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### Topic: Calculating the center void of a simple cubic unit cell  (Read 4459 times)

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#### Zym

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##### Calculating the center void of a simple cubic unit cell
« on: March 05, 2007, 03:08:05 PM »
Ok, heres the question I'm answering :

Hard spheres of radius r are arranged in contact in the form of a simple cubic unit cell. Calculate in terms r the radius of the largest sphere that can fit in the hole in the very center of this unit cell.

Heres what I've come up with :
Let R = the radius of the corner atoms
Let D = the diagonal measurement across the face of the unit
Let H = the diagonal measurement through the unit

Edge length = 2R         (2R)^2 + (2R)^2 = D^2   =   4R^2 + 4R^2 = D^2     D^2 = 8R^2   D = ?8R^2

(2R)^2 +  D^2 = H^2     4R^2 + (?8R^2)^2 = H^2         H^2 = 4R^2 + 8R^2     ?H^2 = ?12R^2  H = 2R?3

The radius of the largest sphere that can fit in the hole is (2R?3 – 2R) / 2

How does this look? Any comments?

#### Bakegaku

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##### Re: Calculating the center void of a simple cubic unit cell
« Reply #1 on: March 05, 2007, 07:21:23 PM »
Quote
(2R?3 – 2R) / 2
'

This can be simplified.
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#### Zym

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##### Re: Calculating the center void of a simple cubic unit cell
« Reply #2 on: March 06, 2007, 01:24:50 AM »

I can't believe I missed something so simple.

OK, last line should read R31/3 - R

This answer seems to simple for some reason. I feel like I'm missing something.

As a side question, is there a way to insert symbols such as a root sign in text here with pasting from word?

#### Yggdrasil

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##### Re: Calculating the center void of a simple cubic unit cell
« Reply #3 on: March 05, 2007, 11:03:38 PM »
The answer looks fine to me (I'm assuming the 1/3 is a typo since it should be 1/2), although you could phrase it as:

(31/2 -1)R

which makes the answer seem more like a scalar multiple of the radius.

Anyway, I am not aware of a way to insert the root symbol w/o copying and pasting.

#### Zym

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##### Re: Calculating the center void of a simple cubic unit cell
« Reply #4 on: March 06, 2007, 12:16:27 AM »
Opps, yes it was a typo.  (31/2-1)R is the most simplified, and definitely will look better on my lab report. My Chem prof is quite a stickler for the little things (aren't they all?).

Thank you for all your help. This is a great site. I hope I don't get too annoying, I'm a chem major with a lot of classes still to go. Hopefully by next year I will be able to answer some questions here, while asking them over on the organic board