Tip model of cold fission

Nuclear Physics A530 (1991) 27-57 North-Holland

MODEL

C

FISSION

F. GÖNNENWEIN and B. BÖRSIG Physikalisches Institut, Universitat Tübingen, Morgenstelle 14, D-7400 Tübingen, Germany Received 7 December 1990 (Revised 28 January 1991) Abstract: Cold fission is defined to be the limiting case of nuclear fission where virtually all of the available energy is converted into the total kinetic energy of the fragments . The fragments have, therefore, to be born in or at least close to their respective ground states . Starting from the viewpoint that cold fission corresponds to most compact scission configurations, energy constraints have been exploited to calculate minimum tip distances between the two nascent fragments in binary fission. Crucial input parameters to this tip model of cold fission are the ground-state deformations of fragment nuclei. It is shown that the minimum tip distances being compatible with energy conservation vary strongly with both the mass and charge fragmentation of the fission prone nucleus . The tip distances refer to nuclei with equivalent sharp surfaces. In keeping with the size of the surface width of leptodermous nuclei, only configurations where the tip distances are smaller than a few fm may be considered as valid scission configurations. From a comparison with experimental data on cold fission this critical tip distance appears to be 3.0 fm for the model parameters chosen. Whenever the model calculation yields tip distances being smaller than the critical value, a necessary condition for attaining cold fission is considered to be fulfilled. It is shown that this criterion allows to understand in fair agreement with experiment which mass fragmentations are susceptible to lead to cold fission and which fragment-charge divisions are the most favored in each isobaric mass chain. Being based merely on energy arguments, the model cannot aim at predicting fragment yields in cold fission . However, the tip model proposed appears well suited to delineate the phase space where cold fission phenomena may come into sight.

l . Introduction The energy release in binary nuclear fission is shared between the total kinetic and the total excitation energy of the two primary fragments . As a rule, the bulk of the excitation energy of the primary fragments is carried away by neutrons . It is long known, however, that in spontaneous fission or at low compound excitation energies of fission prone actinide nuclei, fission may proceed with no neutrons at all being emitted . The probability for neutronless events is rather small and does not exceed a few percent in the lighter actinides ',2 ). These events may be brought about by neutron-gamma competition in favor of gamma emission, though in the majority of cases the excitation energy of a fragment has to fall below the neutron separation energy by neutron evaporation before gamma emission takes over. Still, another compelling reason from the energetics for neutronless fission may just be a situation where from the outset the fission fragments do not carry sufficient excitation energy to overcome the binding forces of neutrons . From the energy balance the total energy has in this case to show up mainly in the kinetic energy of the fragments . Fission events with the kinetic energies of the two fragments coming 0375-9474/91/$03 .50 © 1991 -- Elsevier Science Publishers B.V. (North-Holland)

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F Ginnenwein, B. Bbrsig / Tip model of cold fission

close to the total available energy or Q-value of the reaction have indeed been observed quite early in the history of fission research . As already stated in 1961 by Milton and Baser) fission events showing up in this region are running out of energy and the fragments have to be formed nearly in their ground states . For many years this aspect of fission was considered as, a mere curiosity and drew no further attention. Instead, the interest was focused on a companion observation, viz. the fine structure of fragment mass distributions for fixed kinetic energy bins ;'4). The structure becomes the better pronounced, the higher the kinetic energy is. From an analysis of the fine structure it was realised that it is correlated to the preferential formation of (even, even) fragments ") . This prompted the idea that in low energy fission a superfluid component may have a chance to survive in the flow of nuclear matter from the saddle to the scission point 7). Whatever mechanism leading to high kinetic energy events is invoked, either a compact scission configuration with a large Coulomb repulsion or a large prescission kinetic energy, it is to be expected that these events should offer the best chance to preserve superfluidity . The conjecture has incited Signarbieux et ad "') to explore in more detail fragmentations at experimentally the highest feasible kinetic energies of the fragments . A fission process where, as indicated above, the fragments are born in or close to their ground states is a cold rearrangement of many nucleons . This is especially true for the spontaneous reaction where initially all nucleons together make up a single nucleus in its ground state, while in the final state the nucleons are located in two separate fragment nuclei, both again in their respective ground states. This limiting case of fission where all of the Q-value is converted into fragment kinetic energies has, therefore, been called "cold fission" . The close analogy of cold fission to a-decay or, more generally, cluster radioactivity has been put forward by the Frankfurt-Bucharest group 9- ") since it is a characteristic feature of these decays that virtually all of the available energy is carried away by the kinetic energy of the decay products . A further viewpoint has been pushed by Armbruster' 2) . Cold fission is considered here to be the inverse reaction to cold fusion, the reaction having led to the discovery of very heavy elements 13). It is argued that from a detailed study of cold fission one may find guidelines for a proper selection of reaction partners best suited for observing cold fusion. Rom an experimental point of view, the low yields and hence count rates to be expected for cold fission are a drawback. On the other hand, the fact that in these processes no neutrons are emitted from the fragments, greatly simplifies the analysis of coincidence experiments on the fragments' energies and/or velocities . The fragment masses are indeed uniquely determined upon applying mass and momentum conservation laws to the reaction at hand. Provided the detector resolution is appropriate, the masses of the fission fragments may be identified one by one. The first ones to fully exploit this possibility by a time-of-flight technique were Signarbieux et aL '). The discovery that gas ionization chambers operated with pure alkanes as counting gases exhibit intrinsic resolutions coming close to SE = 100 keV for

F. Gönnenwein, B. Börsig / Tip model of cold fission

29

fission fragments '4) opened up the possibility to use a back-to-back twin ionization chamber for studying cold fission by simply measuring the kinetic energies of correlated fragments '5, '6 ). The ionization chamber technique could even be pushed to identify the nuclear charges of the fragments in cold fission' 1-'7). In parallel to the above detector techniques cold fission has been investigated on the fragment mass separator Lohengrin of the Institut Laue-Langevin in Grenoble ' g,'9) . Basically, the separator allows to determine with perfect resolving power the masses and energies of fragments . In addition, with ®E-E detecting devices placed in the focal plane, it has become feasible to establish also the different nuclear charges in a given mass chain 2°,21) . Comprehensive data on fragment masses and charges showing up in cold fission are by now available for the more common reactions, viz. thermal neutron-induced fission of the compound nuclei 234U , 236U and 24°pu, and spontaneous fission of 252Cf. In the present work it is proposed to analyze which features of cold fission may be understood from a static model of the scission configuration. 2. Model for cold fission For any given mass and charge fragmentation in fission one observes a distribution of the kinetic energies of the fragments . The distribution is near-gaussian and may be specified by the average kinetic energy and the width or variance of the distribution. In a dynamical theory of nuclear fission at least two terms contributing to the kinetic energy have to be taken into account, viz. the Coulomb repulsion between the nascent fragments at scission and the prescission kinetic energy the fragments may already have acquired while sloping down the potential energy surface from the saddle to the scission point 22-24) . Correspondingly, the variance of the distribution has to be traced back to fluctuations in the scission configurations affecting the Coulomb interaction between fragments and to fluctuations in the prescission kinetic energies with, as an additional difficulty, the two types of fluctuations considered not being independent of each other 23) . Cold fission is characterized by the maximum feasible kinetic energy release . The question then is which combination of the two components of the kinetic energy invoked above will bring about the largest kinetic energies . Evidently, the mutual Coulomb energy will be maximised for the most compact scission configurations . Since in this case the scission point is pushed as close as feasible towards the saddle point, not much potential energy is liberated and, hence, also the prescission kinetic energy is expected to be small . In fact, it may be argued that this energy is close to nil. The reasoning is best explained by referring to a scission-point model of fission . The different energy terms coming into play at scission in such a model are visualised conceivable scission configurations are for simplicity schematically . . parametrised by two almost-touching prolate spheroids representing the fragments . In fig. 1 it has been assumed that scission configuration with zero deformation

30

F. Gbnnenwein, B. Bbrsig / Tip model of cold fission

Deformation

00

Fig. 1. Schematic presentation ofpotential energies in a scission-point model offission versus deformation of fragments . V, is the mutual Coulomb energy, VD the total deformation energy of fragments, V, the total potential energy and Q the Q-value of the reactions .

