Structure of ^{14}C and ^{14}O nuclei calculated in the variational approach
aa r X i v : . [ nu c l - t h ] A p r NUCLEI AND NUCLEAR REACTIONS
ISSN 2071-0186. Ukr. J. Phys. 2016. Vol. 61, No. 8 doi: 10.15407/ujpe61.08.0674
B.E. GRINYUK, D.V. PIATNYTSKYI
Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine (14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: [email protected], [email protected])
STRUCTURE OF C AND O NUCLEICALCULATED IN THE VARIATIONAL APPROACH
PACS 27.20.+n, 21.60.Gx,21.10.Ft, 21.10.Gv
The structure of mirror C and O nuclei has been studied in the framework of the five-particle model (three α -particles and two nucleons). Interaction potentials are proposed, whichallowed the energy and radius of C nucleus, as well as the energy of O one, to agree withexperimental data. On the basis of the variational approach with the use of Gaussian bases, theenergies and wave functions for five-particle systems under consideration are calculated. Thecharge radius of O nucleus, as well as the charge density distributions and the form factorsfor both nuclei, are predicted.K e y w o r d s : root-mean-square radius, density distribution, charge form factor, C nucleus, O nucleus.
1. Introduction
Each of the radioactive C and O nuclei can beimagined as composed of three α -particles and twoextra nucleons. The experience of the theoretical re-searches of nuclei with two extra nucleons such as He and Li [1–3] or more complicated Be and Cones [4, 5] showed that the accuracy of this approx-imation can compete with that of other approaches,in which all nucleon degrees of freedom are takeninto consideration [6]. If the number of nucleons ina nucleus is large, the reduction of the number ofvariables in the corresponding problem by makingallowance for the α -clustering becomes even morejustified.In this work, we consider C and O nuclei assystems consisting of three α -particles and two extraneutrons (in C case) or two extra protons (for Onucleus). The problem of five particles is solved in thevariational approach with the use of the Gaussian ba-sis [7–9], which allows the systems of several particleswith various kinds of interaction between them to bestudied with a rather high accuracy. c (cid:13) B.E. GRINYUK, D.V. PIATNYTSKYI, 2016
The applied models of generalized interaction po-tentials between α -particles, as well as between nu-cleons and α -particles, are similar to those used byus in the research of lighter nuclei [3, 5, 10]. However,for the agreement of the energy and charge radius of C nucleus and the energy of O nucleus with ex-perimental values to be exact, the parameters of po-tentials are slightly modified. This gives us a groundto hope for that the density distributions and chargeform factors predicted for C and O nuclei – inparticular, the charge radius of O – will agree withfuture experimental data.
2. Statement of the Problem
The model Hamiltonian for O nucleus contains,besides the one-particle operator of kinetic energy,the potentials of pair interaction between the parti-cles, which are generated by nuclear forces, and theCoulomb repulsion potential, ˆ H = X i =1 p i m p + X i =3 p i m α + U pp ( r )+ X j>i =3 ˆ U αα ( r ij ) ++ X i =1 5 X j =3 ˆ U pα ( r ij ) + X j>i =1 Z i Z j e r ij . (1) tructure of C and O Nuclei
Here, m p and m α are the masses of a proton and an α -particle, respectively; Z = Z = 1 and Z = Z = Z = 2 are the charges of particles divided by theelementary charge unit e . Note that the Hamiltonianfor C nucleus differs from Eq. (1) in that the neu-tron charges equal zero ( Z = Z = 0 ) and the pro-ton mass m p is substituted by the neutron one m n ,although this change of mass practically does not af-fect the result. The Hamiltonians for both nuclei arealso characterized by a small difference between theeffective nuclear interaction of a neutron with an α -particle in C nucleus and the effective interaction ofa proton with an α -particle in O one, because thedistribution of protons in the α -particle has a littlelarger radius than the corresponding distribution ofneutrons (see, e.g., work [11]).The choice of models for the pair interaction po-tentials between the particles is based on the crite-rion that our model should simultaneously describethe experimental energies of the examined nuclei andthe charge radius of C nucleus (the experimentalcharge radius for O is unknown). The pair interac-tion potential between the neutrons in C nucleus ischosen in the form of a local potential that describesboth the low-energy singlet neutron-neutron param-eters