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-£0(q)
(8)
is equal to the Hartree-Fock energy corrected by the zero-point energy:
The matrix elements in formulae (6-9) have to be evaluated microscopically and depend on the single-particle potential used in calculation. We have used the mean field Woods-Saxon hamiltonian consisting of the kinetic energy term T, the potential energy V w s , the spin-orbit term V™s and the Coulomb potential Vboui for protons: Hws
= T + V w s ( f ; P) + V^s(r; P) + | ( 1 + r3)VCoui(r; P) •
(10)
The central part of the potential has the form of the Fermi function ^>p>-
l+exp[dist(f ; / S)/o]
and the spin-orbit part is taken in the usual form ws
KTC; P) = -A( v v
xp)-s,
(
'
(12)
where dist(f; P) denotes the distance of a point f from the surface of the nucleus and Vo, K, a, A are adjustable constants. The Coulomb potential Vboui is assumed to be that of the nuclear charge equal to (Z — l)e and uniformly distributed inside the nuclear surface. In our calculations we have used Woods-Saxon hamiltonian with the so-called "universal" set of its parameters (see Ref. 6 ) which were adjusted to the single-particle levels of odd-A nuclei with A>40.
16
The pairing residual interaction in the hamiltonian (2) is included within the BCS approximation. In the presented paper we have used the pairing strength constants from Ref. 7 . The collective energy V for a given nucleus is calculated by the macroscopic-microscopic model developed by Strutinsky 8 : V = -EWrOS) + 6Eaheli(P) + <*£pair(/?, A) .
(13)
For the macroscopic part JEmaCr we used the Yukawa-plus-exponential model 9 . The so-called microscopic part, consisted of the shell <S.Esheii and pairing <5^pair corrections, was calculated on the basis of single-particle spectra of Woods-Saxon hamiltonian 6 . The spontaneous-^fission half-life is inversely proportional to the probability of penetration through the fission barrier: T
_ ln2 1
J 2 = ^i> m / ~ nP'
(14)
where n is the number of assaults of the nucleus on the fission barrier per unit time. The number of assaults is given by the frequency of zero-point vibration of the nucleus in the fission degree of freedom and for the vibrational frequency tiwo=lMeV it is equal to n « 10 2 0 , 3 8 s _ 1 . Using the one-dimensional WKB approximation for the penetration probability one obtains the following equation for the fission half-life: in-28.04
^
^
=
" ^ o ~ [1 +
6 X P 2S{L)]
(15)
'
where S(L) is the action-integral calculated along a fission path L(s) in the multi-dimensional deformation space ^BeS(s)[V(s)-E]j
ds.
(16)
An effective inertia associated with the fission mode along the path L(s) is:
where Bki are the q^ and qi components of the inertia tensor and ds denotes the element of the path length in the multi-dimensional space of collective coordinates. The integration limits si and S2 correspond to the classical turning points.
17
3
Coupling of the pairing vibrations with the fission mode
Solving the eigenproblem of the collective hamiltonian (5) for the pairing mode i.e. for q = A one can find the pairing vibrational ground-state wave function *o and the ground state energy EQ at each deformation point. The most probable value of the energy gap Avib corresponds to the maximum of the probability of finding a given gap value in the collective pairing ground state. As it is shown in the left hand side of Fig. 1 the Avib is shifted towards smaller gaps from the equilibrium point A e q determined by the minimum of ^pair (or by the BCS formalism). Such a behavior of the pairing ground state function $o is due to the rapid increase of pairing mass parameter B A A when A decreases. In general the ratio of AVib to A e q is of about 0.7.
104c
l~ 4.0 -A \ 3.0 7 I 2.0 ' \
N = 60 p = 0.2. y=20° V Sl'J'ol2 W*"vJ \
V
/
p a lr[»-»J/
1.0
0.0
B^IIOOAtaj '
0.0
0.2
0.4 0.6 A[/fco 0 ]
i
0.8
0+ 2 + 4 + 6 + 8 + 10 + 0+ 2* 4+ 6* 8 + 1 0 *
Figure 1. Collective pairing potential, inertia and ground state wave function for N=60 particles (left hand side figure) and the effect of coupling of the collective pairing vibrations with the quadrupole vibrations for 104 Ru (right hand side figure).
In order to illustrate the effect of the coupling of the quadrupole and pairing vibrations we have compared in the right hand side of Fig. 1 the energy levels obtained with the traditional generalized Bohr hamiltonian ("old") with those evaluated within the present model ("new"). As one can learn from Fig. 1 the improvement in reproducing the experimental data caused by coupling with the pairing vibrations is really significant n . Similar coupling of the pairing vibrations with the fission mode also takes place in the fission process. The system (fissioning nucleus) searches the
18
most probable path to fission in the multidimensional collective space, i.e. the path L(s) which minimizes the action integral S(L) (eq. 16). Such a dynamical path to fission was postulated already in Ref. 12 and realized in practice in Ref. 13 . But in those papers one has discussed the paths in the multidimensional deformation parameters space only. The role of the coupling of fission with the pairing mode was first pointed in Ref. 14 and quantitatively discussed in Ref. 15 . It was found that the system on its way to fission choose larger proton and neutron pairing gaps then those which minimize the BCS energy.
0.8
1
1.2
r/Ro
1.4
1.6
0.9
1
1.1
1.2
1.3
1.4
1.5
r'Ro
Figure 2. Statical and dynamical fission barriers(left hand side figure) and the effective statical and dynamical mass parameters (right hand side figure) along the fission path of 250 Fm.
This effect increases slightly the fission barriers (see the left hand side of Fig. 2) and reduces significantly the effective mass parameter (Beff) connected with the fission mode. In consequence the action integral along the dynamical path to fission is reduced on average by 20% in comparison to that evaluated along the statical (minimal potential energy) path to fission. Such large decrease of the action integral is mainly due to the coupling of the fission and pairing modes or more precisely speaking due to the strong dependence of all components of the inertia tensor By (eq. 7) on the proton and neutron pairing gaps. Using the described above theoretical model one has performed in Ref. 1 6 , 1 7 an extensive calculation of the spontaneous fission half-life times of isotopes with the proton and neutron numbers: 100
19 petes with the spontaneous fission. In order to estimate the a-decay life times (Tf , 2 ) on basis of the evaluated in our model ground state energies, we have used the phenomenological formula of Viola and Seaborg with the new parameters adjusted in Ref. 10 . Our theoretical estimates of the a-decay half-life times 16 are presented in the right hand side of Fig. 3.
logOj2.fi,,36;V4,)[yr]
"ogT^tyr] {&,}, 7*2.4.6
Figure 3. Spontaneous fission (left hand side) and a-decay life times (right hand side) of the heaviest isotopes .
4
Conclusions:
The following conclusions can be drawn from our investigation: • The pairing degrees of freedom A p and A n reduce spontaneous fission half-lives by about 1-6 orders of magnitude and considerably improve theoretical predictions of Taj. This effect is due to the strong dependence of the inertia tensor on pairing energy-gaps. • We observe an increase of fission lifetimes by one to five orders of magnitude when besides fa and fa the deformation fa is added. This effect improve the agreement with the experimental data. • The deformations fa and fa reduce the width of the static fission barriers but they do not change the fission life times. It is due to the dynamical effects.
20
• The five-dimensional collective space: /? 2 ,Pi, Ps, A p and A n is optimal to describe the spontaneous fission of heaviest nuclei. • The difference between the proton and neutron deformations should be taken into account in future macroscopic-microscopic calculations of masses (a-decay) and fission barriers (T„/). Acknowledgments This work was partially supported by the Polish Committee of Scientific Research KBN No. 2P 03B 115 19. References 1. S. Hofmann et al., Z. Phys. A350, 277 (1995); A350, 281 (1995); A354, 229 (1996). 2. Yu.A. Lazarev et al., Phys. Rev. Lett. 73, 624 (1994); 75, 1903 (1995). 3. V. Ninov et al. Phys. Rev. Lett. 83, 1104 (1999). 4. A. G6zdz, K. Pomorski, M. Brack and E. Werner, Nucl. Phys A442, 26 (1985). 5. A. Gozdz and K. Pomorski, Nucl. Phys. A451, 1 (1986). 6. S. Cwiok, J. Dudek, W. Nazarewicz, J. Skalski and T. Werner, Comput. Phys. Commun. 46, 379 (1987). 7. J. Dudek, A. Majhofer and J. Skalski, J. Phys. G6, 447 (1980). 8. V. M. Strutinsky, Nucl. Phys. A95, 420 (1967); A122, 1 (1968). 9. H.J. Krappe, J.R. Nix and A.J. Sierk, Phys. Rev. C20, 992 (1979). 10. A. Sobiczewski, Z. Patyk and S. Cwiok, Phys. Lett. B224, 1 (1989). 11. K. Pomorski, L. Prochniak, K. Zajac, S.G. Rohozinski and J. Srebrny, Physica Scripta T88, 111 (2000). 12. M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44, 320 (1972). 13. A. Baxan, K. Pomorski, A. Lukasiak and A. Sobiczewski, Nucl. Phys. A361, 83 (1981). 14. L.G. Moretto and R.B. Babinet, Phys. Lett. 49B, 147 (1974). 15. A. Staszczak, S. Pilat, K. Pomorski, Nucl. Phys. A504, 589 (1989). 16. A. Staszczak, Z. Lojewski, A. Baxan, B. Nerlo-Pomorska, K. Pomorski Proc. of Third Int. Conf. on Dynamical Aspects of Nuclear Fission, DANF'96, Casta-Papernicka, Eds. J. Kliman and B.I. Pustylnik, JINR Dubna, 1996 , p.22 17. Z. Lojewski, A. Staszczak, Nucl. Phys. A657, 134 (1999)
21
Fission studies with large detector arrays J.K. Hwang 1 , C.J. Beyer 1 , A.V. Ramayya 1 , J.H. Hamilton 1 , G.M. Ter Akopian 2 ' 3 , A.V. Daniel 2 , 3 , J.O. Rasmussen 4 , S.-C. Wu 4 , R. Donangelo 4 ' 5 , J. Kormicki 1 , X.Q. Zhang 1 , A. Rodin 2 , A. Formichev 2 , J. Kliman 2 , L. K r u p a 2 , Yu. Ts. Oganessian 2 , G. Chubaryan 6 , D. Seweryniak 7 , R.V.F. Janssens 7 , W.C. Ma 8 , R.B. Piercey 8 , and J.D. Cole 9 1 Department of Physics, Vanderbilt University, Nashville, TN37235, USA 2 Flerov Laboratory for Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, Russia 3 Joint Institute for Heavy Ion Research, Oak Ridge, TN37831, USA 4 Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 5 Instituto de Fisica, Universidade Federal do Rio de Janeiro, C.P. 68528, 21945-970 Rio de Janeiro, Brazil 6 Cyclotron Institute, Texas A and M University, Texas 77843-3366, USA 7 Argonne National Laboratory, Argonne, IL 60439, USA 8 Department of Physics, Mississippi State University, MS39762, USA 9 Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID83415, USA The spontaneous fission of 252 Cf has been studied via 7 — 7 — 7 coincidence and 7 - 7-light charged particle coincidence with Gammasphere. The yields of correlated Mo-Ba pairs in binary fission with 0-10 neutron emission have been remesured with an uncompressed cube. The previous hot fission mode with 8-10 neutron emission seen in the MoBa split is found to be smaller than earlier results but still present. New On binary SF yields are reported. By gating on the light charged particles detected in AE-E detectors and 7—7 coincidences with Gammasphere, the relative yields of correlated pairs in alpha ternary fission with zero to 6n emission are observed for the first time. The peak occurs around the a 2n channel. A number of correlated pairs are identified in ternary fission with 10 Be as the LCP. We observed only cold, On 10 Be and little, if any, hot, xn 10 Be channel.
22
1
Introduction
Studies of prompt gamma rays emitted in spontaneous fission with large detector arrays have given new insights into the fission process [1-5]. From 7 — 7 — 7 coincidence studies of the prompt 7-ray emitted in the spontaneous fission of 252 Cf with Gammasphere, yields of individual correlated pairs in binary and ternary fission were determined for the first time for 0 to 10 neutron emission. [2-4]. Earlier we reported an ultra hot fission mode in the Mo-Ba split in the 252 Cf SF [2]. By using an uncompressed 7 — 7 — 7 cube, problems in the fission data analysis from complexity in the spectra in the 8-10 neutron emission yields were overcome. The new Mo-Ba yields show a reduced yield for the ultra hot mode. These data also allowed the extraction of more accurate zero neutron emission yields in cold binary fission. A new experiment were carried out in which the light charged particles (LCP) involved in ternary fission were detected in a LCP-7 — 7 coincidence mode. From gating on the alpha particles and a gamma ray, the relative 0 to 6n yields associated with a ternary fission were extracted. Gamma rays associated with new correlated pairs in coincidence with high energy 10 Be particles also were identified. These data give new insights into fission. In contrast to a ternary SF, only the On mode is observed for 10 Be. 2
A n e w d e t e r m i n a t i o n of t h e B a - M o yield m a t r i x for 252
C f
We carried out pioneering work on the quantitative determination of yield matrices, using 7 — 7 and 7 — 7 — 7 coincidence data to extract yields of particular fragment pairs in the SF of 252 Cf [2,3,4]. One interesting finding was that « 14% of the 252 Cf bariummolybdenum split goes via a "hot fission" mode, where, as many as 10 neutrons are emitted [2,3]. This latter feature stimulated some theoretical speculations and also some skepticism, since the
23
hot fission mode (called Mode 2) has been reported only in the Ba-Mo pairs in 252Cf and not in 248 Cm spontaneous fission [6] . There have been some theoretical efforts to understand how this hot fission could arise [4,7,8]. In the present work we used our 1995 Gammasphere data, taken by the GANDS95 collaboration [2]. The analysis was carried out with uncompressed triple coincidence spectra. This differs from the previous analyses where either uncompressed double coincidence spectra or compressed triple coincidence spectra were used. In both of these methods one faces problems, because of the vast number of 7-rays in the spectrum and particulary because of the degeneracy of several 7-rays in the 8-10 neutron emission yields for Mo-Ba. In this new analysis, using
Double gate on 376.7 ond 457.4 keV
«°Xe g 1800 o
8.
woo
*
c
3 O CJ
200 120
JuijJUUl 140
160 160
E r (KeV)
180
200
220
Figure 1: Mo-Ba yields vs. neutron numbers.
the uncompressed 3D data, we remeasured the pair yields of barium (Z=56) with molybdenum (Z=42) partners. Because 104Mo and 108Mo have 2 + —»• 0 + transitions that are too close in energy to resolve and their 4 + —> 2 + transitions are barely resolvable with peak-fitting routines, we have generally chosen to double-gate on
24
the Ba fragments and measure the 2 + —» 0 + intensities in the Mo partners (and 4 + —>• 2+ where the 2 + —> 0 + are unresolvable.) The barium double gates are on the 4 —>• 2 —>• 0 cascade and the 3 —> 2 —» 0 cascade, the latter being significant in the heavier bariums where octupole deformation is reported [9]. The odd-A nuclei are special cases discussed in a separate publication. Their yields in our triple-coincidence analysis fall rather smoothly into the yield patterns of their even-even neighbors. In the yield calculations we have taken into account that Compton suppression is not complete and that, also, Compton scattering on the walls of the chamber and into a detector occur and that true continuum gammas are simultaneously present. Rather than using one of the existing gamma efficiency curves for Gammasphere, as determined off-line with radioactive standards in singles mode, we checked the efficiency curves with rotational cascades in the actual experiment, double-gating on two transitions high in the rotational band and measuring the intensities of the lower transitions in the band. Thus, these efficiency measurements involved coincidence efficiencies and take into account Compton suppression, "time-walk," and other factors at the high count rates of the actual experiment. Fig. 1 shows semi-log plots of the summed Ba-Mo fission yields vs. neutron-emission number found in our work and in the previous work 2,3]. One sees that the hot fission mode is still present but its intensity is reduced about a factor of 3 from the 14% reported earlier [2]. Since work was completed [10], Biswas et al. [11] also reported analogous data that show a similar small irregularity around 8 neutrons lost. They reported they could not observe a 10-neutron loss. We do report one such cell, 104Mo-138Ba, but with a large standard deviation as 0.003 ± 0.002 %.
25
3
Cold Binary Spontaneous Fission
Since the neutronless binary events are much smaller than those with neutrons emitted, double gating techniques have been employed to extract the yields for the cold binary fission. No direct measurements of yields of correlated pairs in cold binary fission had been made prior to our work. Earlier we reported the first results for the correlated pairs in cold binary fission in 252Cf [1,3] and 242 Pu [12]. Subsequently we extracted additional and more accurate yields of cold binary fission [4] from Gammasphere data with 72 detectors. Table 1: Average cold binary fission yields from gates on two light fragment and two heavy fragment transitions AL/
Zr/Ce
Mo/Ba
Tc/Cs Ru/Xe
Pd/Te
AH
100/152 102/150 103/149 104/148 104/148 106/146 107/145 108/144 109/143 110/142 111/141 112/140 114/138 116/136
Y •* exp 0.010(2) 0.020(4) 0.030(6) 0.010(2) 0.010(2) 0.040(8) 0.070(14) 0.030(6) 0.090(18) 0.060(12) 0.10(2) 0.020(4) 0.020(4) 0.050(20)
Ythe
0.38 2.82 4.21 1.03 0.47 0.61 3.07 7.45 11.03 3.78 7.12 0.59 1.17 2.35
v(ren) 1
the
0.004 0.033 0.049 0.012 0.005 0.007 0.036 0.087 0.128 0.044 0.083 0.007 0.014 0.027
The cold binary fission yields are shown in Table 1 along with the theoretical values predicted by Sandulescu et al. [13]. In Table 1, the first reports of the cold binary fission of an odd-Z - odd-Z
26
fragmentation is shown for the Tc and Cs pair and of an even Zodd A pair. The over all agreement with theory is generally good, including the predicted enhancement of the odd Z and odd A cases.
4
Light Charged Particle Ternary Fission
More recently, another experiment was performed incorperating charged particle detectors to detect ternary particles in coincidence with 7 rays in Gammasphere. The energy spectrum of charged particles emitted in the spontaneous fission of 252Cf was measured by using two AE-E Si detector telescopes installed at the center of the Gammasphere array at Argonne National Laboratory. With the
Gated on Alpha
350
450
E, (keV)
Figure 2: a gated spectrum
position resolution of the strip detector (4 mm wide strips and 1 mm resolution along each strip), the AE-E telescopes provided unambiguous Z and A identification for all the light charged particles of interest. The energy calibration of the telescopes was performed
27 Gated on Ternary Alpha and 287.1keV( M2 Xo)
g400 3
Sf
IMJ^
9*
in in
V^viV 400
(keV)
Figure 3: Coincident spectrum gated on a and 287.1 keV( 142 Xe)
with 224 Ra and 228 Th radioactive sources. The 7—ray spectrum in coincidence with ternary a—particles is shown in Fig. 2. In this spectrum, one can easily see the 7—transitions for various partner nuclei where a ternary a—particle is emitted. For example, Xe and Mo isotopes are partners where a and xn are emitted. Now, imposing an additional condition that the CM—gated 7—spectrum should be also in coincidence with the 2+—>-0+ transition in 142Xe, one gets a very clean spectrum as shown in Fig. 3. Various a, xn fission channels are marked on the spectrum. From the analysis of the 7—ray intensities in these types of spectra, one can calculate the yield distributions. The yield distributions both for binary and ternary a—channel from 0 to 6n emission are shown Fig. 4 for two particular channel. These are the first relative 0-6n yields for any ternary a SF. Note the peak of the neutron emission yields for Ba-a- 102 Zr is shifted up about half an AMU from the Ba-106Mo binary yield and so the average neutron emission in this a ternary
28
SF channel is shfted down by about 0.4n. About 5-20 % of the a yield is from 5He ternary fission in 252Cf SF.
?
I1
5
(o)Gat«l an S88.9( ™Xe) k>V and aoted goted on "Bs
I £,
M I i L u l u iLiiBiJiiliiiitiiBim
h n •£ 1
uo
li I i •—LJ—L (b)Gated an 48MC"Xe) keV and gated on "B«
• 3
AkJ^J^Ui;».iuJL.iiL. mI Hill 111 I I
5
„ §
„ I
5 $
I
I
I
fi i L i n I 100
III 300
'
III
llll »
I I I ||
(c) "ANOT goto of 5BB5 and 481B(™Xo) koV, and gated on ""Bo
II I I
500
1
700
L 900
•
'
1100
1300
E7 (keV)
Figure 4: Yield spectrum for a
5
Identification of the cold
10
Be ternary SF pairs of
252
Cf
Ternary fission is very rare process that occurs roughly only once in every 500 spontaneous fissions (SF) dominated by a ternary fission. Roughly, the 10Be particles are emitted once per 105 spontaneous fissions. The maximum yield in the binary spontaneous fission is located around 3 to 4 neutrons. We now find that the a ternary fission is, mostly, accompanied by w 2 to 3 neutrons. In neutronless ternary spontaneous fission (SF), the two larger fragments have very low excitation energy and high kinetic energies. Experimentally it is not easy to identify the 7 transitions of the cold or hot 10Be ternary SF pair because it is very rare process. The first case of neutronless 10Be ternary spontaneous fission (SF) in 252Cf was reported from 7 - 7 - 7 coincidence spectrum where
Figure 5: Gamma spectrum gated on 10 Be particles. Several peaks are marked with several isotopes related to the present work. But, it is hard to assign the right isotopes to each peak because of the expected 7-ray multiplicity in this spectrum.
the pairs are 96Sr and 146 Ba without neutrons emitted [5]. In our LCP-7 — 7 data, the cover foils allowed only the high energy tail of the 10Be energy spectrum to be observed in the particle detector and their partners established from the 3D cube data. In the present work, the neutronless (cold) 10Be ternary spontaneous fission (SF) pairs of 252Cf are identified for two other fragment pairs of 104Zr-138Xe and 106Mo-136Te from the analysis of the 7 - 7 matrix gated by the 10Be particles. Also, several isotopes related to the 10 Be ternary SF are observed. From the AE-E plot, the 10Be charged particles are selected as a gate to make the 7 — 7 matrix. Here we selected a narrow time gate of width ?ss80 nsec between the gamma-rays and the 10Be charged particles. Also, we did not subtract the background spectrum from the full projection of the 7 — 7 matrix because of poor statistics.
30 (o)Gated on 240.7( m Ru) keV
winiuiuunii•
o SK
Si
n mi i i
(b)GatecJ on 192.0( ""Ma) keV
x.j.iMliUilitLuA Inn lili
\
• nun
4450
ai
I HI H I J "
I I I'
(cjGated on 376.7(,40Xe) keV
w4
III ll • l 4 2 0
.2860
llll I
I
ULL
(djGated on 212.6[1D°Zr) keV
jjf
.. iJiiiy^W^wJii..JyLilMiml 700
II ill n 900
in n i 1100
1300
(keV)
Figure 6: (a)Coincidence spectrum gated on the 10 Be particle and 240-7 keV ( 110 Ru) transition, (b)Coincidence spectrum gated on the 10 Be particle and 192.0 keV (104Mo) transition, (c)Coincidence spectrum gated on the 10 Be particle and 376.7 keV ( 140 Xe) transition, (d)Coincidence spectrum gated on the 10 Be particle and 212.6 keV ( 100 Zr) transition.
The high efficiency of Gammasphere enables coincidence relationships to be established even with the low statistics data associated with a small 10Be ternary SF yield. The gamma spectrum gated on the 10Be charged particles is shown in Fig. 5. Several peaks are marked with isotopes related to the present work. It is sometimes hard to assign the right isotopes from the energies of the peaks alone because the same energy transition may be present in one or more isotopes. For example, the strong 212 keV peak in the spectrum can come from several sources such as 100Zr, i n ' U 3 R h , and 147 La. So next, in the 10Be gated 7 — 7 matrix we set a gate on the 212.6 keV energy in 100Zr. There we can see the 352.0 (4+->2+) and 497.0 (4+—>2+) keV transitions in 100Zr. Four examples are shown in Figs. 6 to identify 110Ru(or 108 Ru), 104Mo(or 108Mo), 140Xe and
31
Figure 7: Coincidence spectrum double-gated on 376.7 and 457.4 keV transitions in 140 Xe. See 151.0 ( 102 Zr) and 212.0 ( 100 Zr). 100
Zr, respectively. But the identification of the gamma-transitions belonging to these partner fragments is not clear in those spectra. From the 7 — 7 — 7 cube we could clearly eatablish coincidence for iooZr_i42Xe a n d i02 Zr _i40 Xe A l s 0 ; b y double gating on the 376.7 and 457.3 7-rays in 140Xe (Fig. 7), we can see clearly the zero neutron channel 102Zr and probably the 100Zr 2n channel which is weaker by a factor of 5-10 if present. The identification of several isotopes related with the 10Be emission is made by the observation of two or three transitions in coincidence belonging to each isotope and from the 7 - 7 — 7 cube. All isotopes and the related 7 transitions identified in the present work are tabulated in Table 2. In Table 2, partner fragments pertaining to the cold (neutronless) channel are shown some of which are confirmed as noted. Quadrupole deformations for each isotopes are taken from Refs. [14,15]. From these examples, we can see that the statistics of the coincident spectrum with a single gate on the lowest gamma tran-
32
sition does not depend on the statistics of the gated peak shown in Fig. 1 because of complexity of the gamma-ray multiplicity and the enhanced population of the lowlying levels in the 10Be SF. Two fragment pairs, 138Xe-104Zr and 136Te-106Mo with no neutrons emitted show 7 rays produced from both of pair fragments in the 10Be gated coincidence spectrum with a single 7 gate. In other words, the 171.6 keV transition of 106Mo is observed in the coincidence spectrum with a single gate on the 606.6 keV transition of 136Te. Also, the 140.3 and 312.5 keV transitions in 104Zr are observed in the coincidence spectrum with a single gate on the 588.9 keV transition of 138 Xe. For a single gate set on the 588.9 keV transition (2+—>-0+) in 138Xe, the coincidence spectrum is shown in Fig. 8. The 4+-S-2+ and 2+->-0+ transitions in 104Zr and the 483.8 and 482.9 keV doublet transitions (6+->-4+ and 4+->2+) in 138Xe show up clearly. To find the real peaks coincident with both the 588.9 and 483.8 keV transitions we set the "AND" gate of 588.9 and 483.8 keV transitions as shown in Fig. 8. This logical "AND" gate takes arithmetic minimum of two spectra for each channel in the Radware program [16]. Then only three transitions of energies 140.3 keV (2+-»0 + ) and 312.5 keV (4+^2+) in 104Zr and 482.1 keV (6+—»4+) transition in 138Xe show up clearly in Fig. 8. Although the three peaks in Fig. 4c contain only two counts, the background is less than 0.01/channel in Fig. 8. The 140.3 and 312.5 keV transitions do not exist in the level scheme of 138Xe. Since the 7 — 7 matrix is gated by 10Be particles, 140.3 and 312.5 keV transitions belong to the partner nucleus 104Zr. However, 109.0 and 146.8 keV transitions in 103Zr ( 10 Be+ln channel) and the 151.8 and 326.2 keV transitions in 102Zr ( 10 Be+2n channel) do not show up clearly. In another case of 136Te-106Mo, also, the 1 0 Be+ln and 10 Be+2n channels are not observed. This could be caused by the larger feeding to ground state but more likely by small yields in the 10 Be+n and 10 Be+2n channels. The hot fission mode can excite the fragments up to higher level energies than the cold fission.
33
f
(o)Gatad on 5sa9( ™Xe) keV and gated on "Be
I?
H I I Hi Hill I I I lllllB I HIM III llllllll 111 llll
II ] I 1
H , LiiiyiiiiiluiHkJLjL^JJ,L Jl l "
LJ
II
I
U_
(b)GatBd on i a i ^ ^ X e ) VeV and gated on °Be
" i l l II
•
I I
U1LJ
l_l_
(c) " A W gats of 5M.9 and 48M( ^Xe) keV.
II I H I I I
I
I I I 300
I
III 500
III 700
900
1100
1300
E„ (keV)
Figure 8: (a)Coincidence spectrum gated on the 10 Be particle and 588.9 keV (138Xe) transition, (b)Coincidence spectrum gated on the 10 Be particle and 483.8 keV ( 138 Xe) transition, (c)Coincidence spectrum with "AND" gate of 588.9 and 483.8 keV ( 138 Xe) transitions, and gated on 10 Be. This logical "AND" gate takes arithmetic minimum of two spectra for each channel in the Radware program [16].
Therefore, SF yields of the 10 Be+n and 10Be-t-2n channels have to be smaller than the neutronless (cold) 10Be SF yield. The present results indicate that the cold (neutronless) process is dominant in the ternary SF accompanying a heavy third particle such as 10Be with high kinetic energy. In our work, we are gating only on the high kinetic energy part of the 10Be particles. The 104Zr isotope is highly deformed with a /32 value of around 0.4 [15,16] and the 138Xe nucleus is very spherical. Therefore, the 10Be particle seems to be emitted from the breaking of 148 Ce= 138 Xe+ 10 Be at scission which would enhance the 10Be kinetic energy. Increased deformation at the scission point increases excitation energy for the third ternary particle and two heavy frag-
34
Table 2: Fragments identified from the coincidence relationship between 7-rays and 10 Be ternary particle. * : identified in 7 — 7 — 7 data and ** : LCP-7 — 7 data.
Identified Isotopes (fc [14,15]) i{j°Zr (0.321 ) ™2Zr (0.421 ) l^Zr (0.381 ) *°4Mo (0.325 ) (or ™8Mo) IfMo (0.353 ) \l°Ru (0.303 ) (or i°8Ru) 6 ^ Te (0.000 ) IfXe (0.0309 ) ^°Xe (0.1136 )
Observed 7 rays (keV) 212.6, 352.0, 497.0 151.8, 326.2 140.3, 312.5 192.0, 368.5 171.6 with 606.6 (136Te) 240.7, 422.2 606.6, 424.0 588.9, 483.8, 482.1 376.7, 457.4, 582.5
Partner isotopes (ft [14,15]) If Xe (0.145])* £f Xe (0.1136 )* \fXe (0.0309 )** ^ 8 Te (0.000 ) (or ^ T e ) \fTe (0.000 )** *g2Sn (0.000 ) (or ^ S n ) 6 J° Mo (0.353)** *°4Zr (0.381 )** ^ 2 Zr (0.421 )*
ments. Therefore the possibility of observing the exciated levels in both the fragments increases when both of them are deformed at scission point such as 104Zr(deformed)—148Ce(138Xe+10Be) (deformed). Actually, the neutronless binary fission yield for 148Ce—104Zr pair is as high as 0.05(3) per 100 SF of 252Cf [17]. These cases are very similar to the one we reported earlier for the pair 96Sr (spherical shape) and 146Ba (deformed shape) [5]. In the a teranry fission we see the cold, zero, neutron fission but 2n and 3n channels are much stronger. However, for the cold 10Be ternary SF pairs identified from the 7 — 7 matrix gated on 10Be charged particles and the 3D data, we find the zero neutron channel clearly much stronger than In and 2n. This is a very unique discovery in the study of the cold (zero neutron) fission processes.
35
6
Acknowledgment
Research at Vanderbilt University and Mississippi State University is supported in part by the U.S. Department of Energy under Grants No. DE-FG05-88ER40407 and DE-FG05-95ER40939. Work at Idaho National Engineering and Environmental Laboratory is supported by the U.S. Department of Energy under Contract No. DE-AC07- 76ID01570. Work at Argonne National Laboratory is supported by the Department of Energy under contract W-31109-ENG-38. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
J.H. Hamilton et al, J. Phys. G20 (1994) L85 G.M. Ter-Akopian et al., Phys. Rev. Lett. 77 (1996) 32 G.M.Ter-Akopian et al., Phys. Rev. C55 (1997) 1146 A.V. Ramayya et al., Heavy elements and related new phenomena, Volume /World Scientific (1999) 477 A.V. Ramayya et a l , Phys. Rev. Lett. 81 (1998) 947 N. Schultz, (Private communication). Yu.U. Pyatkov et al., Nucl. Phys. A624 (1997) 140 R. Donangelo et al., Int. J. Mod. Phys. E7 (1998) 669 J.H. Hamilton et al., Proc. Nuclear Structure 98, Gatlinburg (1999) 473 S.-C. Wu et al, Phys. Rev. C62 (2000) 041601 D.C. Biswas et al., Eur. Phys. J. A7 (2000) 189 Y.X. Dardenne et al., Phys. Rev. C54 (1996) 206 A. Sandulescu et al., Int. J. Modern Phy. E7 (1998) 625 S. Raman et al., Atomic Data and Nucl. Data Tables 36 (1987) 1 P. Moller et al., Atomic Data and Nucl. Data Tables 59 (1995) 185 D.C. Radford, Nucl. Instr. Meth. Phys. Res. A361 (1995) 297
36
17. J.H. Hamilton et al., Prog, in Part, and Nucl. Phys. 35 (1995) 635
37
A D I A B A T I C A N D N O N - A D I A B A T I C D Y N A M I C S OF I O N S I N T H E F O R M A T I O N OF ELECTRON EXCITATION W I D T H S I N METAL CLUSTERS A. V. SOLOV'YOV ° Institute for Theoretical Physics, Frankfurt am Main University, Robert-Mayer Str. 8-10, D60054 Frankfurt am Main, Germany E-mail: [email protected] The dynamical jellium model for metal clusters, which treats simultaneously the vibrational modes of the ionic jellium background in a cluster, the quantized electron motion and the interaction between the electronic and the ionic subsystems beyond the adiabatic approximation, is applied for the calculation of widths of electron excitations in metal clusters caused by multiphonon transitions and the investigation of their temperature dependencies. The decay time and the energy relaxation time of electron excitations in metal clusters is estimated.
1
Introduction.
In this paper the influence of the dynamics of ions on the motion of delocalized electrons in metal clusters is considered on the basis of the dynamic jellium model suggested in 1 and further advanced in 2 . This model generalizes the static jellium model 3 ' 4 ' 5 ' 6 , which treats the ionic background as frozen, by taking into account vibrations of the ionic background near the equilibrium point. The dynamic jellium model treats simultaneously the vibrational modes of the ionic jellium background, the quantized electron motion and the interaction between the electronic and the ionic subsystems. This paper gives a brief survey of the ideas developed in many more details in 1 , z . An example of the effect, originating from the interaction of the ionic vibrations with delocalized electrons, is the broadening of electron excitation lines. There are two mechanisms of the electron excitation line broadening, namely adiabatic and non-dynamic ones. The dynamic jellium model 1 allows one to calculate widths of the electron excitations in metal clusters caused by these two mechanisms and investigate their temperature dependence. The adiabatic mechanism is connected with the averaging of the electron excitation spectrum over the temperature fluctuation of the ionic background in a cluster. This phenomenon has been studied in a number of papers 7 ' 8 ' 9 ' 1 0 ' 1 1 . The mechanism of dynamic or non-adiabatic electron excitation line broaden°On leave from: A.F.Ioffe Physical-Technical Institute, Russian Academy of Sciences, Politechnicheskaya 26, 194021 St. Petersburg, Russia E-mail: [email protected]
38
ing has been considered for the first time in 1. Numerically it was further advanced i n 2 . This mechanism originates from the real multiphonon transitions between the excited levels of electrons. Therefore the dynamic line-widths characterize the real lifetimes of the electronic excitations in a cluster. The adiabatic broadening mechanism explains the temperature dependence of the photo-absorption spectra in the vicinity of the plasmon resonance via the coupling of the dipole excitations in a cluster with the quadrupole deformation of the cluster surface. The photo-absorption spectrum was calculated within the framework of deformed jellium model using either the plasmon pole approximation 7 ' 8 or the local density approximation 9,10 ' 11 ' 12,13 . The interest to the problem of the electron excitation line-widths formation in metal clusters was stimulated by numerous experimental data on photoabsorption spectra, most of which were addressed to the region of dipole plasmon resonances 14 ' 15 ' 16 . The non-adiabatic line-widths characterize the real lifetimes of cluster electron excitations. The information about the non-adiabatic electron-phonon interactions in clusters is necessary for the description of electron inelastic scattering on clusters 17>18, including the processes of electron attachment 1 9 ' 2 0 and bremsstrahlung 2 1 ' 2 2 , the problem of cluster stability and fission. The non-adiabatic line-widths determined by the probability of multiphonon transitions are essential for the treatment of the relaxation of electronic excitations in clusters and the energy transfer from the excited electrons to ions, which occurs after the impact- or photo-excitation of the cluster. Following 1 ' 2 , in this work we elucidate the role of the volume and the surface vibrations of the ionic cluster core in the formation of the electron excitation line-widths and demonstrate that the volume and surface vibrations provide comparable contributions to the adiabatic line-widths, but the surface vibrations are much more essential for the non-adiabatic multiphonon transitions than the volume ones. 2
Dynamical jellium model.
Let us consider the quantized motion of the delocalized electrons and the oscillatory motion of ions near the equilibrium points in one approach. The Hamiltonian of the entire cluster in this case can be represented as a sum of the ionic Hamiltonian Hi, which includes the kinetic energy plus the electrostatic energy of the ionic background, and the Hamiltonian He of the quantized motion of electrons in the frozen field of ions: H = Hi(q)+He(q,r).
(1)
39
Here q and r represent sets of the ionic and the electronic coordinates of the cluster respectively. Let us solve the Schrodinger equation, HVX = EXVX,
(2)
with the Hamiltonian (1). The wave function of the entire cluster can be expressed as a sum of products of the electron, ^„(g,r), and the phonon, $An(<7), w a v e functions: *A = E * A » f o W „ ( g , r ) .
(3)
n
Here ipn(q,r) is the many-electron wave function for the frozen ionic background satisfying the following Schrodinger equation: He(q, r)ipn(q, r) = e„{q)ipn(q, r).
(4)
Substituting (3) in (2), multiplying this equation by ip^(q,T) and integrating it over the electronic coordinates, we derive the set of equations for the wave functions $An(g):
(E (£+^)+^<«>-*)#*.«= Here qa, pa, £la and ma are the generalized coordinate, momentum, frequency and effective mass corresponding to the a-th normal oscillatory mode respectively. The term m a fi£<j£/2 describes the dependence of the cluster ground state energy upon the deformation and hu>n(q) = en(q) — £o(o) denotes the electron excitation energy. For a small displacement of the ionic background, when the energy shifts of the excited electron levels are smaller than the inter-level energy distances, the dependence of the excitation energies upon the ionic coordinate q can be described by the linear term of the expansion hu)n(q) over q : M , ) = b„ + ^ g j W , (6) a a
where V"( ' = dHe(q, r)/dqa action and u>„ = w n (0).
plays a role of operator of electron-phonon inter-
40
The terms at the right hand side of equation (5) are responsible for the so-called non-adiabatic coupling. In the adiabatic (or the so-called BornOppenheimer) approximation these terms are neglected. In this case, equation (5) reduces to the set of non-coupled equations describing the oscillations of the ions in the cluster. The electron-phonon spectrum in this approximation is of the form:
* = «f + ?(«W'r- + 5»-^)-
(7)
The corresponding wave functions read *A(0,r) = ^
o )
(r)n*"-(O.
(8)
a
where $JV„ (<j£J are eigen functions of the shifted oscillators, which describe the phonon excitations. This approximation is valid if the shifts of the electron energy levels are much smaller compared to the inter-level energy distances: f>(<*)2
"" 2man2a
«Ae„.
(9)
The excitation spectrum of the cluster can be described as a set of the oscillator strengths corresponding to all electron-phonon transitions from the ground state of the cluster to its excited states (8). Each electron excitation line contains a series of satellite lines corresponding to the excitation of different number of phonons. The distribution of the oscillator strengths determines the electron excitation spectrum pattern and its width T. Calculations of this kind have been performed in 8 ' 9 for the electron coupling with a single surface quadruple vibration mode. Note, that the approximation, treating electron coupling with a single phonon mode is known in solid state theory as HuangRhys model 2 3 . Below we use this approximation and therefore omit the index of the chosen phonon mode a. The generalization of the Huang-Rhys model to the case of finite number of phonon modes is straightforward. Calculating the adiabatic width of an electron excitation line one derives 8,1,2.
r
/4Mn2 „ , M ! V
° = V -^rrtfc(2*f >
Vnn
(10)
In the case of more than one phonon mode, the total adiabatic line-width is given by the square root of the sum of squares of the single-mode line-widths
41
performed over all the modes. Note that for spherical clusters, there is an additional mechanism of the lines broadening, which is connected with the breaking of selection rules under deformations of the ionic background u . Now let us consider effects arising from the non-adiabatic coupling between the electronic and the ionic subsystem in a cluster. The non-adiabatic terms appear at the right hand side of equation (5). The significance of these terms is different for small and large clusters. Indeed, the ionic subsystem has a discrete vibrational spectrum in small clusters due to the limited number of degrees of freedom. The non-adiabatic terms in this case can be treated as a perturbation, providing small corrections to the energies (7) and the wave functions (8). The density of the vibrational states grows with increasing cluster size. The average distance between two neighboring vibrational lines in the spectrum decreases and becomes comparable with the width of each of these lines, resulting in the transformation of the discrete spectrum into a continuous one. Thus for large clusters, the physical consequences of the non-adiabatic electron-ion coupling are much more essential. The non-adiabatic terms in (5) cause the multiphonon transitions between the electron-vibration levels (7) with close energies and thus determine their finite lifetimes. The finite lifetime of the excited levels is connected with the widths of the corresponding excitation lines in the spectrum. This mechanism of the line-width formation we shall call the dynamic broadening. Let us evaluate the width of an excited electron-vibration level in first order perturbation theory for the electron-phonon coupling. For the sake of simplicity, again we consider coupling of the electronic motion only with a single vibrational mode and, below, omit the index of the phonon mode a. The probability of a multiphonon transition from an excited cluster state with electronic and phononic quantum numbers n and N, respectively, to all possible states (n', AT') define the width of the excited level. It is given by: 2
£$, \
\J
6 ( av - "n + Sli(N' -N)-
d
Q<* V
rrn
n ™ 2 ' J • nJ"' 2mft
I
X
(11)
The sum of delta functions over all vibrational levels defines the total density of vibrational states, p. When considering a single vibrational mode, p = I/O. Due to their large mass, the motion of ions in a cluster can be treated semiclassically. Therefore, $jv()/{V„'n> — Vnn)
42 24
. If the condition (9) is fulfilled the tangent point qo lies in the classically forbidden region of the motion of the ions far from the turning points q„ and qni. In this case, the evaluation of the matrix elements can be performed using the Landau method 2 ' 2 4 ' 2 5 . Calculations lead to the following final expression for the probability of the multiphonon transition x ' 2 :
r„-£W-n
2 HI. n'n
v(qo)(V„>„' - V„„)
exp(2^)-2^°)).
(12)
Here Hn>n is the half of the inter-level distance between en{q) and en'(q) in the tangent point, v(qo) is the ion velocity in the tangent point and/2mhtt3: (0)
Zny/Zl-2N
where Zn = (I - S)/V2S
1
1
27V + 1 , Zn + yJZl - 27V - 1 ; In 2/V + l
(13)
and Zn. = {I + 5 ) / \ / 2 5 .
Figure 1 shows that the adiabatic width decreases with the growth of the vibration frequency. Thus the soft vibration modes give the dominating contribution to the total excitation width. The rapid temperature growth of the non-adiabatich line-width is connected with the increase of the number of phonons in the initial cluster state, i.e. with the thermo-activation of the multiphonon transition. At the room temperature the non-adiabatic line-width is about 1.5meV. This value corresponds to the decay time of the plasmon excitations about r w 1/T « 0.4ps. One can estimate the energy relaxation time as rE = e/(de/dt) ~ Tu>„/Ae, where Ae « O.leV' is the average interlevel distance. This estimate gives for the following value TE W 12ps. 3
Conclusions.
The dynamical jellium model provides a powerful tool for further investigations of the influence of the ion dynamics on the electronic motion beyond the adiabatic approximation. The model describes lifetimes of electron excitations and the energy relaxation time. Significant difference between the adiabatic and dynamic widths of the electron excitations in metal clusters is reported. This fact is essential for studying inelastic collision processes involving clusters.
43
_! 100
1
1 MO
T, K*
i
1 300
1
1 400
0.01 I
'
" 100
'
' 200
'
L
-
300
T. K°
Figure 1: Left: temperature dependence of the adiabatic broadening width calculated according to (10) for the dipole electron excitation with the energy uin = 3.013eV in the JVcMo cluster. Dashed lines marked by numbers 1,2 and 3 show the adiabatic width corresponding to the electron coupling with the three first volume vibration modes respectively. Dashed-dotted line shows the adiabatic width arising from the electron coupling with surface vibrations of the cluster. Solid thick line shows the total adiabatic line-width, which is equal to the square root of the sum of the squared widths corresponding to the three volume and one surface phonon modes. Right: temperature dependence of the non-adiabatic width calculated according to (12) for the dipole excitation with the energy u>n = 3.013eV in the Afa4o cluster.
Acknowledgments I am grateful to the Deutsche Forschungsgemeinschaft, Volkswagen Foundation and INTAS for the support of this work. References 1. L.G. Gerchikov, A.V. Solov'yov and W. Greiner, International Journal of Modern Physics E 8, 289 (1999). 2. L.G. Gerchikov, A.N. Ipatov, A.V. Solov'yov and W. Greiner, J.Phys.B: At.Mol.Opt.Phys., in print (2000). 3. W. Ekardt, Phys. Rev. B 32, 1961 (1985). 4. C. Guet and W.R. Johnson, Phys.Rev. B 45, 283 (1992). 5. V.K. Ivanov, A.N. Ipatov, V.A. Kharchenko, M.L. Zhizin, JETP Lett 58, 629 (1993); Phys.Rev. A 50, 1459 (1994). 6. V.K. Ivanov, A.N. Ipatov, in Correlations in clusters and related szstems. New perspectives of the many-body problem, ed. J.-P. Connerade, (World Scientific Publishing, Singapore, 1996, p.141-168); in Physics of clusters, ed. V.D. Lakhno and G.N. Chuev (World Scientific Publishing, Singapure, 1996, p.224-273).
44
7. 8. 9. 10. 11. 12. 13. 14. 15.
16. 17.
18. 19. 20. 21. 22.
23. 24. 25.
J.M. Pacheco and R.A. Broglia, Phys. Rev. Lett. 62 , 1400 (1989). G.F. Bertsch and D. Tomanek, Phys. Rev., B 40, 2749 (1989). Z. Penzar, W. Ekardt, A. Rubio, Phys. Rev. B 42, 5040 (1990). B. Montag, T. Hirshmann, J. Mayer, P.J. Reinhard, Z. Phys. D 32, 124 (1994). B. Montag and P.J. Reinhard, Phys. Rev. B 5 1 , 14686 (1995). Y. Wang, C. Lewenkopf, D. Tomanek, G. Bertsch, S. Saito, Chern. Phys. Lett. 205, 521 (1993). J.M. Pacheco , W.D.Schone, Phys. Rev. Lett. 79, 4986 (1997). W.A. de Heer, Rev. Mod. Phys. 65 , 611 (1993). H.Haberland (ed.), Clusters of Atoms and Molecules, Theory, Experiment and Clusters of Atoms (Springer Series in Chemical Physics 52, Berlin, Heidelberg, New York, Springer, 1994). U.Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, Berlin, Heidelberg, 1995). L.G. Gerchikov, A.N. Ipatov, A.V. Solov'yov, J.Phys.B: At.Mol.Opt.Phys. 30, 5939 (1997). L.G. Gerchikov, A.N. Ipatov, A.V. Solov'yov and W. Greiner, J.Phys.B: At.Mol.Opt.Phys. 3 1 , 3065-3077 (1998). J.P. Connerade, L.G. Gerchikov, A.N. Ipatov, A.V. Solov'yov, J.Phys.B: At.Mol.Opt.Phys. 3 1 , L27-L34 (1998). J.P. Connerade, L.G. Gerchikov, A.N. Ipatov, A.V. Solov'yov, J.Phys.B: At.Mol.Opt.Phys. 32, 877-894 (1999). L.G. Gerchikov and A.V. Solov'yov, Z.Phys. D 42, 279-287 (1997). L.G. Gerchikov, A.N. Ipatov, A.V. Solov'yov, J.Phys.B: At.Mol.Opt.Phys. 3 1 , 2331 (1998). K. Huang and A. Rhys, Proc. R. Soc. A 204, 406 (1950). L.D. Landau and E.M. Lifshitz, Quantim Mechanics (Pergamon, London, 1965). V.N. Abakumov, LA. Merkulov, V.I. Perel, I.N. Yassievich, JEPT 62, 853 (1985).
45
FISSION, D E C A Y OF N U C L E I A N D T H E E X T E N S I O N OF THE PERIODIC SYSTEM WALTER GREINER Institut fur Theoretische Physik, J. W. Goethe- Universitat, D-60054 Frankfurt, Germany The extension of the periodic system into various new areas is investigated. Experiments for the synthesis of superheavy elements and the predictions of magic numbers are reviewed. Different channels of nuclear decay are discussed like cluster radioactivity, cold fission and cold multifragmentation, including the recent discovery of the tripple fission of 2 5 2 Cf.
There are fundamental questions in science, like e. g. "how did life emerge" or "how does our brain work" and others. However, the most fundamental of those questions is "how did the world originate?". The material world has to exist before life and thinking can develop. Of particular importance are the substances themselves, i. e. the particles the elements are made of ( baryons, mesons, quarks, gluons), i. e. elementary matter. The vacuum and its structure is closely related to that. On this I want to report today. I begin with the discussion of modern issues in nuclear physics. The elements existing in nature are ordered according to their atomic (chemical) properties in the periodic system which was developped by Mendeleev and Lothar Meyer. The heaviest element of natural origin is Uranium. Its nucleus is composed of Z — 92 protons and a certain number of neutrons (N = 128—150). They are called the different Uranium isotopes. The transuranium elements reach from Neptunium (Z = 93) via Californium [Z = 98) and Fermium (Z — 100) up to Lawrencium (Z = 103). The heavier the elements are, the larger are their radii and their number of protons. Thus, the Coulomb repulsion in their interior increases, and they undergo fission. In other words: the transuranium elements become more instable as they get bigger. In the late sixties the dream of the superheavy elements arose. Theoretical nuclear physicists around S.G. Nilsson (Lund)1 and from the Frankfurt school2'3'4 predicted that so-called closed proton and neutron shells should counteract the repelling Coulomb forces. Atomic nuclei with these special "magic" proton and neutron numbers and their neighbours could again be rather stable. These magic proton (Z) and neutron (N) numbers were thought to be Z = 114 and N = 184 or 196. Typical predictions of their life times varied between seconds and many thousand years. Fig.l summarizes the expectations at the time. One can see the islands of superheavy elements
46 Decay Modes
Spontaneous, Nuclear Fission
INSTABILITY
Figure 1: The periodic system of elements as conceived by the Frankfurt school in the late sixties. The islands of superheavy elements (Z = 114, N = 184, 196 and Z = 164, N = 318) are shown as dark hatched areas.
around Z — 114, N = 184 and 196, respectively, and the one around Z = 164, N = 318. The important question was how to produce these superheavy nuclei. There were many attempts, but only little progress was made. It was not until the middle of the seventies that the Prankfurt school of theoretical physics together with visiting scientists (R.K. Gupta (India), A. Sandulescu (Romania) f theoretically understood and substantiated the concept of bombarding of double magic lead nuclei with suitable projectiles, which had been proposed intuitively by the russian nuclear physicist Y. Oganessian6. The two-center shell model, which is essential for the description of fission, fusion and nuclear molecules, was developped in 1969-1972 together with my then students U. Mosel and J. Maruhn 7 . It showed that the shell structure of the two final fragments was visible far beyond the barrier into the fusing nucleus. The collective potential energy surfaces of heavy nuclei, as they were calculated in the framework of the two-center shell model, exhibit pronounced valleys, such that these valleys provide promising doorways to the fusion of superheavy nuclei for certain projectile-target combinations (Fig. 4). If projectile and target approach each other through those "cold" valleys, they get only minimally excited and the barrier which has to be overcome (fusion barrier) is lowest (as compared to neighbouring projectile-target combinations). In this way the correct projectile- and target-combinations for fusion were predicted. Indeed,
47 20 2 f e _
Sfi
SU
x'J&
• ; .
ffi
***
•«** :
mag-
;
2t3
Figure 2: The shell structure in the superheavy region around Z = 114 is an open question. As will be discussed later, meson field theories suggest that Z = 120, TV = 172,184 are the magic numbers in this region.
Gottfried Munzenberg and Sigurd Hofmann and their group at GSI 8 have followed this 'approach. With the help of the SHIP mass-separator and the position sensitive detectors, which were especially developped by them, they produced the pre-superheavy elements Z = 106, 107, . . . 112, each of them with the theoretically predicted projectile-target combinations, and only with these. Everything else failed. This is an impressing success, which crowned the laborious construction work of many years. The before last example of this success, the discovery of element 112 and its long os-decay chain, is shown in Fig. 5. Very recently the Dubna-Livermore-group produced two isotopes of Z = 114 element by bombarding 2 4 4 Pu with 4 8 Ca (Fig. 3). Also this is a cold-valley reaction ( i n this case due to the combination of a spherical and a deformed nucleus), as predicted by Gupta, Sandulescu and Greiner 9 in 1977. There exist also cold valleys for which both fragments are deformed 10 , but these have yet not been verified experimentally. The very recently reported Z = 118 isotope fused with the cold valley reaction 12 5 8 Kr 4- 2 § 8 Pb by Ninov et al. 13 yields the latest support of the cold valley idea. Studies of the shell structure of superheavy elements in the framework of the meson field theory and the Skyrme-Hartree-Fock approach have recently shown that the magic shells in the superheavy region are very isotope dependent 14 (see Fig. 6). According t o t h e s e Investigations Z = 120 b e i n g
m (T1/2i)
lnsf l|isf lmsf lsf lksf
ls f lksf lMst-
(Ti/2,a)
150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 Neutron Number Figure 3: The Z = 106 — 112 isotopes were fused by the Hofmann-Miinzeiiberg (GSI)-group. The two Z = 114 isotopes were produced by the Dubna-Livermore group. It is claimed that three neutrons are evaporated. Obviously the lifetimes of the, various decay products are rather long (because they are closer to the stable valley), in crude agreement with early predictions 3»4 and in excellent agreement with the recent calculations of the Sobicevskygroup 11 . The recently fused Z = 118 isotope by V. Ninov et al. at Berkeley is the heaviest one so far.
a magic p r o t o n n u m b e r s e e m s t o b e as p r o b a b l e as Z = 114. Additionally, recent investigations in a chirally symmetric mean-field theory (see also below) result also in the prediction of these two magic number# 7 , 2 8 . The corresponding magic neutron numbers axe predicted to be N = 172 and - as it seems to a lesser extend - N = 184. Thus, this region provides an open field of research. E.A. Gherghescu et al. have calculated the potential energy surface of the Z = 120 nucleus. It utilizes interesting isomeric and valley structures (Fig. 8). The charge distribution of the Z = 120, N = 184 nucleus indicates a hollow inside. This leads us to suggest that it might be essentially a fullerene consisting of 60 Qf-particles and one additional binding neutron per alpha. This is illustrated in Fig 7. The protons and neutrons of such a superheavy nucleus are distributed over 60 a particles and 60 neutrons (forgetting the last 4 neutrons). The determination of the chemistry of superheavy elements, i. e. the calculation of the atomic structure — which is in the case of element 112 the shell structure of 112 electrons due to the Coulomb interaction of the electrons and in particular the calculation of the orbitals of the outer (valence) elec-
49
<3ft)Sn+-pln
'SSn + 'SCe
Figure 4: The collective potential energy surface of 2 6 4 108 and 1 8 4 114, calculated within the two center shell model by J. Maruhn et al., shows clearly the cold valleys which reach up to the barrier and beyond. Here R is the distance between the fragments and TJ = —
denotes the mass asymmetry: 77 = 0 corresponds to a symmetric, 7j = ± 1 to an
extremely asymmetric division of the nucleus into projectile and target. If projectile and target approach through a cold valley, they do not "constantly slide off" as it would be the case if they approach along the slopes at the sides of the valley. Constant sliding causes heating, so that the compound nucleus heats up and gets unstable. In the cold valley, on the other hand, the created heat is minimized. The colleagues from Freiburg should be familiar with that: they approach Titisee (in the Black Forest) most elegantly through the Hollental and not by climbing its slopes along the sides.
50 70
Zn
2O8 +
Pb-277112
+
ln
l/ll.65MeV,400|is
ru.45McV,280|U
"no
"no
«..*9.73 MeV, 170 ms
-jr 11.08 MeV, 110 us
108
3«-23 MeV, 19.7 s
is.12 MeV, 7.1 • ""106
•"106
'4.60 MeV (escape), 7.4 •
?8.77 MeV, 24.1 « 104 feiMeV,
-X8.52 MeV, 4.7 s
32.7 s
No «"8.34 MeV, 15.0 s •°Fr
Figure 5: The fusion of element 112 with 7 0 Zn as projectile and 2 0 8 P b as target nucleus has been accomplished for the first time in 1995/96 by S. Hofmann, G. Miinzenberg and their collaborators. The colliding nuclei determine an entrance to a "cold valley" as predicted as early as 1976 by Gupta, Sandulescu and Greiner. T h e fused nucleus 112 decays successively via a emission until finally the quasi-stable nucleus 2 5 3 F m is reached. The a particles as well as the final nucleus have been observed. Combined, this renders the definite proof of the existence of a Z — 112 nucleus.
trons — has been carried out as early as 1970 by B. Fricke and W. Greiner15. Hartree-Fock-Dirac calculations yield rather precise results. The potential energy surfaces, which are shown prototypically for Z = 114 in Fig 4, contain even more remarkable information that I want to mention cursorily: if a given nucleus, e. g. Uranium, undergoes fission, it moves in its potential mountains from the interior to the outside. Of course, this happens quantum mechanically. The wave-function of such a nucleus, which decays by tunneling through the barrier, has maxima where the potential is minimal and minima where it has maxima. This is depicted in Fig. 9. The probability for finding a certain mass asymmetry rj = — — of Ax + Ai the fission is proportional to ^>*{j\)^)(j])^.r]. Generally, this is complemented by a coordinate dependent scale factor for the volume element in this (curved) space, which I omit for the sake of clarity. Now it becomes clear how the socalled asymmetric and superasymmetric fission processes come into being. They result from the enhancement of the collective wave function in the cold valleys. And that is indeed, what one observes. Fig. 10 gives an impression of it. For a large mass asymmetry (77 « 0.8, 0.9) there exist very narrow valleys.
51
Neutron Number N Figure 6: Grey scale plots of proton gaps (left column) and neutron gaps (right column) in the N-Z plane for spherical calculations with the forces as indicated. The assignment of scales differs for protons and neutrons, see the uppermost boxes where the scales are indicated in units of MeV. Nuclei that are stable with respect to /? decay and the two-proton dripline are emphasized. The forces with parameter sets SkI4 and PL-40 reproduce the binding energy of fig 108 (Hassium) best, i.e. \6E/E\ < 0.0024. Thus one might assume that these parameter sets could give the best predictions for the superheavies. Nevertheless, it is noticed that PL-40 predicts only Z = 120 as a magic number while SkI4 predicts both Z = 114 and Z = 120 as magic numbers. The magicity depends — sometimes quite strongly — on the neutron number. These studies are due to Bender, Rutz, Biirvenich, Maruhn, P.G. Reinhard et al. 1 4 .
52
Figure 7: Typical structure of the fullerene 60C7. The double bindings are illutsrated by double lines. In the nuclear case the Carbon atoms are replaced by a particles and the double bindings by the additional neutrons. Such a structure would immediately explain the semi-hollowness of that superheavy nucleus, which is revealed in-the mean-field calculations within meson-field theories. (Lower picture by H. Weber.)
53
They are not as clearly visible in Fig. 4, but they have interesting consequences. Through these narrow valleys nuclei can emit spontaneously not only a-particles (Helium nuclei) but also 1 4 C, 2 0 O, 24 Ne, 28 Mg, and other nuclei. Thus, we are lead to the cluster radioactivity (Poenaru, Sandulescu, Greiner 16 )By now this process has been verified experimentally by research groups in Oxford, Moscow, Berkeley, Milan and other places. Accordingly, one has to revise what is learned in school: there are not only 3 types of radioactivity (a-, /?-, 7-radioactivity), but many more. Atomic nuclei can also decay through spontaneous cluster emission (that is the "spitting out" of smaller nuclei like carbon, oxygen,...). Fig. 11 depicts some nice examples of these processes. The knowledge of the collective potential energy surface and the collective masses Bij(R,rj), all calculated within the Two-Center-Shell-Modell (TCSM), allowed H. Klein, D. Schnabel and J. A. Maruhn to calculate lifetimes against fission in an "ab initio" way 1 7 .
Figure 8: Potential energy surface as a function of reduced elongation (iZ — Ri)/(Rt and mass asymmetry rj for the double magic nucleus 3 0 4 120. 3 0 4 120i84.
— Rt)
54
Utilizing a WKB-minimization for the penetrability integral V = e~l,
I = minvpaths | fs \/2m(V(R, 1
rj) - E) ds
2mgij(V(xi{t)-E)
f minv paths ^ Jo
dt
dt
(1)
\ where ds2 — gijdxidxj and gij - the metric tensor - is in the well-known fashion related to the collective masses Z?y = 2mgij, one explores the minimal paths from the nuclear ground state configuration through the multidimensional fission barrier (see Fig. 12). The thus obtained fission half lives are depicted in the lower part of figure 12. Their distribution as a function of the fragment mass A2 resembles quite well the asymmetric mass distribution. Cluster radioactive decays correspond to the broad peaks around A2 = 20, 30 (200, 210). The confrontation of the calculated fission half lives with experiments is depicted in Fig. 13. One notices "nearly quantitative" agreement over 20 orders of magnitude, which is - for an ab-initio calculation - remarkable! Finally, in Fig. 14, we compare the lifetime calculation discussed above with one based on the Preformation Cluster Model by D. Poenaru et a l . 1 8 and recognize an amazing degree of similarity and agreement. A
v(R f l x e d / 1 1 )
potential
superasymmetric fission asymmetric fission Figure 9: The collective potential as a function of the mass asymmetry rj •
Ai deAi +A2 notes the nucleon number in fragment i. This qualitative potential V(i£fi xec |,7/) corresponds to a cut through the potential landscape at R = iifi xe d close to the scission configuration. The wave function is drawn schematically. It has maxima where the potential is minimal and vice versa.
55
The systematics for the average total kinetic energy release for spontaneously fissioning isotopes of Cm and No is following the Viola trend, but 258 Fm and 2 5 9 Fm are clearly outside. The situation is similar also for 260 Md, where two components of fission products (one with lower and one with higher kinetic energy) were observed by Hulet et al. 1 9 . The explanation of these interesting observations lies in two different paths through the collective potential. One reaches the scission point in a stretched neck position (i.e. at a lower point of the Coulomb barrier - thus lower kinetic energy for the fragments) while the other one reaches the scission point practically in a touching-spheres-position (i.e. higher up on the Coulomb barrier and therefore highly energetic fragments) 2 0 . The latter process is cold fission; i.e., the fission fragments are in or close to their ground state (cold fragments)and all the available energy is released as kinetic energy. Cold fission is, in fact, typically a cluster decay. The side-by-side occurence of cold and normal (hot) fission has been named
b
_ 1
\
'
Fission of Nuclei with A * 200u Usp = 7-8MeV
""Pm n,0 B(A) = B a, = 6
10' -as 10°
superasymmetric [fission
-! •§
10"'
rJ
\ \
10 J 10* 10*
T = 05MeV
10* _Li
I
,
-0.6 -0.4 -02
0.0 02
,iJȣ
'_ 0.4 J. 0.6 M? h
4.6 -0.4 -02
0.0
02
0.4
0.6
Figure 10: Asymmetric (a) and symmetric (b) fission. For the latter, also superasymmetric fission is recognizable, as it has been observed only a few years ago by the russian physicist Itkis — just as expected theoretically.
56
Figure 11: Cluster radioactivity of actinide nuclei. By emission of 1 4 C , 2 0 O , . . . "big leaps" in the periodic system can occur, just contrary to the known a, /?, 7 radioactivities, which are also partly shown in the figure.
bi-modal fission20. There has now been put forward a phantastic idea 2 1 in order to study cold fission (Cluster decays) and other exotic fission processes (ternary-, multiple fission in general) very elegantly: By measuring with e. g. the Gamma-sphere characteristic 7-transitions of individual fragments in coincidence, one can identify all these processes in a direct and simple way (Fig. 15). First confirmation of this method by J. Hamilton, V. Ramaya et al. worked out excellently 22 . This method has high potential for revolutionizing fission physics! With some physical intuition one can imagine that triple - and quadriple fission processes and even the process of cold multifragmentation will be discovered - absolutely fascinating! We have thus seen that fission physics (cold fission, cluster radioactivity, ...) and fusion physics (especially the production of superheavy elements) are intimately connected. Indeed, very recently, tripple fission of 2 5 2 C f _> 1 4 6 B a U2
+
96gr 130
+
lOgg
-> R y + Sn + 10 Be ->• ... has been identified by measuring the various 7-transitions of these nuclei in coincidence (see Fig. 16). Even though the statistical evidence for the 10 Be line is small (ss 50 events ) the various coincidences seem to proof that spontaneous
57
P" _
V ^ = 9,752 MeV E ^ = 1 MeV 0.8
0.6
232
92
0.4
02
u
0.0
-02
c, = 0.20/fm -0.4
-0.6
-0.8 X\
O
50A2
180 20
40
60
80
100 120 140 160 180 200
220
Figure 12: The upper part of the figure shows the collective potential energy surface for %^\3 with the groundstate position and various fission paths through the barrier. The middle part shows various collective masses, all calculated in the TCSM. In the lower part the calculated fission half lives are depicted.
58
10
10
Figure 13: Fission half lives for various isotopes of Z - 92 ( • ) , Z = 94 ( A ) , Z = 96 (O), Z = 98 (V) and Z = 100 (o). The black curves represent the experimental values. The dashed and dotted calculations correspond to a different choice of the barrier parameter in the Two Center Shell Model (C3 « 0.2 and 0.1 respectively).
10
10
232
236
240
244 A
248
252
256
tripple fission out of the ground state of 252 Cf with the heavy cluster 10 Be as a third fragment exists. Also other tripple fragmentations can be expected. One of those is also denoted above. In fact, there are first indications, that this break-up is also observed. The most amazing observation is, however, the following: The cross coincidences seem to suggest that one deals with a simultaneous three-body breakup and not with a cascade process. For that one expects a configuration as shown in Fig. 17. Consequently the 10 Be will obtain kinetic energy while running down the combined Coulomb barrier of 146 Ba and 96 Sr and, therefore, the 3368 keV line of 10 Be should be Doppler-broadened. Amazingly, however, it is not and, moreover, it seems to be about 6 keV smaller than the free 10 Be 7-transition. At first one thought that perhaps the free transition energy of 3368 keV had not been measured accurately, but a new measurement by C Rolfs et al. (Bochum University), using Ge-detectors, confirmed this value. If this turns out to be true, the only explanation will be that the Gamma is emitted while the nuclear molecule of the type shown in Fig. 17 holds. The molecule has to live longer
59 0.8
20
0.6 0.4
40
60
0.2 0.0 -0.2 -0.4 -0.6 -0.8 T\
80 100 120 140 160 180 200 220
He. 92 u 142
| 20
30 •
40
20
40
60
80
100 120 140 160 180 200 220 A,
Figure 14: Comparison of the fission half lives calculated in the fission model (upper figure see also Fig. 12) and in the Preformation Cluster M o d e l 1 8 . In both models the deformation of t h e fission fragments is not included completely.
than about 1 0 - 1 2 sec. The nuclear forces from the 146 Ba and 96 Sn cluster to the left and right from 10 Be lead to a softening of its potential and therefore to a somewhat smaller transition energy. Thus, if experimental results hold, one has discovered long living (w 1 0 - 1 2 sec) complex nuclear molecules. This is phantastic! Of course, I do immediately wonder whether such configurations do also exist in e.g. U + Cm soft encounters directly at the Coulomb barrier. This would have tremendous importance for the observation of the spontaneous vacuum decay 2 6 , for which "sticking giant molecules" with a lifetime of the order of 1 0 - 1 9 sec are needed. The nuclear physics of such heavy ion collisions at the Coulomb barrier (giant nuclear molecules) should indeed be investigated! As mentioned before there are other tri-molecular structures possible; some with 10 Be in the middle and both spherical or deformed clusters on both sides of 10 Be. The energy shift of the 10 Be-line should be smaller, if the outside clusters are deformed (smaller attraction •£> smaller softening of the potential)
60
Figure 15: Illustration of cold and hot (normal) fission identification through multiple 7coincidencs of photons from the fragments. The photons serve to identify the fragments.
and bigger, if they are spherical. Also other than 10 Be-clusters are expected to be in the middle. One is lead to the molecular doorway picture. Fig. 18 gives a schematic impression where within the potential landscape cluster-molecules are expected to appear, i.e. close to the scission configuration. Clearly, there will not be a single tri-molecular configuration, but a variety of three-body fragmentations leading to a spreading width of the tri-molecular state. This is schematically shown in Fig. 19. Finally, these tri-body nuclear molecules are expected to perform themselves rotational and vibrational (butterfly, whiggler, fi-, 7-type) modes. The energies were estimated by P. Hess et a l 2 4 ; for example rotational energies typically of the order of a few keV (4 keV, 9 keV, ...). A new molecular spectroscopy seems possible! Finally, I mention our recent theoretical investigations of the potential energy surface of the tri-body nuclear molecules, utilizing M3Y-
61
977.5
332.6
3368.0
814.7
181.1 *Ba
6
Sr
Figure 16: The 7 transitions of the three fission products of 2 5 2 Cf measured in coincidence. Various combinations of the coincidences were studied. T h e free 3368 keV line in 1 0 Be has recently been remeasured by Burggraf et al. 2 3 , confirming the value of the transition energy within 100 eV.
