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An exact microscopic multiphonon approach to nuclear spectroscopy

An exact microscopic multiphonon approach to nuclear spectroscopy. N. Lo Iudice Universit à di Napoli Federico II Naples (Andreozzi, Lo Iudice, Porrino) Prague (Knapp, Kvasil) Collaboration Tokyo 07. Eigenvalue problem in a multiphonon space. H | Ψ ν > = E ν | Ψ ν >

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An exact microscopic multiphonon approach to nuclear spectroscopy

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  1. An exact microscopic multiphonon approach to nuclear spectroscopy N. Lo Iudice Università di Napoli Federico II Naples(Andreozzi, Lo Iudice, Porrino) Prague (Knapp, Kvasil) Collaboration Tokyo 07

  2. Eigenvalue problem in a multiphonon space H | Ψν > = Eν| Ψν > | Ψν > H= Σn Hn (n= 0,1.....N ) where Hn ∋|n, α > ~ | ν1 ν2 …..νn > | νi > = Σph cph(νi ) a†p ah |0> First goal: Generate the basis states |n, α > We do it byconstructing a setof equations of motionand solvingthemiteratively

  3. EOM: Construction of the Equations • Crucialingredient <n; β |[H, a†p ah]| n-1; α> • Preliminary step: Derive <n; β |[H, a†p ah]| n-1; α> = ( Eβ(n) - Eα(n-1))<n; β |a†p ah| n-1; α> (LHS) (RHS) TheRHS comes from * property <n; β|a†pah|n’; γ > = δn’,n-1< n; β |a†pah| n-1; γ > ** request < n; β|H|n; α > = Eα(n)δαβ

  4. Equations of Motion: LHS Commutator expansion <n; β| [H, a†p ah] | n-1; α>= (εp- εh) <n; β |a†p ah| n-1; α>. + 1/2 Σijp’ Vhjpk<n; β|a†p’ aha†i aj| n-1; α > + … Linearization <n; β| [H, a†p ah] | n-1; α >= Σp’h’γ[(εp- εh) δγβ+ 1/2 Σij Vhjp’k’ <n-1; γ|a†i aj| n-1; α> + ………]<n; β|a†p’ ah’| n-1; γ> = Σp’h’γAαγ(n)(ph;p’h’)<n; β|a†p’ ah’| n-1; γ> Î = Σγ|n-1; γ><n-1; γ|

  5. LHS=RHS AX = EX jγAαγ(n)(ij) Xγβ(n)(j)= (Eβ(n)- Eα(n1)) Xαβ(n)(i) where Xαβ(n)(i ) = < n; β |a†p ah| n-1; α> Aαγ(n)(ij)= [εp–εh]δij (n-1)δαβ(n-1) + [VPHρH+VHPρP +VPPρP + VHHρH ]αiβj ρH≡{< n,γ|a†hah’|n,α>} ρP≡ {< n,γ|a†pap’|n,α>} n =1 (ρP= 0ρH=δhh’) A(1)Xα(1) = (Eα(1) - E0 (0)) Xα(1) Tamm-Dancoff A(1)(ij)= δij[εp–εh] + V(p’hh’p)

  6. Structure of multiphonon states: Overcompletness The eigenvalue equations yield states of the structure |n; β> =Σαphcαph a†p ah| n-1; α > • Problem: The multiphononstates are not fully antysymmetrized !!! a†p ah| n-1; α > ≡  p h p h Themultiphononstates form an overcompleteset

  7. Solution of the redundancy γAαγ(n)Xγβ(n)= (Eβ(n) - Eα(n-1 ))Xαβ(n) A X = E X Reminder Insert |n; β> =Σα phCαph a†p ah| n-1; α > Xαβ(n)(ph)= < n; β | a†p ah| n-1; α > X = DC(AD)C= H C= E DC where Dij= < n-1; α’| ah’a†p’ a†p ah| n-1; α> overlapormetric matrix Problem due to redundancy Det D = 0

  8. Solution of the redundancy *Removal ofredundancy: Choleski decomposition(no diagonalization) DĎ ** Matrix inversion Exact eigenvectors |n; β>=Σ αphCαph a†p ah|n-1; α>  H n(phys) ● Now compute in Hn i. X(n) • ρ(n) recursive formula HC = (Ď-1AD)C = E C X=D C ρ(n) = C X (n)+ Cρ(n-1)X (n)

  9. Iterative generation of phonon basis Starting point|0> Solve Ĥ(1)C(1) = E(1)C(1) |n=1, α> X(1) ρ(1) Solve Ĥ(2)C (2) = E (2)C (2) |n=2,α> X(2) ρ(2) ……… X(n-1) ρ(n-1) Solve Ĥ(n)C (n) = E (n)C (n) X(n) ρ(n) |n,α.> The multiphonon basis is generated !!!

  10. H: Spectral decomposition, diagonalization H= Σ nαE α(n)|n; α><n;α| + (diagonal) + Σnα β|n; α><n;α| H |n’;β><n’;β| (off-diagonal) n’= n ±1, n±2 Off-diagonalterms: Recursive formulas < n; α | H| n-1; β > = Σphγϑαγ(n-1)(ph)Xγβ(n)(ph) < n; α | H| n-2; β > =ΣV pp’hh’Xγβ(n) (ph)Xγβ(n-1) (p’h’) Outcome of diagonalization H |Ψν> = Eν|Ψν> |Ψν> = Σnα Cα(ν) (n) | n;α> |n;α> = Σγ Cγ(n) | n-1;γ>

  11. Numericaltest: A = 16 ∙ Calculation up to 3-phonons and 3ħω Hamiltonian H = H0 + V = Σi hNils(i)+ Gbare ( VBonnA ⇨Gbare) • CM motion (F. Palumbo Nucl. Phys. 99 (1967)) H H+Hg Hg= g [P2/(2Am) + (½) mA ω2 R2] • Consistent choice of ph space: It must includes all ph configurations up to 3ħω

  12. Ground state |Ψ0> = C(0)0 |0> + ΣλCλ(0) |λ, 0> + Σλ1λ2Cλ1λ2(0) |λ1λ2, 0> |λ, 0> |λ1λ2, 0> 1 = < Ψ0|Ψ0> = P0 + P1 + P2

  13. 16O negative parity spectrum • Up to three phonons

  14. IVGDR |1->IV ~ |1(p-h) (1ħω)>

  15. ISGDR |1->IS ~ |1(p-h) (3 ħω)> + |2(p-h) (1ħω + 2ħω)> + |3(p-h) (1ħω )> Toroidal

  16. Octupole modes |3->IS ~ |1(p-h) (3 ħω)> + |2(p-h) (1ħω + 2ħω)> + |3(p-h) (1ħω )> Low-lying

  17. Effect of CM motion

  18. Effect of the CM motion

  19. Concluding remarks • The multiphonon eigenvalue equations - have a simple structure • yield exact eigensolutions of a general H • The 16O test shows that • an exact calculation in the full multiphonon space is feasible at least up to 3 phonons and 3 ħω. • To go beyond • Truncation of the space needed!!! • Truncation is feasible (the phonon states are correlated). • Ariformulation for an efficient truncation is in progress

  20. THANK YOU

  21. E2 response up to 3 ħω: Running sum Sn = Σn(En – E0(0) )Bn(E2,0→2+n) Rn= Sn/SEW(E2) • It is necessary to enlarge the space!!

