Relativistic continuum-continuum coupling in the dissociation of halo nuclei

0.18 MB·English

by  C. A. Bertulani

Relativistic continuum-continuum coupling in the dissociation of halo nuclei
C.A. Bertulani1,∗
1Department of Physics, University of Arizona, Tucson, Arizona 85721
(Dated: February 9, 2008)
A relativistic coupled-channels theory for the calculation of dissociation cross sections of halo
nuclei is developed. A comparison with non-relativistic models is done for the dissociation of 8B
projectiles. It is shown that neglecting relativistic effects leads to seizable inaccuracies in the
extraction of theastrophysical S-factor for theproton+beryllium radiative capturereaction.
PACSnumbers: 25.60.+v;25.70.De;25.70.Mn
Reactionswithradioactivenuclearbeamshavequickly been used extensively in the study of breakup reactions
5 becomeamajorresearchareainnuclearphysics. Among with stable [8] and unstable nuclear projectiles [3, 9, 10]
0 the newly developed techniques, the Coulomb dissoci-
Special relativity, an obviously important concept in
0
ation method is an important tool to obtain electro-
2 physics, is quite often neglected in the afore mentioned
magneticmatrixelementsbetweencontinuumandbound
dynamical calculations. Most rare isotope facilities use
n statesofrarenuclearisotopes[1]. Thesematrixelements
projectiledissociationat100MeV/nucleon. Attheseen-
a
playanessentialroleinnuclearastrophysics. Atlowcon-
J ergies, relativistic effects are expected to be of the order
7 tinuum energies, they are the same as the ones involved of10%. Relativisticeffects areaccountedforinthe colli-
in radiative capture processes of astrophysical interest.
2 sion kinematics, in first-order perturbation calculations,
In particular, the Coulomb dissociation of 8B projectiles
but not in dynamicalcalculations used sofarin the anal-
2 allows to extract information on the radiative capture ysis of experiments. They also enter the dynamics in a
v reaction p+7Be 8B+γ, of relevance for the standard
non-linear, often unpredictable, way. The reason why
→
0 solarmodel and the production of high-energyneutrinos
these effects have not been considered before is due to
1 in the sun [2].
the inherent theoretical difficulty in defining a nuclear
0
2 Thedissociationofweakly-boundnuclei,orhalonuclei, potential between many-body relativistic systems. Due
1 is dominated by the Coulomb interaction, although the to retardation, an attempt to use a microscopic descrip-
4 nuclearinteractionwiththetargetcannotbeneglectedin tion starting from binary collisions of the constituents is
0 most cases. The final state interaction of the fragments not possible. A successful approach, known as “Dirac
/
h withthetargetleadstoimportantcontinuum-continuum phenomenology”, has been achieved for nucleon-nucleus
t andcontinuum-bound-state couplingswhich appreciably scattering [11]. But for nucleus-nucleus collisions a rea-
-
l modify the reaction dynamics. Higher-order couplings sonable account of these features has not yet been ac-
c
u are more relevant in the dissociation of halo nuclei due complished. In the present letter, the case of dissocia-
n to their low binding. A known example is the “post- tion of 8B projectiles is studied. The major contribution
: acceleration” (or “reacceleration”)effect observedin the comesfromtheCoulombinteractionwithwell-knownrel-
v
dissociation of 11Li projectiles [3, 4, 5, 6]. ativistic transformation properties. A coupled-channels
i
X method based on the eikonal approximation with rela-
Twomethods havebeendevisedto study higher-order
r tivistic ingredients is developed and compared to semi-
a effects inprojectiledissociation. The methodintroduced classical methods [6]. Here I omit the consideration of
inref. [6]usesthedirectsolutionoftheSchr¨odingerequa-
other correctionsin the eikonaltreatment of the scatter-
tion (DSSE) by space-time discretization. One starts
ing involving halo nuclei which have also been shown to
with aground-statewavefunction,propagatesitthrough
play a relevant role (see, e.g., refs. [12, 13]).
