Momentum Distribution of a Fragment and Nucleon Removal Cross Section in the Reaction of Halo Nuclei

Commun. Theor. Phys. Beijing, China) 40 2003) pp. 693 698 c International Academic Publishers Vol. 40, No. 6, December 5, 2003 Momentum Distribution of a ragment and Nucleon Removal Cross Section in the Reaction of Halo Nuclei ZHAO Yao-Lin, MA Zhong-Yu,,2,3 and CHEN Bao-Qiu,2 China Institute of Atomic Energy, P.O. Box 275-8, Beijing 0243, China 2 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China 3 Institute of Theoretical Physics, Academia Sinica, Beijing 00080, China Received April 0, 2003) Abstract Recently the research on the halo structure of drip-line nuclei has shown some interesting properties of the existence of one or more halo nucleons. In the framework of few-body Glauber model, the momentum distribution of a fragment and nucleon removal cross section in the reaction of halo nuclei is presented and extended to nuclei having more than one halo nucleons. The reaction mechanism is treated with and without taking account of the final-state interaction. The wave function of removal halo nucleons in the continuum state is modified by imposing an orthogonal condition to the bound state. An analytical expression of the longitudinal momentum distribution of the fragment is derived when the bound state wave function of halo nucleons is taken as a Gaussian-type function. This is useful in the further investigation on the structure of halo nuclei. PACS numbers: 25.60.Gc, 25.70.Mn, 24.0.-i Key words: halo nuclei scattering, momentum distribution, Glauber theory Introduction The investigation on the halo structure of nuclei is one of the most important subject and has attracted much attention both in the experiment and theory. [] The peculiar properties of halo nuclei are such that the halo nucleons have very low separation energies and the density distribution of nucleus is largely dispersed. This causes large increase of the reaction and nucleon removal cross section in the halo nucleus scattering, and a very narrow momentum distribution of the fragment compared with those for stable isotopes or isotones. [2,3] Up to now, the experimental observations of the halo structure are all restricted to those nuclei with one or two halo nucleons. However the recent theoretical investigation predicts that some neutron-rich nuclei might have a halo nucleon cluster. or example, the relativistic mean-field RM) theory proposed that 26 O might be a halo nucleus with four halo neutrons, and 28 O with six halo neutrons. [4] In addition, the relativistic Hartree ork approach RH) [5] and relativistic Hartree Bogoliubov RHB) [6] model have also proposed the existence of a halo nucleon cluster. At present, the experiment search for the halo nucleon cluster is being carried out actively. The exploration of halo neutron clusters in oxygen isotopes is recently performed at GANIL. [7] It is well known that a Glauber model analysis on reaction cross sections of the nucleus-nucleus scattering is an important and feasible tool to study the nuclear size. The optical limit OL) Glauber theory has been successfully applied to account for the reaction cross section data of stable nucleus-nucleus scattering, [8,9] but it failed to describe the reaction data for halo nuclei. And also in the OL Glauber model the momentum distribution of a fragment and nucleon removal cross section for the halo nucleus scattering cannot be analyzed. Recently Ogawa [0] has proposed a few-body B) Glauber model. It takes account of the intrinsic few-body characteristics of halo nuclei implying strong spatial correlations among the constituents, which results in a smaller calculated reaction cross section than those obtained in the Glauber analysis with the optical limit approximation. [] This has a significant implication for the deduced size and ground state structure of the halo nuclei. In the case of stable nuclei, the momentum distribution of the core fragment in high energy collision reflects the nuclear ermi motion [2] of the single nucleon in the nuclear surface and does not much depend on the nuclei involved. [3] The nucleon removal