The effect of quantum correction on plasma electron heating in ultraviolet laserinteractionS. Zare, E. Yazdani, R. Sadighi-Bonabi, A. Anvari, and H. Hora Citation: Journal of Applied Physics 117, 143303 (2015); doi: 10.1063/1.4916373 View online: http://dx.doi.org/10.1063/1.4916373 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/117/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dual effects of stochastic heating on electron injection in laser wakefield acceleration Phys. Plasmas 21, 083103 (2014); 10.1063/1.4892262 Ponderomotive acceleration of electrons by a laser pulse in magnetized plasma Phys. Plasmas 16, 043103 (2009); 10.1063/1.3098537 Influence of a preplasma on electron heating and proton acceleration in ultraintense laser-foil interaction J. Appl. Phys. 104, 103307 (2008); 10.1063/1.3028274 Determination of electron-heated temperatures of petawatt laser-irradiated foil targets with 256 and 68 eVextreme ultraviolet imaging Rev. Sci. Instrum. 79, 093507 (2008); 10.1063/1.2987683 Strong kinetic effects in cavity-induced low-level saturation of stimulated Brillouin backscattering for high-intensitylaser-plasma interaction Phys. Plasmas 12, 043101 (2005); 10.1063/1.1862246

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The effect of quantum correction on plasma electron heating in ultravioletlaser interaction

S. Zare,1 E. Yazdani,2 R. Sadighi-Bonabi,1,a) A. Anvari,1 and H. Hora3

1Department of Physics, Sharif University of Technology, P.O. Box 11365-9567, Tehran, Iran2Department of Energy Engineering and Physics, Amirkabir University of Technology,P.O. Box 15875-4413, Tehran, Iran3Department of Theoretical Physics, University of New South Wales, Sydney 2052, Australia

(Received 10 November 2014; accepted 18 March 2015; published online 14 April 2015)

The interaction of the sub-picosecond UV laser in sub-relativistic intensities with deuterium is

investigated. At high plasma temperatures, based on the quantum correction in the collision fre-

quency, the electron heating and the ion block generation in plasma are studied. It is found that due

to the quantum correction, the electron heating increases considerably and the electron temperature

uniformly reaches up to the maximum value of 4.91� 107 K. Considering the quantum correction,

the electron temperature at the laser initial coupling stage is improved more than 66.55% of the

amount achieved in the classical model. As a consequence, by the modified collision frequency,

the ion block is accelerated quicker with higher maximum velocity in comparison with the one by

the classical collision frequency. This study proves the necessity of considering a quantum mechan-

ical correction in the collision frequency at high plasma temperatures. VC 2015 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4916373]

INTRODUCTION

The laser-driven acceleration of plasma is based on the

ability to create picosecond or sub-picosecond laser pulses

and high power lasers.1,2 Interaction of such laser pulses

with solid targets can produce accelerating gradients several

orders of magnitude higher than conventional techniques.1

The accelerated ions are highly attractive due to its potential

applications in numerous fields of scientific research, e.g.,

fast ignition, proton radiography, and medical applica-

tions.3–9 Various mechanisms have been proposed to explain

laser-driven ion acceleration, including: Target Normal

Sheath Acceleration (TNSA),10 Coulomb Explosion (CE),11

Skin Layer Ponderomotive Acceleration (SLPA), and

Radiation Pressure Acceleration (RPA).12 In SLPA regime,

the ponderomotive force near critical plasma surfaces

(induced by inhomogeneity of the laser field) efficiently

accelerates the electrons and ions. In SLPA mechanism, the

electrostatic field caused by charge separations due to

the ponderomotive force accelerates the ions.13 A part of the

absorbed laser energy is converted into directed plasma

beams containing ions surrounded by an electron cloud. As

almost whole directed energy of the plasma beams is carried

by ions, this beam is usually called the ion block or the

plasma block.14,15 The nonlinear force creates two ion blocks,

a block with negative velocity moves against the laser light,

which could be used for some applications of ion sources.16,17

Another block with positive velocity penetrates into the target

interior. This ion block could be utilized to ignite very high

efficiency nuclear fusion reactions in uncompressed fuels.18

Because of deep penetration in plasma, short wavelength

lasers are attractive for compressing fuel cores in the fusion

reactions. Furthermore, the short wavelength pulses result in

higher ablation pressure, greater hydrodynamic efficiency,

and less laser-plasma instabilities including the Rayleigh-

Taylor instability in comparison with the long wavelength

pulses.19 These results present outstanding application of

short wavelength lasers in fast ignitions.