each time-step, and after sufficiently long time the it-
erated wavefunction is projected into a specific channel Let us consider the Klein-Gordon(KG) equation with
of interest. Another method discretizes the continuum a potential V0 which transforms as the time-like com-
wavefunctions c and uses them as input to calculate ponent of a four-vector [11]. For a system with total
the matrix elem| ients c′ V c and c V b , where energy E (including the rest mass M), the KG equa-
int int
b denotes a bound sthate| [3].| Tihe mahtr|ix ele|mients are tion can be cast into the form of a Schr¨odinger equa-
t|hienusedinacoupled-channelscalculationfortransition tion (with ~ = c = 1), 2+k2 U Ψ = 0, where
∇ −
amplitudes to dissociation channels. This is known as k2 = E2 M2 and U =(cid:0)V0(2E V0).(cid:1)When V0 M,
− − ≪
continuumdiscretizedcoupled-channels(CDCC)method and E(cid:0) M, (cid:1)one gets U = 2MV0, as in the non-
≃
and was introduced by Rawitscher [7] to study nuclear relativistic case. The condition V0 M is met in pe-
≪
breakupreactions of the type a+A b+c+A. It has ripheralcollisionsbetweennucleiatallcollisionenergies.
→
Thus,one canalwayswrite U =2EV . Afurther simpli-
0
fication is to assume that the center of mass motion of
the incoming projectile and outgoing fragments is only
∗Electronicaddress: [email protected] weakly modulated by the potential V0. To get the dy-
2
namical equations, one discretizes the wavefunction in withS (b)= (z = ,b). The setofequations3and
α α
S ∞
terms of the longitudinal center-of-mass momentum k , 4 are the relativistic-CDCC equations (RCDCC).
z
using the ansatz I have used the RCDCC equations to study the dis-
sociation of 8B projectiles at high energies. The ener-
Ψ= (z,b) exp(ik z) φ (ξ) . (1) gies transferred to the projectile are small, so that the
Sα α kα
Xα wavefunctions can be treated non-relativistically in the
projectile frame of reference. In this frame the wave-
In this equation, (z,b) is the projectile’s center-of-mass functions will be described in spherical coordinates, i.e.
coordinate, with b equal to the transverse coordinate. α = jlJM , where j, l, J and M denote the angular
φ(ξ) is the projectile intrinsic wavefunction and (k,K) m| oimen|tum niumbers characterizing the projectile state.
istheprojectile’scenter-ofmassmomentumwithlongitu- Eq. 3 is Lorentz invariant if the potential V transforms
0
dinalmomentumk andtransversemomentum K. There as the time-like component of a four-vector. The matrix
are hidden, uncomfortable, assumptions in eq. 1. The element α V α′ is also Lorentz invariant, and we can
0
separationbetweenthecenterofmassandintrinsiccoor- thereforehca|lcu|latei them in the projectile frame.
dinates is not permissible under strict relativistic treat- The longitudinalwavenumberk (E2 M2)1/2 also
α
ments. For high energy collisions we can at best justify ≃ −
defines how much energy is gone into projectile exci-
eq. 1 for the scattering of light projectiles on heavy tar-
tation, since for small energy and momentum transfers
gets. Eq. 1isonlyreasonableifthe projectileandtarget ′ ′
k k = (E E )/v. In this limit, eqs. 3 and 4 re-
closelymaintaintheirintegrityduringthe collision,asin α− α α− α
duce to semiclassical coupled-channels equations, if one
the case of very peripheral collisions.
uses z = vt for a projectile moving along a straight-
Neglecting the internal structure means φkα(ξ) = 1 line classical trajectory, and changing to the notation
andthesumineq. 1reducestoasingletermwithα=0, (z,b) = a (t,b), where a (t,b) is the time-dependent
α α α
theprojectileremaininginitsground-state. Itisstraight- eSxcitationamplitudeforacollisionwitimpactparameter
forward to show that inserting eq. 1 in the KG equa- b (see eqs. 41 and 76 of ref. [14]). Here I use the full
tion 2+k2 2EV0 Ψ=0,andneglecting 2 0(z,b) version of eq. 4.
