Transcript R - UiO

Diffraction methods and
electron microscopy
Outline and Introduction to
FYS4340 and FYS9340
FYS4340 and FYS9340
• FYS4340
– Theory based on ”The theory and practice of analytical electron
microscopy in material science” by Arne Olsen
• Chapter: 1-10, 12 + sample preparation
– Practical training on the TEM
• FYS9340
– Theory same as FYS4340 + additional papers related to TEM and
diffraction.
– Teaching training.
– Perform practical demonstrations on the TEM for the master students.
Basic TEM
Electron gun
Electrons are deflected by both
electrostatic and magnetic fields
Force from an electrostatic field
F= -e E
Sample position
Force from a magnetic field
F= -e (v x B)
Electron transparent samples
Introduction
EM and materials
Electron microscopy are based on three
possible set of techniqes
Imaging
With spatial resolution
down to the atomic level
(HREM and STEM)
Spectroscopy
Chemistry and elecronic
states (EDS and EELS).
Spatial and energy
resolution down to the
atomic level and ~0.1 eV.
Electrons
BSE
AE
SE
X-rays (EDS)
Diffraction
From regions down to a
few nm (CBED).
E<Eo
(EELS)
Bragg diffracted
electrons
E=Eo
Basic principles, electron probe
Electron
Auger electron or
x-ray
Valence
M
M
3d6
3p4
3d4
2p2
Electron
shell
L
3s2
2
2p4
3p
2s2
K
L
1s2
K
Secondary electron
15/1-08
Characteristic x-ray emitted or Auger
electron ejected after relaxation of inner
state.
Low energy photons (cathodoluminescence)
when relaxation of outer stat.
MENA3100
Introduction
EM and materials
The interesting objects for EM is not the average
structure or homogenous materials but local
structure and inhomogeneities
Defects
Interfaces
Precipitates
Defects, interfaces and precipitates determines the
properties of materials
Introduction
History of EM: from dream to reality
•
•
•
•
•
•
•
1834
1876
1897
1924
1925/26
1926/27
1927
• 1928
William Rowan Hamilton
Ernst Abbe
J.J. Thomson
de Broglie
E. Schrӧdinger
Hans Busch
C. Davisson and L.H. Germer/
G. Thomson and A. Reid
Max Knoll and Ernst Ruska
The first electron microscope
• Knoll and Ruska
• By 1933 they had produced a TEM
with two magnetic lenses which gave
12 000 times magnification.
Ernst Ruska: Nobel Prize in physics 1986
The first commersial microscopes
• 1939 Elmiskop by Siemens Company
Elmiskop I
• 1941 microscope by Radio corporation of America (RCA)
– First instrument with stigmators to correct for astigmatism. Resolution
limit below 10 Å.
Developments
Realized that spherical
aberration of the magnetic
lenses limited the possible
resolution to about 3 Å.
•
r2
α r1
Spherical aberration coefficient
ds = 0.5MCsα3
M: magnification
Cs :Spherical aberration coefficient
α: angular aperture/
angular deviation from optical axis
r2
α r1
2000FX: Cs= 2.3 mm
2010F: Cs= 0.5 nm
Disk of least confusion
Chromatic aberration
Disk of least confusion
Chromatic aberration coefficient:
v - Δv
v
dc = Cc α ((ΔU/U)2+ (2ΔI/I)2 + (ΔE/E)2)0.5
Cc: Chromatic aberration coefficient
α: angular divergence of the beam
U: acceleration voltage
I: Current in the windings of the objective lens
E: Energy of the electrons
Thermally emitted electrons:
2000FX: Cc= 2.2 mm
2010F: Cc= 1.0 mm
Force from a magnetic field:
ΔE/E=kT/eU
F= -e (v x B)
Developments
~ 1950 EM suffered from
problems like: Vibration of the
column, stray magnetic fields,
movement of specimen stage,
contamination.
Lots of improvements early
1950’s.
Still far from resolving crystal
lattices and making direct
atomic observations.
Observations of dislocations and
lattice images
• 1956 independent observations of dislocations by:
Hirsch, Horne and Wheland and Bollmann
-Started the use of TEM in metallurgy.
• 1956 Menter observed lattice images from materials
with large lattice spacings.
• 1965 Komoda demonstrated lattice resolution of
0.18 nm.
– Until the end of the 1960’s it was mainly used to test
resolution of microscopes.
Menter, 1956
Use of high resolution electron
microscopy (HREM) in crystallography
• 1971/72 Cowley and Iijima
– Observation of two-dimensional lattice images of complex oxides
• 1971 Hashimoto, Kumao, Hino, Yotsumoto and Ono
– Observation of heavy single atoms, Th-atoms
1970’s
• Early 1970’s: Development of energy dispersive x-ray
(EDX) analyzers started the field of analytical EM.
• Development of dedicated HREM
• Electron energy loss spectrometers and scanning
transmission attachments were attached on
analytical TEMs.
– Small probes making convergent beam electron diffraction (CBED)
possible.
1980’s
• Development of combined high resolution and
analytical microscopes.
– An important feature in the development was the use of increased
acceleration voltage of the microscopes.
Last few years
• Development of Cs corrected microscopes
– Probe and image
• Improved energy spread of electron beam
– More user friendly Cold FEG
– Monocromator
Electron beam instruments
• Transmission Electron microscope (TEM)
– Electron energies usually in the range of 80 – 400 keV. High voltage
microscopes (HVEM) in the range of 600 keV – 3 MeV.
•
•
•
•
•
Scanning electron microscope (SEM) early 1960’s
dedicated Scanning TEM (STEM) in 1968.
Electron Microprobe (EMP) first realization in 1949.
Auger Scanning Electron Microscopy (ASEM) 1925, 1967
Scanning Tunneling Microscope (STM) developed 1979-1981
Because electrons interact strongly with matter, elastic and
inelastic scattering give rise to many different signals which
can be used for analysis.
Electron waves
• Show both particle and wave properties
Charge e
Restmass mo
Wave ψ
Wave length λ
λ = h/p= h/mv
de Broglie (1925)
• Electrons can be accelerated to provide sufficient
short wave length for atomic resolution.
λ = h/(2emoU)1/2
U: pot. diff.
• Due to high acceleration voltages in the TEM
relativistic effects has to be taken into account.
λ = h/(2emoU)1/2 * 1/(1+eU/2moc2)1/2
The Transmission Electron Microscope
U
(Volt)
k = λ-1 (nm-1)
λ
(nm)
m/mo
v/c
1
0.815
1.226
1.0000020
0.0020
10
2.579
0.3878
1.0000196
0.0063
102
8.154
0.1226
1.0001957
0.0198
104
81.94
0.01220
1.01957
0.1950
105
270.2
0.00370
1.1957
0.5482
2*105
398.7
0.00251
1.3914
0.6953
107
8468
0.00012
20.5690
0.9988
Relations between acceleration voltage,
wavevector, wavelength, mass and velocity
c
Simplified ray diagram
b
a
Parallel incoming electron beam
3,8 Å
Si
Sample
1,1 nm
PowderCell 2.0
Objective lense
Diffraction plane Objective aperture
(back focal plane)
Selected area
Image plane
aperture
MENA3100 V08
Microscopy and diffraction condition
Focal plane
Image plane
Intermediate
lens
Projector
lens
JEOL 2000FX
Wehnelt cylinder
Filament
Anode
Electron gun 1. and 2. beam deflectors
1. and 2. condenser lens
Condenser aperture
Condenser lens stigmator coils
Condenser lens 1. and 2. beam deflector
Mini-lens screws
Specimen
Intermediate lens
shifting screws
Projector lens
shifting screws
Condenser mini-lens
Objective lens pole piece
Objective aperture
Objective lens pole piece
Objective lens stigmators
1. Image shift coils
Objective mini-lens coils (low mag)
2. Image shift coils
1., 2.and 3. Intermediate lens
Projector lens beam deflectors
Projector lens
Screen
The requirements of the illumination system
• High electron intensity
– Image visible at high magnifications
• Small energy spread
– Reduce chromatic aberrations effect in obj. lens
• High brightness of the electron beam
– Reduce spherical aberration effects in the obj. lens
• Adequate working space between the illumination
system and the specimen
The electron microscope
Additional literature and web resources
• http://nanohub.org/resources/3777
– Eric Stach (2008), ”MSE 528 Lecture 4: The instrument,
Part 1, http://nanohub.org/resources/3907
• D.B. Williams and C.B. Carter, Transmission Electron
Microscopy- A textbook for Material Science, Plenum Press
New York. Second edition 2009
Repetition from 1st lecture
• What type of techniques can be done in an
analytical TEM?
