Transcript Problems and Ideas at the Dawn of Three
8 時 11 分 ECT* workshop “Three-Nucleon Forces in Vacuum and in the medium” Trento, Italy July 11 (11-15), 20 11
Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model
Takaharu Otsuka University of Tokyo / MSU
Outline 1. Monopole problem in the shell model 2. Shell evolution in exotic nuclei 3. Solution by three-body force Introduction to talks by J. Holt, A. Schwenk and T. Suzuki
Spectra of Ca isotopes calculated by most updated NN interaction microscopically obtained By Y. Tsunoda and N. Tsunoda N3LO V low-k with L =2.0 fm -1 2 nd and 3 rd order Q-box 4hw and 6hw s.p.e. used Present GXPH1A KB3G GXPF1A KB3G for comparison
Two-body matrix elements (TBME) may be calculated to a rather good accuracy 40 Ca core is not very stable yet -> 0+ energy lowered
48 Ca
As N or Z is changed to a large extent in exotic nuclei, the shell structure is changed (evolved) by • Monopole component of the NN interaction
Linearity : Shift
Averaged over possible orientations
n j’
: # of particles in
j’
<n
j’
> can be ~ 10 in exotic nuclei -> effect quite relevant to neutron-rich exotic nuclei
Strasbourg group made a major contribution in initiating systematic use of the monopole interaction.
(Poves and Zuker, Phys. Rep. 70, 235 (1981))
j = j’
What’s this ?
j = j’ T=1 monopole interactions in the pf shell GXPF1A G-matrix (H.-Jensen) Tensor force ( p + r exchange)
Basic scale ~ 1/10 of T=0 Repulsive
corrections to G-matrix
j = j’ j = j’
T=1
monopole interactions in the sd shell SDPF-M (~USD) G-matrix (H.-Jensen) Tensor force ( p + r exchange) Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix
T=0 monopole interaction
The correction is opposite !
T=0
monopole interactions in the
pf
shell Tensor force ( p + r exchange) GXPF1A G-matrix (H.-Jensen)
f-f p-p f-p
“Local pattern” tensor force
T=0
monopole interactions in the pf shell Tensor force ( p + r exchange) GXPF1A shell-model int.
G-matrix (H.-Jensen)
Tensor component is subtracted
Correction is attractive
Outline 1. Monopole problem in the shell model 2 . Shell evolution in exotic nuclei 3. Solution by three-body force
Treatment of tensor force by V low k
and
Q box (3 rd order) Monopole component of tensor interactions in pf shell Bare ( AV8’ ) short-range correlation by V low k in-medium correction with intermediate states (> 10 hw, 3 rd order) only for comparison
Systematic description of monopole properties of exotic nuclei can be obtained by an extremely simple interaction as Parameters are fixed for all nuclei monopole component of tensor force
in nuclear medium
almost equal ?
monopole component of tensor force
in free space
Shell evolution due to proton-neutron tensor + central forces Changes of single-particle properties due to these nuclear forces
T=1
NN interaction more relevant to ls splitting change stable nucleus exotic nucleus with neutron skin proton neutron r d r /dr
ls
splitting smaller
From RIA Physics White Paper
Neutron single-particle energies at N=20 for Z=8~20 p 3/2 low Z 8 14 16 20 solid line : full VMU (central + tensor) dashed line : central only 16 Tensor force makes changes more dramatic.
20 These single-particle energies are “normal” f 7/2 -p 3/2 2~3 MeV N=20 gap ~ 6 MeV d 5/2 s 1/2 more exotic d 3/2 Z PRL 104, 012501 (2010)
Increase of 2+ excitation energy Neutron number
Outline 1. Monopole problem in the shell model 2 . Shell evolution in exotic nuclei 3. Solution by three-body force
Nuclear Chart - Left Lower Part Why is the drip line of Oxygen so near ?
Neutron number
Single-Particle Energy for Oxygen isotopes by microscopic eff. int.
G-matrix+ core-pol. : Kuo, Brown by phenomenological eff. int.
