a new equation of motion method for multiphonon nuclear spectra
DESCRIPTION
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 ). - PowerPoint PPT PresentationTRANSCRIPT
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