Ion Temperature Evolution in an Ultracold Neutral Plasma

0.38 MB·English

by  P. McQuillen

Ion Temperature Evolution in an Ultracold Neutral Plasma
P. McQuillen,1,a) T. Strickler,1 T. Langin,1 and T. C. Killian1
Rice University, Department of Physics and Astronomy, Houston, Texas 77005
(Dated: 29 January 2015)
We study the long-time evolution of the ion temperature in an expanding ultracold neutral plasma using
spatially resolved,laser-induced-fluorescencespectroscopy. Adiabatic coolingreduces the iontemperature by
an order of magnitude during the plasma expansion, to temperatures as low as 0.2K. Cooling is limited by
heatexchangebetweenionsandthemuchhotterelectrons. Wealsopresentevidenceforanadditionalheating
5
mechanismanddiscusspossiblesources. Dataaredescribedbyamodeloftheplasmaevolution,includingthe
1
effects of ion-electronheat exchange. We show that for appropriateinitial conditions, the degree of Coulomb
0
2 coupling of ions in the plasma increases during expansion.

n
a
I. INTRODUCTION significantcontributionstotheiontemperaturefromion-
J
electronthermalization,aswellasanadditionalsourceof
7
2 Ultracold neutral plasmas (UNPs), formed by pho- ionheatingweascribetodeviationsoftheplasmadensity
toionizing laser-cooled atoms near threshold, have ion from an ideal spherical Gaussian.

] temperatures around 1K and tunable electron temper-
h Thisstudyaddstopriorstudiesoffundamentalplasma
atures from 1K to 1000K.1 One of the most important
p properties conducted using UNPs, such as the plasma
topicsstudiedinUNPsisthephysicsofstronglycoupled
- creationprocess,3,4,13,14 collectivemodes ofions16,17 and
m plasmas,whichhavearatioofnearestneighborpotential
electrons,13,18,19 and formation and ionization of Ryd-
s energy to kinetic energy larger than one. This is quanti- berg atoms in the plasma.20–22
a fied by the Coulomb coupling parameter
l
p There is long-standing interest in the problem of a
e2
s. Γ= /(kBT), (1) plasma expanding into vacuum, which typically dom-
4πε a
c 0 ws inates the dynamics of plasmas created with pulsed
si where a (4πn/3)−1/3 is the average interparticle lasers,23suchasinexperimentspursuinginertialconfine-
y ws ≡ ment fusion,24 x-ray lasers,25 or the production of ener-
spacing (i.e. Wigner-Seitz radius) and T is the temper-
h getic(>MeV)ionsthroughirradiationofsolids,26,27 thin
p ature of the particle species of interest. Ions in UNPs foils,28 and clusters.29
[ have Γ 2 4, and they have been used to study the
i
≈ −
1 equilibrationofstronglycoupledCoulombsystemsaftera For ultracold neutral plasmas, an early hydrodynamic
v potentialquench2–8 andcollisionratesbeyondtheregime model and numerical study, which included the ef-
4 of validity of Landau-Spitzer theory.9 Strong coupling fects of three-body recombination and other inelastic
4 is of interest in many different plasma environments,10 processes,30 accurately described experimental observa-
9 andtheequilibrationdynamicsobservedinUNPsinpar- tions of the ion density evolution.13 An adiabatic, self-
6 ticular has been connected to laser-produced plasmas similar solution of the Vlasov equations15,31–34 describes
0 formedfromclustersandsolids11 andproposedasapos- theexpansiondynamicswellacrossawiderangeofinitial
.
1 siblelimitationinthebrightnessofphotoemittedelectron conditions,and this is the starting point for understand-
0 beams.12 ing the experiments described here. This model predicts
5 A central focus of research on UNPs is the creation adiabatic cooling of electrons, which has been confirmed
1
of more strongly coupled plasmas to study collisions, throughtheexpansiondynamics,15,35electronloss,36and
:
v transport, collective modes, and many other phenomena the TBR rate.37 The model predicts adiabatic cooling of
Xi across a wider variety of coupling strengths into the liq- the ions as well, and it was pointed out that this should
ar puliadsmreagimafete.rPitrsopcoresaatlsiotnoairnecrseeansseitΓiviebtyo mthoedilfoynign-gtetrhme trheseudlteninsitaynanindcrceoallsiesioonf craotreredlartoiponssotmouΓchi t≈ha1t0cobrerfeolrae-
evolution of the ion temperature. Here, we report mea- tions freeze out.38,39 Small effects of ion correlations on
surementandmodellingoftheion-temperatureevolution the expansionhave been discussedin conjunction with a
ofexpandingUNPs,whichhasneverbeenexperimentally comprehensive model.38 Our work provides the first test
studied before because of the difficulty in distinguishing of these predictions of the evolution of the ion tempera-
the small thermal motions of ions from the large, hydro- ture.
dynamic ion expansion velocity.13–15 Our results show
significant ion adiabatic cooling, with ion temperatures Adiabatic cooling has been observed in trapped non-
decreasinguptoanorderofmagnitude. Wealsoobserve neutralplasmaswhenreducingconfinementforbothpure
ion40 and antiproton plasmas.41,42 It is important in the
interstellar medium and solar wind43–45 and it has been
discussedforionsinaplasmacreatedbylaserirradiation
a)[email protected] of a solid.46,47
2
II. EXPERIMENTAL METHODS
electrons ions hydrodynamic
UNPs are created by photoionizing laser-cooled 88Sr LTE/GTE LTE phenomena
iactaolmGsainusasimanagdneentsoi-toypdtiicsatrlitbruatpi.o1nT(hinisitrieasluelt−s1i/n2adsepnhseitry- ωp−,e1 ≈10ns ωp−,i1 ≈1µs τexp≥5µs
radius σ0 1mm to 2mm) of a few hundred mil-
lion ion≡s (and∼electrons) with a peak density of up to 1e−9 1e−8 1e−7 1e−6 1e−5 1e−4
5 1016m−3. time since t0 (s)
×
Spatially and temporally resolved laser induced fluo-
rescencespectroscopyonthe ion’sprincipaltransitionat FIG. 1. Three main stages of UNP evolution. Electrons
422nmpermitsin situprobingofthelocalkineticenergy reach global thermal equilibrium (GTE) on a timescale ap-
and density of the ions.15,48 We excite fluorescence in a proximately equal to the inverse electron plasma frequency,
1mm thick sheet that bisects the plasma. Fluorescence ωp−e1 =pε0me/nee2. Ions come to local thermal equilibrium
(LTE) on a timescale of the inverse ion plasma frequency.
emitted perpendicular to the sheet is collected by a 1:1
Hydrodynamiceffectssuchasplasmaexpansionoccuronthe
opticalrelayand4 objective/ocularmagnificationstage
× longest timescale.
that images onto an intensified charge coupled device
(ICCD), with a pixel size of 13µm. This allows regional
analysisofsmallvolumesoftheplasmawithroughlycon-
stant density and bulk expansionvelocity.48,49 The total on a very fast timescale compared to the physics of in-
system resolution is mainly limited by the ICCD, which terest in this paper at a temperature determined by1,30
suffers from blooming during the amplification process.