- G-matrix + fit SDPF-M Utsuno, O., Mizusaki, Honma, Phys. Rev. C
60
, 054315 (1999) V low-k : Bogner, Schwenk, Kuo USD-B Brown and Richter, Phys. Rev. C
74
, 034315 (2006) trend
What is the origin of the repulsive modification of T=1 monopole matrix elements ?
The same puzzle as in the pf shell A solution within bare 2-body interaction is very unlikely (considering efforts made so far) Zuker, Phys. Rev. Lett. 90, 042502 (2003) 3-body interaction
The clue : Fujita-Miyazawa 3N mechanism
(D
-hole excitation)
D
particle m=1232 MeV S=3/2, I=3/2
Miyazawa, 2007 D p N p N N
Renormalization of NN interaction due to
D
excitation in the intermediate state
D Modification to bare NN interaction (for NN scattering) T=1 attraction between NN effectively
Pauli blocking effect on the renormalization of single-particle energy
m m single particle states m’ D m’ m’ D m Renormalization of single particle energy due to D -hole excitation more binding (attractive) m Another valence particle in state m’ Pauli Forbidden
The effect is suppressed
m
Inclusion of Pauli blocking
m m’ m’ m’ D m Pauli forbidden (from previous page) D m’ m This Pauli effect is included automatically by the exchange term.
m m Most important message with Fujita-Miyazawa 3NF m m’ Effective monopole repulsive interaction m’ D + D Renormalization of single particle energy m’ m m Pauli blocking
same
Monopole part of Fujita-Miyazawa 3-body force m’ D m’ m
(i) D -hole excitation in a conventional way (ii) EFT with D D -hole dominant role in determining oxygen drip line -> J.Holt, A. Schwenk, T. Suzuki (iii) EFT incl. contact terms (N 2 LO)
O, Suzuki, Holt, O, Schwenk, Akaishi, PRL 105 (2010)
Ground-state energies of oxygen isotopes NN force + 3N-induced NN force (Fujita-Miyazawa force) Drip line
N What was wrong with “microscopic theories” ?
N D N N
Observed in NN scattering
(Effective) two-body interaction N
present picture
N N N If the origin is “forgotten”, constant change of single-particle energy or This is what happened in “microscopic theories”, leading to wrong drip line.
For neutron matter : attractive k k states below Fermi level k k repulsive Brown and Green, Nucl.Phys. A137, 1 (1969 Fritsch, Kaiser and Weise, Nucl. Phys. A750, 259 (2005); Tolos, Friman and Schwenk, Nucl.Phys. A806}, 105 (2008); Hebeler and Schwenk, arXiv:0911.0483 [nucl-th]
For valence neutrons: states outside the core Attractive (single-particle energy renormalization) repulsive (valence neutron interaction)
Quick Summary
More from J. Holt, A. Schwenk and T. Suzuki
Major monopole forces are due to FM 3NF
V
+ + m =1 fm basic binding (T=0), repulsive (T=1) except for j=j’ variation of shell structure limit of existence, shell structure at far stability
Casablanca mechanism Love = attractive force* This love is reduced by the presence of Rick This love is reduced by the presence of Victor Victor Rick repulsion *This equation has no proof.
E N D
The central force is modeled by a Gaussian function
V = V 0 exp( -(r/
m
) 2 ) (S,T dependences)
with V 0 = -166 MeV, m =1.0 fm, (S,T) factor (0,0) (1,0) (0,1) (1,1) ------------------------------------------------- relative strength 1 1 0.6 -0.8
Can we explain the difference between f-f/p-p and f-p ?
Eigenvalues of HO potential 5h
w
4h
w
3h
w
2h
w
1h
w
Spin-orbit splitting 126 82 50 28 20 8 2 Magic numbers Mayer and Jensen (1949)
density saturation + short-range NN interaction + spin-orbit splitting
Mayer-Jensen’s magic number with rather constant gaps
(except for gradual A dependence)
robust feature -> nuclear forces not included in the above can change it -> tensor force
Brief history on our studies on tensor force Magic numbers may change due to spin-isospin nuclear forces Tensor force produces unique and sizable effect Tensor and central forces -> Weinberg-type model
Tensor Interaction by pion exchange
V T = (
t
1
t
2
) ( [s
1
s
2
]
(
2 ) Y ( 2
)
(
W
) )
Z(
r
)
contributes only to
S=1
states
relative motion
p
meson : primary source
s .
p s .
r
meson
Yukawa
(~
p
+
p
) : minor (~1/4) cancellation
Ref: Osterfeld, Rev. Mod. Phys. 64, 491 (92)
How does the tensor force work ?
Spin of each nucleon is parallel, because the total spin must be S=1 The potential has the following dependence on the angle
q
with respect to the total spin S .
V ~ Y
2,0
~ 1 – 3 cos
2
q q
S relative coordinate
q
=0 attraction
q
=
p
/2 repulsion
Monopole effects due to the tensor force - An intuitive picture -
wave function of relative motion spin of nucleon
large relative momentum small relative momentum
attractive
j
> = l + ½, j < = l – ½
repulsive
TO et al., Phys. Rev. Lett. 95, 232502 (2005)
wave function when two nucleons interact - approx. by linear motion -
k 2 k 1
large relative momentum k strong damping k = k
1
– k
2
, K = k
1
+ k
2 k 2 k 1
small relative momentum k loose damping wave function of relative coordinate
k 2 k 1 k 1 k 2
wave function of relative coordinate
General rule of monopole interaction of the tensor force
TO.
et al
., Phys. Rev. Lett. 95, 232502 (2005)
j
<
= l – ½ neutron
j
>
= l + ½
j’
<
proton
j’
>
Identity for tensor monopole interaction
(2
j
> +1) (
j’ v m,T j
> ) + (2
j
< +1) (
j’ v m,T j
< ) = 0
v m,T
: monopole strength for isospin
T
The central force is modeled by a Gaussian function
V = V 0 exp( -(r/
m
) 2 ) (S,T dependences)
with V 0 = -166 MeV, m =1.0 fm, (S,T) factor (0,0) (1,0) (0,1) (1,1) ------------------------------------------------- relative strength 1 1 0.6 -0.8
Can we explain the difference between f-f/p-p and f-p ?
T=0
monopole interactions in the
pf
shell Tensor force ( p + r exchange) GXPF1 G-matrix (H.-Jensen) Central (Gaussian) - Reflecting radial overlap -
f-f p-p f-p
Similarity to Chiral Perturbation of QCD S. Weinberg, PLB 251, 288 (1990)
Central force: strongly renormalized
In nuclei
Tensor force is explicit
finite range (Gaussian) p + r exchange
Central part changes as the cut-off L changes T=0 T=1
j-j’
Tensor (reminder)
Measured spectroscopic factors from Dickhoff short-range + in-medium corrections Tensor force remains almost unchanged !
Higher order effects due to the tensor force yield renormalization of central forces.
Multipole component of tensor forces - diagonal matrix elements -
Test by experiments
An example with 51 Sb isotopes with VMU interaction g 7/2
Z
=51 (= 50 + 1) isotopes
change driven by neutrons in 1h 11/2
tensor force in VMU (splitting increased by ~ 2 MeV)
h 11/2 h 11/2 h 11/2 - g 7/2
repulsive attractive No mean field theory, (Skyrme, Gogny, RMF) explained this before.
Consistent with recent experiment - Position of p 3/2 One of the Day 1 experiments at
RIBF
by Nakamura et al.
From Grawe, EPJA25, 357 Proton single-particles levels of Ni isotopes Crossing here is consistent with exp. on Cu isotopes g 9/2 occupied N Central Gaussian + Tensor solid line: full VMU effect dotted line: central only shaded area : effect of tensor force
Zr Shell structure of a key nucleus 100 Sn solid line : full VMU (central + tensor ) dashed line : central only shaded area : effect of tensor force Sn Exp. d5/2 and g7/2 should be close Seweryniak et al.
Phys. Rev. Lett. 99, 022504 (2007) Gryzywacz et al.
d 3/2 s 1/2 d 5/2 proton f neutron 7/2 Z=28 gap is reduced also Potential Energy Surface 42 14 Si 28 full
Si isotopes SM calc. by Utsuno et al.
exp.
Strong oblate Deformation ?
