x-ray diffraction - synchrotron radiation · 2020-01-16 · marco milanesio x-ray diffraction 5...

$: X-ray Diffraction - Synchrotron Radiation · 2020-01-16 · Marco Milanesio X-ray Diffraction 5 Short history of diffraction X-rays discovered by Röntgen in 1895 Sommerfeld (1904)$
Marco Milanesio X-ray Diffraction 1

X-ray Diffraction

Marco Milanesio

Università del Piemonte OrientaleE-Mail: [email protected]: http://www.mfn.unipmn.it/~marcomi

From slides and chapter (à SNLS book pag 345) by Prof. Davide Viterbo


Why X-ray diffraction?

X-rays

Crystal

Crystals diffract also electrons and neutrons

3D StructureDiffraction pattern


Xray ‘microscope’

Fourier Transform is the

“mathematical lens” allowing to “see the atoms”!


Diffraction can also beapplied on very largemoleculestructures


Shor t history of diffractionX-rays discovered by Röntgen in 1895

Sommerfeld (1904) demonstrates ∼ 0.4 Å

Laueand Ewald (1912): crystals can diffract X-rays (demonstratedexperimentally by Friedrich and Knipping)

Bragg (1913) solves the first crystalstructures (NaCl, KBr, KCl, KI)


Inter ferencebetween scattered waves – The concept

∼

Incidentbean

Interference

Diffractionpattern

Periodic lattice withopening at d ~


The great intuition by Laueand Ewald

X-rays have∼∼∼∼ 0.4 Å

(Sommerfeld)

Crystals are “ lattices” with “ openings” (the scatter ing atoms) located at suitable distancesto give a diffraction pattern containinginformation on the atomic structure itself

Interatomicdistances about 1Å

(Atomic theory)


Outline

Mathematical background and basic concepts

Introduction to powder diffraction

Bragg’s law and its practical applications

Structure factor and electron density

Interaction between X-rays and matter

Diffraction of X-rays by a single crystal


What is the definition of “per iodic lattice” ?

What is a Dirac δδδδ function?

What is a Four ier Transform?

What is a convolution between two functions?

Why Dirac δδδδ, Four ier transform, convolution are basic concept to understand X-ray diffraction?


What are Miller indicesof lattice planes?


2D latticeUnit cell

T = ua+ vb

u, v integersb

a


Crystal planesand Miller indeces

(1 0)

(0 1)

(1 1)

(2 1)

Crystal planes are defined as the planes passing through the lattice

nodes. They are defined by the Miller indexes h k l

h: number of part resulting bythe intersection of family of

planes and the a vector

k: the same for b

l: the same for c (3D caseonly)

b

a

2D case


3D lattices

Unit cell

a, b, c, ,

Lattice constants, cellparameters

Crystallographicaxes

T = ua+ vb +wc

u, v, w integers


Miller indices in 3D


(x - x o)= 0 per x ≠ x o

= ∞ per x = x o

One dimension case (1D)

For instance: an infinitely sharp gaussian function

=−−−

2

2

)2

)(exp

2

1lim(σπσ

δ oo

xxxx

à 0

3D Case: vector czbyaxr ++=)()()()( oooo zzyyxxrr −⋅−⋅−=− δδδδ

0 xo

1)( =−∞

∞−

dxxx oδ

Dirac function


Dirac functions: Proper ties

=−S

o Orfrdrrrf )()()( δ

−=−−S

rrrdrrrr )()()( 2112 δδδ

)()()()( ooo rrrfrrrf −=− δδ

)()( rrrr oo −=− δδ

Why is it useful to define the Dirac δ function?


