Structure of Light Exotic Nuclei
within Fermionic Molecular Dynamics
Hans Feldmeier, Thomas Neff, Robert Roth
GSI Darmstadt - TU Darmstadt
14
14
01
-5
0
x [fm]
0.001
1.0
z [fm]
0.0
0.5 .1
0
0.101
0.
0.0
-5
EXON 2004 - July 5, 2004
1
0.5
0
Be - n
0.01
0.
01
0.0 .01
0
5
01
Be - p
0.
0.0 01
01
5
-5
0
x [fm]
5
Nuclear Structure – ab initio
Aim
➼ describe low energy properties of nuclear many-body system
in terms of a Hamiltonian
and many-body states
Ingredients
➼ solve Schrödinger equation
H
∼
bn
Ψ
=
H =T +V
∼ ∼ ∼ NN
Ψ̂
bn = En Ψ
bn
Ψ
H
∼
Hamiltonian with realistic NN-potential V
∼ NN
(phase shifts, deuteron)
many-body Hilbert space
bn is terribly complicated for realistic interaction
~r1 σ
~ 1 τ1 ,~r2 σ
~ 2 τ2 , · · · ,~rA σ
~ A τA Ψ
main problem: short-range repulsive and tensor correlations induced by V
∼ NN
EXON 2004 - July 5, 2004 –2
Realistic NN-potential
VNN (~r12 , ~p12 = 0, σ
~ 1, σ
~ 2 , T = 0)
200
AV18
VNN [MeV]
100
r12
• V
repulsive at small distances
∼ NN
➼ strong short-range central correlations
nucleons cannot get closer than ≈ 0.6 fm
• V
depends strongly on orientation of σ
~1, σ
~2 with respect to ~r12
∼ NN
0
➼ tensor correlations
protons and neutrons want to align their spins with ~r 12
r12
-100
ag replacements
-200
0.0
0.5
1.0
1.5
2.0
r12 [fm]
Argonne potential
2.5
3.0
Problem:
Slater determinants cannot describe these
correlations
EXON 2004 - July 5, 2004 –3
Realistic NN-potential
VNN (~r12 , ~p12 = 0, σ
~ 1, σ
~ 2 , T = 0)
200
AV18
VNN [MeV]
100
r12
• V
repulsive at small distances
∼ NN
➼ strong short-range central correlations
nucleons cannot get closer than ≈ 0.6 fm
• V
depends strongly on orientation of σ
~1, σ
~2 with respect to ~r12
∼ NN
0
➼ tensor correlations
protons and neutrons want to align their spins with ~r 12
r12
-100
ag replacements
-200
0.0
0.5
1.0
1.5
2.0
2.5
Problem:
3.0
r12 [fm]
Argonne potential
Slater determinants cannot describe these
correlations
Solution: include short-range correlations by unitary transformation (UCOM)
b
Φ = C
Φ = C
C Φ
∼
∼Ω ∼r
• C
central correlator shifts nucleons out of repulsive core
∼r
• C
tensor correlator aligns spins along ~r12
∼Ω
EXON 2004 - July 5, 2004 –4
Unitary Transformation
Unitary Transformation
transform eigenvalue problem
bn = En
Ψ
H
∼
with the unitary operator C
∼
bn = C Ψn ,
Ψ
∼
bn
Ψ
“pre-diagonalization”
include typical effects
common to all states
−1
†
C
=C
∼
∼
into the equivalent eigenvalue problem
b Ψn = (C † HC ) Ψn = En Ψn H
∼
∼ ∼ ∼
Idea: nuclear system has different scales:
b
effective Hamiltonian H
∼
has same phase shifts and energies
as realistic original H
!
