  Resonant ultrasound spectroscopy and magnetic susceptibility experiments have
been used to characterize strain coupling phenomena associated with structural
and magnetic properties of the shape-memory Heusler alloy series
Ni$_{50+x}$Mn$_{25-x}$Ga$_{25}$ ($x=0$, 2.5, 5.0, and 7.5). All samples exhibit
a martensitic transformation at temperature $T_M$ and ferromagnetic ordering at
temperature $T_C$, while the pure end member ($x=0$) also has a premartensitic
transition at $T_{PM}$, giving four different scenarios: $T_C>T_{PM}>T_M$,
$T_C>T_M$ without premartensitic transition, $T_C\approx T_M$, and $T_C<T_M$.
Fundamental differences in elastic properties i.e., stiffening versus
softening, are explained in terms of coupling of shear strains with three
discrete order parameters relating to magnetic ordering, a soft mode and the
electronic instability responsible for the large strains typical of martensitic
transitions. Linear-quadratic or biquadratic coupling between these order
parameters, either directly or indirectly via the common strains, is then used
to explain the stabilities of the different structures. Acoustic losses are
attributed to critical slowing down at the premartensite transition, to the
mobility of interphases between coexisting phases at the martensitic transition
and to mobility of some aspect of the twin walls under applied stress down to
the lowest temperatures at which measurements were made.
