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Moleculardynamicssimulationofelectron–iontemperaturerelaxationindensehydrogen:AschemeoftruncatedCoulombpotential

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High Energy Density Physics 13 (2014) 34e39

Contents lists avai

High Energy Density Physics

journal homepage: www.elsevier .com/locate/hedp

Molecular dynamics simulation of electroneion temperaturerelaxation in dense hydrogen: A scheme of truncatedCoulomb potential

Qian Ma a, Jiayu Dai a, *, Dongdong Kang a, Zengxiu Zhao a, *, Jianmin Yuan a,Xueqing Zhao b

a Department of Physics, College of Science, National University of Defense Technology, Changsha 410073, People's Republic of Chinab Northwest Institute of Nuclear Technology, Xi'an 710024, People's Republic of China

a r t i c l e i n f o

Article history:Received 26 May 2014Received in revised form30 August 2014Accepted 16 September 2014Available online 28 September 2014

Keywords:Temperature relaxationMolecular dynamicsCoulomb logarithm

* Corresponding authors.E-mail addresses: [email protected], jydai203@16

gmail.com (Z. Zhao).

http://dx.doi.org/10.1016/j.hedp.2014.09.0041574-1818/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

Molecular dynamics (MD) simulations are performed to investigate the temperature relaxation betweenelectrons and ions in a fully ionized, dense hydrogen plasma. We used HM (J. P. Hansen and I. R.McDonald) potential and introduced a truncated Coulomb interaction, which can avoid Coulomb ca-tastrophe by choosing an appropriate cutting radius. The calculated results are compared with thosefrom theoretical models (LS, GMS, BPS), whose applicability is also discussed. The effect of the interactionbetween ions and electrons on the temperature relaxation process is also investigated in the strongcollision region. Finally, we discuss the effect of exchange interaction of electrons to the temperaturerelaxation.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Temperature relaxation between electrons and ions in denseplasma is a pivotal issue in understanding the physics of laser-plasma and shockwaveematerial interactions. In particular for in-ertial confinement fusion (ICF) [1], the exact energy transfer ratesare necessary in order to model properly the hydrodynamicalphysics such as energy depositions [2,3]. Recently, the experimentsfor diagnosing the process of temperature relaxation are advancingrapidly [4e6], thanks to powerful lasers such as the OMEGA, x-rayfree electron lasers and the National Ignition Facility [7e9]. Severalrelative experiments are being performed using methods such aslaser-accelerated protons [10e13]. Their systems focused on thecreating non-equilibrium states from sold materials such asgraphite [10], where the electronephonon interactions and defectsshould be important for the mechanism of energy depositions. Upto now, one of the critical issue is still that it involves with anessential non-equilibrium process, in which electrons and ionshave unequal temperatures and thus the equation of state atequilibrium states is not applicable. To design and analyze the

3.com (J. Dai), zhao.zengxiu@

experiments, a simple and costless computing simulation isrequired, by applying classical and semiclassical approximation todeal with the Coulomb collisions in the regime of interest.

The first theory about temperature relaxation in plasmas wasestablished by Landau and Spitzer (LS) [14,15]. The relaxation be-tween the electron temperature (Te) and the ion temperature (Ti) isgoverned by dTe/dt ¼ �nei(Te � Ti), dTi/dt ¼ �nie(Ti � Te), andnei ¼ n0lnL, where lnL is called Coulomb logarithm, and usuallywritten as the ratio of maximum tominimum impact parameters asln(bmax/bmin). In applications of LS, bmax is taken to be the electronDebye length lD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBTe=4pnee2

p, while bmin is chosen to be the

classical distance of closest approach, b0 ¼ Ze2/kBTe, or the electronthermal de Broglie wavelength L0 ¼ Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p=mekBTe

p. Most

theories give the same prefactor n0 ¼ 8ffiffiffiffiffiffiffi2p

pnie4Z2=3mM

ðkBTe=mþ kBTi=MÞ�3=2, where m(M), e(Ze), and ne(ni) are the mass,charge, density of the electrons (ions) respectively, and kB is theBoltzmann constant. LS theory is usually applicable to weaklycoupled, fully ionized plasma where the collisions are weak andbinary with Rutherford's Coulomb scattering.

However, it is not precise to deal with systems whose particlesinteract via Coulomb interactions with ad hoc cutoffs, since thelong-range Coulomb interaction leads to logarithmic divergenceand the infinite of the attractive potential in the strong collisionregion provokes the Coulomb catastrophe. In order to avoid thecutoffs and consider more physical processes, some improved

Delta:1_-Delta:1_given nameDelta:1_surnameDelta:1_given nameDelta:1_surnameDelta:1_given namemailto:[email protected]:[email protected]:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.hedp.2014.09.004&domain=pdfwww.sciencedirect.com/science/journal/15741818http://www.elsevier.com/locate/hedphttp://dx.doi.org/10.1016/j.hedp.2014.09.004http://dx.doi.org/10.1016/j.hedp.2014.09.004http://dx.doi.org/10.1016/j.hedp.2014.09.004https://www.researchgate.net/publication/202924422_Physics_of_Fully_Ionised_Gases?el=1_x_8&enrichId=rgreq-a4938dae-a120-4424-b3c3-a2f218270e72&enrichSource=Y292ZXJQYWdlOzI2NjIwNDMwNDtBUzoxNjg1MjM5MDA5MjM5MDVAMTQxNzE5MDYzODAwMg==

Fig. 1. Difference between ion and electron temperatures as a function of scaled time.Different colors represent different ionic mass scalings: a ¼ 1.0 (brown), 0.4 (red), 0.2(blue), 0.1 (green), 0.05 (pink), in which the time axis has been scaled t / t/a. (Forinterpretation of the references to colour in this figure legend, the reader is referred tothe web version of this article.)

Q. Ma et al. / High Energy Density Physics 13 (2014) 34e39 35

approaches have been developed [16e21]. Dharma-wardana andPerrot derived the coupled-mode (CM) theory [16], and later Vor-berger et al. developed it so that it involves the important contri-butions of collective excitations [17e19]. At the same time,electron-ion coupling was included in the T-matrix approachfrom Gericke, Murillo and Schlanges (GMS) [20], and the dimen-sional continuation was applied by Brown, Preston, and Singleton(BPS) [21]. The essence of these two theories is the consideration ofthe strong scattering. Furthermore, the validation for these modelsis always an open question for the related community, so that thereare huge demands for developing accurate models to obtain theintrinsic physics in these non-equilibrium processes.

2. Molecular dynamics simulation

MD simulation is an effective tool to investigate the propertiesof hot dense matter such as the thermal properties [22e24] andnonequilibrium process of temperature relaxation [22,27e35].Here, we performed MD simulations for a fully ionized densehydrogen plasma to study the energy exchange processes betweenelectrons and ions. In the simulation, the electron number densityvaries from 1.0 � 1022 cm�3 to 1.0 � 1024 cm�3, and the initial ionictemperature Ti0 is set to be 10 eV while the initial electronic tem-perature Te0 varies from 20 eV to 200 eV. The particle charges Zi andZe are ±1 and the number density for the electrons (ne) and ions (ni)are identical. In this way, the plasma coupling parameter, in theclassical regime, G ¼ Zie2/(rskBT), defined as the ratio of theCoulomb energy to the kinetic energy where rs ¼ [3/(4pn)]1/3 is theWignereSeitz radius of an electron or a proton and kBT ¼ kBTe,varies fromweakly coupled plasma to moderately coupled plasma.At the beginning of the simulation, we adopt the isokinetic ther-mostat to thermalize the plasma system. The electron system andthe ion system are approaching equilibrium state respectively formore than 104 time steps according to the initial conditions withdifferent time steps of 1 femto-second (fs) for ions and 1 atto-second (as) for electrons. After that, we take away the thermostatand relax the system freely within the conditions of energy con-servations. The Verlet-velocity method is used to integrate theNewton equations of motion, and the simulations are performed inthe microcanonical ensemble. The time step is chosen from2.4 � 10�5 fs to 1.2 � 10�3 fs in order to conserve the total energy.DE/E between adjacent time steps remains less than 5 � 10�4, andafter long runs, the total energy change remains less than 6 � 10�3with truncated Coulomb potential, which is within the statisticerrors.

