We have proposed a new nuclear cosmochronometer 138 La(T 1/2 = 102 Gyr)-138 Ce-136 Ce for measuring the time elapsed from a supernova ν process episode [1]. This chronometer is applied to evaluate the age of a presolar grain originated from a supernova by measuring isotope ratios of three p nuclei, 138 La, 138 Ce, and 136 Ce; 138 La is synthesized by the ν process in He-and C-rich layers [2,3] and 136,138 Ce are synthesized by the γ process in O/Ne layers [4][5][6][7]. We pointed out that the nuclear structure of 138 La may affect the performance of this cosmochronometer and the ν process origin of 138 La [1]. If a 1 + excited state exists at energy lower than 72 keV that is the energy of the first excited state, the 1 + state may be a β unstable isomer and 138 La synthesized by the ν process is destroyed via the isomer. Recently, Byelikov et al. have measured Gamow-Teller strength distributions in 138 La and 180 Ta with a high-resolution method for the study of their ν process origin [8]. It has been presented that this experiment shows that the existence of a low-energy 1 + state in 138 La is extremely unlikely [9].In the mass region around 138 La, the last proton (neutron) can occupy either a d 5/2 or g 7/2 (d 3/2 or s 1/2 ) orbit around the Fermi surface. Since there exists only a single 1 + state with the (πd 5/2 g 7/2 ) ⊗ (νd 3/2 s 1/2 ) −1 configuration, it seems reasonable to assume that the lowest-lying 1 + state mainly consists of the (πd 5/2 ) ⊗ (νd 3/2 ) −1 configuration. It is still a concern that the ground state in the adjacent isotone 140 Pr has J π = 1 + [10], while 5 + in 138 La. This Z-dependence of the ground-state spin may be ascribed to the Z = 58 subshell in part, giving (πg 7/2 ) −1 ⊗ (νd 3/2 ) −1 ((πd 5/2 ) ⊗ (νd 3/2 ) −1 ) for the ground state of 138 La ( 140 Pr). However, the spacing between the lowest 7/2 + and 5/2 + levels is only 166 keV (145 keV) in 139 La ( 141 Pr) [10], and therefore significant excitation from πg 7/2 to πd 5/2 is expected. Moreover, even with (πd 5/2 ) ⊗ (νd 3/2 ) −1 , it is not straightforward to uniquely * hayakawa.takehito@jaea.go.jp identify for the ground-state spin of 140 Pr. The (πd 5/2 ) ⊗ (νd 3/2 ) −1 configuration derives 2 + , 3 + , and 4 + states, in addition to 1 + . The general character of the coupling between the proton particle and the neutron hole leads to an almost inverted level sequence in 140 Pr. In practice, a previous shell model calculation fails to reproduce the ground-state spin of 140 Pr [12]. This indicates that there is a certain mechanism through which a 1 + level becomes lower. A possible question is whether or not the same mechanism may give rise to a 1 + isomer in 138 La. To answer this question we have carried out a shell model calculation in a relatively large model space and successfully described the ground-state spin of 140 Pr, in which the π (g 7/2 ) −2 (d 5/2 ) 3 and π (g 7/2 ) −4 (d 5/2 ) 5 configurations play a significant role. Applying the same interaction, we have obtained a single 1 + state below 500 keV in 138 La ...