# Visualizing elusive quantum-mechanical concepts through multi-ionization experiments

### G. Stefani

Quantum mechanics has revolutionized the interpretation of the world: from neutron stars, to simple metals, to DNA strand breaks. On its reliability are based many of the technologies that we have nowadays at our disposal in fields as different as electronics, magnetism, solar cells and cryptography, to name a few. In spite of these unquestionable successes, after over a century there are fundamental problems still waiting for a solution. Among them, one nut hard to crack is the correlated motion of electrons in an external field. In the Hamiltonian of a system as simple as two electrons moving in the central Coulomb potential, i.e. the He atom, there is a term that still is not possible to treat exactly: the electron correlation due to electron-electron Coulomb-exchange interactions. In many-electron materials, that are the specific object of this paper, it is this term, hard to deal with, that determines chemical and physical properties. Among them, some are technologically relevant as, for example, high-temperature superconductivity, colossal magnetoresistence, self-assembly and magnetism.

A variety of models have been, and still are being, developed in order to overcome the difficulties of the theory while retaining the complexity of the many-body interaction effects.

In the mid '60s Hohenberg and Kohn and then Kohn and Sham put the basis for a theory that became the cornerstone for modern theoretical and computational approaches, whose essential ideas are that: a) every observable quantity of a quantum system can be calculated from the density of the system; b) the density of particles interacting with each other can be calculated as the density of an auxiliary system of non-interacting particles; c) the total energy, from which many system’s properties can be calculated, is a functional of the density, whose form is universal, with a term indeed related to the exchange-correlation interaction. It is known as the Density Functional Theory (DFT) and accounts for the ground state only.

While DFT and its extension to the time domain (TDDFT), developed to describe excited states, are both exact in principle, their accuracies in practice are limited by the approximations for the exchange-correlation contribution to the total-energy functional and the corresponding interaction term in TDDFT. These approximations move from the local density approximation, in which the solid is considered locally as a uniform electron gas (LDA-DFT), to semi-local and semi-empirical functionals which deal with better description of the exchange. In practice, an adiabatic (i.e. time-averaged) interaction approximation is made to construct the exchange-correlation term of contemporary applied TDDFT, meaning that excitations are described in a theory where the density at time $t$ is not influenced by the density at previous times and relies, practically always, on a local approximation of the density. Several “Beyond DFT” theories have been developed in the last decades that greatly improve the description of the ground and excited states of a system, but rely on a perturbative approach that in most cases has a DFT starting point, inheriting several limitations of the latter.

A key question is whether the approximations adopted by the theories, that are crucial to describe highly correlated systems, are only mental abstractions or can actually lead to detectable features in experiments. In spite of the large number of experiments of ever increasing finesse so far performed, answering this question is still one of the most demanding tasks of the solid state experimental physics. This is because effects due to correlation remain rather elusive for spectroscopies such as photoemission, electron energy loss, Auger, etc., whose spectral response is primarily determined by independent quasi-particle behaviour. Conversely, correlation is related to the interplay of at least two active electrons, that has marginal influence on the response of this class of conventional spectroscopies. Hence the effort devoted in the past 30 years to developing a new class of experiments whose spectral response carries evident signs of the correlated behaviour of electron pairs.

The idea that a photoemission event could affect more than one electron of the target was in embryo already in a 1905 Einstein’s paper: “However, we will not exclude the possibility that the electrons absorb only a part of the energy of the light quanta”. It is spontaneous to exploit the ejection of more than one electron upon absorption of a single photon to gain knowledge about electron correlation. For this purpose, photoionization reactions whose final state features two valence holes and two unbound electrons in the continuum must be used: in short, photoionization events in which a single photon strikes the target and two electrons are ejected (1 photon-in, 2 e-out). It is exactly through interaction of these holes and electron pairs that the correlation shapes the cross-section of double photoemission, thus providing an experimental reference to assess the goodness of the theories.

