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A Question What is Magnetism ? ESM Cluj 2015 Basic Concepts: Magnetostatics J. M. D. Coey School of Physics and CRANN, Trinity College Dublin Ireland. 1. Introduction 2. Magnets 3. Fields 4. Forces and energy 5. Units and dimensions Comments and corrections please: [email protected] www.tcd.ie/Physics/Magnetism This series of three lectures covers basic concepts in magnetism; Firstly magnetic moment, magnetization and the two magnetic fields are presented. Internal and external fields are distinguished. The main characteristics of ferromagnetic materials are briefly introduced. Magnetic energy and forces are discussed. SI units are explained, and dimensions are given for magnetic, electrical and other physical properties. Then the electronic origin of paramagnetism of non-interacting electrons is calculated in the localized and delocalized limits. The multi-electron atom is analysed, and the influence of the local crystalline environment on its paramagnetism is explained. Assumed is an elementary knowledge of solid state physics, electromagnetism and quantum mechanics. Books Some useful books include: • J. M. D. Coey; Magnetism and Magnetic Magnetic Materials. Cambridge University Press (2010) 614 pp An up to date, comprehensive general text on magnetism. Indispensable! • Stephen Blundell Magnetism in Condensed Matter, Oxford 2001 A good treatment of the basics. • D. C. Jilles An Introduction to Magnetism and Magnetic Magnetic Materials, Magnetic Sensors and Magnetometers, CRC Press 480 pp : • J. D. Jackson Classical Electrodynamics 3rd ed, Wiley, New York 1998 A classic textbook, written in SI units. • G. Bertotti, Hysteresis Academic Press, San Diego 2000 A monograph on magnetostatics • A. Rosencwaig Ferrohydrodynamics, Dover, Mineola 1997 A good account of magnetic energy and forces • L.D. Landau and The definitive text E. M. Lifschitz Electrodynamics of Continuous Media Pergammon, Oxford 1989 ESM Cluj 2015 1 Introduction 2 Magnetostatics 3 Magnetism of the electron 4 The many-electron atom 5 Ferromagnetism 6 Antiferromagnetism and other magnetic order 7 Micromagnetism 8 Nanoscale magnetism 9 Magnetic resonance 10 Experimental methods 11 Magnetic materials 12 Soft magnets 13 Hard magnets 614 pages. Published March 2010 Available from Amazon.co.uk ~€50 14 Spin electronics and magnetic recording 15 Other topics Appendices, conversion tables. www.cambridge.org/9780521816144 ESM Cluj 2015 1. Introduction ESM Cluj 2015 Magnets and magnetization 0.03 A m2 3 mm 1.1 A m2 17.2 A m2 10 mm 25 mm m is the magnetic (dipole) moment of the magnet. It is proportional to volume m = MV magnetic moment volume magnetization Magnetization is the intrinsic property of the material; Magnetic moment is a property of a particular magnet. ESM Cluj 2015 Suppose they are made of Nd2Fe14B (M ≈ 1.1 MA m-1) What are the moments? Magnetic moment - a vector Each magnet creates a field around it. This acts on any material in the vicinity but strongly with another magnet. The magnets attract or repel depending on their mutual orientation ↑↑ Weak repulsion ↑↓ Weak attraction Nd2Fe14B ← ← Strong attraction ← → Strong repulsion tetragonal easy axis ESM Cluj 2015 Units What do the units mean? m m – A m2 M – A m-1 1A Ampère,1821. A current loop or coil is equivalent to a magnet m2 ∈ m = IA 1A ∈ 10,000 A Permanent magnets win over electromagnets at small sizes area of the loop m = nIA number of turns ESM Cluj 2015 C 10,000 turns m ! Right-hand corkscrew Magnetic field H – Oersted’s discovery I H δℓ r The relation between electric current and magnetic field was discovered by Hans-Christian Øersted, 1820. H I C ∫ Hdℓ = I Ampère’s law H = I/2πr If I = 1 A, r = 1 