Diff Razi One

STUDIAMO LA STRUTTURA

LA DIFFRAZIONE DI RAGGI X.

W. H. Zachariasen X-Ray Diffraction in CrystalsMermin Aschcroft Solid State Physics cap 6

Maria Grazia Betti

Trattazione semi-classica dell' interazione radiazione materia

Ipotesi: il campo e.m. una perturbazione debole e e possiamo scrivere l' hamiltoniana come

H= Hs+Hp+Hint

Processi elastici: diffusione elastica e diffrazione

H= Hs+Hp+Hint Hs rimane invariata E=E' k2=k'2

L'energia e il modulo quadro del vettore k si conservano. Il vettore momento pu cambiare direzione

Processi anelastici: assorbimento od emissione di un fotone

H= Hs+Hp+Hint Hs varia per assorbimento od emissione di un fotone varia lo stato elettronico nell'atomo

molecola o solido

X-ray scattering cross sections are weak

the intensity of the scattered beam is negligible compared to that of the incident one.

the incident wave is constant in the whole diffracting volume.

multiple scattering is not considered.

The kinematic approximation of single scattering is valid (1st Born approximation)

scattering vector, ki and kd wave vectors for the incident and scattered waves

id kkq =pi== /2di kk

( ) = rr rq 3.)( deqA itot Introducing the electron density in the material tot(r), the total scattered amplitude for

elastic scattering is given by the coherent addition of the waves scattered by the electrons:

A(q) with

Classical approximation: Scattering AmplitudeClassical approximation: Scattering Amplitude

( ) ( ) jj ij

jii

jj

ijj

jefededeqA rqrquqrq uurrr ..3.3. )( ===

The material is a collection of j atoms at positions rj

( ) ( )jj

jtot rrr =

( ) ( ) = uuq uq 3.0 def ijjIntroducing the atomic form factor :

|q|

where j(r) is the electron distribution for atom j

Intensity DistributionAtoms are never residing at fixed lattice sites, they are thermally vibrating around an

average position. If we include a Debye-Waller factor in the structure factor to consider thermal vibrations:

where Mj is the Debye Waller factor associated to the j-th atom:

( ) ( ) ( ) ( )( )

=

=

=

++==1

0

1

0

1

033221

3,2,1

1

1

2

2

3

3

expexpN

n

N

n

N

ne

nnnef nnniFAiFAA aaaqqRqq 1n

Each sum is peaked at 2npi/ai and tends in the limit of large Ni value, to a periodic array of -functions with a spacing of 2pi/ai.

The diffracted intensity from a crystal has the special property of being confined along specific, well-defined directions and is the product of three orthogonal, periodic -function arrays. The momentum transfer q has to simultaneuosly meet the three following conditions for the intensity to be at the maximum:

with h, k and l integer values.

The three conditions can be simultaneously satisfied by Q vectors which represent the reciprocal lattice points. Since the Q vectors are a set of translation vectors in the reciprocal space corresponding to the real space crystal structure given by Rn, the Laue condition simply states that the maximum scattered intensity occurs at the reciprocal lattice points of the real space crystal structure.

( )( )

=

=++

=pi

pi

nnN

nnniN

n 202

exp 11

033221

1

1 n

n1 Rq

Rqaaaq

=

=

=

lkh

pi

pi

pi

222

3

2

aqaqaq 1

Intensity DistributionAtoms are never residing at fixed lattice sites, they are thermally vibrating around an

average position. If we include a Debye-Waller factor in the structure factor to consider thermal vibrations:

where Mj is the Debye Waller factor associated to the j-th atom:

( ) ( ) ( ) ( )jjrqqq MifF Ncj

j = =

expexp1

2

2

2

2228

pisinBsinuM jj ==j

( ) ( ) ( ) ( )( )

=

=

=

++==1

0

1

0

1

033221

3,2,1

1

1

2

2

3

3

expexpN

n

N

n

N

ne

nnnef nnniFAiFAA aaaqqRqq 1n

Bragg's law (1913):Interference pattern of X-rays scattered by long range ordered structures.

