Eric Pierre Simon
email: eric.simon@univ-lille.fr

Pierre Laly

Joumana Farah

Emmeric Tanghe

Davy Paul Gaillot

Measurement of the V2I Channel in Cell-free Vehicular Networks with the Distributed MaMIMOSA Channel Sounder

Keywords: cell-free, MIMO, channel sounding

In this paper, we present a small, yet realistic, vehicular cell-free massive MIMO (multiple-input multiple-output) architecture deployed at the University of Lille in a typical suburban environment under both Line-of-Sight (LOS) and obstructed LOS (OLOS) shadowing conditions. The radio channels were acquired with a distributed RF-over-Fiber (RoF) upgrade of the real-time channel sounder MaMIMOSA. The system operates at 5.89 GHz with an 80 MHz bandwidth, which corresponds to the ITS frequency band offered by the ITS-G5 and C-V2X technologies. Four omnidirectional receive antennas were placed on the roof of a van moving at a speed of 25 km/h. The propagation channel was measured for various transmit antenna configurations, ranging from co-located antennas to fully distributed antennas. The measurement results show a significant gain in the signal-to-noise ratio (SNR) as well as a more uniform coverage and smaller delay spread values with the distributed scenarios compared to the centralized ones. Finally, the path loss measurement results obtained for the cell-free network provide deployment guidelines for the distributed antennas.

I. INTRODUCTION

5G mobile networks are currently built according to a cell architecture with a base station (BS) at its center. The BS is equipped with an array of many antennas called massive multiple-input multiple-output (MIMO) which is one of many cornerstones of the 5G technology. The benefits of massive MIMO are numerous and have been widely studied in the literature [START_REF] Björnson | Massive MIMO networks: Spectral, energy, and hardware efficiency[END_REF]. However, 5G networks still suffer from the limitations attributed to their inherent cellular architecture. On the one hand, there is an extreme variation in the received power level between the center and edge of the cell. This results in a substantial variation in the signal-to-noise-ratio (SNR) and, therefore, quality of service (QoS). On the other hand, users located at the cell-edge suffer from interference from neighboring cells. Hence, a cellular architecture is not compatible to meet the challenges of future wireless communications applications and provide ubiquitous wireless connectivity.

A new network architecture without the concept of cells appeared a few years ago: the cell-free mobile network [START_REF] Interdonato | Ubiquitous cell-free massive MIMO communications[END_REF], [START_REF] Chen | Channel hardening and favorable propagation in cell-free massive MIMO with stochastic geometry[END_REF], [START_REF] Farah | Energy-efficient techniques for combating the influence of reactive jamming using non-orthogonal multiple access and distributed antenna systems[END_REF], [START_REF] Farah | Efficient combinations of NOMA with distributed antenna systems based on channel measurements for mitigating jamming attacks[END_REF]. In the cell-free context, an important number of access points (APs) distributed over a large geographical area cooperate to optimally serve the users (UE). These APs are equipped with a small number of antennas and are connected to a central processing unit (CPU) via a wired or wireless backhaul link. The cell-free concept encompasses the 5G massive MIMO technique wherein many antennas are densely distributed over the area. As a result, cell-free massive MIMO can be viewed as a generalization of 5G massive MIMO. Due to the proximity and diversity of APs, it offers a far more uniform coverage and increased connectivity. In particular, critical life-safety services considered in upcoming vehicular networks specifically rely on a highly uniform QoS.

Although the cell-free massive MIMO has recently received a lot of attention at the theoretical level, there is only one recent experimental study in the literature that investigates the attractive features of vehicular cell-free massive MIMO [START_REF] Löschenbrand | Towards Cell-Free Massive MIMO: A Measurement-Based Analysis[END_REF]. In that work, the topology under investigation consisted in distributing 32 transmit (Tx) antennas along an axis of 46.5 m on a building rooftop with different configurations. In contrast, in this paper, it is considered that the antennas are fully deployed over the area of interest at the street level, as imagined for road-side units (RSU) with V2I applications. Furthermore, different possible Tx antenna system topologies are investigated, i.e. the number of APs and antennas per AP altogether with their locations. Indeed, recent articles showed that the semi-distributed multi-antenna APs is the best configuration [START_REF] Choi | Co-located vs distributed vs semi-distributed MIMO: measurement-based evaluation[END_REF], [START_REF] Guevara | Weave and conquer: A measurement-based analysis of dense antenna deployments[END_REF], [START_REF] Fager | Comparison of Co-located and Distributed MIMO for Indoor Wireless Communication[END_REF] from system-level performance perspective. Nonetheless, these results are solely based on an indoor measured massive MIMO dataset. Hence, the validation of this approach for outdoor vehicular applications is simply missing in the literature. To tackle vehicular networks, the real-time channel sounder MaMIMOSA has been upgraded to measure channels in such distributed configurations. Initially, MaMIMOSA was jointly developed by the University of Lille (France) and Ghent (Belgium) to measure co-located massive MIMO channels for vehicular scenarios [START_REF] Laly | Flexible real-time mimo channel sounder for multidimensional polarimetric parameter estimation[END_REF]. In particular, it was recently set up to measure the time-varying 64 × 16 radio channel at 5.89 GHz with an 80 MHz bandwidth. The upgrade consists in geographically distributing the transmit antennas using up to 500 meter-long RF-over-fiber (RoF) links. This enables the real-time channel between a vehicle and a network of distributed antennas to be measured. The current MaMIMOSA configuration supports up to 8 singleantenna APs or L APs equipped each with a N -antenna array, with L × N = 8. Based on the measurements, the performance of the different topologies are compared in terms of the achievable mean SNR, delay spread, and path loss characteristics.