∞

−∞=

−=n

nxxxL )()( δxn = na

n integera constant

L(x) ≠ 0 for x = nan: -∞ à ∞

For instance: one dimension lattice of

periodicity “a”

Lattice function L – One dimensional case (1D)


Lattice function L – Three dimensional case (3D)

a, b, c base vectors of the lattice

∞

∞−

−=u v w

wvurrrL )()( ,,δ

cwbvaur wvu ++=,, u, v, w integers

Vector identifying the non-zero points of the lattice function : L(r) ≠ 0


Four ier Transform – The definition

Given a function (r), its Fourier transform is:

⋅=S

rdrrirrF )*2exp()()*( πρ

r* is a vector in the space where the Fourier transform is defined

The Fourier transform is an operationrelating the space defined by the r

vector to another space defined by a second vector named r*


Four ier transform (FT)

It can bedemonstrated that the inverse transformexists and is defined as:

⋅−=*

*)*2exp()*()(S

rdrrirFr πρ

In short form:

[ ])()*( rTrF ρ= FT from r space to r* space

[ ])*()( 1 rFTr −=ρ FT-1 from r* space to r space


⋅=S

rdrrirrF )*2exp()()*( πρ

Four ier transform - continued

In general F(r* ) is a complex function

)*()*()*( riBrArF +=

⋅=S

rdrrrrA )*2cos()()*( πρ

⋅=S

rdrrrrB )*2sin()()*( πρ


Four ier transform of a gaussian function

Gaussianfunction

−==

2

2

2exp

2

1)0,()(

σπσρρ x

Nx

∞

∞−

= dxxixxxF )*2exp()(* )( ρ

[ ] 222 *2exp*)()( xxFxT σπρ −==The FT of a gaussian is again a gaussian but with a

width depending on the inverse of


Four ier transform of a Gaussian – The example

(x) T[ (x)]= 1 is sharper = 1 is wider


Four ier transform of a Dirac function

(x)

(x) is infinitelysharp

T[ (x)]F(x* )=T[ (x)] isinfinitely wide

(x) = (x)

∞

∞−

== 1)*2exp()(* )( dxxixxxF πδ

* )2exp(* )( iaxxF π=(x) = (x-a) 0


Four ier transform of a finite 1D lattice

−=

−=p

pn

naxx )()( δρ

With N = 2p + 1 nodes

*)sin(

* )sin(* )2exp(* )(

-n ax

axNinaxxF

p

p=

==

A function with main max/min of height equal to ± N, located at ax* = h (h integer) and width equal to 2/N


…….T[finite lattice] graphically

−=

−=p

pn

nxxxL )()( δ

T[1D Lattice with Nà ] à Lattice in x*


Four ier transform–1D infinite lattice

N à ∞

∞

−∞=

−==n

naxxLx )()()( δρ

* )sin(

* )sin(lim*)(

ax

axNxF

N ππ

∞→=

( )∞

−∞=

∞

−∞=

−=

−=hh

haxaa

hx

axF *

1*

1*)( δδ

is a normalization factora

1


Four ier transform–3D finite lattice

−= −= −=

−=1

1

2

2

3

3

)()( ,,

p

pu

p

pv

p

pw

wvurrr δρ N1 = 2p1 + 1 N2 = 2p2 + 1 N3 = 2p3 + 1

)*sin(

)*sin(

)*sin(

)*sin(

)*sin(

)*sin(* )( 321

rc

rcN

rb

rbN

ra

raNrF

⋅⋅⋅

⋅⋅⋅

⋅⋅=

ππ

ππ

ππ

are the base vectors of the direct lattice:cba ,,

h, k, l integer

With max/min: ,* krb =⋅ lrc =⋅ *,* hra =⋅


Four ier transform–3D finite lattice (ct.d)

The base vectors of the direct lattice are: cba ,,

A new lattice (Reciprocal) can beassociated with base vectors: *,*,* cba

1*0*0*

0*1*0*

0*0*1*

=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅=⋅

ccbcac

cbbbab

cabaaa Reciprocity and orthogonality

conditions

Then the max of F(r* ) are defined by the vector:

**** clbkahr H ++= ),,( lkhHT

=


Four ier Transform –3D infinite lattice

∼

(N1, N2, N3) à ∞

( ) ∞

−∞=

∞

−∞=

∞

−∞=

−=u v w

wvurrr ,,)( δρ

∞

−∞=

∞

−∞=

∞

−∞=

−=

h k l

lkhrrV

rF ,,**1

)*( δ

( ) ∞

−∞=

∞

−∞=

∞

−∞=

−=h k l

HrrV

**1 δ

It is again a 3D infinite lattice, defined at the nodes of the Reciprocal Lattice


Reciprocal space• Definition:

The reciprocal lattice is a periodic set of points constructed in such a way that the direction of a vector from the origin to a lattice point coincides with the direction of the normal to a real space plane and the length of the vector is equal to the reciprocal of the interplanar distance in the real space.

