Tian Ma
email: matian56@sina.com

Shouhong Wang
email: showang@indiana.edu

QUANTUM MECHANISM OF CONDENSATION AND HIGH TC SUPERCONDUCTIVITY

Keywords: Mathematics Subject Classification. 82D50, 82D55, 81P10 PACS numbers. 74.20.-z, 74.20.Fg, 03.65.Ta, 03.75.Kk new interpretation of quantum mechanical wave functions, Copenhagen interpretation, superconductivity, condensates, Bose-Einstein condensation, Cooper pair, BCS theory, high Tc superconductivity, electronphonon interaction, topological phase transition, PID interaction potential, average energy level formula of temperature, electron-pairs, binding energy

The main objective of this paper is to introduce a new quantum mechanism of condensates and for superconductivity based on new interpretation of quantum mechanical wave functions, and on recent developments in quantum physics and statistical physics. First, we postulate that the wave function ψ = |ψ|e iϕ is the common wave function for all particles in the same class determined by the external potential V (x), |ψ(x)| 2 represents the distribution density of the particles, and m ∇ϕ is the velocity field of the particles. Although the new interpretation does not alter the basic theories of quantum mechanics, it is an entirely different interpretation from the classical Bohr interpretation, removes all absurdities, and offers new insights for quantum physics and for condensed matter physics. Second, we show that the key for condensation of bosonic particles is that their interaction is sufficiently weak to ensure that a large collection of boson particles are in a state governed by the same condensation wave function field ψ under the same bounding potential V . For superconductivity, the formation of superconductivity comes down to conditions for the formation of electron-pairs, and for the electron-pairs to share a common wave function. Thanks to the recently developed PID interaction potential of electrons and the average-energy level formula of temperature, these conditions for superconductivity are explicitly derived. Furthermore, we obtain both microscopic and macroscopic formulas for the critical temperature. The field and topological phase transition equations for condensates are also derived.

The main objective of this paper is to attempt to derive the quantum mechanism of condensates and high temperature superconductivity, based on the new interpretation of quantum mechanical wave functions, and on the recent developments of fundamental interactions and statistical physics. The main results of the paper include the following:

(1) new interpretation of wave functions in quantum mechanics, (2) quantum mechanism of condensates and (high Tc) superconductivity, (3) new theory of electron-pairs, and (4) microscopic and macroscopic formulas of the critical temperature for superconductivity.

Motivations

The study of this paper is strongly motivated by the lack of physical mechanism of condensates and high temperature of superconductivity. The phenomenon of superconductivity was first discovered by Kamerlingh Onnes in 1911, in metallic mercury below 4K. Since then, the main goals of study have been to attempt to observe superconductivity at increasing temperatures, and to derive its physical mechanism.

The classical BCS theory [START_REF] Bardeen | Theory of superconductivity[END_REF], developed by John Bardeen, Leon Cooper, and John Schrieffer for which they shared the 1972 Nobel Prize, is a microscopic theory based on the Cooper electron-pair mechanism, which was used to explain superconducting currents, the Meissner effect, the isotope effect, and to calculate specific heats and penetration depths. The Cooper pair theory provides the mechanism of electronphonon interaction in low temperature superconductors, and a physical mechanism form the Bose electron-pair as required in (4.1). However, the BCS theory is applicable only for T < 35K.

In 1986, Bednorz and Müller [START_REF] Bednorz | Possible high Tc superconductivity in the Ba-La-Cu-O system[END_REF] discovered superconductivity in a lanthanum-based cuprate perovskite material, which had a transition temperature of 35 K (Nobel Prize in Physics, 1987). Bednorz and Müller's work opened the door for high Tc superconductors. The current highest temperature of superconductors reaches T c = 130K, and T c = 160K under high pressure.

The discovery of High Tc superconductors shows the limitation of the BCS theory. There have been other attempts for the understanding of the nature of superconductivity; see among others [START_REF] Anderson | The Resonating Valence Bond State in La2CuO4 and Superconductivity[END_REF][START_REF]Is there glue in cuprate superconductors[END_REF][START_REF] Ma | Bifurcation and stability of superconductivity[END_REF][START_REF] Cohen | Superconductivity in d-and f-band metals[END_REF][START_REF] Nagamatsu | Superconductivity at 39 k in magnesium diboride[END_REF][START_REF] Varma | Phenomenology of the normal state of Cu-O hightemperature superconductors[END_REF][START_REF] Hussey | Universality of the mottioffe-regel limit of metals[END_REF][START_REF] Fradkin | Theory of intertwined orders in high temperature superconductors[END_REF][START_REF] Tsuei | Pairing symmetry in cuprate superconductors[END_REF][START_REF] Miyake | Spin-fluctuationmediated even-parity pairing in heavy-fermion superconductors[END_REF][START_REF] Beal-Monod | Possible superconductivity in nearly antiferromagnetic itinerant fermion systems[END_REF][START_REF] Dahm | Strength of the spin-fluctuation-mediated pairing interaction in a high-temperature superconductor[END_REF]136,[START_REF] Vaknin | Antiferromagnetism in La2CuO4-y[END_REF][START_REF] Chakravarty | Low-temperature behavior of two-dimensional quantum antiferromagnets[END_REF][START_REF] Doiron-Leyraud | Quantum oscillations and the fermi surface in an underdoped high-tc superconductor[END_REF][START_REF] Sebastian | Normal-state nodal electronic structure in underdoped hightc copper oxides[END_REF][START_REF] Zaanen | Charged magnetic domain lines and the magnetism of high-tc oxides[END_REF][START_REF] Emery | Frustrated electronic phase separation and high-temperature superconductors[END_REF][START_REF] Tranquada | Evidence for stripe correlations of spins and holes in copper oxide superconductors[END_REF][START_REF] Raghu | Superconductivity in the repulsive hubbard model: an asymptotically exact weak-coupling solution[END_REF][START_REF] Scalapino | A common thread: the pairing interaction for unconventional superconductors[END_REF][START_REF] Liu | Phases of the infinite u hubbard model on square lattices[END_REF][START_REF] Lee | Doping a mott insulator: physics of high-temperature superconductivity[END_REF][START_REF] Paramekanti | High-tc superconductors: a variational theory of the superconducting state[END_REF][START_REF] Georges | Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions[END_REF][START_REF] Stoudenmire | Studying two-dimensional systems with the density matrix renormalization group[END_REF][START_REF] Corboz | Competing states in the t-j model: uniform d-wave state versus stripe state[END_REF][START_REF] Tacon | Intense paramagnon excitations in a large family of hightemperature superconductors[END_REF][START_REF] Dean | Persistence of magnetic excitations in La2-xSrxCuO4 from the undoped insulator to the heavily overdoped non-superconducting metal[END_REF][START_REF] Fong | Spin susceptibility in underdoped YBa2Cu3O6+x[END_REF][START_REF] Dai | Evolution of the resonance and incommensurate spin fluctuations in superconducting YBa2Cu3O6+x[END_REF][START_REF] Carbotte | Bosons in high-temperature superconductors: an experimental survey[END_REF][START_REF] Leggett | A [ldquo]midinfrared[rdquo] scenario for cuprate superconductivity[END_REF][START_REF] Abrikosov | Contribution to the theory of superconducting alloys with paramagnetic impurities[END_REF][START_REF] Garg | Strong correlations make hightemperature superconductors robust against disorder[END_REF][START_REF] Uemura | Universal correlations between Tc and ns/m* in high-Tc cuprate superconductors[END_REF][START_REF]Importance of phase fluctuations in superconductors with small superfluid density[END_REF][START_REF] Chatterjee | Electronic phase diagram of high-temperature copper oxide superconductors[END_REF][START_REF] Matsui | Bcs-like bogoliubov quasiparticles in high-tc superconductors observed by angle-resolved photoemission spectroscopy[END_REF][START_REF] Fedorov | Temperature dependent photoemission studies of optimally doped Bi2Sr2CaCu2O8[END_REF][START_REF] Feng | Signature of superfluid density in the single-particle excitation spectrum of Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Warren | Cu spin dynamics and superconducting precursor effects in planes above Tc in YBa2Cu3O6.7[END_REF][START_REF] Alloul | Y NMR evidence for a Fermiliquid behavior in YBa2Cu3O6+x[END_REF][START_REF] Homes | Optical conductivity of c axis oriented YBa2Cu3O6.70: evidence for a pseudogap[END_REF][START_REF] Puchkov | The pseudogap state in high-tc superconductors: an infrared study[END_REF][START_REF] Bucher | Influence of the spin gap on the normal state transport in YBa2Cu4O8[END_REF][START_REF] Ito | Systematic deviation from T-linear behavior in the in-plane resistivity of YBa2Cu3O7-y: evidence for dominant spin scattering[END_REF][START_REF] Hashimoto | Energy gaps in high-transition-temperature cuprate superconductors[END_REF][START_REF] Marshall | Unconventional electronic structure evolution with hole doping in Bi2Sr2CaCu2O8+[dgr]: angle-resolved photoemission results[END_REF][START_REF] Yang | Emergence preformed Cooper pairs from the doped Mott insulating state in Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Hoffman | Imaging quasiparticle interference in Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Mcelroy | Relating atomic-scale electronic phenomena to wave-like quasiparticle states in superconducting Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Kohsaka | How Cooper pairs vanish approaching the Mott insulator in Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Fujita | Simultaneous transitions in cuprate momentum-space topology and electronic symmetry breaking[END_REF][START_REF] Vishik | A momentum-dependent perspective on quasiparticle interference in Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Vershinin | Local ordering in the pseudogap state of the high-Tc superconductor Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Parker | Fluctuating stripes at the onset of the pseudogap in the high-Tc superconductor Bi2Sr2CaCu2O8+x[END_REF][START_REF] Howald | Coexistence of periodic modulation of quasiparticle states and superconductivity in Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF]Ubiquitous interplay between charge ordering and high-temperature superconductivity in cuprates[END_REF][START_REF] Comin | Charge order driven by Fermi-arc instability in Bi2Sr2-xLaxCuO6+[dgr[END_REF][START_REF] Loeser | Excitation gap in the normal state of underdoped Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Corson | Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Li | Diamagnetism and cooper pairing above tc in cuprates[END_REF][START_REF] Dubroka ; R | Evidence of a precursor superconducting phase at temperatures as high as 180 K in RBa2Cu3O7[END_REF][START_REF] Kaiser | Optically induced coherent transport far above Tc in underdoped YBa2Cu3O6+[dgr[END_REF][START_REF] Norman | Destruction of the fermi surface in underdoped high-tc superconductors[END_REF][START_REF] Reber | The origin and non-quasiparticle nature of Fermi arcs in Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Lee | Spectroscopic fingerprint of phase-incoherent superconductivity in the underdoped Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Chakravarty | Hidden order in the cuprates[END_REF][START_REF] Scalapino | Stripe structures in the t-t[prime]-J model[END_REF][START_REF] Tajima | c-axis optical response in the static stripe ordered phase of the cuprates[END_REF][START_REF] Li | Two-dimensional superconducting fluctuations in stripe-ordered La1.875Ba0.125CuO4[END_REF][START_REF] Arai | Incommensurate spin dynamics of underdoped superconductor YBa2Cu3O6.7[END_REF][START_REF] Ulbrich | Neutron scattering studies on stripe phases in non-cuprate materials[END_REF][START_REF] Hinkov | Spin dynamics in the pseudogap state of a high-temperature superconductor[END_REF][START_REF] Wu | Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy[END_REF][START_REF] Tabis | Connection between charge-density-wave order and charge transport in the cuprate superconductors[END_REF][START_REF] Ghiringhelli | Long-range incommensurate charge fluctuations in (Y,Nd)Ba2Cu3O6+x[END_REF][START_REF] Chang | Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67[END_REF][START_REF]Inelastic X-ray scattering in YBa2Cu3O6.6 reveals giant phonon anomalies and elastic central peak due to charge-density-wave formation[END_REF]132,[START_REF] Blanco-Canosa | Resonant X-ray scattering study of charge density wave correlations in YBa2Cu3O6+x[END_REF][START_REF] Kivelson | Electronic liquid-crystal phases of a doped mott insulator[END_REF][START_REF] Ando | Electrical resistivity anisotropy from self-organized one dimensionality in high-temperature superconductors[END_REF][START_REF] Daou | Broken rotational symmetry in the pseudogap phase of a high-tc superconductor[END_REF][START_REF]Electronic liquid crystal state in the high-temperature superconductor[END_REF][START_REF] Kivelson | How to detect fluctuating stripes in the high-temperature superconductors[END_REF][START_REF] Lawler | Intra-unit-cell electronic nematicity of the high-tc copperoxide pseudogap states[END_REF][START_REF] Da | Detection of electronic nematicity using scanning tunneling microscopy[END_REF][START_REF] Fauque | Magnetic order in the pseudogap phase of high-tc superconductors[END_REF][START_REF] Varma | Non-fermi-liquid states and pairing instability of a general model of copper oxide metals[END_REF][START_REF] Xia | Polar Kerr-effect measurements of the high-temperature YBa2Cu3O6+x superconductor: evidence for broken symmetry near the pseudogap temperature[END_REF][START_REF] Martin | Normal-state transport properties of Bi2+xSr2-yCuO6+[dgr] crystals[END_REF][START_REF] Chien | Effect of Zn impurities on the normal-state Hall angle in single-crystal YBa2Cu3-xZnxO7-[dgr[END_REF][START_REF] Valla | Temperature dependent scattering rates at the Fermi surface of optimally doped Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Sachdev | Quantum phase transitions[END_REF][START_REF] Zaanen | Superconductivity: why the temperature is high[END_REF][START_REF] Tallon | Energy and length scales in the superconducting phase diagram for htsc cuprates[END_REF][START_REF] Ramshaw | A quantum critical point at the heart of high temperature superconductivity[END_REF][START_REF] Castellani | Singular quasiparticle scattering in the proximity of charge instabilities[END_REF][START_REF] Vignolle | Quantum oscillations in an overdoped high-tc superconductor[END_REF][START_REF] Plate | Fermi surface and quasiparticle excitations of overdoped Tl2Ba2CuO6+[dgr[END_REF][START_REF] Wakimoto | Direct relation between the low-energy spin excitations and superconductivity of overdoped high-tc superconductors[END_REF][START_REF]Dispersive spin excitations in highly overdoped cuprates revealed by resonant inelastic x-ray scattering[END_REF][START_REF] Kaminski | Crossover from coherent to incoherent electronic excitations in the normal state of Bi2Sr2CaCu2O8+[dgr[END_REF][START_REF] Gozar | High-temperature interface superconductivity between metallic and insulating copper oxides[END_REF][START_REF] Frieman | Dark energy and the accelerating universe[END_REF][START_REF] Keimer | From quantum matter to high-temperature superconductivity in copper oxides[END_REF][START_REF] Božović | Dependence of the critical temperature in overdoped copper oxides on superfluid density[END_REF][START_REF] Cyr-Choinière | Two types of nematicity in the phase diagram of the cuprate superconductor yba2cu3oy[END_REF][START_REF] Lubashevsky | Optical birefringence and dichroism of cuprate superconductors in the thz regime[END_REF][START_REF] Zhang | Anomalous thermal diffusivity in underdoped yba2cu3o6+x[END_REF][START_REF] Torchinsky | Fluctuating charge density waves in a cuprate superconductor[END_REF][START_REF] Doiron-Leyraud | Berry phase in cuprate superconductors[END_REF][START_REF]Electronic liquid crystal state in the high-temperature superconductor yba2cu3o6.45[END_REF][START_REF] Mesaros | Topological defects coupling smectic modulations to intra-unitcell nematicity in cuprates[END_REF][START_REF] Phillabaum | Spatial complexity due to bulk electronic nematicity in a superconducting underdoped cuprate[END_REF][START_REF] Fernandes | What drives nematic order in iron-based superconductors?[END_REF][START_REF] Beekman | Dual gauge field theory of quantum liquid crystals in two dimensions[END_REF][START_REF] Božović | Atomic-layer engineering of superconducting oxides: yesterday, today, tomorrow[END_REF][START_REF] He | High-precision measurement of magnetic penetration depth in superconducting films[END_REF][START_REF] Wu | Hall effect in quantum critical charge-cluster glass[END_REF][START_REF] Abdel-Jawad | Anisotropic scattering and anomalous normal-state transport in a high-temperature superconductor[END_REF][START_REF] Kokalj | Transport properties of the metallic state of overdoped cuprate superconductors from an anisotropic marginal fermi liquid model[END_REF][START_REF] Vašek | Intrinsic pinning and guided motion of vortices in high-tc superconductors[END_REF].

