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Scattering and diffraction
Based on chapert 4 + some crystallography
Repetition and continuationThe probability of scattering is described in terms of either an “interaction cross-section” (σ) or a mean free path (λ).
Differential scattering cross section (dσ/dΩ).i.e. the probability for scattering in a solid angle dΩ
100keV: σelastic = ~10-22 m2
σinelastic = ~10-22 - 10-26 m2
Is almost always the dominant component of the total scattering.
Scattering
• ElasticElectron-nucleon
High angle scatteringRutherford scattering
Electron-electron Low angle scattering
• InelasticElectron-nucleon
Bremsstrahlung
Electron-electron X-rays SEPlasmons
Electron-atomsPhonons
Inelastic scatteringelectron-nucleus interaction
• Kramers cross section– To predict bremsstrahlung production
N(E)=KZ(E0 – E)/E, N(E): number of bremsstrahlung photons, K: konstant
E<~2 keV is absorbedin the specimen and detector
Ineleastic scatteringelectron-electron interaction
100 200 300 400 Incident beam energy (keV)
σ (m2)
10-21
10-23
10-25
Cross sections in Alassuming θ~0o
P
EL
K
SE
Electron transitions and X-ray notation
• K, L, M, N, O shells– Subshell L1, L2,…
What is the effect of differentIonization cross sections?
Given ”weights” within a family
Inelastic scattering → IonizationTotal ionization cross section (Bethe -1930):
ns: number of electrons in the ionized subshellbs and cs: constants for that shell
- The differential form show that the scattered electron deviate through very small angles (<~10 mrad).
-The resultant characteristic X-ray is a spherical wave emitted uniformly over 4π sr
-with relativistic correction (Williams -1933):
Relativistic factor β=v/c
Ec: Critical ionization energy -Shell and Z dependent (Measured by EELS)
Critical ionization energy
Ec is generally <20 keV
Difference between Ec and the X-ray energy
Cascade to ground
• An ionized atom returns to ground state via a cascade of transitions.
Fluorescence yield, ωProbability of X-ray versus Auger electrons
Auger electrons
E~few hundred eV- a few keVand strongly absorbed in the specimen.
Differential cross section for plasmon exitation
a0: Bohr radius, θE=EP/2E0, Ep~15-20 eV σ→ 0, θ> 10 mrad
Phonon scattering
• Scatter electrons to 5- 15 mrad– Diffuse background in diffraction pattern– Energy loss < 0.1 eV– Scattering increases with Z (~ Z3/2)
Beam damage
• Three principal forms– Radiolysis
• Inelastic scattering breaks chemical bonds
– Knock-on damage or sputtering• Displacement of atoms from the crystal lattice → point defects
– Heating• Source of damage to polymers and biological tissue.
Effect of HT?
Electron dose : Charge density (C/m2) hitting the specimen
Specimen heating • Depends on thermal conductivety of the specimen and beam current
Knock-on damage• Directly related to the incident beam energy
– Primary way metals are damaged• Frenkel pair
– Bond strength is a factor • Related to the displacement energy
Threshold energy for dispacement of an atoms with atomic weight A:
Maximum transferable energy –Dispalcements threshold energy
If more than the threshold energy is transfered to an atom it will dispalce from its site
Elastic scattering-Rutherford
Elastic scattering-Rutherford
Elastic scattering- small angles (<~3o)
• Rutherford cross section can not be used• Scattering-factor approach is complementary
– Wave nature of electrons
Amplitudes:– Atomic scattering factor f(θ)– Structure factor F(θ)
Lattice properties of crystals • The crystal structure is described by specifying a repeating
element and its translational periodicity– The repeating element (usually consisting of many atoms) is replaced by a lattice point
and all lattice points have the same atomic environments.
Repeating element in the example
Crystals have a periodic internal structure
Lattice point
Point lattice
Repeting element 1 2 3
What is the repeting element in example 1-3?
Repeting element
1 2 3
Enhetscellen: repetisjonsenheten 1 2 3
Valgfritt origo!
Point lattice repeting element unit cell
Atoms and lattice points situated on corners, faces and edges are shared with neighbouring cells.
Unit cell
- Defined by three non planar lattice vectors: a, b and c
-or by the length of the vectors a, b and c and the angles between them (alpha, beta, gamma).
Elementary unit of volume!
a
c
bα
βγ
The origin of the unit cells can be described by a translation vector t:t=ua + vb + wc
The atom position within the unit cell can be described by the vector r:r = xa + yb + zc
Axial systems
The point lattices can be described by 7 axial systems (coordinate systems)
x
y
z
a
b
c
α
γ
β
Axial system Axes Angles
Triclinic a≠b≠c α≠β≠γ≠90o
Monoclinic a≠b≠c α=γ=90o ≠ β
Orthorombic a≠b≠c α= β=γ=90o
Tetragonal a=b≠c α= β=γ=90o
Cubic a=b=c α= β=γ=90o
Hexagonal a1=a2=a3≠c α= β=90o
γ=120o
Rhombohedral a=b=c α= β=γ ≠ 90o
Bravais lattice
The point lattices can be describedby 14 different Bravais lattices The point lattices can be describedby 14 different Bravais lattices
Hermann and Mauguin symboler:P (primitiv)F (face centred)I (body centred) A, B, C (bace or end centred) R (rhombohedral)
Lattice planes
• Miller indexing system– Miller indices (hkl) of a plane is found
from the interception of the plane with the unit cell axis (a/h, b/k, c/l).
– The reciprocal of the interceptions are rationalized if necessary to avoid fraction numbers of (h k l) and 1/∞ = 0
– Planes are often described by their normal
– (hkl) one single set of parallel planes– {hkl} equivalent planes
Z
Y
X
(010)
(001)
(100)
Z
Y
X
(110)
(111)
Z
Y
X
y
z
x
c/l
0a/h b/k
Hexagonal axial system
a1
a2
a3
a1=a2=a3
γ = 120o
(hkil)h + k + i = 0
Directions• The indices of directions (u, v and w) can be found from the
components of the vector in the axial system a, b, c.
• The indices are scaled so that all are integers and as small as possible
• Notation– [uvw] one single direction or zone axis– <uvw> geometrical equivalent directions
• [hkl] is normal to the (hkl) plane in cubic axial systems
uaa b
x
z
c
y
vb
wc
[uvw]
Determination of the Bravais-lattice of an unknown crystalline phase
Tilting series around common axis
0o
10o
15o
27o
50 nm
50 nm
Tilting series around a dens row of reflections in the reciprocal space
0o
19o
25o
40o
52o
Positions of the reflections in the reciprocal space
Determination of the Bravais-lattice of an unknown crystalline phase
Bravais-lattice and cell parameters
From the tilt series we find that the unknown phase has a primitive orthorhombic Bravias-lattice with cell parameters:
a= 6,04 Å, b= 7.94 Å og c=8.66 Å
α= β= γ= 90o
6.0
4 Å
7.94 Å8.66 Å
a
bc
100
110
111
010
011
001 101
[011] [100] [101]
d = L λ / R
Resiprocal latticeImportant for interpretation of ED patternsImportant for interpretation of ED patterns
Defined by the vectors a*, b* and c* which satisfy the relations:a*.a=b*.b=c*.c=1 and a*.b=b*.c=c*.a=a*.c=……..=0
Solution:
Vbac
Vacb
Vcba
/)(
/)(
/)(
*
*
*
V: Volume of the unit cell
V=a.(bxc)=b.(cxa)=c.(axb)
a* is normal to the plane containing b and c etc.
Unless a is normal to b and c, a* is not parallel to a.
Orthogonal axes:a* = 1/IaI, b*=1/IbI, c*=1/IcI
Reciprocal vectors, planar distances
• Planar distance (d-value) between planes {hkl} in a cubic crystal with lattice parameter a:
222 lkh
ad hkl
–The resiprocal vector
is normal to the plane (hkl).
andthe spacing between the (hkl) planes is given by
*** clbkahg hkl
hklhkl gd /1
Convince your self !
What is the dot product beteen the normal to a (hkl) plane with a vector In the (hkl) plane?
Unit normal vector: n= ghkl/IghklI