11.21
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 it…
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 ‘nearest neighbors’. 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 system, which includes primitive, body-centered, and face-centered lattices.
Assuming all lattice points are occupied by identical atoms, a primitive cubic unit cell contains one atom. This atom contacts four atoms from the same layer, one in the layer above, and one in the layer below for a coordination number of six.
A body-centered cubic unit cell has two atoms owing to the additional atom at the center of the unit cell. Each atom contacts four atoms in the layer above and four in the layer below for a coordination number of eight.
A face-centered cubic unit cell has four atoms owing to the atoms at all six faces, which corresponds to three additional atoms being assigned to the unit cell. Each atom contacts four atoms in the vertical plane in front of it, four in its own plane, and four in the plane behind it for a coordination number of twelve
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Q1: What is the difference between a primitive lattice and a centered lattice?
A primitive lattice has lattice points only at the corners of its unit cell, while centered lattices have additional lattice points either at the center, on faces, or on specific faces. Centered lattices include body-centered, face-centered, and base-centered types. Centered lattices achieve higher packing efficiencies and coordination numbers than primitive lattices.
Q2: How many atoms are contained in a face-centered cubic unit cell?
A face-centered cubic unit cell contains four atoms. This comes from one-eighth of an atom at each of the eight corners, totaling one atom, plus one-half of an atom on each of the six faces, totaling three atoms. Face-centered cubic structures pack atoms most efficiently, occupying 74% of the total volume.
Q3: What is coordination number and how does it differ among cubic lattice types?
Coordination number is the count of nearest neighbor atoms in direct contact with a given atom. In primitive cubic structures, coordination number is six. Body-centered cubic atoms have a coordination number of eight, while face-centered cubic atoms have a coordination number of twelve, reflecting increasingly efficient packing arrangements.
Q4: Why do body-centered cubic structures pack more efficiently than simple cubic structures?
Body-centered cubic structures have an additional atom positioned at the center of the unit cell. This central atom contacts all corner atoms, creating more efficient space utilization. Body-centered cubic solids occupy approximately 68% of total volume compared to only 52% for simple cubic structures, demonstrating superior packing efficiency.
Q5: How are atoms arranged in a simple cubic lattice?
In a simple cubic lattice, atoms are positioned at the corners of a cubic unit cell, with each atom contacting four neighbors in its own layer, one above, and one below. A simple cubic unit cell contains one atom total. This arrangement represents the least efficient packing, filling only about 52% of the container volume.
Q6: What determines the number of atoms in a cubic unit cell?
The number of atoms in a cubic unit cell depends on lattice centering. Corner atoms contribute one-eighth each since eight unit cells share each corner atom. Face-centered atoms contribute one-half each since two unit cells share each face atom. Center atoms contribute fully. Primitive cubic has one atom, body-centered has two, and face-centered has four atoms.
Q7: How does packing efficiency relate to lattice structure type?
Packing efficiency increases with lattice centering. Primitive cubic structures pack at 52% efficiency. Body-centered cubic structures achieve 68% efficiency through central atom placement. Face-centered cubic structures achieve maximum efficiency at 74%, with atoms occupying nearly three-quarters of available volume, making them the most densely packed cubic arrangement.