Halos and related structures

The understanding of halos that has emerged after the first two decades of studies has changed very little since the latest major review [13], and I shall rely on that for much of the general description and refer to it for a more detailed exposition.

The defining feature of a halo was from the beginning understood to be a large spatial extension caused by neutrons tunneling out from a nuclear core. This picture, based on the first established cases such as ¹¹Li, still prevails but must be refined in order to use the concept consistently also in other physics disciplines [13, 15]. It is not sufficient that a system be large; the tunneling that arises from the quantum wave nature must also be a prominent feature of the system in order that we obtain structures that are truly universal. As an example, Rydberg atoms are therefore excluded since most of the wavefunctions here reside in a classically allowed region; a long-range attractive potential will, in general, not give halos. The system should be dividable into a core and one or more halo particles that can tunnel out, making cluster models or few-body models a natural first choice for describing halos. Tunneling will occur in all quantum systems but it should be significant before one can expect a system to reach the regime of universal structure. One counter-example is the periphery of heavy nuclei where the neutron density will extend further out than the proton density—seen e.g. elegantly through antiprotonic ²⁰⁸Pb and ²⁰⁹Bi atoms [16] (the term 'halo' is, unfortunately, also employed traditionally in those studies, but the physics is, of course, quite different)—without having any significant dynamical effect on the overall system in contrast to light halo nuclei.

A few more technical comments may be useful before proceeding; a more complete discussion can again be found in [13]. A short-range potential is one that falls off with distance more rapidly than r⁻², i.e. r²V (r) → 0 for r → ∞. The obvious measure of the range of the potential, given a halo particle of a certain binding energy, is the classical turning point R of the particle where its potential energy equals its total energy. This definition assumes that we are dealing with a two-body system; the case of one two-body subsystem of a many-body system is not so straightforward. In practice, potentials fall off sufficiently rapidly that we may approximate R by using quantities that are more easily accessible experimentally. For two-body halo nuclei, one can use the equivalent square-well radius that is related to the mean-square radii of the two components by 3/5R² = 〈r²〉₁ + 〈r²〉₂ + 3.3 fm², where the last term reflects the finite range of the nuclear force. The good halos found early on in nuclear physics, such as ¹¹Li and ¹¹Be, have probabilities of being clustered and of having halo particles outside of R that are all above 50% and this value could be used as a yardstick for singling out well-developed halo systems.

Another question that will return in the next subsection is how does one, in practice, decide how many 'clusters' a halo state should be divided into? For most systems this is not a practical problem, but a challenging example is given by the hypertriton, ³_ΛH = n + p + Λ, which is bound by about 0.14 MeV with respect to break-up into d + Λ but where the deuteron in turn is bound by only 2.2 MeV and therefore in itself is quite a good halo. Even though three-body models must be used to give a good description of ³_ΛH, we shall see that it is most naturally considered a two-body halo.

Now, if a system is well clustered and can be described in terms of a short-range potential, a halo state will appear once the binding energy is sufficiently small. The tricky part of the question of when halos will form is therefore whether sufficient clustering is present. In general, a component can only be expected to remain inert if the energy required to excite it or break it apart is higher than the interaction energy between components. On the molecular scale, where atoms are the natural 'building blocks' this is almost automatically fulfilled and one can expect molecular physics to provide interesting halo test cases. On the atomic scale, where electrons are added as active ingredients, the electrons in atom components are easily perturbed, but one may look for halos in systems of the type alkali-atom + e + nobel-gas-atom or electrons bound to a molecule with permanent dipole moment. Atomic halos should be rare and no good case has so far been identified in experiment. Finally, on the nuclear scale, where nucleons could serve as halo particles around a core, the conditions for clustering are less clear and therefore in a sense more interesting. Nuclei may provide us not only with well-developed halo states but also with systems intermediate between normal nuclei and halo nuclei. The key question is to what extent the core will remain inert? This subject is closely related to nuclear cluster physics [17, 18] and we shall return to it in section 4. Further adding to their interest is that, as will also become clear in section 3, halo states in nuclei are characterized by their special dynamical behavior.

There is as yet no clear answer to where nuclear halo states will occur. Nevertheless, several calculations have been made, the most recent ones [19–21] within Hatree–Fock–Bogoliubov theory. The criteria for defining halos in heavier nuclei are typically different from the ones given above, see the more extended discussion in [20], but at least for even–even nuclei it appears that halos will be less numerous (and on a relative scale less extended) than for light nuclei. Part of this is due to pairing that has an important, but intricate, effect on halos [19], in certain cases decreasing sizes by mixing orbitals, in others enhancing them. Some of the necessary general conditions have been identified and will be mentioned in the next subsection. An important restriction is the mixing with other states that cannot be avoided if the local density of states is too high. This limits the occurrence of halos in excited states; the estimates in [22] for an excitation energy E* use a level distance of $D_0\exp (-2\sqrt {aE^*})$ (with D₀ = 7 MeV and a = A/7.5 MeV) and conclude that s-wave neutron halo states should have binding energy $B< 270\,\mathrm {keV}\,\left ( A/Z \right )^2\exp (-4\sqrt {aE^*})$ and that the restriction for p-waves is $Z < 0.44 A^{4/3}\exp (-2\sqrt {aE^*})$. For now, the main inference from these estimates is that nuclear halos will predominantly occur in ground states or at low excitation energy and therefore is a dripline phenomenon.

Several contributions to this symposium deal with the challenges posed by describing nuclear systems. For halo systems the requirements are to describe correlations in the systems well and to describe the large-distance behavior accurately. Not all theoretical frameworks can do this easily (as an example some work better in momentum space, whereas halos are described more naturally in configuration space). I shall focus here on lighter nuclei where few-body models can be used and start by commenting on the classification of few-body systems.

Knot theory in mathematics gives a well-established classification of linked systems including the famous Borromean rings: three linked rings where each pair is unlinked. A general system of N rings is called Borromean if each subsystem of two rings is unlinked and Brunnian if all subsystems with N − 1 rings is unlinked (this can be generalized further [23]). Following [24], this nomenclature was taken over in physics by replacing 'geometrically (un)bound in three dimension' with 'energetically (un)bound'. Although mainly used for three-body systems, one could therefore speak of N-body Borromean systems if all two-body subsystems are unbound. Examples of nuclear Borromean systems are ⁶He (α + n + n), ⁹Be (α + α + n), ¹¹Li (⁹Li + n + n) and ⁴⁵Fe (⁴³Cr + p + p), but one can also find five-body Borromean nuclei such as ⁸He and ¹⁹B that consist of a core nucleus and four neutrons. Similar structures will be abundant for heavier dripline nuclei.

One can gain a first overview of the possible behaviors of an N-body system by going from the 3N spatial coordinates $\vec {r}_i$ to generalized hyperspherical coordinates [5, 25]. After separating out the three center-of-mass coordinates $\vec {r_{\mathrm {cm}}}$, a single radial coordinate ρ is defined through

$$\begin{equation} m\rho^2 = \sum_i m_i (\vec{r}_i-\vec{r_{\mathrm{cm}}})^2 = \sum_{i<k} \frac{m_im_k}{M} (\vec{r}_i-\vec{r}_k)^2, \end{equation} \tag{ 1 }$$

where $M = \sum _i m_i$ and m is chosen as a typical mass scale of the system (for nuclear halos: a nucleon mass). The remaining 3N − 4 coordinates are dimensionless and are generalized angles that incorporate the information on relative distances in the system. From the kinetic energy term one can extract a term depending only on ρ that appears in the same functional shape as a centrifugal barrier:

$$\begin{equation} \frac{\hbar^2}{2m}\frac{\ell^*(\ell^*+1)}{\rho^2} ,\quad \ell^* = \frac{3}{2}(N-2) . \end{equation} \tag{ 2 }$$

