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Giant resonances in exotic nuclei
Nuclear Physics A 693 (2001) 448–462 www.elsevier.com/locate/npe
Giant resonances in exotic nuclei H. Sagawa a,∗ , H. Esbensen b a Center for Mathematical Sciences, University of Aizu, Aizu-Wakamatsu, Fukushima, Japan b Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
Received 29 November 2000; accepted 30 January 2001
Abstract We discuss theoretical studies of the Giant Dipole (GDR) and Giant Quadrupole (GQR) Resonances in nuclei far from stability, based on self-consistent Hartree–Fock and RPA calculations. The soft dipole mode that has been observed in light halo nuclei is predicted to diminish in skin nuclei, due to couplings to the isovector GDR. However, some low-lying dipole strength is predicted to exist in the isoscalar channel as a compression mode. The isoscalar GQR is expected to be much broader in skin nuclei than in stable nuclei, and substantial strength is predicted below the isoscalar GQR. The isovector GQR strength is spread over a wide range of excitation energies, both in skin and stable nuclei. 2001 Elsevier Science B.V. All rights reserved. PACS: 24.30.Cz; 25.70.De; 21.60.Jz Keywords: Giant resonances; Coulomb excitations; Halo nuclei; Skin nuclei
1. Introduction The experimental study of giant resonances in nuclei far from stability is still a rather unexplored territory. The most exciting development in the past decade is the observation of a large E1 strength just above threshold in light dripline nuclei. The large dipole strength has been revealed in breakup reactions on high-Z targets by observing the core like fragment [1]. The large cross sections that have been observed can only be explained by a large contribution from Coulomb dissociation. By measuring the momenta of the emitted particles it has been possible to construct the decay-energy spectrum, i.e., the excitation energy spectrum of a dripline nucleus prior to its decay. Such measurements have been performed for 6 He [2], 8 He [3] 11 Li [4], 11 Be [5], and 19 C [6], which are all neutron halo nuclei, and for 8 B [7–9], which has a proton halo. A large low-lying dipole strength is unusual because it does not ordinarily appear in stable nuclei. Some weak strength may appear and it is often referred to as Pigmy * Corresponding author.
E-mail address: [email protected] (H. Sagawa). 0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 0 6 4 9 - 2
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resonances [10,11]. Coulomb dissociation of a halo nucleus into a nucleon and a core fragment can be viewed as an absorption of virtual photons generated by the charge-current density of the target nucleus [12]. This process is closely related, via detailed balance, to the inverse reaction of radiative nucleon capture on the core nucleus [13]. The relationship that exists between the two processes makes Coulomb dissociation a powerful tool to infer radiative capture rates in cases where direct capture measurements are difficult or impossible, as for example when the core nucleus is short-lived. We discuss in Section 2 how the 7 Be + p → 8 B radiative capture rate can be determined from 8 B → 7 Be + p Coulomb dissociation measurements. An interesting question is why a large low-lying dipole strength may exist in dripline nuclei, and why it tends to disappear as one moves towards stability. Detailed experimental information about this trend is not yet available but some theoretical studies have been performed, based on self-consistent Hartree–Fock (HF) calculations combined with the random-phase approximation (RPA) [14,15]. In Section 3, we discuss this method and the results that have been obtained. Another application of Coulomb excitation is the extensive study of low-lying quadrupole transitions in exotic nuclei, which is discussed by Glasmacher [16]. Experimental studies of giant resonance excitations in nuclei far from stability have just started [11] and detailed results are not yet available. In the mean time, RPA calculations may provide some guidance for what to expect and what to look for. This paper is organized as follows. In Section 2 we discuss how the dipole response of dripline nuclei can be extracted from Coulomb dissociation experiments. We illustrate this method for 8 B and discuss how the results can be used to infer the rate of the inverse capture reaction. In Section 3 we describe the RPA Green’s function method in coordinate space taking into account the coupling to the continuum. The results for the dipole and quadrupole responses are also presented in Section 3. Discussions and a summary will be given in Section 4.
2. Coulomb dissociation of halo nuclei Here we consider a semiclassical description of the Coulomb dissociation of halo nuclei. We assume for simplicity that the halo nucleus (Z, A) can be described as a two-body system, consisting of a valence nucleon (Zx , Ax = 1) and an inert core (Zc , Ac = A − 1). The Coulomb field from a target nucleus (ZT , AT ) acting on the core and the valence nucleon is Zx e Zc e + , (1) VCoul = ZT e Rt + 1 r Rt − A−1 r A
A
where r is the position of the valence nucleon with respect to the core, and Rt is the target position with respect to the halo nucleus. For distant collisions (i.e., Rt > r) one can use the multipole expansion
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VCoul =
∗ (R t ) 4πZT e Yλµ λµ
2λ + 1 Rtλ+1
Mλµ ,
where Mλµ = eλ r λ Yλµ ,
is the electric multipole operator and eλ is the effective multipole charge, A−1 λ −1 λ eλ = Z x e + Zc e . A A
(2)
(3)
The amplitude af i for exciting a halo nucleus, from the initial state |i to a final state |f , is calculated in first-order, time-dependent perturbation theory. It has the form 1 1 Fλµ f |Mλµ |i, (4) dt eiωt f |VCoul |i = af i = i h¯ i λµ
where h¯ ω = Ef − Ei is the excitation energy. The last expression in Eq. (4), which is valid for distant collisions, shows a nice separation into multipole matrix elements, which depend on the structure of the halo, and factors Fλµ , which are independent of the structure; they can be obtained by inserting the expansion (2) into Eq. (4). In a relativistic description one obtains for a straight-line trajectory, Rt = b + vt, the following expression [17] (ω/v)λ Gλµ ZT e 16π ωb Fλµ = i λ+µ √ Kµ , (5) γv h¯ vγ 2λ + 1 (λ + µ)!(λ − µ)! where γ = 1/ 1 − (v/c)2 . The factors Gλµ are equal to one in the nonrelativistic limit. They can be extracted from Ref. [17] for relativistic energies. For dipole and quadrupole excitations one finds that G10 = G20 = G2±2 = 1/γ , G1±1 = 1, and G2±1 = 12 (1 + 1/γ 2). The Coulomb dissociation of halo nuclei is usually dominated by dipole transitions. The p effective dipole charge is e1 = e(A − Z)/A for a proton, and e1n = −Ze/A for a neutron halo, cf. Eq. (3). The effective quadrupole charges are Z − 1 + (A − 1)2 Z e, e2n = 2 e. 2 A A The quadrupole charge of a neutron is small and quadrupole transitions can therefore often be neglected when analyzing Coulomb dissociation measurements of neutron halos [5,6]. For a proton halo, on the other hand, quadrupole transitions may play a significant role. p
e2 =
2.1. Single-particle response of halo nuclei To determine the multipole matrix element in Eq. (4) one needs a model to calculate the initial and final states. The simplest approach is to adopt a single-particle Hamiltonian, based on a Woods–Saxon plus spin–orbit interaction, and adjust it to reproduce the separation energy and other known properties of the halo. The initial state is of the form |i = u0 (r)/r|&0 l0 m0 . The final state f |, which has the asymptotic momentum k of the emitted valence nucleon with respect to the core fragment, is given by the sum
uk&j (r) f | = (−i)& eiδ&j (k) kˆ &j m &j mrˆ , (6) kr &j m
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Nuc is the total phase shift due to the Coulomb and nuclear where δ&j (k) = σ&Coul + δ&j interactions. Without final state interactions this final state would just be a plane wave, f | = exp(−ikr)/(2π)3/2 . The radial wave functions (6) must therefore be normalized so that 2 sin kr + δ&j (k) , for r → ∞. uk&j (r) → π Inserting the final state (6) into the multipole matrix element we obtain
1 iδ&j (k) ˆ k&j ||Mλ ||&0 j0 , (7) e k &j m j0 m0 λµ|j m f |Mλµ |i = √ k 2j + 1 &j m
where k&j ||Mλ ||&0 j0 is a reduced single-particle matrix element. The momentum distribution for the relative motion of the emitted nucleon and the core fragment is equal to the square of the excitation amplitude (4) at a given impact parameter. Integrating over the impact parameter range of interest, say b > bmin , we obtain the differential cross section 2 ∞ ∞ dσ CD 2 . 2πb db |a | = 2πb db F f |M |i (8) = f i λµ λµ d 3k bmin
bmin
λµ
The amplitudes for different multipole transitions are here added coherently. This leads to an interference, for example, between dipole and quadrupole transitions, which can produce an asymmetry in the momentum distribution. If we integrate over all orientations of the momentum k, and take the average over the initial magnetic quantum number m0 , we obtain by inserting Eq. (7) into Eq. (8) ∞ dσ CD 1 dB(Eλ) = 2πb db |Fλµ |2 , (9) dE 2λ + 1 dE λµ
bmin
where the multipole strength function is dB(Eλ) dk | k&j ||Mλ ||&0 j0 |2 = . dE dE 2j0 + 1
(10)
&j
The cross section (9) and the multipole strength functions (10) have here been converted into differentials in the relative kinetic energy of the two fragments, E = (h¯ k)2 /(2m0 ), where m0 is the reduced mass. Let us also mention that the integration over impact parameter can be performed analytically, as shown in Ref. [17]. 2.2. Radiative capture The cross section for the radiative capture, c + p → A + γ , is related to the photoabsorption cross section for the inverse reaction, A + γ → c + p, by detailed balance [13]. For a given electric multipole transition the relation is Eγ 2 2(2IA + 1) (γ ) (rc) σ . (11) σEλ = h¯ ck (2Ic + 1)(2Ip + 1) Eλ
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Here Eγ is the photon energy, and h¯ k is the momentum for the relative motion of c and p. The expression contains the ratio of the phase space factors for the photon and for the relative motion of c and p, and the statistical weights associated with the spins in the initial and final states. The factor of two accounts for the two possible polarizations of the photons emitted in the capture process. The photoabsorption cross section can be expressed in terms of the multipole response as follows [12] (2π)3 (λ + 1) Eγ 2λ−1 dB(Eλ) (γ ) . (12) σEλ = dEγ λ((2λ + 1)!!)2 hc ¯ The two expressions, Eqs. (11), (12), show that the radiative capture cross section for a given multipolarity is determined in a simple way by the multipole strength function dB(Eλ)/dE. The same strength function appears in the decay energy spectrum for Coulomb dissociation, Eq. (9). Thus, if one could extract this strength distribution from a measured decay energy spectrum, one would be able to predict the radiative capture cross section. 2.3. Applications to 8 B Here we illustrate how Coulomb dissociation experiments can be used to extract the dipole strength distribution of 8 B and infer the radiative capture rate of protons on 7 Be at low energies. The capture rate is a determining factor for the solar abundance of 8 B, which, in turn, is responsible for the production of high energy neutrinos from the sun. Direct capture measurements have been performed in the past but one would like to test the results by other methods and also reduce the uncertainty. To illustrate the problem we show in Fig. 1(a) the decay energy spectrum, Eq. (9), calculated in a particular single-particle model [18] at 46.5 MeV/nucleon on a lead target. Also shown are the results of a measurement [7]. Although E1 transitions dominate, the calculation predicts a significant contribution from E2 transitions, which is shown by the dashed curve. The analysis of the measurement, on the other hand, indicated that E2 transitions did not play any role. More recent measurements of the decay energy spectrum [8], and also at much higher beam energies [9], came to the same conclusion, namely, that E2 transitions do not have a significant influence. A more direct way to probe the E2 strength is to analyze the asymmetry of measured momentum distributions, which could be produced by the interference of E1 and E2 transitions, as discussed in connection with Eq. (8). This is illustrated in Fig. 1(b), where we show longitudinal momentum distributions of 7 Be fragments measured at 44 MeV/nucleon on a Pb target [19]. The distributions were obtained for different acceptance angles, which corresponds to probing the Coulomb dissociation over different ranges of impact parameters. The solid curves in Fig. 1(b) are calculated distributions, which reproduce the measurements at the smallest acceptance angles. This was achieved by reducing the E2 matrix elements, predicted by the model of Ref. [18], by a factor of 0.7. The top solid curve was calculated for full acceptance, where no data exist. There is some
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Fig. 1. Decay energy spectrum (a) and longitudinal momentum distributions of 7 Be fragments (b) obtained from Coulomb dissociation of 8 B on a Pb target at 46.5 and 44 MeV/nucleon, respectively. The data in (a) are from Ref. [7]. The data in (b) are from Ref. [19]; they were obtained for different acceptance angles of the 7 Be fragments, namely, 1.5◦ , 2.4◦ , 3.5◦ , and 5.4◦ . The curves are the results of calculations discussed in the text.
discrepancy with the data at the largest acceptance angle, where the observed asymmetry tends to disappear. This could be caused by nuclear and/or higher-order processes. A new measurement of the 8 B decay energy spectrum has recently been performed [20], which gates on smaller scattering angles of the excited 8 B center-of-mass system, i.e., on larger impact parameters. The uncertain influence of nuclear processes has therefore been suppressed. Another advantage is that the relative contribution from E2 transitions is smaller at larger impact parameter. The data have been analyzed by employing the E2 strength that was extracted from the asymmetry of the longitudinal momentum distributions shown in Fig. 1(b). The result of this analysis has not yet been published.
3. RPA linear response of exotic nuclei 3.1. Linear response theory in the coordinate space The RPA theory of the collective response of nuclei is based on the time dependent HF theory (TDHF). It is formulated in coordinate space [21] as the self-consistent linear response of the HF ground state to an external field. The first step is to determine the induced density in an external field. Let us adopt the time dependent HF field, h(t) = h0 (ρ) + h1 (t) = h0 (ρ) + f r e−iωt + f ∗ r eiωt ,
(13)
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where h0 is the static HF Hamiltonian and h1 (t) = f (r )e−iωt + f ∗ (r )eiωt is the time dependent external field. Let ϕi denote the eigenfunctions to the static HF Hamiltonian, h0 ϕi = 1i ϕ. From the time-dependent Schrödinger equation, ∂ ψi r, t = h(t)ψi r, t , (14) ∂t one can calculate how the occupied single-particle states ϕi evolve under the influence of the perturbation h1 (t). The result is that the induced density to first order in the external field can be written as (15) ρ1 = δρ0 r e−iωt + h.c., i
where h.c. implies the complex conjugate of the first term, and the density δρ0 r = d r G0 r, r , ω f r , is expressed in terms of the unperturbed Green’s function G0 ; G0 r, r : ω 1 1 r ϕi r . = ϕi ∗ r r − ω + iη − h0 + 1i ω − iη + h0 − 1i
(16)
(17)
i∈occupied
The inverse operator equation on the rhs of (17) is nothing but the one-body Green’s function in the coordinate space representation. We can use the standard technique to solve the Green’s function taking into account the coupling to the continuum [22]. The next step is to express the induced density self-consistently in terms of the HF field. The point is that the induced density δρ will generate an induced potential, δv = δv δρ δρ, leading to the TDHF Hamiltonian
δv h(t) = h0 (ρ0 ) + (18) δρ + f r e−iωt + [h.c.]. δρ The self-consistent equation for the induced density is, in analogy to Eq. (16),
δv δρ + f r . δρ = G0 δρ
(19)
Here one can isolate the induced density and write it in terms of the RPA Green’s function, GRPA , as δρ r = d r GRPA r, r , ω f r , (20) where, in operator form, δv −1 G0 . GRPA = 1 − G0 δρ
(21)
Finally, the transition strength S(ω), which describes how the HF ground state responds to the external field, is given by
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S(ω) =
Im π
Im d r f ∗ r δρ r = π
d r d r f ∗ r GRPA r, r , ω f r .
