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W. Lynch, Monday T. Otsuka, Wednesday A. Wuosmaa, Thursday T. Aumann, Thursday. The origin of strong proton-neutron correlations revealed in break-up reactions. Mihai Horoi Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA - PowerPoint PPT Presentation
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M. Horoi CMU
The origin of strong proton-neutron correlations revealed in break-up
reactions
Mihai Horoi Department of Physics, Central Michigan University,
Mount Pleasant, Michigan 48859, USA
Supported by NSF grant PHY-1068217 and DOE grant DF-FC02-09ER41584
UNEDFNN2012 San Atonio, May 31, 2012
- W. Lynch, Monday
- T. Otsuka, Wednesday
- A. Wuosmaa, Thursday
- T. Aumann, Thursday
Staggering of Ca Isotopes Charge Radii: A Shell Model Approach
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Valence space: d3/2 , s1/2 , f7/2 , p3/2
€
Hvalence ≡ H2−body
€
r2
ch= r2
sdnsd
p + r2
pfnpf
p
E. Caurier et al, PLB 522, 240 (2001)
I. Talmi NPA 423, 189 (1984)
Evolution of lowest 0+ and 1+ states in N=Z nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Notice the energy scales!
€
3+
€
3+
€
5+
M. Horoi CMU
Shell Model Effective Hamiltonians
core polarization: Phys.Rep. 261, 125 (1995)
PRC 74, 34315 (2006), 78, 064302 (2008)
€
USDA
€
USDB
€
ME
NN2012 San Atonio, May 31, 2012
€
Hvalence = H2−body
can describe most correlations
around the Fermi surface!
empty
valence
frozen core
€
HvalenceΨ = EnΨ
N=Z sd-Nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Decoupling the singlet pn-pairing
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
26 Al
€
H = HUSDB
Decoupling the singlet pairing
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
28 Al
€
H = HUSDB
Decoupling the singlet pairing
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
30 Al
€
H = HUSDB
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Spectroscopic Factors
One particle removal cross section: e.g. (p,d)
€
σ j− ≈ S j σ j( )
sp
€
S f , j , i =|< Ψf
A −1 J f || ˜ a j || ΨiA Ji >|2
2Ji +1
€
S f , j , i =< n j >if −
∑
Sum rule: s.p. occupation probability
€
σ j+ ≈
2J f +1
2Ji +1S j σ j( )
spe.g. (d,p)
Are SFs (reliable) “observables”?
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
[Jenny Lee et al, PRL 2010][Jenny Lee et al, PRL 2010]
[Gade et al, PRL 93, 042501] [Gade et al, PRL 93, 042501] g.s. SFs seem to be more reliable
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Spectroscopic factors in the pf-shell
J. Lee et al. Phys. Rev. C 79, 054601 (2009)B. Tsang et al. Phys. Rev. Lett. 102, 062501
(2009)
Spectroscopic Factors: (d,p) consistent with (p,d)
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
B. Tsang, J. Lee, W. Lynch Phys. Rev. Lett. 95, 222501 (2005)
€
σ j− ≈ S j σ j( )
sp
€
σ j+ ≈
2 J f +1
2 Ji +1S j σ j( )
sp
pf-Nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Spectroscopic Factors: Independent Particle Model (IPM)
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
IPM Model:
- n identical nucleons in
- One single j-shell: e.g. f7/2
- Only pairing interaction
€
S j = n n − even
S j =(2 j +1) − (n −1)
(2 j +1)n −odd
B. Tsang, J. Lee, W. Lynch Phys. Rev. Lett. 95, 222501 (2005)
Can one measure neutron S7/2 for N=28 isotones?
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
S7 / 2 0+ ↔7
2
− ⎛
⎝ ⎜
⎞
⎠ ⎟= ?
The Magic of Proton-Neutron Correlations
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
New/MSU
2010/MSU
Data from J. Lee and B. Tsang
€
54Fe
€
56Ni
S7/2 for N=28 isotones and f7/2 n-occupation
NN2012 San Atonio, May 31, 2012
M. Horoi CMU€
S f , j , i =< n j >if −
∑€
2654Fe28
S7/2 for N=28 isotones and f7/2 n-occupation
NN2012 San Atonio, May 31, 2012
M. Horoi CMU€
S f , j , i =< n j >if −
∑
€
in 53Fe
€
T =1
2
€
T =3
2
€
SF ↔ C2S
S7/2 for N=28 isotones described in the f7/2 shell
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
J T V(JT) Vpn
0 1 -2.068 0.0
1 0 -2.061 0.0
2 1 -0.757
3 0 -0.989
4 1 0.119
5 0 -0.811
6 1 0.333
7 0 -2.218 0.0
f748 int: KUTCHERA et al. NUOVO CIM, VOL-1 NO-12
The case of 46Ar
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
SDPF-U: PRC 79, 014301 (2009)
€
monopoles Vij ijT ≡ Vi j
T =
(2J +1)Vij ijTJ
J
∑
(2J +1)J
∑
€
monopole sensitivity :Vd 3/2 f7/2
T →Vd 3/2 f7/2
T =0,1 − 0.6
€
Vd3/2 f7/2
T =0 =− 2.475 Vd3/2 f7/2
T =1 = −0.928 €
N = 28
€
N = 28
?