∼
• long range (low momenta) – can be described by mean-field
(Slater determinant)
• short-range (high momenta) – cannot be described by mean-field
➼ treat long range correlations in Ψ
➼ short range correlations in unitary C
=C
C
∼
∼ Ω∼ r
b = C Ψ contain both
Result: many-body states Ψ
∼
EXON 2004 - July 5, 2004 –5
Correlations in Nuclei
C
~r
1
z [fm]
C~ Ω
0
−1
−2
−2
−1
0
1
2 −2
−1
x [fm]
0
2 −2
1
x [fm]
−1
0
1
2
➼ central correlator
shifts density out of
the repulsive core
➼ tensor correlator
aligns density with
spin orientation
x [fm]
[A MeV]
ρ(2)
r1 − ~r2 ) S = 1, MS = 1, T = 0
S,T (~
➼ both,
central and tensor
correlations are
essential for
binding
energies
two-body densities
2
60
4
40
16
He
20
0
O
40
Ca
T
∼
H
∼
V
∼
−20
− 40
− 60
C
∼r
C
∼Ω
EXON 2004 - July 5, 2004 –6
Hilbert Space: Fermionic Molecular Dynamics
Fermionic
Slater determinant
Q = A q1 ⊗ · · · ⊗ q A
∼
Antisymmetrization
➼ antisymmetrized A-body state
Molecular
single-particle states
(~x − ~b )2 X
i
~x q =
⊗ χi ⊗ ξ
ci exp −
2ai
i
➼ Gaussian wave-packets in phase-space,
spin is free, isospin is fixed
Dynamics in Hilbert space
➼ Hilbert space contains
shell-model, clusters, halos
spanned by one or several non-orthogonal Q(a)
X
Ψ =
ψa Q(a)
a
variational principle → Q(a) = { q(a)
ν , ν = 1 · · · A }, ψa
EXON 2004 - July 5, 2004 –7
NN Interaction
Effective Correction to the Interaction
24
O
48
Ca
b=C HC
➼ correlated two-body interaction H
∼
∼ ∼ ∼
is lacking three-body forces
†
➼ instead of three-body force use additional momentumdependent and spin-orbit two-body correction term
ρ(1) (~r) [ρ0 ]
Effective two-body interaction
➼ fit correction term to binding energies and radii of
“closed-shell” nuclei
➼ altogether a 15% correction to the ab-initio two-body
potential
16
40
O
Ca
16
O and 40 Ca are not
“closed shell” nuclei !
EXON 2004 - July 5, 2004 –8
FMD
Variation of single determinant
Nuclear Chart
20
Ca36 Ca37 Ca38 Ca39 Ca40 Ca41 Ca42 Ca43 Ca44 Ca45 Ca46 Ca47 Ca48 Ca49
-0.5 0 0.5 1 1.5 2
K35 K36 K37 K38 K39 K40 K41 K42 K43 K44 K45 K46 K47 K48 K49
18
Ar32 Ar33 Ar34 Ar35 Ar36 Ar37 Ar38 Ar39 Ar40 Ar41 Ar42 Ar43 Ar44 Ar45 Ar46 Ar47
1 Gaussian per
single-particle state
16
(E − Eexp )/A [MeV]
Cl31 Cl32 Cl33 Cl34 Cl35 Cl36 Cl37 Cl38 Cl39 Cl40 Cl41 Cl42 Cl43 Cl44 Cl45
S29 S30 S31 S32 S33 S34 S35 S36 S37 S38 S39 S40 S41 S42 S43 S44
P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38 P39 P40 P41
14
Si22
Si24 Si25 Si26 Si27 Si28 Si29 Si30 Si31 Si32 Si33 Si34 Si35 Si36 Si37 Si38 Si39
Al24 Al25 Al26 Al27 Al28 Al29 Al30 Al31 Al32 Al33 Al34 Al35 Al36
12
Mg20Mg21Mg22Mg23Mg24Mg25Mg26Mg27Mg28Mg29Mg30Mg31Mg32Mg33Mg34
Na20 Na21 Na22 Na23 Na24 Na25 Na26 Na27 Na28 Na29 Na30 Na31
10
Ne17 Ne18 Ne19 Ne20 Ne21 Ne22 Ne23 Ne24 Ne25 Ne26 Ne27 Ne28
F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26
8
O13 O14 O15 O16 O17 O18 O19 O20 O21 O22 O23 O24
N12 N13 N14 N15 N16 N17 N18 N19 N20 N21 N22
6
C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20
B8
4
B10 B11 B12 B13 B14 B15 B16 B17
Be7 Be8 Be9 Be10 Be11 Be12 Be13 Be14
Li5
2
B9
Li6
Li7
Li8
Li9 Li10 Li11
He3 He4 He5 He6 He7 He8
H2
H3
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
EXON 2004 - July 5, 2004 –9
FMD
Variation of single determinant
