1

(PACS: 74.20.Mn 74.25.F- )

(Keywords: ion crystal, ion chain, valence electrons, superconductivity mechanism, electron pairing)

Electron-pairing in ionic crystals and mechanism of superconductivity

(Author: Q. LI)

Abstract

The behaviors of valence electrons and ions, particularly ion chains, in some ionic crystals are important to understanding of the mechanism of superconductivity. The author has made efforts to establish a candidate mechanism of electron-pairing and superconductivity in ionic crystals.

Analyses are first made to a one-dimensional long ion lattice chain model (EDP model), with the presence of lattice wave modes having frequency ω. A mechanism of electron pairing is established.

Analyses are then extended to scenarios of 3D ionic crystals, particularly those with a donor/acceptor system, with emphasis being given to the interpretation and understanding of binding energy of electron pairs formed between electrons at the top/bottom of donor/acceptor band and the bottom/top of conducting/full band.

It is established that once the lattice/EM wave modes are established in its range, which can be long or even macroscopic, electron pairs are produced in the crystal’s electron system over the same range by stimulated transitions induced by the EM wave mode. The lattice wave mode having the maximum frequency ωM is of special significance with respect to superconductivity, for electron pairs produced by it can be stabilized in the context of a combination of some special factors (including energy level structure featured by donor/acceptor band and ωM) with a binding energy typically no smaller than hωM/(2π). A candidate mechanism of electron pairing in ion crystals and therefore of superconductivity is provided. Introduction

The behaviors of valence electrons and ions, particularly ion chains, in some ionic crystals are important to understanding of the mechanism of superconductivity. The author has made efforts to establish a candidate mechanism of electron-pairing and superconductivity in ionic crystals.

Analyses are first made to a one-dimensional long ion lattice chain model (EDP model), with the presence of lattice wave modes having frequency ω. A mechanism of electron pairing is established.

Analyses are then extended to scenarios of 3D ionic crystals, particularly those with a donor/acceptor system, with emphasis being given to the interpretation and understanding of binding energy of electron pairs formed between electrons at the top/bottom of donor/acceptor band and the bottom/top of conducting/full band.

2

It is established that once the lattice/EM wave modes are established in its range, which can be long or even macroscopic, electron pairs are produced in the crystal’s electron system over the same range by stimulated transitions induced by the EM wave mode. The lattice wave mode having the maximum frequency ωM is of special significance with respect to superconductivity, for electron pairs produced by it can be stabilized in the context of a combination of some special factors (including energy level structure featured by donor/acceptor band and ωM) with a binding energy typically no smaller than hωM/(2π). A candidate mechanism of electron pairing in ion crystals and therefore of superconductivity is provided.

Generalized analyses of 1-D long ion lattice chain model

It has been established that for a one-dimensional long ion lattice chain, under the assumptions that only the interactions between neighboring ions are considered and that the interaction energy are approximated up to its quadratic term, the general solutions of lattice waves have the form of [1]:

ω±

2=β(M+m)/Mm){1±[1-4Mmsin22πaq/(M+m)2]} (B/A) ± =-(mω±

2-2β)/ (2βcos2πaq) with -1/4a<q≤1/4a, where a is the equilibrium distance between neighboring ions, A and B are the magnitude of the first and second ions respectively, M and m are the mass of the first and second ions respectively, and β is the tension of interaction between neighboring ions. With Born–Karman boundary condition exp(-2πi2Naq)=1, we have: q=n/(2Na), with n=±1, ±2,…. ±N/2.

The above solution of ω+ peaks at q=0, so the optical waves with q=±1/(2Na) has the maximum ω+ value of the system, with value of ω- being always smaller than that of ω+. There will be a total of 2N lattice waves for a total of N ions in the chain, which therefore include all the oscillating modes of the chain.

Thus, the time-dependent potential field can be written as: V(x,t)=V0(x)+ G(x) Σsinωt

where the summation is over all the lattice waves ω, V0(x) is the static potential field of the dipole chain without vibration, and G(x)= G(x+a) is a periodic function of x.

With H=H0 + G(x) Σsinωt and H0=V0(x). We have special solutions: ψn(x,t)= φn (x)exp (iEnt/h) where φn (x) being the static solution of static periodic filed V0(x). With perturbation G(x)sinωt, ψ(x,t)= Σan(t) φn(x) exp(iEnt/h), with a n = a n0+a n1+a n2+…. a n0=δnk, and a n

k1 ∝ ∫Vnk(t) exp (i(En-Ek)t/h)dt

Vnk(t)= ∫φn*(x) G(x)Σsin(ωt) φk(x)dx

with Enk =En-Ek, we have: a n

k1 ∝Σ(exp(2πi(Enk+hω)t/h)/ (Ep+Enk)-exp(-2πiωt(Enk- hω)t/h)/ (Ep-Enk) (Equ. 1-3)

Here we can see that the first term on the right side in Equ.1-3 corresponds to

the probability that the electron absorbs a photon (or phonon and etc.) to transit from

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En to Ek, while the second term corresponds to the probability that the electron emits a photon to transit from Ek to En.

a nk1 has a total of 2N peaks at Enk=±hω/(2π) corresponding to q=n/(2Na), with

n=±1, ±2,…. ±N/2. For illustrative purpose, we identified the maximum one of all ω values as ωM, which corresponds to the “optical” wave at q=±1/(2Na).

