Calculating Modes in a Cavity: Why Use a Spherical Volume?

[Conservation]

My question stems form the section "How Many Modes in a Cavity?" in the following derivation of Rayleigh-Jean Law:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/rayj.html#c2"

In here, they count the number of modes as represented by volume of an eighth of a sphere. What's the mathematical justification behind using a spherical volume? Also, I understand why they divided by 8 since the n's must be positive, but aren't the n's here also supposed to be integers only?

 

[MisterX]

My question stems form the section "How Many Modes in a Cavity?" in the following derivation of Rayleigh-Jean Law:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/rayj.html#c2"

In here, they count the number of modes as represented by volume of an eighth of a sphere. What's the mathematical justification behind using a spherical volume? Also, I understand why they divided by 8 since the n's must be positive, but aren't the n's here also supposed to be integers only?

We may be interested in the mode count within a sphere. It may relate to questions like how many modes have energy less that some particular value or other calculations.

There is an error here introduced by pretending like the points have a uniform density when actually there are discrete grid points. It's an approximation. It's possible to make a more accurate expression. You might want to look at the first chapter of https://www.amazon.com/dp/0123821886/?tag=pfamazon01-20if you are interested.