In the laser interaction with high temperature plasmas, if

the collisional absorption is dominant, a quantum correction

in the collision frequency is required. The effect of this mod-

ification has been confirmed experimentally and theoreti-

cally. Grieger et al. in 1981 reported that in stellarators

working with deuterium at temperature of 800 eV, the diffu-

sion across the magnetic field was 20 times larger than the

classical value.20 This enhancement is due to improvement

of Ohmic heating for tokomak at higher temperatures than

what expected before.21 The electrical conductivity of fully

ionized plasmas in tokamaks is based on a coulomb collision

frequency between the electrons and ions. Comparing the

evaluations with experimental results, it could be concluded

that the Tokamak works only because of this quantum cor-

rection of the collisions (anomalous resistivity) and plasmon

effects.22

In this work, based on two-fluid model, the interaction

of the sub-picosecond UV laser of 248 nm wavelength with

near critical density plasma is studied. Considering the quan-

tum corrected collision frequency, the evolution of the elec-

tron heating and the ion block generation is presented.

THEORY AND METHOD

One of the fundamental problems of wave propagation

in plasmas is the fact that these plasmas are nearly always in-

homogeneous (Sec. 7 of Ref. 23). Mathematically, this is

related to the solution of the ordinary differential equation

for the spatial-dependent electric field amplitude, E by the

wave equations,23a)Sadighi@sharif.ir

0021-8979/2015/117(14)/143303/5/$30.00 VC 2015 AIP Publishing LLC117, 143303-1

JOURNAL OF APPLIED PHYSICS 117, 143303 (2015)

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d2

dx2Eþ2

dE

dx

� �d ln nð Þ

dx

� ��x2

c2n2þ2

c

x

� �2 d2ln nð Þdx2

" #E¼ 0:

(1)

x and c are the laser angular frequency and the vacuum

speed of light, respectively, and n is the refractive index of

the plasma, introduced by Lust,24

n2 ¼ 1�x2

p

x2 � ivx: (2)

xp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pe2ne=me

pand v are the electron plasma frequency

and the electron-ion collision frequency, separately, where

ne, me, and e represent the plasma density, the mass, and

charge of the electron, respectively. In inhomogeneous plas-

mas, the solution of Eq. (1) would be complicated. However,

in 1880, Rayleigh25 reported a solution for an inhomogene-

ous medium with a refractive index of the form,23,26

n ¼ 1

1þ b x: (3)

b is a constant parameter. The solution of Eq. (1) was exactly

expressed by elementary functions and amplitude of E0,23

E xð Þ ¼ 1

1þ b x

� �1=2

E0

� exp 6i

2ln 1þ b xð Þ 2x

cb

� �2

� 1

( )1=224

35: (4)

It was clarified by Hora27 that there exist only two such exact

solutions in the inhomogeneous optical medium, one for a

wave moving to the x direction and another for a wave mov-

ing to the �x direction.

Using plasma hydrodynamics for a one-dimension

plasma in the x direction, the force density could be repre-

sented as follows:23

f ¼ � @

2 @xne kB Tð Þ � @

@x

E2 þ H2

8p

� �: (5)

kB and T represent the Boltzmann constant and the plasma

temperature, respectively. The first term of Eq. (5) represents

the gasdynamic force based on the gas-dynamical pressure

and the second term is the nonlinear ponderomotive force,

fNL, due to the electric field, E and magnetic field, H of the

laser pulse.23

The laser energy in plasma is absorbed by the electrons

which are partially thermalized via the collisions in the laser

fields. The thermal collision effect due to the laser absorption

is entirely included in the following complete hydrodynamic

computations.28 In the classical model, the impact parameter,

r0 ¼ Ze2=meV2 and v are concluded by considering the hy-

perbolical motion of an electron with velocity of V around

Ze potential of the ion (Z shows the charge of the collided

ion). Therefore, the electron-ion collision frequency is,21,23

v ¼ Zp e4ne

3kBTð Þ3=2m1=2e

: (6)

