-
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 [email protected]@ 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.
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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.
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ABOUT THE BOOKSABOUT THE BOOKS--MOLECULAR MAGNETISMMOLECULAR
MAGNETISM
Chapter 5
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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
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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
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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
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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
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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
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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.
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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
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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)
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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
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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: [email protected] Joel S. Miller is in the Department
of Chemistry, University of Utah, Salt Lake City, Utah 84112-0850,
USA. e-mail: [email protected] 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).
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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.
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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)
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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 , ,
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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)
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STRUCTURE OF THE CYANOMETALATES FAMILYFAMILY
Metal ion
CN-group
An example of the octahedral metal complex, O t l( )Oh symmetry:
Metal(CN)6
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eg-orbitals
zzx
x2-y2
t2g-orbitalsy3z
2-r2x -y
zx yz xy
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Shape of d orbitals and splittingShape of d-orbitals and
splitting
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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)
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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
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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
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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.
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Five d-orbitals
ddxzdxz
two dumbbells iny
(3z2-r2, x2-y2)two dumbbells in each: (yz,xz,xy)
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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
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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
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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
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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+)
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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
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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