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Dynamical Microscopic and Macro-Micro Approaches to Nuclear Fission
FIESTA2017
• M.D.Usang,C.Ishizuka,Y.Iwata,S.Okumura,A.Etori,T.Nishikawa(TokyoTech)• F.A.Ivanyuk(TokyoTech.,Kiev)• J.A.Maruhn(FrankfurtU.)• A.Ono(TohokuU.)• M.Kimura(HokkaidoU.)• T.Kawano(LANL)
SatoshiChibaProf.,TokyoInsQtuteofTechnology(TokyoTech)Director,AtomicEnergySocietyofJapan(AESJ)
Contents
• IntroducQonandBackground• NuclearfissionbyLangevinequaQon:Macr-micro• NuclearfissionbyanQsymmetrizedmoleculardynamics(AMD)
• NuclearfricQonbyTDHF• Summary
2
Introduc;on and Background
• Eventhoughthehistoryofresearchesinnuclearfissionisalreadymorethan70years-long,manymysteriessQllexistactuallywhatisgoingonduringtheprocessofnuclearfission,especiallybeforescission
• DuetothedifficulQes,accuracyofthefission-relatednucleardataissQllconsiderednothighenough
• ToresolvethissituaQonandtounderstandmanydifferentaspectsofnuclearfissioninaconsistentway,wethinkaccuratedynamicaltreatmentofnuclearfissionisnecessary
• WearemakingresearchesinfissionbydynamicalapproacheswithLangevinequaQon,AMD(anQsymmetrizedmoleculardynamics)andTDHF
• Hereweexplainthembriefly 3
Origin of extra spin that fission fragments have
Spinoffissi
on
fragment
148Pm
7ℏ
4
spinof148Pmfromfissionofn+235UsystemasafuncQonofexcitaQonenergyofcompoundnucleusAumannetal.,Phys.Rev.C16,254-265(1977).
5
タイトルタイトル
PRC21,204(1980)Maruhnetal.,Phys.Rev.C74,027601(2006).
Challenges
Dofissionfragmentsrotate?Ifso,whatistheoriginoftherotaQonalmoQonandhowmuchangularmomentatheyhave?
TwisQng
Bending
TwisQngmodeinfusionreacQons
TwisQng&Bendingmodesinfission
TDHForAMDapproach
6
AnomalyintheaverageTotalKineQcEnergyofFissionFragments
7
Our approaches (CN to scission)Macro-microapproach:Langevinmodel• PESfrommacro-micro,transportcoefficientsfrommacroormicro+randomforce• 3DLangevinwithmacroscopictransportcoefficients(withF.Ivaniuk)• 3DLangevinwithmicroscopictransportcoefficients(bylinearresponsetheory)• 4DLangevinwithmacroscopictransportcoefficients• 4DLangevinwithmicroscopictransportcoefficients• 5DLangevin--talkbyA.SierkMicroscopicapproachesAnQsymmetrizedMolecularDynamics(AMD)• SlaterdeterminantwithGaussianwavepackets(coherentstate)asbasisfuncQon• meanfieldcalculatedwithSLy4interacQon• StochasQcnucleon-nucleoncollisionstoexpressbranchingofSlaterdeterminant
anddeformaQonofGaussianwavepacket
Time-dependentHartreeFock(TDHF)• NorestricQononsingle-parQclewavefuncQon(3DmeshcalculaQon)• meanfieldbySV-BasinteracQon• basedonOAK3DandSKY3D(nopairing)• toobtainnuclearfricQoncoefficientfromdynamicalmodel
8
neutron
235U
SimulaQonofnuclearfission(235U+140MeVn)byJQMD
