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Magnetism, Angular Momentum, and Spin Chapter 19 Magnetism, Angular Momentum, and Spin P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin In 1820 Hans Christian Ørsted discovered that electric current produces a magnetic field that deflects compass needle from magnetic north, establishing first direct connection between fields of electricity and magnetism. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Biot-Savart Law Jean-Baptiste Biot and Félix Savart worked out that magnetic field, B⃗, produced at distance r away from section of wire of length dl carrying steady current I is 휇 I d⃗l × ⃗r dB⃗ = 0 Biot-Savart law 4휋 r3 Direction of magnetic field vector is given by “right-hand” rule: if you point thumb of your right hand along direction of current then your fingers will curl in direction of magnetic field. current P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Microscopic Origins of Magnetism Shortly after Biot and Savart, Ampére suggested that magnetism in matter arises from a multitude of ring currents circulating at atomic and molecular scale. André-Marie Ampére 1775 - 1836 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Magnetic dipole moment from current loop Current flowing in flat loop of wire with area A will generate magnetic field magnetic which, at distance much larger than radius, r, appears identical to field dipole produced by point magnetic dipole with strength of radius 휇 = | ⃗휇| = I ⋅ A current Example What is magnetic dipole moment induced by e* in circular orbit of radius r with linear velocity v? * 휋 Solution: For e with linear velocity of v the time for one orbit is torbit = 2 r_v. Taking current as charge, *qe, passing point per unit time we have q v I = *q _t = * e e orbit 2휋r Taking area of current loop as A = 휋r2 we obtain magnetic dipole moment of ( ) *q v q 휇 = IA = e .휋r2/ = * e r v 2휋r 2 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Electron Orbital Magnetic Dipole q Recalling 휇 = * e r v, for electron with angular momentum, rearrange L⃗ = m⃗r × v⃗, to get 2 ⃗ q ⃗ L ⃗ ⃗ ⃗휇 e L 훾 ⃗ 훾 ≡ = r × v and then obtain = * = * LL where L qe_.2me/ m 2 me 훾 L is called orbital gyromagnetic ratio : ratio of magnetic moment to orbital angular momentum magnetic Angular momentum vector is antiparallel to magnetic dipole dipole vector due to negative electron charge. Don’t forget : current flow is defined in opposite direction of electron flow. radius ⃗휇 훾 ⃗ Despite simple derivation, = * LL is general and holds for current non-circular orbits as long as angular momentum is conserved. Angular ⃗휇 훾 ⃗ = * LL is independent of size, period, and even shape of momentum orbit. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Electron Orbital Magnetic Dipole Operator From e* orbital motion the magnetic dipole moment operator is q 휇 ⃗̂휇 e ⃗̂ B ⃗̂ l = * L = * ` L 2me 휇 B is defined as Bohr magneton (an atomic unit for dipole moments) and given by q ` 휇 ≡ e : *24 B = 9 274009992054043 × 10 J/T 2me ù 휇 휇 l l Magnitude of dipole moment is l = B . + 1/ Operator for z component of electron orbital magnetic dipole moment is 휇 ̂휇 B ̂ z = * ` Lz For z (or x or y) components only observed with quantized values of 휇 휇 휇 B ê̂ ë B ` 휇 z = * ` Lz = * ` .ml / = * Bml P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical magnetic dipole in a uniform magnetic field Energy of classical magnetic dipole in uniform magnetic field is x Potential energy is N ⃗ ⃗ V = * ⃗휇 ⋅ B = *| ⃗휇||B| cos 휃 z S Note: 휃 = 180◦ is an unstable equilibrium point, i.e., potential is a maximum at 휃 = 180◦. Take kinetic energy as K = 1 I!2, then total energy is 2 1 1 E = I!2 * ⃗휇 ⋅ B⃗ = I!2 * | ⃗휇||B⃗| cos 휃 2 2 With the magnetic dipole and magnetic field vectors in the x-z plane, magnetic dipole experiences torque about y given by [ ] ⃗휏 ⃗휇 ⃗ | ⃗휇||⃗| 휃 ⃗ = × B = B sin ey In absence of friction total energy remains constant and compass needle oscillates about direction of magnetic field. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Magnetic dipole with angular momentum Spinning top precesses in gravitational field. Magnetic dipole with angular momentum precesses in magnetic field Precession Direction N S ⃗ ⃗ dJ ⃗휏 ⃗ ⃗ dJ = = R × Mg = ⃗휏 = ⃗휇 × B⃗ dt dt P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin e*’s magnetic dipole from orbital motion in uniform magnetic field Zeeman interaction 휇 ⃗ For e* with orbital motion substitute ⃗̂휇 B ̂ into ⃗휇 ⋅ ⃗ and obtain l = * ` L V = * B 휇 ̂ B ⃗̂ ⋅ ⃗ V = * ` L B Coupling of e*’s magnetic dipole moment to external magnetic field is called Zeeman interaction. ⃗ ⃗ Taking magnetic field as along z axis, B = Bzez, expression simplifies to 휇 ̂ B ̂ V = * ` LzBz Zeeman interaction lifts degeneracy of ml levels of electron in H-atom. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical magnetic dipole with angular momentum in uniform magnetic field Place classical magnetic dipole with anti-parallel angular momentum in magnetic field and magnetic moment experiences torque ⃗휏 ⃗휇 ⃗ 훾 ⃗ ⃗ 훾 ⃗ ⃗ = × B = * LL × B = LB × L Torque causes change in angular momentum. Newton’s 2nd law says dL⃗ = ⃗휏 = 훾 B⃗ × L⃗ dt L ⃗ ⃗ Taking B = Bzez, and defining 훾 Ω0 = * LBz we write cross product in determinant form | | | e⃗ e⃗ e⃗ | ⃗ | x y z | dL | | ⃗ ⃗ = | 0 0 −Ω0 | = LyΩ0ex * LxΩ0ey dt | | | Lx Ly Lz | Determinant gives 3 coupled differential equations... P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical magnetic dipole with angular momentum in uniform magnetic field dL dLy dL x = Ω L ; = −Ω L ; z = 0 dt 0 y dt 0 x dt Coupled differential eqs. involving Lx and Ly can be decoupled by defining L, = Lx , iLy Then we find dL dL dLy + = x + i = Ω L * iΩ L = *iΩ .L + iL / = *iΩ L dt dt dt 0 y 0 x 0 x y 0 + dL dL dLy * = x * i = Ω L + iΩ L = iΩ .L * iL / = iΩ L dt dt dt 0 y 0 x 0 x y 0 * or dL, = -iΩ L whose solution is L .t/ = L .0/e-iΩ0t dt 0 , , , ⃗휇 훾 ⃗ Converting L, back to Cartesian and taking = * LL we find ... P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical magnetic dipole with angular momentum in uniform magnetic field 휇 휇 휇 x.t/ = x.0/ cos Ω0t + y.0/ sin Ω0t 휇 휇 휇 y.t/ = y.0/ cos Ω0t * x.0/ sin Ω0t 휇 휇 z.t/ = z.0/ These equations for magnetic dipole vector components describe precession motion of vector around magnetic field direction at angular frequency of Ω0. When magnetic dipole vector has angular momentum torque does not cause dipole to align with B⃗, but rather causes it to precess about direction of B⃗. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical Magnetic dipole in a Non-Uniform Magnetic Field P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical Magnetic dipole in a Non-Uniform Magnetic Field If magnetic dipole is in non-uniform magnetic field, B⃗.⃗r/, then forces acting on 2 poles will not cancel and dipole starts to translate along magnetic field gradient direction. N In this case net force is ( ) ⃗ ⃗ ⃗ ⃗휇 ⋅ ⃗ ⃗ Fnet = F+ + F* = ( B.r/ S Dipole oriented in magnetic field gradient is pulled up. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical Magnetic dipole in a Non-Uniform Magnetic Field S Differently oriented dipole in same magnetic field gradient is pulled down. N P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical Magnetic dipole in a Non-Uniform Magnetic Field When dipole is perpendicular to lines of force S there is no translation of dipole. N P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical Magnetic dipole in a Non-Uniform Magnetic Field S N S N N S Note that magnetic field lines also include constant magnetic field component. Also expect this magnetic field to exert torque on all three dipoles causing them to re-orient their north poles upwards. All three dipole orientations would eventually be pulled up in this magnetic field. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Magnetic dipoles with angular momentum On the other hand, if the magnetic dipoles also have angular momentum... S N S N N S ...then torque from constant magnetic field component causes ⃗휇 to precess about constant magnetic field direction instead of reorienting. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Classical Magnetic dipole in a Non-Uniform Magnetic Field Now with Angular Momentum Take example of magnetic field given by ⃗ ⃗ ⃗ ⃗ B.r/ = *gxex + .B0 + gz/ez g is gradient strength. Maxwell’s equations require ( ⋅ B⃗ = 0, so need gradient along x to satisfy this constraint. Thus we find ( ) ( ) ( ) ⃗ ⃗휇 ⋅ ⃗ ⃗ ⃗휇 ⋅ ⃗ ⃗ 휇 ⃗ 휇 ⃗ Fnet = *(V = ( B.r/ = ( .*gxex + .B0 + gz/ez = ( *gx xex + .B0 + gz/ zez 휇 ⃗ 휇 ⃗ = *g xex + g zez Unlike compass needle, we want to consider magnetic dipole with angular momentum.
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