Basic Magnetism Thorsten Glaser, University of Münster. Basic Magnetism. 1. Paramagnetism and Diamagnetism (macroscopic)

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1 asic Mageis Thorse Glaser Uiversiy of Müser asic Mageis. Paraageis ad Diaageis (acroscopic) exeral ageic field H r diaageic saple paraageic saple r : ageic iducio (ageic r r field r iesiy iside saple) H 4π M M r : iesiy of ageizaio (ageic oe r / ui volue) i r o i < o i > o r : relaive ageic pereabiliy r < r > i o χ v o χ v : volue suscepibiliy (ui less) χ v < 0 χ v > 0

2 asic Mageis Thorse Glaser Uiversiy of Müser easuree: Faraday balace QUID... χ es : olar suscepibiliy χ es χ para χ dia χ dia fro ables Pascal s cosas. Paraageis.. Microscopic ierpreaio χ para χ es - χ dia us relae acroscopic suscepibiliy o he ageic dipole oes r of N A olecules classical olza disribuio oal eergy of syse lowered by exeral ageic field M H saple araced io field

3 asic Mageis Thorse Glaser Uiversiy of Müser quau echaics olecule wih eergy levels (...) i he presece of ageic field H r icroscopic ageizaio H acroscopic olar ageizaio M by pariio fucio: (o approxiaio) H M N A χ M H i a weak field: χ is idepede of H χ M H.. The va Vleck equaio (0) () approxiaio: H H (... ) M N A () () () H H () () ( H ) (0) (... ) () H (0) () () H H () H 3

4 asic Mageis Thorse Glaser Uiversiy of Müser furher approxiaios: H ( x ) x for x H ( () ) for 0 M 0 () H 0 (0) 0 () () 0 M H N A () () (0) (0) χ N A () () (0) (0) eed eergies of he syse uder ivesigaio plug io equaio (Va Vleck or - beer - he pariio fucio) calculae χ copare o erie ad gai isigh io he elecroic srucure of he syse uder ivesigaio 4

5 asic Mageis Thorse Glaser Uiversiy of Müser 5.3. ½ syse Zeea effec Curie law free elecro spi ½ s / s -/ deeried: ud z udeeried: x ud y z g ˆ ˆ H z ) ( ˆ ˆ ˆ g g z g ˆ g g z g

6 asic Mageis Thorse Glaser Uiversiy of Müser - M g plug io Va Vleck equaio χ N A () () (0) (0) (0) () () (0) () () ; 0 M g 0 ( M g) N A M χ usig i ( )( 3 ) i N A C N A χ g ( ) i C g ( 3 T 3k Curie law ) 6

7 asic Mageis Thorse Glaser Uiversiy of Müser effecive ageic oe eff : eff 3k χ T.8795 χ T g ( ) e N A spi-oly!!! graphical represeaio calculaed for ½ ad g.00 χt / c 3 K ol - eff / / / /

8 asic Mageis Thorse Glaser Uiversiy of Müser.4. auraio effecs ad rilloui fucio geeral isoropic syse wih spi sae M g plug io he pariio fucio for M M M N... N A A g g M s g M M s g M s M s ( x) wih x s g wih ( x) ( ) coh[ x( ) ] coh ( x) low x values: liear - Curie behavior Noe: Curie law: M high x values: sauraio - oly M s - populaed 8

9 asic Mageis Thorse Glaser Uiversiy of Müser.5. Zero-field spliig: O couplig i syse < cubic syery ± 5/ ergie 5/ 4D D ± 3/ ± / zero-field spi Hailoia ( ˆ ( ( ) )) ( ˆ ˆ ) ˆ D z 3 x y H cobied effec of so couplig ad syery lowerig: fro secod order perurbaio heory i λl i D 4h : D 8λ (0Dq) 9

10 asic Mageis Thorse Glaser Uiversiy of Müser.6. Teperaure Idepede Paraageis TIP. Order Zeea effec χ es χ para χ dia (χ para : posiive; χ dia : egaive) χ para C / T χ TIP χ TIP posiive!!! ad sall xaple: Cu II i ocahedral syery χ TIP 4 k N c ol 0Dq 0000 wih N Dq i c - ad χ i c 3 ol - (cgs eu) copare o χ para ~ 500 c 3 ol - a 300 K faous exaple: Co III l.s. (d 6 ) χ TIP ~ 600 c 3 ol - ( 0) Noe: χ TIP is eperaure idepede i χ!!!! effecs o eff : eff for χ.88 χ T.88 χ T cos T TIP eff c 3 ol c 3 ol T. bei 300 K 0