Nuclear Chart
20
Ca36 Ca37 Ca38 Ca39 Ca40 Ca41 Ca42 Ca43 Ca44 Ca45 Ca46 Ca47 Ca48 Ca49
-0.5 0 0.5 1 1.5 2
K35 K36 K37 K38 K39 K40 K41 K42 K43 K44 K45 K46 K47 K48 K49
18
1 Gaussian per
single-particle state
16
(E − Eexp )/A [MeV]
Ar32 Ar33 Ar34 Ar35 Ar36 Ar37 Ar38 Ar39 Ar40 Ar41 Ar42 Ar43 Ar44 Ar45 Ar46 Ar47
Cl31 Cl32 Cl33 Cl34 Cl35 Cl36 Cl37 Cl38 Cl39 Cl40 Cl41 Cl42 Cl43 Cl44 Cl45
S29 S30 S31 S32 S33 S34 S35 S36 S37 S38 S39 S40 S41 S42 S43 S44
P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38 P39 P40 P41
14
Si22
Si24 Si25 Si26 Si27 Si28 Si29 Si30 Si31 Si32 Si33 Si34 Si35 Si36 Si37 Si38 Si39
Al24 Al25 Al26 Al27 Al28 Al29 Al30 Al31 Al32 Al33 Al34 Al35 Al36
12
Mg20Mg21Mg22Mg23Mg24Mg25Mg26Mg27Mg28Mg29Mg30Mg31Mg32Mg33Mg34
Na20 Na21 Na22 Na23 Na24 Na25 Na26 Na27 Na28 Na29 Na30 Na31
10
12
Ne17 Ne18 Ne19 Ne20 Ne21 Ne22 Ne23 Ne24 Ne25 Ne26 Ne27 Ne28
Mg24
F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26
8
10
O13 O14 O15 O16 O17 O18 O19 O20 O21 O22 O23 O24
Ne20
N12 N13 N14 N15 N16 N17 N18 N19 N20 N21 N22
6
C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20
B8
4
B10 B11 B12 B13 B14 B15 B16 B17
Be7 Be8 Be9 Be10 Be11 Be12 Be13 Be14
Li5
2
B9
8
Li6
Li7
Li8
6
O14
O15
O16
O17
O18
O19
O20
O21
O22
O23
N12
N13
N14
N15
N16
N17
N18
N19
N20
N21
N22
C19
C20
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
B8
B9
B10
B11
B12
B13
B14
B15
B16
B17
Be7
Be8
Be9
Be10 Be11 Be12 Be13 Be14
Li6
Li7
Li8
O24
Li9 Li10 Li11
He3 He4 He5 He6 He7 He8
H2
O13
4
H3
Li5
2
4
6
8
10
12
14
16
18
20
2
22
Li9
Li10
Li11
2 Gaussians per
single-particle state
He3
24He4 He526He6 He7
28 He8 30
H2
H3
2
4
6
8
10
12
14
16
EXON 2004 - July 5, 2004 –10
How to improve ?
Projection After Variation (PAV)
➼ mean-field may break symmetries of Hamiltonian
➼ restore reflection and rotational symmetry by
Jπ
parity and angular-momentum projection P
∼ MK
X
K0
X
π π
π
P J Q · c 0
J
J
0 = E
·
c
QH
P
Q
0 Q
K
K
KK
K
∼∼
∼ KK 0
K0
Variation After Projection (VAP)
➼ effect of projection can be large
➼ perform VAP applying constraints on radius,
dipole moment, quadrupole moment or octupole moment and minimize the energy in the
projected energy surface
Multiconfiguration Calculations
➼ diagonalize Hamiltonian in a set of projected
intrinsic states
π J (a)
P
, a = 1, · · · , N
Q
∼ KK 0
EXON 2004 - July 5, 2004 –11
FMD - Projection, Variation after Proj., Multiconfiguration
PAV
6
Radius and Quadrupole Moment
as Generator Coordinates
4
0
VAP
-2
6
-4
4
-6
2
-4
-2
0
x [fm]
2
4
6
y [fm]
-6
0
-2
-6
-6
-4
-2
0
x [fm]
2
4
rcharge [fm]
Q [fm2 ]
B(E2) [e2 fm4 ]
2.39
2.49
2.74
-6.25
-8.02
-11.88
9.31
15.36
30.39
PAV
VAP
Multiconfig
-4
6
[MeV]
6
6
4
4
2
2
y [fm]
y [fm]
Multiconfig
0
−2
8
-40.0
4+
−4
-45.0
−4
−6
−4
y [fm]
−6
6
2
−2
0
0 x [fm]
2
4
6 −6
−4
−2
0
x [fm]
6
4
2
4
6
4
−4
0
4
−2
−6
−4
−2
y [fm]
2
−4
−6
0
x [fm]
2
4
2
4
0
0+
−4
0
−6
−4
−2
0
x [fm]
2
4
6
4+
-47.7
2+
-51.0
0+
-54.8
2+
-54.2
2+
-53.5
-55.0
0+
-57.2
0+
-56.5
−6
−6
−4
−2
0
x [fm]
2
4
6
Variation
-45.4
-52.7
−6
−46
4+
-49.6
6
−2
−2
0
x [fm]
2+
-50.0
2
6
−6
y [fm]
2
−2
-41.0
-46.3
−2
−4
4+
0
4
−6
−6
-40.6
Be
−2
6
y [fm]
y [fm]
2
PAV
VAP
Multiconf
Exp
EXON 2004 - July 5, 2004 –12
FMD - Projection, Variation after Proj. , Multiconfig.
