a. n. tkachev and s. i. yakovlenko- on the recombination heating of ultracold laser-produced plasmas

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977

Laser Physics, Vol. 11, No. 9, 2001, pp. 977–981.Original Text Copyright © 2001 by Astro, Ltd.

English Translation Copyright © 2001 by MAIK “Nauka/ Interperiodica” (Russia).

1. INTRODUCTION

Killian et al.

[1] have reported the formation of anultracold plasma with unique parameters: the chargedensity N

e

~ 2

× 10

9

cm

–3

, the electron temperature T

e

~0.1 K, the ion temperature T

i

~ 10 µ

K, and the ioniza-tion degree of about 0.1. Such a plasma was producedthrough a two-stage ionization of laser-cooled metasta-ble-state xenon. The authors of [1] also reported that themeasured plasma lifetime is anomalously large, being~100 µ

s.

In our papers [2, 3], the results of experiments [1]were analyzed in terms of the theory summarized inreview papers [4–7]. It was shown that the observed

lifetime is consistent with these nontraditional con-cepts.

Later experiments by Kulin et al.

[8] were devotedto the study of the expansion of an ultracold-plasmaplume. This paper, in particular, presents the data con-cerning temporal variations in plasma density. Theseresults allow a fuller analysis of the experimental situa-tion in terms of conventional concepts to be performed.We start our analysis with the following precedent.Panchenko et al.

[9, 10] have observed a long-termemission of a plasma bubble produced by laser radia-tion evaporating the surface of a metal target into a gas.However, recombination deceleration under these con-

ditions was attributed to a recombination heating of electrons [10, 11]. It would be natural to try to explainthe anomalously large lifetime of ultracold plasmas interms of recombination heating. This paper is devotedto the analysis of this issue.

2. COMPUTATIONAL MODEL

Experiments [8] have demonstrated that the tempo-ral variation in the mean electron density approxi-

mately follows the law corresponding to hydrodynamic

expansion:

(1)

Here, n

is the total number of photoelectrons, σ

0

is the

initial radius of a Gaussian plasma plume, and v

0

is the

expansion velocity. Then, relying on the data of [8], wecan employ the following approximation for the expan-sion velocity within a broad range of experimentalparameters:

(2)

Here, E

e

is the initial energy of electrons produced

through photoionization, m

i

is the mass of xenon ions,

and α

= 1.7 is the fitting parameter.

Using this circumstance, we can write the followingequations for the temperature and density of electronsaveraged in space (cf. [10, 11] and [5, p. 500]):

(3a)

(3b)

Here,

N n / 4π σ0

2v 0

2t

2+( )[ ]

3/2.=

v 0

30–50( ) m/s, for E e / k B 70 K<

E e / αmi, for E e / k B 70 K.>⎩⎨

⎧=

dN e

dt ---------

C

T e9/2

-------- N e3

– f t / t 0( )

t 0---------------- N e,–=

dT e

dt ---------

2

3---ε T e+⎝ ⎠

⎛ ⎞ C

T e9/2

-------- N e2 2

3---

f t / t 0( )t 0

----------------T e.–=

C = 29/2π3/2

e10( ) / 45me

1/2( )Λ 1.3 107–

cm6

s1–

K9/2×≈

≈ 1.7 1025–

cm6

s1–

eV9/2×

INTERACTION OF LASER RADIATIONWITH MATTER

On the Recombination Heatingof Ultracold Laser-Produced Plasmas

A. N. Tkachev and S. I. Yakovlenko

General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia

e-mail: [email protected]

Received May 10, 2001

Abstract

—It is shown that the time dependences of the electron density measured by Kulin et al.

(Kulin, S.D.

et al.

, 2000, Phys. Rev. Lett.

, 85

, 318) do not allow one to conclude whether three-body recombination occursduring the expansion of an ultracold laser-plasma plume. The assumption that recombination is suppressed issupported by the fact that no splash in the signal related to outgoing electrons was detected. Such a splashshould be observed in the regime of recombination heating of plasma electrons.



