In Chapter 1 symmetric group Sn ; the group of permutations of n objects, was introduced as an example of a finite group. An interesting feature of the symmetric group is that every finite group of order n is isomorphic to some subgroup of Sn . This group plays an important role in very many branches of physics and mathematics and has been investigated in great detail right from the early days of group theory. Here we confine ourselves to discussing some known facts and results concerning Sn mainly to illustrate the concepts introduced in Chapter 1. The proofs are quite intricate, and involve ingenious algebraic arguments which could occupy a whole volume. Even to see these results described can be illuminating.

Cycle Structure Notation

In Chapter 1 we learnt two useful notations for an element p ∈ Sn :

While the first form is convenient for stating the composition rule for Sn , the rule for computing inverses and such, as in (1.4), the second, referred to as the cycle structure notation, proves to be extremely useful in delivering a great deal of information about the symmetric group as we shall see. To go from the first form to the second, start from 1 and list the sequence of numbers it visits under repeated applications of p until one is back to the starting point, namely 1. This yields the cycle of 1. Next pick a number j from 1, 2, · · · , n which has not already appeared in the first cycle and construct its cycle. Then pick a number which has not appeared in the first two cycles and so on. Repeat this process until all numbers are exhausted. Thus, for instance, the element

of S6 reads, in the second form, as

Conversely one can go from the second form to the first by viewing (ijk · · · l ) as a set of replacements: i → j , j → k , · · · , l → i.

In the pattern of fundamental UIR's for the four classical families, we found an unusual feature for the real orthogonal groups Dl = SO (2l) and Bl = SO (2l + 1). The fundamental UIR's are not all obtainable from the defining ‘vector’ representation D by tensorial constructions. At the level of elementary UIR's this is seen even more clearly. For Al = SU (l + 1) and Cl = USp (2l) , we have only one elementary UIR in each case, namely D itself. But for both Bl and Dl , the elementary UIR's consist of the ‘vector’ UIR D , and either one or two spinor UIR's (These are actually double valued UIR's of the respective groups SO (2l + 1), SO (2l) .) We study them briefly in this chapter.

Cartan had found the spinor UIR's by around 1913. Independently, Dirac found them in 1928 for the Lorentz group, and then they entered physics. Of course even earlier spinors for SU (2) and SO (3) were used in Pauli's description of spin in the non relativistic framework. In 1935 Brauer andWeyl [Brauer andWeyl, 1935] gave an elegant account of Cartan's fundamental spinor UIR's for Bl and Dl using the Dirac approach, i.e., via the algebra of γ matrices which we develop below.

Spinor UIR's forDl =SO (2l)

We see from the discussion in Section 5.3 that the defining vector representation D of SO (2l) of dimension 2l , has the following highest weight and other weights making

All these weights are simple. It is clear that in any UIR formed out of tensors over D , all weights present will be (positive or negative) integer linear combinations of the ea.

Magnetization and magnetic susceptibility are the physical quantities that describe the response of a magnetic material to the application of an external magnetic field. Magnetic susceptibility χ is defined as the ratio between magnetization M and the intensity of the externally applied magnetic field B ,

These magnetic properties can be studied by determining the force on a magnetized sample of the material under study, or the magnetic induction or a perturbation of the field in the neighbourhood of the sample.

There is a generalized approach for the magnetic measurements techniques [1]. In an externally applied magnetic field, the material under study produces a force (F), or magnetic flux (ϕ), or an indirect signal (I). These phenomena are usually sensed by a detector, which results in output usually in the form of an electrical signal, either DC or AC. These techniques are in general termed magnetometry. The heart of a magnetometer is the detector, and it defines the principle involved in a particular type of magnetometer. A magnetic sample placed in a uniform magnetic field affects the magnetic flux distribution. This change can be sensed by a flux detector, which is usually in the form of a coil but can also be detected through a variety of sensors. On the other hand, a sample placed in a non-uniform magnetic field experiences a force, which can be detected by a force transducer. Such experimental techniques involving force or flux detections are classified as direct techniques, and they measure macroscopic or bulk magnetic properties of the material. These experimental techniques assume only the validity of Maxwell's electromagnetic equations and the thermodynamic equilibrium of the sample. The magnetic properties can also be measured by various indirect techniques, namely the Hall effect, magneto-optical Kerr/Faraday effects, NMR, FMR, M¨ossbauer, Neutron Scattering, μSR, and others, which take advantage of known relationships between the phenomenon detected and the microscopic magnetic properties of the specimen. In this chapter, we will focus on the direct techniques, and the indirect techniques will be the subject of later chapters in the book. It may be mentioned here that each magnetic measurement technique has distinct merits as well as limitations, and no single technique is universally applicable to study all types of magnetic phenomenon [1].