PAV
12
4
y [fm]
2
C
0
-2
-4
-4
-2
0
x [fm]
2
Eb [MeV]
rcharge [fm]
B(E2) [e2 fm4 ]
V/PAV
VAP
Multiconfig
84.7
91.9
93.4
2.33
2.38
2.50
24.7
40.0
Exp
92.2
2.47
39.7 ± 3.3
4
VAP
4
12
2
y [fm]
0+1
[MeV]
0
3–
-2
4+
-70.0
-4
-4
-2
0
x [fm]
2
4+
4+
4
4+
-75.0
4
2
2
y [fm]
0+1
y [fm]
Multiconfig
4
0
-2
-4
-4
-2
0
x [fm]
2
4
-4
4
4
2
2
y [fm]
0+2
y [fm]
-4
0
-2
-4
-4
-2
0
x [fm]
2
4
-2
0
x [fm]
2
2+
0++
2
4+
-80.0
0+
0+3
2–
1–
3–
2++
4
4+
0+
2+
0+
+
-85.0 0
4
0
-2
-4
3−1
0
-2
C
2–
1–
3–
-2
0
x [fm]
2
4+
2+
0+
2–
1–
3–
0+
2+
2+
2+
0+
0+
2+
-90.0
0+
-4
2+
0+
4
Variation
VAP
Multiconfig (4)
Multiconfig (25)
Exp
EXON 2004 - July 5, 2004 –13
Helium Isotopes
4
4
He
1
0.001
0.0
He
5
7
7
He
1
1
5
10
5
ρ(r) [ρ0]
0.01
0.1
z [fm]
total
neutron
proton
ρ(r) [ρ0]
0.1
8
2
4
6
r [fm]
8
8
He
1
2
4
6
r [fm]
0
10
He
-5
10-2
10-3
-4
10
8
1
10
-5
0
x [fm]
5
0
total
neutron
proton
2
4
6
r [fm]
8
10
He
-4
10
5
soft-dipole mode
neutrons are oscillating
against α-core
10-2
10-3
0
x [fm]
0.001
0.01
0
10-1
-5
-5
He
5
total
neutron
proton
10-1
6
He
0.01
0.001
0
0
1
5
0
x [fm]
ρ(r) [ρ0]
0.01
z [fm]
ρ(r) [ρ0]
5
6
-5
1 001
0.0 0.
0.01
10
00
10
10-2
-4
0.
8
10-1
-5
0
x [fm]
-4
5
10-3
-5
6
r [fm]
1
1
0.00
0.1
0
4
01
10
5
He
0.001
5
2
-5
10-2
10-3
0.0
0.1
5
0
total
neutron
proton
z [fm]
0
x [fm]
0
1
-4
-5
z [fm]
10-2
10-3
-5
10-1
0.0
ρ(r) [ρ0]
0.1
0
0.01
0.001
z [fm]
10-1
He
0
total
neutron
proton
2
4
6
r [fm]
8
10
EXON 2004 - July 5, 2004 –14
Helium Isotopes
-22
PAV
Multiconf
Exp
Binding energies
@MeVD
-24
-26
3232- 32-
-28
0+
0+
0+
320+
320+
0+
32-
0+
-30
0+
0+
-32
He4
3.5
He5
Matter radii
@fmD
3.0
He6
He7
He8
320+
2.5
0+
32- 32-
0+
0+
0+
0+
32-
2.0
1.5
0+
0+
He4
0+
He5
He6
He7
He8
Exp: Ozawa,Suzuki,Tanihata, NPA693(2001)32; Raman,Nestor,Tikkanen, Atomic Data and Nucl. Data Tables 78(2001)1
EXON 2004 - July 5, 2004 –15
Helium Isotopes
-22
PAV
Multiconf
Exp
Binding energies
@MeVD
-24
-26
3232- 32-
-28
0+
0+
0+
320+
320+
0+
32-
0+
-30
0+
zero-point oscillations of the
soft-dipole mode essential for
He5 description ofHe6
binding energies He7
and radii
-32
He4
3.5
Matter radii
@fmD
3.0
0+
He8
320+
2.5
0+
32- 32-
0+
0+
0+
0+
32-
2.0
1.5
0+
0+
He4
0+
He5
He6
He7
He8
Exp: Ozawa,Suzuki,Tanihata, NPA693(2001)32; Raman,Nestor,Tikkanen, Atomic Data and Nucl. Data Tables 78(2001)1
EXON 2004 - July 5, 2004 –16
Beryllium Isotopes
7
0.
1.0
0
0.5
z [fm]
0.5
1.0
z [fm]
1
1.0
0.5 .1
0
1
0. 1
0
0.
1
0. 1 1
0.00.00
01
1
00
1
1
0. 1
0
0.
0.0
-5
Be - n
0.0
0.
00
-5
0.001
0.0
1
0.5
0
5
00
11
Be - p
0.
01
5
11
Be - n
0.0
Be - p
01
7
0.0
0.
1
01
0.