As indicated by Equ. 1-3, a nk1 converges to Enk=±hω/(2π) along with time t, and

after some time t, almost all electrons in the system will transit with Enk=±hω/(2π) (where ω has N discrete values), that is: a n

k1 →Σδ(Enk-hωm/(2π)), where m=1, 2, 3…..denotes the different lattice/EM wave modes of the ion chains, with ωM being the greatest one among them. Electron-pairing

However, a well-established fact is that all electrons in a crystal are in energy bands, and in many ion crystals electrons form full bands. Thus, for typical hω/(2π) of lattice wave modes, most (if not all) electrons in energy bands cannot normally transit as indicated by (Equ. 1-3).

The way for the electrons to cope with this is that they form themselves into “pairs”, so that both of the two electrons in each pair, having energy En and Ek respectively (here we can safely assume that En>Ek), can transit by exchanging their states, with the electron originally at energy En emitting a photon (or phonon or the like) of energy hω/(2π)=En-Ek, which is directly absorbed (virtual photon emission/absorption) by the other electron, which is originally at Ek.

With a nk1 →Σδ(Enk-hωm/(2π)) with time t, only transitions corresponding to

En-Ek= hωm/(2π) will exist in the system after sufficient time t. This process results in that each energy level in the bands of the system become distinguishable during stimulated transitions of electrons.

It is to be noted that whether an electron absorbs/emits a phonon or photon in

the above transition does not affect any of the conclusions of this paper, for these absorptions/emissions involved in electron-pairings and/or stimulated transitions are virtual; they do not need to actually happen. But as the above discussed electron transitions and pairings in the ionic crystals are generated by the oscillating field of EM wave modes, it is photons that are absorbed/emitted during these transitions and pairings. Electron-pairing/exchange in 3D ionic crystals

A crystal with N primitive cells has 3nN oscillating modes, where n is the number of atoms/ions in one primitive cell. As according to a report of neutron non-elastic scattering experiment on KBr crystal [2], ω values of different wave modes have the relation: LO>TO>LA>TA. The report also shows that, for each crystal orientation, the maximum of ω is at q→0 of the LO (longitudinal optical) modes; the report further shows that for KBr crystal the maximum of ω in crystal orientation [111] is greater than that in crystal orientation [100], so electron pairs corresponding to ω in [100] will be broken by some of the phonon/photons in [111]. This indicates that only a crystal orientation with the maximum ωM of all possible crystal orientations may correspond to the direction of prospective superconductivity, for it is the direction corresponding to the ωM of the surviving electron pairs.

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Due to limitation of Pauli Principle, electrons in the same pair in system

ψ(t)=U(t,t0)ψ(t0) have opposite spins. Generally, both spontaneous transition and stimulated transition exist in a

system of ψ(t)=U(t,t0)ψ(t0). (Spontaneous transition may be limited by occupancy conflicts in crystals.) In ionic crystals, lattice is formed by ion chains, so vibration modes of lattice generate oscillating electromagnetic (EM) wave modes of the same frequencies as the lattice vibration modes. Thus, the stimulated transitions in an ion crystal are those driven by such oscillating electromagnetic wave modes.

For two electrons in the electron system of such an ion crystal, if their energy difference matches the frequency of one of the lattice vibration modes, the stimulated transitions of the two electrons in the pair can be in the form of their exchange of states between themselves, that is, by pairing themselves with each other. In such pairing, the stimulated transitions of the two electrons become “virtual”- the stimulated transitions need not to happen in reality, especially in the sense that the electrons concerned are non-distinguishable. Under complete “occupancy conflict” (that is, all prospective targeted states for transition of the electrons concerned have been occupied by other electrons,) such electron-pairing/exchange becomes the only way for the electrons to perform the stimulated transitions as required by ψ(t)=U(t,t0)ψ(t0) with U(t,t0) →Σδ(Enk-hωm/(2π)).

Electron pairing and binding energy in an acceptor-doped system If, in the energy band system of the crystal orientation corresponding to ωM, an

acceptor energy band with energy levels Ei1<Ei2<Ei3… is introduced in a full band system of an ionic crystal (see FIG. 1), with Ei1-Esmax equals to or slightly smaller than hωM/(2π), where Esmax being the highest energy level in the full band and ωM being the greatest frequency of the oscillating electromagnetic wave modes associated with the ion chains in the crystal, then, since there are stimulated transitions corresponding to Enk=hωM/(2π), electrons on Esmax level of the full band can transit to Ei1 by stimulated transition of Enk=hωM/(2π), thus forming a new system including the acceptor energy level Ei1 and the original system ψ(t), and this new system (ψ(t)+{Ei1}) is conductive.

