Introduction IJSER. spinless nucleus of charge Ze, where Z is the number of. magnetization densities of nucleus. [1].

Transcription

1 Ieaioal Joual o Scieiic & Egieeig Reseach, Volume 6, Issue, Decembe Ielasic logiudial eleco scaeig C om acos i Ni Fias Z. Majeed 1 ad Fadhel M. Hmood* 1 1Depame o physics, College o Sciece, Baghdad Uivesiy, Baghdad, Iaq. * Coespodig Auho: hmoodadhel@gmail.com Absac: Ielasic logiudial C om acos i Ni ucleus have bee sudied usig he uclea shell model ha caied ou i ems o coiguaio mixig wih limiig umbe o obis i he model-space (esiced model space (p3/, p1/,15/ ad icludig he eecs o he discaded space (coe obis highe obis ouside he model-space, which is called coe-polaizaio eecs, hough a micoscopic heoy as is-ode peubaio heoy ha cosides paicle-hole exciaios om he coe obis ad via he model-space obis io he highe obis wih ħ exciaios. The shell-model wave ucios obaied wih he coiguaio, wih a umbe o acive euos ouside he ie coe 56Ni. The eecive ieacio F5PVH poeial is used o geeae he model space wave ucio. The simple hamoic oscillao (HO poeial is used o geeae he sigle paicle wave ucios. Two ealisic ieacios had bee adoped as a esidual ieacios o couple he paicle-hole pai epeseed by he wo body Michiga sum o hee age Yukawa poeial (M3Y-P ad Gogy. The obaied heoeical esuls had bee compaed wih available expeimeal daa. Keywod: coe polaizaio, Ni, om aco. Ioducio Eleco scaeig is a excelle ool o sudyig he The eleco scaeig pocess ca be explaied accodig uclea sucue because o may easos. Sice he o he is Bo appoximaio as a exchage o a viual ieacio bewee he eleco ad he age ucleus is phoo cayig a momeum bewee he eleco ad elaively weak o ode α =1/137, he ie-sucue ucleus. The is Bo appoximaio is beig valid oly i cosa, ad kow whee he eleco ieacs Zα<<1, whee α is he ie sucue cosa [7]. elecomageically wih he local chage, cue ad The scaeig coss secio o elaivisic elecos om mageizaio desiies o ucleus. [1]. spiless ucleus o chage Ze, whee Z is he umbe o A peec ad good iomaio abou he uclea sucue poos i he ucleus, was is deived by Mo (199 [8]. ca be obaied om he eleco scaeig by he ucleus The ickel isoopes had bee descibed i ems o sogly a high eegy. Whe he eegy o he icide eleco is admixed spheical shell-model coiguaios o euos i he age o 100 MeV ad moe, he de Boglie occupyig he p3/, 15/, ad p1/ obis. A se o wavelegh will be i he age o he spaial exesio o eecive-ieacio maix elemes is deduced which he age ucleus, Thus wih hese eegies, he eleco accuaely epoduces he speca o he Ni isoopes om epeses a bes pobe o sudy he uclea sucue [][3]. Ni58 o Ni67. The wave ucios esulig om he The elecos scaeig om a age ucleus ca occu i calculaios o he eegy levels ae he used o calculae wo ypes: is, he ucleus is le i is goud sae ae he sigle-ucleo specoscopic acos. These ae i aily he scaeig ad he eegy o he elecos is uchaged, good ageeme wih he expeimeal specoscopic his pocesses is called Elasic Eleco Scaeig. I he acos oud i pickup ad sippig expeimes. he E secod he scaeed eleco leaves he ucleus i diee asiio pobabiliies i he eve-mass isoopes o Ni ae excied sae which has a ial eegy educed om he calculaed ad oud o be i ageeme wih expeimeal iiial jus by he amou ake up by he ucleus i is acs [9]. excied sae, his pocesses is called Ielasic Eleco A epo o expeimeal o a se o ielasic om-aco Scaeig [4][5]. measuemes o he is excied sae o 58Ni, Ni, ad I is kow ha he ielasic eleco scaeig has pove 6Ni, which, ogehe wih he accuae Coulomb-exciaio o be a good echique o sudyig he popeies o B(E measuemes by Selso ad McGowa, povide o excied saes o uclei, i paicula hei spis, paiies, he is ime a accuae expeimeal check o he ad he segh ad sucue o he asiio opeaos disoed-wave calculaio o Giy e al. The expeimeal coecig he goud ad he excied saes.[6]. measuemes wee caied ou a he Yale Liea Eleco The om aco ca be oud expeimeally as a ucio Acceleao Laboaoy usig icide eegies agig o he momeum ase (q by kowig he eegies o om 45 o 65 MeV ad scaeig agles om 70 o 130 he icide ad scaeed eleco ad he scaeig agle. [10]. Usig he expeime o sudy he ielasic scaeig o elecos om Ni wih a ove-all eegy esoluio o % by he use o 183 MeV ad 50 MeV eleco beams om he Tohoku 300 MeV liea acceleao [11]. 015 Ielasic eleco scaeig coss secios have bee measued up o a momeum ase q=3.9 m-1, deemiig vey pecisely he asiio chage desiy o he is excied ( 1 sae o 58Ni. The esuls have bee

2 Ieaioal Joual o Scieiic & Egieeig Reseach, Volume 6, Issue, Decembe iepeed i a ully sel-cosise heoeical eame o boh he goud sae ad he ( 1 asiio chage desiy o 58Ni []. Elasic ad ielasic eleco-scaeig om acos o mulipolaiies up o = 7 ad some asiio-segh disibuios ae calculaed wih shell-model wave ucios o abou e age uclei i he mass age A = 5-6 icludig 5,53C, 54,56Fe, 53C, 55M, 59Co ad 58,,6Ni. I is oud ha he seghs o he calculaed mageic asiios ae always less ha abou 50% o he pue sigle-paicle values [13]. A micoscopic descipio o daa o he ielasic scaeig o acos o he 0 as well as 0 4 asiios i some doubly eve Ti, C, Fe, Z, ad Ni isoopes icludig 58,,6Ni is aemped i ems o he pojeced Haee-Fock-Bogoliubov wave ucios esulig om ealisic eecive ieacios opeaig i he p-1 shell [14]. The eecive ieacio GXPF1 o shell-model calculaios i he ull p shell had bee esed i deail om vaious viewpois such as bidig eegies, elecomageic momes ad asiios, ad exciaio speca. The semi magic sucue is successully descibed o N o Z=8 uclei, 53M, 54Fe, 55Co, ad 56,57,58,59Ni, suggesig he exisece o sigiica coe exciaios i low-lyig model ove coveioal calculaios i cases whee ull-space calculaios sill emai oo lage o be pacical [15]. The expeimeal sigle-paicle eegies ad occupaio pobabiliies o euo saes ea he Femi eegy i 58,,6,64 Ni uclei had sudied ad obaied om joi evaluaio o he daa o ucleo sippig ad pickup eacios o he same ucleus. The esulig daa had bee aalyzed wihi a mea_ield model wih dispesive opical_model poeial. Good ageeme had bee obaied bewee he calculaed ad expeimeal [16]. High-pecisio educed elecic-quadupole asiio pobabiliies B(E; 01 1 have bee measued om sigle-sep Coulomb exciaio o semi-magic 58,,6,64Ni (Z = 8 beams a 1.8 MeV pe ucleo o a aual cabo age. The eegy loss o he ickel beams hough he cabo age wee diecly measued wih a zeo-degee Bagg deeco ad he absolue B(E values wee omalized by Ruheod scaeig. The B(E values disagee wih ece lieime sudies ha employed he Dopple-shi aeuaio mehod [17]. Calculaed elasic ad ielasic om acos ad o he asiio om he goud sae o J1 (L = J =, 4 sae i 58 68Ni ad 4Mg, caied ou he saig poi o he mehod wee a se o Haee-Fock-Bogoliubov wave ucios geeaed wih a cosai o he axial quadupole mome ad usig a Skyme eegy desiy ucioal [18]. The Coe Polaizaio (CP eecs deivaio wih highe coiguaio i he is ode peubaio heoy ad he wo-body maix elemes o hee pas o ealisic ieacio: ceal, spi obi ad eso oce which ae belog o M3Y-P ad Gogy as a esidual ieacios i a sepaae picues will be sudied i he pese wok A compue pogam is wie i FORTRAN 90 laguage o iclude ealisic ieacio M3Y ad Gogy i he oigial code which is wie by Po. D. R.A.Radhi, which he modiied by assisace Po. D. Fias Zuhai Majeed o eceive ew ems ad iig paamees [19]. Theoy May paicle maix elemes o he eleco scaeig opeao TˆΛ ae expessed as he sum o he poduc o he oe-body desiy maix elemes (OBDM imes he sigle-paicle asiio maix elemes [0]: Γ Tˆ Λ Γi = OBDM Γ Γ α β α TΛ β i,,, ˆ α, β (1 whee Λ = JT is he muli-polaiy i spi ad isospi Γ especively, ad he saes i J it Γ i J T ad ae he iiial ad ial saes o he ucleus, while α ad β deoe he ial ad iiial sigle-paicle saes, especively (isospi is icluded. The educed maix T elemes o he eleco scaeig opeao Λ cosis o wo pas, oe is he "Model space" maix elemes ad he ohe is he "Coe-polaizaio" maix elemes [1]. Γ ˆ Γi = Γ T ˆ Λ Λ Γi Γ δt ˆ Λ Γi MS CP ( T Γ Tˆ Λ Γ i Γ T Γ i δ ˆ Λ elemes. Γi Γ ad ucios. MS CP is he model-space maix elemes. is he coe-polaizaio maix ae descibed by he model-space wave The model-space maix elemes ae expessed as he sum o he poduc o he oe-body desiy maix elemes (OBDM imes he sigle-paicle maix elemes which ae give by: ˆ Γ TΛ Γi = OBDM ( Γi, Γ, α, β α TΛ β MS α, β MS (3 The coe-polaizaio maix eleme i equaio ( ca be wie as ollows [1]: ˆ 015

3 Ieaioal Joual o Scieiic & Egieeig Reseach, Volume 6, Issue, Decembe Γ δtˆ Λ Γi = OBDM Γi Γ α β α δtˆ ( c ( SE TE ( SO ( TO (,,, Λ β P SE P TE P SO P e TO cp α, β cp (4 ( LS ( LSE ( LSO ( LS (.( P TE PTO L s 1 s Accodig o he is ode peubaio heoy, he siglepaicle maix eleme o he highe-eegy ( TN ( TNE ( TNO ( TN ( P TE PTO S coiguaios is give by []: Q Q α δtˆ β = α Tˆ V β α V Tˆ [ 1 β ( SE ] [ 1 ( TE ( DD DD. ( a DD. ( ] a ( PSE PTE δ (10 J J (0 es es (0 J E H E H (5 Ad o Gogy eq.(9 becomes: V whee es is adoped hee as a esidual ucleo-ucleo ( c ( SE TE ( SO ( TO ( ( P SE P TE P SO PTO e ieacio. ( LS ( LSE ( LSO (.( P TE P δ TO L s 1 s The sigle-paicle eegies ae calculaed accodig o []: ( TN ( TNE ( TNO ( TN 1 ( l 1 ( j= l 1 ( P TE PTO S l o 1 e = ( l ω lj 1l ( o j= l 1 [ 1 ( SE ] [ 1 ( TE ( DD DD. ( a DD. ( ] a ( PSE PTE δ l (11 (6 (SE wih: (SO (TO whee, (TE,, ae he segh paamee ( 0A / 3MeV i ceal pa o (sigle-eve, (sigle-odd, (iple-odd l (LSE (LSO ω= 45A 1/ 3 5A / 3 ad (iple-eve, ad, ae he segh (7 paamee i he spi obi pa o (sigle-eve, (sigleodd, The ealisic M3Y ad Gogy eecive ucleo-ucleo ieacio, which is used i (TNE (TNO eleco scaeig (Ves =v,, ae he segh paamee i eso ( C v pa o (eso eve, (eso-odd ad DD(SE, DD(TE is expessed as a sum o he ceal poeial pa, ae he segh paamee i desiy depedece pas o ( LS v (sigle-eve, (iple-eve especively. These paamee spi-obi poeial pa, log age eso values ae give i Table (1 [3]. ( TN v ( DD pa v, ad desiy depedece pa as ollows [3]: ( c ( LS ( TN ( DD v v v v (8 The ou poeials ae expessed as [3]: ( c ( SE TE ( SO ( TO ( C ( P SE P TE P SO PTO ( LS ( LSE ( LSO ( LS C C (.( P TE PTO L s 1 s ( TN ( TNE ( TNO ( TN ( P TE PTO S (.[ ( 1 ( SE ].[ ( 1 ( TE DD DD a DD ] a ( PSE PTE δ (9 whee ( = e o M3Y ieacio, : age paamee C ( ( = e LS (, = δ epeseed Gogy ieacio, he o M3Y, eq. (9 becomes: 015

4 Ieaioal Joual o Scieiic & Egieeig Reseach, Volume 6, Issue, Decembe Table (1. Shows he values o he bes i o he poeial paamees o M3Y-P [3]. Oscillao maix elemes (Chael Ceeal Sigle-Eve (SE Ceal Tiple-Eve (TE Ceal Sigle-Odd (SO Ceal Tiple-Odd (TO Teso-Eve (TNE Teso-Odd (TNO Spi-Obi Eve (LSE Spi-Obi Odd (LSO Desiy- sigle eve (SE R1=0.5 m R=0.40 m 1 3 R3=1.414 m eecs ae ake io accou hough is ode peubaio heoy, which allows paicle-hole exciaio om shell coe obis 1s1/, 1p3/, 1p1/, 1d5/, s1/, 1d3/ ad 17/ (shell model space havig 56Ni as a ie coe. The eecive ieacio o model space F5PVH poeial has bee used o give he (15/p3/p1/ shell model wave ucios o Ni. The sigle paicle wave ucios o he hamoic oscillao (HO wih size paamee b= m ae used [4]. Two ealisic ieacios M3Y-P ad Gogy as a esidual ieacio ae used wih he logiudial C om acos o Ni om he goud sae (JπT=0 o he excied sae (JπT= a Ex=1.404 MeV have bee calculaed wih coe coibuio oly, sice he model space euos has o coibue o he chage om aco, because hey ae eual paicles, he oly coe poos will be play his ole. Figue (1 usig M3Y ieacio as a esidual ieacio shows ha a ageeme is obaied o he is maximum ( o he om aco o momeum ase age 0 q 1.75m-1, bu he secod maximum ( o he om aco o he q age 1.75 q 3 m-1 had bee queched, ha is clea he calculaios udeesimae he expeimeal daa o he is maximum, ad oveesimae o secod maximum, hee ae a clea deviaio i diacio miimum om he heoeical calculaios o he expeimeal daa appoximaely 0.7 m-1 wih espec o he q values. Figue ( usig Gogy ieacio shows ha he om aco value abou 4x10-3 epeseed he is maximum o he age 0 q 1.75 m-1, which is a ageeme is obaied wih expeimeal daa up o q=1 m-1, bu he secod maximum ( o q age 1.75 q 3 m-1 have bee queched, clealy he calculaios udeesimae he expeimeal daa o he is maximum ad oveesimae o he secod maximum, The OBDM elemes o his asiio ae show i Table (. The expeimeal daa ae ake om Re. [18] MeV m- MeV m MeV m- MeV m MeV m Desiy- Tiple (TE eve 1139 MeV m Table (: Values o he OBDM elemes o he Resuls ad Discussio I his wok, he model space adoped which is p3/ 15/ p1/ coiguaio o Ni ucleus. Coe-polaizaio logiudial C asiio o he 1, is sae o Ni usig F5PVH model space eecive ieacio o M3Y ad Gogy a EX=1.404 MeV. 015

5 Ieaioal Joual o Scieiic & Egieeig Reseach, Volume 6, Issue, Decembe OBDM OBDM Ji J ( T=0 ( T=1 5/ 5/ / 3/ / 1/ / 5/ / 3/ / 1/ / 5/ / 3/ Fig. (: Ielasic logiudial C om acos o he 1 sae a Ex=1.404 MeV i Ni wih esidual ieacio Gogy (solid cuve, he expeimeal daa (illed cicles wee ake om Re.[18]. Figue (3 shows he quadupole C om aco o he secod asiio om he goud sae (JπT=0 o he ial sae (JπT= a Ex=.151 MeV, wih a esidual ieacio M3Y, oly oe lobe ae obaied wih om aco value equal o 10-3 o he momeum ase age 0 q 3 m-1, while he expeimeal daa have wo lobes, wih om aco values equal o o q age 0 q 1 m-1, ad o q age 1 q m-1. The calculaios esuls ae o coicidece wih he expeimeal daa. Figue (4, usig Gogy as a esidual ieacio, oly oe lobe was obaied o om aco, which is equal o wih momeum ase age 0 q 3 m-1, ad wo lobes o expeimeal daa, wih he om aco value equal o o q age 0 q 1 m-1, ad o q age 1 q m-1. The calculaios esuls ae o coicidece wih he expeimeal daa. Fig. (1: Ielasic logiudial C om acos o he 1 The OBDM elemes o his asiios ae show i Table (3. The expeimeal daa ae ake om Re. [18]. sae a Ex=1.404 MeV i Ni wih esidual ieacio M3Y (solid cuve, he expeimeal daa (illed cicles wee ake om Re. [18]. Table (3: Values o he OBDM elemes o he, secod sae o logiudial C asiio o he Ni usig F5PVH model space eecive ieacio o M3Y ad Gogy a EX=.151MeV. Ji J OBDM ( T=0 OBDM ( T=1 5/ 5/ / 3/

6 Ieaioal Joual o Scieiic & Egieeig Reseach, Volume 6, Issue, Decembe / 1/ / 5/ / 3/ / 1/ / 5/ / 3/ Fig. (4: Ielasic logiudial C om acos o he sae a Ex=.151MeV i Ni wih a esidual ieacio Gogy (solid cuve, he expeimeal daa (illed cicles wee ake om Re. [18]. Figue (5 usig M3Y ieacio shows he calculaio esuls o he C asiio o hid case 3 om he goud sae (JπT=0 o he ial sae (JπT= a Fig. (3: Ielasic logiudial C om acos o he Ex=.95 MeV, hee lobes wee obaied om his case, he is maximum o om aco equal o a he sae a Ex=.151MeV i Ni wih a esidual ieacio momeum ase egio 0 q 0.9 m-1, he secod M3Y (solid cuve, he expeimeal daa (illed cicles maximum equal o a he momeum ase wee ake om Re. [18]. 0.9 q.3 m-1, ad he hid maximum is a q age.3 q 3 m-1, hee is a clealy a ageeme bewee he esuls ad expeimeal daa i shape ad magiude, o he is maximum, he esuls udeesimae by 7 wih espec o he om aco value, while o he secod maximum, he esuls oveesimae by 8 wih espec o he om aco value. Figue (6 usig Gogy ieacio shows he calculaio o he C asiio o he hid case 3 om he goud sae (JπT=0 o he ial sae (JπT= a Ex=.95 MeV, hee lobes wee obaied om his case, he is maximum value o om aco equal o a he momeum ase egio om 0 o 1 1, i is vey clea hee is a good ageeme bewee he heoeical ad expeime daa i shape ad magiude whe he esuls udeesimae by 5 wih espec o he om aco value, he secod maximum o om aco equal o a q age 1 q. m-1, which is oveesimae wih he expeimeal daa by 1.5 wih espec o om aco value, ad he hid maximum o om aco equal o o

7 Ieaioal Joual o Scieiic & Egieeig Reseach, Volume 6, Issue, Decembe q age.3 q 3 m-1. I geeal hee is a good ageeme wih he expeimeal daa. The OBDM elemes o his asiios ae show i Table (4. The expeimeal daa ae ake om Re. [18]. Table (4: Values o he OBDM elemes o he logiudial C asiio o he 3, hid sae o Ni usig F5PVH model space eecive ieacio o M3Y ad Gogy a EX=.95 MeV. Ji J OBDM ( T=0 OBDM ( T=1 5/ 5/ / 3/ / 1/ / 5/ / 3/ / 1/ / 5/ / 3/ Fig. (6: Ielasic logiudial C om acos o he sae a Ex=.95MeV i Ni wih a esidual ieacio Gogy (solid cuve, he expeimeal daa (illed cicles wee ake om Re. [18]. Ackowledgeme: The Auhos ae vey gaeul o Po. D. Raad A. Radhi o his assisace o povide he oigial copy o his wo codes o calculaio o om acos ad esidual ieacio. Coclusios: Fom above calculaios o ielasic logiudial C om acos, he bes oe ha close he expeimeal daa is Fig. (5: Ielasic logiudial C om acos o he sae a Ex=.95 MeV i Ni wih esidual ieacio M3Y(solid cuve, he expeimeal daa (illed cicles wee ake om Re. [1]. 3 3 epeseed by he hid asiio usig Gogy ieacio. The ealisic poeial M3Y, ad Gogy as a esidual ieacio used o calculae coe polaizaio eecs has impove he calculaio, i geeal, he esuls owads he ageeme wih he expeimeal daa. The coe-polaizaio eec ehaces he om acos ad makes he heoeical esuls o he ielasic logiudial om acos close o he expeimeal daa i he C. The δ, σ, π ad Ω mesos i M3Y ieacio makes he esuls moe eal ad cosiue wih expeimeal esuls bu he desiy depedece o Mesos δ i spi obi pa o Gogy ieacio will ehaces he esuls vey well beside he Gausse poeial o ceal pa. Reeeces [1] J. D. Walecka, Cambidge Uivesiy Pess, Cambidge, (

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