11 asic Mageis Thorse Glaser Uiversiy of Müser.7. Coplee reae of oouclear syses Hˆ g H ˆ g H ˆ g x D ( ˆ ( ( ) )) ( ˆ ˆ ) z diagoalizaio of he spi Hailoia arix eergies plug io equaio for χ x fi paraeers o erieal χ vs. T x 3 y ierpree spi Hailoia paraeers o gai isigh io he elecroic ad hus geoeric srucure of he coplex ageizaio easurees: effecs of zero-field spliig exaple: 5/ g.00 y y x z H y z ˆ z D 0 correspods o rilloui fucio

12 asic Mageis Thorse Glaser Uiversiy of Müser variable field - variable eperaure (VTVH) easurees i soe cases o oly agiude bu also sig of zero-field spliig accessible by bulk ageizaio easuree 3. Iraolecular exchage coupligs 3.. Iroducio ex.: [Cu II (OAc) 4 (H O) ]

13 asic Mageis Thorse Glaser Uiversiy of Müser eff g ( ) ( ).58 for g. bu erie: a 0 K o ageizaio us have diaageic groud sae aiferroageic couplig: ferroageic couplig: 3

14 asic Mageis Thorse Glaser Uiversiy of Müser Heiseberg-Dirac-va Vleck (HDvV) Operaor Ĥ < 0: aiferroageic > 0: ferroageic... ( ) ( ) ) ( wih ( ) Ĥ ( ) Ĥ ( ) [ ] ) ( ) ( ) (

15 asic Mageis Thorse Glaser Uiversiy of Müser 5 Geeral pi-yse ( ) [ ] ( ) [ ] [ ] ( ) ( ) [ ] [ ] [ ] ) ( 3 ) 3 ( ) ( ] [ ) 3 ( ) )( ( ] [ ) ( Lade Ierval-Regel

16 asic Mageis Thorse Glaser Uiversiy of Müser } spliig is o equidisa bu icreases wih icreasig T 5/ ad 5/ (Fe III - Fe III ) spi syse ( ) [ ( ) ] ( 0) [ 0(0 ) ] 0 ( ) [ ( ) ] ( ) [ ( ) ] 6 ( 3) [ 3(3 ) ] ( 4) [ 4(4 ) ] 0 ( 5) [ 5(5 ) ] spi ladder 6

17 asic Mageis Thorse Glaser Uiversiy of Müser ex. Fe III - O - Fe III diers (g.00) - OH bridge ~ - c - - OH - bridge ~ -0 c - - O - bridge ~ -00 c - Mageis - rucure - Relaioship Why? echais of ieracio 7

18 asic Mageis Thorse Glaser Uiversiy of Müser 3.3. leaey-owers quaios. leaey K. D. owers Proc. R. oc. Lod. 95 A4 45 Goal: equaio for calculaio of χ as fucio of ad T siulaio of easured eperaure-depede suscepibiliies by usig e. g. xcel H va Vleck equaio: oe:!!! e.g.: Cu II Cu II dier (0) () H () H 8

19 asic Mageis Thorse Glaser Uiversiy of Müser 9 leaey-ower equaio for ½ ½ syse; oher spi syses: C..O Coer Prog. Iorg. Che A N (0) (0) () () χ g g g g g g () 4 (0) 4 4 () 3 (0) 3 3 () (0) () (0) g g N A χ g g N A 3 χ g N A 3 χ

20 asic Mageis Thorse Glaser Uiversiy of Müser 4. uper xchage Mechais 4.. Fudaeal Cosideraios ieracios bewee paraageic ceers hrough space hrough bods dipolar spi-spi elecrosaic ieracios ieracios (ageic ieracio (o a ageic ieracio relaive sall r -3 < c - ) relaive large 00 c - ) super exchage hrough overlap of eal d-orbials wih orbials of he bridgig ligads direc exchage direc overlap of eal d-orbials 0

21 asic Mageis Thorse Glaser Uiversiy of Müser odel copoud: Cu II (A) Cu II () L L Cu Cu L L ψ A ψ acive orbials (coai soe ligad characer) eglec all oher orbials (acive orbial approxiaio) ψ A ψ 4.. Aderso Model orhogoalizaio of he wo localized orbials ψ A ad ψ : a b

22 asic Mageis Thorse Glaser Uiversiy of Müser groud sae cofiguraio: a b ur o elecro-elecro Coulob repulsio: a b e - -e - GC 3 GC k "Hud s i he olecule": ferroageic ieracio as i O charge rasfer excied cofiguraios excied cofiguraios by iraolecular elecro rasfer a b a b Cu I Cu III Cu III Cu I hese cofiguraios are of course a higher eergy here are oly sigle cofiguraios ur o elecro-elecro Coulob repulsio syerize

23 asic Mageis Thorse Glaser Uiversiy of Müser Γ g Γ g CTC aa oder bb Γ u Γ u U Γ g GC ab k 3 Γ u - 3 Γ u - 4β U Γ g ferro aiferro 4k 4β U poeial exchage kieic exchage β: rasfer iergral: rasfer of a elecro fro oe sie o he oher hrough bods β Ψ A Ĥ Ψ i Ψ Ψ A : A : A 3

24 asic Mageis Thorse Glaser Uiversiy of Müser he sroger he bods he larger β he larger he aiferroageic coribuio β : overlap iegral are he ageic orbials orhogoal 0 β 0 oly ferroageic coribuio reais 4.3. Goodeough-Kaoori-Rules. The overlap of ageic orbials of wo eal ceers wih a overlap iegral 0 yields a aiferroageic ieracio.. Are he overlappig ageic orbials orhogoal he overlap 0 ad he ieracio is ferroageic. 3. The o-orhogoal overlap of a ageic orbial of oe eal ceer wih a filled or epy orbial of he oher eal ceer yields a ferroageic ieracio 4.4. Kah s Model of he Naural Mageic Orbials NMO s exchage ieracios represes he lii of weak cheical bodig he cheis like o lai cheical bodig by he overlap of localized orbials orhogoalized ageic orbials are o localized 4

25 asic Mageis Thorse Glaser Uiversiy of Müser aural ageic orbials: NMO a is he sigly occupied MO i he AX frage NMO b is he sigly occupied MO i he X frage xchage couplig arises of overlap of NMO s: k 4β T excied saes of Aderso s odel eergeically oo high (e.g. Cu I Cu III Fe II Fe IV ); here: groud cofiguraio pheoeo srog aiferroageic couplig would eed lower eergy excied saes 5

26 asic Mageis Thorse Glaser Uiversiy of Müser 4.5. xaples Fe-OH -Fe syses.8 Å Fe O Fe.0 Å Fe H O Fe. Å Fe H O Fe -00 c - -0 c - - c - Fe 3d Fe 3d Fe 3d O p O p O p 6

27 asic Mageis Thorse Glaser Uiversiy of Müser Cu(OH) Cu Hafield ad Hodgso ageo-srucural correlaio N N Cu H O α O H Cu N N Cu(OH) Cu coplex Cu-O-Cu / / c - [Cu(bipy)OH] (NO 3 ) [Cu(bipy)OH] (ClO 4 ) [Cu(bipy)OH] (O 4 ). 5H O β-[cu(daep)oh] (ClO 4 ) [Cu(eaep)OH] (ClO 4 ) 98.8; [Cu(iz)OH] (ClO 4 ). H O α-[cu(daep)oh] (ClO 4 ) [Cu(e)OH] (ClO 4 ) [Cu(e)OH] (NO 3 ) α-[cu(ee)oh] (ClO 4 ) β-[cu(ee)oh] (ClO 4 ) [Cu(e)OH] r

28 asic Mageis Thorse Glaser Uiversiy of Müser / c Cu-O-Cu / -74 α for α 97.5 < 0 for α > 97.5 aiferroageic > 0 for α < 97.5 ferroageic k 4β α 90 4 β 0 k α 4 β sroger aiferroageic coribuio 8