00
1
-5
0
x [fm]
8
5
-5
0
x [fm]
Be - p
5
-5
0
x [fm]
Be - n
12
5
-5
0
x [fm]
Be - p
12
5
Be - n
0.
0.5
z [fm]
5
0
x [fm]
01
0.
0.0
1
1.0
5
0. 1
0.
0.
0.
0.0
00
0.
1
01
1
1
-5
01
5
Be - n
1
0.5
0
13
cluster structure
changes with
addition of neutrons
01
z [fm]
-5
Be - p
0.0
0.5
13
00
0.0
01
0
x [fm]
0.0
-5
5
0.5
z [fm]
5
Be - n
0.1.01
0
0.11
0.0
0.0
-5
9
0.0
0
0. 1
01
01
0
0.5
0
x [fm]
01
-5
Be - p
0.0
5
-5
0.0
5
9
01
1
0
x [fm]
0. 0.1
01
0.0
01
0.
-5
0
0.
1
0. 1 1
0.00.00
1
0. 1 1
0
0.00.0
-5
0.5
0
0.5
z [fm]
1
00
01
01
0.001
0.0
1
5
0.0
0.0
5
8
0.0
01
01
0
x [fm]
5
Be - n
0.001
1.0
0.5
14
1
z [fm]
0.01
0.5 .1
0
0.0
-5
0
x [fm]
0.
0
-5
5
Be - p
0.101
0.
0
x [fm]
-5
0.5
-5
1
0.5 .1
0 1 01
0 0
0. 0.
5
14
5
01
0.0 .01
0
00
1
1
0
x [fm]
5
0.
00
00
-5
0
x [fm]
01
1
0. 1
0
0.
0.
-5
-5
Be - n
0.
1.0
0
10
5
1.0
Be - p
0
x [fm]
01
-5
0.
5
10
5
0.0
0
x [fm]
z [fm]
-5
0.
0.0 01
01
5
-5
0
x [fm]
5
EXON 2004 - July 5, 2004 –17
Beryllium Isotopes
-30
PAV
Multiconf
Exp
Binding energies
3232- 32-
@MeVD
-40
-50
0+
0+
0+
3232- 32-
-60
12+
0+
0+
12-
1212+
Be7
Be8
Be9
Be10
Be11
0+
0+
-70
3.4
12+
0+
Be12
Matter radii
0+
Be13
12+
Be14
3.2
3.0
0+
@fmD
0+
2.8
12+
12+
2.6
0+
0+
12-
32-
2.4
0+
3232-
32-
0+
32-
0+
0+
32-
2.2
Be7
Be8
Be9
Be10
Be11
Be12
Be13
Be14
Exp: Ozawa,Suzuki,Tanihata, NPA693(2001)32; Raman,Nestor,Tikkanen, Atomic Data and Nucl. Data Tables 78(2001)1
EXON 2004 - July 5, 2004 –18
1/2
11
–
+
1/2
11
Be - p
11
Be - n
0.5
0.001
z [fm]
0.1
0.5
z [fm]
0.1
0
0.1
0.01
0.1
01
0.0.0
01
0
5
Be - n
0.001
5
11
Be - p
0.0
1
0.
00
0.0
-5
1
1
-5
0.
0
y [fm]
5
11
-5
0
y [fm]
5
11
Be - p
1
-5
-5
5
11
-5
0
y [fm]
5
11
Be - p
Be - n
0.001
0.5
0.001
0.1
z [fm]
0.1
0
0.1
0.01
0.5
5
0.1
01
0.0.001
z [fm]
0
y [fm]
Be - n
5
0
FMD @ GSI
00
FMD @ GSI
0.0
1
0.
00
01
0.
-5
-5
1
0.
00
1
FMD @ GSI
-5
0
x [fm]
5
11
-5
0
x [fm]
5
11
Be - p
-5
5
11
Be - n
-5
0
x [fm]
5
11
Be - p
Be - n
5
0
01
0. 0.1
01
0.5
0.0
0.1
y [fm]
0.0
0
0.0 1
1
5
y [fm]
0
x [fm]
FMD @ GSI
1
0.
0
0.
0.
01
0.0
01
1
0.
01
0.0
01
-5
-5
FMD @ GSI
-5
0
x [fm]
5
-5
0
x [fm]
5
FMD @ GSI
-5
0
x [fm]
5
-5
0
x [fm]
5
16
C PAV only
Variation
16
rmatter [fm]
10-1
PAV 1g
PAV
99.1
105.0
2.52
2.49
2.88
2.60
ρ(r) [ρ0]
0.
5
0.1
0.01
0.001
10-2
Exp
110.8
10-3
10
0
x [fm]
5
16
5
2
C-p
8
C-n
01
0.01
1.0
0.0
0.1
0
1
5
-5
0
x [fm]
5
E2+ [MeV]
B(E2) [e2 fm4 ]
PAV
1.29
4.6
Exp
1.77
3.15 ± 0.95
Global Best Fit1
1.77
82 ± 14
0
1.
y [fm]
01
0.
0.
-5
6
0.5
0.1
0.01
0.
0. 1
0.001
01
-5
4
r [fm]
16
1
2.70 ± 0.03
2.76 ± 0.06
total
neutron
proton
0
00
y [fm]
-4
5
0.
0
C
01
-5
16
1
0.0
-5
1.0
y [fm]
0
rcharge [fm]
C
0.001
0.01
5
Eb [MeV]
-5
0
x [fm]
Raman et al, Atomic Data and Nuclear Data Tables 78
(2001) 1
5
➼ calculated B(E2) consistent with
anomalously long lifetime of 2+
state measured at RIKEN
Imai et al, PRL in print
EXON 2004 - July 5, 2004 –19
Summary and Outlook
y [fm]
5
0
-5
16
0.
0.
0. 1
0.001
01
1.0
5
0.
01
C-p
16
1
00
y [fm]
5
0
-5
0.0
1.
0
0.01
C-n
0.1
0.0
16
5
➼ No assumption about core, mean-field, core-nucleon interaction, ...
0
x [fm]
➼ Projection, configuration mixing → masses, radii, halos, spectra, transitions
shell structures, pairing, ...
01
0.5
0.1
0.01
for 3 ≤ A ≤ 60
-5
b H
➼ FMD Calculations with the same H+
0.
5
Feldmeier, Neff, Roth, Schnack, NP A632 (1998) 61
Neff, Feldmeier, NP A713 (2003), 311
bcorr
C
0
x [fm]
for many-body methods like HF, shell model, FMD
∼ ∼ ∼
-5
➼ Unitary Correlation Operator Method
describes short ranged central and tensor correlations
b = C† H C
➼ UCOM provides ab-initio correlated interaction H
01
➼ Predictions for many exotic light nuclei (before measurement)
➼ Correlated transitions: M1, β-decay, quenching
O
Outlook
16
➼ Resonances, width of many-body states
➼ Ab-initio correlated interaction in HF, RPA or similar for A > 60
bcorr
➼ Investigate long ranged tensor correlations – 3-body forces – H
Dr. T. Neff, A. Cribeiro, Prof. R. Roth, A. Ehrlich, H. Hergert
EXON 2004 - July 5, 2004 –20
Radial Correlations
BECAUSE 1.) realistic interactions have short-range repulsion
r = |~r1 − ~r2 |
shell model
correlated
➼ probability density of nucleons in the repulsive core strongly suppressed
potential & two-body density of 4 He in S=0, T=1 channel
✘ correlations in the relative distance of nucleons
cannot be described by Slater determinants (product of single-particle states)
EXON 2004 - July 5, 2004 –21
Tensor Correlations
BECAUSE 2) tensor interaction is an essential
component in any realistic interaction
➼ exchange of pions
✘ without tensor force no bound nuclei !
tensor operator
!
!
!
~r ~r
~r
~r
s12 , = 3 σ1 ·
σ2 · − (σ1 ·σ2 )
r r
r
r
Deuteron S = 1, T = 0
deuteron has L = 2 admixture due to tensor force
b b
w
u(r) b(r) L = 0 +
L = 2
rd =
r
r
couples spins and the relative spatial orientation
of nucleons
✘ correlations between the orientation of the PSfrag replacements
spins and the relative orientation of nucleons
cannot be described by Slater determinant
0.5
û(r)
0.4
0.3
0.2
ŵ(r)
0.1
0.0
0
2
6
4
8
10
r [fm]
↑↑
√1
2
↑↓ + ↓↑
EXON 2004 - July 5, 2004 –22
pr
Radial and Tensor Correlations
PSfrag replacements
p
r
pΩ
~p = ~pr + ~pΩ
n o
1 ~r ~r
~pΩ =
~pr = 2 r r ~p + ~p ~rr ~rr ,
C
=C
C
∼
∼ Ω∼ r
