Humblonium: Classical Atoms and the Earnshaw Plasma
HHumblonium: Classical Atoms and the Earnshaw Plasma
Clifford ChafinOctober 19, 2018
Abstract
It is shown that electrostatic and diamagnetic forces can combine to give long lastingmetastable bound dimers of macro and mesoscopically sized objects for a physicallyattainable material regime. This can be a large enough effect to support itself in a trapagainst Earth’s gravity and they can stable at very high temperatures. For a morerestricted material parameter set, we investigate the possibility of stable many particlecollections that lose their identity as bound pairs and create a kind of plasma. Thesewould constitute a kind of transitional state between fluids and granular materials but,unlike usual approaches, the fluid is a gas rather than a liquid.
The stability of matter was a puzzle of the nineteenth century. It was already believed thatcollections of positive and negative charges were combined to give net neutral matter insome integer ratios in the form of small constituents. It was also assumed that Newton’slaws reigned at all scales. Indeed it was Newton’s triumph to show that the laws of thecosmos were the same as on Earth so that extension to smaller scales was believed. Aproblem arose when Earnshaw showed that there was no stable way to arrange charges ormagnets against collapse [4]. The ultimate resolution of this had to await the 1920’s andquantum mechanics when the oscillations of the electron wavefunction generated enoughquantum pressure to provide stability to planets, stones and atoms.There are dynamic and active approaches to magnetic levitation in the form of theLevitron, magnetic bearings and the like. To achieve static levitation there is a workaroundin the form of diamagnetism. The very tiny diamagnetic forces most notably, in graphiteand bismuth, can provide enough force for a permanent magnet to levitate small objects inEarth’s gravity. Superconductors expel all magnetic field so, while they are not diamagneticper se, they can generate strong levitating forces. In the case of Type II superconductors,flux pinning can cause this levitation to be stable. Once again, stable support of bodiesrequires a quantum effect. However, unlike the case of bulk matter, the quantum fieldsneed not fill the void between the objects and the stabilization is mediated only by the1 a r X i v : . [ phy s i c s . a t m - c l u s ] S e p lectromagnetic fields themselves. It is known that no analog to diamagnetism exists fordielectric response [5] so magnetic methods are essential.It would be quite interesting if we could combine electric and magnetic fields to givebound states of positive and negative pairs that exist as static states of macroscopicallysized bodies. These would be separated enough to be noninteracting in a thermal sense(except by radiation) so that the thermal motions of the bodies would not be damped bythe internal atomic motions of the bodies as happens with granular liquids. Additionally,there would be no delocalization of the bodies necessary to create this support. As such,these entities would be the closest approximation to classical atoms. The practical uses ofthese would depend on what ranges of forces could be manifested, the lifetime of them as(presumably) metastable entities, and if they could exist in large scale collections to definea fluid or granularly obstructed material with no rapid damping in collisions and a possiblytunable angle of repose.The purpose of this article is to demonstrate that such pairs are possible and that thereis some reason to believe that reasonably stable large collections are as well. In the caseof pairs we dub them “Humblonium” in homage to their lowly status as classical boundstates with no important quantum character in the regions between the constituents. Wewill demonstrate that micron sized magnetite and superconducting spheres with sustain-able bulk charge can give bound pairs strong enough to support themselves against gravity.Some estimates for collections are considered. Unlike true atoms and molecules, the distinc-tion of bound pairs or other subcollections are immediately lost leading to an “Earnshawplasma” of mesoscopic particles of uncertain lifetime. Consider the case of two equal sized spheres of radius R as in fig. 1. One of them is a strongferromagnet with surface field B S and opposite charges ± q = ± ne . The second sphere is aType I superconductor which excludes all fields. The charges are assumed to be localizedin domains so they cannot move freely within the balls in response to external fields.For these to give a bound state we need the magnetic repulsion to provide enoughclose repulsion to overcome the electrostatic attraction and the induced ion-induced dipoleattraction and the work function to remove the electrons be more than the potential energyto bring the electrons in from infinity. For these to be of interest we also desire therepulsion to be stronger than the gravitational forces that would pull them downwards ina trap. Finally, we would like the bound states to be deep enough that they can existat interesting temperatures. If the bound states are several times the radii in separation, r ≫ R , then we can approximate the magnetic repulsion by volume of excluded magneticenergy from the presence of the superconducting sphere. This separation also facilitates asimilar approximation of the ion-induced dipole energy.2igure 1: Electrostatically bound magnet-superconductor pair.The conditions we seek to satisfy can then be expressed in the following equations U H + U E ≈ ⎛⎝ πR µ ( B S R r ) ⎞⎠ − π(cid:15) n e r stationary (1)14 π(cid:15) ne R < U Work (2)0 ≈ F E + F g = − π(cid:15) n e r + ρ πR g (3) U H + U E ≪ k B T room (4) U induce dipole ≪ U H (5)Setting the binding energy of the last electron equal to the work function, extremizingthe binding energy and choosing the electric repulsion to repulsion to be equal to the forceof gravity we find: R = ( (cid:15) µ g ρ e U B S ) / (6) r = ( (cid:15) U B S µ e g ρ ) / (7) n = π ( (cid:15) µ U g ρ e B S ) / (8)(9)3onsider the case of a magnetite sphere with ρ = , (cid:15) m ≈
50, and a surfacefield of B S ≈ U = g = . R = . µ m, r = . µ m, and n = r / R = . r ≫ R is valid. The net binding energy is U net = . T b = . × K. Taking U induce dipole ≈ (cid:15) m (cid:15) πR E ( r ) (10)we see U induce dipole = . U H = ∼ χ ≈ × − we can compute a new radius R = . µ m, separation r = µ m and charge number n = It is now natural to ask what happens when we combine such pairs. Unlike the case of realatoms where covalent bonds can exist with large barrier potentials to create new bonds,large collections of such dimers must give more generally coordinated collections with noclear distinction of which pair is bound. The magnetite particles have the same sign chargeso have a long range repulsion. Similarly for the superconducting spheres. A concern hereis that the magnets can flip to be antiparallel and, at short ranges the magnetic attractionbetween them can overwhelm the electrostatic repulsion. This reminds us that such anensemble is, at best, metastable. Indeed, one should be concerned that a pair of magnetscould be drawn together with a superconducting sphere attenuating the net electrostaticrepulsion leading to a kind of three-body collapse reminiscent of the losses in cold gastraps. When the magnets and superconducting spheres are close, the form of the repulsionchanges so that this may provide some brake on such a process.To obtain a binding likely to be stable enough in the case of many particles we needadequate repulsion to keep the magnetic spheres apart at close range so that the electro-static repulsion is never overcome by oppositely oriented poles. This distance must be onthe order of the particle radii themselves so that multipolar fields can be set up in thesuperconductors strong enough to keep multiple such magnetic spheres from binding. Thiscan be estimated as above assuming r ≈ R and allowing F E ≫ F g so that R = . n = .
6. Unfortunately, this does not give a large enough radius to be larger than4he coherence length for any Type I superconductor. In the case ceramic superconductorsthere are coherence lengths this small but the London penetration depth tends to be muchlarger. It might be that some such material could be effective as a superconductor on suchscales to give a strong repulsive force and make the plasma stable. Since it is unclear ifsuch an Earnshaw plasma of clusters can exist stably for any particular material, we give afew properties that would characterize such a collection but do not develop it extensively.These pairs of “atoms” can have a broad range of internal excitations so there is no reason-able chance of any two being “identical.” The quantum pressure they would exert wouldbe negligible in any case.We can compute the pressure of the plasma in our above example at T = R wecan assume that the density is one eighth that of the solid material and the pressure isdetermined by the energy density of the fields between them. The “temperature” of sucha gas can be excited by jiggling from mechanical or field induced means. Due to the poorcoupling of the translational motions and the internal atomic motions such a dampingwill take a very long time. This allows us to obtain very high temperatures with thermalvelocities that are very small. In our previous example v th ≈ T b ∼ K. Forspherical bodies, the transfer of linear to angular momentum motion can be quite slow. Ifwe introduce angular bodies, this can be enhanced and lead to a specific heat of c V = k B T .There is a collective binding between the particles so we expect analogs to surface tensionand vapor pressure. It is rare to find thermodynamic systems that have dynamics that are slow enough to betracked and observed. Some may argue diffusion of large particles qualify but they aredriven by small motions where this is not possible. The existence of such classically boundpairs, and possibly plasma, strangely seems to be a novel realization given the advancesthat have taken place in the quantum world. Even more interesting is that such pairs seemto be actually makable and fairly easily so.Potential uses of such pairs range from mesoscopic probes of quantum-classical tran-sitions to granular-fluid transitions and as imageable slow thermodynamic systems thatcan be driven far from equilibrium. Hopefully, the arguments here will convince someoneskilled to make a few and see what their real possibilities are. Should these be stableand resilient as suggested there are a number of ways they can be extended to richer ob-jects. By virtue of these being classical objects, we could consider modifying the spheresto become hinged and flexibly joined structures. This allows a controlled introduction ofinternal degrees of freedom. Interestingly, these bodies can all be viewed as distinguishableyet thermodynamics based on kinetics almost certainly exists and no Gibbs paradox seemsrelevant. 5ne might naturally wonder if we could utilize the diamagnetic response to bind anatom to a charged superconducting cluster in a kind of mesoscopic analog of the Penningtrap. It seems that the electrostatic force is too strong even for a single electron’s worthof charge to make a stable system this way. In the quantum regime there are a largezoo of unusual bound states with diffuse electron clouds as in Rydberg matter and dipolebound anions. Hyperfine interactions in dilute gases can be tuned magnetically to giveEfimov states and a crossover transition between BCS and BEC behavior [1]. Geonium [2]is a kind of simple atom where one “half” of it is the earth itself and the binding force isgravity. Here we have a classical collection where the delocalization time is very long. Thisdoes not mean that it has to be effectively infinite. The quantum-classical transition is ofgreat interest due to interest in quantum computing and the stability of such complicatedentangled collections. Such small particles in intense isolation could slowly delocalize. Onecould also consider manufacturing them from a delocalized gas in a trap capable of holdingboth the gas and the transitional larger clusters forming the solid despite the impulses ofphoton emission and evaporation. If the ejected particles were not allowed to collapse theresulting cluster should remain in a delocalized state. Macroscopic superposition effectsthen could be measured and provide more fuel to the subject of quantum measurement.
References [1] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, S. Stringari, “Theory of Bose-Einstein con-densation in trapped gases”, Rev. Mod. Phys., , 3, 463, (1999).[2] L. S. Brown, and G. Gabriels, “Geonium theory: Physics of a single electron or ion ina Penning trap”, Rev. Mod. Phys. , 233, (1986).[3] R. Driver, “An amazing invention, and a patent failure”, Providence Journal, Septem-ber 22, (1999).[4] S. Earnshaw, ”On the Nature of the Molecular Forces which Regulate the Constitutionof the Luminiferous Ether”. Trans. Camb. Phil. Soc.
97, (1842).[5] L. Landau and E.M. Lifshitz,