Laser-cooled Be-9(+) ions confined in two-dimensionally extended Latti
ce planes were directly observed, and the images were used to characte
rize the structural phases of the ions. Five different stable crystall
ine phases were observed, and the energetically favored structure coul
d be sensitively tuned by changing the areal density of the confined i
ons. The experimental results are in good agreement with theoretical p
redictions for the planar (infinite in two dimensions) one-component p
lasma. Qualitatively similar structural phase transitions occur, or ar
e predicted to occur, in other experimentally realizable planar system
s.