Phonon spectrum measured in a 1D Yukawa chain

John Goree & Bin Liu

Modes in 1-D chains

Colloids:

• Polymer microspheres

trapped by counter-

propagating laser beams

• Lowest-order modes

(sloshing & breathing

modes) observed

experimentally

Tatarkova, et al., PRL 2002 Cvitas and Siber, PRB 2003

Carbon nanotubes:

• Xe atoms trapped on a tube

• Theory: phonon spectrum

Transverse mode

Modes in a 1-D chain

Longitudinal mode

Experimental system: dusty plasma

Like a colloidal suspension:

• polymer microspheres

• electrically charged

• suspended in medium that

provides screening

• colloidal crystals

• optical methods include:

• direct imaging of particles

• laser manipulation

Experimental system: dusty plasma

The medium is a plasma:

• a low-pressure gas

• partially ionized by applying high voltage

Experimental system: dusty plasma

Medium is low density:

• gas instead of a solvent

• microspheres are underdamped

Suspension is very soft:

• shear modulus of a 3D crystal is

1019 smaller,as compared to

metals

What’s special about plasma:

Temperature can be varied:

• not in this talk

Microspheres

Melamine formaldehyde

diameter 8.09 m

introduced into plasma by shaking a dispenser

Pair potential

Particles suspended as a monolayer interact with a repulsive Yukawa

potential:

In this experiment:

charge Q - 7600 e

screeninglength D 0.86 mm

spacing a 0.80 mm >> particle radius 4 m

Suspension of Microspheres

Microspheres : • have no buoyancy

• levitated by electric field a few mm above electrode substrate

• form horizontal monolayer

• no out-of-plane buckling is observed

• ordered lattice

QE

mgelectrodesubstrate

Setup:video camera

(top view)

lower electrodeRF

Ar laser beam 2 Ar lase beam1

microsphere scanningmirror

Ar laser beam 1

Making a one-dimensional chain

“Channel” on substrate to confine a chainMicrospheres are trappedabove the groove

lowerelectrode

groove mg

QE

Groove-shaped channel in lower electrode shapes the E field that confines particles

0.1 Hz

3 Hz

15 Hz

resonant frequency

groove

Image of chain in experiment

particle’s x,y position measured in each video frame

Vibrational Excitation

Elastic vibrations can be excited by:

• Brownian motion in gas• Laser manipulation

momentum imparted to microsphere

incident laser beam

Experiment:Natural motion of a 1-D chain (no manipulation)

central portionof a 28-particle chain

1 mm

Measuring phonon spectrum

Method:

• Video microscopy• Particle tracking x(t) & v(t):• Calculate current correlation function C(q,t)• Fourier transform C(q,)

Phonon spectrum

Color corresponds to energy

Energy is concentrated in a band that corresponds to a dispersion relation

Symbols indicate peaks

Phonon spectrum

Color corresponds to energy

Energy is concentrated in a band that corresponds to a dispersion relation

Symbols indicate peaks

Excitation with laser manipulation

Wave propagatesto two ends of chain

modulated beam-I0 ( 1 + sint )

continuous beamI0

Net force I0 sint

1 mm

Dispersion relation - natural & externally excited

longitudinal transverse

○ excitednatural

○ excitednatural

N = 28 N = 28

Summary• We used direct imaging

to observe particle motion in a 1-D chain

• We characterized the phonons by:

• Power spectra• Dispersion relation

More details & theory:Liu, Avinash & Goree PRL

2003Liu & Goree PRE

2005

Images of one-dimensional chains

scanningmirror

Modulating the laser power

Ar laser beam

Experiment result

wave:• is excited in the middle of chain• propagates to two ends of chain

Argon laser beam

Argon laser beam

Thermal motion

Gas temperature = room temperature

Particle kinetic temperature was computed from particle velocities

230 K from mean kinetic energy:390K from fit of velocity distribution function: