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## Lncs 651 - an introduction to halo nuclei

**An Introduction to Halo Nuclei**
Department of Physics, University of Surrey, Guildford, GU2 7XH, UK

**Abstract. **This lecture will not aim to provide an exhaustive review of the ﬁeld of

halo nuclei, but rather will outline some of the theoretical techniques that have been

used and developed, both in structure and reaction studies, over the past decade to

understand their properties. A number of review articles have recently appeared in

the literature [1–10] which the interested reader can then go to armed with a basic

understanding of how the theoretical results were produced.

**What Is a Halo?**
The ﬁeld of halo nuclei has generated much excitement and many hundredsof papers since its discovery in the mid-1980s. While early

*β*- and

*γ*-decaystudies of many of these nuclei yielded information about their lifetimes andcertain features of their structure, credit for their discovery should go mostlyto Tanihata [11,12] for the work of his group at Lawrence Berkeley Labora-tory's Bevalac in 1985 on the measurement of the very large interaction crosssections of certain neutron-rich isotopes of helium and lithium, along withHansen and Jonson for their pioneering paper two years later in which theterm ‘halo' was ﬁrst applied to these nuclei [13]. Of course, it is worth men-tioning that the ﬁrst halo nucleus to be produced in the laboratory was 6He,as long ago as 1936, using a beam of neutrons on a 9Be target [14] just a fewyears after the discovery of the neutron! In contrast, the discovery of 11Li,now regarded as the most famous halo nucleus, was not made until thirtyyears later [15], although its remarkable features had to wait a further twodecades to be appreciated. We should begin therefore by deﬁning just whatconstitutes a halo nucleus and under what conditions it will manifest itself.

The halo is a

*threshold *eﬀect arising from the very weak binding of the
last one or two valence nucleons (usually neutrons) to, and hence decouplingfrom, a well-deﬁned inert ‘core' containing all the other nucleons. Textbookquantum mechanics states that the combination of weak binding and shortrange nuclear force (since the core is relatively compact) means that theneutron(s) can tunnel out into a volume well beyond the nuclear core and intothe ‘classically-forbidden' region. Consider for instance the eigenfunctions ofa particle bound in a ﬁnite 1-D square well potential. Deeply-bound states aremostly conﬁned within the potential and have very little extension beyondits walls. But states with eigen-energies just below the surface of the well willhave slowly decaying exponential tails extending well beyond the range of
J. Al-Khalili, An Introduction to Halo Nuclei, Lect. Notes Phys.

**651**, 77–112 (2004)

* *Springer-Verlag Berlin Heidelberg 2004
the potential. Quantum mechanically, this means that there is a signiﬁcantprobability of ﬁnding the particle outside of the well.

In halo nuclei, the potential well corresponds to the mean ﬁeld potential of
the rest of the nucleons in the nucleus. The valence nucleon (we will restrictthe discussion to one halo nucleon for now) has a good chance of ﬁndingitself outside the core. The Uncertainty Principle ensures that such boundstates have a relatively short lifetime, of the order of a few milliseconds to afew seconds. We will see that this is quite long enough for such nuclei to beformed and used in nuclear reactions in order to study their unusual features.

The accepted deﬁnition of a halo nucleus (typically in its ground state)
is therefore that the halo neutron is required to have more than 50% of itsprobability density outside the range of the core potential. In such an openstructure, it is not surprising that shell model and mean ﬁeld approachesto describe such systems break down, and that few-body (cluster) modelsof core plus valence particles can account for the most general properties ofthese nuclei, such as their large size and breakup cross sections.

In addition to the decoupling of core and valence particles and their small
separation energy, the other important criterion for a halo is that the valenceparticle must be in a low relative orbit angular momentum state, preferablean

*s*-wave, relative to the core, since higher

*l*-values give rise to a conﬁningcentrifugal barrier. The conﬁning Coulomb barrier is the reason why protonhalos are not so spatially extended as neutron halos.

Since halo nuclei are short-lived they must be studied using radioactive
beam facilities is which they are formed and then used to initiate a nuclearreaction with a stable target. Indeed, most of what is know about halo nucleicomes from high energy fragmentation reactions in which the halo projectileis deliberately broken up and its fragments detected.

**Examples of Halo Nuclei**
The three most studied halo nuclei are 6He, 11Li and 11Be. However, a fewothers have also now been conﬁrmed, such as 14Be, 14B, 15C and 19C. All theabove are examples of neutron halo systems, and all lie on, or are close to, theneutron dripline at the limits of particle stability. Other candidates, awaitingproper theoretical study and experimental conﬁrmation include 15B, 17B and19B, along with 22C and 23O. Proton halo nuclei are not quite as impressivein terms of the extent of their halo, due to the conﬁning Coulomb barrierwhich holds them closer to the core. Nevertheless, examples include 8B, 13N,17Ne and the ﬁrst excited state of 17F. We will deal for the most part herewith the neutron halos. Another special feature is that most halos tend tobe manifest in the ground states of the nuclei of interest. Indeed, most of theknown halo nuclei tend to only have one bound state; any excitation of sucha weakly bound system tends to be into the continuum, with the notableexception of 11Be which has two bound states.

An Introduction to Halo Nuclei

**Fig. 1. **The two most studied cases are the two-neutron halo nucleus 11Li and the

one-neutron halo nucleus 11Be.

Excited state halos are less well studied. There is a danger of thinking
that many nuclei will have excited states just below the one-neutron breakupthreshold that exhibit halo-like features. After all, if the only criterion is thatof weak binding then surely excited state halos would be everywhere. This isnot the case, however, since, in addition, the core nucleons must be tightlybound together and spatially decoupled from the valence neutron.

**Experimental Evidence for Halos**
The ﬁrst hint that something unusual was being seen came from the measure-ment of the electric dipole transition between the two bound states in 11Be.

Firstly, a simple shell model picture of the structure of 11Be would suggestthat its ground state should consist of a single valence neutron occupyingthe 0

*p*1

*/*2 orbital (the other six having ﬁlled the 0

*s*1

*/*2 and 0

*p*3

*/*2 orbitals).

However, it was found that the 1

*s*1

*/*2 orbital drops down below the 0

*p*1

*/*2and this ‘intruder' state is the one occupied by the neutron, making it a
1 + ground state. The ﬁrst excited state of 11Be, and the only other parti-
cle bound state, is the 1
state achieved when the valence neutron occupies
the higher 0

*p*1

*/*2 orbital. The very short lifetime for the transition betweenthese two bound states was measured in 1983 [16] and corresponded to anE1 strength of 0.36 W.u. It was found that this large strength could onlybe understood if realistic single particle wavefunctions were used to describethe valence neutron in the two states, which extended out to large distancesdue to the weak binding. Thus the radial integral involved in calculating the

**Halo or skin?**
**2-n halos (Borromean)**
**Fig. 2. **A section of the Segre chart showing the halo nuclei.

transition had to be extended to a large distance, evidence of a long rangetail to the wavefunction: the halo.

The Berkeley experiments carried out by Tanihata and his group in the
mid-1980s involved the measurement of the interaction cross sections ofhelium and lithium isotopes and were found, for the cases of 6He and 11Li,to be much larger than expected. These corresponded to larger rms matterradii than would be predicted by the normal

*A*1

*/*3 dependence. Hansen andJonson [13] proposed that the large size of these nuclei is due to the halo ef-fect. They explained the large matter radius of 11Li by treating it as a binarysystem of 9Li core plus a dineutron (a hypothetical point particle, implyingthe two neutrons are stuck together – of course the n-n system is unbound)and showed how the weak binding between this pair of clusters could forman extended halo density.

During the late eighties and early nineties, both theorists and experi-
mentalists seemed satisﬁed with simple estimates of various halo propertiesby reproducing experimental reaction observables such as total reaction andCoulomb dissociation cross sections and momentum distributions followingnuclear breakup. The high beam energies – the Berkeley experiments in-volved nuclear beams of about 800 MeV/nucleon – meant that semi-classicalapproaches could be reliably used in reaction models. More sophisticated nu-merical calculations, of both structure and reactions, have since been carriedout over the past few years. Much of this article will be devoted to describ-ing some of these models and showing how many of the formulae used tocalculate certain observables are derived.

