Computational study of heat transport in compositionally disordered binary crystals

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1 Act Mterili 54 (6) Computtionl study of het trnsport in compositionlly disordered inry crystls John W. Lyver IV,, Estel Blisten-Brojs, * Computtionl Mterils Science Center, College of Science, George Mson University, Firfx, VA 3, USA Office of Sfety nd Mission Assurnce, Ntionl Aeronutics nd Spce Administrtion, Wshington, DC 46, USA Received Mrch 6; received in revised form 6 My 6; ccepted 7 My 6 Aville online 4 August 6 Astrct The therml conductivity of compositionlly disordered inry crystls with toms intercting through Lennrd-Jones potentils hs een studied s function of temperture. The two species in the crystl differ in mss, hrd-core tomic dimeter, well depth nd reltive concentrtion. The isoric Monte Crlo ws used to equilirte the smples t ner-zero pressure. The isoenergy moleculr dynmics comined with the Green Kuo pproch ws tken to clculte the het current time-dependent utocorreltion function nd determine the lttice therml conductivity of the smple. The inverse temperture dependence of the lttice therml conductivity ws shown to fil t low tempertures when the tomic dimeters of the two species differ. Insted, the therml conductivity ws nerly constnt cross tempertures for species with different tomic dimeters. Overll, it is shown tht there is drmtic decrese of the lttice therml conductivity with incresing tomic rdii rtio etween species nd moderte decrese due to mss disorder. Ó 6 Act Mterili Inc. Pulished y Elsevier Ltd. All rights reserved. Keywords: Binry solids; Compositionl disorder; Monte Crlo; Therml conductivity; Moleculr dynmics 1. Introduction It is known tht het is trnsported etter through solid mterils tht re pure nd crystlline. Any type of impurity, defect, doping or internl oundry within the mteril nominlly increses the resistnce to het trnsport, nd thus reduces the ility to conduct therml energy. With the growing interest in nnotechnology, the study of therml conduction properties of systems with reduced dimensions, thin films, nnotues nd superlttices hs incresed. In nnomterils nd nnostructures, phenomen re highly dependent on the length scle where virtions etween nerest-neighor toms occur. The use of moleculr dynmics (MD) nd the Green Kuo (GK) methods for clculting the therml conductivity hve shown promise s tomistic pproches for understnding nnosystems t the nnometer scle. For exmple, there * Corresponding uthor. Tel.: E-mil ddress: listen@gmu.edu (E. Blisten-Brojs). re severl recent clcultions on pure nole gses with Lennrd-Jones interctions nd fce-centered cuic (fcc) structures in which MD ws the method of choice [1 5]. For inry crystls the literture is not so undnt; worth noting is the MD clcultion for crystlline -SiC with point defects [6]. In crystl the therml conductivity is composed of two dditive contriutions: lttice nd electronic. The lttice contriution j ph cptures phenomen ssocited with lttice virtions nd phonon scttering nd is dominted y the structurl chrcteristics of the crystl. The electronic contriution j e is proportionl to the electric conductivity r e through the Wiedemnn Frnz lw [7,8]. The composition of crystl ffects the lttice symmetry chrcteristics nd consequently the lttice virtions. Therefore, j ph, the lttice contriution to the therml conductivity in crystl, should reflect chnges ccording to its composition. In contrst, since j e is function of r e, the conduction properties re expected to remin lmost constnt for fmilies of solids with similr compositionl components. A phenomenon /$3. Ó 6 Act Mterili Inc. Pulished y Elsevier Ltd. All rights reserved. doi:1.116/j.ctmt.6.5.5

