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Dechanneling in single-crystals due to in-flight secondary decay: a method for measuring short nuclear lifetimes
-_ -_ l!!fEJ
&A
Nuclear Instruments and Methods in Physics Research B 129 ( 1997) 341-348
NlNilB
Beam InteractIons with Materials 8 Atoms
ELSEVIER
Dechanneling in single-crystals due to in-flight secondary decay: a method for measuring short nuclear lifetimes Franc0 Malaguti aV *, Giorgio Giardina b, Paola Olivo
’
a Dipartimento di Fisica and Istituto Nazionale di Fisica Nucleare, via Irnerio 46, I-40126 Bologna, Italy ’ Dipartimento di Fisica, Universith di Messina. and Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Catania, Italy ’ Istituto di Matematica Generale e Finanziaria, Universith di Bologna, Bologna, Italy
Received 4 February 1997; revised form received 11 April 1997 Abstract The propagation
of excited heavy ions in single crystals is complicated by their possible in-flight decay, and the subsequent random perturbation on their path (dechanneling). We develop here a theory of the dechanneling valid when the decay occurs at the very beginning of the ion path. This permits us to calculate analytically the distribution of initial conditions for ion motion, that is expressed in terms of elliptic integrals. A large sample of ion trajectories is then constructed by a vectorized Monte Carlo code not containing in-flight decay. These trajectories can be appropriately weighted, and used many times, to simulate dechanneling due to different decay functions. All this produces a variant of the blocking method of measuring short interaction times between nuclei. This new method can be applied when the composite system, formed by the interaction of an ion beam with a crystal target, emits primary fragments in a too short time to be measured by the standard blocking technique. In this case, the blocking dip is sensitive to the secondary time delay, i.e. the lifetime of the primary fragment that decays in flight. Times measurable in this way range from some 10eur s to lo-l6 s. As an example, we re-analyse the data of a previous blocking experiment on the I60 +‘*Si system, that revealed unexpected long reaction times. We show that, if interpreted as delays associated to in-flight decay, such results do not contradict standard statistical model calculations.
1. Introduction The well-known Crystal Blocking Technique (CBT) to measure nuclear reaction times [l-4] has been used for many years, and recently it has been extended to heavy-ion interactions [5-71. In its standard version [2], the one used up to now, CBT can be simply summarised as follows. A beam of particles hits a single-crystal target at an angle 6, with respect to a major axis, producing a Compound Nucleus (CN) that recoils in the same direction, then it decays emitting an ion nearly parallel to the axis. The ion is strongly deflected during its motion through the crystal by the high matter density along the axis, and a detector placed in this direction outside the crystal will register a minimum in the ion angular distribution, the blocking dip. Clearly, the intensity (depth) of the dip depends on the transverse distance from the string
x=vtsini+,,
(1)
recoil velocity, and t its decay time. The larger x is, the weaker the blocking dip. The strongest blocking dips are those due to elastic scattering, for which both r and x are almost zero. To be more precise, in elastic scattering x is never quite zero. In fact, a small (but finite) recoil displacement takes place before the beam particle reaches its closest approach, i.e., before t = 0. However, this displacement is much smaller than the Thomas-Fermi screening length and so a strong blocking dip is indeed observed ‘. Since the blocking dip depends on x hence t, see Eq. (1). comparing the measured dip with the one calculated as a function of t allows us to find the decay time. The calculation is usually made in the framework of Lindhard’s theory [l], though Monte Carlo simulation is sometimes preferred because it can he more easily adapted to the experiment (see Ref. [3] as an example). Distances x that can be measured in this way range around the ThomasFermi screening length of interatomic potentials, that has
at which the ion is emitted from the CN. Here, v is the CN
* Corresponding author. email: [email protected] 0168-583X/97/$17.00
0 1997 Elsevier Science B.V. All rights reserved.
f’fl SO168-583X(97)00306-6
’ We thank the Referee for this observation.
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typical values 0.1-0.2 A. For nuclear reactions with nonrelativistic projectiles, this corresponds to decay times t from lo-‘* s to lo-l6 s. In recent years, the method has been applied to reactions induced by heavy ions, where the observed particle is a complex excited nucleus (often a Fission Fragment (FF)). Unfortunately, here the greater complexity of the physical processes makes interpretation of the data more difficult, if not ambiguous. As pointed out by Karamyan [6], Gomez de1 Campo et al. [7], and HoemlC et al. [8], the fragment can decay in flight, and the random impulse imparted to it by the emitted particle (or y ray) produces a smean’ng of the blocking dip (&channeling) that simulates a non-existing time delay in the intermediate nuclear system. This seems a serious difficulty for the blocking method [8], at least in its standard version. A more optimistic approach points out that dechanneling can be used to measure the secondary time between the formation of the observed fragment from the intermediate nuclear system and its decay in flight, in place of the primary time of the Composite System (CS), that is no longer observable. To do that, one needs a theory of dechanneling due to secondary decay. In Ref. [7] this was made easy by the fact that the observed fragments were projectile-like, and their direction of motion very near the one of the beam, so the standard theory based on Eq. (1) could be used, with the primaIy fragment in place of CN. In Ref. [8], the secondary decay was y emission, resulting in a small recoil angle, and this allowed a theory based on Lindhard’s continuum model [ 11. The model cannot be applied to larger recoil angles, like those due to particle decay, and in this case one is forced to use the Monte Carlo method. In Ref. [9] direct simulation was used, sampling the angles $ and x from their distribution (see Fig. 1 and Ref. [8]), and introducing them explicitly in the calculation of the fragment path. The drawback of this method is that most of the trajectories
2,P.F.
end up at large angles, and are useless for the blocking dip, that is limited to small angles. So, in spite of the big sampling (200000 trajectories), the computed blocking dips were of poor accuracy. In the present paper we discuss a different and more efficient approach, again based on the Monte Carlo method, but that makes use of the path weighting procedure [3,10] together with an analytical treatment of secondary decay.
