# Statistical mechanics problem about a paramagnet

• Clara Chung

#### 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.

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?...

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?

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...

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
$$\tanh(x) \approx x - \frac{1}{3} x^3$$

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
$$\tanh(x) \approx x - \frac{1}{3} x^3$$
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