However, after more than thirty years of intensive research, the origin of high-temperature superconductivity is still not completely understood. Fortunately, the recent developments of quantum physics and statistical physics, together with a new interpretation of quantum wave functions, provide the needed new insights for this challenging problem, which is the main aim of the paper.

New interpretation of wave functions in QM

It is inevitable that a better understanding of the nature of superconductivity should built upon the fundamental level understanding of quantum physics. In classical quantum mechanics, a micro-particle is described by a complex-valued wave function Ψ, satisfying such a wave equation as the Schrödinger equation (2.1) with external interaction potential V (x). The Shrödinger equation (2.1) conserves the energy, and the wave function Ψ can be expressed as Ψ = e -iEt/ ψ(x), where E is the energy, and ψ is the time-independent wave function.

The classical Born statistical interpretation of quantum mechanics amounts to saying that without constraints, the motion of a microparticle is random and there is no definite trajectory of the motion. Also |ψ(x)| 2 stands for the probability density of the particle appearing at the particular point x. The Born interpretation of the wave function is treated as a fundamental postulate of quantum mechanics. This leads to the classical Einstein-Bohr debates, and is the origin of absurdities associated with the interpretation of quantum mechanics; see among many others [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF][START_REF] Bohr | Can quantum-mechanical description of physical reality be considered complete?[END_REF][START_REF] Bell | On the Einstein-Podolsky-Rosen paradox[END_REF][START_REF] Griffiths | Introduction to elementary particles[END_REF].

Our new interpretation, presented in detail in Section 2, says that ψ = |ψ|e iϕ is the common wave function for all particles in the same class determined by the external potential V (x), |ψ(x)| 2 represents the distribution density of the particles, and

(1.1) m ∇ϕ
is the velocity field of the particles. The trajectories of the motion of the particles are then dictated by this velocity field, as shown in (2.10).

The observed particles are the particles in the same class described by the same wave function, rather than a specific particle in the sense of classical quantum mechanics. This is an entirely different interpretation from the classical Bohr interpretation. Also this new interpretation of wave functions does not alter the basic theories of quantum mechanics, and instead offers new understanding of quantum mechanics, and plays a fundamental role for the quantum theory of condensed matter physics and quantum physics.

It is worth mentioning that the Landau school of physics was the first who noticed that the relation between the superfluid velocity v s and the wave function ψ = |ψ|e iϕ of the condensate is given by (1.1); see [81, (26.12) on page 106]. However they fail to make an important connection between (1.1) and the basic interpretation of quantum mechanics, and consequently they were not able to derive the new quantum mechanism of condensates presented in this paper.

Quantum condensation mechanism and superconductivity

In view of the above new interpretation of quantum mechanic wave functions and the Pauli exclusion principle, under the condition of very weak interactions between particles, a large collection of Bose particles can undergo a motion governed by the same wave function field when the particles are all under the same bounding potential field. This gives rise to the physical picture of a condensate: a condensate is a large collection of bosonic particles with the same wave function field ψ, called the condensation wave function or the condensation field, determined by the same bounding potential field V .

The physical picture results immediately the quantum condensation mechanism, stared in Condensation Mechanism 3.1. One key ingredient of this mechanism is that the condition for condensation of bosonic particles is that their interaction is sufficiently weak to ensure that the a large collection of boson particles are in a state governed by the same condensation wave function field ψ under the same external bounding potential V . Another ingredient is that the condensation wave function ψ is the smallest allowed energy state of the wave equation. Hence the motion of particles in a condensate system under the same wave function field resembles the motion of stars in a galaxy, and can last permanently.

With this quantum condensation mechanism applied to superconducting systems, we obtain the following common physical mechanism of superconductivity:

• there is a critical temperature T c > 0 such that for T < T c , a large number of free electrons in the system form bosonic electron-pairs; and • the interactions between the bosonic electron-pairs and other particles in the system are negligible, so that the electron-pairs share a common interaction potential Φ. This common potential Φ enables the electron-pairs to occupy a single wave function field ψ with the velocity field (1.1), forming the condensate, and Φ is the potential given by the interaction between the lattice of the system and the external field.