To minimize this effect, we experimentally obtained the 3k T
pointspreadfunctionoftheICCD50 anddeconvolveeach B e =~ωl Ei. (2)
2 −
image in post analysis.51
Creation and probing of the plasma is repeated with
Here, ~ω is the combined energy of the photoionizing
a 10Hz repetition rate, and the LIF-laser frequency is l
photonsandE istheionizationenergyofagroundstate
scanned to build up an excitation spectrum. The spec- i
atom. We can create UNPs with initial electrontemper-
trum of the region of interest is fit to a Voigt profile.
atures of 1K to 1000K, however we typically stay above
WeassumeaLorentzian-componentwidthdominatedby
40K to avoid three body recombination effects, which
thelaserandnaturallinewidths(6MHz,20MHz,respec-
become important when Γ 0.1.35 We ionize a max-
tively) and the Doppler-broadened Gaussian width is a e ≥
fit parameter related to a local ion temperature.48 The imum of about 25% of the atoms for Te . 150K, but
this approaches 100% near an auto-ionizing resonance
signal is proportional to density and can be calibrated
using absorptionimaging,52 with a resulting uncertainty (aTreet≈rap4p3e0dKb)y.53thAelslpbaucte-achfaerwgepfieerlcdenotf tohfethioense.l5e4ctrons
in density of approximately 20%.
±
To minimize the contamination of ion temperature Next, the ions come to local thermal equilibrium on
measurements by expansionvelocity,48 temperatures are a time scale comparable to the inverse of their fun-
measured for regions that are only one pixel wide along damental oscillation frequency, ωp−i1 = ε0mi/nie2 ∼
the LIFbeamaxis and1mmperpendicular toit. Toim- 1µs. Initial ion velocities and positions are inherited
p
prove statistics, we analyze 79 regions covering a 1 mm2 from the laser-cooled atoms, resulting in low kinetic en-
area,andcomputeanaveragetemperaturefromthemea- ergy and high potential energy because the spatial dis-
surements. Errorbarsontemperaturemeasurementsare tribution is completely random. Coulomb energy is con-
statisticaluncertaintyintheresultingmean. Theaverag- vertedintoionkineticenergyinaprocesscalleddisorder-
ing over multiple regions introduces a maximum relative induced heating (DIH) or correlation heating,2,52 which
density variation of e−1/8 = 0.88 from the center region yields a density-dependent, equilibrium ion temperature
to the outermost region, for σ0 =1mm. All densities we (TDIH ∝n−1/3 ∼ 1K). Disorder-induced heating limits
quote are an average of the regional densities, 95% of the ion’s Coulomb coupling parameter to 2 . Γi . 4,
the peak density. ≥ with the variation determined by electron screening.8
Strictly speaking ions never achieve globalthermal equi-
librium in our experiments because ofthe long timescale
III. OVERVIEW OF ULTRACOLD PLASMA for thermal transport.55
EVOLUTION TIME SCALES In the next stage of evolution, the plasma expands
intothesurroundingvacuuminahydrodynamicfashion,
The plasma is unconfined and undergoes a complex driven by the electron thermal pressure. For a quasi-
evolution as it expands into the surrounding vacuum,1 neutral, perfect Gaussian density distribution, assuming
with dynamics that can be divided into three distinct globalthermalequilibriumofionsandelectronsandneg-
time scales as shown in Figure 1. ligible inelastic collision processes and electron-ion ther-
Globalthermalequilibriumfor electrons is established malization, the expansion is described by a self-similar
3
expansion plasma,is set by density ( n1/3) andelectron tempera-
ture (via electron-screenin∝g). It can be calculated from2
σ2(t)=σ2(1+t2/τ2 ), (3)
0 exp
Ti,e(t)=Ti,e(0)/(1+t2/τe2xp), (4) T 2 e2 U˜ + κ (5)
DIH
≡ 3k 4πε a 2
B 0 ws
where the expansion timescale, τexp = (cid:12) (cid:12)
(cid:12) (cid:12)
miσ02/kB[Te(0)+Ti(0)] ≈ 10µs, is set by initial using tabulated molecular dynamics(cid:12) simula(cid:12)tion results56
size and temperatures after electrons and ions reach for U˜, the excess particle energy in units of e2/4πε a .
p 0 ws
global and local equilibrium respectively.15,32 The κ a /λ is the screening parameter for electron De-
ws D
≡
electron temperature dominates, which allows one to bye length λ = (ε k T /n e2)1/2. A closed-form de-
D 0 B e e
indirectly measure Te(0) by fitting the evolution of scription of KEO dynamics does not exist, and it can
the cloud size. Equations 3 and 4 also show that only be predicted with molecular dynamics simulations.
Te,iσ2 = constant, reflecting the adiabatic cooling of The oscillation frequency has been shown by particle-
both species as the cloud expands. We will discuss in-cell Yukawa simulations,7 tree-code algorithms12 and
below how these equations must be modified to account full molecular dynamics simulations57 to agree well with
for heat exchange between electrons and ions, but they ω , although electron screening softens the ion-ion in-
pi
provide good intuition for plasma dynamics. teraction and slows the oscillations.7,58 This deviation is
small but observable for our conditions. As seen in Fig.
(a) 2 (b) 1.2 2(b),whentimeisscaledby2π/ωpi,andiontemperature
isscaledbyT ,the variousplasmaexhibitsimilarbe-
DIH
1 havior, and oscillations damp to a scaled temperature
1.5
close to unity.
