Arkadiusz Frąckowiak
email: arkadiusz.frackowiak@ifmpan.poznan.pl

Roman Świetlik

Lucky Maulana

Di Liu

Martin Dressel

Olivier Jeannin

Marc Fourmigué

Pressure-Induced Neutral-Ionic Phase Transition in the Mixed-Stack 2:1 Charge-Transfer Complex (EDT-TTF-I 2 ) 2 TCNQF

We explored the neutral-ionic phase transition in (EDT-TTF-I 2 ) 2 TCNQF by measuring the electrical conductivity as a function of temperature and hydrostatic pressure (up to 11.8 kbar). At about 4.3 kbar the compound undergoes the transition to quasi-ionic phase at the temperature about 100 K; the transition temperature increases as the pressure grows. We propose a pressuretemperature phase diagram, discussed in comparison to analogous data obtained for mixed-stack 1:1 charge-transfer complexes which also undergo a neutral-ionic transition. We suggest that the transition smearing can be related with the fact that in this 2:1 complex, not only the Madelung energy but also the ionization potential of donors grow as the crystal lattice contracts on temperature decreasing.

Introduction

Temperature-induced neutral-ionic (N-I) transition is a very rare phenomenon, observed in organic 1:1 charge-transfer (CT) complexes composed of alternating π-electron donor (D) and π-electron acceptor (A) molecules arranged in mixed one-dimensional stacks (...DADADA…).

Uniqueness of this phenomenon is that in the simplified picture a single material undergoes transition between van der Waals molecular crystal (neutral phase) and ionic crystal (ionic phase). Abrupt or continuous change of the degree of CT between molecules (ρ), is observed when the difference between the ionization energy of donor (I D ) and the electron affinity of acceptor (E A ) becomes comparable with the increment of electrostatic Madelung energy (M) of the ionized lattice, induced by thermal contraction of the crystal (I D -E A ~ M). The application of hydrostatic pressure also yields a lattice contraction and modifies intermolecular interactions and thus Madelung energy; therefore the N-I transition can be also induced by pressure. The change of ρ is usually accompanied by dimerization of the regular mixed-stack chains, therefore the phase transition is to be considered as both N-I transition and Peierls distortion. Because of sizable interactions between frontier orbitals in the DA stacks, the neutral (ρ=0) and ionic (ρ=1e) configurations are always mixed to some extent. Consequently, in real crystals the N-I transition rather takes place between quasi-neutral (ρ<0.5e) and quasi-ionic (ρ>0.5e) ground states [START_REF] Torrance | Discovery of a Neutral-to-Ionic Phase Transition in Organic Materials[END_REF][START_REF] Torrance | Anomalous Nature of Neutral-to-Ionic Phase Transition in Tetrathiafulvalene-Chloranie[END_REF][START_REF] Girlando | Charge fluctuations and electronphonon coupling in organic charge-transfer salts with neutral-ionic and Peierls transitions[END_REF][START_REF] Horiuchi | Chemical approach to neutral-ionic valence instability, quantum phase transition, and relaxor ferroelectricity in organic charge-transfer complexes[END_REF][START_REF] Delchiaro | Towards first-principles prediction of valence instabilities in mixed stack charge-transfer crystals[END_REF] .

The first example of the N-I transition was reported in the 1:1 complex formed by the donor tetrathiafulvalene (TTF) and the acceptor chloranil (CA), and this complex was the most thoroughly and widely studied and serves as the prototype of this ordering phenomenon [START_REF] Torrance | Discovery of a Neutral-to-Ionic Phase Transition in Organic Materials[END_REF][START_REF] Torrance | Anomalous Nature of Neutral-to-Ionic Phase Transition in Tetrathiafulvalene-Chloranie[END_REF][START_REF] Girlando | Charge fluctuations and electronphonon coupling in organic charge-transfer salts with neutral-ionic and Peierls transitions[END_REF][START_REF] Horiuchi | Chemical approach to neutral-ionic valence instability, quantum phase transition, and relaxor ferroelectricity in organic charge-transfer complexes[END_REF][START_REF] Horiuchi | Ionic versus Electronic Ferroelectricity in DonorAcceptor Molecular Sequences[END_REF][START_REF] Tomić | Ferroelectricity in molecular solids: a review of electrodynamic properties[END_REF] . On temperature decreasing, at T=81 K the TTF-CA crystal shows an abrupt transition from the quasi-neutral (ρ=0.3e) to quasi-ionic (ρ=0.6e) phase. The ionicity change is accompanied by a dimerization of the mixed stacks and appearance of electronic ferroelectricity or antiferroelectricity of the complex [START_REF] Horiuchi | Ionic versus Electronic Ferroelectricity in DonorAcceptor Molecular Sequences[END_REF][START_REF] Tomić | Ferroelectricity in molecular solids: a review of electrodynamic properties[END_REF] . Moreover, the N-I transition in TTF-CA can be also realized by pressure [START_REF] Tokura | Pressure-induced neutral-to-ionic phase transition in TTF-p-chloranil studied by infrared vibrational spectrocopy[END_REF][START_REF] Mitani | Electric conductivity and phase diagram of a mixed-stack charge-transfer crystal: Tetrathiafulvalene-pchloranil[END_REF][START_REF] Hanfland | Raman study of the pressureinduced neutral-to-ionic transition in tetrathiafulvalene chloranil[END_REF][START_REF] Dengl | Evolution of ferroelectricity in tetrathiafulvalene-p-chloranil as a function of pressure and temperature[END_REF] or light [START_REF] Koshihara | Photoinduced valence instability in the organic molecular compound tetrathiafulvalene-p-chloranil (TTF-CA)[END_REF][START_REF] Okamoto | Photoinduced phase transition in tetrathiafulvalene-p-chloranil observed in femtosecond reflection spectroscopy[END_REF][START_REF] Uemura | Direct Detection of the Ultrafast Response of Charges and Molecules in the Photoinduced Neutral-to-Ionic Transition of the Organic Tetrathiafulvalene-p-Chloranil Solid[END_REF] too. The temperature-induced N-I transition was also discovered in some other organic CT complexes of 1:1 stoichiometry with mixed stacks under ambient or moderate pressures [START_REF] Girlando | Charge fluctuations and electronphonon coupling in organic charge-transfer salts with neutral-ionic and Peierls transitions[END_REF][START_REF] Horiuchi | Chemical approach to neutral-ionic valence instability, quantum phase transition, and relaxor ferroelectricity in organic charge-transfer complexes[END_REF] . There are also reports about the N-I transition at very high pressures (p>2GPa) in a few 2:1 complexes with a local DAD trimer structure [START_REF] Matsuzaki | Pressure-induced neutral-ionic transition in a 2:1 charge transfer crystal of tetrathiafulvalene and iodanil[END_REF][START_REF] Sadohara | Neutral-Ionic Transition Of Charge Transfer Complexes Of TTF And Tetrahalo-p-Benzoquinones[END_REF] but these studies were not followed up in more detail. Since the most distinct N-I transitions were A c c e p t e d m a n u s c r i p t observed in 1:1 complexes, the vast majority of the investigations were performed for these molecular systems.

The N-I transitions in molecular DA complexes were studied using a variety of experimental techniques. Very powerful methods are IR and Raman spectroscopies since some intra-molecular vibrations are strongly charge sensitive; in a good approximation their frequency depends linearly on charge density (ρ) on molecules [START_REF] Torrance | Anomalous Nature of Neutral-to-Ionic Phase Transition in Tetrathiafulvalene-Chloranie[END_REF][START_REF] Hanfland | Raman study of the pressureinduced neutral-to-ionic transition in tetrathiafulvalene chloranil[END_REF][START_REF] Dengl | Evolution of ferroelectricity in tetrathiafulvalene-p-chloranil as a function of pressure and temperature[END_REF][START_REF] Matsuzaki | Pressure-induced neutral-ionic transition in a 2:1 charge transfer crystal of tetrathiafulvalene and iodanil[END_REF][START_REF] Sadohara | Neutral-Ionic Transition Of Charge Transfer Complexes Of TTF And Tetrahalo-p-Benzoquinones[END_REF][START_REF] Ranzieri | Temperature-induced valence and structural instability in DMTTF-CA: Single-crystal Raman and infrared measurements[END_REF] . The ρ estimated from vibrational spectra is more accurate than that one determined from crystallographic data and agree very well with NMR data. The N-I transition induced by both pressure and temperature can be also observed with the help of electrical conductivity methods -this technique clearly shows modifications of both ionicity and crystal lattice that are usually detected as resistivity anomalies [START_REF] Horiuchi | Chemical approach to neutral-ionic valence instability, quantum phase transition, and relaxor ferroelectricity in organic charge-transfer complexes[END_REF][START_REF] Mitani | Electric conductivity and phase diagram of a mixed-stack charge-transfer crystal: Tetrathiafulvalene-pchloranil[END_REF][START_REF] Hasegawa | Neutral-ionic phase transition of (BEDT-TTF)(ClMeTCNQ) under pressure[END_REF][START_REF] Horiuchi | Quantum Phase Transition in Organic Charge-Transfer Complexes[END_REF] . Because of the mixed-stack structure the crystals exhibit semiconducting properties. A characteristic feature is that the activation energy of conductivity estimated from Arrhenius plot (usually smaller than 0.15 eV) is much less than the optical energy gap (about 0.5 eV). This observation gives evidence that lower-energy excitations are responsible for the conductivity. These excitations were identified as N-I domain walls in the neutral phase [START_REF] Nagaosa | Theory of Neutral-Ionic Transition in Organic Crystals. III. Effect of the Electron-Lattice Interaction[END_REF][START_REF] Nagaosa | Theory of Neutral-Ionic Transition in Organic Crystals. IV. Phenomenological Viewpoint[END_REF] and spin solitons in the ionic phase [START_REF] Nagaosa | Theory of Neutral-Ionic Transition in Organic Crystals. III. Effect of the Electron-Lattice Interaction[END_REF][START_REF] Mitani | Soliton Formation at the Neutral-to-Ionic Phase Transition in the Mixed-Stack Charge-Transfer Crystal Tetrathiafulvalene-p-Chloranil[END_REF] .

Recently, a temperature-induced N-I conversion was discovered in the mixed-stack complex (EDT-TTF-I 2 ) 2 TCNQF (where EDT-TTF-I 2 is diiodoethylenedithio-tetrathiafulvalene and TCNQF is monofluoro-tetracyano-p-quinodimethane) [START_REF] Lieffrig | Charge-Assisted Halogen Bonding: Donor-Acceptor Complexes with Variable Ionicity[END_REF][START_REF] Frąckowiak | Temperature-Induced Neutral-Ionic Phase Transition in the (EDT-TTF-I 2 ) 2 TCNQF Mixed-Stack Charge-Transfer Salt[END_REF] . To the best of our knowledge, this is the first observation of such effect in a 2:1 complex under ambient pressure. The compound crystallizes in the triclinic system (space group

) with one TCNQF acceptor on inversion 1 P center and one EDT-TTF-I 2 donor in general position in the unit cell, hence the 2:1 D 2 A stoichiometry. The molecules are arranged in mixed stacks …ADDADDA… and the parallel stacks interact to each other to form layers. The neighboring layers are connected by highly directional C-IN≡C halogen bonds. Both crystallographic and spectroscopic (IR and Raman) studies showed that the degree of CT (ρ) between donors and acceptor is continuously increasing from about zero at room temperature to about 1e at T = 8 K (with a regime change at about 100 K), i.e. the complex undergoes a N-I conversion. It is remarkable that the spectroscopic data indicate that ferroelectric domains, created in stacks by the N-I transition, fluctuate between A c c e p t e d m a n u s c r i p t opposite directions [START_REF] Frąckowiak | Temperature-Induced Neutral-Ionic Phase Transition in the (EDT-TTF-I 2 ) 2 TCNQF Mixed-Stack Charge-Transfer Salt[END_REF] . It was suggested that such effect should be common for the 2:1 mixed-stack complexes.

Here, we present DC electrical conductivity measurements of (EDT-TTF-I 2 ) 2 TCNQF single crystals as a function of both hydrostatic pressure and temperature. Our goal is to investigate the N-I transformation in this mixed stack 2:1 complex. We show that the enhancement of ρ has an influence on the crystal conductivity. Our results will be discussed in comparison with analogous data obtained for 1:1 complexes.

Experimental Section

Single crystals of (EDT-TTF-I 2 ) 2 TCNQF were prepared following the procedure described previously [START_REF] Lieffrig | Charge-Assisted Halogen Bonding: Donor-Acceptor Complexes with Variable Ionicity[END_REF] . The DC resistance in the direction of stacking axis was measured by the four-probe method at ambient pressure and by the two-probe method at high hydrostatic pressures. The electrodes were prepared with a gold paste. In order to measure the electrical resistivity under pressure we used a clamp-type cupper beryllium cell and Daphne 7373 oil as a pressure transmitting medium. After clamping the cell at a selected pressure (up to 11.8 kbar) at room temperature the sample resistivity was measured as a function of temperature. Due to thermal contraction the actual pressure is significantly reduced (ca. 1.5 kbar) on temperature decreasing down to 10 K, therefore suitable pressure corrections were done taking into account the data for Daphne 7373 oil [START_REF] Yokogawa | Solidification of High-Pressure Medium Daphne 7373[END_REF][START_REF] Tateiwa | Evaluations of pressure-transmitting media for cryogenic experiments with diamond anvil cell[END_REF][START_REF] Beyer | Piston pressure cell for low-temperature infrared investigations[END_REF][START_REF] Rösslhuber | Pressure cell for radio-frequency dielectric measurements at low temperatures[END_REF] . We estimate that after corrections the real pressure at low temperatures is known within an accuracy ± 0.5 kbar. For each mounted sample all the pressure and temperature measurements were performed without opening the pressure cell.

Results

The room-temperature electrical resistivity of (EDT-TTF-I 2 ) 2 TCNQF at ambient pressure (p=0) is rather high (about 5 ×10 3 Ωcm), which is typical for mixed stack complexes in the neutral phase. Temperature dependence of the resistivity is displayed in Fig. 1. At 100 K and 90 K the resistivity shows small and sharp jumps (they are better seen in Fig. 2). Analogous kinklike resistivity anomalies were also observed for other complexes and correlated with N-I transition. In the case of TTF-CA the kink is about one order of magnitude and occurs at the N-I transition temperature [START_REF] Mitani | Electric conductivity and phase diagram of a mixed-stack charge-transfer crystal: Tetrathiafulvalene-pchloranil[END_REF] but in the case of (EDT-TTF-I 2 ) 2 TCNQF both kinks are much smaller.

The kink at 100 K well corresponds with the temperature of regime change indicated by IR and

A c c e p t e d m a n u s c r i p t

Raman data [START_REF] Frąckowiak | Temperature-Induced Neutral-Ionic Phase Transition in the (EDT-TTF-I 2 ) 2 TCNQF Mixed-Stack Charge-Transfer Salt[END_REF] . Evidently, at 100 K and 90 K the crystal undergoes an abrupt but small modification which can be related to a weak change of ionicity and/or deformation of the crystal lattice. Modifications of intermolecular interactions (e.g. halogen bond interaction) should be also taken into account. On closer examination one can also distinguish three other smaller resistivity anomalies at temperatures 96, 92 and 91 K (see Fig. 2). The resistivity reduction seen at the lowest temperatures in Fig. 1 is an artifact due to limits of the sensitivity of our apparatus. No thermal hysteresis was observed on temperature cycling (see Fig. S1 -Supplementary Material).

The ambient pressure resistivity data show that the complex is a semiconductor with two activation energies: 109 and 86 meV above and below 200 K, respectively. In our previous studies for crystals from other batches, these energies were determined as equal to 108 and 72 meV, respectively [START_REF] Lieffrig | Charge-Assisted Halogen Bonding: Donor-Acceptor Complexes with Variable Ionicity[END_REF] . Temperature dependence of the crystal resistivity under various pressures is displayed in Fig. 2. The application of pressure strongly decreases the resistivity. It should be noted that the pressure applied at room temperature, indicated in the right part of the figure, is reduced on temperature lowering due to thermal contraction of the pressure medium. Nevertheless, in Fig. 2 we show original results, without any suitable corrections of pressure which should be done for lower temperatures. These data are analogous to those reported for the 1:1 complexes [START_REF] Mitani | Electric conductivity and phase diagram of a mixed-stack charge-transfer crystal: Tetrathiafulvalene-pchloranil[END_REF][START_REF] Hasegawa | Neutral-ionic phase transition of (BEDT-TTF)(ClMeTCNQ) under pressure[END_REF][START_REF] Horiuchi | Quantum Phase Transition in Organic Charge-Transfer Complexes[END_REF] . The resistivity jumps -those observed at 100 and 90 K for p=0 (Fig. 1) -are also seen at higher pressures. The pressure dependence of the resistivity measured at the 90 K dip is displayed in Fig. S2 (Supplementary Material). Temperatures of both jumps are nearly pressure independent (Fig. 3), whereas in TTF-CA the jump of resistivity shifts towards higher temperatures when pressure grows [START_REF] Mitani | Electric conductivity and phase diagram of a mixed-stack charge-transfer crystal: Tetrathiafulvalene-pchloranil[END_REF] . The jump at about 100 K disappears at about 4 kbar but the jump at 90 K is seen up to the highest pressures; the other smaller jumps at 96, 92 and 91 K also disappear above 4 kbar. The pressure dependence of the electrical resistivity at various temperatures, after correcting the thermal modification of pressure, is shown in Fig. 4. For each temperature, as pressure grows the resistivity considerably decreases by several orders of magnitude and then is nearly pressure independent. Such regime change is qualitatively similar to the 1:1 complexes with N-I transitions, where the turning point was associated with the pressure-induced N-I transition [START_REF] Horiuchi | Chemical approach to neutral-ionic valence instability, quantum phase transition, and relaxor ferroelectricity in organic charge-transfer complexes[END_REF][START_REF] Mitani | Electric conductivity and phase diagram of a mixed-stack charge-transfer crystal: Tetrathiafulvalene-pchloranil[END_REF][START_REF] Tokura | Pressure-induced neutral-to-ionic phase transition in TTF-p-chloranil studied by infrared vibrational spectrocopy[END_REF][START_REF] Horiuchi | Quantum Phase Transition in Organic Charge-Transfer Complexes[END_REF] . Assuming a similar interpretation, one can conclude that for lower temperatures the pressure, which is necessary to transform our samples into the ionic phase, is strongly reduced. At about 100 K the N-I transition occurs at p=4.3 kbar and above 200 K the pressure required to transform the crystal into the ionic phase is nearly temperature independent and equals to approximately 7.7 kbar. Since for each temperature the regime change is rather smeared, the pressure, at which the change occurs, was evaluated by crossing two lines, approximating the dependence below and above the turning point (see Fig. 4). After estimating the transition to the ionic phase by this method, a pressure-temperature phase diagram of (EDT-TTF-I 2 ) 2 TCNQF can be composed (Fig. 5). We believe that this diagram shows the transition between quasi-neutral and quasi-ionic phases of the complex; the transition is only observed above 4.3 kbar. Qualitatively the diagram is similar to analogous diagrams for the 1:1 salts 9,18 , i.e. when pressure grows the transition temperature between both phases also grows. However, it should be stressed that it is not clear whether there is a back bending of the line separating the phases (at about 7.7 kbar), since this bending lies within the limits of experimental errors. It is reasonable to assume that above the pressure 7.7 kbar the crystal is in the quasi-ionic phase in the whole temperature range. Our previous spectroscopic studies under ambient pressure revealed that the degree of charge transfer continuously grows on cooling down to the lowest temperatures, reaching at T = 10 K a value of about 1e [START_REF] Lieffrig | Charge-Assisted Halogen Bonding: Donor-Acceptor Complexes with Variable Ionicity[END_REF][START_REF] Frąckowiak | Temperature-Induced Neutral-Ionic Phase Transition in the (EDT-TTF-I 2 ) 2 TCNQF Mixed-Stack Charge-Transfer Salt[END_REF] , whereas the present pressure studies show a more or less abrupt transition. This difference can be either related to a specific crystal modifications induced by pressure or the difference between properties of bulk material and microscopic observations by spectroscopic techniques. Pressure dependence of the activation energies (E a ) as determined from the resistivity curves (Arrhenius plots) is displayed in Fig. 6. Fig. S3 (Supplementary Material) illustrates the method of E a determination. As mentioned above, there are two activation energies at ambient pressure: 109 meV (E a1 ) and 86 meV (E a2 ) above and below 200 K, respectively. Both energies depend on the applied pressure. According to the phase diagram shown in Fig. 5, the crystal is in the quasi-neutral phase up to about 4.3 kbar in the whole studied temperature range. Therefore, we assume that both E a1 and E a2 are related to the quasi-neutral phase and when pressure grows up to 4.3 kbar these energies increase. For high pressures above about 7.7 kbar the crystal is in the quasi-ionic phase and only one E a = 125 meV is seen in the whole studied temperature range.

In the intermediate pressure range p = 4.3 -7.7 kbar (see Fig. 5), depending on temperature the crystal is either in quasi-neutral or quasi-ionic phase, therefore two E a were estimated corresponding to these phases. When pressure increases the E a1 of quasi-neutral phase grows and then in the intermediate range above 4.3 kbar decreases, whereas the E a2 of quasi-neutral phase increases and then at 4.3 kbar -it is remarkable -smoothly passes into the E a of quasi-ionic phase. For this reason one can assume that the E a2 in quasi-neutral phase below 4.3 kbar is most probably related with the quasi-ionic state. It is important that all the observed activation energies for neutral and ionic states (86-170 meV) are much smaller than the CT excitation energy at ambient pressure (about 500 meV).

Discussion

In 1:1 mixed-stack complexes with N-I transitions, the stacks in the neutral phase are usually uniform and they are distorted in the ionic phase (dimerization). However, in the 2:1 charge-transfer complex (EDT-TTF-I 2 ) 2 TCNQF, the stacks are non-uniform already in the quasineutral phase at ambient temperature and the molecules are arranged in centrosymmetric units …DAD…. (Fig. 7a). Evidently, donor-donor interactions between trimers (Fig. 7c) are different in comparison with intra-trimer donor-acceptor interactions (Fig. 7d). It is remarkable that in (EDT-TTF-I 2 ) 2 TCNQF the molecular long axes are parallel, whereas a characteristic feature of other mixed-stack complexes is that the axes are not parallel [START_REF] Mayerle | Mixed-stack complexes of tetrathiafulvalene. The structures of the charge-transfer complexes of TTF with chloranil and fluoranil[END_REF][START_REF] Hasegawa | Mixed-stack organic charge-transfer complexes with intercolumnar networks[END_REF] . The mixed-stacks form molecular slabs within the ab plane and the neighboring slabs are connected by halogen bonds (Fig. 7b). This interaction is essentially electrostatic in nature and does not include any sizeable overlap interaction which could link the slabs electronically. In a way it can be compared with the weak C-H‧‧‧O hydrogen bonds observed in TTF-CA between the hydrogen atoms of TTF and A c c e p t e d m a n u s c r i p t the oxygen atoms of CA, which become activated upon the N-I transition [START_REF] Batail | Structural Characterization of the Neutral-Ionic Phase Transition in Tetrathiafulvalene-Chloranil: Evidence for C-H•••0 Hydrogen Bonding[END_REF] . For this reason, we assume that the crystal is quasi-one-dimensional.

There is another important difference between 1:1 and 2:1 complexes with mixed-stacks showing the N-I transition. In the 1:1 complexes the transition occurs when the difference between the ionization potential of donor (I D ) and the electron affinity of acceptor (E A ) becomes comparable with the electrostatic Madelung energy (M). Both I D and E A are molecular constants, therefore they are independent of temperature but only the energy M is modified due to thermal contraction of the crystal lattice. However, in the case of 2:1 complex one should take into account that the parameter I D can be also dependent on temperature.

At the N-I transition an electron is transferred from the donor HOMO orbital of EDT-TTF-I 2 to the acceptor LUMO orbital of TCNQF. However, due to interaction between neighboring trimers the HOMO of donors splits into bonding and antibonding orbitals. An electron is transferred from the bonding orbital, whose energy is lower in comparison with the undistorted monomer, therefore the value of I D increases. When the crystal lattice contracts, the intra-stack transfer integrals grow, both within and between trimers, therefore both M and I D increase. It is therefore possible that the interplay between M and I D is a reason why -at ambient pressure in (EDT-TTF-I 2 ) 2 TCNQF -the N-I transformation is continuous, as shown by spectroscopic data [START_REF] Lieffrig | Charge-Assisted Halogen Bonding: Donor-Acceptor Complexes with Variable Ionicity[END_REF][START_REF] Frąckowiak | Temperature-Induced Neutral-Ionic Phase Transition in the (EDT-TTF-I 2 ) 2 TCNQF Mixed-Stack Charge-Transfer Salt[END_REF] . Such competition between M and I D should be considered as a specific feature of the 2:1 complexes. On the other hand, there exist a disorder in the (EDT-TTF-I 2 ) 2 TCNQF crystal due to the fluorine atom occupancy in TCNQF which is 50 %. This disorder may have also an influence on smearing of the N-I transformation over broad temperature range.

Previous IR studies showed that the charge distribution among EDT-TTF-I 2 molecules is not uniform below T  100 K [START_REF] Frąckowiak | Temperature-Induced Neutral-Ionic Phase Transition in the (EDT-TTF-I 2 ) 2 TCNQF Mixed-Stack Charge-Transfer Salt[END_REF] . On the other hand, the X-ray data provided evidence that the TCNQF molecules are located on the inversion center down to T=20 K without any indication of structural phase transition [START_REF] Lieffrig | Charge-Assisted Halogen Bonding: Donor-Acceptor Complexes with Variable Ionicity[END_REF] . That is why, on basis of these observations it was suggested that the polarization of donor-acceptor domains in mixed-stacks fluctuates between opposite directions. Within this scenario, taking into account the trimers DAD, one can speculate that the polarization of trimers fluctuates, i.e. the charge density fluctuates between donor molecules. The amplitude of fluctuations increases on cooling down and thus the degree of dynamical trimer distortion grows, as indicated by the intensity increase of IR features related with the electron-molecular A c c e p t e d m a n u s c r i p t vibration coupling effect [START_REF] Lieffrig | Charge-Assisted Halogen Bonding: Donor-Acceptor Complexes with Variable Ionicity[END_REF][START_REF] Frąckowiak | Temperature-Induced Neutral-Ionic Phase Transition in the (EDT-TTF-I 2 ) 2 TCNQF Mixed-Stack Charge-Transfer Salt[END_REF] . Obviously, the fluctuations in neighboring trimers should be strongly correlated, yielding thus short-living ferroelectric domains with polarization fluctuating between opposite directions. The electrical conductivity of 1:1 mixed-stack complexes in the quasi-neutral phase is determined by the charged N-I domain wall (NIDW) dynamics [START_REF] Nagaosa | Theory of Neutral-Ionic Transition in Organic Crystals. III. Effect of the Electron-Lattice Interaction[END_REF][START_REF] Nagaosa | Theory of Neutral-Ionic Transition in Organic Crystals. IV. Phenomenological Viewpoint[END_REF] . The neutral and ionic states are almost degenerated, therefore ionic domains can be spontaneously created in neutral stacks.

Nevertheless, in the quasi-neutral phase the energy of ionic region per molecule is higher than that one of the neutral region, therefore the ionic domains are of limited size. Such ionic domains of limited size are regarded as NIDW pairs which are responsible for the charge transport.

Moreover, it was proposed that the charge transport is primarily determined by thermally activated motions of the existing NIDW pairs but not by the density of existing charge carriers [START_REF] Tokura | Domain-Wall Dynamics in Organic Charge-Transfer Compounds with One-Dimensional Ferroelectricity[END_REF] .

Similarly, in our compound it is reasonable to assume that both neutral and ionic states coexist in the quasi-neutral phase and the charged NIDW are responsible for the electrical conductivity. Within this picture the remarkable decrease of resistivity under pressure can be attributed to an increase of the density of NIDW pairs (Fig. 4), similarly as in 1:1 complexes [START_REF] Horiuchi | Chemical approach to neutral-ionic valence instability, quantum phase transition, and relaxor ferroelectricity in organic charge-transfer complexes[END_REF][START_REF] Mitani | Electric conductivity and phase diagram of a mixed-stack charge-transfer crystal: Tetrathiafulvalene-pchloranil[END_REF][START_REF] Hasegawa | Neutral-ionic phase transition of (BEDT-TTF)(ClMeTCNQ) under pressure[END_REF] . Due to the crystal contraction induced by pressure, the energy difference between neutral and ionic state decreases, therefore the density of ionic domains should grow. When pressure is high enough the saturation of the NIDW density occurs and the crystal is transformed from the mixed-state to a predominantly ionic state, that is observed from the saturation of the resistivity (Fig. 4). The turning point can be related to N-I transition, as in 1:1 complexes. Nevertheless, above the turning point, the resistivity of the 1:1 complexes increases, while in our salt it is more or less constant. On the other hand, in the quasi-ionic state the charge transport is mainly due to spin solitons which are kink-like defects created between domains of opposite polarization. There are two kinds of such solitons with spin "0" (neutral molecule D 0 or A 0 ) and spin "½" (charged molecule D + or A -) and they both can contribute to the electrical conductivity because in real crystal they bear a fractional charge [START_REF] Tokura | Domain-Wall Dynamics in Organic Charge-Transfer Compounds with One-Dimensional Ferroelectricity[END_REF] .

As mentioned above, the temperature variations of the electrical conductivity should be attributed to variations of the mobility of existing charged NIDW. Therefore, we can assume that in (EDT-TTF-I 2 ) 2 TCNQF the change of E a at ambient pressure at about 200 K is due to a change of the NIDW mobility. It is possible that the mobility change is due to modification of intermolecular interactions in the stacks, nevertheless in our compound the influence of intercolumnar halogen bonding should be also taken into account. When pressure increases up to about 4.3 kbar, the density of NIDW grows but their thermally activated motion can be hindered, therefore both E a1 and E a2 increase in the quasi-neutral phase (Fig. 6). In the quasi-ionic phase above about 7.7 kbar the spin solitons should be responsible for charge transport, as in 1:1 complexes [START_REF] Nagaosa | Theory of Neutral-Ionic Transition in Organic Crystals. III. Effect of the Electron-Lattice Interaction[END_REF][START_REF] Nagaosa | Theory of Neutral-Ionic Transition in Organic Crystals. IV. Phenomenological Viewpoint[END_REF] , therefore the E a in this phase is to be related to thermally activated motion of these charge carriers (E a is rather independent on pressure). Within this picture, in the intermediate pressure region (4.3-7.7 kbar), depending on temperature range, the charge transport should be realized either by the charged NIDW or by the spin solitons. As already noticed above, when pressure grows the activation energy E a2 of the quasi-neutral phase passes smoothly into the activation energy of the quasi-ionic phase. Such behavior could suggest that the E a2 corresponds to soliton transport in the whole pressure range. If it is true one could speculate that even at ambient pressure below about 200 K the charged solitons can also contribute to the electrical conductivity.

Conclusions

We have shown here that the hydrostatic pressure can induce a N-I transition in the 2:1 mixed-stack complex (EDT-TTF-I 2 ) 2 TCNQF. On basis of anomalies in the pressure-dependence of electrical resistivity the pressure-temperature phase diagram was determined. In comparison with TTF-CA -a prototypical material exhibiting the N-I transition -in our complex the transition is not so sharp and takes place rather smoothly. The transition smearing can be related with the fact that in the 2:1 complex not only the Madelung energy M but also the ionization potential of donors I D can grow as the crystal contracts on temperature decreasing. Furthermore, the charge transport mechanism in (EDT-TTF-I 2 ) 2 TCNQF appears to be similar as in 1:1 complexes, i.e. by charged NIDW pairs in the quasi-neutral phase and charged solitons in the quasi-ionic phase. However, our analysis of the resistivity activation energies for various pressures suggests that solitons might also contribute to charge transport below 200 K even at ambient pressure, what is reasonable if we take into account that in our compound the N-I transition undergoes gradually.

Supporting Information

Comparing of electrical resistivity of (EDT-TTF-I 2 ) 2 TCNQF measured upon cooling and heating, pressure dependence of resistivity of (EDT-TTF-I 2 ) 2 TCNQF at the dip at about 90 K, linear fits of electrical resistivity of (EDT-TTF-I 2 ) 2 TCNQF which were made to extract values of activation energies for selected values of hydrostatic pressure.

Fig. 1 .

 1 Fig. 1. Temperature dependence of the electrical resistivity of (EDT-TTF-I 2 ) 2 TCNQF determined by the four-probe method at ambient pressure (p=0).

  Fig. 2. Temperature dependent electrical resistivity of (EDT-TTF-I 2 ) 2 TCNQF measured by the two-probe method in a clamp-type high-pressure cell without the thermal modification of pressure -see text. Jumps of the electrical resistivity are indicated by arrows.

Fig. 3 .

 3 Fig. 3. Temperature of the resistivity jumps as a function of pressure (thermal modifications of pressure are corrected).

Fig. 4 .

 4 Fig. 4. Resistivity of (EDT-TTF-I 2 ) 2 TCNQF as a function of pressure at different temperatures as indicated (thermal change of pressure is corrected). The crossed dotted lines at 280 K illustrate the method used for estimation of the pressure of regime change.

Fig. 5 .

 5 Fig.5. Pressure-temperature phase diagram of (EDT-TTF-I 2 ) 2 TCNQF as derived from the electrical resistivity data (thermal change of pressure is corrected). The white area corresponds to a gradual neutral-ionic transformation.

Fig. 6 .

 6 Fig. 6. Pressure dependence of the activation energies of (EDT-TTF-I 2 ) 2 TCNQF determined in three regions for: the quasi-neutral phase (0 -4.3 kbar) two E a are observed in temperature ranges (300-200 K and 200-100 K); the intermediate phase (4.3-7.7 kbar), i.e. in quasi-neutral phase (upper points) and quasi-ionic phase (bottom points); the ionic phase (above 7.3 kbar) -see text.

Fig. 7 .

 7 Fig. 7. The (EDT-TTF-I 2 ) 2 TCNQF structure of the mixed-stacks with trimeric …DAD… units (a); halogen bonding between the stacks perpendicular to a-b plane (b); donor-donor overlap between neighboring trimers (c); intra-trimer donor-acceptor overlap (d).

  A c c e p t e d m a n u s c r i p t

A c c e p t e d m a n u s c r i p t 8

 8 i s t i v i t y ( Ωc m )T e m p e r a t u r e ( K )

  p e r a t u r e ( K ) P r e s s u r e ( k b a r )

  i v a t i o n E n e r g y ( m e V ) P r e s s u r e ( k b a r )

  A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p t

Acknowledgements

One of us (A. F.) is grateful for support from German Academic Exchange Service (DAAD). We are also very indebted to Alberto Girlando for fruitful discussion and Gabriele Untereiner for help in high-pressure setup preparation.


Romulus Breban

Electromagnetism from 5D gravity: Beyond the Maxwell equations

Ever with the work of Kaluza, it has been known that 4D Einstein-and Maxwell-type equations can emerge from the field equations of 5D gravity in Ricci-flat space-times, having a space-like Killing vector. We revisit these equations and compare them with the Maxwell equations and the Ohm's law. Although 5D gravity and traditional electromagnetic theory are mathematically related, a paradigm shift in traditional electromagnetic concepts is required for unification. First, both the Maxwell equations and the Ohm's law are found among the field equations for 5D gravity, which gives Ohm's law the status of a field equation. Second, the concept of 4-current density, bringing together electrical charge and current density, is not fundamental for the electromagnetic theory resulting from 5D gravity. Third, the 5D gravity field equations suggest a generalization for the model of the perfect conductor, to describe a broader range of electromagnetic phenomena and bypass the concept of 4-current density. To see these ideas at work in a simple context, we propose an amendment to the Maxwell equations, based on the field equations of 5D gravity, and discuss a few applications. Namely, we discuss the propagation of electromagnetic waves through an imperfect conductor and topics in electrostatics and magnetostatics, touching ground with the Hall effect and the phenomenon of persistent current.

Introduction

The history of the electromagnetic theory is marked by philosophical choices, standardizations and leaps in understanding [START_REF] Whittaker | A history of the theories of aether and electricity: From the age of Descartes to the close of the nineteenth century[END_REF]. These may be revisited with progress in physics, to shed new light on the exceptional body of work in electromagnetism. The core equations for explaining the phenomenology are the Maxwell equations, written in space and time. The electromagnetic field is defined using three spatial coordinates, x j (j, k, l, ... = 1, 2, 3), in 3D Euclidian geometry, while the time t stands only as a parameter. However, the very structure of the Maxwell equations defines a 4D Lorentzian geometry, where time appears as a coordinate, x 0 . In CGS units, the modern formulation of the Maxwell equations is (e.g., Refs. [2, Ch. 1] and [START_REF] Jackson | Classical Electrodynamics[END_REF]Ch. 6])

∇D = 4πρ, (1.1)
∇B = 0, (1.2)
∇ × E = - 1 c ∂B ∂t , (1.3)
∇ × H = 1 c ∂D ∂t + 4π c j, (1.4)
supplemented by constitutive/material equations

D = E + 4πP , (1.5
) B = H + 4πM , (1.6)
(n.b., in vacuum, P = M = 0) and Ohm's law j = ςE.

(1.7)

For mathematical convenience, the potentials A 0 and A are introduced for the electromagnetic field

E = -∇A 0 - 1 c ∂A ∂t , (1.8)
B = ∇ × A. (1.9)
A recent critique of the Maxwell equations can be found in Ref. [START_REF] Tran | Evidence for Maxwell's equations, fields, force laws and alternative theories of classical electrodynamics[END_REF]. We make two additional notes. First, the Maxwell equations require the concepts of both field and source. Briefly, source generates field; the observer can act directly on the source, but not on the field. It is conjectured that electrical charge in motion is the source of the electromagnetic field. Second, as evidenced by Eqs. (1.5)-(1.7), the Maxwell equations make explicit assumptions about the structure of matter. In fact, many concepts such as conductor, dielectric, electron, were developed in parallel with the equations for the electromagnetic field, itself. Electromagnetic phenomena are attributed to particles, crystalline/amorphous structures and domain dynamics of matter.

The relationship between electromagnetic phenomenology and the Maxwell equations remains complex; e.g., Ref. [START_REF] Kronmüller | Handbook of magnetism and advanced magnetic materials[END_REF]. Here, our discussion is limited to fields, sources and particle electromagnetism in vacuum.

Features of the Maxwell equations can be recovered from a 5D geometry without compact dimensions. In 1921, Kaluza [START_REF] Kaluza | Zum Unitätsproblem der Physik[END_REF] proposed to unify the description of electromagnetic field in vacuum, with that of the 4D gravitational field, using a 5D space-time geometry where the metric is independent of the fifth dimension, x 5 . Later, Wesson [START_REF] Wesson | Space-time-matter[END_REF] elaborated the work of Kaluza, inspired by progress in 4D gravitation theory, particularly resisting Klein's idea [START_REF] Klein | Quantentheorie und fünfdimensionale Relativitätstheorie[END_REF] that the fifth dimension be compact. Specifically, Wesson [START_REF] Wesson | Space-time-matter[END_REF] proposed an interpretation for the covariance breaking in the gravitational field variables, from five to four dimensions. We adopt the Kaluza-Wesson interpretation of induced matter, whereby sourceless 5D gravity can be regarded, in four dimensions, as electromagnetic field in vacuum, onto a space-time hosting 4D gravity. Thus, an empty (i.e., Ricci-flat) 5D space-time appears to contain active matter (i.e., sources of fields), when it is interpreted in four dimensions. Several authors published field equations resulting for 5D Ricci-flat space-times, which are compatible with the induced-matter interpretation [START_REF] Kaluza | Zum Unitätsproblem der Physik[END_REF][START_REF] Thiry | The equations of Kaluza's unified theory[END_REF][START_REF] Overduin | Kaluza-Klein gravity[END_REF][START_REF] Lessner | Unified field-theory on the basis of the projective theory of relativity[END_REF][START_REF] López | Einstein-Rosen solutions from Kaluza-Klein theory[END_REF]. However, covariance breaking of the 5D gravitational field, to obtain field equations of type Einstein and Maxwell, is not unique. There are many ways to parameterize the 5D metric and still obtain 4D field equations with desired properties. Here, we impose two requirements, which make the 5D field parameterization unique. First, the parameterization submits to the geometric picture of the foliation of the 5D metric along x 5 and, second, it provides an anomaly-free theory for single-particle propagation in 4D space-time, according to the principles of general relativity [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF][START_REF] Breban | A five-dimensional perspective on the Klein-Gordon equation[END_REF][START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF][START_REF] Breban | On spinless null propagation in five-dimensional space-times with approximate space-like Killing symmetry[END_REF].

Geometrically speaking, there is nothing peculiar about the fifth dimension. Rather than using an x 5independent 5D metric, we can use an x 0 -independent 5D metric, foliate along x 0 , and obtain Einsteinand Maxwell-type equations for 4D fields from the condition of 5D Ricci flatness; 1 It is claimed that timeindependent particle propagation can be interpreted within a statistical physics formalism [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF][START_REF] Breban | A five-dimensional perspective on the Klein-Gordon equation[END_REF][START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF]. However, the Maxwell-type equations resulting in this case are associated with a 4D Euclidian metric rather than a 4D Lorentzian metric and it remains unclear how they link to the current phenomenology of electromagnetism. An alternative is to maintain the parameterization induced by the foliation along x 5 , when discussing x 0independent physics, as it guarantees an interpretation for the field variables [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF][START_REF] Breban | A five-dimensional perspective on the Klein-Gordon equation[END_REF][START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF]. These topics remain for further work.

The paper is structured as follows. In Sec. 2, we briefly review the theory for 5D null-propagation of passive particles; i.e., material particles which are not sources of fields. 2 In Sec. 3, we explain how, in the case where the 5D metric is x 5 -independent, 5D physics can undergo covariance breaking, to be interpreted as 4D mechanics. Indeed, the resulting 4D theory of passive particle propagation is anomaly free, in full agreement to the traditional equations of 4D, classical and quantum, relativistic mechanics. The reader familiar with these developments may proceed directly to Sec. 4, where we discuss the field equations resulting in Ricci-flat 5D gravity, from the perspective of the Maxwell equations and Ohm's law. As with previous work [START_REF] Kaluza | Zum Unitätsproblem der Physik[END_REF][START_REF] Thiry | The equations of Kaluza's unified theory[END_REF][START_REF] Overduin | Kaluza-Klein gravity[END_REF][START_REF] Lessner | Unified field-theory on the basis of the projective theory of relativity[END_REF], we make the case that 5D gravity is a successful physical theory, in touch with experiment. However, the traditional electromagnetic theory can gain, as well, from the parallel with field equations for 5D gravity. In particular, it can gain sharper concepts and, perhaps, a richer basis, as phenomenology finds reason to be included with the fundamentals. These aspects and others are discussed in Secs. 4.1 and 5. In Sec. 6, we conclude our work.

5D physics for 4D formalisms

We postulate that reality is 5D, yet an observer perceives/acknowledges only 4 dimensions out of 5. For a complete physical picture or formalism, the observer relies on 4D geometry and an array of non-geometrical concepts. The physical pictures employed by a 4D observer result from different foliations of a 5D geometry [START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF]. It is postulated that, if the 5D geometry is independent of the fifth coordinate then the 5D physics appears as 4D mechanics, while if the 5D geometry is independent of time then the 5D physics appears as 4D statistical physics. Throughout the manuscript, we use bold face to indicate that some indices of the corresponding mathematical objects are suppressed, particularly for 3D vectors. We start with a 5D Lorentzian space-time, M (5) , with metric h M N (M, N, • • • = 0, 1, 2, 3, 5), having the signature (-, +, +, +, +), that is sufficiently smooth and subject to the principle of general covariance. We further consider that M (5) is Ricci flat; i.e., R

M N = 0 and the 5D gravity fields have no sources. This is just a minimalist choice, not required by the induced matter principle.

We assume that the observed particle is passive, not a source of field interacting with the space-time metric. Particle propagation can be described using suitable equations defined on (M (5) , h). We are particularly interested in null propagation, therefore 5D optics, defined in two flavors. First, we can define classical or geometrical optics, based on 5D null geodesics. Second, we can define quantum optics. A spinless formulation is provided by the 5D wave equation, where the wavefunction Ψ(•, •) may be regarded as the number of null paths between any two given points of M (5) . A spin-1/2 quantum optics requires a wavefunction Ψ, which is a pair of bispinors or a biquaternion [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF], a vector with two quaternionic components, at every point of M (5) . It can be argued that Ψ represents two tangent null vectors, one incoming and one outgoing, at every point of M (5) [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF]. The wavefunction equation is of Dirac-type, based on a fünfbein field3 and a special Clifford algebra; see Eq. ( 14) in Ref. [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF]. The focus is, thus, solving null-propagation problems for passive particles, given the 5D gravity field. 4 The propagation problems are invariant to conformal transformations of h M N , which cannot be fully recovered by observing only particles in null propagation. We add tilde to the metric symbol, hMN , to denote a conformally transformed metric h M N .

Among space-times suitable for applications are 5D black holes, where the null structure has been a topic of major interest [START_REF] Wald | General Relativity[END_REF][START_REF] Reall | Black Holes in Higher Dimensions[END_REF][START_REF] Frolov | Black holes, hidden symmetries, and complete integrability[END_REF][START_REF] Emparan | A rotating black ring solution in five dimensions[END_REF]. It is interesting to note that, while the event horizon of a 4D black hole is always of spherical topology, the event horizon of a 5D black hole can have spherical, toroidal, or other topologies [START_REF] Emparan | A rotating black ring solution in five dimensions[END_REF]. Here, we propose 4D interpretations of 5D space-times and passive-particle null-propagation in 5D space-times [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF][START_REF] Breban | A five-dimensional perspective on the Klein-Gordon equation[END_REF][START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF].

4D classical and quantum mechanics from 5D physics

Interpretation of the 5D geometry and dimensional reduction are greatly facilitated by symmetry. The isometries of (M (5) , h) form a Lie group, denoted Iso(M (5) , h). The generators of this group are called Killing vectors, ξ, satisfing the Killing equation ∂ M ξ N + ∂ N ξ M = 0. We select a nontrivial subgroup of Iso(M (5) , h), which we call the formalism isometry group, FIso(M (5) , h), and its elements, formalism isometries. Here, we consider that (M (5) , h) has only one formalism isometry: the translation along a space-like dimension, x 5 . 5 We foliate (M (5) , h) along the formalism Killing field, into 4D space-times, using the Arnowitt-Deser-Misner procedure [START_REF] Arnowitt | Republication of: The dynamics of general relativity[END_REF][START_REF] Misner | Gravitation[END_REF], and indicate the lapse and shift fields of the foliation in the 5D metric. In turn, this yields a geometrical parameterization of h M N , which we further write in adapted coordinates, so it does not depend on x 5 . We thus obtain

h M N = g µν N µ N ν N ρ N ρ + N 2 ⇔ h M N = g µν + N µ N ν /N 2 -N µ /N 2 -N ν /N 2 1/N 2 ,
where (M (4) , g) is the foliated 4D manifold and, N and N µ (µ, ν, ... = 0, 1, 2, 3), respectively, are the lapse and shift of the 4D foliation. We introduce physical notation

N µ ≡ - q c 2 A µ , N ≡ Φ -1 ,
where A µ represents the electromagnetic potential in CGS units; q is specific charge and c is the speed of light in vacuum. As it turns out, the 4D metric, which yields the well-known laws of 4D particle propagation, is not g µν , but gµν ≡ g µν /Φ 2 [START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF]. The 5D metric observed through particle propagation in 4D mechanics is hMN = h M N /Φ 2 ; this gives the second requirement for the parameterization of the 5D metric. We write the metric h M N using the fields that are observed in particle propagation; see Sec. 4. The 4D observer may perform any cylindrical transformation of coordinates, consisting of diffeomorphisms of (M (4) , g) leaving x 5 unchanged; i.e.,

x µ → F µ (x ν ), x 5 → x 5 .
(3.10)

Other coordinate transformations, not changing 4D physical observables, are gauge transformations of the shift of the foliation, N µ , or the electromagnetic potential, A µ

x µ → x µ , x 5 → x 5 -q c 2 A(x µ ), ⇔    gµν → gµν , A µ → A µ + ∂ µ A(x ν ), Φ → Φ, (3.11)
where A(x ν ) is the gauge function for A µ . It is unclear how the 4D observer would use other coordinate transformations and what is their physical interpretation within 4D mechanics. The symmetry of (M (5) , h) to translation along x 5 is key for describing 5D passive particles of null mass (i.e., 5D photons), in four dimensions, as 4D passive particles carrying mass and electrical charge. The 4D observer considers himself/herself uncharged, which leads to important choices in the process of dimensional reduction. For classical propagation, 5D null geodesics in (M (5) , h) are given 4D interpretation [START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF]. If the uncharged 4D observer adopts the path element of (M (4) , g) for the infinitesimal proper time, then 5D null geodesics are represented as 4D time-like Randers geodesics [START_REF] Randers | On an asymmetrical metric in the four-space of general relativity[END_REF][START_REF] Cheng | Finsler geometry: an approach via Randers spaces[END_REF] in (M (4) , g), where the eccentricity of the path element is determined by A µ . Furthermore, since the x 5 -translational symmetry yields a conserved quantity for 5D geodesic motion, we say that the 4D particle propagates with constant passive mass [START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF]. For quantum propagation, we perform Fourier transforms with respect with x 5 of the 5D wave equation and the 5D Dirac-type equation, which, in addition, requires a foliated fünfbein field [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF]. 6 It is further postulated that the coordinate x 5 is conjugated with the inverse Compton wavelength λ -1 = mc/ [START_REF] Breban | The 4D Dirac equation in five dimensions[END_REF][START_REF] Breban | A five-dimensional perspective on the Klein-Gordon equation[END_REF]. In this fashion, 5D photons are interpreted as 4D on-shell massive quantum particles.

A priori, there is no reason for the 5D space-time be symmetric, in which case, we do not offer a physical picture for the 5D geometry. However, if the 5D space-time is nearly symmetric, then a interpretation can be constructed based on the alleged symmetry and a top-down hierarchy of physical concepts. The first order concepts are those of the physical picture assuming that the symmetry is exact. They provide a satisfactory description for particle propagation in manifold fields, for a small enough proper-time interval and space-time neighborhood. Next to the first order physical concepts, second order concepts can be added for improving the description of particle propagation, so it holds for longer proper-time intervals and larger space-time neighborhoods. In general, the physical concepts of second order rely on those of first order. Therefore, the hierarchical physical interpretation is not covariant. Reference [START_REF] Breban | On spinless null propagation in five-dimensional space-times with approximate space-like Killing symmetry[END_REF] demonstrates how particle propagation in a 5D space-time with approximate x 5 -translation symmetry can be given a physical picture. In the first order, we use 4D mechanics [START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF]. For the second order, we invoke additional phenomena such as Stokes friction and non-hermitian quantum decoherence [START_REF] Breban | On spinless null propagation in five-dimensional space-times with approximate space-like Killing symmetry[END_REF].

Field equations in 4D mechanics

Several authors [START_REF] Thiry | The equations of Kaluza's unified theory[END_REF][START_REF] Overduin | Kaluza-Klein gravity[END_REF][START_REF] Lessner | Unified field-theory on the basis of the projective theory of relativity[END_REF][START_REF] López | Einstein-Rosen solutions from Kaluza-Klein theory[END_REF] published the field equations for x 5 -independent 5D gravity, R

M N = 0, in the parametrization given by g µν , A µ and Φ. We performed an additional conformal transformation and rewrote the equations for 5D gravity in the fields gµν , A µ and Φ, which have more physical content because gµν is the 4D metric observed for 4D passive particle propagation [START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF] R [START_REF] Tran | Evidence for Maxwell's equations, fields, force laws and alternative theories of classical electrodynamics[END_REF] µν -

1 2 gµν R(4) = q 2 2c 4 T EM µν + T Φ µν , (4.12)
˜ µ Fµν = -3 Fµν ˜ µ ln Φ, (4.13)

1 Φ ˜ µ ˜ µ Φ = q 2 4c 4 Fµν F µν + 1 6 T Φ µν gµν , (4.14)
where R(4) µν and ˜ µ are, respectively, the 4D Ricci tensor and the Levi-Civita connection of the metric gµν . Furthermore, Fµν ≡

F µν = ∂ µ A ν -∂ ν A µ and T EM µν = gνρ Fµσ F σρ + 1 4 gµν Fρσ F ρσ , (4.15)
T Φ µν = 1 Φ ˜ µ ˜ ν Φ - 2 Φ ( ˜ µ Φ)( ˜ ν Φ) - 1 Φ gµν gρσ ˜ ρ ˜ σ Φ - 2 Φ ( ˜ ρ Φ)( ˜ σ Φ) , (4.16)
are the stress-energy tensors for the electromagnetic and Φ fields, respectively. These equations are interpreted according to the induced matter principle. An empty 5D space-time, equipped with a space-like Killing field, is regarded as a 4D space-time containing electromagnetic and Φ fields, as sources of 4D gravity. Equation (4.12) represents the 4D Einstein equations where the dichotomy between fields and sources occurs only with the 4D induced-matter interpretation of the 5D geometry. The RHS of Eq. (4.12) is separated into electromagnetic and Φ sources. It is important to note that the 4D gravity sources are different in nature: T EM µν can be inferred from passive particle propagation, while T Φ µν cannot. Therefore, from this perspective, Φ appears like a dark field [START_REF] Roszkowski | Wimp dark matter candidates and searches-current status and future prospects[END_REF][START_REF] Bertone | History of dark matter[END_REF].

The Maxwell equations and Ohm's law from the equations for 5D gravity

Equation (4.13) is of type Maxwell and helps recover Eqs. (1.1) and (1.4), considered in vacuum. 7 For this, we break Eq. (4.13) to introduce the 4-current density, j ν Eq. (4.17) corresponds to the Maxwell equations (1.1) and (1.4); the RHS of Eq. (4.17) expresses the sources for the electromagnetic field in the LHS. Equation (4.18) further explicates j µ in terms of Φ and the electromagnetic field, and is interpreted in the same way: the RHS designates the sources for the LHS. The alternative interpretation, where the LHS designates the source for the electromagnetic and Φ fields in the RHS, yields a contender to the Maxwell equations, and a line of argument far from the traditional electromagnetic theory. We will not pursue it here. Applying ˜ ν on both sides of Eq. (4.18) and using Eq. (4.13) yields the 4D continuity equation for j ν as an identity 8 4π c

˜ µ Fµν ≡ 4π c j ν , (4.17
˜ ν j ν = -3( ˜ ν Fµν )( ˜ µ ln Φ) -3 Fµν ˜ ν ˜ µ ln Φ = 6 Fµν ˜ ν ˜ µ ln Φ = 0. (4.19)
It is significant that j ν depends on Φ, which does not appear in the LHS of Eq. (4.17). Still, as a divergencefree vector field, j ν retains more freedom than Φ. From the perspective of a 4D field theory, j ν requires no 7 The concept of vacuum is vague in terms of Lorentzian geometry. According to the induced matter principle, Fµν is an active matter field defined on 4D space-time. However, the reference, empty 4D space-time is Ricci flat, with field equations R (4) M N = 0, where Fµν and Φ do not not exist. Therefore, the reference 4D space-time is not sufficient for describing particle propagation in electromagnetic field, in vacuum. We are lead to postulate that vacuum represents 5D emptiness rather than 4D emptiness; it is unclear how this relates to experiment. Another possibility is to associate vacuum to the limit of weak fields evolving in a 4D space-time. The term electrovacuum is often used for solutions of both Einstein and Maxwell equations.

8 This is also obtained by applying ˜ ν on both sides of Eq. (4.17).

further analysis. It is very important to note that, in the theory of 5D gravity, j ν results from a dispensable ansatz (i.e., Eqs. (4.17) and (4.18)), of no particular benefit for solving the 5D field equations. This is in stark contrast with the fundamental role that j ν plays in the theories of electromagnetism and condensed matter, where a fundamental SI unit is dedicated to measurements of electrical current. These theories further describe j ν , postulating conduction mechanisms which rely on the structure of matter. For example, in solid state physics (e.g., Refs. [27, Chs. 1, 2] and [28, Ch. 6]), the current density is described using classical or quantum formalisms for passive particles/quasiparticles in motion, trapped within matter, subject to the Lorentz force. In addition, the particles/quasiparticles are subject to collisions or phenomenological friction forces, due to additional electromagnetic interactions associated to the atomic structure of matter. 9 In plasma physics (e.g., Refs. [29, Ch. 1] and [START_REF] Bittencourt | Fundamentals of Plasma Physics[END_REF]Chs. 5,[START_REF] Kaluza | Zum Unitätsproblem der Physik[END_REF]), conduction mechanisms have similar building principles, although, sometimes, they simply postulate that the Lorentz force is zero for particles within plasma. A 3D velocity field for charged carriers is often defined using 4D active matter v/c ≡ j/j 0 .10 There are two main uses for the microscopic pictures of j ν . First, they explain the Laplace force [START_REF] Henry | Mini Manuel d'Electromagnétisme[END_REF], the force acting on a conductor placed in electromagnetic field, in terms of the Lorentz force;11 see e.g., [START_REF] Jackson | Classical Electrodynamics[END_REF]Ch. 5.2] for an argument based on Ampére's work [START_REF] Ampére | Mémoire sur l'action mutuelle entre deux courants électriques, un courant électrique et un aimant ou le globe terrestre, et entre deux aimants[END_REF]. However, the Laplace force does not require an explanation via the Lorentz force, because the Laplace force can be obtained from the divergence of the electromagnetic stress-energy tensor, ˜ µ T EM µν [START_REF] Jackson | Classical Electrodynamics[END_REF]Ch. 11.11]. Second, they provide microscopic explanations for the Ohm's law and other laws in condensed matter physics. However, as we explain below, a conduction mechanism is not required to explain Ohm's law, which here appears as a field equation. To see this, we provide a physical interpretation for Eq. (4.18), breaking 4D covariance in the limit of weak fields.

A minimal amendment to the Maxwell equations

The rigorous approach to Eqs. (4.12)-(4.14) is through exact solutions, analyzed for their relevance to experiment. Many exact solutions for empty 5D space-times without compact dimensions have already been found [START_REF] Reall | Black Holes in Higher Dimensions[END_REF][START_REF] Iguchi | Exact solutions II: Systematic solution-generation of fivedimensional black holes[END_REF][START_REF] Tomizawa | Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method. II. Production of the black ring solution[END_REF][START_REF] Billyard | Class of exact solutions in 5D gravity and its physical properties[END_REF][START_REF] Dotti | Exact solutions for the Einstein-Gauss-Bonnet theory in five dimensions: Black holes, wormholes, and spacetime horns[END_REF][START_REF] Riveros | Generalized Helmholtz equation[END_REF][START_REF] Alexeyev | Kerr-Gauss-Bonnet black holes: Exact analytical solution[END_REF][START_REF] Anabalón | Kerr-Schild ansatz in Einstein-Gauss-Bonnet gravity: an exact vacuum solution in five dimensions[END_REF], but they primarily describe time-independent geometries, which are not invariant to translation along x 5 , and do not satisfy Eqs. (4.12)-(4.14). Perhaps, the only exception is a class of time-dependent solutions, obtained through the Wick rotation of the 5D Myers-Perry black-hole metric [START_REF] López | Einstein-Rosen solutions from Kaluza-Klein theory[END_REF][START_REF] López | Time-dependent solution from Myers-Perry[END_REF][START_REF] Bretón | 5D gravitational waves from complexified black rings[END_REF]. 12An alternate approach is using the 5D field equations for an amendment to the Maxwell equations, which are already close to experiment. Evidently, the physics of the amended Maxwell equations is superseded by that of corresponding exact 5D solutions. We propose a strong approximation of the 5D gravity equations (4.12)-(4.14) in the limit of weak fields. We consider that the 4D metric, gµν , is approximately the Minkowski metric, η µν ; n.b., (q/c 2 ) 2 T EM 2 + T Φ ≈ 0. We further break covariance to distinguish 3D space and time, and write the 4D connection as ˜ ν ≡ ( ∂0 , ∇). In (pseudo)cartesian coordinates, the 3D space metric is diag(1, 1, 1), so we no longer distinguish covariant and contravariant spatial components of tensors. Furthermore, we use the Levi-Civita symbol, jkl , as a cartesian tensor, and introduce the 3D curl operator and vector product. Setting x 0 ≡ -ct, and using Eqs. (1.8) and (1.9), we introduce the 3D vector field variables

E = F0j , B j = jkl Fkl /2.
(5.20)

Next to Eqs. (1.2) and (1.3), Eq. (4.13) yields

∇E = 4π c E, (5.21)
∇B = 0, (5.22)

∇ × E = - 1 c ∂B ∂t , (5.23)
∇ × B = 1 c ∂E ∂t + 4π c (ςE + × B), (5.24)
where the fields

≡ - c 4π ∇ ln Φ 3 , ς ≡ 1 4π ∂ ln Φ 3 ∂t , (5.25)
are regarded to be material properties, satisfying

- 1 c ∂ ∂t = ∇ς. (5.26)
Alternatively, one can directly choose a model for the field Φ. 13 Electrical charge and current densities are not needed for the amended Maxwell equations (5.21)-(5.24), but they result from Eq. (4.18) as

ρ = E/c, (5.27) j = ςE + × B.
(5.28)

Equations (5.27) and (5.28) can be regarded as a generalization of the Ohm's law. Conceptually, a perfect conductor has very high conductivity, does not hold charge density in volume, nor allows for magnetic field inside the material [START_REF] Jackson | Classical Electrodynamics[END_REF]Ch. 8.1]. This model is recovered in the above equations by assuming that Φ is approximately independent of space (i.e., ∇ ln Φ 3 ≈ 0 or ≈ 0) in some region, where we say the perfect conductor is found. According to Eq. (5.27), the charge density is approximately zero, and Eq. (5.28) yields the Ohm's law (1.7). Equations (5.27) and (5.28) thus describe a more realistic, imperfect conductor, which holds electric charge in volume and allows penetration by the magnetic field. Hence, the conduction current j has an ohmic component and a non-ohmic component, which depends on the local magnetic field. These phenomena are characterized by the 3D vector field , which is regarded here as a material property, next to the conductivity ς.

Taking an imperfect conductor for being perfect can have important consequences for the interpretation of experimental data. The ordinary Hall effect [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Kittel | Introduction to Solid State Physics[END_REF][START_REF] Hall | On a new action of the magnet on electric currents[END_REF] clearly demonstrates that an external magnetic field penetrates conductor materials, in contradiction to the model for the perfect conductor. 14 Still, if the model of a perfect conductor is used to interpret experimental data, then all conduction current is considered to be ohmic and, apparently, the conductivity depends linearly on the local magnetic field; c.f., Eqs. (5.28). Furthermore, if the external magnetic field is large compared to the magnetic field generated by the electric current j, then it appears that the conductivity depends linearly on the external magnetic field. These considerations seem to explain experimental observations by Kapitza [START_REF] Kapitza | The change of electrical conductivity in strong magnetic fields. Part I. Experimental results[END_REF][START_REF] Kapitza | The change of electrical conductivity in strong magnetic fields. Part II. The analysis and the interpretation of the experimental results[END_REF][START_REF] Kapitza | The study of the specific resistance of bismuth crystals and its change in strong magnetic fields and some allied problems[END_REF], regarding a large number of chemical elements. It may however be possible to reinterpret Kapitza's experimental data using the model of an imperfect conductor (c.f., Eq. (5.27) and (5.28)), with the benefit of a simpler picture for the material properties. The phenomena by which an applied magnetic field changes the conductivity15 of a material are known under the generic name of magnetoresistance [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Kittel | Introduction to Solid State Physics[END_REF]. This is a large and complex chapter in condensed matter physics. The imperfect conductor model may help disentangle this complexity, while making no assumptions about the structure of matter, only using 5D field theory.

Another example of a phenomenon where the concept of imperfect conductor may be important is that of persistent current [START_REF] Kokurin | Electronic states and persistent currents in nanowire quantum ring[END_REF][START_REF] Gréget | Magnetic properties of gold nanoparticles: A room-temperature quantum effect[END_REF][START_REF] Bleszynski-Jayich | Persistent currents in normal metal rings[END_REF], which was discovered in gold and other metals. Consider an imperfect conductor in zero electric field and nonzero external magnetic field; this may be, for example, the magnetic field of a probe. According to Eq. (5.28), the external magnetic field causes a current density × B inside the conductor, where B is the local magnetic field inside the conductor. In turn, this current density causes an additional component to the external magnetic, as described by the Maxwell equations. In the model of the perfect conductor, this looks like an instantaneous field response, otherwise, one is lead to assume a strange ohmic current inside the perfect conductor, even in the absence of electric field. We hope that the model of an imperfect conductor presented here can help provide a simple and natural explanation for the phenomenon of persistent current, within the framework of classical field theory.

Lastly, we briefly comment on the Hall effect [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Kittel | Introduction to Solid State Physics[END_REF][START_REF] Hall | On a new action of the magnet on electric currents[END_REF]. Using the perfect conductor model, the effect can be described that applying magnetic field transverse to ohmic current flow triggers an additional current component, transverse to both the initial ohmic current and the applied magnetic field. The traditional explanation of the Hall effect is based on conduction mechanisms, violating both the perfect and imperfect conductor models. It is assumed that the external magnetic field penetrates the conductor material and changes the motion of the conducting particles inside. Consequently, the conducting particles, considered as being passive, deviate from their initial trajectories, creating an additional current component inside the conductor. 16 However, within an imperfect conductor, even without applying a magnetic field, the conducting particles form a current, and already propagate within the magnetic field which they, themselves, create (Ampère's law). Therefore, the conducting particles within an imperfect conductor constitute active matter and cannot be described as passive particles. An alternate explanation of the Hall effect, free of conduction mechanisms, based only on the classical field theory and the concept of the imperfect conductor, would not have these difficulties. In fact, the phenomenon of persistent current and the Hall effect are very much related, both regarding penetration of magnetic field inside an imperfect conductor. The difference is that, for the Hall effect, the conductor is subject to ohmic current, next to the external magnetic field. In the remaining of this section, we discuss several applications of the amended Maxwell equations (5.21)-(5.24) to traditional problems formulated for the Maxwell equations.

On the propagation of electromagnetic waves within a conducting material

The most natural application of Eqs. (5.21)-( 5.24) may be to describing the propagation of electromagnetic waves through conducting materials. Traditionally, the problem is formulated by assuming a perfect conductor and merging the Maxwell equations with the Ohm's law. This results in Eqs. (5.21)- (5.24), where ς is a constant and is zero; see e.g., Refs. [START_REF] Jackson | Classical Electrodynamics[END_REF]Ch. 7.7], [START_REF] Griffiths | Introduction to Electrodynamics[END_REF]Ch. 9.4] and [START_REF] Born | Principles of Optics[END_REF]Ch. 13]. We slightly generalize this application to the case where both ς and are constants. 17 We proceed by deriving an equation for the electric field, E. 18 First, we take a curl on both sides of Eq. (5.23) and use Eq. (5.24)

to substitute ∇ × B ∇(∇E) -∇ 2 E = - 1 c ∂ ∂t 1 c ∂E ∂t + 4π c ςE + 4π c × B . (5.29)
Second, in Eq. (5.29), we substitute ∇E using Eq. (5.21) and ∂B/∂t using Eq. (5.23)

4π c ∇(E ) = ∇ 2 E - 1 c 2 ∂ 2 E ∂t 2 - 4πς c 2 ∂E ∂t + 4π c × (∇ × E).
(5.30)

We now search for plane wave solutions of type

E = E 0 exp(ikx -iωt), (5.31)
where i is the complex unit, E 0 and ω are real constants, but we allow k to be a complex constant. We thus obtain the following dispersion equation in the propagation vector k

ω 2 c 2 -k 2 + 4πi c ςω c -k = 0. (5.32)
In the case where = 0, we simply recover the formula for a perfect conductor [START_REF] Jackson | Classical Electrodynamics[END_REF]Ch. 7.7]. The propagation vector k is complex, indicating that the electromagnetic wave can propagate within a perfect conductor, but it is attenuated over the length scale 1/ Im |k| ∝ ς -1/2 , called skin or penetration depth. Equation (5.32) also shows that an imperfect conductor behaves like a perfect conductor, provided k = 0. This represents an anisotropy in the electromagnetic wave propagation. Notably, the anisotropy does not occur in the model of a perfect conductor and can help measure , assessing how imperfect the conductor material is. Wave propagation through an imperfect conductor is intrinsically anisotropic, where the anisotropy relates to , or the dark field Φ. This is different from other sources of anisotropy, generally associated to the tensor nature of the relative electric permittivity or magnetic permeability, which here are taken for unit scalars. It is remarkable that, according to Eq. (5.32), electromagnetic waves can propagate loss-free within an imperfect conductor given that

|k| = ω/c, cos θ = ς/| |, (5.33)
where θ is the angle between and k, which has no imaginary part. The propagating wave solution requires ς ≤ | |, which cannot occur for a perfect conductor. Finally, we note that recent studies of electromagnetic wave propagations [START_REF] Jalali-Mola | Electrodynamics of tilted Dirac and Weyl materials: A unique platform for unusual surface plasmon polaritons[END_REF][START_REF] Ginzburg | Electromagnetic waves in isotropic and crystalline media characterized by dielectric permittivity with spatial dispersion[END_REF][START_REF] Brand | Enhanced electromagnetic wave transmission through narrow conducting channels[END_REF][START_REF] Carcione | Forbidden directions for TM waves in anisotropic conducting media[END_REF][START_REF] Baqir | Investigation of uniaxial anisotropic chiral metamaterial waveguide with perfect electromagnetic conductor loading[END_REF][START_REF] Farko | Propagation of electromagnetic waves in a conducting magnetic medium[END_REF] take precautions to model realistic features of matter, yet do not go beyond the concept of perfect conductor.

On electrostatics and magnetostatics

We submit that Eqs. (5.21)-(5.24) can describe the electromagnetic field in a large number of applications, not just wave propagation. To discuss electrostatic and magnetostatic solutions, we consider a 5D metric which is independent of both x 5 and x 0 . Therefore, Φ is time-independent, the conductivity ς is zero, and the ohmic component of the electric current density is zero, as well. Consequently, the framework cannot address the traditional magnetostatic theory [START_REF] Jackson | Classical Electrodynamics[END_REF][START_REF] Ampére | Mémoire sur l'action mutuelle entre deux courants électriques, un courant électrique et un aimant ou le globe terrestre, et entre deux aimants[END_REF], unless the ohmic component of the electric current density is zero. This is too restrictive to recover the law of Biot and Savart, or that of Ampère. The field equations become

∇E = 4π c E, (5.34)
∇B = 0, (5.35)
∇ × E = 0, (

∇ × B = 4π c × B, 5.36)
where, based on the definitions (5.25), ≡ -(c/4π)∇ ln Φ 3 has zero curl; i.e., ∇ × = 0. The electrostatic and magnetostatic sources depend on the very fields they determine

ρ = E/c, (5.38
) j = × B. (5.39)
However, these concepts are not strictly needed for the field equations. The equations for the electric and magnetic fields are determined by the same vector field, 19 , yet they are decoupled. Either E or B can be zero, irrespective of the other. There are, of course, solutions where both E and B are nonzero, determined by . The following results through straightforward calculations.

Proposition. Under the change of variables

E → E * Φ -3 , B → B * Φ -3 , (5.40)
Eqs. (5.34)-(5.37) remain invariant, where

(E, B, ) → (B * , E * , -). (5.41)
In consequence, it is sufficient to solve the equations for either electrostatics (i.e., Eqs. (5.34) and (5.36)) or magnetostatics (i.e., Eqs. (5.35) and (5.37)) for a complete solution. For example, the electrostatic equations can be combined into a single one, for the electrostatic potential,

∇ 2 A 0 = 4π c ∇A 0 , (5.42)
that can be used to formulate homogeneous problems of type Dirichlet and Neumann; see Appendix A for alternate formulations of the Eq. (5.42). The solutions for A 0 generate all solutions of Eqs. (5.34)-(5.37).

The typical problem in electrostatics and magnetostatics is finding field configurations due to localized sources; it is assumed that the resulting field is also localized, thus vanishes at infinity. We discuss two elementary solutions for Eqs. (5.34)- (5.37). If all the vector fields are spherically symmetric, we obtain B = 0 and E = (E r , E θ = 0, E φ = 0), where

E r (r) = CΦ -3 (r) r 2 , (5.43)
and C is an integration constant. The radial component of the spherically symmetric, Coulomb electric field, resulting from the Maxwell equations, is qualitatively the same [3, Ch. 1]

E M r (r) = Q(r) r 2 , (5.44)
where Q(r) = 4π r 0 ρ(r )r 2 dr is the electric charge contained by the sphere of radius r, centered at the origin and ρ(•) denotes a spherically symmetric, electrical charge density. Comparing Eqs. (5.43) and (5.44) yields CΦ -3 (r) = Q(r), addressing the integration constant C and providing an interpretation for Φ, in this restricted geometry. The amended Maxwell equations recover the concept of point-like source electric charge, which is remarkably successful with the model of the hydrogenic atom.

In magnetostatics, we say E = 0. If = ( ρ , φ = 0, z = 0), in cylindrical coordinates, then the solution of Eq. (5.37) is B = (B ρ = 0, B φ , B z = 0), where

B φ (ρ) = CΦ -3 (ρ) ρ , (5.45)
is reminiscent of Ampère's law. The solution predicts nonzero magnetic field in the absence of ohmic current.

In fact, taking a curl on both hand sides of Eq. (5.37) yields a generalized Helmholtz equation

∇ 2 B = (∇ )B. (5.46)
We assume that ∇ is zero outside a compact domain and say that the magnetic sources, ×B, are localized. Thus, Eq. (5.46) becomes the Laplace equation outside the compact domain, so we can choose magnetostatic solutions vanishing at infinity; localized magnetic sources yield localized magnetostatic fields. Traditionally, we acknowledge only ohmic current and spin magnetic moments as the sources of magnetic field. However, our 5D theory suggests that sources of magnetic field can also be represented as ×B. Given suitable models for , Eq. (5.37) may serve as a basic tool to model field configurations due to magnetic materials [START_REF] Kronmüller | Handbook of magnetism and advanced magnetic materials[END_REF]. It is very important to note that, the field equations of 4D mechanics (4.12)-(4.14), and, implicitly, Eq. (5.37), cannot describe the thermodynamics of the electromagnetic field response, addressing topics such as the Curie temperature, the Curie-Weiss law, etc. According to the principles of our 5D theory, this requires the translation along x 0 be formalism isometry, and a statistic interpretation of the electromagnetic field.

A new class of field-static problems

In contrast to the traditional electrostatic and magnetostatic equations, (5.34)-(5.37) allow for another class of problems, besides that of localized field configurations. We consider problems of field configurations, where static, constant field is applied from spatial infinity; n.b., the applied field has zero curl and divergence. These problems cannot be formulated using the Maxwell equations (1.1)- (1.4), where sources are considered to be independent of field, and applying field from spatial infinity has no consequence for the localized field component. In electrostatics, we write E = E (0) + E (1) , where E (0) is the field at infinity and E (1) is localized field. Using Eqs. (5.34) and (5.36), and E (1) ≡ -∇A

(1) 0 , we get

∇ 2 A (1) 0 = 4π c ∇A (1) 0 - 4π c E (0) , (5.47)
a non-homogenerous equation in A

(1) 0 , to be solved using Green's function, starting from the homogeneous equation (5.42). In magnetostatics, we write B = B (0) +B (1) , where B (0) is the field at infinity and B (1) is a localized. Introducing B (1) ≡ ∇×A (1) , under the gauge condition ∇A (1) = 0, we obtain a non-homogeneous equation in A (1) ,

-∇ 2 A (1) = 4π c × (∇ × A (1) ) + 4π c × B (0) , (5.48)
which, again, can be approached through the method of Green's function. The RHS of Eq. (5.48) has two terms: × (∇ × A (1) ) represents a current density due to the localized magnetic field B (1) , and × B (0) represents the contribution to magnetic sources by the magnetic field applied from spatial infinity. Hopes are that Eq. (5.48) can serve to model penetration of magnetic field inside materials and permanent currents. For B (0) = 0, we choose the solution B (1) = 0, while for B (0) = 0, we solve Eq. (5.48) to obtain B (1) .

Another 5D space-time geometry for electrostatics and magnetostatics

Field configurations can appear static to the 4D observer, even though the 5D geometry does not have a Killing field along x 0 . In particular, the gravitational (i.e., gµν ) and electromagnetic fields can be observed as being static through observing passive particle propagation. However, time dependence can occur in the lapse field Φ, which is a dark field. The amended Maxwell equations admit time-independent solutions, if ς and are time-independent. Equation (5.26) further implies that ς must have zero gradient, thus the conductivity ς is a constant field, but remains a zero-curl 3D field. 20 The field equations become

∇E = 4π c E, (5.49
)

∇B = 0, (5.50)
∇ × E = 0, (

∇ × B = 4π c ςE + 4π c × B, 5.51)
accommodating better the traditional magnetostatics [3, Ch. 5], which results assuming a perfect conductor; i.e., = 0. However, because translation along x 0 is not a 5D isometry, our 5D framework implies that the statistics of the electromagnetic field is ill-defined. In fact, experiment shows that, as long as conductivity is nonzero, the electromagnetic field continuously loses energy through ohmic heating. It may thus be intuitive that, in this case, the electromagnetic field cannot be in thermodynamic equilibrium, even though it is time-independent. We comment on the original Hall effect, using Eqs. (5.49)-(5.52) to sketch a theory for the timeindependent phenomenon, even though ς and provide just a crude model for the material properties. In particular, constant conductivity over the entire 3D space necessarily implies that the conducting material fills up the entire 3D space. We assume that the magnetic field has two components, B = B (0) + B (1) , where B (0) is a constant magnetic field, applied from infinity, and B (1) is the localized magnetic field. Equation (5.52) yields

-∇ 2 A (1) = 4π c ςE + 4π c × (∇ × A (1) ) + 4π c × B (0) , (5.53)
with the same notation as in Eq. (5.48). The electric field E is an input in Eq. (5.53), and results from Eq. (5.47), where we also distinguish two components for the electric field E = E (0) + E (1) . The typical conceptualization of the field configuration for the Hall effect [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Kittel | Introduction to Solid State Physics[END_REF] considers B (0) pointing along the z-axis and E (0) pointing along the x-axis. Equation (5.53) predicts the existence of a non-ohmic current component due to the magnetic field at infinity, × B (0) , next to a contribution from × (∇ × A (1) ), which depends implicitly on B (0) and E. These additional current components could be translated into electric field components using Ohm's law, within the perfect conductor model. Still, this is very different from the position of the traditional theory of the Hall effect [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Kittel | Introduction to Solid State Physics[END_REF], that the electric field E changes with B (0) . The Hall coefficient R H = E y /(j x B z ) quantifies this effect and is regarded as a measure of electron density within the conductor material. These developments, based on conduction mechanisms and the perfect conductor model, 20 The field ln Φ 3 depends linearly with time.

are incompatible to our theory. It is very important to note that the experimental measurements, themselves, provide readings of electric current and potential, based on Ohm's law and the perfect conductor model, as well. Therefore, as anticipated in Sec. 5, giving up the model of the perfect conductor, poses problems not only for theory, but also for measurement principles and data interpretation. Our field theoretic approach would require measurement principles based on the principles of general relativity; i.e., field probes using passive particle propagation, rather than measurements of electrical current and potential. Corroborating Eq. (5.53) with experimental results, remains a topic for further work.

Discussion and conclusions

The electromagnetic theory and experiment developed in parallel during the nineteenth century, both sides contributing to sharpening much needed physical concepts such as field, perfect conductor, and electron. Meanwhile, condensed matter physics also made considerable progress, particularly through combining elements of thermodynamics, electromagnetism and atomic theory. We submit that, owing to the choice of fundamental concepts in electromagnetism, some actual field phenomena21 might have been misclassified for describing properties of matter. In consequence, traditional electromagnetism seems very accessible and successful, wherever applicable by itself, whereas condensed matter physics struggles with describing complex properties for matter. A revision of the fundamental concepts in electromagnetism and condensed matter physics may be in order, involving both theory and experiment. Kaluza's proposal, that electromagnetism and gravity should be unified within the 5D general relativity framework, has inspired very much work on the theoretical fundamentals of electromagnetism and gravity. On the experimental side, a very broad set of measurement techniques became available, and may be critically evaluated from the fundamental point of view, taking into account philosophical choices in electromagnetism and general relativity.

In this work, we analyzed the equations of 5D gravity from the perspective of the Maxwell equations and Ohm's law. The covariance of the 5D metric field can be broken to distinguish 4D gravitational field, electromagnetic field and a dark scalar field. Among the field equations resulting from the 5D condition of Ricci flatness, we recognized Maxwell-type equations and a generalization of Ohm's law, modeling an imperfect conductor, which holds electric charge in volume and can be penetrated by an exterior magnetic field. Furthermore, our analysis makes it clear that the density of electrical charge and current should not be considered as fundamental concepts in electromagnetism. Still, the notion of electrical charge retains fundamental status. Indeed, passive specific charge occurs as a factor converting the electromagnetic field into 5D gravitational field; see Eq. (3.10). Active charge may occur as a parameter of the 5D metric. For example, black hole metrics have parameters such as mass, angular momenta, etc., often called conserved charges. Active electric charge may be found among these parameters.

Based on the 5D gravity equations, we proposed an amendment to the Maxwell equations, in the limit of weak fields, and commented on a few, direct applications. Notably, we discussed electromagnetic wave propagation through an imperfect conductor, electrostatics, magnetostatics of magnetic materials, permanent currents and the Hall effect. The goal of this manuscript is to convey that, even though the electromagnetic theory resulting from 5D gravity differs in many regards from the traditional theory, it is able to recover much of the formalism, albeit on a slightly different conceptual basis. We did not provide a mathematical definition for the limit of weak fields, rather, we capitalized on the resemblance between the Maxwell equations and 5D gravity equations. There are three main reasons why our discussion of physical applications is limited. First, the 5D space-time may not be Ricci-flat, and a stress-energy tensor representing 5D active matter may be postulated. Second, the x 5 -translational symmetry of the 5D space-time may be only approximate, and the field formalism could be adjusted using perturbation methods to account for the symmetry breaking. Third, the final word is left to exact solutions of the field equations. Therefore, the physics of electromagnetism resulting from 5D gravity could be significantly more complex.

In conclusion, the geometry of an empty 5D space-time with a space-like Killing field can serve to reconsider the fundamentals of electromagnetism and condensed matter theory. If the 5D space-time has a time-like Killing field, then it may be possible to construct a field thermodynamics. This remains a topic for further work.

) 4π c j ν ≡ - 3

 3 Fµν ˜ µ ln Φ.(4.18)

This type of foliation is performed in the Hamiltonian formulation of 4D general relativity, for which the Arnowitt-Deser-Misner foliation[START_REF] Arnowitt | Republication of: The dynamics of general relativity[END_REF][START_REF] Misner | Gravitation[END_REF] was originally intended.

[START_REF] Born | Principles of optics[END_REF] We do not consider the propagation of active massive (charged) particles propagating within the gravitational (electromagnetic) field that themselves produce.

The fünfbein field specifies a choice for a local frame at each point of the 5D manifold.

The

5D observer has no notion of particle charge or mass and does not distinguish between electric and gravitational fields.[START_REF] Kronmüller | Handbook of magnetism and advanced magnetic materials[END_REF] Mathematics allows for much more general theories. For example, we may consider a space-time with a time-like Killing vector or several Killing vectors as formalism isometries, possibly requiring foliation over several dimensions of the 5D manifold. The difficulty resides in the physical interpretation and link to experiment.

The choice for the foliated fünfbein rests with the uncharged 4D observer[START_REF] Breban | A five-dimensional perspective on the Klein-Gordon equation[END_REF].

These atomic electromagnetic fields are not included in the Maxwell equations with the sources jν .

Our theory does not require 4D active matter satisfy the 4D on-shell constraint, designed for 4D passive particles.

The Lorentz force acts on passive particles.

Identifying z with x 5 in Eq. (1) of Ref.[START_REF] López | Einstein-Rosen solutions from Kaluza-Klein theory[END_REF] shows that the 5D metric has nontrivial fields gµν , Aµ and Φ. The induced matter interpretation of 5D metric remains for further work.

The weak field limit of Eq. (4.14) may provide an equation for Φ. Here, we adopt a more general view, where the choice of Φ is only limited by its success in applying Eqs. (5.21)-(5.24) to experiment.

Hall discovered the effect in gold, which does not behave like a perfect conductor. This raises questions about the applicability of the model of the perfect conductor.

[START_REF] Breban | Interpretation of the five dimensional quantum propagation of a spinless massless particle[END_REF] The conductivity remains defined by the Ohm's law, within the perfect conductor model.

It is intriguing that magnetoresistance never enters the discussion about the Hall effect. According to the experiments of Kapitza[START_REF] Kapitza | The change of electrical conductivity in strong magnetic fields. Part I. Experimental results[END_REF][START_REF] Kapitza | The change of electrical conductivity in strong magnetic fields. Part II. The analysis and the interpretation of the experimental results[END_REF][START_REF] Kapitza | The study of the specific resistance of bismuth crystals and its change in strong magnetic fields and some allied problems[END_REF], magnetoresistance always occurs next to the Hall effect.

It remains unclear how applicable to experiment is this choice for ς and .

The same results are obtained by analyzing the equations for the magnetic field.

Taking = 0 accommodates all field configurations where E and B are independent constant vectors.

Hall regarded his discovery as a field effect, going against the Maxwell's conceptualization of electromagnetism[START_REF] Hall | On a new action of the magnet on electric currents[END_REF].

A Alternate forms for the electrostatic potential equation Equation (5.42), describing the electrostatic field versus , is not found among the popular partial differential equations of mathematical physics. However, the equation can be reformulated to resemble more popular equations. For example, in the field variable α ≡ A 0 Φ 3/2 , Eq. (5.42) becomes a generalized Helmholtz equation

Evidently, if A 0 vanishes at infinity, then so does α. The generalized Helmholtz equation has been previously studied, in both two and three dimensions [START_REF] Riveros | Generalized Helmholtz equation[END_REF][START_REF] Osnabrugge | A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media[END_REF][START_REF] Shaw | A generalized Helmholtz equation fundamental solution using a conformal mapping and dependent variable transformation[END_REF]. However, general solutions for the 3D case are yet to be formulated. Another possibility is to transform Eq. ( 5.42) into an integral equation, using the Green's function of the Laplace operator. We obtain

which, under the assumption that ∇A 0 is a localized field, can be integrated by parts to give

where

Therefore, A 0 appears as a fixed point (i.e., eigenvector of unit eigenvalue) of the integral operator with kernel K(•, •). Fixed points of integral operators have been previously studied [START_REF] Datko | A fixed-point theorem for a class of integral operators[END_REF][START_REF] Ali | Existence of fixed points for an integral operator via fixed point theorem on gauge spaces[END_REF][START_REF] Nazam | On a fixed point theorem with application to integral equations[END_REF][START_REF] Rao | On the eigenvalues and eigenfunctions of some integral operators[END_REF].