• Why are electrons suitable for imaging with
atomic resolution?
• What is changing when one goes from
diffraction to imaging mode?
c
Simplified ray diagram
b
a
Parallel incoming electron beam
3,8 Å
Si
Sample
1,1 nm
PowderCell 2.0
Objective lense
Diffraction plane
(back focal plane)
Image plane
MENA3100 V08
Objective aperture
Selected area
aperture
Eric Stach (2008), ”MSE 528 Lecture 4: The instrument,
Part 1, http://nanohub.org/resources/3907
JEOL 2000FX
Wehnelt cylinder
Filament
Anode
Electron gun 1. and 2. beam deflectors
1. and 2. condenser lens
Condenser aperture
Condenser lens stigmator coils
Condenser lens 1. and 2. beam deflector
Mini-lens screws
Specimen
Intermediate lens
shifting screws
Projector lens
shifting screws
Condenser mini-lens
Objective lens pole piece
Objective aperture
Objective lens pole piece
Objective lens stigmators
1. Image shift coils
Objective mini-lens coils (low mag)
2. Image shift coils
1., 2.and 3. Intermediate lens
Projector lens beam deflectors
Projector lens
Screen
Eric Stach (2008), ”MSE 528 Lecture 4: The instrument,
Part 1, http://nanohub.org/resources/3907
The requirements of the illumination system
• High electron intensity
– Image visible at high magnifications
• Small energy spread
– Reduce chromatic aberrations effect in obj. lens
• Adequate working space between the illumination
system and the specimen
• High brightness of the electron beam
– Reduce spherical aberration effects in the obj. lens
Brightness
• Brightness is the current density per unit solid
angle of the source
• β = ie/(πdcαc)2
The electron source
• Two types of emission sources
– Thermionic emission
• W or LaB6
– Field emission
• W Cold FEG ZnO/W Schottky FEG
The electron gun
• The performance of the gun is characterised by:
–
–
–
–
Beam diameter, dcr
Divergence angle, αcr
Beam current, Icr
Beam brightness, βcr
at the cross over
d
Cross over
α
Image of source
The electron gun
Thermionic gun
FEG
Wehnelt
cylinder
Bias -200 V
Cathode
-200 kV
Equipotential lines
Anode
Ground potential
dcr Cross over
αcr
Thermionic guns
Filament heated to give
Thermionic emission
-Directly (W) or
indirectly (LaB6)
Filament negative
potential to ground
Wehnelt produces a
small negative bias
-Brings electrons to
cross over
Thermionic guns
Thermionic emission
• Current density:
Jc= AcT2exp(-φc/kT)
Richardson-Dushman
–
–
–
–
Ac: Richardson’s constant, material dependent
T: Operating temperature (K)
φ: Work function (natural barrier to prevent electrons to leak out from the surface)
k: Boltzmann’s constant
Maximum usable temperature T is determined
by the onset of the evaporation of material.
Field emission
• Current density:
Maxwell-Boltzmann
energy distribution
for all sources
Fowler-Norheim
Field emission
• The principle:
– The strength of an electric field E is considerably increased
at sharp points.
E=V/r
• rW < 0.1 µm, V=1 kV → E = 1010 V/m
– Lowers the work-function barrier so that electrons can
tunnel out of the tungsten.
• Surface has to be pristine (no contamination or oxide)
– Ultra high vacuum condition (Cold FEG) or poorer vacuum if tip is
heated (”thermal” FE; ZrO surface tratments → Schottky emitters).
Characteristics of principal electron sources at 200 kV
LaB6
FEG Schottky
(ZrO/W)
FEG cold (W)
Current density Jc (A/m2)
2-3*104
25*104
1*107
Electron source size (µm)
50
10
0.1-1
0.010-0.100
Emission current (µA)
100
20
100
20~100
Brightness B (A/m2sr)
5*109
5*1010
5*1012
5*1012
Energy spread ΔE (eV)
2.3
1.5
0.6~0.8
0.3~0.7
Vacuum pressure (Pa)*
10-3
10-5
10-7
10-8
Vacuum temperature (K)
2800
1800
1800
300
* Might be one order lower
Advantages and disadvantages of the
different electron sources
W Advantages:
LaB6 advantages:
FEG advantages:
Rugged and easy to handle
High brightness
Extremely high brightness
Requires only moderat
vacuum
High total beam current
Long life time, more than
1000 h.
Good long time stability
Long life time (500-1000h)
High total beam current
W disadvantages:
LaB6 disadvantages:
FEG disadvantages:
Low brightness
Fragile and delicate to
handle
Very fragile
Limited life time (100 h)
Requires better vacuum
Current instabilities
Long time instabilities
Ultra high vacuum to
remain stable
Electron lenses
Any axially symmetrical electric or magnetic field have the properties
of an ideal lens for paraxial rays of charged particles.
• Electrostatic
F= -eE
– Require high voltage- insulation problems
– Not used as imaging lenses, but are used in modern monochromators
• Magnetic
– Can be made more accurately
– Shorter focal length
F= -e(v x B)
General features of magnetic lenses
• Focus near-axis electron rays with the same accuracy as a glass lens
focusses near axis light rays
• Same aberrations as glass lenses
• Converging lenses
• The bore of the pole pieces in an objective lens is about 4 mm or less
• A single magnetic lens rotates the image relative to the object
• Focal length can be varied by changing the field between the pole pieces.
(Changing magnification)
http://www.matter.org.uk/tem/lenses/electromagnetic_lenses.htm
Strengths of lenses and focused image of the source
http://www.rodenburg.org/guide/t300.html
If you turn up one lens (i.e. make it stronger, or ‘over- focus’ then
you must turn the other lens down (i.e. make it weaker, or ‘underfocus’ it, or turn its knob anti-clockwise) to keep the image in focus.
Magnification of image,
Rays from different parts of the object
http://www.rodenburg.org/guide/t300.html
If the strengths (excitations) of the two lenses are changed, the
magnification of the image changes
The transmission electron
microscope
Chapter 2 The TEM (part 2)
Chapter 3 Electron Optics
Some repetition
• What characterizes the performance of an
electron gun?
• What kind of electron sources are used in EM?
• What kind of lenses can be used in a TEM?
• In what way does the trajectory of an electron
differ from an optical ray through a lens?
• What are the deflection coils used for?
• What is the focal length for a lens and how can it
be changed in the TEM?
The Objective lens
• Often a double or twin lens
• The most important lens
– Determines the reolving power of the TEM
• All the aberations of the objective lens are magnified by
the intermediat and projector lens.
