Course T2 Electron Microscopy and Analysis

8/10/2019 Course T2 Electron Microscopy and Analysis

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T2: Electron Microscopy and Analysis

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Course T2: Electron Microscopy and Analysis

Synopsis

Aim

The purpose of this course is to introduce the student to electron microscopy and itscapabilities. The structure of an electron microscope will be introduced to develop anunderstanding of the operation of scanning and transmission electron microscopes in theformation of images and diffraction patterns. The signals generated in the electronmicroscope are discussed, and their use for analysis of different materials problems ishighlighted.

Scanning electron microscopy (4 lectures)

1. Introduction

Basic design of SEM; choice of filament; structure of the electron gun; action ofelectron lenses; scanning system; lens aberrations; spatial resolution; image contrast;choice of objective aperture

2. Beam-specimen interactionsSignals generated and detected in the SEM; electron range; electron detectors

3. Image formationSample preparation; image formation with secondary electrons; image formation

with backscattered electrons; channelling contrast; backscattered diffraction patterns;cathodoluminescence; voltage contrast

4. X-ray microanalysisX-ray generation by electron beams; electron transitions in atoms; chemical analysiswith X-rays; energy dispersive spectroscopy; wavelength dispersive spectroscopy;common artefacts; quantitative analysis and ZAF corrections; detectability limits

Transmission electron microscopy (4 lectures)

5. IntroductionDesign of a TEM; sample requirements and preparation

6. Beam-specimen interactions in a thin filmBrief overview of beam-specimen interactions; atomic scattering factor; scattering

from a unit cell; elastic scattering; diffraction patterns from polycrystalline andamorphous materials

7. Image formationFormation of images and diffraction patterns; bright field and dark field images;

selected area and convergent beam electron diffraction patterns; kinematic scatteringapproximation; bend contours; thickness fringes; defects in materials; stacking faults;dislocations; high resolution TEM


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8. Microanalysis in the TEM/STEMEnergy dispersive X-ray spectroscopy; electron energy loss spectroscopy; comparisonof techniques for chemical analysis and detection

Recommended text books

Two books cover most aspects of the course:

Electron Microscopy and Analysis (3rdEdition) by P. J. Goodhew et al. (Mb262)Electron Beam Analysis of Materials (2ndEdition) by M. H. Loretto (Mb231)

The Mb section of the departmental library contains many other, more specialised textsthat may be useful for aspects of this course, including:

SEM

Practical Scanning Electron Microscopyby J. I. Goldstein et al. (Mb141)Scanning Electron Microscopyby L. Reimer (Mb237)

TEM

Transmission Electron Microscopy A Textbook for Materials Science by D. B.Williams and C. B. Carter (Mb309-312, 4 volumes)

Electron Microscopy of Thin Crystalsby P. B. Hirsch et al. (Mb165)

Supplementary information

DoITPoMS Teaching and Learning Packages www.doitpoms.ac.ukMatter software (available on the computers in the teaching laboratory and in collegecomputer rooms), with some modules available online at www.matter.org.uk

Acknowledgements

Thanks to Dr. P. Midgley, Dr. J Barnard and Dr. S. Friedrichs for providing images thathave been reproduced in this handout.


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Scanning Electron Microscopy

1. Introduction

Electron microscopy techniques provide very powerful methods for examiningmaterials to much higher resolution than can be achieved using light microscopy. Theinteraction between the electron beam and the specimen generates many signals that can

be used to develop a greater understanding of the chemical and physical properties ofmaterials.

The structure of an electron microscope can be compared to that of a light microscope,with many similar features:

Light microscope Electron microscope

We can also use ray diagrams to illustrate schematically the operation of electronmicroscopes.


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Basic structure of a scanning electron microscope (SEM)

Schematic diagram of an SEM.

The main components of an SEM are the electron gun, which generates a beam ofelectrons that are accelerated by an electrostatic potential of ~ 1 - 40 kV, and electronlenses (usually electromagnetic) are then used to focus the electron beam into a finespot on the surface of the specimen. The interaction between the electron beam and thespecimen generates a wide range of signals that can be collected by detectors and usedto analyse the properties of the specimen. The beam is rastered over the surface of the

specimen using electrostatic coils to create a 2-D image formed using the chosen signal.

1.1The electron gun

The electron gun is chosen to produce as bright a source of electrons as possible tomaximise the number of electrons hitting the specimen surface (per unit area). Thisgenerates the maximum signal for detection.

The brightness, , of the electron source is dependent on the emission current, ie, thesource size,D0, and the angle, 0, that the source makes with the anode:

20

20

20

2 4 ee J

Di == (1.1)


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Schematic diagram of a thermionic electron gun

For high brightness, an electron source with a high emission current ( ie) and smallsource size (D0) are required. A small D0 leads to a small spot on the sample, andtherefore a high spatial resolution.

Thermionic emission sources are typically made from either a tungsten wire or a LaB 6

crystal. The current can be high, but the current density (Je) is relatively low due tolarge source size (D0).

A field emission gun uses a strong electric field to enable electrons to tunnel out of avery fine W needle. They have very high brightness with high emission current butsmall source size.

1.2The condenser lens

Electron lenses used in modern electron microscopes are usually electromagnetic, andact to deflect a beam of electrons by the Lorentz force experienced by a moving,charged particle in a magnetic field. They can be used to focus a beam of electrons into

a probe, and the strength of the lens (focal length) can be changed by varying themagnitude of the electromagnetic field.

In an SEM, the condenser lens is used to choose the spot size. This lens collectselectrons emitted by the gun that are subsequently used to form a probe on the specimensurface. By increasing the strength of the condenser lens, electrons can be selectedfrom progressively smaller regions on the filament, thereby reducing the source size

D0


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(demagnifying). This allows a smaller spot to be formed, thus increasing the spatialresolution.

For fixed gun brightness, this reduces the current in the beam, and therefore theavailable signal will reduce in strength. [There is always a compromise betweenresolution and signal intensity.]

1.3 The objective lens

This lens is used to focus the beam onto the specimen surface, and should be adjusted toachieve the minimum spot size (as chosen by the condenser lens). This gives themaximum spatial resolution in the final image.

The magnification is given by the ratio of the image distance, v, to the object distance,

u, i.e.u

vM =

Lens diagram showing the demagnifying effect of the electromagnetic lenses modelledas a simple convex lens.

The distance v is known as the working distance and should be kept to a minimum toachieve the best spatial resolution. The working distance is approximately the distancefrom the bottom of the objective lens to the specimen, and is typically 10 mm.

1.4Scanning

The scan coils used in any electron microscope are electrostatic and act in pairs to shiftor tilt the electron beam. The coils raster the beam, usually in a rectangular array, suchthat an image is built up sequentially. The electron beam is aligned such that it always

passes through the centre of the objective lens. For a TV rate image (30 Hz) of 1200 800 pixels, the beam dwell time is only ~ 35 ns at each pixel.

u v

f


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1.5Aberrations

Electromagnetic lenses are far from perfect, and suffer from many aberrations which actto reduce the minimum achievable probe size, and therefore reduce the spatialresolution. Chromatic and spherical aberrations can only be minimised by carefuldesign of the objective lens, but astigmatism is usually corrected by the operator during

the experiment.

We can consider the contributions of different aberrations to the total spot size byadding in quadrature (assuming that all contributions are Gaussian):

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

N.B. through the action of the lenses, d0is about 10 smaller than D0. This is simplybecause not all of the electron beam which passes through the condenser lens can enterthe objective lens (see 1.7).

In these equations is the convergence semi-angle (half-angle) onto the specimen ofthe beam with current i. The solid angle of the beam, 2, is 2.

Typical values for an SEM are:= 10-2 rad.

E= 40 kVCs= 20 mmCc= 8 mm

For a thermionic source: ~ 109Am-2sr-1For a LaB6source: ~ 10

10Am-2sr-1

For a FEG source: ~ 10

13

Am

-2

sr

-1

1.6 Beam current

To achieve the best spatial resolution, the signal strength must be sacrificed. Inpractice, when imaging in the SEM, the contrast, C, defined as S/S, must be optimised(where Sis the signal and Sis the change in signal).

(1.7)

22220 dsctot ddddd +++=

220

4

id =

cc CE

Ed

=

3

21

ss Cd =

22.1=dd

Probe size (see earlierequation for brightness)

Chromatic aberration

Spherical aberration

Diffraction limit

S

SC

=


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The changes in signal must be greater than 5 times the noise level, , in the image to beable to detect the difference.

(1.8)

For Poisson statistics, the noise, , can be expressed as n, where n is the averagenumber of electrons detected at a particular point (i.e. the signal, S). This leads to the

equation:(1.9)

and therefore (1.10)

If we start to consider the formation of a scanned image, with a frame rate off(Hz) andNppixels, then the beam dwells on each pixel for a time t given by:

(1.11)

In the time t that the beam dwells on each pixel, the number of electrons entering thespecimen, n0, is

(1.12)The number of electrons actually detected, n, depends on the beam-specimen interactionand the efficiency of the detector, say n = qn0. Therefore, to discern a contrast, C, in thespecimen, we require a minimum current of

(1.13)

Therefore the spatial resolution (small spot size, small I) must be compromised forimage contrast (better quality images, largeI).

