Probing proton-neutron pairing with Gamow-Teller strengths ...
Transcript of Probing proton-neutron pairing with Gamow-Teller strengths ...
Probingproton-neutronpairingwithGamow-Tellerstrengthsintwo-
nucleonconfigura8ons
YutakaUtsunoAdvancedScienceResearchCenter,JapanAtomicEnergyAgency
CenterforNuclearStudy,UniversityofTokyo
Y.UtsunoandY.Fujita,arXiv1701.00869
Introduc8on:Fujita-sanโsseminarinJan.2016โขโฏ Systema8csofGamow-Tellerdistribu8onsforN=Z+2โN=Znuclei
โโฏ Usingchargeexchangereac8on(3He,t)
ObservedGTstrengthโขโฏ Concentra8oninthe1+1levelfor
42Caโ42Sc
โโฏ B(GT;0+1โ1+1)=2.2
Y.Fujitaetal.,Phys.Rev.Le`.112,112502(2014).
โlow-energysuperGTstateโ
Ikedasumruleโขโฏ WhenB(GT)isdefinedas๐ต(GTโยฑโ;๐โ๐)= |๐ผโ๐โ๐ฝโ๐โ |๐๐กโยฑโ| ๐ผโ๐โ๐ฝโ๐โโ|โ2โ/2๐ฝโ๐โ+1โ=โ๐๐โ๐โโโ|๐ผโ๐โ๐ฝโ๐โ๐โ๐โ๐โ๐โ๐กโยฑโ ๐ผโ๐โ๐ฝโ๐โ๐โ๐โโ|โ2โโ,โ๐โโ๐ต(GTโโโ;๐โ๐)โโโ๐โโ๐ต(GTโ+โ;๐โ๐)=3(๐โ๐)โissa8sfied.(with ๐กโโโ| ๐โฉ=| ๐โฉ)
โขโฏ IfGT+transi8onishinderedbyPauliblocking,โ๐โโ๐ต(GTโโโ;๐โ๐)โiscloseto3(๐โ๐).
โโฏ โ๐โโ๐ต(GTโโโ;๐โ๐)โfor42Cashouldbecloseto6,ifprotonexcita8ontothepfshellissmall.
proton
neutron
SU(4)?single-par8cletransi8on?โขโฏ Thesitua8onissimilartowhatisexpectedfromtheSU(4)
symmetry(nospinorisospindependentforces).
โขโฏ SU(4)isstronglybrokenbytheexistenceofspin-orbitsplifng.
โขโฏ Single-par8clestructure?
f7/2โf7/2f7/2โf5/2
ฮต(f5/2)-ฮต(f7/2)
B(GT)distribu8onwithpureconfigura8ons
โขโฏ Concentra8oninasinglestatecannotbeaccountedforbyasimpleshell-modelargument.
Fromthesimplesingle-par8clepicture
Experimentalstrength
0 10
Essen8alroleofconfigura8onmixing
Y.Fujitaetal.,Phys.Rev.C91,064316(2015)
โขโฏ RPAcalcula8onโโฏ Isoscalarpairingisessen8al.
โขโฏ CoherenceintheGTtransi8onisalsoobtainedbyshell-modelcalc.
โขโฏ Whyiscollec8vitycreated?
C.L.Baietal.,Phys.Rev.C90,054335(2014).Shell-modelanalysis
Two-par8clevs.two-holeconfigura8onsโขโฏ Aques8onbysomeone(sorry,Idonotrememberwhoraised)
โโฏWhatabouttwoholesystems?
twohole๏ผ14Cโ14NlogA=9.1
p3/2
p1/2
two-par8cle๏ผ6Heโ6Li
logA=2.9
protons neutrons
B(GT) Two-par0cle Two-holep sd pf p sd
A=6 A=18 A=42 A=14 A=381+1 4.7 3.1 2.2 3.5ร10-6 0.0601+2 0.13 0.10 2.8 1.5
Distribu8onofknownlogA
6Heโ6Li 14Cโ14N
Num
bero
fcases
Ques8onsโขโฏ Whydoestheโlow-energysuperGTstateโappearfortwo-par8cle
configura8ons?
โขโฏ WhyisthecorrespondingB(GT)valuesfortwo-holeconfigura8onsverysmall?
โขโฏ Whatdoesthoseproper8estellusaboutisovectorandisoscalarpairingproper8es,sincecorrelatedtwonucleonsformaโCooperpairโ?
