DXC 2013DXC 2013

Peter Y. ZavalijPeter Y. Zavalij

X-ray Crystallographic CenterDepartment of Chemistry and Biochemistry

University of Maryland at College Park, Maryland, USA

HANDS-ON RIETVELD ANALYSISHANDS-ON RIETVELD ANALYSIS

Topics & Scope:Topics & Scope:o Recommended refinement strategieso Structure models and instrumental parameterso How to refine: lattice parameters, site occupancy factors,

thermal parameters, preferred orientationo How to properly consider instrumental parameterso Accuracy and precisiono Refinement indices and what they meano How to use the program to model a hypothetical patterno Quantitative analysis and amorphous contento What to do for an incomplete structure model

This is not attempt to give you all details how torefine crystal structure (this would require much moretime) and which buttons and in which sequences topress (this is rather impossible) but to give you guidanceabout what you need to know and consider whenworking on the structure analysis.

Where to go for help?Where to go for help?

1. V. Pecharsky & P. Zavalij “Fundamentals of Powder Diffraction and Structural Characterization of Materials. Second Edition” Springer, 2009, ISBN: 978-0-387-09578-3- Corrections, Color Figures, Examples,

Problems, Web link are available online*

2. Online materials:- links in the textbook, search- IUCr Teaching Pamphlets:

http://www.iucr.org/education/pamphlets

3. Online discussion groups: a) Rietveld mailing list - to subscribe e-mail to

listserv@ill.fr : SUB Rietveld_l “Your_Name “b) LinkedIn.com groups, e.g. Crystallography

* http://www2.chem.umd.edu/facility/xray/Zavalij/FPDSCM2

Unit Cell

Atomic Structure

Sample & Instrument

Powder Pattern

+

+

=

Powder Pattern = Unit Cell + Atomic Structure +Sample & Instrument

Powder Pattern = Unit Cell + Atomic Structure +Sample & Instrument

Inte

nsity

2θ

2θ

2θ

2θ

1)

2)

3)

Inte

nsity

Inte

nsity

h 1k 1

l 1

h 2k 2

l 2

h 3k 3

l 3

h 4k 4

l 4h 5

k 5l 5

h 6k 6

l 6h 7

k 7l 7

…

Understanding of Powder PatternUnderstanding of Powder Pattern

Patterncomponent

Structureparameters

Specimen property

Instrumental parameter

Peak position

Unit cell :(a,b,c,α,β,γ)

AbsorptionPorosity

Radiation (wavelength)Instrument/sample alignmentAxial divergence of the beam

Peak intensity

Atomic:(x, y, z, B, etc.)

Preferred orientationAbsorptionPorosity

Radiation Geometry and configuration(Lorentz, polarization)

Peak shapeDisorderDefects

Grain sizeStrainAbsorption

Radiation (spectral purity) Geometry, Beam conditioning (Slits)

Table: Powder diffraction pattern as a function of various crystal structure, specimen and instrumental parameters

Other Factors Affecting Peak PositionsOther Factors Affecting Peak Positions

θ2θ2θ2 Δ+= calcobs

654321 θcosθ2sinθtanθ2sinθ2tan

θ2 pppppp

+++++=Δ

22

2

221

2

1 3 ;

3 RKh

pRKh

p −=−=

3

2

3αK

p −=

Rp

eff2μ1

4 =

Rsp 2

5 −=

Axial divergence of the incident beam(Soller slits decrease this effect)

In-plane divergence of the x-ray beam

Transparency shift (absorption effect)

Specimen displacement

Zero shift p6

POaxis

T⊥/T||= 2.5

φhkl

Thkl

T||= 1/τ

T ⊥=

1

d*hk

l

d*Thk

l

a)

T||= 1/τ3

T ⊥=

τ3/2

b)

T||=0.67

T ⊥=1

c)

Fig. 8.21. Ellipsoidal (a) and March-Dollase (b) functions with the magnitude T⊥/T|| = 2.5, and the two functions overlapped when T⊥/T|| = 1.5

[ ]2/1

1i

22 cos)1(11 −

= φ−+=

Nihklhkl N

T τ2/3

1i

222 sin1cos1 −

=

φ

τ+φτ=

Nihkl

ihklhkl N

T

Preferred orientationPreferred orientation

Z

X

Y

Fig. 8.22. Spherical harmonic preferred orientation function for the (100) reflection

