Competing analysis of α and 2p2n-emission from compound nuclei formed in neutron induced reactions

Available online at www.sciencedirect.com

ScienceDirect Nuclear Physics A 957 (2017) 274–288 www.elsevier.com/locate/nuclphysa

Competing analysis of α and 2p2n-emission from compound nuclei formed in neutron induced reactions Amandeep Kaur, Manoj K. Sharma ∗ School of Physics and Materials Science, Thapar University, Patiala-147004, India Received 1 August 2016; received in revised form 17 September 2016; accepted 19 September 2016 Available online 22 September 2016

Abstract The decay mechanism of compound system 61 Ni∗ formed in fast neutron induced reactions is explored within the collective clusterization approach of the Dynamical Cluster-decay Model (DCM) in reference to a recent experiment over an energy spread of En = 1–100 MeV. The excitation functions for the decay of the compound nucleus 61 Ni∗ formed in the n + 60 Ni reaction show a double humped variation with incident beam energy where the peak at lower energy corresponds to α-emission while the one at higher energy originates from 2p2n-emission. The experimentally observed transmutation of α-emission at lower energy into 2p2n-emission at higher incident energies is explained on the basis of temperature dependence of the binding energies used within the framework of DCM. The cross-sections for the formation of the daughter nucleus 57 Fe after emission of α-cluster from the 61 Ni∗ nucleus are addressed by employing the neck length parameter (R), finding decent agreement with the available experimental data. The calculations are done for non-sticking choice of moment of inertia (IN S ) in the centrifugal potential term, which forms the essential ingredient in DCM based calculations. In addition to this, the effect of mass (and charge) of the compound nucleus is exercised in view of α and 2p2n emission and comparative study of the decay profiles of compound systems with mass A = 17–93 is employed to get better description of decay patterns. © 2016 Elsevier B.V. All rights reserved. Keywords: n-induced reactions; DCM; Temperature-dependent binding energy

* Corresponding author.

E-mail address: [email protected] (M.K. Sharma). http://dx.doi.org/10.1016/j.nuclphysa.2016.09.009 0375-9474/© 2016 Elsevier B.V. All rights reserved.

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1. Introduction Reactions involving the use of fast neutrons as projectiles are becoming an issue of huge significance at theoretical as well as experimental front of nuclear physics. The limited occurrence of fissile isotopes put an enigma on what would be the basis of future nuclear power generations. But the ever-rising advents in the field of reactor technology has solved this problem by making use of fast neutrons as projectiles and by supplementing the limited available high fissile material by less fissile but abundantly available nuclear isotopes [1,2]. Proper understanding of the associated reaction dynamics not only helps to enhance the efficiency of such reactors, but also guides to predict the reaction cross-sections for those isotopes for which direct measurements are not explored as yet. This drives us to investigate the decay mechanism of the compound nucleus 61 Ni∗ formed in n + 60 Ni induced reaction over an energy spread of En = 1–100 MeV in accordance with an experiment performed at Los Alamos Neutron Science Center’s Weapons Neutron Research (LANSCE-WNR) facility [3]. The innovation behind this investigation lies in the fact that the excited compound nucleus (61 Ni∗ ) formed in the above-mentioned reaction follows various decay paths forming the same daughter nucleus. Such low mass compound systems decay by light particles (LPs) and γ -ray emission, with small heavy mass fragment emission component [4]. It is relevant to mention that at higher incident energies relatively more phase space is available for multi-nucleon emission, and hence 2p2n-emission starts replacing the α-emission [5]. In the present work, an effort is made to investigate the decay mechanism of the compound nucleus 61 Ni∗ within the collective clusterization approach of Dynamical Cluster-decay Model (DCM) [6–15]. The calculations are done for the optimized orientations of the hot-deformed fragments and the experimental data is fitted using the only parameter of the model, the necklength parameter ‘R’. It is worth noting that firstly, the calculations are done by using sticking limits of moment of inertia, IS (discussed in section-2) but the data could be addressed only for limited range of incident beam energy i.e. En ∼ 1–10 MeV. Consequently, the choice of sticking limits of moment of inertia (IS ) is replaced with non-sticking limits (IN S ) and comparison with experimental data improves significantly. Another point to be noted here is that the transmutation observed in the emission of α-particles at lower energies and 2p2n-emission at higher energies is explained on the basis of temperature dependance of binding energy exercised within the frame-work of DCM [16]. In addition to it, the effect of mass (and charge) of the compound nucleus on α and 2p2n-emission is investigated and a comparative analysis is done for various systems involving light to heavy mass nuclei formed in reactions induced via fast neutrons. Comparisons are made on the basis of the fragmentation structure signifying the decay pattern of these compound systems, and the effects of temperature/excitation energy, angular momentum etc. are analyzed for better description of dynamical evolution of the chosen reactions. This paper is organized as follows: the methodology used for the calculations is described in Section-2. The effects of deformations, temperature and angular momentum in expressions of radius and potential terms are duly incorporated and are briefly explained in this section. Section-3 includes the discussion of the results drawn and finally the outcomes are briefly summarized in section-4. 2. The Dynamical Cluster-decay Model (DCM) The dynamical cluster-decay model (DCM) [6–15] is based on the well known Quantum Mechanical Fragmentation Theory (QMFT) [17–19]. It is relevant to mention here that DCM is based upon the collective clusterization approach in which LPs, intermediate mass fragments

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(IMFs) and fission fragments are treated on equal footing. On the other hand, the statistical models (and its different versions) are employed to study the LP, intermediate mass fragments (IMF) or fission decay modes individually. The ER decay mode (including n, p, α and γ -rays) is considered as the fusion evaporation process from the CN through HF analysis which has been extended to include IMF in light compound nuclei with A ≤ 80 in EHFM model [20] and transition state model [21]. DCM works in terms of the collective coordinates of mass and charge asymmetry ηA =

A1 − A2 Z1 − Z2 ; ηZ = A1 + A2 Z1 + Z2

(1)

the relative separation R, the multipole deformations βλi (λ = 2, 3, 4) and orientations θi of two nuclei or fragments, where i = 1, 2 refer respectively to heavier and lighter fragments. The transfer of the kinetic energy of the incident channel (Ec.m. ) to the internal excitation (total excitation energy, TXE or total kinetic energy, TKE) of the outgoing channel, relates as E∗CN + Qout = TKE(T) + TXE(T), where the compound nucleus excitation energy reads as: E∗CN = Ec.m. + Qin = (ACN /9)T 2 − T . In DCM, the decay cross-sections in terms of partial wave analysis are given as: 

max

max  2μEc.m. π  σ= σ = 2 (2 + 1)P0 P ; k = (2) k h¯ 2

=0h¯

=0h¯

Here, P0 is the preformation probability of the decaying fragments and refers to the motion in η-coordinates and P is penetrability referring to motion in R-coordinates, both depending upon

, T, βi and θi . The deformations (βi ) used here are the quadrupole deformations with λ = 2 only and orientations (θi ) optimized for ‘hot’ configuration [22]. The values of max are fixed, where cross-sections for α-emission falls to negligibly small value. The preformation probability is the solution of Schrödinger wave equation in η-coordinates at fixed R i.e. R = Ra , which reads as: ∂ h¯ 2 1 ∂ {− √ + V (η, R, T )}ψ ν (η) = E ν ψ ν (η), √ ∂η 2 B ηη B ηη ∂η

(3)

with ν = 0, 1, 2, . . . referring to ground (ν = 0) and excited state solutions. The solution of this equation after normalization gives the value of preformation probability which reads as:  2 P0 =| ψ(η(Ai )) |2 Bηη . (4) ACN The mass parameters Bηη , used here are the classical hydrodynamical mass parameters of Kröger and Scheid [23]. The structural information of the preformation of fragments before penetrating the barrier is provided through the T-dependent fragmentation potential given by eq. (5) and such information is absent in the various statistical models. VR (η, T ) =

2  i=1

[VLDM (Ai , Zi , T )] +

2  [δUi ] exp(−T 2 /T02 ) + VC (R, Zi , βλi , θi , T ) i=1

+ VP (R, Ai , βλi , θi , T ) + V (R, Ai , βλi , θi , T )

(5)

where VC , VP and V are respectively, the Coulomb, nuclear proximity and centrifugal potential terms. The shell effects δU are obtained from empirical estimates of Myers and Swiatecki [24].

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The temperature dependent LDM-binding energies is as suggested by Davidson et al. [25] which is based on semi-empirical mass formula of Seeger [26]: η(T ) I 2 + 2|I | Z2 )×( )+ 1/3 A A r0 (T )A1/3 0.7636 2.29 f (Z, A) × (1 − 2/3 − ) + δ(T ) (6) 1/3 2 Z [r0 (T )A ] A3/4

VLDM (A, Z, T ) = α(T )A + β(T )A2/3 + (γ (T ) −

with I = aa (Z − N ), aa = 1, and respectively, for even–even, even–odd and odd–odd nuclei, f (Z, A) = (−1, 0, 1). For further details please refer to [16] Note that binding energies of xnyp-cluster with x neutrons and y protons are calculated as 5/3

B(A2 = xnyp) = xBn + yBp − ac AL

(7)

where Bn is the binding energy of one neutron and Bp is the binding energy of one proton and 5/3 these values are taken to be Temperature-dependent. The additional term ac AL is due to the repulsive Coulomb forces [24] between the protons. The T-dependent centrifugal potential V is given by: V (R, Zi , βλi , θi , T ) =

h¯ 2 ( + 1) , 2I (T )

(8)

where I(T ) represents the moment of inertia and its two limits are defined as IS (T ) = μR 2 + 2 2 2 2 5 A1 mR1 (α1 , T ) + 5 A2 mR2 (α2 , T), the moment of inertia in sticking limit, or, alternatively, the one calculated in non-sticking limit I = IN S = μR 2 . In general, the value of angular momentum are higher for the moment of inertia calculated in the sticking limit. Evidently, the sticking choice of moment of inertia provides larger values of moment of inertia as compared to non-sticking limit, and consequently provides smaller values of centrifugal potential energy to the fragments at comparable -values. So the IS approach is suitable for addressing fusionfission data probably due to larger range of angular momentum while IN S approach seems better for neutron evaporation or light charged particle emission. It is relevant to mention that sticking limit is generally preferred for low-energy dynamics of light mass nuclei as saddle point closely resembles the scission point [4]. The barrier penetration probability P is given by the WKB integral 2 P = exp[− h¯

Rb (2μ{V (R) − Qeff })1/2 dR]

(9)

Ra

with V (Ra , T ) = V (Rb , T ) = T KE(T ) = Qeff for the two turning points. Qeff is the effective Q-value of the decay process. The first turning point of the penetration path is shown in Fig. 1 for both choices of moment of inertia and is defined as: Ra = R1 (α1 , T ) + R2 (α2 , T ) + R(T )

(10)

where R is the only parameter of the model called the neck-length parameter which assimilates the neck-formation effects between the two nuclei. The radius vectors Ri (αi , T ) is defined as    (0) Ri (αi , T ) = R0i (T ) 1 + βλi Yλ (αi ) (11) λ

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Fig. 1. Scattering potential V plotted in terms of Range R (fm) for the decay 61 N i ∗ → 57 Fe + α for both the choices of moment of inertia (sticking as well as non-sticking) at their respective R. The curves are plotted at Ec.m. = 4.91 MeV (equivalently En = 5.00 MeV) and = 0h¯ and the first turning point Ra , barrier height VB and V (Ra ) are also marked.

where 1/3

R0i (T ) = [1.28Ai

−1/3

− 0.76 + 0.8Ai

](1 + 0.0007T 2 )

(12)

3. Results and discussions The present work deals with the study of decay of the compound nucleus 61 Ni∗ formed in neutron induced reaction over an incident beam energy range of En = 1–100 MeV. It should be noted that in some previous works [27,28], the decay of 56 Ni∗ formed in heavy-ion reaction has been investigated. It would be interesting to compare the fragmentation structure of 56 Ni∗ with that of 61 Ni∗ (present work). For this purpose, the fragmentation potential for the decay of compound nucleus 61 Ni∗ is plotted as a function of fragment mass at T = 0 MeV ( = 0h¯ case) as shown in Fig. 2(a) and is compared with the fragmentation structure of the decay of 56 Ni∗ (Fig. 3 of [27]). It has been found that the structure of fragmentation potential is similar for both the systems at T = 0 MeV which signifies that the LPs are favored at =0h¯ as they possess lower magnitudes of fragmentation potential. The strong dips can be seen in the fragmentation curve for both the compound systems corresponding to 4n-nuclei (α-structuring) at T = 0 MeV. Further it has been noted that the compound nucleus 61 Ni∗ formed in the reaction n + 60 Ni undergoes an iso-product exit channel forming 57 Fe as the residual nucleus. It decays through cluster emission (α-emission) at lower incident beam energies and multi-nucleon emission (2p2n-emission) at higher incident energies. The cross-sections for α or 2p2n-emission are fitted within the framework of DCM by varying the only free parameter of the model, the neck-length parameter (R). Firstly, the sticking limits of moment of inertia (IS ) are used in the centrifugal potential term (eq. (8) in Section-2) to investigate the decay pattern of the hot-rotating compound nucleus 61 Ni∗ formed in fast neutron induced reaction. Within this approach, the data could be addressed only up to some lower incident beam energy range En = 1–10 MeV. To analyze the decay behavior of hot-rotating compound nucleus 61 Ni∗ , the fragmentation potential is plotted as a function of light fragment mass at Ec.m. = 4.91 MeV (T = 0 MeV case) at the best fit values