JQMD:JAERIQuantumMolecularDynamics=asemiclassicalmoleculardynamicsfornuclearreacQons(meanfield+NNcollision)
9
JQMD(分子動力学)による140MeV中性子+238Uの核分裂の時間発展
K.Niita, T. Maruyama, Y. Nara, S. Chiba and A. Iwamoto, JAERI-Data/Code 99-042(1999)
TimeevoluQonof235U+140MeVnreacQonbyJQMD
QMDsimulaQonofnuclearfission(235U+140MeVn)
Nuclear fission by Langevin equa;on
These2differentd.o.fhavedifferent2mescales:
BrowningmoQon
• nucleonmo2on:1to10fm/c• shapemo2on:~>10,000fm/c
NuclearshapeevoluQonisdrivenbyrandomkicksbynucleonsinthermalequilibrium(microscopicd.o.f.)giventothenuclearsurface(macroscopicd.o.f)frominsidethesurface
10
qi : deformation coordinates (nuclear shape) in two-center shell-model parametrization
pi : : momentum conjugate to : mass tensor : Hydrodynamical mass (Werner-Wheeler)
( )
( ) ( ) )(21 11
1
tRgpmppmqq
Vdtdp
pmdtdq
jijkjkijkjjkii
i
jiji
+−∂
∂−
∂
∂−=
=
−−
−
γ
ijjkk
ik
ijjii
Tgg
tttRtRtR
γ
δδ
=
−==
∑)(2)()( ,0)( 2121
( ) 21*int )(
21 aTqVppmEE jiij =−−= −
paramterdensity level:energycorrectionshellStrutinsky:)(
)()()(:n deformatioat potential : )V(re temperatu: energy, excitation : energy, intrinsic : *
int
aqV
qVqVqVqqTEE
sh
shLDM +=
多次元ランジュバン方程式 Langevin Equations for nuclear fission
Einstein relation
orfromLinearResponseTheory(microscopic)
iq
ipijm
iq
ijγ : friction tensor : Wall and Window formula (one-body dissipation)
Intrinsic energy and Temperature :
Shell correction to the potential energy :
White noise
Friction Random force
Random force :
11
Macroscopictransportcoefficients
{ } },,,{ 2104 αδδZZq D ={ } },,{ 2103 αδδ == ZZq D
Two-center oscillator model (Maruhn and Greiner, Z. Phys. 251(1972) 431)
Shape parametrization
: Radius of compound nucleus
Massasymmetry
ElongaQon
DeformaQonoffragmentsatouterQps2,1,2
)(3=
+
−= i
baba
ii
iiiδ
●
●
●
RzZZ 0
0 =
21
21
AAAA
+
−=α
R
12 11222
2
2 2 1 1
1
b1b1
b2b2 b1
35.0=ε neck parameter : fixed●
Rneck
Collective coordinates (4 dynamical variables)
12
4D : are independent, 3D : 21, δδ
{ } },,,{ 2104 αδδZZq D =
fragmentleft theof mass : fragmentright theof mass :
2
1
AA
3/12.1 CNA=
{ } },,{ 2103 αδδ == ZZq D
21 δδ =
Notethatneckradiusdependsonallthe5parameters
13
タイトルタイトル
IniQalcondiQonofLangevincalculaQon
PotenQ
alene
rgy
ElongaQon
GroundState
Shapeisomer
Neutron-inducedfission
Tunnelingfission
Fissionfromisomer
Spontaneousfission
Westartfromhereorherewithzeroini2almomenta(thereisnodistribu2onintheini2alstate)
Manypeoplestartfromthisregion
{ } { }( ) { } { }( ) { }( )pqqpqP δδ 0, −=
{ }0q{ }0q
Exampleof4DLangevintrajectories(236U)
(MeV)V
RzZZ 0
0 =14
234U 238Np
240Pu
242Am
MassDistribuQonofFissionFragmentsatEx=20MeV