0.8
K) IH All presented temperatures are fit parameters assum-
T(i 1 /TD 0.6 ing a Voigtspectralprofilewith Maxwellianvelocity dis-
Ti tribution. Ithasbeenshown,7 however,thatthevelocity
0.4 4×1015m−3
distribution shows a non-Maxwellian, high-velocity tail
0.5 2×1015m−3 that relaxes on the same timescale as the oscillations, so
0.2
9×1014m−3 Voigtfits will underestimate the ionRMS kinetic energy
0 0 until the ions have achieved local thermal equilibrium.
0 1 2 0 0.5 1 1.5
Therefore, to interpret our early-time measurements we
time(µs) time/(2π/ω )
pi compare them to a library of ion velocity distributions
from molecular dynamics simulations of a homogeneous
FIG.2. Earlytimemeasurementsofionkineticenergywidths plasma.59 We use the numerical data to produce simu-
for various density UNPs with Te = 430K (a) Disorder in- lated LIF spectra that are fit with the same procedure
duced heating temperatures vary with density as shown. (b) as experimental data (Fig. 2 (b)). Assuming an electron
Scaling temperature by the predicted DIH magnitude and temperature givenby the photoionizing-laserphotonen-
time by the ion plasma oscillation period results in close to
ergy we can allow density to be a fit parameter, and the
universalbehavior. Thesolidlinesareresultsfrommolecular
fit is highly constrainedbecause the KEOfrequency and
dynamics simulation of an equilibrating plasma which are fit
thetemperaturebothdependonthedensity. Goodconfi-
to experimental data to determineion density.
denceinthemeasureddensityandT isimportantfor
DIH
accuratesimulationofthe fulliontemperatureevolution
asdiscussedbelow. Notethatthemeasuredtemperature
only approaches T as the oscillations damp.
IV. INITIAL ION EQUILIBRATION: DIH
Fits of experimentaldata to numericalresults arepre-
DISORDER-INDUCED HEATING, KINETIC-ENERGY
OSCILLATIONS, AND DENSITY CALIBRATION sented as insets throughout the paper. A full study of
this density-fitting method will be given elsewhere.60 It
ismoreaccuratethanusingLIFintensitycalibratedwith
The initial equilibration of the ions has been exten-
absorption imaging,48 but the two methods agree well,
sively discussed previously,2–8,11 but we describe it here
yielding an absolute uncertainty in density on order of
in order to give a complete picture of the temperature
10%.
evolution and because early-time heating dynamics pro- ±
videthe mostaccuratedeterminationofthe plasmaden-
sity.
Figure2showstheincreaseofionkineticenergydueto V. ION ADIABATIC COOLING
initial disorder (DIH) and the subsequent kinetic energy
oscillations (KEOs) at close to twice the ion plasma fre- On a hydrodynamic timescale of 10µs the plasma
∼
quencyω thatresultfromtheevolutionofspatialcorre- cloudwillexpandduetothethermalpressureoftheelec-
pi
lationsintheequilibratingstronglycoupledplasma. The trons, leading to adiabatic cooling of electrons and ions.
equilibrium temperature from DIH for a homogeneous When ion temperatures are measured for longer evolu-
4
tion times we clearly see cooling of the ions by up to an
T = 430 K T = 80 K T = 50 K
order of magnitude, as demonstrated in Figs. 3 and 4. e e e
n (0) = 4e15 n (0) = 3e15 n (0) = 2e15
Toestablishthatweareobservingadiabaticcoolingof
the ions, we check the scaling predicted by Eq. 4. Ac- (a) 2.5 (b)1.5
cording to this simple model, the only two parameters
2
determining the ion temperature are the initial temper-
1
a3t(ubr)e,ssheotwbsythDeIHev,oaluntdiotnheofexthpeaniosinontetmimpeerτaetxupr.eFscigaulerde (K) 1.5 TDIH
by those two parameters for various initial electron tem- Ti 1 T/i
0.5
peratures holding plasma size and initial peak density
0.5
fixed. Figure 4 (b) shows a similar study varying the
plasma size for similar initial electron temperature and 0 0
iondensity. Inbothcases,thescaleddataapproximately 0 1 2 0 .5 1 1.5
collapse onto universal curves, showing that adiabatic time(s) x 10−6 time/(2π/ωpi)
cooling, described by Eq. 4, dominates the ion temper- (c) 2 (d)1.2
ature evolution. We observe temperatures as low as 0.2 1
K, which are the lowest temperatures ever measured for 1.5
0.8
ions in local thermal equilibrium in an ultracold neutral H
plasma. (K) 1 TDI 0.6
Ti T/i
0.4
0.5
0.2
VI. MODELLING UNP DYNAMICS
0 0
0 1 2 0 2 4
Tomoreaccuratelymodelthedata,weuseahybridap- time(s) −5 time/τexp
x 10
proachfirstdevelopedinRef.61thatcombinesahydrody-
namic treatment of the evolution of the plasma size and
FIG.3. IontemperatureevolutionforUNPswithsimilarini-
electronandiontemperatureswithtermsderivedfroma
tial plasma size (σ = 1mm) and comparable ion densities
0
kinetic and numerical treatment of the effects of correla- (ni(0) = 2×1015m−3 to 4×1015m−3) and different initial
tions. The model takesthe formofdifferentialequations
electron temperatures given in the legend. (a) Lower ini-
describing the evolution of the Gaussian size parameter tial electron temperature produces stronger electron shield-
σ, the electronand ion temperatures, and the expansion ing and lower temperature after DIH. (b) Scaling the time
parameter γ. The hydrodynamic expansion velocity for and temperature axes yields universal curves that only vary
the ions at position r with respect to plasma center, is withelectronscreening. Thecomparisonofexperimentaldata
v(r,t) = γ(t)r. The differential form of the equations with molecular dynamics simulations (solid lines) is used to
allows inclusion of new terms to treat electron-ion ther- determine the local ion density. (c) Lower electron tempera-
ture yields slower expansion and ion cooling. (d) Scaling the
malization.
temperature by TDIH and time by the expansion time τexp
approximatelycollapses thedataontoauniversalcurve,con-
firmingthatadiabaticcoolingdominatesthelong-timeevolu-
∂σ2(t)
=2γ(t)σ2(t) (6) tion. Thelineis theprediction oftheion temperaturemodel
∂t (Eqs.6-10)includingcorrelation effectsandcloudexpansion
∂γ(t) = kBTe(t)+Uii(t)/3 γ2(t) (7) only,38 omitting electron-ion thermalization
∂t m σ2(t) −
i
∂T (t) 2 ∂U (t)
i = 2γ(t)T (t) γ(t)U (t)+ ii + Eqs. 6-9 can be solved exactly by the analytic solution
i ii
∂t − − 3 ∂t presented in Eqs. 3 and 4.