and the singlet phase of the neutron scatteringwith a qualitative accuracy. In view of the charge in-variance of nuclear forces, the interaction potentialbetween the protons in O nucleus, U pp ( r ) , is as-sumed to be the same. This potential was successfullyused, while studying He [3], Be, and C [5, 10]nuclei.The interaction potential between a neutron andan α -particle is chosen in the form of a generalizedinteraction potential with the local and nonlocal com-ponents, which models most successfully the interac-tion of a nucleon with the α -cluster [1, 3] and sim-ulates the Pauli exclusion principle. In work [5], inorder to study Be nucleus, the parameters of thepotential between a neutron and an α -particle thathad been used to study He nucleus [3] were slightlychanged. In the present work, analogously to this pro-cedure, some parameters of the same potential arechanged a little in order to simultaneously describethe energy and charge radius of C with a high ac-curacy: ˆ U nα ( r ) = − V exp( − ( r/r ) ) + gπ / R | u ih u | , (2) where the local attraction is characterized by the pa-rameters V = 43 .
95 MeV and r = 2 . fm, andthe separable repulsion has the form factor u ( r ) == exp (cid:16) − ( r/R ) (cid:17) with the radius R = 2 .
79 fm andthe repulsion intensity g = 140 λ ( R ) MeV (hereafter, λ ( x ) ≡ (cid:2) / ( πx ) (cid:3) / ).The interaction potential between a proton andan α -particle that was used by us to study Onucleus had the same form (2), but with slightlychanged parameters. This circumstance is related tothe known fact that the protons and neutrons in He nucleus are not distributed absolutely identi-cally (the details of distributions can be found inwork [11]), so that the potential of nuclear interac-tion ˆ U nα should not exactly coincide with the po-tential ˆ U pα . We select the parameters for the poten-tial ˆ U pα so that the energy of O nucleus could bedescribed by assuming the other potentials of nu-clear interaction between the corresponding particlesin O and C nuclei to be identical. In this work,the attraction intensity V = 44 . MeV and theattraction radius r = 2 . fm are used for the po-tential ˆ U pα . The separable repulsion component ofthis potential is taken the same as for the potential ˆ U nα (2).By its form, the interaction potential between α -particles is also very similar to that used in work [5]: ithas local and nonlocal components, but with slightlymodified parameters: ˆ U ( r ) = − U exp (cid:16) − ( r/ρ ) (cid:17) ++ U exp (cid:16) − ( r/ρ ) (cid:17) + gπ / ρ α | v ih v | , (3)where the attraction intensity U = 43 . MeV, therepulsion intensity U = 240 . MeV, and the cor-responding radii equal ρ = 2 . fm and ρ == 1 . , respectively. The separable repulsion hasthe form factor v ( r ) = exp (cid:16) − ( r/ρ α ) (cid:17) with theradius ρ α = 1 . fm, and the repulsion intensity g = 60 λ ( ρ α ) MeV.Note that the selection of parameters for the po-tentials became possible only after a multiply re-peated procedure that includes the solution of thefive-particle problem and the comparison of the val-ues obtained for the energy and the charge radius withthe experimental data. The method of calculation on
ISSN 2071-0186. Ukr. J. Phys. 2016. Vol. 61, No. 8 .E. Grinyuk, D.V. Piatnytskyi the basis of the variational approach with the use ofGaussian bases will be briefly described in the nextsection. The resulting potential parameters are givenabove, and the energies and radii calculated on thebasis of those potentials for both analyzed nuclei arecompared with the corresponding experimental datain Table 1. In the latter, the energies of nuclei areshown subtracting − . MeV per each α -particle,and the charge radii were calculated with regard forthe non-point character of the particles in the Helmapproximation: R = R α + R (cid:0) He (cid:1) in the case of C nucleus (a small squared chargeradius of a neutron is neglected) and R = 34 (cid:0) R α + R (cid:0) He (cid:1)(cid:1) + 14 (cid:0) R p + R ( p ) (cid:1) for O one. Here, R α designates the root-mean-square radius for “point-like” α -particles distribution,which was calculated in the framework of the modelwith Hamiltonian (1), and the charge radius of α -particle R ch (cid:0) He (cid:1) = 1 . fm was taken from theexperiment (as an average value between the moderndata [12] and the data of work [13]). Analogously, R p means the calculated root-mean-square radius of the“point-like” proton distribution (in the case of Onucleus), and R ch ( p ) = 0 . fm was taken from theexperimental data of work [14].
3. Calculation Technique
To solve the problem of bound states in the sys-tem of five particles, we use the variational methodin the Gaussian representation [7–9], which allowsthe wave function of the system to be obtained inthe explicit convenient form of a gaussoid superposi-tion. Omitting the details of this well-known method,
Table 1.
Energies (MeV) and charge radii (fm)of C and O nuclei. The energies are reckonedfrom the threshold of the nucleus decay into three α -particles and two nucleons Nucleus
E E exp R ch R ch , exp14 C − . − .
398 2 .
500 2 . [17] . [12] O − . − .
845 2 . — we only recall that the Schr¨odinger equation withHamiltonian (1) can be reduced to a system of lin-ear algebraic equations (the Galerkin method): K X m =1 C m D ˆ Sϕ k (cid:12)(cid:12)(cid:12) ˆ H − E (cid:12)(cid:12)(cid:12) ˆ Sϕ m E = 0 , k = 0 , , ..., K. (4)Here, all required matrix elements are expressible inan explicit form if one uses the Gaussian basis. Forthe ground symmetric J π = 0 + state, the wave func-tion Φ has a simple form in the Gaussian repre-sentation: Φ = ˆ S K X k =1 C k ϕ k ≡≡ ˆ S K X k =1 C k exp − X j>i =1 a k,ij ( r i − r j ) , (5)where ˆ S is the symmetrizing operator. Note alsothat the wave function symmetrization can be ei-ther performed or not performed explicitly, because,as was shown in work [15], the symmetric form isrestored automatically, when the non-symmetrizedbasis is expanded. Both opportunities are used inthis work. The explicit symmetrizations for three α -particles and two extra nucleons (in this case, thereare only × terms in the symmetrizedfunction) allowed a Gaussian basis with consider-ably smaller dimensionality to be used than that re-quired in the case of the wave function with no ex-plicit symmetrization, provided the same calculationaccuracy.
4. Density Distributionsand Form Factors
The one-particle density distribution for the j -th par-ticle in a system of particles with the wave function | Φ i is defined as follows: n i ( r ) = h Φ | δ ( r − ( r i − R c . m . )) | Φ i , (6)where R c . m . is the radius vector of the center of massof the system. Hereafter, all density distributions arenormalized to 1: R n i ( r ) d r = 1 . The expressions for ISSN 2071-0186. Ukr. J. Phys. 2016. Vol. 61, No. 8 tructure of C and O Nuclei n i ( r ) have the explicit form in terms of the param-eters a k,ij and the coefficients C k of the linear ex-pansion of the wave function in the Gaussian repre-sentation (5). The resulting one-particle density dis-tributions multiplied by r are shown in Fig. 1 bydashed curves. They illustrate the density distribu-tions for “point-like” α -particles and extra nucleonsin C nucleus. The corresponding distributions of“point-like” particles in O nucleus are very similarto those depicted in Fig. 1, so that they are not ex-hibited separately. The attention is attracted by thefact that the extra nucleons are mainly located inside C cluster formed by α -particles (with a probabilityof about 0.86 for neutrons in C nucleus and 0.84for protons in O nucleus). However, another smallmaximum in the density distribution curve for ex-tra nucleons testifies that the extra nucleons can alsobe found outside C cluster, although with rather alow probability (approximately 0.14 for C and 0.16for O).Note that the extra nucleons move much morerapidly than the α -particles. In particular, the cal-culated average kinetic energy amounts to about32.66 MeV for each of the extra neutrons in C nu-cleus. At the same time, the corresponding value foreach α -particle equals about 6.83 MeV, which is al-most five