Moden Enegy Funconal fo Nucle and Nuclea Mae By: lbeo noosa Teas &M Unvesy REU Cycloon 008 Meno: D. Shalom Shlomo
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.
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.
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
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.
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
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
Skyme neacon To model he nuclea foce V NN he Skyme effecve nucleonnucleon neacon s used.. 6 ] [ 0 3 3 0 0 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.
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
nd he Skyme enegy densy funconal s Η Skyme 0 3 eff fn so sg
[ ] 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
δ δ δ δ d J W U m E h whee δ δ δ δ [ ] " δ δ J
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
4 4 m m h h [ ] 3 8 3 8 6 4 4. 0 3 3 3 3 3 3 0 0 0 0 d e J J W U ch δ [ ] ] [ 8 8 0 J J W W Defnons
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 3. 36 5 / fm n f 3. 99 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.
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.
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
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
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
Skyme Paamees Resuls Paamee 0 MeV fm 3 MeV fm 5 MeV fm 5 KDE0 F -56.5 40.63 430.94 6.67-398.38 7.3 KDEXFCORR -49.83 4.68 309.4 8.79-7.96 3.9 3 MeV fm 3 435.5 680.73 0465.4 33.9 0 0.7583 0.0655 0.474 0.00437 3-0.3087 0.065-0.0857 0.0046-0.9495 0.079-0.644 0.059.445 0.088 0.097 0.0006 W 0 MeV fm 5 8.96 3.33 0.676 0.063 98.90.7 0.4989 0.003
Nucle B ep ΔB B ep -B h KDE0 KDEX Bndng Eneges MeV 6 O -7.60 0.394 3.0 4 O -68.384-0.58 4.58 34 S -83.47-0.656.868 40 Ca 48 Ca 48 N -34.050-45.990-347.36 0.005 0.88 -.437 0.699.59 4.946 G. ud e al Nucl. Phys. 79 337 003 56 N -483.99.09.853 68 N -590.408 0.69.53 78 N -64.940-0.5.597 88 S -768.468 0.86.985 90 Z -783.89-0.7 0.93 00 Sn -84.800-3.664 0.80 3 Sn -0.850-0.4.75 08 Pb -636.430 0.945-5.584
Chage RMS Rad fm E. W. Oen n Tease onn eavy-ion Scence Vol 8 989. Nucle 6 O Epemen.73 KDE0.77 KDEX.73. D. Ves e al. Daa Nucl. Tables 36 495 987. 40 Ca 48 Ca 56 N 3.49 3.48 3.75 3.490 3.50 3.768 3.456 3.485 3.848 F. Le Blanc e al Phys. Rev. C 7 034305 005. 88 S 90 Z 4.9 4.58 4. 4.66 4.3 4.6 3 Sn 4.709 4.70 4.77 08 Pb 5.500 5.489 5.499
Obs Ep Poons KDE0-35.97 s / -50 ± -39.40 p 3/ -6.95-5.53 p / -34 ±6 -.93 -. d 5/ -4.49-3.787 s / -0.9-9.48-8.358 d 3/ -8.3-7.59-8.7067 f 7/ -.4 -.38 -.0307 Neuons KDEX s / -47.77-44.04 p 3/ -34.90-33.074 p / -30.78-9.958 d 5/ -.08 -.344 s / -8. -7.00-5.74 d 3/ -5.6-4.97-6.97 f 7/ -8.3-9.60-9.673 p 3/ -6. -4.98-4.3458 Sngle-pacle Eneges fo 40 Ca MeV
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.
Wok done a: cknowledgmens
Wok suppoed by: Gan numbes: PY-035500 PY-4639-0000 Gan numbe: DOE-FG03-93ER40773