ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: of the next shock after one pulsation period. Any dust formed at 1 may thus not accumulate and grow over several pulsation periods in the shocked photosphere. The situation is different at larger radii, where the post-shock gas temperature drastically decreases. 3.2.1. Alumina The alumina clusters chiefly form in the post-shock gas at 1 , and their abundances in the first oscillation are illustrated in Fig. 6 . The formation of AlO involves the dimerisation of AlO through termolecular recombination while AlO forms from the reaction of Al with OH in the first place. The dimerisation of AlO is also described by termolecular recombination to form the AlO dimer, (AlO), which has a binding energy of eV ( Li Cheng 2012 ), and can thus withstand the high temperatures of the post-shock gas. This dimer is considered as the seed that will further condense into alumina grains. Over one oscillation resulting from the first shock at 1 , only a small quantity of alumina dimers forms, reaching an abundance of at . A large fraction of Al stays in the form of molecular rings (e.g., AlO and AlO). Because the dimerisation of AlO and AlO is termolecular and thus dependent on the background pressure, cluster formation occurs over a reduced range of radii in the inner wind and the formation of AlO cluster ceases after 2 . The grain size distributions for alumina in one oscillation at 1 is presented in Fig. 7 as a function of pulsation phase. Because of the high post-shock gas temperatures after the shock passage and the extremely low abundance of alumina dimers, no grains form at . Very small grains start forming after , but coagulation starts to be efficient for dust growth in the phase range , for which gas temperatures span the range K and the alumina cluster abundance reaches a maximum. The distribution peaks at grains in the size range Å, despite the small fraction of AlO clusters available. However, the post-shock gas densities are very high at 1 , as seen from Table 2 , and the clusters coagulate and condense with high efficiency and lead to the formation of AlO grains of small size over one oscillation and pulsation period. Under the assumption that small alumina grains may not survive the passage of periodic shocks, the results show that there exists a steady population of small size (m) alumina grains located in the shocked photosphere at 1 . If we consider that grain growth may also proceed through surface deposition when the grain radius has reached several tens of Å (at and after), the grain size will increase, but the growth will be time-limited and proceed only between . We now consider the extreme case where alumina grains survive the periodic shocks at each shock passage, while molecules are destroyed and reform. We may then investigate grain growth over several pulsation periods. For a gas drift velocity of 0.5 km s, the gas layer at 1 takes 12 pulsations to reach 2 . We thus consider the build-up of dust grains over a number of 12 pulsations for