THE ELECTROMAGNETIC AND THERMOMECHANICAL EFFECTS OF ELECTRON BEAM ON THE SOLID BARRIER

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1 MATHEMATICA MONTISNIGRI Vol XXXIX (07) MATHEMATICAL MODELING THE ELECTROMAGNETIC AND THERMOMECHANICAL EFFECTS OF ELECTRON BEAM ON THE SOLID BARRIER K.K. INOZEMTSEVA M.B. MARKOV F.N. VORONIN Keldysh Insue of Aled Mahemas Mosow Russa e-mal: Summary: Subsane behavor under hgh-nensve radave eosure s onsdered. The hermomehanal and eleromagne roesses whh hgh-urren eleron beam naes n a sold barrer are modeled n her relaonsh. In addon o eleron ransor equaon mahemaal model nludes Mawell equaons wh onveve urren and Euler equaons wh he Lorenz fore and Joule heang. I allows o nvesgae he omle radaon effe of eleron beam. The eressons for he Lorenz fore densy dealng wh onzed subsane and Joule heang n subsane due o he eleromagne feld are onsrued. Conservave fne dfferene aromaon s aled o he desron of he eleromagne feld ma on onzed subsane. The relmnary resuls of he hermomehanal and eleromagne effes neraon are reresened.. INTRODUCTION Invesgaon of subsane roeres n ereme ondons s an aual roblem. The resuls are moran for new maerals desgn roeon roblems e. Laser radaon and eleron beams reae he ereme ondons n laboraory eermens. Lasers are used for surfae eaon of nvesgaed barrer whle eleron beams are alable when volumer energy release s needed []. The eermens wh owerful soures of radaon are raher eensve. Measurng equmen anno rovde full quanave daa for dealed desron of he neraon beween radaon and subsane. Therefore mahemaal modellng beomes an effeve suor for eermenal nvesgaon [-6]. Radave hermomehanal and eleromagne effes aomany he neraon beween an eleron beam and a sold barrer. All of hem arse from eleron ransor and saerng. The beam neras wh he barrer hrough elas saerng bremssrahlung ma onzaon and eaon. [7-9]. I s ma onzaon ha s he man hannel of energy ransfer from elerons o he barrer. The dfferenal onzaon ross seon s nversely rooronal o he square of he energy ransfer magnude. So a lo of low energy seondary elerons ross o so-alled onzaon serum [0-]. They degrade o equlbrum wh energy ransfer o saerng medum []. Eess harge arrers ause radaon-ndued onduvy energy ransfer leads o medum heang [3]. Heang auses subsane evaoraon ressure nrease and sho wave generaon [4]. Nonunform hermal and mehanal felds arouse hermomehanal effes: deformaon melng evaoraon e [4]. 00 Mahemas Subje Classfaon: 78M0 78A5 76M 35Q3. Key words and Phrases: Eleron beam Saerng Thermomehanal effe Eleromagne feld. 79

2 K.K. Inozemseva M.B. Marov F.N. Voronn. Eleron movemen reaes urren densy whh generaes he eleromagne feld. The nonunform urren densy auses bul eler harge [5]. Eleromagne feld s a reason for eleromagne effes: eler urren of onduvy eler breadown e. Suh ollson roess as bremssrahlung should be onsdered searaely [9]. Mos of he esng hgh-urren aeleraors generae elerons wh he energy less han 0 MeV. Corresondng bremssrahlung hoons reeve energy near MeV [6]. The free ah of suh hoon eeeds he elerons one by wo orders of magnude. As a resul he sze of onzed regon eeeds he eleron free ah suffenly. Bremssrahlung hoons eerene Comon and oheren saerng hoo absoron rodue eleron-osron ars [6]. Consequenly bremssrahlung hoons urn no he flu of harged arles onversely. Radave hermomehanal and eleromagne felds affe eah oher. Densy redsrbuon durng he dumng of he mehan sresses hanges he saerng roeres of subsane. Ionzaon durng radave heang enhanes he onduvy whh redues he eler feld. The Eler feld and he bul harge reae he onderomove fore. I moves subsane along wh he ressure graden. Joule heang of onduve subsane n he eler feld auses addonal energy release. Keldysh Insue researhers have develoed a rogram aage REMP (Radaon and EleroMagne Pulse) for mahemaal modellng of hysal effes whh aomany he neraon of elerons and hoons wh omle ehnal objes. I nludes omuaonal modules and he user nerfae onneed by unfed daa ommunaon roool. The lass ransor equaon desrbes free elerons saerng. The Mone-Carlo mehod rovdes means for he dre modellng of arle ollsons. The lass ne equaons desrbe arle roagaon n vauum and gases. The Mawell equaons desrbe he eleromagne feld evoluon. The menoned equaons form he arsenal o onsder he ransor roesses he eleromagne feld boh eernal and self-onssen and her neraon. The omuaonal modules are onneed hrough a rogram sr wh he hydrodynam ode MARPLE-3D [7]. I enables o onsder he dynams of subsane eosed o radaon. The quanum ne equaons for onduvy elerons and holes of valene band desrbe he radaon-ndued onduvy of semonduors. The free eleron energy release serves as a soure n. The ollson negral n he quanum equaon desrbes saerng on honons of rysal lae. The sasal arles mehod solves numerally boh he quanum and he lass ne equaons. The mehod ombnes sohas saerng smulaon wh he equaons of moon n eleromagne feld whh are solved beween ollsons [8]. The rogram aage s oeraed a he heerogeneous luser К-00. Ths arle s devoed o he sr develomen for smulaon of nerang hermomehanal and eleromagne felds of radaon geness.. PROBLEM STATEMENT The followng ne equaon desrbes fas elerons ransor and saerng: fe + dv( хfe) + edv + [ ] fe + σ eυ fe = с d σ h e( ) f h( ) + E х B ' ' ' + d' σ ( ) ' ( ) ( ) ' ( ) e- e ' υ fe ' + d' σ e ' υ f ' + Qe () 80

