On obtaining the approximate-analytic energy Eq. (18), we can proceed to obtain the partition function and other thermodynamic functions. The partition function \(Z\left( \beta \right)\) of the DFP at finite temperature \(T\) is obtained using the Boltzmann factor as4,58,59;
$$Z\left( \beta \right) = \sum\limits_{n = 0}^{\omega } {e^{{ – \beta E_{n} }} }$$
(21)
with \(\beta = \frac{1}{kT}\) and with \(k\) is the Boltzmann constant.
Substituting Eq. (18) in (21), we have;
$$Z\left( \beta \right) = \sum\limits_{n = 0}^{\omega } {e^{{ – \beta \left( {\Omega_{0} – \Omega_{1} \left( {\frac{{\Omega_{2} – \left( {n + {\rm K}} \right)^{2} }}{{\left( {n + \tilde{\rm K}} \right)}}} \right)^{2} } \right)}} }$$
(22)
where \(n\) is the vibrational quantum number, \(n = 0,1,2,3…\omega\) , \(\omega\) denotes the upper bound vibration quantum number. We have introduce the following notations:
$$\begin{aligned} \Omega_{0} & = \frac{{\hbar^{2} \alpha^{2} }}{2\mu }\left( {\left( {m + \xi } \right)^{2} – \frac{1}{4}} \right);\Omega_{1} = \frac{{\hbar^{2} \eta^{2} }}{8\mu }; \\ \Omega_{2} & = \frac{{4\mu D_{e} b}}{{\hbar^{2} \eta^{2} }} + \frac{{2\mu D_{e} b^{2} }}{{\hbar^{2} \eta^{2} }} + \frac{{\mu^{2} \omega_{c}^{2} }}{{\hbar^{2} \eta^{2} }} – \left( {\left( {m + \xi } \right)^{2} – \frac{1}{4}} \right); \\ \end{aligned}$$
(23)
The maximum value \(n_{\max }\) can be obtained by setting \({\raise0.7ex\hbox{${dE_{n} }$} \!\mathord{\left/ {\vphantom {{dE_{n} } {dn}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${dn}$}} = 0\) ,
$$n_{\max } = – \tilde{\rm K} \pm \sqrt {\Omega_{2} }$$
(24)
Replacing the summation in (22) by an integral, we have;
$$Z\left( \beta \right) = \int\limits_{0}^{\omega } {e^{{ – \beta \left( {\Omega_{0} – \Omega_{1} \left( {\frac{{\Omega_{2} – \left( {n + \tilde{\rm K}} \right)^{2} }}{{\left( {n + \tilde{\rm K}} \right)}}} \right)^{2} } \right)}} } dn$$
(25)
If we set \(\Lambda = n + \tilde{\rm K}\), we can rewrite the above integral in Eq. (25) as follows;
$$Z\left( \beta \right) = \int\limits_{{q_{1} }}^{{q_{2} }} {e^{{\beta \left( {\frac{{\Omega_{1} \Omega_{2}^{2} }}{{\Lambda^{2} }} + \Omega_{1} \Lambda^{2} – \Omega_{3} } \right)}} d\Lambda }$$
(26)
$${\text{where}}\,\,q_{1} = \tilde{\rm K},\,\,q_{2} = \omega +\tilde{\rm K}\,\,{\text{and}}\,\,\Omega_{3} = 2\Omega_{2} \Omega_{1} + \Omega_{0}.$$
(27)
On evaluating the integral in Eq. (26), we obtain the partition function of the DFP in magnetic and AB fields as follows;
$$Z\left( \beta \right) = – \frac{{e^{{ – 2\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } + \beta \,\Omega_{3} }} \sqrt \pi \left( \begin{gathered} Erf\left[ {q_{1} \sqrt { – \beta \,\Omega_{2} } – \frac{{\sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }}{{q_{1} }}} \right] + e^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }} Erf\left[ {q_{1} \sqrt { – \beta \,\Omega_{2} } + \frac{{\sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }}{{q_{1} }}} \right] – Erf\left[ {q_{2} \sqrt { – \beta \,\Omega_{2} } – \frac{{\sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }}{{q_{2} }}} \right] \hfill \\ – e^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }} Erf\left[ {q_{2} \sqrt { – \beta \,\Omega_{2} } + \frac{{\sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }}{{q_{2} }}} \right] \hfill \\ \end{gathered} \right)}}{{4\sqrt { – \beta \,\Omega_{2} } }}$$
(28)
Free energy
