X-rays
The electric field
E(r,t) is given as
a cosine function.
E (r, t )  cos(kr  t )
X-rays
In formal derivations the
vector potential A is used.
The electric field E(r,t) is
directly related to the vector
potential A(r,t).
E (r , t ) 

 t
A(r , t )
Interaction of x-rays with matter 1
The photon moves towards the atom
Interaction of x-rays with matter 1
The photon meets an electron and is annihilated
Interaction of x-rays with matter 1
The electron gains the energy of the photon
and is turned into a blue electron.
Interaction of x-rays with matter 1
The blue electron (feeling lonely) leaves the atom
and scatters of neighbors (cf. EXAFS)
or escapes from the sample (cf. XPS)
Interaction of x-rays with matter 1
The probability of photon annihilation determines
the intensity of the transmitted photon beam
I
I0
Ek
Interaction of x-rays with matter 2
The photon moves towards the atom
Interaction of x-rays with matter 2
The photon meets an electron and is scattered
Interaction of x-rays with matter 2
The photon leaves the atom under a different angle.
(Interference between scattering events yields XRD)
Interaction of x-rays with matter
Energy  Spectroscopy
Direction  Structure
Polarization  Magnetism
I’(’,k’,q’)
I(,k,q)
I”(Ek,k”,)
Interaction of x-rays with matter
HINT(1) describes the interaction of the vector field A on
the momentum operator p of an electron, or in other
words the electric field E acting on the electron
moments.
The momentum operator p is given as the electron
charge q times the displacement operator r.
T1  H INT (1) 
e
mc
p A
Interaction of x-rays with matter 1
The photon meets the electron and is annihilated
p=q•r
A
Interaction of x-rays with matter
HINT(1) describes the interaction of the vector field A on
the momentum operator p of an electron, or in other
words the electric field E acting on the electron
moments.
The momentum operator p is given as the electron
charge q times the displacement operator r.
T1  H INT (1) 
e
mc
p A
Interaction of x-rays with matter
HINT(2) describes the second order interaction of the
vector field A.
This gives rise to the elastic scattering of the x-rays by
the electrons.
This is the basis for x-ray diffraction (XRD) and small
angle x-ray scattering (SAXS)
H INT ( 2) 
2
e
2
2 mc
A
2
Interaction of x-rays with matter
• Elastic
scattering
(Thompson)
• Inelastic
scattering
• (Compton)
Intensity (log)
• XAFS studies
photoelectric
absorption
100
Mn
Photoelectric
10
Thompson
Compton
1
100
1k
10k
Energy (eV)
100k
X-ray absorption and X-ray photoemission
Excitation of core electrons to empty states.
Spectrum given by the Fermi Golden Rule
I XAS ~  f  f T1  i
2
 E f  Ei 
X-ray absorption and X-ray photoemission
I(FIXED)
X-ray absorption and X-ray photoemission
X-ray emission: core hole decay
Basis for X-ray Fluorescence (XRF) and
Energy Dispersive X-ray analysis (EDX)
Interaction of x-rays with matter
H INT (1) 
e
mc
H INT ( 2) 
p A
2
e
2 mc 2
A
2
Photoelectric effect:
(annihilation of photon)
XAS, XPS
XES, XRF, EDX
X-ray scattering:
(photon-in photon-out)
XRD, SAXS
Interaction of x-rays with matter
T2  H INT (2)  H
1
INT (1) Ei  H i / 2
H INT (1)
X-ray scattering:
- with Hint(2)
- with Hint(1) via a (virtual) intermediate state
= Resonant X-ray scattering
Interaction of x-rays with matter 3
The photon moves towards the atom
Interaction of x-rays with matter 3
The photon meets an electron and is annihilated
Interaction of x-rays with matter 3
The electron gains the energy of the photon
and is turned into a virtual blue electron.
Interaction of x-rays with matter 3
The virtual blue electron loses a photon with
exactly the same energy as gained
Interaction of x-rays with matter 3
The photon leaves the atom
Resonant X-ray scattering
Combination of XAS and XES [only Hint(1)]
- RXES
- Resonant Inelastic X-ray Scattering (RIXS)
(also called Resonant X-ray Raman Spectroscopy)
Combination of Hint(1) and Hint(2)
- Resonant XRD (also called: anomalous)
- Multi-wavelength anomalous Diffraction (MAD)
- Resonant SAXS (ASAXS)
- TEDDI