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Magnetism

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Published in: Physics
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Magnetic Properties of Solid

Aliva C / Kolkata

4 years of teaching experience

Qualification: B.Tech/B.E. (Heritage Institute of Technology - 2015)

Teaches: All Subjects, English, Mathematics, Science, Biology, Chemistry, Physics

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  1. Magnetic Properties of Materials Introduction: Three important facts with respect to magnetic materials which require explanations are . (i) Some materials are magnetic even without the application of any magnetic field and become more magnetic when a weak magnetic field is applied to them. (ii) Many other materials lose their initial strong magnetism when heated above a certain critical temperature and completely weakly magnetized. (iii) Some materials show magnetic response in a direction opposite to that of any externally applied field. The magnetic effects in magnetic materials are due to atomic magnetic dipoles in the materials. These dipoles result from the effective current loops of electrons in atomic orbits, from effects of electron spin and from the magnetic moments of atomic nuclei. 1. What are the characteristics of magnetic materials? (i) Magnetic substances like iron, cobalt, nickel are attracted by a magnet. (ii) Like poles of magnets repel each other and unlike poles attract each other. (iii) Magnetic monopole does not exist. They always exist in pairs. 2. What is magnetic permeability? In an unmagnified bar of a magnetic materials is placed in a uniform magnetic field, it gets magnetized by induction and gets a polarity. s s The lines of magnetized bar opposes the lines of original field outside the magnet and favour inside the magnet. As a result, the magnetic field strength (H) is increased inside the bar and decreased outside it. Similarly the magnetic flux density (B) is directly proportional to the magnetic field strength (H), i.e. where, is a constant of proportionality and is unknown as absolute permeability of the medium. If flux density is established in air or vacuum or in a non magnetic material, then the above equation may be written as, = VOH. The ratio — is known as relative permeability of the medium and is expressed by gr. 11 B The relative permeability of air or nonmagnetic material is unity, but for certain nickel-iron alloys, its value is as high as æ105
  2. 3. What is magnetization? The term magnetization may be defined as the process of converting a non magnetic bar into magnetic bar. The flux density, Again, 4. What is magnetic susceptibility? It is defined as the ratio of the intensity of magnetization (M) at any point inside a material to the magnetic field (H). i.e. Xm Definition: The property which determines how easily and up to what extent a specimen can be magnetized is referred to as the magnetic susceptibility of the material. The material is supposed to be made of large number of tiny dipoles when the substance is placed in a magnetic field (H), the tiny dipoles of the magnetic substance start orienting along the direction of applied field and thus contribute to the intensity of magnetization (M). 5. Relation between Magnetic susceptibility and permeability? We have the relation Xm Again, Combining the above two equations, B = go(H + xmii) = + xm)H = + xm)H 11 r = 1 -k Xm. 5. (a) Starting from the Ampere's law, show that, for linear media B = pH. Ampere's law is given by, j. dl = 1101 (Integral form) f] (V x F). da = If j. da (Applying Stokes' theorem) •V x B = (Differential form). The bound current within a material is given by, = V x M; where, M is the magnetization of the material . The field produced due to the magnetization of the material is just the field produced by this bound current. The free current UT) might flow through the wires embedded in the magnetized substance or if the substance is a conductor itself. The total current is written as,