At high temperatures, the impact parameter for 90� scatter-

ing can no longer be described by the classical mechanics

where the de Broglie wavelength, kdB ¼ h=ð2meEeÞ1=2(h and

Ee represent the Planck’s constant and the electron energy,

respectively) is larger than the classical impact parameter for

90� scattering. In this condition introduced by Bethe, the

electron performs a 90� wave mechanical diffraction around

the ion,23,29 and as a consequence, the quantum correction in

the collisions appears and the impact parameter becomes

equal to,21,23

r0 ¼rBohr

2Z

1

1þ 4T�=Tð Þ1=2 � 1

¼

rBohr

4ZTT� ¼ Ze2

3kBTT � T�

rBohr

4Z

ffiffiffiffiffiT�

T

rT � T�

;

8>>><>>>:

(7)

where rBohr ¼ �h2=mee2 is the Bohr radius, and the threshold

temperature for changing from the classical to the quantum

regimes is known as,21,23

T� ¼ 4Z2mec2a2

3kB¼ 4:176� 105 Z2 K: (8)

a ¼ e2=�hc characterizes the fine structure constant. After

that, the collision frequency is generalized as30

v ¼ p ne r2Bohr

4Z3

ffiffiffiffiffiffiffiffiffiffi3kBT

me

s1

1þ 4T�=Tð Þ 1=2 � 1� �

2

¼

Zp nee4

3kBTð Þ3=2m1=2e

T � T�

T

T�Zp nee4

3kBTð Þ3=2m1=2e

T � T�;

8>>>><>>>>:

(9)

for temperatures below T� ¼ 35:6 Z2 eV, Eq. (9) results in

the identical value with Eq. (6) (Chap. 2.6 of Ref. 23).

Similarly, the classical Spitzer value of the electrical conduc-

tivity, jei is modified to,23

jei ¼e ne

2mev¼ eZ3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3kBTmerBohr

p ð ð1þ 4T�=TÞ 1=2 � 1 Þ 2

¼

3kBTð Þ3=2

2Z m1=2e e2

T � T�

T�

T

3kBTð Þ3=2

2Z m1=2e e2

T � T�;

8>>>><>>>>:

(10)

the thermal conduction in a tokamak parallel to the mag-

netic field measured by Razumova31 is rather close to the

quantum corrected results (Fig. 2.6 of Ref. 23). However,

this amount is almost 20 times less than the one resulted

from the classical electrical conduction. Consequently, it is

necessary to express the thermalization of the quivering

electrons on the basis of the quantum modification of the

collision frequency.

143303-2 Zare et al. J. Appl. Phys. 117, 143303 (2015)

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COMPUTATIONAL RESULTS AND DISCUSSION

Computations for plane geometry interaction with con-

sidering the nonlinear force and thermal laser absorption by

collisions and equipartition processes are presented for inter-

action of KrF laser irradiating deuterium with Z¼ 1.

In order to simulate the laser-plasma interaction, a one-

dimensional two-fluid hydrodynamic code is used.32–36 This

code in contrast to the usual two-fluid hydrodynamics derived

by Schluter37 and completed by Hora23,38 includes the gener-

ation of electric double layers, longitudinal electric field

effects and collisions. The plasma treatment is recognized by

solving the Poisson equation and the equations of continuity,

motion, and energy conservation for electrons and ions.32

Using the Rayleigh profile for the inhomogeneous plasma

density and considering b ¼ 3:78� 104 cm�1,14,26 the previ-

ous equations are solved.

Considering the time dependence of the laser-plasma

interaction, computations are performed for the initial den-

sity of the plasma including initial temperature and initial

zero macroscopic velocities.40 By the quantum correction in

the collision frequency, attention is mainly focused on pro-

ducing the plasma block directed inside the plasma with less

distortion in the laser propagation direction.

The initial temperature of the deuterium plasma located

between �10 lm and þ10 lm is assumed to be 400 eV and

the critical density is considered to be 1.85� 1022 cm�3.39

The KrF laser with intensity of 5� 1016 W cm�2, wave-

length of 248 nm, and 300 fs pulse duration is irradiated

from the left-hand side of the plasma within 300–1000 fs.

Based on the above mentioned evaluations, the theoretical

calculations are organized, and the electron and ion accelera-

tion by considering the quantum modified collision fre-

quency are compared by the classical cases. It is noticed that

the initial temperature is higher than the critical temperature

of 35.9 eV for deuterium atoms (Z¼ 1).

