Thermodynamic Properties of a Diatomic Molecule Under Effects of Small Oscillations in an Elastic Environment

Abstract

In this paper, we analytically investigate a diatomic molecule subject to the Morse potential under the small oscillations regime, immersed in a medium with a point defect representing impurities or vacancies in an elastic system. Initially, we apply the small oscillations method to the Morse potential to obtain an analogue to the harmonic potential, and then we solve the generalized Schrödinger equation considering the geometric effects of the defect. The solutions obtained for the bound states reveal that the energy levels and the radial stability point of the molecule are modified by the presence of the defect, depending on the parameters associated with the geometry of the medium. In a second step, we analyze the thermodynamic properties of the system in contact with a thermal reservoir at finite temperature. We derive analytical expressions for the internal energy, Helmholtz free energy, entropy, and specific heat, showing that all these quantities are influenced by the presence of the point defect. The results demonstrate how structural defects alter the quantum and thermodynamic behavior of confined molecules, contributing to the understanding of systems in non-trivial elastic media.

1. Introduction

The scientific literature offers a variety of potential energy models that accurately or approximately describe the interactions of simple molecules, such as diatomic molecules [1,2]. Among these models, the Mie-type potential [3,4] is one from which the Kratzer–Fues potential [5,6] can be derived. Another potential energy model that closely approximates experimental results is the Morse potential energy or Morse potential (MP) [7,8,9].

Proposed in 1919, the MP is a useful interatomic interaction model because it provides a very accurate description of the potential energy of a diatomic molecule. Compared to the quantum harmonic oscillator model, MP offers a better approximation of the molecular vibrational structure, as it explicitly accounts for anharmonic effects and atomic bond breaking influences [8,9]. There have been numerous studies focused on the MP, including investigations of its algebraic [10,11,12,13] and numerical [14] solutions, as well as analyses of its bound state solutions [15]. MP has been examined in the classical context to determine its exact solution [16]. More recently, the MP has been investigated by the Lagrange-mesh method [17] and in a relativistic context [18].

In Ref. [19], the authors examined the MP in a non-trivial background characterized by the presence of a topological defect. Specifically, they analyzed the MP in a regime of small oscillations. This analysis yielded bound state solutions, where the energy levels of the system were modified by the topological defect present in the environment. This modification resulted in a 32 percent decrease in the allowed energy values compared to the original (in the flat or Minkowski spacetime). Additionally, the authors considered a limitation in the radial coordinate, restricting it to the range from the point of minimum energy to infinity, and excluding the interval from the origin to the point of minimum. This consideration particularizes the results obtained.

The topological defect (TD) examined in Ref. [19] is the global monopole (GM) [20], which is one of the TDs most likely to be observed in cosmology, according to theoretical predictions [21]. In the field of mathematics, TDs are solutions to nonlinear differential equations, and from a physical perspective, they represent regions, interfaces, or edges that can separate two or more physical states in a system. In addition to the GM, there are also cosmic strings [22,23,24,25] and domain walls [26,27], all of which are associated with the curvature present in spacetime.

These topological defects have analogues in condensed matter systems, particularly in crystallography. They are often constructed using the Volterra process [28], also known as the cut-and-paste process. In the spherical context, the Volterra process is characterized in three stages: (i) consider a sphere of any material; (ii) cut it in half and remove its interior; (iii) then shrink the sphere until it forms a point. By adding a little extra matter inside the medium, that is, the reverse order, an impurity can be imagined [29]. For example, declination [30], a structural defect of linear nature, is analogous to the cosmic string, as described in Ref. [31]. Similarly, point defects such as vacancies or impurities are analogous to the global monopole, as discussed in Ref. [29]. These point defects have been investigated in various quantum mechanical systems, including the harmonic oscillator [32,33], Kratzer potential [19], Hulthén potential [34], hard-wall potential [35], and Mie-type potential [36]. In all these scenarios, the effects of the GM on the bound state solutions of the quantum system are evident, with the allowed energy values depending on the parameter associated with the GM.

