Single Molecule Magnets: Single Molecule Magnets: Exploring Quantum Magnetization Exploring Quantum Magnetization

Dynamics at the Nanoscale Dynamics at the Nanoscale

Quantum versus classical magnetic clustersQuantum versus classical magnetic clusters

Single molecule magnets (SMMs)Single molecule magnets (SMMs) • In particular, MnIn particular, Mn1212 and Fe and Fe88

Quantum magnetization dynamicsQuantum magnetization dynamics• Quantum tunneling, hysteresis, and the role of the Quantum tunneling, hysteresis, and the role of the

environmentenvironment

High frequency Electron Paramagnetic High frequency Electron Paramagnetic ResonanceResonance• Solution to the tunneling mechanism in MnSolution to the tunneling mechanism in Mn1212-acetate-acetate• Quantum coherence in a supramolecular dimer of SMMsQuantum coherence in a supramolecular dimer of SMMs

MESOSCOPIC MAGNETISMmacroscale nanoscale

permanentmagnets

micronparticles

nanoparticles clusters molecularclusters

Individualspins

S = 1023 1010 108 106 105 104 103 102 10 1

multi - domain single - domain Single molecule nucleation, propagation andannihilation of domain walls

uniform rotation quantum tunneling,quantum interference

-1

0

1

-40 -20 0 20 40

M/M

S

H(mT)

-1

0

1

-100 0 100

M/M

S

H(mT)

-1

0

1

-1 0 1M

/MS

H(T)

Fe8

1K0.1K

0.7K

MnMn1212-ac-ac

FerritinFerritin

1 nm10 nm100 nmsuperparamagnetism

Classical Quantum

size

Magnetic anisotropy bistability, hysteresis

0

2sinK E

nerg

y de

nsity

E = K × Volume anisotropy × # of spins

K = energy density associated with the anisotropy

The case of a classical, single domain, ferromagnetic particle:

EUpUp downdown

Stoner-WohlfarthStoner-Wohlfarth

•Anisotropy due to crystal field - in this case, uniaxial.Anisotropy due to crystal field - in this case, uniaxial.•Energy barrier prevents switching - hence bistability....Energy barrier prevents switching - hence bistability....

....unless ....unless EE < < kkBBTT superparamagnetismsuperparamagnetism [ [ = = ooexp(exp(EE//kkBBT)T)]]

Magnetic anisotropy bistability, hysteresis

0

2sinK E

nerg

y de

nsity

E = K × Volume anisotropy × # of spins

K = energy density associated with the anisotropy

The case of a classical, single domain, ferromagnetic particle:

•Anisotropy due to crystal field - in this case, uniaxial.Anisotropy due to crystal field - in this case, uniaxial.

EUpUp downdown

Stoner-WohlfarthStoner-Wohlfarth

•Switching involves rotation of all spins together.Switching involves rotation of all spins together.•Classical continuum of states between "up" and "down".Classical continuum of states between "up" and "down".

•Energy barrier prevents switching - hence bistability....Energy barrier prevents switching - hence bistability....

....unless ....unless EE < < kkBBTT superparamagnetismsuperparamagnetism [ [ = = ooexp(exp(EE//kkBBT)T)]]

The first single molecule magnet: Mn12-acetate

[Mn[Mn1212OO1212(CH(CH33COO)COO)1616(H(H22O)O)44]]··2CH2CH33COOHCOOH··4H4H2200

Mn(IV)

Mn(III)

Oxygen

Carbon

S S = 2= 2

S S = 3/2= 3/2

Lis, 1980

•Easy-axis anisotropy due to Jahn-Teller distortion on Mn(III)Easy-axis anisotropy due to Jahn-Teller distortion on Mn(III)

•Crystallizes into a tetragonal structure with SCrystallizes into a tetragonal structure with S44 site site symmetrysymmetry

•Organic ligands ("chicken fat") isolate the moleculesOrganic ligands ("chicken fat") isolate the molecules

