QUANTIFIEDLARGE-SCALEDENSITYFUNCTIONALTHEORY(DFT) PREDICTIONSOFNUCLEARPROPERTIES By YuchenCao ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof PhysicsDoctorofPhilosophy 2020 ABSTRACT QUANTIFIEDLARGE-SCALEDENSITYFUNCTIONALTHEORY(DFT) PREDICTIONSOFNUCLEARPROPERTIES By YuchenCao Re˛ection-asymmetricshapesoftheatomicnucleusarerelevanttonuclearstability, nuclearspectroscopy,nucleardecaysand˝ssion,andthesearchfornewphysicsbeyond thestandardmodel.CPviolationinthestandardmodelistooweaktoberesponsible fortheobservedmatter-antimatterasymmetry.Beyondstandardmodeltheoriesrequire additionalsourceofCPviolation,whichcouldbefoundifnon-zeroatomicelectricdipole moment(EDM)isobserved. ThenuclearquantitythatinducestheatomicEDMistheSchi˙moment,whichisen- hancedinoctupole-deformedodd-massorodd-oddnucleiwhereparitydoubletsexist.This callsfortwotasks:First,aglobalsurveyofoctupole-deformedeven-evennucleitodetermine thenuclearregionswithstrongoctupoleinstability;second,Schi˙momentcalculationsin theodd-massandodd-oddinthevicinityofstronglyoctupole-deformedeven-evennuclei. ThecalculatedSchi˙momentswillthenhelpusdeterminethebestcandidatesforatomic EDMmeasurements.Thesetwotasksconstitutethe˝rstpartofthisdissertation. Thetoolofchoiceforalargescalecalculationontheentirenuclearlandscapeisnu- clearDFT.WithintheDFTframework,theSkyrmeHFBmethodwillbeusedtoperform calculationsinthisdissertation. AlthoughnuclearDFTisapowerfultool,itlackstheabilitytoprovidequalityuncertainty estimatesforitspredictions.Inthesecondpartofthisdissertation,weexploreseveral Bayesianmachinelearningtechniquestofurtherincreasethepredictivepowerofnuclear DFT,andtoprovidefullBayesianuncertaintyquanti˝cationforDFTpredictions. Dedicatedtomyfamily,whohavealwaysbeensupportingme. ToClaireandSamantha,mayyoubehealthyandhappyforever. iv ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,WitekNazarewicz,forprovidingguidance,counsel,and numerousopportunitiesthroughoutmytimeingraduateschool.Thankyouforbeingpatient withme,andtolerantofmymistakes,ithasbeenanhonortolearnfromthebest. Iwouldliketoacknowledgemycollaborators,ErikOlsen,LéoNeufcourt,SamuelGiuliani, andSylvesterAgbemavafortheirpatiencewithallmyquestionsandtheirgeneroussupport. Iamalsogratefultoothermembersofmyresearchgroupwhoprovidedendlesshelp alongtheway.IwouldliketorecognizeZacharyMatheson,ChunliZhang,SiminWang, XingzeMao,MengzhiChen,TongLi,BastianSchütrumpf,KevinFossez,JimmyRotureau, YannenJaganathen,NicholasMichel,NobuoHinohara,andFutoshiMinato. IamfortunatetocrosspathwithmanywonderfulandbrilliantgraduatestudentsatMSU, whoaccompaniedmethroughtheearlyanddaysofgraduateschool:JustinEstee, DavidTarazona,KirillMoskovtsev,IvanPogrebnyak,ForrestPhillips,JustinLietz,Joshua Isaacson,ChaoyueLiu,SamuelMarinelli,XueyinHuyan,FaranZhou...Iwasluckytohave sharedmytimeatNSCLwithmyo˚cematesBrentGlassman,HaoLin,andXingzeMao, whosecompanyIwillforevermiss. Iwouldliketothankseveralprofessors:ScottPratt,whohasbeenagoodfriendand mentorinbothphysicsandtheAmericanwayoflife;VladimirZelevinsky,forgivingwon- derfullecturesonphysics;MortenHjorth-Jensen,forshowingextraordinarypassionand patienceinteaching;andShan-GuiZhou,who˝rstshowedmethepathtonuclearphysics. Iwouldliketoexpressgratitudetowardsmycommitteemembers:ScottBogner,Sean Couch,JaideepSingh,andKendallMahn.Specialshout-outtoKendallwhoo˙eredtremen- doushelpinmyjobsearchandgreattipsonbabycare. v IappreciatethehelpfromDanielLayineditingthisdissertation. Iwouldliketoextendmythankstoothersta˙andfacultymembersatNSCLandMSU whowerealwaystheretosupport,especiallyElizabethDeliyski,GillianOlsen,KimCrosslan, andDebbieBarratt. Iamgratefultomyparentsfortheirunconditionalloveandsupportthroughoutmylife. Mom,thankyouforbeingstrictwithmeandneverloweredyourstandards,IhopeIhave madeyouproud.Dad,thankyouforcultivatingmypassiontowardsmathandscience,I knowyouwantedmetobecomeanengineerlikeyou,buthey,Iaccomplishedsomething evenbetter. IamforeverindebtedtomywifeSiqiongChen.Thankyouforallthesacri˝cesyou've made,andforbringingClaireandSamanthaintomyworld. ThankyouNorah,forbeingagoodcatandstayedupwithmethroughcountlessnights ofwriting,andfornevertryingtodeletemy˝les. 2020isaroughyear,butIamgladtobesurroundedbysomanywonderfulandlovely people. vi TABLEOFCONTENTS LISTOFTABLES .................................... ix LISTOFFIGURES ................................... xi Chapter1Introduction ............................... 1 1.1Overviewofnuclearproperties..........................1 1.2Nucleardeformations...............................2 1.3Schi˙moments..................................3 1.4ApplicationofBayesianmachinelearningtonuclearstructuremodels....4 1.5Organizationofthisdissertation.........................5 Chapter2Nucleardensityfunctionaltheory .................. 6 2.1Densityfunctionaltheory.............................6 2.1.1Generalformalism............................6 2.1.2Skyrmeenergydensityfunctional(EDF)................8 2.1.3Hartree-Fock-Bogoliubovmethod(HFB)................11 2.1.3.1Hartree-Fockmethod......................11 2.1.3.2BogoliubovtransformationandtheHFBequations.....12 2.1.4Otherconsiderations...........................14 2.1.4.1Constrainedcalculations....................14 2.1.4.2Deformations..........................15 2.1.4.3Blockingcalculationforoddsystems.............17 2.2Schi˙momentandtheatomicelectricdipolemoment.............18 Chapter3Globalsurveyofoctupole-deformedeven-evennuclei ...... 20 3.1Technicaldetailsoftheglobalcalculation....................21 3.1.1Potentialenergysurface(PES)andtheHFBground-state......21 3.1.2PESgridselection............................23 3.1.3Kick-o˙modeandcharacteristicsofthecalculation..........24 3.1.4Computationalaspects..........................27 3.2Globalresults...................................28 3.2.1Comparisonwiththe2012quadrupolesurvey.............28 3.2.2Octupoledeformation 3 .........................30 3.2.3Octupoledeformationenergy E oct ..................31 3.2.4Octupolemultiplicity:JointanalysiswithcovariantEDFs......33 3.2.5Singleparticleorbitalsinoctupole-deformedregions..........33 3.3Localregionsofoctupoleground-statedeformations..............36 3.3.1Actinideregion..............................37 3.3.1.1Radon ( Z =86) ........................37 3.3.1.2Radium ( Z =88) ........................38 vii 3.3.1.3Thorium ( Z =90) .......................41 3.3.1.4Uranium ( Z =92) .......................41 3.3.1.5Plutonium ( Z =94) ......................42 3.3.1.6Veryneutron-richactinidesaround 288 Pu..........43 3.3.2Lanthanideregion.............................44 3.3.2.1Barium ( Z =56) ........................44 3.3.2.2Cerium ( Z =58) ........................45 3.3.2.3Neodymium ( Z =60) .....................46 3.3.2.4Proton-richnucleiaround 112 Ba................46 3.3.2.5Veryneutron-richlanthanidesaround 200 Gd.........47 3.4Summary:Octupole-deformednuclei......................47 Chapter4IntrinsicSchi˙momentcalculations ................ 50 4.1IntrinsicSchi˙momentsinactinidenuclei...................53 4.1.1Ra ( Z =88) ................................53 4.1.2Ac ( Z =89) ................................54 4.2Summary:IntrinsicSchi˙moments.......................55 Chapter5Bayesianmachinelearning ...................... 59 5.1The S 2 n residualmodel..............................59 5.2Bayesianstatisticalmodels............................64 5.2.1Gaussianprocess.............................67 5.2.2Bayesianneuralnetwork.........................69 5.2.3Inputre˝nement.............................70 5.3Resultsofthe S 2 n residualmodel........................72 5.3.1Trainingset:AME2003;testingset:AME2016-AME2003.......72 5.3.2Trainingsets:AME2003-H,AME2016-H,testingset:JYFLTRAP-201775 5.3.3Two-neutrondrip-lineofSn ( Z =50) ..................75 5.4Neutrondrip-lineintheCaregionusingBayesianmodelaveraging......77 5.5Protondrip-lineanalysisandtwo-protonemitters...............83 5.5.1Modi˝edGaussianprocessandBayesianmodelaveraging.......85 5.5.2Two-protonemitters...........................86 5.6Quanti˝edlimitsofthenuclearlandscape...................90 5.7Summary:Bayesianmachinelearning......................94 Chapter6ConclusionsandOutlook ....................... 96 6.1OctupoledeformationsandintrinsicSchi˙moments..............96 6.2Bayesianmachinelearning............................97 6.3Outlook......................................98 APPENDICES ...................................... 100 APPENDIXA.....................................101 APPENDIXB.....................................113 BIBLIOGRAPHY .................................... 116 viii LISTOFTABLES Table4.1:CandidatesforatomicEDMmeasurementwith 86 Z 94 andhalf- life t 1 = 2 1second. D ^ S z E fromevaluating(4.1)hasbeenaveraged overthe˝veSkyrmeEDFs(exceptfor 223 Frand 229 Npwhichareonly calculatedforfourSkyrmeEDFs).Experimentalenergysplitting betweenparitydoublets E p : d : areshownwheredataexists[21]. Thelastcolumnistheaverage D ^ S z E dividedby E p : d : .Theparity doubletin 224 Achasnotbeenfullyestablished,thelistedvalueis onlyforreference. 229 Paisalsolisted,intheeventthatthelowlying paritydoubletoflessthan1keVfromthegroundstateiscon˝rmed.57 Table5.1:Rootmeansquarevaluesof ( Z;N ) , BNN ( Z;N ) ,and GP ( Z;N ) (in MeV)forvariousnuclearmodelswithrespecttothetestingdataset consistingoftheAME2016-AME2003 S 2 n values.ThetrainingAME2003 andAME2003-Hdatasetswereusedtocomputetheemulators BNN ( Z;N ) and GP ( Z;N ) .Thetwonumberslistedunderthemodel'sname inthe˝rstcolumnaretheuncorrected rms modelvalueswithre- specttoAME2003andAME2003-Hdatasets,respectively.Therms residualscorrectedbyastatisticalmodelareshownintheremaining columns.Foreachmodel,GPresults GP rms aregivenintheupper rowandtheBNNresults BNN rms arelistedinthelowerrow.The numbersinparathensesindicatetheimprovementinpercent.The fourstatisticalvariantsarelisted:Stdisthestandardstandardin- put x =( Z;N ) ;Tindicatesresultsinvolvingthenon-lineartrans- formation ~ x i =( d N ( x i ) ;p ( x i )) ;Hisbasedonthereduceddataset AME2003-Hpertainingtoheavynucleiwith Z 20 .(Tabletaken fromRef.[158]).............................74 Table5.2:SimilarasinTable5.1exceptforthermsvaluesof ( Z;N ) , BNN ( Z;N ) , and GP ( Z;N ) (inMeV)forvariousnuclearmodelswithrespectto thetestingdatasetconsistingofthefourJYFLTRAP S 2 n values.The secondcolumnshowstheuncorrectedrmsvalue rms .Foreachmodel, thetrainingdatasetsAME2003-H(thirdcolumn)andAME2016-H (fourthcolumn)wereusedtocompute GP rms (upperrow)and BNN rms (lowerrow)usingtheT+Hvariantofstatisticalcalculations.(Table takenfromRef.[158])..........................76 ix Table5.3:ModelposteriorweightsobtainedinthevariantsBMA( n )andBMA( p ) ofourBMAcalculations.Forcompactness,thefollowingabbrevi- ationsareused:UNEn=UNEDFn(n=0,1,2)andFRDM=FRDM- 2012.(TabletakenfromRef.[108])..................92 TableA.1: E oct (MeV)and 3 (inparentheses)valuescalculatedusing˝ve SkyrmeEDFs:UNEDF0,UNEDF1,UNEDF2,SLy4,andSV-min. See(2.34)and(3.2)forde˝nitionsof 3 and E oct ,respectively. NucleiwithatleastthreeSkyrmeEDFspredictingthemasoctupole- deformedareshown...........................101 TableA.2:Protonquadrupole Q 20 (fm 2 ) andoctupole Q 30 (fm 3 ) moments(in parentheses)foroctupole-deformedeven-evennucleiwithpredicted 3 0 : 01 from˝veSkyrmeEDFs:UNEDF0,UNEDF1,UNEDF2, SLy4,andSV-min.See(2.33)forde˝nitionsof Q 20 , Q 30 .Theproton multipolemomentscloselyresemblechargemultipolemoments,and canbeusedtocomparewithexperimentaldataderivedfromtran- sitionstrengths.Averagevaluesareshownintherightmostcolumn. Allvaluesroundedtointegers.....................106 TableA.3:ReferencetabletoFig.5.10:even- Z elements.Foreachatomicele- mentwitheven- Z shownare:theneutronnumber N 0 ofthelightest isotopeforwhichanexperimentalone-ortwo-protonseparationen- ergyvalueisavailable;theneutronnumber N obs ofthelightestiso- topeobserved;theneutronnumber N drip ofthepredicteddripline isotopeinBMA-I;andtheneutronnumber N FRIB markingthereach ofFRIB.(TabletakenfromRef.[109])................111 TableA.4:ReferencetabletoFig.5.10:odd- Z elements.Foreachatomicele- mentwithodd- Z shownare:theneutronnumber N 0 ofthelightest isotopeforwhichanexperimentalone-ortwo-protonseparationen- ergyvalueisavailable;theneutronnumber N obs ofthelightestiso- topeobserved;theneutronnumber N drip ofthepredicteddripline isotopeinBMA-I;andtheneutronnumber N FRIB markingthereach ofFRIB.(TabletakenfromRef.[109])................112 x LISTOFFIGURES Figure3.1:Potentialenergysurfaceof 224 RacalculatedusingSkyrmeenergy densityfunctionalSLy4.Thewhitecirclemarksthelowestbinding energy,whichservesasthebaselineenergyvalue...........22 Figure3.2:BindingenergyresidualsforSkyrmeEDFsUNEDF0,UNEDF1,UN- EDF2,SV-min,SLy4,andSkPmass-tables.Reddotsarefrom quadrupole-deformedcalculation[98],bluedotsarefromthecurrent octupole-deformedmass-table.Allrmsdeviationsareimprovedin thelatestcalculations.Thelargeshiftinbindingenergyresidualsin theSkPmass-tableisduetothedi˙erentpairingstrengthsusedin thesetwocalculations.Consequently,resultsfromSkPcalculations havenotbeenconsidered........................26 Figure3.3:Totalg.s.quadrupoledeformation 2 ofeven-evennucleiinthe ( Z;N ) planepredictedwiththeSEDFsUNEDF0,UNEDF1,UN- EDF2,SLy4,andSV-minfromthecurrentre˛ection-asymmetriccal- culations.Sphericalshellclosurescanbeeasilyseen.........29 Figure3.4:Totalg.s.octupoledeformations 3 ofeven-evennucleiinthe ( Z;N ) planepredictedwiththeSEDFsUNEDF0,UNEDF1,UNEDF2,SLy4, andSV-min.Themagicnumbersareindicatedbydashedlines.(Fig- uretakenfromRef.[92])........................31 Figure3.5:SimilartoFig.3.4butfortheoctupoledeformationenergy E oct . (FiguretakenfromRef.[92])......................32 Figure3.6:Thelandscapeofg.s.octupoledeformationsineven-evennuclei.Cir- clesandstarsrepresentnucleipredictedtohavenonzerooctupole deformations.Themodelmultiplicity m ( Z;N ) isindicatedbythe legend.Theboundaryofknown(i.e.,experimentallydiscovered)nu- cleiismarkedbythesolidgreenline.Forsimplicity,thisboundary isde˝nedbythelightestandheaviestisotopesdiscoveredforagiven element.Theaveragetwo-nucleondriplinesfromBayesianmachine learningstudies[108,109]aremarkedbydottedlines.Primordial nuclides[110]areindicatedbysquares.(FiguretakenfromRef.[92])34 xi Figure3.7:ComparisonbetweenSEDFsandCEDFspredictions.Dotsmarkthe SEDFspredictionswith m 3 ,squaresshowtheCEDFspredictions with m 2 ,anddiamondsmarktheoverlapregionbetweenSEDFs andCEDFsresults.Thebordersofknownnucleiandtwo-particle driplinesareasinFig.3.6.(FiguretakenfromRef.[92])......34 Figure3.8:Single-particleenergysplitting e betweentheunusual-parityin- trudershell ( `;j ) andthenormal-parityshell ( ` 3 ;j 3) for˝ve nucleirepresentingdi˙erentregionsofoctupoleinstability.Thes.p. canonicalstateswereobtainedfromsphericalHFB/RHBcalculations. Theneutron(proton)splittingsareindicatedbythesolid(dashed) lines.(FiguretakenfromRef.[92])..................35 Figure3.9:Predicted 2 , 3 ,and E oct valuesforeven-evenRn ( Z =86) iso- topes...................................38 Figure3.10:Predicted 2 , 3 ,and E oct valuesforeven-evenRa ( Z =88) iso- topes...................................39 Figure3.11:Proton Q 20 ofRaisotopesfromSEDFscalculations(seeTableA.2for acomprehensivelistofvalues)comparedwithmeasured E 2 intrinsic moments Q 2 for 2 + ! 0 + transitions(blacksquareswitherrorbars) ofRef.3,124].(FiguretakenfromRef.[125])........40 Figure3.12:Proton Q 30 ofRaisotopesfromSEDFscalculations(seeTableA.2for acomprehensivelistofvalues)comparedwithmeasured E 3 intrinsic moments Q 3 for 3 ! 0 + transitions(blacksquareswitherrorbars) ofRef.[111,112,121,124].(FiguretakenfromRef.[125])......40 Figure3.13:Predicted 2 , 3 ,and E oct valuesforeven-evenTh ( Z =90) iso- topes...................................41 Figure3.14:Predicted 2 , 3 ,and E oct valuesforeven-evenU ( Z =92) isotopes.42 Figure3.15:Predicted 2 , 3 ,and E oct valuesforeven-evenPu ( Z =94) iso- topes...................................43 Figure3.16:Predicted 2 , 3 ,and E oct valuesforeven-evenBa ( Z =56) iso- topes...................................44 Figure3.17:Predicted 2 , 3 ,and E oct valuesforeven-evenCe ( Z =58) iso- topes...................................45 Figure3.18:Predicted 2 , 3 ,and E oct valuesforeven-evenNd ( Z =60) iso- topes...................................46 xii Figure4.1:Predicted 2 , 3 ,and D ^ S z E valuesforRa ( Z =88) isotopes.....53 Figure4.2:Predicted 2 , 3 ,and D ^ S z E valuesforAc ( Z =89) isotopes.....54 Figure4.3: D ^ S z E plottedagainst 2 3 foralloddsystemsintheneutron-de˝cient actinidesregion.Twolinear˝tsweremade,onewithoutintercept (dashedline),andonewithintercept(solidline). R 2 scoreforthe linear˝tsarelistedaccordingly.....................58 Figure5.1:Leftpanel:BindingenergyresidualsofSEDFSkP(calculatedw.r.t. experimentalmassesfromtheAME2016massevaluation[99,160]). Resultsfromoctupole-deformed(bluedots)andquadrupole-deformed (reddots)surveycalculationsareshown.Rightpanel:Sameasinleft panelbutfortwo-neutronseparationenergy S 2 n .Notethattherange ontheenergyaxisis [ 10 ; 10] MeVfor S 2 n comparedto [ 20 ; 20] MeV forBE,illustratingasigni˝cantreductionofsystematicuncertainty in S 2 n ...................................60 Figure5.2:Theexperimental S 2 n ( Z;N ) datasetsforeven-evennucleiusedinour study:AME2003[166,167](lightdots,537points),additionaldata thatappearedintheAME2016evaluation[99,160](darkdots,55 points),andJYFLTRAP[168](stars,4points).(Figuretakenfrom Ref.[158])................................62 Figure5.3:Toppanel:Residualsof S 2 n ( Z;N ) forthesixglobalmassmodelswith respecttothetestingdatasetAME2003.Thermsvaluesof ( Z;N ) inMeVaremarkedforAME2003(uppernumber)andAME2003-H (lowernumber).Bottompanel:Sameastoppanelbutfor S 2 n ( Z;N ) residualssmoothedwithGaussianfoldingfunctiontoemphasizelong- rangesystematictrends.(FigureadaptedfromRef.[158]).....63 Figure5.4:PosteriordistributionsoftheGPparameters f ;ˆ Z ;ˆ N g inthe caseoftheDD-PC1model,withtheposteriormeanandstandard deviationlisted.(FiguretakenfromRef.[158])...........69 Figure5.5:Residualsof S 2 n ( Z;N ) forthesixglobalmassmodelswithrespect tothetestingdataset(AME2016-AME2003): ( Z;N ) (dots)andthe GPemulator GP ( Z;N ) (circles).(FiguretakenfromRef.[158])..71 Figure5.6:Extrapolationsof S 2 n fortheeven-evenSnchaincalculatedwithDD- PC1usingstatisticalGP ( T + H ) andBNN ( T + H ) approaches.One- sigmaand1.65-sigmaCIsaremarked.(FiguretakenfromRef.[158]) ......................................77 xiii Figure5.7:One-neutronseparationenergyfor 55 ; 57 Ca(left)andtwo-neutronsep- arationenergyfor 56 Ca(right)calculatedwiththenineglobalmass modelswithstatisticalcorrectionobtainedwithGPtrainedonthe AME2003(GP+2003)andAME2016*datasets.Therecentdata fromRef.[179](RIKEN2018)andtheextrapolatedAME2016val- ues[99,160]aremarked.Theshadedregionsareone-sigmaerror barsfromRef.[179];errorbarsontheoreticalresultsareone-sigma credibleintervalscomputedwithGPextrapolation.(Figuretaken fromRef.