Multipole Expansions of Charge Density, Electric, and Magnetic Field Generated by Ground State of some 4f Mössbauer ions

Abstract

Multipole expansion was employed to obtain expansions for charge densities, electric field and magnetic field generated by ground state of Europium Eu3+, Gadolinium Gd3+, TerbiumTb3+, Dysprosium Dy3+, Holmium Ho3+, Erbium Er3+ and YtterbiumYb3+ ions using Maple 15 software. And the electric fields and magnetic fields components were determined and are compared. The charge density generated by ground state of Europium Er3+ and Gadolinium Gd3+ were found to be symmetrically distributed around the nucleus while the charge density of TerbiumTb3+, Dysprosium Dy3+, Holmium Ho3+ and Erbium Er3+ were found to be non-symmetrically distributed around the nucleus. The electric fields generated by the ground state of Europium Eu3+ and Gadolinium Gd3+ were found to be symmetric about the nucleus, while the electric field generated by ground states of other ions were found to be nonsymmetrical about the nucleus. That was attributed to be due to the monopole electric field of Europium Eu3+ and Gadolinium Gd3+ and multipole field of Terbium Tb3+, Dysprosium Dy3+, Holmium Ho3+ and Erbium Er3+ ions. The magnetic field generated by ground state of Gadolinium Gd3+ ion was yielding from spin current density only and no orbital effect because the orbital angular momentum of this ion is zero. While the magnetic field generated by ground states for Dysprosium Dy3+, Holmium Ho3+ and YtterbiumYb3+ yield from spin–orbit coupling. All these ions are ferromagnetic because they have unpaired electrons in their outer shells.