ELECTROSTATICPARTICLEBASEDMODELINGANDSIMULATIONOFULTRA
COLDPLASMA
By
MayurJain
ATHESIS
Submittedto
MichiganStateUniversity
inpartialentoftherequirements
forthedegreeof
ElectricalEngineering-MasterofScience
2015
ABSTRACT
ELECTROSTATICPARTICLEBASEDMODELINGANDSIMULATIONOF
ULTRACOLDPLASMA
By
MayurJain
Wemodelmoderatelycoupledultracoldplasmabasedonexperimentalsetupsandin-
vestigatetheofexternalelectricandmagneticbysimulatingtheinteraction
ofthisplasmawithconstantmagneticandradiofrequencyelectricintheformof
continuousapplicationandshortpulses.Adensitydependentresonantresponseisobserved
throughthesesimulationsandweinferthecausetoberapidenergytransfertoindividual
electronsfromelectricthroughthecollectivemotionoftheelectroncloudratherthan
acollisionbasedmechanismsincecollisionaltimescalesarefoundtobelargerthanthe
responseperiod.ItisalsoobservedthatelectronevaporationncestheUCPexpan-
sionbyreducingtheelectrontemperaturetly.Theseargumentsarecorroborated
byexperimentalresults.Wereportdiagnosticssuchastemperature,potentialanddensity
evolution,electronandionpaircorrelationfunctions,andestimatethesizeoftheUCPwith
varyinginitialionizationenergiesfortheultracoldplasmathroughoutcompletesimulation.
ACKNOWLEDGMENTS
ThisworkwouldnothavebeenpossiblewithoutmyadvisorsDr.JohnVerboncoeurandDr.
AndrewChristliebwhohavebeenexcellentmentorsandgrantedmethisgreatopportunity
toworkonanexcitingresearchproject.Iamimmenselygratefultobothofthemfortheir
guidancethroughoutthecompletionofmywork.Iwouldalsoliketothankmycommittee
memberDr.PremChahalforhisvaluablesuggestionsandcommentsonmywork.
Iwouldliketoexpressmyappreciationtomyfellowgraduatestudents,GauthamDharu-
manforhisintellectualdiscussionsandconstantsupportandGuyParseyforallthetechnical
helphehasome.Ithankthemandeverybodyinthegroupforcreatingaveryhealthy
researchenvironment.
Finally,Iwouldliketothankmyfamilyforbelievinginmeandencouragingmewithall
theirloveandmotivation.
iii
TABLEOFCONTENTS
LISTOFFIGURES
...................................
v
Chapter1Introduction
...............................
1
Chapter2BoundaryIntegralTreecode
.....................
3
2.1Treecodealgorithm................................5
2.2Point-clusterinteractions.............................6
2.3Regularizingthekernel..............................10
Chapter3SimulationParameters
.........................
11
3.1Initialconditions.................................11
3.2EarlyevolutionoftheUCP...........................12
Chapter4Results
...................................
15
4.1ContinuousRFresponse.............................17
4.2ResponsewithRFpulses.............................19
4.3Diagnostics....................................20
Chapter5ConclusionAndFutureWork
....................
27
BIBLIOGRAPHY
....................................
28
iv
LISTOFFIGURES
Figure2.1:Structureofasampleparentquadtreewithchildrennodes.Empty
quadrantsarepossible..........................6
Figure2.2:Illustrationofnodetraversalforclustermonopoleapproximation...7
Figure2.3:Relativeerroronpotentialvaryingwithmultipoleorderforrandomly
distributedparticlesinsideacube...................9
Figure2.4:Timingcomparisonfortreecodewithvaryingmultipoleordervs.di-
rectsummationforrandomlydistributedparticlesinsideacube..9
Figure3.1:Initialplasmadistribution.......................12
Figure3.2:Scaledtemperaturevsscaledtimedisplayingkineticenergyoscillation12
Figure3.3:Electronpaircorrelationfunctionforearlyevolutioncorresponding
to
'
1.................................13
Figure3.4:Ionpaircorrelationfunctionforearlyevolutioncorrespondingto
'
113
Figure3.5:SystemenergyinKforcompletesimulationdisplayingCoulombpo-
tentialenergyisgreaterthankineticenergyandmingconser-
vationofthetotalenergy........................14
Figure4.1:Typicalelectronescapesignalfromtheultracoldneutralplasma
(1

10
5
electron-ionsystem)......................16
Figure4.2:Plasmaresponse
E=k
B
=200
K
,1

10
5
electron-ionsystem,
6
V
p

p
=m
)withcontinuousRFapplicationof17
MHz
........17
Figure4.3:Theplasmaresponse
E=k
B
=200
K
,1

10
5
electron-ionsystem,
6
V
p

p
=m
)withRFturnedonat5

s(bottom)atanappliedfre-
quencyof
˘
17
MHz
..........................18
Figure4.4:UCPelectroncloudresponse
E=k
B
=200
K
,5