mm H = 159 A m-1 Right-hand corkscrew Earth’s field ≈ 40 Am-1 ESM Cluj 2015 Magnets and currents – Ampere and Arago’s insight A magnetic moment is equivalent to a current loop. Provided the current flows in a plane m = IA In general: m = ½ ∫ r × j(r) d3r Where j is the current density; I = j.a jm = ∇ x M So m = ½ ∫ r × Idl = I∫ dA = m ESM Cluj 2015 2 long solenoid with n turns. m = nIA Space inversion Polar vector -j Axial vector m Magnetization curves - Hysteresis loop M spontaneous magnetization remanence coercivity virgin curve initial susceptibility H major loop The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to a magnetic field . It reflects the arrangement of the magnetization in ferromagnetic domains. A broad loop like this is typical of a hard or permanent magnet. ESM Cluj 2015 This is slightly misleading because the positions of the fundamental and auxiliary fields are reversed in the two equations. The form (2.27) H = B/µ0 − M is preferable as it emphasizes the relation of the auxiliary magnetic Another insofar Question field to the fundamental magnetic field and the magnetization of the material. Maxwell’s equations in a material medium are expressed in terms of all four fields: 2.4 Magnetic field calculations What is Magnetostatics ? ∇ · D = ρ, Electromagnetism (2.46) In magnetostatics there is no time dependence of B,with D no or timeρ. We only ∇ · B = 0, (2.47) dependence have magnetic material and circulating conduction currents in a steady state. E = −∂ B/∂t, (2.48) Conservation of electric charge∇is×expressed by the equation, Maxwell’s Equations ∇∇ × j + ∂ D/∂t. · jH== −∂ρ/∂t, (2.49) (2.50) In we have only magnetic material andiscirculating currents and inmagnetostatics, aρsteady state ∂ρ/∂t =charge 0. The world ofand magnetostatics, we explore displacement current. Here is the local electric density ∂ D/∂t the which in iniswriting atherefore steadythe state. The fields areofproduced by the magnets further in Chapterall7,of described by three simple equations: A conductors, consequence basic equations electromagnetism in this neat & the currents and easy-to-remember form is that the constants µ0 , ϵ 0 and 4π are invisible. ∇ · j = 0 ∇ · B = 0 ∇ × H = j. But they inevitably crop up elsewhere, for example in the Biot–Savart law (2.5). ESM Cluj 2015 In order solve problems in solid-state to know theappear response The fieldstonaturally form two pairs: Bphysics and E we areneed a pair, which in the 2. Magnetization ESM Cluj 2015 Magnetic Moment and Magnetization The magnetic moment m is the elementary quantity in solid state magnetism. Define a local moment density - magnetization – M(r, t) which fluctuates wildly on a subnanometer and a sub-nanosecond scale. Define a mesoscopic average magnetization δm = MδV M = δm/δV The continuous medium approximation M can be the spontaneous magnetization Ms within a ferromagnetic domain A macroscopic average magnetization is the domain average M (r) M = ΣiMiVi/ ΣiVi Ms atoms The mesoscopic average magnetization ESM Cluj 2015 Magnetization and current density The magnetization of a solid is somehow related to a ‘magnetization current density’ Jm that produces it. Since the magnetization is created by bound currents, ∫s Jm.dA = 0 over any surface. Using Stokes theorem ∫ M.dℓ = ∫s(∇ × M).dA and choosing a path of integration outside the magnetized body, we obtain ∫s M.dA = 0, so we can identify Jm Jm = ∇ × M We don’t know the details of the magnetization currents, but we can measure the mesoscopic average magnetization and the spontaneous magnetization of a sample. ESM Cluj 2015 B and H fields in free space; permeability of free space µ0 When illustrating Ampère’s Law we labelled the magnetic field created by the current, measured in Am-1 as H. This is the ‘magnetic field strength’ Maxwell’s equations have another field, the ‘magnetic flux density’, labelled B, in the equation ∇. B = 0. It is a different quantity with different units. Whenever H interacts with matter, to generate a force, an energy or an emf, a constant µ0, the ‘permeability of free space’ is involved. In free space, the relation between B and H is simple. They are numerically proportional to each other B = µ0H B-field Units Tesla, T Tesla Permeability of free space Units TmA-1 H-field Units A m-1 µ0 depends on the definition of the Amp. It is precisely 4π 10-7 T m A-1 ESM Cluj 2015 In practice you can never mix them up. The differ by almost a million! (795775 ≈ 800000) Field due to electric currents We need a differential form of Ampère’s Law; The Biot-Savart Law δℓ I δB B j C Right-hand corkscrew (for the vector product) ESM Cluj 2015 Field due to a small current loop (equivalent to a magnetic moment) A BA=4δBsinε ε r sinε=δl/2r m δl m = I(δl)2 B I A So at a general point C, in spherical coordinates: C m msinθ θ mcosθ B ESM Cluj 2015 Field due to a magnetic moment m The Earth’s magnetic field is roughly that of a geocentric dipole tan I = Br /Bθ = 2cotθ dr/rdθ = 2cotθ m Solutions are r = c sin2θ Equivalent forms ESM Cluj 2015 Magnetic field due to a moment m; Scaling m θ Why does magnetism lend itself to miniaturization ? P r HA = A 2Ma3/4πr3; H = (m/4πr3){2cosθer- sinθeθ} Just like the field of an electric dipole What is the average magnetization of the Earth ? •A If a = 0.01m, r = 2a, M = 1 MA m-1 HA = 2M/16π = 40 kA m-1 Magnet-generated fields just depend on M. They are scale-independent ESM Cluj 2015 a m 3. Magnetic Fields ESM Cluj 2015 Magnetic flux density - B Now we discuss the fundamental field in magnetism. Magnetic poles, analogous to electric charges, do not exist. This truth is expressed in Maxwell’s equation ∇.B = 0. This means that the lines of the B-field always form complete loops; they never start or finish on magnetic charges, the way the electric E-field lines can start and finish on +ve and -ve electric charges. The same can be written in integral form over any closed surface S ∫SB.dA = 0 (Gauss’s law). The flux of B across a surface is Φ = ∫B.dA. Units are Webers (Wb). The net flux across any closed surface is zero. B is known as the flux density; units are Teslas. (T = Wb m-2) Flux quantum Φ0 = 2.07 1015 Wb ESM Cluj 2015 (Tiny) The B-field Sources of B • electric currents in conductors • moving charges • magnetic moments B = µ0I/2πr • time-varying electric fields. (Not in magnetostatics) In a steady state: Maxwell’s equation ex ey ez (Stokes theorem; ∂/∂x ∂/∂y ∂/∂z Bx BY I ) BZ Field at center of current loop ESM Cluj 2015 Forces between conductors; Definition of the Amp F = q(E + v x B) Lorentz expression. Note the Lorentz force does no work on the charge Gives dimensions of B and E. [T is equivalent to Vsm-1] If E = 0 the force on a straight wire arrying a current I in a uniform field B is F = BIℓ The force between two parallel wires each carrying one ampere is precisely 2 10-7 N m-1. (Definition of the Amp) The field at a distance 1 m from a wire carrying a current of 1 A is 0.2 µΤ B = µ0I1/2πr Force per meter = µ0I1I2/2πr ESM Cluj 2015 The range of magnitude of B Su e th at d ry el ta le So Ea rt h' no s id Fi ne 1E-6 µT e ac Sp ce pa te rp la lla 1E-9 In In te rs te H an um 1E-12 pT rS rt ea in ra B H an um H 1E-15 rf Pe ac rm e Su a p n H e r c ent yb o M Pu rid n d ag u n Ex lse Ma c t i et pl M gn n g os a g et M ag iv ne e t ne Fl t ux C om N eu pr es tr on si on St ar M ag ne ta r • The tesla is a very large unitcontinuous laboratory field 45 T Largest • Largest continuous field acheived in a lab was 45 T (Tallahassee) 1E-3 1 T ESM Cluj 2015 1000 1E6 MT 1E9 TT1E12 1E15 Typical values of B Electromagnet 1 T Human brain 1 fT Magnetar 1012 T Permanent magnets 0.5 T Earth 50 µT Helmholtz coils 0.01 T Electromagnet 1 T ESM Cluj 2015 Superconducting magnet 10 T Sources of uniform magnetic fields Long solenoid Helmholtz coils B =µ0nI B = (4/5)3/2µ0NI/a Halbach cylinder r2 B =µ0M ln(r2/r1) r1 ESM Cluj 2015 The H-field. Ampère’s law for the field is free space is ∇ x B = µ0( jc + jm) but jm cannot be measured ! Only the conduction current jc is accessible. We showed on slide 16 that Jm = ∇ × M Hence ∇ × (B/µ0 – M) = µ0 jc We can retain Ampère’s law is a usable form provided we define H = B/µ0 – M Then ∇ x H = µ0 jc And B = µ0(H + M) ESM Cluj 2015 The H-field. The H-field plays a critical role in condensed matter. The state of a solid reflects the local value of H. Hysteresis loops are plotted as M(H) Unlike B, H is not solenoidal. It has sources and sinks in a magnetic material wherever the magnetization is nonuniform. ∇.H = - ∇.M The sources of H (magnetic charge, qmag) are distributed — in the bulk with charge density -∇.M — at the surface with surface charge density M.en Coulomb approach to calculate H (in the absence of currents) Imagine H due to a distribution of magnetic charges qm (units Am) so that H = qmr/4πr3 [just like electrostatics] ESM Cluj 2015 Potentials for B and H It is convenient to derive a field from a potential, by taking a spatial derivative. For example E = -∇ϕe(r) where ϕe(r) is the electric potential. Any constant ϕ0 can be added. For B, we know from Maxwell’s equations that ∇.B = 0. There is a vector identity ∇. ∇× X ≡ 0. Hence, we can derive B(r) from a vector potential A(r) (units Tm), B(r) = ∇ × A(r) The gradient of any scalar f can be added to A (a gauge transformation) This is because of another vector identity ∇×∇.f ≡ 0. Generally, H(r) cannot be derived from a potential. It satisfies Maxwell’s equation ∇ × H = jc + ∂D/∂t. In a static situation, when there are no conduction currents present, ∇ × H = 0, and H(r) = - ∇ϕm(r) In these special conditions, it is possible to derive H(r) from a magnetic scalar potential ϕm (units A). We can imagine that H is derived from a distribution of magnetic ‘charges’± qm. ESM Cluj 2015 Relation between B and H in a material The general relation between B, H and M is B = µ0(H + M) H i.e. H = B/µ0 - M M B We call the H-field due to a magnet; — stray field outside the magnet — demagnetizing field, Hd, inside the magnet ESM Cluj 2015 Boundary conditions Conditions on the fields It follows from Gauss’s law ∫SB.dA = 0 that the perpendicular component of B is continuous. (B1- B2).en= 0 It follows from from Ampère’s law ∫ H.dl = I = 0 that the parallel component of H is continuous. since there are no conduction currents on the surface. (H1- H2) x en= 0 Conditions on the potentials Since ∫SB.dA = ∫ A.dl (Stoke’s theorem) (A1- A2) x en= 0 The scalar potential is continuous ϕm1 = ϕm2 ESM Cluj 2015 Boundary conditions in linear, isotropic homogeneous media In LIH media, Hence B = µ 0 µ rH B1en = B2en H1en = µr2/µr1H2en So field lies ~ perpendicular to the surface of soft iron but parallel to the surface of a superconductor. Diamagnets produce weakly repulsive images. Paramagnets produce weakly attractive images. Soft ferromagnetic mirror Superconducting mirror ESM Cluj 2015 Field calculations – Volume integration Integrate over volume distribution of M Sum over fields produced by each magnetic dipole element Md3r. Using Gives (Last term takes care of divergences at the origin) ESM Cluj 2015 Field calculations – Ampèrian approach Consider bulk and surface current distributions jm = ∇ x M and jms = M x en Biot-Savart