R. Sessoli et al. JACS 115, 1804 (1993)

•Ferrimagnetically coupled magnetic ions (Ferrimagnetically coupled magnetic ions (JJintraintra 100 K) 100 K)Well defined giant spin (Well defined giant spin (SS = 10) at low temperatures (T < 35 K) = 10) at low temperatures (T < 35 K)

"up""up" "down""down"

EE1010

EE99

EE88

EE77

EE1010

EE99

EE88

EE77

EE66

EE55

Spin projection - ms

EE66

EE55

Ene

rgy

Ene

rgy

EE44EE44

smE

21 discrete 21 discrete mmss levels levels

Quantum effects at the nanoscale (S = 10)Simplest case: axial Simplest case: axial (cylindrical) crystal field(cylindrical) crystal field

2ˆ ˆ ( 0)zDS D H

Eigenvalues given by:Eigenvalues given by:2( )s sm D mE

•Small barrier - Small barrier - DSDS22

•Superparamagnet at Superparamagnet at ordinary temperaturesordinary temperatures

E E DS2 10-100 K10-100 K

|D | 0.1 1 K for a typical single molecule magnetThermal activationThermal activation

"up""up" "down""down"

EE1010

EE99

EE88

EE77

EE1010

EE99

EE88

EE77

EE66

EE55

Spin projection - ms

EE66

EE55

Ene

rgy

Ene

rgy

EE44EE44

smE

ThermallyThermallyassistedassisted

quantumquantumtunnelingtunneling

Break axial symmetry:Break axial symmetry:2

Tˆ ˆ ˆ

zDS H HT interactions which do not commute with Ŝz

Quantum effects at the nanoscale (S = 10)

EEeffeff < < EE

• mmss notnot good quantum # good quantum #

•Mixing of Mixing of mmss states states

resonant tunneling resonant tunneling (of (of mmss) through barrier) through barrier

•Lower effective barrierLower effective barrier

"up""up" "down""down"

EE1010

EE99

EE88

EE77

EE1010

EE99

EE88

EE77

EE66

EE55

Spin projection - ms

EE66

EE55

Ene

rgy

Ene

rgy

EE44EE44

smE

Pure quantum tunnelingPure quantum tunneling

Strong distortion of the Strong distortion of the axial crystal field:axial crystal field:

•Temperature-independent Temperature-independent quantum relaxation as Tquantum relaxation as T00

•Ground states a mixture of Ground states a mixture of pure "up" and pure "down".pure "up" and pure "down".

"down""down""up""up"

±±1

2{ }

•Ground state degeneracy Ground state degeneracy lifted by transverse lifted by transverse interaction: interaction:

splitting splitting ( (T)n

Tunnel splittingTunnel splitting

Quantum effects at the nanoscale (S = 10)

•Temperature-independent magnetization relaxation as Temperature-independent magnetization relaxation as T T 0 0 KK..

•Highly axially symmetric systems show the slowest Highly axially symmetric systems show the slowest relaxation, e.g. Mnrelaxation, e.g. Mn1212-ac (-ac ( ).).

The physics behind this short-time relaxation requires some detailed explanation (to follow).

W. Wernsdorfer, G. Christou, et al., cond-mat/0306303

Mn30 - the largest single molecule magnet displaying a temperature independent relaxation rate

How is this evidenced?

( )1

sat short

M t t

M

Prokof'ev and Stamp Phys. Rev. Lett. 80, 5794 (1998)

Fe3+

S=5/2

S=10

[Fe[Fe88OO22(OH)(OH)1212(tacn)(tacn)66]Br]Br88..9H9H2200

Another well characterized single molecule magnet

Wieghardt, 1984

How do these two S = 10 systems differ?

