Search for long lived heaviest nuclei beyond the valley of stability
  The existence of long lived superheavy nuclei (SHN) is controlled mainly by spontaneous fission and $\alpha$-decay processes. According to microscopic nuclear theory, spherical shell effects at Z=114, 120, 126 and N=184 provide the extra stability to such SHN to have long enough lifetime to be observed. To investigate whether the so-called "stability island" could really exist around the above Z, N values, the $\alpha$-decay half lives along with the spontaneous fission and $\beta$-decay half lives of such nuclei are studied. The $\alpha$-decay half lives of SHN with Z=102-120 are calculated in a quantum tunneling model with DDM3Y effective nuclear interaction using $Q_\alpha$ values from three different mass formulae prescribed by Koura, Uno, Tachibana, Yamada (KUTY), Myers, Swiatecki (MS) and Muntian, Hofmann, Patyk, Sobiczewski (MMM). Calculation of spontaneous fission (SF) half lives for the same SHN are carried out using a phenomenological formula and compared with SF half lives predicted by Smolanczuk {\it et al}. Possible source of discrepancy between the calculated $\alpha$-decay half lives of some nuclei and the experimental data of GSI, JINR-FLNR, RIKEN are discussed. In the region of Z=106-108 with N$\sim$ 160-164, the $\beta$-stable SHN $^{268}_{106}Sg_{162}$ is predicted to have highest $\alpha$-decay half life ($T_\alpha \sim 3.2hrs$) using $Q_\alpha$ value from MMM. Interestingly, it is much greater than the recently measured $T_\alpha$ ($\sim 22s$) of deformed doubly magic $^{270}_{108}Hs_{162}$ nucleus. A few fission-survived long-lived SHN which are either $\beta$-stable or having large $\beta$-decay half lives are predicted to exist near $^{294}110_{184}$, $^{293}110_{183}$, $^{296}112_{184}$ and $^{298}114_{184}$. These nuclei might decay predominantly through $\alpha$-particle emission. 
P. Roy Chowdhury, C. Samanta, D.N. Basu