The new results reported here mainly include: 1) recognition that phonon is carrier of electromagnetic interaction between its lattice wave mode and electrons; 2) recognition that binding energy of electron pairs of high-temperature superconductivity is due to escape of optical threshold phonons, of electron pairs at or near Fermi level, from crystal by direct radiation; 3) recognition that binding energy of electron pairs of low-temperature superconductivity is possibly due to escape of non-optical threshold phonons by anharmonic crystal interactions; and, 4) recognition of a possible mechanism explaining why some crystals never have a superconducting phase. While electron pairing is phonon-mediated in general, HTS should be associated with electron pairing mediated by optical phonon at or near Fermi level (EF), so the rarity of HTS corresponds to the rarity of such pairing match.

1. Phonon as carrier of electromagnetic interaction between lattice
wave modes and electrons and its role in superconductivity
Qiang LI
Jinheng Law Firm
1004, Quantum Plaza, 23, Zhichun Road, Beijing 100191, China
lq@jinheng-ip.com
Abstract
The new results reported here mainly include: 1) recognition that phonon is
carrier of electromagnetic interaction between its lattice wave mode and electrons; 2)
recognition that binding energy of electron pairs of high-temperature
superconductivity is due to escape of optical threshold phonons, of electron pairs at or
near Fermi level, from crystal by direct radiation; 3) recognition that binding energy
of electron pairs of low-temperature superconductivity is possibly due to escape of
non-optical threshold phonons by anharmonic crystal interactions; and, 4) recognition
of a possible mechanism explaining why some crystals never have a superconducting
phase. While electron pairing is phonon-mediated in general, HTS should be
associated with electron pairing mediated by optical phonon at or near Fermi level
(EF), so the rarity of HTS corresponds to the rarity of such pairing match.
Keywords: the essence of phonon; origin of binding energy of electron pair;
superconductivity; Heisenberg Uncertainty Principle; threshold phonon; stability of
electron pair; non-stationary stable state; anharmonic crystal interactions; direction of
electron pairing; multiple-pairing; graph theory
PACS numbers: 74.20.Mn 74.25.F-
The central roles of electromagnetic (EM) wave modes generated by lattice
wave modes and the mechanism of “electron pairing by virtual stimulated transitions”
were proposed with strong suggestions that such electron pairing would result in a
threshold (binding) photon released by the electron at the excited state [1], but the
origin of binding energy of electron pairs was not clearly identified in spite of
attempts to attribute it to Heisenberg Uncertainty Principle and the threshold photon.
While phonon is widely believed to be the mediator of electron pairing relating
to superconductivity, it has a somewhat awkward status in physics; although it is
typically defined as “a quasiparticle characterized by the quantization of the modes of
lattice vibrations of periodic, elastic crystal structures of solids” and/or “a quantum
mechanical description of a special type of vibrational motion” [2], what kind of
interaction (electromagnetic, gravitational, or etc.) phonon is associated with seems
not having been well-addressed so far? If phonon is the carrier of electromagnetic
interaction, then phonon should be essentially the same as photon. In this paper it is
explained that phonons are indeed the carriers of electromagnetic interaction between
electrons and lattice wave modes, and that photon can be regarded as a special kind of
phonon.
With the recognition of identity of phonon to photon, some detailed
1

2. interpretations of the origin of binding energy of electron pairs in crystals are given
hereinbelow. The electron pairs are generally phonon-mediated, but some special ones
are “optical phonon-mediated”, and it seems that this difference constitutes a water
parting between high-temperature superconductivity (HTS) and lower temperatures
superconductivity (LTS).
For discussing some details of non-stationary behavior of electron, let us begin
with examining the exemplary dual energy system with one electron as shown in Fig.
1, which is subject to the interaction by an electromagnetic (EM) wave mode hν=E2-
E1.
