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NEW DEVELOPMENTS IN MICROCANONICAL MONTE CARLO SAMPLING OF NUCLEAR
FRAGMENTATION
D. Gross
To cite this version:
D. Gross. NEW DEVELOPMENTS IN MICROCANONICAL MONTE CARLO SAMPLING OF NUCLEAR FRAGMENTATION. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-119-C2-123.
�10.1051/jphyscol:1987218�. �jpa-00226483�
JOURNAL DE PHYSIQUE
Colloque C2, suppl6ment au n o 6 , Tome 4 8 , j u i n 1987
NEW DEVELOPMENTS I N MICROCANONICAL MONTE CARLO SAMPLING OF NUCLEAR FRAGMENTATION
Bereich Kernphysik des Hahn-Meitner Instituts Berlin and
FPS "Hadron- und Schwerionen- induzierte Kernreaktionen"
Fachbereich Physik der Freien Universitst Berlin, Glienickerstr. 100, 0-1000 Berlin 39, F.R.G.
ABSTRACT: Important steps towards a realistic modeling of statistical fragmentation of hot nuclei are taken. The long-range Coulomb interaction and the excluded volunie of the fragments are treated rigorously. The fragments are allowed to explore their isospin degree of freedom. The experimental binding energies of the fragments are taken for all masses in the 1985-Wapstra table and the experimental excited states are used for A 5 40 wherever available. Unbound neutron resonances of the fragments are treated in semiclassical approximation t o the second virial coef- ficient. .41lgular momentum is incorporated.
Various approaches to simulate the statistical decay of hot nuclei have been suggested. As 1s well known from fission, the fragmentation of big nuclei is controlled by the balancing of Coulomb- and surface energies. The long-range Coulomb interaction was first treated selfconsistently in the mean-field approach[l], similarly to the molecular field approximation to ferromagnetism. One of the most important results of this approarh was the excellent reproduction of the mass-yield for many systems over the entire range of fragment masses (U-shaped disribution, or U-shape
~ l u s fission ~ e a k for 238U).~nother important result was the jump in the E " ( T ) curve for 19'Au a t a temperature of about 5MeV[2]. Later still rough and relatively uncertain Monte Carlo simulations[3] indicated a peak in the heat capacity near to this temperature. Bondorf et.al.[4]
found a plateau of T[E'] at T = 5MeV for a hypothetical system with A = 100 and Z = 50.In their Monte Carlo simulation the isospin degree of freedom was artificially frozen and, what may be more serious,the fluctuations of the Coulomb field were ignored.
Here we report about new results of our complete microcanonical Metropolis sampling ,per- formed along the lines described in detail in ref.[5].The binding energies of the fragments are
("1, collaboration with H. MASSMANN (from Universidad de Chile, supported by a grant of the Alexander van Humboldt Foundation). and Zhang Xiao-ze. Zheng Yu-ming. and Xu Shu-yan (on leave of absence from the Institute of Atomic Energy Peking. P.R. China)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987218
C2-120 JOURNAL DE PHYSIQUE
the experimental ones wherever they were available in the Wapstra mass table[6].Fig.l shows the nuclides offered as fragments in the calculation and the nuclei t h a t are actually produced i n the decay of 13'Xe a t E" = 36OMeV.For nuclei with A 5 40 energy levels and spin values are taken whenever available,otherwise and for nuclei with A > 40 t h e number of excited states of the secondary fragments were calculated using the level density as given by[7] including shell and pairing effects. Sequential neutron evaporation of the fragments was treated semiclassically, in a n approach, which is similar t o t h e calculation of the second virial coefficient, cf.ref.[5].Charged- particle decays were assumed t o be fast a t excitations of the fragments above the barrier and as closed otherwise [S].This simplification is neccessary due t o technical reasons. T h e overall density
Q = A/Volume of t h e system is taken t o be [2.0S3 * 4 ~ 1 3 1 - ' a m u l fm3.
In fig.] we see t h a t in the decay of I3'Xe predominately nuclei a t the neutron-rich side of the stability line are produced.This is of course due t o the neutron excess of the system. We show i11 fig.2 the thermodynamic temperature T ( E X ) a t various excitation energies E' for a 131Xe nucleus.
The most interesting result is the plateau in T ( E " ) a t T - 5 MeV.Again this indicates a 'phase transition' in l3'Xe a t excitation energies 320 M e V 5 E m < 480 MeV.More details of this will be described elsewhere,(D.H.E.Gross a n d H.Massmann,to be published).
There are arguments that this may not be the liquid-gas transiti0n.A good model for phase transitions of macroscopic systems with nearest neighbor interaction is the study of percolations:
N particles occupy V 2. N points of a cubic lattice a t random. For small densities Q = N I V there will be several clusters (occupied neighbor lattice points) of various size a.The mean multiplicity
< m ( a ) > of these clusters will decrease exponentially with a. This corresponds t o a real gas
above its critical temperature T > T,. We call this situation supercritical. With rising density
Q = N / V t h e system will pass a critical density Q, a t which one of t h e clusters reaches throughout the whole lattice. Above Q, this will always be the case. This situation corresponds t o a real gas below its critical temperature T < T,, where a macroscopic piece of liquid coexists with the gas and with small droplets. This region is called the subcritical region. Consequently,Campi[8]
suggested t o study the correlation of lna,,, V.S. In < Sz > (a,,, is t h e mass of t h e largest fragment i n a given decay-channel, S2 = C 1 7 ~ is the second moment of the other masses in that channel,and < S2 i is the average of SZ over all cha~inels with t.he same a,,, ). Near a phase transition it should changq from a nearly horizontal 1,ranch at. high a,,, for the subcrit,ical region to a second branch leading downwards 1.0 the origi~~,represent.i~tg the supercritical region.We don't see that bending at. E x = 360 M e V (fig.d),but. we d o see it, clearly at. E^ > 6OOMeV (fig.4).
More insight into what happens a t the t.ransit,ion call be obtained by analysing the decay- channels according t o the number of heavy fragments produced.Fig.5 shows the relative yield of events where only one of the produced fragments is bigger than 9 (pseudo evaporation,E), of events
Chart o f the Nucleides
...
20 ... ... ... +, .... tnlculated results
t
Fig.1 Nuclides offered in the sampling (within the boundaries) and nuclei produced in t,he decay of l3'Xe at an excitation of E* = 360MeV
Fig.2 Thermodynamic temperature T as function of the excitation energy E*
in a microcanonical sampling of t,he decay-channels of 131Xe
Fig.3 Campi-correlations In h,, vs. In < S2 > at E' = 360hdeV, t at the 'phase transition' Fig.4 Same as fig.3 but. for E' = 6OOMeV
where two of the fragments are bigger than 9 (pseudo fission,F) and finally of events where at least three of the fragments are bigger than 9 (cracking,C).In general there will be many other small fragments with A 5 9. We see a rapid decrease of the yield of E-events at E' 2: 360MeV where the 'phase transition' occurs in favour of the F-events.The F-channel is not the normal binary fission channel as the average number of charged fragments in the F-channels is about 9,the two
JOURNAL DE PHYSIQUE
I3'Xe
s o t
Fig.5 Relat~ve yield of decay-channels where only one of the fragments is bigger than A = 9,(E);
only two of the fragrnvnts are bigger than A = 9,(F); and finally at least three of the fragments have a mass bigger than 9 (cracking,C)
biggest fragments have a mass of about 16 and 86 on the average.Tliere is a second change-over from E,F-events t o cracking events at E' - 6OOMeV where again a slight steepening of E ' ( T ) appears. It is still an open question whether these changes are the reason of the transition or an accidental signal that something changes here.Similarly,we observe that a t 3 6 0 M e V the average number of fragments with A 1 9 is sharply increasing from 1 to 1.6 (fig.6). We hope to get more insight into these problems from studies of their dependence on the density.
T
Fig.6 Number of fragments with A > 9 for 13' Xe
at various excitation energies.
....:
" . h ' ' 50 1 * + - -
Fig.7 Distribution of the angular-rnomentun~ 6 f l ~ h l tor 8 6 M o at El = 4 9 0 M e V in the normal Monte Carlo sampling with < F>= 0
There is one global conservation law which is still violated in our microcanonical samp1ing:the conservation of angular rnomentum.In fig.7 we show the distribution of fi for the decay of 8 6 M o at E- = 490MeI.'.The mean value of < 1 - > is of course zero.If we force the system to have a rotational energy of 1 2 / 2 0 with 1' being uniformly distributed between 0 and l:,, = (65h)',and O being the actual moment of inertia in each channe1,the mass distribution for the decay of 8 6 M ~ at E' = 490MeV is hardly changed.
[l] V.H.E.Gross,L.Satpalhy,Meng Ta-chung,M.Sntpathy,Z.f.Phys. A 3 a (1982) 41
[2] A.Ecker and D.H.E.Gross, J.Comput.Phys.64 (1986) 47
Lg .S(I Hnn-hao and D. H. E. Gross, N u c 1 . P h y s . M (1985) 643
[q l . f'. I~u~~clorf.H.l~oi~angelo,I..~ishustin,and H.Schulz , Nucl.Phys.A444 (1985) 34
