80% and 93% Li beta-alumina single crystals are prepared from Na beta-alumina by ion exchange method. Temperature dependent T-1 relaxation data for both the samples indicates the presence of a local motion of Li ion in beta-alumina crystal. T-1 relaxation data of Na-23 in Na beta-alumina was also studied and compared with Li beta-alumina. Temperature dependent quadrupole interaction was observed for Li-7 in beta-alumina crystal. A simple model suggested that, an out of plane position of Li ion may be possible with small energy imbalance which corresponds to the activation energy for local motion of Li ion. The T-1 relaxation data and the quadrupole data also indicate that mixed alkali effect might be present in the microscopic scale. (C) 2013 The Authors. Published by Elsevier B.V. All rights reserved.