a r X i v : 1 0 1 1 . 5 6 3 5 v 1 [ p h y s i c s . p l a s m  p h ] 2 5 N o v 2 0 1 0
Radiation Reaction E
ﬀ
ects on Electron Nonlinear Dynamicsand Ion Acceleration in Lasersolid Interaction
M. Tamburini
a,
∗
, F. Pegoraro
a
, A. Di Piazza
b
, C. H. Keitel
b
, T. V. Liseykina
c
, A. Macchi
a,d
a
Dipartimento di Fisica “E. Fermi”, Universit`a di Pisa, Largo Bruno Pontecorvo 3, I56127 Pisa, Italy
b
MaxPlanckInstitut f¨ ur Kernphysik, Saupfercheckweg 1, D69117 Heidelberg, Germany
c
Institute of Computer Technologies, SDRAS, Novosibirsk, Russia and Institute of Physics, University of Rostock, Germany
d
Istituto Nazionale di Ottica, CNR, research unit “A. Gozzini”, Pisa, Italy
Abstract
Radiation Reaction (RR) e
ﬀ
ects in the interaction of an ultraintense laser pulse with a thin plasma foil are investigated analyticallyand by twodimensional (2D3P) ParticleInCell (PIC) simulations. It is found that the radiation reaction force leads to a signiﬁcantelectron cooling and to an increased spatial bunching of both electrons and ions. A fully relativistic kinetic equation including RRe
ﬀ
ects is discussed and it is shown that RR leads to a contraction of the available phase space volume. The results of our PICsimulations are in qualitative agreement with the predictions of the kinetic theory.
Keywords:
Radiation Reaction, Ion Acceleration, LaserPlasma Interaction, Radiation Pressure
1. Introduction
Current laser systems may deliver intensities up to10
22
Wcm
−
2
[1] and intensities up to 10
26
Wcm
−
2
are expectedat the Extreme Light Infrastructure (ELI). In such ultrahighintensity regime and for typical laser wavelength
λ
∼
0
.
8
µ
mthe motion of electrons in the laser ﬁeld is ultrarelativistic andRadiation Reaction (RR) e
ﬀ
ects may become important. TheRR force describes the backaction of the radiation emitted byan accelerated electron on the electron itself and accounts forthe loss of the electron energy and momentum due to the emission of such radiation. Apart from the need of including RRe
ﬀ
ects in the dynamics of laserplasma interactions in the ultrarelativistic regime, the latter also o
ﬀ
ers for the ﬁrst time theopportunity to detect RR e
ﬀ
ects experimentally [2, 3].
In this paper we present an approach to a kinetic description of laserplasma interactions where RR e
ﬀ
ects are includedvia the LandauLifshitz (LL) force [4]. Some properties of thekinetic equation with RR are discussed and in particular it isproved that the RR force leads to a
contraction
of the phasespace volume. Then, PIC simulations are used to study RR effects on the acceleration of a thin plasma foil in the regime of Radiation Pressure dominance [5]. Numerical simulations [5]
suggested that Radiation Pressure Acceleration (RPA) becomesthe dominant mechanism of ion acceleration at intensities exceeding 10
23
Wcm
−
2
. Such RPA regime is attractive becauseof the foreseen high e
ﬃ
ciency, the quasimonoenergetic features expected in the ion energy spectrum and the possibilityto achieve a potentially “unlimited” acceleration [6]. PreviousParticleInCell (PIC) simulations [7] showed signatures of RR
∗
Corresponding author
Email address:
tamburini@df.unipi.it
(M. Tamburini)
e
ﬀ
ects at intensities exceeding 5
×
10
22
Wcm
−
2
and increasingnonlinearly with the laser intensity. More recent simulationsstudies of RPA both for thick targets [8, 9] and ultrathin tar
gets [10] suggested that the inclusion of the RR force cools theelectrons and may improve the quality of the ion spectrum.Our approach to the inclusion of RR e
ﬀ
ects in a PIC codehas been discussed in detail in Ref.[11] where onedimensional(1D)simulationsofRPA havebeenalso reported. Inthepresentpaper we report both additional 1D simulations and ﬁrst twodimensional (2D) simulations using parameters similar to thoseof Ref.[12] where, in particular, the impact of a RayleighTaylorlike instability on a thin foil acceleration was studied.Inclassical electrodynamics,thee
ﬀ
ectofRR canbeincludedby means of the LL force [4]
F
R
=
−
4
π
3
r
e
λ
·
γ
∂∂
t
+
v
·∇
E
+
v
×
∂∂
t
+
v
·∇
B
−
E
+
v
×
B
×
B
+
v
·
E
E
+
γ
2
E
+
v
×
B
2
−
v
·
E
2
v
,
(1)where
v
is the electron velocity,
γ
is the relativistic factor,
r
e
≡
e
2
/
mc
2
≈
2
.
8
×
10
−
9
µ
m is the classical electron radius,
λ
=
2
π
c
/ω
is the laser wavelength and we use dimensionlessquantities as in the PIC code: time, space and momentum arenormalized in units of
ω
−
1
,
c
ω
−
1
and
mc
, respectively. Consequently, EM ﬁelds are normalized in units of
m
ω
c
/

