Pairing Effect on Nuclear Level Density of 56Fe Using Statistical Mechanics With Single Particle States (SMSPS)

The SP Partition Function

Starting from the basic definition of the Hamiltonian

where

here $t\left({\overrightarrow{r}}_j\right)$ is the single nucleon kinetic energy

and the second term $v\left({\overrightarrow{r}}_j\mathrm{,}{\overrightarrow{r}}_i\right)$ is the nucleon-nucleon interaction potential for nucleons at ${\overrightarrow{r}}_j$ and ${\overrightarrow{r}}_i$. By adding and subtracting the mean field potential ${\sum }_{j=1}^{A}v\left({\overrightarrow{r}}_j\right)$, one obtains

The first two terms in the right-hand side of Eq. (4) represent the mean field Hamiltonian ${\hat{H}}_m$, whereas the last two terms are called the is the residual interaction Hamiltonian, or simply the pairing Hamiltonian. Thus, the Hamiltonian $\hat{H}$ is

where

and

Here ${\hat{H}}_m$ is the mean field Hamiltonian, $h\left({\overrightarrow{r}}_j\right)$ which represents the single-nucleon Hamiltonian, and ${\hat{H}}_P$ is the residual or (effective) two-body interaction Hamiltonian between the valence nucleons required to be explicitly taken into account for a description of nuclei with two or more valence nucleons, where “residual” refers to the part of the interaction that is not absorbed into the central potential. This leads to the mixing of various configurations and removes the degeneracy of the states that belong to the same configuration.^[5] For open shell nuclear systems, such as ⁵⁶Fe, the nuclear pairing Hamiltonian^[6] is particularly important. For our specific application, we consider the pairing interaction contributions to the states above the Fermi level. The SP partition function Z₁ defined as^[3]

The first term in the right-hand side is the discrete single-particle states (SPS) partition function,

produced by the mean field Hamiltonian. The degenerate substates are labeled by the index k. The index α denotes the set of quantum numbers for each degenerate state. The second term in the right-hand side of Eq. (8) is the continuum (classical) partition function.

E_max stands for an energy cutoff defining the mean field Hamiltonian’s maximum resonance state, and V stands for the volume of the continuum state. The integral can be computed to give the value of Eq. (10), which represents the sum of the continuum states to yield

The total summation of discrete states can be implemented as long as the adopted mean field possesses spherical symmetry (which we adopt here), meaning that there is degeneracy with regard to the angular momenta projection $m_{J_{\alpha }}$ as^[3]

where $E_{\alpha }$ is the SP energy of the level $\alpha$. The degeneracy of the ${\alpha }^{th}$ state is

$g_{\alpha }={\sum }_k\left({\psi }_{\alpha }^{\left(k\right)}|{\psi }_{\alpha }^{\left(k\right)}\right)$

The partition function in Eq. (12) represents the partition function of SPS plus pairing. Within the framework of the SMSPS, the effects of pairing and spin-orbit splitting are averaged by arranging the nucleons using a technique, which we call configuration-restricted recursion.^[3] We implement this technique for a fixed species of protons or neutrons by constructing the partition function for a subgroup of that species that limits the values of the magnetic projection of the total angular momentum quantum number $m_J$ to positive ($+J_{\alpha }$’s) and negative ($-J_{\alpha }$’s) values. Thus, we obtain the two SPS partition functions of those restricted ranges, denoted as ${\left(Z_1^{SPS}\right)}^{(+)}$ and $\left(Z_1\right)$, given by

where $E_{\alpha }=E_{\alpha }+E_F$. Making use of Eq. (11) and Eq. (13) within Eq. (8), we obtain $Z_1^{(\pm )}$

Thermal Evolution of Excitation Energy and the Effect of Pairing Strength

In the finite-temperature BCS (FT-BCS) theory,^[10] the nuclear pairing Hamiltonian is similar to the one given in Eq. (21), except for the interaction strength is assumed to be constant for simplicity.^[11] Thus, the pairing Hamiltonian (21) is written as

$H=\sum _{\alpha }{E}_{\alpha }{a}_{\alpha }^{†}{a}_{\alpha }+A_T\sum _{\alpha \mathrm{,}{\alpha }^{\prime }}{a}_{\alpha }^{†}{a}_{\overline{\alpha }}^{†}{a}_{\overline{\alpha }\text{'}}{a}_{{\alpha }^{\prime }}\mathrm{,}$

where $E_{\alpha }$ are the SP energies, A_T is the pairing strength. At temperature T = β⁻¹, the gap equation is^[10]

