Effects of tensor forces in nuclear spin–orbit splittings from ab initio calculations

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Effects

of

tensor

forces

in

nuclear

spin–orbit

splittings

from

ab

initio

calculations

Shihang Shen (

)

a

,

b

,

Haozhao Liang (

)

b

,

c

,

Jie Meng (

)

a

,

d

,

e

,

∗

,

Peter Ring

a

,

f

,

Shuangquan Zhang (

)

a

a_{StateKeyLaboratoryofNuclearPhysicsandTechnology,SchoolofPhysics,PekingUniversity,Beijing100871,China} b_{RIKENNishinaCenter,Wako351-0198,Japan}

c_{DepartmentofPhysics,GraduateSchoolofScience,TheUniversityofTokyo,Tokyo113-0033,Japan} d_{YukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,Japan}

e_{DepartmentofPhysics,UniversityofStellenbosch,Stellenbosch,SouthAfrica} f_{Physik-DepartmentderTechnischenUniversitätMünchen,D-85748Garching,Germany}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received18November2017

Receivedinrevisedform17January2018 Accepted17January2018

Availableonline3February2018 Editor:W.Haxton

A systematic and specific pattern due to the effects of the tensor forces is found in the evolution of spin–orbit splittings in neutron drops. This result is obtained from relativistic Brueckner–Hartree–Fock theory using the bare nucleon–nucleon interaction. It forms an important guide for future microscopic derivations of relativistic and nonrelativistic nuclear energy density functionals.

©2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3_.

The understanding of nuclear density functionals in terms of thenucleon–nucleon(N N)interaction isone ofthepresent fron-tiersinnuclearphysics.Asmanifestedbythequadrupolemoment ofthedeuteron

[1]

,thetensorforceisanimportantcomponentin theN N interaction.Intheformofthetwopionexchangethe ten-sorforcealsoprovidesthemainpartofthenuclearattraction

[2]

, whichistakenintoaccountbythescalar

σ

mesonin phenomeno-logical models [3]. However, the role of the tensor force on the spinpropertiesinfinitenucleiismuchlessclear.

Inconfigurationinteraction (CI)calculationsithasbeenfound that the tensor force plays an important role in the shell struc-ture far away from stability [4]. On the other side, in nearly all ofthesuccessfulapplicationsofphenomenologicalnuclearenergy densityfunctionals

[5]

,tensorforceshavebeenneglectedformany years.

This has changed recently and much work has been done to investigatetheimpact oftensorforces inphenomenological non-relativistic

[6–24]

,andrelativisticdensityfunctionals

[25–33]

.Still, itisdifficulttofindsignificantfeaturesinexperimentaldatawhich are onlyconnected totensor forcesandtherefore suitable foran adjustment of their parameters. In a fit to nuclear masses and radii,forexample,withrelativisticdensityfunctionaltheory

[29]

, one obtains thebest fitforvanishing tensorforces.Onthe other

*

Correspondingauthor.

E-mailaddress:mengj@pku.edu.cn(J. Meng).

hand it hasbeenfound, that the single particle energies [4,7,34]

depend inasensitive wayontensorforces.However, inthe con-text ofdensityfunctionaltheory,single particleenergiesare only definedasauxiliary quantities [35]. Inexperimentthey are often fragmentedandthereforeonlyindirectlyaccessible.The fragmen-tation is caused by effects going beyond mean field, i.e., by the admixture of complicatedconfigurations,such asthecoupling to low-lyingsurfacevibrations

[36–41]

.

Obviously, the attempts to determine precise values for the strength parameters of thetensor forcesin universal nuclear en-ergydensityfunctionalsbyaphenomenologicalfittoexperimental datain finitenucleiisstilla difficultproblem[15].Insuch a sit-uation we propose to determine these strength parameters from microscopic abinitio calculations based on the well known bare nucleon–nucleonforces.Infact,muchprogresshasbeenachieved inthemicroscopicdescriptionofnuclearstructureinrecentyears

[42–50]. However, these are calculations of extreme numerical complexityandthereforetheycouldbeapplied,sofar,onlyinthe regionoflightnucleiorfornucleiclosetomagicconfigurations.

Fortheinvestigationofheavynucleiallovertheperiodictable, one is still bound to various versions of phenomenological nu-clear densityfunctionals and their extensionsbeyond meanfield

[51–53].Of course the ultimate goal is an abinitio derivation of such functionals. At present, such attempts are in their infancy

[54–56]. In Coulombic systems, where there exist very success-fulmicroscopicallyderiveddensityfunctionals,onestartsfromthe https://doi.org/10.1016/j.physletb.2018.01.058

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

infinitesystemandthe exact solutionof an electron gas[57].In nuclei,thereare attemptstoproceedina similarwayandto de-riveinafirststepsemi-microscopicfunctionals.Modernrelativistic andnonrelativisticabinitio descriptionsofsymmetricnuclear mat-teratvariousdensities are usedasmeta-datain orderto reduce thenumberofphenomenologicalparametersofthedensity func-tionalsconsiderably

[58–60]

.However, microscopiccalculationsof nuclear matter give usno information aboutthe effectivetensor forceinthenuclearmedium,becausethisisaspin-saturated sys-temand their influence is therefore negligible. In order to learn thetensorforce weproposeinthislettertostartfrommeta-data forafinitesystem,neutrondropsconfinedinanexternalpotential tokeeptheneutronsbound.

A neutron drop provides an ideal and simple system to in-vestigate the neutron-rich environment. Because of the missing proton–neutron interaction, the equations for neutron drops are much easier to be solved than those for finite nuclei. Therefore they have been investigated in the literature by many different abinitio methods [61–68] andalso by phenomenologicaldensity functionaltheory

[69]

.

Starting from bare nuclear forces, Brueckner–Hartree–Fock (BHF)theoryprovidestheG-matrix,adensitydependenteffective interactioninthenuclearmediumandthebasisof phenomenolog-icaldensityfunctionaltheoryinnuclei

[70]

.Asanabinitio theory, the nonrelativistic version of BHF with 2N forcesfailed [71] be-causeof the missing 3N forces, but it has been shown that the relativisticversionallowstoderivethesaturationpropertiesof in-finitenuclearmatterfrombare2Nforcesonly

[72]

.

In this Letter we use the relativistic Brueckner–Hartree–Fock (RBHF) theory to study the effects of tensor forces in neutron drops.Thistheory has recentlybeendeveloped todescribe finite nucleiself-consistently

[49,50]

withresultsinmuchbetter agree-mentwithexperimentaldata thanthenonrelativisticcalculations basedon2Nforcesonly.

Westartfromtherelativisticbarenucleon–nucleoninteraction Bonn A [73] andinvestigateneutron drops confined inan exter-nalharmonicoscillatorpotentialusingtheRBHFtheory.Westudy drops with an even number of neutrons from N

=

4 to 50 and comparetheirenergies andradii withother nonrelativisticab ini-tio calculaini-tions.Specialattentionispaidonthepossiblesignature ofthetensorforceintheneutron–neutroninteraction,thatis,the evolutionofspin–orbit(SO)splittingwithneutronnumber.

Westartwitharelativisticone-boson-exchangeN N interaction whichdescribesthe N N scatteringdata

[73]

.TheHamiltoniancan beexpressedas: H

=



kk



k

|

T

|

k



b_k†bk

+

1 2



klkl



kl

|

V

|

kl



b_k†b†_lblbk

,

(1)

wheretherelativisticmatrixelementsaregivenby



k

|

T

|

k

 =



d3r

¯ψ

k

(

r

) (

−

i

γ

· ∇ +

M

) ψ

k

(

r

),

(2)



kl

|

Vα

|

kl

 =



d3r1d3r2

¯ψ

k

(

r1

)

(1)α

ψ

k

(

r1

)

×

Dα

(

r1

,

r2

) ¯

ψ

l

(

r2

)

(2)α

ψ

l

(

r2

).

(3)

The indices k

,

l run over a complete basis of Dirac spinors with positiveandnegative energies, as, forinstance,over the eigenso-lutionsofaDiracequationwithpotentialsofWoods–Saxonshape

[50,74].

The two-body interaction V α contains the exchange contri-butions of different mesons

α

=

σ

,

δ,

ω

,

ρ

,

η

,

π

. The interaction vertices



α for particles 1 and 2 contain the corresponding

γ

-matricesforscalar

(

σ

,

δ)

,vector

(

ω

,

ρ

)

,andpseudovector

(

η

,

π

)

couplingandtheisospinmatrices

τ



fortheisovectormesons

δ,

ρ

, and

π

.FortheBonn interaction

[73]

,aformfactorof monopole-type isattachedtoeachvertexandDα

(

r1

,

r2

)

representsthe

cor-respondingmesonpropagator.Retardationeffectsweredeemedto besmallandwereignoredfromthebeginning.Furtherdetailsare foundinRef.[50].

The matrix elements of the bare nucleon–nucleon interac-tion are very large and difficult to be used directly in nuclear many-bodytheory. WithinBrueckner theory,the bareinteraction is therefore replaced by an effective interaction in the nuclear medium,theG-matrix.Ittakesintoaccounttheshort-range corre-lationsbysummingupalltheladderdiagramsofthebare interac-tion

[75,76]

anditisdeducedfromtheBethe–Goldstoneequation

[77],

¯

Gabab(W

)

= ¯

Vabab

+

1 2



cd

¯

VabcdG

¯

cdab(W

)

W

−

εc

−

ε

d

,

(4)

where in the RBHF theory

|

a

,

|

b



are solutions of the relativis-ticHartree–Fock(RHF)equations,V

¯

abab aretheanti-symmetrized

two-bodymatrixelements

(3)

andW is the starting energy.The intermediatestatesc,d runoverallstatesabovetheFermisurface with

ε

c

,

ε

d

>

ε

F.

The single-particlemotion fulfillstheRHF equation inthe ex-ternalfieldofaharmonicoscillator(HO):

(

T

+

U

+

1

2M

ω

2_r2

₎

_|

_a

_{ =}

_e

a

|

a

,

(5)

whereea

=

ε

a

+

M isthesingle-particleenergywiththerestmass

of the nucleon M and h

_¯

ω

=

10 MeV. The self-consistent single-particlepotential U isdefinedbytheG-matrix

[50,78,79]

:



a

|

U

|

b

 =

N



c=1



ac

| ¯

G

|

bc

,

(6)

wheretheindexc runsover theoccupiedstatesintheFermisea (no-sea approximation). In contrast to the RBHF calculations for self-bound nuclei in Refs. [49,50],a center ofmass correction is notnecessaryintheexternalfield.

ThecoupledsystemofRBHFequations

(4)

,

(5)

,and

(6)

issolved byiteration.TheinitialbasisisaDiracWoods–Saxonbasis

[74]

ob-tained by solving the sphericalDirac equation in a boxwith the size Rbox

=

8 fm anda meshsizedr

=

0

.

05 fm. DuringtheRBHF

iterationitisgraduallytransformedtotheself-consistentRHF ba-sis asexplained inRef.[50].The Bethe–Goldstoneequation (4)is solved inthesamewayasinRef. [50],exceptthat nowonly the isospinchannelTz

=

1 isincluded.

Fig. 1showsthetotalenergyE inunitsof

_¯

h

ω

N4/3_andtheradii

ofN-neutrondrops(withN from4to50)inaHOtrapcalculated bytheRBHFtheoryusingthebareinteractionBonn A

[73]

.Forthe casesofopenshells,thefillingapproximationisused.

TheresultsarecomparedwiththequantumMonte-Carlo(QMC) calculations

[64,66]

basedonthe2NinteractionAV8’

[80]

(without andwiththe3NforcesUIXandIL7),withtheno-coreshellmodel (NCSM) calculations[66,67] based onthe chiral 2N

+

3Nforces, andthe force JISP16. The factorh

_¯

ω

N4/3 _{takes into consideration}

thatintheThomas–Fermiapproximation

[81]

thetotalenergyfor anon-interactingN-FermionsysteminaHOtrapisgivenby

E

=

3 4/3 4

¯

h

ω

N

4/3

_≈

₁

_.

_082h

_¯

_ω

_N4/3

_.

₍₇₎

Withincreasingneutronnumberofthedropsweobservea satura-tionofE

/

h

¯

ω

N4/3forN

≥

20,incontrasttothenuclearcasewhere thebindingenergypernucleonsaturatesforlargemassnumber A.

Fig. 1. (Coloronline)(a)Totalenergydividedby

¯

hωN4/3_{and(b)radiiofN-neutron}

dropsinaHOtrapcalculatedbytheRBHFtheoryusingtheinteractionBonn A[73], incomparisonwithothernonrelativisticabinitio calculations.Seetextfordetails.

BycomparingwiththeQMCandNCSMcalculationsinpanel (a), theresultsoftheRBHFwiththeinteractionBonn Aaresimilarto thoseobtainedwiththeJISP16 interaction.ForN

≤

14,Bonn Ais alsosimilar toAV8’

+

IL7, butgettingcloser to AV8’afterwards. This resultis favorable asJISP16 isa phenomenological nonlocal N N interaction which reproduces the scattering data as well as it gives a good description forlight nuclei[82,83]. On the other hand,AV8’

+

IL7givesa muchbetter descriptionforlight nuclei upto A

=

12 thanAV8’orAV8’

+

UIX,butthethree-pionrings in-cludedinIL7givetoomuchover-bindingforpureneutronmatter athigherdensities

[66,84]

.

Panel (b) of Fig. 1 shows the corresponding radii. While the energies ofRBHF withBonn A are similarto those ofthe JISP16 interaction,theradiiofRBHFaresmaller.Incomparisonwiththe results ofAV8’

+

UIX andchiral force, the energies and radii of RBHF with Bonn A are smaller, except when N approaches 18, wheretheradiibecomeclosetochiral2N

+

3Nresults.

In

Fig. 2

,weshowtheSOsplittingsofN-neutrondropsfor1p, 1d,1 f ,and2p inaHO trapcalculatedby theRBHFtheory using theBonn A interaction.Theyare comparedwithresults obtained byvariousphenomenologicalrelativisticmean-field(RMF) density functionals, includingthe nonlinear meson-exchange models NL3

[85] and PK1 [86],the density-dependent meson-exchange mod-elsDD-ME2

[87]

andPKDD

[86]

,andthenonlinearpoint-coupling model PC-PK1 [88]. This figure shows the evolution of the vari-ousSOsplittingswithneutronnumber.ForthemicroscopicRBHF resultswe finda clearpattern: The SOsplitting of aspecific or-bitwithorbitalangularmomentuml decreasesasthenexthigher j

=

j>

=

l

+

1

/

2 orbitisfilled andreachesaminimumwhenthis

orbitisfullyoccupied.Asthenumberofneutroncontinuesto in-crease,the j

=

j<

=

l

−

1

/

2 orbitbeginstobeoccupiedandtheSO

splittingincreases.

Otsuka etal.

[4]

havefound a similar effectbetweenneutron andprotoninnuclei. Theyexplaineditintermsofthe monopole effectofthetensorforce,whichproducesanattractiveinteraction betweenaprotoninaSOalignedorbitwith j

=

j>

=

l

+

1

/

2 and

aneutronina SOanti-alignedorbitwith j

=

j_<

=

l

−

1

/

2 anda

Fig. 2. (Coloronline)Fromtoptobottompanel,1p,1d,1 f ,and2p spin–orbit split-tingsofN-neutrondropsinaHOtrap(¯hω=10 MeV)calculatedbytheRBHFtheory usingtheBonn Ainteraction,incomparisonwiththeresultsobtainedbyvarious RMFdensityfunctionals.

repulsiveinteractionbetweenthesameprotonandaneutronina SOalignedorbitwith j

=

j_>

=

l

+

1

/

2.

AsdiscussedinRef. [4]asimilarmechanism, butwithsmaller amplitude,exists alsoforthetensorinteractionbetweenneutrons withT

=

1.ThereforewecanexplainthebehavioroftheSO split-ting in

Fig. 2

ina qualitative way:we consider, forinstance, the decrease ofthe 1d SOsplitting ifwego from N

=

20 to N

=

28. Becauseoftheinteractionwiththeneutronsfillingthe1 f7/2 shell

above N

=

20,the 1d5/2 orbitisshifted upwardandthe 1d3/2 is

shifteddownward,thusreducingthe1d SOsplitting.AboveN

=

28 we fillin neutrons into 2p1/2 and1 f5/2. They interactwith the

1d-neutronsintheoppositewayandincrease theSO-splittingfor the1d configuration.

Ontheotherhand,thisspecificevolutionofSOsplittingisnot significantforanyofthephenomenologicalRMFdensity function-alsin

Fig. 2

,whichdonotincludeatensorterm.Inordertoverify that thisspecific patternisindeedcausedbythetensorterm, we showin

Fig. 3

thesamecalculationbutwiththeRHFdensity func-tional PKO1

[25]

,whichincludes thetensorforce induced by the pioncouplingthroughtheexchangeterm.Withoutreadjustingthe other parameters of this functional, we have multiplied a factor

λ

in front of the pion coupling to investigate the effects of the tensorforces.Itisremarkabletoseethat theevolutionoftheSO splitting is influenced by the strengths of the tensor forces sig-nificantly. For

λ

=

1 wehavetheresultsofthedensityfunctional PKO1.Theyshowalreadytherightpattern, butthesizeofthe ef-fect is somewhat too small. This can be understood by the fact, that it is difficultto fit thestrengths of the tensorforces justto bulk propertiessuch asbindingenergies andradii [29]. The gen-eralfeature oftheseSOsplittingsfound inourRBHFcalculations with Bonn A canbe well reproduced withPKO1 simply by mul-tiplying a factor

λ

=

1

.

3 in front of the pion coupling.One may wonderifthetensorforcediscussedhereistoostrong,asthe

orig-Fig. 3. (Coloronline)SimilartoFig. 2,butincomparisonwiththeRHFdensity func-tionalPKO1[25]withdifferentstrengthofpioncouplingcharacterizedbyλ.

inalfunctionalPKO1 canwell reproducetheexperimentaldataof SO splitting reduction in realistic nuclei, e.g., from 40_{Ca to} 48_Ca [26].However,aspointedout intheintroduction,itisnevereasy tocomparedirectlytotheexperimentalsingle-particleenergiesin realistic nuclei, ascomplicated beyond-mean-field effects are in-volved.

Ofcourse, finally one should carry out a complete fit, taking intoaccountatthesame timetheabinitio meta-datafornuclear matter as well as these meta-data for SO splittings in neutron drops together with a fine-tuning of a few final parameters to massesandradii.Workinthisdirectionisinprogress,butitgoes definitelybeyondthescopeofthisletter.

Finally we would like to mention two important aspects of theseresults:

(a)The specific patternof increasing anddecreasing SO split-tingswith neutronnumber is not restrictedto a specific j-shell, i.e., to a specific region. It seems to be generally valid for all the neutron numbers 4

≤

N

≤

50 under investigation and it can bereproduced byreadjusting asingle parameter

λ

forthetensor strengthinthe densityfunctionalPKO1. Thereforewe can expect thatasimilarfeatureisvalidalsoforrealnucleialloverthe peri-odictable.

(b) In relativistic Brueckner–Hartree–Fock theory, there are no higher-orderconfigurations [89]. This means, that effects like particle-vibrationalcouplingare not includedinthese meta-data. Thisallowstoadjustthetensorforce tothesemeta-datawithout theambiguityofadditionaleffectsofparticlevibrationalcoupling. Theyare neitherincludedinthepresentconcept ofdensity func-tionaltheory,norinthepresentRBHFcalculations. Inthecaseof realistic calculationsinfinite nucleicomparable with experimen-taldata,theyhavetobeincludedbygoingbeyondmeanfieldina similarwayasinRef.[39].

In summary, we have studied neutron drops confined in an externalfield of oscillatorshape usingthe relativisticBrueckner– Hartree–Focktheory withthe bare N N interaction. It was found

thattheSOsplittingdecreasesasthenext j

=

l

+

1

/

2 orbitbeing occupied, andincreasesagainasthe next j

=

l

−

1

/

2 orbit being occupied. This is similar to the effects of tensor forces between neutronandproton ashasbeenfound inRef. [4].The patternof the evolution of SO splittings cannot be reproduced by the RMF densityfunctionals,whileitcanbewellreproducedwiththeRHF densityfunctionalPKO1whichincludestensorforces.Thisimplies thatthestrengthsoftensorforcesinneutrondropscanbederived fromabinitio calculations andusedasa guideforfutureabinitio derivationsofnucleardensityfunctionals.

Acknowledgements

We thankPengwei Zhaofor discussions andproviding his re-sults. This work was partly supported by the Major State 973 Program of China No. 2013CB834400, Natural Science Founda-tion of China under Grants No. 11335002, No. 11375015, and No. 11621131001, the Overseas Distinguished Professor Project from Ministry of Education of China No. MS2010BJDX001, the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20110001110087, and the DFG (Ger-many)clusterofexcellence“OriginandStructureoftheUniverse” (www.universe-cluster.de).S.S.wouldliketothanktheshort-term Ph.D. student exchange program of Peking University and the RIKEN IPAproject,andH.L. wouldliketo thanktheRIKEN iTHES projectandiTHEMSprogram.

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