• Relaxation energy refers to the energy released when the electrons in an atom or molecule adjust to a change in the electronic environment, such as the creation of a core hole.(4) It is a final state effect that occurs after a photoemission process, where a core electron is ejected from an atom or molecule, leaving a positive charge. This positive charge causes the remaining electrons to rearrange, lowering the total energy of the system. The relaxation energy is the difference in energy between the initial state with a core hole and the final, relaxed state.(3)
   
  Relaxation energy is a critical concept in understanding core-level spectroscopy. It is essential for the interpretation of both X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES) data. Shifts in core-level binding energies and Auger parameters can be related to changes in the relaxation energy, which can provide information about the chemical environment, the charge state of the atom, and the electronic structure of the material.
   

  There are several types of relaxation energies: #

   
  • Atomic or Intra-atomic Relaxation Energy: This type of relaxation is due to the rearrangement of the core and valence electronic levels of the photoemitter atom.
  • • It can be further divided into contributions from core electrons and valence electrons.
    • The core electron contribution is usually constant, while the valence electron contribution depends on the chemical environment.
    • Intra-atomic relaxation energy is the energy gained due to the electrons in the remaining atomic shells relaxing as a photoelectron leaves.
    •  
  • Extra-atomic Relaxation Energy: This energy arises from the polarization of neighbouring atoms or molecules in the system in response to the core hole.
  • • It is caused by the sudden production of a positive charge within the crystal lattice where the photoexcited atom is located, which provokes a rearrangement of the electron density around it.
    • This energy depends on the polarizability of the surrounding atoms and the distance between them and the core-ionized atom.
    • Extra-atomic relaxation is also called polarization energy.
    •  
  • Total Relaxation Energy: The sum of all contributions from the different relaxation processes. It is generally expressed as:
  • • R(C) = R^a(core electrons) + R^a(valence electrons) + R^ea(extra-atomic contribution).

   

  The magnitude of the relaxation energy can be significant, typically in the range of a few eV.(1,5) The relaxation energy contributes to a shift in the measured binding energy of a core electron.(2,3) This is because the relaxation process stabilizes the final state of the photoemission process, decreasing the observed binding energy.(2,6) The measured binding energy (EB) can be written as: E_B = – εi – Ri, where ε_i is the orbital eigenvalue, and R_i is the relaxation energy.(6)
   
   
  It is important to note that the relaxation process occurs on a very short time scale (approximately 10^-17 s), much faster than the relaxation of atomic positions (approximately 10^-13 s).(3) Therefore, atomic relaxation is not considered in the analysis of relaxation energy.
   
  Different models and approximations have been developed to quantify and estimate relaxation energies. These approaches include:
   
  1. The use of Hartree-Fock orbital energies and relaxation energy corrections to calculate extra-atomic relaxation.(1)
  2. The polarization potential model.(3)
  3. The equivalent core approximation (Z+1 approximation), where the core-ionized atom is approximated by the next element in the periodic table.(7)

Thinking of a simple photoionization process as:

hv + Z ⇒ Z⁺ + e^–

The energy of the photon is equal to the ionization energy of the atom plus the kinetic energy of the photon, we know this from our photoelectric effect fundamentals.

The energy of Z⁺ above the ground state is the ‘binding energy’

E_B(Z⁺) = hv – E(e^–)

According to Koopmans theorum, the binding energy is equal to the energy of the emitting electron shell in the ground state, as determined by Hartree-Fock calculation.

One effect that affects this relationship greatly, is the atomic relaxation – or rearrangement – energy.

Koopmans theorum falls slightly in this aspect, since it is a frozen orbital approximation, and therefore ignores the relaxation effects that follow photoemission.

