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Folding angle and excitation energy of fragments from 235U(nth,f) and 252Cf(sf) reactions
Nuclear Physics A572 (1994) 294-316 North-Holland
NUCLEAR PHYSICS A
Folding angle and excitation energy of fragments from 235U~n~~,f)and 252Cf(sf) reactions T. Haninger, F.J. Hartmann, P. Hofmann +, Y.S. Kim, M.S. Lotfranaei, T. von Egidy Physik-Department, Technische Universitiit Miinchen, D-8.5748 Garching, Germany
H. h&ten
‘, A. Ruben
Institut fiir Kern- und Atomphysik, Technische U~i~~rsit~tDresden, D-01049 Dresden, Germany
Received 19 October 1993
Abstract
Coincident fragments from ?J(n,,,,f) and 252Cf(sf) reactions were investigated with a doublearm fission-fragment spectrometer and PIN-diode arrays. Based on the measurement of kinetic energy, velocity and direction of complementary fragments the total kinetic energy, the total mass, the individual fragment masses, the total number of emitted neutrons as well as the folding angle were deduced event by event. A nearly linear correlation between average folding angle and average total excitation energy of the fragments (and, consequently, the average number of neutrons) was found. Fragment deflection by prompt neutron emission is a~ordingly described by a complex statistical evaporation model in connection with a semi-empirical calculation of energy partition in nuclear fission. In addition, the folding-angle distribution due to ternary fission is estimated. All experimental fragment distributions and correlations are well reproduced by the model calculations.
lvey words: NUCLEAR REACTIONS 235U(n,Fl, E = subthermai; measured (~agmentxfragmen&coin, neutron spectra, E,,,, 0 of both fragments; deduced correlations between fragment mass, kinetic energy, velocity. Statistical evaporation model; enriched targets; PIN-diode arrays RADIOACTIVITY 252Cf (SF); measured (fragmentXfragment)-coin, neutron spectra; deduced average folding angle, total excitation energy linear correlation. Statistical evaporation model
1. Introduction
Slow-neutron-induced and spontaneous fission is known since more than fifty years ago. Mass, charge and energy distributions of fragments have been measured very carefully applying a large variety of experimental methods [1,2]. In spite of this r Present address: Stresemannplatz ’ Deceased.
llb, D-01309 Dresden, Germany.
0375-9474/94/$07.~ 0 1994 - Elsevier Science B.V. All rights reserved ~~D~O375-9474(93)EO686-3
T. Haninger et al. / Excitation energy of fragment
295
Fig. 1. Schematic and exaggerated demonstration of fragment deflection by neutron and y-ray emission (upper part) and by ternary fission with emission of a light charged particle (LCP). Angular distributions (in centre-of-mass system) are shown by dashed curves. Fragments (F), neutrons (n) and angles ((u) are marked with L and H indicating the light or heavy fragment, respectively.
effort, there are still many open questions in fission. For instance, the folding angle of the fragments has not yet been measured together with energies and masses, as far as we know. It is the purpose of the present publication to show that the folding angle of fission fragments contains very valuable information on the fission process, especially on the total excitation energy of the fragments. The fission axis in binary fission is defined by the directions of both primary fragments which have a primary folding angle of 180”. However, recoil effects due to prompt emission of neutrons and gamma rays change both directions in a stochastic way. Only the final folding angle is a direct observable. Multiple-neutron emission from both fast-moving and highly excited fragments gives the most significant recoil effect (l-3 neutrons per fragment). Multiple y-emission has a very small effect on the angle. Light charged particles (LCPs) emitted in ternary fission processes cause the largest deviations of the folding angle from 180”. All these processes are demonstrated in Fig. 1. The fragments, their energies, velocities and angles are measured with a newly constructed double-arm fission-fragment spectrometer using PIN-diode arrays [3]. This instrument was originally designed to investigate antiproton-induced fission [4,51. It is very efficient and precise for the present purpose. Finally, a recently developed computer code [6,7] is available which uses a statistical evaporation model with an energy partition based on a detailed energy balance at the scission point. This program yields all relevant information on the fission process. Thus, comparison of experiment and theory should give further insight into the physics of the fission process. The present investigation provides also useful nuclear data on real angular dispersions of fission fragments.
296
T. Haninger et al. / Excitation energy of fragment
2. Experiment The double-arm fission-fragment spectrometer, the electronics and the data evaluation is described in detail in ref. [31. Fig. 2 shows the arrangement for the measurement of thermal-neutron-induced fission at the Munich research reactor FRM. A subthermal neutron beam from a guide tube with a flux of nearly lo7 n/cm2 s enters the vacuum chamber with the target in the centre. Two circular 235U targets were used: 100 pg/cm2 U,O, with 1 cm diameter on 30 kg/cm2 carbon foil and 39.3 kg/cm2 metallic U with 0.5 cm diameter on 40 Fg/cm2 carbon foil. Most experiments for the present publication are made with the second thinner and smaller target. The source for the spontaneous fission experiment was produced by self-sputtering of 252Cf on a 94.6 kg/cm2 Ni foil. The active area had a diameter of 0.8 cm, but was covered by a diaphragm of 0.6 cm diameter. The fission rate was about 1000/s. A channel-plate detector close to the target gave the start signal for the time-of-flight (TOF) measurement. Two PIN-diode arrays (each 12 x 12 diodes, each diode 1 cm2, thickness 500 urn> are placed on opposite sides of the target. In order to avoid edge effects of the diodes, they were covered by copper-plate masks with 8.56 X 8.56 mm* holes. The distance from the target to a PIN-diode array was 120.9 cm, yielding an angular resolution of 0.54” and a solid angle of 0.12% of 4~. A distance of 60.8 cm was used for the time calibration of the spectrometer. In order to reduce periodicity effects due to the granular structure of the PIN-diode arrays and to enhance the information for very
n
neutron
beam tube
Fig. 2. Schematic drawing of the experimental arrangement. In this case the distance from the target to the PIN-diode array is 60.8 cm.
T. Haninger et al. / Excitation energy of fragment
297
small angles, both arrays were shifted in the same direction from the s~et~ axis by 1.7 cm. The 23SU(nI,,,f) measurement with the thin target and a distance of 120.9 cm from target to PIN-diode array gave 33000 coincident events which were used to determine the folding-angle distribution. The 252Cf measurement with the same geometry yielded 675000 coincident events for the folding-angle distribution. Additional measurements were made with various geometries to determine time and energy calibration, the pulse-height defect, the plasma delay and the energy loss in the target and in the backing (see ref. [31 for details).
3. Data evaluation 3.1. Masses and kinetic energies The coincident fission events were corrected for the pulse-height defect, for energy loss in the target and in the backing and for plasma delay as described in ref. [3]. The pulse-height defect and the mass resolution were determined together by a fit to radiochemical mass distributions [8,9] according to our new method [3]. The mass resolution was about 5 u FWHM. The results of both reactions, 235U(n,f) and 2s2Cf(sf), for average values of masses and kinetic energies of the light and of the heavy fragment, for post-neutron and pre-neutron emission, respectiveIy, are given in ref. 131.They are in very good agreement with previous data [lO,ll] and have frequently smaller errors. 3.2. Folding-angle distribution In the following discussion we use the folding angle 0 = 180” -real folding angle. The folding angIes of coincident fragments were calculated using three points, one on each PIN diode and one on the target. These precise points on each diode and on the target were determined for each event by a random-number generator in order to smooth the distributions. Angular distributions have to be corrected for the angular resolution of our spectrometer and for the angular efficiency. The correction for the angular resolution was obtained in the following way. The measured angular distribution G,(O,) has to be converted into the original one F(Oj). This correlation is purely geometrical and can be calculated as a transformation matrix M(Oi, Oi> with Monte Carlo methods. The original angular distribution was approximated with the function F(Qj)
=pl@,!‘”
eXp( -P3@j
+Pd@T)*
(1)
T. Haninger et al. / Excitation energy of fragment
298
With this function an approximative measured function G,(OJ can be determined,
Oi),
= f: I;(Oj)M(Oj,
G,(Oi)
(2)
j=l
The parameters pr, p2, p3 and p4 have been obtained by a least-squares G1(Oi) to G,(Oi) with x2
=
fit of
C [G,(@i)-Gl(@i)12 i=l
.
G,(@i)
Z and J are the numbers of angular bins. The fit results showed that the assumed function E;(Oj> reproduces the experimental data very well. The efficiency of detecting a given angle 0 depends also on the geometry of the spectrometer and can be calculated with Monte Carlo techniques. This angular efficiency is shown in Fig. 4 of ref. [3]. For the distance of 120.9 cm from the target to the PIN diodes the angular efficiency decreases nearly linearly from 100% at 0 to nearly 0% close to 8”. 3.3. Multiple scattering in the target and backing Multiple scattering in the target material and in the backing influences the folding-angle distribution and its width V. Several test measurements were performed using the target with d, = 39.3 ug/cm2. For instance, additional 238U foils (200 and 400 pg/cm2) were placed on one side of the target and the influence on the light (L) and heavy (H) fragment was separated. The results of the test measurements are shown in Table 1. In order to deduce the original width a,, of the distribution from the data in Table 1, the following assumptions were made:
a2=u,2+a2=u,2+a2 s
s,L +
Table 1 Average folding angles and widths statistical errors are given Test measurements
of the 235U(n, f) reaction
Additional 0
?3(ded
&l,
238U foils. Only
(@/cm’) 200
400
1.25 (1)
1.54 (2)
(deg)
1.10 (1)
1.28 (2)
0, (deg) u (deg) q_ (deg) uH (deg)
1.38 (2) 0.83 0.69 0.90
1.80 (3) 1.03 0.83 1.13
8,
0.90
238U foil
with additional
0.57
some
T. ~aninge~
et al. / ~citation energy of ~ag~ent
299
with the measured width CFand the additional broadening by multiple scattering of the tight and heavy fragments a$,, and uS,n. a, ’ increases linearly with the foil thickness d, i.e., Cr2=ad 9 s
(5)
where a is a constant. We can select those events where only light or heavy fragments suffer multiple scattering in additionai foils. In the case that only light fragments pass through the foils, the measured width rr,. can be expresses as
=ai+aH-+
d O+a,($d,+d)
=a,f+(a,+a,)~~d,+a,d,
(6)
where d, is the thickness of the target and the fission is assumed to occur on average at the centre of the target. Now aL can be obtained from the two a=-values at d = 200 and 400 pg/cm2 in Table 1. Exactly the same procedure for only heavy fragments can be followed to obtain aH: a:(d)
=~$+(a~+a,J-$d,,+a,d.
