What is the c/a Ratio in Hexagonal Close-Packed Structures?

[TFM]

Homework Statement

The hexagonal close-packed structure (hcp) is the hexagonal Bravais lattice with a basis of two identical atoms associated with each lattice point, one at (0,0,0) and the second at

$$r = \frac{2}{3}a_1 + \frac{1}{3}a_2 + \frac{1}{2} a_3$$

where $$\left|a_1\right|$$ = $$\left|a_2 \right|$$ and $$a_1$$ and $$a_2$$ are at 120 degrees in the basal plane and $$a_3$$ is parallel to the c-axis. By considering the atoms to be hard spheres with nearest neighbour contact, show that the “c/a” ratio $$\frac{a_3}{a_1}$$ is 1.633.

Homework Equations

The Attempt at a Solution

I am not quite sure what to do with this question. I am not quite sure how to get a3 and a1.

Any ideas?

many thanks,

TFM

 

[Jhero]

Hello TFM,

Thank you for your question. To find the values of a3 and a1, we can use the information given in the problem. First, we know that a1 and a2 are at 120 degrees in the basal plane, meaning they are at the same length and form an equilateral triangle. This means that |a1| = |a2| = a, where a is the length of the sides of the equilateral triangle.

Next, we can use the information given for the position of the second atom, r, to find the value of a3. Using the Pythagorean theorem, we can find the length of r, which is the distance from the origin (0,0,0) to the second atom. This can be written as:

|r| = √(r1^2 + r2^2 + r3^2)

where r1, r2, and r3 are the components of r in the x, y, and z directions respectively. Substituting in the values given in the problem, we get:

|r| = √(2/3 a^2 + 1/3 a^2 + 1/2 a^2) = √(2/3 a^2 + 1/2 a^2) = a√(5/6)

Since r is the distance between the two atoms, we can also write it as the sum of a3 and a, the length of the sides of the equilateral triangle:

|r| = a3 + a

Substituting in the value we found for r, we get:

a√(5/6) = a3 + a

Solving for a3, we get:

a3 = a(√(5/6) - 1)

Now, we can substitute in the value of a from the equilateral triangle:

a3 = (√(5/6) - 1)a

Finally, we can find the ratio of a3 to a1 by dividing a3 by a1:

a3/a1 = (√(5/6) - 1)a/a = (√(5/6) - 1)

Since we know that a1 = a2 = a, we can simplify this ratio to:

a3/a1 = (√(5/6) - 1) = (√(5/6) -