Production of EM Surface Waves by Superconducting Spheres: A New Type of Harmonic Oscillators

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1 Pogess In Electomagnetics Reseach Symposium Poceedings, Moscow, Russia, August 19 23, Poduction of EM Suface Waves by Supeconducting Sphees: A New Type of Hamonic Oscillatos A. Meessen Institute of Physics, Catholic Univesity of Louvain, Louvain-la-Neuve 1348, Belgium Abstact It is shown that a supeconducting sphee o spheical shell can poduce EM suface waves that ae stationay at any low fequency. This applies in paticula to magnetic dipole oscillations, geneated by a cuent density J that oscilllates aound a given axis on the suface of the sphee. In the quasistatic appoximation, it ceates a synchonously oscillating magnetic field B, while the esulting electic field E povides feedback to sustain the cuent. This yields a new type of oscillatos, whee the magnetic enegy is not completely tansfomed into electic enegy as in classical LC cicuits. EM enegy is conseved, howeve, by means of enegy fluxes. The pupose of this theoy is to account fo evidence of vey intense magnetic dipole fields, poduced by unconventional flying objects of unknown oigin. These obsevations would make sense and suggest that supeconductivity is possible at nomal atmospheic tempeatues. We also examine the popagation of plasma waves along supeconducting sufaces and conside possible paiing mechanisms. 1. INTRODUCTION This eseach was motivated by the fact that the populsion system of Unconventional Flying Objects (UFOs) can be explained when we assume that they ae able to poduce vey intense dipola magnetic fields, oscillating at low fequencies [1]. We want thus to find out how these fields could be poduced and if they have a finite ange as well as othe special popeties. Thee is also an intinsic, physical eason. Indeed, EM waves wee initially poduced as suface waves, since Heinich Hetz tied to veify the validity of Maxwell s theoy by coupling a long metal wie to an LC cicuit [2, 3]. The HF oscillating cuent, suounded by oscillating electic and magnetic fields, was eflected at the end of the wie. This should poduce a standing wave patten and allow fo an indiect measuement of the velocity c, but the EM suface wave was also eflected by laboatoy walls. This petubed the expeiment so much, that Hetz pefeed to use an oscillating electic dipole, emitting feely popagated EM waves. Since Leche found that two paallel wies poduce much moe concentated EM suface waves, he could ealize the initially pojected measuement [4]. Consideable effot was then devoted to the theoetical undestanding of wave guides, fomed by a single wie [5] and two paallel wies of finite conductivity [6]. Zenneck [7] consideed the popagation of EM suface waves along an infinite plane suface, sepaating ai fom a mateial of given conductivity σ and dielectic constant ε. This poblem was also teated by othe authos [8 10]. EM suface waves wee then consideed fo thin metal films, to account fo chaacteistic electon enegy losses [11] and optical esonance absoption [12]. These effects esult fom the ceation of suface plasmons. Hee, we conside stationay EM suface waves fo a supeconducting sphee and EM waves that popagate along a supeconducting plane. Since UFOs ae topologically equivalent to a sphee, we will adopt this model to concentate on essential featues. The natue of the suface mateial is unknown, but it is sufficient that it contains a high density of electon pais to be a supeconducto, obeying known physical laws. To evaluate the intinsic popeties of this system, we assume hee that the outside medium contains no fee chages. 2. BASIC EQUATIONS The quasistatic appoximation is valid fo electic fields E and magnetic fields B that oscillate at exta-low fequencies (ELF) in a potion of space that is small compaed to the coesponding wavelength. Retadation effects can then be neglected, as if c wee infinite. Fo nonmagnetic and electically neutal media, Maxwell s equations ae thus educed to B = µ o J and B = 0 (1) E = t B and E = 0 (2)

2 530 PIERS Poceedings, Moscow, Russia, August 19 23, 2012 J is the cuent density, acting as a souce. We can also use the vecto potential A, since B = A and E = t A (3) A = µ o J and A = 0 (4) We will solve these equations inside and outside a supeconducting sphee o spheical shell of adius R. The outside medium is unionized ai, whee the cuent density J = 0, but the supeconducting mateial contains fee electon pais (of mass 2 m and chage 2e). Being bosons, they emain in thei gound state of momentum p = 2(mv ea) = 0. When n is the density of electons in the supeconducting state, the cuent density J = env, since thee ae n/2 electon pais of chage 2e. Thus, µ o J = α 2 A whee α 2 = µ o ne 2 /m = µ o s (5) Combining (5) with (4), we get two equations fo A alone: A = α 2 A with A = 0 (6) They ae valid outside and inside the sphee, whee α is eithe zeo o vey geat. This paamete is essential fo the cuent density J, which is not only the souce of E and B, but depends also itself on these fields. Howeve, Ohm s law is not valid anymoe inside a supeconducto. It has to be eplaced by the London equations t J = se and J = sb (7) Since electon pais cannot leave the state p = 0, thei equation of motion would be 2m t v = 2eE when B = 0. The second Equation (7) accounts fo the Meissne effect and the finite penetation depth of magnetic fields inside supeconductos. We ae actually consideing a macoscopic quantum state with a athe stable wavefunction. 