Generalized potential for confined positronium atom immersed in plasmas

Journal Pre-proofs Generalized potential for confined positronium atom immersed in plasmas M.K. Bahar, A. Soylu PII: DOI: Reference:

S0301-0104(19)31103-6 https://doi.org/10.1016/j.chemphys.2019.110584 CHEMPH 110584

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Chemical Physics

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Please cite this article as: M.K. Bahar, A. Soylu, Generalized potential for confined positronium atom immersed in plasmas, Chemical Physics (2019), doi: https://doi.org/10.1016/j.chemphys.2019.110584

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Generalized potential for confined positronium atom immersed in plasmas M.K. Bahar1∗ and A. Soylu2† 1 2

Department of Physics, Faculty of Sciences, Sivas Cumhuriyet University, 58140, Sivas, Turkey,

¨ Department of Physics, Faculty of Arts and Sciences, Ni˘ gde Omer Halisdemir University, 51240, Ni˘ gde, Turkey

Abstract We suggest, for the first time, the more general exponential cosine screened Coulomb (MGECSC) potential to investigate the positronium (Ps) atom immersed in Debye and quantum plasmas. Ps atom immersed in Debye and quantum plasmas under spherical confinement is examined by solving the corresponding Schr¨ odinger equation via the tridiagonal matrix method. The MGECSC potential has four different sets to model Debye and quantum plasmas. But, we consider only two sets as the screened Coulomb (SC) potential for Debye plasma and the MGECSC potential for quantum plasma. The results of the confinement and the plasma screening effects on the energies, radial matrix elements, dipole polarizabilities and oscillator strengths of Ps atom are obtained, discussing in detail. It is worthwhile to specify that the spectroscopic features of Ps atom located in only plasma environment enable detailed information about plasma environment. The MGECSC potential is proposed to model various interactions of Ps atom immersed in plasma environment. In this study, the necessity and details of the MGECSC potential is discussed. PACS numbers: 03.65.Ge, 52.77.Bn, 87.53.Bn Keywords: Positronium atom, Debye and quantum plasma, MGECSC potential

∗

Electronic address: E-mail:[email protected]

†

Electronic address: E-mail:[email protected]

1

I.

INTRODUCTION

Hydrogen atom model may be applied to exotic atoms in which proton or electron replaces with some other particles, by performing some trivial modifications. For examples, muonic hydrogen (p+ µ− ), pionic hydrogen (p+ π − ) or positronium (e+ e− ). These exotic atoms are unstable, but the most of them has a long enough lifetime to exhibit a well-defined spectra. Specially, Ps atom that is a quasistable system consisting of an electron and positron allows a good test possibility for the quantum electrodynamics before it annihilates by emitting gamma-ray radiations. The gamma-ray radiations emitted by positron-electron annihilation have been detected in condensed objects [1]. In this manner, positron annihilation lifetime spectroscopy is one of the most powerful methods in examinating of the defects in polymer, metal and semiconductors [2–4]. The positron lifetime in solid, liquid and gas materials depends on electronic environment closely. This dependence enables information about the structure of the environment (or material) that it is too small to be seen using a microscope. On the other hand, the creation-annihilation cases of positron in positron-electron plasmas have also gained remarkable interest in astrophysical and laboratory plasma researches [5, 6]. Besides, Ps atoms in Rydberg states have been observed in solar radiation spectra [7] and interstaller ambient [8]. Ps atom in Rydberg states has been also employed for antihydrogen creation in anti-matter experiments [9]. Collision processes related to positron in dense plasmas have been interesting subjects in recent years due to wide applications above mentioned in physics and astrophysics [10, 11]. The plasma screening effects on Ps formation through positron-hydrogen collisions in Debye plasma have been examined by using the screening approximation model [12]. The influence of dense quantum plasmas on Ps formation through scattering of positron from ground states of hydrogen atoms has been analysed within framework of the distorted wave theory by considering modified DebyeH¨ uckel potential to model interactions in plasma [13]. Another study on Ps formation in Debye plasma has been carried out by using the second-order distorted-wave approximation within the adiabatic dipole polarization potential [14]. In addition to these studies, the investigation of cross-sections concerning Ps formation for positron-lithium collisions via the screening approximation model [15], the investigation of the Debye plasma screening effect on the scattering process of positron from ground states of hydrogen atom via the distorted wave theory [16], the investigation of cross-sections for estimating Ps formation

2

for plasma applications via the classical trajectory Monte Carlo approximation [17] are some of the latest studies on Ps formation. The most of such plasma-immersed atom/positron studies contains positron-atom elastic collision, exciting, rearrangement processes and resonance investigations. The general result of such examinations is that plasma environments affect closely Ps formation, in other words, collision dynamics and plasma parameters have a special ranging for specific properties of collision processes. For example, the threshold of Ps formation decreases with increasing of the screening effect in Debye plasmas, which denotes that low scattering energy range is more significant for Ps process. The effects of plasma environment on structural features of confined Ps atom immersed in plasmas as well as collision calculations involving Ps atom have also been studied in detail. The influences of Debye plasma medium depicted by Debye-H¨ uckel potential on energy levels and structural properties of the compressed Ps atom have been examined through radial set obtained by using linear combination of Slater-type orbitals [18]. Considering same potential model, the effects of Debye plasma on electron affinity of Ps− have been analysed by using multiterm correlated basis sets [19]. The structural properties of Ps atom immersed in spherical confined plasmas represented by the screened Coulomb (Debye-H¨ uckel) and exponential cosine screened Coulomb potentials have been examined using 8th order central finite difference method [20], in which study has been observed that the degeneration disappears as a result of increasing confinement. The common result of these studies exerted on plasma-immersed Ps atom is that there are remarkable influences of confinement effect and plasma environments on energy levels and structural features of Ps atom such as oscillator strength, dipole polarizability, transition probability. These remarkable influences can be illustrated as the decrease of bound state energies with increasing confinement, and the decrease of the energies with increasing Debye shielding parameter. The plasma screening effects create crucial effects not only for Ps atom but also for other plasma-immersed atomic systems [21–24]. Atomic exciting and ionization processes play an important role in interpreting of various scenes related to hot plasma physics, astrophysics and experiments done by using positive charged ions. Also, the emission line formed by excitation gives remarkable information about plasma observables. In this connection, we should underline that the most convenient medium for exciting atomic systems to radiate is the plasma environment. Considerable screening effects presented by plasmas on structural properties of atomic systems in addition to the significance of plasma for experimental researches make something important 3

