Isotopic equilibrium constants for very low-density and low-temperature nuclear matter
J. B. Natowitz, H. Pais, G. Roepke, J. Gauthier, K. Hagel, M. Barbui, R. Wada
aa r X i v : . [ nu c l - e x ] S e p Isotopic equilibrium constants for very low-density and low-temperature nuclearmatter
J. B. Natowitz , H. Pais , G. R¨opke , J. Gauthier , K. Hagel , M. Barbui , and R. Wada Cyclotron Institute, Texas A & M University, College Station, Texas 77843 Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal and University of Rostock, FB Physik, 18059 Rostock, Germany (Dated: September 14, 2020)Yields of equatorially emitted light isotopes, 1 ≤ Z ≤
14, observed in ternary fission in thereaction
Pu( n th ,f) are employed to determine apparent chemical equilibrium constants for low-temperature and low-density nuclear matter. The degree of equilibration and role of medium mod-ifications are probed through a comparison of experimentally derived reaction quotients with equi-librium constants calculated using a relativistic mean-field model employing a universal mediummodification correction for the attractive σ meson coupling. The results of these comparisons in-dicate that equilibrium is achieved for the lighter ternary fission isotopes. For the heavier isotopesexperimental reaction quotients are well below calculated equilibrium constants. This is attributedto a dynamical limitation reflecting insufficient time for full equilibrium to develop. The role ofmedium effects leading to yield reductions is discussed as is the apparent enhancement of yields for He and other very neutron rich exotic nuclei.
I. INTRODUCTION
A high quality nuclear equation of state (EOS) applica-ble over a wide range of density and temperature is an es-sential ingredient for reliable simulations of stellar matterand astrophysical phenomena. In recent decades manynuclear theory efforts have been devoted to developingsuch equations and many are available in the literature,see Refs. [1–14] and references therein. The validation ofthese equations of state usually rests on careful compar-isons between the results of theoretical simulations andastrophysical observations.At the same time, laboratory studies of nuclear mat-ter at different densities, temperatures and isospin con-tent offer some unique possibilities to address specific as-pects of the nuclear equation of state. Exploiting a vari-ety of projectile energies, projectile-target combinationsand reaction mechanisms, nuclear experimentalists haveprobed cluster formation and the composition of nuclearmatter at different densities, caloric curves and phasetransitions, the density dependence of the symmetry en-ergy and medium effects on nuclear binding energies, seeRefs. [12–19] and references therein.While isotope mass fractions are commonly used topresent the results of EOS composition calculations, ref-erences [13, 14, 18, 19] employed chemical equilibriumconstants for production of Z = 1 (H) and Z = 2 (He)derived from the experimental isotope yields. These aremore robust quantities for testing different equations ofstate since, at least in the low-density ideal limit, they areless dependent upon the choice of isotopes included in theEOS model calculations and upon the source asymmetry.The thermodynamic reaction quotient Q for the for-mation of an isotope A Z with mass number A , atomicnumber Z , and neutron number N = A − Z , is definedsuch that, Q = { A Z }{ p } Z { n } N (1)where curly brackets denote the fugacities of the chem-ical species, i.e. the isotope A Z as well as the protons( p ) and neutrons ( n ). Fugacity depends on temperature,pressure and composition of the mixture, among otherthings. The fomulation in terms of fugacities arises be-cause components in non-ideal systems interact with eachother. In nuclear EOS models these interactions are mod-eled in a variety of ways [1, 2, 4, 6–11, 13, 14, 19]. Theright-hand side of this equation corresponds to the reac-tion quotient for arbitrary values of the fugacities. Thereaction quotient becomes the equilibrium constant, K ,if the system reaches equilibrium. The equilibrium con-stant is related to the standard Gibbs free energy changefor the reaction, ∆ G , as∆ G = − RT ln K (2)where T is the temperature and R is the gas constant.If deviations from ideal behavior are neglected, the fu-gacities may be replaced by concentrations or densities.Employing square brackets to indicate concentrations ordensities at equilibrium we can designate this ratio aschemical constant K c , K c = [ A Z ][ p ] Z [ n ] N . (3) K c is defined in an equivalent way to the thermodynamicequilibrium constant but with concentrations or densitiesof reactants and products, denoted by square brackets,instead of fugacities.The experimental equilibrium constants reported inreferences [18, 19] demonstrated clearly that, even atdensities in the 0.003 to 0.03 nucleons/fm range, in-teractions are important and experimental equilibriumconstant data may be employed to evaluate the varioustheoretical models. II. ANALYSIS OF TERNARY FISSION YIELDS
In this paper we report extensions of the measurementsof isotopic equilibrium constants to a broader range ofisotopes at even lower temperature and densities. Specifi-cally we derive isotopic equilibrium constants for isotopesproduced in ternary fission processes which occur in ap-proximately 0.3 % of decays during the spontaneous orthermal neutron induced fission of a heavy nucleus [20–45]. Such ternary fission is characterized by emission ofan energetic light particle or fragment in a direction per-pendicular to the axis defined by the separating massivefragments, signaling their origin in the region between thetwo nascent heavy fragments at or near the time of scis-sion. Collectively, such isotopes, are typically identifiedin the ternary fission literature as ”scission” or ”equato-rially” emitted particles.This well identified isolated mechanism facilitates ex-ploration of yields with minimal perturbations from col-lision dynamics. This allows an experimental test of thechemical equilibrium hypothesis. If that hypothesis issupported, derived equilibrium constants provide infor-mation against which various proposed equations of statemay be tested in the low-density limit. In this regardthey constitute the experimental counterpart of theoret-ical virial equations of state which serve as a low-densitytheoretical baseline for EOS calculations [4, 10]. Dataof sufficient accuracy would allow a careful evaluationof the density dependence of fragment-fragment interac-tions and in medium modifications of cluster properties.See reference [46] for a recent discussion of such effects.The experimental results of Koester et al. , obtainedwith an on-line mass spectrometer, provide a compre-hensive data set for ternary fission yields for 42 isotopesdetermined in the reaction
Pu( n th ,f) [28, 29]. In ad-dition, 17 upper limits are also reported for yields ofother isotopes. In reference [45], these yields were com-pared to results of calculations made using a model whichassumes a nucleation-time-moderated chemical equilib-rium [47–49] in the low-density matter which constitutesthe neck region of the scissioning system. Nucleationapproaches have much in common with thermal coales-cence approaches previously applied to clustering in low-density nuclear systems [50, 51] but explicitly incorpo-rate consideration of cluster formation rates. Coalescenceof nucleons into clusters is a dynamic process requiringtime, while the fissioning system exists for a limited timespan. A reasonably good fit to the Pu( n th ,f) experi-mental data from references [28, 29] was obtained withthe following parameters: Temperature, T = 1 . ρ = 4 × − fm − , proton fraction, Y p = 0 .
34, nu-cleation time t nuc = 6400 fm/ c and critical cluster mass, A cr = 5 .
4. We note that various previous attempts toevaluate the temperatures appropriate to thermal neu- tron induced ternary fission have led to temperatures inthe range of 1.0 to 1.4 MeV [52, 53]. For the
Pu com-pound nucleus the proton fraction, Y p , is 0.388. The de-rived value of 0.34 indicates that the region between theseparating fragments, which dominates the production ofthe ternary particles, is neutron enriched [45, 54]. III. EQUILIBRIUM CONSTANT FOR He Since all yields in the Koester
Pu( n th ,f) data are ref-erenced to the yield of He particles we began by estab-lishing the correspondent equilibrium constant for thisparticle. Determining this equilibrium constant requiresaccurate yields of the neutrons, protons and He ejectedat the time of scission. The equatorial emission originof these particles must be well defined and contributionsfrom other sources (e.g. pre-scission emission, polar emis-sion, secondary particle emission) to the total yields becarefully removed. Establishing the yields of equatorialemission requires careful exploration of the particle an-gular distributions relative to the scission axis. Thesemeasurements are difficult, particularly for the neutronsbecause subsequent evaporation from the fission frag-ments dominates the neutron yield. Fortunately, veryprecise measurements of these yields have been made bya number of extremely competent experimental groupsand absolute yields for many fissioning isotopes are, infact, available and tabulated in the literature [20–44].The systematics of ternary fission yields have been ex-tensively analyzed in various evaluations and review arti-cles [20–22, 26]. Focusing particularly on values reportedfor Pu isotopes we have adopted for our calculations theexperimental values indicated in column 4 of Table I.The adopted value for the He particle yield includes a17 % correction to remove He particles resulting fromthe decay of He nuclei emitted at scission [37]. Theadopted value for protons includes a 14.5 % correctionto remove polar emission protons [39–41]. For neutrons,the adopted value is that determined for scission neu-trons [36].Applying the thermal coalescence model ofMekjian [50] to these data allows extraction of thecoalescence volume, 2937 fm . With the absolute yieldsand this coalescence volume, the relevant densitiesand the experimental equilibrium constant K c ( He)for direct formation of the He in its ground stateis 3 . ± . × fm . As indicated above, thelargest contributor to the uncertainty is the neutronscission yield. Note, however, that the apparent effective K eff c for the total experimentally observed He yield(column 2, Table I), which includes the He contribution(as well as possible smaller contributions from otherparticle unstable isotopes) is 3 . ± . × fm . Byconvention, relative yields in ternary fission are typicallynormalized to the total He yield.In reference [13] Pais et al. reported a study of in-medium modifications on light cluster properties, within particle total yield/fission equatorial scission emission adopted yield n . ± .