In this analytical analysis, our intention is to investigate a diatomic molecule immersed in an environment with a point defect (PD) (vacancy or impurity), which is subject to the effects of small oscillations originating from the MP. Moving beyond the scope of Ref. [19], we will consider the entire range of the radial coordinate, including the stable equilibrium point, in order to obtain the most general results possible.

In this paper, we first apply the method of small oscillations to the MP to obtain an analogous potential energy to that of a harmonic oscillator. Next, through a purely analytical analysis, we derive the bound state solutions for a diatomic molecule subject to small MP oscillations in an environment with a PD. Subsequently, we consider an ensemble of N non-interacting diatomic molecules undergoing small MP oscillations in the presence of a thermal reservoir at finite temperature, and investigate the effects of impurities or vacancies on their thermodynamic properties. Finally, we present our conclusions.

2. Morse Potential Under Effects of Small Oscillations

A potential energy function with a minimum point exhibits the characteristics of a harmonic oscillator. These properties define the regime of small oscillations [37]. Mathematically, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{dV\left(r\right)}{dr}}{|}_{r=r_0}=0;{\displaystyle \frac{d^{2}V\left(r\right)}{d{r}^2}}{|}_{r=r_0}>0,\end{array}$$

(1)

where $V\left(r\right)$ is the potential energy function and $r_0$ is the minimum point in the radial coordinate.

The MP is given in the form [7,16]

$$\begin{array}{c}\hfill V\left(r\right)=D{[1-e^{-a(r-r_0)}]}^2-D,\end{array}$$

(2)

where $D>0$ and $a>0$ are the parameters related to the depth and width of the potential well, respectively, and $r_0$ represents a fixed point (minimum point) on the r axis. Applying the first condition given in Equation (1) to the MP, we can show that its extreme point (maximum or minimum) is given at $r=r_0$; when we apply the second condition given in Equation (1), it is possible to show that the second derivative of the MP calculated at the extremum point, $r=r_0$, gives us ${\displaystyle \frac{d^{2}V\left(r\right)}{d{r}^2}}{|}_{r=r_0}=2a^{2}D$, that is, as $2a^{2}D$ is positive, we have the information that $r_0$ is a minimum point. Expanding the MP in a Taylor series around the stability point $r_0$, and adopting the definitions given in the Equation (1) while disregarding the terms of derivatives greater than two, we obtain

$$\begin{array}{c}\hfill V\left(r\right)\approx a^{2}D{(r-r_0)}^2-D.\end{array}$$

(3)

To obtain the most general analytical solution, we will consider the entire radial axis, where $0\le r<\infty$, rather than restricting the radial coordinate as done in Ref. [19] through variable change ($r^{\prime }=r-r_0$). This allows us to include the interval between the origin and the stability point of the MP. Accordingly, (3) can then be rewritten in the form

$$\begin{array}{c}\hfill V\left(r\right)\approx a^{2}D{r}^2-2a^2{r}_{0}Dr+a^{2}D{r}_0^2-D.\end{array}$$

(4)

The MP under the effects of small oscillations exhibits an oscillator term as well as a linear term. The linear term serves to maintain the cohesion of the atoms within the diatomic molecule.

3. Solutions for Bound States

Let us consider a continuous medium that contains a point-type defect, which can be viewed as an impurity or a vacancy. This type of environment is described by the following line element [35,36]

$$\begin{array}{c}\hfill d{l}^2={\displaystyle \frac{d{r}^2}{{\alpha }^2}}+r^{2}d{\theta }^2+r^2{sin}^2\theta d{\phi }^2,\end{array}$$

