Atomic Radius and Lattice Parameter Relationship in HCP Crystals?

[darkelf]

Hello, can anyone help me out the the relationship of the atomic radius R with the lattice parameter a for a Hexagonal close packed cystals (HCP) and how to prove the ratio of a:C is 1.633.

Any information on this issues would be appreciated. Thanks

 

[Gokul43201]

This should get you going: https://www.physicsforums.com/showpost.php?p=1176371&postcount=2

If you have any trouble, show what you've done and which part you are stuck at, and someone will help you from there.

 

[oswaler]

Sure, I'd be happy to provide some information on the relationship between atomic radius and lattice parameter in HCP crystals.

First, let's define what atomic radius and lattice parameter are. The atomic radius is the distance from the center of an atom to its outermost electron. This can vary depending on the element and its state (solid, liquid, gas). The lattice parameter, on the other hand, is the distance between the centers of two adjacent atoms in a crystal structure.

In HCP crystals, the atoms are arranged in a hexagonal pattern with each atom surrounded by six other atoms. This results in a close-packed structure, where the atoms are packed as efficiently as possible. The lattice parameter in HCP crystals is related to the atomic radius through the following equation:

a = 2R√(3)

This means that the lattice parameter is equal to twice the atomic radius multiplied by the square root of 3. This relationship can be derived from the geometry of the HCP structure.

To prove that the ratio of a:C is 1.633, we can use the formula for the coordination number (C) in HCP crystals:

C = 12/(√(3))

Substituting the value of a from the previous equation, we get:

C = 12/(2R√(3))

Simplifying this, we get:

C = 6/R

We know that the coordination number in HCP crystals is 12, so we can set up the following equation:

12 = 6/R

Solving for R, we get:

R = 6/12 = 0.5

So, the atomic radius is equal to 0.5 times the lattice parameter. Substituting this into the original equation for a, we get:

a = 2(0.5)√(3) = √(3)

Therefore, the ratio of a:C is equal to √(3)/12 = 1.633.

I hope this helps to explain the relationship between atomic radius and lattice parameter in HCP crystals and how to prove the ratio of a:C. Let me know if you have any further questions.