Neutron Skins and Halo Orbits in the $sd$ and $pf$ hells

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by  J. Bonnard

Neutron Skins and Halo Orbits in the sd and pf shells
J. Bonnard1, S. M. Lenzi1,2 and A. P. Zuker2,3
1 Istituto Nazionale di Fisica Nucleare, Sezione di Padova, 35131 Padova, Italy
2 Dipartimento di Fisica e Astronomia, Universita` degli Studi di Padova, I-35131 Padova, Italy
3 Universit´e de Strasbourg, IPHC, CNRS, UMR7178, 23 rue du Loess 67037 Strasbourg, France
(Dated: May 18, 2016)
The strong dependence of Coulomb energies on nuclear radii makes it possible to extract the
latter from calculations of the former. The resulting estimates of neutron skins indicate that two
mechanisms are involved. The first one—isovector monopole polarizability—amounts to noting
that when a particle is added to a system it drives the radii of neutrons and protons in different
directions, tending to equalize the radii of both fluids independently of the neutron excess. This
6
mechanismiswellunderstoodandtheDuflo-Zuker(small)neutronskinvaluesderived14yearsago
1
are consistent with recent measures and estimates. The alternative mechanism involveshalo orbits
0
whose huge sizes tend to make the neutron skins larger and have a subtle influence on the radial
2
behavior of sd and pf shell nuclei. In particular, they account for the sudden rise in the isotope

y shifts of nuclei beyond N = 28 and the near constancy of radii in the A = 40−56 region. This
a mechanism,detectedhereforthefirsttime,isnotwell understoodandmaywellgobeyondEfimov
M physicsusually associated to halo orbits.

7
1 Mirror nuclei in which proton and neutron numbers
] Nth,eZisoasrpeininbtreerachkainnggeindtehraavcetiodniffse(rIeBnIt)ednoemrgineatitcesddbuyethtoe TTA=BL1/E2Im.DirirsoprlancuecmleeinotfemnearsgsieAsbdeetfiwneeednatsheMgDroEun=dEstJa(tZeo>f
h N) − EJ(Z < N). Experimental, full IBI, Coulomb (C)
t Coulomb force. It affects both the spectra (MED for andschematicCoulomb (Eq.(1),sC)contributionsaregiven
-
cl MforirMroirrrEonreDrgiyspDlaicffeemreennctesE)naenrgdietsh)e.gAropurnidmsetaexteasm(pMleDoEf iNn3MLOeV[2.7N]opoctoerneti0a~lωwcitahlccuulattoiffonλw=it2h.0Vflomw−-k1.form [8] of the
u
theMEDisfoundin13Niwherethe1s protonorbitis
[n depressed by about 750 keV with resp1e/c2t of its neutron 1A5 14~.ω67 1/J2π− M3D.5E3e7xp M3D.5E7I4BI M3.D47E4C M3D.6E24sC
2 analoguein 13C.This behavioris referredto as Thomas- 17 13.38 5/2+ 3.543 3.388 3.377 3.514
v Ehrman shift (TES) because it was first studied via R- 39 10.89 3/2+ 7.307 7.120 6.970 7.212
matrix theories by J. Ehrman [1] and R. G. Thomas [2] −
8 41 10.61 7/2 7.278 6.683 6.679 6.675
who also considered the pair 17F-17O.
6
1 The s orbits are the essential ingredients of halo
2 physics [3] and have a decisive influence in the spec- a variety of neutron radii [13, 14] and calculations ap-
0
troscopy of nuclei at A = 16±1 [4, 5], which will be peared in which the NSA was almost absent [15]. The
.
1 shownto extend to higher masses,including the pf shell NSA does not seem to have attracted much attention
0 because of the halo nature of the p orbits. lately but neutron radii are a very hot subject whose
6
The TES can be viewed as an overbinding of orbits— connection with displacement energies—hitherto some-
1
: with respect to naive expectations—because their large what neglected—is worth examining. It follows by not-
v radii reduce the Coulomb repulsion. For the binding en- ingthatisospinconservationimpliesthattheprotonrms
Xi ergies, the naive assumption is that a closed shell core radius ρπ> = hrπ2>i of a nucleus with Z > N equals
r is unperturbed by the addition of a particle. The MDE the neutron rmps radius of its mirror, ρν< with Z < N.
a wouldthenbeduetothecoreCoulombfieldactingonthe AssumingaschematicCoulombcontributionofthe form
extraproton. The resultis oftena severeunderestimate, CZx =0.67Z(Z−1)/ρx we have (disregarding other IBI
as in A = 41: the Nolen-Schiffer anomaly (NSA) [6] il- terms)
lustrated in Table I. While the TES is due to a proton
MDE=C −C =C −C (1)
radius larger than expected for the extra particle, the Z+1π> Zπ< Z+1ν< Zπ<
NSA may be thought to demand the opposite i.e., a re-
Therefore, if we know, say, MDE(17F-17O) and ρ , the
duction of the radius of the added particle but this is π<
proton radius of 17O, we also know its neutron radius
ruled out experimentally [7].
ρ . This simple idea lead to a general estimate of the
ν<
Though Shlomo had noticed that equalizing the total
neutronskinsbyDufloandZuker(DZ)[16]. Theystarted
neutronandprotonradiiwouldeliminatetheanomaly[9]
by fitting the proton mean square radii to experiment
ittooksometime beforethis basicallysoundideagained
through (t=N −Z)
traction: Hartree-Fock (HF) calculations routinely pre-
dicted proton radii in agreement with experiment and ζ t υ t
hr2i=ρ =A1/3 ρ − − ( )2 e(g/A) (2)
substantiallylargerneutronradii[10],thoughexperimen- π π (cid:18) 0 2A4/3 2 A (cid:19)
tal evidence did not support the latter [11, 12]. Then it p
+λ[z(D −z)/D2 ×n(D −n)/D2]A−1/3 (3)
was shown that good proton radii were compatible with π π ν ν
2
wheren, zarethenumberofactiveparticlesbetweenthe A unitary (HF) transformation solves exactly H but
0
EImagicnumbers[17,Sec. IC]atN, Z =6,14,28,50...; onlyapproximatelyH1 becausethetermint−U++t+U−
D =8,14,22...arethecorrespondingdegeneracies. By demands a more refined treatment, ignored here. The
x
fittingknownradiiforA≤60oneobtainsrmsdeviations results can be visualized in Fig. 1. The shaded area cor-
of about 42 mf for a 4 parameter fit with λ=0 reduced responds to the unperturbed Hamitonian bounded by a
to about 18 mf when varying λ. (Including all known parabola, while the heavy lines represent parabolic seg-
radii the rms deviation goes down, with little change in ments with ~ω > ~ω , the situation in which the NSA
ν π
the parameters). In principle the neutron skin (in fm) disappears as the radii tend to equalize i.e., reduce the
neutron skin with respect to the ~ω = ~ω value. The
ν π
ζt sign of v determines whether radii equalize or move
∆r =ρ −ρ = eg/A, (4) 1
νπ ν π A apart. Within this elementary mean field approach all
orbits behave in the same way. A more refined approach
couldbeexpectedtocomeoutofthe fit. However,fixing would allow different polarizabilities for different orbits.
ζ to values between 0.4 and 1.2 did not alter the quality Moreover, the model ignores threshold effects i.e., cou-
of the fit. A useful reminder that the neutron radii are plingtothecontinuumthatcouldplayanimportantrole.
independent of the proton ones. Hence, the authors re- Nonetheless the model has the advantage of suggest-
sorted to Eq. (1) using a form of the Coulomb potential ing the computational strategy that generalizes the DZ
close to the exactone for oscillatororbits. We adoptthe approach. We shall work in 0~ωno-core spaces with
set g =0.985, ρ0 =0.944,λ=5.562, υ =0.368, ζ =0.8, Vlow-k [8] precision potentials: AV18 [25], CDB [26], and
rmsd=0.0176. Allunitsinfmexceptg. Withthesevalues N3LO [27] which produce almost indistinguishable re-
of g and ζ Eq. (4) yields the estimates of Table II where sults according to our checks. They incorporate effects
they are seen to agree with numbers of diverse origin: a not treated in DZ (such as electromagnetic spin-orbit
recentmeasure[18],estimates basedoncomparisonwith coupling)arefullyIBIandwillmakeitpossibletodocon-
electric dipole polarizability αD [19] and an “ab initio” figuration mixing. Saturation is treated in the standard
calculation [20]. It should be noted (stressed) that the shell model way by fixing ~ωat a value consistent with
the observed radius. It is here that Fig. 1 comes in: For
each nucleus, calculations are done for a different ~ωfor
TABLE II. Comparing ∆rνπ from Eq. (4) with estimates neutronsandprotons: ~ω isknownthroughEqs.(2,3,7)
π
(ests) [19, 20] and measure (exp) [18] (fm). for N > Z (and hence ~ω for N < Z). Then ~ω for
ν ν
48Ca 68Ni 120Sn 208Pb 128Pb N >Z and~ω forN <Z followfromζ treatedasafree
π
Eq. (4) 0.14 0.14 0.13 0.17 0.17 parametertoreproducetheexperimentalMDEorMED.
ests-exp 0.135(15) 0.17(2) 0.14(2) 0.16(3) 0.15(3) Torelate~ω to the radiiweadaptfrom[28,Eq.(2.157)]
π
ref. [20] [19] [19] [19] [18] Eq. (7), where the sum runs over occupied proton orbits
in oscillator shells of principal quantum number p, and
results of Eq. (4) also square nicely with those obtained a similar expressionfor neutrons, leading asymptotically
from two other sources analyzed in [23]: they are very to Eq. (8).
close to the GognyD1S force [21] andnot far fromthose
41.47
of Sly4 [22]—which gives slightly bigger skins. It ap- ~ω = z (p +3/2)/Z, (7)
pears that a general mechanism, that we sketch next, is π hrπ2i Xi i i
at play. Think of a model space in which an extra par- ~ω 35.59 ~ω 35.59
π ν
ticle (dot in Fig 1 taken to be a neutron) associated to = ; = . (8)
(2Z)1/3 hr2i (2N)1/3 hr2i
number n and isospint polarizesthe system by inducing π ν
particle-holejumps fromthe closed coreof particles h to The form of ~ωas a function of A is obtained through a
the open shells of particles p 2~ωabove. While H rep-
0
resents isoscalar monopole polarizability, responsible for
an overall increase in radius, its isovector counterpart,
π ν
H takes care of a differential contraction-dilationof the
1
fluids. The model could be termed the “degree zero” of
the mean field [24]:
1
H0 =εS0+v0n(S++S−), ε= (εp−εh), (5)
2
S =nˆ −nˆ , S =a+a +b+b +hc,
0 p h + p h p h
1
H1 =εS0+v1(t−U++t+U−+ t0U0), (6)
2
U =a+a −b+b +a+a −b+b ,
0 p h p h h p h p FIG. 1. (color online) Illustrating the solution of Eq. (6).
U+ =a+pbh+a+hbp, U− =b+pah+b+hap. Explained in text.
3
14 3.5
Naive radii s K exp
Correlated radii pe K th
o Ca exp
13 K isot 3.45 K nCaai vthe
d
n
a
a
π 12 C 3.4
h/ω s for
11 shift 3.35
e
p
o
ot
s
10 I 3.3
16 20 24 28 32 36 40 44 48 52 56 60 40 42 44 46 48 50 52
A A
FIG. 2. (color online) Values of ~ωπ (MeV) for T = 0 and FIG. 3. (color online) Radii ρπ in fm from isotope shifts in
1/2 extracted from Eq. (7) using ρπ from Eqs. (2,3) with theKandCaisotopes[29]incorporatingrecentmeasures[30,
parameters g =0.985, ρ0 =0.944, λ=5.562, υ =0.368, ζ = 31](label exp) compared with estimates from Eqs.(2,3) (pa-
0.8, for the correlated radii and λ = 0. for naive radii. All rameters as in Fig. 2, label naive is for λ = 0). The corre-
unitsin fm except g. lated numbers (label th) have been shifted down by 30 mf,
to restore translation invariance and allow for experimental
uncertainties in the extraction of radii ρπ from δhrπ2i isotope
shifts. ForclaritytheKandCavaluesareshifted by±55mf
term by term (nucleus by nucleus) evaluation of Eq .(7).
respectively
Two variants are chosen: λ = 0 in Eqs. (2) (the naive
fit) and λ 6= 0—the correlated fit—leading to the inter-
esting pattern in Fig 2. Its meaning may not be evident
TABLEIII.MDEandMED∆E forT =1/2mirrornucleiof
at first, but clarification comes in Fig. 3—showing the massA,~ωπ,ν inMeVandthecorrespondingskinparameters
isotope shifts of the K and Ca isotopes, including recent andradiiinfm. NotethattheradiicorrespondtotheN >Z
measures [30, 31]—which make it clear that Duflo’s λ nuclei, theyare interchanged for themirror partners. Exper-
term has a deep physical grounding: The abrupt raise imental and calculated ∆E values coincide by construction.
of radii after A = 47 i.e., the N = 28 is an open prob- Interaction N3LO [27] with cutoff Λ=2fm−1.