978

LASER PHYSICS

Vol. 11

No. 9

2001

TKACHEV, YAKOVLENKO

is the rate constant characterizing three-body recombi-nation,

is the Coulomb logarithm [

γ

≡

e

2

(2

N

e

)

1/3

/

T

e

], f

(

τ

) =

3

τ

/(1 + τ

2

) is the function characterizing variations inthe density and temperature of electrons due to plumeexpansion, t

0

=

σ

0

/ v0

is the characteristic expansiontime of the plasma plume, and ε is the energy releasedin the electron gas per single recombination event.

We restrict our analysis to time intervals muchshorter than the time of electron–ion energy exchange:

For N e = 109 cm–3 and T e0 = 4.2 K, we have τei ≈ 200 µs.Consequently, for time intervals on the order of theexpansion time of a plasma plume, t 0 ≈ 5 µs, electron–ion collisions with energy exchange can be neglected.

We will determine the recombination energy releaseby comparing the frequency of Coulomb collisionsaccompanied by a transfer of the energy ∆ε [12] withthe inverse plasma-expansion time t 0:

Using this relationship to determine ∆ε and invoking

the expressions ε = Ry/ and ∆ε = 2Ry/ (Ry =

Λ γ ( ) 1/2( ) 1 9/ 4πγ 3( )+( ), for γ 0.5<ln

1, for γ 0.5>⎩⎨⎧

=

τei

mi

3me

---------3

4 2π--------------

meT e3/2

Λe4 N e

------------------- .=

πT e /2me( )1/2e

2 / T e( )

2 N e T e / ∆ε( )3

1/ t 0.∼

nε2

nε3

mee4 /22 and nε is the principal quantum number corre-

sponding to the given value of ε), we derive

(4)

For example, with N e ~ 109 cm–3 and t 0 ≈ 5 µs, we haveε ≈ 5T e.

3. RESULTS OF CALCULATIONS

Below, we will compare time dependences of plasma parameters obtained from the solution of the setof equations (3) with the data presented in Fig. 2 in [8].

The results of calculations for the minimum initialelectron temperature T e0 = 4.2 K are shown in Fig. 1.The minimum initial temperature was determined inthe following way. According to our previous results[2, 3, 6, 7, 13, 14], electrons are heated due to collectiveinteractions within a time interval approximately corre-sponding to half the inverse Langmuir frequency,

≈0.5 , ω L = (4πe2 N e / me)1/2. The phase trajectory of a

system of many Coulomb particles becomes mixedwithin this period of time. The mixing time is charac-terized by the Lyapunov index L ≈ 2.4ω L [6, 7, 15, 16].While at the initial moment of time, the nonidealityparameter γ = e2(2 N e)

1/3 / T e is much higher than unity,mixing decreases this parameter down to γ lim ≈0.35−0.5. For the considered conditions ( N e ~ 109 cm–3),

the heating time, 0.5 ≈ 0.3 ns, is less than the time

required to ionize the gas with a laser pulse (~10 ns).

ε T eme

4e

4

28π6

--------------1

N et 0( )2-----------------

⎝ ⎠ ⎜ ⎟ ⎛ ⎞

1/9

.≈

ω L

1–

ω L

1–

1 × 109

1 × 108

1

×107

1 × 106

1 × 105

1 10 100

100

10

1

0.1 1 10 100

t 0

t , µst , µs

N e(t ), N (t ) 100[ N (t ) – N e(t )]/ N (t ), T e(t )/ T e0(b)(a)

Fig. 1. Temporal evolution of plasma parameters: (a) the solid curve shows the electron density, the dotted curve corresponds toexpansion with no recombination (1), and the dots represent the experimental data for the electron density; (b) the solid curve showsthe percentage of electrons undergoing recombination and the dotted line represents the electron temperature in units of T e0. The

initial parameters are σ0 = 220 µm, v 0 = 50 m/s, N e0 = 109 cm–3, and T e0 = 4.2 K ( E e = 6.3 K).



