candidate mechanisms of electron pairing near EF at T=0, binding energy of the electron pairs thus formed, and superconductivity in mono-atom crystals have been proposed. Once EM wave modes are established in the ranges of their associated lattice chains of a crystal concerned, which ranges can be long or even macroscopic, electron-pairs are produced in the crystal’s electron system over these ranges. As EM wave modes with frequencies below certain value (corresponding to an energy value Δ) may have little contribution to stimulated transitions of electrons and electron-pairing, at T=0 each of the electrons at and near EF pairs with one of the electrons at energy levels of EF-hωM/(2π)≤E≤EF-Δ (where ωM is the maximum frequency of lattice wave modes of the system, which is often associated with a specific crystal orientation), resulting in a binding energy of at least Δ for each of these pairs at T=0.

1. (PACS: 74.20.Mn 74.25.F- )
(Keywords: mechanism of binding energy of electron pair at T=0, mechanism of
superconductivity of metals, electron pairing, low-temperature superconductivity)
Mechanism of superconductivity in metals
Author: Q. LI
Affiliation: JHLF
Finish date: 08 Match 2010
Abstract
It has been established [1] that at least some valence electrons in lattice may
undergoes constant virtual stimulated transitions driven by EM wave modes coupled
to and produced by corresponding lattice wave modes. Do and how do such virtual
stimulated electron transitions have something to do with electron-pairing and
mechanism of superconductivity in mono-atom crystals like metals?
With efforts made by the author in addressing these questions, candidate
mechanisms of electron pairing near EF at T=0, binding energy of the electron pairs
thus formed, and superconductivity in mono-atom crystals have been proposed.
Once EM wave modes are established in the ranges of their associated lattice
chains of a crystal concerned, which ranges can be long or even macroscopic,
electron-pairs are produced in the crystal’s electron system over these ranges. As EM
wave modes with frequencies below certain value (corresponding to an energy value
∆) may have little contribution to stimulated transitions of electrons and
electron-pairing, at T=0 each of the electrons at and near EF pairs with one of the
electrons at energy levels of EF-hωM/(2π)≤E≤EF-∆ (where ωM is the maximum
frequency of lattice wave modes of the system, which is often associated with a
specific crystal orientation), resulting in a binding energy of at least ∆ for each of
these pairs at T=0.
Therefore, for mono-atom crystals, the critical parameter like Tc is related to
the characteristics of lattice/EM wave modes, particularly the strength of the EM
wave modes at ω→0.
Introduction
It has been established [1] that at least some valence electrons in lattice may
undergo constant virtual stimulated transitions driven by EM wave modes coupled to
and produced by corresponding lattice wave modes. Do and how do such virtual
stimulated electron transitions have something to do with electron-pairing and
mechanism of superconductivity in mono-atom crystals like metals?
With efforts made by the author in addressing these questions, candidate
mechanisms of electron pairing near EF at T=0, binding energy of the electron pairs
thus formed, and superconductivity in mono-atom crystals have been proposed.
1

