II. Kinematical Scattering Theory and the Reciprocal Lattice
dΘ
Θ Θ
d sinΘ
Bragg-Gleichung 2d sin = nΘ λ
Netzebenen
Bragg‘s Law (William Lawrence Bragg, 1912)
2d nsin Θ = λ
• Only possible for λ < 2d, i.e., we need radiation with short wavelengths (in the non-visible regime)
• geometrical consideration Diffraction condition results from periodic array of crystal lattice
Kinematical Scattering Theory and the Reciprocal Lattice
‘X-Rays‘:Electromagnetic radiation with wavelengths in the Ångstrøm (10-10 m) regime
[ ][ ]keVE
398.12
E
hcÅ ==λWave vector:
λ
π=
2k
Elastic Scattering of Waves:
Scattering vector
Wave vector transfer
Inelastic Scattering: Compton-Effect
)trk(i0eE)t,r(E ω−⋅=
vvvvv
k'kQv
hv
hr
h −=
k
k’Q
2Θ
Scattering Angle
λ
π2' == kk
Elastic
Inelastic
k ≠ 'k
Basic Elements
Kinematical Scattering Theory and the Reciprocal Lattice
II. Kinematical Scattering Theory and the Reciprocal Lattice
dΘ
Θ Θ
d sinΘ
Bragg-Gleichung 2d sin = nΘ λ
Netzebenen
Bragg‘s Law (William Lawrence Bragg, 1912)
2d nsin Θ = λ
• Only possible for λ < 2d, i.e., we need radiation with short wavelengths (in the non-visible regime)
• geometrical consideration Diffraction condition results from periodic array of crystal lattice
Kinematical Scattering Theory and the Reciprocal Lattice
Which effects are not described yet by Bragg‘s law?What has been neglected?
• Intensity of diffracted wave• Crystal structure (spatial distribution of atoms within unit cell)• Atomic structure (spatial distribution of electrons within atom)• Thermal vibration of crystal lattice• Shape/Size of scattering object (e.g. crystal)
• Absorption• Refraction• Multiple Scattering Processes•....
On what objects “X-ray waves” will be scattered?Elementary interaction process ?
Bragg’s Law
Kinematical Scattering Theory and the Reciprocal Lattice
• Incoming X-ray wave (frequency ω) stimulates (weakly bounded) Electron
• Oscillation of Electron (frequency ω) electron acts as oscillating Hertz dipole
• Re-emission of electromagnetic waves (frequency ω)
• Amplitude of scattered wave in the far field (from Maxwell’s equations)
R
eePErRrE
ikRrki
rad
vvv00),( =
[ ]
−
−
+
=
ertunpolarisi
onPolarisatip
onPolarisatis
)2(cos1
)2cos(
1
22
1 θ
θP
Å1082.2mc4
er 15
20
2
0−×=
πε=
Classical electron radiusThomson scattering length
p-Polarisation s-PolarisationObserver
Observer
2θ
2θincoming x-ray wave
incoming x-ray waveele tronc
ele tronc
Scattering by a Single Electron
Thomson Scattering
Kinematical Scattering Theory and the Reciprocal Lattice
For an arbitrary distribution of the electrons (with density ρ(r)) and apart from resonant absorption edges we can assume:
• the electrons can be treated as free electrons
• they equally take part in the scattering process
• the total scattering amplitude is then proportional to the electron density ρ(r)
• the total scattering amplitude at the observer’s position R can be calculated through linear superposition of all partial waves taking into account the phases:
∫−
ρ=
−
Rr
ee)r(dVPEr)R(E
Rrikrki
00rad vvvv
vvvv
Kirchhoff‘s Law
Electron Distribution (r)ρ
dVk
r
Rr-R
Observer
2Θ
0
Scattering by an Ensemble of Electrons
Kinematical Scattering Theory and the Reciprocal Lattice
Observer and object are far away: R >> r
r-R is pointing in the same direction. We name this direction as k‘/k‘
( )rQi
Rikrkir'ki
Rikrki
RrRikrki
Rrik
eR
eee
Re
eR
ee
Rr
e vvvvvvvv)v
vvvv
vv⋅−
⋅⋅⋅−
⋅⋅
⋅−⋅⋅
−⋅
=≈≈−
Fraunhofer Approximation
∫⋅−ρ= rQi
ikR
00rad e)r(dVR
ePEr)Q,R(E
vvvvdV
k
r
R
r-R
Observer
2Θ
k’
λ
Θπ=
sin4Qr
k'kQvvr
−=
k
k’Q
2Θ
Scattering by an Ensemble of Electrons
Fraunhofer Approximation
Kinematical Scattering Theory and the Reciprocal Lattice
∫⋅−ρ= rQi
ikR
00rad e)r(dVR
ePEr)Q,R(E
vvvv
Some remarks:
• We “see“ changes in the electron density (Fourieranalysis)
• Expression above does not hold for crystalline materials only
• It is valid also for non-periodic systems (e.g. amorphous materials, liquids)
• Experimentally relevant value is the scattered intensity
pair correlation functionR)dRrρ()Rρ()rg( 3rrrr
+= ∫
∫−
∝∝rQi)erdVg(
2rad
EIrr
r
Kinematical Scattering Theory and the Reciprocal Lattice
Home work: Derivation of Eq.(1)
(1)
Scattering by an Ensemble of Electrons
Fraunhofer Approximation
Differential Scattering Cross Section
Scattered intensity measured by a detector Isc
Isc = photons per second (detectors usually count single photons)
= energy per second (power) flowing through the area of the detector divided by the energy of each photon
I0 = incident beam intensity = .. |E0|2
= energy per second within a cross sectional area A0
AD = area of detector = R2∆Ω
0
2
20
2
0
),(
A
R
E
RQE
I
I radsc ∆Ω⋅=
v
Kinematical Scattering Theory and the Reciprocal Lattice
∆Ω: solid angle of detectorA0: cross section of incident beamR: distance of sample to detector
Differential Scattering Cross Section
)flux ) ( (Incident into second per scattered photons of Numbers
∆Ω
∆Ω=
Ωd
dσ
( )20
electronPr
dd
=
Ω
σ
( )( )
22
02
02
0
02
00)(
),(
/ ∫−=∆Ω⋅
⋅∆Ω
⋅=
∆Ω=
ΩdVerPrR
AE
ARQE
AI
I
d
d rQiradscvvv
v
ρσ
Kinematical Scattering Theory and the Reciprocal Lattice
It is usual to normalize the scattered intensity Isc by
• the incident flux I0/A0
• the solid angle ∆Ω of the detector
2
0
220
0
22
2
220
0
2
20
2
0)()(
),(∫∫
⋅−⋅− ∆Ω=
∆Ω⋅=
∆Ω⋅= rQirQiradsc erdV
A
Pr
A
RerdV
R
Pr
A
R
E
RQE
I
I vvvvvv
v
ρρ
Differential Scattering Cross Section
Electron density of a single atom: ρA(r)
)Q(fR
ePEre)r(dV
Re
PEr)Q,R(EikR
00rQi
A
ikR
00radvvvv vv
=ρ= ∫⋅−
atomic scattering factor (amplitude)
atomic form factor
∫⋅−ρ= rQi
A e)r(dV)Q(fvv
vv
Scattering VectorA
tom
ic S
catt
eri
ng
Fa
cto
r0
Z
‘delocalized’density
‘localized’density
experimental/calculated values are listed in:
International Tables of Crystallography
or
http://henke.lbl.gov/optical_constants/asf.html
Scattering by a Single Atom
Kinematical Scattering Theory and the Reciprocal Lattice
Kinematical Scattering Theory and the Reciprocal Lattice
Properties of atomic scattering factor
• f(0) = Z strong forward scattering• f(∞) = 0
• The stronger the delocalization of the electron density the stronger the Q dependency
• Point like localization ρA(r) = Z δ(r) f(Q) = Z = const
• Spherical symmetry: Qr
QrrrdrQfQf A
sin)(4)()( 2
∫== ρπr
Home Work: Calculate the atomic form factor of a hydrogen atom (Solution: f(Q) = 16 / (4 + Q2a0
2)2) (a0: Bohr radius)
Scattering by a Single Atom
A 3D lattice can be created by a set of vectors Rn :
a1, a2 and a3: Basic lattice vectors defining the unit cell
n1, n2, and n3 : integer numbers
332211n nnn aaaR ++=
Within the unit cell: Atoms with corresponding atomic form factors fj(Q) located at positions inside unit cell r1, r2, ..., rk.
Scattering by a Crystal Lattice
The atoms
• are thus sitting at positions: n1a1 + n2a2 + n3a3 + rj
• exhibit the (complex) atomic scattering factor fj(Q)
The total scattered amplitude is the sum of all partial amplitudes taking into account the relative phases !