The path difference is 2dsin where is the incidence angle

Von Laueei(k-k)R=1 se k-k=G(k-k)R=2pim

where R is a Bravais lattice vector

PROBE: ions, electrons, neutrons, protons, with a wavelength comparable to the distance between the atomic or molecular structures.

n =2dsin

3D reciprocal lattice

In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors Ksuch that

Cella primitiva di Wigner-Seitz

Cella unitaria: puo riempire tutto il cristallo per operazioni di traslazione.

La cella primitiva la pi piccola cella unitaria ossia la cella unitaria di volume minimo.

I vettori di traslazione primitivi vengono usati per definire gli assi cristallografici,

Diffraction techniques require a long-range translational symmetry of the system giving access to the reciprocal lattice.

The diffraction process satisfies conservation of energy and momentum but for the addition of any reciprocal lattice vector.For a 3D-system:

k 2 = k '( )2

k ' =

k + g hkl

Conservation of Energy

Conservation of Momentum

The diffracted beams are characterized by the points of the points of the reciprocal lattice.

The wavelength of the projectile particle must be of the same order of magnitude of the interplanar spacings of the solid.

Fundamentals of Diffraction Techniques in 3D Space

Onda sferica Onda piana

qqFWHMFWHM=2/=2/LLcc LLcc =2/ =2/ qqFWHMFWHM

D =2=2pipi//qqFWHM FWHM = = pipiLLccq

I(q)

F hklhklI2)(

Z

Detec

tor

Measuring the peak maximum is not reliable

One measures the integrated intensity byRocking the sample in front of the detector

Requirements for a good data set

Diffractometer degrees of freedom grazing angle sample rotation axis Detector in-plane rotation Detector out of plane rotation

Accurate data normalization Instrument resolution function Sample size effects

Strategy

++=j

lzkyhxiMj

calchkl eefF jjjj )(2pi

q = ha* + kb* + lc* where h,k are discrete and l continuous variables ; xj yj zj are the coordinates of atom j in the surface unit cell

Mj is the Debye-Waller factor for atom j related to the thermal vibrations amplitudes

atoms included in the surface cell: all those shifted from ideal bulk location several atomic layers = more than 20-30 atoms = large number p of free parameters

large number N of independent data (not symmetry related)

Model refinement by least-square methods

( )

=

hkl2obs

hkl

hkl2obs

hkl

F

FR

F calchkl

Reliability factor

( )

=

2

22calc

hklobshkl FF1

hklpN

Least-square residue

Building and testing a model

LA DIFFRAZIONE DI RAGGI X.

Dai sistemi ordinati ai sistemi disordinati e alle proteine

Maria Grazia Betti

SEMINARI Lunedi 12-14 AULA 4 Mercoledi 14-16 AULA 4 Giovedi 16-18 AULA 4OTTOBREG 10 G 17G 24G 31 C. Mariani Radiazione di sincrotrone, cenni al free electron laser

NOVEMBREG 7 C. Mariani Applicazioni radiazione di sincrotrone G 14 F. Sciarrino Laser,tecnologie ottiche per informazione quantistica G 21 A. Polimeni Assorbimento e luminescenza nei semiconduttoriG 28 M.G. Betti La spettroscopia di fotoemissione

DICEMBREG 5 N. Saini Assorbimento dei raggi XG 12 P. Postorino Diffusione Rayleigh, Spettroscopia RamanG 19 S. Lupi Assorbimento nell' infrarosso

Reticolo di BravaisUn reticolo di Bravais specifica larrangiamento periodico in cui le unit elementari del cristallo sono disposte.

Tali unit possono essere singoli atomi, gruppi di atomi, molecole, ioni, ecc.

Definizioni:

-Un reticolo di Bravais una schiera infinita di punti discreti con una disposizione e unorientazione che appare la stessa da qualsiasi dei punti la schiera sia vista.

