research article polaronic mechanism of superconductivity...

Hindawi Publishing CorporationInternational Journal of SuperconductivityVolume 2013 Article ID 581025 13 pageshttpdxdoiorg1011552013581025

Research ArticlePolaronic Mechanism of Superconductivity in Cuprates

Pradeep Chaudhary1 Anuj Nuwal2 S C Tiwari3 R K Paliwal4 and S L Kakani5

1 Department of Physics Government Polytechnic College Chittorgarh Rajasthan 312001 India2Department of Physics Sangam University NH No 79 Bhilwara By-Pass Chittor Road Bhilwara Rajasthan 311001 India3 Department of Physics MLV Government College Bhilwara Rajasthan 311001 India4Department of Physics Mewar University Gangrar Chittorgarh Rajasthan 312901 India5 4-G-45 Shastri Nagar New Housing Board Bhilwara Rajasthan 311001 India

Correspondence should be addressed to S L Kakani slkakani28gmailcom

Received 15 April 2013 Accepted 13 June 2013

Academic Editor Zigang Deng

Copyright copy 2013 Pradeep Chaudhary et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A strong polaron pairing model of high-temperature cuprate superconductors is presented The normal and anomalous one-particle Greenrsquos functions are derived from a system with strong electron-phonon coupling Self-consistent equation for thesuperconducting order parameter (Δ) is derived using Greenrsquos function technique and following Lang and Firsov transformationsExpressions for specific heat density of states free energy and critical field based on this model have been derived The theoryis applied to explain the experimental results in the system YBa

2Cu

3O

7minus119909 There is convincing evidence that the theory is fully

compatible with the key experiments

1 Introduction

Strikingly after 26 years of enormous experimental andtheoretical efforts followed by the discovery there is still littleconsensus on the pairing mechanism of high-temperaturesuperconductivity (HTSC) in cuprates [1ndash12] HTSCs haveunique physical properties in both the normal state andsuperconducting one To comprehend the physics of thesecomplex compounds is one of the main tasks of the theory ofsuperconductivity whose solution may allow one to explainthe pairing mechanism ensuring HTSC At present thereexists no mechanism which would explain the totality ofthermodynamicalmagnetic and superconductive propertiesof HTSCs from a single point of view

The electron-phonon pairing mechanism [13ndash16] beingthe principal one in low-temperature superconductorsmakes a considerable contribution to the establishment ofthe superconducting state in HTSCs But in order to obtainproper description it is necessary to consider the othermechanism inherent in HTSCs [3 5 7 8 10]

To explain HTSC a lot of models and mechanismsof this unique phenomenon have been proposed [1ndash5 10

11] The key question is the nature of the mechanism ofpairing of carriers There are many different models ofsuperconductivity available for example magnon modelexciton model model of resonant valence bonds bipolaronicmodel bisoliton model anharmonic model model of localpairs and plasmon model [17] All these models use theconcept of pairing with a subsequent formation of a Bosecondensate at temperature119879

119862irrespective of the nature of the

resulting attraction Some recent theoreticalmodels postulatethemechanismof antiferromagnetic spin fluctuations [18 19]so that the electron scattering on them can be the reason forthe pairing of electrons

In order to comprehend the nature of the superconduct-ing state it is necessary to construct a consistent microscopictheory which should be able to describe superconductive andthe normal properties of HTSCs We have proposed thatthe pairing mechanism in cuprate superconductors can beunderstood on the polaronic model of the charge carriers[20ndash22]

In the present work following Alexandrov and Ranninger[21] we have developed a microscopic theory for HTSCcuprates by generalizing the Holstein [22] and Lang Firsov

2 International Journal of Superconductivity

[23 24] and Gorkov Nambu formalisms in order to evaluatethe Greenrsquos functions for electrons coupled to phonons andconsidering the range of couplingwhich corresponds to smallpolaron formation (120582 ≫ 1) [25 26]

2 Model Hamiltonian

The model Hamiltonian for our system can be expressed as[21]

119867 = 119867119890+ 119867ph + 119867

119890minusph + 119867119890119890 (1)

where119867119890is the kinetic energy in the initial Bloch band119867ph is

the vibration energy of the lattice 119867119890minusph is electron-phonon

interaction and119867119890119890is the Coulomb electron-electron corre-

lationsIn one band approximation119867

119890has the form

119867119890= sum

119896119904

119864 (119896) 119862+

119896119904119862119896119904 (2)

where 119896 and 119904 denote the state with quasi momentum andspin respectively 119864(119896) is bare band energy 119867ph can beexpressed in terms of phonon operators 119889

119902 119902 = (119902 ]) where

] is the type of vibrational mode

119867ph = sum

119902

120596 (119902) 119889+

119902119889119902 (3)

where 120596 is the phonon dispersionThe electron-phonon interaction is described by the

Frohlich Hamiltonian

119867119890minusph = sum

119896119902119904

120596 (119902) 120574 (119902)1

radic2119873119862+

119896+119902119904119862119896119904119889119902+ 119867119862 (4)

in which 120596 (119902) and 120574 (119902) are the phonon frequency and theinteraction matrix element in a parent crystal without chargecarriers respectively Correspondingly one obtains

1205742(119902) =

41205871198902

1199022Ω1205960

(1

120576infin

minus1

1205760

) (5)

In the case of optical longitudinal phonon with frequency120596(119902) = 120596

0and 120576

0 120576

infinare the dielectric constants of the crystal

with andwithout taking ionic part into considerationΩ is thevolume of the unit cell and119873 is their number

For acoustic phonons one finds

]2 (119902) = 1198642

119863

119902

119906119872

120596 (119902) = 1205960

(6)

where119864119863is the deformation potential 119906 is the sound velocity

and 119872 is the mass of an elementary cell For intermolecularphonons

1205742(119902) = 120574

2(0) (7)

The combined Hamiltonian can be expressed as

119867 = sum

119896119904

119864 (119896) 119862+

119896119904119862119896119904

+ sum

119902

120596 (119902) 119889+

119902119889119902

+ sum

119896119902119904

120596 (119902) 120574 (119902) 119889119902

1

radic2119873119862

+

119896+119902119862119896119904

+ 119867119862 + 119881119862

(8)

Here 119881119862is the Coulomb repulsion This Hamiltonian

includes electron-phonon and electron-electron correlationsTo diagonalize the main part of the Hamiltonian the siterepresentation is more convenient

One can express the previous Hamiltonian as

119867 = sum

119898119899

119879 (119898 minus 119899)119862+

119898120590119862119899120590

+ sum

119902

120596 (119902) 119889+

119902119889119902+

1

2

+1

radic2119873sum

119898119902

120596 (119902) 120574 (119902) 119899119898120590

119889119902119890119894119902119898

+ 119889+

119902119890minus119894119902119898

+1

2sum

1198981198991205901205901015840

119881119862(119898 minus 119899) 119899

+

1198981205901198991198991205901015840 + sum

119902

120596 (119902) 119889+

119902119889119902+

1

2

(9)

where

119879 (119898) =1

119873sum

119896

119864 (119896) 119890119894119896119898

119899119898120590

= 119862+

119898120590119862119898120590

(10)

In the small-polaron regime 120582 ge 1 the kinetic energyremains smaller than the interaction energy and a self-consistent treatment of a many-body problem is possiblewith the 1120582 expansion technique [27] Following Lang andFirsov [23 24] and applying canonical transformations todiagonalize the Hamiltonian [28] one obtains

= 119890119878119867119890

minus119878 (11)

where

119878 = sum

119898119904

119862+

119898119904119862119898119904

1

radic2119873120574 (119902) 119889

119902119890119894119902119898

minus 119889+

119902119890minus119894119902119898

(12)

The electron operator transforms as

119889119902= 119889

1199021015840 + [119878 119889

1199021015840]

1

2[119878 [119878 119889

1199021015840]] + sdot sdot sdot

[119904 1198891199021015840] = [sum

119898119904

119862+

119898119904119862119898119904

1

radic2119873120574 (119902) 119889

119902119890119894119902119898

minus 119889+

119902119890minus119894119902119898

1198891199021015840]

[119904 1198891199021015840] =

1

radic2119873120574 (119902)

times [sum

119902119898119904

119862+

119898119904119862119898119904

119889119902119890119894119902119898

1198891199021015840 minus 119889

+

119902119890minus119894119902119898

1198891199021015840

minus 1198891199021015840119889

119902119890119894119902119898

+ 1198891199021015840119889

+

119902119890minus119894119902119898

]

(13)

International Journal of Superconductivity 3

Using 1198891199021015840119889

+

119902= 119889

+

1199021198891199021015840 + 120575

1199021199021015840

[119904 1198891199021015840] =

1

radic2119873120574 (119902) [sum

119902119898119904

119862+

119898119904119862119898119904

1205751199021199021015840119890minus119894119902119898

] (14)

When 119902 = 1199021015840 we have

[119904 1198891199021015840] =

1

radic2119873120574 (119902) [sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898]

lfloor119904 lfloor119904 1198891199021015840rfloorrfloor = 0

(15)

Thus

1198891199021015840 = 119889

1199021015840 +

1

radic2119873120574 (119902)sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898

119889+

1199021015840 = 119889

+

1199021015840 minus

1

radic2119873120574 (119902)sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898

(16)

Hence

119867 = sum

119898119899119904119898 = 119899

119879 (119898 minus 119899)119862+

119898119904119862119899119904

times exp[sum

119902

119889119902

1

radic2119873120574 (119902) 119890

119894119902119898+sum

119902

120596 (119902) (119889+

119902119889119902+1

2)]

+ sum

119898119899119904

119881119862(119898 minus 119899) minus sum

119902

120596 (119902)1

21198731205742(119902) 119890

119894119902(119898minus119899)

times 119862+

119898119904119862+

119899119904119862119899119904119862119898119904

(17)

In obtaining (17) we have omitted the term containingthe on-site interaction119898 = 119899 for parallel spins

3 Greenrsquos Functions

Wedefine the following one particle temperature electron (119866)

and anomalous (119865) Greenrsquos functions

119866 (119896 120596119899) = minus

1

2sum

119898

int

120573

minus120573

d119897119890119897120596119899119890+119894119896119898 ⟨⟨119897119897119862119888120590

(119897) 119862+

119898120590(119900)⟩⟩

119865 (119896 120596119899) = minus

1

2sum

119898

int

120573

minus120573

d119897119890119897120596119899119890+119894119896119898 ⟨⟨119897119897119862119888120590

(119897) 119862119898120590

(119900)⟩⟩

(18)

For convenience dropping spin and applying the Lang-Firsov canonical transformation and neglecting the residualpolaron-polaron coupling and following equation of motionmethod for the evaluation of electron part and Feynmanmethod for the evaluation of phonon part one finally obtains[28]

119866 (119905) = minus119894119890120573Ω119890119897 Tr [119890minus119894V(119898minus119899)119905

119862119862+119890minus120573119890119897]

times Tr [119890120573Ω119901ℎ119890minus120573ph119883(119905)119883+(0)]

(19)

After evaluating the electron part and phonon part of thetrace we obtain the total Greenrsquos function as

119866 (119896 120596119899)

= 119890minus1199022

[1199062

119896

119894120596119899minus 120576

119899

+V2119896

119894120596119899+ 120576

119899

+1

119873

infin

sum

119897=1

1198922119897

119897

times sum

1198961015840

1199062

1198961015840 (1 minus 119899

1198961015840)

119894120596119899minus 119897120596

0minus 120576

1198961015840

+V211989610158401198991198961015840

119894120596119899minus 119897120596

0+ 120576

1198961015840

+1199062

11989610158401198991198961015840

119894120596119899+ 119897120596

0minus 120576

1198961015840

+V21198961015840 (1 minus 119899

1198961015840)

119894120596119899+ 119897120596

0+ 120576

1198961015840

]

(20)

119865 (119896 120596119899)

= 119890minus1199022

[

[

119906119896V119896(

1

119894120596119899minus 120576

119896

+1

119894120596119899+ 120576

119896

) +1

119873

infin

sum

119897=1

(minus1)1198971198922119897

119897

times sum

1198961015840

1199061198961015840V

1198961015840

(1 minus 1198991198961015840)

119894120596119899minus 119897120596

0minus 120576

1198961015840

minus1198991198961015840

119894120596119899minus 119897120596

0+ 120576

1198961015840

+1198991198961015840

119894120596119899+119897120596

0minus120576

1198961015840

minus(1 minus 119899

1198961015840)

119894120596119899+119897120596

0+120576

1198961015840

]

]

(21)

where

1199062

119896=

1

2(1 +

120585119896

120576119896

) V2119896=

1

2(1 minus

120585119896

120576119896

)

119906119896V119896= minus

Δ

2120576119896

119899119896= 119899 (120576

119896)

(22)

With

119899 (119909) = (119890119909119896119861119879+ 1)

minus1

120576119896= radic120585

2

119896+ Δ2

(119896)

Δ (119896) = minus1

2sum

119896

119881(119896 minus 1198961015840)

Δ (1198961015840)

1205761198961015840

tanh120576119896

2119896119861119879

119881 (119896) =1

119873sum

119898

119881 (119898) 119890119894119896119898

(23)

The energy dispersion for the polaronic band is given by

120585119896= sum

119898

120590 (119898119900) 119879 (119898) 119890119894119896119898

minus 120583 (24)

having a narrow band half width 119882 ≪ 119863 where 119863 =

119885119879(119898)


4 Correlation Function

The correlation functions are defined as

⟨119862+

119901119862119901⟩ =

1

2120587int

+infin

minusinfin

119868119866(120596

119899) 119889120596

119899 (25)

⟨119862119901119862119901⟩ =

1

2120587int

+infin

minusinfin

119868119865(120596

119899) 119889120596

119899 (26)

where

119868119866(120596

119899) = 119894(119890

120573120596119899 + 1)

minus1

[11986611

(120596119899+ 119894120576) minus 119866

11(120596

119899minus 119894120576)]

119868119865(120596

119899) = 119894(119890

120573120596119899 + 1)

minus1

[11986511

(120596119899+ 119894120576) minus 119865

11(120596

119899minus 119894120576)]

(27)

where 119866 and 119865 are Green functions given by (20) and (21)respectively

Using the identity

lim120576rarr0

1

2120587[

1

120596 + 119894120576 minus 119864119896

minus1

120596 minus 119894120576 minus 119864119896

] = 119894120575 (120596 minus 119864119896)

int

infin

minusinfin

119891 (120596119899) 120575 (120596

119899minus 120596

plusmn) 119889120596

119899= 119891 (120596

plusmn

119899)

(28)

With the following relations

1199062

119896=

1

2(1 +

120585119896

120576119896

) V2119896=

1

2(1 minus

120585119896

120576119896

) (29)

(25) and (26) become

⟨119862+

119901119862119901⟩

=1

2+

1

2

120585119896

120576119896

tanh120573120576

119896

2

+1

119873

infin

sum

119897=1

1198922119897

119897[1

2+

1

2

1205851198961015840

1205761198961015840

tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus1198991198961015840

2

1205851198961015840

1205761198961015840

tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus tanh120573 (119897120596

0minus 120576

1198961015840)

2]

(30)

⟨119862119901119862119901⟩

= minusΔ (119896)

2120576119896

tanh120573120576

119896

2

+1

119873

infin

sum

119897=1

(minus1)119897

1198971198922119897sum

1198961015840

Δ (1198961015840)

21205761198961015840

times [tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus 119899119896tanh

120573 (1198971205960+ 120576

1198961015840)

2+ tanh

120573 (1198971205960minus 120576

1198961015840)

2]

(31)

5 Superconducting Order Parameter (Δ)

The order parameter of a superconducting state is given by

Δ = 119892sum

119896

⟨119862119896119862119896⟩ (32)

Substituting correlation function given by (31) in (32)and changing summation into integral using the followingrelation

sum

119896

= 119873 (119900) int

ℎ120596119863

0

119889120585119896 (33)

the gap equation becomes

Δ = 119892119873 (0) int

ℎ120596119863

0

119889120585119896

[[

[

minusΔ (119896)

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

minus 1198921

119873

infin

sum

119897=1

(minus1)119897

1198971198922119897

1119873(0) int

ℎ120596119863

0

sum

1198961015840

119889120585119896

minusΔ (1198961015840)

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanh119897120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)2119896

119861119879 + 1

times

tanh(119897120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840))

2119896119861119879

+ tanh(119897120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840))

2119896119861119879

]]

]

(34)

Right-hand side of (34) has two terms which are quiteindependent First term varies with 119896 whereas second termvaries with 119896

1015840 hence one can define two superconductingorder parameters for the YBa

2Cu

3O

7minus119909system The two

independent terms finally yield the two equations as

1

119892119873 (0)= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(35)


with ℓ = 1 the other equation is1

10038161003816100381610038161198921003816100381610038161003816 119873 (0) [119892

2

119897]

= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+Δ2 (1198961015840)2119896

119861119879 + 1

times

tanhℓ120596

0+radic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minusradic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

]]

]

(36)

With the help of (35) and (36) one can study the behaviorof superconducting order parameters with temperature

6 Physical Properties

61 Electronic Specific Heat (119888119890119904) The electronic specific

heat per atom of a superconductor is determined from thefollowing relation [3 4]

119888119890119904

=120597

120597119879[

1

119873sum

119896

2120585119896⟨119862

+

119896119862119896⟩] (37)

where ⟨119862+

119896119862119896⟩ is the correlation function We have obtained

this correlation function in (30) Substituting the correlationfunction from (30) in equation (37) One obtains

119888119890119904

=119873 (0)

2119873int

ℎ120596119863

0

2120585119896

1198961198611198792

119889120585119896

times [minus1

2120585119896sec ℎ2 (

120576119896

2119896119861119879)

+

infin

sum

119897=1

1198922119897

119897minus

1

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 minus 120576

1198961015840)

times sec ℎ2 (1205960119897 minus 120576

1198961015840

2119896119861119879

) ]

(38)

Right-hand side of (38) has two terms which are quiteindependent from each other First term varies with 119896whereas second term varies with 119896

1015840 hence one can study thebehaviour of electronic specific heat of superconductors withtemperature

62 Density of States Function [119873(120596)119873(119900)] For 120596 gt 0 thefunction can be defined as [5]

119873(120596) = lim 1

2120587[119866

11(119896 120596 + 119894120578) minus 119866

11(119896 120596 minus 119894120578)] (39)

Using the following identity

lim120578rarr0

1

2120587[

1

120596 + 119894120578 minus 120596+

119899

minus1

120596 minus 119894120578 minus 120596+

119899

] = 119894120575 (120596 minus 120596+

119899) (40)

changing the summation over ldquo119870rdquo into an integration replac-ing 120576

119896byminus120576

119896 and combining the terms and using the relations

1199062

119896+ V2

119896=

1

2(1 +

120585119896

120576119896

) +1

2(1 minus

120585119896

120576119896

) = 1 (41)

one obtains

119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(42)

63 Free Energy It is well known that free energy of nor-mal paramagnetic phase always exceeds the free energy ofsuperconducting diamagnetic phase The entropy decreasesremarkably on cooling the superconductors below the criticaltemperature The free energy can easily be defined for thesuperconducting transition as it is related by the entropyhence it also exhibits a similar behavior [3] Obviously theentropy as well as the free energy difference in the normalstate is always greater than the entropy in the superconduct-ing state

The free energy difference of a superconductor for itsnormal and superconducting state is given by the followingrelation [27]

119865119904minus 119865

119873

119881= int

infin

0

119889119892(1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (43)

where ldquo119892rdquo is the interaction parameter and ldquoΔrdquo is thesuperconducting order parameter Equation (43) can also beexpressed as

119865119904minus 119865

119873

119881= int

Δ

0

119889Δ119889

119889Δ(

1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (44)


From superconducting order parameter expression wehave

Δ (119896) = 119892119873 (0) int

ℎ120596119863

0

[[

[

minusΔ (119896)

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

1

119892= 119873 (0) int

ℎ120596119863

0

[[

[

1

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

(45)

Equation (44) becomes

[119865119878119873

119881]phonon

= int

Δ(119896)

0

Δ2(119896) 119889Δ (119896)

119889

119889Δ (119896)

times[[

[

119873 (0) int

ℎ120596119863

0

1

2radic1205852

119896+ Δ2

(119896)

times tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

119889120585119896

]]

]

(46)

Since

1205762

119896= 120585

2

119896+ Δ

2(119896)

2120576119896119889120576

119896= 2Δ (119896) 119889Δ (119896)

(47)

Integrating by parts we get

2 [119865119878119873

119881] =

Δ2(119896)

119892minus 119873 (0)

times int

ℎ120596119863

0

2119889120585119896int

120576119896

120585119896

tanh(120573120576

119896

2) 119889120576

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

[ln119890

1205731205761198962(1 + 119890

minus120573120576119896)

1198901205731205851198962 (1 + 119890minus120573120585119896)] 119889120585

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

ln (119890120573(120576119896minus120585119896)2

) 119889120585119896

minus4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120576119896) 119889120585

119896

+4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120585119896) 119889120585

119896

(48)

2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(49)

With the help of (49) we can calculate the free energydifference

64 Critical Field (119867119888) The critical field is related to the free

energy difference as

119867119888= 8120587 (119865

119878minus 119865

119873)

12

(50)

Using (49) we obtain

119867119888= 8120587(

119873 (0) Δ2(119896)

4minus

4119873 (0)

120573

times119890minus120573Δ(119896)

4(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12))

12

(51)

7 Numerical Calculations

Now we evaluate numerically the physical properties of high-temperature cuprate superconductor YBa

2Cu

3O

7minus119909 using

the various relations derived that is (35) (36) (38) (42)(49) and (51)

Values of various parameters appearing in the mentionedrelations are cited inTable 1Using these values we havemadestudy of various parameters related to the physical propertiesfor the system YBa

2Cu

3O

7minus119909

71 Superconducting Order Parameter (Δ) For the studyof superconducting order parameter (Δ) for the systemYBa

2Cu

3O

7minus119909 we have calculated the contributions due

to phonons and polarons separately and also obtained thecombined effect of phonons and polarons

(i) Superconducting order parameter (Δ1)

(When only electron-phonon interaction is considered)


Table 1 Values of various parameters for HTSC cuprate superconductor for YBa2Cu

3O

7minus119909

S no Property Value1 Superconducting transition temperature (Tc) 88 K2 Density of states119873(0) at the Fermi surface 495 times 10

26 per ergs Cu atom3 Phonon energy ℎ120596

11986313 times 10

minus21 J4 Polaron frequency 120596pl 072 eV5 Polaron density (119873 (0) exp (119892

2)) 8 stateseV spin

6 Fermi energy 023 eV7 Crystal structure Orthorhombic8 Cell parameters 119886 = 038 nm 119887 = 039 nm and 119888 = 117 nm9 Number of atoms per unit volume 5 times 10

28m3

10 Boltzmann constant (119896119861) 138 times 10

minus23 JK11 Mass of electron 91 times 10

minus31 kg

We have (35)

1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(52)

Solving the previous equation numerically we get

1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

119910=1

119910=0

119889119910[[

[

1

2radic1199102 + 059171199092

times tanh47099radic1199102 + 059171199092

119879

]]

]

(53)

With the help of the previous equation one can studythe variation of superconducting order parameter Δ

1with

temperature when only electron-phonon interaction is con-sidered

Values of superconducting order parameter obtained atvarious temperatures are given in Table 2 and variation ofΔ

1

with temperature is shown in Figure 1(ii) Superconducting order parameter (Δ

2)

(When only polaron interaction is considered)We have (36)

1

119892119873 (0) [1198922

119897]= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)119896

119861119879 + 1

times

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

]]

