Comparison of binding energy from Atomic Mass Evaluation (AME-2020) with statistical model calculations

. Realistic model calculations are fundamental to any physical problem. The statistical model continues to remain one of the important theoretical basis formalism in the nuclear reaction and structure field, which is based on the theory of compound nuclei. In this work, we have compared the recently measured binding energies from The Atomic Mass Evaluation 2020 with statistical model calculation. This comparison becomes very important from the fact that code calculates everything from excitation energy Q-value of reaction to cross-sections based on binding energy and is used frequently by nuclear research communities throughout the world. We have given a brief explanation of how code calculates and comparison of calculated Binding energy with recently measured binding energies from The Atomic Mass Evaluation 2020. A total of 254 stable nuclides binding energies are compared. Statistical model calculations are presented for spin distribution of fusion cross-section for di ff erent di ff usive parameters for two di ff erent reactions are shown and nuclear level density using di ff erent formalisms are also presented

The atomic nucleus is a quantum many-body system, which when produced in nuclear reaction experiments with different projectile + target combinations may undergo different processes namely fusion, fission, and evaporation residues with the decay of n, p, α. With the advent of heavy ion accelerators in the late 60s, the nuclear community focused on heavy ion reactions. With the establishment of the theory of compound nuclei by the famous experiment of Ghoshal [1], the theoretical interpretation of compound nuclei took a giant leap. It has also led to the development of a statistical model which successfully describes the experimental data. The statistical model is based on the assumption that the compound nucleus (CN) formed above the coulomb barrier with certain excitation energy is fully equilibrated in all degrees of freedom (i.e. energy, angular momentum, etc.) before the decay takes place. The formation of CN does not depend upon the entrance channel. The statistical model code known as CAS-CADE was first developed by F. Puhlhofer and is based on the FORTRAN language. Over the years many groups have made substantial changes to the model and also to the code [2,3]. Like many parameters, Q-value is one of the important ingredients of a statistical model, which determines whether the nuclear reaction process will take place or not. It may be defined as the mass difference between initial reactants and products. In terms of binding energy, it can be defined as the difference in the binding energy of products and reactants. Q-value of nuclear reaction can be given by: a + * e-mail: In terms of binding energy, the Q value of the reaction can be defined as: where B is binding energy. CASCADE calculates the Qvalue according to binding energies. The binding energy of nuclei in CASCADE is calculated by using Myers and Swiatecki [5] where the first term is volume energy, the second term is surface energy, the third and fourth are Coulomb energy terms and the last term is an even-odd correction. c 1 , c 2 , c 3 , c 4 are different coefficients and are given by: where a 1 , a 2 , k are constants. N, Z, and A are the number of neutrons, protons, and mass numbers respectively. CASCADE has many input parameters, target projectile atomic and mass number, and lab energy at which the reaction takes place to name a few. We can choose the type of fission barrier (Sierk's macroscopic model or RLDM fission barrier) and level density approach whether to use the traditional Puhlhofer approach or use Ignatyuk [6] method to handle shell corrections by modifying the level density parameter. Statistical model (CASCADE) calculations have been a benching mark in the nuclear reaction community. So it is important to use parameters as realistic as possible. CASCADE calculates the Q value of the reaction using the binding energy of reactants and products. So, comparing experimental binding energy values with calculated ones is crucial. The National Nuclear Data Center, Brookhaven National Laboratory which is a repository of nuclear data e.g. nuclear structure, decay, and reaction data publishes atomic masses from time to

Analysis
In this section, results from the statistical model calculation are presented.

I. Fusion Cross-section
In a nuclear reaction, the target nuclei are bombarded with projectile nuclei to create the compound nucleus. The projectile and target have to overcome the repulsive coulomb barrier in order to create the compound nucleus. In addition to the coulomb barrier, the Q-value threshold of the system has to be crossed. Therefore, it is an act of precise attention to select the bombarding energy of projectiles. Heavy-ion beams carry a lot of angular momentum, which when transferred to the target nuclei creates a hot and rotating compound nucleus at high excitation energy. The formula for the compound nucleus's maximal angular momentum is: where, r = r 0 (A 1/3 , A t and A p are mass number of target and projectile respectively. E cm is centre of mass energy of compound nucleus, V cb is the coulomb barrier between target and projectile. DCASCADE uses the same formula for angular mo-  mentum and calculates the angular momentum through the function "CLFUS" and distributes cross-section according to angular momentum. The fusion cross-section (σ f usion ) is calculated by: where T l is the transmission coefficient, and k is the wave number. According to the classical concept, fusion takes place only above the Coulomb barrier, and the probability to penetrate through the coulomb barrier is given by the transmission probability coefficient (T l ) : where l g is grazing angular momentum. In DCASCADE code, transmission probability coefficient (T l ) is given by: where d is the diffuseness parameter. Depending on the diffusiveness parameter, the spin distribution shape changes. Fig 2, 3 shows the different shapes depending on the values of 'd' for two different reactions. This angular momentum fusion cross-section is used in the calculations for the decay of the compound nucleus. So, it is important to choose the diffuseness parameter very carefully. One can also use the experimental data in the present calculation using an external file named "CASPOP.DAT". But in the absence of experimental data, normally which is the case, angular momentum (J) dependent fusion is generated internally for the compound nucleus and is further used in particle, gamma spectra generation, evaporation residue, and fission cross-section calculations.

II. Nuclear Level Density
One of the important and indispensable factors which decide the particle, gamma, and fission widths and lifetimes is nuclear level density. It is defined as the number of quantum states available at a particular excitation energy. The state density (ω) is different from nuclear-level density in the sense that for any angular momentum J, there are (2J+1) degenerate magnetic states M. State densities include all states (J, M), whereas nuclear level (ρ) densities only include states with various J's. [4]. Bethe was the first to describe the nuclear level density using simple statistical mechanics, using the idea that the nucleus is made up of a group of fermions that do not interact. But it lacked to describe the measured discrepancy between nuclei level densities. A modified analytic formula of Bethe's formula known as The "back-shifted Fermi-gas model" is used widely to include an energy shift to account for these differences. The nuclear level density at excitation energy (E) and angular momentum (J) is given by : where U is available resonance and θ ′ =θ(1+δJ 2 +δ ' J 4 ) and δ, δ ' are deformation coefficients and θ is moment of inertia. Little 'a' is called the level density parameter and  traditionally or even today many physicists use a = A/k where k is a positive integer number and normally k is taken as a number that fits the experimental data well. Its value in general lies somewhere between 8-10. The little 'a' parameter is also given by a = π 2 g/6, where g is the sum of neutrons and the protons single particle state density near the Fermi energy. In this part, DCASCADE is utilized, and different level density formalisms are included in the code. Nuclear density of 134 Ce ( 19 F + 115 In) has been reproduced using both traditional Puhlhoper ap-proach [7] as mentioned above and modern BJK formalism [8] and shown in fig 5.

Summary
In summary, we have worked on the statistical model (CASCADE) and calculated the binding energy of several isotopes across the periodic table, and compared them with experimental results. It calculates binding energy in accordance with experimental results and it is fruitful to use this code in nuclear reactions analysis. The results of the statistical model, namely fusion cross-section angular momentum distribution with different diffusive parameters are shown and nuclear level density results with different level density parameters and different formalisms are also presented.