∇ − ∇ S
relat(cid:0)ive to ik∂ZS0(z(cid:1),b), one gets ik∂ZS0(z,b) = If the state α is in the continuum (positive
EV0S0(z,b), which leads to the center of mass scatter- proton+7Be energ|y)ithe wavefunction is discretized ac-
ing solution (z,b) = exp iv−1 z dz′ V (z′,b) , cording to α;E = dE′ Γ(E′) α;E′ , where the
with v = k/SE0. Using thish−resultRi−n∞the L0ippmanin- functions Γ|(Eα)αiare Rassumαed toα be| strαoingly peaked
around the energy E with width ∆E. For convenience
Schwinger equation, one gets the familiar result for α
the eikonal elastic scattering amplitude, i.e. f = the histogram set (eq. 3.6 of ref. [3]) is chosen. The
0
i(k/2π) db exp(iQ b) exp[iχ(b)] 1 ,wherethe inelasticcrosssectionisobtainedbysolvingthe RCDCC
−eQiko=nalKp′hRaseKisisgivthene ·btryane{sxvpe[riχse(bm)]om=e−nSt0u}(m∞,tbra),nsafenrd. equTahteiopnostaenndtiaulsiVn0gcdoσn/tdaΩindsEcαon=tr|ifbαut(iQon)s|2frΓo2m(Etαh)e. nu-
Therefore,−theelasticscatteringamplitude inthe eikonal clear and the Coulomb interaction. The nuclear po-
approximation has the same form as that derived from tentials are constructed along traditional lines of non-
the Schr¨odinger equation in the non-relativistic case. relativistic theory. The standard double-folding approx-
For inelastic collisions we inserteq. 1 in the KGequa- imation V(aT)(R) = ρ (r)v (s)ρ (r′) is used, where
N a 0 T
tion and use the orthogonality of the intrinsic wavefunc- v0(s) is the effectiveRnucleon-nucleon interaction, with
′
tions φ (ξ). This leads to a set of coupled-channels s=r+R r. The ground-state densities for the pro-
kα −
equations for : ton, 7Be (ρ ), and Pb target (ρ ), are taken from ref.
α a T
S
[15]. The M3Y effective interaction [16] is used for
2+k2 αeikαz = α U α′ α′ eikα′z, (2) v0(s). The nucleus-nucleus potential is expanded into
(cid:0)∇ (cid:1)S Xα h | | i S l = 0, 1, 2 multipolarities. These potentials are then
transformedas the time-like component ofa four-vector,
with the notation |αi = |φkαi. Neglecting terms of the as described above (see also ref. [14]). The multipole
expansion of the Coulomb interaction in the projectile
form 2 (z,b) relative to ik∂ (z,b), eq. 2 reduces frame including retardationand assuminga straight-line
α Z α
to ∇ S S motion has been derived in ref. [14]. The first term
(monopole) of the expansion is the retarded Lienard-
iv∂Sα∂(zz,b) = hα|V0|α′i Sα′(z,b) ei(kα′−kα)z. (3) Wcitiaetcihoenr,tbpuottetnotitahlewcheicnhtedrooefsmnoatsscosnctartitbeuritnegt.o Dthueeetxo-
Xα′
its long range, it is hopeless to solve eq. 3 with the
The scattering amplitude for the transition 0 α is Coulomb monopole potential, as α(z,b) will always
given by → diverge. This can be rectified by Susing the regulariza-
tion scheme S (b) exp[i(2ηln(kb)+χ (b))] S (b),
α a α
ik where η = Z Z /~→v and the S (b) on the right-hand
f (Q)= db exp(iQ b) [S (b) δ ], (4) P T α
α −2π Z · α − α,0 side is now calculated without inclusion of the Coulomb
3
160
1.00
V]
e 120
M
] 0.10 b/
d m
b/ra E = 0.5 - 0.75 MeV E [ 80
[ 0.01 , /d 40
sd
qd
/
sd 0
.0.100 0 0.5 1 1.5 2
e E [MeV]
0.010
E = 1.25 - 1.5 MeV FIG.2: Crosssectionsforthedissociationreaction8B+Pb→
p+7Be+Pb at 83 MeV/nucleon and for θ8 < 1.80. Data are
fromref. [23]. Thedottedcurveisthefirst-orderperturbation
0 2 4 6 8 10
q result. ThesolidcurveistheRCDCCcalculation. Thedashed
[degrees] curve is obtained with the replacement of γ by unity in the
nuclear and Coulomb potentials.
