geometry, polarization and absorption corrections · 2018-11-09 · geometry, polarization and...

$: Geometry, Polarization and Absorption Corrections · 2018-11-09 · Geometry, Polarization and Absorption Corrections for Two-dimensional X-ray Diffraction Bob He Bruker AXS, Inc$
Geometry, Polarization and Absorption Corrections for Two-dimensional X-ray Diffraction

Bob He Bruker AXS, Inc.

23.02.2010Bruker Confidential1

This document was presented at PPXRD -Pharmaceutical Powder X-ray Diffraction Symposium

Sponsored by The International Centre for Diffraction Data

This presentation is provided by the International Centre for Diffraction Data in cooperation with the authors and presenters of the PPXRD symposia for the express purpose of educating the scientific community.

All copyrights for the presentation are retained by the original authors.

The ICDD has received permission from the authors to post this material on our website and make the material available for viewing. Usage is restricted for the purposes of education and scientific research.

ICDD Website - www.icdd.comPPXRD Website – www.icdd.com/ppxrd

http://www.icdd.com/

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XRD2: What do we know about two-dimensional XRD?

Most recognized features of XRD2 for pharmaceutical: High speed: 102 higher than XRD with a point detectorHigh speed: 10 higher than XRD with a point detectorReliable info: Integration in γ (ring) directionMicro scale sample: Point beam and 2D patternp p

Most concerns with XRD2:R l i d d f iResolution and geometry defocusingRelative intensity is different from Bragg-BrentanoHigher instrument cost (but much higher productivity)Higher instrument cost (but much higher productivity)

This presentation interprets the differences and s p ese tat o te p ets t e d e e ces a dsuggests the best use of XRD2 systems

X-ray Applications for typical pharmaceutical samplesyp p p

XRD & XRD2 Single Crystal

Several Grains Powder Finished

Product Solutions

Qualitative Phase ID Ф⊕ Ф⊕ Ф Ф ФQualitative Phase ID Ф⊕ Ф⊕ Ф Ф ФQuantitative RietveldanalysisQuantitative analysis with standards

X-ray movie, Non-Ambient Ф Ф Ф Ф Ф

Structure solution, Indexing Ф

Microdiffraction/ Mapping Ф⊕ Ф⊕ Ф⊕/ pp g

Shape analysis Ф Ф ФHTS Ф⊕ Ф⊕ Ф⊕

Grain-Size det. Ф Ф

%Crystallinity Ф⊕ Ф⊕ Ф⊕ Ф⊕

- can be performed by either XRD or XRD2

Ф – better with XRD2


Ф better with XRD⊕ - accept performance and accurate results only with XRD2

Bragg-Brentano-GeometryBragg Brentano GeometryAdvantages

Mono-

Resolution

Less expensive

Divergence slit Antiscatterslit

Monochromator

Disadvantages

⌧Detector-slitTube

Sample

⌧ Requires flat sample surface

⌧ Requires bulky p ⌧ Requires bulky powder sample

⌧ Slow


Soller slits are used to control axial divergenceg

l fl l

Most part of the diffraction ring is

clipped off by:large flat sample clipped off by:

In Bragg-Brentano geometry, the line focus beam can be considered as a superposition of point beams. All in parallel with the diffractometer plane and the same


geometry condition separated by soller slit foils.

XRD2: Two-dimensional X-ray DiffractionXRD : Two dimensional X ray DiffractionAdvantages

Hi h dHigh speed

Micro scale sample

Virtual oscillation (large grain size & preferred orientation)

Disadvantages

⌧ Resolution is limited by the detector PSFby t e detecto S

⌧ Defocusing at low angle in reflection


⌧ Expensive

XRD2: Data Collection:Acetaminophen powder

5 second data collection 30 second data collectionLi

n (C

ps)

0

1

2


2-Theta - Scale9 10 20 30 40

XRD2: Diffraction vector approach

Applications Vector approaches

Phase Identification: Polarization and absorption correction

T t A l i O i t ti i lTexture Analysis: Orientation mapping angles;Data collection strategy (scheme)

Stress Measurement: Fundamental equation derived by second order tensor transformation;Data collection strategy (scheme)

Crystal Size Analysis: Equations for the effective volumecalculation at both reflection and


transmission modes.