= −=+

+=L

l

l

lm

ml

ml hkC

lT(h)

2

)(12

π41

= −=+

+=L

l

l

lm

mlC

lJ

2

2

1211

Preferred orientation correction:

Magnitude:

Preferred orientationPreferred orientation

x = 2θj - 2θk (deg.)-0.4 -0.2 0.0 0.2 0.4

G(x

); L(

x) (a

rb. u

nits

)

Lorentz Gauss ∞

∞−

∞

∞−= dxxLdxxG )()(

Normalized:

Gauss:

Lorenz:

( )22/1

expπ

)()( xCH

CxGxy G

G −==

( ) 122/1

1π

)()( −+′

== xCH

CxLxy L

L

Peak Shape Functions:Peak Shape Functions:

Pseudo-Voigt:

Pearson VII:( )

( ) ( ) β22/1

12/1β

β)(VII)( −+π−Γ

Γ== xCH

CxPxy PP

- Caglioti formula: ( ) 2/12 θtanθtan WVUH ++=θtanθcos/ YXH +=′

( ) ( ) ( ) 122/1

22/1

1π

η1expπ

η)()( −+−+−== xCH

CxCH

CxPVxy LL

GG

‘

Commonly Used Peak Shape Functions:Commonly Used Peak Shape Functions:

Bragg angle, 2θ (deg.)44.0 44.2 44.4 44.6 44.8 45.0

Rel

ativ

e in

tens

ity, Y

(arb

. uni

ts)

0

20

40

60

80

100 Kα1

Kα2

Kα1 +Kα2

BackgroundYobs-Ycalc

Fig. 8.13. Using Pearson-VII function to fit experimental data (open circles) representing a single Bragg peak containing Ka1 and Ka2 components.

Fundamental Parameters ApproachFundamental Parameters Approach

Peak shape function (PSF) = convolution* of 3 different functions:

- Ω - instrumental broadening,

- Λ - wavelength dispersion,

- Ψ - specimen function.

+ b – background function

∞

∞−

∞

∞−

−=−=⊗ τ)τ()τ(τ)τ()τ()()( dtfgdtgftgtf*

)θ()θ()θ()θ()θ( bPSF +Ψ⊗Λ⊗Ω=

∗ ∗ ∗Ω =

Slitwidth

X-rayfocus

In planedivergence

Axialdivergence

Fig. 8.15. Graphical representation of most common fundamental functions defining instrumental broadening.

)θ()θ()θ()θ()θ( bPSF +Ψ⊗Λ⊗Ω=

Fundamental Parameter Approach for Instrumental Broadening

Fundamental Parameter Approach for Instrumental Broadening

Specimen Peak Broadening (β):Specimen Peak Broadening (β):

cosθτλβ

⋅=

tanθεβ ⋅⋅= k

Average crystallite size (τ):

Microstrain (ε)

a*

b*

1/d = 2sinθ/nλ1/d(32)

(00)

0

The Indexing ProblemThe Indexing Problem

Fig. 14.1. The illustration of a two-dimensional reciprocal lattice (top) and its one-dimensional projection on the 1/d axis (bottom).

*** cbad lkhhkl ++=*

) , , , , , , , ,( γβα= cbalkhfdhkl

obshkl

obshkld

θλ=

sin2

Fig. 14.2. Three indistinguishable one- and two-dimensional projections obtained from three different three-dimensional objects.

The Indexing ProblemThe Indexing Problem

(CH3NH3)2Mo7O22

Bragg angle, 2θ (deg.)10 15 20 25 30 35 40 45 50

2θob

s - 2

θ cal

c (1

0-3 d

eg.)

-50

-40

-30

-20

-10

0

10

20

30

40

No correction

s/R = -0.00042(1)

Fig. 14.23. Differences between the obs. & calc. Bragg angles after LSQ refinement.