An Introduction to Halo Nuclei
Many of the general features of one-neutron halo nuclei can be studied usinga simple 2-body (cluster) model of core + valence neutron bound by a shortrange potential. If the internal degrees of freedom of the nucleons in the coreare decoupled from that of the single remaining valence neutron then we cansimplify the many-body nuclear wavefunction,

*ΦA ≈ φ*core(

*ξ*)

*ψ*(

*r*)

*,*
where

*ξ *denotes the core's intrinsic coordinates and

*ψ*(

*r*) is the bound state
wavefunction of relative motion of core and valence neutron. One of the cri-teria for a halo state to exist is if the total probability for the neutrons tobe found outside the range of the potential is greater than the correspondingprobability within the potential (i.e. the neutron is most likely to be foundbeyond the reach of the potential that is binding it to the core). Outside thepotential, the wavefunction has a simple Yukawa form

*ψ*(

*r*) =

*N*
which describes its asymptotic behaviour and depends only on the binding(or ‘separation') energy of the neutron via
where

*µ *is the reduced mass of the core-neutron system and

*Sn *is the sepa-ration energy. Clearly, the closer

*Sn *is to zero, the slower the wavefunctionfalls to zero (see Fig. 3).

For halo nuclei, therefore, the dominant part of the wavefunction lies
outside the potential and most of the physics comes from the behaviour ofits tail. Indeed, its properties depend little on the shape of the potential. Themean square radius of such a wavefunction is thus

*r*4

*dr *(

*e−κr/κr*)2

*r*2

* *=

*r*2

*dr *(

*e−κr/κr*)2
That is, the rms radius of the halo is inversely proportional to the squareroot of the separation energy. Such a diverging radius as the separation en-ergy tends to zero is only true for orbital angular momenta

* *= 0

*, *1. Thisexplains why halo states require low relative angular momentum for the va-lence particle, as well as weak binding. For

* ≥ *2 the radius converges withdecreasing separation energy since the centrifugal barrier pushes the boundstate into the potential. Necessary and suﬃcient conditions for the formation

**Fig. 3. **A plot of the matter radii of isotopes of He, Li and Be as predicted by

reaction cross section measurements and deduced from Glauber model calculations

[17,18]

of a halo have been investigated [19] and universal scaling plots that relateradii to binding energies can be used to evaluate possible halo candidates [20].

Quite realistic wavefunctions for one-neutron halo nuclei such as 11Be
can be modelled by solving the 2-body bound state problem with a Woods-Saxon binding potential of appropriate geometry and with the depth chosento produce the correct separation energy.

An obvious question is whether a halo is assumed to have formed whenever
the last valence neutron is weakly bound (one MeV or less) and in a relative

*s*or

*p *state. Clearly, while examples of ground state nuclides with this featureare rare, there must be many examples of excited state just below thresholdthat have this feature. Many such states are unlikely, however, to representclear halo signatures due to the high density of states in those regions. Thecore is unlikely to be tightly bound, inert and with internal degrees of freedomdecoupled from the valence neutron. There are likely to be exceptions to thisof course and one possible candidate is the 2

*− *state in 10Be, which can bedescribed roughly as an

*s*-wave neutron bound by just 0.5 MeV to a 9Be core.

Whether the core is mainly in its ground state is questionable however.

Such simple models of one-neutron halo nuclei in terms of the neutron's
single particle wavefunction are often not accurate enough to account forthe physics that can now be accurately measured experimentally. Instead,one must go beyond this picture in which the core remains inert and in its
An Introduction to Halo Nuclei
ground state. It is well accepted for instance that the loosely bound neutronin the 1

*/*2+ ground state of 11Be is mainly in an

*s*1

*/*2 state, but there is alsoa signiﬁcant core excited 10Be(2+) component coupled to a

*d*5

*/*2 neutron.

Similar results are found in many other halo and exotic light nuclei.

We will deal in the next section with two-neutron halos, which have rather
special features that deserve theoretical investigation. However, we end thissection by considering brieﬂy whether multineutron halos can exist. The dif-ference in Fermi energies between neutrons and protons in nuclei leads to amore extended neutron density distribution than that of the protons. Thisdiﬀerence is called the neutron skin and is a feature of most heavy nuclei.

However, in such nuclei the neutron distribution has the same bulk and sur-face features (diﬀuseness) as the proton distribution. This is quite diﬀerentto the halo, which is characterised by its long range and dilute nature. Anexample of a nucleus with features that are on the boundary between a haloand a skin is 8He. This nucleus is well-described as an alpha core plus fourvalence neutrons. While it has a similar matter radius to its halo sister, 6He,its valence neutron distribution does not extend out so far. It has thereforebeen remarked that, while 8He is not a halo system, it is surrounded bysuch a thick neutron skin that it is akin to a mouse with the skin of anelephant. Clearly, the more valence neutrons there are outside the core, themore strongly their mutual attraction will hold them together, preventing adramatic halo from forming.

**Three-Body Systems – The Borromeans**
Two-neutron halo nuclei, such as 6He and 11Li have the remarkable propertythat none of their two-body subsystems are bound. Thus, 6He can be mod-elled as a bound three-body

*α *+

*n *+

*n *system despite there being no boundstates of

*α *+

*n *(5He) or

*n *+

*n *(the dineutron). Such nuclei have been dubbed‘Borromean' [21] and their wavefunctions require rather special asymptoticfeatures to account for this behaviour.

The relative motion of the core and two neutrons is deﬁned in terms of the
Jacobi coordinates (

*y*) as in Fig. 5. An extention of the one-neutron halo
case suggests we can once again simplify the full many-body wavefunction bywriting

*ΦA ≈ φ*core(

*ξ*)

*ψ*(

*x, y*)

*,*
where the relative wavefunction,

*ψ*, is a solution of a three-body Schr¨
equation. While it is a non-trivial problem to calculate

*ψ*, we can neverthelessreduce this 6-D equation to a one-dimensional ‘radial' equation using hyper-spherical coordinates (

*ρ, α, θx, ϕx, θy, ϕy*) where

*ρ *=

*x*2 +

*y*2 is the

*hyperradius*and

*α *= tan

*−*1(

*x/y*) is the hyperangle. Just as the 3-D equation describingthe hydrogen atom (a 2-body system) is reduced to a radial one by sepa-rating out the angular dependence as spherical harmonics (eigenfunctions of
Bound state in a square well potential
Density (arb. units)

**Fig. 4. **The dependence of the wavefunction tail of a particle bound inside a square

well potential on separation energy (the distance from the top of the well).

the angular momentum operator), we can again reach a 1-D equation in

*ρ*by separating out all angular dependence within ‘hyperspherical harmonics'[21]. Outside the range of the potential, the radial equation has the form
(

*K *+ 3

*/*2)(

*K *+ 5

*/*2)

*− *2

*mE χ*(

*ρ*) = 0

*,*
where the new quantum number

*K *is called the

*hypermoment *and is thethree-body extension of the orbital angular momentum quantum number.

However, an important diﬀerence between this and the two-body case is thateven for

*K *= 0 (corresponding to relative

*s*-waves between the two neutronsand between their centre of mass and the core) there is still a non-zero eﬀec-tive centrifugal barrier. The asymptotic behaviour of the ‘radial' wavefunctionis now of the form

*χ*(

*ρ*)

*∼ e−κρ ,*
which is a generalisation of the Yukawa form (2) for the case of three-body
asymptotics, where here

*κ *=

*h *involving the nucleon mass

*m *and
the two-neutron separation energy

*S*2

*n*. Note here that we do not talk of the
An Introduction to Halo Nuclei
separation energy for a single neutron since the Borromean nature of suchsystems means that if one of the halo neutrons is removed, the other will also‘fall oﬀ'.

The hyperradius,

*ρ*, provides a useful measure of the extent of the halo
for the case of Borromean nuclei since it depends on the magnitudes of bothJacobi coordinates. The overall matter radius of such systems is deﬁned as

*r*2

* *=
(

*A − *2)

*r*2

core +

*ρ*2

* ,*
where

*r*2

core is the intrinsic mean square radius of the core. A typical ex-ample with numbers is 11Li: the radius of 9Li is 2.3 fm while the root meansquare hyperradius, describing the relative motion of the valence neutronsrelative to the core, is about 9 fm. Together, these give a mass-weightedoverall radius for 11Li of about 3.5 fm.

It is of course questionable whether the above approach is a sensible way
of deﬁning the size of a halo nucleus. No one would suggest that the size ofan atom be deﬁned as the mass weighted sum of the sizes of its electron cloudand its nucleus. This is why many popular accounts of halo nuclei describe11Li as being the same size as a lead nucleus rather than, say, 48Ca, whichalso has a radius of about 3.5 fm.

