Introduction to magnetism Part I - Moments Olivier Fruchart Institut Néel (CNRS-UJF-INPG) Grenoble - France http://neel.cnrs.fr Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
INTRODUCTION TO MAGNETISM – Currents, magnetic fields and magnetization Oersted field Magnetic dipole Magnetic material μ µ = 0 B π 3 µ 4 r I = B 0 3 π 2 r × − ( μ.r ) r μ 2 r Magnetization: A.m -1 Magnetic dipole: A.m 2 Magnetic moment: A.m 2 Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.2 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
INTRODUCTION – Hysteresis and magnetic materials Manipulation of magnetic materials: Application of a magnetic field S pontaneous ≠ S aturation = − µ E H.M Zeeman energy: Z 0 s Spontaneous magnetization M s Remanent magnetization M r Another notation M M J s = 0 M s Coercive field H c Hext Hext Losses ∫ = µ E H d M 0 ext Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.3 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
→ Introduction to magnetism – Moments ToC 1. Magnetic moments 2. Magnetism of single atoms 3. Moments in fields 4. Magnetic ordering 5. Magnetism in metals 6. Magnetic anisotropy Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.4 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
Fundamental references Repository of lectures of the European School on Magnetism: http://esm.neel.cnrs.fr Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.5 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
1. MAGNETIC MOMENTS – Angular momentum Classical physics Quantum mechanics Heisenberg uncertainty principle: General definition − 34 J.s h = 6.626 × 10 r . p ≥ℏ/ 2 − 3 4 J.s ℓ = ∭ r . r × v r . d r ℏ= h / 2 = 1.0546 × 10 is a natural measure for angular ℏ momenta. Niels Bohr's postulate, Electron orbiting around a quantization of angular momentum charged nucleus ∣ r × p ∣ ∈ ℏ ℤ Figures 2 / 4 0 r 2 = m e v 2 / r Newton's law: Ze Quantized l : ℓ = m e r v = l ℏ 2 with Bohr radius a 0 = 4 0 ℏ 2 a 0 / Z r = n 2 m e e a 0 = 0.529 ˚ A ℓ = r × p ℓ = m e r v =ℏ/ m e a 0 c ≈ 1 / 137.04 (fine structure cte) − 31 kg Electrons are largely not relativistic m e = 9.109 × 10 Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.6 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
1. MAGNETIC MOMENTS – Orbital magnetic moment Classical physics Quantum mechanics ∣ ℓ ∣= l ℏ with l ∈ ℤ General definition l is an orbital magnetic quantum = 1 2 ∭ r × j r d r number. Electron orbiting around B =ℏ=− e ℏ a charged nucleus 2 m e is Bohr magneton, the quantum for magnetic moments = I S Charge − e z = m ℓ B = m ℓ 2 m e ℏ − e 2 I Mass m e = r 2 − ev / 2 r = r m ℓ ∈[− l ; l ] − 19 C e = 1.6 × 10 = − erv / 2 = ℓ Figures with : gyromagnetic ratio =− e − 24 A.m 2 B = 9.274 × 10 for orbital motion of electrons 2 m e Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.7 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
1. MAGNETIC MOMENTS – Spin magnetic moment Overview Spin angular momentum Landé factor g is a dimensionless Spin = intrinsic quantized angular momentum = g ∣ ℓ ∣ =− eg version of : ∣∣ or 2 m e B ℏ Electrons are fermions (half-integer spin) s =± 1 =− e with spin quantum number g = 1 Orbital moment: 2 2 m e ≈− e s ℏ=±ℏ g ≈ 2 Electron spin: Angular momentum m e 2 Figures 28 atoms / m 3 n ≈ 8.4 × 10 With (for Fe) Spin magnetic moment 5 A / m n B ≈ 7.8 × 10 − 15 m Electron radius r e ≈ 2.818 × 10 6 A / m ∣ ℓ ∣= r.p =ℏ/ 2 Measured: M s,Fe ≈ 1.73 × 10 Relativistic, Dirac equation e = 2m e m s − 1 ≈ 2.2 B atom ( Z =26) − eg m s =± 1 with 2 Few electrons involved in ferromagn. and g = 2.0023 ≈ 2 Landé factor ∣∣≈ e / m Electrons carry a spin magnetic moment Nucleon spin moment is weak ≈ B and will be neglected here Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.8 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
1. MAGNETIC MOMENTS – Spin-orbit coupling Semi-classical picture Figures The nucleaus is orbiting in the sitting Spin-orbit is larger for heavy framework of the electron, inducing elements Current of the Reminder: 'orbiting' nucleus' − 7 S.I. 0 = 4 × 10 B so = 0 I n I n = Zev / r − 24 A.m 2 B = 9.274 × 10 m e r v ~ n ℏ 2 r a 0 = 0.529 ˚ A 2 For the spin of one electron: 0 B 4 ~4.2 K / Z 2 Z 3 4 4 a 0 so =− B B so ~ 0 B 4 3 ~0.36 meV / Z 4 a 0 Screening and details of electrons in atoms are required In the range 10-1000meV for magnetic elements Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.9 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
→ Introduction to magnetism – Moments ToC 1. Magnetic moments 2. Magnetism of single atoms 3. Moments in fields 4. Magnetic ordering 5. Magnetism in metals 6. Magnetic anisotropy Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.10 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
2. MAGNETISM IN ATOMS – Shells and quantum numbers Framework Charged nucleus + Z electrons N-body problem → untractable Seek one electron solutions + perturbation theory (→ Hund's rules etc.) 2 2 H =− ℏ Ze 2 − Schrödinger equation H i = i i with hamitonian ∇ 2 m e 4 0 r r , , = R r Seek solutions: Elecrtonic orbitals ℓ r # of m ℓ , = P ℓ m ℓ e i m ℓ R n and Y ℓ with: states n ∈ ℕ Principal quantum number → Sets the main energy levels l n − 1 Secondary quantum number → Orbital moment m ℓ 2 l 1 Magnetic quantum number → z component of orbital moment m s 2 Spin quantum number Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.11 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
2. MAGNETISM IN ATOMS – Filling shells Labeling the levels Series Usage K (n=1), L (n=2), M (n=3), N (n=4) etc. Spectroscopy s ( l =0), p ( l =1), d ( l =2), f ( l =3), g ( l =4) etc. Chemical-physical properties n and l filling (Hartree-Fock) Examples # orbitals 2 6 10 14 18 l =0 l =1 l =2 l =3 l =4 Fe, Z=26 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 6 1s K n=1 1s 2 2 6 2 6 2 6 L n=2 2s 2p Ion Fe2+ 2 2s 2 2p 6 3s 2 3p 6 4s 0 3d 6 3 1s M n=3 3s 3p 3d N n=4 4s 4p 4d 4f Ion Fe3+ 2 2s 2 2p 6 3s 2 3p 6 4s 0 3d 5 3 1s O n=5 5s 5p 5d 5f 5g Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.12 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
2. MAGNETISM IN ATOMS – Momenta and Hund's rules Combination of momenta and moments Angular and spin momenta from all electrons add up: 2 = S S 1 S L = ∑ ℓ i L = ∑ ℓ i S = ∑ s i J = L S 2 = L L 1 and L 2 = J J 1 J Resulting magnetic moment: Landé factor µ = − B /ℏ L 2 S g = 3 2 [ S S 1 − L L 1 ]/ 2J J 1 = g − B /ℏ J Only partly filled shells may display angular momentum and magnetic moment. Hund's rules Or: empirical rules for populating the last partly filled shell Rule Arises from... 1. Maximize S Coulomb interaction and Pauli principle 2. Maximize L consistent with S Coulomb; electrons orbiting same sense 3. J=|L-S| for less than half-filled shells Spin-orbit coupling J=L+S for more than half-filled shells M J ∈[− J , J ] Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.13 Institut Néel, Grenoble, France http://perso.neel.cnrs.fr/olivier.fruchart/slides/ http://perso.neel.cnrs.fr/olivier.fruchart/slides/
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