Note that this contribution also appears for a system where all particles are in relative s-waves; the effective angular momentum given in table 1 therefore suggests that systems with more particles will be more confined. This can be understood from the following argument: if a wavefunction is allowed to extend away from origo it will spread out much faster in a higher-dimensional space (the volume increases more rapidly with distance) and the overlap with the binding potential close to the origo will therefore decrease; to counteract this and keep the total system bound it needs to stay small. The obvious loophole in this argument is that correlations in a subsystem may ensure binding; mathematically, this corresponds to having ρ large but one (or more) of the distances $\vec {r}_i-\vec {r}_k$ small (giving important contributions also from the 'angle' part of the Hamiltonian). I shall look at this in detail below for three-particle systems where one often introduces the hypermomentum K as a generalization of the two-body angular momenta, and where K = 0 corresponds to the most symmetric state with s-waves in all subsystems.

There is no unique best way to measure the spatial extent of a halo. The tradition is to use the mean-square (or root-mean-square) radius; this is with the above choice of coordinates given as

$$\begin{equation} M \langle r^2 \rangle = m \langle \rho^2 \rangle + \sum_i m_i \langle r^2 \rangle_i , \end{equation} \tag{ 3 }$$

where the last term contains the contributions from the size 〈r²〉_i of each of the components in the total system. This measure of extent emphasizes the tails and will therefore be quite sensitive to the dilute tails in halos. In several cases a more appropriate measure is the probability of finding a particle outside a certain distance (e.g. in a classically forbidden region) as we shall see when discussing the experimental probes of halos.

Since an N-body system without internal correlations (all particles in relative s-waves) appears as a two-body system with an effective angular momentum, we can start by considering two-body systems in the limit of a small binding energy B. This can be solved in the general case [26] and the expectation value of rⁿ will go as

$$\begin{equation} \langle r^n \rangle \sim \begin{array}{l} (\mu B)^{(2\ell-1-n)/2} \\ \ln(\mu B) \\ \mathrm{constant} \end{array} \mathrm{for} \;n \begin{array}{c} > \\ = \\ < \end{array} 2\ell-1 , \end{equation} \tag{ 4 }$$

i.e. it diverges in the zero-energy limit unless n < 2ℓ − 1. An angular momentum, of course, confines a system, but this can be compensated for by a higher weighting of the tail region. From the above analysis, we therefore deduce that the mean-square radius will go to infinity for vanishing binding for two-body systems with relative s- and p-waves and for three-body systems with K = 0 whereas all other systems remain finite. However, it should be noted that the only really pathological system is the two-body s-wave where the probability of being outside the binding potential can go to 1. This does not happen in any other case.

If a long-range repulsive interaction is present, e.g. the Coulomb field for nuclear proton halos, the tail of the wavefunction will be more suppressed at larger distances than for an angular momentum barrier. This makes halo formation more difficult and will prevent it altogether for nuclear charges above 10–20 (none of the identified one- or two-proton emitters will have halo character). On the other hand, it has been shown explicitly that nuclear deformation will not prevent the formation of halos [27].

We are interested in being able to compare also systems at finite binding. To do this across physics fields it is useful to introduce dimensionless scaling variables [15, 28]. For a two-body system we use the classical turning point that in practice is approximated by the R discussed above. The dimensionless mean-square radius is then simply 〈r²〉/R². The momentum corresponding to a range R is of order ℏ/R giving a kinetic energy of order ℏ²/(2μR²), so a natural dimensionless binding energy is therefore μBR²/ℏ². With this set of variables one has indeed scaling at low binding for two-body states with a good angular momentum ℓ. The resulting plot for identified halos will be shown later in figure 2.

