FIESTA2017 0JørgenRandrup
236U
FIESTA2017FissionExperimentsandTheore2calAdvances
Fissiondynamicswithmicroscopicleveldensi4es
JørgenRandrup1,DanielWard2,GillisCarlsson2,ThomasDøssing3,PeterMöller4,SvenÅberg2
1NuclearScienceDivision,LawrenceBerkeleyNaHonalLaboratory,Berkeley,California94720,USA
2MathemaHcalPhysics,LundUniversity,S-22100Lund,Sweden3NielsBohrInsHtute,CopenhagenUniversity,DK-2100CopenhagenØ,Denmark4TheoreHcalDivision,LosAlamosNaHonalLaboratory,LosAlamos,NewMexico87545,USA
SantaFe,18-22September,2017
Editors’Sugges4on:Phys.Rev.C95,024618(2017)
FIESTA2017
N.Bohr&J.A.Wheeler,PhysRev56(1939)426:TheMechanismofNuclearFission
Nuclearfissionisaresultofshapedynamics
JørgenRandrup 1
JohnA.Wheeler(1911-2008) NielsBohr(1885-1962)
LiseMeitner(1878-1968)O`oR.Frisch(1904-1979)
L.Meitner&J.A.O.R.Frisch,Nature143(1939)239:Disintegra4onofUraniumbyNeutrons:ANewTypeofNuclearReac4on
JørgenRandrup FIESTA2017 2
TheshapemoHonishighlydissipa4ve:SmoluchowskiequaHon:0=Fcons+Fdiss
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
Distance between Mass Centers r (Units of R0)
Frag
men
t Elo
ngat
ion σ
(Uni
ts o
f R0)
Frag
men
t Elo
ngat
ion σ
(Uni
ts o
f R0)
Family of shapes considered
LangevinequaHon:dp/dt=Fcons+Fdiss
Nuclearshapedynamics->randomwalk
TheHmeevoluHonofthenuclearshapeparameters,q(t):
PaulLangevin(1872-1946)
MarianSmoluchowski(1872–1917)
M(q) U(q) γ(q)
U(q) γ(q)
BrownianmoHon
IfP(Af)is≈insensi4vetoγ(q):RandomwalkontheenergylandscapeU(q)
J.RandrupandP.Möller,PRL106,132503(2011):
236U
Metropoliswalk…
Startatground-state(orisomeric)minimum
Walkun4ltheneckhasbecomethin…
ElongaHon
Asym
metry
Pup=exp(-ΔU/T)Pdown=1
! ?
…onthepoten4al-energysurface:
FIESTA2017 3JørgenRandrup
P.Möller,Nucl.Phys.A192(1972)529
ΔU:ChangeinpotenHal
T:Localtemperature
Metropolis,Rosenbluth2,Teller2,J.Chem.Phys.21(1953)1087
NicholasC.Metropolis(1915-1999)
…thenbinthemassasymmetry
NuclearshapeevoluHonasarandomwalkonthe5DpotenHalenergylandscape
FIESTA2017 4JørgenRandrup
J.RandrupandP.Möller,Phys.Rev.Le`.106,132503(2011)
ElongaHonQ2NeckradiuscEndcapdefsεf1&εf2Massasymmetry α
Q2
45 Q2 ~ Elongation (fission direction)
15 εf1 ~ Left fragment deformation
εf1 εf2
15 εf2 ~ Right fragment deformation
15
⊗
⊗
⊗
⊗d ~ Neck
d
35 αg ~ (M1-M2)/(M1+M2) Mass asymmetry
Five Essential Fission Shape Coordinates
M1 M2
⇒ 5 315 625 grid points − 306 300 unphysical points⇒ 5 009 325 physical grid points
RayNix1969
5Dshapefamily
U(q)=Umacro(q)+Umicro(q)
q
a b
c d
Exp. 233U(n,f) Calc. (6.54 MeV) 234U
30 40 50 60
0
5
10
15
20
25
Exp. 239Pu(n,f) Calc. (6.84 MeV) 240Pu
0
5
10
15
20
25
