Physical Principles of Electron Microscopy
2. Electron Optics
Ray Egerton
University of Alberta and
National Institute of Nanotechnology
Edmonton, Canada
www.tem-eels.ca
[email protected]
Properties of an ideal image (Maxwell’s rules)
1.
For each point in the object there
is an equivalent point in the image.
Image defects: defocus (first-order focusing)
aberrations (point à disk of confusion)
2.
The object pattern and image pattern
are similar.
Defect: distortion (barrel, pincushion, spiral)
A triangle imaged with magnification
and inversion, showing ray paths.
Points A,B,C are equivalent to a,b,c
and the triangles are similar.
M(r) increase M(r) decrease
f(r) non-zero
3. If object is planar and perpendicular
to the optic axis, so is the image.
Defect: curvature of field
Imaging in Light Optics is based on refraction
Snell’s law:
n1sinθ1=
n2sinθ2
Refraction by a glass prism
For small angles, deviation α ~ (n-1)φ
Focusing by a convex lens à
(think of it as a prism whose angle
increases with distance from the
optic axis)
Fermat’s Principle of least time (ray & wave optics)
Total time T for travel between O and I is a minimum
(otherwise the phases of alternative paths are almost random,
so the net amplitude becomes zero).
T = 4f/c – t/c + n t /c
path A
= (2/c)[(2f)2 + r2]1/2 path B
Making these two times equal
gives a formula for the
focal length f , which can
be compared with the
Lensmakers Formula:
path B
r
path A
t
1/f = (n-1) [2/R2 - (1-1/n)t/R2)]
R is the radius of curvature
for both lens surfaces and
r2 = Rt - t2/4 from geometry.
2f
2f
Ray diagrams with a few special ray paths:
Thin-lens ray diagram (geometric optics) showing first-order focusing
A ray parallel to the optic axis defines the focal length of the lens
Thin-lens equations:
1/u + 1/v = 1/f (real image: u and v are positive)
M = xi/x0 = v/u (usually take M > 0 for a real image)
Lens aberrations (higher-order focusing)
Axial aberrations:
Spherical aberration: f depends on distance from optic axis
Chromatic aberration: f depends on radiation wavelength (dispersion)
Both of these give a circular disk of confusion.
Axial astigmatism: f varies with azimuthal angle φ.
Gives elliptical “line foci” from a “non-round” lens.
Off-axis aberrations (less important in electron optics):
Coma: each object point à comet-like effect in the image
Third-order astigmatism à line foci, even for a round lens
Electron optics
Ray bending cannot use solid lenses (too much scattering)
1. Electrons are charged particles (electrostatic deflection)
Electrostatic force: F = (-e) E
Used for beam deflection and focusing (axial symmetry)
2. Moving electron is equivalent to an electrical current,
so there is a magnetic (Lorentz) force: F = -e (v x B)
More effective when the electron speed v is high
(limit is speed of light)
Also used for deflection and focusing.
Electrostatic (einzel) lens
Used in CRT display tube (TV, oscilloscope),
ion optics (e.g. FIB machine, v is lower than for electrons)
Advantages: lightweight, low power, no image rotation,
self-compensating for voltage fluctuations
Disadvantages: high voltage (insulation, surface breakdown),
higher aberrations, no immersion lens for conducting specimen
Magnetic lens (usually electromagnetic)
1. Uniform field: point-to-point focusing —>
but cannot focus a parallel beam.
B
2. Localised field (axial symmetry) :
coil
carrying
direct current
rotation through angle φ
1 rev.
Electron trajectories
bell-shaped
towards axis
Div(B) = 0 à Br = -(r/2)(dBz/dz)
Fφ = - e (vzBr) + e (Bzvr)
produces angular speed vφ , then
Fr = - e (vφ Bz)
gives radial force towards axis
Fz= (vφ Br) ß net effect = 0
Comments on magnetic focusing
No force in direction of electron travel,
so electron speed v remains constant at all times.
Kinetic energy (KE) always stays the same.
Rotation effect is ignored in e-ray diagram (plot of r versus z).
Radial field Br causes force Fr towards axis (focusing)
and is present only where dBz/dz is non-zero
(e.g. in fringing field at the entrance and exit of a solenoid).
So a long uniform field (as in a solenoid) is unnecessary.
For efficiency, keep the field as short as possible, using polepieces.
Practical lenses
Soft magnetic material
surrounds windings
except at the lens gap
For mechanical stability,
the electron column is vertical
with O-ring seals between lenses.
Cooling water ensures constant
temperature and low thermal drift.
Focusing power of a magnetic lens
in the thin-lens approximation (a << f ):
electron KE
For Lorentzian field Bz = B0 /(1 + z
ß much lower for ions
Reversing the lens current reverses the image rotation
but does not change the focusing power
Defects in electron lenses:
Spherical aberration
mid-plane of thin lens
Focus change Δf increases
with distance x from axis.
Axial symmetry implies
Δf = c2 x2 + c4 x4 + …
But x = f1 tanα ~ f α
and rs = Δf tanα ~ Δf α
so rs ~ c2 (fα)2 α = Cs α3
where Cs is the coefficient
of spherical aberration
For a broad beam (radius x),
rs is the radius of the
disk of confusion.
Paraxial focus is at F (Gaussian image plane)
Screen could be moved to the left to display a
smaller disk of least confusion.
Practical cases of spherical aberration
demagnification
e.g. SEM objective
magnification
(electron rays
reversed)
magnification
(Rayleigh
criterion)
Chromatic aberration concerns the kinetic energy E0 of the electrons
1/f = (π/16)(aB02)[e2/(8mE0)] à f = c E0, Δf = c ΔE0 = (f/E0) ΔE0
If E0 decreases, f decreases à focus closer to lens (dashed rays below)
object
plane
image
plane
Rayleigh
criterion
M = v/u = tanα / tanβ ~ α/β
ri = Δv tanβ ~ β Δv, equivalent to rc ~ β Δv/M ~ α Δv/M2
1/u + 1/v = 1/f à
Δv = (v2/f2) Δf and M>>1 à u ~ f, v ~ Mf
So Δv = M2 Δf and rc ~ α (Μ2Δf)/M2 = α Δf = α (f/E0) ΔE0
More generally, rc = Cc α (ΔE0/E0) where Cc = f approximately
Calculations for an objective lens
with a = 1.8 mm and E0 = 200keV
using thin-lens approximation and
more accurate theory (Glaeser, 1952)
1. Cs and Cc are roughly equal to f
2. Small rs and rc require a strong lens
(small focal length ~ 1 or 2 mm)
Causes of chromatic aberration:
1. Spread of kinetic energy from electron source
2. Fluctuations in accelerating voltage (drift and ripple)
3. Energy losses in specimen, due to inelastic scattering
Axial astigmatism
arises from different 1/f
in the x-z and y-z planes
à two elliptical line foci
It is a parasitic aberration,
due to imperfection in
lens-polepiece material
or machining
Solution: use a weak
quadrupole lens
(called a stigmator)
Electrostatic (a) and magnetic (b) stigmators
viewed along the optic axis
Need to adjust amplitude and rotation (mechanically or electrically)
while sweeping through focus.
In light optics, a weak cylindrical lens is used to correct astigmatism of the eye
Distortion
R = Mr + Cd r3
Cd > 0 gives pincushion distortion,
Cd < 0 gives barrel distortion.
Related to spherical aberration, Cs > 0 à Cd > 0.
Arises mainly from final (projector) lens in TEM.
Distortion is more visible at low magnification.
Important in high-resolution (lattice) imaging of crystals
but does not lead to loss of resolution.
Curvature of field is unimportant in TEM
because the depth of focus is large
Correction of lens aberrations
Scherzer (1936) proved that any electron-optical system will suffer from
spherical and chromatic aberration if all of the following are true:
1. The optical system has rotational (axial) symmetry
2. The system produces a real image of the object
3. The focusing fields are time-independent
4. No charge is present on the electron-optical axis
3. is violated in particle accelerators in order to correct chromatic aberration
and is being pursued in femtosecond electron optics.
[also allows correction by an electrostatic mirror, where v is reversed]
4. has been investigated in the TEM by using foils or mesh,
one problem being hydrocarbon contamination
2. seems a necessary requirement (intermediate virtual images don’t help)
1. has been the main target of development, using weak multipole lenses
Multipoles
E∝r
Dipole is used for beam
deflection (scan coils)
Weak quadrupole used to
correct axial astigmatism
Strong quadrupole lenses
are used in particle
accelerators
Sextupoles & octupoles
are used for aberration
correction in the TEM
and SEM
Diagram by A. Bleloch & Q. Ramasse
in Aberration-Corrected Analytical
Electron Microscopy (Wiley, 2011).
E=constant
Multipoles
r2
E∝
E∝r3
Dipole is used for beam
deflection (scan coils)
Weak quadrupole used to
correct axial astigmatism
Strong quadrupole lenses
are used in particle
Quadrupole-Octupole corrector (Scherzer 1947, Krivanek 1994, Nion STEM)
y-cor.
4-fold
stig. cor.
x-cor.
Diagram by A. Bleloch & Q. Ramasse in Aberration-Corrected Analytical Electron Microscopy (Wiley, 2011).
Aberration-corrected STEM (Nion Company, Seattle)
Hexapole (sextupole) corrector (Hawkes 1965, Rose 1981,
Haider 1998, CEOS TEM/STEM corrector)
transfer lenses
long Hex1
Hex2 rotated 60deg
Diagram by A. Bleloch & Q. Ramasse in Aberration-Corrected Analytical Electron Microscopy (Wiley, 2011).
CEOS sextupole corrector