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

Sample position

Electrons are deflected by both electrostatic and magnetic fields

Force from an electrostatic field F= -e E

Force from a magnetic field F= -e (v x B)

Electron transparent samples

IntroductionEM and materials

Electron microscopy are based on three possible set of techniqes

Imaging

Diffraction

SpectroscopyWith spatial resolution down to the atomic level (HREM and STEM)

Chemistry and elecronic states (EDS and EELS).Spatial and energy resolution down to the atomic level and ~0.1 eV.

From regions down to a few nm (CBED).

Electrons

E<Eo(EELS)

BSE

SEAE X-rays (EDS)

E=Eo

Bragg diffracted electrons

15/1-08 MENA3100

Basic principles, electron probeValence

K

L

M

Electronshell

Characteristic x-ray emitted or Auger electron ejected after relaxation of inner state. Low energy photons (cathodoluminescence)when relaxation of outer stat.

K

L

M

1s2

2s2

2p2

2p43s2

3p2

3p4

3d4

3d6

Auger electron or x-ray

Secondary electron

Electron

Introduction EM and materials

The interesting objects for EM is not the average structure or homogenous materials but local

structure and inhomogeneities

Defects

Precipitates

Interfaces

Defects, interfaces and precipitates determines the properties of materials

• 1834 William Rowan Hamilton• 1876 Ernst Abbe• 1897 J.J. Thomson• 1924 de Broglie • 1925/26 E. Schrӧdinger• 1926/27 Hans Busch• 1927 C. Davisson and L.H. Germer/ G. Thomson and A. Reid• 1928 Max Knoll and Ernst Ruska

Introduction History of EM: from dream to reality

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

• 1941 microscope by Radio corporation of America (RCA)– First instrument with stigmators to correct for astigmatism. Resolution

limit below 10 Å.

Elmiskop I

Developments

• Spherical aberration coefficient

ds = 0.5MCsα3

M: magnificationCs :Spherical aberration coefficientα: angular aperture/ angular deviation from optical axis

2000FX: Cs= 2.3 mm2010F: Cs= 0.5 nm

r1

r2

Disk of least confusion

α

r1

r2

α

Realized that spherical aberration of the magnetic lenses limited the possible resolution to about 3 Å.

Chromatic aberration

vv - Δvdc = Cc α ((ΔU/U)2+ (2ΔI/I)2 + (ΔE/E)2)0.5

Cc: Chromatic aberration coefficientα: angular divergence of the beamU: acceleration voltageI: Current in the windings of the objective lensE: Energy of the electrons

2000FX: Cc= 2.2 mm2010F: Cc= 1.0 mm

Chromatic aberration coefficient:

Thermally emitted electrons:ΔE/E=kT/eU

Force from a magnetic field:F= -e (v x B)

Disk of least confusion

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.

• Development of Cs corrected microscopes– Probe and image

• Improved energy spread of electron beam– More user friendly Cold FEG – Monocromator

Last few years

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

• Electrons can be accelerated to provide sufficient short wave length for atomic resolution.

• Due to high acceleration voltages in the TEM relativistic effects has to be taken into account.

Charge eRestmass mo

Wave ψWave length λ

λ = h/p= h/mv de Broglie (1925)

λ = h/(2emoU)1/2 U: pot. diff.

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

MENA3100 V08

Objective lense

Diffraction plane(back focal plane)

Image plane

Sample

Parallel incoming electron beamSi

a

b

cP

ow

derC

ell 2.0

1,1 nm

3,8

Å

Objective aperture

Selected area aperture

Simplified ray diagram

Mic

rosc

opy

and

diffr

actio

n co

nditi

on

Intermediatelens

Projectorlens

Image plane

Focal plane

JEOL 2000FX Wehnelt cylinderFilamentAnode

Electron gun 1. and 2. beam deflectors

1.and 2. condenser lensCondenser apertureCondenser lens stigmator coilsCondenser lens 1. and 2. beam deflector

Condenser mini-lensObjective lens pole pieceObjective apertureObjective lens pole pieceObjective lens stigmators1.Image shift coilsObjective mini-lens coils (low mag)2. Image shift coils

1., 2.and 3. Intermediate lens

Projector lens beam deflectorsProjector lensScreen

Mini-lens screws

Specimen

Intermediate lensshifting screws

Projector lensshifting screws

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?