005 0 . ± .
015 0 . ± . p . × − ± .
41 3 . × − ± .
35 3 . × − ± . He 2 . × − ± .
20 2 . × − ± .
20 1 . × − ± . He yields [21, 22, 35–44]. References are the primary sources. Measurementsand systematics of other data for adjacent isotopes were also employed in establishing these values. Uncertainties are 1 σ . the relativistic mean-field approximation, where explicitbinding energy shifts and a modification on the scalarcluster-meson coupling were introduced in order to takethese medium effects into account. The interactions ofthe clusters with the surrounding medium are describedwith a phenomenological modification, x i,σ , of the cou-pling constant to the σ meson, g i,σ = x i,σ A i g σ . Using theFSU Gold EOS [12] and requiring that the cluster frac-tions exhibit the correct behavior in the low-density viriallimit [4, 9, 10], they obtained a universal scalar cluster-meson coupling fraction, x i,σ = 0 . ± .
05, which couldreproduce both this limit and the equilibrium constantsextracted from reaction ion data [18, 19] reasonably well.The results are qualitatively similar to the ones obtainedwith other approaches [4, 6–9, 19]. Employing the modelof reference [13] with T = 1 . ρ tot = 4 × − fm − ,and a scalar cluster-meson coupling fraction x i,σ = 0 . K c ( He)= 2 . × fm for direct production,and K eff c ( He) = 3 . × fm .In a more recent work [14], Pais et al. comparedtheir model results to equilibrium constants calculatedfrom a new analysis, where in-medium modificationsare addressed, for experimental data measured in in-termediate energy Xe + Sn collisions. This comparisonlead to a higher scalar cluster-meson coupling constant x i,σ = 0 . ± . K c ( He) becomes 3 . × fm , andfor K eff c ( He) = 4 . × fm . IV. EXTENSION TO OTHER ISOTOPES
Using the adopted values of the equatorial neutron andproton yields together with the measured yields for allisotopes we have calculated the effective experimental re-action quotients, Q eff c for formation of the observed iso-topes from the nucleons, i. e., Q eff c = [ A Z ][ p ] Z [ n ] N (4)where eff denotes total observed yields including all con-tributions from gamma decaying and particle decayingexcited states. Here we employ Q because in our previoustreatment of these same data within the framework of anucleation time modulated statistical equilibrium modelwe have presented evidence that statistical equilibrium is not achieved for the heaviest isotopes [45]. The termeffective is used in recognition of the fact that the finalobserved ground state yields include contributions fromde-excitation of short lived gamma or particle decayingstates initially present in the primary isotope distribu-tion. The relative importance of such contributions willvary with temperature and density. For a system at equi-librium Q eff c = K eff c , the effective equilibrium constant. Adirect comparison between the experimental results andthose of theoretical calculations requires that the contri-butions from relevant excited states be included in thetheoretical treatment.In the original formulation by Pais et al. , only groundstates including particle unstable ground states were in-cluded in the calculation. For the present calculationwe have included experimentally identified (excitationenergy and spin) gamma decaying excited states [57]which can have a significant population at T = 1 . n/p ratio to the experimentally observed free n/p ratio fordifferent assumed total densities indicated a density of2 . ± . × − fm − . This value, which is somewhatlower than the 4 × − fm − derived from nucleationmodel fits, has been adopted for the present calculations.In the recent treatment of the emisssion of Z = 1 , Cf a differ-ent approach suggests quite similar values [46].The experimentally derived reaction quotients are pre-sented in Figure 1. To more clearly present the data,we plot Q eff c against the isotope identifier parameter pro-posed by Lestone [30], i.e., A + 8( Z − Q eff c values, we also present theo-retically calculated equilibrium constants, K eff c , obtainedusing the model of Pais et al. [13] with a scalar cluster-meson coupling constant x i,σ of 0.92. To carry out thesecalculations we fixed the temperature to be 1.4 MeV,the total density to be 2 . × − nucleons/fm andthe proton fraction of the matter to be 0.34. Both theexperimental and theoretical values are tabulated in Ap-pendix A of this paper. Unlike the data employed forthe previous comparisons with this model, the presentdata include isotopes as heavy as Si. Therefore therole of excited states should be much more important in ( - A ) ) - , (f m e ff C Q FIG. 1: Q eff c values vs A + 8( Z − K eff c with T = 1 . Y p = 0 . ρ = 2 . × − fm − and coupling constant x i,σ = 0 . determining the observed isotope yields. This is partic-ularly true for nuclei with lower energy gamma decayingexcited states with high degeneracies. Particle decayingexcited states are also included but many generally occurat relatively higher excitation energies and thus are lesspopulated at low temperature.As is observed in Figure 1, the experimental and theo-retical trends are quite similar. For the heaviest isotopesthere is, however a clear indication that the experimen-tal Q eff c values fall well below the theoretically calculatedequilibrium constants. To better appreciate these differ-ences we plot, in Figure 2 the ratios of the values of theexperimentally derived reaction coefficients to the K eff c values calculated theoretically using the Pais et al. for-mulation [13]. Ratios for isotopes for which measured ex-perimental yield values exist are identified by triangles.Those for which only upper limits to the experimentalyields are available are not included in this figure.In Figure 2 we see that for the lighter isotopes there issome scatter about the ratio R exp / theo = Q exp c /Q theo c = 1,but a general overall accord between the data and thetheoretical values, suggesting that chemical equilibriumhas been achieved for the isotopes with Z ≤
5. The ex-perimental K c value reported for the H is well below thetheoretical value. This appears to reflect the very weakbinding of the deuteron. Such reductions in deuteronyield are a general feature in the production of deuteronsin heavy ion collisions [13, 18, 56]. Interestingly, for thelight neutron rich isotopes He, Li, Be and Be the − − − −
10 110 ( ca l c ) e ff C ( ex p t) / K e ff C Q FIG. 2: Ratio Q eff c (experiment)/ Q eff c (theory) vs A +8( Z − ratio indicates significant excesses relative to the calcu-lated values in the region where there is a reasonableagreement for the other isotopes.Above that point the plotted ratios drop rapidly fallingto R exp / theo ∼ − for the heaviest isotopes. Sincethe Pais calculation includes medium effects through thecluster coupling constant this decrease does not appear toreflect calculated medium effects. Rather, the observeddecline in the ratio of experimental value to theoreticalvalue indicates that equilibrium is not reached for theheavier isotopes. This is entirely consistent with the con-clusion reached in reference [45] where it is attributed toa time moderated nucleation effect. The possibility thatfinite size effects may also contribute to this decline isnot ruled out. V. Z = 1 AND 2 ISOTOPES AND MEDIUMMODIFICATIONS
Given the recent detailed analysis of Z = 1 (H) and 2(He) isotope production for Cf ternary fission [46] itis interesting to focus explicitly on these results for thepresent case. In Table II, the available measured equi-librium constants for these isotopes are presented andcompared to the theoretical values calculated using ascaler cluster-meson coupling constant x iσ = 0 .
92. Asalready noted above, the experimental Q c value basedon the observed yield for the H is well below the theo- particle Q eff c (expt) K eff c (calc) H 5 . ± . × . × H 2 . ± . × . × He 1 . × He 3 . ± . × . × He 1 . × He 5 . ± . × . × He 2 . × He 2 . ± . × . × TABLE II: Chemical constants for the isotopes of the lightelements H, He. The experimental values Q eff c (expt) are com-pared to calculated values K eff c (calc). retical value. (This is also true in the Cf case [46].)This suggests a clear medium effect for this very weaklybound nucleus [55, 58]. He was not observed in theKoester experiment nor has a He yield been reported inany other ternary fission experiment [37, 38]. The theo-retically calculated He and H equilibrium constants inTable II are similar, as expected, the difference arisingfrom the small binding energy difference for these A = 3isotopes. While some similar medium effect may operateon the A = 3 yields, the non-observation of He reflectsthe very small free proton to free neutron ratio at equilib-rium indicated in Table 1. Given that ratio, the He yieldshould be about four orders of magnitude below the Hyield. This low yield, together with possible additionalfactors specific to individual experiments. e.g., separa-tion, identification and background discrimination, offersa natural explanation for the absence of He yield datain the literature.For H, He and He the tabulation indicates reason-able agreement (within experimental errors) between ex-periment and theory.In contrast the experimental value for the very neu-tron rich He is an order of magnitude higher than thatcalculated. The large experimental yield of He is a gen-eral feature of ternary fission experiments. This spe-cial nature of He may reflect some feature of dynam-ics, e.g., time dependent density or temperature fluctua-tions or feeding from parent nuclei, or of detailed struc-tural features not yet understood. As noted in the pre-vious section the comparison of the experimental equi-librium constants with those of the calculation (Figures1, 2, Tables IV(a), IV(b)) also indicates yield enhance-ments for the other neutron rich isotopes Li, Be and Be. The cluster structure of such neutron rich nu-clei has been discussed in the framework of an extendedIkeda diagram [59]. Particularly intriguing is the possi-bility that the yield enhancement reflects the existence ofstrong neutron correlations in the disassembling matter.In this regard He is of special interest as experimen-tal evidence for a possible alpha-tetra-neutron structurehas been published [60] and some theoretical work sug-gests that a tetra-neutron condensate might be formed in low-density neutron rich stellar matter [61]. This subjectwarrants further investigation.In a recent related paper on the spontaneous ternaryfission of
Cf [46], we explored an alternative informa-tion entropy based analysis to characterize the emissionof Z = 1 , λ T , λ n , λ p such that Y rel A,Z ∝ R rel A,Z g A,Z (cid:18) π ¯ h Amλ T (cid:19) − e [ B A,Z +( A − Z ) λ n + Zλ p ] /λ T (5)where g A,Z denotes the degeneracy of the nucleus { A, Z } in the ground state, B A,Z its binding energy, m is theaverage nucleon mass. The Lagrange parameters λ i arenon-equilibrium generalizations of the equilibrium ther-modynamic parameters T , µ n , µ p . Different approxima-tions to treat the Hamiltonian of the many-nucleon sys-tem lead to different values for these parameters. In par-ticular all relevant excited states and continuum stateshave to be taken into account, and in-medium mean-fieldand Pauli blocking effects must be included. These ef-fects are collected in a prefactor R rel A,Z which, in general,depends on the Lagrange parameters λ i .The relevant primary isotopic distribution is relatedto the observed distribution via a non-equilibrium evo-lution, which is described in simplest approximation byreaction kinetics where unstable nuclei feed the observedyields of stable nuclei. As detailed in Ref. [46], takinginto account all bound states below the edge of contin-uum states, the primary distribution Y rel ,γA,Z can be ob-tained, with Lagrange parameters λ γi obtained from aleast squares fit to the observed final yields of H, H, He, He, and He. The correct treatment of continuumstates gives the virial expansion which is exact in the low-density limit. Using measured scattering phase shifts,virial expansions have been determined for H, H, Heand Be (which feeds He), see Ref. [4, 62]. For the otherisotopes estimates are given in [46]. Such a treatment in-cluding the continuum states leads to a significant reduc-tion in the calculated yields of the unbound nuclei He, He, He, and He. As was shown in the ternary spon-taneous fission of
Cf [46], the observed yield of He isoverestimated, and the observed yield of He is underes-timated. A possible explanation may be in-medium cor-rections, in particular Pauli blocking. He is only weaklybound (the edge of continuum states is at 0.975 MeVwhich is small compared even to 2.225 MeV for H) sothat Pauli blocking may dissolve the bound state at in-creasing density. To reproduce the observed yields, wehave determined an effective pre-factor R rel , eff A,Z . Both, theeffective pre-factor and the relevant primary yields re-quired to reproduce the observed yields are shown in Ta-ble III. In detail, the pre-factor R rel , eff A,Z which representsthe internal partition function was taken from the virialexpansion for H, H, He, He, He, Be, as well as theestimates for He and He. The corresponding observedyields are used to determine the three Lagrange param-eters λ T , λ n , λ p . To reproduce the observed (weaklybound) He, the effective factor R rel , eff He was determined.For this, the contribution of the primary yields of He and He must be known. We used the value Y He /Y He = 0 . Cf in [35]. It would be of interest to ver-ify these predictions of Y rel , eff A,Z by measurements for
Puas were done for
Cf.Interpreting the effective pre-factors R rel , eff A,Z as reflect-ing in-medium corrections, we can use these inferred val-ues to estimate the density. These in-medium correc-tions are single-nucleon self-energy shifts which may beabsorbed into the Lagrange parameters λ n , λ p if mo-mentum dependence of the single-particle self-energy isneglected. Then, the density dependence of R rel , eff A,Z isgoverned by the Pauli blocking effects which reduce thebinding energies. A global reduction of the binding ener-gies is described in the generalized RMF approximation( x i,σ = 0 .
92) given above by the effective cluster couplingto the mesonic field. Within a more individual calcula-tion, the Pauli blocking acts stronger for weakly boundstates, eventually dissolving them, denoted as the Motteffect [55, 58]. We have performed an exploratory calcula-tion assuming that Pauli blocking is essential for He be-cause of its small binding but neglect the Pauli blockingshift for the stronger bound nuclei. The reduction factor R rel , eff He derived for He is smaller than the expected value R rel , vir He = 0 .
945 according to the virial expansion. Thisleads to a shift of the binding energy of about 0.9 MeVand a correspondent density value of about n n = 0 . − . Note that this value has a large error because ofuncertainties in the observed yield of He as well as theestimation of the energy shift of He due to in-mediumcorrections. Large deviations from the simple NSE arepredicted for the primary yields of He, and it would beof interest to observe it like in the case of
Cf [46].A paper in which this approach is followed more con-sistently, considering also the Pauli blocking shifts forstrongly-bound nuclei, and calculating the ternary fis-sion yields for the Z = 3 −
14 isotopes observed in
Pu( n th ,f) [28, 29] is currently in preparation [63]. VI. SUMMARY AND DISCUSSION
In conclusion, experimentally determined reaction quo-tients have been determined for equatorially ejected iso-topes of Z ≤
14 observed in the ternary fission of
Pu.The emission is characterized by T = 1 . Y p = 0 . ρ = 2 . × − nucleons/fm . It should be noted thatsince at equilibrium the reaction coefficients are primarilysensitive to temperature and to density through mediumeffects extraction of accurate densities remains a difficultproblem. Here we have used the observed free neutronto free proton ratio to establish the density.A comparison of the reaction quotients with those cal- isotope Y exp A,Z B
A,Z A [MeV] g A,Z R rel , eff A,Z Y rel , eff A,Z λ T - - - - 1.2042 λ n - - - - - 2.9954 λ p - - - - -16.633 n - 0 2 - 1588200 H - 0 2 - 19.16 H 42 1.112 3 0.98 42 H obs
786 2.827 2 - 786 H - 2.827 2 0.99 779.51 H - 1.720 5 0.0606 6.46679 He - 2.573 2 0.988 0.004972 He obs He - 7.073 1 1 8485.89 He - 5.512 4 0.7028 1508.81 He obs
260 4.878 1 - 260 He - 4.878 1 0.8827 14.868 He - 4.123 4 0.6235 45.122 He obs
15 3.925 1 - 15 He - 3.925 1 0.9783 14.72 He - 3.349 2 0.2604 0.27 Be- - 7.062 1 1.07 2.65TABLE III: Properties and relative yields of the H, He and Beisotopes from ternary fission
Pu( n th ,f) which are relevantfor the observed yields of H, He nuclei, (denoted by the su-perscript ’obs’.) Experimental yields Y exp A,Z [29] are comparedto the yields calculated as described in the text. Observedyields - column 2, binding energy B A,Z /A - column 3, groundstate degeneracy - column 4. prefactor - column 5, calculatedprimary isotope distribution - column 6. culated using the EOS model of Pais et al. [13] with ascaler cluster-meson coupling constant of x i,σ = 0 .
92 in-dicates a reasonable agreement between the experimentalresults and the model calculations for the lighter isotopes,indicating that chemical equilibrium is achieved for thoseisotopes and that medium effects are quite small at thistemperature and density. A more detailed evaluation ofpossible medium effects at these densities, addressing theproperties of individual isotopes, is presented in refer-ence [46]. The experimental yield of He is much higherthan predicted in the calculation. Other very neutronrich isotopes, Li, Be and Be, also give evidence ofbeing underestimated in the calculation. Whether thisreflects the particular structural characteristics of theseexotic nuclei warrants careful investigation [60, 61]. Forthe heavier isotopes, the ratio of the measured reactioncoefficient to the theoretically predicted equilibrium con-stant exponentially decreases with increasing mass. Thisis attributed to a dynamical limitation, reflecting insuf-ficient time for full equilibrium to develop [45]. An im-portant point to be emphasized is that valid comparisonsof calculated equilibrium constants to those derived fromexperimental data demand that the actual experimentalensemble of competing species be replicated as fully as isotope Q eff c expt. Q eff c calc. upper limit H 6.61E+03 2.42E+04 H 3.39E+09 3.30E+09 H 3.63E+18 4.74E+18 He 7.12E+25 5.46E+25 He 3.09E+33 3.76E+32 Li 1.54E+32 2.52E+32 Li 2.66E+36 1.80E+36 Li 1.44E+41 3.18E+40 Li 5.88E+46 7.87E+46 Be 1.41E+34 2.33E+32 ** Be 2.34E+44 2.76E+44 Be 6.72E+49 2.20E+49 Be 2.37E+53 1.46E+53 Be 3.08E+57 1.14E+57 Be 2.24E+63 1.68E+63 B 1.34E+50 1.08E+50 ** B 1.97E+56 2.65E+56 B 3.37E+60 6.49E+60 B 3.30E+68 1.34E+69 B 3.21E+72 4.22E+72 B 5.26E+79 2.05E+79 ** C 9.82E+73 1.65E+74 C 9.20E+77 2.70E+78 C 2.94E+82 4.87E+82 C 1.03E+86 4.58E+86 C 1.24E+90 3.19E+90 C 3.04E+92 9.05E+93 C 1.20E+97 4.77E+97TABLE IV(a): Experimental [26,27] and calculated equi-librium constants for light isotopes observed in the ternaryfission of
Pu. Assigned upper experimental limits are indi-cated by **. See text for details. possible in the calculation.
VII. ACKNOWLEDGEMENTS
This work was supported by the United StatesDepartment of Energy under Grant
VIII. APPENDIX A
Tables IV(a) and IV(b) contain the Q eff c values pre-sented in Figure 1 of this paper. We note that Q c val- isotope Q eff c expt. Q eff c calc. upper limit N 2.89E+79 8.24E+80 ** N 1.42E+84 3.26E+85 N 1.68E+89 1.92E+90 N 2.17E+93 2.74E+94 N 9.68E+97 2.56E+98 N 2.96E+100 1.22E+103 N 7.01E+104 1.77E+107 O 2.41E+83 5.06E+79 ** O 2.97E+101 8.11E+102 O 3.45E+106 4.22E+107 O 1.98E+110 3.20E+112 O 2.83E+114 1.14E+117 O 1.42E+123 1.77E+125 ** F 7.00E+103 1.37E+104 F 1.92E+107 4.74E+109 ** F 5.54E+112 1.09E+115 F 2.17E+118 1.31E+120 F 1.64E+126 1.88E+129 Ne 6.69E+129 7.84E+132 Ne 2.77E+142 8.46E+145 ** Na 5.13E+131 3.36E+134 ** Na 5.30E+145 4.68E+149 ** Na 1.75E+150 1.14E+154 Na 1.75E+158 3.94E+162 ** Mg 3.90E+148 2.19E+152 ** Mg 1.61E+154 4.45E+157 ** Mg 6.05E+162 1.66E+167 Al 4.12E+165 5.20E+169 ** Si 3.94E+186 1.53E+192 ** Si 2.95E+191 6.67E+196 ** Si 1.08E+196 2.12E+201 **TABLE IV(b): Continued: Experimental [26,27] and calcu-lated equilibrium constants for light isotopes observed in theternary fission of
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