b. barbara, r. giraud, i. chiorescu*, w. wernsdorfer, lab. louis néel, cnrs, grenoble....
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B. Barbara, R. Giraud, I. Chiorescu*, W. Wernsdorfer, Lab. Louis Néel, CNRS, Grenoble.
Collaborations with other groups:
D. Mailly (Marcoussis)
D. Gatteschi (Florence)
A. Müller (Bielefeld)
G. Christou (Gainsville)
A.M. Tkachuk (St Petersburg)
S. Miyashita (Tokyo)
*Present adress Delft University of Technology
Quantum Magnetism: from large spin molecules to single ions
Magnetization reversal in nanoparticlesLarge (submicrometer), Small (nanometer)
Magnetic tunneling in moleculesLarge spin molecules (Mn12-ac, Fe8)
Tunneling, Berry phases, Quantum dynamics
Low spin molecules (V15)Adiabatic LZS with and without dissipation
Case of nearly isolated Ions Rare-earth ions: (Ho3+ in Y0.998Ho0.002LiF4, Y0.999Ho0.001Cu2Si2)
Entangled electro-nuclear states, co-tunneling
OUTLINE
Particles from micrometers to 100 nanometers Obtained by: Lithography, Electro-deposition
Measurements: Micro-Squids
100 nm
50 nm x 1m
1m x 2 m
Small ellipse Large ellipseNanowire
-1
0
1
-40 -20 0 20 40M
/M
S
H(mT)
MULTI – DOMAIN: nucleation, pinning,
propagation and annihilation of domain
walls
-1
0
1
-100 0 100
M/M
S
H(mT)
SINGLE - DOMAINSingle Nucleation
Curling
2(dH/dt,T)
<Hsw(dH/dt,T)>counts
H
Nanometer scale
NanoparticleCluster
20 nm3 nm1 nm 2 nm
Magnetic ProteinSingle Molecule
50S = 10 103 106
Barrier in Zero Field
H= - DSz2 - BSz
4 - E(S+2 + S-
2) - C(S+4 + S-
4) + gBSxHx
spin down spin up
|S,S-2> |S,-S+2>
Ground state tunneling
|S,S-1> |S,-S+1>
|S,S> |S,-S>
SZ
En
erg
y
Thermally activated tunneling
Multi-Orbach process
Thermal Activation
Mn12-ac : D = 0.56 K, E = 5 mK, B = 1.18 mK, C = 3 10-5 K,
Fe8 : D = 0.23 K, E = - 47 mK, B = 0.03 mK,
Tunnel splitting
Resonant Tunneling in Mn12-ac in Large // Fields
0 1 2 3 4 5 6
-80
-60
-40
-20
0
20
E(K)
m=±5m=±6m=±7
m=±8
m=±9
m=±10
-2-10
+1
+2
+3
+4
+5
+6
+7+8+9+10
n=11n=10
n=9n=8
n=7n=6
n=5n=4
n=3n=2
n=1n=0
longitudinal field (T)
m=6
m=6 m=-10
m=-10
Top of the barrier
Ground-state Avoided level crossing
Energy scheme in B//
H= - DSz2 - BSz
4 - E(S+2 + S-
2) - C(S+4 + S-
4) + gBSHz
- Landau-Zener Mechanism
- Resonance fields
Dynamics: Landau-Zener Transition (isolated system)
en
erg
y
magnetic field
²
| S, -m >
| S, m-n >
1 P
1 - P
| S, -m >
| S, m-n >
Tunneling Probability :
• General result for a single level crossing
• Solution of the Schrödinger equation
H A h B h
H i
t
L. Landau, Phys. Z. Sowjetunion 2, 46 (1932); C. Zener, Proc. R. Soc. London, Ser. A 137, 696, (1932); E.C.G. Stückelberg, Helv. Phys. Acta 5, 369 (1932).