• What is changing when one goes from diffraction to imaging mode?

• Why are electrons suitable for imaging with atomic resolution?

MENA3100 V08

Objective lense

Diffraction plane(back focal plane)

Image plane

Sample

Parallel incoming electron beamSi

a

b

cP

ow

derC

ell 2.0

1,1 nm

3,8

Å

Objective aperture

Selected area aperture

Simplified ray diagram

Eric Stach (2008), ”MSE 528 Lecture 4: The instrument, Part 1, http://nanohub.org/resources/3907

JEOL 2000FX Wehnelt cylinderFilamentAnode

Electron gun 1. and 2. beam deflectors

1.and 2. condenser lensCondenser apertureCondenser lens stigmator coilsCondenser lens 1. and 2. beam deflector

Condenser mini-lensObjective lens pole pieceObjective apertureObjective lens pole pieceObjective lens stigmators1.Image shift coilsObjective mini-lens coils (low mag)2. Image shift coils

1., 2.and 3. Intermediate lens

Projector lens beam deflectorsProjector lensScreen

Mini-lens screws

Specimen

Intermediate lensshifting screws

Projector lensshifting screws

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 ZnO/WCold FEG 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

Cross over

α

d

Image of source

The electron gun

Bias -200 V

Ground potential

-200 kV

Anode

Wehneltcylinder

Cathode

dcr Cross over

αcr

Equipotential lines

Thermionic gunFEG

Thermionic gunsFilament heated to give Thermionic emission-Directly (W) or indirectly (LaB6)

Filament negativepotential to ground

Wehnelt produces a small negative bias-Brings electrons to cross over

Thermionic guns

Thermionic emission

• Current density:

– 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

Jc= AcT2exp(-φc/kT)Richardson-Dushman

Maximum usable temperature T is determined by the onset of the evaporation of material.

Field emission

• Current density: Fowler-Norheim

Maxwell-Boltzmann energy distribution

for all sources

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

• Electrostatic– 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= -eE

F= -e(v x B)

Any axially symmetrical electric or magnetic field have the properties of an ideal lens for paraxial rays of charged particles.

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

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 ‘under-focus’ it, or turn its knob anti-clockwise) to keep the image in focus.

http://www.rodenburg.org/guide/t300.html

Magnification of image, Rays from different parts of the object

If the strengths (excitations) of the two lenses are changed, the magnification of the image changes

http://www.rodenburg.org/guide/t300.html

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

• 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)

The objective lens

Apertures

A.E. Gunnæs MENA3100 V08

Use of aperturesCondenser 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).

BF image

Objectiveaperture

Objective aperture: Contrast enhancement

All electrons contributes to the image.

Si

Ag and Pb

glue(light elements)hole

Only central beam contributes to the image.

Bright field (BF)

Small objective aperture Bright field (BF), dark field (DF) and weak-beam (WB)

BF image

Objectiveaperture

DF image Weak-beam

Dissociation of pure screw dislocation In Ni3Al, Meng and Preston, J. Mater. Scicence, 35, p. 821-828, 2000.Diffraction contrast

Large objective aperture High Resolution Electron Microscopy (HREM)

HREM image

Phase contrast

Use of aperturesCondenser 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

Objective lense

Diffraction pattern

Image plane

Specimen with two crystals (red and blue)

Parallel incoming electron beam

Selected area aperture

Pattern on the screen

Diffraction with no aperturesConvergent beam and Micro diffraction (CBED and µ-diffraction)

Convergent beam Focused beam

Convergent beam Illuminated area less than the SAD aperture size.

CBED pattern µ-diffraction pattern

C2 lens

Diffraction information from an area with ~ same thickness and crystal orientation

Small probe

Shadow imaging (diffraction mode)

Objective lense

Diffraction plane(back focal plane)

Image plane

Sample Parallel incoming electron beam

Magnification and calibration

Microscope Lens Mode Magnification

JEM-2010 Objective MAG 2 000-1 500 000

LOW MAG 50 - 6 000

Philips CM30 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

Resolution of the photographic emulsion: 20-50 µm

Magnification depends on specimen position in the objective lens

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/andimaging 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

StereomicroscopyA 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.

