Table Of ContentStructure effects on fission yields
BharatKumara,b,M.T.Senthilkannanc,M.Balasubramaniamc,B.K.Agrawalb,d,S.K.Patraa,b∗
aInstituteofPhysics,SachivalayaMarg,Bhubaneswar-751005,India.
bHomiBhabhaNnationalInstitute,AnushaktiNagar,Mumbai-400094,India.
cDepartmentofPhysics,BharathiarUniversity,Coimbatore-641046,India.
dSahaInstituteofNuclearPhysics,1/AF,Bidhannagar,Kolkata-700064,India.
7Abstract
1
0 Thestructureeffectsofthefissionfragmentsontheiryieldsarestudiedwithinthestaticaltheorywiththeinputs,like,excitation
2energiesandleveldensityparametersforthefissionfragmentsatagiventemperaturecalculatedusingthetemperaturedependent
relativisticmeanfieldformalism(TRMF).Forthecomparison,theresultsarealsoobtainedusingthefiniterangedropletmodel.At
n
atemperaturesT =1−2MeV,thestructuraleffectsofthefissionfragmentsinfluencetheiryields. ItisalsoseenthatatT =3MeV,
Jthefragmentsbecomesphericalandthefragmentsdistributionpeaksatacloseshellornearcloseshellnucleus.
3
Keywords: Nuclearfission,Leveldensity,Heavyparticledecay,Bindingenergiesandmasses
]PACS:25.85.-w,23.70.+j,21.10.Ma,21.10.Dr
h
t
-
l
c1. Introduction influences,towhatextentofthetemperatureitaffectstheprob-
u ablefragmentation.
n Nuclear fission is still not completely understood, although
Theletterisorganizedasfollows. InSection2,wepresenta
[itwasdiscoveredeightdecadesago. Thestudyoffissionmass
briefdescriptionofthestatisticaltheoryandtheTRMFformal-
1 distribution is one of the major insight of the fission process. ism. In Section 3, we discuss the variation of the quadrupole
vConventionally,therearetwodifferentapproaches,thestatisti-
deformation parameter β of the fragments with the tempera-
1caland thedynamicalapproachesforthe studyoffission pro- 2
ture and study the structure effects of the fission fragments in
3
cess [1, 2]. The latter is a collective calculations of the po-
7 themassdistribution. Thesummaryandconclusionsofourre-
tential energy surface and the mass asymmetry. Further, the
0 sultsaregiveninSection4.
0fission fragmentswere determinedeitheras the minimain the
.potentialenergysurfaceorbythemaximumintheWKBpene-
1
trationprobabilityintegralforthefissionfragments.Thestatis-
0 2. Themethod
7ticaltheory[2],beginswiththestatisticalequilibriumisestab-
1lishedfromthesaddletoscissionpointandtherelativeproba-
:bilityofthefissionprocessdependsonthedensityofthequan- We constraint the charge to mass ratio of the fission frag-
v
mentstothatoftheparentnucleusi.e.,Z /A ≈Z/A,withA ,
itum states of the fragments at scission point. The mass and P P i i P
X Z andA,Z (i=1and2)correspondtomassandchargenum-
the charge distribution of the binary and the ternary fission is P i i
rstudied using the single particle energies of the finite range ber of the parent nucleus and fission fragments, respectively
a
[3]. The constraints, A + A = A and Z + Z = Z are im-
dropletmodel(FRDM)[3,4,5]. InFRDM [6]formalism,the 1 2 1 2
posed to account for the mass and charge conservations. For
energiesat giventemperatureare calculatedusing the relation
242Pu, which is a representative case in the present study, the
E(T) = nǫ with n and ǫ are the fermi-distribution func-
i i i i i
binarychargenumbersarerestrictedtoZ ≥26andZ ≤66by
tionandtPhesingle-particleenergycorrespondingtotheground 2 1
consideringtheexperimentalyield[9]. Accordingtostatistical
state deformation[4]. The temperaturedependenceof the de-
theory [2, 10], the probability of the particular fragmentation
formationsofthefissionfragmentsandthecontributionsofthe
is directly proportional to the folded level density ρ of that
pairing correlationsare also ignored. In the presentletter, we 12
fragmentswiththetotalexcitationenergyE∗.
