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Transverse optical mode in a 1-D Yukawa chain
J. Goree, B. Liu & K. Avinash
Example of 1-D chain
Applications:
• Quantum computing • Atomic clock
WaltherMax-Planck-Institut für Quantenoptik
linear ion trap
image of ion chain
(trapped in the central part of the linear ion trap)
Examples of 1-D chains in condensed matter
Colloids:
Polymer microspheres
trapped by laser beams
Tatarkova, et al., PRL 2002 Cvitas and Siber, PRB 2003
Carbon nanotubes:
Xe atoms trapped in a tube
plasma = electrons + ions Plasma
+
-
+
+
+
+
+
+
+
- -
-
-
--
-
+
-
What is a dusty plasma?
D
• Debye shielding
small particle of solid matter
• becomes negatively charged
• absorbs electrons and ions
& neutral gas
polymer microspheres
8.05 m diameter
Q - 6 103 e
Particles
Solar system• Rings of Saturn• Comet tails
Fundamental science• Coulomb crystals• Waves
Manufacturing• Particle contamination
(Si wafer processing)• Nanomaterial synthesis
Who cares about dusty plasmas?
Electrostatic trapping of particles
Equipotentialcontours
electrode
electrode
positive
potential
electrode
electrode
With gravity, particles sediment to high-field region monolayer possible
Without gravity, particles fill 3-D volume
QE
mg
Chamber
top-viewcamera
laser illumination
side-viewcamera
vacuum chamber
Comparison ofdusty plasma & pure ion plasmas
Similar:
• repulsive particles
• lattice, i.e., periodic phase
• 3-D, 2-D or 1-D suspensions
• direct imaging
• laser-manipulation of
particles
Different - dusty plasma has:
• gaseous background
• 105 charge• no inherent rotation• gravity effects
D
a
D
r
r
QrU
exp
4)(
0
• Yukawa potential
Confinement of a monolayer
– Particles repel each other
– External confinement by bowl-shaped electric sheath above lower electrode
Confinement of 1-D chain
Vertical: gravity + vertical E
lowerelectrode
groove mg
QE
Horizontal:sheath conforms to shape of groove in lower electrode
Setup
Argon laser pushes particles in the monolayer
H eN e lase rho riz o nta ls he e t
v ideo cam e ra(to p v iew )
lo wer e lec tro deR F
two -axiss te e ring
m ic ro sphe res
m o d ula tio n
A r lase rbe a m
x
y
f ram egra bb e r
Radiation Pressure Force
transparent microsphere
momentum imparted to microsphere
Force = 0.97 I rp2
incident laser intensity I
Ar laser
mirror
scanning mirrorchopsthe beam
beam dump
Choppingchopped beam
scanningmirror
Scanningmirror
Ar laser beam
scanning mirror partially blocksthe beam
sinusoidally-modulated beamSinusoidalmodulation
beam dump
Two-axis scanning mirrors
For steering the laser beam
Experiments with a 1-D Chain
lowerelectrode
groove mg
QE
Image of chain in experiment
Confinement is parabolicin all three directions
lowerelectrode
x 0.1 Hz
groove y 3 Hz
z 15 Hz
Measured values of single-particle resonance frequency
Modes in a 1-D chain: Longitudinal
restoring force interparticle repulsion
experiment Homannet al. 1997
theory Melands “dust lattice wave DLW”1997
Modes in a 1-D chain: Transverse
Vertical motion:
restoring force gravity + sheath
experiment Misawa et al. 2001
theory Vladimirov et al. 1997
oscillation.gif
Horizontal motion:
restoring force curved sheath
experiment THIS TALK
theory Ivlev et al. 2000
Properties of this wave:
The transverse mode in a 1-D chain is:• optical• backward
Terminology: “Optical” mode
not optical
k
k
optical
k
Optical mode in an ionic crystal
Terminology:“Backward” mode
forward
kbackward
k
“backward” = “negative dispersion”
Natural motion of a 1-D chain
Central portionof a 28-particle chain
1 mm
Spectrum of natural motion
Calculate:
• particle velocities
vx
vy
• cross-correlation functions
vx vxlongitudinal
vy vytransverse
• Fourier transform power spectrum
Longitudinal power spectrum
Power spectrum
negative slope
wave is backward
Transverse power spectrum
No wave at = 0, k = 0
wave is optical
Next: Waves excited by external force
Setup
Argon laser pushes only one particle
video camera(top view)
lower electrodeRF
Ar laser beam 2 Ar lase beam1
microsphere scanningmirror
Ar laser beam 1
Radiation pressure excites a wave
Wave propagatesto two ends of chain
modulated beam-I0 ( 1 + sint )
continuous beamI0
Net force: I0 sint
1 mm
Measure real part of k from phase vs x
fit to straight lineyields kr
0 5 100.00
0.01
0.02
0.03
0.04
0.05
0.06
exponential fitting
Am
plit
ud
e (
mm
/s)
position (mm)
Measure imaginary part of k from amplitude vs x
fit to exponentialyields ki
transverse mode
0 1 2 30
10
20
30
N = 10 N = 19 N = 28
(s-1)
kr a
CM
Experimental dispersion relation (real part of k)
Wave is:backwardi.e., negative dispersion
smaller N larger a
larger
0 1 2 30
10
20
30
N = 10 N = 19 N = 28
(
s-1 )
ki a
Experimental dispersion relation (imaginary part of k) for three different chain lengths
