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Magnetism in Matter Electric polarisation (P) - electric dipole moment per unit vol. Magnetic ‘polarisation’ (M) - magnetic dipole moment per unit vol. M magnetisation Am-1 c.f. P polarisation Cm-2 Element magnetic dipole moment m When all moments have same magnitude & direction M=Nm N number density of magnetic moments Dielectric polarisation described in terms of surface (uniform) or volume (non-uniform) bound charge densities By analogy, expect description in terms of surface (uniform) or volume (non-uniform) magnetisation current densities Magnetism in Matter Electric polarisation P(r) P(r ).nˆ jpol (r ).nˆ dt 0 jpol (r ) p P(r ) t r (r )dr allspace p electric dipole moment of localised charge distribution Magnetisation M(r) 1 M(r ) r x j(r ) 2 jM (r ) x M(r ) 1 m r x j(r ) dr 2 all space m magnetic dipole moment of localised current distribution Magnetic moment and angular momentum • Magnetic moment of a group of electrons m • Charge –e mass me j(r ) qi v i (r ri ) i v5 1 m qi r x v i (r ri ) dr 2 i all space v4 r5 O r4 r3 1 m qi ri x v i 2 i v3 v1 r1 r2 v2 i me ri x v i angular momentum m -e 2me i i -e L 2me L i L total angular momentum i Force and torque on magnetic moment Fi qi v i x B F Lorentz force on point charges (r ) v(r ) x B(r ) dr Continuous current distributi on j(r ) x B(r ) dr j(r ) (r ) v(r ) all space all space Bk (r ) Bk (0) r.Bk (0) ... Taylor expansion F j(r ) x B (0) r.B (0) ... dr 0 j(r ) x r.B (0) dr ... k k k all space F m.B(0) suggests all space Um -m.B(0) F U Um -m.B(0) c.f. Up -p.E(0) Torque T r x j(r ) x B(r ) dr m x B(0) all space Diamagnetic susceptibility Induced magnetic dipole moment when B field applied Applied field causes small change in electron orbit, inducing L,m Consider force balance equation when B = 0 (mass) x (accel) = (electric force) meo2a Ze ω o 2 3 4oa 4omea Ze 2 2 Ze 2 me a eaB 2 4oa 2 1 2 Ze eB 3 2me 4omea eB o o L 2me 2 B -e 1 2 quadratic in ev B eaB me Z B oa3 2 L is the Larmor frequency Diamagnetic susceptibility Pair of electrons in a pz orbital m B a -e m = o + L |ℓ| = +meLa2 m = -e/2me ℓ v -e v x B v -e = o - L |ℓ| = -meLa2 m = -e/2me ℓ Electron pair acquires a net angular momentum/magnetic moment -e v x B Diamagnetic susceptibility Increase in ang freq increase in ang mom (ℓ) Increase in magnetic dipole moment: m B -e e 2meL a 2 2me eB 2 e e 2a 2 e 2a 2 a m 2me B m B 2me 2me 2me 2me Include all Z electrons to get effective total induced magnetic dipole moment with sense opposite to that of B e2 m Zao2 B 2me ao2 : mean square radius of electron orbit ~ 10 -27 for Z 12 B 1T c.f. 1B 9.274.10 -24 Am 2 1B Intrinsic ' spin' magnetic moment for one electron m Paramagnetism Found in atoms, molecules with unpaired electron spins Examples O2, haemoglobin (Fe ion) Paramagnetic substances become weakly magnetised in an applied field Energy of magnetic moment in B field Um = -m.B Um = -9.27.10-24 J for a moment of 1 B aligned in a field of 1 T Uthermal = kT = 4.14.10-21 J at 300K >> Um Um/kT=2.24.10-3 Boltzmann factors e-Um/kT for moment parallel/anti-parallel to B differ little at room temperature This implies little net magnetisation at room temperature Ferro, Ferri, Anti-ferromagnetism Found in solids with magnetic ions (with unpaired electron spins) Examples Fe, Fe3O4 (magnetite), La2CuO4 When interactions H = -J mi.mj between magnetic ions are (J) >= kT Thermal energy required to flip moment is Nm.B >> m.B N is number of ions in a cluster to be flipped and Um/kT > 1 Ferromagnet has J > 0 (moments align parallel) Anti-ferromagnet has J < 0 (moments align anti-parallel) Ferrimagnet has J < 0 but moments of different sizes giving net magnetisation Uniform magnetisation Electric polarisation p i C.m -2 P i ( Cm ) 3 V m I z x M y IyΔz I xyΔz x Magnetisation M m i i V A.m2 -1 (Am ) 3 m Magnetisation is a current per unit length For uniform magnetisation, all current localised on surface of magnetised body (c.f. induced charge in uniform polarisation) Surface Magnetisation Current Density Symbol: aM a vector current density Units: A m-1 Consider a cylinder of radius r and uniform magnetisation M where M is parallel to cylinder axis Since M arises from individual m, (which in turn arise in current loops) draw these loops on the end face Current loops cancel in interior, leaving only net (macroscopic) surface current m M Surface Magnetisation Current Density magnitude aM = M but for a vector must also determine its direction aM M n̂ aM is perpendicular to both M and the surface normal Normally, current density is “current per unit area” in this case it is “current per unit length”, length along the Cylinder - analogous to current in a solenoid. aM M nˆ c.f. d pol P.dS Surface Magnetisation Current Density Solenoid in vacuum Bv ac oNI With magnetic core (red), Ampere’s Law integration contour encloses two types of current, “conduction current” in the coils and “magnetisation current” on the surface of the core B.d I o encl BL o NLI a ML B o NI a M B v ac B > 1: aM and I in same direction (paramagnetic) < 1: aM and I in opposite directions (diamagnetic) is the relative permeability, c.f. the relative permittivity Substitute for aM B o NI M L I Magnetisation Macroscopic electric field EMac= EApplied + EDep = E - P/o Macroscopic magnetic field BMac= BApplied + BMagnetisation BMagnetisation is the contribution to BMac from the magnetisation BMac= BApplied + BMagnetisation = B + oM Define magnetic susceptibility via M = cBBMac/o BMac= B + cBBMac EMac= E - P/o = E - EMac BMac(1-cB) = B EMac(1+c) = E Diamagnets Para, Ferromagnets Au Quartz O2 STP BMagnetisation opposes BApplied BMagnetisation enhances BApplied cB -3.6.10-5 -6.2.10-5 +1.9.10-6 cB < 0 cB > 0 0.99996 0.99994 1.000002 Magnetisation Rewrite BMac= B + oM as BMac - oM = B LHS contains only fields inside matter, RHS fields outside Magnetic field intensity, H = BMac/o - M = B/o = BMac/o - cBBMac/o = BMac (1- cB) /o = BMac/o c.f. D = oEMac + P = o EMac = 1/(1- cB) Relative permeability =1+c Relative permittivity Non-uniform Magnetisation Rectangular slab of material with M directed along y-axis M increases in magnitude along x-axis z I1-I2 I2-I3 My x I1 Individual loop currents increase from left to right There is a net current along the z-axis Magnetisation current density jM z I2 I3 Non-uniform Magnetisation dx dx Consider 3 identical element boxes, centres separated by dx If the circulating current on the central box is My dy Then on the left and right boxes, respectively, it is My My My dx dy and My dx dy x x Non-uniform Magnetisation My My 1 M M dx My dx My dy 2 y y x x Magnetisation current is the difference in neighbouring circulating currents, where the half takes care of the fact that each box is used twice! This simplifies to M My My 1 2 y dx dy dxdy jMz dxdy jMz 2 x x x Non-uniform Magnetisation Rectangular slab of material with M directed along x-axis M increases in magnitude along y-axis My z -Mx z y x jMz My x I1-I2 I2-I3 x jMz Mx y I1 I2 I3 My Mx x y Total magnetisation current || z jMz Similar analysis for x, y components yields jM M Types of Current j B o j o o jP E t j jf jM jP Total current P t k jM x M M = sin(ay) k j i jM = curl M = a cos(ay) i Polarisation current density from oscillation of charges as electric dipoles Magnetisation current density from space/time variation of magnetic dipoles Magnetic Field Intensity H Recall Ampere’s Law B.d I o encl or B o j Recognise two types of current, free and bound B o j o jf jM o jf M B M jf H jf o B where H M or B o H M o Electric D oE P Magnetic B H M .D f H jf o Magnetic Field Intensity H B o j o o 1 o E t B jf jM jP o jf M E t P E o t t B D M jf oE P H jf t t o D/t is displacement current postulated by Maxwell (1862) to exist in the gap of a charging capacitor In vacuum D = oE and displacement current exists throughout space Boundary conditions on B, H For LIH magnetic media B = oH (diamagnets, paramagnets, not ferromagnets for which B = B(H)) .B 0 B.d S 0 S B1cos1 S B 2cos 2 S 0 B1 B 2 H.d I enclf ree H1sin1 L H2sin 2 L I encl f ree 0 H1|| H2|| B H .d 1 1 1 2 B2 S - H1 sin 1 1 A B1 2 1 1B 2 1 dℓ1 C A H .d H1 2 A 2 B I enclfree H2 dℓ2 2 H2 sin 2 2 Boundary conditions on B, H H||1 H||2 H1sin1 H2sin 2 B 1 B 2 B1cos1 B2cos 2 r1 oH1cos1 r2 oH2cos 2 H1sin1 H2sin 2 r1 oH1cos1 r2 oH2cos 2 tan 1 r1 tan 1 r1 c.f. tan 2 r2 tan 2 r2