Binding energy of electron pairs relating to acceptor band

More generally, for example, assuming that ωM, Esmax, and, say, Ei2 satisfy hωM/(2π)=Esmax- Ei2, a new system (ψ(t)+{Ei1}+{Ei2}) is then formed including the acceptor energy levels Ei1 and Ei2 and the original system ψ(t).

If some holes (such as those left by electrons transiting to the acceptor band) exists in the full band, the electrons pair like (φij +φsmax) (with j=1,2,…) can be broken by transition of the lower electron in the pair to any of the holes. So with the presence of even one hole in the full band, the electron pairs like (φij +φsmax) could not be stable. But a pool of electron pairs (φij +φsmax) in dynamic equilibrium could possibly be maintained across the top of the full band and the bottom of the acceptor band.

However, as an insulator is easily charged, especially under external electrical field/voltage, if the ionic crystal is negatively charged, electrons are injected into the system, thus filling the holes in the full band; in such a scenario, the above electron pairs (φij +φsmax) will be stabilized up to a possible binding energy.

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Then,, a key and subtle factor here is determination of the binding energy of

such an electron pair (φij +φsmax). Due to the limitation of the particular energy structure of this scenario, if the

electron pair is to be broken by transition of an electron in the pair, then at least on electron in the pair has to transit to the energy level Ei3 or higher.

: : : :

As the macroscopic energy of the combined system is, by definition, the average

of the measured energy values over long time, the contribution to the macroscopic energy by the electron that still remains on (φi2 +φsmax) after the other electron transits to Ei3 (or higher) is one half of hωM/(2π) (that is, the average value of the energy values at φi2 and φsmax), while that by the electron that transits to Ei3 (or higher) is some value greater than hωM/(2π), so the change in the macroscopic energy is an increase of at least hωM/(4π).

However, the half photon energy seems strange and ridiculous. An alternative approach is by the argument that the electron transiting to Ei3

actually does not have the energy hωM/(2π) at the moment just before it makes the transition. A model for this is that the electron pair includes the two electrons plus a photon with an energy hωM/(2π), which binds the two electrons together to form the electron pair. This is phenomenologically in conformity with that virtual photon exchange happens when the two electrons exchange their states, as indicated by the expression of Equ. 1-3 discussed above. The two electrons in the pair co-occupy the correlated states of (φij +φsmax) without specifying which electron is in which of the two states. As the pair is broken in a general situation (without the limitation of band

FIG. 1: p-doped system

full band

electron

: : :

Ei1

Esmax

acceptor band

Ei2

Ei3

Ei1- Esmax =hωM/(2π)-Δ

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structure as in the present system), the photon might be taken by any of the two electrons, taken by the corresponding EM wave mode, or even emitted as a free photon. But in the particular situation of the energy level structure under consideration, one of the two electrons must take one photon of hωM/(2π) (not necessarily the photon originally within the pair, though) to go to an energy level at or above Ei3, which is the only way for it to go; the remaining electron will take another photon (which can be the original one) of hωM/(2π) to stay at and transit between (φij +φsmax).

(So the question here is, in the scenario that the electron alternatively emits and absorbs such a photon of hωM/(2π), whether the EM wave “spends” the energy of the photon entirely for the electron transition or reserves half of the photon’s energy as part of EM wave’s own energy? If the former is true, the binding energy of the electron pair concerned is hωM/(2π), otherwise, the binding energy would be halved.)

Thus, where the one half of photon energy is missed is recognized: the photon

associated with the remaining electron at (φij +φsmax) is omitted. Therefore, the energy of the combined system as discussed above should be

increased by hωM/(2π) after the electron pair is broken, which should be the binding energy of this electron pair when there is not hole in the full band.

We then consider the distribution function of Gibbs’ canonical ensemble of the

combined system of the electrons, the lattice, and the EM wave modes associated with the lattice. The proportion ρ of members of the ensemble before the transition of the electron to Ei3 being [4] ρ(E1) ∝ exp(-βhωM/(2π)), while that after the transition of the

electron to Ei3 being ρ(E2)∝ exp(-2βhωM/(2π)) (where 1/β=kT). So ρ(E2)/ρ(E1)=exp(-βhωM/(2π)). This is in fact the probability that the electron pair is broken by any transition (in this particular energy band structure), with hωM/(2π) being the binding energy of the pair against destruction by transition of an electron in the pair.

As an estimation of the stability of such an electron pair with such a binding energy, ωM/(2π)≈1013-1014/s, at T=100K there will be (hωM/(2π))/(kT)≈4.65-46.5. Thus, for ωM above, say, 5x1013/s, such an electron pair can rarely be broken by a phonon even at T=100K [3].

Similarly, Ei1 can also form a superconducting electron pair with an electron on

a corresponding energy level below Esmax, with a binding energy no smaller than hωM/(2π).

