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Volume 2013, Issue No.8, August 16, 2013, Time: 4h36m.PM.

TOPOLOGICAL RANGADHAMA QUANTA

HIGH-Tc SUPERCONDUCTIVITY

by

PROFESSOR Dr. KOTCHERLAKOTA LAKSHMI NARAYANA,

General Physics Labs, Shivaji University, Kolhapur-416004. and

{Retd. Prof. of Physics, SU, Kolhapur}, 17-11-10, Narasimha Ashram, Official

Colony, Maharanipeta.P.O, Visakhapatnam-530002 cell no: 9491902867

Key words: High-Tc, Force Fields, Entropy, Topological Quanta, 

   dressed Ragnons, Renormalized phonon frequency, Rangadhama Effect.

A B S T R A C T

                         A new formulation to explain the High-Tc super-Conductivity is proposed. It emphasizes the role of force field modifications due to defects, dopants etc lattice disorders and perturbations. Entropy, spinorial polarization states, excitation of modes of vibrations typical of a Superconducting state are suggested, to give rise to the spin 0, charged Rangadhama dressed Quasiparticles, and quanta as that which leads to High-Tc and lower Fermi Energy E_F of Superconducting phenomena. At finite temperature long-range order is destroyed by topological soliton. This paper outlines the concept of dressed Ragnons a virtually new concept to account for the manifest 0 and charge ±1 spins that play a role in accounting for the Plasmon-like behavior.

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TOPOLOGICAL RANGADHAMA QUANTA: 

HIGH-Tc SUPERCONDUCTIVITY

KOTCHERLAKOTA L. NARAYANA,

General Physics Labs, Shivaji University, Kolhapur-416004.

{Retd. Prof. of Physics, SU, Kolhapur}, 17-11-10, Narasimha Ashram, Official

Colony, Maharanipeta.P.O, Visakhapatnam-530002 cell no: 9491902867

Key words: High-Tc, Force Fields, Entropy, Topological Quanta, dressed Ragnons,

                            Renormalized phonon frequency, Rangadhama Effect.

INTRODUCTION

          

                  High temperature superconductivity in the pseudo-tetraphenol

and distorted perovskites is well established, in X₂ Y Z₂Cu₃ 0 _8-x

(where   X= Bi….,    Y= Ca, Ti…….,    Al…...,    Z= Sr, Ba…...). Rich

literature on aspects of photon and solitons excitonic mechanisms

as to the origin of the high-Tc in these materials is available and more evidence is continuously being generated in to present the details of how electron-phonon and electron-soliton coupling mechanisms play main roles in the systems appropriate with both soft phonons and high frequency phonon mechanisms. Electron-Electron and Excitonic interactions would then being surmised as only to enhance the phonon induced Tc.

Renormalized Phonon Frequency         

 In this work, I report a possible quantum and semi classical formalism to the explanation of occurrence of High-Tc. The Model of Kivelson, Rokshor and Sethna (1987) revives the Anderson (1987) concept of resonating-valence bond (RVS) state of quantum liquid of valence bonds. Pairs of valence bonds in a large density of next nearest-neighborhood bond pairs, can resonate between horizontal and vertical configurations, with effective tunnel splitting J _res. Out-of-plane buckling of intermediate oxygen atoms are presumed to be expressed in the realization of superconducting state and as well RVB state of valence bonds. Given U on-site electron repulsive energy, α-the electron–phonon coupling. 

M mass of Cu,

                   J _res= ω* exp [- A (t² /U) / ђ ω *]

A≈1,    ω* =   √(α²/KU)   √(K/M);   

 K-force constant of hopping Electron matrix element is the renormalized phonon frequency. Doping can stabilize RVB state where

                               J^’ _res = J_res  χ, 

                 where  χ reflects the soliton density.

