We study low-lying multipole strength in neutron-halo nuclei. The strength
depends only on a few low-energy constants: the neutron separation energy, the
asymptotic normalization coefficient of the bound state wave function, and the
scattering length that contains the information on the interaction in the
continuum. The shape of the transition probability shows a characteristic
dependence on few scaling parameters and the angular momenta. The total E1
strength is related to the root-mean-square radius of the neutron wave function
in the ground state and shows corresponding scaling properties. We apply our
approach to the E1 strength distribution of 11Be.Comment: 4 pages, 1 figure (modified), additional table, extended discussion
  of example, accepted for publication in Phys. Rev. Let

A. B. Migdal

G. Baur

J. M. Blatt

P. G. Hansen

S. Typel

English

arXiv

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Effective-Range Approach and Scaling Laws for Electromagnetic Strength in Neutron-Halo Nuclei

Typel, S.

Baur, G.

GSI Repository

We study low-lying multipole strength in neutron-halo nuclei. The strength depends only on a few low-energy constants: the neutron separation energy, the asymptotic normalization coefficient of the bound-state wave function, and the scattering length that contains the information on the interaction in the continuum. The shape of the transition probability shows a characteristic dependence on few scaling parameters and the angular momenta. The total E1 strength is related to the root-mean-square radius of the neutron wave function in the ground state and shows corresponding scaling properties. We apply our approach to the E1 strength distribution of Be-11