• The most important aberrations
– Asigmatism
– Spherical
– Chromatical
Astigmatism
Can be corrected for with stigmators
The objective lens
• Cs can be calculated from information about
the shape of the magnetic field
– Cs has ~ the same value as the focal length (see
table 2.3)
• The objective lens is made as strong as possible
– Limitation on the strength of a magnetic lens with an iron core
(saturation of the magnetization Ms)
– Superconductiong lenses (gives a fixed field, needs liquid
helium cooling)
Apertures
Use of apertures
Condenser aperture:
Limit the beam divergence (reducing the diameter of the discs in the
convergent electron diffraction pattern).
Limit the number of electrons hitting the sample (reducing the intensity),
.
Objective aperture:
Control the contrast in the image. Allow certain reflections to contribute to
the image. Bright field imaging (central beam, 000), Dark field imaging (one
reflection, g), High resolution Images (several reflections from a zone axis).
A.E. Gunnæs
MENA3100 V08
Objective aperture: Contrast enhancement
Bright field (BF)
glue
hole
(light elements)
Ag and Pb
Objective
aperture
Si
BF image
All electrons contributes to the image. Only central beam contributes to the image.
Small objective aperture
Bright field (BF), dark field (DF) and weak-beam (WB)
Objective
aperture
BF image
DF image
Diffraction contrast
Weak-beam
Dissociation of pure screw dislocation In Ni3Al, Meng
and Preston, J. Mater. Scicence, 35, p. 821-828, 2000.
Large objective aperture
High Resolution Electron Microscopy (HREM)
HREM image
Phase contrast
Use of apertures
Condenser aperture:
Limit the beam divergence (reducing the diameter of the discs in the
convergent electron diffraction pattern).
Limit the number of electrons hitting the sample (reducing the intensity),
.
Objective aperture:
Control the contrast in the image. Allow certain reflections to contribute to
the image. Bright field imaging (central beam, 000), Dark field imaging (one
reflection, g), High resolution Images (several reflections from a zone axis).
Selected area aperture:
Select diffraction patterns from small (> 1µm) areas of the specimen.
Allows only electrons going through an area on the sample that is limited by
the SAD aperture to contribute to the diffraction pattern (SAD pattern).
Selected area diffraction
Parallel incoming electron beam
Specimen with two crystals (red and blue)
Objective lense
Pattern on the screen
Diffraction pattern
Image plane
Selected area
aperture
Diffraction with no apertures
Convergent beam and Micro diffraction (CBED and µ-diffraction)
Convergent beam
Focused beam
C2 lens
Convergent beam
Illuminated area less
than the SAD aperture
size.
Small probe
CBED pattern
Diffraction information from an area with
~ same thickness and crystal orientation
µ-diffraction pattern
Shadow imaging
(diffraction mode)
Parallel incoming electron beam
Sample
Objective lense
Diffraction plane
(back focal plane)
Image plane
Magnification and calibration
Resolution of the photographic emulsion: 20-50 µm
Magnification depends on specimen position in the objective lens
Microscope
Lens
Mode
Magnification
JEM-2010
Objective
MAG
2 000-1 500 000
LOW MAG
50
-
Twin
TEM
25
- 750 000
Super twin
TEM
25
- 1 100 000
Twin
SA
3 800 - 390 000
Super twin
SA
5 600 - 575 000
Philips CM30
6 000
Magnification higher than 100 000x can be calibrated by using lattice images.
Rotation of images in the TEM.
The imaging and recording system
Fluoresent screen consisting of ZnS or
ZnS/CdS powder
Fine grained photographic film or/and
imaging plates
TV or CCD
Specimen holders and goniometers
• Specimen holders
– Rotation holders
– Double tilt holders
– Heating holders
• Up to 800oC
– Cooling holders
• N: -100 - -150oC
• He: 4-10K
– Strain holders
– Environmental cells
• Goniometers:
- Side-entry stage
- Most common type
- Eucentric
- Top-entry stage
- Less obj. lens aberrations
- Not eucentric
- Smaller tilting angles
Stereomicroscopy
A perception of depth can be obtained
if an object is seen along two slightly
different directions.
TEM: Two micrographs taken with
slightly different orientation (~7 o) are
looked at in a steroscopic viewer.
Ray diagram
δ
x
ΔZ= p/(2M sin(θ/2))
θ
ΔZ
p: Parallax, M: total magnification
θ/2
S1
p= M(S1-S2)
S2
Main use of stereoscopic TEM:
-Quatitative visualizing the depth distribution of structural features in the specimen
-Determination of specimen thickness
-Quantitative determinaton of spatial distribution of defects and particles
Electron optics continuation
• Lens formula: 1/f = 1/v + 1/u
– Valid only for monochromatic radiation and
beams close to the optical axis.
– Geometrical approximation : sin θ ~ θ.
– If the approximation is not valid a series of lens aberrations occur.
– Better approximation for larger θ: sinθ ~ θ- θ3/3!
• Five lens aberrations:
– Spherical aberration, coma, astigmatism, curvature of field and
distortion.
– If the radiation is not monochromatic there will in addition be lateral
and longitudal chromatic aberrations
• See fig. 3.1 textbook
Spherical aberration
Gaussian image plane
r2
α r1
Highest intensety in the
Gaussian image plane
Plane of least confusion
ds = 0.5MCsα3 (disk diameter, plane of least confusion)
ds = 2MCsα3 (disk diameter, Gaussian image plane)
M: magnification
Cs :Spherical aberration coefficient
α: angular aperture/
angular deviation from optical axis
2000FX: Cs= 2.3 mm
2010F: Cs= 0.5 nm
Chromatic aberration
Diameter for disc of least confusion:
dc = Cc α ((ΔU/U)2+ (2ΔI/I)2 + (ΔE/E)2)0.5
v
v - Δv
Cc: Chromatic aberration coefficient
α: angular divergence of the beam
U: acceleration voltage
I: Current in the windings of the objective lens
E: Energy of the electrons
Thermally emitted electrons: ΔE/E=kT/eU,
Disk of least confusion
LaB6: ~1 eV
The specimen will introduce chromatic aberration.
2000FX: Cc= 2.2 mm
2010F: Cc= 1.0 mm
The thinner the specimen the better!!
Correcting for Cc effects only makes sense if you
are delaing with specimens that are thin enough.
Lens astigmatism
Due to non-uniform magnetic field
as in the case of non-cylindrical lenses.
Apertures may affect the beam if not
precisely centered around the axis.
This astigmatism can not be
prevented, but it can be
corrected!
y
• Loss of axial asymmetry
x
y-focus
x-focus
Disk of least confusion
Diameter of disk of least confusion:
da: Δfα
Depth of focus and depth of field (image)
• Imperfection in the lenses limit the resolution but gives a
better depth of focus and depth of image.
– Use of small apertures to minimize their aberration.
• The depth of field (Δb eller Dob) is measured at, and refers to,
the object.
– Distance along the axis on both sides of the object plane within which
the object can move without detectable loss of focus in the image.
• The depth of focus (Δa, or Dim), is measured in, and referes to,
the image plane.
– Distance along the axis on both sides of the image plane within which
the image appears focused.
Depth of focus and depth of field (image)
1
dob
1
2
dim
2
βim
αim
Dim
Dob
Ray 1 and 2 represent the extremes of the ray paths that remain in
focus when emerging ± Dob/2 either side of a plane of the specimen.