S. Miyashita, J. Phys. Soc. Jpn. 64, 3207 (1995).
P=1 – exp[-(/ħ)2/c] where c = dH/dt
P ~1 whenpassc~ os= ħ/
Resonance Width and Tunnel Window Effects of Magnetic couplings and 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) Prokofiev and Stamp (1998)
Measured and Calculated Resonance Fields above 0.4 K
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=0
n=1
n=2
n=3
n=4
n=6
n=7
n=8
n=9
n=10
Bn (T
)
T (K)
QT
TA
Three Regimes
Barbara et al, ICM Warsaw (1994); JMMM 140-144, 1891 (1995); JMMM-200, (1999)Paulsen, et al, JMMM 140-144, 379 (1995); NATO, Appl. Sci. 301, Kluwer (1995)
TAQT
Tc-o
Tblocking
In a Transverse Field
1 10 100 1000-1,0
-0,8
-0,6
-0,4
-0,2
0,0
0,2
0,4 0,6 0,8 1,0 1,2
0,4
0,8
1,2
1,6
2,0M / M
S
M|| / M
S
T = 0.5 Kn = 0B
T = 4.42 T
T = 0.9 Kn = 8B
L = 4.02 T
no
rmal
ized
mag
net
izat
ion
t (s)
exponential regime
square root regime
0
2
3L
T
(1/s)
10
10 T (K)
0 1 2 3 4 5 6 7 8 9 10-140
-120
-100
-80
-60
-40
-20
0
20
E (
K)
transverse field (T)
Emin
Emax
Chiorescu et al, PRL, 83, 947 (1999); Barbara et al, J. Phys. Jpn. 69, 383 (2000)
Calculated Energy Spectrum Measured relaxation
Quantum phase interference (Berry phase)in Fe8
Wernsdorfer and Sessoli, Science 284, 133 (1999)
Z
Y
XH
A
B
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°
2 ~ -1
Prokofiev and Stamp PRL 80, 5794 (1998)
Parity Effect: Odd vs. Even Resonances
-1 -0.5 0 0.5 1
0.1
1
10
²tu
nn
el(
10
-8 K
)
µ0Htrans(T)
n = 0
n = 1
n = 2
0°
W. Wernsdorfer and R. Sessoli, Science, 1999.
From Large to Low Spin Molecules
Large spins Low spins Mn12 , Fe8 V15
Order Parameter Ferro. Antiferro. (S = 10) (N =15/2, S=1/2) Barrier DS2 Large Small
Tunnel Splitting Small Large
Dipolar interactions 50mT 1mT
V15: a Gapped Spin ½ Molecule (DH=215)
Dzyaloshinsky-Moriya interactions: HDM= - DijSixSj
The Multi-Spin Character of the Molecule (15 spins)
+
Time Reversal Symmetry = 0 (Kramers Theorem)
Exchange interactions: Antiferromagnetic ~ several 102 KMüller, Döring, Angew. Chem. Intl. Engl., 27, 171 (1988)
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)
More details on the D-M gap of multi-spin systems
Calculation of energy spectra with antisymmetrical interactions:
-3 -2 -1 0 1 2 3-4
-2
0
2
4
6
0 = 80 mK
ener
gy (K
)
magnetic field (T)
H = JSiSj - ijSixSjz - gBBzSiz
ij = ji =0 (2 Kramers doublets)
ij ≠ ji ≠ 0 (2 pairs of singlets)
Miyashita, Nagaosa, Prog. Theo. Phys.106, 533 (2001). Barbara et al, cond-mat / 0205141 v1 and Prog. Theo. Phys. Jpn, Supp. 145, 357 (2002).
LZS transition at finite Temperature At low sweeping rate / strong coupling to the cryostat
( small)
• n1/n2 = exp ( /kTs)
• Phonon-bath bottleneck model: h ± Abragam, Bleaney, 1970; Chiorescu et al, 1999.
M(t)/M() = x(t) is given by
-t/B =x(t) – x(0) + ln [(x(t) –1) /[(x(0)-1)]
Bott=(/2)th2(/2kT)
• Nuclear spin-bath level broadening: 30mK
(Stamp, Prokofiev, 1998).
S. Miyashita, J. Phys. Soc. Jpn. 64, 3207 (1995); V.V. Dobrovitski and A.K. Zvezdin, Euro. Phys. Lett. 38, 377 (1997); L. Gunther, Euro. Phys. Lett. 39, 1 (1997); G. Rose and P.C.E. Stamp, Low Temp. Phys. 113, 1153 (1999); M. Leuenberger and D. Loss, Phys. Rev. B 61, 12200 (2000); …
Spin-phonons transitions (dissipation) Irreversible M(H)
n1
n2
Low sweeping rate / Strong coupling to the cryostat
« Non-Isolated V15 » : A two-level system with dissipationButterfly hysteresis loop
LZS transition at Finite Temperature (dissipative)
1 ~ botl > meas
Hysteresis (≠Orbach process).