A number of elaborate techniques have been used to calculate the three-
body wavefunctions of Borromean nuclei [21–26]. Such approaches assumedtwo-body pairwise potentials between the three constituents. It is importantto treat the three-body asymptotic behaviour of the wavefunctions correctlyin order to reproduce the basic features of these nuclei as well as the variousreaction observables described in the next section.

Of course, projecting the full many-body wavefunction onto two- or three-body model spaces as was done in (1) and (5) is just an approximation. Thefew-body models of the structure of halo nuclei suﬀer from several shortcom-ings, namely that antisymmetrisation is often treated only approximatelyand that excitation and polarisation eﬀects of the core are often ignored, al-though a number of studies are in progress to improve on these deﬁciencies.

In favour of such cluster models of course is that the important few-bodydynamics and asymptotics are included correctly. A number of studies arecurrently developing fully microscopic (

*ab initio*) structure models. These arefully antisymmetric, start from a realistic NN interaction and can even in-clude 3-body forces. The standard shell model fails to describe many of theessential features of halo nuclei (although it has proved to be of importancein providing spectroscopic information on a number of exotic nuclei) andmany theorists acknowledge that there is a real need to go beyond the con-ventional shell model. For instance, the Continuum Shell Model and GamowShell Model [27] are showing promising early results. For very light systems,
progress is being made with the No-Core Shell Model [28], and the hope isthat there will be a convergence of these two methods. But will they be ableto predict a matter radius for 11Li of 3.5 fm?
The acknowledged front runner among such

*ab initio *microscopic struc-
ture models is the Greens Function Monte Carlo method (GMC) [29]. Thisapproach involves calculating an approximate

*A*-body wavefunction usingthe variational Monte Carlo method then using Greens function projectionmethods to obtain the desired bound state wavefunction. To date, the GMCmethod has been applied to describe the bound states of nuclei up to

*A *= 12,including the halo states. However, it will have problems going to systemsany higher in mass.

Another promising approach is the Coupled Cluster Method [30]. This
has been used widely in a number of other ﬁelds such as chemistry andatomic and condensed matter physics and has only recently been appliedseriously to nuclear structure. While still in its early stages, it has beentested successfully against GMC for 4He. Its supporters are hopeful that itwill be more successful in reaching heavier dripline nuclei than GMC.

No more will be said about such sophisticated methods here and the
interested reader, as always, is directed to the listed references for furtherdetails.

While there are many ways of probing the structure of nuclei through observ-ing how they decay – and much has been learnt about halo nuclei from betadecay studies – I will focus here on nuclear reaction studies, which is the mainactivity in the ﬁeld. It must be stressed that since halo nuclei are shortlived(less than a second) they cannot be used as a nuclear target. Instead theymust form the beam that interacts with a stable target. Of course the physicsis still the same and we can think of the reaction as taking place in ‘inversekinematics'.

During the late eighties and early nineties, both theorists and experimen-
talists seemed satisﬁed with quick and dirty estimates of various halo proper-ties by reproducing experimental reaction observables, such as total reactionand Coulomb dissociation cross sections and momentum distributions fol-lowing nuclear breakup. For instance, the rms matter radii were deduced bycomparing calculated reaction cross sections with experimentally measuredinteractions cross sections [for such loosely-bound systems, these two quanti-ties are essentially equal]. The high beam energies meant that semi-classicalapproaches, such as the Glauber model [31], could be reliably used. It will beshown later on how many useful formulae for reaction observables can be de-rived starting from the Glauber model, but it is worth pointing out that thebasic expression for the total reaction cross section was around long beforeGlauber's work in the late 1950s and can be traced back to Hans Bethe in
An Introduction to Halo Nuclei
1940. It relies on a simple geometric picture of the reaction process in whichthe target presents a circular disc blocking the path of the projectile. Thereaction cross section is then an integral over impact parameter

*σR *= 2

*π*
*b db *(1

*− T *(

*b*))
where

*T *(

*b*) is the probability of transmission (or the ‘transparency'). It is thisquantity that, in more sophisticated approaches, contains information aboutthe density distributions of halo projectile and target nuclei. In fact, if onewere to assume simple Gaussian density distributions for both nuclei, thenan analytical expression can be evaluated for the reaction cross section [32].

The rms matter radius thus enters through the Gaussian parameter in thedensity. Such an analysis was used widely in the early work on halo nucleiand the deduced radii were known as ‘experimental' ones on the assumptionthat the calculated reaction cross section was model independent. It has sincebeen shown that this is not correct since the densities of halo nuclei are farfrom Gaussian shaped.

The momentum distribution of the fragments, following nuclear breakup
of the halo projectile, was another observable analysed within a simple semi-classical (geometric) picture. By assuming that the target nucleus representeda fully absorptive black disk, then making a Serber (or sudden) approximation[33,34] (in which it is assumed that the surviving – non-absorbed – fragmentcontinues with the same velocity as that of the incident beam) and neglectingany reaction mechanisms or ﬁnal state interactions, it could be shown thatthe momentum distribution of the fragments was a good approximation tothe momentum distribution of those clusters in the initial bound nucleus.

The ground state wavefunction of the halo nucleus is thus just the FourierTransform of the measured momentum distribution. Indeed the narrow dis-tributions that were found for many of the halo candidates was conﬁrmationof their large spatial extent.

It is indeed easy to show that, for a Yukawa wavefunction of the form of
(2), its Fourier Transform gives a longitudinal momentum distribution of the(Breit-Wigner) form

*p*2 +

*κ*2
where

*κ *was deﬁned in (3). Clearly, the smaller the separation energy, andthus the smaller

*κ *is, the narrower the momentum distribution. Figure 5shows momentum distributions for a range of light nuclei. It is easy to spotthose that have an extended spatial distribution due to a halo.

But of course, such simplistic models can only tell us basic information,
and we need to work harder. In particular, in modelling reactions involvinghalo nuclei, it is important to note that the few-body correlations that arebuilt into these structure models have to be retained. An important con-sideration in the study of reactions with halo nuclei is that they are easily

**Fig. 5. **Top: The Jacobi vectors of a three-body system. Bottom: A correlation

density plot of the ground state of 6He against the two Jacobi coordinates.

broken up in the nuclear and Coulomb ﬁelds of the target nucleus. Therefore,excitations of the halo nucleus into the continuum must be included in thereaction model. Such intermediate state coupling rules out ‘one-step' modelssuch as DWBA for many reactions of interest. It has been shown that anyreliable reaction model must also take into account the few-body nature ofthese nuclei.

In parallel with advances in developing few-body structure models, there-
fore, theorists also developed few-body reaction models. At high (fragmen-tation) energies at which many of the experiments have been performed, anumber of simplifying assumptions can be made to make the calculation ofthe reaction observables both tractable and transparent.

The most precise method of dealing with this problem is to map the
continuum onto a discrete square-integrable basis that is orthogonal to the
An Introduction to Halo Nuclei

**Fig. 6. **The lngitudinal momentum distributions for the core fragments following

single neutron removal from a range of neutron-rich nuclides on a carbon target

[35]. The narrow the distribution, the larger the size of the nucleus.

bound states. This amounts to "chopping up" the continuum into energy binsthat act as eﬀective discrete excited states of the projectile and allows theproblem to be solved within a (ﬁnite number of) coupled channels approach.

This is the so called coupled discretised continuum channels (CDCC) method[36,37] that will be introduced in Sect. 4.6.

Another common approach is to make use of the adiabatic, or ‘sudden',
approximation [38] whereby it is assumed that the interaction time betweenthe projectile and target is suﬃciently short that the halo degrees of freedomcan be regarded as frozen. This will be discussed in more depth in Sect. 3.7.

The most successful few-body approach for calculating probabilities and
cross sections for a range of reactions involving halo nuclei has been based onGlauber's multiple scattering diﬀraction theory for composite systems [31,39]. This model requires making an eikonal assumption in addition to thesudden approximation. This will be discussed in more detail next.

Few-body reactions at lower incident energies are far more diﬃcult to
treat consistently. Not only do nuclear and Coulomb interactions need to betreated within the same model to account correctly for interference eﬀects,but multistep processes are even more important than at higher energies.

An advantage of the CDCC method is its applicability at low energies whereapproximation schemes used in many other few-body approaches break down.