αim≈ tan αim= (dim/2)/(Dob/2)
Angular magnification: MA= αim/ βob
Transvers magnification: MT= dim/ dob
βob≈ tan βob= (dob/2)/(Dim/2)
MT= 1/MA
Depth of focus: Dim=(dob/ βob)MT2 Depth of field: Dob= dob/ βob
Depth of field
Depth of field: Dob= dob/ βob
Carefull selection of βob
• Thin sample: βob ~10-4 rad
• Thicker, more strongly scattereing specimen: βob (defined by
obj. aperture) ~10-2 rad
Example: dob/ βob= 0.2 nm/10 mrad = 20 nm
Example: dob/ βob= 2 nm/10 mrad = 200 nm
Dob= thickness of sample
all in focuse
Depth of focus
Depth of focus: Dim=(dob/ βob)MT2
Example: To see a feature of 0.2 nm you would use a
magnification of ~500.000 x
(dob/ βob)M2= 20 nm *(5*105)2= 5
km
Example: To see a feature of 2 nm you would use a
magnification of ~50.000 x
(dob/ βob)M2= 2 nm *(5*104)2= 5
m
Focus on the wieving screen
and far below!
Fraunhofer and Fresnel diffraction
• Fraunhofer diffraction: far-field diffraction
– The electron source and the screen are at infinit distance
from the diffracting specimen.
• Flat wavefront
• Fresnel diffraction: near-field diffraction
– Either one or both (electron source and screen) distances
are finite.
Electron diffraction patterns correspond closely to the Fraunhofer case
while we ”see” the effect of Fresnel diffraction in our images.
Airy discs (rings)
• Fraunhofer diffraction from a circular aperture will give a
series of concentric rings with intesity I given by:
I(u)=Io(JI(πu)/ πu)2
http://en.wikipedia.org/wiki/Airy_disk
Interaction between electrons
and the specimen
Elastic and inelastic scattering
Analytical methods
Electron scattering
• What is the probability that an electron will be scattered when it passes
near an atom?
– The idea of a cross section, σ
• If the electron is scattered, what is the angle through which it is deviated?
• What is the average distance an electron travels between scattering
events?
– The mean free path, λ
• Does the scattering event cause the electrons to lose energy or not?
– Distinguishing elestic and inelastic scattering
Some definitions
• Single scattering: 1 scattering event
• Plural scattering: 1-20 scattering events
• Multiple scattering: >20 scattering events
• Forward scattered: scattered through < 90o
• Bacscattered: scattered through > 90o
Energy distribution of BSE-SE
C
• Region A: BSE that have lost less
than 50% of E0.
• Region B: BSE which travel greater
distances, losing more energy within
the specimen prior to backscattering.
A
N(E)
B
0
E/Eo
• Region C: at very low energy, below 50 eV, the number of electrons
emitted from the specimen increases sharply. This is due to emission of
secondary electrons.
Backscattered electron coefficient η :
η=numer of BSE/number of primary electrons
BSE used in SEM
1.0
BE coefficient
Goldstein 11, Newbury DE, Echlin P,
Joy DC, Romig Jr D, Lyman CE, et al.
Scanning electron microscopy and
X-ray microanalysis. 2nd ed. New York:
Plenum Press; 1992. 819 p.
Multi element phase:
Ci : mass fractions
Contrast between constituents in BSE images in
the SEM can be calculated as:
The angle of scattering
Scattering fro a single isloated atom
The scattering angle θ is a
semi-angle, and not a total
angle of scattering.
Incident beam
The total solid angle is Ω.
The solid angle Ω is the two-dim angle
in three dimensional space that an
object subtends at a point.
It is a measure of how large that object
apears to an observer looking form that point.
Scattered electrons
θ
dθ
θ is often assumed to be small
sinθ ≈ tanθ ≈ θ
Ω
dΩ
Small angle: <10 mrad (~10o)
Unscattered
electrons
Interaction cross section and
its differental
The chance of a particular electron undergoing any kind of interaction with
an atom is determined by an interaction cross section (an area).
Cross section represents the probability that a scattering event will occure.
σatom=πr2
r has different value for each scattering process.
It is of interest to know wheter or not the scattering process deviates the incident
beam electrons outside a particular scattering angle θ such that, e.g., they do
not go through the aperture in the lens or they miss the electron detector.
The differential cross section dσ/dΩ describes the angular distribution of scattering
from an atom, and is a measure of the probability for scattering in a solid angle dΩ.
Scattering form the specimen
Total scattering cross section/The number of scattering events
per unit distance that the electrons travels through the specimen:
σtotal=Nσatom= Noσatom ρ/A
N= atoms/unit volume
No: Avogadros number, ρ: density of pecimen,
A: atomic weight of the scattering atoms
If the specimen has a thickness t the probability of scattering
through the specimen is:
tσtotal=Noσatom ρt/A
Mean free path λ
The mean free path for a scattering process is the average
distance travelled by the primary particle between
scattering events.
λ = 1/σtotal = A/Noρσatom
Material
10kV
20kV
30kV
40kV
50kV
100kV
200kV
1000kV
C (6)
5.5
22
49
89
140
550
2200
55000
Al (13)
1.8
7.4
17
29
46
180
740
18000
Fe (26)
0.15
0.6
2.9
5.2
8.2
30
130
3000
Ag (47)
0.15
0.6
1.3
2.3
3.6
15
60
1500
Pb (82)
0.08
0.34
0.76
1.4
2.1
8
34
800
U (92)
0.05
0.19
0.42
0.75
1.2
5
19
500
Mean free path (nm) as a function of acceleration voltage for elastic electron
scattering more than 2o.
Electron scattering
The probability of scattering is described in terms of either
an “interaction cross-section” or a mean free path.
Mote Carlo simulations: http://www.matter.org.uk/TEM/electron_scattering.htm#
• Elastic
– The kinetic energy is unchanged
– Change in direction relative to incident electron beam
• Inelastic
– The kinetic energy is changed (loss of energy)
– Energy form the incident electron is transferred to the electrons and
atoms in the specimen
Elastic scattering
• Major source of contrast in TEM images
• Scattering from an isolated atom
– From the electron cloud: few degrees of angular deviation
– From the positive nucleus: up to 180o
Elastic scattering process
• Rutherford scattering (Coulomb scattering)
– Coulomb interaction between incident electron and the electric charge of the
electron clouds and the nuclei.
– Elastic scattering
Differential scattering cross section
i.e. the probability for scattering in a solid angle dΩ:
Solid angle:
Ω= 2π(1- cosθ)
dσ/dΩ = 2πb (db/dΩ)
b=
(Ze2/4πεomv2)cotanθ/2
Impact parameter: b
dσ/dΩ = -(mZe2λ2/8πεoh2)2(1/sin4θ/2)
A diagram of a scattering process
http://en.wikipedia.org/wiki/File:ScatteringDiagram.svg
Inelastic scattering processes
• Ionization of inner shells
– Auger electrons
– X-rays
– Light
Localized processes
• Continuous X-rays/Bremsstrahlung
• Exitation of conducton or valence electrons
• Plasmon exitation
• Phonon exitations
Collective oscillations
Non- localized
Non- localized
SE
Ionization of inner shells
Electron
Auger electron or
x-ray
Valence
M
M
3d6
3p4
3d4
L
3s2
2
2p4
3p
2p2
2s2
1s2
Electron
shell
K
L
K
Characteristic x-ray emitted or Auger
electron ejected after relaxation of inner
state.
Low energy photons (cathodoluminescence)
when relaxation of outer stat.
Auger electrons or x-rays
EELS?