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): Irreversible
Equilibrium (Reversible) M(H)=Msth{(2+H2)1/2/2kT}
Landau-Zener transition at « Zero Kelvin » Fast sweeping rate / Weak coupling to the cryostat
(large)
.... But out of equilibrium.
• n1/n2 = exp ( /kTs)
• Ground-State M(H)
• Nuclear Spin-Bath Level broadening
30mK Overlap near zero field .
No spin-phonon transition (no dissipation) Reversible M(H)...
n1
n2
(Stamp, Prokofiev, 1998).
Bott=(/2)th2(/2kT) >> meas
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= (1/2)(gB)2H/[(2+(gBH)2)]1/2
Fast sweeping rate / Weak coupling to the cryostat
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
Chiorescu et al, submitted to PRB, Cond-mat / 0205141 v1Barbara et al , Prog. Theo. Phys. Jpn, Supp. 145, 357 (2002),
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
Relaxation Experiments Outside and Within the Mixing Region (
Fit of M(t) to the Bottleneck model B (B,T). Phonon bath
0 2000 4000 6000 8000 100000,00
0,05
0,10
0,15
0,20
0,25
0,30
0 2000 4000 6000 8000 100000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
M/M
S
t (s)
B0=0.014 T
0.15 K
H: fit=551s / th=1323s
0.05 K
H: fit=1507s / th=8716s
M/M
S
t (s)
B0=0.07 T
0.15 K
H: fit=970s / th=997s
0.05 K
H: fit=3883s / th=3675s
Barbara et al, Prog. Theo. Phys. Jpn, Supp. 145, 357 (2002).
Inside : Outside : Fit not good; B << calculated value Good fit; B (B,T) ~ calculated value Spin-bath (nuclear spins) Phonon bath
A new direction…Mesoscopic Physics of Rare-earth Ions: Ho3+ in Y0.998Ho0.002LiF4
Entangled electro-nuclear states, co-tunneling,…
• Dipolar interactions between different Ho3+ a few ~ mT
• HCF-Z = -B20 O2
0 - B40 O4
0 - B44 O4
4 - B60O6
0 - B64O6
4 - gJBJH
• Blm : 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)
Tetragonal symmetry (Ho in S4)
J = L+S = 8; gJ=5/4
-6 -4 -2 0 2 4 6-200
-150
-100
-50
0
50
100
150
-9 -6 -3 0 3 6 9-240
-200
-160
-120
-80
-40b)a) E (K)
<Jz>
E (K)
0H
z (T)
CF energy barriers: a comparizon between Mn12-ac and Ho3+
spin down spin up
|S,S-2> |S,-S+2>
Ground state tunneling
|S,S-1> |S,-S+1>
|S,S> |S,-S>
SZ
Ener
gy
Short-cuts Lowest energy levels Giraud et al, PRL, 87, 057203-1 (2001) ground-state: Ising doublet
Mn12-ac S = 10 D = 0.56 K, E ~ 5 mK, B = 1.18 mK, C = 0.030 mK
Ho3+ J = 8 B20 = 0.606 K, B40 = -3.253 mK, B44 =- 42.92 mK, B60 =-8.41mK, B64 =- 817.3mK
Hysteresis loop of Ho3+ ions in YLiF4
Thomas et al, Nature (1996) Giraud et al, PRL, 87, 057203-1 (2001)
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)
-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
Ising CF Ground-state + 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)
Avoided Level Crossings between |, Iz and |+, Iz’ if I= (Iz -Iz
’ )/2= odd
-200 -150 -100 -50 0 50 100 150 200
-180,0
-179,5
-179,0
-178,5
I = 7/2
E (
K)
0H
z (mT)
-7/2
7/2
7/2
5/2
3/2
-7/2
Co-Tunneling of Electronic and Nuclear Spins: Electro-nuclear entanglement
Acceleration of slow quantum dynamics associated with co-tunneling of I and J in a transverse field:
Fast increase of the splitting of entangled electro-nuclear states of single Ho 3+ ions.
-60 -40 -20 0 20 40
-1,0
-0,5
0,0
0,5
1,0
-30-20-10 0 10 20 30
-180,2
-180,0
-179,8
-179,6
T = 50 mKdH/dt < 0
< 5 mT 50 mT 100 mT 200 mT
M/M
S
0H
z (mT)
E (K)
dB/dt = 0.55mT/s
-75 -50 -25 0 25 50 75-1.0
-0.5
0.0
0.5
1.0
T = 30 mKv = 0.6 mT/s
HT=190 mT
HT=170 mT
HT=150 mT
HT=130 mT
HT=110 mT
HT=90 mT
HT=70 mT
HT=50 mT
HT=30 mT
HT=10 mT
M/M
S
0H
z (mT)
dB/dt ~ 1 mT/s
In fast ssweeping field
Two différent relaxation regimes
LiY1-xHoxF4 (x~0.1% at.)
Slow ( ~ 1 mT/s) Fast ( ~ 1 T/s)
Thermodynamical and thermal equilibrium Out of thermodynamical equilibrium
-75 -50 -25 0 25 50 75-1.0
-0.5
0.0
0.5
1.0
T = 30 mKv = 0.6 mT/s
HT=190 mT
HT=170 mT
HT=150 mT
HT=130 mT
HT=110 mT
HT=90 mT
HT=70 mT
HT=50 mT
HT=30 mT
HT=10 mT
M/M
S
0H
z (mT)
-15 -10 -5 0 5 10 15 20-180-120-60
060
120180240
-300 -200 -100 0 100 200 300-1.0
-0.5
0.0
0.5
1.0
n entier n demi-entier
0H
n (
mT
)
2n
ajustement linéaire
0H
n = n*23 mT
v = 0.28 T.s-1
T = 50 mK
M/M
S
0H
z (mT)
Fast sweeping rate ... a different regime
50 mK0.3 T/s
120 160 200 240
0
4
8
-150 -75 0 75 150 225
0
20
40
60
-300 -200 -100 0 100 200 300-1,0
-0,5
0,0
0,5
1,0
-8 -6 -4 -2 0 2 4 6 8 10-180
-120
-60
0
60
120
180
240
n = 6
n = 7n = 8
n = 9
b)
dH/dt<0
n=1
n=0
1/ 0
dm
/dH
z (1/
T)
0H
z (mT)
a)M
/MS
0H
z (mT)
integer n half integer n
linear fit
0H
n = n x 23 mT
0H
n (
mT
)
n
Giraud et al, PRL 87, 057203 1 (2001)
Additional steps at fields: Bn = (23/2).n (mT)
Tunneling, Co-tunneling and Cross-spin reversal of Ho3+ pairs
50 mK0.3 T/s
. Hysteresis from bottleneck (B)
and barrier (1)TBl ~ 200 mK > Tmea= 50mK
. Cross-spin relaxations2<< Time-scale<< 1 = C.exp(10/T) (Orbach)
-½, Iz ½, I’z
0 100 200 300
0.0
1.3x10-6
2.5x10-6
T = 1.75 K
f = 801 Hz'' (
emu
/G)
0H
z (G)
Single ion tunneling
Co-tunneling of two ions
Cross-tunneling
Unambiguous observation of different types of tunneling
Ac-susceptibility at high temperature
R. Giraud and B. Barbara, to be published
Exchange-biased tunnelling between two molecules (Mn4 dimer)
W. Wernsdorfer et al, Nature 416, 406 (2002)
S=9 S=0
For a dimer
+>
+>
->
- >
A
B
C
A
A’
B
C
Energy
H > 0H < 0
+ +>
- ->
+ -> + - +>
+ -> - - +>
Bias tunneling and co-tunneling
Simple 2-spins model:singlet / triplet
B. Barbara, News and Views , Nature to appear
Last new direction: Effect of free electrons on tunneling …
Ho3+ in YRu2Si2 (Hiro Susuki, Tsukuba)
-50 -45 -40 -35-1.0
-0.5
0.0
0.5
1.0
-60 -40 -20 0
-1.0
-0.5
0.0
0.5
1.0
v=2.2 mT/s
v=4.4 mT/s
v=8.8 mT/s
v=17 mT/s
v=35 mT/s
v=70 mT/s
v=140 mT/s
T = 40 mK
M/M
S
0H
z (mT)
Ho in YRu2Si2 (Same matrix as in CeRu2Si2)Non-trivial behaviour when v decreases
-50 -40 -30 -20
-1.0
-0.5
0.0
0.5
1.0
n = 1
v=0.07 mT/s
v=0.14 mT/s
v=0.27 mT/s
v=0.55 mT/s
v=1.1 mT/s
v=2.2 mT/s
T = 40 mK
M/M
S
0H
z (mT)
-80 -60 -40 -20 0 20-1.0
-0.5
0.0
0.5
1.0v = 0.14 mT/s
n = 2
n = 1
HT = 0
HT = 10 mT
T = 40 mKM
/MS
0H
z (mT)
Ho in YRu2Si2 (Same matrix as in CeRu2Si2)
Continuous M(H)… and jumps at well defined fields !
Tunneling seems possible, in the presence
of free electrons !
Ho3+ in YRu2Si2Hiro Susuki (Tsukuba)
Y1-HoRu2Si2
~ 0.1%
10 20 30 40 50
-0,6
-0,3
0,0
Y2Ru
2Si
2: Ho3+ <0.1%
T = 40 mK n=2
n=1
dM/dB
B (mT)
Same hyperfine constant.
Tunneling of electro-nuclear states in a
metal…simple cross-spin relaxations…
-80 -60 -40 -20 0 20 40 60 80-1,0
-0,5
0,0
0,5
1,0
-80 -60 -40 -20 0 20 40 60 80
-180,0
-179,5
v = 0.11 mT/s
b)
M/M
S
0H
z (mT)
a)
E (
K)
0H
z (mT)
Y0.998Ho0.002LiF4
Conclusion
Evidence for tunneling of single ions of rare-earth Ho3+
Avoided level crossings result from crystal field and hyperfine interactions
Entangled electro-nuclear states
Ho3+ , Mn4 pairs:Cross-spin and Spin-phonon transitions
co-tunnelingTunneling in metals ?
Ho3+
* Adiabatic LZS transition with or without dissipation.
* Multi-spin molecule spin ½ gap
V15
Resolved resonance linesQuantum classicalQuantum Dynamics
Mn12-ac
Berry Phases, Quantum Dynamics
Collaborations with other groups:
D. Mailly (Bagneux)
A. Caneschi, R. Sessoli, D. Gatteschi(Florence)
A. Müller, H. Bögge
(Bielefeld)
G. Christou(Gainsville)
A.M. Tkachuk(St Petersburg)
Resonant Tunneling of Magnetization in Mn12-acSingle Crystal
-0,008
-0,004
0
0,004
0,008
-4 -3 -2 -1 0 1 2 3 4
1.5 K1.6 K1.9 K2.4 K
M (
emu)
H (T)
Hysertesis loop
102
104
106
-2 -1 0
(s
ec)
H (T)
(sec
)
T(K)103
105
2 3
0 T0.44 T0.6 T0.88 T1.32 T1.76 T2.2 T2.64 T
Magnetic relaxation
L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. BarbaraNature, 383, 145 (1996).
M(H) (Measured) Lowes Energy Levels (Calculated)
Chiorescu et al, JMMM, 221, 103 (2000); JAP 87, 5496 (2000). Barbara et al , Prog. Theo. Phys. Jpn, Supp. 145, 357 (2002); cond/mat. 0205141 v1.
-3 -2 -1 0 1 2 3-4
-2
0
2
4
6
0 = 80 mK
ener
gy (K
)
magnetic field (T)
S=1/2
S=3/2
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3
-2
-1
0
1
2
3
T = 100 mK
|3/2, -3/2>
|3/2, 3/2>
|1/2, 1/2>
|1/2, -1/2>
M (
µB)
Magnetic Field (T)
S= -1/2 S=1/2
S=1/2 S=3/2
Spin reversal within the « three spins » molecule V15, at equilibrium (no barrier)
A model system for the adiabatic Landau-Zener model : two limiting cases
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