3 K.K. Inozemseva M.B. Marov F.N. Voronn. where fe = fe ( r ) f = f ( r ) f h = f h ( r ) are he dsrbuon funons of elerons osrons and hoons resevely n he hase sae of oordnaes r and momenum х s arle veloy с s he seed of lgh e s an eleron harge E = E( r ) B = B ( r) are he eler feld srengh and he magne nduon ' s he arle momenum before ollson dv s he dvergene oeraor n momenum sae σ e s he oal ross seon of eleron adsoron σ h e σ e e σ e are he dfferenal ross seons of hoon eleron and osron saerng wh eleron generaon Q = Q r s he eleron soure. e e ( ) The followng ne equaon desrbes osrons and hoons whh are generaed n asade roesses: f + dv( хf ) edv + [ ] f + σ υ f = с d σ h ( ) f h ( ) + E х B ' ' ' f + d' σ ( ' ) υ ' f ( ' ) + dv( Щf ) + σ f = d' σ ( ' ) f ( ' ) + d' σ ( ' ) υ ' f ( ' ) + h e h e h h h e- h e + d' σ ( ' ) υ ' f ( ' ) h where Щ s he un veor of hoon veloy adsoron σ s he oal ross seon of osron adsoron () (3) σ h s he oal ross seon of hoon σ h σ are he σ h h dfferenal ross seons of hoon and osron saerng wh osron generaon σe h σ h are he dfferenal ross seons of hoon eleron and osron saerng wh hoon generaon. The equaons () () (3) are onsdered n he sae of fnary generalzed funons [9]. The deals of he mehods of equaon solvng are gven n [8 0 ]. Les onsder he followng mahemaal onsruons: ( ) ( ) ( Δ) e e ( ) e e ( ) ( ) ( ) ( Δ) ( ) ( ) ( ) ( ) ( Δ) h h ( ) h h ( ) e Qε r d dr ε W r r σ υ f r d' σ ( ' ) υ ' f r (4) Qε r d dr ε W r r σ υ f r d' σ ( ' ) υ ' f r (5) h Qε r d dr ε W r r σ υ f r d' σ ( ' ) υ ' f r (6) 3 where ε ( ) energy of arle nfnely dfferenable funon W ( r r Δ) Δ > 0 sasfes ondons W 3 ( Δ ) d = lm d W ( Δ ) ϕ ( ) = ϕ ( ) 3 r r r r r r Δ 0 r r r r r r for every nfnely dfferenable funon from he vo sae. The followng relaons onne he dfferenal saerng ross seons n (4) (5) (6) wh summands of ollson negrals n he equaons () () (3): 8

4 K.K. Inozemseva M.B. Marov F.N. Voronn. The eressons (4) (5) (6) deermne he energy ower densy whh elerons osrons and hoons ransm o saerng medum. The oal energy release s: Les onsder onsruons: where m s eleron mass and he veor σ ( ' ) = σ ( ' ) + σ ( ' ) e e e e h σ ( ' ) = σ ( ' ) + σ ( ' ) + σ ( ' ) e h σ ( ' ) = σ ( ' ) + σ ( ' ) + σ ( ' ). h h e h h h ε Q = Q + Q + Q. (7) e h ε ε ε e ( ) e d d W ( Δ) fe ( ) f ( ) qe ( ) e d d W ( Δ) fe ( ) f ( ) ( ) e d d W ( ) σ eυ fe ( ) d σ e( ) υ ' fe ( ) + σ υ f ( r ) d' σ ( ' ) υ ' f ( r ) + + σ hυ f h ( r ) d σ h( ) υ ' f h ( ) ' ' r ( r) r ( r r ) σ eυ e ( ) σ e( ) υ ' e ( r ' ' r ) + σ υ f ( r ) d σ ( ) υ ' f ( ) ' ' r j r r х r r r r (8) r r r r r r (9) Δ + Q r r r r r ' ' r ρ Q d d mw Δ f d f + Q has omonens Q = 3. The formulae (8) (9) eress harge and eernal eler urren densy. The formulae (0) () eress momenum and mass ransfer o saerng medum. The reresened defnons of he model arbues rovde model alably for heavy arle flues. Energy ransfer o he barrer auses heang melng or evaoraon. Every roess aes lae n aual ases. There are regons of barrer where energy release eeeds mahes or s less han he hea of sublmaon. Dfferen equaons desrbe hs suaons. Bu s mossble o era domans of alably for hs equaons a ror. We use deal hydrodynam Euler equaons [] for all ases. Comle abular equaons of sae desrbe dealed roeres of subsane [3]. The followng Euler equaons are lass: where ρ s he densy of barrer subsane s ressure T s emeraure u s nernal energy v s he sef veloy of barrer subsane wh omonens v = 3. The Summaon onvenon s aled. (0) () ρ + ρ dvρv = Q () ρv + ( ρvv + δ ) = Q (3) v v ε ρ + u +dv vρ + u + v = Q (4) 8