No correction:a = 23.0875(9) Å b = 5.5191(5) Åc = 19.5789(9) Å β = 122.924(3)º

Sample shift:a = 23.0641(8) Å b = 5.5131(2) Åc = 19.5601(6) Åβ = 122.930(1)º

Lattice Parameters RefinementLattice Parameters Refinement

14.4.1. The FN figure of merit

=

θ−θ=

θΔ×= N

i

calci

obsiposs

possN

N

NN

NF

1

2

____

222

1

==

____θ−θ=θΔ=θΔ

N

i

calci

obsi

N

ii NN 11

22121|2|

)|,2(| .possN NValueF_____

θΔ=

14.4.2. The M20 (MN) figure of merit

=

−=

Δ×= 20

1

20____20

2010

||2

1

i

calci

obsiposs

poss QQN

Q

Q

QN

M 22 1* ddQ ==

=

===

−=Δ=Δ20

111

20

1

____

201

201||

i

calci

obsi

ii QQQQ

=

−=

Δ×= N

i

calci

obsiposs

NN

possN

QQN

NQ

Q

QN

M

1

____

2||2

1

Indexing Figures of MeritIndexing Figures of Merit

Crystal Structure Determination FlowchartCrystal Structure Determination Flowchart

Preliminary processing:2θj, Ij

Indexing:hj, kj, lj; a, b, c, α, β, γ

Symmetry/diffraction class

(systematic absences)

Unit Cell Content(gravimetric density)

Full pattern decomposition:hj, kj, lj, Ijobs

Direct space approaches:

Reciprocal space approaches:

Model PhaseanglesOther: Fourier

ChooseSpace group symmetry

Suitablemodel

Rietveld

No

Yes

Solving Crystal Structure Crystal Structure Refinement

The Rietveld RefinementThe Rietveld Refinement

• Where to start & when to finish refinement?

• Problems in refinement from powder data

• Refinement of individual atoms

• Constraints, Restraints & Rigid-body

• Structure Data Representation & Publication

Hugo M. Rietveld (b. 1932).

The following two papers are considered seminal:

1. H.M. Rietveld, Line profiles of neutron powder-diffraction peaks for structure refinement, Acta Cryst. 22, 151 (1967);

2. H.M. Rietveld, A profile refinement method for nuclear and magnetic structures, J. Appl. Cryst. 2, 65 (1969).

Typical Rietveld refinement plot (shown small range)

Typical Rietveld refinement plot (shown small range)

Typical Rietveld refinement plot (shown full range)

Typical Rietveld refinement plot (shown full range)

obsn

calcn

obscalc

obscalc

kYY

kYY

kYY

=

=

=

...22

11

Non-linear least squares minimization of:

The minimized function, Φ

=

−=Φn

i

calci

obsii YYw

1

2)(

Fundamental Equations (I)Fundamental Equations (I)

= =

+−=Φn

ijj

m

jji

obsii xyIKbYw

1

2

1)])([(

Fundamental Equations (II)Fundamental Equations (II)

For single phase crystalline material

yj(xj) - the peak shape function

Ij - the integrated intensity of the jth Bragg reflection

xj = 2θjcalc – 2θi.

bj - the background coefficients

1][ −= obsii YwThe weighting scheme:

= =

Δ+++−=Φn

ijjjjj

m

jji

obsii xxyxyIKbYw

1

2

1)}])(5.0)({[(

Fundamental Equations (III)Fundamental Equations (III)

For single phase and dual wavelength (Kα1 + Kα2) data

Δxj - the difference in positions of Kα1 and Kα2

= = =

Δ+++−=Φn

i

p

ljljljljljl

m

jjlli

obsii xxyxyIKbYw

1

2

1,,,,,

1, ]))}(5.0)({[(

For a mixture of several (p) phases and dual wavelength (Kα1 + Kα2) data

Κl – individual scale factors for each phase(proportional to the content of the phase)

Refined Parameters (I)Refined Parameters (I)

• Background represented by 1 to 12 parameters (may be more)• Sample displacement, sample transparency or zero-shift corrections.• Peak shape function parameters, which usually include:

- full width at half maximum (FWHM parameters: U,V,W,P,X,Y), - asymmetry and other relevant variables.In a multiple phase diffraction pattern, these may be maintained identical or refined independently for each phase present (generally except for the asymmetry), if warranted both by the quality of the data and considerable differences due to the physical state of various phases in the specimen.