also theoretical investigations about plasmas. Plasmas can be categorized according to their thermodynamic equilibrium, pressure, ionization degree, neutrality, magnetization or frequency. The most of plasmas in nature and mentioned above categorization are the weakly coupled (Debye) plasmas. Plasmas can be classified more generally in two different ways, in which classification there are two significant parameters: plasma temperature T and plasma number density n. Therefore, plasmas can be classified as classical and quantum plasmas in case with and without collision [25, 26]. The Coulomb coupling parameter scales to n1/3 /T is defined as the ratio of the average energy to the kinetic energy. The quantum degeneration parameter (γ) defined as the ratio the Fermi temperature to the particle temperature scales to n2/3 /T . While, when γ  1, plasmas are defined by the classical statistic within framework of the Maxwell distribution, they are defined by the quantum statistic when γ ≥ 1. The Coulomb coupling parameter in quantum plasmas compared to classical plasmas scales to n−1/3 instead of n1/3 . While collisions in quantum plasmas are important for higher densities, ones in classical plasmas are notable for lower densities [27]. In this connection, as well-known, while the screened Coulomb (SC) potential is utilized to describe Debye plasma interactions, the exponential cosine screened Coulomb (ECSC) potential is utilized in order to characterize quantum plasmas represented by low temperature and high density [28]. However, in this study, for the first time, the more general exponential cosine screened Coulomb (MGECSC) potential has been proposed in order to investigate Ps atom immersed in Debye and quantum plasmas. The MGECSC potential for Ps atom immersed in plasmas is given by  ar  −e2 (1 + br) exp(−r/λ)cos , VM GECSC (r) = r λ

(1)

where a, b and λ are the screening parameters [29]. Also, λ can be taken into consideration as Debye shielding parameter for weakly coupled plasmas. There are four different sets constituted by a and b parameters of the MGECSC potential: the SC potential when a = b = 0; corresponding potential when a = 0, b 6= 0; the ECSC potential when a 6= 0, b = 0; the MGECSC potential when a 6= 0, b 6= 0. But, in this study, while the SC potential that is the most essential one of the MGECSC potential is considered to Debye plasma interactions, the MGECSC potential that is the most general form is considered to quantum plasma interactions. However, it is worthwhile to note that the MGECSC potential can also be reduced to pure Coulomb potential for a = b = 0, λ −→ ∞, as well as it reduces to the SC 4

and the ECSC potentials. Since Ps atom is surrounded to a certain region in experimental applications, Ps atom under spherical confinement has been considered in this study. This paper is organised as follows: in Section II, the formalism employed to analyse Ps atom in plasmas is outlined. In Section III, computational details are briefly presented. In Section IV, discussions are given. Finally, Section V is assigned to our summary and conclusion.

II.

FORMALISM

Hamiltonian for Ps atom immersed in plasma environment and located in centre of spherical confinement is given by H = Hcm + Hrm , where Hcm and Hrm are Hamiltonians related to the center-of-mass motion and relative motion, respectively. Hcm and Hrm are given by P2 , 2M

(2)

p2 + VM GECSC (|r|) + Vc (|r|), 2µ

(3)

Hcm =

Hrm =

where, being P1,2 and r1,2 the momentum and position vectors of the electron and positron, respectively; while R = (r1 + r2 )/2 and P = (p1 + p2 ) are the center-of-mass coordinates, r = (r1 − r2 ) and p = (p1 − p2 )/2 are the relative motion coordinates. Being m the electron and positron mass, M = 2m and µ = m/2 are the total and reduced mass of the system, respectively. Vc (|r|) denotes spherical confinement potential, and it is defined by   0, if |r| < r 0 Vc (|r|) =  ∞, if |r| ≥ r0 ,

(4)

where r0 is the spherical confinement radius. The total wave function and energy of system, respectively, are Ψ(r, R) = ϕ(r)Θ(R) and E = Ecm + Erm . As seen from Eq.(3), only Hrm includes influences of plasma environment. Therefore, eigenvalue equation to be solved is Hrm ϕ(r) = Erm ϕ(r), where ϕ(r) and Erm are the wave function and energy related to the relative motion. ϕ(r) wave function for Schr¨odinger equation in the relative coordinates may be proposed as

ϕ(r) =

un` (r) Y`m (ˆ r), r 5

(5)

where Y`m (ˆ r) is the spherical harmonics. In the relative coordinates, the radial Schr¨odinger equation including MGECSC potential is obtained through Eq.(5) in the following form: u00n` (r) +

m e2 `(` + 1)
E + (1 + br)exp(−r/λ)cos(ar/λ) − un` (r) = 0, n` ~2 r r2

(6)

where En` = Erm . After this, unless indicated otherwise, atomic units (~ = 1 as well as m = e = 1) will be used throughout the study. As seen, analytical solution of the wave equation containing the MGECSC potential is quite difficult at the present time. So, we have focused on numerical solution of Eq.(6). The tridiagonal matrix algorithm has been used to solve Eq.(6) [30]. The calculations are performed in order to obtain the transition energies, oscillator strengths, and dipole polarizabilities. The oscillator strength of the dipole transition to a final state (nf `f ) (`f = `i ± 1) from an initial state (ni `i ) is given by [31, 32] 2 `max f (ni `i −→ nf `f ) = 4E |Dif |2 , 3 2`i + 1

(7)

where `max = max(`i , `f ), and 4E is the transition energy (4E = Ef − Ei ). Dif is the dipole transition matrix element, and being Rn` (r) = un` (r)/r the radial part of the wave function, it is given by [31, 32] Z

r0

Dif =

Rni `i (r) rRnf `f (r) r2 dr.