## 2 The mechanisms of multiple ionization

The archetypal processes by which a single photon ($h\nu$) is absorbed by a target $( A )$ and creates doubly ionized final states $( A^{++} (v^{–1} v^{–1} ) )$ are:

the direct valence double photoionization: $h\nu +A \rightarrow A^{++} (v^{–1} v^{–1} ) +e_{1} + e_{2}$
and
the core hole Auger decay: $h\nu +A \rightarrow A^{+} ( c^{-1} )e_{1} \rightarrow A^{++} (v^{–1} v^{–1} ) +e_{1} + e_{2}$

In both cases, the Coulomb interaction between valence electrons $( v )$ is responsible either for the direct promotion of the valence electrons pair $(e_{1,2} )$ to the continuum (DPE) or for the core-hole $(c^{–1} )$ autoionizing decay with emission of an Auger electron $( e_2 )$ paired to the core photoelectron $(e_1 )$ (APECS). It is to be noted that the two processes share a similar, often identical, final doubly charged ionic state. Independently of the interaction mechanism, it is exactly through these doubly ionized states that are unravelled electron correlation effects otherwise undisclosed. In both cases, the experiment implies detecting in coincidence two free electrons $(e_{1,2} )$ and in measuring their energy or linear momentum.

High photon energy (several keV) DPE experiments on solids were firstly proposed in the late 1970s to investigate electron correlation in the ground state. Then, using the He atom as a test bed, it was shown that even at intermediate photon energies (few hundred eV) the DPE cross-section factorizes in such a way that from the distribution in energy of the electron pairs the spectral density of the two-particle Green’s function can be derived, therefore providing a spectroscopy of the doubly ionized states. Later it was argued that even at low photon energies (tens of eV) DPE provides detailed information on correlation in solids. Evidence of the influence of correlation on the DPE spectrum was also predicted in the case of semiconductors and nanostructures such as quantum dots and the correlated behaviour of the electron pair was predicted to influence the diffraction from the crystal lattice. All of these theoretical results established DPE as a suitable tool to study electron correlation and many-body effects at large in solids.

When the energy of the photon exceeds the core ionization threshold, DPE is no more the only channel to generate doubly ionized states, because the APECS channel is open. Also in this case, the most complete way to study the process is to detect in time coincidence the Auger-photoelectron pair and to analyze the two partners in energy and in ejection angle (AR-APECS). The feasibility of the APECS experiment was first demonstrated in the late 1970s.

While in DPE the fraction of photon energy that exceeds the double ionization threshold is continuously shared between the two final electrons, in APECS the energetics of the final electron pair is mostly determined by the core-hole lifetime as compared to the electron-electron Coulomb interaction time. For long enough lifetime the energy of the core-hole intermediate state is well defined and the so-called “two step model” is adequate; i.e. core ionization and Auger auto ionization are treated as fully independent processes. As a consequence, the kinetic energy of the two final electrons will be uniquely defined and readily identifiable: the photoelectron energy is determined by the balance of the photon energy and the core state binding energy; the Auger energy is determined by the difference in total energy between the core-hole and the final two valence-hole state. On the contrary, for short enough lifetime, the energy of the core-hole is ill defined and the energy conservation holds for the initially neutral and the final doubly ionized states only. Under these circumstances, Auger and core ionization are no more independent processes and the generation of the correlated electron pairs must be described beyond the independent particle scheme, considering photoemission and Auger decay as a unitary coherent process. Consequently, the final electron pair shares continuously the excess energy and the process becomes similar to a DPE that proceeds through an intermediate core excited resonant state. In this latter case, a “one-step model” must be used to describe the APECS process; the intermediate state becomes virtual and spans over all excited states, including the continuum, and the Coulomb operator, responsible for the Auger decay, acts on the complete many- body system involving both final electrons.

## 3 Instrumentation for multiple ionization experiments

The scheme of principle for experiments on solids based on two-electron emission via single-photon absorption is shown in the inset of fig. 1. A monochromatic beam of photons with controlled polarization ε impinges onto the sample surface. The two final electrons are detected within the solid angles $\Omega_1$ and $\Omega_2$ and selected in the energies $E_1$ and $E_2$ by the electron spectrometers. The electronic signals generated by the detectors are fed to a coincidence circuitry correlating them in time in order to discriminate pairs generated in the same ionization event (true coincidences) from those generated in independent events (false). To see a wide application of this method we must wait for the end of the past century while conventional photoemission spectroscopy had already flourished in the 1960’s.