**The Glauber Model**
The theoretical study of scattering and reactions involving halo nuclei atthe relatively high fragmentation beam energies at which so many of the
experiments have been conducted has led to a renaissance of a number ofsemi-classical reaction models. In particular, the Glauber model, which wasﬁrst applied to nuclear scattering in the late 1950s, provides a remarkablysimple framework for calculating various important observables arising fromexperiments involving loosely-bound projectiles such as halo nuclei. Its cen-tral assumption is the eikonal approximation: that the projectile travels alonga deﬁnite straight line trajectory through the ﬁeld of the target nucleus allowsfor the derivation of a number of crucial yet simple cross section formulae.

It forms the basis for the reaction cross section expression that was used toanalyse the early interaction cross section measurements which conﬁrmed thelarge halo size. Later, a few-body generalisation of the Glauber model wasused to provide a more accurate prediction for this size, and provided theﬁrst realistic calculation - one that included the important few-body struc-ture information of the halo - of elastic scattering angular distributions fora number of nuclei such as 6He, 8He, 11Be, 11Li and 14Be. Even today, it issuch an important tool in reaction theory studies that it is worth discussingin some detail.

Let us ﬁrst examine conditions for the validity of the eikonal approx-
imation. Consider the scattering of a point particle from a potential withstrength

*V*0 and range

*a*. We can deﬁne a quantity

*τ*1 as the time spent bythe projectile in the interaction region:
where

*v *= ¯

*hk/m *is the classical velocity of the projectile. Also,

*τ*2 =

* V*0

* ,*
is the time necessary for the scattering potential to have a signiﬁcant eﬀecton the projectile. The ratio of these two times, a ‘coupling parameter', is thus

* V*0

* a*
We therefore have two simple limits of this coupling parameter:

* *1

*⇒ *weak coupling limit (Born condition)

* *1

*⇒ *strong coupling limit (WKB condition)

*.*
In addition to these two limits, another way of distinguishing between theBorn and WKB approximations is to think of the former as describing thescattering of waves (wavelike nature of projectile) whereas the latter describesparticle scattering and is semi-classical in the sense of the projectile followinga deﬁnite path or trajectory.

An Introduction to Halo Nuclei
So what does the eikonal picture suggest? In common with other semi-
classical approaches, the eikonal method is useful when the wavelength of theincident particle is short compared with the distance over which the potentialvaries appreciably. This short wavelength condition is expressed in terms ofthe incident wave number,

*k*, and the range of the interaction,

*a*, such that

*ka *1

*.*
However, unlike the short wavelength WKB approximation, the eikonal ap-proximation also requires high scattering energies, such that

*E V*0

* .*
It is helpful to re-express the coupling parameter of (13) in terms of thesetwo conditions noting that we can write ¯

*h*2

*k/m *= 2

*E/k*. Thus

* V*0

* *
So while the eikonal approximation holds when the ﬁrst factor is small andthe second factor large, it says nothing about their product! In that sense,the eikonal approximation spans both the Born and WKB limits and con-tains elements of both. In practice, and when

*V *is complex, the high energycondition is not critical and the eikonal approximation works well even when

*E ≈ V*0

* *provided the ﬁrst condition, (15), holds and we restrict ourselvesto forward angle scattering. The reason for this is simple: a short range ab-sorptive part to the scattering potential removes ﬂux from the interior (smallimpact parameters) where the magnitude of the potential is large.

Since the potential varies slowly on the length scale of the incident wave-
length, it is reasonable to extract the free incident plane wave from the scat-tering wave function as a factor, i.e.

*ψ*(

*R*) =

*ei k· *
*R ω*(

*R*)

*,*
where

*ω*(

*R*) is a modulating function and

*R *is the projectile-target separation
vector. The eikonal approximation can be derived starting from either theSchr¨
odinger equation or the Lippmann-Schwinger equation. Here we follow
the ﬁrst approach. The scattering wave function of (18) is substituted in theSchr¨

*∇*2 +

*k*2

*− *2

*µV ei k· R ω*(

*R*) = 0

*,*
where

*µ *is the reduced mass. Using the eikonal conditions of (15) and (16)and with the coordinate

*z*-axis along the incident wave vector

*k*, (19) reducesto the ﬁrst order equation for

*ω*
*∂ω *=

*− iµ V ω .*
The solution of this equation, with the incident wave boundary conditionrequirement that

*ω*(

*z → −∞*) = 1 , is

*ω*(

*V *(

*x, y, z*)

*dz*
*h*2

*k −∞*
and yields the eikonal approximation to the wave function

*ψeik*(

*V *(

*x, y, z *)

*dz ,*
Thus, the modulating function introduces a modiﬁcation to the phase of theincident plane wave that involves an integration along the direction of theincident beam and, as such, assumes that the eﬀects of

*V *are accuratelyaccounted for by assuming the projectile traverses a straight line path. Theeikonal method is therefore more accurate at forward scattering angles.

The scattering wavefunction of (22) has incorrect asymptotics since it does
not look like incident plane wave plus outgoing spherical wave at

*R → ∞*.

So, to calculate amplitudes and observables, it can only be used within atransition amplitude. For two-body elastic scattering via a central potential

*V *(

*R*) the transition amplitude is

*T *(

*k, *
*k *) =

* *
*k V ψeik .*
This leads to the well-known form of the scattering amplitude

*f *(

*θ*) =

*−iK*0

*b db J*0(

*qb*) [

*S*0(

*b*)

*− *1 ]

*,*
where

*q *= 2

*k *sin(

*θ/*2),

*θ *is the cm scattering angle and

*S*0(

*b*) = exp [

*iχ*(

*b*)]is the eikonal elastic

*S*-matrix element at impact parameter

*b*. The eikonalphase shift function,

*χ*(

*b*), is deﬁned as

*χ*(

*b*) =

*− *1

*V *(

*R*)

*dz .*
In order to apply the Glauber model to reactions involving composite
projectiles such as halo nuclei, we generalise the eikonal approach to what iscalled the Few-Body Glauber (FBG) model.

The FBG scattering amplitude, for a collision that takes a composite

*n*-
body projectile from an initial state

*φ*(

*n*) to a ﬁnal state

*φ*(

*n*)
following the same steps as those in the two-body (point particle projectile)case. The post form transition amplitude is

*T *(

*Kα*) =

*φ*(

*n*)

*ei *
*R U*(

*{ *
*j }*)

* Ψ eik*
where

*U *is the sum of projectile constituent-target interactions

*U *(

*R*1

*, · · · , *
*VjT *(

*Rj*)

*,*
An Introduction to Halo Nuclei
and

*Ψ eik *is the eikonal approximation to the full (

*n *+ 1)-body scattering
wavefunction deﬁned as the generalisation of the eikonal wavefunction of(18) to

*Ψ eik *=

*ei *
*K*0

*· *
*R ω*(

*{r*
*i}*)

*.*
By substituting for the eikonal wavefunction in the transition amplitude

*f *(

*n*)(

*Kα*) =

*− iK*0

*db ei q· b φ*(

*n*)

* S*(

*n*)(

*b*1

*, · · · ,bn*)

*− *1

* φ*(

*n*)

*,*
*S*(

*n*) = exp

*i*
*j *(

*bj *)
Thus the total phase shift is the sum of the phase shifts for the scattering ofeach of the projectile's constituents. This property of phase shift additivityis a direct consequence of the linear dependence of eikonal phases on theinteraction potentials

*Vjt*.

Corrections to the straight line assumption of the eikonal approximation
have been calculated and allow the FBG approach to be applied at consider-ably lower energies than expected (below 20 MeV/A). The most straightfor-ward approach is to replace the eikonal S-matrices by the physical ones, ob-tained by solving the Schr¨
odinger equation exactly for each cluster-target (2-
body) subsystem, and then retaining the simplicity of the impact parameterframework of the model [40] by utilising the semi-classical limit

*bk *=

* *+ 1

*/*2.

The model generalises in a natural way when Coulomb forces are included
in the projectile constituent-target potentials,

*VjT *.

**The Optical Limit of the Glauber Model**
The Glauber model can be simpliﬁed considerably at high energies when theinteraction between each projectile constituent and the target is purely ab-sorptive. In this case, each constituent S-matrix,

*Sj*(

*bj*), is calculated withinthe optical limit of the Glauber model [41]. Here, the eikonal phase shiftsare calculated assuming a ‘

*tρρ*' approximation to the optical potentials,

*VjT *,using one-body densities for each

*j *constituent and the target and an eﬀec-tive nucleon-nucleon amplitude,

*fNN *. The optical limit

*S*-matrices are thuswritten as

*SOL*(

*b*) =

*exp i*
*r*2

*ρj*(

*r*1)

*ρT *(

*r*2)

*fNN *(

* *
*R *+

*r*1

*− r*2

* *)
For an absorptive zero range NN amplitude and an isospin zero target wehave

*fNN *(

*r*) = (

*i*¯

*σNN *is the average of the free nn and np total cross sections at the
energy of interest, which enter through the use of the optical theorem.

It is important to note that we have not thrown away here the few-body
correlations in the projectile since at this stage it is only the constituents'scattering via their individual

*Sj*'s that have been treated in OL. The few-body

*S*-matrix is still deﬁned according to (30). However, if all few-bodycorrelations are also neglected then

*S*(

*n*) is replaced by

*SOL*, deﬁned as forthe individual

*SOL *but with

*ρ*
*j *replaced by the one-body density for the whole
projectile. In this case it can easily be shown that the full projectile-targetOL

*S*-matrix is equivalent to neglecting breakup eﬀects in (29), i.e.,

*SOL*(

*b*) = exp

*φ*(

*n*)

* i*

*.*
*j *(

*bj *)

* φ*(

*n*)
This is discussed in more detail in [39].

**Cross Sections in Glauber Theory**
The Glauber model provides a convenient framework for calculating inte-grated cross sections for a variety of processes involving peripheral collisionsbetween composite projectiles and stable targets. In particular, stripping re-actions have been studied using approaches developed by Serber [33]. Variantsof such methods are still in use today due to the simple geometric propertiesof the reaction processes at high energies.

In the few-body Glauber model, the diﬀerential cross section for the scat-
tering process deﬁned by (29) is
=

* f *(

*n*)(

and the total cross section for populating the ﬁnal state

*α *is thus

*dΩ f *(

*n*)(

*Kα*)

* *2

*db φ*(

*n*)

* S*(

*n*)

* φ*(

*n*)

* − δ*
*α*0

* *2

*.*
It should again be noted however that such an expression is only valid athigh beam energies and low excitation energies since energy conservation isnot respected in this model. When

*α *= 0, the total elastic cross section is

*db *1

*− φ*(

*n*)

* S*(

*n*)

* φ*(

*n*)

* *2

*.*
An Introduction to Halo Nuclei
The total cross section is also obtained from the elastic scattering amplitude,employing the optical theorem, to give

*db *1

*− φ*(

*n*)

* S*(

*n*)

* φ*(

*n*)

* .*
Hence, the total reaction cross section, deﬁned as the diﬀerence between theabove two cross sections, is

*db *1

*− φ*(

*n*)

* S*(

*n*)

* φ*(

*n*)

* *2

*db *1

*− S*proj

* *2

*,*
which can be compared with (9). Note that above, the projectile

*S*-matrix isin the form

*SF B*(

*b*) =

*φ*(

*n*)

* S*(

*n*)

* φ*(

*n*)

and is referred to as the projectile's few-body (FB) S-matrix. In the opticallimit (OL), however,

*S*proj could be replaced by the form in (33). As we shallsoon see, the FB and OL

*S*-matrices give diﬀerent answers. The simplestdistinction between the two is when each projectile constituent cluster

*S*-matrix

*Sj *is calculated in optical limit, as in (31). In that case the elasticS-matrix

*SF B *will contain eﬀects due to breakup of the projectile, whereas

*SOL*, which contains the projectile wavefunction in the exponent, does not.

For a projectile of total angular momentum

*j*, (38) is more correctly writ-
1

*− φ*(

*n*)

* *2

*.*
0

*m S*(

*n*)

* φ*(

*n*)
For projectiles with just one bound state, any excitation due to interaction
with the target will be into the continuum. For such nuclei, which includethe deuteron and many of the neutron halo nuclei (such as 6He and 11Li),it is possible to describe elastic breakup channels in which the target andeach cluster in the projectile remain in their ground states. For simplicity ofnotation, we assume a two-body projectile with continuum wave function

*φ *,
where

*k *is the relative momentum between the two clusters and, from (30),

*S*(2)(

*b*1

*, b*2) =

*S*1(

*b*1)

*S*2(

*b*2) is understood. Elastic breakup, also referred to asdiﬀractive dissociation, has amplitudes

*f *(

*k, θ*) =

*−iK*0

*db ei q· b φ S*(2)

* φ*
Making use of the completeness relation (when there is only one bound state)

*k φ φ *= 1

*− φ*
0

*mφ*0

*m *
the total elastic breakup cross section is

*φ*0

*m S*1

* *2

* S*2

* *2

* φ*0

*mδm,m − φ*0

*m S*1

*S*2

* φ*0

*m *2
The diﬀerence between the reaction and elastic breakup cross section is theabsorption cross section,
1

*− φ*0

*m S*1

* *2

* S*2

* *2

* φ*0

*m ,*
which represents the cross section for excitation of either the target or oneor both of the projectile clusters.

The above formula can be understood by examining the physical meaning
of the product

* S*1

* *2

* S*2

* *2. The square modulus of each cluster

*S*-matrix ele-ment, represents the probability that it survives intact following interactionwith the target at impact parameter

*bj*. That is, at most, it is elasticallyscattered. At large

*bj Sj *2

*→ *1 since the

*j*th constituent passes too far fromthe target. The quantity 1

*− Sj *2 is therefore the probability that cluster

*j*interacts with the target and is absorbed from the system. Such a simple pic-ture is useful when studying stripping reactions in which one or more of theprojectile's clusters are removed by the target while the rest of the projectilesurvives. Thus, the cross section for stripping cluster 1 from the projectile,with cluster 2 surviving, is given by

*φ*0

*m S*2

* *2[1

*− S*1

* *2]

* φ*0

*m.*
This cross section is seen to vanish if the interaction

*V*1

*T *of constituent 1with the target is non-absorptive, and hence

* S*1

* *= 1.

**The Binary Cluster Model**
One of the uses of the Glauber approach is that it enables us to consider, andsolve easily, reaction calculations that give insights into the underlying struc-ture of halo nuclei. We mention here, for pedagogical reasons, an analyticalextension of this approach.

Consider a simple composite projectile consisting of

*A *nucleons that can
be modelled as a two-cluster system: an

*Ac*-nucleon ‘core' cluster and an

*Av*-nucleon ‘valence' cluster (

*Ac *+

*Av *=

*A*). The intrinsic matter distributionsof the two clusters are described by one-body Gaussian densities deﬁned as

*ρc,v*(

*r*) =

*ac,v e−r*2

*/α*2

*c,v ,*
*ac,v *=

*√*
(

*παc,v*)3
An Introduction to Halo Nuclei
with normalisations such that

*r ρc,v*(

*r*) =

*Ac,v*. We also assume that the
relative motion wavefunction of the two clusters is a 0

*s *oscillator state suchthat

*φ*0(

*r*)

* *2 =

*√*
(

*πα*rel)3
We can then use such a model to construct simple formulae for the reactioncross section within both few-body (FB) and optical limit (OL) approaches(by using both forms of the projectile S-matrix deﬁned in (40,33) in (39).

By convoluting the individual cluster densities with their motion about
the projectile centre of mass, we obtain the overall one-body density of theprojectile

*ρp*(

*r*) = ˆ

*ac e−r*2

*/*ˆ

*α*2

*c *+ ˆ

*av e−r*2

*/*ˆ

*α*2

*v ,*
*ac,v *=

*√*
with Gaussian range parameters

*α*2

*.*
Since individual mean square radii of Gaussian densities have a simple
analytical expression:

*r*2

*i *=

*α*2

*,*
(

*i *=

*c, v, *rel)

*,*
we can write the overall projectile mean square radius as

*r*2

*p *=

*r*2

*c *+

*r*2

*v *+
What is important to note here is the following. Given any split in nucleonsbetween the two clusters, a ﬁxed choice of the two component ranges, ˆ

*αv *ﬁxes the overall projectile density (49) and its radius (51). However,
crucially, this does not ﬁx the projectile's underlying structure since ˆ

*αc *and ˆ
each depend on two variables: the intrinsic cluster size (through

*αv*,

*αv*) andtheir separation (through

*α*rel). This gives rise to an important distinctionbetween the OL and FB calculations of the reaction cross section and thededuced projectile radius. By choosing spatially extended clusters with smallrelative separation of their centres we can obtain the same overall projectiledensity as one containing small intrinsic clusters that are highly separated.

