Thermodynamically stable lithium silicides and germanides from density-functional theory calculations

Density-functional-theory (DFT) calculations have been performed on the Li-Si and Li-Ge systems. Lithiated Si and Ge, including their metastable phases, play an important technological r\^ole as Li-ion battery (LIB) anodes. The calculations comprise structural optimisations on crystal structures obtained by swapping atomic species to Li-Si and Li-Ge from the X-Y structures in the International Crystal Structure Database, where X={Li,Na,K,Rb,Cs} and Y={Si,Ge,Sn,Pb}. To complement this at various Li-Si and Li-Ge stoichiometries, ab initio random structure searching (AIRSS) was also performed. Between the ground-state stoichiometries, including the recently found Li$_{17}$Si$_{4}$ phase, the average voltages were calculated, indicating that germanium may be a safer alternative to silicon anodes in LIB, due to its higher lithium insertion voltage. Calculations predict high-density Li$_1$Si$_1$ and Li$_1$Ge$_1$ $P4/mmm$ layered phases which become the ground state above 2.5 and 5 GPa respectively and reveal silicon and germanium's propensity to form dumbbells in the Li$_x$Si, $x=2.33-3.25$ stoichiometry range. DFT predicts the stability of the Li$_{11}$Ge$_6$ $Cmmm$, Li$_{12}$Ge$_7$ $Pnma$ and Li$_7$Ge$_3$ $P32_12$ phases and several new Li-Ge compounds, with stoichiometries Li$_5$Ge$_2$, Li$_{13}$Ge$_5$, Li$_8$Ge$_3$ and Li$_{13}$Ge$_4$.Comment: 10 pages, 5 figure

Structural and electronic properties of Si/Ge nanoparticles

Results of a theoretical study of the electronic properties of (Si)Ge and (Ge)Si core/shell nanoparticles, homogeneous SiGe clusters, and Ge$|$Si clusters with an interphase separating the Si and Ge atoms are presented. In general, (Si)Ge particles are more stable than (Ge)Si ones, and SiGe systems are more stable than Ge$|$Si ones. It is found that the frontier orbitals, that dictate the optical properties, are localized to the surface, meaning that saturating dangling bonds on the surface with ligands may influence the optical properties significantly. In the central parts we identify a weak tendency for the Si atoms to accept electrons, whereas Ge atoms donate electrons.Comment: To appear in Phys. Rev.

Stellar laboratories: new Ge V and Ge VI oscillator strengths and their validation in the hot white dwarf RE 0503-289

State-of-the-art spectral analysis of hot stars by means of non-LTE model-atmosphere techniques has arrived at a high level of sophistication. The analysis of high-resolution and high-S/N spectra, however, is strongly restricted by the lack of reliable atomic data for highly ionized species from intermediate-mass metals to trans-iron elements. Especially data for the latter has only been sparsely calculated. Many of their lines are identified in spectra of extremely hot, hydrogen-deficient post-AGB stars. A reliable determination of their abundances establishes crucial constraints for AGB nucleosynthesis simulations and, thus, for stellar evolutionary theory. In a previous analysis of the UV spectrum of RE 0503-289, spectral lines of highly ionized Ga, Ge, As, Se, Kr, Mo, Sn, Te, I, and Xe were identified. Individual abundance determinations are hampered by the lack of reliable oscillator strengths. Most of these identified lines stem from Ge V. In addition, we identified Ge VI lines for the first time. We calculated Ge V and Ge VI oscillator strengths to consider their radiative and collisional bound-bound transitions in detail in our non-LTE stellar-atmosphere models for the analysis of the Ge IV - VI spectrum exhibited in high-resolution and high-S/N UV spectra of RE 0503-289. We identify four Ge IV, 37 Ge V, and seven Ge VI lines. Most of these are identified for the first time in any star. We reproduce almost all Ge IV, Ge VI, and Ge VI lines in the observed spectrum of RE 0503-289 (Teff = 70 kK, log g = 7.5) at log Ge = -3.8 +/- 0.3 (mass fraction, about 650 times solar). Reliable measurements and calculations of atomic data are a prerequisite for stellar-atmosphere modeling. Our oscillator-strength calculations have allowed, for the first time, Ge V and Ge VI lines to be successfully reproduced in a white dwarf's spectrum and to determine its photospheric Ge abundance.Comment: 54 pages, 8 figure

Degree Sequences and Long Cycles in Graphs

Let $G$ be a graph on $n$ vertices with degree sequence $\delta=d_1\le d_2\le...\le d_n$. Let $c$ be the circumference - the order of a longest cycle and $p$ the order of a longest path in $G$. In 1952, Dirac proved: (i) every graph with $2d_1\ge n$ is hamiltonian; (ii) in every 2-connected graph, $c\ge \min\{p,2d_1\}$. Recently, the bounds $2d_1\ge n$ and $c\ge \min\{p,2d_1\}$ in (i) and (ii) are improved to $2d_\delta\ge n$ and $c\ge \min\{p,2d_\delta\}$, respectively, by Koulakzian, Mosesyan and Nikoghosyan. In this paper we present two new sharp bounds $d_\delta+d_{\delta+1}\ge n$ and $\min\{2d_{\delta+1},d_\delta+d_{\delta+2}\}\ge n$ instead of $2d_\delta\ge n$, as well as two new sharp bounds $c\ge \min\{p,d_\delta+d_{\delta+1}\}$ and $c\ge \min\{p,2d_{\delta+1},d_\delta+d_{\delta+2}\}$ instead of $c\ge \min\{p,2d_\delta\}$.Comment: 9 page

Essays on variance risk

My PhD thesis consists of three papers which study the nature, structure, dynamics and price of variance risks. As tool I make use of multivariate affine jump-diffusion models with matrix-valued state spaces. The first chapter proposes a new three-factor model for index option pricing. A core feature of the model are unspanned skewness and term structure effects, i.e., it is possible that the structure of the volatility surface changes without a change in the volatility level. The model reduces pricing errors compared to benchmark two-factor models by up to 22%. Using a decomposition of the latent state, I show that this superior performance is directly linked to a third volatility factor which is unrelated to the volatility level. The second chapter studies the price of the smile, which is defined as the premia for individual option risk factors. These risk factors are directly linked to the variance risk premium (VRP). I find that option risk premia are spanned by mid-run and long-run volatility factors, while the large high-frequency factor does not enter the price of the smile. I find the VRP to be unambiguously negative and decompose it into three components: diffusive risk, jump risk and jump intensity risk. The distinct term structure patterns of these components explain why the term structure of the VRP is downward sloping in normal times and upward sloping during market distress. In predictive regressions, I find an economically relevant predictive power over returns to volatility positions and S&P 500 index returns. The last chapter introduces several numerical methods necessary for estimating matrix-valued affine option pricing models, including the Matrix Rotation Count algorithm and a fast evaluation scheme for the Likelihood function