潮位曲線表 - 東京都建設局潮位曲線表 令和2年4月 時刻 潮位 時刻 潮位 時刻 潮位 時刻 潮位 時分 m 時分 m 時分 m 時分 m 1日 水 小 8:09 1.44
Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka...
-
Upload
aleah-jerrett -
Category
Documents
-
view
217 -
download
0
Transcript of Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka...
![Page 1: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/1.jpg)
Problems and Ideasat the Dawn of
Three-Body Force Effects in the Shell Model
Takaharu OtsukaUniversity of Tokyo / MSU
午前 12:49
ECT* workshop “Three-Nucleon Forces in Vacuum and in the medium”Trento, Italy
July 11 (11-15), 2011
![Page 2: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/2.jpg)
Outline
1. Monopole problem in the shell model
2. Shell evolution in exotic nuclei
3. Solution by three-body force
Introduction to talks byJ. Holt, A. Schwenk and T. Suzuki
![Page 3: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/3.jpg)
Spectra of Ca isotopes calculated by mostupdated NN interaction microscopically obtained
By Y. Tsunoda and N. Tsunoda
N3LO
Vlow-k with =2.0 fm-1
2nd and 3rd order Q-box
4hw and 6hw
s.p.e. used
PresentGXPH1A
KB3G
GXPF1AKB3G
for comparison
![Page 4: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/4.jpg)
Two-body matrix elements (TBME) may be calculatedto a rather good accuracy
40Ca core is not very stable yet -> 0+ energy lowered
![Page 5: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/5.jpg)
![Page 6: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/6.jpg)
48Ca
![Page 7: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/7.jpg)
![Page 8: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/8.jpg)
![Page 9: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/9.jpg)
• Monopole component of the NN interaction
Averaged over possible orientations
As N or Z is changed to a large extent in exotic nuclei, the shell structure is changed (evolved) by
Strasbourg group made a major contribution in initiating systematic use of the monopole interaction. (Poves and Zuker, Phys. Rep. 70, 235 (1981))
<nj’ > can be ~ 10 in exotic nuclei -> effect quite relevant to neutron-rich exotic nuclei
nj’ : # of particles in j’Linearity: Shift
![Page 10: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/10.jpg)
![Page 11: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/11.jpg)
T=1 monopole interactions in the pf shell
j = j’ j = j’
Tensor force (+exchange)
G-matrix (H.-Jensen)
GXPF1A
What’s this ?
Basic scale ~ 1/10 of T=0
Repulsive corrections to G-matrix
![Page 12: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/12.jpg)
T=1 monopole interactions in the sd shell
SDPF-M (~USD) G-matrix
(H.-Jensen)
Tensor force (+exchange)
j = j’ j = j’
Basic scale ~ 1/10 of T=0
Repulsive corrections to G-matrix
![Page 13: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/13.jpg)
T=0 monopole interaction
The correction is opposite !
![Page 14: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/14.jpg)
T=0 monopole interactions in the pf shell
GXPF1A
G-matrix (H.-Jensen)
Tensor force (+exchange)
f-f p-p f-p
“Local pattern” tensor force
![Page 15: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/15.jpg)
T=0 monopole interactions in the pf shell
GXPF1Ashell-model int.
G-matrix (H.-Jensen)
Tensor force (+exchange)
Tensor component
is subtracted
Correction isattractive
![Page 16: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/16.jpg)
Outline
1. Monopole problem in the shell model
2. Shell evolution in exotic nuclei
3. Solution by three-body force
![Page 17: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/17.jpg)
Treatment of tensor force by V low k and Q box (3rd order)
Monopole component of tensor interactions in pf shell
short-range correlationby V low k
in-medium correctionwith intermediate states(> 10 hw, 3rd order)
Bare (AV8’)
only for comparison
![Page 18: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/18.jpg)
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 monopole component of
tensor force in free space
almost equal ?
![Page 19: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/19.jpg)
Shell evolutiondue to proton-neutron tensor + central forces
Changes of single-particle properties due to these nuclear forces
![Page 20: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/20.jpg)
ls splitting smaller
exotic nucleus with neutron skin
proton
neutron
d/dr
stable nucleus
T=1 NN interaction more relevant to ls splitting change
![Page 21: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/21.jpg)
From RIA Physics White Paper
![Page 22: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/22.jpg)
Neutron single-particle energies at N=20 for Z=8~20
Z
8 14Z
16 20
These single-particleenergies are “normal”
f7/2-p3/2 2~3 MeV N=20 gap ~ 6 MeV
energ
y (
MeV
)
d5/2 s1/2 d3/2
16
20
solid line : full VMU (central + tensor)
dashed line : central only
p3/2 low
Tensor force makeschanges more dramatic.