]

(54)


1

10038161003816100381610038161198921003816100381610038161003816119873 (0) [119892

2

119897]

= int

119910=1

119910=0

119889119910

2radic1199102 + 059171199092

times[[

[

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

minus1

exp 942radic1199102 + 059171199092119879 + 1

times

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

+ tanh3623 (8 minus 13radic1199102 + 059171199092)

119879

]]

]

(55)


Table 2 Superconducting order parameter (Δ) for YBa2Cu

3O

7minus119909system

S no Temperature(K)

Δ1= 119909 times 10

minus21 J(when only

electron-phononinteraction is present)

Δ2= 119909 times 10

minus21 J(when only polaron

interaction isconsidered)

Δ = Δ1+Δ

2(119909times10

minus21 J)(in the presence of bothphonon and polaron

interactions)

1 5 24869 0114015 2600965

2 10 24869 0114015 2600965

3 15 24867 0114015 2600795

4 20 24865 0114015 2600515

5 25 24838 0114015 2597865

6 30 24761 0114015 2590145

7 35 24600 0114015 2574075

8 40 24325 0114015 2546595

9 45 23914 0114015 2505415

10 50 23340 0114001 2448001

11 55 22580 0113991 2371991

12 60 21608 0113970 2274770

13 65 20382 0113914 2152114

14 70 18851 0113823 1998923

15 75 16932 0113683 1806883

16 80 14457 0113468 1559168

17 81 13868 0113425 1500225

18 82 13242 0113370 1437570

19 83 12571 0113300 1370450

20 84 11852 0113235 1298435

21 85 11060 0113177 1219177

22 86 10190 0113100 1132100

23 87 09228 0113030 1035830

24 879 08245 0112970 0937470

With the help of the previous equation one can studythe variation of superconducting order parameter (Δ

2) with

temperature when only polaron interaction is consideredValues of superconducting order parameter obtained at

various temperatures are given in Table 2 and variation ofΔ2

with temperature is shown in Figure 1(iii) Superconducting order parameter (Δ = Δ

1+ Δ

2)

(in the presence of combined phonon and polaron inter-actions)

The superconducting order parameter in the presenceof both phonon and polaron interactions can be studiedby taking a sum of the order parameters due to phononand polaron effects Values of order parameters obtained atvarious temperatures are given in Table 2

The behaviour of superconducting order parameter (Δ =

Δ1+ Δ

2) (combined phonon and polaron interactions) is

shown in Figure 1

72 Electronic Specific Heat (119862119890119904) We have obtained the

expression (38) for electronic specific heat putting

120576119896= 120585

2+ Δ

212

Δ = 119909 times 10minus21

119873 (0) = 05eV ℎ120596119863asymp 13 times 10

minus21 J

ℎ120596119863

2119870119861119879

= 47099T

(56)


0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)

Phonon and polaronPhononPolaron

Figure 1 Behaviour of superconducting order parameter for thesystem YBa

2Cu

3O

7minus119909

Equation (38) reduces to

119862es = 28985 times 169 times 10minus49

times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)

One can study the behaviour of electronic specific heat(119862

119890119904) with temperature (119879) with the help of (57) Values of

119862119890119904at various temperatures obtained from (57) are given in

Table 3 and variation of 119862119890119904

with 119879 is shown in Figure 2

Table 3 Electronic specific heat (119862es) for YBa2Cu3O7minus119909system

S no Temperature(K)

119862es times 10minus49

Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000

Variation of 119862es119879 with 119879 (Table 4) and (119862 minus119862es)119879 (Table 5)for a particular range of temperature is shown in Figures 3and 4 respectively A comparisonwith available experimentaldata is also shown in Figures 3 and 4 [29] Agreement betweentheory and experimental results is quite encouraging

73 Density of States Function 119873(120596)119873(0) Density of statesfunction for the polaron case is given by

119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces

Figure 2 Variation of 119862es with temperature for the systemYBa

2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T

Ces T theoreticalCes T experimental

Figure 3 Variation of 119862119890119904119879 with temperature

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)

(C minus Ces )T theoretical(C minus Ces )T experimental

(CminusC

es)T

Figure 4 Variation of (119862 minus 119862119890119904)119879 with temperature

Table 4 119862esT for YBa2Cu

3O

7minus119909system

S no Temperature(K)

119862esTJoulemole-K2

(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170

Table 5 (119862 minus 119862es)119879 for YBa2Cu

3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670

Using the following values

120596119899= 119910 times 10

minus21 J Δ = 119909 times 10minus21J 119897 = 1

1205960= 8 times 10

minus21 J 1198922= 1

(60)

The previous equation reduces as

119873(120596)

119873 (0)=

(119910 minus 8)

radic(119910 minus 8)2minus 1199092

minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)

The values of density of states 119873(120596)119873(0) for the chosenvalue of 119909 = 1106 18851 2334 and 24761 are shown inTable 6 The variation of density of states with 120596 at differenttemperatures is shown in Figure 5


Table 6 Density of states119873(120596)119873(0) at various temperatures for YBa2Cu

3O

7minus119909system

S no Frequency(y) Density of states119879 = 30K119909 = 24761

119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)

Figure 5 Variation of density of states with frequency (120596) atdifferent temperatures for the system YBa

2Cu

3O

7minus119909

74 Free Energy Difference Theexpression for the free energydifference can be expressed as

2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F

Figure 6 Variation of free energy difference with temperature forthe system YBa

2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092

times [1 minus minus001625[119879

119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)

The values of free energy difference at different tempera-tures are given in Table 7 and behaviour is shown in Figure 6respectively


Table 7 Free energy difference for YBa2Cu

3O

7minus119909system

S no Temperature(K)

Free energy differenceJoulemole

1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129

75 Critical Field (119867119888) The low temperature critical field is

given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573

times119890minus120573Δ(K)

4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)

Solving the previous equation numerically we can obtainthe values of low temperature critical field which aredescribed in Table 8 The behavior of the low temperaturecritical field (119867

119888) with temperature is shown in Figure 7

8 Discussion and Conclusions

In the foregoing sections we have developed the polaronicpairing mechanism of HTSC in cuprates and applied itto study physical properties of YBa

2Cu

3O

7minus119909 The idea of

polaron is based on the assumption about the autolocalizationof an electron in the ion crystal due to its interaction withlongitudinal optical vibrations under the local polarizationwhich is caused by the electron itself The electron is con-fined to the local-polarization-induced potential well andconserves it by its own field [30]

Table 8 Critical field for YBa2Cu

3O

7minus119909system

S no Temperature(K)

Critical field119867119862

Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)

Figure 7 Variation of critical field with temperature for the systemYBa

2Cu

3O

7minus119909

Following Greenrsquos function technique and equation ofmotion method we have obtained expressions for supercon-ducting order parameterWe found that phonon contributionis small enough 119879

119862obtained for the system YBa

2Cu

3O

7minus119909is

88 K which is in a good agreement with experiments [1 31]Making use of various parameters given in Table 1 we have


closely studied the density of states at various temperaturesspecific heat free energy and critical field

The specific heat behaviour with temperature is reason-ably in a good agreement with experimental data [32]

The study of density of states free energy and critical fieldprovides great insight into the physics of high-119879

119862cuprates

In the absence of experimental results we could not compareour results with experiments

The investigations reported here aim to establish severalof the properties of polaronic pairing mechanism of HTSCcuprates Only a detailed comparison with future experi-ments will clarify whether this pairing mechanism is or nota good approximation to describe HTSC cuprates

References

[1] T Tohyama ldquoRecent progress in physics of high-temperaturesuperconductorsrdquo Japanese Journal of Applied Physics vol 51no 1 Article ID 010004 2012

[2] S Kruchinin H Nagao and S Aono Modern Aspects ofSuperconductivity Theory of Superconductivity World Scien-tific River Edge NJ USA 2011

[3] S L Kakani and S Kakani Superconductivity Anshan KentUK 2009

[4] K P Sinha and S L Kakani High Temperature Superconduc-tivity Current Results amp Novel Mechanisms Nova Science NewYork NY USA 1995

[5] K P Sinha and S L Kakani ldquoFermion local chargedBoson model and cuprate superconductorsrdquo Proceedings of theNational Academy of Sciences vol 72 pp 153ndash214 2002

[6] K H Bennemann and J B Ketterson SuperconductivityConventional and Unconventional Superconductors Vol 1 2Springer New York NY USA 2008

[7] J R Schrieffer Theory of Superconductivity Westview PressOxford UK 1999

[8] M R Norman ldquoCupratesmdashan overviewrdquo Journal of Supercon-ductivity andNovelMagnetism vol 25 no 7 pp 2131ndash2134 2012

[9] A S Alexandrov Theory of Superconductivity from Weak toStrong Coupling IOP publishing Bristol UK 2003

[10] N PlakidaHigh Temperature Cuprate Superconductors Experi-ment Theory and Applications Springer Heidelberg Germany2010

[11] I Askerzade Unconventional Superconductors vol 153 ofSpringer Series in Material Science Springer Berlin Germany2012

[12] O Gunnarsson and O Rosch ldquoInterplay between electron-phonon and Coulomb interactions in cuprates rdquo Journal ofPhysics vol 20 no 4 Article ID 043201 2008

[13] J Bardeen L N Cooper and J R Schrieffer ldquoTheory ofsuperconductivityrdquo Physical Review vol 108 no 5 pp 1175ndash1204 1957

[14] G M Eliashberg Zhurnal Eksperimentalrsquonoi i TeoreticheskoiFiziki vol 38 p 966 1960 Engkish translation in The Journalof Experimental andTheoretical Physics Soviet Physics vol 11 p696 1960

[15] W L McMillan and J M Rowell ldquoLead phonon spectrumcalculated from superconducting density of statesrdquo PhysicalReview Letters vol 14 pp 108ndash112 1965

[16] W L McMillan ldquoTransition temperature of strong-coupledsuperconductorsrdquo Physical Review vol 167 no 2 pp 331ndash3441968

[17] P Brusov Mechanisms of High Temperature SuperconductivityRostov State University Publishing 1999

[18] TMorya and K Ueda ldquoSpin fluctuations and high-temperaturesuperconductivityrdquo Advances in Physics vol 49 no 5 pp 556ndash606 2000

[19] D Manske Theory of Unconventional Superconductors Spr-inger Heidelberg Germany 2004

[20] A Alexandrov and J Ranninger ldquoTheory of bipolarons andbipolaronic bandsrdquo Physical Review B vol 23 no 4 pp 1796ndash1801 1981

[21] A S Alexandrov and J Ranninger ldquoPhotoemission spectro-scopu of the superconducting and normal state for polaronicsystemsrdquo Physica C vol 198 no 3-4 pp 360ndash370 1992

[22] T Holstein ldquoStudies of polaron motion part I The molecular-crystal modelrdquo Annals of Physics vol 8 pp 325ndash342 1959

[23] L G Lang and Y A Firsov Zhurnal EksperimentalrsquoNoi iTeoreticheskoI Fiziki vol 43 p 1843 1962

[24] I L G Lang andY A Firsov ldquoKinetic theory of semiconductorswith low mobilityrdquo Journal of Experimental and TheoreticalPhysics Soviet Physics vol 16 p 1301 1963

[25] G D Mahan Many Particle Physics Plenum Press New YorkNY USA 1982

[26] D N Zubaroev ldquoDouble-time green functions in statisticalphysicsrdquo Uspekhi Fizicheskikh Nauk vol 71 pp 71ndash116 1960

[27] A L Fetter and J DWaleckaQuantamTheory ofMany ParticleSystem McGraw-Hill New York NY USA 1971

[28] S C Tiwari A Kashyap A K Surana RK Paliwal and S LKaKani ldquopolaronic mechanism of superconductivity in dopedfulleride systemsrdquo International Journal of Modern Physics Bvol 23 p 615 2009

[29] R Lortz T Tomita Y Wang et al ldquoOn the origin of the doublesuperconducting transition in overdoped YBa

2Cu

3O

119909rdquo Physica

C vol 434 pp 194ndash198 2006[30] S I Pekar Studies on the ElectronTheory of Crystals Gostekhiz-

dat Moscow Russia 1951[31] B J Ramshaw J Day B Vignolle et al ldquoVortex lattice melting

andH1198882in underdoped YBa

2Cu

3O

119910rdquo Physical Review B vol 86

no 17 Article ID 174501 6 pages 2012[32] M J W Dodgson V B Geshkenbein H Nordborg and

G Blatter ldquoThermodynamics of the first-order vortex latticemelting transition in YBa

2Cu

3O

7minus120575rdquo Physical Review B vol 57

no 22 pp 14498ndash14506 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



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Journal of



Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014


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Biophysics


ThermodynamicsJournal of


[23 24] and Gorkov Nambu formalisms in order to evaluatethe Greenrsquos functions for electrons coupled to phonons andconsidering the range of couplingwhich corresponds to smallpolaron formation (120582 ≫ 1) [25 26]

2 Model Hamiltonian

The model Hamiltonian for our system can be expressed as[21]

119867 = 119867119890+ 119867ph + 119867

119890minusph + 119867119890119890 (1)

where119867119890is the kinetic energy in the initial Bloch band119867ph is

the vibration energy of the lattice 119867119890minusph is electron-phonon

interaction and119867119890119890is the Coulomb electron-electron corre-

lationsIn one band approximation119867

119890has the form

119867119890= sum

119896119904

119864 (119896) 119862+

119896119904119862119896119904 (2)

where 119896 and 119904 denote the state with quasi momentum andspin respectively 119864(119896) is bare band energy 119867ph can beexpressed in terms of phonon operators 119889

119902 119902 = (119902 ]) where

] is the type of vibrational mode

119867ph = sum

119902

120596 (119902) 119889+

119902119889119902 (3)

where 120596 is the phonon dispersionThe electron-phonon interaction is described by the

Frohlich Hamiltonian

119867119890minusph = sum

119896119902119904

120596 (119902) 120574 (119902)1

radic2119873119862+

119896+119902119904119862119896119904119889119902+ 119867119862 (4)

in which 120596 (119902) and 120574 (119902) are the phonon frequency and theinteraction matrix element in a parent crystal without chargecarriers respectively Correspondingly one obtains

1205742(119902) =

41205871198902

1199022Ω1205960

(1

120576infin

minus1

1205760

) (5)

In the case of optical longitudinal phonon with frequency120596(119902) = 120596

0and 120576

0 120576

infinare the dielectric constants of the crystal

with andwithout taking ionic part into considerationΩ is thevolume of the unit cell and119873 is their number

For acoustic phonons one finds

]2 (119902) = 1198642

119863

119902

119906119872

120596 (119902) = 1205960

(6)

where119864119863is the deformation potential 119906 is the sound velocity

and 119872 is the mass of an elementary cell For intermolecularphonons

1205742(119902) = 120574

2(0) (7)

The combined Hamiltonian can be expressed as

119867 = sum

119896119904

119864 (119896) 119862+

119896119904119862119896119904

+ sum

119902

120596 (119902) 119889+

119902119889119902

+ sum

119896119902119904

120596 (119902) 120574 (119902) 119889119902

1

radic2119873119862

+

119896+119902119862119896119904

+ 119867119862 + 119881119862

(8)

Here 119881119862is the Coulomb repulsion This Hamiltonian

includes electron-phonon and electron-electron correlationsTo diagonalize the main part of the Hamiltonian the siterepresentation is more convenient

One can express the previous Hamiltonian as

119867 = sum

119898119899

119879 (119898 minus 119899)119862+

119898120590119862119899120590

+ sum

119902

120596 (119902) 119889+

119902119889119902+

1

2

+1

radic2119873sum

119898119902

120596 (119902) 120574 (119902) 119899119898120590

119889119902119890119894119902119898

+ 119889+

119902119890minus119894119902119898

+1

2sum

1198981198991205901205901015840

119881119862(119898 minus 119899) 119899

+

1198981205901198991198991205901015840 + sum

119902

120596 (119902) 119889+

119902119889119902+

1

2

(9)

where

119879 (119898) =1

119873sum

119896

119864 (119896) 119890119894119896119898

119899119898120590

= 119862+

119898120590119862119898120590

(10)

In the small-polaron regime 120582 ge 1 the kinetic energyremains smaller than the interaction energy and a self-consistent treatment of a many-body problem is possiblewith the 1120582 expansion technique [27] Following Lang andFirsov [23 24] and applying canonical transformations todiagonalize the Hamiltonian [28] one obtains

= 119890119878119867119890

minus119878 (11)

where

119878 = sum

119898119904

119862+

119898119904119862119898119904

1

radic2119873120574 (119902) 119889

119902119890119894119902119898

minus 119889+

119902119890minus119894119902119898

(12)

The electron operator transforms as

119889119902= 119889

1199021015840 + [119878 119889

1199021015840]

1

2[119878 [119878 119889

1199021015840]] + sdot sdot sdot

[119904 1198891199021015840] = [sum

119898119904

119862+

119898119904119862119898119904

1

radic2119873120574 (119902) 119889

119902119890119894119902119898

minus 119889+

119902119890minus119894119902119898

1198891199021015840]

[119904 1198891199021015840] =

1

radic2119873120574 (119902)

times [sum

119902119898119904

119862+

119898119904119862119898119904

119889119902119890119894119902119898

1198891199021015840 minus 119889

+

119902119890minus119894119902119898

1198891199021015840

minus 1198891199021015840119889

119902119890119894119902119898

+ 1198891199021015840119889

+

119902119890minus119894119902119898

]

(13)


Using 1198891199021015840119889

+

119902= 119889

+

1199021198891199021015840 + 120575

1199021199021015840

[119904 1198891199021015840] =

1

radic2119873120574 (119902) [sum

119902119898119904

119862+

119898119904119862119898119904

1205751199021199021015840119890minus119894119902119898

] (14)

When 119902 = 1199021015840 we have

[119904 1198891199021015840] =

1

radic2119873120574 (119902) [sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898]


(15)

Thus

1198891199021015840 = 119889

1199021015840 +

1

radic2119873120574 (119902)sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898

119889+

1199021015840 = 119889

+

1199021015840 minus

1

radic2119873120574 (119902)sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898

(16)

Hence

119867 = sum

119898119899119904119898 = 119899

119879 (119898 minus 119899)119862+

119898119904119862119899119904

times exp[sum

119902

119889119902

1

radic2119873120574 (119902) 119890

119894119902119898+sum

119902

120596 (119902) (119889+

119902119889119902+1

2)]

+ sum

119898119899119904

119881119862(119898 minus 119899) minus sum

119902

120596 (119902)1

21198731205742(119902) 119890

119894119902(119898minus119899)

times 119862+

119898119904119862+

119899119904119862119899119904119862119898119904

(17)





119866 (119896 120596119899) = minus

1

2sum

119898

int

120573

minus120573

d119897119890119897120596119899119890+119894119896119898 ⟨⟨119897119897119862119888120590

(119897) 119862+

119898120590(119900)⟩⟩

119865 (119896 120596119899) = minus

1

2sum

119898

int

120573

minus120573

d119897119890119897120596119899119890+119894119896119898 ⟨⟨119897119897119862119888120590

(119897) 119862119898120590

(119900)⟩⟩

(18)



119862119862+119890minus120573119890119897]


(19)


119866 (119896 120596119899)

= 119890minus1199022

[1199062

119896

119894120596119899minus 120576

119899

+V2119896

119894120596119899+ 120576

119899

+1

119873

infin

sum

119897=1

1198922119897

119897

times sum

1198961015840

1199062

1198961015840 (1 minus 119899

1198961015840)

119894120596119899minus 119897120596

0minus 120576

1198961015840

+V211989610158401198991198961015840

119894120596119899minus 119897120596

0+ 120576

1198961015840

+1199062

11989610158401198991198961015840

119894120596119899+ 119897120596

0minus 120576

1198961015840

+V21198961015840 (1 minus 119899

1198961015840)

119894120596119899+ 119897120596

0+ 120576

1198961015840

]

(20)

119865 (119896 120596119899)

= 119890minus1199022

[

[

119906119896V119896(

1

119894120596119899minus 120576

119896

+1

119894120596119899+ 120576

119896

) +1

119873

infin

sum

119897=1

(minus1)1198971198922119897

119897

times sum

1198961015840

1199061198961015840V

1198961015840

(1 minus 1198991198961015840)

119894120596119899minus 119897120596

0minus 120576

1198961015840

minus1198991198961015840

119894120596119899minus 119897120596

0+ 120576

1198961015840

+1198991198961015840

119894120596119899+119897120596

0minus120576

1198961015840


1198961015840)

119894120596119899+119897120596

0+120576

1198961015840

]

]

(21)

where

1199062

119896=

1

2(1 +

120585119896

120576119896

) V2119896=

1

2(1 minus

120585119896

120576119896

)

119906119896V119896= minus

Δ

2120576119896

119899119896= 119899 (120576

119896)

(22)

With

119899 (119909) = (119890119909119896119861119879+ 1)

minus1

120576119896= radic120585

2

119896+ Δ2

(119896)

Δ (119896) = minus1

2sum

119896

119881(119896 minus 1198961015840)

Δ (1198961015840)

1205761198961015840

tanh120576119896

2119896119861119879

119881 (119896) =1

119873sum

119898

119881 (119898) 119890119894119896119898

(23)


120585119896= sum

119898

120590 (119898119900) 119879 (119898) 119890119894119896119898

minus 120583 (24)


119885119879(119898)




⟨119862+

119901119862119901⟩ =

1

2120587int

+infin

minusinfin

119868119866(120596

119899) 119889120596

119899 (25)

⟨119862119901119862119901⟩ =

1

2120587int

+infin

minusinfin

119868119865(120596

119899) 119889120596

119899 (26)

where

119868119866(120596

119899) = 119894(119890

120573120596119899 + 1)

minus1

[11986611

(120596119899+ 119894120576) minus 119866

11(120596

119899minus 119894120576)]

119868119865(120596

119899) = 119894(119890

120573120596119899 + 1)

minus1

[11986511

(120596119899+ 119894120576) minus 119865

11(120596

119899minus 119894120576)]

(27)


Using the identity

lim120576rarr0

1

2120587[

1

120596 + 119894120576 minus 119864119896

minus1

120596 minus 119894120576 minus 119864119896

] = 119894120575 (120596 minus 119864119896)

int

infin

minusinfin

119891 (120596119899) 120575 (120596

119899minus 120596

plusmn) 119889120596

119899= 119891 (120596

plusmn

119899)

(28)


1199062

119896=

1

2(1 +

120585119896

120576119896

) V2119896=

1

2(1 minus

120585119896

120576119896

) (29)


⟨119862+

119901119862119901⟩

=1

2+

1

2

120585119896

120576119896

tanh120573120576

119896

2

+1

119873

infin

sum

119897=1

1198922119897

119897[1

2+

1

2

1205851198961015840

1205761198961015840

tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus1198991198961015840

2

1205851198961015840

1205761198961015840

tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus tanh120573 (119897120596

0minus 120576

1198961015840)

2]

(30)

⟨119862119901119862119901⟩

= minusΔ (119896)

2120576119896

tanh120573120576

119896

2

+1

119873

infin

sum

119897=1

(minus1)119897

1198971198922119897sum

1198961015840

Δ (1198961015840)

21205761198961015840

times [tanh120573 (119897120596

0+ 120576

1198961015840)

2


120573 (1198971205960+ 120576

1198961015840)

2+ tanh

120573 (1198971205960minus 120576

1198961015840)

2]

(31)



Δ = 119892sum

119896

⟨119862119896119862119896⟩ (32)


sum

119896

= 119873 (119900) int

ℎ120596119863

0

119889120585119896 (33)


Δ = 119892119873 (0) int

ℎ120596119863

0

119889120585119896

[[

[

minusΔ (119896)

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

minus 1198921

119873

infin

sum

119897=1

(minus1)119897

1198971198922119897

1119873(0) int

ℎ120596119863

0

sum

1198961015840

119889120585119896

minusΔ (1198961015840)

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanh119897120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)2119896

119861119879 + 1

times

tanh(119897120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840))

2119896119861119879

+ tanh(119897120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840))

2119896119861119879

]]

]

(34)



2Cu

3O



1

119892119873 (0)= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(35)



10038161003816100381610038161198921003816100381610038161003816 119873 (0) [119892

2

119897]

= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+Δ2 (1198961015840)2119896

119861119879 + 1

times

tanhℓ120596

0+radic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minusradic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

]]

]

(36)





119888119890119904

=120597

120597119879[

1

119873sum

119896

2120585119896⟨119862

+

119896119862119896⟩] (37)

where ⟨119862+



119888119890119904

=119873 (0)

2119873int

ℎ120596119863

0

2120585119896

1198961198611198792

119889120585119896

times [minus1

2120585119896sec ℎ2 (

120576119896

2119896119861119879)

+

infin

sum

119897=1

1198922119897

119897minus

1

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 minus 120576

1198961015840)

times sec ℎ2 (1205960119897 minus 120576

1198961015840

2119896119861119879

) ]

(38)




119873(120596) = lim 1

2120587[119866

11(119896 120596 + 119894120578) minus 119866

11(119896 120596 minus 119894120578)] (39)


lim120578rarr0

1

2120587[

1

120596 + 119894120578 minus 120596+

119899

minus1

120596 minus 119894120578 minus 120596+

119899

] = 119894120575 (120596 minus 120596+

119899) (40)


119896byminus120576


1199062

119896+ V2

119896=

1

2(1 +

120585119896

120576119896

) +1

2(1 minus

120585119896

120576119896

) = 1 (41)

one obtains

119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(42)



119865119904minus 119865

119873

119881= int

infin

0

119889119892(1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (43)


119865119904minus 119865

119873

119881= int

Δ

0

119889Δ119889

119889Δ(

1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (44)



Δ (119896) = 119892119873 (0) int

ℎ120596119863

0

[[

[

minusΔ (119896)

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

1

119892= 119873 (0) int

ℎ120596119863

0

[[

[

1

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

(45)


[119865119878119873

119881]phonon

= int

Δ(119896)

0

Δ2(119896) 119889Δ (119896)

119889

119889Δ (119896)

times[[

[

119873 (0) int

ℎ120596119863

0

1

2radic1205852

119896+ Δ2

(119896)


2

119896+ Δ2

(119896)

2119896119861119879

119889120585119896

]]

]

(46)

Since

1205762

119896= 120585

2

119896+ Δ

2(119896)

2120576119896119889120576

119896= 2Δ (119896) 119889Δ (119896)

(47)


2 [119865119878119873

119881] =

Δ2(119896)

119892minus 119873 (0)

times int

ℎ120596119863

0

2119889120585119896int

120576119896

120585119896

tanh(120573120576

119896

2) 119889120576

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

[ln119890

1205731205761198962(1 + 119890

minus120573120576119896)

1198901205731205851198962 (1 + 119890minus120573120585119896)] 119889120585

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

ln (119890120573(120576119896minus120585119896)2

) 119889120585119896

minus4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120576119896) 119889120585

119896

+4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120585119896) 119889120585

119896

(48)

2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(49)




119867119888= 8120587 (119865

119878minus 119865

119873)

12

(50)


119867119888= 8120587(

119873 (0) Δ2(119896)

4minus

4119873 (0)

120573

times119890minus120573Δ(119896)

4(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12))

12

(51)



2Cu

3O

7minus119909 using



2Cu

3O

7minus119909


2Cu

3O







3O

7minus119909



11986313 times 10


2)) 8 stateseV spin


28m3



minus31 kg

We have (35)

1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(52)


1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

119910=1

119910=0

119889119910[[

[

1

2radic1199102 + 059171199092

times tanh47099radic1199102 + 059171199092

119879

]]

]

(53)


1with



1


2)


1

119892119873 (0) [1198922

119897]= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)119896

119861119879 + 1

times

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

]]

]

(54)


1

10038161003816100381610038161198921003816100381610038161003816119873 (0) [119892

2

119897]

= int

119910=1

119910=0

119889119910

2radic1199102 + 059171199092

times[[

[

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

minus1

exp 942radic1199102 + 059171199092119879 + 1

times

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

+ tanh3623 (8 minus 13radic1199102 + 059171199092)

119879

]]

]

(55)



3O

7minus119909system

S no Temperature(K)

Δ1= 119909 times 10

minus21 J(when only


Δ2= 119909 times 10



Δ = Δ1+Δ

2(119909times10


interactions)

1 5 24869 0114015 2600965

2 10 24869 0114015 2600965

3 15 24867 0114015 2600795

4 20 24865 0114015 2600515

5 25 24838 0114015 2597865

6 30 24761 0114015 2590145

7 35 24600 0114015 2574075

8 40 24325 0114015 2546595

9 45 23914 0114015 2505415

10 50 23340 0114001 2448001

11 55 22580 0113991 2371991

12 60 21608 0113970 2274770

13 65 20382 0113914 2152114

14 70 18851 0113823 1998923

15 75 16932 0113683 1806883

16 80 14457 0113468 1559168

17 81 13868 0113425 1500225

18 82 13242 0113370 1437570

19 83 12571 0113300 1370450

20 84 11852 0113235 1298435

21 85 11060 0113177 1219177

22 86 10190 0113100 1132100

23 87 09228 0113030 1035830

24 879 08245 0112970 0937470


2) with




1+ Δ

2)




Δ1+ Δ


shown in Figure 1



120576119896= 120585

2+ Δ

212


119873 (0) = 05eV ℎ120596119863asymp 13 times 10

minus21 J

ℎ120596119863

2119870119861119879

= 47099T

(56)


0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)



2Cu

3O

7minus119909



times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)







S no Temperature(K)


Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000



119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Using 1198891199021015840119889

+

119902= 119889

+

1199021198891199021015840 + 120575

1199021199021015840

[119904 1198891199021015840] =

1

radic2119873120574 (119902) [sum

119902119898119904

119862+

119898119904119862119898119904

1205751199021199021015840119890minus119894119902119898

] (14)

When 119902 = 1199021015840 we have

[119904 1198891199021015840] =

1

radic2119873120574 (119902) [sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898]


(15)

Thus

1198891199021015840 = 119889

1199021015840 +

1

radic2119873120574 (119902)sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898

119889+

1199021015840 = 119889

+

1199021015840 minus

1

radic2119873120574 (119902)sum

119898119904

119862+

119898119904119862119898119904

119890minus1198941199021015840119898

(16)

Hence

119867 = sum

119898119899119904119898 = 119899

119879 (119898 minus 119899)119862+

119898119904119862119899119904

times exp[sum

119902

119889119902

1

radic2119873120574 (119902) 119890

119894119902119898+sum

119902

120596 (119902) (119889+

119902119889119902+1

2)]

+ sum

119898119899119904

119881119862(119898 minus 119899) minus sum

119902

120596 (119902)1

21198731205742(119902) 119890

119894119902(119898minus119899)

times 119862+

119898119904119862+

119899119904119862119899119904119862119898119904

(17)





119866 (119896 120596119899) = minus

1

2sum

119898

int

120573

minus120573

d119897119890119897120596119899119890+119894119896119898 ⟨⟨119897119897119862119888120590

(119897) 119862+

119898120590(119900)⟩⟩

119865 (119896 120596119899) = minus

1

2sum

119898

int

120573

minus120573

d119897119890119897120596119899119890+119894119896119898 ⟨⟨119897119897119862119888120590

(119897) 119862119898120590

(119900)⟩⟩

(18)



119862119862+119890minus120573119890119897]


(19)


119866 (119896 120596119899)

= 119890minus1199022

[1199062

119896

119894120596119899minus 120576

119899

+V2119896

119894120596119899+ 120576

119899

+1

119873

infin

sum

119897=1

1198922119897

119897

times sum

1198961015840

1199062

1198961015840 (1 minus 119899

1198961015840)

119894120596119899minus 119897120596

0minus 120576

1198961015840

+V211989610158401198991198961015840

119894120596119899minus 119897120596

0+ 120576

1198961015840

+1199062

11989610158401198991198961015840

119894120596119899+ 119897120596

0minus 120576

1198961015840

+V21198961015840 (1 minus 119899

1198961015840)

119894120596119899+ 119897120596

0+ 120576

1198961015840

]

(20)

119865 (119896 120596119899)

= 119890minus1199022

[

[

119906119896V119896(

1

119894120596119899minus 120576

119896

+1

119894120596119899+ 120576

119896

) +1

119873

infin

sum

119897=1

(minus1)1198971198922119897

119897

times sum

1198961015840

1199061198961015840V

1198961015840

(1 minus 1198991198961015840)

119894120596119899minus 119897120596

0minus 120576

1198961015840

minus1198991198961015840

119894120596119899minus 119897120596

0+ 120576

1198961015840

+1198991198961015840

119894120596119899+119897120596

0minus120576

1198961015840


1198961015840)

119894120596119899+119897120596

0+120576

1198961015840

]

]

(21)

where

1199062

119896=

1

2(1 +

120585119896

120576119896

) V2119896=

1

2(1 minus

120585119896

120576119896

)

119906119896V119896= minus

Δ

2120576119896

119899119896= 119899 (120576

119896)

(22)

With

119899 (119909) = (119890119909119896119861119879+ 1)

minus1

120576119896= radic120585

2

119896+ Δ2

(119896)

Δ (119896) = minus1

2sum

119896

119881(119896 minus 1198961015840)

Δ (1198961015840)

1205761198961015840

tanh120576119896

2119896119861119879

119881 (119896) =1

119873sum

119898

119881 (119898) 119890119894119896119898

(23)


120585119896= sum

119898

120590 (119898119900) 119879 (119898) 119890119894119896119898

minus 120583 (24)


119885119879(119898)




⟨119862+

119901119862119901⟩ =

1

2120587int

+infin

minusinfin

119868119866(120596

119899) 119889120596

119899 (25)

⟨119862119901119862119901⟩ =

1

2120587int

+infin

minusinfin

119868119865(120596

119899) 119889120596

119899 (26)

where

119868119866(120596

119899) = 119894(119890

120573120596119899 + 1)

minus1

[11986611

(120596119899+ 119894120576) minus 119866

11(120596

119899minus 119894120576)]

119868119865(120596

119899) = 119894(119890

120573120596119899 + 1)

minus1

[11986511

(120596119899+ 119894120576) minus 119865

11(120596

119899minus 119894120576)]

(27)


Using the identity

lim120576rarr0

1

2120587[

1

120596 + 119894120576 minus 119864119896

minus1

120596 minus 119894120576 minus 119864119896

] = 119894120575 (120596 minus 119864119896)

int

infin

minusinfin

119891 (120596119899) 120575 (120596

119899minus 120596

plusmn) 119889120596

119899= 119891 (120596

plusmn

119899)

(28)


1199062

119896=

1

2(1 +

120585119896

120576119896

) V2119896=

1

2(1 minus

120585119896

120576119896

) (29)


⟨119862+

119901119862119901⟩

=1

2+

1

2

120585119896

120576119896

tanh120573120576

119896

2

+1

119873

infin

sum

119897=1

1198922119897

119897[1

2+

1

2

1205851198961015840

1205761198961015840

tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus1198991198961015840

2

1205851198961015840

1205761198961015840

tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus tanh120573 (119897120596

0minus 120576

1198961015840)

2]

(30)

⟨119862119901119862119901⟩

= minusΔ (119896)

2120576119896

tanh120573120576

119896

2

+1

119873

infin

sum

119897=1

(minus1)119897

1198971198922119897sum

1198961015840

Δ (1198961015840)

21205761198961015840

times [tanh120573 (119897120596

0+ 120576

1198961015840)

2


120573 (1198971205960+ 120576

1198961015840)

2+ tanh

120573 (1198971205960minus 120576

1198961015840)

2]

(31)



Δ = 119892sum

119896

⟨119862119896119862119896⟩ (32)


sum

119896

= 119873 (119900) int

ℎ120596119863

0

119889120585119896 (33)


Δ = 119892119873 (0) int

ℎ120596119863

0

119889120585119896

[[

[

minusΔ (119896)

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

minus 1198921

119873

infin

sum

119897=1

(minus1)119897

1198971198922119897

1119873(0) int

ℎ120596119863

0

sum

1198961015840

119889120585119896

minusΔ (1198961015840)

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanh119897120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)2119896

119861119879 + 1

times

tanh(119897120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840))

2119896119861119879

+ tanh(119897120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840))

2119896119861119879

]]

]

(34)



2Cu

3O



1

119892119873 (0)= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(35)



10038161003816100381610038161198921003816100381610038161003816 119873 (0) [119892

2

119897]

= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+Δ2 (1198961015840)2119896

119861119879 + 1

times

tanhℓ120596

0+radic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minusradic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

]]

]

(36)





119888119890119904

=120597

120597119879[

1

119873sum

119896

2120585119896⟨119862

+

119896119862119896⟩] (37)

where ⟨119862+



119888119890119904

=119873 (0)

2119873int

ℎ120596119863

0

2120585119896

1198961198611198792

119889120585119896

times [minus1

2120585119896sec ℎ2 (

120576119896

2119896119861119879)

+

infin

sum

119897=1

1198922119897

119897minus

1

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 minus 120576

1198961015840)

times sec ℎ2 (1205960119897 minus 120576

1198961015840

2119896119861119879

) ]

(38)




119873(120596) = lim 1

2120587[119866

11(119896 120596 + 119894120578) minus 119866

11(119896 120596 minus 119894120578)] (39)


lim120578rarr0

1

2120587[

1

120596 + 119894120578 minus 120596+

119899

minus1

120596 minus 119894120578 minus 120596+

119899

] = 119894120575 (120596 minus 120596+

119899) (40)


119896byminus120576


1199062

119896+ V2

119896=

1

2(1 +

120585119896

120576119896

) +1

2(1 minus

120585119896

120576119896

) = 1 (41)

one obtains

119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(42)



119865119904minus 119865

119873

119881= int

infin

0

119889119892(1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (43)


119865119904minus 119865

119873

119881= int

Δ

0

119889Δ119889

119889Δ(

1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (44)



Δ (119896) = 119892119873 (0) int

ℎ120596119863

0

[[

[

minusΔ (119896)

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

1

119892= 119873 (0) int

ℎ120596119863

0

[[

[

1

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

(45)


[119865119878119873

119881]phonon

= int

Δ(119896)

0

Δ2(119896) 119889Δ (119896)

119889

119889Δ (119896)

times[[

[

119873 (0) int

ℎ120596119863

0

1

2radic1205852

119896+ Δ2

(119896)


2

119896+ Δ2

(119896)

2119896119861119879

119889120585119896

]]

]

(46)

Since

1205762

119896= 120585

2

119896+ Δ

2(119896)

2120576119896119889120576

119896= 2Δ (119896) 119889Δ (119896)

(47)


2 [119865119878119873

119881] =

Δ2(119896)

119892minus 119873 (0)

times int

ℎ120596119863

0

2119889120585119896int

120576119896

120585119896

tanh(120573120576

119896

2) 119889120576

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

[ln119890

1205731205761198962(1 + 119890

minus120573120576119896)

1198901205731205851198962 (1 + 119890minus120573120585119896)] 119889120585

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

ln (119890120573(120576119896minus120585119896)2

) 119889120585119896

minus4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120576119896) 119889120585

119896

+4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120585119896) 119889120585

119896

(48)

2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(49)




119867119888= 8120587 (119865

119878minus 119865

119873)

12

(50)


119867119888= 8120587(

119873 (0) Δ2(119896)

4minus

4119873 (0)

120573

times119890minus120573Δ(119896)

4(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12))

12

(51)



2Cu

3O

7minus119909 using



2Cu

3O

7minus119909


2Cu

3O







3O

7minus119909



11986313 times 10


2)) 8 stateseV spin


28m3



minus31 kg

We have (35)

1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(52)


1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

119910=1

119910=0

119889119910[[

[

1

2radic1199102 + 059171199092

times tanh47099radic1199102 + 059171199092

119879

]]

]

(53)


1with



1


2)


1

119892119873 (0) [1198922

119897]= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)119896

119861119879 + 1

times

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

]]

]

(54)


1

10038161003816100381610038161198921003816100381610038161003816119873 (0) [119892

2

119897]

= int

119910=1

119910=0

119889119910

2radic1199102 + 059171199092

times[[

[

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

minus1

exp 942radic1199102 + 059171199092119879 + 1

times

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

+ tanh3623 (8 minus 13radic1199102 + 059171199092)

119879

]]

]

(55)



3O

7minus119909system

S no Temperature(K)

Δ1= 119909 times 10

minus21 J(when only


Δ2= 119909 times 10



Δ = Δ1+Δ

2(119909times10


interactions)

1 5 24869 0114015 2600965

2 10 24869 0114015 2600965

3 15 24867 0114015 2600795

4 20 24865 0114015 2600515

5 25 24838 0114015 2597865

6 30 24761 0114015 2590145

7 35 24600 0114015 2574075

8 40 24325 0114015 2546595

9 45 23914 0114015 2505415

10 50 23340 0114001 2448001

11 55 22580 0113991 2371991

12 60 21608 0113970 2274770

13 65 20382 0113914 2152114

14 70 18851 0113823 1998923

15 75 16932 0113683 1806883

16 80 14457 0113468 1559168

17 81 13868 0113425 1500225

18 82 13242 0113370 1437570

19 83 12571 0113300 1370450

20 84 11852 0113235 1298435

21 85 11060 0113177 1219177

22 86 10190 0113100 1132100

23 87 09228 0113030 1035830

24 879 08245 0112970 0937470


2) with




1+ Δ

2)




Δ1+ Δ


shown in Figure 1



120576119896= 120585

2+ Δ

212


119873 (0) = 05eV ℎ120596119863asymp 13 times 10

minus21 J

ℎ120596119863

2119870119861119879

= 47099T

(56)


0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)



2Cu

3O

7minus119909



times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)







S no Temperature(K)


Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000



119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






⟨119862+

119901119862119901⟩ =

1

2120587int

+infin

minusinfin

119868119866(120596

119899) 119889120596

119899 (25)

⟨119862119901119862119901⟩ =

1

2120587int

+infin

minusinfin

119868119865(120596

119899) 119889120596

119899 (26)

where

119868119866(120596

119899) = 119894(119890

120573120596119899 + 1)

minus1

[11986611

(120596119899+ 119894120576) minus 119866

11(120596

119899minus 119894120576)]

119868119865(120596

119899) = 119894(119890

120573120596119899 + 1)

minus1

[11986511

(120596119899+ 119894120576) minus 119865

11(120596

119899minus 119894120576)]

(27)


Using the identity

lim120576rarr0

1

2120587[

1

120596 + 119894120576 minus 119864119896

minus1

120596 minus 119894120576 minus 119864119896

] = 119894120575 (120596 minus 119864119896)

int

infin

minusinfin

119891 (120596119899) 120575 (120596

119899minus 120596

plusmn) 119889120596

119899= 119891 (120596

plusmn

119899)

(28)


1199062

119896=

1

2(1 +

120585119896

120576119896

) V2119896=

1

2(1 minus

120585119896

120576119896

) (29)


⟨119862+

119901119862119901⟩

=1

2+

1

2

120585119896

120576119896

tanh120573120576

119896

2

+1

119873

infin

sum

119897=1

1198922119897

119897[1

2+

1

2

1205851198961015840

1205761198961015840

tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus1198991198961015840

2

1205851198961015840

1205761198961015840

tanh120573 (119897120596

0+ 120576

1198961015840)

2

minus tanh120573 (119897120596

0minus 120576

1198961015840)

2]

(30)

⟨119862119901119862119901⟩

= minusΔ (119896)

2120576119896

tanh120573120576

119896

2

+1

119873

infin

sum

119897=1

(minus1)119897

1198971198922119897sum

1198961015840

Δ (1198961015840)

21205761198961015840

times [tanh120573 (119897120596

0+ 120576

1198961015840)

2


120573 (1198971205960+ 120576

1198961015840)

2+ tanh

120573 (1198971205960minus 120576

1198961015840)

2]

(31)



Δ = 119892sum

119896

⟨119862119896119862119896⟩ (32)


sum

119896

= 119873 (119900) int

ℎ120596119863

0

119889120585119896 (33)


Δ = 119892119873 (0) int

ℎ120596119863

0

119889120585119896

[[

[

minusΔ (119896)

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

minus 1198921

119873

infin

sum

119897=1

(minus1)119897

1198971198922119897

1119873(0) int

ℎ120596119863

0

sum

1198961015840

119889120585119896

minusΔ (1198961015840)

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanh119897120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)2119896

119861119879 + 1

times

tanh(119897120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840))

2119896119861119879

+ tanh(119897120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840))

2119896119861119879

]]

]

(34)



2Cu

3O



1

119892119873 (0)= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(35)



10038161003816100381610038161198921003816100381610038161003816 119873 (0) [119892

2

119897]

= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+Δ2 (1198961015840)2119896

119861119879 + 1

times

tanhℓ120596

0+radic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minusradic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

]]

]

(36)





119888119890119904

=120597

120597119879[

1

119873sum

119896

2120585119896⟨119862

+

119896119862119896⟩] (37)

where ⟨119862+



119888119890119904

=119873 (0)

2119873int

ℎ120596119863

0

2120585119896

1198961198611198792

119889120585119896

times [minus1

2120585119896sec ℎ2 (

120576119896

2119896119861119879)

+

infin

sum

119897=1

1198922119897

119897minus

1

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 minus 120576

1198961015840)

times sec ℎ2 (1205960119897 minus 120576

1198961015840

2119896119861119879

) ]

(38)




119873(120596) = lim 1

2120587[119866

11(119896 120596 + 119894120578) minus 119866

11(119896 120596 minus 119894120578)] (39)


lim120578rarr0

1

2120587[

1

120596 + 119894120578 minus 120596+

119899

minus1

120596 minus 119894120578 minus 120596+

119899

] = 119894120575 (120596 minus 120596+

119899) (40)


119896byminus120576


1199062

119896+ V2

119896=

1

2(1 +

120585119896

120576119896

) +1

2(1 minus

120585119896

120576119896

) = 1 (41)

one obtains

119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(42)



119865119904minus 119865

119873

119881= int

infin

0

119889119892(1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (43)


119865119904minus 119865

119873

119881= int

Δ

0

119889Δ119889

119889Δ(

1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (44)



Δ (119896) = 119892119873 (0) int

ℎ120596119863

0

[[

[

minusΔ (119896)

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

1

119892= 119873 (0) int

ℎ120596119863

0

[[

[

1

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

(45)


[119865119878119873

119881]phonon

= int

Δ(119896)

0

Δ2(119896) 119889Δ (119896)

119889

119889Δ (119896)

times[[

[

119873 (0) int

ℎ120596119863

0

1

2radic1205852

119896+ Δ2

(119896)


2

119896+ Δ2

(119896)

2119896119861119879

119889120585119896

]]

]

(46)

Since

1205762

119896= 120585

2

119896+ Δ

2(119896)

2120576119896119889120576

119896= 2Δ (119896) 119889Δ (119896)

(47)


2 [119865119878119873

119881] =

Δ2(119896)