Yiel
d Y(
Z f) (
%)
Exp. 235U(n,f) Calc. (6.54 MeV) 236U
Exp. 234U(γ,f) Calc. (11.0 MeV) 234U
30 40 50 60 Fragment Charge Number Zf
>5Mshapespernucleus
>5knuclei
JørgenRandrup FIESTA2017 5
P.Möller&J.Randrup,PRC91,044316(2015)
Asymmetric Symmetric0.2 0.4 0.6 0.8
Fission-Fragment Symmetric-Yield to Peak-Yield Ratio
90 100 110 120 130 140 150Neutron Number N
70
80
90
Prot
on N
umbe
r Z
Asymmetric Symmetric0.2 0.4 0.6 0.8
Fission-Fragment Symmetric-Yield to Peak-Yield Ratio
90 100 110 120 130 140 150Neutron Number N
70
80
90
Prot
on N
umbe
r Z
A.N.Andreyevetal.,PRL105,252502(2010)
180Hg
a b
c d
Exp. 233U(n,f) Calc. (6.54 MeV) 234U
30 40 50 60
0
5
10
15
20
25
Exp. 239Pu(n,f) Calc. (6.84 MeV) 240Pu
0
5
10
15
20
25
Yiel
d Y(
Z f) (
%)
Exp. 235U(n,f) Calc. (6.54 MeV) 236U
Exp. 234U(γ,f) Calc. (11.0 MeV) 234U
30 40 50 60 Fragment Charge Number Zf
J.RandrupandP.Möller,Phys.Rev.Le`.106(2011)132503
FIESTA2017 6JørgenRandrup
EnergydependenceofthefissionshapeevoluHon
UseaneffecHveenergylandscapeobtainedbysuppressingthemicroscopicterms
Useshape-dependentmicroscopicleveldensiHestoguidetherandomwalk:ρmicro(q)
J.RandrupandP.Möller,Phys.Rev.C88,064606(2013):
D.E.Ward,B.G.Carlsson,T.Døssing,P.Möller,J.Randrup,S.Åberg,Phys.Rev.C95,024618(2017)[Editors’Sugges4on]
U(q)=Umacro(q)+Umicro(q)xS(E*(q))
è
Leveldensi4esindynamics
FIESTA2017 7JørgenRandrup
PotenHalenergyU(χ)
Shapecoordinateχ
TotalenergyE
LocalstaHsHcalexcitaHonE*(χ)=E–U(χ)
LocalleveldensityρE(χ)
Temperature:
Drivingforce:≈exp(-δU/T)
CollaboraHonforthepurposeofobtainingleveldensiHesforallrelevantfissionshapes:GillisCarlsson,ThomasDøssing,PeterMöller,JørgenRandrup,DavidWard,SvenÅberg
=>
DETAILEDBALANCE
If ρE(χ)=ρ(E(χ))=>Metropolis:
χ≈χ ’
ButgenerallyρE(χ)≠ρ(E(χ))duetostructureeffects:Souserealis4cleveldensi4es!
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
Distance between Mass Centers r (Units of R0)
Frag
men
t Elo
ngat
ion σ
(Uni
ts o
f R0)
Frag
men
t Elo
ngat
ion σ
(Uni
ts o
f R0)
Family of shapes considered
JørgenRandrup FIESTA2017 8
H.Uhrenholt,S.Åberg,A.Dobrowolski,Th.Døssing,T.Ichikawa,P.Möller:NPA913(2013)127
groundstate 2p-2hstate
Combinatorialmethodforthenuclearleveldensity
=>
ConsiderallmulHplep-hexcitaHonsforprotonsandneutronsseparately
CalculateBCSpairingforeachone
E=Ep+En+Erot
Expectedtobeunimportant=>ignored
Erot(I,K)=[I(I+1)-K2]/2Iperp(χ,Δp,Δn)
RotaHonalbandbuiltoneachintrinsicstate:
Intrinsicstates:
Rota4onalenhancement
Vibra4onalenhancement
OBS:HigherI=>LowerEintr
1p-1hstatePairing
Note:Thesingle-parHclelevelsarethesameasthoseusedtogettheshellandpairingenergiesinU(q)!