(5)

with $\alpha$ being the parameter associated with the PD. Now, let us consider a quantum particle immersed in the environment described by Equation (5). To describe its dynamics in this non-trivial medium, let us consider the generalized Schrödinger Equation [19,33,35]

$$\begin{array}{c}\hfill -{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }_{LB}^2\Psi (\overrightarrow{r},t)+V(\overrightarrow{r},t)\Psi (\overrightarrow{r},t)=i\hslash {\displaystyle \frac{\partial \Psi (\overrightarrow{r},t)}{\partial t}},\end{array}$$

(6)

where m is the mass of the quantum particle, $V(\overrightarrow{r},t)$ is the potential energy to which the particle is subjected, and

$$\begin{array}{c}\hfill {\nabla }_{LB}^2={\displaystyle \frac{1}{\sqrt{g}}}{\partial }_i\left(\sqrt{g}{g}^{ij}{\partial }_j\right),\end{array}$$

(7)

is the Laplace–Beltrami operator, $g^{ij}$ is the metric tensor, with $i,j=1,2,3$ and g = det$\left(g^{ij}\right)$. Therefore, the equation in the medium described by the metric given in the previous equation and with a central force potential, $V(\overrightarrow{r},t)=V\left(r\right)$, becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\Psi }{\partial r^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{\partial \Psi }{\partial r}}+{\displaystyle \frac{1}{{\alpha }^2{r}^2}}\left[{\displaystyle \frac{1}{sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(sin\theta {\displaystyle \frac{\partial }{\partial \theta }}\right)+{\displaystyle \frac{1}{{sin}^2\theta }}{\displaystyle \frac{{\partial }^2}{\partial {\phi }^2}}\right]\Psi -{\displaystyle \frac{2mV}{{\alpha }^2{\hslash }^2}}\Psi =-{\displaystyle \frac{2mi}{{\alpha }^2\hslash }}{\displaystyle \frac{\partial \Psi }{\partial t}}.\end{array}$$

(8)

An ansatz can be proposed for Equation (8), by taking into account that the Hamiltonian commutes with the momentum operators ${\widehat{p}}_{\theta }$ and ${\widehat{p}}_{\phi }$ [19]:

$$\begin{array}{c}\hfill \Psi (r,\theta ,\phi ,t)=e^{-i{\displaystyle \frac{\mathcal{E}t}{\hslash }}}{Y}_{l,m}(\theta ,\phi )R\left(r\right),\end{array}$$

(9)

where $Y_{l,m}$ are spherical harmonics, $\mathcal{E}$ is the eigenvalue of energy, and $R\left(r\right)$ is the radial wave function. By substituting Equation (9) into Equation (8), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}R}{d{r}^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{dR}{dr}}-{\displaystyle \frac{2m}{{\alpha }^2{\hslash }^2}}\left[V\left(r\right)+{\displaystyle \frac{{\hslash }^{2}l(l+1)}{2m{r}^2}}\right]R+{\displaystyle \frac{2m\mathcal{E}}{{\alpha }^2{\hslash }^2}}R=0,\end{array}$$

(10)

where ${\widehat{L}}^2{Y}_{l,m}(\theta ,\phi )=-l(l+1)Y_{l,m}(\theta ,\phi )$.

In order to make Equation (10) into a simpler form, let us consider the change of the dependent variable given in [38]

$$\begin{array}{c}\hfill R\left(r\right)={\displaystyle \frac{u\left(r\right)}{\sqrt{r}}}\end{array}$$

(11)

into Equation (10), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d^{2}u}{d{r}^2}}& +& {\displaystyle \frac{1}{r}}{\displaystyle \frac{du}{dr}}-{\displaystyle \frac{[4l(l+1)+{\alpha }^2]}{4{\alpha }^2{r}^2}}u-{\displaystyle \frac{2m{a}^{2}D{r}^2}{{\alpha }^2{\hslash }^2}}u+{\displaystyle \frac{4m{a}^2{r}_{0}Dr}{{\alpha }^2{\hslash }^2}}u-{\displaystyle \frac{2m{a}^2{r}_0^{2}D}{{\alpha }^2{\hslash }^2}}u+{\displaystyle \frac{2mD}{{\alpha }^2{\hslash }^2}}u\hfill \\ & +& {\displaystyle \frac{2m\mathcal{E}}{{\alpha }^2{\hslash }^2}}u=0,\hfill \end{array}$$

(12)

which is identical to the axial wave equation, with harmonic-type potential plus a linear potential, coming from the Schrödinger-type equation in cylindrical coordinates [38].

