Proceedings of the DAE Symp. on Nucl. Phys. 62 (2017) 440 Empirical formula for the most stable isobar against beta decay M. K. Preethi Rajan1, R. K. Biju2, and K. P. Santhosh1,* 1School of Pure and Applied Physics, Kannur University, Payyanur Campus, Payyanur-670327, INDIA 2Department of Physics, Pazhassi Raja N S S College, Mattanur- 670702, INDIA * email: [email protected] Introduction mass number and ZA value, we have developed an empirical formula for the most stable isobar Many of the nuclei produced in the lab are of a given A against beta decay and is given as, radioactive. The degree of instability grows as the distance of a given nuclide increases from the Z A aA b (2) stable nuclide with the same mass number [1]. The parameters are, a 0.33796 and An unstable nuclide attains stability through b 11.97723. We have also compared the present different decay mechanisms such as alpha decay, formula predictions with those obtained from beta decay, gamma decay, cluster radioactivity eqn. 1 and are shown the figure 1. It is found that etc. In the present work we have developed a the present formula predictions are in close formula for finding the most stable isobar for a agreement with the formula predictions of given mass number against beta decay. We have equation 1. We would like to point out that the also studied the mass parabola for different present formula is much simpler as compared to isobars with mass number ranging from 200-223. other empirical formulae. Hence the present It was found that the lowest point in the equation is better to identify the stability of the parabola, which is the Z value of most stable isotopes against beta decay in the heavy region. isobar against β decay, matches well with our formula predictions. Mass parabola We have also studied the mass parabolas Stability of nuclei against beta decay for different nuclides with mass number ranging One of the main applications of Bethe- from 200-223 and are shown in the figure 2 to 4. Weizsacker semi empirical mass formula is the Mass parabola is a plot of atomic masses versus prediction of the most stable isobar of a given A Z for different nuclei of same mass number. The against beta decay. The Z value of such isobar is minimum of the parabola gives the Z value of given by minimizing the atomic mass including most stable isobar against beta decay. It is found the mass of electron from the semi empirical that there is good match between our formula mass formula. This gives prediction of Z and minimum of mass parabola 2 for the considered range of mass numbers. 2aA (mn mp me )c / 2 (1) Z A A 4a a A2/ 3 The mass parabolas for the different isobars A C fall in two categories according to whether A is Where a 23MeV and a 0.72 MeV are A C odd or even. Figure 2 represents the plot for the coefficient of asymmetry term and Coulomb atomic mass versus proton number for the term in the semi empirical mass formula and mn, different odd A isotopes in the heavy region with mp, me are mass of neutron, proton and electron mass number varies from 201-207. It is obvious respectively. from the plots that for nuclei with odd A, a We have computed the ZA value for single parabola is obtained irrespective of different isobars in the heavy region with mass whether the nucleus is odd-even or even-odd. In number varies from 200 to 250. Figure 1 this case the pairing term in the binding energy represents the plot of mass number versus the does not change from isobar to isobar and the atomic number of most stable isotopes (ZA) question of stability relies on the balance against beta decay. It is obvious from the plot between the symmetry term which prefers equal that ZA values show a linear relationship with the numbers of protons and neutrons and the mass number. From the linear dependence of Coulomb term which prefers fewer protons. For Available online at www.sympnp.org/proceedings Proceedings of the DAE Symp. on Nucl. Phys. 62 (2017) 441 such nuclides there is only one stable isobar with 187235 some atomic number ZA. The nuclides falling on 189100 A = 203 187230 A = 201 189095 either side of this isobar are all unstable. The 187225 189090 187220 isobars on the lower Z side have too many 189085 187215 - 189080 neutrons for stability and are β active and those 187210 189075 187205 189070 78 80 82 84 86 88 78 80 82 84 86 88 on the higher Z side have too many protons and + 192830 hence undergo β decay. 190960 A = 205 192825 A = 207 190955 192820 Figures 3 to 4 represent the plot of atomic 190950 192815 190945 mass versus proton number for the different 192810 Atomic Mass (MeV) 190940 even-even and odd-odd isotopes in the heavy 192805 190935 192800 region with mass number varies from 208-222. 78 80 82 84 86 88 80 82 84 86 88 90 For nuclei with even A, because of the influence Z of pairing term in the binding energy, there are two mass parabolas for the same mass number. Fig. 2 Mass parabolas for different isobars with The parabola for the odd-odd nuclei lies above mass numbers 201,203,205 and 207 that of the even-even nuclei. Here all the odd- 193765 195630 A = 208 odd nuclides have larger atomic mass than one of 193760 195625 A = 210 193755 even-even 195620 the adjacent even- even nuclides and the most 193750 odd-odd 195615 193745 195610 stable isobar falls on the lower parabola. For the 193740 195605 193735 195600 nuclide lying at the lowest point in odd-odd 193730 80 82 84 86 88 90 80 82 84 86 88 90 parabola both electron emission and positron 197500 199360 197495 A = 214 A = 212 emission are energetically possible. Thus in 199355 197490 197485 199350 some cases there can be two even-even isobars 197480 199345 for which Z differs by two units. In all these Atomic Mass (MeV) 197475 199340 197470 cases the most stable isobar against beta decay is 197465 199335 80 82 84 86 88 90 92 82 84 86 88 90 92 obtained as the lowest point in the parabola Z which is in good agreement with our formula predictions. The minima in mass parabolas Fig. 3 Mass parabolas for different isobars with shows stability of the isobar against beta decay. mass numbers 208,210,212 and 214 Hence we would like to propose that it will be a 203090 guide to the future experiments in beta decay in A = 216 201220 A = 218 203085 the heavy region. even-even 201215 odd-odd 203080 201210 203075 201205 250 82 84 86 88 90 92 82 84 86 88 90 92 206820 Equation (1) Present A = 222 240 204950 A = 220 206815 230 204945 Atomic Mass (MeV) 206810 204940 220 206805 84 86 88 90 92 84 86 88 90 92 Z Mass Number 210 200 Fig.4 Mass parabolas for different isobars with mass numbers 216,218,220 and 222 78 80 82 84 86 88 90 92 94 96 Atomic Number References Fig. 1 plot of comparison of Z value of most [1] Jozsef Konya, Noemi M. Nagy, Nuclear and stable isobar against beta decay obtained from Radiochemistry, 74 (2012) equation 4 and the present formula. Available online at www.sympnp.org/proceedings.