applied the self consistent temperature dependent relativistic
mean field theory (TRMF) with the well known NL3 param-
E∗
eterset[7,8]. Here,westudythestructureeffectsofthefission ρ (E∗)= ρ (E∗)ρ (E∗−E∗)dE∗. (1)
fragments,whethertheycouldinfluencetheyieldornot? Ifit 12 Z0 1 1 2 1 1
∗Correspondingauthor Thefoldedleveldensityiscalculatedfromthefragmentlevel
Emailaddress:[email protected](S.K.Patraa,b) densities ρi [3, 5] with the excitation energy Ei∗ and the level
PreprintsubmittedtoPhysicsLettersB January4,2017
densityparametersa (i=1and2),whichisgivenas: neutronstates. Theentropyisobtainedby,
i
ρ E∗ = 1 π2 1/4E∗(−5/4)exp 2 aE∗ . (2) S =− [nilnni+(1−ni)ln(1−ni)]. (6)
i i 12 ai! i (cid:18) q i i(cid:19) Xi
(cid:0) (cid:1)
ThetemperaturedependentRMFbindingenergiesandthegap
Here,weusedtheselfconsistenttemperaturedependentrel-
parameterareobtainedbyminimizingthefreeenergy,
ativisticmeanfieldtheory[7,8]tocalculatetheexcitationen-
ergy of the fragments. The excitation energies are calculated F = E−TS. (7)
using the total energyfor a given temperature and the ground
state energy (T = 0) as, E∗ = E(T) − E(T = 0). Further, Inconstantpairinggapcalculation,foraparticularvalueof
i
wecalculatetheexcitationenergiesofthefragmentsusingthe pairing gap △ and force constant G, the pairing energy E
pair
ground state single particle energies of Finite Range Droplet diverges, if it is extended to an infinite configuration space.
Model (FRDM) [6] at the given temperature T with the help Therefore, a pairing window in all the equationsare extended
of the Fermi-Diracdistribution functionkeepingthe total par- up-tothelevel|ǫ −λ| ≤ 2(41A−1/3)asafunctionofthesingle
i
ticle number conserved. The level density parameters a are particleenergy[11,14].
i
evaluated from the excitation energies and the temperature as
a = E∗/T2. Therelativeyieldisestimatedastheratiobetween
i i 3. Resultsanddiscussions
the probability of a given fragmentation and the sum of the
probabilities of all the possible fragmentationsand it is given In this letter we study the fission fragment distributions of
by, 242Pu as a representative case. After generating the possible
P(A ,Z )
Y(A ,Z )= j j . (3) fragmentsbyequatingthechargetomassratiooftheparentsto
j j P(Aj,Zj) thecharge-massratioofthefragments,theyieldvaluesareob-
Here,thesumofthetotalprobaPbilityisnormalizedtothescale tainedusingEq.3. Thefoldeddensitiesarecalculatedfromthe
2. Thecompetingbasicdecaymodessuchasneutronemission individual level density of the fragments which are evaluated
andαdecayarenotconsideredandthepresentedresultsarethe fromthefragmentexcitationenergiesatthegiventemperature
prompt disintegration of a parent nucleus into two fragments T. TheexcitationenergiesarecalculatedwithintheTRMFand
(democraticbreakup). FRDM. In TRMF model, the excitation energy is the differ-
The well knownrelativistic mean field theoryconsidersthe ence between the temperature dependent total energy and the
nucleonnucleoninteractionviaeffectivemesons.Thenucleon- ground state energy. The minimization of free energy of fis-
mesoninteractionLagrangiandensityarefoundin[11,12,8]. sionfragmentswithinTRMFareobtainedbyallowingthemto
The TRMF formalism is explained in Refs. [7, 8]. Here, we beaxiallydeformed. ThesolutionsfortheDiracequationsfor
brieflyrevisittheRMFwiththeBCSformalism.Foropenshell nulceonsandtheKlein Gordonequationsforthe mesonfields
nucleithepairingisimportantfortheshellstructure. Thecon- are obtainedby expandingthem in terms of the sphericalhar-
stant pairing gap BCS approximation is used in our calcula- monicwavefunctionsasbasis. Forthenumericalcalculations,
tions. The pairing interaction energy in terms of occupation thenucleonwavefunctionsandthemesonfieldsareexpanded
probabilitiesv2andu2 =1−v2iswrittenas[13,14]: tothenumberofmajorshellstakentobeNF =12andNB =20,
i i i
respectively. IntheFRDMmethod,theexcitationenergiesare
2 calculatedusingthesingleparticleenergieswhichareretrieved
E =−G uv , (4)
pair i i
Xi>0 0.7
withG isthe pairingforceconstant. Thevariationalapproach 242 Temperature
withrespecttotheoccupationnumberv2 givestheBCSequa- 0.6 Pu 0 MeV 2 MeV
i 1 MeV 3 MeV
tion2ǫuv−△(u2−v2)=0withthepairinggap△=G uv. 0.5
i i i i i i>0 i i
Thepairinggap(△)ofprotonandneutronistakenfromPtheem- 0.4
pirical formula[11], △ = 12× A−1/2 MeV. The temperatureis
0.3
introducedinthepartialoccupanciesintheBCSapproximation 2
0.2
as:
n =v2 = 1 1− ǫi−λ[1−2f(ǫ˜,T)] , (5) 0.1
i i 2" ǫ˜i i # 0.0
with the function f(ǫ˜,T) = 1/(1+ exp[ǫ˜/T]) represents the -0.1
i i
FermiDiracdistribution;ǫ˜i = (ǫi−λ)2+△2 isthequasipar- -0.2
ticleenergiesandtheǫ isthespingleparticleenergies. Inlower
i -0.3
temperatures,thepairingenergiesplayanimportantroleinthe 70 80 90 100 110 120 130 140 150 160 170
structure of open shell nuclei. The chemical potential λ (λ ) Fragment mass number (A1 & A2)
p n
forprotons(neutrons)isobtainedfromtheparticlenumbercon- Figure1:(coloronline)Thevariationofthequadrupoledeformationparameter
straints inZi(niN)=Z(N). Thesumistakenoverallprotonand β2forthefissionfragmentsfor242PuattemperatureT =0-3MeV.
P
2
0.30 (a) 91Br TT == 12 MMeeVV 151Pr 242Pu (b) 91Br TT == 12 MMeeVV 151Pr 242Pu
T = 3 MeV 0.12 T = 3 MeV
TRMF FRDM
0.25
0.10 98 110Mo 132 Te 144
Sr Ba
d0.20 d
yiel 109 133 yiel0.08 113Ru129Sn
e Mo Te e
v 76 v
ati0.15 Zn81Ga 161Eu166Gd ati
el el0.06
R 96Rb 113Ru129Sn 146La R 80Ga 121Ag 162Eu
0.10 77 165Gd
0.04 Zn
121
Ag
0.05 0.02
0.00 0.00
70 80 90 100 110 120 130 140 150 160 170 70 80 90 100 110 120 130 140 150 160 170
Fragment mass number (A1 & A2) Fragment mass number (A1 & A2)
Figure2:(coloronline)Massdistributionof242PuforthethreedifferenttemperaturesT =1,2and3MeVusingtheTRMFandFRDMformalisms. Thesumof
thetotalyieldisnormalizedtothescale2.
fromtheReferenceInputParameterLibrary(RIPL-3)[15]and aresameasthosefortemperatureT =1MeV,alongwithone
thedetailsofcalculationscanbefoundinRefs. [4,5]. of the neutronclosed shellnucleusin 81Ga + 161Eu(for81Ga,
N=50). Analyzing Fig. 2(a) for TRMF results, we notice a
In Fig. 1 we plotthe quadrupoledeformationparameterβ
2 symmetricyieldregionforT = 3 MeV, i.e. frommassregion
as a functionof mass numbers A and A for the binary frag-
1 2 A ∼ 100−133. Mostof theyield fragmentsare concentrated
ments. ForT =1MeV,thequadrupoledeformationparameter
inthisregion. Additionally,afewmoresecondaryasymmetric
of the fragments are prolate/oblate or spherical similar to the
fractionsare also noticed. We see a similar symmetric region
ground state nuclei. The pairing transition of the nuclei oc-
along with few asymmetric yields in case of FRDM calcula-
cur within the temperature T = 1 MeV. For T = 2 MeV, the
tionsalso(seeFig. 2(b)). Theonlydifferenceisthesecondary
fragments with mass number A ≤ 150 become almost spheri-
asymmetric yields spread widely in this case, contrary to the
calexceptfewexceptions. ForhighertemperatureT =3MeV,
concentratedsymmetricpeaksforTRMFcalculations. Ingen-
all the nuclei become perfectly spherical. When the tempera-
eral,ourresultsagreewiththerecentexperimentalobservations
tureincreases,thelevelsabovetheFermisurfacebecomemore
ofRef. [16].
occupiedduetothetransitionofparticlesfromthepartialoccu-
piedlevels(belowtheFermisurface).Forhighertemperatures,
InFRDMmodel,thefavorablefragmentsareinthevicinity
the occupancies become uniform at the vicinity of the Fermi
of the neutron closed shell (N = 82) nuclei or exactly at the
surface. Thedeformedstructuresaremeltedathighertempera-
protonclosedshellnuclei(Z=50). Theothersecondaryfrag-
tures,andtheevolvedfragmentslandupinacloseshellshape
mentations are not necessarily closed shell nuclei, so a good
duetothehigherexcitationenergyoftheclosedshellnuclei.
fractionofnon-magicfragmentsarealsoappearedintheyield
Nowthefissionmassdistributionsof242Puobtainedfromthe values. For higher temperature T = 3 MeV, the most favor-
TRMFandFRDMformalismsforthreedifferenttemperatures able fragments for both models are more or less same. The
T = 1, 2 and 3 MeV are shown in Fig. 2 and the fragment fragmentcombinations109,110Mo+133,132Teand113Ru+129Sn
yieldsarecomparedintable1. ForT =1MeV,themostfavor- are the most favorable fragments. In TRMF model the most
able fragmentationis 91Br +151Pr, for both the models. How- favorable fragments in region I and region II have sharp peak
ever, other fragmentssuch as 96Rb +146La in TRMF and 98Sr yield values. But in FRDM, the larger yields are at region II
+144BainFRDMarealsonoticed. Inaddition,thebinaryfrag- only. Conclusively,forT =3MeV,oneofthemostfavorable
mentations121,122Ag+121,120Agarefavorableforbothmodels. fragmentsareatthevicinityoftheclosedshell(N∼82)orex-
AttemperatureT =2and3MeVtheyieldsoftheminorfrag- actlyattheclosedshell(NorZ=50)nuclei. Asitisreported
mentsevolutioninFRDMmodelhaveincreased.But,inTRMF in many earlier works, that the rare earth nuclei in the range
model, the most favorable fragmentsare mainly from the two Z = 56−66 region shows, peculiar nature, such as shell clo-
regions, I (with mass A ∼ 71−91 and A ∼ 171−151) and sure behavior, most of the fragmentsare foundin that region.
