Extreme states of nuclearmatter (such that feature high spins, large deformations, high density and temperature, or a large excess of neutrons and protons) play an important role in studying fundamental properties of nuclei and are helpful in solving the problem of constructing the equation of state for nuclear matter. The synthesis of neutron-rich nuclei near the nucleon drip lines and investigation of their properties permit drawing conclusions about the positions of these boundaries and deducing information about unusual states of such nuclei and about their decays. At the present time, experimental investigations along these lines can only be performed via the cooperation of leading research centers that possess powerful heavy-ion accelerators, such as the Large Hadron Collider (LHC) at CERN and the heavy-ion cyclotrons at the Joint Institute for Nuclear Research (JINR, Dubna), where respective experiments are being conducted by physicists from about 20 JINR member countries. The present article gives a survey of the most recent results in the realms of super neutron-rich nuclei. Implications of the change in the structure of such nuclei near the nucleon drip lines are discussed. Information about the results obtained by measuring the masses (binding energies) of exotic nuclei, the nucleon-distribution radii (neutron halo) and momentum distributions in them, and their deformations and quantum properties is presented. It is shown that the properties of nuclei lying near the stability boundaries differ strongly from the properties of other nuclei. The problem of the stability of nuclei that is associated with the magic numbers of 20 and 28 is discussed along with the effect of new magic numbers.
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