Magnetic susceptibility

  • Paramagnet Magnetic susceptibility
    • assignment Paramagnetism

      assignment Homework

      Energy Temperature Paramagnetism Thermal and Statistical Physics 2020 Find the equilibrium value at temperature \(T\) of the fractional magnetization \begin{equation} \frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N} \end{equation} 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): \begin{equation} S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N} \end{equation} 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 \begin{equation} S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N}, \end{equation} 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

      Planck distribution blackbody radiation photon statistical mechanics

      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.

    Plot of magnetization vs. B field

    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 \(mB\ll kT\).

  • Media & Figures
    • figures/
    • figures/paramagnetic-susceptibility-magnetization.svg