes visible.

(attenuation due to Tij is reflection dependent!)

• Since the absolute deviation at low diffraction angles is nevertheless bigger than at higher angles, intensity changes for reflections at low Bragg angle have a much bigger influence on quantitative phase analysis. • However, only than problematically if structure data with and without Debye-Waller factors will be mixed in quantitative phase analysis.

Plane multiplicity

Plane multiplicity MHKL The number of planes following from the point group symmetry for a specific {hkl} family is called the plane multiplicity factor.

For all cubic symmetries {110} is called dodecahedron, i.e. 12 different but symmetrically equivalent crystallographic planes exist. The cube plane {100} is similarly given by 6 planes. Thus, the {110} family will have twice the intensity of the {100} family since statistically for a equally distributed crystal orientation {110} will occur twice more often then {100}. MHKL depends on the point group symmetry. However, because of Friedel’s law in x-ray diffraction one has only to distinguish between the 11 Laue groups, except if anomalous dispersion effects occur. In that case the point group symmetry is really the exact one. This is obviously incomplete.

Preferred orientation Many materials but also most prepared powders exhibit preferred orientations as a characteristic property. Preferred orientation is therefore a common reason for deviations between experimental diffraction data and ideal intensity simulations. Preferred orientation can be recognized and compensated for single phase materials, but it is more difficult during quantitative analysis. The typical way of dealing with preferred orientation of know phase is a careful comparison between simulation and experiment. For axial crystals (hexagonal, trigonal, tetragonal, monoclinic) some simple approximations have been developed. These are based on the change of plane multiplicity.

The given March model calculates for all equivalent reflections HKL the angle distance j to a preselected lattice direction and determines so the probability for the experimental weight (effective multiplicity Meff) for each diffracting family {hkl} by variation of a general parameter o.

PLG factor

Polarization, Lorentz and geometry factor Two commonly used integration methods are (1) rocking curve measurements (w,q) describing the Lorentz factor and (2) powder diffraction method considering the geometry.

Lorentz factor (1)

When each lattice point on the reciprocal lattice intersects the diffractometer circle, a diffraction related to the plane represented will occur. The diffractometer typically moves at a constant 2θ rate, the amount of time each point is in the diffracting condition will be a function of the diffraction angle.

Geometry factor (2)

It considers the intensity of a complete powder diffraction ring (cosθ), however, if only a portion of the ring is measured with a detector with an outof-plane acceptance width ∆w, one has to introduce an additional factor which depends on the size of the powder diffraction ring (1/sin2θ).

Polarization factor

It indicates that the incoming non-polarized x-ray is polarized by the scattering process, resulting in a directional variation in the scattered intensity. So a symmetrically weak dependency to 2θ=90° follows.

PLG After multiplication of all factors constant parts can be neglected so that follows:

using a monochromator a…Bragg angle

In dependence on the used diffraction geometry PLG has to be to change.

Sometimes L and G has been mixed so that in this cases the Lorentz polarization term is described as correction term only. This coresponds with the here given PLG.

The minimum of PLG is at about 2q=100°

Absorption

Absorption

• dependent on the path of the radiation through the sample, i.e. from the used geometry (for Bragg-Brentano it can be neglected because of the symmetrical path of incident and diffracted beam) • dependent on the mass attenuation coefficient µ and the wavelength l

Different equations will be used containing the attenuation relationsships and geometrical parameters

Further influences

• use of variable slits to fix the irradiated sample region (~sinq) • consideration of simple but Bragg angle dependent mathematic profile functions to simulate the intensity distribution caused by used slits, grain size of the powder, lattice strain etc.

Conclusions  Be careful! The smallest R-value does not mean simultaneously that the found result is the best.

Take care especially in case of phases where you cannot evaluate the quality of structure data. Please have a look on the remarks given in the structure data entry of the data base!

 Please check the content of the unit cell if the given atomic arrangement is expected. Usually it helps to generate more than a single unit cell.

 Ask yourself why different structure descriptions exist!

Compare the different crystal structure descriptions and check them out if necessary.

 Attend to non-standardized space-group settings!

It is much better to use a standard setting, or to transform the original data.

 Check automatically imported data on correctness! (atomic positions as well as temperature factors)

 Decide to use or to neglect temperature factors! Attend to the size of the given values.

References: P-XRD (general) Bish, D.L., and Post, J.E., eds., 1989, Modern Powder Diffraction, Min. Soc. America Reviews in Mineralogy Vol. 20, 369 p. (Price $28 only from Mineralogical Society of America) Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, Third Edition, AddisonWesley, 664 p. Klein, Cornelis, 2002, Mineral Science (22nd Edition), John Wiley, 641 p.

Klug, Harold P., and Alexander, Leroy E., 1977, X-Ray Diffraction Procedures for Polycrystalline and Amorphous Materials, Second Edition, John Wiley, 966 p. Moore, Duane E., and Reynolds, Robert C., Jr., 1997, X-Ray Diffraction and the Identification and Analysis of Clay Minerals Nuffield, E.W., 1966, X-ray Diffraction Methods, John Wiley & Sons, 408 p.

References: quantitative P-XRD Bish, D.L, and Howard, S.A., 1988, Quantitative phase analysis using the Rietveld method. J. Appl. Cryst., v. 21, p. 86-91.

Bish, D.L., and Chipera, S. J, 1988, Problems and solutions in quantitative analysis of complex mixtures by X-ray powder diffraction, Advances in X-ray Analysis v. 31, (Barrett, C., et al., eds.), Plenum Pub. Co., p. 295-308. Bish, D.L., and Chipera, S.J., 1995, Accuracy in quantitative x-ray powder diffraction analyses, Advances in X-ray Analysis v. 38, (Predecki, P., et al., eds.), Plenum Pub. Co., p. 47-57 Chipera, S.J, and Bish, D.L, 1995, Multireflection RIR and intensity normalizations for quantitative analyses: Applications to feldspars and zeolites. Powder Diffraction, v. 10, p. 47-55. Chung, F.H., 1974, Quantitative interpretation of X-ray diffraction patterns. I. Matrixflushing method of quantitative multicomponent analysis. J. Appl. Cryst., v. 7, p. 519-525. Downs, R.T., and Hall-Wallace, M., 2003, The American Mineralogist crystal structure database. American Mineralogist, v. 88, p. 247-250. Snyder, R.L. and Bish, D.L., 1989, Quantitative Analysis, in Bish, D.L. and Post, J.E., eds., Modern Powder Diffraction, Mineralogical Society of America Reviews in Mineralogy, V. 20, p. 101-144. Young, R.A., 1993, The Rietveld Method, Intl. Union of Crystallographers Monograph on Crystallography V. 5, Oxford University Press, 298 p.