1

The single quantum rotor

CH3 BH ~ 650 µeVCD3 BD ~ 325 µeV

0 50 100 150 200 250 300 3500

2

4

6

8

10 A1

E1

EO

AO

(meV

)

(°)0 50 100 150 200 250 300 350

0.0

0.5

1.0

A

0

(°)

-101

E

O-

-101

E

0+

-101

E

1-

2 2

0 023

2 r

H VI

Tunnelling SpectroscopyA0 E0 << ħph INS

Quantum Rotation of Methyl Groups: Tunnelling

2

Nonlinearity in the quantum regime

Quantum dynamics in complex environments

Isolated dynamics:

Quantum Rotation of Methyl Groups: Tunnelling

Crystal Structure-Methyl dynamics

3

Experiments

Neutron Diffraction: Powders, single X’stals

Spectroscopy: Neutrons, Infrared, Raman

Chemistry: Isotope substitutions

4

j

H H

H

H H

H

H H

H

H H

H

j-1

L

j-2 j+1

LL

The isolated chain of coupled methyl groups

2 2

012

1 cos3 1 cos32 2 2

cj j j

j r j

V VH

I

2 2

20

12

91 cos3

2 2 4c

j j jj r j

V VH

I

Approximation

Hamiltonian

5

4 Å

3.6Å

4-Methylpyridine: Infinite chains of Methyl groups

6

F. Fillaux et al., Phys. Rev. B 68, 224301 (2003)

-Picoline 10

K

7

4-Methylpyridine: Inelastic Neutron Scattering

F.Fillaux, C. J. Carlile and G. J. Kearley, Phys. Rev. B 58 (1998) 11416

8

-Picoline

9

Isotopic mixtures of 4MP-h7 (%) and 4MP-d7

Inte

nsit

y (a

rb. u

nits

)

Energy Transfer (meV)

100% h7

85% h7

65% h7

50% h7

26% h7

20% h7

5% h7

0.20 0.400.30 0.50 0.60

F.Fillaux and C. J. Carlile, Phys. Rev. B 42 (1990) 5990

10

j

H H

H

H H

H

H H

H

H H

H

j-1

L

j-2 j+1

LL

The infinite chain of coupled methyl groups

2 2

012

1 cos3 1 cos32 2 2

cj j j

j r j

V VH

I

Energy band

Hamiltonian

2 2

02

1 cos32 2ip j

j r j

VH i

I

2 2

02

1 cos3 1 cos62 2 2

cop j j

j r j

V VH

I

11

4-Methylpyridine: Inelastic Neutron Scattering

F.Fillaux, C. J. Carlile and G. J. Kearley, Phys. Rev. B 58 (1998) 11416

0 3.66meV

5.46meVc

V

V

12

j

H H

H

H H

H

H H

H

H H

H

j-1

L

j-2 j+1

LL

The isolated chain of coupled methyl groups

2 2

012

1 cos3 1 cos32 2 2

cj j j

j r j

V VH

I

2 2

20

12

91 cos3

2 2 4c

j j jj r j

V VH

I

Hamiltonian

Approximation

13

Quantization (Semi-classical)

Kink-mass renormalization 2

0 0 31 ; 3i 8 5

8q

K K

iM M

Breather mass

2

0 0, 22

3 82 sin ; 1,2,... 1

3316 1

8

q qB l K

l iM M l

ii

Threefold potential: only l=1Threefold potential: only l=1

Dashen et al., Phys. Rev. D 11 (1975) 3424

14

3.66 5.461 cos 3

2 21 cos3 1V j j j jmeV

Quantization (Semi-classical)

15

Kinetic energy 2 2 2,1 ,1 0

q p qB BE E p c

Quantization rule ; 0, 1, 2,...h L

np n

The energy spectrum 2 2 2 2,1 ,1

q n qB B cE E n

2 2 2 2,1 ,1

q n qB B cE E n

F.Fillaux and C. J. Carlile, Phys. Rev. B 42 (1990) 5990

1 0,1 ,1 0.517meVq q

B BE E

Quantization

16

4-Methylpyridine Single X’stal  500 60 eV

17

Quantum sine-Gordon Theory

Free Breathers = Plane Waves

Nonlocal Excitations in Nonlinear Complex Systems

Quantum Rotation of Infinite Chains of Methyl Groups