Modern Energy Functional for Nuclei and Nuclear Matter. By: Alberto Hinojosa, Texas A&M University REU Cyclotron 2008 Mentor: Dr.

Transcription

1 Moden Enegy Funconal fo Nucle and Nuclea Mae By: lbeo noosa Teas &M Unvesy REU Cycloon 008 Meno: D. Shalom Shlomo

2 Oulne. Inoducon.. The many-body poblem and he aee-fock mehod. 3. Skyme neacon. 4. aee-fock Equaon 5. Smulaed nnealng Mehod. 6. Daa. 7. Resuls and Dscusson.

3 Inoducon Impoan ask: Develop a moden Enegy Densy Funconal EDF wh enhanced pedcve powe fo popees of ae nucle. We sa fom EDF obaned fom Skyme N-N neacon. The effecve Skyme neacon has been used n mean-feld models fo seveal decades and many dffeen paameezaons of he neacon have been ealzed o epoduce nuclea masses ad and ohe daa of nucle. Snce moe epemenal daa has become avalable we ae able o f ou esuls o a boade collecon of nucle a and fa fom he sably lne.

4 The many-body poblem In ode o deemne he popees of a nucleus s necessay o solve he me-ndependen Schödnge equaon. Ψ n... E Ψ... The many-body amlonan s gven by h m Unfounaely s dffcul o oban a soluon o he many-body equaon. n n V

5 The aee-fock Mehod aee-fock F s a mehod fo obanng an appomae soluon o he many body poblem n Quanum Mechancs. I uses he mean feld appomaon whee each pacle neacs wh an aveage poenal poduced by s neacon wh all ohe pacles. Effecs due o coelaed moon of many nucleons ae no accouned fo by he F appomaon.

6 Slae deemnans In F he many-nucleon femon wave funcon Φ s appomaed by an ansymmec poduc of sngle pacle wave funcons. Ths wave funcon Φ can be wen as a Slae deemnan whch guaanees an ansymmec wave funcon of he nucleons sysem ! M M M M Φ

7 The aee Fock Equaon The F equaon s deved usng Vaaonal Calculus by mnmzng he Enegy Funconal. Φ Φ oal E ˆ ˆ oal V m p V T. Coul NN V V V The oal amlonan of he nucleus s whee The oal enegy s ˆ dd V dd V d m E oal h Δ Φ Φ

8 Skyme neacon To model he nuclea foce V NN he Skyme effecve nucleonnucleon neacon s used.. 6 ] [ NN k W k P k k P k k P P V s s s δ δ δ δ δ δ whee s he spn echange opeao and whee he gh and lef aows ndcae ha he momenum opeaos ac on he gh and on he lef especvely. P / k / k s s s W 0 ae he Skyme paamees whch need o be deemned.

9 The oal enegy s hen gven by ˆ Φ Φ Φ Φ d V V T E Coulomb oal whee he enegy densy funconal s m m n n p p Knec h h d d e ch ch ch Coulomb Skyme Coulomb Knec

10 nd he Skyme enegy densy funconal s Η Skyme 0 3 eff fn so sg

11 [ ] J J J ch. Now we apply he vaaon pncple o deve he aee-fock equaons. We mnmze Φ Φ oal E ˆ 0 δ ε δ δ δ ε δ δ d E d E

12 δ δ δ δ d J W U m E h whee δ δ δ δ [ ] " δ δ J

13 fe cayng ou he mnmzaon of enegy we oban he F equaons: 4 3 " R R W l l m d d U R m d d R l l R m ε h h h aee-fock Equaons whee and ae he effecve mass he poenal and he spn ob poenal. They ae gven n ems of he Skyme paamees and he nuclea denses. m U W

