Chapter 13, Diamagnetism And Paramagnetism Video Solutions, Elements of Solid State Physics

Problem 1

The wavefunction of the hydrogen atom in its ground state is
$$
\psi=\frac{1}{\left(\pi a_{0}^{2}\right)^{1 / 2}} \exp \left(-\frac{r}{a_{0}}\right)
$$
Assuming the charge density to be given by
$$
\rho(x, y, z)=e|\psi|^{2}
$$
show that
$$
\left\langle r^{2}\right\rangle=3 a_{0}^{2}
$$
and calculate the molar diamagnetic susceptibility of atomic hydrogen.

Dr. Rajveer Singh

Numerade Educator

02:18

Problem 2

Ampere defined classically the magnetic moment of electron owing to its orbital motion as the average over the orbit of $-e / 2(\mathbf{r} \times \mathbf{v})$.

Prove that our definition, $\mu=\partial E / \partial B_{0}$, reduces to this form by showing from (13.47) that
$$
\mu=-\frac{e}{2 m} \sum_{i} \mathbf{r}_{i} \times\left(\mathbf{p}_{i}-\frac{e}{2} \mathbf{r}_{i} \times \mathbf{B}_{0}\right)
$$
and
$$
\mathbf{v}_{i}=\frac{\partial \mathbf{H}}{\partial \mathbf{p}_{i}}=\frac{1}{m}\left(\mathbf{p}_{i}-\frac{e}{2} \mathbf{r}_{i} \times \mathbf{B}_{0}\right)
$$

Chai Santi

Numerade Educator

02:56

Problem 3

(a) Show that the following formulae summarize the Hund's rules for a shell of angular momentum $l$ containing $n$ electrons:
$$
\begin{aligned}
&S=1 / 2[(2 l+1)-|2 l+1-n|] \\
&L=S[2 l+1-n] \\
&J=|2 l-n| S
\end{aligned}
$$
(b) For a given $L S$ -multiplet, verify that
$$
(2 L+1)(2 S+1)=\sum_{|L-S|}^{|L+S|}(2 J+1)
$$
(c) Find the ground state of
(i) $\mathrm{Eu}^{2+}$ in the configuration $4 \mathrm{f}^{7} 5 \mathrm{~s}^{2} \mathrm{p}^{6}$
(ii) $\mathrm{Dy}^{3+}$ in the configuration $4 \mathrm{f}^{9} 5 \mathrm{~s}^{2} \mathrm{p}^{6}$
(iii) $\mathrm{Tm}^{3+}$ in the configuration $4 \mathrm{f}^{12} 5 \mathrm{~s}^{2} \mathrm{p}^{6}$

Guilherme Barros

Numerade Educator

04:38

Problem 4

A magnetic field $B_{0}$, when applied on an atom with a spherically symmetric charge distribution $\rho(r)$, induces a diamagnetic current. Show that the magnetic field produced by the diamagnetic current at the nucleus is
$$
\Delta B_{0}=-\left(\frac{e B_{0}}{3 m}\right) V_{0}
$$
where $V_{0}$ is the electrostatic potential at the nucleus, given by
$$
V_{0}=\int_{0}^{\infty} \frac{\rho(r)}{r} 4 \pi r^{2} d r
$$

Eduard Sanchez

Numerade Educator

02:07

Problem 5

When a system of electron spins is placed in a magnetic field of $2 \mathrm{~Wb} / \mathrm{m}^{2}$ at a certain temperature, the number of the spins parallel to field lines is twice the number of the spins antiparallel to the field. Determine the temperature.

Kratika Bhadauria

Numerade Educator

03:12

Problem 6

The most important contribution to the paramagnetism of copper sulphate comes from the copper ions which have spin $1 / 2$ and may be treated as non-interacting. Show that the magnetization in field $B_{0}$ is given by $$
M=N \mu_{\mathrm{B}}=\tanh \left(\frac{\mu_{\mathrm{B}} B_{0}}{k_{\mathrm{B}} T}\right)
$$
where $N$ is the number of ions per unit volume.

Suzanne W.

Numerade Educator

01:45

Problem 7

In benzene, the carbon atoms form a regular hexagon of side $1 \AA$. One outer electron from each carbon atom has a wavefunction that extends round the whole ring of atoms (the other three are in $\mathrm{sp}^{2}$ atomic orbitals). Make a rough estimate of the contribution of these electrons to the diamagnetic susceptibility of liquid benzene (density $0.88 \mathrm{~g} \mathrm{~cm}^{-3} ;$ molecular wt. $=76$ ).

Lottie Adams

Numerade Educator

09:59

Problem 8

An ion has a partially-filled shell of angular momentum $J$ with $Z$ electrons in the completely filled shells. Show that the ratio of the Curie law paramagnetic susceptibility to the Larmor diamagnetic susceptibility is
$$
\frac{\chi_{\text {para }}}{\chi_{\text {dia }}}=-\frac{2 J(J+1)}{Z k_{\mathrm{B}} T} \frac{\hbar^{2}}{m\left\langle r^{2}\right\rangle}
$$

Hafiz Shahzaib

Numerade Educator

03:58

Problem 9

A magnetic field is applied to a salt containing $\mathrm{Cu}^{2+}$ ions. $\mathrm{Cu}^{2+}$ has 9 electrons in the $3 \mathrm{~d}$ shell. What magnetic field must be applied to the salt when at $1 \mathrm{~K}$, so that 99 per cent of the ions are in the lowest energy state?

Preeti Kumari

Numerade Educator

03:21

Problem 10

For quantized states we cannot use the Langevin function $\mathrm{L}(x)$ given by $(13.22)$ to compute susceptibilities, but instead, the Brillouin function $\mathrm{B}_{J}(x)$ as given by (13.59) must be used. Show that $\mathrm{B}_{J}(x) \rightarrow \mathrm{L}(x)$ as $J \rightarrow \infty$, with $x=\mu B_{0} / k_{\mathrm{B}} T$.

Hafiz Shahzaib

Numerade Educator

05:02

Problem 11

For temperatures very small compared to the Fermi temperature, show that the temperature dependent correction to the Pauli susceptibility is
$$
\chi(T)=\chi(0)\left(1-\frac{\pi^{2}}{6}\left(k_{\mathrm{B}} T\right)^{2}\left[\left(\frac{D^{\prime}}{D}\right)^{2}-\frac{D^{\prime \prime}}{D}\right]\right)
$$
where $D, D^{\prime}$ and $D^{\prime \prime}$ are the density of levels and its derivatives at the Fermi energy. Show that for electrons it reduces to
$$
\chi(T)=\chi(0)\left(1-\frac{\pi^{2}}{12}\left(\frac{k_{\mathrm{B}} T}{\varepsilon_{\mathrm{F}}}\right)^{2}\right)
$$

Keshav Singh

Numerade Educator

12:23

Problem 12

(a) Show that for a crystal containing $N$ paramagnetic ions with $S=1 / 2$ and $g_{0}=2$, the spin entropy can be expressed by
$$
S=\frac{N \Delta}{2 T} \tanh \left(\frac{\Delta}{2 k_{\mathrm{B}} T}\right)+N k_{\mathrm{B}} \ln (2) \cosh \left(\frac{\Delta}{2 k_{\mathrm{B}} T}\right)
$$
with $\Delta=2 \mu_{\mathrm{B}} B_{0}$
(b) For cooling by adiabatic demagnetization, why is it not possible to start with the nuclear demagnetization, without going through the process of electron-spin demagnetization?

Vishal Sharma

Numerade Educator