2 4634 J.W. Lyver IV, E. Blisten-Brojs / Act Mterili 54 (6) tht reduces j ph produces n overll reduction of the therml conductivity if the electric conductivity is not ffected. In dielectrics, nd the nole gses specificlly, chnges in j ph do not simultneously ffect the electronic conductivity. This work focuses on simulting the lttice therml conductivity due to tomic virtions for inry crystls with compositionl disorder. The gol of this work is to identify rnges of comintions of mterils nd disorder conditions which reduce the lttice therml conductivity of the simulted inry solid mixtures nd my wrrnt further experimentl work. Throughout the reminder of this pper, j is used to identify the lttice contriution to the overll therml conductivity. The effect of compositionl disorder on therml conductivity ws investigted using severl simple models of inry Lennrd-Jones (L-J) solids. Compositionl disorder ws investigted due to differences in the vn der Wl rdii (r), intertomic ond strength (e) nd mss (m) of the two types of toms. Severl reltive concentrtions of simulted crystlline inry mixtures were studied s function of selected potentil prmeters nd nlyzed cross vrious tempertures. The computtionl pproch tken ws to perform tomic-level computer simultions employing comintion of isoenergy MD nd NPT Monte Crlo (MC) with constnt numer of toms (N), pressure (P) nd temperture (T) to clculte the j within liner response theory of mny-ody systems. To vlidte the work, results were compred to other reported results nd experimentl dt ville for montomic crystls. This pper is orgnized s follows. Section descries the methodology used to prepre the inry smple, determine its equilirium density nd llow the smple to rech mechnicl equilirium. Section 3 descries the lttice therml conductivity results otined s function of the prmeters used, the lttice disorder models nd the vrious concentrtions of the two tomic species. Section 4 concludes the work with summry.. Methodology A crystlline inry mixture of 5 toms ws simulted in cuic computtionl ox with periodic oundry conditions in ech direction. In the clcultions, r c is the cutoff rdius tken s 49% of the width of the computtionl ox. The composition of the inry crystl uses toms of type A s the host nd toms of type B s the guest. All prmeters were compred reltive to the host A toms. The L-J potentil with prmeters r nd e ws used s prototype interction etween toms. The compositionl disorder introduced in the host lttice due to the guest toms is modeled prmetericlly y chnges of r, e nd mss. Quntities re expressed in reduced units with respect to the host toms L-J prmeters r A, e A nd m A. For exmple, the mss of Ar is u. nd tht of Xe is u. In reduced units, using Ar s the host tom, the mss of Ar would e 1., wheres the mss of Xe would e 3.3. Reduced units of length, energy, temperture, p time nd therml conductivity re r, e, e/k B, t o ¼ ffiffiffiffiffiffiffiffiffiffiffiffi r m=e nd k B /t o s, respectively, where k B is Boltzmnn s constnt. Four compositionl mixture cses in the computtionl ox were considered with the following chrcteristics: 1% of pure A toms, 75% of A toms nd % B toms, 5% of ech type, nd % of A toms nd 75% of B toms. The L-J prmeters for the inry interctions (A B) re otined from the comintion rules: r AB ¼ r A þ r B p ; e AB ¼ ffiffiffiffiffiffiffiffiffi e A e B : ð1þ Simultions strted t reduced temperture of.5 from configurtion with toms plced in perfect fcc lttice. Next, n initil configurtion ws constructed such tht toms were rndomly ssigned s type A or B consistent with the reltive concentrtion of the two types of toms. Throughout this study, to indicte the rtio of prmeters, the symols R r, R e, re used for r B /r A, e B /e A nd m B /m A, respectively. The system ws equilirted y NPT-MC, which llowed for moves of the N toms in rndom directions nd chnges of the entire computtionl ox volume (V). The cceptnce criterion etween old (V o ) nd new (V n ) configurtions is given y [9] ccðo! nþ ¼minð1; expf ½Uðr N ; V n Þ Uðr N ; V o Þ þ PðV n V o Þ ðn þ 1Þ 1 lnðv n =V o ÞŠgÞ: ðþ Here, is 1/T, r N is the vector of the coordintes of ll toms nd the potentil energy is Uðr N ; V Þ¼ 1 X N i X N i6¼j 4e ij ½ðr ij =r ij Þ 1 ðr ij =r ij Þ 6 Š; where r ij is the core rdius, e ij is the ond strength nd r ij is the intertomic distnce of n tomic pir using Eq. (1). The NPT-MC simultions were run etween 1 nd 3 million steps with step eing N single tom movements nd one volume djustment. The verge density nd other clculted quntities were determined s n verge over the finl one-fourth of the NPT-MC trjectory. Therefore, the position of the toms within the ox is consistent with this verge density. The density is defined s N/V irrespective of the two types of toms, which could hve different msses, r, or e vlues. Becuse the computtionl ox is finite, the vlue of the pressure ws djusted y sutrcting the pressure tht would e exerted y structureless infinite-sized smple outside of the computtionl ox [9]. For the montomic system, the equilirium structure ws n fcc structure for ll tempertures. At low tempertures, no stle morphous phse ws found s otined in Ref. [3]. Becuse the NPT-MC clcultion does not include the mss in the simultion, the equilirium q for inry smples with A nd B toms hving only different e is the sme s the density of the montomic system. Therefore, the NPT-MC clcultions were crried out to determine q t different tempertures when R r 6¼ 1. Fig. 1 ð3þ

