Electron scattering in isotonic chains as a probe of the proton shell structure of unstable nuclei

0.32 MB·English

by  X. Roca-Maza

Electron scattering in isotonic chains as a probe of the proton shell structure of
unstable nuclei
X. Roca-Maza1,2,∗ M. Centelles1, F. Salvat1, and X. Vin˜as1
1 Departament d’Estructura i Constituents de la Mat`eria and Institut de Ci`encies del Cosmos,
Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain
2 INFN, sezione di Milano, via Celoria 16, I-20133 Milano, Italy
Electron scattering on unstable nuclei is planned in future facilities of the GSI and RIKEN up-
grades. Motivated by this fact, we study theoretical predictions for elastic electron scattering in
the N = 82, N = 50, and N = 14 isotonic chains from very proton-deficient to very proton-rich
isotones. WecomputethescatteringobservablesbyperformingDiracpartial-wavecalculations. The
charge density of the nucleus is obtained with a covariant nuclear mean-field model that accounts
for the low-energy electromagnetic structure of the nucleon. For the discussion of the dependence
of scattering observables at low-momentum transfer on the gross properties of the charge density,
2
1 wefitHelmmodeldistributionstotheself-consistentmean-fielddensities. Wefindthatthechanges
0 shown by the electric charge form factor along each isotonic chain are strongly correlated with the
2 underlying proton shell structure of the isotones. We conclude that elastic electron scattering ex-
perimentsinisotonescanprovidevaluableinformationaboutthefillingorderandoccupationofthe
l
u single-particle levels of protons.
J
PACSnumbers: 21.10.Ft,25.30.Bf,13.40.Gp,21.60.-n
6

]
h
t I. INTRODUCTION opportunity of studying the structure of unstable exotic
-
nuclei by means of electron scattering.
l
c
Onthetheoreticalside,muchworkhasbeendevotedto
u Since the 1960’s, elastic electron scattering has been
n utilized to obtain accurate information on the size and thestudyofchargedistributionsofexoticnucleithrough
[ shape of nuclei [1–5]. Because electrons and nucleons in- calculationsofbothelectronscattering(seee.g.Refs.[23–

1 teract essentially through the electromagnetic force, the 29]) and proton scattering (see e.g. Ref. [30]). Suda [31]
pointedoutthatinelectronscatteringoffunstablenuclei
v nucleus remains rather unperturbed during the scatter-
themaximaandtheminimaofthechargeformfactorare
9 ing processand the analysis ofthe data is not hampered
7 by uncertainties associated with the strong interaction. verysensitivetothesizeandthediffusenessofthecharge
6 density. This fact has been confirmed by different works
Thus, electron scattering is able to provide very clean
1 thathaveanalyzedthebehaviorofthechargeformfactor
information about the charge distribution of atomic nu-
7. clei [6–8]. along isotopic [23, 24, 27, 28] and isotonic [29] chains.
0 Toprobethechargedistributioninnuclei,theelectron
Low-energy nuclear physics is nowadays moving very
2
beam energy needs to be of the order of a few hundred
1 fasttowardsthedomainofexoticnuclei[9]. Thisisdueto
MeV.Asonedealswithrelativisticelectrons,itismanda-
: the developmentofsuccessivegenerationsofradioactive-
v tory to solve the elastic scattering problem of Dirac par-
isotope beam (RIB) facilities [10–15], such as FAIR
i ticles in the potential generated by the nuclear charge
X and SPIRAL2 in Europe, FRIB in North America, and
density. The simplest approach is the plane-wave Born
r HIRFL-CSR, RARF or RIBF in Asia, which will allow
a studyingthepropertiesofnucleibeyondthestabilityval- approximation(PWBA)wheretheinitialandfinalstates
of the electron are described by Dirac plane waves. The
ley. Many interesting effects have already been discov-
PWBAaccountsfor manyfeatures ofelectronscattering
ered in exotic nuclei, such as neutron and proton halos,
butitcannotprovideanaccuratedescriptionofthe elec-
neutron skins, and new magic numbers. These effects
tricchargeformfactor,inparticularnearthedeeps. The
may be related to the structure of the nucleon distri-
most elaborated calculations of electron-nucleus scatter-
butions far from stability. As with stable nuclei, one
ing are obtained by the exact phase-shift analysis of the
wayofexploringthe structureofexoticnucleiisthrough
Dirac equation. This calculation scheme is known as
the electromagnetic interaction. For this purpose, a new
distorted-wave Born approximation (DWBA) [32] and
generation of electron-RIB colliders using storage rings
has been used to analyze different aspects of the scat-
is under construction by RIKEN (Japan) [15, 16] and
teringofelectronsbynuclei,seee.g.Refs.[23,24,28,33]
at GSI (Germany) [17, 18]. It is expected that in the
and references therein. It is worth noting that for high-
near future the SCRIT project in Japan [19–21] and the
energy electron scattering, the eikonal approximation is
ELISeexperimentatFAIRinGermany[22]willofferthe
a reliable choice to deal with the problem [34]; it has
been applied in [27, 29, 35] to study electron scattering
off proton- and neutron-rich nuclei.
∗Electronicaddress: [e-mail] The charge density of the target nucleus is one of the
2
basic ingredients of the electron-nucleus scatteringprob- aboutthe Dirac partial-waveanalysis,which we perform
lem. Formediumandheavynuclei,thetheoreticalcharge usingtheELSEPAcode[38]adaptedtothenuclearprob-
densities can be calculated in the mean-field approxima- lem. We devoteSectionIII tothe presentationandanal-
tion using non-relativistic nuclear forces or relativistic ysis of our numerical results for elastic electron-nucleus
mean field (RMF) models. It is known that the over- scattering in the N =82,50, and 14 chains. Finally, our
all trends of the elastic scattering of electrons by stable conclusions are laid in Section IV.
medium and heavy nuclei, are well reproduced by the
mean-field charge densities computed with nuclear mod-
elsthathavebeencalibratedtodescribetheground-state II. METHOD
properties (in particular the charge radii) of some se-
lected nuclei. However,different nuclear models differ in Toinvestigateelectronscatteringinisotonicchainswe
the fine details and describe with different quality the follow the method developed in Ref. [28]. For complete-
experimental scattering data. See Ref. [28] for a recent ness,wesummarizeherethemainaspectsofthismethod.
comparison of the elastic electron scattering results pre- The electronbeamenergyin ourinvestigationis fixedat
dicted by different nuclear mean-field models. 500 MeV, which is a typical energy in electron-nucleus
InRef.[28]westudiedelasticelectronscatteringalong scattering experiments. Indeed, rather than discussing
the Ca and Sn isotopic chains in DWBA. In that work directly the differential cross section (DCS), we study
wereportedseveralcorrelationsamongscatteringobserv- the DWBA electric charge form factor because it is al-
ables and some properties of the nuclear charge density mostindependentoftheelectronbeamenergyinthelow-
along the isotopic chains [28]. In the present work, we momentumtransferregime,asitcanbe seenfromFig.5
investigatewhatinformationonnuclearstructurecanbe ofRef.[28]andfromthe lowerpanelofFig.4below. We
gained from the study of elastic electron scattering in compute the electric charge form factor as follows:
the N = 82, 50, and 14 isotonic chains, which cover dif-
dσ dσ −1
ferent regions of the mass table. We aim at extracting |F(q)|2 = Mott , (1)
general trends, according to current mean-field theories, (cid:16)dΩ(cid:17)(cid:16) dΩ (cid:17)
aboutthebehaviorofsomeobservablesthatmaybemea-
where dσ/dΩ is the DCS calculated in DWBA and
sured in experiments performed with unstable nuclei in
dσ /dΩistheMottDCS.WeusetheexactMottDCS
Mott
the low-momentum transfer region. Our choice of these
(i.e., the DWBA for a point nucleus) instead ofthe DCS
isotonicchainsismotivatedbythefollowingreasons. On
calculated in PWBA. Therefore, the charge form factor
the one hand, information about the structure of unsta-
defined by Eq. (1) goes beyond the first Born approxi-
ble nuclei belonging to the N =82 and N =50 isotonic
mation. It will be denoted by F (q) hereinafter.
DWBA
chains may be relevant in the astrophysical context be-
We calculate the charge densities with the relativistic
cause some of these nuclei could correspond to waiting
mean-fieldparametrizationG2[39,40],whichwealsoem-
points in r-processes [36, 37]. On the other hand, scat-
ployed in Ref. [28]. This nuclear model was constructed
tering data for light nuclei, such as e.g. those of N =14,
as an effective hadronic Lagrangian consistent with the
are likely to be obtained in future electron scattering fa-
symmetries of quantum chromodynamics. The nucleon
cilities such as SCRIT [19–21] and ELISe [22].
densitydistributionsareobtainedself-consistentlyatthe
The study of elastic electron scattering along isotopic mean-fieldlevelbynumericalsolutionofthe correspond-
and isotonic chains explores different aspects of the nu- ing variational equations. In contrast to most of the
clear charge density. The electric charge form factor nuclear mean-field models that assume point densities,
alonganisotopicchaingivesinformationabouttheeffect the G2 Lagrangian incorporates the low-energy electro-
of the different number of neutrons on the charge den- magnetic structure of the nucleon through vector-meson
sity, which becomes more and more dilute and extends dominance [39, 40]. This implies that the chargedensity
to largerdistances as the neutron number increases [28]. is obtained directly from the self-consistent solution of
In an isotonic chain, the changes in the charge form fac- the mean-field equations without any extra folding with
tor primarily inform about the effect of the outer proton external single-nucleon form factors. It has been shown
single-particleorbitalsthatarebeing filledasthe atomic [39–41] that the G2 relativistic mean-field interaction is
number increases in the chain. Thus, our previous [28] a reliable parameter set both for calculations of ground-
and present study together provide a survey of the evo- state properties of nuclei and for predictions of the nu-
lution of the charge form factor with the neutron and clear equation of state up to supra-normal densities, as
proton numbers in different mass regions of the nuclear wellasforpredictionsofsomepropertiesofneutronstars.
chart. First calculations of the charge form factor in PWBA
The rest of this article is organized as follows. In Sec- with G2 were reported in Ref. [39].
tionII,wesummarizethemethodemployedinourstudy In our calculations, we assume spherical symmetry.
of electron scattering in isotonic chains. As the basic The suitability of this assumption for the N = 82, 50,
methodology follows that of Ref. [28], we address the and 14 isotones is confirmed, excluding a very few ex-
reader to that work and references therein for more de- ceptions of small deformation, by mass tables computed
tails about the relativistic nuclear mean-fieldtheory and eitherwiththenon-relativisticSkyrmemodel[42]orwith
3
the relativistic model [43]. Pairing correlations are im- where
portantfor describingopen-shellnuclei. We takepairing
intoaccountthroughamodifiedBCSapproachthatsim- fG(r)= 2πσ2 −3/2e−r2/2σ2. (3)
ulatesthecontinuum(neededfornucleiatthedriplines) (cid:0) (cid:1)
through quasi-bound levels which are retained by their The two parameters, R0 and σ2, of the Helm model de-
centrifugalbarrier(neutronlevels)orbytheircentrifugal- termine the charge density as well as the electric charge
plus-Coulomb barrier (proton levels) [44]. The pairing form factor within the PWBA:
interaction in this approach is described by means of a
constant matrix element fitted to reproduce the exper- F(H)(q)= eiq~·~rρ(H)(~r)d~r = 3 j1(qR0)e−σ2q2/2,
Z qR
imental binding energies of some selected isotopic and 0
(4)
isotonic chains as described in Ref. [44]. It is to be men-
where j (x) is the spherical Bessel function. Note that
1
tioned that a mean-field treatment is not expected to be
we use natural units throughout the present paper.
sufficient for light exotic nuclei [25]. Thus, the N = 14
We proceed as suggested originally in Ref. [50] to ob-
isotonic chain studied below (with nuclei from 22O to
tain the Helm parameters associated to a given nucleus.
34Ca) corresponds to a somewhat limiting case, and the
First, we require that the first zero of Eq. (4) coincides
mean-field results should be taken as semiquantitative.
with the first zero of the mean-field PWBA charge form
The calculations with the G2 model predict a relatively
factor (Fourier transform of the charge density obtained
magic character of the neutron numbers N = 14 and
with the G2 model). We will refer to this charge form
N = 16. These neutron numbers have attracted some
factor as F (q) hereinafter. Therefore, the radius of
PWBA
attention in recent theoretical and experimental studies
the equivalent Helm density reads
as possible new magic numbers in exotic nuclei [45–49].
x
Theuseofmodeledchargedensitiesandelectriccharge R = . (5)
0
form factors in the experimental analysis of scattering q0
data has been extensive in the past, and continues to
where x = 4.49341 is the first zero of j (x) and q is
date. This is because in many cases the parameters of 1 0
the momentum transfer corresponding to the first zero
themodeledchargedensitiesaredirectlyrelatedwiththe
of F (q). Second, we determine the variance σ2
size of the bulk and surface regions of the nucleus under PWBA
of the Gaussian distribution such that |F(H)(q )| =
study. Inthisway,theparametrizeddensitieshelptopro- max
|F (q )|, where q is the momentum transfer
vide a clear physical interpretation of the electron scat- PWBA max max
corresponding to the second maximum of |F (q)|
tering data. This is the case ofthe so-calledHelmmodel PWBA
(the first maximum appears always at q = 0 fm). Us-
[50] that we used for some calculations in our previous
ing Eq. (4), one easily obtains
studyofisotopicchains[28]. Asoneofouraimshereisto
predictglobaltrends ofthe electricchargeformfactorin
2 3j (q R )
thelow-momentumtransferregimeinisotonicchains,we σ2 = ln 1 max 0 . (6)
q2 (cid:18)q R F (q )(cid:19)
also will performcalculations with Helm model densities max max 0 PWBA max
fitted to the self-consistent mean-field charge densities.
The PWBA form factor obtained in the Helm model
We briefly summarize this procedure in the next subsec-
neglectsthedistortionoftheelectronwavefunctionsdue
tion.
to the electrostatic potential of the nuclear charge dis-
tribution. The effect of this Coulomb distortion may be
simulated in the Helm model by replacing the momen-
tum transfer q by an effective value [50]. In our study of
A. Equivalent Helm charge densities
electron scattering in isotopic chains [28], we found that
this correction was important because we were compar-
The original version of the Helm model [50] has been ingelectron-nucleusscatteringobservablesofnucleiwith
extended in various ways for a more accurate descrip- the same proton number, and therefore they exhibited
tion of the experimental charge densities [51–53]. In the rather small differences. However,this correctioncan be
simplest version of the model [50], the charge density is neglectedforthe purposesofthe presentworkbecauseit
obtained from the convolution of a constant density ρ is almost totally masked by the much larger effect from
0
in a hard sphere of radius R with a Gaussian distribu- the changing proton number along the isotonic chain.
0
tion having variance σ2. By construction, R gives the
0
effective location of the position of the nuclear surface,
and hence characterizes the size of the density profile, III. RESULTS: N =82, N =50, AND N =14
whereasthe parameterσ is ameasureofthe thicknessof ISOTONIC CHAINS
the surface region of the density distribution. The Helm
charge density is then given by WestartwiththediscussionoftheresultsfortheN =
82isotonicchainwherethedifferentaspectsofourstudy
are described in detail. After that, we extend our study
ρ(H)(~r)= d~r′f (~r−~r′)ρ Θ(R −r), (2)
Z G 0 0 to the N =50 and N =14 isotonic chains.
4
0.08
122 140 146 154
Zr Ce Gd Hf G2
0 3s1/2 Helm
2d
3s 1h3/2 0.06
1/2 11/2
ε (MeV)nlj-1-05 3221sddg1357////2222 Z=5013122hsgdd11753/1///2222/2 Z=5012112hdggd139751////2222/2 Z=5022112ppggd31975/////22222 −3ρ (fm) ch00..0024 111144256024GCHedf
1g 2p Zr
9/2 1/2
-15 Z=50 22pp31//22 2p3/2 1f5/2 00 2 4 6 8
1f
1g 5/2 r (fm)
9/2
2p 1f5/2 1f7/2
-20 1/2
FIG.2: Chargedensitiesfor14202Zr,15480Ce,16446Gdand17524Hfas
FIG. 1: Energy of the proton single-particle levels for 14202Zr, a function of the radial distance to the center of the nucleus
15480Ce, 16446Gdand 17524Hfas computedwith therelativistic nu- according to the covariant model G2 (solid lines) and to the
fitted Helm distributions (dashed lines).
clear mean field interaction G2.
2d ,and3s orbitals. Finally,inthe caseofthe drip-
A. N =82 isotonic chain 3/2 1/2
line nucleus 154Hf, all of the mentioned single-particle
72
wavefunctionscontributesignificantlytothechargeden-
We firstanalyzethe chargedensitiesalongthe N =82 sity. The differences in the charge distribution due to
chain. The ordering and the energy of the different pro- single-particleeffects become evidentinFig.2 where the
ton single-particle levels, mainly the levels closest to the charge densities of 122Zr,140Ce, 146Gd, and 154Hf com-
40 58 64 72
Fermi level, are quite important for the present study. puted with the relativistic mean field model G2 are dis-
This is because the corresponding single-particle wave played as functions of the radial distance.
functions determine, to a large extent, the shape of the TheequivalentHelmchargedensitiesoftheseisotones,
chargedensityatthesurfaceregionaswellastheelectric withparametersdeterminedasexplainedinSectionIIA,
chargeformfactorinthelow-momentumtransferregion. are depicted in Fig. 2 by dashed lines. As in the case of
Fig. 1 displays the energy of the proton single-particle isotopes studied in Ref. [28], the quantal oscillations of
levels of some selected nuclei of the N = 82 isotopic themean-fieldchargedensitiesarenicelyaveragedbythe
chain. They are representative of proton deficient nuclei bulk part of the Helm model densities. In spite of the
(14202Zr), stable nuclei (15480Ce), proton rich-nuclei (16446Gd) fact that the surface fall-off of the Helm densities is of
and proton drip-line nuclei (17524Hf). Gaussiantype,the agreementatthesurfacebetweenthe
The more relevant proton single-particle levels in our mean-field and the equivalent Helm charge distributions
analysis of the N = 82 isotonic chain are, on the one is in general satisfactory.
hand, the 1g , 1g and 2d levels (which appear The fitted parameters of the equivalent Helm densi-
9/2 7/2 5/2
clearly separated in energy) and, on the other hand, the ties along the N =82 isotopic chain, namely R and σ2,
0
nearly degenerate 1h , 2d and 3s levels (which are displayed in the two panels of Fig. 3 as functions of
11/2 3/2 1/2
havea verycloseenergy). The 1h levelshowsenergy A1/3 and A, respectively. The radius R roughly follows
11/2 0
gapsofabout2and4MeVwithrespecttothe2d and a linear trend with A1/3 as it can be expected from the
5/2
1g levels,respectively,andagapofabout9MeVwith increase of the total number of nucleons. The param-
7/2
respecttothedeeper1g level. Thislargeenergygapis eter σ2, which determines the surface thickness of the
9/2
due to the magicity of the proton number Z =50. With charge density, shows a non-uniform variation with the
increasing mass number these relevant levels are shifted mass number A, reflecting the underlying shell structure
up in energy, roughly as a whole, retaining the same or- of the nuclei of this chain.
dering and approximately the same energy gaps. As a In our study of isotopic chains [28], we found a simi-
consequence of this level scheme, in going from the nu- larbehavioroftheσ2 parameterbutwithtwoimportant
cleus 122Zr to 140Ce the charge densities differ basically differences. First, the range of variation of σ2 in iso-
40 58
by the effects offilling up the 1g9/2 and1g7/2 shells, and topic chains is much smaller than the one exhibited by
in going from 140Ce to 146Gd the charge densities differ the N =82 isotonic chain. Second, in the case of the Sn
58 64
by the occupancy the 2d shell. In these proton-rich isotopes (see Fig. 6 of Ref. [28]) σ2 displays local min-
5/2
isotones with mass number above A = 140, the pairing imafor132Snand176Sn,pointingoutthemagicityofthe
correlations play a non-negligible role and therefore the N =82 and N =126 neutron numbers which makes the
charge densities also get contributions from the 1h , chargedensities of these isotopes more compact. In con-
11/2
5
6.4
−21 122
R (fm) 0 6 128Pd 132Sn136Xe140Ce14144S2Nm1d48D146y1G52dY15b0Er154Hf mbarn / sr ) 1100−24 x102 11 45 04 ZCH re f HGGGHe222ellmm
5.6 120Sr 122Zr Ω ( 10−27 Helm
d
4.9 5 5.1 A1/3 5.2 5.3 / astic10−30 x10
el
σ
122 d
Zr
−33
2m) 0.8 120Sr 128Pd 146G1d48D1y50E1r52Y1b54Hf 10 0 10 20θ (deg)30 40
f
2 (0.6 132Sn 1421N44dSm 102
σ 136
Xe 1
<σ2> = 0.69(3) fm2 140Ce 10
0.4
120 130 140 150 1
A 2(q)| 10−1 x102
A
FIG. 3: Upper panel: Mass-number dependence of the Helm WB −2
parameter R0 predicted by the covariant mean field model D10
G2 in theN =82 isotonic chain. Lower panel: Mass-number |F −3 x10
dependence of the Helm parameter σ2. The average value is 10
depicted by a horizontal dashed line. −4
10
−5
10
0 0.4 0.8 1.2 1.6
−1
q (fm )
trast,intheN =82isotonicchain,thekinksshownbyσ2
are rather related with the filling of the different proton
FIG. 4: DCS for elastic electron-nucleus scattering (upper
single-particle orbitals belonging to the major shell be-
tween Z =50 and Z =82. In particular,when the 1g9/2 15p48a0nCeel),aanndd17s52q4uHafraetc5h0a0rgMeefVorcmomfapcuttoerd(ilnowDeWr pBaAn.eTl)hienr14e20s2uZltrs,
and 1g shells are being filled, i.e., between 122Zr and
7/2 40 are shown both for the self-consistent mean-field densities of
140Ce, σ2 decreases almost linearly. The local minimum G2(solid lines) andfor theequivalentHelmdistributionsfit-
58
of σ2 for 140Ce points to some magic character of this tedtotheG2densities(dashedlines). Inthelowerpanel,we
58
nucleus. Thefactthattheσ parametertakesthesmaller alsoshowbyemptysymbolstheresultsobtainedat250MeV
valuesintheregionaround140Ce,indicatesthatthesur- using theself-consistent G2 densities.
58
face of the equivalent charge density is more abrupt at
and around this nucleus. When the 2d level starts to
5/2
be appreciably occupied, σ2 increases again nearly lin-
q = 2Esin(θ/2) is shown in the lower panel of Fig. 4.
early till 146Gd, where a new kink appears. From 146Gd
64 64 The empty symbols in the lower panel of this figure cor-
to the protondrip line (17524Hf), the value of σ2 continues respond to |F (q)|2 computed at an electron beam
DWBA
to increase, but now with a smaller slope as a conse-
energy of 250 MeV. The comparison of the results for
quence of the higheroccupancy of the 1h11/2, 2d3/2, and |F (q)|2 at E = 500 MeV and E = 250 MeV shows
DWBA
3s1/2 levels. Therefore, the Helm model parameter σ2 is that the electric charge form factor defined in Eq. (1) is
extremely sensitive to the evolution of the proton shell largelyindependentoftheenergyofthebeaminthelow-
structure along an isotonic chain. momentum transfer domain. Therefore, the analysis of
We next inspect the main properties of the differen- |F (q)|2 contains the essential trends of the elastic
DWBA
tial cross sections and electric charge form factors of the electron-nucleus scattering in this regime.
N =82 isotones. Inthe upper panelof Fig.4 we display The dashed lines in the two panels of Fig. 4 corre-
for three representative nuclei of the N = 82 chain the spondtotheDWBAresultbutusingtheequivalentHelm
DCSasafunctionofthescatteringangleθ. Theelectron charge distributions, fitted as explained previously, in-
beam energy is 500 MeV. The DCS is computed in the stead of the self-consistent mean-field densities. One
DWBA using both the self-consistent mean-field charge notes an excellent agreement at low-momentum trans-
densities obtained with the G2 model (solid lines) and fers up to 1.5–2 fm−1 between the results from the orig-
the equivalentHelmchargedensities (dashedlines). The inal mean-field densities and from the equivalent Helm
square modulus of the DWBA electric charge form fac- charge densities. This fact reassures one of the ability of
tor|F (q)|2 asafunctionofthe momentumtransfer the parametrized Helm distributions to describe global
DWBA
6
154Hf First inflection 152Yb 154Hf
10 10 150Er 1h
2(q)|WBA 8 140C13e6X1e32 point 2( q )|AIP 8140Ce1g142Nd131624X4dSe5m/2 1461G48dDy a2lsdo3/ 2c oa1nn1td/r2 i3bsu1t/e2
FD 6 Sn WB 7/2 132Sn
3 | 128Pd FD
10 124 3 | 6 128Pd
Mo 0
4 N=82 122 1 1g9/2 122
Zr N=82 Isotones Zr
0.75 0.8 0.85 0.9 0.95 1 4 500 MeV 120Sr
2p
−1 1/2
q (fm ) 0.3 0.4 0.5 0.6 0.7
σ2q2
FIG. 5: Evolution of thesquare modulusof theDWBA elec- IP
tric charge form factor with the momentum transfer q along
FIG. 6: Square modulus of the electric charge form factor in
the N = 82 isotonic chain as predicted by G2 at an elec-
tronbeamenergyof500MeV.Themomentumtransfercorre- DWBAatthefirstinflectionpoint(qIP)asafunctionofσ2qI2P
predicted by theRMF model G2 for theN =82 isotones.
spondingtothefirstinflectionpointforeachisotoneisshown
by circles.
due to the fact that
trends of elastic electron-nucleus scattering at low q, as 3
hr2i= 5σ2+R2 . (8)
it was also found in Ref. [28] for isotopes. H 5 0
(cid:0) (cid:1)
Inmediumandheavymassnuclei,thefirstoscillations
However, the correlation suggested by this approxima-
of the DCS and of the square charge form factor com-
tion is not fullfiled by the DWBA calculations in the
puted withinthe DWBA usuallydo notshowcleanlocal
relevant region of momentum transfers for our study.
minimabuttheyrathershowinflectionpoints. Aswecan
Forthis reason,we havefurther investigatedthe relation
see in Fig. 4, this is the situation for the first oscillation
of |F (q )|2 with q2 R2 and σ2q2 separately. We
of the DCS and of |F (q)|2 in the N = 82 isotonic DWBA IP IP 0 IP
DWBA show in Fig. 6 the behavior of |F (q )|2 as a func-
chain. In the absence of an explicit minimum, the first DWBA IP
tionof the value of σ2q2 since it will be very instructive
inflectionpoint(IP)isthe bestcandidatetocharacterize IP
to understand the influence of the proton shell structure
the relevant properties of the electric charge form factor
on elastic electron scattering in the isotonic chains.
at low q as we discussed in Ref. [28]. In Fig. 5 we plot
|F (q)|2 fortheN =82isotonesinamagnifiedview The non-uniform variations seen in Fig. 6 along the
DWBA
aroundthefirstIP.Thevalueof|F (q)|2 atthe first horizontal axis are basically due to the Helm parame-
DWBA ter σ2 rather than to q2 . This is because of the fact
IP is depicted by circles for each nucleus. In agreement IP
thatif wecomparethe relativechangealongthe isotopic
withearlierliterature[29],themomentumtransferatthe
chain found in the quantities σ and q (cf. Figs. 3 and
first inflection point (q ) shows an inward shifting and IP
IP
the value of |F (q )|2 shows an upward trend with 5, respectively), it is much larger in the case of the σ
DWBA IP
parameter. Hence, the information along the horizontal
increasing mass number along the isotonic chain.
axisofFig.6is sensitivetothe filling orderofthe single-
Let us discuss possible correlations of the DWBA
particle levels contributing to the charge density at the
charge form factor at low-momentum transfer with the
surface region.
parametersR andσ ofthe equivalentHelmchargeden-
0 The non-uniform variation shown by |F (q )|2
sity, as we did in our previous analysis of isotopic chains DWBA IP
along the vertical axis in Fig. 6 can be qualitatively un-
[28]. If we first look at the analytical expression of the
derstood in terms of the single-particle contributions to
charge form factor predicted by the Helm model, cf. Eq.
the PWBA electric charge form factor. To this end we
(4), it suggests to use qR and σ2q2 as the natural vari-
0 plotin Fig.7 the contributionto the PWBA formfactor
ables to investigate the variation of this quantity. In the
from the individual proton orbitals:
q →0 limit, Eq. (4) can be written as
f (q)≡ d~r |ψ (~r)|2 eiq~·~r, (9)
1 nlj nlj
F(H)(q →0)=1− q2 5σ2+R2 +O[q4]. (7) Z
10 0
(cid:0) (cid:1)
where ψ (~r) is the wavefunction of a protonlevel with
nlj
This result points towards a linear correlation with the quantum numbers n, l, and j. Note that (9) does not
mean square radius hr2i of the Helm distribution [50] include the occupation probability factors (v ) and de-
H nlj
7
1 12
1g 9/2 0.84 0.88 0.2
154
0.8 1g 7/2 Hf
0.1 150
f (q)nlj00..46 2123dhds 1513//1/222/∆2q --000..21 2( q )|WBAIP 1 08 136Xe 1421N46dG14d01C4e4ES1rm48D1y52Yb
0.2 IP FD 132Sn
3 | 6 128Pd N=82 Isotones
0 0
1 500 MeV
120
Sr
-0.2
0 0.5 1 1.5 4 122
Zr
−1
q (fm ) 28 30 32 34 36 38 40
2 2
R (fm )
0
FIG. 7: Shell contribution to the form factor of the last oc-
cupied levels in PWBA [see Eq.(9)]as a function of themo-
mmeenantufimeltdramnsofdeer,l.caTlchuelastheaddiendtrheeginounclienudsic17a524teHsftwhiethratnhgeeGo2f FDIWGB. 8A: aStquthaerefimrsotdiunlfluesctoifonthpeoeilnetct(rqiIcPc)haasrgaeffuonrcmtiofanctoofrRin02
predicted by theRMF model G2 for theN =82 isotones.
observed qIP values in the calculations of the form factor in
DWBA for theN =82 isotonic chain.
orbital also is negative for q values near q .
generacies (2j+1), i.e., IP
The discussed theoretical results pinpoint the impor-
1
F (q)= (2j+1)v f (q). (10) tanceofthefillingorderoftheprotonsingle-particlelev-
PWBA nlj nlj
Z
Xnlj els in elastic electron scattering off exotic nuclei. There-
fore, future experiments such as those planned in the
In Fig. 7 we depict fnlj(q) for the orbitals close to the upgrades of the GSI and RIKEN facilities may become
Fermi level in the nucleus 17524Hf, which can be consid- excellentprobesoftheshellstructureofexoticnucleiand
ered as representative of the level scheme of the N =82 may also confirm or refute the theoretical predictions of
isotopic chain. First, one notes that the single-particle new magic numbers.
contributions f (q) to the PWBA electric charge form
nlj Regarding the relation of |F (q )|2 with q2 R2,
factor do not have the same sign in the range of mo- DWBA IP IP 0
we have not found a simple behavior. In spite of this,
mentum transfers of interest in our analysis. Therefore,
we have observed that |F (q )|2 and the square of
stronginterferenceeffectsmayoccuramongthesesingle- DWBA IP
the Helm radius R2 show a rather similar behavior as a
particlecontributions. Inparticular,wecanseeinFig.7 0
function of the mass number in the isotonic chain. This
that in the regionaround q the contributions from the
IP suggests plotting |F (q )|2 against R2, which we
1g , 1g , and 1h orbitals are negative, while the DWBA IP 0
9/2 7/2 11/2
do in Fig. 8. One observes a good linear correlation
contributionsfromthe3s ,2d ,and2d orbitalsare
1/2 5/2 3/2
between both quantities. This correlation indicates that
positive. The PWBA form factor corresponding to the
the parameter of the Helm model which measures the
underlying Z =40 core is negative. Therefore, when the
sizeofthe bulkpartofthe density profileofeachisotone
1g and 1g orbitals are occupied—in passing from
9/2 7/2
governs the magnitude of the electric charge form factor
122Zr to 140Ce—the square modulus of the PWBA form
40 58 at low momentum transfer.
factor increases. When on top of this configuration, the
2d orbitalisfilledin146Gd,thesquaremodulusofthe To conclude this section, we would like to note that
5/2 64
PWBA charge form factor decreases due to the positive some of the details of the predicted single-particle ener-
sign of the contribution of this level around q . This gies,energy gaps,and filling orderof the orbitalschange
IP
simple pattern in the uniform filling picture is slighlty to some extent if in our calculations we use other RMF
modified due to the pairing correlations that introduce models or Skyrme forces instead of the G2 interaction.
additionalmixing with the contributions fromthe 2d , In particular, this is due to the fact that we are explor-
3/2
3s , and 1h orbitals. In spite of this, the simple ing regions of the nuclear chartbeyond the regionwhere
1/2 11/2
PWBA description is quite useful to help us interpret theparametersoftheseeffectivenuclearinteractionshave
the changes of |F (q )|2 from 122Zr to 146Gd. The beencalibrated. However,thebasicconclusiontobeem-
DWBA IP 40 64
subsequentincreaseshownby|F (q )|2 from146Gd phasized, i.e., the manifest sensitivity of some electron
DWBA IP 64
to154Hfcanalsobeunderstoodinthisschematicpicture scatteringobservablestotheprotonshellstructureofthe
72
since the PWBA charge form factor of 146Gd is globally isotones, is a robust feature that comes out regardlessof
64
negative and the additional contribution of the 1h the effective nuclear model.
11/2
8
0
70 84 90 100
Ca Se Zr Sn 100
(MeV)nlj--2100 ZZZ===52208012211fppgf57319//2///2222 ZZZ===2258001221211gppdsff751913//////222222 ZZ==22801221211dppgsff7513319///////2222222 ZZ==220811221211ddppgsff75153319////////22222222 R (fm) 0454...4284.170C74aC4r.278N8i0Z4n8.23Ge84SAe8461.K/43r 88Sr940Z.5r92M9o4R9u46P.6d98CdSn
ε 12ds1/2 3/2 1d5/2 Z=8
-30 3/2 1d5/2 Z=8 90Zr 92Mo94Ru96
Z=8 1d5/2 Z=8 1p1/2 11pp31//22 2m) 0.8 70Ca 88Sr Pd98Cd
FIG.-94:0Energy of1pt1h/2e proton11pps13i//22ngle-parti1cpl3/e2 levels for 7200Ca, 2σ (f0.6 74Cr 78Ni 80Zn82Ge84S86eKr <σ2> = 0.71(3) 1 f0m0S2n
8344Se,9400Zr,and15000SnascomputedwiththeG2parameterset. 70 80 90 100
A
FIG.10: Upperpanel: Mass-numberdependenceoftheHelm
B. N =50 isotonic chain parameter R0 predicted by the covariant mean field model
G2 in theN =50 isotonic chain. Lower panel: Mass-number
dependence of the Helm parameter σ2. The average value is
The more relevant proton single-particle orbitals for
depicted by a horizontal dashed line.
our study of the N = 50 chain are the 1f , 1f ,
7/2 5/2
2p , 2p , and 1g orbitals. They cover two major
3/2 1/2 9/2
shells betweenCa andSn. This setofprotonenergylev-
els computed with the G2 parametrization is displayed N = 82 isotones, the behavior of σ2q2 is dominated by
IP
in Fig. 9 for some selected isotones. We can see that the Helm parameter σ. This is because the relative vari-
these levelsmoveupinenergy,roughlyas awhole,when ation of σ2 (see Fig. 10) is much larger than the relative
themassnumberincreasesingoingfromproton-deficient variation of q2 along the isotonic chain. The change
IP
nuclei (70Ca) to stable nuclei (84Se, 90Zr) and to proton of |F (q )|2 along the N = 50 chain shows, glob-
20 34 40 DWBA IP
drip-line nuclei (100Sn). ally, an increasing trend with the mass number. We
50
The parameters R and σ2 of the Helm model distri- can appreciate in Fig. 11 that although the variation
0
butions fitted to the mean-field charge densities of the of |F (q )|2 is almost linear when a specific pro-
DWBA IP
N = 50 isotones are displayed in Fig. 10. The global
featuresaresimilartothe caseofthe N =82chain. The
R parameter, which represents the effective location of 100
0 6 Sn N=50 Isotones
the surface of the nucleus, approximately follows a lin- 98
Cd 500 MeV
ear trend with A1/3. The mass-number dependence of 1g
fiσl2lianggaoinf tdhisepdlaiffyesraenntopnr-uotnoifnorsmingtlree-npda,rtoircilgeinoarbteitdalbsy. Wthee 2 )|P5 2p3/2 96Pd94Ru 9/2
see that σ2 decreases in filling the 1f7/2 shell from 7200Ca ( qAI4 84Se 86Kr 88Sr 92M90oZr
itacsornedb72a88esN2ienpsig,1u/in2tfitlrsilelehmdtehlaleusinppasrrotreotoouon83gc44hcdSulreypi,pice-ioltdinnistentialcnlrnue94tca00lwsZeeuhrs,senw15a00nht0dheSennit1itfsht5hre/ee2an2clpehd3ve/eed2-l 30 |FDWB3 82G80eZn78N1fi5/2 2p1/2 1f7/2
by filling the 1g level. Therefore, the more abrupt 1
9/2 2 74
(smaller σ) equivalent charge densities predicted by the Cr
G2 model in the N = 50 isotonic chain correspond to 70
Ca
nuclei between the doubly-magic, proton-deficient 78Ni 1
nucleus and the more stable 84Se nucleus, where ma2i8nly 0.5 0.6 0.7 0.8 0.9 1
the 1f shell has been fille3d4. In these nuclei, the oc- σ2q2
5/2 IP
cupancy of the 2p , 2p , and 1g levels due to the
3/2 1/2 9/2
pairing correlations is rather small.
FIG.11: Squaremodulusoftheelectricchargeformfactorin
ThesquaremodulusoftheDWBAelectricchargeform DWBAatthefirstinflectionpoint(qIP)asafunctionofσ2qI2P
factor for an electron beam energy of 500 MeV is dis- predicted by theRMF model G2 for theN =50 isotones.
played against σ2q2 in Fig. 11. As in the case of the
IP
9
10 22 24 26 28 30 32 34
O Ne Mg Si S Ar Ca
6 N=55000 I MsoetoVnes 100Sn 0 N=201f7/2N=201f7/2 1f
3210 |F( q )|DWBAIP453 74Cr 78N8i0Zn 8862KG89re40SZe8r8Sr92M9o4R9u6P9d8Cd ε (MeV)nlj---321000 NN==8211112ppdds13153/////22222NN==8211112ddpps13513/////22222NNN===228111120ddpps713513//////222222NN==281111210dppdsf713315//////222222NN==280111121ddppsf713513//////222222NN==280111121ddppsf713513//////222222NN==280111121dppdsf71315/////22222
2 -40 1s1/2 N=2 N=2 3/2
70Ca 1s1/2 N=2 N=2
1s
1/2
1 20 22 24 26 28 30 -50 Neutrons 1s1/2 1s1/2 1s1/2 1s1/2
2
R (fm)
0 10
22 24 26 28 30 32 34
O Ne Mg Si S Ar Ca
FDIWGB.1A2:atSqtuhaerfiermstoidnuflleucstoiofnthpeoeinletc(tqriIcP)chaasrgaefufonrcmtiofanctoofrRin02 0 Z=201f7/2Z=201f7/2Z=201f7/2Z=201f7/2Z=201f7/2Z=201f7/2Z=201f7/2
predicted by theRMF model G2 for theN =50 isotones. 12ds13//22 12ds13//22 12ds13//22 12ds13//22 12ds13//22 12ds13//22 12ds13//22
V) -10 1d5/2 1d5/2 1d5/2 1d5/2 1d5/2 1d5/2 1d5/2
ton orbital is being occupied, drastic changes of slope Me Z=8 Z=8 Z=8 Z=8 Z=8 Z=8 Z=8
toEacqkcseu.p(pi9el)adc.aendRweh(c1ea0nl)l,ianwgneetfiwhnedsshitemhllaptlsitifinaerdtthsPetWoreBgbieAonspiogifcntiqufircveaa,nlutcelfys. ε (nlj-20 11pp31//22 11pp13//22 11pp13//22 11pp31//22 11pp13//22 11pp13//22 11pp31//22
-30 Z=2
faarcotuonrdfrqoImP tthhee 1cofnatrnidbu1tgioonrbtoitatlhseiselneecgtraitcivceh,awrgheilefotrhme Z=2 Z=2 Z=2 Z=2 Z=2 Z=2
1s
contribution from the 2p orbitals is positive. This fact -40 1s 1s1/2 1/2
is consistent with the behavior shown by |FDWBA(qIP)|2 1s1/2 1s1/2 1s1/2 1s1/2 1/2 Protons
in Fig. 11. That is, |F (q )|2 increases in pass-
DWBA IP
ing from 7200Ca to 8344Se, basically due to the filling of the FIG. 13: Energy of the neutron (upper panel) and proton
1f7/2 and 1f5/2 shells, and then its value is practically (lower panel) single-particle levels for 282O, 2140Ne, 2162Mg, 2184Si,
quenchedupto9400Zrbecausethe2p3/2 and2p1/2 orbitals 3106S,3128Ar and 3240Ca as computed with theG2 parameter set.
contribute with opposite sign to the 1f orbitals. When
the 1g level is appreciably occupied in approaching
9/2
theprotondripline,thevalueof|F (q )|2 increases
DWBA IP
again with a nearly constant rate. Finally, in Fig. 12 we levelsbecome moreboundwithincreasingmassnumber.
