## Magnetic susceptibility

• Paramagnet Magnetic susceptibility
• assignment Paramagnetism

assignment Homework

##### Paramagnetism
Energy Temperature Paramagnetism 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.
• assignment Extensive Internal Energy

assignment Homework

##### Extensive Internal Energy
Energy and Entropy 2021 (2 years)

Consider a system which has an internal energy $U$ defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where $\alpha$, $\beta$ and $\gamma$ are constants. The internal energy is an extensive quantity. What constraint does this place on the values $\alpha$ and $\beta$ may have?

• assignment Exponential and Logarithm Identities

assignment Homework

##### Exponential and Logarithm Identities
Static Fields 2022 (2 years)

Make sure that you have memorized the following identities and can use them in simple algebra problems: \begin{align} e^{u+v}&=e^u \, e^v\\ \ln{uv}&=\ln{u}+\ln{v}\\ u^v&=e^{v\ln{u}} \end{align}

• 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 Power Series Coefficients 2

assignment Homework

##### Power Series Coefficients 2
Static Fields 2022 (6 years) Use the formula for a Taylor series: $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n$ to find the first three non-zero terms of a series expansion for $f(z)=e^{-kz}$ around $z=3$.
• face Thermal radiation and Planck distribution

face Lecture

120 min.

##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• assignment Series Notation 2

assignment Homework

##### Series Notation 2

Power Series Sequence (E&M)

Static Fields 2022 (6 years)

Write (a good guess for) the following series using sigma $\left(\sum\right)$ notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

1. $1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots$

2. $\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots$

• assignment_ind Normalization of the Gaussian for Wavefunctions

assignment_ind Small White Board Question

5 min.

##### Normalization of the Gaussian for Wavefunctions
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students find a wavefunction that corresponds to a Gaussian probability density.
• group Fourier Transform of a Gaussian

group Small Group Activity

10 min.

##### Fourier Transform of a Gaussian
Periodic Systems 2022

Fourier Transforms and Wave Packets

• assignment Zapping With d 1

assignment Homework

##### Zapping With d 1
Energy and Entropy 2021 (2 years)

Find the differential of each of the following expressions; zap each of the following with $d$:

1. $f=3x-5z^2+2xy$

2. $g=\frac{c^{1/2}b}{a^2}$

3. $h=\sin^2(\omega t)$

4. $j=a^x$

5. $k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)$

• 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 $mB\ll kT$.