MD simulation of the temperature relaxation is time-consumingbecause of the large mass ratio of ions to electrons (M/m~1836),which requires extremely small time step and large number ofsimulation steps (more than 107). To reduce the time scale, weperform MD simulations of temperature relaxation with a tech-nique of scaled ion masses [32], in which the physical masses arereduced by a fraction, i.e.,mj / amj. If the relaxation rate is inverseproportional to the ionic masses, the scaling law holds and we candivide the simulation time by a to derive the relaxation time for thephysical masses after the simulations. However, the factor n0 andlnL are both weakly dependent on mi, so that the results may notbe the simple formula given by LS theory and the scaling schememight fail. Examinations are carefully taken by employing various aranging from 0.01 to 1.0 to check the validation of the reducedmasses technique, as shown in Fig. 1, and finally we choose a ¼ 0.1so that the results are well convergent in the conditions consideredhere. If the relaxation rate is inverse proportional to the ionicmasses, the scheme included mi appears only in the first factor inthe formula of nei. In this case, the reduced mass technique for LSrelations would be valid.

The number of the particles used in MD simulations should besufficient to well describe the related physics statistically. In thiswork, 686 electrons and 686 ions are included and its accuracy hasbeen verified from the comparisons of various simulations. It isshown that the convergent result is in great agreement with cur-rent theories in their well-known valid regimes. Statistical fluctu-ations for each simulation are considered and averaged bycomputing the relaxation process with different initial configura-tions obtained after every 104 � 106 time steps.

When using the bare Coulomb potential in the simulations, thecatastrophe will occur due to the infinite depth of the potential foroppositely charged particles. Here we perform MD simulations in afully ionized hydrogen plasma with a ad hoc correctional Coulombinteraction, shown in Fig. 3. We simply truncate the pure Coulombinteraction at an appropriate cutting radius which varies with bothelectron and ion temperatures based on a unified formula, rC¼ Zie2/kBT, called the Landau length in which kBT ¼ 0.5(kBTi þ kBTe). TheLandau length is also called the classical distance of closestapproach, and the probability that two particles get closer than theLandau length is minuscule. Both ion and electron temperature aretaken into account in the cutting radius. Then a tangent line is madesimply for radius less the rC which decelerates the enormouslyincrease of Coulomb interaction. In this way, the Coulomb catas-trophe can be avoided. As the cutting radius become smaller, theinteraction is more close to the pure Coulomb interaction. To mimicthe quantum behaviors of electrons in finite temperature, pseu-dopotentials are usually constructed [28,30,33]. One of the widelyused pseudopotentials is the HM potential [25e27] which is givenby

VabðrÞ ¼ZaZbe

2

r

�1� expð � 2p=LabÞ

�

þ kBT ln 2exp�� 4pr2

.�L2ab ln 2

��daedbe

(1)

Fig. 2. Difference between ion and electron temperatures as a function of scaled time.The electron number density ne ¼ 1.0 � 1023 cm�3, initial electron (ion) temperatureTe ¼ 60 eV (Ti ¼ 10 eV). Different colors represent different cutting radii: electronDeBroglie wavelength (green), Landau length with the ion temperature effect (blue),Landau length without the ion temperature effect (red), 0.2 a.u. (brown), 0.1 a.u. (pink).(For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

Q. Ma et al. / High Energy Density Physics 13 (2014) 34e3936

where Lab ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pZ2=mabkBT

q, in which kBT ¼ kBTe when one of the

interaction particles is electron, and kBT ¼ kBTi when the twointeraction particles are ions. mab is the reducedmass. The first termin Eq. (1) is a diffraction contribution and the second term in Eq. (1)takes the electron symmetry into consideration.