Unifiedtreatmentofp-pandh-hsystemsโขโฏ Thefollowingtwodescrip8onsareiden8cal
โโฏ Par8cleHamiltonian๐ป=โ๐โโ๐โ๐โ๐โ๐โโโโ โ๐โ๐โโ+ 1/4โโ๐,๐,๐,๐โโ๐ฃโ๐๐;๐๐โโ๐โ๐โโโโ โ๐โ๐โโโโ โ๐โ๐โ๐โ๐โ
โโฏ HoleHamiltonian๐ป=โ๐โโ๐โโ๐โ๐โ๐โโโโ โ๐โ๐โโ+ 1/4โโ๐,๐,๐,๐โโ๐ฃโโ๐๐;๐๐โโ๐โ๐โโโโ โ๐โ๐โโโโ โ๐โ๐โ๐โ๐โ
when๐ฃโ๐๐;๐๐โ= ๐ฃโโ๐๐;๐๐โand ๐โโ๐โ=๐ธ( ๐โโ1โ)aresa8sfied.
Two-bodymatrixelementsarethesame,butthesingle-par8cleenergiesareinthereversedorderforthehole-holeHamiltonian.
p3/2
p1/2
GTstrengthin2pand2hconf.:p-shellcaseโขโฏ Asshowninthelastslide,boththe6Heโ6Li(2p)and14Cโ14N
(2h)decayscanbedescribedastwo-nucleonsystemswiththesametwo-bodymatrixelements.
โขโฏ Theonlydifferencebetweenthemisthesingle-par8clesplifngฮ๐โ๐โ=๐( ๐โ1/2โ) โ๐( ๐โ3/2โ):โโฏ Forthepar8clecase:ฮ๐โ๐โ=0.1 MeV
โโฏ Fortheholecase:ฮ๐โ๐โ=โ6.3 MeV
โขโฏ Itisinteres8ngtoplottheGamow-Tellermatrixelement๐(๐บ๐)= ๐ฝโ๐โ|๐๐กโโโ| ๐ฝโ๐โโasafunc8onofฮ๐โ๐โforgiventwo-bodyinterac8onsinordertoseethedependenceonthesingle-par8cleenergies.
TakenfromtheCohen-KurathโsCKIIinterac8on
(1)Cohen-KurathโsCKIIinterac8onโขโฏ Oneofthemostpopularempiricalinterac8onsforthepshell.
โโฏ 15two-bodymatrixelementsarededucedbyfifngenergylevelsofA=8-16nuclei.
(2)Pairinginterac8onโขโฏ Isoscalar-andisovector-pairinginterac8onsaredefinedas๐โIVpairโ= ๐บโIVโโ๐โโ๐โ๐โโ โโ๐โ๐โ๐โISpairโ= ๐บโISโโ๐โโ๐ทโ๐โโ โโ๐ทโ๐โwhere๐โ๐โโ โ=โ1/2โโ๐๐โโ(โ1)โ๐โโ2๐+1โ[๐โ๐๐โโ โร ๐โ๐๐โโ โ]โโ๐โ๐ฟโ=0,๐โ๐โ=0,๐โ๐โ=๐โ๐ฟ=0,๐=0,๐=1โand ๏ฟฝ๐ทโ๐โโ โ=โ1/2โโ๐๐โโ(โ1)โ๐โโ2๐+1โ[๐โ๐๐โโ โร ๐โ๐๐โโ โ]โโ๐โ๐ฟโ=0,๐โ๐โ=๐, ๐โ๐โ=0โ๐ฟ=0,๐=1,๐=0โ.
โขโฏ Isovector-paircrea8onoperator ๐โ๐โโ โandisoscalar-paircrea8onoperators ๐ทโ๐โโ โaresymmetricintermsofSandT.
M(GT)withthepairinginterac8on
โขโฏ M(GT):Enhancedforsmallฮ๐โ๐โandnovanishingforฮ๐โ๐โ<0.โโฏ Largerthansingle-par8clelimitsonthebothsides
โขโฏ StrengthsofGISandGIVaredeterminedsoastoreproducethemeansquare(J,T)=(1,0)and(0,1)matrixelementsofCKII.
p3/2limit:โ10/3โโ1.83
p1/2limit:โ2/3โโ0.82
CoherenceoftheGTmatrixelementโขโฏ Whentwo-nucleonwavefunc8onsaredecomposedintobasis
statesas| 0โ๐โ+โโฉ=โ๐๐โโ๐ผโ๐๐โIVโ(๐)โ| ๐๐ ๐ฝ=0 ๐=1โฉand| 1โ๐โ+โโฉ=โ๐๐โโ๐ผโ๐๐โI๐โ(๐)โ| ๐๐ ๐ฝ=1 ๐=0โฉ,theGamow-Tellermatrixelementiswri`enas๐(๐บ๐;1โ๐โ+โ)=โ๐๐๐๐โโ๐โ๐๐๐๐โ( 1โ๐โ+โ)โwhere๐โ๐๐๐๐โ(1โ๐โ+โ)= ๐ผโ๐๐โI๐โโโโ(๐)๐ผโ๐๐โIVโ(1)๐๐ ๐ฝ=1 ๐=0|๐๐กโโโ| ๐๐ ๐ฝ=0 ๐=1โ.
โขโฏ Signsof๐โ๐๐๐๐โ(1โ๐โ+โ)โโฏ Construc8veordestruc8veinterferenceโessen8alfordetermininglargeor
smallB(GT)
Signsof๐โ๐๐๐๐โ(1โ๐โ+โ):CKII
โขโฏ Destruc8veinterferenceforฮ๐โ๐โ=โ5 MeV.
๐โ๐๐๐๐โ(1โ๐โ+โ)= ๐ผโ๐๐โI๐โโโโ(๐)๐ผโ๐๐โIVโ(1)๐๐ ๐ฝ=1 ๐=0|๐๐กโโโ| ๐๐ ๐ฝ=0 ๐=1โ.
ฮ๐โ๐โ=5 MeV ฮ๐โ๐โ=โ5 MeV
Signsof๐โ๐๐๐๐โ(1โ๐โ+โ):pairing
โขโฏ Itlooksthatthepairinginterac8onalwaysgivesaconstruc8veinterference,butrealis8cinterac8onsdonot.
๐โ๐๐๐๐โ(1โ๐โ+โ)= ๐ผโ๐๐โI๐โโโโ(๐)๐ผโ๐๐โIVโ(1)๐๐ ๐ฝ=1 ๐=0|๐๐กโโโ| ๐๐ ๐ฝ=0 ๐=1โ.
ฮ๐โ๐โ=5 MeV ฮ๐โ๐โ=โ5 MeV
Theoremforthesignsof๐โ๐๐๐๐โ(1โ๐โ+โ)
Whenthe(J,T)=(0,1)and(1,0)two-bodymatrixelementsaregivenbytheisovector-andisoscalar-pairinginterac8onswith
nega8veGIVandGIS,respec8vely,allthesignsof๐โ๐๐๐๐โ(1โ๐โ+โ)intwo-nucleonsystemsarethesameforanyvalenceshell
(includingmul8-jshell)andforanysingle-par8clesplifng.
๐ปโpairโ=โ๐โโ๐โ๐โ๐โ๐โโโโ โ๐โ๐โโ+ ๐โpairโ
๐โ๐๐๐๐โ(1โ๐โ+โ)= ๐ผโ๐๐โI๐โโโโ(๐)๐ผโ๐๐โIVโ(1)๐๐ ๐ฝ=1 ๐=0|๐๐กโโโ| ๐๐ ๐ฝ=0 ๐=1โโฅ0
๐โpairโ=๐บโIVโโ๐โโ๐โ๐โโ โโ๐โ๐โ+ ๐บโISโโ๐โโ๐ทโ๐โโ โโ๐ทโ๐โ
Proofofthetheorem(1)โขโฏ Writedownj-jcoupledtwo-bodymatrixelements:๐๐ ๐ฝ=0 ๐=1๐โpairโ ๐๐ ๐ฝ=0 ๐=1โ= ๐บโIVโ๐โ๐๐โIVโ๐โ๐๐โIVโ๐๐ ๐ฝ=1 ๐=0๐โpairโ ๐๐ ๐ฝ=1 ๐=0โ= ๐บโI๐โ๐โ๐๐โISโ๐โ๐๐โISโwith๐โ๐๐โIVโ= (โ1)โ๐โ๐โโโ๐โ๐โ+1/2โ๐ฟโ๐๐โ๐โ๐๐โISโ=โ2/1+ ๐ฟโ๐๐โโโ(โ1)โ๐โ๐โโ1/2โโ(2๐โ๐โ+1)(2๐โ๐โ+1)โ{โ1/2&๐โ๐โ&๐โ๐โ@๐โ๐โ&1/2&1โ}๐ฟโ๐โ๐โ๐โ๐โโ๐ฟโ๐โ๐โ๐โ๐โโ
โขโฏ Onecaneasilyshowthatthesignof๐โ๐๐โIVโis(โ1)โ๐โ๐โโandthatthatof๐โ๐๐โISโis(โ1)โ๐โ๐โโ1/2โbyusingtheexactformof{โ1/2&๐โ๐โ&๐โ๐โ@๐โ๐โ&1/2&1โ}.
โขโฏ Whentheconven8onsof| ๐๐ ๐ฝ=0 ๐=1โฉโ= (โ1)โ๐โ๐โโ| ๐๐ ๐ฝ=0 ๐=1โฉand | ๐๐ ๐ฝ=1 ๐=0โฉโ= (โ1)โ๐โ๐โโ1/2โ| ๐๐ ๐ฝ=1 ๐=0โฉaretaken,allthetwo-bodymatrixelementsarenonposi8ve.
Proofofthetheorem(2)โขโฏ MatrixelementoftheHamiltonian๐ปโ๐๐โpairโ= ๐ฟโ๐๐โ( ๐โ๐(๐)โ+ ๐โ๐(๐)โ)+ ๐โ๐๐โpairโ
โขโฏ Alltheoff-diagonalmatrixelementsarenonposi8vewhenthepresentphaseconven8on.
โขโฏ FromaversionofthePerron-Frobeniustheorem,thecomponentsofthelowesteigenvectorarecompletelyofthesamesign.
Foramatrix[โโโ11โ&โฏ&โโ1๐โ@โฎ&โฑ&โฎ@โโ๐1โ&โฏ&โโ๐๐โโ]with โโ๐๐โโค0(๐โ ๐),thelowesteigenvector๐ฃโ1โโ=[โ๐ผโ1โ@โฎ@๐ผโ๐โโ]sa8sfies ๐ผโ๐โโฅ0forany๐.
Proofofthetheorem(3)โขโฏ Gamow-Tellermatrixelementsfortwo-nucleonconfigura8ons
Condon-Shortleyconven8on
presentconven8on
Completelythesamesign!
Proofofthetheorem(4)โขโฏ Wegobackto๐(๐บ๐;1โ๐โ+โ)=โ๐๐๐๐โโ๐โ๐๐๐๐โ( 1โ๐โ+โ)โwith ๐โ๐๐๐๐โ(1โ๐โ+โ)= ๐ผโ๐๐โI๐โโโโ(๐)๐ผโ๐๐โIVโ(1)๐๐ ๐ฝ=1 ๐=0|๐๐กโโโ| ๐๐ ๐ฝ=0 ๐=1โ.
โขโฏ Allof๐ผโ๐๐โI๐โโโโ(1), ๐ผโ๐๐โIVโ(1),and ๐๐ ๐ฝ=1 ๐=0|๐๐กโโโ| ๐๐ ๐ฝ=0 ๐=1โhavefixedsignswhenthephaseconven8onsof| ๐๐ ๐ฝ=0 ๐=1โฉโ= (โ1)โ๐โ๐โโ| ๐๐ ๐ฝ=0 ๐=1โฉand | ๐๐ ๐ฝ=1 ๐=0โฉโ= (โ1)โ๐โ๐โโ1/2โ| ๐๐ ๐ฝ=1 ๐=0โฉaretaken.Therefore,thesignsof๐โ๐๐๐๐โ(1โ1โ+โ)arethesameforallpossible(๐,๐,๐,๐).
โขโฏ Thisistheoriginofthecoherence(โlow-energysuperGTstateโ)obtainedfortwo-par8cleconfigura8ons.
(rough)ProofofthePerron-Frobeniustheoremโขโฏ Considerareal-symmetricmatrixwith๐ป=( โโ๐๐โ)withโ โโ๐๐โโค0
(๐โ ๐)andlet๐ฃโ1โโ=(๐ผโ1โ, โฆ, ๐ผโ๐โ)bethelowesteigenvector.โขโฏ Since ๐ฃโ1โโisthelowesteigenvector,thereisnovectorthat
sa8sfies ๐ฃ๐ป๏ฟฝ๐ฃโ< ๐ฃโ1โ ๐ป๏ฟฝ ๐ฃโ1โโ.โขโฏ If ๐ฃโ1โโcontainsposi8veandnega8vecomponents,onecan
generallyassume ๐ผโ1โโฅ0,โฆ,๐ผโ๐โโฅ0, ๐ผโ๐+1โ<0,โฆ,๐ผโ๐โ<0.Let ๐ฃโ1โโฒโ be(๐ผโ1โ, โฆ ๐ผโ๐โ, โ ๐ผโ๐+1โ, โ๐ผโ๐โ).
โขโฏ Since ๐ฃโ1โ ๐ป๏ฟฝ ๐ฃโ1โโ=โ๐๐โโ๐ผโ๐โโโโ๐๐โ๐ผโ๐โ,onegets๐ฃโ1โ ๐ป๏ฟฝ ๐ฃโ1โโโ ๐ฃโ1โโฒ ๐ป๏ฟฝ ๐ฃโ1โโฒโ=โ๐โค๐, ๐>๐โโ2โ๐ผโ๐โโโ๐๐โ๐ผโ๐โ>0.Thiscontradictsthat๐ฃโ1โโisthelowesteigenvector.
GraphicalimageoftheP-Ftheoremโขโฏ Representanoff-diagonalmatrixelementโโ๐๐โ
asabondthatconnectsthesite๐and๐.โขโฏ Foreachsite๐,aposi8ve(nega8ve)๐ผโ๐โis
representedasanupward(downward)arrow.
โขโฏ Therightfigureshowsfavoredsigns,whichareanalogoustoferromagne8smandan8ferromagne8sm.
โขโฏ Thesitua8onofP-Ftheoremislikeparallelspinalignmentinferromagne8smthatmakesenergystable.
โ๐ ๐
โ๐ ๐
โ
โ
โ
โโ
โ
Goingbacktophysicsโขโฏ ThecoherenceofGTmatrixelementsisduetotheโalignmentโof
allthetwo-nucleonconfigura8ons,wherealignmentstandsforthatcoefficientsofbasisvectorsareofthesamesign.
โขโฏ Sincethisisindependentofsingle-par8cleenergies,suchanโalignedโstatemustbemoststablealsofortwo-holeconfigura8ons.
โขโฏ Theactualsitua8onisdifferent.Thismeansthatsomeoftheoff-diagonalmatrixelementsinthe(J,T)=(1,0)and/or(0,1)channelsareopposite.
Realis8c(J,T)=(0,1)and(1,0)matrixelementsโขโฏ Ques8ons
โโฏ Isovectororisoscalar?โโฏ Whichmatrixelementsaredifferent?
โโฏ Anyruleaboutthedifferentsigns?
โขโฏ Examiningoff-diagonalmatrixelementsinrealis8cinterac8onsโโฏ Empiricalandmicroscopic(Gmatrix)
โโฏ Phaseconven8onsof| ๐๐ ๐ฝ=0 ๐=1โฉโ= (โ1)โ๐โ๐โโ| ๐๐ ๐ฝ=0 ๐=1โฉand | ๐๐ ๐ฝ=1 ๐=0โฉโ= (โ1)โ๐โ๐โโ1/2โ| ๐๐ ๐ฝ=1 ๐=0โฉ
CKII Kuop
-1.56 -1.75
-3.55 -5.06
+1.70 +2.31
CKII Kuop
-4.86 -3.96
(J,T)=(0,1)off-diagonal (J,T)=(1,0)off-diagonal
sdshell
USD Kuosd
-0.72 -1.62
-2.54 -3.17
+0.57 +0.04
+1.10 +0.24
-1.18 -0.60
-1.71 -1.91
-2.10 -1.71
+0.40 +0.80
-0.03 +0.21
-1.25 -0.31
USD Kuosd
-3.19 -3.79
-1.32 -0.97
-1.08 -0.74
(J,T)=(0,1)off-diagonal (J,T)=(1,0)off-diagonal
Someisoscalarmatrixelementshavetheoppositesign.
Pairingvs.realis8cinterac8ons:p-shellcase
โขโฏ Samesignsforisovectormatrixelements.