- Un reticolo di Bravais formato da tutti i punti con vettori posizione della forma:

interi332211 inanananR

++=

ia

sono detti vettori primitivi

3D crystal structures

simple cubic

body centered cubic

face centered cubic

NaCl, 2 at/un.cell: (000)a, (1/2,1/2,1/2)a

Cu, Ag, Au,, Ni, Pd, , Ne, Ar, ,1 at/un.cell

ZnS 2 fcc : (000)a, (1/4,1/4,1/4)a

Li, Na, , Cr, Nb, V, W, 1 at/un.cell

CsCl, 2 at/un.cell: (000)a, (1/2,1/2,1/2)a

3D crystal structuresface centered cubic

C, Si, Ge 2 fcc : (000)a, (1/4,1/4,1/4)a

GaAs, ZnS 2 fcc : (000)a, (1/4,1/4,1/4)a

Oblique (p) net|a1||a2| 90

Rectangular (p) net|a1||a2| =90

Square (p) net|a1|=|a2| =90

Hexagonal (p) net|a1|=|a2| =120

Rectangular (c) net|a1||a2| =90

a1

a2

a1

a2

a1

a2

a1a2

a2

a1a1

a2

Primitive cell

Unit cell|a1||a2| 90

2D Bravais Nets andUnit Meshes

Cella primitiva di Wigner-Seitz

Cella unitaria con cui si puo riempire tutto il cristallo per operazioni di traslazione.

La cella primitiva la pi piccola cella unitaria ossia la cella unitaria di volume minimo. Per costruzione, contiene un solo punto reticolare e i soli atomi della base.

I vettori di traslazione primitivi vengono usati per definire gli assi cristallografici,

A volte si usano assi non primitivi, quando sono pi convenienti o pi semplici.

Reticolo di Bravais in 3D:celle unitarie convenzionali e celle unitarie primitive

real and reciprocal space

2D Crystallography: 2D Point Groups

Simmetrie

1 2 3 4 6 m 2mm 3m 4mm 6mm

10 gruppi puntuali 1,2,3,4,6,

2m,3m,4m,4mm,6mm

SIMMETRIE TRASLAZIONALE E ROTAZIONALE

120o

THREE-FOLD

Dal solido al liquido( ) ( ) jj i

jj

iij

j

ijj

jefededeqA rqrquqrq uurrr ..3.3. )( ===

Da sistemi semplicia sistemi complessi

1937 Perutz : Use X-ray diffraction to uncover the biological function of haemoglobin, the protein in red blood cells responsible for transporting oxygen. No protein structure had been solved using X-ray crystallography. Perutz's photographs were almost picture perfect, with characteristic regular arrays of sharp spots indicating a regular, repeating pattern of crystals.

Protein Crystallography: more than 1000 atoms per unit cell!!!!

We know the amplitude, but we do not know the phase: the undulating cycle of movement from one wave peak to the next The number of reflections and interactions that occur within crystals of haemoglobin is so complex that to translate X-ray patterns into molecular structures requires more than just knowledge about the intensity of waves. The phase of the wave could be calculated by knowing the phase angle, the distance between a point on a wave and a specified reference point.. Researchers called the case of this missing information 'the phase problem', and at the time solving this was the ultimate goal for anyone interested in trying to determine protein structures from X-ray crystallography. Lindo Patterson creates a contour map that can be used to define the distances between atoms in a crystal, through knowing only the position and intensity data.

http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1962/perutz-lecture.pdf

Phase problem and Patterson map

Maria Grazia Betti

Le mappe di Patterson

In un esperimento di diffrazione si misura la I(q) che e' proporzionale al modulo quadro | F(q) | 2 . Il fattore di struttura si puo' esprimere come :

F(q)= (r) e-iqr dr

dove la (r) e' la densita' di carica elettronica in tutta la cella.

