MSE 421/521 Structural Characterization
Optics
A lens is a device, transparent to the incident radiation, which is used to focus and/or magnify
the image of an object.
Principal
image plane
Focal
plane
Object size, S1
Plane of
lens
Focal point
Image size, S2
f
v
u
Direction
Information
Position
Information
As the diagram shows, rays passing through the centre (axis) of the lens are unchanged, while all
rays parallel to the lens axis pass through the focal point in the focal plane. Rays leaving the
object in a parallel direction are brought to focus in the focal plane, forming a diffraction pattern.
Rays leaving the same point in the object plane are brought to focus on the image plane, where
an inverted image is formed. The magnification, M, is the ratio of image size to object size, and
f is the focal length.
M =
S2 v
=
S1 u
and u , v, and f are related by
1 1 1
= +
f u v
Maxwell’s well-known equations give rise to three criteria for the perfect lens. First, all rays
leaving one point on the object must meet at one point in the image. Second, if points in the
object lie on a plane perpendicular to the lens axis then points in the image must also lie in a
plane perpendicular to the lens axis. Third, the ratio of distances in the object must be the same
as the ratio of distances in the image, i.e., the image must be a true representation of the object.
Resolution
There are three requirements for image formation. First, the intensity of radiation must be
sufficient to be detected (brightness). Second, there must be an adequate variation in brightness
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from point to point so that objects can be recognised (contrast). Third, the image must enable
two separate objects to be distinguished separately (resolution).
Resolution is defined as the smallest distance between two points at which they can still be
recognised as two separate entities.
resolvable
unresolvable
The resolution of a microscope is a function of diffraction of the radiation used, the wavelength
of the radiation, perfection of the lenses, and scattering events in the sample.
The example of diffraction through a circular aperture is of great importance because the eye and
many optical (including electron optical) instruments have circular apertures. All microscopes
diffract radiation because, even if an aperture is not deliberately introduced, the physical
dimensions of the lenses are restricted and so act as apertures themselves. When light from a
point source passes through a small circular aperture, it does not produce a bright dot as an
image, but rather a diffuse circular disc known as the Airy disc† surrounded by much fainter
concentric circular rings sometimes called a jinc distribution. Most of the intensity (84%) is
contained in the central disc which has a diameter of d1. If two points are imaged, each of them
will form an Airy disc.
Resolved
Rayleigh
Criterion
Unresolved
d1/2
d1
d1
†
named for George Biddell Airy (1801 – 1892) who first described the phenomenon.
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Moving the objects closer together will move the intensity patterns together until they overlap.
As the overlap increases, a condition is reached at which the two points cannot be distinguished
and appear like a single point.
The mathematical description of optical resolution was laid down separately by Lord Rayleigh
and Ernst Abbe in the latter part of the 19th century. Although both approaches lead to
essentially similar conclusions, each has a somewhat different way of describing the underlying
rules of resolution.
The Rayleigh criterion states that two points can just be resolved when the first diffraction
minimum of the image of one source point coincides with the maximum of the other. The
intensity of the resultant wave drops by about 15% in the centre, which is just sufficient for the
two points to be resolved. Distances less than this are unresolvable. This minimum resolution is
often called the diffraction barrier.
For a uniformly illuminated object, the minimum resolvable distance (d1/2) is then:
d 1 2 = rd =
0.61λ
µ sin β
where µ is the refractive index of the medium between the lens and object and β is semi-angle
of acceptance of the lens (also called half-angle of acceptance). The product µsinβ is referred to
as the numerical aperture. It is important to note that the magnification does not enter into this
expression. In fact, resolution is independent of magnification.
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aperture
object
β
lens
The Abbe criterion takes a complementary approach to describe resolution. Consider a
diffraction grating that splits the incident light into three diffracted orders: the (undiffracted)
zeroth order and the +1 and –1 diffracted orders. If the grating is coarse, then the diffracted
orders are diffracted through a small angle and pass through the objective lens L1, forming
diffraction-limited points in the so-called Fourier plane. These are then re-imaged by lens L2 to
form an image on the eye or camera; however, if the grating is very fine, these diffracted orders
are diffracted through such a large angle that only the zeroth order passes through L2 and the
image that is formed shows no trace of the grating. The Abbe criterion states that the finest
grating that can be imaged (which corresponds to the diffracted orders just passing through L2)
has a period of:
rd =
0.5λ
µ sin β
Both the Rayleigh and Abbe criteria are often referred to as the “diffraction limit” of
microscopic imaging, and they provide convenient ways of thinking about many of the
techniques used to achieve resolution beyond the diffraction limit.