II. DISTRIBUTED MAMIMOSA

The propagation channels are measured with the MaMI-MOSA radio channel sounder [START_REF] Gaillot | Measurement of the V2I massive radio channel with the MaMIMOSA sounder in a suburban environment[END_REF], [START_REF] Laly | Massive radio channel sounder architecture for 5G mobility scenarios: MaMIMOSA[END_REF]. Data are modulated using orthogonal frequency-division multiplexing (OFDM) with 8192 subcarriers using a 12.21 kHz subcarrier spacing. The streaming mode of MaMIMOSA is selected for the vehicular context. It corresponds to the frame structure illustrated in Fig. 1. Each frame is emitted every 510 ms and starts with a 51.2 µs preamble, followed by 4096 OFDM symbols. This yields a duration of around 500 ms. Each Tx antenna transmits a pilot every eight sub-carriers, with one sub-carrier per antenna shift to interleave the Tx antenna signals. Note that one OFDM symbol out of 32 was recorded for this campaign. The purpose of this work does not include a complete Doppler analysis due to the lack of space but this will be addressed in subsequent work. For the time being, eight 500 meter-long RoF links with 1 dB/km attenuation loss are connected to MaMIMOSA and TX as shown in Fig. 2. The RF-optical converters integrate a power amplifier to compensate for the insertion losses at the expense of a 27 dB noise figure. The output power was set to 0 dBm for each RF chain and a 15 dB gain power amplifier was placed in between the optical/RF module and antenna to reach a 15 dBm transmit power signal with 40 dB SNR. The length of all coaxial cables was 2.5 m. A hybrid calibration procedure using one of the RoF links was performed since both RoF and coaxial links are used. A picture of the MaMIMOSA transmission system, playing the role of the cell-free CPU, is shown in Fig 3 . 6.5 dBi patch antennas were used at the transmitting APs (Fig. 4). The receiver was a van equipped with four 2 dBi EM6116 omnidirectional antennas on its rooftop spaced apart by more than 10λ (i.e. more than 0.5 m with λ ∼ 5.1 cm) to ensure a decorrelated reception, as illustrated in Fig. 5. The receiving (Rx) MaMIMOSA system was carried inside the van. Both Tx and Rx units were powered with Li-ion batteries that provide up to 8 hours of continuous measurements. The measurement scenario consisted in driving the van on a section of the campus boulevard from point A to point B over a 275 m distance with a roundabout in-between at an average speed of 25 km/h (Fig. 6). Four AP configurations were investigated in this campaign from fully co-located to fully distributed antennas. The first one consists in one AP equipped with 8 antennas, the second one 2 APs equipped with 4 antennas each, the third one 4 APs equipped with 2 antennas each, and the fourth one 8 single-antenna APs. These configurations are called Config. 1 to 4 throughout the text, respectively. The deployment of the APs is shown in Fig. 6 7 shows, for each configuration, the maximum SNR obtained among the eight Tx antennas. In practice, this would be equivalent to an antenna selection criterion based on the SNR. This choice is motivated to reduce the implementation cost by limiting the system to a unique radio-frequency (RF) chain. The results clearly demonstrate the efficiency of distributing the Tx antennas over the coverage area. Indeed, for Config. 1, the reception quality starting from A is observed to be rather poor since Rx is relatively far from Tx. Quality is only improved when Rx gets nearer to the end of its path (i.e. point B) and, therefore, closer to Tx. This proves that the receiver does not always benefit from the multi-antenna diversity due to Moreover, still considering the aforementioned antenna selection criterion, the channel root mean square (RMS) delay spread was computed for the first Rx and the best corresponding Tx (see Fig. 9). Only the van positions that yield an SNR higher than a threshold of 10 dB are considered. The delay spread values are found to be varying between 12.5 (minimum measurable rms) to 150 ns and are in line with values reported in the literature for a similar scenario and shadowing conditions (see [START_REF] Zhao | Propagation characteristics for wideband outdoor mobile communications at 5.3 ghz[END_REF] for example). It is noticed that distributing the antennas allows a higher number of van positions with low RMS delay spread values. These low values generally correspond to LOS links between transmit and receive antennas, which are more frequent in distributed scenarios. This effect is further confirmed in Fig. 10 which shows the corresponding CDF. The results show a clear advantage of the three distributed configurations with respect to the centralized Config. 1, while the other three exhibit a similar behavior in terms of RMS delay spread.

To further investigate these results, Fig. 11 presents, for each configuration, the CDF of the RMS delay spread of the link corresponding to the first Rx and:

• the Tx yielding the maximum RMS delay spread (dashed lines), • the Tx yielding the minimum RMS delay spread (dotted lines), • the Tx yielding the highest SNR (solid lines). A first observation is that distributing the antennas (ex: Config. 4) naturally produces higher values of the maximum RMS delay spread, compared to less distributed configurations. However, the minimum RMS delay spread is the smallest for Config. 4. Additionally, our antenna selection criterion based on SNR yields very close RMS delay spread values to those obtained with a decision based on the observed RMS delay spread. Optimized deployment strategies of the distributed cellfree network are currently missing in the literature for a realistic complex propagation scenario. Here, the dependence of the AP location in the scenario on the propagation fading properties was investigated for the fully distributed Config. 4. This analysis is critical as it provides some guidelines to distribute the APs with respect to the foliage, buildings, and the AP antenna orientation with respect to the receivers. To this end, the path loss exponent n and lognormal shadow fading χ g were derived for each AP location from the received gain averaged across the receivers, when driving from B to A. Since the receiver crosses the AP during the drive-test, the analysis was split into two datasets (a) and (b). (a) corresponds to the trip from B to the AP, whereas (b) corresponds to the trip from the AP to A. AP locations 5 to 7 were omitted because the corresponding datasets were not of sufficient quality to compute the path loss characteristics. The results presented in

V. CONCLUSION

For the first time to our knowledge, this paper deals with radio channel measurements in a realistic vehicular cell-free network. The measurements were performed with an 80 MHz bandwidth at 5.89 GHz using an upgraded version of the real-time channel sounder MaMIMOSA. The upgrade consists in using RoF links to distribute the Tx antennas in different locations to emulate the distributed APs of the cell-free network in a large suburban area. Although a limited number of transmit antennas (eight) was considered, which is below the expected number in an operational distributed massive MIMO network, the study experimentally validates the promises of cell-free networks compared to co-localized antennas, with a higher and spatially more uniform SNR. The analysis of the path loss characteristics for the cell-free network also provides valuable guidelines for deploying APs in the scenario. Moreover, an important advantage of the proposed framework using MaMIMOSA lies in its high level of flexibility for the deployment of many different distributed configurations. It is also straightforwardly adaptable to include a higher number of additional APs and antennas for emulating denser cell-free networks.

Fig. 1 .

 1 Fig. 1. MaMIMOSA Streamshot Frame structure

Fig. 2 .

 2 Fig. 2. Cell-free sounding architecture using 8 RoF links and one coaxial link

Fig. 3 .

 3 Fig. 3. MaMIMOSA transmission system with the 500m long RoF links

  , where each position has been numbered. The orientation of the Tx patch antennas is depicted by an arrow pointing toward the emission direction. The Tx antennas are positioned at a height of 2.5 m and are vertically polarized. The mean distances between APs are around 225 m, 85 m, and 40 m for the Config. 2, 3, and 4, respectively.For each transmission configuration, the radio channel was measured with the streaming mode using ∼ 70 frames giving a complete recording of ∼ 35 seconds.IV. MEASUREMENT RESULTSFirst, the SNR values are estimated for each Tx -Rx link as a function of time. Then, the obtained SNRs are averaged over the four Rx antennas for each Tx.

Fig. 4 .

 4 Fig. 4. Setup of the investigated linear array configurations with 1, 2, and 8 patch antennas per AP. The red box indicates a 4-antenna AP as a subset of the 8-antenna AP

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Top-view of the measurement campaign on the Lille University Cité Scientifique Campus presenting the four AP configurations

4 Fig. 9 .Fig. 10 .

 4910 Fig. 9. RMS delay spread for the four AP configurations

Fig. 11 .

 11 Fig. 11. CDF of the maximum (dashed line), minimum (dotted line) and highest SNR (solid line) RMS delay spread for the four AP configurations

  Table I indicate that n varies between 1.85 and 4.89, with the χ g values being below 3 dB, which are in line with values

VI. ACKNOWLEDGMENT

This work was supported through the CPER RITMEA project co-financed by the European Union with the European Regional Development Fund, the French state, and the Hautsde-France Region Council.


C D Dashwood

A Geondzhian

J G Vale

A C Pakpour-Tabrizi

C A Howard

Q Faure

L S I Veiga

D Meyers

S G Chiuzbăian

A Nicolaou

S G Chiuzbȃian

N Jaouen

R B Jackman

A Nag

M García-Fernández

Ke-Jin Zhou

A C Walters

K Gilmore

D F Mcmorrow

M P M Dean
email: mdean@bnl.gov

Probing electron-phonon interactions away from the Fermi level with resonant inelastic x-ray scattering

published or not. The documents may come

Probing electron-phonon interactions away from the Fermi level with resonant inelastic x-ray scattering

Interactions between electrons and lattice vibrations are responsible for a wide range of material properties and applications. Recently, there has been considerable interest in the development of resonant inelastic x-ray scattering (RIXS) as a tool for measuring electron-phonon (e-ph) interactions. Here, we demonstrate the ability of RIXS to probe the interaction between phonons and specific electronic states both near to, and away from, the Fermi level. We performed carbon Kedge RIXS measurements on graphite, tuning the incident x-ray energy to separately probe the interactions of the π * and σ * electronic states. Our high-resolution data reveals detailed structure in the multi-phonon RIXS features that directly encodes the momentum dependence of the e-ph interaction strength. We develop a Green's-function method to model this structure, which naturally accounts for the phonon and interaction-strength dispersions, as well as the mixing of phonon momenta in the intermediate state. This model shows that the differences between the spectra can be fully explained by contrasting trends of the e-ph interaction through the Brillouin zone, being concentrated at the Γ and K points for the π * states, while being significant at all momenta for the σ * states. Our results advance the interpretation of phonon excitations in RIXS, and extend its applicability as a probe of e-ph interactions to a new range of out-of-equilibrium situations.