(8)

0

As well-known, when Enf `f > Eni `i , the oscillator strength is positive, and it is defined as absorption oscillator strength. When Enf `f < Eni `i , the oscillator strength is negative, and it is defined as emission oscillator strength. When an atom or molecule interacts with an external electric field, it becomes polarized. Then, a new charge distribution is described by inducing the electric multiple moments. The polarizabilities are important for research areas such as electromagnetic field-matter interactions, interatomic interactions, and collision processes. The dipole polarizability of a system may be defined as lowest-order response of the system to an external electric field. The static dipole polarizability of an atom located in ground state under the external electric field in direction of z-axis, is given by αD = 2e2

| < Ψn`m |z|Ψ100 > |2 , En − E1 6

(9)

where summation is for all states including discrete and continuum wave functions. Therefore, the analytical computation of the integrals is quite complicated. However, for oneelectron atoms in free space, there are some expressions obtained by using variational methods in order to compute the dipole polarizability [33, 34]. The static dipole polarizability statement proposed by Buckingham [35] is expressed as B αD

  2 6 < r2 >3 +3 < r3 >2 −8 < r >< r2 >< r3 > = , 3 9 < r2 > −8 < r >2

(10)

where the integrals are calculated using the ground state wave functions. The dipole polarB izability in this study are determined through αD .

III. A.

COMPUTATIONAL DETAILS Debye Plasma Case

a = 0, b = 0 case in MGECSC potential is well-known SC potential that characterizes Debye plasma interactions. In Table I, the energies of some quantum states of Ps atom in Debye plasma represented by SC potential with λ = 5 − 200 for r0 = 10 spherical confinement radius have been shown in atomic units (a.u.). In Table II, the influences of spherical confinement (r0 = 5, 10, 20, 50) on energies of some quantum states of Ps atom in Debye plasma represented by SC potential with λ = 100 have been shown in a.u. In Table III, for r0 = 5, the dipole matrix elements related to some transitions for Ps atom in Debye plasma represented by SC potential with λ = 50 have been presented in a.u. In Table IV, influences of confinement effect (r0 = 5, 10, 20) and Debye shielding parameter (λ = 50, 100, 200) on dipole polarizability of Ps atom in Debye plasma have been presented in a.u. In Fig.1, effects of λ parameter (λ = 1 − 100) and spherical confinement (r0 = 5, 10, 20) on oscillator strengths of 1s − 3p, 1s − 4p, 2p − 5d transitions have been introduced in a.u. In Fig.3, for 1s-3p transition, the change of |Dif |2 as a function of r0 for SC potential with λ = 10 and MGECSC potential with a = 1, b = 1, λ = 10 has been shown in a.u.

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B.

Quantum Plasma Case

a 6= 0, b 6= 0 case in MGECSC potential is an alternative potential to ECSC potential in order to examine quantum plasma interactions. In Table V, for r0 = 10, the energies of some quantum states of Ps atom in quantum plasma modeled by MGECSC potential with λ = 5 − 200, a = 0.1 − 20, b = 0.1 − 20 have been shown in a.u. In Table VI, spherical confinement effects (r0 = 5, 10, 20, 50) on energies of some quantum states of Ps atom in quantum plasmas modeled by MGECSC potential with a = 1, b = 1, λ = 100 have been presented in a.u. In Table VII, for r0 = 5, the dipole matrix elements related to some transitions for Ps atom in quantum plasma modeled by MGECSC potential with a = 1, b = 1, λ = 50 have been introduced in a.u. In Table VIII, the influence of confinement effect (r0 = 5, 10, 20) and potential parameters (λ = 50, 100, 200; a = 0.1, 1, 10; b = 0.1, 1, 10 ) on dipole polarizability of Ps atom in quantum plasma have been introduced in a.u. In Fig.2, effects of λ parameter (λ = 1 − 100), a parameter (a = 0.1 − 20), b parameter (b = 0.1 − 20), and r0 parameter (r0 = 5, 10, 20) on oscillator strengths of 1s − 3p, 1s − 4p, 2p − 5d transitions have been demonstrated in a.u.

IV.

DISCUSSIONS

The charged particles in plasma environment interact even over very long distances. The SC and ECSC potentials describe Coulombic interactions between charged particles. However, when considering the diversity of particles in plasma environment, and quite involved correlation between them, presence of some screening effects (cannot be modeled by λ; for example, a perturbation or the effect of effective mass for cold plasmas in particular) that cause the interactions more complicated can be discussed. Else, an external perturbation can be affected on the system. Such effects can be categorized as energy-increasing or energydecreasing effects. Therefore, a and b parameters in MGECSC potential can be proposed to describe energy-increasing/decreasing some external or screening influences that cannot be depicted by SC and ECSC potentials [29]. Also, MGECSC potential displays a stronger screening effect compared to ECSC potential [29]. We have investigated in detail plasma screening and spherical confinement effects on spectroscopic properties such as the energy values, oscillator strengths for allowed-transitions,

8

and dipole polarizability of Ps atom immersed in Debye and quantum plasmas characterized by MGECSC potential. There are four significant parameters on structural properties of Ps atom in a such quantum system: plasma screening parameters (a, b, λ) and spherical confinement parameter r0 . λ parameter is taken into consideration as Debye length in Debye plasma, and it depends on plasma temperature T and plasma number density n with respect to λ = (ε0 kB T /e2 n)1/2 , where ε0 , kB , e are, respectively, the dielectric constant of vacuum, the Boltzmann constant, and electron charge. Quantum plasmas include quantum mechanical effects as a result of overlapping of the bound state wave functions of particles close to each other due to their lower-temperature high-density characteristic. Namely, the temperature and particle density are also major physical properties for quantum plasmas, and so λ parameter in quantum plasma environments relates to the quantum wave number of electrons. Various values of λ parameter from 1 to 200 for Ps atom have been considered (see Table IV and Figures). The increase of λ parameter decreases bound and pseudocontinuum states energies of Ps atom immersed in Debye and quantum plasmas (see Table I,V). Increasing λ parameter increases the attraction of the interaction potential [29], and so it lifts down the localizations of bound and pseudocontinuum state energies. Also, it should be underlined that, as seen from Table I, the energy values obtained in this study have a very good agreement with the results of Ref.[20]. There is no the influence of the increase of λ parameter on the number of bound or pseudocontinuum states in Debye plasma. Because, for illustrative example; when λ → ∞, E2s = 0.0706270376 a.u. in Table I. However, in quantum plasma, the increment of λ parameter leads to increase the number of bound or pseudocontinuum states. The reason of this case is stronger screening effect exhibited by MGECSC potential. Moreover, it is worthwhile to note that the increase of λ parameter doesn’t display same effect on transition energy for all allowed-transitions. Because, according to data in Table I, while λ parameter increases from 5 to 200; E1s−2s transition energy increases from 0.291547 a.u. to 0.318793 a.u., but E2p−3p transition energy decreases from 0.352608 a.u. to 0.350049 a.u. That similar case is also valid for data of Table V. On the other hand, while the increment of λ parameter decreases dipole polarizability of Ps atom immersed in Debye plasma (see Table IV), it leads to increase dipole polarizability in quantum plasma due to stronger screening effect of MGECSC potential (see Table VIII). However, for quantum plasma, it can be mentioned a critical value of ”a” parameter that changes the trend of λ parameter on dipole polarizability. This critical value arises from 9

oscillation of cos(ar/λ) term in MGECSC potential. When r0 = 5, b = 0; the critical value of ”a” is around 1.05. But, when considering different values of b and λ parameters, this critical value of ”a” can show a change due to different plasma screening effects. Namely, the criticality of ”a” depends on other plasma screening parameters. When considering the dependence to plasma temperature T and number density n of λ, it is evident that dipole polarizability increases due to the fact that the increase of plasma number density (n) increases the number of Coulombic interactions of Ps atom for a given T (see Table IV). At the same time, the decrease of plasma temperature (T ) exhibits same effect for a given n. Because, the decrease of T decreases kinetic energies of plasma particles, which in turn leads to more effective Coulombic interactions. On the other hand, since the general trend of energies of system in quantum plasma tends to create bound state, the dipole polarizability becomes more meaningful in greater values of λ, as can be seen from Table V. There is remarkable influence of λ parameter on oscillator strengths of Ps atom immersed in Debye and quantum plasmas characterized by both sets of MGECSC potential. As may be gathered from Figs.1,2, the influence of λ parameter on oscillator strengths varies with respect to types of transitions. It is worthwhile to specify that there are critical values of λ parameter that leads to a peak in oscillator strengths, as may be seen from Figs.1,2. For quantum plasma, the increase of b parameter lifts down the localizations of energy levels (see TableV). MGECSC potential becomes more attractive by increasing b parameter [29], which in turn means a decrease in energies, as confirmed in Table V. As increasing b decreases energies of Ps atom immersed in quantum plasmas, it may be considered in modeling of energy-decreasing external or screening effects. The localizations of energy levels of Ps atom display a tendency to move from pseudocontinuum states to bound states, due to increasing b parameter, as can be seen in Table V. In that case, it can be said that the increase of b parameter decreases the instability of Ps atom in plasmas. However, b parameter has not same effect on transition energy for all allowed-transitions, as also in case of λ. For illustrative example, as may be gathered from Table V, for increasing b from 0.1 to 20, while E1s−2s transition energy increases from 0.32056 a.u. to 0.676287 a.u., E4d−5d transition energy decreases from 0.663594 a.u. to 0.638055 a.u. Moreover, in quantum plasma, the increase of λ parameter increases the number of bound states thanks to lifting down influence of b parameter on localizations (see TableV). When considering the dominant effect of b parameter on energies, its dominant effect on dipole polarizability may be also predicted. 10