1.7The objective aperture

The objective aperture controls the convergence angle and is in a plane conjugate withthe objective lens. A small aperture will reduce the effect of spherical aberration(through a small) and increase the depth of focus, but will limit the beam current.

Illustration of how theobjective aperturechanges the depth offocus for a fixedresolution. The depth

of focus is thedistance from the

plane of optimumfocus within whichthe beam diverges byno more than the discof least confusion.

5>S

nC 5>

2

25C

n =

fNt

p

1=

feN

I

e

tIn

p

==.

0

2min

25

qC

feNI

p=

Objective

aperture

Disc of least

confusion

Disc of least

confusion

Depth of focus


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2 Electron beam-specimen interactions

Electrons are charged particles and as such interact much more strongly with matterthan X-rays or neutrons. A variety of signals can be collected from the interaction of anelectron beam with a specimen, and these are summarised in the schematic diagram

below.

A schematic diagram illustrating a range of signals that may be generated when anelectron beam is incident on a specimen.

For SEM, the strongest signals are usually the backscattered and secondary electrons,and therefore only these signals will be discussed further in this section.

2.1The scattering process

The incident electron beam suffers many elastic and inelastic scattering events. Theelastic scattering of the primary beam leads to a spread of the incident electrons withinthe specimen.

(a) A Monte Carlo simulation of theelectron trajectories in Cu at 30 kV(b) Illustration of the variation of theelectron scattering with atomic number andvoltage.

Low Z High Z

Low

voltage

Highvoltage

R

(a) (b)


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The electron range is defined as the average total distance travelled by the electronwithin the specimen and is approximately the diameter of the spread. The table belowshows the range,R(in m) as a function of incident energy for different materials.

2.5 kV 5 kV 10 kV 15 kV

Al 0.12 0.4 1.25 2.4Cu 0.046 0.15 0.47 0.9

Au 0.027 0.088 0.28 0.54

2.2Backscattered electrons

Backscattered electrons (BSE) are those of the primary beam that have escaped thesurface, usually after elastic and inelastic scattering events.

Elastic scattering results from the deflection of the electron beam by the positive

charges of atomic nuclei in the sample. This can be analysed simply using the classicRutherford scattering model. In this model, the differential scattering cross section,d/d, is given as a function of the scattering angle and the atomic number,Z, by theexpression

(2.1)

To estimate the fraction of electrons that are backscattered, , we must consider the factthat the electrons must reach the surface after undergoing scattering events. Monte-Carlo simulations are a very powerful method for modelling such electron beam-specimen interactions.

The dependence of the backscattered electron yield on the accelerating voltage and theatomic number is illustrated in the figures below.

Variation of as a function of atomic number.

2sin4

2

Z

d

d


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Variation of backscattered coefficient, , as a function of beam energy.

The yield of backscattered electrons is also dependent on the angle of incidence of theelectron beam on the specimen surface. Empirically, we can express this dependenceas

(2.2)

where is the tilt angle and (Z, 0)is the value of at zero tilt angle ((Z, 0) ~ 0.25from example below).

Variation of with tilt angle.

Beam energy(keV)

Tilt angle(deg.)

( ) ( )

cos

89.00,

89.0,

=

ZZ


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Angular distribution of BSE for different incident beam directions.

2.3Secondary electrons

At the other end of the energy spectrum, low energy secondary electrons (SE) are

generated as a result of the excitation of the loosely bound atomic (valence) electrons.The energy of secondary electrons lies in the range 0-50 eV. The secondary electronyield, , can be related to the primary energy of the incident electron, the number ofouter shell electrons and to the atomic radius. The angular distribution is similar to thatfor BSE.

The probability of escape of a secondary electron,

where z is the depth from the surface and is the mean free path of the secondaryelectrons. For metals ~ 5 and for insulators 100 . Thus, nearly all of thesecondary electrons that escape originate near the specimen surface.

As the primary beam energy increases, the number of secondary electrons generatedincreases, but more are generated deeper in the sample, and therefore the secondaryelectron yield actually falls (for incident electrons of energy greater than 2 3 keV)with increasing beam energy.

(2.3)

5 kV 20 kV 50 kV

Al 40% 10% 5%

Au 70% 20% 10%

( )

z

ep

8.0E


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2.4Specimen charging

At energies > 2 - 3 keV, the total electron yield + < 1, i.e. more electrons enter thespecimen than leave it by backscattering and secondary electron emission.

The total electron yield, + is dependent on the incident beam energy

The backscattered yield, , is always smaller than 1 The secondary electron yield, , shows a peak at ~ 2 - 3 keV and then decays

rapidly with increasing beam energy This results in a peak in the total electron yield that rises above 1

Schematic diagram illustrating the effect of accelerating voltage on the total electronyield.

If there is a good conducting path to earth, a current flows within the specimen to enablethe specimen to remain electrically neutral. However, if the specimen is an insulator,

the surface may become charged:For + = 1 No specimen charging occurs

+ < 1 The number of electrons that enter the specimen is largerthan the number that leaveSpecimen develops a negative charge

+ > 1 The number of electrons that enter the specimen is smallerthan the number that leaveSpecimen develops a positive charge

At low energies it is possible to choose an accelerating voltage where + = 1 to giveconditions where the specimen (even if insulating) will not charge. Modern SEMs are

optimised to operate in this low energy range.

E1 E2

+

1

Accelerating

voltage


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2.5Detecting electrons

The detection system for electrons in the SEM depends on the energy of the electronsemitted from the specimen.

Energy distribution of the secondary (S) and backscattered (B) electrons generated by

an incident electron beam of energy E0.

2.5.1 Backscattered electron detection

Two types of electron detector are commonly used to collect the high-energy BSEemitted in an SEM.

(a)Scintillator detectorThese are composed of a scintillator (phosphor) with a light pipe and a

photomultiplier. They have a fast response time and high gain, which makes themsuitable for use at TV rates. However, they are bulky and may restrict the workingdistance of the microscope.

(b)Solid state detectorThese comprise of a Si p-n junction which forms electron-hole pairs on theimpingement of a BSE. These are separated by the application of a reverse biasacross the junction, giving a detectable current. The detector is much smaller than ascintillator, and therefore can be permanently attached to the objective lens withoutobstructing normal microscope operation. These detectors are cheap to make, buttheir slow response time makes them unsuitable for operation at TV scan rates.

Schematic diagram illustrating (a) a scintillator and (b) a solid state BSE detector.


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2.5.2 Secondary electron detection

The secondary electron signal is detected using anEverhardt-Thornley detector, whichis based on a scintillator-photomultiplier system.

Schematic diagram illustrating an Everhardt-Thornley detector.

The energy of the secondary electrons is too low to be detected directly, and thereforethey are accelerated by applying a bias of ~ +10 kV to a thin Al film on the scintillator.A collector grid which is biased to ~200 V is also used to attract the secondary electronsThis collector screens the scintillator from the incident beam and improves thecollection efficiency of the detector.

When the secondary electrons strike the scintillator, light is produced that is guided intoa photomultiplier. The light is converted into pulses of electrons which are amplifiedand used to modulate the intensity on a CRT.


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3 Image formation

3.1 Topographic images

One of the principal uses of an SEM is to form an image of the surface topography.

For both SE and BSE the yield is at a minimum when the specimen is perpendicular tothe beam (= 0o). As the specimen is tilted, electrons are more likely to be scattered outof the specimen rather than further into it.

For SE

where 0is the yield at = 0oand is the tilt of the specimen. To improve topographic

contrast specimens are often tilted 20 - 40otowards the detector.

We can draw an analogy betweentopographic imaging in the SEM andimaging with light:

(a) Diffuse and (b) direct illuminationwith light, viewed from above.

Secondary electron imaging (c) isequivalent to (a).

Backscattered electron imaging (d) isequivalent to (b).

Topographic images formed using secondary electrons.

sec0=


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Topographic images formed using BSE will, by analogy to viewing an object withdirect light, contain many more shadows than an equivalent image formed using SE asthey are formed only by BSE that are emitted towards the detector, i.e. line of sightBSE. The multi-element nature of the backscattered detector allows the topography ofalmost flat specimens to be enhanced by looking at a difference signal.

The sampling volume for BSE is much larger than that for SE, and therefore the spatialresolution is much lower.

3.2Compositional images

The secondary electron signal is not very sensitive to composition, but may be affectedby the surface condition and the electronic structure of the material.

The backscattered electron yield, , is strongly dependent on the atomic number and isalmost independent of the accelerating voltage (see 2). It is found empirically that foran element of atomic number,Z, the backscattered yield can be calculated using

(3.1)The backscattered yield for a compound can be calculated using a rule of mixtures

based on weight fraction, i.e.

(3.2)

where Wxrefers to the weight fraction of element x. The atomic number contrast, C,can be determined between two phases or elements using

(3.3)

Phase 1 Z1 Phase 2 Z2 1 2 Contrast(%)

Al 13 Mg 12 0.153 0.141 7.6Al 13 Cu 29 0.153 0.304 49.4Al 13 Pt 78 0.153 0.485 68.4Cu 29 Zn 30 0.304 0.310 2.3

-brass 29.4 -brass 29.5 0.305 0.306 0.2

The contrast is, in general, relatively small and therefore to obtain good compositionalimages the surface topography must be kept to a minimum by polishing to a fine finish.

Unetched, polished metallographic specimens are ideal.

For compositional images (formed using BSE), the signals from the two (or four) solidstate detectors are added, e.g. A+B, to optimise the contrast arising from different

backscattered yields across the specimen. If the contrast is low (e.g. brass), then ahigh beam current is required to maximise the signal, and therefore the spatialresolution is reduced.

3724

1030.81086.1016.00254.0 ZZZ

++=

( ) 2111 1 WWcompound +=

1

21

=C


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Backscattered images from a polished silver soldered joint(a) Topographic contrast arising from different abrasion resistance obtained from the

difference in the signals received by detectors A and B(b) Compositional contrast showing atomic number Z differences obtained from thesum of the signals in detectors A and B.

(a)

(b)


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3.3Crystallographic images

3.3.1 Channelling contrast

The backscattered coefficient, , for a crystalline specimen, is dependent on theorientation of the electron beam with respect to the crystal. This effect is known aschannelling.

Channelling contrast is low and may only be obtained satisfactorily from a specimenwith a surface that is free from large surface deformations. The beam must bereasonably parallel (by using a small aperture and a large working distance). A largecurrent is required to obtain reasonable contrast, but this results in poor spatialresolution

BSE intensity as a function of crystal orientation (rocking curve) for Si at 30 kV

Crystallographic channelling contrast from fine grained material. Such contrast is verysensitive to the specimen tilt


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3.3.2 Diffraction patterns

The orientation of a crystal can be determined from a selected area channelling pattern(SACP). By varying the angle of the incident beam, the variation in backscatteredelectron yield, , can be mapped out as a function of angle, thereby forming a SACP.The electron beam is scanned through a rocking angle whilst staying focused on a single

area of the specimen. However, due to aberrations in rocking the beam, the area fromwhich the pattern is formed tends to be quite large (~13 m), and can be larger than asingle grain.

An alternative geometry is also used for obtaining diffraction patterns in the SEM wherethe electron beam is fixed at a point, and the backscattered electrons are collected on a

position sensitive detector (e.g. film or CCD camera) called electron backscattered diffraction (EBSD) the pattern that is formed is known as an electron backscattered

diffraction pattern (EBSP)

The variation of with angle is recorded directly without needing to rock the electronbeam. The specimen is tilted to 70oto increase the backscattered electron yield. Thistechnique can demonstrate better resolution (~1 m) than SACP.

The series of lines observed in an EBSP are Kikuchilines and are similar in appearanceto the channelling contrast observed in SACPs. Kikuchi lines are formed by a series ofscattering events in the crystal:

(i) the electrons are inelastically scattered(ii) they are subsequently Bragg scattered by a series of parallel planes (as

shown schematically in the figures overleaf). If the specimen is thick then

some of the forward scattered electrons will be re-scattered back towards thesurface or will be inelastically scattered again. (For thin specimens, forwardscattered electrons may be seen in a transmitted signal see later in thecourse.)

SACP of a grain in a copper specimen

Schematic diagram showing theformation of a SACP


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Diagrams showing the origin of Kikuchi lines and the cones of rays due to theelastic scattering of diffuse inelastically scattered electrons.

By scanning the beam across the sample and recording and analysing the pattern at each

point a data set is produced in which each point can be coded according to itsorientation. This is the basis of orientation imaging microscopy(OIM). Such analysiscan give the position of grain boundaries, twins and show the relative orientation ofeach grain to reveal any preferred growth or texture.

3.4Other imaging modes

Many other images can be formed from signals that arise from the electron beaminteraction. Some are listed below:

(i) Cathodoluminescence(CL) images

These are formed from visible or near-visible radiation that is emitted from aspecimen. This signal is very sensitive to specimen strain and the presenceof defects. CL imaging has been used extensively in the study of diamond,silicon and other semiconductors.

(ii) Magnetic contrast imagesThe internal magnetic field in a ferromagnetic specimen will deflect theelectron beam and under certain conditions this can lead to image contrastwhich can be used to examine the magnetic structure of the specimen.

(iii) Electron beam induced current imagesThe specimen current is detected as the beam is scanned across the specimen

surface. This current is sensitive to the electrical properties of the materialand is good for imaging semiconductor junctions, for example.

(iv) Voltage contrast imagesThe SE yield is altered by the potential of the specimen. This method isused in the semiconductor industry to examine voltages on interconnects, forexample.


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4 X-ray microanalysis

The X-ray signal that is emitted from a specimen contains characteristic peaks whoseenergy can be related to an atomic transition and thus to a particular chemical species.Hence X-ray microanalysis can be used to investigate the chemistry of a specimen.

There are two electron-beam specimen interactions that must be considered when usingX-ray microanalysis: core scattering and inner shell ionisation.

Core scatteringoccurs when an incident electron excites an X-ray without knocking outan inner shell electron. When this occurs, the electron can lose any amount of energy(up to its total kinetic energy), and the X-ray is not characteristic of any particular atom.This is called Bremsstrahlung and leads to a background of X-rays in any electrongenerated X-ray spectrum.

Inner shell ionisationgives rise to the characteristic peaks in an X-ray spectrum. Thefigure below shows a schematic diagram illustrating the process of X-ray emission byinner shell ionisation.

Schematic diagram illustrating the ionisation process whereby an inner shell electron isejected from the atom by a high energy incident electron. The hole in the K shell is

filled by one from the L shell, and an X-ray is emitted.

If an incident electron beam has sufficient energy to knock an inner shell electron outinto the vacuum, this creates an excited atom in an energetically unstable state. Theatom is relaxed by an electron in a higher energy shell falling into the lower shell and a

photon is emitted. The energy of this photon is the difference between the two energylevels and will typically be in the X-ray spectrum.

The Auger process may also occur. In this process the emitted photon knocks anotherelectron out from its shell. These Auger electrons can be detected under UHVconditions and form the basis of another analytical technique Auger ElectronSpectroscopy (AES).

Energy

EL3EL2EL1

EK K

L1L2

L3

Valence band

Conduction band

Vacuum

Characteristic

X-ray

Energy loss electrons

Incident

electron beam


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Some of the more common transitions are shown in the diagram below. Of thenumerous possible transitions, the strongest are the lines for each series, e.g. Kwhich is 7-8 times stronger than the Kline.

The electron transitions between the K, L, M, N and O shells leading to K, L and Mcharacteristic X-ray emission.


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4.1X-ray nomenclature

Atomic energy levels can be classified according to their quantum numbers.Accordingto the Pauli exclusion principle, each electron state has a unique set of quantumnumbers, n, l, mland ms.

The principle quantum number, n, describes the quantised energy and gives thevolume of an orbital, with n= 1, 2, 3 etc.

The orbital angular momentum quantum number, l, describes the angularmomentum of an orbital (i.e. its geometry), with l= 0, 1, 2, 3. n-1 (when l= 0, theorbital is spherically symmetrical and denoted as an s-state)

The magnetic quantum number, ml, gives the orientation of the angular momentum ofthe orbital, with ml= -l, -l+1, -l+2,., 0, ., l-2, l-1, l

The spin quantum number, ms, describes the spin of the electron, and can take thevalue of +1/2 or -1/2

According to these rules, each electron shell can contain only a limited number ofelectrons, because no two electrons are allowed to have the same quantum numbercombination,

e.g. the n = 1 shell has a l = 0 electron state (the 1s state) which can be occupied by amaximum of 2 electrons, one with ms= 1/2 and one with ms= -1/2

When discussing atomic spectra, it is convenient to define allowed combinations ofquantum numbers as n, l, j and mj, where j = l ms is the total angular momentumquantum number, and mj= -j, -j+1, -j+2, 0,j-2,j-1,j.

Electrons with the same n, land jhave the same energy (in the absence of an externalfield). For each given l, the jstate ranks the relative energy of the energy level thestate of lowestjlies lowest in energy.

Allowed transitions between energy levels must obey electric-dipole selection rules: The principle quantum number must change in order to create an excited

state, n 0 Angular momentum must be conserved l= 1 only Spin-orbit coupling must be considered j= 0, 1 only

The possible electron transitions and associated X-rays can be assigned quantum

numbers, and the total number of electrons in each level can be specified:


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X-ray

energy

level

n l j No. ofelectrons

s-p-d

designation

K 1 0 2 1s1/2L1 2 0 2 2s1/2L2 2 1 2 2p1/2L3 2 1 3/2 4 2p3/2M1 3 0 2 3s1/2M2 3 1 1/2 2 3p1/2M3 3 1 3/2 4 3p3/2M4 3 2 3/2 4 3d3/2M5 3 2 5/2 6 3d5/2

The designation of lines in terms of 1, 2etc. is a recognition of the relative intensitiesof the X-ray lines. The intensities depend on the relative probabilities of the transitions

and the extent to which a particular state is filled with electrons.

4.2Practical aspects of X-ray analysis

The most efficient production of X-rays generally occurs when the incident electronbeam has about 3 times the energy of the characteristic X-ray of interest. The table inappendix 1 shows that there is at least one strong characteristic peak for all elements inthe 0 - 10 keV range and therefore there is no problem for an SEM operating at25 30 kV.

X-rays (perhaps generated by electrons) can themselves excite atoms and generatefurther X-rays of lower energy, a process known as fluorescence. It is a rather

inefficient process and only a few % of X-rays are generated in this way, but it can alterthe relative amounts of characteristic radiation generated from alloys and compounds,

particularly if elements of similar atomic number are present, e.g. -brass.

The interaction volume is smallest for low energy electrons and heavy elements. It isdifficult to reduce the sampling volume to less than 1 m3without reducing the beamenergy to a level at which few useful X-rays are emitted. The volume of the specimenfrom which X-rays are generated is similar to that sampled by the electron beam, but iffluorescence occurs, then the sampling volume can be larger than the interactionvolume.

The volume analysed and the fraction of X-rays emitted depends critically on(i) The energy of the electron beam(ii) The energy of the X-ray studied(iii) The local atomic weight

Values of the energy and associated wavelengths of the strongest K, L and M lines ofthe elements are shown in appendix 1. X-rays are electromagnetic rays that travel at thespeed of light, and therefore E (keV) = 1.24/.


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4.3X-ray detection

There are two types of X-ray detector systems using either energy dispersivespectroscopy (EDS) or wavelength dispersive spectroscopy (WDS).

4.3.1 Energy dispersive spectroscopy (EDS)

This is by far the most widely used detector system. The detector is based on a p-i-njunction in silicon. Each incoming X-ray creates a number of electron-hole pairs wherethe number generated is proportional to the energy of the X-ray.

e.g. each electron-hole pair in Si has an energy of 3.8 eV, and therefore, for example, anAl KX-ray will give rise to 392 electron-hole pairs.

By putting the junction under an applied bias, these electron-hole pairs generate acurrent, which is amplified and analysed. The junction is doped with Li and cooled to77 K to ensure that the natural conductivity of the Si does not swamp the X-ray

generated signal.

The outer surface of the detector is normally coated with a polymer film (or Be) toprevent contamination (mainly hydrocarbons) from condensing onto the cold detector,but these films absorb low energy X-rays, making EDS detectors very poor at detectingthese.

Each X-ray pulse only lasts for a very short time (~1 s), and the current arising fromthe pulse is amplified and passed to a multi-channel analyser which decides into whichof the 1024 channels to place the signal. The channels typically represent energies inthe ranges of 0-10, 0-20 or 0-40 kV.

Several thousand pulses can be processed per second, and therefore an EDS spectrumcan be obtained in a short time. The detector resolution is only at best 100 eV, andtherefore fine spectral features cannot be discriminated, and lines from differentelements can overlap.

A typical X-ray spectrum taken on a SEM from the high-T superconductorBi-Sr-Ca-Cu-O.


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4.3.2 Wavelength dispersive spectroscopy (WDS)

If higher energy resolution is required, then wavelength dispersive spectroscopy (WDS)is used. In WDS, X-ray radiation is filtered such that only X-rays of a certainwavelength are incident on the detector. The X-rays are filtered using a crystalspectrometer arranged to allow diffraction to separate the X-rays according to their

wavelength (i.e. using Braggs law).

To count as many X-rays as possible, the sample-crystal-detector geometry must beengineered to fall on the arc of a circle the Rowland circle. To cover the range ofX-ray wavelengths, typically 4 crystals with different d-spacings are used because therange of accessible is limited by the geometry of the system.

A crystal X-ray spectrometer for WDS. X-rays are collimated by the two slits S1andS2, diffracted by the crystal and focused onto the detector.

Although the energy resolution is excellent (~ 10 eV), the collection efficiency is verypoor (~2% of an EDS system). However, the background intensity is negligible andvery high count rates are possible. Light elements can be detected very easily (unlikeEDS).

WDS systems are generally found on dedicated electron probe microanalysers (EPMAs)and rarely on conventional SEMs.

Electronbeam

Detector

S3

S1S2

X-rays

Rowlandcircle

R

Curvedcrystal


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4.4Quantitative X-ray mapping in the SEM

To estimate the amount of an element present in a sample, the number of counts in afixed time interval, Nspec, can be compared with those from a standard of knowncomposition,Nstd.

If the concentration of an element within a compound is C, then

Each value of N represents the net count in the window of interest (after backgroundsubtraction). For this to hold true, the conditions must be identical when collecting each

N.

However, complications arise because the specimen is unlikely to be a pure element,and therefore 3 correction factors are needed for

(i) Atomic number (Z)(ii) Absorption (A)(iii) Fluorescence (F)

Together, these contributions are known as the ZAF corrections:

kZ, kAand kFall require some knowledge of the specimen concentration, and thereforethe process is iterative. These factors are usually set to a value of 1, and Cspec isestimated. This is then used to estimate the kfactors and determine a better value forCspec. This refinement usually only takes 3 - 4 iterations.

(i) Z-correction

This is a correction which accounts for the differences in the efficiency of X-raygeneration which depends on

(a) how far the electrons can penetrate the specimen before losing too muchenergy to excite X-rays(b) how many electrons are backscattered without exciting any X-rays.

For analysis of a heavy element in a light element matrix, kZ> 1 (and vice versa)e.g. Sulphur (Z = 16) in stainless steel (mean Z = 27), kZ= 0.87

(ii) A-correction

The amount of absorption by a specimen depends strongly on the elements present in aspecimen through their mass absorption coefficient, .

ReII = 0

whereI0is the intensity at a depth Rin the specimen, andIis the intensity of the X-rayat the surface.

stdstd

std

spec

spec kCCN

NC ==

stdFAZspec CkkkkC =


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It is likely that the specimen and standard will have different absorption coefficients,and therefore this must be considered when quantifying X-ray emission. The magnitudeof this correction can be large, particularly for K lines of light elements emitted fromspecimens also containing heavy elements.

(iii) F-correction

Only a small fraction of high energy X-rays excite low energy fluorescent radiation. Itis an inefficient process but can be important in samples with elements of similar atomicnumber, e.g. Cr in Fe (kF= 0.85).

To determine these factors accurately, the specimen-detector geometry must be wellcharacterised, and samples must be polished flat. An accuracy of ~ 3% in EDS and ~1% in WDS is achievable.

4.5Artefacts and problems

Three common situations can occur where a false reading of local composition can beobtained:

4.6Detectability limits

If the mean background count is B, then assuming Poisson statistics, the noise is B.We can define a peak as being detectable if it rises above the background by 2B. If thecount rate of the background gives b counts per second, and the counting time is t, the

background count, btB = , giving a smallest detectable peak height,

( ) btBP 2= above the background level.

If pis the peak count rate from the standard of known composition, it follows that theminimum detectable concentration (MDC) is given by

Therefore,(weight %)

For most elements, MDC is ~ 0.1% for EDS and ~0.01% for WDS

In a region apparentlyconsisting only ofelement A, an area of

element B below thesurface can emit B X-rays.

Near the boundary of Aand B, both types of X-ray are excited even

though the beam isfocused onto the surfaceof A.

In a rough specimen (e.g.fracture surface), B X-rays can be excited by

fluorescence from the Aregion.

( )

( ) ( )tbpbt

BP

BPMDC

std

spec

=

=

1002100

( ) tbp

bMDC

=

200


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4.7X-ray mapping

If the electron beam is scanned across the specimen surface and an X-ray spectrum isacquired at each point then a 2-D map can be built up. Subsequent data analysis wouldallow the chemical composition and distribution of elements to be determined to highspatial resolution. Many signals can be acquired at once, allowing a multi-dimensionaldata set to be acquired containing X-ray spectra, backscattered and secondary electroninformation.

Digital X-ray maps obtainedsimultaneously from a polished

specimen of an alloy (the data wereacquired in 60 mins.).

(a)Si K(b)Mo L(c)Cr K(d)Co K(e)Detail of (d) showing the

individual pixels.(f) Backscattered image of the

same area.


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Transmission electron microscopy

5. Introduction

The upper half of a TEM is very similar to an SEM. There is an electron gun, an

accelerating voltage (now much larger, typically 100 - 400 kV) and electron lenses tocontrol, collimate and focus the electron beam. There is a condenser aperture that has asimilar function to the objective aperture in the SEM. In an SEM the final lens beforethe specimen focuses the electron beam onto the surface. In the TEM, the specimen isvery thin, and the objective lens is placed after the specimen where it is used to form animage from the electrons that have passed through the specimen. In a TEM, the firstcondenser lens is typically used to adjust the spot size and the second condenser lens isused to either form a focused probe on the specimen surface (as in SEM) or a parallel

beam (for conventional TEM imaging).

There are many additional lenses and apertures in a TEM (as illustrated below) whichdo not have an analogue in an SEM. These will be described in 3.

A cross-section through a modern analytical TEM showing the main components and anumber of detectors.

If scan coils are fitted to the TEM then it can be used as a high voltage, high resolutionSEM if detectors are fitted to examine the backscattered and secondary electron signals.


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If the transmitted signal is detected from a TEM with a scanned probe, then we call themode of operationscanning transmission electron microscopy (STEM).

Many different signals are generated in a TEM (as illustrated on p 9), and if we examinea thin specimen, the elastically and inelastically scattered electrons that are transmittedcan be analysed to deduce the chemical and physical properties of the specimen.

However, to obtain the high resolution that is often required in a TEM, the specimenmust be placed within the objective lens, and this limits its maximum size, usually to ~3mm in diameter, and 1 mm in depth (although the area of interest must be muchthinner). A variety of holders are available for inserting the specimen into the TEM,including straining, heating and cooling stages.

A variety of TEM specimen holders

5.1 Specimen preparation

For TEM we need a specimen that is electron transparent with a thickness and geometrythat are suited to the analysis required. Specimen thicknesses are typically in the rangeof 10 - 300 nm depending on choice of material and technique required.

Many different sample preparation techniques are available to prepare such thinmembranes, however none is particularly easy! These include argon ion beam thinning,mechanical polishing, electropolishing and focused ion beam milling.

Most of these techniques thin a circular disc of material in the centre at a shallow angleuntil a hole just forms. The area surrounding the hole is usually electron transparent.

A cross-section through a conventionally thinned specimen

If the material is conducting, electropolishing can be used. However, if the material isinsulating or semiconducting, or if it has a composite structure with differentelectrochemical potentials (which would selectively electropolish), then argon ionthinning is often used. In this method, a low energy beam of argon ions is fired at a


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low angle at the centre of a rotating disc of material to sputter away material and createa region of electron transparency.

However, many examples of materials analysis now requires the preparation of amembrane at a particular point in a specimen, for example a gate structure in asemiconductor device, quantum dots, or targeting a particular grain in a material.

Focused ion beam (FIB) milling allows a specimen to be imaged (using secondaryelectrons) and milled simultaneously to select exactly the area required for TEMexamination. The beam is typically formed of Ga+ ions that are accelerated to~ 5 - 30 kV and these are used to sputter material to leave a thin membrane suitable forTEM examination lying in the area of interest.

(a) A SE image taken in a FIB workstation of a trench prepared in a silicon wafer withthree different thickness membranes separated by trenches (b) A silicon MOSFET

device prepared and imaged in the FIB (c) A SE image taken in a FIB workstation of aAl polycrystalline film on a Si substrate.

(a) (b)

(c)


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6. Beam-specimen interactions

The transparent nature of specimens examined in the TEM means that electrons thathave been forward scattered can be detected, and these are the basis of most TEManalysis. The high accelerating potential means that most of the electrons are scatteredin the forward direction or unscattered by the specimen, with only a very small fraction

being backscattered for typical TEM specimen thicknesses.

The electrons that are transmitted by the specimen can be scattered or unscattered. Theelectrons that are scattered can be elastically scattered or inelastically scattered (if wethink of them as particles), and they can also be termed as coherent or incoherent (if wethink of them as waves).

Elastically scattered electrons are usually coherent (i.e. in phase with the incident,unscattered electron wave) in thin specimens, and are scattered to small angles (1o - 10o)in the forward direction. At higher angles, elastic scattering becomes incoherent.

Inelastically scattered electrons are almost always incoherent, and are scattered to lowangles (


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This gives a relativistic wavelength for electrons,

+

=

20

0 212

cm

eVeVm

h

(6.4)

A typical electron wavelength is 0.0025 nm for 200 kV electrons.

6.2The atomic scattering factor

The atomic scattering factor, fel() is a measure of the amplitude of an electron wavescattered from an isolated atom. For X-rays,

( ) ( ) ( ) rK.r dirmc

ef

atom

X = 2exp22

(6.5)

where(r)is the electron charge density in the crystal and Kis the scattering vector inreciprocal space, with |K|=2sin/.For electrons,

( ) ( )( )

xfZ

h

emf

=

2

2

20

sin2 (6.6)

where Z is the atomic number. The first term in the brackets is due to Rutherfordscattering from the nucleus and the second term is due to scattering from the electroncloud.

Atomic scattering factors plotted as a function of sin/

6.3Scattering from a unit cell

We can examine scattering from a crystalline solid by considering scattering from eachunit cell. We simply add up each atomic contribution but must take into account therelative positions, ri, of the atoms (relative to an origin) using a phase factor 2(K.ri),where Kis a diffraction vector. This is called the structure factor,F(), and is given by

( ) ( ) ( )ii

i ifF K.r 2exp = (6.6)


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In general we are only interested in the value ofFat the reciprocal lattice points, i.e. forK= g, and so we would normally write

( ) ( )ii

ighkl ifFF g.r 2exp == (6.7)

The potential of the crystal as seen by the electrons can then be written as( ) ( )i

g

g iFem

hV g.rr

2exp

2 0

2

= (6.8)

The choice of a non-primitive Bravais lattice (e.g. F or I) means that, for certainreflections,Fhkl= 0. For example, in Cu (c.c.p.),Fhkl= 0 unless h, kand lare all odd oreven.

A quantity related to the structure factor is the extinction length, g, given by

g

C

g F

V

cos= (6.9)

where is the Bragg angle and VC is the volume of the unit cell. For example, for100 kV electrons, 111(Au) = 159 , 111(Si) = 602 and 111(Ge) = 430 .

6.4Elastic scattering from a perfect crystal

Consider elastic scattering from a crystal unit cell in the far-field condition (withparallel illumination by the incident electron beam). If we define the wave vectorK = k-k, with |k| = |k| =1/. The path difference between the two scattered wavesfrom O and B is (OC-DB) = (K.r)., leading to a phase difference of 2(K.r).

Thus, if ris a lattice vector, such that rn= n1a+n2b+n3cwhere n1, n2 and n3 are integers,

we find that for constructive interference K.rn= N, whereN is an integer. This is theLaue criterion, which is equivalent to Braggs law in reciprocal space.


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6.5Polycrystalline materials

Diffraction from polycrystalline materials, assumed to be a set of randomly orientedpolycrystals, can be described by rotating the reciprocal lattice about all axes to producea set of nested spheres. When intersected by a relatively flat Ewald sphere, we get ringsof reflections. If the specimen is textured, the nested spheres will break up into sections

of preferred orientation, and this will lead to arcs of intensity in the diffraction pattern.

6.6Amorphous materials

A diffraction pattern from an amorphous material will also be a ring pattern, but therings will be diffuse. The Fourier transform of such a pattern can reveal the radialdistribution function (rdf): the probability of finding an atom at a certain distance awayfrom another. The diffuseness of the rings indicates the lack of long-range order.

7.Image formation

The lenses and apertures in a TEM can be used in different ways to form either animage or a diffraction pattern.

Ray diagram illustrating the action of the post-specimen lenses to form either an imageor a diffraction pattern on the viewing screen.


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7.1Bright field and dark field image formation

The objective lens focuses the scattered beams into the back focal plane and forms adiffraction pattern in that plane. As we will see later, typically an image is formed witha number of beams (often only one) selected using an objective aperture which is placedin the back focal plane of the objective lens. Only those beams are then used to form animage. The intermediate and projector lenses magnify the image onto the viewingscreen so that the plane of the viewing screen is conjugate with that of the image.

Ray diagrams illustrating how the objective lens and aperture are used to produce (a) abright-field (BF) image, (b) a dark-field (DF) by displacing the aperture and (c) a DFimage by tilting the beam. The microscope alignment in (c) is preferred to (b) because

the beam of interest passes along the optic axis and will be less susceptible to lensaberrations.

(a) (b) (c)


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7.2Diffraction pattern formation

To view a diffraction pattern on the viewing screen the intermediate lens (and veryweakly the projector lens) is changed so that the plane of the screen is now conjugatewith the back focal plane of the objective lens. A selected area (SA) aperture is insertedin a plane that is conjugate with the specimen plane and is used to select the area from

which the diffraction pattern is formed. The SA aperture is demagnified back to thespecimen plane by ~50 times so that a 50 m aperture selects a region of only 1 m insize.

Ray diagram showing (a) selected area and (b) convergent beam electron diffraction

(CBED)If the electron beam is focussed to a spot on the specimen a convergent beam electrondiffraction pattern is formed in the back focal plane of the objective lens. This is oftencalled micro- or nano-diffraction as the volume of the specimen sampled by the electron

beam is much smaller than conventional SA diffraction. The diffraction pattern iscomposed of discs (rather than spots) and the detail within the discs is sensitive to thecrystal symmetry and to any strain in the crystal arising from defects.

(a) SAED and (b) CBED pattern from axis of Si taken at 200 kV.

(a) (b)


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7.3Excitation error and finite crystals

An electron passing through a crystal may be unscattered, it may undergo one scatteringevent or many. For the purposes of analysing the intensity present in images anddiffraction patterns, it is useful to make an approximation of single scattering, called thekinematic approximation. In this approximation, the intensity of a reflection in adiffraction pattern, Ig|Fg|2. In general this approximation is not valid, but much of thecontrast seen in TEM images can be described (at least qualitatively) within thisapproximation. Unless stated otherwise, this approach will be used for the followinganalysis.

