How to explore a system? Photons Electrons Atoms Electrons Photons Atoms

How to explore a system?

Photons

Electrons

Atoms

Electrons

Photons

Atoms

How to explore a system?

Photons

Electrons Electrons

With differentkinetic energies

Detect the charged particles fora given energy range with good energy (and space) resolution

Electron Energy Analyzers

Retarding Field Analyzer (RFA)Cylindrical Mirror Analyzer (CMA)Hemispherical Analyzer (HA)

Separate the electrons with a defined energy band

Electron Energy Analyzers

Electron Energy (eV)

N(E)

E0E0 + ΔE

Energy distribution curve: response of the system

Retarding Field Analyzer (RFA)

V0


N(E)

V0 = Retarding potential

E0 = eV0 = Pass energy

Current to screen

0

)()()( 0 EEdENEI

pE

EEdENEI

0

)()()( 0

To obtain N(E) one has to differentiate

V0


N(E)

E0E0 + ΔE

V V V k sin t

...!4!3

!243

2

VVI

VVI

VVI

VVIVIVVI


Modulation

VV 25.0

tkVIkVI 2cos...

484

42

tk

VIkVI sin...

83

VIVVI 0

First harmonic

Second harmonic

N(E)

dN(E)/dE

RFA: poor sensitivity and energy resolutionNo angular resolution

Electrostatic deflection analyzers

Energy band pass

Cylindrical Mirror Analyzer (CMA)Hemispherical Analyzer (HA)

Dispersing field

Deflection is functionof electron energy

v1

v1

v2

-

+

Electrons have angular spread around the entrance direction

Electrons with same v willbe deflected by different amounts

Degradation of energy resolution

Concept

Cylindrical Mirror Analyzer (CMA)

202

1mveVE e

1

20 log

r

r

V

VK e

V

rr

eKE

1

2

0

log

V=deflecting voltage between cylinders

e- energy

consider an e- arriving at an angle 0 with

e- energy inside cylinder

)(reVU work of the e.m. field on e-

10

ln2

)(r

r

L

QrV

V

rr

LCVQ

1

2

0

ln

2

V

rr

L

r

r

L

ereVU

1

2

0

10 ln

2ln

2)(

10

1

1

2

loglog

logr

rU

r

r

rr

eVU

00

1

2

1

2

0

loglogK

E

K

eV

V

V

rr

eV

rr

eVU e

e

e

e- cross the inner cylinder through a slitexperience the field –V of the outer cylindergo to second slit and and arrive in F


Electrons with same E willbe deflected by different amountsdepending on the entrance angle

The trajectory for which the e- is focussedis a solution of the equation of motion (cyl. coord.)

The maximum deflection depends on the entrance angle,

00 r

Urm

02

0 sin21

Km err

and shows that K0 depends also on

focussing condition

V

rr

eKE

1

2

0

log

1

20 log

r

r

V

VK e

00 K

EU

A single focussing length L correspondto different acceptance angles (see curve (c))

High sensitivity with one pass energy

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 twofocal distances merge into one


L = 6.130 r1


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

L

For small and small E, the shift in the axis crossing point is

(Taylor series)

)(3.10)(4.156.5 13

11

EE

rrEE

rL

One looks for L 0

3)(75.2 EE

Base resolution

3)(5.5

EE

E

EB


r1 = inner cylinder

Neglecting the product

0.0873rad 5

EE

xE

E

B

B

0037,0

0037,0.087305.5 3

Transmission = fraction of space in front of the sample intercepted by the analyser

analyser transmission

346.1sin2T

12.0T 0037.0 5

E

Efor B

3255.2 TE

EB


For a fixed slit, T does not depend on energy

3)(5.5 E

EB

For a given slit, T does not depend on energy while EB E

peak area (T x EB)

3

1

)(45.518.0

W

rE

EB


N(E)

E0E0 + ΔEBPeak area N(E)xE


So the energy resolution is not constant with E

The spectrum contains the intensity-energy response function of the analyser

but (T x EB) E

The image of the source can be reduced by inserting a slit before the focus that reduces the coefficientsof 3 by a factor of 4

Finite source + ring slit of width W and radius r1

006.0.08730100

318.0

mm 3 w mm; 100r0.0873rad; 5

3

1

xEE

Two spherical electrodes

Concentric Hemispherical Analyzer (CHA)

1/r electrostatic potential

Electrons are injected with energy eV0 at slit S

in the point corresponding to radius R0

0

22

0

110 22 R

RV

RR

VV

The condition to allow e- to describe the central orbit is (point source)

2

002

1

001

23

23

RR

VV

RR

VV

2

1

1

2012 R

RRR

VVV 120 VVkV

Focussing condition in F

For R1=115 R2=185 mm K = 1.013

221

0

RRR

e- forming angle with tangential direction


200 22 R

EE

RR

The resolution is mainly determined bythe central hemispherical radius

R0=150 mm W1 = W2 = 3 mm

e- with energy E with respect to E0

Considering two slits of width W1 and W2

Base resolution

rad096.05

2

0

21 )(2

RWW

EEB

Shift in the radial position

R0=150 mm W1 = W2 = 1 mm 015.0EE

028.0EE

Worse than CMA (lower transmission and resolution)???