S1S2

ΔZxδ

θ/2

θ

Ray diagram ΔZ= p/(2M sin(θ/2))

p= M(S1-S2)

p: Parallax, M: total magnification

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

ds = 0.5MCsα3 (disk diameter, plane of least confusion)

ds = 2MCsα3 (disk diameter, Gaussian image plane)

M: magnificationCs :Spherical aberration coefficientα: angular aperture/ angular deviation from optical axis

r1

r2

Plane of least confusion

α

Gaussian image plane

2000FX: Cs= 2.3 mm2010F: Cs= 0.5 nm

Highest intensety in the Gaussian image plane

Chromatic aberration

vv - Δv

Diameter for disc of least confusion:

dc = Cc α ((ΔU/U)2+ (2ΔI/I)2 + (ΔE/E)2)0.5

Cc: Chromatic aberration coefficientα: angular divergence of the beamU: acceleration voltageI: Current in the windings of the objective lensE: Energy of the electrons

2000FX: Cc= 2.2 mm2010F: Cc= 1.0 mm

Thermally emitted electrons: ΔE/E=kT/eU, LaB6: ~1 eV

Disk of least confusion

The specimen will introduce chromatic aberration.

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• Loss of axial asymmetry

y-focus

x-focusy

x

This astigmatism can not be

prevented, but it can be

corrected! Disk of least confusion

Diameter of disk of least confusion:da: Δfα

Due to non-uniform magnetic fieldas in the case of non-cylindrical lenses. Apertures may affect the beam if not precisely centered around the axis.

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.

αim

DobDim

dob dim

1 12 2

Depth of focus and depth of field (image)

βim

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) βob≈ tan βob= (dob/2)/(Dim/2)

Angular magnification: MA= αim/ βob

Transvers magnification: MT= dim/ dob MT= 1/MA

Depth of focus: Dim=(dob/ βob)MT2 Depth of field: Dob= dob/ βob

αim≈ tan αim= (dim/2)/(Dob/2) βob≈ tan βob= (dob/2)/(Dim/2)

Angular magnification: MA= αim/ βob

Transvers magnification: MT= dim/ dob MT= 1/MA

Depth of focus: Dim=(dob/ βob)MT2 Depth of field: Dob= dob/ βob

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

Depth of field

Example: dob/ βob= 0.2 nm/10 mrad = 20 nmExample: dob/ βob= 0.2 nm/10 mrad = 20 nm

Example: dob/ βob= 2 nm/10 mrad = 200 nmExample: dob/ βob= 2 nm/10 mrad = 200 nm

Dob= thickness of sample all in focuseDob= 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 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

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 mFocus on the wieving screenand far below!Focus on the wieving screenand 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 scatteringAnalytical 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

• 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.

• 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.

N(E)

E/Eo0 1.0

A

B

C

BSE used in SEMBackscattered electron coefficient η :

η=numer of BSE/number of primary electrons

BE coefficientGoldstein 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:

Contrast between constituents in BSE images in the SEM can be calculated as:

Ci : mass fractions

The angle of scatteringScattering fro a single isloated atom

The scattering angle θ is a semi-angle, and not a total angle of scattering.

The total solid angle is Ω.

θ is often assumed to be small sinθ ≈ tanθ ≈ θ

Small angle: <10 mrad (~10o)

θ

dθ

Ω

Incident beam

Scattered electrons

Unscattered electrons

dΩ

The solid angle Ω is the two-dim anglein 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.

The solid angle Ω is the two-dim anglein 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.

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).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 scatteringfrom an atom, and is a measure of the probability for scattering in a solid angle dΩ.

Scattering form the specimenTotal 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 volumeNo: 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/Atσtotal=Noσatom ρt/A

Mean free path λ

λ = 1/σtotal = A/Noρσatom

The mean free path for a scattering process is the average distance travelled by the primary particle between scattering events.