14 4 4 m m h h [ ] d e J J W U ch δ [ ] ] [ J J W W Defnons

15 Fed daa - The bndng eneges fo 4 nucle angng fom nomal o he eoc poon o neuon ones: 6 O 4 O 34 S 40 Ca 48 Ca 48 N 56 N 68 N 78 N 88 S 90 Z 00 Sn 3 Sn and 08 Pb. - Chage ms ad fo 7 nucle: 6 O 40 Ca 48 Ca 56 N 88 S 90 Z 08 Pb. - The spn-ob splngs fo p poon and neuon obs fo 56 N εp / - εp 3/ neuon.88 MeV εp / - εp 3/.83 MeV poon. - Rms ad fo he valence neuon: n he d 5/ ob fo 7 O n d / fm n f fm 7 n he f 7/ ob fo 4 Ca / - The beahng mode enegy fo 4 nucle: 90 Z 7.8 MeV 6 Sn 5.9 MeV 44 Sm 5.5 MeV and 08 Pb 4.8 MeV. Noe: Bold face ndcaes daa aken n ou f fo he fs me.

16 SM Smulaed annealng mehod The SM s a mehod fo opmzaon poblems of lage scale n pacula whee a desed global eemum s hdden among many local eema. We use he SM o deemne he values of he Skyme paamees by seachng he global mnmum fo he ch-squae funcon χ N d Nd N p M ep M h N d s he numbe of epemenal daa pons. N p s he numbe of paamees o be fed. ep M h M and ae he epemenal and he coespondng heoecal values of he physcal quanes. s he adoped unceany.

17 Implemenng he SM o seach he global mnmum of funcon: χ. W. Defne 0 ae wen n ems of B / K nm nm... v B / K m / m E J L G 0 W nm nm s κ 0 χ old 3. Calculae fo a gven se of epemenal daa and he coespondng F esuls usng an nal guess Skyme paamees.. 4. Deemne a new se of Skyme paamees by he followng seps: Use a andom numbe o selec a componen v of veco Use anohe andom numbe η o ge a new value of v v v v dη Use hs modfed veco o geneae a new se of Skyme paamees. v

18 5. Go back o F and calculae χ new 6. The new se of Skyme paamees s acceped only f χ χ old new P χ ep > T 0 < β < β 7. Sang wh an nal value of T T we epea seps 4-6 fo a lage numbe of loops. 8. Reduce he paamee T as and epea seps - 7 T T k 9. Keep dong hs way unl hopefully eachng global mnmum of χ

19 Vaaon of he aveage value of χ he conol paamee T fo he KDE0 neacon fo he wo dffeen choces of he sang paamee. T as a funcon of he nvese of

20 Skyme Paamees Resuls Paamee 0 MeV fm 3 MeV fm 5 MeV fm 5 KDE0 F KDEXFCORR MeV fm W 0 MeV fm

21 Nucle B ep ΔB B ep -B h KDE0 KDEX Bndng Eneges MeV 6 O O S Ca 48 Ca 48 N G. ud e al Nucl. Phys N N N S Z Sn Sn Pb

22 Chage RMS Rad fm E. W. Oen n Tease onn eavy-ion Scence Vol Nucle 6 O Epemen.73 KDE0.77 KDEX.73. D. Ves e al. Daa Nucl. Tables Ca 48 Ca 56 N F. Le Blanc e al Phys. Rev. C S 90 Z Sn Pb

23 Obs Ep Poons KDE s / -50 ± p 3/ p / -34 ± d 5/ s / d 3/ f 7/ Neuons KDEX s / p 3/ p / d 5/ s / d 3/ f 7/ p 3/ Sngle-pacle Eneges fo 40 Ca MeV

24 Conclusons We developed a new Skyme neacon. Uses coelaon-effec coeced daa. Bee epoduces 6 O and 08 Pb chage ms ad. Chage ms adus fo 6 O ncompable wh monopole eneges n ou model. Possble mpovemens: Moe wok on opmzaon. Dffeen ses of daa.

25 Wok done a: cknowledgmens

26 Wok suppoed by: Gan numbes: PY PY Gan numbe: DOE-FG03-93ER40773