Hexagonal Closest Packed Unit Cell Height

In summary, the conversation discusses attempts to derive the height as a function of radius of the spheres packing the unit cell, using trigonometry and symmetry arguments. The trick is to realize that the middle plane of atoms occupies positions directly above the centroids of the triangles in the base plane. Using Pythagoras and the packing efficiency, the height can be calculated as a ratio to the radius.
  • #1
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My attempts to derive the height as a function of radius of the spheres packing the unit cell have failed. The best attempt so far was to create a diagonal line from the top atom to one of the middle ones. With that distance I thought I could use trig to find the component parallel to the height and multiply by two. Of course that didn't work. Any hints or suggestions? Even good 3d visualizations would help.
 
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  • #2
The trick is in realizing that the middle plane of atoms occupy positions directly above the centroids of the triangles in the base plane. This follows directly from a symmetry argument.

#1. Each basal plane has nearest neighbor atoms making equilateral triangles. So, a=2R (where R is the sphere radius).

#2. Each atom at height c/2 above the basal plane is positioned directly above the centroid of the triangles in the base plane. For an equilatreral triangle, the distance from a vertex to the centroid is two-thirds the length of the median.

#3. Each atom in the base plane has a nearest neighbor in this middle plane. So, the distance from the corner atom in the base plane to the nearby atom in the mid-plane is 2R.

#4. Finally, use Pythagoras to write (2R)^2 = (c/2)^2 + (dist. calculated in #2)^2. That should take you to the required ratio.
 
  • #3
I tried to find it with the help of packing efficiency (i.e. 0.74). the number of contributing spheres inside one unit cell is 6. therefore volume of 6 spheres/volume of hexagonal unit cell=0.74
volume of hexagonal unit cell is 6*sqrt(3)*r*r*h
then u will get answer of "h".
 
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What is a hexagonal close packed unit cell?

A hexagonal close packed unit cell is a type of crystal structure in which atoms are arranged in a close-packed hexagonal lattice. This means that the atoms are packed as closely together as possible, with each atom touching six other atoms in a hexagonal pattern.

What is the height of a hexagonal close packed unit cell?

The height of a hexagonal close packed unit cell is equal to the distance between two consecutive layers of atoms in the hexagonal lattice. This distance, also known as the "c-axis" length, is determined by the size of the atoms and the angle between the atomic bonds.

How is the height of a hexagonal close packed unit cell calculated?

The height of a hexagonal close packed unit cell can be calculated using the formula h = c * √(2/3), where h is the height and c is the c-axis length. This formula takes into account the angle between the atomic bonds, which is 120 degrees for a hexagonal lattice.

What is the relationship between the height of a hexagonal close packed unit cell and the atomic radius?

The height of a hexagonal close packed unit cell is directly related to the atomic radius. As the atomic radius increases, the height of the unit cell also increases. This is because larger atoms require more space between layers in order to maintain the hexagonal lattice structure.

Why is the height of a hexagonal close packed unit cell important in materials science?

The height of a hexagonal close packed unit cell is an important factor in determining the properties of a material. It affects the density, strength, and other physical and chemical properties of the material. By understanding the height of the unit cell, scientists can better predict and control the behavior of materials in various applications.

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