               The doping helps to create charged soliton by combining the added electrons or holes that bind to the free spins essentially the dangling bonds, created in pairs by breaking a bond and that would act as free particles.  Quasi-particles are realized from the statistics of many body wave functions, by considering transformation of the wave function under the exchange of two solitons as

         Ψ (Q, R) = [Φ (Q-Qo) Φ(R – Ro) ± Φ (Q – Ro) Φ(R – Qo)] /√2   

where Q, R are quasiparticle co-ordinates, and Qo, Ro the soliton localized near points in an External potential.  Features of the theory are (1) Binding Energy of Elementary Bosons is set by electronic energy

                                                -2 to 2/ U 

rather than by Debye frequency. 

(2) Effective mass of Bosons can be very small, 

                      M* = M (v/a)² approximately

                      

                      u= t_o / (6α), α= 3eV/R

                    t_o= 0.5eV and a= 3.79 Å yields M*/M ≈ 5x 10 ^-5.

                  

                Electron-phonon interaction are used to stabilize RVB state, which on a bipartite lattice has a topological long range order, with both spin and charge excitations. Secondly electronic excitations have charge statistics and changes spin relations. (Neutral spin ½ fermions and charge ±e spinless bosons). At finite temperature long-range order is destroyed by topological soliton. Thirdly charged soliton of the charge defect have 

                     spin 0 and charge ±1, 

  with   size extending over several sites.

CHEMICAL PICTURE          

                  

                   In the chemical picture, the substantial addition of large divalent atoms, results by charge compensation in replacing  Cu²⁺ in,

                      La ₂ Cu O₄  by  La _2-x M _x [ Cu _x Cu_1-x] O₄

with    x   Cu ³⁺     and      1-x Cu ²⁺ ions. Thus charge variations (fluctuations) or mixed valence are a natural feature of the ground state. Their coupling to the charge fluctuations induced by breathing mode phonon appears to be strong and important as per the model by Fu and Freeman (1987). For High-Tc these charge fluctuations are cited as a possible mechanism and are therefore found to be resonantly enhanced by the response, of electron to the lattice distortion.  The phonon mode, involving the motion of oxygen atoms, against the directional bonding ( Cu (d  _x²_-y² ) – O(1) P_{x ; y}) is expected to a large restoring force and high frequency, the best candidate being the breathing mode which induces two inequivalent Cu sites in the Cu-O plane. The effect of this on state A which is of type

            

                            [ Cu ((d  _x²_-y² ) – O P_{x ; y}) ]

causes interstice change fluctuations between Cu (I) and Cu (II). For state B (which has a larger inter layer component) not only in plane but also out-of-plane Cu d_z² - O(2) p_z orbital's) the interactions produce ‘resonant’   type change fluctuations.

          

                In addition, a resonance type enhancement of ground state. Cu²⁺  ±  Cu³⁺ charge fluctuations in La _2-x M _x Cu O₄  by the high frequency oxygen breathing mode, and hence enhancement of Tc is possible. The doping of divalent materials lowers the E_F and leads to a maximum Tc as E_F coincides with energy of state B. Thus the high frequency mode (high in plane Debye temperature Θ_D) and large electron-phonon interaction energy contributes to the observed high-Tc.

          Third paper essentially represents the study of the role of topological R- quanta (proposed by the author 1982) to observe the High-Tc.  The model envisages spin polarization states of different electronic structural equilibrium geometries, such as for example, Sr₂ Cu₃ O _8-x with N formula units of elements or complex structures, with quantization condition for the spinorial polarization given by 

                                             no   = Np²

where p is of the order of the area of force field ellipse of vibration modes of the formula units. The generated Topological R-quanta cannot simply be regarded as analogues to neutral solitons of Anderson (1987) and Kivelson, Rokshan and Sethna (1987). R-quanta arise due to 

          1) The ligand perturbation 

          2) Co-ordination changes(conformal or configurational) changes 

          3) Covalent bonding   

          4) Ionicity   

          5) Electronegativity 

          6) structural deformations 

          7) pressure of external stresses and strains 

          8) Lone pair electron contributions 

          9) Isotopic changes etc. in a collective fashion.

         

The main difference between the topological R-quanta and the neutral solitons lies not only in their mode of origin but as well on the role they play to form the midgap split states as in microelectronic circuits’ materials, or to account for High – Tc superconducting properties. [K.L.Narayana 1990, Ref.No.13].

