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zezima1

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An einstein solid is a solid composed of N quantum harmonic oscillators, which can store evenly spaced energy units q.

Now suppose we have an einstein solid of N oscillators with q energy units, where q>>N. My book wants to derive an expression for the multiplicity of this solid, but I can't quite follow the approximations leading to the final expression:

We have in general that that the multiplicity, Ω, of an einstein solid is given by the binomial:

Ω = C(q+N-1,q) , where C() is the binomial coefficient. I won't verify this expression, because it is the approximations of it that bother me:

In general we have from the definition of the binomial coefficient:

Ω = C(q+N-1,q) = (q+N-1)!/((N-1)!*q!)

My book approximates this to:

Ω = (q+N)!/(q!*N!)

with the argument that:

*"The ratio of N! to (N-1)! is only a large factor (N).*

I don't see what this has to do with the approximation. The multiplicity above gets far bigger with the approximation because it all depends on q.

For instance, say q=100 and N=2. Then you get:

Ωreal = 101!/100! = 101

Ωapprox = 102!/2! ≈ 50*Ωreal

or i general for N=2:

Ωapprox = q/2 * Ωapprox

So in general how is this approximation valid? Am I plugging the numbers in the wrong way?