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arXiv:1307.4810 · Physical Review Letters

Stable He− can Exist in a Strong Magnetic Field

Alexander V. Turbiner, Juan Carlos Lopez Vieyra

Published Oct 18, 2013• 10 citations
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Abstract

The existence of bound states of the system ($\ensuremath{\alpha}$, $e$, $e$, $e$) in a magnetic field $B$ is studied using the variational method. It is shown that for $B\ensuremath{\gtrsim}0.13\text{ }\text{ }\mathrm{a}.\mathrm{u}.$ this system gets bound with total energy below the one of the ($\ensuremath{\alpha}$, $e$, $e$) system. It manifests the existence of the stable ${\mathrm{He}}^{\ensuremath{-}}$ atomic ion. Its ground state is a spin doublet $^{2}(\ensuremath{-}1{)}^{+}$ at $0.74\text{ }\text{ }\mathrm{a}.\mathrm{u}.\ensuremath{\gtrsim}B\ensuremath{\gtrsim}0.13\text{ }\text{ }\mathrm{a}.\mathrm{u}.$ and it becomes a spin quartet $^{4}(\ensuremath{-}3{)}^{+}$ for larger magnetic fields. For $0.8\text{ }\text{ }\mathrm{a}.\mathrm{u}.\ensuremath{\gtrsim}B\ensuremath{\gtrsim}0.7\text{ }\text{ }\mathrm{a}.\mathrm{u}.$ the ${\mathrm{He}}^{\ensuremath{-}}$ ion has two (stable) bound states $^{2}(\ensuremath{-}1{)}^{+}$ and $^{4}(\ensuremath{-}3{)}^{+}$.

