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.
1

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 nk1 ∝ ∫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 nk1 ∝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
2

3. 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 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.
3

4. 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.
4

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.
:
:
:
:
acceptor band
Ei3
Ei2
Ei1
electron
Ei1- Esmax =hωM/(2π)-Δ
Esmax
full band
:
:
:
FIG. 1: p-doped system
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
5

6. 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
6

7. 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.
:
:
:
:
conducting band
Esmin
Esmin-Ei1=hωM/(2π)-Δ
electron
Ei1
Ei2
Ei3
donor band
FIG. 2: n-doped system
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).
7

8. 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.)
8

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noitcaretni fo noisnet eht si β dna ,y levitcepser snoi dnoces dna tsrif eht fo ssam
eht era m dna M ,ylevitcepser snoi dnoces dna tsrif eht fo edutingam eht era B dna
A ,snoi gnirobhgien neewteb ecnat sid muirbi l iuqe eht si a erehw ,a4/1≤q<a4/1- htiw
)qaπ2socβ2( /)β2- ωm(-= )A/B( ± ±
2
}] )m+M(/qaπ2 nismM4-1[±1{)mM/)m+M(β= ω ±
2 2 2
:]1[ fo mrof eht evah sevaw ecittal fo snoitulos
l areneg eht ,mret citardauq sti ot pu detamixorppa era ygrene noitcaretni eht taht dna
deredisnoc era snoi gnirobhgien neewteb snoitcaretni eht ylno taht snoitpmussa eht
rednu ,niahc ecittal noi gnol lanoisnemid-eno a rof taht dehsi lbatse neeb sah tI
ledom niahc ecittal noi gnol D-1 fo sesylana dezilareneG
.dedivorp si ytivitcudnocrepus fo erofereht dna slatsyrc
noi ni gniriap nortcele fo msinahcem etadidnac A .)π2(/Mωh naht rellams on y l lacipyt
ygrene gnidnib a htiw )Mω dna dnab rotpecca/ronod yb derutaef erutcurts level
ygrene gnidulcni( srotcaf laiceps emos fo noitanibmoc a fo txetnoc eht ni dezi libats
eb nac ti yb decudorp sriap nortcele rof ,ytivitcudnocrepus ot tcepser htiw ecnaci f ingis
laiceps fo si Mω ycneuqerf mum ixam eht gnivah edom evaw ecittal ehT .edom evaw
ME eht yb decudni snoitisnart detalumits yb egnar emas eht revo metsys nortcele
s’latsyrc eht ni decudorp era sriap nortcele ,cipocsorcam neve ro gnol eb nac hcihw
,egnar sti ni dehsilbatse era sedom evaw ME/ecittal eht ecno taht dehsi lbatse si tI

