Dirac Bubble Research Articles

Time delay of slow electrons-endohedral elastic scattering

We discuss the temporal picture of electron collisions with fullerene. Within the framework of a Dirac bubble potential model for the fullerene shell, we calculate the time delay in slow-electron elastic scattering by it. It appeared that the time of transmission of an electron wave packet through the Dirac bubble potential sphere that simulates a real potential of the C60 reaches up to 104 attoseconds. Resonances in the time delays are due to the temporary trapping of electron into quasi-bound states before it leaves the interaction region. As concrete targets we choose almost ideally spherical endohedrals C20, C60, C72, and C80. We present dependences of time-delay upon collision energy.

Time delay in electron-C60 elastic scattering in a dirac bubble potential model

Within the framework of a Dirac bubble potential model for the C60 fullerene shell, we calculated the time delay in slow-electron elastic scattering by C60. It appeared that the time of transmission of an electron wave packet through the Dirac bubble potential sphere that simulates a real potential of the C60 cage exceeds by more than an order of magnitude the transmission time via a single atomic core. Resonances in the times delay are due to the temporary trapping of electron into quasi-bound states before it leaves the interaction region.

On the thermophysical and transport properties of 3He and 4He: A bubble interaction potential versus state of the art.

Three keynote thermophysical and transport properties of 3He and 4He, namely, the second virial coefficient, the shear viscosity, and the thermal conductivity, are reported for the "extended Dirac bubble potential" (EDbp), a novel model for He-He [M. Chrysos, J. Chem. Phys. 146, 024106 (2017)]. Comparisons with the experiment as well as with potentials with a proven track record and with the oversimplified Dbp are being made in the range 0.1-500 K to analyze the performance of the EDbp, which is shown here to emerge as a promising analytic model for He-He. A flowchart of how to treat the "buffer" in scattering cross section measurements is designed and conducted, offering a route to EDbp optimization. An impressive consistency with state-of-the-art calculations (which is just striking for such a simple analytic model) is found, essentially thanks to the performance of the phase-shift expression cotδl = λkr0ξlyl(kr0)-yl(krc)λkr0ξljl(kr0)-jl(krc), ξl=jl(kr0)yl(krc)-jl(krc)yl(kr0). A Multimedia view of δl(k,rc) versus k and rc is part of the material presented in this article. Data for the "best" rc(k) is given as a supplementary material.

Dirac bubble potential for He-He and inadequacies in the continuum: Comparing an analytic model with elastic collision experiments.

We focus on the long-pending issue of the inadequacy of the Dirac bubble potential model in the description of He-He interactions in the continuum [L. L. Lohr and S. M. Blinder, Int. J. Quantum Chem. 53, 413 (1995)]. We attribute this failure to the lack of a potential wall to mimic the onset of the repulsive interaction at close range separations. This observation offers the explanation to why this excessively simple model proves incapable of quantitatively reproducing previous experimental findings of glory scattering in He-He, although being notorious for its capability of reproducing several distinctive features of the atomic and isotopic helium dimers and trimers [L. L. Lohr and S. M. Blinder, Int. J. Quantum Chem. 90, 419 (2002)]. Here, we show that an infinitely high, energy-dependent potential wall of properly calculated thickness rc(E) taken as a supplement to the Dirac bubble potential suffices for agreement with variable-energy elastic collision cross section experiments for 4He-4He, 3He-4He, and 3He-3He [R. Feltgen et al., J. Chem. Phys. 76, 2360 (1982)]. In the very low energy regime, consistency is found between the Dirac bubble potential (to which our extended model is shown to reduce) and cold collision experiments [J. C. Mester et al., Phys. Rev. Lett. 71, 1343 (1993)]; this consistency, which in this regime lends credence to the Dirac bubble potential, was never noticed by its authors. The revised model being still analytic is of high didactical value while expected to increase in predictive power relative to other appraisals.

Brane gravitational extension of Dirac's ‘extensible model of the electron’

A gravitational extension of Dirac's ‘extensible model of the electron’ is presented. The Dirac bubble, treated as a three-dimensional electrically charged brane, is dynamically embedded within a four-dimensional Z2-symmetric Reissner–Nordstrom bulk. Crucial to our analysis is the gravitational extension of Dirac's brane variation prescription; its major effect is to induce a novel geometrically originated contribution to the energy–momentum tensor on the brane. In turn, the effective potential which governs the evolution of the bubble exhibits a global minimum, such that the size of the bubble stays finite (Planck scale) even at the limit where the mass approaches zero. This way, without fine-tuning, one avoids the problem so-called ‘classical radius of the electron’.

Inner-shell photoionization of fullerenes C20 and C60 within the "Dirac bubble" model

The "Dirac bubble" (δ-potential) model is applied to description of the structure of fullerenes C20 and C60 and photoelectron states. Energy spectra for σ valence orbitals for both fullerenes are obtained. Photoionization from two innermost delocalized electrons states (1s and 2p) is considered. The s-,p-,d- scattering phaseshifts are calculated, they demonstrate clear resonance properties which have oscillation in energy. Partial and total photoionization cross sections are obtained, the latter reveal an interesting "serrated" structure. Intriguing high-energy asymptotics is deduced, σ ∼ ω−5/2 sin2 (kR).

Broken-Symmetry States of Dirac Fermions in Graphene with a Partially Filled High Landau Level

We report on numerical study of the Dirac fermions in partially filled N=3 Landau level (LL) in graphene. At half-filling, the equal-time density-density correlation function displays sharp peaks at nonzero wave vectors +/-q*. Finite-size scaling shows that the peak value grows with electron number and diverges in the thermodynamic limit, which suggests an instability toward a charge density wave. A symmetry broken stripe phase is formed at large system size limit, which is robust against perturbation from disorder scattering. Such a quantum phase is experimentally observable through transport measurements. Associated with the special wave functions of the Dirac LL, both stripe and bubble phases become possible candidates for the ground state of the Dirac fermions in graphene with lower filling factors in the N=3 LL.

Confinement resonances in photoelectron angular distributions from endohedral atoms

Both dipole and nondipole angular distribution parameters for photoelectrons emitted from the endohedral $\mathrm{Ne}@{\mathrm{C}}_{60}$ atom are investigated within both the framework of the Dirac bubble potential model and spherical, short range potential well of finite thickness $\ensuremath{\Delta}T$. It is demonstrated, for $1s$, $2s$, and $2p$ photoelectrons of $\mathrm{Ne}$, that because the fullerene shell acts as a resonator in this process, the energy dependence of the photoelectron angular distribution parameters differ considerably from that of the free $\mathrm{Ne}$ atom. For confined $\mathrm{Ne}$, in contrast to the isolated $\mathrm{Ne}$ atom, these parameters as a function of photon energy acquire a significant resonance structure, owing to oscillations in the photoionization matrix elements induced by the fullerene shell.

Charge transfer in hyperthermal surface collisions of C60 and C60−: Experiment and model calculations

In this paper we address the issue of electron exchange between a large molecular projectile and a surface during a hyperthermal collision. Hyperthermal neutral C60 molecules with well-defined average vibrational energy Ev=8±0.5 eV were scattered from a graphitized (monolayer-covered) nickel surface for both near normal and near grazing incidence angles. The yield of C60− negative ions was measured for the impact energy range 8–33 eV and was found to scale exponentially with the inverse of the normal component of the scattered molecule velocity. Normal and tangential velocities of the scattered C60 are very low, (1.3–5.6)×10−4 and (0.8–3.1)×10−4 a.u. correspondingly. The fact that nearly the same slope (characteristic velocity) was extracted from the semilog plots for both near normal and near grazing incidence angles shows that tangential velocity effects are negligible. The rate of electron tunneling from C60− to the surface was calculated quasi-classically assuming that the excess (active) electron in C60− is moving in the highly screened short range attractive potential of the neutral C600 core, which is approximated by a radial delta function (Dirac bubble). The transmission coefficient was calculated for C60− located at the center of a spherical metallic cavity with image charge potential barrier for the active electron. The preexponential frequency factor was obtained by solving the quasi-stationary Shrödinger equation for the same system but without the image charge barrier. Finally, the decay rate was corrected for the case of C60− in front of a conducting plane. The experimental results were analyzed using the calculated tunneling rates and a critical ion formation distance was obtained. Independence of the scattering dynamics and negative ion yield on the incoming charge state is demonstrated by using incident C60− at the impact energy range of 40–100 eV. The same relative energy losses were measured for incident C60− ions as for neutral C600 thus extending the range of linear dependence of the scattered kinetic energy on impact energy up to 100 eV.

Delta function model for the helium trimer: origin of three-body forces

The Dirac bubble potential, previously used to model the helium dimer, is applied to the trimer. It is shown that the quantum mechanical kinetic energy operator for a three-body system contains terms over and above the analogues of classical pairwise contributions. The additional terms, the ‘Borromean couplings’, are responsible for a dramatic increase in the binding energy of the trimer. For example, 4He3 is bounded by two orders of magnitude more strongly than 4He2, 279 mK versus 1.31 mK, respectively, according to our calculations. Moreover, the trimer is considerably more compact than the dimer, with (r12) decreasing to 9.01 Å from 51.9 Å.

Deltafunction model for the helium dimer

AbstractThe helium dimer 4He2 has recently been detected, confirming earlier ab initio predictions of stability for a single bound state with binding energy of 1.310 mK. The predicted potential minimum is at 2.96 Å, with a radial distribution function peaking at 6.96 Å. We model this system using a Dirac bubble potential, which also admits just one bound j = 0 state. With the bubble located at 6.96 Å, an overlap of 0.9994 with the ab initio wave function is obtained. An average internuclear distance of 52.6 Å is calculated, in good agreement with the ab initio result. The root mean square deviation from the mean, 48.0 Å, indicates an enormous spread of the radial wave function. Also consistent with our model is the absence of bound states for the isotopic variants 3He4He and 3He2. Cross sections for helium‐helium scattering are also computed, using both a partial‐wave expansion and the Born approximation. General trends in the energy dependence of the total cross section are accounted for, in qualitative agreement with experimental results. © 1995 John Wiley & Sons, Inc.

Electron photodetachment from a Dirac bubble potential. A model for the fullerene negative ion CC 60−

A model for the fullerene anion C 60 − is proposed in which the outer electron moves in an attractive deltafunction potential in the shape of a spherical bubble. Exact solutions for this “Dirac bubble potential” were previously worked out. On the basis of this model, the photodetachment cross-section and its angular distribution are computed as functions of photoelectron energy, for the electron in both S-like and P-like bound states. Secondary maxima and minima in the cross section appear at higher energy values, somewhat analogous to the scattering of radiation from a conducting sphere.

Schrödinger equation for a dirac bubble potential

The quantum-mechanical problem of a particle moving in a “Dirac bubble potential” U( r) = (λ/ r o)δ( r - r o) is solved exactly for both bound and continuum states by making use of partial wave Green's functions G l ( r, r 0, k). Phase shifts are expressed in a compact form related to those for an impenetrable sphere.