5 K.K. Inozemseva M.B. Marov F.N. Voronn. Elerons neraon wh he barrer s aomaned by generaon of eler harge j r denses. Eler urren n s urn generaes eleromagne q ( r ) and urren ( ) e e feld. I reaes onderomove fore whh ses onzed subsane n moon. So balanes of momenum (3) and energy (4) n onzed subsane should be ransformed n he orresondng relaon for he sysem nludng subsane and eleromagne feld [4]. The erns urren wor je E should be subraed from he eernal energy gan. A new soure q E j B с ang on beam elerons. of momenum emerges s he Lorenz fore [ ] The followng Euler equaons desrbe onzed subsane and he feld: where for soro subsane T ' ' ' ' ([ ] [ ] ) 4 E D H B δ π v v = + E'D' + H'B' E' H' + E' H' (8) veors E P D H M B wh omonens E P D H M B reresen he eler feld srengh he olarzaon he eler dslaemen he magne feld srengh he magnezaon and he magne nduon resevely S s he Poynng veor g s he eleromagne feld momenum densy Veors E' P' D' H' M' B' wh omonens e ρ + m dvρv = Q (5) ( ρv + g ) = ( ρvv + ( sr ) δ T ) + Q qe E + [ je B] (6) v E B v ε ρ + u + + M'B + [ M' E' ] = Q jee 8π 8π v E' B' dv vρ + u + S + v sr P' E M' B e sr s he sron ressure. g = [ E ] 4π H (9) S = [ E H ]. 4π (0) E ' P ' D ' H ' M ' B ' reresen he eler feld srengh he olarzaon he eler dslaemen he magne feld srengh he magnezaon and he magne nduon n he subsane's res frame. (7) v D' = D + H' () v E' = E + B' () 83

6 K.K. Inozemseva M.B. Marov F.N. Voronn. v P' = P M' (3) v H' = З D' (4) v B' = B E' (5) v M' = M + P' We roose he followng onneons beween E' and D' B' and H': (6) D' = εe' (7) B' = μh' (8) where he deler ermvy ε and ermeably μ deend on hermodynam arameers only. Ths deendene auses he sron ressure sr E ' ε H ' μ = ρ + ρ. 8π ρ 8π ρ sr [4]: Thermodynam relaons [4] deermne he followng onsequene for delers. Sef nernal energy an be reresened as a sum of omonens. One of hem u 0 s ndeenden from eler feld: (9) E'P' B'M' E ' ε H ' μ u = u T + T. (30) ρ ρ 8πρ T 8πρ T Les onsder Mawell equaons [5] for eleromagne feld n onnuous meda: D 4π * roh = + ( je + j + qv ) (3) B roe = (3) dv D = 4π ( q + qe ) (33) * where q s harge densy n onzed subsane j s he urren densy of onduvy elerons. In he frame of Cauhy roblem for equaons (3-33) Coulomb law (33) s equvalen o harge onnuy equaon: ( q q ) + e * ( e q ) + dv j + j + v = 0. The law of eleromagne feld energy onservaon follows from Mawell equaons (3-33): 84

7 K.K. Inozemseva M.B. Marov F.N. Voronn. where j s he oal urren densy: E B P B + + E M = dvs je (34) 8π 8π * j = j + qv + j e. (35) Les mully Ошибка! Источник ссылки не найден. by v and sum over nde : v ( ρv + g ) = vq v qe E + [ e ] v ( ρvv ( sr ) δ T ). + j B (36) Afer erm rearrangng n (36): ρ v v v + ρ = v ρv ρv v qee + [ je B] g v ( ρvv + ( sr ) δ T ) + vq v. The equaon for ne energy balane follows from (37) and (5): v v g ρ v ( ρvv ( sr ) δ T ) v ρ v = + v qee + [ je B] + vq. Relaons (7) (30) (34) (38) deermne he law of nernal energy onservaon. One an use nsead of (7): d E ' ε H ' μ ε ρ u0 + T + T = ( sr ) dvv + ζ + Q vq + ( j je ) E + d 8π T 8π T g + v v T + v qe E + [ je B] where d = + v s he oal dervave d dv( ( )) v E'P' B'M' ζ = M'B [ M' E' ] + v P' ( E E' ) + M' ( B B' ) + P B + E M. The dervaon of he equaon (39) s founded on he followng assumon. Thermodynam arameers deermne deler ermvy unquely. I means ha only loally equlbrum and reversble roesses hange ermvy. Subsane dynams equaons (6) (7) are formulaed as laws of onservaon. Therefore equaons valdy s (37) (38) (39) (40) 85

8 K.K. Inozemseva M.B. Marov F.N. Voronn. onneed wh he alably of he hermodynam relaons for subsane under onsderaon (for eamle (30)). Equaons (5) (6) (7) hold rue f he relaaon mes of he roesses whh are onneed wh ermvy hange are small n omarson wh he haraers maroso me. The law of eleromagne energy onservaon (34) doesn hange so he equaon (39) holds rue. One an smlfy (39) wh he hel of momenum onservaon law [4]: where g T = qe E + [ je B ] + F (4) E' B' v F = qe + ( e ) ([ ] [ ] j j B + P' + M' + P' B' M' E' ) + (4) ρ d + + d ([ ' B' ] [ m' E' ] ) ( P'E' M'B' ) The ne relaon follows from (39) and (4): ' = P' ρ (43) m' = M' ρ (44) d E ' ε H ' μ ε ρ u0 + T + T = ( sr ) dvv + ζ + Q vq + ( j je ) E vf (45) d 8π T 8π T Usng he followng relaons v d v E' P' = dv (( P'E' ) v) + E' P' ρe' ' (46) r d d v B' M' = dv (( M'B' ) v) + B' M' ρb' m' (47) r d v ([ P' B' ] [ M' E' ]) = dv ( v [ P' B' ] [ M' E' ]) + v ([ P' B' ] [ M' E' ]) (48) r and negleng erms of he order of d ρv ' B' m' E' d v we oban: ([ ] [ ]) E' B' P B vf = dv P' E + M' B + + ( ) v E M M'B v ( [ ]) q d d * ' m' v M' E' + Ev + ρ ρ j B E' B' d d Subsung vf from (49) no (39) one an oban: (49) 86

9 K.K. Inozemseva M.B. Marov F.N. Voronn. d ε E ε * v * ρ u0 + T = ( sr )dvv + Q vq + j E d T 8π j B E'P' M'B' ( E'P' ) ( M'B' ) d' dm' + dv v + v + ρe' + ρb'. d d (50) Usng (7) (8) one an smlfy (50): d ε E v ρ u0 T ( )dv Q d T 8π E ' dε H ' dμ π d 8π d ε * * + = sr v + vq + j E j B + One an oban anoher form of momenum onservaon law for he sysem of subsane and feld nsead of (6) wh he hel of (3): dv * v ρ = ( sr ) + qe ' + ([ ] [ ]) d j B' + P' B' M' E' E ' P' B' M' ρ d + ' ([ ] [ ] ). P' E + M' B' + ' B' m' E' d The eresson for urren densy of onduvy elerons omlees equaons (-4): v v (5) (5) * j = σe' = σ E + B' = σ E + B (53) where σ s oal hermal and radave onduvy. One an oban he relaon for hermal onduvy from he Wedemann-Frans law. I onnes eler and hea onduvy. The eleron hea onduvy λ s onsdered n aromaon [3]: λ 3 = 4aθ l 3 (54) where a 3 3 = π /5. l = λ + λ θ 3 [см] (55) S H where λ S and λ H are he hermal onduvy oeffens for deal nondegenerae and ghly degenerae lasma: λ eθ 5/ 3 S = 7 0 (56) Zeff L 3/ 4 eθθf λh = Z L (57) eff 87

10 K.K. Inozemseva M.B. Marov F.N. Voronn. The faors e and e rovde orreon for lasma nondealy. Aom uns are used for subsane θ and Ferm θ F emeraures. The Coulomb logarhms of eleron-on ollsons an be alulaed n he ne aromaons: ( π eff ) ( ) /3 / L = 0.5ln Z Г (58) / Zeff Z 0 Z eff L = 0.5ln 9 ma + Г 3Г ( + Z0 ) (59) where Г s he arameer of nondealy: Г Z Z mn. (60) eff 0 = r0 θ θ F The ne aromaon s used for effeve harge: { } Z = ma Z / Z (6) eff 0 0 where Z 0 s on average harge n aordane wh he Harree-Fo-Slaer model [3]. Equaons () () (3) (5) (6) (or (5)) (7) (or (39) or (5)) (3) (3) (33) (53) ouled wh he abular equaons of sae and homogeneous nal ondons mae u he Cauhy roblem for ne hydrodynam and Mawell equaons. Problem s osed for he nvesgaon of neraon beween eleron beam and olarzable and magnezable subsane. The man orrelaons beween eleromagne and hermomehanal effes an be esmaed whou onsderng olarzaon and magnezaon when v. In addon boh of ermves should no hange vsble a sales of eleron song ah and haraers hydrodynam me. The followng smlfed equaons should be onsdered for nonolarzed and nonmagnezed subsane nsead of (6) (7): ρ v + ( v ) v = grad + Fl + Q (6) v v ε ρ u 0 dv + = vρ + u0 + v + A + Q (63) where F l s he Lorenz fore affeng he onzed subsane A s he ower of he sef oal eleromagne feld wor. Les defne he energy * Fl = q E + [ v B] + j B (64) * A = qev + j E = ( j je ) E. (65) A H whh feld sends on subsane heang: 88

11 K.K. Inozemseva M.B. Marov F.N. Voronn. A H ( ) * * = A Fl v = j E v j B. (66) Relaons (64) (65) (66) deermne he eleromagne feld nfluene on onzed subsane. Subsung (53) o (64) and (66) one an reeve he eressons for Lorenz fore and he heang ower densy: σ σ Fl = q E + [ v B] + [ E B] + ( vb B( vb )) (67) σ σ AH = σ E ( v[ E B] ) ( v B ( vb ) ). (68) Toal medum heang by he feld elerons osrons and hoons s equal o: AН = A + Q ε vq. (69) Н The Coulomb law (33) enables o eress he Lorenz fore hrough he eleromagne feld omonens only: dv D σ σ Fl = E + [ v B] qe + [ E B] + ( vb B( vb )). (70) 4π 3. NUMERICAL ALGORITHMS The numeral algorhm for hydrodynam equaon s based on he onservave fnedfferene sheme of he nreased order of auray [5]. I s he Kolgan sheme whh s modfed and generalzed for 3D unsruured grds. The redor-orreor sheme negraes grd equaons wh rese o me varable. Sne he sheme s el he Couran FredrhsLewy ondon should be mlemened. The fne-dfferene sheme for Mawell equaons s reresened n [0]. I s also el; rovdes he seond order of auray on homogeneous Caresan grd mlemens he dsree analogue of he energy onservaon law. Le's onsder he sheme for he sysem of eleromagne and hydrodynam equaons. I s neessary o onsru fne-dfferene analogues for Lorenz fore (70) and heang (68) whh rovde onservaveness of he general sheme. Followng relaons [0] deermne omuaonal grd for Mawell equaons (3) (3) (33) (53) wh rese o varable : = + Δ = N 0 = mn N = ma + / = ( + + ) / = 0... N / = 0 N + / = N = / / = 0... N δ = Δ / δ = Δ / δ / N = Δ N. δ + Comuaonal grds wh rese o varables y z are nrodued n a smlar manner. Grd arameers rovde he loalzaon of oeffens dsonnuy on surfaes = y = y j z = z. The oeffens values are defned a he grd ons wh half-neger saal ndees. These ons onde wh eners of reangular aralleleeds whh are organzed by 89

12 K.K. Inozemseva M.B. Marov F.N. Voronn. nerseon of lanes = + y = y j y j + z = z z +. The urren densy and eler feld omonens are defned n he eners of orresondng arallel aralleleed edges. The magne feld omonens are defned n he eners of hose aralleleed faes o whh hey are normal. The onveve urren and medum movng are no under onsderaon n [0]. Tme grd onsss of ons n wh nervals τ n = n+ / n / ; n =... N. The Grd y z y z funons E E E are defned n he neger momens of me n H H H are defned a he half-neger ons n + /. Les onsder grd analogous for Mawell equaons (3) (3) [0]: z n+ / z n+ / y n+ / y n+ / ( + / + / + / / ) δ ( + / + / + / / ) H H y H H δ z = j j j j j n+ n = ( D + / j D + / j ) τ n+ / + I + / j. n+ / n+ / z n+ / z n+ / ( + / + / + / / ) δ ( + / + / / + / ) n The followng desgnaons wll be used. Symbol s sgnfes funon value s on me layer wh number n. Symbol s + / j sgnfes he followng weghed value he of grd funon u : s u j + / j = + / j / + / δ j δ Δ Δ Δ Δ Δ Δ + s + s + s j j j + / j / + / + / j+ / + / + / j+ / /. δ j δ δ j δ δ j δ Δ 4π H H z H H δ = j j j j 4π y n+ y n = ( D j+ / D j+ / ) τ n+ / + I j+ /. y n+ / y n+ / n+ / n+ / ( + / + / / + / ) δ ( + / + / / + / ) H H H H δ y = j j j j j 4π z n+ z n = ( D j + / D j + / ) τ n+ / + I j + / y n+ y n+ z n+ z n+ ( + / + + / ) ( + + / + / ) n+ 3/ n+ / = ( H / / / / ). j+ + H j+ + τ n+ z n+ z n+ n+ n+ ( + + / + / ) ( + / + + / ) yn+ 3/ yn+ / = ( H / / / / ). + j + H+ j + τ n+ n+ n+ y n+ yn+ ( + / + + / ) ( + + / + / ) z n+ 3/ z n+ / = ( H / / / / ). + j+ H+ j+ τ n+ E E Δz E E Δ y = j j j j j E E Δ E E Δ z = j j j j E E Δy E E Δ = j j j j j Δ y z. 90

13 K.K. Inozemseva M.B. Marov F.N. Voronn. The nduon veor s deermned as D + / j = ε + / j E+ / j he oal urren densy s denoed as I+ / j = σ + / j E+ / j + J+ / j where J + / j s he erns urren densy. The followng fne-dfferene law of eleromagne feld energy onservaon [0] aes lae n omuaonal doman: + + ( ) fd n n fd n n fd fd fd fd W (E H ) W (E H ) τ + Q + A + K + S = 0 (7) n fd n n where W (E H ) s he quadra form omosed from values of he eler and he magne felds srengh omonens a me layer wh number n. Ths quadra form s a grd analogue of eleromagne feld energy. Fne-dfferene analogues of all urren fd fd fd fd wors eress values A Q and K. The value S eresses energy flow hrough he boundary doman. The eresson for fne-dfferene wor of he erns urren n omuaonal doman s: N N y Nz N N y Nz fd y y 4π + / j + / j δ jδ 4π j+ / j+ /δ jδ = 0 j= 0 = 0 = 0 j= 0 = 0 (7) A = J E Δ + J E Δ + where / j N N y Nz + 4 π J E δ δ Δ = 0 j= 0 = 0 z z j + / j+ / j E + J + / j are he eler feld srengh and he urren densy a he on = + / y = y j z = z. The oal urren wor s eressed analogously. The fne-dfferene wor of he erns urren n he ell wh he ener a he on = + / y = y j+ / z = z + / s equal o: fd ell π j ( + / j + / j + / j+ + / j+ + / j + + / j + + / j+ + + / j+ + A = Δ Δ Δ J E + J E + J E + J E + + J E + J E + J E + J E + (73) y y y y y y y y j+ / j+ / + j+ / + j+ / j+ / + j+ / + + j+ / + + j+ / + + J E + J E + J E + J E z z z z z z z z. j+ / j+ / + j + / + j+ / j+ + / j+ + / + j+ + / + j+ + / The fne-dfferene wor of he onduve urren n he omuaonal doman s а sum of ell wors n edges: fd ell π j ( + / j + / j + / j+ + / j+ + / j+ + / j+ Q = Δ Δ Δ J E + J E + J E + + J E + J E + J E + J E + y y y y y y + / j+ + + / j+ + j+ / j+ / + j+ / + j+ / j+ / + j+ / + + J E + J E + J E + J E + y y z z z z z z + j+ / + + j+ / + j + / j+ / + j + / + j + / j+ + / j+ + / where: + J z z + j+ + / E + j+ + / ) ) (74) 9

14 K.K. Inozemseva M.B. Marov F.N. Voronn. J + / j = σ+ / j E + / j + v+ / j B + / j+ / + /. (75) Le he subsane veloy n he hydrodynam fne-dfferene sheme be defned n ell eners. The harge densy s defned n ell eners oo. We defne onveve urren densy n edge ener as: J = q v Δ Δ + q v Δ Δ + ( + / j + / j+ / + / + / j+ / + / j + / j / + / + / j / + / j 4δ jδ + / j+ / / + / j+ / / j + / j / / + / j / / j ) / j + q v Δ Δ + q v Δ Δ = J + where J+ / j q + / j+ / + / v + / j+ / + / are he -omonen of onveve urren densy he harge densy and he subsane seed n ell ener. fd The wor K и K of he onveve urren eressed by (76) s defned by analogy fd ell wh he erns one (7) (73). If deler ermvy s onsan only onveve and onduve urrens fulfls he energy ehange beween subsane and feld n aordane wh (34) (65). Les defne sef Lorenz fore omonen for he ell wh ener a he on = y = y z = z aordng o eresson (67). I s neessary o nerolae values + / j+ / + / of he eleromagne feld srengh omonens n he ell ener. We shall alulae he magne feld as an average value wh rese o all ell faes wh wegh δ. The Eler feld wll be alulaed wh rese o all ell edges wh wegh δ δ j : l F + / j+ / + / = q + / j+ / + / E+ / j+ / + / + + / j+ / + / + / j+ / + / + v B l where + / j+ / + / σ + / j + / + / E+ / j+ / + / B+ / j+ / + / + + σ + / j + / + / v / j / /B / j + / + / + σ + / j + / + / + / j+ / + / ( + / j+ / + / + / j+ / + / ) B v B F s he fne-dfferen analogue of he sef Lorenz fore. The wor of onveve and onduve urrens doesn aear n he algorhm elly. So s neessary o onsru he fne-dfferene analogue of wor whh s eended on heang only n oher words on hange of he nernal energy n ell. To do hs les mully Lorenz fore by hydrodynam veloy salarly. Then les subra he resul from he oal wor of onveve and onduve urrens: A = Q K E v q Н fd fd + / j+ / + / ell ell + / j+ / + / + / j+ / + / + / j+ / + / ( + / j+ / + / + / j+ / + / + / j+ / + / ) (76) (77) σ v E B (78) 9

15 K.K. Inozemseva M.B. Marov F.N. Voronn. ( v + ( ) ) / j + / + B / + / j + / + / + / j + / + / + / j + / + / σ v B. Fnally s neessary o eress harge densy n feld omonens. The defnon of oal harge densy a he grd on follows from he fne-dfferen sheme [0]: q o j where / j D D D D D D = + + 4π δ δ j δ + / j / j j+ / j / j + / j / (79) D + s he omonen of eler dslaemen. I s defned n he ener of ell edge. The edge s arallel o he omonen. We defne he harge densy as a dfferene beween oal harge averaged wh all ell ons wh wegh δδ jδ and he harge densy of fas elerons e q : q = q δ δ δ + q δ δ δ + q δ δ δ + ( o o o + / j+ / + / j j + j + j j+ j+ 8V + / j+ / + / + q δ δ δ + q δ δ δ + q δ δ δ + q δ δ δ + (80) o o o o j + j + + j+ + j+ + j + + j + j+ + j+ + + q + q q o o e + j+ + δ + δ j+ δ + + j+ + δ + δ j+ δ + + / j+ / + / where V + / j + / + / s ell volume: V + / j+ / + / = ( δδ jδ + δ+ δ jδ + δδ j+ δ + δδ jδ δ δ δ + δ δ δ + δ δ δ + δ δ δ. + j+ + j + j+ + + j+ + Les subsue (80) n (77) (78). In hs way we eress he Lorenz fore and subsue heang n erms of eleromagne feld and hydrodynam veloy. They an be deermned by solvng Mawell and hydrodynam equaons. Formulae (77) and (78) eress he fne-dfferene momenum and he nernal energy. They are ransmed n he hydrodynam fne-dfferene sheme as a soure. I rovdes he onservaveness of he fne-dfferene sheme n general. Now les onsder he fne-dfferene hydrodynam equaons. Grd funons n hese equaons are arbued o ell eners and are desrbed n [5]. The ranson o he ne me level s erformed n wo sages. A he frs sage ressure fore s onsdered. The nermedae values of he subsane veloy v and he sef energy densy w are alulaed: ρ v ρ n n n n + / j+ / + / + / j+ / + / + / j+ / + / + / j+ / + / τ v n + / j+ / + / ) ) = V ( F + j+ + F j+ + F + j+ + F + j + F + j+ + F + j+ ) + + ρ / / / / / / / / / / / / w ρ n n n n + / j+ / + / + / j+ / + / + / j+ / + / + / j+ / + / τ w = V n + / j+ / + / (8) (8) (83) 93

16 K.K. Inozemseva M.B. Marov F.N. Voronn. ( A + j+ + A j+ + A + j+ + A + j + A + j+ + A + j+ ) + + / / / / / / / / / / / / V s he volume of he ell wh he ener + / j + / + / F s where / j / / he fore ang on a fae A s he wor of he fore F. w = u + v (84) + / j+ / + / + / j+ / + / + / j+ / + / where u + / j + / + / s he nernal subsane energy. Fore and wor are eressed n erms of veloy and ressure n he followng way: F = P S (85) ( ) A = S v P (86) where S s he area of he orened fae whh s dreed o he ell wh ne number P s he ressure n he ell ener. Veloy v s defned n he fae ener. n Conveve ransor s onsdered a he seond sage. The Fnal values ρ + n+ n v w + on he uer me layer are alulaed: ρ ρ ρ n+ n + / j+ / + / + / j+ / + / τ ρ = ( F + j+ / + / V n + / j+ / + / F + F F + F F ρ ρ ρ ρ ρ j+ / + / + / j+ + / + / j + / + / j+ / + + / j+ / v ρ n+ n+ n n + / j+ / + / + / j+ / + / + / j+ / + / + / j+ / + / τ v = V n + / j+ / + / ρv ρv ρv ρv ρv ρv ( F + j+ + F j+ + F + j+ + F + j + F + j+ + F + j+ ) + + ρ / / / / / / / / / / / / w ρ n+ n+ n n + / j+ / + / + / j+ / + / + / j+ / + / + / j+ / + / τ w = V n + / j+ / + / ρw ρw ρw ρw ρw ρw ( A + j+ + A j+ + A + j+ + A + j + A + j+ + A + j+ ) + + where F ρ fae: / / / / / / / / / / / / ρ v F ) (87) (88) (89) w F ρ are he onveve flows of ρ ρ v ρ w hrough he orresondng F ρ = ( S v )( ρuv + Δρuv ) F ρ v = S v ρvuv + Δρv ρw F = S v ρw + Δρw. ( )( uv ) ( )( uv uv ) Inde uv means ha he value s defned for wo ells whh adjon o fae from whh he veloy v s dreed. The symbol Δ means an amendmen o orresondng value whh rovdes monoony and nreased auray order. Le ell wh nde + / j + / + / s wndward for one wh he nde + / j + /. Then he amendmen wll be equal o mnmum modulus of wo values: (90) 94

17 K.K. Inozemseva M.B. Marov F.N. Voronn. Δ ρ = mn[ ρ ρ ρ ρ ]. (9) uw + / j+ / + / j+ / + / + / j+ / + / + / j+ / + The amendmen for ρ v and ρ w s alulaed analogously. If dfferenes n (9) have dfferen sgns amendmen s equal o zero. The amendmens o veloy and he nernal energy ondoned by he Lorenz fore and energy of subsane heaed by he eleromagne feld: Δ v = F τ ρ + (9) l n + / j+ / + / + / j+ / + / n + / j+ / + / Н n Δ u / j / / A / j / /τ n ρ = / j+ / + /. (93) Boh he elerodynam and he hydrodynam fne-dfferene shemes have he seond order of auray. The general fne-dfferene sheme has he seond order of auray oo. 4. TEST CALCULATION The effeveness of he algorhm has been nvesgaed n es alulaon. Condons were hosen n aordane wh he eermenal nsallaon. The beam of elerons wh energy of 00 ev falls normally on he eoy resn barrer. The ulse duraon was 50 ns 6 he eleron fluene N е = 3 0 /m he beam ross-seonal area was m. Suh eermens wh eleron beams are ondued a he Naonal Researh Cener Kurhaov Insue on he aeleraor CALAMARY [6]. Full 3D modellng of eleron ransor subsane dynams and he eleromagne feld was arred ou on luser K-00 a Keldysh Insue of Aled Mahemas. The energy release urren densy and eleromagne feld dsrbuons were alulaed by REMP aage. Fgure reresens he deendene of energy release on he oordnae normal o barrer surfae. The oordnae orgn s loaed on he rradaed barrer surfae n he ener of he beam ross seon. The energy release reahes he value of 50 J/m 3 and eeeds sublmaon hea o a deh of 0.0 m. Beyond hs dsane he energy release dereases sharly. One an see he regons of evaoraon melng and heang. 95

18 K.K. Inozemseva M.B. Marov F.N. Voronn. Fg. : Energy release as a funon of normal oordnae Fgure reresens he deendene of he eler feld srengh on he me varable a a deh of 0.0 m n barrer. The hermal and he radaon onduves were alulaed n he framewor of he aromaons onsdered n Seon. Fg. : The eler feld srengh (ESU CGS) as a funon of me varable The alulaed dsrbuons of energy release urren densy and eleromagne felds have been used for he smulaon of hermomehanal effes by he MARPLE aage. No ereble deendene of her arameers on he eler feld has been deeed. The smulaon has brough o lgh he hgh sensvy of resuls o he value of onduvy. I should be noed ha all models n Chaer are founded on lass fundamenal equaons. Only boh of onduves are onsdered whn he framewor of he emral model. I nvolves an nadmssble omuaonal load requred for he modelng of onzaon serum degradaon whn he framewor of ne heory. The fneness of he degradaon me leads o he delay n he develomen of onduvy relave o he eler urren. As a resul eler feld nreases. The ne omuer eermen was arred ou wh onsderaon of he fa ha one an esablsh he majoran esmae of eler feld equang onduvy o zero. Fgure 3 reresens he deendene of he eler feld srengh on he normal oordnae for nonondung barrer. 96

19 K.K. Inozemseva M.B. Marov F.N. Voronn. Fg. 3: The eler feld srengh (ESU CGS) as a funon of he normal oordnae n nonondung barrer Is obvous ha he eler feld wh he srengh of he order of 0 7 SGSE an be observed n a real eermen. Ths value eeeds sgnfanly deler rgdy of all nown subsanes. Neverheless les onsder he nfluene of suh feld on barrer subsane dynams. Fgure 4 reresens he deendene of he followng values on normal oordnae. The red lnes show he deendene of subsane densy. The blue lnes show he deendene of sefed seed normal veloy omonen. Herenafer sold lnes refer o he alulaon eermen n whh eler feld s aen no aoun. Dashed lnes refer o he alulaon eermen where he feld nfluene s negleed. One an see ha eler feld densy ea value dereases from.5 o.3 g/m 3 and he sefed seed nreases from o m/s. Fg. 4: The subsane densy and normal omonen of sefed veloy as a funon of normal oordnae. Sold lnes wh eler feld onsderng dashed lnes wh eler feld negleng 97

20 K.K. Inozemseva M.B. Marov F.N. Voronn. Fg. 5: The ressure and emeraure as a funon of normal oordnae. Sold lnes wh eler feld onsderng dashed lnes wh eler feld negleed Fgure 5 shows he ressure ea value derease from o Pa. The emeraure nreased from 0.05 o 0. ev. The eler feld of unrealsally large amlude hanged he hermodynam arameers only by ens of eren. 5. CONCLUSIONS Classs ne equaons for eleron hoon and osron dsrbuon funons n oordnae-momenum hase sae desrbe he radaon ransor n barrer subsane. Collson negrals model he medum ma onzaon and eaon by elerons elas saerng and bremssrahlung. Comon and oheren saerng hooabsoron and ar roduon of bremssrahlung hoons omlemen he se of hysal effes under onsderaon. Mawell equaons desrbe eernal and self-onssen eleromagne felds. Euler equaons model he subsane dynams under he nfluene of he eleromagne feld and energy release. Hydrodynam onsderaon unes he omuaonal doman regons of heang melng and evaoraon whn he framewor of he ommon hermodynamal mahemaal model. The ne equaons are solved n he sae of fne generalzed funons. The ower of energy release and eler urren densy are defned as a lnear funonal n hs sae. I enables he arle-n-ell mehod usage for ne equaons numeral smulaon [8] whn he soe of REMP aage. Prevously develoed fully onservave fne dfferene shemes for Mawell and hydrodynam equaons have formed he bass of numeral algorhm. The Fne-dfferene analogues of Joule heang he Lorenz fore and he onveve urren rovde he mlemenaon of fne dfferene energy onservaon law for he sysem of he uned eleromagne feld and onzed subsane. The neraon beween hermodynam and eleromagne felds s nvesgaed n he smulaon of eermen wh he eleron beam of CALAMARY aeleraor. The energy release of he hgh-urren eleron s omarable wh he Lorenz fore n he nononduve medum. The radave and hermal onduvy of he barrer hanges he 98