119892minus 119873 (0)

times int

ℎ120596119863

0

2119889120585119896int

120576119896

120585119896

tanh(120573120576

119896

2) 119889120576

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

[ln119890

1205731205761198962(1 + 119890

minus120573120576119896)

1198901205731205851198962 (1 + 119890minus120573120585119896)] 119889120585

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

ln (119890120573(120576119896minus120585119896)2

) 119889120585119896

minus4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120576119896) 119889120585

119896

+4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120585119896) 119889120585

119896

(48)

2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(49)




119867119888= 8120587 (119865

119878minus 119865

119873)

12

(50)


119867119888= 8120587(

119873 (0) Δ2(119896)

4minus

4119873 (0)

120573

times119890minus120573Δ(119896)

4(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12))

12

(51)



2Cu

3O

7minus119909 using



2Cu

3O

7minus119909


2Cu

3O







3O

7minus119909



11986313 times 10


2)) 8 stateseV spin


28m3



minus31 kg

We have (35)

1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(52)


1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

119910=1

119910=0

119889119910[[

[

1

2radic1199102 + 059171199092

times tanh47099radic1199102 + 059171199092

119879

]]

]

(53)


1with



1


2)


1

119892119873 (0) [1198922

119897]= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)119896

119861119879 + 1

times

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

]]

]

(54)


1

10038161003816100381610038161198921003816100381610038161003816119873 (0) [119892

2

119897]

= int

119910=1

119910=0

119889119910

2radic1199102 + 059171199092

times[[

[

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

minus1

exp 942radic1199102 + 059171199092119879 + 1

times

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

+ tanh3623 (8 minus 13radic1199102 + 059171199092)

119879

]]

]

(55)



3O

7minus119909system

S no Temperature(K)

Δ1= 119909 times 10

minus21 J(when only


Δ2= 119909 times 10



Δ = Δ1+Δ

2(119909times10


interactions)

1 5 24869 0114015 2600965

2 10 24869 0114015 2600965

3 15 24867 0114015 2600795

4 20 24865 0114015 2600515

5 25 24838 0114015 2597865

6 30 24761 0114015 2590145

7 35 24600 0114015 2574075

8 40 24325 0114015 2546595

9 45 23914 0114015 2505415

10 50 23340 0114001 2448001

11 55 22580 0113991 2371991

12 60 21608 0113970 2274770

13 65 20382 0113914 2152114

14 70 18851 0113823 1998923

15 75 16932 0113683 1806883

16 80 14457 0113468 1559168

17 81 13868 0113425 1500225

18 82 13242 0113370 1437570

19 83 12571 0113300 1370450

20 84 11852 0113235 1298435

21 85 11060 0113177 1219177

22 86 10190 0113100 1132100

23 87 09228 0113030 1035830

24 879 08245 0112970 0937470


2) with




1+ Δ

2)




Δ1+ Δ


shown in Figure 1



120576119896= 120585

2+ Δ

212


119873 (0) = 05eV ℎ120596119863asymp 13 times 10

minus21 J

ℎ120596119863

2119870119861119879

= 47099T

(56)


0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)



2Cu

3O

7minus119909



times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)







S no Temperature(K)


Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000



119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





10038161003816100381610038161198921003816100381610038161003816 119873 (0) [119892

2

119897]

= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+Δ2 (1198961015840)2119896

119861119879 + 1

times

tanhℓ120596

0+radic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minusradic120585

2

11989610158401015840+Δ2 (1198961015840)

2119896119861119879

]]

]

(36)





119888119890119904

=120597

120597119879[

1

119873sum

119896

2120585119896⟨119862

+

119896119862119896⟩] (37)

where ⟨119862+



119888119890119904

=119873 (0)

2119873int

ℎ120596119863

0

2120585119896

1198961198611198792

119889120585119896

times [minus1

2120585119896sec ℎ2 (

120576119896

2119896119861119879)

+

infin

sum

119897=1

1198922119897

119897minus

1

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 + 120576

1198961015840)

times sec ℎ2 (1205960119897 + 120576

1198961015840

2119896119861119879

)

+1198991198961015840

2

1205851198961015840

1205761198961015840

(1205960119897 minus 120576

1198961015840)

times sec ℎ2 (1205960119897 minus 120576

1198961015840

2119896119861119879

) ]

(38)




119873(120596) = lim 1

2120587[119866

11(119896 120596 + 119894120578) minus 119866

11(119896 120596 minus 119894120578)] (39)


lim120578rarr0

1

2120587[

1

120596 + 119894120578 minus 120596+

119899

minus1

120596 minus 119894120578 minus 120596+

119899

] = 119894120575 (120596 minus 120596+

119899) (40)


119896byminus120576


1199062

119896+ V2

119896=

1

2(1 +

120585119896

120576119896

) +1

2(1 minus

120585119896

120576119896

) = 1 (41)

one obtains

119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(42)



119865119904minus 119865

119873

119881= int

infin

0

119889119892(1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (43)


119865119904minus 119865

119873

119881= int

Δ

0

119889Δ119889

119889Δ(

1

10038161003816100381610038161198921003816100381610038161003816

2)Δ

2 (44)



Δ (119896) = 119892119873 (0) int

ℎ120596119863

0

[[

[

minusΔ (119896)

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

1

119892= 119873 (0) int

ℎ120596119863

0

[[

[

1

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

(45)


[119865119878119873

119881]phonon

= int

Δ(119896)

0

Δ2(119896) 119889Δ (119896)

119889

119889Δ (119896)

times[[

[

119873 (0) int

ℎ120596119863

0

1

2radic1205852

119896+ Δ2

(119896)


2

119896+ Δ2

(119896)

2119896119861119879

119889120585119896

]]

]

(46)

Since

1205762

119896= 120585

2

119896+ Δ

2(119896)

2120576119896119889120576

119896= 2Δ (119896) 119889Δ (119896)

(47)


2 [119865119878119873

119881] =

Δ2(119896)

119892minus 119873 (0)

times int

ℎ120596119863

0

2119889120585119896int

120576119896

120585119896

tanh(120573120576

119896

2) 119889120576

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

[ln119890

1205731205761198962(1 + 119890

minus120573120576119896)

1198901205731205851198962 (1 + 119890minus120573120585119896)] 119889120585

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

ln (119890120573(120576119896minus120585119896)2

) 119889120585119896

minus4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120576119896) 119889120585

119896

+4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120585119896) 119889120585

119896

(48)

2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(49)




119867119888= 8120587 (119865

119878minus 119865

119873)

12

(50)


119867119888= 8120587(

119873 (0) Δ2(119896)

4minus

4119873 (0)

120573

times119890minus120573Δ(119896)

4(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12))

12

(51)



2Cu

3O

7minus119909 using



2Cu

3O

7minus119909


2Cu

3O







3O

7minus119909



11986313 times 10


2)) 8 stateseV spin


28m3



minus31 kg

We have (35)

1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(52)


1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

119910=1

119910=0

119889119910[[

[

1

2radic1199102 + 059171199092

times tanh47099radic1199102 + 059171199092

119879

]]

]

(53)


1with



1


2)


1

119892119873 (0) [1198922

119897]= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)119896

119861119879 + 1

times

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

]]

]

(54)


1

10038161003816100381610038161198921003816100381610038161003816119873 (0) [119892

2

119897]

= int

119910=1

119910=0

119889119910

2radic1199102 + 059171199092

times[[

[

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

minus1

exp 942radic1199102 + 059171199092119879 + 1

times

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

+ tanh3623 (8 minus 13radic1199102 + 059171199092)

119879

]]

]

(55)



3O

7minus119909system

S no Temperature(K)

Δ1= 119909 times 10

minus21 J(when only


Δ2= 119909 times 10



Δ = Δ1+Δ

2(119909times10


interactions)

1 5 24869 0114015 2600965

2 10 24869 0114015 2600965

3 15 24867 0114015 2600795

4 20 24865 0114015 2600515

5 25 24838 0114015 2597865

6 30 24761 0114015 2590145

7 35 24600 0114015 2574075

8 40 24325 0114015 2546595

9 45 23914 0114015 2505415

10 50 23340 0114001 2448001

11 55 22580 0113991 2371991

12 60 21608 0113970 2274770

13 65 20382 0113914 2152114

14 70 18851 0113823 1998923

15 75 16932 0113683 1806883

16 80 14457 0113468 1559168

17 81 13868 0113425 1500225

18 82 13242 0113370 1437570

19 83 12571 0113300 1370450

20 84 11852 0113235 1298435

21 85 11060 0113177 1219177

22 86 10190 0113100 1132100

23 87 09228 0113030 1035830

24 879 08245 0112970 0937470


2) with




1+ Δ

2)




Δ1+ Δ


shown in Figure 1



120576119896= 120585

2+ Δ

212


119873 (0) = 05eV ℎ120596119863asymp 13 times 10

minus21 J

ℎ120596119863

2119870119861119879

= 47099T

(56)


0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)



2Cu

3O

7minus119909



times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)







S no Temperature(K)


Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000



119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Δ (119896) = 119892119873 (0) int

ℎ120596119863

0

[[

[

minusΔ (119896)

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

1

119892= 119873 (0) int

ℎ120596119863

0

[[

[

1

2radic1205852

119896+Δ2

(119896)

tanhradic120585

2

119896+Δ2

(119896)

2119896119861119879

]]

]

119889120585119896

(45)


[119865119878119873

119881]phonon

= int

Δ(119896)

0

Δ2(119896) 119889Δ (119896)

119889

119889Δ (119896)

times[[

[

119873 (0) int

ℎ120596119863

0

1

2radic1205852

119896+ Δ2

(119896)


2

119896+ Δ2

(119896)

2119896119861119879

119889120585119896

]]

]

(46)

Since

1205762

119896= 120585

2

119896+ Δ

2(119896)

2120576119896119889120576

119896= 2Δ (119896) 119889Δ (119896)

(47)


2 [119865119878119873

119881] =

Δ2(119896)

119892minus 119873 (0)

times int

ℎ120596119863

0

2119889120585119896int

120576119896

120585119896

tanh(120573120576

119896

2) 119889120576

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

[ln119890

1205731205761198962(1 + 119890

minus120573120576119896)

1198901205731205851198962 (1 + 119890minus120573120585119896)] 119889120585

119896

2 [119865119878119873

119881] =

Δ2(119896)

119892minus

4119873 (0)

120573

times int

ℎ120596119863

0

ln (119890120573(120576119896minus120585119896)2

) 119889120585119896

minus4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120576119896) 119889120585

119896

+4119873 (0)

120573int

ℎ120596119863

0

ln (1 + 119890minus120573120585119896) 119889120585

119896

(48)

2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(49)




119867119888= 8120587 (119865

119878minus 119865

119873)

12

(50)


119867119888= 8120587(

119873 (0) Δ2(119896)

4minus

4119873 (0)

120573

times119890minus120573Δ(119896)

4(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12))

12

(51)



2Cu

3O

7minus119909 using



2Cu

3O

7minus119909


2Cu

3O







3O

7minus119909



11986313 times 10


2)) 8 stateseV spin


28m3



minus31 kg

We have (35)

1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(52)


1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

119910=1

119910=0

119889119910[[

[

1

2radic1199102 + 059171199092

times tanh47099radic1199102 + 059171199092

119879

]]

]

(53)


1with



1


2)


1

119892119873 (0) [1198922

119897]= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)119896

119861119879 + 1

times

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

]]

]

(54)


1

10038161003816100381610038161198921003816100381610038161003816119873 (0) [119892

2

119897]

= int

119910=1

119910=0

119889119910

2radic1199102 + 059171199092

times[[

[

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

minus1

exp 942radic1199102 + 059171199092119879 + 1

times

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

+ tanh3623 (8 minus 13radic1199102 + 059171199092)

119879

]]

]

(55)



3O

7minus119909system

S no Temperature(K)

Δ1= 119909 times 10

minus21 J(when only


Δ2= 119909 times 10



Δ = Δ1+Δ

2(119909times10


interactions)