10
5
electron-ion
system,8
V
p

p
=m
)withasingleRFpulse20
MHz
..........19
v
Figure4.5:Plasmaresponsetoa8
V
p

p
=m;
20
MHz
,2cycleRFsignalapplied
tothe5

10
5
electron-ionsystem,
E=k
B
=200
K
,showingapeak
soonaftertheappliedpulse.......................20
Figure4.6:Desnityevolutionindicatingexpansionoftheplasmafor
N
=5

10
5
electron-ionUCPsystem........................21
Figure4.7:TemperatureevolutionoftheUCPfordensitydependentresonant
response.................................22
Figure4.8:PotentialatthecenterofthecubefortheUCPdensitydependent
resonantresponse............................22
Figure4.9:CoulombcollisionfrequencyintheUCPovertimeforthesimulation23
Figure4.10:RMSsizeoftheUCPfortimeofresonanceforerentfrequencies
ofcontinuousRF(

)andtwocycleRF(

)..............24
Figure4.11:UCPresonancetimevariationwithamplitudeofRF......25
Figure4.12:Electronpaircorrelationfunctionattimeofresonantresponse...26
Figure4.13:Ionpaircorrelationfunctionattimeofresonantresponse......26
vi
Chapter1
Introduction
Ultracoldneutralplasmawereproducedbyrapidlyphoto-ionizingsmalllasercooledclouds
ofatoms.Thesenovelplasmapresentinterestingtheoreticalchallenges.Thisworkfocuseson
thedevelopmentofnewmodelingandsimulationtoolsforstudyingstronglycoupledplasma
astheyfromtraditionalplasmainthatthepotentialenergyislargerthanthekinetic
energy.Theseincludedustyionosphericplasma,plasmafromultra-fastlasers/materials
interactions,plasmageneratedbyconventionalexplosives,molecularplasmaforchemical
investigation,andevenexoticwarmdensematterfoundinhighenergydensityandastro-
physicalcontexts.Forexample,considerthecaseofdustintheionosphere.Intheiono-
sphere,dustparticlesaretypically100to1000timesmoremassivethanthetypicalions.
Thelightmobileelectronsimpactthenon-conductingdust,whichchargesupnegatively.
Onaverage,thedielectricdustcarriesordersofmagnitudemorechargethanthechargeof
abackgroundion.Withahighenoughdensityofdust,theheavyimmobiledustformsa
latticeofstronglyinteractingboundchargethathastheofmodifyingthepermittivity
oftheplasma.Astandardquasineutralplasmaapproximationisinadequateinthiscase.
Inadditiontothepossibilityofquantumthestandardquasineutralplasmamodel
doesnotaccountfortwomajorchangeinthepermittivityformodelingEMwaves
andimpactonrelaxationofchargedparticlesundergoingCoulombcollisionsinasystem
withweaklyshieldedlongrangeinteractions.Theuniqueaspectofstronglycoupledplasma
(SCP)isthatthepotentialenergyexceedsthekineticenergy.Strongcouplingisin
1
termsofthedimensionlessparameter,oftenreferredtoastheCoulombcouplingparameter,
=
(
Ze
)
2
kTa
(1.1)
whichistheratioofthepotentialenergytokineticenergy.Intheaboveexpression;
Z
is
thechargenumberoftheionspecies,
e
istheunitcharge,
k
isBoltzmann
'
sconstant,
T
isthetemperatureofthespeciesinKelvin,
a
=[3
=
(4
ˇn
)]
1
=
3
istheWigner-Seitzradius
(meaninter-particledistance),and
n
isthedensityofthespecies.Couplingparameter
elycorrelation.When
<<
1,thechargedspeciesintheplasmahasno
longrangecorrelationandbinarycollisionscharacterizeCoulombscatteringforthatspecies.
For
˘
1,theplasmaspeciesinquestionbeginstoexhibitlongrangecorrelation.As
increasesinthesesystems,theplasmaexhibitsacollectivebehavior,givingthesystem
propertiesresemblingliquidsandsolids.Collectiveoscillationsareafundamentalfeatureof
ultracoldplasma(UCP)andhelpdetermineit'sresponsetoanexternalperturbation.Fora
uniformdensityplasma,ifthethermalmotionofelectronsisignored,theplasmafrequency
isexpressedas
!
p
=
p
e
2
n
e
=m
e

0
,where
e
istheelectriccharge,
m
e
istheemass
oftheelectron,
n
e
isthechargedensityand

0
isthefreespacepermittivity.SinceUCP
cannotbeconsideredtohaveuniformdensities,theresonantfrequencyconditionisnot
applicabledirectly,however,it'sresonantresponsetoRFisasubjectofexperimental
andtheoreticalwork.UCPexpansionratescouldbemeasuredthroughapplicationofan
externalRFeldsinceplasmaoscillationsaredensitydependent.Here,weexplorethis
densitydependentresonantresponsebyapplyingRFcontinuouslyandinshortbursts.
Ourfocusinthisworkisonthecollectivemotionoftheelectronsintheseresonantresponses.
2
Chapter2
BoundaryIntegralTreecode
Thegrid-freeLagrangianapproachstartsbycastingPoisson
'
sequationinintegralform,
˚
(
y
)=
Z