In this part of the book we will study various experimental techniques currently employed in the study of magnetism and magnetic materials. We shall start with a very popular experimental technique called magnetometry in Chapter 6. This technique is used to study macroscopic or bulk magntic properties of materials in research laboratories all over the world. This is a versatile technique to investigate magnetic properties over a wide range of sample environments that differ from each other in terms of temperature, pressure and applied magnetic field. Magnetometry is usually the first experiment to perform for the general magnetic characterization of materials, which apart from providing a plethora of useful information enables one to identify and focus on the particular temperaure, pressure and magnetic field regime of interest for deeper investigation using more specialized experimental techniques.

Electromagnetic (EM) radiation is an interesting probe to study magnetic properties of materials. A material containing magnetic dipoles is expected to interact with the magnetic component of EM radiation. If the material of interest is subjected to a static magnetic field, it is possible to observe a resonant absorption of energy from an EM wave tuned to an appropriate frequency. This phenomenon is known as magnetic resonance. Depending upon the type of magnetic moment involved in the resonance, a number of different experimental techniques can be employed to study magnetic resonance. In Chapter 7 we shall discuss such experimental techniques as nuclear magnetic resonance (NMR), electron paramagnetic resonance (EPR), ferromagnetic resonance (FMR), and Mössbauer spectroscopy, along with muon spin rotation (μSR).

In Chapter 8 we will study experimental techniques employing interaction of EM waves with magnetic material without any resonant absorption. Magneto-optical effects can arise due to an interaction of electromagnetic radiation with magnetic material having either spontaneous magnetization or magnetization induced by the presence of an external magnetic field. Experiments based on the magneto-optical Kerr effect (MOKE) and scanning near-field optical microscopy (SNOM) will be discussed. In addition we will discuss Brillouin light scattering (BLS) or Brillouin spectroscopy, which is based on the phenomenon of inelastic scattering of light. Brillouin spectroscopy provides insights to spin waves in ferromagnetic material without exciting them explicitly.

We have introduced Gaussian optics and used a matrix formalism to describe light rays through optical systems in Chapter 1. Light rays are based on the particle nature of light. Since light has a dual nature, light is waves as well. In 1924, de Broglie formulated the de Broglie hypothesis, which relates wavelength and momentum. In this chapter, we explore the wave nature of light, which accounts for wave effects such as interference and diffraction.

The first interaction between magnetic moments, which is expected to play a role in magnetism, is of course the interaction between two magnetic dipoles μ1 and μ2 separated by a distance r. The energy of this system can be expressed as:

The order of magnitude of the effect of dipolar interaction for two moments each of μ ≈ 1μB separated by a distance of r ≈ 1 ˚A can be estimated to be approximate μ2 /4πr3 ∽10−23 J, which is equivalent to about 1 K in temperature. This dipolar interaction is too weak to explain the magnetic ordering observed in many materials at much higher temperatures, even around 1000 K.

Coupling between Spins

Before we look for suitable interactions between two magnetic moments to explain the magnetic ordering observed in various materials, we shall first discuss the coupling of two spins. We now consider two interacting spin- 1/2 particles represented by a Hamiltonian:

Here S and S represent the spin operators of the two particles. Combining the two particles as a single entity, the total spin operator can be expressed as:

This leads to:

A combination of two spin-1/2 particles gives rise to a single entity with quantum number s = 0 or 1. This leads to the eigenvalue of (STotal)2 as s(s + 1), which is 0 for s = 0 and 2 for s = 1. Now the eigenvalues of both (S)2 and (S)2 are 3/4 [4]. Hence, from Eqn. 4.4 we can write:

The system has two energy levels for s = 1 and 0 with energies as follows:

Each state will have a degeneracy given by (2s + 1). The s = 0 state is a singlet and the z-component of the spin ms of this state takes the value 0. On the other hand, s = 1 state is a triplet and ms takes one of the three values -1, 0, and 1.

