The correlation between quarter point angle and nuclear radius

2.2 MB·

by  W. H. Ma

The correlation between quarter-point angle and nuclear radius
W. H. Ma,1,2,3,∗ J. S. Wang,1,† S. Mukherjee,1,4 Q. Wang,1 D. Patel,4
Y. Y. Yang,1 J. B. Ma,1 P. Ma,1 S. L. Jin,1 Z. Bai,1 and X. Q. Liu1
1Key Laboratory of High Precision Nuclear Spectroscopy and Center for Nuclear Matter Science,
Institute of Modern Physics, Chinese Academy of Science, Lanzhou 730000, People’s Reublic of China
2University of Chinese Academy of Science, Beijing, 100049, People’s Reublic of China
3Lanzhou University, Lanzhou 730000, China
4Physics Department, Faculty of Science, M.S. University of Baroda, Vadodara - 390002, India
(Dated: August 16, 2016)
The correlation between quarter-point angle of elastic scattering and nuclear matter radius has
beenstudiedsystematically. Variousphenomenologicalformulaewithparametersfornuclearradius
are adopted and compared by fitting the experimental data of quarter point angle extracted from
nuclearelasticscatteringreactionsystems. Theparameterizedformularelatedtobindingenergyis
6
recommended, which gives a good reproduction of nuclear matter radii of halo nuclei. It indicates
1
0 that the quarter-point angle of elastic scattering is quite sensitive to the nuclear matter radius and
2 can be used to extract the nuclear matter radius.

g
u I. INTRODUCTION chargedistributioninthecaseofstablenuclei. However,
A for an unstable nuclei being short-lived and difficult to
useasatarget,especiallyforthehalonuclei,oneusually
5 In the recent years, the nuclear reactions with unsta-
usestheisotopemethodorinteractioncrosssectionmea-
1
ble/weakly bound nuclei that have low breakup thresh-
surements to determine the size of nucleus. To be more
old and exotic structure has shown remarkable features
] precise,onecansaythatelectronscatteringisbetterthan
h that is different from those of tightly bound nuclei. It
isotope shift and isotope shift is better than interaction
t will be interesting to understand and revisit in detail,
- cross section. Moreover the interaction cross section is
l thedifferenceinthereactionmechanismsusingatightly,
c much model dependent. Earlier, it was observed that
u weaklyandunboundorhalonuclei. Veryrecently,aphe- the nuclear size is obviously correlated to the quarter-
n nomenological comparison of reduced reaction cross sec-
point angle. This is because the quarter-point angle is a
[ tionsofdifferentreactionsystemswasproposedbyusing
Wong(cid:48)s model [1–3]. Several authors have extracted the function of Rint. This indicates that we may extract the
2 radius of unstable nuclei from the experimental quarter-
quarter-point angle from the elastic scattering angular
v point angle. This could be a new experimental method
1 distribution reaction cross section, in order to compare to determine the nuclear size.
0 theweaklyandtightlyboundprojectiles[4–6]. Thequar-
As introduced above, interaction radius can be ex-
4 ter point angle which is also called the grazing angle or
tracted from the quarter-point angle through the elastic
4 rainbow angle is one of the most conspicuous features
0 scattering angular distribution of the reacting system.
of heavy-ion elastic scattering at above-barrier energies.
. Based on the concept of quarter-point angle, the main
1 Accordingly, the radius of interaction R correlating
int objective of this work is to compare the tightly bound,
0 with quarter-point angle, is the sum of projectile and
weaklybound(stable)andhaloprojectilesusingthephe-
6
target radius and approximately equals to the classical
1 nomenologicalformulaforinteractionradius. Thisdiffer-
apsidal distance, the distance of closest approach, evalu-
: ence of the three kinds of projectiles can be employed to
v ated at the energy for which the experimental cross sec-
find a better understanding of interaction radius. Fur-
i tionisone-quarterofthecorrespondingRutherfordcross
X thermore, nuclear radius can also be obtained from this
section [7]. Earlier evaluatation of the R is given by
r int analysis.
a r (A1/3+A1/3), where A and A are the mass numbers
0 p t p t
of projectile and target, respectively. It has been found
that the value of r ranging from 1.20 to 1.30 fm are the
0 II. PHENOMENOLOGICAL FORMULAE WITH
most appropriate values for the heavy ion interaction at
PARAMETERS FOR RADIUS OF INTERACTION
energies ≥ 10.0 MeV/u (Baluch et al., 1998 and refer-
ences therein).
The theoretical quarter-point angle described else-
Experimentally, the nuclear radius (or nuclear mat- where [6],
ter distribution) can be determined by the measurement
of electron scattering, isotope shift and interaction cross θ =2arcsin[1/(2x−1)]. (1)
1/4
section etc. Since the electron is structureless and the
electromagnetic interaction is very well known, therefore Whichisafunctionofthedimensionlessvariablex,where
the charge distribution of a nucleus can be precisely ob- x = E /V , the ratio of the center of mass en-
cm coul
tained from the electron scattering measurements. The ergy E to Coulomb barrier V . The experimental
cm coul
proton distribution can be deduced from the nuclear values of the quarter-point angle were extracted from
2
Projectile a(fm) b(fm) 180
Tightly bound 1.123 1.00 • (cid:228) Tightlybound
TwtinhAegeaBkbaLlyyvEabfiiIolt.atuTbinnlhdegeaWtefinhxtdeetpaeheadkHranlilyvmaogalubloepulnoraeoutrsajneodldcfitsdaitla11erat..si13nab.83du73otbifofnot11hr..oe00tfh00eetlhateisgthdictilffysecrbaeontuttneiadr-l, Θdeg14 11604000(cid:144)(cid:72)(cid:76)••(cid:196)••••(cid:228)•••••••••••••(cid:228)(cid:228)••(cid:196)(cid:228)•(cid:196)••(cid:196)••••••(cid:228)•••(cid:228)•••••(cid:228)••(cid:228)(cid:228)•(cid:228)•••(cid:196)(cid:228)•••••(cid:228)••••••(cid:228)••••••(cid:228)••(cid:228)•(cid:228)•(cid:196)••••••(cid:228)(cid:228)••••••••(cid:228)•••(cid:196)(cid:228)••(cid:228)•(cid:196)•(cid:228)••••(cid:196)•(cid:228)••(cid:228)(cid:196)•(cid:228)•••••••(cid:228)•••••(cid:228)(cid:228)••(cid:228)(cid:228)•(cid:228)•••(cid:228)••• •(cid:196)••(cid:228)(cid:196)••••(cid:228)(cid:196)•••W•e•••akTHlQyalAbooFund
cross sections with the optical model. The correspond-
0.9 1. 1.11.21.31.41.51.61.71.81.9 2. 2.12.22.3
ing center-of-mass energies can be also obtained from
experiments. The Coulomb barrier was determined by x
Z Z e2/R , where Z and Z are the number of pro-
p t int p t
tonsintheprojectileandtargetrespectively. Inaddition, FIG.1. (Coloronline)Thequarterpointangleasafunction
the value of R can be obtained from the experimental of reduced energy x in the interval from 0.8 to 2.0 for phe-
int
values of quarter-point angle by the following relation- nomenological formula Rint =A1p/3+A1t/3 (PF1). The color
ship. ZpZte2 ·(1+ csc(θ1/4))). Theexperimentalvaluesof points stand for the experimental quarter-point angles. The
2Ecm 2 theoretical quarter-point angle function is labeled as TQAF.
quarter point angle are given in APPENDIX.
Theintroductionofthereducedenergyparameterxis
180
very useful to compare the quarter-point angle of differ-
• (cid:228)
ethnetrceoamctpioanrissoynstoemf qsutaorgteetrheproiinntoannegglreaspohb.tFaiInGe.d1frsohmowas ••••••• (cid:196)• TWigehatkllyybboouunndd
lnaersgseoafmprooujnetctoilfees,xpi.ee.r,imtigenhttalyl dbaotuan,du,swinegakdliyffebroeunntdt,igahntd- deg 140 (cid:76)(cid:228)••••••••(cid:196)(cid:228)•(cid:228)•(cid:228)••(cid:228)••••••••(cid:196)•••••(cid:228)•••(cid:228)•• THQaAloF
hqjHgeluacoelatwoairletrepveesrreoosrpu,jbeocavccilintloeitstuleshsaiselv.nyeeglxlIlyonepwdeogeerfericmnrttehehearenaasstnelea,wlttthhphheroeeeincneeutxtdsrhpvieoffeefrevioqramefulnueattenhrttetoeayfrtlp-hxpveeaoiosolirunffieettxpsiaercnodoa--f.l Θ14 16000(cid:144)(cid:72) •(cid:228)(cid:228)(cid:196)(cid:196)••(cid:228)••(cid:228)•••(cid:228)•(cid:228)•••••(cid:228)••••••(cid:196)(cid:228)(cid:228)••••••••(cid:228)••••(cid:228)•(cid:196)•(cid:228)•••••••(cid:228)••(cid:228)••••(cid:196)•(cid:228)•••••(cid:196)(cid:228)(cid:228)•••(cid:196)(cid:228)(cid:196)••••••(cid:228)•(cid:228)•••(cid:228)(cid:228)•••(cid:228)•(cid:228)•••(cid:228)••••• •(cid:228)(cid:228)•••(cid:196)•
quarter-point angle function (TQAF). This is because
the theoretical R is simply given by phenomenological 0.9 1. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2. 2.1
int
formula A1/3+A1/3 (PF , as employed by L. Jin et al. x
p t 1
[6] to calculate the value of x.
Additionally, from FIG. 1, it can be observed that the FIG. 2. (Color online) Same as FIG. 1, except for phe-
comparison of the three kinds of projectiles is less dis- nomenological formula (PF ).
2
tinguishable when a large amount of experimental data
is taken into account, although the values of the quarter
pointanglefollowasuccessivelydecreasinggeneraltrend a for tightly bound, weakly bound and halo projectiles
from the tightly bound to the halo for a fixed value of x. describe the difference between the reaction systems and
In general, for tightly bound systems for a given value of wellindicatethesizeofprojectilesfromtheexpressiona·
xfrom0.8to2.0,experimentalpointsshowlargerunder- A1/3. Thisisinaccordancewiththenuclearsizeobtained
p
estimation as compared to the higher values of x. This
frompreviousstudies. Thisalsoclearlyindicatesthatthe
anomaly was observed while deriving Rint by using the radius calculated by the expression A1/3+A1/3 in order
usual formula (A1/3 +A1/3). Similar wide distribution p t
p t to compare the different kinds of projectiles result in the
can also be observed in the case of weakly bound and
underestimation of the size of weakly bound projectiles,
halo systems.
and even more in the case of halo projectiles.
According toabove discussion, it isnecessary to intro-
Thus the modified expression of R can reduce the
duce a modified formula for Rint as given by a·A1p/3+ deviation between the experimental diantta and the theo-
b · A1/3 (PF ), where the parameters a and b are fit- retical curve of the quarter-point angle as a function of
t 2
tedbyextractingexperimentalR separatelyfortightly x. In FIG. 2 , the result of the modification is shown.
int
bound, weakly bound and halo projectiles. WhencomparedwiththepointsinFIG.1, thedeviation
Inthefittedvalues,asshowninTableI,theparameter between the experimental data and the theoretical curve
b(b=1)iskeptconstantfortheallthreekindofsystems and among the three kinds of projectiles is diminished.