G
(
x
j
y
)
ˆ
(
x
)
"
0
d
3
x
+
I
@


(
˚
(
x
)
5
x
G
(
x
j
y
)

G
(
x
j
y
)
5
x
˚
(
x
))

:
b
n
dS
x
(2.1)
where
y
2

n
@
and
G
(
x
j
y
)isthefreespaceGreen
'
sfunction.Notethatthevolume
integralistheparticularsolution,
˚
P
(
y
),ofPoisson
'
sequationandtheboundaryintegralis
thehomogeneoussolution,
˚
H
(
y
),i.e.,
˚
(
y
)=
˚
P
(
y
)+
˚
H
(
y
).Dependingontheboundary
conditions,
˚
H
(
y
)canbeeithermodeledas
˚
H
=
I
@


(
x
)
G
(
x
j
z
)
dS
x
or˚
H
=
I
@


(
x
)
@
n
G
(
x
j
z
)
dS
x
(2.2)
whichisasinglelayerordoublelayerpotentialrespectivelyand

(
x
)and

(
x
)arethe
unknowndipolestrengthsthatcanbedeterminedbysolvinganintegralequationforthe
Poisson'sequationontheboundaryorsurface.Imposingconsistency,andappropriately
handlingthesingularityof
@
n
G
ontheboundary,weusetheaboveboundaryintegralfor
˚
H
tonumericallycorrecttheparticularsolution.TheVlasovequationwritteninaLa-
grangianreferenceframetransformsthemodelintoanevolutionequationwhichdescribes
3
thedynamicsofphasespacecontours,
_
x
k
=
v
k
_
v
k
=
5
˚
P
(
x
k
)+
5
˚
H
(
x
k
)
˚
P
(
x
k
)=
X
l

q
l

0
(
ZZ

l
t
G
(
x
l
j
x
k
)
f
l
(
t;
x
l
;
v
l
)
d
x
l
d
v
l
)
where(
x
k
;
v
k
)isthewmapforthephasespacecontoursofspecies
k
,thesumisover
l
speciesand
l
t
isthephasespacevolumeattime
t
.Eachpointinthewisafunctionof
itsinitialphasespacepoint,
x
k
=
x
k
(
t;
x
k
o
;
v
k
o
)and
v
k
=
v
k
(
t;
x
k
o
;
v
k
o
).Sincethesolutions
ofthissystemarevolumepreserving,itiseasytoshowthatunderachangeofvariables,

t
!

t
0
,where
t
0
istheinitialphasespacevolume,thefollowingholds,
f
l
(
t;
x
l
(
t;
x
l
o
;
v
l
o
)
;
x
l
(
t;
x
l
o
;
v
l
o
))
j
J
(
x
l
;
v
l
)
j
=
f
l
(
t;
x
l
o
;
v
l
o
)(2.3)
where
J
(
x
l
;
v
l
)istheJacobian.Giventhisidentity,theLagrangianwmapmaybewritten
as,
_
x
k
(
t;
x
k
o
;
v
k
o
)=
v
k
(
t;
x
k
o
;
v
k
o
)
_
v
k
(
t;
x
k
o
;
v
k
o
)=
5
˚
P
(
x
k
(
t;
x
k
o
;
v
k
o
))+
˚
H
(
x
k
(
t;
x
k
o
;
v
k
o
))(2.4)
˚
P
(
x
k
(
t;
x
k
o
;
v
k
o
))=
X
l
q
l

0
ZZ

l
t
0
G
(
x
l
(
t;
x
l
o
;
v
l
o
)
j
(
x
k
(
t;
x
k
o
;
v
k
o
))
f
l
(
t
0
;
x
l
o
;
v
l
o
)
d
x
l
0
d
v
l
0
(2.5)
Choosing
N
=
N
+
+
N

collocationpointsandapplyingsystematiccollocationtoEq.(2.5)
givesrisetoasystemof
N
coupledODEwhichdescribethediscretewmap.Eq.(2.5)
4
takestheform,
˚
P
(
y
)=

