文档介绍：该【Effects of Λ hyperons on the deformations of even–even nuclei 2022 C. F. Chen 】是由【芳官】上传分享，文档一共【6】页，该文档可以免费在线阅读，需要了解更多关于【Effects of Λ hyperons on the deformations of even–even nuclei 2022 C. F. Chen 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。arXiv:[nucl-th]26Sep2021
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IFORMALISMII.
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N
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ieidensitieskinetic,
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ΛΛ
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)
,
n
=
2
ti-
,
oft
on-
nc-
(1)
ur-
4),
Λ
Λ
Λ
-
2
+EmpCfunctionals[41,50]of
energydensityandΛeffectivemass,Eqs.(6,7,8),usedinthiswork.
ε1ε2ε3ε4ε5ε6ε7µ1µ2µ3
τq,andspin-orbitcurrentsJq,
Nq
ii2i2i∗i
ρq,τq,Jq=∑nq|φq|,|∇φq|,φq(∇φq×σ)/i,(2)
i=1
hihi
i
whereφq(i=1,Nq)aretheself-consistentlycalculatedsingle-
particle(.)wavefunctionsoftheNqoccupiedstatesfor
thespeciesq=n,p,
Schr¨odingerequation,obtainedbytheminimizationoftheto-
talenergyfunctional(1)accordingtothevariationalprinciple,
as
1iii
∇·∇−Vq(r)+iWWq(r)·(∇×σ)φ(r)=eφ(r)
2m∗(r)qqq
q
(3)
inwhichWq(r)isthespin-orbitinteractionpartforthenucle-
onsasgiveninRefs.[45,49].ThecentralmeanfieldsVq(r),
-ΛhypernucleihBΛi≡
correctedbytheeffective-masstermsfollowingtheprocedure−2/3
describedin[40,42,50]areBnΛ/nasfunctionsofAcalculatedbySHFincomparisonwith
theexperimentaldataofsingle-sΛhypernuclei,double-sΛΛhyper-
nuclei,andsingle-
SHF∂εNΛ
VN=VN+fromRef.[7]andreferencestherein.
∂ρN
22/3
∂mΛτΛ3ρΛ(3πρΛ)
+∗−,(4)
∂ρNmΛ(ρN)2mΛ52mΛi
!Theoccupationprobabilitiesnq(fornucleonsonly)in
22/3
∂(εNΛ+εΛΛ)mΛ(3πρΛ)Eq.(2)arecalculatedbytakingintoaccountpairinginterac-
VΛ=−∗−1.(5),thepairing
∂ρm(ρN)2m
ΛΛΛinteractionistakenasadensity-dependentδinteraction[52],
ForthenucleonicpartεNN,weusetheSkyrmeforceSLy5
[47,51],
ρN((r1+r2)/2)
energy-densitycontributionsεNΛ[41,42]andεΛΛ[50]areVq(r1,r2)=V01−δ(r1−r2).(9)
−3−−3
parameterizedas(ρgiveninunitsoffm,εinMeVfm):
2
εNΛ(ρN,ρΛ)=−(ε1−ε2ρN+ε3ρN)ρNρΛForthep-shellnucleiandtheircorrespondinghypernuclei,
25/3thestrengthofthepairingforceissettoV=−410MeVfm3
+(ε4−ε5ρN+ε6ρN)ρNρΛ,(6)0
2forbothneutronsandprotons,whichgivesreasonablebind-
εΛΛ(ρΛ)=−ε7ρΛΘ(NΛ>1),(7)1213
ingenergiesforCandΛC[29,53].Fortheheavier(hy-
3
togetherwithper)nuclei,V0istakenas−999MeVfmforneutronsand
−1146MeVfm3forprotonsasinRef.[54].
∗
mΛ2Inthiswork,wefocusmainlyontheimpurityeffectsof
(ρN)≈µ1−µ2ρN+µ3ρN.(8)
SHFSchr¨odingerequationissolvedincylindricalcoordinates
Theparametersε,...,εinEq.(6)andtheΛeffective-mass
16rzundertheassumptionofaxialsymmetryofthemeanfield
parametersµweredeterminedinBrueckner-Hartree-Fock(,)
i[46,47].Theoptimalquadrupoledeformationparameter
calculationsofhypernuclearbulkmatterwiththeNijmegen
potentialNSC97f[41,42],whiletheempiricalexpressionin-
volvingtheparameterε7inEq.(7)hasbeenproposedbyfit-πh2z2−r2i
tingthebondenergyof6HeinRef.[50].Allparametersβ2=(10)
ΛΛ5hz2+r2i
ofthebindingenergiesofsingle-anddouble-Λhypernuclei
[40,42,50].isdeterminedbyminimizingtheenergy-densityfunctional.
3
10
8+n12+n16+n
0L
BeCO
nLnLnL
8
2L
64L
6L
4
8L
2
0
10
20+n24+n28+n
NeMgSi
nLnLnL
8
6
(MeV)
4
min
2
E-E
0
10
32+n36+n40+n
SArCa
nLnLnL
8
6
4
2
0
----
b
2
-consistentdeformedSHFmethodforeven-
evennucleirangingfrom8Beto40Caandtheircorrespondingmulti-Λ(n=2,4,6,8)
(negative)valuesofβ2correspondtoprolate(oblate)deformation.
-Λor
double-Λhypernucleiisusuallyreflectedbytheshapeshrink-
Duetothelackofexperimentaldataofmulti-Λhypernu-
theimpurityeffectofmulti-Λsystems,
clei,
thecalculatedpotentialenergysurfacesasfunctionsofthe
multi-hyperonhypernuclei,hBi≡B/n,withthatofexper-
ΛnΛquadrupoledeformationβforeven-evennucleirangingfrom
imentalsingle-shypernuclei,double-shypernuclei,and2
ΛΛΛ8Beto40Caandtheircorrespondingmulti-Λ(n=2,4,6,8)
single--Λ
sultsshowthathBidecreaseswithA−2/
Λbindingenergyoftheabsoluteminimumforagivenisotope.
ΛΛinteraction,thehBΛivaluesofdouble-sΛΛhypernucleiare12323616+n
ApartfromC,S,Ar,andthedoublymagicnucleinΛO
veryclosetothoseofsingle-sΛhypernuclei,Foragiveniso-
and40+nCa(n=0,2,4,6,8),allother(hyper)nucleiarewell
tope,hBΛidecreaseswithincreasinghyperonnumber,sincenΛ
12+n28+n
,(n=2,4,6),nΛSi(n=0,2,4,6,8),and
42
thehBΛiof8-Λhypernucleiareclosetothoseofthesingle-6ΛArareoblatelydeformed,whiletheothersareprolatelyde-
,andformed.
experimentalbindingenergiesofmulti-Λhypernucleiarenec-Oneobservesthattheimpurityeffectsbecomestronger
essarytodoastrictevaluationofthecurrenttheoreticalcalcu-withmorehyperonsinvolved,butthedependenceisnotregu-
,astheenergiesofsingle-anddouble-Λlar:For2Λand8Λhypernuclei,theimpurityeffectgivessim-
hypernucleiarereasonablywellreproduced,wecontinuetheilarresultsofdeformationreductionasinthecaseofsingle-Λ
4
+
d3/2,5/2+
[211]3/2+[000]1/2
[200]1/2+
-
[110]1/2
-
-4+
,[202]3/2[101]1/2
8[220]1/2-
)
[202]5/2[101]3/2
+-3
-8
(fm
-
p1/2,3/2L
[101]1/2r
-12
z=0r=0
-
[101]3/2
e(MeV)[110]1/2
-
-162
01234567012345678
r(fm)z(fm)
s1/2
-20[000]1/2+.-
48
Λhyperonsin8ΛCaatβ2=0asfunctionsofr(z=0)andz(r=0).
Thezaxisisthesymmetryaxis.
-24
--
b2
,p,ds.--
48orbits,theirdensitydistributionbecomesspherical.
tionofquadrupoledeformationβ2in8ΛCa.
Amoredetailedvisualizationoftheeffectsofhyperonson
,which
theRMFmodelinRef.[24].However,for4Λand6Λhy-
showstheΛdensitydistributioninthe(r,z)
pernuclei,theoppositeimpurityeffectscanbeseen,namely
theprolateNe,theoblateSi,andthesphericalCahyperiso-
thedeformationsofthehypernucleibecomelargerthanthose
-
ofdouble-Λhypernucleic