XRD2 & Single Crystalsg y

S0

S

λh=−⋅ )( 0ssaLaue

λλ

lk=−⋅

)()( 0ssbequation


λl=−⋅ )( 0ssc

XRD2 & PowdersXRD & Powders

Bragg law

θλ sin2dn =


XRD2: Diffraction pattern with both γ and 2θ information

Debye Cone

Sample

Incident Beam

⎤⎡ −θ 12cos

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

−=

γθγθ

θ

λλcos2sinsin2sin

12cos10ssH


⎥⎦⎢⎣ γθ cos2sin

XRD2: Diffraction Space & Laboratory coordinators

The incident beam is in the direction of the XL axis in the laboratory coordinates so that the unit vectors so t at t e u t vecto sof the incident beam and diffracted beams are given respectively

⎤⎡⎤⎡ 10s ⎤⎡⎤⎡ θ2cosxs

beams are given respectively by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

001

0

0

y

x

ss

0s⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

γθγθ

cos2sinsin2siny

x

ss⎥⎦⎢⎣⎥⎦⎢⎣ 00zs ⎥⎦⎢⎣−⎥⎦⎢⎣ γθ cos2sinzs

XRD2: Diffraction Vector & Unit Diffraction Vector

The diffraction vector is given in laboratory coordinates byin laboratory coordinates by

⎥⎥⎤

⎢⎢⎡−

−=⎥

⎥⎤

⎢⎢⎡

−−

=−

= γθθ

sin2sin12cos

110

00

xx

ssss

ssH

The direction of each diffraction ⎥⎥⎦⎢

⎢⎣−⎥

⎥⎦⎢

⎢⎣ − γθ

γθλλλ

cos2sinsin2sin

0

0

zz

yy

ssssH

vector can be represented by its unit vector given by:

⎤⎡⎤⎡ θih

⎥⎥⎥⎤

⎢⎢⎢⎡−

−=

⎥⎥⎥⎤

⎢⎢⎢⎡

== γθθsincos

sin

L y

x

hh

HHh

⎥⎦⎢⎣−⎥⎦⎢⎣ γθ coscoszhH

XRD2: Sample Space & Eulerian Geometry

The angular relationships between XLYLZL andbetween XLYLZL and S1S2S3 are:

X Y ZL L L

S a a a

S a a a

1 11 12 13

2 21 22 23

The transformation matrix from the diffraction space

S a a a3 31 32 33

⎥⎥⎤

⎢⎢⎡

−−−

−φψ

φωφψω

φωφψω

sincoscossin

sinsincoscoscos

sinsinsinfrom the diffraction space to the sample space is:

⎥⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢⎢

−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

φψφω

φψωφω

φψω

φωφω

coscossinsin

cossincossincos

cossinsin

cossincoscos

333231

232221

131211

aaaaaaaaa

⎥⎥⎦⎢

⎢⎣− ψψωψω sincoscoscossin

XRD2: Sample Space & Unit Diffraction Vector

The components of the unit vector h in the samplevector hS in the sample coordinates S1S2S3 is then given by

⎥⎥⎥⎤

⎢⎢⎢⎡

⎥⎥⎥⎤

⎢⎢⎢⎡

=⎥⎥⎥⎤

⎢⎢⎢⎡

y

x

hh

aaaaaa

hh

232221

131211

2

1

Or in expanded form:

⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ zhaaah 3332313

)ii( iicossincoscos)coscossinsin(sinsin1

φφθψφγθωφωψφθ ++=h

in expanded form: )sincoscossin(sinsincos ωφωψφγθ −−

)sinsincossin(cossincoscoscoscoscos)cossinsinsin(cossin2

ωφωψφγθψφγθωφωψφθ

++−−−=h

h3 = − −sin cos sin cos sin cos cos cos cos sinθ ψ ω θ γ ψ ω θ γ ψ

XRD2: Fundamental Equation for Stress Measurement

As a second order tensor thetensor, the relationship between the measured strains and the strain tensoris given by:

hkl hh ⋅⋅= εε }{

the fundamental equation for stress measurement in XRD2:

jiij hh ⋅⋅= εε φψωγ ),,,(

equation for stress measurement in XRD2:

⎞⎜⎛

+++++

θσσσσσσ

sinl)222

()(

0

2333

2222

211122

13322111

hhhhhh

hhhSS

⎠⎞

⎜⎝⎛=+++

θσσσ

sinln)222 0

322331132112 hhhhhh

XRD2: Fundamental Equation for Texture Analysis

The pole figure angles (α,β) can be calculated

22

21

13

1 cossin hhh +== −−α(α,β) can be calculated from the unit vector components by the pole

i ti

213

⎩⎨⎧

<°<≥°≥

+±= −

0000

cos 2

2211

hifhif

hhh

ββ

βmapping equations: ⎩ <<+ 00 2

22

21

hifhh β

XRD2: Relative Intensity of Powder Pattern

The integrated intensity diffracted from random polycrystalline materials is given by:

Corundum Powder Diffraction

16000

18000

20000

p y y g y

where: 6000

8000

10000

12000

14000

16000

Inte

nsity

( )sthklhkl

Ihkl MMFLPAvpk 22exp)( 23

2 −−= λI

⊕ kI - instrument constant;phkl - the multiplicity of the planes; v - the volume of the unit cell;

0

2000

4000

6000

20 25 30 35 40 45 50

Two Theta

⊕ (LPA) - the Lorentz-polarization and absorption factors;- the structure factor of the crystal plane (hkl) and

exp(-2Mt-2Ms) - the attenuation factor due to lattice thermal vibrations

Two Theta

2hklF

t s

and weak static displacements.

⊕ Denotes the factors which are different between Bragg-Brentano geometry and XRD2 geometry k is determined by the source optics and detector (PPXRD 7)


XRD2 geometry. kI is determined by the source, optics and detector (PPXRD-7)(LPA) will be given in this presentation.

XRD2: Polarization Correction

The polarization factor for Bragg-Brentano geometryBragg Brentano geometry with incident beam monochromator is:

θθ 2cos2cos1 22+

where 2θM is the Bragg M

MIP

θθθ

2cos12cos2cos1

2++

=

angle of the monochromator.

Geometric relationship between the monochromator d d t t i l b t di t

The general polarization factor for the diffracted beam to point P is:

and detector in laboratory coordinates.

MP ρρθθρρθ cossin2cos2cos)sincos2(cos 2222222 +++=beam to point P is:

MGP

θ2cos1 2+=

XRD2: Polarization Correction

The unit vector of the diffraction vector HP and its projection on YL-ZL plane H'P in the laboratory system are given respectively as:ZL plane, H P, in the laboratory system are given respectively as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

θγθ

θsincos

sin

L y

x

hhh

h⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

′′=′

γγ

cossin00

L y

hhh

The unit vector of YL is yL=[0,1,0], then: ⎥⎦⎢⎣−⎥⎦⎢⎣ γθ coscoszh ⎥⎦⎢⎣−⎥⎦⎢⎣ ′ γcoszh

γρ sin)cos(cos LL −=•=′= yy LL hhTherefore, andThe polarization factor for XRD2 can then be given as a function of

γρ sin),cos(cos LL • yy LL hhγρ 22 sincos = γρ 22 cossin =

p gboth θ and γ:

MMPθ

γθθγθθγθ21

cos)2cos2(cossin)2cos2cos1(),( 2

222222 +++=

Mθγ

2cos1),( 2+

XRD2: Sample Absorption Correction

The absorption can be measured by the

Absorption correction of flat slab:be measured by the transmission coefficient:

∫1

where τ is the total

dVeV

AV∫

−= µτ1

(a) reflection (b) transmissionbeam path and A is the average over all the element dV. For

(a) reflection (b) transmission.