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Fig. 2. Study of fragmentation potential [(a) T = 0 MeV and (b) T = 1.45 MeV] and (c) preformation probability P0 minimized in mass coordinate η(A) for the decay of 61 Ni∗ nucleus plotted at center of mass energy Ec.m. = 4.91 MeV (equivalently En = 5.00 MeV) using sticking limits of moment of inertia (IS ) at = 0h¯ and = max values.

of R and corresponding values of max as shown in Fig. 2(b). Fragmentation potential plotted for = 0h¯ and = max values (in Fig. 2(b)) suggests that the emission of light charged particles with A2 ≤ 4 are favored at lower -values and a change in fragmentation pattern is observed when we shift to higher values of angular momentum ( ) which favors the decay of fragments with A2 ≥ 5. Since we are dealing with α and 2p2n-emission we shall confine our analysis to lower -values only. Another point to be observed here is that for 61 Ni∗ the effect of α-structuring seems to disappear at higher T-values while the same trend seems to be consistent at all T-values in case of 56 Ni∗ (comparing Fig. 2(b) of this paper with Fig. 3 of [27]). This may be attributed to the addition of 5 neutrons due to which the α-nucleus structure vanishes for IMF region. Similar results were obtained in earlier work [13], where four isotopes of Barium (114,118,122,126 Ba∗ were studied and the α-structuring was primarily seen only for 114,118 Ba∗ while it vanished for 122,126 Ba∗ isotopes. Along with it, the preformation probability P is plotted at same E 0 c.m. and R as that for Fig. 2(b) and is shown in Fig. 2(c). Note that lower the value of fragmentation potential for a cluster/fragment, consequently higher would be the formation probability of corresponding cluster/fragment. This is further clarified from the preformation probability plot as shown in Fig. 2(c) and it is clear that lower -values (shown by filled circles) favor the emission of light charged particles with A2 ≤ 4 as they have higher P0 values and again when we shift towards max values, the potential energy surface change signifying the decay of the compound nucleus via near symmetric fragments and some intermediate mass fragments (IMFs) as the competing decay mode. Here IMF refers to fragments with A2 = 5–20 and Z2 = 2–10. In addition to this, preformation probability P0 has also been compared with the earlier works [27,28] (comparison of Fig. 2(c) with Fig. 4 of [27]) and it has been observed that similar result is drawn for both isotopic compound systems, that at lower -values ( = 0h¯ ) light particles are highly preformed. Since the sticking limits of moment of inertia (IS ) proves to be inadequate for fitting the experimental data over whole energy range, so we chose to replace the sticking limits (IS ) with non-sticking limits of moment of inertia (IN S ). It is to be noted that the DCM based crosssections (eq. (2)) follow the partial wave analysis involving summation of the cross-sections up to max values. So the choice of limits of inertia is of much significance in context of DCM based calculations. In order to have a comprehensive study of effect of these limits of moment

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Fig. 3. (a) Preformation probability and (b) Penetration probability P as a function of fragment mass A2 for the decay of 61 Ni∗ nucleus plotted at center of mass energy Ec.m. = 4.918 MeV at extreme value of angular momentum max . The solid sphere represents the results using sticking limits of moment of inertia (IS ) and stars represent the use of non-sticking limits of moment of inertia (IN S ).

of inertia, preformation probability P0 and penetrability P are plotted as a function of fragment mass A2 at Ec.m. = 4.91 MeV (equivalently En = 5.00 MeV), for both the choices of limits of moment of inertia sticking (IS ) as well as non-sticking (IN S ). It may be noted that the preformation probability and penetrability values as plotted in Fig. 3 correspond to the best-fit value of R and respective max for the non-sticking choice (IN S ). In Fig. 3, solid spheres represent the values for IS choice and stars represent the variation for IN S choice and this figure clarifies that when the two limits of moment of inertia are interchanged, a change in structure as well as magnitude is observed for both P0 and P. It is seen that the magnitude of P0 of lighter fragments is higher for IS choice as compared to IN S choice, which signifies that IS choice favors the α-emission. But the penetrability P-values for light charged particles is enhanced significantly and this enhancement in the penetrability shows up in the optimization of cross-sections thereby making the non-sticking approach more relevant for the present work. In Fig. 3(b), it is seen that the non-sticking approach IN S gives a relatively smoother variation for penetrability P as compared to that for sticking approach IS . Further in order to justify it, the corresponding values of P0 and P for α-cluster are taken as marked in Fig. 3(a) and (b) by vertical lines and it is noted that for an α-cluster, P0 (IN S ) = 10−6 × P0 (IS ) and P(IN S ) = 109 × P(IS ). So, by interchanging the limits, penetrability plays the lead role in making the non-sticking approach more favorable towards light charged particle emission cross-sections. This observation is in agreement with our earlier studies [14] that non-sticking approach is favorable for light particle and neutron evaporations while sticking approach is favorable for fusion-fission processes. Also, the comparison of Fig. 3(b) with the results of earlier works [27,28], exhibit similar observations, except for the fact that the present analysis is extended for non-sticking limits (IN S ) as well. It is to be added that with the use of non-sticking approach the experimental data is addressed successfully over the whole energy range En ∼ 1–100 MeV for optimum neck-length parameter R and corresponding values of max . The DCM calculated cross-sections are shown along with best-fit values of R and the other terms involved in calculations in Table-1.