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

• 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

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 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

A diagram of a scattering process http://en.wikipedia.org/wiki/File:ScatteringDiagram.svg

Differential scattering cross section i.e. the probability for scattering in a solid angle dΩ:

dσ/dΩ = 2πb (db/dΩ)

b= (Ze2/4πεomv2)cotanθ/2

dσ/dΩ = -(mZe2λ2/8πεoh2)2(1/sin4θ/2) 

Impact parameter: b

Solid angle:Ω= 2π(1- cosθ)

Inelastic scattering processes• Ionization of inner shells

– Auger electrons– X-rays– Light

• Continuous X-rays/Bremsstrahlung

• Exitation of conducton or valence electrons

• Plasmon exitation

• Phonon exitationsCollective oscillationsNon- localized

Localized processes

Non- localized SE

Valence

K

L

M

Electronshell

Characteristic x-ray emitted or Auger electron ejected after relaxation of inner state. Low energy photons (cathodoluminescence)when relaxation of outer stat.

K

L

M

1s2

2s2

2p2

2p43s2

3p2

3p4

3d4

3d6

Auger electron or x-ray

Electron

Ionization of inner shells

Auger electrons or x-rays

EELS?

X-ray spectrum

K

L

M

Photo electron

x-ray x-ray

Fluorescence

Continuous and characteristic x-rays

http://www.emeraldinsight.com/journals.htm?articleid=1454931&show=html

Continous x-rays du todeceleration of incident electrons.

The cut-off energy forcontinous x-rays corresponds to the energy of the incident electrons.

Secondary electrons

Secondary electrons (SEs) are electrons within the specimen that are ejected by the beam 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 eVElectrons from the conduction or valence band.E ~ 0 – 50 eV

Auger electronsAuger electronsThe secondary emission coefficient:

δ=number of secondary electrons/numbers of primary electrons

Dependent on acceleration voltage.

Cathodoluminescence

Valence band

Conduction band

Plasmon excitations

The oscillations are called plasmons. The oscillations are called plasmons.

The incoming electrons can interact with electrons in the ”electron gas”and cause the electron gas to oscillate.

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

The effect in the diffraction patterns:-Reduction of intensities (Debye-Waller factor)-Diffuce bacground between the Bragg reflections

Energy losses ~ 0.1 eV

EELS

Sum of several losses

Thin specimens

Summary

Electron diffraction geometry

Lattice properties of crystalsBravais lattices

Lattice planes and directionsResiprocal lattice

The Laue condition The Bragg condition

The Ewald sphere constructionDifferences between x-ray and ED

Zone axis and Laue zones………….

x

y

z

a

b

c

α

γβ

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– 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

-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)

Macroscopic symmetry elements

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

Crystals can be classified based on symmetry without taking into account their translation symmetry.Crystals can be classified based on symmetry without taking into account their translation symmetry.

Acting at a point since no translations are involved in the symmetry operation.

32 point groups or crystal classesListed in table 5.4

Microscopic symmetry elements

Symmetry elements involving translation within the unit cellSymmetry 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)

• Details about crystal description can be found in International Tables for Crystallography.

– Criteria for filling Bravais point lattice with atoms.

– Both paper books and online

• 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”.

• 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).

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]

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

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

r k’

k

k

The Laue condition

Two lattice points separated by a vector r

r k’

(hkl)

nhkl

nhkl= r*hkl

r*hkl = k’ – k ?

φ= 2πr.(k’-k) = 2πr.r*hkl φ= 2π(ua+vb+wc).(ha*+kb*+lc*) φ= 2π(uh+vk+wl)

Maximum intensity if h,k,l are integers

a.(k’-k) =h

b.(k’-k) =k Laue condition

c.(k’-k) =lThe scattering vector must be oriented in a specific direction in relation to the primitivevectors of the crystal lattice.