II (A ∼ 105− 113 and 137− 129). For T = 2 MeV, region Also, the corresponding neutron numbers for such nuclei act
IIislessprobablethanregionI.Themostfavorablefragments likedeformedshellclosureatN =98−102region.Asaresult,
3
we get many neutron fragmentin that region [17]. Although,
theeven-evenfragmentsaremorepossiblefissionyield,inthe
presentcase,wefindmaximumnumberoffissionyieldisodd
massfragmentsascomparedtotheeven-evencombination.Be-
causeoftheleveldensityoftheoddmassfragmentsarehigher
thantheevenmassfragmentsasreportedinRef. [2]. Forthis
temperature,allnucleibecomespherical,theexcitationenergy
oftheclosedshellnucleiishigherthantheothersphericalnu-
clei. Thus,thedeformationofthefissionfragmentsaffectsthe
most favorable distribution at the temperatures T = 1-2 MeV.
In FRDM model the temperature dependent deformations are
notconsidered,sothemostfavorablefragmentsatT =2and3
MeVaresame.
Table 1: Therelative fissionyield (R.Y.)=Y(Aj,Zj) = PP(A(Aj,jZ,Zj)j) obtained at
temperature T =1, 2and3MeVfromtheTRMFcalculPations arecompared
withtheFRDMpredictionfor242Pu. Thesumofthetotalyieldisnormalized
4. SummaryandConclusions
tothescale2.
T(MeV) TRMF FRDM
The binary fission mass distribution is studied within the
Fragment R.Y. Fragment R.Y.
statistical theory. The level densities and the excitation en-
91Br+151Pr 0.559 91Br+151Pr 0.244 ergies are calculated from the TRMF and FRDM formalisms.
96Rb+146La 0.213 98Sr+144Ba 0.184 The TRMFmodelincludesthe thermalevolutionsof the pair-
76Zn+166Gd 0.182 90Br+152Pr 0.120 ing gaps and deformation in a self-consistent manner, while,
they are ignoredin the FRDM. The excitation energiesof the
1 121Ag+121Ag 0.140 94Rb+148La 0.108
fragmentsareobtainedfromgroundstateresultsintheFRDM
120Ag+122Ag 0.122 97Sr+145Ba 0.088
calculations. So, the study of deformation effects on the fis-
94Rb+148La 0.108 87Se+155Nd 0.086 sion fragmentscan be seen within the TRMF formalism. The
119Pd+123Cd 0.107 99Sr+143Ba 0.082 quadrupoledeformationsoffissionfragmentswiththeincreas-
95Rb+147La 0.098 85As+157Pm 0.076 ingtemperaturearealsodiscussed. Thestructuraleffectsofthe
fissionfragmentsinfluencethemostfavorableyieldsattemper-
91Br+151Pr 0.336 110Mo+132Te 0.140
aturesT =1and2MeV.ForthetemperatureT =3MeV,one
76Zn+166Gd 0.295 98Sr+144Ba 0.110
ofthemostfavorablemassdistributionendsupinthevicinity
81Ga+161Eu 0.282 97Sr+145Ba 0.080 orexactlyatthenucleonclosedshell(N∼82)or(NorZ=50).
2 77Zn+165Gd 0.213 99Sr+143Ba 0.079 Although, the TRMF and FRDM formalisms yield the closed
82Ge+160Sm 0.182 113Ru+129Sn 0.076 shellnucleusasoneofthefavorablefragment,thedetailofthe
fissionyieldsandprobabilityofthemassdistributionsarequite
79Ga+163Eu 0.125 100Y+142Cs 0.068
differentinthesetwoformalisms.
109Mo+133Te 0.329 110Mo+132Te 0.202
The author MTS acknowledge that the financial support
81Ga+161Eu 0.295 113Ru+129Sn 0.152 from UGC-BSR research grant award letter no. F.25-
113Ru+129Sn 0.222 109Mo+133Te 0.128 1/2014-15(BSR)7-307/2010/(BSR) dated 05/11/2015 and
3 77Zn+165Gd 0.120 116Rh+126In 0.087 IOP, Bhubaneswar for the warm hospitality and for providing
thenecessarycomputerfacilities.
79Ga+163Eu 0.095 112Ru+130Sn 0.072
82Ge+160Sm 0.094 77Zn+165Gd 0.059
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