(cid:18) (cid:19)
me The inclusion of the correlation energy in Eqs. 6 - 9
2 γ (t)T (t)
ei e
m allows an approximate treatment of correlation effects
i
(8) such as disorder-induced heating. U can only be calcu-
ii
∂T (t) m lated with a full molecular dynamics description of the
e e
= 2γ(t)Te(t) 2 γei(t)Te(t) (9) plasma.1 It’s evolution can be approximated, however,
∂t − − m
i as38
whereU (t)isthe averageion-ioncorrelationenergyper
ii
particleintheplasma,whichcanberelatedtothespatial ∂U (t) U (t) U (t)
ii ii ii,eq
pair correlation function of the ions.1 U is negative in = − , (10)
ii ∂t − τ (t)
corr
UNPs.
The electron-ion equilibration terms in Eqs. 8 and 9, where the timescale for relaxation of the correlation en-
whichcontainγei willbediscussedbelow. Whentheyare ergy to its equilibrium value, Uii,eq,62 is taken as the
omittedandthecorrelationenergyisneglected(U 0), inverse-ionplasma frequency τ = m ε /e2n .
ii corr i 0 i
≡
p
5
σ0 = 1 mm σ0 = 1.4 mm tahseλio≈nsniknBtωwpoiar2wegs.i5o5nsToefmvpoeluramtuerVe,eiqnutielribfarcaetiaorneabeAt,waenedn
n (0) = 4e15 n (0) = 5e15
separation d is c Vd/λA, where c = 3nk /2 is the
2.5 1.5 ∼ V V B
(a) (b) ionheatcapacitypervolume. Takingalllengthscalesas
the plasma size σ, this yields τ σ2/ω a2 , which
2 is onthe orderof10−3sfor atythpeircmal≈UNP -pmiuwcshlonger
1
K) 1.5 DIH than other dynamic timescales for the plasma.
( T This creates a significant complication for a rigorous
Ti 1 T/i description of the plasma, but we take advantage of the
0.5
extremely long timescale for ion global equilibration to
0.5
make a significant approximation to describe our data.
0 0 Werestrictouranalysistoacentralregionofthe plasma
0 1 2 0 .5 1 1.5 in which the average density is 96% of the peak density.
time(s) x 10−6 time/(2π/ωpi) We consider Eq. 8 as describing an effective tempera-
(c) 2 (d) ture for local thermal equilibribrium within this region.
1 Whileprevioustreatments38 haveapproximatedUii,eq(t)
1.5 as an average over the equilibrium correlation energy in
0.8
H the entire plasma, we take it as the correlation energy
(K) 1 TDI 0.6 for the measured plasma density, which is the average
Ti T/i density of our region of interest. Several additional fac-
0.4
0.5 torsmakeEq.8areasonableapproximationforthe local
0.2 ionconditions. Theactualerrorintroducedwiththisap-
0 0 proximationis smallbecause the temperature after local
0 1 2 0 2 4 equilibration T varies slowly with density. Also,
DIH
time(s) x 10−5 time/τexp the self-simila∼r plasma expansion(Eqs. 3 and 4) leads to
thesamerelativechangeinvolumeandadiabaticcooling
in all regions of the plasma. Finally, the plasma expan-
FIG. 4. Long time ion temperature evolution for UNPs with
similarinitialelectrontemperature(Te(0)=430K)andcom- sion, which controls the adiabatic cooling dynamics, is
parableion densitiesyetdifferentinitial plasma sizes(details dominated by the electron temperature for our condi-
in the legend). (a) Plasma size does not effect local DIH dy- tionsandis relativelyinsensitive to the iontemperature.
namics. (b)Comparisonwithmoleculardynamicssimulations With this interpretation, the equilibrium value of the
isusedtodeterminelocaliondensity. (c)Largerinitialcloud correlationenergy (U <0) is estimated as62
ii,eq
size yields slower expansion and ion cooling. (d) Scaling the
temperature by TDIH and time by the expansion time τexp Uii,eq(Ti,n)=kBTiΓ3/2(Ti,n)
approximately collapses the data ontoa universal curve.
A A (11)
1 3
+ .
A2+Γ(Ti,n) 1+Γ(Ti,n)!
At plasma formation, U is set to zero, but its magni-
ii
tude increases rapidly, raising the ion temperature close where Ti and n pare the instantaneous local ion temper-
to the equilibrium temperature TDIH on a timescale of ature and density, and A1 = −0.9052, A2 = 0.6322, and
eτcqourirl.ibrTiuhme ioisnntoetmppuetraitnutroetahfetemr oesdtealbalisshamneandt hoofclopcaa-l A3E=qs−.6√-130/2ne−glAec1t/i√ngAe2l.ectron-ionequilibrationqualita-
rameter; it emerges naturally from the dynamics. This tively predict the ion temperature evolution, as shown
treatment also includes the retardation of the expan- by rescaling data by TDIH and τexp and overlaying the
sion due to decreasing magnitude of U with decreasing results ofthe model (solidlines in Figs. 3 (b) and4 (b)).
ii
density, which is a small effect for our conditions. The But quantitatively, the theory clearly diverges from the
ion KEOs observed experimentally during the disorder- data after the DIH phase and underestimates the ion
induced-heating phase are not described at this level of temperatureduringthe expansion. Thisimpliesthatthe
treatment. TheslowapproachtoT duringthedamp- modelmissesasignificantamountofenergythatistrans-
DIH
ing of the KEOs mentioned in Sec. IV is also not de- ferredto the ions during the evolution, and we now turn
scribed. to a discussion of the sources of this energy.