FIG. 1. The KrF laser pulse with intensity of 5� 1016 W cm�2 and 300 fs pulse duration is irradiated from the left-hand side of the deuterium plasma located

between �10 and þ10 lm. The initial plasma temperature is assumed to be 400 eV: the electron temperature by considering the quantum corrected collision

frequency (a) and the classical collision frequency (b).

FIG. 2. Comparing the electron temperature at the initial step, by the quan-

tum correction collision frequency (solid line) and the classical (dashed line)

in conditions of Fig. 1. This figure denotes that the electron temperatures by

the modified collision frequency and the classical case are 3.45� 107 K and

2.07� 107 K, respectively.

FIG. 3. The maximum electron temperature at interaction time of 700 fs,

with the corrected collision frequency (star) and the classical collision fre-

quency (square) in conditions of Fig. 1.

143303-3 Zare et al. J. Appl. Phys. 117, 143303 (2015)

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For the interaction time of 700 fs, the electron tempera-

ture evolution is shown in Figure 1. The electron heating

processes for the quantum corrected collision frequency and

the classical collision frequency are presented in Figures 1(a)

and 1(b), respectively. It is realized that with the quantum

correction, the electrons are heating uniformly up to the

maximum temperature of 4.91� 107 K and the temperature

stays almost constant for more than 600 fs of interaction

time. By classical collision frequency, the maximum temper-

ature is 3.62� 107 K, which is about 35% less than the

amount related to the quantum collision frequency. It is

found that by considering the quantum correction, the colli-

sion frequency increases. Consequently, the collisional

absorption of laser radiation and the electron heating are

improved.

Figure 2 represents the comparison of the electron heat-

ing processes with (solid curve) and without (dashed curve)

quantum correction in the collision frequency. It is shown

how the quantum correction results in about 66% increase of

the electron temperature for the interaction time of 700 fs. In

initial stage, the electron temperature by the quantum correc-

tion collision frequency and by the classical case is

3.45� 107 K and 2.07� 107 K, respectively. However, using

Nd:glass laser, this improvement of the electron heating is

only about 15% (Ref. 14) in almost similar condition that

confirms the superiority of the shorter wavelength lasers for

electron heating.

Figure 3 presents the evolution of the maximum electron

temperature for the cases of classical (square) and quantum

corrected (star) collision frequency. At 300 fs, in the begin-

ning of the laser-plasma interaction, the electrons are heated

by the ponderomotive force. As the laser advances through

the plasma and getting closer to the high density plasma, the

collisional absorption mechanism becomes dominant, and

the laser energy absorption by the electrons is fundamentally

via the collisional absorption. Compared to the classical

model, due to the collisional frequency increment in the

quantum correction approach, higher electron temperature is

concluded. In both models, after heating up to the maximum

value, the electron temperature gradually decreases by

energy transfer to the ions.

Figure 4 illustrates the accelerated ion blocks for the

classical and quantum corrected collision frequency. The

block with negative velocity is moving against the laser

pulse and the ion block with positive velocity penetrates into

the plasma. By inclusion of the quantum correction, the ion

block with maximum velocity of 8.30� 106 cm s�1 is

obtained. However, in the classical case, the ion block with

maximum velocity of 3.49� 106 cm s�1 is resulted.

CONCLUSION

Based on the quantum mechanical correction of the

electron-ion collision frequency at comparably high temper-

atures in plasma, the electron heating in dielectrically

increased ion blocks for 248 nm of KrF laser was evaluated.

At high plasma temperatures, i.e., temperature of tokamak

and fusion plasmas, when the thermal absorption is domi-

nant, the laser-plasma interaction requires a revision by the

quantum correction of the collision frequency. The higher

quantum collision frequency with increasing temperature

could improve plasma heating by electromagnetic waves. It

is realized that due to the quantum correction in the men-

tioned plasma conditions, the increment of the electron heat-

ing is accessible where the electron temperature increases

uniformly up to the maximum value. Additionally, by this

correction, the ion block is generated with higher velocity in

comparison to the one produced in the classical case. The

achieved results could explain the available short laser pulse

advantages for fast ignition, which is promising method to

attain desirable high-energy-gain target performance for in-

ertial confinement fusion. Therefore, in fusion researches

when the thermal absorption is dominant this correction

should be considered. These results with new computations

of the fusion ignition dynamics could be also related to the

advanced ignition schemes of laser fusion.14,31

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