e

and densities in units of the critical density
n
c
=
m
ω
2
/
4
π
e
2
.The LL approach holds in the classical framework and quantum e
ﬀ
ects are neglected. As pointed out in [11], the ﬁrst termof the LL force Eq.(1) i.e. the one containing the
derivatives
of the electric and magnetic ﬁelds, should be neglected becauseits e
ﬀ
ect is smaller than quantum e
ﬀ
ects such as the spin force.
Preprint submitted to Nuclear Instruments and Methods Section A November 29, 2010
However, in Sec.(2) we show the e
ﬀ
ect of each term of the LLforce Eq.(1) on the rate of change of the phase space volume.
2. The kinetic equation with Radiation Reaction
In this section, a fully relativistic kinetic equation that includes the RR e
ﬀ
ects is discussed. We show a few basic properties ofthe kineticequationpointingoutthepeculiaritiesoftheRR force whose main new feature is that it
does not
conservethe phasespace volume.Generalized kinetic equations for nonconservative forcesand in particular for the RR force have been known since latesixties forthe LorentzAbrahamDirac(LAD)equation[13, 14]
and late seventies for the LL equation [15]. Recently, the generalized kinetic equation with the LL force included has beenused to study the RR e
ﬀ
ects on thermal electrons in a magnetically conﬁned plasma [16] and to develop a set of closed ﬂuidequations with RR [17–19]. In this paper, we give the kinetic
equation in a nonmanifestly covariant form, see [15, 16] for
the kinetic equation in a manifestly Lorentzcovariant form.The relativistic distribution function
f
=
f
(
q
,
p
,
t
) evolvesaccording to the collisionless transport equation
∂
f
∂
t
+
∇
q
·
(
f
v
)
+
∇
p
·
(
f
F
)
=
0
,
(2)where
q
are the spatial coordinates,
v
=
p
/γ
is the threedimensional velocity,
γ
=
1
+
p
2
is the relativistic factor and
F
=
F
L
+
F
R
is the mean force due to external and collectiveﬁelds (
F
L
≡ −
(
E
+
v
×
B
) is the Lorentz force and
F
R
is givenin Eq.(1)). Physically, Eq.(2) implies the conservation of the
number of particles.The new key feature compared to the usual Vlasov equation is that for the RR force
F
R
we have
∇
p
·
F
R
0. UsingLagrangian coordinates
q
(
t
)
,
p
(
t
), Eq.(2) can be recast in theequivalent form
d
ln
f dt
=
−∇
p
·
F
.
(3)According to Eq.(3),
∇
p
·
F
providesthe percentageof variationofthedistributionfunction
f
withinthecharacteristictimescale
ω
−
1
. IntegratingEq.(3) alongits characteristics,we ﬁndthatthedistribution function
f
remains positive as required.Introducingtheentropydensityin thephasespace
s
(
q
,
p
,
t
)
=
−
f
(
q
,
p
,
t
)ln
f
(
q
,
p
,
t
), from Eq.(2) we get the equation for theevolution of the entropy density
∂
s
∂
t
+
∇
q
·
(
s
v
)
+
∇
p
·
(
s
F
)
=
f
∇
p
·
F
.
(4)Integrating Eq.(4) in the phase space, we get the rate of variation of the total entropy
S
(
t
)
dS
(
t
)
dt
=
d
3
qd
3
p f
∇
p
·
F
.
(5)The Lorentz force
F
L
≡ −
(
E
+
v
×
B
) gives
∇
p
·
F
L
=
0 identically thus
∇
p
·
F
=
∇
p
·
F
R
. Moreover, the distribution function
f
(
q
,
p
,
t
) is always nonnegative
f
≥
0 thus the sign of
dS
/
dt
isgiven by
∇
p
·
F
R
solely.From the LL force Eq.(1) we get [20]
∇
p
·
F
R
=
−
4
π
3
r
e
λ
∇
q
·
E
−
v
·
∇
q
×
B
−
∂
E
∂
t
+
2
E
2
+
B
2
γ
+
4
γ
v
×
E
2
+
v
×
B
2
−
2
v
·
E
×
B
.
(6)In a plasma, the kinetic equation is coupled with the Maxwellequations for the selfconsistent ﬁelds
∇
q
·
E
=
ρ ρ
c
=
1
n
c
j
=
e
,
i
Z
j
d
3
p f
j
(
q
,
p
,
t
) (7)
∇
q
×
B
−
∂
E
∂
t
=
j
j
c
=
1
n
c
c
j
=
e
,
i
Z
j
d
3
p
v
f
j
(
q
,
p
,
t
)
,
(8)where
ρ
c
≡ 
e

