# Grand partition function

1. Jan 13, 2009

### poiuy87

1. The problem statement, all variables and given/known data

a) Suppose particles can be absorbed onto a surface such that each absorption site can be occupied by up to 6 atoms each in single-particle quantum state $$\psi_{\it i}$$ with an absorption energy $$\varepsilon_{i}$$. Write down the grand partition function for one site.

b) If $$\left( \varepsilon_{i} - \mu \right) / k_{B} T = 0.7$$

Show that the expression for the grand partition function is very close to that of a Bose system where any number of particles may occupy the site.

c) Find the probability that there are six particles on the site.

2. Relevant equations

None given

3. The attempt at a solution

For a fermi system only one particle can occupy each state so the following are possible:

Quantum state--------Number of configurations--------Energy--------Number of particles

|0,0,0,0,0,0>--------------------1----------------------0--------------------0
|1,0,0,0,0,0>--------------------6----------------------1--------------------1
|1,1,0,0,0,0>--------------------15---------------------2--------------------2
|1,1,1,0,0,0>--------------------20---------------------3--------------------3
|1,1,1,1,0,0>--------------------15---------------------4--------------------4
|1,1,1,1,1,0>--------------------6----------------------5--------------------5
|1,1,1,1,1,1>--------------------1----------------------6--------------------6

Grand partition function $$\Xi = 1 + 6e^ { - \left( \varepsilon - \mu \right) / k_{B} T} + 15e^ { -2 \left( \varepsilon - \mu \right) / k_{B} T} + 20e^ { -3 \left( \varepsilon - \mu \right) / k_{B} T} + 15e^ { -4 \left( \varepsilon - \mu \right) / k_{B} T} + 6e^ { -5 \left( \varepsilon - \mu \right) / k_{B} T} + e^ { -6 \left( \varepsilon - \mu \right) / k_{B} T}$$

--> $$\Xi = \left( 1 + {e^ { - \left( \varepsilon - \mu \right) / k_{B} T}} \right) ^ 6$$

Up to here I am fine, I know the question just says to write down the grand partition function but I thought I would show how I got to it. Obviously in the quantum states the 1 can take any position hence the number of configurations.

I have no idea how to do this for Bose particles, there are hundreds of different combinations of quantum states as each position can hold 0-6 atoms.

Please can somebody tell me how to find the GPF for bose particles, it must be fairly straight forward as the question is only worth 1 mark on the exam.

Thanks.