Particle-number projection method in time-dependent Hartree-Fock theory: Properties of reaction products
aa r X i v : . [ nu c l - t h ] D ec Particle-number projection method in time-dependent Hartree-Fock theory:Properties of reaction products
Kazuyuki Sekizawa ∗ and Kazuhiro Yabana
1, 2, † Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan (Dated: April 26, 2018)
Background:
The time-dependent Hartree-Fock (TDHF) theory has been successful in describing low-energyheavy ion collisions. Recently, we have shown that multinucleon transfer processes can be reasonably describedin the TDHF theory combined with the particle-number projection technique.
Purpose:
In this work, we propose a theoretical framework to analyze properties of reaction products in TDHFcalculations.
Methods:
TDHF calculation in three-dimensional Cartesian grid representation combined with particle numberprojection method.
Results:
We develop a theoretical framework to calculate expectation values of operators in the TDHF wavefunction after collision with the particle-number projection. To show how our method works in practice, themethod is applied to O+ O collisions for two quantities, angular momentum and excitation energy. Theanalyses revealed following features of the reaction: The nucleon removal proceeds gently, leaving small values ofangular momentum and excitation energy in nucleon removed nuclei. Contrarily, nuclei receiving nucleons showexpectation values of angular momentum and excitation energy which increase as the incident energy increases.
Conclusions:
We have developed a formalism to analyze properties of fragment nuclei in the TDHF theorycombined with the particle-number projection technique. The method will be useful for microscopic investigationsof reaction mechanisms in low-energy heavy ion collisions as well as for evaluating effects of particle evaporationon multinucleon transfer cross sections.
I. INTRODUCTION
The time-dependent Hartree-Fock (TDHF) theory hasbeen successfully applied for studying low-energy nu-clear reactions: fusion reactions, deep inelastic collisions,quasi-fission reactions, extraction of nucleus-nucleus po-tentials and dissipation coefficients, and so on (for arecent review, see Ref. [1]). For reactions producingprojectile- and target-like fragments, expectation valuesof operators are often evaluated in the TDHF wave func-tion after collision to investigate properties of producednuclei.Recently, a particle number projection (PNP) tech-nique has been proposed by C. Simenel [2]. This methodhas made it possible to calculate transfer probabilitiesefficiently from the TDHF wave function after collisionand has been successfully applied [3–5]. Using the PNPtechnique, we studied multinucleon transfer (MNT) pro-cesses in heavy ion reactions at around the Coulomb bar-rier for several systems [5] for which extensive measure-ments are available [6–9]. Comparing calculated crosssections with the measurements, we have concluded thatthe TDHF theory may describe MNT cross sections quan-titatively, in an accuracy comparable to calculations byother existing theories such as GRAZING [10] and Com-plex WKB [11], which are based on semiclassical approx-imation, and a model based on Langevin-type equationsof motion [12, 13]. ∗ [email protected] † [email protected] The PNP technique has been utilized to calculate re-action probabilities which are required to calculate crosssections of specific nucleon numbers. To investigate re-action mechanisms, it will be useful to calculate expecta-tion values of operators in the particle-number projectedwave functions. This is the subject of this article. Wewill consider a general method to calculate expectationvalues of one- and two-body operators.The method will be also useful to investigate deexci-tation effects on the MNT cross sections. The producednuclei through MNT reactions are often highly excited.Measured cross sections are affected by deexcitation pro-cesses such as particle emissions which take place in rel-atively longer timescale. In the existing theories men-tioned above, evaporation effects are usually taken intoaccount employing statistical models. In statistical mod-els of particle evaporation, excitation energy and angularmomentum are the basic inputs. The method to be de-veloped in this paper will be useful to calculate thesequantities.In nuclear structure calculations, it is a routine workto calculate expectation values of operators in particle-number projected wave functions [14]. In this arti-cle, we extend the formalism to the TDHF wave func-tion after collision with the PNP. To illustrate how ourmethod works, we analyze properties of produced nucleiin O+ O collisions.This article is organized as follows. In Sec. II, we de-scribe a general formalism to calculate expectation val-ues of operators in the TDHF wave function after col-lision with the PNP. In Sec. III, we apply the methodto O+ O collisions, as an illustrative example. InSec. IV, a summary is presented.
