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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
/h
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. )( ===
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)
Grazing Incidence X-ray DiffractionGrazing Incidence X-ray Diffraction
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
Grazing Incidence X-ray DiffractionGrazing Incidence X-ray Diffraction
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