With the obtained values aL = 0.91 x 10e3 and aH = 2.4 x 10e3 &g/cm*)-‘, can calculate the contribution of the target (second terms in Eqs. (6) and o$$d,) = 0.27”. The original width is, then, deduced to be a, = 0.50”. Note that broadening of the distribution due to the target thickness 39.3 kg/cm* is negligible compared with the original width.
(7)
one (7)) the not
4. Model calculations The experimental data indicate a remarkable correlation between the average folding angle and the average number of prompt fission neutrons. As suggested above, neutron emission should be the predominant effect in forming the foldingangle distribution. Its adequate theoretical description has to account for the energy partition in nuclear fission in detail, i.e. in dependence on the fragment-mass asymmetry AL/A, and the total kinetic energy TKE of the fragments, in order to consider all measured correlations. 4.1. Energy partition in fission According to energy conservation, the total available energy in the fission process, i.e. the sum of the Q-value (to be calculated from mass tables) and the
I: Hanittger et al. / Excitation energy
300
of fragment
excitation energy Ezn of the fissioning compound nucleus, is equal to the sum of the total excitation energy E* and the total kinetic energy of the fragments:
(8) (The bars indicate the (weighted) averages over charge fragmentations). For given A,/A, and TKE, which are measurable quantities, E* can be deduced directly. However, its partition on both complementary fragments (E* = EC + Eg) is strongly influenced by shell effects of the nascent fragments at scission. The individual excitation energies, which are distributed around mean values, are obviously anticorrelated. Because of the general problems of establishing a firstprinciple theory of nuclear fission, a phenomenological model [7] is used to solve the present ener~-partition problem. In addition, empirical relations serve to describe the widths of the distribution in E”. In order to account for both intrinsic and collective degrees of freedom in the most crucial phase of the fission process, i.e. the descent from saddle to scission point, we start with a rather empirical speci~cation of the asymptotic energy terms TKEZ and E” in relation to certain energy terms at scission: TKE = E,,
+ EM 9
(9)
E* = Edef,L+ Edef,~+ Edis+ EH
T
where: Epre = pre-scission kinetic energy, E cou, = Coulomb repulsion energy corrected for the proximity term, E def = deformation energy of fragment (L, H) at scission,
E,, = dissipative energy released during descent from saddle to scission due to friction forces [6]. E heat is the intrinsic excitation energy (“heat”) at the second saddle (B) of the fission barrier with height E,,. Taking into account the pairing gap A at this point, we have E heat= Ezn -
Ef,a - A,
(11)
with the constraint Eheat 3 0, i.e. it vanishes in the case of spontaneous and subbarrier fission. The sum of Edis and Eheat corresponds to the total intrinsic excitation energy at scission, which can be distributed onto both fragments according to thermodynamical assumptions in conjunction with the Fermi-gas model (cf. ref. 1711,i.e. Edis + E heat = Eint,I. + Eint,H’ The fragments i = L, H with the “asymptotic” excitation energy (i.e. after dissipation of fragment deformation into intrinsic energy)
ET = Ectef,i + Eirnt,i
(12)
T. Haninger et al. / Excitation energy of fragment
0
*
----5
light fragment fragment
heavy
light fragment heavy fragment
I I
301
235U(nth,f) Z52Cf(sf)
,I.’
, ,\,’
0.08
k0 0.06 I 0.04
0.02
o.oq
\c
,III,/,,,,,,,,,,,,,,,,,,,,,,,,I,,,,,,,,,,,,,,,,,/
125
Fig. 3. Calculated slopes (S/aTKE)
135
represented
145
A
155
165
5
as a function of A, for both fragment groups.
de-excite by neutron emission mainly, whereas y-ray emission dominates for E* Iower than the neutron separation energy. Fixing both AL/A, and TIE, the most probable deformation energies can be approximatively calculated by minimizing the sum of the potential energy terms Ecoul + Edef,L + Edef,H in deformation space. For this purpose, the simple two-spheroid model ]6,7], including semi-empirical, temperature-dependent shell-correction energies to describe the defo~abiIi~ of fragments and, hence, their deformation energies at scission, is apphed. A very distinct measure of the she&structure influence on the energy partition is the slope of 8*(TKE:A) or (in equivalence) of the average neutron multiplicity 3(TKE:A). Results for both fission reactions studied in this work are represented in Fig. 3. As emphasized in ref. f71, the calculations reproduce the experimental data 1121in the case of 252Cf(sf). Based on comprehensive measurements of the neutron-multiplicity distributions as functions of A and TKE, Nifenecker et al. [13] studied the initial distribution in excitation energy in detail. Whereas an energy-balance equation for all fragment de-excitation processes yields a linear correlation between ET and the average number of neutrons V quantitatively (cf. ref. ]71), the variance & of the excitation-energy distribution is related to the free energy at scission, i.e. it vanishes at maximum and minimum TKE for a given mass split. The dependence of ui* on TKE for a given mass split is almost parabolic with its maximum at m(A,/A,). The 252Cf(sf) data of Nifenecker et al. [13] was used to parameterize v~*(A, TKE) 1141.This procedure was also adopted to 23sU(nth,f). In summary, the model calculation of the average excitation energy ET as we11 as the parametrization of a$ (both as functions of A and TKE) yield the gaussian-shaped initial distribution P,(E* :A, TKE), which is the starting point for applying the statistical theory to cascade-neutron emission from fission fragments.
T. Haninger et al. / Excitation energy of fragment
302
4.2. Statistical model of neutron emission The scenario of excitation modes in nuclear fission is closely connected with the dynamics of the whole process (ref. [15] and references therein). The evolution of the fissioning nucleus towards scission might cause single-particle excitations and, hence, nucleon emission (“scission neutrons”). A further open question is the role of nonequilibrium during dissipation of the deformation energy into intrinsic excitation which could lead to enhanced nucleon emission during fragment acceleration. Both mechanisms of fission-neutron emission were found to be very secondary [15]. In particular, it was shown [16] that nonequilibrium effects of nucleon emission just after scission (as studied in the case of protons) are not evident. Clearly, the predominant mechanism of neutron emission in fission is the evaporation from fully accelerated fragments [151. Considering the multitude of fragment configurations in the “asymptotic” sense, i.e. after both fragment acceleration and dissipation of deformation into intrinsic excitation (effectively finished at about 3 x 10P2’ s after scission), one can start with a fragment-occurrence probability P(A, 2, TKE, E*, J), where Z is the fragment-charge number and J is the fragment angular momentum. In the present calculations, this distribution is reduced to P(A, TKE, E*) in connection with P(J:A). Further, the charge distribution P(Z:A) [9] is used for averaging all input data (see below). Applying standard statistical theory to cascade-neutron emission (cascade step k) from highly excited nuclei characterized by the initial distribution P,(E*, .I> for fixed A, 2 and TKE (i.e. neglecting charged-particle emission because of their low yield [15]) the emission energy spectrum (PJE) in the centre-of-mass system (c.m.s.1 for step k reads
W,E*,J) (P,JE)= jdE* CC@*, J) rtot(E*,J) +T,fot(E*,J) J r(e,
E*, J) =
J)
pw
J’) ;w’~
’
(13) (14)
9
U=E*-B,-E,
(15)
J=J’+l+s.
(16)
The emission width r is expressed in dependence on the level density p calculated semi-phenomenologically with shell and pairing effects 1171and on the transmission coefficients Tl obtained from optical-model calculations on the basis of the Holmqvist potential [18] found to be best suited for fission neutron calculations in ref. [15]. Pot is the total emission width for neutrons or y-rays. The excitation energy U of the residual nuclei after neutron emission follows simply
T. Haninger
et al. / Excitation
303
energy of frngment
from Eq. (15) with the inclusion of the neutron separation energy I?,.,. Angularmomentum coupling is taken into account by Eq. (16) in order to get the momentum J’ after emission (I and s are the orbital momentum and neutron spin, respectively). Based on Eq. (141, a simplified evaporation ansatz was proposed in ref. 1151: r(e,
E”, j) =
with the parameterized
1 2ap( E* ,J = 0)
EainvP(E*,
J’=O)C(E*,
j),
(17)
correction factor
(18) which accounts appro~matively for spin effects on the neutron spectrum and the neutron/y competition. The parameters C, and C,, which depend on j, were obtained from “exact” calculations. Eq. (17) together with Eq. (18) allows much faster calculations with adequate accuracy. rsi,, is the inverse cross section of compound nucleus formation and corresponds to the weighted sum over the transmission coefficients. Note that the fragment distribution P&Y*, J) for k > 1 has to be deduced from the one of the preceding cascade step, i.e. Pk_,(E*, 11, considering the emission spectrum for step k - 1. The theoretical scheme outlined above is suitable to reproduce the manifold fission-neutron observables like yield, energy and angular distributions as well as their correlations with fragment parameters, e.g. A and TILE. The c.m.s. neutron-emission spectra (pJe) are the basis for full-scale kinematic calculations of recoil effects leading to the folding-angle distributions. Even for this purpose the explicit consideration of the dependence on A and TKE is essential, since these variables define the momentum of the fragment. 4.3. Kinematics The momentum vector of the emitted neutron P,, is defined by the c.m.s. energy E and direction. The neutron-emission anisotropy due to fragment spin as estimated by Gavron [19] is very small and we assume an isotropic emission probabili~ ?a(~, 6) =/&in
&/~P(E),
(19)
where 6 is the c.m.s. polar angle of neutron emission with reference to fragment direction. For given A and TKE, the kinetic energy of the fragment considered before neutron emission is
(20)
T. ~aninger et al / Excitationenergyof cadent
304
A=100 0.6
E,=lOO MeV
,,1,,,,,,,,,,I,,,,,,,,,,,,,,,.
c=1.25 MeV
c=lO
MeV
j
Fig. 4. The distribution in the (polar) deflection angle of a fragment with A = 100 and E = 100 MeV calculated for two emission energies c = 1.25 and 10 MeV.
where A, momentum tively. Due momentum
is the mass number of the fissioning nucleus. Let p and p’ be the vectors of the fragment before and after neutron emission, respecto neutron emission with momentum pn, the fragment has a recoil pR = -pn (c.m.s.). Momentum addition p’ = p + pR leads to
A E’ = --E+ A-l
(21)
E=(A-l)E’+AE-2/mcos
CY,
(22)
where cy is the (polar) deflection angle of the fragment. Its average Z is obtained from Eq. (22) directly by the use of the appro~mations E’ = E and A N A - 1: cos(Yml-E
2E4’
(23)
This relation is useful for estimations. Fig. 4 shows the deflection-angle distribution calculated for fixed E and A and two typical values of E, i.e. E = 125 MeV, that corresponds to the average c.m.s. emission energy approximatively [141, and E = 10 MeV, a rather high value, where the emission probability is already very small. In order to make the full-scale coordinate transformation for all emission steps easily tractable, the deflection-angle distribution calculated in polar coordinates for given A, TKECE) and E” (and average Z(A)) is transfo~ed for all emission steps into an (x, y&plane at a freely elected (large) distance e from the
T. Haninger et al. / Excitation energy of fragment
305
Fig. 5. The typical shape of the deflection-angle distribution for a one-step emission of neutrons with cp(c, S) according to the model described in the text.
point of neutron emission. This is allowed because of the very small values of LY, where sin (YN CY.Hence, we start with the probabili~ distribution Pa(x, y, E*: A, TIE) =6(x - 01%~ - 0) for step k = 0 resulting in P,(x, y, U: E*, A, TKE) for k = 1 and so on. In this way, all emission steps are accounted for, and so all coordinate transformations are performed. For all U G B, at a given step k, the (x, ykcoordinates are the final ones, otherwise the probabihty for the given U is foIded by the deflection-angle distribution again. The total deflection-angle distribution P(x, y: E*, A, TKE) for one fra~ent(!) corresponds to the sum of all Pk(x, y, U: E*,A, TKEZ>for U G II,. Fig. 5 illustrates the shape of the distribution P(x, y) for a one-step emission. Further emission steps broaden this distribution, but do not change its shape qualitatively. Finally, one has to fold the distributions for both complementary fragments (L, H) in order to get the folding-angIe (@ = f(xr, yLt eL, xn, y,, e,>) distribution ET A,
@:,,,A,TKE H H = jdx,jdy,jdx,jdy,P,(nL,
xf’dx,,
YH:
EZi, A,, T=>.
YL:
EZ,
A,,
Tm)
(24
Considering the fragment-occurrence probability P(EE/E&, AL/A,, TKE), we obtain PC@: At/AH), HO: AL/AH, TKF9, I’(@: TIEI), or the total one P(0) by integration/ su~ation. These probabili~ distributions are characterized by the
306
T.Hunhger et aL / Excitation energy of fragment
average folding angle B and variance @2. These parameters comparison between experiment and theory (see sect. 5),
are also used for the
4.4. Tematy fiGon The release of light charged particles close to scission (predominantly perpendicular to the fragment direction) gives rise to the strongest fragment deflection from the fission axis. The corresponding folding-angle distribution has been estimated for the most frequent case of a-accompanied fission on the basis of two model approaches: Approach I three-body decay approach. Application of linear-momentum conservation a~uming the e~erjmental ru-particle spectrum at the most probable angle with reference to the light-fragment direction, Approach 2 ~o~d~rne~io~a~ trajectory calculation. Accounting for: initial a-particle distribution in energy (maxwellian), direction and (iI position (e.g. average deviation of about 1 fm from fission axis corresponding to the neck radius), (ii) initial fragment distribution in energy (pre-scission kinetic energy with an average of about 5 MeV) and elongation, (iii) sn~~sition of CouIomb forces between the three point-like charge centres of the fragments and the light charged particle at any stage of acceleration, absolute yield of ternary fission events for uo~al~~g the folding-an(iv1 gle distribution. The initial parameters of approach 2 were adjusted in order to fit the experimental values of Tm as well as the average energy and variance of the a-energy spectrum. As a consequence of the uncertainties of both approaches, the results of the calculations should be understood as rough estimates of the effect. The calculated average folding angle of the fragments due to ternary fission is about 4” to 6”.
5. ResuIts and discussion
As shown in Figs. 6 and 7 for 252Cf(sf) and 235U(nth, f>, respectively, the total folding-angle distributions are well reproduced by the neutron-emission model calculations in the most probable region up ta about 3”. The agreement between experimental and calculated data concerns the average folding angle as well as the width of the distribution. For this, the consideration of the complex dependence on mass asymmetry and energy was essential. At folding angIes above about 2.8” and 3.6” for W5U&,, f) and 255Cf(sf), respectively, the influence of the ternary
T. Haninger et al. / Excitation energy of fragment
2 J iz
307
.ame.EXPERIMENT THEORY 0.5
2
data (dots) in comparison Fig. 6. The total folding-angle distribution for 252Cf(sf,k experimental calculated results represented as histogram and dashed line (fitted).
with
fission events is predominant. Fig. 8 represents the results of both the neutronemission model and the ternary-fission model calculation (approach 1, cf, sect. 4.4) for the fission reactions considered in this work. However, the statistical errors of the experimental data are too large at high angles for confirming the calculated results. The ternary-fission approach 2 (two-dimensional trajectory calculation) yields very similar data compared to the simple three-body decay calculation, but the fit to all relevant fission data (distributions in fragment kinetic energy,
Fig. 7. The same as Fig. 6, but for ?J(n,,,
f).
T. Haninger et al. / Excitation energy of fragment
308
Fig. 8. Folding-angle
distributions calculated within the neutron-emission model (1) as three-body decay approach to ternary fission (2).
a-energy and angle) is quite unce~ain. Therefore, presentation of rather reliable approach-l results.
as the
we restrict ourselves to the
5.2. Folding-angle distribution in TKE windows The total kinetic energy before neutron emission is given by TKE = E&r + E&2 = $v; v&4,,
tw
with the kinetic energies and velocities of the fragments E~inl, Ezin2, UT, u:, respectively. It is assumed that the fragment velocity does not change, on average, due to neutron emission, i.e. vT = u1 and uz = ZQ, where u1 and u2 are the measured quantities. The mass numbers of the fissioning nuclei A, are in our cases 236 and 252, respectively. The correlation between the folding-angle distribution and the TKE is demonstrated for 235U(nth, f) in Fig. 9. The smaher the TKE, the higher the fragment excitation energy and, consequently, the average number of prompt neutrons, and the larger the average folding angle. The calculated dist~butions for given Ta windows are in good agreement with the measured data. In Table 2, the experimental average folding angles and variances in given TKE bins are listed for both fission reactions studied. 5.3. Mass-number dependence Obviously, the total number of prompt neutrons is strongly correlated with the was deduced according average folding angle. Gtot as a function of mass as~rnet~ to %t =A,, -A,
-A,.
(26)
T. Hminger et at. / Excitation energy of fragment
309
Fig. 9. Folding-angle distributions of 235U(nt,,, f) fragments for the TKE bins indicated, and corresponding theoretical results.
One expects that the dependence on i;,,t determines the O(A) course, Fig. 10 shows that calculated and experimental average angles 0 for various fragment-mass windows are in good agreement confirming the model assumptions. The upper part of Fig. 11 gives is,, together with the fragment-mass yield E&91as a function of A. The very low mass yield in the very asymmetric and in the near-symmetric region explains the larger experimenta errors of i3;,1 and & in these mass ranges. The minimum of @ at A, = 104 corresponds to the position of the minimum of 3,. Again the good agreement. between experiment and theory confirms the model assumptions. The width (+ of the folding-angle distribution is nearly independent of the fragment mass as seen in Fig. 11, lower part. Here, the model calculations slightly underestimate the measured data.
Table 2 Average folding angie and width for several TIE systematic error of B is less than 0.07” CMeV>
All fragments TKE Q 160 160 < TKE < 170 170 < TKE < 180 180
%fn,
windows. f)nly statistical errors are given, The
zs2CfGfI
f3
B (de&
v. (deg)
g Cdeg)
co (degf
0.90 (2) 1.13 (1) 0.98 (1) 0.81 (1) 0.66 (1)
0.50 (6) 0.62 0.56 0.48 0.41
1.040 (4) 1.48 (3) 1,30 (2) 1.15 (1) 1.00 (1) 0.87 (11 0.77KI
0.600 (13) 0.79 (8) 0.70 (4) 0.63 (3) 0.55 (2) 0.49 (31 0.45 (51
0.8
*a..* -
EXPERIMENT THEORY
i
Fig. 10. Average folding angle as a function of the light-fragment
mass number for 235U(nthr f) and
2s2cfw.
As discussed above, TKE and E* are strongly anticorrelated. This is clearly the reason for the decreasing average folding angie as a function of TIE. This dependence is represented in Fig. 12 for 235U(nt,,, f) and 2”2Cf(sf). Within the neutron-emission model, OCTKEZ)drops down to zero at the limit value TIE, = - B,. In reality, there is still secondary influence of prompt y-ray emission at e hiirer TKE, i.e. between TKE, and Q,,. 5.5. Variim correIQtims For obvious reasons, several correlations were investigated. First, Fig. 13 shows the &-G-correlation, where altf measured data as fL4 and f(TKI3 as well as the
T. Haninger et al. / Excitation energy of fragment
/ I
/-’
I--\ \ _
I
\
/
/
/
/
I
.
.
YIELD
\ \ , \ \ \ \ \
+,i I* I
311
. .
‘\ \ + \ \
l
b
j’“““” 70
80
,,,,,,,,,,,,,,,, 1 (,,,,,,,,,,,,
100
90
110
120
A Fig. 11. Upper figure: The average total number of emitted neutrons as a function of A for 235U (dots: experimental neutron-multiplicity data, dashed line: fragment-mass yield in arbitrary units). Lower figure: The distribution parameter o (square root of variance) as a function of the light-fragment mass number A for the reaction 235U(n,,,, f).
total values for both fission reactions are included. With the exception of the 252Cf(sf) data at very low TKE, i.e. very high 5, we obtain a pure linear correlation. An overall fit including all data of both fission reactions studied yields the approximative relation @[deg] = 0.175 + 0.42,
(27)
which should be valid within 10%. This correlation indicates that the average folding angle is a good measure of the average total excitation energy of the
312
T. Haninger et al. / Excitation energy of fragment
IO
0.6
-
0.4
-
.e*..
0.0E 130
140
EXPERIMENT THEORY 150
160
170
180
190
200
210
220
230
TKE (MeV] Fig. 12. Average foldingangle as a functionof TKE for wU(n,,
f’)and uz~.
fragments (see below). Fig. 14 shows the almost linear correlation between B and (T. Linear regression yields a[deg] = 0.49@deg]
+ 0.07.
5.6. Total excitation energy versus average folding angle The average total excitation energy for a given mass asymmetry is given by
(2% where c is the average Q-value for a given mass split, i.e. the average over the relevant charge distribution. A linear correlation between E* and B was found.
T. Haninger et al. / &citation
x
1.6
‘;;I;
energy of fragment
-
TOTAL
313
VALUE
-
1.2-
$ IO
0.8
-
0.4
-
-0 OXXUf X -
0.0
0
1
2
3
f TKE) DATA I A) DATA TOTAL 4
VALUE 5
6
i3
Fig. 13. The correlation between the average folding angle and average number of neutrons shown for both fission reactions studied. The solid line represents the result of an overall fit to all available data (see text).
Fig. 15 shows experimental data for both reactions investigated. At very large folding angle this correlation is disturbed due to ternary fission events. The angle limits discussed above on the basis of the total folding-angle distribution are also visible in Fig. 15. If E*(8) is plotted for different fragment-mass bins, there is a linear correlation for each bin. An approximation by E& = slope X 0 + offset
(30)
yields slopes and offsets of these straight lines as function of the light fragment mass. The results are represented in Fig. 16. The measured data are well reproduced by the neutron-emission model calculations. The observed dependencies are due to the typical energy conditions considered within the energy-partition model
T. Haningeret al. / Excitation energy ornament
314
2.0
Y$
1.6
-
1.2
-
0.8
-
0.4
-
.% IQ
0.0
I 0.0
*
0.2
I
(7
Fig.14. The correlation
I,,,
0.4
0.6
0.8
1.0
k41
between 8 and u deduced from the measured 235LJ(n,,, f) data.
20 l We*
-
30
1
EXPERIMENT THEORY
l ..**
EXPERIMENT
ZO~"""'"""""""".'.""'j t 0 ;
4
j
5
Edeg;
Fig. 15. Average total fragment excitation energy versus folding angle.
315
T. Haninger et al. / Excitation energy of fragment
12
4oy
,,,,,,
EXPERIMENT THEORY
-
80
90
100
---
4.,,...... 80
90
I,,
110
120
EXPERIMENT THEORY
:
- - EXPERIMENT THEORY 110
100
120
130
2OU
80
90
A Fig. 16. Slope and offset of the J?* -O-correlation
100
110
120
130
A function
obtained
from linear fits to the data.
as well as the emission model (e.g. neutron binding energies, level densities). Here, the adequate account for shell effects is essential.
6. Summary Folding-angle distributions and their correlations with TKE and mass asymmetry were investigated for “*Cf(sf) and 235U(n,,, f). In the most probable angular region, it can be understood as due to fragment deflection by prompt neutron emission. At large angles, the influence of ternary fission events is predominant. In summary, it can be stated that the folding angle of coincident fission fragments is a very valuable quantity in fission. It is linearly correlated with the total fragment excitation energy and, consequently, with the average number of prompt neutrons. The quantitatively presented correlation functions @Y) and a(& can be used to estimate folding-angle distributions for any fission reaction (in addition, as func-
316
T. Haninger et al. / Excitation energy
offragment
tion of mass asymmetry and TKE) if the relevant tota number of prompt fission neutrons is known. Finally, it is emphasized that the applied models (energy-partition model together with statistical theory of neutron emission, ternary fission approach) reproduce the bulk of experimental results quite well. The neutron-emission model includes the strict anticorrelation of the individual fragment excitation energies for given mass asymmetry and TIE The emission cascades from the complementary fragments are treated statistically independently. Here, possible angular-momentum effects which might cause a smalf correlation between the emission cascades are neglected. The present work confirms the theoretical assumptions made in describing the folding-angle distribution. We wish to thank F. Gonnenwein for very valuable discussions, H. Daniel for support, V.G. Nedorezov and A.S. Sudov for the 252Cf target, H.J. Maier and P. targets, and the group of the FRM reactor for providing the David for the ?J excellent neutron beam. The work was supported by the Bundesministerium fiir Forschung und Technologie, Bonn.
References [l] R. Vandenbosch, Nucl. Phys. A502 (19891 lc [Z] F. Gonnenwein, Nucl. Phys. A502 (19891 159~ [3] Y.S. Kim, P. Hofmann, H. Daniel, T. von Egidy, T. Haninger, F.J. Hartmann and M.S. Lotfranaei, Nucl. Instr. Meth. A329 (19931403 [4] P. Hofmann, Y.S. Kim, H. Daniel, T. von Egidy, T. Haninger, F.J. Hartmann, P. David, H. Machner, G. Riepe, H.S. Plendl, K. Ziock, B. Wright, D. Bowman, W. Lynch, J. Lieb, J. Jastrzebski and W. Kurcewicz, Sov. J. Nucl. Phys. 55 (19921 713 [5] T. von Egidy, H. Daniel, F.J. Hartmann, P. Hofmann, Y.S. Kim, W. Schmid, H.H. Schmidt, G. Riepe, H. Machner, P. David, H.S. Plendl, A.S. Iljinov, A.S. Botvina, Ye.S. Golubeva, M.V. Mebel, J. Jastrzebski and W. Kurcewicz, Nucl. Phys. A558 (1993) 383~ 161 A. Ruben and H. Mkten, Z. Phys. A337 (19901237 171 A. Ruben, H. M&ten and D. Seeliger, Z. Phys. A338 (1991) 67 [8] E.A.C. Crouch, At. Data Nucl. Data Tables 19 (19771417 [9] A.C. Wahl, At. Data Nucl. Data Tables 39 (1988) 1 [lo] H. Henschel, A. Kohnle, H. Hipp and F. Gonnenwein, Nucl. Instr. Meth. 190 (1981125 [ll] P. Geltenbort, F. Gdnnenwein and A. Oed, Rad. Eff. 93 (19861393 [12] C. Bud&Jorgenson and H.H. Knitter, Nucl. Phys. A490 (19881307 1131H. Nifenecker, C. Signarbieux, R. Babinet and J. Poitou, Physics and chemistry of fission, Proc. IAEA Symp., Rochester, August 1973 (IAEA, Vienna, 1974) [14] H. Marten and D. Seeliger, J. of Phys. GlO (1984) 349 [15] K. Arnold, I. Doring, H. Marten, A. Ruben and D. Seeliger, Nucl. Phys. A502 (19891325~ [16] A. Schubert, H. Kalka and H. M&en, submitted to Nucl. Phys. [17] K.-H. Schmitt, H. Delagrange, J.P. Defour, N. Carjan and A. Fleury, Z. Phys. A308 (19821215 [IS] B. Holmqvist, Ark. Fys. 38 (1968) 403 [19] A. Gavron, Phys. Rev. Cl3 (197612561
NUCLEAR PHYSICS A
Folding angle and excitation energy of fragments from 235U~n~~,f)and 252Cf(sf) reactions T. Haninger, F.J. Hartmann, P. Hofmann +, Y.S. Kim, M.S. Lotfranaei, T. von Egidy Physik-Department, Technische Universitiit Miinchen, D-8.5748 Garching, Germany
H. h&ten
‘, A. Ruben
Institut fiir Kern- und Atomphysik, Technische U~i~~rsit~tDresden, D-01049 Dresden, Germany
Received 19 October 1993
Abstract
Coincident fragments from ?J(n,,,,f) and 252Cf(sf) reactions were investigated with a doublearm fission-fragment spectrometer and PIN-diode arrays. Based on the measurement of kinetic energy, velocity and direction of complementary fragments the total kinetic energy, the total mass, the individual fragment masses, the total number of emitted neutrons as well as the folding angle were deduced event by event. A nearly linear correlation between average folding angle and average total excitation energy of the fragments (and, consequently, the average number of neutrons) was found. Fragment deflection by prompt neutron emission is a~ordingly described by a complex statistical evaporation model in connection with a semi-empirical calculation of energy partition in nuclear fission. In addition, the folding-angle distribution due to ternary fission is estimated. All experimental fragment distributions and correlations are well reproduced by the model calculations.
lvey words: NUCLEAR REACTIONS 235U(n,Fl, E = subthermai; measured (~agmentxfragmen&coin, neutron spectra, E,,,, 0 of both fragments; deduced correlations between fragment mass, kinetic energy, velocity. Statistical evaporation model; enriched targets; PIN-diode arrays RADIOACTIVITY 252Cf (SF); measured (fragmentXfragment)-coin, neutron spectra; deduced average folding angle, total excitation energy linear correlation. Statistical evaporation model
1. Introduction
Slow-neutron-induced and spontaneous fission is known since more than fifty years ago. Mass, charge and energy distributions of fragments have been measured very carefully applying a large variety of experimental methods [1,2]. In spite of this r Present address: Stresemannplatz ’ Deceased.
llb, D-01309 Dresden, Germany.
0375-9474/94/$07.~ 0 1994 - Elsevier Science B.V. All rights reserved ~~D~O375-9474(93)EO686-3
T. Haninger et al. / Excitation energy of fragment
295
Fig. 1. Schematic and exaggerated demonstration of fragment deflection by neutron and y-ray emission (upper part) and by ternary fission with emission of a light charged particle (LCP). Angular distributions (in centre-of-mass system) are shown by dashed curves. Fragments (F), neutrons (n) and angles ((u) are marked with L and H indicating the light or heavy fragment, respectively.
effort, there are still many open questions in fission. For instance, the folding angle of the fragments has not yet been measured together with energies and masses, as far as we know. It is the purpose of the present publication to show that the folding angle of fission fragments contains very valuable information on the fission process, especially on the total excitation energy of the fragments. The fission axis in binary fission is defined by the directions of both primary fragments which have a primary folding angle of 180”. However, recoil effects due to prompt emission of neutrons and gamma rays change both directions in a stochastic way. Only the final folding angle is a direct observable. Multiple-neutron emission from both fast-moving and highly excited fragments gives the most significant recoil effect (l-3 neutrons per fragment). Multiple y-emission has a very small effect on the angle. Light charged particles (LCPs) emitted in ternary fission processes cause the largest deviations of the folding angle from 180”. All these processes are demonstrated in Fig. 1. The fragments, their energies, velocities and angles are measured with a newly constructed double-arm fission-fragment spectrometer using PIN-diode arrays [3]. This instrument was originally designed to investigate antiproton-induced fission [4,51. It is very efficient and precise for the present purpose. Finally, a recently developed computer code [6,7] is available which uses a statistical evaporation model with an energy partition based on a detailed energy balance at the scission point. This program yields all relevant information on the fission process. Thus, comparison of experiment and theory should give further insight into the physics of the fission process. The present investigation provides also useful nuclear data on real angular dispersions of fission fragments.
296
T. Haninger et al. / Excitation energy of fragment
2. Experiment The double-arm fission-fragment spectrometer, the electronics and the data evaluation is described in detail in ref. [31. Fig. 2 shows the arrangement for the measurement of thermal-neutron-induced fission at the Munich research reactor FRM. A subthermal neutron beam from a guide tube with a flux of nearly lo7 n/cm2 s enters the vacuum chamber with the target in the centre. Two circular 235U targets were used: 100 pg/cm2 U,O, with 1 cm diameter on 30 kg/cm2 carbon foil and 39.3 kg/cm2 metallic U with 0.5 cm diameter on 40 Fg/cm2 carbon foil. Most experiments for the present publication are made with the second thinner and smaller target. The source for the spontaneous fission experiment was produced by self-sputtering of 252Cf on a 94.6 kg/cm2 Ni foil. The active area had a diameter of 0.8 cm, but was covered by a diaphragm of 0.6 cm diameter. The fission rate was about 1000/s. A channel-plate detector close to the target gave the start signal for the time-of-flight (TOF) measurement. Two PIN-diode arrays (each 12 x 12 diodes, each diode 1 cm2, thickness 500 urn> are placed on opposite sides of the target. In order to avoid edge effects of the diodes, they were covered by copper-plate masks with 8.56 X 8.56 mm* holes. The distance from the target to a PIN-diode array was 120.9 cm, yielding an angular resolution of 0.54” and a solid angle of 0.12% of 4~. A distance of 60.8 cm was used for the time calibration of the spectrometer. In order to reduce periodicity effects due to the granular structure of the PIN-diode arrays and to enhance the information for very
n
neutron
beam tube
Fig. 2. Schematic drawing of the experimental arrangement. In this case the distance from the target to the PIN-diode array is 60.8 cm.
T. Haninger et al. / Excitation energy of fragment
297
small angles, both arrays were shifted in the same direction from the s~et~ axis by 1.7 cm. The 23SU(nI,,,f) measurement with the thin target and a distance of 120.9 cm from target to PIN-diode array gave 33000 coincident events which were used to determine the folding-angle distribution. The 252Cf measurement with the same geometry yielded 675000 coincident events for the folding-angle distribution. Additional measurements were made with various geometries to determine time and energy calibration, the pulse-height defect, the plasma delay and the energy loss in the target and in the backing (see ref. [31 for details).
3. Data evaluation 3.1. Masses and kinetic energies The coincident fission events were corrected for the pulse-height defect, for energy loss in the target and in the backing and for plasma delay as described in ref. [3]. The pulse-height defect and the mass resolution were determined together by a fit to radiochemical mass distributions [8,9] according to our new method [3]. The mass resolution was about 5 u FWHM. The results of both reactions, 235U(n,f) and 2s2Cf(sf), for average values of masses and kinetic energies of the light and of the heavy fragment, for post-neutron and pre-neutron emission, respectiveIy, are given in ref. 131.They are in very good agreement with previous data [lO,ll] and have frequently smaller errors. 3.2. Folding-angle distribution In the following discussion we use the folding angle 0 = 180” -real folding angle. The folding angIes of coincident fragments were calculated using three points, one on each PIN diode and one on the target. These precise points on each diode and on the target were determined for each event by a random-number generator in order to smooth the distributions. Angular distributions have to be corrected for the angular resolution of our spectrometer and for the angular efficiency. The correction for the angular resolution was obtained in the following way. The measured angular distribution G,(O,) has to be converted into the original one F(Oj). This correlation is purely geometrical and can be calculated as a transformation matrix M(Oi, Oi> with Monte Carlo methods. The original angular distribution was approximated with the function F(Qj)
=pl@,!‘”
eXp( -P3@j
+Pd@T)*
(1)
T. Haninger et al. / Excitation energy of fragment
298
With this function an approximative measured function G,(OJ can be determined,
Oi),
= f: I;(Oj)M(Oj,
G,(Oi)
(2)
j=l
The parameters pr, p2, p3 and p4 have been obtained by a least-squares G1(Oi) to G,(Oi) with x2
=
fit of
C [G,(@i)-Gl(@i)12 i=l
.
G,(@i)
Z and J are the numbers of angular bins. The fit results showed that the assumed function E;(Oj> reproduces the experimental data very well. The efficiency of detecting a given angle 0 depends also on the geometry of the spectrometer and can be calculated with Monte Carlo techniques. This angular efficiency is shown in Fig. 4 of ref. [3]. For the distance of 120.9 cm from the target to the PIN diodes the angular efficiency decreases nearly linearly from 100% at 0 to nearly 0% close to 8”. 3.3. Multiple scattering in the target and backing Multiple scattering in the target material and in the backing influences the folding-angle distribution and its width V. Several test measurements were performed using the target with d, = 39.3 ug/cm2. For instance, additional 238U foils (200 and 400 pg/cm2) were placed on one side of the target and the influence on the light (L) and heavy (H) fragment was separated. The results of the test measurements are shown in Table 1. In order to deduce the original width a,, of the distribution from the data in Table 1, the following assumptions were made:
a2=u,2+a2=u,2+a2 s
s,L +
Table 1 Average folding angles and widths statistical errors are given Test measurements
of the 235U(n, f) reaction
Additional 0
?3(ded
&l,
238U foils. Only
(@/cm’) 200
400
1.25 (1)
1.54 (2)
(deg)
1.10 (1)
1.28 (2)
0, (deg) u (deg) q_ (deg) uH (deg)
1.38 (2) 0.83 0.69 0.90
1.80 (3) 1.03 0.83 1.13
8,
0.90
238U foil
with additional
0.57
some
T. ~aninge~
et al. / ~citation energy of ~ag~ent
299
with the measured width CFand the additional broadening by multiple scattering of the tight and heavy fragments a$,, and uS,n. a, ’ increases linearly with the foil thickness d, i.e., Cr2=ad 9 s
(5)
where a is a constant. We can select those events where only light or heavy fragments suffer multiple scattering in additionai foils. In the case that only light fragments pass through the foils, the measured width rr,. can be expresses as
=ai+aH-+
d O+a,($d,+d)
=a,f+(a,+a,)~~d,+a,d,
(6)
where d, is the thickness of the target and the fission is assumed to occur on average at the centre of the target. Now aL can be obtained from the two a=-values at d = 200 and 400 pg/cm2 in Table 1. Exactly the same procedure for only heavy fragments can be followed to obtain aH: a:(d)
=~$+(a~+a,J-$d,,+a,d.
With the obtained values aL = 0.91 x 10e3 and aH = 2.4 x 10e3 &g/cm*)-‘, can calculate the contribution of the target (second terms in Eqs. (6) and o$$d,) = 0.27”. The original width is, then, deduced to be a, = 0.50”. Note that broadening of the distribution due to the target thickness 39.3 kg/cm* is negligible compared with the original width.
(7)
one (7)) the not
4. Model calculations The experimental data indicate a remarkable correlation between the average folding angle and the average number of prompt fission neutrons. As suggested above, neutron emission should be the predominant effect in forming the foldingangle distribution. Its adequate theoretical description has to account for the energy partition in nuclear fission in detail, i.e. in dependence on the fragment-mass asymmetry AL/A, and the total kinetic energy TKE of the fragments, in order to consider all measured correlations. 4.1. Energy partition in fission According to energy conservation, the total available energy in the fission process, i.e. the sum of the Q-value (to be calculated from mass tables) and the
I: Hanittger et al. / Excitation energy
300
of fragment
excitation energy Ezn of the fissioning compound nucleus, is equal to the sum of the total excitation energy E* and the total kinetic energy of the fragments:
(8) (The bars indicate the (weighted) averages over charge fragmentations). For given A,/A, and TKE, which are measurable quantities, E* can be deduced directly. However, its partition on both complementary fragments (E* = EC + Eg) is strongly influenced by shell effects of the nascent fragments at scission. The individual excitation energies, which are distributed around mean values, are obviously anticorrelated. Because of the general problems of establishing a firstprinciple theory of nuclear fission, a phenomenological model [7] is used to solve the present ener~-partition problem. In addition, empirical relations serve to describe the widths of the distribution in E”. In order to account for both intrinsic and collective degrees of freedom in the most crucial phase of the fission process, i.e. the descent from saddle to scission point, we start with a rather empirical speci~cation of the asymptotic energy terms TKEZ and E” in relation to certain energy terms at scission: TKE = E,,
+ EM 9
(9)
E* = Edef,L+ Edef,~+ Edis+ EH
T
where: Epre = pre-scission kinetic energy, E cou, = Coulomb repulsion energy corrected for the proximity term, E def = deformation energy of fragment (L, H) at scission,
E,, = dissipative energy released during descent from saddle to scission due to friction forces [6]. E heat is the intrinsic excitation energy (“heat”) at the second saddle (B) of the fission barrier with height E,,. Taking into account the pairing gap A at this point, we have E heat= Ezn -
Ef,a - A,
(11)
with the constraint Eheat 3 0, i.e. it vanishes in the case of spontaneous and subbarrier fission. The sum of Edis and Eheat corresponds to the total intrinsic excitation energy at scission, which can be distributed onto both fragments according to thermodynamical assumptions in conjunction with the Fermi-gas model (cf. ref. 1711,i.e. Edis + E heat = Eint,I. + Eint,H’ The fragments i = L, H with the “asymptotic” excitation energy (i.e. after dissipation of fragment deformation into intrinsic energy)
ET = Ectef,i + Eirnt,i
(12)
T. Haninger et al. / Excitation energy of fragment
0
*
----5
light fragment fragment
heavy
light fragment heavy fragment
I I
301
235U(nth,f) Z52Cf(sf)
,I.’
, ,\,’
0.08
k0 0.06 I 0.04
0.02
o.oq
\c
,III,/,,,,,,,,,,,,,,,,,,,,,,,,I,,,,,,,,,,,,,,,,,/
125
Fig. 3. Calculated slopes (S/aTKE)
135
represented
145
A
155
165
5
as a function of A, for both fragment groups.
de-excite by neutron emission mainly, whereas y-ray emission dominates for E* Iower than the neutron separation energy. Fixing both AL/A, and TIE, the most probable deformation energies can be approximatively calculated by minimizing the sum of the potential energy terms Ecoul + Edef,L + Edef,H in deformation space. For this purpose, the simple two-spheroid model ]6,7], including semi-empirical, temperature-dependent shell-correction energies to describe the defo~abiIi~ of fragments and, hence, their deformation energies at scission, is apphed. A very distinct measure of the she&structure influence on the energy partition is the slope of 8*(TKE:A) or (in equivalence) of the average neutron multiplicity 3(TKE:A). Results for both fission reactions studied in this work are represented in Fig. 3. As emphasized in ref. f71, the calculations reproduce the experimental data 1121in the case of 252Cf(sf). Based on comprehensive measurements of the neutron-multiplicity distributions as functions of A and TKE, Nifenecker et al. [13] studied the initial distribution in excitation energy in detail. Whereas an energy-balance equation for all fragment de-excitation processes yields a linear correlation between ET and the average number of neutrons V quantitatively (cf. ref. ]71), the variance & of the excitation-energy distribution is related to the free energy at scission, i.e. it vanishes at maximum and minimum TKE for a given mass split. The dependence of ui* on TKE for a given mass split is almost parabolic with its maximum at m(A,/A,). The 252Cf(sf) data of Nifenecker et al. [13] was used to parameterize v~*(A, TKE) 1141.This procedure was also adopted to 23sU(nth,f). In summary, the model calculation of the average excitation energy ET as we11 as the parametrization of a$ (both as functions of A and TKE) yield the gaussian-shaped initial distribution P,(E* :A, TKE), which is the starting point for applying the statistical theory to cascade-neutron emission from fission fragments.
T. Haninger et al. / Excitation energy of fragment
302
4.2. Statistical model of neutron emission The scenario of excitation modes in nuclear fission is closely connected with the dynamics of the whole process (ref. [15] and references therein). The evolution of the fissioning nucleus towards scission might cause single-particle excitations and, hence, nucleon emission (“scission neutrons”). A further open question is the role of nonequilibrium during dissipation of the deformation energy into intrinsic excitation which could lead to enhanced nucleon emission during fragment acceleration. Both mechanisms of fission-neutron emission were found to be very secondary [15]. In particular, it was shown [16] that nonequilibrium effects of nucleon emission just after scission (as studied in the case of protons) are not evident. Clearly, the predominant mechanism of neutron emission in fission is the evaporation from fully accelerated fragments [151. Considering the multitude of fragment configurations in the “asymptotic” sense, i.e. after both fragment acceleration and dissipation of deformation into intrinsic excitation (effectively finished at about 3 x 10P2’ s after scission), one can start with a fragment-occurrence probability P(A, 2, TKE, E*, J), where Z is the fragment-charge number and J is the fragment angular momentum. In the present calculations, this distribution is reduced to P(A, TKE, E*) in connection with P(J:A). Further, the charge distribution P(Z:A) [9] is used for averaging all input data (see below). Applying standard statistical theory to cascade-neutron emission (cascade step k) from highly excited nuclei characterized by the initial distribution P,(E*, .I> for fixed A, 2 and TKE (i.e. neglecting charged-particle emission because of their low yield [15]) the emission energy spectrum (PJE) in the centre-of-mass system (c.m.s.1 for step k reads
W,E*,J) (P,JE)= jdE* CC@*, J) rtot(E*,J) +T,fot(E*,J) J r(e,
E*, J) =
J)
pw
J’) ;w’~
’
(13) (14)
9
U=E*-B,-E,
(15)
J=J’+l+s.
(16)
The emission width r is expressed in dependence on the level density p calculated semi-phenomenologically with shell and pairing effects 1171and on the transmission coefficients Tl obtained from optical-model calculations on the basis of the Holmqvist potential [18] found to be best suited for fission neutron calculations in ref. [15]. Pot is the total emission width for neutrons or y-rays. The excitation energy U of the residual nuclei after neutron emission follows simply
T. Haninger
et al. / Excitation
303
energy of frngment
from Eq. (15) with the inclusion of the neutron separation energy I?,.,. Angularmomentum coupling is taken into account by Eq. (16) in order to get the momentum J’ after emission (I and s are the orbital momentum and neutron spin, respectively). Based on Eq. (141, a simplified evaporation ansatz was proposed in ref. 1151: r(e,
E”, j) =
with the parameterized
1 2ap( E* ,J = 0)
EainvP(E*,
J’=O)C(E*,
j),
(17)
correction factor
(18) which accounts appro~matively for spin effects on the neutron spectrum and the neutron/y competition. The parameters C, and C,, which depend on j, were obtained from “exact” calculations. Eq. (17) together with Eq. (18) allows much faster calculations with adequate accuracy. rsi,, is the inverse cross section of compound nucleus formation and corresponds to the weighted sum over the transmission coefficients. Note that the fragment distribution P&Y*, J) for k > 1 has to be deduced from the one of the preceding cascade step, i.e. Pk_,(E*, 11, considering the emission spectrum for step k - 1. The theoretical scheme outlined above is suitable to reproduce the manifold fission-neutron observables like yield, energy and angular distributions as well as their correlations with fragment parameters, e.g. A and TILE. The c.m.s. neutron-emission spectra (pJe) are the basis for full-scale kinematic calculations of recoil effects leading to the folding-angle distributions. Even for this purpose the explicit consideration of the dependence on A and TKE is essential, since these variables define the momentum of the fragment. 4.3. Kinematics The momentum vector of the emitted neutron P,, is defined by the c.m.s. energy E and direction. The neutron-emission anisotropy due to fragment spin as estimated by Gavron [19] is very small and we assume an isotropic emission probabili~ ?a(~, 6) =/&in
&/~P(E),
(19)
where 6 is the c.m.s. polar angle of neutron emission with reference to fragment direction. For given A and TKE, the kinetic energy of the fragment considered before neutron emission is
(20)
T. ~aninger et al / Excitationenergyof cadent
304
A=100 0.6
E,=lOO MeV
,,1,,,,,,,,,,I,,,,,,,,,,,,,,,.
c=1.25 MeV
c=lO
MeV
j
Fig. 4. The distribution in the (polar) deflection angle of a fragment with A = 100 and E = 100 MeV calculated for two emission energies c = 1.25 and 10 MeV.
where A, momentum tively. Due momentum
is the mass number of the fissioning nucleus. Let p and p’ be the vectors of the fragment before and after neutron emission, respecto neutron emission with momentum pn, the fragment has a recoil pR = -pn (c.m.s.). Momentum addition p’ = p + pR leads to
A E’ = --E+ A-l
(21)
E=(A-l)E’+AE-2/mcos
CY,
(22)
where cy is the (polar) deflection angle of the fragment. Its average Z is obtained from Eq. (22) directly by the use of the appro~mations E’ = E and A N A - 1: cos(Yml-E
2E4’
(23)
This relation is useful for estimations. Fig. 4 shows the deflection-angle distribution calculated for fixed E and A and two typical values of E, i.e. E = 125 MeV, that corresponds to the average c.m.s. emission energy approximatively [141, and E = 10 MeV, a rather high value, where the emission probability is already very small. In order to make the full-scale coordinate transformation for all emission steps easily tractable, the deflection-angle distribution calculated in polar coordinates for given A, TKECE) and E” (and average Z(A)) is transfo~ed for all emission steps into an (x, y&plane at a freely elected (large) distance e from the
T. Haninger et al. / Excitation energy of fragment
305
Fig. 5. The typical shape of the deflection-angle distribution for a one-step emission of neutrons with cp(c, S) according to the model described in the text.
point of neutron emission. This is allowed because of the very small values of LY, where sin (YN CY.Hence, we start with the probabili~ distribution Pa(x, y, E*: A, TIE) =6(x - 01%~ - 0) for step k = 0 resulting in P,(x, y, U: E*, A, TKE) for k = 1 and so on. In this way, all emission steps are accounted for, and so all coordinate transformations are performed. For all U G B, at a given step k, the (x, ykcoordinates are the final ones, otherwise the probabihty for the given U is foIded by the deflection-angle distribution again. The total deflection-angle distribution P(x, y: E*, A, TKE) for one fra~ent(!) corresponds to the sum of all Pk(x, y, U: E*,A, TKEZ>for U G II,. Fig. 5 illustrates the shape of the distribution P(x, y) for a one-step emission. Further emission steps broaden this distribution, but do not change its shape qualitatively. Finally, one has to fold the distributions for both complementary fragments (L, H) in order to get the folding-angIe (@ = f(xr, yLt eL, xn, y,, e,>) distribution ET A,
@:,,,A,TKE H H = jdx,jdy,jdx,jdy,P,(nL,
xf’dx,,
YH:
EZi, A,, T=>.
YL:
EZ,
A,,
Tm)
(24
Considering the fragment-occurrence probability P(EE/E&, AL/A,, TKE), we obtain PC@: At/AH), HO: AL/AH, TKF9, I’(@: TIEI), or the total one P(0) by integration/ su~ation. These probabili~ distributions are characterized by the
306
T.Hunhger et aL / Excitation energy of fragment
average folding angle B and variance @2. These parameters comparison between experiment and theory (see sect. 5),
are also used for the
4.4. Tematy fiGon The release of light charged particles close to scission (predominantly perpendicular to the fragment direction) gives rise to the strongest fragment deflection from the fission axis. The corresponding folding-angle distribution has been estimated for the most frequent case of a-accompanied fission on the basis of two model approaches: Approach I three-body decay approach. Application of linear-momentum conservation a~uming the e~erjmental ru-particle spectrum at the most probable angle with reference to the light-fragment direction, Approach 2 ~o~d~rne~io~a~ trajectory calculation. Accounting for: initial a-particle distribution in energy (maxwellian), direction and (iI position (e.g. average deviation of about 1 fm from fission axis corresponding to the neck radius), (ii) initial fragment distribution in energy (pre-scission kinetic energy with an average of about 5 MeV) and elongation, (iii) sn~~sition of CouIomb forces between the three point-like charge centres of the fragments and the light charged particle at any stage of acceleration, absolute yield of ternary fission events for uo~al~~g the folding-an(iv1 gle distribution. The initial parameters of approach 2 were adjusted in order to fit the experimental values of Tm as well as the average energy and variance of the a-energy spectrum. As a consequence of the uncertainties of both approaches, the results of the calculations should be understood as rough estimates of the effect. The calculated average folding angle of the fragments due to ternary fission is about 4” to 6”.
5. ResuIts and discussion
As shown in Figs. 6 and 7 for 252Cf(sf) and 235U(nth, f>, respectively, the total folding-angle distributions are well reproduced by the neutron-emission model calculations in the most probable region up ta about 3”. The agreement between experimental and calculated data concerns the average folding angle as well as the width of the distribution. For this, the consideration of the complex dependence on mass asymmetry and energy was essential. At folding angIes above about 2.8” and 3.6” for W5U&,, f) and 255Cf(sf), respectively, the influence of the ternary
T. Haninger et al. / Excitation energy of fragment
2 J iz
307
.ame.EXPERIMENT THEORY 0.5
2
data (dots) in comparison Fig. 6. The total folding-angle distribution for 252Cf(sf,k experimental calculated results represented as histogram and dashed line (fitted).
with
fission events is predominant. Fig. 8 represents the results of both the neutronemission model and the ternary-fission model calculation (approach 1, cf, sect. 4.4) for the fission reactions considered in this work. However, the statistical errors of the experimental data are too large at high angles for confirming the calculated results. The ternary-fission approach 2 (two-dimensional trajectory calculation) yields very similar data compared to the simple three-body decay calculation, but the fit to all relevant fission data (distributions in fragment kinetic energy,
Fig. 7. The same as Fig. 6, but for ?J(n,,,
f).
T. Haninger et al. / Excitation energy of fragment
308
Fig. 8. Folding-angle
distributions calculated within the neutron-emission model (1) as three-body decay approach to ternary fission (2).
a-energy and angle) is quite unce~ain. Therefore, presentation of rather reliable approach-l results.
as the
we restrict ourselves to the
5.2. Folding-angle distribution in TKE windows The total kinetic energy before neutron emission is given by TKE = E&r + E&2 = $v; v&4,,
tw
with the kinetic energies and velocities of the fragments E~inl, Ezin2, UT, u:, respectively. It is assumed that the fragment velocity does not change, on average, due to neutron emission, i.e. vT = u1 and uz = ZQ, where u1 and u2 are the measured quantities. The mass numbers of the fissioning nuclei A, are in our cases 236 and 252, respectively. The correlation between the folding-angle distribution and the TKE is demonstrated for 235U(nth, f) in Fig. 9. The smaher the TKE, the higher the fragment excitation energy and, consequently, the average number of prompt neutrons, and the larger the average folding angle. The calculated dist~butions for given Ta windows are in good agreement with the measured data. In Table 2, the experimental average folding angles and variances in given TKE bins are listed for both fission reactions studied. 5.3. Mass-number dependence Obviously, the total number of prompt neutrons is strongly correlated with the was deduced according average folding angle. Gtot as a function of mass as~rnet~ to %t =A,, -A,
-A,.
(26)
T. Hminger et at. / Excitation energy of fragment
309
Fig. 9. Folding-angle distributions of 235U(nt,,, f) fragments for the TKE bins indicated, and corresponding theoretical results.
One expects that the dependence on i;,,t determines the O(A) course, Fig. 10 shows that calculated and experimental average angles 0 for various fragment-mass windows are in good agreement confirming the model assumptions. The upper part of Fig. 11 gives is,, together with the fragment-mass yield E&91as a function of A. The very low mass yield in the very asymmetric and in the near-symmetric region explains the larger experimenta errors of i3;,1 and & in these mass ranges. The minimum of @ at A, = 104 corresponds to the position of the minimum of 3,. Again the good agreement. between experiment and theory confirms the model assumptions. The width (+ of the folding-angle distribution is nearly independent of the fragment mass as seen in Fig. 11, lower part. Here, the model calculations slightly underestimate the measured data.
Table 2 Average folding angie and width for several TIE systematic error of B is less than 0.07” CMeV>
All fragments TKE Q 160 160 < TKE < 170 170 < TKE < 180 180
%fn,
windows. f)nly statistical errors are given, The
zs2CfGfI
f3
B (de&
v. (deg)
g Cdeg)
co (degf
0.90 (2) 1.13 (1) 0.98 (1) 0.81 (1) 0.66 (1)
0.50 (6) 0.62 0.56 0.48 0.41
1.040 (4) 1.48 (3) 1,30 (2) 1.15 (1) 1.00 (1) 0.87 (11 0.77KI
0.600 (13) 0.79 (8) 0.70 (4) 0.63 (3) 0.55 (2) 0.49 (31 0.45 (51
0.8
*a..* -
EXPERIMENT THEORY
i
Fig. 10. Average folding angle as a function of the light-fragment
mass number for 235U(nthr f) and
2s2cfw.
As discussed above, TKE and E* are strongly anticorrelated. This is clearly the reason for the decreasing average folding angie as a function of TIE. This dependence is represented in Fig. 12 for 235U(nt,,, f) and 2”2Cf(sf). Within the neutron-emission model, OCTKEZ)drops down to zero at the limit value TIE, = - B,. In reality, there is still secondary influence of prompt y-ray emission at e hiirer TKE, i.e. between TKE, and Q,,. 5.5. Variim correIQtims For obvious reasons, several correlations were investigated. First, Fig. 13 shows the &-G-correlation, where altf measured data as fL4 and f(TKI3 as well as the
T. Haninger et al. / Excitation energy of fragment
/ I
/-’
I--\ \ _
I
\
/
/
/
/
I
.
.
YIELD
\ \ , \ \ \ \ \
+,i I* I
311
. .
‘\ \ + \ \
l
b
j’“““” 70
80
,,,,,,,,,,,,,,,, 1 (,,,,,,,,,,,,
100
90
110
120
A Fig. 11. Upper figure: The average total number of emitted neutrons as a function of A for 235U (dots: experimental neutron-multiplicity data, dashed line: fragment-mass yield in arbitrary units). Lower figure: The distribution parameter o (square root of variance) as a function of the light-fragment mass number A for the reaction 235U(n,,,, f).
total values for both fission reactions are included. With the exception of the 252Cf(sf) data at very low TKE, i.e. very high 5, we obtain a pure linear correlation. An overall fit including all data of both fission reactions studied yields the approximative relation @[deg] = 0.175 + 0.42,
(27)
which should be valid within 10%. This correlation indicates that the average folding angle is a good measure of the average total excitation energy of the
312
T. Haninger et al. / Excitation energy of fragment
IO
0.6
-
0.4
-
.e*..
0.0E 130
140
EXPERIMENT THEORY 150
160
170
180
190
200
210
220
230
TKE (MeV] Fig. 12. Average foldingangle as a functionof TKE for wU(n,,
f’)and uz~.
fragments (see below). Fig. 14 shows the almost linear correlation between B and (T. Linear regression yields a[deg] = 0.49@deg]
+ 0.07.
5.6. Total excitation energy versus average folding angle The average total excitation energy for a given mass asymmetry is given by
(2% where c is the average Q-value for a given mass split, i.e. the average over the relevant charge distribution. A linear correlation between E* and B was found.
T. Haninger et al. / &citation
x
1.6
‘;;I;
energy of fragment
-
TOTAL
313
VALUE
-
1.2-
$ IO
0.8
-
0.4
-
-0 OXXUf X -
0.0
0
1
2
3
f TKE) DATA I A) DATA TOTAL 4
VALUE 5
6
i3
Fig. 13. The correlation between the average folding angle and average number of neutrons shown for both fission reactions studied. The solid line represents the result of an overall fit to all available data (see text).
Fig. 15 shows experimental data for both reactions investigated. At very large folding angle this correlation is disturbed due to ternary fission events. The angle limits discussed above on the basis of the total folding-angle distribution are also visible in Fig. 15. If E*(8) is plotted for different fragment-mass bins, there is a linear correlation for each bin. An approximation by E& = slope X 0 + offset
(30)
yields slopes and offsets of these straight lines as function of the light fragment mass. The results are represented in Fig. 16. The measured data are well reproduced by the neutron-emission model calculations. The observed dependencies are due to the typical energy conditions considered within the energy-partition model
T. Haningeret al. / Excitation energy ornament
314
2.0
Y$
1.6
-
1.2
-
0.8
-
0.4
-
.% IQ
0.0
I 0.0
*
0.2
I
(7
Fig.14. The correlation
I,,,
0.4
0.6
0.8
1.0
k41
between 8 and u deduced from the measured 235LJ(n,,, f) data.
20 l We*
-
30
1
EXPERIMENT THEORY
l ..**
EXPERIMENT
ZO~"""'"""""""".'.""'j t 0 ;
4
j
5
Edeg;
Fig. 15. Average total fragment excitation energy versus folding angle.
315
T. Haninger et al. / Excitation energy of fragment
12
4oy
,,,,,,
EXPERIMENT THEORY
-
80
90
100
---
4.,,...... 80
90
I,,
110
120
EXPERIMENT THEORY
:
- - EXPERIMENT THEORY 110
100
120
130
2OU
80
90
A Fig. 16. Slope and offset of the J?* -O-correlation
100
110
120
130
A function
obtained
from linear fits to the data.
as well as the emission model (e.g. neutron binding energies, level densities). Here, the adequate account for shell effects is essential.
6. Summary Folding-angle distributions and their correlations with TKE and mass asymmetry were investigated for “*Cf(sf) and 235U(n,,, f). In the most probable angular region, it can be understood as due to fragment deflection by prompt neutron emission. At large angles, the influence of ternary fission events is predominant. In summary, it can be stated that the folding angle of coincident fission fragments is a very valuable quantity in fission. It is linearly correlated with the total fragment excitation energy and, consequently, with the average number of prompt neutrons. The quantitatively presented correlation functions @Y) and a(& can be used to estimate folding-angle distributions for any fission reaction (in addition, as func-
316
T. Haninger et al. / Excitation energy
offragment
tion of mass asymmetry and TKE) if the relevant tota number of prompt fission neutrons is known. Finally, it is emphasized that the applied models (energy-partition model together with statistical theory of neutron emission, ternary fission approach) reproduce the bulk of experimental results quite well. The neutron-emission model includes the strict anticorrelation of the individual fragment excitation energies for given mass asymmetry and TIE The emission cascades from the complementary fragments are treated statistically independently. Here, possible angular-momentum effects which might cause a smalf correlation between the emission cascades are neglected. The present work confirms the theoretical assumptions made in describing the folding-angle distribution. We wish to thank F. Gonnenwein for very valuable discussions, H. Daniel for support, V.G. Nedorezov and A.S. Sudov for the 252Cf target, H.J. Maier and P. targets, and the group of the FRM reactor for providing the David for the ?J excellent neutron beam. The work was supported by the Bundesministerium fiir Forschung und Technologie, Bonn.
References [l] R. Vandenbosch, Nucl. Phys. A502 (19891 lc [Z] F. Gonnenwein, Nucl. Phys. A502 (19891 159~ [3] Y.S. Kim, P. Hofmann, H. Daniel, T. von Egidy, T. Haninger, F.J. Hartmann and M.S. Lotfranaei, Nucl. Instr. Meth. A329 (19931403 [4] P. Hofmann, Y.S. Kim, H. Daniel, T. von Egidy, T. Haninger, F.J. Hartmann, P. David, H. Machner, G. Riepe, H.S. Plendl, K. Ziock, B. Wright, D. Bowman, W. Lynch, J. Lieb, J. Jastrzebski and W. Kurcewicz, Sov. J. Nucl. Phys. 55 (19921 713 [5] T. von Egidy, H. Daniel, F.J. Hartmann, P. Hofmann, Y.S. Kim, W. Schmid, H.H. Schmidt, G. Riepe, H. Machner, P. David, H.S. Plendl, A.S. Iljinov, A.S. Botvina, Ye.S. Golubeva, M.V. Mebel, J. Jastrzebski and W. Kurcewicz, Nucl. Phys. A558 (1993) 383~ 161 A. Ruben and H. Mkten, Z. Phys. A337 (19901237 171 A. Ruben, H. M&ten and D. Seeliger, Z. Phys. A338 (1991) 67 [8] E.A.C. Crouch, At. Data Nucl. Data Tables 19 (19771417 [9] A.C. Wahl, At. Data Nucl. Data Tables 39 (1988) 1 [lo] H. Henschel, A. Kohnle, H. Hipp and F. Gonnenwein, Nucl. Instr. Meth. 190 (1981125 [ll] P. Geltenbort, F. Gdnnenwein and A. Oed, Rad. Eff. 93 (19861393 [12] C. Bud&Jorgenson and H.H. Knitter, Nucl. Phys. A490 (19881307 1131H. Nifenecker, C. Signarbieux, R. Babinet and J. Poitou, Physics and chemistry of fission, Proc. IAEA Symp., Rochester, August 1973 (IAEA, Vienna, 1974) [14] H. Marten and D. Seeliger, J. of Phys. GlO (1984) 349 [15] K. Arnold, I. Doring, H. Marten, A. Ruben and D. Seeliger, Nucl. Phys. A502 (19891325~ [16] A. Schubert, H. Kalka and H. M&en, submitted to Nucl. Phys. [17] K.-H. Schmitt, H. Delagrange, J.P. Defour, N. Carjan and A. Fleury, Z. Phys. A308 (19821215 [IS] B. Holmqvist, Ark. Fys. 38 (1968) 403 [19] A. Gavron, Phys. Rev. Cl3 (197612561