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Atomic Form Factor Calculations of S-States of Helium

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Atomic Form Factor Calculations of S-States of Helium American Journal of Modern Physics 2019; 8(4): 66-71 http://www.sciencepublishinggroup.com/j/ajmp doi: 10.11648/j.ajmp.20190804.12 ISSN: 2326-8867 (Print); ISSN: 2326-8891 (Online) Atomic Form Factor Calculations of S-states of Helium Saïdou Diallo *, Ibrahima Gueye Faye, Louis Gomis, Moustapha Sadibou Tall, Ismaïla Diédhiou Department of Physics, Faculty of Sciences, University Cheikh Anta Diop, Dakar, Senegal Email address: *Corresponding author To cite this article: Saïdou Diallo, Ibrahima Gueye Faye, Louis Gomis, Moustapha Sadibou Tall, Ismaïla Diédhiou. Atomic Form Factor Calculations of S-states of Helium. American Journal of Modern Physics . Vol. 8, No. 4, 2019, pp. 66-71. doi: 10.11648/j.ajmp.20190804.12 Received : September 12, 2019; Accepted : October 4, 2019; Published : October 15, 2019 Abstract: Variational calculations of the helium atom states are performed using highly compact 26-parameter correlated Hylleraas-type wave functions. These correlated wave functions used here yield an accurate expectation energy values for helium ground and two first excited states. A correlated wave function consists of a generalized exponential expansion in order to take care of the correlation effects due to N-corps interactions. The parameters introduced in our model are determined numerically by minimization of the total atomic energy of each electronic configuration. We have calculated all integrals analytically before dealing with numerical evaluation. The 1S2 11S and 1 S2S 21, 3 S states energies, charge distributions and scattering atomic form factors are reported. The present work shows high degree of accuracy even with relative number terms in the trial Hylleraas wave functions definition. The results presented here, indicate that the highly compact twenty-six variational parameters model will have the quantitative and qualitative applicability for the study of electronic correlation. The correlated wave functions are used to calculate the atomic form factor for the diffusion of electrons by the helium atom. The atomic form factor is evaluated as the Fourier transform of the electron density distribution of an atom or ion, which is calculated from theoretical correlated wave functions for free atoms. Finally, suggestions are made as to the way the atomic form factor of the helium atom may be approximated by a sum of Gaussians for efficiency use. Keywords: Helium, Variational Method, Correlation Effect, Charge Distribution, Scattering parameters must be introduce [1]. The number of parameters 1. Introduction increases with the system size up to several hundreds and the Several theories and methods of calculations have been optimization requires efficient strategies [2]. Several kinds of elaborated these last years to calculate energy levels and trial functions are used to deal with two-electron atomic or electronic charge distribution of atoms and molecules. The ion systems [3-6]. A comparative study of two-electron method of the self-consistent field proposed in 1927 by D. systems with different potentials is done by authors of Ref. Hartree is one of the best resolution processes of the quantum [7]. They discuss the limits of validity of the correlation chemistry problems. The stationary Schrödinger equation can hypothesis. Usually, the method consists of the product of be solved analytically in a very restricted number of case two factors wave functions. The first term includes the (atomic hydrogen, harmonic oscillator, etc.) Most problems electron–nucleus interactions. The second term takes into of the chemistry and quantum mechanics are solved with the account both the electron–electron interaction and the help of approximate methods. In many cases, variational correlation between all the parts of the system elements. This method enables us to solve much more efficiently the second terms can be constructed by a sum with a large Schrödinger equation. number of parameters or by a specific mathematical function. The problem of bound states of two-electron systems has a The Hartree–Fock method, the multiconfiguration Hartree- long history an attracts continuing interest. The variational Fock method and the Ritz method both apply the variational principle of solving this problem is largely used. In order to method. The variational method is sometimes, combined have accurate final wave function and energy, an adequate with other methods to investigate the bound states of two- trial function is chosen and a large number of variational electron system [8]. Highly correlated wave functions can be obtained by including more terms in the polynomial 67 Saïdou Diallo et al .: Atomic Form Factor Calculations of S-states of Helium representation of the correlation function [9-12]. Alternative inter-electronic repulsion potential . Variational methods correlated wave functions are proposed with few terms in the for solving the Schrödinger equation (1) rely on a expansion of the correlation [13-15]. reformulation of the eigenvalue problem. Solving the As illustration, we will discuss two electron systems; this Schrödinger equation for bound states is equivalent to finding includes helium and helium like systems. These are the functions that leave the energy functional simplest systems in which we can study correlation, but they do exhibit many interesting challenges of complex atoms. (3) The and states energies of the helium atom are | |! , = |! calculated1 by2 using Hylleraas basis wave functions that Which is recognized as the expectation value of the energy introduces the coordinates . The principal reason of for a stationary state satisfying the boundary conditions. In introducing these new variables was the volume integral (, , ) addition to the boundary conditions, the variation must also reduction from six to three integrals. A screening parameter have the expected integrability, continuity and has been introduced in our trial wave functions that can be differentiability properties. With the natural Hylleraas seen as an effective nuclear charge of the atom. The main coordinates . We see purpose of the present paper is to extend the calculation of that the variables = + and, = are − always, = |positives − | but the the ground and two first excited states energy of helium. In variable can take both the positive or negative values. The the other hand, the calculation of charge distribution is integrals in equation (3) will contain two parts. The important for the determination of scattering atomic form contribution to the integral from is identical with that factor. The atomic form factor is the scattering contribution from . In this condition, we multiplied(" ) the volume element from a single, isolated atom. Generally, it describes the shape by a factor of 2. The resulting volume element is now found of a scattering object. More specifically, it is the Fourier to be in the following form transform of the atom's spatial distribution. An accurate (4) calculation of wave function is very important to determine the spatial distribution that governs the shape of the atomic The limits of integration#$ = 2% become( − )### form factor. In the most general case of an arbitrary distribution of scattering density, the form factor is computed (5) by integrating over all space. The form of the wave0 'functions ' ' used ' is +( as follows [17] The paper is structured as follows. In section 2 we construct the twenty-six parameters variational wave (6) function. In this section, we have also established all the - (, , ) = )+./ *+,+(, , ) analytical expressions involved in the calculation and From the variational principle, one obtains the matrix minimization of the total energy of the system and the eigenvalue problem: electronic density of each configuration respectively. We (7) derive in section 3, the explicit expression of the scattering atomic form factor. The analytical expressions are obtained The Hamiltonian and( overlap − 0) *matrix = 0 are represented by with the spherical symmetry density. The binomial expansion and respectively. The coefficients are elements of the C is used to evaluate the radial integrals where it’s needed. We matrix.0 The secular equation is: *+ compare our results with some previous calculations done by other authors in section 4. − 20 − 20 3 - − 20 - (8) − 20 − 20 3 - − 20 - 2. Variational Calculations 1 1 = 0 4 4 4 - - - - -- -- The Schrödinger equation for the helium atom, or the Where − is 20 the eigenvalue − 20 matrix3 and the− matrix 20 elements positive ions or the negative hydrogen ion of the . The kinetic energy matrix elements of the isoelectronic sequence is 2 Hamiltonian+5 = 6+5 + $are+5 defined by: (1) (9) = ; : 9 Where the Hamiltonian is given by 6+5 = 7/ # 7/ # 7/ # (8/ + 8 + 8) Where (2) (10) >? A >? A >? A = − − − − + >? @ >? @ >? @ where are the coordinates of the two electrons; is 8/ = ( − <) = >: >: + >B >B + >9 >9 C the distance between them, the charge atomic number, (11) , >? @ >? A >? A >? @ and the eigenfunction and eigenenergy of the system. = 2 8 = ( − ) = >9 >: + >9 >: C The complete description of the S-states requires only three (12) independent coordinates, and [16]. The effects of >? @ >? A >? A >? @ 8 = ( − ) = >9 >B + >9 >B C correlation are globally introduced, in the wave functions by and the potential matrix elements are expressed as $+5 American Journal of Modern Physics 2019; 8(4): 66-71 68 (13) The integration over the coordinate is possible. There ; : 9 are two regions of integration: and . So we can $+5
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