3 J.W. Lyver IV, E. Blisten-Brojs / Act Mterili 54 (6) shows the temperture ehvior of the verge q for equilirted systems t zero pressure for smples with 5:5 reltive concentrtion. The curves correspond to different R r. The vlue of q of pure Ar reported in Ref. [3] compres well with our results. As expected, when R r increses, the volume must lso increse, decresing q. The stndrd devition (SD) of the verge density is very low, of the order of the symol size used in Fig. 1. These smll fluctutions certinly ensure tht the smooth decrese of q with temperture illustrted in Fig. 1 is indeed relistic. The next step ws to initite the isoenergy MD study using the output of the NPT-MC runs. Ech MD tril ws run 35, time steps of Dt =.5 to llow the system first to equilirte t the desired temperture. Next the MD tril continued to run for hlf million time steps to clculte the desired het current opertor vlues from [1] ~J ¼ XN i¼1 E i v * i þ 1= XN i¼1 X N j6¼i ðv * i F * ijþr * ij; where E i is the totl energy of ech tom, ~v i is the velocity of ech tom nd ~F ij nd ~r ij re the force nd intertomic vectors for ech tomic pir. The next step ws to clculte the utocorreltion function C(s) of the het current opertor, which is defined s CðsÞ ¼h~Jðs þ tþ~jðtþi ð5þ where Ææis the time verge, ~J is the het current opertor nd s is the time lg from n origin t chosen from the time trjectory. Ech utocorreltion run typiclly used etween 16 nd 18 time lgs. It ws found tht for R r, R e, nd ner vlue of one required longer times to compute the utocorreltion function thn when disorder sets in. The lttice therml conductivity j ws otined y integrting C(s) over the rnge [,t trj ], where t trj is the totl ð4þ time for which the utocorreltion function ws clculted. This is the GK pproch [11,1]: j ¼ 1 Z 1 CðsÞ ds: ð6þ 3Vk B T The GK pproch works well for oth morphous nd crystlline models s long s the system is homogeneous. GK tkes full ccount of nhrmonic properties ut is clssicl in nture. Ldd et l. [13] were the first to use the GK formlism to clculte therml conductivity for solids with interctions following n inverse-twelfth power lw potentil. Lter, Gilln extended this method for the study of j in plldium doped with hydrogen [14]. More recently, Chen et l. used this sme pproch to study the therml conductivity of pure Ar doped with Xe [15]. Optimlly, it would e est to clculte C(s) out to infinity insted of just the finite trjectory length, ut this is not possile numericlly. We oserved tht C(s) could e pproximted y n exponentilly decying cosine function e ft cos(xt) s shown in Fig. nd fit the prmeters to the numericl MD results. Then the integrtion in Eq. (6) ws done from the ctul simultion dt for 6 s 6 t fit nd used the decying cosine function for t fit 6 s 6 1. The vlue of t fit ws set to e 1. times the period of the fitted cosine function. This time t fit defines the system relxtion time. The NPT-MC smples prepred in the mnner descried in previous prgrphs represent different types of compositionl disorder. For ll vlues of R e or simulted, the structure of the equilirted smple is the fcc lttice. Thus the system disorder is sed on rndom mixture of toms A nd B, which re positioned on perfect lttice. In contrst, when size disorder ws introduced with R r eyond 1.1, the fcc lttice collpses. This is shown in Fig. 3 which depicts the pir correltion function g(r) in which the g AA (r), g AB (r) nd g BB (r) vlues hve een 1.1 = Density.9.8 = 1.1 C( ) Temperture Fig. 1. Density s function of temperture for rdii rtios R r of 1., 1.1 nd 1. for the 5:5 mixture of toms. Crosses re q for pure Ar t zero pressure nd circles represent results from Ref. [3] Time Lg Fig.. MD-clculted C(s) s function of time. The dotted lines show the envelope of the exponentilly decying cosine function otined from the fit of the MD dt.

