Determining Modern Energy Functional for Nuclei And The Status of The Equation of State of Nuclear Matter. Shalom Shlomo Texas A&M University

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1 Deemnng Moden Enegy Funconal fo Nucle And The Saus of The Equaon of Sae of Nuclea Mae Shalom Shlomo Texas A&M Unvesy

2 Oulne. Inoducon. Backgound Enegy Densy Funconal Equaon of Sae Collecve Saes. Enegy Densy Funconal. Haee-Fock Equaons HF Skyme Ineacon Smulaed Annealng Mehod Daa and Consan 3. Resuls and Dscusson. 4. HF-based Random-Phase-Appoxmaon RPA. Fully Self Consen HF-RPA Hadon Excaon of Gan Resonances Compesson Modes and he NM EOS Symmey Enegy Densy 5. Resuls and Dscusson. 6. Conclusons.

3 Nuclea physcs: INTRODUCTION Sudy of sucue neacons and popees of nucle. Am o quanavely undesand and elae vas amoun of popees of nucle n ems of few consuens elemenay laws and pocesses. Cuen suaon: Acve aea of eseach. We have a cean pcue undesandng obaned hough 70 yeas of phenomenologcal eseach qualave consdeaon and applcaon of laws quanum mechancs. Q. M. s vey successful n descbng popees of nucle. Thee s no evdence conadcng Q. M. Recen emphasze: Popees of nucle unde exeme condons of excaon enegy empeaue angula momenum and N-Z asymmey. Relaon o ohe aeas: Asophyscs: souce of enegy sas sucue and evoluon of sas ogn of he elemens neuon sas and supenova. Ohe sysems: aomc cluses meal cluses apped ons and mesoscopc sysems.

4 Mae and chage densy dsbuons a Chage densy dsbuon c fo doubly magc nucle 6 O 40 Ca 90 Z 3 Sn and 08 Pb. The heoecal cuves ae compaed wh he expemenal daa pons uns ae c efm -3 and fm. b Nuclea mae densy dsbuons m fm -3 fo he magc nucle. 0 R a e 0 R / A. 0 fm a fm

5 Nuclea Bndng Wezsacke fomula BE Z N a a 3 Z A /3 a 4 A a N Z A A /3 δ A / a MeV Volume a 7.6 MeV Suface a MeV Coulomb a4 3.7 MeV Symmey. MeV odd-odd The conbuon o B/A. Noe ha he suface asymmey and Coulomb ems all subac fom he bulk em. δ 0. even-odd MeV even-even pang

6 Mean-feld appoxmaon The many-body Schödnge equaon Hψ E ψ s dffcul o solve. In he meanfeld appoxmaon each pacle moves ndependenly fom ohe nucleons n a sngle pacle poenal epesenng s neacons wh all ohe nucleons. H p m < j V j Α Appoxmaon: p H U m H es H h U 0 h hφ E φ Φ Αφ... φ A A Ansymmezaon opeao femons o symmezaon opeao bosons p m

7 Sphecal symmey U d df l U f U A Z N U Coul Z o s Z Wood-Saxon poenal popula: χ φ θ φ ljm Y u.7 /3 fm A R R c 5 0 U U U o s.67 d 0 fm Sphecal symmey U m φ εφ s educed o 0 u s d u d whee m l l U m s ε exp d R f MeV 3 c c R R Ze Ze R c > R c U Coul

8 The spn-ob couplng: descbe eal nucle Fo he hamonc oscllaon spn ob neacon he enegy egenvalues become ε n ls j [ n l 3/ ] ω U0 αδ j Whee δ -l o l f j l / o l-/ Sngle-pacle specum up o N5. he vaous conbuons o he full obal and spn ob splng ae pesened. Paal and accumulaed nucleon numbes ae also gven.

9 Map of he exsng nucle. The black squaes n he cenal zone ae sable nucle he boken nne lnes show he saus of known unsable nucle as of 986 and he oue lnes ae he assessed poon and neuon dp lnes Hansen 99.

