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20 2. Theoretical background
the single-electron picture is expressed as
H = H 0 + H S = 1 ( ) 2
2m i ∇−e c A(r) + V (r), (2.12)
here, H 0 is the Hamiltonian of the unperturbed system, H S is the perturbation operator,
p = /i ∇ denotes the momentum operator, and V (r) is the potential within the solid from the
ionic cores. The single Hamiltonians are written as
H 0 = − 2
2m ∇2 + V (r), H S = 1 (
− e
)
e2
(2A∇ +(∇A)) +
2m ic c 2 |A|2 . (2.13)
Here, we approximate |A| 2 ≈ 0 for small electromagnetic field strengths, and the gradient
∇A ≈ 0, because the field changes within the crystal only very slowly (X-ray penetration of
the crystal).
The ansatz for A is a plane wave with wave vector k γ , the amplitude A 0 , and r is the unit
vector in direction of the electric field:
A(r,t)=A 0 e ik γ r
⇒ H S = − e
mc A 0e ik γ r p. (2.14)
The wave vector k γ = 2π λ
is small with respect to the size of a typical Brillouin zone up to the
XPS regime, therefore e ik γ r ≈ 1, which simplifies the perturbation Hamiltonian in the dipole
approximation as
H S = − e
mc A 0 ·p. (2.15)
We remark that for high-energetic photons (HAXPES), the dipole approximation may be insufficient
and higher order transitions (electric quadrupole E2, or magnetic dipole M1) are
emerging. The Hamiltonian (2.15) depends only on the momentum transfer p between the
plane wave (light) and the electron, which is to be determined now. It is used in Fermi’s
golden rule, which expresses the probability per time that the system is excited by a photon
of the energy hν,
w i→f = 2π 〈 ∣ ∣ f ∣∣H ∣ S 〉 ∣ 2
∣i
∣∣∣
·δ(E f − E i − hν) (2.16)
Here, |i〉 denotes the electron’s initial state with the energy eigenvalue E i , and |f 〉 is the final
state in the single particle model with energy E f , as depicted in Fig. 2.11a. The δ-function
ensures the energy conservation. The transition matrix element in Fermi’s golden rule, M if ,
depends on the symmetries of the final and initial state as
M if = 〈 f ∣ ∣ ∣H
S ∣ ∣ ∣i
〉
∝〈f |p|i〉. (2.17)
Besides the energy conservation, also the conservation of the momentum (∝ k, the wave
The E2 or M1 distributions occur for very short wavelength photons (hν ≫ 1 keV) and for the ejection of electrons
in very large orbits. However, these distortions only alter the angular dependence of photoemission. Since
we conduct angle-integrated HAXPES only at fixed emission angles, the inclusion of higher order transitions is
negligible in this thesis.