This relaxation energy is approximately 75-85% of the square root of the final energy of the ion (in eV, calculated by Gelius1)

A close approximation of the EB of an isolated atom (referenced to the vacuum level) is:

• E_B^V(PE_a) = ε – R

ε = ground state energy

a = isolated atom

PE = photoelectron

R = relaxation energy (atomic or intra-atomic)

The same atom in a molecule (gas) can experience 2 additional effects pertaining to binding energy

1.Bonding to neighbouring atoms will change the atomic energy levels in the ground state by ∆ε

2.Electrons from neighbouring atoms can relax during ionisation to reduce total energy

• E_B^V(PE_m) = ε_a + ∆ε_m – R – R_m^ea

ε = ground state energy

a = isolated atom

m = isolated molecule

PE = photoelectron

R = relaxation energy (atomic or intra-atomic – assumed to not change significantly)

R^ea = Relaxation energy / polarisation energy (extra-atomic)

R^ea is the relaxation energy (or polarising energy) involving electrons from neighbouring atoms

So we can think of a chemical shift between atom and molecule as:

• E_B^V(PE_a→m) = ∆ε_m –  R_m^ea

Finally we consider the chemical shift in moving an isolated atom into the conductive state

Again, we do not expect a significant shift in R, and ∆εc is expected to be relatively small compared to Rcea

• E_B^V(PE_a→c) = ∆ε_c –  R_c^ea

ε = ground state energy

a = isolated atom

c = conductive atom in the solid state

PE = photoelectron

R = relaxation energy (atomic or intra-atomic – assumed to not change significantly)

R^ea = Relaxation energy / polarisation energy (extra-atomic)

In a conductor, we expect Rcea to be related to the ionic charge (e), minimum electron screening distance (r) and dielectric constant (k) by:

R_c^ea = e2/2r (1 – 1/k)

From here we can see that relaxation energies may be influenced by ionic radius, with smaller ions imparting larger effects

Auger KE is the difference between the energies of the initial and final state (note we are discussing here core-type Augers, where the double hole final state vacancies are in core levels only)

For a KLL auger

E^V(Auger_a) = E_B^V(K⁺) – 2E_B^V(L⁺) + R^S – F(x)

E_B^V(K⁺) = binding energy of K core electron (photoelectron)

E_B^V(L⁺) = binding energy of L core electron

R^S = intra-atomic ‘static’ relaxation energy for the L shell

F(x) = electron-electron interaction term required due to the double vacancy in the final state

If we expand this to orbital energies and interatomic relaxation energies we have:

E^V(Auger_a) = ε(K) – R(K⁺) – 2 ε(L) – 2R(L⁺) + R^S – F(x)

Meaning in terms of shifts when an atom is in a molecule (gas) we have:

E^V(Auger_a→m) = ∆E_m^V(K) – 2E_m(L) + R_m^ea(L⁺L⁺) – R_m^ea(K⁺)

Shirley, D. A. “Theory of KLL Auger energies including static relaxation.” Physical Review A7.5 (1973): 1520.

E^V(Auger_a→m) = kinetic energy shift of atomic state into molecular state

ε(K) = ground state energy of K shell

ε(L) = ground state energy of L shell

R(K⁺) = relaxation energy for the ionised K shell

R(L⁺) = relaxation energy for the ionised L shell

R^S = intra-atomic ‘static’ relaxation energy for the L shell

F(x) = electron-electron interaction term required due to the double vacancy in the final state

Since the actual chemical shifts in different shells are experimentally quite close, we can simplify this equation to the below approximation:

E^V(Auger_a→m) = -∆ε_m(L) + R_m^ea(L⁺L⁺) – R_m^ea(K⁺)

Conducting solids

For conductive solids, again – our chemical shifts in the ground state are minimal, so we expect the chemical shift to be dominated by the relaxation energy terms

R_c^ea =e²/2r (1 – 1/k)

E^V(Auger_a→c) = -∆ε_c(L) + ³/₄R_c^ea(L⁺L⁺)

E^V(Auger_a→c) = -∆ε_c(L) + 3R_c^ea(K⁺)

E^V(Auger_a→c) = kinetic energy shift of atomic state into conducting solid state

R^S(LL) = intra-atomic ‘static’ relaxation energy for the L shell

R^T(LL) = total two-hole relaxation energy

R^D(L) = one-hole dynamic relaxation energy

R^S(LL) = R^T(LL) – 2 R^D(L)

This parameter describes the additional relaxational shift of the total energy for a two-hole system relative to two isolated holes

This is dominated by Coulomb interactions and we have:

R^T(LL) = 4 R^D(L)

The Auger parameter is defined as the difference between a photoelectron core line peak, and an auger peak

As such, it is free from any complications due to sample charging, or work functions – since this is a difference, they all cancel out

• α = E^V_{(C’C’’C’’’)}  – E^V_(C)

Modified Auger parameter

This parameter was found to be very useful, but was inconsistent when comparing data between different photon energies.