=e
−i G
−i G
∼Ω
∼r
e
Radial Correlator
gr =
∼
1
2r
n
o
~l × ~r − ~r × ~l
r
r
Tensor Correlations
1
pr s(r∼ )
2 ∼
+ s(r∼ )pr
∼
gΩ =
∼
➼ probability density shifted out of the repulsive core
200
ϑ(r∼ ) 32
n
(σ
· ~p )(σ ·~r) + (σ
·~r)(σ · ~p )
∼1 Ω ∼2 ∼
∼1 ∼ ∼2 Ω
∼
∼
o
➼ tensor force admixes other angular momenta
AV18
S = 0, T = 1
200
S = 1, T = 0
AV18
100
AV18
[fm−3 ]
0.25
0.20
0.35
0
0.30
-100
vc (r)
-200
0.0
0.5
0.05
1.0
1.5
2.0
2.5
3.0
r [fm] PSfrag replacements
0.10
replacements
0.25
PSfrag replacements
0.15
ρ(2) (r)
ρ̂(2) (r)
0.20
ρ̂(2) (r)
0.5
1.0
1.5
r [fm]
2.0
2.5
0
-100
-200
0.0
0.15
0.10
AV18
vc (r)
PSfrag replacements
vt (r)
0.5
1.0
1.5
2.0
2.5
3.0
r [fm]
ρ̂(2)
L=0 (r)
ρ̂(2)
L=2 (r)
0.05
0.0
V [MeV]
0.30
100
[fm−3 ]
V [MeV]
0.35
ρ(2) (r)
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
r [fm]
EXON 2004 - July 5, 2004 –23
GFMC calculations
−20
Energy (MeV)
−30
1/2 −
3/2 − α+2n
0+ α+n
4He
5He
2+
3+
α+d
1+
2+
0+
6He
6He+2n
6Li
−40
7Li
−50
5/2 −
7/2 −
α+t
1/2 −
3/2 −
2+
0+
8He
3+
1+
4+
3+
1+
+
2 6Li+α
0+
α+α
−60
AV6′
AV8′
4+
2+
1+
3+
8Be
AV18
IL2
Exp
10 B
−70
Wiringa, Pieper, PRL 89 (2002) 182501
genuine three-body forces (IL2)
needed for description of real
nuclei
EXON 2004 - July 5, 2004 –24
AV8’ Interaction
preliminary
No-Core Shell Model Calculations
4
He
0
0
Quasi-exact calculations for
light nuclei possible
α
-5
E [MeV]
-10
exact result from PRC64 (2001) 044001
-15
-20
4
placements
-25
10
r [fm]
Nmax
-30
experiment
0
10
• use no-core shell model code from
Petr Navratil (LLNL)
18
20
30
40
~Ω [MeV]
50
60
➼ neglected 3-body correlated terms
same order as genuine 3N interactions
➼ more investigations needed
oscillator width
test of two-body
approximation
EXON 2004 - July 5, 2004 –25
MTV Interaction
No-Core Shell Model Calculations
3
4
He
He
0
0
0
correlated
0
correlated
-5
-2
-10
0
-4
10
-6
-4
0
10
-6
20
30
40
~Ω [MeV]
50
60
70
20
ag replacements
PSfrag replacements
40
-8
0
10
20
30
40
~Ω [MeV]
50
60
70
exact results from PRC52 (1995) 2885
bare
-30
r [fm]
Nmax
-10
40
E [MeV]
E [MeV]
10
-8
r [fm]
Nmax
4
-20
-25
PSfrag
-5 replacements
20
-2
-15
0
PSfrag replacements
r [fm]
Nmax
E [MeV]
E [MeV]
4
-15
0
10
20
10
40
50
60
70
-20
-25
18
bare
-30
r [fm]
Nmax
30
~Ω [MeV]
0
10
20
30
40
~Ω [MeV]
50
60
70
• use no-core shell model code from
Pétr Navratil (LLNL)
only radial correlations
EXON 2004 - July 5, 2004 –26
Other Observables
Nucleon Momentum Distributions
Bonn-A
Argonne V18
1.
placements
1.
16 O
0.1
PSfrag replacements
Bonn A
AV18
α
β
γ
radial
VMC
LDA
0.1
4 He
0.01
central+
tensor
0.001
n̂(k)/A [fm3 ]
n̂(k)/A [fm3 ]
4 He
16 O
Bonn A
AV18
α
β
γ
radial
0.0001
0.00001
central
0
1
2
k [fm−1 ]
3
4
0.01
0.001
0.0001
0.00001
VMC
LDA
0
1
2
k [fm−1 ]
3
4
➼ correlations induce high-momentum components
➼ contributions of tensor correlations very big
➼ different correlator ranges relevant especially at the fermi surface
EXON 2004 - July 5, 2004 –27
Interaction in Momentum Space
1
S 0 channel
[2] 0 0 0 l0 −l Z 3 ?
b[2] b
~x jl0 (k0 x)Yl0 m0 ( x̂)
k l m = i M d x Ylm ( x̂) jl (kx) ~x H
klm H
∼
∼
3
S 1 channel
1
0
PSfrag replacements
correlated
-1
T = 0, 1; L = 0
α
β
γ
Bonn A
AV18 uncorrelated
fully correlated
-2
-3
0.0
0.5
1.0
1.5
k [fm−1 ]
➼ unique effective potential –
identical to Vlowk
2.0
S , T = 1, 0; L = 0
0
∼
uncorrelated
replacements
uncorrelated
correlated
[2] k , k V k [fm]
k Ĥ
∼
[2] k [fm]
k , kV
k Ĥ
∼
∼
1
-1
α
-2
2.0
1.5
-3 1.0
0.0
β
γ
0.5
Bonn A
AV18
1.0
1.5
2.0
k [fm−1 ]
➼ Vlowk Cutoff Λ = 1.0 − 2.0 fm−1
Kuo, Schwenk, nucl-th/0108041
EXON 2004 - July 5, 2004 –28
AV18 Interaction in Momentum Space
3
2
−1
[fm ]
1
0
1
2
k [fm−1 ]
1
0
[2]
b
S 0; k
S 0; k H
∼
replacements
4
1
k0
5
3
2
−1
[fm ]
1
0
4
k0
3
2
−1
[fm ]
1
0
1
2
3
4
5
k [fm−1 ]
bare potential
➼ “pre-diagonalization”
5
3
0
D
;
k
S 1; k V
1
∼
k0
5
5
PSfrag4 replacements
3
3
1
replacements
4
[2]
3
0
b
S 1; k H
D1 ; k
∼
5
1
2
5
PSfrag4 replacements
3
k [fm−1 ]
4
k0
3
1
0
S 0; k V
S
;
k
0
∼
Off-diagonal Matrix Elements
3
2
−1
[fm ]
1
0
1
2
3
4
5
k [fm−1 ]
correlated interaction
EXON 2004 - July 5, 2004 –29
AV8’ Interaction
preliminary
No-Core Shell Model Calculations
Increasing range of tensor correlator
He
0
0
-2
-2
-2
-2
-4
-6
-6
frag replacements
r [fm]
Nmax
10
20
30
40
~Ω [MeV]
r [fm]
60
N50
max
PSfrag replacements
-8
0
10
20
30
40
~Ω [MeV]
r [fm]
N50
max
-4
-6
PSfrag replacements
-8
0
-4
-6
PSfrag replacements
-8
60
-8
0
10
20
30
40
~Ω [MeV]
r [fm]
60
N50
max
0
10
20
30
40
50
60
0
10
20
30
40
50
60
~Ω [MeV]
He
0
0
-5
-5
-5
-5
-10
-10
-10
-10
-15
-20
-25
-20
-25
frag replacements
10
20
30
40
~Ω [MeV]
r [fm]
60
N50
max
-20
10
20
30
40
~Ω [MeV]
r [fm]
N50
max
-20
PSfrag replacements
-30
0
-15
-25
PSfrag replacements
-30
0
-15
-25
PSfrag replacements
-30
r [fm]
Nmax
-15
E [MeV]
0
E [MeV]
0
E [MeV]
E [MeV]
4
-4
E [MeV]
0
E [MeV]
0
E [MeV]
E [MeV]
3
60
-30
0
10
20
30
40
~Ω [MeV]
r [fm]
60
N50
max
~Ω [MeV]
EXON 2004 - July 5, 2004 –30
11
B (3He, t) 11C – Gamov-Teller transitions
11
11
B
1.0
Experiment
6
-2
0.4
5/2-
z [fm]
2
5/23/2-
0.6
1/2-
0
4
3/2-
B(GT)
2
y [fm]
6
0.8
4
C
0
-2
-4
-4
0.2
5
0.6
0.4
4
6
-6
-4
-2
0
2
x [fm]
4
6
-6
-4
-2
0
2
x [fm]
4
6
0
-6
4
5
FMD
15
10
0
Eex [MeV]
-2
3/2-
0
y [fm]
2
-4
-6
1/2-
0.6
3/2-
0.2
0.0
0
5
10
5/2-
3/2-
B(GT)
0
2
x [fm]
-2
0.2
0.4
-2
-4
0.8
FMD with configuration mixing
-4
2
Eex [MeV]
6
0.0
1.0
-6
4
NCSM
15
10
y [fm]
0.8
B(GT)
NCSM: Navrátil, Ormand
no core shell model with 3-body
force, PRC 68(2003)
third 3/2− missing
0
6
5/2-
0.0
1.0
transition: σ · τ+
3/2-
6
5/2
4
5/2-
0
2
x [fm]
-
-2
1/2-
-4
3/2-
-6
-6
3/2-
-6
15
Eex [MeV]
Exp.: Y. Fujita, P. von Bretano et al. to be
published
EXON 2004 - July 5, 2004 –31
FMD
ρ(1) (~r) [ρ0 ]
Silizium - Einteilchendichte des intrinsischen Zustands
Variation → rund
Erdnuss
lokales Minimum
vor Projektion ca. 5 MeV
höher als rund
Ufo
vor Projektion ca. 10 MeV
höher als rund
EXON 2004 - July 5, 2004 –32