16 Magnetic Properties of Solids

Mahavir Singh

Learning Outcomes

After studying this module, you shall be able to

• Learn the cause of magnetism in materials.
• Learn the fact that diamagnetic susceptibility is independent of temperature.
• Learn the cause of paramagnetism in materials.
• Learn the fact that why there was a need of quantum theory to explain the paramagnetism.

    1. INTRODUCTION

Magnetism has been a human curiosity for nearly three thousand years and evolved as one of the most interesting and fascinating branch of science and technology. We know that electrons have an intrinsic magnetic moment associated with the spin angular momentum and an orbit magnetic moment associated with the orbit angular momentum. The nucleus also has a small magnetic moment, but it is insignificant compared to that of the electrons and does not affect the overall magnetic properties. In a large number of elements, the magnetic moment of electrons gets canceled in accordance with the Pauli’s exclusion principle. However, in transition metal atoms, the magnetic moments do not get cancelled, therefore, are common examples of magnetic materials. In transition elements, the spin of the electrons is only responsible for the magnetic moment but in case of rare-earth elements, however, the effect of orbital motion of the electrons does not get cancelled and hence both spin and orbital motion contribute to the magnetic moment.

2. DIAMAGNETISM

Diamagnetism can be regarded as originating from shielding currents induced by an applied field in the filled electron shells of ions. Classically, diamagnetism is related to changes in the orbital motion of electrons known as Larmor precession that occurs when atomic systems are placed in a magnetic field. Atoms or ions in which the orbital and spin angular momenta cancel in pairs have J=0 and have no permanent magnetic moment. It may be recalled that the current induced in closed electrical circuit by a magnetic field is always in such a direction so as to keep the total flux unchanged. Thus, the circuit has a negative susceptibility. The effect is retained even in systems of charges that must be treated by quantum mechanics and is responsible for diamagnetism. Since all atoms or ions produced a diamagnetic contribution to the total susceptibility, although it may be marked by the other types; it is a consequence of magnetic moment induced in the atoms by an external field. In this respect diamagnetism may be compared with the electron polarization in an electric field. Both are essentially independent of temperatures. There exists, however, an essential difference; in the electrical case the induced moment lies along the direction of an applied field leading to the positive electrical susceptibility; in the magnetic case the induced moment produce is a negative susceptibility.

3. CLASSICAL THEORY OF DIAMAGNETISM

Let us consider the Bohr model of the atom in which a central nucleus is surrounded by revolving electrons with some frequency, say ωo as shown in Figure 1.

If and external field is applied, the current changes so as to oppose the change in flux. This appears as a change in the frequency of revolution. In the absence of external field, the force acting on the electron will be:

where r is the radius of the orbit, Ze is the nuclear charge and m is mass of the electron.

Figure 1 An atom under the influence of external magnetic field

In the presence of magnetic field, the Lorentz force acting on the electron will be:

The ± sign on means that those electrons whose orbital moments were parallel to the field and those whose moments were antiparallel are speeded up by an amount 2 . This result is called Larmor theorem. This frequency change gives rise to magnetization. The reason for this is that the frequency change is equivalent to an additional current and this current component in every atom is in the same direction, whereas the original circulating currents were in random directions and cancel each other. In the absence of the field, the electron motions produce no net currents or flux. In the presence of field we can write a current for each electron due to its frequency change. This additional current is given by:

It is evident to equation (16) that the diamagnetic susceptibility is independent of temperature and this result is the classical Langevin result.

4. QUANTUM THEORY OF DIAMAGNETISM

In the quantum-mechanical treatment, we have to consider that the electrons are described by wave functions φ, where φ2 at every point is the probability of finding the electron. Alternatively, we may consider the electron as a charge cloud of intensity φ2 at each point in space. It can be shown that the quantum-mechanical result is correctly given by eq. (16), provided we use the expectation value for the squared electron position parameter r.

For a spherical symmetric system and by first order perturbation theory, the second term on the RHS gives a contribution as below

5. PARAMAGNETISM

Paramagnetism occurs in those atoms which have permanent magnetic moments. Paramagnetic materials have positive magnetic susceptibility. The direction of magnetization is parallel to the applied magnetic field. The permanent magnetic moments of atoms result from the following contributions:

(a)   The intrinsic or spin moment of the electrons.

(b)   The orbital motion of the electrons.

(c)   The nuclear magnetic moment.

Paramagnetism is found in:

(i) All atoms having an odd number of electrons. The atoms of groups I, III, V and VII of the periodic table satisfy this condition. But here I want to mention that if the atom forms a diatomic molecule then it will be out of consideration as in a diatomic molecule the two electrons have a chance to pair up.

(ii) Atoms with unfilled inner shells. The rare-earth elements and the transition elements have atoms with unfilled inner shells.

(iii) Free radicals. Some organic compounds have a single unpaired valence electron. These can be made stable under certain circumstances.

(iv) Metals. The electrons in metals behave in many respects as if they are free to move throughout the lattice like molecules in a gas. They tend to pair up but there are always a few unpaired electrons to produce a weak temperature independent paramagnetism known as Pauli paramagnetism.

6. CLASSICAL THEORY OF PARAMAGNETISM

The classical theory of paramagnetism was developed by Paul Langevin. He considered a paramagnetic gas in which each atom or molecule possesses a permanent magnetic moment. He neglected the mutual magnetic interaction between the gas particles. The dipoles tend to align in the direction of magnetic field when an external magnetic field is applied and this alignment is hampered due to thermal agitations. Thus equilibrium is established and the dipoles orient themselves with respect to the applied field as shown in Figure 2.

The probability that a dipole is oriented at an angle θ to the direction of applied field is proportional to Boltzmann factor exp (-μBcos θ/kBT). According to statistical mechanics, the number of molecules (dn) having inclination θ and θ + dθ and which fall in solid angle 2πsinθdθ is given as

??=????(−??????/???)2???????                                                                                            (23)

where C is a constant. Further each particle out of dn contributes a component of magnetic moment μcosθ to the magnetization, while by symmetry the components which are perpendicular to the direction of the field cancel each other.

Figure 2 Magnetic dipole of moment μ oriented at angle θ to B

Therefore, the average component of the magnetic moment of each atom along the direction of the field multiplied by the number of atoms per unit volume, N gives magnetization M as below:

It can be seen from Figure 3 that for large value of x i.e. for high field strength and for low temperature L(x) approaches unity.

7. QUANTUM THEORY OF PARAMAGNETISM

Before starting the quantum theory of paramagnetism it is necessary to highlight an interesting difference between the classical and quantum theory. In quantum theory the permanent magnetic moment of a given atom or ion is restricted to a finite set of orientations relative to the applied field i.e. the orientations are quantized. Let us consider a medium having N atoms per unit volume and having

the total angular momentum quantum number of each atom ⃗. Magnetic dipole moment   ⃗ in free space and ⃗ are related as:

?⃗=−????⃗                                                               (27)

where is called Bohr magneton, g is the Lande’s factor. Its value is 2 is the total angular momentum of the dipole is due to electron spin and 1 if it is due to orbital motion only. It is given by the relation

SUMMARY

In this module, you studied the followings:

• Diamagnetism is associated with the tendency of electrical charge partially to shield the interior of a body from an applied magnetic field.
• For high field strength and for low temperature the Langevin function L(x) approach to the unity.
• In fact classical theory of paramagnetism suggested the method to calculate the susceptibility but it has major drawbacks too. Since in classical theory the mutual interaction between the gas molecules was neglected but it’s a matter of fact that there is always an interaction between mutual atoms or molecules. Thus, by considering this drawback, it was realized to develop a quantum theory for the paramagnetism.

    Value Addition:

Do You Know?

It’s the 600 b.c. from where history of magnetism started but in the 20th century scientists have given much attention to understand the magnetism so as to develop technologies based on magnetism. Many scientific personalities have contributed a lot to make us comfortable to understand the magnetism. Few of them are William Gilbert, Gauss, Oersted, Ampere and Michael Faraday. The credit to laid theoretical foundation to the physics of electromagnetism was taken by James Clerk Maxwell. The modern understanding of magnetic phenomena originates from the work of two Frenchmen: Pierre Curie and Weiss. Curie examined the effect of temperature on magnetic materials and observed that magnetism disappeared suddenly above a certain critical temperature in materials like iron. Weiss proposed a theory of magnetism based on an internal molecular field proportional to the average magnetization that spontaneously align the electronic micromagnets in magnetic matter. The present day understanding of magnetism based on the theory of the motion and interactions of electrons in atoms (called quantum electrodynamics) stems from the work and theoretical models of two Germans, Ernest Ising and Werner Heisenberg.

1.  Suggested Reading

It will be interesting to go through the Hund’s rule and then after have a detailed knowledge of crystal field splitting as well as quenching of the orbital angular momenta.

For More Details (on this topic and other topics discussed in Text Module) See

1.      C. Kittel, Introduction to Solid State Physics, 7th Ed, John Wiley & Sons.

2.      B. D. Cullity & C. D. Graham, Introduction to Magnetic Materials, 2nd Ed, John Wiley & Sons.

3.      K. H. J. Buschow & F. R. de Boer, Physics of Magnetism and Magnetic Materials, Kluwer Academic Publishers.

Glossary:

Magnetization:

Magnetization is defined as the magnetic moment per unit volume.

Magnetic susceptibility:

It is a measure of the ease with which a unit volume of a material can be magnetized by a magnetizing field.