Relative mass distributions of neutron-rich thermally fissile nuclei within statistical model
Bharat Kumar, M.T. Senthil kannan, M. Balasubramaniam, B. K. Agrawal, S. K. Patra
RRelative mass distributions of neutron-rich thermally fissile nuclei within statistical model
Bharat Kumar , , ∗ M.T. Senthil Kannan , M. Balasubramaniam , B. K. Agrawal , , and S. K. Patra , Institute of Physics, Sachivalaya Marg, Bhubaneswar - 751005, India. Department of Physics, Bharathiar University, Coimbatore - 641046, India. Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata - 700064, India. and Homi Bhabha National Institute, Anushakti Nagar, Mumbai - 400094, India. (Dated: October 2, 2018)We study the binary mass distribution for the recently predicted thermally fissile neutron-rich uranium andthorium nuclei using statistical model. The level density parameters needed for the study are evaluated from theexcitation energies of temperature dependent relativistic mean field formalism. The excitation energy and thelevel density parameter for a given temperature are employed in the convolution integral method to obtain theprobability of the particular fragmentation. As representative cases, we present the results for the binary yieldof
U and
Th. The relative yields are presented for three different temperatures T =
1, 2 and 3 MeV.
PACS numbers: 25.85.-w, 21.10.Ma, 21.10.Pc, 24.75.+i
I. INTRODUCTION
Fission phenomenon is one of the most interesting subjectin the field of nuclear physics. To study the fission properties,a large number of models have been proposed. The fissioningof a nucleus is successfully explained by the liquid drop modeland the semi-empirical mass formula is the best and simpleoldest tool to get a rough estimation of the energy releasedin a fission process. The pioneering work of Vautherin andBrink [1], who has applied the Skyrme interaction in a self-consistent method for the calculation of ground state proper-ties of finite nuclei opened a new dimension in the quantitativeestimation of nuclear properties. Subsequently, the Hartree-Fock and time dependent Hartree-Fock formalisms [2] arealso implemented to study the properties of fission. Most re-cently, the microscopic relativistic mean field approximation,which is another successful theory in nuclear physics is alsoused for the study of nuclear fission [3].From last few decades, the availability of neutron rich nu-clei in various laboratories across the globe opened up newresearch in the field of nuclear physics, because of their exoticdecay properties. The effort for the synthesis of superheavynuclei in the laboratories like, Dubna (Russia), GSI (Ger-many), RIKEN (Japan) and BNL (USA) is also quite remark-able. Due to all these, the periodic table is extended till dateupto atomic number Z = 118 [4]. The decay modes of thesesuperheavy nuclei are very different than the usual modes.Mostly, we understand that, a neutron rich nucleus has a largenumber of neutron than the light or medium mass region of theperiodic table. The study of these neutron-rich superheavy nu-clei is very interesting, because of their ground state structuresand various mode of decays, including multi-fragment fission(more than two) [3]. Another interesting feature of some neu-tron rich uranium and thorium nuclei is that similar to U, U and
Pu, the nuclei − U and − Th are alsothermally fissile, which are extremely important for the energyproduction in fission process. If the neutron rich uranium and ∗ Electronic address: [email protected] thorium nuclei are the viable sources, then these nuclei will bemore effective to achieve the critical condition in a controlledfission reaction.Now the question arises, how we can get a reasonable es-timation of the mass yield in the spallation reaction of theseneutron rich thermally fissile nuclei. As mentioned earlier inthis section, there are many formalisms available in the litera-ture to study these cases. Here, we adopt the statistical modeldeveloped by Fong [5]. The calculation is further extendedby Rajasekaran and Devanathan [6] to study the binary massdistributions using the single particle energies of the Nilssonmodel. The obtained results are well in agreement with the ex-perimental data. In the present study, we would like to replacethe single particle energies with the excitation energies of asuccessful microscopic approach, the relativistic mean field(RMF) formalism.For last few decades, the relativistic mean field (RMF) for-malism [7–11] with various parameter sets have successfullyreproduced the bulk properties, such as binding energies, rootmean square radii, quadrupole deformation etc. not only fornuclei near the β − stability line but also for nuclei away fromit. Further, the RMF formalism is successfully applied to thestudy of clusterization of known cluster emitting heavy nu-cleus [12–14] and the fission of hyper-hyper deformed Ni[15]. Rutz et. al. [16] reproduced the double, triple humpedfission barrier of
Pu,
Th and the asymmetric groundstates of
Ra using RMF formalism. Moreover, the sym-metric and asymmetric fission modes are also successfully re-produced. Patra et. al. [3] studied the neck configuration inthe fission decay of neutron rich U and Th isotopes. The maingoal of this present paper is to understand the binary fragmen-tation yield of such neutron rich thermally fissile superheavynuclei.
