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Phonons in a 2D Yukawa triangular lattice: linear and nonlinear experiments
Dept. of Physics and Astronomy, University of Iowa
supported by DOE, NASA, NSF
V.Nosenko, S.Nunomura, and J.Goree
2D Yukawa triangular lattice
rr
rU exp1
)(
Yukawa interparticle interaction,
where is screening parameter:a
Phonons in a 2D Yukawa triangular lattice
wavenumber ka/
Fre
que
ncy
Theory for a triangular latticeWang et al. PRL 2001
Frequency normalized by :
Qma
Dispersion relation (phonon spectrum)
0
0.5
1
2
2.5
3
0 2 4
compressional
shear
acoustic limit
Plasma
+
-
+
+
+
+
+
+
+
- -
-
-
--
-
+
-
electrons + ions = plasma
Experimental system: Dusty Plasma
• Debye shielding
D
• absorbs electrons and ions
small particles of solid matter
• becomes negatively charged
gas
Ar at 2 mTorr
RF plasma
13.56 MHz
20 W
Experimental conditions
Polymer microspheres
• diameter 8.69 0.17 m
• charge 10000 e
HeNe laserhorizontalsheet
video camera(top view)
micro lens
lower electrodeRF
microspheres Ar laserbeam
servoamp
funcgenscope
framegrabber
scanningmirror
to chopper
chopper
.
.
Experimental setup
2D lattice
External confinement
• natural electric fields in plasma
• gravity mg
Fsheath
D
a
The lattice is characterized
by screening parameter:
Comparison ofdusty plasma & colloids
Similar: Different - dusty plasma has:
• Like-charged particles
• Yukawa potential
• 2D or 3D suspensions
• Direct imaging
• Laser-manipulation of
particles
• Gaseous background
• 105 less dissipation
• 105 less volume fraction
0 1 2 3 40
10
20
30
40
k (mm-1)
0=15.1s-1
/a = 4.05 mm-1
Closed symbol: krOpen symbol: ki
0 1 2 3 40
10
20
30
40
k (mm-1)
0=15.1s-1
/a = 4.05 mm-1
Closed symbol: krOpen symbol: ki
Dispersion relations for both modes
Experiment: S.Nunomura et al.
Theory: Wang et al. PRL 2001
Longitudinal wave Transverse wave
2D lattice can be modeled as a network of masses
connected by springs to the nearest neighbors
2D triangular (hexagonal) lattice
Response:• linear• nonlinear
Triangular (hexagonal) lattice
• separation a = 0.5 -1.0 mm
• areal fraction (0.6 - 2.4) 10-4
2D lattice
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8
CD
g(r
)
normalized distance r/a
experimentfit
Pair correlation function:
• Many peaks in g(r)
• Translation order length 9a
Ordered lattice
These profiles show pulse propagation
Particle velocity profiles
0
0.4
0.8
1.2
1.6
2
2.4
2.8
0 5 10 15 20 25
v (
mm
/s)
x (mm)
0.1 s between curves
Laser off
Theory of nonlinear sound waves in 3D liquid
(Landau & Lifshitz, Fluid Mechanics)
C - wave propagation speed
C0 - sound speed
v - particle speed
- adiabatic coefficient
vCC2
10
Normalization: by Cmin , the pulse propagation speed
for lowest laser power
indication of nonlinearity
Pulse propagation speed vs. pulse amplitude
0.9
1
1.1
1.2
1.3
0 0.02 0.04 0.06 0.08 0.1Peak particle speed / C
1
Pu
lse
pro
pa
ga
tion
sp
ee
d /
Cm
in
min
1
Summary
• In 2D triangular (hexagonal) Yukawa lattice
• Longitudinal and transverse phonons were detected and
their dispersion relations were measured.
• Nonlinear effects in pulse propagation of the longitudinal
wave were observed for large amplitudes (Mach numbers
M > 0.07).
Pulse propagation speed vs. laser power
Pulse propagation speed
depends on:
• particle number density
(i.e. • excitation laser power
indication of
nonlinear effect
10
15
20
25
0.5 1 1.5 2 2.5 3
C (
mm
/s)
Laser power (W)
1<
<
1
<
<
1<
<1
<
<
Deviation from proportionality v n/nis further evidence of nonlinearity
Pulse amplitude (particle speed) vs. Pulse amplitude (number density)
0
0.5
1
1.5
2
0 0.05 0.1 0.15 0.2
v (m
m/s
)
Number density dn/n
insideexcitation region
outside exc. region
0.66 W
2.38 W
0 1 2 3 40
10
20
30
40
k (mm-1)
Closed symbol: krOpen symbol: ki
small
large
middle
0 1 2 3 40
5
10
15
20
k (mm-1)
Closed symbol: krOpen symbol: ki
small large
middle
Dispersion relations: dependence
Experiment: S.Nunomura et al.
Theory: Wang et al. PRL 2001
Longitudinal wave Transverse wave