258Fm
15
258Fm
Distribu;ons of TKE (total kine;c energy of fission fragments)
236U
SuperLong
SuperShort
Q-value
16
Systema;cs of <TKE> by 3D Langevin
17
Extension to 4D Langevin
3DLangevin
4DLangevin
235U+nsystem
{ } },,,{ 2104 αδδZZq D =
18
19
(MeV
)
(MeV)
Present(Macro4D)
1stchance
σTKE by 4D Langevin
3DLangevin
4DLangevin
235U+nsystem
Nuclear fission by An;symmetrized Molecular Dynamics: AMD
• EventhoughtheLangevin-modeldescripQongivessaQsfactoryresultsinmanyaspectsofnuclearfission,ithasonly3to4degrees-of-freedomtoexpressnuclearshape.Itshouldbebemertohavewaystodescribethenuclearfissionbymicroscopictheorieshavingd.o.f.ofallthenucleons
• AMDisaverysuccessfulmicroscopicmodelofnuclearreacQonsandstructures(A.Onoetal,Prog.Theor.Phys.,87(5),(1992),1185-1206)
• WavefuncQonofeachnucleonisexpressedbyaGaussianwavepacketregardedasacoherentstate
• ThetotalwavefuncQonisgivenbyaSlaterdeterminantofthesesingle-parQclewavefuncQons→ unlikeQMD,thisisaproperlyanQsymmetriedquantum-mechanicaltheory
• SkyrmeinteracQonisintroducedasaneffecQvenuclearforce• Ground-stateofnucleiareobtainedbyfricQonalcoolingmethod,whichyieldsgoodBE/Aandnuclearradii
• TheequaQonofmoQonofwavepacketsisgivenbyQme-dependentvariaQonalprinciple
• StochasQcnucleon-nucleoncollision,whichisabsentinTDHFtheory,isintroducedwithPauliblockingeffectinthefinalstate,whichwillactasasourceof2-bodyfricQon. 20
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nucle
on (M
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U,expU,Sly4
BE/A for U isotopes
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0 10 20 3010−3
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100
101
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A
cros
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180MeVp+27Al
180MeVp+27Al
AMD-1 : Basic formula;on
Time-dependentvariaQonalprinciple:
Single-parQclewavefuncQon:Gaussiancoherentstate
21
< 𝒓 | 𝝓↓𝒁↓𝒊 > = ( 𝟐𝝂/𝝅 )↑𝟑/𝟒 𝐞𝐱𝐩[−𝝂(𝒓 − 𝒁↓𝒊 /√𝝂 )↑𝟐 ]
< 𝒓 | 𝝓↓𝒁↓𝒊 > = (𝟐𝝂/𝝅 )↑𝟑/𝟒 𝐞𝐱𝐩[−𝝂(𝒓 − 𝒁↓𝒊 /√𝝂 )↑𝟐 ]𝝌↓𝜶↓𝒊
𝒁↓𝒊 =√𝝂 𝒓↓𝒊 + 𝒊/𝟐ℏ√𝝂 𝒑↓𝒊
|𝜱(𝒓↓𝟏 , 𝒓↓𝟐 ,…, 𝒓↓𝑨 )> = 𝟏/√𝑨! 𝐝𝐞𝐭[𝝋↓𝒊 (𝒓↓𝒋 )]TotalwavefuncQon
𝝌↓𝜶↓𝒊 =𝒑↑,𝒑↓,𝒏↑,𝒏↓
𝜹∫𝒕𝟏↑𝒕𝟐▒𝒅𝒕𝚽𝒊ℏ 𝒅/𝒅𝒕 −𝑯 𝚽 /𝚽𝚽 =𝟎
EquaQonofmoQonofthemeanfield:
𝒊ℏ∑𝒋𝝉↑▒𝑪↓𝒊𝝈,𝒋𝝉 𝒁↓𝒋𝝉 = 𝝏𝓗/𝝏𝒁↓𝒊𝝈↑∗ 𝐧𝐝 𝐜.𝐜.
𝓗= <𝚽|𝑯|𝚽>/<𝚽|𝚽> 𝑪↓𝒊𝝈,𝒋𝝉 = 𝝏↑𝟐 /𝝏𝒁↓𝒊𝝈↑∗ 𝝏𝒁↓𝒋𝝉 𝐥𝐨𝐠<𝜱|𝜱>where
• EffecQveN-NinteracQon(originofnuclearmean-field)• SLy4interacQon
AMD-2 : NN interac;ons
• StochasQcNNcollisionasapartofresidualinteracQon(AtestparQclesfromWignerfuncQon)
mimicsrandomforceinLangevintheoryü In-mediumNNcrosssecQonü PauliBlockingaoerthecollisionü branchinganddeformaQonofwavefuncQons-->tunneling,distribuQon StochasQce.o.m.