Figure 17: Typical linear cluster configuration leading to tripple fission of 252 Cf. T h e influence of both clusters leads to a softening of the l 0 B e potential and thus to a somewhat smaller transition energy. Some theoretical investigations indicate that the axial symmetry of this configuration might be broken (lower lefthand figure).
interactions, which yield broken axial symmetry for the tri-nucleus -molecule: The molecular minimum in the potential corresponds to the 10 Be sitting in the neck of the big fragments - see lower part of Fig. 15. The "cold valleys" in the collective potential energy surface are basic for understanding this exciting area of nuclear physics! It is a master example for understanding the structure of elementary matter, which is so important for other fields, especially astrophysics, but even more so for enriching our "Weltbild", i.e. the status of our understanding of the world around us. I am grateful to Dipl.-Phys. Thomas Biirvenich for helping me in the technical production of these proceedings.
62
deformation molecular states ground state tri - molecular minimum
Figure 18: Cluster molecules: Potential energy curve of a heavy nucleus showing schematically the location of groundstate, shape- and fission-isomeric states and of tri-molecular states.
spreading width
0°0 Oco OX)
146„
llL
96 c
Ba Be Sr 144D
10D
98„
Ba Be Sr ,42
Ba 10 Be 96 Sr Be^
spherical and deformed fragments
+2n
energy shift of characteristic cluster line may depend on its imbedding within the "big brothers"
Figure 19: Microstructure of tri-molecular states: Various tri-cluster configurations are spread out and mix with background states. Thus the tri-molecular state obtains a spreading width.
63
1. S.G: Nilsson et al. Phys. Lett. 28 B (1969) 458 Nucl. Phys. A 131 (1969) 1 Nucl. Phys. A 115 (1968) 545 2. U. Mosel, B. Fink and W. Greiner, Contribution to "Memorandum Hessischer Kernphysiker" Darmstadt, Frankfurt, Marburg (1966). 3. U. Mosel and W. Greiner, Z. f. Physik 217 (1968) 256, 222 (1968) 261 4. a) J. Grumann, U. Mosel, B. Fink and W. Greiner, Z. f. Physik 228 (1969) 371 b) J. Grumann, Th. Morovic, W. Greiner, Z. f, Naturforschung 26a (1971) 643 5. A. Sandulescu, R.K. Gupta, W. Scheid, W. Greiner, Phys. Lett. 60B (1976) 225 R.K. Gupta, A. Sandulescu, W. Greiner, Z. f. Naturforschung 32a (1977) 704 R.K. Gupta, A.Sandulescu and W. Greiner, Phys. Lett. 64B (1977) 257 R.K. Gupta, C. Parrulescu, A. Sandulescu, W. Greiner Z. f. Physik A283 (1977) 217 6. G. M. Ter-Akopian et al., Nucl. Phys. A255 (1975) 509 Yu.Ts. Oganessian et al., Nucl. Phys. A239 (1975) 353 and 157 7. D. Scharnweber, U. Mosel and W. Greiner, Phys. Rev. Lett 24 (1970) 601 U. Mosel, J. Maruhn and W. Greiner, Phys. Lett. 34B (1971) 587 8. G. Miinzenberg et al. Z. Physik A309 (1992) 89 S.Hofmann et al. Z. Phys A350 (1995) 277 and 288 9. R. K. Gupta, A. Sandulescu and Walter Greiner, Z. fur Naturforschung 32a (1977) 704 10. A. Sandulescu and Walter Greiner, Rep. Prog. Phys 55. 1423 (1992); A. Sandulescu, R. K. Gupta, W. Greiner, F. Carstoin and H. Horoi, Int. J. Mod. Phys. E l , 379 (1992) 11. A. Sobiczewski, Phys. of Part, and Nucl. 25, 295 (1994) 12. R. K. Gupta, G. Miinzenberg and W. Greiner, J. Phys. G: Nucl. Part. Phys. 23 (1997) L13 13. V. Ninov, K. E. Gregorich, W. Loveland, A. Ghiorso, D. C. Hoffman, D. M. Lee, H. Nitsche, W. J. Swiatecki, U. W. Kirbach, C. A. Laue, J. L. Adams, J. B. Patin, D. A. Shaughnessy, D. A. Strellis and P. A. Wilk, preprint 14. K. Rutz, M. Bender, T. Biirvenich, T. Schilling, P.-G. Reinhard, J.A. Maruhn, W. Greiner, Phys. Rev. C 56 (1997) 238. 15. B. Fricke and W. Greiner, Physics Lett 30B (1969) 317 B. Fricke, W. Greiner, J.T. Waber, Theor. Chim. Acta (Berlin) 21
64
(1971) 235 16. A. Sandulescu, D.N. Poenaru, W. Greiner, Sov. J. Part. Nucl. 11(6) (1980) 528 17. Harold Klein, thesis, Inst, fur Theoret. Physik, J.W. Goethe-Univ. Frankfurt a. M. (1992) Dietmar Schnabel, thesis, Inst, fur Theoret. Physik, J.W. Goethe-Univ. Frankfurt a.M. (1992) 18. D. Poenaru, J.A. Maruhn, W. Greiner, M. Ivascu, D. Mazilu and R. Gherghescu, Z. Physik A328 (1987) 309, Z. Physik K332 (1989) 291 19. E. K. Hulet, J. F. Wild, R. J. Dougan, R. W.Longheed, J. H. Landrum, A. D. Dougan, M. Schadel, R. L. Hahn, P. A. Baisden, C. M. Henderson, R. J. Dupzyk, K. Summerer, G. R. Bethune, Phys. Rev. Lett. 56 (1986) 313 20. K. Depta, W. Greiner, J. Maruhn, H.J. Wang, A. Sandulescu and R. Hermann, Intern. Journal of Modern Phys. A5, No. 20, (1990) 3901 K. Depta, R. Hermann, J.A. Maruhn and W. Greiner, in "Dynamics of Collective Phenomena", ed. P. David, World Scientific, Singapore (1987) 29
21. 22. 23.
24. 25. 26.
27.
28.
S. Cwiok, P. Rozmej, A. Sobiczewski, Z. Patyk, Nucl. Phys. A491 (1989) 281 A. Sandulescu and W. Greiner in discussions at Frankfurt with J. Hamilton (1992/1993) J.H. Hamilton, A.V. Ramaya et al. Journ. Phys. G 20 (1994) L85 - L89 B. Burggraf, K. Farzin, J. Grabis, Th. Last, E. Manthey, H. P. Trautvetter, C. Rolfs, Energy Shift of first excited state in wBe ?, accepted for publication in Journ. of. Phys. G P. Hess et al., Butterfly and Belly Dancer Modes in g6Sr + 10Be + 146Ba, Journal of Physics G, in print E.K. Hulet et al. Phys Rev C 40 (1989) 770. W. Greiner, B. Miiller, J. Rafelski, QED of Strong Fields, Springer Verlag, Heidelberg (1985). For a more recent review see W. Greiner, J. Reinhardt, Supercritical Fields in Heavy-Ion Physics, Proceedings of the 15th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, World Scientific (1998) P. Papazoglou, D. Zschiesche, S. Schramm, J. Schaffner-Bielich, H. Stocker, W. Greiner, nucl-th/9806087, accepted for publication in Phys. Rev. C. P. Papazoglou, PhD thesis, University of Frankfurt, 1998; C. Beckmann et al., in preparation
65
FISSION IN ( T H E LIGHT OF) A Q U A R K - G L U O N LIQUID THEORY OF T H E N U C L E O N S H. BOHR, Dept. of Physics, DTH, The Technical University of Denmark, B. 423, DK-2800 Lyngby, Denmark. E-mail: [email protected] A plasma theory is attempted to incorporate hadronic strings in a Quantum Chromo Dynamics (QCD) description of the nuclear structure based on quarks and gluons producing a three-body force giving rise to SU(3) singlet nucleon states. The interaction is treated in the mean field theory and the resulting many body system is analysed as a liquid with various phases and with the condition for fission in the water like phase.
1
Introduction
Some years ago a quark theoretical model 1 for the nucleus was proposed. It can explain gross features of the nuclear shell structure with only a few free parameters. The whole nucleus is described as a quark gas inside an MIT bag 2 with radius R. The effective interaction in this model contains two ingredients: 1) An infinite potential wall keeping the quark gas within the boundary of the overall bag. This is assumed to be due to a bag pressure as in the MIT bag model. Inside the bag the particles satisfy the Dirac equation (a • p + pm)ij)ai = etpai,
\x\ < R
With the boundary conditions for the quarks tpai, OL ' X
-T^-lpai = iPi>ai, \x\ = R \x\ The equilibrium condition of the MIT-model (B is the universal bag pressure) gives: _ i £ i = 4 7 rS J R 2 (~ =-) -» R ~ 1.3A1/3{fm}, 9R R
E(A,R)
=
4TTBR3
~ A
(A being the nucleon number) which is to be expected from data. The Dirac equation for the infinite potential well and the boundary condition gives for the single particle spectrum the well-known Mayer-Jensen scheme for nuclear shells with 3A particles.
66
2) A crucial pairing force is derived from instanton contributions in the QCD field theory. The pairing of quark isospins (flavour) i allows only total isospin to be zero, T = 0. For the one-instanton contribution the pairing force is a truely 2-body force. The expression is: Pi = Ocgtpi(x)(l +'y5)tp1(x)ip2{x)(l
+ ^5)t/f2(x)e12
where ey is antisymmetric in isospin and Oc is the SU(3) colour operator. This pairing force removes the far too big degeneracy in the 3A quark system, thus giving a correct shell structure. The normalization of the force, which means fixing the free parameters such as quark mass m and scale- parameter A in QCD, is adjusted to the A-resonance excitation. The total Hamiltonian in this model then becomes: H = HW + # ( 1 ) ,
H(0) = eNop + P2,
# ( 1 ) = Hkin - iNop
Here, H^ is diagonalized exactly while H^ is treated in first order perturbation theory. Nop is a number operator and e is the mean single particle energy. The infinite potential well and the pairing force describe remarkably well the shell structure of the nucleus. However the model should be considered as an approximation to the extreme view of quark structure with no nucleon formation. A mathematical result from ref. 1 means that in such a nuclear bag the vanishing of total colour enforces a division into 3-quark subsystems with vanishing colour each. However such subsystem might have its 3 quarks located at largely separated points in the nuclear bag with no obvious nucleon structure. Shortcomings of the model connected to this problem of missing localized nucleons are, for example, magnetic moments being too large for high spins 3 . 2
The baryonic string force in the nuclear bag
We shall now introduce a three body string force to the Hamiltonian H — H^ + H^ that can be responsible for nucleon formation. A string force proportional to the length between the quarks is a very dominating force so perturbation theory with the string force as a perturbation is not likely to succeed. On the other hand the strings will triplet-wise neutralize the colour leaving us with only dipole or quadropole moments like in water.
67
a)
b)
c)
Figure 1. Baryonic string-configurations, (a) The angles of a 3-quark cluster are all lesss t h a n 27t/3. (b)One of the angles of a 3-quark cluster is greater than 27r/3. (c) A 6-quark cluster.
The best we could hope for is if the previous scheme is basically unaltered, but leaving us with an overall colour effect that reproduces the basic assumption of the bag pressure or infinite potential well, and save the magnetic moment discrepancy. We shall now determine the baryonic string configuration. We have only limited information about nonperturbative string configurations. Lattice QCD gives, in the strong coupling limit, indication of colour tube strings between two heavy quarks. The string force is proportional to the minimal length L of the string: ^ = 2 ^
L
= ^
L
(1)
where a' is the Regge intercept and a is the string tension. The total 3-body force is then V3 = / dxidx2dx3tl)(xi)'ip{x2)rp(x3)Vstilj(xi)ilj(x2)ik{x3)
(2)
The two possible 4 configurations between 3 quarks in SU (3) - QCD are shown below, when we require SU(3) gauge invariance. Mesons are strings between a quark-antiquark pair. Let us dicuss some aspects of the baryonic configurations. Three points define a plane so we only have to consider plane geometry. The minimal configuration with a common vertex has all angles between strings equal 2ir/3 (Fig. 1 (a)) unless the interior angle Lx\x
68
Figure 2. Geometric derivation of the string force: the triangle of the string force and its sub-triangles
It is crucial that the configurations with a common vertex can be constructed in a gauge invariant way 4 . An operator which creates the states of such string configurations is:
M^i)ipj(x2)^k(x3){[[
U}ir{[[ Pi
U}js{Y[
P3
U}kterst
Pa
(Pi is the path from Xi to X4, e r s t is antisymmetric and U is unitary operator.) Colour neutrality gives also other possibilities: (E.g. A 6-body force, see fig. 3,(b,r,g stand for the three colours in SU(3))). 3
Derivation of the baryonic string force
We shall now derive the geometrical expression for the baryonic string force. We start with the triangle between the three quark positions from the configuration in Fig. 2 . The important point is now to express di + cfo +fifoin terms of the trivial lenghts, Zj, I2, hThe baryonic string force was from the start taken to be:
L-(dl+d2+d3)
(3)
69
It is easy to derive the expression for d^ + d2 + d3 using the area A of the triangle of the string force and its sub-areas. We have
co^ili^il
(4,
1A = — {d2d3 + did2 + didz) — l2l3 sin a
(5)
l\ = d\ + d\ + d2d3 l\ = d\ + d\ + drf3 ll = dl + d\ + d2d!
(6) (7) (8)
L 2 = (d1 +d2+
(9)
d3f = V3l2l3 sin a + |(Z 2 + Z2 + Z2)
L 2 = i(Z 2 + Z2 + Z2) + ^ ^ 2 Z | Z 2 + 2Z?Z2 + 2Z2Z2 -l\-l*2-li
(10)
If l\ = I2, +13 + l2l3 a simple calculation shows that L2 = (l3 +12)2. The sring force will therefore become L/(2ira') which is approximately equal to
~[h
+
(ID
h]
if Zi > l2, Zj > l3. If the angle opposite the side of length l\ is greater than 27r/3 we easily obtain the same expression (11) but here, of course, valid exactly. Thus formula (11) is a unifying expression for the minimal length in both configurations.
3.1
Average many-body effects of baryonic string force
In the mean field approximation we write V=
f dr'V(r')n(r')
(12)
We also have to make use of some expansion technics. Expansion: (Potential centered at 0, expansion around r = 0) V(r)=
fdr'V(r-r')n(r') = f V(r')n{r')dr'
+r- f
dr'Vr>V(r')n(r')
+ \ Idr' (r • V r <) 2 V{r')n{r') + •••
(13)
70
« V + ±r 2 ,4, where V = J dr'V(r')n(r') to two examples
(14)
and ~A=\j
dr''d%,V(r')n(r'). This is now applied
V1 = -e~Kr, V2 = ar. (15) r The mean field potential for 1) Yukawa and 2) stringforce then becomes rRo
V2 = n0a / Jo Vi = / dr& -e-Krn(r) J r so, altogether,
4
ATTRA
dr-vrr 3 = ano-j—p ~ a'^^oo o-4
= bn0 [ J0
(16)
drAnr = |bn 0 7ri^ « 6 A ^ - + • • • (17) d Ro
F 2 ( r ) = o'i4' + AAr +•••,
V^r) = b'A'+ • • •
(18)
giving rise to a residual potential. Clearly, V2(r) is sufficient for explaining the overall infinite well bag potential. In perturbation theory we can therefore write the total Hamiltonian decomposed into pairing force, string force, Hun etc: H = #<°> + HW, tf<°> = £Nop + V*airing, H™ = Hkin
- eNop + V2 (r)
or, keep V2{r) as a bag potential,
El = (Um,H™Um), * = £ ^ 4
Nuclear liquid systems
4 1} = {ET-^'
k
* m-
(19)
5
Since the exact many body problem is unsolvable let us try to make use of a "liquid" analogy since the three body configuration of Fig. 1 (a,b) very much look like water molecules. First the Maxwell-Boltzmann energy distribution for solid, liquid and gas is plotted (see Fig. 3) against a typical atomic pair potential. The liquid is most likely to represent the nucleus. In a liquid there are both molecular transport and configurational readjustments and the flow process is charaterized by both configurational and kinetic processes unlike gas and solids but like our nuclear quark model. The liquid has an abillity to form a free bounding surface and a corresponding surface tension. This is calculated below. First we will like to classify liquids to see what class the nucleus could belong to.
71
liquid
solid
Figure 3. Phase plot of the Maxwell-Boltzmann energy distribution.
4-1
Classification of liquids
6
1) Liquid metals Systems with central, symmetrical and non-saturating interaction. Central interaction makes a pair-wise decomposition possible so, for an Nbody system, the interacyion is given by: (Here we exclude three-body forces)
0(1,.... N) = i £ £ 4>(ij) = J2 4(ij) i
j
i>j
Unlike the inert gas, the liquid-metal pseudopotential is temperature and density dependent. 2) Homonuclear diatomics (N,0,H) and heteronuclear diatomics (CO). With negligible dipole moment. Cannot interact centrally. Systematic departure from centro-symmetric systems at low temperature. There are also polar molecules with pronounced electrostatic interaction (SO2) and molecules of great electrical and configurational symmetry such as water. 3) Liquid crystals. Deviation from spherical symmetry. Only compounds
72
with rod-like molecules form liquid crystals. Liquid crystals contain nontrivial topology. 4) Glasses. Frustrated systems. Systems characterized by abnormally high viscosities near the freezing point which hinders molecular rearrangement for the ordered crystaline phase. 5
Phases of nuclear liquid state
At temperatures higher than the quark deconfinement temperature (perhaps obtainable in heavy ion collisions) the string like structure dissolves and nuclear matter contains free quarks within a bounded region. The instanton vacuum structure could persist giving rise to a pairing force. Thus, nuclear matter at high temperature could be represented by a liquid metal phase, while nuclear matter at ordinary low temperature could be represented by a water like liquid phase. Let us therefore list various liquid properties such as surface tension calculated for these two phases. 5.1
Relativistic (quark) liquid metal phase
Degrees of freedom: rf = 2 • 2 • 3. Density: 3A
1 v-^
/ \
4?rfci
p
y 1
Fermi energy: e% = y/mPc4 + h k'pd2 Pairing: § • 2 • 2 • 3 / f c F ($fc1_fc2<5ri_r2<5Tl_T2 ~ a. • $ Surface tension: 7 = | ^ - , 70 = ^l3nomc2 Surface energy: Es = 4irR2i'yo Density profile: When interaction increases then ls decreases 5.2
Molecular (nucleon) water phase
Degrees of freedom: 17* = 2-2 Density: n* = £ = £ £ n ( p ) p < p / =
^j^f
73
valence band conducting band
a)
\
conducting band
b)
c)
Figure 4. T h e three phases pictured as a conduction bands, (a) DiaJectric insulator . (b) Metal, (c) Glass (amorphous), oriented bonding, U 7 < E\ — E2
1/3
Fermi momentum: kF = ( ^2_ . n* j Fermi energy:
Tfkl?
,e-* r »
^ = £F = 42m* T§--32 Fermi pressure. Yukawa String:
• ft2 z*F = a* = F
r
/6TT 2
X ' n*
2m* V */*
/
/6TT 2 X - q[ n*
\V*
J
= an*2/3 - /?n* 1 / 3 e- fcr - + 7TI*- 1 / 3
-fcr,
74
Surface tension:
7* = fiX a
2 Q n * - l / 3 _ P_n*-2/3e-krs
~iW?an -1/3 4 * 3 Surface energy: E* = ATTR\Y5.3
=
+
I n .-4/3
I/>*5/3a 2s
Volume energy: E* = J dAn(A).
Comparison between the two liquid phases Ea
7o
la n0mc22m*
f n* \2/3
_ ls / 3
\
Thus the free surface tension of the metal phase is twice as large the free surface tension of the molecular phase. 5.4
Glass phase
Because of its oriented boundings the nuclear "clustered" matter share similarities with glass. Photons can pass through glass and so can pions through nuclei. Such a glass-phase is obvious in nuclear matter when many strings bind together in a network or cluster (see Fig. 4 (c)). Dialectric insulator: Filled band because every particle participate in bonding. Fig. ?? (a.b.c) represents the electron band distribution in insulator, metal and glass. It shows that photons cannot pass through metal but through the other media. 5.5
Condition for fission
In a liquid quark model of the nucleus it is possible to make a discussion about the fission of nuclei. Fission is very much dependent on the surface tension which varies according to the type of liquid present. The condition for fission is usually written as ECOu\/2EBUrf > 1 and since Esur{ in the metal phase is about 3 times larger than the water phase the latter phase has higher probability for providing fission.
6
Conclusion 1) - String Force: (3 body force) can be nicely approximated by 2 body force and roughly explane the bag potential.
75
2) - Liquid Models: can be used for describing the various possible phases of the quark nuclear system. 3) - Nuclear Fission: described by liquid water drop processes with critical limit given by 2Esurf
Ci A ~
Surface-energy was calculated in the liquid metal quark phase to be approximately 3 times larger than the nucleon water phase and since Ecayi\ is expected (on average, or pr. shell) to be the same in the two cases, the stability limit X is pushed up to a higher value for a (free) quark phase, realised in heavy ion collisions with T > Tc = 170 Mev, than in the lower-energy nucleon (water) phase. Acknowledgments We wish to thank Prof. Kostadinov, Prof. Petry, Prof. Bleuler, Prof. Schutte and, Prof. Yazaki and J. Providencia for helpfull discussions, and the organizing committee of the conference and the physics department secretariate of Coimbra University for help with the manuscript.
References 1. H.R. Petry, K. Narain, H. Bohr and K. Bleuler, Phys. Lett. 159 B ( 1985) 363 2. A. Chodos et al., Phys. Rev. D 9, (1974), 3471 3. H.R. Petry, "Quarks and Nuclear Structure", Lecture Notes in Phys. vol. 197 ed. by K. Bleuler (Springer, Berlin 1983); A. Arima, K. Yazaki and H. Bohr: "A quark Shell Model Calculation of Nuclear Magnetic Moments". Tokyo University Preprint, July 1986. 4. J. Carlson et al., Phys. Rev. D 27, 233(1983); R. Sommer et al., Nucl. Phys. B 267 (1986), 531. 5. Discussions with Dr. Kostadinov are acknowledged. 6. See e.g. J.A. Pryde: "The liquid state" Hutchinson University Library (1966).
76 STATISTICAL M O D E L I N G OF N U C L E A R
Department
J. W . C L A R K of Physics, Washington University, MO 63130 USA E-mail: [email protected]
SYSTEMATICS
St.
Louis,
E . M A V R O M M A T I S , S. A T H A N A S S O P O U L O S , A. D A K O S Physics Department, University of Athens, GR-15771 Athens, Greece E-mail: [email protected]
Department
of Physics,
UMIST, E-mail:
K. G E R N O T H P. O. Box 88, Manchester Kingdom [email protected]
M60 1QD,
United
Statistical modeling of d a t a sets by neural-network techniques is offered as an alternative to traditional semiempirical approaches to global modeling of nuclear properties. New results are presented to support the position that such novel techniques can rival conventional theory in predictive power, if not in economy of description. Examples include the statistical inference of atomic masses and /3-decay halflives based on the information contained in existing databases. Neural network modeling, as well as other statistical strategies based on new algorithms for artificial intelligence, may prove to be a useful asset in the further exploration of nuclear phenomena far from stability.
1
Introduction
There is currently a strong incentive for the development of global models of nuclear properties, driven by the production of many new nuclei at radioactive beam facilities and by the needs of complex reaction-network calculations involved in models of nucleosynthesis. At the same time, recent developments in statistical analysis and inference, especially those based on neural-network and other adaptive techniques, present novel opportunities for global modeling that exploit the rich collection of nuclear data currently available. With suitable coding of input and output variables, multilayer feedforward neural networks with pairwise couplings, trained by backpropagation and related algorithms 1 , 2 , 3 , are capable of learning from examples in the nuclear database and making predictions for properties of "novel" nuclides outside the training set 4 . Neural network models have been developed for a number of properties, including atomic masses 5,6,7 , neutron separation energies 5 , ground-state spins and parities 8 , 6 , branching probabilities into different decay channels 9 , and halflives for f3~ decay 10 . In terms of predictive accuracy, as measured on
77 test nuclei not seen during training, neural-network models can compete with traditional phenomenological and semi-microscopic global models, although t h e number of adjustable parameters (connection weights) is generally much larger. Special attention is called to the study 9 in which neural networks were constructed to learn and predict whether a nuclide is stable in its ground state, and if not, to generate the branching probabilities among a-, /?~-, electroncapture (lumped with /3 + -decay), and fission decay modes. Global network models were created which show remarkable quantitative performance in comparison with the capabilities of standard theoretical approaches. In fitting and prediction of the probabilities of occurrence of the six modes (including stability) in nuclides belonging to the training set and to the test set reserved for prediction, trained networks display errors below 5% and 15%, respectively, as measured by the average, over the relevant sets, of the maximum departure of the o u t p u t branching probabilities from their target values. Here we report improved results for two applications, namely to construction of the atomic-mass table and to determination of /?~-decay lifetimes. T h e procedures involved in neural-network statistical modeling based on multilayer feedforward networks trained by example ("multilayer perceptrons") have been thoroughly documented in Ref. 4, which also discusses the strengths and weaknesses of this approach. Rather t h a n repeat such information, we shall simply indicate the places where essential modifications are made. T h e new experiments on learning and prediction of atomic masses with multilayer perceptrons are based on a new input coding scheme and a modified backpropagation training algorithm t h a t helps the system avoid local m i n i m a of the cost surface. T h e results show much better extrapability (i.e. predictive accuracy for new nuclei away from the stable valley) t h a n has been found in earlier neural-network s t u d i e s 5 ' 6 , 7 . Moreover, this improvement is achieved with a greatly reduced number of connection weights. These findings encourage a redoubled effort toward developing reliable mass predictors. Earlier work 1 0 on /?-decay is also refined, to good effect. Multilayer feedforward networks trained to predict /?-decay halflives from Z, N, and the Q-value of the decay are now performing within the range of accuracy attained by conventional quantum-mechanical models.
2
Global M o d e l s of A t o m i c - M a s s D a t a
A neural-network model will here consist of a collection of neuron-like units arranged in layers, with information flowing only in the forward direction through connections between the units in successive layers. Input d a t a (e.g. values of
78
Z and TV) are encoded in the activities of the input-layer units; the activities of units in the succeeding layers are updated in sequence; and a coded version of the value computed by the network for the property in question (e.g. the mass excess) appears in the activities of the output-layer units. The architecture of a given net is summarized in the notation (I+H1+H2 + • —|- HL + 0)[P], where /, Hi, and O are integers that indicate, respectively, the numbers of neuron-like units in the input layer, the ith intermediate (or "hidden") layer, and the output layer. The total number of connection-weight and bias parameters is denoted by P. Once the gross architecture (number of layers, number of units in each layer) is specified, the behavior of the network - in particular, its response to each input pattern, each example from the database - is entirely determined by the real-number weights of the connections between the units (and the biases of the units). As the system is exposed to a set of training patterns, these weights (and biases) are incrementally altered so as to minimize a cost function. Ordinarily the cost function is taken as the sum of the squared difference between target and actual output activity, the sum being carried out over all output units and over all patterns in the training ensemble. The standard training algorithm accomplishing this goal is called "vanilla" backpropagation, a gradient-descent optimization routine which includes a "momentum" term to permit rapid learning without wild oscillations in weight space. In our most recent mass studies, we institute a modification in the weight-update rule that recursively allows earlier patterns of the current epoch (the current pass through the training set) to exert greater influence on the training than is the case for vanilla backpropagation. Experience has shown that this new learning rule generally yields improved results (though not in all instances). In supervised training of neural networks, there is always the question of when to terminate the process. If the network is trained for too short a period, the training data will be poorly fitted. On the other hand, if the training is continued too long, generalization to new examples (i.e., prediction) will suffer, since the "overtrained" network will be specialized to the peculiarities of the training set that has been employed. Thus some reasonable compromise must be struck between the requirements of an accurate fit and good prediction. In the current round of computer experiments, w.e have adopted the following stopping criterion. A given training run consists of a relatively large number of epochs, specified beforehand. During such a run, we not only record the cost function for the patterns in the training set, we also monitor the cost function for a separate validation set of nuclei whose masses are known. The "trained" network model resulting from a given run is taken as the one with connection
79 weights yielding the smallest value of the cost function on the validation set, over the full course of the run. While the members of the validation set are not used in the weight updates of the backpropagation learning rule, vanilla or modified, they clearly do influence the choice of model. Therefore, accuracy on the validation set cannot strictly be regarded as a measure of predictive performance, although in practice it m a y nevertheless provide a useful (and probably faithful) indicator of this aspect of the model. To obtain an unimpeachable measure of predictive performance, still a third set of examples is needed: a test set t h a t is never referred to during the training process. We are now ready to specify the d a t a sets used in our current mass studies. T h e primary set is a database designated MN consisting of 1323 "old" (O) experimental masses which the 1981 Moller-Nix model n were designed to reproduce, together with 351 "new" (N) experimental masses t h a t lie mostly beyond the edges of the 1981 d a t a collection as viewed in the N — Z plane. This database, with this segmentation, has been used in previous mass-modeling exercises with neural nets. As discussed in Ref. 12, the O and N d a t a sets were formed to quantify the extrapolation capability (extrapability) of different global models of atomic masses. In creating neural-network models, the old (O) masses constitute the training set, while the new (N) masses are employed as a validation set or - in the earlier treatments - as a test set (or prediction set). In addition to the MN database, we make use of another d a t a set composed of the mass-excess values of 158 newer nuclides drawn from the NUBASE evaluation of nuclear and decay properties; generally these nuclides reside still further from the stable valley t h a n those of the N subset. This secondary d a t a set, which we call N B , has been used exclusively as a test set; it is not consulted during the training phase. A third set of examples, which also remains untouched during the training process, consists of 10 "even newer" nuclides of rare-earth elements; their masses have recently been measured with the I S O L T R A P mass spectrometer 1 3 . We label this last set ISO. Three different input coding schemes are relevant to our considerations. In the first 5 , 6 ' s , the input layer consists of sixteen "on-off" units having activity levels 0 or 1. Eight units for Z and eight for N serve to encode the proton and neutron numbers in binary and permit the t r e a t m e n t of Z and N values u p to 255, which is more than sufficient to cover the interesting physical range of input patterns v — (Z, N). This scheme facilitates learning of quantal properties (pairing, shell structure) t h a t depend on the integral nature of Z and N. T h e second scheme utilizes analog coding of Z and N in terms of the activities of only two analog input neurons, which, however, are aided by two further on-off input units t h a t encode the parity (even or odd) of Z and N. T h u s the network is again given information about the integral character of Z
80 and N. In the third scheme, the 16-unit binary-coding input array of the first design is supplemented by two additional units encoding Z and N in analog. For all three choices of input coding, the mass computed by the network is represented by the activity of a single analog output unit. Three different prescriptions have been used to scale the activities of the analog units in the inp u t and o u t p u t layers to the interval [0,1]; the details, not particularly relevant here, will be presented elsewhere. We need only remark t h a t proper attention to the dynamical ranges of the Z, N, and mass-excess variables allows more precise study of new nuclei far from the stability line. Quantitative j u d g m e n t s of network performance in learning, validation, and prediction are m a d e in t e r m s of two different measures, evaluated separately for the three d a t a sets specified above. T h e chief quality measure is the root-mean-square deviation
81 Table 1: Comparison of neural-network models of atomic mass d a t a with other models based on conventional nuclear theory. Learning (fitting) and generalization (prediction) refer to the database MN[1323(0)-35l(N)]. Model Neural Network {0 + H1+H2 + -- + HL)[P] or Conventional (16+10+10+10+1) [401] Z & N in binary (18+10+10+10+1) [421] Z & N in binary and analog (4+10+10+10+l)*[281] Z h N in analog and parity (4+10+10+10+1)** [273] Z &; N in analog and parity ( 4 + 1 0 + 1 0 + 1 0 + l)t[28l] Z & N in analog and parity (18+10+10+10+1) [421] Z & N in binary, A &Z-N in analog (4+40+l)[245] MSller et al. 1 4
Learn ing Mode
Validation (v) or Prediction (p) Mode
°"rms (MeV) 0.393
Recalled Patterns 1172/1323
0.331
1187
2.199 (v)
272
0.491
1141
1.416 (v)
280
0.617
1095
1.209 (v)
284
0.453
1242
1.200 (v)
298
5.981 (p)
0.828
1.068 0.673
— —
3.036 (p) 0.735 (p)
— —
MN database as a training set. T h e best models from two such investigations are included in Table 1. T h e network listed in row 6, having architecture (18+10+10+10+1)[421], was constructed by Gernoth et al. 6 using vanilla backpropagation, with binary encoding of Z and N together with analog encoding of the atomic mass number A and the neutron excess N — Z. T h e three-layer net (4 + 40+-l)[245] was provided by K a l m a n 7 , who employed analog coding of Z and N and auxiliary parity units for these variables. T h e input p a t t e r n s were pre-processed by singular-value decomposition, and the network was trained with a Powell-update conjugate-gradient optimization algorithm. T h e gold standard of quality in global mass modeling is presently set by the macroscopic/microscopic theoretical t r e a t m e n t of Moller, Nix, and collaborators 1 4 , for which rms values are entered in the last row of Table 1. Let us consider the rms error figures for the "cross" network model, in comparison with this standard. T h e result
82
the models we have developed. The value of crTms obtained for the NB set with network (4 + 10 + 10 + 10 + +l) t [281] is 1.462 MeV, which is to be compared with the figure 0.697 MeV attained by the FRDM macroscopic/microscopic model of Ref. 14. Another strong test of the predictive quality of our models is possible for 10 rare-earth nuclides of the ISO set, which are not contained in the O, N, or NB sets. The rms error for these nuclides is found to be 0.963 MeV and 0.500 MeV for the "cross" network and for the FRDM model, respectively. Based on these tests, it should be evident that the current generation of neural-network models of the mass table represents a significant advance toward extrapability levels competitive with those reached by the best traditional global models rooted in quantum theory. 3
Global Models of f3~ -Decay Halflives
Neural-network statistical methodology is being applied to the systematics of nuclear decay, and in particular to the the important problem of predicting the halflives Ti of nuclear ground states that decay 100% by the (3~ mode. A first effort in this direction is described in Ref. 10. We present here some results of a continuation of this work. Since the relevant experimental halflives vary over 26 orders of magnitude, the target variable in these studies is taken to be l n 2 \ . Vanilla backpropagation, involving a mean-square cost function, logistic activation functions, and a momentum term in the update rule is employed for on-line training of a variety of multilayer feedforward models. The cost function is N
1
c=-Y
V-\
2 ^>r ln TTnodel (,,\ J l/2 \V>
(1)
where v indexes the examples (input patterns) in the relevant set of N samples. Each training run involves a pre-set number of epochs, and the weights that are kept after each run are taken as those yielding the best (smallest) value of the "Klapdor" error measure (X)K achieved for the training set during the run. Referring for example to Staudt et al. 1 5 , this error measure is defined as 1
N N
(2)
where the sum is performed over the training or test set as appropriate and the symbol [a, 6]> stands for the ratio formed from its arguments a, b in such a way that it is always larger than 1.
83 Table 2: Assessment of the ability of the selected neural-network model of type ( 1 7 + 10 + 1)[191] to reproduce experimental values of /3 — -decay half-lives, in comparison with traditional models of Moller et al. 1 7 and Homma et al. 1 6 . For good performance, the quantities M< 10 ) = I O ^ M and a i l 0 ' should be as small as possible. friexp
Lear m n g
-M/2 (sec) < 1
< 10
< 100
< 1000
o-o o-e e-e o-o o-e e-e o-o o-e e-e o-o o-e e-e
M< 10 ) 1.15 1.07 1.61 1.17 1.04 1.19 1.18 1.05 1.19 1.19 0.98 1.14
2.27 2.03 1.71 2.25 1.91 2.09 2.18 1.93 1.97 2.13 1.99 1.95
Predi ction M( 10 > 2.05 1.08 1.79 2.26 1.19 1.31 1.76 1.12 0.98 2.22 1.22 0.93
0
M
2.31 2.38 2.71 5.42 2.44 2.30 5.19 3.15 2.67 6.25 5.50 4.78
Moller et al. M( 1 0 ) 0.59 0.59 3.84 0.76 0.78 2.50 2.33 1.11 2.61 3.50 2.77 6.86
2.91 2.64 3.08 8.83 4.81 4.13 49.19 9.45 4.75 72.02 71.50 58.48
Homma et al. 1.75 0.60 1.15 1.89 0.92 1.01 3.15 1.07 1.13 3.02 1.10 1.39
4.96 2.24 2.36 4.60 3.84 2.93 10.51 4.29 3.58 10.25 5.55 6.10
T h e raw database used in the beta-decay modeling experiments consists of all pertinent d a t a available in early 1995 from the Brookhaven National Nuclear D a t a Center, encompassing a total of 766 examples of single-mode decay by the P~ channel. T h e halflives in this collection range from 0.15 x 1 0 - 2 sec for ^ N a to 0.2932 x 10 2 4 sec for ^ C d . In our most recent modeling efforts, we have narrowed attention to the subset of cases in which the decay is from the ground state and has a lifetime not longer t h a n 10 6 y. T h u s we delete all examples with longer lifetimes, as well as a few examples of isomeric decays, arriving at a truncated d a t a set of 692 nuclides, of which 518 are reserved for training and 174 are used to test predictive acuity. This choice of database permits reasonable comparisons to be m a d e with the results from traditional global models of /?"" halflives (see Refs. 16 and 17 and literature cited therein). We implement binary coding of Z and N at the input layer, choosing the same scheme as used in some of the mass models. To the operative bank of 16 on-off neurons, we append an additional analog input unit t h a t encodes the Q-value of the decay as a floating-point variable. A single analog o u t p u t unit generates the coded value of In T±. t h a t the network computes for the input nuclide. % Among the m a n y network models constructed and tested, we single out for further attention a network with architecture (17 + 10 + 1)[191], which demonstrated, overall, the best behavior in the predictive mode. Table 2 pro-
84 vides a detailed comparison of the performance of this model with state-ofthe-art conventional global models recently developed by H o m m a et a l . 1 6 and Moller et a l . 1 7 T h e figures of merit t h a t have been adopted for the models evaluated in this table are those quoted by Moller et a l . 1 7 , namely the quantity M ( i o ) = 1 0 M derived f rom t n e mean error , M
N
= T7 ^2 v=\
T-'modelfjA lo
SlO rv,
r„ =
*£2 , 1/2 I '
(3)
along with the quantity aM ' = 10CTM derived from the standard deviation 1/2 CM
from the mean. Ideally, both measures should be as small as possible. The comparison is broken down according to odd-odd, odd-even/even-odd, and even-even nuclear subclasses and according to different experimental lifetime ranges. T h e level of performance displayed by the neural-network model is similar to (and in some cases better than) t h a t of the traditional models. Two caveats should accompany any appraisal of the relative merits of neural-net and conventional approaches. On the one hand, comparison is hindered by the lack of a clear distinction between the aspects of fitting and prediction in the traditional treatments; and on the other, by the fact t h a t the neural-network model has m a n y more adjustable parameters t h a n the traditional models. At any rate, the good performance of the 1 7 + 1 0 + 1 network model is clearly demonstrated in detailed comparisons with the experimental j3~ halflives of a set of 10 Cu isotopes and of a set of 10 nuclides found on or near the r-process p a t h . Typically, agreement well within an order of magnitude is obtained and usually within a factor 2. These new findings suggest t h a t it m a y be fruitful to seek further improvements of network performance in the /?-lifetime problem and to extend the approach to other decay modes, notably a-decay. Acknowledgments This research has been supported in part by the -U.S. National Science Foundation under Grant No. PHY-9900173. References 1. J. Hertz, A. Krogh, and R. G. Palmer, Introduction to the Theory Neural Computation (Addison-Wesley, Redwood City, CA, 1991).
of
85
2. J. W. Clark, Phys. Med. Biol. 36, 1259 (1991). 3. S. Haykin, Neural Networks: A Comprehensive Foundation (McMillan, New York, 1994). 4. J. W. Clark, T. Lindenau, and M. L. Ristig, Scientific Applications of Neural Nets, Lecture Notes in Physics Vol. 522 (Springer-Verlag, Heidelberg, 1999). 5. S. Gazula, J. W. Clark, and H. Bohr, Nucl. Phys. A540, 1 (1992). 6. K. A. Gernoth, J. W. Clark, J. S. Prater, and H. Bohr, Phys. Lett. B300, 1 (1993). 7. K. A. Gernoth and J. W. Clark, Comput. Phys. Commun. 88, 1 (1995). 8. J. W. Clark, S. Gazula, K. A. Gernoth, J. Hasenbein, J. Prater, and H. Bohr, in Recent Progress in Many-Body Theories, Vol. 3, T. L. Ainsworth, C. E. Campbell, B. E. Clements, and E. Krotscheck, eds. (Plenum, New York, 1992), p. 371. 9. K. A. Gernoth and J. W. Clark, Neural Networks B8, 291 (1995). 10. E. Mavrommatis, A. Dakos, K. A. Gernoth, and J. W. Clark, Condensed Matter Theories, Vol. 13, J. da Providencia and F. B. Malik, eds. (Nova Science Publishers, Commack, NY, 1998), p. 423. 11. P. Moller and J. R. Nix, At. Data Nucl. Data Tables 26, 165 (1981). 12. P. Moller and J. R. Nix, J. Phys. G 20, 1681 (1994). 13. D. Beck et al., Nucl. Phys. A626, 343c (1997). 14. P. Moller, J. R Nix, W. D. Myers, and W. J. Swiatecki, At. Data Nucl. Data Tables 59, 185 (1995). 15. A. Staudt, E. Bender, K. Muto, and H. V. Klapdor, At. Data Nucl. Data Tables 44, 80 (1990). 16. H. Homma, E.Bender, M. Hirsch, K. Muto, H. V. Klapdor-Kleingrothaus and T. Oda, Phys. Rev. C 54, 2972 (1996). 17. P. Moller, J. R. Nix and K. L. Kratz, At. Data Nucl. Data Tables 66, 131 (1997). eject
86 TOROIDAL STRUCTURES IN LIGHT NUCLEI
A. A R R I A G A Centra de Fisica Nuclear da Universidade de Lisboa, 1649-003 Lisboa, Dep. de Fisica da Faculdade de Ciencias da Universidade de Lisboa, Lisboa, Portugal E-mail: [email protected]
Portugal 1749-016
The main purpose of this contribution is to address very interesting nuclear structures that have been predicted for light nuclei, within a nonrelativistic framework, and whose experimental evidence is exhibited by measured reaction observables, and might be further exploited by additional measurements.
1
Introduction
Quantum Chromo Dynamics is vue, at present, as the fundamental theory to describe strong interactions. However, its nonperturbative character at low energies has been halting its ability to quantitatively describe nuclear structure. Hence the development of effective calculable methods is very important, for they may bring insights and estimates useful to progress the fundamental theory. Furthermore, due to quark confinement, the genuine QCD degrees of freedom, quarks and gluons, are not explicit at these regimes of energy, making effective hadronic degrees of freedom the appropriate to work with. Conventional effective nonrelativistic models are formulated in the Schrodinger equation framework and consider an Hamiltonian containing a kinetic energy operator and effective two and three-body potentials. With these interactions, which exhibit an intrincate spin-isospin structure, the Schrodinger equation, for A > 3, is very hard to solve. Several thecniques have been developed recently, namely Faddeev-Yakubovsky 1, hypersphericalharmonics 2 and Quantum Monte Carlo (QMC) methods 3 | 4 . While the first two are restricted to systems with A < 4, results for the binding energies of nuclei with A < 8 have been obtained by Pudliner et al. 5 ' 6 with QMC methods. Although these effective nonrelativistic models involve complex calculations due to the spin-isospin structure of the interactions, they are, in terms of principles, a simple approach to nature. Nevertheless, it is throughoutly documented that the predictions and results obtained in this framework have an impressive agreement with data. The main purpose of this contribution is to evidence the best results obtained so far for light nuclei and to address, in particular, very interesting
87 nuclear structures t h a t have been predicted, and whose experimental evidence is exhibited by measured reaction observables and might be further exploited by additional measurements 7 . In the following sections we present the hamiltonian and wave functions t h a t have been considered, the results obtained using QMC methods, and the nuclear structures induced by the interactions. 2 2.1
The Nonrelativistic Framework The Nonrelativistic
Hamiltonian
Typically, the nonrelativistic hamiltonian is given by
H
"* = E in + E +»* + E *>* • i
i<j
t1)
i<j
where the first term represents the kinetic energy operator and Vij the NN interaction operator. This potential satisfies to general properties determined by the symmetries of the strong interaction, to which in variance and conservation principles are associated: Galilei, spacial translations and rotations, time reversibility and isopsin rotations. We note, however, t h a t the isospin conservation is only approximated and t h a t there are recent interaction models t h a t take into account the small violation 8 . Due to the remarkably small mass of the pion, the long range interaction between nucleons is believed to be mediated only by pion exchange, meaning t h a t the standard asymptotic region of the NN interaction is derived from the nonrelativistic limit of the one-pion exchange diagram. There are some other properties of the interaction known from experiment and interpreted in terms of one-boson-exchange models: the interaction is short ranged, attractive in the asymptotic and intermediate regions, contains a repulsive core and spin-orbit and tensor components. It is the delicate interplay of the repulsive core and of the tensor component t h a t produces the nuclear structures t h a t are presented in Sec. 3. Despite of all the constraints mentioned above, there is still freedom in designing the intermediate and short range parts of the interaction, and different phenomenological models have been considered to account for effective representations of multiple meson exchange, relativistic effects and subnucleonic degrees of freedom. The parameters involved in these models are adjusted through fits to the experimental d a t a of the A = 2 system, both NN phase shifts and deuteron observables. T h e last term in Eq. 1, Vijk, represents the three-body interaction operator. As in the previous case, contains an asymptotic region which is the
88
nonrelativistic limit of the two-pion exchange diagram, and phenomenological intermediate and short range parts,whose parameters are adjusted to the experimental values of the binding energy of 3 H and density of nuclear matter. Due to large cancellation between kinetic and two-body interaction energies, Vijk can have significant effects on binding energies 9 , but its effect on ground state wave functions is much smaller than that of Vij. 2.2
The Nonrelativistic Wave Function
State of the art nonrelativistic variational wave functions have the form
|*„>= 1+ £ Fijk\ [sllFiA \
i<j
)
\
i<j
|#),
1D
:
(2)
J
where a symmetrized product of the two-body correlation operators, Fij, and a sum of three-body correlation operators, F,jk, act on the fully antisymmetric uncorrelated wave function $ . These two- and three-body correlations induce the effects of the interactions on the wave function. Fij involves products of operators, of the same type of the ones present in the interaction, and correlation functions satisfying Schrodinger-like two-body equations with appropriate boundary conditions. Their solutions comprise asymptotic parts which differ very much from nucleus to nucleus, and short range behaviors that are very similar to all nuclei 4 . 2.3
QMC Results for Light Nuclei
Variational Monte Carlo calculations are based on the well known variational principle:
6<»Slg-^l»-> = 0 (*J|*«)
(3)
where Ev and \t„ are the variational energy and wave function respectively. In practice a certain class of wave functions is considered in the minimization process, and here we refer to the class of functions discussed in the previous subsection. These wave functions contain variational parameters which are involved in the correlation functions 10 . The variational wave functions can be further improved by the Green's Function Monte Carlo (GFMC) method 3 . In fact, the true ground state wave function of a given Hamiltonian is obtained in the following way:
89 e-V*-B')T\*v)
| * o ) = lim
(4)
In practice the parameter r is mcresed until convergence is achieved. The ground state energy estimate is then given by:
E(T)
=
(9{T/2)\H\*(T/2))
(5)
<*(r/2)|*(r/2)>
Results for the binding energies of light nuclei are displayed in Tab. 1. These results, obtained by R. B. Wiringa et. al. using the Argonne v\g and the Urbana IX interactions n , have a good agreement with experimental data, although a small systematic underbinding can be seen for A > 6. Table 1. GFMC results for light nuclei, obtained by R. B. Wiringa et. al. with the Argonne «18 and the Urbana IX interactions 1 1 . The results are in MeV.
l
E,exp
3
H -2.2246 -2.2246
'H -8.48 8.47(1)
4
#e -28.30 -28.34(4)
''He -31.99 -28.11(9)
l
-39.24 -37.78(14)
Be -56.50 -54.44(19)
The Nuclear Structures
Although the results for the binding energies shown above might be slightly improved, either by better interaction models or more sophisticated numerical calculations, the existence of a repulsive core and of a tensor term in the NN interaction seems to be well established. Independently of the details of these two pieces of the potential, their main characteristics induce very interesting short range nuclear structures. In the case of the deuteron these structures are most obvious in the one-nucleon density distribution which is related with its wave function by: Pi (?)
16|* Af,, (2f)| 2
(6)
where Mi is the spin projection, v the one-nucleon coordinate in the center of mass reference frame and
90
Figs. 1 and 2 display the density distributions in the xz plan for Md = 0 and Md = ± 1 respectively 7 . The three-dimensional distribution, pMd, can be obtained by rotating these distributions about the z axis. They are represented by equidensity surfaces shown in Fig. 3 for pMd = 0.24 fm - 3 , the two top figures, and for pMi = 0.08 fm~ 3 , the two bottom ones. The figures labeled with A correspond to Md = ±1 and with B to Md = 0. The four figures are in the same scale. The strong anisotropy of these shapes is due to the tensor force, while the central hole in the toroidal structures (figures B) and the existence of disconnected surfaces in the two other structures (figures A) is a consequence of the repulsive core. The calculated diameter of the maximum density torus is d ~ 1 fm and the maximum thickness along the z axis calculated at half maximum density is t ~ 0.88 fm. These density distributions allow an interpretation of the structure of the deuteron electromagnetic form factors, and consequentely of the T20 observable. In fact, it can be shown that the first minimum of T20 is related with d and the known experimental value is consistent with 1 fm. On the other hand, the first maximum, not yet experimentally located, provides a mesure of t '.
Figure 1. The deuteron density pad obtained from the Argonne uig interaction 7 . The peaks locate at z = 0,x = ±d/2
Similar structures can be found in other light nuclei, by considering twocluster density distribution functions. We define the wave function of the cluster configuration A, B inside a nucleus C\ \$M^MB.MC ^ ^ y .
91
Figure 2. The deuteron density p|f obtained from the Argonne i'ig interaction 7 . The peaks locate at x = 0,z = ± d / 2 .
Figure 3. Equidensity surfaces in the deuteron at values of the density of 0.24 fm~ 3 the two top figures, a n d 0.08 f m ~ 3 the two bottom ones. The shapes A correspond to M
< ^
M B
'
M C
( ^ , B ) = ^M,^AiB?A,B)\^c)xAXB
(7)
92
where V%A and * ^ B are the wave functions of clusters A and B respectively, XA and xB the spin functions of clusters A and B respectively, and r^ g the two-cluster relative coordinate. The two-cluster density distribution function is finally given by: MA,MBMC(~
PA,B
\
\rA,B)
MAMB I* A,B
'Mc(rA,B)\2
(8)
As an example, Figs. 4 and 5 show the dp (deuteron-proton) density distribution function inside 3He, for Md = 1, Mp — —1/2, M3He = 1/2 and Ma = 0, Mp = 1/2, Mzne = 1/2 respectively 7 . As before, the threedimensional distribution function is obtained by rotations of these surfaces about the z axis, and are represented by equidensity surfaces. In the first situation, the deuteron cluster having Md — 1 assumes a shape of the type A of Fig. 3, and, as can be concluded from Fig. 4, the proton tends to stay along the xy plan, producing a very compact cluster configuration. In the second situation, the deuteron cluster having Md = 0 assumes a torus shape (type B of Fig. 3), and the proton tends to stay along the z axis, as can be seen from Fig. 5, generating again a very compact cluster configuration.
*dp ( f m )
Figure 4. Density distribution of dp cluster in 3 He, with Md = 1, Mp = —1/2, MzHe — 1/2, obtained from the Argonne v\$ interaction . The peaks locate at Zdp = 0,Xdp ~ ±1 fm.
93
Figure 5. Density distribution of dp cluster in 3He, with Md = 0, M p = 1/2, MzHe = 1/2, obtained from the Argonne v\& interaction . The peaks locate at x^p = 0,2dp ~ ± 1 fm.
References 1. W. Glokle and H. Kamada, Phys. Rev. Lett. 71, 971 (1993). 2. A. Kievsky, M. Viviani and S. Rosati, Nucl. Phys. A 551, 241 (1993); A577, 511 (1994). 3. J. Carlson, in Structure of Hadrons and Hadronic Matter edited by O. Scholten and J. H. Koch, World Scientific, Singapore, 1991, p. 43 4. R. B. Wiringa, Phys. Rev. C 43, 1585 (1991). 5. B. S. Pudliner, V. R. Pandharipande, J. Carlson and R. B. Wiringa, Phys. Rev. Lett. 74, 4396 (1995). 6. B. S. Pudliner, V. R. Pandharipande, J. Carlson, Steven C. Pieper and R. B. Wiringa, Phys. Rev. C 56, 1720 (1997). 7. J. L. Forest, V. R. Pandharipande, Steven C. Peiper, R. B. Wiringa, R. Schiavilla and A. Arriaga, Phys. Rev. C 54, 646 (1996). 8. R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C 51, 38 (1995). 9. V. R. Pandharipande, Nucl. Phys. A 553, 191c (1993). 10. A. Arriaga, V. R. Pandharipande and R. B. Wiringa Phys. Rev. C 52, 2362 (1995). 11. R. B. Wiringa, Steven C. Pieper, J. Carlson, V. R. Pandharipande Phys.Rev. C 62 014001 (2000).
94
NEUTRON HALO OF FISSILE NUCLEI V.I. SEROV, S.N.ABRAMOVICH RFNC - VNIIEF, Sarov, Nizhni Novgorod region, Russia F.F.KARPESHIN Institute of Physics St. Petersburg University, St. Petersburg, Russia
Noticeable anomalies in dependence of fission cross-sections in neutron and charged particle (p,d,t) induced fission of some nuclei and in TKE and v p dependence are observed near thresholds. These data may be explained by production of short-lived loosely bound systems composed of neutrons, protons and a core.
In recent years it has been found that 6He and n Li nuclei could be considered as systems composed of a core (a and 9Li) and dineutron loosely bound with the core [1]. In both cases the binding energy between two neutrons and tOhe core is less than 1.0 MeV. It is of interest to search for these systems among heavy nuclei. The availability of small-amplitude resonance at En=0.3 MeV in the cross-sections of neutrons absorption by even-even nuclei with A>230 that was obtained in the calculations using the method of coupled channels is a basis for such systems existence [2]. In such a situation when the next neutron is absorbed through the energy of coupling (~1.2MeV) there may occur near the nucleus a bound state of two neutrons - the low binding energy dineutron with the central nucleus e2„ ~ -0.9 MeV. Unlike the light nuclei these systems in the heavy nuclei with A>230 will be unstable essentially in the fission, so it is best to seek for them from the anomalies in the fission channel. To find these systems let us try to analyze all existing experimental data on interaction of heavy nuclei with neutrons and other particles. The threshold states are observed both for fissile nuclei and light nuclei [3-7]. In this case a decrease of total kinetic energy of fission fragments (TKE) and change in Vp as well as a change of fission crosssection are the characteristic features of reaction near the neutron threshold.
95 0.9
1 238
0.8
tp* *H?S| 'Mill1 HHW flffl
/*
0.7 0.6 0.5
*"F
/
y
0.6 0.5 .3
0.4 0.3
I
-4 -*
0.2
• *
0 0.4 0.3 0.2
*j «v
••w
0.1
• —^* E -
w»
r
0.9
*w " 1 ^
(jW
**
/
0.7 0.6 0.5 .H>»H*rM
ill
0.9
nw
O.B
^
i*ta [HHtf ^
0.7
/
O.I
0.6
r
i
0
•
•
23'
/ — t)
-
•u*W?"™I * W |
wft * W •*w• * -
1
'1
f .* O 0.4
Mlf
0.8 0.7 0.5
. 233..
^
0.5 0.9
0.6
i
r
0.8
•iM* 9«|W J * * " "
4flLJI|dU|lin
i B f e * * * * * * «m
0.9 O.B 0.7
i
0.6
,»
0.5 0.4 0.3
-
0.2 0.1
0.2 0.1 0
•2
16
20
24
S. , PROTON ENERGY ( M e V )
Figure 1. Relative cross-sections of (p, f) reaction on the uranium isotopes.
The accuracy achieved in measuring TKE is about -0.03%, so it is a good indicator of system's state. Let us start with illustrious example, which shows that mechanisms of threshold phenomena still retain many surprises. In Figure 1 we show the relative cross-section of the (p,f) reaction on uranium isotopes with respect to the total cross-section. For U the threshold of the (p,pn) reaction is 6.14MeV. This energy exactly coincides with a broad maximum in the cross-section. But it is difficult to explain the maximum
96
as a mere consequence of opening of an additional channel, because the energy of the outgoing proton is in this case close to zero, with correspondingly close to zero penetration probability of the Coulomb barrier. This means that the Coulomb excitation by the field of the incoming particle may be the only mechanism responsible for the anomaly. The latter mechanism, however, is expected to be less important in comparison with the strong interaction which acts due to penetration of the incoming proton through the barrier inside the nucleus. Moreover, it is the threshold of the (p, 2n) reaction of 7.14MeV which should be sooner observed in the latter case. We note that the cross-section of Figure 1 is compatible with the presence of a tiny anomaly at the latter energy of 7.14MeV. The question of the broad maximum at 6.14MeV remains therefore not understood. Analogous situation takes place for the 233U isotope cross-section.
3,MeV Figure 2. TKE versus the neutron energy in uranium isotope fission: 233 U - o [4] and ^ U - • [5] Let us consider now TKE dependence near the thresholds where two neutrons are produced as a result of the reaction. The experimental data on TKE dependence in the fast-neutron induced fission of 233U and 235 U are given in Figure 2 [4,5]. One can see that for U(n, f) reaction
97
noticeable TKE minimums are observed at Ea = 0 and 4.7+0.3 MeV. The first minimum is due to the threshold state. For 35U(n, f) reaction the noticeable decrease of TKE is observed at En = 5.0+0.2 MeV according to data of [5] and 4.7+0.2 from [6]. It may be assumed that in both cases the loosely bound systems, that is, halo-states with the binding energy E2n = 1.0610.3 MeV for 233U + n->232U+2n and E2n = - 0.46+0.15 MeV for 235U + n -»234U+2n are produced. Moreover, the less A TKE is observed in 233U(n,f) reaction whose cross-section exceeds Of (E „) of 235U(n,f) reaction by a factor of ~ 1.5. Let us consider in detail the experimental data on mass and kinetic energy distributions of the fission fragments in 235U(n,f,) reaction at various neutron energies as well as the data on neutron and gammaquantum emission. The most detailed and exact data are presented in paper [5]. A considerable part of ATKE is in the mass region of 126 to 138 (close to the magic number 132) and of 146 to 152. In the first case the TKE decrease of 2.2 ± 0.2 MeV and in the second case the TKE increase of 1.2 + 0.2 MeV are observed per fission event with respect to the distribution at Ea = 0 (for thermal neutrons). It is shown in the paper that all distributions can be presented as five groups of fission fragments (two among them correspond to additional fission fragments). Group of fission fragments responsible for the change in TKE has constant parameters at neutron energy of ED ~ 2.5 MeV and E„ > 4.0 MeV. The contribution of this group is about 10% at En of about 2.5*5.0 MeV. As the neutron energy changes from 2.3 to 5.0 MeV, M0i mass increases by 1.0 ± 0.1 MeV. The same neutron thresholds appear in TKE change (see Figure 2). The total energy released by prompt gamma-rays in 235U(n,f) reaction [5], is also subjected to the changes at these neutron energies. However in vp (En) dependence the increased yield of prompt fission neutrons [5] with respect to the initial linear dependence is observed at En > 4.0 MeV only. Thus the given data do not contradict the assumption that there is a halostate at En = 4.8+0.15 MeV, for the facts supporting the low binding energy|e2nl< 1.0 MeV between two neutrons, TKE decrease and vp increase for 235U(n,f) reaction suffice to adopt this statement. Then the neutron energy En = 2.5 MeV corresponds probably either to the onset of the excitation of the second well states. A light but systematic exceeding of Cf(En) of about 2% with respect to the linear dependence at En of about 4.5 MeV can be observed in the dependence of fission cross-sections in
98
neutron induced fission of U and U nuclei. However these changes in 0 f are not convincing while the changes in TKE are evident. When the neutron energy is about 5.0 MeV, the compound system may have the total momentum J of about 6. Either two neutrons providing loosely bound state near the core in the collision of the incident neutron and the external one of the initial nucleus or the core must have this momentum. Most likely two fast-rotating neutrons are produced near the core for their intrinsic momentum is J2a = 0. The half-width of TKE drop in the case of 235U(n,f) reaction is about 0.7 MeV. It may correspond to several states overlapping in J both for neutrons and probably for core. There appears to be stripping of an external neutron to the loosely bound states near the core. Several thresholds are also observed when 233U interacts with neutrons. One among them occurs in v (£„) dependence at En = 2.7 MeV [7]. In this case the decrease of vp (En) and increase of gamma-quantum yield are observed in 233U(n,f) and 233U(d,pf) reactions [8], respectively, over the narrow excitation energy interval AE - 0 . 3 MeV. This threshold may be considered as arising from the excitation of the lower states in the second well. The neutron energy in minimum corresponds to the excitation of the second well state with J = 6+ and E0* = 2.56 MeV. The difference between the energies of 233U+n-»232U+2n and 235U+n-»234U+2n halostates is due to the different parity in the initial states of initial nuclei and different excitation states of the central nuclei. At uranium isotopes fission by protons the noticeable anomalies in the cross-sections of (p, f) reactions are also observed near the thresholds of (p, pn) or (p, 2n) reactions [6]. Along with the gross-structure in c^E„)/f cases is a proof of neutron halo formation at En~4.7MeV in U(n,f) reaction.
99
The three-neutron state shows itself in 232Th(n,f) reaction [10] (see Figure 3). In this case the minimum is observed in a^En) dependence in the fission of 232Th by neutrons with Ea = 9.85 MeV. The energy of e3n = 1.71+0.1 MeV may be assigned to this minimum (0.57 MeV per a neutron). The half-width of minimum if so does not exceed 0.3+0.05 MeV. As the value of Of(En) in this case is considerably lower than in 233U(n,f) 235 U(n,f) reactions, then, the amplitude anomaly has grown noticeably. To demonstrate the existence of three-nucleon systems loosely bound with the core let us compare the production of these systems in 232 Th when neutron is replaced by proton in 232Th(n,f) and 232Th(p,f) reactions. The reaction between 232Th and protons is examined in detail in [11]. The noticeable decrease of cross-section is observed in (p,2n) reaction at £ p = 12.8±0.2 MeV (see Figure 4). It may be considered that 450-
. ft 400-
' •
mb„350-
23J
V
Th(n,f)
V •
"
• 300-
250-
•
*
••
• • •
,
1
'
1
•
1
12
10
ErfMeV Figure 3. Fission cross-section of
232
Th in
232
Th(n,f) reaction [9]
this minimum in (p, 2n) reaction is due to the production of (2np) system near the core and mainly to proton escape rather than to nucleus fission. (Sensitively lower contribution provided by the fission channel as compared with (p,2n) reaction at this proton energy as well as the accuracy of experimental data did not permit observing the anomalies in Cf). The distinct binding energies between 3n and 2np systems and core are explained by the Coulomb repulsion of proton from the core. This distinction is equal to 2.2-5-2.9 MeV.
100
&QD(&mfiy&-
1000
100
10-
- i — | — i — | — i — | — i — | — i — | — i — i — i — | — • — i — i — |
6
8
10
12
14
16
18
20
22
24
ka,u& Figure '4'. The excitation function in proton induced fission of 232Th - o and (p, 2n) reactions - • [11]. The solid curves represent statistical model calculations Table 1. Dependence of fission cross-sections in triton induced fission of 232 Th and 238U Et,
mrh
MeV
CTn, m b 2.65±0.05 5.6±0.7 6.7±0.7 13.7±1.3 30.7+2.5 48.6±3.8 45.0±3.5 65.2+5.0 105.2±8.1 147+11
8.64 8.97 9.31 9.65 10.01 10.26 10.38 10.65 11.04 11.41
ovi/ae Cf2> m b
3.35±0.28 6.810.6 14.2+1.2 23.0+1.9 39.2+3.1 58.8+4.8 70.0±5.4 98.717.6 148+11 209118
0.79 0.82 0.47 0.59 0.78 0.82 0.64 0.66 0.71 0.70
Et, MeV 11.79 .12.17 12.99 13.00 13.20 13.21 13.21 13.4 13.4
«2Th
an, mb 206115 253120 289124 302124 312127 318128 322128 333128 318128
0"fl/0"E
mb 261121 350128 449136 416+33 478138 454137 477138 464+37 476138 OE,
0.79 0.72 0.64 0.72 0.65 0.70 0.67 0.72 0.67
101
However in (t,f) reactions with others uranium isotopes, Np and Am the anomalies in (jf are less noticeable. This fact may be explained by higher fissility of nuclei and larger width of anomalies that did not permit detecting these irregularities in the experiment for its measurement accuracy. Curiously, that in (p,f) and (t,f) reactions the loosely bound three-nucleons systems show themselves in distinct manner in (p,2n), (t,2n) and (t,p) reactions competing with them. In the first case a peculiar reaction of two-neutron stripping by the proton takes place and in the second one there is triton breakup near the core. However both reactions occur with no production of a compound nucleus. Three-nucleon system shows itself apparently in the reaction between neutrons at ED -10.0 MeV and 235U. So when changing as a function of neutron energy at En of 9.4 to 9.8 MeV, TKE decreases by AE = 1.0 MeV. At high neutron energies ED of about 30-5-70 MeV the peaks and minimums are observed in o£En) dependence in the fission of 232Th, 235U, 237 Np, etc. [14]. They may be defined, as halo-state constituted by the core and formations involving several neutrons. The experimental data obtained in neutron induced fission of 237Np at En > 20 MeV are presented in Figure 5 [14]. Well-defined minimums in the cross-section of 237Np(n,f) reaction correspond to the production of loosely bound systems constituted by 5, 6, 7, 9 and 10 neutrons and the core. The neutron thresholds and maximums observed in Gf(Ea) are presented in the following table 2. The lack of detailed experimental data on nuclei at these neutron energies, TKE measurements, mass and energy distribution of fission fragments, neutron and gamma-quanta emission does not permit drawing the conclusion about the nature of anomalies in o£En). The greatest anomalies in c^En) for all nuclei from 232Th to 239Pu are observed at the neutron energies from 50 to 63 MeV which are close to the energy necessary for producing 9n and lOn loosely bound with the core. Similar formations involving protons and several neutrons are of no less interest as in (p,f) and (t,f) reactions at the energies of 8.0 to 13.0 MeV. 243
Table 2. Calculation thresholds of neutrons escape and observed positions of minimums in o"f(En). N neutrons Q.MeV En, MeV for N+l
5 32.78 31.3
6 39.15 38.9
8 53.44 53.8
9 61.44 60.8
102
2,0 XI
— i — i — i
25
'
50
i — • — n — • — F — • — i — • — i — • — i
75
100
125
150
175
200
E,Hff
Figure 5. Fission cross-section of 237Np as a function of neutron energies En > 20.0 MeV [14] The latest data on interesting loosely bound neutron formations observed near the core permit also making a conclusion that there are no bound nuclei constituted exclusively by neutrons with the number of neutrons up to 10, otherwise these formations near the core would be more bound. In conclusion I wish to thank Prof. Yu.C. Oganesyan for the fruitful discussions and L.I.Shizhenskaya, S.M. Taova for the help in the report preparation.