II. FORMULATIONA. Particle-number projection method
We consider microscopic TDHF calculations of low-energy heavy ion collisions in which two fragments, aprojectile-like fragment (PLF) and a target-like fragment(TLF), are produced. In this section, we develop a gen-eral formalism to calculate expectation values of opera-tors for one of the fragments, either PLF or TLF, withthe PNP. We first describe the formalism assuming thatthe system is composed of N identical fermions. An ex-tension to include two kinds of fermions, neutrons andprotons, is straightforward.We assume that the fragments are well separated spa-tially after collision at the final stage of the TDHF cal-culation. We define two spatial regions, V and ¯ V . Thespatial region V includes a fragment to be analyzed. ¯ V isthe complement of V , which includes the other fragment.We denote the TDHF wave function after collision asΨ( x , · · · , x N ), where x denotes a set of the spatial andthe spin coordinates, x ≡ ( r , σ ). The wave function Ψis, in general, not an eigenstate of the particle-numberoperator in the spatial region V but a superposition ofstates with different particle numbers in V . It can beexpressed asΨ( x , · · · , x N ) = N X n =0 Ψ n ( x , · · · , x N ) , (1)where Ψ n denotes a particle-number projected wave func-tion, Ψ n ( x , · · · , x N ) = ˆ P n Ψ( x , · · · , x N ) . (2)Ψ n is a component of Ψ having n particles in the spatialregion V and N − n particles in the spatial region ¯ V . Theoperator ˆ P n is the PNP operator defined by [2, 5]ˆ P n = X s ( { τ i } : V n ¯ V N − n ) Θ τ ( r ) · · · Θ τ N ( r N ) (3)= 12 π Z π dθ e i( n − ˆ N V ) θ , (4)where s ( { τ i } : V n ¯ V N − n ) indicates that a sum over thesequence τ τ · · · τ N should be taken for all possible com-binations that V appears n times and ¯ V appears N − n times. We have introduced a space division function,Θ τ ( r ), and a particle-number operator in the spatial re-gion τ , ˆ N τ , which are defined byΘ τ ( r ) = (cid:26) r ∈ τ, r / ∈ τ, (5)and ˆ N τ = Z τ d r N X i =1 δ ( r − r i ) = N X i =1 Θ τ ( r i ) , (6) where τ represents the spatial region either V or ¯ V .We consider a general operator ˆ O and decompose itinto two operators according to the spatial regions:ˆ O = ˆ O V + ˆ O ¯ V . (7)The operator ˆ O V represents a part of the operator ˆ O acting to the particle when it is in the spatial region V .The operator ˆ O ¯ V represents the remaining part of theoperator ˆ O . Any one-body operator which is local inspace, ˆ O (1) = P Ni =1 ˆ o (1) ( r i , σ i ), can be decomposed asˆ O (1) = N X i =1 (cid:16) Θ V ( r i ) + Θ ¯ V ( r i ) (cid:17) ˆ o (1) ( r i , σ i )= ˆ O (1) V + ˆ O (1)¯ V , (8)where σ i denotes the spin coordinate of a particle i . In the same way, a two-body operator, ˆ O (2) = P Ni We present formulae of expectation values which areuseful for the TDHF wave function Ψ given by a sin-gle Slater determinant composed of single-particle wavefunctions ψ i ( x ),Ψ( x , · · · , x N ) = 1 √ N ! det (cid:8) ψ i ( x j ) (cid:9) . (13)Using the PNP operator of Eq. (4), the probability P n can be calculated as [2, 5] P n = 12 π Z π dθ e i nθ (cid:10) Ψ (cid:12)(cid:12) e − i ˆ N V θ (cid:12)(cid:12) Ψ (cid:11) = 12 π Z π dθ e i nθ det B ( θ ) . (14) B ( θ ) denotes a N -dimensional matrix, (cid:16) B ( θ ) (cid:17) ij = Z dx ψ ∗ i ( x ) ψ j ( x, θ ) , (15)where ψ i ( x, θ ) is defined by ψ i ( x, θ ) ≡ (cid:16) Θ ¯ V ( r ) + e − i θ Θ V ( r ) (cid:17) ψ i ( x ) . (16)Using Eqs. (4) and (10), the expectation value O Vn isexpressed as O Vn = 12 πP n Z π dθ e i nθ (cid:10) Ψ (cid:12)(cid:12) ˆ O V e − i ˆ N V θ (cid:12)(cid:12) Ψ (cid:11) . (17)In the case of one- and two-body operators, ˆ O (1) V andˆ O (2) V , in Eqs. (8) and (9), expectation values can be cal- culated by [15] O V (1) n = 12 πP n Z π dθ e i nθ det B ( θ ) × N X i =1 Z V dx ψ ∗ i ( x ) ˆ o (1) ( x ) ˜ ψ i ( x, θ ) , (18) O V (2) n = 12 πP n Z π dθ e i nθ det B ( θ ) N X i In actual TDHF calculations, the many-body wavefunction Ψ is given by a product of two Slater deter-minants, Ψ = Ψ ν Ψ π , where Ψ ν is for neutrons and Ψ π is for protons. We present formulae of expectation val-ues for this wave function. We denote the PNP operatorfor neutrons (protons) as ˆ P ( n ) N ( ˆ P ( p ) Z ), where N ( Z ) isthe number of neutrons (protons) in the spatial region V . The probability that N neutrons and Z protons arein the spatial region V is then given by a product ofprobabilities for neutrons and protons, P N,Z = (cid:10) Ψ (cid:12)(cid:12) ˆ P ( n ) N ˆ P ( p ) Z (cid:12)(cid:12) Ψ (cid:11) = (cid:10) Ψ ν (cid:12)(cid:12) ˆ P ( n ) N (cid:12)(cid:12) Ψ ν (cid:11) (cid:10) Ψ π (cid:12)(cid:12) ˆ P ( p ) Z (cid:12)(cid:12) Ψ π (cid:11) = P ( n ) N P ( p ) Z . (21)We first consider expectation values for a one-body op-erator. We note that any one-body operator can be writ-ten as a sum of operators for neutrons and for protons,ˆ O (1) V = ˆ O (1 ,n ) V + ˆ O (1 ,p ) V . Thus the expectation value of theone-body operator ˆ O (1) V is given by a sum of two terms.For the fragment nucleus specified by N and Z , we have O V (1) N,Z = (cid:10) Ψ (cid:12)(cid:12) ˆ O (1) V ˆ P ( n ) N ˆ P ( p ) Z (cid:12)(cid:12) Ψ (cid:11)(cid:10) Ψ (cid:12)(cid:12) ˆ P ( n ) N ˆ P ( p ) Z (cid:12)(cid:12) Ψ (cid:11) = (cid:10) Ψ ν (cid:12)(cid:12) ˆ O (1 ,n ) V ˆ P ( n ) N (cid:12)(cid:12) Ψ ν (cid:11)(cid:10) Ψ ν (cid:12)(cid:12) ˆ P ( n ) N (cid:12)(cid:12) Ψ ν (cid:11) + (cid:10) Ψ π (cid:12)(cid:12) ˆ O (1 ,p ) V ˆ P ( p ) Z (cid:12)(cid:12) Ψ π (cid:11)(cid:10) Ψ π (cid:12)(cid:12) ˆ P ( p ) Z (cid:12)(cid:12) Ψ π (cid:11) = O V (1 ,n ) N + O V (1 ,p ) Z . (22) O V (1 ,q ) n is defined by O V (1 ,q ) n = 12 πP ( q ) n Z π dθ e i nθ (cid:10) Ψ q (cid:12)(cid:12) ˆ O (1 ,q ) V e − i ˆ N ( q ) V θ (cid:12)(cid:12) Ψ q (cid:11) , (23)where ˆ N ( q ) V denotes the particle-number operator for neu-trons ( q = n ) and for protons ( q = p ) in the spatial region V . We will use these formulae, Eqs. (22) and (23), to cal-culate expectation values of the kinetic energy operatorincluded in the Hamiltonian and of the angular momen-tum operator.For a two-body operator, expectation values are notsimply given by a sum of neutron and proton contribu-tions, since two-body operators act between neutrons andprotons. Therefore, we apply the PNP operators for bothneutrons and protons simultaneously, O V (2) N,Z = 1(2 π ) P ( n ) N P ( p ) Z Z π dθ Z π dϕ e i( Nθ + Zϕ ) × (cid:10) Ψ (cid:12)(cid:12) ˆ O (2) V e − i( ˆ N ( n ) V θ + ˆ N ( p ) V ϕ ) (cid:12)(cid:12) Ψ (cid:11) . (24)We will use the above formula to evaluate excitation en-ergy of nuclei produced through transfer processes.To evaluate the excitation energy, we need to excludethe energy associated with the center-of-mass motion.For this purpose, we calculate the energy expectationvalue using Eqs. (21)-(24) in the coordinate system whichmoves with the fragment nucleus. In practice, we multi-ply all the single-particle wave functions by e − i K µ · r /A µ ,where K µ is given by K µ = M µ ˙ R µ ( t f ) / ~ , with M µ , A µ , and ˙ R µ ( t f ) being the average mass, the averagenucleon number, and the average velocity of the frag-ment ( µ = PLF or TLF) in the spatial region V attime t f . We calculate the velocity of the fragment by˙ R µ ( t f ) ≡ (cid:2) R µ ( t f + ∆ t ) − R µ ( t f − ∆ t ) (cid:3) / (2∆ t ).We denote the calculated energy expectation value inthe fragment nucleus composed of N neutrons and Z pro-tons as E VN,Z . We separately achieve ground state calcu-lations for the fragment nucleus composed of N neutronsand Z protons, which we denote as E g . s .N,Z . We evaluatean excitation energy of the fragment nucleus by E ∗ VN,Z ( E, b ) ≡ E VN,Z ( E, b ) − E g . s .N,Z , (25)where E and b denote the incident relative energy andthe impact parameter, respectively.In the ground state calculation, we employ a mass cor-rection in the kinetic energy operator, ~ m → ~ m (1 − N + Z ), to take into account the center-of-mass correction.The same correction is applied in evaluating the expecta-tion value of the kinetic energy operator using Eqs. (22)and (23), depending on numbers of neutrons and protons, N and Z , in the fragment nucleus. III. AN ILLUSTRATIVE EXAMPLE: O+ O COLLISION To illustrate usefulness of the PNP method describedin Sec. II, we analyze properties of fragment nuclei in O+ O collisions described by the TDHF theory. For O, pairing correlation may be important. In Ref. [16],the pairing interaction is reported to be negligible inthe ground state, while finite contribution is reported inRef. [17]. In this paper, we restrict ourselves to treat-ments ignoring pairing effects. We note that reactionsincluding neutron-rich oxygen isotopes have been well-studied in the TDHF theory as a typical reaction involv-ing light unstable nuclei [18–21]. We will investigate ex-pectation values of the angular