I. INTRODUCTION

Knowledge of the interactions between electrons and phonons is central to understanding a diverse array of condensed matter phenomena. Basic material properties such as low-field electrical transport [1], as well as emergent collective phases including charge density waves [2] and conventional superconductivity [3], are all crucially dependent on the coupling of phonons to electrons near the Fermi surface. Extensive experimental efforts have therefore been devoted to quantifying this coupling through various techniques. Neutron scattering [4], nonresonant inelastic x-ray scattering (IXS) [5], and Raman spectroscopy [6] all probe e-ph coupling via the lattice vancing theoretical understanding of the phonon contribution to the RIXS cross-section [START_REF] Ament | Determining the electron-phonon coupling strength from resonant inelastic x-ray scattering at transition metal Ledges[END_REF][START_REF] Devereaux | Directly characterizing the relative strength and momentum dependence of electron-phonon coupling using resonant inelastic xray scattering[END_REF][START_REF] Geondzhian | Demonstration of resonant inelastic x-ray scattering as a probe of excitonphonon coupling[END_REF][START_REF] Geondzhian | Generalization of the Franck-Condon model for phonon excitations by resonant inelastic x-ray scattering[END_REF][START_REF] Bieniasz | Beyond the single-site approximation modeling of electron-phonon coupling effects on resonant inelastic x-ray scattering spectra[END_REF]. Much of the excitement around these developments lies in the momentum-resolution that RIXS offers. The other great strength of RIXS -the ability to probe a particular electronic orbital by tuning the incident energy to a specific absorption edge -has thus far been under exploited.

Here, we demonstrate the power of RIXS to probe the interaction of phonons with two distinct electronic states in graphite: the π * state near the Fermi level, and the σ * state well above it. At the resonances of these two states, we find qualitatively different multiphonon excitations in our RIXS spectra, showing a stark difference in the phonon momenta to which they couple. This behaviour cannot be captured by the currently available theoretical models, which assume dispersionless phonons [START_REF] Johnston | Electronlattice interactions strongly renormalize the chargetransfer energy in the spin-chain cuprate Li2CuO2[END_REF][START_REF] Ament | Determining the electron-phonon coupling strength from resonant inelastic x-ray scattering at transition metal Ledges[END_REF][START_REF] Bieniasz | Beyond the single-site approximation modeling of electron-phonon coupling effects on resonant inelastic x-ray scattering spectra[END_REF] or focus on one-phonon processes [START_REF] Devereaux | Directly characterizing the relative strength and momentum dependence of electron-phonon coupling using resonant inelastic xray scattering[END_REF]. We therefore extend a Green's-function approach, previously applied to small molecules [START_REF] Geondzhian | Demonstration of resonant inelastic x-ray scattering as a probe of excitonphonon coupling[END_REF], to treat the full momentum-dependence of phonons in a crystalline lattice. This model accurately reproduces our experimental spectra at both resonances, revealing contrasting trends of the e-ph interaction strength through the Brillouin zone.

II. RIXS MEASUREMENTS ON GRAPHITE

Graphite is a favorable material for this study, as it has a simple electronic and phononic structure, as well as two well-defined excitations to the low-energy π * and high-energy σ * states at the carbon K edge. Coupling to the electronic states near the Fermi level is known to be dominated by the highest optical phonon band above 0.15 eV [5,6,[START_REF] Lazzeri | Phonon linewidths and electron-phonon coupling in graphite and nanotubes[END_REF][START_REF] Park | Electron-phonon interactions in graphene, bilayer graphene, and graphite[END_REF][START_REF] Na | Direct determination of mode-projected electron-phonon coupling in the time domain[END_REF], which is well resolved by our 47 meV-resolution measurements. There are also no other low-energy excitations (such as magnons) that could obscure the phonon excitations in our RIXS spectra. This allows us to validate our methodology at the π * resonance, before extending our treatment to the σ * states. Given their similarities to graphite [6,[START_REF] Lazzeri | Phonon linewidths and electron-phonon coupling in graphite and nanotubes[END_REF][START_REF] Park | Electron-phonon interactions in graphene, bilayer graphene, and graphite[END_REF], our results are also applicable to the other technologicallyimportant allotropes of carbon.

Our carbon K-edge RIXS measurements on graphite are summarized in Fig. 1(a), which shows approximately zone-center RIXS maps around the π * and σ * resonances (see Appendix A for experimental details). Above the elastic line at 0 eV energy loss, both maps show a series of features of decreasing intensity that are reminiscent of the harmonic progression of phonon excitations seen in previous studies [START_REF] Lee | Role of lattice coupling in establishing elec-tronic and magnetic properties in quasi-one-dimensional cuprates[END_REF][START_REF] Lee | Charge-orbital-lattice coupling effects in the dd excitation profile of one-dimensional cuprates[END_REF][START_REF] Moser | Electron-phonon coupling in the bulk of anatase TiO2 measured by resonant inelastic x-ray spectroscopy[END_REF][START_REF] Fatale | Hybridization and electron-phonon coupling in ferroelectric BaTiO3 probed by resonant inelastic x-ray scattering[END_REF][START_REF] Johnston | Electronlattice interactions strongly renormalize the chargetransfer energy in the spin-chain cuprate Li2CuO2[END_REF][START_REF] Meyers | Decoupling carrier concentration and electron-phonon coupling in oxide heterostructures observed with resonant inelastic x-ray scattering[END_REF][START_REF] Rossi | Experimental determination of momentumresolved electron-phonon coupling[END_REF][START_REF] Vale | High-resolution resonant inelastic x-ray scattering study of the electronphonon coupling in honeycomb α -Li2IrO3[END_REF][START_REF] Braicovich | Determining the electronphonon coupling in superconducting cuprates by resonant inelastic x-ray scattering: Methods and results on Nd1+xBa2-xCu3O 7-δ[END_REF][START_REF] Geondzhian | Large polarons as key quasiparticles in SrTiO3 and SrTiO3-based heterostructures[END_REF][START_REF] Li | Multiorbital charge-density wave excitations and concomitant phonon anomalies in Bi2Sr2LaCuO 6+δ[END_REF][START_REF] Peng | Enhanced electron-phonon coupling for charge-densitywave formation in La1.8-xEu0.2SrxCuO 4+δ[END_REF][START_REF] Feng | Disparate exciton-phonon couplings for zonecenter and boundary phonons in solid-state graphite[END_REF]. These features are more intense in the σ * map, suggesting a stronger overall coupling.

We assign contributions to the phonon features through reference to the known phonon dispersion of graphite, shown in Fig. 1(b) [START_REF] Mohr | Phonon dispersion of graphite by inelastic x-ray scattering[END_REF]. The lowest-energy feature occurs at ∼0.19 eV in both maps, which corresponds to the degenerate transverse (TO) and longitudinal optical (LO) modes at the Γ point [whose displacement pat- tern is shown in the final state in Fig. 2(a)]. In a purely local picture, successive peaks would then correspond to higher harmonics of this first peak. The next feature around 0.3-0.4 eV energy loss, however, shows an internal structure that differs markedly between the two maps. At the π * resonance (285.6 eV) the feature is split into two (as can also be seen in the linecuts in Fig. 3), while at the σ * resonance (291.8 eV) a single broader asymmetric peak is visible.