2. The path weighting procedure This technique, described in more detail in Ref. [3], is an example of the so-called randomisurion processes, that permit the transformation of a probability model into another one on the same sample space [lo]. In the present case, this consists essentially in using the Monte Carlo code not to simulate the physical process of actual interest, but to perform a reference calculation of the standard kind, i.e. without secondary decay. This is done for an appropriate distribution of initial conditions for ion motion, and the relevant parameters of each trajectory, viz.
are stored in a file. Here, x is the initial distance between the ion and the string of atoms parallel to the crystal axis, 19,,and ‘p. are its starting direction, while 9 and cpare the final one. Then, in order to simulate the physical process of interest, we read the file and classify the events (2) in a histogram according to the final angle 6, but giving them different weights, viz.
where 9(x, fro, cpO) is the probability density for the initial conditions (x, z?*, rpo), and +a the corresponding density in the reference calculation. N is the sample size, i.e. the number of events stored in the file. In practice, ‘pO does not enter the weight (3), because it is independent from (x, aOo>and uniformly distributed over [0,2~] in both densities. Moreover, if we indicate with D the mixed density of (x, a,,) in the case of interest, and perform the reference calculation assuming standard primary decay as discussed in Section 1, x and a0 are independent random variables in the reference case. Then, assuming for x an exponential distribution with mean sa. for a0 a uniform distribution over [O, AR]. and introducing a multiplier
X =&slneD
Fig. 1. Geometrical relationships for secondary particle dechanneling. The composite system (C.9, that recoils along the beam direction 9,. is assumed to perform a negligible displacement before its (primary) decay; xL is the longihidinal path travelled by the primary fragment PF before its (secondary) decay. The angles JI and x defme the light particle direction after in-flight decay, that occurs at a transverse displacement x from the string.
[(AR + &,I,)/& I’ that will be justified in Appendix B, the weight (3) becomes
SR(AR+
km>2
NAR
D( x,79&xp( +x/s,),
F. Malaguti et al./Nucl. Instr. and Meth. in Phys. Res. B 129 (1997) 341-348
where the normalisation constant before D can be neglected if the whole histogram is finally renormalized to 1. In this way, the same set of N trajectories, calculated once and for all in the reference case and stored in a file, can be used for many different processes, each one being described by the appropriate density D(x, ?Yo). A limitation of this method is that (x, a,,, cp,,) are the initial conditions of ion motion, before it interacts with the crystal lattice. This means that the distance travelled by the ion between its formation (at a lattice site) and its decay should be less than the lattice constant, say 3-4 A. In reality, single interactions between ion and lattice atoms are usually very weak, and some tens are needed to produce a sizeable deflection of the ion path. Therefore, to a good approximation we can assume that the ion continues its rectilinear motion (without interacting with the crystal) for a few interatomic distances, say lo-20 A. For larger distances travelled before secondary decay one should go back to the methods described in Refs. [8,9]. Observe, however, that in the case of secondary decay by particle emission the lifetimes are shorter than lO-‘610-t’ s, and the corresponding distances satisfy the condition mentioned above. Only for gamma decay am the lifetimes so long (typically lo-t4 s) that the present theory is not applicable.
343
ea
++ +
I@o
n =s46tlopsr
=.l A
“.T
R=705flB8ll V.T
&--e-eJ
=
16 A
d,..,, 8 (mrad)
Fig. 3. Example of blocking dips calculated by the present method,
3. The mixed probability density D(x, 6,) The scheme we consider here for dechanneling due to secondary decay is shown in Figs. 1 and 2. The beam interacts with a lattice atom producing a composife system
for longitudinal mean displacements UT ranging from zero to 16 A. The crystal was Si(llO), and the ion was I60 with 60 MeV kinetic energy. Open circles for UT= 0 are prompt (elastic) 2 = 8 experimental data from [ 141. (CS), not necessarily a compound nucleus, whose primary
lifetime is below the lowest sensitivity limit of the blocking method. The CS produces excited primary fragments (PF’s) that eventually decay in flight after a secondary time t. Since the primary time is negligible, the PF’s begin their trajectories at a lattice equilibrium site. Let 6, and 9c be the angles between PF path and string, before and after secondary decay, respectively. Then the transverse displacement x at which secondary decay happens is x = XL?9r= ut?9Qp )
(9
where xL - ut is the longitudinal displacement, with u = PF velocity and t = its decay time. The small-angle approximation will be used from now on, compare with Eq. (1). The angle T?e is given by Eq. (Al) in Appendix A, again in the small-angle approximation, and we see that x and I&, are not independent random variables, because they both depend on 6,. We indicate with D, (ao,/~,) the conditional density of a,,, given 6,, see Appendix A. Then me mixed density of (x,, a,, So> is Fig. 2. The relevant angles of Fig. 1 projected onto the transverse (it,, 4,) plane. 9, is uniformly distributed over the circle of radius A.
Q( xL ,19,,19c) = density of xL. D,( 4,/19~) .density of 19~.
(6)
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We assume an explicit model where xL is independent from (I?&,,‘Yp)and has an exponential distribution of mean ur, while 19, is isotropic in space between 0 and some maximum A. Then, starting from Q and using standard methods of probability transformations [ Ill, one finds
x4( %o/~,)~
(7)
The models for the distribution of xL, i.e. the decay function of the PF, and a,, i.e. its angular distribution, can easily be changed if necessary. We evaluate the integral in Eq. (7) numerically, using the routine QDAGS of the IMSL program library [ 121.
4. Results We display in Fig. 3 some blocking dips calculated with the present method for longitudinal mean displacements (x) = u r, Eq. (5). ranging from zero to 16 A. The calculation refers to the experiment described in Ref. [ 141, where $,,,, Eq. (A4), was guessed to be 92 mrad, see Section 5, below. Reference calculations were made for 9, uniform on [0,30] and [30,40] mrad, then extended to A, = 60 mrad using outward translation, see Ref. [3]. The transverse mean displacement s% was taken to be zero for ur=O,~~=0.1~for~r=1A,andsa=0.4~and0.8 ,&for u r= 4 and 16 A. The Monte Carlo code of Ref. [3]
was used, modified in order to describe the actual crystal, Si( 1lo), and ion beam, 60 MeV 160. We used the Cray C-90 system of the CINECA computing centre in Bologna, that allows a vectorized and very efficient reference calculation: typical computing times were 1.6 ms for trajectories crossing 1420 crystal planes. Reconstruction of secondary-decay blocking dips was much more time-consuming, because use of IMSL routines made vectorization impossible. Typical computing times were around 18 ms per trajectory, mainly spent for the integral contained in Eq. (7). The difficulty was overcome using an interpolation procedure based on the IMSL routine QDZVL [ 121. After appropriate smoothing (see Fig. 4), some 1700 values of D(x, 6,) were calculated according to Eq. (7) and stored in a rectangular grid from which QDZVL performed quadratic interpolation. Typical computing times using interpolation were around 0.5 ms per event. A comparison with Ref. [9], where direct simulation was used, shows that the blocking dips calculated in the present way are much more accurate than the earlier ones. The reason is that now we are making full use of the reference calculation, that is limited to small angles, those really necessary to construct the dip. The lifetimes measurable with this method are easily estimated from the present calculation (Fig. 3), where. one sees that the blocking dip is sensitive to lon itudinal displacements xL = u t ranging from 0.5 to 20 f . Since the velocity of PF’s, like Z = 11 ions having = 50 MeV kinetic energy, is typically 0.2 A/as (1 as - lo-‘* s), the corresponding times t range from some lo-‘* s to lo-l6 S.
