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Ground State, Isoelectronic Ions and Low-Lying Excited States of Lithium Atom in Strong Magnetic Field

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Ground State, Isoelectronic Ions and Low-Lying Excited States of Lithium Atom in Strong Magnetic Field Open Journal of Microphysics, 2021, 11, 37-51 https://www.scirp.org/journal/ojm ISSN Online: 2162-2469 ISSN Print: 2162-2450 Ground State, Isoelectronic Ions and Low-Lying Excited States of Lithium Atom in Strong Magnetic Field Khadija Abdelhassan Kharroube Mathematics Department, Faculty of Science, Lebanese University, Beirut, Lebanon How to cite this paper: Kharroube, K.A. Abstract (2021) Ground State, Isoelectronic Ions and Low-Lying Excited States of Lithium Atom In the framework of the variational Monte Carlo method, the ground states of in Strong Magnetic Field. Open Journal of the lithium atom and lithium like ions up to Z = 10 in an external strong Microphysics, 11, 37-51. magnetic field are evaluated. Furthermore, the two low-lying excited states https://doi.org/10.4236/ojm.2021.113004 2 2 12ss, 12sp−1 and 12sp−−12 3 d of the lithium atom in strong magnetic Received: July 4, 2021 field are also investigated. Simple trial wave functions for lithium are used. Accepted: August 13, 2021 Published: August 16, 2021 Keywords Copyright © 2021 by author(s) and Variational Monte Carlo Method, Atoms in Magnetic Field, Ground States of Scientific Research Publishing Inc. Li, Binding Energy, Total Energy This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ 1. Introduction Open Access The properties of atoms in strong magnetic fields are relevant to several domains such as astrophysics, condensed matter physics, and plasma physics [1]. In the past, only hydrogen and helium atoms have been studied for magnetic fields of arbitrary strength [2]. The influence of a magnetic field on the properties of atoms and molecules is of great interest. The study of strongly magnetized atoms was first treated in a solid-state context. In semiconductors, the absorption of photons can promote electrons from the valence to the conduction band, form- ing an electron-hole pair. The absorption spectra of these excitons, when sub- jected to a magnetic field, display Zeeman effects [3]. The methods of treating the resulting problems arising from these topics have resulted in few studies of many-body effects in strong magnetic fields, i.e., in the field regime where per- turbation theory is no longer applicable. The motivation in this area arises from several sources. On the one hand, this is due to the astrophysical discovery of strong magnetic fields on the surfaces of DOI: 10.4236/ojm.2021.113004 Aug. 16, 2021 37 Open Journal of Microphysics K. A. Kharroube white dwarfs (102 - 105 T) and neutron stars (107 - 109 T) [4] [5]. Also, the com- plex properties of atoms under these extreme conditions are of immediate inter- est from a purely theoretical point of view. In addition, the observations of exci- tons with small effective masses and large dielectric constants in semiconductors, which result in very large effective magnetic fields, give additional impetus for this subject [6]. It is well-known that the study of the electronic structure of atoms in the ex- ternal magnetic field is a quite complicated subject, especially for the strong magnetic field. In a weak magnetic field, which can be treated as a perturbation, the wave function can be expanded in terms of spherical harmonic functions. In a very strong magnetic field, where the Lorentz forces dominate, the wave func- tion can be expanded in terms of Landau-like orbitals. But in the strong mag- netic field, where the Lorentz and Coulomb forces are of nearly equal impor- tance, neither of them can be treated as a perturbation [7]. Hence, it is necessary to develop non-perturbative techniques to solve this problem. The parameter βz can be used to characterize three different regimes of strength: the low −3 −3 (weak, βz < 10 ), the intermediate (strong, 10≤≤βz 1), and the high (very 2 5 strong, βz > 1 ), regimes where βz = B2 BZ0 , B0 = 2.35×10 T , and Z is the charge of the atomic nucleus. So far, considerable effort has been devoted to the theoretical investigations of atoms in magnetic fields with arbitrary strength. Rӧsner et al. [8] have done detailed work on the spectrum of the hydrogen atom in magnetic fields up to 4.7 × 108 T, which has been successfully applied to the identification of the observed spectra from many magnetic white dwarfs [9] [10]. For the hydrogen atom in magnetic fields up to 9.4 × 108 T, Kravchenko et al. [11] [12] presented accurate results with an accuracy of 10−12. In view of this, the issue of the hydrogen atom over a wide range of magnetic fields can be com- pletely solved. Despite this great success, there were also many magnetic white dwarfs whose spectra remain unexplained or could not be completely accounted for with hydrogen atom. It is thus essential to acquire extensive and accurate da- ta of energy levels and resulting transition wavelengths of multi-electron atoms subjected to strong magnetic fields. Hence, the research emphasis of atoms under these extreme conditions focused on the next lightest element, neutral helium. In contrast to the hydrogen atom, the problem of the helium atom is much more in- tricate because of the occurrence of the electron-electron repulsion. Even so, there exist many investigations on helium atoms in the literature. Thurner et al. [13] gave the energies of several triplets excited states of helium atoms for magnetic field strengths in the range 4.7 × 101 to 4.7 × 108 T. Furthermore, they presented a consistent correspondence diagram between the low field and high field do- mains for the first time. Many studies have been recently presented by Doma et al. using the Varia- tional Monte Carlo (VMC) method for the helium, lithium, and beryllium atoms [14] [15] [16] [17] [18]. The success of the VMC method extended also to the light molecules, where similar investigations have been carried out for the hy- drogen molecule and its molecular ion by Doma et al. [19] [20] [21]. DOI: 10.4236/ojm.2021.113004 38 Open Journal of Microphysics K. A. Kharroube In fact, the problem of highly accurate calculations of the bound states in the three-electron atomic systems has attracted continuing attention. For lithium atom and its ions up to Z = 10, in the strong magnetic field and using the VMC method, Doma et al. calculated the energies of these systems [16]. In the present paper, we applied the VMC method and constructed three trial wave functions, which take into consideration the correlation functions due to the interactions between the electrons in the atom, to calculate the ground state of the lithium atom and its ions up to Z = 10 in the presence of an external magnetic field regime with γ = 0 ~ 100 a.u. Furthermore, the two low-lying 2 2 excited states 12ss, 12sp−1 and 12sp−−12 3 d of the lithium atom in the strong magnetic field are also investigated. 2. Method of the Calculations In the present paper, we solved the Schrödinger equation for the lithium atom and the lithium like ions up to Z = 10 in an external magnetic field to find the energy eigenvalues as functions of the field parameter. Furthermore, the case of 2 2 the low-lying excited states 12ss, 12sp−1 and 12sp−−12 3 d of the lithium atom in an external magnetic field is also solved. For this purpose, simple trial wave functions with low number of variational parameters for lithium are used. To find the minimum energy eigenvalues we applied the VMC method. We first constructed the Hamiltonian operator for the three-electron system in the ab- sence of the external field. We made use of the assumption of an infinitely heavy nucleus in the unrestricted Hrtree-Fock (HF) approximation. Hence, the non- relativistic Hamiltonian for the lithium atom, in the absence of the field, under the Born-Oppenheimer approximation of zero order, that is, with the Li nucleus assumed to be of infinite mass, is (in Hartree atomic units) [22] given by: 1133−Z 3 =− ∇+2 + H ∑∑i=11i i= ∑ij< , (2.1) 2 rri ij where ∇i is the 3-vector of the momentum of the ith electron, Z is the nuclear charge (here, Z = 3), ri is the distance between the ith electron and the Li nuc- leus, and rij are the interelectron distances. In our calculations we used the form of H in Hylleraas Coordinates [22] [23] as follows: 22 12 33∂ ∂∂ 3 3 4 ∂ =−++ + H ∑∑i==<<1122i ∑ij2 ∑ij 2 ∂∂rri rri∂∂ i ij rij r ij rrr222+− ∂∂22rrr222+− ++3i ij j 33ij ik jk ∑ij≠ ∑∑ij≠> kj rri ij∂∂ r i r ij rij r ik∂ r ij ∂ r ik 22 3 11∂33 ∂∂cotθ ++∑∑ + ∑i (2.2) ii=11rr22∂∂θϕ= 22sin 2 i= 1 r 2∂θ ii iθi i i i 222 2 3 2crj osθ j rij−− rr i j ∂ + ∑ + cotθi j≠1θθ2 ∂∂ rri ij sin i rri ij ir ij 2 33rjjsinθ ∂− Z 3 1 +∑2 sin (ϕϕij−) ++∑∑ j≠=1 θϕ∂∂ i1 1=ij< rri ijsin i ir ij ri rij DOI: 10.4236/ojm.2021.113004 39 Open Journal of Microphysics K. A. Kharroube According to the assumption that the nuclear mass is infinite, and the mag- netic field is oriented along the z-axis, the non-relativistic Hamiltonian for the Li atom in the presence of the magnetic field (in atomic units (a.u.)) takes the form [7]: 22 γρ γ (LSzZ+ 2 ) =++H . (2.3) 82 2 2 2 2 22 22 22 where ρ=++=+++++ ρρρ1 23( xy 11) ( xy 22) ( xy 33) , γ is the magnetic = 3 = 3 field parameter, Llz ∑i=1 iz and Ssz ∑i=1 iz are the z-components of the γρ22 total angular momentum and spin respectively, is the diamagnetic term, 8 γ ZZZ Lz is the Zeeman term, −−− are the attractive Coulomb interactions 2 rrr123 with the nucleus and γ Sz represents the spin Zeeman term.
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