3. MAGNETIC DIPOLE OSCILLATION Now, we solve Equation (6) to detemine the vecto potential A and the esulting fields E and B when the cuent density J is oscillating at fequency ω on the suface of the supeconducting sphee aound a given z-axis. Using spheical coodinates (, θ, ϕ), we set J = (0, 0, J) cos ωt and A = (0, 0, A) cos ωt (8) Because of (5), µ o J = α 2 A. The second Equation (6) equies that A = A(, θ) and (3) yields E = ω(0, 0, A) sin ωt and B = (B, B θ, 0) cos ωt (9) with B = 1 sin θ θ(sin θa) and B θ = (A) (10) At the poles, J = A = 0. The simplest solution of the fist Equation (6) coesponds thus to A = u() sin θ whee u 2 2 u = α2 u Outside the sphee, α = 0, so that u() deceases like 1/ fo inceasing distances fom the cente of the sphee. Inside the sphee, we could use Bessel functions fo u(), but it is sufficient to conside an exponential decease towads the inside of the sphee (fo R and αr 1). Distinguishing solutions outside and inside the sphee by subscipts + (when > R) and (when < R), we get A + = M 2 sin θ and A = M R eα( R) sin θ (11) This accounts fo the continuity of the tangential component of E, defined by (9). Because of (10), the components of the magnetic field B ae B = 2u() 2 cos θ and B θ = u () sin θ

3 Pogess In Electomagnetics Reseach Symposium Poceedings, Moscow, Russia, August 19 23, Thus, B + = 2M 3 cos θ while B = 2M R 2 eα( R) cos θ (12) B + θ = M 3 sin θ while B θ = αm R eα( R) sin θ (13) Outside the sphee, we get a pefect magnetic dipole field, but inside the supeconducting sphee, the fields E and B decease vey apidly, since the penetation depth 1/α is small compaed to R. The adial component of B is continuous at the suface, but the tangential component is not. This equies a suface cuent density J s = (0, 0, J s ) cos ωt, whee µ o J s = B + θ B θ = α M sin θ (fo αr 1) (14) R2 Inside the sphee, the volume cuent density (8) is also paallel to the suface, but has the opposite sign and fo = R, its magnitude is α times geate than (14). The quasi-infinite suface cuent density J s accounts fo the Meissne effect o pefect diamagnetism. We could also conside othe multipole oscillations, but magnetic dipole oscillations ae sufficient to become awae of emakable facts. 4. ENERGY CONSERVATION AT ANY LOW FREQUENCY LC cicuits ae based on a complete convesion of magnetic enegy into electic enegy and vicevesa. These enegies ae espectively associated with the magnetic field B nea the cuent caying coil and the electic field E inside the chaged condenso. Howeve, thee ae no condensos that could pevent beakdown fo extemely intense electic fields, while EM suface waves aound a spheical supeconducto yield non segegated electic and magnetic fields. Moeove, thee is no eigenfequency as fo LC cicuits. Any low fequency ω is possible and it is not necessay to convet the whole magnetic enegy into electic enegy, although we have to conside wok done by the electic field E. Since it acts on electon pais inside the supeconducto, the dissipated powe pe unit volume is P = J E = JE sin ωt cos ωt It vaies like sin 2ωt, but (1) and (2) lead to a special fom of Poynting s theoem in the quasistatic appoximation: P = S t U whee S = (E B)/µ o and U = B 2 /2µ o S defines the enegy flux, which vaies also like sin 2ωt. The magnetic enegy density U vaies like cos 2 ωt, but its time deivative is popotional to sin 2ωt. Fo adiation in fee space, we would have to conside also the electic enegy ε o E 2 /2, whee ε o = 1/µ 0 c 2, which is negligible when c is quasi-infinite and thee ae no static chages. With the pevious notations, enegy consevation would thus equie that µ o JE = 1 2 ( 2 EB θ ) + 1 sin θ θ(sin θeb θ ) ω(b 2 + B 2 θ ) Setting J = j() sin θ, B = F () cos θ, B = G() sin θ and E = H() cos θ, whee j, F, G and H ae defined by (8), (9), (11), (12), (13) and (14), the enegy consevation would be insued if µ o jh = 1 2 ( 2 HG) + 3 HF + ω(f 2 G 2 ) and 2 2 HF = ωf This is easily veified outside the supeconducto, whee j = 0, but is also tue inside the supeconducto, whee all these functions ae exponentially deceasing towads the cente of the sphee, with αr 1. To account fo the suface cuent density J s we enclose an element of unit suface at the inteface between two infinitely close paallel sufaces. Inside this laye, the dissipated powe is P s = J s E, while the magnetic enegy is zeo. Howeve, the enegy flux S has a discontinuous adial component: S ± = 1 µ o EB ± θ so that J s E = (S + S ) This is equivalent to the definition (14) of J s. The total enegy is always and eveywhee pefectly conseved, but not only because of the absence of esistive and adiative enegy losses. It is also due to the quasistatic appoximation, allowing fo enegy fluxes.

