Helium atom under pressure

Abstract

Abstract. Hard-sphere confinement is used to study helium atoms under pressure. The confined-helium Schrödinger equation is solved with a high accuracy by a Lagrangemesh method.The effects of high pressure on a helium gas can be estimated by studying the helium atom in a hard confinement, i.e. confined at the centre of an impenetrable spherical cavity, for different cavity radii. Contrary to the free helium atom, the confined helium has not been described with a very high accuracy until recently, when we have developed a Lagrange-mesh method to study this system [1]. This method improves by several order of magnitudes the accuracy of previous approaches [2][3][4].The outlines of the model are the following. The assumed-infinite-mass nucleus is fixed and the electrons are characterized by coordinates r 1 and r 2 with respect to this nucleus. In atomic units, the Hamiltonian of the helium atom readswhere r 12 = r 1 − r 2 and Δ 1 and Δ 2 are the Laplacians with respect to r 1 and r 2 . The confinement is introduced by forcing the wave function into some spherical cavity of radius R (r 1 , r 2 ≤ R). The wave function ψ(r 1 , r 2 , r 12 ) of an S state must thus verify the Schrödinger equationand vanishes at r 1 = R and r 2 = R. The coordinates (r 1 , r 2 , r 12 ) are advantageously replaced by the coordinates (u, v, w) defined over [0,1] byThe confinement implies that the wave function ψ(u, v, w) vanishes at u = 1, v = 1, and w = 1. The Schrödinger equation is solved by the Lagrange-mesh method [5][6][7], an approximate variational approach taking the form of a system of mesh equations by computing the Hamiltonian and overlap matrix elements with a Gauss quadrature. Using the coordinates (u, v, w) is essential for an easy treatment of the confinement and an high accuracy of the Gauss quadrature. The ground-state energy and wave function are obtained by diagonalizing a rather large (matrix dimension≈ 10 3 ∼ 2×10 4 ) but sparse matrix. Ground-state energies and mean interparticle distances for several confinement radii R are given in table 1. The pressure acting on the confined helium atom is also given in table 1. It is a

The Hyperspherical Harmonics Method: A Tool for Testing and Improving Nuclear Interaction Models

“…In fact the hyperangular-spin-isospin basis state |K, m is completely antisymmetric. However the sum over the permutations for fixed values of K produces linear dependent states that have to be individuated and eliminated from the basis set [35,46,49]. This procedure could be delicate from a numerical point of view as the number of K increases.…”

“…The first steps to use the HH method without the Suzuki-Lee approach have been shown in Ref. [35], and intense research activity is currently underway. The formalism which will be presented here is in fact quite general, and can be applied also to the A = 5, 6 nuclear systems.Before concluding this section with the outline of this contribution, we would like to make few remarks: (i) the HH method is extremely powerful, and its application to systems up to A ∼ 7, 8 is limited essentially by computing power.…”

“…The first steps to use the HH method without the Suzuki-Lee approach have been shown in Ref. [35], and intense research activity is currently underway. The formalism which will be presented here is in fact quite general, and can be applied also to the A = 5, 6 nuclear systems.…”

Compressed lithium atom under Debye screening

“…Many investigations have been conducted to examine atoms enclosed within spherical and non-spherical cavities, starting from hydrogen [1-4], and extending to helium [5][6][7][8][9][10][11][12][13][14][15] and molecules [16][17][18][19]. However, limited research has been presented on confined many-electron atoms up to neon atom in the literature [20][21][22][23][24][25][26].…”

“…Confined systems serve as valuable models for a wide range of physics problems. When atoms and molecules are spatially confined within either permeable or impermeable cavities, their properties change significantly compared with their unconfined counterparts.Many investigations have been conducted to examine atoms enclosed within spherical and non-spherical cavities, starting from hydrogen [1-4], and extending to helium [5][6][7][8][9][10][11][12][13][14][15] and molecules [16][17][18][19]. However, limited research has been presented on confined many-electron atoms up to neon atom in the literature [20][21][22][23][24][25][26].…”