The eigenstates of this two interacting spin-1/2 particles can be represented as linear combinations of the following basis states: |↑↑〉, |↑↓〉, |↓↑〉and |↓↓〉, where the first (second) arrow corresponds to the z-component of the spin labelled by a(b). The possible eigenstates are presented in Table 4.1.

In 1960 the Systeme International d’Unites (SI) was recommended as the modern version of the metric system, which replaced the old CGS – centimetre, gram, and second – system with seven fundamental or “base” units: the metre, kilogram, second, ampere, kelvin, mole, and candela. Other “derived” units are constructed from these base units. Since then in almost every area of science and engineering, the SI units have been widely used unambiguously.

In the field of magnetism, however, a kind of confusing mixture of SI (in various versions) and CGS units are still being used sometimes. For example, “magnetic field” can mean “B-field” or “H-field”. The SI units for these fields are tesla (T) or amperes per metre (Am−1), whereas in CGS those are gauss (G) and oersted (Oe), all of which are currently in use. Another source of confusion arises due to the different expressions proposed for the magnetic induction B in a polarizable medium by Arthur Kennelly (1936) and Arnold Sommerfeld (1948) [1]. The Kennelly system is traditionally followed by electrical engineers, where B is expressed as:

Here μ0 is the permeability of free space, H is the H field, J is the magnetic polarization, and B0 is the induction of free space that would remain in the absense of the medium. In the Sommerfeld convention, which has been adopted by the International Union of Pure and Applied Physics (IUPAP):

Here M is the magnetization per unit volume. These equations are not in conflict once it is recognized that magnetization and magnetic polarization are different quantities. It is possible to use either the B-field or the H-field when one is dealing with the magnetic field. They can be distinguished by their behaviour at a boundary between media having different relative permeabilities (μ). Across the boundary of the media the normal component of the B-field and the tangential component of the H-field will be continuous. In both the Kennelly and Sommerfeld systems B = μ0Hin the free space. In SI units μ0 = 4π × 10−7 henreys per metre (Hm−1), hence B-field and H-field have different numerical values. On the other hand, in the CGS system μ0 = 1, hence B-field or the H-fields have identical numerical values.

So far, we have assumed that if there are several communities in a network, then those communities are distinct and non-overlapping. In this chapter, we discuss situations in which communities overlap with each other. We describe a number of algorithms for modeling overlapping communities, such as mixed-membership SBMs, link-based clustering, overlapping SBMs, the community-affiliation graph model, and the latent cluster random-effects model.

The magnetite iron ore FeO - Fe2O3 (or Fe3O4), famously known as lodestone, is the first known natural magnet. Folklore is that roughly around 2500 BC a Greek shepherd was tending his sheep in a region of ancient Greece called Magnesia (now in modern Turkey), and the nails that held his shoe together were stuck to the rock he was standing on. There were more such ancient stories about iron parts being pulled out from hulls of the ships sailing past the islands in the south Pacific and ones about the disarming and immobilizing of knights in their iron armor. Depending on the time and places where Fe3O4 or magnetite ore was found, it was variously known as the Magnesia stone, lodestone, the stone of Lydia, l’aimant in France, chumbak in India, or ts’u she in China. The modern name magnet is possibly derived from early lodestones found in the ancient Greek region of Magnesia.

The chronicled history of magnetism dates back to 600 BC. Lodestone's magnetic properties were studied and documented by the famous Greek philosopher Thales of Miletus (Fig. 1.1) in 600 BC [1]. Around the same period, the magnetic properties of lodestone were known in India, and the well-known ancient physician sage Sushruta (see Fig. 1.1) applied it to draw out metal splinters from bodies of injured soldiers [2]. However, Chinese writings dating back to 4000 BC mention magnetite, and indicate the possibility that original discoveries of magnetism might have taken place in China [3]. The Chinese were the first to notice that lodestone would orient itself to point north if not hindered by gravity and friction. The early Chinese compasses, however, were used in fortune-telling through the interpretation of lines and geographic alignments as symbols of the divine. These were also used for creating harmony in a room or building with the alignment of various features to different compass points. The first navigational lodestone compasses emerged from China. They had a unique design with the lodestone being shaped as a ladle (Fig. 1.2). The lodestone ladle sat in the center of a bronze or copper plate/disc. These compasses rotated freely when pushed and usually came to rest with the handle part of the ladle pointing south and so were known as south pointers. The copper/bronze base would be inscribed with cardinal direction points and other important symbols.

