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Formation of singlet Fermion pairs in the dilute gas of boson-Fermion mixture.

1 Introduction

In 1938, the connection between the ideal Bose gas and superfluidity in helium was first made by London [1]. The ideal Bose gas undergoes a phase transition at sufficiently low temperatures to a condition in which the zero-momentum quantum state is occupied by a finite fraction of the atoms. This momentum-condensed phase was postulated by London to represent the superfluid component of liquid [sup.4]He. With this hypothesis, the beginnings of a two- fluid hydrodynamic model of superfluids was developed by Landau [2] where he predicted the notation of a collective excitations so- called phonons and rotons.

The microscopic theory most widely- adopted was first described by Bogoliubov [3], who considered a model of a non-ideal Bose-gas at the absolute zero of temperature. In 1974, Bishop [4] examined the one-particle excitation spectrum at the condensation temperature [T.sub.c].

The dispersion curve of superfluid helium excitations has been measured accurately as a function of momentum [5]. At the lambda transition, these experiments show a sharp peak inelastic whose neutron scattering intensity is defined by the energy of the single particle excitations, and there is appearing a broad component in the inelastic neutron scattering intensity, at higher momenta. To explain the appearance of a broad component in the inelastic neutron scattering intensity, the authors of papers [6-7] proposed the presence of collective modes in superfluid liquid [sup.4]He, represented a density excitations. Thus the collective modes are represent as density quasiparticles [8]. Such density excitations and density quasiparticles appear because of the remaining density operator term that describes atoms above the condensate, a term which was neglected by Bogoliubov [3].

Previously, the authors of ref [9] discovered that, at the lambda transition, there was scattering between atoms of the superfluid liquid helium, which is confirmed by the calculation of the dependence of the critical temperature on the interaction parameter, here the scattering length. On other hand, as we have noted, there are two types of excitation in superfluid helium at lambda transition point [5]. This means it is necessary to revise the conditions that determine the Bose-Einstein condensation in the superfluid liquid helium. Obviously, the peak inelastic neutron scattering intensity is connected with the registration of neutron modes in a neutron-spectrometer which, in turn, defines the nature of the excitations. So we may conclude that the registration of single neutron modes or neutron pair modes occurs at the lambda transition, from the neutron-spectrometer.

In this letter, we proposed new model for Bose-gas by extending the concept of a broken Bose-symmetry law for bosons in the condensate within applying the Penrose-Onsager definition of the Bose condensation [10]. After, we show that the interaction term between Boson modes and Fermion density modes is meditated by an effective attractive interaction between the Fermion modes, which in turn determines a bound state of singlet Fermion pair in a superfluid Bose liquid- Fermion gas mixture.

We investigate the problem of superconductivity presented by Frolich [11]. Hence, we also remark the theory of superconductivity, presented by Bardeen, Cooper and Schrieffer [12], and by Bogoliubov [13] (BCSB). They asserted that the Frolich effective attractive potential between electrons leads to shaping of two electrons with opposite spins around Fermi level into the Cooper pairs [14]. However, we demonstrate the term of the interaction between electrons and ions of lattice meditates the existence of the Frolich singlet electron pairs.

2 New model of a superfluid liquid helium

First, we present new model of a dilute Bose gas with strongly interactions between the atoms, to describe the superfluid liquid helium. This model considers a system of N identical interacting atoms via S-wave scattering. These atoms, as spinless Bose-particles, have a mass m and are confined to a box of volume V. The main part of the Hamiltonian of such system is expressed in the second quantization form as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are, respectively, the "creation" and "annihilation" operators of a free atoms with momentum [??]; [U.sub.[??]] is the Fourier transform of a S-wave pseudopotential in the momentum space:

[U.sub.[??]] = 4[pi]d[h.sup.2]/m (2)

where d is the scattering amplitude; and the Fourier component of the density operator presents as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

According to the Bogoliubov theory [3], it is necessary to separate the atoms in the condensate from those atoms filling states above the condensate. In this respect, the operators [[??].sub.0] and [[??].sup.+.sub.0] are replaced by c-numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within the approximation of the presence of a macroscopic number of condensate atoms [N.sub.0] [much greater than] 1. This assumption leads to a broken Bose-symmetry law for atoms in the condensate state. To extend the concept of a broken Bose-symmetry law for bosons in the condensate, we apply the Penrose-Onsager definition of Bose condensation [10]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