Four-fold axis Tetragonal

Two-fold axis Rhombic

zz-axis is out of -axis is out of the screenthe screen

FeFe88, , SS = 10 = 10MnMn1212-ac, -ac, SS = 10 = 10

Spin projection - ms

Pure quantum tunnelingPure quantum tunneling

Hz

D/E

Spin projection - ms

ThermallyThermallyassistedassisted

quantumquantumtunnelingtunneling

2 2 2ˆ ˆ ˆ ˆ ˆ 'z x yDS E S S H HFeFe88, , SS = 10 = 10MnMn1212-ac, -ac, SS = 10 = 10

2T T

ˆ ˆ ˆ ˆ; ???zDS H H H

}300 GHzor 14.4 kB

}115 GHzor 5.5 kB

D

•To first-order, To first-order, TT mixes states with mixes states with mmss = = ±±kk, where , where kk is the power to is the power to which which SSxx or or SSyy appear in appear in TT..

Example: Example:

for Fefor Fe88, , TT mixes states for which mixes states for which mmss = = ±±22 in first-order, in first-order, mmss = = ±±44 in in second-order, and so on. Thus, the second-order, and so on. Thus, the mmss = = ±±10 10 states interact only in states interact only in 1010thth order order

2 2 2 212

ˆ ˆ ˆ ˆx yS S S S

107 710 so 10 10kHzo

EDS

D

= 0.1 K

•This is why tunneling in high spin (classical) systems is not This is why tunneling in high spin (classical) systems is not observed.observed.•Many potential sources of transverse anisotropy for MnMany potential sources of transverse anisotropy for Mn1212-ac: -ac:

•Internal dipolar, exchange and hyperfine fields; Internal dipolar, exchange and hyperfine fields;

•Higher order crystal field interactions;Higher order crystal field interactions;

•Disorder, lattice defects, etc..Disorder, lattice defects, etc..

•However, after 10 years, the mechanism remains poorly However, after 10 years, the mechanism remains poorly understood. understood.

D D

Below TB 3 K,

2T

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

z B

x x y y z z

DS g B S

B S B S B S B S

H H

Spin projection - ms

•Applied field represents another source of transverse anisotropy

•Zeeman interaction contains odd powers of Ŝx and Ŝy

Several important points to note:

For now, consider only B//z :(also neglect transverse interactions)

2( )s s B sm D m g Bm E

"down""down""up""up"

•Magnetic quantum tunneling is suppressedMagnetic quantum tunneling is suppressed

•Metastable magnetization is blocked ("down" spins) Metastable magnetization is blocked ("down" spins)

System off resonanceSystem off resonanceX

X

Application of a magnetic field

Application of a magnetic field2

Tˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

z B

x x y y z z

DS g B S

B S B S B S B S

H H

Spin projection - ms

System on resonanceSystem on resonance

•Applied field represents another source of transverse anisotropy.

•Zeeman interaction contains odd powers of Ŝx and Ŝy.

Several important points to note:

For now, consider only B//z :(also neglect transverse interactions)

2( )s s B sm D m g Bm E

"down""down""up""up"

•Resonant magnetic quantum tunneling resumesResonant magnetic quantum tunneling resumes

•Metastable magnetization can relax from "down" to "up" Metastable magnetization can relax from "down" to "up"

Increasing fieldIncreasing field

Hysteresis and magnetization steps

Friedman et al., PRL (1996)Thomas et al., Nature (1996)

Mn12-ac

Tunneling "off"Tunneling "off"

Tunneling Tunneling "on""on"

Low temperature H=0 Low temperature H=0 step is an artifactstep is an artifact

This loop represents an This loop represents an ensemble average of the ensemble average of the response of many moleculesresponse of many molecules

2( )s s B sm D m g Bm E

Periodicity of the magnetization steps

+109

+108

+107

+106

+105

+104

+103

Tunneling steps are periodic in Tunneling steps are periodic in HHzz

Both even Both even and oddand odd mmss

steps are steps are observedobserved

•Therefore,Therefore, TT must contain odd powers of must contain odd powers of ŜŜxx and and ŜŜyy..