2. It has been established [1] that some valence electrons in lattice may undergoes
constant virtual stimulated transitions driven by EM wave modes coupled to and
produced by corresponding lattice wave modes.
Stimulated transitions and pairing of electrons
Generally speaking, if an electron is driven by an EM wave mode to perform
stimulated transition, a photon is to be associated with the transition. However, if a
pair of electrons, at energy levels of En and Ek respectively, are driven by an EM wave
mode of frequency of Enk=En-Ek=±hω/(2π) to perform stimulated transitions by
exchanging their states with each other, neither photon emission nor photon
absorption will happen in real, instead a virtual exchange of one photon of
Enk=±hω/(2π) happens between these two electrons. This is the “electron pairing”
under the presence of EM wave modes.
In mono-atom systems like metals, there are no optical lattice wave modes, and
only acoustic lattice wave modes (LA and TA modes) exist. These vibrations of atom
cores cause deviation of charge distribution of positive atom cores with respect to the
background of a sea of negative valence electrons, which results in vibrating dipoles
and corresponding EM wave modes. As each lattice wave mode in all crystals is
directly coupled to an EM wave mode of the same frequency, such “electron pairing”
under stimulated transition is omnipresent in all crystals.
Since at q→0 the vibrations of neighboring atoms in mono-atom system go to in
phase and the frequency of the vibrations go to zero [2], the low frequency
components of the LA and TA modes of lattice produce almost no oscillating EM
waves.
When two electrons, at En and Ek respectively, are paired with each other under
stimulated transitions generated by an EM wave mode of hω/(2π)=En-Ek, they are
bound by one photon of hω/(2π)=En-Ek; or in other words, the two electrons bind the
photon between them.
For the system of ψ(t)=U(t,t0)ψ(t0) as concerned, as indicated by [1]:
k1
a n ∝Σ(exp(i(2πEnk+hωm)t/h)/( hωm/+2πEnk)-exp(-i(2πEnk-hωm)t/h)/(hωm-2πEnk)
(Equ. 1-3)
(where ωm are the frequencies of the lattice wave modes, m=1, 2, 3…..denotes the
different lattice/EM wave modes of the ion chains,) a nk1 converges to Enk=±hωm/(2π)
along with time t, so after sufficient time t, almost all electrons in the system will
perform stimulated transitions with Enk=±hωm/(2π), that is:
a nk1 →ΣAmδ(Enk-hωm/(2π)),
and Am corresponds to the probability of transitions corresponding to hωm/(2π).
Obviously, Am is proportional to the strength of EM wave mode of ωm. However,
as was established with a one-dimensional lattice model [2] and with experimental
result [3], at the limit of ω→0 (q→0), the vibrations of the atom cores go to in phase
so their EM wave modes also goes to zero. Thus, the probabilities of transitions
corresponding to EM wave modes with ω→0 go to zero. In other words, an energy
value ∆ can be set, with only transitions of Enk=±hωm/(2π)≥±∆ actually happening in
the system (no matter it is a mono-, bi- or multi-atom system).
2

3. Electron-pairing in metals at T=0
Considering now for T=0, with a nk1 →Σδ(Enk-hωm/(2π)) after sufficient time t,
only transitions with En-Ek= hωm/(2π)≥±∆ will exist in the system after sufficient time
t. So each of the electrons at E=EF will pair with an electron at an energy level E of
EF-hωM/(2π)≤E≤EF-∆, where ωM is the maximum frequency of all the lattice wave
modes present in the system. Taking E=0, the energy of the pair is ∆.
Thus, if such an electron pair is broken at T=0, the energy of the exiting
electron will be >∆, and that of the remaining electron will be ∆ (for the remaining
electron is still in its state as before the pair is broken and is to make its upward
transition at the moment, so a photon of energy ∆ is with the remaining electron). So
this electron pair has a binding energy of >∆ at T=0.
The minimum binding energy for an electron pair including an electron near and
below EF is (slightly) greater accordingly. Thus, electrons at and near EF are all in
pairs each having a binding energy of >∆.
The electrons at and near EF at T=0 are those contributing to conductivity.
Conclusion
Once EM wave modes are established in the ranges of their associated lattice
chains of a crystal concerned, which ranges can be long or even macroscopic,
electron-pairs are produced in the crystal’s electron system over these ranges. As EM
wave modes with frequencies below certain value (corresponding to an energy value
∆) may have little contribution to stimulated transitions of electrons and
electron-pairing, at T=0 each of the electrons at and near EF pairs with one of the
electrons at energy levels of EF-hωM/(2π)≤E≤EF-∆ (where ωM is the maximum
frequency of lattice wave modes of the system, which is often associated with a
specific crystal orientation), resulting in a binding energy of at least ∆ for each of
these pairs at T=0.
For temperature not too far from T=0, sufficient electron pairs can still be
maintained, so can be the state of superconductivity.
Therefore, for mono-atom crystals, the critical parameter like Tc is related to
the characteristics of lattice/EM wave modes, particularly the strength of the EM
wave modes at ω→0.
[1] “Electron-pairing in ionic crystals and mechanism of superconductivity”, by: Q.
LI,JHLF,
http://www.slideshare.net/edpmodel/100304-affi-electron-pairing-in-ionic-crystals-an
d-mechanism-of-superconductivity#
[2] “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.
[3] See [2], Fig. 5-12, page 113.
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