Question

The atoms of a crystalline solid may occupy either a position of equilibrium, with zero displaced position, with energy >0. To each equilibrium position, energy, or a there corresponds a unique displaced position. Given N atoms and the total energy U, determine 2(U,N) the number of available microstates of this system.

Answer 1

Let us assume that the number of atoms in the equilibrium position is $N_1$. And so, if the total number of atoms is N, then, the number of atoms in the displaced position is
N2NN1
.
And given the total energy , we write this as

U N2€ =(N N1)e

  
N N

  
N N-

And so, the number of micro states is
  
(U, N) NN2!

  
(U,N)N(N N1)! Ni!(N N)!

  
N! U, N) N-U: )!

This is the exact expression for the micro states at an energy U for a given number of atoms N.