𝑣↓𝑖𝑗 =𝑡↓0 (1+ 𝑥↓0 𝑃↓𝜎 )𝛿(𝒓)+ 1/2 𝑡↓1 (1+ 𝑥↓1 𝑃↓𝜎 )[𝛿(𝒓)𝒌↑2 +𝒓𝛿(𝒓)]+ 𝑡↓2 (1+ 𝑥↓2 𝑃↓𝜎 )𝒌・𝛿(𝒓)𝒌+ 𝑡↓3 (1+ 𝑥↓3 𝑃↓𝜎 )[𝜌(𝒓↓𝒊 )]↑𝛼 𝛿(𝒓)
22
jiijB ϕϕ=
( ) ( ) 1
1 1
222/32 −
= =
−−∑∑⎟⎠
⎞⎜⎝
⎛= jiij
A
i
A
jBBe ijRrr ν
πν
ρ ( )jiij ZZR += *
21ν
ji rrr −= ( )ji ppk −=!21where
Ini;al state of 236U (235U + n)AsafirststeptosimulaQonofnuclearfissionbyAMD,herewesimplygiveaboostmomentumtoeachnucleon,totheoppositedirecQonsfornucleonsstayingattheleoregionandrightregion
23
E+~1.3MeV/u
Time evolu;on by AMD, 236U (235U + n)
24
𝐸↑∗ =309.2361[MeV]
Examination of phenomenological assumptions
𝟐𝒍=𝟏𝟏𝒓�𝒍=𝟓.𝟓𝒓
AMD : 𝒍~𝟓𝒓
Brosa, Grossmann & Müller, Phys.Rep.197, 167(1990)
• Brosa's random neck rupture condition due to hydrodynamical Rayleigh instability
2l
r
25
ℓ2
ThisverifiesthatBrosa'sesQmaQonofneckrupturecondiQonissensible
26
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100 110 120 1300.00
0.02
0.04
0.06
0.08
0.10
A mass number
FPY
FPY
Results : FPY (left) and rotational angular momenta (left)
• Due to our specific way of boost, the mass distribution corresponds to symmetric fission, which is categorized as super-long mode in Brosa's terminology (low TKE component)
• Each fragment has, on the average 5 to 10ℏ (on the average around 7ℏ) of rotational angular momentum, which is enough to account for the origin of extra spins of fission fragments as explained in the introduction
File = angular Date = 22:16 06-Sep-2017
100 110 120 130 1400
5
10
15
20
A mass numberL[
h̄] L
236U,boostfissionbyAMD
FragmentmassdistribuQon AngularmomentumdistribuQon
Ac/2
27
千葉先⽣⽤TKE (only exp)と2核⼦衝突ありの動画
80 100 120 140 160120
140
160
180
200
220
Mass, A(u)
TKE(
MEV
)Results : Mass-TKE correlation (left) and excitation energy of each fragment (left)
exp:236UatexcitaQonenergyof12MeV
• Our result, which corresponds to the symmetric components, has the same TKE values deduced from experiment. This is in some sense quite satisfactory as well as surprising if we consider the difference of excitation energy of the data (20MeV) and our calculation (300 MeV).
80 100 120 140 160120
140
160
180
200
220
Mass, A(u)
TKE(
MEV
)
AMD
80 100 120 140 160120
140
160
180
200
220
Mass, A(u)
TKE(
MEV
)File = eng_excitation Date = 20:58 06-Sep-2017
100 110 120 130 140100
150
200
250
A mass numberE*
[MeV
]
E*
• This model has a capability to deduce excitation energy of each fragment, which is not the case in Langevin-type approaches. The results E*〜A denotes equi temperature Fermi-gas scheme
1. ThereseemstobenooscillaQonintheelongaQondirecQonatalleventhoughthefactthatthewaywegavetheiniQalcondiQonisexpectedtoresultinstrongoscillaQon.However,theredoexistsurfaceoscillaQon
2. ThisresultisincontradicQontoLangevintheorypredicQons,butinaccordwithTDHFandQMDresults
3. WenoQcethatneckisthickerthanweexpected,andparQcleexchangethroughtheneckregionseemstobequiteimportant,whichmayresultinrotaQonoffragments
