Charge Exchange of Proton Sodium Atom Collision

Transcription

1 Amerca Joura o Physcs ad Appcatos 7; 5(5: do: 648/jajpa7553 ISSN: (Prt; ISSN: (Oe Chare Echae o Proto Sodum Atom Coso Sabbah A Ekay, Departmet o Mathematcs, Facuty o Scece, Uversty o Dammam, Dammam, Kdom o Saud Araba Departmet o Mathematcs, Facuty o Scece, Kareshekh Uversty, Kareshekh, Eypt Ema address: sabbekay@yahoocom To cte ths artce: Sabbah A Ekay Chare Echae o Proto-Sodum Atom Coso Amerca Joura o Physcs ad Appcatos Vo 5, No 5, 7, pp do: 648/jajpa7553 Receved: Jue, 7; Accepted: Auust 7, 7; Pubshed: October, 7 Abstract: The couped statc appromato s moded or the rst tme to make t appcabe to mut-chaes probem o the coso o the proto by aka atom The possbty o produc more hydroe dur the proto-aka atom coso s vestated The ormato o hydroe H(s ad ected hydroe ( s- ad p-states o the scatter o the proto by sodum atom s treated or the rst tme to test the coverece o our method The moded method s used to cacuate the tota cross-sectos o seve parta waves ( 6, where s the tota auar mometum at cdet eeres betwee 5 ad kev Our p-na resuts ad those determed by prevous authors are reasoabe areemet Keywords: Ieastc Scatter, Proto-Aka, Proto-Sodum, Hydroe Formato, Ected Hydroe Formato, Cross-Sectos Itroducto The appearace o termedate states atomc ad ucear reactos s cosdered as the most terest pheomeo sce the deveopmet o quatum theory The deveopmet o the emtosecod aser has tremedousy ehaced the epermeta detcato o these states chemca ad physca reactos Most theoretca treatmets o eectro-, postro-, ad proto-atom teractos are based o cacuatos o dereta, parta ad tota crosssectos as uctos o cdet eeres us deret appromatos Choudhury ad Sura [] have studed proto-aka atom (Na, K, Rb, Cs coso the wave ormato o mpuse appromato at the eeres ra rom 5 to 5 kev Daee et a [] have bee cacuated the tota cross-sectos or hh eery proto scatter by aka atom us ekoa-appromato Ferrate ad Fordo [3] have bee used the ekoa-appromato to vestate hh-eery proto coso wth aka atom Ferrate et a [4] have aso studed the tota hydroe ormato cross-sectos proto scattered wth aka atom us Oppehemer Brkma Kramers (OBK appromato Twar [5] have bee reported the dereta ad tota cross-sectos o hydroe ormato o the coso o protos by thum ad sodum atoms us the Couomb-projected Bor appromato The am o the preset work s to epore the possbty o produc more hydroe dur the proto-sodum atom coso For ths reaso, t s mportat to vestate the eastc scatter o proto wth sodum atom I the preset paper, the couped statc appromato used by Ekay ([6]-[9] w be moded to make t appcabe to dscuss the mut-chaes probem (=4 o the coso o proto wth sodum atom at termedate eeres o the projecte A umerca method techque w eerazed to sove the obtaed mut-couped equatos Throuhout ths paper Rydber uts have bee used ad the tota crosssectos are epressed uts o π ( = 88 cm ad eery uts o kev a Theoretca Formasm 7 The mut-chaes eastc scatter o protos by aka atoms ca be sketched by(see Fure p A Eastc chae rst chae p A = ( H( A H( ormato chaes ( --chaes where p s the proto, A s aka taret atom, H ( s