DOI
10.1103/physrevlett.111.163003

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The existence of bound states of the system (\ensuremathα\ensuremath{\alpha}\ensuremathα, eee, eee, eee) in a magnetic field BBB is studied using the variational method. It is shown that for B\ensuremath≳0.13  a.u.B\ensuremath{\gtrsim}0.13\text{ }\text{ }\mathrm{a}.\mathrm{u}.B\ensuremath≳0.13  a.u. this system gets bound with total energy below the one of the (\ensuremathα\ensuremath{\alpha}\ensuremathα, eee, eee) system. It manifests the existence of the stable He\ensuremath−{\mathrm{He}}^{\ensuremath{-}}He\ensuremath− atomic ion. Its ground state is a spin doublet 2(\ensuremath−1)+^{2}(\ensuremath{-}1{)}^{+}2(\ensuremath−1)+ at 0.74  a.u.\ensuremath≳B\ensuremath≳0.13  a.u.0.74\text{ }\text{ }\mathrm{a}.\mathrm{u}.\ensuremath{\gtrsim}B\ensuremath{\gtrsim}0.13\text{ }\text{ }\mathrm{a}.\mathrm{u}.0.74  a.u.\ensuremath≳B\ensuremath≳0.13  a.u. and it becomes a spin quartet 4(\ensuremath−3)+^{4}(\ensuremath{-}3{)}^{+}4(\ensuremath−3)+ for larger magnetic fields. For 0.8  a.u.\ensuremath≳B\ensuremath≳0.7  a.u.0.8\text{ }\text{ }\mathrm{a}.\mathrm{u}.\ensuremath{\gtrsim}B\ensuremath{\gtrsim}0.7\text{ }\text{ }\mathrm{a}.\mathrm{u}.0.8  a.u.\ensuremath≳B\ensuremath≳0.7  a.u. the He\ensuremath−{\mathrm{He}}^{\ensuremath{-}}He\ensuremath− ion has two (stable) bound states 2(\ensuremath−1)+^{2}(\ensuremath{-}1{)}^{+}2(\ensuremath−1)+ and 4(\ensuremath−3)+^{4}(\ensuremath{-}3{)}^{+}4(\ensuremath−3)+.
We study the question of existence, i.e., stability with respect to dissociation of the spin-quartet permutation- and reflection-symmetric ${}^{4}{(\ensuremath{-}3)}^{+}{}_{g}$ (${S}_{z}=\ensuremath{-}3/2,M=\ensuremath{-}3$) state of the $(\ensuremath{\alpha}\ensuremath{\alpha}eee)$ Coulomb system: the He2+{{\mathrm{He}}_{2}}^{+}He2​+ molecular ion, placed in a magnetic field $0\ensuremath{\le}B\ensuremath{\le}10\phantom{\rule{0.16em}{0ex}}000$ a.u. We assume that the $\ensuremath{\alpha}$ particles are infinitely massive (Born-Oppenheimer approximation of zero order) and adopt the parallel configuration, when the molecular axis and the magnetic field direction coincide, as the optimal configuration. The study of the stability is performed variationally with a physically adequate trial function. To achieve this goal, we explore several helium-containing compounds in strong magnetic fields, in particular; we study the spin-quartet ground state of the ${\mathrm{He}}^{\ensuremath{-}}$ ion and the ground (spin-triplet) state of the helium atom, both for a magnetic field in $100\ensuremath{\le}B\ensuremath{\le}10\phantom{\rule{0.16em}{0ex}}000$ a.u. The main result is that the He2+{{\mathrm{He}}_{2}}^{+}He2​+ molecular ion in the state ${}^{4}{(\ensuremath{-}3)}^{+}{}_{g}$ is stable towards all possible decay modes for magnetic fields $B\ensuremath{\gtrsim}120\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$ and with the magnetic field increase the ion becomes more tightly bound and compact with a cigar-type form of electronic cloud. At B=1000a.u.B=1000\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.B=1000a.u., the dissociation energy of He2+{{\mathrm{He}}_{2}}^{+}He2​+ into ${\mathrm{He}}^{\ensuremath{-}}+\ensuremath{\alpha}$ is $\ensuremath{\sim}702\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ and the dissociation energy for the decay channel to $\mathrm{He}+\ensuremath{\alpha}+e$ is $\ensuremath{\sim}729\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$, and both energies are in the energy window for one of the observed absorption features of the isolated neutron star 1E1207.4-5209.
We present a detailed study of the ground state of the Coulomb system $(\ensuremath{\alpha}\ensuremath{\alpha}ee)$ which corresponds to the He22+{\mathrm{He}}_{2}^{2+}He22+​ molecular ion in a magnetic field $B=0\ensuremath{-}4.414\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.3em}{0ex}}\mathrm{G}$ in parallel configuration (it is assumed that infinitely massive $\ensuremath{\alpha}$ particles are situated along a magnetic-field line). One uses a variational method, introducing a trial function which includes electronic correlation in the form $\mathrm{exp}(\ensuremath{\gamma}{r}_{12})$, where $\ensuremath{\gamma}$ is a variational parameter. It is shown that the quantum numbers of the lowest total-energy state depend on the magnetic-field strength. This state evolves from the spin-singlet $^{1}{\ensuremath{\Sigma}}_{g}$ metastable state at $0\ensuremath{\le}B\ensuremath{\lesssim}0.85\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ to a repulsive spin-triplet $^{3}{\ensuremath{\Sigma}}_{u}$ state for $0.85\ensuremath{\lesssim}B\ensuremath{\lesssim}1100\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ and, finally, to a stable strongly bound spin-triplet $^{3}{\ensuremath{\Pi}}_{u}$ state at stronger fields $B\ensuremath{\gtrsim}1100\phantom{\rule{0.3em}{0ex}}\mathrm{a}.\mathrm{u}.$ The lowest vibrational energy for the latter case is calculated.
A detailed study of the low-lying electronic states 1 Σ, 3 Σ, 3 Π, 3 Δ of the HeH + molecular ion in parallel to a magnetic field configuration (when α-particle and proton are situated on the same magnetic line) is carried out for B = 0 − 4.414 × 10 13 G in the Born–Oppenheimer approximation. The variational method is employed using a physically adequate trial function. It is shown that for the Σ states the parallel configuration is stable with respect to small deviations. The quantum numbers of the ground state depend on the magnetic field strength. The ground state evolves from the spin-singlet 1 Σ state for small magnetic fields B ≲ 0.5 au to the spin-triplet 3 Σ (unbound or weakly bound) state for intermediate fields and to the spin-triplet strongly bound 3 Π state for B ≳ 15 au. When the HeH + molecular ion exists, it is stable with respect to dissociation.
EXOTIC MOLECULAR IONS ( He H )2+ AND He23+{\rm He}_2^{3+}He23+​ IN A STRONG MAGNETIC FIELD: LOW-LYING STATES
The Coulombic systems (αpe) and (ααe), (αppe), (ααpe) and ( Li 3+ Li 3+e) placed in a magnetic field B ≳ 1011 G are studied. It is demonstrated a theoretical existence of the exotic ion ( He H )2+ for B ≳ 5 × 1012 G in parallel configuration (the magnetic field is directed along internuclear axis) as optimal as well as its excited states 1π, 1δ. As for the exotic ion [Formula: see text] it is shown that in spite of strong electrostatic repulsion of α-particles this ion can also exist for B ≳ 100 a.u. (= 2.35 × 1011 G ) in parallel configuration as optimal in the states 1σg (ground state), 1πu, 1δg. Upon appearance both ions are unstable towards dissociation with He + in the final state but with very large lifetime. However, at B ≳ 10000 a.u. , the ion ( He H )2+ becomes stable, while at B ≳ 1000 a.u. , the ion [Formula: see text] becomes stable. For both ions the vibrational and rotational energies are calculated. With a magnetic field growth, both exotic ions become more and more tightly bound and compact, their lowest rotational and vibrational energies grow drastically. At the edge of applicability of nonrelativistic approximation, B ~ 4.414 × 1013 G , there are indications that three more exotic linear ions ( H – He – H )3+, ( He – H – He )4+ and even [Formula: see text] in parallel configuration may also occur.
The existence and stability of the linear hydrogenic chain H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ in a strong magnetic field is established. Variational calculations for H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ are carried out in magnetic fields in the range ${10}^{11}\ensuremath{\le}B\ensuremath{\le}{10}^{13}$ G with a 17-parametric (13-parametric for $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$), physically adequate trial function. Protons are assumed to be infinitely massive, fixed along the magnetic line. States with total spin projection ${S}_{z}=\ensuremath{-}3/2$ and magnetic quantum numbers $M=\ensuremath{-}3,\ensuremath{-}4,\ensuremath{-}5$ are studied. It is shown that for both H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$, the lowest-energy state corresponds to $M=\ensuremath{-}3$ in the whole range of magnetic fields studied. As for a magnetic field $B\ensuremath{\gtrsim}{10}^{11}$ G, both H3\mathrm{H}{}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ exist as metastable states, becoming stable for $B\ensuremath{\ge}1.9\ifmmode\times\else\texttimes\fi{}{10}^{11}$ G and for $B\ensuremath{\ge}2.7\ifmmode\times\else\texttimes\fi{}{10}^{11}$ G, respectively. The excited states ${}^{4}{(\ensuremath{-}4)}^{+}$, ${}^{4}{(\ensuremath{-}5)}^{+}$ of H3{\mathrm{H}}_{3}H3​ and $\mathrm{H}{{}_{2}}^{\ensuremath{-}}$ appear at magnetic fields $B>7\ifmmode\times\else\texttimes\fi{}{10}^{11}$ and 1012{10}^{12}1012 G, respectively.
The (HeH)2+(HeH)^{2+}(HeH)2+ and He23+He_2^{3+}He23+​ exotic molecular ions can exist in a strong magnetic field
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