Confinement Induced Electron Capture

3p̃µ pµ m2e f f m2e Aµ AµIt has been suggested that the electron capture rate could beenhanced by a factor of 2 3.5 using laser confinement8 . Theypropose that an intense, long wavelength laser could compressthe electronic wavefunction thereby acting like a photon-box,increasing the electron density at the origin, and thereby doubling or tripling the rate of E.C.They derive an effective mass for an atomic electron usinga perturbative treatment of non-relativistic reduction of Diracequation (c e 1), givingcreate a new electron; the Fermi sea is full. So electron capture can occur by bare protons, but, presumably, only underextreme confinement, and with the reverse reaction is suppressed. And there are many studies of numerical rate calculation in high thermal, stellar environments.A complete calculation of the E.C. rate requires numericalintegration of the integrated capture amplitude including fourmomentum integrals for all reaction particles13 . That is, oneneed to consider the full expression for the rate (in terms ofthe LIPS)ΓEC Zd3 p p(2π)3Zd 3 pe(2π)3Zd 3 pn(2π)3Zd3 pν X 2 M (2π)3 spin2Ψe (0) m3e f fwhere the effective mass is proportional to the laser intensity I and inversely proportional to the energy Eγsm3e f f m2e 32παIEγ2The experiment would require a laser intensity of I 1010 Wmm 2 and an energy Eγ 10 3 10 4 eV, or a wavelength of λ 0.01µm (much larger than an Angstrom). δ(E p Ee En Eν )δ3 (p p pe pn pν )We have found no treatment of confined electron captureusing the complete numerical methods like those employed indense astrophysical environments.In this paper, we model the enhanced electron capture process as a particle-in-a-box, but applied to the Femi VA theory,and using the full numerical integration of the full relativisticexpression for the integrated capture amplitude.B.5. Bare Electron Proton Capture and StellarNucleosynthesisAt zero energy, bare proton electron capture is not possiblebecause it violates energy-momentum conservation. Theoretically, a free proton could capture an electron from the continuum, but the interaction energy must be above the threshold for neutron production. This is a huge amount of energy,although this happens regularly in accelerators, and, presumably, in stellar environments. Observing such capture outsideof an accelerator would be an incredibly hard experiment because both final particles are neutral, and the neutrino is extremely weakly interacting.Bare capture is thought to occur in stellar nucleosynthesisbecause the environment is very dense, and the system is inthermal equilibrium. This drives the formation of primordialelements, and occur when neutron stars form. At very hightemperatures, the proton electron collisions have sufficient energy to overcome the reaction barrier. For example, Bahcalland coworkers famously computed the electron capture rateof 7 Be in the Sun11,12 . It took nearly 40 years to resolve thesecalculations, resulting in the 2002 Nobel prize in physics. Itis believed that ionized Hydrogen captures an electron duringthe core collapse supernovae and in neutron stars [14].While we usually characterize a star by it’s temperature,these are also very dense systems, with ρ 106 g cm 3 .In contrast, crudely, the smallest star has density ρ 102 103 g cm 3 . As important, the reverse reaction is prohibited because, inside the dense neutron star, it is impossible toThe Weak Interaction and Femi VA theoryElectron capture is mediated by the Weak Interaction,described most concisely by the Fermi VA (Vector Axial)theory.2,3,14 The VA theory is a simple phenomenological approach, readily amenable to numerical calculations. It is nowunderstood in terms of ElectroWeak Unification and can bederived from the Standard Model.The original paper by Fermi, for which he won the 1938Nobel Prize in Physics, was initially rejected by Nature becauseIt contained speculations too remote from reality to be ofinterest to the reader.15VA theory can be used to compute cross sections for scattering experiments and decay rates for electron capture forvarious atoms, even in different environments, chemical andotherwise. We can use machinery of the VA theory to exploreE.C. in a simple, idealized environment. To properly describeany reaction, however, we need to understand what reactionswe can apply the theory to, and the other, potential competingreactions.1. Electron Capture and other Semi-Leptonic WeakprocessesThe Weak Interaction describes several related, semileptonic processes (those involving both leptons and hadrons)within a single framework16 , including: orbital electron capturep e n0 νe

