Name: crystal packing in cubic unit cells
Uploaded: 2020-09-24T15:06:52.000Z
Duration: 2 min 33 s
Channel: JoVE

K12 Standards

Virginia  Standards of Learning

HIgh School

Physics

PH.9

PH.9.e

lattice-centering-and-coordination-number

Lattice Centering and Coordination Number

The lattice system of a crystalline solid is defined by the shape of its unit cell. Each lattice system represents one or more lattice types, which are distinguished by the positions of the lattice points.
A primitive, or simple, lattice has lattice points only on the corners of its unit cell. Each lattice system has a primitive lattice.
Some lattice systems also have centered lattices, which have additional lattice points on the faces of or entirely enclosed within their unit cells. There are three types of centering: body-centered, base-centered, and face-centered.
Body-centered, base-centered, and face-centered lattices have additional lattice points at the center of the unit cell, on the top and bottom faces, and on each of the six faces, respectively.
The coordination number of an atom in a unit cell is the number of neighboring atoms of interest with which it is in direct contact, or its &#8216;nearest neighbors&#8217;. The nearest neighbors may be within the same unit cell or in other unit cells.
Assuming identical atoms positioned at all lattice points, centered lattices have higher packing efficiencies and coordination numbers than primitive lattices do.
Consider the cubic lattice sys

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 spheres and arranging them uniformly in a container such that one layer is directly above the spheres in the layer below. This arrangement is called a simple cubic structure, and the unit cell is called the simple cubic unit cell or primitive cubic unit cell. In a simple cubic structure, the spheres are not packed as closely as they could be, and they only &#8220;fill&#8221; about 52% of the volume of the container.
An atom in this type of arrangement contacts only the four nearest neighbors in its layer; one atom directly above it in the layer above; and one atom directly below it in the layer below. The number of other particles that each particle in a crystalline solid contact is known as its coordination number. For a polonium atom in a simple cubic array, the coordination number is, therefore, six.
<img alt="Crystallography defect diagram, illustrating point defects in a lattice structure, shown in three stages." src="/files/ftp_upload/11442/11442_Image_1.png" style="text-align: center; height: 250px;" data-altupdatedat="1754904545000" data-alt="Crystal lattice defect diagram; vacancies and interstitials shown in cubic structure; material science." />
Figure 1. Three different crystal lattice structures exhibited by the cubic lattice systems.
In a primitive cubic lattice, the unit cell that repeats in all directions is a cube defined by the centers of eight atoms. Atoms at adjacent corners of this unit cell contact each other, so a cubic unit cell contains only the parts of these atoms that are within it. Since an atom at a corner of a simple cubic unit cell is contained by a total of eight unit cells, only one-eighth of that atom is within a specific unit cell. And since each simple cubic unit cell has one atom at each of its eight &#8220;corners,&#8221; there is 8 &#215; 1/8= 1 atom within one simple cubic unit cell.
Some metals crystallize in an arrangement that has a cubic unit cell with atoms at all of the corners and an atom in the center. This is called a body-centered cubic (BCC) solid. Atoms in the corners of a BCC unit cell do not contact each other but contact the atom in the center. A BCC unit cell contains two atoms: one-eighth of an atom at each of the eight corners (8 &#215; 1/8= 1 atom from the corners) plus one atom from the center. Any atom in this structure touches four atoms in the layer above it and four atoms in the layer below it. Thus, an atom in a BCC structure has a coordination number of eight. Atoms in BCC arrangements are much more efficiently packed than in a simple cubic structure, occupying about 68% of the total volume.
Many other metals crystallize in an arrangement that has a cubic unit cell with atoms at all of the corners and at the centers of each face. This arrangement is called a face-centered cubic (FCC) solid. A FCC unit cell contains four atoms: one-eighth of an atom at each of the eight corners (8 &#215; 1/8= 1 atom from the corners) and one-half of an atom on each of the six faces (6 &#215; 1/2= 3 atoms from the faces). The atoms at the corners touch the atoms in the centers of the adjacent faces along the face diagonals of the cube. Because the atoms are on identical lattice points, they have identical environments. Atoms in an FCC arrangement are packed as closely together as possible, with atoms occupying 74% of the volume. Each atom contacts six atoms in its own layer, three in the layer above, and three in the layer below. In this arrangement, each atom touches 12 near neighbors and therefore has a coordination number of 12.
This text has been adapted from <a href="https://openstax.org/books/chemistry-2e/pages/10-5-the-solid-state-of-matter">Openstax, Chemistry 2e, Sections 10.5 The Solid State of Matter</a>, and <a href="https://openstax.org/books/chemistry-2e/pages/10-6-lattice-structures-in-crystalline-solids">10.6 Lattice Structures in Crystalline Solids</a>.

