Sphere Packings, Lattices and Groups

Foundations Of Time Frequency Analysis

This text is a modern treatment of the theory of solids. The core of the book deals with the physics of electron and phonon states in crystals and how they determine the structure and properties of the solid. The discussion uses density functional theory as a starting point and covers electronic and optical phenomena, magnetism and superconductivity. There is also an extensive treatment of defects in solids, including point defects, dislocations, surfaces and interfaces. A number of modern topics where the theory of solids applies are also explored, including quasicrystals, amorphous solids, polymers, metal and semiconductor clusters, carbon nanotubes and biological macromolecules. Numerous examples are presented in detail and each chapter is accompanied by problems and suggested further readings. An extensive set of appendices provides all the necessary background for deriving all the results discussed in the main body of the text.

Atomic and Electronic Structure of Solids

Harmonic analysis on symmetric spaces and applications

Bats, Grids, and Oscillations Nearly all animals move around in a three-dimensional (3D) world; however, very little is known about the neural circuitry underlying the representation of 3D space (see the Perspective by Barry and Doeller). Using whole-cell patch recordings in slices of entorhinal cortex, Heys et al. (p. 363) found that bat entorhinal stellate cells must generate grid patterns without theta-frequency oscillatory mechanisms. In another study, Yartsev and Ulanovsky (p. 367) used telemetry to record activity from the hippocampus of bats while they were flying around. They found that active pyramidal cells—or place cells—in hippocampal area CA1 fired in positions, depending on where the animals were in the room. The spatial firing properties of neurons were recorded in bats during flight using a wireless neural-telemetry system. [Also see Perspective by Barry and Doeller] Many animals, on air, water, or land, navigate in three-dimensional (3D) environments, yet it remains unclear how brain circuits encode the animal's 3D position. We recorded single neurons in freely flying bats, using a wireless neural-telemetry system, and studied how hippocampal place cells encode 3D volumetric space during flight. Individual place cells were active in confined 3D volumes, and in >90% of the neurons, all three axes were encoded with similar resolution. The 3D place fields from different neurons spanned different locations and collectively represented uniformly the available space in the room. Theta rhythmicity was absent in the firing patterns of 3D place cells. These results suggest that the bat hippocampus represents 3D volumetric space by a uniform and nearly isotropic rate code.

Representation of three-dimensional space in the hippocampus of flying bats (vol 340, pg 367, 2013)

In this paper, we study minimization problems among Bravais lattices for finite energy per point. We first prove that if a function is completely monotonic, then the triangular lattice minimizes its energy per particle among Bravais lattices for any given density. Second, we give an example of convex decreasing positive interacting potential for which the triangular lattice is not a minimizer for a class of densities. We use Montgomery method presented in [L. Betermin and P. Zhang, Commun. Contemp. Math., 17 (2015), 1450049] to prove the minimality of the triangular lattice among Bravais lattices at high density for a general class of potentials. Finally, we deduce the global minimality among all Bravais lattices, i.e., without a density constraint, of a triangular lattice for some Lennard-Jones-type potentials and attractive-repulsive Yukawa potentials.

Two-Dimensional Theta Functions and Crystallization among Bravais Lattices

We prove in this paper that the minimizer of Lennard–Jones energy per particle among Bravais lattices is a triangular lattice, i.e. composed of equilateral triangles, in ℝ2 for large density of points, while it is false for sufficiently small density. We show some characterization results for the global minimizer of this energy and finally we also prove that the minimizer of the Thomas–Fermi energy per particle in ℝ2 among Bravais lattices with fixed density is triangular.

Minimization of energy per particle among Bravais lattices in R^2 : Lennard-Jones and Thomas-Fermi cases

We study the local optimality of simple cubic, body-centred-cubic and face-centred-cubic lattices among Bravais lattices of fixed density for some finite energy per point. Following the work of Ennola (Math Proc Camb 60:855–875, 1964), we prove that these lattices are critical points of all the energies, we write the second derivatives in a simple way and we investigate the local optimality of these lattices for the theta function and the Lennard–Jones-type energies. In particular, we prove the local minimality of the FCC lattice (resp. BCC lattice) for large enough (resp. small enough) values of its scaling parameter and we also prove the fact that the simple cubic lattice is a saddle point of the energy. Furthermore, we prove the local minimality of the FCC and the BCC lattices at high density (with an optimal explicit bound) and its local maximality at low density in the Lennard–Jones-type potential case. We then show the local minimality of FCC and BCC lattices among all the Bravais lattices (without a density constraint). The largest possible open interval of density’s values where the simple cubic lattice is a local minimizer is also computed.

https://hal.archives-ouvertes.fr/hal-01401382/document

Local optimality of cubic lattices for interaction energies

We consider the minimization of theta functions $\theta_\Lambda(\alpha)=\sum_{p\in\Lambda}e^{-\pi\alpha|p|^2}$ amongst lattices $\Lambda\subset \mathbb R^d$, by reducing the dimension of the problem, following as a motivation the case $d=3$, where minimizers are supposed to be either the BCC or the FCC lattices. A first way to reduce dimension is by considering layered lattices, and minimize either among competitors presenting different sequences of repetitions of the layers, or among competitors presenting different shifts of the layers with respect to each other. The second case presents the problem of minimizing theta functions also on translated lattices, namely minimizing $(L,u)\mapsto \theta_{L+u}(\alpha)$. Another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers. The first direction leads to the question of minimization amongst orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we study in detail in two dimensions.

https://hal.archives-ouvertes.fr/hal-01349805/document

Dimension reduction techniques for the minimization of theta functions on lattices

We investigate the local and global optimality of the triangular, square, simple cubic, face-centered-cubic (fcc) and body-centered-cubic (bcc) lattices and the hexagonal-close-packing (hcp) structure for a potential energy per point generated by a Morse potential with parameters (α, r0). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all two-dimensional Bravais lattices with respect to their density. The behavior of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, fcc, and bcc lattices is checked, showing several interesting similarities with the Lennard-Jones potential case. We also show that the square, triangular, cubic, fcc, and bcc lattices are the only Bravais lattices in dimensions 2 and 3 being critical points of a large class of lattice energies (including the one studied in this paper) in some open intervals of densities as we observe for the Lennard-Jones and the Morse potential lattice energies. More surprisingly, in the Morse potential case, we numerically found a transition of the global minimizer from bcc, fcc to hcp, as α increases, that we partially and heuristically explain from the lattice theta function properties. Thus, it allows us to state a conjecture about the global minimizer of the Morse lattice energy with respect to the value of α. Finally, we compare the values of α found experimentally for metals and rare-gas crystals with the expected lattice ground-state structure given by our numerical investigation/conjecture. Only in a few cases does the known ground-state crystal structure match the minimizer we find for the expected value of α. Our conclusion is that the pairwise interaction model with Morse potential and fixed α is not adapted to describe metals and rare-gas crystals if we want to take into consideration that the lattice structure we find in nature is the ground-state of the associated potential energy.We investigate the local and global optimality of the triangular, square, simple cubic, face-centered-cubic (fcc) and body-centered-cubic (bcc) lattices and the hexagonal-close-packing (hcp) structure for a potential energy per point generated by a Morse potential with parameters (α, r0). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all two-dimensional Bravais lattices with respect to their density. The behavior of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, fcc, and bcc lattices is checked, showing several interesting similarities with the Lennard...

Minimizing lattice structures for Morse potential energy in two and three dimensions

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Laurent Bétermin

Author Tools

Chat about Author