4 4636 J.W. Lyver IV, E. Blisten-Brojs / Act Mterili 54 (6) g(r) g(r) r r /ρ 1/3 Fig. 3. Rdil distriution function g(r): () for different R r vlues: 1. (solid), 1.1 (dshed) nd 1. (dotted); () sme s in () plotted s function of scled distnces. summed together. Fig. 3 shows the cse of 5:5 mixture smple t T =.167 with R e =1, = 1 nd three different R r vlues (1., 1.1 nd 1.). To compre directly etween these functions, scling of q 1/3 ws pplied to the rdil dependence. It is very cler tht for R r, the compositionl disorder of the 5:5 smple ffects the structure very significntly nd the crystl collpses into homogeneous morphous solid. The structure of this solid morphous mixture is very different from the structure found in tomic clusters [16], where the toms with smller r segregted nd formed sucluster surrounded y the lrge r toms. 3. Determintion of the lttice therml conductivity A smple with N = 5 t P = with only one type of tom ws prepred, nd j ws otined for severl tempertures using the steps descried in Section. These results llowed vlidtion of our method y comprison with severl clcultions done recently [3,15,17,18] s well s with experimentl results [19]. Fig. 4 shows this comprison, indicting tht our results (lck strs) re in full greement with previous clcultions nd with the experimentl results. In the GK pproch, Eq. (6), there is n explicit dependence of j on the volume of the smple. Smple size effects were studied in Ref. [17] where the uthors considered computtionl ox sizes contining etween 18 nd 4 toms. Those uthors concluded tht in the temperture domin of 7 K, the size effects re irrelevnt for ll prcticl purposes when clculting j for pure Ar system. This is consistent with our findings for computtionl cells contining toms. It ws found tht computtionl oxes smller thn 18 toms were too smll for meningful results. Fig. 5() nd () illustrtes the dependence of the equilirium density nd potentil energy verges s function of the numer of fcc cells (n) on ech Therml Conductivity (W/mK) Experiment [18] Ref [14] Ref [16] Ref [17] This Work Temperture (K) Fig. 4. Therml conductivity j s function of temperture for pure Ar t zero pressure. The results of this work (lck strs) re compred to those of other works: Ref. [3] (dimonds), Ref. [15] (circles), Ref. [17] (tringles), Ref. [18] (crosses), Ref. [19] (squres). ρ Energy/N c Numer of FCC Unit Cells Fig. 5. Effect of computtionl ox sizes. The horizontl xis shows the numer of fcc unit cells long ech side of the cuic computtionl ox. () Density, () j nd (c) energy per tom for the ordered montomic crystl. computtionl ox edge (N =4n 3 ). Fig. 5(c) shows j nd its SD s function of computtionl ox size. Bsed on the continued good greement with oth the previously discussed comprisons offset ginst run times, system size of N = 5 t P = ws selected for ll results reported in this work. The following compositionl mixtures were considered: R r of 1., 1.1, 1., 1.5 nd.; R e of 1., 1. nd 1.5; nd of 1., 1.6,.1 nd 3.3. Additionlly, we studied different reltive concentrtions of A nd B toms rnging from 1% A toms, 75% A with % B, 5% A with 5% B, nd % A with 75% B. For smples with reltive concentrtions of 5:5, t temperture of T =.167, Fig. 6 illustrtes the lttice j

5 J.W. Lyver IV, E. Blisten-Brojs / Act Mterili 54 (6) = 1. = 1.5 = 1. = 1.6 =.1 = c = 1. = 1.1 = Temperture Fig. 6. Therml conductivity s function of the prmeter rtios: () dependence on R r for R e = 1, 1., 1.5, nd = 1; () dependence on R r for = 1, 1.6,.1, 3.3, nd R e = 1; (c) dependence on R e for R r =1, 1.1, 1., nd =1. Fig. 7. Therml conductivity s function of temperture: () structurlly ordered cse R r = 1, nd = 1 with R e = 1 (squres), R e (circles), R e = 1.5 (tringles); () structurlly disordered cses R r = 1.1 (solid lines) nd R r (dshed lines). The top, middle nd ottom curves re for R e = 1., 1. nd 1.5, respectively. s function of one prmeter rtio (R r, R e or ) while the other two prmeter rtios re kept constnt. Fig. 6() nd () shows drmtic decrese of j with incresing R r. In fct, Fig. 6() shows tht j decreses y fctor of over 6 etween R r = 1 nd R r = 1.1 for constnt mss rtio nd vrious vlues of R e. Likewise, Fig. 6() shows drmtic decrese in j etween R r =1 nd R r = 1.1 using different mss rtios. In this cse gin, j decreses y fctors up to 6 depending upon. While Fig. 6() shows sustntil decrese in j etween R r = 1 nd R r = 1.1, the two tom types would hve to e the sme to hve R r = 1 nd = 1, which is very unrelistic cse. On the contrry, Fig. 6(c) shows tht, for R e = 1 nd =1, j increses slightly s function of R r nd incresing R e. This increse lies within the SDs of the j results nd