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Lattice Centering and Coordination Number

Lattice Centering and Coordination Number

The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
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Convolution: Math, Graphics, and Discrete Signals

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In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
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An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a compound is a measure of the strength of this attraction. The lattice energy (ΔHlattice) of an ionic compound is defined as the energy required to separate one mole of the solid into its component gaseous ions. For the ionic solid sodium chloride, the lattice energy is the enthalpy change of the process:

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Updated: Oct 4, 2025

Revealing Neural Circuit Topography in Multi-Color

Revealing Neural Circuit Topography in Multi-Color

Published on: November 14, 2011

Lattice Gauge Equivariant Convolutional Neural Networks.

Matteo Favoni¹, Andreas Ipp¹, David I Müller¹

¹Institute for Theoretical Physics, TU Wien, A-1040 Wien, Austria.

Physical Review Letters

|February 4, 2022

Summary

This summary is machine-generated.

We introduce lattice gauge equivariant convolutional neural networks (L-CNNs) for lattice gauge theory problems. These networks learn gauge invariant quantities that standard convolutional neural networks cannot discover.

Related Experiment Videos

Last Updated: Oct 4, 2025

Revealing Neural Circuit Topography in Multi-Color

Revealing Neural Circuit Topography in Multi-Color

Published on: November 14, 2011

Area of Science:

High Energy Physics
Lattice Gauge Theory
Machine Learning

Background:

Lattice gauge theory is crucial for understanding fundamental physics.
Traditional machine learning methods struggle with gauge symmetries.
Developing novel computational tools is essential for advancing lattice gauge theory research.

Purpose of the Study:

To propose a novel neural network architecture for lattice gauge theory.
To ensure machine learning models respect fundamental symmetries in physical systems.
To enable the learning of gauge-invariant quantities in lattice gauge problems.

Main Methods:

Development of lattice gauge equivariant convolutional neural networks (L-CNNs).
Introduction of a novel convolutional layer preserving gauge equivariance.
Integration of topological information, such as Polyakov loops.

Main Results:

L-CNNs can approximate any gauge covariant function on the lattice.
Demonstrated ability of L-CNNs to learn and generalize gauge invariant quantities.
Outperformance of traditional convolutional neural networks in specific tasks.

Conclusions:

L-CNNs offer a powerful new approach for machine learning in lattice gauge theory.
The proposed architecture effectively handles gauge symmetries, a key challenge in the field.
This work paves the way for more accurate and efficient analysis of lattice gauge models.