References 1. Ikeda K. and Suzuki Y., Strukture and Reactions of Unstable Nuclei, World Scientific, Singapore, 1991, p. 1. 2. Lagrange Ch. Report NEANDC (E)228"Z". France. 3. Serov V.I. In Proceedings of the Prec.Measur. in Nucl. Spectr. XI, (2-6 September, Sarov 1996), VANT, p.17,1997. 4. D'yachenko N.P. et al. VANT, ser.: Yadernye konstanty, vyp 1(40), M.:TsNIIatominform, 1981, p. 53.
103 5. Straede, Ch., Budtz-I0rgensen C and Knitter H., Nucl. Phys., vol. A462, 1987, p.85. 6. Meadows J.W. and Budtz-Jorgensen. Nucl. Data and Meas series, ANL/NDM-64, 1983. 7. Boyce J.R., Hayward T.D., Bass R. et.al. Phys. Rev. CIO, Nl, 1974, p.2. 8. Malinovskii V.V. et al. VANT, ser.: Yadernye konstanty, vyp 5(54), M.:TsNIIatominform, 1983, p. 19. 9. Andreev M.F.et al. Yad. Fiz., vol. 51, 1990, p. 942. 10. Blons J., Mazur C , Paya D. et al. Nucl. Phys. vol. A414, 1984, p.l. 11. Kudo H., Muzamatsu H., Nakahara H., Miyano K. and Kohno I., Phys. Rev., vol. C25, 1982, p. 3011. 12. Andreev M.F., Gladkov V.V., Zavgorodnii V.A., Serov V.I. Preprint, VNIIEF, 6798, Sarov, 1998. 13. Nakanishi T. and Sakanoue M. Radiochem. Acta. 16, 1971, p.24. 14. Lisowski P.W. et al. Proc. Int. Conf. on Nucl. Data for Science and Technology, Julich, 1991, Springer-Verlag, 1992, p. 732. 15. Lisowski P.W., Gawron A. et al., Proc. Specialists' Meeting on Neutron Cross Section Standartsfor the Energy Region above 20 MeV, Uppsala Sweden, 1991 May, 21-23, 1991, NEANDC-305/U, p. 178.
104 SEARCH FOR POSSIBLE ISOMER
m
Pa
E.F. FOMUSHKIN, S.N. ABRAMOVICH, M.F. ANDREEV RFNC-VNIIEF, Sarov, Nizhny Novgorod region, Russia Abstract Basing on the analysis of experimental data on 232Pa (7iQ=31.44 h) fission cross section measurement a hypothesis of 232mPa (T\a~5 h) isomer existence was advanced. To prove the isomer existence and to identify its decay channel an additional measurement cycle has been performed. Isomeric transition into 232Pa basic state followed by p-decay into 232U and direct P-decay into ^ U have been shown to be of low probability. The probability of the supposed isomer spontaneous fission has been practically excluded. More thorough studies of the supposed 232mPa isomer electron capture and p+-decay channels are planned. The measurements are also planned for a2 Pa and 233Pa nuclei accumulation at 231Pa target irradiation with an intense thermal neutron flux. At the conference in Kalmar (Sweden, 1996) we presented a report on the results of measurements of 232Pa fission cross section by thermal neutrons [1]. The main goal of the measurements was to reveal the causes of substantial discrepancy of the previously obtained data [2,3]. In the measurements performed in the 50-s [2] 231Pa samples irradiation with thermal neutrons was used, i.e. 232Pa nuclei were built up in neutron radiation capture reaction. In the work [3], carried out in LANL (Los Alamos, USA), 232Pa samples were built up with 232Th(d,2n) 232Pa reaction use; 232Th target was irradiated with a beam of deuterons accelerated up to the energy J?d = 17-5-18 MeV. In our measurements [1] m P a nuclei were built-up resulting from 232Th(p,n) 232 Pa reaction. Thorium samples were irradiated with a proton beam with the energy of 12MeV. Then from the thorium target a protactinium fraction was chemically extracted and rectified. To produce the 232Pa layer the electrolysis technique was applied. Then the layer was irradiated with neutrons from 9Be(d,n) reaction, moderated in a polyethylene block. The fission fragments were detected by dielectric track detectors. During the experimental results processing a time dependence of the fission count rate in Pa layer was analysed. At the experimental data approximation with a curve corresponding to 232Pa (Tj/2=31.44 h) P-decay a "fast" component with T\a~5 h (Fig.l) was found out. There was advanced a supposition about the unknown isomer existence with the given half-period. It was taken into account that the most favourable situation for isomers to appear among the fissioning nuclei arises for oddodd nuclei [4]. The most well known isomers of heavy odd-odd nuclei are: 234 Pa(r1/2=1.17 min), 236Np(7/1/2=22.5 h), 240Np(r1«=7.22 min), 242Am (Tm=l6.02 h), 244 Am (r,/2=26 min) [5]. It should be mentioned that to obtain the value of 232Pa fission cross-section in our measurements we took into account only a long-lived component (7^2=31.44 h) of fission count rate in the protactinium layer. This cross-section value rjf = (975±70) b - agrees well enough with the result obtained in the 50-s rjf=(700±100) b [2], but is 1.5 times less than the result presented in [3]. This fact indirectly corroborates the supposition of 232raPa isomer existence; this is just the effect, which is to result from neglecting built-up isomer nuclei fission contribution towards the value of Of (if the isomer exists).
105 •
Experiment
.—Approximation
a3
0.020-
y0 A1 11 A2 12
0.00158 ±0 0.01304 ±0.00176 7.63642 ±2.21664 0.0219 ±0.0008 45.358 ±0
•£ 0,010' 0,005 0,000 Time (hour)
Fig.l. Normalised time dependencies of nuclei fission in the protactinium target by thermal neutrons. In 1997-1998 we conducted additional measurements of Pa fission crosssection, which proved the short-lived component presence in 232Pa decay. It can be noted that the short-lived component contribution depended on 232Th irradiated sample chemical etching mode, i.e. on depth of the thorium sample irradiated part etched layer. From this one can infer that the short-lived isomer production probability essentially depends on the energy of a charged particle bombarding a nucleus-target. The supposed isomer can belong to one of protactinium isotopes (231Pa, 232Pa, 233 Pa). Isomers of Th, U or other elements - products of 231Pa decay - can not participate in the play, as even being built up they are to a considerable degree eliminated during the protactinium extraction from the irradiated Th material by the chromatography method. However, the Pa and Pa isotopes fission is a threshold reaction (Ethr ^0-5 MeV) [6]. Their possible isomers ( Pa or Pa) should possess excitation energy of more than 0.5 MeV to provide for a large fission cross-section caused by thermal neutrons. But the mere existence of isomer with such excitation energy and half-life ti/2 > 1 hour seems to be improbable. In the present work there are discussed the researches performed through various techniques for the 232Pa supposed isomer decay channels. If the 232mPa isomer exists, several channels of its decay are possible (Fig.2): a) Pa isomeric transition into basic state followed by (5-decay into U; b) direct p-decay into 232U; c) electron capture with transition into 232Th; d) positron decay into 232Th [5]. Moreover, the experimental data on ^ P a fission by thermal neutrons (Fig.l) do not exclude existence of mmappropriate spontaneously fissioning isomer [4].
106 MeV
0.3331
f
/
jy
1
0.1621 0.0494
1—
1.410,0y
\ P+,EC //
1 1
232
6
\ \
-
Pa
+
\ 1.31 d\
\ M P"
4+
V
2+
Th -\°
232
0.323 0.1565
a 6
+
2*
o —•
0,0476
/ /
A*
72 y
0+
232
u
* a
Fig. 2. A decay scheme of the supposed isomer 232mPa An investigation cycle was carried out to detect the supposed isomer 232mPa and identify its decay channel. A) There has been performed a B 2 Pa buid-up through 232Th(p,n) reaction during one hour at Ep=12 MeV and Ip=0.4 uA. Protactinium fraction extraction and its rectification have been implemented with a technique used in the previous measurements of 232Pa fission cross-sections [1]. Then the solution y-spectrometry has been conducted. The y-spectra under measurement has been searched through for a short-period component (TiaSlO h) in the y-peaks, corresponding to y-transitions between low excitation levels in 232U (£y=47.6; 103.4; 109; 150.1; 165 keV) and in n2 Th(.Ef=49A; 162.1; 171; 323.8 keV) [7]. It should be mentioned that the set and intensities of the y-transitions can differ from transitions realized in 232Pa decay, if the parities of the basic and isomeric states in 232Pa are different and there takes place an isomer direct p-decay into 232U or P+-decay into 232Th. The y-spectra measurements were carried out with two Ge detectors simultaneously: GR-1318 (high-energy spectrum part with £y>150 keV) and GL55PS (low-energy spectrum part with 10<£^200 keV). The measurements total duration was 51 hours and 48 spectra were measured by each detector. The measured y-spectra analysis has not proved the short-period component presence in the analyzed y-spectra; i.e. within the limits of measurements statistical accuracy no specific y-lines with Tic^lO h and E^IO keV have been found. B) There has been carried out a comparison of the number of 232Pa nuclei in the working target, used in 232Pa fission cross-section measurements [1] with the number of 232U nuclei, accumulated in that target due to 232Pa p decay. Their
107 coincidence (within the limits of measurement error of =5%) indicates that if the supposed isomer 232mPa decays into 232U, then such decay probability is small. The measurements of 232U nuclei quantity in the target were carried out with use of the a-spectrometry method. The a-particle spectrometer consisted of a silicon semiconductor detector of a surface-barrier type, placed in a vacuum chamber, and a standard system of electronics. The a-spectrometer energy resolution constituted 40-50 keV for a-particles with the energy within the interval of 4-6 MeV. C) To check a possibility of spontaneously fissioning isotope (SFI) production, two plates of metallic thorium were irradiated with proton and deuteron beams. Irradiation time was four hours, energy of protons and deuterons - 12 MeV, beam current - 0,4 uA. Being irradiated, the thorium plates were kept in contact with dielectric track detectors to detect possible SFI fission fragments built up in the thorium plates. On the supposition that the "fast" component in 232Pa fission cross-section measurements [1] was due to SFI, the expected number of detected tracks of the SFI fission fragments (in the control measurements) should be - 30. But no fission fragments tracks were detected in the measurements. So, the question of possible isomer 232mPa existence with the half-period Ti/2^5 h and its characteristics identification has remained unanswered. We consider that the supposed isomer 232mPa electron capture and p+-decay channels need to be studied more thoroughly. The present report goal is to attract our colleagues attention and probably to awake their interest to this problem. For our part, we are going to perform additional measurements similar to that in [1], paying particular attention to the first section of the curve of protactinium nuclei decay in the working target. Besides, we are planning to conduct measurements of Pa and Pa nuclei accumulation at 231Pa target irradiation with a thermal neutron intense flux. Such experiment would likely exclude a possibility of the isomer production, and at the same time allowed one to measure neutron fission and radiative capture cross-sections for 232Pa nucleus. REFERENCES 1. Fomushkin E.F., Abramovich S.N., Andreev M.F. et al. Proc. 2nd Int. Conf. on Accelerator-Drive Transmutation Thechnologies and Applications, Kalmar, (1966), Uppsala, 326 (1996). 2. Hughes D.J., Magurno B.A., Brussel M.K. Neutron Cross Sections. BNL-325, Suppl. 1,(1960). 3. DanonY., Moore M.S., Koehler P.E. et al. Nucl. Sci. Eng. 124, 482 (1996). 4. Policanov S.M. Isomerism of nuclei forms. Moscow: Atomizdat 27 (1977). 5. Feristone R.B. Table of Isotopes, eighth edition, IIA 151-272 LBNL, (1966). 6. Gorbachev V.M., Zamyatin Yu.S., Lbov A.A. Intaraction of radiation with hard elements nuclei and nuclei fission. Referense-book, Moscow: Atomizdat, 332 (1976). 7. Reus U., WestmeierU. At. Data, Nuclear Data Tables 29, 1 (1983).
108
BROKEN SYMMETRY IN NUCLEAR MATTER A.W. OVERHAUSER AND A.E. POZAMANTIR Department of Physics, Purdue University, West Lafayette, IN 47907, USA E-mail: [email protected] The stability of uniform nuclear matter is investigated by optimizing the energy of a broken-symmetry state having a periodic density wave. The instability is a finite amplitude one. It does not occur until the amplitude is large, and it requires also a severe distortion of the Fermi surface. The binding energy is increased by 1.5 MeV per nucleon. Additionally, an infinite slab of nuclear matter, 12 fm thick, is found to have an extra binding energy of 0.9 MeV, which implies that the surface energy changes sign when the surface curvature becomes small.
The possibility that symmetric nuclear matter might exhibit a nucleon density wave (NDW) was examined in 1960 1 . Compared to a constant density, p0, a sinusoidal modulation generated greater binding. For a single NDW, p& p0(l + \cosQz),
(1)
where, Q ~ 2kp, is the diameter of the Fermi surface, the amplitude, A, was found to be ~ 0.6. All calculations were carried out in the HartreeFock "arena", i.e., many-nucleon wave functions are taken to be single Slater determinants. Short-range ^-function potentials were used for both attractive and repulsive interactions. It is appropriate to revisit the stability of nuclear matter, now that computational facilities allow studies which employ finite-range interactions. Never the less, the need to evaluate the energies of millions of determinants led us to find an effective nucleon-nucleon potential of optimum simplicity. This report summarizes briefly the work leading to such an interaction and its application to nuclear matter, soon to be published in two detailed papers 2 . We discovered a three-parameter potential, V(r), which accounts for many of the systematic properties of special nuclear systems, e.g., even-even nuclei with N = Z and symmetric nuclear matter. (It is intended for use only in a Hartree-Fock arena.) V(r) = -aCr2e~r2/s2
+ Py/T^
63(r).
(2)
Tav is the mean kinetic energy per nucleon. C is merely the normalizing constant, (§7r 3 / 2 s 5 ) - 1 , of the modified Gaussian, and 63(f) is a three-dimensional delta-function. The three parameters are: a = 1690MeV • fm3,
/3 = 255MeV1/2 • fm3,
s = 0.54/m.
(3)
109
Since we consider only configurations for which each spatial orbit is occupied by four nucleons (proton, spin up and down, neutron, spin up and down), there is no need to include tensor forces or spin-orbit coupling, which then sum to zero. Saturation of nuclear forces does not occur in the Hartree-Fock approximation unless the repulsive core increases either with nucleon density or with kinetic energy. The first choice (which is commonly embraced) implies highly excessive three-body forces 3 . The kinetic-energy dependence, which we adopt in (2), describes a greater penetration of the repulsive core resulting from higher kinetic impact 4 . The Coulomb interaction cannot be included in treatments of symmetric nuclear matter. We omit it also when considering even-even nuclei with N = Z; so we delete the Coulomb contribution from all experimental binding energies. With V(r) given by (2) and (3), the following properties are satisfactory: (1). The binding energy, B0 = 15MeV7nucleon, of nuclear matter. (2). The saturation density, p„ = 0.179 nucleons//m 3 , of nuclear matter. This value corresponds to R0 = 1.10/m in the radius formula, R = RgA1^. (A = N + Z). (3). The compressibility modulus, K — 225MeV/fm3, of nuclear matter. (4). The effective mass, M*/M = 0.41, of nucleon excitations. The magnitude of the symmetry term in the semiemperical mass formula, 23MeV(N — Z)21 A, requires an effective-mass ratio between 0.4 and 0.5. (5). The binding energy per nucleon of 4 iJe. (6). The binding energy per nucleon of 1 6 0 . (7). The binding energy per nucleon of i0Ca. Agreement of the last three binding energies with experiment shows that the surface energy is correctly accounted for by the finite range, s, of the attractive term in V(r), Eq.(3). With no asymmetry (N = Z), and with the Coulomb energy extracted, the differences between the three binding energies (of4 He, 1 6 0 , and 40Ca) and B0 are (by definition) the three surface energies. Since, as will be seen below, an NDW has large density gradients, it is crucial that energy contributions which arise from such gradients are implicit in the effective interaction, V(r). Furthermore, since an NDW also involves alternating compression and dilation of the nucleon density, it is noteworthy that V(r) also leads to a compressibility modulus, K, consistent with experiment 5,6
Had we used a ^-function attractive term instead of the finite-range one, we would have obtained a binding energy for 4He too large by a factor, 2.5. Nuclear matter would then immediately crystallize into a face-centered-cubic crystal of a particles. The r2 factor in the attractive term of V(r), Eq.(2), was
110
* xy
0
1
2
3
4
5
6
7
8
m
(k z )
Figure 1. Fermi radii, kXy, of the seven occupied harmonic oscillator levels. The black tops are the washer shaped regions which can be optimized by admixtures from m to m-t-2. (For m = 5 and 6 the entire Fermi circle can be admixed.)
inserted to achieve agreement for 1 6 0 and 40Ca. (A simple Gaussian would overbind ieO by 2 MeV/nucleon.) It is tempting to imagine that the r2 factor mimics the main effect of a Jastrow factor (often employed in going beyond Hartree-Fock) by eliminating attractive contributions that would otherwise occur from pairwise close approach. We computed the energies, nucleon density distributions and intrinsic ellipsoidal deformations of all twenty even-even, N = Z nuclei up to 80Zr. Closed-shell nuclei were found to be spherical. All others were deformed. For example, 6 8 5e was found to be superdeformed oblate, with fa = —0.28, a result which is consistent with a recent measurement 7 . We optimized every reasonable, occupational configuration in a deformed-oscillator basis (1575 of them in mZr alone). Many low-lying shape isomers 8 were found 2 . Our first application to nuclear matter envisions an infinite slab of finite thickness in the z direction. We employed the following one-nucleon orbitals:
^(O^^W,
(4)
are harmonic oscillator functions, and {Am} are normalizing constants. Each m corresponds to a two-dimensional nucleon gas with a radius kxy%m of the
111 1
1
1
I
.
• » — - = * •
•
/
^
—
•
15 -
^ c o fl>
•
14 •
u 3
C
>
13
*—«•
CD 12:
j
i
i_
i
3
4
5
6
i — i — — i
7
8
9
M' Figure 2. Binding energies of a nuclear slab versus t h e number, M', of occupied harmonicoscillator levels. The open points include the extra binding when admixtures from m t o m + 2 are included.
Fermi circle. The variational parameters were the Fermi-circle radii and the "spring constant" of the harmonic-oscillator levels. We also introduced mixing parameters, c m , to modify the oscillator motion: l
P'm(z)
=
(5)
but only for those wave vectors k that are unoccupied in the oscillator level, m + 2. (Mixing m with m + 2 preserves the parity of each state.) This mixing scheme is illustrated in Fig. 1, which shows the final (optimum) radii, kxy<m, of the seven occupied Fermi circles. The binding energy/nucleon is shown in Fig. 2, with and without the mixing, (5), as a function of the number, M', of occupied Fermi circles. The binding saturates at M' — 7 since, for M' > 7, all of the new kxy become zero (after 300 iterations of the variational procedure). The density profile along z is shown in Fig. 3. Its full width at half maximum is ~ 12/m. Since the optimum binding energy shown in Fig. 2 is 15.9 MeV, 0.9 MeV larger than the value, 15MeV, to which V(r), from Eq.(2), gives rise for uniform nuclear matter, it follows that nuclear matter is unstable by fission into slabs 12 fm thick. This observation may be far reaching. The surface energy of the semiemperical mass formula corresponds to
112 0.3
0.25
0.2
•I.
015
o. 0.1
0.05
0 -15
-10
-5
0
5
10
15
2(fm)
Figure 3. Nucleon density profile versus z for an infinite nucleon slab. The optimum density, Po, of nuclear matter is shown.
~ -l-l.lMeV//m 2 . The 0.9 MeV extra binding of a 12 fm thick slab indicates that, for the slab, the surface energy is ~ —1.0MeV/fm2. This change in sign is most likely associated with the reduction in surface curvature from large values, in finite nuclei, to zero for a flat, infinite slab. The extra binding is caused by the peak (in Fig. 3) which rises significantly above p0, the density of uniform nuclear matter. Even for a semi-infinite configuration, the density of a non-interacting, degenerate Fermi gas rises from zero (at the surface) and passes through a peak before it settles back to the mean density 9 . Such behavior will be enhanced by the attractive interactions of a nuclear medium, and can be expected to contribute extra binding. The theory of a NDW requires introduction of several new features. We will assume that the principle wave vector, Q, of the NDW is parallel to z. The wave functions are:
^ n ^
6
^ * . ^ ) '
where k,p are as above, in (4). The cp's here are Bloch functions:
(6)
113 1 —
i
i
1
1.6
o 3 c
>
1 """"—
^* —- .
(13X13)
1.2 a
0.8
2 >
-
(11X11)
/
—i—i—|—i—i
0)
—' CO <
1
. „ . . - - « (15X15) "
c
o 0)
-1
(9X9)
0.4
.
'
/ — i
I—
'.
•-
10
12
M Figure 4. Excess binding, A B (relative to JB 0 ) of an NDW as a function of the number, M, of occupied energy bands and the size of the matrix used to define the Bloch functions. T h e last two points for each size are the same because the highest band allowed prefers to be empty.
¥>*.,»= £
am(kx,n)e«k>+m<»'
(7)
m=i—p
where n is a band index. (2p + 1) is the number of plane-wave components in each Bloch function. kz is the wave number in the fundamental (Brillouin) zone, which runs from — \Q to \Q. Each Bloch function requires (2p + 1) coefficients, am, and these are the eigenvectors of a (2p+1) x (2p+1) matrix, U. The diagonal elements of the matrix are: Um,m = n2(k +
mQ)2/2M.
(8)
The adjacent diagonals are: C'm,m+1 — t/m,m—1 — " ( A ^ ) , Um,m+2 = t/m,m-2 = H{kz).
(9)
G(kz) and H{kz) are variational parameters which determine the shapes of the (2p+l) Bloch functions for each kz. All other matrix elements are taken to
114 FERMI SURFACE PROFILE
-
Q *-
Figure 5. Fermi surface of the NDW for the ten band (15 x 15) calculation. Note that the tenth band is unoccupied. The dashed circle describes the sphere of equal volume. (The central, horizontal line is an axis of cylindrical symmetry.)
be zero, but they could be used to introduce additional variational parameters. The eigenvalues of the matrix are used only to define the energy bands. (All total energies are computed from the occupied wave functions 2 .) Every (kz, n) has its own Fermi circle and Fermi radius, each of the latter being a variational parameter. (All variations were adjusted to conserve nucleon number.) The extra binding energy, AJ3, over and above Bot is shown in Fig. 4 as a function of the number, M, of occupied energy bands allowed, and on the size, (2p + 1 ) , of the matrix. We did not go beyond M = 10 because all Fermi-circle radii of the tenth band decreased to zero after 300 iterations of the variational procedure. The optimum number of occupied energy bands is therefore nine. The fundamental zone was divided into 14 kz values for numerical evaluation of the total energy. Symmetry about kz = 0 reduces the number of variational parameters. For M — 10, there are then 70 Fermi radii, 7 G's, 7 H's, and Q, a total of 85 variational parameters. The severely faceted Fermi surface is shown in Fig. 5. What is of particular interest is the small value of Q (compared to 2kF). NDW instability requires the Fourier transform of V(r) to be significantly attractive for the NDW wave vector, Q, i.e., V(Q) « 0. If the attractive interaction were short range like the repulsive component, V(Q) would remain
115 0.3
, , ..,....,..
0.25
«r
0.2
*-
0.15
E N
— a.^
0.1 0.05 0
i
i
i
i
>. •
•
•
i
i
i
•
zflm)
Figure 6. Density profiles of an NDW and a nucleon slab. pa is t h e density of uniform nuclear matter, and ps is the average density of the NDW.
negative all the way to Q = 2kF, and the optimum Q would be near 2kp 1However, the Fourier transform of a finite range, attractive component falls off rapidly with Q, and can no longer dominate the repulsive component. Consequently Q must be much smaller than 2kF, as shown in Fig. 5. Allowed Q's correspond to NDW wave lengths, ~ 20/m. The results in Fig. 4 show that uniform nuclear matter is a highly excited state, at least 1.5 MeV/nucleon higher than that of a single NDW. (Whether a second NDW can also arise has not been investigated.) The uniform state is metastable, however. We found that the NDW can be reached only if large values of G and H are inserted at the start and, also, only if the Fermi surface is significantly faceted as shown in Fig. 5. If either of these starting features are too small, the variational procedure iterates back to uniform with binding energy B0. An NDW is a finite-amplitude instability, and so has remained hidden until now. Finally, Fig. 6 shows the nucleon density versus z, both for an NDW and for a single infinite slab. It appears that one can regard the NDW as a periodic array of slabs with modest "bridging" regions between them. The similar shape of the peaks for the NDW and the slab is noteworthy. The fact that completely disparate theoretical techniques (harmonic oscillator functions versus Bloch functions) can lead to such compatible profiles indicates
116
that the fundamental physics is "in control". One should appreciate that the energies of an NDW and of a slab can be made even lower by introducing greater variational freedom. The uniform state, however, is an exact solution of the Hartree-Fock equations, so B0 cannot be improved. Further research such as NDW instability in an infinite slab or in a nuclear rope comes quickly to mind. References 1. 2. 3. 4. 5. 6. 7.
8. 9.
A. W. Overhauser, Phys. Rev. Lett. 8 415 (1960). A. E. Pozamantir and A. W. Overhauser, Phys. Rev. C (2001), in press. B. A. Brown, Phys. Rev. C58, 220 (1998). R. Karplus and K. M. Watson, Am. J. Phys. 25, 641 (1957). J. P. Blaizot, J.F. Berger, J. Decharge, and M. Girod, Nucl. Phys. A591, 435 (1995). D. H. Youngblood, H. L. Clark, and Y. -W. Lui, Phys. Rev. Lett. 82, 691 (1999). S. M. Fischer, D.P. Balamuth, P.A. Hausladen, C.J. Lister, M.P. Carpenter, D. Seweryniak, and J. Schwartz, Phys. Rev. Lett. 84, 4064 (2000). J. L. Wood, K. Heyde, W. Nazarewicz, M. Huyse, and P. van Duppen, Phys. Repts. 215, 101 (1992). A. W. Overhauser, Phys. Rev. B33, 1468 (1986).
117 THE DIVERSE MANIFESTATIONS OF FISSION OF CLASSICAL A N D Q U A N T A L DROPLETS
H. J. K R A P P E Hahn-Meitner-Institut Berlin, Glienicker Strafie 100, D-1410 Berlin, Germany, E-mail: [email protected] The stability of rotating, surface-charged, classical droplets with respect to global changes of their shape is investigated and contrasted with the conditions for local, thermodynamical stability of the surface. For systems of microscopic size the appropriate quantum mechanical framework for the dynamics around scission is discussed.
1
Introduction
When Hahn and Strafimann observed nuclear fission for the first time in 1938 1 , this decay mode was quite unexpected and named "fission" by Meitner and Frisch 2 because of a certain analogy to the fission of biological cells. The only decay modes of thermally excited, classical droplets in strong electric fields, known at that time, were the evaporation of neutral and charged molecules and the occurrence of Coulomb bursts, i.e. jets of charged, very small particles from a liquid surface in sufficiently strong electric fields3. There are two main differences between classical and nuclear droplets: First, the latter are the only volume-charged droplets in nature, all others are surface-charged. And second, because of their microscopic size, the dynamics of nuclear droplets is strongly influenced by specific quantum effects. In the following I will discuss two types of instabilities leading to the decay of a charged droplet: It may become unstable against global changes of its shape (Rayleigh instability) or the electrostatic tension of a surface-charged droplet may become larger than the surface tension , locally, at some region of the surface, so that this piece of interface between the droplet and its surrounding medium loses its thermodynamical stability (Rayleigh-Taylor instability). The most widely discussed quantal effects for droplets of microscopic size are shell effects, which influence the stability of metallic clusters and nuclei in similar ways. I shall not talk about this effect, but rather focus on the specific way in which quantum mechanics controls the charge and mass split around the scission configuration.
118
2 2.1
Classical droplets Experimental
results
In a famous paper from 1882 4 Lord Rayleigh investigated the stability of surface-charged, spherical droplets against small-amplitude multipole deformations. He found t h a t the first multipole n to become unstable with increasing charge is the quadrupole n = 2 and t h a t this happens when the Coulomb energy Ecoui — <3 2 /(2R) becomes larger t h a n twice the surface energy Esur — AitR2a, where Q, R, and a are the charge, the radius, and the surface-tension constant of the droplet, respectively. In modern terminology, the spherical drop becomes unstable when its fissility x = Ec0ui/2ESUr is larger t h a n 1. Having derived this result, Rayleigh continues: " When Q is great, the spherical form is unstable for all values of n below a certain limit, the maximum instability corresponding to a great, but still Unite, value ofn. Under these circumstances the liquid is thrown out in fine jets, whose fineness, however, has a limit.1' Considerable experimental efforts were m a d e during the 20th century to observe these jets. T h e first to report on this phenomenon was Zelenyi 5 . He took photographs of drops pending from a faucet in a vertical electric field. When the field strength exceeded a critical value, the drop emits a jet of fine droplets, a process which repeats itself periodically. More recently the Coulomb disintegration of droplets of dielectric liquids was investigated under better defined conditions. Charged droplets with a diameter of a few tens of p.m were kept in a Paul t r a p and their charge and radius were measured continuously over periods of several minutes 6 . 7 . 8 . 9 . 1 0 . One observes the expected decrease of the radius in time because of evaporation of uncharged molecules. The charge remains constant until a critical fissility is reached. T h e n an abrupt loss of a finite fraction of the total charge is seen without any discontinuous change of the radius at the same t i m e . This has been interpreted as an indication for Rayleigh's Coulomb jet. After the charge burst the charge on the droplet remains constant until the next burst occurs after evaporation had again reduced the radius sufficiently much. These observations would fit perfectly into Rayleigh's picture if the bursts would not be observed at a fissility x well below 1. In the following I will discuss possible reasons for a Coulomb instability at x < 1.
2.2
Fission-barrier
heights of rotating,
surface-charged,
spheroidal
droplets
Since the droplets in the Paul t r a p have a finite t e m p e r a t u r e , also their rotational degrees of freedom are excited at thermal equilibrium. If the reduced fissility x = 1 — x is different from zero, but still small, the shape of a surface-
119 charged droplet, which corresponds to the fission barrier, has been shown t o be well described by a prolate spheroid 1 1 . I will therefore discuss the effect of charge and rotation on the equilibrium shape and the fission-barrier height of a droplet within the shape class of prolate spheroids. A spheroid shall be characterized in the following by its numerical eccentricity e and the radius R of a sphere with the same volume, so t h a t the large and the small axis of the spheroid become a = R(l — e 2 ) - 1 / 3 and 6 = R(l — e 2 ) 1 ' 6 , respectively. The Coulomb and surface energies of a prolate spheroid are
* -
= ^ - « " ) " • ' • 1^7
and Esur
= 2itaR2{\
- €2)-l/6
- f2 + \ arcsin e
(Vl
respectively, and the rotational energy for rigid rotation around the small axis is
Emt
5 P (l_e2)-2/3 2 MR?
—
+
( 1
_
,2)1/3
in t e r m s of the angular m o m e n t u m J and the droplet mass M. It is useful to express charges in units of the fissility Q2
\^Cout J sphere
16TTR3
2(E surfsphere
and
12
angular m o m e n t a in units of y
P
\£Z/rot jsphe
MR4a'
16TT
\&sur)sphe
Measuring the deformation energy of a droplet in units of the surface energy of the sphere with equal volume, yields (Esur
Uf{c,x,y)
+ Ecoul
=
+
£,
rot)(spheroicJ.Sphere)
r^—r—^
'-
\^ sur ) sphere
e(\ — e 2 ) 1 / 2 + arcsin e 2e(l-e2)i/6
1+ x
('-""'inl+i-S e
1 -e
•2(l-e2)2/3
+y
2-e2
- 1
(1)
120
This expression does not explicitely contain specific material parameters or the size of the droplet. These quantities are absorbed in the dimensionless parameters x and y and in the energy scale (Esur)sphere- The experiments of ref. 10 were made with droplets of glycol. For glycol one obtains y = 3.4610- 5 AT- 7 / 3 f £ )
1 = 4.15^-,
,
(Esur)sphere
= 3585
kBTN2'3
in terms of the numbers of molecules and elementary charges TV and Z, respectively, and the Boltzmann constant kB • For a glycol droplet with temperature T one finds in equilibrium kBT
(vh
0.002
I
1
,
.
,
.
1.4-10' 4 JV- 2 ' 3 r( K).
8TTR2O-
,
1
I
,
I
,
(2)
,
0.02 • N M
0.001
0.01 • N " 1
Si-0.15/
0.000
*'-~-^
012
'
0.00
NNN/^V^M
\ -0.01 • N m
-0.001
V \ \p.06\
\ 0.03 \
-0.002
\ \ \ o.oo\
1
-0.003 0.0
1
0.1
1
1
1
1
0.2 0.3
1
1
.
l\
-0.02 • Nm
\
\ \ \
l\
\
\ 1 \ .
0.4 0.5 0.6
\
\ 1 1
0.7
Figure 1: Deformation energy as a function of the deformation t2 and the reduced fissility x for y = 0.01. The scale on the left margin gives the deformation energy in units of (ESur)sphere > the right margin is for glycol droplets in units of kgT for T = 300 and particle number N.
Fig. 1 shows £def (e) for y = 0.01 and various values of the reduced fissility x. The deformation energy has a minimum at finite deformation e, which moves
121 t o the right with increasing charge, i.e. decreasing x. It merges with the fission barrier for x m 0.11. For still larger charges there is no fission barrier any more. A more compact representation of the topological structure described by the function (1) is shown in Fig. 2. T h e lines give the deformation e1xtr(x, y) for which idej has an e x t r e m u m with respect to e as function of x and y. Below the thick, almost straight line these extrema are minima (stable equilibrium points), above the line they are m a x i m a (unstable equilibrium points). At the thick line they merge in an inflection point with horizontal tangent.
X Figure 2: Deformation t2extr at the extrema of the deformation energy as a function of S and y-
T h e figure is restricted t o ranges in e and x where the representation of equilibrium shapes of droplets in t e r m s of spheroids is approximately justified. One may however wonder how t h e lines would continue to larger values of e if one formally extends Eq. (1) t o the vicinity of the singularity at e = 1. In fact, one finds another set of m i n i m a , corresponding to very elongated shapes, not necessary of physical interest. They are separated from the m a x i m a in the upper part of Fig. 2 by a second line of inflection points with horizontal tangent. These two lines of inflection points, for which <9c£de/ = d^dej — 0, meet in a cusp at x = 0.1582, e 2 = 0.5912, y = 0.0182. T h e rather simple, closed-form expression (1) is seen to contain a sufficienly rich structure to allow a nice exemplification of some of the results of the theory of catastrophs. T h e difference between the first m i n i m u m and the m a x i m u m is the fission
122 1"
—!
* i
|
i
1
i
1
1
!
'
"1
1
y - 0.00
0.0020
/ 0.002
0.0015
-
/
0.02 • N »
/ 0.004
I I I °-"°6 Sbar
0.0010
/ I I I °-00B / / / 1 /°'°1
0.0005
0.0000 0.00
i
1
0.02
i
i
\
0.04
0.01 • N «
y^^/y^y y y°m2 0.06
0.08
0.10
0.12
0.14
X Figure 3: Barrier height as a function of x and y. The scale on the right margin is for glycol droplets of N molecules and a temperature of 300 K in units of kgT.
barrier £bar in this restricted shape class. This quantity is plotted in Fig. 3 as a function of x and y. On the right of the frame the barrier height is presented in units of ksT, assuming T = 300 K and droplets of glycol. For particle numbers N of the order of 10 14 Eq. (2) shows that one has essentially y = 0 and £bar w 0 in the scale of Fig. 3. For droplets of this size one therefore finds no reduction of the critical fissility from the classical value 1, neither by thermal excitation nor by rotation. The situation is however quite different for clusters with, say, TV = 1000 molecules. Although (J/)T is still of the order of 10~ 6 , the barrier is seen from Fig. 3 to become equal to the temperature for x = 0.09. Such clusters are therefore expected to fission at room temperature with a fissility noticeably smaller than 1. 2.3
Rayleigh-Taylor instability of the spheroid
If one equates the outwards directed Maxwellian tension of a surface-charged, spherical droplet with the inwards directed surface tension 1 8TT
{dnV)l=R = 2a H,
(3)
where dnV is the normal derivative of the electrostatic potential V on the surface and H the mean curvature, one obtains the same stability condition,
123 i = l , which Rayleigh derived for the quadrupole deformation. This somewhat confusing coincidence is rather fortuitous and connected with the high symm e t r y of the sphere. Already for the spheroid, stability against global changes of the shape and the local, mechanical stability of each piece of the surface do not lead to the same stability conditions. To show this one needs t h e mean curvature for the surface of a spheroid as a function of the angle 6 with respect to the s y m m e t r y axis
... [( 2 - g 2 ) - ( 3 -
2g2 f2cos2
)
2
2
2
°][l -
g2cos2
3 2
9 12
}'
2
2R[1 - (2 - e )e cos 0] / [l - e ]V6 and the Maxwellian tension on the surface of a spheroid with total charge Q
el
Q2 (1-e2)1/3 87rJR4(l-e2cos2^)'
Generalizing the notion of the fissility x, one may define a local fissility x(0) r<0\ [)
- ^£L ~2aH~
J(2-f2)-(3-2eVccs2e] 2(l-e2)i/2
3/2 2
2
2
1 - (2 - e )e cos 9
in t e r m s of which the limiting condition (3) is x(6) — 1. T h e ratio x(6)/x is plotted in Fig. 4 for various eccentricities e. For the sphere, e = 0, Rayleigh's result x = 1 for the stability limit is of course recovered. For finite eccentricity however, there are certain ranges of 6 for which x(0)/x > 1, indicating a local loss of stability of the surface for a fissility x < 1 in this angular range. 3
Q u a n t a l effects a r o u n d s c i s s i o n
For two touching, classical, conducting, charged droplets the split of the total charge is unambiuously determined by their shape. This follows from the condition t h a t the surfaces of b o t h nascent fragments should be one equipotential surface at the scission point. For instance for two touching spheres with radii R\ and i? 2 Qi 7 + *(/?) = Q1 + Q2 27 + 2*(/?)+7rcot7T/?' where /3 = R2/{R\ + R2), 7 is the Euler constant, and \t is the derivative of the T f u n c t i o n 1 3 . W h e n the total charge is however only a few times the elementary charge, the granular charge can obviously no longer be described in terms of continuum electrostatics. For not too large metallic clusters the valence electrons m a y be
124 1.3 1.2 1.1
x(e)/x
1 .o
e» = 0.00
0.9
V=0.50~"
0.8
e*-0.75."'
e9 - 0.25
0.7 71/4
71/2
e Figure 4: Angular and eccentricity dependence of the generalized fissility.
treated as independent particles moving in a common potential well. The same is true for the nucleons in a fissile nucleus. When the single-particle potential is cut into two separated, unequal wells by an externally driven potential barrier, each level in the original potential correlates unambiguously to a level in one of the separated potential wells (disregarding accidental degeneracies of levels in the two unequal fragments). For a given volume asymmetry of the separated wells one can therefore label each level by, say, R or L according to whether it eventually correlates to a state in the right or the left well of the separated system. For the Hill-Wheeler box such correlation diagrams are shown in Ref. 14 for various asymmetries. In order to obtain a labelling scheme which is independent of the details of the fission path and depends only on the asymptotic asymmetry, I shall specifically refer to the diabatic level scheme, i.e. across avoided level crossings the levels with the same nodal structure are equally labelled. All along the fission path one can define the number of particles occupying i?-states or L-states. The corresponding particle number operators are
** = X>* H ak,
R
and
NL
5X
where UR and ki run over all states of class "R" or "L", respectively and a£ is a particle creation operator in state k. The operator of the total particle number is N — NL + NR and the operator of the particle-number asymmetry
125
is M = NL — NR. For a given time-dependence of the single-particle potential the evolution of the expectation value of M along the fission path is obtained by solving the equation of motion ihdM/dt = [H, M] in a suitable basis. Both mean-field approaches, the time-dependent shell model and the timedependent Hartree-Fock approach suffer from the deficiency that they cannot properly account for the split of the flux into exit channels with different asymmetry of their associated mean fields since they admit only one mean field at a time. This leads to a drastic underestimate of the charge and mass variances in self-consistent nuclear fission calculations 15 , incorrect branching ratios between exit channels with different charge states in atomic collision calculations 16 and would lead to incorrect charge distributions among the fission fragments of charged metallic clusters, if they were treated in a self-consistent mean-field approach. The remedy for this problem is provided by the Hill-Wheeler method 17 : One extends the mean-field concept by first seeking a stationary solution of the Schrodinger equation in the basis
|*> = X>(m)|#F) m m
at each point along the fission path, where \HF)m is a mean-field solution of the stationary n-body problem with the constraint m(HF\M\HF)m = m, and the Hill-Wheeler amplitude <j>(m) is determined from minimizing the hamiltonian with the constraint (\?|\P) = 1. The time-dependent wave function is then expanded in the basis | ^ ) and propagated in time with the time-dependent Schrodinger equation. The scheme has been implemented for atomic collisions 16 and partly for nuclear fission in Refs. 18,14 . 4
Conclusions
Although there are circumstances under which the critical fissility of a classical, surface-charged droplet may be smaller than the Rayleigh limit, that cannot explain the results reported in Refs. 9 , 1 ° . Arguments are presented that the microscopic treatment of scission of any fermion system has to go beyond the mean-field approximation. The generator-coordinate approach is recommended as the minimal extension of the mean-field theory which is necessary to describe the dynamics of the particle-number asymmetry around scission. References 1. O. Hahn and F. Strafimann, Naturwiss. 27, 11 (1939).
126
2. L. Meitner and O. R. Frisch, Nature 143, 239 (1939). 3. C T. R. Wilson and G. I. Taylor, Proc. Camb. Phil. Soc. 22, 728 (1925). 4. Lord Rayleigh, Phil. Mag. 14, 187 (1882). 5. J. Zeleny, Proc. Camb. Phil. Soc. 18, 71 (1915); Phys. Rev. 10, 1 (1917). 6. M. A. Abbas and J. Latham, J. Fluid Mech. 30, 663 (1967). 7. J. W. Schweizer and D. N. Hansen, J. Colloid Interf. Sci. 35 417 (1971). 8. M. Roulleau and M. Desbois, J. Atmosph. Sci. 29, 565 (1972). 9. D. C. Taflin, T. L. Ward and E. J. Davis, Langmuir 5, 376 (1989). 10. O. Hiibner, Diplomarbeit, Fachbereich Physik, FU Berlin (1997) and T. Leisner, private communication. 11. V. V. Pashkevich, H. J. Krappe, and J. Wehner, Z. Phys. D 40, 338 (1997). 12. S. Cohen, F. Plasil, and W. J. Swiatecki, Ann. Phys. (N.Y.) 82, 557 (1974). 13. H. Buchholz, Elektrische und magnetische Potential/elder (Springer, Berlin 1957), Sec. 5.11. 14. H.J. Krappe and S. Fadeev, Nucl Phys. A645, 559 (1999). 15. Moreau and Heydes, in The Nuclear Fission Process, ed. C. Wagemans (CRC, Boca Raton USA, 1991) p.227. 16. J. Eichler and T. S. Ho, Z. Phys. A311, 19 (1983). 17. P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, New York 1980), chapters 10 and 11. 18. J. F. Berger, M. Girod, and D. Gogny, Nucl. Phys. A428, 23c (1984); Nucl. Phys. A502, 85c (1989).
127 ENERGY-DENSITY FUNCTIONAL APPROACH TO FISSION AND HALF-LIVES OF SUPER-HEAVY ELEMENTS
I. REICHSTEIN School of Computer Science, Carleton University, Ottawa, Ontario K1S 5B6, Canada F. BARY MALIK Physics Department, Southern Illinois University, Carbondale, Illinois, 62901, U. S. A. The energy-density functional theory has been used to calculate nuclear masses with observed density distributions and fission half-lives of some actinides. The theory can account for the masses, and half-lives of these actinides and mass distributions in spontaneous fission of U0P\x. The upper limits of half-lives spontaneous fission of four super heavy elements are presented and found to be short which is in line of the prediction done in 1972.
1
Introduction
In the binary fission process, a parent nucleus undergoes a transformation depicted in Fig. 1, to emit a pair of daughter of nuclei. This clearly involves clearly a dramatic reorganization of the density distribution function of the parent nucleus, since the central density as well as the surface thickness of both the parent and the daughters is about the same. In case, the number of nucleons in the surface region of a nucleus is substantial, as noted by Whetstone [1], configuration E, i.e., a configuration with a neck consisting of nuclear matter with a density lower than the saturation density, is likely to occur prior to scission. The formation of the Whetstone configuration of lowdensity nuclear matter substantially modifies the potential barrier between the saddle and scission points. This was first noted by Bloch et al. [2] and subsequently established by Reichstein and Malik [3, 4] using energy-density functional (EDF) approach. In the next section, we briefly establish that the number of nucleons in the low-density-Whetstone-neck forms a significant fraction of the total number of nucleons and we present a brief outline of the EDF approach to fission. In section 3, we compare the calculated mass distribution of 240Pu, and the half-lives of fission of a number of nuclei with the data. We also present the calculated half-lives of super heavy element in that section.
128
Schematic of Density Reorganization During Fission Black Areas represent zones of nearly constant density
Figure 1. Schematic configurations involving density reorganization during fission. constant density, which gradually drops to zero at the surface.
Black areas arc zones of
2 The EDF Theory of Fission As noted first in [3] and subsequently, in [4,5], it is easy to establish that the number of nucleons in the surface of a nucleus forms a significant fraction of the total. Using an appropriate trapezoidal function to represent the observed nuclear density distribution, one can easily calculate that the fractions of nuclear matter in the central-constant-density zone of a nucleus and in the surface area of a nucleus of mass number A=125 are, respectively, 49.2% and 50.8% for a surface thickness of 2.5 fm. Similarly for A=238 the fractions are 61.5% and 38.5%, respectively. In case the surface thickness is taken to be 3.0 fm, these numbers change to 43.2% and 56.8% for A=125 and to 56.1% and 43.9% for A=238. Clearly, the Whetstone neck contains a significant number of nucleons [3,5]. The potential surface, V(R), may be calculated in the London-Heitler approximation by taking the energy difference between a configuration at a given separation distance R and the asymptotic configuration (F)forR = =[3,4], i.e., V(R)=E(R)-E\R\
= OO)
(1)
129 The energy of a configuration at a separation distance R is generated using EDF theory and the densities U\(R) and U2(R) of a daughter pair. The energy of a well separated daughter pair, E(R==) may either be taken from the data on masses of a daughter pair or calculated using EDF. Thus, V {R)= £'(p.(R>,p2(R))- £(pi(l R l= oo))-£(p 2 (| R |= oo)) (2) In the EDF theory, the energy corresponding to a density U(r) is given by £ = ° . (p(r)>/r (3) We adopt here the energy density £(p(r)) from [6, 7] and it is given by * (p)=T(p)+ PVm(p)+Vg(p)+Vc(p) (4) In (4), T(U) is the contribution from the kinetic energy, VJQ), the mean field contribution, Vg(D), the inhomogeneity term originating from non-uniform density distribution and VC(D), the Coulomb energy and are, respectively, given by
Tip)=(3/5iti2/2M\37C/2fn(l/2tl-ay,3+{l 2
2
4 3
+ ar]p5n
Vm(p)=fci(l + a,a )p+fo(l + a2a )p ' + bi(l + aia2)psn
(5) (6)
V*(p)=r,(fc2/8MXVp)2 V c (p)=(l/2X*/2) 2 (l-a) 2 Jdr'p(r')/lr-r'l
(7)
-1.0636[(l/2Xl-a)] 4/3 p ,/3
(8)
Vm(0) is the parameterized Hartree-Fock potential calculated from the twonucleon potential of Gammel and Thalr using t-matrix method [7]. • is the neutron excess = (N - Z)/A and the second term in (8) is the contribution of the Pauli exclusion principle to the Coulomb potential. Since the central density and surface thickness parameters of a parent and its daughter pair are about the same, the energies of intermediate configurations of Fig. 1 are calculated in the special adiabatic approximation [8, 3] where the half-density radius C and the surface thickness parameter, t have been varied as (x = C or t) Xi(R)= XP(R = o)exp\en(Xd(R = <*>)/XP%RIRc«f\ R < Rem
130
= Xd(R = °°) R > Rcu, (9) The subscripts d and p in (9) stand for a daughter and its parent and Rcu, is the cut off radius where C o r ; achieves their respective values for a daughter nucleus. Finally, the mass number, A, at every stage must be conserved i.e., A = ldr'p(r') (10) At the scission point, the kinetic energy, T, of a daughter pair is equal to its Coulomb repulsion. T may be taken from the data or calculated using the prescription of Terrel [9]. In cases where the kinetic energy of a fission fragment is unknown, it is calculated using the prescription in [3]. Finally, the decay probability of a parent nucleus is equal to (the preformation probability of the daughter) (penetrability through the barrier). The preformation probability calculated in [2] is 1.6 x 10 "* for the even-even parent and 1.6 x 10 "2 for the odd-even parent nucleus. The decay probabilities though the potential surface are calculated in the WKB approximation [3]. 3. RESULTS A crucial test of the EDF approach is to examine the extent to which the theory can calculate observed masses [10] using the observed densitydistributions [11]. This is done in Table 1 and the observed masses can be reproduced, in most cases, with an accuracy of 1% or better i.e., as well as those obtained using the best mass formula based on the liquid drop model [12] which assumes a constant density distribution function. Thus, it is possible theoretically to obtain nuclear masses using observed density distribution. Table 1. Calculated binding energies in MeV, B.E(Th), using observed density distribution functions [11] listed in Column 2 are compared to the experimental ones [10] noted in column 4 for a few elements listed in column 1.
Element li
C
16 Q
'"MR Z8 Si
*Ca iu
Cr
M
Ni
,,4
Cd Sm **Pb
148
•mv
Density Function
2pf 3pf 3pf 2pf 3pf 2pf 2pf 2pf 2pf 2pf 2pf
B.E. (Th) MeV 92.5 125.2 194.5 234.3 340.6 439.9 550.4 984.1 1229.6 1630.1 1808.6
B.E. (Expt) MeV 92.2 127.6 198.3 236.5 342.1 435.0 545.3 972.6 1225.4 1622.3 1801.7
131 DENSITY CONTOUR SPHEROIDAL CASE
234
U —
"*2XE +
92
SR
R= 10.4 fm
R=I2.4 fm
R=14.4 fm
Figure 2. Calculated density contours in the fission of 234 U to 142Xe and 92Sr from separation distance of 10.4 fm to 14.4 fm. The contours are generated assuming each member of the daughter pair to be spheroid of eccentricity 0.48.
In Tables 2 and 3, the calculated half-lives to the fastest decay mode using observed kinetic energies for spherical and spheroidal daughter pair are compared to the observed values. The agreement is reasonable. Table 4 presents the decay of 240Pu to a number of decay modes using observed kinetic energies and once again the theoretical approach can account for the mass distribution which is peaked at the asymmetric mode, as observed.
132 Table 2. Calculated spontaneous fission half-lives in years noted in Column 5 for the fission of elements listed in Column 1 to their fastest decay modes noted in Column 2 using observed total kinetic energies listed in Column 3 [18-20]. The density configurations are generated by superimposing two spheres resulting in spherical daughter pairs. The observed logarithm of half-lives is noted in the last column [18]. Parent
Daughters
234,, U92 ^ , 2 MU Pu M
142y
Cm 9 6
,m
C£*
^Cf98
*i|rin (exp) (MeV)
Ekta (MeV)
Logioti/2 yrs (th)
Logioti/ 2 yrs (exp)
168.5 171.3 175.0
168.5 171.3 175.0
16.2 13.7 11.1
16.2 16.3 11.1
Da S 6 + Zr 40 Ba56+ &40 Ba 5 6 + Z140
185.5 ± 5
183.0 185.0 187.0
8.8 8.4 8.1
7.1
Sm62+ KT36 MI162+ Kr 36 Sm42+ KT36
188.7 ± 1 . 3
190.0 188.0 186.0
2.7 3.0 3.3
3.8
Xes4 + RU44
187.0
187.0
3.3
1.9
«2Q
Ae54+
Sr 38
142Bn56 + !>IlS
Table 3. The same as the one for Table 2, except the density configurations are generated assuming spheroidal configurations of eccentricity • noted in Column 4 resulting into spheroidal daughter pairs. Parent 234,, U92
m
un
""Pi* Cni96 MS Cf98 iW Cf 98
258c„ Fniioo
Daughters x e 5 4 + br 3a vCe54+ Sr3 8 WV SIS,; l44 rsa56+ ,uoM3 8 Ba5 Zr4o 6+
Sm62 + KT36 "" , Ba 56 + "x,Mo42
13Uc„ 13UC„ 0I150+ ^nso
Etui (MeV) 168.1 171.6 175.0 185.5 188.7 186.5
200.0 215.0 235.0
• 0.48 0.44 0.53 0.55 0.54 0.68
Log I0 tl/2 Yrs (th) 16.5 13.2 11.1 8.4 2.2 1.6
Logioti/2 yrs (exp) 16.2 16.3 11.1 7.1 3.8 1.9
0.69 0.56 0.15
-3.7 -4.9 -7.4
-10.4
133 Table 4. Calculated logarithm of half-lives noted in Column 5 of spontaneous decay of ""Pu to a number of spheroidal daughter pairs of eccentricity. • , noted in Column 4. The kinetic energies used are from [20]. Parent
Daughters
•
Logiotm (years)
0.61 0.62 0.58 0.53 0.53 0.44 0.47 0.67
9.1 10.7 11.2 11.1 11.8 10.8 11.4 16.8
(MeV) ""Pu*
SIW
160 c „
S e M + Nd<j<> KJ 3 6 + Ce 5 8 me' 142D„ ar38+ ua56 10U7, 140^"40+
1W
ABJ4
Mo 4 2 + 1 M Te S 2 Ru44+ S1150 Pd46+ Cd 4S
157.9 161.5 167.9 175.0 177.6 185.4 185.5 167.9
In [3, 10, 14] the half-lives of super heavy elements were predicted to be very short However, the isotopes considered were different from the ones now being examined experimentally [14-17]. In Table 5, we present the upper limit of the fission half-lives of isotopes of a few superheavy elements. It confirms the predictions of [13] in 1972 and of [3, 4] that the spontaneous fission half-lives of elements 112,114, 118 and 126 are, indeed, very short. The calculated halflives are the upper-limits since they have been calculated for a scission radius of about 19.5 fm corresponding to spherical daughter pair. Table 5. The predictions of upper limits of half-lives (Column S) in years and in sec (Column 6) and one of the fastest decay modes (Column 2) of some superheavy elements (Column 1). Columns 3 and 4 refer, respectively, to kinetic energy of the daughter pair in MeV and scission radius in fm (A: Mass number, Z: Atomic charge). A(I) Parent »(112) m (114) ,JW (118) Jll) (126)
AandZ Daughter '45(56)+'™(56) Ma (58)+ Iiu (56) "»(59)+' w (59) ,5,, (64)-i-l:,4(62)
Eldn MeV 237 245 298 302
Rsc fm 19.5 19.5 19.5 19.3
Tvi(year) s
2.2x10 1.5x10"* 1.1x10"" 8.3xlO"10
Tvi(sec) 6.95x10* 4.73xl0 +1 3.47xl0"5 2.61x10*
References 1. Whetsone Jr., S. L., Phys. Rev. 114 (1959) pp. 5 8 1 . 2. Block, B., Clark, J. W., High, M . D., Malmin, R. and Malik, F. B . , Ann. Phys. (NY) 6 2 (1971) pp. 464. 3. Reichstein, I. and Malik, F. B., Ann. Phys. (NY) 9 8 (1976) pp. 322. 4. Reichstein, I. and Malik, F. B., Super heavy Element ed. M.A.K. Lodhi, Gordon Breach
134 (1978). 5. Reichstein, I. and Malik, F. B., Clustering Phenomena in Atoms and Nuclei, eds. Brenner Lonroth and Malik (Springer Verlag) (1992) pp. 144. 6. Brueckner, K. A., Buehler, F. R. and Kelly, M. M., Phys. Rev. 173 (1968) pp. 944. 7. Brueckner, K. A., Coon, S. A. and Dabrowski, J., Phys. Rev. 168 (1968) pp. 1184. 8. Reichstein, I. and Malik, F. B., Phys. Lett. 37B (1971) pp. 344. 9. Terrel, J., Phys. Rev. 127 (1962) pp. 880. 10. Wapstra, A. H. and Audi, G., Nucl. Phys. A 432 (1985) pp. 1. 11. H. de Vries, H., de Jaeger, W. and de Vries, C , At. Data, Nucl. Data 36 (1987) pp. 495. 12. MoUer, P., Nix, J. R., Meyers, W. D. and Swiatecki, At. Data Nucl. Data 59 (1995) pp. 185; ibid 66 (1997) pp. 131. 13. Malik, F. B., Magic Without Magic Ed. Klauder, J., (W. H. Freeman and Co., 1972) pp. 47. 14. Yu. Ts. Oganessian, et al. Phys. Lett. 83 (1999) pp. 3154. 15. Yu. Ts. Oganessian et al. Nature (Lond.) 400 (1999) pp. 242. 16. Yu. Ts. Oganessian et al. Eur. Phys. J.A 15 (1999) pp. 63. 17. V. Ninov et al. Phys. Rev. Lett. 83 (1999) pp. 1104. 18. Hyde, E. K., The nuclear properties of the heavy elements. 3 (Prentice Hall, Englewood Cliffs. N J) (1964) pp. 175. 19. Pleasonton, F., Phys. Rev. 174 (1968) pp. 1500. 20. Neiler, J. N., Walter, E. J. and Schmitt, H. W., Phys. Rev. 149 (1966) pp. 894.
135 D E C A Y CHANNELS OF H O T N U C L E I A N D H O T METALLIC CLUSTERS
P. FROBRICH Hahn-Meitner-Institut Berlin, Glienicker Strafle 100, D-14109 Berlin, Germany, also: Institut fiir Theoretische Physik, Freie Universitat Berlin, Arnimalle 14, D-14195 Berlin, Germany E-mail: [email protected] In the following we discuss some aspects of fission (and the acompanying processes) of hot nuclei and hot metallic clusters; we pay particular attention to the time scales of the various processes.
1
T h e d e c a y of h o t n u c l e i
Heavy hot nuclei are mainly produced in heavy-ion fusion reactions. In this case t h e compound nucleus has a well defined spin distribution and a well defined temperature (excitation energy), which determine the various decay routes. We treat the branching into the fission and evaporation residue channels by a Langevin fluctuation-dissipation dynamics, to which the emission of light particles (neutrons, protons, ar-particles, giant dipole -f's) are coupled with a n evaporation model. T h e theory, which is reviewed in Ref. 1 , allows to calculate in particular fission- and survival probabilities, and pre-scission light particle (n, p, a, 7) multiplicities and spectra. A corresponding computer code is available in Ref. 2 . The simultaneous reproduction of the experimental d a t a for fission (respectively survival) probabilities and neutron multiplicities, which has not been possible in any statistical model, was the first big success of the model, and is a proof for the fact t h a t friction (viscosity) plays an i m p o r t a n t role in the decay of hot nuclei. T h e model does not only allow to calculate the measured observables but also t o give a picture of the involved quantities along the fission path; for instance one can calculate which particles with which energy are emitted at which position and at which time. T h u s one obtains considerable physical insight in the decay process. As an example we show in Fig.l the t i m e distributions of fission and of pre-scission n, p, a, and 7 emission events for the decay of 2 0 0 P b produced in a 1 9 F + 1 8 1 T a heavy-ion collision at 125 MeV. Neutrons, protons and a-particles are mainly emitted before and giant dipole 7-quanta after the m a i n peak of the fission time distribution. Pay attention to the very long tail of the t i m e distribution for fission, due to which it is very i m p o r t a n t to distinguish between the mean fission t i m e and the most probable fission time, which
136 is much shorter.
60 . a)
(125MoV)"»F+«lTa SPS —Onion
^•40
J30 |
• -o>-paxticlei
I20 S1
gio
~40
025MeV)i»F+niT» SPS
c)
J 30 -
2 0
60 ~50
60 ~50 £ '40
v>
a25MoV)l»F+iWTa SPS —Anion
Jho
S
60 ~50 ~40
1—'—:
!—•—
(125MeV)i>F+H'T. SPS —limoa • •^quanta
d)
J 30
f20 fcio
S 20
"1
0 1 2 3 4 S 6 7 8 9
lg(time/10-2is) Figure 1: Time-distributions of pre-scission particles (n, p, a, -y, Fig. l(a)-(d))in comparison with the fission-time distribution for 1 9 F + 1 8 1 T a - f 2 0 0 P b at 125 MeV.
The percentage of light particles emitted during fission for the above reaction is given in the table.
Table 1: Percentage of light particles emitted in coincidence with fission.
n 98.60%
P 0.44%
a 0.54%
d 0.02%
7 0.39%
The corresponding results for not too heavy (< 2 5 1 Es) and not too light (> W) at not too high temperatures (T < 3MeV) can be obtained easily by running the code 2 which is written in such a way that various friction form factors can be chosen. 178
2
T h e decay of hot metallic clusters
Multiply charged metallic clusters are either produced by laser irradiation, e.g. Ref. 3 , or by crossing a heavy-ion beam with a cluster beam, e.g. Ref. 4 . In both
137 cases t h e t e m p e r a t u r e of the clusters is not very well defined, which complicates a theoretical analysis. It is, however, clear t h a t the hot clusters produced by heavy-ion i m p a c t are colder t h a n those produced by laser irradiation. 'Fission' in cluster physics is called a separation of charges with a very asymmetric separation of masses, but looks more like the evaporation of the emitted light charged particle. T h e reason for the difference to the nuclear case is the role of the Coulomb interaction: nuclei are homogeneously and metallic clusters surface charged. A quantity of interest is the critical (or appearance) size of multiply charged clusters, which is defined as t h a t size of a cluster with a particular charge below which a cluster becomes unstable against fission, or above which cluster size distributions with a particular charge appear. We determine these critical sizes from an evaporation model as t h a t size for which the r a t e for the emission of neutral monomers becomes equal to the rate of 'fission', i.e. to the rate for the emission of the light charged particle. We apply r a t e formulas based on the Weisskopf theory 5 , see in particular Ref. 6 . Consider the decay of a cluster X^ with charge z and size n into a heavy cluster YnZm of charge z — k and size n — m and a light fragment 3/^ with charge k and size m
+ ykm.
K^Y^
(1)
Rates for this process are determined by Kkm{E)
=
3^ ir2h px{E)
/ 7o
""* deeaF{e)PY(E
- D^m -
c)
.
(2)
Here crp(e) = irR2(l — Vcb/e) is the cross section for the (inverse) fusion process, e t h e kinetic energy of the emitted light fragment, fim the reduced mass of the exit channel, and py(x) a r e the level densities of the final (initial) cluster. T h e level densities for clusters with a high melting point (e.g. for alkaline earth clusters, they are produced in the solid phase) are taken from an Einstein model, whereas the level density for N a clusters (produced in the liquid phase) are calculated from the measured specific heat for the bulk. D^m is the separation (dissociation) energy of the emitted light partner, taken from a liquid drop model for clusters. For the Coulomb potential we use an empirical formula y _ p «(z — k)
rsKn-my/s
+ mV^ + S'
K>
where J is a spill-out factor, C is an universal (approximately the same for all systems) reduction factor of the monopole potential due to dipole polarisation, and rs is t h e Wigner-Seitz radius, which in the case of Na clusters is treated
138 Appearance sizes (experimental)
(theoretical)
0
O
400
*"
• o
1
®
o alkali
O NaXHb.Cs
0 ©
S. 200
8-
o
300
Na
@ O
e o e
200
O
o 100
§
•
alkaline earth < • • *T) Mg,Ca,Sr,Ba
.!.•3 cluster 4 5 6 7 charge z
8 8
8
ft Mg,Ca,Sr,Ba
ft cluster charge z
Figure 2: Comparison of experimental and calculated critical (appearance) sizes.
as temperature dependent, and is calculated from the measured thermal expansion coefficient of the bulk. In Fig.2 we show measured critical (appearance) sizes for Na 3 , 4 and alkaline earth clusters 7 in comparison with theoretical ones 9 . There is reasonable agreement between theory and experiment. Note, however, that the calculated rates for Na (in case of the experiment 4 ) are somewhat too long as compared to the available experimental time window. This might be due to the applied thermal expansion coefficient of the bulk, which certainly is smaller than that of a free cluster, as molecular dynamics calculations suggest 10 . Also shell structures might become important at the lower temperatures for determining the separation energies. In connection with the discussion of time scales we show in Fig.3 the cooling and shrinking of a large neutral Na cluster of size n = 1000 with an initial temperature of T = 1100°K due to sequential emission of monomers. Since there are experiments, e.g. Ref. 8 , concerning the emission of radiation from clusters (which seems to look like black-body radiation), we include the emission of radiation in the equations for shrinking and cooling of the Na clusters. dn ,00 = Rn i l ~dl
139
KUn\ + \el)
j4 dt dT_ ~dt
1
dt
On
J4
dt
~ V
nl>/
M dt
)rad
(4)
dT •
Here i?°° is the emission rate of equation (2). The kinetic energy of the emitted particles is given by (e) = 2T. The expression ,dE.
R2f
(—) v
dtJrad
. = E\-KRT
2
&4r4
=—
(5)
wn3c2
is the Stefan Boltzmann law. We have used an emissivity e — 1 which correponds to black-body radiation. In this way we obtain only an upper limit for the influence of the radiation on the evaporation, because the emissivity of Na is certainly smaller than unity. It remains the problem to find a more realistic value for the emissivity of radiation from Na. In Fig.3 we see in particular that up to a time of t < 10 _ 3 s evaporation is the dominant process. Only after this time radiation takes over when there is not enough energy left for particle emission. The shrinking of the cluster stops but it continues to cool down due to radiation. Cooling and shrinking (black body radiation) Competition between evaporation and radiation
1200
I uiiq m i n i unity I I unq i iiiHfl unity unity mm
unity iiiiail unity unity m i * l l n « | unity I mm
1000
1000 h-
800
| S.
600
K o
400
temperature
200 M|
* 10-1""' *< 1 0 H 0-t &I0-910«10-710<10-S10-« 10-310-210-1 ioo 101 102 103 104 time (sec)
Figure 3: Cooling and shrinking of a Na cluster of size n = 1000 and with an initial temperature T=1100°K as function of time (with and without taking radiation into account).
140
Finally, we investigate the decay of Nafg for which microscopic calculations n show that shell effects lead to a barrier for symmetric fission N a]^" —>• Na^ + No.^ which is of the same size as the barrier for the asymmetric channel Nafg —>• Naf5 + Na^ . From energetic arguments one then would expect that symmetric fission could well compete with the emission of a charged trimer. We show now that this is not the case when viscosity comes into play because it can, as in the nuclear case, slow down the fission process considerably. First, we calculate the fission rate by the statistical (no viscosity) model of Bohr and Wheeler 12 , which reads when using an Einstein model for the level density 1 A
oBohr—Wheeler /—*
=
^BT
(
(1
Bj
^3n-6, )
2^f -(3n-6)*B:r
pE-Bf '
.
kBT
Bf
LlTj^^n^-^-
(6)
This formula leads to fission rates which are larger than those calculated with equation (2) for trimer emission, see Fig.4. We now introduce friction (by using the measured 13 viscosity coefficient H for the bulk) and apply Kramers formula 14 for overdamped motion (overdamped motion turns out 1 6 to be even better justified for fission of hot clusters than for fission of hot nuclei) R^,.(T)=^iJa9(_Bj_y
(7)
The curvatures Vngs(sd) in the frequencies ugs{sd) = \/\Viigs(sd)\/Mgs(sd) for ground state (gs) and saddle (sd) are estimated from the potential of Ref. n . The reduced friction parameter is defined as usual /? = 77/M = 4nRofi/M. If the mass parameter is assumed to be the same at ground state and saddle it drops out of the formula for overdamped motion. Calculating the fission rate with equation (7) it falls below the rate for trimer emission 15 , see Fig.4. Calculations within Langevin dynamics 16 confirm the results obtained with the simplified formula (7). The Langevin equation for overdamped motion reads in discretized form which is suitable for numerical integration
*»+i = *• - ( ^ ^ ) » ' - f ^ r »(*»),
(8)
where q is the distance between the center of masses of the future fission fragments, V(q) is the potential taken from reference 11 , T is the time step for
141
numerical integration, and w(tn) is a Gaussian (^-correlated random variable with zero mean: (w(tn)) — 0, (w(tn)w(tn')) — 26nni. A time dependent fission rate is then calculated by sampling trajectories Af; which have fissioned in the time bin i R(t) =
dNjjt) dt
Ntot-Nj{t)
Ni
N At
Y.U >
Ntl
(9)
Here Ntot are the total number of trajectories, Nf (t) are the number of trajectories which have fissioned at time t, and At is the width of the time bin. The Langevin rates at large times are close to the Kramers rates and are also entered in Fig. 4.
fission: Naf;->NaJ+NaS and evaporation: Na,, Naf, Na; -•
10"
r
10m
F
109
r
1
—
1
107
CD CO
(O CD +•* CO
V
?
A
106 r 105
V •
103 102
i — i —
1 —i
a
V
•
• •
£
V
*a
r——>
A
*
a
8 B Bp
A A
10*
-•—r- —
A
10»
o
'
•n
500
:
}
•
\
R(Kr)
1
v
Naj
!
n
Nat
i
o
Na,
]
•i
o
400
1
A
• o
100 r
300
ft
1
O R(Langevin) < A R(BW)
o
101 r 10-1
|
A
600
700
800
900
1000
1100
1200
temperature (°K) Figure 4: Rates for the decay of NaJ^: Langevin rate for fission (R(Langevin), open diamond), Kramers rate (R(Kr), solid square), Bohr-Wheeler rate (R(BW),solid triangle up), and emission rates for charged monomers (Na^ ) and trimers ( N a j ) and neutral monomers (Na,).
References 1. P. Frobrich, I.I. Gontchar, Phys. Rep. 292 (1998) 131. 2. I. Gontchar, L.A. Litnevsky, P. Frobrich, Comp. Phys. Comm. 107 (1997) 403.
142
3. U. Naher, S. Frank, N. Malinowski, U. Zimmermann, T.P. Martin, Z. Phys. D 31 (1994) 191. 4. F. Chandezon et al., Phys. Rev. Lett. 74 (1995) 3784. 5. V. Weisskopf, Phys. Rev. 52 (1937) 294. 6. P. Frobrich, Ann. Physik 6 (1997) 403. 7. M. Heinebrodt, S. Frank, N. Malinowski, F. Tast, I.M.L. Billas, T.P. Martin, Z. Phys. D 40 (1997) 334. 8. U. Frenzel, U. Hammer, H. Westje, D. Kreisle, Z. Phys. D 40 (1997) 108. 9. P. Frobrich, Czech. J. Phys. 48 (1998) 799. 10. N. Ju, A. Bulgac, Phys. Rev. B 48 (1993) 2721. 11. B. Montag, P.-G. Reinhard, Phys. Rev. B 52 (1995) 16365. 12. N. Bohr, J.A. Wheeler, Phys. Rev. 56 (1939) 426. 13. CRC Handbook of Chemistry and Physics, 49th edition, ed. by R.C. Weast, pp. F-59, and 76th edition, ed. by D.R. Lide, pp. 6-261. 14. H.A. Kramers, Physica 7 1940 284. 15. P. Frobrich, Phys. Rev. B 56 (1997) 6450. 16. P. Frobrich, A. Ecker, Eur. Phys. J. D 3 (1998) 237.
143
CLUSTER R A D I O A C T I V I T Y A.A.OGLOBLIN AND G.A.PIK-PICHAK RRC "Kurchatov Institute", PL I, V.Kurchatov 1, Moscow 123182, Russia E-mail: [email protected] and [email protected] S.P.TRETYAKOVA FNLR, JINR, Dubna E-mail: [email protected] Total mass distributions of the fragments emitted in cold decays of nuclei with A > 100 were calculated. Their analysis shows t h a t the known domain of cluster radioactivity connected with formation of 20&Pb or its neighbours is not distinguished from other possible decay modes. The main factors influencing decay probabilities are shell effects and fragment deformations. A brief review of recent experiments on cluster radioactivity is given. The d a t a and calculations show that the emission mechanism seems to be different in different parts of mass spectra. Light clusters are formed non-adiabatically (like a-particles), and emission of heavy fragments (cold fission) is an adiabatic process. T h e transition between both mechanisms takes place in the vicinity of the fragment mass A=35.
1
INTRODUCTION
The term "cluster radioactivity" is applied for spontaneous emission of light fragments heavier than a-particle in the decays of heavy nuclei. Nowadays there are known 18 nuclides from 221Fr to 242Cm emitting light nuclei from 14 C to 34Si. From this point of view cluster radioactivity occupies an intermediate position between alpha and proton radioactivity on one side and spontaneous fission on the other. The heavy fragments are grouped in the vicinity of the double magic 208Pb, and this allows to speak about the known domain of cluster decays as "lead radioactivity". A number of review papers are dedicated to this phenomenon, including the most recent ones [ 1,2 j. Lead radioactivity is far from being unique. Many other combinations of daughter nuclei are allowed energetically to be emitted including the formation of the products of comparable masses named cold fission. However, cluster radioactivity is a very rare process: the observed partial lifetimes lie in the interval 1 0 u -r- 10 27 sec, which corresponds to probability 10~ 10 -=- 1 0 - 1 7 of those of a-decays. Estimates show that many cluster decays should have lifetimes even more than 10 lo ° sec. In all known cases, except for one, the products of cluster radioactivity are formed in their ground states. Prom this point of view cluster radioactivity is
144
much closer to alpha-decay than to spontaneous fission, the process in which the both fragments are deformed and strongly excited. For this reason correct comparison of both processes can be done only if cold fission is meant because here the fragments are formed in their ground or low-lying excited states. However, cold spontaneous fission itself is studied even worse than cluster radioactivity. The question, what is the mechanism of cluster radioactivity, does it resemble either alpha-decay or fission, was widely discussed ( e.g.,[ 3 ]). The aim of this paper is to give a brief review of recent studies on cluster radioactivity, to get more general insight in different cold processes and to try to find out, how various physical effects influence their probability. We will consider the both cold processes, cluster emission and cold fission, together. The emphasis will be done to some open problems in the field among which we consider as most important the following: * Which domains of cluster radioactivity different from "lead" one can exist ? * What is the connection between cluster radioactivity and cold fission ? * Which nuclear properties ( shell effects, deformations, etc. ) define the probability of cold processes ? * What is the mechanism of cluster emission, is it "a-decay-like" or "fission-like" ? 2
N E W RESULTS A N D C U R R E N T EXPERIMENTAL SITUATION
The main results reported during last few years are: 1) observation of three new cluster emitters 230U, 238U, 242Cm; 2) observation of cold spontaneous fission of 2 5 2 C / ; 3) search for cluster emitter in new expected domain of cluster radioactivity ( 114Ba ). When planning such type experiments, one has to take into account that the kinetic energy of the light fragment is typically (2 — 2.5) -A MeV and its range is comparable to that of alpha particles emitted by the same parent. For cluster indentification by direct detection of the emitted fragment the source must be thinner than ~ 1 - 2 mg • cm2 thick. The decay rate should be high enough to observe at least a few decays per year in a sample of practical dimensions. The lowest rate likely to be detectable without any extraordinary efforts is ~ 10 2 9 sec - 1 . The branching ratio relatively to alpha decay and spontaneous fission should not be so low that the signal is swamped by these backgrounds. The lowest branching ratio detected up to date is ~ 1 0 - 1 6 relatively to alpha decay. A decrease to 10~ 18 seems conceiv-
145
able with improvements in technique [24]. Up to now, three techniques were used for investigation of cluster radioactivity: AE x E telescopes, magnetic spectrometers and solid state track detectors (SSTD). The lowest branching ratio detected up to date by electronic methods is ~ 4 - 1 0 - 1 1 relative to alpha decay ( 224Ra ). Cluster decays of relatively light elements ( 222-224,226^ ) were investigated using both electronic methods and SSTD. Cluster decay of 225 Ac and heavier elements were investigated by SSTD only. SSTD - method is based on detection the latent tracks of heavy particles in plastics or glasses which are insensitive to particles with ionization rate below that of the particles to be detected. After exposition, the tracks are treated by chemical etching and are located by ether manual or automated scanning, after which their dimensions are measured in microscope. In practice, it is possible to measure charge and range at branching ratios at least as low as ~ 1 0 - 1 8 . The method allows to resolve reliably one charge unit. Energy resolution is usually not better than 3 - 5%. For cluster decay study of nuclei with Z > 90 it is necessary to carry out a cluster identification in the conditions of high background of alpha-particles and fission fragments. High a-particles integrated flux (> 10 11 e r a - 2 ) gives rise to the background etchable tracks of recoil nuclei produced by interaction of a-particles with nuclei of the detector material. They hinder the search of events and distore the shape of the tracks being searched for. Therefore, if the spectrometric measurements are being made, a limit exits on the a-particle flux density for polycarbonate (PC), polyethyleneterephtalate (PETP), and phosphate glasses (PG) of the order of 10 10 ,10 12 cmdl0 14 — 10 1 8 cm - 2 , correspondingly. A limit on the fission fragment flux density for these detectors is not more than 10 4 em - 2 . For higher background of fission fragments an absorber with the thickness equal the lightest fission fragment range in the detector matter can be used. In this section we present new results of cluster decay investigations and current experimental situation of this problem. 230 238 ' U The motivation to search the cluster decay of 238U is to look for the dependence of the decay probability on the number of neutrons in the parent nucleus. For uranium isotopes the data exist for 2 3 2 [ / i 2 3 4 U236 U. The cluster decay of 230U -+22 Ne + 2 0 8 Pb has been treated at [ 4 - 6 ]. This isotope has been produced on the isochronous cyclotron of the Research Centre in Orlean (France) under the irradiation of 232Th by protons with the energy of 34 MeV in the reaction 232Th(p, 3n)23°Pa -)• /? -> 2 3 0 U with subsequent chemical separation and deep purification. The conditions and results of these experiments are presented in Table. It was shown [25] that cluster decay probability of 236{J falls out of the existing systematics and is more than one order of magnitude higher than
146 Table 1.
Nucleus 'Jt30jj m
Pa
230 TT
Number nuclei in source 9.6-10 1 3 3 • 10 13 9,6 10 13
Exposition (days) 24 248 130
Detection efficiency -0.5 0.63 0.90
Ad
7.5-10~ 1 4 1.3-10" 1 4 (4.8±2.0)-10~i4
it was expected. Some our extrapolations based on the 236JJ results show that the same kind of anomaly can take place also in the case of 238 J7, and its time of life can reach the values ~ 3 • 10 20 years, what is much shorter than the predictions [13] and [21] . For study of the cluster decay of 238f7 (the Melinex track detectors and the metallic layers of 2 3 8 [ / with enrichment 99.9% (Good Fellow, England) total area 3000 cm2 in 2n geometry were used [Nouvo Cimento]. After two years exposition and scanning of 1000 cm2 area there was found 1 event of Si. This result is consistent with logT\/2 ~ 29.4. The rated data for the 238 [7 cluster decay is ~ 33. 242 C m [8] The main difficulty encountered in this experiment comes from the fact that the spontaneous fission probability of 242Cm is 109 — 10 10 times higher than the expected probability of the cluster decay. 242 Cm was produced from 2 4 1 Am irradiated with thermal neutrons at the Kurchatov Institute reactor. After chemical separation two sources of 242Cm about 0.23 and 0.17 mg. with a thickness ~ Q.lmg/cm2 were prepared. The 242 Cm sources 30 mm in diameter on Pt backing were located in the center of hemispheres 190 mm in diameter. The interior surface of the hemispheres was inlaid with special phosphate glass track detectors. The detectors were covered with 20 /zm thick polyimide film absorbers to prevent the action of the spontaneous fission fragments on the detectors. After penetrating 20^x polyimide 34Si ions must have the residual energy 34MeV and range 9-10/mi. The setup was placed in a vacuum chamber. The total exposition was 292 days. The total number of 242Cm nuclei that decayed during the exposition was 4.29 • 10 17 . The duration of the exposition was fixed to ensure that the a - particle fluence onto detectors was < 10 1 4 a/cm 2 . 15 tracks were found and identified as formed by particles with Z ~ 14. The mean kinetic energy value (81.0±1.9)±2.0 MeV we measured is consistent with the expected one Ek = 82.97 MeV calculated from the ground state Qvalue. The corresponding partial half-life is (1.4 + 0.5/ - 0.3) • 10 23 s. This value is in an agreement with the lower the limit obtained earlier [?]. The
147
branching ratio relatively to alpha decay is 1.0 • 1 0 - 1 6 and the one relatively to spontaneous fission 1.6 • 1 0 - 9 . 252 Cf. The yields of about 20 pairs of fragments corresponding to the maximum possible decay energy of spontaneousfission were observe [9]. This energy resolution was ~ 1 MeV, so transitions to the fragments excited states in this range could contribute to the observed cold fission process. 114 B a . Observation of a new domain of cluster emitters is extremely interesting because it would confirm the common nature of cold decays and give more deep insight into their mechanism. Ba region provides unique possibility to find such emitters resulting in formation of the double magic nuclei 100Sn or its neighbours. The most realistic measurement seems to be the search of 1X4Ba —>12 C + 1 0 2 Sn decay. Different theoretical models (e.g.,[13] and [15]) predict 114Ba partial life-time differing by 6-7 orders of magnitude contrary to what they give for nuclei in the region em A > 220 studied until now ( see below ). Three experiments [12-14] were carried out and gave negative result. The upper limit for the ratio Xci/^a was found to be ~ (3—5)-10~5. However, experimental capabilities are not exhausted, and future measurements are highly desirable.In particular, a search of 112jBa and 114Ba decays is planned by Dubna group. Search of 118 Ce —>16 O +102 Sn decay also seems quite perspective. Future developments. Experimental capabilities based on the methods discussed above almost reach their limits in 242Cm experiment. Further search of heavy cluster emitters seems to have little success. We expect that in the nearest future the emphasis will be given to study of the decays of odd nuclei which are much more sensitive to the details of nuclear structure than the even-even ones. Search of fine structure of 225Ac decay seems to have high priority in this row, and first attempts have been done already. We plan to resume the search of 241Am decay which is a favorable odd-odd nuclide to examine for possible Si emission. Different models predict branching ratios for 34Si emission ranging from 4 • 1 0 - 1 3 to 4 • 1 0 - 1 5 [27] which makes this search feasible. Several groups using different experimental conditions have already looked for 241Am cluster decay with negative results. The lowest upper limit obtained [27] was \ci/Xa = 7.4-10~ 16 . We suppose that using a source containing 100 mg of 241Am and special phosphate glass detectors of total area 1600 cm2 the sensitivity Xci/^a ~ 1 0 - 1 6 could be reached during the exposition ~ 16 months. In Fig. 1 cluster decay probabilities of all known decays are shown as a function of the emitted fragment masses. The dependence is rather irregular, which can partly be explained by incompleteness of the data. Still two tendencies are seen. Firstly, the probability for emission of a particular cluster is biggest when the heavy partner is 2 0 8 P6 ( 222Ra -> 1 4 C, 232U -> 2 4 Ne,
148
0
T T l
1 1 1' 1 1 1 1 1 1 1 1 1 1 1 1 1 T "
* exp —o— GSP
B
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-
-20
mm
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ft
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-
pp
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»
B
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-40
1
-50
~T"
i
l
l
l
l
l
j
j
A
j
'
A
^-CMco^touSob ! 6 ^ -
r-221
A
A
A
A
A
-
'
j
: : : : :
~ •V
CO . 1
CO A 1
CM -
CMCVICVICMCMCMCVlOcOCOCNJCO^-^-CDCOCOCO CM CVIWCMWCMtNICMfOcviCVlCOCOCOCOCOCMCVICM
a j c b c o c b c b i i ^ i t b ^ ^ ^ ^ ^ i i i E"
-u_ a:a:a:Q:a:
Figure 1. Cluster decay probabilities. GSP, CFK, and P P are predictions of models by Greiner-Sandulesku-Poenaru , Chuvilski-Furman-Kadmenski and Pik-Pichak correspondingly.
236
Pu - » 2 8 Mg ) which correlates with the maximum Q-values of these decays. Secondly, the probabilities diminish with light fragment masses at first fast ( approximately by 10 orders of magnitude from 1 4 C to 24Ne ) and than do not change in average ( from 24Ne to 34Si ) . Comparison of the data with predictions of three models (fenomenological by Greiner-Sandulesku-Poenaru [13], semimicroscopic "fission-like" by PikPichak [14] and microscopic " a-decay-like" by Chuvilsky-Furman-Kadmensy [15]) is presented. Though physical grounds of all three models differ very strongly, their predictions coincide among themselves and reproduce the data ( including the tendencies mentioned above ) with the accuracy 1 - 2 orders
149
of magnitude. So, simple comparison of theory with experimental periods of half-lives does not allow to get unambiguous conclusion about the mechanism of the process. 3
M A S S D I S T R I B U T I O N S OF COLD DECAYS
As it was mentioned above, the examples of cluster radioactivity studied until now present only a small part of the whole domain of energetically allowed cold decays. In order to get a more general point of view on the whole problem we calculated complete mass distributions of a series of cold decays of nuclei with A > 112 using a model [14]. This "fission-like" model fits the probabilities of light cluster emission with reasonable accuracy ( see Fig. 1 ) which is expected to increase with fragments masses. The model takes into account deformations of the participating nuclei. Nuclear masses were taken from tables [ 16 ]. For the cases when the masses of mother or daughter nuclei were unknown the values calculated in [17] were employed. The deformations also were taken from [ 17 ]. The yield of a fragment with a particular mass was obtained as a sum of yields of different isobars. We considered only decays into the ground states of the fragments. However, calculations demonstrated strong dependence of the decay probability on the sign of deformation. So the transitions to the excited states with positive deformation can be more probable than those to the ground states with negative deformation. . For this reason we calculated two spectra for each mother nucleus: one corresponds to the accepted sign of deformations of the fragments ground states, and another to the assumption that the deformations are prolate. Such procedure allowed to get qualitative estimation of the role of transitions to the excited states which could be important for fragment masses A- 110-=-120. Some examples of the calculated mass distributions are presented in Figs. 2-8. They demonstrate the yields of light fragments and are prolonged to about 20 mass units in the region of their heavy counterparts. 112 B a (Fig. 2). The most probable decay mode is ll2Ba ->12 C with formation of doubly magic 100Sn ( Z=50, N=50 ). One may call it "tin" radioactivity. The expected lifetime (logTi/ 2 = 5.2 ) is quite measurable, the experimental problem is mainly to synthesize this nucleus. Note, that 112Ba decay is by six orders of magnitude more probable than that of 1 1 4 S a - » 1 2 C ( logTi/2 = 11-1 )• The decay probability falls with the light fragment mass and increases only when approaching symmetric fission. The most probable decay here is to two "semimagic" nuclei 56Ni ( Z=N=28 ). This shell effect is quite prominent.
150
10
0 -
CM
ooO CD CM"?
'o ©CvlOOf.,
oo.® , xJ-LL
40
60
80
Figure 2. Mass distribution of cold decays of 1 1 2 B o . Calculations were done using PikPichak model. Only the light fragments part is shown with prolongation about 20 mass units to the region of corresponding heavy partners. By O a r e marked the results of calculations corresponding to the accepted values of the fragment ground state deformations. By A are denoted probabilities calculated under assumption t h a t the ground state deformations are prolate.
168
Er (Fig. 3). Absolute decay probabilities are very low (logTi/2 < 50 ). One can distinguish three groups in the spectrum: 1) 32Si + 1 3 6 Xe; the heavy fragment has a filled neutron shell ( N=82 ) and a number of protons Z=54 close to a magic one. Here we meet another type of double magic "tin" radioactivity corresponding to the formation of nuclei with Z and N close to Z=50 and N=82. We shall see that this type of decays is prominent also in the other cases. 2) 4SCa+120Cd; relatively high decay probability is connected with magic
151
120
Figure 3. Mass distribution of cold decays of
168
Er.
For explanations see Fig. 2.
properties of 4SCa ( Z=20, N=28 ). 3) Symmetric fission to two 84Se nuclei having N=50. 202 H g (Fig. 4 ) . Absolute decay probabilities are extremely low. Again one can distinguish three groups. The most asymmetric now becomes 4 8 Ca74 group. The most intensive part of the spectrum (6SNi+134Te, Zn+128Sn ) belongs to "tin" radioactivity due to the fact that the heavy fragments have Z and N numbers close to 50 and 82 correspondingly. Relatively high probability of symmetric fission seems to be connected with formation of nuclei with Zvalue close to semimagic Z=40 ( 100Zr + 1 0 2 Zr, 96SV + 1 0 6 Mo, etc. ). Large difference between transition probabilities into the fragments ground states if different signs of deformations is assumed to them is clearly seen for the region A = 80 -r- 90. It is an indication that in reality the transitions to the
152
140
Figure 4. Mass distribution of cold decays of
i02
Hg.
For explanations see Fig. 2.
exited states having prolate deformations in this mass region could be more probable than transitions to the ground states. 222 R a (Fig. 5). 1 4 C emission observed experimentally is connected with formation of double magic 2 0 8 P6 ( Z=82, N=126 ). The probability of "lead" activity is high in comparison with the most of other cases discussed above, but lower than the probability of "tin" activity of 112Ba, the nucleus, which similarly to 222Ra emits carbon cluster. So we see that the studied region of decays named cluster radioactivity does not occupy any exceptional position among cold decays. As the light fragment mass increase, the decay probabilities fall rapidly. New increase begins with 48Ca peak and continues up to appearance the fragments typical for "tin" activity: 8 6 5 e + 1 3 6 X e (Z=54, N=82), 94Sr+128Sn
153
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A Figure 5. Mass distribution of cold decays of
222
Ra.
For explanations see Fig. 2.
( Z=50, N=78 ). In the region of symmetric fission the yield increases by 13 orders of magnitude if prolate deformations of heavy fragments are assumed. 234 U (Fig. 6). "Lead" activity continue to dominate among the other decay modes. Emission of two clusters belonging to it, 24Ne and 28Mg, was observed experimentally. A weak Ca peak and the products of " tin" decays ( Z 50, N 82: 100Zr + 1 3 4 T e , 1 0 4 M o + 1 3 0 Sn, etc. ) are seen as well. A new feature is that there appeared an intense group which is not connected with any filled shell. It corresponds to fragments having large static deformations: 152 Nd, 150Sm ( their light partners are 82Ge, 76Zn ). We shall name intense groups of such origin "deformation" activity. Note, that beginning from 234J7 the probabilities of cluster radioactivity and cold fission become comparable. In the symmetric fission region strong dependence on the signes of deformation
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A Figure 6. Mass distribution of cold decays of 23iU. For explanations see Fig. 2. V denote "quasiexperimental" d a t a based on normalization the experimental d a t a (see text).
manifests itself. An interesting comparison with the experimental mass spectrum of induced cold fission of 2 3 4 [ / [18] can be done. The fragments in the interval A = 76 -f- 102 were observed. We normalized the partial yields obtained in [18] to the known value of Ti/ 2 of 234U spontaneous fission. "Experimental" data obtained in such way agree unexpectedly well with the calculated mass distribution ( Fig. 6). 242 C m (Fig. 7). This nucleus is the heaviest for whom cluster ("lead") decay was observed experimentally. Besides, broad groups corresponding to "tin" ( e.g. 10SRu + 1 3 4 Te, Z=52, N=82 ) and "deformation" ( e.g. 8 6 5 e + 156 Sm ) activities are seen in the mass distribution. Ca peak and symmetric
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A Figure 7. Mass distribution of cold decays of 242Cm.
For explanations see Fig. 2.
decays like 120Cd+120 Cd are also present but with much less probability. 252 C f ( F i g 8 ) Formation of deformed nuclei ( 166Gd, 150Ce, etc., their partners are 8 6 5e, 102Zr ) dominates in 2 5 2 C / spectrum demonstrating that lowering of Coulomb barrier due to prolate deformation of the emitted fragment can be no less important than shell effects. "Tin" activity shifted to the region of symmetric fission. Rather intense is Ca peak. "Lead" activity presented by sulfur and its neighbours becomes suppressed because the corresponding light fragments have too large neutron excess diminishing Q-values of the decays ( the partner of 2 0 8 P6 is 4 4 5 with N/Z=1.75 which is much bigger than N/Z=1.57 for the mother 2 5 2 C / ). The yield of cold fission considerably exceeds that of cluster ( "lead" ) decay. 252Cf is nowadays the only nucleus whose spontaneous cold fission was experimentally studied [ 9 ]. Using
156
80
100
120
140
160
A Figure 8. Mass distribution of cold decays of 2 5 2 C / . For explanations see Fig, 2. V denote "quasiexperimental" d a t a obtained from analysis of experimental dafa (see text).
the published data on mass and energy distributions [19] for normalization we could obtain the absolute values of partial probabilities some of which are shown in Fig. 8. Again as in the case of 234U there is a remarkable agreement with calculations. It is, perhaps, even better because the energy resolution in [9] was ~ 1 MeV, so that transitions to the excited states within this interval contribute to the total yield, and ground state formation probability could be 2-3 orders of magnitude less. Let us summarize. Cold decays are distributed over the whole available range of masses. The phenomenon called to-day "cluster radioactivity" is only a particular case of their more general family. It is not distinguished either by the nature of its origin, or by its probability in comparison with the other modes. One can speak about "lead",
157
"tin" and "calcium" activities depending on the vicinity of Z- and N-values to the corresponding magic numbers. The most wide-spread activity is the " tin" one due to the fact that the ratio 82/50 is close to the average N/Z ratio of decaying parent nuclei ( this provides in average the maximum Q-value ). "Tin" activity drifts from very asymmetric one in the parent mass region A ~ 150 to symmetric fission for 2 6 4 F m . Another source of enhancement of the decay probability is the formation of fragments having prolate static deformations. Orientation of the big axis along the direction of movement results in lowering of the Coulomb barrier and diminishing the path under it. The part of mass spectra attributed to cold fission usually is a combination of "tin" and "deformation" activities. The relation between these types of decay determines the width of the mass distribution and, in particular, if cold fission spectra would be symmetric or asymmetric. So we see that there are no reasons to distinguish between cold fission and different types of cluster radioactivity. However, the dynamics of fragment formation in different parts of mass spectra can be different. 4
T R A N S I T I O N B E T W E E N CLUSTER- A N D FISSION-LIKE T Y P E S OF DECAYS
Alpha-radioactivity and fission usually are described by completely different formalism reflecting different physical picture of what happens, and these extremes are applied to the description of cluster radioactivity. Alpha-decay is considered to be a non-adiabatic process. Its probability is determined by the overlap of the wave function of the parent nucleus with those of both fragments. This is equivalent to a sudden formation of a cluster inside the mother nucleus which then makes attempts to penetrate the barrier. The fission-like process, on the contrary, is described as an adiabatic one. It includes the prescission phase where the matter flow takes place and fragments are overlapping. Their final formation happens only after the system goes through a sequence of geometrical shapes which parameterization is a part of the adopted theoretical approach. Though the calculations based on different assumptions about the mechanism of the decay ( "alpha-like" or "fission-like" ) gave similar results, there are some arguments in favor of non-adiabatic mechanism for emission of, at least, the lightest fragments. The first comes from the observation [ 20 ] of thin structure in 223Ra —>14 C +209 Pb decay, the second comes from validity of Geiger-Nuttol law for cluster decays. Transition probability to the first excited state of 209Pb is several times higher than that to the ground state, contrary to what is expected from taking into account only barrier penetra-
158
r ~ 50
1 60
1 70
,—i 80
-Ln{PJ
Figure 9. Geiger-Nuttol law for known cluster decays with emission of even-even fragment! Cold fission d a t a on emission of l0SRu (see Fig. 8) are included
bilities. It is evident that the structure of participating nuclei which can be expressed by corresponding spectroscopic factors plays important role. An other argument comes from validity of Geiger-Nuttol law for cluster decays (Fig. 9). The distances between the lines corresponding to different fragment masses are proportional to the values of preformation probablity, or spectroscopic factors S. They change with A-values of the emitted fragments in accordance with predictions of non-adiabatic models [21]: S=S(aA-l),3j
is a- particle spectroscopic factor. The first significant deviation from Geiger-Nuttol law was observed for 242 Cm -+34 Si +208 Pb decay [3]. Preformation probability extracted from the positions of Geiger-Nuttol lines is much smaller than one could expect
159
from systematics. Note that the difference between the measured and microscopically calculated lifetimes has the same sign and is the largest also for 242 Cm ( see Fig. 1 ). This is in agreement with finding by Poenaru [ 22 ] that cluster penetration factor through the inner part of Coulomb barrier which in some way could be considered equivalent to the preformation probability behaves according to the predictions of "alpha-like" model [ 21 ] for light fragments and begin to deviates from it at A = 30-35. It is necessary to stress that the very fact of experimental observation of cold fission means that the non-adiabatic mechanism is not valid in the corresponding part of the mass distribution. This results both from formulas of simple " alpha-like" model [ 21 ] and much more complicated microscopical calculations [ 2 3 ] . The decay probability predicted by non-adiabatic mechanism should be several tens of order of magnitude lower than the observed values. This could be easily seen if one applies Geiger-Nuttol law, say, for the 102Zr emission from 252Cf ( Fig. 9 ): a drastic disagreement with the systematics becomes evident. So the existing data indicate that the transition between non-adiabatic and adiabatic mechanisms may exist and occur for fragments with A ~ 35. A well-known signature of spontaneous fission adiabatic mechanism is the dependence of the periods of half-lives on the fissility parameter Z2/A. It reflects the competition between Coulomb and surface tension forces in the oscillation process. A priori, it is not evident that the same systematics will take place in the case of cold fission. The difference between hot and cold fission could be connected with the fact that the latter process corresponds to much smaller deformations of mother nucleus. In order to get an answer to this question we calculated theoretical cold fission probabilities by integrating the parts of mass distributions in the interval A A = 30. The results are presented in Fig. 10. It is seen that the well-known dependence takes place for cold fission as well. So from this point of view cold fission can be considered as "real" one. On the other hand, the same dependence for cluster radioactivity behaves differently (Fig. 11,). For light clusters the periods of half-lives increase, in opposite to what takes place for fission, and for heavier fragments some kind of plateau is reached. For 242Cm even some approaching to "fission-like" pattern is observed. So one can see that two parts of total cold decay mass spectra: the regions of "lead' radioactivity and cold fission, demonstrate different behaviour in wellknown phenomenological systematics, Geiger-Nuttol law and Xi/ 2 — Z2/A dependence. Together with some other arguments discussed above this fact provides evidence that cluster (at least "lead") radioactivity and cold fission have different mechanisms (probably non-adiabatic and adiabatic correspond-
160
22 24 26 28 30 32 34 36 38 40 42 Z2/A Figure 10. Dependence of cold decay life-times on fissility parameter Z2/A of parent nuclei. The points corresponds to the calculated periods of half-lives obtained by integrating the mass spectra over regions A J 4 = 30. Full line is drawn to guide the eye. Dashed line is drawn through the points corresponding to different B a isotopes. The nuclides for which " lead" radioactivity was experimentally observed are denoted by full circles.
ingly). The transition between both mechanisms takes place at the fragment masses in the vicinity A = 35. 5
CONCLUSIONS
* The shapes of cold decay mass distributions mostly depend on two properties of the emitted fragments: 1) the vicinity of their proton and ( or ) neutron numbers to the magic ones, increases
161
Z2/A Figure 11. Dependence of cluster decay life-times on fissility parameter Z2 /A of parent nuclei. The points corresponds to the experimental periods of half-lives of some cluster decaying nuclides.
2) the existence of static deformations ( positive deformation the emission probability ). * Besides well-known "lead radio activity" one can distinguish some others like "tin" or "calcium" ones, and "tin" radioactivity is expected to be the most wide-spread. * Cold fission region of the mass spectra depends mainly on the interplay between deformation and tin shell effects. * Emission of the lightest clusters is governed by non-adiabatic (a-decaylike) mechanism, emission of heavy fragments in cold fission region, is adiabatic. The transition between both mechanisms seems to take place at fragment masses A ~ 35. ACKNOWLEDGMENTS We are grateful to R.Bonetti, Yu.M.Chuvilski, V.M.Furman, F.Goennenwein, V.M.Mikheev, Yu.T.Oganessian, V.A.Shigin for fruitful discussions. The work was partly supported by grant 98-02-06255 of Russian Fond of Fundamental
162
Investigations. References 1. Gupta R.K., Greiner W., Int. Jour. Of Modern Phys.E3, 335, 1994 2. Zamyatin Yu.S. et al.,Particles k Nuclei, 21, 1990, 537. 3. Ogloblin A.A. et al., "Nuclear Shells 50 Years", Dubna, Russia, 21-24 April 1999, ed. By Yu.Ts.Oganessian & R.Kalpakchieva, 124 4. Tretyakova S.P. et al., Radiation Measurements, 28, 1-6, 305-310, 1997 5. PAN Qiang-yan et al. Chin.phys.Lett., !6, 4 251-252, 1999 6. Bonetti R.et al. In: Proceeding of S-P. Conference]. 7. R.Bonetti, private communication 8. Ogloblin A.A. et al., Phys.Rev. C61, 034301-1, 2000 9. Croni M. et al., "Fission and Properties of Neutron-Rich Nuclei", ed. by J.Hamilton k A.Ramayya, 109, 1998 10. Oganessian Yu.Ts.,et al., Z.Phys. A349, 341, 1994 11. Guglielmetti A. et al., Phys.Rev. C52, 740, 1995 12. Guglielmetti A. et al., Phys.Rev. C56, 2912, 1997 13. Poenaru D.N. et al., Atomic Data and Nuclear Data Tables, 48, 231,1991 14. Pik-Pichak G.A., Yadernaya Fisika, 44, 1421, 1986 15. Kadmenski S.G. et al., Yadernaya Fisika, 57, 1981, 1994 16. Audi G., and Wapstra A.H., Nucl.Phys., A595, 409, 1995 17. Moller P. et al., Atomic Data and Nuclear Data Tables, 59,185, 1995 18. Schwab W. et al., Nucl. Phys. A577, 674, 1994 19. Schmitt H.W. et al., Phys.Rev., 141, 1146, 1966 20. Brillard L. et al., C.R. Acad. Sci. Paris, Ser.II, 309, 1105, 1989 21. Blendowsky R. and Walliser H.,Phys. Rev. Lett. 61 (1986) 1930 22. Poenaru D.N., Proc. Int. Conf. on Nuclear Reaction Mechanisms, Varenna, Italy, June 10-15, 1991, p.348 23. Furman V.I., private communication 24. Price P.B. Nuclear Physics A502 (1989) 41-58 25. Tretyakova et al., JETP Let., 59 (1994) 368 26. V.L. Mikheev et al., FLNR Scientific Report 1991-1992, E7-93-57, Dubna, JINR, (1993), p. 48. 27. Moody K.L. et al., Phys. Rev.C36,(1987) 2710.
163
STATIC AND DYNAMICAL PROPERTIES OF SIMPLE METAL CLUSTERS. ANALOGIES WITH ATOMIC NUCLEI Julio A. ALONSO Departamento de Fisica Teorica, Universidad de Valladolid, E-47011 Valladolid, Spain Manuel BARRANCO Departament d'Estructura i Constituents de la Materia, Facultat de Fisica, Universitat de Barcelona, E-08028 Barcelona, Spain Prancesca GARCIAS Departament de Fisica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain Paul-Gerhard REINHARD Institut fur Theoretische Physik, Universitat Erlangen, Staudtstr. 7, D-91058 Erlangen, Germany Eric SURAUD Laboratoire de Physique Quantique, Universite Paul Sabatier, F-31062 Toulouse, France Some features common to the physics of simple metal clusters and atomic nuclei are presented with special emphasis in fission-like processes. Magic numbers, supershell structures and collective dipole excitations are also discussed.
1
Introduction
O n e of t h e most fascinating aspects of physics, and science in general, is to discover relations, similarities and bridges between different subfields. Nuclear Physics is an established subfield of physics a n d Cluster Physics is a relatively recent one. T h e main feature common t o the two subfields is t h a t the systems of interest (nuclei and atomic clusters, respectively) are formed by a finite n u m b e r of fermionic particles. For this reason, it is not surprising t o find t h a t some well established features in nuclear physics have a close parallel in atomic clusters physics. T h e type of clusters t h a t present more similarities with nuclei are the clusters of the alkali elements. Magic numbers of stability were discovered for Sodium clusters in 1984 and these were immediately interpreted as due t o shell-closing effects for a system of electrons moving in a common effective potential with a shape similar to the Woods-Saxon p o t e n t i a l 1 . W h e n t h e clus-
164
ters are subjected to laser pulses of the appropriate wavelength, they absorb photons and the valence electrons perform collective oscillations against the ionic skeleton 2 which are the analogue of the giant dipole resonance 3 in nuclei. The theoretical methods developed to analyze the dynamical response of clusters to external fields rely strongly on the experience accumulated in nuclear physics. Additional similarities occur between the fission of nuclei and the fragmentation of multiply-charged atomic clusters 4 . In this article we concentrate mainly on the later feature, reviewing the theoretical work for metal clusters in close connection to experiment. Along the discussion we emphasize the relation of features and concepts to those in nuclear physics. For this reason, before addressing the fragmentation of charged clusters we present first a brief account of shell and shape effects that helps to visualize the similarities with nuclear physics and also provides a background required for other sections of this paper. The topics we have selected for this contribution do not constitute the complete list of fruitful application of nuclear physics concepts and techniques to the physics of metal clusters. Rather, our choice has been motivated by the work we have done in cluster physics. For this reason, interesting topics such as cluster-cluster collisions for instance, have been left out. For this precise item, and its connection with the physics of heavy-ion reactions, the interested reader may find a thorough discussion in the work carried out by Schmidt and coworkers 5 . 2
Electronic shell and supershell effects
The measured mass spectra of hot alkali-metal clusters 1 generated in vapors by the usual molecular beam techniques have revealed prominent abundance maxima at certain sizes that have been called magic numbers: N = 2, 8, 18, 20, 34, 40, 58, 92... It was correctly realized that the large abundance arises during the evaporative cooling step as a consequence of a higher stability compared to neighbor sizes. Evaporation of atoms from hot clusters enhances the population of those with the largest binding energies. The variations of stability with size were explained by a simple model in which independent electrons move in a common attractive finite potential well with spherical symmetry. Energy levels for electrons bound in a spherically symmetric potential are characterized by principal and angular momentum quantum numbers, n and / respectively (n — 1 is the number of nodes in the radial wave function). For fixed n and I the degeneracies associated to the magnetic quantum number m and to spin s give a total degeneracy of 2(2/ + 1 ) for an (n,l) shell. The detailed form of the confining potential controls the
165
precise ordering of the shells, so a realistic representation of this potential is required. That effective potential well can be calculated selfconsistently within the framework of the Density Functional Theory (DFT) 6 ' 7 VeS (f) - Vbg (f) + VH (f) + Vxc ( r )
(1)
as the sum of a potential representing the attractive effect of the ionic background V^ (f), the classical electrostatic potential of the electronic cloud VH {f), and the exchange-correlation potential Vxc(f). The last one is given by Vxc(r) = ^ 1 on(r)
,
(2)
where £^IC[n] is the exchange-correlation energy functional, often treated in the Local Density Approximation (LDA). The single-particle orbitals Vi are obtained by solving the Kohn-Sham (KS) equations 7
-\v2 + veff(n •0i
= tii>i,
(3)
and from those orbitals the valence electron density is evaluated as n(r) = 5 > ; ( r ) | 2 ,
(4)
i
where the sum is extended over the occupied orbitals. Hartree atomic units (a.u.) are used throughout this paper unless explicitly stated, i.e., h — m = e 2 = 1, length unit ao = 0.53 A, energy unit 1 Hartree = 27.2 eV. A useful approximation for the positive background is the spherical jellium model, that assumes a homogeneous distribution of positive charge that abruptly goes to zero at the cluster surface 8,9 . Considering the cluster as a liquid drop justifies the spherical symmetry. Using this model, the observed magic numbers were reproduced and interpreted as due to the progressive filling of electronic shells: Is, lp, Id, 2s, 1/, 2p, lg, 2d, lh, 3s... Clusters with filled shells present an energy gap between the highest occupied and the lowest unoccupied shells (HOMO-LUMO gap). A theoretical quantity very useful to display the shell effects is A2(N)
= E(N + 1) + E{N - 1) - 2E(N) ,
(5)
where E(N) is the energy of the cluster with N atoms. That energy is given by Eduster
= E[n] + Eself
,
(6)
166
where the second term represents the self-energy of the positive jellium background and the first one contains all the electronic contributions to the energy density functional, that is
E[n] =T+\(
( * J J
n
( r ) n y ) d?dr+ \r — r'\
[n(f) J
Vbg(f) dr + Exc[n] ,
(7)
where the single-particle kinetic energy T is evaluated from the single-particle orbitals
T = -Y,\jPi
(^)VVi(r) dr .
(8)
Notice that &2{N) represents the relative binding energy of clusters with TV atoms compared to clusters with N — l and N+1 atoms. Shell closing results in sharp peaks of A2(N). A thermodynamic analysis shows that A2{N) controls the measured relative abundances x . The existence of magic numbers provides the first relevant similarity with nuclei although the magic numbers are not the same due to the spin-orbit interaction in nuclei. Experiments for noble metal clusters (Cujv, Agw, Au^r) indicate the existence of similar shell effects, that is, the same magic numbers of the alkali clusters are observed. These are reflected in the mass spectrum 10 and in the ionization potentials. The Cu, Ag and Au atoms have an electronic configuration of the type d10s1, so the magic numbers are explained if the DFT jellium model is applied to the s electrons only. As the size of the cluster increases, the number of valence electrons in the cluster increases, and the number of occupied electronic shells also increases. Since the depth of the confining potential remains approximately constant, the forbidden gaps between electronic shells become narrower 8 . Eventually, when N is sufficiently large the discrete energy levels evolve into the quasicontinuous energy bands of the solid. Still, experiments indicate that shell effects remain observable in alkali clusters with a few thousand valence electrons n . Those shell-closing numbers are revealed by measurable drops in the ionization potential. The magic numbers appear at approximately equal intervals when the abundance mass spectrum is plotted on an TV1/3 scale. More precisely, AiV 1 / 3 = 0.6 between two consecutive magic numbers. DFT calculations for clusters with up to a few thousand valence electrons 12,13 have shown that groups of shells bunch together, leaving sizeable energy gaps separating those bunches. This justifies the observation of shell effects in the large size range. The calculations lead to the N1/3 periodicity and predict magic numbers in agreement with experiment.
167 A convenient way of analyzing the results of spherical jellium calculations for large clusters consists in separating the cluster energy into two parts E(N) = Eav(N)
+ EshM{N)
,
(9)
a smooth part Eav(N) and an oscillating part Esheii{N), which is in the spirit of Strutinsky's shell correction theorem 1 4 . The Liquid Drop Model (LDM) can be used to represent the smooth part as the sum of volume, surface and curvature terms Eav(N)
= avN + asN2/3 + acN1'* ,
(10)
and the constants av, as and ac can be fixed by a best fitting of this model expression to the calculated results for E(N). Subtracting the average term from E(N) defines the shell correction contribution. A plot of Esheii (N) versus N shows sharp oscillations with minima at the shell-closing numbers. In the calculations by Genzken and Brack for Sodium clusters 1 3 these minima occur for N = 2, 8, 20, 34, 58, 92, 138, 186, 254, 338, 438-440, 542-556, 676, 758, 832,912, 1074-1100, 1284, 1502, 1760, 2018, 2328, 2654, 3028, 3278, 3690, 4074... These are in reasonable agreement with the experimental magic numbers 1 1 . Another interesting feature in the plots of Esheii{N) is that the amplitude of the shell oscillations varies with N: the shell oscillations are enveloped by a slowly varying amplitude, the supershell. The intensity of the shell effect grows and vanishes periodically, but with a much larger size scale, A./V1/3 = 6. The first supershell node occurs near N — 850. Calculations by Nishioka et al12 using a non-selfconsistent WoodsSaxon potential (instead of the spherical jellium model) give N close to 1000. This node has been observed 11 : results range between 800 and 1000, so the calculations are consistent with the experiments. The supershell structure of Lithium clusters has also been studied by Genzken 13 . The agreement with experiment is even better. The experimental and theoretical first supershell node is found near N = 820. Finite temperature effects lower the intensity of shell 1 5 and supershell effects 16 , but do not shift the positions of the magic numbers. Electronic shell and supershell effects have also been observed in the trivalent metals Aluminium, Gallium and Indium 1 7 , but in order to explain the details it is necessary to go beyond the jellium model. The discovery of supershells confirms the earlier predictions of nuclear physicists, impossible to verify in atoms and nuclei because of their insufficient number of constituents.
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3
Shape deformations
Although the main features of the mass spectrum of alkali-metal clusters, associated to shell-closing effects, are explained by the spherical jellium model, the regions intermediate between shell-closing numbers show also small features that suggest the existence of an interesting structure. Again in analogy with nuclear physics, that structure has been interpreted as arising from the deformation of the cluster shape. If the outer shell is partially filled, the electronic density becomes non-spherical, which in turn leads to a distortion of the ionic background. That is, clusters with open-shell configurations become deformed. This Jahn-Teller type distortion, similar to the distortions that occur in molecules and nuclei 3 , leads to a splitting of the spherical shells into subshells. Deformations were first studied by Clemenger 18 by allowing the spherical shape to change to a spheroid with axial symmetry. For this purpose Clemenger used a model Hamiltonian based on a modified three-dimensional harmonic oscillator potential. The model allows for different oscillation frequencies H z and ilp along the z-axis (chosen as the symmetry axis) and perpendicular to the z-axis, respectively. The Hamiltonian also contains an anharmonic term that serves to flatten the bottom of the potential well to make it more realistic; in the case of a spherical oscillator that term breaks the Z-degeneracy of the (n, I) single-particle level. If the system is cylindrically symmetric, the energy levels are either two- (I = 0) or fourfold (I > 0) degenerate. The model reminds the spheroidal model of nuclei by Nilsson 3,19 . Due to the background deformation, the highly degenerate spherical shells split into spheroidal sub-shells. Denoting the semiaxes of the spheroid by a and b, a distortion parameter 77 can be defined 77 = 2(a - b)/{a + b) = 2(fi p - fi,)/(np + Slz) ,
(11)
that describes the deformation, oblate or prolate, of the positive background. For each cluster 77 is determined minimizing its total energy. An essential feature of the spheroidal model is to produce splittings in the single-electron energy levels which for small distortions are linear in 77. In this way, alkali clusters with N = 3-4 become prolate and those with N = 5-7 oblate. After the spherical cluster with N = 8, again clusters are prolate for N = 9-13 and oblate for N = 14-19, and so on. The quantity A 2 (iV) of eq. (5) predicts peaks at N = 8, 10, 14, 18, 20, 26, 30, 34, 40, 46, 50, 54, 58... In addition to the usual shell-closing numbers of the spherical model these numbers reflect the sub-shell filling. These sub-shell effects have been observed in the mass spectra 1 . In a more fundamental treatment Ekardt and Penzar 2 0 extended the jellium background model to have a deformed, spheroidal shape.
169
Figure 1: Valence electron density for the configuration corresponding to the ground state of Na42.
The advantage is that the model is parameter-free and that the electronic wave functions are calculated selfconsistently. As an example, Figure 1 displays the valence electron density corresponding to the ground state configuration of the non-magic Na42 cluster obtained using the spheroidal jellium model. A splitting of the dipole resonance into three peaks has been observed in some alkali clusters 21 . This observation was interpreted as reflecting collective vibrations of the valence electrons in the directions of the principal axes of a triaxially deformed cluster, and has motivated the extension of the deformed jellium model to ellipsoidal clusters with fully triaxial shapes 2 2 . The model also incorporates a smoothing of the positive background at the surface 23 . The results confirm the predictions of the spheroidal model: prolate clusters after the magic numbers N = 2 and N = 8, and oblate ones before N = 8 and N = 20. However, a transition region of triaxial shapes was found separating prolate and oblate clusters. The width of this transition region is very small between N = 2 and N = 8, and only contains Nas. But it is comparatively broader between N — 8 and N = 20, with N a n , Nai 3 , Na 15 and Nai 7 having triaxial
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2+ 2= 4
2 + 2 + 8 = 12
2 + 8 = 10
2 + 14 = 16
Figure 2: Molecular interpretation of some ground state cluster geometries in the ultimate jellium model. Reproduced by permission from M. Koskinen et al, Z. Phys. D 3 5 , 285 (1995).
ground states. The triaxial character of Nas is strong, but it is extremely soft for the others and thermal effects could wash out the triaxial signature in the dipole resonance spectrum. Proceeding further along these lines, Manninen and coworkers have introduced the "Ultimate Jellium Model" 24 , in which the positive background charge is allowed to be completely deformable, both in shape and in density profile, in order to minimize the energy. The result is that the background adapts itself perfectly to the electronic charge, and the jellium density rn,g{r) becomes everywhere the same as the electron density n(r). Consequently the Coulomb energy of the system cancels out and the electrons move in their own exchange-correlation potential Vxc(f), a model used earlier by March for the metallic surface 25 . Qualitatively the cluster shape has the pattern seen above,
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but the additional freedom adds richness to the variety of shapes. Clusters with N = 3, 4, 6, 7, 9, 14, 21 and 22 have axial and inversion symmetry. N = 10 has only axial symmetry and N = 5, 11, 13 and 15 have only inversion symmetry. N = 12, 16, 17 and 18 are more complex: they have neither inversion nor axial symmetry. Shapes up to N = 8 can be simply understood in terms of the symmetries of the s, px, py and pz single-particle wave functions of a spherical system. An explanation in terms of the filling of d and s levels is still possible between N = 8 and N = 20, at least near the magic numbers, although it is more difficult in the middle of the shell, due to the strong deformation. A striking new feature of the ultimate jellium model is the prediction of some cluster shapes that can be interpreted as "molecules" built from magic clusters. As shown in Figure 2, the 4-electron cluster is formed by two dimers, with a separation energy of only 0.10 eV. The 10-electron cluster can be viewed as a dimer attached to an 8-electron sphere, with a separation energy of 0.15 eV. N = 12 can be interpreted as two dimers attached to an 8-electron cluster, and N = 16 as a composite of the strongly deformed, yet very stable 14electron cluster and a dimer. The clusters without inversion symmetry have symmetric isomers and are extremely soft against deformation. For instance N = 10 has a pear-shape ground state and a prolate isomer with axial and inversion symmetry only 0.5 meV above the ground state in a region of the energy surface that is extremely flat. Related softness of the pear shape has also been observed in nuclear physics 2 6 . These molecular-like features are due to the high stability of the dimer and play a relevant role in the fission of multiply-charged clusters. In that case the fragments corresponding to the most probable fission channel are already "preformed" in the charged parent cluster. Finally, we would like to mention that a Time-Dependent Local Density Approximation (TDLDA) sum-rule approach, inspired in the randomphase approximation plus sum-rule approach 3 commonly employed in nuclear physics, has also been used to describe the plasmon modes of spherical 9>27 and deformed 28 alkali-metal clusters. The Interacting Bose Model originally proposed by Arima and Iachello 29 to describe the low-lying spectrum of openshell nuclei has also found an application to the study of deformed alkali-metal clusters 30 . 4
Fragmentation of charged clusters
The experiments studying the fragmentation of charged metal clusters 4 (sometimes also called "Coulomb explosion") have established that the dissociation of hot multiply-charged alkali-metal clusters is a barrier-controlled process 3 1
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and the same can be expected for other metal clusters 32 . The experiments have also shown that the existence of critical numbers Nc for the observation of charged clusters in mass spectra 33 can be explained by the competition between two fragmentation mechanisms, namely evaporation of a neutral monomer or dimer, and fission (i.e., the dissociation into two charged fragments). The preferred decay channel for hot large clusters is evaporation because the barrier against fission, Fm, is larger than the heat of evaporation, AHe, whereas small hot clusters undergo fission because Fm is in this case smaller than AHe. Preferential emission of fission fragments with a magic number of valence electrons (like NaJ or Kg", which have two and eight valence electrons, respectively) has been observed 34 . Models and calculations of cluster fission Several theoretical approaches have been developed to understand the facts observed in cluster fission experiments. One of the simplest methods to calculate the barrier heights for different fission channels is provided by LDM, in which the metallic character of the fissioning system is taken into account by explicitly concentrating the net charge on the cluster surface 35 . Electronic shell effects have been included in this model by applying Strutinsky's shell correction method 14 to study the symmetric 36 and asymmetric 37 fission of doubly-charged Silver clusters and of highly-charged alkali-metal clusters 3S . However, this type of LDM calculation misses the sizeable spill-out of the electronic density beyond the cluster edge. Spill-out effects have been included in LDM by Yannouleas and Landman 39 to calculate dissociation energies of charged clusters. A more sophisticated description of the fission process, based on DFT calculations in conjunction with molecular dynamics simulations, has been performed for small doubly-charged Sodium and Potassium clusters 4 0 . These calculations, which use the Local Spin-Density Approximation for exchange and correlation, clearly show the influence of electronic shell effects on the fission energetics and barrier heights (predominance of an asymmetric fission channel, double-hump fission-barrier shapes, etc). Unfortunately, such microscopic calculations are not feasible for large clusters. Fragmentation of charged alkali-metal clusters has been thoroughly studied within the much simpler jellium model approach 41 - 42 ' 43 ' 44 ' 45 ' 46 - 47 ' 48 . Jellium-type calculations based on an Extended Thomas-Fermi (ETF) functional containing density gradient terms have been performed for the symmetric and asymmetric fission of doubly-charged Sodium clusters 4 1 ' 4 5 . In the simple ETF formulation, the kinetic energy is expressed as a functional of the
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electron density
T = £ p h ^ /„v3(r-)d?+11 M d r - ,
(12)
where the first term is the local Thomas-Fermi energy and the second one is the Weizsacker quantum correction. The value /3 = 0.5 for the coefficient of the Weizsacker term has been found adequate 4 2 in cluster fission problems. Although this semiclassical ETF model does not account for electronic shell effects, it has the merit of selfconsistently incorporating the electronic spillout. Shell structure effects have been taken into account 4 3 ' 4 8 within the KS framework. The first reliable fission barriers were obtained using the Two-JelliumSphere Model (TJSM) 4 1 , 4 3 . This model is based on the assumption that since Coulomb force is long range, in many cases of interest the fissioning fragments are already preformed before overcoming the fission barrier. This situation is at complete variance with the nuclear one, where due to the short range character of the nuclear force, saddle point (fissioning) configurations are very compact and the fission products are rarely preformed even at the saddle 3 . In the case of clusters this results in an enormous simplification to obtain the fission barrier height, and the barrier structure around the saddle point, since to compute the barrier for the process Mj(+^M#:p1)++M+,
(13)
(the lower index indicates the size of the cluster and the upper index its charge) it is enough to start the calculation with a cluster of N — q valence electrons moving in the mean field created by the electronic cloud plus the trivial Coulomb potential created by two tangent jellium spheres with N — p and p ions respectively (assuming the fission products have spherical symmetry), and the cluster fragmentation path is obtained by increasing the distance between the jellium spheres. To obtain the fission barrier, this path has to be supplemented with the value of the energy of the parent cluster, whose configuration is disconnected from all the others defining the path. To go one step beyond TJSM, an axially-symmetric Deformed Jellium Model (DJM) has been proposed 4 5 ' 4 8 which yields a continuous fission path. In this case, the path has been described by a series of deformed jellium shapes connecting a spherical configuration associated to the parent cluster with the final one corresponding to the two separated fragments. Note that for openshell parent clusters, the initial configuration would not be the ground state one, and the model should be further generalized to allow for spheroidal or
174
elliptical parent configurations. However, we want to point out that the situation for multiply-charged open-shell clusters might be completely different from the nuclear one, and instead of having a deformed initial configuration, as in the neutral case, one could have a molecular-type initial configuration for the parent cluster (supermolecule configuration), see below. In the original formulation of DJM 4 5 ' 4 8 , the positive background of the fissioning cluster is modeled by axially-symmetric shapes 49 corresponding to two spheres smoothly joined by a portion of a third quadratic surface of revolution. This family of shapes is characterized by the values of three parameters: the asymmetry A, the distance parameter p which is proportional to the separation s between the centers of the emerging fragments, and the "deck" parameter A which takes into account the neck deformation. Evidently, asymmetric shapes are necessary to describe asymmetric fission. For a cluster configuration characterized by a set of values of the jellium parameters (A, p, A) and the electron and ion numbers, the density of the valence electrons is selfconsistently calculated minimizing the total energy of the system within the chosen scheme (KS, ETF...). Of course, the density of the jellium background nbg(r) is constant, so the total volume is conserved during the jellium scission process, which occurs before reaching the fission saddle point of the cluster (jellium plus electrons). For Sodium, which is the metal of interest for the examples presented below, nbg = 3.73 x 1 0 - 3 a.u. - 3 ; this corresponds to a Wigner-Seitz radius r s = 4 a.u. We want to mention that other parametrizations have been used to describe the cluster shapes during fission, see for instance the work by Yannouleas and Landman 50 where shapes are defined in terms of four independent parameters, and the work by Vieira and Fiolhais 51 . In DJM, the self-energy of the jellium background and the jellium potential acting on the valence electrons, which are both analytical functions in TJSM, have to be calculated numerically, and it is not a trivial task to obtain them with the accuracy required to calculate the barriers 4 5 . Within DJM, to calculate the barrier for a selected fission channel, for instance the emission of a singly-charged trimer, one has to compare the results obtained following different fragmentation pathways, defined by analytical relations one establishes between the parameters A and p (the asymmetry A is fixed by the size of the final fragments). The simplest choice is a parametrization based on two (intersecting, touching and separated) jellium spheres 5 2 ' 5 3 . Other possibility is to start with a jellium sphere and follow the path corresponding to a cone capped with spheres up to an arbitrary value of p where a concave neck starts to form; after that point one assumes the fastest variation of the neck, and continues with two
175
-38.00
O Oo
OOO
-38.25
> -38.50 CD
W -38.75
-39.00
OO O O Oo • 1 10
15
20
J
25
I
I
L_
30
s (a.u.) Figure 3: Total energy of N a ^ as a function of fragment separation for two different fission pathways: dashed line corresponds to the jellium shapes shown at the top and solid line to those shown at the bottom. Dotted line represents the classical Coulomb barrier.
separated jellium spheres. This fission path seems to be able to extend the dynamical results of Landman and coworkers 40 to larger clusters. We now present two selected examples of cluster fission using the methods described above. The first one is the emission of a charged trimer from NajJ using D J M + K S 4 8 : Na^+ —• Na+ + Na+ . (14) This is the most common fission channel observed for alkali-metal clusters because the charged trimer is a very stable closed-shell cluster with two valence electrons (one could draw a loose parallel with the emission of an alpha particle in nuclear physics). In this particular case the other fragment, Najj, is also a closed-shell cluster, which favors even further this deexcitation channel 31 . Two different parametrizations of the deformed jellium background have been compared, and shown in Figure 3. In the first one (upper shapes, Figure
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-0.20 —
•^ - 0 . 2 5 3
-0.30
-0.35 —
Figure 4: Evolution of one-electron energies with fragment separation for t h e fission of N a ^ . T h e fission pathway is described by the jellium shapes at the top of the figure. Solid and dashed curves correspond to filled and empty levels, respectively. T h e open squares on the left give the level spectrum of the spherical parent and the symbols on the right indicate those of the separated fragments.
3), the parent cluster is forced to elongate up to s = 18.3 a.u. and the jellium scission occurs at s = 23.0 a.u. After jellium scission, the energy barrier (dashed line) tends slowly to the classical Coulomb barrier (dotted curve). This parametrization leads to a barrier height of 0.44 eV, which is essentially due to the deformation energy needed to elongate the parent cluster. This value is obtained as the difference between the energy at the maximum of the barrier (which occurs at s = 18.3 a.u.) and the energy of the minimum at s = 11.8 a.u. This minimum is, in fact, the ground state of the parent cluster for this particular jellium parametrization. The ground state of NajJ is nonspherical because this cluster has the outermost electronic shell only partially filled (see Section 3). In the second parametrization (lower shapes, Figure 3) the neck starts forming at s = 6.1 a.u. Again, the ground state of the parent cluster is
177
deformed and corresponds to s ~ 17 a.u. The configuration of two touching jellium spheres (s = 16.8 a.u.) is very close to the jellium scission point of this fission pathway (s = 17.2 a.u.). Of course, after this point the barrier (solid line) reproduces the results of TJSM. The tendency of the cluster to fission is already apparent in its ground state, since the NaJx and N a J fragments are easily distinguished. We have pointed out in Section 3 that some neutral clusters can be viewed as supermolecules formed by smaller, particularly stable clusters. This is again the case in Na^J. In fact, since this cluster is doubly charged, the electrostatic repulsion between the two components enhances this effect. The evolution in Figure 4 of the single-particle energies with fragment separation, starting from the spherical parent configuration, indicates that the configuration of minimum energy is consistent with the interpretation of it being a cluster supermolecule N a ^ — Na^. We point out that the simpler ellipsoidal jellium model would yield a deformed ground state configuration with a total energy of —38.43 eV, which is above that of the cluster supermolecule. A similar fragment preformation has been found for Na^ J in the Cylindrically Averaged Pseudopotential Scheme (CAPS) 54 , and for K^J in DFT molecular dynamics calculations 40 . Other jellium parametrizations have been studied for Na^J — • N a ^ + NaJ and it has been found that the path described in terms of jellium spheres gives the minimum barrier height. The energy difference between the maximum of the barrier at s ~ 22 a.u. and the ground state at s ~ 17 a.u. gives a fission barrier height of Fm = 0.14 eV. This value can be also obtained from the relationship Fm = Bm + AHf , (15) where Bm is the fusion barrier height, that is the barrier seen by approaching the fragments from large separation, and the heat of fission AHf is the difference between the energy of the fragments at infinite separation and the energy of the parent cluster in its ground state configuration. In this case (we stress that the fragments are spherical, but the parent cluster is deformed) AHf = £?(Na£i) + £(Na+) - £ ( N a & ) = -0.90 eV.
(16)
Since the maximum of the barrier corresponds to a configuration in which the jellium spheres representing the emerging fragments are already disconnected, the fusion barrier height can be obtained from the simpler TJSM. The fusion barrier height Bm has nearly the same value in E T F and KS approaches, Bm ~ 1.04 eV. From eq. (15), this means that the influence of shell structure on the fission barrier height Fm is essentially due to its effect
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on the heat of fission. This is a statement for the most favorable fission path, but maybe not for the others. Let us recall, however, that the most favorable fission path is determined by the interplay between shell effects and liquid drop terms. One arrives at this conclusion by comparing KS 4 8 to E T F 4 5 calculations within DJM. In the ETF approach, two different fission paths characterized by large shape deformations have energy curves rather close to each other. The reason is that the ETF method does not account for shells effects. In contradistinction, in a KS calculation the system can take advantage of the closed-shell nature of NaJ and Najj to minimize its energy all the way along the fission path. For this reason the most favorable fission path is the one that starts from N a ^ split into two preformed magic (i.e., spherical) fragments. Looking at the different components of the total energy as a function of s, one can see that the kinetic energy and the exchange-correlation energy both tend quickly towards their limiting values at large fragment separation. In contradistinction, the electrostatic energy is the dominant term at large separation. The deviation of the true fission barrier from the simple Coulomb repulsion can be considered as a chemical effect of the electronic charge gluing the fragments 44 : the interacting clusters mutually polarize their electronic clouds, giving rise to a bonding contribution which is effective even if the two positive jellium pieces do not overlap. A semi-empirical model has been proposed 47 to express this bonding potential for large highly-charged simple-metal clusters. As a second example, let us summarize the KS results 43 for the symmetric fission of Na 4 J : Na^+ —• Na+ + Na+ . (17) The parent and the fragments are closed-shell clusters, with 40 and 20 valence electrons respectively, and consequently they adopt spherical configurations in the jellium model. Building on our accumulated experience, the fission path (characterized by A = 0) is described by jellium configurations corresponding to two interpenetrating spheres up to jellium scission and to two separated spheres afterwards. The spherical configuration is the ground state of the parent cluster. For this reason the fission barrier has less structure than in other cases. The fission barrier height is Fm = 0.37 eV and the heat of fission is AH/ = —0.58 eV, that is, the reaction is exothermic. The electron density corresponding to the saddle configuration is given in Figure 5 and shows the fragments almost separated 46 ; this confirms a posteriori that use of TJSM is justified to calculate the barrier height. The dramatic influence of shell effects is appreciated when the barriers obtained by KS and ETF methods are compared using the same jellium parametrization (spheres). The fusion barriers are similar, i? m (KS) = 0.95 eV
179
Figure 5: Valence electron density for t h e configuration corresponding to the maximum of the barrier (saddle point) for the symmetric fission of N a 4 j .
and B m (ETF) = 0.84 eV, but the heats of fission are very different, AHf(KS) = —0.58 eV and AH f (ETF) = 2.33 eV. In other words, the reaction is exothermic in the KS case and endothermic in the semiclassical one. The large value of AHf(ETF) is essentially a surface effect. Breaking the spherical parent into two fragments of equal size leads to a substantial increase in the surface area. This increases so much the surface energy that symmetric fission is very unfavorable. On the other hand, in the KS case the two fragments have closed shells (configuration Is 2 , lp 6 , Id 1 0 , 2s 2 ) and the high stability provided by the closed-shell configurations is enough to compensate the increase in surface area. Adding Bm and AHf leads to very different values for the fission barrier: F m (KS) = 0.37 eV and F m (ETF) = 3.17 eV. Having into account that the energy necessary to evaporate a neutral Na atom from clusters or from the metal is approximately 1 eV, the KS calculation predicts the symmetric fission of a hot Na 2 J cluster as more favorable than the evaporation of a neutral atom. The difference between the KS and ETF fission barriers compared here is so large because we have chosen two examples in which shell effects are particularly strong, since both fragments have closed-shell electronic configurations. In other cases, one would expect smaller differences between KS and ETF results.
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5
Optical response along the fission path
This section extends the previous discussion on fission in two respects. First, a detailed ionic background is used instead of a jellium. And second, electronic excitation properties (optical response) are studied at the various stages of fission. The ions are coupled to the electrons by means of pseudopotentials. We have used the simple Ashcroft pseudopotentials 55 . Moreover, we simplify the calculations by employing an angular averaging of the electronic potentials to cylindrical symmetry yielding the CAPS 5 4 . The ionic configurations are optimized by simulated annealing, and the fission path has been mapped by dividing the ions formally into two fragments and enforcing a constraint on a desired fragment separation 56 . The optical response is explored using TDLDA with subsequent spectral analysis 57 ' 58 - 59 . The electrons are excited instantaneously by a dipole shift. The oscillation of the electronic dipole momenta is then recorded. Finally, a Fourier transform into the frequency domain provides the seeked dipole strength distributions. The electronic oscillations are much faster than ionic motion, and the optical response can thus be analyzed for fixed ionic background. We take here as case of study the fission of Na^g" which represents a rare case of strong competition between asymmetric (Na^g" —• Naj"5 + N a J ) and symmetric ( N a ^ —» Nag" + Nag") fission56. Two dominant fission channels compete in the case of NajJ and the basic fission properties of these two channels (Fm around 0.5 eV, AHf around —0.5 eV) are remarkably close. One thus expects an "equilibrated" mixture of symmetric fission into two NaJ and asymmetric fission into Najj + Na^" fragments. We illustrate here the case of symmetric fission. Note that the ground state configuration (configuration 'A', Figure 6) is the same for both fission channels and is naturally asymmetric by virtue of shell effects. As a consequence one can spot a marked change between configurations 'A' and 'B' (Figure 6) where symmetry is restored. This rearrangement corresponds to a large difference in energy between configurations 'A' and 'B'. Ionic scission takes place at stage 'D' only shortly before the saddle while the electron cloud breaks up rather late after the saddle point, around configuration ' F ' (Figure 6). As can be seen from the left column of Figure 6 the optical response provides an enlightening picture of the various stages along the fission path. We focus here on excitations in the linear regime. The ground state configuration 'A' exhibits a pronounced resonance between 2 and 2.2 eV. The low frequency reflects the large elongation of this configuration 60 . With increasing deformation along the fission path we first observe (basically up to 'D') a gradual
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10
a CeV]
15
d
20
c m ( a o)
25
Figure 6: T h e fission of Najg illustrated in equidensity plots of the ionic distribution (upper part), in equidensity plots of the electron distribution (middle part), and in the deformation energy potentials (lower part). Left part shows the optical response at various points along the fission path, as indicated. The letters on the potential curves denote the stages shown above.
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increase in the fragmentation of the spectrum, but still centered around 2 eV. A marked change occurs when the electron densities separate at stage ' F ' . The fragmentation disappears and is replaced by the clean plasmon resonance of Nag" around 2.7 eV. Note that the still existing connectivity of the electron cloud in stages ' C and 'D' suffices to center the dipole spectrum around the low 2 eV, while the ions are already fully separated (stage 'D'). The peak moves towards the typical 2.8 eV in free Nag" with further increasing separation, stage 'G' and beyond. The optical response thus provides an enlightening tool of analysis of the various configurations along the fission path. It might be experimentally feasible to follow such a fission path by time-resolved recording of the electronic response following short laser pulses. One might even hope to have access to fission time scales and thus be able to estimate viscosity effects, in a way somewhat similar to the nuclear case. 6
From fission to fragmentation and Coulomb explosion
In the previous section, the fission process has been described adiabatically. This tell us a lot on both structural and dynamical properties of metal clusters, but they do not exhaust, by far, the dynamical situations accessible to metal clusters, even if it can address the competition between fission and evaporation as discussed by Frobrich 61 to explain the observed critical sizes Nc, calculating the time scale of cluster decay by an evaporation theory similar to that used to describe the statistical deexcitation of compound nuclei 62 . Recent progresses in laser technology have opened new avenues of research in the domain of non-linear cluster dynamics. Lasers actually offer an ideal tool for spanning various regimes of dynamics, ranging from the linear regime with plasmon-dominated dynamics 9 ' 6 0 , 6 3 , to the semi-linear regime of multi-photon processes 64 ' 65 ' 66 and the strongly non-linear regime of Coulomb explosion 67 ' 68 . From the theoretical side, only effective mean-field theories based on DFT have up to now been able to tackle with such different situations and dynamical regimes for clusters. These calculations have been undertaken since the mid 1990's by a few groups around the world, actually exploiting the numerical experience acquired in nuclear physics 57,58,69 ' 70 , see also the work by Calvayrac et al71 for a recent review on the topic. We shall focus here on a few examples dealing with situations very out of equilibrium. Let us first briefly sketch the various steps appearing in the response of highly-excited metal clusters. Experimentally, one can consider in fact two classes of rapid, intense, excitations: collisions with highly-charged energetic ions 72 and irradiation by intense femtosecond laser pulses 6 5 , 7 3 . In both cases the excitation takes place between tens of fs down to below 1 fs.
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This time is directly comparable to characteristic time scales of the valence electron cloud. And not surprisingly, the cluster response will thus be primarily of electronic nature. The first phase of the "reaction" is thus a direct emission of electrons and a collective oscillation of the Mie plasmon. This first phase is characterized by time scales of order 1-10 fs. In a second stage, still of purely electronic nature, damping of collective electronic motion takes place, both by means of Landau damping and by electron-electron collisions. The time scales associated to both effects are variable depending both on cluster size (Landau damping) and deposited excitation energy (electron-electron collisions). Landau damping ranges around 10-20 fs and collisional around 10-100 fs. After that, electronic degrees of freedom will slowly couple to ionic motion, and provoke the possible explosion of the cluster on long times (several hundreds of fs). Two mechanisms are actually at work here: i) the net charge of the cluster following ionization; ii) energy exchanges between the now "hot" electron cloud and the still "cold" ions. Both effects constructively interfere to activate ionic motion and to lead to evaporation, fission or fragmentation. Thermal evaporation of electrons proceeds on a very long time scale, usually slower than ionic processes (monomer evaporation, fragmentation). It can become competitive in the 100 fs range for very hot clusters (T >, 4000 K). In order to illustrate the various stages of the excitation and response of metal clusters in the non-linear regime we consider here two examples, focusing first on the electronic response. As a next step we consider the impact of ionic motion on the cluster response and show how ions can interfere with the excitation process provided the latter is long enough. Figure 7 represents the first stage (i.e. electronic) response of a metal cluster irradiated by an intense laser pulse. We consider here a Nag"3 cluster, treated in the jellium approximation for the ionic background. The cluster is excited by a ramp laser pulse of 100 fs total duration with 10 fs ramps; the intensity is J = 10 10 W/cm 2 and the photon frequency TWJ ~ 3.1 eV slightly above the Mie resonance for this cluster. As was shown in systematic calculations of irradiated clusters the response depends crucially on the actual laser frequency 71>74. For laser frequencies sufficiently far away from the plasmon resonance, the dipole response closely follows the pulse profile and disappears with the laser pulse profile. On the contrary, for laser frequencies close to the Mie resonance, the laser may attach the resonance; this results in a sizable electron emission and the dipole response survives the laser pulse as it generated a true eigenfrequency of the system. The example considered in Figure 7 corresponds to a situation in which the plasmon actually comes into play during the process. For the first 50 fs, the laser pulse remains above resonance and the electronic dipole moment D(t) (upper panel) follows the profile of
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L
120
t i m e Cfs] Figure 7: Electronic response of a Nag 3 cluster, as a function of time (in fs), to a 100 fs laser pulse of peak intensity 10 1 0 W / c m 2 ; upper panel: dipole moment along the axis of laser polarization (in ao); lower panel: number of emitted electrons.
185
the laser pulse. Still, the intensity is sufficient to provoke a small but steady ionization: the remaining cluster acquires a higher charge state, the electron cloud is thus more compressed, which shifts the plasmon resonance towards higher frequencies, and thus closer to the laser frequency. Finally, from about 50 fs on, the Mie plasmon moves into resonance with the laser, which leads to a jump in ionization, as known from systematic investigations 71 ' 74 . The process reaches a peak until the violent electron emission dampens the signal substantially and further blueshift detunes the plasmon from the laser. Still, even after the pulse has been switched off at t — 100 fs and electron emission has levelled off, the remaining electron cloud continues to perform collective oscillations at the actual plasmon frequency of the system (namely, for the net charge of the cluster). This example nicely demonstrates the role of the plasmon resonance in triggering ionization of metal clusters. For the moderately short pulse investigated, ions essentially remain fixed and do not interfere with the ionization process. However, this is not the case anymore when longer pulses are considered. Indeed, as was recently suggested experimentally 68 in Platinum clusters, the highly-charged cluster rapidly undergoes a Coulomb expansion, with a time scale around 100-500 fs. An interference can thus occur between the laser pulse and ionic motion, which may lead to enhanced ionization. In order to investigate this effect on a microscopic basis we consider here the simpler case of a N a ^ cluster subjected to a long laser pulse (T p u i s e = 240 fs) with uvaser = 2.86 eV. As the seeked effect relies on a matching of time scales it may be expected that changing material (to another metal) should not qualitatively alter it. The results of a TDLDA calculation are presented in Figure 8. The third panel (Nesc) shows that in this case ionization proceeds in various different steps (more than in the case of Figure 8, in particular, although on short time scales similarities do exist between the two cases). Indeed, again, during a first phase (<, 80 fs), the cluster experiences a fully electronic response characterized by a small but steady ionization. In turn, the enhanced charge state shifts the plasmon resonance upwards until it comes into resonance with the laser. This results in a sudden increase in ionization around 100 fs leaving the cluster with a charge state of about 5+. Up to that stage, we qualitatively recover the same situation as in the case of Nag~3. Prom then on, ionization proceeds at a slower pace until another burst of electrons shows up around 250 fs stripping again about 5 electrons. To further analyze the ionization process let us consider the other panels of Figure 8. The lowest panel displays the electronic dipole signal D(t). It is obvious from this observable that large slopes in ionization (NeSc) are correlated with large D(t) amplitudes, which again reflects resonant conditions 71,74 , and
186
100
200
300
time [fs] Figure 8: Excitation of Najj with a laser of frequency u>\a = 2.1 eV, intensity / = 9 x 10 9 W / c m 2 and pulse length X p u i s e = 240 fs. From top to bottom: global extension of the ionic distribution in z- (along laser polarization) and axial r-direction (in the direction transverse to the laser polarization); average resonance frequency w re s {t) for the actual (timedependent) net charge and ionic extension; number of escaped electrons; dipole signal.
187
viceversa. To unambiguously establish a link between both observables, we have obtained a collective estimate of the instantaneous plasmon frequency tabulating Wpiasmon as a function of the ionic extension y/< z2 >, \ / < r 2 > and ionization Nesc, and have deduced the instantaneous ai res (i) by interpolation for the actual values of these three key observables. The results are shown in the second panel of Figure 8 in comparison to the laser frequency (dashed horizontal line). The correlation between large slopes in iVesc and resonant conditions is striking. The first coincidence at time 100 fs reflects the blueshift due to the first stage of ionization and thus corresponds to an electronic effect. It also triggers the time at which a sizeable Coulomb expansion of the ionic distribution starts (see also uppermost panel). And it is noteworthy that this occurs very soon (about 50 fs) after the violent initial charging. But the Coulomb expansion in turn leads to a redshift of the resonance (the Mie plasmon being inversely proportional to rs ), which is responsible for the second coincidence at around 230 fs. The system thus acquires a high charge state and ends up in a violent Coulomb explosion. This case of study exhibits very nicely two different phases in the ionization process: a first enhancement of purely electronic origin followed by a second one of ionic origin. It also points the importance of a proper non-adiabatic account of both electronic and ionic degrees of freedom 7 5 . Acknowledgments This work has been supported by DGESIC (Spain) under grants PB98-0345, PB98-1247 and PB98-0124, by the Freeh-German exchange program PROCOPE, number 99074, and by the Institut Universitaire de France.
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192 T H E ROLE OF EXOTICA IN S T U D Y I N G N U C L E A R
FISSION
F.F. Karpeshin Universidade de Coimbra, Departamento de Fisica, Coimbra-3000, Portugal, and St. Petersburg University, Dept. of Physics, 198904 St. Petersburg, Russia E-mail: [email protected] Advantages of applying muons for studying the fission dynamics are briefly reviewed. 1
Introduction
Many hopes were related with p r o m p t fission of nuclei by negative m u o n s . T h e process was first proposed by Wheeler l at the fall of the first decade after discovery of fission. It occurs as a result of a radiationless transition of a m u o n 2s —> I s . T h e transition energy is transferred to the nucleus, giving it chance to undergo fission. As distinct from electron atoms, in muonic a t o m s the 2s level is n o t a m e t a s t a b l e one, and is mainly deexcited via 1.2-MeV electric dipole transition to t h e 2p s t a t e . B u t the m a i n reason which makes this process of the radiationless 2s —> Is transition difficult to observe is a small probability of population of the 2s level in /J. atomic cascade (less than 10%). Fortunately, as it was shown by Zaretsky 2 , there is a great probability of nuclear excitation also in electric dipole transitions 2p —> I s , 3p —> I s , which is of the order of tens percent. These works induced the discovery of the radiationless 2p —> Is transition in muonic atoms of uranium by observing missing intensity of the 2p —> I s line in comparison with lead a t o m 3 . Double-magic nucleus of 208Pb has low level density at the energy of the muonic transition, which fact makes the radiationless transitions unlikely. Furthermore, as it was shown by Teller and W e i s s 4 , Karpeshin and Nesterenko 5 , the probability of radiationless excitation of the giant quadrupole resonance in a muonic a t o m of u r a n i u m is also great, about 25%, which makes m u o n a multipole meter in exciting t h e nucleus by monoenergetic 7 quanta of a certain m u l t i p o l a r i t y 4 . And, according to the papers by Karpeshin and Nesterenko, the probability of radiationless excitation of the low-lying giant octupole resonance in an E3 m u o n transition 3d —>• 2p is of the same order of m a g n i t u d e . Characteristic t i m e of p r o m p t fission is determined by the t i m e of the \i atomic cascade. Most often the muons are captured t o high atomic orbits, when forming a muonic atom, with the principal q u a n t u m number n w 14. T h e t i m e of cascading down in heavy a t o m s is 1 0 - 1 3 to 1 0 - 1 5 second. Along with p r o m p t fission, radiationless excitation gives rise t o other p r o m p t processes,
193 which were studied in experiment: p r o m p t neutrons 6 , g a m m a s , which are of great interest for studying nuclear structure, but which we owe to remain beyond t h e scope of the present short review paper. As distinct from p r o m p t fission, delayed fission occurs due to nuclear capture of t h e muon after t h a t reaches the Is level. Characteristic time of the delayed fission is therefore determined by the lifetime of the muon in the orbit, which is a b o u t 70 ns in u r a n i u m a t o m . This gives a possibility to distinguish the two kinds of fission in experiment by means of electronics. In p r o m p t fission, t h e muon having induced the process remains its close spectator up to end. It is entrained on one of the fragments, mainly the heavy one, forming a muonic a t o m . This presents a unique opportunity to use the m u o n as a probe for studying b o t h dynamics of fission, as well as properties of t h e neutron-rich nuclei of the fragments. Barrier a u g m e n t a t i o n which takes place in the presence of the muon enables one to better know its structure 4>7>8'9. Muon a t t a c h m e n t probability to the light fragment is sensitive to the dynamics of fission. Experiments are also performed to study muonic conversion of the 7 q u a n t a in the fragments ( l 0 ' 1 1 . 1 2 and refs. cited therein), as well as [i c a p t u r e in b o t h the fragments a n d mother u r a n i u m nuclei 6 . 1 3 . 1 4 . A possibility of registration of the characteristic muonic X rays from the fragments is of undoubted interest. A step in this direction has been m a d e in paper 1 5 . A m o n g the possible mechanisms of the muon X ray emission, resonance conversion mechanism is most probable, when the m u o n s are first virtually thrown u p to the 2p state, and then make t h e radiative transition back to the Is o r b i t 1 6 . Let us consider some of these processes in finer detail, outlining t h e present s t a t u s of the problem, and giving prospects of future investigation. Let us s t a r t with the process itself of the radiationless excitation of an actinide nucleus.
2
Radiationless excitation
Most simple and heuristic approach is to consider radiationless excitation of the nucleus as an inverse conversion process 5 . Advantage can be maid of using internal conversion coefficients (ICC), in this case - muonic conversion coefficients ( M C C ) . ICC in electron a t o m s are as a rule rather weakly sensitive to the nuclear structure. This allows one to factorise the probability of a conversion transition in the two independent multipliers: radiative transition probability and properly I C C , reducing the problem to their independent consideration. This is d u e to small probability t o find electron in the nucleus. In our case, penetration effects in muonic atoms of the fragments are not small, a n d their influence on the calculated probabilities m a y be estimated as
194 20 to 30 p e r c e n t 1 0 , which accuracy is fairly enough for treating the available experimental data. Adequate results have been obtained in a nuclear model of the surface transition currents. Use of another nuclear model, of the volume transition currents, which is founded on opposite physical premises, m a y be m a d e for control of the accuracy of the results. By comparison of the M C C calculated within these and other macroscopic models with a microscopic calculation in the R P A it has obtained t h a t the model of the volume nuclear currents is very appropriate for description of the isovector dipole transitions 17 , while for the other transitions the model of the surface nuclear currents appears more suitable, although b o t h the models work well t o provide the accuracy given above. T h e M C C of the multipolarity EL are defined for the inverse nuclear transition u —> 0 as the ratio of the radiative T 7 and conversion Tc widths:
Expressing T 7 in terms of the electromagnetic strength function, one can straightforwardly obtain t h e expression for the probability of the radiationless transition from the principle of the detailed balancing [5,6] as follows: r r ,, = aW{EL;o -> U )}™V +^yL+H(EL.0
^
u )
t
(2)
where a is the fine structure constant. We use relativistic units fi = h — c = 1, H being the muon mass. In eq. (2), M C C ajf>{EL; 0 -> u) is related t o the a\i '(EL;u
—>• 0) defined in eq. (1) by the detailed balancing principle, aW(EL;0
-+ w) = ^ ± I a ( , " ) ( S L ; W -> 0) ,
(3)
and superscript "d" is added to indicate the circumstance t h a t both the electron states , initial "i" and final "f", belong t o the discrete spectrum. T h e probability of a radiationless transition reads as the branching ratio w
"-
=
FTr-
•
(4)
J-a + i-r.l.
where Ta is the n a t u r a l atomic width of the line. Furthermore, the strength function b(EL; 0 —> w) may be extracted from the experimental d a t a on photoabsorption by the nucleus, as was made in the papers 2 , or calculated in a model for the giant multipole resonances, as was
195 m a d e in ref. 4 . With account of these remarks, all t h e formulae of refs. 2 ' 4 can be obtained on the basis of eq. (2) if t h e nonrelativistic long-wavelength approximation is used for calculating t h e M C C . It is worthy t o note t h a t t h e formfactor B introduced in papers 2 to include the muon penetration effects equals t o unity in the model of t h e surface nuclear currents. T h e Bohr model 18 treating the transition nuclear density as proportional to t h e derivative of the density of t h e nuclear m a t t e r h a s been used in ref. 4 . A process of primordial physical interest was considered in ref. 2 . Namely, Zaretsky and Novikov considered a possibility of reexciting t h e muon back t o the 2p state, following the radiationless 2p —> Is transition, when the muon receives back all its energy before transferred to t h e nucleus. Such a process would be forbidden in the classical mechanics by t h e second principle, e.g. like elastic neutron scattering t h r o u g h t h e c o m p o u n d stage. Zaretsky and Novikov found t h a t the result depends on t h e degree of overlapping of t h e c o m p o u n d resonances s = *f/D, where 7 is t h e average width, and D is t h e m e a n energy separation of the resonances. If s > 1, then the nucleus works as a t h e r m o s t a t , and t h e energy is distributed between many compound-resonances within t h e n a t u r a l width of t h e /x-atomic line. In this case, t h e probability of the inverse promoting the muon m a y be really neglected. In t h e opposite case of s < 1, t h e muon may be lifted u p t o t h e original 2p orbit, which diminishes the resulting probability of t h e radiationless excitation of the nucleus. This result can be dire4ctly checked in experiment by comparing t h e probability of t h e radiationless excitation in t h e muonic a t o m s of 235JJ and 238U. In t h e odd nucleus of 235U, the level density is much higher, resulting in the value of s > l . In 2 3 8 { / t h e value of s < 1 is expected. Moreover, the value of r, varying rapidly with t h e transition energy, is expected t o be essentially different for the transitions between the various components of fine structure, 2pi/2 —> I s and 2^3/2 —>• I s . Measuring t h e relative probabilities of these radiationless transitions offers an independent way of checking the effect of the overlapping of t h e levels. In ref. 4 , t h e Coulomb interaction between t h e m u o n and t h e nuclear transition density was supposed: V(\?-R\)
= -?-=• , \r — R\
(5)
where r and R are the muon and nuclear space coordinates, respectively. Matrix element of interaction (5) m a y be expressed in terms of the M C C (cf. 1 9 ) . In t h e long-wavelength approximation, t h e relation reads 2r\V\2 = Tc = aW(EL;w^0)Ty{EL)
,
(6)
196
Table 1: Calculated probabilities of the radiationless transitions WT ; data12^21. Multip.
Transition
w, MeV
Wr.t., theory SC
VC
0.11 0.19 0.55
0.15 0.26 0.65
2Pi/2 -»• 1* 2P3/2 -> l s 3P1/2 -> 1*
El El
El
6.30 6.53 9.59
3p 3 /2 -+ 1* 3rf3/2 -» I s
El E2
9.65 9.50
0.57 0.19
0.69 0.24
3 4 / 2 "~^ ^ 5 3 4 / 2 -+ 2 p 3 / 2 3 4 / 2 ->• 2 p i / 2 3 4 / 2 ->• 2 P 3 /2
E2 E3 E3 E3
9.57 2.97 3.27 3.04
0.25 0.11 0.07 0.05
0.32 0.13 0.08 0.007
Wri
and experimental %, experim.
21.6±3.2 31.1 ± 2.8 88.9 ± 4 . 3 °
1 2 . 8 ± 1,4°
which follows the definition (1). Reducing the giant multipole resonance to the constant D may be determined from the nances, as in ref. 4 . In the first approximation, weighted sum rule. As a result, one obtains an V
1
{
a
T* »
(d)
(EL;UJ
->
0)w 2L + 1
one state at the energy of En, systematics of the giant resouse can be m a d e of the energy analytic expression as follows:
L + 1 [(2L-1)!!]2
ZaR2 ER
V-
(7)
where R is the nuclear radius. Numerical values of V obtained by m e a n s of eq. (6) are in reasonable agreement with those of ref. 4 within tens percent. Microscopic calculations of the probabilities of radiationless transitions of various multipoles for a muonic atom of 2 3 8 £/ have been performed in refs. 1 7 by use of the quasiphonon-quasiparticle nuclear model. T h e model was proved to work well for description of both the low-lying nuclear states and giant resonances, in deformed as well as spherical nuclei. Nuclear strength functions have been calculated in RPA as an average over many one-phonon states: b(EL; 0 - > w ) =
^B{EL;Q -> u)a
A/27T ( c - u , s ) 2 + (A/2)2
(8)
where u>g is the energy of an one-phonon state, and A is the averaging p a r a m eter. T h e results of the calculation in comparison with the experimental d a t a are presented in Table 1.
197 Note t h a t contributions of the transitions of several higher multipoles (E2, E3) t o the radiationless excitation probability turns out to be of the s a m e order of m a g n i t u d e . This is partly due to increasing the I M C C with t h e multipole order 1 0 ' 5 : <*H \n^,uj
-rv)
( W _R)2J+1
On the other hand, the radiative width
a
~ [(2L + 1)!!]2 '
where we allowed for t h a t the Bohr radius of a muonic orbit in heavy a t o m s is close to the nuclear radius. Therefore, the radiative w i d t h is expected to slowly decrease with the multipole order as ~ 1/L 2 . In this respect, spectaculas is a large calculated probability of t h e E3 radiationless transition3d —¥ 2p, which successfully competes even with the radiative electric dipole transition between the same states. T h i s is due to two reasons. First, the above mentioned increase of the IMCC for higher multipoles and low-energy transitions. T h e second reason is in the nuclear structure, because the transition energy coincides with the low-lying giant octupole resonance (LEOR). Therefore, radiationless transitions can be used for studying fine structure of the L E O R . T h e latter arises 2 2 due to interplay between the energies and spreading widths of the one-phonon states, a n d its manifestation is in contrast with the high-energy giant multipole resonances. Due t o fine structure of the L E O R , the probability of its radiationless excitation can considerably vary in different actinoid atoms, or for the muon transitions between various components of the atomic spectra of a muonic a t o m , as in the case of the fine structure of the 2p —> Is transition. This variation is expected due to a possibility for the m u o n transition energies to coincide or not to coincide with an energy of one of the nuclear L E O R component.
3
P r o m p t - t o - d e l a y e d fission ratio.
P r o m p t fission presents a unique possibility to vary fission barrier. This is due to decreasing muon binding energy with increasing deformation of the fissile nucleus. This leads to effective increasing the barrier by t h e corresponding value of 0.5 to 1 M e V 8 . This effect has been studied in m a n y experiments ( e _ g-i 20,7,23,24,25,9,26,27 a n d r e f s c i t e d therein). Thus, in n a t o m of 23SU total fission probability is 0.070 ± 0.08 per a t o m 1 1 . On the other h a n d , the ratio
198 of p r o m p t to delayed fission is 0.0876 ± 0.010 [38]. W i t h the allowance for the radiationless excitation probability to be of ~ 0.25, the prompt fission probability of ~ 2 % can be deduced. This probability is by an order of m a g n i t u d e less t h a n e.g. in photofission at the same excitation energy, in which case the ratio of the neutron and fission widths Tn/Tf « 4. We note t h a t in ref. 2 0 photoemulsions have been used for the observation of prompt fission, which simple and clear m e t h o d can be successively used in the contemporary investigations (e.g. 2 8 )Suppression of the p r o m p t fission probability by an order of m a g n i t u d e is also observed in the E2 transitions of 3d —» I s in 237Np, although the transition energy exceeds the fission barrier by 3 MeV in this case. However, suppression of p r o m p t fission in this case can be explained in terms of diminishing the level density in the fission channel, which occurs due to the augmentation of the barrier. In contrast, in 237Np the augmentation of the barrier does not influence the p r o m p t fission probability, which remains at the same level of about 60% 26 , as for t h e other fission reactions. T h e same holds for the plutonium a t o m s . This p a r a d o x deserves very careful further study, both experimental and theoretical one.
4
M u o n attachment probabilities
May be, m o s t heed was paid to the question of the muon a t t a c h m e n t probabilities to t h e fragments. A great deal of the papers is devoted to the problem (12,29,30,31,32,33,34,36,36,37,38,42,39,40,41 a n d r e f s c i t e d therein), much more t h a n the experimental ones 6>43'44>45>46. T h e interest is due to possibility of studying fission dynamics through the muon attachment probabilities to the light fragments. In the beginning of p r o m p t fission, the muon is in the Is s t a t e of the fissile nucleus. Relative velocity of the collective motion towards fission is about six t i m e s lower t h a n the mean muon velocity in the orbit, b u t their mass is three orders of magnitude as large as the muon mass. Therefore, motion of the fragments may be considered quasiclassically, and the muons most often remain in t h e ground \soo. There is, however, a possibility for the m u o n to be promoted to a higher quasimolecular s t a t e , which correlates with a state at the light fragment for R —> oo. Most of the competition comes from the 2pa level, going over the Is s t a t e of the light fragment. Moreover, these states have a point of avoided crossing, which plays an i m p o r t a n t part in the muon fate.
199 Table 2: Muon attachment probabilities Pi{Z) 4 6 depending on the mass of the light fragment M i (in a.m.u.) and the total kinetic energy of the fragments T K E . ML
T K E , MeV 100 - 170
9.5 ± 3.7
170 - 250 100 - 170
6.9 ± 2.5 6.2 ± 1.9
170 - 250 100 - 170
4.9 ± 1.3 5.2 ± 1.3
170 - 250
3.1 ± 1.3
PL%
1 0 4 . 5 - 111.5
97.5 - 104.5
90.5 - 97.5
For uranium, the average probability of the m u o n a t t a c h m e n t to the light fragment P^ is about 5%. In experiment, the m u o n distribution of the fragments is studied by electronic methods 9.43,44,46^ a s w e j j a g ^y m e a n s 0 f p n 0 . t o e m u l s i o n s 1 3 , 4 5 . In the latter case, emission of the light charged particles (p, d, t, a) from the end points has been observed, induced by the muon capture in the stopped fragments. A problem of this m e t h o d is t h a t the latter probability itself depends on the Coulomb and centrifugal barriers, isotope yield of the elements, energy stored in the fragments, and other factors precise influence of which is difficult to determine. More directly, the muon a t t a c h m e n t probability can be measured by detecting the electrons from the muon decay in the orbit. In this way, in refs. 9 ' 4 6 the Pi values have been measured at the meson factory of the PSI facility at SINDRUM detector. Along with the mean a t t a c h m e n t probability < Pr, > , its correlation with the mass and total kinetic energy of the fragments has been observed in this experiment for the first time. T h e latter influence is crucial for studying the dynamics. It is worthy to remind the two salient features of the d a t a which are of great interest from the viewpoint of studying the dynamics. First, this is divergence of the slopes of the charge dependence of the theoretical and experimental probabilities P L ( Z ) for fission with great a s y m m e t r y of the mass split, with a general agreement concerning the mean value of P^. Second, experimental d a t a give evidence t h a t the PL{Z) values increase with decreasing total kinetic energy of the fragments, which also does not correspond to a simple quasimolecular picture. This is shown in Table 2. Theoretical calculations do not give a u n a n i m o u s answer concerning sensitivity to fission dynamics ( e . g . , 3 8 , 4 0 , 4 2 and refs. cited therein). This question can be clearly solved within the Landau m e t h o d of complex trajectories 5 1 .
200 According t o the m e t h o d , the real trajectory of fragments R(t) is bent into the upper half-plane of complex R, except singular points. Such points are the starting point at the saddle Rs, the scission point Rsc, and the point of avoided crossing of the two lowest t e r m s Rc. Thus, the final transition amplitude is comprised by separate contributions of the domains around these three points. Let us designate them ps, psc and pc, respectively, then, the ps amplitude is expected to be proportional to V(RS) = 0 36 , and therefore small. T h e avoided crossing contribution pc is mainly determined by the value of V(RC), i.e. by the C o u l o m b repulsion of the fragments rather t h a n by the prefission dynamics. Dependence on the prefission dynamics is therefore concentrated in psc. Were the point of scission a regular point of the trajectory, the sensitivity to prefission dynamics would be therefore minimal, if any. Calculation shows t h a t in the case of asymmetrical fission of 2 3 8 [ / with normal charge division, Z = 52 + 40, the psc value is within 10% of the avoided crossing contribution pc. On the other h a n d , in the case of larger asymmetry, Z = 57 + 35, relative value of psc raises u p to about 30%. Therefore, a higher sensitivity to the fission dynamics is predicted for more asymmetric fission. This result has been explained earlier on the physical ground , as the point of avoided crossing is closer to the scission point, remaining beyond it. Results of the calculation are presented in refs. 4 1 ' 4 7 > 4 8 ] and shall be further published elsewhere. It should be noted t h a t a contribution of a nonadiabatic shake-off mechanism due to a b r u p t rupture and contraction of the remains of the neck has been calculated in refs. 49 > 50 . Its probability was found to be of about 0.5%. One should also expect manifestation of shake-up effects in the muon a t t a c h m e n t probabilities.
References 1. Wheeler, J. A., Phys. Rev. 7 3 , 1252 (1948); Rev. Mod. Phys. 2 1 , 133 (1949) 2. Zaretsky, D. F . , Novikov, V. M., Nucl. Phys. 1 4 , 540 (1959); Nucl. Phys. 2 8 , 177 (1961). 3. Balatz, M.Ya., et al. Zhurn. E k s p . Teor. Fiz., 3 8 , 1715 (1960); 3 9 , 1168 (1960). 4. Teller, E., Weiss, M. S., Trans. N. Y. Acad. Sci. Ser. II Vol. 4 0 , 222 (1980) 5. Karpeshin, F . F., Nesterenko, V. O., Com. of JINR, D u b n a (1982), E 4 8 2 - 6 9 4 ; J. Phys. G: Nucl. P a r t . Phys. 17, 705 (1991) 6. W.U.Schroder et al. Phys. Rev. Lett. 4 3 , 672 (1979). 7. J.A.Diaz. Nucl. Phys., 4 0 , 44 (1963)
201
8. G.Leander and P.Moller. Phys. Lett. B57, 1975 (1975) 9. David P. et al. In: Int. School on Heavy Ions, Alushta, 1989. Dubna, JINR, 1990; Int. Conf. "50-th Anniversary of Nuclear Fission", St. Petersburg, 1989. Vol. 1, p. 300. 10. Karpeshin, F.F., Band, I.M., Listengarten, M.A. and Sliv, L.A. Izv. Akad. Nauk SSSR, ser. fiz., 1976, 40, 1164. (Engl, transl. Bull. Acad. Sci. USSR, Phys. Ser., 40, 58 (1976)) 11. Belovitsky G.Ye., Batusov V.A. and Sukhov A.I. Pis'maZh. Eksp. Teor. Fiz., 1980, 32, 380. (Engl, transl. JETP Lett. 1978, 27, 625); Belovitsky G.Ye., Sukhov A.I. and Petitjean K. In: Lecture Notes in Physics, Ed. P.David, T.Mayer-Kuckuk and A. van der Woude. Vol. 158, p. 71. Berlin, Heidelberg, New-York: Springer, 1982; Belovitsky G.Ye., Baranov V.N., Petitjean K. and Rosel C. In: Int. Conf. "50-th Anniversary of Nuclear Fission", St. Petersburg, 1989. Vol. 1, p. 313. 12. Karpeshin, F. F. , J. Phys. G: Nucl. Part. Phys. 16, 1195 (1990) 13. Belovitsky G.Ye. and Petitjean C. Pis'ma Zh. Eksp. Teor. Fiz., 1983, 38, 212. (Engl, transl. JETP Lett. 1983, 38) 14. Belovitsky G.Ye. et al. Yad. Fiz. 4, 1057 (1986) 15. Ch. R6sel et al. Z. Phys. A345, 425 (1993) 16. D.F.Zaretsky and F.F.Karpeshin. Yad. Fiz., 29, 306, 1979. (Engl. transl. Sov. J. Nucl. Phys., 29, 151, 1979.) 17. Karpeshin, F.F. and Starodubsky, V.E. Yad. Phys., 1982, 35, 1365. (Engl, transl. Sov. J. Nucl. Phys. 35, 795 (1982)) 18. Bohr A. and Mottelson B.R. Nuclear Structure, Vols. 1 and 2 (Benjamin, New-York, 1969,1975) 19. F.F.Karpeshin, I.M.Band, M.B.Trzhaskovskaya, B.A.Zon and M.A.Listengarten. JETP (1990), 98, 401; Phys. Lett. 1992, B282, 267; Can. J. Phys. 1992, 70, 623; Phys. Lett. 1996, B372, 1. 20. Belovitsky G.Ye. et al. Zhurn. Eksp. i Teor. Fiz., 4 1 , 66 (1961) 21. Ch. RBsel, P.David et al. Z. Phys. A340, 199 (1992) 22. J.M.Moss et al. Phys. Rev. C18, 741 (1978) 23. V.Cojocaru et al. Phys. Lett., 20, 53 (1966) 24. D.Chultemet al. Nucl. Phys., A247, 452 (1975) 25. B.Budick et al. Phys. Rev. Lett. 24, 604 (1979) 26. W.Schrieder, P.David, H.Hanscheid et al. Z. Phys. A339, 445 (1991) 27. Ch. Rosel, H.Hanscheid, J.Hartfiel et al. Z. Phys. A345, 89 (1993) 28. F.F.Karpeshin, G.Ye.Belovitsky, V.N.Baranov and O.M.Steingrad. Yad. Fiz., 62 (37) 1999. (Engl, transl. Physics of Atomic Nuclei (USA), 62, 32 (1998))
202
29. Karpeshin F.F. Yad. Fiz. 55, 29 (1992). (Engl, transl. Sov. J. Nucl. Phys. 55, 18 (1992)) 30. Yu.N.Demkov, D.F.Zaretsky, F.F.Karpeshin, M.A.Listengarten, and V.N.Ostrovsky. JETP Lett. 28, 263 (1978); Sov. J. Nucl. Phys. 28, 621 (1978). 31. Karnaukhov V.A. Sov. J. Nucl. Phys., 28, 1204 (1978). 32. P.Olanders, S.G.Nilsson and P.Moller, Phys. Lett. 90B, 193 (1980) 33. J.A.Maruhn, V.E.Oberacker, and V.Maruhn-Rezwany. Phys. Rev. Lett. 44, 1576 (1980). 34. Z.Y.Ma, X.Z.Wu, G.S.Zhang, Y.C.Cho, Y.S.Wang, J.H.Chiong, S.T.Sen, F.C.Yang and J.Rasmussen. Nucl. Phys. A348, 446 (1980). 35. Z.Y.Ma, X.Z.Wu, G.S.Zhang, Y.Z.Zhuo and J.Rasmussen. Phys. Lett. 106B, 159 (1981) 36. F.F.Karpeshin and V.N.Ostrovsky. J. Phys. B: At. Mol. Phys. 14, 4513 (1981). 37. Karpeshin F.F., Kaschiev M. and Kaschieva V.A. Yad. Fiz., 36, 336, 1982.(Engl, transl. Sov. J. Nucl. Phys. (USA), 36, 195, 1982) 38. L.Bracci and G.Fiorentini, Nucl. Phys. A423, 429 (1984) 39. David P., Rosel, Ch., Karpeshin F.F. et al. Proc. of the Workshop on Muonic Atoms and Molecules, Monte Verita, Ascona, April 5 - 9 , 1992. 40. Oberacker V.E. et al. Phys. Lett., B293, 270 (1992). 41. F.F.Karpeshin. In: Proc. of the XXXV-th Winter Meeting on Nuclear Physics, Bormio (Italy), 3 - 8 February 1997. 42. Karpeshin F.F., Kaschiev M. and Kaschieva V.A. Yad. Fiz. 1987, 45, 1556. (Engl, transl. Sov. J. Nucl. Phys. 45, 965 (1987)) 43. Ganzorig Dz., Hansen P.G., Johansson T., Jonson B., Konijn J., Krogulski T., Kuznetsov V.D., Polikanov S.M., Tibell G., Westgaard L. Nucl. Phys., 1980, A350, 278. 44. P.David et al. Invited talk at the XVIII-th Int. Symp. "Nuclear Physics and Chemistry of Fission", 1988. - In: Proc. of the XYIII th Int. Symp. on Nucl. Phys. devoted to the Fiftieth Anniversary of the Discovery of Nucl. Fission, Gaussig, 21-25 November 1988. Ed. Marten H., Seeliger D. 45. Belovitsky G.Ye., Baranov V.N. and Petitjean K. In: Physics and Chemistry of Fission. Proc. of theXYHl" 1 Int. Symp. on Nucl. Phys. devoted to the Fiftieth Anniversary of the Discovery of Nucl. Fission, Gaussig, 21-25 November 1988. Ed. Marten H., Seeliger D., p. 249. 46. Risse F., Bertl W., David P.,Hanscheid H., Hermes E., Konijn J., de Laat C.T.A.M., Pruys S., Roosel C , Schrieder W., Taal A., Vermeulen D. Z. Phys. A - Atomic Nuclei, 1991, 339, 427.
203
47. F.F.Karpeshin. In: Proc. of the NDST-97 Intern. Conf. on Nuclear Data for Science and Technology, Trieste (Italy), 19-24 May 1997. 48. Karpeshin F.F. Yad. Fiz., 2000. 49. Karpeshin F.F. Yad. Fiz. 55, 2893 (1992). (Engl, transl. Sov. J. Nucl. Phys. 55, 1618 (1992)) 50. Karpeshin F.F. Z. Phys. A344, 55 (1992) 51. Karpeshin F.F. To be published.
204
PARTICLE-ACCOMPANIED FISSION M. Mutterer V, Yu.N. Kopatch x
1]2) a
, D. Schwalm 3 \ and F. Gonnenwein 4)
> Institut fur Kernphysik, Technische Universitat, 64289 Germany 2 > Frank Laboratory of Neutron Physics, JINR, Dubna, z > Max-Planck-Institut fur Kernphysik, 69115 Heidelberg, 4 ) Physikalisches Institut der Universitat, 72076 Tubingen,
Darmstadt, Russia Germany Germany
A ternary fission experiment was carried out on 2 5 2 Cf(sf), using the DarmstadtHeidelberg 47r Nal(Tl) Crystal Ball spectrometer as highly efficient 7-ray and neutron detector, additionally equipped with detectors for fission fragments and ternary particles. The experiment has covered, besides the relatively abundant aparticle-accompanied fission, also the rare fission modes with emission of ternary particles up to carbon nuclei. The novel results achieved in this study on various aspects of the ternary fission process are summarized and discussed.
1
Introduction
In particle-accompanied fission of actinides, also known as ternary fission (TF), a light charged-particle (LCP) cluster is ejected intermediately from the fission fragments, near the instant of scission. The ternary mode of fission was discovered already in 1946 after exposing uranium-loaded photographic plates to a neutron flux 1 . In low-excitation fission processes, as thermalneutron induced and spontaneous fission, TF occurs with a moderate yield of 3 - 6 x 10~ 3 compared to binary fission, in about 90% of cases with aparticle emission. The probability of a splitting into three fragments of nearly equal masses (so-called "symmetric tripartition" or "true ternary" fission) is in comparison low, with measured upper limits in the order of 10~ 9 (Refs. 2 ' 3 ). The high mean ternary a-particle energy of 16 MeV, and the main emission direction orthogonal to the fission axis, were soon qualitatively understood by imagining creation of the a cluster at the neck that joins the fragments prior to scission, the fragments' Coulomb field having high focusing strength at that moment. Thus, many ternary fission studies have aimed at getting experimental access to the short (~ 10~ 21 s) scission stage in fission, i.e. for probing fragment scission-point configuration and prescission dynamics. This goal has not been fully met to date, though considerable progress has been achieved in recent years by multiparameter studies on the "present address: Gesellschaft fur Schwerionenforschung, 64291 Darmstadt, Germany
205
three-body kinematics with the detection system DIOGENES 4 , by thorough measurements of LCP probabilities and energy spectra with LOHENGRIN (e.g., 5 ' 6 ) , and by high-resolution triple 7-ray correlation experiments with GAMMASPHERE (e.g., 7 ) . The present report is primarily concerned with a recent experiment which has aimed at a more detailed study on the kinematics in LCP-accompanied spontaneous 2 5 2 C / fission, including registration of prompt fission 7-rays and neutrons. References to other closely related work will be given, but no attempt is made at giving a comprehensive survey. Review papers on TF were published most recently by Wagemans (1991) 8 , and Mutterer and Theobald (1996) 9 . 2
Experiment
In the experiment a combination of high-efficiency angle-sensitive detectors was applied for determining not only the kinematical parameters of the reaction, i.e. kinetic energies and mutual emission angles of the ternary particle and the correlated pair of fission fragments, but also the excitation state of the products by registering simultaneously neutrons and 7-rays from their disintegration. The measurement was performed at the MPI-K, Heidelberg, using the Darmstadt-Heidelberg Crystal Ball (CB) spectrometer 10 - 11 , being a homogeneous 4ir detection system made up of 162 large Nal(Tl) crystals of high detection efficiency (> 90% for 7- rays, and ~ 60% for fission neutrons) 12 . Neutron registration is separated from the prompt 7-rays by a time-of-flight method. The 2 5 2 C / source and the detector system "CODIS" for fragments and particles 13 were mounted at the center of the CB. The set of measured parameters has allowed to determine, for each fission event, the following quantities and their mutual correlations: fragment masses and kinetic energies; multiplicity and angular distribution of fission neutrons; multiplicity, energy and angular distributions of fission 7-rays; energy, nuclear charge (mass), and emission angle of the LCP from ternary fission. 3 3.1
Results and Discussion Fragment Excitation Energy and LCP Emission Probability
From fission-fragment energy and mass distributions obtained with CODIS, fragment total excitation energy TXE could be deduced, for the first time in the rarer LCP-fission modes with particles heavier than He isotopes. It was interesting to show that LCP emission proceeds in expense of a considerable
206
amount of fragment TXE (35 MeV, on average, for binary 2 5 2 C / fission), with the required energy increasing with LCP mass and energy. As an example, average TXE decreases from 27 MeV to 15 MeV when instead of an a-particle a ternary C-isotope is emitted. In this sense TF with emission of heavier LCPs features a rather cold large-scale re-arrangement of nuclear matter. o -2
o
••
-i
A D
>; Si
&
-6
® x
>^" £
H3 He4 He6 He8 Li Be B C
-8
-12 -U
0
5
10
15
20
25
30
35
ATXE [MeVl Figure 1. Emission probability (logarithmic scale) of light charged particles, as a function of the difference between mean total excitation energies TXE in binary and ternary fission. Average TXE in binary fission is 35 MeV. The dotted line is a linear fit through the data points without 4He (and C). The solid line is a quadratic fit through the same data points.
The relation between emission probability and loss in fragment TXE is depicted in Fig. 1, for the different LCPs measured. There is an obvious discontinuity in the data for the ternary a's as compared to the other LCPs which, having already been noticed by Halpern 14 , may constitute a hint for preformed a-clusters in the fissioning nucleus, taking up separation energy from the fission dynamics in the saddle-to-scission stage 15 . The significance of cc-cluster preformation in a-TF was recently discussed also by Serot and Wagemans 16 . For the majority of other LCPs the clue for their emission might primarily be in the dynamics of neck-rupture 14>17'18. 3.2
Neutron-Unstable Ternary Particles
For most LCPs the results on TXE deduced from kinematics were found to be in close correspondence with the mean number v of prompt fission neutrons
207
registered by the CB. However, slightly enhanced neutron yields emerge in the case of 4 ' 6 ' 8 iJe isotopes, and Li. The explanation was found in the formation of short-lived neutron-unstable "primary" ternary particles which disintegrate by neutron emission, either from the ground state or an excited state, into the registered "secondary" ones 19>28.
-ISO
-60
60
ISO
-60
60
„ (cleg.)
Figure 2. Different steps in the analysis of the neutron angular correlations in 5He, 7He and Li accompanied fission. Left-hand side: Projections wzx of total neutron intensity, dominated by the prompt neutrons from the fission fragments, on the zx plane in the coordinate system xyz (see text). Arrows indicate mean LCP emission angles where neutron yield is slighly enhanced compared to the opposite direction. Center. Respective wxy distributions perpendicular to the fission axis, with the contribution from prompt fission neutrons (dashed line) included. Right-hand side: Calculated angular distributions wxy of neutrons from LCPs decaying in flight, without (dotted lines), and with account of experimental resolution (full lines), used for fitting the data on wxy in the central patterns.
For the relatively abundant ternary 4H, 6He and Li particles, counting statistics has allowed to evaluate distinct angular distributions for the emitted neutrons 19 . Figure 2 displays neutron emission probabilities as a function of the azimuthal angles 0 in a coordinate system xyz, with the polar axis (z-axis) in the direction of motion of the light fission fragment, and zero 0-angle in the direction of the LCP (x-axis). Enhanced neutron yield appears along the di-
208
rection of these particles (> = 0°), the width of angular contributions getting narrower in the case of 6He and Li. The result for 4He qualitatively agrees with former work 20 in which yields and energy spectra of neutrons were measured within limited angular ranges around 0° and 180° relative to a-particle motion. Comparison with trajectory calculations for primary LCPs decaying in flight in secondary LCPs and neutrons, shown in the right-hand patterns of Fig. 2, unambiguously identifies ternary emission of the short-lived 5He and "'He nuclei (life-times: 1 x 10~ 21 s, and 4 x 10~ 21 s, respectively). Since the 5He and 7He nuclei decay in the Coulomb field of the nascent fragments, complete neutron angular distributions are important to be measured, since they permit relating LCP decay times and break-up Q-values to the primary fragments' velocity. Accordingly, the neutrons emitted along Li-particle motion could be attributed to the decay of 8Li from excited state (E* = 2.26 MeV, life-time: 2 x 10~ 20 s). The probabilities for 5He ( 7 ffe) accompanied fission were determined to be 0.23 (0.21) relative to 4He (6He) accompanied fission. Ternary sLi nuclei are estimated to be emitted in excited state with 33 ± 20 % probability. o,
w"
252
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~6?
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Mass number A
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Figure 3. Left-hand side: Yields of helium isotopes (4 < A < 8) in ternary fission of 252 Cf (dashed line, to guide the eye). Right-hand side: Theoretical analysis of helium isotope yields (3 < A < 8) in ternary fission of 2 3 5 f / ( n 4 f t , / ) , with (squares) and without (points) taking LCP excited states and spins into account (Ref. 1 9 ) . Experimental values (open symbols) are normalized to 10 4 a-particles.
Together with known data on the ratios 6He/4He and 8He/4He of particle-stable even-N helium isotopes the yield systematics in the helium isotope chain could be completed, within mass numbers from 4 to 8 (see Fig. 3). Apparently, there is no neutron even-odd effect on the helium particle
209 yields, against expectation; as an example, current theories 14-17>18 postulate only about 5% 5He relative to 4He. It is suggested that TF theory needs to be supplemented by effects due to LCP nuclear structure. In a statistical approach 21 , the introduction of a statistical-weight factor, (2/j + 1) (with Ii being the spin of the LCPs in states i) would increase theoretical yields ratios 5He/4He, and 7He/6He to closely agree with the present data. The increase is due to the 3/2~ spin of the 5<7He ground states, compared to the 0+ spin of 4'6He. Indeed, a recent re-analysis of measured LCP yields (Z < 8) from thermal-neutron induced fission of 235 U, with inclusion of LCP spins and population of LCP excited states (shown at the right-hand side of Fig. 3), has supported this assumption 19 . As a consequence, precise measurements of LCP yields may be used for providing valuable information on the spin assignment in light neutron-rich nuclei. In the meantime other decay channels from particle-unstable LCPs have been identified, such as the sBe decay into two a-particles, and the decay of 4.63 MeV excited 7Li into an a and a triton 22>23. These sequential decays mimic a true quaternary fission process, with two LCPs being coincidently emitted right at scission. Studying "double-LCP" emission in fission requires highly intense fission sources, what has been achieved in a recent experiment on triple correlations in 233>235{7 fission, induced by an intense polarized cold neutron beam at the high-flux reactor of ILL, Grenoble 2 2 .
3.3
Gamma-Rays from Ternary Particles
The 7-ray pulse-height distribution for Be-accompanied fission (Fig. 4) provides clear evidence for the formation of 10Be in its first excited state at 3.368 MeV. Amazingly, an important fraction of the 10Be radiation appears, inspite of the 0.18 ps life time, as a non-Doppler broadened 7-line in the 7-ray spectrum summed over full solid angle, indicating 7-decay from 10Be at rest. A narrow line was also registered in a recent high-resolution triple 7-ray correlation experiment with GAMMASPHERE 7 . Thus, the absence of Doppler broadening observed in both experiments may constitute a hint for a specific long-lived triple nuclear configuration at scission, i.e. a molecular-type of nuclear structure with a ternary 10Be particle held in the potential well between the two fission fragments for a long enough period (~ 10 _ 1 2 s) during which the 10Be nucleus 7 decays before the system breaks up into ternary SF. However, because of the low probability of 10Be emission in fission (~ 6 x 1 0 - 5 per binary fission) present experimental data on the 7-decay from the excited state naturally suffer from still poor statistics, and need further confirmation.
210
10B 105 104 CD CD
103
\ 102 >-
z
10' 10° 10 "1 0
2
4
6
8
ET [MeV] Figure 4. Gamma-ray spectra summed over all CB crystals, for a-particle, Li- and Beaccompanied fission (top to bottom). The spectrum for Li is multiplied by a factor of 20. Dashed lines represent the shape of the spectrum for binary fission. Error bars are statistical errors.
3-4
Anisotropy of Prompt Fission Gamma-Rays
The anisotropy of the 7-ray emission with respect to the fission axis yields information about fragment spin orientation. Previous work found the well known anisotropy in binary fission of 2 5 2 C / 24 to be either removed 25 or characteristically changed 26 when a ternary a-particle accompanies fission. Due to the high detection efficiency and granularity of the CB, the full angular distribution of the 7-rays with respect to the fission axis could be measured, for both the binary and LCP-accompanied fission modes. For 7energy bins from 0.1 to 1.5 MeV, the results on binary and a-accompanied fission 27 ' 28 are depicted in Fig. 5, along with theoretical curves. No significant difference between binary and ternary fission in the angular distribution patterns is observed. The binary fission value for the average anisotropy ratio of both fragments in their respective rest frames 2 r , A = W(0°)/W( 90°) - 1, is 0.0872 (± 0.0002 statistical error), in fair agreement with Ref. 24 , while the present result of A = 0.0903 (± 0.0019) for a-particle accompanied fission is in obvious contradiction to published data, A = 0.015(22) 26 and A = 0.03(2) 25
The present observation means that the emission of ternary a-particles, although taking away some amount of fragment spin n , does not influence or even destroy the spin alignment. The analysis of the 7-ray anisotropy
211 Binary
9 I 0
i 30
. 60
1 90
8(deg.)
. . 120 150
Ternary
1 0
1 30
, 60
, 90
1 , 120 150
Binary
1 180
B(deg.)
09 I 0
, 30
, 60
, 90
, , 120 150
Q(deg.)
Ternary
1 0
, 30
, 60
, 90
, , 120 150
180
6(deg.)
Figure 5. Angular distributions W(0) of 7-rays with respect to the light fission fragment motion, in the laboratory system. Gamma-ray energy intervals are given in MeV. The scale is correct for the lowest plots; others are shifted consecutively by 0.1 units. Solid lines are fitted curves, calculated with assuming the 7 multipolarity to be < 2.
for other LCPs (tritons, 6He, and Li and Be nuclei), which was possible for the first time with the present data, confirms the observation made for the a-particles. It is thus concluded that the difference in 7-ray anisotropy between the ternary and binary fission modes is very small. Projections of the ternary-fission 7-ray angular distributions onto the plane perpendicular to the fission axis, have also been analysed. The 7-ray emission was found to be isotropic, within 1 % of error, with respect to the azimuthal angle cf> , indicating that there is no correlation of the fragment spin orientation with the emission direction of ternary particles, both being roughly perpendicular to the fission axis. 4
Summary and Conclusions
The use of highly efficient angle-sensitive detectors, namely the CODIS detector for fission fragments and LCPs, and the CB for fission 7-rays and neutrons, has provided deeper insight into the rare process of particle-accompanied fis-
212
sion, discovering also previously unexploited features. For the first time it could be determined to what extent the fragment TXE is reduced when LCPs with nuclear charges up to carbon are emitted. This information is important for a correct modeling of ternary scission configurations. Formation of particle-unstable species, such as 5He and 7He, was identified, and found to provide a source of surplus neutrons. These neutrons add, albeit little, to the 252Cf neutron spectrum used as a reference standard in nuclear measurements. The decay of 5He contributes about one neutron in every 1500 binary fission events. The production of LCPs in excited states that decay by neutron emission (in the case of lithium) or by 7-ray emission (in the case of beryllium), is a new finding opening up new possibilities for studying experimentally the energetics of the fissioning system at scission. On the other hand, the anisotropy of prompt fission 7-rays measured in various TF modes shows, contrary to previous work, that LCP emission at scission has weak influence on the alignment of fragment spins. Fragment spin formation and LCP emission seem to proceed quite independently from each other. Evidently, angular momentum generation in fission fragments is still a challenge to theory (see, e.g. 2 9 ) , and even more sophisticated experimental studies might be needed for gaining further insight into the intimate details of the nuclear dynamics at scission. As regards the formation of long-lived complex nuclear molecules in particular ternary scission-point configurations which the 7-decay data in 10Be accompanied 2 5 2 C/(sf) seem to suggest, it is important to say that the observed neutron decay from excited ternary 8Li* nuclei gives no hint for a comparably long delay in LCP emission. The measured neutron angular distribution tells that the break-up of 8Li*, being faster than the 7-decay of 10Be* by seven orders of magnitude, occurs after LCP separation. If the hypothesis of nuclear molecule formation in Be-accompanied fission holds, this phenomenon seems to show up more likely with LCPs of nuclear charges Z > 3, and masses A > 8. The very low yields of these processes make further experiments with spontaneous fission sources rather complicated. Neutron induced reactions at high neutron fluxes would be an alternative. This work was supported in parts by the German Minister for Education and Research (BMBF) under contracts 06DA461, 06DA913 and 06TU669. References 1. T. San-Tsiang et al., C. R. Acad. Sc. 223, 986 (1946). 2. R. Stoenner, and M. Hillman, Phys. Rev. 7 1 , 716 (1966). 3. P. Schall et al., Phys. Lett. B 191, 339 (1987).
213
4. P. Heeg, PhD Thesis, TH Darmstadt, 1990; P. Heeg et al., Proc. Conf. 50 Years with Nuclear Fission, Gaithersburg (1989), Vol I, p. 299. 5. M. Hesse et al., Proc. Int. Conf. DANF96, Casta Papiernicka (1996), (JINR, Dubna, 1996), p. 238. 6. F. Gonnenwein et al., Proc. Seminar on Fission Pont d'Oye IV, Habayla-Neuve (1999), (World Scientific, Singapore, 2000), p. 59 7. A.V. Ramayya et al., Phys. Rev. Lett. 81, 947 (1998) ; also J.H. Hamilton et al., Acta Physica Slovaca 49, 31 (1999). 8. C. Wagemans, in The Nuclear Fission Process, CRC Press, Boca Radon (1991), p. 580. 9. M. Mutterer and J.P. Theobald, in Nuclear Decay Modes, Institute of Physics Publ., Bristol (1996) p. 487. 10. M. Mutterer et al., Proc. Int. Conf. DANF96, Casta Papiernicka (1996), (JINR, Dubna, 1996), p. 250. 11. P. Singer et al., Proc. Int. Conf. DANF96, Casta Papiernicka (1996),(JINR, Dubna, 1996), p. 262. 12. V. Metag et al, Lecture Notes in Physics 178, 163 (1983). 13. P. Singer et al., Z. Phys. A 359, 41 (1997). 14. I. Halpern, Ann. Rev. Nucl. Sci. 21, 245 (1971). 15. N. Carjan, J. Physique 37, 1279 (1976); S. Oberstedt and N. Carjan, Z. Physik 344, 59 (1992). 16. O. Serot and C. Wagemans, Proc. Seminar on Fission Pont d'Oye IV, Habay-la-Neuve (1999), (World Scientific, Singapore, 2000) p. 45. 17. V.A. Rubchenya, and S.G. Yavshits, Z. Phys. A 329, 217 (1988). 18. A. Pik-Pichak, Phys. Atom. Nuclei 57, 906 (1994). 19. M. Mutterer et al., Proc. Conf. Fission and Properties of Neutron-Rich Nuclei, St. Andrews (1999), (World Scientific, Singapore, 2000), p. 316 20. E. Cheifetz et al. Phys. Rev. Lett. 29, 805 (1972). 21. G. Valskii, Sov. J. Nucl. Phys. 24, 140 (1976). 22. P. Jesinger et al., Proc. Seminar on Fission Pont d'Oye IV, Habay-laNeuve (1999), p. I l l ; also Nucl. Inst. Meth. A 440, 618 (2000). 23. F. Gonnenwein et al., these proceedings 24. K. Skarsvag, Phys. Rev C 22, 638 (1980). 25. O.I. Ivanov, Sov. J. Nucl. Phys 15, 620 (1972). 26. W. Pilz and W. Neubert, Z. Phys. A 338, 75 (1991). 27. Yu.N. Kopatch et al., Phys. Rev. Lett. 82, 303 (1999). 28. M. Mutterer et al., Proc. Seminar on Fission Pont d'Oye IV, Habay-laNeuve (1999), (World Scientific, Singapore, 2000), p. 95. 29. I.N. Mikhailov and P. Quentin, Proc. Conf. Fission and Properties of Neutron-Rich Nuclei, St. Andrews (1999), p. 384.
214 A L P H A - P A R T I C L E S T R U C T U R E ON T H E S U R F A C E OF T H E ATOMIC NUCLEUS
Department
of Physics,
M a r t e n Brenner Abo Akademi University,
FIN 20500,
Turku,
Finland
The outer section of the nucleus has been regarded as a layer of alpha matter. The inner part of highly excited silicon and sulfur may consist of oxygene while the outer should be made up of three and four alpha-particles. Non-linear theories which suggest new quantum numbers to describe vibrations, solitons and numbers of bosons are re vie wd. The preformation of fragments in nuclear decay is addressed.
1
Introduction
W h e n nuclei decay by the emission of clusters or by fission the question arises as to whether the disintegration process is preceded by pre-formation of the fragments emitted and how, in t h a t case, it takes place. T h e present article addresses the existence of alpha particles on the surface of the nucleus at high excitation energies bearing in m i n d t h a t these can be compared with preformed clusters in different kinds of disintegration. W h e n alpha particles are scattered from light nuclei at high energies (5MeV < E < 2 5 M e V ) resonance states are observed that may consist of alpha particles on the surface of the nucleus. T h e states can be formed partly in the course of alpha-particle b o m b a r d m e n t but also by the transfer of the particle stripped from a projectile, e.g. from 6 L i 1 , 2 . There is reason to assume the existence of an alpha-cluster state. Their decay takes place primarily to the ground state of the target nucleus b u t also to the first and second excited state. Target nuclei t h a t are assumed to follow this pattern are a m o n g others light Z = N = A / 2 nuclei, namely 2 8 S i 3 and 3 2 S 4 ' 5 , etc. On being scattered they exhibit strong back scattering. T h e angular distributions take the form of squared Legendre polynomials. T h e energy dependence of the cross section, called the excitation function, reveals peaks which have apparently unique L values of the polynomials and indicate the existence of highly excited states with the spin J = L in the nucleus t h a t is formed. 2
Mixed parity bands mean assymetric bosonic structure
T h e energy or mean energy value for several excitation states with the same spin is linearly dependent on the square of the spin J ( J + 1 ) 6,7>8>9>3.i°; it might therefore be possible to speak of a rotational band, t h e members of which have b o t h even and odd parities. W h e n several resonances have the same spin, they
215
o
o
Figure 1: On the left-hand side a 2 8 Si nucleus in an excited state described as a 1 6 0 core surrounded by four alpha particles with the total rotation quantum number L. On the righthand side the same but with the alpha particles forming a layer of alpha matter round the core.
can be regarded as fragments of one and the same rotational state. B a n d s of states with mixed parity TT = 1~L exist in molecular physics where they represent asymmetric molecules. T h e excited nucleus is consequently expected to be a s y m m e t r i c . T h e slope of the line E against 3(3+1) gives a value for the m o m e n t of inertia of the nucleus. It is greater t h a n t h a t of an alpha particle orbiting the nucleus and less than t h a t of an entire rotating nucleus. Only a p a r t of the nucleus can therefore be considered to rotate. For the system a + 2 4 M g , i.e. an excited 2 8 Si nucleus, the moment of inertia corresponds to three alpha particles orbiting the nucleus and four for a + 2 8 S i , i.e. 3 2 S 4 . Microscopic calculations made according to the Resonating Group Method (R.GM) 1 1 results in two different bands, one for even and another for odd states. For t h e s t a t e of a + 2 8 S i the even band lies 1 MeV higher t h a n the odd b a n d 10,12 . T h e split into two bands follows from the fact t h a t in the calculations a system of fermions is considered for which the Pauli principle holds. If, on the other hand, the system consists of alpha clusters, the Pauli principle is violated. This applies when the structure of the highly excited states brought about by scattering or stripping have a bosonic structure 3 . T h e structure of the excited 3 2 S m a y be illustrated as on the left-hand side of Fig. 1. T h e inner p a r t consists of 1 6 0 while the outer is made u p of four alpha clusters. T h e outer section of the nucleus might be regarded as a layer of alpha m a t t e r while the inner m a y be regarded as an entity whose interaction with the outer layer m a y be overlooked in the first instance. T h e inner nucleus (core) consists of nuclear m a t t e r . T h e right-hand half of Fig. 1 might then represent the nucleus as being m a d e up of an outer layer of alpha m a t t e r (hatched) and a
216 inner core of nuclear m a t t e r . This division can be related to the occurrence of alpha m a t t e r at low densities. At high excitation the outer layer of the nucleus may be assumed to be expanded and thinner. T h e transition from dense to thin m a t t e r can b e compared t o the t h e r m o d y n a m i c transition between two phases. " T h e low density on the nuclear surface makes" according to Clark 13,14 " t h e existence of alpha m a t t e r probable there. Nuclear m a t t e r theories imply indeed t h a t below the point of alpha m a t t e r being compressible there is no thermodynamical stable state of it. Below this point the system breaks u p into clusters." If part of the atomic nucleus is assumed to be a structure of alpha particles or of continuous m a t t e r m a d e u p of t h e m , then we are adopting a mesoscopic approach. Brinck 1 5 has correspondingly regarded whole nuclei as structures of alpha particles. A mesoscopic view m a y be more relevant t h a t a microscopic one. Sub-microscopic physics prove t h a t theories previously regarded as microscopic become mesoscopic when experiments at higher energies, i.e. shorter wavelengths, have revealed the existence of increasingly smaller particles in the m a t t e r .
3
Theoretical approaches
Collaboration between Zagreb and Legnaro has placed the resonance s t a t e in a + 2 8 S i in an E against n diagram where n is a q u a n t u m number for an assumed vibrational movement in t h e 3 2 S nucleus formed 16 > 17 . Assuming a non-linear theory for an alpha particle as an extended object a research group from Bucharest and Frankfurt has introduced a t e r m t h a t is linear with respect to a q u a n t u m n u m b e r k. According to the group k is a q u a n t u m number for an assumed soliton on the surface of the nucleus, Fig. 2 1 8 ' 1 9 , 2 0 . In this way no degeneration shown in the E against n diagram, with several resonances with the same rotation q u a n t u m number L and vibration q u a n t u m number n, would exist. In order to explain the split of the states into m a n y fragments Gridnev likewise introduced a new q u a n t u m number n for the number of bosons in a t r a p p e d Bose-gas condensate and a m a i n q u a n t u m number N. This way of approaching the problem was also based on a non- linear theory, namely the solution of t h e Gross-Pitaevsky e q u a t i o n 2 1 , 2 2 . W i t h the exception of some small quadratic t e r m these three analyses give the expression E = A + Bn + C L ( L + 1 ) + eN (n = k) for the excitation energy where the coefficients B and e constitute the q u a n t u m energy for the corresponding q u a n t u m number. All expressions for the a + 2 8 S i d a t a include energies t h a t emerge from a Fournier analysis of the excitation function for a scattering of 173 d e g r e e s 4 , 2 1 , 2 2 . T h e energies B and e are in this way periods in the series t h a t constitute the Fourier transform of the excitation function. T h e
217
Figure 2: A diagrammatic illustration of the breather states on the nuclear surface presented for three quantum numbers k = 0,1,2,. All three solutions have the same envelope of a solitonic type [30] (courtesy of A. Ludu, Louisiana State University and IOP Publishing Ltd).
series m a y also include periods corresponding to harmonic frequencies. T h e diagrams t h a t follow the two theories above have gaps for states not observed in experiments. The diagrams can consequently hardly be regarded as unique. T h e y can probably be drawn in m a n y different ways. 4
More experiments required
These a t t e m p t s to determine the structure of the highly excited states formed in 3 2 S when an alpha particle is captured in 2 8 Si are based on angular distributions only for large peaks. T h e experimental basis is unsatisfactory for several reasons. Firstly, not just the cross-section at the peaks should be considered; secondly, the energy resolution is approx. 20 keV, so that resonances the t o t a l width of which is of the same size as this uncertainty in the energy m a y be missed or their shape impossible to determine. Determining the spin of the resonances should be based on a complete analysis where coupling to other channels, especially the strongest inelastic ones, should be taken into account. T h e technology at present available for taking measurements of this kind require too much machine t i m e to cover large energy ranges in sufficiently small steps. Measurements using steps of at least 5 keV should be the aim. T h e present situation when it comes to controlling the energy dependence of the cross-section and step-length in the angle of scatter is shown in Table 1, where the second and fourth columns show how deficient the determination of the energy dependence of the cross-section has been. All too poor resolution because of a thick target or too great step lengths of the energy has m a d e it
218 Table 1: Experiments on alpha particle scattering from 2 0 Ne, 2 8 Si, 3 2 S and 4 0 C a . The first column gives the target nucleus and the second the target thickness. The asterisk indicates that the measurements are only on peaks and z that? ^g/cm2 1 60 50 thick 27 120 beam 8 keV 5
E Range MeV 3.8-11 12-20 12-17 6.5-19 14.3-15.4 10-17.5 5.3-21.6 5.0-17.5 18-50 4.4-9.1
Step keV
<
15
40 20 small 5 100 small 10 ~3000 4
Ang. Range degr. 53-168 173 25-173 110-173 153-173 25-175 134-174 26.7-176.1
Step degr. 10
3* 3 3 5 3-18 10
Ref. z z z
" 3 4
9,10 Z
2 < 25
5 26 6
93.2-175.5
10
7
difficult to discern the resonance peaks of the excitation function, which are often as narrow as 20 keV. With a thick fixed target (fourth row) or radiation of helium gas with a beam of the nuclide otherwise bombarded in the fixed target (inverse geometry, seventh row) the step-length is indeed small, but straggling in the target means a deterioration in energy resolution, especially for small scattering angles (i.e. larger angles with inverse geometry). In addition, only scattering in a backward direction can be measured and inelastic scattering not at all with present technology. The cross-sections for a + 4 0 C a (rows 8, 10) have been analysed in order to determine the parameters for an optic potential but the peaks observed in the excitation function have not been analysed to find the resonance states. The fifth row in Table 1 gives the good quality of a measurement carried out in a small energy range at Florida State University 24 . The aim was to compare the results with the earlier measurement referred to and which had been used to develop the above theoretical expressions for the energy 9>10>3. Excitation functions in 5 keV steps with approx. 10 keV resolution for elastic and inelastic scattering at nine angles between 150 and 173 degrees were fitted by theoretical curves 27 . These were reproduced considering the coupling between the elastic and the first inelastic (2+) channel. Within the energy range measured 36 resonances were discovered; of these only 8 had been observed in earlier measurements. The spin of two of the earlier reported resonances was determined anew as 8 instead of 7. The energy, total and partial widths were determined for all resonances. The degree of conformity proved to be good at all angles. A remarkably large number of resonances with a spin of 5 were observed. Destructive interference proved to weaken certain peaks so
219
that, despite wide partial scattering width, they could easily be overlooked in an undetailed analysis of the excitation function. Although these latest measurements put the value of the earlier ones in some doubt, it may nevertheless be noted that resonance states with high spin are those that form the rotation bands of states with mixed parity mentioned earlier . They occur in measurements made hitherto alone as stripping states when a silicon target is bombarded with 6 Li 1 , 2 . The conclusions drawn about the nucleus' division into an inner section of nuclear matter and an outer consisting of alpha matter (Fig. 1) therefore still deserve to be taken seriously. This model can continue to be discussed in relation to a number of phenomena revealed in nuclear decay, especially when it comes to the question of pre-formation of the fragments emitted. This includes matters concerning alpha disintegration and fission. Pre-formation in nuclear splitting has been described in several articles 2 8 , 2 9 ' 3 0 that also make reference to earlier studies.
References 1. K.P. Artemov, M.S. Golovkov, V.Z. Goldberg, V.I. Dukhanov, I.B. Mazurov, V.V. Pankratov, V.V. Paramonov, V.P. Rudakov, I.N. Serikov, V.A. Solovyev, V.A. Timofeev, Yad. Fiz. 51, 1220 (1990). 2. K.P. Artemov, M. Brenner, M.S. Golovkov, V.Z. Goldberg, K.-M. Kallman, T. Lonnroth, P. Manngard, V.V. Pankratov, V.P. Rudakov, Yad.Fiz. 55, 884 (1992), Sov. J. Nucl. Phys. 55, 492 (1992). 3. M. Brenner, Z. Phys A 349, 233 (1994). 4. M. Brenner et al., Heavy Ion Physics 7, 355 (1998). 5. V.Z. Goldberg, G.V. Rogachev, T. Lonnroth, M. Brenner, K.-M. Kallman, M.V. Rozhkov, W.H. Trzaska, R. Wolski, to be published in Phys. At. Nucl. (Yad. Fiz.)(2Wti). 6. R. Stock, G. Gaul, R. Santo, M. Bernas, B. Harvey, D. Henrie, J. Mahoney, J. Sherman, J. Steyaert, M. Zisman, Phys. Rev. C 6, 1226 (1972). 7. D. Frekers, R. Santo, K. Langanke, Nucl. Phys. A 394, 189 (1983). 8. M. Brenner in Clustering Phenomena in Atoms and Nuclei, Springer Series in Nuclear and Particle Physics, edd M. Brenner, T. Lonnroth, F.B. Malik, (Springer, Berlin, 1992). 9. K.-M. Kallman, M. Brenner, V.Z. Goldberg, T. Lonnroth, A.E. Pakhomov, V.V. Pankratov, to be published. 10. P. Manngard, Z. Phys. A 349, 335 (1994). 11. J.A. Wheeler, Phys. Rev. 52, 1083 (1937). 12. K. Langanke, R. Stademann, D. Frekers, Phys. Rev. C 29, 40 (1984).
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13. J.W. Clark, Ann. Phys. 40, 127 (1966) and privat communication. 14. M.T. Johnson, J.W. Clark, Kinam 2, 3 (1980). 15. D.M. Brink in Proc. of the Int. School of Physics "Enrico Fermi" 1965, ed. C. Block (Academic Press, New York, 1966). 16. N. Cindro, privat communication 1994. 17. U. Abbondano, N. Cindro, P.M. Milazzo, Nuov. Chim. 110A, 955 (1997). 18. A.E. Antropov, M. Brenner, V.Z. Goldberg, W. Greiner, K.-M. Kallman, T. Lonnroth, A. Ludu, P. Manngard, A.E. Pahkomov, V.V. Pankratov in 7th Int. Conf. Nucl. Red. Mech., Varenna ed. E. Gadioli (Dept. Phys., Milano Univ., 1994). 19. A. Ludu, A. Sandulescu, W. Greiner, K.-M. Kallman, M. Brenner, T. Lonnroth, P. Manngard, J. Phys. G: Nucl. Part. Phys. 2 1 , L41 (1995). 20. A. Ludu, A. Sandulescu, W. Greiner, / . Phys. G: Nucl. Part. Phys. 2 1 , 1715 (1995). 21. K.A. Gridnev, M. Brenner, A.E. Antropov, S.E. Belov, B.Z. Taibin, K.N. Ershov, D.K. Gridnev, M.P. Kartamishev, I.V. Krouglov, T.V. Taroutina in EN AM, 2nd Intern. Conf. on Nuclei and Atomic Masses, Shanty Creek Resort (USA, 1998) Abstracts p. PC 22. 22. K.A. Gridnev, S.E. Belov, K.V. Ershov, M. Brenner, D.K. Gridnev, J.S. Vaagen in Condensed Matter Theories Vol. 15 edd. G.S. Anagnostatos, R.F. Bishop, K.A. Gernoth, J. Ginis, A. Theophilou (Nova Science Publishers Inc., Huntington, NY, 2 23. R. Abegg, C.A. Davies, Phys. Rev. C 43, 2523 (1991). 24. M. Brenner, N.R. Fletcher, J.A. Liendo, S.E. Belov, D.D. Caussyn, T. Kurtukian Nieto, S.H. Myers, Heavy Ion Physics 11, 221 (2000). 25. J.P. Aldridge, G.E. Crawford, R.H. Davis, Phys. Rev. 167, 1053 (1968). 26. C.P. Robinson, J.P. Aldridge, J. John, R.H. Davies, Phys. Rev. 171, 1241 (1968). 27. V.V. Lazarev, S.E. Belov, M. Brenner, K. A. Gridnev, T. Kurtukian Nieto, D.D. Caussyn, N.R. Fletcher, J.A. Liendo, S.H. Myers, presented at The International Conference "Clustering Phenomena in Nuclear Physics" St. Petersburg June 14-17, 20 28. R. Blendowske, T. Fliessbach, H. Walliser, Z. Phys. A 339, 121 (1991). 29. B. Buck et al., Nucl. Phys. A 617, 195 (1997). 30. Yu. Zamyatnin, V.L. Mikheev, S.P. Tret'yakova, I. Furman, S.G. Kadmenskii, Yu.M. Chuvil'skii, Sov. J. Part. Nucl. 2 1 , 231 (1990).
221
SUPERASYMMETRIC FISSION V . A . Rubchenya Department of Physics, University of Jyvaskyla, FIN-40351, Jyvaskyla, Finland V.G.Khloplin Radium Institute, 194021, St.Petersburg, Russia Abstract The existence of superasymmetric fission mode connected with Z=28 and N=50 nuclear shells is analysed in the framework of the scission point model. Calculations of PES near the scission point had shown that the 78Ni fission mode would be manifested in fission of neutron-rich compound nuclei. In the case of fission of superheavy nucleus the superasymmetric fission mode is enhanced by influence of the Z=82 and N=126 nuclear shells in heavy fragment. Enhancement of highly asymmetric mass and charge division in the proton and neutron fission of 238TJ at intermediate energy in comparison with thermal neutron induced fission was described by the model developed for calculating the product yields with inclusion of superasymmetric fission mode. This model was used for the prediction of the formation cross sections of neutron-rich nuclides in fission. 1
Introduction
The single fission fragment mass spectrum has the interesting features as one can see in Fig. 1 where the calculated light fission product mass distribution for the spontaneous fission of 2 4 2 Cm is shown. This spectrum consists of two groups corresponding to the binary fission, calculated according to our model [1], and ternary fission, calculated in the framework of the double rupture of the neck model developed by us [2], The yields of the lightest fission products decrease very rapidly and at about Aj, =70 are comparable with yields of heavy ternary fission particles (see also [3]). In the mass interval AL =34 - 70 the data for fission of actinide nuclei does not exist at all, it is "terra incognita". The big solid diamond shown at A/, =34 in Fig. 1 is recently measured yield of 34 Si in the cluster decay of 2 4 2 Cm [4]. The term "superasymmetric fission" (SAF) will be used for the binary fission of actinide and transactinide nuclei at AL <80. A great deal of interest in the investigation of very asymmetric fission is connected with the search for the superasymmetric fission mode, and clarifying the perspectives to produce exotic neutron-rich nuclides [1,5]. This report is organised as follows. In Sec. II the experimental data on superasymmetric fission will be reviewed. Section III contains the theoretical model analysis of the fission product yields and results of calculations of potential energy
222 r : 242,
Cm sp. decay
• ternary fission
SAF
cluster decay (?)
0
• 1
10 20 30 40 50 60 70 80 90 100 110 120
A, Figure 1: Calculated single fission product yields for the spontaneous fission of big solid diamond is measured yield of 3 4 Si for cluster decay [4].
242
C m ; the
surface near the scission point. The analysis of formation of neutron-rich fission products and 78 Ni, in partucular, will be presented in Sec. IV. The results are summarized in Sec. V. 2
Experimental results
Information on extreme asymmetric fission of heavy nuclei, where the light fragment mass AL <80, has been scarce until recently mainly due to very low yields. The superasymmetric mass division in the spontaneous fission of 252 Cf was reported for the first time in Ref. [6] and confirmed in the work of Ref. [7]. But later investigation of the far asymmetric region in 252 Cf(sf) [8] did not confirm the previous findings. A shoulder at the level about 10 _ 5 % in the mass yield curve near mass 70 had also been observed earlier in the reactor neutron induced fission of 238 U using the radiochemical method for measurement of the fission product yields [9]. But this measured enhancement was two orders of magnitude smaller than in the case of 252Cf(sf) [6]. Interesting results were obtained in the region of very light fragment masses in thermal neutron-induced fission of 2 3 5 U [10], 2 3 9 Pu and 249 Cf [11]. The appearance of shell effects and odd-even variations in the nuclear charge distribution with increasing asymmetry of the mass split was demonstrated for thermal neutron-induced fission. The pronounced fine structure at A i = 7 0 (Y = 6.6 -10 _5 %) in the mass yield
223
Figure 2: Measured independent yields (points with error bars) of fission products in 25 MeV proton induced fission of 2 3 8 U , and the mass yields (solid circles) in the thermal neutron induced fission of 2 3 S U , and the pre-neutron emission mass distribution (open circles) at E p = 35 MeV. The theoretical isotopic yields are shown by thin solid and dashed lines.
curve had been obtained in the thermal neutron induced fission of 2 4 2 m Am [12]. This fine structure correlates with the less pronounced enhancement of the yields near A £,=70 measured in the thermal neutron induced fission of 235 U [10]and239Pu[ll]. For the first time the independent yields of neutron-rich nuclei in the very asymmetric proton induced fission of 238 U at Ep = 25 MeV had been measured using the ion guide isotope separation technique IGISOL [1] The mass distributions in the proton induced fission of 238 U at Ev = 20, 35, 50 and 60 MeV had also been measured by the time-of-flight method using the HENDES setup at K-130 cyclotron in Accelerator Laboratory of University of Jyvaskyla [13]. The results of these experiments are presented in Fig. 2, where IGISOL data are shown by points with the errors bars and HENDES data for pre-neutron emission fragment distribution at Ep = 35 MeV are shown by open circle. The theoretical results for independent yields are presented in Fig. 2 by lines in comparison with IGISOL data. One can see that the yields in the superasymmetric mass region for fission of Np compound nuclei at intermediate excitation energy differ drastically from the ones in low energy fission of 236 U compound nuclei. In a very asymmetric mass division at intermediate excitation energy ( E c > 20 MeV ) there are contributions from three components: the tails
224 10'
10"
g 2 10''
>-
10'2
J
72
i
I
74
•
1
i
76
L
78
80
82
84
Ap„ ( u )
Figure 3: Measured fission fragment mass distributions for (d,f) channel (squares), and for two proton energy bins for (d,pf) channel (circles and triangles) in comparision with theoretical model calculations for two corresponding neutron energies ( lines).
from the symmetric and the second asymmetric modes and the superasymmetric mode. Enhancement of fission fragment yields in far asymmetric region is also supported by the time-of-fiight measurements with HENDES setup in the neutron induced fission of 238 U using reaction 238 U(d,pf) [14]. In this case fission fragment mass distributions were measured in coincidence with protons for definite proton energy bins that effectively corresponds to some neutron energy bins in accordance with the reaction kinematics. The light tail of fission fragment mass distributions for (d,f) channel and for two proton energy bins (E p ± AE P = 20±5 and 40±5 MeV) for (d,pf) channel (symbols) at 9 P =29° together with theoretical calculation for two corresponding neutron energies (lines) are presented in Fig. 3. The enhancement of the elemental fission yields at 28
Theoretical model analisys
The gross structure of fission characteristics found a natural explanation in terms of fission modes, which are result of influence of nuclear shell structure
225
on the potential energy surface (PES) of fissioning nucleus [16]. The fission process is most probably guided by the valleys and bifurcation points of PES from the equilibrium shape to the scission point. For the medium heavy actinides (Th to Cf) the so-called standard fission modes ( symmetric, 132 Sn, and deformed shell) are manifested. The superasymmetric fission modes is attributed to the nuclear shell Z=28 and N=50 (so-called 78 Ni mode). In the case of super heavy nucleus the superasymmetric mode is connected with the nuclear shell Z=82 and N=126 (so-called 2 0 8 Pb mode). We analyzed the influence of shell structure of the fission fragment in the framework of the scission point fission model. In this model the fission characteristics are defined at the averaged scission line corresponding to the minimum of PES in the multidimension space of collective coordinates described of axially symmetric configuration of two nascent fragments. This collective coordinate space comprises the three deformation parameters (quadrupole, octupole, and hexadecapole deformation types) for each of fragments, the fragment mass and charge asymmetry parameters, and the distance between the fragment tips. In our calculations the standard Strutinsky shell correction method [17] had been applied using the parametrization of the nuclear potential according to work [18]. The dependences of the potential energy, minimized over other collective variables, on the charge of light fragment at AL=70 for fission of 236TJ, 243 Am, and 252 Cf are presented in Fig. 4 where points are calculated values and lines are their parabolic approximation. One can see that the ninimum of V 3 C . P . ( Z L ) is at the magic number Zi=28 for all fissioning nuclei from U to Cf that explains the stability of fine structure at AL=70 in the mass yield curve (see Fig. 5 in [12]). The detailed theoretical calculation of the fission product formation cross sections consists of two parts: (i) modelling the reaction mechanism to calculate mass, charge , and excitation energy distributions of compound nuclei and (ii) modelling the fission process itself. For the case of the light-particle induced fission of heavy nuclei the theoretical model for calculation of independent fission product cross sections was proposed and developed in our previous works [1,13,19] The formation cross section of a fission product with mass number A and charge number Z can be expressed in the form
*t{A,Z)=Y.
J
[YinM,Z)dEcda<{At>Zt>A>>'ZZ'E*>A<:'Zc'E<\ dEc
(1)
£ze where subscripts t, p and c refer to target, projectile and compound nuclei, respectively, dcf/dEc is the partial fission cross section of the compound nucleus at the excitation energy Ec for different fission chances over which the summing is carried out and Yinct(A,Z,Ac,Zc,Ec) is the independent yield. The
226
ji
24
,
1
,
26
1
28
. — - >
.
30
•
32
\ Figure 4: Calculated potential energy at the scission point as function of the light fragment charge for the very asymmetric fission ( A t = 7 0 ) of 2 3 6 U , 2 4 3 A m , and 2 5 2 Cf (points ) and their parabolic approximations (lines).
independent yields Yind are defined as the yields of fission products after light particle emission from excited primary fragments:
Yind(A,Z,AClZc,Ec) = J^
(A + Qicp> Z •+" zicpj Ac, Zc,
Ec)
aicptZlcp
Ppre(Z + zicp,A + aiCp,Ac, Zc,Ec)Ypre(A
+ aicp,Ac,
ZC,EC).
(2)
Here Paicp,z,cp(A + o.icp,Z + zicp,Ac,Zc,Ec) is a probability of emission of (ajCp,;zjCp) light particle from fragment with the mass A + aicp and charge Z -f zicp numbers, Ppre(Z + zicP, A + 0( cp , Ac, Zc, Ec) is a charge distribution of the A + aicp isobaric chain and Ypre(A + aicp, Ac, Zc, Ec) is a primary fission fragment mass distribution. At the excitation energies Ec up to about 150 MeV the role of emission of proton and others particle is negligible in comparision with neutron emission. At low excitation energies, the primary fission fragment mass and charge distributions exhibit odd-even staggering. The primary distributions are presented in the factorised form PpTe(Z) = Ppre(Z)Foe(Z),
Ypre(A) = Ypre(A)Foe(A),
(3)
where Ppre(Z) and Ypre{A) are smoothed distributions, and the functions Foe(Z) and ^ ( ^ 4 ) describe odd-even staggering. The method of modelling
227
the smoothed mass distribution is based on the multimodal nature of nuclear fission [16]. The four fission modes: symmetric, standard-I, standard-II and superasymmetric were taken in consideration to approximate the smoothed primary mass distribution: Ypre(A) = Csys(A) + CaXyaX{A) + Ca2ya2(A) + Ca3ya3(A).
(4)
Here ys and ya\, ya2, J/a3. are symmetric and asymmetric components which present contributions from different fission modes. Each asymmetric component consists of two Gaussians representing the heavy and light fragment mass groups. The component ya\ is connected with the magic number Z = 50 and N = 82 in the heavy fragments, and the superasymmetric component ya3 is influenced by the nuclear shells Z = 28 and N — 50 in the light fragments. The asymmetric mode ya2 is supposed to be connected with a "deformed" nuclear shell at N — 86 — 90. The competition between fission modes is determined by fission dynamics and nuclear shells in the fission fragments. The parameters entered in eq. (4) had been approximated as functions of the excitation energy, mass and charge of the compound nucleus. The smoothed pre-neutron emission isobaric chain charge distribution is approximated by a Gaussian distribution, therefore odd-even structure can be described by the parameter defined as a third difference of the natural logarithms of the fractional yields [20]. We consider the proton and neutron odd-even effect separately and one can write Foe(Z) ~ exp((C* + C$)6Z(AC, Zc, Ec)),
(5)
where C* and C% are defined by the parity of the proton number in the two primary fragments (Cp = 1 if Z is even and Cp = — 1 if Z is odd). The proton odd-even difference parameter 6z(Ac,Zc,Ec) is parameterised as a function of excitation energy, charge and mass number of the compound nucleus in accordance with experimental data. Odd-even staggering in the primary mass distribution is described by the combination of proton and neutron odd-even effects. The proton and neutron odd-even difference parameters are taken to be proportional, i.e. 5^(AC, Zc, Ec) = c5z(Ac, Zc, Ec){c < 1). The partial distribution of the compound nucleus ensemble in eq. (1) depends on the fast and pre-equilibrium stages of reaction and fission dynamics. The fission process had been treated by the method developed by us in [21] with inclusion of the influence of nuclear friction. The excitation energy of the primary fission fragments was calculated in the famework of the scission point fission model. The model has been tested and used for the light particle induced fission at energies up to 100 MeV. The results of calculations presented in Fig. 2,3 have been obtained in the framework of this model.
228 10°
iff' 10*
-O
cross section
E
10° 10"
10* 10'
Iff' 10*
68
68
70
72
74
76
7B
80
product mass number Figure 5: Comparison of calculated isotopic distributions of Co,Ni, and Cu products (open symbols) in the 30 MeV proton induced fission of 2 3 8 U with experimental data (points with error bars) [23].
4
Formation of neutron rich nuclides in fission
Calculations with the developed model allow us to predict the product formation cross sections for different types of fission reaction. Fission of heavy nuclei is practically the only method for production of very neutron-rich nuclides. One of the examples is obtaining and investigation of the double magic nucleus 78 Ni which had been detected in the fission induced by the peripherical collision of 750 MeV/nucleon projectiles of 238 U on Be target [22]. The results of calculations of the Co, Ni, and Cu isotope production cross sections in comparison with experimental data [23] for the 30 MeV proton induced fission of 238 U are presented in Fig. 5. The theoretical isotopic distributions are wider than experimental ones. The predicted 78 Ni nucleus production cross section is about 0.8 nb that is close to value of 0.3 nb estimated in work [22]. The extrapolation of experimental data will give value of this cross section which is three orders of magnitude smaller. Fission process is considered to be very promising for the production of neutron-rich nuclides in different Radioactive Nuclear Beam (RNB) projects [24]. Comparative investigations of neutron-rich fission product yields in different reactions are important for the development of RNB facilities. In Fig. 6 the Ni isotope production cross sections are compared for
229 10
I-1-
10" 10' 10"
Io 10*
io*
in
g u
io J 10" 10*
io'° t 10'"
- • — ""U + n(S0 MeV) - * — ^ U + p(50 MeV) - * — ' " l l + Y ( 1 4 MeV)
70
75
mass number
Figure 6: The Ni isotope production cross sections in reactions: 2 3 5 U(n ( / 1 ,f) (open circles), 238 U(n,f) (solid circles), and 2 3 8 U(p,f) (triangles) at Ep(n) = 50 MeV, and 2 3 8 U ( 7 , f ) at Ey = 14 MeV (open triangles).
four fission reactions: 235 U(n t/l ,f), 238 U(p,f) and 238 U(n,f) at Ep{n) = 50 MeV, and 238 U(7,f) at Ey — 14 MeV. One can note that the production of the very neutron rich Ni isotopes in the fast neutron induced fission of 238 U is higher than in the thermal neutron induced fission of 2 3 5 U. Production of extremly neutron rich nuclei will be higher in fission of more neutron-rich compound nucleus. In this sense, the fast neutron induced fission of 238 U should provide a promising method for production and investigation of extremely neutron-rich isotopes. 5
Conclusion
In conclusion, the analysis of the problem of the very asymmetric mass division at the low and intermediate energy fission of heavy nuclei had been done. The super asymmetric fission mode ( 78 Ni mode) is not detected definitly in the low energy fission. But observations of fine structure at AL=70 in the mass yield curve in the thermal neutron fission of 235 U, 2 3 9 Pu, 2 4 2 m Am and 249 Cf can be explained by the manifestation of nuclear shell Z=28. Calculations of PES near
230
the scission point had shown that the 78 Ni fission mode would be manifested in fission of more neutron rich compound nuclei than ones investigated till now. In the case of fission of superheavy nucleus the superasymmetric fission mode can be enhanced by influence of the Z=82 and N=126 nuclear shells in heavy fragment. Some indication of fine structure in fission fragment mass distribution had been obtained in fusion-fission reaction 40 Ar(243 MeV) + 2 3 8 U by HENDES-collaboration in Jyvaskyla [26]. Using the relativistic radioactive beams [25] is very promising for investiation of the superasymmetric fission mode. Enhancement of highly asymmetric mass and charge division in the proton and neutron fission of 238 U in comparison with thermal neutron induced fission had been established. A model for calculating the fission product yields in light particle induced fission of heavy nuclei has been developed. This model can be used for the prediction of the formation cross sections of exotic nuclides and for the evaluation of product yields in fission at intermediate energy. Fast neutron induced fission of 2 3 8 U and 2 4 4 Pu is efficient reactions for the production of high intensities of neutron-rich isotope beams. This work has been supported by the Academy of Finland under the Finnish Center of Excellence Programme 2000-2005 (project No 44875, Nuclear and Condenced Matter Physics Programme at JYFL). References 1. M. Huhta, P. Dendooven, A. Jokinen, G. Lhersonneau, M. Oinonen, H. Pentilla, K. Perajarvi, V. A. Rubchenya, and J. Aysto, Phys. Lett. B405, (1997) 230. 2. V.A. Rubchenya and S.G. Yavshits, Z. Phys. A329 (1988) 217. 3. F. Gonnenwein et al., Seminar on Fission. Pont d'Oye IV. In proc of the IV Seminar on Fission, Castle of Pont d'Oye, Habay-la-Neuve, Belgium, 5-8 October 1999, ed. C. Wagemans, O. Serot, andP. D'hondt, pp. 59-75, World Scientific, 2000. 4. A.A. Ogloblin et al, Phys. Rev. C61 (2000) 2036. 5. F.-J. Hambsch, Seminar on Fission. Pont d'Oye IV. In proc of the IV Seminar on Fission, Castle of Pont d'Oye, Habay-la-Neuve, Belgium, 5-8 October 1999, ed. C. Wagemans, 0 . Serot, andP. D'hondt, pp. 22-31, World Scientific, 2000. 6. G. Barreau et al., Nucl. Phys. A432 (1985) 411. 7. C. Budtz-J0rgensen and H.-H. Knitter, Nucl. Phys. A490 (1988) 307. 8. F.-J. Hambsh, S. Oberstedt, Nucl. Phys. A617 (1997) 347. 9. V.K. Rao et al., Phys. Rev. C19 (1979) 1372. 10. J.L. Sida et al., Nucl. Phys. A502 (1989) 233c.
231
11. R. Hentzschel et al., Nucl. Phys. A571 (1994) 427. 12. I. Tsekhanovich, H.O. Denschlag, M. Davi et al., Nucl. Phys. A658 (1999) 217. 13. V.A. Rubchenya et al., in: Nuclear Fission and Fission-Product Spectroscopy: Second International Workshop, edited by G.Fioni et al.,AIP, CP447, 1998, 453. 14. V.A. Rubchenya et al., in: Fission and Properties of Neutron-Rich Nuclei. Proc. of The Second International Conference, June 28 - July 2, 1999, St Andrews, Scotland, edited by J.H. Hamilton, W.R. Philips, H.K. Carter, World Scientific, 2000, p. 484. 15. P. Armbruster, M. Bernas, S. Czjkowski et al., Z. Phys. A355 (1996) 191. 16. U. Brosa, S. Grossmann, and A. Muller, Phys. Rep. 197 (1990) 167. 17. V.M. Strutinsky, Nucl. Phys. A122 (1968) 1. 18. V.V. Pashkevich, Nucl. Phys. A169 (1971) 275. 19. V.A. Rubchenya, in: Nuclear Methods for Transmutation of of nuclear Waste, Proc. of International Workshop, Dubna, Russia, 29-31 May 1996, edited by M. Kh. Khankhasayev, Z.B. Kurmanov, H.S. Plendl, World Scientific, 1997, p. 110. 20. B.L. Tracy et al., Phys. Rev. C5 (1972) 222. 21. V.A. Rubchenya et al., Phys. Rev. C58 (1998) 1587. 22. Ch. Engelmann et al, Z. Phys. A352 (1995) 351. 23. K. Kruglov, B. Bruyneel, S. Dean et al., in Proc. of 5th international Conference on Radioactive Nuclear Beams, RNB 2000, Divonne, France, 3-8 April, 2000, Nucl. Phys. A, in press. 24. A.C. Mueller, in "Research with Fission Fragments", Proc. of Int. Workshop, Benediktbeuern, Germany, 28-30 Oct. 1996, eds. T.von Egidy, F.J. Hartmann, D. Habs, K.E.G. Lobner, and H. Nifenecker, World Scientific, Singapore, 1997, pp. 48-53. 25. K.-H. Schmidt, S. Stenhauser, C. Bockstiegel et al., Nucl. Phys. A665 (2000) 221. 26. Yu.V. Pyatkov, Yu.E. Penionzhkevich, O.I. Osetrov et al., Preprint E-799-253, Dubna, 1999; submitted to Eur. Phys. J A.
232 QUATERNARY
FISSION
F. GONNENWEIN, P. JESINGER Physikalisches Institut, Auf der Morgenstelle 14, 72076 Tubingen,
Institut fur Kemphysik,
M. M U T T E R E R TU Darmstadt, 64289 Darmstadt,
A.M. GAGARSKI, G.A. P E T R O V PNPI of RAS, Gatchina, Leningrad district 188350, W.H. TRZASKA University of Jyvaskyld, 40500 Jyvaskyld,
Germany
Germany
Russia
Finland
V. NESVISHEVSKY, O. ZIMMER ILL, BP 156, 38042 Grenoble, France In experiments at the ILL in Grenoble on ternary fission induced by cold neutrons also quaternary fission events were observed. By definition quaternary fission is a process where besides the two main fission fragments two more light charged particles are formed. The reactions studied were 233 U(n,f) and 2 s 6 U(n,f). The main fission fragments were detected by two multi-wire-proportional-counters. Lighter charged particles were intercepted by two arrays of PIN diodes, each array comprising up to 16 diodes. The granularity of the diode arrays allowed to search for coincidences of two light charged particles with the fission fragments and also to investigate angular correlations between the reaction products. Besides measuring energies also the formation of the charge pulse as a function of time was analysed. The technique enables to unambiguously separate a-particles from H-isotopes. With a binary fission count rate of 106 events/s, and a ternary rate of nearly 10s events/s, about 5 X 10a quaternary processes were recorded in 6 weeks of measuring time. This is the largest sample of quaternary fission events ever collected. The results for the two U-isotopes studied are very similar. By far the most abundant light particles in quaternary fission are a-particles. As deduced from the pattern of the angular correlations between two coincident a-particles, most of these events are due to the decay in flight of a primary 8 Be into two as, 8 Be being created as a ternary particle either in its ground or in an excited state. The yield for this "pseudo quaternary" process is 2 X 10~ 7 per fission. But there is also clear evidence for a "true quaternary" process where two a-particles appear in coincidence without being correlated in emission angle. The yield for true quaternary fission is 5 X 10—8 per fission. Likewise a-triton coincidences were observed. The yields are much lower than for the a-a events. Nonetheless, also here a pseudo and a true quaternary component could be disentangled. For both processes the yields are about 6 X 10—9 per fission. No clear signature for quaternary events other than a-a or a-triton coincidences could be disclosed.
1
Introduction and Experimental Setup
Though discovered some fifty years ago 1 , low energy fission with four charged particles in the exit channel has barely been investigated since. For both, thermal neutron induced fission of fissile isotopes 2 and for spontaneous fission3'4, it is known that as a variant to binary fission sometimes not only one additional light charged
233 particle, but even two light charged particles might appear. The light particles accompany two heavy fragments with masses and charges similar to those from binary fission. These processes are called ternary and quaternary fission, respectively. In the majority of cases the lighter particles are a-particles and hydrogen isotopes, with tritons being dominant among the H-isotopes. The experimental results on quaternary fission to be presented here were obtained while probing symmetry laws in ternary fission5. The reactions studied were fission of the two U-isotopes, 233TJ and 2 3 5 U, induced by cold neutrons. The experiments were performed at the high flux reactor of the Institut Laue-Langevin in Grenoble/France. A cold neutron beam with an average wavelength of A = 4.5 A and with a flux of 3 x 10 8 neutrons/cm 2 • s was available. Both the U-targets were made from highly enriched material and had a total mass of 5 mg evaporated uniformly onto a thin Ti-foil with diameter 8 cm.
PIN-diodes ternary particles n-beam MWPC
MWPC fission fragments
fission fragments
PIN-diodes | ternary particles Figure 1. Schematic view of the detector system. The neutron beam is running horizontally. The detectors for fission fragments (MWPC) are positioned in a horizontal plane together with the target and the neutron beam, and left and right to the beam. The detectors for ternary particles (PIN diodes) are mounted on top and on bottom. The target is at the center, oriented parallel to the MWPCs.
The experimental setup is sketched in Fig. 1. The charged reaction products from the interaction of the neutron beam with the target were intercepted by two different types of detectors. For the fission fragments two Multi-Wire-ProportionalCounters (MWPC) with CF4 as the counting gas were used, while the light charged particles were analysed by two sets of PIN-diodes. The target and all the detectors facing the target were mounted in a common plane oriented perpendicularly to the neutron beam, the beam running horizontally. From the MWPCs a time signal
234 was deduced whenever a heavy ion was passing by. From correlated signals in the two MWPCs the time-of-flight (TOF) difference for the two heavy ions was determined. The TOF spectrum exhibits two well separated bumps corresponding to asymmetric Ission with either the light fragment lying in Fig, 1 to the detector on the left and the heavy fragment to the right, or vice versa. A TOF event in a befitting time window is the fingerprint for a fission process. A binary fission countrate of more than 106 events/s was achieved. m 60 j — - — r ^ -
0
5
10
15 20 25 30 35 energy of ternary particle/MeV
Figure 2. Example of an energy vs. risetlme scatterplot. The a-particles are well separated from the tritons and protons.
The light charged particles (LCP) being of prime interest in the present experiments, much effort was put into their analysis. As already stated, the detectors for the LOPs, shown in Fig. 1 on top and bottom, were arrays of PIN diodes, each array comprising l i diodes and each diode being square shaped with side length 3 cm. It should be underlined that, due to the granularity of the arrays, angular correlations between the LCPs may be readily taken. For a detailed analysis of the LCPs an energy signal was derived from any diode by integrating the charge set free when ions are intercepted. In addition, the majority of diodes was equipped with specifically designed electronics that allowed to measure the time duration of the current low whenever a LCP is being stopped in the diode. For a given energy this duration is characteristic for the particle type. Physically this has to be traced to the fact that the stopping range in any material for an ion of given energy depends on the charge and mass of the ion. Since in a pn-diode the range is transformed into the time lapse of current flow, one expects a correlation between duration of the current flow (or risetime of the charge signal) and energy, which should be spe-
235 cine for any particle type. Recently the combination of pn-diodes with injection of ions from the rear side together with fast low noise electronics has allowed a breakthrough in particle identification6. An example from the present experiment is on display in Fig. 2. In the scatter plot of charge rise time versus energy the separation between a-particles and tritons is seen to be perfect. As to the triton signals it has to be noted that some of the tritons have an energy too large to be stopped in the diodes. These particles lose only a fraction of their energy in the diodes. They become visible in Fig. 2 in the lower part of the triton band bending backwards. Evidently these high energy tritons spoil the particle identification for deuterons and protons.
x103 ~
5000
3
3000 2000 1000 5
10
15
20
25
30 energy/MeV
Figute 3. Fit of a Gaussian distribution (dashed line) to the measured energy spectrum (histogram) of ternary a-particles. The spectrum is corrected for energy loss in the target and the aluminum foils.
Ternary fission count rates for single LCP detection in coincidence with fission fragments were measured in the present setup to be less than 10 3 /s and, hence as anticipated, much smaller than binary fission rates. Therefore the PIN-diodes for LCP detection had to be protected from the much more abundant fission fragments by thin Al-foils placed in front of the diodes. The energy losses in these foils and in the target material have to be taken into account as corrections to the energy deposited in the diodes. The corrected energy spectrum for a-particles is given in Fig. 3 as a fine grained histogram. As it is evident from the figure, the low energy cutoff is at almost 10 MeV. The dotted line in Fig. 3 is a Gaussian fit to the data at higher energies. From the fit an average energy of (15.7 ± 0.3) MeV and a width (FWHM) of (9.8 ± 0.3) MeV are found for the energy spectrum of ternary a-particles from the reaction 233 U(n,f). These figures are in very good agreement with data compiled from previous experiments 7 .
236 2
Quaternary Fission
in
per
c
in •«-»
u u o
8000: 7000 r 6000 5000 :
4000 ': 3000 r 2000 : 1000 °
0
10
20
30
40
time/ns Figure 4. Time distribution of double hits in the P I N diode a r r a y s . T h e figure shows t h e time difference between hits from two different P I N diodes.
With the arrangement of detectors on display in Fig. 1 the fingerprint for a quaternary fission event are simultaneous hits in two PIN diodes in coincidence with two fission fragments seen by the MWPCs. A time-to-digital converter was used to measure the time between two impacts in the diodes. The time spectrum obtained is represented in Fig. 4. The distribution of timing differences between two diodes exhibits a pronounced peak in the time range from 0 ns to about 5 ns. It should be noted that time delays up to 5 ns in the arrival times of LCPs from the same fission event are to be expected due to different velocities and/or different path lengths of these particles to the diodes. There is, hence, clear evidence for fission processes producing two long range particles in one single event. However, on account of the extremely high fission rates in excess of 10 6 events/s, there is also a non-negligible background of double hits at larger time differences. Unfortunately quaternary fission events with two coincident LCPs being intercepted by the same diode could not be distinguished from the more common ternary events with only one single LCP. On the other hand, together with the energy deposited and the charge rise time of the diode signals, also the positions of the diodes being hit were recorded event-by-event. It is thus possible to determine the opening angles between two coincident LCPs as seen from the target. The distribution of opening angles measured in the present experiments is shown in Fig. 5. The histogram in the figure does not cover continuously all angles because it represents the count rates in diode pairs of finite size, either for diodes from the same or from opposite arrays, shown in Fig. 5 either to the left or right, respectively.
237 to
g o
2500 2000 1500 1000 500
"0
20
40
60
80
100 120 140 160 180 angle / degrees
Figure 5. The measured angular distribution of double hits (solid line) together with a MonteCarlo simulation (dashed line).
Furthermore, as pointed out, double hits in one single diode are not identified as quaternary events and, therefore, no count rates at small opening angles near 0° can be given. Nevertheless, it is noteworthy to observe in Fig. 5 that the angular distribution is neither uniform nor symmetric around 90° and that in the majority of quaternary processes the opening angles between the two LCPs are rather small as judged from the peak at 20°. On the other hand, there are definitely events at very large opening angles close to 180°. The dashed histogram in Fig. 5 is a Monte Carlo simulation of the angular correlations in quaternary fission. The simulation is based on the results of a more detailed analysis given below. 2.1
a-a
Coincidences
The particle identification for the LCPs detected by the PIN-diodes has proven to be crucial for disclosing several distinct quaternary fission processes. In by far most of these processes the fragments are accompanied by two a-particles. Therefore, firstly a-a coincidences are considered. The two a s could be due, either to the independent emission of two a-particles, or the emission of a primary 8 Be which decays into two a s before reaching the detectors. Only the former process should be called "true" quaternary fission since in the process mediated by 8 Be it is a basically ternary fission process which mimics a quaternary one. The characteristics of this latter "pseudo" quaternary process can be inferred from the decay properties of 8 Be. The 8 Be nucleus is unstable in the ground state and decays with a half life of T 1 / 2 = 0.07 fs into two as. The Q-value for the reaction is Q - 0.092 MeV. The lifetime being long compared to the acceleration time of the LCP in the Coulomb field of the fission fragments, upon decay the 8 Be ion has virtually attained its final
238 energy but is still close to the target. The small decay energy then only allows for small opening angles between the two a s as seen from the target position. Simulations indicate that these decays can indeed not contribute significantly to events with opening angles larger than 20°. Unfortunately, most of the quaternary events stemming from the decay of 8 Be in the ground state escape detection in the present experiments, since for two as with small opening angles the chances are high that they will impinge on one and the same PIN diode and will, hence, be mistaken as a single particle. Nevertheless, the peak in the distribution of opening angles between two LCPs near 20° in Fig. 5 is attributed to pseudo quaternary fission with two as being the decay products of ground state 8 Be. It is conjectured that in experiments where opening angles smaller than 20° could be measured, the enhancement would become even more prominent. However, the primary 8 Be nucleus may also be created in an excited state. The first excited state at 3.04 MeV disintegrates with a half life of Ti/ 2 = 3 x 1 0 - 2 2 s and a Q-value of 3.13 MeV into two a-particles. Due to the higher Qvalue, a s from this decay can exhibit larger opening angles. More delicate for the analysis and evaluation of data is the short half life which prevents the excited 8 Be* nucleus to reach the final velocity before decay. The majority of the 8 Be* nuclei will break up while being still in the accelerating field of the main fission fragments. A reliable simulation of the processes of acceleration and disintegration of 8Be* and the subsequent acceleration of two as would require four body trajectory calculations. These trajectory calculations have so far not been performed and, even worse, in view of the still existing ambiguities for ternary trajectories it is questionable whether the more demanding quaternary trajectory calculations can be made at all to be trusted. In experiment the presence of coincident a s from the decay of an excited 8Be* nucleus is deduced from the enhanced yield for opening angles near to but definitely larger than 20° (s. Fig. 5). On the other hand, there is no indication from experiment for an enhancement in the yields of pseudo quaternary events beyond opening angles of about 50°. Partly this may be due to the fact that the creation of 8 Be* in an excited state is expected to be less probable than the creation of 8 Be in the ground state. As argued below, this is giving true quaternary fission a chance to become the dominating process at large opening angles up to 180°. A further LCP from ternary fission which can simulate a quaternary fission event is the isotope 9 Be. In its ground state this isotope is stable. However, the excited states of 9 Be*, and here especially those at excitation energies of 2.78 and 3.05 MeV, are known to decay with high probability into a neutron and an 8 Be nucleus in the ground state. The half lifes of these states are 4.2 x 1 0 - 2 2 s and 1.6 x 10~ 21 s, respectively. In both cases the lifetimes are shorter than the acceleration time of the 9 Be* in the Coulomb field of the main fission fragments. The 8 Be produced subsequently decays into two a s which upon detection are interpreted as stemming from a quaternary event. The contribution to the pseudo quaternary yield by 9Be* is hence included in the yields ascribed to an intermediary 8 Be. Unfortunately, it is not possible to assess from experiment which fractions of the total measured 8 Be yield are due to a primary 8 Be or 9Be* light charged particle. Obviously the next task is to search for true quaternary fission with generally
239
quaternary (a, a) 8
Be
233TJ
235TJ
(0.24 ±0.04) io-*
(0.23 ±0.04) 10~*
(0.94 ±0.07) 10-*
(0.83 ±0.07) io-*
Table 1. Yields for q u a t e r n a r y fission and for s B e , all yields are relative t o t h e yields for 4He> Errors shown are statistical. Systematic errors estimated t o be ± 3 0 % .
two LCPs or, in the present particular case, two as being born simultaneously with the fragments. Since it is generally accepted nowadays that in ternary fission most of the ternary particles come into view only right at scission, it is natural to assume that the same feature also holds for the LCPs in quaternary fission. Nevertheless, compared to ternary fission this is a novel disintegration process. Indeed, e.g. in models of LCP accompanied fission, where the LCPs emerge from the neck of a dumbbell-like scission configuration, ternary fission corresponds to a double neck rupture, while quaternary fission has to be viewed as a triple neck rupture setting free two LCPs besides the fission fragments. In such a scenario one expects the two a-particles to follow the same angular distribution relative to the fission fragments with a maximum of emission probability at a polar angle of 83° with respect to the light fragment, as observed for as from ternary fission. By contrast, with the light fragment momentum as polar axis, the distribution of azimuthal angles should be isotropic and it is moreover conjectured that there is no correlation between the azimuthal angles of the two as. This allows for both, small and large opening angles between the two a s with equal probability. Let us recall that in pseudo quaternary fission the correlation is conspicuous for small opening angles while it is lost out of sight for angles larger than some 50°. Therefore, in the framework sketched, virtually all events at large opening angles are due to true quaternary fission. In a rather good approximation, hence, double hits in the same array of PIN diodes shown to the left of Fig. 5 are a mixture of true and pseudo quaternary events, while double hits from opposite arrays (s. Fig. 1) on the right of Fig. 5 are liable to originate from true quaternary fission only. Evidently, background events with two as in random coincidence from two distinct fission processes should follow the same angular patterns as just outlined for true quaternary fission. For the evaluation of data a Monte Carlo simulation of the experiment has been set up. The simulation takes into account the arrangement of target and detectors for calculating relative count rates of pseudo quaternary fission, true quaternary fission and background. For each of the three processes the either known or calculated or hypothesised angular correlations between outgoing reaction products serve as inputs. The simulation adjusts the relative weights of the processes. It turned out to be necessary to include at least in an approximate way the decay from the first excited state in 8 Be* for pseudo quaternary fission to get good agreement with the experimental findings. The result of the simulation is shown in Fig. 5 as the dashed histogram. Detector by detector the experimental data are rather well reproduced. The yields obtained from this analysis for true and pseudo quaternary fission with an intermediate 8 Be nucleus are given in Table 1 relative to the ternary yield for
240 4
He. The two reactions studied, viz. 233 U(n,f) and 235 U(n,f), appear to behave very similarly. The errors quoted are only the statistical ones. Due to the assumptions which had to be made for the evaluation, the systematic errors are certainly large and unfortunately difficult to assess. Conservatively these errors are estimated to be ±30%. Prom the analysis also the fractional contribution of excited 8 Be* to the total 8 Be yield can be estimated. Awaiting reliable trajectory calculations for excited 8 Be* decaying in flight before being fully accelerated, the figure of 35% deduced for the 8Be'* contribution to the total 8 Be yield should, however, be considered with reservation. To find the yields per fission reaction, the figures in the table have to be multiplied by the 4 He ternary yield of 2 x 1 0 - 3 . The yields found for true quaternary fission into two a s are, hence, 5 x 1 0 - 8 , while the yields for pseudo quaternary fission with two os in the exit channel are 2 x 1 0 - 7 per fission. Both types of quaternary fission are seen to be rare processes. The yields derived in pseudo quaternary fission for a-a coincidences are to reasonably good approximation the ternary 8 Be yields. The approximation made here is the neglect of the contribution to 8 Be from the neutron decay of excited 9 Be*. The yields of ground state 9 Be are well known for the fission reactions under study 7 . To measure also the yields of excited 9 Be* disintegrating into a neutron and a Be is, at least in principle, feasible. Yet, the low count rates for neutron accompanied pseudo quaternary fission are prohibitive and it is unlikely that this experiment will be undertaken any time soon. The ratios of yields for the formation of 9Be* to 9 Be in ternary fission have, therefore, to be estimated. Recently first experiments evaluating the ratios of probabilities for excited to ground state generation of LCPs have been reported 8 . These data should serve as a guidance in a statistical model calculating the relative populations of LCP levels in fission. This analysis has not yet been performed. The yields of Be-isotopes taken from a compilation 7 together with the 8 Be yields determined here for the first time in thermal neutron induced (n,f) reactions in Uisotopes are plotted in Fig. 6. The 8 Be yields are seen to fit smoothly into the yield systematics. However, it should be underlined that the contribution of 9Be* to a Be has been disregarded and, hence, the yields shown for 8 Be are upper limits while those quoted in the literature for 9 Be are lower limits. Besides the yields, a further interesting feature of quaternary fission are the kinetic energy distributions of the LCPs involved. The energy distributions of aparticles from a-a coincidences are displayed in Fig. 7, on the left for true and on the right for pseudo quaternary fission. All energies are corrected for energy losses in the Al-foils protecting the PIN diodes and in the target. The dashed histograms are measured data while the histograms with full lines are data corrected for counting losses. For true quaternary fission there are mainly counting losses for events with energies close to the cut-off energy at about 10 MeV. These losses were assessed from the losses observed for ternary a-particles shown in Fig. 3. For pseudo quaternary fission additional losses occur for events with two a s entering the same diode, these events being not recognised as double hits by the electronics of the diodes. The corrections were based on the Monte Carlo simulation elaborated for the fit of angular distributions presented in Fig. 5. As is evident from the right panel of Fig. 7, the corrections are quite substantial and were, therefore, not pushed to energies
241
r
I O T—I
101
"c?
10c 10"
0) x
• 233U ' • 235U ]
**
**
•
-I-
r
-2
10
10"
r
8
10 12 mass number
14
Figure 6. Yield for 8 B e compared to t h e yields of other Be isotopes for
233
U and
235
U.
'.
•2 4000
c o o
A
3 O
° 3000
'
/ / /
2000
\\
/
1000 ; / 20
30 energy/MeV
0
. . ."-^TlO
10
20
.....'
30 energy/MeV
Figure 7. Left panel: Energy distribution of or-particles from q u a t e r n a r y fission into two aparticles. Right panel: Energy distribution of a-particles from the decay of 8 B e .
near the cut-off were measured count rates are low and statistically fluctuating. The continuous curves, finally, in Fig. 7 are Gaussian fits to the corrected data. From the Gaussian fits mean kinetic energies and widths (FWHM) of the aenergy distributions were inferred for a-a quaternary fission of the 233TJ and 235 U isotopes studied in (n,f) reactions. The data are summarised in Table 2. Very surprising in the Table are the large differences between the mean energies of as from true and pseudo quaternary fission, respectively. Compared to ternary fission, the
242
mean Energy E
quaternary (a, a) 8
Be
width (FWHM)
233JJ
235TJ
233TJ
235y
14.2 MeV
13.8 MeV
8.5 MeV
9.4 MeV
9.0 MeV
9.9 MeV
8.9 MeV
8.3 MeV
Table 2. Mean energy and with (FWHM) of a-particles from quaternary fission into two orparticles and 8 B e . Uncertainties are estimated to be typically ± 0 . 5 MeV in all cases.
mean a-energies are only down by less than 2 MeV for true quaternary fission. But for pseudo quaternary fission the mean a-energies are further down by an additional 4 to 5 MeV per a-particle. Yet, the latter small average energies fit reasonably well into the systematics of mean LCP energies in ternary fission. In fact, guided by trajectory calculations, mean energies of LCPs have been fitted to experimental data and extrapolated to nuclides where no measurements are available 9 . For 8 Be as the LCP mean kinetic energies of about 20 MeV are predicted. After breakup of 8 Be, the two as should have kinetic energies close to 10 MeV since, as already stated, the decay energy is small. Qualitatively at least, this is in good agreement with the present findings where the average energies are found to be 9.0 ± 0.5 MeV and 9.9 ± 0.5 MeV for the two reactions under study (s. Table 2). It is an unexpected and most remarkable feature of a-a quaternary fission that the kinetic energy spectra for the two modes, true and pseudo, are that different. It gives additional credit to the analysis of angular correlations which led to the discovery of two distinct modes. 2.2
cx-t Coincidences
Besides quaternary fission with two as also coincidences between an a-particle, a hydrogen isotope and two fission fragments could be identified. In PIN diodes of adequate thickness where a more detailed specification of the hydrogen isotopes was feasible, it was found that the lions share of the yield was due to tritons. This finding was taken as justification to consider all hydrogen events to be tritons also in those diodes where no unambiguous distinction between the different hydrogen isotopes could be made. As in the quaternary a-a case, also the a-t coincidences can be conjectured to emerge from two different processes, pseudo and true quaternary ones. A pseudo quaternary process could be mediated by the isotope 7 Li. Though the nucleus r Li is stable in its ground state, the second excited state at an excitation energy of 4.63 MeV is particle unstable and disintegrates into an a-particle and a triton with a Q-value of 2.17 MeV. The half life for this decay is 4.9 x 1 0 - 2 1 s. Created as a LCP in its second excited state, 7Li* thus decays during the acceleration phase in the Coulomb field of the fragments and gives rise to pseudo quaternary fission. Possible contributions from still higher levels have been discussed in the past 1 0 , but it is questionable whether these states are actually populated. The third excited level in 7 Li* comes at an energy of 6.68 MeV and it is unlikely that such large
243 233|J
quaternary (a, t) 7
Li*
235TJ
(3.1 ± 1.0) • 10- 6 (2.5±1.0)-10- 6 (3.4±1.0)-10- 6
(3.7±1.0)-10- 6
Table 3. Yields for 7 Li* and quaternary fission with triton a n d a as light charged particles. Ail yields are relative to the yield of 4 H e .
excitation energies of LCPs play a role in low energy fission. Therefore only the second excited state of 7Li* has been retained. Also the population of this level in a bypass through neutron evaporation from excited states in 8 Li* has been discarded since again excitation energies in excess of 6 MeV would be requested for this nucleus. In experiment one is faced with the same difficulty as discussed above in the preceding section, viz. how to distinguish between the two modes of quaternary fission. In the pseudo mode there is an interposed ternary 7 Li*, while in the true mode an or-particle and a triton are formed right at scission. Similar to the aa coincidences, the procedure followed was to inspect the distribution of opening angles between a-particle and triton. The decay products of 7Li* are conjectured to exhibit angular correlations at small opening angles, while a-particles and tritons from true quaternary fission are assumed to be ejected independently and, hence, distributed evenly on cones around a polar axis given e.g. by the light fragment momentum. The average half angle of the cones being close to 90° for both LCPs, this leads to a-t opening angles evenly distributed between 0° to 180°. As with the a - a events, the ansatz means that a-t coincidences at large relative angles, as detected in opposite diode arrays (s. Fig. 1), can only stem from true quaternary fission. By contrast, coincidences in the same diode array sense smaller a-t opening angles and represent therefore a mixture of the two modes. The experimental data were analysed along the above lines and the results for the two (n,f) reactions studied are summarised in Table 3. The yields are given relative to the ternary a-yield. It is read from the Table that, firstly, the two U-isotopes behave similarly and that, secondly, true and pseudo quaternary a-t fission contribute with equal weight to the total yield. Compared to the a-a quaternary yields the a-t yields are smaller by more than an order of magnitude. Relative to fission this means that the a-t yields are extremely small, viz. 6 X 10"~9 per fission for each of the two modes. As to the errors given in the Table, it has been tried to include estimates for systematic errors. The yield for the production of the ternary 7Li* in its second excited state from the present analysis, shown in Table 3, may be compared to the production of 7Li in its ground state as known from literature 7 , 1 1 . The data are put together in Table 4 with all yields evaluated relative to the yields of ternary 4 He scaled to 10 4 . The creation of 7 Li in the excited state has a remarkably small probability of barely 1% compared to the one in the ground state. In view of the high excitation energy of 4.63 MeV in question this result appears to be not unreasonable. Finally, the first two moments of the kinetic energy distributions for quaternary
244
7
233TJ
235TJ
(4.4±1.3)-10-4
(4.1±0.3)-10-4
Li*
(3.4± 1.0) • 1 0 - 6
(3.7± 1.0) • 10" 6
Li*/ 7 Li gr. st.
(0.77 ±0.35)%
(0.90 ±0.25)%
Li ground state 7
7
Table 4. Comparison of yields:
L i * / 7 L i g.s.. All yields are relative to t h e yield of 4 H e .
mean Energy E
width (FWHM)
12.0 MeV
9.4 MeV
11.3 MeV
8.7 MeV
6 MeV
5 MeV
6 MeV
5 MeV
a-particles: quaternary fission 7
Li*
tritons: quaternary fission 7
Li*
Table 5. Mean energy and width ( F W H M ) for tritons a n d a-particles from q u a t e r n a r y fission into a and triton and the decay of 7 Li*. Uncertainties are e s t i m a t e d to be ± 0 . 5 MeV for a-particles a n d ± 1 MeV for tritons.
a-particles and tritons are summarised in Table 5 for the reactions studied. The errors are large because the statistics is poor and because, in addition, tritons did not always come to rest in the diodes since not for all of them the depletion depths were thick enough. This made the analysis of kinetic distributions for tritons delicate. No significant differences in the results for the reactions 233 U(n,f) and 235 U(n,f) having been observed, averages from the two reactions are presented in the table. The comparison of the two modes of quaternary fission indicates that for a-particles and tritons alike, both the mean energies and the widths of the distributions are quite similar, if not identical. This is contrary to the observation in quaternary a-a processes. For the a-particles it is noteworthy that the mean energies of 12 MeV in (a, t) accompanied fission are by 2 MeV lower than the 14 MeV in (a, a) accompanied fission. This could possibly point to different deformations of the scission configuration for the emission of either two a-particles or one a-particle and one triton. It should be recalled here that, as outlined in the preceding section, the mean a-energies from ternary fission are close to 16 MeV. As to the tritons, their mean energies of 6 MeV are surprisingly low. But keeping in mind that also in ordinary ternary fission the mean triton energy is only 8.4 MeV the energy observed in quaternary fission appears to be sensible. Quaternary fission processes with a combination of LCPs other than (a, a) or (ct,t) could not be identified with certainty.
245 3
Summary
Quaternary fission with two heavy fisision fragments and two light charged particles in the exit channel has been investigated for thermal neutron induced fission of 233 U and 235 U as target nuclei. For the first time, both energy distributions and angular correlations between the two outgoing light particles have been studied with good statistics (more than 5 x 10 3 events) for opening angles from 20° to 180°. The data allow to distinguish between "true" quaternary fission with four charged particles right at scission on one hand, and "pseudo" quaternary fission on the other hand. In the latter case, at scission first a ternary particle is formed which is then accelerated in the Coulomb field of the fission fragments and subsequently decays in flight into two lighter charged particles before reaching the detectors. Together with the two fission fragments these latter two particles mimic quaternary fission. Particle identification in the PIN diodes for the light charged particles further enabled to take data separately for as and hydrogen isotopes. Among the hydrogen isotopes by far the most abundant particles are tritons. In the following, results for all hydrogen isotopes are, hence, taken together and compounded under the heading "tritons". The results for the two U-isotopes studied are very similar. Quaternary fission accompanied by (a, a) pairs was observed to be the most probable process, followed in probability by (a, t) coincidences. Other combinations of LCPs could not be identified with sufficient statistical accuracy and for this reason are not reported here. The most prominent quaternary fission process is pseudo quaternary fission mediated by a primary 8 Be nucleus. When produced in its ground state, 8 Be has a lifetime of less than 1 0 - 1 6 s for decay into two as. The decay energy is 92 keV. The lifetime being long compared to the acceleration time of 8 Be, the nucleus disintegrates having been fully accelerated but being still close to the target. On average the final kinetic energy of 8 Be, estimated to be close to 20 MeV, is much larger than the decay energy. In experiment the fingerprint for 8 Be is, therefore, the small angle subtended by the two as being detected in the lab frame. In the present setup the smallest opening angle, as seen from the target, between the center-of-masses of two neighboring PIN diodes was almost 20° (s. Fig. 5) and thus many 8 Be events were missed. Based on a Monte Carlo simulation corrections for events having escaped detection were calculated in the evaluation of data. In addition to 8 Be decay from its ground state, the angular correlation data exhibit also contributions from the decay of 8 Be* in its first excited state at E* = 3.04 MeV. Due to the short half life of T1/2 = 3 x 1 0 - 2 2 s of this state, the excited 8 Be* nuclei disintegrate before being fully accelerated. This complicates the evaluation of data. The ratio of formation of 8 Be* / 8 Be has been estimated by a simplified approach in the present work. A further complication in the assessment of 8 Be yields is connected with the production of the neighboring isotope 9 Be. From excited states 9 Be* can decay into a neutron and 8 Be. The present technique does not permit to disentangle the formation of 8 Be and 9 Be*, both nuclei having the same signature with two a s in the exit channel. True quaternary fission with two a s as the light particles has been evaluated
246 starting from events at large opening angles between the two as. For large angles close to 180° (s. Fig. 5) the contamination by pseudo quaternary events should be minimised. In experiment these large angles correspond in Fig. 1 to a-a coincidences between diodes from opposite arrays. Assuming independently for both particles an angular distribution which is identical to the one known from ordinary ternary fission, the (a, a) count rates can then be simulated for opening angles down to 0°. At smaller opening angles the two modes of quaternary fission, true and pseudo, contribute in parallel. Small opening angles correspond in the setup of Fig. 1 to coincidences between diodes of the same array. Since the two PIN diode arrays are identical, the procedure to disentangle the two modes of quaternary fission is simply to subtract the count rate spectrum of aenergies observed in opposite arrays from the corresponding spectrum for the same array. The difference spectrum is then entirely due to pseudo quaternary fission. This specrum has to be corrected further for counting losses as already outlined above. Finally, from a Monte Carlo simulation based on the above assumptions the yields for quaternary fission were calculated independently for the two modes by a fit to the measured data. As demonstrated in Fig. 5, the simulation allows for an excellent fit of the observational data thus giving credit to the assumptions and approaches which were made in the evaluation. The yields determined in the present work for true and pseudo (a, a) quaternary fission are, within statistical uncertainties, virtually identical for the two (n,f) reactions studied. Typically the true (a, a) mode has a probability of 5 x 1 0 - 8 per fission, while for the pseudo mode the probability is 2 x 1 0 - 7 per fission. Recalling that the probability for ternary fission accompanied by a single a-particle is 2 x 1 0 - 3 per fission, the yield for the true (a, a) mode shows that the emission probability for the second a-particle in quaternary fission is not just again 2 x 1 0 - 3 but instead down by almost two orders of magnitude. As to pseudo quaternary fission it is noteworthy that the yield found in the present work is qualitatively in good agreement with a crude estimate reported from a study of the 235 U(n,f) reaction at thermal neutron energies 12 . From a total of 33 (a, a) quaternary events with an interposed 8 Be, a pseudo quaternary yield of about 1 0 - 7 per fission was reported. It is further remarkable that in (a, a) accompanied fission the a-particles from the true or the pseudo mode have very different kinetic energy distributions. While in the true mode the energy spectrum is shifted by a moderate 2 MeV to lower energies compared to ternary a-spectra, in the pseudo mode the downward shift amounts up to 7 MeV. These low a-energies from the breakup of an intermediate 8 Be are, however, fully in line with the observed systematics of energy distributions of LCPs in ternary fission and with trajectory calculations 9 ' 11 . The data measured for a-particles and tritons in coincidence with two fission i fragments were analysed along similar lines as sketched in the foregoing for aa coincidences. Again a true and a pseudo mode of quaternary fission can be disentangled. The pseudo mode is mediated by 7 Li* produced in an excited state as primary ternary particle and decaying in the end into an a-particle and a triton. Again results for the two U-isotopes under study are almost undistinguishable but, by contrast to the (a, a) case, also the two modes are similar, both as to yields
247 and as to characteristics of the kinetic energy distributions. The yields are down to 6 x 1 0 - 9 per fission. For true quaternary fission this means that the (a,t) process is almost an order of magnitude less probable than the (a, a) process. The ratio is remarkably close to the ratio of triton to a-particle yields in ternary fission. The mean kinetic energies of a-particles and tritons are about 12 MeV and 6 MeV, respectively. Compared to ternary fission both energies are lower by about 25%. Having observed several thousands of quaternary fission events, the question arises whether there is any chance to discover quinary fission with 5 charged particles in the exit channel. Let us hypothesise that a highly probable quinary process could be fission accompanied by three a-particles. For a true (3a) quinary process the probability may be guessed by extrapolation from the observed true (2a) quaternary decay to be at best 1 0 - 1 2 per fission. Similar orders of magnitude for the probability of pseudo quinary fission are expected for processes mediated by (a, 8 Be) quaternary fission or 12 C* ternary fission with 12 C* decaying via (a+ 8 Be) into 3QS. TO detect events with a probability of 1 0 - 1 2 per fission is out of reach of present day techniques. Quaternary fission has in the past not been discussed in any detail by theory. It is hoped that the present work with many novel and partly startling results from experiment will stimulate research in the theory of this rare process probing extreme configurations at scission of a fissioning nucleus. Support of this work by the DFG (Bonn), RFBR (Moscow) and INTAS (Brussels) is gratefully acknowledged. References 1. E.W. Titterton, Phys. Rev. 83, 1076 (1951) 2. S.S. Kapoor et al., Proc. "Nuclear Physics and Solid State Physics Symposium", Chandigarh, India, 1972, Vol. 15b, p 107 3. S.K. Kataria, E. Nardi, S.G. Thompson, Proc. Conf. "Physics and Chemistry of Fission", Rochester, USA, 1973, IAEA Vienna 1973, Vol. II, p 389 4. A.S. Fomichev et al., Nucl. Instr. and Meth. A 384, 519 (1997) 5. P. Jesinger et al., Nucl. Instr. and Meth. A 440, 618 (2000) 6. M. Mutterer et al., IEEE Trans. Nucl. Sci.,47, 756 (2000) 7. C. Wagemans in "The Nuclear Fission Process", CRC Press, USA, 1991, C. Wagemans ed. 8. M. Mutterer, Proc. "Seminar on Fission, Pont d'Oye IV", World Scientific, C. Wagemans et al. eds., 2000, p 95 9. W. Baum, PHD thesis, TU Darmstadt, Germany, 1992 10. N. Feather, Proc. Roy. Soc. Edinburgh, 71, 323 (1974) 11. F. Gonnenwein et al., Proc. "Seminar on Fission, Pont d40ye IV", World Scientific, C. Wagemans et al. eds., 2000, p 59 12. V.N. Andreev, V.G. Nedopekin, V.I. Rogov, Sov. J. Nucl. Phys. 8, 22 (1969)
248 BREMSSTRAHLUNG EMISSION IN ALPHA-DECAY
D.M. BRINK Department of Physics, Theoretical Physics, 1 Keble Road, Oxford 0X1 3NP, England E-mail: [email protected] We report on t h e current status of the theory of bremsstrahlung emission associated with alpha decay. The photon emission amplitude can have contributions from outside t h e Coulomb barrier and from t h e tunnelling region. It might also be influenced by t h e alpha formation process. Recent quantum mechanical and semi-classical calculations find t h a t acceleration of t h e alpha particle outside t h e Coulomb barrier gives t h e major contribution t o t h e amplitude. There are subtle interference effects between the sub-barrier and outside regions which influence the theoretical results.
1
Introduction
In a-decay the ct-particle appears on the surface of the emitting nucleus, it penetrates the Coulomb barrier and is accelerated by t h e Coulomb field outside t h e barrier. Bremsstrahlung (photon) emission can take place during t h e acceleration phase outside t h e barrier. This contribution should be described well by classical electromagnetic theory. T h e photon emission amplitude can also have a contribution from t h e sub-barrier region and might also b e influenced by t h e a-formation process. B o t h of these are essentially q u a n t a l processes. T h e experiment of Kasagi et al. 3 on bremsstrahlung emission in t h e a-decay of P o was motivated by a desire t o u n d e r s t a n d t h e relative importance of t h e t h r e e contributions. T h e results of t h e experiment were in disagreement with the predictions of classical theory a n d with t h e simplest semi-classical model. Papenbrock a n d Bertsch 4 made a q u a n t u m mechanical calculation using t h e Fermi golden rule for t h e photon emission with modified Coulomb wave functions for t h e initial and final s t a t e of the a-particle. Their theory was in agreement with t h e d a t a within the experimental errors. Subsequently Dyakonov obtained satisfactory results w i t h a semi-classical theory. T h e Sommerfeld p a r a m e t e r r; = Zeffe2/hv ~ 22 ^> 1 under the condition of t h e experiment of Kasagi et al. and a semi-classical calculation would b e expected t o give reasonably accurate predictions. In this contribution we discuss t h e relation between these different theoretical approaches a n d t r y t o understand why t h e classical semi-classical approach fails. It t u r n s out t h a t t h e sub-barrier contributions to t h e photon emission amplitude are important for a consistent theoretical description.
249 2
Classical t h e o r y
According t o classical electromagnetic theory t h e energy spectrum of t h e photons emitted during t h e acceleration outside t h e Coulomb barrier is 2<*Z2efftT,
dP
^
= 3 ^
,,2
m
| 7 ( W ) I
(1)
where I{oS) is t h e Fourier transform of t h e alpha particle acceleration a(t), /•OO
I{LU) = I Jo
a(t)eiutdt
(2)
Here Ey = fvui is t h e photon energy. T h e alpha velocity is zero at t i m e t = 0 when it emerges from t h e barrier. For high frequency photons t h e Fourier transform I{w) h a s an asymptotic expansion in inverse powers of u> which can be obtained from e q . ( l ) by repeated integration by parts. T h e leading t e r m is J(w) ~ - i o ( 0 )
(3)
T h e relation (3) predicts a high 7-energy spectrum dP/dE1 oc E~3. T h e experiment shows a much faster decrease with energy. T h e r e is a classical low frequency theorem which is in agreement with t h e experimental bremsstrahlung results I{IM) —• Voo as LJ —* 0, (4) where v^ is t h e asymptotic alpha velocity at a large distance from t h e parent nucleus. 3
T h e theory of Papenbrock and Bertsch
T h e q u a n t u m calculation of Papenbrock a n d Bertsch was based on a specific model for t h e alpha decay. T h e alpha particle was assumed t o move in a potential which is a constant inside t h e parent nucleus and is a Coulomb potential outside, V(r) = — Vb,
r < r$
and
V ( r ) = Zzer/r,
r > VQ.
(5)
T h e depth Vo a n d radius 7"o were adjusted to give t h e experimental binding energy of t h e alpha particle in Po. Bertsch a n d Papanbrock calculated t h e photon emission using Fermi's golden rule for t h e transition rate 27T
Wi->f
/
v 2
e
= — (/ —A \ me
0
-pi) I
pf
(6)
250
and made the dipole approximation, Ao = constant. The identity ^2
2m
+ V(r)
,A„-p = -ihAo • W ( r )
(7)
allows the matrix element of the alpha momentum p in eq.(6) to be replaced by a matrix element of the gradient of the potential W ( r ) ; - A o p i) = -ift
dP d£L
2 AZl<=// 1 ffe c 2 2 3m c £ 7
**
dV dr
$,
(9)
Here <&i{r) is the radial wave function of the initial state with / = 0 and &f{r) is the wave function of the final state with 1 = 1. The initial state $>i(r) was chosen to be a Gamow state with an outgoing boundary condition. In the case of the decay of 2 1 0 Po the life time is so long that the wave number ki can be taken to be real. The final state 3>/(r) is a scattering state in the potential V(r) with wave-number kf. Both the initial and final states are combinations of regular and irregular Coulomb wave functions for r > TQ. Bertsch and Papenbrock calculated the radial matrix element (y |dV/dr| <&i) by a combination of analytical and numerical methods and obtained a photon emission spectrum which was in agreement with the data of Kasagi et al. to within the the experimental error bars. The dominant contribution came from the imaginary part of the radial matrix element. This part could be calculated analytically and was insensitive to the sub-barrier part of the wavefunction. The real part of the radial matrix element was calculated numerically and was found to be much smaller. It contributed only a few percent to the final result. With the potential (5) the derivative dV/dr has a delta-function at r = ro- There is a strong cancellation between the contribution of this delta-function to the real part of the radial matrix element of dV/dr and the contribution coming from r > rg. It is this cancellation which makes Re ( $ / \dV/dr\ <&i) < < Im ( $ / \dV/dr\ $;). Takigawa et. al. s repeated the calculation of Papenbrock and Bertsch and studied the role of the tunnelling region in more detail. They performed
251 a quantum mechanical analysis in a way which has a close relation to semiclassical theory. They also studied the effect of different choices of the potential parameters Vo and r^ which gave a different number of nodes in the internal wave function but always the same resonance energy. They found a weak dependence on the internal wave function and stressed that it was essential to make a proper treatment of the delta-function in dV/dr at ro in order to obtain an accurate result. 4
A low energy theorem
Papenbrock and Bertsch calculated the low frequency limit of the radial matrix element and found |<*/ \dV/dr\ *i)| = y/mEa/irh
= - ^ _ V2nh
as
u
_> o.
(10)
in agreement with the classical low frequency limit in eq.(4). This low-frequency behaviour (10)of the photon emission matrix element is very general. It can also be derived from the matrix element on the left hand side of eq.(8). Then it depends only on the long-range behaviour of the initial and final wave functions and is quite independent of the exact form of the a-nucleus potential. We give a derivation starting from the radial matrix element of dV/dr and assuming that the initial state has angular momentum 1 = 0 and the final state has 1 = 1. We also assume, for simplicity, that dV/dr = 0 for r > R. Using the relation (7) and integrating by parts we have fR <*/ \dV/dr\ *<) = {£i - Ef) /
VfXSidr R
(11) 0
where X = d/dr — 1/r. For r > R the initial and final radial wave functions are *i = A0 exp(ik0r + 60),
/
$ / = Ax I cosfar + Sx)
sinlfci7* -I— Ot ) \
-^—
J
(12)
Now we substitute the expressions (12) for <&i and $ / into eq.(ll) and let (ei—Ef) —> 0 and the second term tends to a limit independent of R. Altogether ($/ \dV/dr\ $i> — -A\A0Ea
exp(£0 - 6x)
(13)
With the normalization of Papenbrock and Bertsch AQ = ^Jmjhki and Ax \j2m,/-nW-kf and eq.(13) is consistent with the low energy limit (10).
252 5
Semi-classical theory
T h e r e is a simple semi-classical extension of (1) which is inspired by p a t h integral theory in which the alpha-particle propagates in imaginary t i m e und e r n e a t h t h e Coulomb barrier . T h i s suggests replacing t h e Fourier transform in e q . ( l ) by /•oo
Ji(w) = /
a{t)eiwtdt
(14)
JlT
where ir ( r > 0) is a suitable imaginary t i m e related t o a real starting radius rT under t h e barrier. T h e integral in (14) is taken along a contour C in the complex t-plane running from t=ir t o t = 0 along t h e imaginary f-axis and from t = 0 t o t = oo along t h e real axis. T h e large-w behaviour is improved by this substitution but t h e low frequency relation (4) is changed to
Ii(w) —* UQO +ivo
as
w —* 0,
(15)
where WQ is the imaginary velocity at TQ. This violates the low-energy theorem a n d t h e resulting 7-spectrum is very dependent on t h e choice of rT. Now we discuss a resolution of this difficulty which is related t o the work of Dyakonov. T h e analysis of Papenbrock a n d Bertsch a n d of Takigawa et al have shown t h a t it is important t o treat t h e photon emission under the barrier in a consist e n t way. T h e potential (5) has a discontinuity a t r$ which has been shown t o b e very i m p o r t a n t 5 but is not easy t o t r e a t by semiclassical methods. In order t o avoid this difficulty we replace t h e potential (5) which has a discontinuity a t TQ by a smooth potential where d e p t h inside the barrier is adjusted t o give a b o u n d s t a t e at the experimental energy. W i t h this new potential t h e r e is an internal turning point at ro and an external turning point at re. We suppose t h a t it is smooth enough for t h e semi-classical connection formula t o b e used at b o t h of these turning points. We use eq.(14) with r = TQ corresponding t o t h e internal turning point. T h e alpha particle velocity is zero at b o t h the internal and external turning points so t h a t VQ = 0 in eq.(15) and we recover t h e correct low energy behaviour. Integrating eq.(14) by parts we get /•OO
I1(u)=l
POQ
a{t)eiwtdt
= -viiroe^0
- iu /
v{t)eT*dt
(16)
T h e b o u n d a r y term is zero because t h e alpha-velocity is zero at the internal t u r n i n g point and eq.(16) reduces t o />oo
h (u) = -iwv
(w) = -iu
I •lira
v(t)eiwtdt
(17)
253
In both eqs.(16) and (17) u> should have a small positive imaginary part to ensure that the integrals converge as t —* oo. Eq.(17) is the semi-classical analogue of the quantum mechanical relation between the matrix elements in eq.(8 ). The conclusion of this discussion is that the Fourier transform of the acceleration, I(w), in the formula for the photon spectrum (1) should be replaced by Ii(w) from eq.(17). Now we make a link with the semi-classical calculation of Dyakonov . He replaced I(u>) by A(w) = — iuw(u>) in the expression eq.(l) for the energy spectrum of the bremsstrahlung, then he used a pure Coulomb potential to calculate the Fourier transform i>(u>). He argued that, because of the exponential damping of the integral in eq.(17) under the barrier, the contribution from small values of r is not very important and assumed that the alpha particle starts at ro = 0. The dominant contribution to the bremsstrahlung spectrum comes from the real part of Ii{u>). The imaginary part of Ii(ui) has contributions from outside the barrier and from under the barrier. These have opposite signs and there is a significant cancellation so that the imaginary part accounts for less than 20% of the total photon emission probability for the highest photon energies studied in the experiment of Kasagi et al, and a smaller percentage for lower photon energies. If the sub-barrier contribution is emitted one gets the classical spectrum which is larger than the quantum result at the highest energies by a factor of 50. This is in agreement with the conclusions of Papenbrock and Bertsch and Takigawa et al. 6
Derivation of the semi-classical result
We conclude this discussion by making a connection with the semi-classical analysis in ref.5. The aim is to show how eq.(17) follows from making a semiclassical approximation to the wave functions in the matrix element in eq.(9). Before starting we note that the classical bremsstrahlung formula (1, 2) and the semi-classical formulae (16, 17) assume that the motion of the alpha particle in the final state is in the radial direction so that its orbital angular momentum I is zero. In the quantum calculation the initial state is an s-state and the final state is a p-state with 1 = 1. This angular momentum is small compared with the Sommerfeld parameter r] « 22 and its contribution is not very important. It is included in the semi-classical analysis in ref.5 but it will be neglected here in order to simplify the discussion. With this simplification the radial wave functions of the initial and final states are # r W ~ W - ^ y exp (i £ fe(r)dr) ,
(18)
254
r(r)
Ai
sin
* ~ \f^ (JC
kf{r)dr+
(19)
f)'
Here ki(r) and fc/(r) are the local wave numbers of the initial and final states corresponding to energies Ei and Ef and &o and &j are the corresponding asymptotic wave numbers. The product of the semi-classical wave functions in the matrix element of dV/dr is the sum of two terms. One with a rapidly oscillating phase will be neglected. The other term with a slowly varying phase gives a contribution to the matrix element from the external region
dF
/* ($/ \
L\
rdv
dr
MP, ) ~ / — dr\ 7 e x t Jrs dr hy/kfirMr)
f
/.fuu
• ru(\J\
= exp i I ki(r)dr — i I \ Jrei * ' Jr.,
kf(r)dr ' J
f{
This is just the amplitude (2) for bremsstrahlung emission in the classical region outside the barrier. In making the step from eq.(20) to eq.(21) we have assumed that AE =ftu>is small enough to be able to neglect the difference between the external turning points in the initial and final states. In the preexponential term fe/ P=S ki wfe,while in the exponent ki(r) — kf(r) w AE/Kv(r) where v(r) is the particle velocity outside the barrier. The semi-classical sub-barrier contribution can be evaluated in a similar way. The initial and final wave functions under the barrier are
*r(r) ~ vw)exp (r7i(r)dr)' 1 exp ( - J " &}c(r) ~ - 7 =4== Vlf(r)
7/
(r)dr ) ,
{22) (23)
and the sub-barrier contribution to the transition matrix element is $
dV dr
*L~r™-'(-f^*)-jC'^
Combining eqs.(21) and (24) leads to the ampltude I\ in eq.(16, 17). 7
(24)
Conclusions
In this talk I have made a connection between the quantum mechanical calculations of Papenbrock and Bertsch and Takigawa et al; and the semi-classical theory of Dyakonov. In quantal calculation there is a strong cancellation between
255 the sub-barrier and external contributions t o t h e real p a r t of t h e transition matrix element and t h e dominant contribution to photon emission amplitude comes from t h e imaginary part of t h e matrix element. There is a similar effect in t h e semi-classical theory a n d t h e dominant contribution comes from t h e real part of the Fourier transform (16). This is just the real p a r t of t h e classical amplitude (2), which agrees with t h e discussion in Papenbrock a n d Bertsch. There are several points which have not been addressed in t h e above analysis. One is t o study t h e effect of t h e orbital angular m o m e n t u m (I = 1) in t h e final state. Another is t o t r y t o extend t h e semi-classical argument t o include the case of a discontinuous potential like the one used by Papenbrock a n d Bertsch. One can use t h e exact wave functions in t h e internal region but there are ambiguities in t h e t h e matching relations with t h e semi-classical wave functions at r = ro- A third is t o study t h e contribution of t h e internal region 0 < r < n>. A fourth point concerns t h e choice of t h e initial state. I have followed Papenbrock a n d Bertsch and used a Gamow s t a t e . Bertulani et a l 1 have argued t h a t t h e correct choice is a non-stationary s t a t e confined t o t h e internal region of t h e alpha-nucleus potential and t h a t a time-dependent description should be used t o calculate t h e bremsstrahlung emission. We conclude this discussion with a remark about t h e external t u r n i n g point in eq.(20) a n d also in t h e derivation of eq.(24). In fact t h e r e a r e two different turning points r e j a n d ref for t h e initial and final s t a t e s 5 . T h e y are approximated by an average value r e in going from eq.(20) t o (21). There is a similar approximation in t h e derivation of eq.(24). T h e W K B wave functions are not accurate approximations t o t h e t r u e wave functions near t h e turning point. However this difficulty can be avoided by using an analytical continuation argument t o bypass the t u r n i n g point in t h e complex plane. 8
Acknowledgments
I would like t o t h a n k G.F. Bertsch, K. Hagino, T. Papenbrock and V.G. Zelevinsky for interesting a n d helpful discussions relating t o t h e problem of bremsstrahlung emission in alpha decay. 1. C.A. Bertulani, D . T . de P a u l a a n d V.G. Zelevinsky: Phys.Rev. C 6 0 , 031602 (1999) 2. M.I. Dyakonov: Phys. Rev. C 6 0 , 037602-1 (1999) 3. J. Kasagi, H. Yamazaki, N. Kasajima, T. Ohtsuki a n d H. Yuki: Phys. Rev. Lett. 7 9 , 371 (1997) 4. T. Papenbrock a n d G.F. Bertsch : P h y s . Rev. Lett. 8 0 , 4141 (1998) 5. N. Takigawa, Y.Nozawa, K. Hagino, A. O n o and D.M. Brink: Phys. Rev. C 59, R593 (1999)
256
A MODEL FOR PARTICLE INDUCED FISSION F. BARY MALIK Physics Department, Southern Illinois University, Carbondale, IL 62901 U.S.A. The decay widths in the particle induced fission are calculated with the barrier used to explain spontaneous and isomer fissions and the corresponding mass and charge distributions. The observed change in mass distributions with increasing incident energy of incident neutron and alpha particle is well accounted for. In addition, the theoretical calculations of TKE can reproduce the observed magnitudes and shapes, thus confirming the need for a barrier between the saddle and the scission points.
1
Introduction
Since late 1950's and early 1960's, it has been well known that percentage yields curves in particle induced fission change dramatically with the increase of incident energy of a particle like neutrons and alpha-particles. A typical example [1] is shown in Fig. 1, where the percentage yields curves have been plotted for thermal and 14 MeV neutron induced fission of U as a function of mass number, A of a daughter nucleus. The yields to symmetric modes increase by almost two orders of magnitude for the 14 MeV incident neutrons compared to that for thermal neutron. A similar situation occurs for other incident light projectiles, e.g. alpha-particles [2].
257
701
80
90
100 110 120 MASS NUMBER
130
140
150
Figure 1. Observed percentage fission yields as a function of mass number for thermal and 22 MeV neutron incident on 235U [l\.
In mid-1960's, studies of total kinetic energy, TKE along with the percentage yields curves as a function of mass, A, of daughter nuclei [3-5] in the neutron induced fission of Pu and U reveal that, whereas, the change in percentage yields curves near symmetric decay modes is substantial with increasing neutron energies, the change in the magnitude of TKE is not very dramatic. A careful analysis of the neutron-induced fission data of 5U led Facchini and Saetta-Menichela [6] to conclude that the observed phenomenon is indicative of a barrier between the scission and the saddle points in the potential energy barrier, similar to the one proposed by Block and Malik [8] and others [9,10]. Since then extensive calculations of fission barrier using energy-density functional theory [11,12] have supported the existence of such a barrier. Inclusion of such a barrier in theoretical treatment of fission dynamics has been successful in describing (a) spontaneous and isomer fission half-lives [11] (b) the appropriate mass and charge distributions in spontaneous and thermal neutron induced fission using observed TKE spectrum [12-14]. In particular, the TKE for isomer fission was predicted [12, 13] and later confirmed [15]. Compani-Tabrizi and his collaborators'
258
investigations [16-19] have, since then, confirmed the importance of such a barrier in explaining the observed change in percentage yields curves and TKE with the increase in the incident projectile kinetic energy in induced fission. In this paper, we present a short synopsis of these studies. The next section presents the outline of the theory followed by a subsequent section presenting the results. 2
The Theory
2.1 Decay Width The theory used in calculating the percentage yields is discussed in the review article of Ericson [19]. The basic concept of the theory is that the yields to a particular daughter pair of a certain excitation are the formation probability of a composite system comprising of the projectile and the target and subsequent decay of this system to a particular daughter pair, (AiZj) and (A2 Zj) where A; (i=l,2) and Zj (i=l,2) are, respectively, the mass and atomic numbers. The computation of percentage yields involves taking a ratio and hence, the formation probabilities at a given incident energy cancels out and one is to calculate the decay width, T(Ai, A2, IUE) from a composite system of mass and atomic numbers, A and Z, spin I and excitation energy U to a daughter pair. The expression for decay width is, then, given by E-E
r(A,A2, IUE)=T(E) jp^U^C-E-U^dU,
(1)
where T(E) is the transmission function through an appropriate barrier in the exit channel involving interaction between two members of a daughter pair, with kinetic energy E ®i and ®2 are the level density functions of each member of a daughter pair and Ui and (C-E-Ui) are their respective excitation energies. G is the maximum available energy given by the energy conservation. € = M(A -X)C2+ M(X)C2 +EX- M{A)c2 - M(A,)c2 (2) where c is the velocity of light in vacuum, M is the mass of the particle noted in its argument and Ex is the kinetic energy of the projectile of mass number x.
259
A\, Ai and A are mass numbers of a daughter pair and the composite system, respectively. Thus, the maximum available energy € is given by C=Q+Ex+S(x) =E+U,+U2 (3) In (3), Q, and S(x) are, respectively, the Q-value of the process and the separation energy of the particle with mass number x from the composite system. E, Uh and [/? are, respectively, the kinetic energy and excitation energies of the daughter pair. The expression (1) is distinct from the theory used by Fong [19] who neglects the final state interaction between the daughter pair and as such, sets T(E)=1. In our calculation T(E) is calculated using the same barrier between the daughter pair, as the one used to compute the spontaneous and isomer fission half-lives, and the corresponding charge and mass distributions [11, 12, 14, 18]. The transmission coefficient, T(E) in (1) is calculated using the WKB approximation. In addition, the auxiliary condition that the change in entropy should be greater than or equal to zero, has been imposed in evaluating the integral in (1). The level density expression of Gadiola and Zetta with a=0.127 (MeV1) has been used. 2.2 Calculation of TKE In principle, TKE, the most probable kinetic energy, can be determined from (1). It is the kinetic energy E for which the decay width exhibits the maximum. The extracted experimental information from the thermal neutron induced fission, shown in the left insert of Fig. 2, exhibits that the observed decay width as a function of the kinetic energy of a daughter pair, has a Lorentzian shape [6]. The location of the maximum of this curve is the observed decay width for this particular decay mode.
260
10
i-
Thermal n <• U
io
.... -j.-WH.Misi
ISO
1 |6o EKIN(CM)
1 170 M.V
LLL2
,_ |eo
.
Figure 2. The left insert exhibits extracted yields in the thermal neutron induced fission of 23!U to the daughter pair (96 + 146) as a function of their kinetic energy [6]. The right insert presents theoretical calculations of yields as a function of the kinetic energies of three daughter pairs in the thermal neutron induced Fission of 235 U.
3
Results and Discussion
Calculated percentage yields [16,17] shown in solid dots have been plotted as a function of the mass of the number of the heavy member of a daughter pair in Fig. 3 for the case of thermal and 22 MeV neutrons incident on 2 5U. TKE used in the calculations are shown as solid dots in the upper inserts, along with the observed percentage yields, experimental TKE and respective error bars [3]. The theoretical percentage yields near symmetric modes increase substantially for 22 MeV incident neutrons from the ones with thermal neutrons, thus accounting for the observed phenomena. The TKE's used for the calculations are very close to the observed values, particularly taking into account that the measured error bars have a 2 to 5 MeV uncertainties. In the
261
left insert of Fig. 3 the triangles are the Q-values used in the calculation and the broken lines are those obtained from Myers and Swiatecki's mass formula [20].
Figure 3. Observed percentage yields [3-5| as a function of the heavy member of daughter pairs and the corresponding measured TKE, shown as solid lines are compared to the respective calculations shown as dots. Open circles are theoretically calculated TKE Triangles and broken line in the left insert are, respectively, the Q-values used in the calculation and those obtained from the mass formula [20). Typical error bars in the data are shown in the right insert.
Solid lines in lower and upper inserts of Fig. 4 represent the observed percentage mass yields [21] and TKE spectra in 30.8 and 30.7 MeV alpha induced fission of 226Ra. Calculated yields and the TKE used are shown by solid dots [16] in the same figure. Thus, the theory can reasonably account for the change in yields with the increase in the incident energy of projectiles, along with the observed fact that the TKE as a function of mass number does not change substantially in the process. The additional energy associated with the increase in projectile energy primarily goes into the excitation of the daughter pair.
262
30.8(MeV)<E + 226 Ra
120
130
140
ISO
so
160
M
38.7 (MeV) o c + a s R a
-170
2 160 ~c 150 UJ | 4 0 120
120
140
i;o
10
< ,0*
BUI/2)
Figure 4 Observed percentage yields and T K E [2], shown in solid lines arc compared to the respective calculations, noted as solid dots in the 30.8 and 38.7 M e V alpha induced fission of 226 Ra. Open circles are calculated T K E .
The TKE for the decay to a particularly decay mode may also be
263
calculated by plotting F(AiA2, IUE) as a function of kinetic energy E and then determining the location of its maximum. In the right insert of Fig. 2, such calculations for the thermal neutron induced fission of 235U are presented for three daughter pairs. For each pair the yields from all isotopes for the given mass number has been summed over. Clearly, the observed shape is reproduced, although the calculated widths are somewhat narrower than the observed ones. From the location of the maximum of the calculated yields, TKE is determined to be 174 MeV, which compares favorably with the observed value of 172 MeV for the (140 + 96) pair. These calculated TKE's have been plotted in Fig. 3 and 4 as open circles and are very close to the observed TKE.
264 n w "• u
Figure 5. Upper and lower inserts are, respective, calculated yields of three sets of daughter pairs in the fission of USV by 22 MeV neutron using a barrier between the saddle and scission points between the daughter pair and without such a barrier between them.
In the absence of a barrier between the saddle and the scission point, T(E)=1, the spectrum of yields is that of evaporation, i.e. exponential decay having a maximum at zero kinetic energy. This is, indeed, the case for all cases. A typical example for T(E)=1 is shown in the lower insert of Fig. 5 for 22 MeV neutron induced fission of 235U to three daughter pairs. In the presence of an appropriate interaction between two members of a daughter
265
pair, the structure of the percentage yields as a function of the kinetic energy, E, changes dramatically as shown in the upper insert of Fig. 5. Thus, the very fact that the observed kinetic energy for each decay mode is non-zero and substantially well above 100 MeV is the manifestation of the existence of a barrier between the saddle and scission points. Further credence to this point is given from the observed shape of the decay width, an example of which is shown in the right insert of Fig. 2. The observed shapes are similar to the calculated ones, shown in the left insert. References 1. Block, B. and Malik, F. B., Phys. Rev. Lett. 19 (1967) pp. 239. 2. Block, B., Clark, J. W„ High, M. D., Malmin, R. and Malik, F. B., Ann. Phys. (NY) 62 (1971) pp. 464. 3. Compani-Tabrizi, B., Ph.D. Dissertation, Indiana University at Bloomington Unpublished (1976). 4. D'yachenko, P. P., Kuz'minov, B. D. and Tarasko, M. Z., Sov. J. Nucl. Phys. 8 (1969) pp. 165. 5. D'yachenko, P. P. and Kuz'minov, B. D., Sov. J. Nucl. Phys. 7 (1968) pp. 27. 6. Facchini, U. and Saetta-Menichelah, E., Acta Phys. Polonica A 38 (1970) pp. 537. 7. Fong, P., Statistical Theory of Nuclear Fission (Gordon and Breach, New York 1969). 8. Fontela, C. A. and Fontela, D. P., Phys. Rev. Lett. 44 (1980) pp. 1200. 9. High, M., Block, B., Clark, J. W. and Malik, F. B., Bull. Am. Phys. Soc. 15 (1970) pp. 646. 10. Hooshyar, M. A. and Malik, F. B., Helv. Phys. Acta 45 (1972) pp. 567. 11. Hooshyar, M. A. and Malik, F. B., Phys. Lett. 38B (1972) pp. 495. 12. Hooshyar, M. A. and Malik, F. B., Helv. Phys. Acta 46 (1973) pp. 720, 724. 13. Katcoff, S., AWeow'csl8(1960)pp. 201. 14. Malik, F. B. and Hooshyar, M. A. and Compani-Tabrizi, B., Proc. Int'l Conf. On Interaction of Neutrons with Nuclei, ed. E. Sheldon (Energy Research and Development Office No. 1976) pp. 725. 15. Malik, F. B., Hooshyar, M. A. and Compani-Tabrizi, B., Proc. V. Int'l Conf. On Nucl. Reaction Mechanism, ed. Gadioli, E. (The University of Milan Press) (1988) pp. 310 and 50 years with Nuclear Fission Eds. Behrens, J. W. and Carson, A. D., (Am. Nuclear Society Publication) (1989) pp. 650. 16. Myers, W. D. and Swiaticki, W. J., Nucl. Phys. 81 (1966) pp. 1. 17. Reichstein, I. and Malik, F. B., Bull. Am. Phys. Soc. 16 (1970) pp. 516. 18. Reichstein, I. and Malik, F. B., Ann. Phys. (N. Y.) 98 (1975) pp. 322. 19. Roginaki, T. C , Davies, M. E. and Cobble, J. W., Phys. Rev. C 4 (1961) pp. 1361. 20. Vorobeva, V. C , D'yachenko, P. P., Kuz'minov, B. D. and Tarasko, M.Z., Sov. J. Nucl. Phys. 4 (1967) pp. 234.
266 ON THE HALF-LIVES OF TRINUCLEAR
MOLECULES
A. S i N D U L E S C U 1 - 2 , F . C A R S T O I U 1 - 2 , I. B U L B O A C i , W . G R E I N E R 2 1
) National ) Institut
Institute
of Nuclear Physics and Engineering, P.O. Box MG-6, 76900 Bucharest-Magurele, Romania fur Theoretische Physik der J. W. Goethe Universitdt, D-60054, Frankfurt am Main, Germany
Recent discoveries of 1 0 Be and C accompanied cold fission in the spontaneous fission of 2 s 2 C f lead to the surprising result t h a t long living trinuclear molecules may exists. For the description of t h e dynamics and decay of such molecules, we used a coplanar three body cluster model (two deformed fragments and an or particle) with a three body potential computed by a double folding potential generated by M3Y effective interaction. A repulsive compression term was included. The computed a ternary cold fission yields are in agreement with the experiment. The energy and angular distributions of the three clusters at infinity and the half lives are strongly dependent of the initial positions of the a particle relative to the two fragments and of mass asymmetry of the fragments. The evaluated lifetimes of such trinuclear molecules are quite large of the order of one second.
1
Introduction
Cluster radioactivity 1, cold binary fission 2 | 3 , cold ternary fission 4>5'6, and cold fusion 7 ' 8 are well established phenomena. These processes were predicted based on the idea that cold rearrangement of large groups of nucleon is possible 7 . Using multiple 7-coincidence techniques, the correlations between the two heavy fragments and the light particle were observed unambiguously. In the case of 10 Be cold ternary spontaneous fission of 252 Cf only the triple 7-coincidence was used. The 7-ray corresponding to the decay of the first 2 + state in 10 Be was detected in coincidence with the 7-rays of the fission partners 96 Sr and 146 Ba. The surprising result is that the 7 transition in 10 Be is not Doppler broadened as one would expect if the system separates immediately into three clusters and 10 Be decays in flight. The only possible interpretation is the existence of a long living nuclear molecule where the three nuclei stick together with a half-life larger than 1 0 - 1 2 sec. 5 . In this paper we describe a dynamical three cluster model appropriate for the evaluation of a cold ternary fission process at a quantitative level. Initial configurations, dynamical trajectories in the classically forbidden region, penetrabilities, trajectories of the fragments in the asymptotic region, angular and energy distributions of the decaying fragments are deduced from a three body interaction potential. We obtain a set of coupled differential equations
267
which satisfy the minimum action principle, appropriate for the description of the trajectories under the barrier. From the calculated penetrabilities and the barrier assault frequency we estimate the lifetimes for the giant trinuclear molecules. 2
T h e reaction m o d e l
About 90% of the light particles emitted in the ternary spontaneous fission are a particles. That prompted us to study the cold (neutronless) a ternary fission of 252Cf. A coplanar three cluster model consisting of two deformed fragments and a spherical a particle was considered. Three main coordinates were retained for the description of the dynamics during the rearrangement and penetration through the barrier processes. These are (xa,ya), the coordinates of the c m . of the a particle, and a collective coordinate R describing the separation distance between the heavy fragments along the fission axis. Center of mass correlations are fully taken into account in all calculations. We assume that the fission axis is conserved during the penetration process. No preformation factors were included in our description. Realistic deformations for the description of the fragment shapes were included. The three body potential was computed with the help of a double folding potential generated by M3Y- NN effective interaction 13 . Due to the lack of any explicit density dependence in the M3Y effective interaction, which leads to unphysically deep potentials for large density overlaps in the following we introduce a compression term in order to correct the short range dependence of the calculated interaction potentials which read: V(R) = / dfidf2pi{fi)p2(f2)v(\s\)
+ Vcompo / df1df2pi{fi)p2{f2)6{s),
(1)
where i>(|s|) is the effective interaction (depending on the separation distance of the interacting nucleons s = f\ +R — rjj) which includes the usual pseudo-5 knock-on exchange term 14 , Pi(fi), i = 1,2 are the ground state one body densities taken as deformed two parameter Fermi distributions in the intrinsic frames, Pi(fi) i = 1,2 are the corresponding sharp fragment densities with Rsharp given by the equation (r 2 ) = 3/5-R 2 harp . The strength of the compression term is largely uncertain. Throughout the paper we have used a value Vcompo = 300 MeV, close to the value used by Uegaki 16 for molecular resonances in medium-weight nuclei. The Coulomb component was calculated also by double folding using appropriate (deformed) charge densities. The three body potential assumed to be the sum of all two body components displays a quasi-molecular pattern with two minima in the equatorial
268
Figure 1. The two body components of the interaction potential calculated with a compression strength 14 o m p o = 300 MeV are displayed in the upper panel. The adiabatic (continuous line) and dynamic (dashed line) barrier (see text for details) are given in the middle panel. The corresponding adiabatic (continuous line) and dynamic (dashed line) a particle trajectories are given in the bottom panel. The fragmentation channel is indicated on the figure.
region and two polar minima. The minima in the equatorial region are equivalent due to axial symmetry. The potentials for nose to nose configurations in each two body channel are displayed in Fig. 1 (upper panel). The minima in the a particle components are situated at 7 and 8.5 fm with respect to the light and heavy fragment respectively, which means that quasi-stable configurations are formed in the three body system with the a particle situated at rather large distances. We assume first that the movement of the three particles is so slow that the system adjusts adiabatically to stay in a configuration corresponding to the minimum in the three body potential. The corresponding barrier and the a particle trajectory are displayed in Fig. 1 (middle and bottom panels) with continuous lines. Also the total available reaction energy is indicated (Qz). We distinguish three regions: the minimum where the system oscillates corresponding to a cold rearrangement process, the classically forbidden region (the barrier) and the asymptotic region (beyond the outer turning point) where the system decays into three fragments. In the minimum region we solved the classical equations of motion thus obtaining the oscillation time Tosc, the corresponding a particle trajectory and the inner turning point. The oscillation time is of the order of 0.6 x 1 0 - 2 1 sec. and the barrier assault frequency of the order v ~ 1.7 x 10 21 sec _1 . The zero point energy is estimated
269
from the uncertainty principle giving EQ ~ 0.9 MeV for highly deformed fragments and slightly larger for spherical splittings (~ 1.2 MeV). In the classically forbidden region we solve the semiclassical trajectory equations for the relevant degrees of freedom 12 . For completeness we write them here in a simpler but equivalent form: M
*
(&.TL _ j_ an
2(Q-V) \ mi dx\
(2)
mi dxi) '
with M = mi + Y^mj{x'j)2 the effective mass (or the effective inertia) and x' = •££-. mi are the masses for the corresponding degrees of freedom, Q is the reaction energy. The coupled differential equations (2) describe the motion of the system in the barrier region and satisfy the minimum action principle. It can be shown that the set of equations (2) is fully equivalent with the classical equations of motion in imaginary time r = it. The reduced action is given by
i _ / N 2 ( V - « ( . » l + X>,(£)
(3)
2 Sp
and the penetrability factor is simply P = e » . A close examination of Eq. (2) shows that the semiclassical approximation is not valid on the surface Q = V. In order to have a solution we must require that the factor in parenthesis in Eq. (2) vanishes on that surface. This gives us the so called transversality conditions or the steepest descent equations 17 : mjx'j _ mi_ dxi
dxi
which are solved near the turning points. 3
Results
An example of dynamical barrier and a particle trajectory is given in Fig.l ( middle and bottom panels) with dashed lines. The dynamical trajectory should be contrasted with the adiabatic (static) one. In the dynamical case the a particle gets away from the interfragment axis with large ya values at the outer turning point. Having solved the dynamical equations of motion, using the reaction energies (Qs) calculated from the experimental masses or in few cases taken from the theoretical predictions of Moller and Nix 18 we evaluate the penetrabilities as described above. The relative production yields are evaluated from
270
the simple formula YL(AL,ZL)
=
PL{AL,ZL)
(5)
Evidently we assume that the preformation factors are the same for all cold splittings and consequently in the expression (5) this factor cancels. The calculated yields ( normalized to the experimental ones 4 are displayed in Fig. 2.
•:;,il,i,l,:,:,hls,l:«°':
,.,., . . i . l l l l . l .
i •
ll
Figure 2. Comparison of experimental yields [4] (upper panel) with t h e dynamic (middle) and adiabatic (bottom) results. The fragmentation channel is indicated by t h e charge number (on the x-axis) and by the mass number ( on top of t h e stacks) of the light fragment.
The mass distribution is rather flat, showing nonvanishing and comparable mass yields for a large range of mass asymmetry in agreement with the experiment. Another remark concerns the energy and the scattering angle of the a particle. The energy is larger for large asymmetry mass splittings and the scattering angle is smaller. The average values, taking the calculated dynamical yields as weighting factors are: < E^L >=111.0±1.3 MeV,< EkH >=80.6±4.5 MeV, < Eka >=19.3±0.5 MeV and < 6a > = 82.5±1.9°. We shall show below that the variances become much larger if quantum fluctuations are taken into account. According to Heeg 19 , the mean experimental angle, measured with respect to the fission axis oriented towards the light fragment (opposite to our convention, see bellow) is 83°. The adiabatic scenario gives ternary yields only for splittings with one of the fragment spherical, as we have shown in our previous paper 12 . The corresponding yields are evidently dominated by Q-value effects. Only high Q value channels contribute to the total yield. In the dynamical calculation there is a delicate
271
interplay between deformation, Q-value and penetration path effects. Note that the theoretical yields are calculated strictly at zero excitation energy and we neglected the level density close to the ground state of the fragments and as a result the odd-even effects, noticeable in the experiment around the masses AL=101,103,107 are not well reproduced. We evaluate the lifetime for the decaying molecule using the relation 7\/2 = \n2/vP, where v is the barrier assault frequency and P the penetrability. Next we consider fluctuations in the entrance point since the calculated action along the penetration path should be minimized not only with respect to the path , but also with respect to the initial conditions. If we neglect the a particle coordinates , then the lowest quasimolecular state of the system in the potential minimum is of the form \I> ~ exp(—\a{R — Ro)2). From ft2
the estimated energy for the lowest state we have EQ = ^ ^ and the estimated value for the distribution width is ^4JJ = 0.7 fm. Therefore we started a Monte Carlo evaluation of the trajectories using as inner turning point Ri = Rn + a x gauss(seed) where i t t l is the turning point given by the minimization procedure, gauss(seed) is a random gaussian generator with zero mean value and variance 1. a is the desired variance of the generated distribution, taken here of the order of 0.1 fm, a rather conservative value. For all generated Ri values we search for the solutions of the equation for energy conservation V(Ri,xa(Ri),ya(Ri)) — Q 3 at the entrance point. In practice, we relaxed this condition, searching for solutions (xai, yai) in a band Q3 ± 50 KeV. No initial velocity distributions were considered in this simulation, since in our particular system of coordinates it is difficult to conserve the total linear momentum. In this way a number of some 8000 initial positions were generated. The calculation was completed by solving the dynamical equations in the barrier up to the outer turning point. From then on the system disintegrates. The final energy and angular distribution of the fragments were obtained by solving the classical equation of motion for a three body system. Since the exit point locates at rather large distances (Rf ~ 18 fm) and yaf ~ 8/m, the nuclear forces are completely negligible, and we used only the monopole-monopole part of the Coulomb interaction. We have shown 20 that quadrupole-monopole and other higher multipolarity terms have little influence on the final distributions. The results of the simulations are displayed in Figs. 3 and 4 for the splitting 252 Cf-» a + 9 2 K r + 1 5 6 N d (77 = 0.26). We have also considered the more symmetric splitting 252Cf—» a + 1 1 6 P d + 1 3 2 S n (rj = 0.06) shown by shadowed histograms in Figs. 3 and 4. In left panels we have selected all events in the band 5 < yai < 6 fm and
272
Figure 3. Initial and final distributions for dynamic variables obtained in a Monte Carlo simulation. Left panels include selected events in the band 5 < yai < 6 fm; right panels are for events in t h e band 4 < yai < 5 fm. The last two rows are for the distribution of the decimal logarithm of t h e penetrability (P) and for the half lives (T]y 2 ). Filled histograms are for t h e splitting 2 5 2 Cf—» 4 He+ 1 1 6 Pd+ 1 3 2 Sn. The other histograms are for the splitting 252Cf^4He+92Kr+156Nd
Figure 4. Energy distributions for t h e decaying light (L), heavy (H) and alpha (a) particle. Angular distribution (9a) is given in the last row. See caption to the Fig. 3.
4 < yai < 5 / m in the right panels. Much lower initial values yai correspond to an a particle in overlap with the fragments and lead to trajectories traversing one of the heavy fragments and with penetrabilities of the order 1 0 - 1 0 0 or smaller and they are disregarded. The higher the initial position of the a particle, the wider the distribution in the final yQf coordinate. Also the energy distribution of the light fragment (JE^L) is wider. The mean kinetic energy for the a particle is 21(18) MeV and the mean scattering angle is —78° (-88°) for large (small) 77. The angle is negative since we have chosen a system of coordinates with the light fragment placed on the left side, and the a particle is deviated strongly by the heavy fragment, so the scattering angle is always in the second quadrant (0a > 7r/2). For initial positions in the lower band (right panels in Fig. 3 and 4) all distributions are much narrower, showing a strong focussing effect. The final kinetic energy Eka lowers to 19(17.5) MeV and the scattering angle is —79° (—88°). In the upper position band the penetrabilities are small (< 10 - 2 2 ) and the lifetime is of the order of tens of seconds. In the lower band the penetrabilities increase (10 - 2 1 ) and the lifetime varies between 1 and 10 seconds. As we must minimize the reduced action, one should choose
273
the trajectory with maximum of penetrability. Therefore our model predicts lifetimes for the trinuclear molecule of the order of 1 second. Of course, all distributions will be much larger if averaged over all fragmentation channels. Note the strong dependence of all distributions on the mass asymmetry of the heavy fragments (and implicitly on Q value and deformation) (Figs. 3 and 4). This is especially evident for exit configuration (Rf,yaf) and for the kinematic variables of the fragments(jEjfca, Eku, &u)4
Conclusions
In conclusion we designed a coplanar three cluster model which describes reasonably well the cold ternary spontaneous fission process. The complicated penetration process in a multidimensional barrier was described dynamically by solving a set of semiclassical coupled differential equations which satisfies the minimum principle action. Minimum action with respect to the initial conditions was found in an extensive Monte Carlo calculation. The asymptotic distributions of kinematical variables were obtained by solving the classical equations of motion for the three fragments. The three body interaction potential was generated from a G-matrix density independent effective interaction in a double folding procedure. The short range behaviour of the interacting potential is greatly improved by introducing a compression term which suppresses large density overlaps. The obtained three body potential shows a typical quasimolecular pattern and relatively stable configurations are predicted with the a particle laying at rather large distances from the heavy fragments. The lifetime predicted by the model is of the order of 1 second. The model could be tested experimentally since the final energy and angular distributions of the decaying fragments are strongly correlated with the initial configurations prior to the penetration process and with the mass asymmetry of the fragments. References 1. A. Sandulescu and W. Greiner, Rep. Progr. Phys. 55, 1423 (1992). 2. A. Sandulescu, A. Florescu, F. Carstoiu, W. Greiner, J. H. Hamilton, A. V. Ramayya and B. R. S. Babu, Phys. Rev. C 54, 258 (1996). 3. A. Sandulescu, S. Misicu, F. Carstoiu, A. Florescu, and W. Greiner, Phys. Rev. C 57, 2321 (1998). 4. A. V. Ramayya et al., Phys. Rev. C 57, 2370 (1998). 5. A. V. Ramayya et al., Phys.Rev. Lett. 81, 947 (1998).
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6. A. Sandulescu, F. Carstoiu, S. Misicu, A. Florescu, A. V. Ramayya, J. H. Hamilton and W. Greiner, J. Phys. G 24, 181 (1998). 7. A. Sandulescu and W. Greiner, J. Phys. G 3, L189 (1977). 8. P. Ambruster, Rep. Prog. Phys. 62, 465 (1999) 9. J. K. Hwang, A. V. Ramayya, J. H. Hamilton and G A N D S 9 5 Collaboration, to be published 10. F. Ajzenberg-Selove, Nucl. Phys. A490,l(1988) 11. B. Burggraf, K. Farzin, J. Grabis, Th. Last, E. Manthey, H. P. Trautvetter and C. Rolfs, J. Phys.G25,L71(1999) 12. A. Sandulescu, F. Carstoiu, I. Bulboaca and W. Greiner, to appear in Phys. Rev. C 60. 13. G. Bertsch, J. Borysowicz, H. McManus and W. G. Love, Nucl. Phys. A284,399(1977) 14. G. R. Satchler and W. G. Love, Phys. Rep.55,183(1979) 15. F. Carstoiu and R. J. Lombard, Ann. Phys.(N.Y.) 217, 279 (1992). 16. E. Uegaki, Prog. Theor. Phys. Suppl.132, 135(1998) 17. M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky, and C. Y. Wong, Rev. Mod. Phys. 44,320(1972) 18. P. Moller, J. R. Nix, W. D. Myers and W. J. Swiatecki, At. Data Nucl. Data Tables 59, 185 (1995). 19. P. Heeg, Thesis, T. H. Darmstadt (1990) (unpublished) 20. S. Misicu, A. Sandulescu, F. Carstoiu, M. Rizea and W. Greiner, II Nuovo Cimento A112,300 (1999)
275
FUSION-FISSION REACTIONS OF HEAVIEST NUCLEI. SYNTHESIS OF SUPERHEAVY ELEMENTS WITH Z=114 AND 116 YU.TS. OGANESSIAN, M.G. ITKIS, V.K. UTYONKOV Joint Institute for Nuclear Research, 141980 Dubna, Russian E-mail: [email protected]. ru
Federation
The process of fusion-fission of superheavy nuclei with Z= 102-122 formed in the reactions with 22Ne, 26Mg, 48Ca, 3, Fe and 86Kr ions at energies near and below the Coulomb barrier has been studied. The experiments were carried out at the U-400 accelerator of the Flerov Laboratory of Nuclear Reactions (JINR) using a time-of-flight spectrometer of fission fragments CORSET and a neutron multi-detector DEMON. As a result of the experiments, mass and energy distributions of fission fragments, fission and quasi-fission cross sections, multiplicities of neutrons and gamma-rays and their dependence on the mechanism of formation and decay of compound superheavy systems have been studied. The paper presents results of the experiments aimed at producing long-lived superheavy elements located near the spherical shell closures with Z > 114 and yV > 172. For the synthesis of superheavy nuclei, we used a combination of neutron-rich reaction partners, with a 244Pu target and a 48Ca projectile. The sensitivity of the present experiment exceeded by more than two orders of magnitude previous attempts to synthesize superheavy nuclides in reactions of 48Ca projectiles with actinide targets. We observed new decay sequences of genetically linked adecays terminated by spontaneous fission. The large measured a-particle energies, together with the long decay times and spontaneous fission terminating the chains, offer evidence of the decay of nuclei with high atomic numbers. The decay properties of the synthesized nuclei are consistent with the consecutive a-decays originating from the parent nuclides 288,289 114, produced in the 3n and 4n-evaporation channels with cross sections of about a picobarn. The present observations can be considered an experimental evidence of the existence of the "island of stability" of superheavy elements and are discussed in terms of modem theoretical approaches.
1
Introduction
The stability of heavy nuclei is largely determined by nuclear shell structure whose influence is considerably increased near closed proton Z and neutron N shells. Beyond the domain of the heaviest known nuclei the macroscopic-microscopic nuclear theory predicts the existence of an "island of stability" of long-lived superheavy elements. Calculations performed over more than 30 years with different versions of the nuclear shell model predict a substantial enhancement of the stability of heavy nuclei when approaching the closed spherical shells Z=114 and 7V=184. Neutron and proton shell closures are expected to occur there, resulting in formation of spherical superheavy nuclei, next to 208Pb. However, more generally, enhancement of nuclear binding energy can be observed also in deformed nuclei, in particular, in the theoretically predicted intermediate region of increased nuclear stability in the vicinity of the deformed
276
shell closures Z=108 and 7V=162 (see, e.g., reviews [1-3]). These predictions were corroborated by the experimental observation of a new region of nuclear stability near Z=108 and JV=162 [4] and synthesis of the heaviest elements up to Z=l 12 [4-6]. These results gave more credibility to the predicted existence of spherical superheavy elements, thus opening prospects for the production of the heaviest nuclei and the study of their physical and chemical properties [7]. Superheavy nuclei close to the predicted magic neutron shell N=1S4 can be synthesized in complete fusion reactions of target and projectile nuclei with significant neutron excess. The most neutron rich isotopes of element 114 with neutron numbers 174, 175 and, consequently, relatively stable, are expected to be produced in the fusion reaction of 244 Puwith 48 Caions[8]. In this reaction at the 48Ca energy close to the Coulomb barrier the 292114 compound nuclei could be expected to deexcite by emission of 3 or 4 neutrons. According to the macroscopic-microscopic calculations by Smolanczuk et al., who reproduce adequately radioactive properties (a-decay and spontaneous fission) of the known heavy nuclei [1,9], the even-even isotopes 288114 and 290114 are expected to have partial a-decay half-lives r a =0.14 s and 0.7 s, respectively. Their predicted spontaneous fission (SF) half-lives are considerably longer: r SF =2xl0 3 s and 4xl0 5 s, respectively. For their daughter nuclei - isotopes of element 112 - the main decay mode should still be a-decay, although differences between Ta and TSF are considerably less: TSr/Ta«4 for 284112 and about 70 for 286112. The a-decay granddaughters - the isotopes of element 110 - are expected to decay primarily by spontaneous fission. For the odd isotopes, in particular for 289114, the predictions are less definite; the odd neutron can lead to hindrance of a-decay and, especially, spontaneous fission. Here one expects competition between the two decay modes in the daughter products with Z<112 and somewhat longer chains of sequential a-decays with longer half-lives than in the case of the neighbouring even-even isotopes. Furthermore, all these nuclei are located close to the area of beta-stability [10]. We note that the macroscopic-microscopic Ta calculations by Moller et al. [10] for 288_290 ll4 give values exceeding those of [1,9] by orders of magnitude (e.g., Ta of 7xl0 4 s for 289114). This, however, does not change the expected decay pattern for these isotopes of element 114 and their daughters. One could expect a sequence of two or more a decays terminated by spontaneous fission as the decay chain recedes from the stability region around 7V=184. With the doubly magic 48Ca projectile, the resulting compound nucleus 292114 should have an excitation energy of about 33 MeV at the Coulomb barrier. Correspondingly, nuclear shell effects are still expected to persist in the excited nucleus, thus increasing the survival probability of the evaporation residues (EVRs), as compared to "hot fusion" reactions {E «45 MeV), which were used for the synthesis of heavy isotopes of elements with atomic numbers Z=106, 108 and 110 [4]. Additionally, the high mass asymmetry in the entrance channel should
277
decrease the dynamical limitations on nuclear fusion arising in more symmetrical reactions [11]. In spite of these advantages, previous attempts to synthesize new elements in Ca-induced reactions with actinide targets gave only upper limits for their production [12]. In view of the more recent experimental data on the production of the heaviest nuclides (see, e.g., [4-6] and Refs. therein), it became obvious that the sensitivity level of the above experiments was insufficient to reach the goal. Our present experiment with the 244Pu+48Ca reaction was designed to attempt the production of element 114 at the picobarn cross-section level, thus exceeding the sensitivity of the previous experiments by at least two orders of magnitude. Interest in the study of the fission process of superheavy nuclei produced in the reactions with heavy ions is connected first of all with the possibility of obtaining most important information for the synthesis, namely, concerning production cross sections of compound nuclei at the excitation energies of «15-30 MeV (i.e., when the influence of the shell effects on the fusion of reaction partners and decay of the composite system is considerable). This, in turn, allows predicting the survival probabilities of nuclei after evaporation of 1, 2 or 3 neutrons, i.e. in "cold" or "warm" fusion reactions. However, for this problem to be solved, there is a need for a much more penetrating insight into the fission mechanism of superheavy nuclei and for a knowledge of such fission characteristics as the fission - quasi-fission cross section ratio in relation to the ion-target entrance channel mass asymmetry and excitation energy, the mutiplicity of the pre and postfission neutrons, the kinetic energy of the fragments and the peculiarities of the mass distributions of the fission and quasi-fission fragments etc. Undoubtedly all these points are of great independent interest to nuclear fission physics. We present here results of the experiments aimed at the synthesis of nuclei with Z=114 in the vicinity of predicted spherical nuclear shells in the complete fission reaction 244Pu+48Ca. This work presents also the first preliminary results of the experiments on the fission of superheavy nuclei produced in the reactions 208 Pb+48Ca-» 256No, 248 Cm+ 22 Ne^ 270Sg, 248Cm+26Mg-> 274Hs, 238 U+ 48 Ca^ 286112, 244 248 208 Pu+48Ca->292114, Cm+48Ca->.296116, Pb+58Fe->. 266Hs, 248 58 306 208 86 294 Cm+ Fe-> 122, Pb+ Kr-> 118 carried out at FLNR JINR in the last year. The choice of the indicated reactions has undoubtedly been inspired by the T M
^R7
results of the recent experiments on producing the nuclides 112, 114, 289 114 at Dubna [13-16] and 293118 at Berkeley [17] in the same reactions. 2
^KS
114,
Experimental Technique
The production of an intense ion beam of the rare isotope 48Ca (0.187% of abundance in natural Ca), which was extremely important for achieving the high sensitivity in these experiments, required the upgrade of the U400 cyclotron and
278
development of an external multi-charge ion source (ECR-4M). A 48Ca+5 beam was extracted from the ECR-4M ion source and injected into the Dubna U400 heavy ion cyclotron operated in a continuous mode. The typical intensity of the ion beam on the target was 4xl0 12 pps at the material consumption rate of about 0.3 mg/h. The beam energy was determined with a precision of ~1 MeV, by measuring the energies of scattered ions, and by a time-of-flight technique. Another important aspect of experiments was using a target of the unique isotope 244Pu (98.6%) that was provided by LLNL. The target material (a total of 12 mg in the form of Pu0 2 ) was deposited onto 1.5-p.m Ti foils to a thickness of -0.37 mg/cm2, so that heavy recoil atoms would be knocked out of the target layer and transported through the separator to the detectors. The target was mounted on a disk that was rotated at 2000 rpm across the beam direction in hydrogen gas filling the volume of the separator. This reduced the thermal and radiation load of the target. In the course of the experiment, the target withstood 48Ca beam intensities up to 7xl0 12 pps and accumulated a total beam dose of 2xl0 19 ions without damage or significant loss of target material. We used a 48Ca bombarding energy of -236 MeV at the middle of the 244Pu layer. Taking into account the energy loss in the target (-3.4 MeV), some difference in the thickness of nine target sectors, the beam energy resolution and the variation of the beam energy during the long-term irradiation, we estimated the excitation energy of the compound nucleus 292114 to be in the range 31.5-39 MeV [18]. With this excitation energy the compound nuclei would deexcite by the evaporation of 3 or 4 neutrons and ^-emission. For each particular recoiling nucleus we could determine the sector of the target where the reaction occurred and a 48Ca bombarding energy at this time. That allowed us to restrict the excitation energy interval for each event. EVRs recoiling from the target were separated in flight from the primary beam, scattered target and beam particles and various transfer reaction products by the Dubna Gas-filled Recoil Separator [19] consisting of a dipole magnet and two quadrupole lenses. A rotating entrance window (1.5-um Ti foil) separated the hydrogen-filled volume of the separator (at a pressure of 1 Torr) from the vacuum of the cyclotron beam line. The average charge state of recoil Z=114 atoms in hydrogen was estimated to be about 5.6 [20]. The recoils passed through a Mylar window (-1 um), which separated the hydrogen-filled volume from the detector module filled by pentane (at -1.5 Torr), then through a time-of-flight (TOF) system, and were finally implanted in the detector array installed in the focal-plane of the separator. The TOF detector was used to measure the time of flight of recoiling nuclei (with a detection efficiency of -99.7%) and to distinguish the signals arising in the focal-plane detector due to particles passing through the separator from those due to the radioactive decay of previously implanted nuclei. The focal-plane detector consisted of three 40x40 mm2 silicon Canberra Semiconductor detectors, each with
279 1000 2n
800 O 600
2"8Fm
R n , At 211p0
252NO
244Cf
400 200
four 40-mm-highx 10-mm-wide strips having position sensitivity 200 . in the vertical direction. The 252 No detection efficiency for full150detection efficiency for a's escaping the focal-plane detector, we arranged 8 15 20 detectors of the same type, but Energy, MeV without position sensitivity, in a box surrounding the focal-plane detector. Employing these side detectors increased the otparticle detection efficiency to -87% of An. a)
\Ul 9 10
The principal sources of events with a TOF signal are the scattered target nuclei and target-like transfer reaction 160 180 200 220 240 Total deposited energy, MeV products. Background events without a TOF signal, which can Figure 1. a) Energy spectrum of a particles detected in the 20( 48 Tb+ Ca reaction at the bombarding energy -217 MeV. imitate a particles from decay of Long-lived activities of 211Rn, 211At, and 2,1Po were produced implanted nuclei, can be due to in the natYb+48Ca calibration reaction, b) The spectrum of fast light particles produced in total deposited energies of fission fragments of 2:,2No direct nuclear reactions. A set of implants measured by both focal-plane and side detectors. The histogram presents the 252No total kinetic energy 3 similar "veto" detectors was distribution [21] obtained with an external source. Open and mounted behind the detector solid arrows show the total measured deposited energies of array in order to eliminate fission events assigned to 244miAm, 277Hs and 280110 produced signals from low-ionizing light in the 244Pu+48Ca reaction. particles, which could pass through the focal-plane detector (300 um) without being detected in the TOF system. Alpha-energy calibrations were periodically performed using the a peaks from nuclides produced in the bombardments of natYb and enriched 204'206-208pb targets with 48Ca ions [19]. The reaction 206Pb+48Ca is convenient for calibration purposes, since the known nuclide 252No, decaying by both a emission and SF, is produced in it with 0.5-ub cross section. In-beam energy spectrum of a particles registered by focal-plane detector in the 206Pb+48Ca reaction is shown in Fig. la. In the right part of Fig. la the energy spectrum of the 252No EVRs is shown, corresponding to the events correlated in position and time with subsequent a particles of this nuclide.
280 Ca + Pb -» No
" - .
250 200 150 100
'•Ca* ^Cm -^ "'lie
-v
^•B
U.t 0.4
SM
0.2
--
•" ^ " j j l
Hki
120 150 181
RPJSB ^Si ST*^ "•
1M
ifpfl
Pr
50 100 150 200 250
50 100 150 200 250
mass, u Figure 2. Two-dimensional TKE-Mass matrices (left-hand side panels) and mass yields (right-hand side panels) of fission fragments of 256No, 286 112, 292 114 and 296116 nuclei produced in the reactions with 48 Ca at the excitation energy E*=33 MeV.
Note that the experimental energy spectrum of 252No recoils measured by semiconductor detectors is distorted by the pulse-height defect, which is about one third of the initial implantation energy of the heavy nuclei. The energy resolution for the detection of a-particles in the focal-plane detector was «50keV. For a's escaping from the focal-plane detector at different angles and absorbed in the side detectors, the energy resolution was «190 keV, because of energy losses in the
281 58
>
248.
F&(325 MeV) + " " C m -•
122
350 300 250
MM
200
0 60
80 100 120 140 160 180 200 2 2 0 2 4 0 260 mass, u
Figure 3. A two-dimensional TKE-Mass matrix, the mass yield and average TKE as a function of the mass of 306122 fission fragments.
entrance windows and dead layers of both detectors and the pentane. We determined the position resolution of the signals of correlated decays of nuclei implanted in the detectors in the experiments of 1998: For sequential a-a decays the FWHM position resolution was 1.0 mm; for correlated EVR-a signals, 1.4 mm; and for correlated EVR-SF signals, 1.2 mm. Values of 1.1 mm, 0.8 mm, and 0.5 mm, respectively, were obtained in later experiments due to improvement of the detection system. For the fission-energy calibration we measured an energy spectrum of fission fragments from the SF of 252No implanted in the focal-plane detector. Fission fragments of 252No implants produced in the 206Pb+4SCa reaction were absorbed with their full energy in sensitive layers of detectors with a probability of 65%. The initial measured total deposition energies should be corrected for the pulse-height defect of detectors and energy losses of escaping fragments in the entrance windows, detectors' dead layers and pentane. With this aim in view, we compared the average
282
"•Ca+208Pb-»
No ( E = 33 MeV)
>
100
150
mass, u
200
50
100
150
200
mass, u
Figure 4. Two-dimensional TKE-Mass matrices and mass yields of fission fragments for the reactions 208Pb+48Ca, 208Pb+58Fe, 208Pb+86Kr at an excitation energy of »30 MeV.
measured deposited energy with the total kinetic energy (TKE) value of 194.3 MeV measured for SF of ^No in [21]. The measured total deposited energy distribution for SF of 252No implants are shown in Fig. lb together with the TKE distribution obtained with an external source [21]. For the reaction 48Ca+244Pu at a beam intensity of 4xlOI2pps, the overall counting rate of the detector system was about 15 s"1. The collection efficiency of the separator was estimated from the results of test experiments mentioned above. About 40% of the recoiling Z=l 14 nuclei formed in the 244Pu target would be implanted in the focal-plane detector.
283
3
Characteristics of Mass and Energy Distributions of SHE Fission Fragments
Fig. 2 shows the data on mass and energy distributions of fissionfragmentsof 256No, 286 112, 292114 and 296116 nuclei produced in the reactions with 48Ca at the same excitation energy £*«33 MeV. The main peculiarity of the data is the sharp transition from the predominant compound nucleus fission in the case of 256No to the quasi-fission mechanism of 58
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Figure S. Two-dimensional TKE-Mass matrices and mass yields of fission fragments for the reaction 208Pb+58Fe-> 266Hs at different excitation energies.
284
"Ca-^Pu
292
114
75
100 125 150 175 200 225
m, u Figure 6. Two-dimensional TKE-Mass matrices and mass yields of fission fragments for the reaction 244Pu+48Ca-» 292114 at different excitation energies.
decay in the case of the 286112 nucleus and more heavy nuclei. It is very important to note that despite a dominating contribution of the quasi-fission process in the case of nuclei with Z=l 12-116, in the symmetric region of fission fragment masses 04/2±20) the process of fusion-fission of compound nuclei, in our opinion, prevails. It is demonstrated in the framings (see the right-hand panels of Fig. 2) from which it is also very well can be seen that the mass distribution of fission fragments of compound nuclei is asymmetric in shape with the light fission fragment mass of about 132-134. Fig. 3 shows similar data for the reaction 5SFe+248Cm leading to the formation of the heaviest compound system ever studied by us, namely, 306122 (/V=184), i.e., to the formation of the spherical compound nucleus, which agrees well with theoretical
285
predictions [22]. As seen from Fig. 3, in this case we observe an even stronger manifestation of the asymmetric mass distribution of 306122 fission fragments with the light fragment mass of 132. The corresponding structures are also well seen in the dependence of the TKE on the mass (the lower panel of Fig. 3). Fig. 4 shows mass and energy distributions of fission fragments for compound nuclei 256No, 266Hs and 294118 formed in the interaction of 208Pb target with 48Ca, 58 Fe and 86Kr projectiles at the excitation energy of «30 MeV. It is important to note that in the case of the reaction 208Pb+86Kr the ratio between the fragment yields in
"Ca + ^ C m V ^ l i e E, = 234 MeV lab 5 1 -3
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Figure 7. Two-dimensional TKE-Mass matrices and mass yields of fission fragments for the reaction 248Cm+4SCa-» 296116 at different excitation energies.
286 Mass asymmetry in binary fission 1
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Figure 8. The dependence of the light and heavy fragment masses on the compound nucleus mass.
the region of asymmetric masses and those in the region of masses A/2 exceeds by about 30 times a similar ratio for the reactions with 48Ca and 58Fe ions. It signifies to that fact that in the case of the 208Pb+86Kr->294118 reaction in the region of symmetric fragment masses the mechanism of quasi-fission prevails. Figs. 5, 6 and 7 show similar data for the reactions z 5Pb+ Fe-> Hs, 244 Pu+48Ca-»292114 and 248Cm+48Ca->296116 obtained at different excitation energies. In analyzing the data presented in Figs. 2-7 one can notice two main regularities in the characteristics of mass and energy distributions of fission fragments of superheavy compound nuclei: 1. Fig. 8 shows the dependence of the light and heavy fragment masses on the compound nucleus mass. It is very well seen that in the case of superheavy nuclei
287 Fission Fragment Total Kinetic Energy in Heavy Ion Induced Reactions 300
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Figure 9. The dependence of TKE on the Coulomb parameter Z2/Am.
the light spherical fragment with masses of 132-134 plays a stabilizing role, in contrast to the region of actinide nuclei. Fig. 9 shows the TKE dependence on the Coulomb parameter Z2/Am, from which it follows that for the nuclei with Z>100 the TKE value is much smaller in the case of fission as compared with the quasi-fission process. 4
Capture and fusion-fission cross sections
288 -J—I—I—I
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E\ MeV Figure 10. The capture cross section
Figs. 10 and 11 show the results of measurements of the capture cross section ac and the fusion-fission cross section erff for the studied reactions as a function of the initial excitation energy of the compound systems. Comparing the data on the cross sections crff at £*«14-15 MeV (cold fusion) for the reactions 208Pb+58Fe and 208Pb+86Kr, one can obtain the following ratio: CTff (108) / Off (118) > 102. In the case of the reactions from 238U+48Ca to 208Cm+58Fe at E «33 MeV (warm fusion) the value of Z changes by the same 10 units as in the first case, and the ratio crff (112)/ crff(122) is «4-5 which makes the use of asymmetric reactions for the synthesis of spherical superheavy nuclei quite promising.
289 10'
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E,MeV Figure 11. The capture cross sectionCTCand the fiision-fission cross section as for the reactions 48Ca, 58Fe, 86Kr+208Pb as a function of the excitation energy.
Another interesting result is connected with the fact that the values of <% for No and 266Hs at £*=14-15 MeV are quite close to each other, whereas the evaporation residue cross sections crm [23] differ by almost three orders of magnitude (crff/ o^) which is evidently caused by a change in the r{/ ra value for the above mentioned nuclei. At the same time, for the 294118 nucleus formed in the reaction 208Pb+86Kr, the compound nucleus formation cross section is decreasing at an excitation energy of 14 MeV by more than two orders of magnitude according to our estimations (Ofi«500 nb is the upper limit) as compared with crff for 256No and 268 Hs produced in the reactions 208Pb+48Ca and 208Pb+58Fe at the same excitation energy. But when using the value of «2.2 pb for the cross section
290 Ca(233MeV) + ™Pb - V N O
A
18
£
15 V 12
3 60 90 120 150 180 210 240
m, u
V
60 90 120 150 180 210 240
m, u
Figure 12. Two-dimensional TKE-Mass matrices (top panels) and the mass yields and neutron (the diamonds) and gamma ray (the circles) multiplicities in dependence on the fission fragment mass (bottom panels) for the reactions 238 U+ 48 Ca-» 286 112 and 208 Pb+48Ca-> 256No.
In one of me recent works [24] it has been proposed that such unexpected increase in the survival probability for the 294118 nucleus is connected with the sinking of the Coulomb barrier below the level of the projectile's energy that, as a consequence, leads to an increase in the fusion cross section. However, our data do not confirm this assumption. Neutron and gamma-ray multiplicities in the fission of superheavy nuclei Emission of neutrons and gamma-rays in correlation with fission fragments in the decay of superheavy compound systems at excitation energies of near or below the Coulomb barrier had not been properly studied before this publication. At the same time such investigations may be extremely useful for an additional identification of fusion-fission and quasi-fission processes and thus a more precise determination of the cross sections of the above mentioned processes in the total yield of fragments. On the other hand, the knowledge of the value of the fission fragment neutron multiplicity may be used in the identification of SHE in the experiments on their synthesis. The first results of such investigations are presented in Figs. 12, 13 for the reactions 208 Pb+ 48 Ca^ 256No, 238U+48Ca-> 286112, 244Pu+48Ca-* 292114 and 248 Cm+4SCa—» 296116 at energies near the Coulomb barrier. As seen from the figures, in all the cases the total neutron multiplicity vtot is considerably lower (by more than twice) for the region of fragment masses where the mechanism of quasi-fission
291 *Ca(24SMeV) + " C m -* 116 300 270 240
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Figure 13. The same as in Fig. 11, but for the reactions ""Cm+^Ca-* 296116 and 244 Pu+ 48 Ca-+ 292 114.
predominates as compared with the region of fragment masses where, in our opinion, the process of fusion-fission prevails (in the symmetric region of fragment masses). Another important peculiarity of the obtained data is the large values of vtot«(9 and 10.5 for the fission of 292114 and 2%116 compound nuclei, respectively. As well as for vtot noticeable differences have been observed in the values of ^-ray multiplicities for different mechanisms of superheavy compound nucleus decay. 6
Experimental Results on the Synthesis of Superheavy Nuclei
The experiments were performed during November and December, 1998, and from June till October, 1999. Over a time period of 94 days a total of 1.5 xio 19 48Ca projectiles of energy -236 MeV was delivered to the target. In the analysis of the experimental data, we searched for new a-decay sequences with £ a >8 MeV [1,9]. Note that according to the concept of the "stability island" of superheavy elements, as long as any a-decay chain leads to the edge of the stability region, it should be terminated by spontaneous fission. In the course of the 48Ca+244Pu bombardment we observed five spontaneousfission events, all of which could be genetically linked to preceding events, so that we could trace their origin. These events can be classified by their nature in two distinct groups. First, we point out those SF decays that occur within milliseconds following the implantation of the heavy recoil. Two such events, with measured energies £=149 MeV and £=153 MeV, were detected 1.13 ms and 1.07 ms, respectively,
292
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288 114
288i
292 114*
/V SF
9.21 MeV 10.3 s (37 s) 30.1 mm
280
110
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284
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112
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292 114*
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289,
] 4
292, , 4 *
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b)
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108
/V SF
83 MeV (4.04+4.79) 1.6 min (1 - 12 ram) 17.0 mm 172 MeV (120+52) 16.5 min 17.1 mm
Figure 14. Time sequences in the observed decay chains. The expected halflives corresponding to the measured Ea values for the given isotopes are shown in parentheses following the measured lifetimes. Hindrance factors of 1 and 10 were assumed for a decay of nuclei with odd neutron numbers. Positions of the observed decay events are given with respect to the top of the strip.
after the implantation of corresponding position-correlated recoil nuclei. For one of them fission fragments were registered by both the focal-plane (£=141 MeV) and side (£=12 MeV) detectors. Based on the lifetime, we assign these events to the spontaneous fission of the 0.9-ms 244mfAm isomer, a product of transfer reactions with the 244Pu target. Such transfer-reaction products are expected to be suppressed by the gas-filled separator by a factor of ~105 [19]. The measured total deposited energies for SF of the 244mfAm implants are shown in Fig. lb by open arrows. Three other SF events terminated the a-decay sequences of relatively long-lived nuclei. Two such SF events were observed as two coincident fission-fragment signals with energies £=221 (156+65) MeV and £=213 (171+42) MeV. The items in each sum indicate energies deposited in the focal-plane and side detectors,
293 respectively (see Fig. lb). We searched the data backwards in time from these events for preceding a particles and/or EVRs, in the same positions. The latter were defined as the events characterized by the measured energies, TOF signals and estimated resulting mass values, that were consistent with those expected for a complete-fusion EVR, as determined in the calibration reactions. The full decay chains including these two fission events are shown in Fig. 14a. Both decay chains are consistent with one another, taking into account the energy resolution of the detectors and statistical uncertainty in lifetimes determined from a few detected events. The first a-particles have similar energies £ a =9.87 MeV and £ a =9.80 MeV, and were detected in the focal-plane detector 0.77 s and 4.58 s after the implantation of the recoil nuclei in strips 2 and 8, respectively. The second a-particles in corresponding chains, having the energies Ea=9.2l MeV and £ a =9.13MeV, were observed at the same locations after 10.34 s and 18.01s. Finally, 14.26 s and 7.44 s later, the SF events were observed. All events in the two decay chains appeared within time intervals of 25.4 s and 30.0 s and position intervals of 0.5 mm and 0.4 mm (Fig. 14a), respectively [16]. By applying a Monte Carlo technique [25] and the procedure described in Ref. [26] (see below) we calculated probabilities that these decay sequences were caused by the chance correlations of unrelated events at any position of the detector array and at the positions in which the candidate events occurred. The results of the two calculations were similar; the probability that both decay chains consist of random events is less than 5x10~ n , calculated in the most conservative approach. In this case, we observed two identical three-member decay sequences. If we assume that they actually consisted of four decays, the probability of missing one aevent in both decay chains would be less than 3%. The formation of the nuclei which initiated the observed decays resulted from "instant" 48Ca beam energies of 237.6 and 237.0 MeV in the middle of the target. Taking into account die target thickness and beam energy resolution, this corresponds to excitation energy ranges of 33.6-39.7 and 33.2-39.1 MeV for the 292 114 compound nucleus, respectively. This would favor deexcitation of the compound nucleus by evaporation of 4 neutrons and ^-emission, which finally leads to the even-even nucleus 288114. Indeed, the observed chains, including two a-decays and terminated by SF, match the decay scenario predicted for the even-even nuclide 288114 [1,9]. The detected sequential decays have Tm vs. Ea values that correspond well to the decays of the even-even isotopes of elements 114 and 112. To illustrate this, Fig. 14a presents the expected half-lives corresponding to the measured a-particle energies for the genetically related nuclides with the specified atomic numbers. For the calculation of half-lives with given Qa values, the formula by Viola and Seaborg with parameters fitted to the Ta values of 58 even-even nuclei with Z>&2 and N> 126, for which both Ta and Qa were measured [9], has been used. The calculated a-decay half-lives are in agreement with the detected decay times. Conversely, substituting
294
T\a and Ea values corresponding to the detected decays in this formula, results in atomic numbers of 114.4 ^ 0 ' g and 110.2 * o g for the mother and daughter nuclides, respectively. The measured total energies deposited in the detector array for both fission events exceed the average value measured for 252No by about 40 MeV (see Fig. lb). Despite the relatively wide distributions of the total kinetic energies in spontaneous fission, this also indicates the fission of a rather heavy granddaughter nucleus, with Z>106 (see, e.g., Fig. 9). From the above considerations, we can conclude that the detected decay chains originate from the parent even-even nuclide 288114, produced in the 244Pu+48Ca reaction via the 4«-evaporation channel. The next SF event (the first, in chronological order) was also observed as two coincident signals (two fission fragments) with energy deposited in the focal-plane detector £=120 MeV and in the side detector £=52 MeV; £ tot =172MeV (see Fig. lb). The entire position-correlated decay chain is shown in Fig. 14b. An aparticle was detected in the focal-plane detector 30.4 s after the implantation of a recoil nucleus in the middle of the 8- strip. The energy of this first a-particle was £ a =9.71 MeV. A second a-particle, having an energy £ a =8.67 MeV, was observed at the same location 15.4 min later. A third a-particle, escaping the front detector leaving an energy £ al =4.04 MeV and absorbed in the side detector with £a2=4.79 MeV (£tot= 8.83 MeV), was measured 1.6 min later. Finally, 16.5 min later, the SF event was observed [14]. All 5 signals (EVR, ai, a2, a3, SF) appeared within a position interval of 1.6 mm (Fig. 14b), which strongly indicates that there is a correlation among the observed decays. Assuming that the decay sequence for a valid event will terminate with SF, we developed a Monte Carlo technique to estimate the probability of the candidate event being due to random correlations [25]. Artificial SF events (~105) were inserted into the data distributed at random positions and times over the entire detector array and entire experiment duration. We searched the 34 min preceding each random fission for three a-particle-like signals with energies 8.5-10.0 MeV and one EVR-like event preceding the a-events. All four of these events had to be within 2.0 mm of the artificial fission and meet the position criteria at greater than 95% confidence level to be considered a possible random correlation. The probability per fission of finding such a correlated event was determined to be Pcrr=0.006. With the given energy window and no time restriction within the 34-min interval, we found that the majority of these random sequences preceding the artificial SF events could not be proposed as the decay of Z=l 14 or nearby elements. By applying the GeigerNuttall relationship, we imposed a lifetime window for each a-event. Requiring that the hindrance factor must be between 1 and 10 for each a-energy reduced Pm to 6xl0"4. Another Pen calculation was performed for strip 8 at the position in which the candidate event occurred, following the procedure described in Ref. [26]. For a
295
1S6
160
164
168
Neutron number N Figure 15. Alpha-decay energy vs. neutron number for isotopes of even-Z elements with Z>100 (solid circles) [47,23,28-30]. Open circles show data from Ref. [17]; triangle, from Ref [15]; solid squares and diamonds, data from the present work. Open circles connected with solid lines show theoretical Qa values [1,9] for even-even Z=106-114 isotopes.
position-correlation window of 1.6 mm the signals from EVRlike events were observed with a frequency of 1.3 h"1. The signals of a-like events with £=8.110.5 MeV occurred with a frequency of 1 h"1. Thus, calculated from event rates alone even without applying the Geiger-Nuttall relationship, the probability that this decay sequence was caused by the chance correlation of unrelated events in strip 8 is 6x 10"3. In this experiment we observed a four-member decay sequence. If we assume that it actually consisted of five decays (the spontaneous fission was due to 273106), the probability of missing any one of the four ocevents is about 34%, but the probability of missing any particular a event in the chain
and observing the other three is only about 8.5%. All events of the decay chain are correlated in time and position and match the decay pattern of a superheavy nucleus that is predicted by theory. For the whole decay chain the basic rule for a-decay, defining the relation between Qa and Ta, is fulfilled. This can be seen in Fig. 2b where the expected half-lives are shown which correspond to the measured oc-particle energies for the specified nuclides. The halflives were calculated using the formula by Viola and Seaborg with the same parameters as above [9], with hindrance factors of 1 and 10 for the a-decay of odd nuclei. This decay sequence evidently originates from a different parent nucleus than the chains that were assigned to the decay of 288114. Most probably, this decay chain arises from the neighbouring even-odd isotopes of element 114. The excitation energy of the 292114 in our experiment was insufficient to evaporate 5 neutrons, so the best candidate for the parent nucleus is the even-odd isotope 289114, produced in the 3n-evaporation channel. Indeed, the a-decaying nuclides in this chain are characterized by lower decay energies than the corresponding members of the chain attributed to the decay of288114, while SF terminates the decay sequence at a later
296 stage. The decay properties of the observed nuclei are also in agreement with calculations [1,9] (see Fig. 15). A priori, one cannot exclude that the investigated excitation energy range of 31.5-39 MeV was not optimal for the production of this isotope and that the probability of the evaporation of three neutrons from the compound nucleus could be even higher at a lower excitation energy. However, the excitation function for the 3«-evaporation channel should be quite sensitive to the actual fusion barrier, and reducing the energy in the subbarrier region could substantially decrease the complete-fusion cross section. To check this assumption, we performed an experiment in November-December, 1999, using a lower projectile energy to search for additional decays of 289114. In the 31-day bombardment by -231 MeV 48Ca projectiles a total beam dose of 4.6xl0 18 was accumulated. The corresponding excitation energy of the 292114 compound nuclei was in the range of 28.5-34.5 MeV. Only one fission event, the 0.9-ms 244mfAm isomer with £tot=156 MeV, was detected 2.26 ms after the implantation of corresponding position-correlated recoil nucleus in this bombardment. The measured total energy for this SF event is also shown in Fig. lb by open arrow. In the present series of experiments we observed three decay sequences: one was attributed to the decay of the odd-even isotope 289114 and two to the decay of the even-even nuclide 288114. Recent semiempirical calculations [27] predict the cross section maxima for emission of 3 and 4 neutrons at the 292114 compound nucleus excitation energies of 30 MeV and 38 MeV, respectively. From the present observations we estimate the cross sections for producing both nuclides in this reaction to be about a picobarn. The bombardment performed at the lower projectile energy results in only an upper production limit of 2 pb (95% confidence level), thus indicating that the maximum 3»-evaporation cross section practically does not exceed the cross section of the 4« channel.
7
Discussion
The lifetimes of the new isotopes, in particular 285112 and 281110, appear to be approximately 106 times longer than those of the known nuclei 277112 [5,23] and 273 110 [4,5], which have eight fewer neutrons. We can also note that 289114, 285112 and 281110 are about 104-105 times more stable than 285114,28l112 and 277110, the adecay products of 293118 [17] that was recently produced in the bombardment of 208 Pb with 86Kr ions using the Berkeley separator BGS. The isotopes 288114 and 284112 are the heaviest known cc-decaying even-even nuclides, following the production of 260,266Sg (Z=106) [28,4,7] and the observation of a-decay of 264Hs (2=108) [29]. The radioactive decay properties of the newly observed nuclides are in qualitative agreement with macroscopic-microscopic nuclear theory [1,9], which
297 predicts both a-decay and spontaneousfission properties of the heavy nuclei. Alpha-decay energies of synthesized nuclei and previously known isotopes of even-Z elements with Z>100 together with theoretical Qa values [1,9] for eveneven isotopes Z=106-114 elements are shown in Fig. 15. The properties of the new nuclides also agree with those of the neighboring odd-mass isotope 287
174 171 173 Neutron number N
175
Figure 16. Comparison of experimental (solid diamonds) and calculated Qa values for the adecay chains of 28S114 and 289114. Circles show data from Refs [1,9] (mean values for neighbouring even-even nuclei are used for oddN isotopes); squares, from Ref [31]; open diamonds, from Ref. [18]; triangles up, from Ref. [32]; and triangles down, from Ref. [10],
superheavy
nucleus
288
114
114 [15] that was produced in MarchApril, 1999, in the bombardment of a 242 Pu target with 7.5xl018 48Ca ions at the separator VASSILISSA. The experimental data exactly reproduced the decay scenario predicted for 288114, i.e., two consecutive a-decays terminated by spontaneous fission. Comparison of the measured decay properties of the new even-even
(£a=9.84±0.05 MeV,
2"1/2=1.9tog s ) '
2Ml12
(£a=9.1710.05 MeV, Tm=9.St\7&9 s), and 280110 (r1/2=7.5 t£g s) with theoretical calculations [1,9] indicates that nuclei in the vicinity of spherical shell closures with Z=l 14 and JV=184 could be even more stable than is predicted by theory. It can be seen in Fig. 15 that a-decay energies of the heaviest new even-even nuclides with Z=l 12 and 114 are 0.4-0.5 MeV less than the corresponding predicted values. The heaviest even-odd nuclides follow this trend as well. Such a decrease in Qa values leads to an increase of partial a-decay lifetimes by an order of magnitude. Calculations are far less definite regarding spontaneous fission; however, we note that the observed spontaneous fission half-life of 280110 exceeds the predicted value [9] by more than two orders of magnitude. The a-decay properties of the synthesized nuclei can be also compared with predictions of other theoretical models, in particular with calculations performed in Hartree-Fock-Bogoliubov approach with different Skyrme forces [31], and relativistic mean-field calculations [32]. Alpha-decay energies of the synthesized isotopes together with theoretical Qa values [1,9,10,18,31,32] are shown in Fig. 16. Some theoretical calculations using macroscopic-microscopic models [1,9,10] predict the
184
114 to be the next spherical doubly magic nucleus, however recent
self-consistent models [31,32] give preference to the more proton-rich nuclei
298 292
310
120 or even 184 126. While the macroscopic-microscopic models explain the relatively small Qa values and corresponding long half-lives of the synthesized nuclei by the influence of spherical Z=114 and #=184 shell closures, the selfconsistent Skyrme-Hartree-Fock-Bogoliubov model predicts interesting shell structure in the neutron system, but no shell effect at Z=114. The last model reproduces well the measured Qa values for the decay chain originating from 289114 172
and passing through the [611] for #=173). The ground state of
+
levels (excited for #=175, 171 and ground state
289
114 is calculated to be a high-Q isomeric state,
[707] y " [31]. The relativistic mean-field model [32] describes well the observed a-decay chain of 289114 (the break in the measured Qa values, missed in the calculations, was explained by assuming the decay of 289114 to one of the numerous low-lying excited states in 285112). This model perfectly reproduces the measured Qa values for the decay chain of the even-even nuclide 288114, suggesting the influence of deformed Z=114 and #=174 shell closures, for its explanation, although a spherical Z=114 shell cannot be excluded. All the above theoretical approaches predict the existence of the "island of stability" in the region of superheavy elements. The principal result of the present work is the observation of the considerable increase in lifetimes of superheavy nuclei with Z>110, with increasing their neutron number. Comparison of the present data with calculations shows that theoretical predictions agree with experimental results not only qualitatively, but also quantitatively. In this respect, the decay properties of the new nuclides observed in present experiments confirm theoretical expectations and can be considered the proof of the existence of enhanced stability in the region of superheavy elements. As a result of the experiments carried out, for the first time the properties were studied of the fission of the compound nuclei 256No, 270Sg, 266Hs, 271Hs, 274Hs, 286112, 292 114, 295116,294118 and 306122, produced in reactions with ions 22Ne, 26Mg, 48Ca, 58 Fe and 86Kr at energies close to and below the Coulomb barrier. On the basis of those data a number of novel important physics results were received: a) it was found, that the mass distributions of fission fragments for compound nuclei 112, 114, 116 and 122 are asymmetric one, whose nature, in contrast to the asymmetric fission of actinides, is determined by the shell structure of the light fragment with the average mass of 132-134. It was established that TKE, neutron and ^ r a y multiplicities for fission and quasi-fission of superheavy compound nuclei are significantly different; b) the dependence of the capture (crc) and fusion-fission (crff) cross sections for nuclei 256No, 266Hs, 274Hs, 286112, 292114, 296116, 294118 and 306122 on the
299 excitation energy in the range of 15-60 MeV has been studied. It should be emphasized that the fusion-fission cross sections for the compound nuclei produced in the reactions with 48Ca and 58Fe ions at excitation energy of «30 MeV depend only slightly on reaction partners, that is, as one goes from 286 112 to 306122, the as changes no more than by the factor of 4-5. This property seems to be of considerable importance in planning and carrying out experiments on the synthesis of superheavy nuclei with Z>114 in reactions with 48Ca and 58Fe ions. In the case of the reaction Kr+^u Pb, leading to the production of the 294 '118, contrary to reactions with 48Ca and 58Fe, the composite system contribution of quasi-fission is 292H6 296H6' dominant in the region of the 01:21 July 19, 2000 fragment masses close to A/2; <Xi /To 56 MeV 28.3 mm c) the phenomenon of multimodal out of beam 46.9 ms 288H4 fission was first observed and 28.4 mm studied [33,34] in the region of 0(2 9.81 MeV superheavy nuclei 256No, 270Sg, 266 2.42 s Hs, 271 Hsand 274 Hs. 284H2 28.9 mm
0(3
280HO
y9.09i0.46 MeV 53.9 s
8
Addendum
On June 14, 2000, we started an experiment aimed at die synthesis of superheavy nuclei with Z=116 in the Figure 17. Time sequence in the decay chain observed complete fusion reaction 248 in the 2 4 8 Cm+ 4 S Ca reaction. Cm+48Ca [35]. After an integrated beam dose of 6.6x10 was delivered to the targets, we observed a decay chain consisting of three consecutive a-decays and a spontaneous fission that can be assigned to the implantation and decay of the heavy nuclide with Z=l 16 (see Fig. 17). Implantation of a heavy recoil in the focal-plane detector was followed, after 46.9 ms, by an a-particle decay with £o=10.56 MeV. This sequence switched the ion beam off, and further decays - two oc-particles and a spontaneous fission - were detected under low-background conditions. All events in this decay chain appeared within a time interval of 63.3 s and a position interval of about 0.5 mm, which points to a strong correlation between them. The probability that the decay chain consists of random events is less than 10"10. The energies and decay times of the descendant nuclei are in agreement with those observed in the decay chains of even-even isotope 288114 produced in the 244pu+48Ca r e a c t j o n T n u S i t n e f ^ a .decay with Ea=\0.56 MeV should be attributed to the parent nuclide 292116, produced in the 248Cm+48Ca reaction via the 4n-evaporation channel. Experiments are in progress. v\ oF
197 MeV (194+3) 6.93 s 28.5 mm
300
9
Acknowledgements
This paper presents the results obtained by a large group of physicists, many of them are co-authors of the original publications. We express our gratitude to all of them. We are grateful to the JINR Directorate, in particular to Profs. V.G. Kadyshevsky, Ts. Vylov and A.N. Sissakian for the help and support we got during all stages of performing the experiment. This work has been performed with the support of the Russian Foundation for Basic Research under Grant No. 99-02-17981 and INTAS under grants No. 96-662 and 11929. Much of support was provided through a special investment of the Russian Ministry of Atomic Energy. These studies were performed in the framework of the Russian FederationAJ.S. Joint Coordinating Committee for Research on Fundamental Properties of Matter. References 1. 2. 3. 4. 5. 6. 7. 8.
9.
10. 11. 12.
Smolanczuk R., Phys. Rev. C 56 (1997) p.812. Moller P. and Nix J.R., J. Phys. G 20 (1994) p.1681. Cwiok S. et al, Nucl. Phys. A611 (1996) p.211. LazarevYu. A. et al, Phys. Rev. Lett. 73 (1994) p.624; Phys. Rev. Lett. 75 (1995) p.1903; Phys. Rev. C 54 (1996) p.620. Hofrnann S et al, Z. Phys. A 350 (1995) p.277; Nachrichten GSI 02-95 (1995) p.4; Z. Phys. A 350 (1995) p.281; Z. Phys. A 354 (1996) p.229. Ghiorso A. et al, Phys. Rev. C 51 (1995) p.R2293. Turler A. et al, Phys. Rev. C 57 (1998) p. 1648. Oganessian Yu.Ts., Proc. Int. Conf. on Nuclear Physics at the Turn of the Millennium "Structure of Vacuum & Elementary Matter", Wilderness, South Africa, 10-16 March 1996. Singapore: World Scientific (1997) p.l 1. Smolanczuk R., Skalski J., Sobiczewski A., Proc. Int. Workshop XXIV on Gross Properties of Nuclei and Nuclear Excitations "Extremes of Nuclear Structure", Hirschegg, Austria, 15-20 January 1996. GSI Darmstadt (1996) p.35. Moller P., Nix J.R., KratzK.-L., Atom. Data and Nucl. Data Tabl. 66 (1997) p.131. Swiatecki W.J., Nucl. Phys. A 376 (1982) p.275; Blocki J.P., FeldmeierH., Swiatecki W.J., Nucl. Phys. A 459 (1986) p.145. Hulet E.K. et al, Phys. Rev. Lett. 39 (1977) p.385; Illige J.D. et al, Phys. Lett. B 78 (1978) p.209; OttoRJ. et al, J. Inorg. Nucl. Chem. 40 (1978) p.589; Ghiorso A. et al, LBL Report (1977) LBL 6575; Oganessian Yu.Ts. et al, Nucl. Phys. A 294 (1978) p.213; ArmbrusterP. et al, Phys. Rev. Lett. 54 (1985)p.406.
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13. 14. 15. 16. 17. 18. 19.
20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
30. 31. 32. 33. 34. 35.
Oganessian Yu.Ts. et al, Eur. Phys. J. A 5 (1999) p.63. Oganessian Yu.Ts. et al, Phys. Rev. Lett. 83 (1999) p.3154. Oganessian Yu.Ts. et al, Nature 400 (1999) p.242. Oganessian Yu.Ts. et al, Phys. Rev. C 62 (2000) p.041604(R). Ninov V. et al, Phys. Rev. Lett. 83 (1999) p.l 104. Myers W. D. and Swiatecki W. J., Nucl. Phys. A 601 (1996) p.141. Oganessian Yu.Ts. et al, Proc. 4 ^ Int. Conf. on Dynamical Aspects of Nuclear Fission, Casta-Papiernicka, Slovak Republic, 19-23 October 1998. Singapore: World Scientific (2000) p.334; Lazarev Yu.A. et al, Proc. Int. School-Seminar on Heavy Ion Physics, Dubna, 10-15 May 1993, JINR Report E7-93-274, Dubna (1993) v.IIp.497. Lazarev Yu.A. et al, JINR FLNR Scientific Report 1995-1996, E7-97-206, Dubna (1997) p.30. Wild J.F. et al, J. Alloys Compounds. 213/214 (1994) p.86. Patyk Z. and Sobiczewski A., Nucl.Phys. A 533 (1991) p.132. Hofinann S. and Miinzenberg G., Rev. Mod. Phys. 72 (2000) p.733. Myers W.D. and Swiatecki W.J., Phys. Rev. C 62 (2000) p.044610. StoyerN.J. et al, LLNL Preprint UCRL-JC-136927 (1999); Elsevier Science (to be published). Zlokazov V.B., Eur. Phys. J. A 8 (2000) p.81. Sagaidak R.N., JINR FLNR Scientific Report 1997-1998, Dubna, 2000 (to be published). Demin A.G. et al, Z. Phys. A 315 (1984) p.197; Miinzenberg G. et al, Z. Phys. A322(1985)p.227. Oganessian Yu.Ts. et al, Z. Phys. A 319 (1984) p.215; Miinzenberg G. et al, Z. Phys. A 324 (1986) p.489; HePberger F.P. et al., Proc. Tours Symposium on Nuclear Physics III, Tours, France, 1997. American Institute of Physics, Woodbury, New York (1998) p.3. Firestone R.B., Shirley V.S. (editor), Table of Isotopes 8th edition, John Wiley & sons, inc. New York, Chichester, Brisbane, Toronto, Singapore (1996). Cwiok S., Nazarewicz W., Heenen P.H., Phys. Rev. Lett. 83 (1999) p. 1108. Bender M., Phys. Rev. C 61 (2000) p.031302(R). Itkis M.G. et al, Phys. Rev. C 59 (1999) p.3172. Itkis M.G. et al, Proc. 7-th Int. Conf. Clustering Aspects of Nuclear Structure and Dynamics (Claster'99) Rab Island, Croatia, 1999 (WS, 2000) p.386. Oganessian Yu.Ts. et al, Phys. Rev. C (2001) (to be published).
302
P U Z Z L I N G RESULTS ON N U C L E A R SHELLS F R O M T H E PROPERTIES OP FISSION C H A N N E L S K.-H. SCHMIDT, A.R. JUNGHANS Gesellschaft fur Schwerionenforschung, Planckstrafie 1, 64291 Darmstadt, Germany E-mail: [email protected] J. BENLLIURE Universidad de Santiago de Compostela, 15706 Santiago de Compostela,
Spain
C. BOCKSTIEGEL, H.-G. CLERC, A. GREWE, A. HEINZ, M. DE JONG, S. STEINHAUSER Institut fur Kernphysik, TU Darmstadt, Schloflgartenstr. 9, 64289 Darmstadt, Germany At the secondary-beam facility of GSI, the fission properties of short-lived neutrondeficient nuclei have been investigated in inverse kinematics. Detailed features of the measured element distributions and total kinetic energies seem to contradict the present understanding according to which the neutron shells at N = 82 and N KB 90 are decisive for the asymmetric fission channels.
1
Introduction
Nuclear fission is one of the most intensively studied types of nuclear reaction 1 ' 2 . All nuclei investigated from about 234 U to 2 5 6 Fm were found to fission into fragments with strongly different mass. Symmetric fission is suppressed. The mean mass of the heavy component is almost stationary. Obviously, shell effects in the heavy fragment control this asymmetric fission. The most important shells are considered to be the spherical N = 82 shell and a shell at N « 90 at large deformation (/3 « 0.6) 3 . But asymmetric fission dies out on both extremes of the mass range. There is a dramatic change of the mass distribution to a narrow single-humped distribution found in 2 5 8 Fm 4 . This is explained by the formation of two spherical nuclei close to the doubly magic 132 Sn. Selected nuclei in this range are accessible to experiment because they decay by spontaneous fission. But also for lighter nuclides one observes single-humped distributions, e.g. for 213 Ac. However, these are much broader. The present work reports on the first systematic study of the transition form asymmetric to symmetric fission below 2 3 4 U. Previously, only a few mass distributions from low excitation energies could be measured by use of radioactive targets 2 2 6 Ra and 227 Ac (see
303
e.g. 5 ). Some other nuclei in the suspected transition region between 2 2 5 Ac and 213 At had been produced with excitation energies around 30 MeV by fusion reactions 6,7 ' 8 . 2
The Secondary-beam Experiment
In a conventional fission experiment, a target nucleus is excited. The fission fragments reach the detectors with a kinetic energy given by the fission process. The available target materials limit the experiments on low-energy fission. Up to now, spontaneous fission offers the only possibility to overcome this limitation for those nuclei of interest which can be produced e.g. by heavy-ion fusion reactions. The secondary-beam facility of GSI allows now becoming independent of available target nuclides. By fragmentation of a 238 U beam at 1 A GeV, many short-lived radioactive nuclei are produced. After isotopic separation in the fragment separator, several hundred fissile nuclei are available for nuclear-fission studies 9,10 ' 11 . In the present experiment, fission was induced by Coulomb excitation of the secondary beam in a lead-target. The atomic numbers and the velocity vectors of both fission fragments were determined, and the element distributions and the mean total kinetic energies were deduced. The experimental technique is described in detail in Ref.12. The electromagnetic field of a lead target nucleus as seen by the secondary projectiles can be formulated as a flux of equivalent photons according to Ref.13. At relativistic energies as employed here, the spectrum is hard enough to excite giant resonances in the secondary projectiles. With the calculated equivalent photon spectrum and the systematics of the photo-absorption cross sections, one can calculate the energy-differential cross section for electromagnetic excitation. It peaks at about 11 MeV and is very similar for all nuclides investigated. 3
Results and Discussion
The data acquired in the secondary-beam experiment allow for the first time to systematically analyse the fission properties of nuclei in a large continuously covered region on the chart of the nuclides. Fig. 1 shows the elemental yields after electromagnetic-induced fission, covering the transition from a single-humped element distribution at 221 Ac to a double-humped element distribution at 2 3 4 U. In the transitional region, around 2 2 7 Th, triple-humped distributions appear, demonstrating comparable weights for asymmetric and symmetric fission.
304
L.
o
i c O
J-IXMMM »i AAAMMMMMM AAAAAWMM Z yields AAAAAA
92
Pa
90
Th
t t B9
Ac
132
133
134
135
136
137
138
139
140
U1
U2
Neutron number Figure 1. Measured fission-fragment element distributions in the range Z = 24 to Z = 65 after electromagnetic excitation of 28 secondary beams between 2 2 1 Ac and 2 3 4 U are shown on a chart of the nuclides.
The transition seems to be governed by the mass number of the fissioning system as the ordering parameter: Systems with constant mass show similar charge distributions. Another important parameter deduced from the data is the mean position of the heavy fission-fragment component shown in Fig. 2. From previously measured mass distributions, a roughly constant position of the heavy fission component in mass number had been deduced 14 . Due to the long isotopic chains investigated and the high precision of the data, we obtain a much more comprehensive view. It becomes very clear that the position of the heavy component is almost constant in atomic number Z w 54 and moves considerably in neutron number. This also means that the position accordingly moves in mass number. It is not expected that any polarisation in N/Z which is neglected here due to the UCD (unchanged charge density) assumption can explain the observed variation of five units in neutron number from N = 79 to N = 84. Both findings, the mass as the ordering parameter of the transition and the constant position at Z = 54 are unexpected, since the asymmetric fission component is usually traced back to the influence of neutron shells in the heavy component (e. g. 3 ). According to the present understanding of the fission process, the different components which appear in the yields and in the kinetic-energy distributions of the fission fragments are attributed to fission channels 15,16 ' 17,18 which are assigned to valleys in the potential-energy surface of the highly deformed sys-
305
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st
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Th Pi U
A
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Figure 2. Pull symbols: Measured mean position of the heavy asymmetric component in nuclear-charge number Zu (upper part) and neutron number Nn (lower part). While the charge number was measured, the neutron number was estimated by the UCD assumption: Nn = Zu * NCN/ZCNOpen symbols: Result of t h e model calculation described in the text.
tem due to shell effects. Since it is not well understood, how the yields of the different fission channels are determined in the dynamic evolution of the fissioning system, it has become a standard to determine the parameters of the fission channels from a fit to the data by assuming that each of the independent fission channels is characterised by a Gaussian-like peak in the mass or element distribution and a specific elongation of the scission configuration which determines the total kinetic energy. Figure 3 shows the result of a fit to six selected systems, covering the transition from asymmetric fission to symmetric fission. Obviously, the measured data can well be represented by the superposition of the three independent fission channels which also appear in heavier systems. However, we observe two remarkable features. Firstly, the positions of both asymmetric fission channels appear to be astonishingly constant in proton number at Z = 53 and Z = 55, respectively, although the neutron number of the fissioning system varies. Secondly, the kinetic energy of the super-long fission channel approaches that of the standard II channel for the lightest systems. This shows that the scission-point configuration of the super-long channel becomes more compact with decreasing mass number of the fissioning nucleus. This is a sign for the influence of shell effects also in symmetric fission. The theoretical work on structure effects in fission presently concentrates
306
Figure 3. Element yields (left part) and average total kinetic energies (right part) as a function of the nuclear charge measured for fission fragments of several fissioning nuclei after electromagnetic excitations. T h e data points are compared to the result of a simultaneous fit (full lines) with 3 fission channels. The yields are defined as t h e sum, and t h e total kinetic energies are defined as the mean value of the individual contributions of the different channels. T h e super-long, standard I and standard II channels correspond to the symmetric, the inner asymmetric and the outer asymmetric peaks (dashed lines), respectively, in the yields and to the lower, upper and middle curve (dashed lines), respectively, in the total kinetic energies.
on the most realistic description of the shape-dependent potential-energy surface (e.g. Refs. 19 ' 20 ). The results look complicated, and the minimisation
307
with respect to higher-order shape distortions even introduces hidden discontinuities. These discontinuities make it even more difficult to perform full dynamical calculations in order to obtain quantitative predictions of the isotopic distributions of fission fragments. Up to now, these calculations rather serve as a guide to qualitatively relate the structures in the data to the structures in the potential-energy landscape. Since theory cannot yet provide us with a quantitative prediction, we tried to understand the data with a semi-empirical approach. The basic idea of our approach has been inspired by considerations of Itkis et al.21. We consider the fission barrier under the condition of a certain mass asymmetry. The height of the fission barrier V{A) is calculated as the sum of a liquid-drop barrier and two shells. The liquid-drop barrier is minimum at symmetry and grows quadratically as a function of mass asymmetry. The shell effects appear at N = 82 and N » 90. A more detailed description of the model is given in Ref.22. This picture provides us with an explanation for the predominance of asymmetric fission of the actinides. In 234 U like in most of the actinides, the lowest fission barrier appears for asymmetric mass splits. Approaching 264 Fm, the shell effects at N = 82 in both fragments join, giving rise to a narrow symmetric mass distribution. In lighter nuclei, the influence of these shells on the fission process is weakened, because they add up to the higher liquid-drop potential at larger mass asymmetry. In 2 0 8 Pb, the fission barrier is definitely lowest for symmetric mass splits. A more quantitative description of this schematic model is given in Fig. 3. The mass yield Y(A) is assumed to be proportional to the phase space p(A) available above the fission barrier at a certain mass split. The initial excitation energy E* above the mass-dependent barrier V(A) is available for intrinsic excitations. The shell effect in the level density is washed out with energy as proposed by Ignatyuk et al.23. The stiffness of the underlying liquiddrop potential is deduced from a systematics of the width of measured mass distributions 24 . The shells are modelled in a way that the calculated yields Y(Z) for 2 2 7 Th are reproduced. Now the model is applied to other nuclei ( 224 Ac and 2 3 0 Pa) without any further adjustment. The shells move up and down on the liquid-drop potential just a little bit due to the shift in neutron number of the fissioning nucleus. These tiny variations are sufficient to substantially modify the shape of the element distribution just as much as the experimental distributions change. This good reproduction of the data is a strong argument that the global variations of the potential-energy surface as a function of the fissioning system give the correct explanation for the basic features of the transition from asymmetric to symmetric fission.
308
N=90
N=82 N=82
Figure 4. Measured element yields compared to t h e model predictions (upper parts), and the assumed variation AV of the fission barrier as a function of the nuclear charge of one fission fragment with respect to the fission barrier for symmetric splits (lower parts).
Figure 5 presents the element distributions, calculated with the same model, for all measured fissioning systems. There is an astonishingly good agreement with the experimental data for the whole systematics shown in Fig. 1. This success of the very simple model might indicate that the dynamics of the fission process tends to wash out the influence of the details of the potential-energy landscape. It is to be expected that due to the inertia of the collective motion the process does not feel every wiggle in the potential energy but rather takes a smooth trajectory. Two specific features of the data, however, are not reproduced. Firstly, the ordering parameter of the calculated element distributions is not the mass but rather the neutron number. Secondly, the heavy fission component is not found to be constant at Z = 54 is as indicated in Fig. 2. This remarkable finding puts an important constraint on the theoretical description of the fission process. It may indicate that the shell effects in the proton subsystem play a more important role in asymmetric fission than currently assumed. 4
Summary
Nuclear fission is a unique laboratory due to a very specific feature which is not found in other systems: The electric charge in nuclei is homogeneously
309 -
-
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1
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' - I
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-t->
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89
- Pa
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132
133
134
135
136
137
•
i
i
138
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Neutron number Figure 5. Calculated element distributions of fission fragments from electromagneticinduced fission of 28 systems from 2 2 1 Ac to 234 U. See text for details.
distributed over the whole volume. This gives rise to a true fission process which is essentially symmetric. Shell effects in the order of a few MeV lead to very strong structural effects in the yields and in the kinetic energies of the fragments. Nuclear fission is thus a sensitive tool to investigate shell effects at large deformations. Experiments with secondary beams using elaborate experimental installations available at GSI opened up new possibilities for experimental studies of nuclear fission. Element yields and total kinetic energies have been determined for 70 fissioning systems from 2 0 5 At to 2 3 4 U. In this way, new systematic results for a continuous region of fissioning systems have been obtained. The transition from symmetric to asymmetric fission has been traced back to the global features of the potential-energy landscape in the vicinity of the fission barrier. As a puzzling result the element distributions scale with the mass number of the fissioning system, and the heavy component of asymmetric fission is found to be centred at Z = 54 in all systems. In contrast to previous understanding, the data seem to indicate that shell effects in the proton subsystem play a major role in the fission process. Moreover, the super-long fission channel becomes more compact with decreasing mass of the fissioning system, demonstrating the influence of shell effects in symmetric mass splits, too.
310
References 1. R. Vandenbosch, J. R. Huizenga, Nuclear Fission (New York: Academic), 1973. 2. The Nuclear Fission Process, C. Wagemans, ed., CRC Press, London, 1991. 3. B. D. Wilkins, E. P. Steinberg, R. R. Chasman, Phys. Rev. C 14 (1976) 1832. 4. D. C. Hoffman, M. R. Lane, Radiochimica Acta 7 0 / 7 1 (1995) 135. 5. H. J. Specht, Phys. Scripta 10A (1974) 21. 6. I. Nishinaka et al, Phys. Rev. C 56 (1997) 891. 7. I. V. Pokrovsky et al., Phys. Rev. C 60 (1999) 041304. 8. I. V. Pokrovsky et al., Phys. Rev. C 62 (2000) 014615. 9. K.-H. Schmidt et al., Phys. Lett. B 325 (1994) 313. 10. H.-G. Clerc et al., Nucl. Phys. A 590 (1995) 785. 11. A. R. Junghans et al., Nucl. Phys. A 629 (1998) 635. 12. K.-H. Schmidt et al., Nucl. Phys. A 665 (2000) 221. 13. G. Baur, C. A. Bertulani, Phys. Rev. C 34 (1986) 1654. 14. K. F. Flynn et al., Phys. Rev. C 5 (1972) 1725. 15. A. Turkevich, J. B. Niday, Phys. Rev. 84 (1951) 52. 16. V. V. Pashkevich, Nucl. Phys. A 169 (1971) 275. 17. M. G. Mustafa, U. Mosel, H. W. Schmitt, Phys. Rev. C 7 (1973) 1519. 18. U. Brosa, S. Grossmann, A. Mller, Phys. Rep. 197 (1990) 167. 19. V. V. Pashkevich, Nucl. Phys. A 477 (1988) 1. 20. P. Moller, A. Iwamoto, Phys. Rev. C 61 (2000) 047602. 21. M. G. Itkis et al., Sov. J. Part. Nucl. 19 (1988) 301. 22. J. Benlliure et al., Nucl. Phys. A 628 (1998) 458. 23. A. V. Ignatyuk, G. N. Smirenkin, A. S. Tiskin, Yad. Fiz. 21 (1975) 485 (Sov. J. Nucl. Phys. 21 (1975) 255). 24. S. I. Mulgin et al., Nucl. Phys. A 640 (1998) 375.
311
EMISSION OF LIGHT C H A R G E D PARTICLES F R O M FISSION F R A G M E N T S OF U R A N I U M N U C L E I I N D U C E D B Y SLOW N E G A T I V E M U O N S A N D P I O N S , P R O T O N S W I T H T H W E N E R G Y OF 153 A N D 1000 M E V A N D N E G A T I V E P I O N S A T 1700 M E V G.E.BELOVITZKIY, O.M.SHTEINGRAD Institute for Nuclear Research of RAS, Moscow, Russia Thefissionof nuclei induced by high energy hadrons is accompanied by the emission of particles with charge Z=l and 2. These particles are emitted at different stages of fission (before, during and after fission from fragments), and their investigation can provide new information on fission mechanism and the properties of excited fragments. In this paper presented investigation of emission of LCP from fragments of fission Uranium nuclei by slow negative muons 1' and pions , protons with 153 and 1000 MeV 2 - 4 and 1700 MeV negative pions 5 . Early the emission from fragments of particles with charge Z=l and 2 was observed in the spontaneous fission of 252Cf. In this work it was observed that protons were emitted from completely accelerated light fragments.
1
Experiment
We used the Uranium-filled self-supporting photo-emulsion of thickness up to 300/xm. They had a low sensitivity and mostly detected protons with energy less than 15 MeV. These emulsions were exposed by: beam of stopping negative muons and pions in the PSI (Switzerland); protons beam of energy 153 MeV in ITEP (Moscow),and of energy 1000 MeV (Gatchina); pion beam of 1700 MeV in ITEP. The photo-chemistry procedure of emulsion and method of microscopy were described anywhere 1 , s . The ranges of light (J?j) and heavy fragments {Rh) and LCP were measured for fission accompanied by the LCP emission. Both the angles between a track and initial beam and the angles between themselves tracks, as well as the ionization produced by LCP were measured. To determine the masses and energies of LCP we analyzed the tracks of LCP which stopped in the emulsion. The measured ranges and ionization allowed to determine masses and energies LCP using calibration curves 1 , s . If LCP were not stopped in the emulsion the tracks of these particles were used to obtain various angular distributions. All results were recorded by a computer data acquisition. The special code calculated the true ranges and energies of LCP, the fragment ranges and all angles under consideration. These data stored in the direct access files
312
wjuaa
y*
Figure 1. Angular distribution of protons (Vp) with respect to the motion direction of the heavy fragment (V^) for different values of fission asymmetry: a) Mh/Mi < 1.3; b) Mh/Mi > 1.3; c) Mh/Mi > 1.0.
were analyzed programs allowing to reconstruct various angular and energy distribution taking into account several additional parameters. The fragment range relations Ri/Rh is proportional to the fragment mass relations Mh/Mj (Mh and Mi are the masses of heavy and light fragments). This fact allows to investigate the dependence of LCP angular distribution via the fission asymmetry. 2 2.1
Results The emission of protons from fragments of Uranium fission by slow negative muons
Fission by slow negative muons is occurred in results by one or two nucleon absorption with average excitation energy 16 MeV and 60 MeV. As a result fission can have a low or high excitation energy. Experiment. 3 • 105 fission were analyzed. 1056 events accompanied by LCP emission, and among them, 226 protons of energy less than 10 MeV were observed. Fig. 1 a shows that for symmetric fission Mh/Mi < 1.3, the protons mainly are emitted in the direction of motion of light fragments. In this case we suppose that average excitation energy of a nucleus is 16 MeV, not fare
313
Figure 2. Angular distribution of LPC with respect to the motion direction of the heavy fragment (V),) for different values of fission asymmetry: a) M^/Mi < 1.3; b) Mh/Mi > 1.3; c) Mh/Mt > 1.0.
from spontaneous fission energy. Fig. 1 b shows that for asymmetric fission Mh/Mi > 1.3 and the protons mainly are emitted in direction of motion of heavy fragments. In this case probably the average excitation energy of a nucleus is 60 MeV. Total probability of proton emission from fragments is equal to 8 • 1 0 - 4 . 2.2
LCP Emission from fragments of Uranium fission by slow negative pions and 153 MeV protons 2 ~ 4
In these two experiments we obtained the similar results. From this fact we concluded that an emission of LCP from fragments mainly depends on an energy introduced in nuclei. An excitation energy of nucleus lies between zero and 140 (153) MeV for slow pions and 153 MeV protons and average value between 60 and 70 MeV. Therefore we presented the data for fission by 153 MeV protons. About 40000 fission by 153 MeV protons were analyzed. 2900 fission, accompanied with one LCP emission were found. Fig. 2 a - 2c show that number of LCP emitted to direction of motion of the heavy fragments increases with the increasing the Mh/Mi. Fig. 2 c shows that in asymmetric fission LCP are emitted predominantly in the direction of motion of heavy fragments. The
314
Figure 3. Angular distribution of LPC with respect to the motion direction of the heavy fragment (Vj,) for different values of fission asymmetry: a) M^/Mi < 1.3; b) Mh/Mi > 1.3; c) for all values of Afh/Afj. The histograms are plotted for fission by 1700 MeV pions (solid line) and by 153 protons (dashed line).
probability of emission of LCP from heavy fragments is equal to (2.4 ± 0.1) • 1 0 - 2 per a fission. This is 30 times more than in fission by slow muons. If LCP were emitted isotropically in the c.m.s. from completely accelerated fragments, then the ratio of the number of LCP emitted in the direction of motion of heavy fragments in the (l.s.) to the number of LCP emitted in the opposite direction would be 1.7. In experiment, we find this ratio to be equal to 4. We think that the predominance of LCP emitted in the direction of motion of the heavy fragment is due mainly to a large deformation of the this fragment and not to the kinematics. According 6 a nucleus deformation favors the emission of LCP in the direction of the deformation axis. We observe the emission of LCP only in one direction. This fact indicates that heavy fragments undergo a large and asymmetric deformation whose magnitude increases with increasing mass of the fragment. According 7 the total excitation energy of a fragment (the sum of the deformation energy and the internal energy) is considerably higher for heavy fragments. This leads to predominant emission of LCP from heavy accelerated fragments.
315
Wffh
lia....;..!.!*..-^ .»,.T$.^,.l'lr.i..» i,,..,.1
4
$
n
EP, MeV Figure 4. Proton energy distribution for fission by pions (solid line) and by 153 protons (dashed line).
2.3
Emission LCP from fragments of uranium fission by 1000 MeV protons and 1700 MeV negative pions 5
In this two experiments were obtained equal results about angular distribution of LCP. Therefore we presented only data for 1700 MeV pions. We analyzed 2280 fissions. Among them, 1265 fission were accompanied by the LCP emission. The number of LCP per fission ranged from one to ten. The total number of detected LCP was 2210.The angular distributions for all LCP accompanying the fission by pions are presented in Fig. 3 a, b, c (solid histograms) at different values of fission asymmetry. Almost all distributions are isotropic. For comparison the data for fission of uranium nuclei by 153 MeV protons (from Fig. 2) are shown in the figures by dashed lines. In the latter case, we observe the prevailing LCP emission along the direction of motion of heavy fission fragments 3 ' 4 . For the pion-induced fission, the isotropic angular distribution of LCP (the same is for protons less then 10 MeV) emitted by fragments can be explained, if we assume that LCP are emitted from non-accelerated fragments immediately after fission, or on earlier stage .This conclusion appears to be very important since in all previous experiments LCP were emitted from accelerated fission fragments i-2-3-4.5.8.
316 Table 1. Data concerning the number of LCP emitted from fragments at different values of multiplicity m. Multiplicity of LCP m 0 < m < 10 1 <m< 3 m > 4
Number of fissions 1575 690 182
Total number of LCP 1093 635 458
Number of LCP per fission 0.70 ± 0.03 0.90 ± 0.04 2.50 ± 0.20
Additional argument presented on Fig. 4. On this figure we shown the energy distributions of protons accompanying the fission by 1700 MeV pions (solid histogram) and 153 MeV protons (dashed histogram). The dashed line in Fig. 4 illustrates that the peak in this distribution is shifted by about 2 MeV toward higher energies because LCP emitted from acceleration fragments. From fig.4 we concluded that 70% of all LCP emitted from fragments of pion fission. The number of LCP emitted from pion Gssion fragments. The data concerning the number of LCP emitted from fragments at different values of multiplicity m are presented in the following table. The number of LCP emitted from fragments increases with multiplicity, and at m > 4, each fission is accompanied by the emission of more than two LCP. Discussion. After fission neutrons are emitted from the accelerated fragments on fission induced both by low- and high-energy particles 9 , 1 °. In fission by high-energy particles (1000-1700 MeV), in contrast to neutrons, LCP are emitted from non-accelerated fragments. During the interaction of uranium nucleus with 1700 MeV pion, the nucleus emits more than 20 nucleons before the fission. Therefore the produced fragments have a large excess of protons which have less binding energy. These facts and others cause in combination with the high excitation energy of fragments can cause the LCP emission from non-accelerated fragments. 3
Conclusion
In the spontaneous fission LCP are predominantly emitted from the accelerated light fission fragments and their yield is about I O - 5 per fission event 8 . On fission by slow negative muons LCP are emitted from the accelerated light and heavy fragments with the yield of 8 • 1 0 - 4 l'2. On fission by slow pions and 153 MeV protons, LCP are mainly emitted from the accelerated deformed heavy fission fragments, and their yield about 2 • 10~ 2 per fission
317
event 2 ' 3 ' 4 . Finally, the fission by 1000 MeV protons and 1700 MeV pions is accompanied by the LCP emission from the non-accelerated fragments with yield about 0,6 and 0.7 per fission accordingly 5 . We see that the characteristic parameters of the fission fragments such as excitation energy, composition (the number of protons and neutrons), deformation, the time and number of emitted LCP and their angular distribution, etc. vary significantly with the increase in excitation energy of residual nuclei and nuclei undergoing fission. Study emission of LCP from fragment of fission by particles different energy open possibility obtain new information about properties nuclei far from region of /3-stability. References 1. G.E.Belovitzky, V.N.Baranov, V.N.Valishina, N.V.Maslennikova, K.Petitjean, Yad. Fiz. 43, 1057 (1986) 2. G.E.Belovitzky, V.N.Baranov, and O.M.Shteingrad, Yad. Fiz. 57, 2140 (1994) 3. G.E.Belovitzky, V.N. Baranov, and K. Petijean, Yad. Fiz.55, 2139 (1992) 4. G.E.Belovitzky et al., Yad.Fiz. 58, 2131 (1995) 5. G.E.Belovitzky and O.M.Shteingrad, Izv.RAN. Ser. Phys. 64,68(2000); Kratk. Soobshch.Fiz.lO, 9 (1998) 6. A.Jwamoto et a/., Z.Phys. A:At.Nucl. 338, 303 (1991) 7. E. Cheifetz et al, Phys. Rev. C 2, 250 (1970) 8. A.Shubert et al, Z.Phys.A: At. Nucl. 338, 115 (1991) 9. E.Mordhorst et aZ., Phys. Rev. C 43, 716 (1991) 10. W.Schmid et al, Phys. Rev. C55, 2965 (1997)
318
POSSIBLE E X P L A N A T I O N OF THE D I F F E R E N C E IN N U C L E A R FISSION I N D U C E D B Y THE I N T E R M E D I A T E ENERGY P R O T O N S A N D N E U T R O N S . V.E.BUNAKOV Petersburg Nuclear Physics Institute, 188350 Gatchina, L.V.KRASNOV Physical Institute of Petersburg State University,
Russia
198904,St.Petersburg,Russia
A.V.FOMICHEV Khlopin Radium Institute, 194021, St.Petersburg,
Russia
A possible explanation is given of the experimentally observed differences between the the fission cross-sections in the reactions induced by the intermediate energy (hundreds MeV) neutrons and protons.
The experimental studies of nuclear fission stimulated in the recent years by the development of the transmutation program demonstrated (see e.g. V.P.Eismont et al. 1) that the cross section of fission induced by the intermediate (hundreds MeV) protons is usually higher than that caused by the same energy neutrons. While this difference is rather small (30 -=- 40%) for highly fissioning nuclei (e.g. U or Th), it might reach a factor of about 3 for lighter nuclei (like Pb or Bi) - see Fig. 1. The physics of the nucleon-induced fission seems to be fairly well understood and reasonably well described by the cascade-evaporation model. At the initial fast stage of the process the incident nucleon performs a few pair-wise collisions with the target nucleons, sharing its energy and creating secondary fast particles. Since the nuclear radius is comparable with the mean free path of these particles, they usually carry the major part of the incident energy away from the target. The remaining part of the incident energy is left in the residual nuclei in the form of the holes in the Fermi-sea and is rapidly thermalized. Thus by the end of the initial fast stage we have a number of excited nuclides, which cool down at the next evaporation stage by the competing processes of fission or particle evaporation. According to the experimental data evaluation of V.S.Barashenkov and V.D.Toneev 2 the reaction cross-sections of the intermediate energy protons and neutrons on, say, 208Pb are practically equal, which is a reflection of the charge-independence of nuclear forces. Thus one might expect that the fast stage of the process goes in the same way
319
100
E , , MeV
Figure 1: Comparison of fission cross-sections of heavy nuclei induced by protons and neutrons with the same energy. Left scale gives the absolute fission cross-sections. Open circles are the (n,f) reaction. Solid circles are the (p,f) reaction. Dashed lines and the right scale represent the ratios of (p,f)/(n,f) cross-sections. All the d a t a are the evaluation of the different experimental results performed by A.Prokofiev and published in V.P.Eismont et al.1.
for neutrons and protons. Therefore it seems rather puzzling that the fission cross-sections for them differ by the factor of 3. The only seemingly obvious difference between the incident neutrons and protons is the possibility for a proton to produce the additional excitation of he giant resonances in the target by its Coulomb field. The detailed theoretical
320
analysis C.Djalali et. al. 3 of the inelastic scattering of protons on 208Pb shows that the major contribution to the excitation of Giant Dipole Resonances comes indeed from the Coulomb (and not the nuclear) forces. Therefore a fraction w =
= 1
(W*f(E*
~ Bj) ~ l)exp(2 v /a / (i?* - B})) Anaf exp(2^aQE*)
[
'
Here E* is the excitation energy of fissioning nucleus, ao and aj are the parameters of level density of this nucleus at equilibrium deformation and at the saddle-point, correspondingly. For the estimates of fission barrier Bf one can use different approximations of the liquid drop model together with various "irregular" quantum corrections. For our qualitative estimates of the fissility increase due to the additional GDR excitation we assumed the liquid drop approximation formula: Bf = lA2'3f(x)
(2)
Here 7 ~ 15 4- 20 MeV; the fissility parameter x — (Z2/A)/(Z2/A)crit — {Z2/A9A); f(x) = 0.728(1 - xf - 0.661(1 - x)4 + 3.330(1 - xf. The distribution of excitation energies of the residual nuclei after the fast stage of the process for 200 MeV protons and neutrons interacting with 208Pb calculated in CEM is shown in Fig. 2 by open circles and crosses, correspondingly. The additional excitation of 13 MeV increases in this case the fission cross-section from 22.2 mb up to 46.0 mb, but the small value of w — 0.54/1521 fa 4 • 1 0 - 4 makes this increase quite negligible. However the calculations with CEM code without any modifications produced for 208Pb the ratio of (T^/er^ m 2.3, which is quite close to the experimental data. This unexpected result demanded to find what kind of physics allows the cascade-evaporation model to reproduce such a large difference in the fission cross-sections induced by protons and neutrons. As we see from Fig. 2, the excitation energies of the residual nuclei in the proton and neutron cases are the same. Therefore our next step was to compare the distributions of the residual
321 0
50
100
Exitation energy (MeV) Figure 2: Excitation energy distribution for the residual nuclei after the fast stage of reactions p + 2 0 8 Pb (circles) and n -f208 Pb (crosses). AiV — number of the residual nuclei in the 5 MeV bin.
nuclei in those two cases along A and Z. Those distributions are shown in Tables 1 and 2 for incident neutrons and protons, respectively a. C o m p a r i n g the tables, we see t h a t the two distributions strongly resemble each other. Their major difference is the presence of residual nuclei with Z — 83 (Bi isotopes) in the case of proton-induced reactions and their absence in neutron-induced process. The origin of these isotopes is quite obvious - they are formed when the incident fast proton looses the major part of is energy and gets stuck in the target after the emission of one or more neutrons. T h e fissility p a r a m e t e r s x for these isotopes are larger than for all the other residual nuclei, which means t h a t their fission barriers are the lowest. We have checked their contribution into the total nssioncross-section, changing all the Z = 83 residual nuclei in the C E M calculations by the Z = 82 ones. In spite of the fact t h a t Bi isotopes make u p only 14% of all the residual nuclei, such a change reduces the fission cross-section by a factor of 1.7, thus covering the major p a r t of the initial difference in fission cross-sections. T h e rest of t h e difference is explained by the corresponding decrease by 14% of the yields of the Z < 82 isotopes with higher fission barriers in the case of incident protons ( 4 1 % instead of 55% in the neutron case). Consider now t h e case of 200 Mev nucleons interacting with the 238U. As a
fr means fraction.
322 Table 1: A, Z distribution of residual nuclei for n + 2 0 8 Pb. 199 19 12 2 0 0
200 28 95 17 5 0
201 86 150 122 16 0
202 90 520 325 181 0
203 157 532 957 348 0
204 87 905 1141 1105 0
205 55 403 1648 1224 0
206 3 338 1312 2381 0
207 0 31 1063 2398 0
208 0 0 191 989 3
fr
209 A/Z 0 79 0 80 0 81 0 82 5 83
0.03 0.16 0.36 0.46 0.00
209 A/Z 0 79 0 80 0 81 2 82 0 83
0.01 0.11 0.29 0.45 0.14
Table 2: A, Z distribution of residual nuclei for p -f208 Pb. 199 27 39 9 1 0
200 23 118 42 19 2
201 58 176 200 60 4
202 59 427 384 351 21
203 74 343 934 592 111
204 31 562 980 1516 200
205 17 234 1310 1345 431
206 0 178 902 2051 450
207 0 0 741 1894 735
208 0 0 0 797 677
fr
Table 3: A, Z distribution of residual nuclei for n -f238 U. 229 29 31 12 3 0
230 39 137 43 19 0
231 98 236 251 42 0
232 88 597 443 332 0
233 127 487 1077 515 0
234 65 715 983 1404 0
235 45 308 1465 1257 0
236 4 278 997 2339 0
237 0 46 952 2293 0
238 0 0 248 962 4
239 A/Z 0 89 0 90 0 91 1 92 0 93
fr 0.03 0.15 0.34 0.48 0.00
Table 4: A, Z distribution of residual nuclei for p + 2 3 8 U. 229 21 47 37 7 0
230 20 150 81 60 1
231 52 163 302 160 22
232 40 430 428 636 82
233 52 310 963 796 283
234 20 410 755 1605 318
235 12 137 1060 1235 615
236 0 120 657 1918 530
237 0 0 719 1612 785
238 0 0 0 815 741
239 A/Z 0 89 0 90 0 91 1 92 2 93
fr 0.01 0.09 0.26 0.46 0.18
we see in Fig. 1, the experimental ratio Cp/
323
In order to understand, why don't they cause the significant changes as in the lead case, consider the ratio of fission probabilities R for the nuclei A, Z+1 and A, Z. Using Eq. (1) and denoting the difference of the barrier heights as A = Bf(Z + 1) - BS{Z), we obtain: R; x exp 2Va
E*
y/a(E*-Bf(Z)) Bf (Z + 1) - yJE* -Bf(Z
•y/aA lE*-Bf(Z + l) exp y/E*-Bf(Z+l) E*-Bj{Z)
+
l)-A
(3)
If we consider now the x dependence of the liquid drop barrier function f(x) (see, e.g. Fig. 6.56 in A.Bohr and B.Mottelson 5 ), we shall see that both the barrier heights Bf and t h e differences A are much larger in the Pb region (x xi 0.66) than in the U one (x f« 0.73). In the lead region Bf « 18-=-20 MeV and A « 3 -^ 4 MeV. In the U region Bf w 8 -=- 9 MeV, while A ra 1 -=- 2 MeV, which is comparable with the value of the quantum corrections to the liquid drop barrier. Therefore the ratios R in U region are much smaller than in the lead one (according to the simplified Eq. (3) for E* = 40 MeV they are 6 times smaller). This explains the experimental fact that the proton induced fission cross-section for the U region is much closer to the neutron induced one. The work was partially supported by ISTC (project 609). References 1 V.P. Eismont et al., in Up-to-Date Status and Problems of the Experimental Nucleon-Induced Fission Cross-Section Data Base at Intermediate Energies. Proc. 3rd Int. Conf. On Accelerator Driven Transmutation Technologies and Applications, Praha (Pruhonice), Czech Republic, June 7-11, 1999 (CD ROM publication, paper P-C23) 2. V.S. Barashenkov and V.D. Toneev,Interaction of High Energy Particle and Nuclei with Atomic Nuclei. (Atomizdat, Moscow, 1972) 3. C.Djalali et.al.. Nucl.Phys. A 380 42 (1982) 4. S.G.Mashnik, User Manual for Code CEM95 (IAEA 1247/01 1995) 5. A.Bohr and B.Mottelson, Nuclear Structure, V.II, (Benjamin, New York 1974)
324
SCISSION C O N F I G U R A T I O N S IN T H E COLD FISSION §. MI§ICU Institut fur Theoretische Physik der J. W. Goethe Universitat, Frankfurt am Main, Germany, [email protected] P. QUENTIN AND N.PILLET Centre d'Etudes Nucleaires de Bordeaux-Gradignan, Universite Bordeaux I and IN2P3/CNRS, France The scission configuration in the cold fission is studied in the frame of a molecular model consisting of two aligned fragments interacting by means of Coulomb and nuclear proximity forces. As a study case we choose the binary cold splitting of 252 Cf in two even-even nuclei, i.e. 148 Ba and 104 Mo. Their deformation energies are computed via the constrained Hartree-Pock + BCS formalism. For a fixed tip distance between the two fragments, and a fixed excitation energy one are lead to one or several fission channels, which we interpret as being high or low probable depending on how the excitation energy is distributed among the fragments.
1
Introduction
In the cold fission of a heavy nucleus no neutrons are emitted and the energy released in the reaction is converted almost entirely in the kinetic energy of the fragments. Using cluster like models 1,2 , some features of this process have been very recently explained satisfactory . The basic assumptions of such models is that at scission the fragments have very compact shapes, close to the ground state and thus they are carrying very small excitation energy. The scission configuration consists of two co-axial fragments with a certain distance d between their tips. In the model proposed by Gonnenwein et al. 3 , the cold fission was studied by determining the distance dmin of the closest approach between the two fragments, when the decay energy Q equals the interaction energy. This model predicted the smallest tip distance (bellow 3 fm) for fragments, with mass numbers between 138 and 158 and around the double magic 132 Sn, emerging in the sf of 252Cf. Small tip distances were interpreted as a sign of cold fission due to the higher interaction energies at scission. In the past the scission-point model succeded also to explain roughly some basic observables of low-energy fission 4 such as the relative probabilities of formation of complementary fission fragment pairs, and the general features of the distributions of mass, nuclear charge and" kinetic energy in the fission of various nuclides, ranging from Po to Fm. Bellow we generalize the static scission-point concept of nuclear fission
325
model in such a way that instead of describing the fragments as two deformed nearly touching liquid drops with shell corrections taken into account, we incorporate the fragments shell structure by means of the self-consistent Hartree-Fock method with BCS pairing correlations (HF+BCS). For the given binary splitting 252 Cf—> 104 Mo + 148 Ba we computed the fragments deformation by constraining their quadrupole moments and considering a coaxial configuration of the two fragments with distance d between their tips. We are also employing the approximation that the interaction energy at scission is transformed into kinetic energy of the fragments at infinity. Thus, all the excitation energy present in the fissioning system is accounted by the deformation energy. This amounts to neglect that part of the energy released at the descent from saddle to scission which is spent on heat, i.e. we are confined to the low-energy domain of sf including the limiting case of cold fission. By using the above mentioned constraints we were able to deduce the possible shapes of the fragments and excitation energy sharing for various tip distances and total excitation energies E*. 2
Energy balance at scission
In the sf of 252 Cf the fragments are born with a certain deformation and will carry a total excitation energy E*, gained during the descent from saddle to scission which will be dissipated by means of neutron and gamma emission 5 Q = Vsciss + TKEpte
+ J^ Edef(l) 1,2
+ X ] Eint(l), 1,2
(1)
where V^ciss = Vcoui + Kiuci represents the fragments interaction energy at scission. For Vcoui we choose the form corresponding to two diffuseless deformed homogenously charged nuclei with collinear symmetry axes with a distance R between their centers 6 , whereas for the attractive nuclear potential we choose the proximity formula for two nuclei with a finite surface thickness 7 . The prescission kinetic energy TKEpTe is taken to be zero, an assumption which proved to be reasonable for low-energy fission 4 . Also, that part of the excitation energy which is transformed into internal excitation energy E*nt, is neglected. According to time dependent quantum many-body calculations, the intrinsic excitation energy accounts for less than 15% of the collective energy gain in going from the saddle to the scission 8 . That part of the excitation energy which goes into the deformations of the fragments was denoted in eq.(l) by Edef • In the study of sf properties, 2?def is taken usually as a sum of the liquid drop model (LDM) energy, and the shell and pairing corrections 9 . In this paper the deformation energy Edef of the fissioning system at scis-
326
sion is referred to the HF+BCS energy of the two fragments in their ground states (g.s.) 2
Ede{ = J2
{£HF+BCS(M,
Zu Pi) - EHF+BGs(Ni, Zupis)}
.
(2)
i=l
The LDM, which is based on a semiclassical description of the nuclei, supplemented by the shell-effect corrective energy, is only a poore substitute for a self-consistent calculation 10 . One of the main advantages of the self-consistent HF+BCS calculation is that it provides simultanously both the single-particle and semiclassical properties of nuclei. The general properties of the HartreeFock method were reviewed in n . In our study for the HF part of the interaction we choosed the Skyrme interaction SIII 12 , which succeded to reproduce satisfactory the single-particle spectra of even-even nuclei. The difference between the binding energy computed with SIII and the experimental one appears to be, for a large number of nuclei, « 5 MeV 13 . It also produces a fairly well N — Z dependence of the binding energy 14 . The present study envisages nuclei that are not in a closed shell configuration. Thus, the level occupations will have a large effect on the solution of the HF equations. In the deformed HF calculations one have to optimize the basis which is choosen to correspond to an axial symmetric deformed harmonic-oscillator with frequencies wj_ and wz. Such a basis is characterized by the deformation parameter q = U>±/(JJZ and harmonic oscillator length b = ^/muJo/h, with w o — w l ^ z - The basis is cut off after Nmax major shells, where 10< Nmax <12 is a reasonable choice for the nuclei emerging in the sf of 252 Cf 15 . Eventually, the deformation energy curve is obtained by adding a quadratic constraint f-(<2 — Qo)2 to the energy functional 16 . Here Q0 is a specified targed value of the mass quadrupole moment. In Fig.l we represented the deformation energy curves of the nuclei 104 Mo and 1 4 8 Ba produced in the sf of 252 Cf for a range of deformations including the first prolate and oblate minima. 3
The distribution of excitation energy in the fission fragments
The scope of this section is to seek the configuration of the system at scission for a fixed excitation energy E*. According to eq.(l), the interaction energy of two fragments, with deformations 0i and fa .at scission, is related to the excitation energy through the relation
V(01,/32,d)=Q-E*,
(3)
327 148
104*
1184
sq
-0.5 -0.25
0
0.25
0.5
0.75
P2 Figure 1. The deformation energy curves of the nuclei 1 4 8 B a and 1 0 4 M o computed in the frame of the H F + B C S method with quadratic constraint for the mass quadrupole.
where d is the tip distance and enters in the theory as a parameter. We equate this last quantity with the asymptotic kinetic energy TKE(po). This relation is a consequence of the approximations that we made earlier, i.e. we neglected the prescission kinetic energy TKEpre and we forced all the available excitation energy to be stored into deformation E*(/31,p2) = ^def(A) + Ede{{(32).
(4)
Thus, for a given excitation energy we obtain two non-linear equations, i.e. eqs. (3) and (4). In Fig.2 we represented the excitation energy landscape (4), for the pair ( 104 Mo, 1 4 8 Ba). The deepest minimum corresponds to the prolate-prolate configuration ((3\ > 0,/32 > 0). At this point E* — 0 and fission proceeds by
328
Figure 2. Three-dimensional plot of the excitation energy E* for the pair ( 148 Ba, computed in the frame of the HF-fBCS method.
104
Mo)
means of only one channel, customarly known as true cold fission. This configuration has deformations &(U8Ba)=:0.270 and ^2(10*Mo)=0.364, differing by 10% from those computed in the frame of the finite-range liquid-drop model 17 . The non-linear equations, quoted above, admit this solution only for the tip distance d = 2.95 fm, a value very close to the border of 3 fin, allegated by the Tubingen group, bellow which cold fission occurs 2 . When we increase the excitation energy, an infinity of solutions arise according to eq.(4). They have to be identified with the geometrical locus of points with equal excitation energy. However, the second constraint (3) is limiting drastically the number of {Pi,fh) pairs. As an example we give in Fig. 3 the contour plots of the excitation energy and superposed on them the curve relating ft to 02 for the tip distance d = 3.25 fm at E* — 2 MeV, deduced
329
Figure 3. Graphical solution of the non-linear equations (3) and (4). The intersection of the solid curve with the contour lines provides two solutions in the particular case of the pair ( 1 4 8 Ba, 1 0 4 Mo), with tip distance d — 3.25 fm and total excitation energy E* = 2 MeV.
from eq.(3). The intersection of such curves with the contour lines of equal excitation energy will give the physical solutions to our fission problem, i.e. for a certain tip distance one get different scission configurations or channels. One generally get two ore more solutions which are located, mainly for low-excitation energies, in the quadrant with (3i, ft > 0. For pure cold fission (E* — 0) one get a solution for only one d, whereas for E* > 0 one get solutions for several values of d. As a matter of fact our investigation points
330
to different regions of the tip distance. Grossly they are spreaded between 2.65 fm and 5.5 fm for the neutronless fission. Naturally, one may ask next if all these solutions are likely to occur. For that one should look at the ratio of excitation energies between the two fragments. Calculations based on the cascade evaporation model predicted a ratio of the mean excitation energies E^lEl « 0.5 around the splitting 104/148 when approaching the limiting case of cold fission 18 . According to the same reference, disproportions in sharing of the excitation energy should be expected only in the vicinity of magic numbers, when one of the fragments, due to its shell closure, cannot be excited bellow a certain threshold of the excitation energy. Another interesting result is that for large tip distances (d =5.05 fm) one of the fragments emerges with oblate shape which means that the corresponding solution of the non-linear equations (3-4) is located in the second quadrant (the upper-left in Fig.3). 4
Conclusions
Based on a molecular model in which the scission configuration has to fulfill two main energetic constraints, namely that the interaction between the fragments is converted totally into asymptotic kinetic energy and that the excitation energy of the fissioning system is accounted only by the deformation energy, we carried constrained HF+BCS calculations at zero temperature for the nuclei emerging in the low-energy fission reaction. For a fixed excitation energy we varied the distance between the tips of the fragments. Each case admits one, two, three and even four solutions for the fragments deformations. The qualitative criteria which allowed us to select the valid scission configuration was based on the excitation energy distribution between the fragments. We discarded those configurations with a disproportionate ratio between the excitation energis of the two fragments as long as we do not deal with fragments close to magic numbers. Our analyse is predicting roughly two types of scission configurations, depending on the excitation energy present in the system. Prolate-prolate deformations are expected for tip distances between 2.65 and 3.25 fm. Around 5 fm prolate-oblate fragments deformations seems to be favoured. The present study was limited to only one of the observed splittings occuring in the cold fragmentation of 252Cf. By extending these calculations to some other tenths of binary splittings recorded in this reaction it will be possible also to compute the yields for different excitation energies. Untill now only yields at E* = 0 MeV have been reported theoretically 1 although the experiment provides yields up to several MeV of excitation energies.
331
References
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