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An atom in an excited atomic state may decay to a lower energy level through spontaneous photon emission. The probability, A, of this spontaneous transition has units of s−1. This quantity is also usually designated as “transition rate” or “transition probability per unit of time”. To avoid terminology confusion, in the present work, this quantity is always referred to as “transition probability”. Since the energy width Γ and the mean life τ of the excited atomic state are related through Heisenberg uncertainty principle Γ × τ = ~ , the transition probability of the state is therefore given as A = 1/τ = Γ /~ [40]. The excited

state can also transit to lower energy states through a radiationless transition. In this case, the energy of the transition is transferred to another atomic electron of an higher energy shell or sub-shell, resulting in its emission. If the electron transition occurs between shells, the radiationless transition is commonly denominated as Auger transition, and the emitted electron is the Auger electron. Other cases of radiationless transitions include the Coster-Kronig transition, in which the electron transition is intra-shell, and super Coster-Kronig transition, where, in addition to the previous case, the Auger electron is emitted from the same shell.

The most probable radiative transitions in the relaxation process are the so-called “allowed transitions”, while the so-called “forbidden transitions” are much more unlikely to occur. These transitions differ from each other through the selection rules they follow, i.e., the sets of changes in the quantum numbers. In Fig. 2.2 the characteristic radiative transitions are presented, using the most common notations, the “Siegbahn notation”, and the IUPAC (International Union of Pure and Applied Chemistry) notation.

After the atomic transition occurs, the atomic system may still be excited. In fact, the atomic system usually undergoes a sequence of transitions until it reaches the de-excited state, emitting photons and Auger electrons in the process. Such sequence is typically denominated as relaxation cascade or de-excitation cascade. The cascade is depicted in Fig. 2.3.

For a radiative transition where the electron transits from sub-shell j to sub-shell i, with binding energies Ei and Ej, respectively, the energy of the emitted photon is given

as:

Eγ = Ej−Ei. (2.2)

Since binding energies are specific for each sub-shell and element, the emitted photon energy is also specific for each element and each transition. For this reason, photons emit- ted in the atomic relaxation are usually termed as characteristic X-rays. Some different radiative transitions lead to the emission of characteristic X-rays with similar energies. As such, their corresponding measured spectrum lines are often indistinguishable, for which it is common to use a notation where multiple transitions are encompassed, for example, K-M4,5encompasses the K-M4and K-M5transitions.

In an experimental context, the particles that are used to excite the atoms are usually referred to as primary particles, and therefore, the characteristic photons emitted from the consequent atomic relaxation are frequently termed as secondary photons. In the specific cases where the primary particles are X-ray photons, such as in the X-ray fluorescence technique, the secondary photons are also usually referred to fluorescence photons.

For a radiationless transition where the electron transits from thej sub-shell to the i

sub-shell, the Auger electron, emitted from theh sub-shell, has energy given by:

E_e= E_j−_{(Eh+ Ei), (2.3)}

Figure 2.2: Radiative transition representation and nomenclature. Figure adapted from [41].

For an atomic state with an initial vacancy in thei sub-shell, the total radiative width

Γ_i(TR) is the energy width associated with all radiative transitions that can fill the i sub- shell, and the total radiationless width Γ_i(TA)is the energy width associated with all radia- tionless transitions (including Auger, Coster-Kronig, and super Coster-Kronig transitions) that can fill the i sub-shell. The total state width is given as Γi= Γ_i(TR)+ Γ_i(TA). Likewise,

the total transition probability of the state is given as A(T)_i = A(TR)_i + A(TA)_i , where A(TR)_i is the total radiative transition probability, and A(TA)_i is the total radiationless transition probability. Both A(TR)_i and A(TA)_i are given as:

A(TR)_i =X j(>i) A(R)_ij , (2.4a) A(TA)_i =X j(>i) X k(≥j) A(A)_ijk, (2.4b)

Figure 2.3: Schematic representation of atomic relaxation cascade. In this example a vacancy in the K-shell is filled through the K-L2 transition, leaving a vacancy in the

L2 sub-shell. The vacancy in the L2 sub-shell is subsequently filled through the L-N2

transition, leaving a vacancy in the N₂ sub-shell. Since N₂is the valence sub-shell, the atomic system is in a de-excited state and the relaxation cascade finishes. Figure adapted from [41].

where A(R)_ij is the transition probability of the radiative transition where the vacancy in sub-shell i is filled with an electron from sub-shell j, and A(A)_ijk is the transition proba- bility of the radiationless transition where the vacancy in the i sub-shell is filled with an electron from the j sub-shell and an Auger electron is emitted from the k sub-shell. The fluorescence yield, ω_i, is the probability that an atomic state with a vacancy in the i sub-shell de-excites through a radiative transition. It can be given by:

ω_i=Γ (TR) i Γ_i(T) = Γ (TR) i Γ_i(TR)+ Γ_i(TA) . (2.5) or, ωi= A(TR)_i A(T)_i = A (TR) i A(TR)_i + A(TA)_i = P j(>i)A (R) ij P j(>i)A (R) ij + P j(>i) P k(≥j)A (A) ijk . (2.6)

On the contrary, the probability that the atomic state deexcites through radiationless transition is the radiationless yield (or Auger yield), ai, which can be obtained from

analogous equations to Eqs. 2.5 or 2.6. The sum of fluorescence yield and Auger yield is unity: ωi+ ai= 1. As presented in Fig. 2.4, the fluorescence yield increases with atomic

number, while the Auger yield decreases. The fluorescence yield is higher for sub-shells

Figure 2.4: K-shell and Auger yields as function of atomic number.

with lower binding energies, i.e., for any element, the following condition is verified:

ωK< ωL< ωM< ... (2.7)

The partial fluorescence yield of a specific radiative transition (or set of transitions) can be calculated in a similar way as the fluorescence yield of the sub-shell. The partial fluorescence yield ω_ij of the transition ij, in which the vacancy in a i sub-shell is filled with an electron from the j sub-shell, with respective transition probability A(R)_ij , is written as: ωij= A(R)_ij A(T)_i = A (R) ij A(TR)_i + A(TA)_i = A(R)_ij P j(>i)A (R) ij + P j(>i) P k(≥j)A (A) ijk , (2.8)

representing the probability that the atom will de-excite through the specific transition

ij instead of all other possible radiative and radiationless transitions. As expected, for

any excited state, the sum of all partial radiative transition fluorescence yields equals the fluorescence yield of the sub-shell:

X

j

ωij= ωi. (2.9)

Frequently, in literature [35–37], the radiationless transition rates A(A)_ijk are not included in Eq. 2.8. The values calculated this way have varying designations among the literature,

such as “transition probability”, “transition intensity” or “partial fluorescence yield”. In the present work, it is used the designation “partial fluorescence yield (normalized with- out accounting radiationless transitions)” and the symbol “ω(NA)_ij ” (where NA stands for “No Auger”) to represent it. The radiative transition partial fluorescence yield (normal- ized without accounting radiationless transitions) “ω_ij(NA)”, for the radiative transition ij, is given as: ω_ij(NA)= A (R) ij A(TR)_i = A (R) ij P j(>i)A (R) ij