R-quanta states

In broader terms we have ascertained previously that the shift in the square of vibration frequencies is determined by an order parameter  

       < S³ (T) > at a given  R-quanta states characterized by

              

                        Δa/b = 1/ [R_p τ √(n_o)]

where R_p τ is a dimensional coupling constant 0.167x 10^-3.

Dynamical characterization involves that the topological mode frequency ω²_R  is given by

                   ω²_R  -  ω²₀ = 2 * Vo * N < σ^z > =   < S³ (T) >.    

        For a frequency of 700 cm^-1  to  500 cm^-1 we may get Tc  as 419.65 ^oK  to 359.7 ^oK considerably higher than the room temperature with N= 10¹² .

                   

             Hence we arrive at the fact that the coupling constant R_p for the ferroelectric and superconductivity states are drastically different. The Green’s function is given by    

Gss= 2*ђω_o 3

(T) >/ [ђ(ω²_R  -  ω²₀) ]= 2*V_o*N < σ^z > ђω_o

                   

 with  V = V_o*N = 4.864 cm^-2  oC.

If      (ω²_R  -  ω²) = (Tc  - T) ~ < σ^z > V_o * N 

then T is determined by the ratio of effective mass energy relative to thermal energy times the coupling constant. 

          Our estimate revealed S of the order of 40.179 mole^-1 0K that accounts of a relation 

                               S= Constant * R_p, 

        where R_p is the Rangadhama Coupling constant. 

                 

            The constant is a characteristic of the material chosen for study, since it involves the factor (2s+1) where s is the spinorial polarization of the Superconducting material.

Conclusions

Present model is superior in the sense that we directly involve the spinorial states in the formulation. Electron lattice (distorted) interactions are used to stabilize the superconducting state, that on a force field order, with both spin and change excitations, involves the

generation of Rangadhama Quanta [Ref. Nos. 5 to 14]. At finite temperature the force field order gets modified by the R-quanta that, in turn, dress the spinorial and charge excitations.  The dressed Ragnons (analogous to Plasamons, but distinctly different) reflect spin 0 and charge ±1, with their optimum extending over several force field orders of magnitudes.

From the chemical picture view point, our model envisages modifications of force fields due to charge fluctuations and has induced by typical modes of vibrations (not necessarily breathing modes) that may respond, yet times resonantly with the lattice distortion electronic excitations.[Ref. No. 15-17 deal with topological excitations a forethought by the present author].

Apart from oxygen atom motion, our model succeeds to account for the typical motions of other atoms as well. The best candidate to go with the conventional superconducting models is of course, the breathing mode giving rise to the two inequivalent Cu sites. Our model success, partly lies in the fact that we have incorporated entropy consideration that lowers the Fermi Energy etc. and leads to High-Tc.

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A special reference

K. L. Narayana, “Spinorial Optics of Structural Vibrations of Ions (Atoms) and

   the Quasi-Particle Quantization of Rangadhama Effect”,

        Ind. Sci. Cong. Part III, 70^th Session Sri Venkateswara University,

       Tirupati, 3-8^th January, Paper No. 147, Chemistry Section, IV, 1983.

Note: Prof. K. R. Rao D.Sc. (Madras) D.Sc. (London) was the architect and founder

of the Sri Venkateswara University in 1954 at Tirupathi, when present author

was with him for three months before joining the Intermediate Course

in Visakhapatnam under the auspicious Chief Minister of AP, Prakasam Pantulu.

The drafts he prepared and the style of creation and establishment of that

University, follows his earlier experience with Andhra University, striving

the funds for it from Jeypore Vikram Dev Raja of Orissa, in 1932, after

Rao’s return from England with a D.Sc. (London). The credit of Higher

Education in Andhra Pradesh, devoid of England authority, rests solely with

Prof. K. R. Rao.

REFERENCES

1). S. A. Kivelson, D. S. Rokshar, J. P. Sethna,

      Phys. Rev. Vol.5, No.16, p.8865, 1987.

2). P. W. Anderson, Science, Vol.235, p.1196, 1987.

3). J. Ruvalds, Physical Review Vol.B35, No.16, p.8869, 1987.