With this mechanism, the formation of superconductivity comes down to two important aspects: 1) the physical mechanism and condition for the formation of electron-pairs, and 2) the condition for the electron-pairs to share a common wave function. Hereafter we address these two questions in turn.

Formation of electron-pairs

The key issue is to reveal in the microscopic level the needed attractive potential/force between two electrons to counteract the repulsive Coulomb potential/force. In the classical BCS theory [START_REF] Bardeen | Theory of superconductivity[END_REF], the Cooper pair theory provides the mechanism of electron-phonon interaction in low temperature superconductors, and a physical mechanism form the Bose electron-pair as required. However, the BCS theory is applicable only for T < 35K. In fact, in the BCS theory, the Cooper electron-pair is explained as the electron-phonon interaction. This interaction is reminiscent to the Yukawa interaction mechanism in quantum field theory. The electron-phonon interaction means that by exchanging phonons, electrons induces an attractive interaction. However, the physical picture of such an interaction is not clear. The reason is that phonon is a hypothetical pseudo-particle, and is introduced in the quantum field theory as a quantized energy to compute the binding energy ∆ of the Cooper pair. In other words, the phonon is essentially a hypothesis, introduced for computing ∆, that in addition to the repulsive Coulomb potential between electrons, the deformation of lattices induces an attractive interaction potential.

Recently, in [88], using the principle of interaction dynamics (PID) and the SU (N ) gauge theory for multi-particle systems, the authors have obtained a layered electromagnetic interaction potential formula, which can be used here to derive conditions to ensure the validity of the mechanism (4.1) and (4.2) for superconductivity. For an electron pair, this new interaction potential, modifying the classical Coulomb potential, takes the form:

(1.2) Φ e = e 0 r -e 0 A 0 (1 + kr)e -kr .
Here e 0 is the electric charge of an electron, the first term in the right hand side is the Coulomb potential, and the second term represents the effect of the dual field φ of the electromagnetic potential induced from PID. Notice also that there is also spin magnetic moment interaction between two electrons, which can lead to attractive potential between the electrons. We can show that the only stable spin arrangement is the anti-parallel electron-pair, with attracting spin magnetic moment potential given by -αµ 2 0 /e 0 r. Then we derive the interaction force formula of an anti-parallel electron-pair is given by

F = Coulomb + magnetic-moment + effect of PID dual field (1.3) = e 2 0 r 2 1 - α 2 4m 2 c 2 -k 2 A 0 re -kr e 2 0 .
The interval, on which F < 0, defines the attracting shell region of two electrons: r 0 < r < r 1 , whose existence is ensured by the condition:

(1.4)

A 0 > e 3 3 k 1 - α 2 4m 2 c 2 .
A similar condition (4.30) is also derived to ensure that an anti-parallel electron-pair possesses attracting negative potential.

Binding energy of electron-pairs

One key ingredient for a theory of superconductivity is, as in the BCS theory, to examine the binding energy 2∆(T ) of electron pairs. In a recent paper [START_REF]Statistical theory of heat[END_REF] on statistical theory of heat, we have derived an average energy level formula of temperature T . This, together with potential associated with the force (1.3), leads to the following new formula for the binding energy:

(1.5) ∆ = e 2 0 2r 0 A(1 + kr 0 )e -kr 0 + α 2 4m 2 c 2 -1 -ε(T ),
where A = r 0 A 0 is a parameter as in (1.2), and ε(T ) is the average kinetic energy of the two-electrons. One consequence of (1.5) is the equation for critical temperature T c , i.e. ∆(T c ) = 0:

(1.6) ε(T c ) = e 2 0 2r 0 A(1 + kr 0 )e -kr 0 + α 2 4m 2 c 2 -1 .
With the above formulas, one can then easily verify the two conditions on the formation of electron pairs and on the existence of a global wave function ψ for the condensate, as required in the physical mechanism of superconductivity stated earlier.

Critical temperature formulas

There are two approaches to find the critical temperature formula. One is to use the standard model of thermodynamics based on the thermodynamic potential (free energy) under no external magnetic field; see [START_REF] Liu | Thermodynamical potentials of classical and quantum systems[END_REF]:

(1.7) F = -g 0 |ψ| 2 + 1 2 g 1 |ψ| 4 - α 0 2 T S 2 -α 1 T S|ψ| 2 -ST,
where g 0 > 0 is the binding potential of condensates, g 1 is the coupling constant of particles, α 0 and α 1 are entropy coupling constants. Then the equilibrium state solves the Euler-Lagrangian equation of the potential functional and transition from the normal state to the superconducting state occurs at the critical temperature T c , which makes the first eigenvalue of the linearized equilibrium state equation vanishing.

We derive then the following macroscopic formula for T c :

(1.8)

T c = g 2 0 g 1 ∆C V (1 -α 0 ∆C V ).
The microscopic critical temperature formula is derived using on (1.6) and the average energy level formula of temperature [START_REF]Statistical theory of heat[END_REF]:

(1.9) k B T c = N 1 N 2 (1 + θ 0 ) N 2 E 0 ,
where E 0 = ε(T c ) is the intrinsic energy of electron-pairs given explicitly by (1.6), N = N 1 + N 2 is the total number of particles in the system, N 1 , N 2 are respectively the numbers of free electrons and lattice particles, and θ 0 is the following ratio at the critical temperature:

(1.10) θ = average energy level of lattice particles average energy level of free electrons .

The paper is organized as follows. Section 2 introduces the new interpretation of quantum mechanical wave functions, Section 3 derives 1) the quantum mechanism of condensates and superconductivity, and 2) the field and pattern formation equations of condensates. In Section 4, we derive 1) the physical mechanism of superconductivity, 2) the conditions on formation of electron-pairs, 3) binding energy, and 4) the microscopic and macroscopic formulas for the critical temperature.

New Interpretation of Quantum Wave Functions 1). Classical interpretations of wave functions

The classical quantum mechanics tells us that a micro-particle can be described by a complex-valued wave function Ψ : Ω → C, which satisfies a wave equation such as the Schrödinger equation:

(2.1) i ∂Ψ ∂t = - 2 2m ∆Ψ + V (x)Ψ, x ∈ Ω,
where Ω ⊂ R 3 is the region that the particle occupies, m is the mass of the particle, and V (x) is the external interacting potential. The system (2.1) conserves the energy, and hence the wave function Ψ can be expressed as

(2.2) Ψ = e -iEt/ ψ(x),
where E is the energy, and ψ is the time-independent wave function. Putting (2.2) into (2.1), we derive that

(2.3) - 2 2m ∆ψ + V (x)ψ = Eψ.
The solution of (2.3) can be expressed as

(2.4) ψ = |ψ|e iϕ ,
where ϕ is the phase of ψ.

The Born interpretation of quantum mechanics amounts to saying that Let the momentum density of ψ be expressed as momentum density of the particle = m|ψ| 2 v, where v is the velocity of the particle. Then we infer from (2.4) and (2.7) that

(2.8) v = m ∇ϕ.
Namely the velocity is the gradient of the phase of the wave function (2.4). (2.8) is consistent with the classical physical conclusion that the current density J is given by (2.9)

J = ie 2m (ψ∇ψ -ψ ∇ψ).

2). New interpretation of the wave function

Notice that (2.8) is the velocity field of particles, rather than the velocity of a single particle. This leads to the new interpretation of the wave function ψ as follows:

(1) Under the external potential field V (x), the wave function ψ in (2.3) is the field function for the motion of all particles with the same mass in the same class determined by the external potential V (x). In other words, it is not the wave function of a particular particle in the classical sense; (2) When a particle is observed at a particular point x 0 ∈ Ω, then the motion of the particle is fully determined by the solution of the following motion equation with initial position at x 0 :

(2.10)

dx dt = m ∇ϕ(x), x(0) = x 0 ,
where ϕ is the phase of the wave function ψ in (2.4); (3) With ψ being the field function, (2.11) |ψ(x)| 2 = distribution density of particles at x;

(4) The energy E in (2.3) represents the average energy level of the particles and can be written as

(2.12) E = Ω 2 2m |∇|ψ|| 2 + 2 2m |∇ϕ(x)| 2 + V (x)|ψ| 2 dx,
where in integrand on the right-hand side, the fist term represents the non-uniform distribution potential of particles, the second term is the average kinetic energy, and the third term is the potential energy of the external field. Here ∇|ψ| is characteristic of quantum mechanics and there is no such a term in classical mechanics.

3). Physical meaning of new quantum interpretation

The above new interpretation of wave functions does not alter the basic theories of quantum mechanics. Instead it offers new understanding of quantum mechanics, and plays a fundamental role for the quantum theory of condensed matter physics below.

The classical Born statistical interpretation of quantum mechanics amounts to saying that without constraints, the motion of a microparticle is random and there is no definite trajectory of the motion. Also |ψ(x)| 2 stands for the probability density of the particle appearing at the particular point x. This leads to the classical Einstein-Bohr debates, and is the main reason for absurdities associated with the interpretation of quantum mechanics.

The new interpretation says that ψ = |ψ|e iϕ is the common wave function for all particles in the same class determined by the external potential V (x), |ψ(x)| 2 represents the distribution density of the particles, and m ∇ϕ is the velocity field of the particles. The trajectories of the motion of the particles are given by (2.10). The observed particles are the particles in the same class described by the same wave function, rather than a specific particle in the classical quantum mechanics. Hence the initial position of a particle is determined by the distribution |ψ| 2 , and its motion trajectory is fully determined by (2.10). This is an entirely different interpretation from the classical Bohr interpretation, and will lead to impact to quantum physics and condensed matter physics.

In summary, there are field functions ψ 1 , ψ 2 , |ψ| and ϕ in the wave function

ψ = ψ 1 + iψ 2 = |ψ|e iϕ ,
whose physical meanings are given as follows:

(1) ψ 1 and ψ 2 are the conjugate functions of the following Hamilton system

∂ψ 1 ∂t = k δ δψ 2 H(ψ), ∂ψ 2 ∂t = -k δ δψ 1 H(ψ);
(2) |ψ(x)| 2 represents the distribution density of particles;

(3) ∇|ψ| is the potential describing the non-uniform distribution of particles, and 2 2m |∇|ψ|| 2 is potential energy for the non-uniform distribution of particles; and (4) m ∇ϕ is the velocity field of particles.