PRL 104, 012501 (2010)more exotic
![Page 23: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/23.jpg)
Increase of 2+ excitationenergy
Neutron number
2+
level (M
eV
)
![Page 24: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/24.jpg)
Outline
1. Monopole problem in the shell model
2. Shell evolution in exotic nuclei
3. Solution by three-body force
![Page 25: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/25.jpg)
Pro
ton
nu
mb
er
Neutron number
Nuclear Chart- Left Lower Part -
Why is the drip line of
Oxygen so near ?
![Page 26: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/26.jpg)
Single-Particle Energy for Oxygen isotopes
Utsuno, O., Mizusaki, Honma, Phys. Rev. C 60, 054315 (1999)
G-matrix+ core-pol. : Kuo, Brown
SDPF-M
- G-matrix + fit -
V low-k : Bogner, Schwenk, Kuo
trend
trend
by microscopic eff. int. by phenomenological eff. int.
USD-B Brown and Richter, Phys. Rev. C 74, 034315 (2006)
![Page 27: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/27.jpg)
What is the origin of
the repulsive modification of
T=1 monopole matrix elements ?
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 same puzzle as in the pf shell
![Page 28: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/28.jpg)
The clue : Fujita-Miyazawa 3N mechanism-hole excitation)
particle m=1232 MeV S=3/2, I=3/2
N N N
Miyazawa, 2007
![Page 29: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/29.jpg)
Renormalization of NN interaction due to excitation in the intermediate state
T=1attraction
between NNeffectively
Modification tobare NN interaction(for NN scattering)
![Page 30: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/30.jpg)
Pauli blocking effect on the renormalization of single-particle energy
Pauli blocking effect on the renormalization of single-particle energy
Pauli ForbiddenThe effect is
suppressed
m
m
m’ m’
Renormalization of single particle energy
due to -hole excitation more binding
(attractive)
m
m
m’
singleparticle states
Another valenceparticle in state m’
![Page 31: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/31.jpg)
Pauli forbidden(from previous page) This Pauli effect is
included automaticallyby the exchange term.
m
mm’
m’
m
m’ m’
m
Inclusion of Pauli blocking
![Page 32: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/32.jpg)
Most important message with Fujita-Miyazawa 3NFMost important message with Fujita-Miyazawa 3NF
+
Renormalizationof single particle
energy
m
m
m’
same
Monopole part ofFujita-Miyazawa3-body force
Pauli blockingm
mm’
m’
Effective monopole repulsive interaction
![Page 33: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/33.jpg)
(i) -hole excitation in a conventional way
-hole dominantrole in determiningoxygen drip line
(ii) EFT with
(iii) EFT incl. contact terms (N2LO)
-> J.Holt, A. Schwenk, T. Suzuki
![Page 34: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/34.jpg)
Ground-state energies of oxygen isotopes
Drip line
NN force + 3N-induced NN force (Fujita-Miyazawa force)
O, Suzuki, Holt, O, Schwenk, Akaishi, PRL 105 (2010)
![Page 35: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/35.jpg)
(Effective) two-body interaction
constant change ofsingle-particle energy
If the origin is“forgotten”,
This is what happenedin “microscopic theories”,leading to wrong drip line.
N N
NN
N
NN
N
or
What was wrong with “microscopic theories” ?
Observed in NN scattering
presentpicture
![Page 36: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/36.jpg)
For neutron matter : states below Fermi level
attractive
repulsivek
k
k
k
Brown and Green, Nucl.Phys. A137, 1 (1969Fritsch, 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]
![Page 37: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/37.jpg)
For valence neutrons: states outside the core
Attractive (single-particle energy renormalization)
repulsive
(valence neutron interaction)
![Page 38: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/38.jpg)
Major monopole forces are due to
+V
=1 fm
Quick Summary More from J. Holt, A. Schwenk and T. Suzuki
+
FM 3NF
basic binding (T=0),repulsive (T=1) except for j=j’
variation ofshell structure
limit of existence,shell structure at far stability
![Page 39: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/39.jpg)
Casablanca mechanism
RickVictor
This love is reduced by the presence of Rick
This love is reduced by the presence of Victor
repulsion
Love = attractive force*
*This equation hasno proof.
![Page 40: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/40.jpg)
E N D
![Page 41: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/41.jpg)
![Page 42: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/42.jpg)
The central force is modeled by a Gaussian function
V = V0 exp( -(r/) 2) (S,T dependences)
with V0 = -166 MeV, =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 ?
![Page 43: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/43.jpg)
Spin-orbit splitting
Eigenvalues of HO potential
Magic numbers
Mayer and Jensen (1949)126
8
20
28
50
82
2
5h
4h
3h
2h
1h
![Page 44: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/44.jpg)
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
![Page 45: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/45.jpg)
Magic numbers may change due to spin-isospin nuclear forces
Tensor force produces unique and sizable effect
Tensor and central forces -> Weinberg-type model
Brief history on our studies on tensor force
![Page 46: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/46.jpg)
VT = (2) Y(2 Z(r)
contributes only to S=1
states
relative motion
meson (~ +) : minor (~1/4) cancellation
meson : primary source
Ref: Osterfeld, Rev. Mod. Phys. 64, 491 (92)
Tensor Interaction by pion exchange
Yukawa
![Page 47: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/47.jpg)
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 with respect to the total spin S.
V ~ Y2,0 ~ 1 – 3
cos2
attraction=0
repulsion=/2
S
relativecoordinate
![Page 48: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/48.jpg)
Monopole effects due to the tensor force
- An intuitive picture -
wave function of relative motion
large relative momentum small relative momentum
attractive repulsive
spin of nucleon
TO et al., Phys. Rev. Lett. 95, 232502 (2005)j> = l + ½, j< = l – ½
![Page 49: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/49.jpg)
wave function when two nucleons interact
k1k2
k1k2
- approx. by linear motion -
k = k1 – k2 , K = k1 + k2
large relativemomentum k
strong damping
wave functionof relative coordinate
k1k2
wave functionof relative coordinate
small relativemomentum k
loose damping
k1k2
![Page 50: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/50.jpg)
j> = l + ½
j< = l – ½
j’<
proton
neutron
j’>
General rule of monopole interaction
of the tensor force
Identity for tensor monopole interaction
(2j> +1) vm,T + (2j< +1) vm,T = 0( j’ j>) ( j’ j<)
vm,T : monopole strength for isospin T
TO. et al., Phys. Rev. Lett. 95, 232502 (2005)
![Page 51: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/51.jpg)
The central force is modeled by a Gaussian function
V = V0 exp( -(r/) 2) (S,T dependences)
with V0 = -166 MeV, =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 ?
![Page 52: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/52.jpg)
GXPF1
G-matrix (H.-Jensen)
Central (Gaussian)- Reflecting
radial overlap -
Tensor force (+exchange)
T=0 monopole interactions in the pf shell
f-f p-p f-p
![Page 53: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/53.jpg)
S. Weinberg, PLB 251, 288 (1990)
Tensor force is explicit
Central force:strongly renormalized
finite range(Gaussian)
In nuclei
+ exchange
Similarity to Chiral Perturbation of QCD
![Page 54: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/54.jpg)
Central part changes as the cut-off changes
T=0 T=1
j-j’
Mon
opol
e in
t. (M
eV)
Tensor (reminder)
![Page 55: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/55.jpg)
from Dickhoff
Rat
io to
naï
ve s
ingl
e-pa
rtic
le m
odel
Measured spectroscopic factors
short-range +in-mediumcorrections
Tensor force remains almost unchanged !
Higher order effectsdue to the tensor forceyield renormalization of central forces.
![Page 56: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/56.jpg)
Multipole component of tensor forces- diagonal matrix elements -
![Page 57: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/57.jpg)
Test by experiments
![Page 58: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/58.jpg)
Z =51 (= 50 + 1) isotopes
No mean field theory, (Skyrme, Gogny, RMF) explained this before.
An example with 51Sb isotopes with VMU interaction
tensor force in VMU (splitting increased by ~ 2 MeV)
change driven by neutrons in 1h11/2
g7/2
![Page 59: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/59.jpg)
Consistent with recent experiment
One of the Day 1 experiments at RIBF by Nakamura et al.
- Position of p3/2
![Page 60: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/60.jpg)
FromGrawe, EPJA25,357
g9/2 occupied
E (
MeV
)
N
Central Gaussian+ Tensor
solid line: full VMU effect
dotted line: central only
Crossing hereis consistentwith exp. onCu isotopes
Proton single-particles levels of Ni isotopes
shaded area : effect of tensor force
![Page 61: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/61.jpg)
solid line : full VMU (central + tensor)
dashed line : central only
shaded area : effect of tensor force
Shell structure of a key nucleus 100Sn
Exp. d5/2 and g7/2 should be closeSeweryniak et al. Phys. Rev. Lett. 99, 022504 (2007) Gryzywacz et al.
Zr Sn
![Page 62: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/62.jpg)
Potential Energy Surface
s1/2
Z=28 gap is reduced alsoproton
neutronf7/2d3/2
d5/2
full
Tensor force removedfrom cross-shell interaction
Strong oblateDeformation ?
Otsuka, Suzuki and Utsuno,Nucl. Phys. A805, 127c (2008)
exp.
4214Si28
42Si
Other calculationsshow a variety of shapes. 42Si: B. Bastin, S. Grévy et al.,
PRL 99 (2007) 022503
Si isotopesSM calc. by Utsuno et al.
![Page 63: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/63.jpg)
Spectroscopic factors obtained by (e,e’p)
on 48Ca and the tensor force
Collaboration with Utsuno and Suzuki
![Page 64: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/64.jpg)
Spectroscopic factor for 1p removal from 48Ca
• d5/2 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)
Same interaction as theone for 42Si
![Page 65: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/65.jpg)
What happens, if the tensor force is taken away ?
no tensor in the cross shell part
s1/2
d3/2
d5/2
with full tensor force
![Page 66: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/66.jpg)
Summary
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 48Ca suggests the importance of the tensor force, which is consistent with exotic feature of 42Si.
Direct reactions with RI beam should play important roles in exploring structure of exotic nuclei driven by nuclear forces.
1. Changes of shell structure and magic numbers in exotic nuclei are a good probe to see effects of nuclear forces. Such changes are largely due to tensor force, as have been described by VMU. Transfer reactions have made important contributions.
4. Fujita-Miyazawa 3N force can be the next subject for the shell evolution.
![Page 67: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/67.jpg)
Summary1. 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, 42Si, 78 Ni, 100Sn, Sb, 132Sn, Z=64,…
Tensor force in nuclear medium is very similar to the bare 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 spacings between neutron single-particle levels can become wider as N increases, and new magic numbers may arise.Examples are shown for O and Ca isotopes with visible effects.
<--> shell quenching 4. Structure change on top of the shell evolution -> diagonalization with super computer
![Page 68: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/68.jpg)
Collaborators
T. Suzuki Nihon U.M. Honma AizuY. Utsuno JAEAN. Tsunoda TokyoK. Tsukiyama Tokyo M. H.-Jensen Oslo
A. Schwenk DarmstadtJ. Holt ORNLK. Akaishi RIKEN
![Page 69: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/69.jpg)
E N D
![Page 70: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/70.jpg)
SPE : GXPF1 f7:-8.62 f5: -1.38 p3: -5.68 p1: -4.14
Ca ground-state energy cont’d
![Page 71: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/71.jpg)
Ca 2+ level systematics
2+ of 48Ca rises by 3N becomes about right by using GXPF1A SPE
N=32, 34 higher 2+ levels
![Page 72: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/72.jpg)
Spin quenching factor 0.8
8-13MeV
GXPF1
spe : GXPF1
48Ca M1 excitation 10
![Page 73: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/73.jpg)
Dominant monopole forces are due to
+V
=1 fm
Summary-2
+
FM 3NF
basic binding variation ofshell structure
limit of existence
![Page 74: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/74.jpg)
古典力学での三体問題と三体力
(有効三体力)
MG
GM
EG
GE
EM
ME
G
G
M
M
E
E
r
mGm
r
mGm
r
mGm
m
P
m
P
m
PH
222
222
),,( GME rrrV
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?
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(もちろん相対論効果もあるが)
酒井(英)氏より拝借
![Page 75: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/75.jpg)
Ground-state energies of oxygen isotopes
Drip line
NN force + 3N-induced NN force (Fujita-Miyazawa force)
![Page 76: Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model Takaharu Otsuka University of Tokyo / MSU 3時26分 3時26分 ECT* workshop “Three-Nucleon.](https://reader036.fdocuments.net/reader036/viewer/2022062404/551ae80255034606048b5c95/html5/thumbnails/76.jpg)
Collaborators
T. Suzuki Nihon U.M. Honma AizuY. Utsuno JAEAN. Tsunoda TokyoK. Tsukiyama Tokyo M. H.-Jensen Oslo
A. Schwenk TRIUMF/DarmstadtJ. Holt ORNLK. Akaishi RIKEN