A TEM sample can be approximated to a rectangular parallelpiped with sides of lengthA, B and C:

If we consider the amplitude of a scattered electron wave, g, and sum over all of the nunit cells within a crystal, then

ni

n

gg eFK.r 2= (7.1)

whereFnis the scattering amplitude from the nth unit cell and rn= n1a+ n2b+ n3candn1, n2and n3are integers.

In general, the reflection g from which an image is taken does not lie exactly on theEwald sphere but slightly away from it. We can define the scattering vector, K= g+ sfor some small value of s that describes the deviation of the vector of the selecteddiffracting beam g from the Ewald sphere. Thus we can redefine the scatteredamplitude as

( ) nn i

n

g

i

n

gg eFeFs.r.rsg 22 + == (7.2)

since all the g.rn are integers. Approximating the summation to an integral (since sissmall and slowly varying),

321 rs.r deFV

crystal

i

g

C

gn

= (7.3)

Defining r= xa+yb+zc and s= ua*+ vb*+ wc*, the integral may be evaluated toshow

( ) ( ) ( )

w

Cw

v

Bv

u

Au

V

F

C

g

g

sinsinsin= (7.4)

The intensity distribution (the shape of the diffraction spot) is given by I = |g|2and is

a sinc2function.

C

A B


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The intensity distribution along wfor u= v= 0. The FWHM w~1/C where C is thespecimen thickness (defined in the diagram above).

Schematic diagram showing the scattering geometry for a thin film normal to theincident electron beam. The spike represents the central peak in the sinc2function and

shows how that function is mapped out as svaries.

Thus, as the crystal is tilted, the Ewald sphere sweeps through the Bragg condition from sto + s(or vice versa).

N. B.: The sign of s can be determined using Kikuchi lines (that are generated in asimilar way as in the SEM), as illustrated in the diagram below.

Consider again the parallelpiped and let B and C . Then, for a crystal of thickness t(i.e. A = t) and for sperpendicular to the crystal surface and perpendicular to g, we find(for s= |s| = w)

( )

( )22

222

2sin

cos),(

s

ts

Vk

FtsI

BC

g

g

= (7.5)

C

1

C

2

C

1

C

20 w

w ~ 1/C


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This can be re-expressed in terms of the extinction length, g,

( )

( )22

2

sin),(

s

tstsI

g

g

= (7.6)

In this kinematic 2-beam condition, only a single diffracted beam is strong. If theintensity of the diffracted beam is Ig, then we can say that the intensity of the

undiffracted beam, I0= 1-Ig(for an incident beam of unit intensity), and it is assumedthatI0>>Ig.

7.4 Breakdown of the kinematic approximation

If we examine the 2-beam condition as we approach the exact Bragg condition,

i.e. s 0, then we see from equation (7.6) that

2

g

g

tI

.

If t > g/thenIg> 1, and this is absurd! The kinematic approximation is valid at largevalues of s, but at small values of swe must use the more exact, dynamical form of

equation (7.6):( )

( )222

2222 sin),(

+

+

=

g

g

g

gs

st

tsI

(7.7)

( )

( )22

2

sin),(

s

tstsI

g

g

=

( )

( )2222222 sin

),(

+

+

=

g

g

g

gs

sttsI

Plots of intensity as a function of deviation parameter, s, close to the Bragg conditionfor both the kinematic approximation and dynamical theory for a specimen of fixed

thickness.

N. B. For the dynamical theory at s= 0:

=

g

g

tI

2sin andIg= 0 when t/g= n, an integer.


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7.5Bend contours

For a thin TEM specimen of approximately constant thickness, if it is bent or buckled(e.g. like a saucer or a saddle) then bend contours are observed in the TEM. Theintensity observed in a TEM image can be described using equation (7.6) as thedeviation parameter, s, varies with the bending of the specimen.

(a) Schematic diagram showing scattering at the Bragg angles from a thin, bentspecimen. (b) A bright field image taken from a bent Al crystal near the [215] axis

illustrating strong scattering when the planes are bent at the Bragg angle.

End-on(hkl)plane

sg< 0

sg= 0sg= 0

Increasingsg

Increasingsg

{hkl}planes

{hkl}planes

Diffractedbeams

-G G0-2G 2G

(a) (b)


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7.6 Thickness fringes

Thickness fringes are very commonly observed in the TEM because most specimens arevery shallow wedges (from the thinning process). The intensity of the diffracted beams,

Ig, oscillates as a function of t(sis fixed for a given specimen orientation).

(a) At the Bragg condition, intensities of the direct and diffracted beams oscillate in acomplementary fashion.

(b) For a wedge shaped specimen, the fringe separation shown in (c) is determined bythe wedge angle and the extinction distance g.

(d) A BF image of thickness fringes. The arrows mark the position of the specimenedge.

As above, for the dynamical theory at s= 0:

=g

gtI

2sin andIg= 0 when t/g= n, an integer. Therefore, the intensity in the dark

field image is a minimum when the thickness, t, is an integer number of extinctionlengths for a given diffraction condition.

(a) (b)

(c) (d)


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7.7 Contrast from imperfect crystals

7.7.1.Stacking faults

Illustration of a stacking fault running across a specimen causing a displacement Rbetween the two parts of the crystal.

The stacking fault introduces a relative shift, R, between the atomic positions of the tworegions of crystal that bound the fault, i.e. rn= rp + R,where rp is the perfect crystal.This creates a phase shift between the electron beams above and below the stacking

fault and leads to an image composed of fringes whose intensity distribution is given by

( ) ( )

+

+

+= sztsts

sI

g

g

2cos2

sin2

sin22

sin2

sin1 22

2 (7.8)

where z is the distance from the specimen centre to the fault and is the phase shiftgiven by = 2g.R. Therefore, for a fixeds, t, and R, the image is composed of fringeswith a periodicity of z= 1/s.

(a) BF and (b) DF images of a stacking fault with = 2/3 in a Cu-Al alloy near the zone axis

The stacking fault is invisible if g.R = 0 or an integer value. Thus, Rcan be determined

experimentally by finding DF images using three linearly independent g vectors thatshow no stacking fault contrast.

(a) (b)

111 g[111]- -

[020]

[101]-

Fault

planeCrystal 1

Crystal2

R

Column is displaced

by distance R


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[The form of equation (7.8) can be appreciated by imagining the propagation of theelectrons through thin slices of the crystal. After propagation through thickness t, theamplitude of the scattered wave can be expressed (following equation (7.1)) as

( )

( )( )r

r

rRsg

rsg

de

de

p

n

i

i

g

++

+

2

2

(7.9)

If R= 0 for t< t1(the depth of the stacking fault at the position of interest), and if weignore the small term s.R, letting s.rp= sz then we can express equation (7.9) as

dzeedze

t

t

iisz

t

isz

g +

1

1

2

0

2 (7.10)

As before, the intensity,Ig|g|2, and from this equation (7.8) can be obtained.

7.7.2. Dislocations

Contrast arising from the presence of dislocations in a crystal can be interpreted in asimilar way to that for stacking faults. The displacement vector, R, varies as a functionof position with respect to the dislocation.

For a screw dislocation:

The phase angle is given by = 2g.R, where

(7.11)

If g.b= 0, = 0 and the dislocation is invisible. As with stacking fault contrast, bcanbe determined by using three different dark field images with 3 linearly independent gvectors that show no dislocation contrast.

R

xy

tz

u

b bis the Burgers vectoruis the line vectorFor a screw dislocation,b // uFor an edge dislocation, b u

==

x

yz1tan22

bbR


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Two BF images taken with different diffraction conditions (a) g = 200 and (b) g = 313of screw dislocations in Si.

Edge dislocations are not as straightforward because even if g.b= 0, then there mightstill be some component of displacement causing diffraction from a g.(bxu) term.

A schematic illustration of the distortion of a crystal structure around a dislocation core.

On one side of the dislocation, the planes might be bent closer to the Bragg condition(i.e. s is smaller) such that the intensity, Igis higher than the background. On the otherside of the dislocation, the planes would be bent away from the Bragg condition,therefore Ig~ Ibackground.

Schematic diagram illustrating the position of the dislocation image as a function of thesign of gand s.

bu

bu

Dislocationcore

Image line of thedislocation


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If the crystal is tilted through the Bragg condition, the distorted planes sweep throughthe Bragg condition, and the bright image of the dislocation moves from one side of thedislocation core to the other. Reversing either g or s will reverse the position of theimage of the dislocation relative to the dislocation core.

We can also observe contrast from dislocation loops, where the Burgers vector b is

constant but the dislocation line, u, varies around the loop. Thus, even if g.b= 0 we seesome contrast except for the positions where g.bxu = 0, where no contrast would beobserved.

7.8 High resolution TEM (HRTEM)

So far we have generally considered images formed from one beam, either a diffractedor undiffracted beam, and these are examples of amplitude contrast. If more than one

beam is selected in the objective aperture, interference between these beams leads toimage contrast arising from phase differences between the beams. High resolutionimages are formed by using the objective aperture to select two or more beams in the

back focal plane of the electron microscope.