• Unit cell dimensions, usually from 1 to 6 independent parameters for each phase .• Preferred orientation, and if necessary, absorption, porosity, and extinction

parameters, which usually are independent for each phase.• Scale factors, one for each phase (Kl) ,

and in the case of multiple patterns, per pattern.• Positional parameters (x,y,z) of all independent atoms in the model of the crystal

structure of each crystalline phase, usually from 0 to 3 per atom.• Population parameters, if certain site positions are occupied partially or by

different types of atoms simultaneously, usually one per site.• Atomic displacement parameters, which may be treated as:

- an overall displacement parameter (one for each phase or a group of atoms) or - individual atomic displacement parameters, with 1-6 of independent variables

Parameter or group of parameters Linear Stable Sequence

Phase scale Yes Yes 1 Specimen displacement No Yes 1 Linear background Yes Yes 2 Lattice parameters No Yes 2 More background Generally no Yes 2 or 3 Peak Shape Parameter (W, or X, or Y) No Poorly 3 or 5 Coordinates of atoms No Fairly 3 Preferred orientation No Fairly 4 or not Population and isotropic displacements No Varies 5 Other profile parameters No No Last Anisotropic displacements No Varies Last Zero shift No Yes 1, 5 or not

Refined Parameters (II)Refined Parameters (II)

Non-linear least squares techniquemay fail in finding the best solution

x xx0 x0 xtruextrue

Min

imiz

ed fu

nctio

n, Φ

(x)

Min

imiz

ed fu

nctio

n, Φ

(x)

Wy)(A)WA(AΔx TT 1λ −+= DMarquardt dumping:

Correlation coefficients: 111 )()()( −−−=ρ jjiiijij WAAWAAWAA TTT

Profile residual, Rp

Weighted profile residual, Rwp

Bragg residual, RB,

Figures of merit and quality of refinement (I)Figures of merit and quality of refinement (I)

%100R

1

1p ×

−=

=

=n

i

obsi

n

i

calci

obsi

Y

YY

( )%100

)(R

21

2

1

1

2

wp ×

−=

=

=n

i

obsii

n

i

calci

obsii

Yw

YYw

%100R

1

1B ×

−=

=

=m

j

obsj

m

j

calcj

obsj

I

II

Expected residual Rexp

Goodness of fit, χ2

The Durbin-Watson d-statistic

Figures of merit and quality of refinement (II)Figures of merit and quality of refinement (II)

%100)(

R

21

2

1

exp ×

−=

=

n

i

obsii Yw

pn

=

= −

−

σΔ

σΔ−σ

Δ

=n

i i

i

n

i i

i

i

i

Y

YY

d

1

22

2

1

1

( ) 2

exp

wp1

2

2

RR

χ

=

−

−=

=

pn

YYwn

i

calci

obsii

Good fit

Inte

nsity

, Y(1

03co

unts

)

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

I calc too high

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

I calc too low

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

Quality of profile fitting: IntensityQuality of profile fitting: Intensity

Good fit

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

2θcalc too high

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

2θcalc too low

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

Quality of profile fitting: 2ΘQuality of profile fitting: 2Θ

Good fit

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

FWHM too large

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

FWHM too low

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

Quality of profile fitting: FWHMQuality of profile fitting: FWHM

Asymmetry underestimated

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

Good fit

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

Bragg angle, 2θ (deg.)67.6 67.8 68.0 68.2 68.4

-4

-2

0

2

4

6

8

10

12

Inte

nsity

, Y(1

03co

unts

)

Asymmetry overestimated

Quality of profile fitting: AssymetryQuality of profile fitting: Assymetry

I calc and FWHM too large

Bragg angle, 2θ (deg.)21.0 21.2 21.4 21.6

Inte

nsity

, Y(1

03co

unts

)

-2

-1

0

1

2

3

4

5

Good fit

Bragg angle, 2θ (deg.)21.0 21.2 21.4 21.6

-2

-1

0

1

2

3

4

5

Inte

nsity

, Y(1

03co

unts

)

I calc and assym. too small

Inte

nsity

, Y(1

03co

unts

)

Bragg angle, 2θ (deg.)21.0 21.2 21.4 21.6

-2

-1

0

1

2

3

4

5

Quality of profile fitting: CombinationQuality of profile fitting: Combination

HANDS-ON EXAMPLE NiMoO2(OH)