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Table 1 DCM based cross sections for the decay of compound system 61 Ni∗ via α and 2p2n-emission along with R and the

max values. The experimentally measured cross-sections are also tabulated. S. No. 1 2 3 4 5 6 7 8 11 12 13 14 15 16 17

En (MeV)

Ec.m. (MeV)

T (MeV)

R (fm)

max (h¯ )

Cross-sections σα

σ2p2n

5.00 5.70 6.50 7.50 8.80 10.40 12.60 15.40 29.30 34.30 40.80 49.30 61.00 74.90 98.00

4.91 5.60 6.39 7.37 8.65 10.22 12.39 15.14 28.81 33.73 40.13 48.49 59.99 73.67 96.39

1.45 1.49 1.53 1.58 1.64 1.71 1.81 1.92 2.40 2.55 2.74 2.96 3.24 3.37 3.99

0.53 0.70 0.79 0.80 1.67 2.05 2.25 1.95 2.05 2.40 2.40 2.45 2.45 2.20 2.20

26 26 27 29 30 32 33 33 33 34 35 35 35 33 33

2.12 5.05 6.39 5.88 8.03 12.10 12.80 8.34 8.59 10.80 13.50 13.30 10.40 6.69 4.62

0.62 0.67 0.89 0.98 3.31 6.57 5.72 3.38 6.40 8.67 11.90 14.70 12.10 8.27 7.33

σexpt. ± σ 2.00 ± 1.1 4.50 ± 1.1 6.00 ± 1.1 6.10 ± 1.1 8.00 ± 1.1 12.60 ± 1.3 12.50 ± 1.4 8.50 ± 1.4 8.20 ± 1.1 11.50 ± 1.1 14.20 ± 1.1 15.80 ± 1.1 12.10 ± 0.9 8.90 ± 1.0 7.30 ± 0.9

In Fig. 4(a) and (b) fragmentation potential is plotted as a function of fragment mass A2 for the three energies Ec.m. = 4.91, 49.30 and 96.39 MeV covering the whole spread of experimentally available data respectively for = 0h¯ and = max values. From Fig. 4(a) it is clear that the light charged particles with A2 ≤ 4 correspond to lower magnitudes of fragmentation potential as compared to that for higher fragments. It seems to suggest that the compound nucleus 61 Ni∗ prefers to decay via emission of light charged particles at lower angular momentum values. On the other hand when we transmute towards higher values of angular momentum, a change in fragmentation structure is observed indicating that heavier fragments tend to have lower values of fragmentation potential. Moreover, if we compare the magnitude of fragmentation potential for α-emission for all the three energies (as marked in Fig. 4(a)) it is seen that the fragmentation value is least for minimum energy Ec.m. = 4.91 MeV. The minimum value of fragmentation potential for α-cluster at Ec.m. = 4.91 MeV signifies that the emission of α-cluster is favored at lowest energy. The effect of angular momentum and center of mass energy (Ec.m. ) on the decay of compound nucleus is also analyzed via the preformation probability pattern as shown in Fig. 3(c) and (d). Here Fig. 3(c) and (d) are same as Fig. 3(a) and (b) but plotted for preformation probability. Same results are evident here and one may clearly see that the decay of light charged particles is favored at lower angular momentum. Also, it is observed that the decay pattern of compound nucleus 61 Ni∗ gets modified at different center of mass energies. From Fig. 4(d); it is clear that at Ec.m. = 4.91 MeV the preformation probability is higher for heavier fragments signifying that the compound nucleus undergoes near symmetric decay at high -values. As the center of mass energy Ec.m. is increased, a change in potential energy surface (PES) is observed and a drift from symmetric to asymmetric configuration is observed. At Ec.m. = 48.49 MeV, the compound nucleus 61 Ni∗ starts following the asymmetric decay mode with fragments of intermediate mass region (with A2 = 5–20 plus their complimentary fragments) forming the decay path. And at Ec.m. = 96.39 MeV, the fragmentation pattern of the compound nucleus becomes relatively more asymmetric, with IMFs forming the preferred decay mode. The novelty of DCM based calculations lies in the fact that it allows us to predict the preferred decay path of

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Fig. 4. Fragmentation potential and preformation probability plotted as a function of fragment mass A2 for the decay of 61 Ni∗ nucleus plotted for = 0h and =

¯ max case at energies Ec.m. = 12.39 MeV, 48.49 MeV and 96.39 MeV using non-sticking (IN S ) approach.

a hot rotating compound nucleus at different Ec.m. and -values which are otherwise difficult to detect using statistical approaches or experimentally. In order to justify α-emission at lower energy and 2p2n-emission at higher energy, the preformation probability P0 of the respective clusters is plotted as a function of angular momentum ( ) in Fig. 5 at energies corresponding to the peaks in excitation function plot. Fig. 5(a) is plotted at Ec.m. = 12.39 MeV while part (b) is plotted for Ec.m. = 48.49 MeV and these energies lie across the point where transmutation is observed experimentally [3]. It is clear that Ec.m. = 12.39 MeV, the preformation probability of α-emission (filled spheres) has much higher magnitude than that of 2p2n-emission (hollow circles) and the trend gets reverted as we shift towards higher energy signifying 2p2n-emission to be more prominent. In other words the formation probability of 2p2n-emission starts superseeding the α-emission with increase in incident energy. Also, it is seen that α and 2p2n-emission show maximum probability of emission around angular momentum = 10h¯ while for ≤ 5h¯ and ≥ 18h¯ , both α and 2p2n-emission seems to be equally probable. So it can be concluded that light cluster or multinucleon emission from a hot excited compound nucleus is highly sensitive towards energy as well as angular momentum of the decaying system. Note that in DCM based calculations, α and 2p2n-emission are differentiated on the basis of binding energies and the transmutation is associated with the temperature dependence of the

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Fig. 5. Preformation probability P0 as a function of angular momentum ( ) plotted simultaneously for α and 2p2n-emission at the two peak energies (a) Ec.m. = 12.39 MeV and (b) Ec.m. = 48.49 MeV at corresponding values of R and max .

binding energies, which forms the essential part of the fragmentation potential. The various terms involved in liquid drop potential term (VLDM ) (eq. (6) in Section-2) for the compound nucleus 61 Ni∗ formed in fast neutron induced reaction n + 60 Ni are plotted as a function of compound nucleus temperature T (in MeV) in Fig. 6. The surface energy term and the volume energy term are plotted in Fig. 6(a) while asymmetry energy term and the Coulomb energy term are plotted in Fig. 6(b). It is clear that all these terms are highly sensitive towards the temperature of the decaying compound nucleus and this temperature dependance enters through the temperaturedependent coefficients involved in these terms [16,25]. Thus, the total binding energy of the fragment, which is the sum of all these terms, gets modified significantly with respect to temperature there by suggesting that the compound nucleus follows different decay paths at different beam energies. In addition to this, the binding energies of α-cluster and 2p2n-multinucleon system is plotted as a function of compound nucleus temperature T (MeV) in Fig. 6(c) where solid spheres represent the binding energy values of α-cluster and stars represent those for 2p2n-multinucleon system. It may be noted that binding energy of α-cluster increases with rise in temperature of the compound nucleus while that of 2p2n shows an opposite behavior and decrease with rise in temperature. It has been observed that within the Seeger’s semi-empirical formula, the mass dependency vanishes for proton and neutron A = 1 and for the case of neutron the contribution due to Coulomb term also becomes zero as Z = 0. Due to this, the LDM potential term gets reduced to the following form for a proton (A = 1, Z = 1): VLDM (A, Z, T ) = α(T ) + β(T ) + 3(γ (T ) − η(T )) + + δ(T )

1 2.29 × (1 − 0.7636 − ) r0 (T ) r0 (T ) (13)

while for a neutron it becomes; VLDM (A, Z, T ) = α(T ) + β(T ) + 3(γ (T ) − η(T )) + δ(T )

(14)

It is clear that the VLDM term comprises of the Seeger’s constants only. Since these constants α(T ), β(T ), γ (T ), η(T ) and δ(T ) decrease with temperature, consequently, VLDM also de-

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Fig. 6. The ingredients of liquid drop model (LDM) potential terms are plotted as a function of excitation energy (temperature) of the compound nucleus 61 Ni∗ . Here (a) shows the surface term and volume term and (b) shows the variation of asymmetry and Coulomb component and (c) shows the comparative binding energy for α and 2p2n-emission.

