Hyperfine Interactions 21(1985)265-273
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ANOMALONS? Denys WILKINSON
Department of Physics, University of Sussex, Brighton, UK
1.
Introduction
Kenzo Sugimoto's interests in the fields of nuclear and atomic physics are very widespread and he has been involved in many novel advances. When, therefore, I wondered how I might most appropriately respond to the happy invitation to help honour him in this volume, my thoughts naturally turned to those matters of my own concern that touch upon wide ranges of physical concepts and novel ideas. The matter that I have chosen to discuss here indeed involves concepts and speculations both directly derivative of traditional nuclear physics and remote from it. It is the so-called phenomenon of the anomalons. It is a matter of some contention, into which I do not intend to go, as to whether or not the phenomenon has actually been discovered yet, but that should not inhibit discussion of it: many wild geese have had high heuristic value as both Kenzo Sugimoto and I know full well in relation to second-class currents, but that is another story. 2.
Anomalons?
It has been rumoured for many years [ 1] that nuclear collisions of relativistic heavy ions give rise to a class of multiply-charged nuclear fragments that display anomalously-short mean free paths against their own subsequent nuclear interactions. It has also been suggested that these anomalons' can pass on this property after their first or even subsequent collisions so that there are sometimes several generically-related anomalously short free paths. These early reports were derived from cosmic-ray studies using nuclear emulsions. More recently [2] there have been extensive studies, still using nuclear emulsions but with incident heavy ions (16 O, 4~ and s6 Fe of energy about 2A GeV) from the Berkeley Bevalac. These experiments appear to establish the phenomenon with high statistical likelihood. The phenomenon is consistent with the production of the anomalons in a few percent of the primary collisions and with the anomalons having a mean free path a few times shorter than that to be expected of a normal nucleus of the same value of Z. Alternatively, the proportion of anomalons produced might be some tens of percent and the mean free path shortened by a factor 9 J.C. Baltzer A.G., Scientific Publishing Company
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of less than 2 below normal; in that case the 'anomalon' property would have to be lost by some electrically neutral mechanism after a few times 10 -11 sec. The lifetime of the anomalons against reversion to a normal nuclear state is not clearly indicated, although it must be at least 10 -~~ sec or so for the phenomenon to have been detected. It is obviously highly desirable that such an unexpected and striking phenomenon of possibly fundamental importance should be confirmed by independent methods of experimentation. Two such further approaches have been reported: the first [3] also employing heavy ions from the Bevalac, replaced the nuclear emulsion by etched track detectors and reported confirmation of the phenomenon; the second [4], an electronic experiment also at the Bevalac, gave no evidence for the phenomenon and appeared to eliminate most, but not all, of the previous interpretive hypotheses at a high level of confidence. So whether or not the existence of anomalons has been established is not yet clear, but what is clear is that we should turn our thoughts towards states of nuclear matter that might be excited in the novel ranges of energy and ion species that now become available through relativistic heavy-ion accelerators. There is, indeed, no point whatever in investing in such accelerators other than the expectation, or hope, that they will reveal to us modes of organisation of hadronic matter towards which we can now but speculate. The report of the existence of anomalons is therefore most welcome if only to set our minds working towards ways in which hadronic matter of a given Zvalue might manifest a propensity for nuclear interaction significantly enhanced over that for a regular nucleus of the same Z-value. Of course, before one embarks upon the search for an anomalously-enhanced hadronic interaction one must be clear that the phenomenor/is not due to some longlived high excitation of a product fragment whose decay in flight simulates a nuclear interaction. Such an excitation, for it to persist for 10 -~~ sec or so, would not be of a normal nucleonic kind but might be some sort of hypemuclear state which might then naturally have an appropriate lifetime against decay and therefore an apparently appropriate mean free path against nuclear interaction. Such an interpretation appears to be excluded [2]. We therefore ask what sorts of hadronic matter might exist that present strong interaction cross sections increased by a factor of several over that of a normal nucleus of the same charge. 3.
Preamble
Before embarking upon our search for the anomalon, it is important to disabuse ourselves of the idea that it must be found within the framework of our traditional descriptions of nuclear structure. The nucleus, including the excited states into which our usual methods of experimentation raise it, is a compact object of more-orless constant density never very far from spherical in form, that can be well described by a one-centre shell model, spherical or non-spherical in its basis. For certain special
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purposes such as our description of fission or our discussion of quasi-molecular states, we move to a two-centre description but still one in which the objects related to those two centres remain closely similar to, individually, or derive from, ordinary one-centre nuclei. Higher energy interactions, particularly those initiated by heavy ions, cause major disruptions to the tranquil nuclei of traditional experimentation and traditional thought; and much imagination is now being devoted to possibilities for new gross forms of nucleonic matter that may be produced in such collisions associated with, on the one hand, abnormally high densities with possible phenomena such as pion condensation, new sorts of hadronic matter of the Lee-Wick kind and the moves towards the quark-gluon plasma and, on the other hand, abnormally low densities at which various classes of phase change might possibly occur. Anomalons might be born of any of these novel states of nuclear matter and might be a critical clue to their existence and properties. But it seems to me that, so far, insufficient attention has been paid to the possibility that the aftermath of a high-energy collision, following the production of a low-density expanded gas of nucleons, will be the recondensation of that gas of nucleons, ultimately into ordinary nuclei but passing through intermediate configurations, perhaps of significant lifetime, and perhaps not of very high excitations relative to ordinary nuclei of the same N and Z, that are geometrically remote from the nuclei of .our normal vision. It may be the most natural thing that such geometrically-unusual nuclei should be formed indirectly by condensation out of the kind of state left behind by a high-energy collision, while it being most unlikely that the same configuration should be reached by direct excitation from a normal nucleus: the indirect, twostage excitation may be a much easier approach than the direct transition. That is the avenue I wish to explore in my own quest for the anomalon, but first I wish to discuss other suggestions that have been made. 4.
Possible origins o f the a n o m a l o n
1. Conventional nuclear structure considerations. Bayman, Ellis and Tang [5] have pointed out that lighter nuclei tend to be proportionately rather larger, on the scale of A1/3, than heavier nuclei and that this will lead, for them, to an interaction mean free path somewhat shorter than expected from systematics that do not take this into account. They also suggest that the heavier anomalons might be quasi-molecular resonances such as are now familiar from heavy-ion physics both experimental and theoretical. 2. Hidden colour excitations. Romo and Watson [6] consider the properties of 6-quark bags in which, unlike the situation in the deuteron, all six quarks interact strongly but with distinct spatial wave functions and hence of mixed spatial symmetry and with the repulsive force between quarks of the same colour just balancing the attractive force between quarks of different colour. The dimensions of this system, and hence the associated interaction cross section, could be large.
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3. Di-quark deuteron. Frederiksson and J~ndel [7] also consider a 6-quark bag but one whose ingredients factor into three di-quarks, each in a p-orbital but with L = 0 overall and so with overall spatial anti-symmetry; each di-quark has TSJ ~ = 000 § with the overall 'demon' deuteron of 000-. It is suggested that an enhanced nuclear interaction could come from violent colour fluctuations at the surface of the 'demon', resulting in much stronger 'van der Waals' forces between the 'demon' and other hadronic objects than is familiarly the case for ordinary baryons. 4. Low density quark blobs. St6cker, Graebner, Maruhn and Greiner [8] consider the sorts of quark matter that might result from relativistic heavy-ion collisions and conclude that under certain conditions the quark-matter ground state might have a density as low as 1/30 of that of normal nuclear matter so that a blob of such stuff would have an interaction cross section some 10 times greater than for a normal nucleus of the same baryonic number. 5. Pineuts. McHarris and Rasmussen [9], following suggestions by several authors (see Garcilazo [10]), consider the properties of p i n e u t s - hadronically bound states of one or more negative pions and several neutrons. They suggest that an anomaIon might be an ordinary nucleus surrounded by a loosely-bound halo of pineuts that would lower the overall Z-value and increase its size at the same time. 6. Vacuum perturbation outside pion-condensed nuclei. Fowler, Raha, Weiner and Wheeler [11] ask how far outside a nucleus in which pion condensation has been induced the order parameter of the pion field might extend. They find a solution in terms of the chiral sigma model outside the nucleus that shows that there might be a significant value for the pion field even out to some tens of fermis beyond the nuclear surface; they suggest therefore that an anomalon might be such a pion-condensed nucleus with this extensive pionic penumbra. 7. Nuclear Overhauser states. MacGregor [12] suggests that nuclei might develop Overhauser-type wave functions [ 13] under appropriate anisotropic perturbations and that the resulting laminar structures might be the anomalons. 8. Unconfined gluon fields. Chapline [14] considers the possibility that quantum chromo-dynamics (QCD) is broken by gluons of finite mass in a way that would permit the existence of free quarks with an unconfined gluon field. Such quarks would bind strongly to nuclear matter with the associated gluon field, for gluon masses of 1 0 - 5 0 MeV, reaching out to distances adequate to account for anomalon behaviour.
5.
A l p h a - n e u t r o n rings
My own candidate for the anomalon derives from the above remark that it may be much easier to reach a remote and unusual geometrical configuration in two stages than in one. I expressed this idea several years ago [15] in the very context of high-energy heavy-ion collisions in which the anomalon has been reported. Such a
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collision produces ahot expanded low-density gas of nucleons; parts of this gas disperse, together with possibly large numbers of mesons, and parts of it may recondense, depending on the violence of the collision and on the details of the energy distribution in the collision process. The recondensation is moderated by the attractive effects of the nuclear forces, the dispersive effects of the Coulomb forces together with the kinetic energies and the centrifugal effects of the possibly very large amount of angular momentum in play among the consituents under consideration. As condensation proceeds, the cooling mechanism being the evaporation of the more energetic nucleons, it will be alpha-particles that first form in numbers because the alpha-particle is a very stable and dense object (its central density is uniquely high, almost twice that of lead) and it also, as is well-known from the story of stellar nucleosynthesis, forms a barrier against the build-up of heavier nuclei owing to the instability of the A -- 5 systems. So we have a gas of alpha-particles and neutrons with the protons captured to form the alpha-particles. The nuclear forces that tend to bring about final collapse into a normal nucleus are opposed by the Coulomb forces which may indeed win out and disperse the gas into the continuum so that the result as observed is a shower of alpha-particles such as is not infrequently found. But if the nuclear forces begin to bring about collapse of a spherically-symmetrical non-rotating gas, they quickly overcome the repulsive Coulomb force because the repulsive Coulomb energy increases more slowly as the radius of the system decreases than the attractive nuclear energy increases, as may be shown in model calculations. But if the gas contains a great deal of angular momentum, as will be likely as the result of an energetic heavy-ion collision, the situation is different. As the swirling gas contracts it will flatten out into a pancake, as a less energycostly way of containing the overall angular momentum; it will finally form into a ring to maximize the moment of inertia. We are therefore led to enquire into the stability of a rotating ring of alphaparticles held together by an abundance of neutrons that run around and between the alpha-particles. Although this ring configuration would be one that it would be very difficult to reach by direct excitation from the ground state of an ordinary nucleus, it is seen to be a very natural configuration to reach via the two-stage process of condensation from a hot swirling gas of nucleons at low density. As a specific model I have considered a ring of N alpha-particles plus 2N neutrons so that with the neutrons uniformly distributed around the ring there are two neutrons associated with ('between') every pair of alpha-particles as in 1~ Now argue crudely that each neutron pair constitutes a 2n-bond whose energy value is equal to that of the pair of neutrons 'between' the two alpha-particles in the alphaparticle model of 1~ when the alpha-particles in t~ are separated by the same distance as separates the alpha-particles in the Na-ring. Of course, in the case of the Na-ring the neutrons do not 'belong to' the two alpha-particles between which they are notionally situated; any one alpha-particle is simultaneously 'used by' the pair of neutrons to its one side as well as by the pair of neutrons to its other side. This is no
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qualitative problem because two alpha-particles can bind a multiplicity of neutrons: the binding energy of a pair of neutrons to 8Be to make X~ is 8.48 MeV, while the further binding energy of another pair of neutrons to 1~ to make 12Be is another 3.67 MeV; all the neutrons remain in the lp-shell so that Pauli principle problems do not arise. (14 Be is also heavy-particle stable.) Therefore, although this simple neutron bonding model is very crude, it should not be wholly unrealistic. The stability of 12Be and 14Be suggests that an Na-ring will be able to bind substantially more than 2N neutrons. Neutrons will attach to the ring from the condensing gas cloud until no more can be held. The last neutrons will therefore be weakly bound and will form an extensive penumbra around the Na-ring that may be of importance in discussing the subsequent nuclear interaction of the ring as a whole. However, we perform all our stability calculations in terms of a chain of N a + 2Nn. Of course, when the alpha-particles of the Na-ring are close together they do not retain their literal identity. In the case of 1~ that we are using roughly to estimate the energy of the 2n-bonds in our Na-ring, we must have proper anti-symmetrization of the neutrons 'in the alpha-particles' with those of the 2n-bond. As the repeat distance around the Na-ring increases, as the separation of the centres in the alphaparticle model of 1~ increases, objects more and more closely resembling literal alpha-particles emerge. This process has been calculated in detail by Seya, Kohno and Nagata [16] using the method of molecular orbitals with full overall anti-symmetrization. This calculation [16] is very ~uccessful and presents us with the binding energy of 1~ as a function of the separation of the 'alpha-particle' centres and hence the stability of the system against break-up into 2a + 2n as a function of the a - a separation distance. This energy of stability we use to derive the energy of the 2n-bond in our Na-chain as a function of the a - a separation distance along the chain. The calculations of Seya et al. [16] do not include the Coulomb energy; we therefore adjust the overall energy of their theoretical ~~ system so that its minimum corresponds to the experimental total binding of ~~ against break-up into 2a + 2n minus the calculated mutual a - a electrostatic energy at the separation corresponding to the minimum in the total energy; for this mutual a - a Coulomb energy we use that of two interpenetrating Gaussian distributions of the correct experimental rms size. This constructed curve of binding energy of ~~ against 2a + 2n versus a - a the separation distance therefore includes the internal Coulomb energies of the individual alpha-particles but not their mutual a - a Coulomb energy. The total energy of the Na-ring is now computed (up to N = 20) as a function of the radius R of the ring, by adding the N 2n-bond energies as a function of the a - a separations around the ring plus the N ( N - 1)/2 mutual a - a Coulomb energies. To this energy we now add the rotational kinetic energy defined as J ( J + 1 ) h 2 / 2 N M R 2, where M = rn~ + 2m n to find, in Born-Oppenheimer spirit, an effective potential to confine the ring as a function of its rotational angular momentum J. A problem arises for small values of R. The calculations of Seya et al. [16] refer, of course, to only two alpha-particles; they show an increase of the total energy
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for small a - a separation distances because of the Pauli principle. But in the case of the Na-ring, the increase of energy will be more rapid for small a - a separations because more than two alpha-particles begin to overlap, so the Pauli effect is stronger than is contained in the calculation of Seya et al. [16]. This is not a very important problem because for the high values of J with which we are chiefly concerned the centrifugal potential, rising for small values of R, means that the equilibrium radius of the Na-ring is at values of R where multiple-alpha-particle overlap is of no concern. However, in order to discuss the effective ring potential for all R-values, the following procedure was adopted: Use of the potential due to Seya et al. [16] plus the Coulomb energy (for J = 0) gives an Na-ring whose energy minimizes at a certain value of R such that multiple overlap of alpha-particles is negligible. For smaller values of R, where multiple-overlap sets in, treat the ring as a continuum, describing it as a toms of major radius R and cross-section radius X. Choose X such that the mean nucleon density within the toms is equal to that characteristic of light nuclei (0.091 fm -3); this yields X = 2.70 fro. Describe the nuclear fluid within the toms using the standard parameters of the semi-empirical mass equation [17] slightly adjusted as necessary to give equilibrium for both the R-value of the ring as given by the Na-procedure and the mean nucleon density of 0.091 fm -3. (These necessary adjustments, in fact, amount to 3% or less to the volume energy and surface tension terms.) In applying the constants of the semi-empirical mass equation to the configuration of the toms, the formulae presented by Wong [18] are used.~ To describe the compressibility of nuclear matter, and hence the change in the volume energy term as the toms is compressed at values of R below that of the equilibrium minimum, use the results for the equation of state of nuclear matter of the appropriate neutron-to-proton ratio presented by Irvine [19]. Take the surface energy term as proportional to the nucleon density. The details of this procedure do not affect any of the significant results. The overall potential is now one within which the (effectively one-dimensional) Schroedinger equation may be solved for the radial wave function of the Na-ring. The result is that the Na-ring for J = 0 has a radius Ro, given rather accurately by R o = 0.46N fro. This means that the a - a spacing around the ring is about 2.9 fm which is to be compared with the a - a spacing of about 2.5 fm found for ~~ by Seya et al. [16] ; the spacing in the Na-ring is somewhat larger because of the distending effect of the multiple Coulomb forces. We shall not be able to speak of a fully-developed ring until R > X which occurs for N = 6, but 3 ~< N < 6 will be considered shortly when the question of fission arises.
*In the second reference [18], Wong has considered the rotational stability of a nuclear torus of normal nuclear fluid (not related to the specific Na-rings of our present consideration) and concludes that such objects should be stable over ranges of J very similar, particularly for larger values of N, to those suggested by the present calculations.
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As the J-qalue rises the a - a bond stretches and the energy increases. At a certain Jmax the ring becomes unstable against break-up into Na + 2Nn although it will remain metastable to considerably higher J-values because of the Coulomb barrier. We f'md for N = 6, 10, 15, 20; Jmax = 25, 61, 127,210, respectively. For these Jmax we find Rmax --- 0.52N fm so that the rings stretch centrifugally by a little more than 10% before becoming unstable. It is important to enquire into the absolute stability of the rings against fission, symmetrical and asymmetrical, into two smaller rings. For sudden fission the new rings have J-values proportional to their N-values, symmetrical fission is then energetically favoured and becomes possible when J falls to about 0.3 Jmax independent of N. For slow fission in which flux around the deformed rings and the tension around the rings are constant, the new rings have J-values proportional to their N 2values. Asymmetric fission is now favoured and becomes possible when J falls to about 0.5 Jmax for large N-values or to less than this for smaller N. However, both modes of fission face large Coulomb barriers and must be very improbable. Stability against fission is therefore absolute for any mode and for all J-values above 0.5 Jmax, It is obvious that these Net-rings will present large cross sections for nuclear interactions. To estimate their cross section for relativistic collisions we may compute the cross section of a randomly oriented Net-ring of radii R and X, as defined above, and compare it with the cross section for a normal nucleus of the same Z-value, defining the latter as the interaction cross section as measured using neutrons of 20 GeV [20]. For Z-values of 8, 16, 24, 32, 40 we find that the rings present effective cross sections greater by the following factors than ordinary nuclei of the same Z-value: 1.9, 2.2, 2.6, 3.2, 3.7, respectively. However, these enhancement factors ignore the effect of the loosely-bound neutron penumbra to which reference was made above and which may considerably increase the effective interaction cross section. This penumbra could also explain the reported persistence of the short free paths following successive interactions since the removal of a few peripheral neutrons from the anomalon would leave the ring intact for further enhanced interactions. The question of the electromagnetic lifetimes for de-excitation down the Jchain remains open at present - recall that anomalons must have lifetimes of 10 -1~ sec or so - but it is quite possible that their high degree of circular symmetry will inhibit fast radiation of low order. 6.
Envoi
Whether or not the phenomenon of the anomalon has already been established, it has caused a great deal of wide speculation covering a remarkable range of nuclear and particle physics; I am sure it is the kind of quest that will give Kenzo Sugimoto interest and amusement. I should be surprised if he did not have his own ideas about it.
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