BenBen--Gurion University of the NegevGurion University of the Negevy gy gDepartment of ChemistryDepartment of Chemistry

MOLECULAR MAGNETISM MOLECULAR MAGNETISM AND MATERIALSAND MATERIALS--

THEORY AND APPLICATIONSTHEORY AND APPLICATIONSTHEORY AND APPLICATIONSTHEORY AND APPLICATIONS

Professor Boris Tsukerblattsuker@bgu.ac.il@ g

BeerBeer--ShevaSheva20062006

SYLLABUSSYLLABUS

I. Scope of molecular magnetism. Diversity of the field. Main kinds of magnetic systems and the main types of the

ti d imagnetic ordering.II. Spin, fundamental equations in molecular magnetism.

Magnetic susceptibility magnetic moments Curie-Weiss lowMagnetic susceptibility , magnetic moments. Curie-Weiss low, magnetization. Electron paramagnetic resonance.

III. Magnetic properties of a free ion, molecules containing a i ti t ith t fi t d bit l tiunique magnetic center without first-order orbital magnetism

and EPR of transition metal ions and rare-earths, spin-orbital interaction.

IV. Effects of crystal field. Group-theoretical introduction. Ground terms of the transition metal ions in the crystal fields. Anisotropy of the g-factor. Zero-field splitting: qualitative and py g p g qquantitative approaches. Covalence and orbital reduction. EPR of the metal ions in complexes.

V. Exchange interaction in clusters. Exchange effect, the nature of the potential exchange. Magnetic properties ofnature of the potential exchange. Magnetic properties of binuclear compounds, dimers of Cu(II) , EPR, magnetic anisotropy.

VI Heisenberg Dirac Van Vleck model of the exchangeVI. Heisenberg-Dirac-Van Vleck model of the exchange interaction. Concept of spin-Hamiltonian. Many-electron problem of the exchange. Spin-coupling scheme for the polynuclear compounds Kambes approach Trimericpolynuclear compounds, Kambe s approach. Trimeric and tetrameric clusters: basic chromium and iron acetates. EPR spectra of polynuclear compounds.

VII Si l l l h i l i i lVII. Single molecule magnets, physical principles- quantum tunneling, relaxation. Mn12-ac molecule. Applications in molecule-based devices.

VIII. Mixed-valence compounds. The phenomenon of mixed valence. Spin-dependent delocalization-double exchange- classical and quantum-mechanicalexchange classical and quantum mechanical description (Andersons theory). Robin and Day classification of mixed-valence compounds. Intervalence light absorption (light induced electron transfer)light absorption (light induced electron transfer). Magnetic properties.

SOURCESSOURCES DD2)THE MAIN BOOKS AND2)THE MAIN BOOKS AND 3) 3) REFERENCES REFERENCES

SOURCES:SOURCES: 1) 1) CDCD (Power Point file)(Power Point file)2)THE MAIN BOOKS AND2)THE MAIN BOOKS AND 3) 3) REFERENCES REFERENCES

THROUGHOUT THE FILETHROUGHOUT THE FILE Oliver Kahn Molecular Magnetism VCH NY(1993)Oliver Kahn, Molecular Magnetism, VCH,NY(1993). Alessandro Bencini, Dante Gatteschi, Electron Paramagnetic

Resonance of Exchange Coupled Systems, Springer-Verlag, Berlin (1990)Berlin (1990).

F.A.Cotton, Chemical Application of Group Theory, 2nd Edition, Interscience, New York (1971).

B.S.Tsukerblat, Group Theory in Chemistry and Spectroscopy. A Simple Guide to Advanced Usage, Academic Press, London (1994).

J.J.Borras-Almenar, J.M.Clemente-Juan, E.Coronado, A.V.Palii, B.S.Tsukerblat, Magnetic Properties of Mixed-Valence Clusters:Theoretical Approaches and Applications, in: M ti M l l t M t i l Magnetism: Molecules to Materials, ( J.Miller, M.Drillon, Eds.), Wiley-VCH (2001) p.p.155-210.

ABOUT THE BOOKSABOUT THE BOOKS--MOLECULAR MAGNETISMMOLECULAR MAGNETISM

Chapter 5

ABOUT THE BOOKSABOUT THE BOOKS--GROUP THEORYGROUP THEORY

Exceptionally clear presentation !

YOU ARE EXPECTED TO KNOW:YOU ARE EXPECTED TO KNOW:YOU ARE EXPECTED TO KNOW:YOU ARE EXPECTED TO KNOW:

Main concept of quantum mechanics:Main concept of quantum mechanics: Schrdinger fequation, wave-functions, hydrogen atom, many-electron

atoms, some knowledge of the perturbation theory. Orbital and spin angular momenta. Pauli principle.p g p p

Group theory for chemistsGroup theory for chemists (standard course for chemists): how to determine the point symmetry group that the molecule belongs to concept of the reducible andmolecule belongs to, concept of the reducible and irreducible representations, classification of the molecular energy levels, selection rules. Classification of molecular vibrationsvibrations.

Background of the crystal field theoryBackground of the crystal field theory for transition metal ions: general idea of the crystal field splitting and some results for the transition and rare earth ionsresults for the transition and rare-earth ions.

Molecular orbital approachMolecular orbital approach the main concepts.

YOU ARE EXPECTED TO LEARNYOU ARE EXPECTED TO LEARN Magnetic substancesMagnetic substances.. The main kinds of magnetic

behavior. Basic concepts: magnetic moments, magnetic p g , gsusceptibility. Spin, free ions, spin-orbit coupling, g-factors. Electron paramagnetic resonance.C t l fi ld thC t l fi ld th l f th li d ti Crystal field theoryCrystal field theory, role of the ligands, magnetic properties of complex compounds, zero-field splitting, magnetic resonance, anisotropy of g-factors. g , py g

Exchange interactionExchange interaction in clusters. Properties of polunuclear compounds. Magnetic anisotropy,

i i l l l tnanoscience -single molecular magnets. Concept of mixedConcept of mixed--valencyvalency and electron transfer double

exchange ferromagnetic effect of the double exchangeexchange, ferromagnetic effect of the double exchange, role of the electron-vibrational interactions-localization vs. delocalization. Spin-dependent delocalzation in iron-

l h t isulphur proteins.

LIST OF THE MAIN NOTATIONSLIST OF THE MAIN NOTATIONS((to be used as necessaryto be used as necessary))((to be used as necessaryto be used as necessary))

)arial"-"(fontvalueabsolute-fieldmagnetic-H(vector), fieldmagnetic - H

H )It li "Rti"(f tH ilt i

.matrices and Vectors).arial-(font valueabsolute-fieldmagnetic -H

bold

.,H

H

etc : cap""by marked are Operators

).Italic"Roman,new times"-(fontnHamiltonia

S

S

S

),cap""and-operator(vectoroperator spin

vector, spin classical- number), (quantum spin -

bold

L.S,S,S zyx S

momentumangularorbitaltheofnumberquantum operator vector the of components

),pp(pp

L

LL operator). (vector operator vector, classical- momentum,angularorbitaltheof numberquantum-

.L,L,L zyx L operator vector the of components

operator)( ectoroperatorectorclassical

momentum, angular total the of number quantum -

J

JJ operator vector the of components

operator),(vector operatorvector, classical-

.J,J,J zyx JJJ

momentum. angular total and momentum angular orbital spin, of numbers quantummagnetic ,- , , MMM JLS

state. aoffactor electron, free a of factor

LSJgggg

J

e

vector), (classicalmomentmagnetic lity.susceptibimagnetic -

ggJ

value)(absolutemomentmagnetic operator), (vector momentmagnetic

),(g

parametersplittingfieldzeron.Hamiltonia splitting fieldzero

value) (absolutemomentmagnetic

DHZFS

parameter exchange parameter.splittingfield-zero

J D

Chapter IpScope of molecular magnetism.Diversity of the field. The main kinds of magnetic systems and the main typesmagnetic systems and the main types

of the magnetic ordering

SCOPE OF MOLECULAR MAGNETISMSCOPE OF MOLECULAR MAGNETISM Magnetic properties of isolated atoms ions and Magnetic properties of isolated atoms ,ions and

molecules ( in particular, metal complexes) containing one magnetic center. gExample: complex ion coordination compound

MetalMetal--(Ligand)(Ligand)66( g )( g )66[Cr (NH3)6]3+ or [Cr(NH ) ]ClNH

NH3[Cr(NH3)6]Cl3

Central metal ion- Cr3+ Cr

NH

NH3

NH Central metal ion Crsurrounded by six ammonia molecules, Cr3+ contains three

NH3NH3

NH3unpaired d-electrons

NH3

NH3

Octahedral symmetry, Oh point group

Magnetic properties of the molecules containing more than one magnetic centers polynuclear compounds, magnetic clusters or exchange clusters.Example: binuclear Co cluster, bi-octahedral edge-shared geometry- oxygen bridged system

[(H3N)4Co(OH)2Co(NH3)4]4+

NH NH

NH3

C 2+(d7 h ll)OH

NH3Co

NH3

NH3 Co2+(d7-shell)-bearer of

CoNH3

NH3

OH

magnetism

NH

NH3 NH3OH

Point symmetry D2hNH3

POLYOXOANION POLYOXOANION [Ni3 Na(H2O)2(AsW9O34]11-

WO6 3Ni2+-

magnetic gfragment

NiO6Na

AsO4

Na

Polyhedral representation Ball and stick representation

Inorg. Chem. ,2003, 42, 5143-52Polyhedral representation Ball and stick representation

POLYOXOANION POLYOXOANION [Ni6 As3W24O94(H2O)17]-

T 3Ni2+WO6

Two 3Ni2+-magnetic

f tNiO6 fragments

AsO44

P l h d l t ti B ll d ti k t tiPolyhedral representation Ball and stick representationInorg. Chem. ,2003, 42, 5143-52

Assemblies of molecules with the magnetic interactions b t th l l titi di i l tbetween the molecular entities, one-dimensional systems

Structure of donor-acceptor compound (TTF)+[CuS4C4(CF3)4]-compound (TTF) [CuS4C4(CF3)4]with TTF+= tetrathiafulvalinium. Phys.Rev.Lett.,35(1975) 744

Structure of the ferrimagnetic chain MnCu(pba)(H2O)32(H2O) with b 1 3 il bi ( )pba= 1,3-propilene-bis(oxamato).

Inorg.Chem.,26(1987)138.