law gives For uniform M, the Bulk term is zero since ∇ x M = 0 ESM Cluj 2015 Field calculations – Coulombian approach Consider bulk and surface magnetic charge distributions ρm = -∇.M and ρms = M.en H field of a small charged volume element V is δH = (ρmr/4πr3) δV So For a uniform magnetic distribution the first term is zero. ∇.M = 0 ESM Cluj 2015 Hysteresis loop; permanent magnet (hard ferromagnet) M spontaneous magnetization remanence coercivity virgin curve initial susceptibility major loop A broad M(H) loop ESM Cluj 2015 H Hysteresis loop; temporary magnet (soft ferromagnet) Ms Slope here is the initial susceptibility χi > 1 M H ’ Applied field Susceptibility is defined as χ = M/H A narrow M(H) loop ESM Cluj 2015 Paramagnets and diamagnets; antiferromagnets. Only a few elements and alloys are ferromagnetic. (See the magnetic periodic table). The atomic moments in a ferromagnet order spontaneosly parallel to eachother. Most have no spontaneous magnetization, and they show only a very weak response to a magnetic field. M paramagnet H Here |χ| << 1 χ is 10-4 - diamagnet 10-6 Ordered T < Tc A few elements and oxides are antiferromagnetic. The atomic moments order spontaneously antiparallel to eachother. Disordered T > Tc ESM Cluj 2015 Magnetic Periodic Table 1H 1.00 66Dy Atomic Number 3 Li 4 Be 6.94 1 + 2s0 9.01 2 + 2s0 11Na 12Mg 19K 20Ca 22.99 1 + 3s0 38.21 1 + 4s0 Typical ionic change Antiferromagnetic TN(K) 40.08 2 + 4s0 Sr 21Sc 44.96 3 + 3d0 39 Y 22Ti 23V 47.88 4 + 3d0 40 Zr 50.94 3 + 3d2 41 Nb 24Cr 52.00 3 + 3d3 312 42 88.91 2 + 4d0 91.22 4 + 4d0 92.91 5 + 4d0 95.94 5 + 4d1 55Cs 56Ba 57La 72Hf 73Ta 74W 87Fr 88Ra 89Ac 58Ce 59Pr 226.0 2 + 7s0 138.9 3 + 4f0 178.5 4 + 5d0 227.0 3 + 5f0 180.9 5 + 5d0 140.1 4 + 4f0 13 90Th 232.0 4 + 5f0 25Mn 26Fe 55.85 2 + 3d5 96 Mo 43 Tc 87.62 2 + 5s0 223 5B 6C 7N 8O 13Al 14Si 15P 183.8 6 + 5d0 140.9 3 + 4f2 91Pa 231.0 5 + 5f0 97.9 55.85 3 + 3d5 1043 44 Ru 27Co 58.93 2 + 3d7 1390 45 Rh 28Ni 58.69 2 + 3d8 629 46 Pd 29Cu 31Ga 32Ge 33As 63.55 2 + 3d9 47 Ag 12.01 14.01 9F 10Ne 16S 17Cl 18Ar 34Se 35Br 36Kr I 54Xe 16.00 19.00 30Zn 65.39 2 + 3d10 48 Cd 69.72 3 + 3d10 49 In 28.09 72.61 50 Sn 30.97 74.92 51 Sb 32.07 78.96 52 Te 35.45 79.90 53 39.95 83.80 101.1 3 + 4d5 102.4 3 + 4d6 106.4 2 + 4d8 107.9 1 + 4d10 112.4 2 + 4d10 114.8 3 + 4d10 118.7 4 + 4d10 121.8 127.6 126.9 75Re 76Os 77Ir 78Pt 79Au 80Hg 81Tl 82Pb 83Bi 84Po 85At 60Nd 61Pm 62Sm 63 Eu 64Gd 65Tb 66Dy 67Ho 68Er 69Tm 70Yb 95Am 96Cm 97Bk 98Cf 99Es 100Fm 101Md 102No 103Lr 186.2 4 + 5d3 144.2 3 + 4f3 19 92U 238.0 4 + 5f2 190.2 3 + 5d5 192.2 4 + 5d5 145 150.4 3 + 4f5 105 93Np 94Pu 238.0 5 + 5f2 244 195.1 2 + 5d8 152.0 2 + 4f7 90 197.0 1 + 5d10 157.3 3 + 4f7 292 243 247 200.6 2 + 5d10 204.4 3 + 5d10 158.9 162.5 3 + 4f8 3 + 4f9 229 221 179 85 247 251 207.2 4 + 5d10 164.9 3 + 4f10 132 20 252 209.0 167.3 3 + 4f11 85 20 257 209 168.9 3 + 4f12 56 210 173.0 3 + 4f13 258 Nonmetal Diamagnet Ferromagnet TC > 290K Metal Paramagnet Antiferromagnet with TN > 290K Magnetic atom Antiferromagnet/Ferromagnet with TN/TC < 290 K Radioactive 20.18 35 26.98 3 + 2p6 85.47 1 + 5s0 137.3 2 + 6s0 4.00 10.81 Ferromagnetic TC(K) 24.21 2 + 3s0 37Rb 38 132.9 1 + 6s0 Atomic symbol Atomic weight 162.5 3 + 4f9 179 85 2 He BOLD ESM Cluj 2015 259 83.80 86Rn 222 71Lu 175.0 3 + 4f14 260 Susceptibilities of the elements ESM Cluj 2015 Magnetic fields - Internal and applied fields In ALL these materials, the H-field acting inside the material is not the one you apply. These are not the same. If they were, any applied field would instantly saturate the magnetization of a ferromagnet when χ > 1. M Consider a thin film of iron. iron Ms Field applied || to film Field applied ⊥ to film Hʹ′ Ms substrate H = H’ + Hd Internal field Demagnetizing