What do X-ray powder diffraction patterns look like? What do X-ray powder diffraction patterns look...

What do X-ray powder diffraction patterns look like?What do X-ray powder diffraction patterns look like?

Each peak in pattern is a reflection from a different set of planes

By determining 2 for a peak, can use Braggs' law to get d for that set of planes

Powder patterns - what information available in pattern?

1. peak positions2. peak intensities3. peak shape4. background structure

1. peak positions - give unit cell shape, size, and symmetry1. peak positions - give unit cell shape, size, and symmetry

Simple example - P cubic:Simple example - P cubic:

Naming each peak with Miller indices is called " indexing the pattern" Naming each peak with Miller indices is called " indexing the pattern"

Indexing Indexing

Message: indexing cubic patterns is usually pretty easyespecially, if done visually

Why:usually, relatively few reflectionsreflections evenly spacedonly one lattice parameter to guess

Indexing Indexing

In cubic:

dhkl = a/(h2 + k2 + l2)1/2 = a/s1/2

a = s1/2 dhkl

s is an integer …… 1, 2, 3, 4….

In cubic:

dhkl = a/(h2 + k2 + l2)1/2 = a/s1/2

a = s1/2 dhkl

s is an integer …… 1, 2, 3, 4….

Since = 2dhkl sin hkl, smallest means largest dhkl

Since a = s1/2 dhkl, largest dhkl means smallest s value

Thus, lowest 2reflections in pattern are those with small s values

Since = 2dhkl sin hkl, smallest means largest dhkl

Since a = s1/2 dhkl, largest dhkl means smallest s value

Thus, lowest 2reflections in pattern are those with small s values

Indexing Indexing Fill in the (hkl)s for these s valuesFill in the (hkl)s for these s values

IndexIndex

Indexing Indexing

(321)(222) (400)

IndexIndex

Indexing Indexing

Slight complication: 3 different Bravais lattices in cubic

P cubic

I cubic

F cubic

Slight complication: 3 different Bravais lattices in cubic

P cubic

I cubic

F cubic

Translational symmetry gives missing planes - thus missing reflections

All planes, reflections possibleAll planes, reflections possible

(hkl) present only if h + k + l = even(hkl) present only if h + k + l = even

(hkl) present only if h,k,l all even or all odd(hkl) present only if h,k,l all even or all odd

Indexing Indexing

These are "extinction rules"These are "extinction rules"

Indexing Indexing Fill in the (hkl)s for these I cubic (h + k + l = even) & F cubic (h,k,l all even or all odd)

Fill in the (hkl)s for these I cubic (h + k + l = even) & F cubic (h,k,l all even or all odd)

Indexing Indexing Indexing Indexing

IndexIndex

Indexing Indexing

How to know indexing is correct?How to know indexing is correct?

Indexing Indexing

Correct indices give consistent set of lattice parameters

Example for cubic:

d(Å) (hkl) a(Å)

3.157 (100) 3.1571.931 (110) 2.7311.647 (111) 2.8521.366 (200) 2.732

Correct indices give consistent set of lattice parameters

Example for cubic:

d(Å) (hkl) a(Å)

3.157 (100) 3.1571.931 (110) 2.7311.647 (111) 2.8521.366 (200) 2.732

(hkl) a(Å)

(111) 5.468(220) 5.462(311) 5.462(400) 5.464

(hkl) a(Å)

(111) 5.468(220) 5.462(311) 5.462(400) 5.464

Other types of translational symmetry also give missing planes

glide planes and screw axes

Other types of translational symmetry also give missing planes

glide planes and screw axes

Indexing Indexing

Ex: F 41/d 3 2/m

(0kl) present if k + l = 4n

(hhl) present if h + l = even

(hkl) if h + k + l = odd or 4n(for some structures w/ this space group)

Ex: F 41/d 3 2/m

(0kl) present if k + l = 4n

(hhl) present if h + l = even

(hkl) if h + k + l = odd or 4n(for some structures w/ this space group)

What if it's not cubic?

Manual indexing methods usually very tedious - use computer indexing programs such as "Ito", "Treor", "Crysfire", "Dicvol", others

What if it's not cubic?

