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How to explore a system?
Electrons
Photons
Atoms
Electrons
Photons
Atoms
How to explore a system?
Electrons
Electrons
Photons
With different
kinetic energies
Electron Energy Analyzers
Detect the charged particles for
a given energy range with good
energy (and space) resolution
Retarding Field Analyzer (RFA)
Cylindrical Mirror Analyzer (CMA)
Hemispherical Analyzer (HA)
Electron Energy Analyzers
Energy distribution curve: response of the system
N(E)
E0
E0 + ΔE
Electron Energy (eV)
Separate the electrons with a defined energy band
Retarding Field Analyzer (RFA)
V0
Retarding Field Analyzer (RFA)
N(E)
V0
V0 = Retarding potential
E0 = eV0 = Pass energy
Current to screen
I (E 0 ) 

E
0
I (E0 ) 
N (E )d (E )
Ep
E
0
N (E )d (E )
N(E)
E0 E0 + ΔE
Electron Energy (eV)
To obtain N(E) one has to differentiate
Retarding Field Analyzer (RFA)
Modulation
V  V  V  k sint 
V  0.5  2V

I V 
V 2
I V  V   I V   I V V 
2!
I V 
I V 
3
V  
V 4  ...

3!
4!
I V  V   I0 V  
First harmonic
I V  3
 



I
V
k

k

...
sint  


8

N(E)
Second harmonic
 I V k 2 I V k 4



 ... cos2t 
4
48


dN(E)/dE
RFA: poor sensitivity and energy resolution
No angular resolution
Electrostatic deflection analyzers
Cylindrical Mirror Analyzer (CMA)
Hemispherical Analyzer (HA)
Dispersing field
Concept
Energy band pass
v1
v2
Deflection is function
of electron energy
Electrons have angular spread
around the entrance direction
v1
+
Electrons with same v will
be deflected by different amounts
Degradation of energy resolution
Cylindrical Mirror Analyzer (CMA)
e- cross the inner cylinder through a slit
experience the field –V of the outer cylinder
go to second slit and and arrive in F
V=deflecting voltage between cylinders
consider an e- arriving at an angle 0 with
E  eVe 
1
mv 02
2
work of the e.m. field on e-
U  eV (r )
r
V (r ) 
ln 
2 0 L  r1 
Q
U  eV (r ) 
r
r
eV
log   U 0 log 
r
r 
 r1 
log 2   1 
 r1 
eVe V eVe
eV
E
U0 



K0 K0
r 
r V
log 2  log 2  e
 r1 
 r1 
U
e- energy
Q  CV 
 r  2 0 L
ln 
V
2 0 L  r1   r2 
ln 
 r1 
e
e- energy inside cylinder
E
K0 
r 
Ve
log 2 
V
 r1 
2 0 L
V
 r2 
ln 
 r1 
K 0e
V
 r2 
log 
 r1 
Cylindrical Mirror Analyzer (CMA)
The trajectory for which the e- is focussed
is a solution of the equation of motion (cyl. coord.)
mr 
U0
0
r
The maximum deflection depends on the entrance angle
rm  r1e
2 K0 sin 2  0
and shows that K0 depends also on 
Electrons with same E will
be deflected by different amounts
depending on the entrance angle
U0 
E
K0
E 
K 0e
V
 r2 
log 
 r1 
focussing condition
K0 
r 
Ve
log 2 
V
 r1 
Cylindrical Mirror Analyzer (CMA)
The numerical solution shows that for
a single K0 there are two values of entrance angle
This means that in general
there are two focussing distances
For K0 = 1.31 the two
focal distances merge into one
A single focussing length L correspond
to different acceptance angles (see curve (c))
L = 6.130 r1
High sensitivity with one pass energy
Cylindrical Mirror Analyzer (CMA)
L
The emission angle determines three main factors
(a) source-image distance on the common cylinders axis (L)
(b) the deflecting voltage for particles with energy E
(c) the required ratio between the cylinders radii
For 0= 42°18.5'
the first spherical aberration term = 0
Cylindrical Mirror Analyzer (CMA)
For small  and small E,
the shift in the axis crossing point is
(Taylor series)
r1 = inner cylinder
L  5.6r1
E
E
 15.4r1 (  )  10.3r1
3
One looks for L  0
E
E
Base resolution
E
(  )
Neglecting the product
E B
E

 5.5(  )3
E
E
 2.75(  )3
  5  0.0873rad
E
E B
 5.5x 0.0873  0,0037
E
E B  0,0037E
3
Cylindrical Mirror Analyzer (CMA)
Transmission = fraction of space in front of the sample
intercepted by the analyser
T  2 sin  1.346 
analyser transmission
for   5
E B
E
 0.0037
For a fixed slit, T does not depend on energy
E B
 5.5(  )3
E
E B
E
 2.255T
3
T  0.12
Cylindrical Mirror Analyzer (CMA)
The image of the source can be reduced by inserting
a slit before the focus that reduces the coefficients
of 3 by a factor of 4
Finite source + ring slit of width W and radius r1
EB 0.18
5.5

W
( )3
E
r1
4
  5  0.0873rad;
r1  100 mm; w  3 mm
E
E

0.18x 3
3
 0.0873  0.006
100
For a given slit, T does not depend on energy while EB  E
So the energy resolution is not constant with E
peak area  (T x EB)
but (T x EB)  E
Peak area  N(E)xE
The spectrum contains the intensity-energy
response function of the analyser
N(E)
E0 E0 + ΔEB
Electron Energy (eV)
Concentric Hemispherical Analyzer (CHA)
Two spherical electrodes
1/r electrostatic potential
Electrons are injected with
energy eV0 at slit S
in the point corresponding
to radius R0
R1
R2
V0  V1
V2
2R0
2R0
The condition to allow e- to describe the central orbit is (point source) R0 

R 
V1 V0  3  2 0 
R1 


R 
V2 V0  3  2 0 
R2 

R R 
V2 V1 V0  2  1 
 R1 R2 
V0  k V2 V1 
Focussing condition in F
For R1=115 R2=185 mm
K = 1.013
R1  R2
2
Concentric Hemispherical Analyzer (CHA)
e- forming angle  with tangential direction
e- with energy E with respect to E0
Shift in the radial position
R  2R0
E
E
 2R0 2
Considering two slits of width W1 and W2
Base resolution
EB W1 W2

 ( )2
E
2R0
  5  0.096rad
R0=150 mm W1 = W2 = 3 mm
R0=150 mm W1 = W2 = 1 mm
E
E
E
E
 0.028
 0.015
The resolution is mainly determined by
the central hemispherical radius
Concentric Hemispherical Analyzer (CHA)
Transmission
If the sample is at the position of the slit W1, we assume W1 = 0 and neglect 
so the angular acceptance in the plane depends on the slit W2
2 
W2
2R0
T  2 sin  2
We also have to consider the angular acceptance in the plane perpendicular to the screen ()
T 
2W2 
Transmission of the analyser
R0 
R0  150mm
W2  3 mm
In analogy to the CMA
EB W1  2W2  T


E
2R0
2 2
2
  57(1 rad )
2
T 
E B
E
1
2
0.04  0.1
 0.02
W2  1 mm
1
T 
0.04  0.06
2
E B
E
Worse than CMA (lower transmission and resolution)???
 0.007
Concentric Hemispherical Analyzer (CHA)
Problem: sample cannot be at position of slit 1
solutions
Use lenses to focus
beam at slit 1
Reduce the analyzer
angle to 150°
r = source radius
 = cone semiangle of source
E = e- energy at the source
rp = source image at entrance slit W1
 = cone seminagle of image
Ep = e- energy at the image
Helmoltz-Lagrange equation
r E  rp E p
rp
M
r
R
Lens magnification
E
Ep
Retarding ratio