Quantum Mechanism of Condensation

3.1. Quantum condensation mechanism. Based on the new interpretation of quantum mechanics presented in the previous section, we now establish the quantum mechanism for the micro-structure and its formation of condensates. This new mechanism serves as a unified physical picture and formation mechanism for superconductivity, superfluidity and Bose-Einstein condensation.

1). Physical picture of the formation of condensates

We start with the mechanism of the formation of condensates. The new interpretation of quantum mechanics shows that the wave function is no longer the state function of a specific particle, and is now considered as the field function governing the motion of particles. The motion of particles in the field of a wave function is similar to the motion of macroscopic objects in a gravitational field. This new viewpoint enables us to describe precisely the physical picture of condensates.

By the Pauli exclusion principle, under the condition of very weak interactions between particles, a large collection of Bose particles can undergo a motion governed by the same wave function field, as long as all the particles are in the same external potential field. Hence the physical picture of a condensate is as follows:

(3.1)
a condensate is a large collection of Bosons in the same wave function field ψ determined by the same external potential field V . We call ψ the condensation wave function or the condensation field.

2). Microscopic quantum mechanism

Condensation occurs at low temperatures. The main reason is that at higher temperature, particles possess relatively high kinetic energy, causing them to collide (scattering) with each other. Collision are interactions. Hence under such collisions, each particle is situated at an independent interaction field with its own wave function, and consequently the condensation picture (3.1) does not happen.

However, at a relative low temperature, the kinetic energy of particles decreases since temperature is the average energy level of particles. Hence the possibility of collision decreases, leading to the formation of a common external potential field. Also, since particles with the same wave function field do not collide, the system energy decreases. By the minimum potential principle, the system has the tendency toward condensation of particles. This is the mechanism of the physical condensation depicted in (3.1).

3). Collisionless

We now use the motion law (2.10) to verify that particles of a condensate do not collide with each other.

Since the large collection of particles share a common wave function field ψ = |ψ|e iϕ , their motion is dictated by the velocity field m ∇ϕ as shown in (2.10). For different particles, their initial positions are different so that they will not collide with each other. For example, consider two different particles A an B with initial positions at x 1 and x 2 , i.e.

x A (0,

x 1 ) = x 1 , x B (0, x 2 ) = x 2 ,
where x A (t, x 1 ) and x B (t, x 2 ) are the trajectories of particles A and B, which are solutions of

dx dt = m ∇ϕ(x)
with initial data x 1 and x 2 . It is then clear that if

x 1 = x 2 , then x A (t, x 1 ) = x B (t, x 2 ), ∀t > 0.
In other words, particles A and B can never collide with each other. The motion of particles in a condensate resembles the motion of planets around the nucleus of a galaxy.

4). Minimal energy

For N particles on a state described by one wave function field ψ 0 , by (2.12), their total energy is given by

N E 0 = N 2 2m |∇|ψ 0 || 2 + 2 2m |∇ϕ 0 | 2 + V 0 |ψ 0 | 2 dx.
On the other hand, for N particles such that each particle occupies its own state described by its own wave function ψ i , their total energy is

N i=1 E i = N i=1 2 2m |∇|ψ i || 2 + 2 2m |∇ϕ i | 2 + V i |ψ i | 2 dx.
When condensation occurs, the particles collectively occupy the minimal energy state, so that

E 0 ≤ E i (1 ≤ i ≤ N ).
Hence the total energy of the system satisfies:

N E 0 < N i=1 E i ( if there is an E i such that E i > E 0 ).
Namely, when the condensate particles occupy the minimal energy levels, the total energy of the system is less than the total energy of the particles in the normal states. By the minimal potential principle [START_REF]Dynamic law of physical motion and potential-descending principle[END_REF], the system will be in a condensate state.

5). Condensation mechanism of superconductivity and superfluidity

In summary, we have derived the following condensation formation mechanism. This is the mechanism of superconductivity, superfluidity and BEC.

Condensation Mechanism 3.1.

(1) The condition for condensation of bosonic particles is that their mutual interaction is sufficiently weak to ensure that they are in a state governed by the same wave function field under the commom bounding potential V ;

(2) a large collection of bosonic particles occupy the same wave function ψ, called condensation wave function, that is determined by the wave equation under the same potential V ; (3) the condensation wave function ψ is the smallest allowed energy state of the wave equation; and (4) the motion of particles in a condensate system under the same wave function with velocity field (2.8) resembles the motion of stars in a galaxy, and can last permanently.

Field equations of condensates.

There are two cases of condensates: The first is the case where the system is at near the critical temperature T c , the condensation is in its early stage and the condensed particle density is small. The second is the case where away from the critical temperature, the system enters a deeper condensate state. The field equations for these two cases are different and will be addressed separately below.

1). Condensation field equations near T c

At this stage, the field equations are the variational equations of the thermodynamic potential. Let F (ψ, Φ) be the thermodynamic potential, ψ be the wave function, and Φ = (S, H, E), where S is the entropy, H is the electromagnetic field, and E is the electric field. Then the field equations are the equilibrium equations of the thermodynamic phase transition:

(3.2) δ δψ * F (ψ, Φ) = 0, δ δΦ F (ψ, Φ) = 0.

2). Fully developed condensation field equations

For the fully developed condensation away from critical temperature, the system obeys both the principle of Lagrangian dynamics and the principle of Hamiltonian dynamics, leading to two equivalent field equations.

Let H be the Hamiltonian energy functional. Then the Lagrangian dynamical equation is given by

(3.3) i ∂Ψ ∂t = δ δΨ * H(Ψ)
. Also, the Hamiltonian dynamic equations are

(3.4) 2 ∂Ψ 1 ∂t = δ δΨ 2 H(Ψ), 2 ∂Ψ 2 ∂t = - δ δΨ 1 H(Ψ).
where

Ψ = Ψ 1 + iΨ 2 .
The equations (3.3) and (3.4) are equivalent, and both will be used in describing different properties of quantum systems of condensates. Hence we reduce (3.3) to the steady state form given by (3.8).

1). Gaseous and liquid condensate systems

For gaseous and liquid systems, (3.2) and (3.8) can be written in a unified form as follows:

(3.9) -

2 2m ∆ψ + f (|ψ|)ψ = 0,
where f (x) is a function of x ∈ R. If ψ is a spinor, then f (x) is a matrix-valued function of x ∈ R 3 , and ψ can be written as

ψ = ζe iϕ , ζ = (ζ + , ζ 0 , ζ -),
as the three types of particles in the system possess the same mass, and consequently share the same velocity fieldm ∇ϕ.

In view of (3.5), we infer from (3.9) the following pattern formation equation for gaseous and liquid condensate systems:

div(ζ 2 ∇ϕ) = 0, (3.10) - 2 2m [∆ζ -ζ|∇ϕ| 2 ] + f (ζ)ζ = 0. (3.11)

2). Superconducting systems

For superconducting systems, (3.2) and (3.8) can be written in the following unified form:

(3.12) 1 2m s i ∇ + e s c A ψ + f (|ψ|)ψ = 0,
with superconducting currents:

J s = - e 2 s m s c 2 |ψ| 2 A - e s 2m s c i(ψ * ∇ψ -ψ∇ψ * ).
In view of (3.5), J s becomes (3.13)

J s = - e 2 s m s c 2 ζ 2 A + e s m s c ζ 2 ∇ϕ.
Also by (3.5), (3.12) is rewritten as On the other hand, the pattern formation equations are fundamental equations for studying quantum phase transitions. In fact, quantum phase transition deals with the phenomena of structure changes in the physical space of pattern formation equations (3.16) of condensate systems at critical control parameters.

div ζ 2 ∇ϕ - e s c ζ 2 A = 0, (3.14) - 2 2m s ∆ζ -ζ|∇ϕ| 2 + e 2 s 2m s c 2 A 2 ζ + f (ζ)ζ = 0. ( 3
4. Theory for High-Temperature Superconductivity 4.1. Physical mechanism of superconductivity. In the previous section, we introduced Quantum Condensation Mechanism 3.1. For superconducting systems, this mechanism can be explicitly stated as follows:

(4.1)

• there is a critical temperature T c > 0 such that for T < T c , a large number of free electrons in the system form bosonic electron-pairs; and

(4.2)

• the interactions between the bosonic electron-pairs and other particles in the system are negligible, so that the electron-pairs share a common interaction potential Φ. This common potential Φ enables the electron-pairs to occupy a single wave function field ψ, forming the condensate, and Φ is the potential given by the interaction between the lattice of the system and the external field.

This is the common physical mechanism of superconductivity. The classical BCS theory [START_REF] Bardeen | Theory of superconductivity[END_REF] can be used to explain superconductivity in low temperatures. The Cooper pair theory provides the mechanism of electron-phonon interaction in low temperature superconductors, and a physical mechanism form the Bose electron-pair as required in (4.1). However, the BCS theory is applicable only for T < 35K.

However, in 1986, Bednorz and Müller [START_REF] Bednorz | Possible high Tc superconductivity in the Ba-La-Cu-O system[END_REF] discovered superconductivity in a lanthanum-based cuprate perovskite material, which had a transition temperature of 35 K. Bednorz and Müller's work opened the door for high Tc superconductors. The current highest temperature of superconductors reaches T c = 130, and T c = 160K under high pressure.

The discovery of high Tc superconductors shows the limitation of the BCS theory. In fact, in the BCS theory, the Cooper electron-pair is explained as (4.3) the electron-phonon interaction, whose physical meaning (picture) is hard to understand. This interaction is reminiscent to the Yukawa interaction mechanism in quantum field theory. The electron-phonon interaction (4.3) means that by exchanging phonons, electrons induce an attractive interaction; the physical picture of such an interaction is not clear. Phonon is a hypothetical pseudo-particle. In fact, in quantum field theory, to compute the binding energy ∆ of the Cooper pair, a quantized energy needs to be introduced as the binding energy, and is called phonon. In other words, phonon is essentially a hypothesis, written in the form (4.3), and is introduced for computing ∆. This hypothesis can be expressed in the following easier understandable fashion:

(4.4)

In addition to the repulsive Coulomb potential between electrons, the deformation of lattice induces an attractive interaction potential.

