Quasiclassical Approximation in the Non-Relativistic and Relativistic Problems of Tunneling Ionization of a Hydrogen-Like Atom in a Uniform Electric Field

EPJ Web of Conferences 108, 0 2 0 39 (2016 ) DOI: 10.1051/epjconf/ 2016 10 8 0 2 0 39  C Owned by the authors, published by EDP Sciences, 2016 Quasiclassical Approximation in the Non-Relativistic and Relativistic Problems of Tunneling Ionization of a Hydrogen-Like Atom in a Uniform Electric Field O. K. Reity1 , a , V. K. Reity1 , b , and V. Yu. Lazur1 , c 1 Department of Theoretical Physics, Uzhhorod National University, 54 Voloshyna Street, Uzhhorod 88000, Ukraine Abstract. A recurrent scheme for finding the quasiclassical solution of the onedimensional equation obtained after the separation of variables in the Schrödinger equation in parabolic coordinates is derived. The method of quasiclassical localized states is developed for the Dirac equation with an arbitrary axially symmetric potential of barrier type which does not allow complete separation of the variables. By means of the proposed quasiclassical methods the non-relativistic and relativistic wavefunctions for hydrogenlike (H-like) atoms in an external uniform electrostatic field of intensity F are constructed in the classically forbidden and allowed regions. The general analytical expressions of the leading term of the asymptotic behaviour (at small F) of the ionization rate of an H-like atom in the uniform electrostatic field are obtained for the non-relativistic and relativistic cases. 1 Introduction The problem of the hydrogen atom in an electric field plays a fundamental role in quantum mechanics and atomic physics and has many applications (see, e.g., [1–3] and the references therein). Since the twenties (see, e.g., the review [4]), the properties of the energy spectrum of the hydrogen atom and other atoms in external fields were rather intensively studied in the framework of the Schrödinger equation. At the same time the logic of development of the studies of highly ionized atomic systems demands the formulations of new problems, similar to those already solved only for neutral (or weakly ionized) atomic systems. The relativistic character the of electron motion in fields created by multiply charged ions (the characteristic velocity of the electron in H-like ions with nuclear charge Z is ∼ αZc; α is the fine structure constant, c is the velocity of light) distinguishes them drastically from the neutral atoms. Thus, the consistent theory of the tunneling ionization of such systems should be essentially relativistic since the relativistic effects are not small in this case, and moreover they determine the order of magnitude of the spectral characteristics. In order to construct such a theory one should employ the solution of the relativistic problem of the electronic motion in the field created by the nucleus and a constant uniform electric field. Since the a e-mail: b e-mail: c e-mail:                4                   Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/201610802039 EPJ Web of Conferences Dirac equation with such superpositional potential does not permit complete separation of variables in any orthogonal system of coordinates, the given problem has not exact analytical solution, and numerical methods are needed which demand significant computational efforts. Relativistic calculations of the linear Stark effect were carried out by means of perturbation theory [5, 6], and the quadratic Stark effect was treated by means of the RCGF (Relativistic Coulomb Green Function) method in the form of the expansion in powers of Zα [7]. However, the publications in this field are basically devoted to the calculation of the position of a quasistationary level, and there are only rare cases of calculation of level widths Γ = w (w is the tunneling ionization rate) in the relativistic case. In our previous paper [8] a hybrid version of a spherically symmetrical model of the Stark effect with account the Lorentz structure of the interaction potential has been studied in the quasiclassical approximation. The ionization rate of the s-level, the binding energy of which can be of the order of the rest energy in electric and magnetic fields has been calculated by means of a generalization of the imaginary time method (ITM) [9] and the so-called ADK-theory [10]. However, in the general case, the widths of the quasistationary states have not been found until now. Due to such situation in the theory and the intensive experimental researches during the last years, asymptotic methods for the calculation of ionization rates, which are based on clear physical ideas describing the under-the-barrier electronic transition, become especially important. From this point of view it is worthwhile to use the quasiclassical approximation which enables one to find the approximative analytical solutions of the relativistic problem and to express the required ionization probability in terms of the quantum penetrability of the potential barrier which separates the domains of discrete and continuous spectra. As it is known, this method has a rather high accuracy even for small quantum numbers. In the present paper we apply the quasiclassial approximation to both the non-relativistic and relativistic problems of tunneling ionization of H-like ions in a constant uniform electric field. The first problem is much simpler than the second one due to the separability of the Schrödinger equation in parabolic coordinates. In this problem we can use the expansion in powers of the Planck constant . For the relativistic problem we apply the method of quasiclassical localized states for the Dirac equation with axially symmetrical potential the basics of which were described in [11]. 2 Quasiclassical solutions of the non-relativistic problem of the atom in a constant uniform electric field The potential of an H-like atom with charge Z in a constant uniform electric field (the intensity vector F of which is opposite to the axis z) can be represented in the form ( = e = me = 1) V = −Z/r − Fz. (1) As it is known [1], the Schrödinger equation with the potential (1) permits complete separation of variables in parabolic coordinates ξ = r + z, η = r − z, φ = arctg(y/x). For this purpose we seek the solution in the form Ψ = (ξη)−1/2 ϕ(ξ)χ(η)e±imφ , (2) where m = 0, 1, 2, . . . is the absolute value of the magnetic quantum number. Substituting (2) into the Schrödinger equation, we obtain the following equations for the unknown functions ϕ(ξ) and χ(η):   d2 ϕ E β1 1 − m2 F + + + + ξ ϕ = 0, (3) 2 ξ 4 dξ2 4ξ2   E β2 1 − m2 F d2 χ + − (4) + + η χ = 0, β1 + β2 = Z. 2 η 4 dη2 4η2 02039-p.2 Mathematical Modeling and Computational Physics 2015 For the energy E of a quasistationary level we shall use the known expansion [12] E=− Z2 3n(n1 − n2 )F − + O(F 2 ), 2Z 2n2 (5) where n = n1 + n2 + m + 1 is the principal quantum number. With (...truncated)