assuming the target to be stable. The fitted parameter With the enlightenment of reducing the difference in
aincreasesfromtightlyboundtohaloprojectiles. Itcan the quarter point angle values among the three kinds of
bedistinctlyobservedthatthefittedvaluesofparameter projectiles by using modified R , it is feasible to find a
int
3
betterphenomenologicalformulatogiveabetterdescrip-
tion of the strong absorption radius. To begin with, the 180 • a (cid:228)• WTigeahktllyybboouunndd
nAtCtcpwhluaeodiercancdaelkrseinmtniasdiRereotpesrtnrrrsoeiaaornnldtwslggo=yiarun,tfs0soh,naraeburcn(cid:88)sa1eicsdss,,luoeertmndsoRh2pneeeouiadn;ntdnimridsitosnyhptnsamrer,tzioebmlavaiiuqsffeedtudtegiroicoidyetvnfxeedtdupnhhrnraoaeebivptfqsyoisudnmri(aoigmPslnotldfiyFidnoen3figbl)istuyRtwe[ri8saiitn]bhssh,tuueeurstwfsinaboeitcudnehe--.. Θdeg14 11604000(cid:144)(cid:72)(cid:76)•(cid:196)•••••(cid:228)•••••••••(cid:228)(cid:228)•(cid:196)••••••(cid:228)(cid:196)•••••(cid:228)(cid:196)•••••••••(cid:228)•(cid:228)••(cid:228)•••(cid:196)(cid:228)•(cid:228)•••(cid:228)••••(cid:228)(cid:228)••••••(cid:228)•••••••(cid:228)••(cid:228)••(cid:196)••(cid:228)••(cid:228)••••(cid:228)••••(cid:196)•(cid:228)•(cid:228)••••••(cid:196)•(cid:228)•••(cid:196)(cid:228)•(cid:228)(cid:196)•(cid:228)••••••••(cid:228)••(cid:228)•(cid:228)(cid:228)••••(cid:228)•(cid:196)•(cid:228)•(cid:228)•••• THQa•AloF(cid:196)•(cid:228)(cid:228)••(cid:196)••
int i (2)
wanhderetaRrZigse=tta.b(l(erT0ih==+ep,Atv1ra2i.19/l8u3+e+s0.A0oAi1f6r4i·2A/t3h2i/)e3+;paaizraZdmieen−toeAtZreissstarbp0lre,o)rAje11ic,/t(3ir3l;2e) Θdeg14 11160480000(cid:144)(cid:72)(cid:76)•••••••(cid:228)••(cid:196)•••••••(cid:228)•(cid:228)•••••(cid:228)•••••(cid:228)•••••(cid:196)(cid:196)•••••(cid:196)(cid:228)•(cid:228)••(cid:228)••(cid:228)•••(cid:228)••(cid:228)••(cid:228)(cid:228)•••••••(cid:196)(cid:228)•••••••(cid:228)••(cid:228)••••(cid:228)•••(cid:228)(cid:196)••••(cid:228)••••(cid:228)•••(cid:228)(cid:196)•••••(cid:228)•••(cid:196)•(cid:228)•(cid:228)•(cid:196)(cid:228)(cid:196)•••b•••••(cid:228)•(cid:228)•(cid:228)(cid:228)••••••(cid:228)•(cid:228)(cid:196)(cid:228)••(cid:228)••••WTigeahTHkt•QllayyAlobbFo•o(cid:228)(cid:228)uu••n••n(cid:196)dd
and a obtained by fitting the experimental data, were
z 0.9 1. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2. 2.1
1.0152fm, 0.6383fm, −1.2781fm and −0.2981fm re-
spectively. However, as shown in FIG. 3 (a), the quarter x
point angle values are still inconsistent with respect to
the three types of projectiles, although the deviation be-
FIG.3. (Coloronline)TheoreticalandexperimentalQuarter
tween the experimental data and the theoretical curve
point angle values for three types of projectiles using PF
show a decrease. Thus the modified formula given by 3
(a) and PF (b). The color points are for the experimental
PF is not adequate to accurately describe the nuclear 4
3 quarter-pointanglevalues. Theexperimentaldatawereused
size, especially for the halo nuclei. In order to obtain a
in the interval of x from 0.8 to 2.0. The theoretical quarter-
consistent description for nuclear size, it is important to point angle function is labeled as TQAF.
discuss this phenomenological formulation with further
improvement by considering the binding energy of the
nuclei. Accordingtothequantummechanics,thenuclear
rmsradiusisinverselyproportionaltothebindingenergy.
The relation between the rms radii and the binding en-
ergycanbeobtainedfromthesimplifiedNsingle-particle
Schr¨odinger equation as, R(B) = √4.04 [9]. However,
B(A)
this relation is not sufficient for a good agreement with
the experimental data. A modified quantitative formula and λ = 10.7186fm/MeV. The experimental values of
3
of the nuclear rms radius as a function of binding energy binding energy were taken from literature [11] and were
per nucleon was introduced and discussed by Wang et used to determine B . As a result, the deviation be-
i
al., [10]. It is important to take into account the binding tween the experimental and the theoretical curve for all
energies and based on this fact, the theoretical Rint is the three types of projectiles (mainly between the tight
given by (PF4) bound and the halo) were diminished as can be seen in
FIG.3(b). ThegoodnessoffitforPF andPF wasob-
(cid:88) 3 4
R = R ; tained by square of regression fit (R-Square). R-Square
int i (4)
i=p,t is a number that indicates how well data fit a curve. An
R-Square of 1 indicates that the regression line perfectly
fitting the data, while an R-Square of 0 the line does not
R = λ A1/3+λ +λ √Ii +(λ √Ii )2; (5) fitthedataatall. InthepresentworkR-SquareforPF3
i 0 i 1 2 B 3 B is0.998585andforPF 0.9986632. SoPF showsbetter
i i 4 4
fit than PF . Using the results from the above method
3
whereIi = Ai−A2iZi, denotesthesymmetryparameter, Bi and comparing them with the fitted data as shown in
is the binding energy per nucleon and subscript denotes FIG.2,onecanemphasizetheimportanceofthebinding
projectile and target. The experimental data of quarter- energy to understand the nuclear size, especially in the
point angle were fitted to obtain the parameters λ0 = case of halo nucleus. For a clear justification, the next
0.9776fm, λ1 = 0.2475fm, λ2 = −0.1492fm/MeV1/2 section gives a more detailed comparison.
4
PF PF PF PF
1 2 3 4 0.15 (cid:159) (cid:159) (cid:227) (cid:159)
(cid:52)η 0.0374 0.0077 0.0342 0.0059 (cid:159) (cid:227) (cid:159) (cid:159)
TanAdBPLEF4I.I.(cid:52)ThηeTBvalu0e.0s5o6f (cid:52)0η.0a4n0d (cid:52)0.η0T3B9fo0r.0P3F31, PF2, PF3 Η 000...001050 (cid:245)(cid:237)(cid:231)(cid:227)(cid:236)(cid:159)(cid:236)(cid:159)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:236)(cid:230)(cid:244)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:236)(cid:227)(cid:230)(cid:244)(cid:159)(cid:236)(cid:230)(cid:244)(cid:159)(cid:245)(cid:245)(cid:237)(cid:237)(cid:231)(cid:231)(cid:227)(cid:227)(cid:236)(cid:230)(cid:244)(cid:159)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:236)(cid:230)(cid:244)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:236)(cid:230)(cid:244)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:236)(cid:245)(cid:237)(cid:245)(cid:230)(cid:237)(cid:236)(cid:231)(cid:231)(cid:244)(cid:230)(cid:244)(cid:227)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:236)(cid:230)(cid:244)(cid:159)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227) (cid:245)(cid:237)(cid:231)(cid:227) (cid:236)(cid:230)(cid:244)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)(cid:245)(cid:237)(cid:231)(cid:227)
(cid:45)0.05 (cid:230)(cid:244)(cid:230)(cid:244)(cid:245)(cid:237)(cid:231) (cid:245)(cid:237)(cid:231) (cid:159) H_PF1 (cid:227) TB_PF1
(cid:230) H_PF2 (cid:231) TB_PF2
III. COMPARISON OF THE FOUR (cid:45)0.10 (cid:236) H_PF3 (cid:237) TB_PF3
PHENOMENOLOGICAL FORMULAE (cid:244) H_PF4 (cid:245) TB_PF4
(cid:45)0.15
40 60 80 100 120 140
In this section, we give a quantitative comparison of Θ deg.
1 4
the four phenomenological formulae (PF to PF ). We
1 4
defineagoodnessoffitratioη = x−xPF thatwillestimate
xPF FIG. 4. (Color online) Compar(cid:144)ison of the four phenomeno-
thedeviationbetweentheexperimentalquarter-pointan- (cid:72) (cid:76)
logicalformulaewiththeexperimentaldatausedintheinter-
gle and the curve of (TQAF). In other words, smaller is
val of x from 0.8 to 2.0. The solid symbols are for the halo
the value of η, less will be the deviation. In this rela- nuclei and the open symbols for tightly bound nuclei.
tion x is determined by the center-mass energy and
PF
the Coulomb barrier via the phenomenological formulae,
0.20
x = E /V . However, x is directly calculated by
PF cm coul
usingtheformula(1), namely, usingthecurveofTQAF.
