17 Magnetic Properties of Solids 1

Mahavir Singh

Learning Outcomes

After studying this module, you shall be able to

• Learn the cause of ferromagnetism in materials.
• Learn about the spontaneous magnetization in ferromagnetic materials and its cause.
• Learn about antiferromagnetism

    1. HUND RULES

The Hund rules as applied to electrons in a given shell of an atom affirm that electrons will occupy orbitals in such a way that the ground state is characterized by the following:

(a)    The maximum value of the total spin S allowed by the exclusion principle.

(b)    The maximum value of the orbital angular momentum L consistent with this value of S.

(c)     The value of the total angular momentum J is equal to | − | when the shell is less than half filled and to L+S when the shell is more than half filled. When the shell is just half full, the application of the first rule gives L + 0, so that J = S.

2. CRYSTAL FIELD SPLITTING

The difference in behaviour of the rare earth and the iron group salts is that the 4f shell responsible for the paramagnetism in the rare earth ions lies deep inside the ions, within the 5s and 5p shells, whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell. The 3d shell experiences the intense inhomogeneous electric field produced by neighbouring ions. This inhomogeneous electric field is called the crystal field. The interaction of the paramagnetic ions with the crystal filed has two major effects: the coupling of L and S vectors is largely broken up, so that the states are no longer specified by their J values; further, the 2L+1 sublevels belonging to a given L which are degenerate in the free ion may now be split by the crystal field.

3. INTRODUCTION

Ferromagnetic materials exhibit parallel alignment of magnetic moments to one another resulting in a large net magnetization (known as spontaneous magnetization) even in the absence of magnetic field. The atomic dipole moments in ferromagnetic materials is characterized by very strong exchange coupling caused by exchange fields which results in a parallel alignment of atomic moments. Ferromagnetism is strongly temperature dependent and the magnetization of a ferromagnetic material is inversely proportional to temperature in accordance with Curie-Weiss law. Therefore, ferromagnetic materials have their highest magnetization at 0 K and the magnetization decreases up to the Curie temperature, where it disappears.

4. WEISS THEORY OF FERROMAGNETISM

Ferromagnetism is the existence of a spontaneous magnetization, even in the absence of an external field. Weiss suggested that magnetic moments are ordered not only by an external magnetic field B but also by the cumulative action of all magnetic moments i.e. elementary moments interact with one another. This field is named as exchange field and is treated as equivalent to a magnetic field BE. This field is proportional to the magnetization, M and hence

?_?=??                                         (1)

where λ is a constant and is independent of temperature. The effective magnetic field Beff on an atomic dipole is therefore

Since M must satisfy both equations (3) and equation (6), its value at a given temperature can be obtained from the point of intersection of the two corresponding M versus α as shown in Fig. 1.

Figure 1 Graphical solution for spontaneous magnetization at a temperature

where μ is the total magnetic moment per atom. Equation (8) is known as Curie-Weiss law.

It can be observed from Fig. 1 that M has only non-vanishing value when T<TF and for T≥TF the curve does not intersect the lines i.e. spontaneous magnetization vanishes.

5. HEISENBERG MODEL FOR THE WEISS FIELD

In 1928, Heisenberg had shown that the cause of internal field may be because of the exchange interaction between the atomic spins. Therefore, the exchange energy of a given atom and its neighbors can be written as

?=−2?_?Σ??.??                                                            (9)

Stoner assumed that the instantaneous values of the neighboring spins should be replaced by their time averages.

?=−2??_??⃗.〈?⃗〉                                                        (10)

where 〈?⃗〉 is the average spins of adjoining neighbor atoms. If we assume that the magnetization M is along z-axis then

6. ANTI FERROMAGNETISM

In anti ferromagnetism the spin moments of the neighboring atoms are aligned antiparallel to each other hence completely cancel each other. In this case the exchange integral Je is negative. In these materials there is no net magnetization however, when a field is applied a small magnetization appears in the direction of the field. This magnetization further increases with the temperature and becomes maximum at a critical temperature known as the Neel temperature. Above Neel temperature, the magnetization goes on decreasing which indicates the paramagnetic state.

Figure 3 Alignment of atomic moments in a antiferromagnetic material

A crystal exhibiting antiferromagnetism may be considered as made up of two sublattices, so that the spin moments of atoms in one sublattice array have an opposite sense from the spin moments of atoms in the other sublattice array. If A and B are two sublattice arrays therefore effective field at sites A and B will be:

?_?=?−??_?−??_?                                     (14)

?_?=?−??_?−??_?                                     (15)

It is assumed that beside an antiferromagnetic AB interaction, there are also AA and BB antiferromagnetic interactions. α and β are Weiss constants and MA and MB is the magnetization at A and B lattices such that total magnetization is the sum of MA and MB. When the temperature is above than Neel temperature then at this temperature paramagnetic state will exist therefore magnetization of the two sublattice is given by as below

Equation (19) is the analogous of Curie-Weiss law. In the absence of magnetic field (B = 0) and when T= TN, then

Hence it is evident that Neel temperature increases as the AB interactions (β) become stronger, whereas it decreases with increasing AA and BB interactions (α).

For temperature below than Neel temperature if we consider only A-B interactions then as a result of anisotropy there may be more than one direction along which the spins will tend to align themselves. Therefore there are two cases of interest (1) when magnetic field is applied perpendicular to the natural spin direction and (2) when magnetic field is applied parallel to the natural spin direction. The susceptibility in the case of perpendicular is independent of temperature and is equal to the susceptibility at Neel temperature when approached from the high temperature region. When T = TN, then the susceptibility in case of parallel is equal to the susceptibility in case of perpendicular.

SUMMARY

In this module, you studied

• About the Hund rules
• Crystal field splitting
• About the ferromagnetism
• That ferromagnetic materials have spontaneous magnetization
• That what is the cause of spontaneous magnetization
• That in antiferromagnetic materials the spin moments of the neighboring atoms are aligned antiparallel to each other hence completely cancel each other.
• That Neel temperature increases as the AB interactions (β) become stronger, whereas it decreases with increasing AA and BB interactions (α).