Kinematical Scattering Theory and the Reciprocal Lattice
Scattering by a Crystal Lattice
Kinematical Scattering Theory and the Reciprocal Lattice
For the summation over n1,n2,n3, and j it is very helpful that:
fj(Q) exp [-iQ(n1a1 + n2a2 + n3a3 + rj)] = exp [-iQ(n1a1 + n2a2 + n3a3)] x fj(Q) exp(-iQrj)
S(Q): Structure factor G(Q): Lattice factor
)Q(G)Q(SPRr
E
ee)Q(fRr
PEEE
00
Atoms
G
n,n,n
)ananan(Qi
S
k
1j
rQij
00jcrystal
321
332211j
vv
4444 34444 21444 3444 21
v vvvvvv
××=
=××== ∑ ∑∑ ++−
=
−
We can thus independently perform the summation:
The total scattered amplitude is the sum of all partial amplitudes taking into account the relative phases !
∑ ++−=
321
332211
n,n,n
)ananan(QieGvvvv
• The sum over n1,n2 and n3 depends on the shape and size of the crystal
it might by complicated to perform the summation
• For simplicity we assume that the crystal shape is a parallelepiped (for 90°angle cuboid), with the edges parallel to the basic vectors a1, a2 and a3
)aQinexp()aQinexp()aQinexp(G1N
0n33
1N
0n22
1N
0n11
3
3
2
2
1
1
∑∑∑−
=
−
=
−
=
−⋅−⋅−=vvvvvv
)aQ(sin
)aQN(sin
)aQ(sin
)aQN(sin
)aQ(sin
)aQN(sinG
3212
33212
2212
22212
1212
11212
2vr
vr
vr
vr
vr
vr
⋅⋅=
and thus for the intensity:
Laue Function
Scattering by a Crystal Lattice: The Lattice Factor
Kinematical Scattering Theory and the Reciprocal Lattice
• Maxima appear at Q = n ⋅2π/a Bragg‘s Law
• The maxima of the Laue function define the reciprocal lattice
• Width and height of the peaks (Bragg reflections) depend on crystal shape and size (macroscopic symmetry)
• Widths and intensity of Bragg reflections are independent of n: ∆Q = 2π/aN
0
10
20
30In
tens
ity
Qa
2 /Nπ
N2
0−2π−3π 2π 3π−π π
Laue Function: N = 5 N = 10
0
5000
10000
Inte
nsity
Qa0−2π−3π 2π 3π−π π
Laue Function: N = 100
1 1
Kinematical Scattering Theory and the Reciprocal Lattice
Scattering by a Crystal Lattice: The Lattice Factor
|G(Q
)|2
|G(Q
)|2
The Reciprocal Lattice
Kinematical Scattering Theory and the Reciprocal Lattice
• The Laue function exhibit maxima at
Q1 = h ⋅2π/a1, Q2 = k ⋅2π/a2, Q3 = l ⋅2π/a3
(Laue Equations)
• The Maxima of the Laue function define the reciprocal lattice
• A possible (primitive) basis of reciprocal space is given by
• Every (arbitrary) reciprocal lattice vector Ghkl can be expressed as
ba a
a a ab
a a
a a ab
a a
a a a
2 3
1 2 3
2
3 1
1 2 3
3
1 2
1 2 3
1 2 2 2=×
⋅ ×=
×
⋅ ×=
×
⋅ ×π π π
321 bbbG lkh ++=hkl
hklGQ = Miller Indices
Please note: Diffraction is caused by net-planes of type (hkl)
Indexing of Net-planes
Kinematical Scattering Theory and the Reciprocal Lattice
Example: cubic lattice
• Determine intersection of the plane with the axes a1, a2,a3 (in units of the lattice parameters)
• Calculate reciprocal values and search for the smallest integers – h,k,l –, being in the same relationship
• Please note the difference between directions [hkl] and planes (hkl)
• General case: directions and planes are not perpendicular to each other (but is valid for cubic crystal)
Bragg Condition: Reciprocal Space versus Angular Space
Kinematical Scattering Theory and the Reciprocal Lattice
In reciprocal space we have found the following relationship for constructive interference:
Since |Ghkl |= 2π/dhkl
and Q = 4π sinθΒ/λ
we can write this as
hklGQ =
λsinθ2d Bhkl =⋅
Homework: Prove the relationship: |Ghkl|= 2π/dhkl
Equivalent!