  3. j=jf+Jb . = go(jf +1b) = go(jf + V x M) 1 110 1 Now, the intensity of magnetic field is defined as, 1 B = go(ii + M) = go(Fi + xFi) = + 6. What do you mean be magnetic dipole and dipole moment? s Magnetic dipole: A bar magnet of finite separation between the two poles of it is called a magnetic dipole. Let m be the pole strength and 21 be the separation between poles of a bar magnet, then dipole moment is = 21m. 11m Here, is a vector directed from the north to the South Pole. 7. Discuss the atomic theory of magnetic dipole moment and define Bohr Magnetron. Let's consider the simplest model of hydrogen atom in which one electron moves the proton, say on a circular loop of radius r. The moving electron is equivalent to a current loop. If v is the velocity of the electron, it V will make — — revolutions per second and it is equivalent to a current, I 2 ltr 2 It The circular current loop is equivalent to a magnetic dipole and the magnetic moment due to this orbiting electron is 2 eor = IA = 2 The angular momentum associated with the electronic motion is given by mor2. Hence, we can relate the magnetic dipole moment and the angular momentum as, k) e x angular momentum. The —ve sign indicates that the dipole moment points in a direction opposite to a vector representing the vector angular momentum. A substance therefore possesses permanent magnetic dipoles if the electron of its constituent atom has a net non-vanishing angular momentum. The ratio of the magnetic dipole moment of the electron due to its orbital motion and the angular momentum of the orbital motion of the electron represented by y. Magnetic moment 2 — 2m' Angular momentum mor This (y) is known as orbital gyromatic ratio. According to the quantum theory, the angular momentum of an electron in the orbit is determined by the orbital quantum number I which is restricted to the set of values
  4. (n — 1). n = Principal quantum number. The angular momentum of electrons associated with a particular value of I is given by I Therefore, the permanent dipole moment is given by, —11B l. 2m 27T 4mn = is an atomic unit called Bohr magnetron and has a value of eh The quantity 4Ttm 2m 9.27 x 10 24amp/m2 , and this represents the magnetic moment of elementary permanent dipole. 8. Quantum theory of magnetism: The magnetic behaviour of atoms, molecules, solids etc. is actually related to the orbital and spin motion of electrons. The motion of the electrons are govern by three well known quantum numbers, namely- (i) Principal quantum number (n) with values 1,2,3,4 • (ii) Orbital quantum number (l) with values 0,1,2,3, (n — 1) and magnetic quantum number (ml) with values - l, —l + 1, —l + 2 I — 1, l. Here, n determines the energy of an electron, and I gives the magnitude of the angular momentum vector L. Orientation of L is determined by mt. The magnitude of the electron's orbital angular momentum, L is quantized and the value is given by, The orbitals are designated by s, p, d, f for I = 0, 1, 2, 3 etc. The directions of L is also quantized with respect to an external field. This fact is often referred to as space quantization. If the magnetic field direction be along the Z axis, the component of L in this direction is Lz = mih. The possible values of ml for a given for a given value of I range from - I through 0 to +1, in a magnetic field is (21 + 1). If 0 is the angle between L and the magnetic field direction, which may be taken as Z direction, then, cos9 = mih -2 1
  5. Space quantization of orbital angular momentum An electron also possesses spin angular momentum represented by vector S, magnitude is quantized in terms of h, such that, whose 1 With s = — is the spin quantum number. Thus, S 2 2 The space quantization of electron spin is described by the spin magnetic quantum number ms which can take values +- and — The component Sz of the spin angular momentum of an electron along a magnetic field in Z direction is determined by spin magnetic quantum number, so that, 1 Sz = msh = + _ h. The gyromatic ratio, characteristic of the electron spin is almost twice that of electron orbital motion. Taking the ration equal to 2, the spin magnetic moment of an electron is related to its spin angular momentum S by, eh usz 2m is the Bohr magnetron. The orientation of spin space with respect to Z axis The constant — (magnetic field direction) are determined by the relation cosp ms h s scs -k 1) 1 2 1 2
  6. current loop is, For electrons, Two possible orientations of spin angular momentum 9. Theory of magnetism in electrons: The electrons of an atom move round the nucleus in some specific orbits. The motion of the electrons in an orbit can be considered as small current loops. The magnetic moment of a ML = (Ttr2) 2 ML = iA = inr 2 2m V 2 evr 2 The angular momentum of the electron is, — mr — mvr. . rv Putting this equation in the above equation, Since quantum mechanically, L eh The negative sign indicates that ML and L are in opposite directions. The quantity — is a 2m constant and known as Bohr Magnetron (VB). Similarly, magnetic moment corresponding to the spin of the electron is, The net angular momentum of the atom corresponding to L and S is, The effective magnetic moment is obtained by summing the components ML and Ms along J. The resultant magnetic moment is M = + gscosp. From the figure, - 2JLcosO 2JL
  7. Similarly, In terms of quantum numbers, cosp 2JL 11B L2 + J 2 — 2 — 11B] 1 -k + 2S. + s2 + 12 2 2j(j -k 1) Where, 2j(j -k 1) and g is called the Lande g factor. 10. Based on the magnetic properties, how the materials are classified? Write their differences. Materials are classified into three basic categories, (i) Diamagnetic material, (ii) Paramagnetic material, (iii) Ferromagnetic material; besides there are two more categories (iv) Anti-ferromagnetic material and (v) Ferrimagnetic material. Diamagnetic (i) Weakly repelled by the magnetic field. (ii) Magnetic susceptibility does not depend on temperature. (iii) Magnetic susceptibility is negative. Paramagnetic (i) Weakly attracted by the magnetic field. (ii) Magnetic susceptibility depends on the temperature. (iii) Magnetic susceptibility is positive. Ferromagnetic (i) Strongly attracted by the magnetic field. (ii) Magnetic susceptibility depends on the temperature. (iii) Magnetic susceptibility is positive. 11. Write down a note on classical theory of diamagnetism. Consider a circular orbit of radius r in which an electron revolves with an angular velocity (Do around the nucleus of charge Ze. Then, 2 ze2 2 47T€or ze2 2 47T€or ze2 3 47T€omr
  8. 4TT€omr The orbital magnetic moment of the electron is eoor I-IL iA = If an external magnetic field is applied to the atom, then the Lorentz force acting on the electron is, Bev = —Beor. Now the equation of motion is, 47T€or 47T€omr 2m 2m Therefore the change in frequency of the electron is 2m -9 27TAV — 2m 471m Therefore, the corresponding change in the magnetic moment of the electron is, enr2 (eB) Al-IL = AiA = —eAvnr 471m e2r2B On summing over all electrons in the atom, the induced moment per atom becomes, e2B AUL Where, the summation extends over all the orbital electrons (z) in the atom, since the core electrons have different radii, we may write, — z(r2)• Where, I (r 2) IE is the average radius of the orbitals. If the orbitals lie in xy plane, then If (ro ) represents the average distance of the electrons from the nucleus, When the atom has spherical symmetry, 02) = (y2) (r 2) = 2(x2) and (rd) - 3(x2); ...
  9. Observations: (i) X is independent of temperature. (ii) X is negative. 12. Discuss on classical theory of paramagnetism. For the analysis of variation of the susceptibility of a paramagnetic spin system with temperature, the following assumption have been made. (i) Only the effect electron spin magnetic moment is considered. (ii) Individual spin moment can accept only two possible components along the field direction +1 or -1 Bohr magnetron, i.e. spin moments in the presence in the presence of field are either parallel or anti-parallel with the field. (iii) The interaction between individual spin system is negligible, so that, the field at any point in the spin system is the applied field. On the basis of equipartition of energy, the number of molecules whose potential E is proportional to e - F, where k is the Boltzmann constant and T is the absolute temperature. Let N be the number of molecules in unit volume. If dN is the number of molecules having inclinations 0 and 0 + d0 and energy value E, then dN oc e-ÆT sin0d9 The torque on the dipole is , T = mB(21sin0) = gmBsin0 = 2ml is the dipole moment of the dipole. Where, Therefore, the potential energy of the dipole is given by, Therefore, e2Bz 2 AVIL If N be the number of atoms per unit volume, then the magnetization is given by, gmBsin9d0 = —gmBcos0 M goHze2N(r02) 6m goze2N(r02) 6m 9 900 Therefore, then Let a - kBT' Td0 = 9 900 gmBCOS9 e kBT sin0d0, dlV = ce acos9 sin9d0 Therefore, the total number of molecules (N) having inclinations between 0 to TT is given by, acos9 )sin0d9. 0 Let, x = cosO, dx = —sin9d9,