FIG. 1: Angular distributions for the dissociation reaction
8B+Pb → p+7Be+Pb at 50 MeV/nucleon. Data are from
of widths ∆E =250 keV, centered at E =1.25, 1.5...,
ref. [24] and are corrected for the detection efficiency ε. The α α
2.0 MeV, and bins of width ∆E = 0.75 MeV, centered
dottedcurveisthefirst-orderperturbationresult ofref. [25]. α
ThesolidcurveistheRCDCCcalculation. Thedashedcurve at Eα = 2.50, 3.25, ..., 10.00 MeV. Each state α is a
isobtainedwith thereplacement ofγ byunityinthenuclear combinationof energyand angularmomentum quantum
and Coulomb potentials. numbers α = E ,l,j,J,M . Continuum s, p, d and f
α
{ }
waves in 8B were included.
ThecalculationswiththeRCDCCequationswerecom-
monopole potential in eq. 3. The purely imaginary ab- pared to the data of refs. [24] and [23]. The results were
sorptionphase,χa(b), wasintroducedto accountfor ab- folded with the efficiency matrix as well as the energy
sorption at small impact parameters. It has been cal- averagingprocedures explained in ref. [24] and provided
culated using the imaginary part of the “tρρ” interac- by the RIKEN collaboration [24]. At 83 MeV/nucleon,
tion [17], with the 8B density calculatedwith a modified the angular distribution was chosen to match the same
Hartree-Fock model [18]. The E1 and E2 interactions, scattering angles referred to in ref. [23].
taken from ref. [14] replacing vt z. Explicitly,
Figure 1 shows the angular distributions for the
→
dissociation reaction 8B+Pb p+7Be+Pb at 50
VE1µ =r23πξY1µ(cid:16)ˆξ(cid:17)(b2γ+ZγT2ez12e)3/2 (cid:26)√∓2bz ifif µµ==±01, MeneeVrg/ynuEcleboentw. eeDnattahearperoftroonmarn→edf.7B[2e4]i.s avTehraegreedlaotviveer
(5) the energy intervals E=0.5-0.7 MeV (upper panel) and
for the E1 (electric dipole) field, and E = 1.25 1.5 MeV (lower panel). The dotted curve is
−
the first-order perturbation result reported in ref. [25].
3π γZ e e The solid curve is the result of the RCDCC calculation.
VE2µ =r10ξ2Y2µ(cid:16)ˆξ(cid:17)(b2+γT2z22)5/2 The dashed curve is obtained with the replacement of γ
(Lorentz factor) by unity in the nuclear and Coulomb
b2 if µ= 2 potentials. The dashed curve is on average 3-6% lower
±
 2γ2bz if µ= 1 (6) than the solid curves in figure 1. Using non-relativistic
× 2/3∓2γ2z2 b2 if ±µ=0 . potentials yields results always smaller than the full
−
p (cid:0) (cid:1) RCDCC calculation. It is a non-trivial task to predict
forthe E2(electric quadrupole)field, wheree = 3e and whattherelativisticcorrectionsdoinacoupled-channels
1 8
e = 53e are effective charges for p+7Be. For γ 1 calculation.
2 64 →
these potentials reduce to the non-relativistic multipole Figure 2 shows the relative energy spectrum between
fields (see, e.g., eq. 2 of ref. [19]) in distant collisions. the proton and the 7Be after the breakup of 8B on lead
Asingle-particlemodelwasusedfor8BwithaWoods- targets at 83 MeV/nucleon. The data are from ref. [23].
Saxonpotentialadjustedtoreproducethebindingenergy In this case, the calculation was restricted to b>30 fm.
of 0.139 MeV [20, 21, 22]. I follow the method of ref. [3] The dotted curve is the first-order perturbation calcu-
anddividethecontinuumintobinsofwidths∆E =100 lation, the solid curve is the RCDCC calculation, and
α
keV,centeredatE =0.01,0.11,0.21,...,1.01MeV,bins the dashed curve is obtained with the replacement of γ
α
4
by unity in the nuclear and Coulomb potentials. The crease the cross sections. At 250 MeV/nucleon they can
difference between the solid and the dashed-curve is of reacha 15% value. This has been treated before in first-
the order of 4-9%. I have also used the DSSE method, order perturbation theory, but not in the dynamical cal-
described in ref. [6], to compute the same spectrum, culationswithcontinuum-continuumcouplingusedinthe
with the same partialwaves,assuming a largelowercut- experimentalanalysis[23, 26]. The consequence of using
off in the parameter of b = 30 fm. This would justify these approximations on the extracted values of the as-
a semiclassical limit. The same relativistic nuclear and trophysicalS-factorsforthe7Be(p,γ)reactioninthe sun
Coulombpotentials have been used in both calculations. isnoteasytoaccess. Itmightbenecessarytoreviewthe
The differencebetweenthe tworesults(the RCDCC and results of some of these data, using a proper treatment
the DSSE) is very small (less than 2%) for the whole of the relativistic corrections in the theory calculations
rangeofthespectrum. Tomyknowledgesuchacompar- used in the experimental analysis. Other improvements
isonhasneverbeenmadebefore. Thisisaproofthatthe of the present formalism needs to be assessed. The rela-
twomethodsyieldthesameresult,ifthesamepotentials tivisticeffectsinthenuclearinteractionhastobestudied
are used, and as long as large b’s are considered. Such a inmoredepth. TheeffectofcloseCoulombfields[27,28]
comparison is only possible because the same potential should also be considered in the case of dissociation of
was used for the bound-state and the continuum. The halo nuclei.
DSSE method [6] does not allow for use of different po-
tentials for p+7B. This is not a problem in the RCDCC Lab energy ∆ ∆ ∆
method, since the states b and c can be calculated
[MeV/nucleon] E =0.1 MeV E =1 MeV E =2 MeV
| i | i
within any levelof sophistication,beyond the simple po-
50 1.5% 4.2% 3.4%
tential modeladopted here. Inthis respect, the RCDCC
is superior than the DSSE method and is more suitable 80 3% 5.5% 4.1%
for an accurate description of dynamical calculations. 250 5.3% 14.6% 6.9%
The conclusions drawn in this work are crucial in Table 1: Relativistic corrections in the dissociation of
the analysis of Coulomb breakup experiments at high 8Bprojectilesimpingingonleadtargetsatdifferentbom-
bombarding energies, as the GSI experiment at 254 barding energies. E is the relative energy of the proton
MeV/nucleon [26]. In table 1 I show the calculations for and 7Be.
the correction factor ∆ = (σRCDCC σCDCC)/σCDCC I would like to thank T. Aumann, H. Esbensen, K.
for the dissociation of 8B on lead targ−ets at 3 bombard- Suemmerer and I. Thompson for beneficial discussions.
ing energies. E is the relative energy of the proton and ThisworkwassupportedbytheU.S.DepartmentofEn-
7Be. Oneseesthattherelativisticcorrectionstendtoin- ergy under grant No. DE-FG02-04ER41338.
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