Planar Concentration of Nb Atoms in BCC Crystal

In summary: So, one must be more careful, and a diagram may help. The base has length ##\sqrt{2}##a. The vertical height, if one measures from the center of the base of the (111) plane to the corner of the cell, is ##\sqrt{6}##a/3. The answer is correct. In summary, the conversation discusses the calculation of planar densities of atoms in the (100), (110), and (111) planes in both bcc and fcc crystal structures. The correct calculations involve using the Pythagorean theorem and finding the area of the triangle formed by the base of the plane and the upper corner of the unit cell. For the (111) plane
  • #1
elcotufa
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Homework Statement


Niobium Nb has the BCC crystal with a lattice parameter of a= .3294 nm

Find the planar concentrations of atoms per m^2 of the (100, (110) and (111) planes.

The 100 plane has 1 atom on the plane (1/4 times 4)

the 110 plane has 2 atoms on the plane and using pythagorean theorem the long side is a*sqrt(2)

http://www.gly.uga.edu/schroeder/geol6550/111.jpeg [Broken]
best pic I could find for the 111 plane
Now for the last plane (111) this is what I did

the area would be 1/2 base * height
the base is [tex]a*\sqrt2[/tex] and

pythagorean theorem to find the height is a^2 and half of the long side from plane (110) squared [tex]\sqrt{a^2+\frac{2a^2}4}=.4034nm[/tex]

[tex]
\frac12(.4034)10^{-9}(.3294)10^{-9}
[/tex]
Now how many atoms are on the plane? and are there mistakes in what I have done so far?

And how many atoms would that (111) plane have in a FCC model?
Answer would have number of atoms over area
Thanks for the help
 
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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.

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

One calculates the number of atoms per unit area, where area may be given in terms of nm2.

For the (111) plane in bcc, note that the plane does not pass through the atom in the center of the cubic structure. The base is the diagonal in one of the 6 faces, so the length is as one shows, ##\sqrt{2}##a.

The height of the triangle (hypotenuse) formed by the base of the (111) plane to the upper corner of the cell is found from the Pythagorean theorem, realizing one measures from the center of the diagonal to the upper corner of the cubic cell. One's answer is correct.

For an fcc cell, note that the base of the (111) plane (the diagonal of the unit cell) does intersect an atom.
 

1. What is planar concentration of Nb atoms in BCC crystal?

Planar concentration refers to the number of Nb atoms that are present in a specific plane within a BCC (body-centered cubic) crystal lattice. This measure is used to understand the distribution of Nb atoms in the crystal structure.

2. How is planar concentration calculated?

Planar concentration is calculated by dividing the number of Nb atoms in a specific plane by the total number of atoms in the same plane in the crystal lattice. This value is then multiplied by 100 to obtain a percentage.

3. What factors affect planar concentration of Nb atoms in BCC crystal?

The planar concentration of Nb atoms in BCC crystal can be influenced by various factors such as the temperature at which the crystal is formed, the presence of impurities or defects, and the atomic size of Nb compared to the other elements in the crystal.

4. Why is planar concentration important in materials science?

Planar concentration is an important factor to consider in materials science because it can affect the mechanical, electrical, and thermal properties of a material. It can also impact the performance and stability of the material in various applications.

5. How can planar concentration of Nb atoms be controlled?

The planar concentration of Nb atoms can be controlled by adjusting the composition of the material, the processing conditions, and the crystal growth methods. Additionally, the use of doping or alloying elements can also help to manipulate the planar concentration in a BCC crystal.

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