In Fig. 3, we have made the comparison of our truncatedCoulomb potential with the HM potential. It can be seen that bothare dependent on the electron and ion temperatures. The potentialcurves in Fig. 3 are presented for given electron and ion

Fig. 3. Electrons and ions interacted forces with different temperatures. Differentcolors represent different initial electron temperatures, the dotted lines denote thetruncated Coulomb interaction, the cutting radii are the Landau length, rC ¼ Zie2/kBT,where kBT ¼ 0.5(kBTi þ kBTe), the dashed lines denote the HM potential interaction, thesolid lines denote the pure Coulomb interaction, the dottededashed line denotes thetruncated Coulomb interaction with the cutting radius with a fixed value of 0.5 a.u..

temperatures. In the temperature relaxation process, the HMinteraction varies with time as the electron and ion temperaturesare changed, but the truncated interactions are fixed once the initialelectron and ion temperatures are determined. Furthermore, theinfluence of electron exchange interaction is also discussed in thepresent work by ignoring the second term in HM potential. Thesimulation results are compared with different theoretical modelsat different conditions, for example, LS model, GMS model [36] andBPS model. Glosli et al. [28] compared these three theoreticalmodels with MD results, and found that GMS and BPS agree withthe simulation for lnL � 1. For lnL < 1, GMS model is better to thesimulation results compared with BPS model which increasinglyunderestimates the relaxation rate. We extend the simulation casesand try to show the intrinsic physics for the differences of the threemodels from MD simulation.

3. Simulation results

In Table 1 all simulated cases at different initial conditions arelisted. The average temperature relaxation time t is calculated bydDT/dt ¼ DT/t, where DT ¼ Te � Ti is obtained directly from thesimulation results, and the standard deviation s is derived from 8equivalent samples by changing initial electron and ion configu-rations. Initial electron temperature ranks from 20 eV to 200 eVwhile initial ion temperature is kept 10 eV. The electron numberdensity changes from 1.0 � 1022 cm�3 to 1.0 � 1024 cm�3. Here wefocus on the cases that the electron temperature is higher than theion temperature.

After testing several cases, we choose the Landau length,rC ¼ Zie2/kBT where kBT ¼ 0.5(kBTi þ kBTe), as the cutting radii. Thecutting radius decreases as the initial electron temperature in-creases from 20 eV to 200 eV. This simple treatment is consistentwith physics that while the electron temperature increases, theelectrons can reach closer to the ions in statistics. When two par-ticles run closer, their interacted forces between electron and ionare stronger so that the cutting radius is required to be smaller. Acutting radii study is shown in Fig. 2. The curves of the differencebetween electron and ion temperature is shown in the figure. Theelectron number density is 1.0 � 1023 cm�3, the time stepdt ¼ 0.01 a.u., and the initial electron temperature (ion tempera-ture) is 60 eV (10 eV). Different colors represent different cuttingradii. We choose electron DeBroglie wavelength (green), Landaulength considering the ion temperature effect (blue) and thatwithout the effect (red), and two smaller radii (0.2 a.u. (brown) and0.1 a.u. (pink)) respectively, as the cutting radii. It is obviously that ifthe cutting radius is smaller, the interaction potential will be closerto the pure Coulomb potential, but the time step needs to be shorterto control the number simulation errors, and it costs a huge of time.So a appropriate cutting radius is requisite to make the simulationsufficient and valid. From Fig. 2, we can see that the differencebetween the brown curve and the red curve is small, even thecutting radius of the red (Landau length)is twice more than that ofthe brown curve (Rcut¼ 0.2 a.u.). The radius of green curve (electronDeBroglie wavelength)is too long that it decreases the effect of theCoulomb interaction, and the radius of the pink curve(Rcut ¼ 0.1 a.u.) is too short that the number simulation errors cannot be neglected, and to avoid this problem, the time step need tobe reduced, which may cost a huge of time. So we choose theLandau length, rC ¼ Zie2/kBT where kBT ¼ 0.5(kBTiþkBTe), as thecutting radii, taking the ion temperature influence into consider-ation. Using a simply truncated Coulomb potential in MD simula-tions is appropriate and efficient in the conditions we focus on, andwe can directly investigate the effect of strong collision and theprocess of Coulomb catastrophe.