Further, in some samples, in a range of Δ=hωM/(2π)-(Ei1-Esmax) there can be a plurality of energy levels Ei1<Ei2<Ei3… in the acceptor band, and each of these energy levels may have an electron forming a stabilized electron pair with an electron at a corresponding level in the acceptor band. But if Δ increases to the extent as making hωM/(2π)-Δ=Ei1-Esmax≤maximum frequency of LO modes corresponding to any other crystal orientation, superconductivity may never happen.

Electron pairs in donor band system The mechanism of superconductivity in a system having a donor band

(Ei1>Ei2>Ei3……) is similar to that having a acceptor band, except that the donor band is beneath a conducting band, with the lowest level Esmin in the conducting band being

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higher than the highest level Ei1 in the donor band by a difference equal to or slightly smaller than hωM/(2π). (see FIG. 2)

For illustration, we assume Esmin-Ei2=hωM/(2π). Then, electrons on donor energy levels Ei1 and Ei2 may enter the conducting band by stimulated transition by the EM wave mode of ωM, and may then form electron pairs with electrons which later transit to Ei1 and Ei2. But these electron pairs are unstable as far as any hole(s) (particularly the holes left by the electrons transiting to the conducting band) exists in the donor band, for the electron at the lower energy of each of the electron pairs can easily transit to such a hole.

If, however, the hole(s) in the donor band are somehow filled, those pairs formed by electrons on the conducting band with the electrons on levels Ei1 and Ei2 will become stabilized, with the similar mechanism as explained above with respect to acceptor band.

: : : :

A scenario is that external electrons may enter the system, at a relatively high

energy level (particularly under external electric field/voltage), and transit to the donor band or levels, so that holes in the donor band are filled and the electron pairs are stabilized.

Similar to a system with an acceptor band, the energy level range of Δ=hωM/(2π)-( Esmin-Ei1) can be increased to accommodate a plurality of energy levels so that a plurality of electron pairs can be formed between respective energy levels at the bottom of the conducting band and those at the top of the donor band (see FIG. 2). But if Δ is increased to make hωM/(2π)-Δ=Esmin-Ei1≤maximum of frequency of LO modes of other crystal orientation, superconductivity might never happen.

Also similar to a system with an acceptor band, each of such electron pairs,

formed between respective energy levels at the bottom of the conducting band and those at the top of the donor band, has a binding energy no smaller than hωM/(2π).

Esmin-Ei1=hωM/(2π)-Δ

donor band

electron

Ei1 Ei2 Ei3

Esmin

conducting band

FIG. 2: n-doped system

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In summary, once EM wave mode of ωM is established in the range of its

associated ion chain, which can be long or even macroscopic, electron-pairing is correspondingly produced in the crystal’s electron system over the same range. As some electron pairs in the donor/acceptor band system of a suitable ionic crystal have a binding energy no smaller than photon energy hωM/(2π) of the highest frequency of the EM wave/lattice wave modes of the ionic crystal, these electron pairs can hardly be broken by phonons or stimulated excitation in the crystal, and superconductivity can therefore be established. The destruction of the electron pairs may be due to other interactions, particularly many-phonon interactions, and/or destruction of domination of lattice wave mode of ωM over a sufficiently long range, and etc.

For single-atom crystal like metals, vibrations of atom cores generated by

acoustic wave modes of lattice might cause deviation of charge distribution, resulting dipole chains and EM wave modes corresponding to the lattice wave modes, which promote electron pairing. While factors like energy band structure features may not be present in metals, those such as flattened shape of Fermi face might serve similar function in limiting possible transitions by electrons in pairs, stabilizing electron pairs and resulting in a corresponding binding energy of the electron pairs. Conclusion It is established that once the lattice/EM wave modes are established in its range, which can be long or even macroscopic, electron pairs are produced in the crystal’s electron system over the same range by stimulated transitions induced by the EM wave mode. The lattice wave mode having the maximum frequency ωM is of special significance regarding superconductivity, for electron pairs produced by it can be stabilized in the context of a combination of some special factors (including energy level structure featured by donor/acceptor band and ωM) with a binding energy typically no smaller than hωM/(2π). A candidate mechanism of electron pairing in ion crystals and therefore of superconductivity is provided. [1] “Solid State Physics”, by Prof. HUANG Kun, published (in Chinese) by People’s

Education Publication House, with a Unified Book Number of 13012.0220, a publication date of June 1966, and a date of first print of January 1979, page 106, Equ. 5-40.

[2] See [1], Fig. 5-13, page 114. [3] Physics constants taken from “Introduction to Statistical Physics”, by Professor

WANG, Zhuxi, published in Chinese by People’s Education Publication House with a Unified Book Number of 13012.0131, second edition, August 1965, printed in February 1979, Appendix I.

[4] “Introduction to Statistical Physics”, by Professor WANG, Zhuxi, published in Chinese by People’s Education Publication House with a Unified Book Number of 13012.0131, second edition, August 1965, printed in February 1979, pages 52-54.)