$1=A_T\sum _{\alpha }\frac{1-2f_{\alpha }}{2E_{\alpha }}\mathrm{,}$

with

$f_{\alpha }=\frac{1}{1+e^{\beta E_{\alpha }}}\mathrm{,}{E}_{\alpha }=\sqrt{{(E_{\alpha }-\mu )}^2+{\Delta }^2}\mathrm{,}$

where $f_{\alpha }$ is the Fermi-Dirac distribution, and µ is the chemical potential (in our case, it is $E_F$). As the temperature increases, the pairing gap ∆ gradually decreases and collapses at a critical temperature T_c, signaling the breakdown of pairing correlations, and the gap vanishes ($\Delta =0$). The thermal excitation energy is defined by

$E_x\left(T\right)={\langle H\rangle }_T-{\langle H\rangle }_{T=0}\mathrm{,}$

which, in the FT-BCS framework, can be expressed as

$E_x\left(T\right)=\sum _{\alpha }\left(\begin{array}{c}(E_{\alpha }-\mu )\left(1-\frac{E_{\alpha }-\mu }{E_{\alpha }}\tanh \left(\frac{\beta E_{\alpha }}{2}\right)\right)\\ +\frac{{\Delta }^2}{E_{\alpha }}\tanh \left(\frac{\beta E_{\alpha }}{2}\right)\end{array}\right)-E_{\text{gs}}\mathrm{,}$

where E_gs is the ground-state energy at T = 0. The relationship describing the behavior of the BCS energy gap near the critical temperature $T_c$ is given by the empirical relation^[12]

$\Delta \left(T\right)\approx 3.06k_B{T}_c\sqrt{1-\frac{T}{T_c}}\mathrm{.}$

The validity of the BCS energy gap equation near the critical temperature implies that $T\le T_c$. At low temperature $T\ll T_c$, the energy gap can be approximated to first order.

$\Delta \left(T\right)\approx 3.06k_B{T}_c\left(1-\frac{T}{2T_c}\right)\mathrm{.}$

The typical critical temperature in the region has been estimated to be approximately 0.6 MeV.^[13] As reported in Ref.,^[3] for ⁵⁶Fe, the pairing gap remains nearly constant at Δ = 2.01 MeV when T ≤ 0.885 MeV. In this temperature regime, the thermal excitation is dominated by contributions from the valence nucleons, while the effect of the core nucleons is negligible. Consequently, the present work restricts the model space to the four valence-shell orbits.

Using the BCS theory, pairing correlations are related to thermal effects through a common description. The pairing Hamiltonian in BCS

can be incorporated in the partition function of SPS plus pairing given by Eq. (12). The thermal population of particles among available states has been addressed using various approaches, most notably through Monte Carlo techniques, to extend model applicability to hot nuclei.^[4,14,15] However, even when restricted to a limited set of orbits near the valence shell, such methods demand substantial computational resources. The Hartree–Fock–Bogoliubov (HFB) theory^[16,17] provides the theoretical foundation for an alternative framework that incorporates pairing correlations. Nevertheless, the HFB approach is constrained by issues related to particle-number fluctuations and quasiparticle parity mixing.^[6,18] In the present work, we employ the SMSPS method to describe the thermal population of states.

Two-nucleon matrix-element

The two-nucleon interaction V is formulated in terms of the antisymmetrized two-nucleon matrix element ${\overline{\upsilon }}_{\alpha \beta \gamma \delta }$,^[19] given as

The anti-symmetric matrix element can be calculated as

where

are the normalized and anti-symmetrized two-nucleon states relative to the chosen vacuum state $|0\rangle$JQ29_087 - Copy.wmf], usually are |CORE). The normalized angular-momentum coupled of two-nucleon states a and b is given by^[20]

Here $\left(j_a{m}_{\alpha };j_b{m}_{\beta }|JM\right)$ is the Clebsch-Gordan coefficient, and $N_{ab}\left(J\right)$ is the normalization factor given by

where

${\delta }_{ab}=\left(\begin{array}{cc}0,& \text{for}a\ne b\\ 1,& \text{for}a=b\end{array}\right)$

The coupled-creation operator

We can obtain uncoupled operators by multiplying Eq. (29) by the conjugate of the Clebsch-Gordan coefficient and sum over all possible J and M states, and using the unitarity property, we get

The uncoupled two-body operators

and

operate with this on the vacuum and account for normalization in Eq. (27)

Make use of Eq. (33) and its conjugate in Eq. (24), one can determine the antisymmetric matrix element

