Tme-dependen Mean-feld approaces o nuclear response and reacon Takas Nakasukasa RIKEN Nsna Cener Inerfaces beween srucure and reacons for rare soopes and nuclear asropyscs, INT, Seale, Aug. 8 - Sep.2, 2
Conens Fne amplude meod FAM for TDHFB A feasble alernave approac o QRPA Codes developed so far HF3DFAM 3D coordnae-space rep. HFBRADDFAM D radal coordnae rep. HFBTHO2DFAM 2D HO-bass rep. Pygmy dpole resonances n lg o medum-eavy nucle Sell effecs/magc numbers/neuron skn Glauber calculaon of reacon cross secon Densy npu from e mean-feld calculaon Sell effec smlar o e PDR
Tme-dependen Harree-Fock TDHF [ ],, 2 r r r r { } [ ], ex ex Tme-dependen Harree-Fock equaon
TDHFB for superflud sysems [ ] R H H { } [ ], ex ex R H R H Ψ Ψ Tme-dependen Harree-Fock-Bogolubov equaon Ψ Ψ Ψ R U κ κ
Small-amplude lm Random-pase approxmaon [ ], ex [ ] [ ] ex,, e e e e One-body densy operaor under a TD exernal poenal Assumng a e exernal poenal s weak, Le us ake e exernal feld w a fxed frequency, e e ex ex ex Te densy and resdual feld also oscllae w,
[ ] [ ] ex,, Te lnear response RPA equaon Noe a all e quanes, excep for and, are non-erman. A ψ ψ A X Ts leads o e followng equaons for X and : { } { }Q Q X X ˆ ˆ ex ex ε ε A Q ˆ Tese are nong bu e RPA lnear-response equaons. X and are called forward and backward ampludes.
Marx formulaon If we expand e X and n parcle orbals: > > A m m m A m m m X X, { } { }Q Q X X ˆ ˆ ex ex ε ε A Q ˆ m m m m X A B B A ex ex jn m nj m nj m j mn m nj m B A ε ε,, Takng overlaps of Eq. w parcle orbals In many cases, seng ex and solve e normal modes of excaons: Dagonalzaon of e marx
Small-amplude approxmaon --- Lnear response RPA equaon --- A B B A X m m ex m ex m A B m, nj εmε m, nj m jn mn j m nj Tedous calculaon of resdual neracons Compuaonally very demandng, especally for deformed sysems. However, n prncple, e self-conssen sngle-parcle Hamlonan sould conan everyng. We can avod explc calculaon of resdual neracons.
Fne Amplude Meod [ ],, ' ' ψ ψ ψ ψ X { } { }Q Q X X ˆ ˆ ex ex ε ε Resdual felds can be esmaed by e fne dfference meod: Programmng of e RPA code becomes very muc rval, because we only need calculaon of e sngle-parcle poenal, w dfferen bras and kes. T.N., Inakura, abana, PRC76 27 2438. Sarng from nal ampludes X and, one can use an erave meod o solve e followng lnear-response equaons.
. Se e nal ampludes X and 2. Calculae e resdual felds by e FAM formula 3. Now, we can calculae e l..s. of e followng equaons: 4. Updae e amplude o X, by an erave algorm, suc as e conjugae graden meod and s dervaves Sep-by-sep numercal procedure [ ],, ' ' ψ ψ ψ ψ X, ex ex ex ex ε ε b X x b Ax X r r r r
TDHFB for superflud sysems [ ] R H H { } [ ], ex ex R H R H Ψ Ψ Tme-dependen Harree-Fock-Bogolubov equaon Ψ Ψ Ψ R U κ κ
Fne amplude meod for superfludsysems [ ] { } [ ] { },, U U U UX U, Resdual felds can be calculaed by QRPA equaons are 2 2 2 2 ~ µν µν µν ν µ µν µν µν ν µ F H E E F H X E E ~ ~ U U W W W H H µν µν Avogadro and TN, PRC 84, 434 2
Implemenaon of e Fne amplude meod TDHF 3D coord. FAM Implemenaon by Tsunenor Inakura Inakura, T.N., abana, PRC 8, 443 29; arxv:6.368 Spercal HFB radal coord. FAM Implemenaon o HFBRAD by Paolo Avogadro Tme-odd felds are added Avogadro and T.N., PRC 84, 434 2 Deformed HFB FAM Implemenaon o HFBTHO by Maro & Markus Tme-odd felds are added Sosov e al, arxv:7.353
HFBRADFAM Tes calculaon: IS monopole Our resul: Red lne qp cu-off a 6 Me All 2qp saes are ncluded. Calculaon by Terasak e al. PRC7, 343 25: Green lne Lnearzaon parameer 9 ~ 5
HFBTHOFAM N sell 5 Comparson w Losa e al. PRC 8 2 6437 N sell 2 Requred memory szes
Cal. w N sell 2 Zr 24 Pu g.s. & f.. Calculaon was performed on a lapop PC.