momentum operator andaverage excitation energies.We consider reactions in which two fragments are gen-erated after collision. We call the O-like fragment nu-cleus as the PLF and the O-like fragment nucleus asthe TLF. We describe the collision in the center-of-massframe. We choose xy -plane as the reaction plane set-ting the incident direction parallel to the x axis. Theprojectile, O, moves towards the negative- x direction,while the target, O, moves towards the positive- x di-rection. The impact parameter vector is set parallel tothe positive- y direction. A. Computational details We use our own computational code of TDHF calcu-lation for nuclear collisions, as in Ref. [5]. Our code uti-lizes a three-dimensional uniform-grid representation forsingle-particle wave functions without any symmetry re-strictions. The 11-point high-order finite difference for-mula is used for the spatial derivatives. For the timeevolution, we use fourth order Taylor expansion method.The spatial grid points of N x × N y × N z = 90 × × x direction. The initial wavefunctions of projectile and target nuclei are prepared ina box with N x × N y × N z = 40 × × 26 grid points. Wecalculate time evolution until a distance between the cen-ters of the PLF and the TLF exceeds 32 fm. For the PNPanalysis, integrals over θ are performed by employing thetrapezoidal rule discretizing the interval [0 , π ] into M equal grids. We find that M = 30 is sufficient for the O+ O system. All the results reported here are cal-culated using the Skyrme SLyIII.0.8 parameter set [22]. B. Ground states We calculate ground states of O and O nuclei,which are both spherical in the self-consistent solutions.Figure 1 shows single-particle energies of neutrons (redsolid lines) and protons (green dotted lines) in O in -45-40-35-30-25-20-15-10-50 S i ng l e - p a r ti c l e e n e r gy ( M e V ) (a) Neutron Proton O s d p p s p p s (b) Neutron Proton O s d p p s s d p p s FIG. 1. (Color online) Single-particle energies of occupiedorbitals for neutrons (thick red solid lines) and protons (thickgreen dotted lines) in O and O are shown in the panels (a)and (b), respectively. Single-particle energies of unoccupiedorbitals are also shown by thin dotted lines. panel (a) and in O in panel (b). Occupied orbitalsare shown by thick lines, while unoccupied orbitals areshown by thin lines. As recognized from the figure, thereare neutron orbitals characterized by small binding ener-gies in neutron-rich O nucleus. All proton orbitals in O are deeply bound. C. Reaction dynamics We first provide an overview of the reaction dynamicsin O+ O collisions. In Fig. 2, the deflection angle Θ inthe center-of-mass frame and the total kinetic energy loss(TKEL) are shown in the panels (a) and (b), respectively,as functions of the distance of closest approach, d . Weevaluate Θ and TKEL from the momenta of two fragmentnuclei and the Coulomb energy between them at the finalstage of the TDHF calculation where two nuclei are wellseparated.We employ the distance of closest approach d , insteadof the impact parameter b . They are related by d = Z P Z T e E + r(cid:16) Z P Z T e E (cid:17) + b , (26)where E denotes the incident relative energy. Z P and Z T denote the proton numbers of the projectile and thetarget, respectively. We consider it is useful to use d , be-cause transfer reactions take place at around the distanceof closest approach. For head-on collisions, calculatedresults are indicated by b = 0 and are plotted against d which is related to the incident relative energy E by d = Z P Z T e /E .We find the fusion reaction takes place at d = 9 . b = 0) which corresponds to theincident energy of E lab ∼ . E lab = 2, 4, and8 MeV/nucleon, the fusion reaction is found to take placeat d = 8 . 7, 8.3, and 7.5 fm, respectively.The deflection angle is positive for reactions at the inci-dent energy of 2 MeV/nucleon due to the Coulomb repul- -100-75-50-25 0 25 50 Q ( d e g ) (a) 7 8 9 10 11 12 T K EL ( M e V ) d (fm) (b) E =8 E =4 E =2 b =0 FIG. 2. (Color online) Deflection angle Θ in the center-of-mass frame (a) and total kinetic energy loss (b) are shown asfunctions of the distance of closest approach, d = d ( E, b ). sion, as seen in Fig. 2 (a). As the distance of closest ap-proach decreases, the nuclear attractive interaction actsto decrease the deflection angle. It becomes negative for d < d region where we observed negative deflection angles. D. Transfer probability In Fig 3, we show transfer probabilities calculated us-ing Eq. (21). Red circles show probabilities for head-oncollisions ( b = 0) with several values of d . Green trian-gles, blue squares, and purple diamonds show probabil-ities as functions of d for incident energies E lab = 2, 4,and 8 MeV/nucleon, respectively.In the calculations, we adopted two choices for the spa-tial region V . For the probabilities observing a PLF,which are shown in the right panels of Fig. 3, we adopteda sphere with a radius of 16 fm around the PLF for thespatial region V . For the probabilities observing a TLFshown in the left panels of Fig. 3, a sphere with a radiusof 16 fm around the TLF is used. We have confirmedthat obtained results are almost independent of the cho-sen radius R of the spatial region V , if R is taken in therange of 15 fm < R < 20 fm. We will use this radius forevaluation of expectation values of angular momentumand excitation energies.Figure 3 (a) and (b) show probabilities of one-protontransfer processes, while (c) and (d) show probabilitiesof two-proton transfer processes. We note that, from theabove choices of V for the PLF and the TLF, the prob-abilities of proton removal from O ((a) and (c)) shouldbe coincide with the probabilities of proton addition to O ((b) and (d)), if the breakup processes can be ne-glected. As seen from the figure, (a) and (b) are veryclose to each other, indicating that the breakup processes -4 -3 -2 -1 7 8 9 10 11 12 P N , Z d (fm) Target-like fragment (a) N (-1 p , 0 n ) -4 -3 -2 -1 7 8 9 10 11 12 d (fm) Projectile-like fragment (b) F (+1 p , 0 n ) -8 -7 -6 -5 -4 -3 -2 -1 7 8 9 10 11 12 P N , Z d (fm)(c) C (-2 p , 0 n ) -8 -7 -6 -5 -4 -3 -2 -1 7 8 9 10 11 12 d (fm)(d) Ne (+2 p , 0 n ) -4 -3 -2 -1 7 8 9 10 11 12 P N , Z d (fm)(e) O (0 p , +1 n ) -4 -3 -2 -1 7 8 9 10 11 12 d (fm)(f) O (0 p , -1 n ) -8 -7 -6 -5 -4 -3 -2 -1 7 8 9 10 11 12 P N , Z d (fm)(g) O (0 p , +2 n ) -8 -7 -6 -5 -4 -3 -2 -1 7 8 9 10 11 12 d (fm)(h) O (0 p , -2 n ) -3 -2 -1 7 8 9 10 11 12 P N , Z d (fm)(i) O (0 p , 0 n ) -3 -2 -1 7 8 9 10 11 12 d (fm)(j) E =8 E =4 E =2 b =0 O (0 p , 0 n ) FIG. 3. (Color online) Transfer probabilities with respect tothe TLF (left) and the PLF (right) are shown as functions ofthe distance of closest approach, d = d ( E, b ). are indeed negligible. We also find that (c) and (d) areclose to each other. On the other hand, in the case ofneutron transfer channels, one-neutron transfer in panels(e) and (f) and two-neutron transfer in panels (g) and(h), we find that the probability of neutron removal from O is much larger than that of neutron addition to O,especially for reactions at E lab = 8 MeV/nucleon. Thisfact indicates that there are substantial probabilities ofbreakup processes for neutrons.In Fig. 3, we find that transfer probabilities decreaseas the incident energy increases. Comparing probabil-ities of neutron and proton transfer processes, neutrontransfer probabilities are much larger than proton trans-fer probabilities at the same distance of closest approach and the same incident energy. We also find that the slopeof probabilities for protons against the distance of closestapproach is much steeper than that for neutrons. Thesefeatures are consistent with orbital energies of the twocolliding nuclei in their ground states which are shown inFig. 1. Since there are neutrons bound weakly in O,transfer probabilities of neutrons are much larger thanthose of protons. Since these weekly bound neutrons arespatially extended in O, we find a long tail of neutrontransfer probabilities.At the highest incident energy of E lab =8 MeV/nucleon, the proton transfer probability ismaximum around d = 8 fm. The probability decreasesas the distance of closest approach decreases. Thedecrease at the small- d region indicates the increase ofprobabilities for other channels with transfers of a largernumber of protons. E. Angular momentum In this subsection, we investigate expectation valuesof the angular momentum operator in the fragment nu-clei. We will use the same definition for V as that inthe previous subsection, spheres with a radius of 16 fmaround the center-of-mass of the PLF and the TLF. Weconsider the angular momentum operator in the spatialregion V , ˆ J V = ˆ J ( n ) V + ˆ J ( p ) V . The operator ˆ J ( q ) V denotesthe angular momentum operator for neutrons ( q = n )and for protons ( q = p ) in the spatial region V , given byˆ J ( q ) V = P i ∈ q Θ V ( r i )ˆ j i = P i ∈ q Θ V ( r i ) (cid:2) (ˆ r i − R µ ) × ˆ p i +ˆ s i (cid:3) . R µ is the center-of-mass coordinate of the fragment( µ = PLF or TLF).Figure 4 shows expectation values of the angular mo-mentum operators in the PLF and the TLF composedof specific numbers of neutrons and protons. A com-ponent perpendicular to the collision plane is shown.Left panels show expectation values in the TLF, whileright panels show those in the PLF. For reactions at E lab = 8 MeV/nucleon, expectation values at the small- d region, d < d region ( d > O or O. This fact may be under-stood from properties of orbitals. For O, orbitals ofthe smallest binding energy are 1 p / for both neutrons J z , N , Z ( h ) d (fm) Target-like fragment (a) - N (-1 p , 0 n ) d (fm) Projectile-like fragment (b) F (+1 p , 0 n ) J z , N , Z ( h ) d (fm) (c) - C (-2 p , 0 n ) d (fm) (d) Ne (+2 p , 0 n ) J z , N , Z ( h ) d (fm) (e) - O (0 p , +1 n ) d (fm) (f) O (0 p , -1 n ) J z , N , Z ( h ) d (fm) (g) - O (0 p , +2 n ) d (fm) (h) O (0 p , -2 n ) J z , N , Z ( h ) d (fm) (i) - O (0 p , 0 n ) d (fm) (j) E =8 E =4 E =2 O (0 p , 0 n ) FIG. 4. (Color online) Expectation values of the angular mo-mentum operator for fragment nuclei in each transfer channelare shown as functions of the distance of closest approach, d = d ( E, b ). and protons. For O, they are 2 s / for neutrons and1 p / for protons. We thus find that the orbitals of thesmallest binding energy are characterized by small angu-lar momenta. Since nucleon removals from spatially ex-tending single-particle orbitals are expected to take placefor orbitals with the smallest binding energy, removal ofnucleons from these orbitals may not leave large valuesof angular momentum in nucleon removed nuclei.In nucleon addition channels ((b), (d), (e), and (g)),we find finite positive values of angular momentum in allchannels. The expectation values increase as the inci-dent energy increases. They do not depend much on the l z ( h ) d (fm) - E =8 E =4 E =2 FIG. 5. (Color online) The angular momentum carried into O by an added nucleon evaluated by Eqs. (27) and (28)is shown as a function of the distance of closest approach, d = d ( E, b ). distance of closest approach d . These features may beunderstood by the following intuitive considerations. Letus consider a transfer of one nucleon from O to O.We assume that the nucleon transfer takes place whentwo nuclei are at the distance of closest approach. Ignor-ing the interaction potential by nuclear force, the relativevelocity of two nuclei is approximately given by v rel = s µ (cid:18) E − Z P Z T e d (cid:19) , (27)where E and µ denote the incident relative energy andthe reduced mass, respectively. In the rest frame of Onucleus, we assume that the transferred nucleon has thesame velocity as the relative velocity v rel , ignoring theinternal motion in O. This may be reasonable, since weobserved very small expectation values of the angular mo-mentum in nucleon removed fragments, as seen in Fig. 4(a), (c), (f), and (h). If the transferred nucleon staysat the surface of O, the transferred nucleon brings theangular momentum, l z = Rmv rel , (28)into O, where m is the nucleon mass and R is theradius of O which we estimate by a simple formula, R = r A / , with r = 1 . A = 16.In Fig. 5, we show the angular momentum l z evaluatedusing Eqs. (27) and (28) as functions of the distance ofclosest approach d for several energies. The estimatedvalues of the angular momentum coincide quantitativelywith the calculated results in channels of one-neutron ad-dition to O, shown in Fig. 4 (e). The estimated angularmomentum depends little on the distance of closest ap-proach d , since the Coulomb potential in Eq. (27) givesonly a minor effect except for a case of very low incidentenergy. The angular momentum is roughly proportionalto the square root of the energy. In the case of two-nucleon transfer, the angular momentum carried into Ois given by twice of l z . This reasonably explains the ob-servation in the panel (g). E N , Z * ( M e V ) d (fm) Target-like fragment (a) N (-1 p , 0 n ) 7 8 9 10 11 12 d (fm) Projectile-like fragment (b) F (+1 p , 0 n ) E N , Z * ( M e V ) d (fm) (c) C (-2 p , 0 n ) 7 8 9 10 11 12 d (fm) (d) Ne (+2 p , 0 n ) E N , Z * ( M e V ) d (fm) (e) O (0 p , +1 n ) 7 8 9 10 11 12 d (fm) (f) O (0 p , -1 n ) E N , Z * ( M e V ) d (fm) (g) O (0 p , +2 n ) 7 8 9 10 11 12 d (fm) (h) O (0 p , -2 n ) E N , Z * ( M e V ) d (fm) (i) O (0 p , 0 n ) 7 8 9 10 11 12 d (fm) (j) E =8 E =4 E =2 b =0 O (0 p , 0 n ) FIG. 6. (Color online) Average excitation energies of fragmentnuclei in each transfer channel are shown as functions of thedistance of closest approach, d = d ( E, b ). F. Excitation energy In Fig. 6, we show excitation energies of fragment nu-clei evaluated using Eq. (25) as functions of the distanceof closest approach d . Left panels show the excitationenergies of the TLF, while right panels show the excita-tion energies of the PLF. As in previous figures, thereare two kinds of calculations: Red circles show resultsof head-on collisions ( b = 0) varying the incident en-ergy. Green triangles, blue squares, and purple diamondsshow results for fixed incident energies, E lab = 2, 4, and8 MeV/nucleon, respectively, changing the impact pa- rameter b .As we mentioned below Eq. (25), we take into accountthe center-of-mass correction in calculating energies offragment nuclei and reference energies of ground statesin Eq. (25), while we ignore it in the time evolution cal-culations. For the quasi-elastic channels without nucleontransfer, we find very small average excitation energiesat large- d region, d > d region, d < d region, d > O in (f) or one-proton re-moval from O in (a), the average excitation energy isless than 3 MeV. This indicates that the nucleon is re-moved dominantly from the highest occupied orbital. Intwo-nucleon removal channels ((c) and (h)), the excita-tion energy becomes somewhat large, about 5-10 MeV intwo-proton removal from O in (c). The excitation ener-gies after nucleon removal are almost independent of theincident energy. This suggests that nucleons are removedgently even at higher incident energies.Contrarily, in nucleon addition channels ((b), (d), (e),and (g)), we find that excitation energies depend muchon the incident energy. A similar feature was also seenin the angular momentum shown in Fig. 4, where theadded nucleon carries an angular momentum associatedwith the translational relative motion into the fragment.The expectation values of the angular momentum werealso found to increase as the incident energy increases.This fact may be related to the increase of the excitationenergies as the incident energy increases in nucleon addi-tion channels: The transferred nucleons must stay at or-bitals of higher angular momenta as the incident energyincreases. The energies of orbitals with higher angularmomenta are high.For nucleon addition channels ((b), (d), (e), and (g)),we observe an increase of excitation energies as the dis-tance of closest approach increases. One may considerthat this fact contradicts to an intuitive picture thatan excitation energy will be smaller as the distance ofclosest approach increases since two nuclei cannot collideviolently. We examine this behavior for head-on colli-sions ( b = 0). As shown by red circles in the panels (b),(d), (e), and (g), the excitation energies are very smallat d = 9 . O and Oare rather different because of the excess neutrons inneutron-rich O. When a nucleon is transferred at a largedistance of closest approach which is much larger thanthe sum of the radius of two colliding nuclei, the nucleontransfer is expected to take place between orbitals whichare close in energy. The energy-conserving transfer pro-cesses must cause excitations of produced fragments if aneutron-rich nuclei is included in the collision.Let us consider one-proton transfer from O to Oin head-on collisions, which are shown by red circles inthe panel (b). The transfer takes place dominantly fora proton in the highest occupied orbital of O, 1 p / at − . s / at − . Ois 1 p / at − . d / unoccupiedorbitals at − . d / orbitalsis occupied in the ground state of F, we expect theexcitation energy, E ∗ ∼ ε ( O; π s / ) − ε ( O; π d / ) =5 . F shown in the panel (b)at the large- d region.We next consider one-neutron transfer from O to Oin head-on collisions, which are shown by red circles inthe panel (e). The highest occupied neutron orbital in O is 2 s / at − . Oat a similar energy, 2 s / at − . O is 1 d / orbital at − . O,we expect the excitation energy, E ∗ ∼ ε ( O; ν s / ) − ε ( O; ν d / ) = 3 . O shownin the panel (e) at the large- d region.In the above considerations, we may understand thetransfer mechanism in terms of orbital properties in theground state: the highest occupied orbitals dominantlycontribute to the transfer process. We note that, inRef. [21], single-particle transfer dynamics in O+ Ocollision has been examined analyzing density contribu-tions from individual orbitals. The result reported inRef. [21] is consistent with the above conclusion.We make a final comment on an abrupt increase ofexcitation energy seen at the largest d value, 12 fm, andthe highest incident energy, 8 MeV/nucleon in panels (b)and (d). We consider that they are due to a numericalfailure. We note that probabilities of these processes arevery small, as confirmed in Fig. 3. IV. SUMMARY In the time-dependent Hartree-Fock theory, low-energyheavy ion collisions are described by a time evolution ofa single Slater-determinant wave function. At the finalstage of calculation, the wave function may be regardedas a superposition of a number of channels with differ-ent particle numbers and quantum states. To obtain de-tailed information on reaction products, projection op-erator techniques will be useful. In this paper, we pro-posed a method to calculate expectation values of oper-ators with the particle-number projection to investigateproperties of projectile- and target-like fragments aftercollision.To demonstrate usefulness of our method, we appliedthe method to one- and two-nucleon transfer processes in O+ O collisions. We analyzed expectation values ofthe angular momentum operator and average excitationenergies of produced nuclei. For fragment nuclei after nu-cleon removal, we found small values of angular momen-tum and excitation energy, suggesting a gentle removalof nucleons. For fragment nuclei with added nucleons,we found substantial expectation values of angular mo-mentum and average excitation energies. We have foundthat the expectation value of the angular momentum ofproduced nuclei is proportional to the relative velocity ofthe two colliding nuclei at the turning point. The excita-tion energy can be understood by a transfer of nucleonsbetween approximately degenerate orbitals of projectileand target nuclei.The above example clearly shows the usefulness of thepresent method for microscopic investigations of reactionmechanisms in heavy ion collisions. The formalism willalso be useful to estimate effects of particle evaporationafter multinucleon transfer processes, which are difficultto describe directly in the time-dependent Hartree-Fockcalculation because of the very long timescale of the evap-oration processes [23]. ACKNOWLEDGMENTS K.S. would like to thank K. Washiyama and S.A. Satofor valuable comments and discussions. K.S. greatly ap-preciates the organization of the TALENT course [1] C. Simenel, Eur. Phys. J. A , 152 (2012).[2] C. Simenel, Phys. Rev. Lett. , 192701 (2010).[3] M. Evers, M. Dasgupta, D.J. Hinde, D.H. Luong, R.Rafiei, R. du Rietz, and C. Simenel, Phys. Rev. C ,054614 (2011).[4] G. Scamps and D. Lacroix, Phys. Rev. C , 014605(2013).[5] K. Sekizawa and K. Yabana, Phys. Rev. C , 014614(2013).[6] L. Corradi, J.H. He, D. Ackermann, A.M. Stefanini, A.Pisent, S. Beghini, G. Montagnoli, F. Scarlassara, G.F.Segato, G. Pollarolo, C.H. Dasso, and A. Winther, Phys.Rev. C , 201 (1996).[7] L. Corradi, A.M. Stefanini, J.H. He, S. Beghini, G. Mon-tagnoli, F. Scarlassara, G.F. Segato, G. Pollarolo, andC.H. Dasso, Phys. Rev. C , 938 (1997).[8] L. Corradi, A.M. Vinodkumar, A.M. Stefanini, E.Fioretto, G. Prete, S. Beghini, G. Montagnoli, F. Scar-lassara, G. Pollarolo, F. Cerutti, and A. Winther, Phys.Rev. C , 024606 (2002).[9] S. Szilner, L. Corradi, G. Pollarolo, S. Beghini, B.R.Behera, E. Fioretto, A. Gadea, F. Haas, A. Latina, G.Montagnoli, F. Scarlassara, A.M. Stefanini, M. Trotta,A.M. Vinodkumar, and Y. Wu, Phys. Rev. C , 044610(2005).[10] A. Winther, Nucl. Phys. A572 , 191 (1994); A594 , 203 (1995).[11] E. Vigezzi and A. Winther, Ann. Phys. (N.Y.) , 432(1989).[12] V. Zagrebaev and W. Greiner, J. Phys. G , 825 (2005).[13] V. Zagrebaev and W. Greiner, J. Phys. G , 1 (2007).[14] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod.Phys. , 121 (2003).[15] S. Shinohara, H. Ohta, T. Nakatsukasa, and K. Yabana,Phys. Rev. C , 054315 (2006).[16] S. Hilaire and M. Girod, Eur. Phys. J. A , 72(2002).[18] K.-H. Kim, T. Otsuka, and M. Tohyama, Phys. Rev. C , R566 (1994).[19] K.-H. Kim, T. Otsuka, and P. Bonche, J. Phys. G ,1267 (1997).[20] A.S. Umar and V.E. Oberacker, Phys. Rev. C , 054607(2006).[21] A.S. Umar, V.E. Oberacker, and J.A. Maruhn, Eur.Phys. J. A , 245 (2008).[22] K. Washiyama, K. Bennaceur, B. Avez, M. Bender, P.-H. Heenen, and V. Hellemans, Phys. Rev. C86