λ
Θπ=
sin4Qr
k'kQvvr
−=
k
k’Q
2Θ
tQ
πθ
λ
θπ 2cos4=∆=∆
θ
λ=θ∆
cost2
Width of Bragg Reflections – Impact of Crystal Size
Kinematical Scattering Theory and the Reciprocal Lattice
∆Q = 2π/aN
Question: Can Bragg reflections become infinitely sharp?
Reciprocal dependence of ∆Q and thickness
Answer: No! Penetration depth inside crystal is finite!
Numerical Values: t = 5 µm (penetration depth into Silicon, CuKα radiation, λ = 0.154 nm, θ = 45° (Bragg angle)
∆θ = 2·10-5 = 0.0011° Sharp reflections !
Consequence: Small Crystals exhibit broad Bragg reflections !
k
k’No refraction
No extinction
Single Scattering
∫⋅−ρ= rQi
ikR
00rad e)r(dVR
ePEr)Q,R(E
vvvvv
• Refraction Re(χ0), Absorption Im(χ0) und Diffraction (χg)
• Re(χ) = 10-5..10-6; Im(χ) = 10-7..10-8 weak interaction
• Bragg Reflection: strong multiple diffraction
• Bragg Reflection: strong extinction t = 1 .. 10 µm
• Intrinsic (Darwin) width of Bragg reflections ∆θD ∝λ-1·χg (1 .. 10 arcsec)
See Lecture Dynamical Theory
Interaction between Material and X-rays is mediated by dielectric polarizability (susceptibility) χ(r)
Solving of Maxwell‘s equations
( ) ∑ ⋅−⋅χ=χg
rgig er
v
vvvv
Width of Bragg Reflections – Dynamical Theory
Darwin Curve
(b) Transmission Geometry(Laue Case)
λ = 1.54 Å, Silicon crystal (thickness = 300 µm), 422 reflection
(a) Reflection Geometry (Bragg Case)
Kinematical Scattering Theory and the Reciprocal Lattice
Width of Bragg Reflections – Dynamical Theory
S(Q): Structure factor G(Q): Lattice factor
)()(
)(
00
,,
)(
1
00
321
332211
QGQSPR
rE
eeQfR
rPEEE
Atoms
G
nnn
anananQi
S
k
j
rQi
jjcrystal
j
vv
444 3444 21444 3444 21
v vvvvvv
××=
=××== ∑ ∑∑ ++−
=
−
Scattering by a Crystal Lattice: Structure Factor
Kinematical Scattering Theory and the Reciprocal Lattice
• Lattice factor G(Q) Position of Bragg reflections
|G(Q)|2 independent of (hkl)
• Structure factor S(G) Summation over Atoms within unit cell
relative Intensity of Bragg reflections
∑=
−=
k
1j
rQij
je)Q(f)Q(Svvvv
Structure factor (Structure amplitude) S(Q) = S(G)
Crystal structure determines actual value of S(G)
Selection rules forbidden/weak reflections
High symmetry a few allowed reflections only
Kinematical Scattering Theory and the Reciprocal Lattice
Defined at Bragg reflections
Scattering by a Crystal Lattice: Structure Factor
Kinematical Scattering Theory and the Reciprocal Lattice
( )[ ]∑ ++−===j
jyjjhkl lzkyhxifSSS π2exp)( GG
When expressing the coordinates of the atoms within the unit cell as rj = xja1 + yja2 + zja3 = (xj, yj, zj) und using the Miller indices (h,k,l)
we end up with
321 bbbG lkh ++=
Homework: Show that the periodic electron density ρ(r) in a crystal can be expressed by the structure factors S(G) through (V: Volume of primitive unit cell)
∑ ⋅⋅=ρ
G
rGiG eS
V1
)r(v
vvvv
Comment:
In the literature various notations can be found for the term “Structure factor“. Most common are:
Shkl or Fhkl
Scattering by a Crystal Lattice: Structure Factor
S = 0 for h+k+l = odd
S = 2f for h+k+l = even
S = 4f if all h, k, l are either even or odd
S = 0 else
Structure factor – Selection Rules
Kinematical Scattering Theory and the Reciprocal Lattice
( ) ( )[ ] S hkl f i h k l= + − + +1 exp π
bcc-lattice (body centered cubic): identical atoms at r1 = 0 and r2 = (½ , ½ , ½ )
fcc-Gitter (face centered cubic):identical atoms at r1 = 0, r2 = (0, ½, ½ ), r3 = (½, 0, ½ ), r4 = (½ , ½, 0)