X-ray spectrum
Fluorescence
x-ray
x-ray
M
L
K
Photo electron
Continuous and
characteristic x-rays
The cut-off energy for
continous x-rays corresponds
to the energy of the incident
electrons.
Continous x-rays du to
deceleration of incident
electrons.
http://www.emeraldinsight.com/journals.htm?articleid=1454931&show=html
Secondary electrons
Secondary electrons (SEs) are electrons within
the specimen that are ejected by the beam electrons.
Electrons from the conduction or valence band.
E ~ 0 – 50 eV
Auger electrons
The secondary emission coefficient:
δ=number of secondary electrons/numbers of primary electrons
Dependent on acceleration voltage.
Cathodoluminescence
Conduction band
Valence band
Plasmon excitations
The incoming electrons can interact with electrons in the ”electron gas”
and cause the electron gas to oscillate.
The oscillations are called plasmons.
Plasmon frequency: ω=((ne2/εom))1/2
Energy: Ep=(h/2π)ω
Ep~ 10-30 eV, λp,100kV ~150 nm
n: free electron density, e: electron charge, εo: dielectric constant, m: electron mass
Phonon excitation
Equivalent to specimen heating
Energy losses ~ 0.1 eV
The effect in the diffraction patterns:
-Reduction of intensities (Debye-Waller factor)
-Diffuce bacground between the Bragg reflections
EELS
Sum of several losses
Thin specimens
Summary
Electron diffraction geometry
z
c
β
a
x
α
γ
b
Lattice properties of crystals
Bravais lattices
Lattice planes and directions
Resiprocal lattice
The Laue condition
y
The Bragg condition
The Ewald sphere construction
Differences between x-ray and ED
Zone axis and Laue zones
………….
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.
Point lattice
Repeating
Lattice element
point
in the example
Crystals have a periodic internal structure
Repeting element
1
2
3
What is the repeting element in example 1-3?
Repeting element
1
2
3
Enhetscellen: repetisjonsenheten
1
2
Valgfritt origo!
3
Point lattice
repeting element
unit cell
Atoms and lattice points situated on corners, faces and edges
are shared with neighbouring cells.
Unit cell Elementary unit of volume!
– The smallest building blocks.
– The whole lattice can be described by repeating a unit cell in all three
dimensions.
- Defined by three non planar lattice
vectors:
a, b and c
c
α
β
γ
b
-or by the length of the vectors a, b and c
and the angles between them (alpha,
beta, gamma).
a
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)
z
c
β
α
γ
a
x
b
y
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 described
by 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)
Macroscopic symmetry elements
Crystals can be classified based on symmetry without taking
into account their translation symmetry.
Macroscopic symmetry element
Hermann-Mauguin symbol
1, 2, 3, 4 and 6-fold rotation
1, 2, 3, 4 and 6
Plane of symmetry
m for mirror plane
Rotation-inversion axes
1, 2, 3, 4 and 6
Center of symmetry
1
Acting at a point
since no translations are involved in the symmetry operation.
32 point groups or crystal classes
Listed in table 5.4
Microscopic symmetry elements
Symmetry elements involving translation within the unit cell
Glide plane and screw axes
If we take into account all symmetry elements
(macroscopic and microscopic) crystals can be
classified according to 230 space groups
Crystal classification and data
• Crystals can be classified
according to 230 space groups.
– A space group can be referred
to by a number or the space
group symbol (ex. Fm-3m is
nr. 225)
• Structural data for known
crystalline phases are available in
books like “Pearson’s handbook
of crystallographic data….” but
also electronically in databases
like “Find it”.
• Details about crystal description
can be found in International
Tables for Crystallography.
• Pearson symbol like cF4 indicate
the axial system (cubic), centering
of the lattice (face) and number
of atoms in the unit cell of a
phase (like Cu).
– Criteria for filling Bravais
point lattice with atoms.
– Both paper books and online
Lattice planes
z
• 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).
c/l
a/h
0
b/k
y
x
– The reciprocal of the interceptions are
rationalized if necessary to avoid
fraction numbers of (h k l) and 1/∞ = 0
Z
(110)
Y
– Planes are often described by their
normal
Z
Z
(001)
X
(111)
(010)
Y
– (hkl) one single set of parallel planes
– {hkl} equivalent planes
Y
(100)
X
X
Hexagonal axial system
a1=a2=a3
γ = 120o
a2
a1
a3
(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.
wc
[uvw]
c
ua
• 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
z
a
b
vb
y
x
Resiprocal lattice
Important 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:
a *  (b  c) / V
b *  (c  a ) / V
c  ( a  b) / V
*
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.
V: Volume of the unit cell
V=a.(bxc)=b.(cxa)=c.(axb)
Orthogonal axes:
a* = 1/IaI, b*=1/IbI, c*=1/IcI
Reciprocal vectors, planar distances
–The resiprocal vector
g hkl  h a *  k b*  l c *
is normal to the plane (hkl).
Convince your self !
What is the dot product beteen the
normal to a (hkl) plane with a vector
In the (hkl) plane?
and
the spacing between the
(hkl) planes is given by
d hkl  1 / g hkl
Unit normal vector: n= ghkl/IghklI
• Planar distance (d-value)
between planes {hkl} in a cubic
crystal with lattice parameter a:
d hkl 
a
h2  k 2  l 2
Scattering from two lattice points
• Path difference for waves scattered
from two lattice points separated by
a vector r.
• The path difference is the difference
between the projection of r on k’
and the projection of r on k.
• The scattered waves will be in phase
and constructive interference will
occur if the phase difference is 2π.
Phase difference: φ= 2πr.(k’-k)
Constructive interference when φ= 2πn
Two lattice points
separated by a
vector r
k
r
k’
The Laue condition
nhkl= r*hkl
Two lattice points
separated by a
vector r
r*hkl = k’ – k ?
φ= 2πr.(k’-k) = 2πr.r*hkl
k
φ= 2π(ua+vb+wc).(ha*+kb*+lc*)
φ= 2π(uh+vk+wl)
nhkl
r
k’
Maximum intensity if h,k,l are integers
a.(k’-k) =h
b.(k’-k) =k
c.(k’-k) =l
(hkl)
Laue condition
The scattering vector must be oriented in a
specific direction in relation to the primitive
vectors of the crystal lattice.
Bragg’s law
r*hkl
k’
k
2θB
IkI = Ik’I = 1/λ
Ik’- kI = (2/λ) sinθB
d hkl  1 / g hkl
r*hkl = k’ – k
1/dhkl= (2/λ) sinθB
λ= 2dhklsinθB
y
θ
d
x
θ
The path difference: x-y
Y= x cos2θ and x sinθ=d
cos2θ= 1-2 sin2θ
• nλ = 2dsinθ
– Planes of atoms responsible
for a diffraction peak behave
as a mirror
The Ewald Sphere is flat (almost)
Cu Kalpha X-ray:  = 150 pm => small k
Electrons at 200 kV:  = 2.5 pm => large k
ED and form effects
Figure 5.7 and 5.8
Real space
Resiprocal space
Zone axis and Laue zones
Excitation error:
sg
Zone axis [uvw]
(hkl)
uh+vk+wl= 0
Lattice plane spacings and
camera constant
R=L tan2θB ~ 2LsinθB
2dsinθB =λ
↓
R=Lλ/d
Film plate
Indexing diffraction patterns
The g vector to a reflection is normal to the
corresponding (h k l) plane and IgI=1/dnh nk nl
(h2k2l2)
-
Measure Ri and the angles between
the reflections
-
Calculate di , i=1,2,3
-
Compare with tabulated/theoretical
calculated d-values of possible phases
-
Compare Ri/Rj with tabulated values for
cubic structure.
