Critical stability and quantum phase transition of two-electron system under exponential-cosine-screened-Coulomb interac
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THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Critical stability and quantum phase transition of two-electron system under exponential-cosine-screened-Coulomb interaction Anjan Sadhukhan1,2 , Sujay Kr. Nayek3 , and Jayanta K. Saha2,a 1 2 3
Department of Physics, Adamas University, Barasat, Kolkata 700126, India Department of Physics, Aliah University, Newtown, Kolkata 700 160, India Department of Mathematics, Netaji Nagar Day College, Kolkata 700 092, India Received 9 June 2020 / Accepted 4 September 2020 Published online 13 October 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. The stability of two electron system under the influence of the exponential-cosine-screenedCoulomb (ECSC) potential has been studied as a function of continuously varying nuclear charge (Z) and screening constant (µ). Inter-electronic correlation has explicitly been considered by adopting Hylleraas type basis set under Ritz variational framework. The correlated basis integrals originating from the elements of Hamiltonian and overlap matrices are evaluated analytically. Critical nuclear charges (Zc ), the minimum amount of nuclear charge required to form at least one bound state, corresponding to different µ values have been predicted. Different structural properties along with two-particle densities (TPD) for all possible orientations of the electrons have been estimated using the optimized wave function. Prominent signature of quantum phase transition (QPT) of the system around the Zc values has been reported. A comparative study has been done on the critical stability of the system under the influence of ECSC potential, screened Coulomb (SC) potential and pure Coulomb (PC) potential.
1 Introduction
˜ E(λ) =
The Hamiltonian of the two-electron system with Z nuclear charge (referred as Zee system) for infinite nuclear mass approximation is given by, H=−
2 X 1 i=1
2
∇2i
Z + ri
+
1 r12
(1)
where, ri represents the electron-nucleus distance and r12 represents the inter-electronic separation. Under the scaling transformation r → Zr , (1) will reduce to [1], ˜ =− H
2 X 1 i=1
2
∇2i
= H0 + λH 0
1 + ri
+
1 1 Z r12 (2)
with, the energy eigen value E˜ = E/Z 2 , where E is the eigen value of (1). In (2), 1/Z factor is considered to be the perturbation parameter (λ), the inter-electronic interaction 1/r12 is considered as the perturbing operator (H 0 ) and rest of the Hamiltonian is considered as the unperturbed one (H0 ). The energy eigen value of (2) can be expressed as the power series, a
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∞ X
n λn
(3)
n=0
where n ’s are the different order of perturbation corrections. It’s a convergent series and Kato [2] proved that, the series (3) possess a finite radius of convergence (λ∗ ) on the positive real axis of the complex λ plane. There exists a value of λ, say λc for which the energy eigen value will be degenerate with the threshold energy, i.e. the energy of the corresponding one-electron (Ze) system. The value
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