Unit Cell - Linear Density - BCC & FCC

In summary, the linear densities for the most densely packed directions in BCC and FCC are the same, with both having a value of 2/4R. However, the close-packed planes and directions are different in BCC and FCC, with BCC having {110} planes and <111> directions, and FCC having {111} planes and <110> directions. This difference in close-packed planes and directions results in a difference in the linear density calculations between BCC and FCC, even though the final numerical values are the same.
  • #1
General_Sax
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Unit Cell -- Linear Density -- BCC&FCC

Homework Statement


Calculate the linear density for the most densely packed directions in BCC and FCC. Note the difference in linear density. Note the difference in linear density.

Homework Equations


LD = #of atoms/length


The Attempt at a Solution



I arrive at an answer, however my answers for BCC and FCC are the same. I chose vector [110] for the BCC and got this for the LD equation: LD = 2/4R

For FCC I chose vector [101] ... LD = 2/4R

I was hoping someone could see my error.
 
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  • #2
One should find diagrams and discussion in W.D. Callister, Materials Science and Engineering, An Introduction, 7th Ed., John Wiley and Sons, 2007, or later editions.

In BCC, the close-packed planes are the {110} planes, and the close-packed directions are the <111>
In FCC, the close-packed planes are the {111} planes, and the close-packed directions are the <110> direction
Ref: http://monoceros.physics.muni.cz/~j...xty/Krystaly/CrystalStructures&Properties.pdf

Other notes
https://faculty.ksu.edu.sa/sites/default/files/chapter3.pdf

https://chem.libretexts.org/Bookshe...iquids/7.08:_Cubic_Lattices_and_Close_Packing

Problems with planar densities of fcc and bcc are found in this assignment.
http://maecourses.ucsd.edu/~jmckittr/mae20-wi11//Assignment 4 solutions.pdf
 

FAQ: Unit Cell - Linear Density - BCC & FCC

1. What is a unit cell and why is it important in crystal structures?

A unit cell is the smallest repeating unit in a crystal structure. It is important because it allows us to understand the overall structure and properties of a material, as well as predict its behavior under different conditions.

2. What is linear density and how is it calculated for BCC and FCC crystal structures?

Linear density is the number of atoms per unit length along a specific crystallographic direction. For BCC structures, the linear density is calculated by dividing the number of atoms in a unit cell by the length of the unit cell along the specific direction. For FCC structures, the linear density is calculated by dividing the number of atoms in a unit cell by half of the unit cell length along the specific direction.

3. How does the linear density differ between BCC and FCC crystal structures?

The linear density is higher in FCC structures compared to BCC structures. This is due to the fact that FCC structures have a higher number of atoms per unit cell, resulting in a higher linear density along the same crystallographic direction.

4. How does the number of atoms in a unit cell affect the linear density?

The number of atoms in a unit cell directly affects the linear density. As the number of atoms in a unit cell increases, the linear density also increases because there are more atoms per unit length along a specific direction.

5. Can the linear density change in different crystal structures?

Yes, the linear density can vary in different crystal structures. This is because the number of atoms and the unit cell length can differ in different crystal structures, resulting in different linear densities along the same crystallographic direction.

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