n
c
,
j
c
≡ 
e

n
c
c
,
d
3
qd
3
p f
j
(
q
,
p
,
t
)
=
N
j
is thetotal number of particles for each species (
j
=
e
electrons,
j
=
i
ions) and
Z
j
is the charge of the particle species in units of

e

(for electrons
Z
e
=
−
1). For a plasma, Eq.(6) can be recast as
∇
p
·
F
R
=
−
4
π
3
r
e
λ
ρ ρ
c
−
v
·
j
j
c
+
2
E
2
+
B
2
γ
+
4
γ
v
×
E
2
+
v
×
B
2
−
2
v
·
E
×
B
.
(9)The terms of Eq.(9) proportionalto the charge density
ρ
and tothe current density
j
come from the ﬁrst term of the LL forceEq.(1) i.e. the term containing the derivatives of the ﬁelds. Ingeneral, these terms can give either a positive or negative contribution to
∇
p
·
F
R
. The second term of Eq.(9) i.e. the termproportional to (
E
2
+
B
2
) has always a negative sign, its effect decreaseswith increasingelectronenergyandit is typicallynegligible. The third term of Eq.(9) comes from the stronglyanisotropic “friction” term of the LL force i.e. the term proportional to
γ
2
in Eq.(1) (see [11] for a detailed discussion of this
term) and dominates in the ultrarelativistic limit
γ
≫
1.It is possible to prove [20] the following statement: for any
v
such that

v
 ∈
[0
,
1[ then
(
v
×
E
)
2
+
(
v
×
B
)
2
−
2
v
·
(
E
×
B
)
+
E
2
+
B
2
2
γ
2
≥
0
,
(10)therefore according with Eqs.(5, 9), the terms of the LL forceEq.(1) that
do not
depend on the derivativesof the ﬁelds alwayslead to a
contraction
of the available phase space volume. In afew special cases, the e
ﬀ
ect of the terms of the LL force Eq.(1)that depend on the derivatives of the ﬁelds (i.e. the terms proportional to
ρ
and
j
in Eq.(9)) might lead to an expansion of thephase space volume. Anyway, their e
ﬀ
ect should be negligiblecompared to quantum e
ﬀ
ects as discussed in [11].We show explicitly the contraction of the phase space in thespecial case of a small bunch of electrons interacting with aplane wave where collective ﬁelds are assumed to be negligible compared with the plane wave ﬁelds. Assuming an initialdistribution
f
=
g
(
q
)
δ
3
(
p
−
p
0
), from Eqs.(5, 9) we have
dS
(
t
)
dt
=
−
4
π
3
r
e
λ
d
3
q g
(
q
)
2
E
2
+
B
2
γ
(
p
0
)
+
4
γ
(
p
0
)
··
(
v
0
×
E
)
2
+
(
v
0
×
B
)
2
−
2
v
0
·
(
E
×
B
)
,
(11)2
where
v
0
=
p
0
/γ
(
p
0
). If the electron bunch counterpropagateswith the plane wave ([
v
0
·
(
E
×
B
)]
<
0) or propagates in thetransverse direction ([
v
0
·
(
E
×
B
)]
=
0), from Eq.(11) it is clearthat RR leads to a contraction of the phase space. In particular,in thecase ofcounterpropagation(using

E

=

B

,
E
·
B
=
0)wehave
∇
p
·
F
R
=
−
(4
π
r
e
/
3
λ
)4
E
2
2
γ
(
p
0
)

v
0

(1
+

v
0

)
+
1
/γ
(
p
0
)
.On the other hand, if the bunch propagates in the same direction of the plane wave (
v
0
parallel to
E
×
B
), then the contribution of the friction term (proportional to
γ
in Eq.(9)) becomes comparable with the contribution of the second term(proportional to (
E
2
+
B
2
) in Eq.(9)) and we have
∇
p
·
F
R
=
−
(4
π
r
e
/
3
λ
)
4
E
2
/
(1
+

v
0

)
2
γ
3
(
p
0
)
which still leads to a contraction of the phase space but with a rate
γ
4
smaller than thecase of counterpropagation. This reinforcesthe evidenceofthestrongly
anisotropic
featuresoftheLLforceEq.(1)(see[11]for
further details).The physical interpretation of the above properties is that theRR force acts as a cooling mechanism for the system: part of the energy and momentum are radiated away and the spread inboth momentum and coordinate space may be reduced. Thisgeneral prediction is conﬁrmed by our PIC simulations (seeSec.3) where we foundthat RR e
ﬀ
ects lead to bothan increasedbunching in space and to a noticeable cooling of hot electrons.Finally, it is worthwhile mentioning that Eq.(2) is more general than the Vlasov equation but the PIC approach is still validi.e. the PIC approach provides a solution for Eq.(2) and it notlimited to the Vlasov equation [20].
3. PIC simulations
Suitable approximations to the LL force and our approachto its inclusion in a PIC code are described in Ref.[11]. Thenumerical approach is based on the widely used Boris particlepusher and it can be implemented in codes of any dimensionality. Inclusion of RR e
ﬀ
ects via this method in PIC simulationsleads to only a
∼
10% increase in CPU time, which may beessential to perform largescale simulations with limited computing power.
3.1. 1D simulations
We ﬁrstreportonedimensional(1D3P)PIC simulationswithlaser and plasma parameters similar to Ref.[5]. Previous 1Dsimulations in this regimehave been reportedin Ref.[11] wherea detailed comparison with other work is also made. In thepresent paper we review the basic observations in the 1D caseand we include results at intensities higher than those investigated in Ref.[11].The target is a plasma foil of protons with uniform initialdensity
n
0
=
100
n
c
and thickness
ℓ
=
1
λ
where
λ
=
0
.
8
µ
m isthe laser wavelength and
T
=
λ/
c
≈
2
.
67fs is the laser period.In these simulations, the laser pulse front reaches the edge of the plasma foil at
t
=
0, the proﬁle of the laser ﬁeld amplitudehas a “trapezoidal” shape in time with onecycle, sin
2
functionrise and fall and a ﬁve cycles constant plateau. We consideredthree intensities
I
=
2
.
33
×
10
23
Wcm
−
2
,
I
=
5
.
5
×
10
23
Wcm
−
2
and
I
=
10
24
Wcm
−
2
for both Circular (CP) and Linear (LP)polarization of the laser pulse.In the CP case, we found that RR e
ﬀ
ects on the ion spectrumare negligible even at intensities of
I
=
10
24
Wcm
−
2
as shownin Fig.1. For CP, electrons pile up and the numerical densitygrows exceeding thousand of times the critical density
n
c
. Thelaser pulse does not penetrate deeply into the target (i.e. thee
ﬀ
ectiveskindepthis a verysmall fractionofthe foil thickness)and electrons move in a ﬁeld much weaker than the vacuumﬁeld.In Ref.[5] it was expected that RR e
ﬀ
ects in the radiationpressure dominated acceleration of the thin foil would havebeen weak because in this regime the whole foil becomesquickly relativistic, hence in the foil frame the laser wavelength
λ
′
increasesandthe typicalstrengthof theRR parameter
∼
r
e
/λ
[seeEq.(1)]decreases. ThepresentcaseofaccelerationwithCPpulses appears to conﬁrm this picture. The weakness of RR effects may also be explained on the basis of the LL equation foran electron moving into a plane wave [21]. As electrons movein the forward direction coherently with the foil (while rotatingin the transverse plane in the CP ﬁeld) and the amplitude of thereﬂected wave is weak when the foil is strongly relativistic, thesituation is similar to an electron copropagatingwith the planewave at a velocityclose to
c
, forwhichthe LLforcealmost vanishes [11]. The relativistic motion of the foil also prevents theonset of SelfInduced Transparency by increasing the opticalthickness parameter
ζ
=
π
n
0
ℓ/
n
c
λ
in the foil frame (see [22]and references therein) . For smaller target thickness, breakthrough of the laser pulse occurs and RR e
ﬀ
ects are greatly enhanced also for CP [11].It may be worth noticing that, at the highest intensity considered
I
=
10
24
Wcm
−
2
, in principle one would expect the classical approach to RR to break down due to the onset of quantumelectrodynamics (QED) e
ﬀ
ects, as discussed in Ref.[11]. However, it can be shown by a direct analysis of the simulation datathat the thresholdconditionforQED e
ﬀ
ect is not violatedin theCP case.
Figure 1: Ion energy spectra at
t
=
66
T
with (top) and without (bottom) RRfor Circular Polarization. The laser intensity
I
is 2
.
33
×
10
23
Wcm
−
2
(yellow),5
.
5
×
10
23
Wcm
−
2
(blue), 10
24
Wcm
−
2
(red) and the target thickness is
ℓ
=
1
λ
.
3
Figure 2: Ion energy spectra at
t
=
14
T
with (top) and without (bottom) RRfor Linear Polarization. The laser intensity
I
is 2
.
33
×
10
23
Wcm
−
2
(yellow),5
.
5
×
10
23
Wcm
−
2
(blue), 10
24
Wcm
−
2
(red) and the target thickness is
ℓ
=
1
λ
.
For linear polarization(LP), di
ﬀ
erentlyfrom the CP case, wefound that RR e
ﬀ
ects are important leading to a reduction of the maximum achievable ion energy and to some narrowing of the width of the ion spectrum as shown in Fig.2. This di
ﬀ
erent dynamics for LP is correlated with the strong longitudinaloscillatory motion driven by the oscillating component of the
j
×
B
force which is suppressed in the CP case. This allows adeeper penetration of the laser pulse into the foil with a signiﬁcantfractionofelectronsonthefrontsurfacemovinginastrongelectromagnetic ﬁeld of the same order of vacuum ﬁelds [11].The relative reduction in the ion energy when RR is included isclose to the percentage of the laser pulse energy which is lostas highenergy radiation escaping from the plasma.The results for LP (Fig.2) are shown for the same intensityvalues of the CP case (Fig.1) for a direct comparison. However, at least for the highest intensity case, the LP results mustbe taken with some caution as the condition for the validity of a classical approach tends to be signiﬁcantly violated. In suchregime, an analysis based on quantumRR e
ﬀ
ects might be necessary [23, 24].
3.2. 2D simulations
We report preliminary twodimensional (2D3P) PIC simulations with laser and plasma parameters similar to Ref.[12]. Tothebest ofourknowledge,thisis theﬁrst paperreportingresultsof twodimensional PIC simulations with RR e
ﬀ
ects included.The target is a plasma slab of fully ionized deuterium(
Z
/
A
=
1
/
2) of width 40
λ
, density
n
0
=
169
n
c
and thickness
ℓ
=
0
.
5
λ
.The size of the computational box is 95
λ
×
40
λ
with a spatialresolution
∆
x
= ∆
y
=
λ/
80 and 625 quasiparticles per cellcorresponding to a total of 8
×
10
7
quasiparticles. The laserpulse is spolarized with the electric ﬁeld along the
z
axis. Itsnormalizedamplitude is
a
0
=
320 correspondingto an intensity
I
=
1
.
4
×
10
23
Wcm
−
2
with a wavelength
λ
=
1
.
0
µ
mandperiod
T
=
λ/
c
≈
3
.
3fs. Thepulsehas a Gaussian transverseproﬁleof width 20
λ
FWHM and a sin
2
longitudinal proﬁle of length 40
λ
Figure 3: Plots of the 2D PIC simulations at
t
=
70
T
. The laser pulse is spolarized with an intensity
I
=
1
.
4
×
10
23
Wcm
−
2
and the target thickness is
ℓ
=
0
.
5
λ
. From top to bottom, ion
n
i
and electron
n
e
density distributions with(left column) and without (right column) RR, longitudinal
E
x
(ﬁrst row) andtransverse
E
z
(second row) electric ﬁeld, ion and electron spectrum with (red)and without (blue) RR.
FWHM. In these simulations, the frontof the lase pulse reachesthe foil at
t
=
