Phosphorus donors in silicon are widely used in silicon electronics. since phosphorus has one more electron that silicon, when it gets incorporated into the silicon lattice it gives up an electron. since the phosphorus atom is now ionized, the electron it gave up tends to hang around the ionized atom (we say the atoms is occupied). however temperature and diffusion can work to remove the electron from the ionized phosphorus atom (we call this unoccupied). let's try to understand this situation by applying the gibbs factor. let an unoccupied atom have energy zero and an occupied atom have energy - e. write down the grand partition function, the average occupancy of the phosphorus atom and probability that the phosphorus atom is occupied.