The purpose of this contribution is double. The first aim is an attempt to understand the positron's behavior in diamond, i.e. the positron annihilation rate in its delocalized states and the trapping process to vacancy-type defects. The second purpose is to qualify diamond membranes which have been produced for the application of field-assisted positron moderation. The results obtained from this comprehensive study of diamond single crystals and diamond membranes by using positron annihilation techniques are presented. The proposed mechanism for the trapping of positrons to vacancy-type defects, namely through positronium formation, gives a new picture for understanding the trapping process in insulators. It can be used to interpret the discrepancies in the earlier experimental results. The obtained results for diamond membranes indicate that they have sufficient quality to be used as the main body of field-assisted moderators.