Thermal expansion of ${\mathrm{Ce}}_3$${\mathrm{Bi}}_4$${\mathrm{Pt}}_3$ at ambient and high pressures

${\mathrm{Ce}}_3$ ${\mathrm{Bi}}_4$ ${\mathrm{Pt}}_3$ is a cerium compound that exhibits a hybridization gap and concomitant semiconducting behavior in the coherent ground state. Using neutron powder diffraction, we have determined the variation of the lattice constant with temperature at two pressures (P=${10}^{\mathrm{-}3}$ and 17.7 kbar), and determined the bulk modulus and atomic mean-square displacements for ${\mathrm{Ce}}_3$ ${\mathrm{Bi}}_4$ ${\mathrm{Pt}}_3$ and its normal analog, ${\mathrm{La}}_3$ ${\mathrm{Bi}}_4$ ${\mathrm{Pt}}_3$. The thermal expansion Δβ in ${\mathrm{Ce}}_3$ ${\mathrm{Bi}}_4$ ${\mathrm{Pt}}_3$ exhibits a maximum at ${\mathit{T}}_{\max }$=50 K at ambient pressure. Below 50 K, the bulk modulus ΔB is proportional to TΔβ. We apply a Grüneisen analysis, under the assumption that the free energy exhibits T/${\mathit{T}}_0$(V) scaling. The Grüneisen parameter deduced from the ratio ΔB/TΔβ is consistent with the value Ω=36 deduced from the pressure variation of ${\mathit{T}}_{\max }$. The analysis allows us to predict the temperature dependence of the 4f specific heat (ΔC∝Δβ/Ω), which thus has a maximum at 50 K and a high-temperature entropy nearly equal the expected value R ln6. We argue that the maximum at 50 K reflects an (indirect) hybridization gap of order 100 K. We show further that the lattice constant anomaly Δ${\mathit{a}}_0$ is proportional to the effective moment Tχ; this suggests that a relationship known to be valid for antiferromagnets, namely ∂(Tχ)/∂T∝ΔC where ΔC is the specific heat, may be valid for ${\mathrm{Ce}}_3$ ${\mathrm{Bi}}_4$ ${\mathrm{Pt}}_3$. Finally, we show that the temperature dependence of the mean-square atomic displacements is nearly identical for ${\mathrm{Ce}}_3$ ${\mathrm{Bi}}_4$ ${\mathrm{Pt}}_3$ and ${\mathrm{La}}_3$ ${\mathrm{Bi}}_4$