To conserve momentum, a single-phonon excitation in a zone-centre spectrum can include only a zone-centre mode in the final state. The final states of a twophonon feature, however, can conceivably consist of pairs of opposite-momenta phonons at any point in the Brillouin zone. The split feature at the π * resonance has peak energies of 0.32 eV and 0.39 eV, which correspond to double the energies of the TO mode at K and Γ respectively (see the Supplemental Material). Meanwhile, the broad feature at the σ * resonance spans double the full bandwidth of the TO mode. Phonons of different momenta therefore appear to be excited at each of the resonances. ωi and wavevector ki excites an electron from the 1s to the π * or σ * bands, leaving a core hole. In the intermediate state, the altered charge density perturbs the positions of surrounding ions and generates nq phonon modes. The excited electron then relaxes via the emission of a photon of energy ω f , leaving n q phonons with total energy (ωiω f ) in the final state (the zone-center TO mode is depicted [START_REF] Mohr | Phonon dispersion of graphite by inelastic x-ray scattering[END_REF]). (b) Electronic band structure of graphite from Ref. [START_REF] Marinopoulos | Ab initio study of the optical absorption and wavevector-dependent dielectric response of graphite[END_REF], with the low-energy π * bands shaded in orange and the high-energy σ * bands in blue.

III. ELECTRON-PHONON INTERACTIONS IN RIXS

We have identified the modes that contribute to the RIXS spectra, but why do phonons of different momenta appear at the π * and σ * resonances? To answer this, we need to understand how phonons are generated in a RIXS measurement.

Figure 2(a) shows a schematic of the RIXS process at the carbon K edge. An incident x-ray photon excites an electron from the 1s to the π * or σ * bands, depending on the energy, where it experiences a partially-screened potential from the resulting core hole. The altered charge density of this intermediate state perturbs the positions of the surrounding ions, generating multiple phonons. Finally, the excited electron relaxes to fill the core hole through the emission of a photon, with the difference in incident and scattered photon energies equal to the energy of the phonons remaining in the final state.

The composition of the intermediate electronic state can therefore have a significant impact on the phonon modes that are excited. As both resonances involve the same core-hole, this cannot explain the difference between the RIXS spectra. We instead focus on the band structure of the excited electron, shown in Fig. 2(b). For electrons excited into the low-energy π * states, the steepness of the bands confines them to a small region around the Dirac points at K. These states can only interact with phonons with momenta close to Γ, which scatter electrons within a single Dirac cone, or K, which scatter electrons between different Dirac cones. By contrast, the high-energy σ * bands are flatter and electrons excited into them can span the Brillouin zone, suggesting interactions with a wide range of phonon momenta. This provides an intuitive understanding of our earlier assignment of the RIXS features, where the split two-phonon feature at the π * resonance is dominated by phonons at Γ and K, while the single broad feature at the σ * resonance has contributions from across the TO mode dispersion.

To put this on a more rigorous footing, we need a theoretical model that can account for interactions of the full excitonic intermediate state with multiple phonons of different momenta. Within the exact-diagonalization approaches proposed thus far [START_REF] Johnston | Electronlattice interactions strongly renormalize the chargetransfer energy in the spin-chain cuprate Li2CuO2[END_REF][START_REF] Ament | Determining the electron-phonon coupling strength from resonant inelastic x-ray scattering at transition metal Ledges[END_REF][START_REF] Devereaux | Directly characterizing the relative strength and momentum dependence of electron-phonon coupling using resonant inelastic xray scattering[END_REF], this requires a computationally-intractable expansion of the Hilbert space. We therefore employed a Green's-function approach which implicitly accounts for vibronic effects in the intermediate state (full details can be found in Appendix B). We address e-ph interactions within the framework of a linked-cluster (cumulant) expansion using a second-order Fan-Migdal-type self-energy in the time domain. RIXS cross-sections were obtained by considering the non-diagonal elements of an exciton Green's function, as described elsewhere [START_REF] Geondzhian | Demonstration of resonant inelastic x-ray scattering as a probe of excitonphonon coupling[END_REF]. As well as fully accounting for the phonon dispersion, this approach allows a proper treatment of mode mixing in the intermediate state, which leads to significant renormalization of the spectral features.

In order to focus on the effect of the interactionstrength dispersion, we treat both π * and σ * intermediate states as localised excitons, which is a good approximation over the intermediate-state lifetime of graphite [START_REF] Brühwiler | π * and σ * excitons in c 1s absorption of graphite[END_REF]. The intermediate-state lifetime itself was determined using the core-hole lifetime from x-ray photoemission spectroscopy [START_REF] Sette | Lifetime and screening of the C 1s photoemission in graphite[END_REF], adjusted to mimic the presence of the extra electron (see the Supplemental Material). Given the weak inter-layer coupling of graphite, the optical phonon modes of interest have a minimal dependence on the out-FIG. 3. e-ph interactions for low-energy π * excitations. (a) Modeled momentum dependence of G through the 2D Brillouin zone (orange), compared to g determined by IXS (blue) [5], Raman spectroscopy (yellow) [6] and time-resolved ARPES (purple) [START_REF] Na | Direct determination of mode-projected electron-phonon coupling in the time domain[END_REF]. (b) Normalized experimental (black points with error bars) and calculated (orange line, including experimental broadening) RIXS spectra on resonance (285.6 eV). The contribution from each phonon generation process is indicated by the vertical orange lines, grouped above by the number of phonons. (c) Incident-energy dependence of the experimental (black points with error bars) and calculated (red to orange lines) spectra, plotted alongside the XAS with arrows indicating the energies.

of-plane momentum [START_REF] Mohr | Phonon dispersion of graphite by inelastic x-ray scattering[END_REF] and we can safely restrict our analysis to the 2D-projected Brillouin zone. We used the phonon dispersion calculated with density functional perturbation theory, which is in good agreement with IXS measurements [START_REF] Mohr | Phonon dispersion of graphite by inelastic x-ray scattering[END_REF]. The only free parameter of the model is then the momentum-dependent e-ph interaction strength, G(q), which can be determined by fitting the experimental RIXS spectra.

IV. ELECTRON-PHONON INTERACTIONS NEAR THE FERMI LEVEL

We turn first to our data around the π * resonance. From our earlier assignment of the RIXS features, it appears that significant contributions come only from the TO mode at the K point, and degenerate TO/LO modes at the Γ point (see the Supplemental Material). We therefore include only the TO mode dispersion in our modelling, with a minimal set of momentum points: the Γ and K points, the experimental in-plane momentum transfer q exp ≈ (0.08, 0.00) r.l.u., and Kq exp . Inclusion of the latter two effectively captures the width over which G(q) falls off away from the high-symmetry points. The intensity of the one-phonon peak is sensitive to the fact that q exp = Γ, but for the multi-phonon features it is sufficient to approximate the total momentum transfer as zero.

Figure 3(b) shows a fit of this model to the spectrum on resonance, with the corresponding momentum dependence of G(q) plotted in Fig. 3(a). We see good agreement between the experimental and calculated spectra, including the striking split two-phonon feature, despite our simple parameterization of G(q). This confirms that the splitting arises due to the dominant contributions from pairs of phonons at K and Γ, indicated by the vertical lines at 0.32 eV and 0.39 eV respectively. Intensity above 0.45 eV is attributed to overlapping three-and four-phonon features, which are also consistent with contributions coming from pairs of ±K-momenta phonons with additional Γ modes.

We further tested our model through its ability to reproduce spectra with the incident x-ray energy detuned from the maximum of the XAS. Figure 3(c) shows the energy dependence of the experimental and calculated spectra through the resonance. As well as the persistent splitting of the two-phonon feature below the resonance, our model captures the changing relative intensities of the one-and two-phonon features. The non-monotonic energy dependence of these intensities, including the offset of the maximum of the one-phonon peak intensity from zero-detuning [START_REF] Feng | Disparate exciton-phonon couplings for zonecenter and boundary phonons in solid-state graphite[END_REF], arises from a combination of strong e-ph interaction strength and long core-hole lifetime [START_REF] Geondzhian | Generalization of the Franck-Condon model for phonon excitations by resonant inelastic x-ray scattering[END_REF]. We emphasise that the only parameter being varied between the spectra in Fig. 3(c) is the incident energy, providing a strong validation of our model.

As the incident energy is raised through the upper tail of the XAS peak, the composition of the intermediate electronic state will change (picking up contributions from the M point, see Fig. 2(b)] [START_REF] Olovsson | Vibrational effects in x-ray absorption spectra of two-dimensional layered materials[END_REF]. This in turn allows scattering by phonons of different momenta, changing the momentum-dependence of G and explaining the discrepancies between the experimental and calculated spectra above 285.6 eV. Agreement can be improved by allowing G(q) to vary slightly with incident energy (see the Supplemental Material).

We can now compare our results with those from other techniques. Alongside our best-fit G(q), Fig. 3(a) also shows the e-ph interaction strength determined by IXS [5] and Raman spectroscopy [6], as well as a novel time-resolved ARPES technique [START_REF] Na | Direct determination of mode-projected electron-phonon coupling in the time domain[END_REF]. We see reasonable agreement between all the techniques. Note, however, that we use a lower-case g for the coupling probed by these other techniques, to highlight that they do not involve a core hole. As discussed above, RIXS measures a related but inequivalent coupling, G, of phonons to an excitonic intermediate state consisting of both the excited electron and core hole [START_REF] Geondzhian | Demonstration of resonant inelastic x-ray scattering as a probe of excitonphonon coupling[END_REF]. The concept of exciton-phonon coupling is not new, being relevant to Raman and other optical spectroscopies [START_REF] Jiang | Excitonphoton, exciton-phonon matrix elements, and resonant raman intensity of single-wall carbon nanotubes[END_REF][START_REF] Antonius | Theory of the exciton-phonon coupling[END_REF] (the only difference here being that the hole is often in the valence band rather than core level). Recent work in these fields has shown that, under certain approximations, the two quantities G and g can be related by [START_REF] Chen | Excitonphonon interaction and relaxation times from first principles[END_REF]

G ν nm (Q, q) = k   h,c,c A m(Q+q) * h(k),c(k+Q+q) A n(Q) h(k),c (k+Q) g ν c c (k + Q, q) - c,h,h A m(Q+q) * h(k-q),c(k+Q) A n(Q) h (k),c(k+Q) g ν hh (k -Q, q)   . (1)
This expression describes an exciton in initial state n, with centre-of-mass momentum Q, and wave-function

|S n = hc A n hc |hc
, where h (c) indicate hole (conduction) states. The exciton is scattered by a phonon in mode ν with momentum q, to a final state m with momentum Q + q and wave-function

|S m = h c A m h c |h c .
It can be seen that Eq. 1 is separated into two terms, the first accounting for electron-phonon scattering with the hole-state unaffected, and the second for hole-phonon scattering with a passive electron, each weighted by the coefficients of the excitonic wavefunction. In cases where the core hole is strongly screened, RIXS reflects the single-particle coupling of the excited electron. When the core hole is weakly screened, RIXS accesses exciton-phonon coupling, which is the quantity of interest for the many out-of-equilibrium situations discussed in the Introduction.

From Fig. 3(a), we can also see how the various techniques probe different components of g ν cc (Q, q) and G ν nm (Q, q). Standard ARPES resolves g by electronic momentum Q for bands c below the Fermi energy, but integrates over phonon modes ν and momenta q [START_REF] S.-I. Tanaka | An investigation of electron-phonon coupling via phonon dispersion measurements in graphite using angle-resolved photoelectron spectroscopy[END_REF]. By contrast, IXS [5] and neutron scattering [4] resolve the phonon momentum q throughout the Brillouin zone for all modes ν, but average over the equilibrium electronic state. Raman spectroscopy [6] and the novel timeresolved ARPES [START_REF] Na | Direct determination of mode-projected electron-phonon coupling in the time domain[END_REF] technique can access low-energy electronic excitations, but are restricted to particular phonon momenta (q = Γ and K for graphite). By contrast, we have shown that RIXS can resolve G by electronic state n, phonon mode ν and phonon momentum q, integrating only over the electronic momentum Q.

By modelling the contributions to the multi-phonon features, we have therefore determined the interactions between the π * electronic state and TO phonons throughout the Brillouin zone, all from a single zone-center spectrum. This method will be invaluable, as the soft x-ray edges at which phonons are enhanced (due to the long intermediate-state lifetimes) and energy resolutions are maximised, also suffer from restricted momentum transfer [see Fig. 1(c)].

V. ELECTRON-PHONON INTERACTIONS AWAY FROM THE FERMI LEVEL

Having successfully reproduced the π * spectra, we now come to the high-energy σ * states. Based on our assignment of the RIXS features, we again restrict our analysis to the TO mode (see the Supplemental Material). In this case, however, the broad profile of the two-phonon peak suggests G(q) is finite over a wide range of momenta. In order to constrain our parameter space, we therefore assume a linear dependence between the high-symmetry points Γ, K, and M . Given the more uniform variation of G(q) around Γ, we can also approximate the experimental momentum transfer as zero for the whole spectrum.

Figure 4(a) shows the best-fit dispersion of G(q), which we find to be significant at all the high-symmetry points. The resulting spectrum on resonance is shown in Fig. 4(b), and again we see that a simple parameterization of reciprocal space is able to describe the experimental data well. Unlike the splitting seen at the π * resonance, here our model reproduces the subtle broadening and asymmetry of the two-phonon feature, confirming that these characteristics arise from the varying contributions of pairs of opposite-momenta phonons which disperse gradually through the Brillouin zone.

In order to determine the sensitivity of our fit to the values at the high-symmetry points, we also show spectra in Fig. 4(b) calculated with G(M ) = 0 and G(K) = 0 (G(Γ) must be finite to produce a significant one-phonon peak). We can see that the calculated spectra are most sensitive to the value of G(K), which strongly influences the intensities of all the multi-phonon peaks. While the value of G(M ) has little impact on the three-, four-and five-phonon peaks, we see that a finite value on the order of G(Γ) is required to accurately capture the intensity and shape of the two-phonon peak. We also show in the Supplemental Material how a previous RIXS measurement of G(q) at the σ * resonance of graphite [START_REF] Feng | Disparate exciton-phonon couplings for zonecenter and boundary phonons in solid-state graphite[END_REF], which assumed zero coupling between the high-symmetry points, is unable to reproduce the multi-phonon features.

Our best-fit G(q) is also able to reproduce the changing intensities of the features on detuning below 291.8 eV, as shown in Fig. 4(c). As for the π * states, we expect agreement between the model and experiment to degrade above the resonance due to changes in the intermediate electronic state. Further, just above the σ * 1 resonance, that we are focused on, lies the σ * 2 resonance [see Fig. 2(b)], which arises from zero-point vibrations [START_REF] Olovsson | Vibrational effects in x-ray absorption spectra of two-dimensional layered materials[END_REF] that are not included in our modelling.