InherentinthetraditionalinterpretationofEq.8,how-
ever, is an assumption of global thermal equilibrium for
the ions thatis notwell-justified. The equilibriumcorre- 1. Electron-Ion Thermalization
lationenergydepends onthe localdensity, anddisorder-
induced heating only leads to local thermal equilibrium. Collisional energy transfer between electrons and ions
The characteristic timescale for global ion thermal equi- is typically neglected in theory and experiments on
libration τ is set by the thermal conductivity for UNPs.1 Due to the large mass difference, complete ther-
therm
the strongly coupled ions, which can be approximated malizationbetweenthetwowouldrequire10to100times
6
the expansion time (τ ) of an UNP. Thus, the energy the density drops and collisions become negligible. In
exp
lost by electrons due to collisions with ions is insignif- Fig. 5(b), electron-ion thermalization contributes 0.5K
icant compared to the electron temperature,54 and the and doubles the ion temperature at later times. For the
expansion of the plasma and the electron temperature oppositeextremeoftheaccessibleregimeoftheseparam-
evolution can be described by Eqs. 3 and 4. eters in UNPs, the effect is very small.
However,ifoneisconcernedwiththeiontemperature,
electron-ionthermalizationmust be consideredsince the n=1.1e15m−3 n=2.3e15m−3
ions are typically orders of magnitude colder than elec- (a) Te=50K,τexp=15µs (b) Te=50K,τexp=15µs
1.5 2 1.5
trons. This is particularly important for proposed ef- 1
1
forts to increase Coulomb coupling by laser cooling63,64 0.5 1.5 0.5
or circumventing DIH57 because a small transfer of elec- 1 0 0
K) 0 0.5 1 0 0.5 1
tron energy to thermal ion energy heats the ions signif- ( 1
icantly. In general, thermal relaxation in a plasma has Ti
0.5
beenalong-standingfundamentalinterest65–72 aswellas 0.5
a critical aspect of many areas of study such as inertial
confinementfusion,73–77 warmdensematter,78 andspace 00 0.5 1 1.5 2 00 0.5 1 1.5 2
plasmas.79–81 n=1.3e15m−3 n=2.7e15m−3
(c) (d)
Themeasurementspresentedherearethefirsttofollow Te=80K,τexp=12µs Te=80K,τexp=12µs
the ion temperature on the timescale of the expansion 1.5 1.5 2 1.5
1 1
and observe the effects of electron-ion thermalization in 0.5 1.5 0.5
an UNP. To describe the data, and help us identify the 1 0 0
contributionofthisheatingmechanism,westartwiththe (K) 0 0.5 1 1 0 0.5 1
classicelectron-ioncollisionfrequency ina singly ionized Ti
0.5
plasmas, assuming n =n , m >>m and T >>T ,82 0.5
i e i e e i
4√2πn e4ln[Λ] 0 0
γ = i 0 1 2 3 0 1 2
ei 3(4πε0)2m1e/2(kBTe)3/2 (e) nTe==14.83e01K5,mτ−ex3p=5µs (f) nTe==44.300e1K5,mτ−ex3p=5µs
2 2 2 2 2
= Γ3/2ω ln[Λ], (12)
3π e pe 1 1
r 1.5 1.5
and the ion heating and electron cooling rates due to K) 00 0.5 1 00 0.5 1
electron-ion collisions (EICs), ( 1 1
Ti
dT dT m 0.5 0.5
i e e
= =2 γ T . (13)
ei e
dt − dt m
i 0 0
0 1 2 3 4 0 1 2 3 4
The argument of the Coulomb logarithm is Λ = time/τexp time/τexp
1/√3Γ3/2. The Coulomb coupling parameter for the
e
electrons is never greater than 1 for our experiments, FIG.5. Comparisonofdatawithmodelsincludingandnotin-
so strong coupling effects are small,83 as has been cludingelectronioncollisions. Thelinesaresimulationresults
shown by comparison of Eq. 12 with quantum T-matrix of the ion temperature model (Eqs. 6-10) including correla-
calculations84 in this regime of coupling. tion effects and cloud expansion38 with (solid) and without
Theheatingandcoolingtermsareincludedintheevo- (dotted) electron-ion thermalization. The insets show early
lution of ion and electron temperatures (Eqs. 8 and 9). time KEOs with time normalized by 2πωp−i1. The solid lines
in the insets are results from MD simulations fit to the data
We use the measured, averageplasma density for the re-
in order to determine experimental density.
gion of interest in these expressions. Following a similar
line ofreasoningas discussedin the contextof the corre-
lation energy, this provides a reasonable approximation
of the evolution of the local ion temperature in the cen-
tral regionof the plasma because of the very slow global 2. Additional Heating/Ion Acoustic Waves
equilibration of the ions.
Withtheinclusionofelectron-ionthermalizationinthe It is evident from Fig. 5 that the simulation still un-
model, it is possible to determine the significance of the derestimates the measured ion temperature during the
effectforiontemperatureevolution. Figure5showsdata expansionbyasmuchasseveralhundredmillikelvin,and
and simulation for various regimes of UNP parameters. that the excess energy appears on the timescale of τ .
exp
ThesimulationresultsshowthattheheatingfromEICsis Weobservethatthiseffectisverysensitivetoexperimen-
more significant for higher density, smaller electrontem- talparameterssuchaslaseralignmentandbeamquality.
perature,andlargerinitialplasmasize. Theseconditions But in general the additional energy is more significant
increase the collision rate and the time ( τ ) before for larger density and electron temperature.
