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Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS Sub-atomic – pm-nm Mesoscale – nm- m Macro – m-mm Module 1 – 01/02/2001 – Introduction

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Context Magnetism is a physical phenomenon that intrigued scientists and laymen alike for centuries. Some materials attract or repel each other, depending on their orientation. Experimentally, it became soon clear that magnetism was related to the motion of charges. But how exactly? And why? Classical physics gives us a basic framework, but doesn’t help us much in developing a coherent and comprehensive bottom-up picture The advent of QM provided some answers Relativity provided some additional answers How far can we go? Can we understand what “magnetism” is, and how a magnet works? Yes we can… Magnetism is a physical phenomenon that intrigued scientists and laymen alike for centuries. Some materials attract or repel each other, depending on their orientation. Experimentally, it became soon clear that magnetism was related to the motion of charges. But how exactly? And why? Classical physics gives us a basic framework, but doesn’t help us much in developing a coherent and comprehensive bottom-up picture The advent of QM provided some answers Relativity provided some additional answers How far can we go? Can we understand what “magnetism” is, and how a magnet works? Yes we can… Magnetism: A tangible macroscopic manifestation of the quantum world

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Meet your teachers KLCFMBMFHBMAJBHLTKJOB

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Timeline Tuesday, Feb 113:00-16:45 IntroductionMB Friday, Feb 408:15-12:00 Isolated magnetic momentsMB Tuesday, Feb 813:00-16:45 Crystal fieldsMB Friday, Feb 1108:15-12:00 InteractionsMB Tuesday, Feb 1513:00-16:45 Magnetic orderMB The basics Friday, Feb 18 08:15-12:00 Micromagnetism IMB Tuesday, Feb 2213:00-16:45 Micromagnetism IIMB Friday, Feb 2508:15-12:00 Macroscopic magnetsMB The mesoscale Friday, Mar 1108:15-12:00 Bulk measurements/dynamics KL Tuesday, Mar 15 13:00-16:45 Nanoparticles ICF Friday, Mar 1808:15-12:00 Nanoparticles IICF Tuesday, Mar 2213:00-16:45 Magnetization measurements MFH Hard disks MFH Friday, Mar 2508:15-12:00 Imaging and characterization CF Advanced topics Tuesday, Mar 2913:00-16:45 ThermodynamicsLTK GMR and spintronicsJBH Friday, Apr 108:15-12:00 Magnetism in metalsBMA Advanced topics 30/3 or 4/4 Oral exam (KU students) 4/4-9/5 Oral exam (DTU students) 26/5 or 27/5 Oral exam (DTU students) Evaluation Experimental methods and applications Tuesday, Mar 813:00-16:45 Order and broken symmetryKL Feb 18A1 Mar 11A2 Mar 26A3 May 9PW (DTU only) Assigments deadlines

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Workload Your homework will be: Go through what you have learned in each Module, and be prepared to present a “Flashback” at the beginning of the next Module Carry out home-assignments (3 of them) Self-study the additional reading material given throughout the course Your group-work will be: Follow classroom exercise sessions with Jonas DTU only: project work Your final exam will be: Evaluation of the 3 home-assignments Oral exam DTU only: evaluation of the written report on the project work

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This course will be successful if… Macroscopic magnets, how they work (MB) In depth (QM) explanation of bound currents (ODJ) I know why some things are magnetic (JJ) Know more about magnetic monopoles (ODJ) Lorentz transformations of B and E (MB). Students’ feedback to be gathered in the classroom – 01/02/2011

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Intended Learning Outcomes (ILO) (for today’s module) 1.Describe the logic and structure of this course, and what will be learned 2.List the electron’s characteristics: charge, mass, spin, magnetic moment 3.Predict the main features of electron motion in presence of an applied field 4.Calculate the expression and values of Larmor and cyclotron frequencies 5.Define the canonical momentum, and explain its usefulness 6.Describe the connections between magnetism and i) QM, ii) Relativity 7.Write down simple spin Hamiltonians, and solve them in simple cases 8.Manipulate consistently spin states (spinors) with spin operators

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Meet the electron Mass: m e =9.10938215(45) 10 -31 Kg Charge: e=-1.602176487(40) 10 -19 C Spin: 1/2 Magnetic moment: ~1 B Size: <10 -22 m (from scattering) Classical radius: 2.8 fm (little meaning) Calculate the classical electron radius

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Electron in motion and magnetic moment L v I S Calculate the classical electron velocity for some hypothetical l=1 state with R=a 0.

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Precession Since magnetic moment is linked with angular momentum… Einstein-De Haas Barnett coil Ferromagnetic rod Calculate the Larmor precession frequency B, z

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Electron motion in applied field The Lorentz force Calculate the cyclotron frequency B, z y x - x y z Left or right? B

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More in general: canonical momentum To account for the influence of a magnetic field in the motion of a point charge, we “just” need to replace the momentum with the canonical momentum in the Hamiltonian ClassicalQuantum mechanical

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Connection with Quantum Mechanics A classical system of charges at thermal equilibrium has no net magnetization. The Bohr-van Leeuwen theorem “It is interesting to realize that essentially everything that we find in our studies of magnetism is a pure quantum effect. We may be wondering where is the point where the h=/=0 makes itself felt; after all, the classical and quantum Hamiltonians look exactly the same! It can be shown […] that the appearance of a finite equilibrium value of M can be traced back to the fact that p and A do not commute. Another essential ingredient is the electron spin, which is a purely quantum phenomenon.” P. Fazekas, Lecture notes on electron correlation and magnetism Perpendicular to velocity No work No work = no change in energy = no magnetization Z independent of B, ergo M=0

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Connection with Relativity B and E, two sides of the same coin. No surprise we always talk about Electromagnetism as a single branch of physics. From: M. Fowler’s website, U. Virginia (a) (b) (a) (b) Hint: Lorentz contraction Restore relativity and show that the force experienced in (a) and (b) is the same, although in (b) the force is electric

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Quantum mechanics of spin Quantum numbers: n,l,m l,s,m s Orbital angular momentum: l,m l ; l(l+1) is the eigenvalue of L 2 (in hbar units) m l is the projection of L along an axis of choice (e.g. L z ) The resulting magnetic moment is 2 =l(l+1) B and z =-m l B Spin angular momentum: s,m s ; s(s+1) is the eigenvalue of S 2 (in hbar units) m s is the projection of S along an axis of choice (i.e. S z ) The resulting magnetic moment is 2 =gs(s+1) B and z =-gm s B Zeeman splitting: E=gm s B B (remember the Stern-Gerlach experiment) The g-factor (with a value very close to 2) is one difference between oam and spin Another difference is that l can only be integer, while s may be half-integer Also, spin obeys a rather unique algebra (spinors instead of “normal” vectors) Other than that, they behave similarly. But there are consequences… [exercise on EdH effect]

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Pauli matrices and spin operators s=1/2 Generic spin state Commutators Ladder operators

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Stern Gerlach What is the final state? Will the final beam split?

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The simplest spin Hamiltonian: coupling of two spins Combining two s=1/2 particles gives an entity with s=0 or s=1. The total S 2 eigenvalue is then 0 or 2. Hence, the energy levels are: Possible basis: Consider symmetry of wave function for Fermions Eigenstates:triplet and singlet

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Sneak peek Diamagnetism Paramagnetism Hund’s rules

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Wrapping up Next lecture: Friday February 4, 8:15, KU Isolated magnetic moments (MB) Magnetic moment Electron motion under an applied field Precession of magnetic moments Magnetism as a quantum-relativistic phenomenon Einstein-de Haas effect Orbital and spin angular momentum Spin behaves strangely Stern-Gerlach Coupling of spins

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