times lower and can be explained, mainly,by the larger mass of the latter. Concerning the par-ticle velocities, it turns out that the extra neutronsin C nucleus move approximately 4.4 times fasterthan the α -particles. The same ratio between the ve-locities of extra nucleons and α -particles is charac-teristic of O nucleus. For the latter, the calculatedaverage kinetic energy amounts to about 31.77 MeVfor each extra proton and to about 6.62 MeV for each α -particle.To calculate the charge density distribution in thenuclei, the non-point nature of α -particles, as well asprotons in the case of O nucleus, has to be takeninto consideration. For this purpose, we use the Helmapproximation [16]. In particular, the charge densitydistribution for C nucleus, n ch ( r ) = Z n α ( | r − r ′ | ) n ch , He ( r ′ ) d r ′ , (7)looks like the convolution of the one-particle den-sity distribution n α found for “point-like” α -particleswith the charge density of an α -particle itself, n ch , He ,which follows from the experimental form factor [18]. r n (r) r, fm
12 3
Fig. 1.
Charge density distribution in C nucleus multipliedby r (solid curve ). Dashed curves correspond to the densitydistributions (multiplied by r ) of “point-like” particles: extraneutrons (curve ) and α -particles (curve ) In expression (7), we neglect a small contributionof extra neutrons to the charge density distribu-tion. Note that the Helm approximation (7) is ob-tained, by supposing that the wave function of a nu-cleus is an (approximate) product of the wave func-tion obtained for the Hamiltonian that describes therelative motion of “point-like” particles with the wavefunctions of He nuclei ( α -clusters).A similar expression is obtained for O nucleus, inwhich, besides the contribution of α -particles to thedensity distribution, the contribution of extra protonsis also made allowance for: n ch ( r ) = 34 Z n α ( | r − r ′ | ) n ch , He ( r ′ ) d r ′ ++ 14 Z n p ( | r − r ′ | ) n ch ,p ( r ′ ) d r ′ . (8)Here, the charge distribution for a proton, n ch ,p , istaken from work [19]. The coefficients 3/4 and 1/4(their sum equals 1) are proportional to the totalcharges of α -particles and extra protons, respectively,in O nucleus.The distributions of charge density in C and Onuclei obtained on the basis of expressions (7) and(8) are shown in Fig. 2. A considerably higher chargedensity at short distances in O nucleus is explainedby the presence of additional proton charges locatedat rather small distances in this nucleus. For this rea-son, the charge radius of O nucleus turns out to be
ISSN 2071-0186. Ukr. J. Phys. 2016. Vol. 61, No. 8 .E. Grinyuk, D.V. Piatnytskyi n c h (r) , f m - r, fm Fig. 2.
Charge density distributions in C and O nuclei.The dashed curve demonstrates the density distribution for“point-like” α -particles in C nucleus. The distributions arenormalized to 1 | F c h ( q ) | q , fm -2 Fig. 3.
Charge form factor for C nucleus (solid curve )in comparison with the same quantity calculated in the “point-like”-particle approximation (dashed curve ). Curve demon-strates the form factor for He nucleus [12] smaller than that of C one: R (cid:0) O (cid:1) = Z r n ch , O ( r ) d r << R (cid:0) C (cid:1) = Z r n ch , C ( r ) d r , (9)although all distributions for “point-like” particles in O nucleus have, on the contrary, larger radii incomparison with their counterparts in C. This cir-cumstance and a lower binding energy in O nucleusare explained mainly by the additional Coulomb re-pulsion due to extra protons. In order to confirm | F c h ( q ) | q , fm -2
12 3
Fig. 4.