We will now define the change of independent variable $s={\displaystyle \frac{{\left(2m{a}^{2}D\right)}^{1/4}}{\sqrt{\alpha \hslash }}}r$, which transforms Equation (12) into

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}u}{d{s}^2}}+{\displaystyle \frac{1}{s}}{\displaystyle \frac{du}{ds}}-{\displaystyle \frac{b^2}{s^2}}u-s^{2}u+csu+du=0,\end{array}$$

(13)

where we also define the new parameters

$$\begin{array}{c}\hfill b^2={\displaystyle \frac{4l(l+1)+{\alpha }^2}{4{\alpha }^2}};c=2r_0{\displaystyle \frac{{\left(2m{a}^{2}D\right)}^{1/4}}{{(\alpha \hslash )}^{1/2}}}d={\displaystyle \frac{\mathcal{E}}{\alpha \hslash a}}\sqrt{{\displaystyle \frac{2m}{D}}}+{\displaystyle \frac{{\left(2mD\right)}^{1/2}}{\alpha \hslash a}}-{\displaystyle \frac{r_0^{2}a{\left(2mD\right)}^{1/2}}{\alpha \hslash }}.\end{array}$$

(14)

By analyzing the asymptotic behavior of Equation (14), we can propose a finite solution $[s\to 0(r\to 0\left)\right]$ both at the origin and at infinity $[s\to \infty (r\to \infty \left)\right]$, given in the form of

$$\begin{array}{c}\hfill u\left(s\right)=s^{\left|b\right|}{e}^{-{\displaystyle \frac{1}{2}}s(s-c)}f\left(s\right),\end{array}$$

(15)

where $f\left(s\right)$ is a known function. Substituting Equation (15) into (13), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{2}f}{d{s}^2}}+\left[{\displaystyle \frac{\left(2\right|b|+1)}{s}}-2s+c\right]{\displaystyle \frac{df}{ds}}+\left[g+{\displaystyle \frac{h}{s}}\right]f=0,\end{array}$$

(16)

with

$$\begin{array}{c}\hfill g=d-2\left(\right|b|+1)+{\displaystyle \frac{c^2}{4}};h={\displaystyle \frac{c}{2}}\left(2\right|b|+1).\end{array}$$

(17)

which is the biconfluent Heun differential Equation [38,39], and $f\left(s\right)$ is the biconfluent Heun function [38]:

$$\begin{array}{c}\hfill f\left(s\right)=H_B\left(2\left|b\right|,c,d+{\displaystyle \frac{c^2}{4}},0;s\right).\end{array}$$

(18)

To solve the Equation (16), we must impose that the Heun biconfluent function is a solution given in terms of a power series around the origin, as it is a regular singular point. Accordingly, the Equation (18) is written in this form:

$$\begin{array}{c}\hfill f\left(s\right)=\sum _{j=0}^{\infty }{\gamma }_j{s}^j.\end{array}$$

(19)

Subsequently, by substituting Equation (19) into Equation (16), we obtain

$$\begin{array}{c}\hfill {\gamma }_{j+2}={\displaystyle \frac{-[h+c(j+1)]{\gamma }_{j+1}-(g+2j){\gamma }_j}{(j+2)(j+2+2|b\left|\right)}},\end{array}$$

(20)

with

$$\begin{array}{ccc}\hfill {\gamma }_1& =& -{\displaystyle \frac{c}{2}}{\gamma }_0;\hfill \\ \hfill {\gamma }_2& =& {\displaystyle \frac{{\gamma }_0}{4(1+|b\left|\right)}}\left[{\displaystyle \frac{(h+c)c}{2}}-g\right].\hfill \end{array}$$

(21)

To construct the biconfluent Heun polynomials of finite degree n, we must truncate the biconfluent Heun series by imposing the conditions:

$$\begin{array}{c}\hfill {\gamma }_{n+1}=0;g=2n,\end{array}$$

(22)

where $n=1,2,3,\dots$. Since we have two truncation conditions, we need to analyze each of them. To analyze the condition ${\gamma }_{n+1}=0$, we must specify values for n. For the ground state, which corresponds to the lowest allowed energy values, we consider $n=1$. In this case, the condition ${\gamma }_{n+1}=0$ reduces to ${\gamma }_2=0$, which leads to the expression

$$\begin{array}{c}\hfill r_0^{l,1}={\displaystyle \frac{1}{a}}\sqrt{\left(2\right|b|+3){\displaystyle \frac{{\mathcal{E}}_{l,1}}{D}}-{\displaystyle \frac{2\alpha \hslash a\left(\right|b|+1)\left(2\right|b|+3)}{\sqrt{2mD}}}+\left(2\right|b|+3)}.\end{array}$$

(23)

Equation (23) arises from the choice of the stability point $r_0$ as a tuning parameter to provide a physical result for the condition ${\gamma }_2=0$. From a physical perspective, Equation (23) represents the allowed values for the stability point of the MP in the small oscillation regime, which are defined in terms of the allowed energy values corresponding to the ground state of the system, ${\mathcal{E}}_{l,1}$. Therefore, we have renamed the stability point as $r_0=r_0^{l,1}\left({\mathcal{E}}_{l,1}\right)$. It is worth noting that the choice of $r_0$ as a tuning parameter will be valid for all n, and not only for $n=1$, that is, $r_0=r_0^{l,n}\left({\mathcal{E}}_{l,n}\right)$. This choice was inspired by the quantization of the atomic radius in terms of the Bohr radius in the hydrogen atom [37].

The condition $g=2n$ leads to

$$\begin{array}{c}\hfill {\mathcal{E}}_{l,n}=\alpha \hslash a\sqrt{{\displaystyle \frac{2D}{m}}}(1+n+|b\left|\right)-D,\end{array}$$

(24)

which represents the energy profile of the quantum system. The energy levels are influenced by the PD. This influence is evident from the dependence of the parameter associated with the eigenvalues of the angular momentum, $b=\sqrt{{\displaystyle \frac{l(l+1)}{4{\alpha }^2}}+{\displaystyle \frac{1}{4}}}$, and the redefinition of the angular frequency ${\omega }_{\alpha }=\alpha \omega$, where ${\omega }_{\alpha }=\alpha a\sqrt{{\displaystyle \frac{2D}{m}}}$. We can interpret the angular frequency values as follows: in the presence of an impurity, the oscillation frequency becomes lower $(0<\alpha <1)$; in the presence of vacancies or holes, the oscillation frequency becomes higher $\alpha >1$. Similarly, for the energy levels, we observe that a particle subject to small oscillations of the MP in the presence of an impurity will have a decrease in its allowed energy values, while in the presence of a vacancy, the allowed energy values increase. By taking $\alpha \to 1$,we recover the energy profile of a diatomic molecule under the effects of Morse potential small oscillations in a flat medium.

However, it is important to note that our analysis is not yet complete. Specifically, Equation (23) depends on the allowed values of the lowest energy of the system, ${\mathcal{E}}_{l,1}$, in order to obtain the allowed values of the stability point of the diatomic molecule subject to small oscillations. Therefore, by considering the radial mode $n=1$ in Equation (24), we can derive the following expression:

$$\begin{array}{c}\hfill {\mathcal{E}}_{l,1}=\alpha \hslash a\sqrt{{\displaystyle \frac{2D}{m}}}(2+|b\left|\right)-D,\end{array}$$

(25)

which represents the lowest allowed energy values of a particle in a region of MP small oscillations in an environment with a PD. By substituting Equation (25) into Equation (23), we have

$$\begin{array}{c}\hfill r_0^{l,1}={\displaystyle \frac{2^{1/4}}{a}}\sqrt{{\displaystyle \frac{\alpha \hslash a\left(2\right|b|+3)}{\sqrt{mD}}}}.\end{array}$$

(26)

Equation (26) gives us the allowed values of the stability point of the diatomic molecule in a regime of MP small oscillations in the presence of a PD. We can notice that the radial equilibrium point of the diatomic molecule is strongly influenced by the presence of the PD. This influence can be noticed by dependency $r_0^{l,1}\propto {\displaystyle \frac{\left|b\right|\sqrt{\alpha }}{a}}\propto {\displaystyle \frac{{\alpha }^{1/4}}{a}}$. We can see that the radial equilibrium point of the diatomic molecule in a regime of MP small oscillations decreases in the presence of an impurity, while it increases if it is in the presence of a vacancy. By making $\alpha \to 1$, we obtain the allowed values of the stability point of the diatomic molecule in a regime of MP small oscillations in a flat environment.

Figure 1 and Figure 2 show how the allowed values of energy of the ground state of the analyzed system are influenced by the PD. For the allowed values for the parameter $\alpha$, it means that we have a vacancy. In this sense, there is an increase in the allowed values of energy of the ground state, as demonstrated in Figure 1. In Figure 2, we have that the allowed values of energy for the ground state behave as an affine function, when the defect is considered as an independent variable.

Figure 3 and Figure 4 show us the permitted values of the radial stability point for the ground state which are influenced by the presence of the PD. For the values attributed to the parameter associated with the presence of the defect, it is then a vacancy. In this sense, we can notice an increase in the values of $r_0^{l,1}$. In Figure 3, we consider the values for the angular momentum, while in Figure 4, we consider the generic angular momentum.

4. Thermodynamic Properties

The thermodynamic properties of confined quantum systems in nontrivial backgrounds have been extensively studied in the literature. For instance, the literature has analyzed a spin-0 particle subjected to exponential-type molecule potentials in D dimensions [40], a 2D-charged particle confined by magnetic and quantum flux fields under the radial scalar power potential [41], a neutral particle in a magnetic cosmic string background [42], the radial harmonic oscillator [33], a quantum particle subjected to the discontinuous potential in the global monopole spacetime [36], and a Dirac particle under the effects of Landau-type quantization induced by Lorentz symmetry violation [43].

Let us now consider a fixed number N of diatomic molecules in contact with a thermal reservoir of finite temperature T, through a fixed and impermeable diathermic wall. These diatomic molecules are subjected to MP small oscillations in a medium defined by the length element given in Equation (4) Furthermore, we will assume that these molecules are localized and do not interact with each other. By considering a fixed b $\left(\mathrm{fixed}l\right)$, the microscopic states of the quantum system are characterized by the set of principal quantum numbers $n_1,n_2,\dots ,n_N$ [44]. Then, for a microscopic state $n_j$, the energy of that state can be written in the form

$$\begin{array}{c}\hfill \mathcal{E}\left\{j\right\}=\sum _{j=1}^N(\delta n_j+\lambda ),\end{array}$$

(27)

where

$$\begin{array}{c}\hfill \delta =\alpha \hslash \omega ;\lambda =\alpha \hslash \omega (1+|b\left|\right)-D;\omega =a\sqrt{{\displaystyle \frac{2D}{m}}}.\end{array}$$

(28)

Accordingly, the canonical partition function is expressed as

$$\begin{array}{c}\hfill Z=\sum _{n_j}{e}^{-\beta \mathcal{E}\left\{j\right\}}=\sum _{n_1,n_2,n_3,\dots ,n_N}{e}^{-{\sum }_{j=1}^N(\delta n_j+\lambda )\beta },\end{array}$$

(29)

where $\beta =1/k_{B}T$ and $k_B$ represents the Boltzmann constant. Due to the absence of interactions between the molecules, the partition function can be rewritten as

$$\begin{array}{c}\hfill Z={\left[\sum _{\nu =1}^{\infty }{e}^{-(\delta \nu +\mu )\beta }\right]}^N=Z_1^N,\end{array}$$

(30)

where

$$\begin{array}{c}\hfill Z_1=\sum _{\nu =0}^{\infty }{e}^{-(\delta \nu +\mu )\beta }=e^{-\mu \beta }\sum _{\nu =0}^{\infty }{e}^{-\beta \delta \nu },\end{array}$$

(31)

with the new parameters

$$\begin{array}{c}\hfill \nu =n-1;\mu =\lambda +\delta ,\end{array}$$

(32)

and $Z_1$ denotes the canonical partition function per diatomic molecule.

To solve the sum in Equation (31), we will use the sum of a geometric progression, considering l fixed, that is, [33,44]:

$$\begin{array}{c}\hfill Z_1^l\left(\beta \right)=e^{-\mu \beta }(1+e^{-\beta \delta }+e^{-2\beta \delta }+e^{-3\beta \delta }+\dots )={\displaystyle \frac{e^{-\mu \beta }}{1-e^{-\beta \delta }}},\end{array}$$

(33)

or

$$\begin{array}{c}\hfill Z_1^l\left(T\right)={\displaystyle \frac{e^{-\frac{\alpha \hslash \omega (2+|b|-D)}{k_{B}T}}}{1-e^{-\frac{\alpha \hslash \omega }{k_{B}T}}}}\end{array}$$

(34)

Equation (34) represents the partition function per diatomic molecule subjected to MP small oscillations in the presence of a PD. We can observe that the partition function per molecule is influenced by the topology of the environment, as evidenced by the presence of the parameter associated with the PD in the redefinition of the angular frequency ${\omega }_{\alpha }=\alpha \omega$.

Through Equation (34), it is possible to determine the Hemholtz free energy per molecule, $f^{\left(l\right)}\left(T\right)$, that is,

$$\begin{array}{c}\hfill f^{\left(l\right)}\left(T\right)=-{\displaystyle \frac{1}{\beta }}ln\left(Z_1^{\left(l\right)}\right)=\alpha \hslash \omega (2+|b\left|\right)-D-k_{B}Tln\left(1-e^{-{\displaystyle \frac{\alpha \hslash \omega }{k_{B}T}}}\right).\end{array}$$

(35)

With Equation (33), we can define the internal energy per molecule, $u^l\left(\beta \right)$,

$$\begin{array}{c}\hfill u^l\left(\beta \right)=-{\displaystyle \frac{d\left[\ln Z_1^l\left(\beta \right)\right]}{d\beta }}=\mu +{\displaystyle \frac{\delta e^{-\beta \delta }}{(1-e^{-\beta \delta })}},\end{array}$$

(36)

or

$$\begin{array}{c}\hfill u^l\left(T\right)=\alpha \hslash \omega (2+|b\left|\right)-D+{\displaystyle \frac{\alpha \hslash \omega e^{-\frac{\alpha \hslash \omega }{k_{B}T}}}{(1-e^{-\frac{\alpha \hslash \omega }{k_{B}T}})}}.\end{array}$$

(37)

The entropy per molecule, ${\int }^{\left(l\right)}\left(T\right)$, can be determined from the expression for the Helmholtz free energy per molecule given in Equation (35)

$$\begin{array}{c}\hfill {\int }^{\left(l\right)}\left(T\right)=-{\displaystyle \frac{d{f}^l\left(T\right)}{dT}}=k_{B}ln\left(1-e^{-{\displaystyle \frac{\alpha \hslash \omega }{k_{B}T}}}\right)+{\displaystyle \frac{\alpha \hslash \omega e^{-\frac{\alpha \hslash \omega }{k_{B}T}}}{T\left(1-e^{-\frac{\alpha \hslash \omega }{k_{B}T}}\right)}}.\end{array}$$

(38)

We can go further, calculating the specific heat, $c^{\left(l\right)}\left(T\right)$, through the entropy per molecule

$$\begin{array}{ccc}\hfill c^{\left(l\right)}\left(T\right)& =& T{\displaystyle \frac{d{\int }^l\left(T\right)}{dT}}={\displaystyle \frac{{\alpha }^2{\hslash }^2{\omega }^2}{k_B{T}^2}}{\displaystyle \frac{e^{-\frac{\alpha \hslash \omega }{k_{B}T}}}{{\left(1-e^{-\frac{\alpha \hslash \omega }{k_{B}T}}\right)}^2}}-{\displaystyle \frac{2\alpha \hslash \omega e^{-\frac{\alpha \hslash \omega }{k_{B}T}}}{T\left(1-e^{-\frac{\alpha \hslash \omega }{k_{B}T}}\right)}}.\hfill \end{array}$$

(39)

Equations (35) and (37)–(39) show us that the thermodynamic properties of a diatomic molecule are influenced by the defect present in the environment. This influence occurs due to the dependence of the calculated thermodynamic quantities on the angular frequency of small oscillations redefined in terms of the parameter associated with the impurity or vacancy in the material, that is, ${\omega }_{\alpha }=\alpha \omega$. By taking $\alpha \to 1$ into Equations (35) and (37)–(39), we recover the thermodynamic properties of a diatomic molecule under the effects of MP small oscillations in a flat medium.

We can observe in Figure 5, Figure 6, Figure 7 and Figure 8 that the thermodynamic quantities calculated in this analysis are influenced by the presence of PD. For $\alpha =1$, we have the case of an environment without defect, that is, without curvature (a half-plane). For $\alpha >1$, we have the presence of a point defect, a vacancy. In this case, by redefining the frequency of small oscillations, ${\omega }_{\alpha }=\alpha \omega$, the frequency increases with the presence of vacancy in the environment. This fact directly influences the thermodynamic quantities, as illustrated in Figure 5, Figure 6, Figure 7 and Figure 8. For example, although the behavior of the functions is the same, as the temperature varies, the presence of the vacancy (or the increase in the frequency of small oscillations) causes an increase in the Helmholtz free and internal energies; in entropy and specific heat, the presence of vacancy causes them to become increasingly sensitive in a given range of temperature variation, making the curve more abrupt.

5. Conclusions

We have analytically investigated a diatomic molecule undergoing small oscillations originating from the MP in an environment characterized by the presence of a PD, which can be viewed as an impurity or a vacancy. Our initial step was to analyze the MP for small oscillations around its stability point. We then examined the effects of these small oscillations in the search for bound state solutions, from which we obtained the energy levels of the system. While the potential energy for small oscillations is analogous to the energy of the quantum harmonic oscillator, the energy levels are entirely distinct. This difference begins with the radial mode corresponding to the ground state, which, in our case, is given by $n=1$, rather than $n=0$. Another quantum effect observed is that the radial stability point is quantized, that is, it depends on the quantum numbers of the system.

The energy levels are influenced by the presence of PD in the environment. This influence arises from the correction applied to the angular frequency of the small oscillations. Specifically, if the quantum system is immersed in a background with an impurity, the frequency of the small oscillations decreases. Conversely, if the quantum system is in the presence of a vacancy, the frequency of the small oscillations increases. Since the allowed energy values of the system depend linearly on the frequency of the small oscillations, they are affected accordingly.

With the energy levels determined, we next consider the diatomic molecule interacting with a thermal reservoir at finite temperature T. By calculating the partition function, we can obtain thermodynamic quantities such as the Helmholtz free energy, internal energy, entropy, and specific heat, all of which depend on the presence of the point defect.