An exact microscopic multiphonon approach to nuclear spectroscopy
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Transcript of An exact microscopic multiphonon approach to nuclear spectroscopy
An exact microscopic multiphonon approach to nuclear
spectroscopyN. Lo Iudice
Università di Napoli Federico II
Naples(Andreozzi, Lo Iudice, Porrino) Prague (Knapp, Kvasil)
Collaboration
Tokyo 07
Eigenvalue problem in a multiphonon space
H | Ψ ν > = Eν | Ψ ν >
| Ψ ν > H = Σn Hn ( n= 0,1.....N )where
Hn ∋ |n, α > ~ | ν1 ν2 …..νn >
| νi > = Σph c ph(νi ) a
†p ah |0>
First goal: Generate the basis states |n, α >
We do it by constructing a set of equations of motion and solving them iteratively
EOM: Construction of the Equations • Crucial ingredient < n; β | [H, a†
p ah] | n-1; α>
• Preliminary step: Derive< n; β | [H, a†
p ah] | n-1; α> = ( Eβ
(n) - Eα
(n-1)) < n; β | a
†
p ah| n-1; α >
(LHS) (RHS)
The RHS comes from
* property < n; β | a†
pa
h | n’; γ > = δ
n’,n-1 < n; β | a†
pa
h | n-1; γ >
** request
< n; β | H | n; α > = E α
(n) δαβ
Commutator expansion
< n; β | [H, a†p ah] | n-1; α > =
(εp- εh) < n; β | a†p ah| n-1; α >. +
1/2 Σijp’
Vhjpk < n; β | a†p’ ah a
†i aj | n-1; α > + …
Linearization
< n; β | [H, a†p ah] | n-1; α > =
Σp’h’γ [(εp- εh) δγβ
+ 1/2 Σij
Vhjp’k’ < n-1; γ | a†i aj | n-1; α > +
………] < n; β | a†p’ ah’ | n-1; γ >
= Σp’h’γ
Aαγ(n)(ph;p’h’) < n; β | a†
p’ ah’ | n-1; γ >
Î = Σ γ |n-1; γ >< n-1; γ|
Equations of Motion : LHS
LHS=RHS AX = EX
jγ Aαγ(n) ( ij) Xγβ
(n)(j) = (Eβ
(n) - Eα(n1)) Xαβ
(n)(i) where
Xαβ(n) (i )
= < n; β | a†
p ah| n-1; α >
Aαγ(n) ( ij) = [εp–εh] δij
(n-1) δαβ (n-1)
+ [VPHρH + VHPρP + VPPρP + VHHρH ]αiβj
ρH ≡ {< n,γ|a†
hah’|n,α>} ρP ≡ {< n,γ|a†
pap’|n,α>} n =1 (ρP = 0 ρH = δhh’)
A(1) Xα(1) = (Eα
(1) - E0 (0) ) Xα
(1) Tamm-Dancoff
A(1) (ij)= δij[εp–εh] + V(p’hh’p)
Structure of multiphonon states: Overcompletness
The eigenvalue equations yield states of the structure
|n; β> = Σ α ph cα
ph a†
p ah | n-1; α >
• Problem: The multiphonon states are not fully antysymmetrized !!!
a†p ah | n-1; α > ≡
p h p h
The multiphonon states form an overcomplete set
Solution of the redundancy Reminder
Insert |n; β> = Σα ph Cαph a†
p ah | n-1; α >
Xαβ
(n)(ph)
= < n; β | a†
p ah | n-1; α >
X = DC (AD)C = H C = E DC
where
Dij = < n-1; α’| ah’ a†
p’ a†
p ah | n-1; α> overlap or metric matrix
Problem due to redundancy Det D = 0
γ Aαγ(n) Xγβ(n)= (Eβ
(n) - Eα(n-1 ) )Xαβ(n)
A X = E X
Solution of the redundancy *Removal of redundancy: Choleski decomposition(no
diagonalization)
D Ď ** Matrix inversion
Exact eigenvectors
|n; β> = Σ αph Cαph a
†
p ah | n-1; α > H n (phys)
● Now compute in Hn
i. X(n)
ii. ρ(n) recursive formula
X= D C
ρ(n) = C X (n) + C ρ (n-1) X (n)
HC = (Ď-1AD)C = E C
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 = 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
Off-diagonal terms: Recursive formulas
< n; α | H| n-1; β > = Σphγ
ϑαγ
(n-1)(ph) Xγβ
(n)(ph)
< n; α | H| n-2; β > = Σ V pp’hh’
Xγβ
(n) (ph)
Xγβ
(n-1) (p’h’)
Outcome of diagonalization H |Ψν> = Eν |Ψν>
|Ψν> = Σnα Cα(ν) (n) | n;α> |n;α> = Σγ Cγ
(n) | n-1;γ>
Numerical test: A = 16 ∙ Calculation up to 3-phonons and 3ħω
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 ] • Consistent choice of ph space: It must includes all ph
configurations up to 3ħω
Ground state
|Ψ0> = C(0)
0 |0>
+ Σλ C
λ
(0) |λ, 0>
+ Σ λ1λ2 Cλ1λ2
(0) |λ 1 λ
2, 0
>
|λ, 0> |λ 1 λ
2, 0
>
1 = < Ψ0|Ψ
0> = P
0 + P
1 + P
20
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
16O negative parity spectrum • Up to three phonons
IVGDR
|1->IV ~ |1(p-h) (1ħω)>
ISGDR |1->IS ~ |1(p-h) (3 ħω)> + |2(p-h) (1ħω + 2ħω)> + |3(p-h) (1ħω )>
Toroidal
Octupole modes |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
THANK YOU
E2 response up to 3 ħω: Running sum
Sn = Σn (En – E0(0) ) Bn(E2,0→2+
n)
Rn = Sn/SEW(E2)
• It is necessary to enlarge the space!!
E2 response up to 3 ħω
S(ω,E2) = Σn Bn(E2,0→2+n) ρΔ(ω-ωn)
ρΔ(x) = (Δ/2π) / [x2 + (Δ/2)2]
M(E2μ) = Σ(p)ep rp2 Y 2μ
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>, Q2×Q2×Q3|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 D × D |0>
From TDA to RPA
A(n)
B(n)
X(n)
= (E(n)
- E(n-1)
)
B(n)
A(n)
-Y(n)
Aαγ(n) (ph;p’h’)=δhh’δpp’δαγ(n-1)[εp–εh]
+ Σh1 V(p’h1h
’p) ραγ
(n-1) (h1h)
+ Σp1 V(p’hp
1h’) ραγ
(n-1) (pp1)
+ δhh’ [ Σp2p3 V(p’p2pp3)ραγ
(n-1) (p2p3)
+ Σh2h3 V(p’h2ph3)ραγ
(n-1)(h2h3)]
+ δpp’ [ Σh2h3 V(hh2h
’h3)ραγ
(n-1)(h2h3)
+ Σp2p3 V(hp2h
’p3)ραγ
(n-1)(p2p3)]
Bαγ
(n) (ph;p’h’)=
Σh2 V(p p’h2 h
’)ραγ
(n-1)(h2h)
+ Σp2 V(h’hp2p
’)ραγ
(n-1)(pp2)
AX = EXwhere
Aαγ
(n)
(ph;p’h’)=δhh’δpp’δαγ
(n-1)[εp–εh]
+ Σh1 V(p’h1h’p) ραγ
(n-1)(h1h)
+ Σp1 V(p’hp1h’) ραγ
(n-1)(pp1)
+ δhh’1/2 Σp2p3 V(p’p2pp3) ραγ
(n-1)(p2p3)
+ δpp’1/2 Σh2h3 V(hh2h’h3) ραγ
(n-1)(h2h3)
(ραγ
(n)(ij) = <n,α|a†
iaj|n,α>)
Choleski decomposition• Any real non negative definite
symmetric matrix can be written as
D = L LT
( L {lij} lower triangular matrix) Det{D} = (Det{L})2
= l112 l22
2 ...lii2…..
= λ21 …λ2
i… (D | λi > = λi | λi > )
Definition of L: Recursive formulas
l211 = d11
l11 lj1 = dj1 j=2,….,nl2
ii = dii – Σk=1,i-1 l2
ik
lii lji = dji – Σk=1,i-1 lik ljk
(It reminds the Schimdt orthogonalization method)
The decomposition goes on until lnn = 0
lnn = 0 → Det{L} =0→Det{D} =0 ⇨
⇨ | λn > (D | λi > = λi | λi > ) linearly dependent ⇨ to be discarded
• For numerical stability we need maximum overlap
λj ≤ λi j > i
• Sequence order
lii ≤ ljj j > i
Once lnn = 0 → lii = 0 i >n
1) We can stop at the nth step
2) We get Ď (Nn < N
r) with maximum determinant
(overlaps)
Elimination of CM spuriosity F. Palumbo Nucl. Phys. 99 (1967)
• HSM H = HSM + Hg
• Hg = g [ P2/(2Am) + (½) mA ω2 R
2 ] = g/A [ Σi hi + Σi<j vij ]
hi = pi
2/(2m) + (1/2) m(Aω)2ri
2
vij = (1/m) pi· pj + m (Aω)2ri · rj
Hg effective only in Jπ = 1
- ph channel.
H Ψ = (HSM + Hg) Ψ =E Ψ
In the full space
Ψ = ψ
int Φn
CM
For the physical states
Ψn = ψn Φ0
CM
with CM energy
E0
CM = 3/2 g ħω
For the spurious ones
Ψn
(1)
= ψn
Φ1
CM
at the very high CM energy
E1
CM
- E0
CM
= g ħω
They can therefore be tagged and eliminated