Question

The energy of a magnetic moment $\vec{\mu}$ in a magnetic field $\vec{B}$ is $-\vec{\mu}\cdot \vec{B}$ . A certain paramagnetic salt contains 10²⁵ magnetic moments per m³. Each one has a value

, due to the atom's spin.

As the spin is 1/2, there only are two possible states and the magnetic moments can be parallel or antiparallel to the field. Each magnetic moment belongs to one distinguishable atom. A 1 cm³ sample of this salt is placed in a electromagnet producing a uniform magnetic field of 1T in the z direction.

(a) Analyze this system directly with the Maxwell-Boltzmann statistics. Show that the ratio between the parallel $n_{\uparrow}$ and antiparallel $n_{\downarrow}$ populations and the sample's magnetization, defined as

$M\equiv \mu(n_{\uparrow}-n_{\downarrow})$ , are respectively

$\frac{n_{\uparrow}}{n_{\downarrow}}=e^{2\beta\mu B}$ and $M = M_s \frac{n_{\uparrow}-n_{\downarrow}}{N}$

, where M_s is the saturation magnetisation, where all the moments point towards the same direction. What is the magnetisation at 300K? 4K?

(b) Now, let's analyze the system with the micro canonical ensemble. Calculate the partition function in the micro canonical ensemble, and show that the system's internal energy is given by

What are the asymptotic values for $T\rightarrow 0$ and for T>>0? Compare the results with the part (a) results.

Show that in this ensemble, we have

$n_i=-Nk_BT(\frac{\partial}{\partial{\epsilon}_i}\ln Z)_T$

Show that, as in (a), the magnetization is given by

$M=N\mu\tanh(\frac{\mu B}{k_BT})$

Plot a graph of the magnetization as a function of B.

μ 9.274 × 10-J /T 24 kBT

Answer 1

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