The challenge for double photoemission experiments lies in the smallness of the multiply differential cross-section that is at least three orders of magnitude smaller than the conventional single photoemission one. Furthermore, the need to reveal two coincident electrons decreases the luminosity of a coincidence apparatus by additional orders of magnitude with respect to a photoemission one. This handicap has been partially overcome on the one hand by the advent of high brilliance, tuneable and polarization controlled lasers and third-generation synchrotron radiation sources, and on the other hand by the development of setups that allow the simultaneous detection of many pairs of electrons at various energies and in a wide solid angle. Multicoincidence spectrometers have been realized in different ways in the United States, Germany, Japan and Italy. The latter, used here as a prototypical example of multicoincidence spectrometer, is the end station of the ALOISA beamline at the ELETTRA synchrotron radiation facility, that is pictorially sketched in fig. 1. Six hemispherical analyzers are hosted inside an ultra-high vacuum chamber and can be independently tuned on either one of the $E_1$ or $E_2$ energies, thus allowing to simultaneously collect a plurality of coincidence spectra. The electron spectrometers are mounted on two independently rotatable frames (analyzers A and analyzer B) thus providing ample flexibility in setting them with respect to the experiment quantization axes: the electric vector $\epsilon$ of the linearly polarized light beam, and the sample surface normal.

The output of the experiment is efficiently represented by a tridimensional plot of the coincidence count rate versus the kinetic energies $E_1$ and $E_2$: in short the 2D energy spectrum. It is to be kept in mind that in the dipole approximation and in the absence of electron-electron interaction the probability for generation of electron pairs is zero. Hence, a 2D energy spectrum, such as the one depicted in fig. 2, is primarily modulated by correlation effects. APECS and DPE events are expected to display markedly different features that in essence can be reduced to the ones sketched with white ellipses in fig. 2. For DPE the single-step model holds, the energy in excess of the double ionisation threshold determines the sum energy $E_{sum} = E_1 + E_2$ and is continuously shared between the pair. The coincidence events are then expected to be distributed as a ridge of coincidence counts at constant $E_{sum}$ as indicated by the long white ellipse in fig. 2. The width $\Gamma_1$ of the ridge is linked to the lifetime of the two-hole final state. For the two-step model, appropriate for the APECS events, the coincidence rate is expected to be localized only in areas where $E_1$ and $E_2$ assume the well specific photoelectron and Auger values $( E_{pe}, E_{Au} )$, like in the twin white ellipses in fig. 2. Also in this case $E_{sum}$ is a constant but the width $\Gamma_2$ is now related to the core-hole lifetime. In both cases, DPE and APECS, the energy balance between photon energy and $E_{sum}$ is an observable related to the energy of the doubly ionized state $( E ( A^{++} (v^{–1} v^{–1} ) )$ through the relation

$E_{sum} = h\nu - [ E ( A ) - E ( A^{++} ( v^{-1} v^{-1} ) ) ]$

Summing up all the coincidence events characterized by an identical $E_{sum}$ it is then possible to measure a binding energy spectrum of the doubly ionized final states, which is not achievable by conventional photoemission.

The correctness of this schematism is demonstrated by the results of an experiment conducted on a Cu surface with 125 eV photons, that are also shown in false colour in fig. 2.

The spectrum displays an onset at states with two holes in the Fermi level (${}^{2h}E_{F}$ full line in fig. 2). The increase in coincidence rate inside the white ellipse labelled DPE corresponds to states with two holes in the localized Cu $3d$ band. This feature is correctly described by DPE and shows an almost even distribution of probability of the $E_1$ and $E_2$ energies.

This is not the case for the sharp structures in the twin ellipses. It is easily verifiable that those peaks of intensity happen when $E_1$ and $E_1$ correspond to a $3p$ photoelectron and to a $M_{2,3}$ VV Auger. In addition, there is a continuum of coincidence rate underlying these two APECS structures, extending all along the diagonal at $E_{sum}$ constant, that conversely can be explained only an a DPE framework. These findings provide evidence for inadequacy of a sharp distinction between the two aforementioned mechanisms of creating correlated electron pairs and for the need of a unitary description of the electron pair generation in terms of a double photoemission that resonates at the core ionization energies.

It is the aim of the following sections to exploit the features induced by correlation in the double photoionization cross section in order to discuss value and limitation of the approximations widely used in describing quantum-mechanical properties of highly correlated solids.

## 4 The exchange-correlation hole

The so-called exchange correlation hole ( XChole ) is a model widely adopted in condensed matter in order to embed in theories the correlation effects due to exchange and Coulomb interactions. In essence, it assumes that in an otherwise uniform gas of electrons, each one of them is surrounded by a sphere of depleted probability of finding a second electron, i.e. of depleted charge density. The pair correlation function yields the probability of finding a second electron at a given distance from the first one. Taking into account electron correlation, this function turns out to vanish at short distances and tends to one starting from distances of few Ångstroms. It thus establishes a sphere of reduced probability that represents the XChole in the direct space. The question to be answered is whether the XC hole is a computational artifice or it is liable to experimental testing.

There is a close connection between the XChole and the angular distribution of the electron pair in a DPE event. This occurs because in any scattering process, features of the target charge distribution in direct space do appear as features of the scattered particles probability distribution in the conjugate momentum space. Therefore, a depletion of charge density in the target should modulate the pair probability in the DPE angular distribution as well.

This “hand waving” reasoning was put on firmer grounds by a DFT calculation of the DPE cross section performed for a Cu surface, that takes explicitly into account mutual exchange and Coulomb interaction between the two electrons involved in the photoemission. The DPE intensity, i.e. the probability of detecting electron pairs, is reported in a 2D representation in fig. 3. The calculation is carried out at a photon energy of 42.4 eV and equal electron energies of 16 eV. One electron is directed along the direction normal to the surface (fig. 3a) and at 30° from it (fig. 3b), while the second electron angle is varied and the double photoemission intensity is plotted versus its momentum components parallel to the surface. From the distribution represented in false colour, the presence of a circular zone of depleted probability is evident whose centroid coincides with the direction of the fixed in angle electron (white dot in fig. 3a and b), followed by a ridge of maximum whose radius is related to the XChole radius.

These predictions make a good account of an experiment conducted on a Cu surface under similar experimental conditions, with 50 eV photons and energies of 23 eV for one electron and of 12 eV for the other. Parallel detection of electron pairs with different momentum (direction of flight) was obtained by the use of time-of-flight spectrometers with wide accepted solid angles. The results of the experiment are shown in fig. 3c-f. The fast electron was detected at –1rad off-normal (fig. 3c) or 1rad off-normal (fig. 3e) within the solid angle marked by the vertical dashed lines. The slow electron spanned over the full solid angle accepted by the experiment (in the figures $\Theta$ and $\Phi$ are the electron take-off angles). In panels c and e the intensity for correlated emission of electron pairs is reported in false colour and a zone of depleted probability is evident surrounding the direction of the fast electron (indicated with the white dot). This effect is even more evident if the coincident events are summed along $\Phi$ and reported as a function of $\Theta$, as in fig. 3d and 3f. In both cases, the coincidence rate displays a clear minimum in correspondence to the fast electron ejection angle and a ridge of intensity for electrons emitted at $\sim 1.2$ radians away from each other, in agreement with the predictions.

Hence, the reduced probability of finding electron pairs with small relative momentum implies a very low number of strongly interacting electrons within a sphere of a few Ångstroms radius. In other words, the depletion of coincidence rate measured as a function of the ejection angles, i.e. in the momentum space, reflects the existence of a XChole in the direct space.

## 5 The core relaxation dynamics

In recent years, much attention shifted from understanding the behaviour of strongly correlated solids in equilibrium to their behaviour when suddenly excited out of equilibrium. In a wide variety of time-resolved experiments a many-body system is rapidly driven out of equilibrium and then its electronic structure is investigated. Among them are the successful pump-probe experiments, that were made possible by the availability of extremely intense and ultrashort (femto- or pico-seconds) light pulses. Ultra-short light pulses are not the only possible experimental approaches at our disposal. Upon core ionization, the charge rearrangement gives rise to a dynamical screening of the hole whose time scale (from atto- to femto-seconds) depends on the strength of the interaction. Charge screening happening on different time scale influences differently the photoemission and Auger energy spectra. For this reason APECS experiments could give access, through energy selection rather than time selection, to the fast relaxation dynamics of the fermionic gas in a solid.

When the two step (quasi-particle) model collapses, the core-hole state is not any more a true “stationary” state, then the one-step model is more appropriate and both photoelectron and Auger spectra become so broad in energy that the distinction between the two is fuzzy. Under these conditions, the energy distribution of the final electrons of an APECS experiment resembles a DPE distribution resonant through a virtual core hole state rather than a pure APECS one.

The Ag $4p$ photoemission peak is fairly broad, while the $3d$ is rather sharp. Hence, Ag is a good candidate to induce, through ionization of different core levels, either fast or slow screening effects within the same valence band. This possibility was exploited by an experimental study on Ag that features a $3d$ photoemission peak (binding energy 378 and 372 eV for 3/2 and 5/2 spin-orbit components) that is only 0.3 eV wide (long-lived core state), while the $4p$ (binding energy 64 and 58 eV for 1/2 and 3/2 spin orbit) is 13 eV wide (short-lived core state). Energy spectra were recorded for electron pairs associated to the processes $4p^{–1}\rightarrow 4d^{-1} 4d^{-1}$ and $3d^{–1} \rightarrow 4d^{–1}4d^{–1}$, that have identical final states but sharply different intermediate core-hole lifetime. The 2D energy distributions for the aforementioned $3d$ and $4p$ double ionization are reported in fig. 4a and b, respectively. While for the $3d$ ionization electron pairs are found only at well-defined energies of the Auger and photoelectron (the four intensity maxima in region II of fig. 4a), in the case of $4p$ a continuous sharing of energies is measured (the intensity ridge of region II of fig. 4b). It is therefore evident that the $3d^{–1}\rightarrow 4d^{–1} 4d^{–1}$ proceeds via a two-step process, while the $4p^{–1} \rightarrow 4d^{–1} 4d^{–1}$ proceeds via a single-step one. To compare the two cases, it is convenient to project the coincidence rate on the variable $E_{sum} = ( E_1 +E_2 )$, that is related to the binding energy of $4d^{–1} 4d^{–1}$ states, measured in relation to the binding energy of the state with two holes at the Fermi level $({}^{2h}E_{F} )$, see fig. 4c.

It is evident that the two spectra share the same pattern of final states in region II (two valence holes in the same atomic site) and in region I (holes in different sites): a same multiplet of quasi-atomic, strongly correlated, states dominated by the ${}^{1}G^{3}_{F}$. Taking into account the difference of the spectra, normalized at the same total area, reported in fig. 4d, clearly emerges that for the $4p$ ionizations there is a transfer of transition strength from the relaxed region II to the unrelaxed single and double interband excitations of region III (peaks P1 and P2 of fig. 4d). This is because the $4p^{–1}$ state is degenerate in energy with the $4d^{–2} 4f$ and fast fluctuation between the two happens. The fluctuation time is of the order of tens of attoseconds, as deduced from the width of the energy distribution of the final electrons, tens of eV in the case of $4p^{–1}\rightarrow 4d^{–1} 4d^{–1}$ (see fig. 4b).

In conclusion, by the APECS experiment we learn notions about the dynamical screening of core holes happening on the attosecond time scale, not obtained by conventional Auger and photoelectron spectroscopy. Core-resonant double photoemission is a powerful tool to investigate ultrafast charge screening dynamics that is relevant to modelling the transport of hot electrons in solids.

## 6 The exchange-correlation energy

To accurately describe the electronic structure of strongly correlated solids, careful control must be gained of the correlation energy $U$ whose knowledge is crucial in order to perform reliable calculations beyond mean-field approximations. The Auger line shape has always been considered to be very sensitive to correlation. Cini and Sawatzky’s work (CS) evidenced that the Auger line shape, simply determined by the autoconvolution of the valence band Density Of States (SC-DOS) in the absence of correlation, is profoundly modified if the correlation energy $U$ is taken into account. In their models, as $U$ increases and becomes comparable to the valence bandwidth $W$, the Auger line shape changes from an almost featureless SC-DOS to a sharp atomic-like structure determined by two-hole resonances in the Auger final state. Hence, the Auger spectrum is the ideal candidate to provide evidence of concreteness and value of $U$, as clearly shown in fig. 5 where the Auger profile, calculated for the majority band of a thin iron film deposited on a copper surface, evolves from purely band-like when $U=0$ (black dashed line) to strongly atomic-like when $U=8$ eV (red line).

Unfortunately, in most cases Auger spectra originating from different core holes do overlap, partially or totally, in energy thus hampering the possibility to derive unique information on $U$ from their line shape. This drawback is overcome if the Auger electron is measured in coincidence with a specific photoelectron in an APECS experiment.

Provided the two-step model holds, by these experiments a different Auger profile is measured for each specific and well defined core hole. This claim is exemplified by a paradigmatic study on S atoms adsorbed on the Cu ( 100 ) surface. The result is depicted in the 2D energy distribution of the coincidences shown in fig. 6a. The energy chosen for the photon allowed both the $L_2$ and $L_3$ photoelectrons and the $L_2$VV and $L_3$VV Auger to fall into the energy window accepted by the spectrometer. Using the notions deduced from fig. 2 about the pair generation mechanism, it is immediately clear that the distribution of coincident events is typical of APECS processes, and therefore the experiment can be interpreted in the framework of the two-step model. Selecting only events characterized by an identical photoelectron, the overlap in energy of the Auger spectra is erased on physical basis. Selecting pairs with one of the energies equal to the $L_3$ photoelectron, the pure $L_3$VV spectral line (bright ridges of high coincidence rate inside the white rectangles in fig. 6) is measured, and the result is reported in the linear energy graph of fig. 6b. Unlike conventional Auger spectroscopy, the APECS spectrum shows several features that can be interpreted with the help of theoretical predictions. The sulfur $L_3$VV line shape has been predicted computing within an LDA-DFT approximation the SC-DOS in the presence of a Cu surface, and then applying the CS model for different $U$.

The results obtained for three $U$ values (0, 0.3 and 0.4 eV) are reported, convoluted with the experimental broadening, in fig. 6b together with the experimental spectrum. The first observation is that from the onset, and down to an energy of 145 eV, the APECS line shape is well reproduced by the model with $U=0.3$ eV. This is a clear evidence of the APECS capability to give direct experimental access to the $U$ value, even in case of moderate correlation energy as the present one. The second observation is that the low-energy side of spectrum ($E < 145$ eV) corresponds to final states with two holes located below the bottom of the valence band. The peak appearing at about 143 eV in the experiment-theory difference (fig. 6c) is sharp with respect to the two-hole final state spectrum and so extrinsic excitations, such as energy losses suffered by the electrons while leaving the target, are to be ruled out. The structure is to be interpreted as Auger intrinsic satellites, perhaps due to extra hole creation within the autoionization process, also resulting from correlation in the valence band and not accounted for by the CS model.

It can therefore be concluded that by two-step APECS it is possible to associate the theoretical parameter $U$ with observables experimentally quantifiable.

## 7 The spin-dependent exchange-correlation energy

The magnetic properties of solids are explained in terms of exchange interactions, that split the valence band into “majority” and “minority” spin bands unevenly populated, thus giving rise to spin polarization, i.e. magnetization. APECS experiments resolved in the ejection angles of the electron pair (AR-APECS) have disclosed a spin dependence of the exchange-correlation term $U$.

Let us consider an AR-APECS event initiated by a linearly polarized photon where the photoelectron is detected along a direction aligned with the light polarization vector. Only the subset of core-hole states with magnetic quantum number $m_l = 0$ is selected. By detecting in coincidence the subsequent Auger electron, it is possible to focus on autoionizing events that originate not from a statistical population of core-hole states but from a given subset, hence from an aligned core state. By selecting in angle the Auger, a further constraint is imposed on the $m_l$ of the Auger wave function. Taking into account these discriminations and the selection rules valid for photoemission and Auger processes, a propensity rule for the creation of specific final states with two valence holes is established. According to these propensity rules, when both the final electrons are aligned with the light polarization (AA geometry) low-spin final states are preferentially populated. On the contrary, when only the photoelectron is aligned and the Auger is not (AN geometry), high-spin final states are preferentially accessed. In practice, AR-APECS measurements are selective in the total spin of the final state without having to perform an experimentally inefficient spin selection on the detected electrons, as it is done in the conventional spin-selected spectroscopies.

