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Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS

Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS. Sub-atomic – pm-nm. Macro – m m-mm. Mesoscale – nm- m m. Module 1 – 01/02/2001 – Introduction. Context. Magnetism is a physical phenomenon that intrigued scientists and laymen alike for centuries.

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Magnetism and Magnetic Materials DTU (10313) – 10 ECTS KU – 7.5 ECTS

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

  2. 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: A tangible macroscopic manifestation of the quantum world

  3. Meet your teachers KL CF MB MFH BMA JBH LTK JOB

  4. Timeline The basics Advanced topics Advanced topics Tuesday, Feb 1 13:00-16:45 Introduction MB Friday, Feb 4 08:15-12:00 Isolated magnetic moments MB Tuesday, Feb 8 13:00-16:45 Crystal fields MB Friday, Feb 11 08:15-12:00 Interactions MB Tuesday, Feb 15 13:00-16:45 Magnetic order MB Tuesday, Mar 8 13:00-16:45 Order and broken symmetry KL Tuesday, Mar 29 13:00-16:45 Thermodynamics LTK GMR and spintronics JBH Friday, Apr 1 08:15-12:00 Magnetism in metals BMA Experimental methods and applications Assigments deadlines Friday, Mar 11 08:15-12:00 Bulk measurements/dynamics KL Tuesday, Mar 15 13:00-16:45  Nanoparticles I CF Friday, Mar 18 08:15-12:00  Nanoparticles II CF Tuesday, Mar 22 13:00-16:45 Magnetization measurements MFH Hard disks MFH Friday, Mar 25 08:15-12:00 Imaging and characterization CF Feb 18 A1 Mar 11 A2 Mar 26 A3 May 9 PW (DTU only) The mesoscale Evaluation Friday, Feb 18 08:15-12:00 Micromagnetism I MB Tuesday, Feb 22 13:00-16:45 Micromagnetism II MB Friday, Feb 25 08:15-12:00 Macroscopic magnets MB 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)

  5. 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

  6. 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

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

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

  9. Electron in motion and magnetic moment S L v I m Calculate the classical electron velocity for some hypothetical l=1 state with R=a0.

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

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

  12. 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 Classical Quantum mechanical

  13. Connection with Quantum Mechanics A classical system of charges at thermal equilibrium has no net magnetization. The Bohr-van Leeuwen theorem Perpendicular to velocity No work No work = no change in energy = no magnetization “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 Z independent of B, ergo M=0

  14. 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. (a) (a) (b) (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 From: M. Fowler’s website, U. Virginia

  15. Quantum mechanics of spin Quantum numbers: n,l,ml,s,ms Orbital angular momentum: l,ml; l(l+1) is the eigenvalue of L2 (in hbar units) ml is the projection of L along an axis of choice (e.g. Lz) The resulting magnetic moment is m2=l(l+1)mB and mz=-mlmB Spin angular momentum: s,ms; s(s+1) is the eigenvalue of S2 (in hbar units) ms is the projection of S along an axis of choice (i.e. Sz) The resulting magnetic moment is m2=gs(s+1)mB and mz=-gmsmB Zeeman splitting: E=gmsmBB (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]

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

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

  18. 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 S2 eigenvalue is then 0 or 2. Hence, the energy levels are: Possible basis: Consider symmetry of wave function for Fermions Eigenstates: triplet and singlet

  19. Sneak peek Paramagnetism Diamagnetism Hund’s rules

  20. Wrapping up • 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 Next lecture: Friday February 4, 8:15, KU Isolated magnetic moments (MB)

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