# What's does square root of number of atoms mean?

Number of atoms $$N$$ was counted in 3 dimension. $$x,y,z$$. However, when calculate it, i.e. in many cases such as refractive index , people take the square root of it, i.e. in reflective index $$n=\sqrt{1+(\propto \frac{N}{Volume})}$$.(The volume here was pulled out by another $$[meter^{-1}]$$ term.)

On one side, there's no contradiction from dimensional analysis because number of atoms are just scalars. However, the more I think about it, the more twisting it felt like. Because even though you can still count atoms, you can't always count them in one direction.

I feel like there's something to do with the 2 norm if we put the atoms as dots in vector space. However, it's a bit confusing what I'm looking at. Could you help me to understand what's does square root of number of atoms physically mean?

• why do you think total number of atoms has to be associated with some direction? – wcc Apr 25 at 15:18
• @IamAStudent It's going to be more complicated, but:"Suppose $N_1$ number of type I atoms and $N_2$ number of type II atoms." There's a difference if we consider statistical ensemble.(and it's an exterior quantity) Further, we are still count number of atoms in 3 space(in periodic lattice), if one project it to a 2d surface, they may cover each other. This seemed to be suggestive that $N$ may be characterized by some sort of amplitude, which, in quantum, the sum of probability amplitude squared. Anyway I was wondering it may be viewed as a "vector" like object in special settings. – ShoutOutAndCalculate Apr 25 at 16:35