Highresolution Electron Microscopy

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Do we know the arrangement of atoms ?
High-resolution Electron Microscopy
HREM
2
Introducing the optical transfer function of the
lens system
r1Mr0
r0r0
S0(r0)
S1(r1)
System corresponding function
3
Fourier transform
Object diffraction pattern
Object transmission function
F
F
Image diffraction
Image wave function
T(u)
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Effects of aberrations and astigmatism on the
transfer function
Defocus
Define q
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Spherical aberration
?r
a
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Chromatic aberration
V0 ?V0
V0
?ri
a
ai
U
V
?V
For electro-magnetic lens
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Astigmatism
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The transfer function of the objective lens
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Explanation of the high-resolution image

• Amplitude contrast vs. phase contrast
• Amplitude contrast
• Thick sample
• Single beam or two beam condition
• Wave amplitude modified
• Phase contrast
• Very thin sample
• Many beam condition
• Only phase modified

Goal to reconstruct the atomic structure
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Phase object
For an ideal concept, when electron wave passing
through an object, only its phase but not the
amplitude changes
Change of the phasefield of potential
f (x,y,z)
z
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Defining interaction constant
Integrate the whole thickness z
Projected potential along z direction
The transmission function of an ideal phase object
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Weak phase object approximation
For light element/thin specimen
On the back focal plane, introduce the transfer
function
Then on the image plane
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Contrast on the image plane
Ignore the second order terms of sf
Ignore the chromatic aberration in the transfer
function
O-Phase contrast diffraction sF
Phase contrast diffraction AC
F
O-Phase difference sf
F
Phase contrast C
A(u)
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About
F
V
Rayleigh
Resolution defined by the diffraction limit
The effect is usually ignored
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About
Contrast transfer function
However, if we take
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• In practice, the constant CTF is only an
  approximation due to the complicated form of Sinc
• A wide constant band only occurs when Df lt 0 and
  c(u,v) is close to -p/2

Scherzer defocus
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• The wide bandsimilar phase shift on q(x,y),
  makes it possible to reconstruct the details of
  material structure without much distortion
• Point resolution
• The 1st zero crossing at Scherzer defocus
• It corresponds to intuitively interpretable
  resolution for very thin sample
• Image not at Scherzer defocus does not
  necessarily show the information up to point
  resolution correctly

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Defocus effect
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Spherical aberration effect
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Electron wave length effect
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Projected charge density approximation
If we ignore the OBJA and spherical aberration
Assuming ?f ltlt1
Knowing that
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For phase object
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Ignore higher order terms of s and ?f
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Projection of electron density

• ?f 0, zero contrast in the absence of
  aberration
• ?f gt0, underfocus condition, C(x,y) lt0, the
  higher r is, the darker the area
• ?f lt0, overfocus condition

Assumption ?f is not very large The contrast
is linear proportional to the defocus value
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PCD vs. WPO
Thicker specimen Heavier elements
However, the Cs is ignored in the above analysis
Explanation of HREM image

• The origin of HREM image
• Interference of scattered (by the material)
  waves
• Not necessarily corresponding to atomic
  structure
• Projection of the potential
• System imperfections

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Coherence

• Temporal coherence the electron wave is not
  monochromatic
• native energy difference
• Instability of the high-voltage system

Path difference between the two scattered waves
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Spacial coherence the source is not an ideal
geometrical point
P1
P
P2
Coherence
Incoherence
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Coherence effect on high-resolution imaging
K00
K0 wave vector (incident) in the reciprocal
space ri a point on the image plane (real
space) F(K0) distribution of the incident wave
vector
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Define
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Ideal coherence
Incoherence
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Partial coherence
The information limit
when the damping envelope is too small, the
information is considered not transferable any
more
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The concept of resolution

• Rayleigh resolution (diffraction limit)
• Not applicable
• Resolution defined by lens aberrations and
  defocus value Normally referred to as the point
  resolution
• Can be extended to smaller values through
  defocus series
• Resolution defined by spatial coherence of the
  illumination source
• A more precise concept is the information limit
  for the microscope