  The large magnetocaloric effect (MCE) observed in Ni-Mn based shape-memory
Heusler alloys put them forward to use in magnetic refrigeration technology. It
is associated with a first-order magnetostructural (martensitic) phase
transition. We conducted a comprehensive study of the MCE for the
off-stoichiometric Heusler alloy Ni$_{2.2}$Mn$_{0.8}$Ga in the vicinity of its
first-order magnetostructural phase transition. We found a reversible MCE under
repeated magnetic field cycles. The reversible behavior can be attributed to
the small thermal hysteresis of the martensitic phase transition. Based on the
analysis of our detailed temperature dependent X-ray diffraction data, we
demonstrate the geometric compatibility of the cubic austenite and tetragonal
martensite phases. This finding directly relates the reversible MCE behavior to
an improved geometric compatibility condition between cubic austenite and
tetragonal martensite phases. The approach will help to design shape-memory
Heusler alloys with a large reversible MCE taking advantage of the first-order
martensitic phase transition.
