• Support PF! Buy your school textbooks, materials and every day products Here!

Grand partition function

  • Thread starter poiuy87
  • Start date
  • #1
1
0

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 [tex]\psi_{\it i}[/tex] with an absorption energy [tex]\varepsilon_{i}[/tex]. Write down the grand partition function for one site.

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

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.

Homework Equations



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 [tex]\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} [/tex]

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


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.
 

Answers and Replies

Related Threads for: Grand partition function

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
0
Views
2K
Replies
8
Views
7K
Replies
0
Views
1K
Replies
1
Views
5K
Replies
1
Views
2K
Replies
0
Views
2K
Top