Transverse structural modulation in SrAl$_4$ and elucidation of its origin in the BaAl$_4$ family of compounds

At ambient conditions SrAl$_4$ adopts the BaAl$_4$ structure type with space group $I4/mmm$. It undergoes a charge-density-wave (CDW) transition at $T_{CDW}$ = 243 K, followed by a structural transition at $T_{S}$ = 87 K. Temperature-dependent single-crystal X-ray diffraction (SXRD) leads to the observation of incommensurate superlattice reflections at $\mathbf{q} = \sigma\,\mathbf{c}^{*}$ with $\sigma = 0.1116$ at 200 K. The CDW has orthorhombic symmetry with the superspace group $Fmmm(0\,0\,\sigma)s00$, where $Fmmm$ is a subgroup of $I4/mmm$ of index 2. Atomic displacements represent a transverse wave, and they are mainly along one of the diagonal directions of the $I$-centered unit cell. The orthorhombic phase realized in SrAl$_4$ is analogous to that found in EuAl$_4$, albeit with the presence of higher order satellite reflections (up to $m = 3$) and a shorter modulation wave vector. Electronic structure calculations and phonon calculations by density functional theory (DFT) have failed to reveal the mechanism of CDW formation. SrAl$_4$ remains incommensurately modulated at the structural transition, where the symmetry lowers from orthorhombic to $\mathbf{b}$-unique monoclinic. The present work draws a comparison on the modulated structures of non-magnetic SrAl$_4$ and magnetic EuAl$_4$ elucidating their similarities and differences, and firmly establishing that although substitution of Eu to Sr plays little to no role in the structure, the transition temperatures are affected by the atomic sizes. We have identified a simple criterion, that correlates the presence of a phase transition with the interatomic distances. Only those compounds $X$Al$_{4-x}$Ga$_x$ ($X$ = Ba, Eu, Sr, Ca; $0 < x <4$) undergo phase transitions, for which the ratio $c/a$ falls within the narrow range $2.51 < c/a < 2.54$