Charge form factor for O nucleus (solid curve ),form factor corresponding to the density distribution of α -particles in this nucleus in the “point-like”-particle approxima-tion (curve ), and analogous form factor obtained for extraprotons (curve ) this fact, we quote the calculated root-mean-squarerelative distances between extra nucleons, r NN , nu-cleon and α -particle, r Nα , and α -particles, r αα , aswell as the root-mean-square radii for the distribu-tions of “point-like” nucleons, R N , and “point-like” α -particles, R α , in both analyzed nuclei in Table 2. Wehope for that the value of charge radius R ch (cid:0) O (cid:1) =2 .
415 fm predicted by us for O nucleus will be ex-perimentally confirmed in the future. With regardfor the experimental error for the α -particle radius[12, 13] and the restricted accuracy of our model, inwhich the Helm approximation was used, we evalu-ate the calculation error for the charge radius of Onucleus to equal about ± . fm.Note that the integration (convolution) in Eqs. (7)and (8) substantially “smoothes” out the one-particledistributions obtained for “point-like” particles. As aresult, the charge distribution at short distances doesnot contain a “dip” typical of distributions for “point-like” α -particles in both nuclei. Table 2.
Root-mean-square relativedistances and radii (fm) of C and O nuclei
Nucleus r NN r Nα r αα R N R α R ch14 C .
621 2 .
667 3 .
189 1 .
786 1 .
852 2 . O .
732 2 .
750 3 .
239 1 .
864 1 .
882 2 . ISSN 2071-0186. Ukr. J. Phys. 2016. Vol. 61, No. 8 tructure of C and O Nuclei
Characteristic features in the behavior of densitydistributions manifest themselves in the correspond-ing form factors, the Fourier transforms of the den-sity. In particular, convolution (7) transforms into theproduct F ch , C ( q ) = F α, C ( q ) F ch , He ( q ) , (10)and expression (8) into the sum of products F ch , O ( q ) = 34 F α, O ( q ) F ch , He ( q ) ++ 14 F p, O ( q ) F ch ,p ( q ) . (11)In the products in expressions (10) and (11), the firstmultipliers correspond to the form factors obtained byus from the wave function of the five-particle prob-lem in the “point-like”-particle approximation. Thesecond multipliers are the experimental form factorsof an α -particle F ch , He ( q ) [18] or proton F ch ,p ( q ) [19]. If a form factor becomes zero at a definite trans-ferred momentum squared, the absolute value of theform factor has a “dip” at this q . Owing to the rep-resentation of the charge form factor for C nucleusin the form of product (10), the form factor abso-lute value has “dips” at those q , where each of themultipliers has its own “dip”, which is illustrated inFig. 3. Note that the “dip” located at the squaredtransferred momentum q ≃ fm − and originatingfrom the form factor of an α -particle is also observedin the form factors of He [3] and Be [5] nuclei, aswell as in all other cases where the charge distribu-tion in cluster nuclei is driven only by the chargesof α -particles. Concerning the first dip in the chargeform factor of C nucleus in the interval slightly be-low q ∼ fm − , which is connected with the “dip”in the first multiplier in Eq. (10), it is explained bya substantial decrease in the density distribution atshort distances, which was obtained in the approxi-mation of “point-like” α -particles (in Fig. 2, this dis-tribution is depicted by a dashed curve). It can beshown that, after the Fourier transformation, this dis-tribution transforms into a form factor that changesits sign in the momentum representation, and the ab-solute value of the form factor has a “dip”.At the same time, in the case of O nucleus, owingto the second term in Eq. (11), the corresponding“dips” do not manifest themselves per se in the chargeform factor of the nucleus, but only slightly affect the change of regimes in its behavior, as is shown inFig. 4. Hence, the form factors of O and C nucleiconsiderably differ from each other owing to the roleof extra protons in O instead of neutrons in C.
5. Conclusions
In the framework of the five-particle model (three α -particles and two extra nucleons) and on the basis ofvariational calculations in the Gaussian representa-tion, the density distributions and the form factors for C and O nuclei are calculated. The extra nucle-ons in both nuclei are found to move predominantlyinside C cluster, although they can also be foundat its periphery with a low probability. The chargeradius of O nucleus is predicted. It is shown that,owing to the extra protons that are located closer tothe nucleus center, this radius is smaller than thatof C nucleus, despite that all relative distances be-tween the corresponding particles in O nucleus arelarger than in C one. We hope for that the more de-tailed ideas of the structure of C and O nuclei andthe character of motion of their component particlescan be obtained on the basis of calculations and theanalysis of such structural functions as the pair cor-relation functions and the momentum distributions.
1. V.I. Kukulin, V.N. Pomerantsev, Kh.D. Razikov et al. ,Nucl. Phys.
A 586 , 151 (1995).2. M.V. Zhukov, B.V. Danilin, D.V. Fedorov et al. , Phys.Rep. , 151 (1993).3. B.E. Grinyuk and I.V. Simenog, Yad. Fiz. ,10 (2009).4. Y. Ogawa, K. Arai, Y. Suzuki, and K. Varga, Nucl. Phys.A , 122 (2000).5. B.E. Grinyuk and I.V. Simenog, Yad. Fiz. , 443 (2014).6. A.V. Nesterov, F. Ariks, Ya. Brukkhov, and V.S. Vasilevs-kii, Elem. Chast. At. Yadro , 1337 (2010).7. V.I. Kukulin and V.M. Krasnopol’sky, J. Phys. G , 795(1977).8. N.N. Kolesnikov and V.I. Tarasov, Yad. Fiz. , 609(1982).9. Y. Varga and K. Suzuki, Stochastic Variational Approachto Quantum-Mechanical Few-Body Problems (Springer,Berlin, 1998).10. B.E. Grinyuk and I.V. Simenog, Ukr. J. Phys. , 635(2011).11. B.E. Grinyuk, D.V. Piatnytskyi, and I.V. Simenog, Ukr. J.Phys. , 424 (2007).12. I. Angeli and K.P. Marinova, At. Data Nucl. Data Tables , 69 (2013).13. I. Sick, Phys. Rev. C , 041302 (2008). ISSN 2071-0186. Ukr. J. Phys. 2016. Vol. 61, No. 8 .E. Grinyuk, D.V. Piatnytskyi
Review of Particle Physics , J. Phys. G , 1–1232 (2006).15. I.V. Simenog, M.V. Kuzmenko, and V.M. Khryapa, Ukr.Fiz. Zh. , 1240 (2010).16. A.I. Akhiezer and V.B. Berestetskii, Quantum Electrody-namics (Interscience, New York, 1965).17. L.A. Schaller, L. Schellenberg, T.Q. Phan et al. , Nucl.Phys.
A 379 , 523 (1982).18. R.F. Frosch, J.S. McCarthy, R.E. Rand, and M.R. Yearian,Phys. Rev. , 874 (1967).19. P.E. Bosted et al. , Phys. Rev. Lett. , 3841 (1992).Received 17.03.16.Translated from Ukrainian by O.I. Voitenko Б.Є. Гринюк, Д.В. П’ятницький
СТРУКТУРА ЯДЕР C ТА OУ ВАРIАЦIЙНОМУ ПIДХОДIР е з ю м еВ рамках п’ятичастинкової моделi (три α -частинки i два до-датковi нуклони) дослiджено структуру дзеркальних ядер C та O. Запропоновано потенцiали взаємодiї, якi дозво-лили узгодити з експериментом енергiю та радiус ядра C,а також енергiю ядра O. На основi варiацiйного пiдходуз використанням гаусоїдних базисiв розраховано енергiї iхвильовi функцiї дослiджуваних п’ятичастинкових систем.Передбачено зарядовий радiус ядра O, а також зарядовiрозподiли густини i формфактори обох ядер.680