A new equation of motion method for multiphonon nuclear spectra.

N. Lo IudiceUniversità 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)

From mean field to multiphonon approaches

Anharmonic features of nuclear spectra:

Experimental evidence of multiphonon excitations

Necessity of going beyond mean field approaches

A new (in principle exact) multiphonon method

A successful microscopic (QPM) multiphonon approach

Collective modes: anharmonic featuresMean 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 triple dipole giant resonances

- 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

Proton-neutron mixed-symmetry states (N. Pietralla et al. PRL 83, 1303 (1999))

 

A microscopic multiphonon approach: QPM Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonons, Bristol, (1992)

H = Hsp + Vpair + Vpp + Vff

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

α†α† O†λ

α α Oλ

Oλ† = Σkl [Xkl(λ) α†

k α†l – Ykl(λ) αk αl]

HQPM= Σnλ ωn (λ) Q†λ Qλ + Hvq

Ψν = Σncn Q†ν(n) |0> + Σij Cij Q† (i) Q†(j) |0>

+ Σijk Cijk Q†(i) Q†(j) Q†(k)|0>

π-ν MixedSymmetry • Symmetric |n, ν>s = QS

n |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

n=2

n=3

n=1

Sym

E2

E2

E2

E2

E2

M1

M1

n=2

MS n=1

Scissors multipletS | n,J> = (Jp – Jn) |nJ>

= ΣJ’ |n J’> <nJ’| S|nj>

Bsc(M1) = ΣJ’ |<nJ’|M(M1)|nJ>|2

~ 1.5 - 2 μN2

J

J-1J

J+1

M1

A QPM calculation: N=90 isotonesN. L, Ch. Stoyanov, D.Tarpanov, PRC 77, 044310 (2008)

M1 E2

4+ state in Os isotopesN. L. and A. V. Sushkov submitted to PRC

Hexadecapole one-phonon?

Ψ |n=1,4+>

E4

S(t,α)

Double-g ?

|> E2

E4 2 Eg 2E

R4(E2) = B(E2,4+→ 2+)/B(E2,2+ → 0+)

2

QPM

Ψ 0.60 |n=1,4+> + 0.35 |>

4+

0+

2g+

4+

2g+

0+

4+: QPM versus EXP

Successes and limitations of the QPMSuccessesIt is fully microscopic and valid at low and high energy.including the double GDR (Ponomarev, Voronov)

Limitations: Valid for separable interactions Antisymmetrization enforced in the quasi-boson approximation, Ground state not explicitly correlated (QBA)

Temptative improvementsMultistep Shell model (MSM) (R.J. Liotta and C. Pomar, Nucl. Phys. A382, 1 (1982))They expand and linearize <α|[[H, O†],O†]|0> ( O† = Σph X(ph) a†

p ah )

Multiphonon model (MPM) (M. Grinberg, R. Piepenbring et al. Nucl. Phys. A597, 355 (1996) Along the same linesBoth MSM and (especially) MPM look involved

A new (exact) multiphonon approachEigenvalue problem in a multiphonon space

| Ψ ν > H = Σn Hn Hn |n; β> ( n= 0,1.....N )

An obvious (but prohibitive!!) choice

|n; β> = | ν1, ν2,… νi ,…νn>

where (TDA) | νi > = Σph cph(νi ) a†

p ah |0>

A workable choice |n=1; β> = | νi > = Σph cph (νi ) a†

p ah |0> ( TDA)

H | Ψν > = Eν | Ψν >

Generation of the |n; β> (basis states)

|n; β> = Σαph C(n)αph a†

p ah | n-1; α >

EOM: Construction of the Equations

< n; β | [H, a†p ah] | n-1; α>

< n; β |[H, a†p ah]| n-1; α> = ( Eβ

(n) – Eα (n-1) ) < n; β | a†p ah| n-1; α >

(LHS) (RHS)

* 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 >

It follows from

Preliminary step:

Crucial ingredient

Î = Σ γ |n-1; γ >< n-1; γ|

Equations of Motion : LHS

< 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’ ah a†

i aj | n-1; α > + not linear

Commutator expansion

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

A (n ) X

(n) = E (n) X

(n)

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 > = Σph cph a†p ah |0>

n> 1 |n; β> = Σαi Cα(β) (i ) a†

p ah | n-1; α >

Aαγ(n) ( ij) = [(εp–εh ) + Eα

(n-1) ] δij(n-1) δαβ

(n-1) + [VPH ρH + VHPρP + VPPρP + VHHρH ]αiβj

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

A(n) X(n) = E(n) X(n) X(n) = D(n) C(n)

Problem i) how to compute D

Problem ii) redundancy Det D = 0

(AD)C = H C = E DCEigenvalue

Equation

Dαα’ (ij) = < n-1; α’|( a†h’ ap’)( a†

p ah) | n-1; α> overlap matrix

General Eigenvalue Problem

• Solution of problem i)

(AD)C = H C = E DC

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!!!!

Aαγ(n) ( ij) = [εp–εh + Eα

(n-1) ] δij(n-1) δαβ

(n-1)

+ [VPH ρH + VHPρP + VPPρP + VHHρH ]αiβj

Dαα’ (ij) = < n-1; α’|( a†h’ ap’)( a†

p ah) | n-1; α> overlap matrix

General Eigenvalue Problem

Solution of Problem ii) (redundancy)

*Choleski decomposition

** Matrix inversion

Exact eigenvectors

HC = (Ď-1AD)C = E C

(AD)C = H C = E DC

D Ď

|n; β> = Σαph Cα(β) (i ) a†

p ah | n-1; α > Hn (phys)

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

Outcome

H |Ψν> = Eν |Ψν>

Off-diagonal terms: Recursive formulas

< n; β | H| n-1; α > = Σphγ ϑαγ (n-1) (ph) Xγ(β) (ph)

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

(α) (p’h’)

H = Σ nα E α(n) |n; α><n;α| + (diagonal)

+ Σ nα β |n; α><n;α| H |n’;β><n’;β| (off-diagonal)

n’ = n ±1, n±2

|Ψν> = Σnα Cα(ν) (n) | n;α>

|n;α> = Σγ Cγ(α) a†

p ah | n-1;γ>

E.m. response

C3 | λ1 λ2 λ3 > + C1 | λ1 >

C2 | λ1 λ2> + C0 |0>

Ψ0 Ψν(n)

EMPM

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

|λ> = Σph cph (λ) a†p ah|0>

W.F.

e.m. operator

Мλμ = rλ Yλμ

Strength Function

S(Eλ) = Σ Bn (Eλ) δ(E- En)

Bn (Eλ) =|<Ψnν|| Мλ ||Ψ0>|2

16O: TDA (CM free) response and SM space dimensions

SM spaceAll particle-hole (p-h) configurations up to 3ħω

(2s,1d,0g)

(1p,0f)

(1s,0d)

0p

0s

EMPM : Exact implementation in 16O for N=4

Free of CM spurious admixtures

Mean field versus EMPM E response

NEW running sums

EW running sums

CM motion

Hamiltonian

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

H H + Hg

Hg = g [ P2/(2Am) + (½) mA ω2 R2 ]

For g>>1 E CM >> Eintr

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

CM motion in TDA: Isoscalar E1

CM motion in EMPM

Spectra

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†p ah

λ’γ Aαγ(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) ] δλ λ’ δαγ + ρλλ’ V ραγ(n-1)

ρλλ’ (kl) ≡ < λ’| a†kal |λ> ραγ

(n-1) (kl) = < n-1,γ| a†kal |n-1,α>

|n; β> = Σαλ C(β)αλ O†

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

C (β) {λi} = Σ C(β)αλ1

C(α) γλ2 C (γ)

δλ3

Vertices

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

-------- ---------

Vph’hp’ ρλλ’ V ραγ(n-1)

h p

p’h’ λ’

λ α

γ

Aαγ(n) (λ,λ’) = [ Eλ + Eα

(n-1) ] δλ λ’ δαγ + ρλλ’ V ραγ(n-1)

THANK YOU

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

0

10

20

30

40

50

60

70

80

0ph 2ph 4ph 6ph

EMNocoreHJ

0

10

20

30

40

50

60

70

80

0ph 1ph 2ph

P(n) %noCMcorr

|Ψ0> = C0(0) |0>

+ Σλ Cλ(0) |λ, 0>

+ Σ λ1λ2 Cλ1λ2

(0) |λ1 λ2, 0 >

1 = < Ψ0|Ψ0> = P0 + P1 + P2

16O negative parity spectrum • Up to three phonons

IVGDR

Мλμ = τ3 r Y1μ ≈ Rπ - Rν

|1- > IV ~ |1 (p-h) (1ħω)>(TDA)

TDA

ISGDR

Мλμ = r Y1μ ≈ RCM !!!

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

Мλμ = r3 Y1μ

Toroidal

Octupole modes

Мλμ = r3 Y1μ

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

Low-lying

Effect of CM motion

Effect of the CM motion

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).

- A riformulation for an efficient truncation is in progress

Implications of the redundancy

Σj [ <i|H|j> - Ei <i|j> ] Cj = 0

|n; β> = Σ j Cj |i>where

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

Eigenvalue problem of general form

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

p h p h

|n; β> = Σ αph Cαph a†p ah | n-1; α >

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

a†p ah | n-1; α > ≡

The multiphonon states are not linearly independent

and form an overcomplete set.

16O as theoretical lab

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 ħωW.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 spaceT. 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)

Structure of 16O: A theoretical challenge