โขโฏ Oppositesignfor ๐โ3/2โ๐โ1/2โ ๐๏ฟฝ ๐โ1/2โ๐โ1/2โโinthe(J,T)=(1,0)channel.Thiscausesโfrustra8onโ,whichcannotuniquelydeterminethesigns.Theactualsignsdependonthediagonalmatrixelements.โโฏ Two-par8cleconfig.:coherentduetothedominanceof๐ฃโ1โand ๐ฃโ2โโโฏ Two-holeconfig.:noncoherentduetodominanceof๐ฃโ2โand ๐ฃโ3โ.
Originofdifferenceinsignโขโฏ Weconsiderthematrixelementsofthedeltainterac8onasashort-
rangecentralinterac8on.
โขโฏ Thej-jcoupledmatrixelementsarewri`enas๐๐๐ฝ๐ ๐ฟ(๐) ๐๐๐ฝ๐โ= ๐ถโ0โโ๐ฟ๐ (๐ฟ+๐+๐=odd)โโ๐โ๐๐โ(๐ฟ๐๐ฝ)๐โ๐๐โ(๐ฟ๐๐ฝ)โwith ๐โ๐๐โ(๐ฟ๐๐ฝ)=โ1โ (โ1)โ๐+๐โ/1+ ๐ฟโ๐๐โโโ๐พโ๐ฟ๐โ(๐ฝ)โ(๐,๐)(โ1)โ๐โ๐โ+ ๐โ๐โโโ(2๐โ๐โ+1)(2๐โ๐โ+1)โ(โ๐ฟ&๐โ๐โ&๐โ๐โ@0&0&0โ)and ๐พโ๐ฟ๐โ(๐ฝ)โ(๐,๐)=โ(2๐โ๐โ+1)(2๐โ๐โ+1 )(2๐ฟ+1)(2๐+1)โ{โ๐โ๐โ&1/2&๐โ๐โ@๐โ๐โ&1/2&๐โ๐โ@๐ฟ&๐&๐ฝโ}.
โขโฏ ๐ถโ0โdependson(๐โ๐โ, ๐โ๐โ)etc.,butweassumeaconstant๐ถโ0โwhichiscalledthesurfacedeltainterac8on.
โขโฏ Inthiscase,the๐ฟ=0contribu8onsareexactlythesameasthepairingmatrixelements.
L=0and2contribu8onsโขโฏ ๐๐๐ฝ๐ ๐โSDIโ ๐๐๐ฝ๐โ=๐ถโ0โโ๐ฟ๐ (๐ฟ+๐+๐=odd)โโ๐โ๐๐โ(๐ฟ๐๐ฝ)๐โ๐๐โ
(๐ฟ๐๐ฝ)โโขโฏ Restric8ons
โโฏ ๐ฝโ= ๐ฟโ+ ๐โ;๐=0or1.
โโฏ ๐ฟisonlyevenwhenoneconsidersasingle-majorshell,since๐โ๐๐โ(๐ฟ๐๐ฝ)โ(โ๐ฟ&๐โ๐โ&๐โ๐โ@0&0&0โ).
โขโฏ Possible๐ฟ๐โโฏ Only๐ฟ๐=00for(๐ฝ,๐)=(0,1)
โโฏ ๐ฟ๐=01and21for(๐ฝ,๐)=(1,0)
Onemusttakethe๐ฟ๐=21termintoaccountwhenfullyevalua8ngthematrixelementsofshort-rangecentralforces.
Cancella8onduetoL=2โขโฏ ๐๐๐ฝ๐ ๐โSDIโ ๐๐๐ฝ๐โ=๐ถโ0โโ๐ฟ๐ (๐ฟ+๐+๐=odd)โโ๐โ๐๐โ(๐ฟ๐๐ฝ)๐โ๐๐โ
(๐ฟ๐๐ฝ)โโขโฏ Exactexpressionof ๐โ๐๐โ(๐ฟ ๐=1 ๐ฝ=1)dividedby๐ = (โ1)โ๐โ๐โ
โ1/2โ
โขโฏ Clearly,cancella8onduetoL=2occursfor ๐โ>โ๐โ>โ ๐๏ฟฝ ๐โ>โ๐โ<โโand ๐โ<โ๐โ<โ ๐๏ฟฝ ๐โ>โ๐โ<โโ.
๐โโฒโ=๐โ2
Quan8ta8veargumentโขโฏ ๐๐ ๐ฝ=1 ๐=0๐โSDIโ ๐๐ ๐ฝ=1 ๐=0โ/๐withtheconven8onof(โ1)โ๐โ๐โโ1/2โ| ๐๐ ๐ฝ=1 ๐=0โฉโCondonโShortleyโ
โขโฏ ๐โ<โ๐โ<โ ๐๏ฟฝ ๐โ>โ๐โ<โโwithlow-๐arestronglycancelledbyL=2.โขโฏ Finite-rangeinterac8onscanmake๐โ<โ๐โ<โ ๐๏ฟฝ ๐โ>โ๐โ<โโposi8ve.
Frustra8oncausedbyฮl=2mixingโขโฏ Matrixelementsconcerning| ๐โ<โ ๐โ>โโฒโโฉwith ๐โโฒโ=๐โ2(suchas| ๐โ3/2โ๐ โ1/2โโฉ)doesnotappearwiththeL=0terms,andcanbeanaddi8onalsourceoffrustra8on.
โขโฏ Thesignof โโ12โโโ23โโโ31โisinvariantunderphasetransforma8on.Whenitisposi8ve,thefrustra8onoccursamong ๐ฃโ1โ, ๐ฃโ2โand ๐ฃโ3โ.๐โ๐๐โ(๐ฟ ๐=1 ๐ฝ=1)/ (โ1)โ๐โ๐โโ1/2โ
โโ
โ
( ๐โ<โ ๐โ>โโฒโ)
( ๐โ<โ ๐โ>โโ)( ๐โ>โ ๐โ>โโ)
Tensor-forcecontribu8ons
Matrixelement(MeV)
diagonal +1.40
+0.59
-0.78
off-diagonal +1.16
-0.22
+0.43
Matrixelement(MeV)
diagonal +1.05
+0.51
-0.23
off-diagonal +0.76
-0.25
+0.21
โขโฏ PointedoutbyTalmiintermsoftheoriginoflonglife8meof14C.
โขโฏ Evalua8onwiththeฯ+ฯmesonexchangetensorforceusingthe(โ1)โ๐โ๐โโ1/2โ| ๐๐ ๐ฝ=1 ๐=0โฉโCondonโShortleyโconven8on
pshell sdshell
Pairtransfermatrixelementsโขโฏ Itiso{endiscussedthatpairtransferprobabilityisagoodmeasure
ofpairingcorrela8on.
โขโฏ Isoscalarpaircrea8onandremovalprobabili8es:|๐ฝโ๐โ๐โ๐โ ||๐ทโโ โ|| ๐ฝโ๐โ๐โ๐โโ|โ2โand |๐ฝโ๐โ๐โ๐โ ||๐ท|| ๐ฝโ๐โ๐โ๐โโ|โ2โ
Paircrea0ononvacuum
Pairremovalfromclosure
p 8.1 5.3ร10-3
sd 15.7 1.7
Possibleeffectsonpaircondensateโขโฏ OnthebasisofstrongT=0a`rac8veforce,isoscalarpair
condensatesareexpectedtooccurespeciallyaroundN=Znuclei,similartowell-knownisovectorpaircondensates.โโฏ | ๐นโฉ= (๐ฌโโ โ)โ๐โ| โโฉwiththeCooperpair๐ฌโโ โ=โ๐๐โโ๐โ๐๐โ[๐โ๐โโ โ
ร๐โ๐โโ โ]โ๐ฝ=1 ๐=0โโ
โขโฏ Thereseemstobenostrongevidenceforisoscalarpaircondensatesintheactualnuclei.
โขโฏ Asshowninthistalk,thesignsoftheisoscalarpairarenotuniquelydeterminedbecauseoffrustra8on.Thisisapossiblereasonwhyisoscalarpaircondensatesarenotwellestablished.
Summaryโขโฏ WehaveexactlyproventhattheGTmatrixelementsintwo-nucleon
configura8ons(2pand2h)arealwaysinphasewiththeisovetor-andisoscalar-pairinginterac8ons,whichaccountsforโlow-energysuperGTstatesโ.
โขโฏ TheobservedhindranceoftheGTmatrixelementsintwo-holeconfigura8onsisduetoโnoncoherenceโofrealis8c(J,T)=(1,0)matrixelements,whichcannotgivedefinitesignsinโCooperpairsโ.
โขโฏ Differenceinsignbetweenisoscalar-pairingandrealis8cinterac8onsispredominantlycausedbyL=2centralforcesandtensorforces.
โขโฏ Thiseffectcanpreventcorrelatedproton-neutronpairfromformingisoescalar-paircondensates,whicharenotestablishedinexperiement.