-
g1,hkl+ g2,hkl=g3,hkl (vector sum must be ok)
-
Perpendicular vectors: gi ● gj = 0
Orientations of corresponding
planes in the real space
(=K/Ri)
Zone axis: gi x gj =[HKL]z
All indexed g must satisfy: g ● [HKL]z=0
Determination of the Bravais-lattice of an
unknown crystalline phase
Tilting series around common axis
27o
50 nm
15o
10o
0o
Determination of the Bravais-lattice of an unknown
crystalline phase
Tilting series around a dens row of
reflections in the reciprocal space
0o
50 nm
19o
Positions of the
reflections in the
reciprocal space
25o
40o
52o
Bravais-lattice and cell parameters
011
111
001
c
101
b
010
a
110
100
[011]
[100]
[101]
d=Lλ/R
6.04 Å
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
7.94 Å
α= β= γ= 90o
Chemical analysis by use of EDS and EELS
Ukjent fase
BiFe2O5
BiFeO3
CCD
counts
x 1000
CCD
counts
x 1000
1_1evprc.PICT
Nr_2_1evprc.PICT
40
14
35
12
O-K
Fe - L2,3
30
10
25
8
20
6
BiFeO3
Ukjent fase
4
15
2
10
-0
5
500
1 eV pr.600
kanal 800
-0 eV forskyvning,
200
400
-0
200
Energy
(eV)
400 Loss 600
Energy Loss (eV)
800
1000
1000
Published structure
A.G. Tutov og V.N. Markin
The x-ray structural analysis of the antiferromagnetic Bi2Fe4O9 and the isotypical combinations Bi2Ga4O9 and Bi2Al4O9
Izvestiya Akademii Nauk SSSR, Neorganicheskie Materialy (1970), 6, 2014-2017.
Romgruppe: Pbam nr. 55,
celleparametre: 7,94 Å, 8,44 Å, 6.01Å
Bi
O
Bi
O
Bi
Fe
Fe
O
O
O
O
4g
4h
4f
4g
8i
4h
2b
x
0,176
0,349
0
0,14
0,385
0,133
0
y
0,175
0,333
0,5
0,435
0,207
0,427
0
O
Bi
z
0
0,5
0,244
0
0,242
0,5
0,5
O
Bi
Fe
O
O
Fe
Fe
O
O
Fe O
O
Fe
O
O
Fe
O
Fe
O
O
Fe
O
Fe
O
O
O
Fe
Fe
O
O
Fe
Bi
O
Bi
O
O
Bi
O
Bi
c
b
Po wd erCell 2 .0
a
Celle parameters found with electron diffraction (a= 6,04 Å, b= 7.94 Å and c=8.66 Å) fits reasonably well with the
previously published data for the Bi2Fe4O9 phase. The disagreement in the c-axis may be due to the fact that we
have been studying a thin film grown on a crystalline substrate and is not a bulk sample. The conditions for
reflections from the space group Pbam is in agreement with observations done with electron diffraction.
Conclusion: The unknown phase has been identified as Bi2Fe4O9 with space group Pbam with cell parameters a=
6,04 Å, b= 7.94 Å and c=8.66 Å.
Kinematical theory
Solutions of the Schrödinger
equation
Schrödinger equation
HΨ(r) = EΨ(r)
Ψ(r) Wave function
E = eU
Try solutions
Total energy of the electrons
H= p2/2m – eV(r)
P = (h/2πi) 
Kinetic
Potential
p2/2m = (mv)2/2m= ½ mv2
2 – eV(r)) Ψ(r) = eU Ψ(r)
((-h2/4π2m) 
Transmission through a thin specimen
2 – eV(r)) Ψ(r) = eU Ψ(r)
((-h2/4π2m) 
Vaccum; eV(r)=0
Try solution: Ψ(r) = Ψo exp(2πikr)
(h2/2m)k2 Ψ(r)= eU Ψ(r)
Solution if:
k= 1/λ =(2meU/h2)1/2
Constant potensial; eV(r)=eV0
Ψ(r) = Ψo exp(2πik’r)
((h2/2m)k2 –eV0 )Ψ(r)= eU Ψ(r)
Solution if:
k’= 1/λ =(2me(U+V0)/h2)1/2
V0<< U k’= (k2(1+V0/U))1/2 ~ k (1+ V0/2U)
k’ = k + V0/2λU= k + (σ/2π) V0
Scrödinger equation
2 – eV(r)) Ψ(r) = eU Ψ(r)
((-h2/4π2m) 
2 /4π– k2) Ψ(r) = -U(r) Ψ(r)
(
K2 = 2meU/h2
U(r) = 2meV(r)/h2
Modified potential
Solution of the Scrödinger equation
Ψ(r) = exp(2πik.r) + (πexp(2πikr)/r) f(θ)= Ψi + Ψs
f(θ)=∫exp(-2πik’.r’)U(r’) Ψ(r’) dV’
Weak scattering: 1. Born or kinematical approximation:
Ψ(r’) = exp(2πik.r’) Plane incoming wave
fB(θ)= ∫exp(-2πis.r’)U(r’) dV’
s = k’ - k
Scattered wave:
Ψs = (πexp(2πikr)/r) ∫exp(-2πis.r’)U(r’) dV’
U(r’) = 2meV(r’)/h2
Solution of the Scrödinger equation
Scattered wave:
Ψs = (πexp(2πikr)/r) ∫exp(-2πis.r’)U(r’) dV’
fB(θ)= ∫exp(-2πis.r’)U(r’) dV’
U(r’) = 2me V(r’) /h2
s=k’-k
IsI =2sinθ/λ
X-ray scattering factor
Scattering from an atom: fB(θ)= me2/8πεoh2)(λ/sinθ)2[Z –fx]
Scattering from a Coulomb potential: fB(θ)= mZe2λ2/8π2εoh2sin2θ
Scattering cross section: σ(θ)=IfB(θ)I2 = Ze2/8πεomv2)2(1/sin4θ) ≈ 1/θ4
Rutherford scattering equation
Scattering from one unit cell
Scattering from one atom: Ψs(r)= (πexp(2πikr)/r) f(θ)
Scattering from two atoms:
k
o
rj
k’
Phase difference between rays from two atoms:
2π(k’-k)rj = 2πs.rj
At large distance from O the amplitude is:
Ψs(r)= (πfo(θ)/r) exp(2πikr)+ (πfj(θ)/r) exp(2πikr)exp 2πs.rj
Scattering from all the atoms in the unit cell:
Ψs(r)= (π/r) exp(2πikr) Σfj(θ)exp(2πs.rj ) = (π/r) exp(2πikr) F(θ)
Structure factor: F(θ)
Ψs(r)= (π/r) exp(2πikr) Σfj(θ)exp(2πs.rj ) = (π/r) exp(2πikr) F(θ)
F(θ)= Σfj(θ)exp(2πs.rj ) The structure factor of the unit cell
j
Scattering from all unit cells in the crystal:
.r ),
Ψs(r)= Σ(π/r)
exp(2πikr)
f
(θ)exp(2πs
i
i
i
ri = rn + rj
Ψs(r)= (π/r) exp(2πikr)Σn fn(θ)exp(2πs.rn ) Σ fj(θ)exp(2πs.rj )
j
The same for all unit cells
O
rj
rn
Scattering from all unit cells in the crystal
Ψs(r)= (π/r) exp(2πikr)Σ fn(θ)exp(2πis.rn ) Σ fj(θ)exp(2πis.rj )
Ψs(r)= (π/r) exp(2πikr)F(θ) Σ fn(θ)exp(2πis.rn ),
s= ra*+ sb*+ tc*,
When is Ψs(r)≠0?