[159]).............................80 Figure5.8:Extrapolationsof S 1 n and S 2 n fortheCachaincorrectedwithGPand one-sigmaCIs,combinedforthreerepresentativemodels.Thesolid linesshowtheaveragepredictionwhiletheshadedbandsgiveone- sigmaCIs.Theinsertshowstheposteriorprobabilityofexistencefor theCachain.The p ex =0 : 5 limitismarkedbyadottedline.(Figure takenfromRef.[159]).........................82 Figure5.9:Posteriorprobabilityofexistenceofneutron-richnucleiintheCa regionaveragedoverallGPcorrectedmodels.Top:Uniformmodel averaging.Bottom:AveragingusingposteriorweightsEq.(5.18)con- strainedbytheexistenceof 52 Cl, 53 Ar,and 49 S.Therangeofnuclei withexperimentallyknownmassesismarkedbyayellowline.The redlinemarksthelimitofnucleardomainthathasbeenexperimen- tallyobserved;nucleitotherightoftheredlineawaitdiscovery.The estimateddriplinethatseparatesthe p ex > 0 : 5 and p ex < 0 : 5 regions isindicatedbyablueline.(FiguretakenfromRef.[159])......83 Figure5.10:Probabilityofexistence p ex thatanucleiisboundwithrespectto protondecayforproton-richnucleiwith 16 Z 82 .Calcula- tionsusingBMA-I(top)andBMA-II(bottom)variantsofmodel averagingareshown.Foreachprotonnumber, p ex isshownalong theisotopicchainversustherelativeneutronnumber N 0 ( Z ) N , where N 0 ( Z ) ,listedinTablesA.3andA.4,istheneutronnumber ofthelightestproton-boundisotopeforwhichanexperimentalone- ortwo-protonseparationenergyvalueisavailable.Thedomainof nucleithathavebeenexperimentallyobserved(bothproton-bound andproton-unbound)ismarkedbyopenstars;thosewithinFRIB's experimentalreacharemarkedbydots.(FiguretakenfromRef.[109])87 xiv Figure5.11: Q 2 p valuespredictedinBMA-Iforeven-evenisotopeswith 16 6 Z 6 80 .Thethicksolidlinesmarkthelifetimerange(5.24).Themass numbersofselectedisotopesareshown.Thenucleiwiththeproba- bility p 2 p > 0 : 4 areindicatedbydots.Here,weusedthisvalueof p 2 p ratherthan p 2 p > 0 : 5 becausethecriterion(5.25)ofthe true 2 p emis- sionisslightlymorerestrictivethantheenergycriterionpreviously adoptedinRef.[202].(FiguretakenfromRef.[109])........89 Figure5.12:Thequanti˝edlandscapeofnuclearexistenceobtainedinourBMA calculations.High-resolutionofthis˝gurecanbefoundinRef.[217]. Foreverynucleuswith Z;N > 8 and Z 6 119 theprobabilityofexis- tence p ex (5.27),i.e.,theprobabilitythatthenucleusisboundwith respecttoprotonandneutrondecay,ismarked.Thedomainsofnu- cleiwhichhavebeenexperimentallyobservedandwhoseseparation energieshavebeenmeasured(andusedfortraining)areindicated.To providearealisticestimateofthediscoverypotentialwithmodernra- dioactiveion-beamfacilities,theisotopeswithinFRIB'sexperimental reacharemarked.Themagicnumbersareshownbystraight(white) dashedlines,andtheaveragelineof -stabilityde˝nedasinRef.[216] ismarkedbya(black)dashedline.Inourestimates,weassumedthe experimentallimitforthecon˝rmationofexistenceofanisotopeto be1event/2.5days.(FiguretakenfromRef.[108]).........93 xv Chapter1 Introduction 1.1Overviewofnuclearproperties Theatomicnucleuscametolightin1911withtheformulationofthemodelfromErnest Rutherford[1],followinghisinterpretationoftheGeiger-Marsdenexperiments(morefa- mouslyknownastheRutherfordgoldfoilexperiment)in1909[1].Fromtheseriesof scatteringexperiments,the˝rstnuclearpropertystudiedwasthechargeandsizeofthe nucleus. Withmanyunknownstobediscovered,theatomicnucleussoonbecameatestingground fortherapidlydeveloping˝eldofquantummechanics.Subsequentstudiesrelatedtothe stabilityand˝ssionofthenucleus,calledforanaccuratedescriptionofthenuclearmass,or itsbindingenergy,whichresultedintheliquiddropmodel(LDM)formulatedbyWeizsäcker in1935[2]. Asoneofthe˝rsttheoriestodescribethenuclearmass,theLDMprovidedgoodapproxi- mationoftheexperimentalnuclearmasses,andbecamethepredecessorformanytheoretical approachesusedtoday However,theshortcomingoftheoriginalLDMisitsassumptionthatthenucleusre- semblesthemacroscopicsphericaldroplet.Microscopicdescriptionthattakesintoaccount thevariousquantummechanicale˙ectsofthenucleusisthusrequiredtofurtheradvance nucleartheory.Manymicroscopictheoriesthatweredevelopedovertheyearscanbedivided 1 intothreemaincategories: abinitio [6,7],con˝gurationinteraction(shellmodel)[8,9],and nucleardensityfunctionaltheory(DFT)[10,11]. Inordertostudythenuclearpropertiesofnucleiacrosstheentirenuclearlandscape,the toolofchoiceisDFT,whichreplacesprotonsandneutronswithnucleonicdensities.Theuse ofnucleonicdensitiesenablesustoscaleupthenuclearmany-bodyproblemwithrelatively lowcomputationalcost,andalsoprovidesanintuitivepathtodescribingandinvestigating theshapeofthenucleus. Studyingvariousglobalnuclearproperties,suchassize,mass,andshape,canhelpus betterunderstandtheboundariesofthenuclearlandscape,˝ssionprocess,nucleosynthesis, andalsogivesusthetoolstoprobethefundamentalsymmetriesofouruniverse. 1.2Nucleardeformations Theatomicnucleuswasinitiallybelievedtobesphericallyshaped.Thiscanbeseeninthe formulationoftheliquiddropmodel(LDM)[2]in1935,whichassumedthenucleustobea droplet-likesphere.Aroundthesametime,˝rstevidenceforanon-sphericalnuclearshape camefromtheobservationofaquadrupolecomponentinthehyper˝nestructureofoptical spectra,whichshowedthattheelectricquadrupolemomentsofthenucleiweremorethan anorderofmagnitudelargerthanthemaximumvaluethatcouldbeattributedtoasingle proton[12,13].Theseobservationssuggestedacollectivedeformationofthenucleus,which isaresultofthenuclearJahn-Tellere˙ect Mostofthequadrupole-deformednucleipreservere˛ectionsymmetryintheirground states.However,inrarecases,thenucleuscanspontaneouslybreakthisintrinsicre˛ection symmetry,andacquirenon-zerooctupolemomentsassociatedwithpear-likeshapes[1 2 (seeRefs.[21,22]forcomprehensivereviews). Duetotheintrinsicre˛ectionasymmetry,thecomputationalrequirementisdramatically increasedforcalculationsinvolvingself-consistentmethods.Consequently,earlycalculations oftheseshapeswereperformedusingmacroscopic-microscopic(MM)modelsbasedonthe shellcorrectionmethod[3,4,23,24].Thosewerefollowedbyself-consistentstudiesusing nuclearDFT,whichmadeuseofdi˙erentfamiliesofenergydensityfunctionals(EDFs), suchasGogny28],BCP[26,29],Skyrme[30,31],andcovariant Thesestudiesmostlyfocusedonthreespeci˝cregions:proton-richactinides,neutron-rich lanthanides,andneutron-richheavyandsuperheavynucleiwhichareimportantformodeling heavy-elementnucleosynthesis,andonlyahandfulofthemwereglobalsurveys[4,25,31,34, 36].Inordertobetterunderstandthesystematictrendofoctupoleinstabilityacrossthe nuclearlandscapeandreducethebiasduetoachoiceofaparticularmodel,itisimportant tocarryoutinter-modelcomparisons.Aglobalsurveyofoctupole-deformedeven-evennuclei usingDFTispresentedinCh.3,andconstitutesanimportantpartofthisdissertation. 1.3Schi˙moments Thesearchforadditionalsourcesofcharge-parity(CP)violationbeyondthestandardmodel isanongoingexcitingtopic,oneofthereasonsbeingthattheCPviolationinthestandard modelistooweaktoaccountforthematterandantimatterasymmetryinouruniverse[37]. IfoneassumestheCPTtheoremtohold,violationoftime-reversal(T)isequivalenttoCP violation.OneofthebestwaystoobserveTviolation(togetherwithparity(P)violation) istomeasureanon-zerostaticelectricdipolemoment(EDMS)ofelectrons,neutrons,or atoms[38]. 3 CurrentatomicexperimenthassetthelowerlimitofatomicEDMusing 199 Hg,at j d 199 Hg j < 3 : 1 10 29 e cm [39].Duetothescreeninge˙ectcausedbytheatomic electrons,thenuclearquantitythatinducestheatomicEDMistheSchi˙moment[40]. IthasbeenknownforsometimethatthesensitivitybetweenthemeasuredEDM andthestrengthofP,TviolationcouplingconstantsisenhancedinnucleiwithlargeSchi˙ moments.Thelargestenhancementsarepredictedtobeinnucleiwhichhavelow-lyingparity doublets,andisacommoncharacteristicofoctupole-deformednuclei. Thus,theglobalsurveyofoctupole-deformedcalculationservesalsoasaprecursorfor ˝ndingnucleiwithpotentiallylargeSchi˙moments,whichcouldpredictbettercandidates intheatomicEDMmeasurements. 1.4ApplicationofBayesianmachinelearningtonuclear structuremodels Machinelearningisoneofthemostrapidlydevelopingcomputerscience/appliedmathe- maticsareasinthepastdecade.Overtheyears,machinelearningtechniqueshaveventured beyondthescopeofcomputerscienceandopenedupnewpathsandbroughtbreakthroughs inmultipledisciplines. Itisthusnaturaltoalsoincorporatethesetechniquesintonuclearphysics.Varioussuch attemptshavebeenmadeinnuclearphysics[4Comparedtoanarti˝cialneuralnetwork whichdeterminesthemodelparametersusuallyviatheminimizationofsomecostfunction, aBayesianneuralnetwork(BNN)adoptsBayesianinferencetechniquesanddeterminesthe probabilitydistributionsforthemodelparameters,whichinturnprovidepredictionsin termsofprobabilitydistributions.NotonlycanaBayesianmachinelearningapproach,such 4 astheBNN,improvethepredictabilityofournuclearphysicsmodels,italsoaddressesthe burningquestionofuncertaintyquanti˝cationsforpredictions. 1.5Organizationofthisdissertation Thisdissertationisorganizedasfollows.FormulationofthenuclearDFTandrelatednu- clearpropertiescanbefoundinCh.2.Theglobalsurveyofoctupole-deformedeven-even nucleiwillbepresentedinCh.3,followedbyCh.4,whichcontainsresultsofSchi˙moment calculationsintheoctupole-deformedactinideregion.Chapter5describesvariousapplica- tionsofBayesianmachinelearningtonuclearbindingenergy.Finally,conclusionsandfuture prospectsarepresentedinCh.6. 5 Chapter2 Nucleardensityfunctionaltheory 2.1Densityfunctionaltheory Thedensityfunctionaltheory(DFT)wasoriginallydevelopedtoinvestigatethestructure oftheelectronicmany-bodysystems,andwaslateradaptedtoinvestigatethestructureof nuclearsystems. InnuclearDFT,protonsandneutronsaredescribedusingnucleonicdensitiesandcur- rents.Theadvantageofdoingsocomparedwithsolvingthemany-bodySchrödingerequation directlyisthatinthelatter,theproblemwillquicklybecomecomputationallyinfeasibleas thenumberofparticlesincreases,whereasthishaslittleimpactintheDFTframework.By replacingtheinteractionsbetweennucleonswithamean-˝eld,theexactstrongnuclearforce doesnothavetobeknown,furthersimplifyingtheproblem.Forthesereasons,thenuclear DFTiscapableofdescribingallnuclearsystemsinthenuclearlandscape,andbecomeses- peciallyusefulinheavysystemswhereothertheoriescannotbeused.ThismakesDFTthe preferredtheoreticaltooltoforlarge-scalesurveysofnuclearproperties. 2.1.1Generalformalism DFTis˝rmlyrootedintheHohenberg-Kohntheorems(H-K)[50].The˝rstH-Ktheorem statesthattheenergyofan N -bodysystemcanbeuniquelydeterminedbythelocalparticle 6 density ˆ ( r ) ,whichonlydependsonthe 3 N spatialcoordinates,thusgreatlysimplifying themany-bodyproblem.ThesecondH-Ktheoremsaysthatforanynondegeneratesystem ofparticlesputintoalocalexternal˝eld,thereexistsauniversalenergyfunctionalofthe particledensity E [ ˆ ] ,whichisminimizedatthecorrectgroundstatedensity ˆ g : s : ( r ) .This minimizedenergywillcorrespondtothegroundstatetotalenergyofthesystem E min [ ˆ g : s : ]= E g : s : . TosolvethenucleargroundstateusingtheH-Ktheorems,weareleftwithtwoproblems totackle.First,weneedto˝ndthecorrectenergyfunctional.Sincewedonotknow theexactnuclearinteractions,we'llhavetoresorttocertainapproximationsmethods.In practice,anenergydensityfunctional(EDF) H ( r ) isconstructed,whichisareal,scalar, isoscalarfunctionoflocaldensitiesandtheirderivatives.Integrating H ( r ) overspacegives thetotalenergyofthenucleus: E [ ˆ ]= Z H ( r ) d 3 r: (2.1) ExamplesofnuclearEDFsaretheSkyrmeEDFs(Sec.2.1.2),GognyEDFsand covariantEDFswhichare,respectively,basedonthezero-rangeSkyrmeinterac- tion[58,59],the˝niterangeGognyinteraction[10,60],andthemesonexchangeforces[61]. Inmywork,SkyrmeEDFsareused,duetothesimplicitieso˙eredbythezero-rangeinter- action,andgoodpredictivepowerwhenitcomestoexperiment. Thesecondproblemistosolveforthegroundstateparticledensity.Intheelectronic system,thisissolvedbymeansofvariationalmethodprescribedintheKohn-Shamtheorem (KS)[62].Inanuclearsystem,sincethenucleusisaself-boundsystem,thisexternal˝eld presentintheelectronicKSformulationisabsentandcreatesconceptualproblems.The 7 correctivetreatmentsarecomplicatedinpracticalimplementation,thusinpractice,nuclear DFTreliesonmean-˝eldvariationalmethodssuchastheHartree-FockandHartree-Fock- Bogoliubovmethodtosolveforthegroundstatedensityofthenucleus.Thesemethodswill bediscussedinSec.2.1.3.1and2.1.3.2. 2.1.2Skyrmeenergydensityfunctional(EDF) WithintheHFBframework,thetotalenergydensity H ( r ) inEq.(2.1),usingthezero-range SkyrmeEDF,canbeexpressedas: H ( r )= H kin + H int + H Coul + H pair H corr ; (2.2) wherethetermsontherighthandsidecorrespond,respectively,tothekineticenergydensity, interactionenergydensity,Coulombenergydensity,pairingenergydensity,andcorrection forthespuriouscenterofmassmotion[10]. Thekineticenergydensitycanbewrittenas: H kin ( r )= ~ 2 2 m 1 1 A ˝ p ( r )+ ˝ n ( r ) ; (2.3) wherethekineticdensities ˝ ( r ) canbeexpressedwiththenonlocaldensities ˆ r ; r 0 = D a y r 0 a r E as: ˝ ( r )= r r r r 0 ˆ r ; r 0 r 0 = r ; (2.4) Thesymbol A representsthemassnumber(totalnumberofnucleons)and 1 1 A isthe centerofmasscorrection.Thesubscripts q = n;p correspondtoneutrons( n )andprotons ( p ),andwillbeusedintheremainderofthisdissertation. 8 TheinteractionenergydensityoftheSkyrmeEDFisbasedonthezero-rangeSkyrme force[58,59],andcanbewrittenasasumofthetime-evenandtime-oddterms: H int ( r )= X t =0 ; 1 E even t + E odd t ; (2.5) E even t ( r )= C ˆ t ˆ 2 t + C ˆ t ˆ t ˆ t + C ˝ t ˆ t ˝ t + C J T J 2 T + C r J t ˆ t r J t ; (2.6) E odd t ( r )= C s t s 2 t + C s t s t s t + C T t s t T t + C j t j 2 t + C r j t s t ( r j t ) ; (2.7) where ˆ t , ˝ t aretheparticleandkineticenergydensity,respectively; J t isthespin-current tensorenergydensitiesand J t thespin-orbitcurrentdensity; s t and T t arethespindensity andspinkineticdensity; j t isthemomentumdensity.Thesymbols t =0 and t =1 correspondstotheisoscalarorisovectordensities,respectively, ˆ 0 = ˆ n + ˆ p and ˆ 1 = ˆ n ˆ p . AdditionaldiscussionofthesedensitiescanbefoundinRef.[10]. Inmanycases,oneisinterestedinthegroundstatesofeven-evennuclei,wheretime reversalsymmetryisconservedandthetime-oddterms(2.7)vanish.Mostofthetime-even couplingconstants C arerealnumbers,exceptfor C ˆ t ,whichisadensity-dependentfunction: C ˆ t = C ˆ t 0 + C ˆ tD ˆ : (2.8) TheCoulombenergydensityfromtheprotonscanbedividedintothedirecttermand theexchangeterm,whichisaresultoftheanti-symmetrization: 9 H Coul = H dir Coul + H exc Coul ; (2.9) H dir Coul ( r )= e 2 2 Z d r 0 ˆ p ( r ) ˆ p r 0 r r 0 ; (2.10) H exc Coul ( r ) ˇ 3 e 2 4 3 ˇ 1 = 3 ˆ 4 = 3 p ( r ) : (2.11) TheaboveformofCoulombexchangetermiscalculatedintheSlaterapproximation[58,63] toavoidinvolvingnon-localdensities. Therefore,thestandardSkyrmeinteractionenergydensity H int ( r ) canbedescribedby 13parameters: n C ˆ t 0 ;C ˆ tD ;C ˆ t ;C ˝ t ;C J t ;C r J t o t =0 ; 1 and : (2.12) Furthermore,someoftheparametersin(2.12)canberepresentedintermsofthenuclear matter(NM)properties:thetotalenergypernucleon E=A anddensity ˆ c atequilibrium; theisoscalarandisovectore˙ectivemasses M s and M v ,respectively;thenuclearmatter incompressibility K ;thesymmetryenergycoe˚cient a sym ;andthedensitydependenceof thesymmetryenergy L sym .(2.12)canthenberepresentedbythefollowing,morecommonly usedparameters: n ˆ c ;E NM =A;M s ;M v ;K NM ;a NM sym ;L NM sym o ; n C ˆ t ;C J t ;C r J t o t =0 ; 1 : (2.13) Thepairingenergydensity H pair istoaccountforthesuper˛uidcorrelationthatinvolves nucleonsthatoccupyorbitalswiththesamequantumnumbersbutoppositespin;such nucleonstendtocoupleintoCooperpairs.Thepairingenergydensityisusuallywrittenas: 10 H pair = X q = n;p V q 0 2 1 ˆ 0 ( r ) 2 ˆ c ~ ˆ 2 q ( r ) ; (2.14) where V q 0 isthepairingstrength,andcanhavedi˙erentvaluesforprotonsandneutrons. ˆ c ˇ 0 : 16fm 3 isthesaturationdensity. ˆ 0 ( r ) and ~ ˆ q ( r ) aretheisoscalarparticledensity andthepairingdensity,respectively. 2.1.3Hartree-Fock-Bogoliubovmethod(HFB) 2.1.3.1Hartree-Fockmethod Thegroundstatedensityofanuclearsystemisobtainedbyusingthevariationalmethods suchastheHartree-Fock(HF)andHartree-Fock-Bogoliubovmethod(HFB). TheHFequationisobtainedbyvaringtheenergyfunctional E [ ˆ ] (2.1)withrespectto theparticledensity ˆ : E =0 ; (2.15) with E = h j E [ ˆ ] j i h j i : (2.16) TheHFgroundstatewavefunction j i isrepresentedbyaSlaterdeterminant[64]ofsingle particle(KS)states j i = A Y i =1 ^ a y i j 0 i ; (2.17) wheretheHFcreationandannihilationoperators ^ a y k , ^ a k actontheHFvacuum,andrep- resentthesingleparticlewavefunctions ˚ k .Thesingleparticlewavefunctionsarethe 11 eigenfunctionsofthesingleparticle(KS)Hamiltonian ^ h : ^ h ( i ) ˚ k ( i )= k ( i ) ˚ k ( i ) ;i = f r i ;s i ;q i g ; (2.18) whichsumsuptobethetotalHamiltonianoftheHFsystem ^ H HF = A P i =1 ^ h ( i ) .Inpractice,we usuallysolvetheHFequationsinthecon˝gurationspaceonsomecompleteandorthogonal setofbasisfunctions f ˜ l g : ˚ k = X l D lk ˜ l : (2.19) Foreach ˜ l ,wecande˝netheircorrespondingfermionoperators ^ c y l ; ^ c l ,andexpressthesingle particleoperators ^ a y k , ^ a k as: ^ a y k = X l D lk ^ c y l ; ^ a k = X l D lk ^ c l : (2.20) 2.1.3.2BogoliubovtransformationandtheHFBequations TheHFwavefunctiondoesnottakeintoaccounttheshort-rangeparticle-particlecorrela- tions.Inpractice,theHFmethodisoftenusedinparallelwiththeBCSpairingmodel[65], whichisananalogoftheBCStheoryforsuperconductivity[66].TheHFBmethodcombines andgeneralizesboththeHFandBCStheory.TheHFBgroundstatewavefunctioncanbe writtenas: j i = Y k ^ y k j 0 i ; (2.21) where ^ y k ; ^ k arethequasi-particle(q.p.)operatorswhichobeythefermionanti-commutation 12 relation,andrelatestotheoperators ^ c y l ; ^ c l in(2.20)viatheBogoliubovtransformation: ^ k y = X l U lk ^ c y l + V lk ^ c l ; ^ k = X l U lk ^ c l + V lk ^ c y l : (2.22) Wecande˝neaunitarytransformationmatrix W y , W y 0 B @ U y V y V T U T 1 C A ; (2.23) andrewritethetransformation(2.22)as 0 B @ ^ ^ y 1 C A = W y 0 B @ ^ c ^ c y 1 C A : (2.24) Usingthematrices U , V ,theparticledensity ˆ ll 0 = h j ^ c y l 0 ^ c l j i andtheHFBpairingtensor = h j ^ c l 0 ^ c l j i canbewritteninthematrixformas: ˆ = V V T ; = V U T : (2.25) Thepairingtensor issimplyrelatedtothepairingdensity ~ ˆ (2.14)[67].Thedensities ˆ and ~ ˆ (or )uniquelyde˝netheenergyofthesystemaccordingtotheH-Ktheorems[68]. Variationofthetotalenergy2.1withrespectto ˆ and ~ ˆ yieldstheHFBequations[69]: 0 B @ h h + 1 C A 0 B @ U k V k 1 C A = E k 0 B @ U k V k 1 C A ; (2.26) 13 with h ij = t ij + X kl v iljk ˆ kl ; ij = 1 2 X kl v ijkl kl ; (2.27) where h istheself-consistentHF˝eld, istheself-consistentpairing˝eld,and isthechem- icalpotential.TheHFBequationeigenvalues E k representq.p.energies,and ( U k ;V k ) T are theHFBeigenvectorsoftheHFBgroundstate.Duetothedensitydependenceofthe mean-˝elds,theHFBequationsarenon-linearandthusneedtobesolvedusingaself- consistentapproach,suchasiterativediagonalization.VariousHFBsolversexists,e.g.HF- BTHO(v3.00)[70]andHFODD(v2.73y)[71].TheformeristhemainsolverthatIusedfor calculationsinthisdissertation,andthelattercontainsmorefunctionalitiesandtheability tobreakallself-consistentsymmetries. 2.1.4Otherconsiderations 2.1.4.1Constrainedcalculations TheBogoliubovtransformation(2.22)doesnotconserveparticlenumber,thusweneed tocontrolthecorrectparticlenumberbyintroducingaconstraint.Acommonmethodto introduceconstraintsistousetheLagrangianmultipliermethod.Themodi˝edtotalenergy, orRouthian, E 0 canbewrittenas: E 0 = E n D ^ N n E p D ^ N p E ; (2.28) where ^ N n , ^ N p aretheneutronandprotonparticlenumberoperators, n and p arethe correspondingFermienergies,whicharedeterminedoncetheconditions D ^ N q E = N q aremet. 14 Othermoreadvancedrestorationschemes,suchastheLipkin-Nogamimethod[69,are alsocommonlyused. ConstraintsonthedeformationofthenucleuscanalsobeintroducedviaLagrangian multipliers C : E 0 = E X C ^ Q E Q 2 ; (2.29) wherethe ^ Q arethemassmultipolemomentoperatorsand Q thedesiredvaluesofthe multipolemoments.Anothercommonlyusedmethodforconstrainingthedeformationsis thelinearconstraintmethod[76], E 0 = E X C ^ Q E Q ; (2.30) wheretheLagrangeparameters C arereadjustedateveryiterationfollowingtheproce- duresinRef.[76].Thismethodforthemultipolemomentconstraintisimplementedin recentversionsoftheHFBsolversHFBTHO[70,77]andHFODD[71,78]. 2.1.4.2Deformations Onewaytodescribethedeformationofthenucleusistousethelengthoftheradiusvector atagivenpointonthenucleus'surface.Itcanbeexpandedusingtheorthonormalized sphericalharmonics[68]: r ( ;˚ )= R 0 0 @ 1+ 00 + 1 X =1 X = Y ( ;˚ ) 1 A : (2.31) In(2.31) R 0 istheradiusofaspherethathasthesamevolumeasthenucleus.Theconstant 00 servesto˝xthevolumeofthenucleusto V = 4 3 ˇR 3 0 .Inthesmalldeformationlimit, 15 the =1 termsmostlydescribethetranslationofthenucleusasawhole,andareusually ˝xedbyconstrainingthecenterofmassofthenucleustotheorigin: Z V r d 3 r =0 : (2.32) Thisconditionisautomaticallysatis˝edifthesystemisre˛ectionsymmetric,inwhichcase theexpansion,equation2.31,onlycontainseven-valued terms.Insystemswithnon-zero odd-value > 1 terms,e.g.inre˛ectionasymmetricnuclei,oneneedstoconstrainthe corresponding Q 1 (2.29)tozeroinordertocorrectlydescribethenucleardeformations. Inanaxiallysymmetricsystem,all 6 =0 termsof becomezero,andtheremaining non-zero 0 areusuallycalled .The =2 ; 3 ; 4 ;::: termsdescribethequadrupole, octupole,andhexadecapoledeformationsetc.,respectively. Anotherwaytodescribethedeformations,whichrelatetothe values,arethemass multipolemoments Q 0 .Unlikethedimensionless values,themultipolemomentsare subjecttoanarbitraryconstantfactor.Acommonde˝nitionforthequadrupole( =2 ) andoctupole( =3 )momentsis: Q 20 = D 2 z 2 x 2 y 2 E ; Q 30 = D z 2 z 2 3 x 2 3 y 2 ; (2.33) wheretheyrelatetothedeformationparameters 2 and 3 via: 2 = Q 20 = r 16 ˇ 5 3 4 ˇ AR 2 0 ! ; 3 = Q 30 = r 16 ˇ 7 3 4 ˇ AR 3 0 ! : (2.34) 16 For R 0 oneusuallyadoptsthesemi-empiricalexpression R 0 =1 : 2 A 1 = 3 fm . Inpractice,oneneedstobewaryofthepotentiallydi˙erentde˝nitionsofthemultipole momentswhencomparingresults.Asomewhatsaferoptionistocompare parameters. 2.1.4.3Blockingcalculationforoddsystems Inaneven-oddorodd-oddnucleus,theunpairednucleonscarrynon-zeroangularmomen- tum.Thenon-zeroangularmomentumbreakstime-reversalsymmetry,whichresultsinthe presenceoftime-odd˝elds(2.7).IntheHFBframework,theso-calledparticleblockingis required,makingcalculationsinoddsystemsmorecomputationallyinvolvedcomparedto theireven-evenneighbors. InHFB,anodd- A nucleuscanbeviewedasaoneq.p.excitation ^ y 0 withrespect tothegroundstateofitseven-evenneighbor.Forinstance,inthecaseofaeven-proton, odd-neutronnucleus ( Z;N ) ,itsgroundstatewavefunctioncanbeexpressedas: j Z;N ) i = ^ y 0 j Z;N 1) i ; (2.35) where 0 representsthequantumnumbersoftheblockedstate[79].Afullblockingmethod treatsthetime-reversedcomponent 0 oftheq.p.state 0 properlywhenmodifyingthe densitymatrixandpairingtensor(2.25).Forthedetaileddescriptionofblocking,see Refs. Anapproximateapproachtoblockingistheso-calledequal˝llingapproximation(EFA)[82, 83].EFAtreatstheKramers-degeneratestates 0 and 0 withequalweightswhentheyenter thedensitymatrixandpairingtensor(2.25),thusconservingtime-reversalsymmetry.This allowsoddsystemstobecalculatedintheabsenceofthetime-odd˝elds(2.7). 17 IthasbeenshownthatEFAispracticallyequivalenttotheexactblockingwhentime- odd˝eldsaresettozero,andevenwhentheyareswitchedon,theimpacttotheground stateenergyisrathersmall[77,81].ThusEFAcanbeusedtoexploreoddsystems,without needingtodealwiththetime-oddterms(2.7). 2.2Schi˙momentandtheatomicelectricdipolemoment TheSchi˙moment[40]servesasabridgebetweentheatomicelectricdipolemoment(EDM) andtheparity( P )andtime-reversal( T )symmetryviolatingnucleon-nucleon( ˇNN )inter- actionmediatedbythepion.SeveralestimatesoftheatomicEDMexpressedintermsof theSchi˙momenthavebeengiven,e.g. 129 Xeodd-massisotopesofRn,Fr,Ra,Ac, andPa[87],and 239 Pu. Becauseofatomicelectrons'screening[40],thenuclearquantitythatinducesthethe atomicEDMisnotthenucleardipolemomentbutrathertheSchi˙moment(tothe˝rst order[88]), S h 0 j ^ S z j 0 i = X i 6 =0 h 0 j ^ S z j i ih i j ^ V PT j 0 i E 0 E i +c : c :; (2.36) where j 0 i isthememberofthegroundstatemultipletwiththemaximum z -axisprojection oftheangularmomentum J ,andthesumrunsoverallexcitedstates j i i . ^ S z isthethird componentoftheSchi˙operator,writtenas: ^ S z = e 10 X p r 2 p 5 3 r 2 ch z p : (2.37) TheenergydenominatorinEq.(2.36)impliesthatthelowestexcitedstatesdominate 18 thesumin(2.36).Thismakesoctupole-deformednucleiofparticularinterest.Anoctupole- deformednucleuswithodd-numberedneutronsor/andprotons,iscommonlyaccompanied bytheexistenceofaparitydoublet,whichisanear-degeneratestatewiththesameangular momentumasthegroundstatebutwithreversedparity[21,22,89].Forinstance,in 225 Ra theenergysplittingbetweenthegroundstatewith J ˇ =1 = 2 + anditsoppositeparitypartner with J ˇ =1 = 2 is E ˇ 55 keV[90].Intherigid-deformationlimitofDFT,theground state j 0 i = J + anditsoppositeparitypartner 0 = J areprojectionsontogood parityandangularmomentumofthesameintrinsicstate,andtheSchi˙moment(2.36)can beapproximatedas[91]: S ˇ 2 h 0 j ^ S z 0 0 ^ V PT j 0 i E = 2 J J +1 D ^ S z ED ^ V PT E E ; (2.38) where D ^ S z E and D ^ V PT E aretheexpectationvaluesof S z and V PT intheintrinsicframeof thenucleus. TheworkthatIhavedoneinthisdissertationfocusonevaluatingtheintrinsicSchi˙ moment D ^ S z E expression(2.37)intheframeworkofHFB.Assumingaxiallysymmetric (around z -axis)deformations,itcanbeexpressedintheintegralform: D ^ S z E e 10 Z ˆ p r 2 p z p d 3 r: (2.39) Thisexpression(2.39)willbeevaluatedforodd- A andodd-oddsystemsintheactinide region(Ch.4)thathaveonenucleonmorethantheireven-evenneighborspredictedtobe octuple-deformedinCh.3. 19 Chapter3 Globalsurveyofoctupole-deformed even-evennuclei Inthischapter,I'lldiscusstheprocedureand˝nalresultsoftheglobalsurveyofoctupole- deformedeven-evennuclei.PartoftheresultshavebeenpublishedonarXiv[92]andsub- mittedtoPhysicalReviewC. Calculationsfortheoctupole-deformedeven-evennucleiareperformedusingamodi˝ed version(Sec.3.1.4)oftheHFBTHO(v3.00)solver.Particlenumbersymmetryisrestored usingtheLipkin-Nogamiprescription,andthelinearconstraintmethodisusedforthe deformationconstraints(Sec.2.1.4.1). Weselected˝veSkyrmeenergydensityfunctionals(SEDFs,seeSec.2.1.2):UNEDF0[93], UNEDF1[94],UNEDF2[95],SLy4[96],andSV-min[97]toperformthecalculationof octupole-deformedeven-evennuclei.Theselectionoftheseinteractionscomesnaturallyas thisworkisanextensiontothequadrupolemass-tablecalculatedin2012thatusedtheabove SEDFs[98].Theroot-mean-square(rms)errorofBE,whencomparingtotheexperimental massesinthemassevaluationAME2016[99],rangesfrom1.7MeV(UNEDF0)to5.3MeV (SLy4). Re˛ection-asymmetriccalculationswereperformedinalleven-evennucleiwith Z 120 , startingwithneutronnumbersaround(lessthan)theprotondrip-lineandafewbeyond 20 theneutrondrip-line,toensureallboundsystemsareincluded,uptoneutronnumber N =300 .Atotalof2836nucleiwerecomputedforeachoftheSEDFs,resultingin˝ve octupole-deformedmass-tables. Intheresultsreportedhere,weremovedallsystemswith Z 112 ,duetothefact thatCoulombfrustrationatthisregioncouldintroduceexoticshapessuchasbubblesand toriandmultipolemomentsconstraintsareinsu˚cientindescribingthesenuclear shapes.Theresultsinthisregiondisplayfrequentcrossingsamongtwo-neutronseparation energylinesbetweendi˙erentisotopicchains,andirregularjumpsinthedeformationswithin agivenisotopicchain,whichindicatesinstabilityoftheresults. 3.1Technicaldetailsoftheglobalcalculation 3.1.1Potentialenergysurface(PES)andtheHFBground-state WithinthenuclearDFTframework,acommonstrategyto˝ndtheshapeofanucleusisto performasetofconstrainedcalculationstocreatethepotentialenergysurface(PES). ByemployingtheLagrangemultipliermethod,wecanconstrainthedeformationof thenucleusintermsofitsmultipolemoments,i.e.quadrupolemoment Q 20 andoctupole moment Q 30 etc.(Sec.2.1.4.2),to˝ndtheenergyofthenucleusatthisparticularshape. Wecanthencreateatwo-dimensionalgrid,whereeachmeshpointwouldcorrespondtoa valueof ( Q 20 ;Q 30 ) ,andperformconstrainedcalculationsfortheentiresurface.Afterthese calculationsconverge,each ( Q 20 ;Q 30 ) pairwillhaveacorrespondingenergy,inourcase wheretheHartree-Fock-Bogoliubovmethodisused,theHFBenergy.Theglobalminimum oftheseenergieswillbechosenasthenucleus'ground-state(g.s.)bindingenergy(BE)in thePES,andthusthecorresponding ( Q 20 ;Q 30 ) areitsg.s.deformations. 21 Figure3.1isanexampleofaPESforthe 224 RanucleususingtheSLy4SkyrmeEDF. Thewhitecircle, ( Q PESg : s : 20 ;Q PESg : s : 30 )=(1711fm 2 ; 3312fm 3 ) ,representsthepositionof theg.s.deformations,andits BE PESg : s : (-1710.705MeV)servesasthezero-energyofthe colorcontour. Figure3.1:Potentialenergysurfaceof 224 RacalculatedusingSkyrmeenergydensity functionalSLy4.Thewhitecirclemarksthelowestbindingenergy,whichservesasthe baselineenergyvalue. Sincethesizeofthegridis˝nite,valuesareinterpolatedinbetweencalculatedmesh points.Inaddition,the˝nitegridimpliesthatthetrueg.s.ofthesystemisbelieved tobeinthevicinityof ( Q PESg : s : 20 ;Q PESg : s : 30 ) .Thus,oneneedstoperformanadditional unconstrainedcalculation,byremovingtheLagrangianmultipliertermsusedtoconstrain themultipolemoments,andrestarttheHFBcalculationfromtheconvergedg.s.many- bodywavefunctions.SincetheHFBequationisrootedinthevariationalprinciple,we areguaranteedto˝ndacon˝gurationwithalowerenergycomparedto BE PESg : s : .This 22 resultingcon˝gurationfromtheunconstrainedcalculationisthenthetrueHFBg.s.with deformation ( Q HFBg : s : 20 ;Q HFBg : s : 30 ) . 3.1.2PESgridselection Ideally,onewouldchooseaPESgridasdenseandaslargeastheirtimeandbudgetallows. Thechoicefordensityofthegridsisarbitraryandsubjecttobenchmarks.ThePESfor anucleusinaHFBmass-tablecalculationusuallyconsistsaround150meshpoints,as thishasbeenprovene˙ectivein˝ndingtheHFBg.s.withbenchmarksusingthetwo-step constrained + unconstrainedmethodmentionedabove. Verylarge Q 20 and Q 30 valuescorrespondtoextremeandunstableshapesina˝ssion process,whereonewouldnotexpectag.s.tobelocated.Thislimitstherange,orarea, ofthePESwehavetocalculate.Althoughthedeformationconstraintsareimposedonthe expectationvaluesofmultipolemomentsusingLagrangianmultipliers,thesemomentsscale withthesizeofthenucleus.Fora˝xedshape,wehave: Q 20 / R 2 0 / A 2 = 3 ; Q 30 / R 3 0 / A: (3.1) Thus,whensettingupthePESforaglobalmass-tablecalculationthatincludesmasses rangingfrom4to420,therangeof Q 20 and Q 30 di˙ersgreatly,andinstead,oneshould de˝nethePESgridsacrossallmassrangesusingthedeformationparameters 2 and 3 , suchthatforaxiallysymmetricnucleiwithidentical values,theirshapesareidentical. Inourlatestoctupolemass-tablecalculations,therangeof 2 and 3 is [ 0 : 35 ; 0 : 35] and [0 ; 0 : 4] ,respectively,andthestep-size0.05and0.1,respectively,creatingaPEScontaining 23 15 5=75( 20 ; 30 ) meshpointsforeachnucleus.IntheactualHFBcalculations,the valuesareconvertedtothemoments Q 0 usingEq.(2.34). 3.1.3Kick-o˙modeandcharacteristicsofthecalculation Theconventionaltwo-stepprocedureof˝ndingthenuclearg.s.iscumbersome,inthesense thatateverypointonthePESwe˝rstneedtogeneratea˝le,whichstoresinfor- mationofthecurrentnuclearcon˝gurationinordertostarttheunconstrainedcalculations. Tosavestorage,onecouldperformaPESwithoutcreatingallthe˝les,identify the ( Q PESg : s : 20 ;Q PESg : s : 30 ) ,thenredotheconstrainedcalculationforthisdeformationwhile allowingittogeneratea˝le. Animprovedsolutionistocombinethetwostepsofconstrainedandunconstrained calculations,anduseaso-calledkick-o˙mode.ThiswasintroducedintheHFBsolver HFBTHO(v2.00d)[77]andsubsequentlyimprovedinthelatestHFBTHO(v3.00)[70],where aderivedversionfromthelatterwasusedtoperformouroctupolemass-tablecalculation. Theideaofkick-o˙modeistostarttheHFBcalculationwithmomentconstraints, continuethisforthe˝rst N kick iterationsoruntiltheconvergencecriteria " kick ismet, thenreleasetheconstraintsandallowtheHFBcalculationtosmoothlytransitintoan unconstrainedcalculation.Byexperimentingwiththe N kick and " kick parameters,we wereabletodeterminethatwith N kick =20 and " kick =1 : 0 thenumberofPESpoints neededto˝ndtheg.sisreduced.ThiswastestedinallboundO,ThandFmisotopes. Onecanthinkofthispre-releasingoftheconstraintsasgrantingthecalculationasortof hwherethecalculationisallowedtoiterateinthe˝rst20stepsto˝ndroughly thedensitycon˝gurationnearthegivenconstraints,thenreleasetheconstraintsandletthe variationalprincipletakeustotheminimum.Usingtoosmalla N kick valuedoesnot 24 thecalculationlongenoughtogetcloseenoughtotheinitialmultipolemoments, andusingtoolargea N kick value,intheextremeofin˝nity,isequivalenttothecombined two-stepprocedure,anddoesnotreducethenumberofPESpoints. Thee˙ectivenessofkick-o˙modehasbeenthroughlytestedandbenchmarkedpriorto theglobalcalculations.Theadoptedmeshis i 2 2 [ 0 : 35 ; 0 : 35] , i +1 2 i 2 =0 : 05 and i 3 2 [0 ; 0 : 4] ; i +1 3 i 3 =0 : 1 .Thisselectionreproducesg.s.BE(within10eV)ofallbound nucleiinO,Ca,Gd,Dy,Th,andFmisotopicchainswhencomparedwithPEScalculations thatusedensepopulatedgrids,wherethekick-o˙calculationsareabletoconvergewithin 200iterationsinmostcases.Wealsoperformedthesekick-o˙calculationsbydoublingthe numberofmeshpointsin 2 and 3 ,andfoundnosigni˝cantdecreaseinthecomputedg.s. energies. Althoughthekick-o˙modeisabletoreducethenumberofpointsbyatleast50%,the disadvantageofkick-o˙modeisthatwecannolongercreateaPEScontourplotsuchas Fig.3.1,sincetheconvergedcalculationshave ( Q 20 ;Q 30 ) thatarerandomlyspacedand sometimesclustered.However,ifoneisonlyinterestedintheglobalnuclearg.s.properties, kick-o˙modecanprovideuswithamoree˚cientsolution,whileprovidingequallyhigh qualitypredictions. Followingthecompletionoftheoctupolemass-table,weselectednucleithathavezero octupoledeformationandthenperformedre˛ection-symmetriccalculations.Theaverage g.s.BEdi˙erencebetweenthesebenchmarksandthere˛ectionasymmetricoctupolemass- tablesis4eV,withthelargestdi˙erenceat26eV,whichdemonstratesthatHFBTHO(v3.00) solverhandlesthebreakingofre˛ectionsymmetrywithhighprecision. Theg.s.ofeachnucleusisselectedtobetheoutputthathasthelowestbindingenergy amongresultsfromthe75inputs.Inpractice,notallinputswillconverge,aswelimitedthe 25 maximumnumberofHFBiterationstobe1000.Inlimitedcaseswherethenuclearg.s.come fromtheHFBcalculationthatrequiresmorethan1000iterations,theadditionaliterations decreasetheg.s.BEontheorderof10eV,whichdoesnotjustifythecostofincreasingthe iterationlimit. Welimitthenumberofharmonicoscillatorshellsinthebasisto20,asincreasingthis numberto30shellsdidnotbringanyimprovements.Outofthe75 ( Q 20 ;Q 30 ) points, thepercentageofconvergedmeshpointsvariesbetweenEDFs:UNEDF0(78%),UNEDF1 (31%),UNEDF2(62%),SV-min(94%),andSLy4(95%).Webelievethispercentagedoes nothaveanyimpactonthequalityoftheg.s.BEpredictions,asonecouldseeinFig.3.2 thatthermsdeviationofBEpredictedbyUNEDF1issimilartoUNEDF2. Figure3.2:BindingenergyresidualsforSkyrmeEDFsUNEDF0,UNEDF1,UNEDF2,SV- min,SLy4,andSkPmass-tables.Reddotsarefromquadrupole-deformedcalculation[98], bluedotsarefromthecurrentoctupole-deformedmass-table.Allrmsdeviationsareim- provedinthelatestcalculations.ThelargeshiftinbindingenergyresidualsintheSkP mass-tableisduetothedi˙erentpairingstrengthsusedinthesetwocalculations.Conse- quently,resultsfromSkPcalculationshavenotbeenconsidered. Inafewcases,thelowestbindingenergywasobtainedatdeformationsthatclearlywent 26 beyondthe˝ssionbarrier,suchas 2 aslargeas1.5.Hence,wediscardedalloutputswith 2 and 3 greaterthan0.5.Allnucleiwithdeformation 2 > 0 : 4 havebeenmanuallyinspected, theoneswith 2 > 0 : 5 arepronetohavesuspiciouslylowminima,creatingsharpkinksin theseparationenergy.Werealizethatthisisbynomeansthemostrigorousapproachto dealwithlargedeformations,asoneshouldlookatthePESineachcasetobecertain,but giventhescaleofcalculationinvolvedwe˝ndthiscompromisereasonable. Finally,asareminder,onealwaysneedtocheckifthedipolemomentofthenucleusis constrainedtozero(Sec.2.1.4.2),whichisessentiallyrequiringthecenterofmassofthe nucleustobethesameastheoriginoftheintrinsicreferenceframe. 3.1.4Computationalaspects The2836nucleiineachmass-table,togetherwiththe75deformationmeshpointsforeach nucleus,resultedinatotalof212,700entriesforeachofthe˝veSEDFs.Eachentryis itselfanindividualHFBcalculation,whichtakesanywherefrom10minutesto4hoursto complete,withanaverageof2hoursfortheUNEDFfamilyofSEDFsand40minutes forSLy4andSV-min,usingtheIntelXeonE5-2680processoratiCER[106].Thisputs usatroughly2millionCPU-hourforthese˝vemass-tablesandtheirrelatedbenchmarks. AlthoughHFBTHO(v3.00)hasOpenMPcapability,noOpenMPwereusedintheseglobal studyaswewouldliketoachieve100%e˚ciency. AdjustmentstotheMPIofHFBTHO(v3.00)wasmade.Theparallelmass-tablemode isnowmodi˝edtoallowforoctupoledeformationconstraintsintheinput˝le.Thedefault MPIusesthestaticschedulingscheme,whereeachMPItaskwillbeassigneda˝xedsetof gridstocalculateatthebeginningofaparallelrun.Thisquicklyprovestobeine˚cientfor globalcalculations,sincethetotaltimeneededforallMPItasksisdeterminedbytheslowest 27 MPItask.Additionally,inalmosteverybatchofcomputation(1000MPItasks),therewill betaskthatfreezes,likelycausedbyhardwareissues,creatingfurthercomplications.We havethusreplacedthisstaticschedulingschemewithdynamicscheduling,whereaMPItask wasdesignatedasamanager.Onceaworkertaskcompletesitscalculationandbecomes available,itwouldsendamessagetoamanagertoacquireanewtask.Inthisdynamic schedulingscheme,theworst-casescenariowasforallMPItaskstowaitforonesingletasks tocomplete.Thisgreatlyreducedtheloadimbalanceinthislarge-scalecomputation. Byimplementingdynamicscheduling,accordingtoconservativeestimates,roughly50% ofcomputationalcostwassaved.TogetherwiththereducedPESsizesrequiredbyusing thekick-o˙mode,anestimated6millionCPU-hoursweresavedattheconclusionofour octupoleproject. 3.2Globalresults 3.2.1Comparisonwiththe2012quadrupolesurvey Figure3.2istheBEresidual(di˙erencewithrespecttoexperimentalmassesfromAME2016[99]) obtainedinthecurrentoctupolemass-tables(bluedots)andthe2012quadrupolemass-tables (reddots)[98].TheoverallrmserrorandtheimprovementsareinunitsofMeV.ForUN- EDF0,UNEDF1,UNEDF2,SLy4,andSV-minaslightimprovementinthermserrorhas beenobtained,likelyduetotheadditionaloctupoledegreeoffreedom;thiscanbeseenin theseparationofblueandreddotsaround N =130 ,whichiswheremostoftheoctupole- deformednucleiarelocated.Theoctupolemass-tableofSkP(lowerright)andSkM*(not displayedhere)wasleftoutofthecurrentstudyduetothedramaticimprovementinthe rmserrorcomparedwiththeir2012mass-table[98].Thisimprovementisclearlysystematic, 28 andnotjustintheoctupole-deformedregion.Webelievethishasbeencausedbydi˙erent pairingparametersusedthanthepreviousmass-table.Unfortunately,sincetheoriginalpair- ingparametersofSkPandSkM*usedforthe2012mass-table[98]couldnotberetrieved, andthecurrentlyusedpairingparametershavenotbeenbenchmarked,wedecidedtoleave thesetwoSEDFsoutofthecurrentstudy.ThepairingstrengthsforSLy4andSV-min wereassumedtobe-258.2MeVand-214.28MeV,respectively,assumingthesamevaluefor neutronsandprotons,consistentwiththe2012mass-table[98]. Figure3.3:Totalg.s.quadrupoledeformation 2 ofeven-evennucleiinthe ( Z;N ) plane predictedwiththeSEDFsUNEDF0,UNEDF1,UNEDF2,SLy4,andSV-minfromthecur- rentre˛ection-asymmetriccalculations.Sphericalshellclosurescanbeeasilyseen. Figure3.3showsthelandscapeofquadrupoledeformationfromtheourcalculations(note: unboundnucleihavenotbeenremovedfromthis˝gure).Thepredictionofquadrupole deformationisidenticaltothe2012survey,whichcanbefoundonMassExplorer[107], adatadistributionplatformfortheoreticalnucleardataofourresearchgroup.Inmost 29 cases,the Q 20 valuesobtainedinthere˛ection-symmetriccalculationsaresimilartothatin ourre˛ection-asymmetriccalculations,whichcouldalsobeseenintheexampleofFig.3.1. However,inrarecases,particularlyneartheendoftherangeofoctupoledeformationinan isotopicchain,thisisnotguaranteed,astheoctupoleminimumbecomesshallower. 3.2.2Octupoledeformation 3 Theg.s.octupoledeformations 3 obtainedinourcalculationsaredisplayedinFig.3.4. Thereisagoodinter-modelconsistency,withlargeoctupoledeformationspredictedaround 146 Ba(neutron-richlanthanides), 200 Gd(veryneutron-richlanthanides), 224 Ra(neutron- de˝cientactinides),and 288 Pu(neutron-richactinides),i.e.,intheregionsofstrongoctupole collectivityde˝nedbythepresenceofclose-lyingprotonandneutronshellswith ` = j = 3 [21].This˝ndingisconsistentwithpreviousglobalstudies[4,25,31,34,36]. Ineachregionofoctupole-deformednuclei,themagnitudeofoctupoledeformationin- creaseswiththenumberofvalencenucleons.All˝veSEDFspredictneutron-de˝cientand neutron-richactinidestoexhibitstrongoctupoledeformations,whilepredictionsinthelan- thanideregionarelessuniformregardingwhichnucleiaredeformedandhowdeformedthey are.Ingeneral,UNEDF2andSLy4predictthelargestnumberofoctupole-deformednuclei andalsolargervaluesof 3 .Inbothmodels,proton-richnucleiaround 112 Baareexpected tobere˛ection-asymmetric.ThefunctionalUNEDF0predictstheleastamountofoctupole- deformednucleiandsmaller 3 deformationsoverall. 30 Figure3.4:Totalg.s.octupoledeformations 3 ofeven-evennucleiinthe ( Z;N ) plane predictedwiththeSEDFsUNEDF0,UNEDF1,UNEDF2,SLy4,andSV-min.Themagic numbersareindicatedbydashedlines.(FiguretakenfromRef.[92]) 3.2.3Octupoledeformationenergy E oct Themagnitudeofoctupoledeformation,i.e. 3 ,aloneisinsu˚cientindeterminingwhether robustoctupoledeformationispresentsinceitdoesnotprovideanyinformationonthe softnessofthepotentialenergysurfaceintheoctupoledirection.Toaddressthispoint,we alsolookatthegaininbindingenergy E oct duetooctupoledeformation: E oct = E a ( 2 ; 3 ) E s 0 2 ; 0 3 =0 ; (3.2) where E a istheabsolutebindingenergyobtainedinre˛ection-asymmetriccalculations,and E s isthebindingenergyminimumfromre˛ection-symmetriccalculations.Thesetwominima donotnecessarilyhavethesamequadrupoledeformation,butasmentionedinSec.3.2.1, 31 theyareexpectedtobeveryclose. Figure3.5:SimilartoFig.3.4butfortheoctupoledeformationenergy E oct .(Figure takenfromRef.[92]) Theoctupoledeformationenergies E oct predictedinourmass-tablecalculationsare showninFig.3.5.Wecanseethatlanthanidenucleihaveappreciablysmaller E oct values ascomparedtotheactinidesinspiteofsimilaroctupoledeformations.Thisindicatesthat mostofthere˛ection-asymmetriclanthanidenucleiarepredictedtohaveverysoftPESsin theoctupoledirection,regardlessoftheequilibriumvalueof 3 . Thevaluesof E oct and 3 ofnucleiwithatleastthreeSEDFspredictingitasoctupole- deformedaredisplayedinTableA.1. 32 3.2.4Octupolemultiplicity:JointanalysiswithcovariantEDFs Inane˙orttoobtainamorerobustpictureofoctupoledeformations,wecombinedtheoc- tupolepredictionsfromour˝veSEDFscalculatedusingtheHFBmethodandfourcovariant energydensityfunctionals(CEDFs)DD-ME2[54],DD-PC1[57],NL3*[55],andPC-PK1[56] usingtherelativisticHartree-Bogoliubovmethod(RHB),inFig.3.6.MostoftheCEDFs resultscanbefoundinRef.[35].Wede˝nethemodelmultiplicity m ( Z;N )= k ifanu- cleus ( Z;N ) ispredictedby k models ( k =1 ;::: 9) tohaveanonzerooctupoledeformation. NucleipredictedbyallnineEDFsasoctupole-deformed(i.e., m =9 )areshownbystars. Theseare: 146 Ba, 224 ; 226 Ra, 226 ; 228 Th,and 228 Puintheregionsexperimentallyaccessible; 288 ; 290 Pu, 288 ; 290 Cm,and 288 ; 290 Cfintheveryneutron-richactinidesregion.Apartfrom theoverallagreementbetweenSEDFsandCEDFswhenitcomestothepredictedregionsof octupole-instability,weseesystematicshifts(by2-4neutrons)betweentheregionsof E oct and 3 obtainedbythesetwoenergydensityfunctionals(EDF)families.Thissystematic e˙ectisillustratedinFig.3.7,wheredotsmarktheSEDFs'predictionswith m 3 ,squares showtheCEDFs'predictionswith m 2 ,anddiamondsmarktheoverlapofthetwo.This shifthasbeennoticedinRef.[35]pertainingtosuperheavynuclei. 3.2.5Singleparticleorbitalsinoctupole-deformedregions Microscopically,octupoledeformationscanbetracedbacktoclose-lyingpairsofsingle- particle(s.p.)shellscoupledbytheoctupole˝eld[21].Eachpairconsistsoftheunusual- parityintrudershell ( `;j ) andthenormal-parityshell ( ` 3 ;j 3) .Consequently,the regionsofnucleiwithstrongoctupolecorrelationscorrespondtoparticlenumbersnear34 ( g 9 = 2 $ p 3 = 2 coupling),56( h 11 = 2 $ d 5 = 2 ),88( i 13 = 2 $ f 7 = 2 ),134( j 15 = 2 $ g 9 = 2 ),and196 33 Figure3.6:Thelandscapeofg.s.octupoledeformationsineven-evennuclei.Circles andstarsrepresentnucleipredictedtohavenonzerooctupoledeformations.Themodel multiplicity m ( Z;N ) isindicatedbythelegend.Theboundaryofknown(i.e.,experimentally discovered)nucleiismarkedbythesolidgreenline.Forsimplicity,thisboundaryisde˝ned bythelightestandheaviestisotopesdiscoveredforagivenelement.Theaveragetwo- nucleondriplinesfromBayesianmachinelearningstudies[108,109]aremarkedbydotted lines.Primordialnuclides[110]areindicatedbysquares.(FiguretakenfromRef.[92]) Figure3.7:ComparisonbetweenSEDFsandCEDFspredictions.DotsmarktheSEDFs predictionswith m 3 ,squaresshowtheCEDFspredictionswith m 2 ,anddiamonds marktheoverlapregionbetweenSEDFsandCEDFsresults.Thebordersofknownnuclei andtwo-particledriplinesareasinFig.3.6.(FiguretakenfromRef.[92]) 34 Figure3.8:Single-particleenergysplitting e betweentheunusual-parityintrudershell ( `;j ) andthenormal-parityshell ( ` 3 ;j 3) for˝venucleirepresentingdi˙erentregionsof octupoleinstability.Thes.p.canonicalstateswereobtainedfromsphericalHFB/RHBcal- culations.Theneutron(proton)splittingsareindicatedbythesolid(dashed)lines.(Figure takenfromRef.[92]) ( k 17 = 2 $ h 11 = 2 ). Figure3.8showstheenergysplitting e = e ( `;j ) e ( ` 3 ;j 3) ; (3.3) betweens.p.canonicalshellsobtainedfromsphericalHFB/RHBcalculations.Ingeneral, 35 thereisasystematicdecreaseof e withmass,whichtogetherwiththeincreaseddegen- eracyofs.p.orbits(andmatrixelementsoftheoctupolecoupling)resultsinenhanced octupolecorrelationsinheavynuclei.However,whilethisgeneraltrendisrobust,themagni- tudeof e isnotagoodindicatorofoctupolecorrelationswhencomparingdi˙erentmodels. Indeed,whencomparingdi˙erentmodelsonealsoneedstoconsiderotherfactorsrelated toeachmodel'sstructure.Forinstance,theisoscalare˙ectivemassofSLy4iscloseto0.7, whiche˙ectivelyincreasesthes.p.splittingascomparedtoUNEDFmodels(whichhave e˙ectivemassclosetoone).Asaresult,althoughinmostcasesSLy4haslarger e than UNEDF1,itpredictsmoreoctupole-deformednucleiandlarger E oct values.Itissaferand moreinstructivetocomparepredictionsoftheUNEDFfamilyofSEDFs,astheirproperties arenotverydi˙erent.Here,theUNEDF2parametrization,constrainedtothespin-orbit splittingsinseveralnuclei,yieldsthelowestvaluesof e forneutronsandpredictsthe strongestoctupolecorrelations,seeFigs.3.5and3.6. 3.3Localregionsofoctupoleground-statedeformations Themajorityofnucleiwithg.s.octupoledeformationarefoundneartheintersectionbetween neutronnumbers88,134,and194andprotonnumbers56and88.Thispatternismore pronouncedinheavynuclei,duetotheirlowervaluesof e ,seeFig.3.8. Iwillbediscussingtheoctupolecollectivityintheactinideregion( Z ˇ 88 )insec- tion3.3.1,andthelanthanideregion( Z ˇ 56 )insection3.3.2,withafocusonthecases robustlypredictedtobeoctupole-deformedinourSEDFcalculations.OurSEDFs'results willbecomparedwithoctupolepredictionsusingothertheoreticalmethods:macroscopic- microscopic(MM)methodusingFRLDMinteraction[22],generatorcoordinatemethod 36 (GCM)usingGognyinteraction[23],Hartree-Fock+BCS(HF+BCS)methodwithSkyrme interactionSkM*[3],andtheRHBmethoddiscussedearlier(Sec.3.2.4).Exceptforthe MMapproach[22]thatmadepredictionsforodd-oddandodd-Anuclei,allothermentioned methodsarecalculatedforeven-evennucleionly. WenotethatallEDFsusedinthisstudyproviderobustandconsistentpredictionsfor quadrupolemoments,whichgenerallyagreewellwithavailableexperimentaldata Thissuggeststhatthequadrupolecollectivityiswelldeveloped.Ontheotherhand,in manynuclei,theoctupoledeformationenergyhasamodestvalueoflessthan500keV.Such smallvaluesof E oct indicatesoftPESsresultingintheoctupolecollectivityoftransitional character,i.e.,betweenoctupolerotationalandvibrationalcollectivemotions[21].While inthisworkwerefertoanucleusasoctupole-deformedwhenithas 3 6 =0 ,thisdoesnot meanthatthisoctupoledeformationisstatic.Foroctupole-soft,transitionalnuclei,beyond mean-˝eldmethodsareneededtodescribethesystem[26,28,29,36, 3.3.1Actinideregion Becauseoflargeoctupolecorrelatione˙ectsandexperimentalaccessibility,neutron-de˝cient actinideshavetraditionallybeeninthespotlightofoctupoledeformationstudies.Asseen inFig.3.6,thisregionisexpectedtobeabundantinoctupole-deformednuclei,withmany systemspredictedrobustlybyseveralmodels,i.e.,havinghighoctupolemultiplicity. 3.3.1.1Radon ( Z =86) Theisotopes 218 ; 220 Rnand 224 ; 226 Rnhavebeenfoundexperimentallytobeclosetotheoc- tupolevibrationallimit[111,AsseeninFig.3.9, j E oct j reachesitsmaximumfor 220 Rn,withanaveragevaluearound0.5MeV.Theseshallowoctupoleminimasuggestthat 37 Figure3.9:Predicted 2 , 3 ,and E oct valuesforeven-evenRn ( Z =86) isotopes. neutron-de˝cientRnisotopesaretransitionalsystems,consistentwithexperiment.Interest- ingly,UNEDF0predictedthelargest j E oct j forRnisotopesamongtheSEDFs.Asweshall seelater,UNEDF0tendstopredictthesmallest j E oct j forotherrobustoctupole-deformed nuclei. MorethanhalfoftheSEDFspredicted 218 224 Rnasoctupole-deformed,aswellas GCM[25]andMM[4],wherethelatterextendeditsoctupolepredictionallthewayto 232 Rn.TheHF+BCSmodel[31]reported 220 ; 222 Rnintheiroctupolelist.Nooctupole deformationispredictedbyCEDFsforRnisotopesinRef.[34]. 3.3.1.2Radium ( Z =88) Thesearchforoctupoleinstabilityinneutron-de˝cientRaisotopeshasbeenofgreatinter- est[22,111,121],alsobecauseofatomicEDMstudies[122,123].Accordingtonumerous theoreticalcalculations, 224 Rahasthelargestoctupoledeformation[22,121],andisoften predictedtohavethelargest E oct amongtheRaisotopes.Itisthereforehardlysurprising that 224 Ra,alongwith 226 Ra,ispredictedtobeoctuple-deformedbyallnineEDFsstudied. 38 Figure3.10:Predicted 2 , 3 ,and E oct valuesforeven-evenRa ( Z =88) isotopes. WithintheSEDFs,thevaluesofPredicted 2 , 3 ,and E oct valuesappeartobevery consistentfor 220 ; 224 Ra,cf.Fig.3.10.Thelargest j E oct j ispredictedfor 222 Ra,followed by 220 Raand 224 Ra.InCEDFspredictions, j E oct j islargestfor 224 Raduetotheshiftin neutronnumbersdiscussedearlier(Sec.3.2.4). Even-even 222 230 RaarecalculatedbyatleasthalfoftheCEDFstobeoctupole- deformed.BothMM[4]andGCM[25]modelsreportedoctupolemomentsineven-even 218 228 Ra,incompleteagreementwiththemajorityoftheSEDFs;HF+BCS[31]reported 222 ; 224 Raasoctupole-deformed. Recentexperimentssuggest 222 RahasthelargestoctupoledeformationamongtheRa isotopesfollowedby 226 Ra, 228 Ra,and 224 Ra[112,121,124].Figs.3.11and3.12showthe proton Q 20 and Q 30 fromourSEDFs'predictionscomparedwithchargemultipolemoments derivedfromexperimentaldata.Thesuddendropintheproton Q 20 of 224 Raispredictedby noneoftheSEDFsandCEDFs,andcurrentlywedonothaveanytheoreticalexplanationfor thise˙ect.Additionalmeasurementsmightbeneededtocon˝rmthisparticularbehavior. 39 Figure3.11:Proton Q 20 ofRaisotopesfromSEDFscalculations(seeTableA.2fora comprehensivelistofvalues)comparedwithmeasured E 2 intrinsicmoments Q 2 for 2 + ! 0 + transitions(blacksquareswitherrorbars)ofRef.[111124].(Figuretakenfrom Ref.[125]) Figure3.12:Proton Q 30 ofRaisotopesfromSEDFscalculations(seeTableA.2fora comprehensivelistofvalues)comparedwithmeasured E 3 intrinsicmoments Q 3 for 3 ! 0 + transitions(blacksquareswitherrorbars)ofRef.[111,112,121,124].(Figuretakenfrom Ref.[125]) 40 Proton Q 20 and Q 30 valuesforoctupole-deformedeven-evennucleifromSEDFspredictions arecompiledinTableA.2. 3.3.1.3Thorium ( Z =90) Figure3.13:Predicted 2 , 3 ,and E oct valuesforeven-evenTh ( Z =90) isotopes. Experimentally,even-even 222 226 Thexhibitmanysignaturesofstableoctupoledefor- mation[118,126,127],inagreementwiththeSEDFs'predictionsshowninFig.3.13.All SEDFspredictoctupoledeformationsineven-even 220 228 Th.Thesenucleiareidenticalto MM[4]andGCM[25]predictions,whileHF+BCS[31]onlyreported 220 ; 222 Thasoctupole deformed.TheCEDFsmodels[34]predictedoctupoledeformationineven-even 224 232 Th, ashiftoffourneutronscomparedtotheSEDFs.FromourSEDFscalculation,wepredict thenucleus 222 Thand 224 Thtohavestrongg.s.octupoledeformation. 3.3.1.4Uranium ( Z =92) ThemajorityofSEDFspredicteven-even 222 228 Utobeoctupole-deformed.Asseenin Fig.3.14,thelargestoctupoledeformationenergyexceeding2MeViscalculatedfor 224 U, 41 Figure3.14:Predicted 2 , 3 ,and E oct valuesforeven-evenU ( Z =92) isotopes. followedby 222 ; 226 U.Experimentally,thenucleus 226 Uhassimilaroctupolecharacteristics as 222 Raand 224 Th[128]. MM[4]predictedtheoctupole-deformedUisotopestobe 220 226 U,GCM[25]listed 222 232 UandHF+BCS[31]listed 220 ; 224 ; 226 Uasoctupole-deformed.TheCEDFs[34]pre- dictedoctupoledeformationsin 226 234 U. AccordingtoourSEDFsstudy,thenuclei 222 ; 224 ; 226 Uarestrongcandidatesforpear- shapeddeformations,with 224 Ubeingmostpromising. 3.3.1.5Plutonium ( Z =94) Neutron-de˝cientPuisotopeshavereceivedlittleattentioninexperimentalsearchesfor octupoleinstabilityastheyareextremelydi˚culttoaccess.Thelightest-knownPuiso- tope, 228 Pu,hasahalf-lifeof1.1s[129]butspectroscopicinformationaboutthissystem isnonexistent.Likewise,virtuallynothingisknownabout 230 ; 232 ; 234 Pu,exceptfortheir g.s.properties[110].Interestingly,theisotope 228 Puispredictedbyallourmodelstobe octupole-deformed,followedby 226 Pu( m =7 )and 230 Pu( m =8 )(seeFig.3.6).Thelarge 42 Figure3.15:Predicted 2 , 3 ,and E oct valuesforeven-evenPu ( Z =94) isotopes. valuesof j E oct j in 224 ; 226 ; 228 Pu(Fig.3.15)calculatedbySEDFsaresimilartothoseRa, Th,andUisotopesthatshowevidenceforstableoctupoledeformations. TheGCMstudy[25]listed 228 234 Puasoctupole-deformed,whileMMmodel[4]reported 222 228 Pu,withthelargest j E oct j =1 : 09 MeVin 224 Pu.ThemajorityoftheCEDFsof Ref.[34]predictedeven-even 228 232 Puasoctupole-deformed. ThelightestCmisotopeknownexperimentallyis 233 Cm,whichissigni˝cantlyheavier thanourmostpromisingCmcandidatesforpear-likeshapes: 228 ; 230 Cm.AsseeninFig.3.6, inneutron-de˝cientactinideswith Z 98 ,mostofthebestcandidatesforoctupoledefor- mationliewellbeyondthecurrentdiscoveryrange,andsomeappeartobeclose,oroutside, thepredictedtwo-protondripline[108]. 3.3.1.6Veryneutron-richactinidesaround 288 Pu Manyextremelyneutron-richactinideandtransactinidenucleiwith 184 184 ,butthemagnitudeofthishindrancestrongly dependsonpredicted˝ssionbarriers. 49 Chapter4 IntrinsicSchi˙momentcalculations TheresultspresentedinCh.3gaveusabetterunderstandingofdi˙erentregionsofoctupole deformations.Thestrongestandmostrigidoctupole-deformedeven-evennucleiarefound intheneutron-de˝cientactinideandthesuperheavyactinideregion,thelatterhowever,are farbeyondexperimentalreach. Inthischapter,wediscusstheevaluationoftheintrinsicSchi˙moment, D ^ S z E e 10 Z ˆ p r 2 p z p d 3 r: (4.1) ThisevaluationwascarriedoutwithintheHFBframework,usingamodi˝edversionof thesolverHFBTHO(v3.00)[70].Blockingcalculationforoddsystemsemploystheequal ˝llingapproximationmethod(Sec.2.1.4.3),hencetime-oddtermsarenotincluded.Particle numberiskeptonaverageusing(2.28),insteadoftheLipkin-Nogamiprescription(LN) usedfortheeven-evenoctupolesurvey,becausetheimplicationofLNapproximationinthe contextofblockingcalculationsisstillunclear.SincetheLNprocedureisnotusedhere, thegroundstatedeformationsofeven-evennucleiareslightlydi˙erentthanfromtheglobal octupolesurvey,thusthesenucleiwererecalculated.Intotal,120deformationmeshpoints wereused,de˝nedas: i 2 2 [ 0 : 35 ; 0 : 35] ; i +1 2 i 2 =0 : 05 ; i 3 2 [0 ; 0 : 35] ; i +1 3 i 3 =0 : 05 : (4.2) 50 Thesame˝veSkyrmeEDFs(SEDFs)astheglobaloctupolesurveywereusedforthe Schi˙momentcalculations,namelyUNEDF0,UNEDF1,UNEDF2,SLy4,andSV-min(see Ch.3). TheSchi˙momentexpression(2.38)containstwomajorassumptions.The˝rstoneis theexistenceofanear-degenerateparitydoublet,whichdominatesthesum(2.36)through itssmallenergydenominator,andallowsustoevaluatetheintrinsicSchi˙moment D ^ S z E directlyusingthegroundstateprotondensity(2.25)fromtheHFBcalculation.These paritydoubletsaremorecommonlyidenti˝edinneutron-de˝cientactinideregion[21].Other studiesfoundevidenceofparitydoubletsinthelanthanides 151 Pm, 153 Eu,and 155 Euwhich haveenergysplittingofabout100keVbetweenthedoublet Thesecondassumptionrequiresrigiddeformationinthenucleus.Althoughseveraleven- evennucleiinthelanthanideregion,suchas 144 Baand 146 Ba,exhibitlargeoctupoledefor- mations,theirsmall j E oct j values(Fig.3.16)suggestthattheoctupoledeformationsin thesenucleiaresoft,thuswedonotconsidertheminthecurrentSchi˙momentanalysis. Giventhesereasons,ourcalculationsforintrinsicSchi˙momentsareperformedfor octupole-deformedsystemsintheneutron-de˝cientactinideregionwith 86 Z 94 and N 142 .Neutron-de˝cientisotopeswith Z =95 ; 96 thatareoctupole-deformedarevery closetotheprotondrip-line,hence,theywillnotbepresentedinthischapter. Abriefdescriptionoftheprocedureusedincalculatingtheoddsystemsisasfollows: 1. Createalistofoctupole-deformedeven-evennucleifromtheglobaloctupolesurveyfor eachofthe˝veSEDFs, 2. RecalculatethesenucleiwithnoLNparticlenumberrestoration, 3. Identifytheeven-evengroundstateamongthe120outputsforeachnucleus,store 51 thecorrespondingrestart˝leforlateruse,selectthelowestlyingprotonandneutron quasiparticle(q.p.)statesabovetheFermisurfaceastheblockingcandidates. 4. Performblockingcalculationsusingkick-o˙modeandidentifythegroundstateofthe oddsystem. Severalapproximationshavebeenusedintheoddsystem'scalculation:notime-odd˝elds wereconsidered,approximatetreatmentofblockingusedtheequal˝llingapproximation[82, 83],andidenti˝cationoftheq.p.stateusedNilssonorbitsofthelargests.p.componentof thedesiredq.p.state.Allcreateroomforlargeuncertainties. Sincetheq.p.statetobeblockedischosenfromthelowestq.p.stateabovetheFermi surfaceofitseven-evencore,werequirethattheoddsystem'sdeformationisnottoodi˙erent fromthoseofitseven-evencores.Ifthedeviationinthedeformationsbetweentheoddand evennucleiistoolarge,theorderingoftheq.p.orbitalsintheoddsystemmayhaveshifted, resultingintheblockingofawrongq.p.statethatdoesnotcorrespondtothegroundstateof theoddsystem.Hence,whensearchingforthegroundstateoftheoddsystem,thefollowing conditionswereapplied: j g : s : ( Z odd ;N odd ) g : s : ( Z even ;N even ) 0 : 05 ; =2 ; 3 ; where ( Z odd ;N odd ) 2 [( Z even ;N even +1) ; ( Z even +1 ;N even ) ; ( Z even +1 ;N even +1)] : Inpractice,sincethetypicalrangeof 2 is [0 ; 0 : 35] and [0 ; 0 : 15] for 3 ,theaboveconditions arenottoorestrictive,andthecorrectgroundstateoftheoddsystemisnotlikelytobe 52 excluded. 4.1IntrinsicSchi˙momentsinactinidenuclei IntrinsicSchi˙moments D ^ S z E wereevaluatedinnucleiwith 86 Z 96 and 128 N 142 ,usingEq.(4.1).Asaresult,Schi˙momenineven-evennucleiarealsocalculated andpresented,althoughstrictlyspeaking,theyarenotrelevanttothelaboratory-system EDM(Sec.2.2).Asademonstration,IwillpresentresultsfromtheRaandAcisotopic chainsinthissection.MoreresultscanbefoundinSec.4.2andTable4.1. 4.1.1Ra ( Z =88) Figure4.1:Predicted 2 , 3 ,and D ^ S z E valuesforRa ( Z =88) isotopes. AtthethecenterofoctupoledeformationoftheRaisotopes,i.e. 222 225 Ra,thepredicted valuesfor 2 , 3 ,and D ^ S z E areveryconsistentbetweenthe˝veSEDFs.Theoctupole 53 deformationisbelievedtoberigidintheeven-evenRawith A =220 224 (Fig.3.10), judgingfromthemagnitudeof j E oct j .Assumingthisrigiddeformationalsoholdsfor theirneighboringodd-massRa, 221 ; 223 ; 225 RaarelikelygoodcandidatesfortheatomicEDM search,withaverage D ^ S z E > 30 e fm 3 .Paritydoubletsin 223 ; 225 Rahavebeenidenti˝ed withenergysplittings E ˇ 50 keV,andin 221 Ra,thesplittingisroughlydoubled[21]. Theoctupoledeformationin 227 Raisperhapsnotasrigidasthelighterodd-massRa,as the E oct of 227 Ralikelyliescloseto 226 Raand 228 Ra.Theenergysplittingoftheparity doubletin 227 Rais90keV. 4.1.2Ac ( Z =89) Figure4.2:Predicted 2 , 3 ,and D ^ S z E valuesforAc ( Z =89) isotopes. Paritydoubletswereidenti˝edin 223 ; 224 ; 225 ; 227 Ac,with E ˇ 70keV,22keV,40keV, and27.4keV,respectively[21].Fig.4.2showsthatthepredicted 2 valuesfortheseAcnuclei 54 areconsistentamongtheSEDFs,althoughthe 3 valuesexhibitalargerspread.Assuming theodd- N Acisotopesalsosharesimilartendencytodeveloprigidoctupoledeformationas theireven-evencores,awiderangeof 221 227 Acisotopescouldbeconsideredasexcellent candidatesfortheatomicEDMmeasurements. 4.2Summary:IntrinsicSchi˙moments Inthissection,IhaveshowntheresultsoftheintrinsicSchi˙moment D ^ S z E calculatedusing theexpression(4.1),whichassumesrigiddeformationandtheexistenceofparitydoublets thatareonlyfoundinoctupole-deformednucleiwithodd-nucleonnumber.Oneneedsto keepthisinmindwheninterpretingtheresultsof D ^ S z E .Iftheseassumptionsarenotmet, theinterpretationintermsoftheintrinsicSchi˙momentreportedhereshouldbetakenwith agrainofsalt. OnealsoneedstounderstandthattheSchi˙momentinthelaboratoryframedependson the D ^ V PT E term,andisverysensitivetotheenergysplitting E betweentheparitydoublet (Eq.(2.38)).Nonetheless,theintrinsicSchi˙momentandoctupoledeformationscalculated fortheoddsystemso˙ervaluableinsightstotherelativestrengthofthefullSchi˙moment, sinceparitydoubletsandlarge 3 and D ^ S z E valuestendtooccursimultaneously. Byqualitativelyextrapolatingfromtherigidnessofoctupoledeformationintheodd system'seven-evenneighbors,andonlyconsidertheoddsystemsthatarelikelytohaverigid octupoledeformations,thefollowingnucleiarelikelytohavelargeintrinsicSchi˙moments. Foreven- Z isotopes, 221 ; 223 ; 225 Ra, 221 ; 223 ; 225 ; 227 Th, 223 ; 225 ; 227 U,and 225 ; 227 Pu;forodd- Z isotopes, 223 Fr, 221 227 Ac, 221 227 Pa, 223 229 Np.Amongthesenuclei,paritydoubletshave beenidenti˝edin 221 ; 223 ; 225 Ra, 227 Th, 223 Fr, 223 ; 225 ; 227 Ac[21]. 55 Ifonealsotakesintoaccountthelifetimesoftheseodd- A andodd-oddsystemsinorder toconductatomicEDMmeasurements,andlimittoisotopeswith t 1 = 2 > 1s,thepotential candidatesforatomicEDMmeasurementsarelistedinTable4.1.Amongisotopeswith con˝rmedparitydoublets, 223 Ra, 225 Raand 227 Thsharesimilarstrengthof D ^ S z E avg = E . ThiseighSchi˙momentismuchlargerin 225 Acand 227 Acduetothesmallenergy denominator,with 227 Acalmosttwicethevalueof 225 Ra. Asa˝nalcomment,wenotethatthereisamacroscopicexpression[157]fortheintrinsic Schi˙moment: S intr = eZR 3 0 9 20 ˇ p 35 2 3 : (4.3) Whenattemptingto˝tallcalculated D ^ S z E to ZR 3 0 2 3 intheneutron-de˝cientactinides region,the R 2 scoreforalinearregression˝tisonly0.43.Interestingly,when˝tting D ^ S z E directlyto 2 3 ,the R 2 ˇ 0 : 9 (seeFig.4.3). 56 Table4.1:CandidatesforatomicEDMmeasurementwith 86 Z 94 andhalf-life t 1 = 2 1second. D ^ S z E fromevaluating(4.1)hasbeenaveragedoverthe˝veSkyrmeEDFs(except for 223 Frand 229 NpwhichareonlycalculatedforfourSkyrmeEDFs).Experimentalenergy splittingbetweenparitydoublets E p : d : areshownwheredataexists[21].Thelastcolumn istheaverage D ^ S z E dividedby E p : d : .Theparitydoubletin 224 Achasnotbeenfully established,thelistedvalueisonlyforreference. 229 Paisalsolisted,intheeventthatthe lowlyingparitydoubletoflessthan1keVfromthegroundstateiscon˝rmed. Isotope t 1 = 2 D ^ S z E avg ( e fm 3 ) E p : d : (keV) D ^ S z E E ( e fm 3 keV ) 223 Fr22.00(7)min31.30160.450.195 221 Ra28(2)s33.91103.40.328 223 Ra11.43(5)d33.3550.190.664 225 Ra14.9(2)d39.6955.20.719 227 Ra42.2(5)min35.5590.00.395 222 Ac5.0(5)s33.80 223 Ac2.10(5)min37.6064.60.582 224 Ac2.78(17)h31.65(22.0)(1.439) 225 Ac10.0(1)d37.2740.10.929 226 Ac29.37(12)h38.87 227 Ac21.772(3)y35.8427.41.308 225 Th8.72(4)min40.67 227 Th18.68(9)d46.4367.20.691 225 Pa1.7(2)s44.17 226 Pa1.8(2)min43.51 227 Pa38.3(3)min46.74 229 Pa1.50(5)d50.60 227 U1.1(1)min46.03 228 Np61.4(14)s46.22 229 Np4.0(2)min49.76 57 Figure4.3: D ^ S z E plottedagainst 2 3 foralloddsystemsintheneutron-de˝cientactinides region.Twolinear˝tsweremade,onewithoutintercept(dashedline),andonewithintercept (solidline). R 2 scoreforthelinear˝tsarelistedaccordingly. 58 Chapter5 Bayesianmachinelearning Thischaptersummarizesaseriesofpapersproducedinourgroup,involvingmyself,from 2018to2020[108,109,158,159].ThesestudiespertaintotheapplicationsofvariousBayesian machinelearningtechniquestonuclearmassmodels.Inparticular,thesetechniqueswere usedtoimprovemodelpredictability,andtopresentuncertainty-quanti˝edanalysisofnu- clearinstabilityneartheparticledrip-lines. 5.1The S 2 n residualmodel Thebindingenergy,ormassofanucleusdescribeshowstablethenucleusiscompared toitsneighbors,whichdeterminesthedecaysofthenucleus.Theorymodelsthatpredict thenuclearmasses,fromphenomenologicalmodelssuchastheliquiddropmodelinthe earlydays,tothemodernself-consistentmean-˝eldmodels,allcontainsomeassumptions duetoincompletephysics,andapproximationstoreducecomputationalrequirements;both aspectscontributetothesystematicuncertaintiesofthepredictions.Ontheotherhand, experimentalerrorsand˝ttingofthemodelparametersgiverisetostatisticaluncertainties. Thesetwotypesofuncertaintiesareresponsibleformostofthediscrepanciesbetweentheory predictionsandthemeasurednuclearmasses. Missinginformationcontainedinthemassdi˙erences,orthemassresiduals,isgenerally unknownanalytically.Thispresentsuswiththeperfecttestgroundfork-bomodeling 59 withmachinelearning.Themodeledresidualscanbeaddedbacktotheoriginaltheory predictionstoimprovethem.Moreover,byutilizingBayesianmachinelearningtechniques, onecanprovideuncertaintyquanti˝cationforthenewtheoryestimates. Totestthismethodology,we˝rstfocusonthetwo-neutronseparationenergy S 2 n ,de˝ned as: S 2 n ( Z;N )=BE( Z;N 2) BE( Z;N ) ; (5.1) where BE( Z;N ) isthebindingenergyfornucleuswith Z protonsand N neutrons. Thereasonformodelingtheresidualof S 2 n insteadofthebindingenergyisbecause,by performingasubtractionbetweenneighboringnuclei,someofthesystematictrendinthe predictedmasseswillcancelout(Fig.5.1).Inthissection,welimitourselvestoeven- Z and even- N nucleiand S 2 n . Figure5.1:Leftpanel:BindingenergyresidualsofSEDFSkP(calculatedw.r.t.experimen- talmassesfromtheAME2016massevaluation[99,160]).Resultsfromoctupole-deformed (bluedots)andquadrupole-deformed(reddots)surveycalculationsareshown.Rightpanel: Sameasinleftpanelbutfortwo-neutronseparationenergy S 2 n .Notethattherangeon theenergyaxisis [ 10 ; 10] MeVfor S 2 n comparedto [ 20 ; 20] MeVforBE,illustratinga signi˝cantreductionofsystematicuncertaintyin S 2 n . The S 2 n residualisde˝nedasthedi˙erencebetweenexperimentaltwo-neutronseparation 60 energyandthecorrespondingtheoryprediction: ( Z;N )= S exp 2 n ( Z;N ) S th 2 n ( Z;N;# ) ; (5.2) where # isrepresentstheparametersinthetheorymodel.Ourtaskistoconstructthe emulator em ( Z;N ) usingmachinelearningtechniques,suchastheBayesianneuralnetwork, Gaussianprocess,orfrequencydomainbootstrap49, Uponobtainingtheposteriordistributionof em ( Z;N ) ,wecanprovideanewestimate ofthepredicted S 2 n bycombiningtheoriginaltheorypredictionswiththemeanvalueof theemulatedresiduals: S est 2 n ( Z;N )= S th 2 n ( Z;N;# )+ em ( Z;N ) : (5.3) Becausetheresulting em ( Z;N ) isgivenbyaprobabilitydistribution,wecanassignan errorbarassociatedwithitsmeanvalue,whichcouldbeinterpretedastheerrorbarforthe newestimate S est 2 n ( Z;N ) . Inthisstudy,537experimental S 2 n valuesfromAME2003massevaluation[166,167](along withtheoreticalpredictions)wereusedasthetrainingset.55newdatapointsinAME2016 massevaluation[99,160],i.e. S 2 n previouslyunavailableinAME2003,wereusedasthe testingdatasetandwillbereferredasAME2016-AME2003(Fig.5.2).Thisstrategyaims totestdatapointsoutsidethetrainingregions,whichisimportantforfarextrapolations; thisisdi˙erentthantheinterpolationsdoneinpreviousstudies[45,46,49,164,165]. Wehaveselectedthreegroupsoftheorymodelsrepresentingdi˙erenttheoreticalframe- works.Theselectioncriteriaformodelshavebeende˝nedasfollows.First,themodelneeds 61 Figure5.2:Theexperimental S 2 n ( Z;N ) datasetsforeven-evennucleiusedinourstudy: AME2003[166,167](lightdots,537points),additionaldatathatappearedintheAME2016 evaluation[99,160](darkdots,55points),andJYFLTRAP[168](stars,4points).(Figure takenfromRef.[158]) tobeabletoextrapolatewellintoregionsoftheunknownnuclei,thusitshouldbebased onsoundmany-bodyformalismusingquanti˝edinputsuchasinter-nucleoninteractionor nuclearenergydensityfunctionaletc.Second,thetheoryneedstobeabletoreproduce otherbasicnuclearstructuralpropertiesthatimpactnuclearbindingenergy,suchasshell structureanddeformations.Finally,itshouldbeaglobalmassmodelcapableofpredicting bindingenergyinallmassregions. The˝rstgroupcontainsthemorephenomenologicalglobalmassmodelsFRDM-2012[169] andHFB-24[170],whicharecommonlyusedinastrophysicalnucleosynthesisnetworksimu- lations.FRDMisrepresentativeofasetofwell-˝ttedmicroscopic-macroscopicmassmodels. HFB-24isrootedinaself-consistentmean-˝eldmethodwithseveralphenomenologicalcor- rectionsadded.Theroot-mean-square(rms)deviationof S 2 n frombothmodelcomparedto AME2003isaround0.6MeV(Fig.5.3),whichisaslowasonecanexpectwhileusingrea- 62 Figure5.3:Toppanel:Residualsof S 2 n ( Z;N ) forthesixglobalmassmodelswithrespect tothetestingdatasetAME2003.Thermsvaluesof ( Z;N ) inMeVaremarkedforAME2003 (uppernumber)andAME2003-H(lowernumber).Bottompanel:Sameastoppanelbut for S 2 n ( Z;N ) residualssmoothedwithGaussianfoldingfunctiontoemphasizelong-range systematictrends.(FigureadaptedfromRef.[158]) sonablenumbersofphenomenologicalcorrections.FRDM-2012contains38parametersand was˝ttedto2149experimentalnuclearmasses;HFB-24contains25self-consistentmean- ˝eldparametersand5phenomenologicalcorrectionparametersandwas˝ttedto729nuclear masses.Boththesemodelsincludedodd-oddandodd- A nucleimassesinthe˝ttingsamples. ThesecondgroupcontainssixmicroscopicSkyrme-DFTmodelsbasedonenergydensity functionals(EDF)(Sec.2.3)SkM*[171],SkP[172],SLy4[96],SV-min[97],UNEDF0[93], andUNEDF1[94].Thermsdeviationof S 2 n ofthisgroupcomparedtoAME2003isaround 63 1MeV(Fig.5.3).TheseSkyrmemodelscontainaround12modelparametersandwere˝tted tolessthan10even-evennuclearmassesforthe˝rst3models,andaround70even-even nuclearmassesforthelatter3models. Thethirdgroupcontainsfourmicroscopiccovariant-DFTmodelsbasedonrelativistic energydensityfunctionalsNL3*[55],DD-ME2[54],DD-PC1[57],DD-ME [173].Therms deviationof S 2 n ofthisgroupcomparedtoAME2003isaround1.1MeV,slightlyhigher thantheSkyrmegroup(Fig.5.3).ThecovariantEDFshave6to10parameters,withNL3* andDD-ME2˝ttedto12even-evennuclearmasses,andtheothertwo˝ttedtoaround60 even-evennuclearmasses. AvisualrepresentationoftheresidualdataareshowninFig.5.3,withtwomodelsselected fromeachofthethreetheoreticalframeworks.Thebottompanelisasmoothedoutversionof thetheresidualsintheupperpanelbyusingaGaussianfoldingfunctiontobettervisualize thelong-rangecorrelations.Theselong-rangepatternsaremostnoticeableinSLy4,DD- ME2,andDD-PC1,peakingattheneutronshellgapsanddecreasingwhenmovingfurther away;thisislikelyduetothelowe˙ectivemassesusedinthesetheorymodels.UNEDF1 showsamuchsmoothertrend.Ontheotherhand,themorephenomenologicalmodels FRDM-2012andHFB-24,being˝ttedtolargeamountofexperimentalmasses,providea verygoodreproductionoftheexperimentalvalues. 5.2Bayesianstatisticalmodels Looselyspeaking,aBayesianmethodcanbeseenasastatisticalwayofsolvinganinverse problemofinferringthemodelparametersgivensomeobservations/data.Unlikethe deterministicapproachcommonlyusedintheoptimizationprocessofphysicsmodeling, 64 wherethemodelparametersaredeterminedthroughminimizingcertainpenaltyfunction, inaBayesiansetting,aprobabilitydistributionoftheparametersaregeneratedusingpure statisticalsamplingprocesses,suchasinourcase,theMetropolis-Hastingsalgorithm[174]. WeperformaBayesiananalysisoftheresiduals ( Z;N ) usingtwodi˙erentstatistical models:Gaussianprocesses(GP)andBayesianneuralnetworks(BNN).Weinvestigatethe actualposteriordistributionsofallpredictedquantitiesandparametersintheemulator model.ThisselectionwasmadebecauseGPismorecapableofcapturingshort-rangecorre- lationswhileBNNisexpectedtocapturemorelong-rangetrends.Asgeneralnotation,we denotethestatisticalmodelbyafunction f oftheparticlenumber x i :=( Z;N ) ,andpa- rameters ,whichareunknownandlaterestimatedviaBayesianinference.Wealsoreplace theresidualwith y i := ( Z;N ) .OurBayesianmodelisthenoftheform: y i = f ( x i ; )+ ˙ i ; (5.4) where f iseitherGPorBNNwithparameters .Theaddedterm i isarandomvariable usedtomodeltheerror.WeassumeittobeanindependentstandardGaussianvariables withmeanzeroandunitvariance,and ˙ isanoisescaleparameter. TherelationEq.(5.4)iscalledthelikelihoodequation,whichrelatesthedata y i with theunknownparameters and ˙ .Inthelikelihoodmodel,assuming˝xed and ˙ ,the probabilitydensityof y isdenotedby p ( y j ;˙ ) .For x i =( Z;N ) wherethevaluesof y i ( S 2 n residual)areunknown,i.e.thetestingdataset,weuseEq.(5.4)topredictthem oncetheposteriordensityoftheunknownparameters and ˙ aredeterminedviaBayesian inference.Inordertodoso,wemustassumesomepriordistributionfortheseunknown modelparameters,denotedasajointprobabilitydensity ˇ ( ;˙ ) . 65 AccordingtoBayestheorem,theposteriordensity p ( ;˙ j y ) ,giventhetrainingdata y , theprior ˇ ( ;˙ ) andthelikelihoodmodels: p ( ;˙ j y ) / p ( y j ;˙ ) ˇ ( ;˙ ) : (5.5) Wecanthenusethisposteriordensity p ( ;˙ j y ) topredict y intheregionsof x where y isunknown.Thisrequirescomputingtheconditionaldensity p ( y j y;;˙ ) ,where y isthe knownresidualdata,andintegratingovertheposteriordensity p ( ;˙ j y ) oftheunknown modelparameters: p ( y j y )= Z p ( y j y;;˙ ) p ( ;˙ j y ) d˙: (5.6) Asmentioned,wenotonlyacquirethemeanvaluesofthepredictedresiduals,butalso theiruncertaintiesduetotheprobabilisticdescription(5.5)and(5.6)oftheunknownpa- rametersandpredictions.InBayesianterms,thequantitythatresemblesthe intervinclassicalstatisticsiswhat'scalledthecredibilityintervals.Thecredibilityin- tervals,orCI,isanintervalaroundtheBayesianmeanvalueinwhichcontains,e.g.,68% ofthesimulatedsample,andisapproximatelythesameasthecon˝denceintervalofone standarddeviationiftheposteriordistributionof y issymmetricontheleftandrightof themean.Forsimplicity,inthistext,weshalltreatCIasthecon˝denceinterval.Thisis justi˝edbecausetheposteriordistributionsoftheresidualsarehighlysymmetric. Inmoreprecisenotations,let m bethetotalnumberofsamplesintheposteriordis- tributionof y i foragiven x i =( Z;N ) .Theseposteriordistributionsarecomputedvia MonteCarlotechniques,inwhichthesamplesareobtainedusing100,000iterationsofer- godicMarkovchainproducedbytheMetropolis-Hastingsalgorithm[174].Thepredictionfor theresidual,thecorrectedprediction S em 2 n andtheone- ˙ uncertainty(errorbar)arethus 66 respectively: em ( Z;N )= 1 m P m j =1 y j ( Z;N ) ; S em 2 n ( Z;N )= S th 2 n ( Z;N )+ em ( Z;N ) ; ˙ em ( Z;N )= r 1 m P m j =1 h y j ( Z;N ) em ( Z;N ) i 2 : (5.7) Hereweuse S em 2 n ( Z;N ) ˙ em ( Z;N ) toapproximatelyrepresentatwo-sided68%-CIand S em 2 n ( Z;N ) 1 : 96 ˙ em ( Z;N ) asatwo-sided95%-CI.BelowI'lldiscussthetwospeci˝cBayesian statisticalmodelsthatwereusedtoproducetheposteriorsamples y i := ( Z;N ) byconstruct- ingdi˙erent f ( x i ; ) inEq.(5.4). 5.2.1Gaussianprocess Gaussianprocesseshavebeenusedcommonlyinphysicsandothernaturesciencestomodel theshort-range,orlocal,correlationsofthedata.Inthecontextofnuclearphysics,itis fairtoassumeresidualinformationofneighboringnucleihavestrongerimpactthannuclei furtherawayduetosimilaritiesintheirnuclearstructures,thusGPisasuitablemodelforthe taskathand.WetreataGaussianprocessasaGaussianfunctionalonthetwo-dimensional nuclearlandscape,characterizedbyitsmeanfunctionandcovariancefunction[175].Wetake themeanfunctiontobe0,andthedependenceofneighboringnucleiisdescribed usinganexponentialquadraticcovariancekernel, k ;ˆ x;x 0 := 2 exp " Z Z 0 2 2 ˆ 2 Z N N 0 2 2 ˆ 2 N # ; (5.8) where x =( Z;N ) and f ;ˆ Z ;ˆ N g aretheunknownparametersthatneedtobeacquired throughBayesianinference.Theparameter de˝nesthestrengthofdependencebetween neighboringnuclei, ˆ Z and ˆ N arethecorrelationrangesintheprotonandneutrondirection, 67 respectively.Thecovariancekernel k istheclassicalGaussiankerneluptoalineartransfor- mationofitsparameters,anditissymmetricin x;x 0 .OtherkernelssuchasMatérnkernels andexponentialkernelso˙ersimilarperformanceoncapturingshort-rangecorrelations,but theformerrequiresmoreintensecomputationduetotheembeddedBesselfunctionswhile thelatterhaveheavytailsthatarenotneededhere.Usingthenotation GP asaGaussian vectorhere,wecande˝nethefunction f inEq.(5.4)asarandomvectorwithparameters f ;ˆ Z ;ˆ N g andinput x :=( Z;N ) : f ( x; ) ˘GP 0 ;k ;ˆ ; (5.9) whichmeansthedistributionof f ( x; ) hasmean0andcovariancematrix k ;ˆ .Thekey componentoftheGaussianprocess f ( x; ) isthecovariancematrix k ;ˆ ,whichistrained onthevalues k ;ˆ x i ;x j fromtheknownregionandusedtopredict y accordingtoits Gaussiandistributiongiven y and ,andcanbeexpressedexplicitlyusing k ;ˆ [175].We usepureexperimentaluncertaintytorepresentthenoiseparameter inEq.(5.4),whichis verysmallcomparedtothetheoryuncertainties.Thisvalueis˝xedas0.0235MeV,which istheaverageexperimentalerror. Figure5.4isthecomputedposteriordistributionoftheGPparameters f ;ˆ Z ;ˆ N g inthecaseoftheDD-PC1relativisticDFTmodel.Itcanbeseenthatallthreeparameters arewelldeterminedwithsmallvarianceandsymmetricarounditsmean.Thestrengthof dependencebetweenneighboringnucleiisofthescale =0 : 87 MeV,andthecorrelation rangegivenby ˆ Z and ˆ N alongthe Z and N directionsareprecisely68%concentrated withinthewidthof 2 ˆ and95%withinthewidthof 4 ˆ .Thuswecanconcludethatroughly 90%ofthecorrelatione˙ectsarelocalizedintheregionof Z 4 ;N 2 . 68 Figure5.4:PosteriordistributionsoftheGPparameters f ;ˆ Z ;ˆ N g inthecaseofthe DD-PC1model,withtheposteriormeanandstandarddeviationlisted.(Figuretakenfrom Ref.[158]) 5.2.2Bayesianneuralnetwork ABayesianneuralnetworkisbasicallyanarti˝cialneuralnetwork[175],whichcomputes distributionsforitsmodelparameters/weightsinsteadofacquiringthemusingthefrequen- tistapproachofminimizingcertainpenaltyfunctions,thusprovidinguswithaprobabilistic descriptionofthemodelparameters.Here,wesetthefunction f ( x; ) inEq.(5.4)tohave onlyonehiddenlayerand H =30 hiddenneurons[175]: f ( x; ):= a + H X j =1 b j ˚ c j + X i d ji x i ! ; (5.10) where ˚ isanonlinearactivationfunction,andinourcasechosenasthehyperbolictangent function ˚ ( z )=tanh( z ) .Theunknownparametersaretheweights a;b j ;c j ;d ij of thisfunction,andarebothinternaltoeachneuron j 'sactivationandexternalbasedonthe trainingdatatoformtheinteractingnetworkamongneurons.Thelayersandnumberof neuronsperlayer H wereselectedtobe1and30respectively,duetothesmallnumberof trainingdata( ˘ 500)availableinanuclearphysicssetting. 69 Theposteriordensityof y oftheunknownregionisdeterminedbyintegratingoverthe posteriordensityoftheunknownparameters a;b j ;c j ;d ij asshowninEq.(5.6).To acquirethelikelihoodfunction p ( y j y;;˙ ) ,we˝rstassumethenoiseterm inEq.(5.4) asanormalvectorwithindependentandidenticaldistributedcomponents,withmean0 andunitvariances.WepresumethereisnoinformationgaininourBNNinmakingmore complexassumptionforthisterm.Thus,thelikelihoodfunctionisgivenas: p ( y j ;˙ ) / exp " X i ( y i f ( x i ; )) 2 2 ˙ 2 # ; (5.11) where ˙ isthenoisescaleinEq.(5.4).The y intheaboveprobabilitydensityisacombination ofboththetrainingdata y andtheto-be-predictedvalues y .Therefore,given ( ;˙ ) ,these twodatasets y and y canbeseenasstochasticallyindependent,andwehave p ( y j y;;˙ )= p ( y j ;˙ ) forEq.(5.6). 5.2.3Inputre˝nement Weexperimentedwiththreevariantsofinputfortheabovestatisticalmodelsotherthan thestandard x i =( Z;N ) .Inthe˝rstvariant,denotedasGP(H)andBNN(H),nucleibelow Caareremovedfromthedataset(HstandsforDoingsoallowustocompareour resultswithpreviousstudies49,176]thathavesystematicallydisregardedlightnuclei intheirdatasets. ThesecondvariantisdenotedbyGP(T)andBNN(T),whereTrepresents ThisvariantwasinspiredbyRef.[49]whichsupplementstheinput x i =( Z;N ) withinfor- mationpertainingtothenucleus'distancefromthemagicnumbers.Thisinformationis 70 representedbytwoadditionalinputs: ~ x i ( d N ( x i ) ;p ( x i )) ;p ( x )= d Z ( x ) d N ( x ) d Z ( x )+ d N ( x ) ; (5.12) where d Z ( x ) and d N ( x ) arethedistanceof x totheclosestprotonandneutronmagicnumber, respectively.Thequantity p ( x ) isthepromiscuityfactorthatindicatesthecollectivelyin theopen-shellnuclei[177]. Thethirdvariantisthecombinationoftheabovetwo,denotedasGP(T+H)andBNN(T+H). Aswewillseelater,thesere˝nementswillsigni˝cantlyimprovetheBNNmodel,andprovide onlyminorimprovementsfortheGP. Figure5.5:Residualsof S 2 n ( Z;N ) forthesixglobalmassmodelswithrespecttothetesting dataset(AME2016-AME2003): ( Z;N ) (dots)andtheGPemulator GP ( Z;N ) (circles). (FiguretakenfromRef.[158]) 71 5.3Resultsofthe S 2 n residualmodel Wedividedtheexerciseofmodelingthe S 2 n residualsintothreegroups,eachusingdi˙erent datasets.The˝rstgroupteststhepredictivepowerofourmethodologyandtocompare theperformancebetweenGPandBNN,whichusedthetrainingdatasetAME2003and AME2003-H(heavy)totestonAME2016-2003datapoints(bluedotsinFig.5.2).The resultsareshowninFig.5.5andTable5.1.SinceAME2016containsremeasureddatafor nucleithatwerealreadyinAME2003,ifthedi˙erencesinthetwomeasurementisgreater than30%foranucleus,weremoveditfromtheAME2003trainingdataset-thisincludes 10 He, 24 O, 34 Mg,and 52 Ca,whicharemovedintothetestingdataset. ThesecondgroupusedAME2003-HandAME2016-Hastwodi˙erenttrainingdatasets, andcomparedthetrainedresidualmodelsonthethenrecentlymeasuredmassesatJYFLTRAP in2017.Onlythe(T+H)variantoftheinputswereused.Thisistotesthowmuchthe additionaldatapointsa˙ectmodelpredictions.TheresultsarecomparedinTable5.2. Finally,wetrainedourresidualemulatorsonthecombineddatasetofAME2016-Hand JYFLTRAPtoproduceanestimateforthe S 2 n oftheentirenuclearlandscape,andanalyzed thetwo-neutrondrip-lineoftinisotopeswiththeuncertainty-quanti˝ed S 2 n values. 5.3.1Trainingset:AME2003;testingset:AME2016-AME2003 Theoriginalresidualsfrom S th 2 n ( Z;N ) (blackdots)andtheresidualsfromthenewestimates S est 2 n ( Z;N ) (whitecircles)inEq.(5.3)ofsixrepresentativenuclearmassmodelsareshownin Fig.5.5forGP(T+H).InFig.5.5,nearlyallwhitecircles,whicharetheresidualsfromthe newestimates,movedcloserto0comparedtothecorrespondingblackdots,whicharefrom theoriginaltheorypredictions.ThisistosaythattheGPmodelsystematicallyimprovedthe 72 predicted S 2 n forthesenuclearmassmodels.Additionally,thelocaltrendsoftheresiduals havebeenvisiblydampened,andthecorrectedresidualsareclosertohavingadistribution ofmeanzero. Table5.1liststhermsdeviationsoftheresidualsfromdi˙erentnuclearmassmodels usingthestandardinputandthreedi˙erentinputvariantsdiscussedinSec.5.2.3,and thetwoBayesianmodelsGPandBNNfortheemulators em ( Z;N ) .BothGPandBNN reducedthermsdeviationoftheresidualsnoticeablyfortheSkyrmeandrelativisticDFT models,withGPhavingconsistentlybetterperformance.Theperformanceonrelativistic DFTmodelsarethebest,around50%rmsreduction,followedbytheSkyrmeDFTmodels, around30%rmsreduction.InthemorephenomenologicalmodelsFRDM-2012andHFB- 24,theresultsaremixed.InHFB-24,insteadofdecreasing,thermshasincreasedformost cases,andforFRDM-2012,onlytheGPmodelssystematicallyimproveditsprediction.This isnotsurprising,asweexpecttheresidualsfromthemoremicroscopicmodelstohavemore implicitstructure,whereasthemorephenomenologicalmodelshavebeen˝ttedveryclosely tomassdata,andtheirresidualshaveahigherproportionofstatisticalnoise.Thiscan alsobeseenfromthefactthataftertheimprovements,thermsdeviationsseemtohave reachedalowerboundataround300-500keVfortheGP(T+H)variant,whichhasthe bestperformance,andislikelyfromtheirreduciblestatisticalnoise.Thislowerboundalso suggeststhatourstatisticalmodelscapturedmostoftheresidualstructure,andthatan uncertaintyquanti˝cation(UQ)analysisbecomesnecessarytofurtherassessthemodels' quality. 73 Table5.1:Rootmeansquarevaluesof ( Z;N ) , BNN ( Z;N ) ,and GP ( Z;N ) (inMeV)forvar- iousnuclearmodelswithrespecttothetestingdatasetconsistingoftheAME2016-AME2003 S 2 n values.ThetrainingAME2003andAME2003-Hdatasetswereusedtocomputetheem- ulators BNN ( Z;N ) and GP ( Z;N ) .Thetwonumberslistedunderthemodel'snameinthe ˝rstcolumnaretheuncorrected rms modelvalueswithrespecttoAME2003andAME2003- Hdatasets,respectively.Thermsresidualscorrectedbyastatisticalmodelareshowninthe remainingcolumns.Foreachmodel,GPresults GP rms aregivenintheupperrowandtheBNN results BNN rms arelistedinthelowerrow.Thenumbersinparathensesindicatetheimprove- mentinpercent.Thefourstatisticalvariantsarelisted:Stdisthestandardstandardinput x =( Z;N ) ;Tindicatesresultsinvolvingthenon-lineartransformation ~ x i =( d N ( x i ) ;p ( x i )) ; HisbasedonthereduceddatasetAME2003-Hpertainingtoheavynucleiwith Z 20 . (TabletakenfromRef.[158]) modelStdTHT+H SkM* 1.25/1.01 0.96(23)0.96(23)0.49(52)0.49(52) 0.99(20)0.81(35)0.73(28)0.53(47) SLy4 0.95/0.80 0.82(13)0.82(13)0.52(35)0.52(35) 0.91(3)0.82(14)0.71(11)0.56(30) SkP 0.84/0.62 0.75(11)0.75(11)0.38(39)0.38(39) 0.76(9)0.74(12)0.59(5)0.45(27) SV-min 0.78/0.49 0.70(10)0.70(10)0.32(34)0.33(34) 0.72(8)1.35(-73)0.50(-1)0.43(12) UNEDF0 0.78/0.54 0.73(6)0.73(6)0.34(37)0.34(37) 0.87(-12)0.73(7)0.55(0)0.46(16) UNEDF1 0.66/0.49 0.61(8)0.61(8)0.34(30)0.34(30) 0.79(-20)0.74(-12)0.53(-10)0.32(33) NL3* 1.19/0.86 0.84(29)0.84(29)0.46(47)0.45(47) 1.10(7)0.90(24)0.83(4)0.69(20) DD-ME 1.13/0.96 0.73(35)0.74(35)0.55(42)0.55(42) 1.08(4)0.91(19)0.89(7)0.75(22) DD-ME2 1.04/0.95 0.71(32)0.71(31)0.63(34)0.62(34) 1.00(4)1.32(-27)0.90(5)0.61(36) DD-PC1 1.10/0.91 0.79(28)0.79(28)0.46(50)0.46(50) 1.00(9)1.33(-22)0.85(7)0.54(41) FRDM-2012 0.63/0.49 0.57(9)0.57(9)0.36(25)0.36(26) 0.61(4)0.72(-15)0.48(2)0.45(7) HFB-24 0.40/0.37 0.40(-1)0.40(-1)0.40(-8)0.40(-8) 0.59(-48)0.44(-10)0.37(1)0.35(6) 74 5.3.2Trainingsets:AME2003-H,AME2016-H,testingset:JYFLTRAP- 2017 Thedi˙erenceinperformanceofthemodelstrainedonAME2003-Handthelargerdataset AME2016-Hiscomparedbypredictingtherecent S 2 n valuesfromJYFLTRAPusingthe (T+H)variantofthemodelinput.Thetestingdatasetconsistsof4points(redstarsin Fig.5.2);theresultsareshowninTable5.2.Fromthistable,wecanseethattheGP modelsreducedthermssigni˝cantlyforallcases,whiletheBNNpredictionsofSLy4,SkP, SV-min,UNEDF0,andDD-ME deteriorated.Thiscanbeattributedtotheemphasison long-rangecorrelationsfromtheBNNmodelduetothenon-vanishingtailintheactivation function.Ontheotherhand,theGPmodels,whichuseaGaussian-likekernelandarangeof correlatione˙ectsof Z 4 ;N 2 ,focusmoreontheshort-rangecorrelationsandperformed better.Thiscanbeexplainedbytheobservationthattheresidualsurfaceintheregionof JYFLTRAPdata ( Z ˘ 62 ;N ˘ 100) isfairlysmooth(Fig.5.3). Overall,slightlybetterimprovementshavebeenachievedforthemodelstrainedusing theAME2016-Hdataset.However,thisimprovementisnotsigni˝cant-weseethatonly onenewmeasurementintheextendeddata,i.e.AME2016-2003(Fig.5.2bluedots),isin thevicinityofthetestingdataset,andthuscanonlyimposeweakconstraintsonregionsof thetestingdata. 5.3.3Two-neutrondrip-lineofSn ( Z =50) Inthenextstep,wetrainedtheresidualemulatorsontheentireeven-evennuclearlandscape withthecombineddatasetofAME2016-HandJYFLTRAP,usingthe(T+H)inputvariant. Basedonresultsfromthissetofpredictions,Fig.5.6showstheextrapolated S 2 n values 75 Table5.2:SimilarasinTable5.1exceptforthermsvaluesof ( Z;N ) , BNN ( Z;N ) ,and GP ( Z;N ) (inMeV)forvariousnuclearmodelswithrespecttothetestingdatasetconsisting ofthefourJYFLTRAP S 2 n values.Thesecondcolumnshowstheuncorrectedrmsvalue rms .Foreachmodel,thetrainingdatasetsAME2003-H(thirdcolumn)andAME2016-H (fourthcolumn)wereusedtocompute GP rms (upperrow)and BNN rms (lowerrow)usingthe T+Hvariantofstatisticalcalculations.(TabletakenfromRef.[158]) model rms 2003-H2016-H SkM* 0.91 0.40(56)0.31(66) 0.24(74)0.25(72) SLy4 0.27 0.09(65)0.09(67) 0.42(-57)0.28(-4) SkP 0.19 0.16(14)0.14(26) 0.35(-85)0.36(-92) SV-min 0.14 0.11(18)0.10(29) 0.17(-20)0.26(-86) UNEDF0 0.11 0.11(-3)0.11(1) 0.33(-199)0.22(-97) UNEDF1 0.26 0.17(36)0.14(48) 0.09(64)0.13(50) NL3* 0.32 0.19(39)0.22(32) 0.17(47)0.18(43) DD-ME 0.16 0.08(50)0.09(46) 0.18(-14)0.28(-4) DD-ME2 0.30 0.12(58)0.13(55) 0.28(8)0.29(2) DD-PC1 0.28 0.17(41)0.13(52) 0.25(12)0.27(5) FRDM-2012 0.13 0.10(20)0.09(26) 0.05(60)0.05(58) HFB-24 0.13 0.12(2)0.11(12) 0.07(43)0.10(25) anditscredibilityintervals(CI)oftheSn( Z =50 )isotopesusingGPandBNN,forthe relativisticDFTmodelDD-PC1.Theobjectivehereistopredictthelocationofthetwo- neutrondrip-line.Thisisroughlyequivalentto˝ndingthelargestneutronnumber N such thatthelowerendpointofitsone-sided1.65-sigmaCIbarelytouchesthe S 2 n =0 line,i.e. 76 Figure5.6:Extrapolationsof S 2 n fortheeven-evenSnchaincalculatedwithDD-PC1 usingstatisticalGP ( T + H ) andBNN ( T + H ) approaches.One-sigmaand1.65-sigmaCIs aremarked.(FiguretakenfromRef.[158]) a95%probabilitythatthis N givesthecorrecttwo-neutrondrip-line.InFig.5.6,wecan seethatboththeoriginalDD-PC1predictionandtheDD-PC1+GP(T+H)posteriormean predicted N tobe126,and N =122 and118atthe1-and1.65-sigmaCI,respectively. Similarly,theDD-PC1+BNN(T+H)posteriormeanpredicted N =118,and N =102and 104atthe1-and1.65-sigmaCI,respectively.Onecanimmediatelyseetheadvantageofthis descriptionofthedrip-lineoversimplysaying:predictsthetwo-neutrondrip-line ofSnat N = 5.4Neutrondrip-lineintheCaregionusingBayesian modelaveraging Thestatisticalmodelofresidualsservedasaproofofconcept,whichallowedustotestand analyzetheperformanceofvariousBayesianmodelsandotheraspectsofthemethodology. 77 Inconclusion,wefoundthatusingtheGPasourlikelihoodmodelhastheoverallbest performance,andisalsomorenumericallystable(Ref.[158]).Theperformancefurther increasesbyexpandingtheinputdimensionforthestatisticalmodelstoincludeinformation onanucleus'proximitytotheneutronmagicgaps.Onecouldalsolimittheirdomainof trainingtoheavynuclei,suchasnucleiaboveCatoimprovemodelperformance,iftheyare onlyinterestedinheavierregionsofthenuclearlandscape.Anadditionalbene˝tofusing theBayesianmethodologyistheresultingprobabilisticdescriptionofthemodelparameters. Thisprovidesusaquantitativewaytoestimateourpredictions'uncertainties,andthus quanti˝esthelevelofcon˝denceofourpredictions. Wenextdevotedourselvestoamorespeci˝canalysisontheneutrondrip-lineneartheCa regionusingthetechniquesdevelopedinSec.5.1and5.2,andonlyusedtheGPmodelwhich wasproventobemoree˙ective.Wealsoincludedtheresidualmodelfortheone-neutron separationenergy: S 1 n ( N;Z )= BE ( N;Z ) BE ( N 1 ;Z ) ; (5.13) whichisthedi˙erenceinbindingenergiesbetweenneighboringnucleiwiththedi˙erenceof onlyoneneutron.Theneutronnumberhereisalwaysanoddnumber,whichistosaywe onlystudytheresidualsof S 1 n inasystemwithodd- N and S 2 n forsystemswitheven- N . Thisisduetothefactthatanodd- N systemwilldecay˝rstviaremovingoneneutron, andaneven- N systemwilldecay˝rstviaremovingapairofneutrons,aconsequenceofthe pairingcorrelation. SincemostcalculationsinDFTareperformedfortheeven-evensystemsduetothe additionalblockingprocedurerequiredwhendealingwiththeoddnucleon(Sec.2.1.4.3),we tookasimpleapproachinwhichweusedthebindingenergiesoftheeven-evenneighborand 78 theaverageparinggapstoapproximatethebindingenergyofanodd- A (orodd-odd)system via: BE( Z 1 ;N )= 1 2 [BE( Z;N )+BE( Z 2 ;N )] + 1 2 p ( Z;N )+ p ( Z 2 ;N ) ; (5.14) BE( Z;N 1)= 1 2 [BE( Z;N )+BE( Z;N 2)] + 1 2 n ( Z;N )+ n ( Z;N 2)] ; (5.15) where n and p aretheaverageneutronandprotonpairinggapsfromDFTcalcula- tions[65].The S 1 n residuals 1 n ( Z;N ) arede˝nedsimilarlyto 2 n ( Z;N ) inEq.(5.2),with achangeinallthesubscriptsfrom 2 n to 1 n . Twotrainingdatasetswereusedinthiswork,inordertotesttheimpactofadditional nuclearmassmeasurementsontheperformanceofourresidualmodels,similartowhat wasdoneinSec.5.3.The˝rstsetusestheAME2003experimentaldata,andthesecond setisAME2016*,whichisAME2016supplementedwiththethennewmeasurementof 52 55 TimassesfromexperimentsatTRIUMF[178].Ninetheorymodelswereusedinthis exercise,includingsevenSkyrmeDFTmodelsSkM*[171],SkP[172],SLy4[96],SV-min[97], UNEDF0[93],UNEDF1[94],andUNEDF2[95]andtwomorephenomenologicalmodels FRDM-2012[169]andHFB-24[170]. Figure5.7showsthenewestimatesfor S 1 n= 2 n fromtheninetheorymodels: S est 1 n= 2 n ( Z;N )= S th 1 n= 2 n ( Z;N; )+ em 1 n= 2 n ( Z;N ) : (5.16) Experimentalseparationenergiesof 55 57 CaareprovidedbyAME2016extrapolations. WecanseethatallSkyrmeDFTmodels'predictionsforthe S 1 n valuesof 55 ; 57 Caare 79 Figure5.7:One-neutronseparationenergyfor 55 ; 57 Ca(left)andtwo-neutronseparation energyfor 56 Ca(right)calculatedwiththenineglobalmassmodelswithstatisticalcorrection obtainedwithGPtrainedontheAME2003(GP+2003)andAME2016*datasets.The recentdatafromRef.[179](RIKEN2018)andtheextrapolatedAME2016values[99,160] aremarked.Theshadedregionsareone-sigmaerrorbarsfromRef.[179];errorbarson theoreticalresultsareone-sigmacredibleintervalscomputedwithGPextrapolation.(Figure takenfromRef.[159]) consistentwithexperimentswithintherangeofuncertainties,andthe S 2 n valuesof 56 Ca areconsistentwiththeexperimentaldatafornewpredictionsofUNEDF1,UNEDF2,SLy4, SkPandSkM*.TheGPcorrectionsforthemorephenomenologicalmodelsareverysmall, duetothefactthatthesemodelsarealreadywell-˝ttedinalmostallmassregions,thus leavingverylittleinformationtobeusedforthestatisticalprocess.Thelargedeviation 80 inthenewHFB-24predictionsisduetotheirregularbehaviorofitsneutronseparation energiesinitsoriginalprediction. Sincewe'reusingGPmodel,asdiscussedinSec.5.2.1,thee˙ectiverangeoftheGPis veryshort,meaningthatadditionaldatapointsintheAME2016*trainingsetcomparedto AME2003donothaveahugeimpactonourGPmodel.Thereareslightimprovementsin themeanvaluepredictionsoftheseparationenergies. Wealsointroduceaquantity p ex ( Z;N ) ,whichiscalledtheprobabilityofexistence. p ex ( Z;N ) istheprobabilityofthepredictedseparationenergy S 1 n= 2 n ( Z;N ) tobepositive initsposteriordistribution: p ex ( Z;N ):= p S 1 n= 2 n ( Z;N ) > 0 j S 1 n= 2 n : (5.17) Thisprovidesanotherwayofdescribinganucleus'instabilitytoneutrondecay. The S 1 n= 2 n predictionsneartheCadrip-lineareshowninFig.5.8forthetheorymodels UNEDF0,SV-minandFRDM-2012.Theprobability p ex ( Z;N ) isshowninthe˝gure's insert.Thecoloredbandsaretheone-sigmaCIs.Wecanseethattheposteriorpredictions betweenthethreetheorymodelsareoverallconsistent. Withoutthelossofgenerality,wechose p ex =0 : 5 astheboundaryfortheestimatedone andtwoneutrondrip-lines.Thisismarkedbyadottedlineinthis˝gure. Applyingthesamestrategyforthebroaderregionaroundtheneutron-richCaisotopeswe arriveatFig.5.9.Thetoppanel,Fig.5.9(a)usesuniformaveraging(equalweights)acrossthe ninetheorymodelsaftercorrectionwithEq.(5.16).Underthissimplisticaveragingscheme, the p ex valuesforthealreadydiscoverednuclei[180,181] 49 S, 52 Cl,and 53 Arare0.58,0.45, and0.64,respectively.Theselow p ex valuesshowthattheodd-neutronsystemspresenta 81 Figure5.8:Extrapolationsof S 1 n and S 2 n fortheCachaincorrectedwithGPandone-sigma CIs,combinedforthreerepresentativemodels.Thesolidlinesshowtheaverageprediction whiletheshadedbandsgiveone-sigmaCIs.Theinsertshowstheposteriorprobabilityof existencefortheCachain.The p ex =0 : 5 limitismarkedbyadottedline.(Figuretaken fromRef.[159]) challengeforourtheorymodels,aswasnoticedinRef.[180].Thus,weusetheknowledge thatthesethreeisotopehavebeenobservedtoinformthemodelaveragingprocess,through theposterioraveragingweights: w k := p M k j 52 Cl ; 53 Ar ; 49 S exist : (5.18) Theseweightsre˛ecttheabilityofamodel M k topredicttheexistenceoftheseisotope, i.e.theprobabilityofpredicting S 1 n > 0 aftercorrection.Usingtheposteriorweights,the resulting p ex valuesfor 49 S, 52 Cl,and 53 Arare0.69,0.53,and0.69,respectively(Fig.5.9(b)), slightlybetterthanusinguniformmodelaveraging. BycomparingFig.5.9(a)and(b),theimpactofincludingtheinformationthat 49 S, 52 Cl, and 53 Arexistwereabletoextendthedrip-line( p ex ( Z;N ) 0 : 5 )bytwoneutronnumbers 82 Figure5.9:Posteriorprobabilityofexistenceofneutron-richnucleiintheCaregionaveraged overallGPcorrectedmodels.Top:Uniformmodelaveraging.Bottom:Averagingusing posteriorweightsEq.(5.18)constrainedbytheexistenceof 52 Cl, 53 Ar,and 49 S.Therange ofnucleiwithexperimentallyknownmassesismarkedbyayellowline.Theredlinemarks thelimitofnucleardomainthathasbeenexperimentallyobserved;nucleitotherightofthe redlineawaitdiscovery.Theestimateddriplinethatseparatesthe p ex > 0 : 5 and p ex < 0 : 5 regionsisindicatedbyablueline.(FiguretakenfromRef.[159]) inmostcases.Accordingtotheseaverage p ex ( Z;N ) ,wecansaythat 61 Caand 71 Tiare expectedtobeone-neutronunstablewhilethetwo-neutrondrip-linesareat 70 Caand 78 Ti. Thenucleus 59 K,whichhadoneregisteredeventinRef.[180],ispredictedtobestable againstneutrondecay. 5.5Protondrip-lineanalysisandtwo-protonemitters InthenextprojectweimprovedpredictionsbyusingBayesianmodelaveraging(BMA)tothe proton-richsideofthenuclearlandscape,withanadditionalgoalofidentifyingtwo-proton 83 emitters. BecauseoftheCoulombbarrier,theone-andtwo-protondrip-lineslierelativelyclose tothevalleyofstability.Thusthehalf-livesofproton-unstablenucleibeyondthedrip- linesarerelativelylong,allowingustostudythenuclearstructureanddynamicsinsystems withlow-lyingprotoncontinuum.Oneofthephenomenonthatemergesintheseregionis theexistenceoftwo-protonemitters.Unliketheone-andtwo-neutrondrip-linesshownin Fig.5.8,whichhasthecleartrendthatthe S 1 n liestotheleftofthe S 2 n line,andthus one-neutrondrip-linewillalwaysoccurbeforetwo-neutrondrip-line(alsoseeninFig.5.9), thereisthepossibilitythatinaproton-richnucleus,thetwo-protonseparationenergy S 2 p isnegativewhile S 1 p ispositive.Thisimpliesthatthesystemcanbeone-protonboundbut atthesametimeunstabletotwo-protonemission.Severaltwo-protonemittershavebeen identi˝edbyexperiments: 19 Mg[182], 45 Fe[183,184], 48 Ni 54 Zn[189,190],and 67 Kr[191].Additionally,broadresonancesassociatedwiththetwo-protonemissionwere reportedinseverallightnucleisuchas 6 Be[192]and 11 ; 12 O[193,194]. Inthiswork,weincludedtwoGognyDFTmodelsD1M[53]andBCPM[195]totheset ofmodelsusedinSec.5.4,makingthetotalnumberoftheorymodelsusedtobeeleven.The D1Mmodelisamodernparametrizationofthe˝nite-rangeGognyinteraction,and˝ttedto 2149massesfromtheAME2003datasetandothernuclearproperties.TheBCPMmodel isprimarilygivenbya˝ttotheequationofstateinbothneutronandsymmetricnuclear matter,whichresultedinrelativelysmallnumbersoffreeparameters,andwas˝ttedtothe massesof579even-evennucleifromtheAME2003massdataset[166]. 84 5.5.1Modi˝edGaussianprocessandBayesianmodelaveraging Wemodi˝edtheGPmodelslightlyascomparedtowhatwasdoneinSec.5.2.1,andintro- ducedanadditionalparameter asthemeanoftheGaussianvector: f ( x; ) ˘GP k ;ˆ x;x 0 : (5.19) Thisaddedparameter improvedthermsdeviationoftheresidualbyanadditional15% comparedtotheinitial25%improvementfromthezero-meanversionoftheGP. ThetrainingdatasetusedisAME2016+,whereweaugmentedtheAME2016dataset withmassesfromRef.[168,178,200],andkeptthemostrecentvalueiftherewere repeatmeasurements. Fourvariantsofweightsforthemodelaveragingwereusedinthiswork.The˝rst useduniformweights(BMA-0)withouttheneedofadditionalinformationandthecostly computationoftheposteriors.Theotherthreevariants,BMA-I,-II,and-IIIarebuilton Bayesianmodelaveragingandtheirweightsdependsonhowinformationfromtheknown two-protonemitters x 2 p; known 19 Mg ; 45 Fe ; 48 Ni ; 54 Zn ; 67 Kr isutilized. Thesecondvariant,BMA-I,usedtheconditionalprobabilitythatthecorrectedthe- orymodelisabletopredictthecorrectsignsoftheexperimental Q 2 p and S 1 p valuesfor x 2 p; known .Theweightsaregivenas: w k (I) / p M k j Q 2 p > 0 ;S 1 p > 0 for x 2 p; known ; (5.20) where M k representsthetheorymodel. ThethirdvariantBMA-IIwasbaseddirectlyontheabilityofatheorymodel M k to 85 predictthe Q 2 p valuesofthe˝veknowntwo-protonemitters x 2 p; known : w k (II) / p M k j Q 2 p of x 2 p; known : (5.21) The˝nalvariantBMA-IIIisatrivialversionofBMA-II,consistingoftheGaussian likelihoodof x 2 p; known evaluatedattheposteriormeanandposteriorvariance,assuming thatthesevaluesarestatisticallyindependent.ComparedtoBMA-II,thisvariantignores allcorrelatione˙ectsoftheposteriorpredictionsfor x 2 p; known .TheweightsforBMA-III aregivenas: w k (III) / Y i 2 x 2 p; known 1 q 2 ˇ˙ 2 i e 1 2 ( y i ˙ i ) 2 ; (5.22) where y i arethe Q 2 p residualsof x 2 p; known . Asaresult,wediscoveredthattheBMA-0,BMA-I,andBMA-IIIvariantsachieved betterperformancecomparedtothemoresophisticatedBMA-II,andthebestperformance intermsofrmsdeviationreductioncamefromthesimplestBMA-0andBMA-Ivariants. Theresultingpredictionof p ex forproton-richnucleiwith 16 Z 82 isshowninFig.5.10. 5.5.2Two-protonemitters Althoughtheprotonchemicalpotentialispositivefornucleiwith S 1 p= 2 p < 0 ,theHFB (Sec.2.1.3)calculationsareverystableintherangeofbindingenergiesconsidereddueto theCoulombbarrier'scon˝nemente˙ectontheprotondensitythate˙ectivelypushesthe protoncontinuumupinenergyintheproton-richnuclei.Thus,wecansafelyobtainthe protonseparationenergies S 1 p= 2 p andthecorresponding Q valuesfromthebindingenergies oftheprotonunboundsystems. 86 Figure5.10:Probabilityofexistence p ex thatanucleiisboundwithrespecttoproton decayforproton-richnucleiwith 16 Z 82 .CalculationsusingBMA-I(top)andBMA-II (bottom)variantsofmodelaveragingareshown.Foreachprotonnumber, p ex isshown alongtheisotopicchainversustherelativeneutronnumber N 0 ( Z ) N ,where N 0 ( Z ) ,listed inTablesA.3andA.4,istheneutronnumberofthelightestproton-boundisotopeforwhich anexperimentalone-ortwo-protonseparationenergyvalueisavailable.Thedomainof nucleithathavebeenexperimentallyobserved(bothproton-boundandproton-unbound)is markedbyopenstars;thosewithinFRIB'sexperimentalreacharemarkedbydots.(Figure takenfromRef.[109]) Thisworkisconsideredasanextensionofthepreviousglobalsurveyofprotonemit- ters[201,202],andusesthesamecriteriontoselect true two-protonemitters: Q 2 p > 0 ;S 1 p > 0 ; (5.23) where Q 2 p = S 2 p .Thisconditioncorrespondstoasimultaneousemissionoftwo-proton inthediprotonmodel[185,186],comparedtothesequentialemissionoftwoprotonsinthe direct-decaymodel[163].Weshallrefertothetwo-protonemissioncorrespondingtothe diprotonmodelasthe true two-protonemission. 87 Thereisanadditionalconstraintonthenucleus'lifetimewhenselectingatwo-proton emitter,duetothefactthatverylarge Q valueswillcausethedecaytobetoofasttobe observed.Ontheotherhand,ifthe Q valuesaretoosmall,theproton-decayrateswillbe negligiblecomparedtootherdecaychannelssuchas or decays.Thepracticalrangeof lifetimetoconsideris[201]: 10 7 0 j S 1 p= 2 p ; (5.25) where S 1 p= 2 p arethevaluesfromtheunknownmassregion,and S 1 p= 2 p arewhat'sbeing usedasthetrainingdataset,followingconventionsinSec.5.2. Fig.5.11showsthe Q 2 p valuesfromtheBMA-Ipredictionstogetherwiththerangeof 88 Figure5.11: Q 2 p valuespredictedinBMA-Iforeven-evenisotopeswith 16 6 Z 6 80 . Thethicksolidlinesmarkthelifetimerange(5.24).Themassnumbersofselectedisotopes areshown.Thenucleiwiththeprobability p 2 p > 0 : 4 areindicatedbydots.Here,weused thisvalueof p 2 p ratherthan p 2 p > 0 : 5 becausethecriterion(5.25)ofthe true 2 p emissionis slightlymorerestrictivethantheenergycriterionpreviouslyadoptedinRef.[202].(Figure takenfromRef.[109]) lifetimein(5.24).Thecoloredshadesforthelifetimerangecorrespondstotheuncertainty inthe˝ttedprotonoverlap O 2 ,andisclearlynegligibleinmostcasesexceptfore.g. 41 Cr. Itisimportanttonotethatthelargeerrorbarsinthe Q 2 p predictionscorrespondtoseveral decadesofthetwo-protondecaylifetimeduetotheexponentialenergydependenceinthe WKBintegral.Theknowntwo-protonemitters x 2 p; known 19 Mg ; 45 Fe ; 48 Ni ; 54 Zn ; 67 Kr consistentlyfallwithinthetargetlifetimerange(5.24).Theblackdotscorrespondtonu- cleiwith p 2 p > 0 : 4 ,thisvalueof p 2 p isselectedratherthan p 2 p > 0 : 5 isbecausethe criterion(5.25)isslightlymorerestrictivethanthesingle-valueenergycriterionpreviously adoptedinRef.[201]. Wepredictthemostpromisingcandidatesfor true two-protonemission,otherthan 89 theknownones x 2 p; known ,are: 30 Ar ; 34 Ca ; 39 Ti ; 42 Cr ; 58 Ge ; 62 Se ; 66 Kr ; 70 Sr ; 74 Zr ; 78 Mo , 82 Ru ; 86 Pd ; 90 Cd ; and 103 Te . Thecalculated p 2 p isverylowfornucleiwith Z 54 thatalsofallintothelifetime range(5.24).ThelargeCoulombbarriersandtheconditionof p 2 p > 0 : 4 whichcorresponds tolow Q 2 p values,i.e.verylonglifetimes,resultedinthesmalltwo-protondecaywidths. Manyoftheveryproton-richnucleiwithsmall p 2 p valuessuchas 131 ; 132 Dy , 134 ; 135 Er ,and 144 ; 145 Hf ,areexcellentcandidatesforthedirect-decaymodeloftwo-protonemission[163]. These˝ndingsusingtheBMAmethodsaremostlyconsistentwithotherpredictions.The nuclei 39 Ti and 42 Cr ,areexpectedtobeexcellenttwo-protondecaycandidates[204,205]. The Q 2 p valuepredictedinBMA-Iisnotconsistentwiththeclaimthat 39 Ti primarily decaysvia decaybyRef.[186].Othertwo-protondecaycandidatespredictedbyBMA-I anddiscussedinotherliteratureinclude 26 S , 29 31 Ar [206], 34 Ca [207], 58 Ge , 62 Se ,and 66 Kr [208].Ref.[202]predicts 103 Te toexhibitacompetitionbetweenalphadecayandtwo- protondecay,and 145 Hf toexhibitcompetitionbetween decayanddirect-decaymodelof two-protonemission. 5.6Quanti˝edlimitsofthenuclearlandscape Thelatestprojectintheseriesisonthequanti˝edlimitsofthenuclearlandscape[108]. WithintheDFTframework,limitsofthenuclearlandscapewereprobedbyperforming globalmasssurveysusingseveralenergydensityfunctionals(EDFs):Skyrme[98,209,210], Gogny[53,211],andcovariant[210,Theproblemhowever,isthattheseearly studiesoftenlackeduncertaintyquanti˝cation.Infewcases[98,210,215],systematicuncer- taintieshavebeenestimatedbycombiningpredictionsofseveraldi˙erentsurveys,andby 90 performingsimplemeanaveraging. TheBayesianresidualmodeling,predictionsfortheprobabilityofexistence p ex ( Z;N ) followcloselytothedescriptionsinSec.5.2and5.4.Gaussianprocesswasselectedas ourBayesianmodel,usingthenon-zeromeanvariancede˝nedin(5.19).Fourresidual models:one-andtwo-neutronseparationenergies(forodd- N andeven- N ,respectively), one-andtwo-protonseparationenergies(forodd- Z andeven- Z ,respectively)weremodeled andtrainedseparately. FortheBMA,weconsideredeightmodelsbasedonmassescalculatedusingEDFsunder theHFBframework(Sec.2.1.3).Massesfromtheodd- Z andodd- N nucleiwerecalculated usingEq.(5.14).ThemorephenomenologicalmassmodelsFRDM-2012[169]andHFB- 24[170]arealsoincluded. TwovariantsofweightwereusedfortheBMA,eachfocusingondatafromeitherthe neutron-rich(BMA( n ))ortheproton-rich(BMA( p ))nuclearregions,followingsimilarmeth- odsdescribedinSec.5.4and5.5.Inordertoassessthenuclearlandscape,wealsoapplied athirdBMAvariantwhichassignedlocalmodelaveragingweightsforeachnucleus,called BMA( n + p ): w k ( Z;N )= w k ( n ) H N > N ( Z ) + w k ( p ) H N 0 j S 1 p= 2 p p S 1 n= 2 n > 0 j S 1 n= 2 n ; (5.27) where p S 1 p= 2 p > 0 j S 1 p= 2 p wasobtainedwithBMA( p )and p S 1 n= 2 n > 0 j S 1 n= 2 n with BMA( n ).Inpractice,oneofthesetwoprobabilitiesisalways ˇ 1 ,asonewouldexpectthe protonseparationenergiesforneutron-richnucleitobewellabovezeroandviceversa. Plotting p ex fortheentirenuclearlandscapegaveusthequanti˝edlimitofthenuclear landscapeshowninFig.5.12.Thedrip-linecorrespondsto p ex =0 : 5 ,andtherangesof 92 Figure5.12:Thequanti˝edlandscapeofnuclearexistenceobtainedinourBMAcalculations. High-resolutionofthis˝gurecanbefoundinRef.[217].Foreverynucleuswith Z;N > 8 and Z 6 119 theprobabilityofexistence p ex (5.27),i.e.,theprobabilitythatthenucleusis boundwithrespecttoprotonandneutrondecay,ismarked.Thedomainsofnucleiwhich havebeenexperimentallyobservedandwhoseseparationenergieshavebeenmeasured(and usedfortraining)areindicated.Toprovidearealisticestimateofthediscoverypotential withmodernradioactiveion-beamfacilities,theisotopeswithinFRIB'sexperimentalreach aremarked.Themagicnumbersareshownbystraight(white)dashedlines,andtheaverage lineof -stabilityde˝nedasinRef.[216]ismarkedbya(black)dashedline.Inourestimates, weassumedtheexperimentallimitforthecon˝rmationofexistenceofanisotopetobe1 event/2.5days.(FiguretakenfromRef.[108]) nuclearmassmeasurementsandknownnucleiaremarked.AccordingtotheBMA( n + p ) variantofmodelaveraging,we˝ndthenumberofparticle-boundnucleiwith Z;N > 8 and Z 6 119 tobe7708 534. Ontheproton-richsideofthelandscape,wecanseethatmanyheavyproton-unstable nucleiwillbereachedbyFRIB,whichcouldtestourpredictionsforthetwo-protonemitters (Sec.5.5.2).Ontheneutron-richside,FRIBcouldalsoexamineour p ex predictionsforthe neutronunstablesystemsinthelighttomediummassregion. 93 5.7Summary:Bayesianmachinelearning Inthischapter,westartedbyintroducingthemethodofusingBayesianneuralnetworks (BNN)andGaussianprocess(GP)toemulatetheresidualsofthetwoneutronseparation energyasawaytoimprovetheorymodels'predictability.Theresultingprobabilisticdescrip- tionofthemodeledresidualsgaveusawaytoquantifythesystematicmodeluncertaintyof theseparationenergypredictions.Thisleadtotheconceptoftheprobabilityofexistence p ex ,whichstatestheprobabilityofaneardrip-linenucleustohaveapositiveseparation energy.WealsoconcludedthattheGaussianprocessisabetterapproachcomparedto BNNtoinmodelingtheseparationenergyresiduals,duetoitsemphasisontheshort-range correlations. Usingthenotionof p ex ,weperformedanuncertainty-quanti˝edanalysisoftheneutron drip-lineintheCaregion,andpresentedaprobabilisticdescriptionfortheone-andtwo- neutrondrip-line.WealsointroducedBayesianmodelaveraging(BMA)techniques. Followingtheneutrondrip-lineanalysisintheCaregion,wesetoureyesontotheproton drip-line,andfurtherexperimentedwithvariousBMAweightingmethods.Wealsousedthe predictionfortheone-andtwo-protonseparationenergytoprovidepredictionsforpotential two-protonemitters,inadditiontotheuncertainty-quanti˝edprotondrip-linepredictions inthemedium-massregion.Inthisproject,wealsoincludedanon-zeromeanrandom variableintoourGPforthe˝rsttime,whichreducedthetheorymodels'rmsdeviationby anadditional15%comparedwiththeinitial25%reductionbroughtbytheGP. Finally,weexpandedtheBayesiantechniquesdevelopedinthe˝rstthreeprojectsto theentirenuclearlandscape,andprovidedprobabilityofexistencepredictionsforallnuclei with Z;N > 8 and Z 6 119 .Foralleven-even,even-odd,andodd-oddsystems,ourmodel 94 predicted7708 534nucleihaveprobabilityofexistencegreaterthan0.5.Wehopethis quanti˝edlimitofthenuclearlandscapecanprovideaguideforexperimentalresearchat next-generationrareisotopefacilityincludingtheFRIBfacilityonMSUcampusthatwill soonbecomeoperational. 95 Chapter6 ConclusionsandOutlook Thisdissertationconsistsoftwoparts:theapplicationsofnuclearDFTtopredictground statenuclearpropertiesonalargescale,andtheapplicationsofBayesianmachinelearning techniquesinnuclearphysics.Thecommonthemeislarge-scaleDFTcalculationsofnuclear propertiesacrossthenuclearlandscapeaidedbyhigh-performancecomputing. 6.1OctupoledeformationsandintrinsicSchi˙moments Theglobalsurveyofoctupole-deformedeven-evennucleiwasperformedusingnuclearDFT, inparticulartheSkyrmeHFBtheory.Thisservedasaprecursortothesecondproject, whichwasthecomputationofintrinsicSchi˙momentsinthevicinityofrobustcandidates foroctupole-deformedeven-evennuclei. IntheglobaloctupolesurveydiscussedinCh.3,resultsfrom˝veSkyrmeEDFsandfour covariantEDFswerecombinedtopresentthelandscapeofoctupolemultiplicity;thisgave usalessmodel-biasedpredictionoftheoctupoledeformationsineven-evennuclei.Wehave con˝rmedthatenhancedoctupoleinstabilitymostlyoccurintheneutron-de˝cientactinide regionandthesuperheavyactinideswith N 182 ,wherethelatterisfarfromcurrent experimentalreach.Amongnucleiintheneutron-de˝cientactinideregion,wepredicted thelargestoctupoledeformationsin 224 ; 226 Ra, 226 ; 228 Th, 224 ; 226 ; 228 U, 226 ; 228 ; 230 Pu,and 228 ; 230 Cm. 96 Bylookingattheoctupoledeformationenergy E oct intheoctupole-deformednuclei, wewerealsoabletoqualitativelydeterminetherigidnessofoctupoledeformation.For instance,althoughthepredicted 3 forlanthanidenucleisuchas 144 ; 146 Ba, 146 ; 148 Ce,and 146 ; 148 Ndarelargeandcomparablein 3 toRaandThisotopes,theiraveragevaluesof E oct oflessthan0.5MeVimplyveryshallowoctupoleminima,andsuggestthatoctupole deformationsarenotrigid.However,largeoctupolecorrelationsarestillexpectedforthese lanthanidenuclei,whichcouldbere˛ectedintheenhanced E 1 and E 3 transitionstrengths thatinmanycaseshavebeenmeasured.Wealsorealizethatbeyond-DFTe˙ects,crucial forsoftsystems,canplayanimportantroleinthelanthanideregionnuclei,whichcould enhancethetheoreticalpredictionsofoctupoledeformationinthisregion[218]. Theglobaloctupoledeformationsurveyhelpedustodeterminethebestcandidatesto conductSchi˙momentintheneutron-de˝cientactinideregion.Alistofpromisingcandidates fortheatomicEDMmeasurementarepresentedinTable.4.1. 6.2Bayesianmachinelearning Chapter5isasummaryoftheworksdonebetween2018and2020,whichinvolvedus- ingBayesianmachinelearningtechniquestoproduceuncertainty-quanti˝edpredictionsof nuclearstability. WebeganwithusingtheBayesianstatisticalmodelsBNNandGPtoproduceemulators forthe S 2n residualsofthetheorypredictionscomparedwithexperimentaldata.Thisserved asapilotstudytodeterminewhichstatisticalmodelisbetterformodelingtheseparation energyresiduals,andwhatimprovementscanbemadetoachievebetterperformance.It wasdeterminedthattheGPmodel,whichhasalargeremphasisonshort-rangecorrelations, 97 ismoresuitedforthemodelingoftheseparationenergyresiduals,andisalsomorestable fromacomputationalviewpoint. Thesecondprojectfocusedonthedetailedanalysisoftheneutrondrip-lineinthecalcium regionbymeansofBayesianmodelaveragingtechniques.Theconceptofprobabilityof existence p ex ofanucleuswasintroducedinthisprojectbyevaluatingtheprobabilityofa nucleustohaveapositiveseparationenergy. Wethenstudiedtheprotondrip-lineandmadepredictionsfortwo-proton( 2 p )emitters. Weemphasizedexperimentingwithdi˙erentvariantsoftheweightsforBMA.Anon-zero meanGPwasalsointroducedinthisproject,whichincreasedmodelperformancefroman averageof25%to40%rmsdeviationreduction. Finally,weexpandedthepredictionof p ex toincludeallnucleiwith Z;N > 8 and Z 6 119 ,andperformedBMAforeleventheory+GPmodels.Thisresultedinthequanti˝ed limitofthenuclearlandscape(Fig.5.12)whichpredictedthenumberofparticle-boundnuclei with Z;N > 8 and Z 6 119 tobe 7708 534 .Wealsohopetheestimatesofthedrip-lines couldguideexperimentalresearchforthediscoveryofexoticisotopesatFRIBandother next-generationrareisotopefacilities. 6.3Outlook Despitethemanyexcitingresultspresentedinthisdissertation,muchmoreawaitstobe done. Thecalculationperformedforoddsystemsinthisdissertationemployedapproximated blockingmethods,whichlackskeyinputsfromthetime-oddtermsoftheSkyrmeinteraction. ThiscanbeovercomebyusingthelatestversionofthesolverHFODD(v2.73y).However, 98 thiscanbecomputationallyexpensiveasHFODDissymmetry-unconstrained.Asaversatile solver,HFODDcouldalsobene˝tfromare-designoftheuserinterface,aswellasincor- poratingadditionalparallelprogrammingtopromotewideruseandbroaderapplications. Globalminimacalculationscanalsobene˝tfrommachinelearningtools. TheSchi˙momentcalculationpresentedhereislimitedtotheevaluationoftheintrinsic Schi˙moment.Thiscanbeexpandedtoincludethee˙ectsofP-,T-violatingpotential -suchfunctionalityhasalreadybeenimplementedinHFODD,butwasnotinvestigated inthisdissertation.Thecalculationofparitydoubletsusingbeyondmean-˝eldmethods, includingrestorationofintrinsically-brokensymmetries,isalsocrucialindeterminingthe magnitudeofSchi˙moment,albeitchallenging.Inthe˝rststep,onecouldlimitthescopeof calculationstotherobustoctupole-deformedodd-massandodd-oddsystems,byperforming aglobalsurveyoftheoddsystemswiththefullblockingmethod. TheapplicationofBayesianmachinelearninginnuclearphysicswilllikelybecomea majorfocus,withgreatpotentialfordiscoveries[219].Here,oneshouldalwayskeepaneye openfornewtechnologies,andbewellinformedofthelatestdevelopments. 99 APPENDICES 100 AppendixA:SupplementaryTables TableA.1: E oct (MeV)and 3 (inparentheses)valuescalculatedusing˝veSkyrmeEDFs: UNEDF0,UNEDF1,UNEDF2,SLy4,andSV-min.See(2.34)and(3.2)forde˝nitionsof 3 and E oct ,respectively.NucleiwithatleastthreeSkyrmeEDFspredictingthemas octupole-deformedareshown. NAUNEDF0UNEDF1UNEDF2SLy4SV-min Z=56(Ba) 561120.09(0.07)0.36(0.12)0.08(0.07) 881440.04(0.04)0.56(0.11)0.5(0.11)0.15(0.07) 901460.08(0.06)0.2(0.09)0.48(0.12)0.48(0.12)0.18(0.08) Z=58(Ce) 861440.51(0.09)0.27(0.09)0.05(0.04) 881460.37(0.1)0.9(0.13)0.55(0.12)0.22(0.08) 901480.25(0.11)0.45(0.14)0.14(0.12)0.09(0.08) Z=60(Nd) 861460.24(0.08)0.75(0.1)0.23(0.09)0.05(0.04) 881480.3(0.1)1.15(0.14)0.35(0.11)0.08(0.06) 1361960.03(0.03)0.07(0.06)1.03(0.14)0.28(0.1) 1381980.07(0.06)0.5(0.15)0.3(0.1) Z=62(Sm) 1321940.05(0.03)0.51(0.08)0.62(0.1)0.1(0.05) 1341960.24(0.08)0.79(0.11)1.05(0.14)0.38(0.1) 1361980.35(0.1)0.89(0.12)1.28(0.15)0.46(0.12) 101 TableA.1 ( cont 0 d ) NAUNEDF0UNEDF1UNEDF2SLy4SV-min Z=64(Gd) 1321960.34(0.08)1.06(0.11)0.87(0.12)0.25(0.07) 1341980.59(0.11)1.45(0.13)1.02(0.15)0.53(0.11) 1362000.45(0.13)1.21(0.15)0.81(0.16)0.33(0.12) Z=66(Dy) 1321980.37(0.09)1.36(0.12)0.86(0.12)0.24(0.07) 1342000.52(0.12)1.68(0.14)0.5(0.14)0.29(0.1) 1362020.15(0.14)1.13(0.16)0.28(0.13) Z=68(Er) 1322000.17(0.08)1.21(0.12)0.18(0.1)0.05(0.05) 1342020.17(0.1)1.36(0.15)0.05(0.08) Z=86(Rn) 1322180.86(0.1)0.24(0.07)0.6(0.09)0.52(0.09)0.36(0.08) 1342200.95(0.12)0.31(0.09)0.2(0.11)0.67(0.11)0.47(0.1) 1362220.88(0.12)0.15(0.09)0.56(0.12)0.33(0.1) 1382240.55(0.12)0.26(0.09)0.1(0.07) 1922780.04(0.04)0.19(0.05)0.13(0.05) 1942800.18(0.07)0.01(0.06)0.36(0.1) 1962820.25(0.09)0.36(0.11)0.03(0.04) Z=88(Ra) 1302180.91(0.1)0.47(0.07)0.79(0.09)0.63(0.08) 1322201.24(0.12)1.07(0.11)1.41(0.12)1.58(0.12)1.14(0.11) 1342221.41(0.14)1.27(0.13)1.48(0.14)1.81(0.14)1.33(0.13) 1362241.24(0.14)1.01(0.14)0.84(0.14)1.65(0.15)1.1(0.14) 1382260.77(0.14)0.51(0.13)0.09(0.08)1.17(0.15)0.61(0.13) 1402280.29(0.1)0.12(0.07)0.42(0.11)0.16(0.08) 1922800.65(0.09)0.25(0.05)1.02(0.11)0.58(0.08) 1942820.91(0.11)0.0(0.04)1.51(0.13)0.75(0.11) 1962840.97(0.12)0.28(0.09)1.58(0.14)0.79(0.12) 1982860.79(0.12)0.22(0.08)0.9(0.14)0.49(0.11) 2002880.45(0.12)0.12(0.06)0.19(0.07)0.13(0.07) 102 TableA.1 ( cont 0 d ) NAUNEDF0UNEDF1UNEDF2SLy4SV-min Z=90(Th) 1302200.96(0.11)0.99(0.1)0.38(0.09)1.25(0.11)1.05(0.1) 1322221.21(0.13)1.66(0.13)2.3(0.14)1.91(0.13)1.56(0.13) 1342241.16(0.14)1.72(0.15)2.28(0.16)1.85(0.15)1.54(0.15) 1362260.72(0.15)1.13(0.16)1.2(0.16)1.19(0.17)0.99(0.15) 1382280.18(0.13)0.46(0.15)0.25(0.15)0.63(0.17)0.33(0.15) 1922821.04(0.11)0.45(0.06)1.66(0.12)1.09(0.11) 1942841.25(0.13)0.99(0.12)0.3(0.07)2.11(0.14)1.41(0.13) 1962861.11(0.14)1.14(0.13)1.91(0.16)1.3(0.14) 1982880.62(0.14)0.81(0.13)1.06(0.16)0.75(0.14) 2002900.05(0.12)0.3(0.11)0.11(0.12) Z=92(U) 1302220.79(0.11)1.35(0.12)0.39(0.11)1.46(0.12)1.25(0.11) 1322240.81(0.13)1.8(0.14)2.54(0.15)1.83(0.14)1.49(0.14) 1342260.52(0.14)1.48(0.16)2.08(0.17)1.46(0.16)1.19(0.15) 1362280.73(0.17)0.87(0.17)0.59(0.17)0.51(0.16) 1382300.14(0.15)0.05(0.16)0.15(0.17) 1902820.89(0.1)0.66(0.11)0.16(0.09) 1922841.04(0.12)0.23(0.11)0.53(0.08)2.03(0.14)1.53(0.12) 1942861.04(0.13)1.49(0.13)0.69(0.09)2.24(0.15)1.65(0.14) 1962880.7(0.14)1.51(0.14)0.38(0.09)1.67(0.17)1.21(0.15) 1982900.09(0.13)0.76(0.15)0.8(0.17)0.5(0.15) Z=94(Pu) 1302240.44(0.11)1.61(0.13)1.58(0.12)1.56(0.13)1.29(0.12) 1322260.3(0.11)1.58(0.15)3.26(0.15)1.35(0.15)1.06(0.14) 1342280.04(0.1)0.91(0.16)1.42(0.17)0.85(0.16)0.62(0.15) 1362300.32(0.16)0.33(0.17)0.4(0.16)0.2(0.14) 1902840.9(0.11)0.77(0.09)1.68(0.09)1.93(0.12)1.0(0.11) 1922860.74(0.12)1.71(0.12)2.16(0.11)2.42(0.14)1.92(0.13) 1942880.55(0.13)2.06(0.14)1.76(0.12)2.16(0.16)1.59(0.15) 1962900.19(0.12)1.4(0.16)0.94(0.12)1.24(0.17)0.85(0.16) 1982920.4(0.16)0.43(0.15)0.29(0.18)0.07(0.15) 103 TableA.1 ( cont 0 d ) NAUNEDF0UNEDF1UNEDF2SLy4SV-min Z=96(Cm) 1282240.31(0.07)0.37(0.08)0.6(0.08)0.45(0.09)0.15(0.07) 1302260.05(0.1)1.73(0.14)3.11(0.13)1.57(0.14)1.21(0.13) 1322281.22(0.16)3.23(0.16)0.83(0.15)0.5(0.15) 1342300.36(0.17)0.98(0.18)0.31(0.16) 1882840.69(0.09)1.03(0.09)1.85(0.09)1.6(0.11)0.53(0.08) 1902860.83(0.11)2.4(0.12)3.19(0.12)2.93(0.14)1.92(0.12) 1922880.36(0.12)2.45(0.14)3.62(0.13)2.67(0.15)2.15(0.14) 1942900.07(0.1)2.12(0.15)2.91(0.14)2.05(0.17)1.45(0.15) 1962921.28(0.17)1.92(0.15)1.11(0.18)0.62(0.16) 1982940.29(0.17)0.7(0.16)0.2(0.18) Z=98(Cf) 1282260.26(0.07)0.86(0.1)1.38(0.1)0.96(0.1)0.46(0.08) 1302281.67(0.14)3.32(0.14)1.49(0.14)0.82(0.13) 1322300.71(0.17)2.76(0.17)0.53(0.16) 1862840.01(0.02)0.45(0.07)0.77(0.08)0.86(0.09)0.21(0.06) 1882860.8(0.09)1.82(0.11)2.75(0.11)2.47(0.12)1.25(0.1) 1902880.74(0.11)2.7(0.13)3.71(0.13)3.28(0.14)2.61(0.13) 1922900.07(0.1)2.46(0.15)4.0(0.15)2.68(0.16)2.06(0.15) 1942921.93(0.17)3.1(0.16)1.95(0.17)1.32(0.16) 1962941.29(0.17)1.84(0.17)1.26(0.18)0.55(0.17) 1982960.46(0.17)0.7(0.18)0.35(0.18) Z=100(Fm) 1262260.17(0.05)0.42(0.07)0.33(0.06)0.03(0.03) 1282280.16(0.06)1.05(0.11)1.67(0.11)1.17(0.1)0.58(0.09) 1302301.28(0.15)2.76(0.15)1.22(0.15) 1322320.27(0.17)2.32(0.18)0.35(0.16) 1842840.09(0.04)0.19(0.06)0.45(0.07)0.02(0.02) 1862860.1(0.04)0.75(0.09)1.1(0.09)1.29(0.1)0.48(0.07) 1882880.79(0.09)2.08(0.12)2.74(0.12)2.8(0.13)1.52(0.11) 1902900.61(0.11)2.61(0.14)3.83(0.14)3.21(0.15)2.42(0.14) 1922922.2(0.16)3.86(0.16)2.5(0.17)1.84(0.15) 1942941.59(0.17)2.86(0.17)1.65(0.18)1.09(0.17) 1962961.07(0.18)1.47(0.18)1.17(0.18)0.39(0.16) 1982980.38(0.17)0.47(0.18)0.24(0.17) 104 TableA.1 ( cont 0 d ) NAUNEDF0UNEDF1UNEDF2SLy4SV-min Z=102(No) 1282300.05(0.04)0.87(0.11)1.47(0.11)0.9(0.1)0.45(0.08) 1842860.14(0.05)0.25(0.06)0.48(0.08)0.06(0.04) 1862880.1(0.04)0.76(0.09)1.09(0.1)1.22(0.1)0.49(0.08) 1882900.63(0.08)1.98(0.12)2.47(0.13)2.53(0.13)1.37(0.11) 1902920.38(0.11)2.39(0.15)3.72(0.15)2.92(0.16)2.0(0.14) 1922941.9(0.17)3.61(0.17)1.46(0.18)1.08(0.16) 1942961.07(0.17)2.55(0.18)0.15(0.18)0.29(0.16) Z=104(Rf) 1842880.09(0.05)0.19(0.06)0.29(0.07)0.02(0.02) 1862900.03(0.03)0.57(0.09)0.86(0.09)0.86(0.09)0.32(0.07) 1882920.39(0.07)1.66(0.12)2.1(0.13)1.94(0.13)1.0(0.1) 1902940.1(0.1)1.91(0.15)2.64(0.16)1.25(0.16)0.53(0.14) Z=106(Sg) 1842900.01(0.02)0.06(0.04)0.07(0.05) 1862920.33(0.07)0.57(0.08)0.45(0.08)0.14(0.06) 1882940.15(0.05)0.74(0.12)0.47(0.13)0.4(0.12)0.45(0.09) 105 TableA.2:Protonquadrupole Q 20 (fm 2 ) andoctupole Q 30 (fm 3 ) moments(inparentheses) foroctupole-deformedeven-evennucleiwithpredicted 3 0 : 01 from˝veSkyrmeEDFs: UNEDF0,UNEDF1,UNEDF2,SLy4,andSV-min.See(2.33)forde˝nitionsof Q 20 , Q 30 . Theprotonmultipolemomentscloselyresemblechargemultipolemoments,andcanbeused tocomparewithexperimentaldataderivedfromtransitionstrengths.Averagevaluesare shownintherightmostcolumn.Allvaluesroundedtointegers. NAUNEDF0UNEDF1UNEDF2SLy4SV-minAverage Z=56(Ba) 56112352(524)360(862)351(485)354(623) 88144292(339)306(952)330(932)310(593)309(704) 90146333(496)356(789)359(1058)368(1048)352(705)353(819) Z=58(Ce) 86144248(818)295(776)275(369)272(654) 88146346(950)323(1182)363(1086)347(779)344(999) 90148405(1046)396(1308)425(1150)418(775)411(1069) Z=60(Nd) 86146277(804)227(991)297(834)286(430)271(764) 88148373(1007)308(1344)400(1026)393(608)368(996) 136196258(409)155(640)369(1666)339(1113)280(957) 138198297(743)431(1779)390(1188)372(1236) Z=62(Sm) 132194170(430)166(1019)209(1287)176(573)180(827) 134196223(1024)197(1354)289(1676)264(1201)243(1313) 136198263(1297)207(1532)367(1897)339(1418)294(1536) Z=64(Gd) 132196184(1127)174(1491)186(1516)169(958)178(1273) 134198225(1501)200(1781)279(1893)251(1436)238(1652) 136200264(1744)205(1957)385(2025)357(1551)302(1819) Z=66(Dy) 132198196(1328)177(1738)186(1634)165(1010)181(1427) 134200241(1728)197(2050)310(1892)271(1440)254(1777) 136202307(1947)194(2251)437(1824)312(2007) 106 TableA.2 ( cont 0 d ) NAUNEDF0UNEDF1UNEDF2SLy4SV-minAverage Z=68(Er) 132200223(1195)183(1824)210(1498)199(711)203(1307) 134202307(1529)192(2185)396(1073)298(1595) Z=86(Rn) 132218351(2152)382(1453)293(1781)411(1799)374(1593)362(1755) 134220431(2440)463(1896)414(2270)476(2196)452(1987)447(2157) 136222488(2618)524(1921)531(2378)511(2043)513(2240) 138224538(2573)582(1927)563(1458)561(1986) 192278359(928)226(1198)370(1183)318(1103) 194280443(1703)291(1386)482(2293)405(1794) 196282498(2079)550(2644)482(1055)510(1926) Z=88(Ra) 130218349(2142)362(1531)408(1935)362(1669)370(1819) 132220472(2650)497(2389)445(2617)515(2547)489(2430)483(2526) 134222556(2957)579(2814)559(3068)587(2973)569(2806)570(2923) 136224619(3144)649(3033)637(3145)648(3253)633(2990)637(3113) 138226682(3053)712(2783)692(1821)708(3224)694(2791)697(2734) 140228752(2310)770(1595)756(2376)750(1846)757(2031) 192280504(2442)249(1300)508(2706)442(1990)425(2109) 194282577(2890)336(1072)599(3333)558(2755)517(2512) 196284633(3154)580(2173)668(3674)631(3051)628(3013) 198286683(3264)651(2129)724(3637)694(2902)688(2983) 200288735(3106)716(1589)780(1987)755(1982)746(2166) Z=90(Th) 130220424(2467)430(2240)345(2027)461(2432)434(2283)418(2289) 132222557(2950)564(2919)510(3071)580(3010)562(2915)554(2973) 134224649(3273)660(3352)635(3530)662(3459)652(3294)651(3381) 136226725(3426)735(3592)723(3720)734(3749)726(3493)728(3596) 138228806(3063)809(3475)798(3423)804(3734)798(3338)803(3406) 192282582(3045)177(1461)558(3280)524(2922)460(2677) 194284663(3457)603(3097)235(1722)661(3841)633(3484)559(3120) 196286728(3723)690(3469)740(4235)713(3791)717(3804) 198288788(3801)761(3517)802(4383)782(3811)783(3878) 200290859(3476)831(3168)855(3206)848(3283) 107 TableA.2 ( cont 0 d ) NAUNEDF0UNEDF1UNEDF2SLy4SV-minAverage Z=92(U) 130222473(2640)450(2673)356(2437)477(2782)462(2653)443(2637) 132224628(3058)600(3304)526(3386)619(3344)610(3249)596(3268) 134226739(3301)719(3717)682(3884)726(3782)725(3603)718(3657) 136228819(3940)798(4085)818(4045)820(3735)813(3951) 138230913(3691)898(3747)907(3892)906(3776) 190282503(2884)403(3003)340(2415)415(2767) 192284629(3396)498(3075)159(1982)566(3706)543(3427)479(3117) 194286727(3763)640(3739)185(2285)690(4257)667(3948)581(3598) 196288810(3972)741(4125)187(2292)784(4657)764(4263)657(3861) 198290896(3804)828(4249)863(4838)853(4277)860(4292) Z=94(Pu) 130224509(2667)439(3015)340(2769)458(3058)456(2923)440(2886) 132226711(2760)607(3621)484(3597)633(3592)647(3486)616(3411) 134228857(2384)774(3988)690(4182)793(3940)809(3696)784(3638) 136230915(4011)877(4278)922(3957)939(3314)913(3890) 190284497(3163)316(2714)230(2540)384(3470)362(3069)357(2991) 192286654(3574)490(3639)283(3050)537(4074)525(3814)497(3630) 194288787(3779)627(4213)314(3346)674(4571)664(4303)613(4042) 196290910(3532)741(4587)370(3525)792(4983)788(4603)720(4246) 198292857(4707)688(4276)909(5125)921(4395)843(4625) Z=96(Cm) 128224193(1769)184(2052)155(1916)189(2113)125(1621)169(1894) 130226564(2334)417(3334)345(3174)417(3308)428(3150)434(3060) 132228600(3942)453(3863)629(3855)674(3642)589(3825) 134230821(4165)635(4434)845(3893)767(4164) 188284267(2596)238(2681)243(2834)251(3105)175(2408)234(2724) 190286465(3372)368(3539)330(3514)372(3933)353(3529)377(3577) 192288671(3591)487(4177)378(3961)499(4460)496(4167)506(4071) 194290879(2864)614(4667)420(4290)651(4904)651(4646)643(4274) 196292734(5022)490(4574)788(5311)799(4902)702(4952) 198294873(5111)666(4924)935(5342)824(5125) 108 TableA.2 ( cont 0 d ) NAUNEDF0UNEDF1UNEDF2SLy4SV-minAverage Z=98(Cf) 128226166(1718)195(2568)193(2613)173(2519)139(2119)173(2307) 130228404(3672)361(3606)380(3572)395(3359)385(3552) 132230599(4279)472(4269)615(4113)562(4220) 1862847(575)116(2282)136(2494)125(2709)56(1733)88(1958) 188286228(2789)270(3408)277(3513)255(3675)210(3130)248(3303) 190288415(3523)392(4155)372(4169)362(4374)342(3938)376(4031) 192290708(3223)517(4771)428(4607)482(4896)484(4554)523(4410) 194292652(5229)482(4964)670(5336)659(5011)615(5135) 196294771(5514)581(5341)804(5653)814(5137)742(5411) 198296897(5508)743(5584)936(5568)858(5553) Z=100(Fm) 12622628(1456)57(1887)31(1737)8(719)31(1449) 128228130(1503)176(2834)177(2947)133(2710)112(2289)145(2456) 130230407(4023)384(4041)365(3864)385(3976) 132232606(4556)506(4709)617(4347)576(4537) 18428423(1430)44(1806)47(2317)4(680)29(1558) 18628631(1445)118(2825)133(2981)111(3187)73(2429)93(2573) 188288179(2853)260(3830)276(3956)227(4034)192(3473)226(3629) 190290359(3580)411(4684)401(4702)360(4832)328(4285)371(4416) 192292554(5341)466(5161)493(5415)489(4952)500(5217) 194294704(5718)532(5550)703(5735)682(5292)655(5573) 196296827(5817)663(5963)831(5831)847(5087)792(5674) 198298942(5494)841(5873)949(5420)910(5595) Z=102(No) 12823078(947)140(2878)140(3052)70(2640)61(2196)97(2342) 18428616(1825)27(2101)21(2508)7(1227)17(1915) 18628817(1449)83(2992)97(3154)66(3308)50(2579)62(2696) 188290124(2749)243(4082)271(4264)193(4218)152(3571)196(3776) 190292318(3480)421(5089)420(5131)377(5270)324(4545)372(4703) 192294558(5713)480(5580)496(5835)494(5232)507(5590) 194296733(5883)537(5942)723(5838)719(5249)678(5728) 109 TableA.2 ( cont 0 d ) NAUNEDF0UNEDF1UNEDF2SLy4SV-minAverage Z=104(Rf) 184288-3(1680)-6(2016)-10(2288)0(839)-4(1705) 1862901(923)39(2919)45(3122)15(3165)17(2417)23(2509) 18829264(2437)222(4192)251(4443)162(4254)101(3465)160(3758) 190294303(3160)406(5299)407(5377)371(5516)313(4643)360(4799) Z=106(Sg) 184290-6(867)-22(1504)-19(1592)-15(1321) 1862925(2621)0(2913)-22(2792)-4(1906)-5(2558) 18829415(1830)194(4160)215(4496)128(4152)48(3155)120(3558) 110 TableA.3:ReferencetabletoFig.5.10:even- Z elements.Foreachatomicelementwith even- Z shownare:theneutronnumber N 0 ofthelightestisotopeforwhichanexperimental one-ortwo-protonseparationenergyvalueisavailable;theneutronnumber N obs ofthe lightestisotopeobserved;theneutronnumber N drip ofthepredicteddriplineisotopein BMA-I;andtheneutronnumber N FRIB markingthereachofFRIB.(Tabletakenfrom Ref.[109]) Z Elem. N 0 N obs N drip N FRIB 16S12111110 18Ar14111312 20Ca16151514 22Ti18171817 24Cr21181918 26Fe23192019 28Ni25202220 30Zn28242523 32Ge31272825 34Se33293028 36Kr35313231 38Sr37353533 40Zr40373735 42Mo43393936 44Ru46414138 46Pd48444340 48Cd50464542 50Sn50494745 52Te53525352 54Xe55545554 56Ba58585857 58Ce68636057 60Nd70656260 62Sm73676663 64Gd76716966 66Dy77737269 68Er78767574 70Yb81797874 72Hf84828077 74W86838380 76Os88858684 78Pt90889087 80Hg94919488 82Pb98969793 111 TableA.4:ReferencetabletoFig.5.10:odd- Z elements.Foreachatomicelementwithodd- Z shownare:theneutronnumber N 0 ofthelightestisotopeforwhichanexperimentalone- ortwo-protonseparationenergyvalueisavailable;theneutronnumber N obs ofthelightest isotopeobserved;theneutronnumber N drip ofthepredicteddriplineisotopeinBMA-I;and theneutronnumber N FRIB markingthereachofFRIB.(TabletakenfromRef.[109]) Z Elem. N 0 N obs N drip N FRIB 17Cl14111414 19K16161616 21Sc19181918 23V20202019 25Mn22212221 27Co24232423 29Cu27262725 31Ga30292928 33As33313131 35Br35343333 37Rb37353535 39Y40373737 41Nb42414139 43Tc44434340 45Rh47444542 47Ag49454744 49In51474947 51Sb55525552 53I57555755 55Cs62576157 57La67606158 59Pr69626561 61Pm72676864 63Eu74677167 65Tb76707569 67Ho79737872 69Tm82768175 71Lu85798376 73Ta87828778 75Re91849081 77Ir95879384 79Au97919787 81Tl1029510290 112 AppendixB:Listofmycontributions 1. LéoNeufcourt, YuchenCao ,WitoldNazarewicz,andFrederiViens, "Bayesianap- proachtomodel-basedextrapolationofnuclearobservables" ,Phys.Rev.C. 98 ,034318 (2018) ‹ Producedandcompilednucleartheoryandexperimentaldata, ‹ ProducedFigures1,2,3,5,and6. 2. LéoNeufcourt, YuchenCao ,WitoldNazarewicz,ErikOlsen,andFrederiViens, "NeutronDripLineintheCaRegionfromBayesianModelAveraging" ,Phys.Rev. Lett. 122 ,062502(2019) ‹ Producedandcompiledrawnucleartheoryandexperimentaldata, ‹ ProducedFig.1, ‹ Contributedtoproducingthelistofallobservedisotopesandnucleiwithmasses measured. 3. LéoNeufcourt, YuchenCao ,SamuelGiuliani,WitoldNazarewicz,ErikOlsen,and OlegB.Tarasov, "Beyondtheprotondripline:Bayesiananalysisofproton-emitting nuclei" ,Phys.Rev.C 101 ,014319(2020) 113 ‹ Producedandcompiledrawnucleartheoryandexperimentaldata, ‹ Suggestedintroductionofnon-zeromeanparametertotheGaussianprocess, ‹ Performedcalculationstoestimatethelifetimesoftruetwo-protonemitters, ‹ ProducedFig.5. 4. LéoNeufcourt, YuchenCao ,SamuelGiuliani,WitoldNazarewicz,ErikOlsen,and OlegB.Tarasov, "Quanti˝edlimitsofthenuclearlandscape" ,Phys.Rev.C 101 , 044307(2020) ‹ Producedandcompiledrawnucleartheoryandexperimentaldata. 5. YuchenCao ,SylvesterE.Agbemava,AnatoliV.Afanasjev,WitoldNazarewicz,and ErikOlsen, "Landscapeofpear-shapedeven-evennuclei" ,arXiv:2004.01319(2020) ‹ PerformedglobalSkyrmeHFBcalculationsforoctupole-deformedeven-evennu- clei,includingnecessarycomputationaldevelopments(seep.7), ‹ Performedcalculationsofsingleparticleenergiesintheoctupole-deformedregion, ‹ Wrotethe˝rstdraft(exceptforSec.II(B)), ‹ Producedall˝guresinthispaper. 6. MassExplorer ‹ Developedsourcecodes(withE.Olsen)formainpageof"PlottingTools"using HTML. ‹ Developedsourcecodes(individually)fortheDataSearche/Iso- tone/IsobarChainEnergyandQuadrupoleDefor- mationfunctionalitiesusingJavaScript. 114 ‹ Developedsourcecodes(withE.OlsenandA.Savanur)forthe functionalityusingJavaScript. 7. Othercontributions ‹ ImplementeddynamicMPIschedulinginHFBTHO(v3.00)masstablemode, ‹ IntroducedQ30constraintsinHFBTHO(v3.00)masstablemode, ‹ ImplementedcalculationofLipkin-NogamicorrecteddeformationsinHFBTHO (v3.00), ‹ ImplementedcalculationofintrinsicSchi˙momentinHFBTHO(v3.00), ‹ DevelopedPythoncodeforautomatingblockingcalculationwithHFBTHO, ‹ DevelopedPythoncodeforautomatingblockingcalculationwithHFODD, ‹ DevelopedPythoncodeforautomatingsingleparticleorbitalcalculationswith HFBTHO, ‹ DevelopedPythoncodeforremotestatusmonitoringoflargescalesurveycalcu- lations, ‹ ServedasgraduatementorfortheNationalScienceFoundation ResearchExperi- enceforTeachers (RET)programatMSUinsummer2019. 115 BIBLIOGRAPHY 116 BIBLIOGRAPHY [1] E.Rutherford.Thescatteringofalphaandbetaparticlesbymatterandthestructure oftheatom. Phil.Mag.Ser.6 ,1911. [2] C.F.v.Weizsäcker.Zurtheoriederkernmassen. ZeitschriftfürPhys. ,, July1935. 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