1
2
  MR 
 = cone semiangle of source
defined by the lens
Concentric Hemispherical Analyzer (CHA)
What is defining the transmission of the analyser?
Consider the cone with semiangle 
1.
1
2

1
2
  MR  ;   MR 
2.
The lens defines the transmission in 
and the spectrometer in 
  M R  ;   M R 
3.
The spectrometer defines the transmission
  M R  ;   M R 
The CHA is designed
to accept  of about 4-5° (similar to  = 5)
E
E
 0.027 %
1
2
  MR 
The lens defines the transmission


T 
2W2 
R0 
Electrostatic lenses
Optical ray refraction
Electron refraction
e speed
Refraction
index
Snell’s law:
n1 sin1 = n2 sin2
electrostatic
potential
1
1
mv 12  eV1  mv 22  eV2
2
2
The potential changes abruptly at the interface:
only the perpendicular component of the momentum changes
mv1 sin1  mv 2 sin2
V1 sin1  V2 sin2
Electrostatic lenses
For real lenses there are no abrupt changes in the potential, as shown in the figure
e path
Equipotential lines
But one can assume the asymptotic behavior of the electron trajectories to make use of the lens equations
r2
r1

M  2
1
M
Transverse
magnification
M M  
Angular
magnification
f1 n1
V
  1
f2 n2
V2
M 
f1
q

p
f2
M  
r1 f1
r2 f2
Helmoltz-Lagrange
equation
Conservation of brightness
Electrostatic lenses
Lenses has the effect to change the kinetic energy of the beam
Focussing
Defocussing
Focussing
Electron lenses formed using metallic apertures.
Electrostatic deflection analyzers
CMA
Energy resolution
EB
E0
 As  B n  C n
E0 = pass energy
ΔEB = Emin-Emax transmitted
s = slit width
, angular apertures
CHA
A
B
C
n
Cylindrical mirror
2.2/l
5.55
0
3
Cylindrical deflector 127°
2/r
4/3
1
2
Spherical deflector 180°
1/r
1
0
2
Electrostatic deflection analyzers
Geometry of the acceptance slit is very different
CHA
5°
CMA
6°
42.3°
Small signal
Compatible with simple electrostatic
aperture and tube lenses
Long focal distance
Radius 100 - 150 mm
Res. Power about 1000-5000
Working distance about 25-50 mm
EB 
E0
5000
 2 meV
Large signal
Non compatible with simple electrostatic
aperture and tube lenses
Short focal distance
Cyl diam 100 - 150 mm
Res. Power about 200
Working distance about 5 mm
E B 
E0
200
 0.1 eV
Electrostatic deflection analyzers
Detection mode
Single
Multi
Scan over voltages acquiring
counts at each energy
E(V)
Scan over voltages acquiring
the position and therefore the energy
of the electrons with different trajectories
E1 E2 E3
Electrostatic deflection analyzers
Detection mode
Channelplates
+ ccd camera
Multi Channeltron
Scan over voltages acquiring
the position and therefore the energy
of the e- with different trajectories
Electrostatic deflection analyzers
Mode of operation
No pre-retarding potential
Pre-retarding potential
Vary E0 with Vscan
Vary the pre-retarding potential
and not E0
E
E0
 As  B n  C n
EB is increasing with energy
E
E0
 As  B n  C n
EB is constant
Ek  ER  E 0
Hemispherical Analyzer of
Electron Kinetic Energy
Lay-out of an Electron Spectroscopy Experiment
Based onto a Double-Pass Cylindrical Mirror
Analyzer
Hemispherical
Analyzer of
Electron Kinetic
Energy with
Entrance Optics
Designed for
Lateral Resolution
Electron Source and Energy Monochromator
and
Electron Kinetic Energy Analyzer
HREELS Apparatus
Electron Multiplier
Detection efficiency 80 %
Gain 108
Microchannel plates
Thin Si-Pb oxide glass wafers
Channel density  105 cm-1
Channel walls act as electron multipliers
Detection efficiency 80 %
Gain 105 - 108

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