1

(PACS: 74.20.Mn 74.25.F- )

(Keywords: ion crystal, ion chain, valence electrons, superconductivity mechanism, electron pairing)

Electron-pairing in ionic crystals and mechanism of

superconductivity

(Author: Q. LI)

Abstract The behaviors of valence electrons and ions, particularly ion chains, in some

ionic crystals are important to understanding of the mechanism of superconductivity.

The author has made efforts to establish a candidate mechanism of electron-pairing and superconductivity in ionic crystals.

Analyses are first made to a one-dimensional long ion lattice chain model (EDP model), with the presence of lattice wave modes having frequency ω. A mechanism of electron pairing is established.

Analyses are then extended to scenarios of 3D ionic crystals, particularly those with a donor/acceptor system, with emphasis being given to the interpretation and

understanding of binding energy of electron pairs formed between electrons at the top/bottom of donor/acceptor band and the bottom/top of conducting/full band.

It is established that once the lattice/EM wave modes are established in its range, which can be long or even macroscopic, electron pairs are produced in the crystal’s

electron system over the same range by stimulated transitions induced by the EM wave mode. The lattice wave mode having the maximum frequency ωM is of special

significance with respect to superconductivity, for electron pairs produced by it can be stabilized in the context of a combination of some special factors (including energy

level structure featured by donor/acceptor band and ωM) with a binding energy typically no smaller than hωM/(2π). A candidate mechanism of electron pairing in ion

crystals and therefore of superconductivity is provided.

Introduction

The behaviors of valence electrons and ions, particularly ion chains, in some ionic crystals are important to understanding of the mechanism of superconductivity.

The author has made efforts to establish a candidate mechanism of electron-pairing and superconductivity in ionic crystals.

Analyses are first made to a one-dimensional long ion lattice chain model (EDP model), with the presence of lattice wave modes having frequency ω. A mechanism of

electron pairing is established. Analyses are then extended to scenarios of 3D ionic crystals, particularly those

with a donor/acceptor system, with emphasis being given to the interpretation and understanding of binding energy of electron pairs formed between electrons at the

top/bottom of donor/acceptor band and the bottom/top of conducting/full band.

2

It is established that once the lattice/EM wave modes are established in its range, which can be long or even macroscopic, electron pairs are produced in the crystal’s electron system over the same range by stimulated transitions induced by the EM wave mode. The lattice wave mode having the maximum frequency ωM is of special significance with respect to superconductivity, for electron pairs produced by it can be stabilized in the context of a combination of some special factors (including energy level structure featured by donor/acceptor band and ωM) with a binding energy typically no smaller than hωM/(2π). A candidate mechanism of electron pairing in ion crystals and therefore of superconductivity is provided.

Generalized analyses of 1-D long ion lattice chain model

It has been established that for a one-dimensional long ion lattice chain, under the assumptions that only the interactions between neighboring ions are considered and that the interaction energy are approximated up to its quadratic term, the general solutions of lattice waves have the form of [1]:

ω±

2=β(M+m)/Mm){1±[1-4Mmsin22πaq/(M+m)2]} (B/A) ± =-(mω±

2-2β)/ (2βcos2πaq) with -1/4a<q≤1/4a, where a is the equilibrium distance between neighboring ions, A and B are the magnitude of the first and second ions respectively, M and m are the mass of the first and second ions respectively, and β is the tension of interaction between neighboring ions. With Born–Karman boundary condition exp(-2πi2Naq)=1, we have: q=n/(2Na), with n=±1, ±2,…. ±N/2.

The above solution of ω+ peaks at q=0, so the optical waves with q=±1/(2Na) has the maximum ω+ value of the system, with value of ω- being always smaller than that of ω+. There will be a total of 2N lattice waves for a total of N ions in the chain, which therefore include all the oscillating modes of the chain.

Thus, the time-dependent potential field can be written as: V(x,t)=V0(x)+ G(x) Σsinωt

where the summation is over all the lattice waves ω, V0(x) is the static potential field of the dipole chain without vibration, and G(x)= G(x+a) is a periodic function of x.

With H=H0 + G(x) Σsinωt and H0=V0(x). We have special solutions: ψn(x,t)= φn (x)exp (iEnt/h) where φn (x) being the static solution of static periodic filed V0(x). With perturbation G(x)sinωt, ψ(x,t)= Σan(t) φn(x) exp(iEnt/h), with a n = a n0+a n1+a n2+…. a n0=δnk, and a n

k1 � ∫Vnk(t) exp (i(En-Ek)t/h)dt

Vnk(t)= ∫φn*(x) G(x)Σsin(ωt) φk(x)dx

with Enk =En-Ek, we have: a n

k1 �Σ(exp(2πi(Enk+hω)t/h)/ (Ep+Enk)-exp(-2πiωt(Enk- hω)t/h)/ (Ep-Enk) (Equ. 1-3)

Here we can see that the first term on the right side in Equ.1-3 corresponds to

the probability that the electron absorbs a photon (or phonon and etc.) to transit from En to Ek, while the second term corresponds to the probability that the electron emits a

3

photon to transit from Ek to En. a n

k1 has a total of 2N peaks at Enk=±hω/(2π) corresponding to q=n/(2Na), with

n=±1, ±2,…. ±N/2. For illustrative purpose, we identified the maximum one of all ω values as ωM, which corresponds to the “optical” wave at q=±1/(2Na).

