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Ideal hcp lattice, ratio c/a = 1.633 proof

  1. Feb 9, 2009 #1
    1. The problem statement, all variables and given/known data

    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

    -------------- = 16 R^2

    3a^2 = 2 *16 R^2

    a^2 = 2*16 R^2

    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
    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.
  2. jcsd
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