Example in Chapter 20 includes:• Ab initio indexing of the powder pattern• Solving the crystal structure

1. Rietveld refinement using lab X-ray data 2. Simultaneous refinement of X-ray & neutron data3. Chemical composition & H atom determination

Examples including original experimental data, intermediate and final results can be downloaded as following:

1) http:\www.chem.umd.edu\crystallography2) Click the book icon on the right and then WebPage3) Save zip file with Examples

Example Files:Example Files:

NiMnO2(OH), Chapter 20:− Unit Cell Indexing (TREOR)− Structure Solution using SHELXS (Ch20Ex01.ins and .hkl)o Initial Refinement (Ch20Ex01a.exp and .cif)o X-ray Data Fitting (Ch20Ex01b)wo Combined X-ray and Neutron Data Fitting (Ch20Ex01c)o Chemical Composition and H atom Determination

Experimental Data Files:− X-ray: Ch20Ex01_CuKa.raw− Neutron: Ch20Ex01_Neut.raw

Instrumental Parameters:− X-ray: Ch20Ex01_CuKa.prm− Neutron: Ch20Ex01_Neut.prm

Refinement Sequence for NiMnO2(OH)Refinement Sequence for NiMnO2(OH)

Refined parameters Rp Rwp RB χ2

Initial 36.9 50.8 99.7 350Scale factor only 18.2 25.9 36.0 90.9Scale, background, unit cell dimensions, grain size (X) 14.9 22.4 34.0 68.1All of the above plus preferred orientation (PO) for [010] axis and then adding another PO for [100] axis

9.6 13.5 12.2 24.8

All of the above plus strain (Y) instead of X, PO1/PO2 ratio, asymmetry (α), coordinates of all atoms, Uover

7.4 10.6 8.9 15.3

All of the above plus Mn2 was changed to Ni1 (5 cycles), then individual Uiso for Mn1 and Ni1

7.3 10.5 9.6 15.0

All of the above plus Ss, profile parameters, grain size, strain together with their anisotropy (Xa and Ya)

5.9 8.0 6.8 8.75

Only scale, background, unit cell dimensions and absorption, a1 and a2, in the Suortti approximation

6.0 8.1 6.3 8.79

All of the above plus coordinates, Uiso for Mn1 and Ni1, Uover for O, PO[010], PO[100], X, Xa α.

6.0 8.0 6.7 8.77

All of the above plus U, V, W, Y, Ya. Final (x-ray only), 5.1 6.6 6.7 5.99Combined final: x-ray 5.1 6.7 6.7 n/aCombined final: neutrons 4.0 5.0 24.4 n/aCombined final: total 5.0 6.5 n/a 5.85

NiMnO2(OH), Cu Kα

Bragg angle, 2θ (deg.)10 20 30 40 50 60 70 80 90 100 110

Inte

nsity

, Y (1

03 cou

nts)

0

10

20

30

40

1 µm 2 µm1 µm1 µm 2 µm2 µm

Fig. 20.1. Powder diffraction pattern collected from the NiMnO2(OH) powder using CuKα radiation

Polyrystalline NiMnO2(OH)Polyrystalline NiMnO2(OH)

NiMnO2(OH), Cu Kα

Bragg angle, 2θ (deg.)10 20 30 40 50 60 70 80 90

FWH

M (d

eg.)

0.0

0.1

0.2

0.3

0.4

Fig. 20.2. Full width at half maximum as a function of Bragg angle for the powder diffraction pattern of NiMnO2(OH)

TREOR indexing using 23 peaks gave an orthorhombic C-centered lattice with figure of merit, F20 = 135 (0.003, 27), and unit cell a = 2.8609 Å, b = 14.6482 Å, c = 5.2703 Å.

Solved using SHELXS direct method and intensities of 30 peaks in space group Cmc21.