Introduction

Matrices are often aptly described as key to solve everything in the scientific world. This chapter expounds the usefulness of vectors and matrices that occur in many kinds of problems across the disciplines. They are used to study innumerable physical phenomena such as, motion of rigid bodies, eigen states of a quantum mechanical system, electrical networks and coordinate system conversion.

In Scilab, matrix computation forms the basis of all calculations. This chapter recapitulates the basic Scilab rules that have to be followed for creating and editing matrices. It also summarizes the arithmetic operations that can be performed on matrices. Section 1.2 gives an overview on different ways of generating a matrix and its elements. Some special types of matrices such as row/column vector, diagonal matrix, identity matrix and triangular matrices have been introduced in Section 1.3. Matrix operations such as row/column operation, conjugation, scalar/vector multiplication and division have been explained in Section 1.4. The laws of vector algebra have been outlined in Section 1.5. Some interesting examples of applications and use of matrices in physical sciences have been discussed in Section 1.6.

Creation of a Matrix

Matrices are rectangular arrangements of ‘m’ rows and ‘n’ columns; an arrangement of m rows and n columns is called an (m × n) matrix. If it contains only one row or only one column, then it is called a vector. There are several ways of defining vectors and matrices in Scilab. Some of them have been explained as follows.

• 1. The elements of a matrix are defined by writing them inside a square bracket, such that the elements of a row are separated by a comma or a white space. The elements of consecutive rows are separated by a semi-colon.

• 2. The elements of a matrix can be of several types and have been listed in Table 1.1. As can be seen in this table,

  • a. The elements can be real numbers.

  • b. The elements can be complex numbers. The complex number consists of a real part and/or an imaginary part.

  • c. The elements can be rational numbers, which are defined using the ‘rlist’ command of Scilab.

The aim of this chapter is to provide readers with an introduction to the basic ideas of networks and their representation by graphs. We will be using ideas, definitions, terminology, and notation from graph theory throughout this book.

Magnets have been used in society for centuries. In ancient times they were considered paranormal or mysterious substances. Nobody knew how or why the magnets attracted certain but not all materials. As we have seen in Chapter 1, it was not until the seventeenth century that there was considerable understanding of electromagnetism and a progressive increase in the use of magnetic materials as useful functional materials. Nowadays magnets are all-pervasive in modern society, starting from home to medical applications, to transport, and industrial sectors (Fig. 2.1).

We utilize magnetism all over our homes although it may not be very obvious. Electric motors create force by using electricity and magnetic fields. So nearly all household appliances such as fans, washing machines, vacuum cleaners, and blenders that use electricity to create motion invariably have magnets. Many households have small magnets holding paper notes or small items to the metal refrigerator door. While some magnets are visible, the others are hidden inside various items and appliances such as computers, cellphones, DVDs, iPods, cameras, sensors, doorbells, and toys of children. The dark stripe on the backside of credit cards is a magnetic strip storing the relevant data of the cardholder.

Computers use hard disk drives to store information. Hard disks are memory devices where magnets alter the direction of magnetic material on disk segments. Information is processed in computers in binary language, the base-2 units of which correspond to a magnetic field aligned to either the north or the south. These fields are spun in a hard disk, and a magnetic sensor is used to read these. Inside the small speaker found in computers, televisions and radios, electrical signals are converted into sound vibrations by wire coil and magnet.

Magnets are used profusely in the industrial world. Mechanical energy is converted into electricity with the use of magnets in electric generators. On the other hand, motors use magnets to convert electricity back into mechanical work. Sorting machines using magnets are deployed in mines to separate useful metallic ores from crushed rock. In the food processing industry magnets are used for removing small metallic particles from grains and other food.