455
(22)
where Im implies the imaginary part of the expression. If we had used the unperturbed Green’s function (17) in this expression, we would instead have obtained the unperturbed independent-particle response of the HF ground state. σ τ ( r ), which are applied to probe the response of The transition operators, f ( r ) = fλµ a nucleus may depend on the multipolarity λ, spin σ , and isospin τ . We shall not discuss the spin response here but focus instead on the isoscalar (IS) and isovector (IV) responses for λ = 1, 2. Explicit forms of transition operators are τ =0 fλ=1,µ
=
A
ri3 Y1µ rˆi ,
for IS dipole strength,
(23)
i=1 τ =1 = fλ=1,µ
A
τz (i)ri Y1µ rˆi ,
for IV dipole strength,
(24)
for IS quadrupole strength,
(25)
i=1 τ =0 fλ=2,µ =
A
ri2 Y2µ rˆi ,
i=1 τ =1 fλ=2,µ =
A
τz (i)ri2 Y2µ rˆi ,
for IV quadrupole strength.
(26)
i=1
ri ) generates a spurious center of Since the lowest-order IS dipole operator A i=1 ri Y1µ (ˆ mass motion, the compression mode (23) is the lowest-order physical excitation in this channel. Numerical results of the continuum RPA model have been obtained for the dipole [14], monopole [23], and quadrupole [24] responses in nuclei near the drip lines. The RPA model has also been applied to these nuclei using a discrete harmonic oscillator basis [15]. We discuss in the following the results that have been obtained in the continuum RPA model for the dipole and quadrupole responses, focusing on the low-energy strength below Ex = 10 MeV. This strength is of particular interest to pin down the importance of couplings between the halo (or the skin) and the core wave functions. 3.2. Dipole response The low-lying dipole strength of halo nuclei is usually calculated as the unperturbed independent-particle response to the dipole operator associated with Coulomb dissociation. In RPA calculations the basic mechanism, which may deplete the low-lying dipole strength from the IV channel, is a strong residual coupling to the IV GDR. This coupling is strong if there is a large overlap between the densities of the valence nucleons and the core nucleons. Only if the two densities are well separated in space, the low-lying dipole strength may survive in the IV channel. This is the case in halo nuclei, where the weakly bound valence nucleons occupy low angular momentum states, preferably s-waves. In skin nuclei, on the other hand, where the spatial separation of the two densities is less dramatic, the low-lying
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dipole strength is somewhat quenched in the IV channel. However, a compression mode in the IS channel [14,15] will appear with a large strength. We illustrate here how the coupling between the GDR and the soft dipole excitation is suppressed in halo nuclei. Using a contact IV residual interaction, Vph (r, r )= IV δ(r − r ) [25], the coupling matrix element between the two states is Vph SD|V |GDR ∝ Vph r, r δρGDR (r)δρSD r dr dr IV δρGDR (r)δρSD (r) dr, = Vph (27) where δρGDR (r) and δρSD (r ) are the transitions densities of the GDR and the soft dipole excitation, respectively. The transition density of the GDR is commonly expressed in the Tassie model as dρ0 (r) Y1µ rˆ , (28) δρGDR (r) = α dr where ρ0 (r) is the ground state density and the factor α is determined by the energyweighted sum rule (EWSR). The transition density of the soft dipole excitation is similarly dρhalo(r) Y1µ rˆ , (29) dr where ρhalo (r) is the density of the halo configuration and the factor β is obtained from the cluster sum rule [26]. The radial integral (27) becomes almost zero or even negative for the two densities (28) and (29) because of the cancellation between the contributions from inside and outside the nuclear surface, as can be seen in Fig. 2. In a well bound nucleus, ρSD is replaced by a product of the wave functions of 1h¯ ω particle–hole excitations and the integral (27) becomes large enough to make a strong collective state in the IV dipole δρSD (r) = β
Fig. 2. Calculated transition densities of GDR and soft dipole excitations. The ground state density ρ0 is obtained by the HF calculation with SGII interaction for 11 Li. The halo configuration is assumed to be a pure 1p1/2 configuration. The central part of the HF potential is adjusted so that the separation energy of the halo orbit is 300 keV. The dotted bar shows the rms radius of the 9 Li core.
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channel absorbing most of unperturbed particle-hole strength. This is the essential physical mechanism which explains why the soft dipole mode in halo nuclei is essentially unaffected by RPA correlations, while there is no low-energy dipole strength in stable nuclei. We show in Fig. 3 the calculated results for the RPA IS and IV dipole responses in the stable nucleus 208 Pb and in the dripline nuclei 34 Ca and 60 Ca. The center of mass spurious motion has carefully been subtracted from both dipole operators, Eqs. (23), (24), and also from the response function (22), as explained in Ref. [14]. In order to simulate the coupling to many-particle many-hole configurations, we also show in Fig. 3 the averaged strength functions, (30) S(E) = S(E0 )ρ(E − E0 ) dE0, with the weight function ρ(E − E0 ) =
: 1 . π (E − E0 )2 + :2
(31)
In plotting the figures we have used the value of : = 1 MeV. The multiple peak structure 34 60 can be seen in the IV dipole strength function of 208 82 Pb126 , 20 Ca14 and 20 Ca40 in Fig. 3. In the stable nucleus 208 Pb, there is essentially no strength below Ex = 6 MeV in the IV response and below Ex = 10 MeV in the IS response. In the case of 40 20 Ca, no strength is found below Ex = 10 MeV for both the IS and IV responses [14]. The IV GDR in 208 Pb appears in the energy interval 6 E 20 MeV, centered at E = 13 MeV and x x exhausting about 80% of the energy weighted sum rule (EWSR) value. The IS compression mode is a broad resonance in the energy region 15 Ex 30 MeV, and it is peaked at around 24 MeV. In the dripline nuclei 34 Ca and 60 Ca, the IV GDR peaks are not so much affected by the proton or neutron skin, although an appreciable strength appears below Ex = 10 MeV. On the other hand, the IS compression mode is very much affected by the skin effect. In both dripline nuclei, the strength splits into two large peaks. The main peak appears below Ex = 10 MeV and another broad peak is found in the energy region 20 Ex 40 MeV. It is seen that a considerable part of the IS dipole transition strength is consumed by the threshold strength and lies clearly below the “IV GDR”, while the higher-lying IS dipole strength is observed as a very broad peak with an extremely large tail. Thus, the IV GDR lies energetically between the low-energy IS dipole peaks and the very broad “IS GDR” in the two dripline nuclei. 3.3. Quadrupole response The GQR is one of the most well established collective states in nuclei. Good sets of experimental data exist for the IS GQR and some data are also available for the IV GQR. We would like to address how the neutron (proton) skin effect manifests itself in the GQR of dripline nuclei. In Fig. 4 we show the calculated quadrupole strength function for the β-stable nuclei 40 Ca and 48 Ca. In stable nuclei, in general, the RPA calculation gives a single IS GQR peak as a collective excitation. This is also the case both in 40 Ca and 48 Ca. The IS peak appears
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Fig. 3. Calculated RPA IS and IV dipole strength functions of the nuclei 208 Pb (a), 34 Ca (b), and 60 Ca (c), as functions of the excitation energy. The scale of the IS dipole strength is shown on the right-hand side, while the scale of the IV dipole strength is shown on the left-hand side. The solid line is the IS dipole strength, while the dashed line denotes the IV dipole strength. The thick lines are obtained by averaging the calculated RPA strength (denoted by the respective thin lines) using Eq. (30) with := 1 MeV. The strength appearing below the threshold due to the averaging procedure has no meaning. The interaction SkM* is used consistently in the HF and RPA calculations. The figures are taken from Ref. [14].
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Fig. 4. Calculated RPA quadrupole strength functions of the β-stable nuclei 40 Ca (a) and 48 Ca (b), as functions of excitation energy. The strength carried by two RPA 2+ states below the threshold in 48 Ca are plotted as vertical solid lines. The interaction SkM* is used consistently in the HF and RPA calculations. The IS (IV) response is shown by the solid (long-dashed) curve, while the unperturbed response is given by the dashed curve.
at Ex = 16.2 MeV in 40 Ca. About 80% of the IS EWSR is found in the IS GQR region of 14.9 Ex 17.7 MeV. On the other hand, the IV strength is spread over a wide energy region of 22 Ex 35 MeV. The summed IV strength in this energy region exhausts 68% of the IV EWSR value. The experimental IS GQR peak is found at Ex = 18 MeV having about 40% of the EWSR value. A large E2 strength has also been observed in the energy region 10 Ex 16 MeV exhausting another 40% of the EWSR value [27]. The IV GQR has been observed experimentally at around Ex = 30 MeV by photoreactions [28]. In 48 Ca, the IS GQR is found at Ex = 16.7 MeV which is 0.5 MeV higher in energy than in 40 Ca. Summing up the RPA strength in the energy region of 15 Ex 18 MeV, the IS strength exhausts 77% of the EWSR value. The first 2+ state is found at Ex = 3.61 MeV with an IS strength of 516 fm4 . This IS state has the electric transition strength B(E2) = 89 e2 fm4 . Experimentally, the first 2+ state is observed at Ex = 3.83 MeV with B(E2) = 82 e2 fm4 which is close to the present calculated result. The IV quadrupole strength is again spread over the energy region 20 Ex 40 MeV. The summed strength
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Fig. 5. Calculated RPA quadrupole strength functions of the dripline nuclei 34 Ca (a) and 60 Ca (b), as functions of excitation energy. The interaction SkM* is used consistently in the HF and RPA calculations. The IS (IV) response is shown by the solid (long-dashed) curve, while the unperturbed response is given by the dashed curve.
in this region is 80% of the EWSR. Summing up the RPA strength in the energy region of 15 Ex 18 MeV, the IS moment amounts to 47.8 fm2 , while the IV moment is 9.76 fm2 . The ratio of the IV moment to the IS moment in the IS GQR region is 0.20, which is slightly larger than the value, ((N − Z)/A) = 0.17. The quadrupole responses of the two dripline nuclei 34 Ca and 60 Ca are shown in Fig. 5. The IS GQR peak in 34 Ca is found at Ex = 17.5 MeV. The summed strength between 15 Ex 20 MeV exhausts 68% of the IS EWSR. We can see substantial strength below Ex = 15 MeV in 34 Ca, while there is essentially no continuum strength below Ex = 15 MeV in 40 Ca and 48 Ca. The summed strength below Ex = 15 MeV in 34 Ca is 13% of the EWSR. The two small peaks at Ex = 3.2 MeV and 7.6 MeV are mainly due to the neutron p–h excitations (1d5/2 → 2s1/2 ) and (1d5/2 → 1d3/2), respectively. Besides these two peaks, the continuum strength below Ex = 15 MeV is dominated by the proton excitations. The main IV strength in 34 Ca is found in the energy region 23 Ex 40 MeV. The summed strength in this wide energy region exhausts 66% of the IV EWSR.
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The IS and IV quadrupole response of 60 Ca is shown in Fig. 5(b). The broad IS peak is found peaked at Ex = 14.9 MeV which is 2–3 MeV lower than the peak energies of the other isotopes. The summed strength in the energy region 11 Ex 17 MeV exhausts 82% of the EWSR. Appreciable strength is found below Ex = 11 MeV, which has 5% of the EWSR value. The neutron excitations to the continuum are responsible for the strength below Ex = 11 MeV. The IV strength in 60 Ca is distributed over a wide energy range 22 Ex 35 MeV, having 62% of the IS EWSR. We can see IV strength below the IS GQR peaks both in 34 Ca and 60 Ca. The ratios between the IV moment to IS moment in the IS GQR region are 7.24/23.6 = 0.30 for 34 Ca (17 Ex 18 MeV) and 17.0/46.4 = 0.37 for 60 Ca (14.4 Ex 15.4 MeV), respectively. These ratios are somewhat larger than the values ((N − Z)/A) = 0.18 and 0.33 for 34 Ca and 60 Ca, respectively. These differences might be due to large effects of the continuum in the dripline nuclei, even in the GQR energy region. The widths ΓFWHM of the IS GQR are about 1.6 MeV in 34 Ca and 1.3 MeV in 60 Ca, while the corresponding widths are 0.6 MeV in 40 Ca and 0.3 MeV in 48 Ca, respectively. The strong coupling to the continuum in the dripline nuclei increases substantially the escape width Γ ↑ of the IS GQR. By comparing Figs. 4 and 5 we see that the IS GQR has more strength in dripline nuclei than in stable nuclei at lower excitation energies. The IV GQR is very widely spread out both in stable and dripline nuclei.
4. Discussions and summary We have studied giant resonances in nuclei far from stability. Firstly, the Coulomb dissociation process was studied to extract the dipole response. We applied this method for 8 B to extract not only the dipole strength but also the quadrupole strength from the experimental data. Secondly, the self-consistent HF + RPA model was formulated in coordinate space to take into account the coupling to the continuum. We applied this model to calculate the dipole and quadrupole responses in both the IS and IV channels. A large transition strength below 10 MeV is found in the IS dipole response of the dripline nuclei 34 Ca and 60 Ca, whereas the IV dipole response is suppressed in this energy region. A substantial strength is also found in the IS quadrupole response below the IS GQR peak of these dripline nuclei. Calculations for other neutron dripline nuclei, such as 22 C and 28 O, have also shown that the threshold dipole strength of these nuclei does not appear in the IV channel but rather as a compression mode in the IS channel [14]. Thus, this feature of the dipole response might be characteristic for nuclei with neutron or proton skin. Since Coulomb dissociation probes the IV E1 strength, the large Coulomb dissociation cross sections that have been observed in halo nuclei should disappear rather quickly as one moves away from the dripline, towards skin and stable nuclei. On the other hand, the soft IS dipole compression mode predicted in skin nuclei [14,15] can be excited by the nuclear field from a target nucleus, and this may lead to an enhancement of the nuclear induced breakup. It remains to be seen how important this breakup mechanism is.
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Acknowledgements This work is supported in part by the Ministry of Education, Science, Sports and Culture in Japan by Grant-In-Aid for Scientific Research under the program number (C(2)) 12640284, and the US Department of Energy, Nuclear Physics Division, under Contract No. W-31-109-ENG.
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Giant resonances in exotic nuclei H. Sagawa a,∗ , H. Esbensen b a Center for Mathematical Sciences, University of Aizu, Aizu-Wakamatsu, Fukushima, Japan b Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
Received 29 November 2000; accepted 30 January 2001
Abstract We discuss theoretical studies of the Giant Dipole (GDR) and Giant Quadrupole (GQR) Resonances in nuclei far from stability, based on self-consistent Hartree–Fock and RPA calculations. The soft dipole mode that has been observed in light halo nuclei is predicted to diminish in skin nuclei, due to couplings to the isovector GDR. However, some low-lying dipole strength is predicted to exist in the isoscalar channel as a compression mode. The isoscalar GQR is expected to be much broader in skin nuclei than in stable nuclei, and substantial strength is predicted below the isoscalar GQR. The isovector GQR strength is spread over a wide range of excitation energies, both in skin and stable nuclei. 2001 Elsevier Science B.V. All rights reserved. PACS: 24.30.Cz; 25.70.De; 21.60.Jz Keywords: Giant resonances; Coulomb excitations; Halo nuclei; Skin nuclei
1. Introduction The experimental study of giant resonances in nuclei far from stability is still a rather unexplored territory. The most exciting development in the past decade is the observation of a large E1 strength just above threshold in light dripline nuclei. The large dipole strength has been revealed in breakup reactions on high-Z targets by observing the core like fragment [1]. The large cross sections that have been observed can only be explained by a large contribution from Coulomb dissociation. By measuring the momenta of the emitted particles it has been possible to construct the decay-energy spectrum, i.e., the excitation energy spectrum of a dripline nucleus prior to its decay. Such measurements have been performed for 6 He [2], 8 He [3] 11 Li [4], 11 Be [5], and 19 C [6], which are all neutron halo nuclei, and for 8 B [7–9], which has a proton halo. A large low-lying dipole strength is unusual because it does not ordinarily appear in stable nuclei. Some weak strength may appear and it is often referred to as Pigmy * Corresponding author.
E-mail address: [email protected] (H. Sagawa). 0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 0 6 4 9 - 2
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resonances [10,11]. Coulomb dissociation of a halo nucleus into a nucleon and a core fragment can be viewed as an absorption of virtual photons generated by the charge-current density of the target nucleus [12]. This process is closely related, via detailed balance, to the inverse reaction of radiative nucleon capture on the core nucleus [13]. The relationship that exists between the two processes makes Coulomb dissociation a powerful tool to infer radiative capture rates in cases where direct capture measurements are difficult or impossible, as for example when the core nucleus is short-lived. We discuss in Section 2 how the 7 Be + p → 8 B radiative capture rate can be determined from 8 B → 7 Be + p Coulomb dissociation measurements. An interesting question is why a large low-lying dipole strength may exist in dripline nuclei, and why it tends to disappear as one moves towards stability. Detailed experimental information about this trend is not yet available but some theoretical studies have been performed, based on self-consistent Hartree–Fock (HF) calculations combined with the random-phase approximation (RPA) [14,15]. In Section 3, we discuss this method and the results that have been obtained. Another application of Coulomb excitation is the extensive study of low-lying quadrupole transitions in exotic nuclei, which is discussed by Glasmacher [16]. Experimental studies of giant resonance excitations in nuclei far from stability have just started [11] and detailed results are not yet available. In the mean time, RPA calculations may provide some guidance for what to expect and what to look for. This paper is organized as follows. In Section 2 we discuss how the dipole response of dripline nuclei can be extracted from Coulomb dissociation experiments. We illustrate this method for 8 B and discuss how the results can be used to infer the rate of the inverse capture reaction. In Section 3 we describe the RPA Green’s function method in coordinate space taking into account the coupling to the continuum. The results for the dipole and quadrupole responses are also presented in Section 3. Discussions and a summary will be given in Section 4.
2. Coulomb dissociation of halo nuclei Here we consider a semiclassical description of the Coulomb dissociation of halo nuclei. We assume for simplicity that the halo nucleus (Z, A) can be described as a two-body system, consisting of a valence nucleon (Zx , Ax = 1) and an inert core (Zc , Ac = A − 1). The Coulomb field from a target nucleus (ZT , AT ) acting on the core and the valence nucleon is Zx e Zc e + , (1) VCoul = ZT e Rt + 1 r Rt − A−1 r A
A
where r is the position of the valence nucleon with respect to the core, and Rt is the target position with respect to the halo nucleus. For distant collisions (i.e., Rt > r) one can use the multipole expansion
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VCoul =
∗ (R t ) 4πZT e Yλµ λµ
2λ + 1 Rtλ+1
Mλµ ,
where Mλµ = eλ r λ Yλµ ,
is the electric multipole operator and eλ is the effective multipole charge, A−1 λ −1 λ eλ = Z x e + Zc e . A A
(2)
(3)
The amplitude af i for exciting a halo nucleus, from the initial state |i to a final state |f , is calculated in first-order, time-dependent perturbation theory. It has the form 1 1 Fλµ f |Mλµ |i, (4) dt eiωt f |VCoul |i = af i = i h¯ i λµ
where h¯ ω = Ef − Ei is the excitation energy. The last expression in Eq. (4), which is valid for distant collisions, shows a nice separation into multipole matrix elements, which depend on the structure of the halo, and factors Fλµ , which are independent of the structure; they can be obtained by inserting the expansion (2) into Eq. (4). In a relativistic description one obtains for a straight-line trajectory, Rt = b + vt, the following expression [17] (ω/v)λ Gλµ ZT e 16π ωb Fλµ = i λ+µ √ Kµ , (5) γv h¯ vγ 2λ + 1 (λ + µ)!(λ − µ)! where γ = 1/ 1 − (v/c)2 . The factors Gλµ are equal to one in the nonrelativistic limit. They can be extracted from Ref. [17] for relativistic energies. For dipole and quadrupole excitations one finds that G10 = G20 = G2±2 = 1/γ , G1±1 = 1, and G2±1 = 12 (1 + 1/γ 2). The Coulomb dissociation of halo nuclei is usually dominated by dipole transitions. The p effective dipole charge is e1 = e(A − Z)/A for a proton, and e1n = −Ze/A for a neutron halo, cf. Eq. (3). The effective quadrupole charges are Z − 1 + (A − 1)2 Z e, e2n = 2 e. 2 A A The quadrupole charge of a neutron is small and quadrupole transitions can therefore often be neglected when analyzing Coulomb dissociation measurements of neutron halos [5,6]. For a proton halo, on the other hand, quadrupole transitions may play a significant role. p
e2 =
2.1. Single-particle response of halo nuclei To determine the multipole matrix element in Eq. (4) one needs a model to calculate the initial and final states. The simplest approach is to adopt a single-particle Hamiltonian, based on a Woods–Saxon plus spin–orbit interaction, and adjust it to reproduce the separation energy and other known properties of the halo. The initial state is of the form |i = u0 (r)/r|&0 l0 m0 . The final state f |, which has the asymptotic momentum k of the emitted valence nucleon with respect to the core fragment, is given by the sum
uk&j (r) f | = (−i)& eiδ&j (k) kˆ &j m &j mrˆ , (6) kr &j m
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Nuc is the total phase shift due to the Coulomb and nuclear where δ&j (k) = σ&Coul + δ&j interactions. Without final state interactions this final state would just be a plane wave, f | = exp(−ikr)/(2π)3/2 . The radial wave functions (6) must therefore be normalized so that 2 sin kr + δ&j (k) , for r → ∞. uk&j (r) → π Inserting the final state (6) into the multipole matrix element we obtain
1 iδ&j (k) ˆ k&j ||Mλ ||&0 j0 , (7) e k &j m j0 m0 λµ|j m f |Mλµ |i = √ k 2j + 1 &j m
where k&j ||Mλ ||&0 j0 is a reduced single-particle matrix element. The momentum distribution for the relative motion of the emitted nucleon and the core fragment is equal to the square of the excitation amplitude (4) at a given impact parameter. Integrating over the impact parameter range of interest, say b > bmin , we obtain the differential cross section 2 ∞ ∞ dσ CD 2 . 2πb db |a | = 2πb db F f |M |i (8) = f i λµ λµ d 3k bmin
bmin
λµ
The amplitudes for different multipole transitions are here added coherently. This leads to an interference, for example, between dipole and quadrupole transitions, which can produce an asymmetry in the momentum distribution. If we integrate over all orientations of the momentum k, and take the average over the initial magnetic quantum number m0 , we obtain by inserting Eq. (7) into Eq. (8) ∞ dσ CD 1 dB(Eλ) = 2πb db |Fλµ |2 , (9) dE 2λ + 1 dE λµ
bmin
where the multipole strength function is dB(Eλ) dk | k&j ||Mλ ||&0 j0 |2 = . dE dE 2j0 + 1
(10)
&j
The cross section (9) and the multipole strength functions (10) have here been converted into differentials in the relative kinetic energy of the two fragments, E = (h¯ k)2 /(2m0 ), where m0 is the reduced mass. Let us also mention that the integration over impact parameter can be performed analytically, as shown in Ref. [17]. 2.2. Radiative capture The cross section for the radiative capture, c + p → A + γ , is related to the photoabsorption cross section for the inverse reaction, A + γ → c + p, by detailed balance [13]. For a given electric multipole transition the relation is Eγ 2 2(2IA + 1) (γ ) (rc) σ . (11) σEλ = h¯ ck (2Ic + 1)(2Ip + 1) Eλ
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Here Eγ is the photon energy, and h¯ k is the momentum for the relative motion of c and p. The expression contains the ratio of the phase space factors for the photon and for the relative motion of c and p, and the statistical weights associated with the spins in the initial and final states. The factor of two accounts for the two possible polarizations of the photons emitted in the capture process. The photoabsorption cross section can be expressed in terms of the multipole response as follows [12] (2π)3 (λ + 1) Eγ 2λ−1 dB(Eλ) (γ ) . (12) σEλ = dEγ λ((2λ + 1)!!)2 hc ¯ The two expressions, Eqs. (11), (12), show that the radiative capture cross section for a given multipolarity is determined in a simple way by the multipole strength function dB(Eλ)/dE. The same strength function appears in the decay energy spectrum for Coulomb dissociation, Eq. (9). Thus, if one could extract this strength distribution from a measured decay energy spectrum, one would be able to predict the radiative capture cross section. 2.3. Applications to 8 B Here we illustrate how Coulomb dissociation experiments can be used to extract the dipole strength distribution of 8 B and infer the radiative capture rate of protons on 7 Be at low energies. The capture rate is a determining factor for the solar abundance of 8 B, which, in turn, is responsible for the production of high energy neutrinos from the sun. Direct capture measurements have been performed in the past but one would like to test the results by other methods and also reduce the uncertainty. To illustrate the problem we show in Fig. 1(a) the decay energy spectrum, Eq. (9), calculated in a particular single-particle model [18] at 46.5 MeV/nucleon on a lead target. Also shown are the results of a measurement [7]. Although E1 transitions dominate, the calculation predicts a significant contribution from E2 transitions, which is shown by the dashed curve. The analysis of the measurement, on the other hand, indicated that E2 transitions did not play any role. More recent measurements of the decay energy spectrum [8], and also at much higher beam energies [9], came to the same conclusion, namely, that E2 transitions do not have a significant influence. A more direct way to probe the E2 strength is to analyze the asymmetry of measured momentum distributions, which could be produced by the interference of E1 and E2 transitions, as discussed in connection with Eq. (8). This is illustrated in Fig. 1(b), where we show longitudinal momentum distributions of 7 Be fragments measured at 44 MeV/nucleon on a Pb target [19]. The distributions were obtained for different acceptance angles, which corresponds to probing the Coulomb dissociation over different ranges of impact parameters. The solid curves in Fig. 1(b) are calculated distributions, which reproduce the measurements at the smallest acceptance angles. This was achieved by reducing the E2 matrix elements, predicted by the model of Ref. [18], by a factor of 0.7. The top solid curve was calculated for full acceptance, where no data exist. There is some
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Fig. 1. Decay energy spectrum (a) and longitudinal momentum distributions of 7 Be fragments (b) obtained from Coulomb dissociation of 8 B on a Pb target at 46.5 and 44 MeV/nucleon, respectively. The data in (a) are from Ref. [7]. The data in (b) are from Ref. [19]; they were obtained for different acceptance angles of the 7 Be fragments, namely, 1.5◦ , 2.4◦ , 3.5◦ , and 5.4◦ . The curves are the results of calculations discussed in the text.
discrepancy with the data at the largest acceptance angle, where the observed asymmetry tends to disappear. This could be caused by nuclear and/or higher-order processes. A new measurement of the 8 B decay energy spectrum has recently been performed [20], which gates on smaller scattering angles of the excited 8 B center-of-mass system, i.e., on larger impact parameters. The uncertain influence of nuclear processes has therefore been suppressed. Another advantage is that the relative contribution from E2 transitions is smaller at larger impact parameter. The data have been analyzed by employing the E2 strength that was extracted from the asymmetry of the longitudinal momentum distributions shown in Fig. 1(b). The result of this analysis has not yet been published.
3. RPA linear response of exotic nuclei 3.1. Linear response theory in the coordinate space The RPA theory of the collective response of nuclei is based on the time dependent HF theory (TDHF). It is formulated in coordinate space [21] as the self-consistent linear response of the HF ground state to an external field. The first step is to determine the induced density in an external field. Let us adopt the time dependent HF field, h(t) = h0 (ρ) + h1 (t) = h0 (ρ) + f r e−iωt + f ∗ r eiωt ,
(13)
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where h0 is the static HF Hamiltonian and h1 (t) = f (r )e−iωt + f ∗ (r )eiωt is the time dependent external field. Let ϕi denote the eigenfunctions to the static HF Hamiltonian, h0 ϕi = 1i ϕ. From the time-dependent Schrödinger equation, ∂ ψi r, t = h(t)ψi r, t , (14) ∂t one can calculate how the occupied single-particle states ϕi evolve under the influence of the perturbation h1 (t). The result is that the induced density to first order in the external field can be written as (15) ρ1 = δρ0 r e−iωt + h.c., i
where h.c. implies the complex conjugate of the first term, and the density δρ0 r = d r G0 r, r , ω f r , is expressed in terms of the unperturbed Green’s function G0 ; G0 r, r : ω 1 1 r ϕi r . = ϕi ∗ r r − ω + iη − h0 + 1i ω − iη + h0 − 1i
(16)
(17)
i∈occupied
The inverse operator equation on the rhs of (17) is nothing but the one-body Green’s function in the coordinate space representation. We can use the standard technique to solve the Green’s function taking into account the coupling to the continuum [22]. The next step is to express the induced density self-consistently in terms of the HF field. The point is that the induced density δρ will generate an induced potential, δv = δv δρ δρ, leading to the TDHF Hamiltonian
δv h(t) = h0 (ρ0 ) + (18) δρ + f r e−iωt + [h.c.]. δρ The self-consistent equation for the induced density is, in analogy to Eq. (16),
δv δρ + f r . δρ = G0 δρ
(19)
Here one can isolate the induced density and write it in terms of the RPA Green’s function, GRPA , as δρ r = d r GRPA r, r , ω f r , (20) where, in operator form, δv −1 G0 . GRPA = 1 − G0 δρ
(21)
Finally, the transition strength S(ω), which describes how the HF ground state responds to the external field, is given by
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S(ω) =
Im π
Im d r f ∗ r δρ r = π
d r d r f ∗ r GRPA r, r , ω f r .
455
(22)
where Im implies the imaginary part of the expression. If we had used the unperturbed Green’s function (17) in this expression, we would instead have obtained the unperturbed independent-particle response of the HF ground state. σ τ ( r ), which are applied to probe the response of The transition operators, f ( r ) = fλµ a nucleus may depend on the multipolarity λ, spin σ , and isospin τ . We shall not discuss the spin response here but focus instead on the isoscalar (IS) and isovector (IV) responses for λ = 1, 2. Explicit forms of transition operators are τ =0 fλ=1,µ
=
A
ri3 Y1µ rˆi ,
for IS dipole strength,
(23)
i=1 τ =1 = fλ=1,µ
A
τz (i)ri Y1µ rˆi ,
for IV dipole strength,
(24)
for IS quadrupole strength,
(25)
i=1 τ =0 fλ=2,µ =
A
ri2 Y2µ rˆi ,
i=1 τ =1 fλ=2,µ =
A
τz (i)ri2 Y2µ rˆi ,
for IV quadrupole strength.
(26)
i=1
ri ) generates a spurious center of Since the lowest-order IS dipole operator A i=1 ri Y1µ (ˆ mass motion, the compression mode (23) is the lowest-order physical excitation in this channel. Numerical results of the continuum RPA model have been obtained for the dipole [14], monopole [23], and quadrupole [24] responses in nuclei near the drip lines. The RPA model has also been applied to these nuclei using a discrete harmonic oscillator basis [15]. We discuss in the following the results that have been obtained in the continuum RPA model for the dipole and quadrupole responses, focusing on the low-energy strength below Ex = 10 MeV. This strength is of particular interest to pin down the importance of couplings between the halo (or the skin) and the core wave functions. 3.2. Dipole response The low-lying dipole strength of halo nuclei is usually calculated as the unperturbed independent-particle response to the dipole operator associated with Coulomb dissociation. In RPA calculations the basic mechanism, which may deplete the low-lying dipole strength from the IV channel, is a strong residual coupling to the IV GDR. This coupling is strong if there is a large overlap between the densities of the valence nucleons and the core nucleons. Only if the two densities are well separated in space, the low-lying dipole strength may survive in the IV channel. This is the case in halo nuclei, where the weakly bound valence nucleons occupy low angular momentum states, preferably s-waves. In skin nuclei, on the other hand, where the spatial separation of the two densities is less dramatic, the low-lying
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dipole strength is somewhat quenched in the IV channel. However, a compression mode in the IS channel [14,15] will appear with a large strength. We illustrate here how the coupling between the GDR and the soft dipole excitation is suppressed in halo nuclei. Using a contact IV residual interaction, Vph (r, r )= IV δ(r − r ) [25], the coupling matrix element between the two states is Vph SD|V |GDR ∝ Vph r, r δρGDR (r)δρSD r dr dr IV δρGDR (r)δρSD (r) dr, = Vph (27) where δρGDR (r) and δρSD (r ) are the transitions densities of the GDR and the soft dipole excitation, respectively. The transition density of the GDR is commonly expressed in the Tassie model as dρ0 (r) Y1µ rˆ , (28) δρGDR (r) = α dr where ρ0 (r) is the ground state density and the factor α is determined by the energyweighted sum rule (EWSR). The transition density of the soft dipole excitation is similarly dρhalo(r) Y1µ rˆ , (29) dr where ρhalo (r) is the density of the halo configuration and the factor β is obtained from the cluster sum rule [26]. The radial integral (27) becomes almost zero or even negative for the two densities (28) and (29) because of the cancellation between the contributions from inside and outside the nuclear surface, as can be seen in Fig. 2. In a well bound nucleus, ρSD is replaced by a product of the wave functions of 1h¯ ω particle–hole excitations and the integral (27) becomes large enough to make a strong collective state in the IV dipole δρSD (r) = β
Fig. 2. Calculated transition densities of GDR and soft dipole excitations. The ground state density ρ0 is obtained by the HF calculation with SGII interaction for 11 Li. The halo configuration is assumed to be a pure 1p1/2 configuration. The central part of the HF potential is adjusted so that the separation energy of the halo orbit is 300 keV. The dotted bar shows the rms radius of the 9 Li core.
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channel absorbing most of unperturbed particle-hole strength. This is the essential physical mechanism which explains why the soft dipole mode in halo nuclei is essentially unaffected by RPA correlations, while there is no low-energy dipole strength in stable nuclei. We show in Fig. 3 the calculated results for the RPA IS and IV dipole responses in the stable nucleus 208 Pb and in the dripline nuclei 34 Ca and 60 Ca. The center of mass spurious motion has carefully been subtracted from both dipole operators, Eqs. (23), (24), and also from the response function (22), as explained in Ref. [14]. In order to simulate the coupling to many-particle many-hole configurations, we also show in Fig. 3 the averaged strength functions, (30) S(E) = S(E0 )ρ(E − E0 ) dE0, with the weight function ρ(E − E0 ) =
: 1 . π (E − E0 )2 + :2
(31)
In plotting the figures we have used the value of : = 1 MeV. The multiple peak structure 34 60 can be seen in the IV dipole strength function of 208 82 Pb126 , 20 Ca14 and 20 Ca40 in Fig. 3. In the stable nucleus 208 Pb, there is essentially no strength below Ex = 6 MeV in the IV response and below Ex = 10 MeV in the IS response. In the case of 40 20 Ca, no strength is found below Ex = 10 MeV for both the IS and IV responses [14]. The IV GDR in 208 Pb appears in the energy interval 6 E 20 MeV, centered at E = 13 MeV and x x exhausting about 80% of the energy weighted sum rule (EWSR) value. The IS compression mode is a broad resonance in the energy region 15 Ex 30 MeV, and it is peaked at around 24 MeV. In the dripline nuclei 34 Ca and 60 Ca, the IV GDR peaks are not so much affected by the proton or neutron skin, although an appreciable strength appears below Ex = 10 MeV. On the other hand, the IS compression mode is very much affected by the skin effect. In both dripline nuclei, the strength splits into two large peaks. The main peak appears below Ex = 10 MeV and another broad peak is found in the energy region 20 Ex 40 MeV. It is seen that a considerable part of the IS dipole transition strength is consumed by the threshold strength and lies clearly below the “IV GDR”, while the higher-lying IS dipole strength is observed as a very broad peak with an extremely large tail. Thus, the IV GDR lies energetically between the low-energy IS dipole peaks and the very broad “IS GDR” in the two dripline nuclei. 3.3. Quadrupole response The GQR is one of the most well established collective states in nuclei. Good sets of experimental data exist for the IS GQR and some data are also available for the IV GQR. We would like to address how the neutron (proton) skin effect manifests itself in the GQR of dripline nuclei. In Fig. 4 we show the calculated quadrupole strength function for the β-stable nuclei 40 Ca and 48 Ca. In stable nuclei, in general, the RPA calculation gives a single IS GQR peak as a collective excitation. This is also the case both in 40 Ca and 48 Ca. The IS peak appears
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Fig. 3. Calculated RPA IS and IV dipole strength functions of the nuclei 208 Pb (a), 34 Ca (b), and 60 Ca (c), as functions of the excitation energy. The scale of the IS dipole strength is shown on the right-hand side, while the scale of the IV dipole strength is shown on the left-hand side. The solid line is the IS dipole strength, while the dashed line denotes the IV dipole strength. The thick lines are obtained by averaging the calculated RPA strength (denoted by the respective thin lines) using Eq. (30) with := 1 MeV. The strength appearing below the threshold due to the averaging procedure has no meaning. The interaction SkM* is used consistently in the HF and RPA calculations. The figures are taken from Ref. [14].
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Fig. 4. Calculated RPA quadrupole strength functions of the β-stable nuclei 40 Ca (a) and 48 Ca (b), as functions of excitation energy. The strength carried by two RPA 2+ states below the threshold in 48 Ca are plotted as vertical solid lines. The interaction SkM* is used consistently in the HF and RPA calculations. The IS (IV) response is shown by the solid (long-dashed) curve, while the unperturbed response is given by the dashed curve.
at Ex = 16.2 MeV in 40 Ca. About 80% of the IS EWSR is found in the IS GQR region of 14.9 Ex 17.7 MeV. On the other hand, the IV strength is spread over a wide energy region of 22 Ex 35 MeV. The summed IV strength in this energy region exhausts 68% of the IV EWSR value. The experimental IS GQR peak is found at Ex = 18 MeV having about 40% of the EWSR value. A large E2 strength has also been observed in the energy region 10 Ex 16 MeV exhausting another 40% of the EWSR value [27]. The IV GQR has been observed experimentally at around Ex = 30 MeV by photoreactions [28]. In 48 Ca, the IS GQR is found at Ex = 16.7 MeV which is 0.5 MeV higher in energy than in 40 Ca. Summing up the RPA strength in the energy region of 15 Ex 18 MeV, the IS strength exhausts 77% of the EWSR value. The first 2+ state is found at Ex = 3.61 MeV with an IS strength of 516 fm4 . This IS state has the electric transition strength B(E2) = 89 e2 fm4 . Experimentally, the first 2+ state is observed at Ex = 3.83 MeV with B(E2) = 82 e2 fm4 which is close to the present calculated result. The IV quadrupole strength is again spread over the energy region 20 Ex 40 MeV. The summed strength
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Fig. 5. Calculated RPA quadrupole strength functions of the dripline nuclei 34 Ca (a) and 60 Ca (b), as functions of excitation energy. The interaction SkM* is used consistently in the HF and RPA calculations. The IS (IV) response is shown by the solid (long-dashed) curve, while the unperturbed response is given by the dashed curve.
in this region is 80% of the EWSR. Summing up the RPA strength in the energy region of 15 Ex 18 MeV, the IS moment amounts to 47.8 fm2 , while the IV moment is 9.76 fm2 . The ratio of the IV moment to the IS moment in the IS GQR region is 0.20, which is slightly larger than the value, ((N − Z)/A) = 0.17. The quadrupole responses of the two dripline nuclei 34 Ca and 60 Ca are shown in Fig. 5. The IS GQR peak in 34 Ca is found at Ex = 17.5 MeV. The summed strength between 15 Ex 20 MeV exhausts 68% of the IS EWSR. We can see substantial strength below Ex = 15 MeV in 34 Ca, while there is essentially no continuum strength below Ex = 15 MeV in 40 Ca and 48 Ca. The summed strength below Ex = 15 MeV in 34 Ca is 13% of the EWSR. The two small peaks at Ex = 3.2 MeV and 7.6 MeV are mainly due to the neutron p–h excitations (1d5/2 → 2s1/2 ) and (1d5/2 → 1d3/2), respectively. Besides these two peaks, the continuum strength below Ex = 15 MeV is dominated by the proton excitations. The main IV strength in 34 Ca is found in the energy region 23 Ex 40 MeV. The summed strength in this wide energy region exhausts 66% of the IV EWSR.
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The IS and IV quadrupole response of 60 Ca is shown in Fig. 5(b). The broad IS peak is found peaked at Ex = 14.9 MeV which is 2–3 MeV lower than the peak energies of the other isotopes. The summed strength in the energy region 11 Ex 17 MeV exhausts 82% of the EWSR. Appreciable strength is found below Ex = 11 MeV, which has 5% of the EWSR value. The neutron excitations to the continuum are responsible for the strength below Ex = 11 MeV. The IV strength in 60 Ca is distributed over a wide energy range 22 Ex 35 MeV, having 62% of the IS EWSR. We can see IV strength below the IS GQR peaks both in 34 Ca and 60 Ca. The ratios between the IV moment to IS moment in the IS GQR region are 7.24/23.6 = 0.30 for 34 Ca (17 Ex 18 MeV) and 17.0/46.4 = 0.37 for 60 Ca (14.4 Ex 15.4 MeV), respectively. These ratios are somewhat larger than the values ((N − Z)/A) = 0.18 and 0.33 for 34 Ca and 60 Ca, respectively. These differences might be due to large effects of the continuum in the dripline nuclei, even in the GQR energy region. The widths ΓFWHM of the IS GQR are about 1.6 MeV in 34 Ca and 1.3 MeV in 60 Ca, while the corresponding widths are 0.6 MeV in 40 Ca and 0.3 MeV in 48 Ca, respectively. The strong coupling to the continuum in the dripline nuclei increases substantially the escape width Γ ↑ of the IS GQR. By comparing Figs. 4 and 5 we see that the IS GQR has more strength in dripline nuclei than in stable nuclei at lower excitation energies. The IV GQR is very widely spread out both in stable and dripline nuclei.
4. Discussions and summary We have studied giant resonances in nuclei far from stability. Firstly, the Coulomb dissociation process was studied to extract the dipole response. We applied this method for 8 B to extract not only the dipole strength but also the quadrupole strength from the experimental data. Secondly, the self-consistent HF + RPA model was formulated in coordinate space to take into account the coupling to the continuum. We applied this model to calculate the dipole and quadrupole responses in both the IS and IV channels. A large transition strength below 10 MeV is found in the IS dipole response of the dripline nuclei 34 Ca and 60 Ca, whereas the IV dipole response is suppressed in this energy region. A substantial strength is also found in the IS quadrupole response below the IS GQR peak of these dripline nuclei. Calculations for other neutron dripline nuclei, such as 22 C and 28 O, have also shown that the threshold dipole strength of these nuclei does not appear in the IV channel but rather as a compression mode in the IS channel [14]. Thus, this feature of the dipole response might be characteristic for nuclei with neutron or proton skin. Since Coulomb dissociation probes the IV E1 strength, the large Coulomb dissociation cross sections that have been observed in halo nuclei should disappear rather quickly as one moves away from the dripline, towards skin and stable nuclei. On the other hand, the soft IS dipole compression mode predicted in skin nuclei [14,15] can be excited by the nuclear field from a target nucleus, and this may lead to an enhancement of the nuclear induced breakup. It remains to be seen how important this breakup mechanism is.
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Acknowledgements This work is supported in part by the Ministry of Education, Science, Sports and Culture in Japan by Grant-In-Aid for Scientific Research under the program number (C(2)) 12640284, and the US Department of Energy, Nuclear Physics Division, under Contract No. W-31-109-ENG.
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