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 ...

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Chemistry

Liquids, Solids, and Intermolecular Forces

KEY TERMS AND CONCEPTS

crystal packing in cubic unit cells

Crystal packing in cubic unit cells shows how atoms are arranged in a repeating solid structure. A crystal is best described by its unit cell, which is the smallest repeating part of the lattice. Lattice points mark where atoms or ions are located. The full crystal is built when this unit cell repeats in three dimensions.

The cubic lattice has three common unit cells. A simple cubic structure, also called a primitive cubic structure, can be pictured as identical spheres stacked in layers with each layer directly above the one below it. This arrangement is not packed as tightly as it could be and fills about 52% of the container volume. In this pattern, each atom touches four nearest neighbors in its own layer, one atom above, and one atom below, so the coordination number is six. A polonium atom in a simple cubic array is an example.

In a primitive cubic lattice, the unit cell is a cube formed by the centers of eight corner atoms. Corner atoms are shared by eight unit cells, so only one-eighth of each corner atom belongs to one unit cell. Because there are eight corners, the simple cubic unit cell contains 8 × 1/8 = 1 atom total.

A body-centered cubic, or BCC, solid has atoms at all eight corners and one atom in the center of the cube. The corner atoms do not touch each other. They touch the atom in the center instead. A BCC unit cell contains two atoms in total: one atom from the corners and one atom from the center. Each atom contacts four atoms in the layer above and four in the layer below, so the coordination number is eight. BCC arrangements are packed more efficiently than simple cubic ones and occupy about 68% of the total volume.

A face-centered cubic, or FCC, solid has atoms at all corners and at the centers of each face. This unit cell contains four atoms total. The corners contribute 8 × 1/8 = 1 atom, and the six face atoms contribute 6 × 1/2 = 3 atoms. The corner atoms touch the atoms in the centers of adjacent faces along the face diagonals of the cube. Because the lattice points are identical, the atoms have identical environments. FCC structures are packed as closely together as possible and occupy 74% of the volume. Each atom touches six atoms in its own layer, three above, and three below, so the coordination number is 12. This text is adapted from OpenStax, Chemistry 2e, Sections 10.5 The Solid State of Matter and 10.6 Lattice Structures in Crystalline Solids.

Crystal Packing in Cubic Unit Cells

molecular comparison of gases, liquids, and solids

molecular-comparison-of-gases-liquids-and-solids

How Intermolecular Forces Shape States

Intermolecular forces and particle motion shape the states of matter: solids, liquids, and gases. In a solid, particles are tightly packed and often arranged in a regular pattern. A solid has a fixed shape, and its particles vibrate in place rather than flowing past one another. In a liquid, particles stay close together but do not have a regular arrangement. Liquids have no fixed shape, yet their particles move past each other while remaining in contact. That is why liquids can flow and take...

intermolecular vs intramolecular forces

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intermolecular forces

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comparing intermolecular forces: melting point, boiling point, and miscibility

comparing-intermolecular-forces-melting-point-boiling-point

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surface tension, capillary action, and viscosity

surface-tension-capillary-action-and-viscosity

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phase transitions

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phase transitions: vaporization and condensation

phase-transitions-vaporization-and-condensation

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vapor pressure

Vapor Pressure and Boiling Point

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clausius-clapeyron equation

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phase transitions: melting and freezing

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phase transitions: sublimation and deposition

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heating and cooling curves

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phase diagrams

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structures of solids

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molecular and ionic 
solids

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ionic crystal structures

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metallic solids

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band theory

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network covalent solids

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x-ray crystallography

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Crystal packing in cubic unit cells shows how atoms are arranged in a repeating solid structure. A crystal is best described by its unit cell, which is the smallest repeating part of the lattice. Lattice points mark where atoms or ions are located. The full crystal is built when this unit cell repeats in three dimensions. The cubic lattice has three common unit cells. A simple cubic structure, also called a primitive cubic structure, can be pictured as identical spheres stacked in layers with...

key terms and concepts

Explore how simple cubic, body-centered cubic, and face-centered cubic unit cells differ in atom count, packing efficiency, and coordination number.

Crystal Packing in Cubic Unit Cells | JoVE K12

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Georgia Standards of Excellence

9-12

SC2.

Molecular Structures in Material Design

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PH.9.d

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Crystal Packing in Cubic Unit Cells

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