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

The molecules are regularly arranged in the crystal

Mn(IV)S=3/2

Mn(III)S=2

Total Spin =10

Mn12acetateMn12acetate

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