11. 3
.sriap nortcele gni vivrus eht fo Mω eht ot gnidnopserroc noitcerid eht si ti rof
,ytivitcudnocrepus evitcepsorp fo noitcerid eht ot dnopserroc yam snoitatneiro latsyrc
el bissop l la fo Mω mum ixam eht htiw noitatneiro latsyrc a ylno taht setacidni sihT
.]111[ ni snotohp/nonohp eht fo emos yb nekorb eb l liw ]001[ ni ω ot gnidnopserroc
sriap nortcele os ,]001[ noitatneiro latsyrc ni taht naht retaerg si ]111[ noitatneiro
latsyrc ni ω fo mum ixam eht latsyrc rBK rof taht swohs rehtruf troper eht ;sedom
)lacitpo lanidutignol( OL eht fo 0→q ta si ω fo mum ixam eht ,noitatneiro latsyrc
hcae rof ,taht swohs osla troper ehT .AT>AL>OT>OL :noitaler eht evah sedom
evaw tnereffid fo seulav ω ,]2[ latsyrc rBK no tnemirepxe gnirettacs citsale-non
nortuen fo troper a ot gnidrocca sA .llec eviti mirp eno ni snoi/smota fo rebmun
eht si n erehw ,sedom gnita l licso Nn3 sah sl lec eviti mirp N htiw latsyrc A
slatsyrc ci n o i D3 n i eg n ah cx e/g n iriap - n ortcelE
.sgniriap
dna snoitisnart eseht gnirud dettime/debrosba era taht snotohp si ti ,sedom evaw ME
fo dleif gnital licso eht yb detareneg era slatsyrc cinoi eht ni sgniriap dna snoitisnart
nortcele dessucsid evoba eht sa tuB .neppah y llautca ot deen ton od yeht ;lautriv
era snoitisnart detalumits ro/dna sg niriap-nortcele ni devlovni snoissime/snoitprosba
eseht rof ,repap siht fo snoisulcnoc eht fo yna tceffa ton seod noitisnart evoba eht
ni notohp ro nonohp a stime/sbrosba nortcele na rehtehw taht deton eb ot si tI
.snortcele fo snoitisnart detalumits
gnirud elbahsiugnitsid emoceb metsys eht fo sdnab eht ni level ygrene hcae taht
ni stluser ssecorp sihT .t emit tneici ffus retfa metsys eht ni tsixe l liw )π2(/mωh =kE-nE
ot gnidnopserroc snoitisnart ylno ,t emit htiw ))π2(/mωh-knE(δΣ→ 1kn a htiW
.kE ta y llanigiro s i hcihw ,nortcele rehto eht yb )noitprosba/noissime
notohp lautriv( debrosba yltcerid si hcihw ,kE-nE=)π2(/ωh ygrene fo )eki l
eht ro nonohp ro( notohp a gnitti me nE ygrene ta y l lanigiro nortcele eht htiw ,setats
rieht gnignahcxe yb tisnart nac ,)kE>nE taht emussa ylefas nac ew ereh( y levitcepser
kE dna nE ygrene gnivah ,riap hcae ni snortcele owt eht fo htob taht os ,”sriap“
otni sevlesmeht mrof yeht taht si siht htiw epoc ot snortcele eht rof yaw ehT
.)3-1 .uqE( yb detacidni sa
tisnart y l lamron tonnac sdnab ygrene ni snortcele )lla ton fi( tsom ,sedom evaw ecittal
fo )π2(/ωh lacipyt rof ,suhT .sdnab l luf mrof snortcele slatsyrc noi ynam ni dna ,sdnab
ygrene ni era latsyrc a ni snortcele lla taht si tcaf dehsi lbatse-llew a ,revewoH
g n iriap - n ortcelE
.meht gnoma eno tsetaerg
eht gnieb Mω htiw ,sniahc noi eht fo sedom evaw ME/ecittal tnereffid eht setoned..…3
,2 ,1=m erehw ,))π2(/mωh-knE(δΣ→ 1kn a :si taht ,)seulav etercsid N sah ω erehw(
)π2(/ωh±=knE htiw tisnart l liw metsys eht ni snortcele lla tsomla ,t emit emos retfa
dna ,t emit htiw gnola )π2(/ωh±=knE ot segrevnoc 1kn a ,3-1 .uqE yb detacidni sA
.)aN2(/1±=q ta evaw ”lacitpo“ eht ot sdnopserroc hcihw ,Mω sa seulav
ω l la fo eno mum ixam eht dei fitnedi ew ,esoprup evitartsul li roF .2/N± .…,2± ,1±=n
htiw ,)aN2(/n=q ot gnidnopserroc )π2(/ωh±=knE ta skaep N2 fo latot a sah 1kn a
.nE ot kE morf tisnart ot notohp

