B.C.L¨utf¨uo?glu

Department of Physics,Akdeniz University,07058 Antalya,Turkey

1 Introduction

In recent years,the thermodynamic functions gained popularity in order to understand the physical properties of numerous potentials in relativistic or nonrelativistic regimes.Hassanabadiet al.studied thermodynamic properties of the three-dimensional Dirac oscillator with Aharonov-Bohm field and magnetostatic monopole potential.[1]Pachecoet al.analyzed one-dimensional Dirac oscillator in a thermal bath and they showed that its heat capacity is two times greater than that of the onedimensional harmonic oscillator for high temperatures.[2]Franco-Villafa?neet al.performed the first experimental study on one-dimensional Dirac oscillator.[3]Later,Pachecoet al.also studied three-dimensional Dirac oscillator in a thermal bath.[4]They reported that the degeneracy of energy levels and their physical implications implied that,at high temperatures,the limiting value of the speci fic heat is three times bigger than that of the one-dimensional case.Boumali studied the properties of the thermodynamic quantities of the relativistic harmonic oscillator using the Hurwitz zeta function.He compared his results with those obtained by a method based on the Euler-MacLaurin approach.[5]Boumali also showed that,with the concept of effective mass,the model of a twodimensional Dirac oscillator can be used to describe the thermal properties of graphene under a uniform magnetic field,where all thermodynamic properties of graphene were calculated using the zeta function.[6]He also studied the thermodynamics of the one-dimensional Duffin-Kemmer-Petiau oscillator via the Hurwitz zeta function method[7]in which study,he calculated the free energy,the total energy,the entropy,and the speci fic heat.Larkinet al.have studied thermodynamics of relativistic Newton-Wigner particle in external potential field.[8]Vinczeet al.investigated nonequilibrium thermodynamic and quantum model of a damped oscillator.[9]Ardaet al.studied thermodynamic quantities such as the mean energy,Helmholtz free energy,and the speci fic heat with the Klein-Gordon,and Dirac equations.[10]Donget al.studied hidden symmetries and thermodynamic properties for a harmonic oscillator plus an inverse square potential.[11]

The WS potential well[12]is widely employed to model the physical systems in nuclear,[12?21]atom-molecule,[21?22]relativistic,[23?31]and non-relativistic[32?37]physics problems.

To describe the energy barrier at the surface of atomic nucleus that nucleons are exposed,various type of additional terms to WS potential are proposed to produce GSWS potentials.Such potential wells can be used to model any system,in which a particle is trapped in a finite space,as well as the effects,such as non-zerol,spin-orbit coupling.[38?55]

Our main motivation in this work is to compare physical consequences of the two potentials in context of quantum mechanics and statistical thermodynamics.We consider the physical properties ofαparticle as an application,to reinforce the formal treatment of the two potentials for Bh-270 nucleus.

In Sec.2,we interpret the forms of the WS and GSWS potentials,and corresponding energy eigenvalues,for a massive non-relativistic con fined particle,using the formalism proposed by Ref.[41]In Sec.3,we give a brief summary on the thermodynamic functions,that are calculated in the following section for the two potentials.In Sec.4,the energy spectra ofαparticle in Bh-270 nucleus for the two potentials are presented as an application of the formalism presented,upon which,the thermodynamic functions of the system are plotted and discussed in terms of the parameters of the problem.In Sec.5,the conclusion is given.

2 The Model

The WS potential well in one dimension is described by

whereθ(±x)are the Heaviside step functions,ais the reciprocal of the diffusion coefficient,Lmeasures the size of the nucleus,V0is the depth of the potential,given by[14]

whereAis the atomic number of the nucleus.

According to the assumption that a nucleon suffers a potential barrier when near the surface of its nucleus or being emitted to outside,the WS potential is considered inadequate to explain the dynamics of this type of problems.In order to take the surface effect into account,an additional term to the WS potential is widely used.[39?41]The WS potential combined with the additional terms are called GSWS potential.

here the second terms in the brackets correspond to the energy barrier that nucleon faces at the surface,which is taken as linearly proportional to the spatial derivative of the first term multiplied by the nuclear size.A unitless proportionality multiplier,herebyρ,which is implicitly included inW0,can be calculated via conservation laws.

Because of the symmetry of the potential,evenand oddenergy eigenvalues arise,which are studied extensively in Ref.[41]and evaluated to be

heren′are integers,whereasnstands for the number of nodes,the roots of the wave functions.N1andN2are complex numbers

and implicitly dependent on the energy eigenvalues via the coefficientsa1,b1,andc1

WhenW0=0,N1,andN2remain unchanged because of the symmetry in the multiplication of the Gamma functions under the possible values ofθ,which are either 0 or 1.Moreover,since the whole energy spectrum is negative,μis real,the ordinary solutions for the WS potential are obtained.WhenW0is between 0 andV0,the WS potential well is slightly modi fied because of being narrower,but not yet giving rise to positive energy eigenvalues.WhenW0exceedsV0,the barrier starts to grow and the well keeps narrowing,this alters the energy spectrum,including an extension to positive values.The positive energies are the reason for complex values ofμ,which are responsible for tunnelling in some nuclei.These states are called quasi-bound states.

The energy spectrum of a nucleon under GSWS potential is composed of energy eigenvalues,satisfying

then,νcan take only imaginary values for the entire scope of the spectrum.

3 Thermodynamics of a System

Using the energy eigenvaluesEn,the partition function of the system is given by

andkBstands for the Boltzmann constant,Tis the temperature in the unit of Kelvin.The Helmholtz free energy of the system can be calculated using the equation

The entropy of the system is given by,

4 An Application of the Formalism for Bh-270 Nucleus

In this section we present the thermodynamic treatment of anαparticle within Bh-270 nucleus as an application of the formalism described in previous sections,in order to investigate the effects of the surface term addition to the WS potential.For this nucleus,in Ref.[14]the inverse diffusion parameter is given asa=1.538 fm?1,while the radius is evaluated to beL=8.068 fm.Substituting the atomic numberA=270 into Eq.(2),we haveV0=75.617 MeV and thenW0=215.523 MeV.The corresponding WS and GSWS potentials are shown in Fig.1.

The calculated energy spectra of anαparticle with massm=3727.379 MeV/c2in the nucleus are tabulated in Tables 1 and 2 for WS and GSWS,respectively.Purely real bound state energy eigenvalues in both spectra imply in finite time constants,which mean zero decay probability for the nucleon from the nucleus.Whereas,the quasibound states in the GSWS spectrum have a complex form with finite time constants,which are responsible for the decay probability.[56?57]

Fig.1 The WS and GSWS potential well for an α particle in a Bh-270 nucleus.

Table 1 The energy spectrum of the α particle within Bh-270 nucleus under WS potential well assumption.

Table 2 The energy spectrum of the α particle within Bh-270 nucleus under GSWS potential well assumption.The rightmost column tabulates the quasi-bound energy levels.

Using the partition function given in Eq.(5),the Helmholtz and the entropy functions versus reduced temperature curves of the system,corresponding to the cases of GS and GSWS potentials are presented in Figs.2(a)and 2(b),respectively.The entropy in both cases start from zero,being in agreement with the third law of thermodynamics.The saturation values of the entropy are 2.66×10?10MeV/K and 2.62×10?10MeV/K for WS and GSWS potentials,respectively.Surprisingly the addition of the the surface term does not lead to an increase in the number of available states,contrarily,it results in a decrease in the number of available states from 22 to 21,accompanied by an upward shift in the energy spectrum.This is a consequence of the upper shift of the energy spectrum by squeezing the well with the addition.The increase of the Helmholtz free energy is due to the upward shift in the energy spectrum.

The addition of the surface term leads to increase in the internal energy as observed in Fig.3,since it is the expectation value of the energy eigenvaluesEn.The internal energies initiate at the values?75.283 MeV and?75.166 MeV at 0 K for the WS and GSWS potentials,which are the lowest energy eigenvalues in the spectra,respectively.These internal energies are not distinguishable to the naked eye until the reduced temperature of about 0.007,at which the difference broadens,as observed in Fig.3(b).The limiting values of the internal energies of the two cases goes to the mean values of the?42.586 MeV and?35.535 MeV for the spectra of WS and GSWS,respectively,as the reduced temperature goes to in finity.

Fig.2 Helmholtz energy F(T)(a),entropy S(T)(b),as functions of reduced temperature.

Fig.3 Internal energy U(T)as functions of reduced temperature(a),the initial behavior(b).

In Fig.4,the speci fic heatCv(T)versus reduced temperature curves are demonstrated.The steep linear initial increase is a consequence of the initial convex behavior of the internal energies,which is a common characteristics of the two cases.After that,Cv(T)remains constant,followed by a decay to zero,in the whole scale of the reduced temperature.The speci fic heat function for GSWS potential has higher values during this decay,which veri fies that the internal energy saturates to a higher value at higher reduced temperature.

Fig.4 Speci fic heat Cv(T)as functions of reduced temperature(a),the initial behavior(b).

5 Conclusion

In this study,we analyze the effect of the additional term,representing the surface effect of a nucleus,to WS potential well.We formally discuss how the additional term modi fies the whole non-relativistic energy spectrum by squeezing the well,resulting in an upward shift of the spectrum.GSWS potential does not merely accommodate extra quasi bound states,but also has modi fied bound state spectrum.As an application of the formal treatment,we considerαparticle inside Bh-270 nucleus,modeled with both WS and GSWS potential wells.The thermodynamic functions Helmholtz free energy,entropy,internal energy,speci fic heat are calculated in both approaches and compared.The internal energy and the speci fic heat capacity increase,as a result of upward shift in the spectrum.The shift of the Helmholtz free energy is a direct consequence of the shift of the spectrum.The entropy decreases due to the decrement in the number of available states,which arises as a result of narrowing the well with the additional term.It is concluded that GSWS potential is more realistic to describe the physical properties ofαparticle within Bh-270 nucleus.

The author would like to thank to Dr.M.Erdogan for all scienti fic discussions and the preparation of this manuscript.The author also thanks Profs.I.Boztosun and E.Pehlivan for sharing their valuable comments on the manuscript.

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