1. Akselrod Gennadij Semenovich
Electric tension of a superconducting current
at extreme temperatures
Annotation
On the basis of representations about resonant character electron-phonon interactions in
superconductors the internal mechanism of the electric pressure’s presence of a superconducting
current is revealed. The consideration is conducted from the engineering point of view.
Interpretation of the terms
1. «Resonant character electron - phonon interaction»
The formulation, translating to the engineering level of the concept about potential holes, barriers,
steps, etc. Really, under certain conditions of energy (frequency) of the short-living virtual phonons
and cooper pars electrons can be comparable between themselves.
2. «Virtual (exchange) phonons»
Phonons are short-living quasi particles through which are connected electrons in cooper pars.
3. «Extreme temperatures»
It is the temperature, equal to zero of Calvin’s degrees at which the superconducting current reaches
the highest value, and critical temperature at which the superconducting current stops.
I. Introduction
The internal mechanism of the electric tension’s presence (potential difference) in
superconductors is revealed at the flow in them of a superconducting current. The formal device of
consideration of the superconductivity, based on the assumption of resonant character electron-
phonon interaction is for this purpose applied. This new approach is based nevertheless on
conclusions of theory BCS and two-fluid model of Landau-Ginzburg. Consideration is conducted
from the engineering point of view, at extreme temperatures for a superconducting current: 0ºК and
critical temperatures.
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2. The question’s analysis is conducted from the positions of quantum mechanics and the
relativity theory of Einstein. It is applied the deductive method - from the to the general.
Particularly, it is considered one cooper pair electrons, and conclusions extend on all superconducting
boson condensate. Total results can be interesting both from the point of view of superconductivity
research, and for measurement of quantity and characteristics of a superconducting current in
superconductors.
II. Consideration at 0К
1.1 It is supposed, generally, that there is some part of energy electron-phonon interaction (Еb),
responsible for an electron attraction. We admit, that:
phb TgE ω⋅⋅= )(
(1),
Where: Т - superconductor temperature; ωph - vibration frequency of the virtual (exchangeable)
phonons; g (T) - density of the acoustic wave vectors of the phonon’s vibration , responsible for an
attraction between electrons; ħ - Planck constant
1.2 It is supposed, that at Т=Т0=0К in superconductors are quantized:
- Eleсtron’s binding energy in a cooper pairs (Еbq);
- Vibration frequency (ωphq) of the virtual phonons, creating energy state for cooper pair electrons;
- Coherent length (of the correlation) (ξ0).
Then the equation (1) takes the form:
)};({)();( 00000 TTgTE phqbq ξωξ ⋅⋅= 
(2),
And it is postulated, that:
||}/)({);( 0000 phqphq VTgT

ξξω =
(3)
Where ↓ phqV

- speed virtual phonons, vibrating with frequency
phq↓
ω
;
g (T0)> 0, only an integral number of the vectors, defining the direction of the impulses of the
acoustic phonons , fluctuating with frequency ωphq, and also 1 ≤ g (T0) ≤ ns (T0), where ns (T0) - density
of the cooper pair electrons at T0 0К.
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3. From (3) follows, there are the integral number of wave’s length of the acoustic phonon’s
vibration, corresponding with ωphq at T0 =0К at the length of correlation ξ0.
In such context from (2) and (3) follows, that energy of cooper pairs electron’s connection at
T0 =0ºК is equal to an integer of oscillator (ћ ωphq), fluctuating with the minimum frequency for a
considered superconductor exchange phonons.
Certainly, thus it is meant, that at T0 =0К all electrons in a superconductor have only energy of
attraction Еbq, and there is no energy of electron’s pushing away. In other words, if to excite a
superconducting current under these conditions all electrons will participate in it [1].
2.3 For the concrete analysis we will consider microvolume of a superconductor with the
radius equal ξ0. We will allocate in this microvolume one cooper pair electron in the field of one ionic
lattice through which there is made an emission and absorption of the virtual phonons. The index «1»
will designate characteristics of one pair. Then it is possible to write down:
)};({)();( )1(0)1(0)1()1(0)1()1(0)1(0)1( TTgTE phqbq ξωξ ⋅⋅= 
(4)
|);(|}/)({);( )1(0)1(0)1()1(0)1(0)1()1(0)1(0)1( TVTgT phqphq ξξξω

⋅=
(5)
And g (1) (T0 (1)) ≡ 1 (6)
By definition.
For simplification of the formula’s record is made the assumption, that as properties of this
microvolume are defined by surrounding characteristics of the sample’s geometry, an
electromagnetic field’s figuration and number of particles in the sample, becomes:
.;|;);(|);(| 0)1(00)1(000)1(0)1(0)1( TTTVTV phqphq === ξξξξ

The minimality ωphq (1) and Ebq (1) is defined only from a reality (6).
Then
);();( 00)1(00)1( TTE phqbq ξωξ ⋅=
(4а)
And
00000)1( /|);(|);( ξξξω TVT phqphq

=
(5а)
In the absence of a current (the basic power condition of a superconductor) we believe, that ωphq (1) is
absolute minimum frequency of exchange phonons vibration, which is equal to the frequency 0 (1) of
"zero fluctuations” of electrons in a considered superconductor at Т0=0К. This assumption is a
consequence of resonant character of electron-phonon interaction in superconductors.
As )1(0ω = 2E0 (1) / ћ, where Е0 (1) is the minimum of the electron’s energy in the basic
condition of a superconductor at Т0=0К, from (4а) and (5а) follows, that:
000)1(0)1(0)1( /|);(|2 ξξω TVEE phqbq

 ⋅=== (7),
And
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4. 000)1(0 2/|);(| ξξ TVE phq

⋅=
(8),
At Т0=0К.
Let's excite a superconducting current in a superconductor. Now, considering resonant
character at electrons in cooper pair (superposition of wave functions) at Т0=0К, we assume, that
frequency of these electron’s fluctuations is equal to 2 )1(0ω =2E0 (1) / ћ. (9)
[Expression (9) is absolutely exact, as, according to theory BCS (and projecting on all cooper
pairs) energy «frosted» boson condensate remains former - only an impulse is changing [2].]
Then:
000)1(0)1( 2/|);(|' ξξ TVEE phqbq

⋅== (10),
Where E’bq (1) is energy of an attraction in cooper pair at the flow of a superconducting current
(the raised condition of a superconductor).
From (7) and (10) follows the important principal result at the power barrier between the
raised and basic condition of a superconductor (for energy of an attraction between electrons) at
Т0=0К, for one, separately taken cooper pair electrons, is size of the absolute value of the minimum
energy of electron in a superconductor:
(1) (1) 0(1)| ' | | |bq bqE E E− =
(11)
2.4 Now, for g (T0) =ns (T0) in the basic condition, where g (T0)> 1, but only an integer; ns -
concentration of cooper pair electrons at Т0=0К, it is possible to write down :
)1(0
2
000000 )}({2}/|);(|)({)( ETnTVTnTnE sphqssbq ⋅=⋅⋅⋅= ξξ


(12).
For the raised condition, at Т0=0К and g (T0) =ns (T0):
)1(0
2
000000 )}({}2/|);(|)({)(' ETnTVTnTnE sphqssbq ⋅=⋅⋅⋅= ξξ


(13)
And further, a difference between raised and the basic conditions:
0)}({' )1(0
2
0 <⋅−=−=− ETnEEE sbqbqbq
(14)
That is, superconductivity remains only until the difference of attraction’s energies between
electrons is negative at raised and the basic conditions is (at Т0=0ºК)
Expression (14) is anything to others, as in macroscopic quantity - a potential difference of a
superconducting current in a superconductor.
Really, for example, from the engineering point of view obviously, that if electric resistance of a
superconducting current is equally to zero, also tension )( RJU ×↓=

should be equal to zero. But
after all electric pressure is observed in practice. Now it is clear, that for superconductor current is
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5. responsible the difference of energies of electron’s tension between raised and the basic
superconductor conditions.
From the microscopic point of view, from (14) it is clear, that the more concentration of cooper pair
electrons, the more on absolute value a potential’s difference of a superconducting current, and
accordingly, and a current in a superconductor himself.
III. Consideration at critical temperature.
3.1 As at occurrence of a superconducting current of cooper pair electrons, - that is, the whole
boson condensate , - do not experience electric resistance, caused by dispersion of electrons on knots
and defects of a lattice, this is possible to consider, that they move with the greatest possible speed,
that is, with a velocity of light (cs) in the given material. It is natural, that this speed depends on
temperature of a superconductor and its density (so, and from length of correlation ξ); cs (T; ξ). Then,
it is possible to write for energy of an electronic superconducting condensate:
2
)};({)(2 TcmTnE sesce ξ⋅⋅=
(15),
At 0K≤T≤Tc; where me - weight of electron; Tc - critical temperature
On the other hand, phonons through which cooperate electrons in cooper pairs, possess energy:
2
)};({
)(
2
TVm
TnE
phph
scph
ξ⋅
⋅=
(16),
Where mph - weight of phonon, phV

