Basic Concepts in Magnetism J. M. D. Coey School of Physics and CRANN, Trinity College Dublin Ireland. 1. Magnetostatics 2. Magnetism of multi-electron atoms 3. Crystal field 4. Magnetism of the free electron gas 5. Dilute magnetic oxides Comments and corrections please: jcoey@tcd.ie www.tcd.ie/Physics/Magnetism
2 . Magnetism of multi-electron atoms
2.1 Einstein-de Hass Experiment Demonstrates the relation between magnetism and angular momentum. A ferromagnetic rod is suspended on a torsion fibre. The field in the solenoid is reversed, switching the direction of magnetization of the rod. An angular impulse is delivered due to the reversal of the angular momentum of the electrons- conservation of angular momentum. Three huge paradoxes; — Amperian surface currents — Weiss molecular field — Bohr - van Leeuwen theorem
The electron The magnetic properties of solids derive essentially from the magnetism of their electrons. (Nuclei also possess magnetic moments, but they are ≈ 1000 times smaller) . An electron is a point particle with: mass m e = 9.109 10 -31 kg charge -e = -1.602 10 -19 C intrinsic angular momentum (spin) ħ /2 = 0.527 10 -34 J s The same magnetic m moment, the Bohr Magneton, µ B = 9.27 10 - 24 Am 2 is associated with � � ђ /2 of spin angular I l momentum or ħ of orbital angular momentum (a) (b) Orbital moment Spin On an atomic scale, magnetism is always associated with angular momentum . Charge is negative, hence the angular momentum and magnetic moment are oppositely directed
2.2 Origin of Magnetism 1930 Solvay conference At this point it seems that the whole of chemistry and much of physics is understood in principle. The problem is that the equations are much to difficult to solve….. P. A. M. Dirac
2.3 Orbital and Spin Moment Magnetism in solids is due to the angular momentum of electrons on atoms. Two contributions to the electron moment: • Orbital motion about the nucleus • Spin - the intrinsic (rest frame) angular m momentum. m m = - ( µ B / ħ )( l + 2 s )
2.3.1 Orbital moment Circulating current is I ; I = -e/ τ = -e v /2 π r The moment is * m = I A m = -e vr /2 Bohr: orbital angular momentum l is quantized in units of ħ ; h is Planck’s constant = 6.6226 10 -34 J s; ħ = h/2 π = 1.055 10 -34 J s. | l| = n ħ Orbital angular momentum: l = m e r x v Units: J s Orbital quantum number l, l z = m l ħ m l =0, ± 1, ± 2,..., ± l so m z = - m l (e ħ /2 m e ) The Bohr model provides us with the natural unit of magnetic moment m z = m l µ B Bohr magneton µ B = (e ħ /2m e ) µ B = 9.274 10 -24 A m 2 In general m m = γ l γ = gyromagnetic ratio Orbital motion γ =-e/2m e * Derivation can be generalized to noncircular orbits: m = I A for any planar orbit .
g-factor Ratio of magnitude of m in units of µ B to magnitude of l in units of ħ . ( m m / µ B ) = g( l / ħ ) g = 1 for orbital motion The Bohr model also provides us with a natural unit of length, the Bohr radius a 0 = 4 πε 0 ħ 2 /m e e 2 a 0 = 52.92 pm And a natural unit of energy, the Rydberg R 0 R 0 = (m/2 ħ 2 )(e 2 / 4 πε 0 ) 2 R 0 = 13.606 eV
2.3.2 Spin moment Spin is a relativistic effect. Spin angular momentum s Spin quantum number s s = 1/2 for electrons Spin magnetic quantum number m s m s = ± 1/2 for electrons s z = m s ħ m s = ± 1/2 for electrons m m = -(e/ m e ) s m z = -(e/m e ) m s ħ = ±µ B For spin moments of electrons we have: γ = -e/m e g ≈ 2 More accurately, after higher order corrections: g = 2.0023 m z = 1.00116µ B An electron will usually have both orbital and spin angular momentum m = - ( µ B / ħ )( l + 2 s )
Quantum mechanics of spin In quantum mechanics, physical observables are represented by operators - differential or matrix. e.g. momentum p = -i ħ ∇ ; energy p 2 /2m e = - ħ 2 ∇ 2 n magnetic basis states ⇒ n x n Hermitian matrix Pauli spin matrices Spin operator (for s = 1/2) s = σ ħ /2 Electron: s = 1/2 ⇒ m s = ± 1/2 i.e spin up and spin down states Represented by column vectors: | ↑〉 = | ↓〉 = s | ↑〉 = ( ħ /2 ) | ↑〉 ; s | ↓〉 = - ( ħ /2 )| ↓〉 Eigenvalues of s 2 : s(s+1) ħ 2 The fundamental property of angular momentum in QM is that the operators satisfy the commutation relations: or Where [A,B] = AB - BA and [A,B] = 0 ⇒ A and B’s eigenvalues can be measured simultaneously [ s 2 , s z ] = 0
Quantized spin angular momentum of the electron z - M S 1/2 H - ħ /2 -1/2 s = 1/2 g √ [s(s+1)] ħ 2 2 µ 0 µ B H - 1/2 ħ /2 1/2
The electrons have only two eigenstates, ‘spin up’( ↑ , m s = 1/2 ) and ‘spin down’ ( ↓ , m s = -1/2), which correspond to two possible orientations of the spin moment relative to the applied field. Populations of the energy levels are given by Boltzmann statistics; ∝ exp{-E i / k Β T}. The thermodynamic average 〈 m 〉 is evaluated from these Boltzmann populations. 〈 m 〉 = [µ B exp(x) - µ B exp(-x)] where x = µ 0 µ B H /k B T. [exp(x) + exp(-x)] 〈 m 〉 = µ B tanh(x) In small fields, tanh(x) ≈ x, hence the susceptibility χ = N 〈 m 〉 /H χ = µ 0 Nµ B 2 /k B T This is again the famous Curie law for the susceptibility, which varies as T -1 . In other terms χ = C/T, where C = µ 0 Nµ B 2 /k B is a constant with dimensions of temperature; Assuming an electron density N of 6 10 28 m -3 gives C ≈ 0.5 K. The Curie law susceptibility at room temperature is of order 10 -3 .
2.4 Spin-Orbit Coupling Spin and angular momentum coupled to create total angular momentum j . m = γ j From the electron’s point of view, the nucleus revolves round it with speed v ⇒ current loop I = Ze v /2 π r Which produces a magnetic field µ 0 I /2 r at the centre B so = µ 0 Ze v /2 π r 2 E =- m . B E so = - µ B B so Since r ≈ a 0 /Z and m e vr ≈ ђ E so ≈ - µ 0 µ B 2 Z 4 /4 π a 0 3
2.5 Magnetism of the hydrogenic atom Orbital angular momentum The orbital angular momentum operators also satisfy the commutation rules: l x l = I ђ l and [ l 2 , l z ]=0 Spherical coordinates x = r sin θ cos φ y = r sin θ sin φ z = r cos θ
QM operators for orbital angular momentum (l is the orbital Eigenvalues of l 2 : angular momentum l(l+1) ħ 2 quantum number) l=1 case m l = 1 , 0, - 1 corresponds to the eigenvectors l x , l y and l z operators can be represented by the matrices;: where
Single electron wave functions Schrodinger’s equation: Satisfied by the wavefunctions: l are Laguerre polynomials V 0 (V n 1 =1) Where: (Legendre polynomials) And the combined angular parts are Normalized spherical harmonics:
The three quantum number n,l m l denote an orbital. Orbitals are denoted nx ml , x = s,p,d,f... for l = 0,1,2,3,... Each orbital can accommodate at most two electrons* ( m s = ± 1/2) The hydrogenic orbitals: An orbital can accommodate 2(2 l +1) electrons. * The Pauli exclusion principle: No two electrons can have the same four quant
Hydrogenic orbitals
2.6 Magnetic Periodic Table 1 H 2 He 1.00 4.00 66 Dy Atomic Number Atomic symbol 4 Be 5 B 6 C 7 N 8 O 9 F 10 Ne 3 Li 162.5 Atomic weight Typical ionic change 3 + 4 f 9 6.94 9.01 10.81 12.01 14.01 16.00 19.00 20.18 Antiferromagnetic T N (K) 179 85 Ferromagnetic T C (K) 2 + 2 s 0 1 + 2 s 0 35 11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar 26.98 28.09 30.97 32.07 22.99 24.21 35.45 39.95 1 + 3 s 0 2 + 3 s 0 3 + 2 p 6 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr 55.85 72.61 38.21 40.08 44.96 47.88 50.94 52.00 55.85 58.93 58.69 63.55 65.39 69.72 74.92 78.96 83.80 79.90 1 + 4 s 0 2 + 4 s 0 3 + 3 d 0 4 + 3 d 0 3 + 3 d 2 3 + 3 d 3 2 + 3 d 5 3 + 3 d 5 2 + 3 d 7 2 + 3 d 8 2 + 3 d 9 2 + 3 d 10 3 + 3 d 10 312 96 1043 1390 629 37 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I 54 Xe 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 87.62 88.91 91.22 92.91 95.94 97.9 101.1 102.4 106.4 107.9 112.4 114.8 118.7 121.8 127.6 126.9 83.80 85.47 1 + 5 s 0 2 + 5 s 0 2 + 4 d 0 4 + 4 d 0 5 + 4 d 0 5 + 4 d 1 3 + 4 d 5 3 + 4 d 6 2 + 4 d 8 1 + 4 d 10 2 + 4 d 10 3 + 4 d 10 4 + 4 d 10 56 Ba 57 La 72 Hf 73 Ta 74 W 75 Re 76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 Ti 82 Pb 83 Bi 84 Po 86 Rn 55 Cs 85 At 137.3 138.9 178.5 180.9 183.8 186.2 190.2 192.2 195.1 197.0 200.6 204.4 207.2 209.0 209 210 222 13.29 2 + 6 s 0 3 + 4 f 0 4 + 5 d 0 5 + 5 d 0 6 + 5 d 0 4 + 5 d 3 3 + 5 d 5 4 + 5 d 5 2 + 5 d 8 1 + 5 d 10 2 + 5 d 10 3 + 5 d 10 4 + 5 d 10 1 + 6 s 0 87 Fr 88 Ra 89 Ac 223 226.0 227.0 2 + 7 s 0 3 + 5 f 0 58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu 63 Eu 140.1 140.9 144.2 145 150.4 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0 152.0 4 + 4 f 0 3 + 4 f 2 3 + 4 f 3 3 + 4 f 5 3 + 4 f 7 3 + 4 f 8 3 + 4 f 9 3 + 4 f 10 3 + 4 f 11 3 + 4 f 12 3 + 4 f 13 3 + 4 f 14 2 + 4 f 7 13 19 105 292 229 221 179 85 132 20 85 20 56 90 90 Th 91 Pa 92 U 93 Np 94 Pu 95 Am 96 Cm 97 Bk 98 Cf 99 Es 100 Fm 101 Md 102 No 103 Lr 232.0 231.0 238.0 238.0 244 243 247 247 251 252 257 258 259 260 4 + 5 f 0 5 + 5 f 0 4 + 5 f 2 5 + 5 f 2 Diamagnet Ferromagnet T C > 290K Nonmetal Paramagnet Antiferromagnet with T N > 290K Metal Antiferromagnet/Ferromagnet with T N /T C < 290 K Radioactive Magnetic atom BOLD
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