Structural Analysis

Apurva Mehta

Physics of Diffraction

X-ray Lens not very good

Mathematically

Intersection of

Ewald sphere with Reciprocal Lattice

outline

Information in a Diffraction pattern

Structure Solution

Refinement Methods

Pointers for Refinement quality data

What does a diffraction pattern tell us?

Peak Shape & Width:crystallite sizeStrain gradient

Peak Positions:Phase identificationLattice symmetryLattice expansion

Peak Intensity:Structure solutionCrystallite orientation

Sample Diffraction

a

asin (e

itF(

e it + i

F(

e it + i3

e it + i2F(

F(

e it + in

F(

e it + in

F(A =

A = F e i n

e it

A(e niasin(

= FT(a)

eit

Laue’s Eq.

A = cos(n i

sin(n)

Sample Diffraction

= x *

Diffraction Pattern ~ {FT(sample) } {FT(sample) }

Sample size (S) Infinite Periodic

Lattice (P)

Motif

(M)

Sample Diffraction

FT(Sample) = FT((S x P)*M)

Convolution theorem

FT(Sample) = FT(S x P) x FT(M)

FT(Sample) = (FT(S) * FT(P)) x FT(M)

FT(S)

X

Y

FT(P)

FT (S x P) = FT(S) * FT(P)

*

=

y

x

FT(M)

FT(sample) = FT(S x P) x FT(M)Along X direction

xX

What does a diffraction pattern tell us?

Peak Shape & Width:crystallite sizeStrain gradient

Peak Positions:Phase identificationLattice symmetryLattice expansion

Peak Intensity:Structure solutionCrystallite orientation

Structure Solution

Single Crystal Protein Structure

Sample with heavy Z problems Due to

Absorption/extinction effects

Mostly used in Resonance mode

Site specific valence Orbital ordering.

Powder Due to small crystallite size

kinematic equations valid

Many small molecule structures obtained via synchrotron diffraction

Peak overlap a problem – high resolution setup helps

Much lower intensity – loss on super lattice peaks from small symmetry breaks. (Fourier difference helps)

Diffraction from Crystalline Solid

Long range order ----> diffraction pattern periodic crystal rotates ----> diffraction pattern rotates

Pink beam laue pattern

Or intersection of a large Ewald Sphere with RL

From 4 crystallites

From Powder

Powder Pattern

Loss of angular information

Not a problem as peak position = fn(a, b &

Peak Overlap :: A problem But can be useful for

precise lattice parameter measurements

Peak Broadening

~ (invers.) “size” of the sampleCrystallite sizeDomain size

Strain & strain gradient

Diffractometer resolution should be better than Peak broadening But not much better.

Diffractometer Resolution

Wd2 = M2 x b

2 + s2

M= (2 tan tan m -tan a/ tan m -1)

Where

b

= divergence of the incident beam,

s = cumulative divergences due to slits and apertures

a and

m = Bragg angle for the sample, analyzer and the monochromator

Powder Average

Fixed 2

2

Single crystal – no intensity

Even if Bragg angle right,

But the incident angle wrong

Mosaic width ~ 0.001 – 0.01 deg

beam dvg ~ >0.1 deg for sealed tubes

~ 0.01- 0.001 deg for synchrotron For Powder Avg

Need <3600 rnd crystallites – sealed tubeNeed ~ 30000 rnd crystallites - synchrotron

Powder samples must be prepared carefullyAnd data must be collected while rocking the sample

Physics of Diffraction

No X-ray Lens

Mathematically

Phase Problem

xyz = hkl Fhkl exp(-2i{hx + ky + lz})

Fhkl is a Complex quantity

Fhkl(fi, ri): (Fhkl)2 = Ihkl/(K*Lp*Abs)

xyz = hkl CIhkl exp(-) = phase unknown

Hence Inverse Modeling

Solution to Phase Problem

Must be guessedAnd then refined.

How to guess?Heavy atom substitution, SAD or MADSimilarity to homologous compounds

Patterson function or pair distribution analysis.

Procedure for Refinement/Inverse Modeling

Measure peak positions:Obtain lattice symmetry and point group

Guess the space group. Use all and compare via F-factor analysis

Guess the motif and its placementPhases for each hkl

Measure the peak widthsUse an appropriate profile shape function

Construct a full diff. pattern and compare with measurements

Inverse Modeling Method 1

Reitveld Method Data

Model Refined Structure

40 60 80 100 120 140 160

0

1000

2000

3000

4000

5000

6000

7000

Inte

nsity

Profile shape

Background

Inverse Modeling Method 2

Fourier Method Data

Model Refined Structure

40 60 80 100 120 140 160

0

1000

2000

3000

4000

5000

6000

7000

Inte

nsity

Profile shape

Backgroundsubtract

phases

Integrated Intensities

Inverse Modeling Methods

Rietveld Method More precise Yields Statistically

reliable uncertainties

Fourier Method Picture of the real space Shows “missing” atoms,

broken symmetry, positional disorder

Should iterate between Rietveld and Fourier.Be skeptical about the Fourier picture if Rietveld

refinement does not significantly improve the fit with the “new” model.

Need for High Q

Many more reflections at higher Q.

Therefore, most of the structural information is at higher Q

Profile Shape function

EmpiricalVoigt function modified for axial

divergence (Finger, Jephcoat, Cox)Refinable parameters – for crystallite size,

strain gradient, etc…

From Fundamental Principles

Collect data on Calibrant under the same conditions

Obtain accurate wavelength and diffractometer misalignment parameters

Obtain the initial values for the profile function (instrumental only parameters)

Refine polarization factor

Tells of other misalignment and problems

Selected list of Programs

CCP14 for a more complete listhttp://www.ccp14.ac.uk/mirror/want_to_do.html

GSASFullprofDBWMAUD

Topaz – not free - Bruker – fundamental approach

Structure of MnO

40 60 80 100 120 140 160-1000

0

1000

2000

3000

4000

5000

6000

7000

MnO @ 6530eV

structural fit

72 74 76

0

2000

Inte

ns

ity

2

Inte

nsi

ty

2

Scatteringdensity

fMn(x,y,z,T,E)

fO(x,y,z,T,E)

Resonance Scattering

Fhkl = xyz fxyz exp(2i{hx + ky + lz})

40 60 80 100 120 140 160

0

1000

2000

3000

4000

5000

6000

7000

Inte

nsi

ty

fxyz = scattering densityAway from absorption edge

electron density

Anomalous Scattering Factors

fxyz = fe{fixyzT} fe = Thomson scattering for an

electron

fi = fi0(q) + fi’(E) + i fi”(E)

(E) = E * fi”(E)Kramers -Kronig :: fi’(E) <->

fi”(E) dEEfEfEE

E

0

0 20

2)("2

)('

Resonance Scattering vs Xanes

6440 6460 6480 6500 6520 6540 6560 6580 6600-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

from KK transform of XANES

from resonance scattering

MnO2

f'

Energy(eV)6500 6520 6540 6560 6580 6600

0.0

0.2

0.4

0.6

0.8

1.0

=

f"/

E

Energy (eV)

40 60 80 100 120 140

0

1000

2000

3000

4000

5000

6000

7000

Inte

nsi

ty

2

40 60 80 100 120 140

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Inte

nsi

ty

2

40 60 80 100 120 140

0

1000

2000

3000

4000

5000

Inte

nsity

2

XANE Spectra of Mn Oxides

6540 6560

0.0

0.4

0.8

6540 6560

0.0

0.4

0.8

MnO2

Mn2O

3

Mn3O

4

MnO

6540 6570 6600

0.0

0.4

0.8

6540 6570 6600

0.0

0.4

0.8

Abso

rptio

n

Energy (eV)

Abs

orpt

ion

Energy (eV)

Mn Valence

MnO

MnO2

Avg. Actual

Mn

Mn

Mn(II)?

Mn(I)?Mn(II)?

Mn(I)?

Mn3O4

Mn2O3

F’ for Mn Oxides

6450 6500 6550

-9

-6

-3

f' (e

lect

rons

)

Energy (eV)

6530 6540 6550 6560

-10

-8

-6

MnO2

Mn2O

3:1

Mn2O

3:2

Mn3O

4:2

Mn3O

4:1

MnO

Cromer-Liberman Mn

f' (

ele

ctr

on

s)

Energy (eV)

Why Resonance Scattering?

Sensitive to a specific crystallographic phase. (e.g., can

investigate FeO layer growing on metallic Fe.)Sensitive to a specific

crystallographic site in a phase. (e.g., can investigate the tetrahedral and the

octahedral site of Mn3O4)

Mn valences in Mn Oxides

6540 6544 6548 6552 6556

0

CL

d(f')

/dE

X Axis Title

•Mn valence of the two sites in Mn2O3 very similar

•Valence of the two Mn sites in Mn3O4 different but not as different as expected.

6460 6480 6500 6520 6540

-9

-8

-7

-6

-5

-4

-3

A. Mehta, A. Lawson, and J. Arthur

Mn3O

4

f'

Energy (eV)

Mn1 Mn2