might not e rel effect. The conclusion of the prmeter nlysis is tht t T =.167, oth rdii disorder nd mss disorder impose strong depletion of j. Even slight difference in tomic rdii of only 1% hs mjor effect on decresing j while the mss rtio hs more grdul depleting effect on j. The mss disorder leves the crystlline symmetry intct. In comprison, the rdii disorder llows the solid to cquire incipient morphous chrcteristics s evidenced y the pir correltion function signture illustrted in Fig. 3. In fct for the lrge difference in tomic rdii of %, Fig. 3 indictes tht the fcc symmetry is lredy lost nd the solid is no longer crystl. For montomic crystlline mterils, the expected theoreticl dependence of the therml conductivity with temperture follows n inverse temperture lw [7,8]. While previous MD simultions [15] reported j exhiiting this expected ehvior, our results show deprture for ny of the proposed smples with disorder. Fig. 7 shows the j ehvior for vrious vlues of R e of 1., 1. nd 1.5 nd = 1 for 5:5 concentrtion. In Fig. 7 the inverse temperture dependence is plotted with dotted line to guide the eye. Fig. 7() depicts the temperture dependence for R r = 1 with the squre, circle nd tringle symols identifying the three vlues of R e (1., 1. nd 1.5), respectively. SDs re shown for the R e cse nd re representtive of the other cses. Fig. 7() gives results for systems with R r = 1.1 s solid lines corresponding to R e = 1., 1., nd 1.5 (top, middle nd ottom) nd dshed lines for R r. SDs re out 1 units of j for ll results. It is pprent from these plots tht the ordered crystl with no core rdius disorder follows the 1/T reltionship very closely (Fig. 7()) while ny of the compositionlly disordered systems (Fig. 7()) present nerly constnt j s function of temperture. This degrding of the therml conduction is similr to tht predicted for covlent inry crystls with defects [6] where j ws found to e essentilly temperture independent. In our study it should e rememered tht compositionl disorder in which the tomic rdii differ y only 1% produces drmtic reduction of j to minimum vlue, which keeps firly constnt for the tempertures investigted. In summry, we emphsize tht the rdii disorder hs n extremely strong effect to reduce j, ringing its vlue to e minimum for ll clcultions with widely vrying mteril prmeters. The lst prt of this study pertins to chnges in the reltive concentrtions of the A nd B toms. Reltive concentrtions of A:B toms of :75 nd 75: were nlyzed in ddition to the 1% type A nd the 5:5 mixture cses discussed ove. As the concentrtion chnges, the numer of smller toms increses reltive to the lrger toms hving significnt effect on q s shown in Fig. 8. In nlyzing mixtures with the :75, 5:5 nd 75: reltive concentrtions over the rnge of T, R r, R e nd

6 4638 J.W. Lyver IV, E. Blisten-Brojs / Act Mterili 54 (6) Density Relxtion Time = 1. = 1.5 = 1. = Temperture Fig. 8. Density s function of temperture for different reltive concentrtions in the mixture. Filled symols re for R r = 1.1 nd open symols re for R r. The circle, tringle nd squre re for R e = 1., 1. nd 1.5, respectively. The cse of R r = R e = = 1 is shown s crosses. Fig. 9. Relxtion time s function of prmeter rtios for the 5:5 smple: () dependence on R r with = 1 nd R e = 1 (squres), 1. (circles),.5 (tringles); () dependence on with R r = 1 nd three R e vlues s in ()., the ehvior of j ws very similr to tht of the 5:5 cse. Tle 1 summrizes ll results of j for the vrious disorder cses t five tempertures. Once gin, for these reltive concentrtions studied, the mximum decrese in j is through R r. Additionlly, s is shown in Fig. 6(c) for the 5:5 reltive concentrtions, the effect of incresing R e while R r nd remin constnt, produced n pprent slight increse in j. This effect is lso present for the other reltive concentrtions s reported in Tle 1. To compute the lttice therml conductivity from Eq. (6), the utocorreltion function C(s) ws pproximted y n exponentilly decying cosine function. In Fig. 9, the verticl xis on oth plots depicts the system relxtion time nd is plotted s function of R r in Fig. 9() nd in Fig. 9(). The relxtion time ppers to e directly relted to the mount of core rdii nd mss disorder present in the smple. The chnge in relxtion time due to the e disorder is smll s evidenced y the three curves in Fig. 9() nd (). Tle 1 Lttice therml conductivity for solid mixtures with vrious reltive concentrtions nd t T =.4,.83,.167,.333 nd.5 (top to ottom in ech tle entry) Reltive concentrtion R r =1, =1,R e =1 R r R e % A % A, % B % A, 5% B % A, 75% B For ny prmeter rtio 6¼ 1, the other two prmeter rtios = 1. Vlues re in reduced units.