LOCI10
/ Vad
1
Fig. 4. Plot of the function D(x, 6,) defined in Section 3. It has been smoorhed dividing it by 8,. see Eqs. (A5 and A@, then taking its common Iogarithm. Fixed parameters here are U’T= I A, A = 0.152 rad, and (Im, = 0.092 rad, see Appendices.
F. Maluguti et al./NucI. Insrr. und Meth. in Phys. Res. 3 129 (1997) 341-348
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5. The I60 (100 MeV) + *‘Si( 110) reaction The first reaction investigated using the experimental set-up described in Ref. [9] was induced by I60 ions (100 MeV kinetic energy) on “Si( 110) single-crystals, and observed at Olab- 15” [ 141. Experimental details are to be found in Refs. [9,14]. The most intriguing result was found [ 141for fragments having Z = 11 and 12, whose energy spectrum suggested a fusion-fission process, and the blocking dip, analysed in the standard way, corresponded to a large time delay, of the order of lo-” s. This is in striking contradiction with presently accepted knowledge about fusion-fission and fusion-evaporation processes for light nuclei. See for example Refs. [15-181 for similar reactions. Detailed statistical model calculations show that fusion-fission has a very small probability in the present energy region, while fusion-evaporation is produced with a very small CN lifetime - of the order of 10-22 s. It is also interesting to remember that the same experiment, but at two different beam energies, has been carried out by Prasad and coworkers with the same results [19]. The two time delays observed in [19] are very similar in spite of the different excitation energies of the hypothetical CN, thus suggesting that they do not correspond to the CN lifetime, but to some secondary effect only weakly dependent on CN energy. It seems therefore reasonable to try a different interpretation of the data, assuming that the reaction proceeds through a direct process for the CS (may be deep inelastic scattering), followed by in-flight decay of the PF’s, that have a long lifetime due to their low excitation energy. All this encourages us to try the way of analysis suggested by the former Sections, though a difficulty arises since our experimental set-up does not see the Light Particle (LP) emitted during PF decay. So, we are forced to rely on model guesses for the maximum deflection angle I+&, , see Eq. (‘44). Standard statistical model (SSM) calculations suggest that PF’s are likely to be generated by direct reactions (e.g. break-up processes, transfer reactions, etc.) or enegy-disTable 1 Statisticat model calculations about the most likely formation channels of secondary fragments with Z = 11.12. The primary fragment (PF), having an excitation energy E’ - 27-30 MeV, decays (with lifetime T) emitting a light particle (LP), that induces a maximum recoil angle I,& on the observed fragment Z. W is the probability of the formation channel, normalized to total fragment production = 1. It enters in the average values (II;, > = 92 mrad, and (7) = 11.4 as (1 as = lo-l8 s)
w (%) 6.77 36.26 5.26 24.26 27.45
SL’VT
(ii,
Fig. 5. The calibration curve R(s, = UT) calculated by the present formalism, tracking some 2.5E+5 ions I60 (60 MeV kinetic energy) along a (1 IO) axis of a 28Si crystal, 0.25 pm thick.
sipating processes (e.g. deeply-inelastic collisions, fissionlike reactions, etc.). Then, they are expected to decay mainly by LP emission: n, p, or LY.This means that every observed secondary frasment (SF) can be produced in many different ways, so its I,&,, and the corresponding PF lifetime will be an average among the different possible formation channels, Relevant characteristics of the most likely PF formation leading to the observed final fragments have been calculated by SSM and are collected in Table 1, where the excitation energies E of the PF’s were. assumed to be around 27-30 MeV. Such excitation energies are reasonable either for PF’s produced by a very unlikely (and therefore neglected in the following) CN fission after evaporation of some light particles, or for PF’s produced by an energy-dissipating process, like strongly damped deeply-inelastic collisions and fission-like reactions. The aueraged maximum deflection angle and mean lifetime are then, from Table 1: l
(I,&_>-92mrad
and
(7)=
11.4as.
(8)
Blocking dips corresponding to $,,,, = 92 mrad and different values for s,_ = UT have been calculated using the formalism described above and used to construct the calibration curue R(s,) displayed in Fig. 5. Here, R is the volume 0, of the observed (delayed) dip, normal&d to the prompt (st_ - 0) one, R,. Use of the parameter R in place of 0, is convenient because R is only weakly dependent on crystal defects [4,14]. Then, from the experimental R-value for Z = 11 and 12 (Table 1 of Ref. [ 141) R = 0.48 + 0.21
(9) (to be more precise, R in [14] was misprinted as 0.44 + 0.19), using the curve of Fig. 5 and calculating u from reaction kinematics, we get the delay time 7(Z=
11 and 12) = 22’::
as
(10)
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F. Malaguti et al./Nucl. Instr. and Meth. in Phys. Res. B I29 (1997) 341-348
in reasonable agreement with the theoretical value (8). Obviously, the theoretical calculation could well differ by a factor of 2-4 in view of the uncertainty on the excitation energy of the primary fragments. So, by the present method we have demonstrated that the secondary delay time can be related to the excited primary fragments, but more complete experiments are needed, that also detect the emitted particles in coincidence with the blocked final fragments.
6. Conclusions Very often, the crystal blocking method has produced unexpectedly long lifetimes, much longer than guessed from SSM calculations, and sometimes raising the question about the validity of the model. But, in nearly all cases, a deeper insight has permitted to attribute the discrepancy to fine details of the reaction mechanism, without needing a revision of SSM. See [5] as an example. In recent experiments of this kind [9,14,19], very long reaction times have been observed, that again seem to contradict SSM. For these experiments, we propose here an alternative explanation of the data, based on the assumption that the observed time delay is not that of the CN, but that of in-flight decay of PF’s formed in the interaction between beam and crystal target. We develop a formalism that permits a quantitative extraction of PF lifetimes, and we verify that they do not contradict known predictions of SSM in light nuclei. For a better understanding of the matter, we remember that according to Lindhard’s theory [l], the shape and volume of blocking dips depend on the distribution of transverse energy E,=V(?)+l/2m(~sin6)~
(11)
at exit from the crystal. Here V is the multistring potential energy in the transverse plane for the channeled ion, and vsin0 its transverse velocity. In a very crude but sometimes useful approximation, one can forget about the kinetic part of E I , Eq. (1 1), and state that blocking dips are sensitive to the distribution of initial distances x between the atomic string and the detected ion at the moment of its last decay (so, after that, it is stable). See Section 1, and Eqs. (1) and (5). Transforming such a distance distribution into a time distribution, i.e. a decay function, needs knowledge of the reaction mechanism. The same experimental data, if read assuming different mechanisms, lead to different time delays (not very different: usually, a factor of 2 or 4). In the simplest case, the standard one [2], the reaction proceeds through the formation of a CN that decays emitting a stable light fragment, that is simply steered by the crystal. In this case, x is the CN transverse velocity multiplied by its lifetime (l), and data interpretation is unambiguous. In the just slightly more complicated case of heavy ion interactions, in-flight decay, if it exists, makes the analysis
ambiguous and dependent both on primary and secondary time. There are probably also more complicated cases (evaporation chains, for example, or CN reactions followed by FF decay), for which the ambiguities are even worse. In such cases, the sole observation of the blocking effect is probably useless, and more complete experiments are called for, where all particles emitted in coincidence with the blocked one are detected, and a precise definition of the composite system is reached. Only then will the blocking effect be a precious additional constraint on the data, and only at the end of a global analysis will the time information be recovered. And, as the example discussed in Section 5 suggests, the time information will probably agree with predictions of standard theories, without any need to modify them. The contribution of blocking will be in clarifying the reaction mechanism, not in calling for new theories of nuclear reactions. We believe that the formalism presented here will be a step in such a direction.
Acknowledgements We thank Giovanni Erbacci and the whole staff of CINECA computing center for their precious help and encouragement, and Stefano Alliney for stimulating discussions.
Appendix
A. The conditional
density
DC (aO/
9,)
All directions of space forming a small angle with the string can be represented as vectors in the transverse (8X, J+J plane, see Ref. [3]. The ones relevant in the present case are defined in Fig. 1 and displayed by Fig. 2 in the transverse plane. From the figure we see that 6,=
lY;++~+2~~~cosx,
(Al)
where 6, is isotropic, i.e. uniformly distributed over the circle of radius A, and the azimuthal angle x is uniform over [-rr, 7r]. The probability density for the deflection angle @, assuming isotropic emission from the PF and small deflection angles [8,9], is
(‘9 where the maximum deviation angle I,&, is given by @; = E,2/(2&T)
(A3)
for y decay, or I/J: = M,T,/MT
(‘9
for particle decay. Here, M and T CM,, and T,,) are mass and kinetic energy of the PF (emitted light particle), respectively. For a given 6,. from definition (Al) and
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knowing the densities of $ and x, standard methods of probability transformations [ 1l] allow to find the density of 8o, given 9,. A lengthy calculation, that will not be reported here, permits to express the result in terms of elliptic integrals first [ 131, then of Carlson’s functions, that have been coded in the IMSL program library [12]. Let us define the two functions:
For 19~> A - I),,,, the density decreases, reaching zero at A + &,,, because now redistribution mixes inside (19, < A) and outside (6, > A). The nonnalisation integral of density (B2) over [0, A - I),,,] is (1 - h/A)‘.
(B3)
The model space we are working in is limited by the condition imposed on the reference calculation (see Eq. (4)), viz. 9, E [O,A,].
(‘w
347
(B4)
It is convenient that Go, after secondary decay, is isotropic on the whole model space, so blocking dips can be normalised as usual [l]. Comparing Eqs. (B2) and (B4), this means A=A,+&,,
WI
hence, the normalisation integral (B3) becomes [ A,/(
A, +
h)12
W)
thus justifying the extra factor introduced in Eq. (4). where
RF( -LY,Z) -
dt ‘jm 2 0 [(t+X)(f+y)(l+z)]“z
(A7)
Carlson’s incomplete elliptic integral of the first kind [ 121. Then the final result D, (8,J&,) can be resumed as follows: Case A: IY~- 0, hence fit = $. In this case D, is the same as the density of $, Eq. (A2). and has an integrable divergence for 8,, = I,&. CaseB:O<8P $,,, and 9, E [8r - $,,,, 19~+ $,,,I. In this case D, is given by D,, Eq. (A6), and has no divergences. In all other cases, DC = 0. is
Appendix B. Renormalization Any redistribution of an isotropic yield yields again the same isotropic yield. In Section 3 we assumed 6, isotropic on [0, A], and now we will further assume A>&.
(Bl)
Since secondary decay acts as a redistribution over [0, I,&], see Fig. 2, it follows that &, has an isotropic distribution, with density %
ifaoE[O,A-&,,I.
