Chapter 10: Problem 55
The unit cell present in \(\mathrm{ABCABC}\), closest packing of atoms is: (a) hexagonal (b) tetragonal (c) face centered cube (d) primitive cube
Short Answer
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The unit cell present in \(\mathrm{ABCABC}\) closest packing of atoms is (c) face centered cube.
Step by step solution
01
Understand the Packing Structure
The notation \(\mathrm{ABCABC}\) refers to the sequence of close-packed layers. In each sequence layer, atoms are arranged in such a way that the next layer is offset, creating a repeating pattern. This pattern leads to a three-dimensional structure that must be identified.
02
Identify the Unit Cell Type
Based on the \(\mathrm{ABCABC}\) notation, we can deduce that the atoms do not repeat their positions in the immediate next layer. \(A\), \(B\), and \(C\) represent different layers where atoms in layer \(B\) sit in the depressions of layer \(A\), and layer \(C\) sits in the depressions of layer \(B\). After layer \(C\), the next layer \(A\) would sit in the depressions of layer \(C\), creating a stack. This type of stacking results in a face-centered cubic (fcc) structure, also known as cubic close-packed (ccp) structure.
03
Match the Unit Cell to the Answer Choices
The face-centered cubic (fcc or ccp) structure fits the description of the \(\mathrm{ABCABC}\) stacking sequence. This is different from a hexagonal close-packed (hcp) sequence, which follows an \(\mathrm{ABAB}\) pattern, and is neither a tetragonal nor a primitive cube. Thus, the correct choice is the one that describes a face-centered cubic unit cell.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Face-Centered Cubic Unit Cell
The face-centered cubic (fcc) unit cell is a cornerstone of understanding crystal structures in materials science. It is characterized by atoms located at each corner of a cube and at the centers of each face, summing up to a total of 4 complete atoms per unit cell. The presence of additional atoms on the faces gives this structure a higher packing density compared to simple cubic cells.
This configuration allows for close packing of the atoms, which can be described as a packing efficiency of approximately 74%. It is an arrangement where space utilization is optimized. In the face-centered cubic unit cell, each of the eight corner atoms is shared by eight adjacent unit cells, and each face-centered atom is shared by two unit cells, which is why we only count one-eighth of the corner atoms and one-half of the face atoms when calculating the number of atoms in one unit cell.
The fcc structure is crucial in materials science because many metals such as aluminium, copper, and nickel crystallize in this manner, leading to their characteristic properties such as high ductility and good thermal and electrical conductivity.
This configuration allows for close packing of the atoms, which can be described as a packing efficiency of approximately 74%. It is an arrangement where space utilization is optimized. In the face-centered cubic unit cell, each of the eight corner atoms is shared by eight adjacent unit cells, and each face-centered atom is shared by two unit cells, which is why we only count one-eighth of the corner atoms and one-half of the face atoms when calculating the number of atoms in one unit cell.
The fcc structure is crucial in materials science because many metals such as aluminium, copper, and nickel crystallize in this manner, leading to their characteristic properties such as high ductility and good thermal and electrical conductivity.
Cubic Close-Packed Structure
A cubic close-packed (ccp) structure is another term for the face-centered cubic (fcc) crystal structure. The naming comes from the way atoms pack together in three dimensions to occupy the least amount of space. In a ccp structure, each atom contacts 12 others, which is the maximum fill for a sphere packing without gaps.
The ccp structure has a distinctive layering pattern often described as 'ABCABC', indicating that no two consecutive layers are the same. This layering sequence can be visualized by stacking equilateral triangles of atoms, with each successive layer offset so that its atoms nestle in the gaps of the previous layer. This optimal arrangement contributes to the material's stability and density, which is why it's prevalent amongst metals and alloys.
The ccp structure has a distinctive layering pattern often described as 'ABCABC', indicating that no two consecutive layers are the same. This layering sequence can be visualized by stacking equilateral triangles of atoms, with each successive layer offset so that its atoms nestle in the gaps of the previous layer. This optimal arrangement contributes to the material's stability and density, which is why it's prevalent amongst metals and alloys.