DIVERSITY OF THE FIELD,SELECTED DIVERSITY OF THE FIELD,SELECTED APPLICATIONSAPPLICATIONSL NL N

MaterialMaterial BiologyBiologyMaterialMaterialsciencessciences

Molecular Molecular Molecular Molecular magnetismmagnetism

NNM l lM l l NanoNano--sciencescience

MolecularMolecularelectronicselectronics

SINGLE MOLECULAR MAGNETSINGLE MOLECULAR MAGNET--MAGNET IN ONE MOLECULEMAGNET IN ONE MOLECULEMAGNET IN ONE MOLECULEMAGNET IN ONE MOLECULE

[Mn12O12 (CH3COO)16 (H2O)4] -molecule - Mn12-ac (Mn12-acetate)

NanoNano--ii

MolecularMolecularelectronicselectronics

Mn4+

sciencescienceelectronicselectronics

Mn

Mn 3+

MANGANESEMANGANESE--12 CLUSTER12 CLUSTERMANGANESEMANGANESE 12 CLUSTER12 CLUSTEReight Mn3+ ions (Si =2) and four Mn4+(Si =3/2)

PHYSICAL BACKGROUND PHYSICAL BACKGROUND BRIEFLYBRIEFLYPictures: Michel VerdaguerPictures: Michel Verdaguer

E0

zThermal activation

Pictures: Michel VerdaguerPictures: Michel Verdaguer

0

DSz2y

x ne

r

g

y

DS2

Sz

x

E

n

- Szz

+Sz0-2-4 +2 +4Direction of magnetizationBarrier for anisotropy

If the Mn12-ac molecule is magnetized by an applied

Magnetization vectors

t e ac o ecu e s ag et ed by a app edfield, the molecule retains magnetization for a long time ,

approximately 108 seconds = 3 years at 1.5KA li tiApplications:

quantum computing , memory storage elements in one molecule

MULTIFUNCTIONAL MATERIALSMULTIFUNCTIONAL MATERIALS

MolecularMolecular MaterialMaterialelectronicselectronics sciencessciences

Nature 408, 421 - 422 (2000) Molecular electronics: A dual-action materialMolecular electronics: A dual action materialFERNANDO PALACIO* AND JOEL S. MILLER Fernando Palacio is at the Instituto de Ciencia de Materiales de Aragn, CSIC, Universidad deFernando Palacio is at the Instituto de Ciencia de Materiales de Aragn, CSIC, Universidad de Zaragoza, 50009 Zaragoza, Spain. e-mail: palacio@posta.unizar.es Joel S. Miller is in the Department of Chemistry, University of Utah, Salt Lake City, Utah 84112-0850, USA. e-mail: jsmiller@chemistry.utah.edu In the drive for smaller electronic components, chemists are thinking on a

l l l B bi i i l l l h b id h b d dmolecular scale. By combining two simple molecules, a hybrid has been produced that is both magnetic and an electrical conductor.

DISCOVERY OF MULTIFUNCTIONAL DISCOVERY OF MULTIFUNCTIONAL MOLECULEMOLECULE--BASED MATERIALSBASED MATERIALS

Nature 408, 447 - 449 (2000) Coexistence of ferromagnetism and metallicCoexistence of ferromagnetism and metallic conductivity in a molecule-based layered compound

EUGENIO CORONADO*, JOS R. GALN-MASCARS*, CARLOS J. GMEZ-GARCA* & VLADIMIR LAUKHIN* * Instituto de Ciencia Molecular, Universidad de Valenc ia, Dr. Moliner 50, 46100 Bur jasot, Spainj p Present addresses: Department of Chemistry, Texas A&M University, College Station, Texas, USA (J.R.G.-M.); ICMB-CSIC, Campus de la UAB, 08193 Bellaterra, Spain (V.L.) Crystal engineering the planning and construction of crystalline supramolecularCrystal engineeringthe planning and construction of crystalline supramolecular architectures from modular building blockspermits the rational design of functional molecular materials that exhibit technologically useful behavior such as conductivity and superconductivity ferromagnetism and nonlinear optical properties Because theand superconductivity, ferromagnetism and nonlinear optical properties. Because the presence of two cooperative properties in the same crystal lattice might result in new physical phenomena and novel applications, a particularly attractive goal is the design of molecular materials with two properties that are difficult or impossible to combineof molecular materials with two properties that are difficult or impossible to combine in a conventional inorganic solid with a continuous lattice.

A DUAL ACTION MATERIALA DUAL ACTION MATERIAL

Molecular components:

( )(a) The organic

molecule BEDT TTFBEDT-TTF

bis(ethylenedithio)tetrathiafulvalene

(b) A ferromagnetic bimetallic complex of manganese(ii)tris(oxalato)chromium(iii).

Carbon atoms are in pink, sulphur in blue.

By alternating layers of the molecules in a and b, E. Coronado et al. (Nature)have created a hybrid material that supports both magnetism and conduction.

p p

M- magnetic layers, E-conducting layers

Structures of the hybrid material and the two sublatticesStructures of the hybrid material and the two sublattices. a, View of the [MIIMIII(C2O4)3]- bimetallic layers. Filled and open circles in thevertices of the hexagons represent the two types of metals.b St t f th i l h i th ki f th BEDT TTF l lb, Structure of the organic layer, showing the packing of the BEDT-TTF molecules. c, Representation of the hybrid structure along the c axis, showing the alternating organic/inorganic layers.

BIOLOGICAL SYSTEMSBIOLOGICAL SYSTEMS--TWO EXAMPLESTWO EXAMPLESTriTri--iron clusteriron cluster

DiDi--iron clusteriron cluster

S-cys S

Fe

Schematic structure of the protein with [Fe S ] core

Schematic structure of the two iron (Fe2+ Fe3+ ) ferredoxinprotein with [Fe3S4] core.

S-cys stands for the sulfur atom of a cystein group.

Th ti ll l d F i

two-iron (Fe2+, Fe3+ ) ferredoxin. S-cys stands for the sulfur atom

of a cystein group.T ti ll l d F iThree magnetically coupled Fe ions. Two magnetically coupled Fe ions.

MAIN KINDS OF MAGNETIC SUBSTANCESMAIN KINDS OF MAGNETIC SUBSTANCES

ferromagnetparamagnet antiferromagnetferromagnetparamagnet antiferromagnetDisordered directions of

Long-range collinear

Long-range interactiondirections of

the magnetic moments,

collinear alignment of all moments in the

interaction, moments are

aligned,macroscopic

magnetization

moments in the substance,

spontaneous

aligned antiparallel to each other, no

is zero magnetization magnetization

ferrimagnet weak ferromagnet triangular structure

Antiparallel different

Two sub-lattices with non-collinear

Triangular configuration

magnetic moments,

with non collinear magnetic

moments, weak

configuration of the

magnetic spontaneous macroscopic

magnetization

spontaneous magnetization

gmoments

magnetization

HELICOIDAL MAGNETIC STRUCTURESHELICOIDAL MAGNETIC STRUCTURES

simple helix ferromagnetic complex static longitudinalhelix helix longitudinal spin wave

SCHEMATIC PHASE DIAGRAM OF BULK SCHEMATIC PHASE DIAGRAM OF BULK HOLMIUMHOLMIUMHOLMIUMHOLMIUM

Chapter IISpin, fundamental equations in

molecular magnetism.Magnetic susceptibility magneticMagnetic susceptibility , magnetic moments, electron paramagnetic

resonance

MAGNETIC FIELD, MAGNET IN A FIELDMAGNETIC FIELD, MAGNET IN A FIELDpermanent magnetpermanent magnetpermanent magnetpermanent magnet

NNorth SouthS

Magnetic field H

F

g

N

F

Turning moment acting on a magnetic stickTurning moment acting on a magnetic stick in a homogeneous magnetic field

SPIN OF THE ELECTRONSPIN OF THE ELECTRONElectron Electron elementary bearer of magnetismelementary bearer of magnetismElectron Electron -- elementary bearer of magnetismelementary bearer of magnetism

Elementary magnetic moment :Elementary magnetic moment :

J.gausserg.mc|e| 124120 109274001092840

2 T

B

or

:notations acceptedally conventiontwo,magneton Borh

h,serg.hB

27100512

constant Planck

cgse.e110

10

103

1084

vacuuminlighttheofvelocity

electron, the of charge

g.m

scmc28

110

1019

103

electron, the of mass

vacuum,inlightthe ofvelocity

gauss4101 1TeslaT

SPINSPIN--BEARER OF THE MAGNETIC MOMENTBEARER OF THE MAGNETIC MOMENT

Classical image-rotating spherical charge,Classical image rotating spherical charge, this picture fails in the evaluation of spin magnetic moment.

Adequate description Adequate description --quantumquantum--mechanical mechanical q pq p qqconcept.concept.

MAGNETIC MOMENTMAGNETIC MOMENTMagnetic moment associated with the spin (mechanical

angular momentum) of the electronelectron can have two projectionson the direction of the external magnetic field H:

eon the direction of the external magnetic field H:

MagneticmcS 2

Magnetic field H

mce

S 2spin downS

N

spin up eSS

N

spin upmcS 2

S

ELEMENTS OF QUANTUM MECHANICS OF SPINELEMENTS OF QUANTUM MECHANICS OF SPINfunction spin the for Notation

projection spin of number quantum spin,

MS

M,S

S

!componentsthreeoperatorVectormomentum angular spin of operator- )cap"" (notation operator spin

S,S,S

zyx

S

:PROPERTIES GENERAL

!componentsthree operator Vector

2

S,S,S zyx

12

M,SMM,SS

M,SSSM,SS

SSSz

SS

and

:operators two of functions-eigen the are functions-Spin2 SS

projectionspinofnumberquantum

of and of:valueseigen the with

and

1 2M

SMSSS

SS

zS

z

tymultiplici spin-values,projection spin ofnumber quantum

121

SS,,S,SMM

S

S

THE CASE OF SPIN S=1/2THE CASE OF SPIN S=1/2THE CASE OF SPIN S 1/2THE CASE OF SPIN S 1/2

21

21 ss mm and :sprojection spin Two

1

21

21

21

21 s ,,m,s

ft tiSh t

and functions waveSpin

21

:s

down""spinandup""spin

fornotationsShort

43

23

211

ss, - spin of operator

downspin and upspin

21s

3232 ss:properties Main

21

21

43

43

zz s,s

s,s

22

SPATIAL QUANTIZATION SPATIAL QUANTIZATION -- AN IMPRESSIVE RESULT AN IMPRESSIVE RESULT OF QUANTUM MECHANICS OF QUANTUM MECHANICS -- PHYSICAL PICTUREPHYSICAL PICTURE

23SM

1M21SM 1SM

2SM0SM

21SM

1SM21SM 23SM1S 1S 3S

Classical mechanicsClassical mechanics all directions for the magnetic 21S 1S 23S

C ass ca ec a csC ass ca ec a cs a d ect o s o t e ag et cmoment in the space are allowedare allowed.Quantum mechanicsQuantum mechanics only selected directions for the magnetic moment in the space are allowedare allowed--spatial quantization. Arbitrary z-axis.

VECTOR (CLASSICAL) MODEL FOR THE ANGULAR VECTOR (CLASSICAL) MODEL FOR THE ANGULAR MOMENTA IN QUANTUM MECHANICSMOMENTA IN QUANTUM MECHANICS

ZVector S precesses around arbitrary direction Z at the conical surface so that the meanSZ

S

conical surface, so that the mean values of the projections of S at the plane perpendicular to axis of

SZ

p p pZ are zero (SX , SY).Good quantum numbers:

S and M

S and MS

SY d b l t l 12 SSS

YSX

SY squared absolute value (length) of the vector S.

Spatial quantizationM S

X Classical pictureSpatial quantization

selected directions (MS):Mean values: 0 1 SS

Mcos SMean values: = =0 = Scos

PRECESSING SPIN PRECESSING SPIN CLASSICAL PICTURE ILLUSTRATING CLASSICAL PICTURE ILLUSTRATING CLASSICAL PICTURE ILLUSTRATING CLASSICAL PICTURE ILLUSTRATING

THAT:THAT:Zmean values

0

Z

= =0, but = Scos 0

X Y

Vector S performs precession around arbitrary direction Z atthe conical surface in an external magnetic field thethe conical surface, in an external magnetic field theprecession occurs around the vector of external magnetic field

I f h // i il/ h h /V /h h lImage from: http://www.weizmann.ac.il/chemphys/Vega_group/home.htmlProf. Shimon Vega , Weizmann Institute of Science, Israel

SPATIAL QUANTIZATION: SPATIAL QUANTIZATION: ILLUSTRATION for S=1/2ILLUSTRATION for S=1/2ILLUSTRATION for S=1/2ILLUSTRATION for S=1/2

11121SMZ

1 321

21

21

SS

MMS SS

SS

and 2S

S

1 2Mcos

SS

S

SS

754.

S

754

11

21 .cosM

SS

S

projections 3125.