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VOLUME 93, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R Sweek ending1 OCTOBER 2004Effective-Range Approach and Scaling Laws for Electromagnetic Strengthin Neutron-Halo NucleiS. TypelGesellschaft fu¨r Schwerionenforschung mbH (GSI), Planckstraße 1, D-64291 Darmstadt, GermanyG. BaurInstitut fu¨r Kernphysik, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany(Received 22 June 2004; published 29 September 2004)142502-1We study low-lying multipole strength in neutron-halo nuclei. The strength depends only on a fewlow-energy constants: the neutron separation energy, the asymptotic normalization coefficient of thebound-state wave function, and the scattering length that contains the information on the interaction inthe continuum. The shape of the transition probability shows a characteristic dependence on few scalingparameters and the angular momenta. The total E1 strength is related to the root-mean-square radius ofthe neutron wave function in the ground state and shows corresponding scaling properties. We apply ourapproach to the E1 strength distribution of 11Be.DOI: 10.1103/PhysRevLett.93.142502 PACS numbers: 24.50.+g, 25.20.–x, 25.60.Gc, 27.20.+nElectromagnetic dipole strength in stable nuclei isconcentrated in the giant dipole resonance. In recentyears it was found that there is an appreciable low-lyingE1 strength in light neutron-rich nuclei [1,2]. This phe-nomenon is especially pronounced in neutron-halo nu-clei, where it can be explained essentially as a single-particle effect [3,4]. Examples are 11Be [5,6] and 19C [7].This low-lying strength was observed mainly in Coulombdissociation experiments; for a recent review, see [8].The dissociation cross section directly reflects the di-pole strength distribution or reduced transition probabil-ity dBE1=dE. Photodissociation of and radiativecapture leading to halo nuclei provide equivalent infor-mation. The measured strength distributions show simplefeatures. They tend to be universal when plotted in theappropriate reduced parameters [9] because low-energyprocesses do not depend on certain details of the interac-tion. Consequently, effective field theories are nowadaysused for the description of halo nuclei [10]. In Ref. [11]nonrelativistic two-body scattering by a short range po-tential was studied using the renormalization group. Itwas found that the expansion around the nontrivial fixedpoint is equivalent to the effective-range (ER) expansion.It is the purpose of this Letter to apply the ER approachto one-neutron-halo nuclei and find scaling laws for thetransition strength. This work is similar in spirit to ef-fective field theories: the effects of unknown short dis-tance behavior are parametrized by a few low-energyconstants. It is not our aim to relate these parameters tothe many-body physics.General expressions for electromagnetic strength dis-tributions in halo nuclei were derived before (see, e.g.,[12], and references therein, for proton  core systems),however, often neglecting the nuclear continuum interac-tion. Also expansions for small relative energies wereobtained [13]. The application of ER theory to the studyof halo nuclei was considered before, e.g., in the descrip-0031-9007=04=93(14)=142502(4)$22.50 tion of ground state properties [14], and for radiativecapture cross sections into s-, p-, and d-wave bound statestaking into account the interaction only for s-waves as-suming a zero-range potential [15]. The effect of theinteraction in the continuum states on direct neutroncapture and photodisintegration of 13C was studied in[16,17]. While they find a sensitivity on neutron opticalmodel parameters for s! p capture, this sensitivity isstrongly reduced for the cases of p! s and p! dcapture.The prototype of a neutron-halo nucleus is the deu-teron. Radiative capture or photodissociation at low en-ergies are well described by the binding energy and theeffective range; see, e.g., [18]. Quite similar, the low-lyingE1 strength in neutron-halo nuclei is determined by twoparameters if the continuum interaction is neglected: oneis the binding energy (or neutron separation energy Sn)which determines the overall shape of the dBE1=dEdistribution; the other one is a normalization constant. Inthe analogous case of 19C, the binding energy could bedetermined directly from the shape of the observedstrength distribution [7]. Effects of the neutron-core in-teraction are parametrized efficiently by the ERapproximation.The cross section for photodissociation of a nucleus ainto a neutron n and a core c is given bya ! n c  1232 1!!2E	hc21 dBdE:(1)The reduced transition probability for electromagneticmultipolarity  is denoted by dB=dE. The photonenergy is given by E  E Sn with Sn  	h2q2=2,where q is the inverse decay length of the bound-statewave function and  is the reduced mass. Using detailed2004 The American Physical Society 142502-1VOLUME 93, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R Sweek ending1 OCTOBER 2004balance, the corresponding radiative capture cross sectioncan be obtained from this expression. With the equivalentphoton number, cross section (1) determines the (firstorder) Coulomb dissociation cross section [8].The strength distribution dBE; li ! lf=dE for acertain transition depends on the corresponding matrixelement that contains a radial integral with the wavefunctions of the initial and final states with orbital angu-lar momenta li and lf, respectively. (We consider spinlessneutrons and cores; a generalization is obvious.) At lowenergies E  	h2k2=2, the main contribution to theradial integral arises from radii larger than the radius Rof the nucleus. This is true for all possible values of theangular momenta involved. In the figures of Refs. [16,17]it can very well be seen that the radial integrals aredominated by the outside region. Away from a resonance,a case which we always assume here, the continuum wavefunction is small inside the nuclear radius. In a hardsphere model it is exactly zero.For the neutron  core case the dimensionless reducedradial integral is well approximated byI lfli   ilikq2Z 1Rdrr2h1li iqrcoslf jlf kr sinlf ylf kr (2)with spherical Bessel (jl), Neumann (yl), and Hankel(h1l ) functions, respectively, that describe the behaviorof the radial wave functions beyond the range of thenuclear potential. The phase shift lf contains the infor-mation on the interaction in the final continuum state.Introducing the dimensionless shape functionsS lfli  qkjI lfli j2; (3)the reduced transition probability is given bydBdEE; li ! lf Zeffe22	h22 14li00jlf02jCli j2q23Slfli (4)with the effective charge number Zeff  Zcmn=mn mc and the asymptotic normalization constant (ANC)Cli of the bound-state wave function. For neutron  coresystems the dipole transitions are dominant since E2transitions are suppressed by the small effective charge.The ANC of the actual many-body wave function as itappears in (4) consists of the ANC of a neutron single-particle wave function and a spectroscopic factor whichaccounts for the many-body aspects. The single-particleANC is determined by the normalization of the wavefunction. It depends on q with the scaling behaviorjCli j2 / q for li  0 and jCli j2 / q2li1 for li > 0 asseen, e.g., in a square-well model. Here, we have intro-duced the dimensionless scaling parameter   qR. The142502-2factor jCli j2=q23 in the reduced transition probability(4) depends only on properties of the ground state. Itdirectly shows the scaling with the characteristic parame-ter q. Clearly, for small separation energies of the neutrona large transition strength can be expected.The reduced radial integrals (2) have the general formI lfli   exp2  "21"lf Rlfli  cos" lf Rlfli  sin" lf  (5)with "  kR, where the rational functions Rlfli  canbe found for all relevant values of li and lf by means ofthe recursion relationsRlf1li  1  2"2 1  2  "22lf  1Rlfli   "2  "2dd"Rlfli  Rlfli; (6)Rlfli1  1  22 1  2  "2li   1 lf  Rlfli 2  "2 ddRlfli  (7)starting fromR00 0"; R00 0;R11 0"; R11 0 "212=2: (8)The radial integral involving the regular scattering wavefunction shows a rlilf1 behavior for small values of r;thus it can be extended to R  0. On the other hand, theradial integral involving the irregular function shows arlilf behavior. Thus for   1 we can extend theintegral to zero only for li  0; lf  1 and li  1; lf  0and have convergence. In the other cases, one has to cutoff the integral for a finite R value.For low energies we can use the effective-range ap-proximation k2l1 cotl  1=al  rlk2=2    forthe phase shifts l with the scattering length al and theeffective-range rl in order to take the interaction in thecontinuum state into account. (Note that al and rl have thedimension of a length only for l  0.) We restrict our-selves to the lowest order term, i.e.,tanl  xcl2l1 (9)with the dimensionless reduced scattering length cl de-fined by c2l1l al=R2l1 and the ratio x  "=  k=q E=Snp. We assume that this is a reasonable approxima-tion of the phase shift. We further assume that the scat-tering length has a ‘‘natural’’ value with cl being O1.For example, for s-wave scattering from a hard sphere142502-2TABLE I. Dimensionless quantities cj1, see Eq. (9), and spec-troscopic factors C2S from the fit to experimental data [6] of11Be Coulomb dissociation.Model c3=21 c1=21 C2SEffective-range 0:4186;20 2:7713;14 0.704(15)Woods-Saxon  0:4670;14a 1.87(3)a 0.696(15)aAt energy E  0:4 MeV in the peak of the strengthdistribution.VOLUME 93, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R Sweek ending1 OCTOBER 2004with radius R we have c0  1. In exceptional cases, thisassumption is not fulfilled, e.g., if there are resonances inthe low-lying continuum or subthreshold states. For ans-wave halo nucleus bound by a zero-range force thescattering length is given by a0  1=q; i.e., c0  1=.In such a case the scattering length in the s wave isunnaturally large; it diverges in the halo limit ! 