• Why we use it? It’s a mathematical concept (product of two vectors) that together with the Laueequations allow to easily calculate the reflection positions in real space.


Convolution (C) – Symbol *

∼

Operation involving two generic functions(r) and g(r), defined as:

)(*)()(*)()( rrgrgruC ρρ ==

)]([)]([)](*)([ rgTrTrgrT ×= ρρ

Convolution theorem

)]([*)]([)]()([ rgTrTrgrT ρρ =×

and viceversa

−==S

rdrugrrgruC )()()(*)()( ρρ


Convolution – What’s happening?

∼

dxxugx )()( −ρIs the grey part

g(u-x) inverse of g(x) with respect to u

ρ(x)g(x)

ρ(r) & g(r)

)(*)()( xgxxC ρ=

The sum of all greyareas for different u is


Convolution involving sharp functions

If g(x) is “sharp” :ρ(x)*g(x) is very similarto ρ(x)

ρ(x)

g(x)

What is the result if g(x) isan infinitely sharp δfunction at x=0? ρ(x)remains unchanghed!

ρ(x)*δ(x)


If δ is centered in x=aρ(r)*δ(r) = ρ(r) BUT ρ(r) IS NOW CENTERED in a, i.e. relocation of (x) in x = a

ρ(x) δ(x) ρ(x)*δ(x)

Convolution with a Dirac – 3D case

)()(*)( oo rrrrr −=− ρδρTranslation of (r) by a vector ro

Convolution with a Dirac g(r )= (r -r o) – 1D case


Convolution with a 1D infinite L(x)

∼

( ) )()(*)( rnarfrfrLn

ρ=−= ∞

−∞=

à Function defined in the interval 0 ≤ r ≤ a)(rf

The result is the (r) function, representing the periodic

repetition of f(r)


ρ(x) L (x) ρ(x)*L(x)

…… (r)*L(r) graphically

……theoperation repeats (r) at each nodeof the 1D lattice


Convolution of a function with a 2D lattice

∼

……it’s a repetition of “ toluene” (x,y) at eachnodeof the 2D lattice, i.e. a 2D crystal

The same happens (only more complicated for a graphical

representation) with a 3D lattice

L(x,y) ρ(x,y)*L(x,y)ρ(x,y)


Convolution with a 3D lattice

∼

( ) )()(*)(,,

,, rrrfrfrLwvu

wvu ρ=−= ∞

∞−

à 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c ),,()( zyxfrf =

The (r) function is obtained,

which is a 3D periodic repetition of f(r)


Outline







You are here



∼

Scattering

Elastic (Thompson) =0, =

Inelastic (Compton) ≠0

Absorption

Without photoemission

With photoemission

Kinematic theory of X-ray diffraction

Dynamical theoryNeglected: Interferencebetween scattered and direct beam,

multiple diffraction, absorption and crystal defectivity


X ray features

∼

X ray Electromagnetic radiation

High energy, refraction index ≈ 1 (always<1) in allmedia, v = c = 3 108 m s-1

(Å)

E(keV)

UVX ray 1000.1

∼150 ∼ 0.1

It’s very complicated to makea “ lens” capable of focusing X-rays, similar to thoseavailable for UV-Vis radiationsand for electrons (magnets à TEM)

0.3-2.4 Å


X ray scatter ing without energy loss (coherent) ààààThompson Scatter ing with =0

∼

Scattered intensity

2

2cos1eII

2

42

4

iTh

θ+⋅=

cm

Ii = Incident beam intensitye = particlechargem = particlemass2 = scattering angler = detection distance

Becauseof the 1/m2 factor only electrons scatter !

Polarization factorfor non-polarized

X-rays


)2(sin)2cos(2

' 2 θψτ ⋅=oP Partially polarized light (Synchrotron)detector azimutal angle

The polar ization of synchrotron radiation

2

)2cos(1 θ+=oPNon-polarized light (X-ray conventional tube)

)1()1(

)1()1('

τταττατ

−++−−+=

depends on monochromator crystal:

= cos2 M with M

Bragg angle of the monochromator

⊥

⊥

+−=

II

II

||

||τ

polarization degreewith = 0 for a conventional X ray source (X ray tube), ≠ 0 for synchrotron light


X-ray scatter ing with energy loss(incoherent) àààà Compton Scatter ing

∼

Elastic collision between a photon and anelectron which isdisplaced

= 0.024 (1-cos2 )

Thomson >> Compton becauseThomson ∝ (n. e-)2

whileCompton ∝ (n. e-)


When more than one par ticle interact with X-raysinter ferenceand diffraction occur

Incidentbeam;

Interference

Diffractionpattern

d

Periodic lattice withopening at d ~


Inter ferencebetween scattered waves – Some math

( ) rrrSS o ⋅=⋅−= *22 πλπδ

Scattered amplitudeby O à Ao

Scattered amplitudeby O’ àScattered amplitudeby a freeelectron à ATh

)*2exp(' rriAo ⋅π

fj=Aj/ATh scattering factor à n. of electrons in point j

λϑsin2

|*| =rλ

oSSr

−=*

S, So versorsPhasedifference depending on the optic path difference (o.p.d.)

2 : = : o.p.d.

kr =*

Transferredmomentum


I f thereare N scatter ing points:

∼

Resulting amplitude (with respect to that of the electron located at the origin)

( ) ( )j

n

jjj

n

j Th

j rrifrriA

ArF ⋅=⋅=

==

*2exp*2exp)*(11

Continuous scatterer: described by an electron density continuous function (r), governing the scattered amplitude

)*2exp()( rrirdr ⋅πρ

The volume element dr in r contains (r)dr electrons and

scatters with amplitude:


The total scattered amplitude is the integral over all the scatter ing volume v in which the

continuous function (r ) is defined, i.e.:

∼

( ) [ ])(*2exp)()*( rTrdrrirrF j

v

ρπρ =⋅=

( ) [ ])*(**2exp)*()( 1

*

rFTrdrrirFr j

v

−=⋅−= πρ


Let us consider different scatter ing functions (r)

∼

Electron in one atom.....

Assuming spherical symmetry:

drrr

rrrrfe

∞

=0

e*2

*)2sin()(U*)(

ππ

|)|( rr =

)(4)(U 2e rrr eρπ=

Radial electron density function

2

)()( rre ψρ =


….poly-electronic “ fixed” atom.....

∼

Assuming spherical symmetry:

=

∞

==Z

jea j

fdrrr

rrrrf

10

a *2

*)2sin()(U*)(

ππ

)(4)(U 2a rrr aρπ=

Z=n. of electron (atomic number)


….Atom with isotropic thermal motion.....

∼

−=

2

2

exp*)(* )(λ

θsenBrfrf a

oa

T

28 uB π= à Temperature factor, ADP, DebyeWaller factor

u2 à mean squaredisplacement


Atomic scatter ing factor vs. scatter ing angle

∼

“ fixed” atom

Atom withthermal motion

Growing scattering angle [2 , (sen )/λ…]

Decay with 2depending on the fact

that atoms are notlocated in a

mathematical point àFT not constant


….Atom or molecules in a unit cell

∼

N atom located at Njrj ,1, =

Each atom has its electron density )( jj rr −ρNeglecting bonding

electron density(Independent Atom

Approximation- IAM)

)()(1

j

N

jM rrr

=

−= ρρ

)*2exp()*()*(1

rrirfrFN

jjM ⋅=

=


Outline






Diffraction of X-rays by a single crystalYou are here


…. Diffraction of X-rays by a single crystal….

∼

Unit cell repeated at the nodes of an infinite lattice

)()()( rLrr M ∗=∞ ρρ

∞

∞−

∞ −=lkh

HM rrV

rFrF,,

)**(1

)*()*( δ

**** clbkahrH ++=

Scattered amplitude

???)]([)]([)*( =×=∞ rLTrTrF Mρ


THEN:

∼

∞

∞−

∞ −=lkh

HM rrHFrF,,

)**()()*( δ

which is not zero only at the nodes of the reciprocal lattice where the continuous

function is sampled

)*(rF∞

)*(rFM


Laueequations

∼

**** clbkahrH ++=

λ

λ

λ

lSSc

kSSb

hSSa

o

o

o

=−⋅

=−⋅

=−⋅

)(

)(

)(Discretevaluesof S tohavediffracted beam!