A.Schiller,etal.,PRC63(2001)021306(R)
JørgenRandrup FIESTA2017 9
H.Uhrenholt,S.Åberg,A.Dobrowolski,Th.Døssing,T.Ichikawa,P.Möller:NPA913(2013)127
Combinatorialmodelforthenuclearleveldensity
=>
E.Melby,etal.,PRC63(2001)044309
S.Siem,etal.PRC65(2002)044318
M.Gu`ormsen,etal.,PRC68(2003)064306
JørgenRandrup FIESTA2017 10
Project:
Usethecombinatorialmethodtoobtainthemicroscopicleveldensityforall(>5M)3QSshapesforwhichthepotenHalhasbeentabulated:
ρZA(E,I,shape)foreachindividualfissioningnucleusAZ(UZA(shape)existsfor>5kAZ)
Usethoseasthebasisfortherandomwalk:Pdown:P(U’≤U)=1-->P(ρ’≥ρ)=1Pup:P(U’≥U)=exp(-ΔU/T)-->P(ρ’≤ρ)=ρ’/ρ
ThenthegradualdisappearanceofpairingandshelleffectswithexcitaHonisautoma4callyincludedintheshapeevoluHon
AsymmetricshapesReplace{εn}by3QSGetalls.p.levels(PM)
TrivialcodemodificaHon
Fullyconsistent:sames.p.levelsusedforUandρ(noparameters)
JørgenRandrup FIESTA2017 11
MassyieldsusingmicroscopicleveldensiHes
0
5
10
15
20
25
Cha
rge
yiel
d Y(Z f
) (%
) (a) 234U (6.84 MeV) Exp
ρmicro
0
5
10
15
20
Cha
rge
yiel
d Y(Z f
) (%
) (b) 234U (11 MeV)
30 40 50 60 70Fragment proton number Zf
0
5
10
15
20C
harg
e yi
eld Y(Z f
) (%
) (c) 234U (16 MeV)
è
0
5
10
15
20
25
Cha
rge
yiel
d Y(Z f
) (%
)
0
5
10
15
20
Cha
rge
yiel
d Y(Z f
) (%
)
Exp
ρmicro
20 30 40 50 60 70Fragment proton number Zf
0
5
10
15
20
Cha
rge
yiel
d Y(Z f
) (%
)
(b) 236U (6.55 MeV)
(c) 240Pu (6.53 MeV)
(a)
235U(nth,f)
239Pu(nth,f)
233U(nth,f)
Fissionof234U(E*)
✔
JørgenRandrup FIESTA2017 12
Energydependenceoffissionyields:Non-monotonicbehaviorofthesymmetricyield
[A]L.E.Glendeninetal.,PhysicalReviewC24,2600(1981)[B]M.B.Chadwicketal.,NuclearDataSheets112,2887(2011)
JørgenRandrup FIESTA2017 13
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4Mass asymmetry αg
2
4
6
8
10
12
14
16
Elongationq 2
1
32
1
32
1
mean scission elongation
30 35 40 45 50 55 60Fragment proton number Zf
Fissionof236U:
235U(nth,f)
Jørgen'Randrup' FIESTA:'8'September'2014' 29'
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PotenHal-energysurface Terminalshapes
T.Ichikawa,A.Iwamoto,P.Möller,A.J.Sierk,PhysicalReviewC86,024610(2012)
meanelongaHon
JørgenRandrup FIESTA2017 14
M.E.Goodenetal.,NuclearDataSheets131,319(2016):Non-monotonicenergydependenceofY(Af)from240Pu*
Leveldensityforthreeshapesalongtheridge;allhaveadensesingle-parHclespectrum,henceaposiHveshellenergyandalargepairinggap:
2-qpstates
4-qpstates
OBS:
Energydependenceoffissionyields:Non-monotonicbehaviorofthesymmetricyield
FIESTA2017 15JørgenRandrup
ThenuclearshapeevoluHonisakintoBrownianmoHonandcanbeapproximatelydescribedasarandomwalkonthemulH-dimensionaldeformaHon-energysurface
ThisconceptuallysimpletreatmentmakesitpossibletocalculatefissionfragmentmassandchargeyieldsforanynucleusforwhichsuitablepotenHal-energysurfacesexist
Asymmetric Symmetric0.2 0.4 0.6 0.8
Fission-Fragment Symmetric-Yield to Peak-Yield Ratio
90 100 110 120 130 140 150Neutron Number N
70
80
90
Prot
on N
umbe
r Z
(noadjustableparameters,computaHonallyfast)
(5Dsurfacesexistforover5,000nuclei)
FIESTA2017FissionExperimentsandTheore2calAdvances
UE(shape)=Umacro+Umicro
Ageneral&consistentdescripHonwasobtainedbyusingthemicroscopicleveldensiHescalculatedforeachshapebymeansofarecentlydevelopedcombinatorialmethod;thegradualdisappearanceofshellandpairingeffectsisthenautoma4callyensuredwithoutanynewparameters
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
Distance between Mass Centers r (Units of R0)
Frag
men
t Elo
ngat
ion σ
(Uni
ts o
f R0)
Frag
men
t Elo
ngat
ion σ
(Uni
ts o
f R0)
Family of shapes considered
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
Distance between Mass Centers r (Units of R0)
Frag
men
t Elo
ngat
ion σ
(Uni
ts o
f R0)
Frag
men
t Elo
ngat
ion σ
(Uni
ts o
f R0)
Family of shapes considered 0
5
10
15
20
25
Yie
ld Y
(Zf)
(%)
a) 234U (6.84 MeV) ExpUEρmic
0
5
10
15
20
Yie
ld Y
(Zf)
(%)
b) 234U (11 MeV)
30 40 50 60 70Fragment proton number Zf
0
5
10
15
20
Yie
ld Y
(Zf)
(%)
c) 234U (16 MeV)
Editors’Sugges4on:Phys.Rev.C95,024618(2017)
ρmicro(shape)
Fissiondynamicswithmicroscopicleveldensi4es
Amarriagebetweennuclearstructure&dynamics!
SantaFe,18-22September,2017
