Semi Empirical Formula For Neutrinoless Double Beta Decay

Semi Empirical Formula For Neutrinoless icon genus Beta DecayAbstractA Semi empirical form for some(prenominal) figure seat fixings and Nuclear ground substance section (NME) is authentic for neutrinoless pronged of import decompose, and the verbal expression is used to compute the neutrinoless biramous genus Beta downslope half lives. The computed half lives for neutrinoless parlay genus Beta annihilation ar compargond with the corresponding data-based determine and with those predicted by QRPA model. The rigging empirical reflection foretellings be found to be in good agreement with experimental data. The rig empirical formula is used to predict neutrinoless twice beta change integrity of discordant isotopes Ca, Ge, Se, Zr, Mo, Pd, Cd, Sn, Te, Xe, Nd and Sm that exhibiting single beta molder. As our semi empirical formula predictions agree with the experimental data we consent that the symbolize cream will be useful for the future experiments.Keyw ord Neutrinoless trope beta decay, Nuclear Matrix cistronIntroductionDouble beta decay is a radioactive decay dish out where a karyon releases 2 beta rays as a single process. Here 2 neutrons in the nucleus are converted in to devil protons and in the process two electrons and two electron antineutrinos are emitted. In mark for beta decay to be possible the final nucleus must(prenominal) have larger covert muscularity than the original nucleus. Double beta decay is difficult to study in most practically evoke cases, because both(prenominal) beta decay and range beta decay are possible, with probability favouring beta decay. The ternary beta decay is usually canvass only for beta s set back nuclei. Like single beta decay, duplicate beta decay does not change the business deal number A. more(prenominal) than 60 naturally occurring isotopes are capable of undergoing persona beta decay.Double beta decay is of two types the two neutrino and neutrinoless double beta de cay. The two neutrino double beta decay 2(2) which involves the transformation of two neutrons into two protons preserves not only the electric charge but also the lepton number. On the other hand neutrinoless double beta decay 2(0) violates lepton number preservation and is therefore forbidden in the standard electroweak theory. According to this theory neutrinos are gageless. The observation of neutrino mass and oscillation is a clear example of a phenomenon at variance with the standard model.There are different models for explaining the double beta decay process. Among them, two methods are mainly used to expect Nuclear Matrix Elements (NME) for 2(0) decays. One is the family of Quasi particle haphazard conformation Approximation (QRPA) 1. This method has been used by different groups and varieties of techniques are employed with results for most of the possible emitters 2. The other method concerned to double beta decay process is the interacting shell model (ISM) 3. It h as been shown that as the engagement in deformation between parent and daughter grows, the NMEs of both the neutrinoless and two neutrino mode decreases rapidly.The interest in double beta decay spans more than six decades. In 1937 Racah 4 following the primaeval suggestion of genus genus Majorana 5, discussed the opening of a neutrinoless transformation of two neutrons into two protons plus two electrons. Even earlier Geoppert-Mayer 6 evaluated the decay rate of 2(2) mode and realised that the corresponding half lives could exceed 1020years. Furry 7 shortly afterwards estimated that 2(0) should be much faster than 2(2) decay. Thus the stage was fructify for the realization that observation of the 2(0) decay would establish that the neutrino is a big Majorana particle. In 1982 J. Schechter-Valle while regarding 2(0) decay suggested the existence of Majorana mass of the neutrino in the frame work of Gauge theories 8. In 1984 Fiorini et al 9 introduced a program to develop low te mperature detectors for 2 decay search. conterminous year Doi et al 10 made a fundamental conjectural analysis of 2 decay to obtain the main formulae for probability of decay, energy and angular electron spectra. In 1986 utilise QRPA model Vogel et al 11 gave fitted agreement between theoretical and experimental 2(2) half spirit values.Neutrinoless double beta decay is of great interest for studying the fundamental properties of neutrino beyond the standard electro-weak theory. High sensitivity 2(0) studies are the ludicrous and practical ways for studying the Majorana nature of neutrinos, the neutrino mass spectrum, the unequivocal neutrino mass scale, the majorana CP levels and other fundamental properties of neutrinos in the foreseeable future. The graduation exercise experiment 12 to claim 2(0) is the Klapdor, HM experiment done in the year 2001. Numerous experiments like COBRA, GERDA etc have been carried out to search neutrinoless double beta decay and 48Ca, 76Ge, 82S e, 96Zr, 100Mo, 116Cd, 128Te, 150Nd, 238U are some of the isotopes exhibiting neutrinoless double beta decay 13-17.For the double beta decay processes, two crucial ingredients are the phase property factors and the Nuclear Matrix Elements (NME). A general theory of phase position factors was developed by Doi et al. 18, 19 following the previous work of Primakoff and Rosen 20, and Konopinski 21. It was reformulated by Tomoda 22 by approximating the electron wave functions at the thermonuclear radius and without inclusion of electron screening. The Nuclear Matrix Element depends on the nuclear structure of the nuclei involved in the decay. The expression for Nuclear Matrix Element can be written in general as the hit of three components 23 as (1)Where, , , are the Gamow-Teller, Fermi and tensor components respectively. is the axial vector duo constant and is the vector coupling constant.The present work aims to develop a semi empirical formula for both phase space factor and N uclear Matrix Element for computing the neutrinoless double beta decay half life. By using this formula we would like to predict the possibility of 2(0) decay from various isotopes exhibiting single beta decay. The details of the semi empirical formula are given in Section 2 and results, treatment and conclusion are given in Section 3.The semi empirical formulaIn the standard scenario, when 2(0) decay process occurs by exchange of light Majorana neutrinos between two nucleons inwardly the nucleus, and in the presence of left handed weak interactions, the life fourth dimension expression can be written as a harvest-time of three factors and is given as 24 (2)Where G0 is the phase space factor for this decay mode, is the effective neutrino mass parameter, me is the electron mass and M0 are the Nuclear Matrix Elements depending on the nuclear structure of the nuclei involved in the decay.The phase space factor depends on the energy decay Q and nuclear charge Z and studied the depen dence of phase space factor with ZQ3 and Z2Q6 for various isotopes undergoing neutrinoless double beta decay. From the observed dependence of phase space factor interpreted from ref 25, with ZQ3 and Z2Q6 we have developed a semi empirical formula for the phase space factor. Using ZQ3, Z2Q6 and Z3Q9 as variables, a new formula is obtained and is given as, (3)The constants are, , , Due to the two-body nature of the transition operator, the NMEs can also be expressed as a sum of product of two-body transition densities (TBTDs) and intercellular substance sections of the two-body transition operators for two-particle states. We have studied the dependence of nuclear matrix element values taken from 26 with Z-1/3 for various isotopes undergoing neutrinoless double beta decay and a new formula is obtained by making least-squares fit to the nuclear matrix elements data and is given as, (4)The constants are,, ,,, , The comparison of the computed nuclear matrix elements using the present fo rmula with the values of Ref 26 and comparison of computed phase space factor with the values of Ref 25 are shown in Table 1.Results, discussion and conclusionThe Q value for double beta decay of mother nuclide with mass mm to the daughter nuclide with mass md is given by the mass difference 27 which in turn can be written as a function of frequency ratio and the electron mass me (5)In the present work Q values are computed using the experimental binding energies of Audi and Wapstra 28. The present empirical formula is applied for all the observed neutrinoless double beta decay isotopes. Column 7 of Table 1 represents the computed half-lives for neutrinoless double beta decay of various isotopes and is compared with the experimental values given in column 8 and QRPA values 26 in column 9. It is found from the table that our formula predictions are in good agreement with the experimental values and the QRPA values. The value of is taken as 50meV and is obtained from Rodin et al 25.W e have applied the present formula for computing the phase space factor, Nuclear Matrix Element and half lives for various isotopes that exhibiting single beta decay. Tables 2 represents the computed Q values, Phase space factors, Nuclear Matrix Elements and half lives for neutrinoless double beta decay of various Ca, Ge, Se, Zr, Mo, Pd, Cd, Sn, Te, Xe, Nd and Sm isotopes. As our semi empirical formula prediction agree with the experimental data we hope that our prediction on neutrinoless double beta decay of various Ca, Ge, Se, Zr, Mo, Pd, Cd, Sn, Te, Xe, Nd and Sm isotopes will be a guide for future experiments.Table 1. The computed, and for neutrino less double beta decay of various isotopes and their comparison with the experimental, QRPA and Ref 26 values___________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________Table 2. The computed Q values, Phase space factors, nuclear matrix elements and the predicted half lives for neutrino less double beta decay of various Ca, Ge, Se, Zr, Mo, Pd, Cd, Sn, Te, Xe, Nd and Sm, Gd and Pt isotopes_____________________________________________________________________________________________________________________________________________________________________________________Table 2. go on..________________________________________________________________________________________________________________________103Mo6408.81.10330E-113.476787.83E+23104Mo7759.06.71620E-113.476781.29E+23105Mo8588.01.72446E-103.476785.01E+22106Mo100677.43636E-103.476781.16E+22107Mo114302.37009E-093.476783.65E+21109Pd901.01.92669E-152.568798.22E+27110Pd2004.07.81588E-152.568792.03E+27111Pd3253.52.48567E-142.568796.37E+26112Pd4244.52.49383E-132.568796.35E+25113Pd5359.32.60657E-122.568796.07E+24114Pd6523.91.74643E-112.568799.06E+23115Pd7690.58.20373E-112.5 68791.93E+23116Pd8759.02.73683E-102.568795.78E+22117Pd9895.08.38279E-102.568791.89E+22118Pd112392.67934E-092.568795.91E+21114Cd540.11.30076E-152.371891.43E+28115Cd1945.57.60027E-152.371892.44E+27116Cd2809.11.46594E-142.371891.27E+27117Cd3975.01.46594E-142.371891.25E+26118Cd4947.11.48733E-132.371891.37E+25119Cd6158.41.35555E-122.371891.61E+24120Cd7131.11.15360E-112.371894.01E+23121Cd8144.14.62494E-112.371891.17E+23122Cd9215.91.59314E-102.371893.73E+22123Cd10510.54.97662E-102.371891.12E+22124Cd11526.81.65584E-092.371894.84E+2