By multiply rH* = (S-So)/ by a, b, c the Laueconditions to havenon-zero diffraction are obtained


DiffractOgramanimation

!"# $%%


∼

.

)()()( rrrcr Φ×= ∞ρρShape function )(rΦ

=1 in the crystal

=0 out of the crystal

)(*)()]([*)]([)( *** rDrFrTrTrFcr ∞∞ =Φ= ρ

Ω

⋅= rdrrirD )2exp()( *π

∞

∞−

−=lkh

cr HM rrDHFV

rF,,

*** )()(1

)(

….Finite crystal of gener ic volume

∞

∞−

∞ −=lkh

HM rrHFrF,,

)**()()*( δ


*

*3

*

*2

*

*1* )()()(

)(z

zAsen

y

yAsen

x

xAsenrD

ππ

ππ

ππ ⋅⋅=

A1

A3

A2

Maxima with amplitude (A1-1)•(A2

-1) •(A3 -1)

The reciprocal lattice nodes becomedomains of dimension Aj

-1 (j=1,2,3) in the threedirections (x, y, z)

….Finite crystal of volume : a pr ism with A1 cells along a, A2

cells along b and A3 cells along c.

!!! Note that also the experimental setup concurs to the shape/sizeof the observed signal!!!


The function like (senNx)/x have the samebehaviour of T[finite lattice]

−=

−=p

pn

nxxxL )()( δ

T[lattice in x going to infinite] à lattice in x*


If an ORDERED DOMAIN is considered with a number of unitcell > 1000 the peak becomes a Dirac δδδδ function

à in the “ real world” : the peak width is then defined by the instrumental setup.

Proteine case: cell edges a,b,c ∼ 10 nm:To have a single crystal like diffraction pattern ordered domainsof at least 1÷10 µm are required

Metals and simple inorganic oxides, cell edges a,b,c, ∼ 0.5 nm: crystals of about 50 ÷ 100 nm might give a good diffractionpattern (Powder data) and particles of about 5÷10 nm show widebut still measurable peaks (à by the FWHM analysis the crystalsizemight beestimated, but also stress, strain etc)

A coupleor real “ extreme” casesof T[lattice]


∼

Diffraction pattern of 1D and 2D crystals.

1D

2D


Quasi-1D chains of alkali metal atoms (K or Cs)

K-InAs(110): supercell 2x6 withrespect to the original In-As cell

4 nm x 4 nm

Local order by STM

Imagesby prof. Mariani20 nm x 20 nm

long-rangeorder

Diffraction pattern of a “ column” of K atoms on a 2D InAs sur face: a real example

InAs Disordered K Ordered K


∼

Diffraction pattern of a 3D finite crystal

Peak position à unit cellPeak intensity à atom positionsPeak width à size/shapeof the crystalN.B.: Peak position, I, width à experimental setup


Outline







You are here


∼


∞

∞−

−==lkh

HMcr rrDHFV

rFrF,,

**** )()(1

)()(

Structure factor)(HFM

vector (matrix 1 x 3); iftransposed becomes)(H ),,( lkhH

T

=


∼

Reciprocity of (hkl) and (xyz)

In the sameway

vector (matrix 1 x 3); if transposed becomes

)( jX),,( jjjj zyxX

T

=

components of vector rH* (reciprocal space):

rH* = ha* + kb* + lc*

components of vector rj (direct space): r j = xja + yjb + zjc

vector (matrix 1 x 3); if transposed becomes)(H ),,( lkhH

T

=


Using the defined vectors to represent F(H)

jjjj

T

jH lzkyhxXHrr ++=⋅=⋅*

=

⋅=N

jj

T

j XHifHF1

)2exp()( HH iBA +=

)2cos(1

=

⋅=N

jj

T

jHXHfA

)2(sin1

=

⋅=N

jj

T

j XHfB π

)*2exp()*()*(1

Hj

N

jHjHM rrirfrF ⋅=

=

π)*2exp()*()*(1

rrirfrFN

jjM ⋅=

=


…..GRAPHICALLY

H

H

A

Btg

H=ϕ

)exp(||HHH

iFF Φ=

( )[ ] [ ]==

=++==N

jjj

N

jjjjjhkl iflzkyhxifHFF

11

exp2exp)( απ

2

122

||

+=

HBAF HHAH

BH

DIFFRACTED AMPLITUDE STRUCTURE?


Four ier SYNTHESIS

⋅−=*

*)*2exp()*()(S

rdrrirFr πρ

)*(rF Measured only at the nodes of reciprocal lattice

)2exp(1

)(,,∞

∞−

⋅−=lkh

T

HXHiF

Vrρ Fourier series

( )[ ]∞

∞−

++−==lkh

lzkyhxiFV

zyxr,,

hkl 2exp1

),,()( ρρReal f.,

Friedel law( )[ +++= ∞

=

∞

=

∞

=

⋅

0 0 0hkl 2cos

2)(

h k l

lzkyhxAV

r πρ( )]lzkyhxB +++ ⋅ π2sinhkl


The phase problem

|Fhkl|∝Ihkl1/2 can bederived

from measured Intensities

The hkl -i.e. the phase- islost (phase problem)

( )[ ]∞

=

∞

=

∞

=

−++=0 0 0

hkl 2cos2

)(h k l

hkllzkyhxFV

r ϕπρ

Becuaseof Friedel law and

and becausesin is an odd function, the “sin” termsdisappear and becomes a real function

hklhkl FF = hklhkl ϕϕ −=


The phase problem – in practice

The structurecan not be immediatelyobtained after measuring the intensities Ihkl

à Lesson by M. Nardini

Difffraction pattern

Reciprocal space

3D structure

Direct space

The phaseis lost


Outline







You are here


dH

Optic pathdifference=

Bragg law and its practical applications

Crystal planeswith indices hkl

Incidentbeam

“Reflected”beam

Reflected X-rays are “ in phase” if:

??=+ BCAB ??sin2 =ϑHd λ

Hdr

1sin2|*| ==

λϑ


Reflection sphereand limiting sphere

λϑϑ sin2

sin1

* ==== IOd

rOPH

H

Sphereof radius 1/λ.Origin O of R.L. where

incident beam hits sphere

If a P point of the R. L. is on the reflection sphere then a “ reflection” along AP arises

If OP > 2/λ, P CAN NEVER be on the reflection sphere: then the limiting sphere has a radius = 2/λ


Resolution

minmax d

1sin =

λϑ dmin: resolution

dmin: “smallest thing”that can be “seen” at a

certain

large à largehklà small dmin à high resolution

hx+ky+lx

Sensible to errors on x,y,z and to the fine

details of (r)


Resolution - ExampleFor Cu(K ): = 1.54 Å

λϑ =sin2 Hd 1sin max =ϑ

5.0sin max =ϑ Å54.15.02min =

⋅= λ

d

Atomic resolution??sin2min =

⋅=

MAX

dϑ

λÅ77.0


Resolution – real cases

Protein Resolution of 2-5Å is typical, <1.0Å the limit

Softer X-rays, employed : 0.8-1.6Å

Harder X-rays, employed : 0.4-0.9Å

(Hard) Materials Resolution of 0.9-1.0Å isthe standard, <0.8Å the

limit


From the operational point of view:

( )[ ]∞

=

∞

=

∞

=

−++=0 0 0

hkl 2cos2

)(h k l

hkllzkyhxFV

r ϕπρ

To havea good representation of (r) a largenumber of |Fhkl|should bemeasured and their phasesdetermined in some way

To measure the largestpossiblenumber of

reflections (Ihkl)

To movea single crystal in a mono-

chromatic X-ray beam

Powder in a mono-chromatic X-ray beam

Crystal in a polychromatic X-ray beam (Laue method)

M. Nardini(September 17th)

P. Scardi, September 13th

squaremodulus


Detector to measure the diffracted beam (Ihkl) BM01 at SNBL@ESRF

Ihkl=K |Fhkl|2 K contains many factors:

L, P, 3, Io, e Vcr

( )[ ]∞

=

∞

=

∞

=

−++=0 0 0

hkl 2cos2

)(h k l

hkllzkyhxFV

r ϕπρ

0D-detector

2D detector (Imageplateor CCD)

