An Electron Lens
David Muller 2008
Geometric Optics Geometric Optics – A Simple Lens A Simple Lens
x x
Object
plane
image
plane
Back
focal
plane
front
focal
plane
Lens
at z=0
?
Focusing: angular deflection of ray ? distance from optic axis
David Muller 2008
Geometric Optics Geometric Optics – A Simple Lens A Simple Lens
x x
Object
plane
image
plane
Back
focal
plane
front
focal
plane
Lens
at z=0
?1
?1
Wavefronts in focal plane are the Fourier Transform of the Image/Object
David Muller 2008
X-ray and Electron Diffraction from a Silicon Crystal
?=0.0251?
200 keV Electrons
?=1.54 ?
10 keV x-rays
?
?
sin d n =
In Si d220 = 1.92 ?
Bragg’s Law:
David Muller 2008
Electron Velocity and Wavelength
De Broglie Wavelength: p
h = ?
Where h is Planck’s constant
And p=mv are the momentum,
mass and velocity of the electron
If an electron is accelerated through a potential eV, it gains kinetic energy
eV mv = 2
2
1
So the momentum is meV mv 2 =
V
nm
meV
h 23 . 1
2
2
= = ?
Electron wavelength
( relativistically correct form:
) 2 ( 2
0
2 2
eV c m eV
c h
+
= ?
)
(V in Volts)
David Muller 2008
Electron Wavelength vs. Accelerating Voltage
0.0087189 0.81352 1 MeV
0.019687 0.77653 300 keV
0.025078 0.69531 200 keV
0.037013 0.54822 100 keV
0.12204 0.019194 10 keV
0.38763 0.062469 1 keV
1.2263 0.0062560 100 V
12.264 0.0019784 1 V
?(?) v/c Accelerating
Voltage
0
0.01
0.02
0.03
0.04
0.05
0 200 400 600 800 1000
Relativistic
Non-relativistic
? (Angstroms)
Electron Kinetic Energy (keV)
David Muller 2008
Resolution Limits Imposed by Spherical Aberration, Cs
(Or why we can’t do subatomic imaging with a 100 keV electron)
Lens
3
min 2
1
? s C d =
Plane of
Least Confusion
Gaussian
image plane
Cs=0
Cs>0
For Cs>0, rays far from the axis are bent too strongly and come to a crossover
before the gaussian image plane.
For a lens with aperture angle ?, the minimum blur is
min d
Typical TEM numbers: Cs= 1 mm, ?=10 mrad ? dmin= 0.5 nm
David Muller 2008
Resolution Limits Imposed by the Diffraction Limit
(Less diffraction with a large aperture – must be balanced against Cs)
Lens
0 0
0
61 . 0
sin
61 . 0
?
?
?
?
? =
n
d
Gaussian
image plane
The image of a point transferred through a lens with a circular aperture of
semiangle
?
0
is an Airy Disk of diameter
0 d
(for electrons, n~1, and the angles are small)
?
0
(0.61 for incoherent imaging e.g. ADF-STEM, 1.22 for coherent or phase contrast,. E.g TEM)
David Muller 2008
Balancing Spherical Aberration against the Diffraction Limit
(Less diffraction with a large aperture – must be balanced against Cs)
2
3
0
2
0
2 2
0
2
2
1 61 . 0
??
?
??
? + ?
??
?
? ??
?
= + ?
?
?
?
s s tot C d d d
For a rough estimate of the optimum aperture size, convolve blurring terms
-If the point spreads were gaussian, we could add in quadrature:
1
10
100
1 10
Probe Size (Angstroms)
? (mrad)
ds
d0
Optimal aperture
And minimum
Spot size
4 / 3 4 / 1
min 66 . 0
?
s C d =
David Muller 2008
Balancing Spherical Aberration against the Diffraction Limit
(Less diffraction with a large aperture – must be balanced against Cs)
4 / 3 4 / 1
min 43 . 0
?
s C d =
A more accurate wave-optical treatment, allowing less th