  10. — C (e ax alV Therefore, the resolved components of all the dipole moments inclined at an angle 0 will be, dM = dNgmcos0. Hence the total dipole moment due to all N molecules lying between 0 to 7T contained in unit volume gives intensity of magnetization, acos9 Let x dM = = —cos0d9. . M = cum dNgmcos9 = [xe ax gmcos9Ce CIO. xe x dx e x dx where, L(a) ea—e— - um NL(a), — is known as Langevin's function. In the above equation, um N represents the dipole moment per unit volume when all the spins are turned into the direction of the applied magnetic field and hence it gives the saturation value of M say Ms . This Ms is a constant depending upon the nature of the gas. Thus, — L(a) coth(a) = becomes large, that is, at very low temperature the function reaches a When a ( saturation corresponding to a maximum alignment of the dipoles along the field direction, i.e. at very high filed and low temperature. so, at large a, L(a) * 1, and so, M = = um N. The second characteristic of the curve is important for many practical purposes. That is for high temperature or small filed,
  11. 2 a 1 a Neglecting the higher powers of a. 2 a -1 3 a 3! 2 a 2 2a2 6a(1 -k 2 a 3 a The initial part of the curve (i.e. for small values of a) is sensibly linear and coincides with 1 the tangent of the curve at the origin whose slope is - 3kBT- 3kBT 3kBT Therefore, paramagnetic susceptibility is given by, H - 3kBT 3kB ) 11m Where, C is a constant C = This is known as Curie's law. 13. Discuss Curie-Weiss theory of ferromagnetism. Weiss introduce the concept of intermolecular field in order to explain the complicated type of dependence of susceptibility. In a real gas, the molecules are mutually in influenced by their magnetic moments and consequently, molecular field is developed. The field produced at any point by the neighbouring molecules is proportional to the intensity of magnetization and acting along it. Let this intermolecular field be Hi, Now, Hi = 1M, Where, 1 is the molecular field coefficient. Therefore the net effective filed should be, = H + Hi. From Lagevin's theory we know, M x c c c +1M). MT = + 1M) CH Therefore, the susceptibility is given by, M x c c This is known as Curie-Weiss law, where Octhe ferromagnetic Curie temperature is. 14. What is hysteresis? Draw hysteresis loop for a ferromagnetic material and define coercively, retentively and saturation magnetization. 15. What is soft and hard magnets? 16. Draw — versus T [Where, X is the susceptibility of the material] for paramagnetic, ferromagnetic and anti-ferromagnetic materials on the same graph.
  12. 17. Discuss about Anti-ferroma netism, ferrima netism and ferrites. Ferromagnetism 1. All the spins are aligned parallel to each other and the individual spin magnetic moment has the same magnitude. tttttt 2. The ferromagnetic materials have spontaneous magnetization. 3. Negative susceptibility, independent of temperature. Ex: Fe, co, Ni 18. What are ferrites? Anti-ferromagnetism 1. All the spins are aligned anti-parallel to each other and the individual spin magnetic moment has the same magnitude. 2. The anti-ferromagnetic materials have zero spontaneous magnetization. 3. Positive susceptibility, x Ex. Mn Ferrimagnetism 1. All the spins are aligned anti-parallel to each other and the individual spin magnetic moment has has different magnitude. 2. The ferromagnetic materials have nonzero spontaneous magnetization. 3. 3. Positive susceptibility, c x (T - ocY Ex: Strontium ferrite, Barium ferrite, Cobalt ferrite etc. A ferrite is a type of ceramic compound composed of iron oxide (Fe203) combined chemically with one or more additional metallic elements. They are both electrically nonconductive and ferrimagnetic, meaning they can be magnetized or attracted to a magnet. Ferrites can be divided into two families based on their magnetic coercivity, their resistance to being demagnetized. Hard ferrites have high coercivity, hence they are difficult to demagnetize. They are used to make magnets, for devices such as refrigerator magnets, loudspeakers and small electric motors. Soft ferrites have low coercivity. They are used in the electronics industry to make ferrite cores for inductors and transformers and in various microwave components. Applications: Strontium ferrite, SrFe12019 (SrO• 6Fe203), used in small electric motors, micro- wave devices, recording media, magneto-optic media, telecommunication and electronic industry. Barium ferrite, BaFe12019 (Bao 6Fe203), a common material for permanent magnet applications. Barium ferrites are robust ceramics that are generally stable to moisture and corrosion-resistant. They are used in e.g. loudspeaker magnets and as a medium for magnetic recording, e.g. on magnetic stripe cards. Cobalt ferrite, CoFe204 (CoO•Fe203), used in some media for magnetic recording.

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