Hexagonal Close-Packed Structure
The hexagonal close-packed (hcp) structure contrasts the cubic packing as it follows an 'ABAB' pattern. It is made up of layers of atoms in which every other layer is directly above and below each other, thus forming a hexagonal shape when viewed from above. Each atom in the hcp structure also contacts 12 other atoms, but the arrangement is different from ccp.
The difference in stacking sequences - hcp's 'ABAB' versus ccp's 'ABCABC' - leads to variations in properties such as ductility, crack resistance, and slip systems. This structure is found in metals like magnesium, titanium, and zinc. Understanding the distinction between hcp and ccp is key in material science and engineering because it influences how a material can be processed and used in applications.
The difference in stacking sequences - hcp's 'ABAB' versus ccp's 'ABCABC' - leads to variations in properties such as ductility, crack resistance, and slip systems. This structure is found in metals like magnesium, titanium, and zinc. Understanding the distinction between hcp and ccp is key in material science and engineering because it influences how a material can be processed and used in applications.
Unit Cell Identification
When exploring the microscopic world of crystals, identifying the unit cell is essential. A unit cell is the smallest repeating component that represents the symmetry and structure of the entire crystal lattice. To identify a unit cell, look for repeating patterns in the atomic arrangement.
Techniques such as X-ray diffraction are often used to reveal the geometry of a unit cell. These patterns, when interpreted correctly, can reveal whether the structure is face-centered cubic, body-centered cubic, hexagonal close-packed, or another lattice type. Each type of cell contains specific points called lattice points where an entire crystal can be reconstructed if these points are connected throughout space. Recognizing the type of unit cell helps in understanding the physical properties of the material, how it interacts with light, and its electrical and thermal conductivities.
Techniques such as X-ray diffraction are often used to reveal the geometry of a unit cell. These patterns, when interpreted correctly, can reveal whether the structure is face-centered cubic, body-centered cubic, hexagonal close-packed, or another lattice type. Each type of cell contains specific points called lattice points where an entire crystal can be reconstructed if these points are connected throughout space. Recognizing the type of unit cell helps in understanding the physical properties of the material, how it interacts with light, and its electrical and thermal conductivities.
Packing Structure in Solids
In solid-state physics, the packing structure refers to how atoms or molecules arrange themselves in a crystal lattice. The tightness of the packing affects the solid's density and other physical properties. Depending on the packing efficiency, solids may be classified into simple cubic, body-centered cubic, face-centered cubic, or hexagonal close-packed structures.
Atoms strive for the most stable, energy-efficient organization, which often results in closely packed structures. The more efficiently the atoms are packed, the higher the coordination number, which means more atoms are in direct contact with each other. Materials scientists look at the packing structure to predict how materials will behave under stress, how they will diffuse, and their melting points, as well as to tailor them for specific applications by altering the packing via alloying or other forms of material processing.
Atoms strive for the most stable, energy-efficient organization, which often results in closely packed structures. The more efficiently the atoms are packed, the higher the coordination number, which means more atoms are in direct contact with each other. Materials scientists look at the packing structure to predict how materials will behave under stress, how they will diffuse, and their melting points, as well as to tailor them for specific applications by altering the packing via alloying or other forms of material processing.
Crystal Lattice Structures
Crystal lattice structures constitute the three-dimensional arrangement of atoms within a crystal. Atoms in a crystal lattice vibrate around fixed points known as lattice points, creating a well-ordered pattern that extends in all three spatial dimensions. These structures can be simple and geometric, like the cubic systems, or more complex, like the triclinic systems.
The study of these arrangements, their symmetry, and their intersecting planes is a key aspect of crystallography. The arrangement of atoms in a crystal lattice affects the physical properties of the material, such as hardness, cleavage planes, and electronic properties. By understanding the lattice structure, scientists and engineers can predict how materials will behave in different conditions and design new materials with desired properties for various technologies including semiconductor devices, optical equipment, and metallic components.
The study of these arrangements, their symmetry, and their intersecting planes is a key aspect of crystallography. The arrangement of atoms in a crystal lattice affects the physical properties of the material, such as hardness, cleavage planes, and electronic properties. By understanding the lattice structure, scientists and engineers can predict how materials will behave in different conditions and design new materials with desired properties for various technologies including semiconductor devices, optical equipment, and metallic components.
