TOPOLOGICAL RANGADHAMA QUANTA AND HIGH–Tc SUPERCONDUCTUIVITY by KLN

Friday, August 16, 2013

trusciencetrutechnology@blogspot.com

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

TOPOLOGICAL RANGADHAMA QUANTA AND

 HIGH–Tc SUPERCONDUCTUIVITY

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

Keywords: High-Tc, Force Fields, Entropy, Topological Quanta.

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 Quasi-particles, and quanta as that which leads to High-Tc and lower Fermi Energy E_F Superconducting phenomenon.

Friday, August 16, 2013

trusciencetrutechnology@blogspot.com

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.

INTRODUCTION

                  

                  High temperature Superconductivity in the pseudo-tetraphenol and distorted perovskites is well established, in X₂ Y Z₂ Cu₃ O_8-x (where X= Bi…, Y=Ca, Ti,…..,  Z= Sr, Ba….). Rich literature on aspects of photon and soliton 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 mechanism 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.

                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 Anderson1987 concept of resonating-valence-bond (RVB) state of quantum liquid of valence bonds. Pairs of valence bonds, in a large density of next neatest-neighbor 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 suppressed in the realization of superconducting state and as well RVB state is to be lower that the spin-peierls states( crystalline arrangements) of valence bonds. Given U on-site electron repulsive energy, α- the electron-phonon coupling, M mass of Cu,

                   J _res  = ω* e ^{-A (t^2/U) / (ћ ω*),}

A≃1, ω*= √(α² / (KU) ) √K/M;  K-force constant to hopping electron matrix element, is the renormalized phonon frequency. Doping can stabilize RVB state where J’ _res =   J^x _res, x reflecting the soliton density.

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

Ψ(Q,R) = [ Φ(Q-Q₀) . Φ(R-R₀) ±Φ (Q-R₀) . Φ(R-Q₀)]/√2

where Q, R are quasi-particle coordinates and Q₀ and R₀ the soliton localized near points in an external potential.

Features of the theory are 

(i)             Binding Energy of elementary bosons is set by electronic energy – 2 to 2/U rather than by Debye frequency

(ii)           Effective mass of bosons can be very small, MB*= M (u/a)² approximately  u= t₀/(6α) ; α= 3eV/R; t₀ = 0.5eV; and a=3.79Å yields M*/M =5E-05.

Electron-phonon interactions are used to stabilize RVB state that on a bipartite lattice has a topological long range order, with both spin and charge excitations. Secondly electronic excitations have charge statistics and charge spin relations. (neutral spin ½ fermions and charge ±e spin less bosons. At finite temperature long-range order is destroyed by topological solitons. Thirdly, charged soliton or the charge defect have spin 0 and charge ±1, with their size extending over several sites.

In the chemical picture, the substitutional addition of large divalent atoms, results by charge compensation in replacing Cu²⁺ in   La₂ Cu O₄ by La_2-x M_x [ Cu _x .Cu _1-x] O4 with x Cu ³⁺ and (1-x) Cu²⁺ ions.  Thus charge variation (fluctuations) or mixed valence, are a natural feature of the ground sate. Their coupling to the charge fluctuations induced by breathing mode phonon appears to be a 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 electrons to the lattice distortion. The phonon mode, involving, the motion of oxygen atoms, against the directional bonding ( Cu ( d_x2-y2 -  O(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_{- y}²  -  O(p_x,y)] causes intersite change fluctuations between Cu (I) and Cu (II). For site R ( which has a larger inter layer component) not only in-plane but also out-of-plane Cu (d_z²  - O(2)_pz) orbital’s) the interactions produce ‘resonant’ type charge fluctuations.

   In addition, to a resonance type enhancement of ground state, Cu ²⁺ ± Cu ³⁺ charge fluctuations in La_2-x M_x CuO₄ 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 contribute to the observed High- Tc.

This 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 n_o= Np² where p is of the order of the area of force field ellipse of vibrational mode of the formula units. The generated topological R-quanta cannot simply be regarded as analogous to neutral solitons of Anderson 1987, and Kivelson, Rokshan & Sethna 1987.

 R-quanta arise due to 

(i)              The ligand perturbation.

(ii)           The co-ordination changes (conformal or configurational changes).

(iii)        Covalent bonding.

(iv)         Ionicity.

(v)            Electronegativity.

(vi)         Structural deformations.

(vii)      Pressure of external stresses and strains.

(viii)   Lone-Pair electron contributions.

(ix)         Isotopic changes etc.,

in a collective fashion.   

          The main difference between the topological R-quanta and 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).

        In broader terms we have ascertained previously that the shift of the vibrational frequencies is determined by an order parameter 3

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

Δa/b = ( 1/ { R_p τ √n₀ }) 

where R_pτ is a dimensionless coupling constant 0.167E-03. Dynamical characterization involves that the topological mode frequency ω²_R is given by

                           ω²_R   -  ω²₀ = 2 Vo N < σ^z >; =  3

(T)> 

                  For a frequency of 700 cm^-1 to 500 cm^-1 we may get 
Tc = 419.65⁰K to 359.7⁰K considerably higher than the room temperature with N= 10¹².

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

Gss = [2ћω₀ < S³(T) >]/[ћ(ω²_R -  ω²)] = [2Vo N <σ^z> ћω₀]/(ω²_R   -  ω²)

with V  = V₀N = 4.864 cm^{-2 0}C.

        If ω²_R -  ω²  ≃  (Tc - T ) ≃ Vo N <σ^z>,   

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 0}K that accounts for 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.

        Present model is superior in the sense that we directly involve the spinorial state s in the formulation of Electron lattice (distorted) interactions are used to stabilize the superconducting state, that on a force field order, with both spin and charge excitations, involves the generation of Rangadhama Quanta. 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 Plasmons 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. 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 considerations that lowers the Fermi Energy etc., and leads to a High-Tc.

ACKNOWELDGEMENT

                        The author is deeply indebted to Late Prof. K. R. Rao, D. Sc. (Madras) D.Sc. (London) of Andhra University, Waltair to work at his internationally famous Laboratories (years 1932-1972) and discussions with several people those days intimately and respectfully devoted to him.

REFERENCES

1.S.A. Kivelson, D.S. Rokshar, J.P. Sethan

   Phys Rev. Vol.B35,No.16, p.8865, 1987.

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

3. J. Ruvalds, Phys. Rev. Vol.B35, No.16, p.8869, 1987.

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

5. (*180.) K. L. Narayana,”Optical Micrographic Study and IR-Pyroelectric

         Characteristics of a New Series of Ferroelectric Ceramic Compounds”,

        Paper No. PB-110, 6^th CIMTEC World Congress on High Tech Ceramics,

         June 23-28, Milan, Italy, 1986.

6. (√102.) K. L. Narayana,”The spin polarization of Molecular Vibrations and Peram

        Manifold of XY₄ type Metallic Complexes (Molecules)”, Paper No.74, 69^th Sess. of

        Ind. Sci. Cong. Section.IV, Chemistry Section,p.171 Manasagangotri, 5^th Jan 1982.

7. (√106.) K. L. Narayana, Kolhapur, ”Spinorial Optics of structural vibrations of

        Ions(atoms) and The Quasi-particle Quantization of Rangadhama Effect”,

        Paper No.147, Section IV, Chemistry Section, p.65-66, 70^th Proc. Ind. Sci. Cong.

        Part III, Shri Venkateswara University, 6^th January 1983.

8.  (√*169.) K. L. Narayana, “New Semiconductor device Physics in LLDPE

        Micro-electronic Transistors”, Cochin University, Indian Science Congress,

         February 1990.

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.

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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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