Tensor force removed from cross-shell interaction Other calculations show a variety of shapes.
42
42
Si: B. Bastin, S. Grévy et al.,
PRL 99 (2007) 022503
Si Otsuka, Suzuki and Utsuno, Nucl. Phys. A805, 127c (2008)
Spectroscopic factors obtained by (e,e’p) on
48
Ca and the tensor force
Collaboration with Utsuno and Suzuki
Spectroscopic factor for 1p removal from 48 Ca Same interaction as the one for 42 Si • p d 5/2 – deep hole state More fragmentation • Distribution of strength – quenching factor 0.7 is needed (as usual).
– Agreement between experiment and theory for both position and strength (e,e’p): Kramer et al., NP A679, 267 (2001)
with full tensor force
d 3/2
What happens, if the tensor force is taken away ?
s 1/2 d 5/2 no tensor in the cross shell part
Summary 1. Changes of shell structure and magic numbers i n exotic nuclei are a good probe to see effects of described by VMU. nuclear forces. Such changes are largely due to tensor force, as have been Transfer reactions have made important contributions. 2. The tensor force remain ~unchanged by the treatments of short-range correlations and in-medium correction.
This feature is very unique.
3.
(e, e’p) data on 48 Ca suggests the importance of the tensor force, which is consistent with exotic feature of 42 Si.
Direct reactions with RI beam should play important roles in exploring structure of exotic nuclei driven by nuclear forces.
4 . Fujita-Miyazawa 3N force can be the next subject for the shell evolution.
Summary 1. Monopole interactions : effects magnified in neutron-rich nuclei 2. Tensor force combined with central force : a unified description particularly for proton-neutron monopole correlation. -> N=20 Island of inversion, 42 Si, ７８ Ni, 100 Sn, Sb, is very similar to the 132 bare Sn, Z=64,… Tensor force in nuclear medium one.
This central force may be a challenge for microscopic theories.
3.
Fujita-Miyazawa 3-body force produces repulsive effective interaction between valence neutrons in general.
The wider spacings as N between neutron <--> shell quenching single-particle levels can become increases, and new magic numbers may arise.
Examples are shown for O and Ca isotopes with visible effects.
4. Structure change on top of the shell evolution -> diagonalization with super computer
Collaborators
T. Suzuki Nihon U .
M. Honma Aizu Y. Utsuno JAEA N. Tsunoda Tokyo K. Tsukiyama Tokyo M. H.-Jensen Oslo A. Schwenk Darmstadt J. Holt ORNL K. Akaishi RIKEN
E N D
Ca ground-state energy
cont’d
SPE : GXPF1 f7:-8.62 f5: -1.38 p3: -5.68 p1: -4.14
Ca 2+ level systematics 2 + of 48 Ca rises by 3N becomes about right by using GXPF1A SPE N=32, 34 higher 2 + levels
48 Ca M1 excitation GXPF1 10 8-13MeV spe : GXPF1 Spin quenching factor 0.8
Summary-2 Dominant monopole forces are due to FM 3NF
V
+ m =1 fm basic binding variation of shell structure limit of existence +
古典力学での三体問題と三体力 酒井（英）氏より拝借
H
P E
2 2
m E
P M
2 2
m M
P G
2 2
m G
Gm E m M r EM
Gm E m G r EG
Gm M m G r MG
GPS
の位置をこの方程式を数値的に 解いても正確には求まらない。
GPS
の 効果もあるが） 役割を果たさない！ それは地球が変形するから（もちろん相対論
V
(
r E
,
r M
,
r G
)
（有効三体力） ここでの三体力は、二体力＋超多体問題を回避するための“有効”相互作用 正真正銘の三体力は存在するか？
Ground-state energies of oxygen isotopes NN force + 3N-induced NN force (Fujita-Miyazawa force) Drip line
Collaborators
T. Suzuki Nihon U .
M. Honma Aizu Y. Utsuno JAEA N. Tsunoda Tokyo K. Tsukiyama Tokyo M. H.-Jensen Oslo A. Schwenk TRIUMF/Darmstadt J. Holt ORNL K. Akaishi RIKEN