(B2)
References [l] J. Lindhard, K. Dan, Vidensk. Selsk., Mat. Fys. Medd. 4 (I%51 34. [2] A.F. Tulinov, Sov. Phys. Dokl. 10 (1965) 463. [3] F. Malaguti, Nucl. Instr. and Meth. B 34 (1988) 157. [4] W.M. Gibson, Ann. Rev. Nucl. Sci. 25 (1975) 465; I. Massa and G. Varmini, Riv. Nuovo Cimento (1982) 9. [5] J.U. Andersen, AS. Jensen, K. Jorgensen, E. Laegsgaard, K.O. Nielsen, J.S. Forster, I.V. Mitchell, D. Ward, W.M. Gibson, J.J. Cuomo, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 40 (1980) 1. [6] S.A. Karamyan, Nucl. Instr. and Meth. B 51 (1990) 534. [7] J. Gomez de1 Campo, J. Barrette, R.A. Dayras, J.P. Wieleczco, E.C. Pollacco, S. Saint-Laurent, M. Toulemonde, N. Neskovic’, R. Oistojic’, Phys. Rev. C 41 (1990) 139. [8] R.F.A. HoemIt, R.W. Fearick, J.P.F. Sellshop, Phys. Rev. Lett. 68 (1992) 500. [9] E. Fuschini, F. Malaguti, E. Fioretto, R.A. Ricci, L. Vannucci, A. D’Arrigo, G. Giardina, A. Taccone, A. Moroni, N. Eremin, 0. Yuminov, G. Vannim, Phys. Rev. C 50 (1994) 1964. [lo] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2 (Wiley, New York and London, 1971) pp. 53-55 and 73-74. [l 11 See, for instance: B.W. Lindgmn. Statistical Theory. 3rd Ed. (Collier and MacMillan, New York and London, 1976) Chap. 2.1. [ 121 IMSL MATH/LIBRARY Special Functions: Fortran subroutines for mathematical applications, Version 2.0 (I 99 1) Chap. 9, pp. 172 and 177; Vol. II, Integration and Differentiation, Chap. 4, pp. 686-688; Interpolation and Approximation, Chap. 3, p. 501. [13] I.S. Gradshteyn and I.M. Ryzhik, in: Table of Integrals, Series, and Products, ed. A. Jeffrey (Academic Press, New York and London, 1965) p. 219 [English translation].
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NlNilB
Beam InteractIons with Materials 8 Atoms
ELSEVIER
Dechanneling in single-crystals due to in-flight secondary decay: a method for measuring short nuclear lifetimes Franc0 Malaguti aV *, Giorgio Giardina b, Paola Olivo
’
a Dipartimento di Fisica and Istituto Nazionale di Fisica Nucleare, via Irnerio 46, I-40126 Bologna, Italy ’ Dipartimento di Fisica, Universith di Messina. and Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Catania, Italy ’ Istituto di Matematica Generale e Finanziaria, Universith di Bologna, Bologna, Italy
Received 4 February 1997; revised form received 11 April 1997 Abstract The propagation
of excited heavy ions in single crystals is complicated by their possible in-flight decay, and the subsequent random perturbation on their path (dechanneling). We develop here a theory of the dechanneling valid when the decay occurs at the very beginning of the ion path. This permits us to calculate analytically the distribution of initial conditions for ion motion, that is expressed in terms of elliptic integrals. A large sample of ion trajectories is then constructed by a vectorized Monte Carlo code not containing in-flight decay. These trajectories can be appropriately weighted, and used many times, to simulate dechanneling due to different decay functions. All this produces a variant of the blocking method of measuring short interaction times between nuclei. This new method can be applied when the composite system, formed by the interaction of an ion beam with a crystal target, emits primary fragments in a too short time to be measured by the standard blocking technique. In this case, the blocking dip is sensitive to the secondary time delay, i.e. the lifetime of the primary fragment that decays in flight. Times measurable in this way range from some 10eur s to lo-l6 s. As an example, we re-analyse the data of a previous blocking experiment on the I60 +‘*Si system, that revealed unexpected long reaction times. We show that, if interpreted as delays associated to in-flight decay, such results do not contradict standard statistical model calculations.
1. Introduction The well-known Crystal Blocking Technique (CBT) to measure nuclear reaction times [l-4] has been used for many years, and recently it has been extended to heavy-ion interactions [5-71. In its standard version [2], the one used up to now, CBT can be simply summarised as follows. A beam of particles hits a single-crystal target at an angle 6, with respect to a major axis, producing a Compound Nucleus (CN) that recoils in the same direction, then it decays emitting an ion nearly parallel to the axis. The ion is strongly deflected during its motion through the crystal by the high matter density along the axis, and a detector placed in this direction outside the crystal will register a minimum in the ion angular distribution, the blocking dip. Clearly, the intensity (depth) of the dip depends on the transverse distance from the string
x=vtsini+,,
(1)
recoil velocity, and t its decay time. The larger x is, the weaker the blocking dip. The strongest blocking dips are those due to elastic scattering, for which both r and x are almost zero. To be more precise, in elastic scattering x is never quite zero. In fact, a small (but finite) recoil displacement takes place before the beam particle reaches its closest approach, i.e., before t = 0. However, this displacement is much smaller than the Thomas-Fermi screening length and so a strong blocking dip is indeed observed ‘. Since the blocking dip depends on x hence t, see Eq. (1). comparing the measured dip with the one calculated as a function of t allows us to find the decay time. The calculation is usually made in the framework of Lindhard’s theory [l], though Monte Carlo simulation is sometimes preferred because it can he more easily adapted to the experiment (see Ref. [3] as an example). Distances x that can be measured in this way range around the ThomasFermi screening length of interatomic potentials, that has
at which the ion is emitted from the CN. Here, v is the CN
* Corresponding author. email: [email protected] 0168-583X/97/$17.00
0 1997 Elsevier Science B.V. All rights reserved.
f’fl SO168-583X(97)00306-6
’ We thank the Referee for this observation.
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typical values 0.1-0.2 A. For nuclear reactions with nonrelativistic projectiles, this corresponds to decay times t from lo-‘* s to lo-l6 s. In recent years, the method has been applied to reactions induced by heavy ions, where the observed particle is a complex excited nucleus (often a Fission Fragment (FF)). Unfortunately, here the greater complexity of the physical processes makes interpretation of the data more difficult, if not ambiguous. As pointed out by Karamyan [6], Gomez de1 Campo et al. [7], and HoemlC et al. [8], the fragment can decay in flight, and the random impulse imparted to it by the emitted particle (or y ray) produces a smean’ng of the blocking dip (&channeling) that simulates a non-existing time delay in the intermediate nuclear system. This seems a serious difficulty for the blocking method [8], at least in its standard version. A more optimistic approach points out that dechanneling can be used to measure the secondary time between the formation of the observed fragment from the intermediate nuclear system and its decay in flight, in place of the primary time of the Composite System (CS), that is no longer observable. To do that, one needs a theory of dechanneling due to secondary decay. In Ref. [7] this was made easy by the fact that the observed fragments were projectile-like, and their direction of motion very near the one of the beam, so the standard theory based on Eq. (1) could be used, with the primaIy fragment in place of CN. In Ref. [8], the secondary decay was y emission, resulting in a small recoil angle, and this allowed a theory based on Lindhard’s continuum model [ 11. The model cannot be applied to larger recoil angles, like those due to particle decay, and in this case one is forced to use the Monte Carlo method. In Ref. [9] direct simulation was used, sampling the angles $ and x from their distribution (see Fig. 1 and Ref. [8]), and introducing them explicitly in the calculation of the fragment path. The drawback of this method is that most of the trajectories
2,P.F.
end up at large angles, and are useless for the blocking dip, that is limited to small angles. So, in spite of the big sampling (200000 trajectories), the computed blocking dips were of poor accuracy. In the present paper we discuss a different and more efficient approach, again based on the Monte Carlo method, but that makes use of the path weighting procedure [3,10] together with an analytical treatment of secondary decay.
2. The path weighting procedure This technique, described in more detail in Ref. [3], is an example of the so-called randomisurion processes, that permit the transformation of a probability model into another one on the same sample space [lo]. In the present case, this consists essentially in using the Monte Carlo code not to simulate the physical process of actual interest, but to perform a reference calculation of the standard kind, i.e. without secondary decay. This is done for an appropriate distribution of initial conditions for ion motion, and the relevant parameters of each trajectory, viz.
are stored in a file. Here, x is the initial distance between the ion and the string of atoms parallel to the crystal axis, 19,,and ‘p. are its starting direction, while 9 and cpare the final one. Then, in order to simulate the physical process of interest, we read the file and classify the events (2) in a histogram according to the final angle 6, but giving them different weights, viz.
where 9(x, fro, cpO) is the probability density for the initial conditions (x, z?*, rpo), and +a the corresponding density in the reference calculation. N is the sample size, i.e. the number of events stored in the file. In practice, ‘pO does not enter the weight (3), because it is independent from (x, aOo>and uniformly distributed over [0,2~] in both densities. Moreover, if we indicate with D the mixed density of (x, a,,) in the case of interest, and perform the reference calculation assuming standard primary decay as discussed in Section 1, x and a0 are independent random variables in the reference case. Then, assuming for x an exponential distribution with mean sa. for a0 a uniform distribution over [O, AR]. and introducing a multiplier
X =&slneD
Fig. 1. Geometrical relationships for secondary particle dechanneling. The composite system (C.9, that recoils along the beam direction 9,. is assumed to perform a negligible displacement before its (primary) decay; xL is the longihidinal path travelled by the primary fragment PF before its (secondary) decay. The angles JI and x defme the light particle direction after in-flight decay, that occurs at a transverse displacement x from the string.
[(AR + &,I,)/& I’ that will be justified in Appendix B, the weight (3) becomes
SR(AR+
km>2
NAR
D( x,79&xp( +x/s,),
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where the normalisation constant before D can be neglected if the whole histogram is finally renormalized to 1. In this way, the same set of N trajectories, calculated once and for all in the reference case and stored in a file, can be used for many different processes, each one being described by the appropriate density D(x, ?Yo). A limitation of this method is that (x, a,,, cp,,) are the initial conditions of ion motion, before it interacts with the crystal lattice. This means that the distance travelled by the ion between its formation (at a lattice site) and its decay should be less than the lattice constant, say 3-4 A. In reality, single interactions between ion and lattice atoms are usually very weak, and some tens are needed to produce a sizeable deflection of the ion path. Therefore, to a good approximation we can assume that the ion continues its rectilinear motion (without interacting with the crystal) for a few interatomic distances, say lo-20 A. For larger distances travelled before secondary decay one should go back to the methods described in Refs. [8,9]. Observe, however, that in the case of secondary decay by particle emission the lifetimes are shorter than lO-‘610-t’ s, and the corresponding distances satisfy the condition mentioned above. Only for gamma decay am the lifetimes so long (typically lo-t4 s) that the present theory is not applicable.
343
ea
++ +
I@o
n =s46tlopsr
=.l A
“.T
R=705flB8ll V.T
&--e-eJ
=
16 A
d,..,, 8 (mrad)
Fig. 3. Example of blocking dips calculated by the present method,
3. The mixed probability density D(x, 6,) The scheme we consider here for dechanneling due to secondary decay is shown in Figs. 1 and 2. The beam interacts with a lattice atom producing a composife system
for longitudinal mean displacements UT ranging from zero to 16 A. The crystal was Si(llO), and the ion was I60 with 60 MeV kinetic energy. Open circles for UT= 0 are prompt (elastic) 2 = 8 experimental data from [ 141. (CS), not necessarily a compound nucleus, whose primary
lifetime is below the lowest sensitivity limit of the blocking method. The CS produces excited primary fragments (PF’s) that eventually decay in flight after a secondary time t. Since the primary time is negligible, the PF’s begin their trajectories at a lattice equilibrium site. Let 6, and 9c be the angles between PF path and string, before and after secondary decay, respectively. Then the transverse displacement x at which secondary decay happens is x = XL?9r= ut?9Qp )
(9
where xL - ut is the longitudinal displacement, with u = PF velocity and t = its decay time. The small-angle approximation will be used from now on, compare with Eq. (1). The angle T?e is given by Eq. (Al) in Appendix A, again in the small-angle approximation, and we see that x and I&, are not independent random variables, because they both depend on 6,. We indicate with D, (ao,/~,) the conditional density of a,,, given 6,, see Appendix A. Then me mixed density of (x,, a,, So> is Fig. 2. The relevant angles of Fig. 1 projected onto the transverse (it,, 4,) plane. 9, is uniformly distributed over the circle of radius A.
Q( xL ,19,,19c) = density of xL. D,( 4,/19~) .density of 19~.
(6)
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We assume an explicit model where xL is independent from (I?&,,‘Yp)and has an exponential distribution of mean ur, while 19, is isotropic in space between 0 and some maximum A. Then, starting from Q and using standard methods of probability transformations [ Ill, one finds
x4( %o/~,)~
(7)
The models for the distribution of xL, i.e. the decay function of the PF, and a,, i.e. its angular distribution, can easily be changed if necessary. We evaluate the integral in Eq. (7) numerically, using the routine QDAGS of the IMSL program library [ 121.