lem [30, 31], so far only qualitatively explained by rela- A Jπ ∆E ~ων ~ωπ ζ ∆rνπ rπ rν
tivisticmeanfieldcalculations[32]. Fig.3suggestsavery 15 1/2− 3.537 14.55 14.62 0.358 0.025 2.507 2.532
simplesolution: theraiseisduetothefillingofhugep3/2 3/2− 3.389 14.39 14.66 0.609 0.043 2.503 2.547
orbits. As the filling occurs for neutron orbits, and the 17 5/2+ 3.543 13.62 13.38 0.906 0.056 2.641 2.697
shift measures the behavior of proton orbits, isovector 1/2+ 3.167 12.86 13.51 2.367 0.147 2.628 2.776
polarizabilitymustbeatworkhere: ifonefluidincreases 39 3/2+ 7.307 10.97 10.91 0.258 0.007 3.361 3.368
in size, the other fluid must follow suit. The operation 1/2+ 7.253 10.90 10.89 0.523 0.014 3.365 3.379
ofthe λ termdoes notdepend onζ, whichmay takeany 41 7/2− 7.278 10.78 10.63 0.610 0.015 3.422 3.437
value, but must be fairly constant. To learn some more 3/2− 7.052 10.61 10.59 1.513 0.038 3.427 3.465
−
about the nature of s and p which seem (are) re- 1/2 7.129 10.61 10.59 1.482 0.037 3.428 3.465
1/2 3/2 −
sponsible for the elegant undulating patterns in Fig. 2, 5/2 7.351 10.75 10.61 0.702 0.018 3.424 3.442
−
weexaminethesingleparticleandsingleholestatesbuilt 5/2 7.338 10.75 10.61 0.725 0.018 3.427 3.442
on 16O and 40Ca.
Results are given in Table III and Fig. 4. The values
of ζ have been adjusted so as to obtain the observed en- A = 17. A reasonable value of ζ solves the NSA for
ergies. In the figure,the calculatedζ and∆r arecom- d . Thes orbitistrulylarge: itsrmsradiusisabout
νπ 5/2 1/2
pared with those obtained under the ~ω = ~ω (naive 1.2 fm larger than its d counterpart. No doubt about
ν π 5/2
shell model) assumption, expected to produce too large its halo nature.
skins. However, because of the pronounced shell effects A = 39. Here we find that s is no longer gigantic,
1/2
exhibitedintheplots,fortheholestatesi.e., A=15and butlargeenoughtokeepsomememoryofitshalostatus.
39theskinsremainmoderateorsmall. Afewcomments: A = 41. Most interesting. NSA is solved for f
7/2
A = 15. Independently of the ζ values, ~ω < ~ω viaareasonableζ veryclosetowhatisdemandedbythe
ν π
rules out an isovector polarization mechanism. As there lowestobservedpairoff candidateswhichhaveonlya
5/2
is no simple argumentto treatthese orbits as “halo”,we fractionofthespectroscopicstrength. Bothp andp
3/2 1/2
prefer to leave the question open. are accommodated by the same ζ and have pronounced
4
2.60 1/2+ exact Recentprogresshasbeenmade[18]andtheirconnection
2.20 asymp withtheisovectordipolepolarizabilityαD haveledtore-
1.80 exp liable estimates [19]. Theoretically the problem is much
-
ζ (fm) 11..0400 5/2+ 31//22- saibmilpitlyer[.24I]t, iosrsfuobrsSukmyerdmebytyipseovfeucntcotriomnaolnsobpyolceopnotrlaorlioz-f
-
0.60 3/2- 1/2- 1/2++ 7}5/2/2- tsheveesryalmcmalecturlyateionnersgaypp[1e3a,r1t4o].reAprsondoutceedskafintesrwTeallb.le II,
0.20 3/2
0.14 1/2+ Halo orbits are another matter: no existing calcula-
0.12 tion [30, 31] explains the observed isotope shifts as done
m) 0.10 in Fig. 3. We have interpreted the result as due to an
(fπ 0.08 increase in size of a p orbit. We have also learned from
r∆ν 00..0046 3/2- 5/2-+ 13//22-- TthaebylecIoIuIldanbdeFviige.w4edthaasthsa1/lo2 aonrbditps3/in2 aAre=so1h7uagnedth4a1t,
00..0002 1/2 1/2+3/2+ 75//22-- but their influence extends well beyond. We have also
8 12 16 20 24 28 32 36 40 44 48 52 56 60 learned that at A = 39, s1/2 is no longer huge. We ex-
A pecttolearnmuchaboutitsevolutionthroughfullMED
and MDE configuration mixing calculations now under
way.
FIG. 4. (color online) Values of ζ and ∆rνπ from Table III We close by proposing an alternative to the use of
(dots labeled exp) compared with those obtained for ~ων = Eq. (3) to represent shell effects:
~ωπ form Eqs. (7) (labeled exact) and (8) (labeled asymp).
41.47
hr2i= z (p +3/2+δ )/Z (9)
halo nature. Their rms radiiexceedthose ofthe f orbits π ~ωπ Xi i i i
by some 0.7 fm. Interestingly, orbits of the same l have
the same behavior. where~ω isnowthe“naive”estimateusingEq.(2)alone
π
Old problems come back under new guises: the NSA andthe δ correctionsto the oscillatorvalues replace the
i
as neutron skins, the TES as halo orbits associated to λ term. Eq. (9) could be useful in interpreting the stuc-
subtle shell effects detected in isotope shifts. ture of isotope shifs as reflecting orbital occupancies as-
Neutron skins are difficult to measure experimentally. sociated to given orbital radii.
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