U and
Th are taken for further calculations asthe representative cases.The paper is organized as follows: In Section II, the sta-tistical model and relativistic mean field theory are presentedbriefly. In subsection A of this section, the level density pa-rameter and it’s relation with the relative mass yield are out-lined. In subsection B of II, the equation of motion of thenucleon and meson fields obtained from the relativistic meanfield Lagrangian and the temperature dependent of the equa- a r X i v : . [ nu c l - t h ] S e p tions are adopted through the occupation number of protonsand neutrons. The results are discussed in Section III andcompared with the finite range droplet model (FRDM) pre-dictions. The summary and concluding remarks are given inSection IV. II. FORMALISM
The possible binary fragments of the considered nucleus isobtained by equating the charge to mass ratio of the parentnucleus to the fission fragments as [17]: Z P A P ≈ Z i A i , (1)with A P , Z P and A i , Z i ( i = 1 and 2) correspond to massand charge numbers of the parent nucleus and the fission frag-ments [6]. The constraints, A + A = A , Z + Z = Z and A ≥ A are imposed to satisfy the conservation of chargeand mass number in a nuclear fission process and to avoid therepetition of fission fragments. Another constraint i.e., the bi-nary charge numbers from Z ≥
26 to Z ≤
66 is also takeninto consideration from the experimental yield [18] to gener-ate the combinations, assuming that the fission fragments liewithin these charge range.
A. Statistical theory
The statistical theory [5, 19] assumes that the probabilityof the particular fragmentation is directly proportional to thefolded level density ρ of that fragments with the total exci-tation energy E ∗ , i.e., P ( A j , Z j ) ∝ ρ ( E ∗ ) . Where, ρ ( E ∗ ) = (cid:90) E ∗ ρ ( E ∗ ) ρ ( E ∗ − E ∗ ) dE ∗ , (2)and ρ i is the level density of two fragments ( i = 1, 2). Thenuclear level density [20, 21] is expressed as a function offragment excitation energy E ∗ i and the single particle leveldensity parameter a i which is given as: ρ i ( E ∗ i ) = 112 (cid:18) π a i (cid:19) / E ∗ ( − / i exp (cid:16) (cid:112) a i E ∗ i (cid:17) . (3)In Refs. [17, 22], we calculate the excitation energies of thefragments using the ground state single particle energies of fi-nite range droplet model (FRDM) [23] at a given temperature T keeping the total number of proton and neutron fixed. In thepresent study, we apply the self consistent temperature depen-dent relativistic mean field theory to calculate the E ∗ of thefragments. The excitation energy is calculated as, E ∗ i ( T ) = E i ( T ) − E i ( T = 0) . (4)The level density parameter a i is given as, a i = E ∗ i T . (5) The relative yield is calculated as the ratio of the probabilityof a given binary fragmentation to the sum of the probabilitiesof all the possible binary fragmentations and it is given by, Y ( A j , Z j ) = P ( A j , Z j ) (cid:80) j P ( A j , Z j ) , (6)where A j and Z j are referred to the binary fragmentationsinvolving two fragments with mass and charge numbers A , A and Z , Z obtained from Eq. (1). The competing basicdecay modes such as neutron/proton emission, α decay andternary fragmentation are not considered. In addition to theseapproximations, we have also not included the dynamics ofthe fission reaction, which are really important to get a quan-titative comparison with the experimental measurements. Thepresented results are the prompt disintegration of a parent nu-cleus into two fragments (democratic breakup). The resultingexcitation energy would be liberated as prompt particle emis-sion or delayed emission, but such secondary emissions arealso ignored. B. RMF Formalism
The RMF theory assume that the nucleons interact witheach other via meson fields. The nucleon - meson interactionis given by the Lagrangian density [7–9, 11, 24, 25], L = ψ i { iγ µ ∂ µ − M } ψ i + 12 ∂ µ σ∂ µ σ − m σ σ − g σ − g σ − g σ ψ i ψ i σ −
14 Ω µν Ω µν + 12 m w V µ V µ − g w ψ i γ µ ψ i V µ − (cid:126)B µν . (cid:126)B µν + 12 m ρ (cid:126)R µ . (cid:126)R µ − g ρ ψ i γ µ (cid:126)τ ψ i . (cid:126)R µ − F µν F µν − eψ i γ µ (1 − τ i )2 ψ i A µ . (7)Where, ψ i is the single particle Dirac spinor. The arrows overthe letters in the above equation represent the isovector quan-tities. The nucleon, the σ , ω , and ρ meson masses are denotedby M, m σ , m ω and m ρ respectively. The meson and the pho-ton fields are termed as σ , V µ , R µ and A µ for σ , ω , ρ − mesonsand photon respectively. The g σ , g ω , g ρ and e π are the cou-pling constants for the σ , ω , ρ − mesons and photon fields withnucleons respectively. The strength of the constants g and g are responsible for the nonlinear couplings of σ meson ( σ and σ ). The field tensors of the isovector mesons and thephoton are given by, Ω µν = ∂ µ V ν − ∂ ν V µ , (8) (cid:126)B µν = ∂ µ (cid:126)R ν − ∂ ν (cid:126)R µ − g ρ ( (cid:126)R µ × (cid:126)R ν ) , (9) F µν = ∂ µ A ν − ∂ ν A µ . (10)The classical variational principle gives the Euler-Lagrangeequation and we get the Dirac-equation with potential termsfor the nucleons and Klein-Gordan equations with sourceterms for the mesons. We assume the no-sea approximation,so we neglect the antiparticle states. We are dealing with thestatic nucleus, so the time reversal symmetry and the conser-vation of parity simplifies the calculations. After simplifica-tions, the Dirac equation for the nucleon is given by, {− iα. (cid:53) + V ( r ) + β [ M + S ( r )] } ψ i = (cid:15) i ψ i , (11)where V(r) represents the vector potential and S(r) is the scalarpotential, V ( r ) = g ω ω + g ρ τ ρ ( r ) + e (1 − τ )2 A ( r ) ,S ( r ) = g σ σ ( r ) , (12)which contributes to the effective mass, M ∗ ( r ) = M + S ( r ) . (13)The Klein-Gordon equations for the mesons and the elec-tromagnetic fields with the nucleon densities as sources are, {−(cid:52) + m σ } σ ( r ) = − g σ ρ s ( r ) − g σ ( r ) − g σ ( r ) , (14) {−(cid:52) + m ω } ω ( r ) = g ω ρ v ( r ) , (15) {−(cid:52) + m ρ } ρ ( r ) = g ρ ρ ( r ) , (16) −(cid:52) A ( r ) = eρ c ( r ) . (17)The corresponding densities such as scalar, baryon (vector),isovector and proton (charge) are given as ρ s ( r ) = (cid:88) i n i ψ † i ( r ) ψ i ( r ) , (18) ρ v ( r ) = (cid:88) i n i ψ † i ( r ) γ ψ i ( r ) , (19) ρ ( r ) = (cid:88) i n i ψ † i ( r ) τ ψ i ( r ) , (20) ρ c ( r ) = (cid:88) i n i ψ † i ( r ) (cid:18) − τ (cid:19) ψ i ( r ) . (21)To solve the Dirac and Klein-Gordan equations, we expandthe Boson fields and the Dirac spinor in an axially deformedharmonic oscillator basis with β as the initial deformationparameter. The nucleon equation along with different me-son equations form a set of coupled equations, which canbe solved by iterative method. The center of mass correc-tion is calculated with the non-relativistic approximation. Thequadrupole deformation parameter β is calculated from theresulting quadrupole moments of the proton and neutron. Thetotal energy is given by [10, 26, 27], E ( T ) = (cid:80) i (cid:15) i n i + E σ + E σNL + E ω + E ρ + E C + E pair + E c.m. − AM, (22) with E σ = − g σ (cid:90) d rρ s ( r ) σ ( r ) , (23) E σNL = − (cid:90) d r (cid:26) g σ ( r ) + 12 g σ ( r ) (cid:27) , (24) E ω = − g ω (cid:82) d rρ v ( r ) ω ( r ) , (25) E ρ = − g ρ (cid:82) d rρ ( r ) ρ ( r ) , (26) E C = − e π (cid:82) d rρ c ( r ) A ( r ) , (27) E pair = −(cid:52) (cid:88) i> u i v i = − (cid:52) G , (28) E c.m. = − × A − / . (29)Here, (cid:15) i is the single particle energy, n i is the occupation prob-ability and E pair is the pairing energy obtained from the sim-ple BCS formalism. C. Pairing and temperature dependent RMF formalism
The pairing correlation plays a distinct role in open-shellnuclei. The effect of pairing correlation is markedly seen withincrease in mass number A. Moreover it helps in understand-ing the deformation of medium and heavy nuclei. It has a leaneffect on both bulk and single particles properties of lightermass nuclei because of the availability of limited pairs nearthe Fermi surface. We take the case of T=1 channel of pairingcorrelation i.e, pairing between proton- proton and neutron-neutron. In this case, a nucleon of quantum states | jm z (cid:105) pairswith another nucleons having same I z value with quantumstates | j − m z (cid:105) , since it is the time reversal partner of the other.In both nuclear and atomic domain the ideology of BCS pair-ing is the same. The even-odd mass staggering of isotopes wasthe first evidence of its kind for the pairing energy. Consid-ering the mean-field formalism, the violation of the particlenumber is seen only due to the pairing correlation. We findterms like ψ † ψ (density) in the RMF Lagrangian density butwe put an embargo on terms of the form ψ † ψ † or ψψ sinceit violates the particle number conservation. We apply ex-ternally the BCS constant pairing gap approximation for ourcalculation to take the pairing correlation into account. Thepairing interaction energy in terms of occupation probabilities v i and u i = 1 − v i is written as [28, 29]: E pair = − G (cid:34)(cid:88) i> u i v i (cid:35) , (30)with G is the pairing force constant. The variational approachwith respect to the occupation number v i gives the BCS equa-tion [29]: (cid:15) i u i v i − (cid:52) ( u i − v i ) = 0 , (31)with the pairing gap (cid:52) = G (cid:80) i> u i v i . The pairing gap ( (cid:52) )of proton and neutron is taken from the empirical formula [10,30]: (cid:52) = 12 × A − / . (32)The temperature introduced in the partial occupancies in theBCS approximation is given by, n i = v i = 12 (cid:20) − (cid:15) i − λ ˜ (cid:15) i [1 − f ( ˜ (cid:15) i , T )] (cid:21) , (33)with f ( ˜ (cid:15) i , T ) = 1(1 + exp [ ˜ (cid:15) i /T ]) and ˜ (cid:15) i = (cid:112) ( (cid:15) i − λ ) + (cid:52) . (34)The function f ( ˜ (cid:15) i , T ) represents the Fermi Dirac distri-bution for quasi particle energy ˜ (cid:15) i . The chemical potential λ p ( λ n ) for protons (neutrons) is obtained from the constraintsof particle number equations (cid:88) i n Zi = Z, (cid:88) i n Ni = N. (35)The sum is taken over all proton and neutron states. The en-tropy is obtained by, S = − (cid:88) i [ n i ln ( n i ) + (1 − n i ) ln (1 − n i )] . (36)The total energy and the gap parameter are obtained by mini-mizing the free energy, F = E − T S. (37)In constant pairing gap calculations, for a particular value ofpairing gap (cid:52) and force constant G , the pairing energy E pair diverges, if it is extended to an infinite configuration space.In fact, in all realistic calculations with finite range forces, (cid:52) is not constant, but decreases with large angular momentastates above the Fermi surface. Therefore, a pairing windowin all the equations are extended up-to the level | (cid:15) i − λ | ≤ A − / ) as a function of the single particle energy. Thefactor 2 has been determined so as to reproduce the pairingcorrelation energy for neutrons in Sn using Gogny force[10, 28, 31].