field External field ESM Cluj 2015 Ms = 1.72 MAm-1 Demagnetizing field in a material - Hd The demagnetizing field depends on the shape of the sample and the direction of magnetization. For simple uniformly-magnetized shapes (ellipsoids of revolution) the demagnetizing field is related to the magnetization by a proportionality factor N known as the demagnetizing factor. The value of N can never exceed 1, nor can it be less than 0. Hd = - N M More generally, this is a tensor relation. N is then a 3 x 3 matrix, with trace 1. That is Nx+Ny+Nz=1 Note that the internal field H is always less than the applied field H’ since H = H’ - N M ESM Cluj 2015 Demagnetizing factor N for special shapes. Shapes N = 0 N = 1/3 N = 1/2 N =1 H ’ H ’ ESM Cluj 2015 H ’ The shape barrier N < 0.1 Shen Kwa 1060 Daniel Bernouilli 1743 S N Gowind Knight 1760 T. Mishima 1931 N = 0.5 New icon for permanent magnets! ⇒ ESM Cluj 2015 The shape barrier ovecome ! Hd = - N M working point Ms M (Am-1) Mr Hd Hc Hd ESM Cluj 2015 H (Am-1) Demagnetizing factors N I for a general ellipsoid. Calculate from analytial formulae Nx+Ny+Nz=1 ESM Cluj 2015 Other shapes cannot be unoformly magnetized.. When measuring the magnetization of a sample H is always taken as the independent variable, M = M(H). ESM Cluj 2015 Paramagnetic and ferromagnetic responses Susceptibility of linear, isotropic and homogeneous (LIH) materials M = χ’H’ χ’ is external susceptibility (no units) It follows that from H = H’ + Hd that 1/χ = 1/χ’ - N Typical paramagnets and diamagnets: χ ≈ χ’ (10-5 to 10-3 ) Paramagnets close to the Curie point and ferromagnets: χ diverges as T → TC but χ’ never exceeds 1/N. χ >> χ’ M M M M M H’ M /3 Mss/3 H’ ESM Cluj 2015 H' H Demagnetizing correction to M(H) and B(H) loops. N=0 N=½ The B(H) loop is deduced from M(H) using B = µ0(H + M). ESM Cluj 2015 N=1 Permeability In LIH media B =µ H Relative permeability is defined as B = µ0(H + M) gives defines the permeability µ µr = µ / µ0 µr = 1 + χ ESM Cluj 2015 Units: TA-1m 4. Energy and forces ESM Cluj 2015 Energy of ferromagnetic bodies Torque and potential energy of a dipole in a field, assumed to be constant. Force Γ=mxB ε = -m.B Γ = mB sinθ. ε = -mB cosθ. In a non-uniform field, f = -∇εm B θ m f = m.∇B • Magnetostatic (dipole-dipole) forces are long-ranged, but weak. They determine the magnetic microstructure (domains). • ½µ0H2 is the energy density associated with a magnetic field H M ≈ 1 MA m-1, µ0Hd ≈ 1 T, hence µ0HdM ≈ 106 J m-3 • Products B.H, B.M, µ0H2, µ0M2 are all energies per unit volume. • Magnetic forces do no work on moving charges f = q(v x B) [Lorentz force] • No potential energy associated with the magnetic force. ESM Cluj 2015 Reciprocity theorem The interaction of a pair of dipoles, εp, can be considered as the energy of m1 in the field B21 created by m2 at r1 or vice versa. εp = -m1.B21 = -m2.B12 So εp = -(1/2)(m1.B21+ m2.B12) Extending to magnetization distributions: ε = -µ0 ∫ M1.H2 d3r = -µ0 ∫ M2.H1 d3r ESM Cluj 2015 Self Energy The interaction of the body with the field it creates. Hd. Consider the energy to bring a small moment δm into position within the magnetized body Hloc=Hd+(1/3)M δε = - µ0 δm.Hloc Integration over the whole sample gives The magnetostatic self energy is defined as Or equivalently, using B = µ0(H + M) and ∫ B.Hdd3r=0 For a uniformly magnetized ellipsoid ESM Cluj 2015 Energy associated with a field General expression for the energy associated with a magnetic field distribution Aim to maximize energy associated with the field created around the magnet, from previous slide: Can rewrite as: where we want to maximize the integral on the left. Energy product: twice the energy stored in the stray field of the magnet -µ0 ∫i B.Hd d3r ESM Cluj 2015 Work done by an external field Elemental work δw to produce a flux change δΦ is I δΦ Ampere: ∫H.dl = I So δw = ∫ δΦ H.dl So in general: H = H´+ Hd δw = ∫ δB.H.d3r B = µ0(H + M) Subtract the term associated with the H-field in empty space, to give the work done on the body by the external field; gives ESM Cluj 2015 Thermodynamics First law: dU = HxdX + dQ (U ,Q, F, G are in units of Jm-3) M dQ = TdS Four thermodynamic potentials; ΔG U(X,S) E(HX,S) − ΔF F(X,T) = U - TS dF = HdX - SdT G(HX,T) = F- HXX dG = -XdH - SdT Magnetic work is HδB or µ0H’δM dF = µ0H’dM - SdT dG = -µ0MdH’ - SdT H’ S = -(∂G/∂T)H’ µ0 M = -(∂G/∂H’)T’ Maxwell relations (∂S/∂H’)T’ = - µ0(∂M/∂T)H’ etc. ESM Cluj 2015 Magnetostatic forces Force density on a magnetized body at constant temperature Fm= - ∇ G Kelvin force General expression, for when M is dependent on H is V =1/d d is the density ESM Cluj 2015 Anisotropy - shape The energy of the system increases when the magnetization deviates by an angle θ from its easy axis. θ E = K sin2 θ The anisotropy may be due to shape. The energy of the magnet in the demagnetizing field is -(1/2)µ0mHd For shape anisotropy Ksh = (1/4)µ0M2(1 - 3N ) For example, if M = 1 MA m-1, N = 1, Ksh = 630 kJ m-3 e.g. if a thin film is to have its magnetization perpendicular to the plane, there must be a stronger intrinsic magnetocrystalline anisotropy K1. ESM Cluj 2015 Polarization and anisotropy field Engineers often quote the polarization of a magnetic material, rather than its magnetization. The relation is simple; J = µ0M. For example, the polarization of iron is 2.15 T; Its magnetization is 1.71 MA m-1 The defining relation is then: B = µ0H + J The anisotropy of whatever origin may be represented by an effective magnetic field, the anisotropy field Ha Ha = 2K/µ0Ms The coercivity can never exceed the anisotropy field. In practice it is rarely possible to obtain coercivity greater than about 25 % of Ha. Coercivity depends on the microsctructure, and the ability to impede fprmation of reversed domains Sometimes anisotropy field is quoted in Tesla. Ba = µ0Ha. ESM Cluj 2015 Some expressions involving B F = q(E + v × B) F = B lℓ Force on a charged particle q Force on current-carrying wire E = - dΦ/dt Faraday’s law of electromagnetic induction E = -m.B Energy of a magnetic moment F = ∇ m.B Force on a magnetic moment Γ=mxB Torque on a magnetic moment ESM Cluj 2015 5. Units ESM Cluj 2015 A note on units: Magnetism is an experimental science, and it is important to be able to calculate numerical values of the physical quantities involved. There is a strong case to use SI consistently Ø SI units relate to the practical units of electricity measured on the multimeter and the oscilloscope Ø It is possible to check the dimensions of any expression by inspection. Ø They are almost universally used in teaching Ø Units of B, H, Φ or dΦ/dt have been introduced. BUT Most literature still uses cgs units,You need to understand them too. ESM Cluj 2015 SI / cgs conversions: SI units cgs units B = μ0(H + M) B = H + 4πM m A m2 emu M A m-1 (10-3 emu cc-1) emu cc-1 (1 k A m-1) σ A m2 kg-1 (1 emu g-1) emu g-1 (1 A m2 kg-1) H A m-1 (4π/1000 ≈ 0.0125 Oe) Oersted B Tesla (10000 G) Gauss (10-4 T) Φ Weber (Tm2) (108 Mw) Maxwell (G cm2) (10-8 Wb) dΦ/dt V (108 Mw s-1) Mw s-1 (10 nV) χ (4π cgs) - (1/4π SI) - ESM Cluj 2015 (1000/4π ≈ 80 A m-1) B3.1 Dimensions Mechanical Quantity Symbol Unit m l t i θ Area Volume Velocity Acceleration Density Energy Momentum Angular momentum Moment of inertia Force Force density Power Pressure Stress Elastic modulus Frequency Diffusion coefficient Viscosity (dynamic) Viscosity Planck’s constant A V v a d ε p L I f F P P σ K f D η ν ! m2 m3 m s−1 m s−2 kg m−3 J kg m s−1 kg m2 s−1 kg m2 N N m−3 W Pa N m−2 N m−2 s−1 m2 s−1 N s m−2 m2 s−1 Js 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 2 3 1 1 −3 2 1 2 2 1 −2 2 −1 −1 −1 0 2 −1 2 2 0 0 −1 −2 0 −2 −1 −1 0 −2 −2 −3 −2 −2 −2 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ESM Cluj 2015 Thermal Electrical Quantity Symbol Unit m l t i θ Current Current density Charge Potential Electromotive force Capacitance Resistance Resistivity Conductivity Dipole moment Electric polarization Electric field Electric displacement Electric flux Permittivity Thermopower Mobility I j q V E C R ϱ σ p P E D % ε S µ A A m−2 C V V F " "m S m−1 Cm C m−2 V m−1 C m−2 C F m−1 V K−1 m2 V−1 s−1 0 0 0 1 1 −1 1 1 −1 0 0 1 0 0 −1 1 −1 0 −2 0 2 2 −2 2 3 −3 1 −2 1 −2 0 −3 2 0 0 0 1 −3 −3 4 −3 −3 3 1 1 −3 1 1 4 −3 2 1 1 1 −1 −1 2 −2 −2 2 1 1 −1 1 1 2 −1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 ESM Cluj 2015 Magnetic Mobility µ m V s −1 0 2 1 0 Magnetic Quantity Symbol Unit m l t i θ Magnetic moment Magnetization Specific moment Magnetic field strength Magnetic flux Magnetic flux density Inductance Susceptibility (M/H) Permeability (B/H) Magnetic polarization Magnetomotive force Magnetic ‘charge’ Energy product Anisotropy energy Exchange stiffness Hall coefficient Scalar potential Vector potential Permeance Reluctance m M σ H ' B L χ µ J F qm (BH ) K A RH ϕ A Pm Rm A m2 A m−1 A m2 kg−1 A m−1 Wb T H 0 0 −1 0 1 1 1 0 1 1 0 0 1 1 1 0 0 1 1 −1 2 −1 2 −1 2 0 2 0 1 0 0 1 −1 −1 1 3 0 1 2 −2 0 0 0 0 −2 −2 −2 0 −2 −2 0 0 −2 −2 −2 −1 0 −2 −2 2 1 1 1 1 −1 −1 −2 0 −2 −1 1 1 0 0 0 −1 1 −1 −2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 H m−1 T A Am J m−3 J m−3 J m−1 m3 C−1 A Tm T m2 A−1 A T−1 m−2 ESM Cluj 2015 Thermal 594 Quantity Symbol Unit Enthalpy Entropy Specific heat Heat capacity Thermal conductivity Sommerfeld coefficient Boltzmann’s constant H J S J K−1 C J K−1 kg−1 c J K−1 κ W m−1 K−1 Appendices γ J mol−1 K−1 J K−1 kB B3.2 Examples (1) Kinetic energy of a body: ε = 12 mv 2 [ε] = [1, 2, −2, 0, 0] m l t i θ 1 1 0 1 1 1 1 2 2 2 2 1 2 2 −2 −2 −2 −2 −3 −2 −2 0 0 0 0 0 0 0 0 −1 −1 −1 −1 −1 −1 [m] = [1, 0, 0, 0, 0] 2[0, −1, −1, 0, 0] [v 2 ] = [1, −2, −2, 0, 0] (2) Lorentz force on a moving charge; f = qv × B [f ] = [1, 1, −2, 0, 0] [q] = [0, 0, 1, 1, 0] [v] = [0, 1, −1, 0, 0] [1, 0, −2, −1, 0] [B] = [1, 1, −2, 0, 0] √ (3) Domain wall energy γ w = AK (γ w is an energy per unit area) √ −1 [ AK] = 1/2[AK] [γ w ] = [εA ] √ = [1, 2, −2, 0, 0] [ A] = 12 [1, 1, −2, 0, 0] √ [1, −1, −2, 0, 0] −[ [ K] = 12 - [0,1,2,1,0,−2, 0, 0 0, ] 0] [1, 0, −2, 0, 0] = [1, 0, −2, 0, 0] (4) Magnetohydrodynamic force on a moving conductor F = σ v × B × B (F is a force per unit volume) ESM Cluj 2015 [σ ] = [−1, −3, 3, 2, 0] [F ] = [F V −1 ] √ [ K] = −[ 1, 1, −2, 0, 0] (4) (5) (6) (7) (8) 1 2 [1, −1, −2, 0, 0] [1, 0, −2, 0, 0] = [1, 0, −2, 0, 0] Magnetohydrodynamic force on a moving conductor F = σ v × B × B ( F is a force per unit volume) [σ ] = [−1, −3, 3, 2, 0] [F ] = [F V −1 ] = [1, 1, −2, 0, 0] [v] = [0, 1, −1, 0, 0] 2[1, 0, −2, −1, 0] [0, 3, 0, 0, 0] [B 2 ] = − [1, −2, −2, 0, 0] [1, −2, −2, 0, 0] Flux density in a solid B = µ0 ( H + M) (note that quantities added or subtracted in a bracket must have the same dimensions) [B] = [1, 0, −2, −1, 0] [µ0 ] = [1, 1, −2, −2, 0] [0, −1, 0, 1, 0] [M], [H ] = [1, 0, −2, −1, 0] Maxwell’s equation ∇ × H = j + d D/dt . [j ] = [0, −2, 0, 1, 0] [d D/dt ] = [Dt −1 ] [∇ × H] = [H r −1 ] = [0, −1, 0, 1, 0] = [0, −2, 1, 1, 0] −[ 0, 1, 0, 0, 0] −[0, 0, 1, 0, 0] = [0, −2, 0, 1, 0] = [0, −2, 0, 1, 0] Ohm’s Law V = I R = [1, 2, −3, −1, 0] [0, 0, 0, 1, 0] + [1, 2, −3, −2, 0] = [1, 2, −3, −1, 0] Faraday’s Law E = −∂%/∂t [1, 2, −2, −1, 0] = [1, 2, −3, −1, 0] −[0, 0, 1, 0, 0] = [1, 2, −3, −1, 0] ESM Cluj 2015