As indicated by Equ. 1-3, a nk1 converges to Enk=±hω/(2π) along with time t, and

after some time t, almost all electrons in the system will transit with Enk=±hω/(2π)

(where ω has N discrete values), that is: a nk1 →Σδ(Enk-hωm/(2π)), where m=1, 2,

3…..denotes the different lattice/EM wave modes of the ion chains, with ωM being the

greatest one among them.

Electron-pairing However, a well-established fact is that all electrons in a crystal are in energy

bands, and in many ion crystals electrons form full bands. Thus, for typical hω/(2π) of

lattice wave modes, most (if not all) electrons in energy bands cannot normally transit as indicated by (Equ. 1-3).

The way for the electrons to cope with this is that they form themselves into “pairs”, so that both of the two electrons in each pair, having energy En and Ek respectively (here we can safely assume that En>Ek), can transit by exchanging their

states, with the electron originally at energy En emitting a photon (or phonon or the like) of energy hω/(2π)=En-Ek, which is directly absorbed (virtual photon

emission/absorption) by the other electron, which is originally at Ek.

With a nk1 →Σδ(Enk-hωm/(2π)) with time t, only transitions corresponding to

En-Ek= hωm/(2π) will exist in the system after sufficient time t. This process results in

that each energy level in the bands of the system become distinguishable during stimulated transitions of electrons.

It is to be noted that whether an electron absorbs/emits a phonon or photon in

the above transition does not affect any of the conclusions of this paper, for these absorptions/emissions involved in electron-pairings and/or stimulated transitions are

virtual; they do not need to actually happen. But as the above discussed electron transitions and pairings in the ionic crystals are generated by the oscillating field of

EM wave modes, it is photons that are absorbed/emitted during these transitions and pairings.

Electron-pairing/exchange in 3D ionic crystals

A crystal with N primitive cells has 3nN oscillating modes, where n is the number of atoms/ions in one primitive cell. As according to a report of neutron

non-elastic scattering experiment on KBr crystal [2], ω values of different wave modes have the relation: LO>TO>LA>TA. The report also shows that, for each

crystal orientation, the maximum of ω is at q→0 of the LO (longitudinal optical) modes; the report further shows that for KBr crystal the maximum of ω in crystal

orientation [111] is greater than that in crystal orientation [100], so electron pairs corresponding to ω in [100] will be broken by some of the phonon/photons in [111].

This indicates that only a crystal orientation with the maximum ωM of all possible crystal orientations may correspond to the direction of prospective superconductivity,

for it is the direction corresponding to the ωM of the surviving electron pairs.

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Due to limitation of Pauli Principle, electrons in the same pair in system ψ(t)=U(t,t0)ψ(t0) have opposite spins.

Generally, both spontaneous transition and stimulated transition exist in a

system of ψ(t)=U(t,t0)ψ(t0). (Spontaneous transition may be limited by occupancy conflicts in crystals.) In ionic crystals, lattice is formed by ion chains, so vibration

modes of lattice generate oscillating electromagnetic (EM) wave modes of the same frequencies as the lattice vibration modes. Thus, the stimulated transitions in an ion

crystal are those driven by such oscillating electromagnetic wave modes. For two electrons in the electron system of such an ion crystal, if their energy

difference matches the frequency of one of the lattice vibration modes, the stimulated transitions of the two electrons in the pair can be in the form of their exchange of

states between themselves, that is, by pairing themselves with each other. In such pairing, the stimulated transitions of the two electrons become “virtual”- the

stimulated transitions need not to happen in reality, especially in the sense that the electrons concerned are non-distinguishable. Under complete “occupancy conflict”

(that is, all prospective targeted states for transition of the electrons concerned have been occupied by other electrons,) such electron-pairing/exchange becomes the only

way for the electrons to perform the stimulated transitions as required by ψ(t)=U(t,t0)ψ(t0) with U(t,t0) →Σδ(Enk-hωm/(2π)).

Electron pairing and binding energy in an acceptor-doped system If, in the energy band system of the crystal orientation corresponding to ωM, an

acceptor energy band with energy levels Ei1<Ei2<Ei3… is introduced in a full band

system of an ionic crystal (see FIG. 1), with Ei1-Esmax equals to or slightly smaller than hωM/(2π), where Esmax being the highest energy level in the full band and ωM being

the greatest frequency of the oscillating electromagnetic wave modes associated with the ion chains in the crystal, then, since there are stimulated transitions corresponding

to Enk=hωM/(2π), electrons on Esmax level of the full band can transit to Ei1 by stimulated transition of Enk=hωM/(2π), thus forming a new system including the

acceptor energy level Ei1 and the original system ψ(t), and this new system (ψ(t)+{Ei1}) is conductive.

