Keating Model
Encyclopedia
The Keating Model is a model
introduced by P. N. Keating
in 1966 for the force
s induced on neighboring atom
s when one atom is moved in a solid. It is most often applied to the forces on first- and second-nearest neighboring atoms which arise when an atom is moved in tetrahedrally
-bonded solid
s, such as diamond
, silicon
, germanium
, and a number of other covalent crystal
s with the diamond
or zinc blende structures.
Crystalline solids generally consist of an ordered array of interconnected atoms, generated by repetition of a unit cell in three dimensions, and are of two extreme types; ionic crystals, and covalent crystals, with others which are intermediate: partly ionic and partly covalent. Ionic crystals are made up of quite different ions, such as Na+ and Cl- in common salt, for example, while covalent crystals such as diamond are made up of atoms which share electrons in a covalent bond
.
In either case, there are attractive and repulsive forces which resist moving an atom/ion or a set of them from their equilibrium positions, thus giving solids their rigidity against compressive, tensile, and shear stresses. The nature and strength of these forces is important for the scientific understanding of solids since they determine the way the solid responds to these stresses (elastic constants), the velocity of sound waves in it, its infra-red absorption, and many other properties.
satisfies the requirement that it is invariant under a simple rotation
of the crystal, without deformation. It is a formalism for the way adjacent and close-by atoms respond when one or more atoms move in covalently bonded crystals. It is also a specific parameterization of this response for diamond, silicon, and germanium. (see the article listed under "Further Reading").
The general method is applicable for small atomic displacements to all crystal structures . It has been extended by P. N. Keating
to include anharmonic effects (and calculate third-order elastic constants), and many other researchers have extended it to include forces between the covalent bonds, and augment it in other ways.
The key paper (see suggested reading, below) which introduced the model was determined to be one of the 50 highest-impact papers over a century of Physical Review publications http://arxiv.org/abs/physics/0407137v1). The model has been, and is, used by many research scientists for calculating elastic constants, lattice dynamics, band structure, dislocation strains, atomic configurations at surfaces and interfaces, and other purposes for a wide range of solids, including amorphous (i.e., non-crystalline)materials.
Scientific modelling
Scientific modelling is the process of generating abstract, conceptual, graphical and/or mathematical models. Science offers a growing collection of methods, techniques and theory about all kinds of specialized scientific modelling...
introduced by P. N. Keating
P. N. Keating
Dr. Patrick N. Keating is a theoretical physicist who has contributed to several fields of solid-state physics, including semiconductors, semi-insulators and the basic properties of solid materials, and to other fields including optics, liquid crystals, acoustic holography, and signal processing...
in 1966 for the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s induced on neighboring atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s when one atom is moved in a solid. It is most often applied to the forces on first- and second-nearest neighboring atoms which arise when an atom is moved in tetrahedrally
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
-bonded solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s, such as diamond
Diamond
In mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...
, silicon
Silicon
Silicon is a chemical element with the symbol Si and atomic number 14. A tetravalent metalloid, it is less reactive than its chemical analog carbon, the nonmetal directly above it in the periodic table, but more reactive than germanium, the metalloid directly below it in the table...
, germanium
Germanium
Germanium is a chemical element with the symbol Ge and atomic number 32. It is a lustrous, hard, grayish-white metalloid in the carbon group, chemically similar to its group neighbors tin and silicon. The isolated element is a semiconductor, with an appearance most similar to elemental silicon....
, and a number of other covalent crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
s with the diamond
Diamond cubic
The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group IV also adopt this structure, including tin, the semiconductors silicon and germanium, and silicon/germanium...
or zinc blende structures.
Crystalline solids generally consist of an ordered array of interconnected atoms, generated by repetition of a unit cell in three dimensions, and are of two extreme types; ionic crystals, and covalent crystals, with others which are intermediate: partly ionic and partly covalent. Ionic crystals are made up of quite different ions, such as Na+ and Cl- in common salt, for example, while covalent crystals such as diamond are made up of atoms which share electrons in a covalent bond
Covalent bond
A covalent bond is a form of chemical bonding that is characterized by the sharing of pairs of electrons between atoms. The stable balance of attractive and repulsive forces between atoms when they share electrons is known as covalent bonding....
.
In either case, there are attractive and repulsive forces which resist moving an atom/ion or a set of them from their equilibrium positions, thus giving solids their rigidity against compressive, tensile, and shear stresses. The nature and strength of these forces is important for the scientific understanding of solids since they determine the way the solid responds to these stresses (elastic constants), the velocity of sound waves in it, its infra-red absorption, and many other properties.
Description
The Keating model is the result of a general method proposed to ensure that the elastic strain energyStrain energy
In a molecule, strain energy is released when the constituent atoms are allowed to rearrange themselves in a chemical reaction or a change of chemical conformation in a way that:* angle strain,* torsional strain,* ring strain and/or steric strain,...
satisfies the requirement that it is invariant under a simple rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
of the crystal, without deformation. It is a formalism for the way adjacent and close-by atoms respond when one or more atoms move in covalently bonded crystals. It is also a specific parameterization of this response for diamond, silicon, and germanium. (see the article listed under "Further Reading").
The general method is applicable for small atomic displacements to all crystal structures . It has been extended by P. N. Keating
P. N. Keating
Dr. Patrick N. Keating is a theoretical physicist who has contributed to several fields of solid-state physics, including semiconductors, semi-insulators and the basic properties of solid materials, and to other fields including optics, liquid crystals, acoustic holography, and signal processing...
to include anharmonic effects (and calculate third-order elastic constants), and many other researchers have extended it to include forces between the covalent bonds, and augment it in other ways.
The key paper (see suggested reading, below) which introduced the model was determined to be one of the 50 highest-impact papers over a century of Physical Review publications http://arxiv.org/abs/physics/0407137v1). The model has been, and is, used by many research scientists for calculating elastic constants, lattice dynamics, band structure, dislocation strains, atomic configurations at surfaces and interfaces, and other purposes for a wide range of solids, including amorphous (i.e., non-crystalline)materials.
Further reading
- P. N. KeatingP. N. KeatingDr. Patrick N. Keating is a theoretical physicist who has contributed to several fields of solid-state physics, including semiconductors, semi-insulators and the basic properties of solid materials, and to other fields including optics, liquid crystals, acoustic holography, and signal processing...
, Effect of Invariance Requirements on the Elastic Strain Energy of Crystals with Application to the Diamond Structure, Physical ReviewPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
145, 637 (1966)