(pseudo-) 1D detector


Techniques also involving diffraction:

Surfacediffraction

A. RuoccoSeptember 19th

Small Angle X-ray Scattering

P. RielloSeptember 13th

Photoelectrondiffraction

G. PaolucciSeptember 13th

Diffraction in earth Science

S. Quartieri September 19th


When issynchrotron useful?à Nardini & QuartieriConventional source à Synchrotron

PolycrystallinesampleLow resolution à high resolution

Pattern in 10 ÷ 1000 minute à 10 ÷ 1000 µs, TR-WAXS

Single crystal: crystal 100 m à 5 mWell diffracting crystal à Semi-ordered crystals, solvates

DAFS: impossible à possibleGuinier: “Wehavea fantastic theory about SAXS, but X-ray

1000 times more brilliant are needed”Information on thin films à Surfacediffraction

(GIWAXS)

“ Normal diffraction” àààà resonant diffraction, Lauemethod (Time Resolved-Singlecrystal diffraction)


Outline







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Crystallinepowder

Sample formed by a very largenumber of random-oriented ‘crystals’ (crystallites)

Very large (à infinite) number of randomly orientedreciprocal lattices with a common origin.

T[crystallinepowder]

Then all the reciprocal lattice points within the limiting sphere are always in diffraction condition


The consequence:

Each reciprocal lattice vector assumes all possibleorientationswith a common origin


Then:The locus of each set H of R.L. nodes defines a sphere of radius |rH* |, intersecting the reflection sphere on an circle

Reflections are on an axial (with respect to the incident X-raybeam) conewith opening equal to 4


“ The old times” : collecting data on a film

Pattern features:

dH line position (Å-1) (2 à sin / )

IH : line intensity(X-rays converted intoelectrons, photons…and finally scaledà ArbitraryUnit)

∝2

0

10

20Debye-Scherrer camera and sample in a rotating

cylindrical capillary


…..Modern version of Debye-Sherrer geometry at the synchrotron (ID31 at ESRF)


Bragg-Brentano geometry- parafocusing

Flat sample, powder pressedon the sampleholder

(à preferred orientation?)

Position sensitive detectorsallow to record at the sametime reflection intensitiesfor a certain angular range

Geometry rarely used at the synchrotron where X-raysare already collimated and

brilliant


DIFFRACTION PATTERN ANALYSISPhase recognition (Qualitative analysis)

Each crystal phaseshows a typical diffraction pattern, characterized by dH e IH foreach H=(h k l) reflection

ICDD databases(PDF2,3,4) allow

comparing dH, IH EXP

with dH, IH ICPD torecognize the phase(s)

Profile fitting methods (Rietveld and more) à Quantitative analysis


Structural refinement

ab initio structuresolution(if you solve the phase

problem)

“Rietveld” refinement forstructural analysis

min2

→−=i

Ci

Oii yywSPurple

curve


Phase transition analysis

When a phaseundergoes a phase transition its

diffraction pattern changes

Enviromental chambers are used to condition the sampleenvironment:

Temperature, pressure, atmosphere, irradiation,

laser in situ during XRPD

(a)

4 6 8 10 12

(b)

(040)(202)

(301)(102)

(002) (053)(352)(133)

(303)(501)

(200)(011)

Cou

nts

(a.u

.)

T(K)

2θ(°)

973

573

1÷100000 micro sec to get a XRPD pattern


Size, stress and strain of crystallites …..

“Perfect” crystal Infinite and no defects

“Real” crystal Finite and defectiveFull width at half

maximum (FWHM) (2 ) is related also to

the averagecrystal size(D). For instance:

)2(cos9.0

θθλ∆⋅

=D

Broad diffraction peaks

From the variation of cellparameters of a stressedmaterial the deformation

tensor can bederived


…… preferred orientation/strain

Non-randomly oriented crystals à Non-uniform intensity along the diffraction cone à

pole figureà preferred orientation

Aluminium with preferredorientation

Synthetic olivine without deformation


Thank you for the attention

Enjoy the diffraction

techniques and synchrotron!!!

x-ray diffraction - synchrotron radiation · 2020-01-16 · marco milanesio x-ray diffraction 5...

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