•This, in turn, suggest that transverse fields play a This, in turn, suggest that transverse fields play a crucial role in the tunneling. crucial role in the tunneling. [Chudnovsky and Garanin, PRL [Chudnovsky and Garanin, PRL 8787, , 187203 (2001)]187203 (2001)]

-500

So, what about internal fields?• These are These are too weaktoo weak to explain tunneling at odd resonances in to explain tunneling at odd resonances in

MnMn1212..

• Nevertheless, they play a crucial role in the tunneling mechanism.Nevertheless, they play a crucial role in the tunneling mechanism.

20 mT 0 12 mT

50 mT 10 mT 1.2 mT

Mn12

56Fe8

field dipolar exchange hyperfine

Msat

(1)(1)

1)1) Prepare the sample in a fully magnetized state.Prepare the sample in a fully magnetized state.

(2)(2)

2)2) Put the system in resonance so that the magnetization Put the system in resonance so that the magnetization can begin to relax. Note: due to internal dipolar fields, can begin to relax. Note: due to internal dipolar fields, not all molecules will be in resonance - depends on not all molecules will be in resonance - depends on sample shape.sample shape.

M<Msat

(3)(3)

3)3) Some molecules will relax, thus burning a hole in the Some molecules will relax, thus burning a hole in the dipolar field distribution; this relaxation results in a dipolar field distribution; this relaxation results in a change in the magnetization of the sample. change in the magnetization of the sample.

x

(4)(4)

4)4) The change in magnetization drives those molecules The change in magnetization drives those molecules that were initially in resonance, out of resonance. that were initially in resonance, out of resonance. Meanwhile, other molecules are brought into Meanwhile, other molecules are brought into resonance, and the hole becomes broader.resonance, and the hole becomes broader.

(5)(5)

5)5) Dynamic nuclear fluctuations and electron/nuclear Dynamic nuclear fluctuations and electron/nuclear spin cross-relaxation broaden the magnetic energy spin cross-relaxation broaden the magnetic energy levels (even at mK temperatures). levels (even at mK temperatures).

This combination of "dipolar shuffling" and the This combination of "dipolar shuffling" and the coupling to the nuclear bath governs the time-coupling to the nuclear bath governs the time-evolution of the magnetization, leading to the short evolution of the magnetization, leading to the short time time tt1/21/2 relaxation. relaxation.

[Prokof'ev and Stamp, Phys. Rev. Lett. 80, 5794 (1998)]

• Relaxation is a complex many-body problem.Relaxation is a complex many-body problem.

Here, we see the importance of the environment

The role of dipolar and hyperfine fields was first The role of dipolar and hyperfine fields was first demonstrated via studies of isotopically substituted demonstrated via studies of isotopically substituted versions of Feversions of Fe88..

[Wernsdorfer et al., Phys. Rev. Lett. [Wernsdorfer et al., Phys. Rev. Lett. 8282, 3903 (1999)], 3903 (1999)]

What have we learned thus far?TT provides the tunneling mechanism (tunnel matrix provides the tunneling mechanism (tunnel matrix elements)elements)For Fe8: rhombic term, E(ŜŜxx

22 - - ŜŜyy22), is the dominant source of ), is the dominant source of

transverse anisotropy.transverse anisotropy.

This has been verified via measurements of tunnel This has been verified via measurements of tunnel splittings splittings , using an elegant Landau-Zener , using an elegant Landau-Zener method.method.

[Wernsdorfer, Sessoli, Science [Wernsdorfer, Sessoli, Science 284284, 133 (1999)], 133 (1999)]For MnFor Mn1212: : dominant source of transverse anisotropy not dominant source of transverse anisotropy not known; 4th order term, known; 4th order term, BB44

44((ŜŜ++44 + + ŜŜ--

44), believed to ), believed to play a role.play a role.