This reasoning is a very important factor in the microscopic investigation of the model non-ideal Bose gas because the presence of a macroscopic number of atoms in the condensate means new excitations in the model Bose-gas for superfluid liquid helium:

[N.sub.[??][not equal to]0]/[N.sub.0] = [alpha] [much less than] 1,

where [N.sub.[??][not equal to]0] is the occupation number of atoms in the quantum levels above the condensate; [alpha] is the small number. Obviously, conservation of the total number of atoms suggests that the number of the Bose-condensed atoms [N.sub.0] essentially deviates from the total number N:

[N.sub.0] + [summation over ([??][not equal to]0)] [N.sub.[??][not equal to]0] = N,

which is satisfied for the present model. In this context,

[alpha] = N - [N.sub.0]/[N.sub.0][[SIGMA].sub.[??][not equal to]0] 1 [right arrow] 0,

where [[SIGMA].sub.[??][not equal to]0] 1 [right arrow] [infinity].

For futher calculations, we replace the initial assumptions of our model by the approximation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The next step is to find the property of operators [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by applying (5). Obviously,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Excluding the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the density operators of bosons [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] take the following forms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are, respectively, the Bose-operators of density-quasiparticles presented in reference [8], which in turn are the Bose-operators of bosons used in expressions (6) and (7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Thus, we reach to the density operators of atoms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], presented by Bogoliubov [3], at approximation No-const, which describes the gas of atoms [sup.4]He with strongly interaction via S-wave scattering:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

which shows that the density quasiparticles are absent.

The identical picture is observed in the case of the density excitations, as predicted by Glyde, Griffin and Stirling [5-7] proposing [[??].sub.[??]] in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where terms involving [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are written separately; and the operator [[??].sub.[??]] describes the density-excitations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

After inserting (6) and (7) into (15), the term, representing the density-excitations vanishes because [[??].sub.[??]] = 0.

Consequently, the Hamiltonian of system, presented in (1) with also (12) and (13), represents an extension of the Bogoliubov Hamiltonian, with the approximation [N.sub.0]/N const, which in turn does not depend on the actual amplitude of interaction. In the case of strongly interacting atoms, the Hamiltonian takes the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the velocity of sound in the Bose gas, and which depends on the density atoms in the condensate [N.sub.0]/N.

For the evolution of the energy level, it is a necessary to diagonalize the Hamiltonian [[??].sub.a] which is accomplished by introduction of the Bose-operators [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by using of the Bogoliubov linear transformation [3]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

where [L.sub.[??]] is the unknown real symmetrical function of a momentum [??].

Substitution of (17) into (16) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

hence we infer that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the "creation" and "annihilation" operators of a Bogoliubov quasiparticles with energy:

[[epsilon].sub.[??]] = [[[([p.sup.2]/2m]).sup.2] + [p.sup.2][v.sup.2]].sup.1/2](19)

In this context, the real symmetrical function [L.sub.[??]] of a momentum [??] is found

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

As is well known, the strong interaction between the helium atoms is very important and reduces the condensate fraction to 10 percent or [N.sub.0]/N = 0.1 [5], at absolute zero. However, as we suggest, our model of dilute Bose gas may be valuable in describing thermodynamic properties of superfluid liquid helium, because the S-wave scattering between two atoms, with coordinates P1 and P2 in coordinate space, is represented by the repulsive potential delta-function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The model presented works on the condensed fraction [N.sub.0]/N and differs from the Bogoliubov model where [N.sub.0]/N [approximately equal to] 1.

3 Formation singlet spinless neutron pairs

We now attempt to describe the thermodynamic property of a helium liquid-neutron gas mixture. In this context, we consider a neutron gas as an ideal Fermi gas consisting of n free neutrons with mass [m.sub.n] which interact with N interacting atoms of a superfluid liquid helium. The helium-neutron mixture is confined in a box of volume V. The Hamiltonian of a considering system [[??].sub.an] consists of the term of the Hamiltonian of Bogoliubov excitations [[??].sub.a] in (18) and the term of the Hamiltonian of an ideal Fermi neutron gas as well as the term of interaction between the density of the Bogoliubov excitations and the density of the neutron modes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are, respectively, the operators of creation and annihilation for free neutron with momentum [??], by the value of its spin z-component [sigma] = [+ or -] 1/2; U0 is the Fourier transform of the repulsive interaction between the density of the Bogoliubov excitations and the density modes of the neutrons:

[U.sub.0] = 4[pi][d.sub.0][h.sup.2]/[mu] (22)

where [d.sub.0] is the scattering amplitude between a helium atoms and neutrons; [mu] = m x [m.sub.n]/m + [m.sub.n] is the relative mass.