  22. E2 response up to 3 ħω S(ω,E2) = ΣnBn(E2,0→2+n)ρΔ(ω-ωn) ρΔ(x) =(Δ/2π) / [x2 + (Δ/2)2] M(E2μ) = Σ(p)ep rp2 Y 2μ

  23. Effect of CM on the E2 response

  24. Growing evidence of multiphonon excitations * Low-energy M. Kneissl. H.H. Pitz, and A. Zilges, Prog. Part. Nucl. Phys. 37, 439 (1996); M. Kneissl. N. Pietralla, and A. Zilges, J.Phys. G, 32, R217 (2006) : • Two- and three-phonon multiplets Q2×Q3|0>, Q2×Q2×Q3|0> • Proton-neutron (F-spin) mixed-symmetry states (N. Pietralla et al. PRL 83, 1303 (1999)) [Q2(p)- Q2(n)](Q2(p)+ Q2(n))N|0>, ** High-energy (N. Frascaria, NP A482, 245c(1988); T. Auman, P.F. Bortignon, H. Hemling, Ann. Rev. Nucl. Part. Sc. 48, 351 (1998)) • Double and (maybe) tripledipole giant resonances D×D|0>

  25. From TDA to RPA A(n) B(n) X(n) = (E(n) - E(n-1)) B(n) A(n)-Y(n)

  26. Aαγ(n)(ph;p’h’)=δhh’δpp’δαγ(n-1)[εp–εh] + Σh1V(p’h1h’p) ραγ(n-1) (h1h) + Σp1V(p’hp1h’) ραγ(n-1) (pp1) + δhh’ [ Σp2p3V(p’p2pp3)ραγ(n-1) (p2p3) + Σh2h3V(p’h2ph3)ραγ(n-1)(h2h3)] + δpp’ [ Σh2h3V(hh2h’h3)ραγ(n-1)(h2h3) + Σp2p3V(hp2h’p3)ραγ(n-1)(p2p3)] Bαγ(n) (ph;p’h’)= Σh2V(p p’h2 h’)ραγ(n-1)(h2h) + Σp2V(h’hp2p’)ραγ(n-1)(pp2)

  27. AX = EX where Aαγ(n)(ph;p’h’)=δhh’δpp’δαγ(n-1)[εp–εh] + Σh1V(p’h1h’p) ραγ(n-1)(h1h) + Σp1V(p’hp1h’) ραγ(n-1)(pp1) + δhh’1/2Σp2p3V(p’p2pp3) ραγ(n-1)(p2p3) + δpp’1/2Σh2h3V(hh2h’h3) ραγ(n-1)(h2h3) (ραγ(n)(ij) = <n,α|a†iaj|n,α>)

  28. Any real non negative definite symmetric matrix can be written as D = L LT ( L{lij} lower triangular matrix) Det{D} = (Det{L})2 =l112 l222 ...lii2….. = λ21 …λ2i… (D | λi > = λi | λi > ) Definition of L: Recursive formulas l211 = d11 l11 lj1 = dj1 j=2,….,n l2ii = dii – Σk=1,i-1 l2ik lii lji = dji – Σk=1,i-1 lik ljk (It reminds the Schimdt orthogonalization method) The decomposition goes on until lnn = 0 lnn = 0 → Det{L} =0→Det{D} =0 ⇨ ⇨ | λn > (D | λi > = λi | λi > ) linearly dependent ⇨ to be discarded For numerical stability we need maximum overlap λj ≤ λi j > i Sequence order lii ≤ ljj j > i Oncelnn = 0 → lii = 0 i >n 1) We can stop at the nth step 2) We getĎ (Nn < Nr)withmaximum determinant(overlaps) Choleski decomposition

  29. HSM H = HSM+Hg Hg= g [P2/(2Am) + (½) mA ω2 R2] = g/A [Σi hi + Σi<j vij] hi=pi2/(2m) + (1/2) m(Aω)2ri2 vij=(1/m) pi· pj + m (Aω)2ri · rj Hg effective only in Jπ = 1-ph channel. HΨ = (HSM+Hg)Ψ =EΨ In the full space Ψ = ψintΦnCM For the physical states Ψn = ψn Φ0CM with CM energy E0CM= 3/2 g ħω For the spurious ones Ψn(1) = ψnΦ1CM at the very high CM energy E1CM- E0CM= g ħω They can therefore be tagged and eliminated Elimination of CM spuriosityF. Palumbo Nucl. Phys. 99 (1967)

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