exp
∼
7
One possibility is that hydrodynamic expansion ve- byω /kleadingtoadecreasingenergywithwavevector,
pi
locity is being misinterpreted as thermal velocity. The
plasma is expanding with a velocity that varies in space, 1 ne2 δn 2
∆T (16)
and LIF signal captured from a region with finite width i,max ≈ 6ǫ k k2 n
δx will reflect a spread of expansion velocities that can 0 B (cid:18) 0(cid:19)
broaden the measured spectrum. For the self-similar ex- This effect seems capable of providing the observed
pansion(Eqs.3-4),thevelocityspreadδv =γ(t)δxpeaks heating, and residues of the density-distribution fit to
at t=τexp and then decreases. Assuming anideal single a 2D Gaussian are often on the order of 5% on a
pixel region of width δx= 13µm, δv peaks at a value of 1mm length scale. Similar depth ion holes have bee∼n
1ms−1 correspondingto aDopplerbroadeningof2MHz shown to perturb the ion temperatures by several hun-
or8mKintemperatureunits. Theactualvaluewouldin- dred millikelvin.17,87
creaseiftheexpansiondeviatesfromtheidealself-similar Despite great effort, deviation of the density distri-
expansion or if the point spread function used for image bution from an ideal Gaussian remains a limitation for
deconvolution is inaccurate or not constant throughout achieving lower ion temperatures. For all but the lowest
the fluorescence volume. However, it is unlikely this ac- atom densities, the magneto-optical trap for the atoms
counts for a large fraction of the observed heating. is optically thick to the laser-cooling light, this results
Another possibility is that Eqs. 12 and 13 underesti- innon-Gaussianatomdistributionsthatareinheritedby
mate the amount of ion heating due to electron-ion col- theplasma. Also,theoutputofthepulsed-dyelaserthat
lisions. We do not attempt to use our data to extract ionizes the atoms has spatial intensity modulations that
the electron-ion thermalization rate because of the diffi- are imprinted onto the plasma. Complete saturation of
culty in separating this heat source from other possible the ionizing process could minimize this source of heat
mechanisms. While there are many assumptions that go but it would require larger pulse energies than available
into the derivation of Eqs. 12 and 13, it is unlikely that with our current system.
this can be the source of the anomalous heating because
it appears to scale very differently than heating due to
electron-ion collisions (Fig. 5). VII. INCREASE IN Γi WITH EXPANSION
We hypothesize that the largest contribution to the
observed excess ion energy reflects ion acoustic waves
For the best chance of seeing increasing Γ with
i
(IAWs) that are excited during the ionization process expansion,39 minimal heating and fast expansion is re-
due to deviations from a smooth Gaussian in the initial
quired. Fast expansion is achieved by small cloud size
density distribution of the plasma. Previously we have
and high T . Higher density doesn’t change the ini-
e
used transmission masks on the ionizing beam to inten-
tial coupling parameter, but it increases the DIH tem-
tionally modulate the initial density distribution lead-
perature - minimizing the relative decrease in Γ com-
ing to the excitation of IAWs.16,85,86 However, similar i
ing from additional heating sources. Higher density also
excitations would be caused by any imperfection in the
enhances EIC heating, but for sufficiently high electron
ionizing laser profiles and/or atom density distribution.
temperatures, e.g. near the auto-ionizing resonance, the
Long-wavelength IAW excitations with a broad spread
EIC cross section is so small that the highest achievable
of wavevectors would increase the RMS velocity width
densities should not experience significant EIC heating.
of the ions. Short wavelength excitations should de-
Higher density also allows data collection at later times.
cay quickly throughLandaudamping, increasingthe ion
Therefore,atleastforouraccessibleexperimentalparam-
thermal energy. Bothwould increasethe temperature as
eters, high T , small cloud and high density are optimal
e
measured by our LIF probe.
for observing increasing Γ .
i
To assess the feasibility of this explanation, we esti-
Measurements at later times are limited by plummet-
mate the energy per ion from a single IAW mode of fre-
ing densities and signal-to-noise ratios, but for the best
quency ω, wavevector k, and fractional density modula-
conditions,weseeΓ increaseabove5,asshowninFig. 6.
tion δn as86 i
n0 EICsandothersourcesofheatingpreventΓi fromreach-
ing as high as expected from Eqs. 6-10 neglecting these
2
1 ωδn
∆EIAW mi . (14) effects. The measured value of Γi is calculated from ob-
≈ 4 (cid:18)k n0(cid:19) served ion temperature and density, and we are not able
todetermineifspatialcorrelationsexistcorrespondingto
In the long wavelength limit ω/k can be replaced with
equilibrium at this value.39
the sound speed k T /m . If we assume all the IAW
B e i
energy is transferred into thermal energy, we find that
p
1 δn 2 VIII. CONCLUSION
∆T T . (15)
i,max e
≈ 6 n
(cid:18) 0(cid:19)
We have used spatially resolved fluorescence spec-
Fora δnn0 ofonly5%andTe =430Kthisisnearly200mK troscopy to measure ion temperature evolution during
of energy. In the short wavelength limit ω/k is replaced the expansionofanUNP forvariousinitialplasma sizes,
8
7
8M. Lyon, S. D. Bergeson, and M. S. Murillo,
Phys.Rev.E87,033101(2013).
6
9G. Bannasch, J. Castro, P. McQuillen, T. Pohl, and T. C. Kil-
5 lian,Phys.Rev.Lett.109,185008 (2012).
10S.Ichimaru,ReviewsofModernPhysics54,1017(1982).
4 11M.S.Murillo,Phys.Plasmas14,055702(2007).
Γi 12J.M.Maxson,I.V.Bazarov,W.Wan,H.A.Padmore, andC.E.
3 Coleman-Smith,NewJournalofPhysics15,103024 (2013).
13S.Kulin,T.C.Killian,S.D.Bergeson, andS.L.Rolston,Phys.
2 Rev.Lett. 85,318(2000).
14F. Robicheaux and J. D. Hanson, Phys. Rev. Lett. 88, 55002
1
(2002).
0 15S.Laha, P. Gupta, C.E. Simien,H.Gao, J. Castro, andT. C.
0 1 2 3 4 Killian,Phys.Rev.Lett.99,155001(2007).
time/τ 16J. Castro, P. McQuillen, and T. C. Killian, Phys. Rev. Lett.
exp
105,065004(2010).
17P.McQuillen,J.Castro,T.Strickler,S.J.Bradshaw, andT.C.
FIG.6. IncreaseintheionCoulombcouplingparameterwith Killian,Phys.Plasmas20,043516(2013).
σpla=sm1amexmp.anTshioenlifnoeriTset(h0e)=pre4d3i0ctKio,nno0f=Eq4s×. 61-01105.m−3, and 18KPh.ys.ReAv..Lett.T10w8e,d0t65003a(n2d012). S. L. Rolston,
0 19A. Lyubonko, T. Pohl, and J.-M. Rost,
NewJournalofPhysics14,053039(2012).
