# Powder Diffraction at SSRL by Q4PJ9z

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```									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 size
 Peak Positions:
Phase identification
Lattice symmetry
Lattice expansion
 Peak Intensity:
Structure solution
Crystallite orientation
niasin(a)
A(a) = S e                = FT(a)
Sample Diffraction            A = S cos(nf)
+ i sin(nf)

a
A=   Feiwt
S    e
i nf

A = S F(a)    e
iwt + inf

iwt + inf
F(a) e
a
iwt
F(a) eiwt + i3f
e
F(a) eiwt + i2f
iwt + if
asin (a) = f                F(a) e
iwt
Laue’s Eq.                 F(a)e
Sample  Diffraction
Diffraction Pattern ~ {FT(sample) } {FT(sample) }

=                 x                         *

M
o
t
i
Sample size                              f
(S)              Infinite Periodic
Lattice (P)      (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)

*                  x

=

y
FT(M)

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

x

X
What does a diffraction
pattern tell us?

 Peak Shape & Width:
crystallite size
 Peak Positions:
Phase identification
Lattice symmetry
Lattice expansion
 Peak Intensity:
Structure solution
Crystallite orientation
Structure Solution

 Single Crystal                 Powder
 Due to small crystallite size
 Protein Structure               kinematic equations valid

 Sample with heavy Z            Many small molecule
problems Due to                 structures obtained via
synchrotron diffraction
 Absorption/extinction
effects
 Peak overlap a problem – high
resolution setup helps
 Mostly used in Resonance
mode                           Much lower intensity – loss on
super lattice peaks from small
 Site specific valence
symmetry breaks. (Fourier
 Orbital ordering.            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 & a )

 Peak Overlap :: A problem
 But can be useful for precise
lattice parameter
measurements

 ~ (invers.) “size” of the sample
Crystallite size
Domain size

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

Wd2 = M2 x fb2 + fs2
M= (2 tan q/tan qm -tan qa/ tan qm -1)

Where
f = divergence of the incident beam,
b
f = cumulative divergences due to slits and apertures
s
q, q and q = Bragg angle for the sample, analyzer and the monochromator
a      m
Powder Average
Single crystal – no intensity
Fixed 2q
Even if Bragg angle right,
Q +/- d(Q) = q +/- d(q)
But the incident angle wrong

d(q)                 d(q) = Mosaic width ~ 0.001 – 0.01 deg
d(Q) = beam dvg ~ >0.1 deg for sealed tubes
~ 0.01- 0.001 deg for
synchrotron
q                                         For Powder Avg
2q            Need <3600 rnd crystallites – sealed tube
Need ~ 30000 rnd crystallites - synchrotron
Q
Powder samples must be prepared carefully
And data must be collected while rocking the sample
Physics of Diffraction

No X-ray Lens

Mathematically
Phase Problem

rxyz = Shkl Fhkl exp(-2pi{hx + ky + lz})
Fhkl is a Complex quantity
Fhkl(fi, ri): (Fhkl)2 = Ihkl/(K*Lp*Abs)
= Shkl CIhkl exp(-(f + Df))
 rxyz
Df = phase unknown
Hence Inverse Modeling
Solution to Phase Problem

 Must be guessed
And then refined.

How to guess?
Similarity 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 placement
Phases for each hkl

Measure the peak widths
Use an appropriate profile shape function

Construct a full diff. pattern and compare with
measurements
Inverse Modeling Method 1
7000

6000

Reitveld Method
5000

Data               4000

Intensity
3000

2000

1000

0

40   60   80    100   120   140   160

q

Profile
shape                                                                  Refined Structure
Model

Background
Inverse Modeling Method 2
7000

6000

Fourier Method
5000

Data               4000

Intensity
3000

2000

1000

0

40   60   80    100   120   140   160

q

subtract
Background

Profile
shape

Integrated
Intensities
Refined Structure
Model
phases
Inverse Modeling Methods

 Rietveld Method                     Fourier Method
 More precise                        Picture of the real space
 Yields Statistically reliable       Shows “missing” atoms,
uncertainties                        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

Empirical
Voigt function modified for axial divergence
(Finger, Jephcoat, Cox)
Refinable parameters – for crystallite size, strain

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 list
http://www.ccp14.ac.uk/mirror/want_to_do.html

GSAS
Fullprof
DBW
MAUD

Topaz – not free - Bruker – fundamental
approach
Structure of MnO
MnO @ 6530eV
7000
Scattering
6000                                                                                  density
5000
structural fit
2000

fMn(x,y,z,T,E)
In te n sity

4000
Intensity

3000                           0                                                   fO(x,y,z,T,E)
2000
72              74                   76

2q
1000

0

-1000
40   60                         80   100        120       140        160

2q
Resonance Scattering
Fhkl = Sxyz fxyz exp(2pi{hx + ky + lz})

7000

6000

5000

4000
Intensity

3000

2000

1000

0

40   60   80    100   120   140   160

fxyz = scattering density
q

Away from absorption edge
a electron density
Anomalous Scattering Factors

fxyz = fe{fiexyzT} f = Thomson scattering for an electron
e

fi = fi0(q) + fi’(E) + i fi”(E)
m(E) = E * fi”(E)
Kramers -Kronig :: fi’(E) <-> fi”(E)

2
f ' ( E0 ) =
p  0
f " ( E ) E 2 E E 2 dE
- 0
Resonance Scattering vs Xanes
7000

6000

5000

4000
Intensity

3000

2000
-2          MnO2
5000
1000
-3
4000                            0
from resonance scattering
40            60     80         100          120   140
-4
3000                                                              2q
Intensity

2000                                                                                                  -5
1000
-6
9000

0                                                                                                                                                                     8000

-7                                                                7000

40                  60              80         100        120         140                                                                                        6000
f'
2q
-8                                                                5000

Intensity
4000

3000

-9                                                                2000

1000

0
1.0
-10                                                                       40   60   80        100   120   140

from KK transform of XANES                                                      2q

0.8
-11
0.6
-12
m = f"/E

0.4
-13
0.2                                                                                                                          6440   6460   6480   6500   6520   6540   6560               6580         6600
0.0                                                                                                                                                 Energy(eV)
6500        6520                    6540        6560        6580             6600

Energy (eV)
XANE Spectra of Mn Oxides

0.8
Mn Valence
MnO
MnO2    Mn
bsorption

0.8
A

0.4

Mn3O4                        Mn(II)?
0.0
Mn2O3
6540

Energy(eV)
6570   6600

Mn(II)?
0.4                                                                 Mn2O3
Mn(I)?
Absorption

Mn3O4
MnO2
Mn(I)?
0.0                                                                                     MnO     Mn
6540                              6560
Avg.    Actual
Energy (eV)
F’ for Mn Oxides
-3
f' (electrons)

-6
M n 2O 3:1
M n 2O 3:2
-6
f ' ( e le c t r o n s )

M nO   2

-8                                        M nO
-9
M n 3O 4:2
M n 3O 4:1
-1 0
C r o m e r -L ib e r m a n M n
6530                            6540                  6550                6560
6450                                  6500               6550
E n e rg y (e V )

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

•Mn valence of the two sites in
Mn2O3 very similar
•Valence of the two Mn sites in
Mn3O4 different but not as                                              CL
0

d(f')/dE
different as expected.
-3
Mn3O4
-4

-5

-6
f'

-7

Mn1                                                6540   6544      6548        6552   6556
-8               Mn2
X Axis Title
-9   A. Mehta, A. Lawson, and J. Arthur

6460     6480      6500       6520   6540
Energy (eV)

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