The Helmholtz free energy is a thermodynamic potential that determines an estimate of the useful work obtained from a thermodynamic system that is closed and maintained at a constant temperature59. The free energy is computed using the expression given below17;
$$F\left( \beta \right) = – \frac{1}{\beta }\ln Z\left( \beta \right),$$
(29)
Substituting Eq. (28) into Eq. (29), we obtain the free energy for the TiH diatomic molecule modelled by the Deng-Fan potential as follows;
$$F\left( \beta \right) = – \frac{{\ln \left( { – \frac{{{\text{e}}^{{ – 2\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } + \beta \,\Omega_{3} }} \sqrt \pi \left( \begin{gathered} Erf\left[ {q_{1} \sqrt { – \beta \,\Omega_{2} } – \frac{{\sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }}{{q_{1} }}} \right] + {\text{e}}^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }} Erf\left[ {q_{1} \sqrt { – \beta \,\Omega_{2} } + \frac{{\sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }}{{q_{1} }}} \right] – Erf\left[ {q_{2} \sqrt { – \beta \,\Omega_{2} } – \frac{{\sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }}{{q_{2} }}} \right] \hfill \\ – {\text{e}}^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }} Erf\left[ {q_{2} \sqrt { – \beta \,\Omega_{2} } + \frac{{\sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }}{{q_{2} }}} \right] \hfill \\ \end{gathered} \right)}}{{4\sqrt { – \beta \,\Omega_{2} } }}} \right)}}{\beta }.$$
(30)
Entropy
Entropy is the measure of the amount of a system’s thermal energy per unit temperature that cannot be used to perform any productive work. The amount of entropy in a system can be thought of as a measure of the molecular disorder, or unpredictability, of the system as a whole due to the fact that work is obtained from the orderly motion of molecules. The idea of entropy offers profound insight into the course of spontaneous change for a wide variety of phenomena encountered daily58,60. The entropy is computed using the expression given as60;
$$S\left( \beta \right) = \ln Z\left( \beta \right) – \beta \frac{d\ln Z\left( \beta \right)}{{d\beta }},$$
(31)
Substituting Eq. (28) into Eq. (31), we obtain the entropy for the TiH diatomic molecule modelled by the Deng-Fan potential as follows;
$$S\left( \beta \right) = \frac{{\left( { – 4{\text{e}}^{{2\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \,\Omega_{2}^{2} } }} \left( {{\text{e}}^{{\frac{{\beta \,\Omega_{2} \left( {q_{1}^{4} + \Omega_{1} \Omega_{2} } \right)}}{{q1^{2} }}}} q_{1} – {\text{e}}^{{\frac{{\beta \,\Omega_{2} \left( {q_{2}^{4} + \Omega_{1} \Omega_{2} } \right)}}{{q_{2}^{2} }}}} q_{2} } \right)\sqrt { – \beta \,\Omega_{2} } + \sqrt \pi \left( {{\rm N}_{1} } \right){\rm T}_{1} – {\text{e}}^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }} \sqrt \pi \left( {N_{2} } \right)T_{1} + \sqrt \pi \left( {\left( {N_{3} } \right)T_{2} + {\text{e}}^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }} \left( {N_{2} } \right)T_{2} } \right)} \right)}}{{\left( {2\sqrt \pi \left( { – 1 + T_{1} + {\text{e}}^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }} \left( {T_{1} – T_{2} } \right) + T_{3} } \right)} \right)}} + \Xi$$
(32)
where
$$T_{1} = Erf\left[ {q_{1} \sqrt { – \beta \,\Omega_{2} } – \frac{{\sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }}{{q_{1} }}} \right]$$
(32a)
$$T_{2} = Erf\left[ {q_{2} \sqrt { – \beta \,\Omega_{2} } – \frac{{\sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }}{{q_{2} }}} \right]$$
(32b)
$$T_{3} = Erfc\left[ {q_{2} \sqrt { – \beta \,\Omega_{2} } – \frac{{\sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }}{{q_{2} }}} \right]$$
(32c)
$$T_{4} = Erfc\left[ {q_{2} \sqrt { – \beta \,\Omega_{2} } – \frac{{\sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }}{{q_{2} }}} \right]$$
(32d)
$$N_{1} = 1 + 4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } – 2\beta \,\Omega_{3}$$
(32e)
$$N_{2} = – 1 + 4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } + 2\beta \,\Omega_{3}$$
(32f)
$$N_{3} = – 1 – 4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } + 2\beta \,\Omega_{3}$$
(32g)
$$\Xi = \ln \left[ {\frac{{{\text{e}}^{{ – 2\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } + \beta \,\Omega_{3} }} \sqrt \pi \left( { – T_{1} + T_{2} + {\text{e}}^{{4\sqrt { – \beta \Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }} \left( { – T_{1} + T_{2} } \right)} \right)}}{{4\sqrt { – \beta \,\Omega_{2} } }}} \right] $$
(32h)
Internal energy
The energy contained within a thermodynamic system is referred to as its internal energy. Internal energy is constant in an isolated system. It is the energy required to develop or prepare the system in its current internal state. It does not contain the system’s overall kinetic energy, but it includes the kinetic energy of the particles inside the system. It keeps track of the system’s energy gains and losses due to changes in its internal condition45,59. The internal energy can be evaluated using the expression below36;
$$U\left( \beta \right) = – \frac{d\ln Z\left( \beta \right)}{{d\beta }},$$
(33)
Substituting Eq. (28) into Eq. 33), we obtain the internal energy for the TiH diatomic molecule modelled by the Deng-Fan potential as follows;
$$U\left( \beta \right) = \frac{{\left( { – 4{\text{e}}^{{2\sqrt { – \beta \Omega_{2} } \sqrt { – \beta \Omega_{1} \Omega_{2}^{2} } }} \left( {{\text{e}}^{{\frac{{\beta \Omega_{2} \left( {q_{1}^{4} + \Omega_{1} \Omega_{2} } \right)}}{{q_{1}^{2} }}}} q_{1} – {\text{e}}^{{\frac{{\beta \,\Omega_{2} \left( {q_{2}^{4} + \Omega_{1} \Omega_{2} } \right)}}{{q_{2}^{2} }}}} q_{2} } \right)\sqrt { – \beta \,\Omega_{2} } + \sqrt \pi \left( {N_{1} } \right)T_{1} – {\text{e}}^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\,\Omega_{1} \Omega_{2}^{2} } }} \sqrt \pi \left( { – 1 + 4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } + 2\beta \,\Omega_{3} } \right)T_{1} + \sqrt \pi \left( {\left( {N_{3} } \right)T_{2} + {\text{e}}^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }} \left( {N_{2} } \right)T_{2} } \right)} \right)}}{{\left( {2\sqrt \pi \beta \left( { – 1 + T_{1} + {\text{e}}^{{4\sqrt { – \beta \,\Omega_{2} } \sqrt { – \beta \,\Omega_{1} \Omega_{2}^{2} } }} \left( {T_{1} – T_{2} } \right) + T_{4} } \right)} \right)}}$$
(34)
Specific heat capacity
The specific heat capacity of material in thermodynamics is the heat capacity of a sample of the substance divided by its mass, also known as massic heat capacity. Loosely, the quantity of heat must be added to one unit of mass of the substance to generate one unit of temperature increase. Specific heat capacity frequently changes with temperature and depends on the state of materials56,61. Substituting Eq. (28) into Eq. (33), we obtain the specific heat capacity for the TiH diatomic molecule modelled by the Deng-Fan potential as follows56,61;
$$C\left( \beta \right) = \beta^{2} \frac{{d^{2} \ln Z\left( \beta \right)}}{{d\beta^{2} }},$$
(35)
However, due to the complicated nature of the analytical expressions, the graphic representations are shown.
Magnetic properties at \(T \ne 0\)
In this section, the magnetic properties of TiH diatomic molecule at finite temperature are considered;
Magnetization
Magnetization, also known as magnetic polarisation, is a vector quantity that represents the density of permanent or induced dipole moments in a magnetic material. As we know, magnetization is caused by the magnetic moment, which is caused by the mobility of electrons in atoms or the spin of electrons or nuclei. It is a very important quantity. Substituting Eq. (28) into Eq. (33), the magnetization at finite temperature for the TiH diatomic molecule modelled by the Deng-Fan potential is evaluated using the expression given below56,56,61;
$$M\left( \beta \right) = \frac{1}{\beta }\left( {\frac{1}{Z\left( \beta \right)}} \right)\left( {\frac{\partial }{{\partial \vec{B}}}Z\left( \beta \right)} \right).$$
(36)
Magnetic susceptibility
Magnetic susceptibility is a measure of how much a substance will become magnetized in an applied magnetic field in electromagnetism. This quantity allows a straightforward categorization of most materials’ responses to an applied magnetic field into two categories: alignment with the magnetic field called, paramagnetism, or alignment against the field called diamagnetism. Substituting Eq. (28) into Eq. (37), the magnetic susceptibility at finite temperature for the TiH diatomic molecule modelled by the Deng-Fan potential is evaluated using the expression given below56,56,61;
$$\chi_{m} \left( \beta \right) = \frac{\partial M\left( \beta \right)}{{\partial \vec{B}}}.$$
(37)
Persistent current
A persistent current is a thermodynamic quantity. It is the continuous electric current that does not require an external power source. Such a current is usually said to be unachievable in standard electrical equipment since all commonly used conductors have a non-zero resistance, and any such current would quickly dissipate as heat. However, due to quantum phenomena, persistent currents are feasible and observed in superconductors and some mesoscopic devices. Because of size effects in resistive materials, persistent currents can emerge in microscopic samples. Persistent currents are commonly employed in superconducting magnets. Substituting Eq. (28) into Eq. (38), the persistent current at finite temperature for the TiH diatomic molecule modelled by the Deng-Fan potential is evaluated using the expression given below56,56,61;
$$I\left( \beta \right) = – \frac{e}{hc}\frac{\partial F\left( \beta \right)}{{\partial m}}.$$
(38)
Due to the complicated nature of the analytical expressions, the graphic representations are shown for the magnetic properties at finite temperature.
Magnetic properties at \(T = 0\)
In this section, the expressions for evaluating the magnetic properties at zero temperature are presented below.
Magnetization
The magnetization at zero temperature is evaluated at zero temperature using the expresssion given below and Eq. (18)45.
$$M_{nm} = – \frac{\partial E}{{\partial \vec{B}}}.$$
(39)
Magnetic susceptibility
The magnetic susceptibility at zero temperature is evaluated at zero temperature using the expression given below and Eq. (18)45.
$$\chi_{m} = \frac{{\partial M_{nm} }}{{\partial \vec{B}}}.$$
(40)
Persistent current
The persistent current at zero temperature is evaluated at zero temperature using the expresssion given below and Eq. (18)62.
$$I_{nm} = – \frac{\partial E}{{\partial \phi_{AB} }}$$
(41)
Again, we point out here that due to the complicated nature of the expression, the graphical representations are presented.
Magnetic entropy change
The magneto temperature effect (MCE) is the sensitivity of a magnetic material to an applied magnetic field at a specific temperature. The ability of magnetic materials to regulate their temperature or swap heat with a thermal reservoir in response to a changing magnetic field is a remarkable attribute of magnetic materials. The magnetocaloric effect is studied by considering the magnetocaloric potential (S) and the change in temperature (T). Because of this, we must first determine the entropy both with and without a magnetic field to compute the magnetic entropy change. To calculate S in an isothermal process, one uses the formula. To calculate the magnetic entropy change, the entropy is first calculated with and without magnetic field. For an isothermal process, the quantity ΔS is given by40,56,63:
$$\Delta S = S\left( {B \ne 0,T} \right) – S\left( {B = 0,T} \right).$$
(42)