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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)
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
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
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
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