Hyperspherical three-body calculation for muonic atoms

Abstract

Ground state energies of exotic three-body atomic systems consisting two muons and a positively charged nucleus like: ¹H⁺ μ ⁻ μ ⁻, ⁴He²⁺ μ ⁻ μ ⁻, ³He²⁺ μ ⁻ μ ⁻, ⁷Li³⁺ μ ⁻ μ ⁻, ⁶Li³⁺ μ ⁻ μ ⁻, ⁹Be⁴⁺ μ ⁻ μ ⁻, ¹²C⁶⁺ μ ⁻ μ ⁻, ¹⁶O⁸⁺ μ ⁻ μ ⁻, ²⁰Ne¹⁰⁺ μ ⁻ μ ⁻, ²⁸Si¹⁴⁺ μ ⁻ μ ⁻ and ⁴⁰Ar¹⁸⁺ μ ⁻ μ ⁻ have been calculated using hyperspherical harmonics expansion (HHE) method. Calculation of matrix elements of two body interactions involved in the HHE method for a three body system is greatly simplified by expanding the bra- and ket- vector states in the hyperspherical harmonics basis states appropriate for the partition corresponding to the interacting pair. This involves the Raynal-Revai coefficients (RRC), which are the transformation coefficients between the hyperspherical harmonics bases corresponding to the two partitions. Use of these coefficients found to be very useful for the numerical solution of three-body Schrödinger equation where the two-body potentials are other than Coulomb or harmonic oscillator type. However, in this work the interaction potentials involved are purely Coulomb. The calculated energies have been compared with (i) those obtained by straight forward manner; and (ii) with those found in the literature (wherever available). The calculated binding energies agree within the computational error.