Hence, we note that the Fermi operators [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfy to the Fermi commutation relations [***]+ as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

The density operator of neutrons with spin [sigma] in momentum [??] is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The operator of total number of neutrons is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

on other hand, the density operator, in the term of the Bogoliubov quasiparticles [[??].sub.[??]] included in (21), is expressed by following form, to application (17) into (12):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Hence, we note that the Bose- operator [[??].sub.[??]] commutates with the Fermi operator [[??].sub.[??]] because the Bogoliubov excitations and neutrons are an independent.

Now, inserting of a value of operator [[??].sub.[??]] from (28) into (21), which in turn leads to reducing the Hamiltonian of system [[??].sub.a,n]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Hence, we note that the Hamiltonian of system [[??].sub.a,n] in (29) is a similar to the Hamiltonian of system an electron gas-phonon gas mixture which was proposed by Frolich at solving of the problem superconductivity (please, see the Equation (16) in Frolich, Proc. Roy. Soc. A, 1952, v.215, 291-291 in the reference [11]), contains a subtle error in the term of the interaction between the density of phonon modes and the density of electron modes which represents a third term in right side of Equation (16) in [11] because the later is described by two sums, one from which goes by the wave vector [??] but other sum goes by the wave vector [??]. This fact contradicts to the definition of the density operator of the electron modes [[??].sub.[??]] (please, see the Equation (12) in [11]) which in turn already contains the sum by the wave vector [??], and therefore, it is not a necessary to take into account so-called twice summations from [??] and [??] for describing of the term of the interaction between the density of phonon modes and the density of electron modes Thus, in the case of the Frolich, the sum must be taken only by wave vector w, due to definition of the density operator of electron modes with the momentum of phonon [??].

To allocate anomalous term in the Hamiltonian of system [[??].sub.a,n], which denotes by third term in right side in (29), we apply the Frolich approach [11] which allows to do a canonical transformation for the operator [[??].sub.a,n] within introducing an operator H:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

which is decayed by following terms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

where the operators represent as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

and satisfy to a condition [??] + = -[??].

In this respect, we assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

where [A.sub.[??]] is the unknown real symmetrical function from a momentum [??]. In this context, at application [S.sub.[??]] from (34) to (33) with taking into account [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

In analogy manner, at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

To find [A.sub.[??]], we substitute (29), (35) and (36) into (31). Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] within application a Bose commutation relations as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the form of new operator [??] in (31) takes a following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

The transformation of the term of the interaction between the density of the Bogoliubov modes and the density neutron modes is made by removing of a second and fifth terms in right side of (39) which leads to obtaining of a quantity for [A.sub.[??]]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

In this respect, we reach to reducing of the new Hamiltonian of system (39):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

As result, the new form of Hamiltonian system takes a following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

where [[??].sub.n] is the effective Hamiltonian of a neutron gas which contains an effective interaction between neutron modes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

where [V.sub.[??]] is the effective potential of the interaction between neutron modes which takes a following form at substituting a value of Ap from (40) into (41):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

In this letter, we consider following cases:

1. At low momenta atoms of a helium p [much less than to] 2mv, the Bogoliunov's quasiparticles in (19) represent as the phonons with energy [[epsilon].sub.[??]] [much less than to] pv which in turn defines a value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this context, the effective potential between neutron modes takes a following form:

[V.sub.[??]] [approximately equal to] 2m[U.sup.2.sub.0][N.sub.0]/[V[p.sup.2] = 4[pi][h.sup.2][e.sup.2.sub.1]/[p.sup.2] (45)

The value [e.sup.1] is the effective charge, at a small momenta of atoms:

[e.sub.1] = [U.sub.o]/h [square root of m[N.sub.0]/2V[pi]].

