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2002 Agilent Technologies 2002 Agilent Technologies
Europhysics Prize Lecture Europhysics Prize Lecture on
Bernard Barbara, L. Néel Lab, Grenoble, FranceJonathan R. Friedman, Amherst College, Amherst, MA, USADante Gatteschi, University of Florence, ItalyRoberta Sessoli, University of Florence, ItalyWolfgang Wernsdorfer, L. Néel Lab, Grenoble, France
Budapest 26/08/2002
Quantum Dynamics of Nanomagnets
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the miniaturization process
Single Domain Particles
coherent rotation of all the spins
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Ene
rgy
E
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Quantum effects in the dynamics of the magnetization
First evidences of Quantum Tunneling in nanosized magnetic particles
(difficulties due to size distribution)
0 3 nmQuantum Coherence in ferrihydrite confined in the ferritin mammalian protein
(inconclusive due to distribution of iron load)
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= metal ions = oxygen = carbon
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The molecules are regularly arranged in the crystal
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Mn(IV)S=3/2
Mn(III)S=2
Total Spin =10
Mn12acetateMn12acetate
T. Lis Acta Cryst. 1980, B36, 2042.
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high spin molecules
and low spin molecules
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Uniaxial magnetic anisotropy H=-DSz
2
H=0
M
M=+S
M=-S+1M=S-1
0
E
-S+S
If S is large
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H=-DSz2+gBHzSz
H0
M=-S
M=S
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return to the equilibrium thermal activated thermal activated
mechanismmechanism
H=0
E=DS2
M=-SM=S
=0exp(E/kBT) 010-7 E=63 K
time=0
J. Villain et al. Europhys. Lett.1994, 27, 159
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return to the equilibrium thermal activated thermal activated
mechanismmechanism
H=0
E=DS2
M=-SM=S
=0exp(E/kBT) 010-7 E=63 K
time=
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0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(/
0)]-1
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(/
0)]-1
Sessoli et al. Nature 1993, 365, 141
0=2x10-7 s
E/kB=61 K
Temperature dependence of the relaxation time of
Mn12acetate
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Mn12acetate: Hysteresis loopMn12acetate: Hysteresis loop
-3 -2 -1 0 1 2 3
-20
-10
0
10
20
T=2.1K
M
AG
NE
TIZ
AT
ION
( B
)
MAGNETIC FIELD (T)
magnetic hysteresis without cooperativity
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High Spin Clusters
Single Molecule Magnets
applications?
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0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(/
0)]-1
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(/
0)]-1
0=2x10-7 s
E/kB=61 K
Temperature dependence of the relaxation time of
Mn12acetate
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(/
0)]-1
deviations from the Arrhenius law
Barbara et al. J. Magn. Magn. Mat. 1995, 140-144, 1825
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return to the equilibriumtunnel mechanismtunnel mechanism
H=0
M=S M=-S
terms in Sx and Sy of the spin Hamiltonian
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return to the equilibriumtunnel mechanismtunnel mechanism
H=0
M=S M=-S
terms in Sx and Sy of the spin Hamiltonian
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What is the difference ?
Four fold axis Tetragonal (E=0)
Mn12 Fe8
HH = = BB S.g.BS.g.B - D - D SSzz22 + + E (E (SSxx
22--SSyy22)) + B + BSSzz
44 + + C (C (SS++44++SS--
44))
Stot=10
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What is the difference ?
Four fold axis Tetragonal (E=0)
Two fold axis Rhombic (E0)
Mn12 Fe8
HH = = BB S.g.BS.g.B - D - D SSzz22 + + E (E (SSxx
22--SSyy22)) + B + BSSzz
44 + + C (C (SS++44++SS--
44))
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Hysteresis loops for Mn12
Friedman et al., PRL, 1996; Hernandez et al, EPL, 1996; Thomas et al., Nature, 1996
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-30 -20 -10 0 10 20 30
2.0 K2.2 K2.4 K2.6 K2.8 K
M (
emu)
H (kOe)
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Hysteresis loops for Mn12
0 5 10 15 20
2.0 K2.2 K2.4 K2.6 K2.8 K3.0 K
dM/d
H (
10-5
em
u/O
e)
H (kOe)
Friedman et al., PRL, 1996; Hernandez et al, EPL, 1996; Thomas et al., Nature, 1996
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Uniform spacing between steps
0
5
10
15
20
25
30
0 1 2 3 4 5 6
Hef
f (kO
e)
step number n
Step spacing: ~4.5 kOe
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-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-30 -20 -10 0 10 20 30
2.0 K2.2 K2.4 K2.6 K2.8 K
M (
emu)
H (kOe)
Hysteresis loops for Mn12
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Enhanced Relaxation at Step Fields
10-3
10-2
9.5 kOe9.0 kOe
(Msa
t - M
) (
emu)
t (s)
Higher energy barrier
Yet faster relaxation!
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Enhanced Relaxation at Step Fields
0 5 10 15 20
(s-1)
H (kOe)
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Thermally Assisted Resonant Tunneling
m = -10
m = -9
m = 10
m = 9
Thermal activation
Fast tunneling
Tunneling occurs when levels in opposite wells align.
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Hamiltonian for Mn12
2z BDS g S HH
The field at which (in the left well) crosses (in the right well):
m nm
Bnmm g
DnH
,
Steps occur at regular intervals of field, as observed.
Step occurs every 4.5 kOe D/g = 0.31 K
Compare with ESR data: D = 0.56 K, g = 1.93 D/g = 0.29 K(Barra et al., PRB, 1997)
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Hamiltonian for Mn12
2z BDS g S HH 4
zBSSpectroscopic studies revealed a 4th-order longitudinal anisotropy term B ~ 1.1 mK. (ESR: Barra et al., PRB, 1997 and Hill et al., PRL, 1998; INS: Mirebeau et al., PRL, 1999, Zhong et al., JAP, 2000 and Bao et al., cond-mat, 2000)
Different pairs of levels cross at slightly different fields.
Allows for the Examination of the Crossover from Thermally Assisted to Pure Quantum Tunneling.
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Crossover to Ground-state Tunneling
Abrupt “first-order” transition between thermally assisted and ground state tunneling.
Theory: Chudnovsky and Garanin, PRL, 1997; Exp’t: Kent, et al., EPL, 2000, Mertes et al., JAP, 2001.
2 2, 1 ( )m m
B
Dn BH m m
g D
Level crossing fields:
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Fe8 Hamiltonian in Zero Field
2 2 2 4 4( ) ( )z x yDS E S S C S S H
Easy Axis Hard Axis
Spin wants to rotate in the y-z plane
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Two Paths for Magnetization Reversal
Easy axis
Hard axis
Z
Y
XH
A
B
ClockwiseCounterclockwise
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Destructive Topological InterferenceEasy axis
Hard axis
Z
Y
XH
A
B
Equivalence between paths is maintained when H is applied along the Hard Axis.
Topological (Berry’s) phase depends on solid angle enscribed by the two paths.
Complete destructive interference occurs for certain discrete values of .
Theoretical Prediction: A. Garg., 1993.
Solid Angle
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Destructive Topological Interference
A. Garg., 1993.
Modulation of Tunnel Splitting:
where depends on the field along the Hard Axis.
When S = /2, 3/2, 5/2…, tunneling is completely suppressed!
Interval between such destructive interference points:
cos( ),S
22 ( )
B
H E E Dg
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Measured Tunnel Splitting
0 0.2 0.4 0.6 0.8 1 1.2
0.1
1
10
Tunn
el s
pitti
ng ²(
10-7
K)
Magnetic tranverse field (T)
0°
= 90°50°
30°20°
10°
5°
0 0.2 0.4 0.6 0.8 1 1.2 1.40.1
1
10
Tunn
el s
plitt
ing
²(10
-7 K
)
Magnetic transverse field (T)
M = -10 -> 10
0°
7°
20° 50° 90°
experimentalcalculated with
D = -0.29, E = 0.046, C = -2.9x10-5 K
W. Wernsdorfer and R. Sessoli, Science, 1999.
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Parity Effect: Odd vs. Even Resonances
-1 -0.5 0 0.5 1
0.1
1
10
²tu
nne
l(1
0-8
K)
µ0Htrans(T)
n = 0
n = 1
n = 2
0°
W. Wernsdorfer and R. Sessoli, Science, 1999.
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What Causes Tunneling and Why the Parity Effect in Fe8
• Tunneling is produced by terms in the Hamiltonian that do not commute with Sz.
• For Fe8, these terms are
• Selection rule:• Every other tunneling resonance is
forbidden!
2 2 2 2( ) ( )2x yE
E S S S S
,...)3,2,1(2 ppm
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What Causes Tunneling and Why the Parity Effect in Fe8
n = 1
-9 8
9
10-10
n = 0
-10
9-9
10
2m p 2m p Tunneling Allowed Tunneling Forbidden
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Parity Effect: Odd vs. Even Resonances
-1 -0.5 0 0.5 1
0.1
1
10
²tu
nne
l(1
0-8
K)
µ0Htrans(T)
n = 0
n = 1
n = 2
0°
W. Wernsdorfer and R. Sessoli, Science, 1999.
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Crossover From Classical to Quantum Regime
0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,00,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
n=5
n=0
n=1
n=2
n=3
n=4
n=6
n=7
n=8
n=9
n=10B
n
T (K)
Activated Tunneling
Measured ( ) and Calculated ( ) Resonance Fields
Barbara et al, JMMM 140-144, 1891 (1995) and J. Phys. Jpn. 69, 383 (2000) Paulsen, et al, JMMM 140-144, 379 (1995); NATO, Appl. Sci. 301, Kluwer (1995)
Classical Thermal Activation
Tblocking
Ground-state Tunneling
Tc-o
(Mn12-ac)
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The Tunnel Window: An effect of weak Hyperfine Interactions
• Chiorescu et al, PRL, 83, 947 (1999)• Barbara et al, J. Phys. Jpn. 69, 383
(2000)• Kent et al, EPL, 49, 521 (2000)
3,75 3,80 3,85 3,90 3,95 4,00 4,05 4,10 4,15
0
1
2
3
4
n=8T=0.95 K
dm /
dB0
B0 (T)
8-1 8-0
Inhomogeneous broadening of Two resonances: Dipolar fields
Data points and calculated lines
Level Scheme
0,4 0,6 0,8 1,0 1,2 1,4
3,0
3,5
4,0
4,5
5,0 10-010-1
9-09-1 9-2
8-08-1 8-2
7-07-1 7-2
6-06-1 6-2
Bn (
T)
T(K)3,0 3,5 4,0 4,5 5,0
-30
-20
-10
0
10
20
(n-p) : -S+p S-n-p
9-2 10-1
9-1 10-0
9-0
8-2
8-1
8-0
7-2
7-1
7-0
6-0
6-1
6-2
E (K)
B0 (T)
-0.04 -0.02 0 0.02 0.04 0.06 0.0810-7
10-6
10-5
sqrt(s
-1)
µ0H(T)
M in = -0.2 M s
-0.005 0 0.0054 10-6
6 10-6
8 10-6
10-5
2 10-5 t0=0s
t0=10s
t0=5s
t0=20s
t0=40s
Homogeneous broadening of nuclear spins: Tunnel window
• Wernsdorfer et al, PRL (1999)
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Effects of Strong Hyperfine Interactions:
Tetragonal symmetry (Ho in S4)
J = L+S = 8; gJ=5/4
Dipolar interactions between Ho3+ << mT
Case of Rare-earth ions: Ho3+ in Y0.998Ho0.002LiF4
HCF-Z = -B20 O2
0 - B40 O4
0 - B44 O4
4 - B60O6
0 - B64O6
4 - gJBJH Bl
m : acurately determined by high resolution optical spectroscopy
Sh. Gifeisman et al, Opt. Spect. (USSR) 44, 68 (1978); N.I. Agladze et al, PRL, 66, 477 (1991)
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Hysteresis loop of Ho3+ ions in YLiF4
Mn12-ac
Thomas et al, Nature (1996) Giraud et al, PRL, 87, 057203-1 (2001)
