Takash NAKATSUKASA Theoetcal Nuclea Physcs Laboatoy RIKEN Nshna Cente 009.3.5-6 Mn-WS: Real-space, eal-tme appoaches DFT, TDDFT ((Q)RPA ) Few-body model (CDCC ) Nuclea Chat 70 Los Alamos Natonal Laboatoy's Chemsty Dvson Pesents a Peodc Table of the Elements Goup** Peod 1 IA 1A 18 VIIIA 8A 1 1 H 1.008 IIA A 13 IIIA 3A 14 IVA 4A 15 VA 5A 16 VIA 6A 17 VIIA 7A He 4.003 3 L 6.941 4 Be 9.01 5 B 10.81 6 C 1.01 7 N 14.01 8 O 16.00 9 F 19.00 10 Ne 0.18 8 9 10 3 11 Na.99 1 Mg 4.31 3 IIIB 3B 4 IVB 4B 5 VB 5B 6 VIB 6B 7 VIIB 7B ------- VIII ----- -- ------- 8 ------- 11 IB 1B 1 IIB B 13 Al 6.98 14 S 8.09 15 P 30.97 16 S 3.07 17 Cl 35.45 18 A 39.95 4 19 K 39.10 0 Ca 40.08 1 Sc 44.96 T 47.88 3 V 50.94 4 C 5.00 5 Mn 54.94 6 Fe 55.85 7 Co 58.47 8 N 58.69 9 Cu 63.55 30 Zn 65.39 31 Ga 69.7 3 Ge 7.59 33 As 74.9 34 Se 78.96 35 B 79.90 36 K 83.80 5 37 Rb 85.47 38 S 87.6 39 Y 88.91 40 Z 91. 41 Nb 9.91 4 Mo 95.94 43 Tc (98) 44 Ru 101.1 45 Rh 10.9 46 Pd 106.4 47 Ag 107.9 48 Cd 11.4 49 In 114.8 50 Sn 118.7 51 Sb 11.8 5 Te 17.6 53 I 16.9 54 Xe 131.3 6 55 Cs 13.9 56 Ba 137.3 57 La* 138.9 7 Hf 178.5 73 Ta 180.9 74 W 183.9 75 Re 186. 76 Os 190. 77 I 190. 78 Pt 195.1 79 Au 197.0 80 Hg 00.5 81 Tl 04.4 8 Pb 07. 83 B 09.0 84 Po (10) 85 At (10) 86 Rn () 7 87 F (3) 88 Ra (6) 89 Ac~ (7) 104 Rf (57) 105 Db (60) 106 Sg (63) 107 Bh (6) 108 Hs (65) 109 Mt (66) 110 --- () 111 --- () 11 --- () 114 --- () 116 --- () 118 --- () Lanthande Sees* 58 Ce 140.1 59 P 140.9 60 Nd 144. 61 Pm (147) 6 Sm 150.4 63 Eu 15.0 64 Gd 157.3 65 Tb 158.9 66 Dy 16.5 67 Ho 164.9 68 E 167.3 69 Tm 168.9 70 Yb 173.0 71 Lu 175.0 Actnde Sees~ 90 Th 3.0 91 Pa (31) 9 U (38) 93 Np (37) 94 Pu (4) 95 Am (43) 96 Cm (47) 97 Bk (47) 98 Cf (49) 99 Es (54) 100 Fm (53) 101 Md (56) 10 No (54) 103 L (57)
Hgh-pefomance computng ~ DFT to cove all One-to-one Coespondence Extenal potental Mnmum-enegy state Ψ Densty ρ( ) V ( ) Gound state Ψ V v-epesentatve densty ρ V ( )
The followng vaaton leads to all the gound-state popetes. δ { F[ ρ] ρ( ) v( ) d µ ( ρ( ) d N )} = 0 + In pncple, any physcal quantty of the gound state should be a functonal of densty. Vaaton wth espect to many-body wave functons Vaaton wth espect to one-body densty ρ( ) Physcal quantty A[ ρ( )] = Ψ[ ρ] Aˆ Ψ[ ρ] Ψ( 1, L, N ) Real nteactng system Kohn-Sham Scheme V ( ) Gound state Ψ V densty ρ( ) Vtual non-nteactng system V s ( ) Gound state Ψ S densty ρ( )
ρ Kohn-Sham scheme ( ) φ ( ) Ψ = det{ φ ( )} = S j h φ + v S = m [ ρ] φ ε φ KS canoncal equaton Densty functonal F [ ρ( )] = T [ ρ( )] + ( F[ ρ( )] T [ ρ( )]) = S p φ φ + V m eff S [ ρ( )] V eff [ ρ( )] Mnmzaton of ths densty functonal leads to v S [ ρ]( ) δveff = δρ ( ) S. Goely et al., ENAM 04 Nuclea DFT Global popetes, global calculatons M. Stotsov et al. * Global DFT mass calculatons: HFB mass fomula: m~700kev Takng advantage of hgh-pefomance computes
One-to-one Coespondence Extenal potental Tme-dependent state statng fom the ntal state Ψ( t 0 ) Tme-dependent densty V (, TD state Ψ t ( ) V ( t ) v-epesentatve densty ρ ( V, Real nteactng system TD Kohn-Sham Scheme V (, TD state Ψ( V TD densty ρ (, Vtual non-nteactng system V s (, TD state Ψ( S TD densty ρ (,
Skyme TDDFT n eal space Tme-dependent Kohn-Sham equaton t ψ ( στ, t ) = HF + ex t t 3D space s dscetzed n lattce Sngle-patcle obtal: η ~ ( h [ ρ, τ, j, s, J ]( t ) V ( t )) ψ ( στ, n= 1, LMt ϕ (, = { ϕ ( k, tn)} k = 1, L M, = 1, L, N ( ) y[ fm ] N: Numbe of patcles M: Numbe of mesh ponts Mt: Numbe of tme slces Spatal mesh sze s about 1 fm. Tme step s about 0. fm/c X [ fm ] Nakatsukasa, Yabana, Phys. Rev. C71 (005) 04301 Real-tme calculaton of esponse functons 1. Weak nstantaneous extenal petubaton. Calculate tme evoluton of 3. Foue tansfom to enegy doman db( ω; Fˆ ) dω V ext ( = Fˆ δ ( Ψ( Fˆ Ψ( 1 = Im t F t e π Ψ( ) ˆ Ψ( ) ωt 0 1 3 t [ /MeV ] dt Ψ( Fˆ Ψ( db ( ω; F ˆ ) dω [ MeV ]
Neutons 16 O n ( ) n δρ ( ρ = ρn( 0 Tme-dep. tanston densty > 0 < 0 p ( ) p δρ ( ρ = ρ p ( 0 Potons 18 O 16 O Polate 10 0 30 40
4 Mg Polate 6 Mg Taxal 10 0 30 40 10 0 30 40 8 S 30 S Oblate Oblate 10 0 30 40 10 0 30 40
40 A Oblate 10 0 30 40 44 Ca Polate 40 Ca 48 Ca 10 0 30 10 0 30 40 10 0 30 40
Cal. vs. Exp. Electc dpole stengths Z N SkM* R box = 15 fm Γ = 1 MeV Numecal calculatons by T.Inakua (Unv. of Tsukuba)
Few-body-model calculaton of fuson coss secton Real-tme, eal-space appoach No need fo scatteng bounday condton Altenatve method to the CDCC Wave packet dynamcs of fuson eacton potental scatteng wth absopton nsde a Coulomb bae Radal Schoednge equaton fo l=0 d u, t, h h m d wth ncdent Gaussan wave packet u ( = + V () + W () u( [ ] (, t ) = k γ ( ) 0 exp 0 10Be-08Pb (A,Z=10,4 and 08,8) V0=-50 W0=-10, RV=1.6,RW=1.15, AV=0.44, AW=0.45 E_nc=8 MeV (+Coulomb at R_0), R_0=40fm, gamma=0.1fm- N=400, d=0.5, Nt=10000, dt=0.001 10 Be 08 Pb V ( ) Flux absobed by W() epesents fuson. W ( ) Wave packet dynamcs nclude scatteng nfomaton fo wde enegy egon. Then, how to extact eacton nfomaton fo a fxed enegy?
Fuson pobablty P fuson P ( E) = nt ( E) Pfnal ( E) P ( E) nt dffeental. eq. (statc cal) wave packet method Fuson pobablty fo whole bae egon fom sngle wave-packet calculaton. No bounday condton equed n the wave packet calculaton. Fuson pobablty of thee-body eacton h ψ t h h ( R,, = R + VnC ( nc ) + VCT ( CT ) + VnT ( nt ) ψ ( R,, µ J ul ( ) ( R,, R, t = m ( ) ψ J, Pl cosθ R l Intal ncdent wave Coulomb + Nuclea potental Absopton => C-T fuson Tansfe Elastc Flux loss by absopton FUSION (Complete + Incomplete) Beakup
P fuson ( E) P = ( E) Pf ( E) P ( E) (nc)-t 3-body n C R T C-T -body Enhancement of fuson pobablty at sub-bae eneges Case (): Weakly-bound pojectle (Neuton-halo) y n-c obtal enegy: -0.6 MeV (Halo) neuton 11 Be(n+ 10 Be)- 08 Pb head-on collson (J=0) R Coe Taget x ρ ( R,, = d( cosθ ) ψ ( R,, θ, ρ (, θ, = drψ ( R,, θ, R y x
Fuson pobablty of neuton-halo nucle s suppessed -body no V nt wth V nt Coe ncdent enegy deceases effectvely by neuton beakup E coe M coe M + M coe n E pojectle C n T Why dffeent fom othe studes? l l 70 Conclusons of othe studes Quantum calculatons have been done usng the dscetzed contnuum channels. Hagno et al, PRC61 (000) 03760 Daz-Toes & Thompson, PRC65 (00) 04606 Fuson was enhanced wth a weakly-bound neuton at sub-bae eneges Nuclea couplng was mpotant fo an the fuson enhancement 10 Be n R 08 Pb We need to nclude hgh-patal waves fo n- 10 Be motons. The low-patal-wave tuncaton leads to an opposte concluson!
Fuson Coss Secton of 11 Be Thee body full calculaton of 11 Be + 09 B Fuson pobablty s hndeed by the pesence of the halo neuton 10 3 10 Be + 09 B Fuson coss secton ( mb ) 10 11 Be + 09 B neuton 10 Be R 09 B 10 1 36 40 44 48 E c.m. ( MeV ) Expement C. Sgnon et.al, Nucl. Phys. 735 (004) 39. Theoy M. Ito, M. Ueda, T. Nakatsukasa, K. Yabana, Phys. Lett. B 637, 53(006) Summay DFT/TDDFT Systematc calculatons fo all nucle ncludng those fa fom the stablty lne Descpton of lage ampltude dynamcs, such as fsson Real-tme, eal-space appoach to few-body models Accuate few-body scatteng dynamcs An altenatve appoach to CDCC