0.We can expand the analytical results for Slfli  in termsof the small parameter  and obtain for the most impor-tant cases for dipole transitionsS 101 4x31 x24 1 c311 3x23    ; (10)S 011 x3 x221 x241 4c03 x2    : (11)The term in front of the square bracket in Eq. (10) is thewell-known result without continuum interaction; see,e.g., [5,17]. In general there will be a modification ofthe shape function. It is getting more and more importantfor larger binding energies of the halo nucleus, i.e., larger. Transitions with lf  li  1 are more affected by thisfinal-state interaction than transitions with lf  li  1. Inthe case s! p the first correction to the shape functionfor ! 0 appears only in the 3 term proportional to c31.There are no corrections of lower order in  independentof the final-state interaction. This explains the remarkableuniformity of the characteristic shape for the low-lyingdBE1=dE strength in halo nuclei such as 11Be and 19C.For the p! s transition there is a greater sensitivity tothe final-state interaction. It appears already as a correc-tion linear in  and c0. This is in accord with the results ofRef. [16].The total transition strength can be obtained by inte-grating (4) over the relative energy E. It is given byBE;li! lf  Zeffe22 14li00jlf02jCli j2q21T lfli (12)withT lfli  2Z 10dxxSlfli  2Z 10dxjI lfli j2: (13)For dipole transitions and no final-state interaction (i.e.,cl  0 corresponding to a plane wave in the final state),we findT 011  T 211  145 6 22e2; (14)T 101  141 2 22e2; (15)T 121 1436 37 142  23e2: (16)A finite value of  leads to a reduction of the total142502-3strength in the continuum as compared to the extremehalo limit   0. With final-state interaction, the func-tions T lfli  also depend on the reduced scattering lengthcl; however, rather lengthy expressions are obtained.The value of BE1; li PlfBE1; li ! lf for all E1transitions from a state with orbital angular momentum liis directly related by the non-energy-weighted sum ruleBE1; li  Z1effe234hr2ili (17)to the root-mean-square (rms) radius hr2ili of the samebound state; see, e.g., [2]. The rms radius scales with  fordifferent values of the bound-state orbital angular mo-mentum li as hr2i0 / R2=2, hr2i1 / R2=, and hr2ili / R2for li  2 [4]. On the other hand, the energy-weightedsum rule (Thomas-Reiche-Kuhn sum rule) gives a con-stant independent of li. From the above we see clearly thatthe low-lying strength is most pronounced for s-wavebound states and to a lesser degree for p-wave boundstates in halo nuclei. Considering the q dependence ofBE1; li ! lf in (12) the scaling of hr2ili with  is exactlyreproduced. With no final-state interaction the full tran-sition strength as predicted by the sum rule is found in thecontinuum. However, if there are bound states in theparticular final state the continuum contribution to thetotal strength is reduced and the shape will be distorted.This will correspond to a nonzero value of cl.Let us now apply the general theory to 11Be, whichshows a very pronounced single-particle halo structurewith a neutron bound by 504 keV in the 12 ground statethat can be considered as a 2s1=2 state. Assuming a radiusof R  2:78 fm the neutron separation energy corre-sponds to a small value   0:4132< 1 typical for ahalo nucleus. Unlike the case of the deuteron, where thereis no excited bound state, there is a 12 bound state at320 keV excitation energy. We can regard it to a goodapproximation as a 1p1=2 single-particle state. We per-form a &2 fit of our analytical ER approach (not using theexpansion for small ) to the experimental Coulombdissociation data [6] by adjusting the reduced scatteringlengths c3=21 and c1=21 in the two p-wave channels (seeTable I) and the ground state ANCC0. The quantity c1=21 isdefinitely unnaturally large as expected from the exis-tence of the weakly bound 12 state. In Fig. 1 our model fit142502-30 1 2 3 4 5 6E*   [MeV]0.00.10.20.30.40.50.60.7dB(E1)/dE   [e2 fm2 /MeV] effective-range modelWoods-Saxon modeltransition to 1/2- bound stateFIG. 