The quantum thermodynamic functions of plasma in terms of the Green s function

Size: px

Start display at page:

Download "The quantum thermodynamic functions of plasma in terms of the Green s function"

Winifred Walton
5 years ago
Views:

1 Vol. No (04) Nturl cinc Th quntum thrmodynmic functions of plsm in trms of th Grn s function Ngt A. Hussin Abdl Nssr A. Osmn Dli A. Eis * Rg A. Abbs Mthmtics Dprtmnt Assuit Univrsity Assuit Egypt Mthmtics Dprtmnt outh Vlly Univrsity Qn Egypt Mthmtics Dprtmnt Assiut Univrsity Nw Vlly Egypt; * Corrsponding Author: dli_h@yhoo.com Rcivd 0 Novmbr 0; rvisd 0 Dcmbr 0; ccptd 7 Jnury 04 Copyright 04 Ngt A. Hussin t l. This is n opn ccss rticl distributd undr th Crtiv Commons Attribution Licns which prmits unrstrictd us distribution nd rproduction in ny mdium providd th originl work is proprly citd. In ccordnc of th Crtiv Commons Attribution Licns ll Copyrights 04 r rsrvd for CIRP nd th ownr of th intllctul proprty Ngt A. Hussin t l. All Copyright 04 r gurdd by lw nd by CIRP s gurdin. ABTRACT Th objctiv of this ppr is to clcult th third viril cofficint in Hrtr pproximtion Hrtr-Fock pproximtion nd th Montroll- Wrd contribution of plsm by using th Grn s function tchniqu in trms of th intrction prmtr ξ nd usd th rsult to clcult th quntum thrmodynmic functions for on nd two componnt plsm in th cs h of nλ b whr λ b = is th thrml mbkt D Brogli wv-lngth. W comprd our rsults with othrs. KEYWORD Th Excss Fr Enrgy; Th Two Componnt Plsm; Th Third Viril Cofficint; Th Hrtr Trm; Th Hrtr-Fock Approximtion. INTRODUCTION Th thrmodynmic functions r of grt intrst for undrstnding th proprtis of plsm such s th css fr nrgy nd th prssur. An ccurt dscription of th prssur volum nd tmprtur (P-V-T) bhvior rprsnts on of th most importnt gols of sttisticl thrmodynmics. Bsids simultions th quilibrium bhvior cn b dtrmind ithr from th qution of stt (EO) or from viril pnsion. Th proprtis nd th bhvior of mny prticl systms with Coulomb intrctions r ssntilly dtrmind by th long rng chrctr of th forcs btwn th chrgd prticls. Thrfor systms with Coulomb intrctions r of spcil intrst nd importnc in sttisticl physics []. Mny uthors hv clcultd th css fr nrgy Ebling t l. [] hv clcultd th viril pnsion of th css fr nrgy until th scond viril cofficint from th pir distribution function. Hussin nd Eis [] clcultd th quntum css fr nrgy for two componnt plsm. All ths rsrchs r usd th mthod of ltr um in quntum sttisticl mchnics. Also th thrmodynmic functions which r clcultd from ths rsrchs nr to th clssicl limit. In this ppr w us th Grn s function tchniqu which mns tht w hv to strt from th grnd cnonicl nsmbl. To our knowldg no ttmpt to find th thrmodynmic function until third viril cofficint by using Grn s function tchniqu for on nd two componnt plsm. Th mthod of Grn s functions is on of th most powrful tchniqus in quntum sttisticl physics nd quntum chmistry. Th min dvntg of Grn s functions is tht on cn formult ct qutions of pproximtions nd st up consqunt schms to gin ny highr pproximtion []. Anothr dvntg of such Grn s functions is th istnc of highly ffctiv mthods for thir dtrmintion such s Fynmn digrm tchniqus functionl tchniqus nd th formultion of qutions of motion [4]. Th singl prticl Grn s function in fct contins mor dtild informtion thn th totl nrgy lon to th tnt tht th locl ltr corrltion potntil cn b obtind from it [5]. Mny uthors usd th mthod of Grn s functions such s Dwitt t l. [] nd hv clcultd th low dnsity pnsion of th qution of stt for quntum lctron gs. Rimnn t l. [7] hv clcultd th qution of stt of th wkly dgnrt on-componnt plsm (OCP). Th modl undr considrtion is th two-componnt plsm (TCP); i.. nutrl systm of point lik prticls Copyright 04 cirs.

2 7 N. A. Hussin t l. / Nturl cinc (04) 7-80 of positiv nd ngtiv chrgs which ntisymmtric with rspct to th chrgs = i = nd thrfor symmtricl with rspct to th dnsitis n = n i = n. Also w usd th on componnt plsm modl (th modl of idnticl point chrgs immrsd in uniform bckground whil th continuous chrg dnsity of th bckground is chosn to b qul nd opposit to th vrg chrg dnsity of th point chrgs so tht th systm s whol is lctriclly nutrl) for mpl th lctron gs i = = nd thrfor symmtricl with rspct to th mss m = mi = m. This ppr is orgnizd s follows: In ction w prsnt th Grn s function. In ction w clcult th css fr nrgy until th third viril cofficint for on nd two componnt plsm in quntum form. Also w clult th gnrl formul of th third viril cofficint in Hrtr-Fock pproximtion. Finlly in ction 4 w clcult th prssur for on nd two componnt plsm until th third viril cofficint.. THE GREEN FUNCTION Th n-prticl Grn s function is dfind by [8] in th following form Gn ( n) = n () i Tˆ ψ ψ n ψ n ψ n whr ψ t ψ ψ 0 ˆ > T ψ ψ = ± ψ t < 0 t = t t = t. () T ˆ is th tim-ordring oprtor nd = { r t σ} with r is th vctor of loction t is tim σ is z -projction of spin. o w cn dfind th grtr nd lssr Grn s function by > G ( x x t ) = ψ ψ ( ) (4) < G x x t = ψ ψ. () In th Hrtr nd th Hrtr-Fock pproximtion th two prticl Grn s function is givn by H ( 4) = ( 4) ( ) = ± G G G G 4 G4 G G G 4. (5) Or th vritionl rprsnttion [8] δg ( 4) G ( 4) = G( 4) G( ) ff δv () whr th polriztion function δg π b = ± i ff δv ( 4) V is th ffctiv potntil. Also th thr prticl Grn s function is dfind by [8] ff nd ( 45) = ( 45) ± G G G G G G G (7) THE EXCE FREE ENERGY Th css fr nrgy corrsponds to th prt of th fr nrgy chng in th rl systm tht riss from intrctions mong ions [9]. Th css fr nrgy F cn b clcultd from th mn vlu of th potntil nrgy ccording to gnrl quntum sttisticl formul with th coupling prmtr λ in th following form [0] dλ F = λ v( t). (8) 0 λ N whr. v t = v t v t (9) N b bc v ( t ) = limd d ( 4 ). b b r rv G (0) c( 4) b i v( t) = lim d d d bc ( 45 ). bc r r rv G bc ( 54) () xn i xi 0 i n lim = (.) ( n ) tj tt j > tj j v( t ) b nd v( t ) bc r th mn intrction potntil for two nd thr prticl rspctivly. Aftr prforming th intgrtion with rspct to λ in Eqution (8) w cn gt κ F = VkT Bb Bbc () π whr th scond viril cofficint is givn by [4] Bb = πnn bλb K ( ξb ) lnκλb ξb b κ 9 nλ 9 κλb κλb π 8 s 4 s () nd i Bbc = lim d d d bc ( ) ( 45) r r rv rrr G (4) ( 54) bc Now w will clcult th third viril cofficint; for Coulomb systms it is usful to pply instd of v th s scrnd potntil v in Eq.4 for thr prticl which Copyright 04 cirs.

3 N. A. Hussin t l. / Nturl cinc (04) is givn by th following form v s bc = v 4 5 bc drdrdrv 4 π v 5 df bd df fc Thn w cn gt th Eq.4 in th form whr (5) H B = B B B ; () bc bc bc bc H i H Bbc = lim d d d bc ( 4 5) r r rv G (7) ( 54) bc i B = lim drdr dr v ig π nd { bc bc b bc ( 54) bc G c } ig 5 π 4 π B = drdr drdr drdrv v 4 bc 4 5 bc bd bc vbc G π df (8) (9) vbc = vb vbc vc (0) H H H whr vb vbc vbc π df G( ) G G Bbc Bbc B bc r th binry potntil th triplt potntil th triplt scrnd potntil th triplt polriztion function on prticl Grn s function th two prticl Grn s function in Hrtr pproximtion th thr prticl Grn s function th triplt Hrtr trm th triplt Hrtr-Fock th triplt scrnd rspctivly... Th Hrtr Trm of B bc By substituting from Eq.5 nd 7 into Eq.7 thn th H Hrtr trm of th quntum third viril cofficint B bc bcoms H i Bbc = d d d r r rvbc bc G G 5 G 4 ± G 5 G G 4 () { ( 4) ( 5)} G G G By tking th invrs Fourir trnsformtion of th bov qution nd mking us of th Wignr distribution function [4]; dω G( pω ) = ± if ( p) π () dω G( pω) = ± ifb ( p). π nd using Eqution (0) w obtin H i Bbc = nnn b c ( vb ( 0) vbc ( 0) vc ( 0 )). bc ( π) () whr th numbr dnsity dp n R t = f p Rt. (4) Eq. is vnishs for mpl for on componnt plsm such s th lctron gs... Th Hrtr-Fock Trm of B bc Th polriztion function is dfind rndom phs pproximtion by < < π = ± iδ G G (5) b b b By substituting from Eq.5 into Eq.8 w cn gt th Hrtr-Fock trm of th quntum third viril B bc s follow Bbc = ( s ) dd bc r tv () < < < G G G Th invrs Fourir trnsform of this qution with th hlp of th Eq. givs dpdpdp Bbc = ( s ) 9 ( π) v p p f p f p f p (7) whr f nd v r Frmi functions nd th Fourir trnsform of Coulomb potntil which is dfind by v = D p p KT ( ) (8) with D = εε r is th dilctric constnt whr ε ε r r th vcuum nd th rltiv dilctric constnt. W ssum tht dpdpdp I ( ) = v 9 ( p p) f ( p) f ( p) f ( p) π (9) By pnding f( p) f( p ) nd f ( p ) in powrs of nd using th sphricl polr coordints (s Appdnix A) thn B bc ( )( )( ) 4 bc D KT ( ) ` ( ) ( ) s s s = 4 π π b c b c r r r r r r r = r = rr r = r d ` (0) Copyright 04 cirs.

4 74 N. A. Hussin t l. / Nturl cinc (04) 7-80 Thn lt Th clcultion of K K ( ) ( s )( sb )( sc ) b c = 4 bc 4( π) πd KT ( ` ) B I I d ` bc nlyticlly givs ( ` ) () K = I I d `. () k k k k ( ) ( ) ( ) ( ) = k k k k = k= k k k ( ) k ( [ ] ) ( k k k ) k k k k k = = = k ( k k) k k By substituting from Eq. into Eq. thn w gt () B k k k k ( )( )( ) ( ) s sb sc b c bc = bc 4 k = k = 4( π) πd KT k k k k k k ( ) k ( [ ] ) ( k k k ) k k k k k = = = k ( k k) k k (4) whr I ( ` ) V s µ K nd T r th Frmi intgrl th volum th spin projction th dgnrcy prmtr th chmicl potntil th Boltzmnn s constnt nd th bsolut tmprtur rspctivly. Thn w cn writtn th third viril cofficint in th following form B 7Λ 84π π π bc = κ KT 4 nλ nn bλ s s sb nnn b cλ 9 ( s )( sb )( sc ).. Th crnd Contribution of B bc (5) By substituting from Eqs. nd 7 into Eqs.9 w cn rwrit th scrnd quntum third viril B bc until first thr trms in th following form Bbc = d d d d 4 d 5 d ( ) ( 4 ) r r r r r rvbc vbd π df vbc G G G () by using th pir potntil nd pir polriztion function thn 4 5 ( 5) ( 5) ( 5) ( 4) Bbc = drdrdrdrdrdrvb vbd 4 (7) v G G G G G bc Tking th invrs Fourir trnsformtion of Eq.7 nd using Eq.0 whr v b 4π p ( p) = is th Fourir trnsform of binry potntil thn w gt in th wkly dgnrt cs or low dgnrcy limit ( ) nd th cs of high tmprtur limit or low nλ in th following form dnsity ( β ) B = 9π nnn bc b c b c bc F( ; ; λbl 4) dl ( κ ( ; ; λ 8 )) 0 L L F L b (8) Copyright 04 cirs.

5 N. A. Hussin t l. / Nturl cinc (04) κ = = 4πβn is th invrs Dby lngth nd F dnots th conflunt hyprgomtric function. r whr d Th nlyticl clcultion of th intgrl from Eq.8 is vlutd by solving this intgrl by prts nd using Gmm functions in th rgions of smll κλ (low dnsitis high tmprturs) thn w hv B ( ) 84π π 97π = κ KT κλ κλ κλ ( 4π) bc b bc c By substituting from Eqs. 5 nd 9 into Eq. w gt th css fr nrgy until th third viril cofficint for on componnt plsm s follow. (9) κ κ 9 nλ 9 F = VKT πnλ K ξ lnκλ ξ κλ κλ π π 8 s 4 s 7Λ πn λ ( ) ln K ξ κλ ξ κ KT 84π π π 4 nλ nn Λ nnn Λ s ( s )( s ) 9 ( s )( s )( s ) 84π π 97π ( κλ ) κ KT ( 4π) (40) h whr K is th quntum viril function which givn by [4] nd Λ= is th normlizd thrml wv πmkt lngth. imilrly w cn writ th css fr nrgy until th third viril cofficint for two componnt plsm s follow κ ξ π λ ( ξ ) lnκλ F = VKT n n K π ξii ξi nn i iλii K ( ξii ) lnκλii nn iλi K ( ξi ) lnκλi κ 9 nλ 9 niλi ( κλ ) ( κλii ) ( κλi ) π s si κλ κλi κλii 4 s s 4 s si 4 si si ξi ξi π nn i λλii K ( ξi ) lnκλi π nn i λii λ K ( ξi ) lnκλi ξ ξii λ K ( ξ ) lnκλ πni λii K ( ξii ) lnκλii πn 4 7Λ nλ nn iλ nnn Λ κ β 89 π π s s si 9 s s s nnn i i iλ nniλ ( s )( s )( s ) ( s )( s )( s ) i i i i 84π π 97π 97π 97π κ KT ( κλ ) ( κλi ) ( κλii ). ( 4π) nn i Λ ( s )( s )( s ) i i (4) Copyright 04 cirs.

6 7 N. A. Hussin t l. / Nturl cinc (04) THE EQUATION OF TATE Following th mthod of ffctiv potntils dvlopd by [9] w gt th prssur from th css fr nrgy s follow p F id = p V T Nk. (4) whr id p is th idl prssur nd F F κ =. V κ V By substituting from Eq.40 into Eq.4 w cn gt th qution of stt until third viril cofficint for on componnt plsm; κ πλ ( ξ ) ( lnκλ ) ξ 4π p = nkt KT n K κ 9 nλ 7 κλ κλ 4π s 4 s (4) 4 7Λ nλ nn Λ nnn Λ κ β 409π π s ( s )( s ) 9 ( s )( s )( s ) 84 π π 97 π κ KT ( κλ ) πn λ K ( ξ ) ( ln κλ ) ξ. ( 4π) Also by substituting from Eq.4 into Eq.4 w gt th qution of stt until third viril cofficint for two componnt plsm κ ξ i π λ ( ξ ) ( lnκλ ) p = n KT n KT KT n n K 4π ξ ii ξ i nn i iλii K ( ξii ) ( lnκλii ) nn iλi K ( ξi ) ( lnκλi ) κ 9 nλ 9 niλi ( κλ ) ( κλii ) ( κλi ) 4π s si 7 4( s ) ( s ) 54 7 κλ κλi κλii 4 s si 4 si si ξ i ξ i π nn i λλii K ( ξi ) ( lnκλi ) π nn i λii λ K ( ξi ) ( lnκλi ) ξ ξ ii πn λ K ( ξ ) ( lnκλ ) πni λii K ( ξii ) ( lnκλii ) 7Λ 4 nλ nn iλ nnn Λ κ β 409 π π s ( s )( si ) 9 ( s )( s )( s ) nnn i i iλ nniλ ni nλ 9 ( si )( si )( si ) 9 ( s )( s )( si ) 9 ( si )( si )( s ) 78 π π 97π κ KT 97π 97π κλ κλi κλii. ( 4π) In th numricl clcultion w lt 8 K =.8 0 rg/dg m= m = gm = su n = 0 cm h =.4 0 = 0 7 (44) Copyright 04 cirs.

7 N. A. Hussin t l. / Nturl cinc (04) DICUION To our knowldg thr is no ppr to clcult th third viril cofficint by using Grn s function tchniqu until now; this ppr is th first ppr to clcult th third viril cofficint in Hrtr Hrtr-Fock pproximtion nd th Montroll-Wrd contribution by using th Grn s function tchniqu nd usd it to clcult th quntum thrmodynmic functions. In pst th potntil ws usd s th mn potntil for two prticls only so thir rsults wr until th scond viril cofficint but in this ppr w usd th potntil s th sum of th mn potntil of two nd thr prticls so our rsults wr vlutd until th third viril cofficint. Also th quntum thrmodynmic functions until th third viril cofficint which r clcultd by using th binry ltr sum r nr t th clssicl limit only; thy usd th potntil s th pir potntil only nd nglctd th triplt potntil so thr rsults Is not ctly corrct rsults. W considrd only th thrml quilibrium plsm in th cs of on nd two componnt plsm by using Grn s function mthod. W obtind th gnrl formul of th third viril cofficint in Hrtr-Fock pproximtion nlyticlly (Eq.4). As shown in Figurs - w plottd th comprison btwn th css fr nrgy until th scond viril cofficint for on nd two componnt plsm of Ebling t l. [] Hussin t l. [] until th third viril cofficint nd our rsults until th third viril cofficint t 4 0 smll nrly for smll vlus of ξ this is du to th diffrnc btwn Grn s function tchniqu which ws usd hr nd ltr um tchniqu in Hussin t l. [] nd Ebling t l. [] which r givn in Figurs -. In ths figurs w obsrv tht thr ist diffrnc for smll vlus of ξ for two comopnnt T = 0 nd 5. W noticd tht th curvs r plsm spcilly ordr rison btwn th prssur until th scond viril cofficint for on nd two componnt plsm of Krmp t l. [4] Ebling t l. [] Hussin t l. [] until th third viril cofficint nd our rsults until th third viril 4. Also w plottd th comp- cofficint up to ordr 4 nd s shown in Figurs 4-. In Figurs 5 nd th curvs r fr from ch othr for on nd two componnt plsm. But in Figur 4 w obsrvd tht th rd curv for Ebling t () on componnt plsm (b) two componnt plsm Figur. Th comprison btwn th quntum css fr nrgy until th scond viril cofficint [] (blck solid lin) [] (blu short dshd dottd lin) until th third viril cofficint nd our rsults (grn long dshd lin) up to ordr. () on componnt plsm (b) two componnt plsm Figur. Th comprison btwn th quntum css fr nrgy until th scond viril cofficint [] (blck solid lin) [] (blu short dshd dottd lin) until th third viril cofficint nd our rsults (grn long dshd lin) up to ordr 4. Copyright 04 cirs.

