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Related Concept Videos

Lattice Centering and Coordination Number02:33

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
Imagine taking a large number of identical...
Crystal Density01:19

Crystal Density

The crystal lattice structure of a material allows us to determine how many molecules exist in its unit cell. With this information, alongside the unit-cell parameters - three distance parameters (a, b, c) and three angular parameters (α, β, γ).Density (ρ) = (Z × M) / (a × b × c × NA)where:Z is the number of formula units per unit cellM is the molar mass of the substancea, b, and c are the edge lengths of the unit cellNA is Avogadro’s numberFor a simple cubic lattice, atoms are located only at...
Structures of Solids02:22

Structures of Solids

Solids in which the atoms, ions, or molecules are arranged in a definite repeating pattern are known as crystalline solids. Metals and ionic compounds typically form ordered, crystalline solids. A crystalline solid has a precise melting temperature because each atom or molecule of the same type is held in place with the same forces or energy. Amorphous solids or non-crystalline solids (or, sometimes, glasses) which lack an ordered internal structure and are randomly arranged. Substances that...
Metallic Solids02:37

Metallic Solids

Metallic solids such as crystals of copper, aluminum, and iron are formed by metal atoms. The structure of metallic crystals is often described as a uniform distribution of atomic nuclei within a “sea” of delocalized electrons. The atoms within such a metallic solid are held together by a unique force known as metallic bonding that gives rise to many useful and varied bulk properties.
All metallic solids exhibit high thermal and electrical conductivity, metallic luster, and malleability. Many...
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
Unit Cells01:18

Unit Cells

A crystal's internal structure is an orderly array of atoms, ions, or molecules, and the details of this array significantly influence the solid's properties. In a crystal, periodically repeating 'structural motifs' - which could be atoms, molecules, or groups thereof - create a 'space lattice.' This is essentially a three-dimensional, infinite array of points, each surrounded by its neighbors in an identical way, forming the basic structure of the crystal.A 'unit cell' is a theoretical...

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Related Experiment Video

Updated: May 9, 2026

An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production
07:46

An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production

Published on: March 27, 2017

Efficient linear programming algorithm to generate the densest lattice sphere packings.

Étienne Marcotte1, Salvatore Torquato

  • 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 16, 2013
PubMed
Summary
This summary is machine-generated.

The Torquato-Jiao (TJ) algorithm efficiently finds dense sphere packings in high dimensions. This method significantly outperforms previous techniques, accelerating the discovery of optimal lattice sphere packings.

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A Robust Method for the Large-Scale Production of Spheroids for High-Content Screening and Analysis Applications
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A Robust Method for the Large-Scale Production of Spheroids for High-Content Screening and Analysis Applications

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Related Experiment Videos

Last Updated: May 9, 2026

An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production
07:46

An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production

Published on: March 27, 2017

A Robust Method for the Large-Scale Production of Spheroids for High-Content Screening and Analysis Applications
06:40

A Robust Method for the Large-Scale Production of Spheroids for High-Content Screening and Analysis Applications

Published on: December 28, 2021

Area of Science:

  • Physics
  • Applied Mathematics
  • Computational Science

Background:

  • Sphere packing is a fundamental problem in mathematics and physics with applications in diverse fields.
  • High-dimensional sphere packing presents significant computational challenges due to combinatorial complexity.
  • Existing methods struggle with efficiency and scalability in higher dimensions.

Purpose of the Study:

  • To apply and evaluate the Torquato-Jiao (TJ) packing algorithm for determining densest lattice sphere packings.
  • To assess the computational efficiency of the TJ algorithm compared to prior methods.
  • To investigate suboptimal packing configurations and understand the density landscape.

Main Methods:

  • Utilized the Torquato-Jiao (TJ) packing algorithm, a technique based on solving sequential linear programs.
  • Applied the algorithm to reproduce known densest lattice sphere packings in dimensions 2 through 19.
  • Analyzed suboptimal local density maxima (inherent structures) to explore the packing density landscape.

Main Results:

  • Successfully reproduced the densest known lattice sphere packings for dimensions 2 to 19 using the TJ algorithm.
  • Demonstrated that the TJ algorithm is substantially more efficient than previously published methods.
  • Observed speed improvements of up to three orders of magnitude in certain dimensions.

Conclusions:

  • The Torquato-Jiao (TJ) algorithm offers a robust and highly efficient approach for solving high-dimensional sphere packing problems.
  • The TJ algorithm significantly advances the computational feasibility of finding optimal lattice sphere packings.
  • Further analysis of suboptimal packings provides insights into the complex 'density landscape'.