0.Comparing the results of our simulations with and withoutRR (see Fig.3, we report the results at
t
=
70
T
) it is apparent4
that RR leads to both an increased electron and ion bunchingand to a strong cooling of electrons. These results are qualitatively consistent with our expectations from the kinetic theorythat we have discussed in Sec.2 and in particular with the prediction of a contraction of the electrons available phase spacevolume.A qualitative understanding of these results can be achievedrecalling that the RR force Eq.(1) is mainly a stronglyanisotropic and nonlinear frictionlike force that reaches itsmaximum for electrons that counterpropagate with the laserpulse [11]. The backward motion of electrons is thus impededby RR, more electrons and consequently ions are pushed forward leading to an enhanced clumping that improves the e
ﬃ
ciency of the RPA mechanism. In fact, the ion spectrum withRR shows a region between about three hundred and six hundredMeVwithasigniﬁcantincreaseinthenumberofionscompared to the case without RR (Fig.3). This picture is conﬁrmedby both the enhancement of the longitudinal electric ﬁeld
E
x
and by the formation of denser bunches in the ion density compared to the case without RR (see Fig.3). However, for linearpolarization, hot electrons are always generated by the oscillating component of the
j
×
B
force. The generation of hot electrons providesa competingaccelerationmechanismto RPA andultimatelyleadstothegenerationofthefractionofionswiththehighest energy. The noticeable suppression of the
j
×
B
heatingmechanismdue to the RR forcethereforeleads to a lower maximum cuto
ﬀ
energyboth in the electron and in the ion spectrum(see Fig.3).These preliminary results for twodimensional simulationswith RR e
ﬀ
ects includedsuggest that, in the LP case, the trendsfound in onedimensional simulations hold qualitatively evenfor higher dimensions. More detailed studies and quantitativecomparisons between onedimensional and twodimensionalPIC simulations are left for forthcoming publications.
4. Conclusions
We summarize our results as follows. Radiation Reactione
ﬀ
ects on the electron dynamics in the interaction of an ultraintense laser pulse with a thin plasma foil were studied analytically and by onedimensional and twodimensional PIC simulations. The details of the numerical implementation of the RRforce in our PIC code were described in Ref.[11].In onedimensional simulations, we checked RR e
ﬀ
ects forthree di
ﬀ
erent intensities:
I
=
2
.
33
×
10
23
Wcm
−
2
,
I
=
5
.
5
×
10
23
Wcm
−
2
and
I
=
10
24
Wcm
−
2
comparing the results forCircular and Linear Polarization of the laser pulse. For CP,we found that RR e
ﬀ
ects are not relevant even at intensityof
I
=
10
24
Wcm
−
2
whenever the laser pulse does not break through the foil. In contrast, for LP we found that RR e
ﬀ
ectsare important reducing the ion energy signiﬁcantly.In twodimensional simulations, we found that RR reducesthe
j
×
B
heating mechanism leading to a lower maximum cuto
ﬀ
energy both in the electron and in the ion spectrum. Moreover, RR increases the spatial bunching of both electrons andions which are collected into denser clumps compared to thecase without RR. This might lead to a somewhat beneﬁcial effect with a longer and more e
ﬃ
cient radiation pressure acceleration phase whose signature would be an ion energy spectrumpeaking at an intermediate energy.A generalized relativistic kinetic equation including RR effects has been discussed and we have shown that RR leads to acontraction of the available phase space volume. This prediction is in qualitative agreement with the results of our PIC simulations where we observed both an increased spatial bunchingand a signiﬁcant electron cooling as discussed above.
Acknowledgments
We acknowledge the CINECA award under the ISCRA initiative (project “TOFUSEX”), for the availability of high performance computing resources and support.
References
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