Accordingly that, as can be seen from Table VIII, the increase of b parameter decreases dipole polarizability. If the attractive influence of b parameter on the effective potential is considered, it is evident that the dipole polarizability becomes meaningful for feasible large values of b parameter. Taking into account the dependence to b parameter of oscillator strengths, it may be said that oscillator strengths tend to decrease gradually (see Fig.2). In other words, there is no a critical value of b parameter on oscillator strengths. The localizations of energy levels of Ps atom immersed in quantum plasmas increase monotonically with increasing ”a” parameter (see Table V). Because, increasing ”a” parameter leads to a more repulsive potential [29]. Therefore, ”a” parameter can be utilized in modeling of energy-increasing some external or screening effects for Ps atom immersed in quantum plasmas. When investigating the general trend of ”a” parameter on energies, it may be said that ”a” parameter tends to increase instability of Ps atom immersed in quantum plasma. In addition, the increase of ”a” parameter doesn’t also exhibit same effect on transition energy of all allowed-transitions. For illustrative example, as may be gathered from Table V, for the increase of ”a” parameter from 0.1 to 10; while E1s−2s transition energy increases from 0.335781 a.u. to 0.392598 a.u., E3d−4d transition energy decreases from 0.451941 a.u. to 0.448400 a.u. There is also considerable effect of ”a” parameter on dipole polarizability. The dipole polarizability of Ps atom in quantum plasmas decreases with increasing ”a” parameter (see Table VIII). When examining the influence of ”a” parameter on oscillator strengths, it is observed that there is no a critical value of ”a” parameter (see Fig.2). ”a” parameter decreases oscillator strength, as seen Fig.2. As seen from Table VI, energy values in environment without plasma are higher than that in environment with plasma. But, the energy levels can be made higher or lower through ”a”, b and λ parameters compared to ones in environment without plasma. In addition, when considering the energy-increasing/decreasing effects of a, b, λ and r0 parameters, it is obvious that the incidental degeneracy cases reported in Ref.[20] may be observed thanks to shifting effects of a, b, λ and r0 parameters in this study. The increase of r0 lifts down the localizations of energy levels of Ps atom in Debye and quantum plasma (see Table II,VI), in which case is an expected result for the most of confined atomic systems. Accordingly this result, it is worthwhile to mention that the increase of r0 leads to change from pseudocontinuum states to bound states of energy spectra of Ps atom embedded in Debye and quantum plasmas. But, of course, the ability of r0 to create the bound states changes for values of the potential parameters, which case can 11

understand more clearly by comparing with each other Table II,VI. When considering influence of r0 on the spectra, it is evident that increasing r0 becomes more stable Ps atom. On the other hand, the influence of r0 parameter on transition energy for all allowed-transitions is same in plasma environments characterized by both sets of MGECSC potential, and it increases the transition energy. For illustrative example, from data in Table VI, for the increase of r0 from 5 to 20; E1s−2s transition energy decreases from 1.105718 a.u. to 0.255914 a.u., E3p−4p transition energy decreases from 2.304152 a.u. to 0.132596 a.u., E4d−5d transition energy decreases from 2.708208 a.u. to 0.066128 a.u.,... The influence of r0 parameter on dipole polarizability is considerable and this influence is same for Ps atom immersed in plasmas described by both sets of the MGECSC potential (see Table IV,VIII). The increase of r0 increases dipole polarizability, and the influence of r0 on dipole polarizability is more dominant compared to that of other parameters. However, there are also considerable effects of r0 on oscillator strengths, which is already an expected result when considering the dipole matrix elements. If λ-regimes at which oscillator strengths become stable are considered, it is clear that the increase of r0 increases oscillator strengths (see Figs.1,2). Furthermore, if how to change with r0 the square of the dipole matrix element (|Dif |2 ) is probed, the presence of a critical r0 value (rc ) can be determined. For example, when considering 1s − 3p transition, |Dif |2 increases by rc -value and then decreases. This rc -value is around 25 for SC potential with λ = 10; 11 for MGECSC potential with a = 1, b = 1, λ = 10 (See Fig.3). A critical r0 value (rc ) that exhibits same behaviour (in terms of peaking) is also possible for other transitions, and its value depends on type of transitions and values of potential parameters.

V.

CONCLUSION

In this study, for the first time, the energy values, dipole matrix elements, oscillator strengths and dipole polarizabilities of Ps atom immersed in Debye and quantum plasmas described by MGECSC potential have been examined by solving the Schr¨odinger equation via the tridiagonal matrix method. We have determined some interesting specifications of proposed new model in order to investigate the structural properties of Ps atom in plasma environments, as well as a very good agreement with the results of the models (the SC and ECSC potentials) in other studies. The reason of these new properties is the con12

sideration of ”a” and b potential parameters, in other words, energy-increasing (”a”) and energy-decreasing (b) some external or screening effects. The observed interesting results can be outlined as follows: Firstly, there is a critical value (ac ) of ”a” parameter which in turn leads to change the effects of λ parameter on dipole polarizability of Ps atom immersed in quantum plasma. Secondly, although the increase of λ parameter decreases dipole polarizability in Debye plasma, the increase of λ parameter increases polarizability by stronger screening effect of MGECSC potential in quantum plasma. More specifically, in quantum plasma compared to Debye plasma, the inverse behaviour exhibited by λ parameter on dipole polarizability arises from the more attractive feature of MGECSC potential. The increasing or decreasing general behaviours of ”a”, b and λ parameters on energy values are not affected by each other. But, same case is not valid for the radial wave functions. The attraction-repulsion or narrowing-widening cases on the interaction potential profile, more precisely, a different potential affects the radial probabilites of the wave functions. Namely, any change in potential profile affects closely oscillator strengths and dipole polarizabilities due to steeping/flattening or overlapping cases of the wave functions connected to allowedtransitions. These facts mentioned on the potential profile and the radial wave functions are the reasons of different behaviour of λ parameter on the dipole polarizability in Debye and quantum plasma. In this manner, the oscillator strength behaviours should be evaluated by considering Eqs.(7,8,10) in parallel with above discussions. Actually, these explanations imply more detailed investigation of the dipole matrix element. While the most dominant plasma screening parameter on energies of Ps atom is b parameter, the weakest one is ”a” parameter. Since b parameter increases the number of bound states of Ps atom, it decreases instability of the system. Also, ”a” and b parameters exhibit a more stable behaviour on oscillator strengths compared to λ parameter. The increase of r0 spherical confinement parameter decreases also the energies, and it increases the number of bound states; in which case is same with the function of b parameter. Therefore, r0 and b parameters are alternative parameters to each other in order to increase the energies, number of bound states, and stability. The most dominant parameter on dipole polarizability and oscillator strength is r0 due to the fact that the spherical confinement affects closely the wave functions and their overlapping cases. However, given the fact that increasing r0 decreases always the transition energy, it is understood clearly that why r0 has a critical value on the oscillator strengths (in other words, increase by a certain value and then decrease) relate to the dipole matrix 13

element directly. Namely, it is evident that the probability densities and overlapping cases of the wave functions display same trend by a certain value of r0 . In the light of these observations, the importance of this study may be outlined as follows: i)Given the interactions of Ps atom in environments with or without plasma, the presence of some external or screening effects that cannot be described by the SC and ECSC potentials may be discussed. The results of such effects have been addressed in detail. In a nutshell, the MGECSC potential is more general, physical and detailed compared to other models considering the SC and ECSC potentials. ii)The alternativeness to each other of the external, screening or confinement effects in terms of exhibiting same effect have been examined. iii)The energies of Ps atom in environment with and without plasma have been compared. iv)When considering Ps atom immersed in a solid, if the feasible of the ECSC potential in characterizing of electron-positron interaction is taken into consideration [36], it can be said that the MGECSC potential is a more comprehensive and physical in order to depict some effects in environment without plasma. So, it should be pointed out that the MGECSC potential can be proposed not only for Ps atom but also for interactions in nano-scale objects such as quantum dot, nanowires, semiconductor devices.

[1] W. R. Purcell, in Book of Abstracts Workshop on Low energy Positron and Positronium Physics , Nottingham University, Nottingham, (1997). [2] V. Krsjak, C. Cozzo, J. Bertsch, EPJ Nuclear Sci. Technol., 3, 1-5, (2017). [3] O. Shpotyuk, A. Ingram, Ya. Shpotyuk, Nuclear Inst, and Methods in Physics Research B, 416, 102-109 (2018). [4] L. C. Damonte, G. N. Darriba, M. Renteria, Journal of Alloys and Compounds 735, 2471-2478, (2018). [5] C. M. Surko, M. Leventhal, A. Passner, Phys. Rev. Lett., 62 , 901 -904, (1989). [6] T. S. Pedersen, J. R. Danielson, C. Hugenschmidt, G. Marx, X. Sarasola, F. Schauer, L. Schweikhard, New J. Phys., 14, 03510-03523, (2012). [7] N. N. Mondal, Int. J. Astron. Astrophys., 4, 620-627, (2014). [8] N. Guessoum, R. Ramaty, R. E. Lingenfelter, Astrophys. J., 378, 170-180, (1991). [9] G. Consolati, R. Ferragut, A. Galarneau, F. Di Renzo, and F. Quasso, Chem. Soc. Rev., 42 ,

14

3821-3832, (2013). [10] A. R. Frey, N. B. Reid, Phys. Rev. D, 87, 103508-103517, (2013). [11] A. R. Bell, J. G. Kirk Phys. Rev. Lett. 101, 200403-200407, (2008). [12] J. Ma, Y. C. Wang, Y. J. Zhou, H. Wang, Chin. Phys. B, 27, 013401-013406, (2018). [13] P. Rej, A. Ghoshal, Phys. Plasmas, 24, 043506-043516, (2017). [14] S. Sen, P. Mandal, P. K. Mukherjee, Eur. Phys. J. D, 62, 379388, (2011). [15] Y. Wang, J. Ma, L. Jiao, Y. Zhou, Journal of Physics: Conf. Series, 875, 052017-052018, (2017). [16] P. Rej, A. Ghoshal, J. Phys. B: At. Mol. Opt. Phys., 49, 125203-125214, (2016). [17] T. C. Naginey, B. B. Pollock, E.W. Stacy, H. R. J. Walters, C.T. Whelan, Phys. Rev. A, 89, 012708-012717, (2014). [18] B. Saha, P. K. Mukherjee, Physics Letters A, 302, 105109, (2002). [19] B. Saha, P. K. Mukherjee, T. K. Mukherjee, Chem. Phys. Lett. 373, 218222, (2003). [20] D. Munjal, P. Silotia, V. Prasad, Phys. Plasmas, 24, 122118-122128, (2017). [21] B. Saha, P. K. Mukherjee, G. H. F. Diercksen, Astron. and Astrophys., 396, 337344, (2002). [22] Y. Y. Qi, J. G. Wang, R. K. Janev, Phys. Rev. A, 78, 062511-062522, (2008). [23] M. K. Bahar, A. Soylu, A. Poszwa, IEEE Trans.Plasma. Sci., 44, 2297-2306, (2016). [24] S. Sahoo, Y. K. Ho, Phys. Plasmas, 13, 063301-063311, (2006). [25] M. S. Murillo, J. C. Weisheit, Physics Reports, 302, 1-65, (1998). [26] M. Bonitz, D. Semkat, A. Filinov, V. Golubnychyi, D. Kremp, D. O. Gericke, M. S. Murillo, V. Filinov, V. Fortov, W. Hoyer, S. W. Koch, J. Phys. A: Math. Gen. 36, 5921-5930, (2003). [27] C. Y. Lin, Y. K. Ho, Eur. Phys. J. D, 57, 21-26, (2010). [28] A. Ghoshal, Y. K. Ho, J. Phys. B, 42, 075002-075006, (2009). [29] A. Soylu, Phys. Plasmas, 19, 072701-072709, (2012). [30] M. Hjorth-Jensen, Computational Physics, University of Oslo,(2010). [31] A. Messiah, Quantum Mechanics (Amsterdam: Elsevier), (1961). [32] H. A. Bethe, E. E. Salpeter, Quantum Mechanics of One and Two-Electron Atoms (New York: Plenum), (1977). [33] J.G. Kirkwood, Phys. Z., 33, 57-, (1932). [34] A. Unsold, Z. Phys., 43, 563-574, (1927). [35] R.A. Buckingham, B. A., Proc. R. Soc. London, 160, 94-113, (1937).

15

[36] E. P. Prokopev, Sov. Phys. Solid State, 9, 993-994, (1967).

16

TABLE I: When r0 = 10, the energy values (upper rows) for some quantum states of Ps embedded in Debye plasma depicted by SC potential with λ = 5 − 200, in a.u. Data in bottom rows are the results of Ref.[20]. State

λ=5

λ = 10

λ = 50

λ = 100

λ = 200

1s

−0.095351

−0.161103

−0.228760

−0.238340

−0.243234

−0.095361

−0.161114

−0.228771

−0.23835

−0.243244

0.196196

0.148305

0.089578

0.080359

0.075559

0.196196

0.148306

0.089578

0.080359

0.075559

0.655083

0.605120

0.545570

0.536324

0.531518

0.655109

0.605147

0.545598

0.536352

0.531545

0.127958

0.081619

0.022774

0.013535

0.008729

0.127960

0.081621

0.022776

0.013537

0.008731

0.480566

0.431977

0.372819

0.363584

0.358780

0.480579

0.431991

0.372832

0.363597

0.358793

1.043550

0.993771

0.934150

0.924898

0.920090

1.043606

0.993828

0.934207

0.924955

0.920147

0.279886

0.239235

0.183367

0.174253

0.169480

0.279888

0.239237

0.183370

0.174255

0.169482

0.742381

0.696702

0.638723

0.629531

0.624738

0.742398

0.696720

0.638741

0.629548

0.624755

1.409176

1.361344

1.302477

1.293252

1.288451

2s

3s

2p

3p

4p

3d

4d

5d

TABLE II: The energy values (upper rows) for some quantum states of Ps embedded in Debye plasma depicted by SC potential with λ = 100 under spherical confinement with r0 = 5 − 50, in a.u. Data in bottom rows are the results of Ref.[20]. State

r0 = 5

r0 = 10

r0 = 20

r0 = 50

1s

−0.157552

−0.238340

−0.240108

−0.239905

−0.157556

−0.238351

−0.240148

0.942591

0.080359

−0.046884

0.942612

0.080359

−0.046887

2.844813

0.536324

0.055171

2.844954

0.536352

0.055173

0.435844

0.013535

−0.049852

0.435850

0.013537

−0.049852

1.919010

0.363584

0.034066

1.919062

0.363597

0.034070

4.223703

0.924898

0.167502

4.223932

0.924955

0.167517

1.007826

0.174253

0.005891

1.007834

0.174255

0.005892

2.907527

0.629531

0.110679

2.907599

0.629548

0.110684

5.616859

1.293252

0.270137

2s

3s

2p

3p

4p

3d

4d

5d

17

−0.053058

−0.018455

−0.052986

−0.018551

0.000440

−0.018615

−0.002214

0.020478

TABLE III: The radial matrix elements (| Dif |) (upper rows) of Ps embedded in Debye plasma depicted by SC potential with λ = 50 under spherical confinement with r0 = 5, in a.u. Data in bottom rows are the results of Ref.[20]. State

1s

2s

3s

2p

3p

3d

1s

2.039078

0.907511

0.094102

2.233618

0.063232

2.152296

2.039076

0.907510

0.094106

2.233615

0.063236

2.152296

0.907511

2.582400

0.969595

1.893588

2.200891

2.358933

0.907510

2.582409

0.969592

1.893592

2.200896

2.359931

0.094102

0.969595

2.556699

0.218730

1.936377

0.618335

0.094107

0.969592

2.556654

0.218723

1.936352

0.618316

2.233618

1.893588

0.218730

2.823219

0.807179

2.932976

2.233616

1.893592

0.218723

2.823220

0.807178

2.932976

0.063232

2.200891

1.936377

0.807179

2.679931

1.395954

0.063236

2.200896

1.936352

0.807178

2.679924

1.395941

2.152296

2.358934

0.618335

2.932976

1.395954

3.175002

2.152297

2.358931

0.618316

2.932976

1.395941

3.174995

2s

3s

2p

3p

3d

TABLE IV: Dipole polarizability of Ps embedded in Debye plasma depicted by SC potential with λ = 50 − 200 under spherical confinement with r0 = 5 − 20, in a.u. λ

r0 = 5

r0 = 10

r0 = 20

50

11.124220

54.652722

72.548146

100

11.119932

54.500700

72.147817

200

11.118841

54.461652

72.043766

18

TABLE V: When r0 = 10, the energy values for some quantum states of Ps embedded in quantum plasma modeled by MGECSC potential with a = 0.1 − 20, b = 0.1 − 20, λ = 5 − 200 under spherical confinement with r0 = 10, in a.u. r0 = 10, a = 1, b = 1 State

λ=5

λ = 10

λ = 50

λ = 100

λ = 200

1s

−0.639194

−0.911372

−1.173245

−1.210090

−1.228979

2s

0.002789

−0.359733

−0.801143

−0.864834

−0.897038

3s

0.444855

0.101119

−0.347529

−0.410379

−0.441902

2p

−0.106674

−0.466177

−0.873758

−0.934025

−0.964892

3p

0.289198

−0.065253

−0.518967

−0.582465

−0.614316

4p

0.825344

0.482892

0.039941

−0.022290

−0.053545

3d

0.165570

−0.216339

−0.693950

−0.763951

−0.799542

4d

0.577954

0.219451

−0.247514

−0.313607

−0.346866

5d

1.213552

0.863675

0.411768

0.347882

0.315745

r0 = 10,λ = 150, b = 1 a = 0.1

a=1

a=3

a = 10

a = 20

1s

−1.222922

−1.222646

−1.220436

−1.196976

−1.133049

2s

−0.887141

−0.886285

−0.879394

−0.804378

−0.596357

3s

−0.432253

−0.431394

−0.424457

−0.346404

−0.103883

2p

−0.955291

−0.954552

−0.948617

−0.885816

−0.722725

3p

−0.604548

−0.603698

−0.596837

−0.519653

−0.283326

4p

−0.043964

−0.043123

−0.036357

0.039705

0.277282

3d

−0.788609

−0.787640

−0.779852

−0.695696

−0.465522

4d

−0.336668

−0.335771

−0.328525

−0.247296

0.003285

5d

0.325602

0.326465

0.333428

0.411468

0.653052

b = 0.1

b=1

b=3

b = 10

b = 20

1s

−0.339623

−1.222646

−3.186472

−10.071828

−19.927470

2s

−0.019063

−0.886285

−2.813885

−9.569432

−19.251183

3s

0.436790

−0.431394

−2.360194

−9.106237

−18.736332

2p

−0.086009

−0.954552

−2.886658

−9.669367

−19.402018

3p

0.264089

−0.603698

−2.531655

−9.276099

−18.913280

4p

0.825353

−0.043123

−1.972742

−8.722812

−18.356712

3d

0.075353

−0.787640

−2.706627

−9.436852

−19.088234

4d

0.530256

−0.335771

−2.260169

−8.994154

−18.611770

5d

1.193850

0.326465

−1.600902

−8.344875

−17.973715

r0 = 10,λ = 150, a = 1

19

TABLE VI: The energy values for some quantum states of Ps embedded in quantum plasma modeled by MGECSC potential with a = 1, b = 1, λ = 100 under spherical confinement with r0 = 5 − 50, in a.u. Data in bottom rows are the results in plasma free case, in a.u. State

r0 = 5

r0 = 10

r0 = 20

r0 = 50

1s

−1.137147

−1.210090

−1.211035

−1.210964

−0.167453

−0.248205

−0.249986

−0.249920

−0.031429

−0.864834

−0.955121

−0.955581

0.9327333

0.070627

−0.056402

−0.062494

1.870602

−0.410379

−0.831399

−0.849271

2.835034

0.526601

0.045710

−0.027295

−0.535837

−0.934025

−0.972659

−0.972783

0.425987

0.003796

−0.059430

−0.062501

0.945972

−0.582465

−0.856213

−0.865173

1.909177

0.353854

0.024594

−0.027455

3.250124

−0.022290

−0.723617

−0.787310

4.213986

0.915193

0.158023

−0.008252

0.039695

−0.763951

−0.885867

−0.888034

0.997988

0.164558

−0.003546

−0.027660

1.936196

−0.313607

−0.775967

−0.807990

2.897716

0.619820

0.101220

−0.010927

4.644404

0.347882

−0.620127

−0.741862

5.607190

1.283565

0.260668

0.011730

2s

3s

2p

3p

4p

3d

4d

5d

TABLE VII: The radial matrix elements (| Dif |) of Ps embedded in quantum plasma modeled by MGECSC potential with a = 1, b = 1, λ = 50 under spherical confinement with r0 = 5, in a.u. State

1s

2s

3s

2p

3p

3d

1s

2.009132

0.899917

0.104415

2.203236

0.074550

2.117593

2s

0.899917

2.592234

0.968079

1.903732

2.200090

2.375465

3s

0.104416

0.968079

2.562146

0.214804

1.942512

0.622132

2p

2.203236

1.903732

0.214804

2.805477

0.808356

2.916544

3p

0.074550

2.200090

1.942514

0.808356

2.638287

1.407128

3d

2.117593

2.375465

0.622134

2.916544

0.179087

3.164303

20

TABLE VIII: Dipole polarizability of Ps embedded in quantum plasma modeled by MGECSC potential with a = 0.1 − 10, b = 0.1 − 10, λ = 50 − 200 under spherical confinement with r0 = 5 − 20, in a.u. a = 1, b = 1 λ

r0 = 5

r0 = 10

r0 = 20

50

10.545567

38.416007

41.808740

100

10.827020

45.228152

52.395717

200

10.971552

49.481453

60.428772

λ = 100, b = 1 a

r0 = 5

r0 = 10

r0 = 20

0.1

10.834618

45.547451

53.027920

1

10.827020

45.228152

52.395717

10

10.111098

27.118013

27.494264

λ = 100, a = 1 b

r0 = 5

r0 = 10

r0 = 20

0.1

11.088965

53.396085

69.306124

1

10.827020

45.228152

52.395717

10

8.575400

16.330785

16.427691

21

FIG. 1: Oscillator strengths of Ps atom embedded in Debye plasma characterized by SC potential with λ = 1 − 100 under spherical confinement with a)r0 = 5, b)r0 = 10, c)r0 = 20; for 1s − 3p, 4p and 2p − 5d transitions, in a.u.

22

FIG. 2: Oscillator strengths of Ps embedded in quantum plasma modeled by MGECSC potential with a = 0.1 − 20, b = 0.1 − 20, λ = 1 − 100 under spherical confinement with a)r0 = 5, b)r0 = 10, c)r0 = 20 d)r0 = 10,e)r0 = 10 for 1s − 3p, 4p and 2p − 5d transitions, in a.u.

23

FIG. 3: When considering 1s-3p transition, the behaviour of |Dif |2 as a function of r0 for SC potential with λ = 10 and MGECSC potential with a = 1, b = 1, λ = 10, in a.u.

24