Wave optic and basics of TEM

Transcription

Etienne SNOECK
CEMES - Toulouse
CEMES -CNRS
29, rue Jeanne Marvig
31055 Toulouse
email: snoeck@cemes.fr
http://www.cemes.fr
Dr. Etienne SNOECK - CEMES - Toulouse
CEMES - Toulouse
•
Wave - particles dualism
•
Electron – matter interactions
•
•
•
single atom electron scattering
•
electron diffraction
Electron optics
•
TEM column
•
Magnetic lenses
Image formation and aberration
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The beginning…
1897: J. J. Thomson discovered the electrons by studying the «cathodic beams »
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The beginning…
1925: Louis de Broglie introduced the matter and wave-particle duality
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The beginning…
1927: C. Davisson and L. Germer showed the wave behaviour of electrons
Electron wave lengths vs acceleration voltage
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The beginning…
Electrons are particles
•
Mass me and kinetic energy: 1/2 mev2
•
Electric charge -e and electrostatic energy: eV
•
Submited to electrostatic force : F = -e.E
•
Submited to magnetic force : F = -eVB
•
Interact with the electrons cloud and nucleus of atoms
•
Can be localized
Electrons are waves
•
Wave lenght :  = h/mv
•
Interferences
•
Cannot be localized
•
Diffraction by periodical lattices
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Waves
Spherical wave
Plane wave
k=1/
Wave front
S
Wave front

k
S
k
Ponctual source
« Extented » source



 

 ( r , t )  Ar  exp i t  k  r   ( r )
Wave function
Amplitude

Propagation
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Interferences
Plane wave
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Interferences
Electrons are particles and waves
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•
•
•
•
•
Electron optics
•
TEM column
•
Magnetic lenses
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What do an electron see when entering a crystal ?
atom
k0
nucleus

k’
Electron cloud screening
2
4π
with
4π
2
Atoms = Scattering centers : essentialy the nucleus positively charged
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What do an electron see when entering a crystal
Mean inner
potential
Vi
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 An highly localised positive potential
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Electrons are waves
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Electrons are waves
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Electrons are waves
S1
S2

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Electrons are waves
Electron wave
Pure « Amplitude » object
Pure « Phase » object
I ( x )    *  a 2
2
Measured intensity :
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Huygens-Fresnel principle
Huygens contribution (1678)
The light propagates step by step. Every element of surface diffuses and behaves as a secondary source of spherical waves.
Fresnel contribution (1818)
The complex amplitude of the wave at a specific point is the sum of the complex amplitudes of the vibrations produced by all the secondary sources. These vibrations interfere to form the vibration in the considered point. • wave at point « M »:
 M    0 exp  it
with 0 complex amplitude and  = 2 the wave frequency
S

r
M
dS


n

P
 wave at point « P »:
exp i 2 k r
 ( P )   M Q
dS
r
S
with •
•
•
•
r = MP
k = 1/
dS surface element @M
Q a diffusion coefficient
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Rutherford diffusion
Rutherford diffusion :

X rays (single Si atom) :
Electrons
s (nm-1)
s (nm-1)
f i X ( s )   A j exp( B j s 2 )  C
me 2  2
f ei ( )  2 (
) ( Zi  fi X )
2h sin 
Doyle-Turner coefficient
Huge electron –matter Interaction
4
j 1
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Waves and rays
Electrons are waves
Incident plane wave on a single atom
Incident plane wave on a 1D periodic array of atoms
atom
k0
k’
k0
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
+ interference
a
s (nm-1)
s (nm-1)
f ei ( ) 
me 2  2
(
) ( Zi  fi X )
2h 2 sin 
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
+ interference
a'
s (nm-1)
s (nm-1)
1/a’
me 2  2
) ( Zi  fi X )
f ei ( )  2 (
2h sin 
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Waves and rays
Electrons are waves
Incident plane wave on a periodic 3D array of atoms
k0
kB


Path difference : 2
= 2d.sin
d
 
Constructive interferences : = n.
d.sin
2d.sin n.
k0
d ~ 10-10 m
 ~ 10-12 m
  ~ 10-3 rad (sinB= B )
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Atomic diffusion
The crystal contains n atoms. Each atom j in
as only elastic diffusion is considered

rj creates an elastic diffusion of the wave


k0  k

Incident beam k 0
•
 phase shift relative to the original wave
•
fj atomic diffusion factor


k
Diffused beam
Atom
• The diffused wave function by the atom j is:
:

k

rj

 j ( r , t )   0 exp  i t   r . f j
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Atomic diffusion
Total phase shift of the diffused wave in « r »
P

r
o
 
r  ri

k

k0 
2
rj
1
2 contributions :
 
  
2  2 k  r  rj 

r j : phase shift 1 of the incident wave relative to the origin:


Phase shift 2 of the diffused wave between the atom in r j and the point r
1  2 k0  rj
• 0 
•
Total phase shift:

 
 





 
 r   1  2  2 k0  rj  k  r  rj   2 k  k0  rj  k  r



Diffraction vector: Difference between the diffused wave vector k and the incident wave vector k0
  
K  k  k0
P

r


k0  k

K

k0 
o
1

k
rj
2
 
r  ri

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Atomic diffusion
  
K  k  k0
Wave function in « r » due to the diffusion of the atom in rj
 


i 2  K r j
 j ( r , t )   0 exp i t 2 k r  . f j exp
o

k0 
1
P

r

K

k
rj
 
r  ri
2
• For the cristal (n atoms): the diffracted beam tot is the sum of the diffused waves by the n atoms.
 ( r , t )   j ( r , t )   0 exp i t 2 k r  .  f j exp

n
j 1


n
 
i 2  K r j
j 1
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Diffraction
 ( r , t )   j ( r , t )   0 exp i t 2 k r  .  f j exp


n

j 1
n
 
i 2  K r j
j 1

Diffraction by the crystal with n atoms located on the « j » position :r j

FK   f e j ( ) exp(  B j 2 ) exp( 2iK .rj )
j
B j  8 2 u 2
  
r j  ru  Rm,n , p
Periodic crystal

ru
u2
Debye-Waller due to the atom vibration




 xu , yu , zu  0,1 




R

m
a

n
b

p
c
vector of the lattice:
m ,n , p
vector of the unit cell: ru  xu a  yu b  zu c
with
O
(m, n, p integers)
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Diffraction
FK 
 
 exp i 2 K  R
m ,n , p
m ,n , p
S(K ) 
 
.  f eu ( ) exp(  Bu 2 ) exp i 2 K  ru
u
 
 exp i 2 K  R
m ,n , p
m ,n , p
 
F ( K )   f eu ( ) exp(  Bu 2 ) exp i 2 K  ru
u
Structure Factor (unit cell)
Form Factor (whole crystal)
The total diffracted intensity in a direction « K » is given by


 2
I   ( r , t )*. ( r , t )   ( r , t )
=>
 

I F K
2
 

S K
2
Yes in x-ray diffraction BUT… generally not in TEM
Maximal when K corresponds to a node of the reciprocal lattice


K  K hkl  h a  k b  l c

 Laue conditions
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•
•
•
•
•
Electron optics
•
TEM column
•
Magnetic lenses
•
Diffraction pattern formation
•
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Why microscopy with electrons ?
Diffraction through a circular aperture
•
First minimum at sin  
1, 22 
a
« a » being the aperture diameter
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Resolution
Two ponctual sources could be separated if their diffraction figures do not overlap
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Rayleigh criterium
•
The two images are just separated when the central
maximum of one diffraction figure overlap with the
first minimum of the other
•
The angular switching between the two sources
corresponding to Rayleigh criteria is :
c 
1, 22 
a
Lord Rayleigh (1842-1919)
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c 
1, 22 
a
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c 
1, 22 
a
Increase the resolution  decrease the wavelength of the radiation
(X-Ray : 10-10 - 10-12 m but no lenses !!)

h
mv
 Use « fast » electrons
Gun, electromagnetic lenses, etc…
microscope
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How to focalize electrons ?
Wehnelt and Gabor: using electric and magnetic fields
1927: Beginning of electron optic: by Hans Busch « an
electromagnetic field has the same effect on an electron than an
optic lens on a light beam »
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TEM : the beginning…
Ernst Ruska et Max Knoll worked at TH Berlin, early
30es
Oscilloscope with a small intense spot and a fast scan
1931: Ruska get a first X14.4 magnified image.
Ruska E (translated by Mulvey T). The early development of electron lenses and
electron microscopy. Stuttgart: Hirzel, 1980.
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TEM : the beginning…
D’après Knoll et Ruska, Ann. Phys. 12, 607, 1932
1932: Vertical microscope
Higher magnification than an optical (12000 X )
•
•
•
•
•
Cold cathod (glow discharge) 65kV
Condenser
Objective
Projector
Fluorescent screen
Ruska, E, Z. Physik 87, 580-602 (1934)
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TEM
Optical microscope
TEM
Def
Gun
Source
Def
Beam
Sample
Def
Image
Diff
Optical column
Def
Proj
Gun + High tension
Condensor Part
C1+C2
Condensor aperture
Objective Part
Objective aperture
Diffraction Part
Diff + inter
SAED aperture
Shutter
Projector Part
P1+P2
Detector
Detector
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TEM
Increasing the resolution …


h
mv
v
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The TEM column
A TEM = 6 parts + deflectors to
align each part each other
Def
Gun
Gun + High tension
Part of the column
Bloc
Line tube
Condensor Part
C1+C2
Def
Beam
Condensor aperture
+
Def
Image
Diff
=
+
Objective Part
Polar piece
Objective aperture
Diffraction Part
Diff + inter
Def
Proj
SAED aperture
Projector Part
P1+P2
Shutter
Detector
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The electron source
Def
Gun
Def
Beam
Gun + High tension
Condensor Part
C1+C2
Condensor aperture
Def
Image
Diff
Objective Part
Objective aperture
Def
Proj
Diffraction Part
Diff + inter
SAED aperture
Projector Part
P1+P2
Shutter
Detector
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Thermoionic source
Electron acceleration
Schematic representation
Metallic filament heating
Filament
Vacuum
Increase the
T°K
Exit work = f

E
Fermi Level
Fermi-Dirac distribution
change with T°K
Real design
Filament
Whenelt
8mm
W filament
T  2800 K
  4,5eV
Cross over
LaB6 filament
T  2000 K
  2,5eV
Anode
10 7 Torr
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Cold Field Emission Gun (CFEG)
Real design
Supressor
Tip
Emission process :
Schottky effect
Fermi-Dirac
distribution
at 300K
Tunneling effect
Extractor
10 11Torr
Fowler-Nordheim law
jFN 
4me 2
 b 
d exp  
h3
 d 
d
b  c2
c1 F
 f ( F , )
 3/ 2
F
g ( F , )
W <310>
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CFEG issues
I-a
I-b-c-d
II
II
Arc
I
Solution : flash
to heat the tip with a
current during some
seconds
Technology more
difficult
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Comparison between sources
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Electromagnetic lenses
Def
Gun
Def
Beam
Gun + High tension
Condensor Part
C1+C2
Condensor aperture
Def
Image
Diff
Objective Part
Objective aperture
Def
Proj
Diffraction Part
Diff + inter
SAED aperture
Projector Part
P1+P2
Shutter
Detector
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Principle of electromagnetic lenses
Electrons rotation
+ focalisation
e
Real design of a lens
coil + plugs + cooling + polar piece
Power supply
Water
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Basic geometrical optic
Magnetic lens = thin convergent optical lens
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Magnetic lens focalisation
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Magnetic lens focalisation
Main advantage of electromagnetic lenses : the focal length is tunable with the lens current !!
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Illumination system of the column
Def
Gun
Def
Beam
Def
Image
Diff
Def
Proj
Gun + High tension
Condensor Part
C1+C2
Condensor aperture
Objective Part
Objective aperture
Diffraction Part
Diff + inter
SAED aperture
Shutter
Projector Part
P1+P2
Detector
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1. Spot Size : Condensor 1
C1 Strength
Spot size
2.
Intensity: Condensor 2
C2 Strength change the illumination area and the convergence angle
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Sample illumination and convergence angle of the electron beam change
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Objective lens
Def
Gun
Def
Beam
Def
Image
Diff
Def
Proj
Gun + High tension
Condensor Part
C1+C2
Condensor aperture
Objective Part
Objective aperture
Diffraction Part
Diff + inter
SAED aperture
Shutter
Projector Part
P1+P2
Detector
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Objective lens
The objective lens transfers two informations:
•
 image in the image plane
•
 Fraunhofer diffraction in the focal plane
Object plane
Object
Conjugated
planes
f
Focal plane
Fourier Transform (FT)
of the object : diffraction pattern
Image plane
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The diffraction lens and the projection system
Def
Gun
Gun + High tension
Condensor Part
C1+C2
Def
Beam
Condensor aperture
Def
Image
Diff
Objective Part
Objective aperture
Diffraction Part
Diff + inter
Def
Proj
SAED aperture
Projector Part
P1+P2
Shutter
Detector
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« diffraction » mode
« imaging » mode
1/p1 + 1/p2 = 1/f
objective
Focal plane
p’1
Image plane
Diffraction lens
p1
f
f’
p2
Adjusting the focal lenght of the diffraction lens, one can get either the image plane or the focal plane (diffraction)
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• Diffraction lens : select the image or the diffraction plane (Diffraction control)
• Intermediate lens + Projector 1 and Projector 2 change the magnification or the camera length
Cross over of the Gun
Condensor 1
 Spot size
Condensor 2
 Intensity
Minilens
Objective Condensor
 Focus
Objective Image
Diffraction lens
 Diffraction and image mode
Intermediate
Projector 1
 Magnification
Projector 2
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•
•
•
•
•
•
Electron optics
•
TEM column
•
Magnetic lenses
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High Resolution Imaging
e–
Incident Electron
wave
Crystal
Crystal
2q
Lens
Exit Wave function
e–
Screen
Depends on
•
Aberrations
•
Atomic potential
•
Defocalization
•
Diffusion
•
Beam incoherences
•
Diffraction
•
Phase shift
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High Resolution Imaging : electron wave within the object
e–
 (r , z )  e iKz
 (r)
e–
 (r, z )   (r )eiKz
 exit (r )
Exit Wave function
Diffracted beams
~ ( k )
 exit (r )   k ~( k ) exp( 2ik.r )
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High Resolution Imaging : electron wave in the column
 o  exp(iK.r )
Object
obj  ao (r) expi( K .r  o (r))
Objective
 s  As( r )exp i ( K .r  s( r ))
Focal plane
Intermediate lenses
& projector
I(x,y) s   As2(r)
2
 s (r )
is lost
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High Resolution Imaging
~
s (r)  gexp{2ig.r}
g
Wave function on the detector
Diffracted beams
I (r)  s (r)
2
~ ~
 gg*exp{2i(g  g).r}
Measured intensity
g
g
• The contrast is due to the interferences
between the g and g’ diffracted beams
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Single crystal in zone axis
 
0  g2
 
0  g1
 
0  g3
 
g1 g3
 
0 g2
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High Resolution Imaging: interferences + TEM aberrations
exit(r)  ~gexp{2ig.r}
g
Exit wave function
Diffracted beams
(r)  ~gei (g)exp{2ig.r}
g
Image wave function
on the detector
Tranfert function
I(r)  (r) 2
Image intensity
•
The objective lens aberrations modifies the phase
of the exit wave  phase plate
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Lens Aberration
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Spherical Aberration
Disk of
least
confusion
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In optical microscopy: Convergent lens + divergente lens
Divergent lens in TEM ???
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Spherical Aberration delocalisation
Pb with surface and interface
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Chromatic Aberration
Slow electrons
Fast electrons
Lack of resolution
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Axial Coma and First order Astigmatism
C2
The coma (off-axial aberration)
C2’
I
C1’
C1
O
Astigmatism and field curvature (On and Off axial aberrations) :
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Anisotropic aberrations
Anisotropic coma, anisotropic astigmatism, anisotropic distorsion
Due to the helicoïdal trajectories of the electrons inside the lenses
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Aberrations : wavefront description
• Focalisation (first order effect) C1
C1
• Spherical aberration C3 (Cs), C5, etc … and
• Chromatic aberration (Cc)
C3=Cs
and …
A1
• Off-axial aberrations
• Astignatism and field curvature (A1, A2, etc ..)
• Coma (B2, B4, etc …)
B2
A2
A3
• High order aberrations (S3,D3, …)
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The first order astigmatism correction: the stigmator
Quadrupole lenses :
Huge elliptic beam
Round beam
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The first order astigmatism correction: the stigmator
Size of the line tube
2 quadrupoles  stigmator (correction of A1)
Uncorrected spot
Corrected spot
Qpol X
Qpol Y
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Spherical aberration correction: see Max Haider lecture
Non cylindrical lenses :
Quadrupoles, Octopoles
Hexapoles
Dodecapoles
Two major effects
• Negative C3
• Huge A2
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High Resolution Imaging: interferences + TEM aberrations
(r)  ~gei (g)exp{2ig.r}
g
C1
C3=Cs
B2
A1
C1 + C3
A2
A3
+ A1(astigmatism)
 (g )
+ B1 (coma)
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Phase plate
 (g )
Scherzer focus
Zero focus


Cs = 1.34mm C1 = 0
Cs = 1.34mm C1 = -60nm (@300kV)
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40
Contraste transfert function
sin g 
C1 = 0 nm
C1 = – 60 nm
1
(1.8Å)–1
–1
g
g
Scherzer focus
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Exple : CdSe crystal
C = 0 nm
C = – 60 nm
Cd
S
•
Complicated and non trivial contrast at Gaussian focus
•
The HREM image at Scherzer focus is the negative of the projected
potential
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41
Phase plate: astigmatism
C1 = – 60nm
A1 = 0
+ A1 = 80nm
Scherzer focus + A1 = 80nm
 (g)

Cd
Cd
Cd
S
S
S

The effect of the objective lens is not symetric
 Looks like a bad sample alignement….
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Phase plate: Coma
C1 = – 60nm
B2 = 0
+ B2 = 1000nm
Scherzer focus + B2 = 1000nm
 (g ) 
Cd
Cd
S
S

incident beam
The central beam is not aligned along the optic axis
crystal
 The g and –g diffracted beam have not the
same phase
lens
 The symetry is broken
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42
Cs Aberration corrector
Cs = 1.34 mm - C1 = 0
Cs = 0 nm - C1 = 0


0.5 nm
0.5 nm
5 nm
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Cs Aberration corrector
HubbleTelescope
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43
Advanced TEMs
Gun
Cond
Biprism 1
Condensor aperture 1
Stigm Cond
Condensor 1
Condensor 2
Probe Cs corrector
Condensor aperture 2
Condensor 3
Condensor Objective
Large pole piece gap for in situ (10 mm)
Contrast aperture
Obj
Objective Image
Stigm obj
Transfert lens doublet 1
Short hexapole 1
Image Cs corrector
Long hexapole
Short hexapole 2
Diff
Stigm diff
Biprism 2
SA aperture
Intermediate lens 1
Biprism 3
Proj
Intermediate lens 2
Biprism 4
Intermediate lens 3
Detector
Projector 1
Projector 2
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Advanced TEMs
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44
The end
email: snoeck@cemes.fr
http://www.cemes.fr
CEMES - Toulouse
45

Wave optic and basics of TEM

Transcription

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