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In either case, it is clear that, since rd is inversely proportional to sinβ (and therefore to β), the
resolution is governed by both wavelength (which must be small) and the size of the aperture
(which must be large). The medium is necessarily vacuum (µ ≈ 1) for electron microscopy, but
resolution can also be improved by changing the medium to one with a higher µ (e.g., water has
µ = 1.33, silicone oil has µ = 1.6) where possible. A good optical microscope may have
µsinβ = 1.58 (oil submersion lenses), resulting in a resolution of about 0.2 µm. Using only violet
light (λ = 400nm), β = 80°, and µ = 1.6, the very best resolution possible from an optical
microscope is 150nm, which was already achieved in the 19th century. The resolution of optical
microscopes remains diffraction limited.
In a 200kV electron microscope, λ = 0.025 Å, but β is quite small (~10 mrad), giving us an
ultimate resolution of 1.5 Å. This distance is less than the radius of many atoms; however, this
theoretical limit assumes perfect lenses and focusing, which are never achieved in practice.
Lens imperfections
Astigmatism
Differences in the optical properties of the lens from point to point result in rays being focused at
different focal lengths and the object having a different appearance at different distances from
the lens.
Vertical
focal
line
Disc of
least
confusion
Horizontal
focal
line
lens
Rays travelling in the horizontal plane are focused at a different point to those travelling in the
vertical plane. The difference leads to the formation of a disc of least confusion instead of a
point image and smears the image in a particular direction.
Astigmatism can be seen in an electron microscope when the focus is changed in that the
smearing occurs in one direction and gradually changes to the perpendicular direction as the
focus is changed. It can be completely compensated for by using stigmators, which are small
electromagnets supplying compensating magnetic fields.
Chromatic aberration
Chromatic aberration is when rays of different wavelength (colour in the case of light) are
refracted by different amounts in the lens and hence are brought to focus at different places.
Shorter wavelengths refract more and hence have shorter focal lengths. This aberration arises in
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electron microscopes from an energy (or wavelength) spread in the electron beam which is
caused by instabilities in the accelerating voltage or lens currents but can also be caused by
inelastic scattering in the sample.
blue
green
red
The resolution limit due to chromatic aberration is:
rc = C c β (δE / E o ) 2 + (2δI / I ) 2
where δE is the energy spread, Eo is the accelerating voltage, δI the lens current spread, and I the
lens current. In this case, α must be minimised, unlike for diffraction limited resolution. A
typical value for Cc, the chromatic aberration constant, is about 3mm, and δE ≈ 20 eV.
For relatively thick samples in a transmission optical or electron microscope (e.g., biological
samples) the probability of electrons or photons losing energy while interacting with the sample
is increased. This inelastic scattering changes the wavelength of the electrons, thus causing
chromatic aberration.
Spherical aberration
Spherical aberration is when rays passing near the centre of the lens (optic axis) are focused at a
different point than rays passing through the edges. It is described as:
rs = C s β 3
where Cs is the spherical aberration coefficient. Note the strong dependence on α, again
opposing the requirement for diffraction limited resolution.
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Marginal
focus
Axial
focus
Disc of
least confusion
Distortion
If the magnification of the lens varies from the centre (axis) to the edge, the result is a distorted
image.
Rectangular object
Barrel distortion
Pin-cushion distortion
Overall effect of aberrations on resolution
The overall effect on resolution is a combination of diffraction effects plus the various lens
aberration effects. In optical microscopy it is possible to correct both chromatic and achromatic
aberrations by careful design of lenses; however, it is very difficult to eliminate achromatic
aberrations, especially spherical aberration, in the electron microscope. Assuming that the net
resolution is dopt = rd + rs, an optimal α and d can be found:
1
−1
α opt = 0.67λ C s
3
4
4
1
d opt = 1.21λ 4 C s 4
It turns out that the factor of 1.21 can be reduced to as low as 0.7 in favourable conditions. The
point resolution of a 200kV electron microscope is now about 2 Å, which is below the atomic
spacings of many ceramics and some metals.
Depth of Field
Depth of field is the range of object distances, u, over which the object can be placed without the
eye detecting any change in image sharpness. A related value is the depth of focus, which is the
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range of distances, v, over which the image can be viewed without loss of sharpness. If this
range of distances is Dim, as shown, then the object can be placed anywhere over this distance
Dob without changing resolution. Dob is typically greater than the specimen thickness.
aperture
lens
β
d1
d1
α
Dim
Dob =
0.61λ
0.61λ
or for an electron microscope, Dob ≈
µ sin β tan β
β2
and Dim = Dob M 2
The only way to improve Dob is by increasing λ or reducing β, typically by inserting an aperture.
As β decreases, Dob increases but diffraction-limited resolution is reduced (contrast is also
increased). In fact, optical microscopy is always a compromise between resolution and depth of
field. In electron microscopes, where lens imperfections are the limiting factor in resolution,
reducing β can actually improve resolution by ameliorating the effects of aberrations. Because it
is necessary to keep β small in the electron microscope in order to reduce the effects of spherical
aberration, electron microscopes gain at the same time a large depth of field (despite their lower
λ).
Human eye limited in:
• Range of EM wavelengths detectable (400 – 700 Å)
o Peak sensitivity in green (Hg vapour lamp) – so often use green filters in optical
scopes and TEM image screens fluoresce green.
• Signal intensity needed to trigger “recognition” in brain (~100 photons per “pixel”)
• Resolution in good green light for healthy eyes is ~34µm (text says 200µm)
• Integration time over which an image is recorded by eye (0.1 s)
Resolution in Reflection
In the scanning electron microscope, resolution is limited by the source brightness.
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1/ 2
db =
π  I th 
 
2α  B 
where B is the source brightness (controlled by the electron gun), and Ith is the minimum beam
current required to provide sufficient contrast for a feature to be seen above the signal noise
level. So, for the SEM the only way to improve resolution is to use a brighter electron gun;
hence, for high resolution work an SEM with a field emission electron gun (FEGSEM) must be
used.
Construction of the (Optical) Microscope
2β
β
Three systems:
Illuminating system – light source
Imaging System transfers magnified image to plane of observation
Specimen Stage holds/positions specimen
Illuminating System
Light Source: must be uniformly spread across specimen but also focused for sufficient reflected
intensity
Carbon arc – bright but unstable white light
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Mercury arc lamp – bright green (λ = 546 nm) corresponds well with peak sensitivity of
human eye
Tungsten halide discharge tube – stable and intense white light (3200 K)
Condenser lens assembly – focuses (brings to a point – crossover) an image of the source so that
objective lens “sees” a point source and specimen is illuminated by parallel illumination.
Condenser aperture – limits amount of light (why?) to increase contrast. If too small, diffraction
effects can hurt resolution.
Objective or virtual image aperture (really another condenser aperture) – ensures that only light
from the area under observation is admitted – reduces unwanted background intensity.
Add a central stop and the aperture becomes an annulus for darkfield imaging
Imaging System
Objective Lens – provides most of the magnification
Types of (optical) lenses:
http://www.gonda.ucla.edu/bri_core/lenses.htm
Plan – corrected for flatness of field.
Lenses which are not corrected for flatness of field yield images with focused centres and
outer edges out of focus (or vice versa). A plan lens allows the whole field to be in focus.
Achroplans - best for transmitted light
Epiplans - designed for reflected light
Achromatic -
spherical aberration corrected for one colour and chromatically corrected
for two (middle of spectrum) so best results when used with yellow-green
filters. Budget-priced.
Planachromatic -
achromatic lenses with correction for flatness of field as well as
achromatic colour correction. Some microscope manufacturers list such
lenses as simply "Plan".
Apochromatic most highly colour-corrected objectives: they are chromatically corrected
(plan apochromatic) for three or four colours and spherically for two.
Top of the line ($!) in objective lenses
Highest numerical apertures so best resolution
Best for colour (esp. in blue and green ranges)
Do not transmit UV (so no good for fluorescence)
Semiapochromatic (plan semiapochromatic)
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corrected chromatically for two or three colours and spherically for
two colours. Use fluorite (low n, better dispersion).
e.g., plan neofluar and plan fluotar)
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good for DIC and fluorescence
Fluar – introduced by Zeiss. Not plan, made especially to increase brightness of
fluorescence. Images are ~10% brighter and in UV brightness increases 25-50%.
The objectives of the BSCMCC’s Leica DM6000 M microscope are:
• plan fluotar (semiapochromatic) 1.25x ($2,587), 5x ($883), 10x ($1,343), 20x ($1,664)
• plan apochromatic 50x ($5,655) and 150x ($6,335)
Immersion -
used to increase numerical aperture and so resolution.
Marked for the immersion medium used:
(Oel) or (Oil) for oil
(W) for water immersion
(Imm) Multi-immersion, for oil, water, and glycerin
Darkfield -
If the lens has a phase ring it and can be used for dark-field illumination.
The lens will be marked with a "Ph" followed by a manufacturer’s number for
matching to a ring in the condenser.
Remainder of magnification supplied via:
1. intermediate/tube lens plus eyepiece
2. lenses to focus image onto film or CCD (charge-coupled device) for later enlargement
Image Formation
There are three fundamental ways in which an image of an object can be formed. The simplest is
by projection, essentially casting its shadow. Another way is via lenses to form an optical
image (although such images can also be achieved with electrons or even atoms or ions). Both
of these methods produce all parts of the image simultaneously; however, it is also possible to
produce a scanning image in which each part of the image is produced serially, much as the
picture on a television screen is produced, with the refresh frequency so high that the image
appears complete to the eye.
Electrons vs light
Many courses on electron microscopy include topics on electron optics. In fact, much of the
field of optics (light) can be applied to electrons as well, a fact which has led to a strange hybrid
terminology where we call an electron beam "illumination" and speak of focusing, lenses, and
wavelengths.
Of course, electrons as well as photons can be described as both waves and particles; however,
microscopy relies on the wave nature of both photons and electrons. Visible light can be thought
of a waves with wavelengths between about 400 - 700nm, whilst electrons can also be
considered waves with wavelengths dependent on the voltage used to accelerate them (and thus
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on their energy). Typical electron wavelengths may vary from about 0.0251 Å for 200keV
electrons like those in the JEOL 2100-HR TEM to 0.0859 Å for 20keV electrons typical in an
SEM. The first obvious difference between such electrons and light is that their wavelengths
differ by many orders of magnitude. Due to the very small electron wavelengths, diffraction
angles are also very small, which allows for the approximation sinα = tanα = α, where α is the
half-angle of the lens.
Another important difference between photons and electrons is that electrons are much more
strongly scattered by matter. It is for this reason that electron microscopes typically operate
under vacuum. Additionally, unlike the glass lenses of an optical microscope, lenses for
electrons are magnetic fields created by electromagnets circling the column. The lens is not
composed of solid material and so the refractive index can be assumed to be unity.
With these simplifications in mind, the equation for resolution can be re-written as:
rd =
0.61λ
β
A further difference between electrons and light is that electrons carry a charge.
By contrast with optical microscopy, electron microscopy offers higher resolution, higher
magnification, greater depth of field, and the possibility of crystallographic and chemical
analysis - but at a much higher price.
The medium is vacuum (µ ≈ 1) for electron microscopy.
In a 200kV electron microscope, λ = 0.025 Å, but β is quite small (~10 mrad), giving us
an ultimate resolution of 1.5 Å. This distance is less than the radius of many atoms;
however, this theoretical limit assumes perfect lenses and focusing, which are never
achieved in practice.
Electron generation
Resolution in an electron microscope is controlled in part by the diameter of the incident beam
and the mode of operation (scanning or transmission). The coherence of the source is not as
important as its energy spread, which should be as low as possible to reduce chromatic
aberration. In addition, because images are acquired over some finite time, the stability of the
electron beam should be as high as possible.
The illumination system in any electron microscope consists of a source of electrons (the
electron gun) and a lens system to focus and control emitted electrons. They work in one of two
basic ways. The first method is to overcome the work function by resistance heating; the second
method is to use a strong electric field.
Thermionic emission
A thermionic electron gun consists essentially of a heated wire or crystal from which electrons
are given enough thermal energy to overcome the work function combined with an electric
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potential to give the newly freed electrons velocity. The heated source is usually held at some
potential (~500V to ~100kV) negative relative to ground, so that a sample (as well as the rest of
the microscope) can be kept at ground.
Tungsten
Wires are made of W because it has a very high melting temperature (3422°C), so more thermal
energy can be made available, although its work function (4.5 eV) is fairly high even at the
~2500°C operating temperature. Tungsten hairpin filaments are used in standard machines like
the LEO 1430 VP SEM. A wire of tungsten metal is heated (T > 2500°C) under a large bias
field, stripping electrons off according to the Edison effect.
LaB6/CeB6
It is easier to strip electrons off lanthanum hexaboride, LaB6, or cerium hexaboride, CeB6, and
such crystals can produce beams 30 times as bright as a tungsten one which are stable over long
periods. The disadvantages of using such crystals are that they require a higher vacuum, they
suffer from thermal shock (and so must be heated and cooled slowly), and they are expensive
($1000 compared to $30 for tungsten). The JEOL 2100 uses a LaB6 crystal. LaB6 melts at
2530°C and CeB6 at 2540°C. Both are used at ~1500°C because at this temperature they both
have a low work function (~2.5 eV). Higher temperatures cause increased sublimative loss of
the crystal and premature failure.
Field-effect emission
Field emission guns (FEGs) offer a further increase in brightness, but at the penalty of requiring
an ultrahigh vacuum system. The cold field emission gun, such as the one on the Hitachi 4500,
uses a tungsten single crystal and an intense electric field to extract the electrons. This field
lowers the height of the potential barrier via the Schottky effect. If the electric field is increased
sufficiently, the width of the potential barrier becomes small enough to allow electrons to escape
through the surface potential barrier by quantum-mechanical tunnelling, a process known as
field-emission. A high vacuum is required to reduce contamination of the crystal and the gun
must be flashed (heated to high temperature) every few hours to drive off contaminants. In the
thermally assisted field emission gun (Schottky emitter), a pointed crystal of W is welded to the
end of a V-shaped W filament, coated with ZrO2, and heated to 1500°C. An intense electric field
is still used to extract the electrons, but the stability is greatly improved and the gun does not
require flashing. Such thermal field emitters enhance the pure field emission effect by giving
some thermal energy to the electrons in the metal, so that the required tunneling distance is
shorter for successful escape from the surface. As a result, the emission current density is greatly
increased (compared to LaB6).
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A comparison of electron emission characteristics of W, LaB6, CeB6 and FEG guns
LaB6
CeB6
FEG
FEG
W wire
<100>
<100>
(cold)
(Schottky)
Operating temperature (°C)
2500
1500
1500
27
1500
Work function (eV)
4.5
~2.5
~2.5
4.5
4.5
Emission current (A/cm2)
1
100
100
105
105
Brightness (A cm-2 sr-1)
106
107
107
109
109
30
10
10
0.01
0.01
Crossover diameter (µm)
Short-term beam current
<1
<1
<1
3-5
>5
stability (%RMS)
Energy spread (eV)
2
1.5
1.5
0.2 - 0.4
0.2 - 0.4
Typical service life (hr)
30-100
1000
1,500
1000
100
Operating vacuum (torr)
10-5
10-7
10-7
10-8
10-9
Evaporation rate (g cm-2 sec-1)
NA
2.2 x 10-9
1.6 x 10-9
NA
NA
For a given source, the brightness of the beam increases linearly with accelerating voltage.
Thermionic guns have large areas of emission and are therefore insensitive to changes in
filament current and contamination. They are very stable. The beam size (spot size) is reduced
by 104 times for analysis to just ≈1nm, further reducing the effects of instability. The cold field
emission gun has a very small area of emission and so is highly sensitive to contamination,
resulting in short-term fluctuations in beam current and gradual long-term build-up of
contaminants. The beam size is reduced only 5-10 times from the source, to ≈1nm for analysis;
therefore, the beam is still very sensitive to contamination. Also, as the electrons come from
only a tiny area on the crystal in a field emission gun, the beams produced are nearly coherent
(i.e., same wavelength and phase), which is why they can be so intense.
Beam brightness:
Brightness, B, is a simple function of current density, jc, and convergence angle, α (in radians):
B=
jc
πα
2
[Am-2sr-1]
For thermionic emission: B = 2 × 10 5 TV exp − φ  Am -2 sr -1
kT 

The dose rate is the number of electrons in the spot per second.
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Triode electron gun
In this triode electron gun, the filament is contained in a Wehnelt cylinder with a small (< 1mm)
hole directly below the filament. The electrons thermionically emitted are accelerated towards
the anode as a result of the large potential difference between the filament assembly (cathode)
and the anode. The anode is usually at earth potential so that the rest of the microscope column
is at earth. Consequently, the Wehnelt cap is kept at a large negative potential compared to the
anode. In addition, there is a bias resistor between the filament heating circuit and the Wehnelt
cap so that the cap is biased slightly negative with respect to the filament to ensure that it does
not act as a sink for electrons. This arrangement ensures that most of the electrons are emitted
from the filament tip and that the Wehnelt repels the electrons and causes a focusing action.
The gap between the filament tip and the Wehnelt cap is critical to this focusing action and
filament lifetime. The triode gun produces a beam of electrons which comes to a focus (crossover) at a point just below the Wehnelt cap with a diameter of about 30 µm for W and 10 µm for
LaB6 or CeB6.
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Magnetic Lenses
Magnetic fields can displace electrons and so can be used to focus electron
beams. A magnetic field can be produced by passing a current through a coil
consisting of many turns of wire. The field strength can be increased by using
a ferromagnetic core like Fe. Varying the current to the coil will alter the
strength of the magnetic field generated. The direction and magnitude of the
force felt by the electron is:
v
v v
F = e(v × B )
v
v
where e is the electron charge, v is the velocity of the electron, and B is the magnetic field
strength. The geometry of magnetic lenses in electron microscopes is slightly more complicated,
but the principle is the same.
A coil of wire is wound on an iron core (the pole piece), which has a very small gap across
which the field is produced. By varying the current in the coil, typically 0 – 1A, the magnetic
field strength and hence the focal length of the lens can be varied. The lens is designed to
produce a magnetic field almost parallel to the electron trajectory. This magnetic field will have
components both along the axis of the microscope, Bax, as well as in the radial direction, Brad.
Initially, the electron is unaffected by Bax, which is parallel to its direction of travel, but
experiences a small force, Bradev, from the small radial component, which pushes it out of the
plane of the paper (in fact, since the actual lens is circular, the path is a helix). With this motion,
the velocity of the electron now has a component perpendicular to the axis, and so is affected by
Bax, which pushes it to the right (again, because the lens is circular, motion is actually towards
the axis). The net result of Bax and Brad on the electron is to make it spiral down the gap in a
helical path. Because Brad grows as the electron approaches the middle of the gap, its path
narrows and becomes focussed around the optic axis. This spiralling of electrons through
magnetic lenses is an important difference between these lenses and optical ones. As the
strength of the lens changes, so does the rotation; but this effect can be eliminated (as it is in the
JEOL 2100) by using oppositely poled subsequent lenses. Another difference is, of course, that
these lenses can be switched off simply by stopping the current through them.
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Electromagnetic lenses only approximate to a Maxwell lens when electrons pass close to the lens
axis. It is impossible to produce a perfect electromagnetic lens according to the Maxwell
criteria. There will always be aberrations and distortions present, which ultimately limit the
resolution in transmission electron microscopes.
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Interaction of Electrons with Matter
Electron Scattering
The probability that a particular electron will be scattered in a particular way is defined by its
mean free path, λ, or scattering cross section, σ. This cross section represents the area which
the scattering particle appears to present to the electron. If there are N such particles per
volume, then the probability of a single electron being scattered in a distance dx through the
specimen is related to Nσdx. The mean free path is:
λ = 1/Nσ.
Here, λ is essentially the average distance an electron will travel through the specimen before
being scattered in this way. Mean free paths are often similar to the typical thickness of a TEM
sample, which implies that most electrons are scattered just once or not at all in the specimen.
On the other hand, thicker samples like those used in an SEM provide sufficient opportunity for
an electron to be scattered many times.
The probability of an electron undergoing n scattering events while travelling a distance x is
given by the Poisson equation:
p(n) = (1/n!)(x/λ)nexp(-x/λ)
This equation assumes multiple scattering by the same process (same λ), and so does not very
accurately predict multiple scattering in real materials.
Interaction volume
The primary or incident electron beam enters the specimen and can potentially undergo
numerous scattering events. The scattering means that the sample is excited below its surface
over a region much larger than the incident beam diameter. The region into which the electrons
penetrate the specimen is known as the interaction volume. In a scanning electron microscope
(SEM) sample this interaction volume is much larger than in a transmission electron microscope
(TEM) sample because TEM samples are generally only ≈100nm thick.
Incident (primary) beam
SEM
Secondary electrons
Backscattered electrons
TEM
≈100nm
≈1µm
X-rays
The amount which the beam spreads, b, through the TEM specimen can be estimated as:
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b = 0.198(Z/E)(ρ/A)1/2t3/2
where Z = atomic number, E = electron energy (keV), ρ = density (g/cm3), A = atomic weight,
and t = thickness (nm). Note that, according to this expression, the atomic-weight dependence is
opposite to that of atomic-number. To avoid confusion, this expression can be re-written by
making the following substitution:
ρ/A = 1021(Nv/NA)
where NV = the number of atoms per nm3, which can easily be calculated from crystallographic
data, and NA = Avagadro’s number (6.0221x1023). The beam spread can then be written as:
b = 0.00807(Z/E)Nv1/2t3/2
Assuming that a Gaussian intensity distribution is maintained in the beam throughout, then the
two terms, d (beam diameter at the top surface) and b, can then be combined in quadrature to
yield the beam diameter, D, at any depth t:
D = (d2 + b2)½
The penetration depth is a function of two factors. First, it increases as the accelerating voltage
is increased. Secondly, it decreases as the atomic number of the sample atoms increases.
primary electrons
Backscattered
electron
Characteristic
x-ray
cathodoluminescence
or
Auger
electron
ejected secondary electron
ejected secondary electron
primary electron
primary electron
Inelastic Scattering
Inelastic scattering is a general term which refers to any process which causes the primary
electron to lose a detectable amount of energy, ∆E. These processes all involve an interaction
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MSE 421/521 Structural Characterization
between the primary beam electrons and orbital electrons of the atoms in the sample. Most of
the exchanged energy is lost as heat; however, it may also be released in other ways.
Phonon scattering
Phonons are quanta of atomic vibrations in a solid. Primary electrons can lose energy by
exciting phonons, which result in heating of the solid. The amount of energy arising from each
interaction is small, about 1eV; however, the heating effects can be strong particularly at high
voltages. In addition, the scattering event results in a large deflection of the electrons, typically
10°, which gives rise to increased diffuse intensity in electron diffraction patterns. The heating
amounts to only about 10°C in a good metallic conductor like Al; however, in poor conductors
like polymers or ceramics the heating can be large. It is possible to melt Al2O3 (Tm > 2000°C) in
a TEM.
Plasmon scattering
A plasmon is a wave in the sea of electrons in the conduction band of a metal. This type of
scattering is a result of the interaction of the primary electrons with the free electron gas in a
material. Primary electrons lose 5 – 30 eV per event and have a small mean free path (they do
not travel far into the sample before causing plasmon scattering). These events are common and
are important in Auger and EELS studies. They also contribute to the diffuse intensity around
the transmitted spot in an electron diffraction pattern.
Single valence electron excitation
It is also possible, but less likely, that a primary electron will transfer some energy to a single
conduction-band electron rather than the entire sea. The mean free path for this process is quite
large (microns), the energy loss is small (≈1eV), and the typical scattering angle is also small;
therefore, this process is not exploited in electron microscopy.
Inner shell excitation leading to characteristic x-ray production
A primary electron may transfer enough energy to a K or L shell electron to knock it out of the
atom. Because the binding energies of such electrons are typically high, the amount of energy
lost by the primary beam is quite large. It requires 283eV to remove a carbon K electron and
69,508eV to remove a tungsten K electron. The mean free path for this process is quite large, so
it is relatively infrequent. The probability of this type of scattering occurring decreases both as
the incident voltage increases (high-energy electrons are more likely to pass through the sample
without interacting) and as atomic number increases (the critical energy required to produce an
x-ray increases with atomic number). The hole left in the inner shell can be filled by one of the
outer electrons dropping down, releasing a quantum of energy characteristic of that particular
transition in that particular atom. Virtually all x-rays which are emitted escape from the surface,
and they are produced throughout the entire interaction volume. These x-rays may therefore
have come from well below the sample surface. For this reason, the smallest region which can
easily be analysed in an SEM using x-rays is about 1µm.
Fluorescence yield: w = Z
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(Z
4
+c
)
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MSE 421/521 Structural Characterization
where w is the fluorescence yield, Z is atomic number, and c ~ 106 for K-shell excitation. The
yield of Auger electrons is 1 − w .
Inner shell excitation leading to characteristic (Auger) electron production
An alternative to x-ray emission is the ejection of an outer electron carrying away the transferred
energy of the primary electron as kinetic energy. This process is called Auger emission. Three
electrons are involved: the primary electron (which loses some of its energy to the inner shell
electron), the outer electron (which fills the hole in the inner shell), and the other outer electron
(which carries off the surplus energy). Auger electrons have low energies and are easily
absorbed by the specimen or any gaseous atoms in the vacuum; therefore, their detection requires
very high vacuum systems. Nonetheless, their easy absorption means that those which escape
the sample to be detected have come from the outer ≈1nm of the sample surface and so give
detailed information about this region. Auger electron spectroscopy is an important surface
analysis technique. They are outer electrons and so associated with bonding and can give
information on the local atom environment. The probability of Auger electron emission instead
of x-ray emission increases for light elements.
Outer shell excitation leading to cathodoluminescence
If the primary electron ejects an outer electron (one from between valence and conduction bands)
from an atom, that hole will also be filled by a higher-energy electron (one from the conduction
band); but the difference in energy levels is likely to be small and so the characteristic radiation
emitted will be of low frequency, typically in the visible spectrum. This effect is called
cathodoluminescence. Certain semiconductors and insulators will emit ultraviolet or visible light
when bombarded with high-energy electrons. The intensity of these emissions is modified by the
presence of impurities or defects like dislocations. This imaging technique is used extensively to
examine defects in semiconductors like GaAs and AlN.
Excitation of outer electrons leading to emission of low-energy secondary electrons
Secondary electrons describe those electrons which escape from the specimen surface with
energies below about 50eV. They could conceivably be primary electrons which have
undergone numerous collisions eventually escaping from the surface with a little energy left
over; however, they are much more likely to be electrons ejected by one of the processes above.
This interaction forms the main method of image formation in the SEM. Because secondary
electrons have low energies, they can only escape from regions near the surface (≈10nm), i.e.,
only from a small proportion of the interaction volume; however, the yield of secondary
electrons can be ≥1 (each primary electron produces at least one secondary electron).
primary beam
high yield
shadowing
low yield
specimen
Consequently, sharp edges and corners appear bright, due to their greater surface areas, while flat
regions appear darker. Similarly, parts of the sample tilted towards the detector appear bright
because electrons can escape easily whereas those parts not facing the detector appear dim, not
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MSE 421/521 Structural Characterization
because fewer electrons are emitted, but because the distance the electrons must travel to the
detector is much further. Since these secondary electrons arise from the surface, they produce
what are termed topographical images showing the hills and valleys of a sample’s surface.
Bremsstrahlung x-rays
When the primary electrons slam into the specimen, they are slowed and deflected by the
Coloumb field of the atoms and their kinetic energy is transformed into photons of x-radiation.
These x-ray emissions are made up of a mixture of rays with many wavelengths.
decelerated
electron emits
radiation
The primary electrons can collide with any atom at any angle any number of times, such that a
whole range of energy losses and hence wavelengths of x-rays are produced. These x-rays are of
no use in microanalysis and are simply unwanted background radiation.
Elastic Scattering
Rutherford scattering or back scattering
These electrons have undergone billiard-ball type collisions with nuclei of specimen atoms and
have as much energy as the primary electrons and are scattered from atoms up to 2µm deep in
the sample. About 30% of the primary beam is reflected in this way. The contrast in such
images is very sensitive to the average atomic number, Z, of the phases being imaged. Atoms
with a high Z have large nuclei, so there is a greater chance of scattering an incoming electron.
The more electrons being detected from a given area, the brighter the image will be; therefore,
phases with higher average atomic numbers appear brighter than those with lower Z.
Forward-peaked distribution of scattered electrons. If E0 = energy of primary electron, L is the
thickness of the specimen, and Z is the atomic number of the specimen, then the probability,
p(θ), of the electron being scattered through an angle θ is given by the Rutherford formula:
p (θ ) ∝
LZ 2
E o2 sin 4 (θ 2 )
Diffraction
As we have already seen, electrons can be diffracted by crystal samples. Such interactions
involve no change in electron energy and so are elastic. In a TEM, the result is typically a spot
pattern (or a pattern of discs as in convergent beam electron diffraction). Electron diffraction can
also take place in the SEM, and this technique is called electron backscattered diffraction
(EBSD).
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