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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?

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  • $\begingroup$ why do you think total number of atoms has to be associated with some direction? $\endgroup$ – wcc Apr 25 at 15:18
  • $\begingroup$ @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. $\endgroup$ – user9976437 Apr 25 at 16:35

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