According to well-known time-dependent perturbation approach, the matrix
element to the first order is
a nk=δnk+a nk1 (1)
k
a n 1 =2π/(ih) ∫Vnk(t1) exp (i(En-Ek)t1/h)dt1 (2)
where the integral is carried over [t0,t], and
Vnk(t1)=<φn|Vnk(r,t1) |φk> (3)
The potential of the EM mode (hν) the system shown in Fig. 1 is
V(r,t)=V0(r,t)cos(2πνt) (4)
When the EM mode (hν) is stable, its magnitude remains time-independent as
V0(r,t)=V(r) (5)
So Vnk(t1)=Vnkcos(2πνt1) (6)
with
Vnk=<φn|Vnk(r) |φk> (7)
Then, Equation (2) becomes
a nk1 ~(Vnk/h){exp[2πi(Enk+hν)t/h]-1}/ (Ep+hν)
-{exp[2πit(Enk-hν)t/h]-1}/ (Enk-hν) (8)
According to Equation (8), at Enk=±hν there is a nk1 ∝t, so all a nk components
except those at Enk=±hν will be normalized to zero with increasing time t, resulting in
a nk1=1 for Enk=±hν. Remarkably, the matrix a nk becomes time t-independent because t
can be taken away from the matrix at t→∞; thus, the system becomes stable but is not
at any eigenstate of energy.
It is to be further noted that this result of “stable state” should not be limited by
the approximation of perturbation approach in that all the higher terms of perturbation
should either be normalized off or become time-independent.
While the establishment above is a well-known problem in quantum mechanics,
I would request special attention to the precondition “the EM mode (hν) is stable so
its magnitude is constant”, as this is a critically important factor to stability of electron
pairs in crystals, as will be explained below.
For an EM wave mode of (m ＋ 1/2)hν (with m=0,1,2…), if the number m of
photon fluctuates, the EM wave mode is no longer stable and Equation (5) no longer
valid; conversely, V0(r,t) in Equations (4) and (5) becomes a step function of time t,
the integral of Equation (2) becomes segmented, and the result of Equation (8) is no
longer a single term uniform over the entire range of integration (0,t) but becomes a
summation like
a nk1 ~ΣCjVnkj/h{exp[2πi(Enk+hν)(tj-tj-1)/h]-1}/ (Ep+hν)
-{exp[2πit(Enk-hν)(tj-tj-1)/h]-1}/ (Enk-hν) (9)
where the summation is over index j; in each time segment (tj-1,tj), the number m of
photon of the EM wave mode remains unchanged, but the number m assumes
2

3. different values in different time segments (tj-1,tj), and Cj will denote a random
complex number. Then, Equation (9) becomes a summation of a series of random
complex numbers, so the matrix element a nk1 including such a summation not only
cannot normalize off other matrix elements (a nk) but also goes to zero statistically. In
conclusion, when the photon number m of the EM mode fluctuates, transition of the
electron in the system no longer converges to Enk=±hν.
Let us now consider the energy relation of the system as shown in Fig. 1. Some
interpretation says that at the limit of t→∞ Equation (8) indicates that the electron is
exchanging phonon (photon) with the EM wave mode or the outside. But such an
interpretation is problematic. First, Equation (8) does not indicate the requirement of
real phonon emission/absorption. Second, at finite time t transitions other than
Enk=±hν at sufficiently low temperature (T→0) is allowable according to Equation
(8), but the EM mode and the outside environment will surely not provide such
phonon or photon, indicating that the transitions as indicated by Equation (8) do not
require real emission/absorption of phonon/photon. (Hereinbelow transition not
including real emission/absorption of quantum would be referred to as “virtual
transition”, and that including real emission/absorption of quantum as “real
transition”.)
As I understand it, virtual transition could be explained on two bases: 1) the
significance of “measurement”, and 2) the Heisenberg uncertainty principle.
“Measurement” is usually interpreted as an intervention to the system to be measured,
which makes the system “collapse” to an eigenstate of which the eigenvalue is the
result of the measurement. Apparently, virtual transitions seem not in conformity with
the requirement of energy conservation. But all energy terms we observe are
“observable”, some “non-observable” energy terms can get involved in a virtual
transition, and the relationship of energy conservation shall cover both “observable”
and “non-observable” terms of energy involved in the virtual transition concerned.