In summary, both (4.3) and (4.4) are different expressions of the same hypothesis. For the BCS theory, regardless of the validity of this hypothesis, the key issue is if there is a good agreement between experiments and the computational results derived under the existence assumption of the binding energy ∆ between electrons. Indeed, such agreement is only observed in low temperature cases, but not in high temperature cases. Hence it is worth pointing out that the validity of a result does not ensure the correctness of the hypothesis introduced to derive the result, since reasons for the same result are usually not unique.

PID interaction potential of electrons.

The classical theory of electromagnetism shows that there is a repulsive interaction force between two electrons:

F = e 2 r 2
(e is the electric charge), preventing the paring of electrons, called screening of Coulomb repulsion. The formation of Cooper electron-pairs in the BCS theory makes the hypothesis that the lattice deformation induces the electron-phonon attractive interaction: when free electrons are near the Fermi surface, attractive force F 1 > Coulomb force F.

Notice also that there exists Coulomb repulsive force between two Cooper pairs:

(4.5) F = 4e 2 r 2 , which will cause collision and scattering between Cooper-pairs, preventing condensation and superconducting behaviors of particles. In other words, the repulsive force (4.5) makes the mechanism (4.2) impossible.

In [88], using the principle of interaction dynamics (PID) and the SU (N ) gauge theory for multi-particle systems, the authors obtained a layered electromagnetic interaction potential formula, which can be used here to derive conditions to ensure the validity of the mechanism (4.1) and (4.2) for superconductivity. For the sake of completion, we recapitulate this formula below.

Consider a particle A with electric charge Q = ne 0 with either e 0 < 0 or e 0 > 0. Let there be a test electron e at distance r from particle A. By the field equations [88, (6.5.20) and (6.5.21)] for N particles, the SU (N ) (N = n + 1) electromagnetic potentials generated by particle A acting on the electron:

(4.6) {A a µ | 0 ≤ µ ≤ 3, 1 ≤ a ≤ N 2 -1}
satisfy the following field equations, using the standard SU (N ) generators to ensure that T r(τ a τ + b ) = 2δ ab :

∂ ν A a νµ - e 0 c λ a bc g αβ A b αµ A c β -e 0 Ψγ µ τ a Ψ = [∂ µ - 1 2 k 2 x µ ]φ a , (4.7) iγ µ ∂ µ + ie 0 c A b µ τ b Ψ - c M Ψ = 0, (4.8)
where M is the mass matrix of the particles,

Ψ = (Ψ 1 , • • • , Ψ N ) T , Ψ i (1 ≤ i ≤ N ) represent i wave function (Dirac spinors), (g αβ ) is the Minkowski metric, λ a
bc are the structure constants of SU (N ), φ a are the dual potentials of {A a µ } , γ µ are the Dirac matrices, γ µ = g µα γ α , τ a = τ a , and

A a νµ = ∂ ν A a µ -∂ µ A a ν + e 0 c λ a bc A b µ A c ν .
We note here that the field equations [88, (6.5.20) and (6.5.21)] for N particles are derived from first principles: principle of Lorentz invariance, principle of gauge invariance, Einstein's principle of general relativity, principle of representation invariance (PRI), principle of Lagrangian dynamics and principle of interaction dynamics (PID).

We now derive from (4.7) and (4.8) the electron interaction potential. Thanks to the PRI applied to SU (N ), there is an SU (N ) representation tensor {θ a | 1 ≤ a ≤ N 2 -1} such that (4.9)

A µ = θ a A a µ (µ = 0, 1, 2, 3)
representing the total electromagnetic potential induced by the N electron charges. Namely,

A 0 electric potential, A = (A 1 , A 2 , A 3 ) magnetic potential.
Let the radius of particle be r A , Φ(r) = A 0 (r) be the electric potential acting on the test electron e. The additivity of electric potential shows that the µ = 0 component of (4.7) is linear, taking the following form; see [88] for details:

(4.10) -∆Φ = e 0 Q 0 - 1 4 k 2 cτ φ, -∆φ + k 2 φ = -e 0 div Q,
where cτ represents the wave-length of the electron, k is the electric charge interaction parameter,

φ = θ a φ a , (Q 0 , Q) = Q µ = -θ a Ψγ µ τ a Ψ,
and θ a is as in (4.9). Here (4.11)

Q 0 = nB 0 r e r A 3 δ(r), div Q = B 1 r A δ(r),
where r A , r e are radii the particle A and the electron respectively, δ(r) is the Dirac function, B 0 , B 1 are parameters.

Then with the same procedure and derivation in [88, the layered strong interaction formula (4.5.41)], we obtain from (4.10)-(4.11) the following (4.12)

Φ(r) = nB 0 r e r A 3 e 0 1 r -A 0 (1 + kr)e -kr ,
where B 0 is as (4.11), k is as (4.10), and A 0 can be expressed explicitly as

A 0 = k 2 cτ r 2 A B 1 4nr 3 e B 0 .
Here A 0 depends on the types of particle A and the test electron, and is determined by experiments. Formula (4.12) represents the electric potential induced by particle A, where (4. [START_REF] Blanco-Canosa | Resonant X-ray scattering study of charge density wave correlations in YBa2Cu3O6+x[END_REF] q A = nB 0 r e r A 3 e 0 represents the effective electric charge of A, and B 0 depends on the particle type of A. The physical meaning of the effective charge is apparent in (4.14) and (4.15) below. For two effective charges q 1 and q 2 , the electric potential between them is

(4.14) V 12 (r) = q 1 q 2 1 r -A 12 (1 + kr)e -kr ,
where A 12 is a parameter depending on the types of the two particles.

The force between the two particles with effective charges q 1 and q 2 is then given by (4.15)

F 12 = -q 1 q 2 d dr 1 r -A 12 (1 + kr)e -kr .

4.3.

Condition for formation of electron-pairs. The field equations (4.7) and (4.8), together with the electron interaction potential and force formulas (4.14) and (4.15), form the theoretical basis of the new theory on electron-pairs in superconductivity in this section.

1). Electric charge interaction between two electrons

When both particles are electrons, n = 1, B 0 = 1, r A = r e . Then (4.12) becomes (4. [START_REF] Božović | Dependence of the critical temperature in overdoped copper oxides on superfluid density[END_REF])

Φ e = e 0 r -e 0 A 0 (1 + kr)e -kr , which is the modification of the classical Coulomb potential. The first term in the right hand side is the Coulomb potential, and the second term represents the effect of the dual field φ of A µ , induced from PID. By (4.16), the interaction force between two electrons is given by (4.17)

F e = -e 0 ∇Φ e = e 2 0 r 2 -k 2 A 0 e 2 0 re -kr .
There is a short range attractive force in the right hand side of (4.17)

(A 0 > 0): f = -k 2
A 0 e 2 0 re -kr < 0. It is this force f , together with magnetic moment of electrons, that induces a short range attractive force to counteract the repulsive Coulomb force.

2). Spin magnetic moment interaction of electrons

Electrons possess spin magnetic moment, with numerical value given by (4.18)

µ 0 = - e 0 2mc
, where e 0 < 0 is the electric charge of an electron, and m is the mass of electron. We can define the direction of the spin magnetic moment, as the north pole N , and the opposite direction as the south pole S.

Then there is a spin magnetic moment interaction force (4. [START_REF] Castellani | Singular quasiparticle scattering in the proximity of charge instabilities[END_REF])

F m = ± αµ 2 0 r 2
, where α is the magnetic moment interaction coefficient. Physically, we know that (4.20) 0 < α < e 2 0 /µ 2 0 . There are two electron orientation arrangements that can produce the magnetic moment force given by (4.19): one is the parallel arrangement, as shown in

3). Stability of orientation arrangements of electrons

For the two electron orientation arrangements that produce attractive spin magnetic moment force, the antiparallel electron-pair shown in Figure 4.1(b) is stable, and the one in Figure 4.2 (b) is unstable. There are two main reasons behind the stability and instability of these two cases. First, since the collinear arrangement of an electron-pair possesses electric charge 2e 0 and spin J = 1, it induces a magnetic moment with intensity: µ = -2e 0 /mc. This causes magnetic moment interaction between electron-pairs. Such long arrangement is also the source of instability, and two reversed parallel electron-pairs as shown in This implies that the two electrons in an electron-pair must rotate around a mass center. Hence the rotating arrangement of electrons in Figure 4.1 (b) is stable, and the one in Figure 4.2(b) is unstable. We arrive then at the following conclusion:

(4.21) only electron-pairs with total spin J = 0 is stable.

4). Attracting shell region of electron-pairs

By (4.17) and (4.19), the interaction force formula of two anti-parallel electrons is given by

(4.22) F = e 2 0 r 2 1 - αµ 2 0 e 2 0 -k 2 A 0 re -kr e 2 0 .
The interval where F takes negative values defines the attraction shell region of two electrons: Here the condition for (4.24) having two positive roots is that the maximum of r 3 /e kr is greater than the right-hand side of the above equation:

3 e 3 > k A 0 1 - αµ 2 0 e 2 0
, where e 2.718. By (4.18), the above condition can be rewritten as

(4.25) A 0 > e 3 3 k 1 - α 2 4m 2 c 2
, which is the condition to ensure the existence of the attracting shell region (4.23).

5). Attracting potential of electron-pairs

The attracting interaction potential of magnetic moment for an electronpair is

V m = - αµ 2 0 r = - α 2 4m 2 c 2 r e 2 0 .
By (4.16), the total interaction potential of an anti-parallel electronpair is

(4.26) V = e 2 0 r 1 - α 2 4m 2 c 2 -e 2 0 A 0 (1 + kr)e -kr .
By the relation (4.22) between the potential V and the force F :

F = - d dr V,
The first zero r 0 of F in (4.23) is the minimum point V :

(4.27) V (r 0 ) = min V (r).
The condition for an anti-parallel electron-pair to possess a shell region of attracting potential is V (r 0 ) < 0, i.e.

(4.28)

A 0 > e kr 0 r 0 (1 + kr 0 ) 1 - α 2 4m 2 c 2 .
This is the condition for an electron-pair to possess attracting interaction potential.

It is clear that we can view r 0 as the theoretical radius of electronpairs. Let A > e kr 0 1 + kr 0 1 -

α 2 4m 2 c 2 .
Obviously if this condition holds true, then (4.25) holds true as well for kr 0 > 0.

E(p) = (E p -E F ) 2 + ∆(T ).
where E F is the energy of Fermi level, and E p = 1 2m p 2 is the kinetic energy of the particle. Also, ∆ is determined by the following equation, called energy-gap equation:

(4.33) N (0)V ∞ 0 1 (2π ) 3 E(p) 1 - 2 1 + exp(E(p)/kT ) dp = 1,
where E(p) is as in (4.32), N (0) is the Fermi energy density with temperature at absolute zero, and V is the intensity of attracting portion of potential of electrons. The above three formulas (4.31)-(4.33) are the three main ingredients of the BCS theory, and the remaining components of the BCS theory can be derived from these three formulas. By ∆(T c ) = 0, we derive from the energy-gap equation (4.33) the following formula for the critical temperature T c :

(4.34) kT c = 1.14 ω D exp[-1/N (0)V ],
where ω D is the Debye frequency, N (0) and V are as in (4.33). The highest critical temperature formula (4.34). Hence for high Tc superconductivity, the BCS theory is invalid, and in particular for high Tc superconductivity, the electron-phonon interaction mechanism for electron-pairs is not convincing. We now establish the new theory of binding energy of electrons, suitable for high Tc superconductivity, based on the new interaction theory of electrons established earlier in this article.

1). Binding energy of electron-pairs

The previous section has shown that under condition (4.30), two electrons with reversed parallel orientation can form an electron-pair, with radius r 0 satisfying (4.27).

In the BCS theory, the binding energy 2∆ of electron-pairs is determined by the energy-gap equation (4.33). The binding energy 2∆ can also be expressed using the potential in (4.26) as follows:

(4.35) 2∆ = -V (r 0 ) -mv 2 ,
where mv 2 stands for the total average kinetic energy of the two electrons. By the the average energy level formula of temperature, we can view the average kinetic energy of the two electrons as a function of temperature:

(4.

A(1 + kr 0 )e -kr 0 + α 2 4m 2 c 2 -1 -ε(T ),
where A is a parameter as in (4.29)-(4.30). Formula (4.37) is the formula for the binding energy ∆ of electronpairs of superconductivity, in terms of average kinetic energy of electrons and temperature function ε(T ). At low temperature, ∆ as well as ε(T ) are determined by (4.33), showing that the new theory is consistent with the BCS theory at low temperatures. Also, physically, by (4.35), the critical temperature T c satisfies ∆(T c ) = 0, which amounts to saying that

(4.38) ε(T c ) = e 2 0 2r 0 A(1 + kr 0 )e -kr 0 + α 2 4m 2 c 2 -1 ,
which is the new temperature formula, applicable for both low and high T c cases. 3). Verification of mechanism (4.2) of superconductivity

For T < T c , electron-pairs with radius r 0 emerge in a superconducting system. By (4.13), the effective charge of a superconducting electron-pair is

q 0 = 2B 0 r e r 0 3 e 0 .
The classical radius of an electron is r e = e 2 0 mc 2 = 2.8 × 10 -13 cm, and the radius of an electron-pair r 0 has the same order of magnitude as the diameter of the particle lattice of the system: ρ ∼ 10 -8 cm. Hence q 0 ∼ 10 -13 B 0 e 0 .

In other words, the effective electric charge of an electron-pair is negligibly small, almost electric neutral. Hence according to the interaction force formula of electric charge (4.15), the interaction between superconducting electron-pairs and particles in the system are extremely weak. Consequently this verifies the mechanism (4.2) of superconductivity. 4.5. Formulas for the critical temperature T c . We now compute the critical temperature T c from both the microscopic and macroscopic approaches. The macroscopic approach is based on the thermodynamic potential (free-energy) in the standard model of thermodynamics, and the microscopic approach is based on (4.37). 1). Formula for T c based on the standard model First we recall that the thermodynamic potential for superconductivity. Under no external magnetic field and homogeneous conditions, the potential functional for superconductivity is in the following form; see [START_REF] Liu | Thermodynamical potentials of classical and quantum systems[END_REF] for its detailed derivation:

(4.40) F = -g 0 |ψ| 2 + 1 2 g 1 |ψ| 4 - α 0 2 T S 2 -α 1 T S|ψ| 2 -ST,
where g 0 > 0 is the binding potential of condensates, g 1 is the coupling constant of particles, α 0 and α 1 are entropy coupling constants. The entropy is determined by

(4.41) ∂F ∂S = 0 ⇒ S = - 1 α 0 - α 1 α 0 |ψ| 2 .
On the other hand, we have

(4.42) - ∂ ∂ψ * F (ψ) = g 0 ψ -g 1 |ψ| 2 ψ + α 1 T Sψ.
In view of (4.41) for entropy S, we have

(4.43) - ∂ ∂ψ * F (ψ) = β 1 ψ -g 1 + α 2 1 α 0 T |ψ| 2 ψ,
where β 1 represents the eigenvalue of the linearized operator of (4.43), given by

β 1 (T ) = g 0 - α 1 α 0 T.
By the theory of phase transition dynamics [START_REF]Phase Transition Dynamics[END_REF], β 1 (T ) = 0 provides the critical temperature:

|ψ| 2 = β 1 (T ) g 1 + α 2 1 T /α 0 = α 1 α 0 (T c -T ) g 1 + α 2 1 T /α 0 .
Plugging (4.45) into (4.41), we obtain that

S =      - 1 α 0 - α 1 α 0 2 (T c -T ) g 1 + α 2 1 T /α 0 + S 0 (T ) for T < T C , S 0 (T ) for T C < T.
Therefore, we derive that

(4.46) ∆C v = T c ∂S ∂T T - c - ∂S 0 ∂T = α 1 α 0 2 T c g 1 + α 2 1 T c /α 0 ,
where ∆C V is the increment of heat capacity at T c . By (4.46), we have (4.47)

α 2 0 α 2 1 = T c g 1 ∆C V (1 -α 0 ∆C V ),
where it is required that (4.48)

α 0 ∆C V < 1.
This is the condition for superconductivity to occur. By (4.44) and (4.47), we deduce that (4.49)

T c = g 2 0 g 1 ∆C V (1 -α 0 ∆C V ).
This is the formula for critical temperature T c based on the standard model.

Remark 4.1. The formula (4.49) of the critical temperature T c depends on the parameters g 0 , g 1 , α 0 and ∆C V . Here ∆C V is the increment of heat capacity at the critical temperature T c , which is a theoretical value without taking into consideration of fluctuations and does not reflect the real measurement.

2). Formula for T c based on microscopic theory Formula (4.38) provides the theoretical basis for computing T c . To obtain a formula for T c , we need consider the expression for the function ε(T ). First, ε(T ) is defined by (4.36), and represents the average energy level of electron gas in the superconducting system. By the statistical theory of heat [START_REF]Statistical theory of heat[END_REF], we can derive an expression for ε(T ) as follows: where γ is called the energy ratio of the system, and can be expressed as

(4.51) γ = N 2 N 1 N 2 (1 + θ) .
Here N is the total number of particles in the system, N 1 , N 2 are respectively the numbers of free electrons and lattice particles such that N 1 + N 2 = N , and θ is (4.52) θ = average energy level of lattice particles average energy level of free electrons .

We shall prove (4.50)-(4.52) later. It is clear that θ defined by (4.52) depends on the temperature, and consequently γ depends on the temperature as well. Let γ 0 be the temperature ratio of the electron gas at the critical temperature T c , then by (4.50)-(4.51), the formula for T c in (4.38) can be rewritten as (4.53)

k B T c = E 0 γ 0 = N 1 N 2 (1 + θ 0 ) N 2 E 0 .
Here E 0 is the intrinsic energy of electron-pairs, expressed as 3). Derivation of the expression for ε(T )

First we recall the average energy level temperature formula in [START_REF]Statistical theory of heat[END_REF]:

(4.55) k B T = 1 N n a n ε n 1 + β n ln ε n 1 - a n N ,
where ε n is the n-th energy level, a n is the number of particles on the energy level ε n , N = n a n is the total number of particles. A superconducting system can be considered as a binary system of free electron gases and lattice-particles. Therefore, (4.55) can be written as (4.56)

N 1 N 2 N 2 ε electron 1 + β 1 ln ε electron + N 1 N 2 N 2 ε lattice 1 + β 2 ln ε lattice ,
where N 1 , N 2 are respectively the numbers of free electrons and latticeparticles in the system, ε electron and ε lattice are the average energy levels of electrons and lattice-particles. We approximately take β 1 , β 2 = 0, then the above expression reduces to where ε electron = 1 2 mv 2 is the average kinetic energy. Therefore, (4.58) implies (4.50).

4). Theoretical analysis of T c

From (4.53), we can deduce the following property of high Tc superconducting materials:

the energy level for the lattice-particle is much larger than that of free electrons, and consequently θ 1. We have derived basic analysis of the new microscopic theory of superconductivity, and further analysis needs to consider the effect and physical meaning of β 1 , β 2 = 0 in (4.56).

Also, the formula (4.49) based on the standard model indicates that the critical temperature T c is a decreasing function of the increment of heat capacity. We now demonstrate that the microscopic theory of T c can reveal such property as well. Recall the heat capacity formula The phase transition is due to condensation of the electron gas, and is independent of the lattice. Hence the heat capacity for the lattice at T c is continuous. Hence at T c , γ 0 is given by

C V = ∂E ∂T , E is
γ 0 ∼ ∆C V + δ (δ = γ(T - c )
). We infer then from (4.53) that

k B T c ∼ E 0 ∆C V + δ .
This provides the same property as the macroscopic formula (4.49) based on the standard model. In summary, (4.49) is derived using the standard model of thermodynamics, and the microscopic formula (4.53) is derived using PID interaction theory of electrons and the statistical theory of heat. With different origins, these two formulas possess the same property.

2 .

 2 PID interaction potential of electrons 18 4.3. Condition for formation of electron-pairs 20 4.4. Binding energy of superconducting electron-pairs 24 4.5. Formulas for the critical temperature T c

(2. 5 )

 5 |ψ(x)| 2 represents the probability density of finding the particle at x . Also, basic quantum mechanics shows that (2.6) -i ∇ = momentum operator. Hence (2.7) momentum density of ψ = imaginary part of ∇ψ • ψ * .

3. 3 .

 3 Pattern formation equations of condensates. Let the wave function ψ of a condensate system be expressed as (3.5) ψ = ζe iϕ , ζ = |ψ|. In Section 2, we have derived the physical meaning of ζ and ϕ: (3.6) ρ = ζ 2 distribution density of particles in the condensate, m ∇ϕ the velocity field of the flow of particles. Hence ζ and ϕ describe the structure of condensates in the physical space, and the equations of ζ and ϕ are called pattern formation equations. The pattern formation equations for ζ and ϕ can be derived from the field equations (3.2) and (3.3) as follows.First, near the critical temperature, (3.2) is the steady state equation of(3.5). Away from the criticality, (3.3) are evolution equations. The energy of the Hamiltonian system (3.4) is conserved, i.e. the number of particles is conserved. Hence the solution of (3.3) can be written as: (3.7) Ψ = e -iλt/ ψ(x), ψ is as (3.5), where λ ∈ R represents the chemical potential. By (3.7), we infer from (3.3) that (3.8) λψ = δ δψ * H(ψ).

  .15) In (3.14), we have used the Coulomb gauge divA = 0. By (3.13) and (3.14), we obtain that divJ s = 0. The above two systems of equations: (3.10)-(3.11), and (3.14)-(3.15) are the pattern formation equations in the physical space for superfluidity, superconductivity and BEC. These are important equations. On the one hand, they dictate two important physical structures of condensate particles: (3.16) • physical picture of the distribution function ρ = ζ 2 , and • topological structure of the flow determined by the velocity field m ∇ϕ.

Figure 4 .

 4 1 (a) and (b); the other is collinear arrangement as shown in Figure 4.2 (a) and (b). The plus sign in (4.19) represents repulsive force, corresponding to the cases of Figures 4.1

  (a) and 4.2(a). Hence only Figures 4.1(b) and 4.2(b) may give rise to superconducting electron-pairs.

Figure 4 . 1 .

 41 Figure 4.1. (a) Parallel orientation leads to repulsive magnetic moment force; and (b) anti-parallel orientation leads to attractive magnetic moment force.

Figure 4 . 2 .

 42 Figure 4.2. (a) Anti-collinear orientation leads to repulsive magnetic moment force; and (b) collinear orientation leads to attractive magnetic moment force.

  Figure 4.2(b) can break form two electron-pairs with the structure as shown in Figure 4.1 (b). Second, by the Heisenberg uncertainty relation, for an electron-pair, the distance between the two electrons and the kinetic energy p = mv of the electrons satisfy the following inequality: v ≥ 2mr .

(4. 23 )

 23 r 0 < r < r 1 (r 0 > 0), where r 0 , r 1 are two positive roots of F (r) = 0

4. 4 .

 4 Binding energy of superconducting electron-pairs. The key ingredient of the BCS theory of superconductivity is to postulate the existence of the binding energy of electron-pairs for T < T c : (4.31) binding energy of electron-pair = 2∆(T ) for T < T c , ∆(T c ) = 0. The relation between ∆(T ) and the energy spectrum E(p) of the electronpair is (4.32)

2 ).

 2 Verification of mechanism (4.1) of superconductivity By (4.36), ε(T ) is an increasing function of T : d dT ε(T ) > 0. Hence we infer from (4.37) and (4.38) that (4.39) ∆(T ) = 0 for T ≥ T c , d dT ∆(T ) < 0 for T < T c . This verifies the mechanism (4.1) of superconductivity. Namely, no superconducting electron-pairs emerge for T ≥ T c , superconducting electron-pairs emerge for T < T c . By (4.21), we note that total spin of electron-pairs J = 0.

(4. 44 )

 44 T c = α 0 g 0 /α 1 . Consider T c in (4.44). If T < T c , by (4.43) and ∂ ∂ψ * F (ψ) = 0, we deduce the solution for superconducting phase (4.45)

  (4.50) ε(T ) = γk B T, k B is the Boltzmann constant,

1 ++ α 2 4m 2 c 2 - 1 .

 11 kr 0 )e -kr 0 Formulas (4.53) and (4.54) provide the needed formula for T c based on the microscopic theory.

( 4 .

 4 [START_REF] Hinkov | Spin dynamics in the pseudogap state of a high-temperature superconductor[END_REF])k B T = N 1 N 2 N 2 ε electron + N 1 N 2 N 2 ε lattice . Let θ = ε lattice /ε electron ,then we infer from (4.57) that (4.58)k B T = N 1 N 2 N 2 (1 + θ)ε electron ,

  the internal energy. By (4.50), we haveN 1 k B γ = N 1 ∂ε ∂T= heat capacity of electron gas.

The work was supported in part by the US National Science Foundation (NSF), the Office of Naval Research (ONR) and by the Chinese National Science Foundation.


Yifeng Jiang

Lai Chung Liu

Henrike M Müller-Werkmeister

Cheng Lu

Dongfang Zhang

Ryan L Field

Antoine Sarracini

Gustavo Moriena

Eric Collet

R J Dwayne Miller

Structural Dynamics upon Photoexcitation in a Spin Crossover Crystal Probed with Femtosecond Electron Diffraction

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Structural Dynamics upon Photoexcitation in a Spin Crossover

Crystal probed with Femtosecond Electron Diffraction

Yifeng Jiang 1 , Lai Chung Liu 2 , Henrike M. Müller-Werkmeister 1 , Cheng Lu 2 , Dongfang Zhang 1 , Ryan L. Field [START_REF] Bousseksou | [END_REF] , Antoine Sarracini 2 , Gustavo Moriena 2 , Eric Collet 3 , R. J. Dwayne Miller The current industrial drive towards device miniaturization has created considerable interest in switchable molecular systems. Among these are transition metal complexes, which show great promise in various domains, such as molecular electronics, highdensity magnetic data storage, data displays, and dye-sensitized solar cells. [START_REF] Brady | Spin Crossover in Transmission Metal Compounds III[END_REF] They exhibit a reversible phase transition, known as SCO, between a low-spin (LS) and a high-spin (HS) state that can be triggered externally through pressure, temperature, magnetic fields, or light. [START_REF] Bousseksou | [END_REF]3] In particular, the photoinduced case has been intensely studied in iron(II)-complexes and it has been found to occur in solution [4] and in the solid state (doped lattices, nanocrystals, and bulk crystal). [5][6][7] The ultrafast photoinduced dynamics of solvated SCO complexes have been studied using X-ray spectroscopies [8 -11] X-ray scattering, [12] Raman spectroscopy [13] and transient absorption spectroscopies. [14,15] The relaxation process leading to SCO in these compounds proceeds from an initial excitation from the 1 A1 LS state into the metal-to-ligand charge-transfer (MLCT) manifold, from which the system relaxes to the 5 T2 HS state within <200 fs via multiple intersystem crossings. [15][16][17] From the HS state, the vibrationally hot molecule is believed to relax through IVR within ~1-2 ps. [15,16] In solids, ultrafast photoinduced SCO has been investigated using ultrafast time-resolved XANES [17,18] , transient reflectivity, transient absorption and femtosecond magneto-optical Faraday spectroscopy. [16 -20] Picosecond X-ray diffraction [21] and time-resolved optical spectroscopy [22] have been applied to track further processes much later after photoexcitation. These studies suggest that, for SCO compounds with both thermal and photoinduced SCO, the early <10 ps ultrafast dynamics in crystals are due to photoinduced dynamics in a manner comparable to those observed in solution, while the late-time, >0.1 ns changes in HS fraction are driven by cooperative elastic and thermal processes. [21,22] The structure of a Fe(II) metal compound ([Fe(PM-AzA)2](NCS)2 cis-bis(thiocyanato)-bis(N-2'-pyridyl-methylene)-4-phenyl-azo) aniline iron(II)) is shown in Figure 1a. It undergoes thermally induced SCO at T1/2 = 184 K, [18,19] as shown in Figure S1 in Supplementary Information (SI). The photoinduced SCO in single crystalline [Fe(PM-AzA)2](NCS)2 has previously been identified as a two-step ISC process with a fast (<200 fs) relaxation process through short-lived intermediate states (INT) followed by IVR within the photoinduced HS state on a timescale of ~2 ps. [18,19] Recent ultrafast dynamics studies [9,12,[17][18][19] have concluded that the Fe-N bond elongation is directly correlated in time (200 fs) to the electronic state dynamics observed by changes in the spectral features of the complex, followed by IVR. However, previous picosecond X-ray diffraction studies [21] underlined that the structural rearrangement inside the unit cell, through inter-molecular contact, is a much more complex process and direct femtosecond structural studies on the overall photoinduced SCO are lacking, despite indirect information gained from different experiments [21] .

Femtosecond electron diffraction (FED) directly probes atomic positions by scattering off the Coulomb potential of the nuclei, [23- 27] capable of resolving the above issues related to the structural relaxation associated with spin transitions. FED is employed in a pump-probe protocol, using femtosecond laser pulses to trigger the ultrafast dynamics and femtosecond electron pulses to probe the resulting nuclear motions (Figure 1b). Here, we reveal the structural dynamics upon photoexcitation during the IVR that accompanies the spin-state switching with high spatial precision, which is specific to FED and in contrast to X-ray diffraction due to its shorter carrier wavelength for probing atomic motions.

FED experiments were performed with excitation conditions similar to the previously reported optical pump-probe reflectivity studies. [18,19] Samples were kept at 170 K, pumped at 800 nm (pulse duration = 60 fs, fluence = 1.28 mJ cm -2 ), and probed by 267 fs electron pulses. To allow full recovery of the sample from HS to LS (<1 ms), [19] the repetition rate was set to 100 Hz. At the fluence used, no damage over the duration of the experiment was observed, see SI for additional information and results from control experiments.

