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Surface energy coefficient determination in global mass formula from fission barrier energy

Semi-empirical mass formula of the atomic nucleus describe binding energies of the nuclei. In the simple form of this formula, there are five terms related to the properties of the nuclear structure. The coefficients in each terms can be determined by various approach such as fitting on experimental binding energy values. In this study, the surface energy coefficient in the formula which is a correction on total binding energy has been obtained by a method that is not previously described in the literature. The experimental fission barrier energies of nuclei have been used for this task. According to the results, surface energy coefficient in one of the most conventional formula has been improved by a factor 3.4.


Introduction
The nuclear mass formula is very important for describing nuclear properties and exploring the exotic structure of the nuclei such as halo structure, super-heavy nuclei structures and decays [1]. Liquid drop model clarifies many nuclear phenomena which are unachievable by the shell model of the nucleus. The semi-empirical mass formula based on this model of the nucleus was first proposed in 1935 by Bethe and von Weizsacker [2,3]. According to the formula, the nuclear binding energy is expressed in terms of A and Z numbers of the nuclei. The conventional formula has simply five terms named as volume, surface, Coulomb, asymmetry and pairing energy terms. The surface term is a correction in total binding energy due to deficit of binding energy for nucleons in the surface area. The magnitude of the nuclear surface energy is intimately related to the diffuseness of the nuclear surface and should provide a measure of the thickness of the nuclear surface. Since the surface energy is related to the lack of binding of the particles in the surface, it is clear that any attempt at a quantitative account of the nuclear surface energy will come up against difficulties due to our insufficient understanding of the nature of the effects responsible for nuclear cohesion [4]. Besides, the well-known force inside the nucleus is related to Coulomb energy term. This is a repulsive term among the protons. The coefficient in the Coulomb term can easily be calculated by the formula = 3 2 /5 0 .
Recently, semi-empirical mass formula has been extended by adding extra terms or has been modified slightly or completely [5][6][7][8][9][10][11][12][13]. These attempts have been made in order to obtain binding energies of the nuclei more accurately. In the study of Kim and Cha [14], the coefficients and even the power of the A number have been determined in order to reach experimental values as close as possible. Also in that work, the nuclei are divided into different groups according to their half-lives and it is obtained different coefficients for each group. The coefficients in each term can be determined by fitting the formula to the experimental binding energies on the atomic nuclei.
After the discovery of the fission, this phenomenon was started studying by considering nuclear drop model. If the Coulomb energy does not exceed a critical value, a charged drop is stable against fission. The surface energy in the drop model wants to keep the nucleus spherical, whereas Coulomb energy wants to deform it. Whether there will be a fission or not depends on the balance of these two effects. One can determine fissility parameter (x) that is characterized by the ratio of surface and Coulomb energies. If x exceeds the value of 1, fission occurs immediately [15].
Throughout the years, the constant in the semi-empirical mass formula has been determined many times by using various procedures or on different data sets. Every determined coefficient is different from each other. In this study, we have applied a different approach to obtain a constant in the basic five term formula. We have used experimental fission barrier energies to determine the surface energy coefficient in semi-empirical mass formula. We have taken the measured fission barriers from Myers study [16]. We have considered x to perform this task.
Our aim was to obtain surface energy coefficient from experimental fission barrier energies and hence to reduce the mean square error value between theoretically determined binding energies of the nuclei and experimental ones.

Material and Methods
The most conventional simple semi-empirical mass formula considered in this work has been given in Eq. 2.1. This formula has simply five terms named as volume, surface, Coulomb, asymmetry and pairing. The coefficients in each term are calculated mostly by fitting to experimentally measured masses of nuclei. They usually vary depending on the fitting methodology.
In the formula, k takes the value +1, 0 or -1 for even-even, even-odd or odd-odd nuclei, respectively.
In fission process in which nuclear shape deviates from spherical shape, the surface energy of the nuclei increases and the Coulomb energy decreases because charge density is reduced. The other terms contributing to the total binding energy of the nuclei are not appreciable changed when the nuclei split into two fragments. The total potential energy is determined by the sum of these two terms given in Eqs. 2.3-4.
The ratio of these terms given in Eq. 2.5 has been known as fissility parameter x. Stable, unstable, and metastable states are defined using the fissility parameter, the released energy, and the fission barrier [16].

Results and Discussion
We have used the barrier maximum formula (Eq. 3.1) in order to obtain surface term coefficient in the semi-empirical mass formula.  We have tested semi-empirical mass formula (Eq. 2.1) with Krane coefficient (av=15.5, as=16.8, aC=0.72, aa=23 and ap=34). The mean square error (MSE) value between theoretical masses of the nuclei and the experimental masses has been obtained 100.9 for 3245 isotopes from A=20 to 295. If we use the new surface coefficient as as=16.481 in same formula, the MSE value has been achieved as 29.6 which gives 3.4 factor better result than Krane surface coefficient gives.
In Fig. 1.a, the differences between experimental binding energies (BEexp) and theoretical binding energies (BEtheo) calculated by Krane coefficients have been shown. The deviations from experimental values are lied between about -10 to 40 MeV. In Fig. 1.b, we have also shown these differences by redefined surface coefficient. The deviations are lied between about -10 to 20 MeV.

Conclusion
In this work, the experimental fission barrier energy values exist for 36 nuclei have been used for determination of the surface energy coefficient in semi-empirical mass formula of liquid drop model of the nucleus. We have considered conventional mass formula with Krane coefficient. We have borrowed Coulomb energy coefficient as existing value and calculated surface energy terms and then their coefficients for each 36 nuclei. After obtaining the coefficients, we have calculated the average value. The redefined value of the surface coefficient is as=16.481. Other coefficients remain same, when we used this coefficient in semiempirical formula, the result is 3.4 factor better the result of Krane surface coefficient.