In order to improve this, a shift towards using a new term – the modified auger parameter was adopted.

This uses PE core peak binding energy, instead of kinetic energy.

• α’ = E^V_{B(C’C’’C’’’)} + E^V_B(C) = E^V_{(C’C’’C’’’)} – E^V_(C) + hv

When referring to shifts (∆α or ∆α’), these are going to be identical whether we refer to the Auger, or the modified Auger, parameter.

For the purposes of this explanation, we will stick with the original term, however the shifts themselves are identical for both ∆α and ∆α’.

The chemical shift, from atom to molecule is:

∆a_a→m = R_m^ea (L⁺L⁺) – R_m^ea (K⁺)

And from atom to conducting solid:

∆a_a→c = R_c^ea (L⁺L⁺) – 2R_c^ea (K⁺)

Assuming R_m^ea (L⁺L⁺) – 4R^ea (K⁺)

∆a_a→c = ½R_c^ea (L⁺L⁺) = 2R_c^ea (K⁺)

From:

• E_B(Z⁺) = hv – E(e^–)

• E_B^V(PE_a) = ε – R

• ∆ E^V(Auger_a→m) = ∆ε_m(K) – 2 ε_m(L)+ R_m^ea (L⁺L⁺) – R_m^ea (K⁺)

• α_a = 2ε(K) – 2ε(L) – 2R(K⁺) + 2R(L⁺) – hν – F(x) + R_S

Now – going from a conducting solid, to an insulating materials:

• ∆α_c→i = ½[R_i^ea(L+L+) – R_c^ea(L⁺L⁺)] = 2[R_i^ea(K⁺) – R_c^ea(K⁺) ]

If we then look at these -∆α values for a range of materials, we see that this auger shift scales with polarizability of the atom.

The more polarizable a species (i.e. the more covalency involved in it’s bonding) – the closer the -∆α value will be to 0 (i.e. the conductive solid state)

The change in modified auger parameter can be defined as

• ∆α’ = α’ (x) – α’ (s)

Where α’ (s) = the modified auger parameter of the element in it’s solid form.

And α’ (x) is the modified auger parameter of our sample in question.

Final State Effects: The Auger parameter is primarily sensitive to final-state effects, which arise from the relaxation of the electron cloud after core ionization.(3) These effects are related to the ability of the environment to screen the core hole.(8) The change in the Auger parameter (Δα) is directly related to changes in the extra-atomic relaxation energy (ΔR), with Δα ≈ 2ΔR.(1)

Specifically, the Auger parameter shift with respect to the free atom is related only to final state effects and the ground state valence charge (as discussed in the above maths section).

Initial State Effects: Initial state effects are related to the ground state charge distribution of an atom before core ionization.

In a Wagner plot, compounds that exhibit similar initial state effects will fall on a line with a slope of approximately 3. These shifts reflect the nature of the chemical bonds and the local potential at the core-ionized atom.

Initial state effects can be estimated from the shifts in the core-level eigenvalues.

Deconvolution Process #

Experimental Measurement: Obtain the kinetic energy of a core-core-core Auger transition and the binding energy of a core electron for the same element using XPS. These values should be measured under the same experimental conditions.

Calculate Auger Parameter: Calculate the Auger parameter using α = E_k + E_b or α’ = E_k + E_b if the values are referenced to the Fermi level.(8)

Determine Shifts: Compare the Auger parameter values (and binding energies) for different chemical states.

Wagner Plot Analysis: Plot the Auger kinetic energy against the core-level binding energy for each chemical state.

Compounds with similar final-state relaxation energies will fall on a line with a slope of -1 (or +1, due to the reversed x-axis in the Wagner plot).

The intercept of this line with the ordinate is the Auger parameter, which is related to the final state relaxation energy. Shifts in the Auger parameter between different chemical states are related to changes in the final state relaxation energy.

Extract Final State Contributions: Changes in the Auger parameter (Δα) are directly related to changes in the final-state relaxation energy (ΔR). Δα ≈ 2ΔR.

