2. Lecture: Basics of Magnetism: Paramagnetism Hartmut Zabel Ruhr-University Bochum Germany
Content 1. Orbital moments 2. Spin orbit coupling 3. Zeeman splitting 4. Thermal Properties – Brillouin function 5. High Temperature – low temperature approximation 6. Van Vleck paramagnetism 7. Paramagnetism of conduction electrons 2 H. Zabel, RUB 2. Lecture: Paramagnetism
Orbital moments of the d-shell 5 orthogonal and degenerate H z ( ) m orbital wave functions of 3d shell + l l 1 l 2 1 0 − 1 = + = − − = + m 1 m 1 m 2 2 l l l = µ = µ m m g m = − = m m 2 0 l l z l l l B l B , The upper three wave functions have ( ) = − − µ 2 , 1 , 0 , 1 , 2 maxima in the xy, xz, yz planes, the B lower two have maximia along x,y and z coordinate. In a magnetic field the degenerate sublevels split. 3 H. Zabel, RUB 2. Lecture: Paramagnetism
1. Spin-Orbit-coupling Coupling of spin and orbital moment yields the total H z angular momentum of electrons: = + S J L S The spin-orbit (so) interaction or LS-coupling is L described by: J µ dU ( ) ( ) 1 ⋅ λ = − = λ B E r L S r ; SO 2 m ec r dr e λ > for less than half filled d and f shell 0 λ < 0 for more than half filled d and f shell Spin-orbit coupling is due to the Zeeman – splitting of the spin magnetic moment in the magnetic field that is produced by the orbital moment: ⋅ − E = m B SO S L 4 H. Zabel, RUB 2. Lecture: Paramagnetism
• In rest frame of electron, E and B – fields act on electron due to positive charge of the nucleus: × r p E = B L 2 m c r 0 • The magnetic field is proportional to angular momentum of electrons: B L ~ L: ( ) L ∂ U r 1 1 = B L ∂ 2 m ec r r 0 In magnetic field B L , S precesses with a angular velocity ω L and couples to L: • ( ) µ ∂ U r 1 − ⋅ B E = L S ∂ SO 2 m ec r r e L ⋅ S can be evaluated via: • ( ) ( ) 1 2 = + ⋅ = − − 2 2 2 2 J L S L S J L S ; 2 • Yielding: β [ ] ( ) ( ) ( ) + − + − + E SO = j j l l s s 1 1 1 2 5 H. Zabel, RUB 2. Lecture: Paramagnetism
2. Fine structure Terms with same n and l – quantum numbers are energetially split according to whether the electron spin is parallel or antiparallel to the orbital moment. This is called the fine structure of atomic spectra. Example hydrogen atom: +1/2 β - β The total splitting of 3/2 β increases with the number of electrons in the atom and becomes in the order of 50 meV for 3d metals. LS-splitting lowers the energy for L and S antiparallel. Therefore level filling starts with lowest j-values. 6 H. Zabel, RUB 2. Lecture: Paramagnetism
Why ist λ changing sign? ( ) ⋅ S < = λ E SO r L 0 λ d-electrons More than half filled: L+S Less than half filled: L-S S L S L ⋅ > ⋅ < L S 0 L S 0 ( ) ( ) ⋅ < ⋅ < = λ = λ E r L S E r L S 0 0 SO SO > < λ λ if if 0 0 7 H. Zabel, RUB 2. Lecture: Paramagnetism
LS coupling for light and heavy atoms Russel-Saunders coupling for light atoms (LS-coupling): This approximation assumes that the LS-coupling of individual electrons is weak compared to the coupling between electrons. Orbital moments of all electrons couple to a total angular momentum L and spin moments of all electrons couple to S. Finally L and S couple to J: Z z ∑ ∑ = = = + L l , S s , J L S i i = = i 1 i 1 jj-coupling of heavy atoms : In the limit of big LS – coupling, the spin and orbital moment of each individual electrons couples to j, and all j are added to total angular moment J. z ∑ = + = j l s , J j i i i i = i 1 8
Total angular moment and total magnetic moment = + Total angular moment is: J L S Total magnetic moment is: = − µ + = − µ + = − µ + m ( g L g S ) ( L 2 S ) ( J S ) ( ) + J S B L S B B Total angular moment J and the total magnetic moment are not collinear . However, in an external magentic field, m J+S precesses fast about J , and J precesses much slower about H z . Thus the time average component of the magnetic moment 〈 m 〉 = m || is parallel to J . S S H z L S Total spin : J + J S Total orbital moment L : Total angular moment J : m + J S m || 9 H. Zabel, RUB 2. Lecture: Paramagnetism
Different total angular moments J = + E E E Coulomb SO S S S L L L J J max J = = L S min 3 , 3 2 + ≥ ≥ − L S J L S Which one is the ground state? Next lecture: Hund‘s rule λ < 0 10 H. Zabel, RUB 2. Lecture: Paramagnetism
3. Zeeman-splitting • In an external field the quantization axis is defined by the field axis H z • A state with total angular momentum J has a degeneracy of 2 J +1 without field. These states are labled according to the magnetic quantum number m J : – J ≤ m J ≤ J. • In an external field H z the states with different m J have different energy eigenstates, their degeneracy is lifted: ( ) = + + µ µ E H E E g m H m z SO J B J z 0 0 J m J = 3/2 E m m J z J , = + + µ E E m H SO z J z 0 0 , m J = 1/2 H z • The energy eigenstate are equidistant and linearly proportional to the external field H z. m J =-1/2 = − + + m J , J 1 , J 2 ,...., J m J =-3/2 J µ 0 g µ H + J B z 2 J 1 11 H. Zabel, RUB 2. Lecture: Paramagnetism
LS and Zeemann Splitting for L=3, S=3/2 , λ < 0 Conversation: λ < 0 1000 cm -1 = 0.124 eV 12 H. Zabel, RUB 2. Lecture: Paramagnetism
Landé factor Z-component of the total magnetic moment: = µ m m g z j J J B , H z ( ) m + J J 1 j Landé factor : 3 2 + + + − + J J S S L L ( 1 ) ( 1 ) ( 1 ) = + g J 1 1 2 + J J 2 ( 1 ) + − + − S S L L 3 ( 1 ) ( 1 ) 3 2 = + − + 1 2 J J 2 2 ( 1 ) Notice: g j =1 for J=L and 2 for J=S 13 H. Zabel, RUB 2. Lecture: Paramagnetism
Evaluating the Landé-factor From 1. Lecture we have for the paramagnetic response in an ( ) B external B - field: = µ + ⋅ E L S 2 B Considering L+2S projected onto J and J project onto the B-axis: ( ) ( ) ( ) + ⋅ ⋅ µ L S J J B 2 = µ = + ⋅ + B E L S L S J B 2 B z z 2 J J J This yields: ( ) µ = + ⋅ + 2 2 B E L L S S J B 3 2 z z 2 J ( ) ( ) Using: 3 2 = + ⋅ = − − 2 2 2 2 J L S 3 L S J L S ; 2 ( ) − + 2 2 2 J L S 3 We find: = µ E J B B z z 2 J 2 = µ Which must equal: E g m B J B J z ( ) ( ) + − + S S L L 3 1 1 = + With: g J ( ) + J J 2 2 1 14 H. Zabel, RUB 2. Lecture: Paramagnetism
4. Thermal properties Thermal population of the Zeeman-split levels in the ground state (gs). Example: J =1, m J = -1,0,+1 E E m J m J 1 1 0 0 -1 -1 N N H z =0, T=0 H z >0, T= 0 H z >0, T>0 gs degenerate, all Lifting of At high temperature population atoms in the degeneracy, all of higer energy states same state atoms in the gs ( ) µ 0 µ Discrete energy levels with E m = - g H m J J B z J m J =- J ,… J ( ) µ 0 µ E m = - g H m J J B z J Average thermal energy: ∂ N E N 1 µ M = = g m - µ Magnetization: ∂ J B J V H V z 0 15 H. Zabel, RUB 2. Lecture: Paramagnetism
Thermal average of the magnetization N = µ M g m j B j V Thermal average of the magnetic moment follows from the partition function: ( ) ∑ − α m m exp j j m = ( ) m j ∑ j − α m exp j m j j µ g H α = B z with: k T B N ~ = µ α M T H g jB ( , ) ( ) z j B j V B i is the Brillouin function. The Brillouin function replaces the Langevin function in case of discrete energy levels. 16 H. Zabel, RUB 2. Lecture: Paramagnetism
Brillouin Function ~ + + j j a 2 1 2 1 1 ~ ~ α = α − B j ( ) coth coth j j j j 2 2 2 2 j µ g jH ~ α = α = B z mit j k T B 17 H. Zabel, RUB 2. Lecture: Paramagnetism
Examples for the Brillouin-Function B J : J J J 18 H. Zabel, RUB 2. Lecture: Paramagnetism
4. Low and high temperature approximations Low temperature approximation (LTA) for: j µ g H j ~ α = >> B z 1 k T B ~ ( α → B 1 ) the Brillouin function approaches 1 The thermally averaged magnetization then becomes: N = µ = = = M g j M M T 0 ( ) j B S V This corresponds to the saturation magnetization M S . The saturation magnetization can not become bigger than given by j . It corresponds to a state in which all atoms occupy the ground state. 19 H. Zabel, RUB 2. Lecture: Paramagnetism
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