4 positon emission (β decay) p n0 e νe β decayn0 p e ν eThere are also several related reactions, including reverse electron capturen0 νe p e free neutron decayn0 p e ν e inverse Beta decayp ν e n0 e Let us briefly review these.2.Beta decayBeta (β , or just β) decay is the most familiar Weak process,and is discussed in great detail in numerous articles and texts.In contrast, electron capture, which is can be significantly difficult to describe in detail, is a very rare topic. Indeed, themost recent review is from 1976.23.Positron emissionIn any high energy relativisitic process, there is the possibility of positron emission. As noted above, however, in 7 Be,the competing positron decay reaction can not occur becausethere is not enough energy. Also, positron emission occursat length scales below the (reduced) Compton length of theelectron, which, is smaller than we will need to consider.4.Neutron DecayBy detailed balance, reverse electron capture has the samerate as orbital electron capture–but is more favorable energetically. Indeed, inside the nucleus, the neutron is relatively stable. Free neutron decay has mean lifetime of τ 881.5 1.5 sec, or about 15 minutes.In contrast, orbital electron capture by a free proton isunspoken-of outside of a stellar environments. Even if thebare reaction could proceed, the reverse reaction would stilldominate unless it is suppressed or is kinetically unfavorable.5.Inverse Beta decayElectron capture is also sometimes called inverse β decay,but, here, we mean this to be the scattering of a proton and anelectron anti-neutrino ν̄e . It is characterized by emission of apositron e .C.Higher order corrections1.Orbital effectsHighly accurate rate calculations must treat the electronicstructure of the initial and final electronic states, and it isstrongly affected by their overlap. But when Hydrogen isstrongly confined, the atomic electron effectively detachesfrom the proton, and effectively behaves like a particle-in-abox (depending on the shape of the box).17 So we do not needto consider atomic orbital effects here, and this greatly simplifies the analysis.2.Radiative Electron CaptureThe Weak Interaction, as presented here, does not includehigher order QED contributions. There are 2 dominant effects:positron emission and internal Bremsstrahlung.5In particular, in very rare cases, a gamma ray photon isemitted with the neutrino; this is called Radiative ElectronCapture (REC).5,18–20 This can be thought of as a kind of Internal Bremsstrahlung (or so-called braking) radiation, causedby the electron accelerating toward the nucleus during capture, taking energy away from outbound neutrino.4 It traditionally has been treated as a second order QED correction tothe VA theory.5 REC is 1000X less likely, but does occur. Theresulting gamma (γ) rays are called soft because they do notexhibit sharp spectral lines. Recent, detailed rate calculationshave elucidated the quantum mechanical details.20II.CONFINEMENT INDUCED ELECTRON CAPTUREWe pose the following Gedankenexperiment: We imaginesome protons embedded in a lattice, such that we can say eachbare proton is confined in a Fermi sea of electrons. We modelthis as a classical particle-in-a-box, with volume L3 . We further imagine that the box is transiently compressed, as in Figure 2, such that the box size L is just small enough to ’induce’electron capture.We write this asEbox p e n0 νewhere Ebox represents a confinement energy, which is induced by the box constraints.After the electron capture event, the box contains a coldneutron, with very little kinetic energy and a very small meanfree path. Also, as depicted in Figure 3, the box now wantsto expand because the ’walls’, which are effectively a Fermisea of electrons, are repelling each other and the stabilizingpositive charge is gone.At this point, shown in Figure 4, we imagine the box haseffectively expanded, and the neutron can capture a nearbyproton, forming deuterium, and releasing of 2.2MeV of energy:

5FIG. 2. Confinement Induced Electron Capture: beforeFIG. 4. Confinement Induced Electron Capture: afterthat employ the full machinery of the Weak Interaction, as anillustrative exercise.Before we describe the theoretical approach, we address afew conceptual issues in setting up the problem.1.Compton lengthThe first obvious question is, should we use a classical or arelativisitic box?Most electron capture rate calculations use ab initio classical wavefunctions,2,3 perhaps with some relativisitic corrections to the electronic Hamiltonian.21We argue that we can safely use a classical box as long1as Lmin 2πλe , where λe is Compton wavelength of an22–24electron.The Compton wavelength sets the scale, accounting for both quantum mechanics and special relativity.λe he2 me c me c2FIG. 3. Confinement Induced Electron Capture: afterλe 2.426 10 12 m n p d 2.2MeV0We would like to understand if such a process is reasonablypossible and how to model both the rate of electron captureand the maximum excess power such a process might produce.Here, we examine what do detailed calculations look likeNote, however, for relativisitic calculations, one uses thereduced Compton wavelengthoe .λe 0.4 10 12 m2π

6Thus, for an electron, the minimum L is on the order of0.004 AngstromLmin 0.004 ÅIn any high energy, relativisitic system, positrons can beproduced; here it is through β -decay. This generally occursat or below the reduced Compton length. We are seeking themaximum box size which can induce electron capture, and weassume that, at the maximum, positron emission will be veryrare.If the box is extremely small, and the energy of the electronis of the order of a W boson, we are no longer in the lowmomentum limit and one would need to consider higher ordercorrections to the Weak interaction.25We also assume that the electron wavefunction does notchange appreciably during the interaction, so that we may usea very simplified form for the cross section (σ) and rate (Γ).Again, this is reasonable for boxes L λe .2.FIG. 5. EC particle production processboxes sizes, and it is a general phenomena of confined relativistic particles. Recent experiments on Graphene have reopened the debate27 . Still, we will assume the traditional interpretation and we will ignore the Klein paradox.So we use a classical box, with minimum size Lmin 0.004 Å. We compute the maximum size below.Confined Atomic systems5.Usually electron capture is described using the atomic orbitals of the parent and daughter nuclei.2,3 This is very complicated and the basic physical insight can be lost in the detailsof various angular momentum selection rules and orbital overlap calculations. Moreover, when an H atom is confined, theelectron will detach from the proton and behave like a freeparticle in a box.173.Pair ProductionWe do recognize, however, that in typical cage or other confined environment, there would be significant thermal effectsthat may induce positron emission. Indeed, the electron energy needs to be very fine-tuned in order to allow capture butnot pair creation. In the box with multiple electrons, any finetuning would be overcome by electron-electron collisions,which would induce energy fluctuations that could drive pairproduction. For example, at high density, assuming a thermaldistribution, if there are 800 keV electrons, there would besome 1 MeV electrons, which are enough to cause significantpair creation, probably faster than electron capture.Here, we assume that the confinement is a highly transient,far-from-equilibrium process, and that we can ignore pair production from both thermal noise and electron-electron scattering.4.Neutron post-reactionTo prevent the reverse reaction, we assume that, in the expanded box, the free neutron subsequently combines with another proton, and gives us 2.2 MeV of energy in the process.This post-process contributes to the power output. Realistically, we expect this to happen at the maximum box size, atthe energy threshold, where the outbound neutron has extraordinary low momentum and therefore a very small mean freepath.Still, for illustrative purposes, we will compute the poweroutput, assuming this post-reaction, at all box sizes.III.THEORY AND CALCULATIONSThe electron capture rate can be computed using the FermiVA theory,2,3 .A.Particle production under the Weak InteractionElectron capture is mediated by the Weak Interaction,through the particle production process, given by the 4-pointInteraction (see Figure 5). It requires at least 782 KeV energyto overcome the reaction barrier, which, here, is provided bythe box.Klein Paradox782KeV p e n0 νeKlein noted that a relativistic (Dirac) particle-in-a-box willleak out at box sizes near, or above, the Compton wavelength–this called the Klein paradox.25,26 And while this is usuallytaught as being simply particle-antiparticle creation, it hasbeen suggested that the Klein paradox can occur even at largerWe want to compute the rate of E.C. and the (minimum)power generated, as a function of the box size (L), using thefull Weak Interaction Hamiltonian. We will compute the electronic density classically, using box wavefunctions, and, for

7each box size, determine the incident velocity (as the momentum). We then compute the cross section using the full relativistic kinematics and Dirac spinors.To do this, we need to express the rate in terms of the relativistic differential cross section, and in a form suitable fornumerical calculations. But first, we want to motivate why weuse such a complicated form of the cross section.B.Electron Capture RatesOrbital electron capture and other Beta decay processes follow first order kinetics, so the capture rate is described by asingle number. To calculate the rate, we require a full relativistic, quantum mechanical treatment because the captureprocess involves both creating particles and the kinetic energyspectrum is of order me c2 .The VA theory is based on second order perturbation theory. It assumes an incoherent nuclear process, it is local, andthat the interaction is phenomenological. It is treated as simply a contact potential at the nucleus. Here, this means weneed to compute the nuclear charge–the electron density atthe the nucleus kψe (R 0)k, which we obtain from a classicalparticle-in-the-box wavefunction.The orbital electron capture rate ΓEC can be written by multiplying the cross section σEC by the incident velocity vinep and2(orbital) electron density Ψe (0) at the originWe could also describe the proton-neutron capture this way,but, for simplicity, we will simply assume the cross section forthe post reaction is maximally large, and we ignore the kinematics. We provide the equivalent expression for the protonneutron capture cross section in the appendix.More complicated calculations are used for larger nuclei,second order processes, etc. They only require modificationsto treat either atomic electronic structure of reactant and product atoms, and/or specific considerations for nuclear internalconversion and other second order processes.We write the Lorentz Invariant (LI) differential cross section in the C.M. frame asdσEC 12π!2PfiMfi16 k · (E n k 2k3 pe dΩkk0 pn )pe · (E p pe Ee pe )(1)where M f i is a matrix element of the Weak InteractionHamiltonian, k represents the neutrino momentum components (k0 , k) (Eν , k) p4ν , and we use natural units ( 1, c 1).This can be readily derived from the standard LI differentialcross section for a relativistic (1 2 3 4) elastic scatteringreaction in the C.M. frame. It is not commonly used in theliterature and it is this form that allows us to perform detailednumerical rate calculations.2ΓEC Ψe (0) vinep σEC .Of course, this velocity is not relativistic28 . And that is finefor typical calculations.For example, for something like 7 Be confined in a cage,we assume that vinep and σEC are not changing much, and wecan estimate the ’cage’ rate by n computing the ratio of theconfined to uncaged classical molecular electronic densities.ΓcageEC Ψcage(0) 2e Ψef ree (0) 2f reeΓECAnd in many other cases, we can just estimate the crosssection within a order of magnitude, without worrying aboutthe kinematics.C.Relativistic Cross SectionsIn our Gedankenexperiment, however, we imagine that thebox induces capture, we are at least in the regime of relativistic kinematics. To that end, we use a semi-classical, numericalmethod to describe the electron-capture cross section and perform the rate calculations. Using the Lorentz Invariant (LI)scattering cross section for a relativistic (1 2 3 4) reaction. The most basic calculations require only specifyingthe electronic wavefunctions(s), averaging over the possibleelectron-proton momenta, and numerically integrating overthe outbound neutrino momentum.1.RateThis gives the (differential) rate asdΓEC 12π!2 Pfihi2M f i pep i ) ψep (x) x 016E p Ee k · (E n k k0 pn )k3 dΩk(2)where we explicitly specify the electronic wavefunction.We represent the (e , p ) pair using a 3D box wavefunction,and obtain the Energies and 3-momenta from the relativistickinematics.As usual, we average over the initial spins, and sum overthe final spins. That is, we average over all 8 permutations ofthe incident velocities (really momenta pep ) for the 3D box,and integrate over outbound neutrino solid angle dΩk usingnumerical quadrature. The final rate is computed as, for eachbox size, as a function the kinematics, usingΓEC1X 8 ppeZdΓECdΩkwhere pep is given by the box size (described below).

82.PowerWe estimate the excess power generated by the confinedelectron capture, resulting if/when the outbound neutron reacts with the environment. The power P isBecause the VA theory assumes an incoherent process, theelectron, proton wavefunction is usually factored as an electron wavefunction, with a point-particle in the centerψep (x) ψ p (0)ψe (x)P ΓEC Q ρwhere is the nuclear decay energy, or Q-value, and ρ is thedensity of confined elements.We don’t actually know ρ, and as a placeholder we canchoose the density to that of a typical material, of order Avogadro’s number, NA 6.02 1023 . We will be examiningpower ratios, so this choice is irrelevant.We estimate the power for both the bare proton-electroncapturehiP pe ΓEC (Ee me ) (E p M p ) ρWe only consider the ground state ψ0ep wavefunction.We note, in ab initio electronic structure calculations, it isnow generally possible to treat the Hydrogen proton wavefunction explicitly, and to treat the electron-proton couplingat the level of Hartree Fock21 . This has proved useful, for example, for describing isotope effects on electronic structure.It may also be possible to eventually treat this problem usingfully relativistic ab initio QFT methods, although this has onlybeen applied to positronium so far29 .Here, we treat the confined electron, proton pair in the C.M.frame so that the 3-momentum of the electron and proton arerelated aspe p peand the subsequent neutron post-reactionp p p pePn ΓEC [2.2 (En Mn )] ρand we consider all 8 permutations for each given box size.L:assuming that the post-reaction has maximal efficiency. Wethen define an excess maximum power as p pe (1), p pe (1), p pe (3)PXS P pe Pn4.which we argue is a good measure of the gross maximumpotential reactive power output of the confined E.C. process.Of course, if the outbound neutron has any considerablevelocity vn , the proton-neutron cross-section σ p n would bevery small. The proton-neutron cross section scales inverselywith the velocity (σ p n v1n ). In principle, we could alsocompute this cross section using the Weak Interaction and therelativistic kinematics, as we did for electron capture.But we are only interested in box lengths where the excesspower is significant enough to drive the reaction, where theoutbound neutron is moving very slow, or is ulta cold. This is,of course, when the box is very large.3.While most electron capture calculations assume a specific,bound, atomic electronic wavefunction(s), we perform a muchsimpler calculation; we treat the electron-proton pair as classical particle-in-a-box, and analyze the problem just abovethe Compton scale using the low order VA theory. Write thewavefunction as! 32cos πx LcosFor consistency with the particle production process, wetreat all kinematics and energetics relativistically. Givenenergy-momentum conservationE 2 m2 p2and the 3D particle-in-the-box energy ground state energyEgs we use E Particle-in-the-box Wavefunctions2ψep (x) LRelativistic Kinematics and Energetics πy Lcos πz L3π2,2mL2p2to write2mEe2 m2e 3 π 2L,andE 2p M 2p 3 π 2L.The threshold Kinetic energy in the center of momentum(C.M.) frame is given as

91.EKemin : Ke Ee me (Mn me mν )2 M 2pThe matrix elements M f i of H(x) are given by30 .2(Mn mν )GFM f i ū(pn , sn )(GV G A γ5 )γµ u(p p , s p )2 ū(pν , sν )γu (1 γ5 )u(pe , se ) h.c. (3)which is approximately 781.6 KeV (or 783.1 KeV in theproton rest frame).Of course,

Beta ( ) decay, a nucleus emits an electron with energy of . a nucleus captures a bound, low lying electron, creating in a neutron and an electron neutrino. Electron capture : p e !n0 e Orbital electron capture (E.C.) is a fundamental nuclear process, on pair with the more familiar Beta d