1 (color online). Reduced transition probability as afunction of the excitation energy E ESn compared toexperimental data extracted from the Coulomb breakup of11Be [6].VOLUME 93, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R Sweek ending1 OCTOBER 2004(solid line) is compared to the experimental strengthdistribution [6], including corrections due to the detectorresponse.The obtained ANC C0  0:7248 fm1=2 can betranslated into a spectroscopic factor C2S by comparingto the ANC of a normalized single-particle wave functiongenerated from a Woods-Saxon (WS) potential with adepth adjusted to the correct neutron separation energy.Assuming a radius of R  2:78 fm as above and a dif-fuseness parameter of a  0:65 fm we obtain C2S 0:70415 which compares well with the spectroscopicfactors from experiment and theory as given in Ref. [6].The model ground state wave function has a rms radius of7.172 fm that corresponds to a total strength of BE1 1:645e2 fm2 according to (17). Applying the spectro-scopic factor we obtain a total strength of1:15936e2 fm2. The experimental continuum strength[6] exhausts about 78(5)% of the sum rule. The groundstate transition probability to the first excited state isexperimentally known to be BE1  0:11612e2 fm2[19] corresponding to 10(1)% of the sum rule. Thisbound-state transition strength is indicated in Fig. 1 as adotted line assuming an arbitrary FWHM of 0.160 MeV.The remaining 12(6)% can be attributed to the p3=2 statethat is occupied. Comparing the experimental transitionstrength to results of a plane-wave calculation wouldunderestimate the extracted spectroscopic factor.Our ER approach can be compared to an exact calcu-lation of the matrix elements with wave functions ob-tained from WS potentials. We assume the same shape ofthe potential with R and a as given above. Repeating the&2 fit by adjusting the depth of the potential in the two pwaves independently we find C2S  0:69615 close tothe result of our ER approach. The exact calculation(dashed line in Fig. 1) almost coincides with our ERapproximation. We can also extract the dimensionless142502-4quantities cj1 according to Eq. (9). Their modulus de-creases with relative energy (much more strongly forc1=21 than for c3=21 ). Since the cj1 in our ER approach areconstant they can be considered as an average of theactual values over the peak of the strength distribution.Taking this into account a reasonable agreement of thetwo approaches is found again (Table I).In conclusion we find that low-lying electric multipolestrength in one-neutron-halo nuclei can be describedeffectively by a few low-energy constants: the bindingenergy, the scattering length, and the asymptotic normal-ization constant. We apply our ER approach to the halonucleus 11Be and find remarkable agreement with an exactcalculation. The exhaustion of the strength due to thebound state affects the extraction of the ANC and of thespectroscopic factor. In our approach there is no need todetermine parameters of optical potentials from elasticscattering.The present theoretical approach will be applied toother halo nuclei. It will be extended to higher multi-polarities and to proton halo nuclei in a forthcomingpublication [20]. An effective-range theory of two-neu-tron-halo nuclei like 11Li would be very interesting but iscertainly much more complex [4]. An early work relatedto this problem is [21].We are grateful to T. Aumann and H. Emling for usefuldiscussions.[1] A. Leistenschneider et al., Phys. Rev. Lett. 86, 5442(2001).[2] B. Jonson, Phys. Rep. 389, 1 (2004).[3] P. G. Hansen and B. Jonson, Europhys. Lett. 4, 409(1987).[4] A. Jensen et al., Rev. Mod. Phys. 76, 215 (2004).[5] T. Nakamura et al., Phys. Lett. B 331, 296 (1994).[6] R. Palit et al., Phys. Rev. C 68, 034318 (2003).[7] T. Nakamura et al., Phys. Rev. Lett. 83, 1112 (1999).[8] G. Baur, K. Hencken, and D. Trautmann, Prog. Part.Nucl. Phys. 51, 487 (2003).[9] S. Typel and G. Baur, Phys. Rev. C 64, 024601 (2001).[10] C. Bertulani, H.-W. Hammer, and U. van Kolck, Nucl.Phys. A712, 37 (2002).[11] M. C. Birse et al., Phys. Lett. B 464, 169 (1999).[12] A. M. Mukhamedzhanov and F. M. Nunes, Nucl. Phys.A708, 437 (2002).[13] D. Baye, Phys. Rev. C 70, 015801 (2004).[14] H. Esbensen, G. F. Bertsch, and K. Hencken, Phys. Rev. C56, 3054 (1997).[15] D. M. Kalassa and G. Baur, J. Phys. G 22, 115 (1996).[16] A. Mengoni, T. Otsuka, and M. Ishihara, Phys. Rev. C 52,R2334 (1995).[17] T. Otsuka et al., Phys. Rev. C 49, R2289 (1994).[18] J. M. Blatt and V. E. Weisskopf, Theoretical NuclearPhysics (Springer-Verlag, New York, 1979).[19] T. Glasmacher, Annu. Rev. Nucl. Part. Sci. 48, 1 (1998).[20] S. Typel and G. Baur (to be published).[21] A. B. Migdal, Sov. J. Nucl. Phys. 16, 238 (1973).142502-4

Effective-range approach and scaling laws for electromagnetic strength in neutron-halo nuclei

Effective-range approach and scaling laws for electromagnetic strength
  in neutron-halo nuclei

Effective-range approach and scaling laws for electromagnetic strength in neutron-halo nuclei

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