corresponds to two juxtaposed spherical fragments and that the ground-state shapes of the two fragments are spheres, i.e. the deformation energy VD vanishes for zero deformation . Upon deforming the fragments one reaches scission configurations being more stretched out. The repulsive Coulomb potential Vc will then decrease, while the deformation energy VD of the fragments will increase . The sum of the two potential energies is depicted in fig. I as the total potential energy VP= YC+ VDEnergy conservation imposes a constraint on the admissible ultimate scission configurations where the neck joining the two fragments breaks apart. The potential energy VP of this configuration will have to be smaller than the total available energy or Q-value of the reaction, the latter being fixed for any given mass/charge fragmentation . In general, the energy difference (Q - VP) will be left to the fragments as what has been called "free energy" (see fig. 1). This energy may show up partly as prescission kinetic energy, and partly as intrinsic excitation energy ofthe fragments right as scission. For the most compact scission configuration still being compatible with energy conservation no free energy is available. Hence, in the framework of a static scission-point model one has to conclude that for this configuration also the prescission kinetic energy is squeezed down to zero. The maximum Coulomb repulsion between fragments is, therefore, attained at the full expense of prescission kinetic energy. The next question that may be raised is whether the maximum Coulomb energy in fact corresponds to the maximum total kinetic energy TIDE of fragments observable in experiment, or whether a judicious combination of Coulomb and prescission energies may yield even larger TKE's . As reasoned in the preceding paragraph, the

F. Gönnenwein, B. Börsig / Tip model of cold fission

31

prescission kinetic energy is drained from the free energy. From the schematic dependence of the deformation energy VD on the overall deformation of the scission configuration shown in fig. I it appears that a sizeable free energy involves, however, large deformation energies . In addition, due to nuclear viscosity, part of the free energy will be converted into intrinsic fragment excitation. Since both, the deformation and the excitation energy at scission, contribute to the total excitation energy of the fragments TXE at infinity, the latter energy is expected to become rather large whenever free energy is liberated. The minimum feasible TXE ;, is read from fig. 1 to obtain for the most compact scission configuration being consistent with energy conservation, i.e. for a vanishing free energy. Finally, keeping in mind that the available energy Q has to be shared between the total kinetic and excitation energy TKE and TXE, respectively, the minimum TXE correlates to the maximum TKE. Hence, the maximum kinetic energy release TKEmax should precisely correspond to that compact scission configuration where no free energy and, as a consequence, no prescission kinetic energy is made available. The configuration is labeled as "cold fragmentation" in fig. 1, since the fragments will carry no intrinsic excitation energy at scission or, stated otherwise, they have to be born in a cold state. This configuration is simple in the sense that the two observable quantities TKE and TXE not only reach a maximum or minimum, respectively, but they are both uniquely linked to potential energies at scission, viz. TKE max = VC ,

TXE;n = VD,

Q = VC + Vo -

(1)

As indicated in fig. 1, even in cold fragmentation the maximu=n measurable kinetic energy TKE,ax will not necesEarily fully exhaust the Q-value of the reaction, that is the energy at one's disposal . To meet within the framework ofa static scission-point model the conditions for cold fission, whose definition may succinctly be given as TXE,,; - 0 or TKEmax - Q, one has to postulate that one or both of the fragments have ground-state shapes being deformed. More precisely, taking the schematic model of fig. 1 at face value, prolate ground-state deformations of the fragments are envisaged . In fact, this will shift the zero point of the deformation energy Vp = 0 to more elongated scission configurations . If now by chance, say, the deformation energy vanishes right at the same scission deformation where in fig. 1 the Coulomb energy Vc crosses the reaction Q-value, then the cold fragmentation point will move towards the crossover point Vc = Q. From eq. (_) one then has TXE  ;, = 0 and TKE max = Q, i.e . cold fission as a special case of cold fragmentation becomes feasible. In case the chance coincidence of VD = 0 and Vc = Q is not perfectly valid, the maximum total kinetic energy TKE max will fall more or less short of the Q-value . The schematic model having been expounded showing promising simple features for understanding cold fission, it has evidently to be put on more quantitative grounds . The model being static, it should be stressed that from such a model one may learn something on the limits of the phase space being available in principle . But whether the actual limit, e.g. cold fission, is indeed attained or not has to be

32

F Gbnnenvvein, B. Bbrsig / Tip model ofcoldfission

inferred from a dynamical treatment of fission, or, as will be done here, from a comparison with the experimental evidence . quantitative scission point model w :,-h cold fission in mind will have to start from two complementary fragments of given mass and charge number, both taken in their ground states in order to avoid any deformation energy. A crucial ingredient to the calculations then is the knowledge of the ground-state deformations of nuclei in a wide range of masses and charges. Experimentally, nuclear shapes may be determined from a measurement of ß(E2) values . The available information on for even-even nuclei has been surveyed experimental ß(E2) values and compared .25,26) to various model predictions by Raman et al . The nuclear ground-state deforform mations deduced a valuable set of data but, unfortunately, cover only a small fraction of the deformation data needed. Fcr thu~~ ca!ctdations we had in the m. ajoeity of cases, therefore, to rely on the extensive tables of nuclear shapes having been prepared by M611er and Nix on the basis ofa macroscopic-microscopic approach to nuclear structure. The shapes are calculated by these authors in the E,,-parametrisation and considered to be rotationally symmetric. For the present purpose the deformation parameters (-c, , --,) have been taken into account. In the next step the two fragments are placed on a common axis with the intrinsic symmetry axes of the individual nuclei being aligned parallel to the line joining their center of gravities. The leading term of the interaction will certainly be the Coulomb repulsion between the fragments, but at smaller separation distances an attractive nuclear potential will come into play. For the compact scission configurations presumed to be typical for cold fission this nuclear interaction should not be neglected . Various prescriptions have been given how to calculate the nuclear interaction part 28-30). In some earlier calculations 31,32) the fragment shapes had simply been approximated by spheroids and the parametrisation of the nuclear interaction as given by Krappe et al. 29) was adopted. For the more complex nuclear shapes studied in the present work an elaborate scheme for determining both the Coulomb and the nuclear interaction has been presented by Shi and Swiatecki 33 ). The nuclear interaction is here approximated by the proximity potential of Blocki et al. 28-30) . The interaction energy of the two adjacent fragments was determined following closely the Shi-Swiatecki recipe 33). With the fragment shapes being fixed, the only free parameter left is the distance r between the charge centers, or equivalently the tip distance d between the fragments . It will be sufficient to indicate the main lines of the calculation . For more details the reader should consult the original literature 28,30,33) . In the work of Shi-Swiatecki the equivalent sharp surface of nuclei R(19) is written in the a,,-parametrisation: (0) = (1Z/,A)[ 1 + a2P2(COS 0) + a4P4(COS 0)] ,

(2)

with 0 the polar angle relative to the axis of symmetry, P,,(cos 49) the Legendre

F. Gönnenwein, B. Bürsig / Tip model of cold fission

33

polynomials and A a factor ensuring volume conservation with respect to a sphere is parametrised as R =1 .28A1/3 -0.76-ß-0.8A -D/ 3 with of radius R. The radius the mass number of the nucleus . R is given in units of fm and for typical fragment masses ranging from A = 80 to A =160 corresponds in R = ro A 1 / 3 to a nuclear radius "constant" ro which slightly increases with mass, viz. from ro - -1.15 fm to ro ---1.17 fm, respectively . The expansion parameters an of eq. (2) are found by transforming the E, deformation parameters calculated by M61ler-Nix for the groand-state shapes of nuclei 2') from the E,, to the a presentation . The nuclear radius constant in that study has been taken to be ro =1 .16 fm for all nuclear masses A and is, hence, compatible with the radius "constant" ro adopted by Shi-Swiatecki . Relationships between various parametrizations of the nuclear surface have been conveniently summarised by Hasse and Myers 34) . The Coulomb potential Vc between two deformed homogeneously charged nuclei with collinear symmetry axes and radii R(O- =0") is then readily obtained by slightly generalising a formula derived by Shi-Swiatecki 33), giving the interaction of one deformed nucleus in the (a2 , a4) parametrisation with a sphere, to the case of two deformed nuclei . This has been done in analogy to the work of Cohen and Swiatecki 35). It should be noted, however, that in the form-dependent correction factor F to the Coulomb interaction Vc = Z1Z2e r

2

F,

between two homogeneously charged spheres, the terms containing the mixed deformations of both fragments have been calculated under the approximation E4= 0 and a4 = 0. The attractive nuclear proximity potential VN is introduced for nuclei with a finite surface thickness and written as VN = 4 ,ffRbyO(C) .

(4)

Part of the factors in eq. (4) are directly related to the leptodermous distribution of the nuclear density at the surface of the nucleus . Thus, the factor b represents the difuseness of the surface . Without giving the precise definition of b, which may be found in ref. 34), the quantity 26 may, to good approximation, be pictured as the change in radius when the nuclear density drops from 90% to 10% of its value at the center. The difuseness b is taken to be b -1 fm. Furthermore, the proximity interaction refers to the central radii C (O) of nuclei being related to the equivalent sharp radius R(O) of eq. (2) by C(O)=R(O) - ZKb 2 ,

(5)

with rc the total curvature of the surface at the point in question . For the tip on the symmetry axis the curvature K is K = 2/ Re with Re the curvature radius at the tip. In case of spherical symmetry, for most distribution riiinctions of the nuclear density considered, the central radius C equals the radius R,/2 at one half of the central

F Giinnenwein, B. Börsig / Tip model of cold fission

34

density 34).

e qu

tity

in eq. (4) then is defined as CO 1 C02/ ( CO 1 + C02) ,

and takes into account the curvature of the tips of the two fragments (label 1 and 2, respectively) expressed by the central radii of curvature C® . Explicit formulas for the radii of curvature in the (0-1 , a4) parametrisation have been derived by e reduction of the surface energy whenever two nuclei overlap S i- wiatecki 33 ) . is at the basis of the proximity interaction. In eq. (4) this becomes manifest by the factor y, the surface tension, being parametrised as y = 0 .9517[l -1 .7826(( -Z)/A)2] 1VieV/fm2 . Finai®y, %(C) in eq. (4) stands for the universal nuclear proximity function which governs the strength of the interaction as a function of the distance r between the charge centers . It is given as a function of C = s/ b with s = r - (C, + C2), where C, and C? are the central tip radii of the two fragments. O(C) is defined for ~, 0. For the distance r of closest approach envisaged r is equal to the sum (C, + C2 ) of the two central radii, i.e. s = 0 and C = 0, and O(C) is shown to take on the value (0) = -1 .7817 . For ~ increasing 0(C) rapidly approaches zero, e.g. 0(4)/0(0) 10-2. Again, the full formula for O(C) is to be found in the literature 30). As a final step, the condition for cold fission in the framework of a static scission-point model is imposed by requiring that the reaction Q-value is equal to the total interaction energy of two fragments being in their respective ground states at scission, i .e . with the Coulomb and the proximity potential energy taken from eqs. (3) and (4), respectively . Let us recall that ultimately the interaction energy ( Vc + VN) is converted into the total kinetic energy release TKEnax of the fragments being observed in experiment. The Q-values needed for evaluating eq. (8) were computed from the experimental mass tables having been compiled by Wapstra et cal. 36 ) for any given (mass, charge) fragmentation relevant to fission . For some fragmentations where experimental data are not yet available the mass tables were complemented by theoretical mass values taken from the work of M61ler and Nix 27) . It should be noted that for the thermalneutron-induced fission reactions studied the binding energy ofthe incoming neutron has to be included in the Q-value . As a result of the calculations one obtains a minimum distance rn,i,, of the charge centers of the fragments at scission being compatible with energy conservation . For a physical interpretation of the results it is, however, worthwhile to consider instead the minimum tip distance drain separating the two fragments being defined as drain = rmin -

where

and

2

(R, +

R2)

are the equivalent sharp radii of the tips, or the lengths of the

F. Gönnenwein, B. Börsig / Tip model of cold fission

35

symmetry axes, of the fragments 1 and 2, respectively. The basic idea of the model proposed for cold fission ultimately invokes the size of the minimum tip distance dm; calculated from the condition expressed in eq. (8) as the crucial criterion . In case the tip distance dm ;,, turns out to be rather large, say larger than about 3-4 fm, the corresponding configuration cannot be considered as a valid scission configuration since the two nascent fragments do not overlap . In keeping with a leptodermous nuclear surface and with the difuseness constant b taken to be b =1 fm, a valid scission configuration should have tip distances dm;  not larger than perhaps 2-3 fm. Putting forward a critical tip distance ofabout dm; - 3 fm for discriminating between valid and non-valid scission configurations is intuitively reasonable and, as shown in the following paragraphs, corroborated by experiment as the borderline governing the outcome of cold fission. It should be stressed that in the present context it has not been sought to find a proper definition for a valid scission configuration from theory . Anyhow, by introducing minimum tip distances dm ;,, as a criterion for attaining cold fission the present model may be called a "tip-model of cold fission". A preliminary account of some partial results was already published elsewhere 37) . 3. Potential energy curves

To start with it should be helpful to illustrate the dependence of the Coulomb potential Vc and the nuclear proximity potential VN on the tip distance d between two complementary and collinearly aligned fission fragments, both taken to be deformed into a shape corresponding to their ground states . This is shown in fig. 2 for a fissioning 236U nucleus with the mass and charge ratio for the light/heavy fragment being chosen to be A L/A H =104/132 and ZL/ZH = 42/50, respectively . The repulsive Coulomb potential exhibits of course the expected decrease upon increasing the tip distance and, hence, increasing the distance between the fragment charge centers . The attractive proximity potential partly balances the Coulomb potential. However, at tip distances larger than, say, d - 4 fm the proximity potential has virtually faded away and the interaction between the two fragments becomes purely electrostatic in nature . The interplay between the two types of interactions is depicted in fig. 3 as the energy [(Vc + VN) - Q] versus the tip distance d for the same fragmentation of a 236U nucleus as in fig. 2. In calculating the Q-values, the binding energy of the last neutron has been included as the excitation energy of the 236U compound nucleus . Therefore, the potential energy curve in fig. 3 corresponds to thermal-neutroninduced fission for the reaction 235U( n, f) . Usually, this reaction is held to be over-the-barrier fission since, both from experiment and theory, the standard fission barrier turns out to be lower than the neutron binding energy . The potential barrier showing up in fig. 3 does not contradict this notion . In fact, the occurrence of the barrier is simply due to the rather artificial sequence of different scission configurations adopted in the model. The barrier only tells whether a specific scission

36

iinnenivein, ß. Börsig J Tip model ofcoldfission 260

2360 AL/AH _ 104/132 ZL/ZH = 42/ 50

250 240 230

Fig. 2. Coulomb potential Vc (right scale) and nuclear proximity potential VN (left scale) for the mass and charge fragmentation given in the insert of a fissioning 2360 nucleus versus the tip distance between the two nascent fragments. The deformations of the fragments are kept fixed and taken to be the ones of the ground states .

configuration is prohibited by energy conservation or not. Extremely compact scission configurations like the touching configuration at d = 0 fm are outruled . In the example of fig. 3 the most compact configuration not violating the energy law would be charar :C zed by a minimum tip distance d,,,i =2-6 fm, where the total potential energy ( Vc + VN) just equals the Q-value of the reaction [see eq. (8)]. This tip distance matches in fig. 1 the condition for cold fragmentation under the additional constraint, though, that the deformation energy VD vanishes and, hence, cold fragmentation may become cold fission . The distance of closest approach d,,i indicates the limit of phase space accessible to fission at the highest total kinetic energy or lowest total excitation energy of fragments being feasible. efore exploiting the properties and the dependence of the minimum tip distance dmi  on the fission reaction considered in a more systematic way for interpreting cold fission, let us stress that potential barriers similar to the one shown in fig. 3 have originally been studied in connection with cluster radioactivity 9-" ' '3 ) . There the probability for spontaneous cluster emission from heavy nuclei has been deduced from the lifetimes against tunnelling through a barrier, which for separated daughter nuclei was obtained by Shi-Swiatecki 33,38) along the lines sketched in sect. 2, while the barrier in the initial phase of the deforming mother nucleus was found by smooth extrapolation of the outer barrier into the interior of the nucleus prone to decay . The same basic concept, but with a different procedure for computing the barrier,

F. Gbnnenwein, B. ßörsig / Tip model of'coldfission

37

10 .0

235 U(n,f)

z

AL/AH = 104/132 ZL/ZH =

42/ 50

.r a

3.0

Tip - Distance d / fm Fig. 3. Excess of total interaction energy (Vc + VN ), i.e. sum of Coulomb and nuclear energy, of two ground-state fragments over Q-value for the thermal neutron fission reaction 235 U( n, f) with fragments of given masses and charges versus tip distance between fragments .

was first introduced by the Frankfurt-Bucharest group 9-") to analyze cluster radioactivity and later on extended to also cover cold fission . In some more recent work by this group the cumbersome calculation of the tunnelling probability has been replaced by a barrier height criterion in order to select among the various fragmentations the most propitious cases for cold fission 39,40) . It is argued that the lower the barrier, the higher the chance to observe cold fission should be. In case the ShiSwiatecki formalism is followed to search for a barrier, the criterion, though, is not unambiguous . The criterion is clear cut e.g. for thermal-neutron-induced fission of 235U where for all fragmentations an outer barrier shows up and, hence, a barrier height may readily be deduced. A typical example is given in fig. 3. However, as 252Cf represents a already noted by Sandulescu et al. 40), spontaneous fission of counter example, since there are fragmentations where iiie barïier is pushed to the interior of the mother nucleus and, hence, is not that easy to evaluate. A potential energy curve of this type is presented in fig. 4 for the mass split A, /A H =104/ 148 and the charge split ZL/ZH = 42/56 of 2S2Cf(sf) . Independently from this more technical difficulty, viewing cold fission similarly to cluster radioactivity as a tunnelling phenomenon through a barrier constructed in the way described means that a mechanism is invoked where the fragments assume their final ground-state shapes at a very early stage of fission . Though for the present work the same Shi-Swiatecki prescription for calculating the potential energy curve

