64
P. Krüger
of freedom in the following, focusing on the electronic state of the system. When
an X-ray photon impinges on an atom it can be either absorbed or scattered, energy,
momentum and spin being, of course, conserved in the whole process. In absorption,
the photon vanishes: all its energy is transferred to an electron which is excited to
an empty state above the vacuum level. In scattering, the energy of the photon can
remain the same [elastic (Thomson) scattering]; it can also be partly transferred to the
atom (inelastic scattering) as in Compton scattering, which leads to the ejection of an
electron, or in a Raman-like scattering, in which the energy lost by the photon brings
the atoms in an excited state, without any ionization. Neglecting relativistic effects
and treating the X-ray field classically, the light–electron interaction is obtained
by replacing, in the electronic Hamiltonian, the electron momentum operator p by
p − eA/c, where A(r, t) is the vector potential of the light [1]. In Coulomb gauge
(∇ · A = 0), the interaction Hamiltonian then becomes
H int = −
e
mc
A · p +
e
2
2mc 2 A
2
.
(3.1)
The first-order term in A describes light absorption and stimulated emission while
the second-order term is responsible for (non-resonant) light scattering. Here we
focus on the absorption process and neglect the generally much weaker A
2 term.
First-order perturbation theory (Fermi golden rule) leads to the following expression
for the absorption intensity from an initial state |i of energy E i :
I (ω) ∝
f
|| f |A · p|i|
2
δ(ω − E f + E i ) ,
(3.2)
where the sum runs over all possible final states | f with energy E f . It is common to
make the dipole approximation, i.e. to neglect the spatial variation of the X-ray field
A(r). We may also replace the transition operator A · p, by e · r, where e = A/|A|
is the light polarization vector and the change from p to r is possible by exploiting
commutation relations between r, p and H , and the fact that |i and | f are energy
eigenstates [2]. Equation (3.2) is often interpreted in a single-particle picture, in
which case |i is an atomic core state and | f are unoccupied states above the Fermi
level. However, electrons interact with each other through the Coulomb interaction,
such that the excitation of one electron affects the motion of the others. Therefore,
the correct use of (3.2) is in a many-particle sense, where |i = | g is the manyelectron ground state, and | f = | f are many-electron excited states with a core
hole. Putting the constants we have
I (ω) = 4π
2
αω
f
|| f |e ·
j
r j | g |
2
δ(ω − E f + E g ) ,
(3.3)
where α = e
2
/c is the fine structure constant and j counts the electrons. Having
established the expressions of the absorption intensity, the remaining task is to calculate the eigenstates of the (unperturbed) electronic system, | g and | f , and their
P. Krüger
of freedom in the following, focusing on the electronic state of the system. When
an X-ray photon impinges on an atom it can be either absorbed or scattered, energy,
momentum and spin being, of course, conserved in the whole process. In absorption,
the photon vanishes: all its energy is transferred to an electron which is excited to
an empty state above the vacuum level. In scattering, the energy of the photon can
remain the same [elastic (Thomson) scattering]; it can also be partly transferred to the
atom (inelastic scattering) as in Compton scattering, which leads to the ejection of an
electron, or in a Raman-like scattering, in which the energy lost by the photon brings
the atoms in an excited state, without any ionization. Neglecting relativistic effects
and treating the X-ray field classically, the light–electron interaction is obtained
by replacing, in the electronic Hamiltonian, the electron momentum operator p by
p − eA/c, where A(r, t) is the vector potential of the light [1]. In Coulomb gauge
(∇ · A = 0), the interaction Hamiltonian then becomes
H int = −
e
mc
A · p +
e
2
2mc 2 A
2
.
(3.1)
The first-order term in A describes light absorption and stimulated emission while
the second-order term is responsible for (non-resonant) light scattering. Here we
focus on the absorption process and neglect the generally much weaker A
2 term.
First-order perturbation theory (Fermi golden rule) leads to the following expression
for the absorption intensity from an initial state |i of energy E i :
I (ω) ∝
f
|| f |A · p|i|
2
δ(ω − E f + E i ) ,
(3.2)
where the sum runs over all possible final states | f with energy E f . It is common to
make the dipole approximation, i.e. to neglect the spatial variation of the X-ray field
A(r). We may also replace the transition operator A · p, by e · r, where e = A/|A|
is the light polarization vector and the change from p to r is possible by exploiting
commutation relations between r, p and H , and the fact that |i and | f are energy
eigenstates [2]. Equation (3.2) is often interpreted in a single-particle picture, in
which case |i is an atomic core state and | f are unoccupied states above the Fermi
level. However, electrons interact with each other through the Coulomb interaction,
such that the excitation of one electron affects the motion of the others. Therefore,
the correct use of (3.2) is in a many-particle sense, where |i = | g is the manyelectron ground state, and | f = | f are many-electron excited states with a core
hole. Putting the constants we have
I (ω) = 4π
2
αω
f
|| f |e ·
j
r j | g |
2
δ(ω − E f + E g ) ,
(3.3)
where α = e
2
/c is the fine structure constant and j counts the electrons. Having
established the expressions of the absorption intensity, the remaining task is to calculate the eigenstates of the (unperturbed) electronic system, | g and | f , and their