10 OBJECTIVE. Impoan ask: Develop a moden Enegy Densy Funconal EDF E E[] wh enhanced pedcve powe fo popees of ae nucle.. We sa fom EDF obaned fom he Skyme N-N neacon. 3. The effecve Skyme neacon has been used n mean-feld models fo seveal decades. Many dffeen paameezaons of he neacon have been ealzed o bee epoduce nuclea masses ad and vaous ohe daa. Today hee s moe expemenal daa of nucle fa fom he sably lne. I s me o mpove he paamees of Skyme neacons. We f ou mean-feld esuls o an exensve se of expemenal daa and oban he paamees of he Skyme ype effecve neacon fo nucle a and fa fom he sably lne.

11 Equaon of sae and nuclea mae compessbly 8 ] [ ] [ o o o K E E The symmec nuclea mae NZ and no Coulomb ncompessbly coeffcen K s a mpoan physcal quany n he sudy of nucle supenova collapse neuon sas and heavy-on collsons snce s decly elaed o he cuvaue of he nuclea mae NM equaon of sae EOS E E. o fo d A E d dk A E d k K k f f / 9 / [fm -3 ] 0.6 fm -3 E/A [MeV] E/A -6 MeV 8 ] [ ] [ β β β β β o o o o o ANM K E E ] [ ] [ β β J E E o o v ] [ β β K K K o A Z N / β ESYM[ o ] J A Z y / / / 8 y SYM dy A E d E

12 Macoscopc pcue of gan esonance L 0 L L

13 Moden Enegy Densy Funconal χ φ α m jlm Y R Whn he HF appoxmaon: he gound sae wave funcon Φ ! A A A A A A A A A A A A A φ φ φ φ φ φ φ φ φ Φ In sphecal case Φ Φ oal H E ˆ HF equaons: mnmze

14 The oal Hamlonan of he nucleus ˆ j j A oal V m p V T H. Coul j NN j j V V V whee The oal enegy ' ' ' ' ' ' ' ' ˆ * * * * * dd V dd V d m H E j j j j A j A j A oal α α α α α α α α α α φ φ φ φ φ φ φ φ φ φ Δ Φ Φ

15 4 A j j j j Coul j e V j j 6 ] [ j j j j j j j j j j j j j j j j j j NN k W k P x k k P x k k P x P x V j δ δ δ δ δ δ α we adop he sandad Skyme ype neacon NN V j Fo he nucleon-nucleon neacon Skyme neacon 0 W x α ae 0 Skyme paamees.. Coul j NN j j V V V

16 The oal enegy Φ Φ Φ Φ d H V V T H E Coulomb oal ˆ whee m m H n n p p Knec ' ' ' ' ' ' d d e H ch ch ch Coulomb H H H H Skyme Coulomb Knec

17 H Η H H H H Skyme 0 3 eff fn so H sg

18 φ φ A A φ [ ] A J ' ' * ' φ φ J J ' ' ' * ' φ φ ch. Now we apply he vaaon pncple o deve he Haee-Fock equaons. We mnmze he Enegy E gven n ems of he enegy densy funconal ˆ oal E H H d Φ Φ * 0 δ ε δ δ δ ε δ δ d E d E

19 δ δ δ δ * d J W U m E whee ' * ' ' δφ φ δ ' * ' ' δφ φ δ [ ] '' ' * " ' '' ' φ δφ δ J

20 Afe cayng ou he mnmzaon of enegy we oban he HF equaons: 4 3 * ' * " * R R W l l j j m d d U R m d d R l l R m α α α α α α α α α α α α ε 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

21 4 4 * x x x x m m [ ] ' ' ' d e J J W x x x x x x x x x x x x x U ch δ α α α α [ ] ] [ J x x J W W

22 Wh an nal guess of he sngle-pacle wave funcons example; hamonc oscllao wave funcons we can deemne m* U and W and solve he HF equaon o ge a se of new sngle-pacle wave funcons; hen one can poceed n hs way unl eachng convegence. NOTES:. One should sa close o he soluon.. Accuacy and convegence n hee dmenson 3 Convegence of HFB equaons n hee dmenson?