Binding energy of electron pairs relating to acceptor band More generally, for example, assuming that ωM, Esmax, and, say, Ei2 satisfy

hωM/(2π)=Esmax- Ei2, a new system (ψ(t)+{Ei1}+{Ei2}) is then formed including the

acceptor energy levels Ei1 and Ei2 and the original system ψ(t). If some holes (such as those left by electrons transiting to the acceptor band)

exists in the full band, the electrons pair like (φij +φsmax) (with j=1,2,…) can be broken by transition of the lower electron in the pair to any of the holes. So with the presence

of even one hole in the full band, the electron pairs like (φij +φsmax) could not be stable. But a pool of electron pairs (φij +φsmax) in dynamic equilibrium could possibly be

maintained across the top of the full band and the bottom of the acceptor band. However, as an insulator is easily charged, especially under external electrical

field/voltage, if the ionic crystal is negatively charged, electrons are injected into the system, thus filling the holes in the full band; in such a scenario, the above electron

pairs (φij +φsmax) will be stabilized up to a possible binding energy.

5

Then,, a key and subtle factor here is determination of the binding energy of such an electron pair (φij +φsmax).

Due to the limitation of the particular energy structure of this scenario, if the electron pair is to be broken by transition of an electron in the pair, then at least on

electron in the pair has to transit to the energy level Ei3 or higher.

:

:

:

:

As the macroscopic energy of the combined system is, by definition, the average

of the measured energy values over long time, the contribution to the macroscopic energy by the electron that still remains on (φi2 +φsmax) after the other electron transits

to Ei3 (or higher) is one half of hωM/(2π) (that is, the average value of the energy values at φi2 and φsmax), while that by the electron that transits to Ei3 (or higher) is

some value greater than hωM/(2π), so the change in the macroscopic energy is an increase of at least hωM/(4π).

However, the half photon energy seems strange and ridiculous.

An alternative approach is by the argument that the electron transiting to Ei3 actually does not have the energy hωM/(2π) at the moment just before it makes the

transition. A model for this is that the electron pair includes the two electrons plus a photon with an energy hωM/(2π), which binds the two electrons together to form the

electron pair. This is phenomenologically in conformity with that virtual photon exchange happens when the two electrons exchange their states, as indicated by the

expression of Equ. 1-3 discussed above. The two electrons in the pair co-occupy the correlated states of (φij +φsmax) without specifying which electron is in which of the

two states. As the pair is broken in a general situation (without the limitation of band structure as in the present system), the photon might be taken by any of the two

FIG. 1: p-doped system

full band

electron

:

:

:

Ei1

Esmax

acceptor band

Ei2

Ei3

Ei1- Esmax =hωM/(2π)-Δ

6

electrons, taken by the corresponding EM wave mode, or even emitted as a free photon. But in the particular situation of the energy level structure under

consideration, one of the two electrons must take one photon of hωM/(2π) (not necessarily the photon originally within the pair, though) to go to an energy level at or

above Ei3, which is the only way for it to go; the remaining electron will take another photon (which can be the original one) of hωM/(2π) to stay at and transit between (φij

+φsmax). (So the question here is, in the scenario that the electron alternatively emits and

absorbs such a photon of hωM/(2π), whether the EM wave “spends” the energy of the photon entirely for the electron transition or reserves half of the photon’s energy as

part of EM wave’s own energy? If the former is true, the binding energy of the electron pair concerned is hωM/(2π), otherwise, the binding energy would be halved.)

Thus, where the one half of photon energy is missed is recognized: the photon

associated with the remaining electron at (φij +φsmax) is omitted. Therefore, the energy of the combined system as discussed above should be

increased by hωM/(2π) after the electron pair is broken, which should be the binding energy of this electron pair when there is not hole in the full band.

We then consider the distribution function of Gibbs’ canonical ensemble of the

combined system of the electrons, the lattice, and the EM wave modes associated with the lattice. The proportion ρ of members of the ensemble before the transition of the

electron to Ei3 being [4] ρ(E1) � exp(-βhωM/(2π)), while that after the transition of

the electron to Ei3 being ρ(E2)� exp(-2βhωM/(2π)) (where 1/β=kT). So

ρ(E2)/ρ(E1)=exp(-βhωM/(2π)). This is in fact the probability that the electron pair is broken by any transition (in this particular energy band structure), with hωM/(2π)

being the binding energy of the pair against destruction by transition of an electron in the pair.

As an estimation of the stability of such an electron pair with such a binding energy, ωM/(2π)≈1013-1014/s, at T=100K there will be (hωM/(2π))/(kT)≈4.65-46.5.

Thus, for ωM above, say, 5x1013

/s, such an electron pair can rarely be broken by a phonon even at T=100K [3].

Similarly, Ei1 can also form a superconducting electron pair with an electron on

a corresponding energy level below Esmax, with a binding energy no smaller than hωM/(2π).