• nλ = 2dsinθ– Planes of atoms responsible

for a diffraction peak behave as a mirror

Bragg’s law

d

θ

θ

y

x

The path difference: x-y

Y= x cos2θ and x sinθ=dcos2θ= 1-2 sin2θ

k’

k

r*hkl

2θB

IkI = Ik’I = 1/λIk’- kI = (2/λ) sinθB

hklhkl gd /1

r*hkl = k’ – k 1/dhkl= (2/λ) sinθB

λ= 2dhklsinθB

Cu Kalpha X-ray: = 150 pm => small kElectrons at 200 kV: = 2.5 pm => large k

The Ewald Sphere is flat (almost)

ED and form effectsFigure 5.7 and 5.8

Real space Resiprocal space

Zone axis and Laue zones

Zone axis [uvw]

(hkl)

uh+vk+wl= 0

Excitation error:sg

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

- Measure Ri and the angles between the reflections

- Calculate di , i=1,2,3 (=K/Ri)

- 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

- Zone axis: gi x gj =[HKL]z

- All indexed g must satisfy: g ● [HKL]z=0

(h2k2l2)

Orientations of correspondingplanes in the real space

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

Chemical analysis by use of EDS and EELS

Ukjent faseBiFeO3 BiFe2O5

1_1evprc.PICT

-0 200 400 600 800 10005

10

15

20

25

30

35

40

Energy Loss (eV)

CC

D c

ount

s x

100

0

Nr_2_1evprc.PICT

-0 200 400 600 800 1000

-0

2

4

6

8

10

12

14

Energy Loss (eV)

CC

D c

ount

s x

100

0

Ukjent faseBiFeO3

Fe - L2,3

O - K

500 eV forskyvning, 1 eV pr. kanal

Published structure

A.G. Tutov og V.N. MarkinThe 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

x y zBi 4g 0,176 0,175 0Fe 4h 0,349 0,333 0,5Fe 4f 0 0,5 0,244O 4g 0,14 0,435 0O 8i 0,385 0,207 0,242O 4h 0,133 0,427 0,5O 2b 0 0 0,5

ab

c

O

Bi

Fe

O

Fe

Bi

O

Fe O

O

O

Fe

Fe

O O

O

O

Fe

Bi

O

O

Bi

O

Bi

O

O

Bi

Fe

O

O

O O

Fe

Fe

O

O

O Fe

O

Bi

Fe

O

Fe

Bi

O

PowderCell 2 .0

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)

E = eU

H= p2/2m – eV(r)

Total energy of the electrons

Kinetic Potential

Ψ(r) Wave function Try solutions

p2/2m = (mv)2/2m= ½ mv2

P = (h/2πi)

((-h2/4π2m)2 – eV(r)) Ψ(r) = eU Ψ(r)

Transmission through a thin specimen

Vaccum; eV(r)=0 Constant potensial; eV(r)=eV0

((-h2/4π2m)2 – eV(r)) Ψ(r) = eU Ψ(r)

Try solution: Ψ(r) = Ψo exp(2πikr) Ψ(r) = Ψo exp(2πik’r)

(h2/2m)k2 Ψ(r)= eU Ψ(r)

Solution if: k= 1/λ =(2meU/h2)1/2

((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

((-h2/4π2m)2 – eV(r)) Ψ(r) = eU Ψ(r)

(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’

fB(θ)= ∫exp(-2πis.r’)U(r’) dV’ s = k’ - k

Scattered wave:

Ψs = (πexp(2πikr)/r) ∫exp(-2πis.r’)U(r’) dV’

Scattered wave:

Ψs = (πexp(2πikr)/r) ∫exp(-2πis.r’)U(r’) dV’

U(r’) = 2meV(r’)/h2

Weak scattering: 1. Born or kinematical approximation: Ψ(r’) = exp(2πik.r’) Plane incoming wave

Solution of the Scrödinger equationScattered wave:Ψs = (πexp(2πikr)/r) ∫exp(-2πis.r’)U(r’) dV’

fB(θ)= ∫exp(-2πis.r’)U(r’) dV’

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

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

X-ray scattering factor

Rutherford scattering equation

Scattering from one unit cell Scattering from one atom: Ψs(r)= (πexp(2πikr)/r) f(θ)

Scattering from two atoms: 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

rj

k

k’

o

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)

Scattering from all unit cells in the crystal:Ψs(r)= Σ(π/r) exp(2πikr) fi(θ)exp(2πs.ri ), ri = rn + rj i

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 cellj

O

rn

rj

Ψs(r)= (π/r) exp(2πikr)Σ fn(θ)exp(2πs.rn ) Σ fj(θ)exp(2πs.rj )n j

The same for all unit cells

Scattering from all unit cells in the crystalΨs(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 ), F(θ)= Σ fj(θ)exp(2πis.rj )

s= ra*+ sb*+ tc*, rn=ua + vb + wc

When is Ψs(r)≠0? (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 reflectionsF(θ)=Fg= Σ fj(θ)exp(2πi (hxj + kyj + lzj)) ≠ 0

29/1-08 MENA3100

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.

– Forbidden reflectionsd

θ

θ

y

xx’

y’

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).

From the structure factor of the unit cell one can determine if the hkl reflection it is allowed or forbidden.

29/1-08 MENA3100

Structure factors

N

j

ejhklg fFF

1

)( 2exp( ))( jjj lwkvhui X-ray:

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.

rj

ujaa b

x

z

c

y

vjb

wjc

The intensity of a reflection is proportional to:

ggFF

29/1-08 MENA3100

Example: Cu, fcc• eiφ = cosφ + isinφ • enπi = (-1)n

• eix + e-ix = 2cosx

N

jjhklg fFF

1

2exp( ))( jjj lwkvhui

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))

If h, k, l are all odd then:Fhkl= f(1+1+1+1)=4f

If h, k, l are mixed integers (exs 112) thenFhkl=f(1+1-1-1)=0 (forbidden)

What is the general condition for reflections for fcc?

What is the general condition for reflections for bcc?

Simplified kinematical theory for perfect crystals

Wavefunction at exit surface:Ψ(r) =exp(2πikr)exp(iσV0Δz) (Ψo=Ψo’=1) 6.27σ=2πmλe/h2 6.16

Ψ(r) =Ψoexp(2πikr)exp(2πimeλV(r)dz/h2) 6.139

This sample Taylor series:Ψ(r) =Ψoexp(2πikr) (1+2πimeλV(r)dz/h2)

The specimen has a periodic structure Fourier expansion of crystal potentialV(r)=ΣVg exp(2πig.r)

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

kk’

g

kk’

gsg

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

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:

Ψg(t)= ∫(πi/ξg) exp(-2πisgz)dz, Ψo=1, t: crystal thickess t

0

Intensity of the scattered beam g (dark field):

Ig= l Ψg(t) l2= sin2 πsgt/(ξgsg)2

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

Intensity of the unscattered beam 0 (bright field): I0= 1-Ig= 1- l Ψg(t) l2= 1 - sin2 πsgt/(ξgsg)2

Ψg(t)= (i/ξgsg) exp(-πitsg) sinπsgt

A.E. Gunnæs MENA3100 V10

Sample (side view)

e

000 g

t

Ig=1- Io

In the two-beam situation the intensityof the diffracted and direct beamis periodic with thickness (Ig=1- Io)

Sample (top view)Hole

Positions with max Intensity in Ig

Thickness fringes (sg konstant)

Intensity of the scattered beam g: Ig= l Ψg(t) l2= sin2 πsgt/(ξgsg)2

Intensity of the scattered beam g: Ig= l Ψg(t) l2= sin2 πsgt/(ξgsg)2

A.E. Gunnæs MENA3100 V10

Thickness fringes, bright and dark field images

Sample Sample

DF imageBF image

Bend contours (t constant)

Intensity of the scattered beam g: Ig= l Ψg(t) l2= sin2 πsgt/(ξgsg)2

Intensity of the scattered beam g: Ig= l Ψg(t) l2= sin2 πsgt/(ξgsg)2

Sg varies

BF image

DF image

DF image

Obj. aperture

Obj. lens

sample

Kinematical theory for imperfect crystals

Exs. dislocation

Latt

ice

disp

lace

men

t

If crystal defects give rise to lattice displacementR, the potential at the pont r in the imperfect crystalis 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

The effect of the lattice displacement is to introduce an extra face factorα=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 projectionsSymmetry 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.

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.

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.

• 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).

