In the Half Cent series, all of the dates from 1840-1848 are Proof-only issues. In each of those years, the Mint struck a small number of Proof Half Cents as presentation pieces or for private sales to collectors -- these are so-called Originals. Later, and at various times, the Mint struck additional Proof 1848 Half Cents, presumably because all of the Originals had been sold. Multiple striking events suggest that only a few Proofs were restruck at each event, and more were made each time as demand warranted.
1848 Half Cent Restrikes are found in two varieties: First Restrike and Second Restrike. The First Restrikes used a reverse known as Reverse of 1856. This variety has a recut T in CENT. The Second Restrikes used a reverse known as the Reverse of 1840. This variety has die file lines near the border above ICA of AMERICA.
Again, these were struck in limited quantities and, while exact mintages are unknown, most researchers believe that two dozen coins represents the mintage cap. As might be expected, these are very popular coins today because of their rarity, their Proof-only status, and their fascinating history.
(Reverse of 1840) ESM Collection (PCGS Set Registry)