Landau-Zener studies fail to observe clear tunnel Landau-Zener studies fail to observe clear tunnel splittings; instead, broad distributions are found.splittings; instead, broad distributions are found.

[del Barco et al., Europhys. Lett. [del Barco et al., Europhys. Lett. 6060, 768 (2002)], 768 (2002)]For both: For both: odd tunneling resonances suggest additional odd tunneling resonances suggest additional source of transverse anisotropy, possibly related to source of transverse anisotropy, possibly related to disorder.disorder.

[Chudnovsky and Garanin, PRL [Chudnovsky and Garanin, PRL 8787, 187203 (2001)], 187203 (2001)]•Nuclear and dipolar couplings control magnetization Nuclear and dipolar couplings control magnetization dynamics.dynamics.

•These couplings also represent unwanted source of These couplings also represent unwanted source of decoherence.decoherence.

2( )s s B sm D m g Bm E

Reminder: field//Reminder: field//zzz, S4-axis

Hz

mmss represents spin- represents spin- projection along the projection along the molecular 4-fold molecular 4-fold axisaxis

Single-crystal, high-field/frequency EPR

•Magnetic dipole transitions (ms = ±1) - note frequency scale!

•EPR measures level spacings directly, unlike Landau-Zener

s

z, S4-axis

Hxy

In high-field limit (In high-field limit (gBBB > B > DSDS), ), mmss represents spin- projection represents spin- projection along the applied field-axisalong the applied field-axis

2T

ˆ ˆ ˆ ˆz BDS g B S

H H 21

2ˆ( )s T s B sm D m g Bm E H

Rotate field in Rotate field in xyxy-plane and look for symmetry effects-plane and look for symmetry effects

Single-crystal, high-field/frequency EPR

The first single-crystal studyPRL 80, 2453 (1998)

Sample: 1 Sample: 1 × × 0.2 0.2 × × 0.2 0.2 mmmm33

The advantages are two-fold:The advantages are two-fold:

•One can perform angle dependent One can perform angle dependent studies to look for symmetry of studies to look for symmetry of TT..

•One can also carry out detailed One can also carry out detailed line-shape analyses; this is harder line-shape analyses; this is harder to do for powder averages.to do for powder averages.

Subsequent outcomes:Subsequent outcomes:

Determination of significant Determination of significant DD-strain effect in Mn-strain effect in Mn1212 and Fe and Fe88..[Polyhedron [Polyhedron 2020, 1441 (2001); PRB , 1441 (2001); PRB 6565, 014426 (2002)], 014426 (2002)]

Measurement of dipolar couplings in MnMeasurement of dipolar couplings in Mn1212 and Fe and Fe88, and first , and first evidence for significant intermolecular interactions in Feevidence for significant intermolecular interactions in Fe88..

[PRB [PRB 6565, 224410 (2002); PRB , 224410 (2002); PRB 6666, 144409 (2002)], 144409 (2002)]

Location of low-lying Location of low-lying SS = 9 state in Fe = 9 state in Fe88..[cond-mat/030915; PRB in-press (2003)][cond-mat/030915; PRB in-press (2003)]

Cavity perturbationCavity perturbationCylindrical TE01Cylindrical TE01n n ((QQ ~~ 10104 4 -- 101055))ff = 16 = 16 300 GHz 300 GHzSingle crystal 1 Single crystal 1 × 0.2 × 0.2 mm× 0.2 × 0.2 mm33

T = 0.5 to 300 K, T = 0.5 to 300 K, ooH up to 45 teslaH up to 45 tesla

M. Mola M. Mola et alet al., Rev. Sci. Inst. ., Rev. Sci. Inst. 7171, 186 (2000), 186 (2000)

A u S S

B

•We use a Millimeter-wave Vector We use a Millimeter-wave Vector Network Analyzer (MVNA, ABNetwork Analyzer (MVNA, ABmmmm) as a ) as a spectrometerspectrometerSusumu TakahashiSusumu Takahashi (2003) (2003)two-axis rotationtwo-axis rotationcapabilitycapability

(now 715 GHz!)(now 715 GHz!)

Determination of transverse crystal-field interactions in MnDetermination of transverse crystal-field interactions in Mn1212-Ac-Ac

Hard-plane (Hard-plane (xyxy-plane) -plane) rotationsrotations

ff = 50 GHz = 50 GHz

0

30

60

90

120

150

Ang

le (

degr

ees)

Easy

•Four-fold line shifts due Four-fold line shifts due to a to a quarticquartic transverse transverse interaction in interaction in TT

•Previously inferred from Previously inferred from neutron studiesneutron studiesMirebeau et al., PRLMirebeau et al., PRL 83, 83, 628 (1999)628 (1999)

•BB4444 is the only free is the only free

parameter in our fitparameter in our fit

PRLPRL 90, 90, 217204 (2003)217204 (2003)

4

4

4 4 4142

46(1) K

ˆ ˆ

B

B S S

PRLPRL 90, 90, 217204 (2003)217204 (2003)

four-fold line shape/width four-fold line shape/width modulation due to a modulation due to a quadraticquadratic transverse transverse interaction caused by interaction caused by solvent disordersolvent disorder

Solvent disorder (rogueness)

Organic surrounding (not directly bound to core):4 water molecules2 acetic acid molecules

22yx SSE

-E(Sx2 - Sy

2)

Equal population of molecules with opposite signs of EA. Cornia et al., Phys. Rev. Lett. 89, 257201 (2002)

Disorder lowers the symmetry of the moleculesDisorder lowers the symmetry of the molecules+E(Sx

2 - Sy2) 2 2

0.015K

0.012K

x y

E

D

E S S

Rotation away from the hard plane

10 10 vanishes in vanishes in the first the first degree of degree of rotationrotation

9 does not appear 9 does not appear until 2.5 degrees of until 2.5 degrees of rotationrotation

Rotation away from the hard plane

3 4 5 6 7 8 9

Cav

ity

tran

smis

sion

(ar

b. u

nits

- o

ffse

t)

Magnetic field (tesla)

0.18o increments

Hardplane

2.15o spread10

9

1.25o spread

Implies about Implies about 1.51.5oo distribution distribution in the in the orientations of orientations of the easy axes of the easy axes of the Mnthe Mn1212 moleculesmolecules

This is the solution to the odd tunneling resonance problemThis is the solution to the odd tunneling resonance problem

Returning to issue of decoherence and the environment1.1. MnMn1212 (Fe (Fe88?) probably not ideal quantum system for future ?) probably not ideal quantum system for future

workwork• 1st single molecule magnets - novel, yet contrasting behaviors1st single molecule magnets - novel, yet contrasting behaviors• Physicists have focused almost exclusively on these two systemsPhysicists have focused almost exclusively on these two systems

The molecular approach is the The molecular approach is the keykey• Immense control over the magnetic unit Immense control over the magnetic unit

and its coupling to the environmentand its coupling to the environment

2.2. Timescale assoc. with nuclear and dipolar fluctuations is slowTimescale assoc. with nuclear and dipolar fluctuations is slow• Strong coupling to slow spin dynamics Strong coupling to slow spin dynamics quantum decoherence quantum decoherence• Move tunneling into a "coherence window" - different frequency rangeMove tunneling into a "coherence window" - different frequency range

[Stamp and Tupitsyn, cond-mat/0302015][Stamp and Tupitsyn, cond-mat/0302015]• Larger tunnel splittings via transverse applied field, or smaller spin Larger tunnel splittings via transverse applied field, or smaller spin SS