4). C. L. Fu, A. J. Freeman, Phys. Rev.Vol.B35, No.16, p.8861, 1987.

5). K. L. Narayana, “The Spin Polarization of Molecular vibrations and Peram

      Manifold of XY₄ type metallic complexes (molecules)”, 69^th of Ind. Sci. Cong.

      Manasagangotri, 6th January, Paper No. 74, 1982 and also Math. Section,

      Paper No.54, January, 1982.

6). K. L. Narayana, “Spinorial Optics of Structural Vibrations of Ions (Atoms) and

      the Quasi-Particle Quantization of Rangadhama Effect”, Ind. Sci. Cong. Part III

      70^th Session Sri Venkateswara University, Tirupati, 3-8^th January, Paper No. 147,

      Chemistry Section, IV, 1983.

7). K.L.Narayana, “Renormalization of Vibrational Energy by Rangadhama Quanta and

      Spinorial Phase transition “, Ranchi University, Session of Physics,

       Ind. Sci. Congress, 5^th January, 1984.

 8). K.L.Narayana, “Topological Quantization of Rangadhama Quanta and differential geometric

      description of Spinorial Polarization of vibration of complex atoms (ion) structures”,

       “Algebraic Topological Structures International Symposium”, S. N. Bose Institute,

         of Physical Sciences, Calcutta, December, 1983.

9). K.L.Narayana, Paper No.0-33, Proc. Nat. Symposium on TSL phenomenon,

       PRL, Ahmadabad, 1984.”

       Rangadhama Relaxation time τ decides the dependence of density of final states at on

       energy above the levels within that duration.

10). K. L. Narayana,”Optical micrographic study and IR-pyro-electric-characteristics of a

        new series of Ferroelectric ceramic compounds”, Chem. Soc. (Lucknow) 72^nd Session

        Ind. Sci. Cong. January, 1984 and also 6^th CIMTEC, Faenza, world Congress on High Tech.

        Ceramics VI, Paper No.PB-110, June 23-28, Milan, Italy, 1986.

11). K. L. Narayana, M. Inst. P. (Lond), “Peram Manifold, Topological Windings and

        Rangadhama Quanta of Vibration Radicals of certain molecular complexes”,

        Shivaji University,   Kolhapur.

12). K. L. Narayana, “Novel techniques of Potential distributed spinorial (Rangadhama)

        Optical quanta ratio thermometry”, Paper No.45, (Sect.III), Page.26,

        74th Session of Ind. Sci. Cong, Bangalore, Mathematics section, 1987.

13). K. L. Narayana, 90^th Ind. Sci. Cong. Session, Physics section, University of Cochin,

        Cochin, January 1990.

14). K. L. Narayana,”Topological Quantization of Rangadhama Effect and Differential

        geometric description of Peram Manifold and Spinorial Polarization of Vibrations

        of Atoms (ion) complexes”, Algebraic Topological Structures International

        Symposium, S. N. Bose Institute of Physical Sciences, Calcutta, December 1983.

15). K. L. Narayana, “Topological Helicity Formulation and a new Elucidation of

        Heisenberg Uncertainty Principle”, Paper No.20, Proc. 73rd session of

        Ind. Sci. Cong. University of Delhi, Physics Section, 1986.

16). K. L.Narayana,”Topological Quanta Energy transduction Mechanism and Organic

       Superconductivity”, 32^nd ISTAM, Indian Institute of Technology, Powai, I.I.T.

       Bombay December 17^th, 1987.

17). Narayana L. Kotcherlakota,” Polymer Topological Excitations and the Novel

       Micro-Electronic Devices”, National Symposium on High Polymers and

        Coordination Polymers”, Paper No. HP-9, Nagpur University, February 26-28^th,

        1989.

----------------------------------------------------------------------------------------------------------------------Address:

Prof. Dr. Kotcherlakota Lakshmi Narayana, (Retd. Prof of Physics, SU, Kolhapur-416004),

17-11-10, Narasimha Ashram, Official Colony, Maharanipeta.P.O,

Visakhapatnam-530002.

Andhra Pradesh. Cell No. +919491902867.

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