3125

754

31

21

32

.cosM

.cosM

S

S

3

21S21SM 22

SPATIAL QUANTIZATION: SPATIAL QUANTIZATION: ILLUSTRATION for S=1ILLUSTRATION for S=1ILLUSTRATION for S=1ILLUSTRATION for S=1

1011 M,MMS SSS , 1M

Z

2M

S1SM

S 1 SSMcos S

0SM45 S

9000

4512

1

M

cosM S135

90

1351

9000

21

cosM

cosM

S

S

1M

135

2S

1S1SM

ZEEMAN INTERACTION ZEEMAN INTERACTION --interaction of the electronic spin with the magnetic fieldinteraction of the electronic spin with the magnetic fieldp gp g

s g eS :momentmagnetic spin of Operator

sss

geS

:)components (three operator Vector

222

gs,s,s

e

zzyyxx

:electron free a for factor-g or Lande, factor

222

.ge momentmagnetic the of ninteractio the ofEnergy

200232

H HHHEZ H: fieldmagnetic external the with HHHE yyyyxxZ

:nHamiltonia ZeemanH

HsgH eZ

ZEEMAN INTERACTIONZEEMAN INTERACTION-- ARBITRARY SPIN S>1/2ARBITRARY SPIN S>1/2 S

ion or atom an of spin total,g

i

eS

sS

S

electronsunpairedallover vectors the of Summation i

i

s

shellatomic the in ) symbol theby numbered (electronsunpairedallover"i"

,cosE and vectors between angleHfieldmagnetic the withmomentmagnetic the of nInteractio

H.H:H

:operators theirby values classical ngsubstitutiby obtained be can nHamiltonia

Z SSSggH,HE

HHH

HS zzyyxxeeZ SSSggH HHH HS

ZEEMAN INTERACTION FOR A SPIN STATE ZEEMAN INTERACTION FOR A SPIN STATE -- SS

H

( l )j tiHHH"t ":fieldmagnetic the for Notation

)b ld(H zyxninteractio ZeemanofnHamiltonia

(scalars).sprojectionH H H "vector-" ,, ),bold( H

H

:)H ,HH axis,- along field (magnetic zyx

z

00

zHzeZ SgH Important remark:Important remark:pp

z-axis is chosen arbitrary, free atom is spherically symmetric

tiS h diZEEMAN LEVELS FOR A SPIN STATE ZEEMAN LEVELS FOR A SPIN STATE -- SS

:equationrSchrodinge

,SMESMH SSZ

fi ldifihdS li iHH zz SMMgSMSg

,

SSeSze

SSZ

:levels) (Zeemanfieldmagnetic of actiontheunderSplitting

valueseigen

Hz

12S

MgME SeS values-eigen12SMagnetic field removes (2S+1)-fold degeneracy of spin level

For a free atom ( ion) Zeeman splitting is independent of the direction of the field isotropic in spacethe direction of the field - isotropic in space

Paul Maurice Adrien Dirac

1960

English theoretical physicist known for his work in quantum mechanics and for his theory of the electronic spin . In 1933 he shared the Nobel Prize with the Austrian physicist Erwin Schrdingerthe Austrian physicist Erwin Schrdinger .

ZEEMAN SPLITTING FOR A FREE ELECTRONZEEMAN SPLITTING FOR A FREE ELECTRON HE 1 110 EEH

zz

H

H

e

SSeS

gE

m,mgmE

21

21

21

212121

21

0

0

EE

EE

H

H

1mSE(mS)

21Sm

e

r

g

y

NS=1/2

e

n

e

S 1/2

21Sm

splitting

21Sm N

2S

2

magnetic field HzS

Magnetic field removes (2S+1)-fold degeneracy of a spin level: energy levels become dependent of spin projections mS

Pieter ZeemanPieter Zeeman

Born May 25, 1865,Zonnemaire, Netherland.Died Oct 9 1943 AmsterdamDied Oct. 9, 1943, Amsterdam

Nobel Winner, 1903 :for his discovery of the Zeeman effecty

Zeeman effect in physics and astronomy the splittingsplitting ofof aa spectralspectral lineline intoZeeman effect in physics and astronomy, the splittingsplitting ofof aa spectralspectral lineline intotwo or more components of slightly different frequency when the light sourceis placed inin aa magneticmagnetic fieldfield. It was first observed in 1896 by the Dutch

h i i t Pi t Z b d i f th ll D li f di iphysicist Pieter Zeeman as a broadening of the yellow D-lines of sodium in aflame held between strong magnetic poles. Later the broadening was found tobe a distinct splitting of spectral lines into as many as 15 components.Zeeman's discovery earned him the 1902 Nobel Prize for Physics, which heshared with a former teacher, Hendrik Antoon Lorentz, another Dutchphysicist. Lorentz, who had earlier developed a theory concerning the effect ofp y , p y gmagnetism on light, hypothesized that the oscillations of electrons inside anatom produce light and that a magnetic field would affect the oscillations andthereby the frequency of the light emitted. This theory was confirmed bythereby the frequency of the light emitted. This theory was confirmed byZeeman's research and later modified by quantum mechanics, according towhich spectral lines of light are emitted when electrons change from onediscrete energy level to another Each of the levels characterized by andiscrete energy level to another. Each of the levels, characterized by anangular momentum (quantity related to mass and spin), is split in a magneticfield into substates of equal energy. These substates of energy are revealed byth lti tt f t l li tthe resulting patterns of spectral line components.

Pieter Zeeman and Niels BorhPieter Zeeman, Albert Einstein, Paul Erenfest

Magnet ofMagnet of Pieter Zeeman

John H Van VleckAmerican physicist and mathematician who shared the Nobel Prize for Physics in1977 with Philip W. Anderson and Sir Nevill F. Mott. The prize honoured Van Vleck's

John H. Van Vleck

contributions to the understanding of the behaviour of electrons in magnetic,noncrystalline solid materials.Van Vleck developed during the early 1930s the first fully articulated quantum

h i l th f ti L t h hi f hit t f th li d fi ldmechanical theory of magnetism. Later he was a chief architect of the ligand fieldtheory of molecular bonding. He contributed also to studies of the spectra of freemolecules, of paramagnetic relaxation, and other topics. His publications includeQ ant m Principles and Line Spectra (1926) and the Theor of Electric andQuantum Principles and Line Spectra (1926) and the Theory of Electric andMagnetic Susceptibilities (1932).

ZEEMAN SPLITTING, ZEEMAN SPLITTING, ILLUSTRATION FOR SPIN ILLUSTRATION FOR SPIN S=1S=1ILLUSTRATION FOR SPIN ILLUSTRATION FOR SPIN S=1S=1 101 ,,M,MgME SSeS zH

zHeS gEM 11 zeS g1Sr

g

y

000 EM S1SE n er

zHeS gEM 11M ti fi ldMagnetic field

ELECTRON PARAMAGNETIC RESONANCEELECTRON PARAMAGNETIC RESONANCE--CLASSICAL PICTURECLASSICAL PICTURE

Zeeman splitting- constant magnetic field along Z-axis.Alternating magnetic field in the XY plane:Alternating magnetic field in the XY plane:

field, galternatin the offrequency

fieldtheoffrequency cyclic HH

,tcost XX 2

cycle) one of (time period ,1H0

Hr-rotating fieldH0-constant fieldZ

forcer g

Y

force precessing spin

Rotating field produces a turning momentX

force

Rotating field produces a turning moment-to align spin in the plane XY, i.e. parallel to Hr !

CONDITION FOR THE RESONANCECONDITION FOR THE RESONANCEConstant field Resonance condition:

H0Resonance condition:

frequency of rotating field=frequency of spin precessionfrequency of spin precession

0turning moment 0H field constant the in

precessionofFrequency :moment

00 H or H g

hg 00

HrRotating field

Under the resonance condition the turning g

moment acts in-phase with spin precession and spin

Electron paramagnetic Electron paramagnetic resonance (EPR), or electron resonance (EPR), or electron

i (ESR)i (ESR) rapidly changes orientation.spin resonance (ESR) .spin resonance (ESR) .Eugenii Zavoisky, Kazan, 1944Eugenii Zavoisky, Kazan, 1944

CLASSICAL PICTURE OF THE CLASSICAL PICTURE OF THE ELECTRON PARAMAGNETIC ELECTRON PARAMAGNETIC ELECTRON PARAMAGNETIC ELECTRON PARAMAGNETIC

RESONANCERESONANCE

tcosX H

Spin up Spin down

X

Spin up Spin down

ROTATING PERPENDICULAR MAGNETIC FIELDROTATING PERPENDICULAR MAGNETIC FIELDROTATING PERPENDICULAR MAGNETIC FIELD ROTATING PERPENDICULAR MAGNETIC FIELD OF THE RESONACE FREQUENCYOF THE RESONACE FREQUENCY

REVERSES SPINREVERSES SPINREVERSES SPINREVERSES SPIN

QUANTUM DESCRIPTION OF EPRQUANTUM DESCRIPTION OF EPRninteractioZeemanE

tcosSgH H

field galternatin the withninteractioZeeman

21SMS=1/2

E

XXalt

MM

.tcosSgHstransition induces ninteractio This

H

EPR

transition

S

SS

MME:ME

MM

H levels Zeeman different between

21SM

H0 SS MgME 0Hl lZb t

stransition quantum for rule" selection-" rule Important

0

111 MMMMMM d allowed are levels Zeeman g"neighborin" between stransition theonly

:levels Zeemanbetween

111 SSSSSS MMMMMM:low) onconservati(energy condition Resonance

or and

1SS MEMEfieldgalternatinofquantumofenergy energy spin of increase

QUANTUM RESONANCE CONDITION QUANTUM RESONANCE CONDITION (arbitrary spin )(arbitrary spin )(arbitrary spin )(arbitrary spin )

libitfH

SMMEMEME SS 1

hh i li hidi iThH H H

valuespinarbitrary anforH

gMgMg

SMgME

SS

SS

01

H

:approachmechanical-quantum withinconditionresonanceThe

g 0

H ,transition allowed an forenergy quantumH

gg

0

0

Hfieldmagnetic inprecessionspin

classicaloffrequency cyclic H But .

g

0

00

transitionallowedanforfrequencyquantum:conclusion main The

gpp

0

fieldmagnetic in precession spin offrequency cyclic transition allowedanforfrequency quantum

DETECTION OFDETECTION OF RESONANT ABSORPTIONRESONANT ABSORPTIONSome estimations of the physical values:

for a free electron (g=2) at frequency of 30GHz (Gigahertz) (1GHz=109Hz) the resonant field H0=10,700Gauss.

30GHz-area of microwave frequencies of radiation, 1 1 (1 8 066 1)energy 1cm-1 (1ev = 8,066cm-1).

Case I: the separation of the Zeeman levels is fixed by holding the magnetic field constant; the microwave frequency is then varied until a resonance absorption is found.

00

H g 0

Resonance: = 0

DETECTION OFDETECTION OF RESONANT ABSORPTIONRESONANT ABSORPTIONCase II: the microwave frequency is fixed; the magnetic field isCase II: the microwave frequency is fixed; the magnetic field is then varied. The characteristic aspect of EPR spectroscopy is the variation of the energy level separation by variation of the gy p ymagnetic field until the resonance is reached (at H=Hres ).non-resonance resonance frequency

H Hg21 Resonance equation:frequences

field resonanceHH

res

res

g

H1

: factor- of zationCharacteri g

Hg21EPR line

resHeffgHHres

Preliminary remark: g=2 onlyonly for a free electron

EPR, S>1/2 EPR, S>1/2 -- ISOTROPIC SYSTEMISOTROPIC SYSTEM MgME H

In the case of S>1/2 all allowed

SS MgME 0H0Hg23

transitions have the same resonance 0Hg213Sfields and for this reason give the l EPR lil EPR li Thi0Hg1

2

only EPR lineonly EPR line. This is valid in the case the case

of isotropic Zeemanof isotropic Zeeman

0Hg2Forbidden t iti of isotropic Zeeman of isotropic Zeeman

interactioninteraction (free atoms or the case

0Hg23transitions

of a cubic crystal field).EPR line

H0HresEPR line

RESONANCE FIELDS AND gRESONANCE FIELDS AND g--FACTORSFACTORS:conditionresonancestransitionAllowed 1 MM SS HH:conditionresonance ,stransition Allowed

resres

1

1

MgMg

MM

SS

SS

andstransitionForbiddenH

Hres

res

32

MMMM

gg

HH):conditions resonance two and stransition Forbidden

21

32

MgMg

,MMMM SSSS

H

H

HH )

res

resres

22

21

gg

MgMg SS

HH)and

Hres

32

2

MgMg H

H

HH )

res

resres

33

32

gg

MgMg SS

observed be not can fields"low " in lines two TheseHres3

MAGNETIZATION OF A SUBSTANCEMAGNETIZATION OF A SUBSTANCE121Sm21Sm

A singlesingle spin- up or down 2Sm

Ensemble of N non-interacting spins in a ti fi ld ( i d d )magnetic field (spins up and down):

N- number of spins up, N- number of spins down

N+N=NN :spinsofmomentMagnetic NNNNN

:spinsof momentMagnetic

Main question: numbers N and N - ?

BOLTZMAN DISTRIBUTIONBOLTZMAN DISTRIBUTIONmolecules in the medium (ensemble):

3

molecules in the medium (ensemble):(each molecule having, let say, three levels):

21

Due to interaction with the medium (thermostat or bath) electrons(bolls-) jump up (absorption of heat) and down (emission ofheat) traveling among the levels 1 2 and 3 These jumps are veryheat) traveling among the levels 1, 2 and 3. These jumps are veryfast, so one can say about the distribution of the electrons over thelevels in the thermodynamic equilibrium of the ensemble.

N1-mean number of the molecules with the energy E1N2 -mean number of the molecules with the energy E2N b f th l l ith th EN3 -mean number of the molecules with the energy E3N= N1+ N2+ N3-total number of the molecules.

p1= N1/N- probability to find a molecule with the p1 1 p ypopulated level 1 (i.e. with the energy E1), etc

The main question: what are these probabilities?

BOLTZMAN DISTRIBUTIONBOLTZMAN DISTRIBUTION--GENERAL EXPRESSIONGENERAL EXPRESSIONGENERAL EXPRESSIONGENERAL EXPRESSION

Probability to find a molecule with the populated level i (i e a molecule with the energy E ):(i.e. a molecule with the energy Ei):

kTiEep /1 i eZp k B lt t t T b l t t tk-Boltzman constant, T-absolute temperature,

Z-partition function (important characteristics !!!)

levels all over summation i

kTiEeZi

Probability pi depends on the energy Ei and on the temperature T

BOLTZMAN DISTRUBUTIONBOLTZMAN DISTRUBUTION--ILLUSTRATION FOR TWO LEVELSILLUSTRATION FOR TWO LEVELS

eZ kTE

1E2 excited

countedisenergy EE,E 21 0E

level ground the fromgy

E

1 0 ground

0101 2121

p,p,Tep,p EkTE

E

populatedis"1"levelonly11

ee kTkT p py21

21 pp,Tpopulated)equally are "2" and "1" (levels

2

POPULATION OF THE ENERGY LEVELS INPOPULATION OF THE ENERGY LEVELS INTHE THERMODYNAMIC EQUILIBTIUMTHE THERMODYNAMIC EQUILIBTIUM

E

E5e

r

g

y

E

E4

5

kTiEi eTp 1E

n

e

E34 i eZTp

EE2

population 1

0E1

Population exponentially decreases with the increase of the

p1 Population exponentially decreases with the increase of the

energy; level i is populated significantly if kT Ei ; The ground level is always (at any T ) The ground level is always (at any T )

the most populated level .

PARTITION FUNCTION FOR A SPIN S IN A PARTITION FUNCTION FOR A SPIN S IN A MAGNETIC FIELDMAGNETIC FIELD

i

kTiEeZ S...,,SM,MgME SSeS Hi

systemisotropiclymagneticaln,orientatioarbitrary field,magnetic -H

kTMgexpZ S SHsystemisotropic ly magnetical

kTMgexpZ

SSe

:made) is summation the after ( result Final

H

kT

gx,xSinh

xSSinhZ e2

212 H

eeSinhkTxSinh

21

2

:sine Hyperbolic

eeCosh 21 :cosine Hyperbolic

MAGNETIZATION MAGNETIZATION QUANTUMQUANTUM--MECHANICAL MECHANICAL EXPRESSIONEXPRESSION

magnetic external anby perturbed is sample a whenmechanics,classicalIn

M

through variationenergy its to related is ionmagnetizat its field,E

mechanicsquantumoflanguagethe UsingH

M

spectrumenergy an withmolecule a consider we

qg gg

i ,...,iE 21 definecanwelevelenergyeachFor

H. fieldmagnetic a of presence the in i ,,

as ionmagnetizatc microscopi adefinecanwelevelenergy eachFor

i

E

H

ii E

MAGNETIZATION MAGNETIZATION QUANTUMQUANTUM--MECHANICAL MECHANICAL EXPRESSION, MEAN VALUEEXPRESSION, MEAN VALUE

upsummingbyobtainedthenisMionmagnetizatmolarc macroscopi The

ionsmagnetizatc microscopi theupsummingby obtainedthen

:number)sAvogadro'(low ondistributi Boltzman the toaccordingaveraged

N

M)g(

i ii

kTEexpkTEexpN

then andi i kTEexp

HM

i i

i ii

kTEexpkTEexpEN

magnetism molecular in expression lfundamenta a is This

MAGNETIC SUSCEPTIBILITYMAGNETIC SUSCEPTIBILITY:system)(isotropiclitysusceptibimagneticMolar

HM or H,M

:system)(isotropiclity susceptibimagnetic Molar

:function partition the through sExpressionH

H kTEexpENZl Hfi ldtithff ti

HH

1kTEexp

kTEexpENkTZln

i i

i ii

f ii ihfd i ih hliibii

and ionmagnetizat for sexpression following the to leads ThisHfield,magnetic theoffunctions EEE iii

M

:functionpartitiontheofsderivativethroughlity susceptibimagnetic

ZlnTkNM

HM

2 ZlnTkN

TkN

HH 2TkN

CALCULATION OF THE MOLAR MAGNETIZATIONCALCULATION OF THE MOLAR MAGNETIZATION

kT

gx,xSinh

xSSinhZ e H2

212

xCothxSCothS

kTgZln e

H221212

2 kT:as rewritten be can This

H 2

SyCothSySCothS

kTgZln e

H221212

2

Sge Hwith

eeCosh

kTgy e

eeee

SinhCoshCoth:cotangentHyperbolic

MOLAR MAGNETIZATIONMOLAR MAGNETIZATIONthiiti tlTh

HM:thenisionmagnetizatmolarThe

SgBSN e :asdefinedfunctionBrillouintheis

M yB

kTgy,yBSNg eSe

:as defined function Brillouin the is

yCothySCothSyByB

S

S

111212

eeCosh

yS

CothS

yS

CothS

yBS 2222

:functionBrillouintheforcasesextremeTwo ee

eeSinhCoshCoth

splitting Zeeman e,temperaturlow ):functionBrillouin theforcasesextreme Two

kT,y 11splittingZeeman e,temperaturhigh ) kT,y 12

BRILLOUIN FUNCTIONBRILLOUIN FUNCTION--FIELD AND TEMPERATURE DEPENDENCEFIELD AND TEMPERATURE DEPENDENCE

SB1 S

7/2

5/20

5/2

3/2

1/2

kTge H

kT

Low temperature or/and high magnetic field: BS 1Hi h / d k i fi ld B High temperature or/and weak magnetic field: BS 0

MAGNETIZATION MAGNETIZATION LOW TEMPERATURE LIMITLOW TEMPERATURE LIMITWhen T0 or field is strong , y=gSH/kT becomes large ,

BS(y) tends to unity.L (hi h fi ld) li i f l i iLow temperature (high field) limit of molar magnetization:

220 gSNSNgTM M 220 eesat gSNSNgTM MThis is the saturation value only ground Zeeman

l l M S i l t dlevel MS=-S is populated:

Maximum magnetization all spins along magnetic fieldall spins along magnetic field

.g KJ 423

1010103811

(T l )TTHHMaximum magnetization-all spins along magnetic fieldall spins along magnetic field

gauss,K

..

.TkT

gJK 4

27 10170102792103811 (Tesla)T

THH

T

MOLECULAR MAGNETSMOLECULAR MAGNETS--SPIN ALIGNMENT IN EXTERNAL SPIN ALIGNMENT IN EXTERNAL

MAGNETIC FIELDMAGNETIC FIELDMAGNE IC FIELDMAGNE IC FIELD

H

Paramagnetic-disordered Ordered (parallel to field)g (p )

SATURATION OF MAGNETIZATION, S=1/2SATURATION OF MAGNETIZATION, S=1/2

1/2

MS1/T

-1/2

1/2 1/T

zHegE 2121 zeg22 zHegE 2121

H

PHYSICAL SENCE OF SATURATION (S=1/2)PHYSICAL SENCE OF SATURATION (S=1/2)

zHegE 2121 zHegE 2121 eg H

Boltzman factors for two Zeeman sublevels:

kTeg

ep,p H

H

H

1 11

kTeg

kTeg

ep,

ep HH 11 2

121

Decrease of T ( at fixed field fixed energy gap) increases population of the ground level.population of the ground level. Increase of magnetic field ( at a certain T) increases the Zeeman gap and thus increases population of the ground l l d d l ti f th it d l llevel and decreases population of the excited level.

MAGNETIZATION MAGNETIZATION HIGH TEMPERATURE LIMITHIGH TEMPERATURE LIMITllWe can check that for small y=gH/kT , BS(y) may be

replaced by the first term of the expansion in terms of y :HSg

fi ldld/t thi hHll

H

SgkT

Sgy...,yintermsSSyyBS

1

31 3

)5K 1 T means this conditions alexperiment standard (under

fieldlow and/or etemperaturhighHsmallkT

Sgy

1

HS

SkT

SgyBS

31

HM:limitthis inionmagnetizatMolarSSgSNgySBNg eS 1

HMM

SSk

NgSkT

SNgySBNg S

13

322

general) (more :sexpression all In gg

kTe

3

FIELD DEPENDENCE OF MAGNETIZATIONFIELD DEPENDENCE OF MAGNETIZATION--CLASSICAL PICTURECLASSICAL PICTURECLASSICAL PICTURECLASSICAL PICTURE

HM 122 SSNg HM 13

SSkT

Weak fieldWeak field Strong fieldStrong fieldApplied magnetic fieldApplied magnetic field

disordered partially ordered fully ordered

gg

p y y

MAGNETIC SUSCEPTIBILITYMAGNETIC SUSCEPTIBILITY--CURIE LAWCURIE LAWN 22 SS

kTNg

:litysusceptibiMagnetic

HM 13

22

SSkT

Ng

HM

:litysusceptibiMagnetic

13

22 :T

CkT

etemperatur of function a as varieslity susceptibiMagnetic

H 3

SSk

NgC,TC 1

3

22

kT

mechanicsquantumofonintroductithebeforedata alexperiment from 1910 in proposed was whichlow Curie the is This

3

TT :linestraighthorisontalabeshouldthisfunctionaasmeasureto:onverificati alExperiment

mechanics.quantum ofonintroductithebefore data

CT,TT

:linestraighthorisontalabeshouldthisfunctionaas measure to

EFFECTIVE MAGNETIC MOMENTSEFFECTIVE MAGNETIC MOMENTS22N

2222

13

SSkT

Ng 22 1

:SSSg

spin withparticle a for momentmagnetic theofvaluesquared the is value The

222 1SSg presented be can datality susceptibimagnetic alExperiment

:momentmagnetic effective called-so the of dependence etemperatur the of form the in