creases with temperature for the case of proton and neutron. And, since, the binding energies of 2p2n-multinucleon system is calculated from the binding energies of proton and neutron by eq. (7) in Section-2, so the binding energy of 2p2n-multinucleon system shows the trend as in Fig. 6. It can be seen that at around T = 2.9 MeV a transition is seen and below this point, the magnitude of binding energy of α-cluster is lesser than that of 2p2n while above this point, the trend is reverted. This binding energy goes as an input for fragmentation potential as explained in eq. (5) of section-2 and lower the value of fragmentation potential higher is the probability of decay of that cluster. So, the lower binding energy values of α-cluster for T < 2.9 MeV make it more probable to form the preferred emission channel of the compound nucleus 61 Ni∗ while for T > 2.9 MeV the trend gets reverted and one gets 2p2n multinucleon emission channel of 61 Ni∗ . Finally, the effect of mass (and charge) of the compound nucleus on the relative emission of α and 2p2n emission is seen and for this, the excitation functions of three more systems formed in fast neutron induced reactions, lying in different mass regions (from light to heavy) are exercised. The other three reactions studied in addition to this reaction are n + 16 O, n + 48 Ti and n + 92 Mo forming the compound nuclear systems 17 O∗ , 49 Ti∗ and 93 Mo∗ over the incident beam energy of En = 1–100 MeV. As a matter of fact, it is seen that the novelty of studying neutron induced reactions lies in the point that they tend to convert the even–even target nuclei into odd–even compound nuclear systems there by affecting the decay dynamics significantly. In Fig. 7, the cross-sections for α or 2p2n-emission from compound nuclear systems 17 O∗ , 49 Ti∗ , 61 Ni∗ and 93 Mo∗ are plotted as a function of incident beam energy (E ). Here solid circles represent the n experimental data while solid and dotted lines represent the DCM calculated cross-sections for α and 2p2n-emission respectively. For the compound nuclear systems 49 Ti∗ , 61 Ni∗ and 93 Mo∗ DCM based calculations find reasonable agreement with the experimental data. The calculations are done by using the non-sticking approach in moment of inertia. It is clear that at lower beam energies, the DCM calculated cross-sections for α-cluster match with the experimental data for lower energies while the cross-sections for 2p2n-emission are very less in magnitude as compared to α-cluster. And at higher incident energies, the trend seems to be reverted where 2p2n-emission cross-sections match with the experimental cross-sections (as shown by dotted

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Fig. 7. DCM calculated cross-sections for the decay of compound nuclei 17 O∗ , 49 Ti∗ , 61 Ni∗ and 93 Mo∗ formed in fast neutron induced reactions via x-emission (where x = α and 2p2n cluster) as a function of center of mass energy Ec.m. along with experimental data (shown by filled circles).

line) but the cross-sections for α-cluster (shown solid lines) falls to much smaller values. It may be noted that the cross-sections for α and 2p2n emission are calculated at same values of R (and respective values of max ) for each energy. Another point to be noted here is that for all the three systems the point where the trend of binding energies reverses its magnitude for α and 2p2n-emission lies near the same compound nucleus temperature T 2.9 MeV although it corresponds to different Ec.m. (and En ) values. In summary, the emergence of 2p2n-multinucleon emission at expense of α-emission is visible in all the chosen reactions. In order to have better insight into the transition behavior observed in chosen reactions, it will be of further interest to explore related dynamics and associated nuclear behavior in near future. For the compound system 17 O∗ , the experimental data could be addressed for beam energies En ≤ 15 MeV, but for higher energies, the DCM calculations over-estimate the data by few orders although a transition is seen for α and 2p2n-emission at around En ∼ 25 MeV. The inadequacy in the addressal of the data may be due to the excessive small mass of compound system 17 O∗ . It is relevant to mention that DCM works out in coordinates of mass asymmetry and the collective clusterization involves the distribution of preformation probability amongst all the possible decay channels emerging from a compound system. In other words, collective clusterization means all the possible out-going fragments like light particles, intermediate mass fragments, heavy mass fragments and fission fragments are treated on equal footing. Mass asymmetry (ηA ) (as given by eq. (1) in section-2) for each decay mode from a particular compound nucleus is calculated for α-emission channel and its values come out to be 0.529, 0.836, 0.868 and 0.913 respectively for 17 O∗ , 49 Ti∗ , 61 Ni∗ and 93 Mo∗ systems. Due to small mass of compound system and relatively

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Fig. 8. Preformation probability P0 plotted for the decay of compound nuclear systems 17 O∗ , 49 Ti∗ , 61 Ni∗ and 93 Mo∗ minimized in mass coordinates at the common center of mass energy Ec.m. = 8.80 MeV and = 0h¯ .

much lower mass asymmetry, the dynamics of the lightest system 17 O∗ , do not go in accordance with the dynamics of other heavier systems i.e. 49 Ti∗ , 61 Ni∗ and 93 Mo∗ . This point is further clarified in Fig. 8, where the preformation probability P0 for the decay of all the aforementioned compound systems formed in fast neutron induced reactions is plotted as a function of fragment mass number Ai at common incident beam energy En = 8.80 MeV and = 0h¯ . It is clear from the Fig. 8 that the distribution of preformation probability P0 shows almost similar trend for three heavy mass compound nuclear systems 49 Ti∗ , 61 Ni∗ and 93 Mo∗ except for the case of 17 O∗ . Fig. 8(b), (c) and (d) shows that at =0h¯ , the preformation probability P0 for light particles lie close to 1 making them more probable towards the decay while for heavier fragments i.e. for fragments lying around A2 ± 20, the value of preformation probability is negligibly small and lie close to 0 which signifies that their possibility to form the decay path is negligibly small at lower

-values. While for the case of 17 O∗ , due to very small value of A, the preformation probability of fragments with A2 = 1, 6 and 7 (plus their complementary fragments) lie close to 1 and for fragments with A2 = 2, 3, 4, 5 (plus their complementary fragments) have relatively lesser magnitude which means, the decay of compound nucleus 17 O∗ does not opt for one decay path at a particular value of Ec.m. and angular momentum ( ) and hence the dynamics of decay of 17 O∗ does not seem to be adequately explained under the clusterization approach of DCM. Also, experimentally, it is observed that the probability for an α-cluster emission depends upon the mass (and charge) of the compound nucleus from which it is emitted. The lightest system 17 O∗ ,

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shows the maximum probability of α-emission while the same goes on decreasing with increase in mass number of the compound system. This observation is further analyzed using the preformation probability curve and it is found that magnitude of P0 values for α-cluster emission is maximum for the lightest compound nuclear system 17 O∗ and least for heaviest nucleus 93 Mo* . This shows that the preformation behavior is in accordance with the experimental observations. 4. Summary In the present work, the dynamics involved in the decay of compound nucleus 61 Ni∗ formed in fast neutron induced reactions over an incident beam energy En = 1–100 MeV is studied using the clusterization picture of DCM. The sticking limits of moment of inertia (IS ) in the centrifugal term prove to be inadequate in addressing the data over whole energy range. For appropriate addressal of the experimental data the IS limits are replaced with IN S and the cross-sections for the α-cluster emission or 2p2n-multinucleon emission are estimated by opting appropriate values of neck-length parameter. The fragmentation path shows changes in terms of magnitude as well as structure when the two limits of moment of inertia are interchanged and an enhancement in the order of P is observed which signifies IN S approach to be more relevant for addressing the data. The following points are observed from the fragmentation and preformation behavior: (i) the emission of light charged particles is favored at lower angular momentum values. (ii) the compound nucleus 61 Ni∗ undergoes near symmetric fragmentation at lower center of mass energies Ec.m. and with increase in Ec.m. , the structure becomes highly asymmetric. (iii) With increase in energy, the magnitude of fragmentation potential of α-cluster increases and hence the probability of its decay decreases. In addition to this the transmutation of α-emission at lower energies and 2p2n-emission at higher energies is explained on the basis of temperature dependence of the binding energies used in DCM. The binding energy of α-cluster as emitted from 61 Ni∗ shows lower magnitude at lower temperature (T in MeV) as compared to that of 2p2n-cluster. While above a particular temperature T = 2.9 MeV, the trend gets reverted with 2p2n-cluster having lower values of binding energies. Lastly, the effect of mass (and charge) of the compound nucleus on α and 2p2n-multinucleon emission is analyzed and comparative analysis is done for the decay of various compound nuclei having mass ranging from light to intermediate mass region. DCM based calculations find reasonable comparison with experimental data except for 17O∗ nucleus, where collective clustrization approach seems relatively inadequate due to small mass of compound system. Acknowledgements The financial support from the Department of Science and Technology (DST) INSPIRE Fellowship (Dy. No./JS and FA/359) and DST-SERB sponsored research project (F. No. EMR/2016/000008), Government of India is gratefully acknowledged. References [1] V.V. Orlov, Sov. At. Energy 36 (5) (1974) 435–449. [2] T.B. Cochran, H.A. Fevison, W. Patterson, G. Pshakin, M.V. Ramanna, M. Schneider, T. Suzuki, F.V. Hippel, Fast Breeder Reactor Progress History and Status, Research Report 8, International Panel on Fissile Materials, Feb. 2010. [3] N. Fotiades, et al., Phys. Rev. C 91 (2015) 064614. [4] S.J. Sanders, A. Szanto de Toledo, C. Beck, Phys. Rep. 311 (1999) 487–551. [5] A.C. Xenoulis, et al., Phys. Lett. B 106 (1981) 461.

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