20T. C. Killian, M. J. Lim, S. Kulin, R. Dumke, S. D. Bergeson,
electron temperatures, and densities. We have observed andS.L.Rolston,Phys.Rev.Lett. 86,3759(2001).
severaleffectsinUNPsforthe firsttime, suchasionadi- 21S.D.BergesonandF.Robicheaux,Phys.Rev.Lett.101,073202
(2008).
abaticcoolingandionheatingduetocollisionswithelec-
22M.P.Robinson,B.L.Tolra,M.W.Noel,T.F.Gallagher, and
trons. We have presented a numerical model describing P.Pillet,Phys.Rev.Lett.85,4466(2000).
theevolutionofplasmasizeandelectronandiontemper- 23M.D.PerryandG.Mourou,Science264,917(1994).
atures including the effects of electron-ioncollisions. We 24J.Lindl, Phys.Plasmas2,3933(1995).
alsopresentedevidence foradditionalionheating,which 25H.Daido,ReportsonProgressinPhysics65,1513(2002).
26E. L. Clark, K. Krushelnick, M. Zepf, F. N. Beg, M. Tatarakis,
we hypothesize is dominated by IAWs excited during
A. Machacek, M. I. K. Santala, I. Watts, P. A. Norreys, and
plasmacreation. Weseeincreasingcouplingforionswith A.E.Dangor,Phys.Rev.Lett.85,1654(2000).
time, but it is limited by these heating effects. Attempts 27R. A. Snavely, M. H. Key, S. P. Hatchett, T. E. Cowan,
to access stronger Coulomb coupling, such as through M. Roth, T. W. Phillips, M. A. Stoyer, E. A. Henry, T. C.
proposedschemesto lasercoolionsin UNPs63,64,88–94 or Sangster, M. S. Singh, S. C. Wilks, A. MacKinnon, A. Offen-
berger, D. M. Pennington, K. Yasuike, A. B. Langdon, B. F.
simply through plasma expansion as demonstrated here,
Lasinski, J. Johnson, M. D. Perry, and E. M. Campbell,
should benefit from improved control of the initial den- Phys.Rev.Lett.85,2945(2000).
sity distribution of the plasma. 28A. Maksimchuk, S. Gu, K. Flippo, D. Umstadter, and V. Y.
Bychenkov, Phys.Rev.Lett. 84,4108(2000).
29D.R.Symes,M.Hohenberger,A.Henig, andT.Ditmire,Phys.
Rev.Lett. 98,123401 (2007).
ACKNOWLEDGMENTS 30F.RobicheauxandJ.D.Hanson, Phys.Plasmas10,2217(2003).
31A.V.BaitinandK.M.Kuzanyan,J.PlasmaPhys.59,83(1998).
32D.S.DorozhkinaandV.E.Semenov,Phys.Rev.Lett.81,2691
We thank Thomas Pohl for providing results from (1998).
molecular dynamics simulation of equilibrating plasmas 33V. Kovalev, V. Bychenkov, and V. Tikhonchuk,
and Barry Dunning for the loan of a laser for LIF exci- JournalofExperimentalandTheoretical Physics95,226(2002).
tation. This work was supported by the Department of 34V.F.KovalevandV.Y.Bychenkov,Phys.Rev.Lett.90,185004
(2003).
EnergyandNationalScienceFoundation(PHY-0714603)
35P.Gupta,S.Laha,C.E.Simien,H.Gao,J.Castro,T.C.Killian,
and the Air Force Office of Scientific Research (FA9550- andT.Pohl,Phys.Rev.Lett. 99,75005(2007).
12-1-0267). 36J. L. Roberts, C. D. Fertig, M. J. Lim, and S. L. Rolston,
Phys.Rev.Lett.92,253003(2004).
37R. S. Fletcher, X. L. Zhang, and S. L. Rolston,
1T. C. Killian, T. Pattard, T. Pohl, and J. M. Rost, Physics Phys.Rev.Lett.99,145001(2007).
Reports449,77(2007). 38T.Pohl,T.Pattard, andJ.M.Rost,Phys.Rev.Lett.92,155003
2M.S.Murillo,Phys.Rev.Lett.87,115003(2001).
(2004).
3S. G. Kuzmin and T. M. O’Neil, Phys. Rev. Lett. 88, 65003 39T. Pohl, T. Pattard, and J. M. Rost,
(2002). Phys.Rev.Lett.94,205003(2005).
4S.Mazevet,L.A.Collins, andJ.D.Kress,Phys.Rev.Lett.88, 40G. Li, R. Poggiani, G. Testera, and G. Werth,
55001(2002). ZeitschriftfrPhysikDAtoms,MoleculesandClusters22,375(1991).
5Y. C. Chen, C. E. Simien, S. Laha, P. Gupta, Y. N. Martinez, 41G. Gabrielse, W. S. Kolthammer, R. McConnell, P. Richerme,
P.G.Mickelson,S.B.Nagel, andT.C.Killian,Phys.Rev.Lett.
R. Kalra, E. Novitski, D. Grzonka, W. Oelert, T. Sefzick,
93,265003(2004). M.Zielinski,D.Fitzakerley, M.C. George, E.A. Hessels,C. H.
6E.A.Cummings,J.E.Daily,D.S.Durfee, andS.D.Bergeson,
Storry, M. Weel, A. Mu¨llers, and J. Walz (ATRAP Collabora-
Phys.Rev.Lett. 95,235001(2005). tion),Phys.Rev.Lett.106,073002(2011).
7S. D. Bergeson, A. Denning, M. Lyon, and F. Robicheaux,
Phys.Rev.A83,023409(2011).
9
42G. Manfredi and P.-A. Hervieux, 71B. R. Beck, J. Fajans, and J. H. Malmberg,
Phys.Rev.Lett. 109,255005(2012). Phys.Plasmas3,1250(1996).
43L. Saul, P. Wurz, and R. Kallenbach, 72G. Dimonte and J. Daligault,
TheAstrophysicalJournal703,325(2009). Phys.Rev.Lett.101,135001(2008).
44M. Mark, E. Haller, K. Lauber, J. Danzl, 73M. S. Murillo and M. W. C. Dharma-Wardana,
A. Janisch, H. Bu¨chler, A. Daley, and H.-C. N¨agerl, Phys.Rev.Lett.100,205005(2008).
Phys.Rev.Lett. 108,215302(2012). 74J. N. Glosli, F. R. Graziani, R. M. More, M. S. Murillo, F. H.