III. RESULTS AND DISCUSSIONS
In our very recent work [32], we have calculated the ternarymass distributions for
Cf,
Pu and
U with the fixedthird fragments A = Ca, O and O respectively forthe three different temperatures T = 1, 2 and 3 MeV within the TRMF formalism. The structure effects of binary frag-ments are also reported in Ref. [33]. In this article, we studythe mass distribution of
U and
Th as a representativecases from the range of neutron-rich thermally fissile nuclei − U and − Th. Because of the neutron-rich natureof these nuclei, a large number of neutrons emit during thefission process. These nucleons help to achieve the criticalcondition much sooner than the normal fissile nuclei.To assure the predictability of the statistical model, we alsostudy the binary fragmentation of naturally occurring
Uand
Th nuclei. The possible binary fragments are obtainedusing the Eq. (1). To calculate the total binding energy ata given temperature, we use the axially symmetric harmonicoscillator basis expansion N F and N B for the Fermion andBoson wave-functions to solve the Dirac Eq. (11) and theKlein Gordon Eqs. (14 - 17) iteratively. It is reported [34]that the effect of basis space on the calculated binding energy,quadrupole deformation parameter ( β ) and the rms radii ofnucleus are almost equal for the basis set N F = N B = ∼
200 . Thus, we use the ba-sis space N F =
12 and N B =
20 to study the binary frag-ments up to mass number A ∼ β and the proton(neutron) pairing gaps (cid:52) p ( (cid:52) n ) for the given temperature. Atfinite temperature, the continuum corrections due to the exci-tation of nucleons to be considered. The level density in thecontinuum depends on the basis space N F and N B [35]. It isshown that the continuum corrections need not be included inthe calculations of level densities up-to the temperature T ∼ A. Level density parameter and level density within TRMFand FRDM formalisms
In TRMF, the excitation energies E ∗ and the level densityparameters a i of the fragments are obtained self consistentlyfrom Eqns. (4) to (5). The FRDM calculations are also donefor comparison. In this case, level density of the fragmentsare evaluated from the ground state single particle energies ofthe finite range droplet model (FRDM) of M¨oller et. al. [38]which are retrieved from the Reference Input Parameter Li-brary (RIPL-3) [39]. The total energy at a given temperatureis calculated as E ( T ) = (cid:80) n i (cid:15) i ; (cid:15) i are the ground state sin-gle particle energies and n i are the Fermi-Dirac distributionfunction. The T dependent energies are obtained by vary-ing the occupation numbers at a fixed particle number for agiven temperature and given fragment. The level density pa-rameter a is a crucial quantity in the statistical theory for theestimation of yields. These values of a for the binary frag-ments of U, U, Th and
Th obtained from TRMFand FRDM are depicted in Fig. 1. The empirical estimation a = A/K are also given for comparison, with K , the inverselevel density parameter. In general, the K value varies from 8to 13 with the increasing temperature. However, the level den-sity parameter is considered to be constant up-to T ≈ K = 10 as mentioned L eve l d e n s i t y p a r a m e t e r , T = 1 MeV, T = 2 MeV, T = 3 MeV U A / (a) Fragment mass number (A & A ) U A / (b) Th A / (c) Th A / (d) FIG. 1: (Color online) The level density parameter a for the binaryfragmentation of U, U, Th and
Th at temperature T = in Ref. [40]. The a values of TRMF are close to the empiricallevel density parameter. The FRDM level density parametersare appreciably lower than the referenced a . Further, in bothmodels at T = T = T ≈ a is evaluated in two different ways using excitation energy and , T = 1 MeV, T = 2 MeV, T = 3 MeV Th (b) I n ve r se l eve l d e n s i t y p a r a m e t e r U (a) U (c) Fragment mass number (A & A ) Th (d) FIG. 2: (Color online) The inverse level density parameters K E (solid lines) and K S (dash lines) are obtained for U, U, Thand
Th at temperatures T =
1, 2 and 3 MeV. the entropy of the system as: a E = E ∗ T , (38) a S = S T .
For instance, the inverse level density parameters K E and K S of U, U, Th and
Th within TRMF formalism aredepicted in Fig. 2. Both K S and K E have maximum fluc-tuation upto 30 MeV at T = − MeV at temperature T = 2 MeV or above. It is tobe noted that at T = A ∼
130 in all
60 80 100 120 140 160 18010 (a) U L eve l d e n s i t y Fragment mass number (A & A ) , T = 1 MeV, T = 2 MeV, T = 3 MeV (b) U (c) Th Th (d) FIG. 3: (Color online) The level density of the binary fragmentationsof U, U, Th and
Th at temperature T =
1, 2 and 3 MeVwithin the TRMF (solid lines) and FRDM (dash lines) formalisms. cases. This may be due to the neutron closed shell ( N =
82) inthe fission fragments of
U and
Th and the neutron-richnuclei
U and
Th. The level density for the fission frag-ments of U, U, Th and
Th are plotted as a functionof mass number in Fig. 3 within the TRMF and FRDM for-malisms at three different temperatures T =
1, 2 and 3 MeV.The level density ρ has maximum fluctuations at T = a . The ρ values are substantially lowerat mass number A ∼ for all nuclei. In Fig. 3, one can no-tice that the level density has small kinks in the mass region A ∼ − of U and A ∼ − of U, compar-ing with the neighboring nuclei at temperature T = ρ values. A further inspection reveals that the leveldensity of the closed shell nucleus around A ∼
130 has highervalue than the neighboring nuclei for both , U, but it haslower yield due to the smaller level density of the correspond-ing partners. At T = A ∼
72 and 130 have larger values com-pared to other fragments of
U. On the other hand, the leveldensity in the vicinity of neutron number N = 82 and protonnumber Z = 50 for the fragments of the neutron-rich Unucleus is quite high, because of the close shell of the frag-ments. This is evident from the small kink in the level densityof
Cd ( N = In ( N ∼
82) and
Sn ( Z = Th, the level densities are found to be maxi-mum at around mass number A ∼
81 and 100 for T = Th, the ρ values are found to be large for thefragments around A ∼
78 and 97 at T = T = ρ values of Th frag-ments are notable around mass number A ∼ . Similarly,for Th, the fission fragments around A ∼
78 has higherlevel density at T = N =
82) nucleus.