• The glide with vector a/2 and a reflection, is the symmetry operation.

a/2

a

bc

Element

Operation

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. 55Reference axes for Hermann-Mauguin symbols: table 2.4.1 IT 1983

Orthorhombic: Primary [100], Secondary [010], Tertiary [001] (see p. 63)

m

2/m: Two fold rotation axis with centre of symmetry

21: Screw diad parallel

to the paper

a

Axial glide plane normal to plane of projection

b

a

b

• Look at conditions limiting possible reflections p. 54

Pbam P 21/b 21/a 2/m No. 55

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.

– Reflection– Rotation – Inversion

S*

[uvw]*

[hkl]*

[h’k’l’]*

[h’k’l’]*[hkl]*

S*

Structure determination with electrons

• Imaging

• Diffraction

• Spectroscopy

Which methods can one use to carry out a structure determination?

Kikuchi lines

Origin and use

Kikuchi pattern

http://www.doitpoms.ac.uk/index.htmlhttp://www.doitpoms.ac.uk/tlplib/diffraction-patterns/kikuchi.php

ExcessDeficient

Excess line

Deficient line

2θB

θB

θB

Diffraction plane

Objective lens

1/d

Inelastically scattered electronsgive rise to diffuse background in the ED pattern.

-Angular distribution of inelastic scattered electrons falls of rapidly with angle. I=Iocos2α

Kikuchi lines are due to:-Inelastic + elastic scattering event

Kikuchi maps

http://www.umsl.edu/~fraundorfp/nanowrld/live3Dmodels/vmapframe.htm

000 g-g

Ig=I-g

Sg<0

Sg=0

Effect of tilting the specimen

Practical determination of sg

L1

L2

g0

2θB

φ

Image on the screen/negative Resiprocal space

2θB

g Sg

φ

k

All angles are small:

L1/L2= φ/2θB

φ=sg/g

L1/L2= sg/2gθB

sg= L1/L22gθB

L2=λ L/dλ= 2d θB ~ Bragg’s lawg=1/dsg= L1/L22gθB

Kikuchi lines are used for determination of

-crystal orientation

-lattice parameter/accelerating voltage

-Burgers vector/sg

φ

XY1

L1

Determination of crystal orientation

N

B

A

C

Trace of plane (h3k3l3)

φ1

φ2φ3

Angle between the zone axes A and B

R3 used to calculate d for the plane (h3l3k3)

L2

XY2

L = XY/φL = XY/φ

How to determine N

Determination of crystal orientation

N

B

A

C

φ1

φ2φ3

L = XY/φL = XY/φ

L1 = AB/φ1

L2 = AC/φ2

L3 = BC/φ3

A . B = IAI IBI cosφ1 A . C = IAI ICI cosφ2 B . C = IBI ICI cosφ3

What is L in position N?

LN=1/3 (L1+ L2 + L3)

What is φ?θ1

L = NA/θ1 N . A = INI IAI cosθ1

L = NB/θ2 N . B = INI IBI cosθ2

L = NC/θ3 N . C = INI ICI cosθ1

How to determine N

Determination of crystal orientation

L = NA/θ1 N . A = INI IAI cosθ1

L = NB/θ2 N . B = INI IBI cosθ2

L = NC/θ3 N . C = INI ICI cosθ1

How to determine N

N . A = uuA + vvA + wwA = INI IAI cosθ1 N . B = uuB + vvB + wwB = INI IBI cosθ2 N . C = uuC + vvC + wwC = INI IAI cosθ3

What is INI?

Depends on the crystal system.The length of a vector in the triclinic case is given on page 10.

4 unknown ( u, v, w and INI) and three equations

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

• Accelration voltage detrmination with kikuchi lines rely on knowing the lattice paramers

Cubic case: the ratio λ/a can be found with 0.1% accuracy

Accurate determination of lattice parameter and acceleration voltage

C

A

B

g3

g2

g1

Three sets of kikuchi linse from different zone axis

2 gi . K=-IgiI2

IgiI=1/di

K=1/λ

ΔR

The increment Δλ3 necessary to shift the line g3 to A is determined byλ3=λ+ Δλ3~λ (1+ΔR3/R)

a/λ can be determined

Scattering from defects and disorder

• Order-disorder• Eks. Cu-Au system