“Peridot” is a gemological term for gem-quality forsterite olivine, but it does not differ in any chemical sense from ordinary olivine. Gem-quality olivine is known from several places on Earth, including Peridot Mesa in southeastern Arizona, USA.
The volcanic rocks making up Peridot Mesa are gray vesicular basanites (= feldspathoid olivine basalts) of the Peridot Mesa Flow, a 3 to 35 meter thick lava flow erupted during the Middle Pleistocene (~580,000 years ago).
The Peridot Mesa lavas have common greenish-colored masses of lherzolite. Lherzolite is a variety of peridotite (other varieties of peridotite include dunite, pyroxenite, wehrlite, and harzburgite). Lherzolites are coarsely-crystalline, ultramafic intrusive igneous rocks having mixtures of olivine and pyroxene (>10% each). The lherzolite mass shown below is dominated by forsterite olivine (green). The black specks are pyroxene crystals. There's also some chromian diopside pyroxene, which has a darker green color than the olivine. The lherzolite masses in the Peridot Mesa lavas are samples of mantle rock.
Some of the green olivine crystals are large enough, transparent enough, and richly green enough to be gem quality.
Locality: Peridot Mesa, San Carlos Volcanic Field, San Carlos Apache Reservation, southeastern margin of Gila County, southeastern Arizona, USA.
Lherzolite mantle xenolith (greenish) in vesicular basanite (gray) (7.5 cm across) from the Peridot Mesa Flow (Middle Pleistocene, ~580 k.y.) at Peridot Mesa, Arizona, USA.