# Unit cell structure of ionic crystal

I have a question about the structure of an LiH unit cell, and while this is related to a homework problem, it isn't the problem itself, I'm just looking for conceptual understanding. I've already found that there should be 4 Li+ ions and 4 H- ions in one unit cell. However, I am struggling with how they will be arranged. Because the number of LiH molecules is 4, I am fairly certain that the structure intended for this problem is face-centered cubic lattice. This is where my conflict begins. In my class we have had the face-centered cubic unit cell described as (for one element) an atom at each corner and an atom in each side. From this arrangement we get (1/8)*8 + (1/2)*6 = 4 atoms. Now, I've seen a description online of the face-centered cubic structure for NaCl as "We can think of this as chloride ions forming an FCC cell, with sodium ions located in the octahedral holes in the middle of the cell edges and in the center of the cell. The sodium and chloride ions touch each other along the cell edges. The unit cell contains four sodium ions and four chloride ions, giving the 1:1 stoichiometry required by the formula, NaCl" (Chemistry LibreTexts). For me this makes a certain amount of sense, because this would give 4 cations when you use the same counting principle as before. However, I've seen multiple diagrams online that contradict this, such as: https://images.app.goo.gl/CGAsc4xAJM39tMgi8. This seems to claim that the center image is the unit cell, and while it does have 4 of each ion "on" it, it does not seem to fulfill the same structure as the aforementioned unit cell suggestion. Is the problem here just that they are showing 1/8th of the full unit cell for simplicity? If not, I am even more confused.

Thank you for reading this extraordinarily long question for what could be a very simple answer! And please don't roast me too hard, I've never posted here before.

I looked and there are a lot of questions on the site about units cells, many of which are pertinent. However I couldn't find one that answers your question directly.

First there is no reason to guess that $$\ce{LiH}$$ has a FCC structuree. You can look that up, for instance in Wikipedia.

Second $$\ce{LiH}$$ doesn't really have molecules in the solid. Think of $$\ce{LiH}$$ as an empirical formula rather than a molecular formula.

UNIT CRYSTAL

I lifted the following from website https://opentextbc.ca/chemistry/chapter/10-6-lattice-structures-in-crystalline-solids/

Now the text says:

The unit cell contains four sodium ions and four chloride ions, giving the 1:1 stoichiometry required by the formula NaCl.

That is correct. The unit cell must have the correct stoichiometric ratio. However the section seems to imply that the middle drawing, as show below, is the proper unit cell. It isn't!

First realize that when looking at proper unit cells is that that the lattice points between which the edges are drawn are the nuclei, not points in space that would contain whole atoms. So consider the drawing below. For the cell as shown in the illustration, there are 8 corners with 1/8 atom each. Four corners are Na, and four are Cl. Thus the cell shown only has $$\frac{1}{2}$$ of a Na cation and $$\frac{1}{2}$$ of a Cl anion.

Now down the rabbit hole!

The proper unit cell for NaCl is shown below:

From Wikipedia article on Cubic crystal system:

The face-centered cubic system (cF) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell (​1⁄8 × 8 from the corners plus ​1⁄2 × 6 from the faces). Each sphere in a cF lattice has coordination number 12. Coordination number is the number of nearest neighbors of a central atom in the structure.

So the unit cell contains $$\ce{Na4Cl4}$$ !?!

The reason is that the proper unit cell must also contain all the symmetry elements of the crystal. Look at the drawing below:

I won't get into all of how group theory applies to crystals and molecules, but the red line shows a four-fold rotation axis. In other words all the atoms are identical. You can rotate the crystal 90 degrees about the red axis and the sodium and chlorine atoms are in identical spots. The axis is a four-fold axis because you have to rotate 90 degrees four times to get the identical atoms back to their starting position.

• Thank you so much for your response! This was related to a homework problem where I had to solve for the number of atoms, which is why I said "I am fairly certain" that the structure is FCC, and why I referred to LiH molecules (I combined the molar masses when solving). You perfectly addressed my confusion about this unit cell. The insufficient number of ions in the proposed unit cell that you showed above was why I was conflicted, and the FCC unit cell you showed below was what I was assuming based on the Chem LibreText source. Also, thank you for your comment on the symmetry requirement. Apr 25, 2020 at 21:37