For example, the EM wave mode at its ground state (with m=0) still might “lend” a
threshold photon to the electron for transition from E1 to E2, and the electron then
could return the threshold photon to the EM wave mode in the subsequent transition
from E2 to E1. As such a “lend/return” process is transient, the phonon/energy
exchanges could be “non-observable”, because the system cannot collapse to a state in
which the EM mode has a negative number of photon like m= -1.
According to Heisenberg Uncertainty Principle, the energy of the electron in the
system of Fig. 1 has a spread ΔE, which is related to a lifetime Δt, by which the
system stays in a (stationary) state associated with an energy level, as
ΔEΔt≥h/(4π) (10)
It is important to note here that the energy spread ΔE is associated with the energy of
system (electron) concerned, not with an energy level a stationary state, and it is likely
that the system concerned is not “at” any stationary state at all.
Therefore, in the system as shown in Fig. 1, as long as the electron transits
between energy states of E1 and E2 at a transition frequency of νE with
νE≥4πν (11)
there will be
ΔE≥2(E2-E1) (11.1)
Which means the energy spread ΔE of an electron, which is “originally” at
stationary state of E1, is broad enough to cover the excited energy level E2, so the
electron may transit to E2 without actually absorbing any photon/phonon. In fact, since
the notion “transit” relates to “stationary state”, it becomes somewhat meaningless in
3

4. such a situation where no stationary state exists at all. Hereinbelow, the stabilized
non-stationary state, at which an electron has an energy spread ΔE covering at least
two stationary levels (as E1 and E2 shown in Fig. 1), is referred to as “non-stationary
stable (NSS) state”.
Jan Hilgevoord et al, at discussing ΔEΔt≥h/(4π), make a good comment:
“Evidently a point particle and a point of space are very different things.
Nevertheless they are not always clearly distinguished.” [3] We could make a
similar comment as: an energy eigenvalue and the energy of an electron are very
different things even if the electron is associated with the eigenstate of the energy
eigenvalue, so they are to be clearly distinguished. At a stationary state, an electron
may “steadily” stay at an energy eigenstate with a lifetime of Δt→∞, so its energy
uncertainty spread becomes ΔE→0; then, “the energy of the electron” equates to “the
energy eigenvalue” of the eigenstate. Under a non-stationary state, however, an
electron does stays at any single eigenstate and can be associated with a plurality of
energy eigenstates.
The electron system as shown in Fig. 1 differs from one in real crystals in that
the effect of wavevector selection rule is not taken into account. The wavevector
selection rule relates to the physical significance of phonon as the carrier of
electromagnetic interaction in crystals, just like photon is the carrier of
electromagnetic interaction. This interpretation can be made on the basis of the well-
discussed mathematical operation concerning transition in electron-phonon
interaction, as made in lattice representation by Huang [4], which is summarized in
English in Italic below due to its high relevance:
The first order approximation of potential variation δVn of the atom at the nth
lattice point Rn caused by a lattice wave mode is
δVn≈-μn•▽V(r-Rn) (7-86)
where V(r) is the potential of one atom, and
μn=Aecos2π(q•Rn-νt) (7-87)
represents displacement of the atom by the lattice wave mode, with e being the unit
vector in the wave direction, A being the magnitude of the lattice wave mode, ν being
the frequency of the lattice wave mode, and q being the wavevector of the lattice wave
mode under elastic wave approximation. Then, the potential variation of the entire
lattice is
ΔH=ΣδVn=-(A/2)exp(-2πiνt)Σexp(2πiq•Rn)e•▽V(r-Rn)
-(A/2)exp(2πiνt)Σexp(-2πiq•Rn)e•▽V(r-Rn) (7-88)
where the summation is over all the lattice points (n).
ΔH can be treated as a perturbation. With Equation (7-88), transition from k1 to
k2 has the energy relation of
E2(k2)=E1(k1)±hν (7-90)
The normalized wave function can be written as
Ψk(r)=1/(N)1/2exp(-2πikr)μk(r)
where N is the number of primitive cells in the limited crystal concerned. The matrix
element can be written as
(A/2)(e•RIkk’)[(1/N)Σexp{2πi(k1-k2±q)•Rn}] (7-91)
where the summation is over all the lattice points (n), and
Ikk’ =∫exp{-2πi(k2-k1)•ξ}μ*k’(ξ)μk(ξ)▽V(ξ)dξ (7-92)
Of special importance is the summation in the matrix element:
4

5. (1/N)Σexp{2πi(k1-k2±q)•Rn}
which yields
(1/N)Σexp{2πi(k1-k2±q)•Rn}=1 for
k1-k2±q=-Kn (7-93)
and zero otherwise.
The above presentation is given without introducing the concept of “phonon”;
the two key parameters-the wavevector q and frequency ν of the lattice wave mode-
are taken directly from the deduction of lattice wave modes based on lattice dynamics
[5] [6], where “phonon” is also not introduced.
The dispersion as arising in Equation (7-87) is the attribute of the lattice; it can
be seen from Equations (7-86) to (7-93) that each lattice wave mode endows its
attribute of dispersion, its wavevector q, to its carriers of electromagnetic interaction
(phonons), as particularly indicated by Equations (7-88) and (7-90) to (7-93).
Equations (7-86) to (7-90) indicate that phonon is the (energy) carrier of
electromagnetic interaction between electron and the atoms/ions of oscillating lattice.
Specifically, Equations (7-88), (7-91) and (7-93) also indicate that phonons of some
special lattice wave modes, the optical lattice wave modes, show no difference from
photons. So a photon can be understood as being a special form of phonon.
Moreover, we shall distinguish a phonon from a quantum of oscillation of the
lattice wave mode associating with the phonon. Here, again in analogy to the above
comment by Jan Hilgevoord et al, we should say that phonons are not their lattice
wave modes, although their lattice wave modes are often characterized by
them. As phonons necessarily relate to the time-dependent electric field of their
lattice wave modes, they are always associated with non-stationary process in
crystals.
An optical wave mode can interact with incident electromagnetic wave of the
same wavevector and frequency [7], so an optical phonon can escape its crystal as a
photon, while a non-optical phonon cannot. This important difference is identified as
a candidate water parting between HTS and LTS, as will be explained below.
Hereinbelow “optical phonon” refers to the phonon of an optical lattice wave mode,
“non-optical phonon” refers to the phonon of a non-optical lattice wave mode, and
“phonon” is used to cover both “optical phonon” and “non-optical phonon”.
Back to “measurement” of electrons, it is to be understood as involving not only
intervention by interaction between incident photons and electrons (as in AREPS or
the like) but also electron-phonon interactions in all “real transitions”, particularly the
transitions in the process of electric resistance; therefore, the “real transitions” are
also referred to as “measurement” hereinbelow. If an energy process cannot be
realized by human-performed “measurement”, it also cannot be realized by the
electron-phonon process in electric resistance mechanism. For the system as shown in
Fig. 1, assuming that the system is at ground state E1 at t=0, that the system is
isolated, and that the EM wave mode (hν) is in its ground state, then after time t1, the
energy of the electron can only be “measured” as E 1, as in conformity with the
requirement of energy conservation. But this does not mean that the electron keeps
staying at the eigenstate of E1 all the time, rather it just indicates that “measurement”
can only “collapse” the electron to the eigenstate of E1. Conversely, according to
quantum mechanics, the electron shall transit between the eigenstates of E 1 and E2
during the time period [0, t1], or it may be said that the electron is in an NSS state that
incorporating both energy eigenstates of E1 and E2.
5

6. The condition “the EM wave mode is stable” means “its magnitude (number of
phonons) remains unchanged”. But “number of phonons remains unchanged” can
hardly be ensured unless the lattice wave mode is at its ground state and the system
concerned is at sufficiently low temperature; only at the ground state of a lattice wave
mode can its number of phonons be reliably kept constant (zero). The higher the
frequency of the lattice wave mode is and/or the lower the temperature is, the more
likely and reliable that its phonon number is kept at constant zero.
Let us now consider the two electron system as shown in Fig. 2, which differs
from the system shown in Fig. 1 in that the excited state E2 originally has an electron
too. As I proposed in a previous paper [1], in the system of Fig. 2 “electron pairing by
virtual stimulated transitions” will happen as long as wavevectors k1 and k2 satisfy
wavevector selection rule indicated in Equation (7-93). As has been explained above,
the electron pairing in a system as shown in Fig. 2 is phonon-mediated in general. It
should be easier for the two electron system as shown in Fig. 2 to enter into an NSS
state than the one electron system as shown in Fig. 1, for the electron at the excited
state E2 may emit a threshold phonon, which will balance off the energy deficit as
apparent in the system of Fig. 1. However, once the two electrons in Fig. 2 are in NSS
state, the threshold phonon becomes redundant as far as the NSS state is maintained.
The fate of this threshold phonon is critical in deciding the binding energy of the
electron pairs in crystals and the superconducting attributes of the crystal concerned,
as will be explained shortly later.
For each state (E, k), its pairing candidate could be determined as the
intersections of laminated plot of hν-q dispersion curves [8] of lattice wave modes and
the plot of E-k bands, with the origin of the hν-q plot being placed at the (E, k) point;
for determining pairing candidates for the excited state in the pairing, the hν-q plot
should be placed upside-down. Obviously, each electron usually has more than one
matches of phonon-mediated electron pairing. The collection of all these matches
covers all possible (one phonon)-electron interactions of the subject electron. If all
these phonon-mediated electron pairs can “normally” become superconducting
carriers, HTS would be omnipresent, which is definitely not in conformity with the
rarity of HTS in reality.
Some exemplary scenarios of the phonon-mediated electron pairing by
stimulated transition are shown in Figs. 3 and 4, where exemplary pairings are
indicated by dotted or dashed lines with double arrows. As shown in Figs. 3 and 4, an
electron pairing typically occurs slantingly due to the dispersion of the mediating
phonon. But a few of the pairs are between nearly vertically separated electrons; these
are “optical phonon-mediated” (OPM) pairs, which are indicated by thick dashed lines
in Figs. 3 and 4. The optical threshold phonon of OPM pair of the longitudinal optical
(LO) branch should have the maximum frequency (νM) of all phonons in the crystal.
We now discuss the fate of the threshold phonon and its effect on binding
energy of its electron pair. As mentioned above, the electron originally at excited state
(E2) may emit a threshold phonon, which can be absorbed by the electron originally at
the ground state (E1), so the system of Fig. 2 can easily enter NSS state. Once the
system is in NSS state, the threshold phonon becomes redundant and can be absorbed
by the lattice wave mode (here the significance of distinguishing a phonon from its
lattice wave mode is seen.) But the phonon cannot be emitted to the outside of the
crystal, unless it is an optical phonon. After the phonon is absorbed by the lattice
wave mode, due to the stimulation of the EM wave mode associated with the lattice
wave mode, the phonon is easily taken back by one of the two electrons for that
6

7. electron to perform real transition, and the cycle restarts as the system begins to re-
establish NSS state. As explained above with reference to Equation (9), such frequent
exchanges of the threshold phonon between the lattice wave mode and the pair of
electrons destroy the dominance of matrix elements a 12 and a 21over other matrix
element components, which make the NSS state to collapse into the stationary energy
eigenstates, at which the threshold phonon has to be retrieved by the electron
collapsing to the excited state. As such, a non-optical phonon-mediated electron pair
would not be stable and could not become superconducting carriers.
But an optical phonon-mediated (OPM) pair is different in that the redundant
optical threshold phonon has a definite and substantial probability of escaping the
crystal by radiation, although it can also be absorbed with certain probability by the
lattice wave mode. When the optical lattice mode is at ground state and the
temperature is sufficiently low, once the threshold phonon escapes, the lattice wave
mode will surely be stable and, most notably, each of the two electrons can only
collapse to the ground state (E1) -this is the origin of binding energy of OPM pairs. As
the optical threshold phonon can escape without affecting the stability of the lattice
wave mode, the NSS state will not be affected by the escape of the threshold phonon
and will be maintained until the lattice wave mode received a new optical threshold
phonon, whence the new threshold phonon will allow one of two electrons to collapse
to the excited state. Therefore, the escape of the optical threshold phonon is self-
consistent.
Let us now further consider the fate of a redundant non-optical threshold
phonon. Lattice wave modes may couple with one another by anharmonic crystal
interactions [9] [10], by which a redundant threshold phonon may be taken away from
its lattice wave mode so that each of the two electrons in the pair can only collapse to
the ground state. This might be the origin of binding energy corresponding to low
temperature superconductivity (LTS). However, the probability with which the
threshold phonon escapes by anharmonic crystal interactions must have to compete
with the probability of occurrence of thermal noise phonon of the lattice wave mode.
Thus, even if escape of the threshold phonon by anharmonic crystal interactions can
win over occurrence of thermal noise phonon, non-optical phonon-mediated (NOPM)
electron pairs should be stabilized only below temperatures much lower than those for
OPM pairs. These may indicate that binding energy along may not decide
superconducting temperature (Tc), as the stability of electron pairs is subject to the
effects of a plurality of factors, including the strength of anharmonic crystal
interactions, the presence/absence of optical phonon-mediated pairing matches, and so
on. Of course, if escape of the threshold phonon by anharmonic crystal interactions
cannot win over occurrence of thermal noise phonon, the crystal will not have a
superconducting phase.
While “multiple pairing” is explained above as being common, additional
pairing between E1 and an energy level E3>E1 do not affect stability of pairing
between E1 and <E2 (E3 is not shown in the drawings). We are now explaining this.
Assuming that a third energy level E3 is present in the system shown in Fig. 2, with
E1<E3<E2, and E3 has unstable NOPM pairing with E1 while E2 has OPM pairing with
E1. Then, the matrix elements A13 and A31 will oscillate between 0 and a value having
a modulus less than one, and A12 and A21 will also oscillate, but this will not affect the
NSS states of the electrons on levels E1 and E2. This can be clearly seen from the
composition of the system, which has three electrons, a optical threshold phonon, and
a non-optical threshold phonon; the optical threshold phonon will escape soon or later,
7

8. then at sufficiently low temperature no new optical threshold phonon will enter the
system, so the two electrons originally at E1 and E2 can only stay at NSS states and
collapse to ground state E1 upon “being measured”, no matter what state the third
electron is in.
Now let us consider the fact that both electrons in a stabilized pair can only
collapse to the ground state (E1) upon “being measured”. Obviously, this means that
both electrons condensate to the ground state; the condensation here is in a non-
stationary static (NSS) state, which is a “measured” state, so condensation represents
a “measurement effect”; it does not indicate that the electrons are co-staying on the
stationary ground state (E1); conversely, the electrons are “staying” on a plurality of
stationary states including the original excited state (E2). Moreover, insofar that the
ground state may be the common lower state of multiple pairings as discussed above,
all electrons in these pairings will condensate to the common ground state (E 1) when
their pairs get stabilized.
The effect of an additional pairing between E1 and an energy level E4<E1 varies
(E4 is not shown in the drawings). First, we do not know the exact amount of energy
spread ΔE of any electron. However, if we assume ΔE approximately corresponds to
the energy of the threshold phonon, then the answer to the above question would be
NO in view of the limitation of Pauli Exclusion Principle. This can be explained that,
in the situation shown in Fig. 2, for example, if an electron at NSS state also pairs up
with an electron at E4<E1, its energy spread ΔE might no longer cover the level of E 2
so it could not keep its association with the eigenstate of E2. Thus, the two eigenstates
of E4 and E1 would be co-occupied by the two electrons originally at the states of E4
and E1, but the eigenstate of E1 must also be co-occupied by the electron originally at
E2 if the latter electron is to keep in NSS state, resulting in that the “degree of
occupancy” of eigenstate of E1 would exceed one, which is not in conformity with the
requirement of Pauli Exclusion Principle.
If this explanation is valid, candidate pairings having the same ground state (E1)
have to compete with all its “lower neighbors” (the candidate pairings with eigenstate
E1 being their excited state) to realized themselves. As each candidate pairing can be
characterized by its threshold phonon, whether the electron at a level (such as E1) is
“pairing upward” or “pairing downward” can be said to depend on the competition
between its “upper threshold phonon(s)” and “lower threshold phonon(s)”, with the
definite rule that if one of the “upper threshold phonons” wins then all the “upper
threshold phonons” win (and vise versa).
Obviously, the “upper threshold phonons win” outcome is pro-
superconductivity. It seems that the threshold phonon with greater energy (binding
energy) would have an edge, but magnitude of a matrix element depends on, among
other things, degree of coupling between the two states concerned, and anharmonic
crystal interactions may play a very important role. The question may be that whether
anyone of the upper threshold phonon can eventually dissolve itself into the lower
threshold phonon and something else (as T→0). If it can, the outcome will surely be
“upper threshold phonons win”; but it cannot, the situation could be more
complicated, and superconducting phase (if one exists) could possibly be unstable
and/or uncertain. So no general answer to this question can be given at this time,
except that an optical threshold phonon of LO mode, which corresponds to HTS and
escapes by simple and direct radiation with definite and substantial probability at a
very early stage of the temperature-decreasing process, will definitely win. On the
other hand, it may be likely that all electrons at or near EF cannot get any win in each
8

9. of their candidate pairings. If this happens, the crystal will never have a
superconducting phase.
As explained above, the collection of all candidate threshold phonons of an
electron corresponds to all possible phonon-electron interactions of the electron,
which is also all the candidate pairing options of the electron (except for levels above
EF). Thus, an electron system in lattice can be treated as a network, wherein each
eigenstate E=E(k) is a node with each of its candidate threshold phonons representing
one of its connecting edges, so the electron system can be treated with tools of graph
theory. All electrons in the system might not be in one single graph. And the
characteristic or attribute of the edges (threshold phonons or pairing state) will vary
with temperature, such as that some of them may become “directional” (i.e. it can
only pairing upward or downward, as discussed above) below certain temperature.
As a few more comments on virtual transitions, such as those discussed above
with reference to Fig. 1, a reasonable understanding seems that virtual transition is the
“normal” or “general” form of transition (at least for a system like that shown in Fig.
1) while real transitions are “abnormal” or “special”. In virtual transitions, a process
of virtual lending-returning of boson (phonon) happens in a continuous way while the
system is in a non-stationary stable state; when a real transition occurs, the “normal”
process of lending/returning of boson is interrupted and the system “temporarily”
collapses to an eigenstate; then virtual transitions take place again and the system
begins to re-establish its non-stationary stable state. So an eigenstate corresponds to a
transient process triggered by a real transition (at least in the time-dependent system
as shown in Fig. 1). In other words, like in a time-dependent system at low
temperature, real transitions and associated collapses to eigenstates are occasional
events happening on a continuous background of virtual transitions and non-stationary
stable state, like scattered small islands on a background of vast ocean, while at
sufficiently high temperatures such islands of real transition become so frequent that
they merge into a continent of “common” process of real transitions.