1 5 24869 0114015 2600965

2 10 24869 0114015 2600965

3 15 24867 0114015 2600795

4 20 24865 0114015 2600515

5 25 24838 0114015 2597865

6 30 24761 0114015 2590145

7 35 24600 0114015 2574075

8 40 24325 0114015 2546595

9 45 23914 0114015 2505415

10 50 23340 0114001 2448001

11 55 22580 0113991 2371991

12 60 21608 0113970 2274770

13 65 20382 0113914 2152114

14 70 18851 0113823 1998923

15 75 16932 0113683 1806883

16 80 14457 0113468 1559168

17 81 13868 0113425 1500225

18 82 13242 0113370 1437570

19 83 12571 0113300 1370450

20 84 11852 0113235 1298435

21 85 11060 0113177 1219177

22 86 10190 0113100 1132100

23 87 09228 0113030 1035830

24 879 08245 0112970 0937470


2) with




1+ Δ

2)




Δ1+ Δ


shown in Figure 1



120576119896= 120585

2+ Δ

212


119873 (0) = 05eV ℎ120596119863asymp 13 times 10

minus21 J

ℎ120596119863

2119870119861119879

= 47099T

(56)


0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)



2Cu

3O

7minus119909



times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)







S no Temperature(K)


Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000



119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





3O

7minus119909



11986313 times 10


2)) 8 stateseV spin


28m3



minus31 kg

We have (35)

1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

ℎ120596119863

0

119889120585119896

[[

[

1

2radic1205852

119896+ Δ2

(119896)

tanhradic120585

2

119896+ Δ2

(119896)

2119896119861119879

]]

]

(52)


1

10038161003816100381610038161198921003816100381610038161003816 119873 (0)

= int

119910=1

119910=0

119889119910[[

[

1

2radic1199102 + 059171199092

times tanh47099radic1199102 + 059171199092

119879

]]

]

(53)


1with



1


2)


1

119892119873 (0) [1198922

119897]= int

ℎ120596119863

0

1198891205851198961015840

2radic1205852

11989610158401015840+ Δ2 (1198961015840)

times[[

[

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

minus1

exp radic1205852

11989610158401015840+ Δ2 (1198961015840)119896

119861119879 + 1

times

tanhℓ120596

0+ radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

+ tanhℓ120596

0minus radic120585

2

11989610158401015840+ Δ2 (1198961015840)

2119896119861119879

]]

]

(54)


1

10038161003816100381610038161198921003816100381610038161003816119873 (0) [119892

2

119897]

= int

119910=1

119910=0

119889119910

2radic1199102 + 059171199092

times[[

[

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

minus1

exp 942radic1199102 + 059171199092119879 + 1

times

tanh3623 (8 + 13radic1199102 + 059171199092)

119879

+ tanh3623 (8 minus 13radic1199102 + 059171199092)

119879

]]

]

(55)



3O

7minus119909system

S no Temperature(K)

Δ1= 119909 times 10

minus21 J(when only


Δ2= 119909 times 10



Δ = Δ1+Δ

2(119909times10


interactions)

1 5 24869 0114015 2600965

2 10 24869 0114015 2600965

3 15 24867 0114015 2600795

4 20 24865 0114015 2600515

5 25 24838 0114015 2597865

6 30 24761 0114015 2590145

7 35 24600 0114015 2574075

8 40 24325 0114015 2546595

9 45 23914 0114015 2505415

10 50 23340 0114001 2448001

11 55 22580 0113991 2371991

12 60 21608 0113970 2274770

13 65 20382 0113914 2152114

14 70 18851 0113823 1998923

15 75 16932 0113683 1806883

16 80 14457 0113468 1559168

17 81 13868 0113425 1500225

18 82 13242 0113370 1437570

19 83 12571 0113300 1370450

20 84 11852 0113235 1298435

21 85 11060 0113177 1219177

22 86 10190 0113100 1132100

23 87 09228 0113030 1035830

24 879 08245 0112970 0937470


2) with




1+ Δ

2)




Δ1+ Δ


shown in Figure 1



120576119896= 120585

2+ Δ

212


119873 (0) = 05eV ℎ120596119863asymp 13 times 10

minus21 J

ℎ120596119863

2119870119861119879

= 47099T

(56)


0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)



2Cu

3O

7minus119909



times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)







S no Temperature(K)


Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000



119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





3O

7minus119909system

S no Temperature(K)

Δ1= 119909 times 10

minus21 J(when only


Δ2= 119909 times 10



Δ = Δ1+Δ

2(119909times10


interactions)

1 5 24869 0114015 2600965

2 10 24869 0114015 2600965

3 15 24867 0114015 2600795

4 20 24865 0114015 2600515

5 25 24838 0114015 2597865

6 30 24761 0114015 2590145

7 35 24600 0114015 2574075

8 40 24325 0114015 2546595

9 45 23914 0114015 2505415

10 50 23340 0114001 2448001

11 55 22580 0113991 2371991

12 60 21608 0113970 2274770

13 65 20382 0113914 2152114

14 70 18851 0113823 1998923

15 75 16932 0113683 1806883

16 80 14457 0113468 1559168

17 81 13868 0113425 1500225

18 82 13242 0113370 1437570

19 83 12571 0113300 1370450

20 84 11852 0113235 1298435

21 85 11060 0113177 1219177

22 86 10190 0113100 1132100

23 87 09228 0113030 1035830

24 879 08245 0112970 0937470


2) with




1+ Δ

2)




Δ1+ Δ


shown in Figure 1



120576119896= 120585

2+ Δ

212


119873 (0) = 05eV ℎ120596119863asymp 13 times 10

minus21 J

ℎ120596119863

2119870119861119879

= 47099T

(56)


0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)



2Cu

3O

7minus119909



times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)







S no Temperature(K)


Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000



119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0010203040506070809

1111213141516171819

221222324252627

0 10 20 30 40 50 60 70 80 90

Supe

rcon

duct

ing

orde

r par

amet

er

Temperature (K)



2Cu

3O

7minus119909



times [int

119910=1

119910=0

1199102

1198792119889119910 minus 13(sec ℎ119876)

2

minus119878

119875(sec ℎ119871)2 + 1

119890119876 + 1

119878

119875(sec ℎ119871)2

minus1

119890119876 + 1

119872

119875(sec ℎ119877)2]

(57)

where

119875 = [1199102+ 05917119909

2]12

119876 =47099

119879119875

119878 = 8 + 13119875

119872 = 8 minus 13119875

119871 =3623

119879119878

119877 =3623

119879119872

(58)







S no Temperature(K)


Joulemole-K

1 879 1275236

2 87 125044

3 86 1224406

4 85 1199085

5 84 1174477

6 83 1150882

7 82 1127512

8 81 1104441

9 80 1081521

10 75 9697717

11 70 8596727

12 65 7492361

13 60 6385744

14 55 5285128

15 50 4201019

16 45 3158173

17 40 2192602

18 35 1350073

19 30 6891036

20 25 2565851

21 20 0536428

22 15 0033761

23 10 0000000

24 5 0000000



119873(120596)

119873 (0)=

1

119873

119897=infin

sum

119897=1

1198922119897

119897sum

1198961015840

119894120596119899minus 119897120596

0

(119894120596119899minus 119897120596

0)2

minus Δ212

minus119894120596

119899+ 119897120596

0

(119894120596119899+ 119897120596

0)2

minus Δ212

(59)


020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




020406080

100120140

0 10 20 30 40 50 60 70 80 90 100Temperature (K)

minus20

Elec

troni

c spe

cific

hea

t (C

es)

Ces


2Cu

3O

7minus119909

134

136

138

14

142

144

146

79 80 81 82 83 84 85 86 87 88 89Temperature

Ces

T



0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Temperature (K)


(CminusC

es)T



3O

7minus119909system

S no Temperature(K)


(Theory)


(Experimental)1 80 13510 135202 81 13635 135913 82 13749 136504 83 13866 137325 84 13980 138036 85 14106 138517 86 14236 139458 87 14372 140639 879 14508 14170


3O

7minus119909system

S no Temperature(K)

(119862 minus 119862es)119879

Joulemole-K2

(Theory)

(119862 minus 119862es)119879

Joulemole-K2

(Experimental)1 4 32943048 85422 5 314759423 8333 6 304582297 7924 7 299338593 7255 8 296793776 70836 9 295322975 68307 10 29398128 66708 11 292235587 66709 12 289786723 667010 13 286485315 667011 14 282270186 667012 15 277163961 6670


120596119899= 119910 times 10


1205960= 8 times 10

minus21 J 1198922= 1

(60)


119873(120596)

119873 (0)=

(119910 minus 8)


minus(119910 + 8)

radic(119910 + 8)2

minus 1199092

(61)




3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





3O

7minus119909system


119879 = 50K119909 = 2334

119879 = 70K119909 = 18851

119879 = 85K119909 = 1106

1 115 047791324 039120977 0212929844 0064854772 120 0303783239 0256874065 0149461031 0048095733 125 0214803743 0184567166 011137659 0037080424 130 0161399908 0139994146 00864163 0029433235 135 0126234662 0110167093 00690518 0023900966 140 0101617339 0089063119 0056435285 0019767377 145 0083611432 0073510888 0046958665 0016597348 150 0069997004 0061686732 003965028 0014113439 155 0059430763 0052471393 0033891537 00121316510 160 0051054964 0045142493 0029271859 00105259411 165 0044298094 0039214703 0025509277 00092074912 170 0038765958 0034351053 0022404542 00081122613 175 0034178738 0030311067 0019813414 0007193114 180 0030332928 0026919081 0017629272 000641463

0

005

01

015

02

025

03

035

04

045

05

115 125 135 145 155 165 175 185

Den

sity

of st

ates

T = 30KT = 50K

T = 70KT = 85K

Frequency (120596)


2Cu

3O

7minus119909


2[119865119878119873

119881]phonon

= (minus) [119873 (0) Δ

2(119896)

2minus

4119873 (0)

120573

times119890minus120573Δ(119896)

2(2120587Δ (119896) 120573

minus1)12

+4119873 (0)

120573

1

120573(1205872

12)]

(62)

01

012

014

016

018

02

022

0 20 40 60 80 100

Free

ener

gy

Temperature (K)

F


2Cu

3O

7minus119909

Solving numerically

2[119865119878119873

119881]phonon

= (minus) 0024751199092


119909]

12

119890minus7246119909119879

+12517 times 10minus4[119879

119909]

2

]

(63)




3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





3O

7minus119909system

S no Temperature(K)


1 5 016822 10 017053 15 017444 20 017985 25 018646 30 019377 35 020138 40 020859 45 0214710 50 0219411 55 0221912 60 0221513 65 0217414 70 0208615 75 0193816 80 0170917 85 0137518 86 0129419 87 0120920 879 01129


given as

119867119888= 8120587(

119873 (0) Δ2(119870)

4minus

4119873 (0)

120573


4(2120587Δ (K)

120573)

12

+4119873 (0)

120573

1

120573(1205872

12))

12

(64)





2Cu

3O




3O

7minus119909system

S no Temperature(K)


Tesla1 5 14540922 10 1464103 15 14805374 20 15032055 25 15305136 30 15604747 35 15906748 40 16187629 45 164276110 50 166053911 55 166998012 60 166869813 65 165326914 70 161942915 75 156071516 80 146567617 81 144068518 82 141342219 83 138355720 84 135101321 85 131480022 86 127519323 87 123257324 879 1191492

12

125

13

135

14

145

15

155

16

165

17

0 10 20 30 40 50 60 70 80 90 100Temperature

Criti

cal fi

eld

(Hc)


2Cu

3O

7minus119909



2Cu

3O

7minus119909is






119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119862cuprates



References






























2Cu

3O

119909rdquo Physica




2Cu

3O




2Cu

3O


no 22 pp 14498ndash14506 1998






FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article polaronic mechanism of superconductivity...

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