VI. CONCLUSION

We have demonstrated that, through careful choice of the incident photon energy, RIXS is able to probe the interactions between phonons and electrons both close to, and away from, the Fermi level. Our results for graphite reveal that the momentum dependence of the interaction strength, G(q), can be highly distinct in these two regimes: for low-energy π * states it is concentrated in small regions around the Γ and K points, while for high-energy σ * states it is significant across the Brillouin zone. This establishes RIXS as a uniquely versatile probe of e-ph interactions in crystalline materials, applicable to studies of out-of-equilibrium processes alongside ordered phases in correlated systems. Combined with the recent development of ultrafast time-resolved RIXS at xray free-electron lasers [START_REF] Dean | Ultrafast energyand momentum-resolved dynamics of magnetic correlations in the photo-doped mott insulator Sr2IrO4[END_REF][START_REF] Cao | Ultrafast dynamics of spin and orbital correlations in quantum materials: an energyand momentum-resolved perspective[END_REF][START_REF] Mitrano | Probing light-driven quantum materials with ultrafast resonant inelastic x-ray scattering[END_REF][START_REF] Mazzone | Laser-induced transient magnons in Sr3Ir2O7 throughout the brillouin zone[END_REF], our methodology should prove particularly powerful for determining e-ph interactions in pumped systems.

We find that contributions to the multi-phonon features in a RIXS spectrum can come from phonons away from the experimental momentum transfer, allowing us to constrain G(q) throughout the Brillouin zone from a single zone-center spectrum. This is particularly apparent at the π * resonance, where the rapid variation of G with momentum produces a split two-phonon feature.

These contributions are also important, however, for the σ * states with more uniform G(q). Here, the gradually dispersing phonons cause a broadening and asymmetry of the multi-phonon features, as has been seen in other studies [START_REF] Lee | Role of lattice coupling in establishing elec-tronic and magnetic properties in quasi-one-dimensional cuprates[END_REF][START_REF] Vale | High-resolution resonant inelastic x-ray scattering study of the electronphonon coupling in honeycomb α -Li2IrO3[END_REF], alongside shifts in their peak energies that could be mistaken for anharmonic effects [START_REF] Vale | High-resolution resonant inelastic x-ray scattering study of the electronphonon coupling in honeycomb α -Li2IrO3[END_REF]. As we show in Appendix B, mode mixing in the intermediate state results in the multi-phonon contributions affecting the intensity and resonance behavior of all the phonon features in the spectra, including the one-phonon peak. Neglecting this effect is likely to lead to qualitative errors in the determination of G(q) [START_REF] Feng | Disparate exciton-phonon couplings for zonecenter and boundary phonons in solid-state graphite[END_REF]. These insights from graphite will be important when analysing the phonon excitations in RIXS measurements of more complex materials, such as the much-studied cuprates [14-16, 19, 21, 23, 25, 26].

Our results highlight the importance of developing a robust theoretical description of the complex RIXS process in order to analyse experimental data. The Green's function approach that we employ offers a practical way to incorporate the phonon and e-ph interaction strength dispersions without the need for summation over all possible intermediate states. There are many avenues that remain to be explored, including the effects of itinerancy [START_REF] Bieniasz | Beyond the single-site approximation modeling of electron-phonon coupling effects on resonant inelastic x-ray scattering spectra[END_REF] and changes to the potential energy surface [START_REF] Geondzhian | Demonstration of resonant inelastic x-ray scattering as a probe of excitonphonon coupling[END_REF] in the intermediate state, and how the core hole affects the coupling probed by RIXS compared to other techniques [START_REF] Geondzhian | Demonstration of resonant inelastic x-ray scattering as a probe of excitonphonon coupling[END_REF]. The availability of theoretical methods to treat these effects will only become more pressing as the energy resolution of RIXS improves, and it is used to probe phonons in an ever increasing range of topical materials.

To model our rich experimental data we employed a phenomenological Green's-function formulation of the RIXS cross-section. In principle, final states involving both phonons and low-energy electronic excitations are allowed, the latter of which would result in a continuum of excitations. As such a continuum is not seen above the background in our data, however, we neglect these weak higher-order process and restrict the final states to those with only phonon excitations present.

We account for electron-electron interactions by constructing an effective quasi-particle exciton Green's function. Due to the different time scales, it is reasonable to separately consider electron-lattice interactions in the presence of an exciton which propagates through the material [START_REF] Geondzhian | Demonstration of resonant inelastic x-ray scattering as a probe of excitonphonon coupling[END_REF][START_REF] Antonius | Theory of the exciton-phonon coupling[END_REF]. Expanding the interacting exciton Green's function in a time series, we find two important types of diagrams: closed-loop diagrams where a phonon gets created and destroyed by the same exciton, and those where a phonon connects two excitons and gives rise to the final state phonon population. The former affect the intensities of the RIXS features non-linearly, and can be interpreted as contributions from intermediate state phonons. Evaluating these contributions has proven to be very computationally demanding in exact diagonalization approaches. Here, we account for them using the cumulant representation of the interacting Green's function, which reduces computational cost. The second type of diagram will be treated explicitly up to the number of phonon satellites visible in the experiment.

To simplify the notation, we omit electron-photon matrix elements which affect only the absolute intensity and not the characteristic shape of the phonon features, and drop the index of the excitonic state. The energy-and momentum-dependent cross-section then can be written as

σ(ω i , ω f , q) ≈ - n |Λ n (ω i , q)| 2 D n (ω i -ω f , q) (B1)
where Λ n (ω, q) is the complex off-diagonal part of the exciton Green's function, representing the scattering of an exciton by n phonons of total momentum q = k i -k f , and the presence of phonons in the final state is reflected by the many-body Green's function D n (ω, q). In the limit of infinite phonon life-time D n (ω, q) = -1 π δ(ω -i=1,n ω i )δ(q -i=1,n q i ). Here, we account for phonon-phonon interactions by attributing a small lifetime broadening to the phonon propagator.

Previous work has suggested an approximate form of Λ in the limits of weak [START_REF] Devereaux | Directly characterizing the relative strength and momentum dependence of electron-phonon coupling using resonant inelastic xray scattering[END_REF] and intermediate [START_REF] Geondzhian | Demonstration of resonant inelastic x-ray scattering as a probe of excitonphonon coupling[END_REF] e-ph interactions. For the purpose of this work, however, we use a closed-form solution for Λ in the limit of local excitons coupled linearly to phonons. Although the cross-section in Eq. (B1) is given in the energy domain, from here on we use the time-dependent form. Focusing on the characteristic behavior of the phonon satellites with multiple momentum contributions, we have

Λ n (t, q) = Λ 0 (t) qi...qn iG(q i ) t 0 D > (τ, q i )dτ / √ n!.
(B2) The diagonal part of the exciton Green's function is denoted Λ 0 , and is the same dressed propagator that would appear in the absorption process (I XAS ∼ F {Λ 0 }). The bracketed part of Eq. (B2) represents the phonon contributions to the scattering diagram, with the product over n phonons conserving momentum, q = q 1 +...+q n . G(q) is the momentum-dependant electron-phonon interaction strength, and the greater phonon propagator in the zerotemperature limit is simply D > (t, q) = -i b q (t)b † q (0) = iθ(t)e -iω(q)t . Thus, the non-diagonal part of the exciton Green's function differs from the diagonal part by the interaction term with a certain number of final state phonons. Despite the simple form, in the limit of small excitons Eq. (B2) accounts for all types of diagrams including those with vertex corrections.

The exciton propagator dressed by the exciton-phonon interaction can be expressed as Λ 0 (t) = L(t)e C(t) , where L is the bare exciton Green's function. We accounted for the exciton lifetime by including a damping factor e -γt/2 in L. Note that this expression contains half of the inverse lifetime, γ/2. We used a value γ/2 = 0.15 eV, slightly adjusted from the core-hole lifetime as determined by x-ray photoelectron spectroscopy mea-surements (0.1 eV) [START_REF] Sette | Lifetime and screening of the C 1s photoemission in graphite[END_REF] to account for the presence of the extra electron (see the Supplemental Material).