Peaks used for unit cell indexing and solving the structure

Peaks used for unit cell indexing and solving the structure

NiMnO2(OH), Cu Kα

Bragg angle, 2θ (deg.)10 20 30 40 50 60 70 80 90 100 110

Inte

nsity

, Y (1

03 cou

nts)

-5

0

5

10

15

20

25

30

35

40

Bragg angle, 2θ (deg.)70 75 80 85 90

Inte

nsity

, Y (1

03 cou

nts)

0

5

Rp = 18.16 %Rwp = 25.91 %RB = 36.02 %χ2 = 90.9

Fig. 20.3. The observed and calculated powder diffraction patterns of NiMnO2(OH) after the initial Rietveld refinement with only the scale factor determined

NiMnO2(OH) – scale factor onlyNiMnO2(OH) – scale factor only

NiMnO2(OH), Cu Kα

Bragg angle, 2θ (deg.)10 20 30 40 50 60 70 80 90 100 110

Inte

nsity

, Y (1

03 cou

nts)

-5

0

5

10

15

20

25

30

35

40

Bragg angle, 2θ (deg.)70 75 80 85 90

Inte

nsity

, Y (1

03 cou

nts)

0

3

Rp = 7.37 %Rwp = 10.60 %RB = 8.86 %χ2 = 15.3

Fig. 20.4. The observed and calculated powder diffraction patterns of NiMnO2(OH) after preferred orientation, coordinates of all atoms and the overall displacement parameter were refined in addition to the scale factor, unit cell dimensions, background, grain size and strain effects, and peak asymmetry

NiMnO2(OH) – most parametersNiMnO2(OH) – most parameters

NiMnO2(OH), Cu Kα

Bragg angle, 2θ (deg.)10 20 30 40 50 60 70 80 90 100 110

Inte

nsity

, Y (1

03 cou

nts)

-5

0

5

10

15

20

25

30

35

40

Bragg angle, 2θ (deg.)70 75 80 85 90

Inte

nsity

, Y (1

03 cou

nts)

0

3

Rp = 5.08 %Rwp = 6.63 %RB = 6.68 %χ2 = 5.99

Fig. 20.5. The observed and calculated powder diffraction patterns of NiMnO2(OH) after the completion of Rietveld refinement using only x-ray powder diffraction data (the hydrogen atom is still missing from the model).

NiMnO2(OH) – all possible parameters:switched Mn & Ni, PSF aniso, 2PO, etc.

NiMnO2(OH) – all possible parameters:switched Mn & Ni, PSF aniso, 2PO, etc.

Fig. 20.6. Distributions of the nuclear density in the unit cell of NiMnO2(OH) at x = 0, calculated without H atom.

The nuclear density in NiMnO2(OH) at x = 0:The nuclear density in NiMnO2(OH) at x = 0:

Difference Fourier map using:|ΔF| = |Fobs-Fcalc| and αcalc;

Conventional Fourier map using: |Fobs| and αcalc

NiMnO2(OH), neutrons

10 20 30 40 50 60 70 80 90 100 110

Inte

nsity

, Y (1

02 cou

nts)

10

20

30 λ = 1.392

Rp = 3.99 %Rwp = 5.02 %RB = 24.4 %χ2 = 5.85

Å

a)

NiMnO2(OH), Cu Kα

Bragg angle, 2θ (deg.)10 20 30 40 50 60 70 80 90 100 110

Inte

nsity

, Y (1

03 cou

nts)

0

10

20

30

40

Rp = 5.12 %Rwp = 6.73 %RB = 6.67 %χ2 = 5.85

b)

Fig. 20.7. The observed and calculated powder diffraction patterns of NiMnO2(OH) after the completion of combined Rietveld refinement using neutron (top) and x-ray (bottom) powder diffraction data

NiMnO2(OH): Neutrons & X-ray simultaneouslyNiMnO2(OH): Neutrons & X-ray simultaneously

YX

Z

Mn

Ni O H

Fig. 20.8. The model of the crystal structure of NiMnO3-x(OH)x, x=0.62(5). The covalent O1-H bonds are shown as cylinders, and the H…O3 hydrogen bonds are shown using thin lines

Validation of the Crystal Structure NiMnO2(OH)Validation of the Crystal Structure NiMnO2(OH)

Instead of Conclusions

Thank You!

“The Metamorphosis of Narcissus” where Narcissus (polycrystalline sample) falls in love with his own reflection (diffraction pattern), transforms into an egg (lattice), and then into a flower (crystal structure), which bears his name.