LASER PHYSICS Vol. 11 No. 9 2001

ON THE RECOMBINATION HEATING 979

Therefore, the initial temperature is no lower in itsorder of magnitude than some limiting temperature

In our calculations, we started with this temperature.

As can be seen from our data, more than 10% of electrons recombine within 10 ns. Due to recombina-tion heating, the electron temperature increases by afactor of 3.5 for t ~ 5 µs. We should note, however, thatthe experimental variations in the electron density donot allow us to judge whether recombination occurs ornot.

In the case when the initial energy of photoelectronsis high (see Fig. 2), the scenario of relaxation changes.The electron temperature remains unchanged untilelectron cooling starts due to expansion. Recombina-tion heating is insignificant under these conditions.

4. ON THE UNCOMPENSATED CHARGE

4.1. The Potential Well

The plasma plume contained up to 5 × 105 atomsunder conditions of experiments [1, 8]. Some of theelectrons in such a situation escaped from the plasmavolume. We employed the model described in [2–7] to

simulate the dynamics of many Coulomb particles inorder to reveal the influence of the uncompensatedcharge on plasma parameters.

When modeling a plasma plume, we defined the ini-tial coordinates of n = 512 ions and n = 512 electronsinside some part of the simulation volume (a cube),namely, inside a sphere with a radius R = (4π N e /3n)–1/3 ≈39 µm located at the center of the cube (the cube edgelength l = 800 µm). It was assumed that particles reach-ing the cube walls get frozen on these walls.

T emin N e( ) e2

2 N e( )1/3 / γ lim 4.2 N e 10

9cm

3×( )1/3

K.≈ ≈

As one might expect, only a small fraction of elec-trons escape from the sphere occupied by ions (seeFig. 3). Under these conditions, charge remains uncom-pensated only at the periphery of the sphere (seealso [17]).

Our computational resources allow us to modelplasma plumes with much smaller sizes than the sizesof plasma plumes produced in experiments [1, 8]. Nev-ertheless, even small plasma plumes with parameters[1, 8] efficiently trap electrons, and only a small frac-tion of electrons escape from the initial sphere.

Let us estimate the depth of the potential well usinga very simple approach. The flux of electrons from the

plume is given by 4π1/2 N e(2T e / me)1/2exp(–U b / T e)(1 +

U b / T e). If the number of electrons escaping from theplume within the time ∆t normalized to the total num-

ber of electrons n = N e(4π )3/2 is equal to αesc, then the

potential barrier x = U b / T e is given by

. (5)

With αesc ~ 0.1 and ∆t ~ 0.25 µs, which corresponds tothe results of [1], we have U b ~ 23 K.

The electron flux is highly sensitive to the depth of the potential well and the electron temperature. Whenthe temperature abruptly increases or the depth of thepotential well decreases, the electron flux on the collec-tor should steeply grow. This effect is observed inexperiments, when an electric field linearly growingwith a growth rate of 0.05 (V/cm)/(2 µs) is applied tothe plasma (see Fig. 1 in [1]). When the external field is~0.05 V/cm (for t ≈ 1.5–2 µs), a sharp splash of the sig-nal related to outgoing electrons is observed. Thissplash is detected for close field strengths with different

σ02

σ0

2

e x–

1 x+( )πα es cσ0

2T e / me( )1/2∆t ---------------------------------=

1 × 109

1 × 108

1

×107

1 × 106

1 × 105

1 10 1001 × 10 –5

1 10 100

t 0

t , µst , µs

N e(t ), N (t ) 100[ N (t ) – N e(t )]/ N (t ), T e(t )/ T e0

1 × 10 –4

1 × 10 –3

0.01

0.1

1(b)(a)

Fig. 2. The same as in Fig. 1 for a high initial temperature, v 0 = 108 m/s and T e0 = 211 K ( E e = 316 K).

0.1



980


TKACHEV, YAKOVLENKO

densities of the plasma plume, N e ~ 107–109 cm–3,although the signal detected in a plasma with a higherdensity has a much larger amplitude. This result is con-sistent with the fact that, as can be seen from Eq. (5),the depth of the potential well trapping the electrons is

independent of the density. The external field changesthe potential on the spatial scale on the order of theplume size by 0.05σ0 V/cm ~ 13 K, which is compara-ble with the depth of the potential well (5).

The heating of electrons by an external high-fre-quency electric field with a frequency close to theLangmuir frequency also increases the electron temper-ature, giving rise to a splash in the number of electronsescaping from the plasma. This effect underlies themethod of measurement of the time dependence of theelectron density employed in [8].

Note that, in the regime of recombination heating,the electron temperature also increases by several

times, and the characteristic time scale of this processis also on the order of a microsecond (see Fig. 1). How-ever, in accordance with the data of [8], the outgoingflow of electrons from the plasma plume virtually stopswithin the time ∆t ~ 0.25 µs.

We should note also that the outgoing flow of elec-trons from the plasma promotes the cooling of electronsremaining in the plasma. The energy released perrecombination event under these conditions is on theorder of the energy spent to remove an electron from

the potential well, ε ~ U b. It would be natural to assumethat the flux of electrons from the plasma under theseconditions is comparable with the number of electronsrecombining per unit time. This finding also brings usto a conclusion that about a half of the total number of electrons should escape from the plasma within severalmicroseconds in the presence of recombination (seeFig. 1). However, this is not what is observed experi-

mentally.

4.2. On the Mechanism of Plasma Acceleration

In [17], we considered a Coulomb explosion in alaser-produced plasma with a cylindrical geometry.Performing a similar analysis for a spherical case, wecan derive the following expressions describing thetime dependences of the velocity and the radius of theboundary of a charged plume:

where q = 4πe N i(r ) – N ie(r ))r 2dr is the uncompen-

sated charge of the plasma plume.

To provide some estimates, we will assume that theuncompensated charge is concentrated within a Debyelayer with a thickness r D = (4πe2 N e / T e)

–1/2. Then, with N e ~ 109 cm–3 and T e ~ 4.2 K, we have

As the uncompensated charge is detached from theplasma, electrons should leave the area of uncompen-sated charge, giving rise to a new boundary, whichshould be accelerated again. The characteristic expan-sion velocity of the main fraction of plasma massshould be several times lower than the expansion veloc-ity of the boundary of uncompensated charge, v0 ~150 m/s. This is exactly what was observed in experi-ments [8].

5. CONCLUSIONThe analysis performed above shows that the time

dependences of the electron density measured in [8] donot allow one to conclude whether three-body recombi-nation occurs in the expansion of an ultracold laser-plasma plume. The assumption that recombination issuppressed is supported by the fact that no splash in thesignal related to outgoing electrons was detected [1, 8].Such a splash should be observed in the regime of recombination heating of plasma electrons.

v x( ) v 0 1 1/ x–( ),=

t / t 0 x

2

x–( )

1/2

1/2( ) 2 x 1– 2 x

2

x–( )

1/2

+( ),ln+=

x r / σ0, v 0 2eq / σ0mi( )1/2,= =

(0

σ0∫

q / e 4π / σ0

2r D 3 10

3,×≈=

v 0 4π1/2σ0 / mi( )1/2 N eT e( )1/4 150 m/s.∼=

1 × 108

500 100 150 200

r , µm

1 × 106

1 × 107

1 × 109

1 × 1010

N i, e(r ) –400

ni, e(r ), ue(r )10/K

–200

0

200

400

600

R

(a)

(b)

R

Fig. 3. Distributions of (solid curves) ions and (dotted

curve) electrons over the distance from the center of asphere with a radius R where the initial coordinates of par-ticles were defined: (a) n(r ) is the number of particles withina sphere with radius r , the dashed line shows the electronpotential energy u(r ) in units of K/10 at the point r and (b)

N (r ) is the density of particles at the distance r (in µm).Averaging is performed over the time interval 7T L < t < 8T L(where T L = 2π / ω L is the Langmuir period).




ON THE RECOMBINATION HEATING 981

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a. n. tkachev and s. i. yakovlenko- on the recombination heating of ultracold laser-produced plasmas

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