La funzione di Patterson e' definito come il prodotto di autocorrelazione della densita' di carica elettronica

P(r')= (r+r') (r) dr

I massimi principali si avranno quando r' e' un vettore di reticolo diretto

P(r')= qq' F(q) F(q') e-iq(r+r') e-iq'r drse q'=- q P(r')= 0

P(r')= q| F(q) | 2 e-iqr'

I coefficienti dello sviluppo in serie della funzione di Patterson sono il modulo quadro del fattore di struttura

If there is a peak of electron density for atom 1 at position x1 and a peak of electron density for atom 2 at position x2, then the Patterson map will have peaks at positions given by x2-x1

and x1-x2..

The figure illustrates a Patterson map corresponding to a cell with one molecule. It demonstrates that you can think of a Patterson as being a sum of images of the molecule, with each atom placed in turn on the origin. Because for each vector there is one in the opposite direction, the same Patterson map

is also a sum of inverted images of the molecule, as shown in the bottom representation.

Electron Density Map

Direct Spaceimaging by local probemicroscopy Reciprocal space

Recording the Diffraction Pattern

Metallic tip STMSi tip AFM

non-interacting probe

top layer STM ~ []2

charge density topography

lateral resolution 0.1nm

ElectronsE=p2/2m = (h/)2/2m()=12,265/E0.5(eV)

=1 , E=150 eV LEEDinteracting probe

Abs.length 1 nm (LEED)

a ~ 108 barn

resolution 0.01nm

COMPARISON Microscopy vs. Diffraction

X-rays E=h=hc/

()=12398/E(eV)=1 , E=12,4 keVinteracting probe

Abs.length > 100m

a ~ Z2barn

resolution 0.001nm

n = 1- : GIXD

chhE ==

DE BROGLIEDE BROGLIE

vmh

=

chhE == hk2/2m

electrons

ESPERIMENTO DI

DAVISSON E GERMER Le particelle si comportano come onde con =h/mv e vengono diffratte

Sonda Massa (g) Velocit (m/s) (m)

e- lento 9 x 10-28 1.0 7 x 10-4

e- veloce 9 x 10-28 5.9 x 106 1 x10-10

Inelas

tic

Mea

n Fr

ee P

ath

(nm)

Kinetic Energy (eV)

The inelastic mean free path of excited electrons in solids is very short with respect to the inter-atomic distance.

50 100 500 1000 5000 1000

0.5

1.0

5

10

Inelastic Mean Free Path of Electrons vs. Kinetic Energy

Excited Atom Solid

Excited Electron

0

z

z3

Solid Surface

1-st

2-nd

3-rd

4-th5-th

How the emission of elastically scattered electrons decays as a function of depth (z)

Thereby:

I = I0 + I0 exp[-z1/sin] + I0 exp[-z2/sin] +

+ I0 exp[-z3/sin] + .=

= I0 {1 + exp[-d/sin] + exp[-2d/sin] +

+ exp[-3d/sin] + }

where I is the total emission of electrons which did not undergo inelastic scattering and I0 is the emission from the surface plane.

Thereby only electrons created in the near-surface region can escape into the vacuum with no energy gain or loss.

LEED

1) At surfaces 2D translational symmetry holds thereby only the wave vector parallel to the surface is conserved with the addition of a reciprocal net vector

2) The procedure of Ewald sphere construction is similar to the 3D case

3) The dashed scattered wave vectors propagate into the solid and are not observable

Ewald Sphere Construction in 2D Reciprocal SpaceNotice that the reciprocal lattice is now replaced by infinite reciprocal lattice rods perpendicular to the surface and passing through the reciprocal net points

Fundamentals of Diffraction Techniques

k 2 = k '( )2

k '|| =

k || +

g hk

Conservation of Energy

Conservation of Momentum

k ||

2+ k

2=

k '||2+

k '2

k is not conserved since the translational symmetry normal to the surface is now broken

The indexing of the diffracted beams is, by convention, referenced to the substrate real and reciprocal net.If the selvedge or adsorbate structures have larger periodicities, the surface reciprocal net is smaller than that of the substrate alone. The extra reciprocal net points and associated diffracted beams are dnoted by fractional rather than integral indices.