4 532 PIERS Poceedings, Moscow, Russia, August 19 23, PROPAGATION OF SURFACE PLASMA WAVES Fo a moe complete exploation of this matte, we conside also EM waves that ae popagating along the suface of a supeconducto. Even a small potion of a lage spheical suface can be teated like a plane, when the wavelength is small compaed to the adius of this sphee. Using Catesian coodinates, whee the x-axis is nomal to this suface, situated at x = 0, we conside an EM wave that popagates along the y-axis: A = (A x, A x, 0)e i(ky ωt) and B = (0, 0, B)e i(ky ωt) while J = sa and E = iωa. The divegence and the cul of A yield x A x + ika y = 0 and B = x A y ika x We set A x = a ± u ± (x), A y = au ± (x) and B = b ± u ± (x), whee the + and signs do coespond to x > 0 and to x < 0 (outside and inside the supeconducto). This accounts fo the fact that the tangential component of E has to be continuous, while the amplitude of the oscillations deceases exponentially towads the inside and the outside of the supeconducting mateial: u (x) = e βx and u + (x) = e γx, whee γa + = βa = ika, γb + = (k 2 γ 2 )a and βb = (k 2 β 2 )a The puely tangential magnetic field is continuous, when k 2 = βγ and b ± = (β γ)a. No suface cuent density is equied, but the fist Equation (6) becomes A cta 2 = α 2 A. This yields the dispesion elation (ω/c) 2 = k 2 γ 2 and β 2 = α 2 γ 2. We can set γ 0, β α and ω ck. The suface wave is thus nealy popagating at the velocity c and extending fa outside the supeconducto (a 0), but the nomal component of the electic field is discontinuous at the inteface. This yields oscillating suface chage densities, chaacteistic of suface plasma waves. 6. THE PAIRING MECHANISM Poduction of low fequency stationay EM suface waves should be veifiable by means of low tempeatue supeconductos. Howeve, this theoy applies also to Unconventional Flying Objects of unknown oigin, since thee is evidence that they poduce vey intense magnetic fields, oscillating at exta-low fequencies [13]. These facts seem to imply that supeconductivity is possible at atmospheic tempeatue and even highe ones. That would be of temendous theoetical and pactical impotance, but equies the existence of a yet unknown mechanism, gluing two electons togethe with a paiing enegy kt c, whee T c is the highe tansition tempeatue. Any electon epels othe electons and attacts positive ion coes inside a solid. We know that this leads to Debye-Hückel sceening, but othe pocesses ae also possible. Since ionic motions ae slow, they ceate a wake of positive chage that can attact anothe electon. This was the basic idea of the BCS theoy, whee electon paiing was attibuted to an exchange of vitual phonons. That accounts fo conventional low-tempeatue supeconductivity, but not fo high- T c supeconductivity (HTS), whee the tansition tempeatue T c is of the ode of 100 K. Even 20 yeas afte its discovey, one could say that the physics behind this stange state of matte emains a mystey [14]. Many ideas wee poposed and expeiments yielded so many supizing esults that we can expect the unexpected [15]. This justifies even the seach fo nomal tempeatue supeconductivity. To elucidate the paiing mechanism fo HTS, Dal Conte et al. measued the elaxation times in one type of cupate supeconductos fo vey shot pulses of optical excitation [16]. They concluded that electon-phonon inteactions contibute much less to the glue than collective electonic excitations, such as spin fluctuations fo instance. Gademaie et al. [17] pefomed simila measuements fo pnictides, cupates and bismuthates, but they concluded that thei T c depends mainly on the stength of electon-phonon inteactions. They added even that the expeimental esults ae only consistent with bipolaonic paiing. A polaon is an electon that is accompanied by mobile lattice distosions, coesponding to a cloud of vitual phonons. Optical phonons have highe enegies than acoustic phonons and allow fo the fomation of small bipolaons [18], whee two electons ae bound to one anothe by means of local polaization waves. Today, the theoy of supeconducting bipolaons is well developed [19, 20]. Although it is quite complex, the basic ideas can be explained in tems a simple model [21]. These ideas wee also illustated by the impotance of the lage polaizability of As anions in Fe-based supeconductos [22], involving bound electons.