12. 4
.ygrene gnidnib elbissop a ot pu dezi l ibats eb l liw )xamsφ+ jiφ( sriap
nortcele evoba eht ,oiranecs a hcus ni ;dnab l luf eht ni seloh eht gni l l if suht ,metsys
eht otni detcejni era snortcele ,degrahc y levitagen si latsyrc cinoi eht fi ,egatlov/dlei f
lacirtcele lanretxe rednu y l laicepse ,degrahc y l isae si rotalusni na sa ,revewoH
.dnab rotpecca eht fo mottob eht dna dnab l luf eht fo pot eht ssorca deniatniam
eb y lbissop dluoc muirbi l iuqe ci manyd ni )xamsφ+ jiφ( sriap nortcele fo loop a tuB
.elbats eb ton dluoc )xamsφ+ jiφ( ekil sriap nortcele eht ,dnab l luf eht ni eloh eno neve fo
ecneserp eht htiw oS .seloh eht fo yna ot riap eht ni nortcele rewol eht fo noitisnart yb
nekorb eb nac )…,2,1=j htiw( )xamsφ+ jiφ( ekil riap snortcele eht ,dnab lluf eht ni stsixe
)dnab rotpecca eht ot gnitisnart snortcele yb tfel esoht sa hcus( seloh emos fI
.)t(ψ metsys lanigiro eht dna 2iE dna 1iE slevel ygrene rotpecca
eht gnidulcni demrof neht si )}2iE{+}1iE{+)t(ψ( metsys wen a ,2iE -xamsE=)π2(/Mωh
yfsitas 2iE ,yas ,dna ,xamsE ,Mω taht gni mussa ,elpmaxe rof ,y llareneg eroM
dnab rotpecca ot gnitaler sriap nortcele fo ygrene gnidniB
.evitcudnoc si )}1iE{+)t(ψ(
metsys wen siht dna ,)t(ψ metsys lanigiro eht dna 1iE level ygrene rotpecca
eht gnidulcni metsys wen a g nimrof suht ,)π2(/Mωh=knE fo noitisnart detalumits
yb 1iE ot tisnart nac dnab l luf eht fo level xamsE no snortcele ,)π2(/Mωh=knE ot
gnidnopserroc snoitisnart detalum its era ereht ecnis ,neht ,latsyrc eht ni sniahc noi eht
htiw detaicossa sedom evaw citengamortcele gnital licso eht fo ycneuqerf tsetaerg eht
gnieb Mω dna dnab l luf eht ni level ygrene tsehgih eht gnieb xamsE erehw ,)π2(/Mωh
naht rellams ylthgils ro ot slauqe xamsE-1iE htiw ,)1 .GIF ees( latsyrc cinoi na fo metsys
dnab l luf a ni decudortni si …3iE<2iE<1iE slevel ygrene htiw dnab ygrene rotpecca
na ,Mω ot gnidnopserroc noitatneiro latsyrc eht fo metsys dnab ygrene eht ni ,fI
metsys depod-rotpecca na ni ygrene gnidnib dna gniriap nortcelE
.))π2(/mωh-knE(δΣ→ )0t,t(U htiw )0t(ψ)0t,t(U=)t(ψ
yb deriuqer sa snoitisnart detalumits eht mrofrep ot snortcele eht rof yaw
y lno eht semoceb egnahcxe/gniriap-nortcele hcus ),snortcele rehto yb deipucco neeb
evah denrecnoc snortcele eht fo noitisnart rof setats detegrat evitcepsorp lla ,si taht(
”tci lfnoc ycnapucco“ etelpmoc rednU .elbahsiugnitsid-non era denrecnoc snortcele
eht taht esnes eht ni y llaicepse ,yt ilaer ni neppah ot ton deen snoitisnart detalumits
eht -”lautriv“ emoceb snortcele owt eht fo snoitisnart detalumits eht ,gniriap
hcus nI .rehto hcae htiw sevlesmeht gniriap yb ,si taht ,sevlesmeht neewteb setats
fo egnahcxe rieht fo mrof eht ni eb nac riap eht ni snortcele owt eht fo snoitisnart
detalumits eht ,sedom noitarbiv ecittal eht fo eno fo ycneuqerf eht sehctam ecnereffid
ygrene rieht fi ,latsyrc noi na hcus fo metsys nortcele eht ni snortcele owt roF
.sedom evaw citengamortcele gnitallicso hcus yb nevird esoht era latsyrc
noi na ni snoitisnart detalumits eht ,suhT .sedom noitarbiv ecittal eht sa seicneuqerf
emas eht fo sedom evaw )ME( citengamortcele gnital licso etareneg ecittal fo sedom
noitarbiv os ,sniahc noi yb demrof si ecittal ,slatsyrc cinoi nI ).slatsyrc ni stci lfnoc
y cnapucco yb detim il eb yam noitisnart suoenatnopS( .)0t(ψ)0t,t(U=)t(ψ fo metsys
a ni tsixe noitisnart detalumits dna noitisnart suoenatnops htob ,y l lareneG
.snips etisoppo evah )0t(ψ)0t,t(U=)t(ψ
metsys ni riap emas eht ni snortcele ,elpicnirP iluaP fo noitati m il ot euD

13. 5
owt eht fo yna yb nekat eb thgi m notohp eht ,)metsys tneserp eht ni sa erutcurts
dnab fo noitati m il eht tuohtiw( noitautis lareneg a ni nekorb si riap eht sA .setats owt
eht fo hcihw ni si nortcele hcihw g ni y ficeps tuohtiw )xamsφ+ jiφ( fo setats detalerroc
eht ypucco-oc riap eht ni snortcele owt ehT .evoba dessucsid 3-1 .uqE fo noisserpxe
eht yb detacidni sa ,setats rieht egnahcxe snortcele owt eht nehw sneppah egnahcxe
notohp lautriv taht htiw ytimrofnoc ni y l lacigolonemonehp si sihT .riap nortcele
eht mrof ot rehtegot snortcele owt eht sdnib hcihw ,)π2(/Mωh ygrene na htiw notohp
a sulp snortcele owt eht sedulcni riap nortcele eht taht si siht rof ledom A .noitisnart
eht sekam ti erofeb tsuj tnemom eht ta )π2(/Mωh ygrene eht evah ton seod y llautca
3iE ot gnitisnart nortcele eht taht tnemugra eht yb si hcaorppa evitanretla nA
.suolucidir dna egnarts smees ygrene notohp f lah eht ,revewoH
.)π4(/Mωh tsael ta fo esaercni
na si ygrene cipocsorcam eht ni egnahc eht os ,)π2(/Mωh naht retaerg eulav emos
si )rehgih ro( 3iE ot stisnart taht nortcele eht yb taht elihw ,)xamsφ dna 2iφ ta seulav
ygrene eht fo eulav egareva eht ,si taht( )π2(/Mωh fo flah eno si )rehgih ro( 3iE ot
stisnart nortcele rehto eht retfa )xamsφ+ 2iφ( no sniamer llits taht nortcele eht yb ygrene
cipocsorcam eht ot noitubirtnoc eht ,emit gnol revo seulav ygrene derusaem eht fo
egareva eht ,noiti nifed yb ,si metsys denibmoc eht fo ygrene cipocsorcam eht sA
metsys depod-p :1 .GIF
:
:
:
dnab lluf
xamsE
Δ-)π2(/Mωh= xamsE -1iE
nortcele
1iE
2iE
3iE
dnab rotpecca
:
:
:
:
.rehgih ro 3iE level ygrene eht ot tisnart ot sah riap eht ni nortcele
no tsael ta neht ,riap eht ni nortcele na fo noitisnart yb nekorb eb ot si riap nortcele
eht fi ,oiranecs siht fo erutcurts ygrene ralucitrap eht fo noitati m il eht ot euD
.)xamsφ+ jiφ( riap nortcele na hcus
fo ygrene gnidnib eht fo noitanimreted si ereh rotcaf eltbus dna yek a ,,nehT

14. 6
)2 .GIF ees( .)π2(/Mωh naht rellams
ylthgils ro ot lauqe ecnereff id a yb dnab ronod eht ni 1iE level tsehgih eht naht rehgih
gnieb dnab gnitcudnoc eht ni nimsE level tsewol eht htiw ,dnab gnitcudnoc a htaeneb si
dnab ronod eht taht tpecxe ,dnab rotpecca a gnivah taht ot ral i mis si )……3iE>2iE>1iE(
dnab ronod a gnivah metsys a ni ytivitcudnocrepus fo msinahcem ehT
metsys dnab ronod ni sriap nortcelE
.neppah reven yam ytivitcudnocrepus ,noitatneiro latsyrc
rehto yna ot gnidnopserroc sedom OL fo ycneuqerf mum ixam≤xamsE-1iE=Δ-)π2(/Mωh
gnikam sa tnetxe eht ot sesaercni Δ fi tuB .dnab rotpecca eht ni level gnidnopserroc
a ta nortcele na htiw riap nortcele dezi libats a gnimrof nortcele na evah yam slevel
y grene eseht fo hcae dna ,dnab rotpecca eht ni …3iE<2iE<1iE slevel ygrene fo ytilarulp
a eb nac ereht )xamsE-1iE(-)π2(/Mωh=Δ fo egnar a ni ,selpmas emos ni ,rehtruF
.)π2(/Mωh
naht rel lams on ygrene gnidnib a htiw ,xamsE woleb level ygrene gnidnopserroc a
no nortcele na htiw riap nortcele gnitcudnocrepus a mrof osla nac 1iE ,y lral imiS
.]3[ K001=T ta neve nonohp
a yb nekorb eb ylerar nac riap nortcele na hcus ,s/3101x5 ,yas ,evoba Mω rof ,suhT
.5.64-56.4≈)Tk(/))π2(/Mωh( eb l liw ereht K001=T ta ,s/4101-3101≈)π2(/Mω ,ygrene
gnidnib a hcus htiw riap nortcele na hcus fo yti libats eht fo noitamitse na sA
.riap eht
ni nortcele na fo noitisnart yb noitcurtsed tsniaga riap eht fo ygrene gnidnib eht gnieb
)π2(/Mωh htiw ,)erutcurts dnab ygrene ralucitrap siht ni( noitisnart yna yb nekorb
si riap nortcele eht taht ytil ibaborp eht tcaf ni si sihT .))π2(/Mωhβ-(pxe=)1E(ρ/)2E(ρ
oS .)Tk=β/1 erehw( ))π2(/Mωhβ2-(pxe )2E(ρ gnieb 3iE ot nortcele eht
fo noitisnart eht retfa taht elihw ,))π2(/Mωhβ-(pxe )1E(ρ ]4[ gnieb 3iE ot nortcele
eht fo noitisnart eht erofeb elbmesne eht fo srebmem fo ρ noitroporp ehT .ecittal eht
htiw detaicossa sedom evaw ME eht dna ,ecittal eht ,snortcele eht fo metsys denibmoc
eht fo el bmesne lacinonac ’sbbiG fo noitcnuf noitubirtsid eht redisnoc neht eW
.dnab lluf eht ni eloh ton si ereht nehw riap nortcele siht fo ygrene
gnidnib eht eb dluohs hcihw ,nekorb si riap nortcele eht retfa )π2(/Mωh yb desaercni
eb dluohs evoba dessucsid sa metsys denibmoc eht fo ygrene eht ,eroferehT
.dettimo si )xamsφ+ jiφ( ta nortcele gni niamer eht htiw detaicossa
notohp eht :dezingocer si des si m si ygrene notohp fo f lah eno eht erehw ,suhT
).devlah eb dluow ygrene gnidnib eht ,esiwrehto ,)π2(/Mωh si denrecnoc riap nortcele
eht fo ygrene gnidnib eht ,eurt si remrof eht fI ?ygrene nwo s’evaw ME fo trap
sa ygrene s’notohp eht fo flah sevreser ro noitisnart nortcele eht rof y leritne notohp
eht fo ygrene eht ”sdneps“ evaw ME eht rehtehw ,)π2(/Mωh fo notohp a hcus sbrosba
dna sti me y levitanretla nortcele eht taht oiranecs eht ni ,si ereh noitseuq eht oS(
.)xamsφ+
jiφ( neewteb tisnart dna ta yats ot )π2(/Mωh fo )eno lanigiro eht eb nac hcihw( notohp
rehtona ekat lliw nortcele gni niamer eht ;og ot ti rof yaw ylno eht si hcihw ,3iE evoba
ro ta level ygrene na ot og ot )hguoht ,riap eht nihtiw y l lanigiro notohp eht y lirassecen
ton( )π2(/Mωh fo notohp eno ekat tsum snortcele owt eht fo eno ,noitaredisnoc
rednu erutcurts level ygrene eht fo noitautis ralucitrap eht ni tuB .notohp
eerf a sa dettime neve ro ,edom evaw ME gnidnopserroc eht yb nekat ,snortcele

15. 7
sti fo egnar eht ni dehsi lbatse si Mω fo edom evaw ME ecno ,yram mus nI
.)π2(/Mωh naht rel lams on ygrene gnidnib a sah ,dnab ronod eht fo pot eht ta esoht
dna dnab gnitcudnoc eht fo mottob eht ta slevel ygrene evitcepser neewteb demrof
,sriap nortcele hcus fo hcae ,dnab rotpecca na htiw metsys a ot ral i mis oslA
.neppah reven thgim yt ivitcudnocrepus ,noitatneiro latsyrc rehto fo sedom
OL fo ycneuqerf fo mum ixam≤1iE-nimsE=Δ-)π2(/Mωh ekam ot desaercni si Δ fi tuB
.)2 .GIF ees( dnab ronod eht fo pot eht ta esoht dna dnab gnitcudnoc eht fo mottob eht
ta slevel ygrene evitcepser neewteb demrof eb nac sriap nortcele fo ytilarulp a taht os
slevel ygrene fo yti larulp a etadom mocca ot desaercni eb nac )1iE-nimsE (-)π2(/Mωh=Δ
fo egnar level ygrene eht ,dnab rotpecca na htiw metsys a ot ral imiS
.dezi libats era
sriap nortcele eht dna del l if era dnab ronod eht ni seloh taht os ,slevel ro dnab ronod
eht ot tisnart dna ,)egatlov/dleif cirtcele lanretxe rednu ylralucitrap( level ygrene
hgih ylevitaler a ta ,metsys eht retne yam snortcele lanretxe taht si oiranecs A
metsys depod-n :2 .GIF
3iE
dnab ronod
2iE
1iE
nortcele
E
Δ-)π2(/Mωh=1i -nimsE
nimsE
dnab gnitcudnoc
:
:
:
:
.dnab rotpecca
ot tcepser htiw evoba denialpxe sa m sinahcem rali m is eht htiw ,dezi l ibats emoceb l liw
2iE dna 1iE slevel no snortcele eht htiw dnab gnitcudnoc eht no snortcele yb demrof
sriap esoht ,del lif wohemos era dnab ronod eht ni )s(eloh eht ,revewoh ,fI
.eloh a hcus ot tisnart
y l isae nac sriap nortcele eht fo hcae fo ygrene rewol eht ta nortcele eht rof ,dnab
ronod eht ni stsixe )dnab gnitcudnoc eht ot gnitisnart snortcele eht yb tfel seloh eht
ylralucitrap( )s(eloh yna sa raf sa e lbatsnu era sriap nortcele eseht tuB .2iE dna 1iE ot
tisnart retal hcihw snortcele htiw sriap nortcele mrof neht yam dna ,Mω fo edom evaw
ME eht yb noitisnart detalumits yb dnab gnitcudnoc eht retne yam 2iE dna 1iE slevel
ygrene ronod no snortcele ,nehT .)π2(/Mωh=2iE-nimsE emussa ew ,noitartsul li roF

16. 8
).45-25 segap ,9791 yraurbeF ni detnirp ,5691 tsuguA ,noitide dnoces ,1310.21031
fo rebmuN kooB dei f inU a htiw esuoH noitaci lbuP noitacudE s’elpoeP yb esenihC
ni dehsi lbup ,ixuhZ ,GNAW rosseforP yb ,”sci syhP lacitsitatS ot noitcudortnI“ ]4[
.I xidneppA ,9791 yraurbeF ni
detnirp ,5691 tsuguA ,noitide dnoces ,1310.21031 fo rebmuN kooB dei f inU a htiw
esuoH noitaci lbuP noitacudE s’elpoeP yb esenihC ni dehsilbup ,ixuhZ ,GNAW
rosseforP yb ,”sci syhP lacitsitatS ot noitcudortnI“ morf nekat stnatsnoc scisyhP ]3[
.411 egap ,31-5 .giF ,]1[ eeS ]2[
.04-5 .uqE
,601 egap ,9791 yraunaJ fo tnirp tsrif fo etad a dna ,6691 enuJ fo etad noitaci lbup
a ,0220.21031 fo rebmuN kooB dei f inU a htiw ,esuoH noitaci lbuP noitacudE
s’elpoeP yb )esenihC ni( dehsi lbup ,nuK GNAUH .forP yb ,”scisyhP etatS diloS“ ]1[
.dedivorp si ytivitcudnocrepus fo erofereht dna slatsyrc
noi ni gniriap nortcele fo msinahcem etadidnac A .)π2(/Mωh naht rellams on y l lacipyt
ygrene gnidnib a htiw )Mω dna dnab rotpecca/ronod yb derutaef erutcurts level
ygrene gnidulcni( srotcaf laiceps emos fo noitanibmoc a fo txetnoc eht ni dezi libats
eb nac ti yb decudorp sriap nortcele rof ,ytivitcudnocrepus gnidrager ecnaci f ingis
laiceps fo si Mω ycneuqerf mum ixam eht gnivah edom evaw ecittal ehT .edom evaw
ME eht yb decudni snoitisnart detalumits yb egnar emas eht revo metsys nortcele
s’latsyrc eht ni decudorp era sriap nortcele ,cipocsorcam neve ro gnol eb nac hcihw
,egnar sti ni dehsilbatse era sedom evaw ME/ecittal eht ecno taht dehsi lbatse si tI
n o isu lc n oC
.sriap nortcele eht fo ygrene gnidnib gnidnopserroc a ni gnitluser dna
sriap nortcele gnizi libats ,sriap ni snortcele yb snoitisnart elbissop gniti m il ni noitcnuf
ral i m is evres thgi m ecaf i mreF fo epahs denettal f sa hcus esoht ,slatem ni tneserp
eb ton yam serutaef erutcurts dnab ygrene ekil srotcaf el ihW .gniriap nortcele etomorp
hcihw ,sedom evaw ecittal eht ot gnidnopserroc sedom evaw ME dna sniahc elopid
gnitluser ,noitubirtsid egrahc fo noitaived esuac thgim ecittal fo sedom evaw citsuoca
yb detareneg seroc mota fo snoitarbiv ,slatem ekil latsyrc mota-elgnis roF
.cte dna ,egnar gnol yltneiciffus a revo Mω fo edom evaw ecittal fo
noitanimod fo noitcurtsed ro/dna ,snoitcaretni nonohp-ynam y lralucitrap ,snoitcaretni
rehto ot eud eb yam sriap nortcele eht fo noitcurtsed ehT .dehsilbatse eb erofereht nac
ytivitcudnocrepus dna ,latsyrc eht ni noitaticxe detalumits ro snonohp yb nekorb eb
y ldrah nac sriap nortcele eseht ,latsyrc cinoi eht fo sedom evaw ecittal/evaw ME eht
fo ycneuqerf tsehgih eht fo )π2(/Mωh ygrene notohp naht rel lams on ygrene gnidnib a
evah latsyrc cinoi elbatius a fo metsys dnab rotpecca/ronod eht ni sriap nortcele emos
sA .egnar emas eht revo metsys nortcele s’latsyrc eht ni decudorp y lgnidnopserroc
si gniriap-nortcele ,cipocsorcam neve ro gnol eb nac hcihw ,niahc noi detaicossa