If we consider a two beam-condition at exactly the Bragg condition (s= 0), i.e the directunscattered beam 000 and one diffracted beam, g, which can be described as planewaves, ( )k.riA 2exp1 = and ( ).rk'2exp2 iB = , then the intensity of the imageformed is given by

g.r2sin222 ABBAIg += (7.12)

where ris a real space vector, A and B are the amplitudes of the two waves (assumingthat they are real) and g= k-k. Therefore in the image we see a sinusoidal oscillationin intensity with a periodicity r = 1/|g|.

If we include an array of reflections centered on the 000 beam, the image formed will bea cross-grating of sine fringes generated by the interference between all the beamsincluded in the aperture. These lattice fringes are not necessarily direct images of thestructure because the phase relationship between the beams is distorted by theaberrations of the lenses.

g In the

image

plane:Fringe spacing

1/|g|

In the

back focal

plane:


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Mathematically, if the object is described by a function, g(r) and the image by f(r), then

)()()( rrr hgf = (7.13)

where h(r) is a real space function that describes the effect of the electron lenses as ittransmits the information. In Fourier space, equation (7.13) becomes

)()()( uuu HGF = (7.14)

where H(u) is the contrast transfer function of the electron lenses (mainly dependent onthe aberrations in the objective lens), and can be expressed

)()()()( uuuu BEAH = (7.15)

where A(u) is the aperture function (there is no information transmitted beyondthe edge of the objective aperture)E(u) is the envelope function (attenuation of high frequency information)

B(u) is the wave aberration function

( )[ ]uu iB = exp)( (7.16)

where 43221

)( uCuf s +=u (7.17)

f= defocus. If f < 0, the image is underfocussed, if f > 0, the image is overfocussed.

A characteristic quantity for the resolution of an electron microscope is the objectivelens transfer function, T(u), often referred to as the contrast transfer function.

A plot of sin (u) for a 300 keV microscope with Cs= 0.6 mm at Scherzer defocus.

u[nm-1]

sin(u)


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The contrast observed in a HRTEM image depends strongly on the defocus, f, andspherical aberration, Cs, and varies for different beams (i.e. with different u). Imageinterpretation is therefore not straightforward! It is found that for a particular defocus,called the Scherzer defocus, that the image can be interpreted with relative ease. At thisvalue of defocus, d/du = 0, = -2/3 and

21

34

= s

Sch

Cf (7.18)

which leads to a resolution limit, ( ) 41

366.0 sSch Cr = (7.19)

HRTEM image of a double walled carbon nanotube filled with crystalline KI.


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8. Advanced TEM

8.1 Microanalysis in the TEM and STEM

8.1.1.Energy dispersive spectroscopy (EDS)

As in SEM, the X-ray signal generated by the electron-specimen interaction can bedetected using an EDS system. However, the geometry of the objective lens restrictsthe collection angle of the EDS detector and therefore X-ray detection is very poor inthe TEM. The generation volume for X-rays is also significantly reduced in the thinTEM specimens, therefore giving fewer X-rays but higher resolution, which isdescribed using the beam spread, B (in nm):

23

21

198.0 tAE

ZB

=

(8.1)

whereZis the average atomic number of the specimen,is the density (in g cm-3),Eisthe incident beam energy (in keV),Ais the average atomic weight and tis the thicknessof the specimen (in nm).

To perform EDS in the TEM, the beam is normally focussed to a fine spot. In modernTEMs this can be ~ 0.5 nm, and this is scanned in a raster to produce X-ray line scans ormaps at very high resolution. To achieve a sufficient number of counts from such asmall area then a high brightness gun is required (a FEG is ideal).

STEM X-ray maps taken near a carbide particle in a Al-Ni-Ti alloy

If the film is sufficiently thin, then the three ZAF corrections can be replaced by a singlekfactor, such that

B

AAB

B

A

N

Nk

C

C= (8.2)

where CX is the weight fraction of the element X and NX is the number of countsdetected. Normally kis determined experimentally with respect to a common elementsuch as Si so that

(8.3)

BSi

ASiAB

k

kk =


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8.1.2. Electron energy loss spectroscopy (EELS)

If an inelastic scattering event occurs to an electron passing through a specimen, theelectron loses energy. This energy loss can be measured by using a magnetic field todeflect the electron beam. Electrons of different energies will be deflected throughdifferent angles and therefore by using either a slit to select electrons of a particular

energy, or a position sensitive detector, the energy loss spectrum from a thin specimencan be obtained. The electrons are usually detected using a scintillator and diode arrayor a CCD camera.

A schematic diagram of an EELS spectrometer for parallel detection (PEELS).

A typical EELS spectrum consists of a zero loss peak (containing elastically scatteredelectrons), low loss peaks (mainly valence band excitations) and high energy ionizationedges which correspond to excitations from the inner shell states to the vacuum. Thecross-section for scattering from light elements is large, and so EELS is an excellentway of detecting light elements such as B, C, N and O which may be difficult to detectusing EDS. The position and shape of the edges are used to determine the chemicalcomposition of the specimen.

Electron energy loss spectrum obtained from 316 stainless steel.


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Although it is relatively simple to detect the elements present, it is harder to quantifytheir relative proportions within the specimen. If the specimen is sufficiently thin, wecan assume that only single scattering events occur, and therefore some quantification is

possible. We can remove a background count at a particular edge, and calculate thecounts present after the edge, as illustrated below:

Schematic EEL spectrum showing the low-loss region and a single ionisation edge

The elemental concentration, C, can be calculated from

(8.4)

where tis the thickness of the specimen, is the partial cross-section for energy losseswithin a range of the ionization threshold, and the other quantities are illustrated in

the diagram above.

The low loss spectrum can be used to determine the electronic nature of the sample, anda Kramers-Kronig analysis can be used to determine the real and imaginary parts of thedielectric constant, . The intensity of the EEL spectrum I Im(1/).

EELS can also be used to measure the thickness of a specimen. In most samples,plasmon scattering is dominant and can be described in terms of Poisson statistics wherethe probability of scattering into the nthplasmon is given by:

tn

n et

nP

=

!1

(8.5)

Therefore, by measuring the total spectrum, P = 1, and the zero loss, P0 = e-t/, then thethickness can be found from

=

0

lnI

It total

(8.6)

where is a characteristic mean free path for inelastic scattering which can bedetermined by other means.

( )( ) ( )tI

IC

k

k

=

,,,

I0

I(,,,,)

Ik(,,,,)

Ikl

(,,,,)


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8.1.3.EELS ionisation edge nomenclature

The high energy ionisation edges have a nomenclature that is illustrated schematicallyin the diagram below. Some edges, such as the L2 and L3 edges, may not be resolvableand then are simply called the L2,3 edge. The thresholds for various ionisation edgesare shown in appendix 2.

The range of possible edges and EELS nomenclature arising from inner shell ionisation.

8.1.4.EELS fine structure

The fine structure at the beginning of an edge carries a great deal of information aboutthe chemical state of the species in question. This means that we can distinguish

between different oxidation states, for example, the fine structure in the Cu L-edgearising from metallic copper is different from the fine structure seen from Cu 2+in CuO.Likewise, we can distinguish between hybridisation states of carbon, as illustrated

below.


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(a) EELS spectra obtained from CuO (Cu3d9) and from elemental Cu (Cu 3d10 4s1)

(b) C K-edge from a variety of C-containing compounds.

8.1.5. EDS vs EELS a simplistic comparison for TEM

(a) (b)


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8.2 Other advanced TEM techniques (a small selection!)

(a) Electron tomography

So far we have only considered imaging specimens in two dimensions in the TEM. Bythe very nature of TEM, all images formed are a projection of the properties under

examination through the thickness of the specimen. To characterise the threedimensional structure of a general object, one single image is generally insufficient. Weuse a technique called electron tomography to reconstruct three-dimensionalinformation from a tilt series of TEM images. Electron tomography has only beenimplemented in the physical sciences over approximately the past 5 years duemicroscope developments and the need to characterise novel nanoscale 3-D objectswhose properties depend on their 3-D nature, e.g. quantum dots, catalytic structures.

(b) Magnetic imaging

Many recording media including hard discs and tapes use small magnetic regions, andmore recently, discrete single domain nanostructures to store and process information.The TEM can be used to characterise the local magnetic field to nanometre resolutionwithin a device. However, standard amplitude imaging techniques cannot be used

because the magnetisation of a device does not alter the amplitude of the electron wave,it only alters the phase. We must therefore use a phase imaging technique, i.e. we mustinterfere more than one electron wave together to form the final image. The mostcommonly used techniques are called Fresnel imaging, which allows the interference

between electron waves passing through the specimen by recording an image at aparticular defocus, and electron holography.

