23 results in Experimental Techniques in Magnetism and Magnetic Materials
II - Basic Phenomenology of Magnetism
- Sindhunil Barman Roy
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Summary
The magnetic properties in solids originate mainly from the magnetic moments associated with electrons. The nuclei in solids also carry a magnetic moment. That, however, varies from isotope to isotope of an element. The nuclear magnetic moment is zero for a nucleus with even numbers of protons and neutrons in its ground state. The nuclei can have a non-zero magnetic moment if there are odd numbers of either or both neutrons and protons. However, the magnetic moment of a nucleus is three orders of magnitude less than that of the electron.
The microscopic theory of magnetism is based on the quantum mechanics of electronic angular momentum, which has two distinct sources: orbital motion and the intrinsic property of electron spin [1]. The spin and orbital motion of electrons are coupled by the spin–orbit interaction. The magnetism observed in various materials can be fundamentally different depending on whether the electrons are free to move within the material (such as conduction electrons in metals) or are localized on the ion cores. In a magnetic field, bound electrons undergo Larmor precession, whereas free electrons follow cyclotron orbits. The free-electron model is usually a starting point for the discussion of magnetism in metals. This leads to temperature-independent Pauli paramagnetism and Landau diamagnetism. This is the case with noble metals and alkali metals. On the other hand, localized non-interacting electrons in 3d-transition metals, 4f-rare earth elements, 5f-actinide elements, and their alloys and intermetallic compounds with incompletely filled inner shells exhibit Curie paramagnetism. Many transition metal-based insulating oxide and sulfide compounds also show Curie paramagnetism. In the presence of magnetic interactions, many such systems eventually develop long-range magnetic order if the magnetic interaction can overcome thermal fluctuations in some temperature regimes.
Against the above backdrop, in the next three chapters, we will introduce the readers to the basic phenomenology of magnetism, concentrating mainly on solid materials with some electrons localized on the ion cores. There are some excellent textbooks available on the subject, including those by J. M. D. Coey [1], B. D. Cullity and C. D. Graham [2], D. Jiles [3], S. J. Blundell [4], and N. W. Ashcroft and N. D. Mermin [5].
5 - Magnetically Ordered States in Solids
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Summary
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.
9 - Neutron Scattering
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Summary
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.
8 - Optical Methods
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Summary
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.
Contents
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Appendix C - Demagnetization Field and Demagnetization Factor
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Summary
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).
12 - Nano-Scale Magnetometry with Nitrogen Vacancy Centre
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Summary
We have studied in earlier chapters that spin-based techniques like neutron scattering, muon spin resonance spectroscopy, and nuclear magnetic resonance can give detailed information on the magnetic structure of a material down to the atomic scale. These techniques, however, cannot provide a real-space image of the magnetic structure and are not sensitive to samples having nanometre-scale volumes. On the other hand, techniques like magnetic force microscopy, scanning hall bars, and superconducting quantum interference devices (SQUIDs) enable real-space imaging of the magnetic fields in nanometre-scale samples. But they have a constraint of finite size, and also they act as perturbative probes working in a rather narrow temperature range. A relatively new technique of magnetometry based on the electron spin associated with the nitrogen-vacancy (NV) defect in diamond combines the powerful aspects of both these classes of experiments. A very impressive combination of capabilities has been demonstrated with NV magnetometry, which sets it apart from other magnetic sensing techniques. That includes room-temperature single-electron and nuclear spin sensitivity, spatial resolution on the nanometre scale, operation under a broad range of temperatures from ∽1 K to above room temperature, and magnetic fields ranging from zero to a few tesla, and most importantly it involves a non-perturbative operation [1]. Here we present a concise introduction to NV magnetometry. There are a few excellent review articles [2, 3] and tutorial article [4] on NV magnetometry, and readers are referred to those for a more detailed exposure to the subject.
Physics of the Nitrogen-Vacancy (NV) Centre in Diamond
Figure 12.1 presents an NV centre in the crystal lattice of a diamond. It is a point defect consisting of a substitutional nitrogen atom and a missing carbon atom in the neighborhood. The NV centres can have negative (NV−), positive (NV+), and neutral (NV0) charge states of which NV− is used for magnetometry [2]. An NV− centre has six electrons, five of which come from the dangling bonds of the three neighbouring carbon atoms and the nitrogen atom. The negative charge state arises from one extra electron captured from an electron donor. The NV axis is defined by the line connecting the nitrogen atom and the vacancy. There can be four NV alignments depending on the four possible positions of the nitrogen atom with respect to the vacancy.