23 Infne Nuclea Mae I s mpoan o noe ha he poon and neuon denses ae consans. So ha The EOS of asymmec NM wh wh Noe ha α

24 Deemnng he Skyme neacon usng he HF appoach Appoxmaons Coulomb enegy V d Coul H e Coulomb p ' d ' 3 V ' d Coul Noe ha hee he dec em and exchange Coulomb ems each nclude he spuous self-neacon em. Ineacon: p V ex Coul 3 4 e V ex Coul 3 p KDE0 KDE0v neglec exchange em KDE π nclude exchange em p 3 KDEX nclude conbuons of g.s. coelaons

25 Cene of mass coecon a. Coecon o he oal bndng enegy: We use he hamonc oscllao appoxmaon. The CM enegy s aken as osc 3 KCM ω bu ω 4 Is deemned by usng he mass mean-squae ad ω 3 N ma b. Coecon o he chage ms ad ch The chage mean-squae adus o be fed o he expemenal daa s obaned as N ch p 3 nlj j l HF A p Z n µ. υ Z mc lj

26 Smulaed Annealng Mehod SAM The SAM s a mehod fo opmzaon poblems of lage scale n pacula whee a desed global exemum s hdden among many local exema. We use he SAM 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 exp M N d s he numbe of expemenal daa pons. N p s he numbe of paamees o be fed. M h M exp and ae he expemenal and he coespondng heoecal values of he physcal quanes. h s he adoped unceany.

27 Implemenng he SAM o seach he global mnmum of χ funcon:. x W. Defne α 0 ae wen n em of χ old B / A K nm nm... ' v B / A Knm nm m*/ m Es J L κ G 0 W0 3. Calculae fo a gven se of expemenal daa and he coespondng HF esuls usng an nal guess fo 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 o ge a new value of v Use hs modfed veco o geneae a new se of Skyme paamees. η v v dη v v

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

29 Fed daa - The bndng eneges fo 4 nucle angng fom nomal o he exoc 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 ε p / - εp 3/.88 MeV neuon εp / - εp 3/.83 MeV poon. - Rms ad fo he valence neuon: n he d 5/ ob fo 7 O n he f 7/ ob fo 4 Ca n d / n f / fm fm - 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.

30 . The ccal densy V Landau ph l Consans < 0 < 3 c 0 ' ' F F G G δ l l ' ' Landau sably condon: F F G G > l k m F F Example: K 6 0 F / 3 l l l l l l. The Landau paamee G should be posve a 0 ' 0 3. The quany ds P 3 mus be posve fo denses up o 3 0 d 4. The IVGDR enhancemen faco 0.5 < κ < 0. 5 ES T L NZ E de κ m A

31 Self-conssen calculaon whn consaned HF

32 v v 0 v d B/A MeV K nm MeV nm fm m*/m E s MeV J MeV L MeV Kappa G W 0 MeV fm

33 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

34 Values of he Skyme paamees and he coespondng physcal quanes of nuclea mae fo he KDE0 and KDE0v and KDEX neacons. Paamee KDE0 KDE0v KDEX 0 MeV fm MeV fm MeV fm MeVfm 3α x x x x W 0 MeV fm α B/A MeV K MeV fm m*/m J MeV L MeV κ G'

35 Nucle B exp B B exp -B h KDE0 KDEX Bndng Eneges MeV 6 O O S Ca 48 Ca 48 N G. Aud e al Nucl. Phys. A N N N S Z Sn Sn Pb

36 Chage RMS Rad fm E. W. Oen n Tease onn Heavy-Ion Scence Vol Nucle 6 O Expemen.73 KDE0.77 KDEX Ca H. D. Ves e al A. Daa Nucl. Tables Ca N F. Le Blanc e al Phys. Rev. C S 90 Z Sn Pb

37 Sngle-Pacle Eneges n MeV fo 40 Ca Obs Exp. KDE0* Obs Exp. KDE0* Poons Neuons s / s / p 3/ p 3/ p / p / d 5/ d 5/ s / s / d 3/ d 3/ f 7/ f 7/ p 3/ *TAMU Skyme Ineacon: B. K. Agawal S. Shlomo and V. Km Au Phys. Rev. C

38 GIANT RESONANCES Hadon Scaeng HF-Based RPA Resuls

39 Equaon of sae and nuclea mae compessbly 8 ] [ ] [ o o o K E E The symmec nuclea mae NZ and no Coulomb ncompessbly coeffcen K s a mpoan physcal quany n he sudy of nucle supenova collapse neuon sas and heavy-on collsons snce s decly elaed o he cuvaue of he nuclea mae NM equaon of sae EOS E E. o fo d A E d dk A E d k K k f f / 9 / [fm -3 ] 0.6 fm -3 E/A [MeV] E/A -6 MeV 8 ] [ ] [ β β β β β o o o o o ANM K E E ] [ ] [ β β J E E o o v ] [ β β K K K o A Z N / β ESYM[ o ] J A Z y / / / 8 y SYM dy A E d E

40 The soveco gan dpole esonance The oal phooabsopon coss-secon fo 97 Au llusang he absopon of phoons on a gan esonang elecc dpole sae. The sold cuve show a Be-Wgne shape. Boh and Moelson Nuclea Sucue vol. 975.

41 Macoscopc pcue of gan esonance L 0 L L

42 Hadon excaon of gan esonances χ f Ψ f α V αn Nucleus χ Ψ Theoss: calculae anson sengh SE whn HF-RPA usng a smple scaeng opeao F ~ L Y LM : Expemenalss: calculae coss secons whn Dsoed Wave Bon Appoxmaon DWBA: o usng foldng model.

43 DWBA-Foldng model descpon

44 EWSR enegy weghed sum ule ES E de 0

45 Elasc angula dsbuons fo 40 MeV alpha pacle. Flled squaes epesen he expemenal daa. Sold lnes ae f o he expemenal daa usng he foldng model DWBA wh nucleon-alpha neacon.

46 Snce α pacles have S 0 T 0 hey ae deal fo sudyng elecc S 0 and soscala T 0 Gan Resonances.

47 Hsoy A. ISOSCALAR GIANT MONOPOLE RESONANCE ISGMR: 977 DISCOVERY OF THE CENTROID ENERGY OF THE ISGMR IN 08 Pb E 0 ~ 3.5 MeV TAMU Ths led o modfcaon of commonly used effecve nucleon-nucleon neacons. Haee-Fock HF plus Random Phase Appoxmaon RPA calculaons wh effecve neacons Skyme and ohes whch epoduce daa on masses ad and he ISGMR eneges have: K 0 ± 0 MeV J.P. BLAIZOT 980. A. ISOSCALAR GIANT DIPOLE RESONANCE ISGDR: 980 EXPERIMENTAL CENTROID ENERGY IN 08 Pb AT E ~.3 MeV Jülch PRL ; ~ 9 MeV PRC HF-RPA wh neacons epoducng E 0 pedced E ~ 5 MeV. K ~ 70 MeV fom ISGDR? T.S. Dmescu and F.E. Se [PRC ] poned ou If fuhe measuemen confm he value of.3 MeV fo hs mode he dscepancy may be sgnfcan. Relavsc mean feld RMF plus RPA wh NL3 neacon pedc K 70 MeV fom he ISGMR [N. Van Ga e al. NPA ].

48 Haee-Fock HF - Random Phase Appoxmaon RPA In fully self-conssen calculaons:. Assume a fom fo he Skyme paamezaon δ-ype.. Cay ou HF calculaons fo gound saes and deemne he Skyme paamees by a f o bndng eneges and ad. 3. Deemne he esdual p-h neacon 4. Cay ou RPA calculaons of sengh funcon anson densy ec.

49 Geens Funcon Fomulaon of RPA In he Geens Funcon fomulaon of RPA one sas wh he RPA- Geens funcon whch s gven by G Go VphGo whee V ph s he pacle-hole neacon and he fee pacle-hole Geens funcon s defned as G o ' E ϕ * ϕ ' ho ε E ho ε E whee φ s he sngle-pacle wave funcon є s he sngle-pacle enegy and h o s he sngle-pacle Hamlonan.

50 The connuum effecs such as pacle escape wdh can be aken no accoun usng m U V / W h0 Z h whee < and > ae he lesse and geae of and especvely U and V ae he egula and egula soluon of H 0 -Zψ 0 wh he appopae bounday condons and W s he Wonskan. NOTE he wo ems n he fee pacle-hole geens funcon

51 A We use he scaeng opeao F F f o oban he sengh funcon S E 0 F n δ E En Im[ T π n f G f ] and he anson densy δ RPA E ΔE S E ΔE f ' [ ImG ' E] d π 3 ' RPA δ s conssen wh he sengh n E ± ΔE / S E δ RPA E f d ΔE

52 RMF-RPA: J. Pekaewcz PRC ; Z.Y. Ma e al. NPA Relavsc Mean Feld Random Phase Appoxmaon The seps nvolved n he elavsc mean feld based RPA calculaons ae analogous o hose fo he non-elavsc HF-RPA appoach. The nucleon-nucleon neacon s geneaed hough he exchange of vaous effecve mesons. An effecve Lagangan whch epesens a sysem of neacng nucleons looks lke I conans nucleons ψ wh mass M; ω mesons; he elecomagnec feld; non lnea self-neacons fo he and possbly ω feld. Values of he paamees fo he mos wdely used NL3 neacon ae m MeV m ω MeV m MeV g 0.7 g ω.868 g g fm - and g n hs case hee s no self-neacon fo he ω meson. NL3: K 7.76 MeV G.A.Lalazsss e al. PRC

53 Self-conssen calculaon whn consaned HF

54 Isoscala sengh funcons of 08 Pb fo L 0-3 mulpolaes ae dsplayed. The SC full lne coesponds o he fully selfconssen calculaon whee LS dashed lne and CO open ccle epesen he calculaons whou he ph spn-ob and Coulomb neacon n he RPA especvely. The Skyme neacon SGII [Phys. Le. B ] was used.

55 Isoveco sengh funcons of 08 Pb fo L 0-3 mulpolaes ae dsplayed. SC full lne coesponds o he fully self-conssen calculaon whee LS dashed lne and CO open ccle epesen he calculaons whou he ph spnob and Coulomb neacon n he RPA especvely. The Skyme neacon SGII [Phys. Le. B ] was used.

56 90 Z SE fm 4 /MeV 6 Sn 44 Sm 08 Pb E MeV Isoscala monopole sengh funcon

57 S. Shlomo and A.I. Sanzhu Phys. Rev. C ISGDR f Y M SL neacon K 30 MeV E α 40 MeV Reconsucon of he ISGDR EWSR n 6 Sn fom he nelasc α-pacle coss secons. The mddle panel: maxmum double dffeenal coss secon obaned fom RPA. The lowe panel: maxmum coss secon obaned wh coll dashed lne and sold lne nomalzed o 00% of he EWSR. Uppe panel: The sold lne calculaed usng RPA and he dashed lne ae he aos of he mddle panel cuve wh he sold and dashed lnes of he lowe panel especvely. coll 5 d d

58 Fully self-conssen HF-RPA esuls fo ISGMR cenod enegy n MeV wh he Skyme neacon SK55 SGII and KDE0 ae compaed wh he RRPA esuls usng he NL3 neacon. Noe he coespondng values of he nuclea mae ncompessbly K and he symmey enegy J coeffcens. ω -ω s he ange of excaon enegy. The expemenal daa ae fom TAMU. Nucleus ω -ω Exp. NL3 SK55 SGII KDE0 90 Z ± Sn ± Sm ± Pb ± K MeV J MeV

59 E CEN MeV E CEN MeV ΔE CEN MeV Ca 48 Ca ISGMR T0 E0 C 0.95 C Ca - 40 Ca C 0.3 None of he neacons fall n he Expemenal ange fo 40 Ca E CEN ESEdE SEdE C s he Peason coelaon coeffcen 48 Ca 40 Ca > 0 fo all he neacons whch goes agans he end of deceasng ISGMR wh nceasng A ΔE CEN MeV Ca - 40 Ca K NM MeV Noe ha fo no self-conssen RPA calculaons whch neglec he Coulomb and Spn-Ob pas. Some neacons would fall n he coec 48 Ca 40 Ca ange.

60 E CEN MeV E CEN MeV ΔE CEN MeV Ca 48 Ca ISGDR T0 E C 0.88 C Ca - 40 Ca K NM MeV E CEN MeV E CEN MeV ΔE CEN MeV ISGDR T0 E C C Ca 48 Ca 48 Ca - 40 Ca C 0.3 C m*/m

61 E CEN MeV E CEN MeV ΔE CEN MeV ISGQR T0 E C C Ca 48 Ca C Ca - 40 Ca m*/m

64 E CEN MeV Ca IVGDR T E C -0.7 E CEN MeV Ca C -0.3 No clea value of he symmey enegy J can be deduced. ΔE CEN MeV Ca - 40 Ca C J MeV

65 E CEN MeV Ca IVGDR T E C E CEN MeV Ca IVGDR T E C -0.3 E CEN MeV Ca C -0.4 E CEN MeV Ca C -0. ΔE CEN MeV Ca - 40 Ca C L MeV ΔE CEN MeV Ca - 40 Ca C K sym MeV

66 α D fm Pb C AB n - p fm

67 Conclusons We have developed a new EDFs based on Skyme ype neacon KDE0 KDE KDE0v... applcable o popees of ae nucle and neuon sas. Fully self-conssen calculaons of he compesson modes ISGMR and ISGDR whn HF-based RPA usng Skyme foces and whn elavsc model lead a nuclea mae ncompessbly coeffcen of K 40 ± 0 MeV sensvy o symmey enegy. Sensevy o symmey enegy: IVGDR GR n neuon ch nucle Rn Rp sll open poblems. Possble mpovemens: Accoun fo effec of coelaons on B.E. Rad S.P. eneges Popely accoun fo he sospn dependency of he spn-ob neacon Include addonal daa such as IVGDR J and ISGQR m*

68 Refeences [] A. Boh and B. Moelson Nuclea Sucue Vol. II Benjamn London 975. [] A. deshal and H. Feshbach Theoecal Nuclea Physcs Vol. I: Nuclea Sucue John Wley & Sons Inc. New Yok 974. [3] D. J. Rowe Nuclea Collecve Moon Models and Theoy Mehuen and Co. Ld [4] P. Rng and P. Schuck The nuclea many-body poblems Spnge New Yok- Hedeleg-Beln 980. [5] G. F. Besch and R. A. Bogla Osclaons In Fne Quanum Sysems Cambdge Unvesy Pess 994. [6] G. R. Sachle Dec Nuclea Reacons Oxfod Unvesy Pess Oxfod 983. [7] S. Shlomo and G. F Besch Nucl. Phys. A

69 [8] S. Shlomo and D. H. Youngblood Phys. Rev. C [9] A. Kolomes O. Pochvalov and S. Shlomo Phys. Rev. C [0] S. Shlomo and A. I. Sanzhu Phys. Rec. C [] B. K. Agawal S. Shlomo and A. I. Sanzhu Phys. Rev. C [] B. K. Agawal S. Shlomo and V. Km Au Phys. Rev. C R 003. [3] B. K. Agawal S. Shlomo and V. Km Au Phys. Rev. C [4] N. K. Glendennng Phys. Rev. C [5] B. K. Agawal S. Shlomo and V. Km Au Phys. Rev. C [6] S. Shlomo V. M. Kolomez and G. Colo Eu. Phys. A

70 Acknowledgmens Wok done a: Wok suppoed by: Gan numbe: PHY Gan numbe: DOE-FG03-93ER40773

71 CONCLUSION We have developed a new EDF based on Skyme ype neacon KDE0 applcable o popees of ae nucle and neuon sas. Fully self-conssen calculaons of he ISGMR usng Skyme foces lead a nuclea mae ncompessbly coeffcen of K 40 ± 0 MeV wh sensvy o symmey enegy. I s possble o buld bonafde Skyme foces wh K close o he elavsc value. Fuhe mpovemen Accoun fo he effec of coelaon on B.E. R ch and sngle pacle enegy Popely accoun fo he sospn dependency of he spn-ob neacon Include addonal daa such as IVGDR J and ISGQR m*

72 SUMMARY AND CONCLUSIONS Fully self- conssen calculaons of he compesson modes ISGMR and ISGDR usng moden enegy densy funconals Skyme foces lead o K 40 ± 0 MeV wh sensvy o symmey enegy. Symmey enegy densy IVGDR Rn Rp Open poblem Accounng fo pos- emsson decay allows one o oban conssen values of empeaue of a dsassemblng souce fom he double- ao mehod. 3 Alhough a low denses he empeaue calculaed fom gven yelds changes only modesly f medum effecs ae aken no accoun lage dscepances ae obseved when he nucleon denses ae deemned fom measued yelds 4 Due o clusezaon a low densy nuclea mawe he symmey enegy s much lage han ha pedced by mean feld appoxmaon

73 . Inoducon Oulne Defnons: nuclea mae ncompessbly coeffcen K Backgound: soscala gan monopole esonance soscala gan dpole esonance Hadon excaon of gan esonances. Theoecal appoaches fo gan esonances Haee-Fock plus Random Phase Appoxmaon RPA Commens: self-conssency? Relavsc mean feld RMF plus RPA. Dscusson ISGMR vs. ISGDR Non-elavsc vz. Relavsc

74 Nuclea mae popees fom collecve modes n nucle Shalom Shlomo Cycloon Insue Texas A&M Unvesy

75 New Skyme effecve nucleon-nucleon neacon Shalom Shlomo Cycloon Insue Texas A&M Unvesy

76 Effecs of self-conssence volaons n HF-based RPA calculaons fo gan esonances Shalom Shlomo Texas A&M Unvesy

77 Nuclea mae equaon of sae and gan esonances n nucle Shalom Shlomo Texas A&M Unvesy

78 Inoducon 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 bee epoduce nuclea masses ad and vaous ohe daa. Today hee s moe expemenal daa on nucle fa fom he sably lne. I s me o mpove he paamees of Skyme neacons. We f ou mean-feld esuls o an exensve se of expemenal daa and oban he paamees of he Skyme ype effecve neacon fo nucle a and fa fom he sably lne.

79 The oal enegy [ ] Φ Φ Φ Φ d H H H V V T H E Skyme Coulomb Knec Coulomb oal ˆ Whee m m H n n p p Knec ' ' ' ' ' ' d d e H ch ch ch Coulomb [ ] [ ] [ ] [ ] x x J J J W J x x J J x x x x x x x x x x H n p n n p p n p n n p p n n p p n p Skyme α α

80 MODERN ENERGY DENSITY FUNCTIONAL FOR NUCLEI AND THE NUCLEAR MATTER EQUATION OF STATE Shalom Shlomo Cycloon Insue Texas A&M Unvesy

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

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