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phonon as carrier of electromagnetic interaction between lattice wave modes and electrons and its role in superconductivity

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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 …

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.

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  • 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