4. Nuclearshapeisquitecomplicatedduringfission,whichisoutofreachifweuseonly3,4or5shapeparameters(suchasusedinLangevindescripQon).Inthisrespect,descripQonofnuclearfissionbymicroscopictheoriesisrecognizedtobeveryimportant
Results from AMD calcula;on for 236U (235U + n)
28
skipTDHF
Summary
Nuclear fric;on by TDHF
• We calculate the relative distance of fragment centers
• From its time evolution, we derive average friction coefficient as explained in the next slide
𝜌(𝒓)=∑𝑖↑𝑁▒|𝜑↓𝑖 |↑2 𝑅=|∫↑ ▒𝒓𝜌(𝒓)𝒅𝒓 |𝑖ℏ 𝑑/𝑑𝑡 𝜑↓𝑖 = ℎ ↓𝑖 𝜑↓𝑖
410
86
29
How to obtain fric;on coefficients
Fromthemacroscopice.o.m.forthecenterdistanceR
∫𝐼↑𝐹▒𝜇𝑅 𝑑𝑅 +∫𝐼↑𝐹▒𝑑𝑉/𝑑𝑅 𝑑𝑅 +𝛾∫𝐼↑𝐹▒𝑅 𝑑𝑅 =0
Wecanintegrate
𝜸= (𝟏/𝟐 𝝁𝑹↓𝑩𝑭↑𝟐 + 𝑽↓𝑩𝑭 )−(𝟏/𝟐 𝝁𝑹↓𝑨𝑭↑𝟐 + 𝑽↓𝑨𝑭 )/∫𝑩𝑭↑𝑨𝑭▒𝑹 𝒅𝑹
𝜇𝑅 + 𝑑𝑉/𝑑𝑅 +𝛾𝑅 =0
ThisistheaveragefricQoncoefficients,whichdependsoncollisionenergy
Numbersinredcharacgersinducate 𝛾∕𝐴 at 𝐸↓𝑐𝑚 ∕𝐴 =7MeV
𝐵𝐹:BeforeFusion𝐴𝐹:AoerFission 30
For 236U (=118Pd + 118Pd) system
Wall-and-Windowfric2onisa(sta2candsemiclassical)hydrodynamicalresult,whereasourresultsarepurelydynamicalandquantalones,buttheycoincidequitewell. 31
FricQonbyTDHF
Wallandwindow
32
Possibility of describing ternary and quarterly fission by TDHF
6MeV/A 7MeV/A
208Pb+208Pb
Summary• Wehavedescribed3differentdynamicalapproachestonuclearfissioncurrentlystudiesatTokyoTech:
1. DynamicaldescripQonoffissionbyLangevinequaQonisexplained• CalculaQonsareperformedforacQnidestoFmregionwithboth3Dand4Dversions,macroandmicrotransportcoefficients
• MassandTKEdistribuQonoffissionfragmentsarereproducedwithahighaccuracy,includingtheirsystemaQcaltrends
• Withthisdegreeofaccuracy,thismethodcanbeconsideredtobeoneofthedataproducQontool
2. OurfirstamempttodescribenuclearfissionbyAMDwasdescribed• Quitepromising• RotaQonalangularmomentumcouldbealreadyderivedtobefinite• Itcanaccountfortheextraspinwhichfissionfragmenthasattheonset
3. TDHFapproachwasalsodescribed• todeducenuclearfricQonfromquantaldynamicalmodel• possibilityofdescribingternaryorquarterlyfission
33
34
ThankyouforyouramenQon
35
AsymmetricandfinestructuresofmassdistribuQonofprimaryfissionproductsfromneutron-inducedfission
A=134A=100
post-neutronmassdistribuQon
36
タイトルタイトルn
n
n
γγ
e-e- n
γ
StaQsQcaltheory
DynamicalTheory Brosa-inspired
TDGCMTDHFAMD
RandomWalkLangevinequa2on
MulQ-modalanalyses
Weisskopf-EwingMadland-Nix(LAM)Hauser-Feshbach
GrossTheoryQRPAShellModel
β-decay+staQsQcaltheory
Scission• IsotopedistribuQon• SharingofE*between2fragments,• JπdistribuQon
eν eν
Weisskopf-EwingHauser-Feshbach
StaQcTheory
FongWilkinsCHFRMF
NuclearTheorieswhichcandescribevariousaspectsoffissionCoupled-ChannelsMethod
37
タイトルタイトル
n
n
即発中性子及びγPromptnandγn
γγ
e-
e-n
γ
核分裂片(FissionFragment)
(一次)核分裂生成物(PrimaryFissionProduct)
複合核の形成 Compoundnucleus
サドル点 Saddle,2ndminima
断裂点 Scission
β-decay
eν
独立収率(IndependentYield)
累積収率(CumulaQveYield)
1次収率(PrimaryYield)
NuclearData
Isotopedistribu2on,sharingofexcita2onenergy,spin-paritydistribu2onsdetermineactasini2alcondi2onsforthefollowingprocesses,butarenotknowwell
