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