Finally, the cumulant function to second order in the exciton-phonon interaction can be written

C(t) = [L(t)] -1 t 0 t 0 L(t -τ 1 )Σ FM (τ 1 -τ 2 )L(τ 2 )dτ 1 dτ 2 (B3) where the Fan-Migdal self-energy is Σ FM = -i q G(q) 2 L(t)D(t, q).
Substituting the exciton and phonon Green's functions, and taking the time integrals in Eq. (B3) analytically [START_REF] Langreth | Singularities in the x-ray spectra of metals[END_REF], we get C(t) = q G(q) 2 /(ω(q) 2 N )(e -iω(q)t + iω(q)t -1). It is important to notice that the summation runs over the whole Brillouin zone without restriction, since the total momentum transfer of the contributing diagrams is zero. From the form of Eqs. (B2) and (B3), it is clear that for any final state configuration there are contributions to the cross-section from phonons of all momenta. Therefore, even the intensity of the one-phonon peak depends on G(q) throughout the Brillouin zone, not just at the experimental q point.

To evaluate Eq. (B3), we used the TO phonon dispersion calculated with density functional perturbation theory. Brillouin zone integrations were performed using a reduced set of weighted momentum points which reflect the symmetries of the graphite structure [START_REF] Monkhorst | Special points for brillouin-zone integrations[END_REF]. Having calculated all the contributions to Eq. (B2) in the time domain, we can then perform a Fourier transform at the given incident photon energy and plug it into Eq. (B1) to obtain the RIXS cross-section.

I. SELF-ABSORPTION CORRECTION

Self-absorption -the re-absorption by the sample of previously scattered photons -depends on the energies and angles of both the incident and outgoing x-rays, and must therefore be corrected to reliably compare spectra. We follow previous resonant inelastic x-ray scattering (RIXS) studies [1][2][3] and define a self-absorption correction factor

C , (2Θ, θ, ω i , ω o ) = 1 1 + u(2Θ, θ)t , (ω i , ω o ) . (1)
In this expression, u = sin θ/ sin (2Θθ) is a geometrical term that depends on the scattering angle 2Θ = 154°, and sample angle θ = 20°(90°) for the π * (σ * ) spectra. The other term

t , (ω i , ω o ) = α 0 + α (ω o ) α 0 + α (ω i ) (2)
depends on α 0 , the absorption coefficient before the resonance (taken at 284 eV for the π * resonance and 289 eV for the σ * resonance), and α (ω), the resonant absorption coefficient for x-rays with energy ω and polarization . The absorption coefficients can be obtained from XAS measured in total electron yield, shown in Fig. S1(a) and S2(a). With our incident x-rays polarized in the (h, 0, l) scattering plane (π polarization), we measure the absorption for c ( ⊥ c) using x-rays incident normal (grazing) to the sample surface. From these, we calculate the absorption coefficients as

α π (ω i ) = XAS(ω i , ⊥ c) sin 2 θ + XAS(ω i , c) cos 2 θ, α π (ω o ) = XAS(ω o , ⊥ c) sin 2 (2Θ -θ) + XAS(ω o , c) cos 2 (2Θ -θ), α σ (ω o ) = XAS(ω o , ⊥ c). (3)
The multi-phonon peak. As shown in Fig. S1(c) and S2(c), after normalisation the difference between the corrected spectra is minimal over the energy range of interest, and does not significantly affect the fits to our model. We therefore correct all spectra that the outgoing are π polarised.

II. ASSIGNMENT OF PHONON MODES

In the main text, we assign the contributions to the RIXS spectra based on the known phonon dispersion of graphite [4]. Here we demonstrate this process for the two-phonon features. Fig. S3 shows spectra near the π * and σ * resonances plotted alongside the phonon dispersion, scaled such that the phonon energies line up with twice the energy loss of the spectra. From this, it is clear that the two peaks in the split feature of the π * spectrum at twice the energies of transverse optical (TO) phonons at Γ and K, while the broad asymmetric feature of the σ * spectrum spans twice the bandwidth of the TO mode. Taking into the 47 meV experimental energy resolution, it can be seen that contributions from the acoustic (ZA, TA, and LA) and out-of-plane optical (ZO) modes are negligible, and the longitudinal optical (LO) mode could only have significant contributions when it is nearly degenerate with the TO mode.

III. INTERMEDIATE-STATE LIFETIME

To model our RIXS spectra we adjusted the intermediate-state lifetime from the value γ/2 = 0.1 eV (half-width at half-maximum of the Lorentzian component of the spectral peak) measured by x-ray photoemission spectroscopy [5] due to the presence of the extra excited electron in RIXS. Figure S4 compares best-fit RIXS spectra at the π * and σ * resonances calculated using the bare core-hole lifetime 0.1 eV, to the adjusted value 0.15 eV. We can see that the adjusted lifetime gives better agreement with both experimental spectra, especially for the high-order peaks at the σ * resonance. This adjusted lifetime was used for all calculations shown in the main manuscript, successfully reproducing the energy dependence of the experimental spectra below both resonances.

IV. INDEPENDENT FITTING OF THE ENERGY-DEPENDENT SPECTRA

The calculated spectra shown in Fig. 3(c) of the main manuscript are the result of a simultaneous fit to the experimental spectra around the π * resonance, such that the only parameter of the model that is varied between the spectra is the incident energy.

We have also fitted the spectra at each energy independently to see how this affects the extracted electron-phonon (e-ph) interaction strengths. Figure S5 shows a comparison of the simultaneous and independent fits, with the resulting values of G(q) listed in Table S1.

Both sets of calculated spectra have an acceptable level of agreement with the experimental S1. G(q) extracted from simultaneous and independent fits to the RIXS spectra around the π * resonance, at the momenta Γ, q exp ≈ (0.08, 0.00), Kq exp , and K.

G (eV) q = Γ q = q exp q = Kq exp q = K Simultaneous fit 0. data given the noise level, but the independent fits do offer better agreement especially for the spectrum at 286 eV. As we note in the main text, we expect composition of the intermediate electronic state and therefore the momentum dependence of G to vary in the high-energy tail of the XAS peak, providing a natural explanation for this. While Table S1 shows some variation in the extracted interaction strengths, the general trend of G(q)

through the Brillouin zone -with G(K) > G(Γ) >> G(Kq exp ) ≥ G(q exp ) -is valid irrespective of the fitting procedure.

V. COMPARISON WITH PREVIOUS RIXS MEASUREMENTS ON GRAPHITE

Feng et al. recently presented lower-resolution RIXS measurements around the σ * resonance of graphite [3], interpreting their data with reference to Raman spectroscopy results.

Their analysis implicitly assumed, however, a coupling of phonons to electrons near the Fermi level, leading them to constrain G(q) to be non-zero only at the Γ and K points.

While this is reasonably accurate for the π * resonance, we show in the main text that RIXS at the σ * resonance probes the interaction of phonons with elections away from the Fermi level, whose strength is significant throughout the Brillouin zone. As well as overlooking the possibilities afforded by tuning the incident photon energy, this serves as a warning that careful consideration of the states involved in the RIXS process is essential for correct interpretation of experimental spectra, especially if the energy resolution is lacking.

We compare our best-fit G(q) at the σ * resonance (blue), to that obtained by Feng et al. intensities depend on the ratio of G to lifetime). We therefore see that, even for the relatively flat optical modes studied here, accounting for intermediate-state mixing strongly affects the momentum-dependence of G extracted from the data.

VI. MOMENTUM DEPENDENCE OF THE RIXS SPECTRA

The maximum momentum transfer is limited by the low energy of the carbon K edge.

Additionally, the RIXS cross-section falls off rapidly for sample angles away from θ = 0°( grazing incidence) at the π * resonance, and away from θ = 90°(normal incidence) at the σ * resonance. In the main manuscript, we show how these restrictions can be circumvented by analysing the contributions to the multi-phonon peaks, which can come from phonons throughout the Brillouin zone.