To make the relative intensity comparable to B B t t i t dthe element dV. For

Bragg-Brentano geometry, we have:

A 1/(2 )

Bragg-Brentano geometry, we introduce a normalized transmission coefficient T:

AAAT BB µ2/ ==ABB=1/(2µ) AAAT BB µ2/

D8 DISCOVER with GADDS HTS (IµS):Reflection & Transmission (Pharmaceutical Delight)Reflection & Transmission (Pharmaceutical Delight)

Reflection Transmission convertible Geometry

Reflection mode Transmission mode


Reflection mode Transmission mode

Comparison: IbuprofenIµS & VÅNTEC-2000 vs. Clasical set-upµ p

IµS – XRD2 – focus

2mmX2mm on sample, and 200um spot focused on detector

Sealed Tubefocused on detector

small slice for integration to obtain better resolution

15 sec collection time

• 0.3 mm collimator

• Sample-Detector distance 29 cm

120 sec collection time



For reflection mode diffraction with a thick plate:with a thick plate:

⎥⎥⎥⎤

⎢⎢⎢⎡

= 01

os⎥⎥⎥⎤

⎢⎢⎢⎡−= γθ

θsin2sin

2coss

and⎥⎦⎢⎣0 ⎥⎦⎢⎣− γθ cos2sin

⎥⎥⎤

⎢⎢⎡−

= ψωψω

coscoscossin

n

The normalized transmission⎥⎥⎦⎢

⎢⎣

=ψ

ψωsin

coscosn

The normalized transmission coefficient:

ith( )ηcos2

=Tψωη cossincos =⋅−= nso

ψωθζ cossin2coscos == nswith( )ζη coscos +T ψωθζ cossin2coscos −=⋅= ns

ψγθψωγθ sincos2sincoscossin2sin −−


For transmission mode: ⎤⎡1 ⎤⎡ θ2cos

d⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

00os

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

γθγθ

cos2sinsin2sins

and

⎥⎥⎥⎤

⎢⎢⎢⎡

+−+

= φωφψωφωφψω

cossinsinsincoscoscossinsinsin

n

The normalized transmission coefficient :

⎥⎦⎢⎣ φψ sincos

coefficient :( ) ( )[ ]

ηζζµηµη

secsecsecexpsecexpsec2

−−−−

=ttT

φωφψωη coscossinsinsincos +=⋅= nso

γθφωφψωθφωφψωζsin2sin)cossinsinsin(cos

2cos)coscossinsin(sincos−++=⋅= ns

γθφψγφφψ

cos2sinsincos)(

−

XRD2: Texture Effect and Correction

The integrated intensity with texture is:texture is:

where g() is the normalized ( )sthklhkl

hklIhkl MMFLPA

vpk 22exp),()( 23

2 −−= βαλ gI

g()pole density function.

For the BB geometry, )0,( 2π

hklgThe texture effect for XRD2:

)( γ∆=

mhklc

hkl gII

2 )]([γ

∫ dFor fiber texture:Correct to the B-B equivalent with a texture effect:

)( γ∆hklg

)0,( 2π

=mhklhklBB g II

12

1

)]([)(

γγ

γγγχγ γ

−=∆

∫ dhkl

hkl

gg

For fiber texture:

d 1)( γ∆=

hklhkl g

I and3

1cos h−=χ

XRD2: Summary and Suggestions

Two-dimensional X-ray diffraction has many d t th ti l diff ti ( dadvantages over the conventional diffraction (speed,

completeness and accuracy, micro-sample volume).Th di b t XRD2 d B B tThe discrepancy between XRD2 and Bragg-Brentano

in geometry, polarization, absorption and preferred orientation can be interpreted and correctedorientation can be interpreted and corrected.Fortunately, most of the discrepancies can be ignored

without affecting the specific applicationwithout affecting the specific application. Or the corrections are already built in the software, so

users do have to work on the correction detailsusers do have to work on the correction details.

XRD2: Theory System and Applications

Small Angle X-ray Scattering

XRD : Theory, System and Applications

X-ray Diffraction (XRD)

Micro DiffractionGeometry Conventions

Phase Identification

Mapping

Thi Fil

System and Configuration

Phase Identification

Quantitative Analysis High-Throughput Screening

Thin Films

Texture Forensics and Archaeology

ReferenceStress Reference

XRD2: Fundamentals, theory and applicationsu da e ta s, t eo y a d app cat o s


Finalist of MRS “Science as Art” competition:3D View of Corundum Powder Pattern

Thank You for Your AttentionThank You for Your Attention


geometry, polarization and absorption corrections · 2018-11-09 · geometry, polarization and...

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