Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Lattice Centering and Coordination Number02:33

Lattice Centering and Coordination Number

9.8K
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...
9.8K
Structures of Solids02:22

Structures of Solids

14.4K
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...
14.4K
Molecular and Ionic Solids02:54

Molecular and Ionic Solids

17.3K
Crystalline solids are divided into four types: molecular, ionic, metallic, and covalent network based on the type of constituent units and their interparticle interactions.
Molecular Solids
Molecular crystalline solids, such as ice, sucrose (table sugar), and iodine, are solids that are composed of neutral molecules as their constituent units. These molecules are held together by weak intermolecular forces such as London dispersion forces, dipole-dipole interactions, or hydrogen bonds, which...
17.3K
Crystal Field Theory - Octahedral Complexes02:58

Crystal Field Theory - Octahedral Complexes

27.0K
Crystal Field Theory
To explain the observed behavior of transition metal complexes (such as colors), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It helps to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.
CFT focuses on...
27.0K
Crystal Field Theory - Tetrahedral and Square Planar Complexes02:46

Crystal Field Theory - Tetrahedral and Square Planar Complexes

43.4K
Tetrahedral Complexes
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
43.4K
Trends in Lattice Energy: Ion Size and Charge02:54

Trends in Lattice Energy: Ion Size and Charge

24.1K
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:
24.1K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Correction for Chang et al., Plant pathogenic nematode exosomes remodel vector tracheae to enhance pathogen transmission.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Correction for Cao et al., Cenozoic geoclimatic changes drove the evolutionary dynamics of floristic endemism on the Qinghai-Tibet Plateau.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Hydration and hydrolysis define antibiotic resistance conferred by macrolide esterases.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Alpha rhythms in the left auditory cortex set the speed limit for speech comprehension.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

A population of primary afferent sensory neurons mediates pain relief through nocifensive coping behavior in mice.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Evidence of intertwined pair density and charge density wave orders in UTe<sub>2</sub>.

Proceedings of the National Academy of Sciences of the United States of America·2026

Related Experiment Video

Updated: Aug 6, 2025

Measuring the Densities of Aqueous Glasses at Cryogenic Temperatures
09:50

Measuring the Densities of Aqueous Glasses at Cryogenic Temperatures

Published on: June 28, 2017

8.8K

Zariski density of crystalline points.

Gebhard Böckle1, Ashwin Iyengar2, Vytautas Paškūnas3

  • 1IWR and Mathematical Institute, Ruprecht-Karls-Universität Heidelberg, Heidelberg 69120, Germany.

Proceedings of the National Academy of Sciences of the United States of America
|March 20, 2023
PubMed
Summary

Crystalline points are Zariski dense in the deformation space of p-adic field representations. This density also holds for deformations with a fixed determinant, proven using local methods for all p-adic fields.

Keywords:
Galois representationsdensityp-adic Hodge theory

More Related Videos

Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses
08:55

Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses

Published on: June 7, 2018

8.6K
Fully Autonomous Characterization and Data Collection from Crystals of Biological Macromolecules
07:11

Fully Autonomous Characterization and Data Collection from Crystals of Biological Macromolecules

Published on: March 22, 2019

6.9K

Related Experiment Videos

Last Updated: Aug 6, 2025

Measuring the Densities of Aqueous Glasses at Cryogenic Temperatures
09:50

Measuring the Densities of Aqueous Glasses at Cryogenic Temperatures

Published on: June 28, 2017

8.8K
Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses
08:55

Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses

Published on: June 7, 2018

8.6K
Fully Autonomous Characterization and Data Collection from Crystals of Biological Macromolecules
07:11

Fully Autonomous Characterization and Data Collection from Crystals of Biological Macromolecules

Published on: March 22, 2019

6.9K

Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Galois Representations

Background:

  • Deformation theory studies variations of mathematical structures.
  • Galois groups encode symmetries of field extensions.
  • p-adic fields are fundamental in modern number theory.

Purpose of the Study:

  • To investigate the distribution of crystalline points in the deformation space of Galois representations.
  • To determine the density of crystalline points under specific determinant conditions.

Main Methods:

  • Utilizing the concept of Zariski density.
  • Employing purely local methods for proofs.
  • Analyzing deformation spaces of absolute Galois groups.

Main Results:

  • Crystalline points are shown to be Zariski dense in the deformation space.
  • These points are also dense in subspaces with fixed crystalline determinant characters.
  • The results are applicable to all p-adic fields and residual Galois representations.

Conclusions:

  • The local nature of the proof simplifies the analysis of crystalline points.
  • This work provides a deeper understanding of the structure of deformation spaces.
  • The findings have broad implications for the study of Galois representations.