F(θ)= Σ fj(θ)exp(2πis.rj )
rn=ua + vb + wc
(ur+vs+wt)=integer →for all u,v,w if r, s, t are integers
s is a resiprocal lattice vector
Ψs(r)= (π/r) exp(2πikr)F(θ) Σ fn(θ)exp(2πig.rn )
≠0
F(θ)=Fg= Σ fj(θ)exp(2πig.rj ) =Σ fj(θ)exp(2πi (hxj + kyj + lzj))
Condition for possible reflections
F(θ)=Fg= Σ fj(θ)exp(2πi (hxj + kyj + lzj)) ≠ 0
Allowed and forbidden reflections
•
Bravais lattices with centering (F, I, A,
B, C) have planes of lattice points that
give rise to destructive interference
for some orders of reflections.
y’y
θ
– Forbidden reflections
In most crystals the lattice point
corresponds to a set of atoms.
Different atomic species scatter more
or less strongly (different atomic
scattering factors, fzθ).
29/1-08
MENA3100
d
x’
x
θ
From the structure factor of the unit
cell one can determine if the hkl
reflection it is allowed or forbidden.
Structure factors
N
Fg  Fhkl   f j( e) exp( 2 i (hu j  kv j  lw j ))
X-ray:
j 1
The coordinate of atom j within the crystal unit cell is given rj=uja+vjb+wjc.
h, k and l are the miller indices of the Bragg reflection g. N is the number of atoms
within the crystal unit cell. fj(n) is the x-ray scattering factor, or x-ray scattering
amplitude, for atom j.
wjc
c
a
uja
z
rj
b
vjb
y
x
The intensity of a reflection is
Fg Fg
proportional to:
29/1-08
MENA3100
Example: Cu, fcc
•
•
•
N
eiφ =
cosφ + isinφ
enπi = (-1)n
eix + e-ix = 2cosx
Fg  Fhkl   f j exp( 2 i (hu j  kv j  lw j ))
j 1
Atomic positions in the unit cell:
[000], [½ ½ 0], [½ 0 ½ ], [0 ½ ½ ]
Fhkl= f (1+ eπi(h+k) + eπi(h+l) + eπi(k+l))
What is the general condition
for reflections for fcc?
What is the general condition
for reflections for bcc?
29/1-08
If h, k, l are all odd then:
Fhkl= f(1+1+1+1)=4f
If h, k, l are mixed integers (exs 112) then
Fhkl=f(1+1-1-1)=0 (forbidden)
MENA3100
Simplified kinematical theory for perfect crystals
Wavefunction at exit surface:
Ψ(r) =exp(2πikr)exp(iσV0Δz)
σ=2πmλe/h2
(Ψo=Ψo’=1)
Ψ(r) =Ψoexp(2πikr)exp(2πimeλV(r)dz/h2)
6.27
6.16
6.139
This sample
Taylor series:
Ψ(r) =Ψoexp(2πikr) (1+2πimeλV(r)dz/h2)
The specimen has a periodic structure
V(r)=ΣVg exp(2πig.r)
Fourier expansion of crystal potential
Ψ(r) =Ψoexp(2πikr) (1+2πimeλdz/h2) ΣVg exp(2πig.r)
Simplified kinematical theory for perfect crystals
Ψ(r) =Ψoexp(2πikr) (1+2πimeλdz/h2) ΣVg exp(2πig.r) = Ψi + Σ Ψs
Ψ(r) =Ψoexp(2πikr) + ΣdΨgexp(2πik’.r)
dΨgexp(2πik’.r)=(2πimeλdz/h2)Ψo Vg exp(2πi(k+g).r), introduce:k’=k+g+sg
Simplified kinematical theory for perfect crystals
k
k’
k
k’
g
g
sg
k’=k+g+sg
Simplified kinematical theory for perfect crystals
Ψ(r) =Ψoexp(2πikr) (1+2πimeλdz/h2) ΣVg exp(2πig.r) = Ψi + Σ Ψs
Ψ(r) =Ψoexp(2πikr) + ΣdΨgexp(2πik’.r)
dΨgexp(2πik’.r)=(2πimeλdz/h2)Ψo Vg exp(2πi(k+g).r), introduce:k’=k+g+sg
dΨg=(2πimeλdz/h2)Ψo Vg exp(-2πisgz)dz
= (πi/ξg)Ψo exp(-2πisgz)dz
Extinction distance ξg:
ξg=h2/2meλVg
ξg=1/λUg =V/λFg
Modified potential: U(r) = 2me V(r) /h2
U(r) = ΣUgexp(2πig.r)
Ug=Fg/V,
V:unit cell volume
Simplified kinematical theory for perfect crystals
Basis of kinematical theory of electron diffraction for imperfect crystals:
t
Ψg(t)= ∫(πi/ξg) exp(-2πisgz)dz,
Ψo=1, t: crystal thickess
0
Ψg(t)= (i/ξgsg) exp(-πitsg) sinπsgt
Intensity of the scattered beam g (dark field):
Ig= l Ψg(t) l2= sin2 πsgt/(ξgsg)2
Intensity of the unscattered beam 0 (bright field):
I0= 1-Ig= 1- l Ψg(t) l2= 1 - sin2 πsgt/(ξgsg)2
Thickness fringes (sg konstant)
In the two-beam situation the intensity
of the diffracted and direct beam
is periodic with thickness (Ig=1- Io)
000
e
g
Sample (side view)
Ig=1- Io
t
Hole
Sample (top view)
Intensity of the scattered beam g:
Ig= l Ψg(t) l2= sin2 πsgt/(ξgsg)2
A.E. Gunnæs
MENA3100 V10
Positions with max
Intensity in Ig
Thickness fringes, bright and dark field images
Sample
Sample
BF image
A.E. Gunnæs
DF image
MENA3100 V10
Bend contours (t constant)
Intensity of the scattered beam g:
Ig= l Ψg(t) l2= sin2 πsgt/(ξgsg)2
sample
Obj. lens
Obj. aperture
BF image
DF image
DF image
Sg varies
Kinematical theory for imperfect crystals
Lattice displacement
If crystal defects give rise to lattice displacement
R, the potential at the pont r in the imperfect crystal
is the same as in r-R for the perfect crystal.
The specimen has a periodic structure
Fourier expansion of crystal potential
V(r)=ΣVg exp(2πig.(r-R))
Amplitude of the diffracted beam at the exit surface:
Ψg(t)= (πi/ξg)∫Ψo exp(-2πi(sgz+g.R))dz
Exs. dislocation
The effect of the lattice displacement is to introduce an extra face factor
α=2 πg.R, if α=0 at all points x,y the defect is invisible. Because g is
perpendicular to the reflecting planes, displacements parallell to the planes cannot
Produce contrast.
Crystallography
Stereographic projections
Symmetry elements
Classification of space groups
Macroscopic symmetry elements
• Mirror plane: m
• Rotation axes: 1, 2, 3, 4,
6
_ _ _ _ _
• Inversion-rotation axes: 1, 2, 3, 4, 6
_
– Inversion center: 1
All possible combinations of the macroscopic symmetry
elements and the translation property of crystals give the 32
possible point groups or crystal classes.
