Introduction to Energy Loss SpectrometryHelmut Kohl

Physikalisches Institut Interdisziplinäres Centrum für Elektronenmikroskopie und Mikroanalyse (ICEM)

Westfälische Wilhelms-Universität Münster, Germany

1. Introduction2. The scattering process3. Inner shell losses4. The low-loss regime5. Relativistic effects6. Summary and conclusion

Contents:

1. Introduction

integrated over the energy window and up to the acceptance angle

2

~d

Id dE

Spectrum of BN (Ahn et al., EELS Atlas 1982)

2. The scattering process

Assumptions:

- weak scattering

- non-relativistic

- object initially in the ground state

Fermis golden rule (1. order Born approximation)

Scattering geometry

0 , :n

0k fkplane wave state of the incident and outgoing electron

initial and final state of the object 0 n

interaction between the incident electron and

the electrons in the object

2

0 0

²~ 0 f n

n

dk V k n E E E

d dE

0 , :fk k

0

²

4j j

eV

r r

jr

r

After some calculations (Bethe, 1930)

2 2

04 2

40 nK

nH

dn E E E

d dE K a

��������������

exp :jK

j

iK r ������������������������������������������

kinematics object function

Scattering vector

Å0,529Ha

Fourier transformed density (operator)

Bohrs radius

E

0 fK k k ������������������������������������������

dynamic form factor (vanHove, 1954)

2

0, 0 nKn

S K n ����������������������������

More general case: coherent superposition of two incident waves

Scattering of two coherent waves

How can one calculate the dynamic form factor?

Mixed dynamic form factor (MDFF; Rose,1974)P. Schattschneider, Thursday

'0'

, , 0 0 nK Kn

S K K n n

��������������������������������������������������������

3. Inner-shell losses

Approximations: - free atoms - describe initial and final state as a Slater-determinant of single-electron

atomic wave functions (not valid for open shells 3d, 4d: transition metals; 4f, 5f: lanthanides, actinides)

single-electron matrix element.

SIGMAK (Egerton, 1979), SIGMAL (Egerton, 1981)Hartree-Slater model (Rez et al.)

* 300 exp nK

n r iKr r d r ������������������������������������������������������������������ ����

0 0Kn iK r n ��������������

����������������������������

2 2

02 2

40 n

nH

dz n E E E

d dE K a

2 2 2 2EK k

2Ei

E

E

geometry:

; scattering angle

exp 1iK r iKr ��������������������������������������������������������

2

2 2 2 2

1 8

E H

d mE df

d dE k a dE

For small scattering angles small scattering vectors dipole approximation

Example: - Ionisation of hydrogen

- experiment for carbon

2

02

20 n

n

df mEz n E E E

dE

2

02class

e

mc

E

photo absorption

oscillator strength

generalized oscillator strength (GOS):

In solids the final states are not completely free.near-edge structure (ELNES) analogous to XANESextended fine structure (EXELFS) analogous to EXAFS

2d

d dE

2

, 2,

df K E mES K

dE K

����������������������������

abs class

df

dE

generalized oscillator strength for hydrogen (Inokuti, Rev. Mod. Phys. 43, (1971) 297)

double differential cross-section for carbon (Reimer & Rennekamp, Ultramicr. 28, (1989) 256)

C. Hébert, Wednesday

Spectrum of BN (Ahn et al., EELS Atlas 1982)

4. Low loss spectra

For relatively low frequencies ( low energy losses) the free electron gas

can partly follow the field of the incident electron shielding

Electron causes -field

Acting field:

Absorption: Imaginary part

Relation to dynamic structure factor ?

D��������������

1E D

����������������������������

D ��������������

div

2

02

1, Im

,

KS K

e K

�������������� ��������������

For

In addition: surface plasmon losses

O. Stephan, Thursday

is response function

Dissipation-fluctuation theorem:

0

peaks for : volume plasmons

Why don‘t we use that for higher energy losses ?

0

1 2i

Formally: describes fluctuations in the object (density-density correlation);

1

,K ��������������

,S K ��������������

,S K ��������������

2

1Im

1

dielectric function of Ag (Ehrenreich & Philipp, Phys. Rev. 128 (1962) 1622)

dielectric functions of Cu (Ehrenreich & Philipp, Phys. Rev. 128 (1962) 1622)

5. Relativistic effects

Non-relativistic: Incident electron causes Coulomb field

field is instantaneously everywhere in space

Relativistic: Incident (moving) electron causes an additional magnetic field

fields move in space with the speed of light c ( retardation)

Matrix elements are sums of an electric and a magnetic term

In Coulomb gauge: electric term corresponds to the non-relativistic term,

but with relativistic kinematics

20

202

i

E

i i

E E m c

E E m c

Double-differential cross-section in dipole-approximation

2 2 42

2 2 2 2 2

1~ 1

1E

E E

d

d dE

(Kurata at al., Proc. EUREM-11 (1996) I-206)

6) Summary and conclusions

- quantitative interpretation of EEL-spectra requires knowledge of cross-sections

- cross-section related to dynamic form factor

- for inner-shell ionization these can be calculated using a one–electon model

- large errors may occur when 3d, 4d, 4f, 5f shells are involved

- for small scattering angles (dipole approximation) one obtains a Lorentzian angular shape

- in dipole approximation the cross-section is closely related to the photoabsorption cross-section

- near-edge and extended fine structures can be interpreted as in the X-ray case

- the low-loss spectrum permits to determine the dielectric function

- WARNING: relativistic effects are not included in the commonly used equations