5 Pogess In Electomagnetics Reseach Symposium Poceedings, Moscow, Russia, August 19 23, Nomal tempeatue supeconductvity (NTS) seems to equie anothe paiing mechanism. Thee ae aleady popositions fo oom-tempeatue supeconductivity [23, 24]. Since layeed stuctues ae essential fo known high-t c supeconductos and since the pesent model fo the poduction of vey stong low fequency magnetic fields equies only supeconductivity fo the oute suface of UFOs, it is quite pobable that it involves the Electodynamics of Inhomogeneous Media and Gadient Metamateials. Moeove, the dielectic constant of fee electons and electon pais tends towad, when the fequency becomes vey small, which has also an effect on image foces. Anyway, the theoetical esults pesented hee, combined with empiical data [1, 13], seem to justify futhe eseach of mateials that allow fo supeconductivity at nomal atmospheic tempeatues and even highe ones. 7. CONCLUSIONS We found that it is possible to geneate vey intense, low fequency EM suface waves by means of supeconducting sphees and that such a system has emakable popeties. It is thus impotant to examine evidence that Unconventional Flying Objects of unknown oigin do poduce magnetic dipole oscillations of this type. It suggests that supeconductivity is possible at nomal tempeatue and even highe ones, which justifies the seach of a new paiing mechanism of electons. It could involve vitual plasmons instead of vitual phonons, but we don t know the mateial that constitutes the extenal suface of UFOs. Since the type of supeconductivity we ae consideing has only to exist nea thei oute suface, it may involve suface effects. Anyway, it appeas that an objective study of the UFO phenomenon without peconceptions o beliefs is necessay and useful. It aises questions of a new type, which is always impotant fo basic and applied science. It could stimulate the seach of nomal tempeatue supeconductivity and motivate the conception of new Gaded Metamateials o some othe physical pocess. REFERENCES 1. Meessen, A., Pulsed EM populsion of unconventional flying objects, Accepted by PIERS, Hetz, H., Ann. Physik, Vol. 270, 155, Hetz, H., Electical Waves, 1893, Dove, Leche, E., Phil. Mag., Vol. 30, 128, Sommefeld, A., Ann. Phys., Vol. 67, 233, Mie, G., Ann. Phys., Vol. 6, 201, Zenneck, J., Ann. Phys., Vol. 23, 846, Fank, P. and R. V. Mises, Die Diffeential- und Integalgleichungen de Mechanik und Physik, Vol. 2, , , Vieweg, Epstein, P. S., Poc. Nat. Ac. Sc., Vol. 40, 1158, Stum, K., Z. Physik, Vol. 209, 329, Ritchie, R. H., Phys. Rev., Vol. 106, 874, Steinmann, W., Phys. Stat. Sol., Vol. 28, 437, Meessen, A., Evidence of vey stong low fequency magnetic fields, Accepted by PIERS, Timusk, T., Physics Wold, 31 35, July Cove Stoy, Natue Physics, Vol. 2, Mach Dal Conte, S., et al., Science, 335, , Gademaie, C., et al., AXive: Alexandov, A. S., Phys. Rev. B, Vol. 38, , Alexandov, A. S., AXive: Deveese, J. and A. S. Alexandov, AXive: Komyushin, Y., AXive: Beciu, M., et al., AXive: , Phys. Rev. B, Vol. 78, , Mouachkine, A., Room-tempeatue Supeconductivity, Cambidge Intl. Sc. Publication, Pickett, W. E., AXive: