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Physical Review A

Magnetically bound states of electronic and positronic hydrogen ions: The case of zero orbital angular momentum

Daniel H. E. Dubin

Published Feb 14, 2005• 3 citations
Publisher

Abstract

This paper considers the binding of an electron or a positron to a ground-state hydrogen atom in a uniform magnetic field $B$, concentrating on states with zero orbital angular momentum $J$. It is shown that the affinity of a positron to a hydrogen atom is $2.23{b}^{2}{e}^{2}∕a$, where $a$ is the Bohr radius, $b\ensuremath{\equiv}B∕{B}_{0}$, ${B}_{0}=2.3505\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, $b⪡1$ is assumed, and the proton is assumed to have infinite mass. With these assumptions it is also shown that no $J=0$ singlet or triplet excited state of ${\mathrm{H}}^{\ensuremath{-}}$ exists. These results are compared to a previous estimate of the electron affinity of $J=0$ magnetically bound ions [V. G. Bezchastnov, P. Schmelcher, and L. S. Cederbaum, Phys. Rev. A 61, 052152 (2000)].

DOI
10.1103/physreva.71.022504

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This paper considers the binding of an electron or a positron to a ground-state hydrogen atom in a uniform magnetic field BBB, concentrating on states with zero orbital angular momentum JJJ. It is shown that the affinity of a positron to a hydrogen atom is 2.23b2e2∕a2.23{b}^{2}{e}^{2}∕a2.23b2e2∕a, where aaa is the Bohr radius, b\ensuremath≡B∕B0b\ensuremath{\equiv}B∕{B}_{0}b\ensuremath≡B∕B0​, B0=2.3505\ifmmode×\else\texttimes\fi105T{B}_{0}=2.3505\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.3em}{0ex}}\mathrm{T}B0​=2.3505\ifmmode×\else\texttimes\fi105T, b⪡1b⪡1b⪡1 is assumed, and the proton is assumed to have infinite mass. With these assumptions it is also shown that no J=0J=0J=0 singlet or triplet excited state of H\ensuremath−{\mathrm{H}}^{\ensuremath{-}}H\ensuremath− exists. These results are compared to a previous estimate of the electron affinity of J=0J=0J=0 magnetically bound ions [V. G. Bezchastnov, P. Schmelcher, and L. S. Cederbaum, Phys. Rev. A 61, 052152 (2000)].
We have investigated the elastic scattering of positronium atoms by hydrogen atoms below the excitation threshold at 5.1 eV, limiting the investigation to the spin-zero (singlet) state of the two electrons. Nonresonant SSS-wave phase shifts have been obtained from normalizable trial functions by extracting the center-of-mass wave function and examining its spatial behavior. An effective-range expansion is fitted to the phase shifts and the previously obtained positronium-hydride bound-state energy to yield a scattering length $a\ensuremath{\cong}5.3{a}_{0}$ compared with Fraser's corrected exchange result, a=7.28a0a=7.28{a}_{0}a=7.28a0​ (a0{a}_{0}a0​ is the Bohr radius). The stabilization method suggests that a resonance occurs at a scattering energy of 4.5 eV, about 0.6 eV below the excitation threshold. Using the complex rotation method, we have obtained evidence confirming the existence of the resonance and an estimate of its width of about 6 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ eV.
This paper considers the possibility that two charged particles with an attractive short-ranged potential between them which is not strong enough to form bound states in free space, may bind in uniform magnetic fields. It is shown that in the formal limit where Coulomb repulsion is negligible ($q\ensuremath{\rightarrow}0\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{B}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}\phantom{\rule{0.3em}{0ex}}\text{with}\phantom{\rule{0.3em}{0ex}}q{B}_{0}\phantom{\rule{0.3em}{0ex}}\text{fixed}$, where qqq is the charge and B0{B}_{0}B0​ the field strength) there always exists a bound state for a system of two identical charged particles in a constant magnetic field, provided that there is a short-range uniformly attractive potential between them. Moreover, it is shown that in this limit any potential with an attractive s-wave scattering length will possess bound states provided that the range of the potential is much smaller than the characteristic magnetic length, ${r}_{0}\ensuremath{\equiv}(\frac{q{B}_{0}}{4}){}^{\ensuremath{-}1/2}$. For this case, the binding is computed numerically. We estimate the size of the magnetic field needed to approximately reach a regime where the formal limit considered becomes a good approximation to the dynamics. These numerical estimates indicate that two protons in an extremely strong magnetic field such as might be found in a magnetar will bind to form a diproton.
We investigate the hydrogen molecule in a strong parallel magnetic field using a fully numerical Hartree-Fock approach. We find that for magnetic fields below $4.2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ T the ground state of H2{\mathrm{H}}_{2}H2​ is the strongly bound singlet state ${}^{1}{\ensuremath{\Sigma}}_{g},$ for magnetic fields stronger than $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the ground state of the molecule is the strongly bound triplet ${}^{3}{\ensuremath{\Pi}}_{u},$ and for magnetic fields between $4.2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ T and $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the symmetry of the ground state is the triplet state ${}^{3}{\ensuremath{\Sigma}}_{u},$ which is characterized by repulsion at intermediate internuclear distances and by a weak quadrupole-quadrupole interaction between atoms at large internuclear separation. In this region of magnetic field strength the hydrogen molecule is bound weakly, if at all; the hydrogen atoms behave like a weakly nonideal gas of Bose particles and can form a superfluid phase predicted in earlier works [Korolev and Liberman, Phys. Rev. Lett. 72, 270 (1994)]. For magnetic fields between $\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{5}$ T and $3\ifmmode\times\else\texttimes\fi{}{10}^{6}$ T the triplet state ${}^{3}{\ensuremath{\Pi}}_{u}$ is found to be metastable. This state may be responsible for an unknown excitonic line observed experimentally [Timofeev and Chernenko, JETP Lett. 61, 617 (1995)].
The existence and structure of positronic atoms with a total angular momentum of L=1L=1L=1 is studied with the configuration interaction method. Evidence is presented that there is a 2Po^{2}P^{o}2Po state of e+Ca{e}^{+}\mathrm{Ca}e+Ca and 2,4Po^{2,4}P^{o}2,4Po states of e+Be(3Po){e}^{+}\mathrm{Be}(^{3}P^{o})e+Be(3Po) that are electronically stable with binding energies of 45meV45\phantom{\rule{0.3em}{0ex}}\mathrm{meV}45meV and 2.6meV2.6\phantom{\rule{0.3em}{0ex}}\mathrm{meV}2.6meV, respectively. These predictions rely on the use of an asymptotic series analysis to estimate the angular $L\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of the energy. Incorporating corrections that compensate for the finite range of the radial basis increased the binding energies of e+Ca{e}^{+}\mathrm{Ca}e+Ca and e+Be{e}^{+}\mathrm{Be}e+Be to 71meV71\phantom{\rule{0.3em}{0ex}}\mathrm{meV}71meV and 42meV42\phantom{\rule{0.3em}{0ex}}\mathrm{meV}42meV, respectively.