45J. H. Chen, E. M¨obius, G. Gloeckler, P. Bochsler, Streitz,M.P.Surh,L.X.Benedict,S.Hau-Riege,A.B.Langdon,
M. Bzowski, P. A. Isenberg, and J. M. Soko´, andR.A.London,Phys.Rev.E78,025401R (2008).
JournalofGeophysical Research: SpacePhysics118,3946(2013). 75J. R. Rygg, J. A. Frenje, C. K. Li, F. H. S´eguin,
46S. Betti, F. Ceccherini, F. Cornolti, and F. Pegoraro, Plasma R. D. Petrasso, D. D. Meyerhofer, and C. Stoeckl,
PhysicsandControlledFusion47,521(2005). Phys.Rev.E80,026403(2009).
47F. Ceccherini, S. Betti, F. Cornolti, and F. Pegoraro, 76J. Vorberger, D. O. Gericke, T. Bornath, and M. Schlanges,
LaserPhysics16,594(2006). Phys.Rev.E81,046404(2010).
48J. Castro, H. Gao, and T. C. Killian, Plasma Physics of Con- 77L.X.Benedict,M.P.Surh,J.I.Castor,S.A.Khairallah,H.D.
trolledFusion50,124011(2008). Whitley, D. F. Richards, J. N. Glosli, M. S. Murillo, C. R.
49S. Laha, Y. C. Chen, P. Gupta, C. E. Simien, Y. N. Martinez, Scullard,P.E.Grabowski,D.Michta, andF.R.Graziani,Phys.
P.G.Mickelson,S.B.Nagel, andT.C.Killian,EuropeanPhys- Rev.E86,046406 (2012).
icalJournalD40,51(2006). 78M. Schlanges, T. Bornath, J. Vorberger, and D. Gericke,
50P. McQuillen, High Resolution Sculpting and Imaging of Ultra- ContributionstoPlasmaPhysics50,64(2010).
cold Neutral Plasmas, Master of science thesis, Rice University 79K.Papadopoulos,JournalofGeophysical Research76,3806(1971).
(2012). 80E. Marsch, in Physics of the Inner Heliosphere II, Physics and
51K.R.Castleman,DigitalImage Processing (PearsonEducation, ChemistryinSpace,Vol.21,editedbyR.SchwennandE.Marsch
2007). (SpringerBerlinHeidelberg,1991)pp.45–133.
52C. E. Simien, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, 81O. Y. Gnedin, D. G. Yakovlev, and A. Y. Potekhin,
P.G.Mickelson,S.B.Nagel, andT.C.Killian,Phys.Rev.Lett. MonthlyNoticesoftheRoyalAstronomicalSociety 324,725(2001).
92,143001(2004). 82L.Spitzer,Physics of Fully Ionized Gases,Intersciencetractson
53W. R. S. Garton and K. Codling, physicsandastronomy,3(Interscience Publishers,1956).
JournalofPhysicsB:AtomicandMolecularPhysics1,106(1968). 83J. P. Hansen and I. R. McDonald, Physics Letters A 97, 42
54T. C.Killian,S. Kulin,S. D. Bergeson, L. A. Orozco, C. Orzel, (1983).
andS.L.Rolston,Phys.Rev.Lett.83,4776(1999). 84D. O. Gericke, M. S. Murillo, and M. Schlanges,
55Z.DonkoandP.Hartmann,Phys.Rev.E69,016405(2004). Phys.Rev.E65,036418(2002).
56S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, Phys. Rev. 85P.McQuillen,J.Castro, andT.C.Killian,J.Phys.B44,184013
E56,4671(1997). (2011).
57L. Guo, R. H. Lu, and S. S. Han, Phys. Rev. E 81, 046406 86T. C. Killian, P. McQuillen, T. M. O’Neil, and J. Castro,
(2010). Phys.Plasmas19,055701(2012).
58M. Lyon and S. D. Bergeson, 87P. McQuillen, J. Castro, and T. C. Killian,
JournalofPhysicsB:Atomic,MolecularandOpticalPhysics44,18401J4ou(r2n0a1l1)o.fPhysicsB:Atomic,MolecularandOpticalPhysics44,184013(2011)
59T.Pohl,Privatecommunication(2014). 88A. Gavrilyuk, I. Krasnov, and N. Shaparev,
60T. Langin, P. McQuillen, T. Strickler, and T. Killian, To be TechnicalPhysicsLetters 23,61(1997).
published. 89A.Gavrilyuk,I.Krasnov, andN.Y.Shaparev,LaserPhysics8,
61T. Pohl, T. Pattard, and J. M.Rost, Phys. Rev. A 70,033416 653(1998).
(2004). 90A.P.Gavrilyuk,I.V.Krasnov, andN.Y.Shaparev,JETPLett.
62G.ChabrierandA.Y.Potekhin,Phys.Rev.E58,4941(1998). 76,423(2002).
63S.G.KuzminandT.M.O’Neil,Phys.Plasmas9,3743(2002). 91T.C.Killian,V.S.Ashoka,P.Gupta,S.Laha,S.B.Nagel,C.E.
64T.Pohl,T.Pattard, andJ.Rost,J.Phys.B38,S343(2005). Simien, S. Kulin, S. L. Rolston, and S. D. Bergeson, J. Phys.
65J.M.Dawson,PhysicsofFluids7,419(1964). A:Math.Gen. 36,6077(2003).
66D. Montgomery and C. W. Nielson, 92T. Pohl, T. Pattard, and J. M. Rost,
PhysicsofFluids13,1405(1970). Phys.Rev.Lett.92,155003(2004).
67D.B.BoerckerandR.M.More,Phys.Rev.A33,1859(1986). 93I.Krasnov,PhysicsLetters A373,2291(2009).
68A. W. Hyatt, C. F. Driscoll, and J. H. Malmberg, 94A.Gavriliuk,I.Isaev,S.Karpov,I.Krasnov, andN.Shaparev,
Phys.Rev.Lett. 59,2975(1987). Phys.Rev.E80,056404(2009).
69P.Martin,PlasmaPhysicsandControlledFusion41,A247(1999).
70I. V. Morozov and G. E. Norman,
JournalofPhysicsA:Mathematical andGeneral36,6005(2003).