- #1

MathematicalPhysicist

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**The question:**

A system consists of N sites and N particles with magnetic moment m.

each site can be in one of the three situations:

1. empty with energy zero.

2. occupied with one particle and zero energy (when there isn't magnetic field around).

3. occupied with two particles with anti parallel moments with energy [tex]\epsilon[/tex].

(particles in the same site are indistinguishable).

the system is under the influence of external magnetic field,B, which acts on the particles.

the question is to: 1. find the chemical potential of the system. 2. find the entropy of the system. 3. find the energy of the system in limit of high tempratures and low.

**my solution**:

ok first i need to find the grand canonical partition function, i think this should be:

[tex]Z_G=1+e^{\beta *\mu}+e^{\beta *(2\mu -\epsilon)}[/tex]

(because the particles are indisitiguishable, the partition function equals: [tex]Z_G^{N}/N![/tex]

now, what I think to do is to calculate the energy with this function and then equate it with the energy given which is i think is: [tex]-NmB[/tex]

Now I'm using the next equation:

[tex]U=(\frac{\mu}{\beta}\frac{d}{d\mu}-\frac{d}{d\beta})log(Z_G)[/tex]

After some manipulations, I get to the next equation:

[tex]\frac{e^{2\beta *\mu -\beta *\epsilon}\epsilon}{1+e^{\beta *\mu}+e^{2\beta * \mu-\beta *\epsilon}}=mB[/tex]

after that I need to solve a quadratic equation wrt e^(beta*mu) and choose the positive solution, is this seems plausible?

with finding the entropy I can use the thermodynamic identity, that :

[tex]\mu =-\tau (\frac{d\sigma}{dN})[/tex]

and integrate what i find for the chemical potential which is mu ofcourse (we are using the notation in kittel's which i think is used worldwide).

anyway, what do you think of my attempt at solution?

thanks in advance for any help.