2 74 Sabbah A Ekay: Chare Echae o Proto-Sodum Atom Coso hydroe ormato o ope chaes states ad s the umber o [8] ad ( Vt ( = V ( ρ ccou (5 where V ccou ( = V ccou ( r ad V ( σ,,3,4, t =, are the teracto potetas o the (--hydroe ormato chaes, respectvey ad are ve by: V ( σ t = V r ccou V ( r V ( r =,3,4, ccou ce The tota eeres E o the -chaes are deed by: (6 Fure Courato space o p Atom scatter: ad r are the posto vectors o the projecte proto ad the vaece eectro o the taret wth respect to the ceter o mass o the taret, ρ s the posto vector o the projecte proto wth respect to the vaece eectro o the taret, σ s the posto vector o the ceter o mass o H rom the taret, M T = mass o the uceus o the taret The Hamtoa o the eastc chae s ve by: ( ( H = H = H V ( t T ( = V ( r V ( t r r c T where H T s the Hamtoa o the taret atom T s the reduced mass o the taret atom The Hamtoa o the (--rearraemet chaes are epressed by: H = H = H V σ σ t = V ( σ, t ρ ρ σ =,3,4, where H, =,3,4, are the Hamtoas o the hydroe ormato atoms, H(, respectvey, =,3,4, are the reduced masses o (-- chaes, respectvey V ( r c s a screeed poteta ad ( V ( t s the teracto poteta o the rst chae ad are ve by: ( (3 V ( r = V r V r c ccou ce (4 where V ( r ad V r are the Couomb ad ccou ce echae parts o the core poteta, respectvey(see re where E = E,,,3 k = (7 k s the ketc eery o the cdet proto reatve to the taret ad k, =,3,4,, are the ketc eery o the ceter o mass o the hydroe ormato atoms, H (, respectvey, wth respect to the uceus o the taret E s the bd eery o the taret atom, ad E, =,3,4,, reer to the bd eeres o the hydroe ormato atoms, respectvey I the mut-chaes couped-statc appromato (MCSA t s assumed that the projectos o the vector (H E Ψ oto the boud state o the -chaes are zero Thus, the oow codtos: Φ (H E Ψ =, =,,3,, (8 are satsed The tota waveucto Ψ s epressed by Ψ = Φ ψ, (9 = ψ ( ( ˆ = Y ψ = σ Y ( ˆ,3, σ = ( ( ( where ( ad ( σ, =,3, are the rada waveuctos o the eastc ad the hydroe ormato atoms, respectvey, correspod to the tota auar mometum Y ( ad Y ( ˆ σ =,3,, are the reated spherca harmocs ˆ ad ˆ σ, =,,3,, are the sod aes betwee the vectors ˆ, ˆ σ, =,3, ad the z-as, respectvey ψ, =,,3, are the

3 Amerca Joura o Physcs ad Appcatos 7; 5(5: correspod scatter waveucto o the -chaes, respectvey Φ s the waveucto or the vaece eectro o the taret atom whch s cacuated us re [] Φ, =,3,4,, are the waveuctos o the hydroe ormato atoms, H(, respectvey, whch are deed us hydroe ke waveucto The mut-chaes couped statc appromato (MCSA (eq (8 ca be soved by cosder the - terodereta equatos d ( [ k] ( ( d ( ( = U st ( Q (, α α = d ( [ k] σ dσ σ = Ust σ ( σ 'Q ( σ, α α = =, 3,,, where the prme o the sum s meas that α, ad ( (3 Q ( = (, α d, α K σ σ σ α α α α (4 α =, 3,, Q (σ = (, ( ( d, K σ (5 =, 3,, Q ( σ = (, ( α K d, α σ σ σ σ α α α α (6, α =,3,,, α the Keres K α, =,, 3,,, α are epaded by: K α (, σ α = (8 σα Φ ( r Φα ρα ( α [ ( k V ] o( ˆ o( ˆ ˆ ˆ dd, σ α t Y Y σα σ α α α (7 α =, 3,, K ( σ, = (8 σ Φ ( ρ ( r Φ ( [ ( k ] o( ˆ o( ˆ ˆ ˆ d d, V t Y σ Y σ (8 =,3,, K α ( σ, σ α = (8 σαα Φ ( ρ Φα ρα ( α [ ( V ] o( ˆ o( ˆ d ˆd ˆ, σ kα t Y σy σα σ σ α α α, α =,3,, α (9 The umber eht appear the preced equatos reers to the Jacoba o the trasormatos dr 8 d ad dρ 8 dσ The statc potetas ( U st ( ad Ust ( σ =, 3,, are deed by ( ( U st =< Φ r V Φ ( r > t V Ust σ ρ t ρ =< Φ Φ > ( The mut-chaes couped tero-dereta equatos (, 3 are homoeeous equatos, ad σ, =,, 3,, ad are possess the eera orm where ε s k ( =,,, H s ( ε H χ > = η ( d ( d ( d or, =,3,, dσ σ χ s ( ( or ( σ, =,3,, η s the rht-had sde o the couped terodereta equatos, respectvey The souto o the mut-chae couped terodereta eqs (, 3 are ve (ormay by Lppma- Schwer equato the orm χ χ η > = G > ( where G s Gree operator ( ε H ad χ s the souto o the homoeeous equato (ε H χ =, (3 Us the parta wave epasos o the Gree operator G, the soutos o (, 3 are ve ormay by (, j ( ( (, = { δ [ j j ɶ k ( U k st ( j Q ( ] d } ɶ ( k { ɶ ( k α k α = [ ( (, j ( j U st Q ( ] d } ( k ɶ α α = j =,, 3,, (4 (,j ( σ = {δ [ ( (,j j ɶ (kσ σ ( σ U k st ' Q ( σ ] dσ } ɶ (kσ { ɶ (kσ α k α = (5 ( (,j [ (σ (σ 'Q (j U st (σ ]dσ } (k σ ɶ α α= =,3,, j =,, 3,,

4 76 Sabbah A Ekay: Chare Echae o Proto-Sodum Atom Coso where the deta uctos δ j,,j =,, 3,,, specy two (, j depedet orms o soutos or each o ad (, j ( σ, =, 3,, accord to the chae cosdered The uctos ɶ ( η ad ɶ ( η, η = k, or η = k,, 3,, σ = are reated to the Besse uctos o the rst ad secod kds, e j( η ad y ( η, respectvey, by the reatos ɶ ( η = ηj( η ad ɶ ( η = ηy( η It s obvous rom Eqs (4, 5 that the soutos ca be oy oud teratvey ad the th terato are cacuated by X (, j, ( ( (,, = { δ [ j j k ( ɶ U st ( k X (, Q j ] ( d } ɶ ( k { ɶ ( k [ ( α U st k α = (, j, ( Q ( j, ( ] d } ( k ɶ α α = j =,, 3,,; (6 (,, j (,, ( σ = { δ [ j j k ɶ σ Ust σ ( σ k (, 'Q j ] } { [ σ d σ ɶ k σ ɶ k α U st ( k σ σ (7 α = (, j, ( σ 'Q (j, ( σ ] dσ }, α k ɶ σ α = =, 3,,, j =,, 3,, ; where X,, =, specy the terato rae away rom the uceus over whch the teras at equatos (6, 7 are cacuated us Smpso s epasos (, j, To d the start vaue o (, we cosder the ( Tayor epaso o U ( ɶ ( k ad ɶ st ( k aroud the or (see re [8] Equatos (6, 7 ca be abbrevated to (, j, (, (, = a j ɶ k b j ɶ ( k (8 j =,, 3,; > (, j, σ ( j, ( j, = a ɶ k σ b ɶ ( k σ (9 =,,,j =,, 3,; > The preced coecets o eqs (8, 9 are eemets o the matrces a υ ad bυ whch are ve by: ( aυ = m /k j a ( jυ, ( jυ, ( bυ = /k b, j m, j =,,,, > (3 whch are coected wth the reactace matr, υ R, throuh the reato: { R υ } υ( υ = b a, βγ β,γ =,, 3,, > βγ (3 The parta eastc ad hydroe ormato, H(, cross sectos are determed ( π a : (, 4( T βγ = βγ, k β, γ =,,3,, > σ (3 where k s the mometum o the cdet protos, s the umber o teratos ad T βγ s the eemets o the trasto matr T whch s ve by: ( I R, T = R ɶ >, (33 where R s the reactace matr ad I s a ut matr ad ɶ = Fay, the tota cross sectos ( πa uts are th epressed ( terato by: (, σj = σ, = j,j =,, 3,, > 3 Proto-Sodum Scatter (34 We are o to appy our mut-chaes couped statc appromato (MCSA the case o =4 (our chaes couped statc appromato to the eastc scatter o protos by sodum atoms Our probem ca be sketched by p Na(3 s Eastc chae ( rst chae H(s Na H (s ormato chae ( secod chae p Na(3 s = H(s Na H (s ormato chae ( thrd chae H(p Na H(p ormato chae ( ourth chae (35 ad Φ ( r s the vaece eectro waveucto o the taret (sodum atom whch s cacuated us (Cemet s tabes [] ad Φ ( ρ =,3,4 are the waveuctos o the hydroe ormato whch are ve by: ep, Φ = ρ π ( ρ ep( ρ / Φ 3 = 3 3 3π ρ θ ρ Φ 4 = 4 cos ρ, ep( 4 / 3 4 σ π 4 (36

5 Amerca Joura o Physcs ad Appcatos 7; 5(5: Resuts ad Dscusso We start our cacuatos o p-na scatter by test the varato o the statc potetas U ( ( ad st U ( σ, =,3, 4 o the cosdered chaes wth the crease st o, (,3, 4 σ = Vaues o, (,3,4 σ = have bee 3 5 chose such that = σ =,,,, where h = s the mesh sze (or Smpso's step empoyed or cacuat teras appear tera equatos us Smpso's rue Cacuato o cross sectos o proto-sodum ( p Na scatter has bee proceeded by vestat varato o eemets o R wth crease o terato rae (IR ad the umber o teratos We h at ad have obtaed 6 a the resuts preseted beow wth 5 mesh pots (e IR = 3a It s oud that eceet coverece ca be obtaed wth υ = 5, ths demostrates stabty o our teratve method Fa cacuatos have bee carred out or seve parta waves correspod to 6 at vaues o k represet the ketc eery reo (5 k kev k kev Tabe Preset tota cross sectos ( π o p-na scatter wth the compared resuts ([],[],[4] ad [5] a Preset Choudhury Daee Ferrate Preset Choudhury Twar Preset ChoudhuryH H(s H(s [] H(s [] H(s [4] H(s H(s [] H(s [5] H(p (p[] (MCSA (IP (Ekoa (OBK (MCSA (IP (CPB (MCSA (IP E-3 58E E-3 6E-3 569E- 7549E-4 77E-4 946E-4 E E-3 749E- 7675E-5 87E-5 49E- 6974E-5 683E E-4 335E-4 5E-5 4E-5 87E E-5 7E-5 688E-4 34E-4 47E-4 683E E-6 3E-5 56E E-6 55E E E E E E E E-5 44E E E-5 867E E E E E E-6 356E E-7 9E-7 6E E-8 84E E E E E E E E E E E E-8 58E E E E E E-8 9E E E E-9 575E E E E E E-9 953E E E-9 5E E- Fure Preset tota H ( s ormato cross sectos ( σ π a o p Na scatter wth those o Choudhury ad Sura [] Fure 3 Preset tota H ( s ormato cross sectos ( σ π a 3 o p Na scatter wth compared theoretca Choudhury ad Sura []

6 78 Sabbah A Ekay: Chare Echae o Proto-Sodum Atom Coso Choudhury ad Sura [] We aso otced that the avaabe compared resuts o Daee et a [], Ferrate et a [4], ad Twar [5] are about (45%-999% arer tha our vaues The preset cacuated tota cross sectos have the same tred o the compared resuts ad ve ood areemet wth the avaabe prevous resuts o Choudhury ad Sura [] Our cacuatos aso show that we have more hydroe ormato we ope more ected chaes o hydroe ormato the coso o protos wth sodum atoms 5 Cocusos Fure 4 Preset tota H ( p ormato cross sectos ( σ π a 4 o p Na scatter wth those o Choudhury ad Sura [] Proto-sodum eastc scatter s studed or the rst tme us the mut-chaes couped statc appromato, (MCSA as a our chaes probem (eastc, H(s H(s ad H(p Our terest s ocused o the ormato o roud, H (s, ad ected hydroe, H (s, ad H(p p Na eastc scatter The derece betwee the our chaes probem ad the three or two chaes probems (whch are used by Ekay ([6]-[9] s mprov the tota cross sectos o the cosdered chae by add the eect o more keres o the other three chaes ( two chaes probem, we have oy oe kere ad three chaes, we have two keres Reereces [] K B Choudhury ad D P Sura, J Phys B5, 853( %45%44%43pd Fure 5 Preset tota eastc ad hydroe ormato ( H ( s H ( s ad H ( p cross sectos ( π a o p Na scatter Tabe shows the preset tota cross-sectos o p-na scatter wth those o Choudhury ad Sura o the wave ormato o mpuse appromato [], Daee et a that obtaed us ekoa-appromato [], Ferrate et a us Oppehemer Brkma Kramers (OBK appromato [4], ad Twar us the Couomb-projected Bor appromato [5] the eery rae (5- kev Our resuts ad those o compar resuts at the rae o eery (5- kev are aso dspayed Fures (-4 I Fure 5, we aso show the preset resuts o the tota cross-sectos o the our chaes (eastc ad the hydroe ormato (H(s H(s H(p the rae o eery (5- kev It s see that the preset vaues o the tota cross-sectos o the our chaes have smar treds o the comparso resuts Our vaues o the tota cross-sectos o the our chaes decrease wth the crease o the cdet eeres The cacuated tota cross-sectos σ o H(s are about (54%-79% ower tha the Choudhury ad Sura resuts [] The tota cross-sectos σ 3 o H(s are shty about (67%-88% ower tha those o Choudhury ad Sura [] Our resuts o the tota cross-sectos σ 4 o H(p are about (79%-97% ower tha the avaabe vaues o [] R Daee, G Ferrate ad E Fordo, IL Nuovo Cmeto B54 85( pd [3] G Ferrate ad E Fordo, IL Nuovo Cmeto B57, (98 [4] G Ferrate, E Fordo ad M Zarcoe, IL Nuovo Cmeto B5, 5( pd [5] Y N Twar, Pramaa-J Phys 7, 753( pd [6] S A Ekay, Brz J Phys, 44(6 69( pd [7] S A Ekay, Ch Phys B4(3 334 (5 [8] S A Ekay, Ca J Phys, 93, 83 (5 hubbz/d749e635e4c4de989d6d5a9a9e7/39%4cjp- 5-6pd

7 Amerca Joura o Physcs ad Appcatos 7; 5(5: [9] S A Ekay, Pramaa-J Phys 5, 78 (6 /7%4s ypd [] E Cemet, C Roett, Atomc Data Nucear Data Tabes, 4, 77 ( %874%986-pd