In this chapter, we shall study different types of ordered magnetic states that can arise as a result of various kinds of magnetic interactions as discussed in the previous section. In Fig. 5.1 we present some of these possible ground states: ferromagnet, antiferromagnet, spiral and helical structures, and spin-glass. There are other more complicated ground states possible, the discussion of which is beyond the scope of the present book. For detailed information on the various magnetically ordered states in solids, the reader should refer to the excellent textbooks by J. M. D. Coey [1] and S. J. Blundell [4].

Ferromagnetism

In a ferromagnet, there exists a spontaneous magnetization even in the absence of an external or applied magnetic field, and all the magnetic moments tend to point towards a single direction. The latter phenomenon, however, is not necessarily valid strictly in all ferromagnets throughout the sample. This is because of the formation of domains in the ferromagnetic samples. Within the individual domains, the magnetic moments are aligned in the same direction, but the magnetization of each domain may point towards a different direction than its neighbour. We will discuss more on the magnetic domains later on.

The Hamiltoninan for a ferromagnet in an applied magnetic field can be expressed as:

The exchange interaction Jij involving the nearest neighbours is positive, which ensures ferromagnetic alignment. The first term on the right-hand side of Eqn. 5.1 is the Heisenberg exchange energy, and the second term is the Zeeman energy. In the discussion below it is assumed that one is dealing with a system with no orbital angular momentum, so that L = 0 and J= S.

In order to solve the equation it is necessary to make an assumption by defining an effective molecular field at the ithsite by:

Now the total energy associated with ith spin consists of a Zeeman part gμB_Si._B and an exchange part. The total exchange interaction between the ith spin and its neighbours can be expressed as:

The factor 2 in Eqn. 5.3 arises due to double counting. The exchange interaction is essentially replaced by an effective molecular field Bmf produced by the neighbouring spins.

A neutron is a nuclear particle, and it does not exist naturally in free form. Outside the nucleus, it decays into a proton, an electron, and an anti-neutrino. The scattering of low energy neutrons in solids forms the basis of a very powerful experimental technique for studying material properties. A neutron has a mass mn= 1.675 × 10−27 kg, which is close to that of the proton and a lifetime τ = 881.5 ±1.5 s. This lifetime is considerably longer than the time involved in a typical scattering experiment, which is expected to be hardly a fraction of a second.

A neutron has several special characteristics, which makes it an interesting tool for studying magnetic materials as well as engineering materials and biological systems. It is an electrically neutral, spin-1/2 particle that carries a magnetic dipole moment of μ = -1.913 μN, where nuclear magneton μN = eh/mp = 5.051 ×10−27 J/T. The zero charge of neutron implies that its interactions with matter are restricted to the short-ranged nuclear and magnetic interactions. This leads to the following important consequences:

• 1. The interaction probability is small, and hence the neutron can usually penetrate the bulk of a solid material.

• 2. Additionally, a neutron interacts through its magnetic moment with the electronic moments present in a magnetic material strong enough to get scattered measurably but without disturbing the magnetic system drastically. This magnetic neutron scattering has its origin in the interaction of the neutron spin with the unpaired electrons in the sample either through the spin of the electron or through the orbital motion of the electron. Thus, the magnetic scattering of neutrons in a solid can provide the most direct information on the arrangement of magnetic moments in a magnetic solid.

• 3. Energy and wavelength of a neutron matches with electronic, magnetic, and phonon excitations in materials and hence provide direct information on these excitations.

Neutrons behave predominantly as particles in neutron scattering experiments before the scattering events, and as waves when they are scattered. They return to their particle nature when they reach the detectors after the scattering events.

Magneto-optical Effects

The magneto-optical effect arises in general as a result of an interaction of electromagnetic radiation with a material having either spontaneous magnetization or magnetization induced by the presence of an external magnetic field. Michael Faraday in 1846 demonstrated that in the presence of a magnetic field the linear polarization of the light with angular frequency w was rotated after passing through a glass rod. This rotation is now termed as Faraday rotation, and it is proportional to the applied magnetic field B. The angle of rotation θ(w) can be expressed as [1].

Here V(w) is a constant called the Verdet constant, which depends on the material and also on the frequency w of the incident light; |B| is the magnitude of the applied magnetic field, and l thickness of the sample. The Faraday effect is observed in non-magnetic as well as magnetic samples. For example, the Verdet constant of SiO2 crystal is 3.25 × 10−4 (deg/cm Oe) at the frequency w = 18300cm−1 [1]. This implies that a Faraday rotation of only a few degrees can be observed in a sample of thickness 1 cm in a magnetic field of 10 kOe. A much larger Faraday rotation can, however, be observed in the ferromagnetic materials in the visible wavelength region under a magnetic field less than 10 kOe.

In 1877 John Kerr showed that the polarization state of light could be modified by a magnetized metallic iron mirror. This magneto-optical effect in the reflection of light is now known as the magneto-optical Kerr effect (MOKE), and it is proportional to the magnetization M of the light reflecting sample. Today MOKE is a popular and widely used technique to study the magnetic state in ferromagnetic and ferrimagnetic samples. With MOKE it is possible to probe samples to a depth, which is the penetration depth of light. This penetration depth can be about 20 nm in the case of metallic multilayer structures. In comparison to the conventional magnetometers like vibrating sample magnetometer and SQUID magnetometer which measure the bulk magnetization of a sample, MOKE is rather a surface-sensitive technique.

In photography, the intensity of a 3-D object is imaged and recorded in a 2-D recording medium such as a photographic film or a charge-coupled device (CCD) camera, which responds only to light intensity. Since there is no interference during recording, the phase information of the wave field is not preserved. The loss of the phase information of the light field from the object destroys the 3-D characteristics of the recorded scene, and therefore parallax and depth information of the 3-D object cannot be observed by viewing a photograph. Holography is a technique in which the amplitude and phase information of the light field of the object are recorded through interference. The phase is coded in the interference pattern. The recorded interference pattern is a hologram. It is reminiscent of Young’s interference experiment in which the position of the interference fringes depends on the phase difference between the two sources. Once the hologram of a 3-D object has been recorded, we can reconstruct the 3-D image of the object by simply illuminating the hologram or through digital reconstruction. We record the complex amplitude of the 3-D object in coherent holography, whereas in incoherent holography, we record the intensity distribution of the 3-D object. In this chapter, we discuss the principles of coherent holography.

Appendix

Phenomenology

Maxwell equations in free space are expressed as:

The electric field E(r,t) and magnetic field B(r,t) are averaged over elementary volume ΔVcentred around the position r. Similarly ρ and j represent electric charge density and current density, respectively. Equation C.1 indicates the absence of magnetic charge and Eqn. C.2 represents Faraday's law of indication in differential form. These two equations do not depend on the sources of an electric field or magnetic field, and they represent the intrinsic properties of the electromagnetic field. Eqns. C.3 and C.4 contain ρ and j, and they describe the coupling between the electromagnetic field and its sources.

Let us now consider a sample of ferromagnetic material through which no macroscopic conduction currents are flowing. A ferromagnet is characterized by the presence of spontaneous magnetization that can produce a magnetic field outside the sample. The microscopic current density jmicro producing such a magnetic field can be associated with the electronic motion inside the atoms and electron spins, or elementary magnetic moments of the ferromagnetic materials. Such microscopic currents present in an elementary volume ΔVcentred about a position r gives rise to an average current [1]:

j M is termed as magnetization current and represents the current density in Maxwell Eqn. C.4 for a ferromagnetic material. This magnetization current jM does not represent any macroscopic flow of charges across the sample. It can rather be crudely associated with current loops confined to atomic distances. This, in turn, implies that the surface integral jM over any generic cross section Sof this ferromagnetic sample must be zero:

This, in turn, tells that jM(r) can be expressed as the curl of another vector M(r):

Now inserting Eqn. C.6 into Eqn. C.7 and with the help of Stoke's theorem, one can convert Eqn. C.6 into a line itegral along some contour completely outside the ferromagnetic sample:

The Eqn. C.8 will be satisfied under all circumstances provided M (r) = 0 outside the sample. This latter condition is true if we take M as the magnetization or magnetic moment density of the ferromagnetic sample. It can be seen from Eqn. C.7 that the magnetic field created by the ferromagnetic sample is identical to the field that would be created by a current distribution jM(r) = ∇×M(r).

To have some basic understanding of optical coherence, we discuss temporal coherence and spatial coherence quantitatively in the beginning of the Chapter. We then concentrate on spatial coherent image processing, followed by spatially incoherent image processing. While spatial coherent imaging systems lead to the concept of coherent point spread function and coherent transfer function, spatially incoherent imaging system introduces intensity point spread function and optical transfer function. Scanning image processing is also covered in the chapter, illustrating an important aspect in that a mask in front of the photodetector can change the coherence properties of the optical system. Finally, two-pupil synthesis of optical transfer functions is discussed, illustrating bipolar processing in incoherent imaging systems.

It has rightly been said that the mathematical theory of groups and group representations is a magnificent gift of nineteenth century mathematics to twentieth century physics. While this is particularly true within the framework of quantum mechanics, with the passage of time its relevance within classical physics has also become well understood and greatly appreciated. Today the importance of group theoretical ideas and methods for physics can hardly be overemphasised; and over the past century or so, a veritable profusion of books devoted to this theme, many of them gems of the literature, have appeared.

The present monograph is primarily based on lectures given by one of us (NM) at the Institute of Mathematical Sciences in Chennai, India, in the Fall of 2007. The lectures were prepared and presented at the invitation of Rajiah Simon, to whom both authors are indebted for his support and encouragement.

The course was titled ‘Continuous Groups for Physicists’ and consisted of about 45 extended lectures over a two month period. Its aim was to introduce the basic ideas of continuous groups and some of their applications to an audience of post graduate and doctoral students in theoretical physics. After an introduction to the basic ideas of groups and group representations (mainly in the context of finite groups and compact Lie groups), the course presented a selection of useful, interesting and quite sophisticated specific topics not often included in standard courses in physics curricula. The methods and concepts of quantum mechanics served as a backdrop for all the lectures.

The real rotation groups in two and three dimensions are followed by an account of the structures of Lie groups and Lie algebras, and then a description of the compact simple Lie groups. Their irreducible representations are described in some detail. Some of the ‘non standard’ topics that follow are: spinor representations of real orthogonal groups in both even and odd dimensions; the notion of the ‘Schwinger’ representation of a group with examples, induced representations, and systems of generalised coherent states; the properties and uses of the real symplectic groups, which are defined only in real even dimensions, and their metaplectic covering group, in a quantum mechanical setting; and the Wigner Theorem on the representation of symmetry operations in quantum mechanics.

Introduction

The science of physics generally deals with physical phenomena where one quantity (called an independent variable) is related to another quantity (called a dependent variable) through a mathematical equation. The graphical representation of these data is a convenient tool for deciphering this scientific information. It is of utmost importance that the experimental data should be plotted very carefully so that it is easy to appropriately visualize and interpret the relationship between the dependent and independent variables. For example, it is always advisable to

This chapter introduces the reader to various plotting commands invariably used in this book for developing meaningful graphs. The importance of this chapter lies in the fact that it gives an overview on writing small user-defined functions for generating self-explanatory graphs, instead of writing long codes.

The first method is trivial and is left for the reader to explore. In most of the following chapters, graphs and plots have been formatted using small functions that are executed in a script. The major focus of this chapter is to introduce the reader to this kind of formatting tool. However, for completeness, direct command line instructions have also been mentioned wherever possible.

The layout of this chapter is as follows. The Scilab commands ‘plot’ and ‘plot2d,’ have been used in this book for generating graphs. Section 2.2 starts with highlighting the basic difference between these two commands and manipulating them so that they are on equal footing. This section also focuses on writing small functions for editing the coordinate axes.

The study of SO (3) and SU (2) has shown how elements of a continuous group can be labelled by (a certain number of) real independent continuous coordinates or parameters; how the composition law can be expressed using these coordinates; how in a representation we encounter generators, commutation relations, structure constants; the representation of finite group elements by (products of) exponentials in the generators; and so on. Now we will try to understand all this in a more basic manner and in a general situation.

The work of this chapter and the next one will lead us to a vast generalisation of SO (3) and SU (2) resulting in the so-called classical families of continuous groups which are all, like SO (3) and SU (2), compact. (The concept of compactness will be briefly described in a heuristic manner in Chapter 5.) These are mathematical results from the late nineteenth and early twentieth centuries, associated with the names of Killing, Cartan and Weyl and are truly beautiful.

Local Coordinates, Group Composition, Inverses

Let a Lie group G be given. The dimension of G , also called its order, will hereafter be denoted by r rather than n . (In the development of the theory of compact simple Lie groups, the order is traditionally denoted by r ; and another important property called the rank, which we will come to in Chapter 5, by l . These are the notations used, for instance, in the classic 1951 Princeton lectures by Giulio Racah on Group Theory and Spectroscopy.) In some neighborhood N of e ∈ G , we use r essential real independent parameters to label group elements:

It is understood that a , b , · · · ∈ N ⊂ G. As a convention we always assume

As a ∈ G runs over N , α runs over some open set around the origin in r -dimensional Euclidean space. So in this region and inN , coordinates and group elements determine one another uniquely.