(c) Electron holography

The examination of electrostatic and magnetic potentials within specimens is an issue ofsignificant current interest, with many novel nanoscale materials devices relying on wellcharacterised properties. It is therefore important to be able to measure these potentialsquantitatively on a nanometre scale. Off-axis electron holography is a TEM-basedtechnique that uses a biprism to split the electron beam, such that one half of theelectrons pass through the specimen, and the others pass only through vacuum. Theyare recombined to form an image, a hologram, which is an interference pattern betweenthese electrons. This interference pattern can be used to determine quantitatively the

phase change experienced by the electrons passing through the specimen, and thereforethe electrostatic and magnetic potential within the specimen.

(d) Energy filtered TEM

EELS allows us to obtain chemical information at a high resolution by focussing aprobe on the specimen and relating the electron energy losses to particular electronictransitions and excitations. Energy filtered TEM (EFTEM) uses the same energy losses,

but by using parallel illumination and an energy slit, only electrons that have suffered aparticular energy loss are used to form a 2-D image. A series of images can be acquiredat different energy losses to generate maps of elemental distribution within a specimen.


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Appendix 1: The Energy and Associated Wavelengths of the Strongest K, L and M

Lines of the Elements.Atomic RelativeNumber Atomic K1 L1 M1

Element Z Mass,A E(keV) (nm) E(keV) (nm) E(keV) (nm)

Hydrogen 1 1.0Helium 2 4.0Lithium 3 6.9 0.05Beryllium 4 9.0 0.11 11.40Boron 5 10.8 0.18 6.76Carbon 6 12.0 0.28 4.47Nitrogen 7 14.0 0.39 3.16Oxygen 8 16.0 0-52 2.36Fluorine 9 19.0 0.68 1.83Neon 10 20.2 0.85 1.46Sodium 11 23.0 1.04 1.19Magnesium 12 24.3 1.25 0.99Aluminium 13 27.0 1.49 0.93Silicon 14 28.1 1.74 0.71

Phosphorus 15 31.0 2.01 0.61Sulphur 16 32.1 2.31 0.54Chlorine 17 35.5 2.62 0.47Argon 15 39.9 2.96 0.42Potassium 19 39.1 3.31 0.37Calcium 20 40.1 3.69 0.34 0.34 3.63Scandium 21 45.0 4.09 0.30 0.39 3.13Titanium 22 47.9 4.51 0.27 0.45 2.74Vanadium 23 50.9 4.95 0.25 0.51 2.42Chromium 24 52.0 5.41 0.23 0.57 2.16Manganese 25 54.9 5.90 0.21 0.64 1.94Iron 26 55.8 6.40 0.19 0.70 1.76Cobalt 27 58.9 6.93 0.18 0.77 1.60Nickel 28 59.7 7.48 0.17 0.85 1.46

Copper 29 63.5 8.05 0.15 0.93 1.33Zinc 30 65.4 8.64 0.14 1.01 1.23Gallium 31 69.7 9.25 0.13 1.10 1.13Germanium 32 72.6 9.88 0.12 1.19 1.04Arsenic 33 74.9 10.54 0.12 1.28 0.97Selenium 34 79.0 11.22 0.11 1.38 0.90Bromine 35 79.9 11.92 0.10 1.48 0.84Krypton 36 83,8 12.65 0.10 1.59 0.78Rubidium 37 85.5 13.39 0.09 1.69 0.73Strontium 38 87.6 14.16 0.09 1.81 0.69Yttrium 39 88.9 14.96 0.08 1.92 0.64Zirconium 40 91.1 15.77 0.08 2.04 0.61Niobium 41 92.9 16.61 0.07 2.17 0.57Molybdenum 41 95.9 17.48 0.07 2.29 0.54

Technetium 43 98.0 18.36 0.07 2.42 0.51Ruthenium 44 101.1 19.28 0.06 2.55 0.48Rhodium 45 102.9 20.21 0.06 2.70 0.46Palladium 46 106.4 21.17 0.06 2.70 0.44Silver 47 107.9 22.16 0.06 2.98 0.41Cadmium 48 112.4 23.17 0.05 3.13 0.39Indium 49 114.8 24.21 0.05 3.29 0.38Tin 50 118.7 25.27 0.05 3.44 0.36Antimony 51 121.7 26.36 0.05 3.60 0.34Tellurium 52 127.6 27.47 0.04 3.77 0.33Iodine 53 126.9 28.61 0.04 3.94 0.31


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Atomic RelativeNumber Atomic K1 L1 M1

Element Z Mass,A E(keV) (nm) E(keV) (nm) E(keV) (nm)

Xenon 54 131.3 29.77 0.04 4.11 0.30Caesium 55 132.9 30.97 0.04 4.29 0.29

Barium 56 137.3 32.19 0.04 4.46 0.28Lanthanum 57 138.9 33.44 0.04 4.65 0.27 0.83 1.49Hafnium 72 178.5 55.78 0.02 7.90 0.16 1.64 0.75Tantalum 73 181.0 57.52 0 02 8.14 0.15 1.71 0.73Tungsten 74 183.8 59.31 0.02 8.40 0.15 1.77 0.70Rhenium 75 186.2 61.13 0.02 8.65 0.14 1.84 0.67Osmium 76 190.2 62.99 0.02 8.91 0.14 1.91 0.65Iridium 77 192.2 64.88 0.02 9.17 0.14 1.98 0.63Platinum 78 195.1 66.82 0.02 9.44 0.13 2.05 0.60Gold 79 197.0 68.79 0.02 9.71 0.13 2.12 0.58Mercury 80 200.6 70.81 0.02 9.99 0.11 2.19 0.56Thallium 81 204.4 72.86 0.02 10.27 0.11 2.27 0.55Lead 82 207.2 74.96 0.02 10.55 0.12 2.34 0.53Bismuth 83 209.0 77.10 0.02 10.94 0.11 2.42 0.51

Polonium 84 210.0 79.28 0.02 11.13 0.11 ? ?Astatine 85 210.0 81.50 0.02 11.43 0.11Radon 86 222.0 83.77 0.01 11.73 0.11Francium 87 223.0 96.09 0.01 12.03 0.10Radium 88 226.0 88.45 0.01 12.34 0.10Actinium 89 227.0 90.87 0.01 12.65 0.10Thorium 90 232.0 93.33 0.01 12.97 0.10 3.00 0.41Protactinium 91 231.0 95.85 0.01 13.29 0.09 3.08 0.40Uranium 92 238.0 98.42 0.01 13.61 0.09 3.17 0.30


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Appendix 2: Threshold Energies of Ionisation Edges (in eV) observable by EELS

State 1s 2s 2p1/2 2p3/2 3p 3d 4p

Shell K L1 L2 L3 M23 M45 N23

2 He 24.6h

3 Li 55h4 Be 111h5 B 188h

6 C 284h7 N 400h8 O 532h9 F 685h

10 Ne 867h 18w

11 Na 1072h 32h12 Mg 1305h 52h13 Al 1560h 118h 73d14 Si 1839h 149h 100d

15 P 2149h 189h 135d

16 S 2472h 229h 165d17 Cl 2823 270h 200d18 Ar 3203 320h 246d19 K 3608 377h 294w20 Ca 4038 438h 350w 347w

21 Sc 4493 500h 406w 402w22 Ti 4965 564h 461w 455w 4723 V 5465 628h 520w 513w 4724 Cr 5989 695h 584w 575w 4825 Mn 6539 770h 652w 640w 51

26 Fe 7113 846h 721w 708w 5727 Co 7709 926h 794w 779w 6228 Ni 8333 1008 872w 855w 6829 Cu 8979 1096 951h 931w 7430 Zn 9659 1194 1043 1020d 87

31 Ga 1298 1142 1115d 10532 Ge 1414 1248 1217d 125 3033 As 1527 1359 1323d 144 4134 Se 1654 1476 1436d 162 57h35 Br 1782 1596 1550d 182 70d

36 Kr 1921 1727 1675 214 89h37 Rh 2065 1846w 1804w 239 111d

38 Sr 2216 2007w 1940w 270 134d 20p39 Y 2373 2155w 2080w 300 160 28p40 Zr 2532 2307w 2222w 335 181 32p

41 Nb 2698 2465w 2371w 371 207h 35h42 Mo 2866 2625w 2520w 400 228h 37d44 Ru 3224 2967w 2838w 472 281h 42d


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State 3d3/2 3d5/2 4p 4d 4f 5p 5d

Shell M4 M5 N23 N45 N6, N7 O2, O3 O4, O5

45 Rh 312 308d 4846 Pd 340 335d 5047 Ag 373 367d 59

48 Cd 411 404d 6749 In 451 443d 77

50 Sn 494 485d 9051 Sb 537 528d 99 3252 Te 582 572h 110 4053 I 631 620h 123 5054 Xe 685 672h 147 64

55 Cs 740w 726w 7856 Ba 796w 781w 9357 La 849w 832w 9958 Ce 902w 884w 11059 Pr 951w 931w 114

60 Nd 1000w 978w 11862 Sm 1107w 1081w 13063 Eu 1161w 1131w 13464 Gd 1218w 1186w 14165 Tb 1276w 1242w 148

66 Dy 1332w 1295w 154 30, 2367 Ho 1391w 1351w 161 31, 2468 Er 1453w 1409w 168 31,

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Course T2 Electron Microscopy and Analysis