Calculating Planar Density for FCC {100}, {110}, {111}

In summary, to calculate the planar density for {100}, {110}, {111} planes in a FCC unit cell, you need to know the number of atoms on the plane and the area of the plane. For {100} plane, there are 2 atoms and the area is a^2. For {110} plane, there is 1 atom and the area is a*sqrt(2). And for {111} plane, there are 2 atoms and the area is a^2*sqrt(3)/2. The density is then calculated by dividing the number of atoms by the area of the plane. In the case of Si diamond structure, the density for (110) plane is 9.6*
  • #1
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HOw od you calculate the planar density for {100}, {110}, {111} for FCC?
 
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  • #2
Lets start with [100] plane which is a plane parallel to a face of the unit cell and it looks like a square. There is one atom at the center of the square and a total of 4*1/4 atoms on the coners of the plane. Hence there are a net total of 2 atoms inside the square. Now the area is just the area of the square which is a^2 (where a is the lattice constant). So the suface density is 2/a^2 atoms per unit area.

The [110] plane is the plane which cuts the unit cell diagonally in half and it looks like a square. There are just 4*1/4 atoms on the corners of the square - a net total of 1 atom inside the square. The length of one of the sides of the plane is a*sqrt(2). Hence the surface density is 1/(a*sqrt(2)) atoms per unit area.

The [111] plane is a plane that touches the three far corners of the unit cell and it looks like a triangle. There are then a total 1/6*3 atoms that make up the vertices of the triangle and there are a total of 1/2*3 atoms that make up the three edges of the triangle. So you have a net total of 2 atoms inside the triangle. The triangle is an equlilateral triangle with a leg of length a*sqrt(2). The area of an equilateral triangle is s^2*sqrt(3)/4 which then gives us a^2*sqrt(3)/2 as the area of that trinagle. Hence the density is 2/(a^2*sqrt(3)/2) atoms per unit area.
 
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  • #3
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Thank you very much for explaning that very clearly!
 
  • #4
FCC for (2 0 0)

how do you find the planar density for the (2 0 0) FCC unit cell
 
  • #5
First of all, can you picture the (200) plane in the FCC unit cell? Secondly, across how many atoms does it cut?
 
  • #6
Just in case, in Si diamond structure, the plane (110) includes a total of 4 atoms, which increases the density to 9.6*10^(14).
 
  • #7
I need to know the number at surface atoms in a cube of a FCC lattice of gold atoms knowing that R= 144.2 pm with respect to L( length of the cube) and a ( length of a unit cell )
 

Related to Calculating Planar Density for FCC {100}, {110}, {111}

What is planar density?

Planar density is a measure of the number of atoms per unit area in a crystal lattice.

What is the formula for calculating planar density for FCC crystals?

The formula for calculating planar density for FCC crystals is: PD = 4N / a2, where PD is the planar density, N is the number of atoms in the unit cell, and a is the lattice constant.

How do you determine the number of atoms in a FCC unit cell?

The number of atoms in a FCC unit cell is determined by multiplying the number of atoms at each lattice point (4) by the number of lattice points within the unit cell (1).

What are the three most common {hkl} planes for FCC crystals?

The three most common {hkl} planes for FCC crystals are {100}, {110}, and {111}.

How do you calculate the planar density for specific {hkl} planes in FCC crystals?

To calculate the planar density for specific {hkl} planes in FCC crystals, you can use the formula PD = (h2 + k2 + l2) / a2, where h, k, and l represent the Miller indices for the plane and a is the lattice constant.

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