21

23

NkT

eff

g

28

11250503 .,.kNcgsemu

Nff

to closevery units In

218 Teff

EXPLANATIONEXPLANATION--QUANTUM MECHANICAL BACKGROUNDQUANTUM MECHANICAL BACKGROUNDQUANTUM MECHANICAL BACKGROUNDQUANTUM MECHANICAL BACKGROUND

:momentmagnetic theofOperator

f

gp

gg SS SS 2222

theof valuemeanthemechanicsquantumofrulethetoy Accordingl

f tithith state quantum a in quantity physical

A

:ofoperatorascalculatedbeshould function-wavethewith AA n r notation sDirac'

:ofoperatorascalculatedbe should

nAndAAAA

nn rr

MAGNETIC MOMENTMAGNETIC MOMENT

S :ascalculatedbeshouldspinwith particle a of momentmagnetic squared of value mean The

2222 SMSMgS

SSS

:ascalculated beshouldspinwith

S

122

SMSSSM

SM S of functions-eigen the are

S

S

1

1222

SMSSSMg

SMSSSM

SSS

SSS

1

122

SMSM

SMSMSSg SSconditionionnormalizat1 SMSM SS

:result Finalconditionionnormalizat

11222 SSgSSg SS or

MAGNETIZATIONMAGNETIZATION--FIELD AND TEMPERATURE DEPENDENCE FIELD AND TEMPERATURE DEPENDENCE FIELD AND TEMPERATURE DEPENDENCE FIELD AND TEMPERATURE DEPENDENCE

2spinwithstategroundpossessingmolecules for plotsH versus units in ionMagnetizat

gSkTNM

2spinwithstategroundpossessingmolecules eg,S

Pierre CuriePierre Curie, 1903 Nobel Laureate

in Physics

MAGNETIC MOMENTS OF SOME METAL IONSMAGNETIC MOMENTS OF SOME METAL IONS

Gd3+, S=7/2- Gd-sulphate,

Fe3+, S=5/2- iron-ammonium alum

e

n

t

/

m

o

l

Cr3+, S=3/2- chromium-potash l

m

o

m

e

alum

a

g

n

e

t

i

c

Experimental data:Brillouin

M

a

H/T, Tesla/K

Experimental data:Henry W., Phys.Rev.,88 (1952) 559

H/T, Tesla/K

Strong field and/or low temperature Msat=2SW k fi ld d/ hi h Weak field and/or high temperature M=0

Chapter IIIMagnetic properties of a free ionMagnetic properties of a free ion,

molecules containing a unique magnetic center without first order orbital magnetism and EPR ofwithout first-order orbital magnetism and EPR of transition metal ions and rare-earths, spin-orbital

interactioninteraction.

QUANTUMQUANTUM--MECHANICAL DESCRIPTION MECHANICAL DESCRIPTION OF A FREE ATOMOF A FREE ATOM

Quantum numbers, spinQuantum numbers, spin--orbital orbital coupling, gcoupling, g--factorsfactorscoupling, gcoupling, g factorsfactors

Niels Henrik David Bohr

Erwin SchrodingerErwin SchrodingerBorn: 12 Aug 1887 in Erdberg, Vienna, AustriaDied: 4 Jan 1961 in Vienna, Austria,

Nobel Prize, 1933 fundamentals of QUANTUM MECHANICS

Wolfgang Ernst Pauli , Born: 25 April 1900 in Vienna, AustriaDi d 15 D 1958 i Z i h S it l dDied: 15 Dec 1958 in Zurich, Switzerland

In 1945 he was awarded the Nobel Prize for decisive contribution through his discovery inIn 1945 he was awarded the Nobel Prize for decisive contribution through his discovery in 1925 of a new low of Nature, Pauli exclusion principle. He had been nominated for the Prize by Albert Einstein

QUANTUM NUMBERS QUANTUM NUMBERS FOR ONE ELECTRON IN A SPHERICAL POTENTIAL FOR ONE ELECTRON IN A SPHERICAL POTENTIAL

(HYDROGEN ATOM ONE ELECTRON IONS)(HYDROGEN ATOM ONE ELECTRON IONS)(HYDROGEN ATOM, ONE ELECTRON IONS)(HYDROGEN ATOM, ONE ELECTRON IONS)

numberquantummainthe ...,,nn 321 momentumangular

orbitaltheofnumberquantumn,...,,l

l110

lnumberquantummagnetic

momentumangular

lllllm

n,...,,l

l

1211

110

numberquantumprojectionspin

valuesm

ll,l,...,l,lm

s

l 1211

down""andup""spin

spin :m,s s 2121

, state,quantumtheofparitydown -and up -spin

l1

etc. even,- ,odd"-" states, odd"" and even"" dp

SPECTROSCOPIC NOTATIONS:SPECTROSCOPIC NOTATIONS:SPECTROSCOPIC NOTATIONS:SPECTROSCOPIC NOTATIONS:

l ,...,,,,, 543210hgfdps

even and odd states:even and odd states:

p-odd ( l =1), d even (l=2)seven (l=2),

TRANSITION METAL IONSTRANSITION METAL IONSTRANSITION METAL IONSTRANSITION METAL IONSTypical oxidation degrees and d n:Ions of transition

metals of the iron 3231 VdTid ,,Typical oxidation degrees and d :

group have unfilled 3d shells:

233 VCrd

VdTid,,

,,

2635

234 CrMnd ,,23 ln ,Closed d-shellcontains 10 electrons:

2827

2635

NidCodFedFed ,,

contains 10 electrons:(2l+1)2=10

29 Cud

NidCod ,,

ATOMIC TERMS, SPECTROSCOPIC NOTATIONSATOMIC TERMS, SPECTROSCOPIC NOTATIONSDEFINITION: 2S+1L(or SL)-TERMS

closedshells

Rule of the addition of the angular (spin and orbital) shells

unfilled shells

momenta(vector coupling scheme):

|ll|,....,ll,llL 212121 1 |ss|,...,ss,ssS 212121 1

MOMENTUMANGULARORBITALTHEOFNUMBERQUANTUML

|| 212121

113 SLPTHEOFSPINFULL

SHELLELECTRONICTHEOF MOMENTUMANGULARORBITAL

S

,, 11 SLP 2334 SLF ,

SHELL ELECTRONIC THEOFSPINFULLS

GROUND TERMS OF TRANSITION METAL IONSGROUND TERMS OF TRANSITION METAL IONSELECTRONIC GROUND

IONSELECTRONIC

CONFIGURATIONGROUND

TERM

Ti3+ V 4+ 3d1 2D (L=2 S=1/2)Ti , V 3d D (L 2, S 1/2)

V 3+ 3d2 3F (L=3, S=1)

Cr3+, V 2+ 3d3 4F (L=3, S=3/2)

M 3+ C 2+ 3d4 5D (L 2 S 2)Mn3+ Cr2+ 3d4 5D (L=2, S=2)

Fe3+, Mn2+ 3d5 6S (L=0, S=5/2)( , )Fe2+ 3d6 5D (L=2, S=2)

Co2+ 3d7 4F (L=3, S=3/2)

Ni2+ 3d8 3F (L=3, S=1)Ni 3d F (L 3, S 1)Cu2+ 3d9 2D (L=2, S=1/2)

SOME OBSERVATIONSSOME OBSERVATIONSTransition metal complexes Transition metal complexes

Partiall filled dPartiall filled d shellshell ll 22Partially filled dPartially filled d--shellshell, , llii =2=2,degeneracy of one-electron states= 2(2l+1)=10dn- n electrons , d10-n- n holes in the fully filled d10 shell

dn d d10 n h ll ( li t fi ti ) h dn and d10-n shells (complimentary configurations) have the same ground terms:

3d1 (one electron) and 3d9 (one hole) 2D (L=2, S=1/2),3d2 (two electrons) and 3d8 (two holes)3F (L=3, S=1), etc.3d (two electrons) and 3d (two holes) F (L 3, S 1), etc. d5- half-filled d-shell, 6S- term, L=0 (important case: total orbital angular momentum =0 ) , S=5/2.

DEGENERACY OF THE ATOMIC (IONIC) DEGENERACY OF THE ATOMIC (IONIC) LEVELSLEVELS--REMINDERREMINDERLEVELSLEVELS REMINDERREMINDER

DegeneracyDegeneracy one energy level contains several quantum states (wave-functions):several quantum states (wave-functions):

1) 1s level (n=1, l=0) of H (hydrogen) is orbitally non-degenerate (singlet) this level is doubly degeneratedegenerate (singlet), this level is doubly degenerate over spin projection :

spin up or down m =1/2 or -1/2;spin up or down , ms 1/2 or -1/2; 2) 2p level (n=2, l=1) is orbitally triply degenerate

(m = 1 0 1) The general multiplicity of the(ml = -1, 0, 1). The general multiplicity of thedegeneracy is 6 (ms=1/2 or -1/2);In H atom there is an additional (accidental)In H atom there is an additional ( accidental )

degeneracy. The energy levels are independent ofthe quantum number l, so that the energies of 2s q , gand 2p levels are equal.

MANYMANY--ELECTRON IONSELECTRON IONSIn many-electron atoms the value of L (totalorbital angular momentum of all electrons in the

unfilled shells) is the appropriate quantum number that enumerates the energy levels.

The multiplicity of the orbital degeneracy is (2L+1).The full multiplicity of the LS term is

(2L+1) (2S+1).Example1: Ti3+ ion, 1 electron in the unfilledExample1: Ti ion, 1 electron in the unfilledd-shell( 3d1-ion ).Ground term 2D (L=2, S=1/2)

Example2: Cr3+ ion 3 electrons in the unfilledExample2: Cr3+ ion, 3 electrons in the unfilled d-shell (3d3-ion ). Ground term 4F (L=2, S=3/2,

i i f th l t )maximum spin for three electrons).

NOTATIONS FOR THE NOTATIONS FOR THE WAVEWAVE--FUNCTIONS OF A FREE IONFUNCTIONS OF A FREE ION nnSMLLSM ,..,,,,,..,,

electronstheofscoordinate

2121rrr

rrr

n

n

,..,,,,..,,

) down"" or up"(" variables spin electrons theofscoordinate

21

21 rrr

SL

spintotaltheofnumberquantum momentumangulartotaltheofnumberquantum

LMmomentumangulartotaltheof

projection the of number quantumpq

SM projection the of number quantummomentumangulartotal the of

:numbersquantum-notation)(Dirac notationShortspintotal the of

SLMLSM q)(

EXCERPTSEXCERPTS FROM QUANTUM MECHANICS (REMINDER)FROM QUANTUM MECHANICS (REMINDER) :operators)theforioncap"-notat("operatorsvaluesPhysical

H

etcmomentumtheofoperatornHamiltonia i.e. energy, the of operator

:operators)theforioncap notat(operatorsvalues Physical

p

functions,-eigen values,-eigenvalues Observable

etc.momentum,theofoperator

SMLLSM functions-eigentheare functions-waveThe EH :examplefor

:S,S,L,Lfour zz

SMLLSM

:operators following the of

g22

S

L

squaredspin

squared, momentum angular orbital

2

2

,zL

S

z L operator vector the of projection- squared,spin

.zSz S operator vector the of projection-

EIGENEIGEN--VECTORS AND EIGENVECTORS AND EIGEN--FUNCTIONS:FUNCTIONS:

34 3 SLF:

T

ExampleGeneral rules for the angular momenta operators 234 3 S,LF

:(labels) functions-Eigen Termangular momenta operators

of the arbitrary nature, in particular- orbital angular

2 2323223

3433

3

MM,,MM,,LMM,,

SLSL

SLparticular orbital angular

momentum and spin: SLSL

SLSL

MLSMSSMLSMS

MLSMLLMLSML

1

12

2

23

25

23

232

22

33

3433

MM,,MM,,S

MM,,MM,,L

SLSL

SLSL

SLLSLz

SLSL

MLSMMMLSMS

MLSMMMLSML 23

23

3210123

33

,,,,,,M

MM,,MMM,,L

L

SLLSLz

SLSSLz MLSMMMLSMS

3113 23

23 33

M

MM,,MMM,,S SLSSLz 2222 ,,,M S

ABOUT QUANTUM NUMBERSABOUT QUANTUM NUMBERS

SLMLSM functionsEigen 12 MLSMLLMLSML

112

SLSL

LL

MLSMLLMLSML

length definiteahasvector 2L 12 SLSL MLSMSSMLSMS

g

2 1

SLLSLZ MLSMMMLSML

SS length definite a has vector 2S

0

YXLZ

SLLSLZ

LL,MLZ

MLSMMMLSML

:axis around Precession L

0

SLLSLZSSMSZ

MLSMMMLSMS

:axisaroundPrecession S 0 YXLZ SS,MSZ :axisaround Precession S

MAGNETIC FIELD CREATED BY AN ORBITAL MAGNETIC FIELD CREATED BY AN ORBITAL MOTIONMOTIONMOTIONMOTIONH

Electronic orbit reminds earth orbitElectronic orbit reminds earth orbit , , orbital motion is equivalent to a circular orbital motion is equivalent to a circular electric currentelectric current

that produces a magnetic field that is perpendicularthat produces a magnetic field that is perpendicular to the planeto the plane

CLASSICAL ESTIMATIONCLASSICAL ESTIMATIONMagnetic field H created by the moving

electron: proportional to the orbital angular t Lmomentum L .

Energy of interaction spin-magnetic field

E =-S HHL SHL, S S

Energy of spin orbital coupling=Energy of spin orbital coupling const LS ,

Hamiltonian of spin orbital coupling=Hamiltonian of spin orbital coupling

d

const SL

,operatorsare and SL

SPINSPIN--ORBITAL INTERACTIONORBITAL INTERACTIONInteraction of the spin magnetic moment with the magneticInteraction of the spin magnetic moment with the magnetic field created by the orbital motion (current) of the electron

Magnetic field created byMagnetic field created by the orbital motionOrbital motion

H

spin

:aspresentedcanninteractioorbitalspinofOperator

SOV SL

:aspresentedcanninteractio orbital-spinof Operator

SLSL and operators vector of product scalar

n,interactioorbital-spinofparameter

zzyyxxSO SLSLSLV

SPINSPIN--ORBITAL SPLITTING ORBITAL SPLITTING --QUALITATIVELYQUALITATIVELY

H

spinspin

HEnergy of the system does depend on the

mutual orientation of the full spin and pmagnetic field created by the orbital motion

of the unpaired electron (electrons).p ( )

PARAMETERS OF SPINPARAMETERS OF SPIN--ORBIT COUPLING FOR THE ORBIT COUPLING FOR THE GROUND TERMS OFGROUND TERMS OF SOME TRANSITION METAL IONSSOME TRANSITION METAL IONS

Ion Configuration Term ,cm-1

Ti3+ 3d1 2D 154V 3+ 3d2 3F 104V 2+ 3d3 4F 55

Cr3+ 3d3 4F 87Cr 3d F 87Mn3+ 3d4 5D 85Fe3+ 3d5 6S 0Fe3 3d5 6S 0Fe2+ 3d6 5D -102C 2+ 3d7 4F 180Co2+ 3d7 4F -180Ni2+ 3d8 3F -335Cu2+ 3d9 2D -829

COMMENTCOMMENTCOMMENTCOMMENT Spin-orbital interaction is positive for d1,d2 , p p , ,

d3 , d4 ions. Spin orbital interaction is negative for d6 d7 Spin-orbital interaction is negative for d6,d7,

d3 , d4 ions, for the complimentary fi ti d t t f iconfigurations dn and d10-n constants of spin-

orbital coupling are of the opposite signs. Spin-orbital interaction is zero for d5 , i.e.

for 6S term S state does not carry orbitalfor 6S term - S state does not carry orbital angular momentum.

TOTAL ANGULAR MOMENTUMTOTAL ANGULAR MOMENTUM:momentumangulartotaltheofOperator :momentumangulartotaltheofOperator

SLJ

: and axes the at sprojection the of operators-components threeoperator,typeVector

z yx,

:operatorsmomentaangularallforcommon-Propertiesxxxyyyzzz .SLJ,SLJ,SLJ

:operatorsmomenta angularallforcommonProperties

JJ

,M,JJJM,J 12J

valuesJJJJz

JJ,J,...,J,JMM,JMM,JJ

1211

,projectionand momenumangulartotaldefiniteawith function-eatomic wav the for notation sDirac'

J

J

MJM,J

momentum angular orbital and spin coupled withstates quantum i.e.p jg J

TOTAL ANGULAR MOMENTUMTOTAL ANGULAR MOMENTUM--Wave-FunctionsAtomic term definite L and SAtomic term definite L and S

(Russel-Saunders coupling).All d l f JAllowed values of J:

J = L+S, L+S-1, , | L-S |, , , | || L-S |= L-S if L>S and | L-S |= S-L if S>L

Example: term 3F S=1, L=3 J=4, 3, 2Labeling of the atomic wave-functions:Labeling of the atomic wave functions:

JMJSLImportant:

this state with a definite J and MJ is a state with the definitethis state with a definite J and MJ is a state with the definite L and S but not (!) with the definite projections ML and MS

LABELSLABELS--QUANTUM NUMBERSQUANTUM NUMBERSQQ

Total orbital Total spin angular momentum

pangular momentum

MJSL JMJSL

Total Projection of the totalTotalangular momentum

Projection of the totalangular momentum

CLARIFICATIONCLARIFICATION What does it mean: definitedefinite J and M ?

:JJ andoperatorstheof 2

What does it mean: definitedefinite J and MJ ?This means that the wave-functions are the eigen-functions

:JJ z and operatorstheof JJ ,MJSLJJMJSLJ 12

JJJz MJSLMMJSLJ What does it mean: definitedefinite L and S but not (!)What does it mean: definitedefinite L and S but not (!)

definite projectionsdefinite projections ML and MSThis means that the wave-functions are the eigen-functions

MJSLLLMJSLL 12 This means that the wave functions are the eigen functions

:SL 2 and operators the of 2

JJ

JJ

MJSLSSMJSLS

,MJSLLLMJSLL

1

12

zz SL and of functions-eigen the not but

CLASSICAL PICTURE OF COUPLING OF SPIN CLASSICAL PICTURE OF COUPLING OF SPIN AND ORBITAL ANGULAR MOMENTAAND ORBITAL ANGULAR MOMENTA

Vectors L and S precess at the conical surfaces around vector J so that the

Jsurfaces around vector J, so that the

vector sum is L+ S = J. Because of the rapid precession of LLand S about the direction J it may be said that mean projections of these

t t th l XY Th

L

vectors onto the plane XY are zero. The length of L and the length of S remain

constant [L(L+1)]1/2 and [S(S+1)]1/2 TheS

LS constant, [L(L+1)] and [S(S+1)] . The angles: L (between L and J)and S (between S and J)

S

Sallowed values of J (general quantum-mechanical rule of momenta addition):

J L+S L+S 1 | L S |

XY

J = L+S, L+S-1, , | L-S |

VECTOR MODEL FOR THE ANGULAR VECTOR MODEL FOR THE ANGULAR MOMENTA IN QUANTUM MECHANICSMOMENTA IN QUANTUM MECHANICSMOMENTA IN QUANTUM MECHANICSMOMENTA IN QUANTUM MECHANICS

Vector J is in a precession about arbitrary direction Z at

Zabout arbitrary direction Z at the conical surface, so that the mean values of the

JZ the mean values of the projections of J onto the plane perpendicular to Z

J

p p paxis are zero (JX , JY).Good quantum numbers:

J and MJ

YJX

JY 1 JJJX

YJX 1

Mcos

JJ

JJ

1 JJcos

SPINSPIN--ORBIT COUPLINGORBIT COUPLING--CLASSICAL ILLUSTRATIONCLASSICAL ILLUSTRATIONZZ

L JZVector model:Vector model:

GoodGoodquantum

b

Z

MJSLnumbers:

JMJSLS Y

XVector J is in a precession around Z-axis and at the same time Land S precess around J. Length of |L| and length of |S| have definite values but not their projections M and M on Z axis Vector J has avalues but not their projections MS and ML on Z axis. Vector J has a definite length and projection MJ but mean and vanish.

SPINSPIN--ORBIT SPLITTINGORBIT SPLITTINGSLJ- multipletsp 22221

2222

SLJ SLSL 22221

SLJ:ninteractio orbitalspin of Operator

SL

22221

E

SLJV

J

SO :Eq.thefromfoundbecanvaluesEigen

SL

1

MJSLEMJSLV

E

JJJSO

J :Eq.thefromfound becanvaluesEigen

11121 J

MJSLSSLLJJMJSLV JJSO of function a asenergy - multiplets the of Enegies

111 SSLLJJESL

:) and definite (

1112

SSLLJJEJ

MULTIPLETSMULTIPLETS--TERMS OFTERMS OF dd2 2 ANDAND dd88 IONSIONS

-43 J=4,d2 d8>0

RareRare--Earth IonsEarth Ions--Strong SpinStrong Spin--Orbit CouplingOrbit Coupling

MAGNETIC MOMENT OPERATORMAGNETIC MOMENT OPERATORhasatomoriontheinelectronEach

ttibit lthftV t

.spin and momentum angular orbitalhasatom or iontheinelectronEach

zzyyxxl l,l,l l :momentmagnetic orbitaltheofoperator Vector

Bmce

2 ) (or magneton Borh

zezyeyxexeS sg,sg,sgg s :momentmagnetic Spin

Sl

ee

,.gg

sl 2

00232

:electron free a for factor-g or Lande, factor

Sl sl 2:ionelectron-manyaofmomentmagneticTotal

:ion) electron-(one momentmagnetic Total

i

ii

i ,, sSlLSL 2:ionelectronmany aofmomentmagnetic Total

VECTOR MODEL FOR THE COUPLING OF THE VECTOR MODEL FOR THE COUPLING OF THE ANGULAR MOMENTA IN QUANTUM MECHANICSANGULAR MOMENTA IN QUANTUM MECHANICSANGULAR MOMENTA IN QUANTUM MECHANICSANGULAR MOMENTA IN QUANTUM MECHANICS

Vectors L and S (of a given length) precess around vector J at the conical

zJ precess around vector J at the conical

surfaces, so that the mean values of the projections of L and S at the plane

J

the projections of L and S at the plane perpendicular to axis of J are zero (Lx,Ly and Sx,Sy). At the same time

LLz ( x, y x, y)projection of L and S at the axis of J are non-zero and Jz=Lz+ Sz.S

Sz

111

11121

SSLLJJSSLLJJcosSL

11

111

SSLL

SSLLJJcosx

y

Selected values for (J ) according to: J=L+S, L+S-1,.,|L-S|

ZEEMAN SPLITTING FOR ZEEMAN SPLITTING FOR LSJ LSJ TERMSTERMS-- VECTOR MODELVECTOR MODEL !!!factor-factor:attention:vectordefineusLet gSLM 22

electron)theoffactor(2factortoowing momentum angular the withldirectiona-co not is Vector

!!! factorfactor:attention:vector defineus Let

g

geSLJM

SLM 22

electron).theof factor( 2factor toowing g

2SB

lM2S

C

S- angle between S and J

S

JS L- angle

between L and J

LL

A

Because of the rapid precession of M around the direction ofJ, it may be assumed that the component BC of M averages

t t i fi it ti h th t l th t ACout to zero in any finite time, such that only the component ACof M along J needs to be considered.