Fundamentals of Diffraction Techniques in 2D Space

a1a2 b1

b2

s0 s00

s01

s01s11

How the Surface Meshes in the Real Space and in the Reciprocal Space Correspond Each Other

a1 (s - s0) = n1 a2 (s - s0) = n2

s / = n1 b1 + n2 b2 (ai bj = ij)

Laue Conditions

LEED:

To a first approximation, the single scattering formalism (kinematical approximation) is adopted.The incident electron is described as a plane wave and the amplitude of the outgoing electron is given by the coherent sum of scattering from each atom

Ak= expi k

A ( )

m

n

fjexpi k r j( )j

A

f j

r j

Real surface net vector

Atomic scattering factor

Position vector within a surface unit mesh

A k Amplitude of the outgoing electron

n, m Indices of surface meshes

LEED: A Series of Observations

1) The atomic scattering cross section (fj) involves a phase shift (also dependent on k) and is thus complex

2) the incident wave is exponentially attenuated in the solid

3) Since atom-electron scattering cross sections can be very large ( 1 2, i.e. 1010 times larger than in X-ray diffraction), multiple scattering must be included. This means that each incident electron is treated a a superposition of the primary wave and the scattered waves

Ak= expi k

A ( )

m

n

Fk

Fk = fj expi k r j( )

j Geometrical structure factor

LEED: A Series of Observations

One atom

Two atoms(distance a)

N atoms in a row (regular distance a)

Several (M) groups of N atoms each (regularly spaced) [distance of group centers (N+1/2)a]Several groups of varying size (arranged as in (d))

N atoms randomly distributed over 2N regular sites

Spot Separation Halfwidth (relative to 2/a)

1 1/2

1 1/N

1/[N+(1/2)] 1/{M[N+(1/2)]}

1 dependening onspot size and

mixture

1 1/(2N)

Calculated Diffraction Intensities

GRAFENE: Super-reticoli in 2D

Examples of spot profiles as a function of the primary beam energy

Spot profile analysis reveals the inadequacy of the simple scattering approach. Multiple scattering plays a major role and it must be included in the theory.

LEED

n = 1 - - i

6104

=

pi

51022

2 emc

e pi

The refraction index for X-rays is expressed as:

and are related to the polarisability and absorption

Boundary conditions

)cos( 2)cos( //, iiit kk pi

==

( ) iittt inkkk pi

pi 22222

//,2

, cos12cos2 ===

pi ik it 22

2 2,

pi2nkt =with

For small incidence angle

n=1n z

xi ft

cos (i) = n cos(t )

i < c with cosc = n Total reflexion

Grazing Incidence X-ray DiffractionGrazing Incidence X-ray Diffraction

radc3102

pi

pi iik ciit 2

2222 222, =

By introducing the critical value c : cos c = Re(n) = 1-

For i < c is purely imaginary the cristal wave is evanescent in z ( for =0),tk

( ) ( )ee zkzkki ttt r + ,,////, Im)Re(.The expression for the crystal wave which is the true incident waveFor further diffraction processes is :

A thorough treatment is given in Vineyard, Phys. Rev.B (1982)


0 1 2 3 410

100

1000

Prof

onde

ur d

e p

ntra

tion

i/

c

Penetration depth ( ) pi

=

ikL

ct 2 Im 4 Im 2 1

22i,

Cu15 keV = 7,6 10-6 = 4,4 10-7c = 0.22

L ()

Penetration depth

i : 0.1 c c 2 c

L () : 15 100 1000


q = kd - ki

Z

Incident beam

Specular beam D

iffrac

ted be

am

Detec

tor

Diffra

cted beam p

rojection

I(q)

Collecting Grazing Incidence X-ray Diffraction Data

( )( ) 0for 0

0for 1

>=

AsGa

X [110]Y [110]

_ 1x1

0 1 2 30,01

0,1

1

10

100

1000

10000

100000

pi2/sin2(pil)

1/l2

l

I(l)

Intensity distribution inIntensity distribution inreciprocal space reciprocal space

0 1 2 3 4

10

100

1000

Stru

ctur

e fa

ctor

l

Cu (001) (2,0, l) volumeC

2

After I. K. Robinson Phys. Rev. B33, 3830 (1986).

Optimal sensitivity in anti-bragg position: used to monitor layer by layer growth through pseudo-periodic oscillations

Truncation Rod for a rough surfaceTruncation Rod for a rough surface

X [110]Y [110]

_ 1x1

Intensity distributionFrom a 2D layer:Peaks in-planeDiffuse rods out-of plane

X [110]Y [110]

_ 1x1

Intensity distributionFrom a 2D layer:Peaks in-planeDiffuse rods out-of plane

Intensity distribution from a surface layer on a semi-infinite bulk:

Bulk Bragg peaks

Diffuse Truncation Rods

interfering with the surface rods

Intensity distribution from a surface layer

0 1 2 3 4

10

100

1000

C Cu(001) Surface : rod (2,0, l)

++

++

++

++

--

--

--

--

d/d = +10%d/d = -10%

Stru

ture

Fac

tor

l

d

Influence of surface relaxation on truncation rod scatteringInfluence of surface relaxation on truncation rod scattering dilationdilation oror contractioncontraction of the interplanar spacingof the interplanar spacing

Truncations rods :bare Cu (001) and saturated N-Cu (001)

10

100

103

104

0 0,5 1 1,5 2

ExprienceCalcul

ExprienceCalcul

Am

plitu

de d

iffra

cte

(u. a

.)

L

cuivre nu cuivre satur en N

10

100

103

104

0 0,5 1 1,5 2

ExprienceCalcul

ExprienceCalcul

Am

plitu

de d

iffra

cte

(u. a

.)L

cuivre nu cuivre satur en N

rod (1,1,L) rod (2,0,L)

bare Cu N saturated Cu ( =1)Relaxation d12 -3,16% +13,54%Relaxation d23 -0,54% +1,46%

After B. Croset et al. PRL 88, 56103 (2002 Truncations rods

X [110]Y [110]

_ 1x1 Larger direct unit cell ex c(2x2)

Additional half-order surface rods

Reconstruction or adsorption

Bulk Bragg peaks

Diffuse Crystal Truncation Rods (CTR) interfering with the 1x1 surface rods

X [110]Y [110]

_ 1x1

Pure surface rods

reconstructed surface or adsorbed layer on a semi-

infinite bulk

Pagina 1Pagina 2Pagina 3Pagina 4Pagina 5Pagina 6Pagina 7Pagina 8Pagina 9Pagina 10Pagina 11Pagina 12Pagina 13Pagina 14Pagina 15Pagina 16Pagina 17Pagina 18Pagina 19Pagina 20Pagina 21Pagina 22Pagina 23Pagina 24Pagina 25Pagina 26Pagina 27Pagina 28Pagina 29Pagina 30Pagina 31Pagina 32Pagina 33Pagina 34Pagina 35Pagina 36Pagina 37Pagina 38Pagina 39Pagina 40Pagina 41Pagina 42Pagina 43Pagina 44Pagina 45Pagina 46Pagina 47Pagina 48Pagina 49Pagina 50Pagina 51Pagina 52Pagina 53Pagina 54Pagina 55Pagina 56Pagina 57Pagina 58Pagina 59Pagina 60Pagina 61Pagina 62Pagina 63Pagina 64Pagina 65Pagina 66Pagina 67Pagina 68Pagina 69Pagina 70Pagina 71Pagina 72Pagina 73Pagina 74Pagina 75Pagina 76Pagina 77Pagina 78Pagina 79Pagina 80Pagina 81Pagina 82Pagina 83Pagina 84Pagina 85Pagina 86Pagina 87Pagina 88Pagina 89

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