Magnetism Introduction Introduction magnetism is nothing - PDF document

Basic properties 1 Basic properties 1 Magnetism Introduction Introduction magnetism is nothing new........ magnetism is nothing new........ 1 Paramagnetism 2 Magnetic order 1 2 Basic properties 1 Basic properties 1 Our first contact with magnetism… Technology 3 4 Basic properties 1 Basic properties 1 Plasma spectroscopy, Zeeman effect Magnetic Resonance Imaging 5 6 1

Basic properties 1 Basic properties 1 Quizzzz Ferromagnetism of the elements 1. What are the three sources of atomic magnetism and sort them by size 1.Nuclear spin moment ? 2.Electron spin moment 3.Orbital moment 2. What is Lenz’s law? Lenz’s law (1833): The induced current produced in the conductor always flows in ? such a direction that the magnetic field it produces will oppose the change that produces it. 3. How large is the magnetic field you can generate with: a)A refrigerator magnet Room-temperature magnets b)An iron horseshoe magnet c)A superconducting magnet All other elements are: 4. Which material reacts strongest on an applied magnetic field? Paramagnetic a)Argon or b)Na c)O 2 Diamagnetic d)Fe 3 0 4 7 8 Basic properties 1 Basic properties 1 Names and definitions 5 different types of magnetic materials χ ( T ) H: magnetic field / magnetic field strength Temperature dependence of magnetic susceptibility Temperature dependence of magnetic susceptibility m B: magnetic induction / magnetic field H T C T C Ferromagnetism or Ferromagnetism or B = μ H In vacuum Ferrimagnetism Ferrimagnetism 0 Material in magnetic field H Paramagnetism of of Paramagnetism B = μ H + M ( ) localized electrons localized electrons M = χ H 0 μ ( H , T ) ( T ) M = m M Magnetization tot Antiferromagnetism Antiferromagnetism V Paramagnetism Paramagnetism of free electrons of free electrons T N T N Diamagnetism Diamagnetism M = χ H Magnetic susceptibility m T μ = μ χ = − K K 1 K m = relative permeability m 0 m m 9 10 Basic properties 1 Basic properties 1 Relative permeability @ 20 o C Paramagnetic susceptibility of localized electrons χ m =K m -1 χ m =K m -1 Material Material (x 10 -5 ) (x 10 -5 ) 7 Paramagnetic M = χ H Diamagnetic ( T ) M Iron oxide (FeO) 720 Lage T/hoog veld: Ammonia -.26 μ Verzadigd N Iron amonium alum 66 Bismuth -16.6 B 5 χ > 0 moment Uranium 40 Mercury -2.9 M parallel to H Platinum 26 Silver -2.6 � Attractive force 3 Tungsten 6.8 Carbon (diamond) -2.1 Cesium 5.1 Carbon (graphite) -1.6 Aluminum 2.2 Lead -1.8 1 Lithium 1.4 Sodium chloride -1.4 Magnesium 1.2 0 Copper -1.0 B/T (T/K) Sodium 0.72 Water -0.91 Oxygen gas 0.19 0 1 2 3 4 11 12 2

Basic properties 1 Basic properties 1 localized electrons Champion in localization: RE 4f shell Some valence shells lie deep in the atom Radial expectation value of Rare earth: Radial expectation value of Rare earth: � have little overlap with neighbor atoms hydrogen orbitals hydrogen orbitals Electron configuration 4f N Electron configuration 4f N 5d 5d 1 1 6s 6s 2 2 � atomic orbital moment “survives” band formation Thus three open valence shells! Thus three open valence shells! Which electrons of the elements are these: – 3d transition metals – 4f Rare-earth metals – 5f Early actinides little overlap with neighbor atom http://winter.group.shef.ac.uk/orbitron/AOs/4f/e-density-xzz-dots.html 13 14 Basic properties 1 Basic properties 1 Magnetic dipole moment of atom with 1 electron: Orbital moment in field // z axis μ = − μ l Electron has spin and orbital moment contributing Electron has spin and orbital moment contributing l B l= μ = − μ m each to the magnetic dipole moment each to the magnetic dipole moment l , z B l 0 (s) l = r x p 1 (p) s 2 (d) μ = − μ s e g μ = − r × p = − μ l spin B 0 orbit B 2 m 3 (f) 0 = h e g 2 . 0023 − μ = = l= 1 l= 2 24 9 . 2 x 10 J/T B m l = 2 m 0 1 2 3 15 16 Basic properties 1 Basic properties 1 Spin moment in field // z as Orbital moment in magnetic field − μ . B = μ Bm Zeeman Zeeman energy energy l B l μ = − μ s g s 0 B μ μ = − μ B g m B s , z B 0 s 0 = g 2 . 0023 ≅ ± h h 1 s z 2 − μ B = μ . g 0 Bm Zeeman Zeeman energy energy s B s 17 18 3

Basic properties 1 Basic properties 1 Spin-orbit interaction Spin-orbit splitting • Spin orbit interaction leads to splitting of levels • New eigenstates Electron reference frame: with quantum number Nuclear reference frame rotating proton produces j=l+s or j=l-s magnetic field (Relativistic effect) m j =j,j-1,…-j Spin- -orbit energy orbit energy Spin = λ l ⋅ s for state with quantum number n,l : E − spin orbit • No splitting if l = 0 (s orbitals) • Increases with Z, decreases with n, l 2 Z λ ~ • 10-100 meV for valence shell Labeling: l j + + nl ( l 1 )( l 1 ) • 10-1000 eV for inner shell 2 19 20 Basic properties 1 Basic properties 1 Multiplet splitting of np 2 configuration Total moment of valence shell nl N The total magnetic moment is the sum of all magnetic dipole Configuration Configuration + Coulomb + Coulomb + Spin- + Spin -orbit orbit + magnetic field + magnetic field moments interaction interaction interaction interaction (Zeeman ( Zeeman splitting) splitting) ( ) μ = − μ L + S 2 B ~ 10 eV N ∑ Mulitplet state L = l h h Total atomic orbital-moment i = i 1 N ∑ Hund’s rule S = s h h Total atomic spin-moment i ground state = i 1 0-1 eV (here J=0) Full shells have no dipole moment: ns 2 , np 6 , nd 10 , nf 14 Partly filled shells have permanent dipole moment Spectroscopic notation: 2S+ 1 L J 21 22 Basic properties 1 Basic properties 1 Coupling schemes for adding orbital and spin angular moments Coupling schemes for adding orbital and spin angular momenta 1. Russel Saunders or L-S coupling if spin-orbit interaction is weak � eigenstates of the atom are also eigenstates of 2. j-j coupling if spin-orbit interaction is dominant L 2 and S 2 with eigenvalues L(L+1) and S(S+1). Add up orbital and spin moments of each electron i N ∑ L = l i = + – Add up all orbital moments = J l s i 1 i i i N ∑ = S s – Add up all spin moments N ∑ i = Add these J i together to J = J J i 1 i = – Add these together to total atomic angular i 1 3. Intermediate coupling complicated no fixed rule moment J = L + S 23 24 4

Basic properties 1 Basic properties 1 Hund’s rules for J of ground state Zeeman effect for state with total moment J Procedure for J of the ground state (L-S coupling): • Ground state J is 2J+ 1 times degenerated: J z = -J, -J+ 1, … J • Splits in magnetic field into sublevels 1. Maximize S= Σ s z,i 2. Maximize L = Σ l z,i J z = − μ ⋅ B = − μ ⋅ H B P z 2 3. J=|L-S| for less than half filled shell 1 L+S for more than half filled shell =< >= μ E H g J B 0 P Lande B z z if subshell subshell is exactly half is exactly half- -filled, filled, L=0, L=0, so so J = S J = S if J -1 Δ = μ E g B example: example: -2 + − + 5 L B z 3 L ( L 1 ) S ( S 1 ) S 4.5 = − 3d g Lande 4 L 3d transition metals: 3d transition metals: + 2 2 J ( J 1 ) 3.5 J 3 B Fill the 3d orbital (l= 2) Fill the 3d orbital (l= 2) 2.5 2 1.5 1 • Spectroscopic splitting factor g Landee depends on L, S, and J 0.5 • Splitting at B= 1 Tesla in the order of meV 0 0 5 10 Atom behaves as if it has effective moment: μ eff = -g L μ B J # electrons • 25 26 Basic properties 1 Basic properties 1 Multiplet + magnetic field: Zeeman effect Effective moment often called m Effective moment often called m J J once we have, J J , we can get the maximum component of the once we have, , we can get the maximum component of the Configuration Configuration + Coulomb + Coulomb + Spin- + Spin -baan baan + magnetic field + magnetic field magnetic moment of the atom parallel to the magnetic field: magnetic moment of the atom parallel to the magnetic field: interaction interaction interaction interaction (Zeeman ( Zeeman splitting) splitting) Splitting typically 10 meV pure orbital pure orbital motion = 1, motion = 1, pure spin = 2 pure spin = 2 g , g , Landé Landé splitting factor: splitting factor: Typically 1 eV Here: J= 0 ground state is non-magnetic singlet 27 28 Basic properties 1 Basic properties 1 Temperature dependence Curie’s law 7 J z ⎛ μ ⎞ 7 g BJ ⎜ ⎟ = μ B 2 M ( B , T ) N gJ B ⎜ ⎟ χ M B J ⎝ k T ⎠ ( T ) saturated B 1 m 5 μ moment N M B 0 for not too small T: 5 μ J Curie’s law N -1 B μ μ 3 Δ = μ 2 2 E g B Np -2 L B z χ = B 0 ( T ) m 3 k T magnetic moment of system with N ions at T is magnetic moment of system with N ions at T is 3 B 1 determined by Boltzmann determined by Boltzmann statistics: statistics: [ ( ) ] 2 1 = + J − μ p g J J 1 ∑ B/T (T/K) g BJ − μ J Landee g J exp( B z ) 1 B z k T Effective number of magnetons 0 = − B = J J M z 0 1 2 3 4 1 J − μ T ∑ g BJ exp( B z ) k T = − B J J z 0 ⎛ μ ⎞ 0 1 2 3 4 g BJ = μ ⎜ ⎟ B M ( B , T ) N gJ B ⎜ ⎟ B J B J Brillouin function T ⎝ k T ⎠ B/T (T/K) B 29 30 5