Pygmy dpole resonance PDR Inakura, T.N., abana, PRC n press, arxv:6.368 Srong neuron sell effecs Correlaon w neuron skn ckness
Magc numbers for PDR emergence Z 5 29 Up o Me
Nex magc number: N5 Z24 Z28 Z32
Magc numbers and low-lorbs Magc numbers: N5, 29, 5, Imporance of weakly bound orbs w l,, and 2. s d 3/2 /2 f p f 5/2 g 7/2 3/2,/2 9/2 d 5/2 s, d, g /2 m PDR/m %
Pygmy dpole resonance PDR and neuron skn sknckness Inakura, T.N., abana, PRC n press, arxv:6.368 Renard and Nazarewcz, PRC 8, 533 2 er weak correlaon beween PDR and neuron skn ckness
PDR sreng vs neuron skn ckness m PDR/m % ν-rc sable S n [ Me ] -Z 2 /A 2 R n -R p [ fm ] Pekarewcz, PRC73 26 44325. Weak correlaon conssen w P.-G.&Wek, PRC8
Unversal correlaon w skn ckness PDR fracon/δr np sows a unversal rae, bu for specfc ranges of neuron numbers Te rae s abou.2 /fm.
Reacon cross secon n Glaubereory Reacon cross secon: Pase sf funcon: Many-body operaor, mulple negral Profle funcon: Parameers are fed o reproduce N-N scaerng α: rao of e real and magnary par of e N-N scaerng β: slope parameer of e N-N elasc dfferenal cross secons. Gve a range of e neracon. pp np E < Pon producon resold E > Pon producon resold
Praccal way o calculae pase-sf funcon Cumulan expanson Need λ OLA: Opcal Lm Approxmaon One-body densy dsrbuons are calculaed by e 3D HF calculaon. Odd-A nucle are calculaed w e fllng approxmaon.
Ne soopes a 24AMe Gbeln e al, PRL, 2253 28 26 Ne Exp: M. Takece al., Mod. Pys. Le. A25, 878 2. Mean-feld calculaon for densy provdes a reasonable agreemen, excep for even-odd effecs.
PDR sreng fracon % Mg S Neuron number Knks n σ R and n PDR sreng are due o s-wave conrbuon. Deformaon effec s seen n σ R.
Summary Fne amplude meod FAM provdes an alernave feasble approac o lnear response calculaon. Several codes developed FAM on D-, 2D-HFB, 3D-HF Sysemac analyss on Pygmy Dpole Resonance PDR Magc numbers for PDR N5, 29, 5,, wc are relaed o e occupaon of low-l orbals s, p, d. Unversal correlaon beween e PDR fracon and e neuron skn ckness; m PDR/m.2 / fmδr np. Sysemac calculaons of reacon cross secons for O, Ne, Mg, S soopes Qualave agreemen w expermenal daa Te knk a N4 s conssen w a n PDR fracon
Collaboraors Paolo Avogadro RNC/Mlano, Sucro Ebaa RNC, Tsunenor Inakura RNC, Kazuro abana Tsukuba/RNC Waaru Horuc RNC, asuyuk Suzuk Ngaa/RNC, Markus Korelanen ORNL, Crsna Losa SISSA, Wold Nazarewcz UTK/ORNL, Maro Sosov ORNL RNC: RIKEN Nsna Cener Wako, Japan ORNL: Oak Rdge Na. Lab. Oak Rdge, USA UTK: Unv. Tennessee Knoxvlle, USA SISSA: Scuola Inernazonale Superore d Sud Avanza Trese, Ialy