CLASSICALCLASSICAL VECTOR MODELVECTOR MODEL--PROJECTIONS PROJECTIONS L L AND AND SS

cos LLJLJS 2222

ldfdtThcos

cos

S

L

JSLSLSJSJL

LJLJS

2

2222

:as expressed bemay along and of and components The coscos SL JSLSL

coscos SL JLSJSJSLJL

2

2222

222

axisisthisthatassumewevectortheofdirectiontheonto and vectors the of sprojection the are components These

Z

cos S

JSL

JLSJS 2

y sphericall a for directions special no is there fact, Inaxis.-isthisthatassume we,vectortheofdirectionthe onto ZJ

ionsandatoms freelikesystems,symmetric

ZEEMAN PERTURBATION FOR A ZEEMAN PERTURBATION FOR A LSJLSJ--TERMTERM

HH SL 2Z H:ninteractioZeeman HH SL

21

2Lorb

Z

gggH

t ib tiithff t :oncontributi orbital the for factor-

2Se ggg:as writtenbe thenmay energy Zeeman The

:oncontributispintheforfactor-

HSL22 SLZ

gcoscosE

!!!part"spin"infactor:Attention

JSL 2

.,g

and vectors classical the withdealing are we:Note!!!partspin infactor:Attention

H HJJLS JLSJJSLJ 222222222

223

222

ZE HJJLS 223

ZEEMAN HAMILTONIAN FOR AZEEMAN HAMILTONIAN FOR A LSJLSJ--TERMTERM

and vectors the of function a asEnergy HJJLS JLSE :,Z 222 223 operatorsnotbutvaluesclassicalarevectorsall

,expression )!!! mechanical-quantum a not but ( classical a is ThisZ

mechanicsquantumofrulethetoaccordingexpression mechanical-Quantum

operators.notbutvaluesclassicalarevectors all

:operators theirby dsubstitute be should values classicalmechanics,quantumofrulethetoaccording

222222

nHamiltonia the obtains one way this InJJJJLLSS .,,,,HE ZZ

222222

:energy classical of instead HJJLS H Z 222 223 levels.energy Zeeman the find to have weFinally,

Z

ZEEMAN LEVELS FOR AZEEMAN LEVELS FOR A LSJLSJ--TERMTERM 223 222

Z

HZ:Hfieldmagnetictheofdirectionthealongchosenbecanaxis

JJLS H

223 222 .JHZ- ZZl )(l lTh

H

:Hfieldmagnetic theofdirectionthealongchosen be canaxis

0

0

JLS

JMSLHMJSLE JZJJJM

values) (meanlevelsenergy The

1

112

22

MJJJ,LLSS

andareandofvalues-eigenthe

and are and of values-eigen The

J

LS

1MgE

.MJJJ JZ

H:find can one account into this Taking

andare andofvalues-eigen the

J

113

LLSSg

MgE

J

JJJJM

:Notation

H0

122 JJgJ

gg--FACTORS FOR LSJFACTORS FOR LSJ--TERMSTERMSThe energy sublevels are enumerated by the quantumThe energy sublevels are enumerated by the quantum

numbers MJ (projection of the full angular momentum) and theenergy splitting depends on the field.

gy p g p

The value gJ is the g- factor for the LSJ-term, g- factor for the LSJ-term is the function of L, S and J:

1211

23

JJLLSSgJ 122 JJ

Limiting cases:Limiting cases:pure spin state L=0 and J=S (orbital angular pure spin state L=0 and J=S (orbital angular

momentum=0):g = g =2gJ= gS=2

pure orbital state S=0 and J=L (spin angular momentum=0):momentum=0):

gJ= gL=1

g SHARP DISTINCTION FROM g !gJ -SHARP DISTINCTION FROM ge !Rare-earth ions-strong spin-orbit coupling:Rare-earth ions-strong spin-orbit coupling:

5314 1 JLSFIIICef 2termground 4331

23

24 25

J,L,SF,IIICe,f termground

76

27

252

4322

23

25

g

451422

42 J,L,SH,IIIPr,f 3 term ground

54

4 g

EPR of LSJ STATESEPR of LSJ STATES21M

field resonanceHH

res

res

g

21SM

21S

Hres g

H:electronFree

res 2 egg 25JM

23M

21SM

momentum)angularorbitaltocoupled(spinion earthRare

eg 23JM21JM25J

Hmomentum)angularorbitaltocoupled(spin

res 22 JJJ

g,gg 21JM252F

shifted is line EPR:result main The23JM

25JMfieldstrongaofregionthein

H

MAGNETIC SUSCEPTIBILITY FOR MAGNETIC SUSCEPTIBILITY FOR LSJLSJ--TERMSTERMSLSJLSJ--TERMSTERMS

The derivation of the magnetic susceptibility for LSJ term is rigorously parallel to the derivation in the case of pure spin systems.rigorously parallel to the derivation in the case of pure spin systems.

The final result can be obtained by substitution: SJ, gSgJMagnetic susceptibility for aMagnetic susceptibility for a LSJ termterm:

13

22 JJ

kTNgJ

3kT

13

22

JJk

NgC

TC J

JJ withThis leads to the Curie low: 3kT

Magnetization:Magnetization: yBJNgM JJ

kT

JgyyB

,yBJNgM

JJ

JJ H

withfunction BrillouinkT

Important: the results are valid for a well isolatedwell isolated LSJ term term

VALUES OF VALUES OF gJ AND AND T FORFOR RARERARE--EARTH IONSEARTH IONSSLJ 4fGround terms SLJ for 4f n ions- see previous slide

O.Kahn, Molecular magnetism

RareRare--Earth IonsEarth Ions--Strong SpinStrong Spin--Orbit CouplingOrbit Coupling

MAGNETISM OF RAREMAGNETISM OF RARE--EARTH IONS EARTH IONS -- SOME SOME EXAMPLES (Gd(III) and Eu(II))EXAMPLES (Gd(III) and Eu(II))EXAMPLES (Gd(III) and Eu(II))EXAMPLES (Gd(III) and Eu(II))

shellfilledhalf). -( shell filledpartially contains ions earthRare

444

44477

140

ffIIEufIIIGd

fff

and

:term Groundshell filledhalf

27027

444

8

SJSLJS

ffIIEu,fIIIGd

tithtt ib tibit lthhti li1)Thi

and : ions of features main Two

and

4

270277

27

IIEuIIIGdf

SJS,L,JS

withstate spin pure a to equivalent is vanishes. sticscharacteri

magnetic thetooncontributi orbitalthewhencaseparticular ais1)This

27186

278 SS

l dh llidhlhThi

es.temperatur reasonable all at ly considerab exceeds that

,energy inhighvery are states Excited ) 000302 1278

276

kT

cm,SEPE

isotropicperfectlyislitysusceptibimagneticThe coupling. orbit-spin no is here Since

.populatedthermally is termgroundtheonly that means This0 ,L

level. spin for valid islow Curieisotropic,perfectly islity susceptibimagnetic The

27S

MAGNETISM OF RAREMAGNETISM OF RARE--EARTH IONS EARTH IONS --SOME EXAMPLES (SOME EXAMPLES (Sm(II)Sm(II) and and Eu(III)Eu(III)))SOME EXAMPLES (SOME EXAMPLES (Sm(II)Sm(II) and and Eu(III)Eu(III)))

The main feature of these ions: the dependence T vs. T does not followth C i l t f LSJ d t tthe Curie low as one can expect for a LSJ ground state:

7F0 (L=S=3)In fact the situation is different due to the presence ofIn fact, the situation is different due to the presence of thermally populated excited states (J=1, 2, 3, 4, 5, 6):

7F 7F 7F 7F 7F 7F7F1, 7F2, 7F3, 7F4, 7F5, 7F6The energy levels are given by:

7

12

JJJEi ikidhfhh

0

7

IISmIIIEuF

to close are states excited cases, special and origin.anastakenisstate groundthe ofenergy thewhere

1300 cm one ground the

J6 population)(Boltzmanthermalaccountintotaking

averagedbeshouldsusceptibilityMagnetic

6 JT:levelsexcitedtheof

population)(Boltzmanthermalaccountintotaking

5

0

p

JpT

J

JJ

theofprobability-factorBoltzmantheiswhere

4 1 kTEexpZp:E

JJ

Jenergy thewithleveltheofpopulation

3

122

JJNgJ

JJ

J

:givenawithstatequantumaoflity susceptibi

12

3 60

3

6

,...,JkT

J

:alloversummationincludesfunctionPartition

01

Energy

60

E

kTEexpZJ

J

:levelsEnergyEnergy pattern 211510630 6543210 E,E,E,E,E,E,E

EJ :levelsEnergy

THE MAGNETIC SUSCEPTIBILITYTHE MAGNETIC SUSCEPTIBILITY

:lity susceptibi averagedthermally The

k6

kTJJJ

kTJJexpJJT J

2112

2112

60

theofdegeneracythe of factorThe

tymultipliciJ

kTJJexpJJ

12

21120

functionpartitiontheinsummationtheinandfactor Boltzman the in account into taken is factor This level.

g yJ

yp

p

JJ

LLSSgJ 1211

23

.g

JJ

23

122

to equall are states excited the for factors- All

2

FINAL RESULT FOR MOLARFINAL RESULT FOR MOLAR (T)FINAL RESULT FOR MOLARFINAL RESULT FOR MOLAR (T)MolarMolar ((TT) for) for Sm(II) and Eu(III) ions:Sm(II) and Eu(III) ions:

kTJJexpJJJN

112136

2 MolarMolar ((TT) for ) for Sm(II) and Eu(III) ions:Sm(II) and Eu(III) ions:

kTJJexpJkT

NT J

11243

60

2

.Jg

p

J

J

dsubstitute are andby replaced are all where 230

Note: this expression is strictly valid for a free ion onlyvalid for a free ion only. Influence of surrounding in crystal and complexes-Influence of surrounding in crystal and complexes-a separate question. Crystal field splits (in general)

J-multiplets and affects magnetic properties.p g p p

THE NUMERICAL RESULT:THE NUMERICAL RESULT:T versusversus kT/ PLOT FOR ANPLOT FOR AN Eu(II) COMPOUNDCOMPOUNDT versusversus kT/ PLOT FOR ANPLOT FOR AN Eu(II) COMPOUNDCOMPOUND

T is temperature dependent, i.e. does not follow the Curie low. At T=0 the product T0 due to the fact that (J=0)=0.T increases with the increase of temperature due to thermal

pop lation of the states ith high J that contrib te to thepopulation of the states with high J that contribute to the susceptibility.

Chapter IVEffects of crystal field.

Group-theoretical introductionGroup theoretical introduction. Ground terms of the transition metal

ions in the crystal fields Anisotropy ofions in the crystal fields. Anisotropy of the g-factor. Zero-field splitting:

qualitative and quantitativequalitative and quantitative approaches. Covalence and orbital

reduction EPR of the metal ionsreduction. EPR of the metal ions in complexes.

CRYSTAL FIELDCRYSTAL FIELD--THE MAIN PROBLEMTHE MAIN PROBLEM

MeMe

Free metal ion Men+ in a LS or

LSJ state Coordinated ion M (li d) li dMe(ligand)6 -ligand

surrounding in a complex compound or in a crystalcompound or in a crystal

The main questionThe main question--how the surrounding affects the energy how the surrounding affects the energy l l d th ti til l d th ti tilevels and the magnetic propertieslevels and the magnetic properties

Hans Bethe ,

G b A i

Hans Bethe ,Nobel winner,1967

German-born American theoretical physicist who helped to shape classical physics into quantum physics and increased the understanding of the atomic processes responsible for the p pproperties of matter and of the forces governing the structures of atomic nuclei He received theatomic nuclei. He received the Nobel Prize for Physics in 1967 for his work on the production of energy in stars Moreover he wasenergy in stars. Moreover, he was a leader in emphasizing the social responsibility of science.

J.H. Van Vleck

American physicist and mathematician who shared the Nobel Prize forPhysics in 1977 with Philip W. Anderson and Sir Nevill F. Mott. The prizehonoured Van Vleck's contributions to the understanding of the behaviour ofhonoured Van Vleck's contributions to the understanding of the behaviour ofelectrons in magnetic, noncrystalline solid materials.Van Vleck developed during the early 1930s the first fully articulated

h i l h f i L h hi f hi fquantum mechanical theory of magnetism. Later he was a chief architect ofthe ligand field theory of molecular bonding. He contributed also to studiesof the spectra of free molecules, of paramagnetic relaxation, and othertopics. His publications include Quantum Principles and Line Spectra (1926)and the Theory of Electric and Magnetic Susceptibilities (1932).

SPLITTING OF THE ATOMIC LEVELS SPLITTING OF THE ATOMIC LEVELS IN CRYSTAL FIELDSIN CRYSTAL FIELDSIN CRYSTAL FIELDSIN CRYSTAL FIELDS

Each atomic (ionic) level with a given L or J isEach atomic (ionic) level with a given L or J is split in a crystal field.

STATEMENTS AND RULES DERIVED FROM THESTATEMENTS AND RULES DERIVED FROM THE BACKGROUND OF THE GROUP THEORYGROUP THEORY:

1) (2L+1) wave functions belonging to the atomic level with1) (2L+1) wave-functions belonging to the atomic level with a given L ( LS- term) form the basis of a degenerate irreducible representation of the full spherical symmetryirreducible representation of the full spherical symmetry group R3.

2) This representation is referred to as D(L), basis (wave-2) This representation is referred to as D(L), basis (wavefunctions) is formed by (2L+1) wave-functions of the type of YLM (spherical functions), LM ( p ),

M=-L,-L+1,, L-1, L, (2L+1) - values.

3) Point symmetry of the atom (ion) in a crystal or in a3) Point symmetry of the atom (ion) in a crystal or in aligand surrounding in a complex compound is lower thanthe spherical one (R3). Under this condition thethe spherical one (R3). Under this condition therepresentations D(L) become reducible. Each reduciblerepresentation can be decomposed into irreduciblep prepresentations (in the point symmetry group)possessing low dimensions.

4) Each irreducible representation (in R3 or in the crystalsymmetry group) corresponds to an one energy level.

5) Th h i l f th th ti l5) The physical consequence of these mathematicalconclusions is that each atomic level becomes split (ingeneral) when the atom (ion) is placed in the ligandgeneral) when the atom (ion) is placed in the ligand

surrounding: instead of one ionic terms SL one obtainsseveral crystal field terms (crystal field splitting).y ( y p g)

BOOKS ON GROUP THEORY AND CRYSTAL BOOKS ON GROUP THEORY AND CRYSTAL

1 F A Cotton Chemical Application of Group Theory

FIELD THEORYFIELD THEORY1.F.A.Cotton, Chemical Application of Group Theory,

2nd Edition,Interscience, New York (1971).2. B.S.Tsukerblat, Group Theory in Chemistry and , p y y

Spectroscopy. A Simple Guide to Advanced Usage, Academic Press, London, 1994.

3. Robert L.Carter, Molecular Symmetry and Group Theory, John Wiley, 1998.

4 S S Y T b H K i M lti l t f4. S.Sugano, Y.Tanabe, H.Kamimura, Multiplets of Transition Metal Ions in Crystals, Academic Press, New-York 1970New-York, 1970.

5. C.L. Ballhausen, Introduction to the Ligand Field Theory and its Applications, Pergamon Press, Oxford, 1963.pp , g , ,

THE MAIN PROBLEM IN QUESTION:THE MAIN PROBLEM IN QUESTION:HOW THE FREE ION TERMS (SL) ARE SPLIT IN AHOW THE FREE ION TERMS (SL) ARE SPLIT IN A CRYSTAL FIELD-CRYSTAL FIELDS TERMS TERMS (S) AND CRYSTAL FIELD SPLITTINGSSPLITTINGSCRYSTAL FIELD SPLITTINGS SPLITTINGS Irreducible representations (irreps) of the point group OOhh(cubic group octahedral or cubic surrounding of the ion):(cubic group -octahedral or cubic surrounding of the ion):Even irreps : A1g, A2g - one-dimensional irrep

E bi dimensional irrepEg - bi-dimensional irrepT1g , T2g -tri-dimensional irreps

Odd i A A di i l iOdd irreps: A1u , A2u - one-dimensional irrepsEu - bi-dimensional irrepT T tri dimensional irrepsT1u , T2u - tri-dimensional irreps

Important notation: g-even (gerade), u-odd (ungerade)(parity of the crystal field states for the(parity of the crystal field states, for the point groups with the inversion symmetry)

STRUCTURE OF THE CYANOMETALATES FAMILYFAMILY

Metal ion

CN-group

An example of the octahedral metal complex, O t l( )Oh symmetry: Metal(CN)6

eg-orbitals

zzx

x2-y2

t2g-orbitalsy3z

2-r2x -y

zx yz xy

Shape of d orbitals and splittingShape of d-orbitals and splitting

HOW TO FIND THE SPLITTING OF HOW TO FIND THE SPLITTING OF SLSL IONIC TERMS IONIC TERMS IN THE OCTAHEDRAL LIGAND SURROUNDING?IN THE OCTAHEDRAL LIGAND SURROUNDING?

RESULTS for several values of L: (decomposition in oohh group, even ionic states, d-electrons, l=2 )

LDANSWER: to decompose the reducible D(l) irreps into the irreducible ones (the procedure is well known from the group theory)

b li ll LDsymbolically:irrepsL

singletASAD gg 110

ld bl

tripletTPTD

g

gg

gg

2

111

11

tripletstwosingletTTAFTTAD

tripletdoubletTEDTED

gggggg

gggg

2122123

222

pggggggg 212212each each irrepirrep energy level in crystal field (crystal field splitting)

PHYSICAL PICTURE OF THE CRYSTAL FIELD PHYSICAL PICTURE OF THE CRYSTAL FIELD SPLITTINGSPLITTING

Shapes of the electronic clouds:

functionwave 2

r

densityelectronictheof

ondistributi spatial 2r

)cloud"" electronic the of (shape density electronic theof

withcloudelectronictheof ninteractio field crystal the inEnergy

ligands the of charges the withcloud electronictheof

SHAPES OF THREE SHAPES OF THREE pp--ORBITALSORBITALS

pp--ORBITALS IN THE ORBITALS IN THE OCTAHEDRAL (OOCTAHEDRAL (Ohh) )

dumbbell-shaped electronic clouds

O AHEDRAL (OO AHEDRAL (Ohh) ) CRYSTAL FIELDCRYSTAL FIELD

Positive Positive (black) and(black) and(black) and (black) and

negative negative (light) petals(light) petals(light) petals (light) petals of the waveof the wave--

functionsfunctions

CONCLUSION FROM THE PICTURECONCLUSION FROM THE PICTURECONCLUSION FROM THE PICTURECONCLUSION FROM THE PICTUREThe energies of the interaction of the dumbbell-

h d l t i l d f th bit l ithshaped electronic clouds of three p-orbitals with the ligands of the octahedral surrounding are

lequal.Three p-orbitals form triply degenerate level in the

t h d l t l fi ldoctahedral crystal field. This is the physical sense of the group-theoretical

(1)statement D(1)T1u (the only triply degenerate irrep, this means that there is one triply degenerate level in a cubic crystal field) .p-level remains degenerate in the cubic (octahedral crystal surrounding.

Five d-orbitals

ddxzdxz

two dumbbells iny

(3z2-r2, x2-y2)two dumbbells in each: (yz,xz,xy)

Five dFive d--orbitals in the octahedral fieldorbitals in the octahedral field

zz

yy

x

z

zx

dxz

yy

x

(3z2-r2, x2-y2) E-orbitals(yz,xz,xy) T2-orbitals

ELECTRONIC STATES (TERMS) ELECTRONIC STATES (TERMS) IN CRYSTAL FIELD IN CRYSTAL FIELD LABELSLABELS

12SSpin multiplicity Irreducible representation2T2g - orbital triplet, S=1/2,

2E orbital doublet S=1/2 etc

representation

Eg - orbital doublet, S=1/2, etc.IMPORTANT REMARK: PARITY RULES

1) one electron: p(parity) = (-1)l1) one electron: p(parity) = (-1)p-electron: l=1 (odd states)d-electron: l=2 (even states)( )f-electron: l=3 (odd states)

2) Many (n) electrons: ( but not (-1)L !!! )d h ll ll l 2 ( t t )dn- shells, all li=2 (even states )

p1(odd), p2 (even), p1d1(odd), etc.In the point symmetry groups involving inversion center:In the point symmetry groups involving inversion center:

irrepseven,irrepsodd gu

SPLITTING OF THE dSPLITTING OF THE d--LEVEL IN A CUBIC LEVEL IN A CUBIC FIELD INTO A TRIPLET AND DOUBLETFIELD INTO A TRIPLET AND DOUBLET

YrRd level (l=2) ,YrR lmnld-level (l=2) Five d-functions (angular parts): Y2,-2, Y2,-1,Y2,0,Y2,1 Y2,2

D(2)T2+E (triplet +doublet)T2(xy xz xy) (real) and E( 3z2-r2 x2-y2)(real)T2(xy, xz, xy) (real) and E( 3z r , x y )(real)5-fold degenerate d-level is split into

a triplet and a doublet in a cubic crystal fielda triplet and a doublet in a cubic crystal field.Notations for the d-functions in OOhh:

gzyxgxyxzyz Ed,dTd,d,d 2222 and zyx

CRYSTAL FIELD SPLITTING IN THE CASE OF CRYSTAL FIELD SPLITTING IN THE CASE OF ONE ONE dd--ELECTRON OR ONE HOLEELECTRON OR ONE HOLEONE ONE dd ELECTRON OR ONE HOLEELECTRON OR ONE HOLE

One d-electron, l = 2. tripletdoubletTEDTED gggg 222

T2E2 gT222D

gE

2D10Dq 10Dq

T2 E2gT2

d1 l t (T 3+) d9 h l (C 2+)

gE

d1- electron (Ti 3+) d9-hole (Cu2+)

d 9-hole in the closed shell d 10(reversed order of the levels: (

in Ti3+-ground triplet, in Cu2+-ground doublet ) Physical reason:Physical reason:

electron-negative charge-cloud (repulsion f th li d ) h l iti h l dfrom the ligands), hole-positive charge-cloud

(attraction to the ligands). 10Dq- cubic crystal field parameter =

splitting of the one-electron level (d1) in p g ( )a cubic crystal field

CUBIC CRYSTAL FIELD PARAMETER 10DqCUBIC CRYSTAL FIELD PARAMETER 10DqP i tP i t h d lh d l ff

4reqD

PointPoint--charge modelcharge model forforthe crystal fieldthe crystal field--ligands ligands

are the point charges are the point charges 506R

Dq are the point charges are the point charges (covalency is not taken (covalency is not taken into account):into account):

q* -charge of the ligands (point charges,q*MetalR0-metal-ligand

distances in the

q

octahedral surrounding10Dq- crystal field

splitting of the

R0

splitting of theone-electron d- level

-mean value of r