Stereographic projections
Microscopic symmetry elements
• Screw axes: Nq, N: 2, 3, 4, 6
q: 1, 2, 3, 4, 6
– Angular component αs= 360o/N
– Translational component ts=(q/N)t
• Glide plane (glide-reflection plane): a, b, c, n, d
– Translation component:
• a, b and c: a/2, b/2 and c/2 respectively
• n: (a+b)/2 or (a+c)/2 or (b+c)/2
• d: (a+b)/4, (a+c)/4, (c+b)/4, (a+b+c)/4 (cubic or tetragonal lattices)
When the translation properties, microscopic and macroscopic
symmetry elements are taken into account crystals can be classied
in 230 different space groups.
Symmetry operation and element
• Exs: A rotation is a symmetry operation while the rotation axis is a
symmetry element.
Operation
Element
• A glide plane is a symmetry element. The
orientation of the plane for an ”a” type
glide(-reflection) plane is normal to either
[010] or [001] (figure 7.6, page 51).
a/2
c
b
• The glide with vector a/2 and a reflection, is
the symmetry operation.
a
Hermann-Mauguin symbols
for space groups
• P 21/b 21/a 2/m, F 4/m 3 2/m, ……..
– 1st position: Bravais lattice
– A set of characters indicating symmetry elements of the
space group (1, 2 or 3 kinds of symmetry directions of the
lattice belonging to the space group).
• Symmetry planes are given by their normals. If a normal to a
symmetry plane and a symmetry axis are parallell, the two
symbols are separated by / .
Reference axes for
Hermann-Mauguin symbols
Lattice
Symmetry direction (position in Hermann-Mauguin symbol)
Primary
Secondary
Tertiary
Monoclinic
[010] (unique axis b)
[001] [unique axis c)
Orthorhombic
[100]
[010]
[001]
Tetragonal
[001]
[100]
[010]
[1-10]
[110]
Hexagonal
[001]
[100], [010], [-1-10]
[1-10], [120], [-2-10]
Rombohedral (h) [001]
[100], [010], [-1-10]
[1-10], [120], [-2-10]
Rombohedral (R)
[111]
[1-10], [01-1], [-101]
Cubic
[100], [010], [001]
[111], [1-1-1],
[-11-1], [-1-11]
[1-10], [110], [01-1]
[011], [-101], [101]
• Origin
– Suitable choice of origin
• Centro-symmetric space groups: Inversion center or a point of high
symmetry.
• Non-centrosymmetric space groups: highest site symmetry.
• Asymmetric unit
– Is a part of space from which the whole space can be filled exactly by
operation of the symmetry elements of the space group.
• Positions
– General and special
• For centred space groups the centering translations has to be
added to the listed coordinate triplets.
• The special positions appear because a symmetry operation of the
space group maps a point onto itself.
Reflection conditions
• General conditions
– Apply to all positions of the space group and are always
satisfied and are due to
• Centering (i.e all odd, all even for FCC)
• Glide planes (apply to two dim sets of reflections: hk0, h0l, 0kl, hhl)
• Screw axes (apply to one-dim. sets of reflections: h00, h-h0, 0k0, 001)
– Example Si, table p. 66
• Special conditions
– Apply only to special positions and must be added to the
general conditions of the space group.
– Extra condition is valid only for the scattering contribution of
those atoms which are located in the relevant position.
Pbam
P 21/b 21/a 2/m
No. 55
Reference axes for Hermann-Mauguin symbols: table 2.4.1 IT 1983
Orthorhombic: Primary [100], Secondary [010], Tertiary [001] (see p. 63)
Axial glide plane normal
to plane of projection
m
b
b
a
21: Screw diad parallel
to the paper
a
2/m: Two fold rotation axis
with centre of symmetry
Pbam
P 21/b 21/a 2/m
No. 55
• Look at conditions limiting possible reflections p. 54
Twins
• Two crystals are called twins if they have certain spesific
orientations with respect to each other and this relation can
be described by on of the crystallografic symmetry elements.
[uvw]*
– Reflection
– Rotation
– Inversion
[hkl]*
S*
[h’k’l’]*
S*
[hkl]*
[h’k’l’]*
Structure determination
with electrons
Which methods can one use to carry out a structure determination?
• Imaging
• Diffraction
• Spectroscopy
Kikuchi lines
Origin and use
Kikuchi pattern
Inelastically scattered
electrons
give rise to diffuse background
in the ED pattern.
θB
Excess
θB
-Angular distribution of
inelastic scattered electrons
falls of rapidly with angle.
I=Iocos2α
Kikuchi lines are due to:
-Inelastic + elastic scattering
event
Deficient
2θB
Objective lens
Diffraction plane
Excess
line
Deficient
line
1/d
http://www.doitpoms.ac.uk/index.html
http://www.doitpoms.ac.uk/tlplib/diffraction-patterns/kikuchi.php
Kikuchi maps
-g
000
g
Ig=I-g
Sg<0
Sg=0
Effect of tilting the specimen
http://www.umsl.edu/~fraundorfp/nanowrld/live3Dmodels/vmapframe.htm
Practical determination of sg
All angles are small:
L1/L2= φ/2θB
φ=sg/g
φ
2θB
L1/L2= sg/2gθB
k
2θB
0
g
g
φ
Sg
L1
L2
Image on the screen/negative
Resiprocal space
sg= L1/L22gθB
L2=λ L/d
λ= 2d θB ~ Bragg’s law
g=1/d
sg= L1/L22gθB
Kikuchi lines are used for
determination of
-crystal orientation
-lattice parameter/accelerating voltage
-Burgers vector/sg
Determination of crystal
orientation
L = XY/φ
φ
How to determine N
N
L2
Angle between the
zone axes A and B
L1
φ1
φ2 φ3
B
XY2
XY1
A
Trace of plane (h3k3l3)
R3 used to calculate d
for the plane (h3l3k3)
C
Determination of crystal
orientation
L = XY/φ
How to determine N
L = NA/θ1
N . A = INI IAI cosθ1
What is L in position N?
φ1
φ2 φ3
θ1
L = NB/θ2
N . B = INI IBI cosθ2
L = NC/θ3
N . C = INI ICI cosθ1
L1 = AB/φ1
L2 = AC/φ2
L3 = BC/φ3
N
B
LN=1/3 (L1+ L2 + L3)
What is φ?
A
C
A . B = IAI IBI cosφ1
A . C = IAI ICI cosφ2
B . C = IBI ICI cosφ3
Determination of crystal
orientation
How to determine N
L = NA/θ1
N . A = INI IAI cosθ1
L = NB/θ2
N . B = INI IBI cosθ2
L = NC/θ3
N . C = INI ICI cosθ1
N . A = uuA + vvA + wwA = INI IAI cosθ1
N . B = uuB + vvB + wwB = INI IBI cosθ2
N . C = uuC + vvC + wwC = INI IAI cosθ3
4 unknown ( u, v, w and INI) and three equations
What is INI?
Depends on the crystal system.
The length of a vector in the triclinic case is given on page 10.
Accurate determination of lattice
parameter and acceleration voltage
• Lattice parameter determination with SAD
rely on knowing Lλ (d=Lλ/R)
– Not sufficient accuracy
• Lattice parameters determination with kikuchi
lines rely on knowing λ.
Alternatively
Cubic case: the ratio λ/a can be found with 0.1% accuracy
Accurate determination of lattice
parameter and acceleration voltage
Three sets of kikuchi linse from different zone axis
g2
g3
2 gi . K=-IgiI2
A
ΔR
B
IgiI=1/di
C
K=1/λ
g1
The increment Δλ3 necessary to shift the line g3 to A is determined by
λ3=λ+ Δλ3~λ (1+ΔR3/R)
a/λ can be determined
Scattering from defects and disorder
• Order-disorder
• Eks. Cu-Au system