8 78 N. A. Hussin t l. / Nturl cinc (04) 7-80 () on componnt plsm (b) two componnt plsm Figur. Th comprison btwn th quntum css fr nrgy until th scond viril cofficint [] (blck solid lin) [] (blu short dshd dottd lin) until th third viril cofficint nd our rsults (grn long dshd lin) up to ordr. () on componnt plsm (b) two componnt plsm Figur 4. Th comprison btwn th quntum prssur until th scond viril cofficint [] (blck solid lin) [4] (rd long dshd dottd lin) nd [] (blu short dshd dottd lin) until th third viril cofficint nd our rsults (grn long dshd lin) up to ordr. () on componnt plsm (b) two componnt plsm Figur 5. Th comprison btwn th quntum prssur until th scond viril cofficint [0] (rd long dshd dottd lin) [] (blu short dshd dottd lin) until th third viril cofficint nd our rsults (grn long dshd lin) up of ordr 4. Figur. Th comprison btwn th quntum prssur until th scond viril cofficint (rd dottd lin) [] (blu short dshd dottd lin) until th third viril nd our rsult (grn long dshd lin) up to ordr. Copyright 04 cirs.

9 N. A. Hussin t l. / Nturl cinc (04) l. [] nd grn curv for our rsult r nrly for som vlus of ξ for two componnt plsm. REFERENCE [] Hussin N.A. nd Eis D.A. (0) Th quntum qution of stt of fully ionizd plsms. Contributions to Plsm Physics [] Ebling W. (9) Zur frin nrgi von systmn gldnr tilchn. Annln dr Physik [] Krft W.D. nd Fhr R. (997) lf nrgy nd two prticl stts in dns plsms. Contributions to Plsm Physics [4] Krmp D. chlngs M. nd Krft W.D. (005) Quntum sttistics of non idl plsm. pringr Nw York. [5] Hollboom L.J. nijdrs J.G. Brnds E.J. nd Buijs M.A. (988) A corrltion potntil for molculr systms from th singl prticl Grn s function. Th Journl of Chmicl Physics [] Dwitt H.E. chlngs M. kkur A.Y. nd Krft W.D. (995) Low dnsity pnsion of th qution of stt for quntum lctron gs. Physics Lttrs A [7] Rimnn J. chlngs M. Dwitt H.E. nd Krft W.D. (995) Eqution of stt of th wkly dgnrt on-componnt plsm. Physic A :i::p: [8] Yuklov V.I. (998) ttisticl Grn s functions. Kingston London. [9] Flknhgn H. nd Ebling W. (97) Equilibrium proprtis of ionizd dilut lctrolyts. In: Flkn-Hgn H. nd Ebling W. Eds. Ionic Intrctions from Dilut olution to Fusd lts Acdmic Prss Nw York nd London [0] Krft W.D. Krmp D. nd Ebling W. (98) Quntum sttistics of chrgd prticl systms. Akdmi Vrlg Brlin. [] Hoffmnn H.J. nd Ebling W. (98) Quntnsttistik ds Hochtmprtur-Plsms im thrmodynmischn Glichgwicht. II. Di fri Enrgi im Tmprturbrich 0 bis 0 8 K. Bitrg us dr Plsmphysik Copyright 04 cirs.

10 80 N. A. Hussin t l. / Nturl cinc (04) 7-80 APPENDIX A By pnding f ( p ) f ( p ) nd whr by substituting by rr r f p in powrs of thn r r r ( r r r) rr r (A.) r = r = r = ( ) = ( ) I I ( π) p p I = pdp pd p ln p d p ν = rp w gt rr r ( π) rp rp rp π D KT p p ν rp r rp π D KT r ν p r I = pdp ln d( ) p d p by using th intgrtion by prts w gt without Dirichlt formul π I = du pdp p d p ru r r p rp rr r 4 0 p 0 (π) π D r KT p ν (A.) (A.) (A.4) w hv th intgrl rgion dd up for th innr bounds if w look t th curvs u = p u = which w wnt to solv in trms of u w find tht p u nd finlly w hv π I du pdp p d p rr r ( r r) p rp ru = u ( π) π D r KT ru rp 4 4 ( ) ( ) 0 0 which cn b writtn s Irr d d. r = u p p = π π D r r r KT 4 π π D r r r r r r KT (A.5) (A.) Copyright 04 cirs.

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9 Lctur contnts Bloch thorm -vctor Brillouin zon Almost fr-lctron modl Bnds ffctiv mss Hols Trnsltionl symmtry: Bloch thorm On-lctron Schrödingr qution ch stt cn ccommo up to lctrons: If Vr is priodic function: