Transcript Nessun titolo diapositiva

A new equation of motion method
for multiphonon nuclear spectra.
N. Lo Iudice
Università 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 successful microscopic (QPM) multiphonon approach
A new (in principle exact) multiphonon method
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 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
αα
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 = QSn |0 >
= (Qp + Qn)n |0 >
•
E2
n=3
E2
M1
n=2
MS
|n, ν>MS = QAQS
(n-1)
= (Qp - Qn) (Qp + Qn)
|0 >
E2
n=2
M1
E2
(n-1)
n=1
|0 >
MS
n=1
J+1
Signature
•
Preserving symmetry transitions
M(E2)  QS
•
n → n-1
Changing symmetry transitions
M(M1)  J
n
– Jp n → n
E2
Sym
M1
J
J-1
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)
M1
E2
4+ state in Os isotopes
N. L. and A. V. Sushkov submitted to PRC
Hexadecapole one-phonon?
Ψ  |n=1,4+>
4+
E4
0+
S(t,α)
2g+
Double-g ?
 |gg>
E4  2 Eg
4+
E2
2
E2
+
g
0+
R4(E2) = B(E2,4+→ 2+)/B(E2,2+ → 0+)
2
QPM
Ψ  0.60 |n=1,4+> + 0.35 |gg>
+
4:
QPM versus EXP
Successes and limitations of the QPM
Successes
It 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 improvements
Multistep 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 lines
Both MSM and (especially) MPM look involved
A new (exact) multiphonon approach
Eigenvalue problem in a multiphonon space
H | Ψν > = Eν | Ψν >
| Ψ ν >  H = Σn  Hn
Hn  |n; β>
Generation of the |n; β>
( n= 0,1.....N )
(basis states)
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>

|n; β>
= Σαph C(n)αph a†p ah | n-1; α >
( TDA)
EOM: Construction of the Equations
Crucial ingredient
< n; β | [H, a†p ah] | n-1; α>
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
Commutator expansion
< 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
Linearization
Î = Σ γ |n-1; γ >< n-1; γ|
< 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)
Aαγ(n) ( ij) = [(εp–εh ) + Eα (n-1) ] δij(n-1) δαβ(n-1) + [VPH ρH + VHPρP + VPPρP + VHHρH ]αiβj
Xα(β) (i ) = < n; β | a†p ah| n-1; α>
n =1 TDA
ρH ≡ {< n-1,γ|a†hah’ |n-1,α>}
ρP ≡ {< n-1,γ|a†pap’ |n-1,α>}
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
C=X
n> 1
|n; β> = Σαi Cα(β) (i ) a†p ah | n-1; α >
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)
(AD)C = H C = E DC
Dαα’ (ij) = < n-1; α’|( a†h’ ap’)( a†p ah) | n-1; α>
Problem i)
how to compute D
Problem ii)
redundancy
Eigenvalue
Equation
overlap matrix
Det D = 0
General Eigenvalue Problem
• Solution of problem i)
(AD)C = H C = E DC
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>
recursive
relations
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)
Problem i) solved!!!!
General Eigenvalue Problem
Solution of Problem ii) (redundancy)
(AD)C = H C = E DC
*Choleski decomposition
D
Ď
** Matrix inversion
HC =
Exact
(Ď-1AD)C = E C
eigenvectors
|n; β> = Σαph Cα(β) (i ) a†p ah | n-1; α >
 Hn (phys)
Iterative generation of phonon basis
Starting point
|0>
(1)
Ĥ
Solve
(1)
C
(1)
= E
C
(1)
|n=1, α>
(1)
X
ρ
(1)
Ĥ(2) C (2) = E (2) C (2)
|n=2,α>
Solve
X(2) ρ(2)
………
X(n-1) ρ(n-1)
Ĥ(n) C (n) = E (n) C (n)
Solve
X(n) ρ(n)
The multiphonon basis is generated !!!
|n,α.>
H: Spectral decomposition, diagonalization
H = Σ nα E α(n) |n; α><n;α| +
+ Σ nα β |n; α><n;α| H |n’;β><n’;β|
(diagonal)
(off-diagonal)
n’ = n ±1, n±2
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’
Outcome
H |Ψν> = Eν |Ψν>
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>
E.m. response
W.F.
|Ψν> = Σn{λ} C{λ}(ν) (n) | n;{λ1λ2. λn }>
C3 | λ1 λ2 λ3 >
+
C1 | λ1 >
|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
C2 | λ1 λ2>
+
C0 |0>
Ψ0
EMPM
Ψν(n)
16
O: TDA (CM free) response and SM space dimensions
EMPM : Exact implementation in 16O for N=4
SM space
All particle-hole (p-h)
configurations up to 3ħω
(2s,1d,0g)
(1p,0f)
(1s,0d)
Free of CM
spurious admixtures
0p
0s
Mean field versus EMPM E response
NEW running sums
EW running sums
CM motion
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 ]
For g>>1
E CM >> Eintr
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
Aαγ(n) (λ,λ’) = [ Eλ + Eα(n-1) ] δλ λ’ δαγ + ρλλ’ V ραγ(n-1)
TDA (n=1)
p’
h’
MPEM (n=3)
λ’
---------
-------p
h
Vph’hp’
γ
λ
α
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
|Ψ0> = C0(0) |0>
80
70
+ Σλ Cλ
(0)
|λ, 0>
60
50
P(n) %
noCMcorr
40
+ Σ λ1λ2 Cλ1λ2 (0) |λ1 λ2, 0 >
30
20
10
0
0ph
1ph
2ph
80
70
60
50
EM
Nocore
HJ
40
1 = < Ψ0|Ψ0> = P0 + P1 + P2
30
20
10
0
0ph 2ph 4ph 6ph
16O
negative parity spectrum
• Up to three phonons
IVGDR
Мλμ = τ3 r Y1μ ≈ Rπ - Rν
TDA
|1- > IV ~ |1 (p-h) (1ħω)>(TDA)
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
Мλμ = r3 Y1μ
Low-lying
|3->IS ~ |1(p-h) (3 ħω)> + |2(p-h) (1ħω + 2ħω)> + |3(p-h) (1ħω )>
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
|n; β> = Σ j Cj |i>
where
|i > = a†p ah | n-1; α >
Eigenvalue problem

(not linearly independent )
of general 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
|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 > ≡

p
h
p h
The multiphonon states are not linearly independent
and form an overcomplete set.
16O
as theoretical lab
Structure of 16O: A theoretical 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)