N
+
X
j
=1
q
+
w
j

0
G
(
x
j
j
y
)+
N

X
k
=1
q

w
k

0
G
(
x
k
j
y
)(2.6)
where
w
j;k
arequadratureweightsdeterminedat
t
0
2.1Treecodealgorithm
EvaluatingthesuminEq.(2.6)isanN-bodyproblemandtheCPUtimeisanimportantissue.
Atreecodealgorithmisemployedtoreducetheoperationcountfrom
O
(
N
2
)to
O
(
NlogN
).
Inthisalgorithm,theparticlesaredividedintoahierarchyofclustersandtheparticle-particle
interactionsarereplacedbyparticle-clusterinteractionswhichareevaluatedusingmultipole
expansions.TheBarnes-Huttreecodecleverlygroupsnearbybodiesandrecursivelydivides
setsofbodiesstoringthemintrees(Fig.2.1).Thetopmostnoderepresentsthewhole
spacewhilethechildrenformquadrantsofspace.Particlesarespatiallydividedbasedon
theirphysicallocations.Eachexternalnoderepresentsasinglebodywhileeachinternal
noderepresentsthegroupofbodiesbeneathit,andstoresthecenter-of-massandthetotal
massofallitschildrenbodies.BarnesandHutusedmonopoleapproximationsandadivide-
and-conquerevaluationstrategy.Treecodealgorithmshavebeenverysuccessfulinparticle
simulationsandthereisongoinginterestinoptimizingtheirperformance.
5
Figure2.1:Structureofasampleparentquadtreewithchildrennodes.Emptyquadrants
arepossible.
Fig.adaptedfromTheBarnes-HutAlgorithm,TomVentimigliaandKevinWayne
2.2Point-clusterinteractions
Thepotential
˚
P
(
y
)isexpressedas
˚
P
(
y
)=
X
C
r
X
j
2
C
r
w
j
G
(
x
j
j
y
)=
X
C
r
˚
i
(
y
;C
r
)
˚
i
(
y
;C
r
)=
X
j
2
C
r
w
j
G
(
x
j
j
y
)(2.7)
where
C
r
=

x
j
j
x
j
2
C
r
and
x
j
62f[
s
C
s
n
C
r
g

denotesaclusterofparticlesand
˚
i
(
y
;C
r
)
isthepotentialatpoint
y
duetocluster
C
r
.Theprocedureforchoosingtheclusterswill
beexplainedbelow;fornowitisenoughtoassumeclustersinEq.(2.7)arenon-overlapping
andtheirunionisthewholedistribution.Tocalculatethenetforceonaparticularbody,
traversethenodesofthetree,startingfromtheroot(Fig.2.2).Ifthecenter-of-massof
aninternalnodeistlyfarfromthebody,approximatethebodiescontainedinthat
partofthetreeasasinglebody,whosepositionisthegroup
'
scenterofmassandwhose
massisthegroup
'
stotalmass.Thealgorithmisfastbecausewedon
'
tneedtoindividually
6
Figure2.2:Illustrationofnodetraversalforclustermonopoleapproximation.
Fig.adaptedfromTheBarnes-HutAlgorithm,TomVentimigliaandKevinWayne
examineanyofthebodiesinthegroup.Iftheinternalnodeisnottlyfarfromthe
body,recursivelytraverseeachofitssubtrees.Todetermineifanodeistlyfaraway,
computethequotient
s=d
,where
s
isthewidthoftheregionrepresentedbytheinternal
node,and
d
isthedistancebetweenthebodyandthenode
'
scenter-of-mass.Then,compare
thisratioagainstathresholdvalue

.If
s=d<
,thentheinternalnodeissutlyfar
away.Byadjustingthe

parameter,wecanchangethespeedandaccuracyofthesimulation.
Wealwaysuse

=0
:
5,avaluecommonlyusedinpractice.Notethatif

=0,thenno
internalnodeistreatedasasinglebody,andthealgorithmdegeneratestobruteforce.
7
PerformingaTaylorexpansionoftheGreen
'
sfunctionabouttheclustercenter
x
cr
,
˚
i
(
y
;C
r
)
ˇ
X
j
2
C
r
p
X
l
=0
1
l
!
@
l
y
G
(
x
cr
j
y
)

w
j
(
x
j

x
c
)
l

=
p
X
l
=0
1
l
!
@
l
x
G
(
x
cr
j
y
)
X
j
2
C
r
w
j
(
x
j

x
c
r
)
l
=
p
X
l
=0
T
l
(
x
cr
;
y
)
M
l
(
C
r
)(2.8)
where
p
istheorderofapproximation,
T
l
(
x
cr
;
y
)isthe
l
th
TaylorcotoftheGreen
'
s
function,and
M
l
(
C
)isthe
l
th
momentofthecluster.NotethatCartesianmulti-index
notationisbeingused.Thespeedupoccursbecausetheclustermomentsareindependent
ofthepoint
y
,whilethe
T
l
(
x
cr
;y
)areindependentofthenumberofparticlesin
C
r
.This
formofthefastsummationissuitedtoaregularizedkernel.The
C
r
haveahierarchicaltree
structure.Typically,
C
r
oneachlevelofthetreeareuniformcubesobtainedbybisectingthe
previousgenerationofclustersineachcoordinatedirection.Thepotential
˚
i
isevaluated
usingthetreestructureinarecursivedivide-and-conquerstrategy.
Weexaminetheerrorforthetreecodeincomparisonwithdirectsummation(Figure2.3)
fortheproblemofcomputingpotential
V
foratestcaseofrandomlydistributedparticles
insideacube.Thesetofrepresentativeparametervaluesusedforthetreecodewere

=0
:
5
forthemultipoleacceptancecriterion(MAC),
p
=1:5fororderofTaylorapproximation
andmaximumnumberofparticlesintheleaf
N
leaf
=100.
8
Figure2.3:Relativeerroronpotentialvaryingwithmultipoleorderforrandomlydistributed
particlesinsideacube
Figure2.4showstheCPUtimeasafunctionofthenumberofparticles
N
andthe
approximationoforder
p
.TheCPUtimeis
O
(
N
2
)fordirectsummationandisconsistent
with
O
(
NlogN
)forthetreecode.Forasmallernumberofparticles(
<
10
3
)and

=0
:
5,
mostparticle-clusterinteractionsarecomputedbydirectsummationandsincerelatively
veryfewTaylorapproximationsareevaluated,theoverheadincreases.However,witha
tlylargenumberofparticles
N
,thetreecodeoutperformsdirectsummation.
Figure2.4:Timingcomparisonfortreecodewithvaryingmultipoleordervs.directsumma-
tionforrandomlydistributedparticlesinsideacube
9
2.3Regularizingthekernel
Itisimportanttonotethatin2Dand3D,theelectrostaticforce,
F
j
(thegradientofEq.
(2.1)),becomessingularasthedistancebetweentheparticlestendstozero.Whenwedis-
cretizetheLagrangianformoftheVlasovequation,timesteppingparticlesmaycausetwo
particlestoapproachcloserthantheirminimumseparationinthecontinuouscase.Accu-
racyconstraintsimposedtoavoidthisissuecanplaceasevererestrictiononthemaximum
allowabletimestep.OurapproachtoovercomingthisproblemistoregularizetheGreen
'
s
function,i.e.inthreedimensionsuse
G
d
3
D
(
x
j
y
)=

1
4
ˇ
(
k
x

y
k
2
2
+
d
2
)
where
d
isaparameter,sothatthemaximalforcein3Disproportionalto1
=d
2
.In1D,this
issueariseswhentwotestparticlescross,sincetheforcein1Dhasadiscontinuity.Hence,
toachievehighorderwithexplicittimestepping,eventhe1DGreen
'
sfunctionmustbe
regularized.
10
Chapter3
SimulationParameters
Thenumberofparametersthatoursimulationsisreducedtoaminimumbyusing
lengthscaledbytheWigner-Seitzradius,
a
andtime,
!

1
p
.Withthis,theequationsof
motionandinitialconditionsarespby:massratio
m
i
=m
e
,electrondensity
n
e
,
couplingparameter
e
andtheCoulombpotentialregularizationparameter

.
3.1Initialconditions
Themassratio
m
i
=m
e
istakenas100toensureionshavetimetoparticipateinthesimulation
dynamics.Theinitialnumberofelectronsandionsinoursystemisontheorderof
˘
10
5
,
andwearrangetheboundaryconditionstostartwithauniformsphericalGaussianelectron-
iondensitydistributiondescribedby
n
(
r
)=
n
0
exp
(

r
2
=
2
˙
2
)(Fig.3.1),where
n
0
isthe
peakdensityand
˙
characterizesthespatialextentofthestronglycoupledultracoldplasma.
Theseplasmatypicallyhasapeakplasmadensityof
n
i
=
n
e
=10
13

10
14
m

3
whichis
aboutanorderortwolowerthanmostUCPexperimentsand
˙
˘
1mm.Webeginwith
T
e
'
1Kand
T
i
'
10

K.Theelectron-electronselfequilibrationtimeinthiscaseismore
than1

s.Anexternalelectric(
˘
2

8V/m)isappliedtopulltheescapingelectrons
withtheassistanceofaguidingmagnetic(
˘
7

9G)whichisaxiallysymmetricwith
respecttotheelectrodes.
11
Figure3.1:Initialplasmadistribution
3.2EarlyevolutionoftheUCP
WeobtainahistogramofelectronkineticenergiesandmatchthemtoaMaxwellian.Rapid
heatinginitiallyraises
'
1.Thelongertermslowerheatingmaybeassociatedwiththree
bodyrecombination.
Figure3.2:Scaledtemperaturevsscaledtimedisplayingkineticenergyoscillation
12
Theelectron-electronpaircorrelationfunctionistimeaveragedoverabout
t!
p
=3
to
7
andthecorrelationfunctionstartsoutcorrespondingtorandomlydistributedelectrons
andrelaxesatalaterstage.
Figure3.3:Electronpaircorrelationfunctionforearlyevolutioncorrespondingto
'
1
Theion-ioncorrelationfuntionbehavesinaverysimilarfashionexceptthatitisaveraged
over
t!
p
=65
to
70astherelaxationtimeislongerthanthatfortheelectrons.Thecoupling
hereisabout
˘
1
Figure3.4:Ionpaircorrelationfunctionforearlyevolutioncorrespondingto
'
1
13
Fromthesesimulationresults,weseethatintrinsicrapidheatingpreventsdevelopment
ofastrongcorrelationevenwheninitialelectronandiontemperaturesare0
i:e;

e
(0)=

i
(0)
˘1
.Electronevaporationfromanunboundedcloud,beingacoolingmechanismalso
doesnotcompetewiththeheating.Althoughthesesimulationsonlyfollowearlyevolution
andplasmaexpansionatalaterstagecanbeastrongcoolingmechanismthatcouldreduce
thetemperatureoftheplasma.Lowinitialtemperaturesdonotdirectlyleadtostrong
correlationduringearlytimes.
Figure3.5:SystemenergyinKforcompletesimulationdisplayingCoulombpotentialenergy
isgreaterthankineticenergyandconservationofthetotalenergy
14
Chapter4
Results
Allsimulationsinthisworkareobtainedforaonetoonephysicalrepresentationofthe
chargedspecies.VelocityforthespeciesissampledfromaMaxwell-Boltzmanndistribution.
Therandomnumbersusedforoursimulationsaregeneratedwiththesrandpseudorandom
numbergeneratorwhichisseededwiththesystemtime.Timeintegratorforequationsof
motionisfourthorderRunge-Kuttamethod.Resultsshowninthissectionaresmoothedby
timeaveragingeachpointforover1000runs.
Collectiveoscillationsareoneofthefundamentalfeaturesinanultracoldplasmaand
canhelpcharacterizeit'sdensityastheyfreelyexpand.WeapplyexternalRFelectric
toexcitetheoscillationstomeasuretheUCPexpansionratealsoallowingustoinfer
earlytimetemperatureandit'ssubsequentevolution.Inthissection,resultsfromprevious
experimentalUCPworkarealsodiscussedasabasisforcomparison.Welookatthe
evolutionofaneutralultracoldplasma.
15
Figure4.1:Typicalelectronescapesignalfromtheultracoldneutralplasma(1

10
5
electron-ionsystem)
Fig.4.1showstheplasmaevolutionandexpansionprocess.Webeginwithaninitially
neutralultracoldplasmaandthereforeweseeabunchofelectronsescapeat
t
=0dueto
theenergyofelectronsandnonettduetoabsenceofexternalThe
resultingexcesspositivechargecreatesaCoulombpotentialwellgreaterthantheaverage
electrontemperature,trappingtheremainingelectrons.Electronsmusthaveanenergy
higherthantheinitialformationenergytoescapetheUCPandelectronsnowbeginto
thermalize.Thiscreatesanenergydistributionwhereonlythehighestenergyelectronsare
abletoescapetheUCPloweringtheoveralltemperatureofelectrons.Greatertemperature
ofelectronscomparedtoionswillcausetheelectrondensitytodecreaseinspaceslightly
relativetotheions.Thisdecreaseinelectrondensityproduceselectricthat
theelectronsbutnowdrivetheionexpansion.AstheUCPsizeincreases,thepotentialwell
shallowsallowingmoreelectronstoescape.
16
4.1ContinuousRFresponse
Electronsfromtheultracoldplasmaescapeimmediatelydirectedbytheexternalelectric
sincethereisnonett,andproducethepeakinthesignalresultingin
anexcessofpositivechargeintheplasma,therebycreatingaCoulombpotentialwellwhich
trapstheremainingelectrons.HighenergyelectronsbegintoescapetheUCP.
Figure4.2:Plasmaresponse
E=k
B
=200
K
,1

10
5
electron-ionsystem,6
V
p

p
=m
)with
continuousRFapplicationof17
MHz
InthepresenceofanexternalRFinadditiontotheresponseasseeninFig.4.1,
anadditionalpeakappearsintheelectronescapesignal(Fig.4.2).TheappliedRFexcites
plasmaoscillationsattheresonantfrequencyastheUCPexpandsanddecreasesindensity.
Theamplitudeoftheseoscillationsismuchlessthan
˙
.Previousworkimpliedthattheac-
quiredenergyiscollisionallyredistributedbutthisdoesnotoccurinstantaneouslyasthetime
scaleisassociatedwiththeelectronself-equilibrationtime,whichscaleswithbothdensity
andtemperatureoftheplasmaandisapproximatelyproportionalto
T
3
=
2
e
=n
e
.Theescaping
17
electronshaveahigherenergythantheaverageelectronenergy,sotheenergytransferwould
infactbeslower.ItwasalsoassumedthattheappliedRFwasonlyresonantwithinregions
ofplasmawheredensitiessatisfytheresonancecondition
f
=(1
=
2
ˇ
)
p
e
2
n
e
=m
e

0
where
f
istheappliedRFfrequency.Weobservecollectiveoscillationstobethephenomenon.
Figure4.3:Theplasmaresponse
E=k
B
=200
K
,1

10
5
electron-ionsystem,6
V
p

p
=m
)
withRFturnedonat5

s(bottom)atanappliedfrequencyof
˘
17
MHz
WealsoapplyRFnearresonanceataparticulartimebydelayingtheapplicationuntil
afteracertainpoint(5

s)(Fig.4.3)intheplasmaevolutionandmeasurethetimedelay
associatedwiththecollisions,butthisproducesarapidresponseofelectronescapewithno
realdependenceontemperatureordensity.Theresponsetimehereismuchshorterthan
thecollisionaltimescaleimplyingatmechanismotherthancollisionalredistribution
ofenergyamongtheelectrons.Theresonantresponseisalsoshiftedtolaterintime.The
initialpeaksmaychangeinheightasafunctionoftheappliedfrequency.Thisrapidresponse
canalsobeseenbyexcitingtheplasmawithafewcyclesofRFpulses.
18
4.2ResponsewithRFpulses
WeapplyatwocyclepulsetotheUCP,generatingapeakintheresponseindicatingreso-
nanceattheparticulartimefortheappliedfrequency.ApplyingshortburstsofRFpulses
inducesadensitydependentresonantresponsewithashortdelaybetweentheapplication
andtheresponse,indicatingyetagainthatthemodelisbasedoncollectivemotionofthe
electroncloud(Fig.4.4)whichproducesinternalelectricandisthemainmechanism
forenergytransfercausingindividualelectronstoescapetheplasma.
Figure4.4:UCPelectroncloudresponse
E=k
B
=200
K
,5

10
5
electron-ionsystem,
8
V
p

p
=m
)withasingleRFpulse20
MHz
ThetwocycleresponsecanbecomparedwiththeUCPresponsewhennoRFisapplied
todeterminewhenthesignalresponseoccursrelativetotheinitialonsetofthepulse.The
peakintheresponseisobservedattheresonantfrequencyachievedbysuccessivelychanging
throughmeasurements.
19
Figure4.5:Plasmaresponsetoa8
V
p

p
=m;
20
MHz
,2cycleRFsignalappliedtothe5

10
5
electron-ionsystem,
E=k
B
=200
K
,showingapeaksoonaftertheappliedpulse
4.3Diagnostics
ResponseoftheUCPdependsstronglyonmanyparameters,densitybeingonlyoneofthem.
Thenatureoftheresponseswithcontinuousapplicationandpulsescouldbecomparedunder
similarconditions.TheofdelayintheRFapplicationcausesashiftintheresonance
time,andresponseswithcontinuousRFapplicationchangewiththeamplitudeofthe
appliedastheresonanceshiftstoanearliertimewithanincreasedamplitude,implying
heatingoftheUCPthatdrivesafasterexpansion.
WebeginoursimulationswithaUCPdensityofabout10
7
c
m

3
andasdiscussedearlier,
electronsstartescapingtheplasmaimmediatelyandtheplasmabeginstoexpand.The
appliedRFdriveselectronsoutofthesystemandthereducedelectrondensitydrivesthe
plasmaexpansionduetotheinternalelectricitproduces.Allthesefactorscontributeto
20
areducingUCPdensityasshownbelowin4.6.Theaveragedensitycanbedescribed
by

n
=
N=
[4
ˇ
(
˙
2
0
+
v
2
0
t
2
)]
3
=
2
(4.1)
where
˙
0
istheinitialrmsradius,
v
0
isthermsradialvelocitythatcouldequateto
v
0
=
(
k
B
T
e
=m
i
)
1
=
2
.TheUCPexpansioncanberelatedtotheelectrontemperatureas
˙
(
t
)
2
=
˙
(0)
2
+(
v
0
t
)
2
.
n
isthedensityinresonancewiththeRFassumedtobeequaltothe
averagedensityinplasma.
Figure4.6:Desnityevolutionindicatingexpansionoftheplasmafor
N
=5

10
5
electron-ion
UCPsystem
ThetemperatureoftheUCP(Fig.4.7),afteraninitalrise,dropssteadilywithoscillatory
behaviorattheappliedRF.TheRFuponapplicationinitiallyheatsuptheelectrons
toabout0
:
03eVandaselectronsbeginescapingthesystem,thetemperaturequicklydrops
astheelectronswithhighestkineticenergyleavethesystemandtheslowerelectrons
displayacollectiveoscillationatthefrequencyoftheRFFinaltemperatureofour
UCPsystemapproachesaconsiderablylowvaluewhichispromisingfromtheperspective
ofultracoldplasma.
21
Figure4.7:TemperatureevolutionoftheUCPfordensitydependentresonantresponse
ThepotentialatthecenteroftheUCPsystem(Fig.4.8)isthehighestinitiallyand
beginstodropastheelectronsstartescapingtheUCP.Thenowreducedelectrondensity
furtherdrivestheUCPexpansionandthepotentialcontinuestodropwithanoscillatory
behavioratplasmafrequencywhichisdependentonthedensity.
Figure4.8:PotentialatthecenterofthecubefortheUCPdensitydependentresonant
response
Itisveryimportanttoconsidercollisionprocessesinnon-equilibriumplasmadynamics.
Uponformation,theelectronsandionsarenotinthermalequilibriumwitheachotherand
astheUCPstartsevolvingtheparticleswillworktowardsestablishingaquasi-equilibrium.
22
Themostimportanttimescalesforoursystemwouldinvolveelectron-electroninteractions
orelectron-ioninteractions.Forelectron-electroninteractions,thedeterminationoftimefor
aparticlewith3
=
2
k
B
T
ofkineticenergytoundergothesameamountofenergychanging
collisionsistheelectronself-equilibrationtimegivenby,
t
se
=
0
:
266
T
3
=
2
e
n
e
ln

(4.2)
where
n
e
isin
cm

3
and
T
e
isin
K
.=12
ˇn
e

3
D
where

D
istheDebyescreeninglength.
Forelectron-ioncollisions,wecanthetimeitwouldtakeforanet90degree
fromanelectron
'
soriginaltrajectoryas
t
90
=
2
ˇ
2
0
p
me
(3
K
B
T
)
3
=
2
10
6
n
e
e
4
ln

(4.3)
Theelectron-ioncollisiontimescalealsofallsontheorderofelectron-electroncollisions.
Figure4.9:CoulombcollisionfrequencyintheUCPovertimeforthesimulation
23
ForourUCPsystems,thesetimescalesrangefromtensofnanosecondstoafewmicrosec-
ondsaccordingtothecollisionfrequencycalculatedusingtheaveragedensityandestimated
temperature,whichwillalsoallowustostudythephysicsoftheseultracoldplasmasina
regimewherecollisionsareimportantandinaregimewherewecantreattheUCPasacolli-
sionlessHowever,UCPshaveanon-uniformdensityandstartwithauniformelectron
energydistribution.Thiscomplicatestheexactmeaningofthesecollisiontimescales,and
areusedasonlyestimates.
AdirectcomparisonbetweencontinuousRFandthetwocycleRFmethodcannotbe
madewithoutaccountingforheatingoftheUCPfromtheappliedRF,whichcausesashift
intheresonancetime.Byaccountingforthechargeimbalance

attheextrapolatedtime
ofresonance,wecandeterminethevalueof
!
peak
whichcangiveusameasureofthepeak
density
n
peak
.Fromthisvalueofpeakdensityandthetotalnumberofionsandelectrons
inourUCP
N
ion
,wecancalculatetheRMSsize
˙
ofasphericallysymmetricgaussian
distributiongivenby
˙
=[
N
ion
=
(2
ˇ
)
3
=
2
n
peak
]
1
=
3
.Thisrmssizecanbeusedforanestimate
ofexpansionoftheUCP.
Figure4.10:RMSsizeoftheUCPfortimeofresonancefortfrequenciesofcontinuous
RF(

)andtwocycleRF(

)
24
WeplotthemeasuredRMSsizevaluesusingboththetwocycleandcontinuousRF(Fig.
4.10)andthereisnotcebetweenthetwotechniquesforvaluesof
E=k
B
rangingfrom100Kto400K.Wealsomeasuretheobservedtimeofresonanceasafunction
oftheamplitudeoftheappliedRF.
Figure4.11:UCPresonancetimevariationwithamplitudeofRF
FromtheaboveFig.4.11,itcanbeobservedthatforlower
E=k
B
thepeaktimeas
afunctionoftheappliedRFamplitudeisnotlinear.Theextrapolationforhigherenergies
however,islinearallowingustomakecomparisonswithtwocyclemethod.
Weobservethattheelectronandionpaircorrelations(Fig.4.12andFig.4.13)aremuch
weakercomparedtothecorrelationfunctionsobtainedduringearlyevolutionoftheultra
coldneutralplasma.
25
Figure4.12:Electronpaircorrelationfunctionattimeofresonantresponse
Figure4.13:Ionpaircorrelationfunctionattimeofresonantresponse
Thisimpliesthatthecoupling,particularlyintheelectroncomponentoftheplasmahas
tlyweakenedtly,primarilyduetotheinternalelectricproduced
bytheremainingchargedensities,thusnotallowingstronginteractionsbetweenthetwo
chargedspecies.
26
Chapter5
ConclusionAndFutureWork
ThenatureofresponseoftheUCPtoappliedRF(continuous,delayedandshort
pulses)displayanearlycollisionlessmechanismforenergytransferwithinourlowdensity
UCPsetup.TheappliedRFwasshowntoexciteacollectiveoscillationoftheelectroncloud
intheUCPwitharesonantfrequency.Thiscollectiveresponsetotheexternalshow
thatthisfrequencycanbedeterminedbythepeakdensityandchargeimbalanceinthe
plasmasystem.Themethodsusedinthesesimulationsallowedustocharacterizedt
propertiesoftheUCP.Weachieveconsiderablequalitativeagreementbetweenourmodel
withexistingexperimentsforgiveninitialconditionsalthoughweobservethecorrelationin
theUCPweakensatthetimeofthedensitydependentresonantresponsedenotingweaker
couplingwhichmaybecausedbytheinternalelectricresultinginweakerinteractions
andpossiblethreebodyrecombination.Weaimtoexplorefundamentalphysicsofour
systemincollaborationwithexperimentalistsandsettingupahighercorrelatedgas.One
ofourgoalsissettingupaFermigasforhighercorrelationsasopposedtotheRydberggas
whereahigherdensityofatomsmightbelesslikely.WecouldalsouseanadaptiveYukawa
treecodemodelforcomparisonofYukawaatomswithionelectronatomsandwouldliketo
setupourmodelsasavirtuallaboratorytohelpdesignexperimentsetups.
27
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