7 J.W. Lyver IV, E. Blisten-Brojs / Act Mterili 54 (6) Summry nd conclusions Throughout this work, it hs een shown tht studying the effects of rdii, mss, intertomic interction disorder nd temperture cn e demonstrted in computer-simulted environment. Our work ws performed on PC with single Pentium 4 processor (3. GHz) nd ech NPT-MC nd MD run consumed out 9 nd 4 h of processing time, respectively, per dt point, mking the work resonle to ccomplish. The results of this work show tht compositionl disorder t the nnoscle in crystlline inry mixtures decrese the lttice therml conductivity in drmtic fshion. Findings in this work re importnt for tiloring the synthesis of new mterils with poor het conduction chrcteristics. The reltive properties of L-J solid mixtures re summrized elow in order of importnce for degrding the lttice therml conductivity: (1) vn der Wl rdii. Atoms should hve different rdii. Even 1% difference rings the lttice therml conductivity to minimum constnt vlue nd suppresses the inverse temperture depletion. The reson for the drmtic degrdtion of the het conduction is the dditionl phonon scttering imposed t the nnoscle y toms tht re t the threshold of collpsing the crystl structure of the solid. () Mss. Atoms should hve different msses. Differences of 6% in mss decrese the therml conductivity y out hlf t ny temperture elow the melting point. (3) Intertomic interction strength. Atoms should hve lmost equl vlues. With 5% difference in strength, therml conductivity cn e incresed y out %, which is not desired outcome. (4) Temperture. Temperture is key fctor for ny ppliction serching to deplete het conduction due to tomic virtions. This work ws done for reduced tempertures of up to.5, which re elow the meting points of the L-J compositionlly disordered crystls studied. In this temperture rnge, when rdii disorder exists, the lttice therml conductivity is essentilly temperture independent nd mrkedly degrded due to the enhnced phonon scttering induced y toms with different rdii plced rndomly on n fcc crystl. (5) Composition reltive concentrtion. Reltive concentrtion of the two components in the crystl ppers to hve only minor effect on the therml conductivity. References [1] Lukes JR, Dy L, Ling X-G, Tien C-L. Trns ASME ;1: [] Feng X-L, Li Z-X, Guo Z-Y. Chin Phys Lett ;18: [3] McGughey AJH, Kviny M. Int J Het Mss Trnsf 4;47: [4] Chen Y, Li D, Yng J, Wu Y, Lukes JR. Physic B 4;349:7. [5] Heino P. Phys Rev B ;71:1443. [6] Li J, Porter L, Yip S. J Nucl Mter 1998;5: [7] Bermn R. Therml conduction in solids. Oxford: Clrendon Press; [8] Jonson M, Mhn GD. Phys Rev B 198;1:43 9. [9] Frenkel D, Smit B. Understnding moleculr simultion. New York (NY): Acdemic Press; [1] Blescu R. Equilirium nd non-equilirium sttisticl mechnics. New York (NY): Wiley; 1979 [Chpter 1]. [11] Green MS. J Chem Phys 195;:181 95; Green MS. J Chem Phys 1954;:398. [1] Kuo R. J Phys Soc Jpn 1957;1: [13] Ldd AJC, Morn B, Hoover WG. Phys Rev B 1986;34: [14] Gilln MJ. J Phys C Condens Mtter 1987;: [15] Chen Y, Lukes JR, Yng J, Wu Y. J Chem Phys 4;1: [16] Grzon IL, Long XP, Kwi R, Were JH. Chem Phys Lett 1989;158:5 3. [17] Tretikov KV, Scndolo S. J Chem Phys 4;1: [18] Kurki H, Li J, Yip S. Mtter Res Soc Symp Proc 1998;538:53 8. [19] Christen DK, Pollck GL. Phys Rev B 1975;1: