probing proton-neutron pairing with gamow-teller strengths
TRANSCRIPT
Probingproton-neutronpairingwithGamow-Tellerstrengthsintwo-
nucleonconfigura8ons
YutakaUtsunoAdvancedScienceResearchCenter,JapanAtomicEnergyAgency
CenterforNuclearStudy,UniversityofTokyo
Y.UtsunoandY.Fujita,arXiv1701.00869
Introduc8on:Fujita-san’sseminarinJan.2016• Systema8csofGamow-Tellerdistribu8onsforN=Z+2→N=Znuclei
– Usingchargeexchangereac8on(3He,t)
ObservedGTstrength• Concentra8oninthe1+1levelfor
42Ca→42Sc
– B(GT;0+1→1+1)=2.2
Y.Fujitaetal.,Phys.Rev.Le`.112,112502(2014).
“low-energysuperGTstate”
Ikedasumrule• WhenB(GT)isdefinedas𝐵(GT↑± ;𝑖→𝑓)= |𝛼↓𝑓 𝐽↓𝑓 |𝜎𝑡↑± | 𝛼↓𝑖 𝐽↓𝑖 |↑2 /2𝐽↓𝑖 +1 =∑𝜇𝑀↓𝑓 ↑▒|𝛼↓𝑓 𝐽↓𝑓 𝑀↓𝑓 𝜎↓𝜇 𝑡↑± 𝛼↓𝑖 𝐽↓𝑖 𝑀↓𝑖 |↑2 ,∑𝑓↑▒𝐵(GT↑− ;𝑖→𝑓)− ∑𝑓↑▒𝐵(GT↑+ ;𝑖→𝑓)=3(𝑁−𝑍) issa8sfied.(with 𝑡↑− | 𝑛⟩=| 𝑝⟩)
• IfGT+transi8onishinderedbyPauliblocking,∑𝑓↑▒𝐵(GT↑− ;𝑖→𝑓) iscloseto3(𝑁−𝑍).
– ∑𝑓↑▒𝐵(GT↑− ;𝑖→𝑓) for42Cashouldbecloseto6,ifprotonexcita8ontothepfshellissmall.
proton
neutron
SU(4)?single-par8cletransi8on?• Thesitua8onissimilartowhatisexpectedfromtheSU(4)
symmetry(nospinorisospindependentforces).
• SU(4)isstronglybrokenbytheexistenceofspin-orbitsplifng.
• Single-par8clestructure?
f7/2→f7/2f7/2→f5/2
ε(f5/2)-ε(f7/2)
B(GT)distribu8onwithpureconfigura8ons
• Concentra8oninasinglestatecannotbeaccountedforbyasimpleshell-modelargument.
Fromthesimplesingle-par8clepicture
Experimentalstrength
0 10
Essen8alroleofconfigura8onmixing
Y.Fujitaetal.,Phys.Rev.C91,064316(2015)
• RPAcalcula8on– Isoscalarpairingisessen8al.
• CoherenceintheGTtransi8onisalsoobtainedbyshell-modelcalc.
• Whyiscollec8vitycreated?
C.L.Baietal.,Phys.Rev.C90,054335(2014).Shell-modelanalysis
Two-par8clevs.two-holeconfigura8ons• Aques8onbysomeone(sorry,Idonotrememberwhoraised)
– Whatabouttwoholesystems?
twohole:14C→14NlogA=9.1
p3/2
p1/2
two-par8cle:6He→6Li
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 3.5×10-6 0.0601+2 0.13 0.10 2.8 1.5
Distribu8onofknownlogA
6He→6Li 14C→14N
Num
bero
fcases
Ques8ons• Whydoesthe“low-energysuperGTstate”appearfortwo-par8cle
configura8ons?
• WhyisthecorrespondingB(GT)valuesfortwo-holeconfigura8onsverysmall?
• Whatdoesthoseproper8estellusaboutisovectorandisoscalarpairingproper8es,sincecorrelatedtwonucleonsforma“Cooperpair”?
Unifiedtreatmentofp-pandh-hsystems• Thefollowingtwodescrip8onsareiden8cal
– Par8cleHamiltonian𝐻=∑𝑖↑▒𝜀↓𝑖 𝑎↓𝑖 ↓↑† 𝑎↓𝑖 + 1/4 ∑𝑖,𝑗,𝑘,𝑙↑▒𝑣↓𝑖𝑗;𝑘𝑙 𝑎↓𝑖 ↓↑† 𝑎↓𝑗 ↓↑† 𝑎↓𝑙 𝑎↓𝑘
– HoleHamiltonian𝐻=∑𝑖↑▒𝜀 ↓𝑖 𝑏↓𝑖 ↓↑† 𝑏↓𝑖 + 1/4 ∑𝑖,𝑗,𝑘,𝑙↑▒𝑣 ↓𝑖𝑗;𝑘𝑙 𝑏↓𝑖 ↓↑† 𝑏↓𝑗 ↓↑† 𝑏↓𝑙 𝑏↓𝑘
when𝑣↓𝑖𝑗;𝑘𝑙 = 𝑣 ↓𝑖𝑗;𝑘𝑙 and 𝜀 ↓𝑖 =𝐸( 𝑖↑−1 )aresa8sfied.
Two-bodymatrixelementsarethesame,butthesingle-par8cleenergiesareinthereversedorderforthehole-holeHamiltonian.
p3/2
p1/2
GTstrengthin2pand2hconf.:p-shellcase• Asshowninthelastslide,boththe6He→6Li(2p)and14C→14N
(2h)decayscanbedescribedastwo-nucleonsystemswiththesametwo-bodymatrixelements.
• Theonlydifferencebetweenthemisthesingle-par8clesplifngΔ𝜀↓𝑝 =𝜀( 𝑝↓1/2 ) −𝜀( 𝑝↓3/2 ):– Forthepar8clecase:Δ𝜀↓𝑝 =0.1 MeV
– Fortheholecase:Δ𝜀↓𝑝 =−6.3 MeV
• Itisinteres8ngtoplottheGamow-Tellermatrixelement𝑀(𝐺𝑇)= 𝐽↓𝑓 |𝜎𝑡↑− | 𝐽↓𝑖 asafunc8onofΔ𝜀↓𝑝 forgiventwo-bodyinterac8onsinordertoseethedependenceonthesingle-par8cleenergies.
TakenfromtheCohen-Kurath’sCKIIinterac8on
(1)Cohen-Kurath’sCKIIinterac8on• Oneofthemostpopularempiricalinterac8onsforthepshell.
– 15two-bodymatrixelementsarededucedbyfifngenergylevelsofA=8-16nuclei.
(2)Pairinginterac8on• Isoscalar-andisovector-pairinginterac8onsaredefinedas𝑉↑IVpair = 𝐺↑IV ∑𝜇↑▒𝑃↓𝜇↑† 𝑃↓𝜇 𝑉↑ISpair = 𝐺↑IS ∑𝜇↑▒𝐷↓𝜇↑† 𝐷↓𝜇 where𝑃↓𝜇↑† =√1/2 ∑𝑛𝑙↑▒(−1)↑𝑙 √2𝑙+1 [𝑎↓𝑛𝑙↑† × 𝑎↓𝑛𝑙↑† ] ↓𝑀↓𝐿 =0,𝑀↓𝑆 =0,𝑀↓𝑇 =𝜇↑𝐿=0,𝑆=0,𝑇=1 and �𝐷↓𝜇↑† =√1/2 ∑𝑛𝑙↑▒(−1)↑𝑙 √2𝑙+1 [𝑎↓𝑛𝑙↑† × 𝑎↓𝑛𝑙↑† ] ↓𝑀↓𝐿 =0,𝑀↓𝑆 =𝜇, 𝑀↓𝑇 =0↑𝐿=0,𝑆=1,𝑇=0 .
• Isovector-paircrea8onoperator 𝑃↓𝜇↑† andisoscalar-paircrea8onoperators 𝐷↓𝜇↑† aresymmetricintermsofSandT.
M(GT)withthepairinginterac8on
• M(GT):EnhancedforsmallΔ𝜀↓𝑝 andnovanishingforΔ𝜀↓𝑝 <0.– Largerthansingle-par8clelimitsonthebothsides
• StrengthsofGISandGIVaredeterminedsoastoreproducethemeansquare(J,T)=(1,0)and(0,1)matrixelementsofCKII.
p3/2limit:√10/3 ≈1.83
p1/2limit:√2/3 ≈0.82
CoherenceoftheGTmatrixelement• Whentwo-nucleonwavefunc8onsaredecomposedintobasis
statesas| 0↓𝑘↑+ ⟩=∑𝑎𝑏↑▒𝛼↓𝑎𝑏↑IV (𝑘) | 𝑎𝑏 𝐽=0 𝑇=1⟩and| 1↓𝑘↑+ ⟩=∑𝑎𝑏↑▒𝛼↓𝑎𝑏↑I𝑆 (𝑘) | 𝑎𝑏 𝐽=1 𝑇=0⟩,theGamow-Tellermatrixelementiswri`enas𝑀(𝐺𝑇;1↓𝑘↑+ )=∑𝑎𝑏𝑐𝑑↑▒𝑚↓𝑎𝑏𝑐𝑑 ( 1↓𝑘↑+ ) where𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 .
• Signsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )– Construc8veordestruc8veinterference—essen8alfordetermininglargeor
smallB(GT)
Signsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ ):CKII
• Destruc8veinterferenceforΔ𝜀↓𝑝 =−5 MeV.
𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 .
Δ𝜀↓𝑝 =5 MeV Δ𝜀↓𝑝 =−5 MeV
Signsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ ):pairing
• Itlooksthatthepairinginterac8onalwaysgivesaconstruc8veinterference,butrealis8cinterac8onsdonot.
𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 .
Δ𝜀↓𝑝 =5 MeV Δ𝜀↓𝑝 =−5 MeV
Theoremforthesignsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )
Whenthe(J,T)=(0,1)and(1,0)two-bodymatrixelementsaregivenbytheisovector-andisoscalar-pairinginterac8onswith
nega8veGIVandGIS,respec8vely,allthesignsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )intwo-nucleonsystemsarethesameforanyvalenceshell
(includingmul8-jshell)andforanysingle-par8clesplifng.
𝐻↑pair =∑𝑖↑▒𝜀↓𝑖 𝑎↓𝑖 ↓↑† 𝑎↓𝑖 + 𝑉↑pair
𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 ≥0
𝑉↑pair =𝐺↑IV ∑𝜇↑▒𝑃↓𝜇↑† 𝑃↓𝜇 + 𝐺↑IS ∑𝜇↑▒𝐷↓𝜇↑† 𝐷↓𝜇
Proofofthetheorem(1)• Writedownj-jcoupledtwo-bodymatrixelements:𝑎𝑏 𝐽=0 𝑇=1𝑉↑pair 𝑐𝑑 𝐽=0 𝑇=1 = 𝐺↑IV 𝜒↓𝑎𝑏↑IV 𝜒↓𝑐𝑑↑IV 𝑎𝑏 𝐽=1 𝑇=0𝑉↑pair 𝑐𝑑 𝐽=1 𝑇=0 = 𝐺↑I𝑆 𝜒↓𝑎𝑏↑IS 𝜒↓𝑐𝑑↑IS with𝜒↓𝑎𝑏↑IV = (−1)↑𝑙↓𝑎 √𝑗↓𝑎 +1/2 𝛿↓𝑎𝑏 𝜒↓𝑎𝑏↑IS =√2/1+ 𝛿↓𝑎𝑏 (−1)↑𝑗↓𝑎 −1/2 √(2𝑗↓𝑎 +1)(2𝑗↓𝑏 +1) {█1/2&𝑗↓𝑎 &𝑙↓𝑎 @𝑗↓𝑏 &1/2&1 }𝛿↓𝑛↓𝑎 𝑛↓𝑏 𝛿↓𝑙↓𝑎 𝑙↓𝑏
• Onecaneasilyshowthatthesignof𝜒↓𝑎𝑏↑IV is(−1)↑𝑙↓𝑎 andthatthatof𝜒↓𝑎𝑏↑IS is(−1)↑𝑗↓𝑏 −1/2 byusingtheexactformof{█1/2&𝑗↓𝑎 &𝑙↓𝑎 @𝑗↓𝑏 &1/2&1 }.