B. Relative fragmentation distribution in binary systems
In this section, the mass distributions of U, Th andthe neutron rich nuclei
U and
Th are calculated at tem-peratures T =
1, 2 and 3 MeV using TRMF and FRDMexcitation energies and the level density parameters a as ex-plained in Sec. II. The binary mass distributions of , Uand , Th are plotted in Figs. 4 and 5. The total energyat finite temperature and ground state energy are calculatedusing the TRMF formalism as discussed in the section III A.From the excitation energy E ∗ and the temperature T , the leveldensity parameter a and the level density ρ of the fragmentsare calculated using Eq. 3. From the fragment level densities ρ i , the folding density ρ is calculated using the convolutionintegral as in Eq. 2 and the relative yields are calculated usingEq. 6. The total yields are normalized to the scale 2.The mass yield of normal nuclei U and
Th are brieflyexplains first, followed by the detailed description of the neu-tron rich nuclei. The results of most favorable fragments yieldof , U and , Th are listed in Table I at three differ-ent temperatures T =
1, 2 and 3 MeV for both TRMF andFRDM formalisms. From Figs. 4 and 5, it is shown that themass distributions for
U and
Th are quite different thanthe neutron-rich
U and
Th isotopes.The symmetric binary fragmentation Pd + Pd for
U is the most favorable combination. In TRMF, the frag-ments with close shell ( N =
100 and Z =
28) combinationsare more probable at the temperature T = N ≈
82 and Z ≈ T = Pr + As, Cs + Rb and Ba + Kr are thefavorable combinations at temperature T = T = N =
82, 50 and R e l a t i ve y i e l d Nb Rb Cs Sb Tb A g R h Co U T = 1 MeV T = 2 MeV T = 3 MeV
FRDM Zr Te M o Sn As Pr Zn Sm Ru (b) Ni Nb Gd Nd Sb Kr Pd Pd U T = 1 MeV T = 2 MeV T = 3 MeV
TRMF Pd Ba Ge Y I (a) In Tc Cd Cs Pr As U T = 1 MeV T = 2 MeV T = 3 MeV
TRMF Pr As Ge Ni N d Gd Te Zr Mo Sn (c) In Tc G e Ni Rb Zr As Pr N d G d T b Ag Te Co Rh U T = 1 MeV T = 2 MeV T = 3 MeV
Fragment mass number ( A & A ) FRDM (d)
FIG. 4: (Color online) Mass distribution of
U and
U at tem-peratures T = 1, 2 and 3 MeV. The total yield values are normalizedto the scale 2. Z =
28) have larger yields. From Fig. 5 in TRMF formalism,the combinations Pd + Ru and Xe + Kr are the pos-sible fragments at T = Th. At T = N =
82, 50). Forhigher temperature T = N ∼ Sb + Y is the most favorable fragmentationpair compared with all other yields. Similar fragmentationsare found in the FRDM formalism at T = Sn + Zr isalso quite substantial in the fission process. For T = U the fragment combinations , Te C s Br F e Gd Pr Ga Y Ru Pd Sb Th T = 1 MeV T = 2 MeV T = 3 MeVTRMF Xe Kr Tc Ag Ni Sm (a) (b)(c)(d) Ni R e l a t i ve y i e l d Gd Th T = 1 MeV T = 2 MeV T = 3 MeVFRDM Br Cs Se Ba Tc Ag Fe Y Sb Zr Sn Ga Pr Se Br Zr Sm Ba Cs Sn Rh Th T = 1 MeV T = 2 MeV T = 3 MeVTRMF Y Sb Fe G d Tc Ag Sb Rh Rh Ru Th T = 1 MeV T = 2 MeV T = 3 MeV
Fragment mass number ( A & A ) FRDM Ce Ge Nd Zn Eu Co Y Rh Pd FIG. 5: (Color online) Mass distribution of
Th and
Th at tem-peratures T = 1, 2 and 3 MeV. The total yield values are normalizedto the scale 2. + , Zr have the maximum yields at T = Z = 40 in Zr isotopes [41]. Contrary to this al-most symmetric binary yield, the mass distribution of this nu-cleus in FRDM formalism have the asymmetric evolution ofthe fragment combinations like , Pr + , As, , Nd + , Ge and Cs + Rb. Interestingly, at T = T = Pr + As, Nd + Ge and Gd + Ni are the more probable frag-mentations (see Fig. 4(c)). It is reported by Satpathy etal [42] and experimentally verified by Patel et al [43] that N =
100 is a neutron close shell for the deformed region,where Z = 62 acts like a magic number. In FRDM, Ag + Rh, In + Tc, Te + Zr and Gd + Ni havelarger yield at temperature T = A ≈ − ) which is the blend of vicinity of neu-tron ( N =
82) and proton ( Z =
50) closed shell nuclei at T = Cd + Ru, In + Tc and Sn + Mo are the major yields for
U atT=3 MeV in TRMF calculations. In FRDM method, at T = T = T = T = T = A and A . Here, the dynamics of entire process starting fromthe initial stage upto the scission are ignored. As a result, theenergy conservation in the spallation reaction does not takeninto account. The fragment yield can be regarded as the rel-ative fragmentation probability, which is obtained from Eq.6. Now we analyze the fragmentation yields for Th isotopesand the results are depicted in Fig. 5 and Table I. In this case,one can see that the mass distribution broadly spreads throughout the region A i = 66 − . Again, the most concen-trated yields can be divided into two regions I( A = A = A = A = Th in TRMF formalism at the temperature T = Sn + Zr is ob-tained from region I. The other combinations in that regionhave also considerable yields. In region II, the isotopes ofBa and Cs appears curiously along with their correspondingpartners. Categorically, in FRDM predictions, region I haslarger yields at T = Ce + Ge, Nd + Zn and Gd + Fe (SeeFig. 5 (b,d)). The mass distribution is different with differ-ent temperature and the maximum yields at T = , , Sm + , , Ni. Apart fromthese combinations, there are other considerable yields canbe seen in Fig. 5 for region II. The prediction of maximumprobability of the fragments production in FRDM method are Sb + Y, Eu + Co and Rh + Rh at T = T = Rh + Rh has the largest yield due to theneutron close shell ( N =
82) of the fragment
Rh. The otheryield fragments have exactly/nearly a magic nucleon combi-nation, mostly neutron ( N =
82) as one of the fragment. Aconsiderable yield is also seen for the proton close shell ( Z =
28) Ni or/and ( Z =
62) Sm isotopes supporting our earlierprediction [33]. This confirms the prediction of Sm as a de-formed magic nucleus [42, 43]. Another observation of thepresent calculations show that the yields of the neutron-richnuclei agree with the symmetric mass distribution of Chaud-huri et. al. [44] at large excitation energy, which contradictthe recent prediction of large asymmetric mass distribution of neutron-deficient Th isotopes [45]. These two results [44, 45]along with our present calculations confirm that the symmet-ric or asymmetric mass distribution at different temperaturedepends on the proton and neutron combination of the parentnucleus. In general, both TRMF and FRDM predict maxi-mum yields for both symmetric/asymmetric binary fragmen-tations followed by other secondary fragmentations emissiondepending on the temperature as well as the mass number ofthe parent nucleus. Thus, the binary fragments have largerlevel density ρ comparing with other nuclei because of neu-tron/proton close shell fragment combinations at T = N =
50, 82 and 100 have maximumpossibility of emission at T = U and
Th). This is a general trend, we could expectfor all neutron-rich nuclei. It is worthy to mention some ofthe recent reports and predictions of multi-fragment fissionfor neutron-rich uranium and thorium nuclei. When such aneutron-rich nucleus breaks into nearly two fragments, theproducts exceed the drip-line leaving few nucleons (or lightnuclei) free. As a result, these free particles along with thescission neutrons enhance the chain reaction in a thermonu-clear device. These additional particles (nucleons or light nu-clei) responsible to reach the critical condition much fasterthan the usual fission for normal thermally fissile nucleus.Thus, the neutron-rich thermally fissile nuclei, which are inthe case of − U and − Th will be very useful forenergy production.
IV. SUMMARY AND CONCLUSIONS
The fission mass distributions of β − stable nuclei U and
Th and the neutron-rich thermally fissile nuclei
U and
Th are studied within the statistical theory. The possiblecombinations are obtained by equating the charge to mass ra-tio of the parents to that of the fragments. The excitation en-ergies of fragments are evaluated from the temperature de-pendent self-consistent binding energies at the given T and TABLE I: The relative fragmentation yield (R.Y.) = Y ( A j , Z j ) = P ( A j , Z j ) (cid:80) P ( A j , Z j ) for U, U, Th and
Th, obtained with TRMF atthe temperatures T =
1, 2 and 3 MeV are compared with the FRDM prediction (The yield values are normalized to 2).Parent T (MeV) TRMF FRDM Parent T (MeV) TRMF FRDMFragment R.Y. Fragment R.Y. Fragment R.Y. Fragment R.Y.
U 1
Pd +
Pd 0.949
Pr + As 0.210
U 1
Te +
Zr 1.454
Pr + As 0.248
Pd +
Pd 0.910
Cs + Rb 0.178
Te +
Zr 0.491
Pr + As 0.247
Ba + Kr 0.032
Ba + Kr 0.134
Xe +
Sr 0.014
Pr + As 0.1662
Gd + Ni 0.323
Sb +
Nb 0.216 2
Pr + As 0.348
Ag +
Rh 0.193
Gd + Ni 0.264
Te +
Zr 0.213
Nd + Ge 0.197
In +
Tc 0.168
Gd + Ni 0.0.221
Pr + As 0.210
Pr + As 0.176
Te +
Zn 0.140
Nd + Ge 0.240
Sb + Zn 0.087
Gd + Ni 0.175
Te +
Zn 0.1003
Gd + Ni 0.249
Sb +
Nb 0.283 3
Cd +
Ru 0.565
Ag +
Rh 0.414
Gd + Ni 0.214
Te +
Zr 0.242
In +
Tc 0.255
In +
Tc 0.278
I +
Y 0.143
Te +
Zr 0.102
Ag +
Rh 0.236
Ag +
Rh 0.149
Sb +
Nb 0.114
Sn +
Mo 0.092
Sn +
Mo 0.161
Cd +
Ru 0.083
Th 1
Pd +
Ru 0.773
Cs + Br 0.190
Th 1
Sn +
Zr 0.439
Sb +
Y 0.183
Xe + Kr 0.515
Ba + Se 0.124
Sb +
Y 0.291
Ce + Ge 0.118
Cs + Br 0.174
Ag +
Tc 0.123
Cs + Br 0.176
Sb +
Y 0.115
Ag +
Tc 0.129
Pm + Cu 0.092
Ba + Se 0.139
Nd + Zn 0.0772
Pr + Ga 0.505
Sb +
Y 0.213 2
Sm + Ni 0.370
Sb +
Y 0.161
Sb +
Y 0.334
Te + Sr 0.202
Sm + Ni 0.290
Eu + Co 0.141
Gd + Fe 0.134
Sn +
Zr 0.146
Ba + Se 0.172
Sb +
Y 0.1323
Sb +
Y 0.886
Sb +
Y 0.252 3
Rh +
Rh 0.803
Rh +
Rh 0.325
Te + Sr 0.148
Sn +
Zr 0.207
Pd +
Ru 0.350
Rh +
Rh 0.210
Nd + Zn 0.063
Te + Sr 0.153
Rh +
Rh 0.307
Ag +
Tc 0.120 the ground state binding energies which are calculated fromthe relativistic mean field model. The level densities and theyields combinations are manipulated from the convolution in-tegral approach. The fission mass distributions of the afore-mentioned nuclei are also evaluated from the FRDM formal-ism for comparison. The level density parameter a and in-verse level density parameter K are also studied to see thedifference in results with these two methods. Besides fissionfragments, the level densities are also discussed in the presentpaper. For U and
Th, the symmetric and nearly sym-metric fragmentations are more favorable at temperature T = N = Z =
28) at temperature T = U and
Thwith their accompanied possible fragments at T = Th. This could be due to the deformed close shell in the region Z = 52 − of the periodic table [46]. TheNi isotopes and the neutron close shell ( N ∼ U and
Th at tem-perature T = T = N =
82) is one of the largest yield fragments. The symmetricfragmentation Rh + Rh is possible for
Th due to the N =
82 close shell occurs in binary fragmentation. For
U,the larger yield values are confined to the junction of neutronand proton closed shell nuclei.
V. ACKNOWLEDGMENT
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