The Auger parameter is used to directly measure the extra-atomic relaxation energy.

Estimate Initial State Contributions: Once the final state contribution (relaxation energy) is known, the initial state contribution (ΔV) to the core-level shift can be estimated using the relationship ΔE_b = ΔV – ΔR or ΔE_k = -ΔV + 3ΔR, where ΔEb is the binding energy shift, and ΔEk is the kinetic energy shift.

The initial state shift can be estimated by comparing measured core-level binding-energy shifts with calculated shifts based on core-level eigenvalue.

Theoretical Calculations: The initial and final state effects can be further studied with theoretical calculations.

Ab initio calculations, such as Hartree-Fock or Density Functional Theory (DFT), can provide detailed insights into the initial and final state contributions to the binding energy. These methods can also be used to predict the surface segregation of an element in an alloy.

Additional Considerations: #

Core-level Selection: The specific core levels and Auger transitions chosen for analysis can affect the accuracy of the separation.

For example, in systems with high lattice strain, when Auger transitions involve states with high-lying d-holes, the Auger parameter analysis may not accurately represent the initial or final state contributions due to symmetry effects.(9)

Local vs Non-Local Screening: The Auger parameter can help differentiate between local and non-local screening mechanisms.(10)

In some compounds, the relaxation process is non-local, involving electron transfer from neighbouring ligands to the core-ionized atom.

Multiplet Splitting: The presence of unpaired electron spins can result in observable effects from exchange splitting. These are considered to be final-state contributions.(4)

Limitations #

The accuracy of Auger parameter analysis depends on some approximations. For example, the assumption that the relaxation energy for a doubly core-ionized state created by the Auger process equals 2R may not always be valid.

For insulators and semiconductors, charging problems can affect the semi-quantitative analysis when using Wagner plots (since charge referencing errors can affect the position on the X/Y).(3)

1. Wagner, C. D., and A. Joshi. “The auger parameter, its utility and advantages: a review.” Journal of electron spectroscopy and related phenomena 47 (1988): 283-313. Read it online here.
2. Wagner, C. D. “Chemical shifts of Auger lines, and the Auger parameter.” Faraday Discussions of the Chemical Society 60 (1975): 291-300. Read it online here.
3. Moretti, Giuliano. “Auger parameter and Wagner plot in the characterization of chemical states by X-ray photoelectron spectroscopy: a review.” Journal of electron spectroscopy and related phenomena 95.2-3 (1998): 95-144. Read it online here.
4. Egelhoff Jr, W. F. “Core-level binding-energy shifts at surfaces and in solids.” Surface Science Reports 6.6-8 (1987): 253-415. Read it online here.
5. Broughton, J. Q., and P. S. Bagus. “ΔSCF calculations of free atom—ion shifts.” Journal of Electron Spectroscopy and Related Phenomena 20.1 (1980): 127-148. Read it online here.
6. Hohlneicher, George, Harald Pulm, and Hans-Joachim Freund. “On the separation of initial and final state effects in photoelectron spectroscopy using an extension of the auger-parameter concept.” Journal of electron spectroscopy and related phenomena 37.2 (1985): 209-224. Read it online here.
7. Mejias, J. A., et al. “Interpretation of the binding energy and auger parameter shifts found by XPS for TiO2 supported on different surfaces.” The Journal of Physical Chemistry 100.40 (1996): 16255-16262. Read it online here.
8. Moretti, Giuliano. “The Wagner plot and the Auger parameter as tools to separate initial-and final-state contributions in X-ray photoemission spectroscopy.” Surface science 618 (2013): 3-11. Read it online here.
9. Bagus, Paul S., Andrzej Wieckowski, and Hajo Freund. “Initial and final state contributions to binding-energy shifts due to lattice strain: Validation of Auger parameter analyses.” Chemical physics letters 420.1-3 (2006): 42-46. Read it online here.
10. Moretti, Giuliano, and Horst P. Beck. “Relationship between the Auger parameter and the ground state valence charge of the core‐ionized atom: The case of Cu (I) and Cu (II) compounds.” Surface and Interface Analysis 51.13 (2019): 1359-1370. Read it online here.