# Grand partition function

## Homework Statement

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.

None given

## 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.