It is less obvious how scaling appears in three-body systems. A more thorough discussion of this can be found in [13]; it suffices here to note that if a scaling radius ρ₀ can be found one can introduce the dimensionless variables 〈ρ²〉/ρ²₀ and mBρ²₀/ℏ². There are two slightly differing choices of ρ₀; the first [15] proceeds in analogy to the way the radius ρ was introduced in equation (1) and makes use of two-body scaling radii R_ik to define

$$\begin{equation} m\rho_0^2 = \sum_{i<k} \frac{m_im_k}{M} R_{ik}^2 . \end{equation} \tag{ 5 }$$

The second [28] builds on a deeper analysis of the case when all potentials are square wells and defines

$$\begin{equation} \sqrt{m}\rho_0 = \frac{\sqrt{2}}{3} \sum_{i<k} \sqrt{\mu_{ik}} R_{ik} \end{equation} \tag{ 6 }$$

with μ_ik being the reduced mass of the subsystem. If all masses and two-body scaling radii are the same, both definitions reduce to ρ₀ = R.

The main features of the classification of three-body systems [28] are reproduced in figure 1 (the second definition of ρ₀ is used here). The results are shown for three model systems corresponding to ¹¹Li, ³_ΛH and a three-boson system where the interactions in all cases are varied to give different binding energies; the large triangle and the filled circle correspond to the physical ¹¹Li and ³_ΛH. The uncorrelated case with K = 0 shows good scaling properties. When correlations are allowed in the wavefunctions, the main effect is to increase the binding energy whereas the radii are less affected. The resulting curves therefore lie to the right (or, equivalently, above) the ones for K = 0. The closest curve is the one for Borromean systems. Here again a universal behavior is seen for three different systems. However, the physical hypertriton has a bound subsystem (the deuteron) and is further away. That radii increase (for a given three-body binding energy) as more and more subsystems become bound is likely to be a general effect [29]. The arrows indicate the positions where a two-body subsystem goes from being bound to unbound. If the neutron–proton interaction is weakened the hypertriton system (filled circles) eventually becomes Borromean and approaches the other Borromean systems. The physical ¹¹Li is, of course, Borromean and when the neutron–⁹Li interaction is increased nothing spectacular happens to the ground state (filled triangles), but a very extended excited state (open triangles) appears. This is an Efimov state that appears along the dashed line just before the ¹⁰Li subsystem becomes bound, increases in binding energy and finally increases in size again when becoming unbound with respect to n + ¹⁰Li once the latter has become bound.

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The Efimov effect [30] appears when a two-body subsystem has a very large scattering length a. This induces an effective potential proportional to ρ⁻² in the three-body system for distances between R and a, a potential where successive excited states will scale in energy as well as extension. More details can be found in [25, 31]. These extraordinary states have been searched in various systems, including nuclei and few-atom molecules, and were finally observed indirectly in cold atom gases a few years ago. Most observations rely on the increase of the recombination rate of the many-body system seen when the binding energy of the Efimov state goes to zero; see [32] for a recent overview, but in a few cases states have also been produced directly at finite binding energy ([33] and references therein). So far experiments have probed the binding energy systematics rather than the spatial extension of the states. The one case where experimental information on sizes of loosely bound molecules is available is the ⁴He dimer and trimer where a very elegant experiment [34] has succeeded in determining the sizes as 5.2(4) and 1.1(5) nm, respectively. An excited He trimer Efimov state has been predicted and looked for several decades, but still not identified. In principle, much larger molecular halo systems have already been produced via manipulation of atomic scattering lengths in cold atom gasses and more cases are expected to exist also as isolated systems.

To sum up, the only truly pathological behavior occurs in two-body systems where the scattering length a can be arbitrarily large (and, in the bound case, a halo state forms with arbitrarily small overlap between the two bodies). A very large a induces the Efimov effect where the two-body correlations in a three-body system give rise to excited states with universal scaling properties and sizes reaching up to a. In contrast to Efimov states that appear at the two-body threshold, the Borromean three-body states that occur at the three-body threshold are far more moderate in size for a given binding energy. Three-body systems that appear above the Borromean curve in the scaling plot therefore have substantial two-body correlations.

Going on now to N-body systems, recent calculations for N = 4,5,6 identical bosons [35] show that the systems remain finite even for vanishing binding energy. The most important correlations remain the two-body ones, even though higher-order correlations can also play a role for excited states. The overall sizes seem to decrease as N increases, but one can still observe systems that are significantly larger than the scale set by R. Similar results are likely to be present in the nuclear case and would be relevant for heavier nuclei along the dripline than the ones accessed currently. If the nuclei there are best described as N-body Borromean systems with N larger than three, one would expect halo formation to be quenched. This picture with several 'active' neutrons around a core is probably the reason, seen from few-body models, for the earlier mentioned decreased size of halos when pairing mixes orbitals.

The hypertriton was used above to clarify the classification of three-body systems and is best thought of as a Λ-halo around a deuteron. Hypernuclear physics has probably more halos to offer [36], but now with neutrons forming the halo rather than the Λ-particle. Among the interesting systems are ⁶_ΛHe, where the outer neutron is expected to extend further out than in ⁶He, and ⁷_ΛBe that with a binding energy of around 0.3 MeV below the ⁵_ΛHe + p + p threshold probably is the best two-proton halo one could hope for. A recent experiment [37] indicated that ⁶_ΛH is also bound and quite close to the ⁴_ΛH + n + n threshold.

The typical binding energy scale for nuclear halos is of the order of 100 keV (less than about 1 MeV in light nuclei, decreasing with the mass number as A^−2/3) and for molecular halos 1 μeV. There have been very few studies of the dynamics of molecular halos, so the discussion will from now on concentrate on nuclei.

States with halo structure can be excited in many different ways; the ones of specific interest here are the ones that involve the halo degree of freedoms. Since the halo extension depends so crucially on binding energy, one cannot expect the structure to be maintained under e.g. rotation or isospin changes and we need to carefully look at which states can be reached in excitations.

Strong interactions typically require more dedicated treatments, whereas electroweak interactions can be treated perturbatively with operators that may change spin and/or isospin, and may contain factors r^λ that enhance the tail. As will become obvious from the following section, halo excitations very often are dominated by transitions to the continuum and also often dominated by the tail properties. As will also be seen, in particular from the electromagnetic probes in section 3.3, these are not qualitatively new features in nuclear physics but their magnitude puts them on a quantitatively new level for halos.

Due to the importance of the continuum, the question of the continuum structures seen in excitations must be considered carefully. A good example is the E1 strength distribution of neutron halos that is now understood as being single-particle strength going mainly directly to the continuum [38] and not a semi-collective soft resonance, see also [39]. As shown in detail in [40] for the case of the ¹¹Be E1 strength, one may choose to ascribe some of the strength to resonances in the continuum. This particular work followed the definition of Berggren [41] of a resonant state and his elucidation of how one can go from a description entirely in terms of a continuum to an equivalent description where resonances are included explicitly. An important outcome of Berggren's analysis is that one, in general, may expect a remaining non-resonant continuum contribution; this is very much the case for the ¹¹Be E1 strength and can be expected also to be an important feature of other halo excitations. The research theme of discrete states embedded in a continuum is an important one in contemporary nuclear physics [42] and will also be covered in these proceedings [43].

Due to the extreme sensitivity to binding energies, it is important to know the mass of halo nuclei accurately. In most cases it is sufficient that the binding energy be determined with an uncertainty of 10 keV or less, but this is a rather challenging goal for nuclei close to the driplines. General overviews of the techniques used in mass measurements of radioactive nuclei can be found in [44, 45]. The masses of halo nuclei were at first measured via nuclear reactions or various time-of-flight measurements [46], but Penning traps have by now also been brought into use [47] and give an important step forward in accuracy as well as precision. The lightest halo nuclei have by now been measured with sufficient accuracy and the challenges lie in producing neutron dripline nuclei above Be in amounts that enable determination of the masses of heavier halo candidates. A particularly important challenge is the mass of ¹⁹C that at the moment is deduced from reaction measurement assuming it to be a good halo state, rather than from direct experiments.

Optical measurements on radioactive beams [48] give access to many nuclear ground-state properties, including charge radii, spin and electromagnetic moments. These can give very precise information, often on a level beyond what current nuclear theory can predict. The charge radii are of course very valuable for a direct determination of the sizes of proton halos, the experiments being sensitive to the mean-square radius of the total charge distribution. The best example of this is the charge radius of ¹⁷Ne [49] that was found to be clearly above the radius of heavier Ne isotopes although the magnitude of the effect, an increase in rms radius by less than 0.1 fm, is clearly below what is found for good neutron halo nuclei. This is, of course, consistent with the confining effect of the Coulomb barrier that already in Ne is very noticeable. Accurate measurements of the charge radius of light neutron halo nuclei have also appeared during the last few years [47]. Due to their high accuracy they are important in giving constraints on core modifications, they can furthermore for multi-neutron systems test the correlations in the overall system through the sensitivity of the total charge radius to the movement of the charged core around the center of mass.

Magnetic dipole moments and electric quadrupole moments have also been extracted for some halo nuclei (quite apart from the spin, an even more basic property). The accurate measurements must be coupled to model calculations to give the optimal information on the system; two good examples are the magnetic moment of ¹¹Be that is μ = −1.6816(8)μ_N [50] and the electric quadrupole moment of ¹¹Li whose ratio to that of ⁹Li, |Q/Q_core| = 1.088(15) [51], is similar to that of the rms charge radii for the two nuclei. In general, knowledge of the moments of a halo candidate and the 'bare core' nucleus can give important information on how inert the core is in the halo state.

The half-life of a radioactive nucleus also belongs to its ground-state properties, but will only in exceptional cases carry a clear signature of a halo structure. Beta decay probabilities depend on the overlap between the initial and the final state wavefunction and the effects of a halo on an overlap are in most cases below a factor of two, which means that one needs a good understanding of the core structure of the initial and final states in order to draw conclusions. However, beta decay brings information on halos in other ways (see [14, 52, 53] for more detailed reviews).

Firstly, beta decay may help in pinpointing the exact configuration of a halo. A good example is ¹¹Li where already the total half-life indicates that the two last neutrons must reside partly in the sd-shell rather than only in the p-shell [54]. Individual decay branches, of course, can be used to strengthen this argumentation [52]. Secondly, there are indications that the halo and core beta decays decouple at least in some cases (it is not yet clear whether this will be a general feature for good halo states). The prime example of a core decay that reappears in the decay of the halo nucleus is that of ¹²Be where more than 99% of the decay goes to the ¹²B 1⁺ ground state and one correspondingly finds that most of the ¹⁴Be decays goes to a slightly unbound 1⁺ state in ¹⁴B [12, 52, 55].

An even more convincing case would be a decoupled halo decay. This appears to occur at least for two-neutron halo systems as beta-delayed deuteron emission. The current understanding of this decay mode is that it proceeds directly to the continuum [52, 53], a decay of the two halo neutrons directly to a deuteron in the periphery of the total system. Two cases have been established so far; for ⁶He, the intensity is low due to a cancellation of contributions from the inner and outer parts of the wavefunction, whereas ¹¹Li, where the energy spectrum was measured recently [56], offers the possibility of a more stringent test once the final state interaction between the deuteron and the ⁹Li core is understood better. It will be interesting to see whether other halo nuclei will give similar unique decays. At least for the two known cases it is clear that the contribution from the non-resonant continuum is essential for describing the decays.

Electromagnetic transitions of type E(λ) or M(λ + 1) have operators that contain a factor r^λ which enhances the tail behavior and makes these processes very sensitive to halo formation. It is worth noting that halos in excited states can be probed as well provided that one can single out transitions that involve these states. The emphasis on the wavefunctions at large radii was known also before reaction experiments indicated the existence of halo states, and several earlier papers identified the main features of what we now recognize as halo signatures.

For transitions between bound states the classic case is that of ¹¹Be where the E1 transition between the first excited 1/2⁻ state and the 1/2⁺ ground state could only be reproduced if the spatial extent of the wavefunctions was taken into account [57]. Transitions between a bound state and the continuum were known to be important for the deuteron, but the case of proton radiative capture into a spatially extended state was also clearly understood for ⁸B and the first excited state in ¹⁷F [58, 59]. In these cases, direct capture, i.e. contributions from the non-resonant continuum, are more important than resonance capture in line with the discussion in section 2.3. The proton radiative capture was investigated early on due to its importance for nuclear astrophysics, but a similar sensitivity can be expected in neutron radiative capture [26] and should again give a low-lying non-resonant peak [60].

Electromagnetic dissociation is the inverse reaction to radiative capture and will therefore be as sensitive to halo structures. It is now recognized as a key probe for nuclear halo states and an important dynamical consequence of the special structure, as was first pointed out in [3] and observed experimentally shortly afterwards [61], see also [39]. The prediction in [3] was based on sum rules for the E1 strength. These are easily generalized and give for a two-cluster model [62], where the clusters have charge and mass number Z_i and A_i and distance r between them, a contribution to the energy-weighted sum of

$$\begin{equation} \frac{9}{4\pi} \frac{(Z_1A_2-Z_2A_1)^2}{(A_1+A_2)A_1A_2} \frac{\hbar^2e^2}{2m} \end{equation} \tag{ 7 }$$

and for the non-energy-weighted sum

$$\begin{equation} \frac{3}{4\pi} \frac{(Z_1A_2-Z_2A_1)^2}{(A_1+A_2)^2} \langle r^2 \rangle e^2 . \end{equation} \tag{ 8 }$$

Compared to a normal state, a halo state will therefore give more strength that appears at lower excitation energy than usual and if the specific contribution to the E1 strength from the halo degree of freedom can be extracted, one can derive its spatial extent. The E1 strength will carry the dominant signal for neutron halos, whereas higher orders will also be affected for proton halos. Concerning the strength distribution, the simple structure of halos implies that simple analytic models [63–65] can be expected to give the main features. Simple analytic expressions for the photodissociation cross-sections valid for loosely bound nuclei have also been derived [66].

As discussed above and in complete correspondence to radiative capture, the dissociation reactions mainly involve the non-resonant continuum. Their importance will be clearly seen in the next section; one good example is the identification of ³¹Ne as a halo candidate based on its large one-neutron removal cross-section [67].

Most of the studies on halos have been performed using nuclear reactions. I can here give only a brief overview on this large field that is also covered in several other contributions to these proceedings, in particular [68]. Detailed reviews on the reaction theory of halo nuclei can be found in [4, 13, 69–71].

It is customary to classify the reactions according to the beam energy, low-energy reactions taking place around the Coulomb barrier, intermediate energy reactions around the Fermi energy and above this the high-energy reactions (extending to relativistic energies), where the smaller nucleon–nucleon cross-section and the shorter interaction times make the reaction mechanisms simpler. The reaction mechanisms do evolve gradually as the energy increases, so a strict division into the three regions is not possible, but the rough dividing points are somewhat above 10 MeV u⁻¹ and around 100 MeV u⁻¹.

As realized essentially from the start, the 'halo-removal' channel (one-neutron removal for one-neutron halos, two-neutron removal for two-neutron halos, etc) will, for both strong and Coulomb interactions, have a significant cross-section and will furthermore be clearly influenced by the halo structure. The main focus has been on this channel; this is fully justified but has implied that possible information from other channels has often been neglected.

3.4.1. Interaction cross-section.

3.4.2. High to intermediate energy probes.

3.4.3. Intermediate- to low-energy probes.

In order to use a few-body picture for the halos the amount of 'configuration mixing' should be limited, i.e. there should in the nuclear case be mainly one component in the wavefunction with one or two nucleons around a core (that very often, but not necessarily, will be in the ground state). This appears to be fulfilled for most of the nuclei considered below. Scaling plots updated with the latest experimental information are shown in figures 2 and 3. For the three-body halos, equation (5) is used to define ρ₀; the alternative definition in equation (6) gives a ρ²₀ that for the nuclear two-neutron halos is smaller by about a factor of 1.7. It may be interesting to note that systems where the probability of being outside of the potential range is higher than 50% correspond to a value just below 2 for the scaled square radii. As can be seen in the figures, this limits the number of pronounced halos.

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For easier reference, some properties of selected halo systems are collected in tables 2 and 3. The tables list the root mean square radii for the total system as well as for the core nucleus, the components of the system and the binding energy [93] for the halo particle(s). Except where discussed explicitly, in the following the experimental values for radii are taken from [73, 94]; in the cases when several analyses have been performed, the 'few-body' results (that take correlations in the system into account) are used. For two-body halos the angular momentum of the halo particle is given; for three-body halos the possible angular momenta for a halo neutron with respect to the core is given. The deduced mean square radius of the halo is listed in the end. Since selections between different data had to be done in several cases, the following detailed discussion should be consulted to obtain a complete overview for a specific state. The assumptions made are also discussed there.

4.1.1. The deuteron.

4.1.2. ¹¹Li.

4.1.3. ^6,8He.

4.1.4. ¹¹Be.

4.1.5. ⁸B.

4.1.6. ¹⁷F*.

4.1.7. ¹⁷Ne.

4.1.8. ¹⁴Be.

4.1.9. ¹⁵C.

4.1.10. ¹⁹C.

4.1.11. ^17,19B.

4.1.12. ²²C.

4.1.13. ³¹Ne.

4.1.14. Heavier nuclear states.

4.1.15. Halos in excited nuclear states.

4.1.16. Halos in other systems.

Halo physics has made impressive progress during the 25 years since they were discovered. Nevertheless, there are still several open questions we need to address. The following incomplete list contains the main ones that came to my mind when preparing this paper.

• Where will nuclear halos heavier than the currently known ones occur in the nuclear chart? Somewhat coupled to this is the question of how much single-particle character remains in heavier nuclei?
• Are there dynamical characteristics of skin nuclei?
• Does closeness to continuum promote (or stabilize) halo cluster structures? For the general case of clusters this may be the case [139]; recent work [140–142] is starting to establish quantitative criteria for clustering. The findings so far that most halo candidate states (with low angular momenta) end up having a good halo structure may be a 'selection bias' since all the cases are among the light nuclei, but the continuum could also play a more active role.
• When is the concept of a resonant continuum useful? As seen above, the concept of a non-resonant continuum is essential in order to understand all aspects of electromagnetic processes and beta decays involving halos. Nuclear processes are less transparent, but the final state structures there should also be treated with great care. (Even though a resonance fit may describe the data it will not necessarily correspond to the physical reaction mechanism [84].)
• Can we find ways to experimentally study the hypernuclear halos?
• It would be very instructive to have a detailed experimental characterization of one of the halos in atomic and/or molecular physics.

The first decade or so of halo physics established the main concepts and identified the main cases occurring among light nuclei. Apart from a consolidation and refinement, reflected e.g. in the cross-checks on the spatial extension that have now been successfully made in many cases, the intervening years have also seen the concept being established in molecular physics and the links to the Efimov effect clarified. During the last few years, first experiments at a new generation of radioactive beam facilities have yielded new nuclear cases and more experimental information will soon be available. This may allow us to check our understanding of the conditions for halo formation and thereby complete our picture of the phenomenon.