By far, I have
1) identified phonons as carriers of electromagnetic interaction between
electrons and lattice wave modes; the lattice wave modes act on electrons by the EM
wave modes they generate;
2) recognized a promising origin of binding energy of electron pairs in crystals
for high-temperature superconductivity, as relating to electron pairs mediated by
optical phonons, which may escape by direct radiation;
3) identified a possible origin of binding energy of electron pairs in crystals for
low-temperature superconductivity, as relating to electron pairs mediated by non-
optical phonons, which may escape by anharmonic crystal interactions;
4) established a picture of electron condensation, where all electrons in one or
more pairings condensate to their common ground state when their pair(s) gets
stabilized; the condensation is a non-stationary static (NSS) state, which is a
“measured” state, and represents a “measurement effect”;
5) identified a possible mechanism explaining why some crystals never have a
superconducting phase, as that all electrons at or near EF cannot get any win in pairing
competitions for each of their candidate pairings;
6) investigated some virtual behaviors (particularly relating to “measured”
energy) of one-electron system and two-electron system on the basis of Heisenberg
Uncertainty Principle, and identified non-stationary stable state of electrons engaging
9

10. in “electron pairing by virtual stimulated transitions” and its importance in
establishing superconductivity;
7) recognized the behavior and role of threshold phonon, released in non-
stationary stable state by electron from excited state; recognized the redundancy of the
threshold phonon in non-stationary stable state;
8) identified a mechanism by which the stability of lattice wave mode and/or
temperature affect the stability of electron pairs mediated by the phonon
corresponding to the lattice wave mode;
9) identified and discussed the situations and effects relating to “multiple-
pairing”; recognized “pairing competition” and “direction of pairing” and some
details of the mechanism in which the “direction of pairing” is determined; and
10) proposed that electron pairing, and the electron system in crystal generally,
may be treated with tools of graph theory, with each eigenstate E=E(k) in the system
being a node and each of its candidate threshold phonons representing one of its
connecting edges.
In summary, a promising origin of binding energy of electron pairs in crystals
has been recognized, and the identity of phonon as carrier of electromagnetic
interaction between electrons and lattice wave modes has been recognized. While
electron pairing is phonon-mediated in general, HTS should be associated with
electron pairing mediated by optical phonon at or near EF, so the rarity of HTS
corresponds to the rarity of such pairing match. The origin of binding energy of
electron pairs, and of superconductivity in general, may relate to some attributes of
physics as manifesting in virtual transitions and non-stationary stable process of an
electron system subjected to an EM wave mode.
References:
[1] “A mechanism of electron pairing relating to superconductivity”, Qiang LI,
http://www.paper.edu.cn/index.php/default/releasepaper/content/42033
[2] “Phonon”, http://en.wikipedia.org/wiki/Phonon
[3] “Time in Quantum Mechanics”, Jan Hilgevoord et al,
http://atkinson.fmns.rug.nl/public_html/Time_in_QM.pdf.
[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
201-205.
[5] Pages 102－112 of [4].
[6] Kittel Charles Introduction To Solid State Physics 8Th Edition, pages 95-99 and
Figs, 7, 8(a) and 8(b) of Chapter 4.
[7] Fig. 5-9 and page 108 of [4].
[8] Figs, 7, 8(a), 8(b), and 11 of Chapter 4 of [6]
[9] Pages 119-120 of [6].
[10] Page 120-121 of [4].
10

11. E2(k2)
Stimulated transition of
electron
EM mode hν=E2- E1
E1(k1)
electron
Figure 1 A dual energy system of one electron subject to an electromagnetic (EM)
wave mode hν=E2-E1.
11

12. :
E2(k2)
Electron 2
Electron pairing
EM mode hν=E2- E1
E1(k1)
Electron 1
Figure 2 A dual energy system of two electrons subject to an electromagnetic (EM)
wave mode hν=E2-E1.
12

13. E= E(k) Band E2(k2)
EF
Pairing with
E2(k2)-E1(k1)~hνM Energy Gap
Pairing with
E2(k2)-E1(k1)<hνM Band gap
Peak
Band E1(k1)
k
Figure 3 An exemplary scenario of the phonon-mediated electron
pairing by stimulated transition
13

14. E= E(k) E1(k1) E2(k2)
EF
wi
Gap
Peak
Pairing
k
Figure 4 Another exemplary scenario of the phonon-mediated electron pairing
by stimulated transition
14