The 1845 Half Cent was minted in Proof only, either as Originals (some of which were included in 1845 Proof sets or were sold individually) or as Restrikes which were made to satisfy collector demand even as late as 1860! Originals have large berries on the reverse; restrikes have small berries and microscopic differences that separate them into what are known as First and Second Restrikes. First Restrikes are also known as "Reverse of 1840"; Second Restrikes are known as "Reverse of 1856". All 1845 Half Cent Proofs are rare, although the Restrikes appear to be slightly more "common" than the Originals. Gem, unspotted Proofs are extremely rare and it is doubtful if a full Red Gem example exists.
Cohen PO-6 - Original
Cohen SR-7 - First Restrike
Cohen SR-22 - Second Restrike
Sources and/or recommended reading:
"American Half Cents - The 'Little Half Sisters' (Second Edition)" by Roger S. Cohen, Jr.
"Walter Breen's Encyclopedia of United States Half Cents 1793-1857" by Walter Breen
Relevant collector organizations:
Early American Coppers Club