Both these very diﬀerent structures would give rise to the same overall OLcross section. However, their FB cross sections will be quite diﬀerent (seeFig. 8).

**Fig. 7. **Eikonal scattering. The straight line assumption of the eikonal approxima-

tion can be extended to incorporate a composite projectile, within the few-body

Glauber model, where each constituent travels along a straight line path deﬁned

by its impact parameter with the target.

For the case of a one neutron halo nucleus, the situation is simpler. Now we
set the valence cluster size to be pointlike (

*αv *= 0 but will have a large relativeseparation between core and valence neutron. A given projectile density doesnow correspond to a unique underlying structure (for a given intrinsic coresize) but, interestingly, we ﬁnd that the OL and FB calculations still do notagree (Fig. 9). Indeed, Fig. 8 shows that for a constant overall projectile one-body density, a limiting case is when the valence cluster is pointlike and thecluster separation is a maximum (a halo). This gives rise to the maximumdiﬀerence between the OL and FB calculations of the cross section.

The above general result is a consequence of the nature of the expressions
for the two

*S*-matrices in (33) and (40) and states that the OL cross sectionis always greater than the FB one. This result can be proved in the case ofpurely absorptive interactions between the clusters and the target and followsfrom the Johnson-Goebel inequality [42], which is described here due to itssimplicity. Consider a real variable

*y*. Due to the upward concavity of theexponential function we have
exp(

*y*)

*≥ *1 +

*y .*
An Introduction to Halo Nuclei
Projectile density
rms c−v separation (fm)

**Fig. 8. **The total reaction cross section for a projectile described by the two cluster

model. The cross sections are calculated using two versions of the Glauber model:

few-body (FB) and optical limit (OL). The projectile is assumed to have

*A *=

10 with a mass spilt of

*Ac *= 8

*, Av *= 2. The plot is against increasing cluster

separation. However, since all calculations assume the same overall projectile one-

body density, so increasing cluster separation must be balanced by a shrinking of

the intrinsic cluster size. The OL calculation is not sensitive to this structure change

whereas the FB one is.

This inequality is also valid if

*y *is replaced by the expectation value of anHermitian operator

*Y *,

exp(

*Y *)

* ≥ *1 +

*Y .*
If we now replace

*Y *by another Hermitian operator

*F *such that

*Y *=

*F −F *then

exp(

*F *)

* ≥ *exp

*F .*
Therefore, by replacing

*F *by

*i*
*j *(

*bj *) appearing in (30) and (33), which
is real for absorptive

*Vjt*, and taking its projectile ground state expectationvalue, we see that

*SF B*(

*b*)

*≥ SOL*(

*b*) for all

*b*. It follows therefore that

*σOL ≥*
*σF B*.

So we see that the explicit treatment of the few-body nature of the pro-
jectile in

*SF B *allows the inclusion of the eﬀects of its breakup and resultsin a reduction of the calculated reaction cross section when compared to the
use of the no-breakup

*SOL*. Another way of stating this is that the colli-sion is more transparent and less absorptive in the FB case. This is because,in many conﬁgurations of the spatially separated constituents, they do notoverlap, and hence interact with, the target. This additional transparencydue to the cluster nature of the projectile more than compensates for theadditional absorption due to removal of ﬂux from the elastic channel intothe, now included, breakup channels.

It follows that if one compares measured high energy cross sections with
those obtained from

*σOL*, to deduce interaction radii, or nuclear sizes, then
these sizes will be an underestimate of the actual spatial extent of the nucleiin those cases where the projectile has a well developed few-body internalstructure (see Fig. 8).

**More General Few-Body Reaction Models**
We now take a step back from the Glauber model and ask whether the few-body problem can be solved without making any semi-classical assumptions.

Clearly, solving the full (

*AP *+

*AT *)-body scattering problem exactly is notfeasible. However, it can be reduced dramatically in complexity if we canignore the target's internal structure and simplify the projectile from an

*A*-body system to an

*n*-cluster system as is done in the FBG model. Its groundstate is assumed bound, with eigenfunction

*φ*(

*n*). Each of its

*n *constituents is
assumed to interact with the target nucleus,

*T *, via complex 2-body eﬀectiveinteractions

*VjT *. Thus, the target is allowed to be excited by each constituentseparately. If the vector

*R *joins the centre of masses of

*P *and

*T *then we must
solve the (

*n *+ 1)-body Schr¨

*TR *+

*U *(

*R*1

*, · · · , *
*Rn*) +

*HP − E Ψ *(+)(

*r*
1

*, · · · , *
*rn−*1

*, *
*R*) = 0

*,*
where

*TR *is the kinetic energy operator for the centre of mass of the projectilerelative to the target,

*Hp *is the projectile's internal Hamiltonian and

*U *isthe sum of projectile constituent-target interactions as deﬁned in (27). Note
that the (

*n *+ 1)-body scattering wavefunction

*Ψ *(+) is a function of

*n − *1
independent vectors describing the projectile's internal motion along withthe centre of mass vector

*R*. The

*n*-body projectile intrinsic wavefunction
(

*HP *+

0)

*φ*(

*n*) = 0
(

*HP *+

*k*)

*φ*(

*n*) = 0

*,*
where

*φ*(

*n*) are eigenfunctions of excited states of the projectile. We assume
here that there is only one bound state (the ground state) which is typical ofmost halo nuclei.

An Introduction to Halo Nuclei
We must look for solutions of the full scattering equation of the form

*Ψ *(+)(

*{r*
*R*)

*.*
*R*) =

*φ*(

*n*)

*i}*)

*χ *
The functions

*χ*(

*R*) give the amplitudes for exciting the projectile to unbound
states. Note that the incident channel amplitude

*χ*0(

*R*) also includes the
incident plane wave.

By substituting for this wavefunction in the Schr¨
odinger equation (57) we
can follow standard procedures (see for instance textbooks by Satchler [44]and Feshbach [45]) to arrive at a set of coupled equations:

*E*0

*− TR − V ii*(

*R*)

*χi*(

*Vij*(

*R*)

*χj*(

*R*)

*.*
The coupling potentials are

*Vij*(

*r*1

*dr*2

*· · · φ*(

*n*)

*U φ*(

*n*)

*.*
But this remains an impossible problem to solve exactly. Indeed, for halo
nuclei, which are very weakly-bound and therefore easily broken up, couplingto the continuum is vital. This means there are an inﬁnite number of coupledchannels! Clearly approximation methods are required. We mention two ofthese models brieﬂy here.

**The CDCC Method**
Even with the restricted model space of the (

*n *+ 1)-body problem, the calcu-lation of the full few-body scattering wavefunction

*Ψ *(+) that can be used toobtain various scattering and reaction observables is not feasible. We thereforeneed theoretical schemes to approximate

*Ψ *(+) while retaining the essentialphysics (such as the few-body dynamics and correlations).

The most accurate method available for the case of a two-cluster pro-
jectile (a 3-body problem) is that of the Coupled Discretised ContinuumChannels (CDCC). In this method, the continuum is discretised into a ﬁnitenumber of energy bins, reducing the scattering problem to one involving aﬁnite number of coupled channels. The method has been applied extensivelyto light nuclei that can be treated as two-cluster systems such as the deuteron(

*p *+

*n*), 6Li (

*α *+

*d*), 7Li (

*α *+

*t*), 8B (7Be+

*p*) and 11Be (10Be+

*n*). The methodcannot, however, be extended readily to three-body projectiles, such as theBorromean nuclei (6He and 11Li), although such a four-body CDCC modelis under development.

The CDCC method approximates the three-body Schr¨
as a set of eﬀective two-body coupled-channel equations by constructing asquare integrable basis set

*{φα} *of relative motion states between the two
constituents of the projectile. Projectiles treated using the CDCC methodtend to have very few bound states and the method provides a means ofdescribing excitations to the continuum.

First, the continuum is truncated at a certain maximum,

*k*max and di-
vided up into bins of width

*∆ki *=

*ki − ki−*1, each of which is then regardedas a discrete excited state and represented by a normalised square integrablewavefunction describing the relative motion of the two clusters in the projec-tile. These bin states, together with the ground state, constitute an (

*n *+ 1)state coupled-channels problem for solution of the CDCC approximation to

*Ψ *(+). Thus, (59) is replaced by the simpler

*Ψ CDCC *(

where

*i *= 0 refers to the projectile ground state.

Solution of the coupled equations leads to the calculation of the elastic or
inelastic scattering amplitude required for observables such as the diﬀerentialcross section angular distribution. Nuclear and Coulomb breakup of two-bodyprojectiles can also be calculated with this model.

Convergence of the calculations have to be tested for diﬀerent sizes of the
model space. The number of bins and their upper limit depend on the partic-ular state they are describing and the various parameters must be carefullychosen to describe the projectile continuum. Diﬀerent schemes for construc-tion of the bin states, as well as a more detailed discussion of the formalism,can be found in the literature [39].

**The Adiabatic Model**
This approach is a considerable simpliﬁcation on the CDCC method providedthe incident projectile energy is not too low. It assumes an ‘adiabatic' (slowlyvarying) treatment of the projectile's internal motion while its centre of massmotion relative to the target is fast (a high energy approximation). Alsoreferred to as the sudden approximation, it assumes the projectile's intrinsicdegrees of freedom are frozen during the time taken for it to traverse theinteraction region.

The most natural way of understanding how this approximation comes
about is by considering the time-dependent Shr¨
odinger equation (for a two-
body projectile for simplicity):

*∂Ψ *(

*HΨ *(

*r, t*) =

*i*¯
We then make the transformation

*Ψ *=

*e−i*(

*HP −
An Introduction to Halo Nuclei
where **HP *and 0 were deﬁned earlier. Then, making the substitution intothe Schr¨
*V*1(
*r*(*t*)) + ˜
*V*2(
*r*(*t*)) + 0 *Φ *= *i*¯
*Vj*(
*r*(*t*)) = *ei*(*HP −
*0)*t/*¯*h VjT *(
*r*) *e−i*(*HP −
(**j *= 1*, *2) *.*
In this so called ‘Heisenberg picture' we have removed the projectile Hamil-tonian *HP *from the full *H *at the expense of placing the time-dependence inthe potentials. And since *HP *does not depend on
*R *then we have
*Vj*(
*r*(*t*)) = *VjT *(
*r*(*t*) = *ei*(*HP −
**r e−i*(*HP −
*0)*t/*¯*h . *(67)
Hence, if
*r *varies slowly with time then we can replace
*r*(*t*) by
*r*(0) =
is accurate provided the time spent by the projectile in the presence of theinteractions is small (i.e *HP − *0)*t/*¯*h *1). We see now that we have reacha time-independent Schr¨
*TR *+ *U *(
*r*) *− E *+ 0 *Φ*(
in which the dependence on the projectile's internal coordinate
*r *enters only
as a parameter in the potentials and we have reduced the 3-body problem toa set of 2-body problems (one for each value of
What is not so obvious at ﬁrst sight is how we have dealt with the con-
tinuum of the projectile that is so important in halo scattering. Note that wehave replaced *HP *in the Schr¨
odinger equation by the ground state binding
energy 0. So now all eigenstates of the projectile are degenerate with theground state. This assumption is good provided (1) *E *0 and (2) the im-portant continuum energies that the projectile is most likely to break up toalso involve *k E*.

The adiabatic approximation (often referred to in the context of halo
scattering as the ‘frozen halo' approximation) to the scattering wavefunctioncan be used, as in the case of the CDCC approach, to calculate varioustransition amplitudes for reactions of interest. Further details on this methodcan be found elsewhere [39].

It has been found recently that, contrary to what one might expect, the
frozen halo approximation is valid even at incident energies as low as 10MeV/nucleon or below [46]. This is because ﬁrst order (‘non-adiabatic') cor-rections mainly aﬀect the scattering close to the target, where absorptioneﬀects dominate. Thus, provided the projectile core-target optical potentialhas a strong imaginary part, this approach is valid to quite low scatteringenergies.

Clearly, for two-body projectiles, the full CDCC approach is more accu-
rate than the adiabatic approach, particularly at low energies. However, the
adiabatic model does not suﬀer so much from convergence issues or computa-tional limitations. What is interesting from the point of view of the scatteringof halo nuclei is that the adiabatic approximation allows for certain simpli-fying insights, such as when only one of the projectile's constituents interactwith the target [47] (known as the recoil limit approximation and discussedin the next section) or when the zero range approximation is made in the casewhen the scattering wave function is required only at the point *Ψ AD*(
**The Recoil Limit Approximation**

Consider a one-neutron halo nucleus, such as 11Be scattering elastically from atarget. As described already, such a system can be treated within a three-bodymodel of core + neutron + target. Assuming, in addition to the adiabaticapproximation, that the valence neutron-target interaction is much weakerthan the core-target interaction, we can set *Vn *= 0. In this case only the corefeels the presence of the target. It can be shown [47,39] that in this situationthe scattering amplitude factorises as
*T *(
*K*) = *F *(
*Q*) * *
*K Vc*(
*Rc*)* χ*(+)* *= *F *(
*Q*) *T*
*pt*(
where *χ*(+)(
*c*) is the two-body scattering wavefunction distorted by *Vc*, de-
scribing the scattering of the projectile, assumed pointlike, from this poten-tial. Crucially, however, the important eﬀects of the break-up of the haloprojectile on the elastic scattering are retained in the form factor *F *(
*F *(
*r φ*(2)(*r*)* *2 *ei *
*Q *= (
*K*)*/A *is the momentum transfer to the valence particle
during the scattering process.

This provides us with a useful formula for the cross section, which retains
the important coupling between the weakly-bound halo g.s. wavefunction andthe continuum of breakup channels, while providing a simple expression forthe scattering cross section reminiscent of the expression for Born approx-imation scattering (although there is no Born approximation being madehere):
= * F *(
*Q*)* *2 *×*
The above formula shows how the scattering of an extended halo nucleus
deviates (through the form factor) from the scattering of a point particlefrom the same potential. Figure 10 shows the application of this model tothe elastic scattering of 11Be. The recoil limit approximation clearly does
An Introduction to Halo Nuclei
**Fig. 9. **The calculated reaction cross sections in the few-body model (solid sym-

bols) and the optical limit approximation (open symbols), for 19C+12C at 960

MeV/nucleon, as a function of the rms radius of the 19C. The halo neutron sep-

aration energy is taken to be 0.24 MeV and the two-body wavefunction for the

projectile is calculated with a WS potential between the 18C core and the valence

neutron by searching on the potential depth to give the separation energy. The

potential radius is kept ﬁxed at *r*0 = 1*.*22 fm while the diﬀuseness *a *was varied

to obtain states with diﬀerent rms radii. The horizontal lines represent the mea-

sured interaction cross section with its error bars. Further details can be found in

[43]. The predicted radius is the one required for the theoretical calculation of the

reaction cross section to fall within the error bars of the experimental cross section

very well in this case due to the large mass ratio of halo neutron and core(1/10). The long dashed curve is the point cross section due to scattering of11Be via the 10Be potential only. The solid curve is the product this crosssection and the square modulus of the form factor. The diﬀerence is due tothe eﬀects of breakup. This is highlighted by the folding model cross section(the no breakup limit) which is very close to the point cross section despitethe inclusion of the valence potential.

**Other Models**

A number of other few-body reaction models have been developed and ap-plied to reactions in which the projectile is treated as a core+valence nucleonsystem. These clearly deserve more than the brief mention aﬀorded them inthis lecture, but as always, the selection of topics reﬂects the author's preju-dices. One method is to solve the time-dependent Schr¨
odinger equation after
assuming that the relative motion between the projectile's core and the tar-
Point scatt (with V )
Formfactor squaredProduct of aboveFolding (with V +V )
**Fig. 10. **Calculated diﬀerential cross sections for the elastic scattering of 11Be from

12C at about 50 MeV/nucleon showing the validity of the core recoil limit approx-imation. The curves are discussed in the text.

get can be treated classically and approximated by a constant velocity path.

This method [49,50] treats the time dependence of the reaction explicitly andthus conserves energy, but not momentum. Breakup amplitudes can then becalculated within time dependent perturbation theory [51]. Other time de-pendent approaches [52,53] also treat the projectile-target relative motionsemi-classically but solve the time dependent Schr¨
odinger equation using a
non-perturbative algorithm on a three-dimensional spatial mesh that allowsthe treatment of Coulomb breakup in the nonperturbative regime.

A very recent approach involves a combination of coupled channels ap-
proach such as CDCC and single-step Born Approximation to describe aparticular reaction process such as a transfer reaction involving halo nucle-ons [54].

**Results from Reaction Studies**

**Reaction Cross Sections**

Early estimates of the size of neutron rich isotopes of lithium and heliumemployed the optical limit of the Glauber model [41] in which the nuclearone-body densities were taken to be simple Gaussians. They predicted anenhanced size for these nuclei compared with that obtained from the usual*r*21*/*2 *∝ A*1*/*3 scaling. But by retaining the few-body degrees of freedom inthe projectile wave function, its important structure information is retained,
An Introduction to Halo Nuclei
as described in 3.4. As a consequence, studies that evaluated the reactioncross section within a few-body approach [17,18], rather than take the opticalmodel limit, predicted an even larger matter radius, as shown in Fig. 9. Thismay at ﬁrst sight seem contrary to what we might expect, since such a modelallows for new breakup channels to become available, predicting a largerreaction cross section (and hence a smaller radius to bring the cross sectionback down to the experimental value again). However, the Johnson-Goebelinequality relation [42], discussed earlier, shows that for a given halo wavefunction, the optical limit model always *overestimates *the total reaction crosssection for strongly absorbed particles, thus requiring a smaller halo size thansuggested by the full few-body calculation.

**Elastic and Inelastic Scattering**

Much can been learned about the structure of nuclei from elastic scattering.

But for unstable systems such as halo nuclei the scattering has to be carriedout in inverse kinematics with the nucleus of interest as the beam scatteringfrom a stable nucleus or single proton. Over the past decade, a number ofmeasurements of the angular distribution for the scattering of halo nucleifrom a stable target (often 12C) were unable to distinguish between elasticand inelastic scattering due to the poor energy resolution in the detectors.

Such 'quasielastic' cross sections were thus unable to resolve low-lying excitedstates of the target from the elastic channel and the data were an incoherentsum of elastic and inelastic pieces.

Angular distributions have been measured for the scattering of 6He [55],
8He [56], 8B [57], 11Li [58,59] and 14Be [60]. Since most of these nuclei haveonly one bound state then any excitation they undergo during the scatteringprocess will therefore couple to the breakup channels and the scattering willbe strongly inﬂuenced by their dynamic polarization. For such projectiles,simple folding models based on single particle densities fail to generate theoptical potentials needed to describe the elastic scattering angular distribu-tions.

A more microscopic approach to elastic scattering is to use a few-body
scattering model (based on a CDCC, adiabatic or Glauber approaches) inwhich the few-body correlations of the projectile are retained and breakupeﬀects are included. Here it is the few-body wavefunction of the projectilethat is used directly rather than its one-body density as well the projectileconstituent-target optical potentials. One of the advantages of the Glauberapproach is that breakup is included in a natural way to all continuum en-ergies and angular momenta, and to all orders in breakup, through a closurerelation. In fact, it has been found that higher order breakup terms, such asthose responsible for continuum-continuum coupling, are indeed very impor-tant [61]. This topic is dealt with more carefully and in greater depth in thelecture by Alamanos and Gillibert in this volume.

Be + C elastic at 10 MeV/nucleon
**Fig. 11. **Calculated diﬀerential cross sections for the elastic scattering of 11Be from

12C at 10 MeV/nucleon using various three-body models: Glauber (dashed curve),adiabatic (dot-dashed curve) and CDCC (solid curve) [40,46]. The importance ofboth non-adiabatic and non-eikonal corrections at larger scattering angles can beseen clearly at these relatively low scattering energies. No data exist at this energy.

Halo nuclei are very weakly-bound and consequently easy to break up. Itis therefore not surprising that breakup cross sections are much easier tomeasure than elastic ones. Numerous breakup measurements have been per-formed, even when the radioactive beam intensity was rather low. In parallel,the theoretical community has been attempting to model these reactions ac-curately.

The semiclassical theory for Coulomb excitation was developed in the
early days of nuclear physics [62]. The approach is valid for large impact pa-rameters and relies on the fact that the relative motion between the projectileand target can be treated classically whilst the excitation of the projectile istreated quantum mechanically. In this case, the total breakup cross sectionis a product of the Rutherford cross section by the square of the excitationamplitude. But while such a ﬁrst order semiclassical method is appealing dueto its simplicity, there are many aspects of the problem that are left out. Oneof the debated issues concerned the post-acceleration of the light fragmentin the Coulomb ﬁeld. In order to describe this process properly, one shouldformulate the problem non-perturbatively.

Measurement of the momentum distributions of the fragments (core and va-lence nucleons) following the breakup of halo nuclei on stable targets is now
An Introduction to Halo Nuclei
a well-established method for studying halo properties. While it has beenused for many decades as a tool to access the structure of stable nuclei, itis particularly well-suited to loosely bound systems. The basic physics (de-scribed brieﬂy in Sect. 3) is simple: since very little momentum transfer isrequired in the breakup process to dissociate the projectile fragments, theywill be detected with almost the same velocity as they had prior to breakup,and their relative velocities will be very similar to those within the initialbound projectile. In all reactions with weakly-bound systems the momentumdistributions are found to be very narrowly focussed about the beam velocity.

Many measurements have been made, involving detection of both the valencenucleons and the core fragments, and the halo structure of several light nucleihas been established.

Two types of distributions can be measured: either perpendicular (trans-
verse) or parallel (longitudinal) to the beam direction. Transverse distribu-tions are more diﬃcult to deal with theoretically since they are broadened dueto nuclear and Coulomb diﬀraction eﬀects (elastic scattering of the fragmentsfrom the target). This is why longitudinal momentum distributions are morewidely used. Early on, simple models, based on eikonal assumptions agreedwith measurements rather well. Similar widths were obtained from nuclearbreakup on light targets and Coulomb breakup on heavy targets, supportingthe view that the distributions were no more than the square of the FourierTransform of the projectile ground state wavefunction. However, this view isconsidered too simplistic.

For single valence nucleon systems, the longitudinal inelastic breakup mo-
mentum distributions for the core - at high energies the elastic breakup pieceis small - can be expressed within the Glauber framework as
*s * *√*
*dz eikzz φ*
*dbc Sc*(*bc*)* *2 1 *− Sn*(*bn*)* *2 *,*
where *bn *= * bc *+ *s *and *s *is the projection of the core-nucleon relativecoordinate onto the impact parameter plane, *φlm*(*s, z*) is the valence nucleonwavefunction with orbital angular momentum *l *and projection *m*, and *Sc*,*Sn *are the core and nucleon elastic S-matrices as described in Sect. 3.3. Theintegral over *b *in (72) represents the reaction mechanism and involves theproduct of the core survival probability (in its ground state) and the nucleonabsorption probability by the target. Without this factor, the momentumdistribution is just a Fourier transform of the nucleon wavefunction.

Reactions in which the halo neutron is knocked out of the projectile
(known as knockout reactions) are becoming the standard tool for study-ing features of the halo (as well as the ground state structure of many otherunstable and exotic nuclei). Such reactions involve two diﬀerent mechanisms:diﬀractive dissociation (in which the projectile is broken up elastically with
the target remaining in its ground state) and stripping (in which the tar-get is excited, often by absorbing the halo neutron). Theoretically, each ofthese two contributions is evaluated separately. In particular, the strippingcross section can be calculated within a model in which the projectile com-prises of the stopped neutron plus the surviving fragment. Such a ‘three-body'model (fragment+neutron+target) treats the detected fragment as a ‘specta-tor core' which, at most, interacts elastically with the target. A recent reviewof knockout reactions can be found in [10].

In this lecture, only a brief introduction to the ﬁeld of halo nuclei and theirtheoretical study has been possible. Many topics have not been includedand others given only a cursory mention. Luckily, for the practitioner, manyrecent reviews are now available , describing diﬀerent aspects of the ﬁeldfrom experimental techniques to theoretical models of both structure andreactions. Particular attention was paid here to the interplay between thestructure input and the reaction model since this is, on the whole, how thework in this ﬁeld has developed over the past decade. The lesson has beenthat in order to obtain a successful description of reactions with halo nuclei,speciﬁc features associated with the exotic nature of these systems need tobe included, and this has spurred theoretical developments in both structureand reaction studies.

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Source: http://www.euroschoolonexoticbeams.be/site/files/nlp/LNP651_contrib3.pdf

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