Figure S7 shows the momentum-dependence of the RIXS spectra at the σ * resonance along the (h, 0, l) direction. The in-plane momentum was varied by changing the sample angle θ while keeping the spectrometer angle 2Θ( = 2 × θ) fixed, so the total momentum transfer q ∝ 4h 2 /(3a 2 ) + l 2 /c 2 is constant. While the overall intensity falls off rapidly with decreasing θ, it can be seen that the relative intensities of the phonon peaks change only minimally, supporting our conclusion that G(q) varies gradually away from the Γ point.

The low intensity of the phonon features at the π * resonance, along with the strong falloff in intensity with increasing θ, precluded measurement of a momentum-dependence with reasonable statistics.

FIG. 1 .

 1 FIG. 1. Contrasting RIXS response at the π * and σ * resonances. (a) RIXS maps around both resonances, with XAS above for reference (the dashed vertical lines in the XAS mark the peaks of the resonances at 285.6 eV and 298.1 eV respectively). Both maps show a series of phonon features above the elastic line, with the contrasting resonance behavior most apparent for the two-phonon feature between 0.3-0.4 eV. The intense feature above ∼1 eV in the π * map arises from electronic transitions. (b) Phonon dispersion of graphite from Ref. [33], with the transverse (T), longitudinal (L) and outof-plane (Z) acoustic (A) and optical (O) modes marked. The TO mode with significant coupling is marked in red. (c) The 2D-projected Brillouin zone of graphite, with the momenta accessible in our RIXS measurement indicated by the turquoise circle, and high symmetry positions Γ, K, and M marked.

FIG. 2 .

 2 FIG.2. Phonon generation in RIXS. (a) Schematic of the RIXS process at the carbon K edge. An incident photon of energy ωi and wavevector ki excites an electron from the 1s to the π * or σ * bands, leaving a core hole. In the intermediate state, the altered charge density perturbs the positions of surrounding ions and generates nq phonon modes. The excited electron then relaxes via the emission of a photon of energy ω f , leaving n q phonons with total energy (ωiω f ) in the final state (the zone-center TO mode is depicted[START_REF] Mohr | Phonon dispersion of graphite by inelastic x-ray scattering[END_REF]). (b) Electronic band structure of graphite from Ref.[START_REF] Marinopoulos | Ab initio study of the optical absorption and wavevector-dependent dielectric response of graphite[END_REF], with the low-energy π * bands shaded in orange and the high-energy σ * bands in blue.

FIG. 4 .

 4 FIG. 4. e-ph interactions for high-energy σ * excitations. (a) Best-fit momentum dependence of G (blue), and that with G(M ) = 0 (orange), and G(K) = 0 (yellow). (b) Normalized experimental (black points with error bars) and calculated [blue, orange, and yellow lines, corresponding to those in (a)] RIXS spectra on resonance (291.8 eV). The contribution from each n-phonon process is labeled above. (c) Incident-energy dependence of the experimental (black points with error bars) and calculated (blue to green lines) spectra, plotted alongside the XAS with arrows indicating the energies.

FIG

  FIG. S1. (a) XAS around the π * resonance with incident x-rays polarized parallel ( c) and perpendicular ( ⊥ c) to the c axis. (b) Self-absorption correction factors in the ππ (solid) and πσ (dashed) polarization channels, with colours corresponding to the incident energies labelled in (c). (c) Uncorrected (black) and corrected (red to yellow) RIXS spectra in the two polarization channels, normalized to the intensity of the two-phonon peak. The corrections introduce changes below the noise level to the normalized spectra.

  FIG. S2. (a) XAS around the σ * resonance with incident x-rays polarized parallel ( c) and perpendicular ( ⊥ c) to the c axis. In this work we focus on the σ * 1 resonance at 291.8 eV and ignore the broad σ * 2 resonance above which arises from zero-point vibrations. (b) Self-absorption correction factors in the ππ (solid) and πσ (dashed) polarization channels, with colours corresponding to the incident energies labelled in (c). (c) Uncorrected (black) and corrected (blue to green) RIXS spectra in the two polarization channels, normalized to the intensity of the one-phonon peak.

  photons will depend on the symmetry of the phonon modes being excited, and was not determined in our experiment. As our model only predicts the relative intensities of the phonon peaks, however, we can normalize the spectra to the intensity of the strongest

FIG. S4 .

 S4 FIG. S4. Comparison of best-fit RIXS spectra at (a) the π * and (b) the σ * resonance with half inverse intermediate-state lifetimes of γ/2 = 0.15 eV (orange/blue) and 0.10 eV (yellow).

FIG. S5 .

 S5 FIG. S5. Comparison of independent (solid colored lines) and simultaneous (dashed coloured lines) fits to the experimental RIXS spectra (black points with error bars) around the π * resonance, plotted alongside the XAS with arrows indicating the energies.

  FIG. S6. (a)Our best-fit G(q) for σ * electrons (blue), compared to that obtained by Feng et al.[3] (orange) and our best-fit with G constrained to be finite only at q = Γ and K (yellow). (b) Comparison of calculated RIXS spectra at the σ * resonance, with colours corresponding to those in (a) and the experimental spectrum shown as black points with error bars. All spectra are calculated with an intermediate-state lifetime γ/2 = 0.15 eV.

(

  FIG. S7. Momentum dependence along (h, 0, l) of the experimental RIXS spectra at the σ * resonance.

ACKNOWLEDGMENTS

We thank Matteo Calandra for insightful discussions. C.D.D. was supported by the Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in the Advanced Characterisation of Materials under Grant No. EP/L015277/1. Work at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. A.G. was supported by a National Research Foundation of Korea grant funded by the Ministry of Science and ICT No. 2016K1A4A4A01922028, the U.S. National Science Foundation under Grant No. DMR-1904716, and acknowledges Research Computing at Arizona State University for providing high performance computing resources. K.G. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences as part of the Computational Materials Science Program. R.B.J. acknowledges the support of the European Commission "GREENDIAMOND" project through the H2020 Large Project under Grant No. SEP-210184415. Work at UCL was supported by the EPSRC under Grants No. EP/N027671/1, EP/N034872/1, EP/H020055/1, and EP/N004159/1. We acknowledge SOLEIL for provision of synchrotron radiation facilities at the SEXTANTS beamline under proposals 20150221 and 20130732, and the Diamond Light Source for time on beamline I21 under proposal MM22695.

Appendix A: Experimental details

Our carbon K-edge RIXS measurements were performed at beamline I21 of the Diamond Light Source. A natural graphite single crystal was mounted such that the scattering plane is (h, 0, l), and cleaved in vacuum. We chose an energy resolution of 47 meV (determined from scattering from amorphous carbon tape mounted next to the sample) to balance throughput and resolution. All RIXS spectra were taken with linear horizontal (π) incident x-ray polarisation, at a temperature of 20 K, and with a counting time of 20 min. X-ray absorption spectra in total electron yield were obtained by measuring the sample drain current. Data for energies around the π * resonance were taken with grazing-incident x-rays (θ = 20°) while data around the σ * resonance were taken with normal-incident x-rays (θ = 90°) in order to maximise the intensity of the phonon features. All data have been corrected for incident flux and self-absorption (see the Supplemental Material).

At these soft x-ray energies, the momentum transfer is confined to a small in-plane region of < 0.1 r.l.u. around the zone center (see Fig. 1c). As we used a fixed scattering angle 2Θ = 154°, the momentum transfer varies slightly with incident energy. Around the π * resonance the momentum transfer q = (0.0802 ± 0.0004, 0, 0.164 ± 0.001), while around the σ * resonance q = (0.0220 ± 0.0002, 0, 0.300 ± 0.002).