Further, in some samples, in a range of Δ=hωM/(2π)-(Ei1-Esmax) there can be a plurality of energy levels Ei1<Ei2<Ei3… in the acceptor band, and each of these energy

levels may have an electron forming a stabilized electron pair with an electron at a corresponding level in the acceptor band. But if Δ increases to the extent as making

hωM/(2π)-Δ=Ei1-Esmax≤maximum frequency of LO modes corresponding to any other crystal orientation, superconductivity may never happen.

Electron pairs in donor band system The mechanism of superconductivity in a system having a donor band

(Ei1>Ei2>Ei3……) is similar to that having a acceptor band, except that the donor band

is beneath a conducting band, with the lowest level Esmin in the conducting band being higher than the highest level Ei1 in the donor band by a difference equal to or slightly

smaller than hωM/(2π). (see FIG. 2)

7

For illustration, we assume Esmin-Ei2=hωM/(2π). Then, electrons on donor energy levels Ei1 and Ei2 may enter the conducting band by stimulated transition by the EM

wave mode of ωM, and may then form electron pairs with electrons which later transit to Ei1 and Ei2. But these electron pairs are unstable as far as any hole(s) (particularly

the holes left by the electrons transiting to the conducting band) exists in the donor band, for the electron at the lower energy of each of the electron pairs can easily

transit to such a hole. If, however, the hole(s) in the donor band are somehow filled, those pairs

formed by electrons on the conducting band with the electrons on levels Ei1 and Ei2 will become stabilized, with the similar mechanism as explained above with respect to

acceptor band.

:

:

:

:

A scenario is that external electrons may enter the system, at a relatively high

energy level (particularly under external electric field/voltage), and transit to the donor band or levels, so that holes in the donor band are filled and the electron pairs

are stabilized. Similar to a system with an acceptor band, the energy level range of

Δ=hωM/(2π)-( Esmin-Ei1) can be increased to accommodate a plurality of energy levels so that a plurality of electron pairs can be formed between respective energy levels at

the bottom of the conducting band and those at the top of the donor band (see FIG. 2). But if Δ is increased to make hωM/(2π)-Δ=Esmin-Ei1≤maximum of frequency of LO

modes of other crystal orientation, superconductivity might never happen.

Also similar to a system with an acceptor band, each of such electron pairs, formed between respective energy levels at the bottom of the conducting band and

those at the top of the donor band, has a binding energy no smaller than hωM/(2π).

In summary, once EM wave mode of ωM is established in the range of its

Esmin-Ei1=hωM/(2π)-Δ

donor band

electron

Ei1 Ei2

Ei3

Esmin

conducting band

FIG. 2: n-doped system

8

associated ion chain, which can be long or even macroscopic, electron-pairing is correspondingly produced in the crystal’s electron system over the same range. As

some electron pairs in the donor/acceptor band system of a suitable ionic crystal have a binding energy no smaller than photon energy hωM/(2π) of the highest frequency of

the EM wave/lattice wave modes of the ionic crystal, these electron pairs can hardly be broken by phonons or stimulated excitation in the crystal, and superconductivity

can therefore be established. The destruction of the electron pairs may be due to other interactions, particularly many-phonon interactions, and/or destruction of domination

of lattice wave mode of ωM over a sufficiently long range, and etc.

For single-atom crystal like metals, vibrations of atom cores generated by acoustic wave modes of lattice might cause deviation of charge distribution, resulting

dipole chains and EM wave modes corresponding to the lattice wave modes, which promote electron pairing. While factors like energy band structure features may not be

present in metals, those such as flattened shape of Fermi face might serve similar function in limiting possible transitions by electrons in pairs, stabilizing electron pairs

and resulting in a corresponding binding energy of the electron pairs.

Conclusion

It is established that once the lattice/EM wave modes are established in its range,

which can be long or even macroscopic, electron pairs are produced in the crystal’s electron system over the same range by stimulated transitions induced by the EM

wave mode. The lattice wave mode having the maximum frequency ωM is of special significance regarding superconductivity, for electron pairs produced by it can be

stabilized in the context of a combination of some special factors (including energy level structure featured by donor/acceptor band and ωM) with a binding energy

typically no smaller than hωM/(2π). A candidate mechanism of electron pairing in ion crystals and therefore of superconductivity is provided.

[1] “Solid State Physics”, by Prof. HUANG Kun, published (in Chinese) by People’s Education Publication House, with a Unified Book Number of 13012.0220, a

publication date of June 1966, and a date of first print of January 1979, page 106, Equ. 5-40.

[2] See [1], Fig. 5-13, page 114. [3] Physics constants taken from “Introduction to Statistical Physics”, by Professor

WANG, Zhuxi, published in Chinese by People’s Education Publication House with a Unified Book Number of 13012.0131, second edition, August 1965, printed

in February 1979, Appendix I. [4] “Introduction to Statistical Physics”, by Professor WANG, Zhuxi, published in

Chinese by People’s Education Publication House with a Unified Book Number of 13012.0131, second edition, August 1965, printed in February 1979, pages 52-54.)