cross section is very small due to a large single nucleon separation energy. However for the halo nuclei, it is quite different. The momentum distribution of the fragment and nucleon removal cross section are very important observables, which can provide a direct information about the configurations of the ground state of halo nuclei. The width of momentum distribution is very narrow, and the removal cross section is largely increased. These are all consistent with the characteristics of halo nuclei: a spatially extended density distribution and low single nucleon separation energy. The project supported by National Natural Science oundation of China under Grant Nos. 0075080, 075092, and 0275094, and the State Key Basic Research Development Program of China under Contract No. G2000077400

694 ZHAO Yao-Lin, MA Zhong-Yu, and CHEN Bao-Qiu Vol. 40 In this paper, both the momentum distribution of the fragment and nucleon removal cross section are presented within the B Glauber. Since most previous theoretical description is restricted to the halo nucleus scattering with only one or two halo nucleons, we extend the formalism to describe the scattering of halo nuclei with a halo nucleon cluster. The few-body characteristics of halo nuclei are carefully taken into account. The reaction mechanism is treated with and without taking account of the final-state interaction. The wave function of removal halo nucleons in the continuum state is modified by imposing an orthogonal condition to the bound state when the final-state interaction is taken into account. An analytical expression of the momentum distribution of the fragment can be obtained if the density distribution of halo nucleons in a nucleus is taken as a Gaussian-type function. The paper is arranged as follows. A general formalism of the momentum distribution of a fragment and nucleon removal cross section in the halo nucleus-nucleus scattering is derived in the framework of the B Glauber theory, which is given in Sec. 2. A simple analytic expression of the longitudinal momentum distribution of a fragment is obtained if Gaussian-type wave functions are adopted, presented in Sec. 3. In the final section we give a brief summary. 2 ormalism The momentum distribution of a fragment after the nucleus-nucleus scattering and nucleon removal cross section are derived in the framework of the B Glauber model. We assume that the halo nucleus has m halo nucleons. m could be equal to one, two, or more. The reaction process is described in the project rest frame as P 0, ψ 0 ) + T K, θ 0 ) k = q ) ki, φ 0 + mn k, k 2,..., k m ) + T K q, θ β ), ) where P and T indicate projectile and target, respectively. At the initial stage of the reaction the projectile with the intrinsic ground state wave function ψ 0 is at rest in the projectile rest frame, while the target nucleus with the intrinsic ground state wave function θ 0 approaches the interaction region with the momentum h K. At the final stage of reaction the projectile breaks up into a core and m halo nucleons. The core stands in its ground state φ 0 with momentum h k, where the contribution of its bound excited states is neglected. The m nucleons stand in continuum states with asymptotic momenta h k,..., h k m. h q is the momentum transfer and is perpendicular to incident direction, which is shared by the core and halo nucleons. The target nucleus receives the momentum transfer h q and excites to the state specified by the wave function θ β. Considering the few-body characteristic of halo nuclei, we assume that its ground state and final break-up wave function are given by ψ 0 = ϕ 0 φ 0 and ϕ k,..., k m φ 0, where ϕ 0 and ϕ k,..., k m represent the wave functions of halo nucleons standing in their bound and continuum states, respectively. Therefore in the B Glauber theory the scattering amplitude for the reaction process can be written as ik k,..., k mβ q) = d 2 b exp i q b ) ϕ k,..., 2π k m φ 0 θ β [ Γ ij b i t j )] ϕ 0 φ 0 θ 0, 2) where b is the impact parameter and Γ ij b i t j ) is the profile function of nucleon-nucleon scattering. r i = s i + z i z is the coordinate of the i-th nucleon in the projectile with respect to the core mass center. Among them r,..., r m are coordinates of m halo nucleons. η j = t j + z j z is the coordinate of the j-th nucleon in the target, which is in respect to the target mass center. b, s i, and t j lie in a plane perpendicular to the incident direction. b = b A s i, and A is the mass number of the projectile. The momentum distribution of the core fragment is described by the cross section for the process of observing the core with the momentum k. It could be obtained by integrating the norm of the scattering amplitude over the momentum transfer q, asymptotic momentums k,..., k m of m halo nucleons, and summing over all possible final states of the target, d 3 = k β = 2π) 2 k 2 d2 qd 3 k d 3 k m δ k + d 2 qd 3 k d 3 k m δ k + θ 0 φ 0 ϕ 0 i P j T ) ki q k,..., k mβ q ) 2 ki q) d 2 bd 2 b exp[ i q b b )] [ Γ ij b i t j )] φ 0 ϕ k,..., k m

No. 6 Momentum Distribution of a ragment and Nucleon Removal Cross Section in the Reaction of Halo Nuclei 695 φ 0 ϕ k,..., k m [ Γ ij b i t j )] φ 0 ϕ 0 θ 0, 3) where δ k + m ) ki q is a δ function and indicates the momentum conservation in the reaction process. Therefore the integration of Eq. 3) over the core fragment momentum k will provide the m halo nucleons removal cross section, σ mn = d 3 k d 3. 4) k The momentum distribution in Eq. 3) might be divided into two parts, depending on the final state of the target. ) el ) inel d 3 = +. 5) k d 3 k d 3 k We call the first term as the momentum distribution of elastic dissociation, where the target remains in its ground state β = 0) after the scattering. The second term corresponds to inelastic dissociation, where the target is excited to its excited state β 0). Usually at low energies, the elastic dissociation is comparable to or even exceeds the inelastic dissociation. The momentum transfer is rather small in the process. Therefore one has to consider the final-state interaction during the analysis. On the contrary, the inelastic dissociation is the dominant process at high energies. The final-state interaction can be neglected. In the following derivation, the momentum distribution is treated in two approaches with and without considering the final-state interaction. Same cases also exist for the nucleon removal cross section. 2. Neglecting of the inal-state Interaction irst we discuss a simple form of the momentum distribution, where the final-state interaction is neglected. The continuum wave function of m halo nucleons ϕ k,..., k m is approximated by the plane wave function of free nucleons ϕ k,..., k m r,..., r m ) = 2π) exp m i ) k 3m/2 i r i. 6) Here the wave functions ϕ k,..., k m r,..., r m ) and ϕ 0 r,..., r m ) are not orthogonal to each other. In this approximation the momentum distribution of Eq. 3) can be written as d 3 = k 2π) 2 d 2 qd 3 k d 3 k m δ k + ki q) d 2 bd 2 b exp[ i q b b )] d 3 r d 3 r mϕ 0 r,..., r m)ϕ k,..., k m r,..., r m) d 3 r d 3 r m ϕ k,..., k m r,..., r m )ϕ 0 r,..., r m )D n b, s,..., s m; b, s,..., s m ), 7) where D n is a distorting function, which contains the whole information concerning the reaction dynamics. It is expressed as D n b, s,, s m; b, s,..., s m ) = φ 0 θ 0 [ Γ b i t j )] φ 0 θ 0 φ 0 θ 0 [ Γ b i t j )] φ 0 θ 0 + φ 0 θ 0 [ Γ b i t j )] φ 0 φ 0 [ Γ b i t j )] φ 0 θ 0. 8) Those matrix elements can be expressed by phase-shift functions, which have been derived in Ref. [], φ 0 θ 0 [ Γ b i t j )] φ 0 θ 0 = exp{iχ T b) + iχnt b ) + + iχ NT b m )}, 9) φ 0 θ 0 [ Γ b i t j )] φ 0 φ 0 [ Γ b i t j )] φ 0 θ 0 = exp{ iχ T b ) iχ NT b ) iχ NT b m ) + iχ T b) + iχnt b ) + + iχ NT b m )

696 ZHAO Yao-Lin, MA Zhong-Yu, and CHEN Bao-Qiu Vol. 40 + χ NNT b, b ) + + χ NNT b m, b m ) + χ T b, b)}. 0) The optical phase-shift functions χ T b) and χnt b i ) represent for the core and the i-th halo nucleon scattering with the target, respectively. And χ NNT b i, b i ) and χ T b, b) are cross terms. Substituting the wave function ϕ k,..., k m r,..., r m ) into Eq. 7) and then integrating over the momentum transfer q, one can obtain, d 3 k d 3 k m d 3 r d 3 r m d 3 r d 3 r m d 2 bd 2 b ϕ 0 r,..., r m)ϕ 0 r,..., r m ) d 3 k = 2π) 3m+2 [ exp i k + ki ) b ] b ) exp [ m i ] k i r i r i ) D n b, s,..., s m; b, s,..., s m ). ) Here one should note that k + m ki is a vector located in a plane perpendicular to the incident direction, although each of the momentums is a three-dimensional vector. We define k + m ki = p, thus k = p k m ki. The integration with respect to k, therefore to p will lead to a two-dimensional δ function δ 2 b b s ). The other integrations over k 2,..., k m will give m three-dimensional δ functions δ 3 r i r i + r r ) and i, which would remove all integration over r i, i. Thus the momentum distribution of the core fragment could be written as d 3 = k 2π) 3 d 3 r d 3 r d 3 r m ϕ 0 r,..., r m)ϕ 0 r,..., r m ) exp[ i k r r )] d 2 bd n b, s,..., s m; b, s,..., s m ), 2) where r i = r i + r r and b = b s. The m halo nucleon removal cross section can be obtained by integrating Eq. 2) with respect to k, σ mn = d 3 r d 3 r m ϕ 0 r,..., r m ) 2 d 2 bd n b, s,..., s m ; b, s,..., s m ). 3) The longitudinal momentum distribution of the core fragment can be obtained by integrating over k, dσ = d 2 k d 3 σ dk d 3 = d 3 r d 3 r m W r,..., r m, k ) d 2 bd n k b, s,..., s m ; b, s,..., s m ). 4) W r,..., r m, k ) is no more than the Wigner transform to the wave function of halo nucleons. W r,..., r m, k ) = dz 2π ϕ 0 r, r 2 + r r,..., r m + r r )ϕ 0 r,..., r m ) exp[ ik z z )], 5) where s = s, r i = r i + r r = s i, z i + z z ). 2.2 inal-state Interaction The final-state interaction might not be neglected at low energies, where the momentum transfer is rather small in the elastic dissociation process. The wave function of m halo nucleons in the continuum state ϕ k,..., k m may be modified by imposing an orthogonal condition ϕ k,..., k m = ϕ γ ϕ 0 ϕ 0 ϕ γ, 6) ϕ γ r,..., r m ) = 2π) exp m 3m/2 i=2 i k i r i ). 7) Here ϕ γ is a plane wave function of free nucleons. The modified wave function ϕ k,..., k m is therefore orthogonal to the bound state wave function ϕ 0 of halo nucleons, and also satisfies the completeness relation. Based on the definition of the momentum distribution in the elastic dissociation, it could be expressed as ) el = d 3 k 2π) 2 d 2 qd 3 k d 3 k m δ k + ki q) d 2 bd 2 b exp[ i q b b )] { φ 0 ϕ 0 θ 0 i P j T [ Γ b } i t j)] φ 0 θ 0 [ ϕ γ ϕ 0 ϕ 0 ϕ γ ]

No. 6 Momentum Distribution of a ragment and Nucleon Removal Cross Section in the Reaction of Halo Nuclei 697 { [ ϕ γ ϕ γ ϕ 0 ϕ 0 ] φ 0 θ 0 i P j T } [ Γ b i t j )] φ 0 ϕ 0 θ 0. 8) Due to the orthogonal condition, it brings some additional terms and each term can be derived similarly to before. Thus one can get the final expression of longitudinal momentum distribution, dσ ) el = d 3 r dk d 3 r m W r,..., r m, k ) d 2 bd el b, s,..., s m ; b, s,..., s m ) 2 d 3 r d 3 r m ϕ 0 r,..., r m ) 2 d 3 r d 3 r mw r,..., r m, k ) d 2 b Re D el b, s,..., s m; b, s,..., s m ) + ρk ) d 3 r d 3 r m ϕ 0 r,..., r m) 2 d 3 r d 3 r m ϕ 0 r,..., r m ) 2 d 2 bd el b, s,..., s m; b, s,..., s m ), 9) where D el is the distorting function of elastic dissociation, and defined by D el b, s,..., s m; b, s,..., s m ) = φ 0 θ 0 [ Γ b i t j)] φ 0 θ 0 φ 0 θ 0 [ Γ b i t j )] φ 0 θ 0 = exp{ iχ T b ) iχ NT b ) iχ NT b m ) + iχ T b) + iχnt b ) + + iχ NT b m )}. 20) ρk ) is the probability density of the core in the momentum space, ρk ) = d 3 r d 3 r m W r,..., r m, k ). 2) In a similar way, the nucleon removal cross section to the elastic dissociation process can also be obtained by integrating Eq. 9) over k. σ el mn = d 3 r d 3 r m ϕ 0 r,..., r m ) 2 d 2 bd el b, s,..., s m ; b, s,..., s m ) d 3 r d 3 r m ϕ 0 r,..., r m) 2 d 3 r d 3 r m ϕ 0 r,..., r m ) 2 d 2 bd el b, s,..., s m; b, s,..., s m ). 22) Obviously, when the final-state interaction is taken into account, the calculation of momentum distribution or removal cross section becomes complicated. 3 Analytical Expression of the Longitudinal Momentum Distribution It is found that the longitudinal momentum distribution in the expression of Eq. 4) or Eq. 9) is only a function of the longitudinal coordinates associated with the halo nucleon wave functions, while the distorting function is related with the impact parameter and transverse coordinates of halo nucleons. Therefor when the bound state wave function of halo nucleons ϕ 0 r,..., r m ) is chosen as a separable form, for instance as a Gaussian-type function, we can disentangle the longitudinal momentum distribution of the fragment by integrating the longitudinal coordinates. Assume a Gaussian distribution of the halo nucleons, ϕ 0 r,..., r m ) = m µ exp r2 i ν 2 ) = m ) ) µ exp s2 i ν 2 exp z2 i ν 2. 23) Substituting the above expressions into Eq. 5), one can get an analytic expression for the Wigner transform of the halo nucleon wave function and, dz dz m W r,..., r m, k ) = 2 µ2m υ m+ π ) m exp [ νk ) 2 ] m [ ] exp 2s2 i 2 m 2m ν 2, 24) dz dz m ϕ 0 r,..., r m ) 2 = µ 2m ν m π 2 ) m m [ ] exp 2s2 i ν 2. 25)

698 ZHAO Yao-Lin, MA Zhong-Yu, and CHEN Bao-Qiu Vol. 40 The probability density ρ k ) is expressed as ρk ) = 2 2π)m 2m ν3m+ π ) m µ 4 m exp 2 m [ νk ) 2 2m ]. 26) With those expressions Eqs. 24) 26) the longitudinal momentum distribution of the fragments after the removal of m halo nucleon is obtained straightforward, dσ ) el, dσ exp [ νk ) 2 ]. 27) dk dk 2m The remaining integrations over the impact parameter b and transverse coordinates of halo nucleons s,..., s m or s,..., s m contribute a constant to them. One can easily find the width of the longitudinal momentum distribution of the fragment, = 2 ln 4 m. 28) k ν It is shown that the width is directly related to the parameter ν in the Gaussian distribution of halo nucleons. The larger ν is, which reflects a disperse density distribution of halo nucleon with a long tail, the narrower width of the momentum distribution is. On the contrary, with the increase of the number of halo nucleons, the width would be increased. 4 Summary In this paper, a theoretical description of the momentum distribution and nucleon removal cross section of halo nuclei scattering in the framework of the B Glauber model is presented. We extend the approach to the halo nuclei with more than one halo nucleons. In this approach, the few body characteristic of halo nuclei and the reaction mechanism are carefully taken into account. The reaction mechanism is treated with and without considering the final-state interaction, respectively. And the wave function of halo nucleons standing in the continuum state is taken as a modified free particle wave function by imposing an orthogonal condition. The structure information of the halo nuclear ground state is involved in the Winger function W. When the bound state wave function of halo nucleons is taken as a Gaussian-type function, a simple analysis shows that the width of the longitudinal momentum distribution of the fragment would become narrow as the tail of the halo nucleon wave function becomes longer. Therefore the experimental measurements on the reaction cross section, momentum distribution of the fragment, and nucleon removal cross section in the halo nucleus-nucleus scattering require one to understand the halo structures. The B Glauber model is one of the most powerful theoretical tools to simultaneously analyze those experimental data. The further investigation on the experimental data of the halo nucleus-nucleus scattering is in progress. References [] P.G. Hansen., Nucl. Phys. A533 993) 89c. [2] A. Ozawa, O. Bochkarev, L. Chulkov, et al., Nucl. Phys. A69 200) 599. [3] E. Sauvan,. Carstoiu, N.A. Orr, et al., Phys. Lett. B49 2000). [4] Z.Z. Ren, W. Mittig, B.Q. Chen, and Z.Y. Ma, Phys. Rev. C52 995) R20. [5] B.Q. Chen, Z.Y. Ma,. Grümmer, and S. Krewald, Phys. Lett. B455 999) 3. [6] J. Meng, H. Toki, J.Y. Zeng, et al., Phys. Rev. C65 2002) 04302R). [7] O. Tarasov, R. Allatt, J.C. Angélique, et al., Phys. Lett. B409 997) 64. [8] S.K. Charagi and S.K. Gupta, Phys. Rev. C4 990) 60; ibid. C46 992) 982. [9] ZHAO Yao-Lin, MA Zhong-Yu, and CHEN Bao-Qiu, Commun. Theor. Phys. Beijing, China) 36 200) 33. [0] Y. Ogawa, K. Yabana, and Y. Suzuki, Nucl. Phys. A543 992) 723. [] Zhao Yao-Lin, Ma Zhong-Yu, Chen Bao-Qiu, and Sun Xiu-Quan, High Energy Phys. & Nucl. Phys. 25 200) 506 in Chinese). [2] A.S. Goldhaber, Phys. Lett. B53 974) 306. [3] D.E. Greiner, P.J. Lindstrom, H.H. Heckman, et al., Phys. Rev. Lett. 35 975) 52.