Optimization of semi-empirical mass formula co-efficients using least square minimization and variational Monte-Carlo approaches

Abstract

In this paper, both Least Squares minimization (LSM) and Variational Monte-Carlo (VMC) techniques have been implemented to determine the co-efficients of Semi-Empirical Mass Formula (SEMF). First, the experimental binding energies (BEs) are determined for all the available nuclei from Atomic Mass Evaluation (AME2016) data. Then, LSM technique is implemented in Gnumeric worksheet to obtain the SEMF co-efficients by considering only the first three co-efficients which are deduced from Liquid Drop Model (LDM). The mean squared error (MSE) value, between obtained BEs from the optimized co-efficients and the experimental BEs, has been determined. Then, to emphasize the relevance of empirical terms, they have been introduced successively one after other and the procedure is repeated. A reduction in MSE-value has been observed after each iteration. This same procedure has also been employed using Monte-Carlo approach to obtain SEMF co-efficients by minimizing MSE-value as in variational principle. A comparative analysis has shown that the optimized parameters using VMC have resulted in smaller MSE-value than that of LSM technique.

Approximating the nuclear binding energy using analytic continued fractions

“…The LDM envisions the nucleus as a charged, irrotational spherical liquid drop, with its energy comprising several components, including a 'volume energy' , a 'surface energy' , and a 'Coulomb energy' . Additionally, it includes two more specific terms known as the 'asymmetry' and 'pairing' terms 2,[17][18][19][20] .…”

“…δ(N, Z) is either zero or ±δ 0 depending on the parity of number of neutrons N and protons Z [19][20][21][22] . The free parameters found in the model derive from least squares to fit experimental data and can be found in 20 ; they are the volume coefficient, a V = 15.192 ; the surface coefficient, a S = 16.269 ; the Coulomb coefficient, a C = 0.679 ; the asymmetry coefficient, a a = 21.675 and the previously mentioned called the pairing term, δ 0 = 10.619 . It is known that further terms exist to explain additional phenomena 4,23 and that this model does not fit well the data for low values of A as we will show in our results.…”

Approximating the Nuclear Binding Energy Using Analytic Continued Fractions

“…The LDM envisions the nucleus as a charged, irrotational spherical liquid drop, with its energy comprising several components, including a 'volume energy,' a 'surface energy,' and a 'Coulomb energy.' Additionally, it includes two more specific terms known as the 'asymmetry' and 'pairing' terms 2,[15][16][17][18] .…”

“…known as the pairing term. δ (N, Z) is either zero or ±δ 0 depending on the parity of number of neutrons N and protons Z [17][18][19][20] . The free parameters found in the model derive from least squares to fit experimental data and can be found in 18…”