Table 1The average relaxation time t and standard deviation s are obtained fromMDResults. t1 and s1 are the results of truncated Coulomb potential, the cutting radii are chosen to bethe Landau length, rC ¼ Zie2/kBT, and kBT ¼ 0.5(kBTi þ kBTe) and t2 and s2 are the results of HM potential interaction.

Case neðcm�3Þ TiðeVÞ TeðeVÞ t1ðfsÞ s1ðfsÞ t2ðfsÞ s2ðfsÞA1 1022 10 20 1:06� 103 2:15� 102 9:41� 102 9:95� 101B1 1022 10 40 1:71� 103 1:77� 102 1:79� 103 1:05� 102C1 1022 10 60 2:36� 103 2:22� 102 2:53� 103 2:34� 102D1 1022 10 80 3:11� 103 3:09� 102 3:36� 103 9:24� 101E1 1022 10 100 3:88� 103 1:46� 102 4:17� 103 2:38� 102F1 1022 10 200 9:90� 103 4:70� 102 9:68� 103 5:21� 102A2 1023 10 20 2:87� 102 7:26� 101 2:28� 102 3:07� 101B2 1023 10 40 3:21� 102 3:71� 101 3:34� 102 3:03� 101C2 1023 10 60 3:95� 102 2:40� 101 4:30� 102 1:06� 101D2 1023 10 80 5:08� 102 2:46� 101 6:00� 102 6:34� 101E2 1023 10 100 6:37� 102 2:19� 101 6:59� 102 2:28� 101F2 1023 10 200 1:37� 103 5:74� 101 1:36� 103 8:89� 101A3 1024 10 20 1:06� 102 2:06� 101 8:87� 101 1:41� 101B3 1024 10 40 1:06� 102 6.67 9:55� 101 1:02� 101C3 1024 10 60 1:01� 102 7.86 1:10� 102 1:01� 101D3 1024 10 80 1:16� 102 7.03 1:28� 102 7.10E3 1024 10 100 1:19� 102 7.63 1:41� 102 7.19F3 1024 10 200 1:82� 102 4.63 2:33� 102 1:59� 101


The time evolution curves of electron and ion temperatures arepresented in Fig. 4, in which the blue curves are performed withHM potential and the red curves use the Coulomb potential with atruncation. The electron number density is 1.0 � 1023 cm�3. In thefigure, we can see that electron temperature decreases and iontemperature increases with time increasing, indicating that theelectron subsystem transfers the energy into the ion subsystem.Finally the whole system reaches equilibrium, and then the elec-tron and ion temperatures keep identical. The time evolutioncurves of electron and ion are symmetrical approximatively indespite of some fluctuations. Here we consider the electron relax-ation rate (1/tei ¼ nei) and ion relaxation rate (1/tie ¼ nie) to beidentical, tei ¼ tie ¼ 2t. Generally, the relaxation time becomes

Fig. 4. The time evolution curves of electron and ion temperature for the cases A2, B2,C2, D2, E2, F2 listed in Table 1. The blue curves are results from HM potential inter-action and the pink curves are from the truncated Coulomb potential, the cutting radiiare chosen to be the Landau length, rC ¼ Zie2/kBT, and kBT ¼ 0.5(kBTi þ kBTe). (Forinterpretation of the references to colour in this figure legend, the reader is referred tothe web version of this article.)

longer if initial electron temperature is higher, and becomes shorterif the electron number density is higher. Besides, we can see thatthe curves of electron and ion temperatures obtained with trun-cated Coulomb potential are in good agreement with the resultsfrom HM potential, although some differences are still being therewhich can be explained by the fluctuation. The interactions be-tween ion and electron are plotted in Fig. 3, the dashed lines are theforces derived from HM potential, the dotted lines are those fromtruncated Coulomb potential, and the solid line is that from pureCoulomb potential. Different colors represent different tempera-tures. From the figurewe can see that when R > 1.5 a.u., the forces ofdifferent potentials are consistent, but when R < 0.5 a.u., the

