  The kagome metals display an intriguing variety of electronic and magnetic
phases arising from the connectivity of atoms on a kagome lattice. A growing
number of these materials with vanadium kagome nets host charge density waves
(CDWs) at low temperatures including ScV$_6$Sn$_6$, CsV$_3$Sb$_5$, and
V$_3$Sb$_2$. Curiously, only the Sc version of the $R$V$_6$Sn$_6$
HfFe$_6$Ge$_6$-type materials hosts a CDW ($R = $Gd-Lu, Y, Sc). In this study
we investigate the role of rare earth size in CDW formation in the
$R$V$_6$Sn$_6$ compounds. Magnetization measurements on our single crystals of
(Sc,Lu)V$_6$Sn$_6$ and (Sc,Y)V$_6$Sn$_6$ establish that the CDW is suppressed
by substitution of Sc by larger Lu or Y. Single crystal x-ray diffraction
reveals that compressible Sn-Sn bonds accommodate the larger rare earth atoms
within loosely packed $R$-Sn-Sn chains without significantly expanding the
lattice. We propose that Sc provides the extra room in these chains crucial to
CDW formation in ScV$_6$Sn$_6$. Our rattling chain model explains why both
physical pressure and substitution by larger rare earths hinder CDW formation
despite opposite impacts on lattice size. We emphasize the cooperative effect
of pressure and rare earth size by demonstrating that pressure further
suppresses the CDW in a Lu-doped ScV$_6$Sn$_6$ crystal. Our model not only
addresses why a CDW only forms in the $R$V$_6$Sn$_6$ materials with tiny Sc, it
also advances to our understanding of why unusual CDWs form in the kagome
metals.
