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A new equation of motion method for multiphonon nuclear spectra. N. Lo Iudice Universit à di Napoli Federico II Kazimierz08. Acknowledgments. J. Kvasil , F. Knapp, P. Vesely (Prague) F. Andreozzi , A. Porrino (Napoli) Also Ch. Stoyanov (Sofia) A.V. Sushkov ( Dubna ).

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A new equation of motion method for multiphonon nuclear spectra

A newequationofmotionmethodformultiphononnuclearspectra.

N. Lo Iudice

Università di Napoli Federico II

Kazimierz08


Acknowledgments
Acknowledgments

  • J. Kvasil, F. Knapp, P. Vesely (Prague)

  • F. Andreozzi, A. Porrino (Napoli)

  • Also

    Ch. Stoyanov (Sofia)

    A.V. Sushkov (Dubna)


From mean field to multiphonon approaches
From mean field to multiphonon approaches

Anharmonic features of nuclear spectra:

Experimental evidence of multiphonon excitations

Necessity of going beyondmean field approaches

A successful microscopic(QPM) multiphononapproach

A new (in principle exact) multiphonon method


Collective modes anharmonic features
Collective modes: anharmonic features

Mean field:Landau damping

  • Beyond mean field:

  • * Spreading width

  • * *Multiphon excitations

    - 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 tripledipolegiantresonances

    - 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-phononmultiplets

  • Proton-neutronmixed-symmetrystates

    (N. Pietrallaet al. PRL 83, 1303 (1999))


A microscopicmultiphononapproach: QPMSoloviev, TheoryofAtomic Nuclei: Quasiparticles and Phonons, Bristol, (1992)

H = Hsp + Vpair + Vpp + Vff

H[(a†a), (a†a†),(aa)]  H[(α†α ),(α†α†),(αα ) ]

α†α†O†λ

αα Oλ

Oλ†=Σkl [Xkl(λ)α†k α†l–Ykl(λ)αkαl]

HQPM= Σnλωn (λ)Q†λ Qλ + Hvq

Ψν= ΣncnQ†ν(n) |0> + ΣijCijQ† (i) Q†(j) |0>

+ ΣijkCijkQ†(i) Q†(j) Q†(k)|0>


Mixedsymmetry

Symmetric

|n, ν>s = QSn |0 >

= (Qp + Qn)n |0 >

MS

|n, ν>MS = QAQS (n-1) |0 >

= (Qp - Qn) (Qp + Qn) (n-1) |0 >

Signature

Preserving symmetry transitions

M(E2)  QS n → n-1

Changing symmetry transitions

M(M1)  Jn – Jp n → n

π-νMixedSymmetry

E2

n=3

E2

M1

n=2

E2

n=2

M1

E2

n=1

MS

n=1

J+1

E2

M1

J

J-1

Sym

J

Scissors multiplet

S | n,J> = (Jp – Jn) |nJ>

= ΣJ’ |n J’> <nJ’| S|nj>

Bsc(M1) = ΣJ’ |<nJ’|M(M1)|nJ>|2

~ 1.5 - 2 μN2


A qpm calculation n 90 isotones n l ch stoyanov d tarpanov prc 77 044310 2008
A QPM calculation: N=90 isotonesN. L, Ch. Stoyanov, D.Tarpanov, PRC 77, 044310 (2008)

E2

M1


4 state in os isotopes n l and a v sushkov submitted to prc
4+ state in OsisotopesN. L. and A. V. Sushkovsubmittedto PRC

4+

Hexadecapoleone-phonon?

Ψ  |n=1,4+>

E4

S(t,α)

Double-g ?

 |> E2

E4 2 Eg E2

R4(E2) = B(E2,4+→ 2+)/B(E2,2+ → 0+)

2

4+

2g+

2g+

0+

0+

QPM

Ψ  0.60 |n=1,4+> + 0.35 |>


4 qpm versus exp
4+: QPM versus EXP


Successes and limitations of the qpm
Successes and limitations of theQPM

Successes

Itisfullymicroscopicand valid atlowand highenergy.

includingthe double GDR(Ponomarev, Voronov)

Limitations:

Valid for separable interactions

Antisymmetrization enforced in the quasi-boson approximation,

Ground state not explicitly correlated (QBA)

Temptativeimprovements

MultistepShellmodel (MSM) (R.J.Liotta and C. Pomar, Nucl. Phys. A382, 1 (1982))

Theyexpand and linearize

<α|[[H, O†],O†]|0> ( O†= ΣphX(ph)a†pah )

Multiphononmodel (MPM)

(M. Grinberg, R. Piepenbringet al. Nucl. Phys. A597, 355 (1996)

Along the samelines

BothMSM and (especially) MPM look involved


A new exact multiphonon approach
A new (exact) multiphonon approach

Eigenvalue problem in a multiphonon space

|Ψν > H = Σn HnHn |n; β> ( n= 0,1.....N )

H | Ψν > = Eν| Ψν >

Generationof the|n; β> (basis states)

An obvious (but prohibitive!!) choice

|n; β> = | ν1, ν2,… νi ,…νn>

where (TDA) | νi > = Σphcph(νi )a†p ah|0>

A workable choice|n=1; β> = | νi > = Σphcph(νi )a†p ah|0> ( TDA)

|n; β> =ΣαphC(n)αpha†p ah| n-1; α >


Eom construction of the equations
EOM: Construction of the Equations

<n; β |[H, a†p ah]| n-1; α>

Crucialingredient

Preliminary step:

<n; β |[H, a†p ah]| n-1; α> = ( Eβ(n) – Eα(n-1))<n; β |a†p ah| n-1; α >

(LHS) (RHS)

It follows from

* request

< n; β|H|n; α > = Eα(n)δαβ

** property

<n; β|a†pah|n’; γ > = δn’,n-1< n; β |a†pah| n-1; γ >


Equations of motion lhs
Equations of Motion: LHS

Commutatorexpansion

< n; β | [H, a†p ah] | n-1; α >=

(εp- εh) <n; β |a†p ah| n-1; α >. + linear

1/2 Σijp’Vhjpk<n; β|a†p’ aha†iaj| n-1; α > + not linear

Î = Σγ|n-1; γ >< n-1; γ|

Linearization

< n; β | [H, a†p ah] | n-1; α >=

= Σp’h’γ Aαγ(n)(ph;p’h’)<n; β|a†p’ ah’| n-1; γ>


Lhs rhs
LHS=RHS

A(n ) X(n)= E(n)X(n)

Aαγ(n) (ij)= [(εp–εh ) + Eα(n-1) ] δij(n-1)δαβ(n-1)+ [VPHρH+ VHPρP+VPPρP+ VHHρH]αiβj

Xα(β) (i ) = < n; β |a†p ah| n-1; α>

ρH ≡{< n-1,γ|a†hah’|n-1,α>}

ρP ≡ {< n-1,γ|a†pap’|n-1,α>}

n =1 TDA A(1)X (1) = E (1)X (1)

A(1)(ij) = δij[(εp–εh )+ E(0)] + V(p’hh’p)

n=1 |n=1; β> = | νi > = Σphcpha†p ah|0>

n> 1 |n; β> =ΣαiCα(β) (i ) a†pah| n-1; α >

n=1 C = X

n> 1 X= DC

Dαα’ (ij) = < n-1; α’|(a†h’ap’)( a†p ah)| n-1; α> overlap matrix


General eigenvalue problem
General Eigenvalue Problem

A(n)X(n)= E(n)X(n)

X(n)=D(n)C(n)

(AD)C= H C= E DC

Eigenvalue

Equation

Dαα’ (ij) = < n-1; α’|(a†h’ap’)( a†p ah)| n-1; α> overlap matrix

Problem i)how to compute D

Problem ii) redundancyDetD = 0


General eigenvalue problem1
General Eigenvalue Problem

(AD)C= H C= E DC

  • Solution of problem i)

Aαγ(n) (ij)= [εp–εh + Eα(n-1)]δij(n-1)δαβ(n-1)

+ [VPHρH+ VHPρP+VPPρP+ VHHρH]αiβj

Dαα’ (ij) = < n-1; α’|(a†h’ap’)( a†p ah)| n-1; α> overlap matrix

D = ρH (n-1) – ρP (n-1)ρH (n-1)

ρP(n-1) = C (n-1) X (n-1) – C (n-1) X (n-1)ρP (n-2)

recursive

relations

Problem i) solved!!!!


General eigenvalue problem2
General Eigenvalue Problem

(AD)C= H C= E DC

Solution of Problem ii) (redundancy)

*Choleski decomposition

** Matrixinversion

Exact eigenvectors

HC = (Ď-1AD)C = E C

|n; β>=ΣαphCα(β) (i )a†p ah |n-1; α>  Hn (phys)


Iterative generation of phonon basis
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 !!!


H spectral decomposition diagonalization
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; β | H| n-1; α> = Σphγϑαγ(n-1) (ph)Xγ(β) (ph)

< n; β | H| n-2; α > =Σ V pp’hh’Xγ(β) (ph) Xγ(α) (p’h’)

|Ψν> = Σnα Cα(ν) (n) | n;α>

|n;α> = Σγ Cγ(α)a†p ah | n-1;γ>

Outcome

H |Ψν> = Eν|Ψν>


E m response

C3 | λ1 λ2 λ3>

+

C1 | λ1>

C2 | λ1 λ2>

+

C0 |0>

Ψ0Ψν(n)

EMPM

E.m. response

W.F.

|Ψν> = Σn{λ} C{λ}(ν) (n) | n;{λ1λ2.λn }>

|λ> = Σph cph (λ) a†p ah|0>

e.m. operator

Мλμ= rλ Yλμ

Strength Function

S(Eλ) = Σ Bn (Eλ) δ(E- En)

Bn (Eλ) =|<Ψnν|| Мλ ||Ψ0>|2


16 o tda cm free response and sm space dimensions
16O: TDA (CM free) response and SMspacedimensions


Empm exact implementation in 16 o for n 4
EMPM : Exact implementation in 16O for N=4

SM space

All particle-hole (p-h)

configurations up to3ħω

(2s,1d,0g)

(1p,0f)

(1s,0d)

0p

0s

Free ofCM

spuriousadmixtures


Mean field versus empm e response
Meanfield versus EMPM E response


New running sums
NEW runningsums


Ew running sums
EW runningsums


Cm motion
CM motion

Hamiltonian

H = H0 + V = ΣihNils(i)+ Gbare ( VBonnA  Gbare)

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

H H+Hg

Hg= g [P2/(2Am) + (½) mAω2 R2]

For g>>1 E CM >> Eintr


Cm motion in tda isoscalar e1
CMmotion in TDA: Isoscalar E1


Cm motion in empm
CMmotion in EMPM



Perspectives new formulation
Perspectives: New formulation

< n; β |[H, a†p ah]| n-1; α>  < n; β |[H, O†λ]| n-1; α>

O†λ= Σph cph(λ )a†pah

λ’γAαγ(n) (λ,λ’) X(β)γ λ’= Eβ(n)X (β)γ λ

X(β)αλ= < n; β |O†λ| n-1; α >

Aαγ(n) (λ,λ’) = [ Eλ + Eα(n-1) ] δλλ’δαγ+ ρλλ’Vραγ(n-1)

ρλλ’ (kl)≡<λ’| a†kal|λ>ραγ(n-1) (kl) = < n-1,γ| a†kal|n-1,α>

|n; β> =ΣαλC(β)αλO†λ| n-1; α > = ΣC(β) {λi} |λ1,.…λi….λN >

C (β) {λi} = Σ C(β)αλ1C(α)γλ2C (γ)δλ3


Vertices
Vertices

Aαγ(n) (λ,λ’) = [ Eλ + Eα(n-1) ] δλλ’δαγ+ ρλλ’Vραγ(n-1)

TDA (n=1) MPEM (n=3)

γ

--------

---------

p’

λ’

h’

p

h

λ

α

Vph’hp’

ρλλ’Vραγ(n-1)



Ew sum rule
EW sum rule

SEW (E ) = ½ Σμ <[M (E μ),[H, M (E μ)]>

= [(2  + 1)2/16π ](ħ2/2m) A < r2  -2>

SEW (E 1, τ = 0) = [(2  + 1)2]/16π (ħ2/2m)

x A[11< r4> -10 R2<r2> + 3 R4]


Ground state
Ground state

|Ψ0> = C0(0) |0>

+ ΣλCλ(0) |λ, 0>

+ Σλ1λ2Cλ1λ2(0) |λ1 λ2, 0>

1 = < Ψ0|Ψ0> = P0 + P1 + P2


16 o negative parity spectrum
16O negative parity spectrum

  • Up to three phonons


Ivgdr
IVGDR

Мλμ= τ3 r Y1μ≈ Rπ - Rν

TDA

|1->IV ~ |1 (p-h) (1ħω)>(TDA)


Isgdr
ISGDR

Мλμ= r Y1μ≈ RCM !!!

Мλμ= r3 Y1μ

Toroidal

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


Octupole modes
Octupole modes

Мλμ= r3 Y1μ

Low-lying

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




Concluding remarks
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


Implications of the redundancy
Implications of the redundancy

|n; β> =ΣjCj |i>

where

|i > = a†p ah| n-1; α >(not linearly independent )

Eigenvalue problemofgeneral form

Σj [<i|H|j> - Ei <i|j>]Cj = 0

But (problems again!!)

i.A direct calculation of <i|H|j>and<i|j> is prohibitive !!

ii.The eigenstates would contain spurious admixtures!!

How to circumvent these problems?


Problem overcompletness for n 1
Problem: Overcompletness for n>1

|n; β> =ΣαphCαpha†p ah| n-1; α >

a†p ah | n-1; α > are not fully antysymmetrized !!!

p h p h

a†p ah| n-1; α > ≡

The multiphononstates are not linearly independent

and form an overcompleteset.


16 o as theoretical lab
16O as theoretical lab

Structure of16O: Atheoretical challenge

Pioneering work: First excited 0+as deformed 4p-4h excitations

G. E. Brown, A. M. Green, Nucl. Phys. 75, 401 (1966)

(TDA) IBM (includes up to 4 TDA Bosons)

H. Feshbach and F. Iachello, Phys. Lett. B 45, 7 (1973); Ann. Phys. 84, 211 (194)

SM up to 4p-4h and 4 ħω

W.C. Haxton and C. J. Johnson, PRL 65, 1325 (1990)

E.K. Warbutton, B.A. Brown, D.J. Millener, Phys. Lett. B293,7(1992)

No-core SM (NCSM) Huge space!!!

Symplectic No-core SM (SpNCSM) a promising tool for cutting the SM space

T. Dytrych, K.D. Sviratcheva, C. Bahri, J. P. Draayer, and J.P. Vary, PRL 98, 162503 (2007)

Self-consistent Green function (SCGF)

(extends RPA so as to include dressed s.p propagators and coupling to two-phonons)

C. Barbieri and W.H. Dickhoff, PRC 68, 014311 (2003);

W.H. Dickhoff and C. Barbieri, Pro. Part. Nucl. Phys. 25, 377 (2004)


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