Statistical mechanics problem about a paramagnet

[Clara Chung]

Homework Statement

Attached below

Relevant Equations

Attached below

I don't know how to solve part c and d.
Attempt:
c) B_eff=B+e<M>
Substitute T_c into the equation in part b,
=> (B_eff-B)/e = Nμ_B tanh(B_eff/(N*e*μ_B))
Then?
Thank you.

 

[DrClaude]

Could you show what you got for part b?

 

[Clara Chung]

Could you show what you got for part b?

I just rewrite the equation I found on part A to an equation containing B_eff only... is it what a self-consistency equation mean?...

 

[DrClaude]

is it what a self-consistency equation mean?...

A self-consistency equation for $B_\textrm{eff}$ means that you get an equation for $B_\textrm{eff}$ that depends on itself. You could solve it numerically by guessing the value of $B_\textrm{eff}$ in the hyperbolic tangent, then calculating $B_\textrm{eff}$ on the left-hand side, and iterate until the value of $B_\textrm{eff}$ doesn't change anymore.

Your equation for (b) is correct. For (c), I think you can assume that $B_\textrm{eff}$ is small or, more properly
$$
\mu_\mathrm{B} B_\textrm{eff} \ll k T
$$

 

[Clara Chung]

A self-consistency equation for $B_\textrm{eff}$ means that you get an equation for $B_\textrm{eff}$ that depends on itself. You could solve it numerically by guessing the value of $B_\textrm{eff}$ in the hyperbolic tangent, then calculating $B_\textrm{eff}$ on the left-hand side, and iterate until the value of $B_\textrm{eff}$ doesn't change anymore.

Your equation for (b) is correct. For (c), I think you can assume that $B_\textrm{eff}$ is small or, more properly
$$
\mu_\mathrm{B} B_\textrm{eff} \ll k T
$$

I did part c using your hint

However in part d), how to deal with the cube term T^3/T_c^3?

 

[DrClaude]

I did part c using your hint

You didn't get the expected answer, which is given in the question. There is no need to keep the cubic terms.

 

[Clara Chung]

You didn't get the expected answer, which is given in the question. There is no need to keep the cubic terms.

Thank you, the answer I displayed above is for part d. The (T/T_C)^3 in the last line should be (T/T_C)^3/2. The expect term have no factor (T/T_C)^3/2, but there is one in my expression...

 

[DrClaude]

Above, you wrote
$$
B_\textrm{eff} = e N \mu_\mathrm{B} \left( \frac{B_\textrm{eff} \mu_\mathrm{B}}{kT} - \frac{B_\textrm{eff}^3 \mu_\mathrm{B}^3}{3 k^3T^3} \right)
$$
which means you took
$$
\tanh(x) \approx x - \frac{1}{3} x^3
$$
Try instead $\tanh(x) \approx x$.

 

[Clara Chung]

Above, you wrote
$$
B_\textrm{eff} = e N \mu_\mathrm{B} \left( \frac{B_\textrm{eff} \mu_\mathrm{B}}{kT} - \frac{B_\textrm{eff}^3 \mu_\mathrm{B}^3}{3 k^3T^3} \right)
$$
which means you took
$$
\tanh(x) \approx x - \frac{1}{3} x^3
$$
Try instead $\tanh(x) \approx x$.

However, in part d the question says expand it to 3rd order?

 

[DrClaude]

However, in part d the question says expand it to 3rd order?

Sorry, I was looking at part (c). The solution to (d) is not obvious to me, I'll have to think a bit.

 

[DrClaude]

In post #5, on the last line, you made an error when taking the square root.

To get the equation given in the problem, I had to assume also that $T/T_C \approx 1$, as stated in the problem.