Thermo-magnetic properties of the screened Kratzer potential with spatially varying mass under the influence of Aharanov-Bohm(AB) and position-dependent magnetic fields

https://doi.org/10.1016/j.physe.2021.114710Get rights and content

Highlights

  • The position-dependent mass Schrödinger equation (PDMSE) for the screened Kratzer potential with the (AB)and position-dependent external magnetic fields are investigated using asymptotic iteration are investigated.

  • The energy eigenvalues and the corresponding eigenfunctions are calculated.

  • The partition function, Helmholtz free energy, entropy, are evaluated.

  • And the magnetization and susceptibility of the system are also calculated.

Abstract

In this study, the solution of the position-dependent mass Schrödinger equation (PDMSE) for the screened Kratzer potential with the Aharanov-Bohm and position-dependent external magnetic fields are investigated using asymptotic iteration method. The energy eigenvalues and the corresponding eigenfunctions are obtained in closed form. We evaluated the partition function of the system and calculated the thermodynamic properties of the system such as Helmholtz free energy, entropy, internal energy, specific heat and the magnetization and magnetic susceptibility of the system at zero and finite temperatures.

Introduction

In the last three decades, researchers in physics and chemistry have paid huge attention to the solutions of the Schrodinger equation (SE) [[1], [2], [3], [4], [5]] whose solutions such as energy eigenvalues and the wave function of the system contains all the necessary information required to scrutinize the behavior of such a system [[6], [7], [8], [9], [10]]. Although particular attentions have been paid to the case where the particle has a constant mass, many researchers in recent times have obtained solutions of the SE for different central potentials [[11], [12], [13], [14], [15], [16], [17], [18], [19], [20]]. One of these potentials among many others is the screened Kratzer potential (SKP) [21], recently proposed by Ikot and his collaborators [21]. The SKP is given by [21].V(ρ)=(Aρ+Cρ2)eαρwhere A and C are the potential strength, α represents the screening parameter. The SKP reduced to the well-known Kratzer potential [22,23], when α approaches zero, Yukawa (screened Coulomb) potential [24,25] when C=0, and the Coulomb potential [26], when C=0 and α0. More so, when A=0, the potential reduces to the inversely quadratic Yukawa potential [27,28]. It is explicitly seen that the SKP of equation (1) is a generalized potential composed of Kratzer, screened Coulomb, Coulomb and inverse quadratic Yukawa potentials, respectively. A good number of studies have been carried out with this potential. For instance, Ikot et al. [21] in a first consideration solved the SE [[29], [30], [31]] with the SKP and Amadi et al. [32] studied the information theoretic measures with this potential. Moreover, when the information on the material properties and systems properties is encrypted in the quantum particle's mass, the concept of quantum particles with varying mass is unavoidable. This varying mass concept is what is popularly known as the position-dependent mass (PDM) [33,34]. The concept of PDM is known to play an important role in the energy-density functional approach in quantum many-body problems [[35], [36], [37]]. In recent times, many authors have carried out extensive studies with the position-dependent mass Schrodinger equation (PDMSE) [[38], [39], [40], [41], [42]]. More so, the concept of a PDMSE has gained good attention in the last two decades [43,44], because of its applications in several disciplines of physics and chemistry [45,46], such as semiconductors [47,48], quantum fluids [49,50]. For instance, the study of semiconductor heterostructures [51], the effective interaction in nuclear physics [52] and the dynamical properties of a neutron superfluid in a neutron star [45] and many others are described by the PDMSE. Similarly, the problem of an electron in magnetic (B) [53,54] and Aharanov-Bohm (AB) fields [53,54] are also of great importance in physics and chemistry since the early days quantum mechanics. These problems have been studied both in the non-relativistic and in the relativistic regime for different central potentials by so many authors [[55], [56], [57], [58], [59], [60], [61], [62], [63], [64]]. The applications of external magnetic fields continue to play an important role in modern physics, from the magnetic confinement of plasma in tokomaks [65], the magnetic levitation [66],Stern-Gerlach experiment [67], curving the membrane [68] among others. Even more interestingly, the study of the PDMSE in the presence of the above discussed fields has recently become a great center of attraction. For example, Eshghi et al. [69], solved the PDMSE with the Hulthen plus Coulomb-like potential field and under the actions of the B and AB flux fields. Eshghi et al. [70] also solved the PDMSE for the Morse and Coulomb potentials under the influence of B and AB flux fields. Mustafa and Algadhi [71], considered some PDM charged particles moving in position-dependent (PD) magnetic and AB flux fields. Algadhi and Mustafa [72] generalized the Landau quantization to include PDM neutral particles. Furthermore, an essential and intriguing problem in physics is to study the thermal and magnetic properties of quantum systems. Some researchers have carried investigations in this direction for numerous quantum systems [[73], [74], [75], [76], [77]]. For instance, Eshghi et al. [70] studied the thermal quantities of Morse and Coulomb potentials. Khordad [78] evaluated the specific heat, entropy and magnetic susceptibility of an asymmetric GaAs quantum dot (QD). Very recently, Ikot et al. [79], studied the thermodynamic properties of pseudo-harmonic potential in the presence of B and AB fields.In the present investigation, our objective is to solve the PDMSE with the SKP model in the presence of external magnetic and AB flux fields. The asymptotic iteration method (AIM) [80] will be used to obtain detailed solutions of the PDMSE with the SKP model in the presence of magnetic and AB flux fields. The resulting energy equation will be used to obtain the partition function which will be used to obtained other thermodynamic functions like; entropy, mean free energy, specific heat capacity and magnetic susceptibility. We analyze the effect of the fields on these properties. Furthermore, magnetization and magnetic susceptibility at zero temperature is considered as well.

The outline of our paper is as follows. In section 2, we present the theory and calculations. In section 3, we obtain the analytical expressions for some of the thermodynamics and magnetic (at finite and zero temperature) in the presence of external fields. The behavior of these properties is discussed in section 4. Finally, the paper ends with concluding remarks in section 5.

Section snippets

Theory and calculations

The most broad Hamiltonian with PD effective mass, originally proposed by von Roos [81,82] is given as follows [81]:HVR=14[M(r)αrM(r)βrM(x)γ+M(r)γrM(r)βrM(r)α]+V(r)where M(r)=m0m(r), m0 is the rest mass, m(r) is a PD scalar multiplier. The PDM, M(r), r=(x,y,z), r=r, α, β, and γ are the ordering ambiguity parameters and V(r) is the potential force field. It has been established in literature that these ordering ambiguity parameters α, β, and γ only satisfy the von Roos

Thermal and magnetic properties of screened Kratzer potential with PDM, magnetic and AB fields

The first step in achieving the thermal properties of a system is by obtaining the partition function. We consider the contribution of the bound state to the partition function of the system at a given temperature T to be [70,76];Z(ζ)=n=0χ˜eβEn,β=1kBT

Here, kB is the Boltzmann constant and Enm is the energy of the nth bound state.

We can rewrite eq. (42) to be of the formEn,m,δ=M+P(n+Q)2P=αη;Q=ε1+ε2;M=Aη(116α2H˜)αη

We substitute eq. (51) into eq. (50) to haveZ(ζ)=n=0χ˜eβ[M+P(n+Q)2]where, χ˜=

Discussion of results and applications

In quantum mechanics, it is established that the ground and first excited state contained the necessary information needed to describe any quantum mechanical system. In view of this, we will limit our analysis to the ground and first excited states corresponding to n=0 and n=1.Fig. (1a) and (1b) show the variation of energy for the screened Kratzer potential with PDMSE under the influence of the magnetic field and the AB flux field as a function of external magnetic field with varying α for m=n=

Declaration of competing interest

We write to inform you that the manuscript titled “Thermo-Magnetic Properties of the Screened Kratzer potential with Spatially varying mass under the influence of Aharanov-Bohm(AB) and Position-Dependent Magnetic fields” and submitted to your journal for publication has no conflict of interest.

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