11 - Microscopic Magnetic Imaging Techniques
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Summary
Magnetic imaging techniques enable one to have a direct view of magnetic properties on a microscopic scale. One of the most well-known magnetic microstructures is the magnetic domain. The other example of magnetic microstructures is the nucleation and growth of a magnetic phase across a first-order magnetic phase transition. Such structures can be observed in real space, and their distribution as a function of material and geometric properties can be investigated in a straightforward manner. In this chapter, we will discuss three different classes of magnetic imaging techniques, namely (i) electron-optical methods, (ii) imaging with scanning probes, and (iii) imaging with X-rays from synchrotron radiation sources. There are numerous scientific papers and review articles on these subjects. Instead of going into detail about the individual techniques, this chapter will provide a general overview of the working principles of various magnetic imaging techniques. There are not many specialist books, monographs, or review articles covering all these magnetic imaging techniques under the same cover, but the present author has found the book Modern Techniques for Characterizing Magnetic Materials [1] and the article “Magnetic Imaging” [2] to be quite useful while writing this chapter.
Electron-Optical Methods
Electron-optical methods and electron microscopy encompass a large body of techniques for magnetic imaging. The advanced electron microscopy techniques today can provide images with very impressive resolutions of the order of 1 nm, and show high contrast and sensitivity to detect small changes in magnetization in a material. The particle-like classical picture of the Lorentz force acting on an electron in a magnetic field form the basis of magnetic images of materials observed in various modes of electron microscopy. Electrons are charged particles, and hence electromagnetic fields are utilized as lenses for electrons. A magnetic lens consists of copper wire coils with an iron bore. The magnetic field generated by this assembly acts as a convex lens, which can bring the off-axis electron beam back to focus. Change in the trajectory of an electron in the magnetic field of a magnetic sample results in magnetic contrast and, in turn, provides information on the local magnetization in the material. However, the correct interpretations of the results in many cases involve a wave-like quantum mechanical picture of electrons.
Index
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3 - Magnetic Moment and the Effect of Crystal Environment
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Magnetic Moment and Magnetization
When we examine a magnetic material, it is first essential to identify parameters that characterize the response of the magnetic material to an applied magnetic field. We will see that these parameters are magnetic moment and magnetization.
Magnetic Moment
All of us at some point in our lives have come across magnets and experienced the strange forces of attraction and repulsion between them. These magnetic forces appear to originate in regions called poles, which are located near the ends of, say, a bar magnet. In magnets, poles always occur in pairs, but it is impossible to separate them. A magnetic field is created by a magnetic pole, which pervades the region around the pole [2]. This magnetic field causes a force on a second pole nearby. This magnetic force is directly proportional to the product of the pole strength p and field strength or field intensity H, which can be verified experimentally:
This equation defines H if the proportionality constant k is put equal to 1. A magnetic field of unit strength causes a force of 1 dyne on a unit pole [2]. In CGS units, a field of unit strength has an intensity of 1 oersted (Oe).
Let us now consider a bar magnet with poles of strength p located near each end and separated by a distance l, which is placed at an angle θ to a uniform field B = μ0H(Fig. 3.1). The magnet will experience a torque, which will tend to turn the magnet parallel to the magnetic field. The moment μof this torque is expressed as [2]:
When H= 1 Oe and θ = 900 , the moment is given by μ = pl. The magnetic moment of the magnet is defined as the moment of the torque experienced by the magnet when it is at right angles to a uniform field of 1 Oe. In a non-uniform magnetic field, the magnet will also feel a translational force acting on it. We will see in the subsequent sections that magnetic moment is a fundamental quantity for magnetism in materials.
7 - Magnetic Resonance and Relaxation
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Summary
Electromagnetic (EM) radiation consists of coupled electric and magnetic fields oscillating in directions perpendicular to each other and the direction of propagation of radiation. EM radiation can be an interesting probe to study materials’ properties. It is the electric field component of EM radiation that interacts with molecules and solids in most cases. Two conditions need to be fulfilled for the absorption of EM radiation during such interaction: (i) the energy of a quantum of EM radiation must be equal to the separation between energy levels in the atom/molecule, (ii) the oscillating electric field component must be able to stimulate an oscillating electric dipole in the atom/molecule. EM radiation in the microwave region of the EM spectrum can interact with molecules having a permanent electric dipole moment created by molecular rotation. On the other hand, infrared radiation would interact with molecules in vibrational modes giving rise to a change in the electric dipole moment.
Similarly, a solid or molecule containing magnetic dipoles is expected to interact with the magnetic component of EM radiation. EM irradiation of a molecule over a wide range of spectral frequencies does not normally result in absorption attributable to magnetic interaction. The absorption of EM radiation attributable to magnetic dipole transitions may, however, occur at one or more characteristic frequencies if the material of interest is additionally subjected to a static magnetic field. The application of a magnetic field B can cause precession of a magnetic moment at an angular frequency of |γB|, where γ is the gyromagnetic ratio. A material with magnetic moments placed in a magnetic field can absorb energy at this frequency. It is thus possible to observe a resonant absorption of energy from an EM wave tuned to an appropriate frequency. This phenomenon is known as “magnetic resonance”, and can be studied with a number of different experimental techniques, depending upon the type of magnetic moment involved in the resonance.
The presence of a static magnetic field is a crucial requirement for magnetic dipolar transitions. If there is no static magnetic field, the energy levels will be coincident. The permanent magnetic moments in a material are associated either with electrons or with nuclei. The magnetic dipoles arise from net electronic or nuclear angular momentum.
10 - X-ray Scattering
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Summary
We have seen in the previous chapter that thermal neutrons with their unique combination of wavelength (≈˚A), magnetic moment (≈ μB), and penetration power in materials can be a very sensitive probe to investigate microscopic magnetic properties. Elastic neutron scattering experiments provide information on the magnitudes and direction of the magnetic moment for complex spin arrangements. Inelastic neutron scattering experiments provide information on spin dynamics such as spin waves or magnons, spin fluctuations, crystal-field excitations, etc.
It had been known for quite some time that X-rays being part of the electromagnetic spectrum can have specific interactions with ordered arrangements of spins in magnetic materials. The absorption cross section and the scattering amplitudes of electromagnetic waves are related through the optical theorem [1], hence magnetic effects are also expected to be observed in X-ray scattering. However, that magnetic interaction is very weak, leading to only very small effects, and as a result for many years, X-rays were not considered as a useful probe of magnetism. In the 1970s–80s in a series of elegant pioneering experiments de Bergevin and Brunel [2, 3] and Gibbs and his collaborators [4, 5] demonstrated that X-ray diffraction effects from magnetic materials could be detected experimentally. These important developments along with the relatively easy access to modern synchrotron radiation sources with the capability of probing small samples with a highly collimated and intense X-ray beam, high degree of photon beam polarization, and energy tunability have made magnetic X-ray diffraction studies a complementary and in some cases competitive technique to neutron scattering studies in the study of magnetic materials [6, 7, 8, 9]. These magnetic X-ray diffraction techniques will be discussed below, and the narrative will closely follow the articles by C. Vettier and L. Paolisini [6, 7].
Magnetic and Resonant X-ray Diffraction
In a purely nonrelativistic limit, X-rays interact only with the electronic charges in solids, hence no information is obtained on the spin densities. The first calculation of the amplitude of X-rays elastically scattered by a magnetically ordered solid was carried out by Platzman and Tzoar [10] in 1970 within the framework of the relativistic quantum theory. That calculation predicted that magnetic X-ray scattering could be observed. Within two years De Bergevin and Brunel [2] reported the first experimental observation of such effects in a single-crystal sample of antiferromagnetic NiO.
Appendix A - Magnetic Fields and Their Generation
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A magnetic field is called a steady field if the timescale of the increasing (or decreasing) magnetic field during an experiment is a few minutes or a few tens of minutes. On the other hand, if the magnetic field changes in a very short time, like milliseconds, it is called a pulsed-field [1]. Pulsed-fields with a duration of several hundred milliseconds up to several seconds are sometimes called long pulse fields [1].
Steady Field
Electromagnets are used if the magnetic field requirement is below 2 T. This is because of the saturation of iron cores used in conventional electromagnets. In 1933 the American physicist Francis Bitter introduced a design of an electromagnet to produce a higher magnetic field. This is known as the Bitter magnet where the current flows in a helical path through circular conducting metal plates stacked in a helical configuration with insulating spacers [2]. The schematic of a Bitter magnet is shown in Fig. A.1. The stacked metal plate design helps to withstand the Lorentz force due to the magnetic field acting on the moving electric charges in the plate. The Lorentz force causes an enormous outward mechanical pressure. The other important point in the design of Bitter magnets is about how to remove the huge amount of heat generated in this resistive magnet. This is achieved by flowing water along the axial direction through the holes incorporated in the stacked plates [1]. Bitter magnets are still used today to produce fields of up to 33 T at the National High Magnetic Field Laboratory (NHMFL) at Florida State University, Tallahassee, USA. Fig. A.2 shows a Florida Bitter disk from Tallahassee with highly elongated, staggered cooling holes. In more recent times a magnetic field of 37.5 T is produced at room temperature by a Bitter electromagnet at the High Field Magnet Laboratory in Nijmegen, Netherlands.
Superconducting magnets are more popular nowadays in laboratories worldwide to produce fields up to 20 T. A magnetic field of 7 T using Nb3Sn-based superconducting magnet was first generated in 1961 by Kunzler et al. [3], and the superconducting magnets that produced over 10 T became commercially available in the 1970s [1].
6 - Conventional Magnetometry
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Summary
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].
III - Experimental Techniques in Magnetism
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Summary
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.
4 - Exchange Interactions and Magnetism in Solids
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Summary
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.
Appendix B - Units in Magnetism
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Summary
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.
1 - A Short History of Magnetism and Magnetic Materials
- Sindhunil Barman Roy
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- Book:
- Experimental Techniques in Magnetism and Magnetic Materials
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- 27 October 2022
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- 05 January 2023, pp 3-12
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Summary
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.
2 - Role of Magnetism and Magnetic Materials in Modern Society
- Sindhunil Barman Roy
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- Book:
- Experimental Techniques in Magnetism and Magnetic Materials
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- 27 October 2022
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- 05 January 2023, pp 13-18
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Summary
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.
Preface
- Sindhunil Barman Roy
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- Book:
- Experimental Techniques in Magnetism and Magnetic Materials
- Published online:
- 27 October 2022
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- 05 January 2023, pp xiii-xvi
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Magnetism and magnetic materials have been subjects of considerable interest for more than 3000 years, going back to ancient times. In modern times, myriad are the applications of magnetism and magnetic materials ranging from generation of electrical power to communications and information storage. Magnetic materials are absolutely indispensable in modern technology, and the intensity and importance of their applications are reflected in the multi-billion dollar market for magnetic materials in three broad areas: permanent magnets, soft magnets, and the magnetic recording medium. Continuous evolution in the field of magnetic materials has not, however, remained confined only to these well-identified areas. It is now well recognized that fresh applications are possible through the coupling of magnetism with other physical properties of materials such as magneto-thermal, magneto-elastic, magneto-optic, and magneto-electrical couplings. Newer classes of magnetic materials are being discovered with interesting new functions stimulating further growth of newer technology in various areas including information technology, wireless communication, microelectronics, biotechnology, and such like.
Against this backdrop, it is natural that the subject of magnetism finds a prominent place in solid state/condensed matter physics textbooks taught in the advanced undergraduate and postgraduate courses at universities all over the world. However, this coverage is mostly confined to a basic understanding of the phenomenon of magnetism as one of the physical properties of solid materials within the general framework of quantum mechanics. Detailed theoretical exposition of the subject is left to the specialized books on magnetism, and there are not too many of such books. Unlike high energy and particle physics, magnetism with its huge potential in technological applications is mostly an experimental science, where experimental techniques and related instruments play a very crucial role. The quantum many-body theories of magnetism (and for that matter condensed matter physics in general) are continuously evolving to explain the classes of emerging phenomena being discovered through experimental work on magnetic materials (and other classes of condensed matter). It is experimentation which is leading the field in the case of condensed matter/magnetism rather than theory as in the case of high energy and particle physics.
In the area of high energy and particle physics, the students are at least aware of the necessary theoretical techniques and the relevant experimental methods before they enter the research field at the Ph.D. level.