# Distance between successive tetrahedral voids in FCC

What is the distance between successive tetrahedral voids in a FCC crystal structure?

The distance between a face centered atom and a tetrahedral void is $\sqrt{\frac{3a}{4}}$ if the cube's edge length is $a$. Now If I know the angle made, between the line joining a tetrahedral void and the face centered atom, and the successive tetrahedral void and face centered atom, I can find the distance. But how do I calculate that distance since I don't know the angle.

As shown in figure, if we divide a FCC unit cell into 8 small cubes, then each small cube has 1 Tetrahedral void at its own body centre. Thus, there are total 8 Tetrahedral voids in one unit cell. It can also be seen from the figure that the nearest distance between two Tetrahedral voids is a/2.

• It looks like the illustration have been adapted from a third-party source which you must specify (looks like a scanned textbook, so a complete reference would be ideal), otherwise it might be considered a plagiarism. Dec 6, 2019 at 6:56

The distance between any two successive (adjacent) tetrahedral voids. To calculate it, draw a line joining a tetrahedral void to the corresponding vertex. We know that this distance is $\frac{\sqrt{3}a}{4}$, now take the projection of this on any of the edges. The angle between body diagonal and an edge is cos inverse of $\frac{1}{\sqrt{3}}$, so the projection of the line on an edge is $a/4$, Similarly the other tetrahedral void will also be $a/4$ away from its vertex on this edge. So the distance between these two is $a - a/4 - a/4 = a/2$

• If they are not successive what are the other possible distances between tertrahedral voids....what is the geometry behind them. May 1, 2018 at 15:43

See.. The centre of the tetrahedral void is same as the centre of the smaller cube whose vertices the tetrahedron shares. Use this simple fact to calculate the distance between centres of 2 consecutive tetrahedral void.

Now , clearly distance between centres is a/4 + a/4 = a/2.

• This answer is better suited as a comment since it is more of a hint rather than a solution. Apr 13, 2018 at 15:29
• Corrected the math. What I would like to see, though, is a picture showing how the atoms surrounding the void fit in the smaller cube and how the latter fits in the cell. May 12, 2018 at 10:22

Answer Is Given In This Image With Complete Explanation... Hope It Works...

• Please avoid posting screenshots or photos depicting textual information; type it in instead: there is Markdown and MathJax support allowing you to post nearly everything but complex graphics as searchable text. Also, avoid asking for the feedback or upvotes in the answer. Apr 24, 2020 at 19:08
• Question is asked to find distance between two tetrahedral voids, you have calculated between a tetrahedral and an octahedral void Dec 30, 2020 at 6:01