Magnetism, Magnons, and Magnetic Resonance

Abstract

The first chapter was devoted to the solid-state medium (i.e. its crystal structure and binding). The next two chapters concerned the two most important types of energy excitations in a solid (the electronic excitations and the phonons). Magnons are another important type of energy excitation and they occur in magnetically ordered solids. However, it is not possible to discuss magnons without laying some groundwork for them by discussing the more elementary parts of magnetic phenomena. Also, there are many magnetic properties that cannot be discussed by using the concept of magnons. In fact, the study of magnetism is probably the first solid-state property that was seriously studied, relating as it does to lodestone and compass needles.

Problems

• 7.1 Calculate the demagnetization factor of a sphere.

• 7.2 In the mean-field approximation in dimensionless units for spin 1/2 ferro- magnets the magnetization (m) is given by

  $$ m = \tanh \left( {\frac{m}{t}} \right), $$

  where t = T/T_c and T_c is the Curie temperature. Show that just below the Curie temperature t < 1,

  $$ m = \sqrt 3 \sqrt {1 - t} . $$

• 7.3 Evaluate the angular momentum L and magnetic moment $\mu$ for a sphere of mass M (mass uniformly distributed through the volume) and charge Q (uniformly distributed over the surface), assuming a radius r and an angular velocity $\omega$. Thereby, obtain the ratio of magnetic moment to angular momentum.

• 7.4 Derive Curie’s law directly from a high-temperature expansion of the partition function. For paramagnets, Curie’s law is

  $$ \chi = \frac{C}{T}\quad\left( {\text{The magnetic susceptibility}} \right), $$

  where Curie’s constant is

  $$ C = \frac{{\mu_{0} Ng^{2} \mu_{B}^{2} j\left( {j + 1} \right)}}{3k}. $$

  N is the number of moments per unit volume, g is Lande’s g factor, $\mu_{B}$ is the Bohr magneton, and j is the angular momentum quantum number.

• 7.5 Prove (7.175).

• 7.6 Prove (7.176).

• 7.7 In one spatial dimension suppose one assumes the Heisenberg Hamiltonian

  $$ \mathcal{H} = - \frac{1}{2}\sum \limits_{{R,R^{\prime}}} J\left( {R - R^{\prime}} \right) \varvec{S}_{R} \cdot \varvec{S}_{{R^{\prime}}} ,\quad J\left( 0 \right) = 0, $$

  where R − R′ = ±a for nearest neighbor and J₁ ≡ J(±a) > 0, J₂ ≡ J(±2a) = −J₁/2 with the rest of the couplings being zero. Show that the stable ground state is helical and find the turn angle. Assume classical spins. For simplicity, assume the spins are confined to the (x, y)-plane.

• 7.8 Show in an antiferromagnetic spin wave that the neighboring spins precess in the same direction and with the same angular velocity but have different amplitudes and phases. Assume a one-dimensional array of spins with nearest-neighbor antiferromagnetic coupling and treat the spins classically.

• 7.9 Show that (7.183) is a consistent transformation in the sense that it obeys a relation like (7.195), but for ${\text{S}}_{j}^{ - }$.

• 7.10 Show that (7.158) can be written as

  $$ \mathcal{H} = - J\sum \limits_{j\delta } \left[ {S_{jz} S_{j + \delta ,z} +\frac{1}{2}\left( {S_{j}^{ - } S_{j + \delta }^{ + } +S_{j}^{ + } S_{j + \delta }^{ + } } \right)} \right] - 2\mu_{0} \mu H\sum \limits_{j} S_{jz} . $$

• 7.11 Using the definitions (7.199), show that

  $$ \begin{array}{*{20}c} {\left[ {b_{\varvec{k}} ,b_{{\varvec{k}^{\prime}}}^{\dag } } \right]} & = & {\delta_{k}^{{k^{\prime}}} ,} \\ {\left[ {b_{\varvec{k}} ,b_{{\varvec{k}^{\prime}}} } \right]} & = & {0,} \\ {\left[ {b_{\varvec{k}}^{\dag } ,b_{{\varvec{k}^{\prime}}}^{\dag } } \right]} & = & {0.} \\ \end{array} $$
• 7.12

  • (a) Apply Hund’s rules to find the ground state of Nd³⁺ (4f³5s²p⁶).

  • (b) Calculate the Lande g-factor for this case.

• 7.13 By use of Hund’s rules, show that the ground state of Ce³⁺ is ²F_5/2, of Pm³⁺ is ⁵I₄, and of Eu³⁺ is ⁷F₀.

• 7.14 Explain what the phrases “3d¹ configuration” and “²D term” mean.

• 7.15 Give a rough order of magnitude estimate of the magnetic coupling energy of two magnetic ions in EuO $(T_{c}\cong 69\,\text{K})$. How large an external magnetic field would have to be applied so that the magnetic coupling energy of a single ion to the external field would be comparable to the exchange coupling energy (the effective magnetic moment of the magnetic Eu²⁺ ions is 7.94 Bohr magnetons)?

• 7.16 Estimate the Curie temperature of EuO if the molecular field were caused by magnetic dipole interactions rather than by exchange interactions.

• 7.17 Prove the Bohr–van Leeuwen theorem that shows the absence of magnetism with purely classical statistics. Hint—look at Chap. 4 of Van Vleck [7.63].

• 7.18 Describe how iron magnetizes.