Figure 2a shows a static diffraction pattern of ([Fe(PM-AzA)2](NCS)2 at 170 K. Figure 2b depicts the change of the static electron diffraction patterns corresponding to the structural changes associated with the thermal phase transition, from below T1/2 or 'low-temperature' (LT, T = 170 K) to above T1/2 or 'high-temperature' (HT, T = 300 K). The observed change in the Bragg peak intensities and positions are closely related to those resulting from the photoinduced transition. The difference patterns resulting from photoinduced SCO (measured at 170 K and time delay t = +10 ps) are shown in Figure 2c. They match the expected Bragg peak intensity changes simulated using the LS and HS X-ray structures in Figure 2d. Figure 3a gives the transient behavior of some selected Bragg spots for the short timescale (-5-22 ps) and Figure 3b for the longer timescale (-50-1000 ps).

The most noteworthy feature in Figure 3a is that the intensity of all observed Bragg peaks change exponentially, decaying or rising with almost the same time constant of 2.3 ps. In SI Section 6, further discussion and analysis is presented to support this observation. We also note that, for Bragg peaks (h = k = 0, l ≠ 0) such as , ,and , the shifts in peak positions have the same time constant as the changes in the peak intensity. These displacements are consistent with a local expansion of the unit cell that takes places concurrently with the ultrafast nuclear motions.

Previous literature [17][18][19] suggests an ultrafast Fe-N bond elongation after photoexcitation, but only slow Fe-N bond elongation is observed in our measurement. To unravel the underlying structural changes responsible for this transient behavior, we calculated the structure factor changes for a selection of the Bragg peaks. Two possible pathways for atomic motions were considered. The first model (Figure 4a) involves only the <200 fs ultrafast Fe-N bond elongation. Figure 4b shows the corresponding simulated diffraction intensities. Alternatively, we considered a second model where the Fe-N bond elongation is coupled to ligand motion, and local unit cell expansion (Figure 4c) upon IVR of the hot excited HS state; the calculated intensities for this scenario are shown in Figure 4d. Clearly, the structure values of the former case do not match the experimentally observed difference pattern (shown in Figure 4e) while those of the latter do match relatively well.

We calculated the contributions of different molecular motions to the observed Bragg peak intensities. For select peaks, these values are shown in Figure 5. The ligand motion is the major contributor to the change in intensity: (

) is purely based on ligand motion while all other peaks, most dominantly ( ), , and have weighted contributions of Fe-N bond elongation and ligand motion. The intensities of these Bragg peaks should exhibit bi-exponential behavior due to a fast Fe-N bond elongation, driving intensity change within the ~300 fs time resolution, and the ligand motion occurring in the following 2 ps, an effect which we did not observe experimentally (Figure 3a).

The time constant of our observations matches the timescale of IVR. [28] Previous work using transient UV-VIS spectroscopy observed the narrowing of the excited state absorption feature with a 2 ps time constant, [14,20] while recent XANES work [29] suggests that Fe-N bond distribution after photoexcitation narrows on the same timescale as spectral features associated with IVR. Our results are therefore consistent with previous findings and can be explained by IVR of hot excited HS molecules and energy transfer from Fe-N stretching motions to the rearrangements of the ligands and the unit cell. Indeed, after photoexcitation, the LS molecule converts into the HS state in 200 fs timescale with a great deal of excess energy, activating coherent and incoherent molecular vibrations, as observed by a related XANES study by Lemke et al. [29] Therefore, the energy dissipation through ligand rearrangement associated with this process would occur on the timescale of IVR. Similarly, the motion of the ligands causes extension of the molecule in the unit cell, driving local unit cell expansion through intermolecular contracts. The resulting elastic stress field acts as a chemical pressure to the whole crystal. [30] Thus, the apparent slow Fe-N bond elongation can be explained by the 2 ps bond distribution narrowing during IVR, which modifies the relative weight of Fe-N and ligand rearrangements in diffracted intensity, since the electron diffraction probes the ensemble average of the Fe-N bond lengths.

We should underline that the ultrafast structural transition studied here is a purely light-induced SCO process. Although deposited laser energy in the crystal lattice can also result in thermal SCO and lattice expansion, this only happens with inhomogeneous lattice expansion between 10 ns and 10 µs, followed by homogenized crystal expansion and heat-induced SCO at 50 µs. [21,22] Another noteworthy feature in Figure 3a is the plateau behavior observed for time delay from 5 ps to 24 ps. This is indicative of the formation of the excited HS structure further supporting our model (as illustrated in Figure 4c). To obtain a time-dependent map of the molecular motions during SCO, we applied a structural refinement algorithm based on a parameterized molecular model. [23] The structural transformation from the LS to HS states can be decomposed into three independent dynamical groups: Fe-N bond elongation (N); ligand motion to HS structure (L); unit cell expansion (U). The term ξ is an atomic structural parameter composed of the three time-dependent generalized reaction coordinates (ξN ξL ξU) assigned to the motion of these groups.

Figures 6a-c depict the three key dynamical groups as a function of time, representing the ultrafast structural dynamics. The time constant of these key modes is 2.3 ps. The time-dependent changes of ξ are summarized in Figure 6d. The reaction coordinates of three motions represent relative atomic motions from the initial LS state at ξ = (0, 0, 0) to towards the final HS state at ξ = (1, 1, 1). Previous picosecond X-ray diffraction studies on later processes associated with SCO indicate a local excited HS molecule within a LT unit cell, which subsequently changes to a HT unit cell through molecular rearrangement. [21,22] Here, we find that the photoexcited HS structure, with ξ = (0.9, 0.8, 0.75), is similar but distinct from the thermal HS structure since the photoexcited HS molecule is constrained inside the smaller LT unit cell. We experimentally detect a local unit cell expansion on the ultrafast timescale, which is induced by the Fe-N bond elongation and ligand motion through intermolecular elastic interaction, which was not observed in previous studies. This is of key interest to understand how IVR and light-induced intermolecular forces within the unit cell drive lattice expansion that is subsequently responsible for cooperative elastic spin-state switching. [21] The full relaxation pathway and physical picture based on this atomically resolved structural transition is shown in Figure 7. The initial intersystem crossing occurs through Franck Condon factors, weighted by the overlap in the MLCT manifold, that creates a highly excited vibrational superposition state that then undergoes IVR and narrowing of the Fe-N bond distribution to the fully relaxed configuration on the timescale of IVR. [16,28] In this study, the ultrafast structural changes associated with photoinduced SCO in [Fe(PM-AzA)2](NCS)2 have been directly characterized for the first time using FED with near-uniform sample excitation. The electron probe used is uniquely sensitive to changes in the crystal structure of the sample following photoexcitation. The photoinduced structural changes between LS and HS states are very similar to the ones observed at thermal equilibrium, which is not always the case as symmetry breaking was reported in some photoinduced states. [31][32][33][34] These global changes (Fe-N bond elongation, ligand motions, and local unit cell expansion) were all found to share the same time constant (2.3 ps). The structural dynamics can be fully modelled by three key modes, such that the atomic motions induced by IVR after SCO are monitored with ultrafast temporal resolution and atomic spatial resolution. The picture that emerges is consistent with previous studies while fully depicting the relaxation processes to the HS state.

Figure 1 .

 1 Figure 1. a) Low-spin molecular structure of [Fe(PM-AzA)2](NCS)2;arrangement of the FeN6 complexes within the crystallographic unit cell. b) Schematic of our femtosecond electron diffraction setup.[23]

Figure 2 .

 2 Figure 2. Comparative electron diffraction patterns with peaks of interest highlighted: a) data measured at low temperature (T = 170 K); b) difference between LT and HT data; c) signal induced by photoexcitation, measured after 10 ps time delay; d) difference between LT and HT data, simulated using static X-ray diffraction structures.

Figure 4 .

 4 Figure 4. Model comparison. a) Model 1: Elongation of Fe-N coordination bond by an average of 0.2 Å. c) Model 2: Simultaneous nuclear motions from low-spin to high-spin structure. b), d), Simulated change in electron diffraction from model 1 and 2, respectively. e) Signal induced by photoexcitation.

Figure 5 .

 5 Figure 5. Contribution to relative intensity change. Comparing changes from Fe-N bond elongation versus all other nuclear motions.

Figure 6 .

 6 Figure 6. Molecular movie. a)-c), Time-dependence of refined reaction coordinates. d), Plot of independent refinement over reaction-coordinate space.

Figure 7 .

 7 Figure 7. Schematic representation of the proposed reaction pathway of SCO in configuration space. Photoexcitation of the singlet ground state is followed by sub 200 fs ISC dynamics which put the molecule onto the quintet potential surface. This vibrationally hot HS state has a broad Fe-N bond distribution, which narrows dramatically over 2.3 ps as IVR activates ligand motions and drives further reorganization of the cold unit cell.

  1,2 * 1 Max Planck Institute for the Structure and Dynamics of Matter Luruper Chaussee 149, 22607, Hamburg, Germany 2 Departments of Chemistry and Physics, University of Toronto, 80 St. George Street, Toronto, Ontario, M5S 3H6, Canada 3 Université Rennes 1, CNRS, Institut de Physique de Rennes, UMR 6251, UBL, F-35042 Rennes, France Abstract: Photoexcitation of spin crossover (SCO) complexes can trigger extensive electronic spin transitions and transformation of molecular structure. However, the precise nature of the associated ultrafast structural dynamics remains elusive, especially in the solid state. Here, we studied a single crystal SCO material with femtosecond electron diffraction (FED). The unique capability of FED allows us to directly probe atomic motions and to track ultrafast structural changes within a crystal lattice. By monitoring the time-dependent changes of the Bragg reflections, we observed the formation of a photoinduced structure similar to the thermally induced high-spin state. The data and refinement

calculations indicate the global structural reorganization within 2.3 ps, as the metal-ligand bond distribution narrows during intramolecular vibrational energy redistribution (IVR) driving the intermolecular rearrangement. Three modes are identified to model the structural dynamics upon photoinduced SCO.

Acknowledgements:

The authors thank G. Sciaini and H. D. Pinto for helpful discussions. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC, L. L., R. J. D. M.), the Max Planck Society and the excellence cluster "The Hamburg Centre for Ultrafast Imaging -Structure, Dynamics and Control of Matter at the Atomic Scale" of the Deutsche Forschungsgemeinschaft (R. J. D. M.), by the ANR (ANR-13-BS04-0002, E.C) and the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 623994 (H. M.-W.). We declare no conflict of interest.

Keywords: spin crossover • photochemistry • N ligands • electron diffraction • structure elucidation