2D calculations for atoms and ions in strong magnetic fields of white dwarfs and neutron stars
The term “strong field” characterizes a situation for which the Lorentz force is of the order of magnitude or greater than the Coulomb binding force. For a hydrogen atom in the ground state the corresponding field strength cannot be reached in the laboratory, but only in astrophysical objects like white dwarfs ( B≈ 10–10T) or neutron stars ( B≈ 10–10T). Astrophysicists possess therefore a vivid interest in the behavior and properties of matter in strong magnetic fields: theoretically calculated data of magnetized atoms can be used for the determination of the decomposition and magnetic field configuration of astrophysical objects [1–4]. On the other hand the strong magnetic field regime is accessible in the laboratory if one considers highly excited Rydberg states of e.g. atoms [5,6]. In solid state physics donor states in semiconductors with parabolic conduction bands are systems which possess a Hamiltonian equivalent to the one of hydrogen within an effective mass approximation. Due to screening effects the Coulomb force is much weaker than in the case of hydrogen. The regime where the ground state of the system is dominated by magnetic forces can therefore be reached for certain semiconductors in the laboratory. As an example we mention GaAs for which the effective mass is m∗ = 0.067 me and the static dielectric constant s = 12.53 0. Since the Hamiltonian of the atomic ion and the negative donor are connected through a scaling transformation the values for the energies given in the present work hold for both systems equally. The reader should however keep in mind that they are given in differently scaled units. Apart from the above atoms and molecules in strong magnetic fields are also of interest from a pure theoretical point of view. Due to the competition of the spherically symmetric Coulomb potential and the cylindrically symmetric magnetic field interaction we encounter a nonseparable, nonintegrable problem. Perturbation theory, which is possible in the weak and in the ultrastrong field regime, breaks down in the intermediate field regime. It is therefore necessary to develop new techniques to solve such problems. The neutral hydrogen atom in a strong magnetic field is now understood to a high degree (see [5,7] and references therein). Recently Kravchencko has published an “exact” solution which provides an infinite double sum for the eigenvalues . With the presented method all energy values of bound states could in principle be calculated to arbitrary precision. For two electron atoms the situation is significantly different. The problems posed by the electron-electron interaction and the non-separability on the one-particle level have to be solved simultaneously, which is much harder. The H− ion provides an additional challenge since correlation plays an important role for its binding properties. Without a field it possesses only one bound state . In the presence of a magnetic field and for the assumption of an infinitely heavy nucleus it could be shown  that there exists an infinite number of bound states. For laboratory field strengths these states are, due to the binding mechanism via a one dimensional projected polarization potential, very weakly bound . Some finite nuclear mass effects can be included via scaling relations [7,12,14]. However, the influence of the center of mass motion has not been investigated in detail so far. In the present work we assume an infinitely heavy nucleus which represents a good approximation for the slow H− atomic ion in strong magnetic fields and describes simultaneously the situation of negatively charged donors D− in the field. Relativistic corrections were neglected since they are assumed to be small compared to the electron detachment energy of the system. We will use in the following the spectroscopic notation M for the electronic states of the ion where M and S are the total magnetic and spin quantum numbers. Since states with negative z-parity are not considered here we omit the corresponding label in our notation (see also section II A). Many authors have tackled the quantum mechanical problem of H− in a strong magnetic field. One of the first, who pursued a variational approach to this problem, were Henry et al. . They give first qualitative insights into the weak and intermediate field regime. Mueller et al.  qualitatively described the strong field ground state 3(−1) and the 0 state for high fields (γ ≈ 4 to γ ≈ 20 000, where γ = 1 a. u. corresponds to 2.3554 · 10T).