- phonon’s speed of terragertion frequencies, also depending on
temperature and length of correlation.
Natural condition of resonant electron-phonon interaction is equality specified energies:
cphce EE =
(17).
From here, after simple calculations, follows
2
1
)/(2);(/|);(| phesph mmTcTV =ξξ

(18).
3.2 For a superconductor at critical temperature, considering (10); (14) and (18), we receive:
=⋅⋅−=− }
2
|);(|
{)}({);( 2
c
ccphq
csccbq
TV
TnTE
ξ
ξ
ξ


0)}({)/()};({)/( 22
1
=⋅⋅⋅−= cscccsphe TnTcmm ξξ  (19),
Where: })/(1{)(~)( 4
0 cscs TTTnTn −⋅− (20),
Where Т - current temperature; Т → cT ;
Here it is necessary to consider also, that )2/(~)( 0 css TnTn ↓− (21),
Т0=0К [3].
IV. The Conclusion
Thus, it is opened the internal nature of presence of a potential difference of a superconducting
current in superconductors at extreme temperatures from the point of view of resonant electron-
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6. phonon interactions. It has applied value also, both at research of superconductivity’s effect, and at
measurement of quantity of superconductors.
The literature:
[1] F.Z.Vilf of "The Basis of physics of superconductors», URSS, Moscow, 1988
[2] Collection of articles «The Superconductivity theory» I.L., Moscow, 1960
[3] T.Van Druzer, C.U.Terner «Physical bases of supeconductor knots and chains». Moscow: «Radio
and communication», 1984
The author's abstract
The internal mechanism of the electric tension’s presence (potential difference) in superconductors is
revealed at the flow in them of a superconducting current. The formal device of consideration of the
superconductivity, based on the assumption of resonant character electron-phonon interaction is for
this purpose applied. This new approach is based nevertheless on conclusions of theory BCS and two-
fluid model of Landau-Ginzburg. Consideration is conducted from the engineering point of view, at
extreme temperatures for a superconducting current: 0К and critical temperatures
The question’s analysis is conducted from the positions of quantum mechanics and the
relativity theory of Einstein. It is applied the deductive method - from the particular to the general.
Particularly, it is considered one cooper pair electrons, and conclusions extend on all superconducting
boson condensate. Total results can be interesting both from the point of view of superconductivity
research, and for measurement of quantity and characteristics of a superconducting current in
superconductors.
6

7. phonon interactions. It has applied value also, both at research of superconductivity’s effect, and at
measurement of quantity of superconductors.
The literature:
[1] F.Z.Vilf of "The Basis of physics of superconductors», URSS, Moscow, 1988
[2] Collection of articles «The Superconductivity theory» I.L., Moscow, 1960
[3] T.Van Druzer, C.U.Terner «Physical bases of supeconductor knots and chains». Moscow: «Radio
and communication», 1984
The author's abstract
The internal mechanism of the electric tension’s presence (potential difference) in superconductors is
revealed at the flow in them of a superconducting current. The formal device of consideration of the
superconductivity, based on the assumption of resonant character electron-phonon interaction is for
this purpose applied. This new approach is based nevertheless on conclusions of theory BCS and two-
fluid model of Landau-Ginzburg. Consideration is conducted from the engineering point of view, at
extreme temperatures for a superconducting current: 0К and critical temperatures
The question’s analysis is conducted from the positions of quantum mechanics and the
relativity theory of Einstein. It is applied the deductive method - from the particular to the general.
Particularly, it is considered one cooper pair electrons, and conclusions extend on all superconducting
boson condensate. Total results can be interesting both from the point of view of superconductivity
research, and for measurement of quantity and characteristics of a superconducting current in
superconductors.
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