Manual indexing methods usually very tedious - use computer indexing programs such as "Ito", "Treor", "Crysfire", "Dicvol", others

Indexing Indexing

Computer indexing frequently not easy - sometimes have to use several, or all, indexing programs to figure out right answer

Why index? Why index?

cubic: dhkl2 = a2/(h2 + k2 + l2)

tetragonal: 1/dhkl2 = (h2 + k2)/a2 + l2 /c2

hexagonal: 1/dhkl2 = (4/3)(h2 + k2 +

hk)/a2 + l2 /c2

orthorhombic: 1/dhkl2 = h2/a2 + k2/b2 +

l2 /c2

etcetera

cubic: dhkl2 = a2/(h2 + k2 + l2)

tetragonal: 1/dhkl2 = (h2 + k2)/a2 + l2 /c2

hexagonal: 1/dhkl2 = (4/3)(h2 + k2 +

hk)/a2 + l2 /c2

orthorhombic: 1/dhkl2 = h2/a2 + k2/b2 +

l2 /c2

etcetera

It can be fun! Like solving a puzzleIt can be fun! Like solving a puzzle

Indices needed to calculate lattice parameters

need (hkl) and dhkl for all reflections

use all reflections to get high precision

Indices needed to calculate lattice parameters

need (hkl) and dhkl for all reflections

use all reflections to get high precision

Indexing Indexing

ATCF!!!ATCF!!!

But……小心But……小心

Looks cubicLooks cubic

But it is not!(C orthorhombic)But it is not!(C orthorhombic)

1. peak positions2. peak intensities - how much?3. peak shape4. background structure

Relatively recent increase in use of X-ray diffraction for quantitative analysis

1940 1950 1960 1970 1980 1990 2000 2010

Number of Publications

1940 1950 1960 1970 1980 1990 2000 2010

Number of Publications

Papers publishedon quantitative analysisBy x-ray diffraction

1. peak positions2. peak intensities - how much?3. peak shape4. background structure

Quantitative analysis

Example: 2-phase mixtures of CaF2 & ZnO

Quantitative analysis

Example: 2-phase mixtures of CaF2 & ZnO

67 wt% CaF2/33 wt% ZnO

23 wt% CaF2/77 wt% ZnO

Iia =Iia =KiaXaKiaXa

Main equation for quantitative analysisMain equation for quantitative analysis

Quantitative AnalysisQuantitative AnalysisUse measured intensities ….. but not simpleUse measured intensities ….. but not simple

Iia = intensity of reflection i of phase a

ais density of phase aais density of phase a

Kia contains structure factor, multiplicity, Lorentz-polarization factor, temperature factor + scale factor for reflection i of phase a

Xa = wt fraction of phase a (want this!!)

measure this!!measure this!!

But what do we do with ?But what do we do with ?

Xa = wt fraction of phase a (Want this!!)

Measure this!!Measure this!!

is mass attenuation coefficient for mixture …. unknown, unless we know

mix = X1phase1 + X2phase 2 + X3phase 3 + ………mix = X1phase1 + X2phase 2 + X3phase 3 + ………

for each phase may be calculated from the chemical composition of each phase

for each phase may be calculated from the chemical composition of each phasecompd = X11 + X22 + X33 + ………compd = X11 + X22 + X33 + ………

If sample contains > one phase, then unknown if wt fractions unknownIf sample contains > one phase, then unknown if wt fractions unknown

would also be nice not to calc Kia or get a

Use measured intensities ….. but not simpleUse measured intensities ….. but not simple

Simple example: rutile/anatase polymorphs since polymorphs have same composition:

mix = rutile = anatasemix = rutile = anatase

Quantitative AnalysisQuantitative Analysis

Iirutile =Iirutile =KirutileXrutileKirutileXrutile

rutile rutile

Iirutile =Iirutile =KirutileKirutile

rutile rutilerutile rutile

Suppose we measure Iirutile of pure rutile

can eliminate mix (& other stuff)

Suppose we measure Iirutile of pure rutile

can eliminate mix (& other stuff)

= Xrutile= Xrutile

For mixtures that are not polymorphs, many single & multiple peak procedures have been developed that use known amounts ofinternal standards - mixed with unknownexternal standards - run separate x-ray patternof standard material

Most analysis techniques tell only what elements are present & how much. X-ray diffraction tells how much of what PHASES are presentMost analysis techniques tell only what elements are present & how much. X-ray diffraction tells how much of what PHASES are present

Rietveld analysis of fuel depositRietveld analysis of fuel deposit

Recently, whole pattern analysis methods for quantitative analysis have become popular. Example: Rietveld analysisRecently, whole pattern analysis methods for quantitative analysis have become popular. Example: Rietveld analysis

Rietveld analysis of fuel depositRietveld analysis of fuel deposit

What do X-ray powder diffraction patterns look like? What do X-ray powder diffraction patterns look...

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