2. At high momenta atoms of a helium p [much greater than] 2mv, we obtain [[epsilon].sub.[??]] [approximately equal to] [p.sup.2]/2m + m[v.sup.2] in (19) which in turn defines [L.sub.[??]] [approximately equal to] 0 in (20). Then, the effective potential between neutron modes presents as:

[V.sub.[??]] [approximately equal to] n[U.sup.2.sub.0][N.sub.0]/v[p.sup.2] = 4[pi][h.sup.2][e.sup.2.sub.2]/[p.sup.2] (46)

where [e.sub.2] is the effective charge, at high momenta of atoms:

[e.sub.2] = [U.sub.o]/2h [square root of m[N.sub.o]/V[pi]

Consequently, in both cases, the effective scattering between two neutrons is presented in the coordinate space by a following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

where [e.sub.*] = [e.sub.1], at small momenta of atoms; and [e.sub.*] = [e.sub.2], at high momenta.

The term of the interaction between two neutrons V([??]) in the coordinate space mediates the attractive Coulomb interaction between two charged particles with mass of neutron [m.sub.n], having the opposite effective charges [e.sub.*] and -[e.sub.*] which together create a neutral system. Indeed, the effective Hamiltonian of a neutron gas in (43) is rewrite down in the space of coordinate by following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

where [[??].sub.i] is the Hamiltonian of system consisting two neutron with opposite spin which have a coordinates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

The transformation of considering coordinate system to the relative coordinate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the coordinate of center mass [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

In analogy of the problem Hydrogen atom, two neutrons with opposite spins is bound as a spinless neutron pair with binding energy:

[E.sub.n] = [m.sub.n][e.sup.4.sub.*]/4[h.sup.2][n.sup.2] = -const/[n.sup.2] [([N.sub.0]/V).sup.2], (51)

where n is the main quantum number which determines a bound state on a neutron pair, at const > 0.

Thus, a spinless neutron pair with mass [m.sub.0] = 2[m.sub.n] is created in a helium liquid-dilute neutron gas mixture.

4 Formation of the Frolich electron pairs in superconductivity

We now attempt to describe the thermodynamic property of the model a phonon-electron gas mixture confined in a box of volume V. In this context, we consider an electron gas consisting of n free electrons with mass [m.sub.e] which interact with phonon modes of lattice by constancy interaction [11]. The Frolich Hamiltonian has a following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (54)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are, respectively, the Fermi operators of creation and annihilation for free electron with wave-vector [??] and energy [[epsilon].sub.[??]] = [h.sup.2][k.sup.2]/2[m.sub.e], by the value of its spin z-component [sigma] = [+ or -] 1/2; s is the velocity of phonon; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are, respectively, the Bose operators of creation and annihilation for free phonon with wave-vector [??] and energy hws; [D.sub.w] is the constant of the interaction between the density of the phonon excitations and the density modes of the electrons which equals to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the constant characterizing of the metal; C" is the constant of the interaction; M is the mass of ion); [[??].sub.[??]] is the density operator of the electron modes with wave vector [??] which is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, we note that the Fermi operators [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a satisfy to the Fermi commutation relations [***]+ presented in above for neutrons (23-25).

Obviously, the Bose- operator [[??].sub.[??]] commutates with the Fermi operator [[??].sub.[??],[sigma]] because phonon excitations and electron modes are an independent.

Now, we introduce new transformation of the Boseoperators of phonon modes [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the new Bose operators of phonon excitations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which help us to remove an anomalous term:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (59)

Then, [[??].sub.1] in (56) and [[??].sub.2] in (57) take following forms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (60)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (61)

To allocate anomalous term in the Hamiltonian of system [??] in (54), presented by the term in (63), we use of the canonical transformation for the operator [??] presented by formulae (30). Due to this approach, we obtain new form for operator Hamiltonian [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (62)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (63)

hence [V.sub.[??]] is the effective potential of the interaction between electron modes, which at taking into account [D.sub.w] = [square root of [alpha]hws/v], has the form:

[V.sub.[??]] = -2[D.sup.2.sub.w]V/hws = -2[alpha] (64)

Consequently, the effective scattering between two electrons in the coordinate space takes a following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (65)

at using of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using of the relative coordinate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the coordinate of center mass [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we reach to the Hamiltonian of system consisting two electron with opposite spins:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](66)

To find the binding energy E < 0 of electron pair, we search the solution of the Schrodinger equation with introduction of wave function [psi]([??]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this respect, we have a following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (67)

which may determine the binding energy E < 0 of electron pair, if we claim that the condition [p.sub.f]d/h [much less than to] 1 always is fulfilled. This reasoning implies that the effective scattering between two electrons is presented by the coordinate space:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (68)

where we introduce a following approximation as sin(wr)/wr [approximately equal to] 1 [w.sup.2][r.sup.2]/6 at conditions w [less than or equal to] [w.sub.f] and [w.sub.f] d [much less than to] 1 (wf = [(3[[pi].sup.2]n/V).sup.1/3] is the Fermi wave number). The later condition defines a state for distance r between two neighboring electrons which is a very small [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where 4[pi].sup.3.sub.f]/2V. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (69)

Thus, the effective interaction between electron modes V([??]) = -2[alpha][delta]([??]), presented in (65) is replaced by a screening effective scattering presented by (69). This approximation means that there is an appearance of a screening character in the effective scattering because one depends on the density electron modes. Now, denoting E = [E.sub.s], and then, we arrive to an important equation for finding a binding energy [E.sub.s] of singlet electron pair:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (70)

which we may rewrite down as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (71)

where we take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, introducing the wave function [[psi].sub.s](r) via the Chebishev-Hermit function [H.sub.s](it) from an imaginary number as argument it [15] (where i is the imaginary one; t is the real number; s = 0; 1; 2; ...), the equation (71) has a following solution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

at [theta] < 0, where

[lambda] = [theta] (s + 1/2)

Consequently, the quantity of the binding energy [E.sub.s] of electron pair with mass [m.sub.0] = 2[m.sub.e] takes a following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (72)

at s = 0; 1; 2;....

The normal state of electron pair corresponds to quantity s = 0 which defines maximal binding energy of electron pair:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (73)

This fact implies that the formation of the superconducting phase in superconductor is appeared by condition for density of metal v:

n/V > [([C.sup.2][m.sub.e]/2M[s.sup.2][h.sup.2]).sup.3/2]

At choosing C [approximately equal to] 10 eV [11]; M [approximately equal to] 5 x [10.sup.-26] kg; s [approximately equal to] 3 x [10.sup.3] m, we may estimate density of electron n/v > [10.sup.27] [m.sup.-3] which may represent as superconductor.

Vahan Minasyan and Valentin Samoilov

Scientific Center of Applied Research, JINR, Dubna, 141980, Russia

E-mails: mvahan@scar.jinr.ru; scar@off-serv.jinr.ru

Acknowledgements

We thank Professor Marshall Stoneham for help with the English.

Submitted on April 12, 2010 / Accepted on April 19, 2010

References

[1.] London F. Nature, 1938, v. 141, 643.

[2.] Landau L. Journal of Physics (USSR), 1941, v. 5, 77.

[3.] Bogoliubov N.N. On the theory of superfludity. Journal of Physics (USSR), 1947, v. 11,23.

[4.] Bishop R.F. J. Low Temp. Physics, 1974, v. 15, 601.

[5.] Blagoveshchenskii N.N. et al. Physical Review B, 1994, v. 50, 16550.

[6.] Glyde H.R., Griffin A. Physical Review Letters, 1990, v. 65, 1454.

[7.] Stirling W.G., Griffin A., Glyde H.R. Physical Review B, 1990, v. 41, 4224.

[8.] Minasyan V.N. et al. Physical Review Letters, 2003, v. 90, 235301.

[9.] Morawetz K. et al. Physical Review B, 2007, v. 76, 075116.

[10.] Penrose O., Onsager L. Physical Review, 1956, v. 104, 576.

[11.] Frolich H. Proc. Roy. Soc., 1952, v. A215, 576.

[12.] Bardeen J., Cooper L.N., and Schrieffer J.R. Physical Review, 1957, v. 108, 1175.

[13.] Bogoliubov N.N. Nuovo Cimento, 1958, v. 7, 794.

[14.] Cooper L.N. Physical Review, 1956, v. 104, 1189.

[15.] Lavrentiev N.A. and Shabat B.V. Methods of the theory of a function of a complex variable. Nauka, Moscow, 1972.
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Date:Oct 1, 2010
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