## Paramagnetism

• Energy Temperature Paramagnetism
• face Entropy and Temperature

face Lecture

120 min.

##### Entropy and Temperature
Thermal and Statistical Physics 2020

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.
• assignment Nucleus in a Magnetic Field

assignment Homework

##### Nucleus in a Magnetic Field
Energy and Entropy 2021 (2 years)

Nuclei of a particular isotope species contained in a crystal have spin $I=1$, and thus, $m = \{+1,0,-1\}$. The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, $E=\varepsilon$, in the state $m=+1$ and the state $m=-1$, compared with an energy $E=0$ in the state $m=0$, i.e. each nucleus can be in one of 3 states, two of which have energy $E=\varepsilon$ and one has energy $E=0$.

1. Find the Helmholtz free energy $F = U-TS$ for a crystal containing $N$ nuclei which do not interact with each other.

2. Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

3. Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

• assignment Centrifuge

assignment Homework

##### Centrifuge
Centrifugal potential Thermal and Statistical Physics 2020 A circular cylinder of radius $R$ rotates about the long axis with angular velocity $\omega$. The cylinder contains an ideal gas of atoms of mass $M$ at temperature $T$. Find an expression for the dependence of the concentration $n(r)$ on the radial distance $r$ from the axis, in terms of $n(0)$ on the axis. Take $\mu$ as for an ideal gas.
• assignment Entropy of mixing

assignment Homework

##### Entropy of mixing
Entropy Equilibrium Sackur-Tetrode Thermal and Statistical Physics 2020

Suppose that a system of $N$ atoms of type $A$ is placed in diffusive contact with a system of $N$ atoms of type $B$ at the same temperature and volume.

1. Show that after diffusive equilibrium is reached the total entropy is increased by $2Nk\ln 2$. The entropy increase $2Nk\ln 2$ is known as the entropy of mixing.

2. If the atoms are identical ($A=B$), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

3. Since the Helmholtz free energy is lower for the mixed $AB$ than for the separated $A$ and $B$, it should be possible to extract work from the mixing process. Construct a process that could extract work as the two gasses are mixed at fixed temperature. You will probably need to use walls that are permeable to one gas but not the other.

Note

This course has not yet covered work, but it was covered in Energy and Entropy, so you may need to stretch your memory to finish part (c).

• face Chemical potential and Gibbs distribution

face Lecture

120 min.

##### Chemical potential and Gibbs distribution
Thermal and Statistical Physics 2020

These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.
• assignment_ind Magnetic Moment \& Stern-Gerlach Experiments

assignment_ind Small White Board Question

30 min.

##### Magnetic Moment & Stern-Gerlach Experiments
Quantum Fundamentals 2022 (2 years)

Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.
• assignment Magnetic susceptibility

assignment Homework

##### Magnetic susceptibility
Paramagnet Magnetic susceptibility Thermal and Statistical Physics 2020

Consider a paramagnet, which is a material with $n$ spins per unit volume each of which may each be either “up” or “down”. The spins have energy $\pm mB$ where $m$ is the magnetic dipole moment of a single spin, and there is no interaction between spins. The magnetization $M$ is defined as the total magnetic moment divided by the total volume. Hint: each individual spin may be treated as a two-state system, which you have already worked with above.

1. Find the Helmholtz free energy of a paramagnetic system (assume $N$ total spins) and show that $\frac{F}{NkT}$ is a function of only the ratio $x\equiv \frac{mB}{kT}$.

2. Use the canonical ensemble (i.e. partition function and probabilities) to find an exact expression for the total magentization $M$ (which is the total dipole moment per unit volume) and the susceptibility \begin{align} \chi\equiv\left(\frac{\partial M}{\partial B}\right)_T \end{align} as a function of temperature and magnetic field for the model system of magnetic moments in a magnetic field. The result for the magnetization is \begin{align} M=nm\tanh\left(\frac{mB}{kT}\right) \end{align} where $n$ is the number of spins per unit volume. The figure shows what this magnetization looks like.

3. Show that the susceptibility is $\chi=\frac{nm^2}{kT}$ in the limit $mB\ll kT$.

• assignment Series Notation 1

assignment Homework

##### Series Notation 1

Power Series Sequence (E&M)

Static Fields 2022 (4 years)

Write out the first four nonzero terms in the series:

1. $\sum\limits_{n=0}^\infty \frac{1}{n!}$

2. $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}$
3. $$\sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}}$$

• assignment Entropy, energy, and enthalpy of van der Waals gas

assignment Homework

##### Entropy, energy, and enthalpy of van der Waals gas
Van der Waals gas Enthalpy Entropy Thermal and Statistical Physics 2020

In this entire problem, keep results to first order in the van der Waals correction terms $a$ and \$b.

1. Show that the entropy of the van der Waals gas is \begin{align} S &= Nk\left\{\ln\left(\frac{n_Q(V-Nb)}{N}\right)+\frac52\right\} \end{align}

2. Show that the energy is \begin{align} U &= \frac32 NkT - \frac{N^2a}{V} \end{align}

3. Show that the enthalpy $H\equiv U+pV$ is \begin{align} H(T,V) &= \frac52NkT + \frac{N^2bkT}{V} - 2\frac{N^2a}{V} \\ H(T,p) &= \frac52NkT + Nbp - \frac{2Nap}{kT} \end{align}

• assignment Magnetic Field and Current

assignment Homework

##### Magnetic Field and Current
Static Fields 2022 (3 years) Consider the magnetic field $\vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases}$
1. Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
2. Find a formula for the current density that creates this magnetic field.
3. Interpret your formula for the current density, i.e. explain briefly in words where the current is.
• Thermal and Statistical Physics 2020 Find the equilibrium value at temperature $T$ of the fractional magnetization $$\frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N}$$ of a system of $N$ spins each of magnetic moment $m$ in a magnetic field $B$. The spin excess is $2s$. The energy of this system is given by \begin{align} U &= -\mu_{tot}B \end{align} where $\mu_{tot}$ is the total magnetization. Take the entropy as the logarithm of the multiplicity $g(N,s)$ as given in (1.35 in the text): $$S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N}$$ for $|s|\ll N$, where $s$ is the spin excess, which is related to the magnetization by $\mu_{tot} = 2sm$. Hint: Show that in this approximation $$S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N},$$ with $S_0=k_B\log g(N,0)$. Further, show that $\frac1{kT} = -\frac{U}{m^2B^2N}$, where $U$ denotes $\langle U\rangle$, the thermal average energy.