Fig. 5. Comparison of theoretical results (LS (dotted line), BPS (dashed line) and GMS[36] (solid line)) with data from MD simulations (blue dots are HM potential, red dotsare truncated Coulomb interaction) of lnL (Coulomb logarithm) as a function of initialTe for electron number density ne ¼ 1.0 � 1022 cm�3 and ne ¼ 1.0�1023 cm�3. Thecutting radii are chosen as the Landau length, rC ¼ Zie2/kBT and kBT ¼ 0.5(kBTi þ kBTe).See text for details. (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)

Fig. 6. Comparison of theoretical results (LS (dotted line), BPS (dashed line) and GMS[36] (solid line)) with data from MD simulations (blue dots are HM potential, red dotsare truncated Coulomb interaction) of lnL (Coulomb logarithm) as a function of initialTe for electron number density ne ¼ 1.0 � 1024 cm�3. The cutting radii are chosen as theLandau length, rC ¼ Zie2/kBT. See text for details. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.)

Q. Ma et al. / High Energy Density Physics 13 (2014) 34e3938

differences increase dramatically. The forms of the two potentialsare different and HM potential considers some quantum effectswhile the truncated Coulomb potential is just classical, but in someregimes the quantum effect may be approximately substituted byclassical method which is more simple, just like the relative worksand conditions investigated here.

To make precisely comparative analysis, we display the predic-tion of Coulomb logarithm from LS, GMS and BPS model as well asthe MD results using HM potential interaction and truncated

Fig. 7. The time evolution curves of electron and ion temperature with HM potential and nonelectronic temperatures, the red lines are the results of nonexchange HM potential in diffene ¼ 1.0 � 1024 cm�3. See text for details. (For interpretation of the references to colour in

Coulomb interaction in Figs. 5 and 6, in which electron numberdensities are ne ¼ 1.0 � 1022 cm�3, ne ¼ 1.0 � 1023 cm�3 andne ¼ 1.0 � 1024 cm�3, respectively. Firstly we compare the theo-retical model with the MD results of HM potential. The conclusionthat which model is the best can not be reached simply, and itdepends on the studied conditions. In Fig. 5, where lnL > 2, the BPSand GMS models are both well fitted with MD results with HMpotential. For lnl < 2, the BPS becomes better fitted modelcomparedwith the results of HM potential, but for case A1, the GMSmodel is better. In Fig. 6, for 0.4 � lnl < 1, the MD results are closerto BPS model, and it is the same with the results in Fig. 5. But whenlnl < 0.4, for case A3 (Te ¼ 20 eV), the BPS model increasinglyunderestimates the Coulomb logarithm compared with the MDresults. The BPS model is not fitted in this regime for lnL < 0.4.

To specify the effects of the electron and ion interaction in thestrong collision region on the relaxation process, we compare theinteracted forces between ion and electron plotted in Fig. 3. For thecases when lnL > 2 in Fig. 5 (cases B1,C1,D1,E1,F1,E2,F2), the dif-ferences of forces between truncated Coulomb potential and HMpotential shown in Fig. 3 are visible, but the MD results of Coulomblogarithms are consistent considering the error bar. In these con-ditions, electron and ion may not or rarely encounter in the strongcollision region where the discrepancy of the interaction betweenHM potential and truncated Coulomb potential is obvious. ForlnL � 2, we check all the cases and find that if the electron and ioninteraction is stronger, the Coulomb logarithm is higher. In Fig. 6,we renewedly perform cases A3, B3, C3, D3, E3, F3 in Table 1 anduse the truncated Coulomb interactionwith the cutting radius to be0.5 a.u.. For cases A3, B3, C3, D3, the interaction force with cuttingradius Rcut ¼ 0.5 a.u. is bigger than that with the Landau lengthRcut ¼ rC as the cutting radius, and so the related Coulomb loga-rithmwith cutting radius Rcut ¼ 0.5 a.u. is bigger than that with theLandau length Rcut ¼ rC.

