Ideal hcp lattice, ratio c/a = 1.633 proof

[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.

 

[Astronuc]

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).