In an SN¹ reaction on chiral center, there is

In case of optically active alkyl halides, SN¹reaction is accompanied by racemization by  racemization .The carbocation formed in the slow step being sp² hybridized is planar  and attack of nucleophile may take place from either side resulting  in a mixture of products , one  having the same configuration and having inverted configuration .

The isomer corresponding to inversion is present in slight excess because SN¹also depends upon the degree of shielding of the front side of the reacting carbon.