The document discusses the mechanism of electron pairing in crystals with a binding energy greater than or equal to one band gap. It explains that due to conservation of wavevector, virtual photon absorption/emission inducing electron pairing can only occur across at least one band gap. Therefore, electron pairs formed by virtual stimulated transitions must have a binding energy greater than or equal to the width of the band gap. It also reviews how electron pairing is established and explains why virtual stimulated transitions in low-frequency ranges are negligible.

1. (PACS: 74.20.Mn 74.25.F-)
(Keywords: mechanism of electron pairing across band gap with binding energy in
crystals)
Mechanism of electron pairing in crystals, with binding energy no
smaller than one band gap
Author: Q. LI
Affiliation: JHLF
Date: 26 March 2010
Abstract
Establishment of mechanism of electron pairing with a lower limit of binding
energy is necessary for understating of superconductivity. Due to conservation of
wavevector, photon absorption/emission by an electron in crystal can only be allowed
across at least on band gap, which is also true for virtual photon absorption/emission
inducing electron pairing in crystal. Therefore, it is clearly explained that electron
pairs, formed by virtual stipulated transition, can only exist between electrons across a
band gap, with a binding energy no smaller than the width of the band gap.
Introduction
In one of my previous papers entitled “Electron-pairing in ionic crystals and
mechanism of superconductivity” [1], it was concluded that an electron pairs formed
by virtual stimulated transitions across a band gap could have a binding energy no
smaller than the width of the band gap, where the virtual stimulated transitions were
induced by electromagnetic wave modes generated by lattice wave modes.
In another of my previous papers entitled ““Vertical” pairing of electrons and
origin of superconducting energy gaps”, attempt was made to use the mechanism of
electron pairing by the virtual stimulated transitions to explain experiment results as
presented in Fedorov et al [3].
But a serious problem existed in that the virtual stimulated transitions at the low
frequency range should be suppressed to such an extent that they could be totally
negligible, but a clear mechanism of such negligibility were not identified.
With some research basically concerning the mechanism of semiconductor laser,
I have reached an explanation for the absence of the virtual stimulated transitions in
low-energy/frequency range, which I believe would result in a complete and clear
mechanism of high-Tc superconductivity.
Review of mechanism of electron pairing
As already discussed in [2], considering the situation of Fig. 1-1, where an
electromagnetic wave mode of hω/(2π) =E2- E1 is applied. With static energy levels E2
and E1 are initially occupied by Electrons 1 and 2 respectively, Electrons 1 and 2 have
to perform stimulated transitions (as required by quantum mechanics) by constantly
exchanging their states with each other, with the electron at the higher energy level E2
emitting a photon of hω/(2π) =E2- E1, which is directly absorbed by the electron at the
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2. lower level of E1 as the lower electron transits up to E2. The two electrons engaging in
such a mutual exchange of states are in electron-pairing. It is to be noted that none of
the electrons concerned is in a static state. It is also to be noted that the photon
emission and/or absorption involved in the pairing are virtual, so the electromagnetic
wave mode needs not to spend any energy in driving constant exchanges and pairing.
We could further treat the pair of Fig. 1-1 as a subsystem of the two electrons plus a
binding photon of hω/(2π) =E2- E1, which binds the two electrons in the pair.
The scenario of Fig. 1-1 could be a portrayal of the electrons in a real crystal,
where electromagnetic wave modes are generated by lattice wave modes and are
always present (even at T=0). So electrons in a crystal could also pair up in more or
less the same way as Electrons 1 and 2 shown in Fig. 1-1, as will be discussed later,
and the paired electrons would neither in any static state nor with any determined
energy.
E3
E2
electron
Electron 2
E2- E1 =hω/(2π)
EM wave mode hω/(2π)
E1
:
Electron 1
:
E0
Electron
FIG. 1-1: Vertical pairing of electrons under stimulated transitions. Two electrons, at energy levels
of E1 and E2 respectively, perform stimulated transitions by exchanging their states with each other
under the stimulation of electromagnetic wave of the frequency hω/(2π)=E2-E1, thus reaching a
“vertical” pairing of the two electrons. Photon emissions/absorptions associated with these
stimulated transitions are virtual (not real). And the binding energy of the pair, in this particular
system of levels E1, E2 and E3, is hω/(2π)+(E3-E2).
How does an electron pair establish its binding energy ?
When the pair in Fig. 1-1 is to be broken, one of Electrons 1 and 2 has to go to
E3. It has been shown [1][3] that the binding energy of such a pair, in such an energy
level structure shown in Fig. 1-1, is Ebind=hω/(2π)+(E3-E2). So as long as E3≥E2, there
is Ebind≥hω/(2π).
Here, a remarkable feature as evidenced by Fedorov et al is that electrons at
these upper energy levels do not pair among themselves, for otherwise a peak would
had been detected at EF; this is in line with the suggestion that stimulated transitions
generated by EM wave modes of frequencies below certain value be negligible, which
is a key factor of the mechanism of low-Tc superconductivity.
Wavevector conservation of photon-electron interactions in the
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3. virtual stimulated transition
According to a well-established mechanism of lasing process, a photon emitted
by stimulated transition will have the same frequency and momentum as the incident
photon that induces the stimulated transition.
This should equally applies to the photon virtually emitted during the
above-mentioned virtual stimulated transition, which leads to electron pairing, so the
electron that binds the two electrons concerned in a pair have the same momentum (as
well as the frequency, of course) as the photon of the electromagnetic wave mode that
have induced the virtual stimulated transition.
It has been established [4] that electron-photon interaction in a crystal obeys
momentum conservation:
k'=k±q+Kn (Equ. 1)
where k' and k being the wavevectors of the electron after and before interaction with
the photon respectively, q being the wavevector of the photon, Kn being an inverse
lattice vector, and ± indicating absorption/emission of the photon
Virtual stimulated transitions in low-frequency range cannot happen
It has been well-established, as in the field of semiconductor lasers [5], that a
transition as indicated by Equ. 1 occurs with k'≈k+Kn as wavevector of the photon
concerned is much smaller than that of the electrons, such transitions can only occur
across at least one band gap due to the limitations of wavevector conservation and
energy conservation.
Thus, virtual stimulated transitions in low-frequency range can be negligible,
and, it is established, in association with the description relating to Fig. 1-1, that an
electron pair formed by virtual stimulated transitions must have a binding energy no
smaller than one band gap.
Further discussion
The origin of the energy gap and peak as reported in Fig. 1(a) and (b) of
Fedorov et al can still be explained as that electrons associated with energy levels at
the upper band across the band gap is reported as electrons associated with energy
levels at the lower band.
Fedorov et al also evidences that electrons pairing associated with
low-frequency photon almost did not happen, suggesting that “small” stimulated
transitions be negligible [3].
Conclusion
Electrons can pair up only across a band gap, resulting in a binding energy no
smaller than the width of the band gap.
[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] ““Vertical” pairing of electrons and origin of superconducting energy gaps” , by:
Q. LI, JHLF,
http://www.slideshare.net/edpmodel/amended-vertical-pairing-of-electrons-and-under
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4. standing-of-energy-gaps-relating-to-superconductivity
[3]Phys. Rev. Lett. 82, 2179 (1999): Fedorov et al. Temperature Dependent Photoemission
Studies of Optimally Doped Bi2Sr2CaCu2O8
[4] “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 205,
Equ. 7-93.
[5] “Principles of Laser”, page 260 and Fig. 8.3-1, Optics Group, Dept. of Physics,
Peking University, May 1981.
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