Friedman et al, PRL (1996), Hernandez et al, EPL (1996)
Steps at Bn = 450.n (mT) Steps at Bn = 23.n (mT) Tunneling of Mn12-ac Molecules Tunneling of Ho3+ ion
-80 -40 0 40 80 120
-1,0
-0,5
0,0
0,5
1,0
200 mK 150 mK 50 mK
M/M
S
0H
z (mT)
-20 0 20 40 60 800
100
200
300
n=0n=3
n=1
n=-1
n=2
dH/dt > 0
1/ 0
dm
/dH
z (1/
T)
Ho3+
-1
-0,5
0
0,5
1
-3 -2 -1 0 1 2 3
1.5K
1.6K
1.9K
2.4K
M/M
S
BL (T)
Comparison with Mn12-ac
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Role of Strong Hyperfine Interactions H = HCF-Z + A.I.J
-80 -40 0 40 80 120
-1,0
-0,5
0,0
0,5
1,0
200 mK 150 mK 50 mK
M/M
S
0H
z (mT)
-20 0 20 40 60 800
100
200
300
n=0n=3
n=1
n=-1
n=2
dH/dt > 0
1/ 0
dm/d
Hz (
1/T)
-200 -150 -100 -50 0 50 100 150 200
-180,0
-179,5
-179,0
-178,5
E (
K)
0H
z (mT)
-7/2
7/2
5/2
-7/2
7/2
3/2
5/2
3/2
-5/2 -3/2-1/2
1/2
-5/2 -3/2
Induce Tunneling of Electronic Moments
-1/2 1/2
Avoided Level Crossings between |, Iz and |+, Iz’ if DI= (Iz -Iz’ )/2 integer
Co-Tunneling of Electronic and Nuclear Spins:
Electro-nuclear entanglement
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Exchange-biased quantum tunnelling in a dimer of Mn4 molecule
W. Wernsdorfer et al, Nature 416, 406 (2002)
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V15 : The Archetype of Low spin Molecules
A Mesoscopic Spin S=1/2
Anisotropy of g-factor:
~ 0.6%Ajiro et al, J..Low. Temp.
Phys. to appear (2003)
Barra et al,J. Am. Chem. Soc. 114, 8509 (1992)
Exchange interactions: Antiferromagnetic ~
several 102K
Müller, Döring, Angew. Chem. Intl. Engl., 27, 171 (1988)
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Barbara et al, cond-mat / 0205141 v1; submited to PRL.
0,0 0,2 0,4 0,6 0,8 1,00,0
0,2
0,4
0,6
0,8
1,0
M/M
S
B0(T)
= 130
60 mK 0.14 T/s 0.14 mT/s
V15
M(H) : Reversible and out of equilibrium
80 mK
en
erg
y
magnetic field
²
| S, -m >
| S, m-n >
1 P
1 - P
| S, -m >
| S, m-n >
Adiabatic Landau-Zener Spin Rotation
« Isolated V15 » : A two-level system « without dissipation »
M(H) = dE(H)/dH
Fast sweeping rate / Weak coupling to the cryostat
Nuclear Spin-Bath :Weak Level Broadening
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Low sweeping rate / Strong coupling to the cryostat
« Non-Isolated V15 » : A two-level system « with dissipation »
0,0
0,2
0,4
0,6
0,8
1,0
-0,6 -0,3 0,0 0,3 0,60,00
0,05
0,10
0,15
T=0.1 K
B0 (T)
TS=T
ph (K)
(c)
M (
µB)
M (
µB) T = 100 mK
0.14 T/s 0.07T/s 4.4 mT/s
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,70,0
0,2
0,4
0,6
0,8
1,0(d)
B0 (T)
Measured
Calculated
Chiorescu et al, PRL 84, 3454 (2000)
M(H): IrreversibleLZS transition at Finite Temperature
(dissipative)
Phonon-bath bottleneck modelAbragam, Bleaney, 1970; Chiorescu et al, 1999.