(二次)核分裂生成物(SecondaryFissionProduct)
断裂中性子Scissionneutron
Delayedn・γ
TimeevoluQonoffission
courtesyF.MinatoConstrainedHartree-Fock
38
neutron
235U
SimulaQonofnuclearfission(235U+140MeVn)byJQMD
JQMD:JAERIQuantumMolecularDynamics=asemiclassicalmoleculardynamicsfornuclearreacQons
39
JQMD(分子動力学)による140MeV中性子+238Uの核分裂の時間発展
K.Niita, T. Maruyama, Y. Nara, S. Chiba and A. Iwamoto, JAERI-Data/Code 99-042(1999)
TimeevoluQonof235U+140MeVnreacQonbyJQMD
QMDsimulaQonofnuclearfission(235U+140MeVn)
neutron
235U
40
JQMD(分子動力学)による140MeV中性子+238Uの核分裂の時間発展
K.Niita, T. Maruyama, Y. Nara, S. Chiba and A. Iwamoto, JAERI-Data/Code 99-042(1999)
TimeevoluQonof235U+140MeVnreacQonbyJQMD
NuclearelongaQonseemstoevolvemonotonically(notoscillatory)insemi-classicalmoleculardynamics.ThisisalsoseeninTDHFcalculaQonbyBulgacetal.(Univ.Washington)FissionQmescale~ 10,000fm/c
QMDsimulaQonofnuclearfission(235U+140MeVn)
41
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Skyr
me
Forc
e[M
eV]
Skyrme force
Time evolution of <V> before (left) and after (right) scission
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Essence of computa;onal method• Shape parametrization : Two-center model in 3 and 4
dimensions
• Potential energy -- macroscopic-microscopic method • Macroscopic part : Krappe-Nix (double-folded Yukawa model)• Microscopic part : two-center shell model or two-center
Woods-Saxon model + Strutinsky + BCS
• Excitation energy dependence of the shell damping : Ignatyuk or Randrup-Moeller prescription
• Transport coefficients• Werner-Wheeler method for mass tensor (3D and 4D)
• Wall-and-Window model for friction tensor (3D and 4D)
• Linear response theory for microscopic mass and friction tensors (3D only) (Ivaniuk et al., JNST DOI:
10.1080/00223131.2015.1070111)
•
43
44
𝐩(𝟏𝟖𝟎𝐌𝐞𝐕)+^↑𝟐𝟕↓𝐀𝐥 𝐛𝐲 𝐀𝐌𝐃
impact parameter:0<b<6force:SLy4
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b)
AMDexp
Nuclear fission by An;symmetrized Molecular Dynamics: AMD
• EventhoughtheLangevin-modeldescripQongivessaQsfactoryresultsinmanyaspectsofnuclearfission,ithasonly3to4degrees-of-freedomtoexpressnuclearshape.Itshouldbebemertohavewaystodescribethenuclearfissionbymicroscopictheorieshavingd.o.f.ofallthenucleons
• AMDisaverysuccessfulmicroscopicmodelofnuclearreacQonsandstructures(A.Onoetal,Prog.Theor.Phys.,87(5),(1992),1185-1206)
• WavefuncQonofeachnucleonisexpressedbyaGaussianwavepacketregardedasacoherentstate
• ThetotalwavefuncQonisgivenbyaSlaterdeterminantofthesesingle-parQclewavefuncQons→ unlikeQMD,thisisaproperlyanQsymmetriedquantum-mechanicaltheory
• SkyrmeinteracQonisintroducedasaneffecQvenuclearforce• Ground-stateofnucleiareobtainedbyfricQonalcoolingmethod,whichyieldsgoodBE/Aandnuclearradii
• TheequaQonofmoQonofwavepacketsisgivenbyQme-dependentvariaQonalprinciple
• StochasQcnucleon-nucleoncollision,whichisabsentinTDHFtheory,isintroducedwithPauliblockingeffectinthefinalstate,whichwillactasasourceof2-bodyfricQon. 45