• Whentheconven8onsof| 𝑎𝑏 𝐽=0 𝑇=1⟩ = (−1)↑𝑙↓𝑎 | 𝑎𝑏 𝐽=0 𝑇=1⟩and | 𝑎𝑏 𝐽=1 𝑇=0⟩ = (−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩aretaken,allthetwo-bodymatrixelementsarenonposi8ve.
Proofofthetheorem(2)• MatrixelementoftheHamiltonian𝐻↓𝑖𝑗↑pair = 𝛿↓𝑖𝑗 ( 𝜀↓𝑎(𝑖) + 𝜀↓𝑏(𝑖) )+ 𝑉↓𝑖𝑗↑pair
• Alltheoff-diagonalmatrixelementsarenonposi8vewhenthepresentphaseconven8on.
• FromaversionofthePerron-Frobeniustheorem,thecomponentsofthelowesteigenvectorarecompletelyofthesamesign.
Foramatrix[█ℎ↓11 &⋯&ℎ↓1𝑛 @⋮&⋱&⋮@ℎ↓𝑛1 &⋯&ℎ↓𝑛𝑛 ]with ℎ↓𝑖𝑗 ≤0(𝑖≠𝑗),thelowesteigenvector𝑣↓1 =[█𝛼↓1 @⋮@𝛼↓𝑛 ]sa8sfies 𝛼↓𝑖 ≥0forany𝑖.
Proofofthetheorem(3)• Gamow-Tellermatrixelementsfortwo-nucleonconfigura8ons
Condon-Shortleyconven8on
presentconven8on
Completelythesamesign!
Proofofthetheorem(4)• Wegobackto𝑀(𝐺𝑇;1↓𝑘↑+ )=∑𝑎𝑏𝑐𝑑↑▒𝑚↓𝑎𝑏𝑐𝑑 ( 1↓𝑘↑+ ) with 𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 .
• Allof𝛼↓𝑎𝑏↑I𝑆 ↑∗ (1), 𝛼↓𝑐𝑑↑IV (1),and 𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 havefixedsignswhenthephaseconven8onsof| 𝑎𝑏 𝐽=0 𝑇=1⟩ = (−1)↑𝑙↓𝑎 | 𝑎𝑏 𝐽=0 𝑇=1⟩and | 𝑎𝑏 𝐽=1 𝑇=0⟩ = (−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩aretaken.Therefore,thesignsof𝑚↓𝑎𝑏𝑐𝑑 (1↓1↑+ )arethesameforallpossible(𝑎,𝑏,𝑐,𝑑).
• Thisistheoriginofthecoherence(“low-energysuperGTstate”)obtainedfortwo-par8cleconfigura8ons.
(rough)ProofofthePerron-Frobeniustheorem• Considerareal-symmetricmatrixwith𝐻=( ℎ↓𝑖𝑗 )with∀ ℎ↓𝑖𝑗 ≤0
(𝑖≠𝑗)andlet𝑣↓1 =(𝛼↓1 , …, 𝛼↓𝑛 )bethelowesteigenvector.• Since 𝑣↓1 isthelowesteigenvector,thereisnovectorthat
sa8sfies 𝑣𝐻�𝑣 < 𝑣↓1 𝐻� 𝑣↓1 .• If 𝑣↓1 containsposi8veandnega8vecomponents,onecan
generallyassume 𝛼↓1 ≥0,…,𝛼↓𝑘 ≥0, 𝛼↓𝑘+1 <0,…,𝛼↓𝑛 <0.Let 𝑣↓1 ′ be(𝛼↓1 , … 𝛼↓𝑘 , − 𝛼↓𝑘+1 , −𝛼↓𝑛 ).
• Since 𝑣↓1 𝐻� 𝑣↓1 =∑𝑖𝑗↑▒𝛼↓𝑖 ℎ↓𝑖𝑗 𝛼↓𝑗 ,onegets𝑣↓1 𝐻� 𝑣↓1 − 𝑣↓1 ′ 𝐻� 𝑣↓1 ′ =∑𝑖≤𝑘, 𝑗>𝑘↑▒2 𝛼↓𝑖 ℎ↓𝑖𝑗 𝛼↓𝑗 >0.Thiscontradictsthat𝑣↓1 isthelowesteigenvector.
GraphicalimageoftheP-Ftheorem• Representanoff-diagonalmatrixelementℎ↓𝑖𝑗
asabondthatconnectsthesite𝑖and𝑗.• Foreachsite𝑖,aposi8ve(nega8ve)𝛼↓𝑖 is
representedasanupward(downward)arrow.
• Therightfigureshowsfavoredsigns,whichareanalogoustoferromagne8smandan8ferromagne8sm.
• Thesitua8onofP-Ftheoremislikeparallelspinalignmentinferromagne8smthatmakesenergystable.
⊝𝑖 𝑗
⊕𝑖 𝑗
⊝
⊝
⊝
⊝⊝
⊝
Goingbacktophysics• ThecoherenceofGTmatrixelementsisduetothe“alignment”of
allthetwo-nucleonconfigura8ons,wherealignmentstandsforthatcoefficientsofbasisvectorsareofthesamesign.
• Sincethisisindependentofsingle-par8cleenergies,suchan“aligned”statemustbemoststablealsofortwo-holeconfigura8ons.
• Theactualsitua8onisdifferent.Thismeansthatsomeoftheoff-diagonalmatrixelementsinthe(J,T)=(1,0)and/or(0,1)channelsareopposite.
Realis8c(J,T)=(0,1)and(1,0)matrixelements• Ques8ons
– Isovectororisoscalar?– Whichmatrixelementsaredifferent?
– Anyruleaboutthedifferentsigns?
• Examiningoff-diagonalmatrixelementsinrealis8cinterac8ons– Empiricalandmicroscopic(Gmatrix)
– Phaseconven8onsof| 𝑎𝑏 𝐽=0 𝑇=1⟩ = (−1)↑𝑙↓𝑎 | 𝑎𝑏 𝐽=0 𝑇=1⟩and | 𝑎𝑏 𝐽=1 𝑇=0⟩ = (−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩
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
• Samesignsforisovectormatrixelements.
• Oppositesignfor 𝑝↓3/2 𝑝↓1/2 𝑉� 𝑝↓1/2 𝑝↓1/2 inthe(J,T)=(1,0)channel.Thiscauses“frustra8on”,whichcannotuniquelydeterminethesigns.Theactualsignsdependonthediagonalmatrixelements.– Two-par8cleconfig.:coherentduetothedominanceof𝑣↓1 and 𝑣↓2 – Two-holeconfig.:noncoherentduetodominanceof𝑣↓2 and 𝑣↓3 .
Originofdifferenceinsign• Weconsiderthematrixelementsofthedeltainterac8onasashort-
rangecentralinterac8on.
• Thej-jcoupledmatrixelementsarewri`enas𝑎𝑏𝐽𝑇 𝛿(𝑟) 𝑐𝑑𝐽𝑇 = 𝐶↓0 ∑𝐿𝑆 (𝐿+𝑆+𝑇=odd)↑▒𝜂↓𝑎𝑏 (𝐿𝑆𝐽)𝜂↓𝑐𝑑 (𝐿𝑆𝐽) with 𝜂↓𝑎𝑏 (𝐿𝑆𝐽)=√1− (−1)↑𝑆+𝑇 /1+ 𝛿↓𝑎𝑏 𝛾↓𝐿𝑆↑(𝐽) (𝑎,𝑏)(−1)↑𝑛↓𝑎 + 𝑛↓𝑏 √(2𝑙↓𝑎 +1)(2𝑙↓𝑏 +1) (█𝐿&𝑙↓𝑎 &𝑙↓𝑏 @0&0&0 )and 𝛾↓𝐿𝑆↑(𝐽) (𝑎,𝑏)=√(2𝑗↓𝑎 +1)(2𝑗↓𝑏 +1 )(2𝐿+1)(2𝑆+1) {█𝑙↓𝑎 &1/2&𝑗↓𝑎 @𝑙↓𝑏 &1/2&𝑗↓𝑏 @𝐿&𝑆&𝐽 }.
• 𝐶↓0 dependson(𝑛↓𝑎 , 𝑙↓𝑎 )etc.,butweassumeaconstant𝐶↓0 whichiscalledthesurfacedeltainterac8on.
• Inthiscase,the𝐿=0contribu8onsareexactlythesameasthepairingmatrixelements.
L=0and2contribu8ons• 𝑎𝑏𝐽𝑇 𝑉↑SDI 𝑐𝑑𝐽𝑇 =𝐶↓0 ∑𝐿𝑆 (𝐿+𝑆+𝑇=odd)↑▒𝜂↓𝑎𝑏 (𝐿𝑆𝐽)𝜂↓𝑐𝑑
(𝐿𝑆𝐽) • Restric8ons
– 𝐽 = 𝐿 + 𝑆 ;𝑆=0or1.
– 𝐿isonlyevenwhenoneconsidersasingle-majorshell,since𝜂↓𝑎𝑏 (𝐿𝑆𝐽)∝(█𝐿&𝑙↓𝑎 &𝑙↓𝑏 @0&0&0 ).
• Possible𝐿𝑆– Only𝐿𝑆=00for(𝐽,𝑇)=(0,1)
– 𝐿𝑆=01and21for(𝐽,𝑇)=(1,0)
Onemusttakethe𝐿𝑆=21termintoaccountwhenfullyevalua8ngthematrixelementsofshort-rangecentralforces.
Cancella8onduetoL=2• 𝑎𝑏𝐽𝑇 𝑉↑SDI 𝑐𝑑𝐽𝑇 =𝐶↓0 ∑𝐿𝑆 (𝐿+𝑆+𝑇=odd)↑▒𝜂↓𝑎𝑏 (𝐿𝑆𝐽)𝜂↓𝑐𝑑
(𝐿𝑆𝐽) • Exactexpressionof 𝜂↓𝑎𝑏 (𝐿 𝑆=1 𝐽=1)dividedby𝑠= (−1)↑𝑗↓𝑏
−1/2
• Clearly,cancella8onduetoL=2occursfor 𝑗↓> 𝑗↓> 𝑉� 𝑗↓> 𝑗↓< and 𝑗↓< 𝑗↓< 𝑉� 𝑗↓> 𝑗↓< .
𝑙↑′ =𝑙−2
Quan8ta8veargument• 𝑎𝑏 𝐽=1 𝑇=0𝑉↑SDI 𝑐𝑑 𝐽=1 𝑇=0 /𝑙withtheconven8onof(−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩↓Condon−Shortley
• 𝑗↓< 𝑗↓< 𝑉� 𝑗↓> 𝑗↓< withlow-𝑙arestronglycancelledbyL=2.• Finite-rangeinterac8onscanmake𝑗↓< 𝑗↓< 𝑉� 𝑗↓> 𝑗↓< posi8ve.
Frustra8oncausedbyΔl=2mixing• Matrixelementsconcerning| 𝑗↓< 𝑗↓>↑′ ⟩with 𝑙↑′ =𝑙−2(suchas| 𝑑↓3/2 𝑠↓1/2 ⟩)doesnotappearwiththeL=0terms,andcanbeanaddi8onalsourceoffrustra8on.
• Thesignof ℎ↓12 ℎ↓23 ℎ↓31 isinvariantunderphasetransforma8on.Whenitisposi8ve,thefrustra8onoccursamong 𝑣↓1 , 𝑣↓2 and 𝑣↓3 .𝜂↓𝑎𝑏 (𝐿 𝑆=1 𝐽=1)/ (−1)↑𝑗↓𝑏 −1/2
⊝⊕
⊝
( 𝑗↓< 𝑗↓>↑′ )
( 𝑗↓< 𝑗↓>↑ )( 𝑗↓> 𝑗↓>↑ )
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
• PointedoutbyTalmiintermsoftheoriginoflonglife8meof14C.
• Evalua8onwiththeπ+ρmesonexchangetensorforceusingthe(−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩↓Condon−Shortley conven8on
pshell sdshell
Pairtransfermatrixelements• Itiso{endiscussedthatpairtransferprobabilityisagoodmeasure
ofpairingcorrela8on.
• Isoscalarpaircrea8onandremovalprobabili8es:|𝐽↓𝑓 𝑇↓𝑓 ||𝐷↑† || 𝐽↓𝑖 𝑇↓𝑖 |↑2 and |𝐽↓𝑓 𝑇↓𝑓 ||𝐷|| 𝐽↓𝑖 𝑇↓𝑖 |↑2
Paircrea0ononvacuum
Pairremovalfromclosure
p 8.1 5.3×10-3
sd 15.7 1.7
Possibleeffectsonpaircondensate• OnthebasisofstrongT=0a`rac8veforce,isoscalarpair
condensatesareexpectedtooccurespeciallyaroundN=Znuclei,similartowell-knownisovectorpaircondensates.– | 𝛹⟩= (𝛬↑† )↑𝑁 | −⟩withtheCooperpair𝛬↑† =∑𝑎𝑏↑▒𝜆↓𝑎𝑏 [𝑎↓𝑖↑†
×𝑎↓𝑗↑† ]↑𝐽=1 𝑇=0
• Thereseemstobenostrongevidenceforisoscalarpaircondensatesintheactualnuclei.
• Asshowninthistalk,thesignsoftheisoscalarpairarenotuniquelydeterminedbecauseoffrustra8on.Thisisapossiblereasonwhyisoscalarpaircondensatesarenotwellestablished.
Summary• WehaveexactlyproventhattheGTmatrixelementsintwo-nucleon
configura8ons(2pand2h)arealwaysinphasewiththeisovetor-andisoscalar-pairinginterac8ons,whichaccountsfor“low-energysuperGTstates”.
• TheobservedhindranceoftheGTmatrixelementsintwo-holeconfigura8onsisdueto“noncoherence”ofrealis8c(J,T)=(1,0)matrixelements,whichcannotgivedefinitesignsin”Cooperpairs”.
• Differenceinsignbetweenisoscalar-pairingandrealis8cinterac8onsispredominantlycausedbyL=2centralforcesandtensorforces.
• Thiseffectcanpreventcorrelatedproton-neutronpairfromformingisoescalar-paircondensates,whicharenotestablishedinexperiement.