Chapter 2: Problem 16
Which one of the following is not a close packed arrangement? (a) \(\mathrm{BCC}\) (b) \(\mathrm{ccp}\) (c) \(\mathrm{hcp}\) (d) all of these
Short Answer
Expert verified
(a) BCC is not a close packed arrangement.
Step by step solution
01
Understanding Close Packed Structures
Close packed structures are arrangements where atoms are packed as closely together as possible. The common types are cubic close-packed (ccp) and hexagonal close-packed (hcp). Both ccp and hcp fill space efficiently, with the maximum packing factor of 74%.
02
Identifying Body-Centered Cubic (BCC)
A body-centered cubic (BCC) lattice arranges atoms with one atom at each cube corner and a single atom at the center. This structure has a packing efficiency of 68%, which is less efficient than close-packed structures like ccp and hcp.
03
Analyzing the Options
Let's examine the structures given:
- (a) BCC: Not a close packed structure (as it has 68% packing efficiency).
- (b) ccp: Is a close packed structure (also known as face-centered cubic or fcc).
- (c) hcp: Is a close packed structure.
- (d) all of these: Incorrect as both ccp and hcp are close packed.
04
Selecting the Correct Answer
The only structure listed that isn't close packed is BCC, which means option (a) is correct. The other structures, ccp and hcp, are close packed and should not be included in a list of non-close packed structures.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Body-Centered Cubic (BCC) Structures
The Body-Centered Cubic (BCC) structure is one of the common types of crystal arrangements in solid materials. In a BCC lattice, each cubic unit cell has an atom at every corner of the cube and a single atom in the very center of the cube.
This placement means that each corner atom is shared among eight neighboring cubes, and the central atom is exclusively part of the cell. Thus, the BCC unit cell contains a total of two atoms per cell \(\text{(1 corner atom shared by 8 cubes = \frac{1}{8}\ and 1 center atom completely in-cell)}\).
One of the key characteristics of BCC is its packing efficiency. Due to the way the atoms are arranged, the BCC structure has a packing efficiency of only 68%. This means that 68% of the space in the lattice is filled with atoms, while the remaining 32% is empty. This lower packing efficiency compared to other structures, like CCP and HCP, indicates it is not as closely packed. Despite this, BCC structures are found in several metals, like iron at room temperature, and other materials particularly where the relative strength and ductility are required.
The formula for calculating the packing efficiency of BCC is: \[\text{Packing Efficiency} = \frac{Volume \ of \ atoms \ in \ unit \ cell}{Total \ volume \ of \ unit \ cell} \times 100\]
This placement means that each corner atom is shared among eight neighboring cubes, and the central atom is exclusively part of the cell. Thus, the BCC unit cell contains a total of two atoms per cell \(\text{(1 corner atom shared by 8 cubes = \frac{1}{8}\ and 1 center atom completely in-cell)}\).
One of the key characteristics of BCC is its packing efficiency. Due to the way the atoms are arranged, the BCC structure has a packing efficiency of only 68%. This means that 68% of the space in the lattice is filled with atoms, while the remaining 32% is empty. This lower packing efficiency compared to other structures, like CCP and HCP, indicates it is not as closely packed. Despite this, BCC structures are found in several metals, like iron at room temperature, and other materials particularly where the relative strength and ductility are required.
The formula for calculating the packing efficiency of BCC is: \[\text{Packing Efficiency} = \frac{Volume \ of \ atoms \ in \ unit \ cell}{Total \ volume \ of \ unit \ cell} \times 100\]
Cubic Close-Packed (CCP) Structures
Cubic Close-Packed, also known as Face-Centered Cubic (FCC), is another arrangement where atoms are packed very closely together. This structure plays a crucial role in why metals like aluminum, copper, and silver exhibit high ductility and resilience. Within the CCP lattice, an atom is positioned at each face of the cubic cell, as well as at each of the cube's corners.
This arrangement results in a CCP unit cell having four atoms, calculated as follows: \(\text{8 corner atoms} \times \frac{1}{8} + \text{6 face atoms} \times \frac{1}{2} = 4 \text{ atoms per unit cell}\).
Cubic Close-Packed structures are characterized by their high packing efficiency of 74%, which means they are among the most efficient space-filling structures known. This high packing efficiency is critical in various applications because it results in metals that are stable and dense, often providing good conduction of electricity and heat.
The mathematical depiction using the radius (\(r\)) of the atoms and the lattice parameter (\(a\)) is given by the formula: \[a = 2 \sqrt{2} r\] This equation reflects that the closest neighboring atoms are found along the face diagonal of the cube.
This arrangement results in a CCP unit cell having four atoms, calculated as follows: \(\text{8 corner atoms} \times \frac{1}{8} + \text{6 face atoms} \times \frac{1}{2} = 4 \text{ atoms per unit cell}\).
Cubic Close-Packed structures are characterized by their high packing efficiency of 74%, which means they are among the most efficient space-filling structures known. This high packing efficiency is critical in various applications because it results in metals that are stable and dense, often providing good conduction of electricity and heat.
The mathematical depiction using the radius (\(r\)) of the atoms and the lattice parameter (\(a\)) is given by the formula: \[a = 2 \sqrt{2} r\] This equation reflects that the closest neighboring atoms are found along the face diagonal of the cube.
Hexagonal Close-Packed (HCP) Structures
Hexagonal Close-Packed (HCP) structures are another form of highly efficient atomic arrangement. The distinctive geometry of HCP is a combination of hexagonally arraigned layers where each atom touches twelve neighboring atoms.
The HCP unit cell is made up of two hexagonally arranged layers with a third layer fitting into the grooves of the prior layer, defined often as ABAB... stacking. This particular structure allows the HCP to achieve a packing efficiency of 74%, similar to CCP structures, making it an ideal configuration for metals that need robustness and durability.
Common metals that crystallize in this form include titanium, cobalt, and zinc. Each atom in the HCP lattice is surrounded by 12 other atoms, creating a highly stable and tightly packed formation essential for its mechanical properties.
The geometrical relationship in the HCP structure is characterized by the ideal c/a ratio, which is 1.633. This ratio describes the height and width of the unit cell in relation to the atomic radius, ensuring maximum efficiency in space usage.
The HCP unit cell is made up of two hexagonally arranged layers with a third layer fitting into the grooves of the prior layer, defined often as ABAB... stacking. This particular structure allows the HCP to achieve a packing efficiency of 74%, similar to CCP structures, making it an ideal configuration for metals that need robustness and durability.
Common metals that crystallize in this form include titanium, cobalt, and zinc. Each atom in the HCP lattice is surrounded by 12 other atoms, creating a highly stable and tightly packed formation essential for its mechanical properties.
The geometrical relationship in the HCP structure is characterized by the ideal c/a ratio, which is 1.633. This ratio describes the height and width of the unit cell in relation to the atomic radius, ensuring maximum efficiency in space usage.