In the following, selected findings are reported from an AR-APECS study on a ferromagnetic Fe film deposited on a Cu substrate. The conventional $M_{2,3}$VV Auger spectrum of iron is a featureless band-like structure. It does not bear any direct information on the correlations that instead are expected to be important in magnetic systems. The same spectrum was measured in coincidence with the Fe $3p$ photoionization line and in two geometries, and the fits to the experiment are shown by dashed lines in fig. 7.

The results are now much more interesting in as much as different line shapes are obtained in different geometrical conditions. Precisely, when the experiment was conducted in AN geometry (green dashed line), an additional peak clearly rises at an energy of 38 eV, well below the position of the main structure of the spectrum at 43 eV, that is present in both AN and AA (red dashed line) geometries. Referring to the propensity rules mentioned above, it is conceivable to attribute the two structures to high-spin and low-spin final states, respectively. This qualitative interpretation was put on solid quantitative footing by modelling the AR-APECS line shape taking into account all the allowed final spin configurations in which the system can be left. The electronic structure of an iron film deposited on Cu was calculated within a LDA-DFT approach.

If $U$ is not taken into account, the model predicts the energy feature at 43 eV but not the one at 38 eV. To reproduce the transition observed at 38 eV, the single-particle SC-DOS must be replaced by an interacting two-hole DOS, as proposed by Cini and Sawatzky (green and red continuous line in fig. 7). In particular, for the calculation to reproduce the experiment, the correlation term $U$ must be 2.7 eV for the two holes in the majority band and 0 eV in all other cases. Having introduced the spin dependence of the exchange-correlation, the full spectrum is described by the model in all geometries. The meaning is that different on-site electron-electron interactions exist in the same system, depending on the spin of the electron pair, and the hole-hole interaction is stronger when two majority spins are involved. This should influence photoemission and spin-dependent transport properties predictions for other magnetic systems as well.

## 8 Perspectives

The present concise overview of the double photoemission experiments on solids, performed with a coincidence technique, shows that these methods allow direct verification of the principal models used in describing the correlated properties of solid. In particular, it is possible to obtain in a direct experimental way information on: existence and extension of the exchange-correlation hole; strength of the exchange and correlation interactions; dynamical screening effects on tens of attoseconds scale; spin dependence of exchange-correlation in magnetic materials.

These are concepts introduced by quantum-mechanical models and the fact that they are directly connected to experimentally detectable effects provides a sort of exchange-correlation imaging. All this is of paramount relevance in modelling both microscopic and macroscopic properties of highly correlated solids.

These experiments, however, have still a yield of counts that is too low to be proposed as routine spectroscopies for material characterization. However, It should be borne in mind that, over the past 30 years, the time needed to perform a campaign of such measurements has decreased from a few weeks to a few hours. Such a progress is certainly due to the fast development of the synchrotron radiation and laser characteristics, but mainly to the impetuous development of parallel detection of the photoelectrons pairs, and to the remarkable increase in the throughput of the electron analyzers. A good example is the bursting of time-of-flight methods at the scene of electron spectroscopies. It must ultimately be considered that so far these spectroscopies have mostly been achievable at large synchrotron radiation facilities. However, due to the rapid development of UV and soft X-ray lasers, the realization of efficient on-campus experiments has been already reported and multionization spectroscopies are expected to be carried out in the future at academic or even industrial laboratories.

## Acknowledgments

The ideas presented are the result of a fruitful cooperation with a number of colleagues. It is impractical to mention all of them but it is just impossible not to mention R. A. Bartynski, J. Berakdar, M. Cini, G. DiFilippo, R. Gotter, S. Iacobucci, J. Kirshner, A. Ruocco, F. O. Schumann. A special mention is deserved to F. Da Pieve and F. Offi for also critically reading the manuscript. Partial financial support from PRIN 2015 NEWLI: NEW Light on transient states in condensed matter by advanced photon-electron spectroscopies, Protocol: 2015CL3APH is gratefully acknowledged.