( ) ( )[ ] ( )[ ] ( )[ ] S hkl f i h k i k l i h l= + − + + − + + − +1 exp exp expπ π π
S = 8f for h+k+l = 4n and all h,k,l even (e.g. 220, 400, 444)
S = 8f for all h,k,l odd (e.g. 111, 135)
S = 0 else (z.B. 100, 200, 211)
Kinematical Scattering Theory and the Reciprocal Lattice
Diamond structure: identical atoms atr1 = (0,0,0), r2 = (0,½,½), r3 = (½,0,½), r4 = (½ ,½,0), r5 = (¼,¼,¼), r6 = (¼ ,¾ ,¾), r7 = (¾,¼,¾), r8 = (¾ ,¾ ,¼ )
( )
( )[ ] ( )[ ] ( )[ ]( )[ ] ( )[ ]( )[ ] ( )[ ]
S hkl f
i h k i k l i h l
i h k l i h k l
i h k l i h k l
=
+ − + + − + + − + +
+ − + + + − + + +
+ − + + + − + +
1
2 3 3 2
3 3 2 3 3 2
exp exp exp
exp / exp /
exp / exp /
π π π
π π
π π
Homework: Calculate the structure factor for Zinkblende structure and determine the selection rules! Atoms of sort A at: r1 = (0,0,0), r2 = (0,½,½), r3 = (½,0,½), r4 = (½ ,½,0)Atoms of Sort B at: r5 = (¼,¼,¼), r6 = (¼ ,¾ ,¾), r7 = (¾,¼,¾), r8 = (¾ ,¾ ,¼ )
Structure factor – Selection Rules
What happens to the scattered intensity when we introduce a flat surface (with normal vector parallel zu a3 ) with infinite lateral dimension?
• ‘Rod like‘ intensity distribution which is extended along a3
(perpendicular to surface)
• Envelop of Laue function
a1
a3
a2
a *3 a *3
a *1 a *1
G
Flayer(Q): Scattering amplitude of a two-dimensional atomic layer
3z
3zaiQ
layer
0j
jaiQlayerCTR
e1
)Q(Fe)Q(FF
−== ∑
∞
=
−v
v
( )2/aQsin4
)Q(FFI
3z2
2layer2CTRCTR
v
==
The Influence of a Surface – Crystal Truncation Rods
Crystal Truncation Rods (CTR)
Kinematical Scattering Theory and the Reciprocal Lattice
|F
| C
TR
2
|F
| C
TR
2
10-1
100
101
102
δz/z = 0.00δz/z = +0.05δz/z = -0.05
Qa0−π π 2π−2π
10-1
100
101
102
Qa0−π π 2π−2π
Perfect atomic layers
( )3z2
2layer2CTRCTR
aQsin4
)Q(FFI
v
==
Vertical relaxation ∆z/z of the top atomic layer
)zz
1(aiQlayeraiQ
layerCTR 3z
3ze)Q(F
e1
)Q(FF
∆+−
⋅+−
=v
v
Qza3 Qza3
Kinematical Scattering Theory and the Reciprocal Lattice
The Influence of a Surface – Crystal Truncation Rods
Calculated Intensity Profiles for These Surfaces
Two possible ‘ideal‘(111)Si Surfaces
J.K. Robinson, D.J. Tweet, Surface x-ray diffraction, Rep.Prog.Phys. 55, 599 (1992)
Experimental Reflectivity (dots) along with best fitting (line)
Dimer-Adatom-Stacking Fault Model of the r Si(111)7x7 Surface Reconstruction
The Influence of a Surface – Crystal Truncation Rods
Applications of CTR-Analysis
• Surface structure:
Vertical lattice relaxation
Surface Reconstruction
• Surface Morphology (Roughness)
• Size and Shape of nano structures
• Identification und Characterization of surface facets
• ...
• ...
Kinematical Scattering Theory and the Reciprocal Lattice
The Influence of a Surface – Crystal Truncation Rods
Kinematical Scattering Theory and the Reciprocal Lattice
Identification und Characterization of surface facets
Laue function of truncated pyramid
The Influence of a Surface – Crystal Truncation Rods
(How) Can we determine the crystal structure through measurement of all structure factor?
• The structure factor S(Q) is a complex number
• We measure the diffracted intensity |S(Q)|2
• The complex phases of S(Q) have to be determined
• Things gets easier when the crystal structure are centrosymmetric:
• S(Q) is a real number with unknown sign
∑ ++π−=ρ
l,k,h
)lzkyhx(i2hkleS
V1
)z,y,x(
Structure Factor – Determination of Crystal Structure