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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 > - PowerPoint PPT Presentation

### Transcript of An exact microscopic multiphonon approach to nuclear spectroscopy

• An exact microscopic multiphonon approach to nuclear spectroscopyN. Lo IudiceUniversit di Napoli Federico II

Naples(Andreozzi, Lo Iudice, Porrino) Prague (Knapp, Kvasil) Collaboration

Tokyo 07

• Eigenvalue problem in a multiphonon space

H | > = E | > | > H = n Hn ( n= 0,1.....N )where Hn |n, > ~ | 1 2 ..n > | i > = ph c ph(i ) ap ah |0>

First goal: Generate the basis states |n, >

We do it by constructing a set of equations of motion and solving them iteratively

• EOM: Construction of the Equations Crucial ingredient < n; | [H, ap ah] | n-1; >Preliminary step: Derive< n; | [H, ap ah] | n-1; > = ( E(n) - E(n-1)) < n; | ap ah| n-1; > (LHS) (RHS)The RHS comes from* property < n; | apah | n; > = n,n-1 < n; | apah | n-1; > ** request < n; | H | n; > = E (n)

• Equations of Motion : LHSCommutator expansion

< n; | [H, ap ah] | n-1; > = (p- h) < n; | ap ah| n-1; >. + 1/2 ijp Vhjpk < n; | ap ah ai aj | n-1; > + Linearization

< n; | [H, ap ah] | n-1; > = ph [(p- h) + 1/2 ij Vhjpk < n-1; | ai aj | n-1; > + ] < n; | ap ah | n-1; >

= ph A(n)(ph;ph) < n; | ap ah | n-1; >

= |n-1; >< n-1; |

• LHS=RHS AX = EX

j A(n) ( ij) X(n)(j) = (E(n) - E(n1)) X(n)(i) where X(n) (i ) = < n; | ap ah| n-1; > A(n) ( ij) = [ph] ij (n-1) (n-1) + [VPHH + VHPP + VPPP + VHHH ]ij H {< n,|ahah|n,>} P {< n,|apap|n,>} n =1 (P = 0 H = hh)

A(1) X(1) = (E(1) - E0 (0) ) X(1) Tamm-Dancoff A(1) (ij)= ij[ph] + V(phhp)

• Structure of multiphonon states: OvercompletnessThe eigenvalue equations yield states of the structure |n; > = ph cph ap ah | n-1; >

Problem: The multiphonon states are not fully antysymmetrized !!! ap ah | n-1; > p h p h

The multiphonon states form an overcomplete set

• Solution of the redundancy Reminder

Insert |n; > = ph Cph ap ah | n-1; > X(n)(ph) = < n; | ap ah | n-1; >

X = DC (AD)C = H C = E DC

where Dij = < n-1; | ah ap ap ah | n-1; > overlap or metric matrix

Problem due to redundancy Det D = 0 A(n) X(n)= (E(n) - E(n-1 ) )X(n) A X = E X

• Solution of the redundancy *Removal of redundancy: Choleski decomposition(no diagonalization)

D ** Matrix inversion Exact eigenvectors |n; > = ph Cph ap ah | n-1; > H n (phys)

Now compute in Hn

i. X(n) (n) recursive formulaX= D C(n) = C X (n) + C (n-1) X (n)HC = (-1AD)C = E C

• Iterative generation of phonon basis Starting point |0> Solve (1) C(1) = E(1) C(1) |n=1, > X(1) (1)

Solve (2) C (2) = E (2) C (2) |n=2,> X(2) (2) X(n-1) (n-1)

Solve (n) C (n) = E (n) C (n)

X(n) (n) |n,.>

The multiphonon basis is generated !!!

• H: Spectral decomposition, diagonalization H = n E (n) |n; > = ph (n-1)(ph) X(n)(ph)

< n; | H| n-2; > = V pphh X(n) (ph) X(n-1) (ph)

Outcome of diagonalization H |> = E |>

|> = n C() (n) | n;> |n;> = C(n) | n-1;>

• Numerical test: A = 16 Calculation up to 3-phonons and 3

Hamiltonian H = H0 + V = i hNils(i)+ Gbare ( VBonnA Gbare)

CM motion (F. Palumbo Nucl. Phys. 99 (1967))

H H + Hg

Hg = g [ P2/(2Am) + () mA 2 R2 ] Consistent choice of ph space: It must includes all ph configurations up to 3

• Ground state|0> = C(0)0 |0>

+ C(0) |, 0>

+ 12 C12 (0) | 1 2, 0 >

|, 0> | 1 2, 0 >

1 = < 0|0> = P0 + P1 + P2

• 16O negative parity spectrum Up to three phonons

• IVGDR |1->IV ~ |1(p-h) (1)>

• ISGDR |1->IS ~ |1(p-h) (3 )> + |2(p-h) (1 + 2)> + |3(p-h) (1 )>

Toroidal

• Octupole modes |3->IS ~ |1(p-h) (3 )> + |2(p-h) (1 + 2)> + |3(p-h) (1 )>

Low-lying

• Effect of CM motion

• Effect of the CM motion

• Concluding remarksThe 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 .

To go beyond Truncation of the space needed !!! Truncation is feasible (the phonon states are correlated).

A riformulation for an efficient truncation is in progress

• THANK YOU

• E2 response up to 3 : Running sumSn = n (En E0(0) ) Bn(E2,02+n)Rn = Sn/SEW(E2)It is necessary to enlarge the space!!

• E2 response up to 3

S(,E2) = n Bn(E2,02+n) (-n)(x) = (/2) / [x2 + (/2)2]M(E2) = (p)ep rp2 Y 2

• Effect of CM on the E2 response

• 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>, Q2Q2Q3|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) triple dipole giant resonances