21 K.K. Inozemseva M.B. Marov F.N. Voronn. resul of smulaon dramaally. The energy release n suh meda reeaedly eeeds he Lorenz fore n nononduve medum. The nvesgaon showed ha he onduvy mahemaal model should be he man subje of furher researhes. The auhors are graeful o V. A. Gaslov and O. G. Olhovsaya for he ooruny o use he aage MARPLE3D and useful adve. Russan Foundaon for Bas Researh arally suored hs wor roje REFERENCES [] Vladmr E. Forov Ereme Saes of Maer. Hgh Energy Densy Physs Srnger Seres n Maeral Sene (06). [] P. V. Breslavsy A. V. Mazhun and O. N. Koroleva Smulaon of he Dynams of Plasma Eanson he Formaon and Ineraon of Sho and Hea Waves n he Gas a he Nanoseond Laser Irradaon Mahemaa Monsngr (05). [3] V. I. Mazhun M. M. Demn and A. V. Sharanov Hgh-seed Laser Ablaon of Meal wh Po- and Suboseond Pulses Aled Surfae Sene (04). [4] Yu. M. Mlehn D. N. Sadovnh and A. P. Tyunev Eleral effes n roellan omosons radaed by elerons. Combuson Eloson and Sho Waves 43 (3) (007). [5] D. N. Sadovnh A. P. Tyunev and Yu. M. Mlehn Charge aumulaon n olymehylmearylae radaed by hgh-energy elerons Khm. Vyso. Énerg. 39(3) 8389 (005). [6] M. E. Zhuovsy M. B. Marov S. V. Podolyao I. A. Taraanov R. V. Usov A. M. Chlenov and V. F. Znheno Researhng he serum of bremssrahlung generaed by he RIUS-5 eleron aeleraor Mahemaa Monsngr (06). [7] N. F. Mo H. S. W. Massey The heory of aom ollsons Oford: Clarendon Press (965). [8] H. S. W. Massey E. H. S. Burho Eleron and Ion Ima Phenomena Oford: Clarendon Press (969). [9] Н. Daves Н.А. Behe and L.C. Mamon Theory of bremssrahlung and ar roduon. Inegral ross seon for ar roduon Phys. Rev (954). [0] М. Gryzns Class Theory of Eleron and Ion Inelas Collsons Phys. Rev. 5 () (959). [] Yong-K Km M. E. Rudd Theory for Ionzaon of Moleules by Elerons Phys. Rev (994). [] М. B. Marov S. V. Paron The ne model of radaon-ndued gas onduvy Mahemaal Models and Comuer Smulaons 3 (6) 77 (0). [3] А. V. Berezn Yu. А. Volov М. B. Marov I. А. Taraanov The radaon-ndued onduvy of slon. Mahemaa Monsngr (05). [4] Ya. B. Zel'dovh Yu. P. Razer Physs of Sho Waves and Hgh-Temeraure Hydrodynam Phenomena Aadem Press New Yor (968). [5] L. D. Landau E. M. Lfshz The Classal Theory of Felds 4h Edon Buerworh Henemann Vol. (975). [6] W. Heler The Quanum Theory of Radaon Oford: Clarendon Press (954). [7] V. A. Gaslov A. S. Boldarev S. V. Dyaheno O. G. Olhovsaya E. L. Karasheva G. A. Bagdasarov S. N. Boldyrev I. V. Gaslova M. S. Boyarov V. A. Shmyrov Program Paage MARPLE3D for Smulaon of Pulsed Magneally Drven Plasma Usng Hgh Performane Comung Mah. Mod. 4 () (0). [8] А. V. Berezn А. S. Voronsov М. Е. Zhuovsy М. B. Marov S. V. Paron Parle mehod for elerons n a saerng medum Comuaonal Mahemas and Mahemaal 99

22 K.K. Inozemseva M.B. Marov F.N. Voronn. Physs 55 (9) (05). [9] G. E. Shlov Generalzed Funons and Paral Dfferenal Equaons Gordon and Breah Sene Publshers In. (968). [0] А. V. Berezn А. А. Kruov B. D. Plushhenov The mehod of eleromagne feld wh he gven wavefron alulaon Mah. Mod. 3 (3) 09-6 (0). [] А. N. Andranov А. V. Berezn А. S. Voronsov К. N. Efmn М. B. Marov The radaonal eleromagne felds modelng a he mulroessor omung sysems Mah. Mod. 0 (3) 984 (008). [] L. D. Landau E. M. Lfshz Flud Mehans nd Edon BuerworhHenemann Vol. 6 (987). [3] A. F. Nforov V. G. Novov V. B. Uvarov Quanum-Sasal Models of Ho Dense Maer: Mehods for Calulaon Oay and Equaon of Sae Brhauser Basel-Boson-Berln (005). [4] S.R. de Groo P. Mazur Non-equlbrum Thermodynams Amserdam: Norh-Holland Publ. Co. (96). [5] A. S. Boldarev V. A. Gaslov O. G. Olhovsaya On he Soluon of Hyerbol Equaons usng Unsruured Grds Mah. Mod. 8 (3) 5-78 (996). [6] V. А. Demdov V. P. Efremov М. V. Ivn А. N. Meshheryaov V. А. Perov Effe of nense energy flues on vauum-gh rubber Tehnal Physs 48 (6) (003). The resuls were resened a he 6-h Inernaonal semnar "Mahemaal models &modelng n laser-lasma roesses & advaned sene ehnologes" (5-0 June 07 Perova Monenegro). Reeved May