Non-coherent character of isoscalar pairing probed with Gamow-Teller strength: New insight into $^{14}$C dating $\beta$ decay

0.14 MB·

by  Yutaka Utsuno

Non-coherent character of isoscalar pairing probed with Gamow-Teller strength: New
insight into 14C dating β decay
YutakaUtsunoa,b,,YoshitakaFujitac,d
∗
aAdvancedScienceResearchCenter,JapanAtomicEnergyAgency,Tokai,Ibaraki319-1195,Japan
bCenterforNuclearStudy,UniversityofTokyo,Hongo,Bunkyo-ku,Tokyo113-0033,Japan
cResearchCenterforNuclearPhysics,OsakaUniversity,Ibaraki,Osaka567-0047,Japan
dDepartmentofPhysics,OsakaUniversity,Toyonaka,Osaka560-0043,Japan
Abstract
7
1We investigatethephasecoherenceofisoscalarpairsfromthe B(GT;0+T =1 1+T =0)valuesin two-particleconfigurations
0of A = 6, 18, and 42 nucleiand two-holeconfigurationsof A = 14 and1 38 one→s. W1e find thatthese Gamow-Teller(GT)matrix
2
elementsarealwaysconstructiveandthusenlargedunderisovector-andisoscalar-pairingHamiltonians,whereastheobservedGT

nstrengthsare stronglyhinderedforthe two-holeconfigurations,includingthe famous14C datingβ decay. This indicatesthatthe
a
actual isoscalar pair, unlike the isovector pair, has no definite phase coherence, which can work against forming isoscalar-pair
J
condensates.
3
Keywords: Gamow-Tellertransition,Shellmodel,Isoscalarpairing,Phasecoherence

]
h
t
-
1. Introduction theoriginofelusiveISpairing,butmuchattentionhasnotbeen
l
c
paidtophasecoherence.
u Pairingcorrelationisoneofthemostbasicpropertieswidely
InthisLetter,weshowthattheIS-pairinginteractionalways
n
seen in quantum many-body problems including condensed-
[ causes a specific combination of signs in the lowest (J,T) =
matter physics and nuclear physics. This is quite a common
(1,0)state foranytwo-particle(2p)configurationandfor any
1phenomenon caused by attractive interactions between con-
two-hole (2h) one and that the resulting B(GT;0+T = 1
9vstituentparticles. Innuclei,thesourceoftheattractionisshort- 1+T=0)valueisenhanced. Whilethispropertyw1ellaccoun→ts
range nucleon-nucleon forces, owing to which time-reversal 1
6 forthelow-energysuperGTstateforA = 6,18,and42[9],it
pairs with large spatial overlap gain much energy and then a
8 failstoexplainstronglyhinderedB(GT)valuesforthe2hcon-
0condensateof the Cooperpairs[1] occurs. Whereasisovector figurations,includingthefamous14Cdatingβdecay. Thisisa
0(IV)pairing(like-particlepairing)with(J,T) = (0,1)isfirmly
clearsignaturethattheISpairinrealitydoesnottakeanydef-
1.establishedforinstancebyextrabindingenergiesineven-even inite signs, in contrastto what occurs for the ideal IS pairing.
0nuclei,acondensateofisoscalar(IS)proton-neutronpairswith
The IS pairing is thus fragile in nature, constituting an essen-
7(J,T) = (1,0) appears quite elusive [2]. This is puzzling be-
tialdifferencefromtheIVpairingwhichalwaysfavorsdefinite
1
causemeanattractionintheISchannelismuchstrongerthanin
: signsandisthusrobust.
vtheIVchannel.PossiblesignalsforIS-pairingcorrelationhave
ibeen explored for instance in terms of binding energies, rota-
X
tional responses, Gamow-Teller (GT) β-decay properties, and 2. GTstrengthin2pand2hconfigurations
r
aproton-neutron transfer amplitudes (see [2] for a review). Of
particularinterestisthatIS-pairingcorrelationispredictedtobe WestartwithanoverviewofobservedGTstrengthsin2pand
quitesensitivetodouble-βdecaymatrixelements[3,4,5,6,7]. 2hconfigurationsontopoftheLS-closedshells. Hereafterwe
A condensate of pairs is based on the formation of an en- restrictourselvestotheinitialandfinalstateswiththequantum
ergetically stable pair in two-particle configurations. Using a numbers(J,T,T )=(0,1, 1)and(J ,T ,T )=(1,0,0),re-
i i iz f f zf
simpleattractiveforce,Cooperhasshownthatthebindingen- spectively. In Table 1, expe±rimental B(GT;0+ 1+ ) values
1 → 1,2
ergyof the lowest-energypair is muchlargerthan the onesof aresummarizedforthe p,sd,and pf shells. Forthe2pconfig-
theothereigenstatesandalsothanthescaleoftwo-bodymatrix urations,onecanclearlysee thatthe B(GT) isconcentratedin
elements[1]. Thishappensbecauseacoherentcombinationof the 1+ state for any valence shell considered, which is named
1
pairedconfigurations—witha specificcombinationofsigns— the low-energysuper GT state [9]. In contrast, the 2h config-
cooperativelyworkstolowerenergy[8].Whethersuchacoher- urationshavea strikingdifference: mostoftheGT strengthis
entISpairisformedinnucleimayprovideakeytoelucidating exhaustedbyexcitedstates, especiallythe1+ state. Itisnoted
2
that the A = 14 case is well known for radiocarbon dating,
which utilizes the very long half-life of 14C, 5730 30 yr, to
Correspondingauthor
∗ ±
Emailaddress:[email protected](YutakaUtsuno) determinetheageoforganicmaterials.
PreprintsubmittedtoPhysicsLettersB January5,2017
Table1: ExperimentalB(GT;0+ 1+ )valuesfor2pand2hconfigurations 3
1 → 1,2
inthep,sd,andpf shells.Datatakenfrom[10,11]. (a) CKII
2p 2h
p sd pf p sd ) 2 +
A=6 A=18 A=42 A=14 A=38 GT 1+1
11++1 4.7 03..113 02..120 3.5×2.180−6∗ 01.0.560 M( 1 1 2
2
0
We examinehow this strong asymmetryin GT strength be-
tweenthe2pand2hconfigurationsarisesintheframeworkof -10 -5 0 5 10
theshellmodel.Thecaseofthepshellisnowtakenasanexam-
ple,butsimilardiscussionsareapplicabletoothershells. Inthe
3
shell model, 2p and2h configurationscan be treated in a uni-
(b) pairing
fied way in terms of particle-hole conjugation, since particle-
particletwo-bodymatrixelementsareidenticalwiththecorre- ) 2 +
spondinghole-holematrix elements[12]. The only difference GT 1+1
between2pand2hconfigurationsconcerningtheGTtransition M( 1 1 2
is single-particleenergies. Keepingthisin mind, we calculate
the GT matrixelementsby changing∆εp = ε(p1/2) ε(p3/2), 0
−
whereεstandsforthesingle-particleenergy.Thevaluesof∆ε
p
forthe2p(A=6)and2h(A=14)configurationsare0.1MeV
-1
and 6.3 MeV, respectively, taken from the CKII interaction -10 -5 0 5 10
−
[13].
For the two-body part, we first use the CKII interaction as Figure1: (coloronline). Calculated M(GT;1+ )fortwo-nucleonconfigura-
1,2
a realistic one, and show the calculated GT matrix elements tionsinthepshellasafunctionof∆εp. Noquenchingfactorisused. (a)The
M(GT;1+) = 1+ σt 0+ (k = 1,2) in Fig. 1 (a). When the CKIIinteraction[13]and(b)theIV-andIS-pairinginteractionsareused.
k h k|| ±|| 1i
initialandfinalstatesareexpandedas 0+ = αIV(k)abJT
and|1+ki = abαIaSb(k)|abJfTfi, respe|ctkiviely,PMab(GaTb;1+k|)isiexii-
pressedbytPhesumofsingle-particlecontributionsas as VIVpair = GIV µP†µPµ and VISpair = GIS µD†µDµ, where
M(GT;1+k)= mabcd(1+k), (1) P†µ = √1/2 nl(−P1)l√2l+1[a†nla†nl]LM=L0=,S0,=M0S,T==0P1,MT=µ and D†µ =
aXbcd √1/2 nl(−1)Pl√2l+1[a†nla†nl]LM=L0=,S0,=M1S,T==µ0,MT=0. Those jj-coupled
∆wεhere>m0ab,cdth(1e+k)M(=GTα;Ia1Sb+∗()kv)αalIcVude(1)shtraobnJgflTyf|e|σnht±a|n|ccdeJsi,Twii.ellFroe-r tSwhoo-rbtlPoedyyphmasaetrcixonevleenmtieonntsasa[r1e4,ex1p5r]essed with the Condon-
p 1
producing the observed GT strength for A = 6. This quan- abJT VIVpair cdJT = GIVχIVχIVδ δ , (2)
tity is very close to the sum-rule limit of √6 2.45 up to h i ab cd J0 T1
≃ abJT(cid:12)(cid:12)VISpair(cid:12)(cid:12)cdJT = GISχISχISδ δ , (3)
∆εp 5MeV,andthengraduallydecreasestothe p3/2 single- h (cid:12) (cid:12) i ab cd J1 T0
≃ (cid:12) (cid:12)
particlelimitof √10/3 1.83.For∆ε <0,ontheotherhand, (cid:12) (cid:12)
the M(GT;1+)valuesha≃rplydecreasespwithdecreasing∆ε . It with (cid:12) (cid:12)
1 p
icnrgosfsoerstthhee Mva(nGisTh)in=g0GlTinsetraetn∆gεthpo=bs−e5rv.7edMienVt,htehuβsdaecccaoyunotf- χIaVb = (−1)la ja+1/2δab (4)
p
2
14C. χIaSb = r1+δ (−1)ja−1/2 (2ja+1)(2jb+1)
ab p
1/2 j l
3. GTstrengthwithpairinginteractions ×( jb 1/a2 1a )δnanbδlalb, (5)
To probe pairing properties in the 2p and 2h configura-
wherea, b, c, andd standforsingle-particlestateswithquan-
tions, it is interesting to compare those realistic shell-model
tum numbers (n ,l , j ) etc., and δ is the abbreviation for
a a a ab
calculations to the ones using the IS- and IV-pairing interac-
δ δ δ . The strengths GIV and GIS are negative for at-
tions. TheIV-andIS-pairinginteractionsareequivalenttothe nanb lalb jajb
tractive interactions, and here we set GIV = 3.4 MeV and
L = 0 part of the surface delta interaction (SDI) in the LS −
GIS = 2.8 MeV so that abVIVpaircd 2 and
coupling,hencethesimplestinteractionofshort-rangecentral- − ab,cd|h | | iJ=0,T=1|
forcecharacter. TheIV-andIS-pairinginteractionsaredefined ab,cd|hab|VISpair|cdiJ=1,T=0|P2 becomethose ofthe CKII inter-
aPction.
The results of the above pairing interactions are plotted in
For the 14C 14N β decay. The corresponding B(GT) value for the Fig.1(b). SimilartotheCKIIinteraction,theenhancementof
14O∗ 14Nβdecay→is2.0 10 4. theGTmatrixelementfromthesingle-particlelimitoccursfor
→ × −
2
∆ε > 0. However,thetrendfor∆ε < 0iscompletelydiffer-
p p
ent.TheM(GT;1+1)valuedecreasesrathermildlyas∆εpmoves Tthaebltew2o:-nGucTlemonatrwixaveelefmunecnttisonhsf||iσ(tT±|=|ii1i)natnhdepfa(iTrin=g0p)haarseedceonnovteendtioasn,(awbh)ebrye
awayfromzero.Thisisamonotonicdecreasewhichasymptot- using j> =l+1/2and j< =l 1/2. Wepresentonlythebasisstatesiand f
−
ically approachesthe p single-particlelimitof √2/3 0.82 thatappearwiththepairingHamiltonians.
andnevervanishes. 1/2 ≃ ❍❍❍ f (j j ) (j j ) (j j ) (j j )
The essential feature of the pairing interaction shown in i ❍❍ > > > < < > < <
Fig.1(b)isthattheM(GT;1+)valueenlargescomparedtothe (j j ) 2(2l+3) 2 l 2 l 0
p and p single-particle1limits for ∆ε > 0 and ∆ε < 0, > > −q 2l+1 − q2l+1 − q2l+1
re3s/p2ectivel1y/.2Thisiscausedbytheconstrupctiveinterferepnceof (j<j<) 0 −2q2ll++11 −2q2ll++11 −q2(22ll+−11)
m (1) in Eq. (1), and such an in-phase character has been
abcd
presentedforthe pf-shellcaseof42Ca[9,10,16]onthebasis
of numerical analyses using the shell model and the random- 4. PhasecoherenceintheISpair
phaseapproximation.Itisstillnotveryclear,however,whythe
constructive interference occurs and whether it is realized for As indicated by the above proof, the key to obtaining the
differentvalenceshells. constructiveinterferenceofm (1+)isthatalltheoff-diagonal
abcd 1
Inordertoanswerthisquestion,wefindatheoremconcern- Hamiltonianmatrixelementsareofthesamesignwhichcauses
ingthesignofm (1+). phasecoherenceintheISandIVpairs. Westressthatonlythe
abcd 1
signs are relevant. The simplest case of the phase coherence
Theorem1. Allofthem (1+)valuesareofthesamesignfor is found in the original paper of the Cooper pair [1], where
abcd 1 all the off-diagonal matrix elements between paired electrons
anyvalenceshellandforanysingle-particlesplittingwhenthe
near the Fermi surface are taken to be F and all the diag-
two-bodymatrixelementsaregivenbythepairinginteractions
−| |
ofEqs.(2)and(3)withnegativeGIV andGIS. onal matrix elements are zero. In this case, the lowest eigen-
vector is (α ,α ,...,α ) = (1,1,...,1)/√n, and the corre-
1 2 n
sponding energy eigenvalue is (n 1)F. The enhancement
Proof. We first consider the signs of the two-body matrix el- of eigenenergycomparedto the−off-−diag|on|almatrix elements,
ements of Eqs. (2) and (3). Since the sign of χIV is ( 1)la called pairing gap, is due to the phase coherence. In nuclei
ab −
[see Eq. (4)], all the matrix elements of VIVpair can be nega- it is well known that such a coherent pair is formed between
tive (or zero) when one takes a phase convention abJiTi = like-particles. All the off-diagonal (J,T) = (0,1) matrix ele-
| i
( 1)laabJiTi . Similarly, for the IS pairing one can easily mentsareindeednegativeinrealisticshell-modelHamiltonians
− | i
show that the sign of χIaSb is (−1)jb−1/2, thus obtaining entirely ofCKII,USD[17],KB3[18]andGXPF1[19]. Similarphase
negative matrix elements of VISpair with a phase convention coherenceintheISpairisexpectedtobeformedonthebasisof
abJfTf = ( 1)jb−1/2abJfTf . Hereafter we refer to those theIS-pairingHamiltonian,givingrisetotheconstructiveinter-
| i − | i
phasechoicesaspairingphaseconvention,andthecomponents ferenceofm (1+). Inreality,however,the2hconfigurations
of the eigenvectorsin this conventionare expressed by α¯IV(k) havenearlyvaabncdish1ingB(GT)valuesasshowninTable1,point-
ab
andα¯IS(k). ing to destructive interference. Hence, the coherent IS pairs
ab
Thus, when the IS- and IV-pairing interactions are taken, arenotalwaysformedwithrealisticinteractionsbecauseofthe
their off-diagonal Hamiltonian matrix elements in the pair- opposite sign in some of the (J,T) = (1,0) two-body matrix
ing phase convention are completely negative or zero for any elements.
two-nucleonconfiguration,sincesingle-particleenergiesdonot Takingthepshellasanexample,wepresentinFig.2anintu-
change the off-diagonal matrix elements in the jj-coupling. itivepictureabouttheformationofcoherentandnon-coherent
Forsuchmatriceshavingnon-positiveoff-diagonalmatrixele-
ments,itisgenerallytruethatallthecomponentsofthelowest
eigenvectorare of the same sign accordingto a versionof the
(a) pairing Hamiltonian (b) CKII Hamiltonian
Perron-Frobenius theorem in linear algebra . We thus obtain
†
α¯IaVb(1) ≥ 0andα¯IaSb(1) ≥ 0forany(a,b). AsshowninTable2, v1=(p> p>) v1=(p> p>)
the GT matrix elements between two-nucleon wave functions
satisfy abJ T σt cdJT 0 for any(a,b)and (c,d)con-
f f ± i i
h || || i ≤
cerned,hencethesamesignofm (1+).
abcd 1
or
Thisisamathematicallyexactstatement,andthereforepro-
v2=(p> p<) v3=(p< p<) v2=(p> p<) + v3=(p< p<)
videsarobustbasisfortheoccurrenceofthelow-energysuper
GTstate[9]in2pconfigurations. Figure2: (coloronline). Graphicalillustration ofthesignsof(J,T) = (1,0)
off-diagonaltwo-bodymatrixelements(representedas and )inthepairing
⊕ ⊖
phaseconventionandtheresultingsignsofthelowesteigenstate. Thep-shell
casesusing(a)theISpairingHamiltonian and(b)theCKIIHamiltonianare
One can easily prove this case by showing the expectation value of compared.Theupanddownarrowsstandforpositiveandnegativecoefficients
†
~v = (+α1,...,+αk,−αk+1,...,−αn) is greater than or equal to that of~v′ = of|vii,respectively.
(+α1,...,+αk,+αk+1,...,+αn)foranyαi 0.
≥
3
IS pairs for the (a) IS-pairing and (b) CKII Hamiltonians, re- (a) diagonal
0
spectively, where the pairing phase convention is used. Now
the lowesteigenstateisexpressedas α v byusingthe ba-
i i| ii -0.5
sis states vi . Before proceeding toPdetailed discussions, it
| i
should be reminded that the negative sign of an off-diagonal
-1
matrixelementbetweentwobasisvectors v and v ,denoted
i j
| i | i
as h , favors the same sign of α and α and a positive h fa-
vorsijtheoppositesigninthelowiesteigejnstate. Herewe mijean v/l-1.5
|v1i = |p>p>i, |v2i = |p>p<iand|v3i = |p<p<i, where p> and -2 <j j |V|j j >
p are p and p , respectively. As illustrated in Fig. 2 (a), > > > >
< 3/2 1/2
<j j |V|j j >
the obtainedcoherentpair isquite stabilized bythe IS-pairing > < > <
-2.5
Hamiltonian,sinceanycombinationof(i, j)satisfiestheabove <j j |V|j j >
< < < <
rule. On the other hand, the CKII Hamiltonian has a positive
-3
h andnegativeh andh . Inthiscase,theremustbeatleast 0 2 4 6 8 10
23 12 13
acombinationof(i, j)thatdoesnotcomplywiththeaboverule, l
(b) off-diagonal
asillustratedinFig.2(b). Thisisanalogoustothegeometrical
0
frustration in magnetism [20], although the physical situation
is rather different. The signs of α can no longer be uniquely
i -0.5
determined,andtheactualsignsdependonthediagonalterms.
For the A = 6 system, the valuesofh11, h22 and h33 are close -1
tooneanother,andthenthesamesignofα andα isrealized
1 2
because of h12 h23 h13. In this case v3 has a small -1.5
| | ≫ | | ≃ | | | i
amplitudeof the oppositesign, hencecontributinglittle to the
eigenstate. Thedominanceof v1 and v2 ofthesamesignac- -2 <j j |V|j j >
| i | i > > > <
countsfortheenhancedB(GT)value. Forthe A = 14system,
<j j |V|j j >
h11 ishigherthanh33 bymorethan10MeV,sothattheground -2.5 > > < <
<j j |V|j j >
stateisdominatedby v and v .Theresultingsignsofα and > < < <
2 3 2
| i | i
α are oppositebecauseof h > 0, thusleadingto the nearly -3
3 23 0 2 4 6 8 10
vanishingB(GT)value.
l
In this way the favorable signs for abJ=0T=1 in the IS
| i
pairarenotdefiniteanddependonthecoreassumedforrealistic
Figure3: (coloronline). (a)Diagonaland(b)off-diagonalmatrixelementsof
interactionswhoseoff-diagonal(J,T) = (1,0)matrixelements abVcd J=1,T=0 devided bylforthe SDI,where the dashed and solid lines
h | | i
are notcompletelyofthesame signin thepairingconvention. standfortheL=0andtotalmatrixelements,respectively. Thestrengthofthe
This is an essential differencebetween IV and IS pairing, and interactionisdeterminedsothatitsL=0termcanbethesameastheIS-pairing
interactionwithGIS= 1.
clearlyworksagainstforminganIS-paircondensate. −
We pointoutthatnon-coherentISpairscanbe probedwith
pair-transfer strength, which is regarded as a good measure
of IS pairing correlation [2]. The IS-pair creation and re- withtheSDI,andstrongcancellationbetweenL = 0and2oc-
moval strengths are now defined as |hJfTf|||D†|||JiTii|2 and cursespeciallyforhj<j<|V|j′>j′<iwithlowlandl′.Forl=1,for
|hJfTf|||D|||JiTii|2, respectively, and we considerthe transition instance,hj>j>|V|j>j<istillhasalargenegativevalue,whereas
tothelowest(Jf,Tf) = (1,0)state. ByusingtheCKIIinterac- hj<j<|V|j>j<ivanishes. In addition,the L = 2 termgivesrise
tion,theIS-pairremovalstrengthfrom16Oisonly5.3×10−3, totheoppositesignbetweenhjj|V|j′>j′<′iandhj>j<|V|j′>j′<′ifor
while the IS-pair creation strength on 4He is 8.1. A similar l′ =l′′ 2andn′ =n′′+1,thuscausingfrustrationamong jj ,
− | i
strongasymmetrybetweentheIS-paircreationandremovalis |j>j<iand|j′>j′<′iwhenhjj|V|j>j<i<0issatisfied. Weconfirm
alsoobtainedforthesdshell. thatfinite-rangeinteractionsleadtoessentiallysimilarresultsto
Finally,webrieflysurveytheoriginofdifferenceinthesigns the SDI byusing the VMU interaction[21], butsome quantita-
oftheoff-diagonal(J,T) = (1,0)matrixelementsbetweenre- tivedifferencesappear.Forinstance,the p<p<V p>p< matrix
h | | i
alisticinteractionsandtheIS-pairinginteraction. Firstwecon- element,whichisexaclyzerofortheSDI,ispositivebytaking
sider the central forces. Since the IS-pairing Hamiltonian is onlythe(S,T)=(1,0)termoftheVMU,andthismatrixelement
equivalenttothe(L,S)=(0,1)termoftheSDI,thedominance can be positive or negative depending on the strength of the
oftheL=0centralforceisthesourceofthecoherentISpairs, (S,T) = (0,0)term. Itshouldbenotedthatthe(S,T) = (0,0)
as well as for the usual IV pairing. While the (L,S) = (2,1) termvanishesforzero-rangeinteractions.
term is absentin the (J,T) = (0,1)matrixelements, thisterm Another important source to change the signs of (J,T) =
can modify the (J,T) = (1,0)matrix elements. In Fig. 3, the (1,0) matrix elements is the non-central forces. It has been
effectofthe(L,S)=(2,1)termispresentedforvariousorbital pointedoutbyJancoviciandTalmithatphenomenologicalten-
angularmomentalbyusingtheSDI.Ingeneral, j j V j j sorforcesareneededtoaccountfortheextraordinarylonglife-
h > >| | ′> ′<i
and j j V j j matrix elements due to L = 2 are positive time of 14C [22]. It is worth mentioning that its microscopic
h < <| | ′> ′<i
4
origin has recently been discussed from ab initio approaches [15] A.PovesandG.Martinez-Pinedo,Phys.Lett.B430(1998)203.
[23,24,25,26,27]. [16] C.L.Bai,H.Sagawa,G.Colo`,Y.Fujita, H.Q.Zhang,X.Z.Zhang,and
In the present context, the 14C lifetime problem is a man- F.R.Xu,Phys.Rev.C90(2014)054335.
[17] B.A.BrownandB.H.Wildenthal, Ann.Rev.Nucl.Part.Sci.38(1988)
ifestation of the non-coherent IS pair formed by the positive
29.
sign of p>p<V p<p< due to the L , 0 central forces and [18] A.PovesandA.Zuker,Phys.Rep.70(1981)235.
h | | i
the non-central forces. This idea can readily be applied to [19] M.Honma, T.Otsuka, B.A.Brown, andT.Mizusaki, Phys.Rev.C65
(2002)061301(R);ibid.69(2004)034335.
other cases. For instance, the small B(GT) value for A = 38
[20] Forinstance,R.MoessnerandA.P.Ramirez,PhysicsToday59(2006)24.
(see Table 1) is caused by the positive sign of d d Vd d
> < < < [21] T.Otsuka,T.Suzuki,M.Honma,Y.Utsuno,N.Tsunoda,K.Tsukiyama,
h | | i
madein a similar wayto p p V p p > 0. Incontrast, the andM.Hjorth-Jensen,Phys.Rev.Lett.104(2010)012501.
> < < <
j j V j j matrixelemhentsh|av|ealwaiyslargenegativeval- [22] B.JancoviciandI.Talmi,Phys.Rev.95(1954)289.
> > > <
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uesbothinschematicandrealisticinteractions,thuscausingthe
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WehaveshownthatthestrongasymmetryintheB(GT;0+ [27] A. Ekstro¨m, G.R. Jansen, K.A. Wendt, G. Hagen, T. Papenbrock,
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1
a coherent combination of (J,T) = (1,0) pairs is not neces-
sarily formed. By introducing the pairing phase convention
and the idea of frustration, we have presented a comprehen-
sivebutmathematicallyrobustexplanationastowhytheideal
IS-pairinginteractionalwaysleadsto phasecoherenceregard-
lessofsingle-particleenergiesbutrealisticinteractionsdonot.
This is in sharp contrast to the IV pairing in realistic interac-
tions,andmayprovideakeytoelucidatingtheoriginofelusive
IS-pair condensatesin nature. It is of greatinterestto investi-
gatehowmodernmicroscopiceffectiveinteractionspredictthe
(J,T)=(1,0)matrixelementsinawiderangeofnuclearshells
and how the non-coherenteffect changes observablesin more
complexnuclei,includingdouble-βdecaymatrixelements.
Acknowledgement
Y.U.thanksM.MoriforfruitfuldiscussionsandN.Shimizu
for his careful reading of the manuscript. This work was
supported in part by JSPS KAKENHI, under Grant No.
JP15K05094andNo. JP15K05104.
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