see that |FDWBA(qIP)|2 of the N = 50 isotones shows InadditiontotheprominentenergygapatN =8seenin
a good linear correlation with the square of the Helm thewholechain,itmaybenoticedthatthe1d neutron
5/2
parameter R0. levelbecomes progressivelymoreisolatedwhen the mass
number increases, which points to some magic character
oftheneutronnumberN =14inthecalculationwiththe
C. N =14 isotonic chain G2model. Thismagiccharacterisconfirmedbythevan-
ishing neutronpairing gapfound in our calculation from
26Mg to 34Ca. It may be observed that the G2 model
Inthissectionwediscussthelightestisotonicchainan- 12 20
also predicts a slightly magic character of the neutron
alyzedin our work. Although the presentfindings are to
number N = 16 towards the neutron drip line. Actu-
be taken with some reservations because the mean-field
ally, we see in Fig. 13 that the relatively magic trend of
approach is not best suited for light-mass exotic nuclei,
N =14 increases fromthe proton-poorside (22O) to the
we note that similar general trends to those observed 8
proton-richside (34Ca)ofthe chain,while the somewhat
in the heavier-mass isotonic chains also appear in the 20
magic trend of N =16 decreases from 22O to 34Ca.
N =14 chain. 8 20
In Fig. 13 we display the neutron (upper panel) and The more relevant proton single-particle orbitals for
proton(lowerpanel)single-particlelevelscomputedwith our study of the N = 14 chain belong to the s-d major
theG2interactionfortheN =14isotonicchain,fromthe shell. The energy levels of this proton major shell (see
veryproton-deficientnucleus22Ototheveryproton-rich the lowerpanelofFig.13)lie approximatelyatthe same
8
nucleus 34Ca. As expected, the neutron single-particle energy for all the nuclei from 22O to 34Ca, with roughly
20 8 20
10
10 1
∆q
First minimum min 1d 5/2
0.8 2s 1/2
1d 3/2
2
)| 0.6 1f 7/2
q
( 1f 5/2
A )
B 1 q
DW (nlj 0.4
F 34 f
30 | Ca 32Ar 30S 28 0.2
1 Si
26
Mg
0
24
0.1 N=14 Ne 22
O
-0.2
1.2 1.3 1.4 1.5 0 0.5 1 1.5 2 2.5 3
q (fm−1) q (fm−1)
FIG. 15: Shell contribution to the form factor of the last
FIG.14: EvolutionofthesquaremodulusoftheDWBAelec-
tric charge form factor with the momentum transfer q along occupied levels in PWBA [see Eq. (9)] as a function of the
the N = 14 isotonic chain as predicted by G2 at an elec- momentum transfer. The shaded region indicates the range
tron beam energy of 500 MeV. The momentum transfer cor- of observed qmin values in the calculations of the form factor
in DWBA for theN =14 isotonic chain.
respondingtothefirstminimumforeachisotoneisshownby
circles.
ing trend with A1/3. In turn, the variation of the Helm
parameterσ reflectsthe underlyingshellstructureofthe
constant energy gaps. The 1d and 2s proton lev-
5/2 1/2 mean-fieldchargedensities. InthelowerpanelofFig.16,
els exhibit a considerable energy gap between them in
we see that the value of σ2 remains almost constant be-
thisisotonicchainaccordingtothepredictionsofthe G2
tween 22O and 28Si when mainly the 1d shell is being
model. Itis tobe mentionedthatdue to the pairingcor- 8 14 5/2
filled. From 30S on, the 2s and 1d levels start to
relations, the proton levels 1f and 1f (the latter is 16 1/2 3/2
7/2 5/2 be appreciably occupied and σ2 starts increasing almost
not displayed in Fig. 13) also play some role in our cal-
linearly with A till the proton-drip line nucleus 34Ca.
culation of the mean-field charge densities. These levels 20
simulate to a certain extent the effect of the continuum
duetotheirquasi-boundcharacterowingtotheCoulomb
3.75
and centrifugal barriers [54].
34
achgaIaniinnF.stigWt.he1e4smeweoemtdheiasntptulianmytathrimasnacsghfneariifinfeodorfvtlhioeewwNeorf=m|Fa1Ds4sW,iBstoAhte(oqni)n|i2c- (fm) 03.32.55 24Ne 26Mg 28Si 30S 32Ar Ca
flection point that was found after the first oscillation R
3 22O
of the charge form factor in the heavier chains N = 50
and N = 82 becomes a clearly well defined local min- 2.75
2.8 2.9 3 3.1 3.2
imum. Thus, for the discussion of the N = 14 chain
1/3
we focus on the properties of |F (q)|2 at its first A
DWBA
minimum (q ). The value of |F (q )|2 (shown
min DWBA min 1
by the circles in Fig. 14) increases when the value of
34
the momentum transfer at the first minimum decreases, ) 0.9 32Ar Ca
i.e., |FDWBA(qmin)|2 grows with increasing mass number 2m 30S
iFnigt.h1e4cihsarionu.gThlhyoluingeharthweitihncqrmeians,eoonfe|nFoDtWesBaAk(qinmkina)t|2thine 2σ (f00..78 24O 24Ne 26Mg 28Si
pointcorrespondingtothe2184Sinucleus. Aswecanrealize <σ2> = 0.79(5) fm2
from Fig. 15 in the schematic PWBA picture, this kink 0.6
is originated by cancellation effects between the oppo- 25 30 35
A
site contributions to the charge form factor around q
min
coming from the single-particle wave function of the 1d
FIG.16: Upperpanel: Mass-numberdependenceoftheHelm
protonlevel(negativecontribution)andofthe 2sproton
parameter R0 predicted by the covariant mean field model
level (positive contribution).
G2 in theN =50 isotonic chain. Lower panel: Mass-number
The upper panel of Fig. 16 shows that the parameter dependence of the Helm parameter σ2. The average value is
R of the equivalent Helm charge densities displays, as depicted by a horizontal dashed line.
0
in the heavier isotonic chains, an overall linear increas-