Transmission of the analyser

02.0E

EB

0

22RW

T


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

We also have to consider the angular acceptance in the plane perpendicular to the screen ()

2

22

0

21

222

T

RWW

EEB

Transmission

150mmR

) 1(57

0 rad

In analogy to the CMA

0

22

2RW

1.00.0421

T

mm 3W2 mm 1W2

06.00.0421

T

007.0E

EB

2sin2 T


Problem: sample cannot be at position of slit 1

solutionsReduce the analyzer

angle to 150°Use lenses to focus

beam at slit 1

r = source radius = cone semiangle of sourceE = e- energy at the sourcerp = source image at entrance slit W1 = cone seminagle of imageEp = e- energy at the image

pp ErEr Helmoltz-Lagrange equation

Lens magnificationr

rM p

pEE

R Retarding ratio

21

MR = cone semiangle of sourcedefined by the lens

Concentric Hemispherical Analyzer (CHA)What is defining the transmission of the analyser?

Consider the cone with semiangle

1. The lens defines the transmission

2. The lens defines the transmission in and the spectrometer in

3. The spectrometer defines the transmission

21

21

;

MRMR

RMRM ;

RMRM ;

%027.0EE

The CHA is designedto accept of about 4-5° (similar to = 5)

21

MR

0

22RW

T

Electrostatic lenses

Optical ray refraction Electron refraction

e speed

Refractionindex

electrostaticpotential

The potential changes abruptly at the interface:only the perpendicular component of the momentum changes

2221

21 2

121

eVmveVmv Snell’s law: n1 sin1 = n2 sin2

2211 sinsin mvmv

2211 sinsin VV


For real lenses there are no abrupt changes in the potential, as shown in the figure

But one can assume the asymptotic behavior of the electron trajectories to make use of the lens equations

Equipotential lines

1

2

1

2

M

rr

M

e path

Transverse magnification

Angular magnification

2

1

2

1

2

1

ff

rr

M

fq

pf

M

2

1

2

1

2

1

VV

nn

ff

MM Helmoltz-Lagrangeequation

Conservation of brightness

Electron lenses formed using metallic apertures.


Lenses has the effect to change the kinetic energy of the beam

Focussing Defocussing Focussing


CHA

Energy resolution

nnB CBAsE

E

0

E0 = pass energyΔEB = Emin-Emax transmitted

s = slit width, angular apertures

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

CMA

Electrostatic deflection analyzersGeometry of the acceptance slit is very different

CHA CMA5°6°

42.3°

Small signalCompatible with simple electrostatic

aperture and tube lensesLong focal distance

Radius 100 - 150 mmRes. Power about 1000-5000

Working distance about 25-50 mm

Large signalNon compatible with simple electrostatic

aperture and tube lensesShort focal distance

Cyl diam 100 - 150 mmRes. Power about 200

Working distance about 5 mm

meV 2 5000

0 E

EB eV 0.1 200

0 E

EB


Detection mode

Single

Scan over voltages acquiringcounts at each energy

Scan over voltages acquiringthe position and therefore the energy

of the electrons with different trajectories

Multi

E1 E2 E3E(V)


Detection mode

Scan over voltages acquiringthe position and therefore the energy

of the e- with different trajectories

Multi ChanneltronChannelplates+ ccd camera


Mode of operation

No pre-retarding potential

Vary E0 with Vscan

EB is increasing with energy

Pre-retarding potential

Vary the pre-retarding potentialand not E0

EB is constant

nn CBAsEE

0

nn CBAsEE

0

0EEE Rk

Hemispherical Analyzer ofElectron 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 MonochromatorandElectron Kinetic Energy Analyzer

HREELS Apparatus

Electron Multiplier

Detection efficiency 80 %

Gain 108

Microchannel plates

Thin Si-Pb oxide glass wafers

Channel walls act as electron multipliers

Channel density 105 cm-1

Detection efficiency 80 %

Gain 105 - 108

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How to explore a system? Photons Electrons Atoms Electrons Photons Atoms