Nuclear spin-bath level broadeningStamp, Prokofiev, 1998.
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V15: a Gapped Spin ½ Molecule
Dzyaloshinsky-Moriya interactions: HDM= - DijSixSj
The Multi-Spin Character of the Molecule (15 spins)
+
Time Reversal Symmetry D = 0 (Kramers Theorem)
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Magnetism: From Macroscopic to Single atoms
S = 1020 1010 108 106 105 104 103 102 10 1
clusters spinsmoleculesNano-particlesNano-wiressubmicron
0 3 nm
Ho
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Macroscopic Quantum Tunneling of Magnetization
of Single Nanoparticles easy axis
Barium ferrite Nanoparticle (10 nm)
Wernsdorfer et al. et al, PRL, 79, 4014, (1997)
Tc=0.31 K
Stoner-Wohlfarth astroid
0.2
0.4
0.6
0.8
1
1.2
0° 15° 30° 45° 60° 75° 90°
Tc(
)/T
c(4
5°)
angle
Tc() 0Ha 1/4 cot 1/ 6
1 cot 2/3 1
Miguel and Chudnovsky, PRB (1995)
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Dissipation control of LZSMolecule spins 1/2 : Gapped
V15
Ho3+ , Mn4 pairs:Cross-spin transitions, Co-tunneling
Mn4 Fe8
Quantum DysnamicsBerry Phases
Quantum Classical crossoverQuantum Dynamics, Spin Bath
Mn12-ac
Conclusion and Perspectives
Quantum Tunneling at the Mesoscopic Scale(Environmental Effects on Quantum Mechanics)
Evidence for Quantum Coherence (, Rabbi oscillations, … )
Manipulations of Quantum Spins, Spins Qbits(Quantum Informations and Computers)
Tunneling of single Ho3+ ions Entangled I-J states
Ho3+
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AcknowledgementsUniv. Florence & Modena:Andrea CaneschiClaudio SangregorioLorenzo SoraceAngelo RettoriAnna FortAndrea Cornia
L. Neél Lab. CNRS GrenobleE. BonetI. Chiorescu R. Giraud L. Thomas C. Thirion R. Tiron
CCNY:Myriam SarachikYicheng Zhong
U. Barcelona:Javier TejadaJoan Manel HernandezXixiang Zhang (now Hong Kong)Elias Molins
XeroxRon Ziolo
Lehman College CUNYEugene Chudnovsky
Univ. Rio de Janeiro M. Novak
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Italian INFMA. LascialfariF. BorsaR. CaciuffoG. AmorettiM. Affronte
CNRS GrenobleJ. VillainA. L. BarraC. PaulsenV. Villar A. BenoitCNRS Bagneux D. Mailly
Lehman College CUNYEugene Chudnovsky
Univ. Calif. San Diego:David HendricksonSheila AubinEvan RumbergerE. Yang
Los Alamos:Rob Robinson(now ANSTO, Australia)Tim KelleyHeinz NakotteFrans Trouw (Argonne) Wei Bao
Univ. FloridaN. Aliaga, S. Bhaduri, C. Boskovic, C. Canada, C._Sanudo, M. Soler, G. Christou
Univ. P. et M. Curie, ParisV. Marvaud, M. Verdaguer,
Univ. BielefeldH. Bögge, A. Müller
U. St. PetersburgA.M. Tkachuk
Univ. Manchester R.E.P. Winpenny
Univ. Kyoto H. Ajiro, T. Goto, S. Maegawa
Univ. OkkaidoY. Furukawa
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AcknowledgementsJ. F. Fernandez
A. Garg
M. Leuenberger D. Loss,
A. Mukhin
S. Miyashita
N.V. Prokof'ev
A. Zvezdin
L. J. de Jongh
J. Schweizer
P.C.E. Stamp
I. Tupitsyn