0.15 (cid:248)
Both x and x corresponds to the same quarter-point
PF (cid:248)
atinognleoefxtthraectreedacftrionmg seylastsetmicss.caTttheerincgomanpgaurilsaorndiosftrηibuin- 0.10 (cid:248)(cid:248)(cid:248)(cid:248)(cid:248) (cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248) (cid:248)(cid:248)(cid:248) (cid:248) (cid:248)
terms of the four phenomenological formulae (PF1 to Η 0.05 (cid:225)(cid:225) (cid:225)(cid:225)(cid:225)(cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225)(cid:225)
PF ) is shown in FIG. 4. It is clear from the figure (cid:225) (cid:225)
4
that the modified formulae PF , PF and PF show a 0.00 (cid:225)
2 3 4 (cid:248) Η Η
better agreement as compared to PF so far as the devi- 1(cid:45)TB 1(cid:45)TB
1 (cid:45)0.05 (cid:225) Η Η
ation between the experimental data and the theoretical 2(cid:45)TB 2(cid:45)TB
curve (TQAF) is concerned. The advantages and dis- (cid:45)0.10
advantages of the four methods for calculating R are 40 60 80 100 120 140
int
more clearly understood while comparing the deviation Θ deg.
1 4
between tightly bound and halo projectiles. This devi-
ation between tightly bound and halo projectiles by the FIG.5. (Coloronline)Compar(cid:144)isonofη withtheexperimen-
(cid:72) (cid:76)
four methods may be more clearly obtained by intro- tal data used in the interval of x from 0.8 to 2.0 for the sets
ducing another parameter (cid:52)η = η −η , where, η ofexperimentalquarter-pointanglepointsofthetightbound
H TB H
is the arithmetic mean of η for halo nuclei and ηTB is nuclei near and far from the curve of TQAF for PF1. η1−TB
that for tightly bound nuclei. In addition, the inconsis- is the points of η for that far from the curve of TQAF and
η is the points of η for that near the curve of TQAF.
tencyoftheexperimentalpointscanbefoundfortightly 2−TB
bound systems with two clear groups (taking FIG. 5 as
a sample case) while for the weakly bound systems there
is a scattered distribution. We take the case of tightly In conclusion, the theoretical radius of interaction
bound nuclei to explain this feature. Analogically, the givenbyPF isnotthegoodformulaealthoughitcanbe
1
exact deviation between η and η is given by used to compare the different tightness of systems. The
1−TB 2−TB
(cid:52)η =η −η , where η is the arithmetic PF with parameters separately fitted by tight bound,
TB 1−TB 2−TB 1−TB 2
meanofforthegroupofexperimentalquarter-pointangle weakly bound and halo projectiles is a parameterized
points of tight bound nuclei far from the curve of TQAF method to do the comparison. For finding the unified
and η is that near to the curve of TQAF. The cal- formulafornuclearradius,namelyfortheradiusofinter-
2−TB
culatedvaluesof(cid:52)η and(cid:52)η areshowninTABLEII. action,thePF andPF arecompared. However,theim-
TB 3 4
From these values, it may be observed that the results provementinPF withsymmetrydependenceconsidered
3
given by PF show the minimum values for both (cid:52)η is not good enough to consistently describe the nuclear
4
and (cid:52)η . Therefore, one can conclude that among all size. Finally,thePF withthebindingenergyconsidered
TB 4
thefourphenomenologicalformulae, PF showsthebest is recommended by this work for the phenomenological
4
improvementinpresentworkforallthethreetypesofnu- formula of R , which reduces the deviation not only
int
clei. Thus in the present work, PF is recommended for between the experimental data and the theoretical curve
4
giving better consistency among tightly bound systems, but also among the three kinds of projectiles. And the
besidesreducingthedeviationbetweentheexperimental inconsistency among the tightly bound systems is also
data and the theoretical curve of quarter-point angle. improved by PF .
4
5
Nuclei R /fm Ref.
rms
6He 2.30±0.07 [15]
8He 2.69±0.03 [16]
8B 2.38±0.04 [17]
9C 2.71±0.32 [18]
10Be 2.479±0.028 [19]
10C 2.42±0.10 [20]
11Li 3.34+0.04 [21]
−0.08
11Be 2.73±0.05 [22]
11C 2.46±0.03 [19]
12N 2.47±0.07 [23]
13O 2.53±0.05 [23]
14Be 3.10±0.15 [24]
17B 2.99±0.09 [24]
17F 2.71±0.18 [25]
17Ne 2.75±0.07 [23]
FIG. 6. (Color online) Comparison of the calculated mass
19B 3.11±0.13 [24]
radius (R1, R2, R3 and R4) basing on the four phenomeno-
23Al 2.905±0.250 [25]
logical formulae with nuclear charge radius for tightly-bound
27P 3.020±0.155 [25] nuclei 4He, 12C, 16O, 20Ne and 24Mg.
TABLE III. The experimental values of rms matter radii for
light halo nuclei.
IV. CALCULATION OF NUCLEAR RADIUS
Asdiscussedabove,PF notonlyreducesthedeviation
4
between the experimental data and the theoretical curve
but also among the three kinds of projectiles, when we
comparethefourparameterizedtheoreticalformula(PF
1
toPF )inordertoobtaintheradiusofinteractionR .
4 int
Which means that if we use the recommended formula
PF for calculating R of systems with tightly bound,
4 int
weaklyboundandhaloprojectilestocalculatexandplot
thefigureofθ vsx,wecanseethatallthesystemsare
1/4
ononecurveofTQAF.Thenuclearsizeisobviouslycor-
relatedtothequarter-pointangle,becausexisafunction FIG. 7. (Color online) Comparison of the calculated mass
of R , which indicates that we may extract the radius radius based on the four phenomenological formulae PF to
int 1
of nuclei from the experimental quarter-point angle. PF4 (R1,R2,R3andR4)withnuclearrmschargeradiusfor
weakly-bound nuclei.
For all applications, we calculate radius of nuclei as
projectiles using the formulae fitted from experimental
quarter-point angle and separately compare them with
the experimental charge radius [12] for tightly-bound From the comparison, the calculation for light halo nu-
(FIG. 6) and weakly-bound nuclei (FIG. 7) and with ex- clei based on PF3 does not agree with the experimental
perimental root mean square (rms) matter radius (TA- values.
BLE III) for halo nuclei (FIG. 8). For light nuclei, the As a matter of fact, the elastic scattering of a halo nu-
nuclear experimental charge radius usually agrees with cleusfromastabletargetcangivesimpledirectevidence
the mass radius, but for heavy nuclei having more neu- for the structure of the halo nucleus [13]. The angular
trons than protons, the mass radius might larger than distribution of elastic scattering reactions show a maxi-
the charge radius. As shown in Figs. 7 and 8, the cal- mumdifferenceforincidentenergiesaroundthetopofthe
culation of tightly-bound and light weakly-bound nuclei Coulombbarrier, therebysuggestingthatitisinthisen-
based on PF and PF can be deemed as a better rep- ergy region where the elastic scattering is most sensitive
3 4
resentation for determining nuclear size than PF and tothedetailsofthenuclearstructureoftheexoticprojec-
1
PF . The goal of extracting the radius of exotic nuclei tiles[14]. Thequarter-pointangleasafunctionofradius
2
fromtheexperimentalquarter-pointanglecanbeembod- of interaction R obtained via angular distributions of
int
ied by the calculation based on PF as shown in FIG. 8, elasticscatteringcrosssection,isrelatedtotheactualre-
4
whichclearlyshowsthefeasibilityofacquiringtheradius action mechanisms, which is not only related to the size
of halo nuclei via the experimental quarter-point angle. of the nuclei but also to the elastic scattering reactions
6
nuclei are short-lived and difficult to use as targets. For
the unstable nuclei especially for the halo nuclei, peo-
ple usually use the isotope method or interaction cross
section to measure the size of nuclei. However, extract-
ing the radius of unstable nuclei from the experimental
quarter-point angle could be a useful tool as the new ex-
perimental measurement. Therefore, the PF with more
4
details of structure (spatial extension, isospin symmetry
and binding energy) is recommended by this work for
nuclear radius.
V. CONCLUSIONS
Themotivationofpresentworkistocorrelatequarter-
point angle and nuclear radius. The theoretical radius
of interaction R were obtained and compared. In
int
thiswork,fourphenomenologicalformulae(PF toPF )
1 4
werepresumedandtheparametersfordifferentformulae
werefittedbyusingtheextractedexperimentalvaluesof
R . Consideringthedifferentkindsofreactionsystems,
int
thefourphenomenologicalformulaeareanalyzedanddis-
cussed. Basedontheabovementionedformulae,theradii
ofdifferentkindofnucleiasprojectileswereobtainedand
explained in detail. As a result, the parameterized for-
mula related to binding energy was recommended. In
conclusion, the deviation between the experimental data
and the theoretical curve and among the three kinds of
projectilescanbeminimizedbyappropriatelycalculating
the nuclearradius in orderto determine theradius ofin-
teraction. Thismayleadtoabetterunderstandingofthe
nuclearstructureandtheactualreactionmechanismsus-
FIG. 8. (Color online) Comparison of the calculated mass
radiusbasedonthephenomenologicalformulaePF andPF ingthethreetypesofprojectiles(stronglybound,weakly
3 4
(R3 and R4) with nuclear rms matter radius for halo nuclei. bound and halo nuclei).
VI. ACKNOWLEDGMENTS
with coupling to the channels of inelastic scattering or
other reactions. The nuclear properties, such as, nuclear This work was financially supported by the National
radius,isospinsymmetryandbindingenergypernucleon, Natural Science Foundation of China with Grant No.
will affect the strength of the couplings for different inci- U1432247 and 11575256 and the National Basic Re-
dent energies. Inversely, we can extract the information search Program of China (973 Program) with Grant No.
of the structure from elastic scattering reaction, such as 2014CB845405 and 2013CB83440x. One of the authors
the nuclear size. Experimentally, the nuclear radius can (SM) thanks the Chinese Academy of Sciences for the
be determined by electron scattering, isotope shift and support in the form of Presidents International Fellow-
interaction cross section etc. Since electron is structure ship Initiative (PIFI) Grant No. 2015-FX-04.
less and the electromagnetic interaction is known very
well, therefore the charge distribution of the nuclei can
be precisely measured by electron scattering. However, REFERENCES
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APPENDIX
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8
Reaction Systems E (MeV) θ (deg.) Reaction Systems E (MeV) θ (deg.)
cm cm cm cm
12C+28Si 16.80 99.4 12C+28Si 34.51 31.0
12C+28Si 45.50 24.8 12C+28Si 130.48 6.4
12C+208Pb 59.47 139.3 12C+208Pb 61.36 125.1
12C+208Pb 66.09 102.7 12C+208Pb 70.81 87.7
12C+208Pb 80.27 70.3 12C+208Pb 90.76 55.5
12C+208Pb 111.57 43.7 12C+208Pb 170.18 24.6
12C+208Pb 283.64 14.0 12C+208Pb 397.09 9.6
16O+12C 10.29 96.0 16O+12C 15.43 55.0
16O+12C 18.00 40.0 16O+12C 26.57 30.0
16O+12C 34.29 20.8 16O+12C 56.57 12.3
16O+16O 37.50 22.7 16O+16O 40.50 20.6
16O+16O 43.50 19.5 16O+16O 46.00 18.0
16O+16O 47.50 17.3 16O+16O 58.00 13.6
16O+40Ca 28.57 93.0 16O+40Ca 33.57 74.3
16O+40Ca 42.86 47.3 16O+40Ca 40.55 50.5
16O+40Ca 43.45 45.5 16O+56Fe 31.11 137.6
16O+56Fe 32.67 116.7 16O+56Fe 34.22 102.8
16O+56Fe 35.78 94.0 16O+56Fe 37.33 85.0
16O+56Fe 38.89 79.5 16O+56Fe 40.44 73.8
16O+56Fe 42.00 70.3 16O+56Fe 43.56 65.0
16O+56Fe 45.11 61.3 16O+90Zr 67.92 53.8
16O+90Zr 117.34 26.4 16O+90Zr 165.06 17.8
16O+208Pb 120.25 53.0 16O+208Pb 178.29 31.2
16O+208Pb 77.07 138.2 16O+208Pb 81.71 114.8
16O+208Pb 83.57 107.5 16O+208Pb 87.29 96.4
16O+208Pb 89.14 92.1 16O+208Pb 94.71 81.9
6He+65Cu 17.90 41.5 6He+65Cu 27.50 25.0
6He+120Sn 17.19 75.5 6He+120Sn 18.86 67.0
6He+120Sn 19.52 65.3 6He+197Au 38.82 40.5
6He+208Pb 21.38 110.0 6He+208Pb 26.24 72.0
6He+208Pb 28.77 59.2 6He+208Pb 53.46 27.8
6He+209Pb 21.87 102.8 8B+58Ni 22.23 134.5
8B+58Ni 23.90 102.5 8B+58Ni 25.75 87.1