Specific heat measurements constitute one of the most powerful experimental methods to probe fundamental excitations in solids. After the proposition of Einsteins model, more than one century ago (Annalen der Physik textbf{22}, 180 (1907)), several t
heoretical models have been proposed to describe experimental results. Here we report on a detailed analysis of the two-peak specific heat anomalies observed in several materials. Employing a simple multilevel model, varying the spacing between the energy levels $Delta_i$ = $(E_i$ $-$ $E_{0})$ and the degeneracy of each energy level $g_i$, we derive the required conditions for the appearance of such anomalies. Our findings indicate that a ratio of $Delta_2$/$Delta_1$ $thickapprox$ 10 between the energy levels and a high degeneracy of one of the energy levels define the two-peaks regime in the specific heat. Our approach accurately matches recent experimental results. Furthermore, using a mean-field approach we calculate the specific heat of a degenerate Schottky-like system undergoing a ferromagnetic (FM) phase transition. Our results reveal that as the degeneracy is increased the Schottky maximum in the specific heat becomes narrow while the peak associated with the FM transition remains unaffected.
Charge-ordering phenomena have been highly topical over the last few years. A phase transition towards a charge ordered state has been observed experimentally in several classes of materials. Among them, many studies have been devoted to the family o
f quasi-one dimensional organic charge-transfer salts (TMTTF)$_2$X, where (TMTTF) stands for tetramethyltetrathiafulvalene and X for a monovalent anion (X = PF$_6$, AsF$_6$ and SbF$_6$). However, the relationship between the electron localization phenomena and the role of the lattice distortion in stabilizing the charge-ordering pattern is poorly documented in the literature. Here we present a brief overview of selected literature results with emphasis placed on recent thermal expansion experiments probing the charge-ordering transition of these salts.
We report on synthesis, structural characterization, resistivity, magnetic and thermal expansion measurements on the as yet unexplored $delta$-phase of FeSe$_{1-x}$, here synthesized under ambient- (AP) and high-pressure (HP) conditions. We show that
in contrast to $beta$-FeSe$_{1-x}$, monophasic superconducting $delta$-FeSe$_{1-x}$ can be obtained in off-stoichiometric samples with excess Fe atoms preferentially residing in the van der Waals gap between the FeSe layers. The AP $delta$-FeSe$_{1-x}$ sample studied here ($T_c$ $simeq$ 8.5,K) possesses an unprecedented residual resistivity ratio RRR $simeq$ 16. Thermal expansion data reveal a small feature around $sim$90,K, which resembles the anomaly observed at the structural and magnetic transitions for other Fe-based superconductors, suggesting that some kind of magnetic state is formed also in FeSe. %indicative of a fluctuating magnetic ordering. For HP samples (RRR $simeq$ 3), the disorder within the FeSe layers is enhanced through the introduction of vacancies, the saturated magnetic moment of Fe is reduced and only spurious superconductivity is observed.
We report results of high-resolution measurements of the emph{c$^*$}-axis expansivity ($alpha_{c^{*}}$) at the charge-ordering (CO) transition for the quasi-1D (TMTTF)$_{2}$X compounds with X = SbF$_6$ and Br and make a comparison with previous resul
ts for the X = PF$_6$ and AsF$_6$ salts. For X = SbF$_6$, due to the screening of the long-range Coulomb forces, a sharp $lambda$-type anomaly is observed at $T_{CO}$, which contrasts with the step-like mean-field anomaly at $T_{CO}$ for PF$_6$ and AsF$_6$, where CO occurs in the Mott-Hubbard charge-localized regime. For the latter two salts, a negative contribution to $alpha_{c^{*}}$ is observed above $T_{CO}$. This feature is assigned to the anions rigid-unit modes, which become inactive for $T$ $<$ $T_{CO}$. Our $alpha_{c^{*}}$ results for the X = Br salt, where such rigid-unit modes are absent, reveal no traces of such negative contribution, confirming the model based on the anions rigid-unit modes for the X = PF$_6$ and AsF$_6$ salts.
