Pho.rs: Waves and Phase Transitions in Spin Systems

A1 ^0.30 It is possible to write the ring's magnetic moment $\vec{\mu}$ in terms of its angular momentum $\vec{L}$ as $\vec{\mu}=\gamma \vec{L}$. Find the constant $\gamma$, called the gyromagnetic ratio, of this system in terms of $Q$ and $M$.

1 correct result for $L$ (just magnitude is fine)	0.10
2 correct result for $\mu$ (just magnitude is fine)	0.10
3 correct result for $\gamma$	0.10

A2 ^0.40 Find the angular frequency $\omega_L$ of the angular momentum precession (the so-called Larmor frequency) due to the external magnetic field in terms of $B$ and $\gamma$. Take the positive direction to be counter-clockwise with respect to $+z$.

1 Writing the torque formula	0.10
2 Realizing only the perpendicular component $L\sin \theta$ is changing	0.10
3 Correct magnitude for $\omega_L$	0.10
4 Correct sign for $\omega_L$	0.10

A3 ^0.50 The magnetic interaction energy between the two rings can be written as $U=J_0 \vec{L}_1\cdot \vec{L}_2$, where $J_0$ is a constant and $\vec{L}_i$ is the angular momentum of the $i$th ring. Find $J_0$ in terms of $\gamma, d$ and fundamental constants.

1 Writing the interaction energy correctly	0.10
2 Writing the correct magnetic field magnitude	0.10
3 Writing the correct magnetic field direction	0.10
4 Correct magnitude for $J_0$	0.10
5 Correct sign for $J_0$	0.10

B1 ^0.30 The energy terms containing $\vec{S}_i$ in the sum above can be viewed as the interaction energy between an effective magnetic field $\vec{B}_{i,\text{eff}}$ and the magnetic moment of $\vec{S}_i$. Find $\vec{B}_{i,\text{eff}}$ and express your answer in terms of $J$, the gyromagnetic ratio $\gamma$, and other spins $\vec{S}_j$ (specify the indices $j$ in relation to $i$).

1 Explicit understanding that $i−1$ and $i+ 1$ are the contributors for spin $i$	0.20
2 Correct result for effective magnetic field	0.10

B2 ^0.30 Using the concept of effective magnetic field, express the rate of change of the $i$th spin vector, $d\vec{S}_i/dt$, in terms of $J, \vec{S}_i$, and other spins $\vec{S}_j$ (specify the indices $j$ in relation to $i$).

1 Writing the rate of change of $\vec{S}$ using the effective magnetic field	0.10
2 Correct equation	0.20

B3 ^2.00

Find the relationship between $\omega$ and $k$ (known as the dispersion relation, $\omega(k)$) for the spin waves in terms of $J, S$ and $a$.

Hint: express the position of the $i$th spin as $x=a\cdot i$.

1 Writing traveling waves as function of $kx \pm \omega t$ (either sign is okay, either trig function or complex exponentials is okay)	0.30
2 Same amplitude for $S_x$ and $S_y$	0.20
3 Correct phase relation between $S_x$ and $S_y$ (a difference of $\pi/2$)	0.30
4 Writing the correct explicit equation of motion for either $S_x$ or $S_y$	0.50
5 Explicit use of $S_z \approx S$	0.20
6 Correct final result ($\pm$ is okay)	0.50

B4 ^0.60 For small $k$ ($k\ll1/a$), find the effective mass $m_\text{eff}$ of the spin wave. Express your answer in terms of $J, S, a$ and fundamental constants.

1 Correct Taylor expansion result	0.20
2 Correct relation between momentum and wave vector	0.10
3 Correct relation between energy and angular frequency	0.10
4 Correct identification of effective mass	0.20

B5 ^1.30 Suppose that initially, all the spins in the chain are pointing along the $z$ direction. A neutron with low energy travels on the $x-y$ plane making an incident angle $\theta_{in}$ with the chain and scatters with an angle $\theta_{out}$ as shown in Figure B.3. Assuming the neutron excites a single low wave-vector spin wave, find the effective mass $m_\text{eff}$ of the spin wave, in terms of $\theta_\text{in}, \theta_\text{out}$ and the neutron mass $m_n$. Assume that the chain stays at rest.

1 Conservation of momentum along $y$-axis	0.40
2 Conservation of momentum along $x$-axis	0.30
3 Conservation of energy	0.20
4 Relation between $E_{out}$and $E_{in}$	0.10
5 Correct final answer (should be in either two forms at the end) $$ m_{\text{eff}} = \frac{\sin^2(\theta_{\text{in}} - \theta_\text{out})}{\cos^2 \theta_\text{out} - \cos^2 \theta_\text{in}} m_n = \frac{\sin(\theta_\text{in} - \theta_\text{out})}{\sin(\theta_\text{in} + \theta_\text{out})} m_n $$	0.30
6 Other forms of answer	0.20

C1 ^0.50 First assume that $\tilde{J}=0$. What is the ratio between the probability to find an arbitrary spin aligned to the magnetic field $p_\uparrow$ to being anti-aligned to the magnetic field $p_\downarrow$? Express $p_\uparrow/p_\downarrow$ in terms of $h$, $T$ and fundamental constants.

1 Use of Boltzmann factor	0.20
2 Correct Boltzmann factor for up and down (0.1 each)	2 × 0.10
3 Correct final result for the ratio	0.10

C2 ^1.00 Find the average polarisation of the system $\bar{s}\equiv\frac{1}{N} \sum_i s_i$ for $N\gg 1$ in terms of $h$, $T$ and fundamental constants. If the magnetic field $h$ can range from $-h_0$ to $h_0$, make a sketch of $\bar{s}$ as a function of $h$ for the cases $h_o\gg k_BT$, $h_o\approx k_BT$ and $h_o\ll k_BT$.

1 Deducing $\overline{s} = p_\uparrow - p_\downarrow$	0.20
2 Normalization condition $p_\uparrow + p_\downarrow = 1$	0.10
3 Final result for $\overline{s}$	0.10
4 Correct sketches (0.2 each)	3 × 0.20

C3 ^0.20 What is the energy $E_g$ of the ground state (the lowest energy state)? Express your answer in terms of $\tilde{J}$ and $N$.

1 Realizing the spins minimize their energy by aligning along the same direction	0.10
2 Correct result (both $N −1$ and $N$ are fine)	0.10

C4 ^0.20 Approximate the energy of the system as a sum over all spins
$$E=-\tilde{J}_{\text{eff}} \sum_i s_i$$and express $\tilde{J}_{\text{eff}}$ in terms of $\tilde{J}$ and $\bar{s}$.

1 Realizing $s_{i+1}$ can be replaced with $\overline{s}$	0.10
2 Correct final result	0.10

C5 ^1.20 Using your result from C.2, find an equation that the average polarisation $\bar{s}$ must satisfy. The number of solutions to this equation depends on $T$. Find the critical temperature $T_c$ at which the number of solutions changes. Express your answer in terms of $\tilde{J}$ and fundamental constants.

1 stating $\overline{s} = \tanh (\tilde{J}_\text{eff}/k_BT)$	0.10
2 Replacing $J_\text{eff}$ for $h$ in $\overline{s}$ results from C.2	0.20
3 Realizing that there is one trivial solution for $\tilde{J} \ll k_B T$	0.20
4 Realizing that there are two non-trivial solutions for $\tilde{J} \gg k_B T$	0.20
5 Clearly stating the condition at when number of solutions changes	0.30
6 Correct final result for $T_c$	0.20

C6 ^1.00 Find all possible values of $\bar{s}$ when $T<T_c$ and $T_c-T\ll T_c$. Express your answers in terms of $T$ and $T_c$. Sketch all possible values of $\bar{s}$ for temperatures $T$ in the range $0\leq T\leq 2 T_c$.

1 Using the proper approximation for tanh	0.10
2 Correct non-trivial solutions for $\overline{s}$, even if not fully simplified (0.1 each)	2 × 0.10
3 Sketch $\overline{s}$ as the only solution above $T_c$	0.10
4 The non trivial solutions are vertical at $T_c$	0.20
5 $\overline{s} = 0$ is sketched as a solution for $T < T_c$	0.10
6 The two non-trivial solutions monotonically increase to $\pm 1$ at $T = 0$ (0.1 each)	2 × 0.10
7 Either non-trivial solutions have a zero slope approaching $T = 0$.	0.10

C7 ^0.20 What magnetic phase of matter does $T>T_c$ correspond to? How about when $T<T_c$? Choose between paramagnetic or ferromagnetic.

1 Correct choice for $T > T_c$ (paramagnetic)	0.10
2 Correct choice for $T < T_c$ (ferromagnetic)	0.10

Waves and Phase Transitions in Spin Systems

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