Why Does the Crystal Model Use Squared Combinations for State Count?

[KFC]

In textbook of statistical mechanics, there is an example considering an idealization of a crystal which has N lattice points and the same number of interstitial positions (places between the lattice points where atoms can reside). Let E be the energy necessary to remove an atom from a lattice site to an interstitial position and let n be the number of
atoms occupying interstitial sites in equilibrium. Now try to find the number of state

It is quite easy to think about this: choose n atoms from N atoms to fill n interstitial positions, number of possible configuration is given by combination


C_{N}^n = \frac{N!}{n!(N-n)!}

I think the number of state should be

\Omega = C_{N}^n = \frac{N!}{n!(N-n)!}

but the example just put

\Omega = \left(C_{N}^n\right)^2 = \left(\frac{N!}{n!(N-n)!}\right)^2

without saying why. Do you think it is a mistake?

 

[tiny-tim]

… an idealization of a crystal which has N lattice points and the same number of interstitial positions (places between the lattice points where atoms can reside).
Let E be the energy necessary to remove an atom from a lattice site to an interstitial position and let n be the number of
atoms occupying interstitial sites in equilibrium.
Now try to find the number of state
…
the example just put

\Omega = \left(C_{N}^n\right)^2 = \left(\frac{N!}{n!(N-n)!}\right)^2

without saying why. Do you think it is a mistake?

Hi KFC!

the atoms have to come from somewhere,

and they've left gaps behind them …

so there are ^NC_n ways of choosing where they're from, and ^NC_n ways of choosing where they're going.

 

[KFC]

Hi KFC!

the atoms have to come from somewhere,

and they've left gaps behind them …

so there are ^NC_n ways of choosing where they're from, and ^NC_n ways of choosing where they're going.

Got it. Thanks tiny-tim, you help me a lot.