Make use Eq. (34) into Eq. (23) yields

Make use of Eqs. (31) and (32) to substitute $c_{\alpha }^{†}{c}_{\beta }^{†}$ and $c_{\gamma }{c}_{\delta }$ by ${\left[c_a^{†}{c}_b^{†}\right]}_{JM}$ and ${\left[c_c{c}_d\right]}_{J^{\prime }{M}^{\prime }}$, respectively, utilize the normalization property of the Clebsch-Gordan Coefficients and apply the summation over all SPS, and use ${\tilde{c}}_{\alpha }\equiv {(-1)}^{j_a+m_a}{c}_{-\alpha }$, we obtain

The two-body interaction V in Eq. (36) is a scaler. In spherical tensor language, it is a tensor of rank zero; λ = 0. Using Wigner-Eckart theorem^[21,22]

where

and

$\hat{J}=\sqrt{2J+1}\mathrm{.}$

Thus,

We arrive at the resulting expression for V

The scaler product of two spherical tensors of the same rank^[23]

simplifies Eq. (40) to

Eq. (33) can be modified further by adding the isospin

The relation between coupled and uncoupled two-nucleon interaction matrix elements

where

The normalization factor $N_{ab}\left(JT\right)$JQ29_117 - Copy.eps] is given by

The equivalent form of Eqs. (37) and (38) are given by

$\langle ab;JM\left(V\right)cd;J^{\prime }{M}^{\prime }\rangle ={\delta }_{J{J}^{\prime }}{\delta }_{M{M}^{\prime }}{\delta }_{T{T}^{\prime }}{\delta }_{M_T{M}_{T^{\prime }}}\langle ab;JT\left(\left(V\right)\right)cd;J^{\prime }{T}^{\prime }\rangle \mathrm{,}$

which leads to

The nuclear states of two-particle nuclei with coupled angular momentum

On the other hand, for two-hole nuclei, the coupled angular momentum and isospin quantum state can be written as

The operator $h_{a{m}_{\alpha }\pm \frac{1}{2}}^{†}=-{\tilde{c}}_{a{m}_{\alpha }\pm \frac{1}{2}}$. For one-hole systems, either one-proton

or one-neutron

Schematic Zero-Range Force

A zero-range force assumes that the interaction between two nucleons occurs only when they are at the same point in space. In other words, instead of having a force that depends on distance (like the Coulomb or Yukawa potential), it is treated as acting only when the relative distance r = 0. Such a force is presented by the delta function

where $V_{\delta }$ is the strength constant. Using the expansion of the delta function in spherical coordinates

$V_0=-V_{\delta }\frac{1}{r_1{r}_2}\delta (r_1-r_2)\delta ({\Omega }_1-{\Omega }_2).$

Assuming the central part $V_L^{\delta }$ is involved in calculating the energy gap

The δ-function potential is non-separable. Instead, we consider the SDI defined by

where R is the nuclear radius. V₀ is the potential depth (energy/L⁴). The SDI is a modified, more realistic version of the delta interaction. It assumes that nucleon–nucleon pairing occurs mostly at R, near the nuclear surface, where the valence nucleons reside. So instead of acting everywhere, the delta force is localized at the nuclear surface. The central component of the SDI becomes

We note that the SDI potential becomes separable and has the same radial term for all pole ranks. Here, the rank = 2 ^L. This separability makes SDI easy to use in configuration interaction and pairing models, especially in shell-model calculations.

Investigations show^[24] SDI is realistic because it reproduces the nucleon-nucleon scattering data. The reason is that the outermost nucleons contribute the most to the scattering cross-section, especially at low energy scattering, where the cross-section is maximum.^[25] The SDI effectively captures the pairing correlations among valence nucleons without detailed short-range dynamics. It reproduces empirical energy gaps and spectra for many medium-mass nuclei.^[26] Since valence nuceons tend to occupy orbitals near the surface, the SDI gives a good approximation to the observed trends in nuclear structure.

Two-Body Matrix Element of the SDI

We want to evaluate the matrix element $\langle ab;JT\left(V\right)cd;JT\rangle$ in Eq. (44) using SDI to evaluate the two-body matrix as in Eq. (45). Because the SDI acts only on the surface and depends only on angular coordinates, it’s separable in the angular part. We can write

where $P_L\left(\cos \gamma \right)$​ is a Legendre polynomial of the angle $\gamma$, such that $\cos \gamma ={\hat{r}}_1\cdot {\hat{r}}_2$. The radial functions $f_L\left(r\right)$ are taken from Eq. (55),

