# Ideal hcp lattice, ratio c/a = 1.633 proof

• skyav
In summary, the conversation discusses the calculation for the c/a ratio in an ideal hexagonal close-packed structure, taking into consideration the Pythagorean Theorem and the cosine for 30 degree triangles. The steps for solving the geometric problem are outlined, with a final clarification on the shape of the slice being used.
skyav

## Homework Statement

q: show that for an ideal hcp structure the c/a ratio is equal to (8/3)^(1/2) = 1.633

this question has come up before in the forum but still it has not fully answered:

Kouros Khamoushi
Dec30-05, 12:27 AM
This is the mathematical calculation ^ means to the power of

c/2 = a/2 Then a^2 /2 = c^2/2

a^2 + a^2
----- = (4R)^2
2

2a^2+a^2
-------------- = 16 R^2
2

3a^2 = 2 *16 R^2

a^2 = 2*16 R^2
-----
3

a = 2* square root of 16 divided by square root of 3

a = 8 / 3 = 1.6329 R
Kouros Khamoushi
Jan26-06, 06:48 PM
The assumptions above are roughly correct, but we have to take into the considerations the Pythagorean Theorem.
The Pythagorean Theorem states: b^2=a^2+c^2. as well as c/2 or half of hexagonal crystal structure as well as the cosine for 30 degree triangles generated by the hole in Hexagonal closed packed.
So we write:
c/a half of this c/2.

Cos 30 degree
skyav
Feb9-09, 03:49 PM
Dear all,

i have tried. the last post by Kouros Khamoushi almost worked...

how ever i do not understand where some of the steps...

1. c/2 = a/2? how?

2. 3a^2 = 2 *16 R^2? where in the world did the factor of 3 come from on the LHS of this eqn.

You are prob correct... however please clarify the steps as i am totally baffled.

ps. inha: what shape of slice do u mean? also i hope it is from hexagonal lattice?

Kind regards.

skyav said:
c/2 = a/2 Then a^2 /2 = c^2/2
This is not correct.

To solve the geometric problem, consider a tetrahedron formed by 4 atoms, where 1 atom is equidistant from the other three, with the distance between the centers of the atoms = a = 2R (R = atomic radius). The height of the tetrahedron is simply half the height of the unit cell c/2, and the height of the tetrahedron can be determined in terms of a (or R).

## 1. What is an ideal hcp lattice?

An ideal hcp (hexagonal close-packed) lattice is a type of crystal structure in which the atoms are arranged in a close-packed hexagonal pattern, with each atom surrounded by 12 nearest neighbors. This type of lattice is often found in metals and some types of ceramics.

## 2. What does the ratio c/a = 1.633 mean?

The ratio c/a is the ratio of the lattice parameters c and a, which represent the vertical and horizontal spacing between the lattice points, respectively. In an ideal hcp lattice, this ratio is approximately equal to 1.633, meaning that the vertical spacing is 1.633 times greater than the horizontal spacing.

## 3. Why is the c/a ratio important in an hcp lattice?

The c/a ratio is important because it affects the stability and mechanical properties of the lattice. In an ideal hcp lattice, a c/a ratio of 1.633 results in the most efficient packing of atoms, leading to greater strength and stability.

## 4. How is the c/a ratio determined in an hcp lattice?

The c/a ratio can be determined experimentally by measuring the lattice parameters using techniques such as X-ray diffraction or electron microscopy. It can also be calculated using theoretical models that take into account the atomic arrangements and bonding in the lattice.

## 5. Is an ideal c/a ratio of 1.633 always achieved in hcp lattices?

No, the c/a ratio can vary slightly from the ideal value due to imperfections in the lattice or changes in temperature and pressure. However, an hcp lattice with a c/a ratio close to 1.633 is considered to be the most stable and ideal structure for many materials.

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