Elements of Solid State Physics

J.P. Srivastava

Chapter 14 Ferromagnetism, Antiferromagnetism And Ferrimagnetism - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that the magnon dispersion relation for a ferromagnetic cubic lattice with nearest neighbour interactions is
$$
\hbar \omega=2 J_{e x} S\left[z-\sum_{\delta} \cos (\mathbf{k} \cdot \delta)\right]
$$
where the summation is over the $z$ vectors denoted by $\delta$ that connect the central atom to its nearest neighbours.

Khoobchandra Agrawal

Numerade Educator

05:15

Problem 2

If the effective fields on the two sublattices of an antiferromagnet are written as
$$
\mathbf{H}_{\mathrm{A}}=\left(-\gamma \mathbf{M}_{\mathrm{B}}-\eta \mathbf{M}_{\mathrm{A}}\right) ; \quad \mathbf{H}_{\mathrm{B}}=\left(-\gamma \mathbf{M}_{\mathrm{A}}-\eta \mathbf{M}_{\mathrm{B}}\right)
$$
show that
$$
\frac{\theta}{T_{\mathrm{N}}}=\frac{\gamma+\eta}{\gamma-\eta}
$$

Zachary Warner

Numerade Educator

01:09

Problem 3

Apply the formalism of the two-sublattice model to a ferrimagnetic solid if the magnetic moments on the two sublattices are of differing strengths.

Raj Bala

Numerade Educator

11:58

Problem 4

The Brillouin function $\mathrm{B}_{j}(x)$ is of the form $\left(A x-B x^{3}\right)$ for small $x$, where $A$ and $B$ are positive. Use the mean-field approximation to show that the spontaneous magnetization of a ferromagnet vanishes as $\left(T_{\mathrm{c}}-T\right)^{1 / 2}$ as $T_{\mathrm{c}}$ is approached from below.

Christopher Provencher

Numerade Educator

01:41

Problem 5

Use the mean field theory to derive the following relation for the exchange integral, considering only the nearest neighbour interactions:
$$
J_{e x}=\frac{3 k_{\mathrm{B}} T_{\mathrm{c}}}{2 z S(S+1)}
$$
where $z$ is the number of nearest neighbours.

Narayan Hari

Numerade Educator

03:37

Problem 6

Assuming that iron in its metallic form has a magnetic moment of 2 Bohr magneton per atom, calculate (a) the Curie constant, (b) the Weiss constant, (c) the saturation magnetization, and
(d) the magnitude of the internal field (lattice constant $=2.86 \AA: T_{\mathrm{c}}=1043 \mathrm{~K}$ ).

Ameer Said

Numerade Educator