For case E3, the difference between the two interaction forces issmall, and so the Coulomb logarithms are approximative inmagnitude. But for case F3, interaction force with cutting radiusRcut ¼ 0.5 a.u. is smaller than that with the Landau length Rcut ¼ rC,and so is the Coulomb logarithm.We attribute the discrepancy of LSmodel, GMS model and BPS model for the region lnL � 2 to the

exchange HM potential. The blue lines are the results of HM potential in different initialrent initial electronic temperatures. In Fig. 7(a), ne ¼ 1.0 � 1023 cm�3 and In Fig. 7(b),this figure legend, the reader is referred to the web version of this article.)


difference of interaction forces in the strong collision region, andthe closer to the pure Coulomb potential, the more approachablethe Coulomb logarithmwill be to the GMSmodel and LSmodel. TheCoulomb catastrophe is caused by the infinite of the attractiveCoulomb potential in the strong collision region, and the simplytruncated Coulomb potential with the Landau length as the cuttingradii can avoid the catastrophe, and the results are consistent withthe MD results with HM potential which takes the quantum effectof electrons into account by coincidence.

The neglect of quantum effect of electrons is the most importantlimitation in the presentMD simulation. The semiclassical potentialis used commonly to mimic quantum potential such as HM po-tential. Here we study the electron exchange effects on the relax-ation process by expurgating the Pauli part in HMpotential. We plotthis effect on the time evolution curves of electron and ion tem-peratures in Fig. 7. The blue curves are the results with original HMpotential while the red curves are with the nonexchange HM po-tential. In Fig. 7(b), the discrepancy can be analyzed considering thefluctuations and computational errors. When the electron numberdensity increases and the coupling parameter increases, thediscrepancy become visible, indicating that the effects of electronexchange can not be neglected in Fig 7(b). We derive the Coulomblogarithms of the two situations where the ne ¼ 1.0 � 1024 cm�3,and Te ¼ 20,40,60 eV, Ti ¼ 10 eV. the lnL of no exchange is largerthan that of HM potential by 33.28% at Te ¼ 20 eV (by 18.21% atTe ¼ 40 eV, 10.13% at Te ¼ 60 eV). For the electron densityne ¼ 1.0 � 1023 cm�3 at Te ¼ 20,40,60 eV, the plasma couplingparameter are G ¼ 0.54, 0.27, 0.18, respectively, and G ¼ 1.16, 0.58,0.39, respectively for ne ¼ 1.0 � 1024 cm�3 at Te ¼ 20,40,60 eV. It isobviously that the effects of electron exchange increase as theplasma coupling parameters become larger. This results indicatethat the exchange effect of electrons decreases the rate of energyexchange between electrons and ions. That is to say, the crosssections of electron-ion collisions with exchange effect are lowerthan those without exchange effects. With respect to the fluctua-tions, we can derive that the effect of exchange can not be neglectedwhen G > 0.6. This is reasonable since when the density is higher,the quantum effect or degenerate effect of electrons become moreimportant.

4. Conclusions

We performMD simulation of the HM potential and a truncatedCoulomb interaction. We investigate the effects of interaction ofelectron and ion to the temperature relaxation in the strong colli-sion region for lnL � 2, and find out that the stronger the inter-action of electron and ion, the higher the Coulomb logarithm willbe. The Landau length is chosen as the cutting radii that the resultscan be appropriate to the MD simulation with HM potential. For

weakly or moderately coupled plasmas, where lnL � 2, the BPS,GMS and LS models are all fitted with MD results of HM potential,for 0.4 < lnL < 2, BPS is the best model compared with MD results,but for lnL < 0.4, the BPS model increasingly underestimates theCoulomb logarithm, and GMSmodel is best. We also investigate theeffect of exchange interaction of electrons to the temperaturerelaxation process, and derive that the effect of exchange can not beneglected when G > 0.6.

Acknowledgments

This work is supported by the National Basic Research Programof China (973 Program) under grant no 2013CB922203, the NationalNSFC under Grant Nos. 11422432, 11104351, and 11104350. Fundingsupport (SKLLIM1107) from the State Key Laboratory of LaserInteraction with Matter is acknowledged.

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