4. Results We display in Fig. 3 some blocking dips calculated with the present method for longitudinal mean displacements (x) = u r, Eq. (5). ranging from zero to 16 A. The calculation refers to the experiment described in Ref. [ 141, where $,,,, Eq. (A4), was guessed to be 92 mrad, see Section 5, below. Reference calculations were made for 9, uniform on [0,30] and [30,40] mrad, then extended to A, = 60 mrad using outward translation, see Ref. [3]. The transverse mean displacement s% was taken to be zero for ur=O,~~=0.1~for~r=1A,andsa=0.4~and0.8 ,&for u r= 4 and 16 A. The Monte Carlo code of Ref. [3]
was used, modified in order to describe the actual crystal, Si( 1lo), and ion beam, 60 MeV 160. We used the Cray C-90 system of the CINECA computing centre in Bologna, that allows a vectorized and very efficient reference calculation: typical computing times were 1.6 ms for trajectories crossing 1420 crystal planes. Reconstruction of secondary-decay blocking dips was much more time-consuming, because use of IMSL routines made vectorization impossible. Typical computing times were around 18 ms per trajectory, mainly spent for the integral contained in Eq. (7). The difficulty was overcome using an interpolation procedure based on the IMSL routine QDZVL [ 121. After appropriate smoothing (see Fig. 4), some 1700 values of D(x, 6,) were calculated according to Eq. (7) and stored in a rectangular grid from which QDZVL performed quadratic interpolation. Typical computing times using interpolation were around 0.5 ms per event. A comparison with Ref. [9], where direct simulation was used, shows that the blocking dips calculated in the present way are much more accurate than the earlier ones. The reason is that now we are making full use of the reference calculation, that is limited to small angles, those really necessary to construct the dip. The lifetimes measurable with this method are easily estimated from the present calculation (Fig. 3), where. one sees that the blocking dip is sensitive to lon itudinal displacements xL = u t ranging from 0.5 to 20 f . Since the velocity of PF’s, like Z = 11 ions having = 50 MeV kinetic energy, is typically 0.2 A/as (1 as - lo-‘* s), the corresponding times t range from some lo-‘* s to lo-l6 S.
LOCI10
/ Vad
1
Fig. 4. Plot of the function D(x, 6,) defined in Section 3. It has been smoorhed dividing it by 8,. see Eqs. (A5 and A@, then taking its common Iogarithm. Fixed parameters here are U’T= I A, A = 0.152 rad, and (Im, = 0.092 rad, see Appendices.
F. Maluguti et al./NucI. Insrr. und Meth. in Phys. Res. 3 129 (1997) 341-348
345
5. The I60 (100 MeV) + *‘Si( 110) reaction The first reaction investigated using the experimental set-up described in Ref. [9] was induced by I60 ions (100 MeV kinetic energy) on “Si( 110) single-crystals, and observed at Olab- 15” [ 141. Experimental details are to be found in Refs. [9,14]. The most intriguing result was found [ 141for fragments having Z = 11 and 12, whose energy spectrum suggested a fusion-fission process, and the blocking dip, analysed in the standard way, corresponded to a large time delay, of the order of lo-” s. This is in striking contradiction with presently accepted knowledge about fusion-fission and fusion-evaporation processes for light nuclei. See for example Refs. [15-181 for similar reactions. Detailed statistical model calculations show that fusion-fission has a very small probability in the present energy region, while fusion-evaporation is produced with a very small CN lifetime - of the order of 10-22 s. It is also interesting to remember that the same experiment, but at two different beam energies, has been carried out by Prasad and coworkers with the same results [19]. The two time delays observed in [19] are very similar in spite of the different excitation energies of the hypothetical CN, thus suggesting that they do not correspond to the CN lifetime, but to some secondary effect only weakly dependent on CN energy. It seems therefore reasonable to try a different interpretation of the data, assuming that the reaction proceeds through a direct process for the CS (may be deep inelastic scattering), followed by in-flight decay of the PF’s, that have a long lifetime due to their low excitation energy. All this encourages us to try the way of analysis suggested by the former Sections, though a difficulty arises since our experimental set-up does not see the Light Particle (LP) emitted during PF decay. So, we are forced to rely on model guesses for the maximum deflection angle I+&, , see Eq. (‘44). Standard statistical model (SSM) calculations suggest that PF’s are likely to be generated by direct reactions (e.g. break-up processes, transfer reactions, etc.) or enegy-disTable 1 Statisticat model calculations about the most likely formation channels of secondary fragments with Z = 11.12. The primary fragment (PF), having an excitation energy E’ - 27-30 MeV, decays (with lifetime T) emitting a light particle (LP), that induces a maximum recoil angle I,& on the observed fragment Z. W is the probability of the formation channel, normalized to total fragment production = 1. It enters in the average values (II;, > = 92 mrad, and (7) = 11.4 as (1 as = lo-l8 s)
w (%) 6.77 36.26 5.26 24.26 27.45
SL’VT
(ii,
Fig. 5. The calibration curve R(s, = UT) calculated by the present formalism, tracking some 2.5E+5 ions I60 (60 MeV kinetic energy) along a (1 IO) axis of a 28Si crystal, 0.25 pm thick.
sipating processes (e.g. deeply-inelastic collisions, fissionlike reactions, etc.). Then, they are expected to decay mainly by LP emission: n, p, or LY.This means that every observed secondary frasment (SF) can be produced in many different ways, so its I,&,, and the corresponding PF lifetime will be an average among the different possible formation channels, Relevant characteristics of the most likely PF formation leading to the observed final fragments have been calculated by SSM and are collected in Table 1, where the excitation energies E of the PF’s were. assumed to be around 27-30 MeV. Such excitation energies are reasonable either for PF’s produced by a very unlikely (and therefore neglected in the following) CN fission after evaporation of some light particles, or for PF’s produced by an energy-dissipating process, like strongly damped deeply-inelastic collisions and fission-like reactions. The aueraged maximum deflection angle and mean lifetime are then, from Table 1: l
(I,&_>-92mrad
and
(7)=
11.4as.
(8)
Blocking dips corresponding to $,,,, = 92 mrad and different values for s,_ = UT have been calculated using the formalism described above and used to construct the calibration curue R(s,) displayed in Fig. 5. Here, R is the volume 0, of the observed (delayed) dip, normal&d to the prompt (st_ - 0) one, R,. Use of the parameter R in place of 0, is convenient because R is only weakly dependent on crystal defects [4,14]. Then, from the experimental R-value for Z = 11 and 12 (Table 1 of Ref. [ 141) R = 0.48 + 0.21
(9) (to be more precise, R in [14] was misprinted as 0.44 + 0.19), using the curve of Fig. 5 and calculating u from reaction kinematics, we get the delay time 7(Z=
11 and 12) = 22’::
as
(10)
346
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in reasonable agreement with the theoretical value (8). Obviously, the theoretical calculation could well differ by a factor of 2-4 in view of the uncertainty on the excitation energy of the primary fragments. So, by the present method we have demonstrated that the secondary delay time can be related to the excited primary fragments, but more complete experiments are needed, that also detect the emitted particles in coincidence with the blocked final fragments.
6. Conclusions Very often, the crystal blocking method has produced unexpectedly long lifetimes, much longer than guessed from SSM calculations, and sometimes raising the question about the validity of the model. But, in nearly all cases, a deeper insight has permitted to attribute the discrepancy to fine details of the reaction mechanism, without needing a revision of SSM. See [5] as an example. In recent experiments of this kind [9,14,19], very long reaction times have been observed, that again seem to contradict SSM. For these experiments, we propose here an alternative explanation of the data, based on the assumption that the observed time delay is not that of the CN, but that of in-flight decay of PF’s formed in the interaction between beam and crystal target. We develop a formalism that permits a quantitative extraction of PF lifetimes, and we verify that they do not contradict known predictions of SSM in light nuclei. For a better understanding of the matter, we remember that according to Lindhard’s theory [l], the shape and volume of blocking dips depend on the distribution of transverse energy E,=V(?)+l/2m(~sin6)~
(11)
at exit from the crystal. Here V is the multistring potential energy in the transverse plane for the channeled ion, and vsin0 its transverse velocity. In a very crude but sometimes useful approximation, one can forget about the kinetic part of E I , Eq. (1 1), and state that blocking dips are sensitive to the distribution of initial distances x between the atomic string and the detected ion at the moment of its last decay (so, after that, it is stable). See Section 1, and Eqs. (1) and (5). Transforming such a distance distribution into a time distribution, i.e. a decay function, needs knowledge of the reaction mechanism. The same experimental data, if read assuming different mechanisms, lead to different time delays (not very different: usually, a factor of 2 or 4). In the simplest case, the standard one [2], the reaction proceeds through the formation of a CN that decays emitting a stable light fragment, that is simply steered by the crystal. In this case, x is the CN transverse velocity multiplied by its lifetime (l), and data interpretation is unambiguous. In the just slightly more complicated case of heavy ion interactions, in-flight decay, if it exists, makes the analysis
ambiguous and dependent both on primary and secondary time. There are probably also more complicated cases (evaporation chains, for example, or CN reactions followed by FF decay), for which the ambiguities are even worse. In such cases, the sole observation of the blocking effect is probably useless, and more complete experiments are called for, where all particles emitted in coincidence with the blocked one are detected, and a precise definition of the composite system is reached. Only then will the blocking effect be a precious additional constraint on the data, and only at the end of a global analysis will the time information be recovered. And, as the example discussed in Section 5 suggests, the time information will probably agree with predictions of standard theories, without any need to modify them. The contribution of blocking will be in clarifying the reaction mechanism, not in calling for new theories of nuclear reactions. We believe that the formalism presented here will be a step in such a direction.
Acknowledgements We thank Giovanni Erbacci and the whole staff of CINECA computing center for their precious help and encouragement, and Stefano Alliney for stimulating discussions.
Appendix
A. The conditional
density
DC (aO/
9,)
All directions of space forming a small angle with the string can be represented as vectors in the transverse (8X, J+J plane, see Ref. [3]. The ones relevant in the present case are defined in Fig. 1 and displayed by Fig. 2 in the transverse plane. From the figure we see that 6,=
lY;++~+2~~~cosx,
(Al)
where 6, is isotropic, i.e. uniformly distributed over the circle of radius A, and the azimuthal angle x is uniform over [-rr, 7r]. The probability density for the deflection angle @, assuming isotropic emission from the PF and small deflection angles [8,9], is
(‘9 where the maximum deviation angle I,&, is given by @; = E,2/(2&T)
(A3)
for y decay, or I/J: = M,T,/MT
(‘9
for particle decay. Here, M and T CM,, and T,,) are mass and kinetic energy of the PF (emitted light particle), respectively. For a given 6,. from definition (Al) and
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knowing the densities of $ and x, standard methods of probability transformations [ 1l] allow to find the density of 8o, given 9,. A lengthy calculation, that will not be reported here, permits to express the result in terms of elliptic integrals first [ 131, then of Carlson’s functions, that have been coded in the IMSL program library [12]. Let us define the two functions:
For 19~> A - I),,,, the density decreases, reaching zero at A + &,,, because now redistribution mixes inside (19, < A) and outside (6, > A). The nonnalisation integral of density (B2) over [0, A - I),,,] is (1 - h/A)‘.
(B3)
The model space we are working in is limited by the condition imposed on the reference calculation (see Eq. (4)), viz. 9, E [O,A,].
(‘w
347
(B4)
It is convenient that Go, after secondary decay, is isotropic on the whole model space, so blocking dips can be normalised as usual [l]. Comparing Eqs. (B2) and (B4), this means A=A,+&,,
WI
hence, the normalisation integral (B3) becomes [ A,/(
A, +
h)12
W)
thus justifying the extra factor introduced in Eq. (4). where
RF( -LY,Z) -
dt ‘jm 2 0 [(t+X)(f+y)(l+z)]“z
(A7)
Carlson’s incomplete elliptic integral of the first kind [ 121. Then the final result D, (8,J&,) can be resumed as follows: Case A: IY~- 0, hence fit = $. In this case D, is the same as the density of $, Eq. (A2). and has an integrable divergence for 8,, = I,&. CaseB:O<8P
Appendix B. Renormalization Any redistribution of an isotropic yield yields again the same isotropic yield. In Section 3 we assumed 6, isotropic on [0, A], and now we will further assume A>&.
(Bl)
Since secondary decay acts as a redistribution over [0, I,&], see Fig. 2, it follows that &, has an isotropic distribution, with density %
ifaoE[O,A-&,,I.
(B2)
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