Wave damping is weakest in the frequency band
Experimental parameters
To determine Q and D from experiment:
We used equilibrium particle positions & force balance
Q = 6200 e
D = 0.86 mm
Theory
Derivation:
• Eq. of motion for each particle, linearized & Fourier-transformed
• Different from experiment:
• Infinite 1-D chain
• Uniform interparticle distance
• Interact with nearest two neighbors only
Assumptions:
• Probably same as in experiment:
• Parabolic confining potential
• Yukawa interaction
• Epstein damping
• No coupling between L & T modes
Wave is allowed in a frequency band
Wave is:backwardi.e., negative dispersion
R
L
0 1 2 30
10
20
(s
-1)
k a
I
II
III
CM
L
(
s-1)
Evanescent
Evanescent
Theoretical dispersion relation of optical mode (without damping)
CM = frequency of sloshing-mode
0 1 2 30
10
20
30
ki
kr
(s
-1)
k a
C
M
L
I
II
IIIsmall damping
high damping
Theoretical dispersion relation (with damping)
Wave damping is weakest in the frequency band
Molecular Dynamics Simulation
Solve equation of motion for N= 28 particles
Assumptions:
• Finite length chain
• Parabolic confining potential
• Yukawa interaction
• All particles interact
• Epstein damping
• External force to simulate laser
Results: experiment, theory & simulation
Q = 6 103e = 0.88a = 0.73 mmCM = 18.84 s-1
real part of k
Damping:theory & simulation assume E = 4 s-1
0 1 2 30
10
20
30 experiment MDsimulation theory 3
(s-1)
ki a
imaginary part of k
Results: experiment, theory & simulation
Why is the wave backward?
k = 0Particles all move togetherCenter-of-mass oscillation in confining
potential at cm
Compare two cases:
k > 0Particle repulsion acts oppositely to
restoring force of the confining potentialreduces the oscillation frequency
Conclusion
Transverse Optical Mode• is due to confining potential & interparticle repulsion• is a backward wave• was observed in experiment
Real part of dispersion relation was measured: experiment agrees with theory
Possibilities for non-neutral plasma experiments
Ion chain(Walther, Max-Planck-Institut für Quantenoptik )
Dust chain
2-D Monolayer
triangular lattice with hexagonal symmetry
2-D lattice
Dispersion relation (phonon spectrum)
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4
wavenumber ka/
Fre
quen
cy
Theory for a triangular lattice, = 0°Wang, Bhattacharjee, Hu , PRL (2000)
compressional
shear
acoustic limit
Longitudinal wave
4mm
k Laser incident here
f = 1.8 Hz
Nunomura, Goree, Hu, Wang, Bhattacharjee Phys. Rev. E 2002
Random particle motion
No Laser!
= compression + shear
4mm
S. Nunomura, Goree, Hu, Wang, Bhattacharjee, AvinashPRL 2002
Phonon spectrum
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
6.0
4.0
2.0
0.0
Longitudinal mode6.0
4.0
2.0
0.0
k (mm-1)
f (H
z)f (
Hz)
ka/-2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0
/
0
3.0
2.0
1.0
0.0
4.0
/
0
3.0
2.0
1.0
0.0
4.0
5
10
15
En
erg y
den
s ity
/ k B
T (
10-
3m
m2s)
k
a
= 0°
Transverse mode
& sinusoidally-excited waves
S. Nunomura, Goree, Hu, Wang, Bhattacharjee, AvinashPRL 2002
Phonon spectrum
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
6.0
4.0
2.0
0.0
Longitudinal mode6.0
4.0
2.0
0.0
k (mm-1)
f (H
z)f (
Hz)
ka/-2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0
/
0
3.0
2.0
1.0
0.0
4.0
/
0
3.0
2.0
1.0
0.0
4.0
5
10
15
En
erg y
den
s ity
/ k B
T (
10-
3m
m2s)
k
a
= 0°
Transverse mode
& theory
S. Nunomura, J. Goree, S. Hu, X. Wang, A. Bhattacharjee and K. AvinashPRL 2002
Damping
With dissipation (e.g. gas drag)
method of excitation k
natural complex real
external real complex
(from localized source)
laterthis talk
earlier this talk
incident laser intensity I
Radiation Pressure Force
transparent microsphere
momentum imparted to microsphere
Force = 0.97 I rp2
How to measure wave number
• Excite wavelocal in xsinusoidal with timetransverse to chain
• Measure the particles’ position:x vs. t, y vs. tvelocity: vy vs. t
• Fourier transform: vy(t) vy()
• Calculate k
phase angle vs x kr
amplitude vs x ki
Analogy with optical mode in ionic crystal
negative positive + negative
external confining potential
attraction to opposite ions
1D Yukawa chain ionic crystal
charges
restoring force
M m
+ -- + -- + ---- -- --
m mM >> m
Electrostatic modes(restoring force)
longitudinal acoustic transverse acoustic transverse optical (inter-particle) (inter-particle) (confining potential)
vx vy vz vy
vz
1D
2D
3D
groove on electrode
x
y
z
Confinement of 1D Yukawa chain
28-particle chain
Ux
x
Uy
y
Confinement is parabolicin all three directions
method of measurement verified:
x laser purely harmonic
y laser purely harmonic
z RF modulation
lowerelectrode
x 0.1 Hz
groove y 3 Hz
z 15 Hz
Single-particleresonance frequency