# Magnetic Susceptibility and Curie Temperature

## Homework Statement

Part(a): Derive susceptibility
Part(b): Find field experienced by neighbour.
Part(c): State temperature range. What explains temperature dependence beyond curie temperature? Why is curie temperature so high?
Part(d): In practice, measured magnetic moment is far lower than theoretical. Why? ## The Attempt at a Solution

Part (a)[/B]
Hamiltonian for an electron is given by ##H = g \mu_B \vec B \cdot \vec \sigma##. Thus, partition function is given by
$$Z = e^{-\beta \mu_B B} + e^{\beta \mu_B B}$$
$$m = -\frac{\partial F}{\partial B} = \mu_B tanh(\beta \mu_B B)$$
$$\chi = \frac{\partial M}{\partial H} = \frac{n \mu_0 \mu_B^2}{k_B T}$$

Part(b)
$$H = \approx \frac{m}{4\pi r^3}$$
$$\frac{B}{\mu_0} \approx \frac{e\hbar}{m_e r^3}$$
$$B \approx 0.2 T$$
This gives temperature of about ##0.13 K##.

Part(c)
I suppose this material is a ferromagnet. Therefore, is the temperature range simply ##0 < T < T_C##? I know that curie temperature is defined as the point where material loses its permament magnetization and instead has induced magnetization.
Not sure what they mean by "outline a simple model". Do they simply mean the Ising Model? The paramagnetic susceptibility is calculated to be ##\chi \propto (T-T_C)## in accordance to "Curie-Weiss Law".
Not sure why for some materials curie temperature is so high at ##T_C \approx 1000K##.

Part(d)
I suppose due to non-zero temperature, thermal fluctuations interfere with its permament magnetic moments, as higher temperatures make permament magnets weaker.

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Why is curie temp so high?

Any physical explanation as to why curie temperature in some metals are higher than others?

bump?

curie temperature? anyone?

BruceW
Homework Helper
Hi, sorry slow reply. Yes, if we just consider magnetic field, it would seem the Curie temperature should be much lower. So there must be some other kind of interaction which causes the measured Curie temperature to be much higher. What kind of interaction could this be? hint: you have been using a semi-classical treatment so far.

Also, yeah, I'm not sure what they mean by outline a simple model... Maybe you can just state the model in part a). I guess they are asking for any model which fits the curve above Tc.
Edit: actually, no the model in part a) is not good enough, unless you put in some shift in the temperature... ah I'm not sure about this one.

For part d) I don't think that's the right answer... Presumably, they are talking about a theoretical model which already takes temperature into account. (although they don't specifically mention it). What are some other possible reasons. For example, how might the theoretical Ising model be different from a real-life crystal?

Hi, sorry slow reply. Yes, if we just consider magnetic field, it would seem the Curie temperature should be much lower. So there must be some other kind of interaction which causes the measured Curie temperature to be much higher. What kind of interaction could this be? hint: you have been using a semi-classical treatment so far.

Also, yeah, I'm not sure what they mean by outline a simple model... Maybe you can just state the model in part a). I guess they are asking for any model which fits the curve above Tc.
Edit: actually, no the model in part a) is not good enough, unless you put in some shift in the temperature... ah I'm not sure about this one.

For part d) I don't think that's the right answer... Presumably, they are talking about a theoretical model which already takes temperature into account. (although they don't specifically mention it). What are some other possible reasons. For example, how might the theoretical Ising model be different from a real-life crystal?
Is it because of the presence of "islands" of magnetic domains where each island points in such a way that the overall magnetization is close to zero? BruceW
Homework Helper
that sounds like a good answer. Although, the magnetisation would not be close to zero, it is just less than the theoretical prediction. So, perhaps some of the domains get nudged out of place, and become non-aligned to the majority of domains which are pointing in the same direction. I think this is the typical explanation for how a permanent magnet can lose its magnetisation when you knock it on a hard surface a few times.

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