3.3. Other ways to beat decoherence Other ways to beat decoherence • Isotopically label with Isotopically label with II = 0 nuclei = 0 nuclei• Focus on antiferromagnetic Focus on antiferromagnetic SS = 0 systems (immune to dipolar effects) = 0 systems (immune to dipolar effects)• Isolate molecules, either by dilution, or with "chicken fat"Isolate molecules, either by dilution, or with "chicken fat"

Ni4Ni4

OH OH,

Chicken FatChicken FatSS = 4 Ni = 4 Ni44 system with strong quartic ( system with strong quartic (ŜŜ++

44 + + ŜŜ--44) anisotropy) anisotropy

(Ni also predominantly (Ni also predominantly II = 0) = 0)

E-C. Yang, JACS (submitted, 2003)

O

Mn

O

Mn

OC

R

O

MnMn

O

Mn

N

Mn

O

Mn

O

MnO

NN

Mn

N

Mn

O

Mn

O

MnO

co

Mn

O

Mn

O

Mn

O

MnO

N

O O

Mn

O

Mn

O

Mn

O

MnO

C HHH

Mn

Cl

Mn

O

Mn

O

MnO

Mn

F

Mn

O

Mn

O

MnO

Mn

Br

Mn

O

Mn

O

MnO

MnMn44 SS = = 99//22

-1

-0.5

0

0.5

1

-1.2 -0.8 -0.4 0 0.4 0.8 1.2

0.04 K0.3 K0.4 K0.5 K0.6 K0.7 K0.8 K1.0 K

M/M

sµ0H (T)

0.14 T/s

aExchange coupling in a dimer of Exchange coupling in a dimer of SS = = 99//22 Mn Mn44 clusters clusters

1 1 2 2 1 2ˆ ˆˆ ˆ ˆ( , ) ( , )H H S B H S JS SB

No H = No H = 0 0 tunnelintunnelingg

To lowest order, the exchange generates a bias which To lowest order, the exchange generates a bias which each spin experiences due to the other spin within the each spin experiences due to the other spin within the dimerdimerWolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409

Including the full exchange into the calculationIncluding the full exchange into the calculation 1

1 2 1 2 1 2 1 22ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ

D S S z zH H H JS S J S S S S

Clear evidence for coherent transitions involving both moleculesClear evidence for coherent transitions involving both molecules

ExperimentExperiment SimulationSimulation

(submitted, 2003)(submitted, 2003)

UF PhysicsUF Physics

Rachel EdwardsRachel EdwardsAlexey KovalevAlexey Kovalev

Susumu TakahashiSusumu TakahashiSabina KahnSabina KahnJon LawrenceJon LawrenceAndrew BrowneAndrew Browne

Shaela JonesShaela JonesNeil BushongNeil BushongSara MaccagnanoSara Maccagnano

UF ChemistryUF Chemistry

George ChristouGeorge Christou

Nuria Aliaga-Alcalde Nuria Aliaga-Alcalde Monica SolerMonica SolerNicole ChakovNicole ChakovSumit BhaduriSumit Bhaduri

UCSD ChemistryUCSD Chemistry

David HendricksonDavid Hendrickson

En-Che YangEn-Che YangEvan RumbergerEvan Rumberger

FSU ChemistryFSU Chemistry

Naresh DalalNaresh Dalal

Micah NorthMicah NorthDavid ZipseDavid ZipseRandy AcheyRandy Achey

NYU PhysicsNYU Physics

Andy KentAndy Kent

Enrique del BarcoEnrique del Barco

Many collaborators/students to acknowledge...Many collaborators/students to acknowledge... ...illustrates the interdisciplinary nature of this work...illustrates the interdisciplinary nature of this work

Also:Also: Kyungwha Park (NRL) Kyungwha Park (NRL)Wolfgang Wernsdorfer (Grenoble) Wolfgang Wernsdorfer (Grenoble) Mark Novotny (MS State U)Mark Novotny (MS State U)Per Arne Rikvold (CSIT - FSU)Per Arne Rikvold (CSIT - FSU)