33

F. Gbnnenwein, B Börsig / Tip model ofcoldfission 10.0

252Cf

Tip - Distance d / fm Fig. 4. Same as fig. 3 but for the reaction `'`'Cf(sfy

of separated fragments is made use of as in the study of Sandulescu et al. 4°), the physical interpretation fostered here is quite different and more in accordance with standard fission theory . Indeed, the criterion Q = Vc + VN proposed is introduced to specify the most compact scission configuration compatible with energy conservation without claiming that the fissioning nucleus has to tunnel through the barrier calculated in the one-dimensional parametrization of scission configurations chosen. From this merely SiàiiC consideration 't iç neither possible to infer how a fissioning nucleus manages to reach the scission configuration in question, nor whether the configuration is reached at all. possible dyna ical path leading within the framework ofstandard fission theory to very compact scission configurations has been outlined by Berger et al. 41,42) . In a microscopic calculation of the potential energy surface of a fissioning nucleus with two deformation parameters (intuitively speaking the overall elongation of the nucleus and the neck constriction) taken into account, two valleys come into sight: a first one, called the fission valley, for a not fragmented nucleus, and a second one, called the fusion valley, for separated fragments . The two valleys are kept apart by a potential ridge which, however, for sufficiently strong elongations eventually vanishes . Ordinary fission is thought to choose this latter "exit point" for passing from the fission into the fusion valley, i.e. to undergo scission. Nevertheless, there may be a finite chance for the nucleus to tunnel through the potential ridge or "constriction barrier" at much smaller than standard elongations . Thereby the

F. Gönnenwein, B. Bdrsig j Tip model of coldfission

39

scission point is shifted towards a compact configuration being characteristic for cold fission. Besides tunnelling through a constriction barrier other mechanisms within established fission theory are conceivable which could 'Pad to cold fission 42). We will not go into more details here, but we want to stress that the results to be presented in the following being based on an unequivocal energy criterion are quite independent from the specific mechanism leading to cold fission. 4. Results

Results for the minimum tip distance dmi between two fission fragments at scission having been calculated in the framework ofthe model outlined in sect. 2 are depicted in fig. 5 for two reactions: thermal-neutron-induced fission of 236U* and spontaneous fission of 252 Cf. The tip distance is plotted as a function of the light-fragment mass, with the heavy-fragment mass being the complement to the respective compoundnucleus mass . For the calculations presented in fig. 5 the ground-state deformations of the fragments were taken to be those given in the tables of öller-Nix 27). Some more detailed information concerning nuclear shapes, as deduced in the framework of the above theory, may be found in ref. 43 ). Theory predicts that, besides mass regions with prolate deformations, there exists an CÄtcnded region of nuclei around

235 U(n,f)

W U C

v

m

4

C

70 80 90 100110120

70 80 90 100110120130

Mass of light fragment AL/u

235U(n, f) with Fig . 5. Minimum tip distances as a function of light-fragment mass AL for the reactions 252 Cf(sf) (right) . The ground-state deformations of fragments have been thermal neutrons (left) and taken from theor .271 .

40

F. G6nnenwein, B. Bbrsig / Tip model oî cold fission

= 45 and N = 70 where the ground states are pronouncedly oblate in shape . These nuclei turn up as fission fragments . For oblate fragments, the minimum tip distances at scission being compatible with the condition Q = Vim + VN [see eq. (8)] will become large in comparison to prolate fragments . This is simply due to the fact that for any tip fixed distance the fragment charge centers are closer by in case of oblate rather than prolate deformations . The energy condition ofeq. (8) will, hence, impose larger tip distances for oblate compared to prolate fragments . This feature is indeed manifest in fig. 5 for mass numbers A of the light fragment around A = Z + N 45+70=115 . Still, one may wonder whether oblate deformations come into play in the process of fission . In addition, in many cases where the theoretical potential energy surfaces of nuclei as a function of deformation exhibit an absolute minimum for oblate shapes, there is also a marked minimum not far away in energy for prolate shapes 43) . For the nuclei in question here, we scanned the potential energy surfaces made available to us through the courtesy of Mbller"). In all cases where a (relative) minimum of the potential energy for prolate deformations was within I MeV of the (absolute) minimum for oblate deformations, the calculations of the minimum tip distance were repeated for prolate fragment shapes. This prescription was followed for all calculational results to be presented below. An example for the minimum tip distances obtained with this slight modification for of the scheme is given in fig. 6 the thermal-neutron-induced reaction 235 U(n, f) . The tip distances as a function oflight-fragment mass A L reveal a striking dependence on the fragmentation of the compound nucleus . Tip distances around d.i,, = 3 fm are obtained for light-fragment masses ranging from A L - 80 to A L --106, with minimum values reached around A L --104. Both towards symmetric and very asymmetric mass splits the tip distances increase sharply . In the spirit of the proposed tip model for cold fission discussed in sect. 2, cold or near-cold fission should become observable in experiment only in the mass range A L - 80 through A L --106, with the most promising case being the fragmentation A L/AH = 104/132 . The physical reason why the latter fragmentation emerges as the most favourable for cold fission is readily seen : the doubly-magic-heavy fragment AH =132 with ZH = 50 and NH = 82 boosts the Q-value, while the strong prolate deformation of the light fragment A L =104 lowers the Coulomb interaction Vc at any fixed tip distance ; large Q-values and small Coulomb potential energies Vc act coherently to allow for short tip distances under the condition Q = Vc + VN of cold fission . The special importance of the magic cluster nucleus 132 Sn for the outcome of cold fission recalls the role played by the magic 208 Pb nucleus for cluster radioactivity, which is likewise a cold process with virtually no excitation energy being imparted to the daughter nuclei. Starting from a different physical picture this analogy has been fully exploited by the Frankfurt-Bucharest group" 1,39,4°). A survey of minimum tip distances at scission as a function of tight-fragment mass A L is depicted in fig. 7 for several thermal-neutron-induced fission reactions

F. Gbnnenwein, B. Bbrsig / Tip model of cold fission

41

5.0

U235(n,f)

W 4.5 c

0 C

3.0-

2.5

60

70

80

90

160

110

120

Mass of light fragment AI

Fig. 6. Minimum tip distances as a function of light-fragment mass AL for the reaction 235U(n, f) induced by thermal neutrons . In contrast to Fig. 5 oblate ground-state deformations of fragments have been rejected and replaced by prolate deformations wht:.ever feasible (see text).

ranging from 229Th(n, f) up to 245Cm(n, f). The general appearance of the curves is very similar. From the gross structure of the curves the message is that the highest probability for cold fission to be attained is reached for asymmetric mass divisions roughly coinciding with the mass range where the yields ofthese typically asymmetric fission reactions reach their peak values. In other words, for very asymmetric and symmetric mass splits cold fission is expected to be inhibited from the present model. At closer inspection it should, however, be noticed that with larger masses A or fissilities Z2/ A of the compound nuclei the tip distances for both, asymmetric and symmetric fission, decrease systematically . In the tip model being put forward this should indicate that cold fission is to become observable in a broader mass range and possibly with higher yields . An interesting observation, most clearly pronounced for the 245Cm(n, f) reaction in fig. 7, is further that the smallest tip distances are 235 not necessarily correlated with magic cluster fragments as discussed for the U(n, f) case. For 245 Cm(n, f) the closest approach between the tips is found for fragmentations with the light- and heavy-fragment mass being typically AL = 98 and A H =148, respectively. The reason for the smallness of the tip distances is here due to the large prolate deformations of both fragments . The typical Q-values for the above mass splits are almost 20 MeV below the maximum Q-value reached for the fragment mass ratio AL/AH =120/126. This specific example shows that, as to be anticipated, the size of the minimum tip distance depends on a subtle balance between the

42

F. G6nnenwein, B. Börsig / Tip model of coldfission

i

'

l

-T

80

z

100

_

120 60

80

100

Mass of light fragment AL/u

120

Fig. 7. Survey of minimum tip distances as a function of light-fragment mass A L for some thermal neutron reactions .

-value of the fragmentation being considered and the intrinsic deformations of the outcomi g fragments . A sample of minimum tip distances having been calculated for some spontaneous fission reactions are displayed in fig. 8 . As already pointed out, in contrast to fig. 5 only prolate ground-state or near-ground-state shapes of the fragments have been made allowance for. As a consequence, the spike in the tip distance for light masses around A L =115 being prominent in fig. 5 for the 252 Cf(sf) reaction has disappeared in fig. 8. Otherwise, the trend for the tip distance to decrease for increasing fissility of the compound nucleus, to be read from fig. 7 ijr the thermal neutron (n, f) reactions, persists in fig. 8 for spontaneous fission of still heavier nuclei. Still, two finer details are noteworthy . Firstly, for the heavier systems the tip distances

F. Gönnenwein, B. Börsig / Tip model of cold fission

43

6 5

C

4 3 2 5 260Md( sf)

4 3 2 1 60

80

100

12060

80

100

120

Mass of light fragment AL/ u Fig . 8. Survey of minimum tip distances as a function of light-fragment mass AL for some spontaneous fission reactions .

portrayed in fig. 8 fall below dmi = 3 fm close to symmetric mass divisions. This feature has obviously to be traced back to outstandingly large Q-values whenever fragmentations into two near-magic cluster nuclei with both the light and heavy fragment coming close to 132Sn become feasible . Secondly, for the Md isotopes a conspicuous dip of the minimum tip distance is observed for light-fragment masses A L near A L =105 and, hence, mass ratios A L/AH =105/ 155 . Similarly to the 245 Cm(n, f) case discussed in connection with fig. 7, also for the d isotopes two strongly prolate fragments come together for the above mass splits . The same situation obtains, in fact, for the 26°Fm(sf) reaction studied . However, the respective Q-values for the Md isotopes are larger by more than 8 MeV in comparison to

260Fm . This is sufficient to bring down the calculated minimum tip distance to values

approaching d =1 fm, where the influence of the nuclear proximity potential becomes strongly felt (see fig. 2). Accordingly, the dramatic decrease of the tip distance in the Md isotopes mainly reflects the steep dependence of the nuclear potential on fragment separation at close distances . The fragmentation of a fissioning nucleus is fully defined by specifying both the mass and the charge ratio of the light to heavy fragment AL/AH and ZL/ZH, respectively. So far, for any given mass ratio A L/A H only results for the smallest tip distances as a function of the charge ratio ZL/ZH were discussed since these should indicate the most favorable candidates for cold fission . However, for all

44

E Gbnnenwein, B. Bdrsig / Tip model of cold fission

54-

w

3

J

AL/AH=86/150

AL/AH=87/149

i

1

235

U(n,f) I

AL/AH=88/148

I

I I I I I I 32 34 36 , 38

I

.

40

Charge Fig. 9.

I

I

I

I

I

I

I

I

I

I

AL/AH= 104/132

.

I

42 of

I

.

I

I

44

I

light

I

.

32

.

34

I I - I , I r---r 36 , 38 40 42 44

fragment

ZL

Dependence of minimum tip distance dm; on light-fragment charge ZL for some mass fragmentations indicated in the inserts for thermal neutron induced fission 235U(n, f) .

AL/AH =92 / 160

J

AL/AH=110/142

E

AL/AH=98/154

2

32

34

36

38

40

Charge Fig . 10 .

42 of

44

4:0

light

4~2

4:4

4:6

fragment

4:8

50

~2

ZL

Dependence of minimum tip distance d, i ,, on light-fragment charge ZL for mass fragmentations 252Cf(Sf) . specified in the inserts for

F. G6nnenwein, B. B6rsig / Tip model of coldfission

45

reactions studied there are many mass fragmentations where several charge splits compete to yield small tip distances. Some examples are given in figs . 9 and 10 for the reactions 235 U( n, f) induced by thermal neutrons and 252Cf(sf), respectively. For the mass ratios specified in the inserts the minimum tip distance is plotted as a function of the charge number ZL of the light fragment. There are obviously mass fragmentations where in figs . 9 and 10 the smallest tip distance is well defined in the sense that the nearest competitor is 0.5 fm away. Yet there are also mass splits where two or even three charge numbers ZL or charge ratios ZL/ZH yield the same tip distances within a few tenths of a fm. The uncertainties in the input data to the calculations, viz. the Q-values and, above all, the deformation parameters adopted for the fragments, rule out a safe selection of the smallest tip distance under these circumstances . On the other hand, it should be stressed that, of course, for a fixed mass ratio several charge ratios of the fragments may indeed have very similar minimum tip distances, and in case these are sufficiently small, may allow for several charge combinations in nature to show up in cold fission. The different competing charge splits are visualized as the light-fragment charge ZL as a function of the light-fragment mass A L in figs . 11 and 12, again for the reactions 235 U( n, f) with thermal neutrons and 252Cf(sf), respectively . It has been chosen to select for the presentation of favorable charges a range of minimum tip

235 U(n,f)

N 45

0

o"

60000 0 00 004100 0041000041P

J

-Ir

C N

v t

v'

40 o

40 35 30

06000

Oessoeee

1 00 ® 00000

0000 0686900 O®00 00000 000 00004100 000 0

0 000 0000 00606

20

60

70

80

90

100

110

Mass of light fragment AL / u

120

1,50

Fig . 11 . Light-fragment charges ZL with tip distances within 0.26 fm of smallest minimum tip distance as a function of light-fragment mass A L for 235U(n, f) induced by thermal neutrons . Full heavy points: charges with smallest minimum tip distance. Open points : charges with 2nd smallest tip distances . Full light points : charges with 3rd smallest dip distance.

F. Gbnnenwein, B. Bbrsig / Tip model of cold frssion 5&

N

2 52C(s f)

45

MOW 4DOSOW 6~ go 0000 CCCOCCO

Q)

Qm

E 4V

0000040 000

moo

q()GOGWO 0000 Go*

35-

140

(D 0)

00

30

0~ LI+L 30 OCCOO

0 a

5

20

62

72

62

92

102

112

122

Mass of light fragment AL / u

Fig . 12. Same as fig . I I but for the reaction

252Cf(Sf) .

distances covering 0.26 fin. In the figures the full heavy points stand for the fragment charges corresponding to the smallest feasible tip distances at any given mass split. The actual tip distances may be read from figs . 6 and 8 (for some masses two full heavy points have been plotted indicating that the respective tip distances coincide within 0.01 fin) . The light-fragment charges are seen in figs . I I and 12 to increase stepwise with the light-fragment mass, even charge numbers being definitely favored compared to odd ones for the compound nuclei with even charge numbers on display . This pronounced odd-even effect in the charges liable to lead to cold fission is the more striking when confronted to minimum tip distances as a function of fragment mass (see figs. 7 and 8) where no clear-cut odd-even staggering is discernible. Since for the compound vuclei 236 U and 252Cf under discussion also the neutron number is even, the two observations on the odd-even staggering of charge and mass number taken together suggest that the neutron numbers of the fragments switch almost regularly back and forth between odd and even as the fragment mass is increased step by step. Finally, in figs . I I and 12 the open points and the full light points represent the second and third charge, respectively, appearing in the window of tip distances specified in the foregoing . It is observed that in the majority of cases more than just one charge split emerges from the calculation . For all mass fragmentations where the minimum tip distances are small enough, the present tip model, hence, predicts that several charge splits at given fragment masses may compete in cold fission.

F. Gönnenwein, B. Börsig / Tip model ofcoldfission

47

iscusslon As already emphasized, the tip model for cold fission having been presented can from the energy arguments invoked, only predict whether cold fission is feasible but not whether it is realized in nature . A major uncertainty within the model is, in addition, the precise size of the minimum tip distance, below which cold fission may be anticipated and above which cold fission is outruled . From the width of the leptodermous surface of nuclei this critical tip distance may at best be guessed to lie somewhere between 2 and 3 fm. It is more important to compare the minimum tip distances from the calculations to the evidence for cold fission from experiment. The critical tip distance may then be inferred from this comparison. One of the best studied reactions with a view to cold fission is thermal-neutron fission of the 236U compound . The maximum total kinetic energies T E..X of both fragments observed in experiment are plotted in fig. 13 as a function of the lightfragment mass A L (open points) . The data have been obtained with a twin ionization chamber by Signarbieux 45) . In addition, the maximum Q-values Qm.R for any mass fragmentation are given in fig. 13 as full points. Evidently, the maximum kinetic energies TKEmax approach the maximum Q-values Qneax to within the neutron separation energy for a rather broad mass range from A L ---80 up to AL --108. It is striking that both, for very asymmetric fission with A L < 80 and for symmetric fission

Mass of light fragment AL/u

Fig . 13. Maximum Q-value Q x (full points) and maximum total kinetic energy of fragments235TKEm, observed in experiment (open points) as a function of light-fragment mass A L for the reaction U(n, f) with thermal neutrons .

F. Giinnenwein, B. Bbrsig / Tip model of cold fission

48

with A L ~ 108, the kinetic energies TKE,,,.,., fall short of the available energies Qmax y the way, the difference in maximum kinetic energies for the mass ratio A L /A ti = IN/132 and for symmetric mass divisions in fig. 13 comes very close to the well es'tablished kinetic energy dip, ix. the THerence between the largest average kinetic energy release observed for the same mass ratio AL/A :{ =104/ 132 and the average 46) the energy dip kinetic energy at mass symmetry. According to Signarbieux el cal. -'3s U(n, f) amounts to 21 .2 _+ 0.8 MeV while the for thermal-neutron fission in difference in maximum kinetic energies is read from fig. 13 to be -23 MeV. For a more convenient juxtaposition with the results of the tip model, the energy differences (Qmax - TKEm;,,) from the data points in fig. 13 have been displayed in fig. 14, again as a function of the light-fragment mass A L . The energy difference Emax) just equals the minimum total excitation left to the final fragments in case the charges of the fragments detected are those maximizing the Q-values . The mass region of cold fission is readily identified to lie in the mass bin given in the foregoing . Upon comparing the experimental results on the outcome of cold fission in fig. 14 to the calculational results on the minimum tip distances at scission for the same reaction in fig. 6, a conspicuous similarity in the overall structure of the two curves as a function of mass division catches the eye. Cold fission in the less stringent sense that not sufficient deformation or excitation energy is left to the fragments to evaporate neutrons, that is to say less than about 6 MeV, is observed to be correlated to minimum tip distances being smaller than dm ;,, - 3 .5 fm. Large 25.0 40

235 U(n,f)

X 0

1

15 .0 10 .0

X 0

E

CY

5 .0

n n

60

70

80

90

100

110

Mass of light fragment AL/u

120

fig. 14. Difference between maximum Q-value Qm.,x and maximum total kinetic energy release TKErnax as a function of light-fragment mass A L for the reaction 235 U(n, f) with thermal neutrons .

F. Gönnenwein, E. Börsig / Tip model of cold fission

49

tip distances for both, very asymmetric and symmetric mass splits, match large excitation energies which no longer meet the conditions of cold fission. The basic idea of the present tip model, with the size of the minimum tip distance playing a pivotal role for the outcome of cold fission, is hence seen to be corroborated by experiment . As repeatedly stressed, the model only . specifies necessary conditions for attaining cold fission . On the other hand, assuming that the model correctly outlines the limits of phase space to fission at compact scission configurations, one of the main messages of experimental cold fission studies is that in fission all of the available phase space is invaded. Cold fission in a more rigorous sense, viz. with the total kinetic fragment energy virtually exhausting all of the available energy Qrnax or (Qma" - TKEmaR)^- 0, is inferred from figs . 6 and 14 to correspond to tip distances smaller than about ;,, - 3.0 fm. This critical tip distance, which could not be derived from the static dm model under study, appears nonetheless to be quite compatible with the notion of leptodermous nuclear surfaces and surface width parameters b -1 fm, as made use of in the model. Concerning the question whether true cold fission with TKE maR = 235 U( n, f) reaction with thermal neutrons, it should Qmax is indeed attained in the be kept in mind that the uncertainty on the absolute values of the energy TKE from the calibration of the ionization chambers employed to take the data in figs. 13 and 14 is about t 1 MeV [ref. 471]. Thus, from experiment this question is Pit yet fully settled . On closer inspection of fig. 14 one realizes that the minimum excitation energy ( Qmax - TKEmax) left to the final fragments exhibits an odd-even staggering, with odd mass fragmentations having smaller excitation energies than even ones, while in fig. 6 the fine structure in the minimum tip distances may be shown from the input data to the calculations to be before all induced by variations in the shapes of the fragments . Still, in some small mass bins an odd-even effect is also perceived in the tip distances, e.g. around the mass ratio A L/AH =104/132 in fig. 6. Rather deceivingly, however, the odd-even effect in the calculated tip distances is anticorrelated to the experimental minimum excitation energies : close to the above mass ratio, in figs . 6 and 14 small tip distances coincide with large excitation energies and vice versa, i.e. opposite to the general trends . These shortcomings clearly point to limitations of the model to reproduce finer details, especially those connected .37,48) that in order to cover also these with odd-even effects . It has been argued 3' latter effects the notion of level densities of the nascent fragments should be implemented into the model . Since especially at low excitation energies the level densities of nuclei strongly depend on their odd-even nature, differences in level densities should come into play when the fissioning system probes the limits of phase space in cold fission . For energy bins at small excitations the yield of detected fragments should increase upon moving from (e, e) through (e, o) and (o, e) to (o, o) nuclei, provided the necessary conditions for cold fission are fulfilled. Stated otherwise, the influence of level densities may overcompensate conflicting tendencies

50

F-. Gbnnenwein, B. Bbrsig / Tip model of cold fission

following from the criterion of minimum tip distances. This is indeed borne out in a detailed analysis of cold fission events where besides the fragment masses also 48) . the charges were identified hile from the simple ansatz chosen the tip model cannot cope with fluctuations of the yield in cold fission being affected by level densities of nuclei, nevertheless not only the masses but also the fragment charges competing in cold fission are correctly predicted. This is demonstrated in fig. 15 for thermal-neutron fission of the compound nucleus `36 U*. In the top part of the figure the light-fragment charges maximizing the -value in each isobaric chain are plotted as a function of the light-fragment mass A L . The middle part shows the charge numbers obtained from experiment for those fragments with the largest observable kinetic energies 45,49)_ e experimental charges ZL increase stepwise with mass A L . Evidently, the steps are due to the pronounced predominance of even charges . At least part of this

Mass of light fragment AL/u

Fig. 15 . Light-fragment charges ZL as a function of light-fragment mass AL for the reaction ...U(n, f) with thermal neutrons . Top part : charges Z(Qma,,) maximizing the Q-values. Middle part: charges Z(Exp) observed in experiment at the highest kinetic energies of fragments . Bottom part: charges Z(d,;,,) predicted from the tip model for cold fission; only charges with tip distances within 0.14 fm of the smallest tip distance or with tip distances dm; < 3.0 fm are presented ; the meaning of the symbols is similar to that explained in the caption of fig . 11 .

F. Gönnenwein, B. Börsig / Tip model ofcold fission

51

odd-even staggering may incidentally be brought about by the bias introduced upon selecting in each mass chain the charges with the highest kinetic energies. In fact, charges with the largest Q-values, i.e. the Q..,, values, are favored by this procedure, and these latter charges exhibit in fig. 15 (top part) a similar stepwise increase or odd-even staggering as the experimental charges . Most experimental charge numbers are indeed identical to those anticipated from the maximum Q-values . However, interestingly there are exceptions to this rule, viz. at the light-mass numbers AL = 88, 97, 98 and 99. The exceptions are quite well understood in the framework of the tip model. In the bottom part of fig. 15 the charge numbers ZL which should come into view in cold fission according to the model calculations are presented as a function of the mass AL of the isobaric chains . In contrast to fig. 11 the range of minimum tip distances accepted for the presentation has been narrowed from 0.26 fm down to 0.14 fm. ®n the other hand, the critical minimum tip distance d.; = 3.0 fm derived in the foregoing discussion of fig. 14 has been taken into account by showing in fig. 15 all charges meeting the condition dm; , 3 .0 fm. The symbols, full heavy points, open points and full light points, indicate in decreasing order the probabilities calculated for charges to appear in cold fission. From a comparison with the measured charges in the middle part of fig. 15 it emerges that, out of a total of 29 charge numbers, 24 charges are assigned the highest probability by the nodel, while the remaining 5 charges come forth with second (4 cases) or third priority (1 case) . In view of both, possible slight uncertainties of the input data to the calculations and odd-even shifts induced by level densities, this close agreement between experiment and prediction is considered as a success of the tip model. The deviations from the rule that the charge maximizing the Q-value should be the charge to be observed at the highest kinetic energies (see discussion above) is traced back bar the model to the strong ground-state deformations of nuclei, in the present context especially those with mass numbers around AL =100. It should further be noted that, not only from the model, but also from experiment it is often ambiguous to decide which charge appears first at the highest kinetic energies. An example for this situation is provided in fig. 15 for the light mass AL = 87 . It is comforting to see that this ambiguity in experiment is again well understood, since also the model gives equal probabilities to a choice of two charges for mass AL = 87. More generally, whenever several charges are predicted in the bottom part of fig. 15, one would like to check the predictive power of the model against expelmental evidence. Unfortunately, results from recent measurements giving the sequence of fragment charges in each mass chain, have not yet been made available. Still, one example for the fragmenta48). tion A L /A" =104/132 of the 236U* compound nucleus has been published in ref. At the highest kinetic energies detected the charge split ZL/ZH = 42/50 is singled out. But, at somewhat lower kinetic energies and as soon as the Q-value allows, the odd charge split ZL/ZH =41/51 comes into sight, in full agreement with the charges anticipated from the tip model (see fig. 15).

F Gbnnenwein, B. B&sig / Tip model of coldfission

52

Similar to the 235 Wn, f) case, a survey of charge data has been assembled in fig. 16 for the 252Cf/4,sf) reaction. Again, both the experimental charges in cold fission (middle part of fig. 16) and the charges maximizing the isobaric Q-values (top part of fig . 16) increase stepwise as a function of the light fragment mass A L . The kinetic experimental charges at the highest energies have been made accessible prior to publication from two independent measurements by courtesy of SignarbieuX 45) and Khitter et al. '0), respectively . For eight mass chains the experimental charges do not coincide with those for Qtnax- In the bottom section of fig. 16 the most favorable charges as calculated from the tip model are displayed as a function of the mass of the light-fragment isobaric chain. Up to three charges in each chain have been included, provided either the corresponding minimum tip distances at scission do not deviate more than 0.14 fm from the smallest one, or are inferior to 3 .0 fin. The meaning of the symbols is the same as in fig. 15. Virtually all measured charges are correctly predicted by the model, most of them as the first choice (31 cases out of a total of 39 cases) . Only for the mass chains A L = 85 and 86 the charges L = 35 seen in experiment fall just outside: `he tight criteria adopted in preparing fig. 16 and are, hence, missing in the plot on bottom of the figure. On the whole, this is again regarded as a striking success of the tip model. 55 45 N

35 25

E al 0

4-

o 25 45 35 25 70

80

90

100

110

120

Mass of light fragment AL/u

dig. 16 . Same as fig. 15 but for the reaction

252Cf(Sf) .

130

F. Gönnenwein, B. Börsig / Tip model ofcold fission

53

We should like to emphasize that before all the charge measurements in cold fission of 252 Cf(sf) have been important to reach the conclusion that oblate fragmF ni deformations are not relev nt to the fission process. Coming back to fig. 5 (right part) we recall that the spike in the minimum tip distances for fragments from 252 Cf(sf) with light masses h the vicinity of A =115 is due to oblate ground-state L deformations being forecast from theory for this mass region. Using these oblate deformation parameters as input data to the calculations it turned out that the charge assignments from the model were erroneous . Only upon replacing oblate by prolate deformations, with the selection of deformation parameters being again guided by theory (see sect. 2), the close agreement between experiment and model prediction could be reached. A final comment should address cold fission phenomena in spontaneous fission of very heavy actinides like the neutron rich Fm, Md :end No isotopes. The discovery of bimodal fission by Hulet et al. 5') in these isotopes has been acknowledged as a major advance in the fission research of recent years. The term "bimodal fission" is meant to indicate that two modes compete in producing the characteristics of fragment mass and kinetic energy distributions. Though both modes lead to symmetric mass distributions, the two modes are differentiated by a small width of the mass distribution and an exceptionally high total kinetic energy release on one hand, and a large mass width and a more standard lower kinetic energy on the other hand. The high energy mode of bimodal fission fits perfectly into the scheme of cold fission. In the isotopes where bimodal fission has been put into evidence, cold fission is even a by far more manifest effect than it the lighter actinides: it took typically less than a total of 103 events to discern bimodal and, hence, cold fission, while in the lighter actinides typically 108 events had to be accumulated for the analysis of cold fission. This is corroborated by recent measurements of neutron-emission numbers. While for the 252C f(sf) reaction it is well known that only in 0.2% of all events fission proceeds with no neutrons at all being emitted, this figure rises to a surprising 9.1% for the 260 Md(sf) reaction 52). However, to avoid confusion, one should bear in mind that in bimodal fission the high energy mode with prominent cold fission is an independent "new" mode which appears to be limited to an island of fissioning compound nuclei where a decay into two almost identical magic fragments around '32Sn is feasible. The more standard low energy mode may as well display cold fission, but at a yield level more comparable to those encountered in the lighter actinides. Unfortunately, due to the lack of statistics this conjecture c-?not be probed experimentally at the present time. The minimum tip distances calculated =on! the tip model for two of the isotopes 260Md, were given in where bimodal fission could be established 5' ), vie. 259Md and fig. 8. It is observed from the figure that, for both isotopes, near mass symmetry the tip distances are smaller than the critical distance dm; = 3.0 fm having been deduced as the limiting value for cold fission in the foregoing . Hence, from the model the necessary condition for detecting cold fission in the high energy mode of bimodal fission is fulfilled. Still, much smaller minimum tip distances are reached in fig. 8

$4

F. Gbnnenwein, B. Mrsig / Tip model of cold fission

for light-fragment masses near A L =105, where in bimodal fission only the low-energy mode contribute . The extremely small tip distances are at the basis of the above conjecture that cold fission should also have a chance in this mode to become observable. Since cold fission in the low energy mode is predicted for fragment masses lying outside the range of the high energy mode, cold fission phenomena in the two modes could in principle be disentangled . Anyhow, confronting the model predictions in fig. 8 to the experimental evidence from bimodal fission, it is obvious that the tip model does not anticipate the high yield of cold fission in the high energy mode. Not surprisingly, energy arguments invoked at scission as the only input to a model are not sufficient to explain fission yields depending on both, the whole potential energy surface between the compound stage and scission, and the dynamics of the process. On the other hand, the propensity of an independent and non-standard high energy mode towards cold fission is self-evident. oncluding remarks Froan the analysis of experimental results in the light of the tip model proposed in the present study we conclude that the model provides an appropriate framework for discussing cold fission piienomena . The model falls into the general class of scission-point models which have been widely investigated in fission theory . Yet, only the limiting case of most compact scission configurations is considered here. Thus many of the more delicate problems of scission-point models do not have to be tackled : let us quote the deformabilities of fragments including shell and pairing corrections, questions concerning the thermal equilibrium and, more generally, all issues related to the dynamics of the process . Hopefully, the simplifying features of the tip model should give credit to the reasoning . As argued in the preceding chapters, these simplifying features are thought to precisely apply to cold fission . Similar arguments have been put forward in the past by Clerc et al. ") and Montoya et aL `53 ) . Specific to the present tip model are the inclusion of the deformation properties of nuclei in their ground states, and a proper treatment of the nuclear interaction between the nascent fragments at scission. Before all the ground-state deformations of nuclei turn out to play an eminent role in cold fission . A noteworthy by-product of the analysis has been that, not unexpectedly, in the course of fission the fragments do not assume oblate shapes, even in those cases where oblate ground-state deformations are predicted from theory. The model reproduces fairly well the experimental findings, firstly, concerning the mass range where the fragment kinetic energies stretch into the limit of cold fission and, secondly, concerning the fragment charges being prominent in each mass chain. Once more it has to be underlined that the model can only specify necessary conditions for attaining cold fission, but cannot predict the probabilities or yields of cold fission . However, if the view is accepted that the limits of phase space are correctly outlined by the mode!, then from experiment the most important

F. Gönnenwein, B. Börsig / Tip model of cold fission

55

conclusion to be reached is that in fission all available phase space is invaded. This constitutes an experimental proof to the basic premises of any scission-point model. The reliability of the tip model hinges on the correctness of the input parameters, viz. the Q-values and deformations of fission fragments . The situation concerning Q-values has definitely improved in recent years, with most of the Q-values entering the calculations being nowadays quite precisely known from experiment. In contrast, the experimental information concerning ground-state deformations of nuclei is still scanty . Fortunately, use could be made of the extensive calculations of nuclear ground-state deformations by Möller-Nix 27 ). The success of the model implies that the deformations predicted from this theory present an excellent basis also in those regions of the chart of nuclides being far away from the line of stability. Comparing the theoretical to the available experimental deformation data having been compiled by Raman et al. 25), one notices likewise a generally good agreement, with a tendency, however, for the measured deformations to be somewhat larger than the predicted ones. In the tip model, larger (prolate) deformations will yield smaller tip distances. Upon repeating the present calculations with nuclear deformations taken from experiment, the tip distances decreased in fact by as much as 0.3 fm in some cases. The critical minimum tip distance having been derived from the theoretical deformations to be 3 .0 fm for "true" cold fission should, therefore, not be taken without due reserve . Of course, it would be extremely helpful to have more experimental data on nuclear ground-state deformations at hand. Likewise, the minimum tip distances calculated should be expected to depend crucially on a proper choice of the nuclear radius constant. Unfortunately, this dependence could not be studied in more detail since both the proximity potential and the nuclear deformations made use of in the calculations prescribe well-defined radius constants. The ultimate limit of true cold fission corresponds to a process where both binary fragments emerge directly in their respective ground states . In analogy to cluster radioactivity, true cold fission in the above sense has been advocated from theory by the Frankfurt-Bucharest group 9-11,40) as a special mechanism leading to fission . From experiment the question is still open whether true cold fission has been observed or not and, despite this uncertainty, the present modcl analysis contends that all experiments having become known so far are fully in line with standard fission theory . Anyhow, the tantalizing conjecture of a special mechanism for cold fission certainly deserves to be further scrutinized in future experiments where the challenge will be to further improve the statistics and to solve the ambiguities with the energy calibration of detectors. In passing it may be noted that, on the other hand, the present calculations have for some nuclei been extended to very low fragment masseb covering those encountered in cluster radioactivity 54). The clusters having been observed in the spontaneous decay of heavy nuclei are indeed also identified by the tip model as the most favorable candidates . In the framework of the tip model it is mainly the Q-value of the reaction which is steering the outcome of cluster radioactivity . 4 final comment should address the complementarity

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eyesn cold Rshon and cold fusion. From the niodel analysis of fission experiments at low compound excitation energies it has become apparent that nuclei with strong prolate deformations are propitious for reaching cold fission, with both receding . Jso in the fragments being in their ground states. One may, therefore, suggest that inverse reaction of two approaching nuclei the probability for fusion at minimum excitation energy the compound should be high in case at least either the target or projectile are suitably deformed. A pertinent example for the potential energy experienced by two nuclei prone to fission or fusion was given in fig. 3 for the 116U compound nucleus at an excitation energy corresponding to thermal-neutron capture and for a fragment mass ratio of A L /Atj =104/132. It has to be stressed, however, that the potential energy was obtaincd under the constraint that the two nuclei remain properly aligned during the fission or fusion phase.

a

It is a pleasure to acknowledge the support and the permission given by F.J. ambsch, H .H . Knitter, P. M61ler, S. Raman and C . Signarbieux to make use of unpublished data. Discussions with V. Pashkevich, A. Sandulescu and above all C. Signarbieux have been very helpful to elucidate some of the ideas expounded. G. artinez and G. Barreau contributed much to a preliminary version ofthe tip model. Last but not least we thank M. Khalil for the skilful preparation of the manuscript. his work was supported by the Bundesminister für Forschung and Technologie, onn, Germany, under the contract number 06 TO 243 . e rances 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)

B.C . Diven, H.C . Martin, R.F. Taschek and J. Terrell, Phys. Rev. 101 (1956) 1012 J.W. Boldeman, Nucl . Sci. Eng. 55 (1974) 188 J.C .D. Milton and J.S . Fraser, Phys . Rev. Lett. 7 (1961) 67 W.M . Gibson, T.lJ Thomas and G.L . Miller, Phys . Rev. Lett . 7 (1961) 65 T.D . Thomas, W.M . Gibson and G.J. Safford, Proc. Symp . Physics and chemistry of fission, 1AEA Vienna 1966, vol . 1, p467 W.N . Reisdorf, J.P. Unik and L.E . Glendenin, Nucl . Phys . A205 (1973) 348 S. Bjornholm, Phys . Scr. A10 (1974) 110 C. Signarbieux, M. Montoya, M. Ribrag, C. Mazur, C. Guet, P. Perrin and M . Maurel, J. de Phys . Lett. 42 (1981) L437 W. Grainer, M. lvascu, D.N . Poenaru and A. Sandulescu, Treatise on heavy-ion science, vol. 8 (P!f,num, New York ; 1989) p. 641 A. Sandulescu, J. of Phys . G15 (1989) 529 11K Poenaru, M.S . Ivascu and W. Grainer, Particle emission from nuclei, CRC Press 1989, vol. 111, p. 203 P. Armbruster, Lecture notes in physics 158, (Springer, Berlin, 1982) p. 1 G. Miinzenberg, Rep. Progr. Phys . 51 (1988) 57 A. Oed, P. Geltenbort, F. G6nnenwein, T. Manning and D. Souque, Nucl . Instr. Meth . 205 (1983) 455 C. Signarbieux, G. Sir i . - i, J. Trochon and F. Brisard, J. de Phys. Lett. 46 (1985) L 1095 C. Budtz-Jorgensen, H.-H. Knitter, Ch . Straede, F.J . Hamosch and R. Vogt, Nucl . Instr. Meth . A258 (1987) 209 A. Oed, P. Gehenbod and F. Gdnnenw6n, Nud. Instr. Meth. A205 (1983) 451

F. Gönnenwein, B. Börsig / Tip model ofcoldfission

57

18) P. Armbruster, U. Quade, K. Rudolph, H.-G. Clerc, M. Mutterer, J. Pannicke, C. Schmitt, J.P. Theobald, W. Engelhardt, F. Gönnenwein and H. Schrader, Proc. Int. Conf. on nuclei far from stability, Helsingor, Denmark, CERN 81-09 (1981) 675 19) H.-G. Clerc, W. Lang, M. Mutterer, C. Schmitt, J.P. Theobald, U. Quade, K. Rudolph, P. Armbruster, F . Gönnenwein, H. Schrader and D. Engelhardt, Nucl. Phys. A452 (1986) 277 20) H.-G. Clerc, W. Lang, H. Wohifarth, K.-H. Schmidt, H. Schrader, K.E. Pferdekiimper and R. Jungmann, Z. Phys. A274 (1975) 203 21) J.P. Bocquet, R. Brissot and H .R. Faust, Nucl. Instr. Meth. A267 (1988) 466 22) J. R. Nix, A.J. Sierk, H. Hofmann, F. Scheuter and D. Vautherin, Nucl. Phys. A424 (1984) 239 23) G.D . Adeev and I.I. Gonchar, Z. Phys. A322 (1985) 479 24) S. Grossmann, U. Brosa and A. Möller, Nucl. Phys. A481 (1988) 340 25) S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor and P.H. Stelson, At. Data Nucl. Data Tables 36 (1987) 1 26) S. Raman, C.W. Nestor, S. Kahane and K.H. Blatt, At. Data Nucl. Data Tables 42 (1989) 1 27) P. Möller and J.R . Nix, At. Data Nucl. Data Tables 26 (1981) 165, and Los Alamos Preprint LA-UR-86-3983, 1986 28) J. Blocki, J. Randrup, W.J. Swiatecki and C.F. Tsang, Ann. of Phys. 105 (1977) 426 29) H.J. frappe, J.R. Nix and A.J. Sierk, Phys. Rev. C20 (1979) 992 30) J. Blocki and W.J. Swiatecki, Ann. of Phys. 132 (1981) 53 31) F. Gönnenwein, Proc. Seminar on fission, Pont d'Oye, Belgium 1986, ed. C. Wagemans, SCK/CEN Mol BLG 586, p. 106 32) F. Gönnenwein, Proc. Int. school seminar on heavy ion physics, Dubna, USSR 1987, p.232 33) Y.-J. Shi and W.J. Swiatecki, Nucl. Phys. A464 (1987) 205 34) R.W . Hasse and W.D. Myers, Geometrical relationships of macroscopic nuclear physics (Springer, Berlin, 1988) 35) S. Cohen and W.J. Swiatecki, Ann . of Phys. 19 (1962) 67 36) A.H. Wapstra, G. Audi and R. Hoekstra, At. Data Nucl. Data Tables 39 (1988) 281 37) F. Gönnenwein and B. Börsig, Proc. Conf. 50 years with nuclear fission, Gaithersburg, USA, 1989, Amer. Nucl. Soc. Vol . II, p. 515 38) Y.-J. Shi and W.J. Swiatecki, Nucl. Phys. A438 (1985) 450 39) K. Depta, Hou ji. Wang, J.A. Maruhn, W. Greiner and A. Sandulescu, J. of Phys. G14 (1988) 891 40) A. Sandulescu, A. Florescu and W. Greiner, J. of Phys. G15 (1989) 1815 41) J.F. Berger, M. Girod and D. Gogny, Proc. Fifty years research in nuclear fission, Berlin, 1989, Nucl. Phys. A502 (1989) 35C 42) J.F. Berger, M. Girod and D. Gogny, Nucl. Phys. A428 (1984) 23C 43) R. Bengtsson, P. M611er, J.R. Nix and J.-Y. Zang, Phys. Scr. 29 (1984) 402 44) P. Möller, private communication 45) C. Signarbieux, private communication 46) C. Signarbieux, M. Ribrag and H. Nifenecker, Nucl. Phys. A99 (1967) 41 47) G. Simon, J. Trochon, F. Brisard and C. Signarbieux, Nucl. Instr. Meth. A286 (1990) 220 48) G. Simon, J. Trochon -nd C. Signarbieux, Proc. Conf. 50 years with nuclear fission, Gaithersburg, USA, 1989, Am. Nucl. Soc. vol. I, p. 313 49) G. Simon, J. Trochon and C. Signarbieux, Proc. Journées d'Études sur la Fission, Arcachon, France, 1986, ed. B. Leroux CENBG 8722, p. B21 50) H.H. Knitter, F.J . Hambsch and C. Budtz-Jorgensen, private communication 51) E.K. Hulet, J.F. Wild, R.J. Dougan, R.W. Lougheed, J.H . Landrum, A.D. Dougan, P.A. Baisden, C.M . Henderson, R.J. Dupzyk, R.L. Hahn, M. Schiidel, K. Summerer and G.R . Bethune, Phys. Rev. C40 (1989) 770 52) J.F. Wild, J. van Aarle, W. Westmeier, R.W. Lougheed, E.K. Huiet, K.J. Moody, R.J. Dougan, E.A. Koop, R.E. Glaser, R. Brand and P. Patzelt, Phys. Rev . C41 (1990) 640 53) M. Montoya, R.W. Hasse and P. Koczon, Z. Phys. A325 (1986) 357 54) F. Gönnenwein, B. Börsig and H. Löfller, Proc. Int . Symp. Dynamics of collective phenomena, ed. P. David (World Scientific, Singapore, 1988) p.29