Thus, the SDI becomes

Using the addition Theorem for spherical harmonics for two directions ${\hat{r}}_1=\left({\Omega }_1\right)$[GETIMGFILENAME=D:\ZAZA\SS\JQUS\XML\JQUS_29_2025\JQUS_29_2025\JQUS_29_2025_R30_InDesign\Links\ and ${\hat{r}}_2=\left({\Omega }_2\right)$[GETIMGFILENAME=D:\ZAZA\SS\JQUS\XML\JQUS_29_2025\JQUS_29_2025\JQUS_29_2025_R30_InDesign\Links\, the SDI potential given in Eq. (55) can be expanded as follows:

Defining the multipole operator $Q_{LM}$ as

Thus, expansion (59) becomes

The multipole operator obeys the parity relation

$Q_{LM}^{\ast }=(-{1)}^M{Q}_{L\mathrm{,}-M}\mathrm{.}$

Using expansion (61) into Eq. (23), where ${\overline{\upsilon }}_{\alpha \beta \gamma \delta }=\left(\alpha \beta \left|{\sum }_{LM}{Q}_{LM}^{\ast }\right({\overrightarrow{r}}_1\left)Q_{LM}\right({\overrightarrow{r}}_2\left)\right|\gamma \delta \right)$ in Eq. (24) yields

Since $Q_{LM}$ is a one-body spherical tensor, we turn to the Wigner-Eckart theorem in Eq. (37), and one can express the operator $Q_{LM}$ in terms of the reduced matrix element.

$Q_{LM}=\sum _{\alpha \beta }\left(\alpha \right)Q_{LM}\left(\beta \right)c_{\alpha }^{†}{c}_{\beta }={\hat{L}}^{-1}\sum _{ab}\left(a\left(Q_L\right)b\right)\left(c_{\alpha }^{†}{\tilde{c}}_{\beta }\right)\mathrm{,}$

where ${\tilde{c}}_{\alpha }\equiv {(-1)}^{j_a+m_a}{c}_{-\alpha }$. Making use of Eq. (62) yields

The operator $Q_{LM}$ is analogous to the electric operator,^[8] therefore, the reduced matrix element

where the quantity $R_{ab}^L$ is

The function $g_{n_a{l}_a}\left(r\right)$ is the SP radial wave function for nucleon a with principal quantum number $n_a$ and orbital angular quantum number $l_a$. Making use of Eq. (57) in Eq. (64), we obtain

or

Define the parameter ${\kappa }_{ab}$ as

making the radial parameter $R_{ab}^{\left(L\right)}$ to be

Define

Consequently, Eq. (64) becomes

Make use of Eq. (71) in Eq. (63), assuming the nucleons couple to total angular momentum J, and thus replace L by J yields the two-body matrix for SDI as^[27]

The two-body matrix element in the isospin representation can be directly obtained, as the Hamiltonian contains no isospin-dependent term. By generalizing Eq. (72), we obtain:^[28]

The value of the oscillator wave function g_nl(r) at the nuclear surface r = R is independent of quantum numbers n and l, with a phase factor depending on n.^[29,30] This makes the parameter K_abcd independent of a, b, c, and d. Hence

We utilize a simplified constant magnitude parameter K_abcd for T = 0,1^[28,29]

Here, A_T=0 and A_T=1 is chosen for best best-fit parameters. Special cases for Eq. (73) when a = b and c = d

$\begin{array}{c}\left(a^2;JT\right)V_{SDI}\left(c^2;JT\right)=-A_T[1-{(}^{-}]{\widehat{j_a}}^2{\widehat{j_c}}^2\\ \times \left(\begin{array}{c}\frac{1}{2}[1+{(-1)}^T]\left(\begin{array}{ccc}{j}_a& j_a& J\\ \frac{1}{2}& \frac{1}{2}& -1\end{array}\right)\left(\begin{array}{ccc}{j}_c& j_c& J\\ \frac{1}{2}& \frac{1}{2}& -1\end{array}\right)-\\ {(-1)}^{l_a+l_c+j_a+j_c}\left(\begin{array}{ccc}{j}_a& j_a& J\\ \frac{1}{2}& -\frac{1}{2}& 0\end{array}\right)\left(\begin{array}{ccc}{j}_c& j_c& J\\ \frac{1}{2}& -\frac{1}{2}& 0\end{array}\right)\end{array}\right)\mathrm{.}\end{array}$

More special cases when a = c yields

The two-nucleon interaction matrix element for the SDI of T = 1 in the 0f_{7 /2} shell is given by

Similarly, the two-nucleon interaction matrix element for the SDI of T = 1 in the 1p_{3 /2} shell is given by