2011-01-26, 01:00 AM
XTOTHEL Wrote:Simply put, you want d: because along that line is where your spheres are touching.
That's what I have in the first post.
XTOTHEL Wrote:you get D - 2R = 2r, 0.5(D-2R) = r
About this. The first time I did this problem, I set the sides of the cube to 1. That makes the diagonal of the diagonal of the cube (what you're talking about) sqrt(3).
2R + 2r = sqrt(3)
(sqrt(3) is what you have for D in the formula).
When you do it this way, you need to make R=kr so that the function can be written in terms of k, the ratio of the two radii.
2kr+2r = sqrt(3)
r = sqrt(3) / (2k+2)
2R +2R/k = sqrt(3)
R = sqrt(3) / (2+2/k)
4/3 pi [sqrt(3) / (2k+2)]^3 + 4/3 pi [sqrt(3) / (2+2k)]^3 / 1^3
That simplifies exactly to what I have in the first post, so it's still wrong. >_>"
Cyadd Wrote:[COLOR="Red"]The easiest answer is going straight to body centered cubic from material science.
Body Centered cubic.
It's approx. 0.68[/COLOR]
That's actually where I got this problem from. I want a formula that can tell me the efficiency of the packing for any k (ratio of the radii). Unfortunately, Wikipedia doesn't provide the formula. It only provides the formula for when the outer and the inner formulas are equal.
It also gives approximate solutions for k = 0 and k = 1, but I already know those. I'm just looking for a formula. By logic, the formula should spit out the same numbers for k and 1/k. In mathematical terms, f(k) = f(1/k). The formula I derived does that, but it's not giving me the efficiency anywhere except at k=1. I'm not sure why.