Basic Concepts in Magnetism; Many-electron atoms J. M. D. Coey - PowerPoint PPT Presentation

Basic Concepts in Magnetism; Many-electron atoms J. M. D. Coey School of Physics and CRANN, Trinity College Dublin Ireland. 1. Spin-orbit interaction 2. Magnetism of single-electron atom 3. Magnetism of many-electron atoms 4. Paramagnetism 5. Crystal field Comments and corrections please: jcoey@tcd.ie www.tcd.ie/Physics/Magnetism

1 Introduction 2 Magnetostatics 3 Magnetism of the electron è 4 The many-electron atom 5 Ferromagnetism 6 Antiferromagnetism and other magnetic order 7 Micromagnetism 8 Nanoscale magnetism 9 Magnetic resonance 10 Experimental methods 11 Magnetic materials 12 Soft magnets 13 Hard magnets 14 Spin electronics and magnetic recording 614 pages. Published March 2010 15 Other topics Appendices, conversion tables. www.cambridge.org/9780521816144 ESM Cluj 2015

1 . Spin-orbit interaction ESM Cluj 2015

Spin-Orbit Coupling Spin and angular momentum coupled to create total angular j momentum j . J = l + s s m = γ j l From the electron ’ s point of view, the nucleus revolves round it with speed v ⇒ current loop. It is a relativistic effect 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 /4 π r 2 [~10 T for B or C] E = - m . B E so = - µ B B so E so ≈ - µ 0 µ B 2 Z 4 /4 π a 0 3 Since r ≈ a 0 /Z and m e vr ≈ ħ The spin – orbit Hamiltonian for a single electron is of the form: in general H so = (1/2m e 2 c 2 r)d V /dr l.s H so = λ ˆ l · ˆ s , Here the two ħ s have been assimilated into λ , making it an energy (c.f. exchange) ESM Cluj 2015

2 . Single-electron atom ESM Cluj 2015

Orbital angular momentum The orbital angular momentum operators also satisfy the commutation rules: l x l = I ħ l and [ l 2 , l z ] = 0 Electron -e Spherical polar coordinates x = r sin θ cos φ Nucleus Ze y = r sin θ sin φ z = r cos θ = × ˆ l = r x p = l = − i¯ h ( y ∂ / ∂ z − z ∂ / ∂ y ) e x − i¯ h ( z ∂ / ∂ x − x ∂ / ∂ z ) e y − i¯ h ( x ∂ / ∂ y − y ∂ / ∂ x ) e z . (3.14) ESM Cluj 2015

Orbital angular momentum operators Eigenvalues of l 2 : l is the orbital angular momentum quantum number z l(l+1) ħ 2 l =1 case m l � m l = 1 , 0, - 1 corresponds to the eigenvectors √ [ l ( l +1)] � l x , l y and l z operators can be represented by the matrices: where ESM Cluj 2015

Solution of Schrodinger’s equation Schrodinger ’ s equation: Satisfied by the wavefunctions: (V n l are Laguerre polynomials V 0 1 =1) Where: (Legendre polynomials) And the combined angular parts are Normalized spherical harmonics: ESM Cluj 2015

One-electron hydrogenic states n l m l m s No of states The three quantum number n ,l m l denote an orbital. 1s 1 0 0 ±1/2 2 Orbitals are denoted nx ml , 2s 2 0 0 ±1/2 2 x = s, p, d, f... for l = 0,1,2, 3,... 2p 2 1 0,±1 ±1/2 6 Each orbital can accommodate at 3s 3 0 0 ±1/2 2 most two electrons* ( m s = ± 1/2) 3p 3 1 0,±1 ±1/2 6 3d 3 2 0,±1,±2 ±1/2 10 4s 4 0 0 ±1/2 2 4p 4 1 0,±1 ±1/2 6 4d 4 2 0,±1,±2 ±1/2 10 4f 4 3 0,±1,±2,±3 ±1/2 14 * The Pauli exclusion principle: No two electrons can have the same four quantum numbers. ⇒ Two electrons in the same orbital must have opposite spin. ESM Cluj 2015

Single-electron orbitals s electrons P electrons d electrons 0.6 10 0.5 0.4 ρ 2 | R nl ( ρ )| 2 0.3 21 20 0.2 32 31 30 0.1 43 42 41 40 0.0 10 15 20 30 0 5 25 ρ = r /a 0 ESM Cluj 2015

Periodic Table 1 H 2 He 1.00 4.00 66 Dy Atomic Number Atomic symbol 3 Li 4 Be 5 B 6 C 7 N 8 O 9 F 10 Ne 162.5 Atomic weight Typical ionic change 3 + 4 f 9 9.01 10.81 12.01 14.01 16.00 19.00 20.18 6.94 179 85 2 + 2 s 0 Antiferromagnetic T N (K) Ferromagnetic T C (K) 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 32 Ge 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 33 As 34 Se 35 Br 36 Kr 72.61 38.21 40.08 44.96 47.88 50.94 52.00 55.85 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 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 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 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn 55 Cs 137.3 138.9 178.5 180.9 183.8 186.2 190.2 192.2 195.1 197.0 200.6 204.4 207.2 209.0 209 222 210 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 Magnetic atom Antiferromagnet/Ferromagnet with T N /T C < 290 K Radioactive BOLD ESM Cluj 2015

3 . Many-electron atom ESM Cluj 2015

The many-electron atom Hartree-Fock approximation • No longer a simple Coulomb potential. • l degeneracy is lifted. • Solution: Suppose that each electron experiences the potential of a different spherically-symmetric potential. ESM Cluj 2015

Magnetic Periodic Table 1 H 2 He 1.00 4.00 66 Dy Atomic Number Atomic symbol 3 Li 4 Be 5 B 6 C 7 N 8 O 9 F 10 Ne 162.5 Atomic weight Typical ionic change 3 + 4 f 9 9.01 10.81 12.01 14.01 16.00 19.00 20.18 6.94 179 85 2 + 2 s 0 Antiferromagnetic T N (K) Ferromagnetic T C (K) 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 32 Ge 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 33 As 34 Se 35 Br 36 Kr 72.61 38.21 40.08 44.96 47.88 50.94 52.00 55.85 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 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 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 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn 55 Cs 137.3 138.9 178.5 180.9 183.8 186.2 190.2 192.2 195.1 197.0 200.6 204.4 207.2 209.0 209 222 210 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 Magnetic atom Antiferromagnet/Ferromagnet with T N /T C < 290 K Radioactive BOLD ESM Cluj 2015

Addition of angular momenta First add the orbital and spin momenta l i and s i to form L and S. Then couple them to give the total J S J J = L + S ⎢ L-S ⎢ ≤ J ≤ ⎢ L+S ⎢ Different J -states are termed multiplets , denoted by; L 2 S +1 X J X = S, P , D, F, ... for L = 0,1,2,3,... Hund ’ s rules To determine the ground-state of a multi-electron atom/ion. 1) Maximize S 2) Maximize L consistent with S . 3) Couple L and S to form J. Less than half full shell J = L-S • More than half full shell J = L+S • ESM Cluj 2015

Hund ’ s rules; examples Fe 3+ 3d 5 S = 5/2 L = 0 J = 5/2 ↑ ↑ ↑ ↑ ↑ 6 S 5/2 2 1 0 -1 -2 Note; Maximizing S is equivalent to maximizing M s = Σ m si , since M s ≤ S ESM Cluj 2015

Co 2+ 3d 7 ↑ ↑ ↑ ↑ ↑ ↓ ↓ S = 3/2 L = 3 J = 9/2 2 1 0 -1 -2 4 F 9/2 Note; Maximizing L is equivalent to maximizing M L = Σ m li , since M L ≤ L ESM Cluj 2015

Ni 2+ 3d 8 ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↓ S = 1 L = 3 J = 4 2 1 0 -1 -2 3 F 4 ESM Cluj 2015