M. Horoi CMU
Shell Model Effective Hamiltonians
core polarization: Phys.Rep. 261, 125 (1995)
PRC 74, 34315 (2006), 78, 064302 (2008)
€
USDA
€
USDB
€
ME
NN2012 San Atonio, May 31, 2012
€
Hvalence = ε ja j+a j +
j
∑ V2−body
can describe most correlations
around the Fermi surface!
empty
valence
frozen core
€
HvalenceΨ = EnΨ
The case of 46Ar
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
SDPF-U + pn “monopole” modifications
Occupation probabilities in 46Ar:
€
Vd 3/2 f7/2
T =0 →Vd 3/2 f7/2
T =0 − 0.6
€
all J : Vd 3/2 f7/2
JT =1( )
pn→ Vd 3/2 f7/2
JT =1( )
pn− 0.6
€
46Ar
€
N = 28
M. Horoi CMU
Summary and Outlook Spectroscopic factors could be a good tool to identify the
enhanced proton-neutron correlations in N~Z nuclei. Large fluctuations of the neutron SF vs proton number are
predicted by the shell model and observed in experiments: strong proton-neutron correlations are essential in nuclei!
These correlations are challenging for mean field theories, but seems to be accommodated by the Green’s function approach!
Parts of the effective H responsible for these strong neutron-proton correlations were identified.
This seems to be just the tip of the iceberg for proton-neutron correlation! More work necessary.
NN2012 San Atonio, May 31, 2012
S7/2 for N=28 isotones described in the f7/2 shell
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
J T V(JT) Vpn
0 1 -2.068 0.0
1 0 -2.061 0.0
2 1 -0.757
3 0 -0.989
4 1 0.119
5 0 -0.811
6 1 0.333
7 0 -2.218 0.0
f748 int: KUTCHERA et al. NUOVO CIM, VOL-1 NO-12
IPM
Short Overview
• Few more examples of proton-neutron correlations effects in nuclei
• Proton-neutron correlations effects observed in transfer reactions
• Which pieces of the shell model Hamiltonian seem to be responsible for these effects
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
pn-pairing vs pp/nn pairing: Why the difference?
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Experimental positive parity levels
sd-Nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Which pieces of H2 are responsible? The case of S5/2 for N=14 isotones
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
f →0d5 / 2
r → 0d1/ 2, 1s1/ 2
€
κ =1.1
A. Zuker, PRL 90 (2003)
Which pieces of H2 are responsible? The case of S5/2 for N=14 isotones
NN2012 San Atonio, May 31, 2012
M. Horoi CMU€
(5 /2, 5 /2)01 |V | (5 /2, 5 /2)01 = −2.5 →− 8.5
(5 /2, 5 /2)01 |V | (3/2, 3/2)01 = −1.1 →− 0.1
(5 /2, 5 /2)01 |V | (1/2, 1/2)01 = −1.6 →−0.6
€
No sensibilty to JT =10 matrix elements
€
f →0d5 / 2
r → 0d1/ 2, 1s1/ 2
Proton-Neutron Pairing: A mean field approach ?
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Staggering of charge radii of Sn isotopes, MH, PRC 50, 2834 (1995)
€
H3 ∝ Pn+Pnρ p ← Zawischa, PRL 61, 149(1988)
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
W. Dickhoff’s talk
Can one measure S7/2 for N=27 isotones?
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
2754Co27
€
2552Mn27
€
2350V27
€
2148Sc27
€
S7 / 2 0+ ↔7
2
− ⎛
⎝ ⎜
⎞
⎠ ⎟= ?
sd-Nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
26Al Spectroscopic Factors vs (H01)pn
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
26 Al : n = 5 ⇒ S5 / 2(IPM) =6 − 4
6= 0.33
€
H = H(USDB) +ε H01(USDB)[ ]t3 =0
€
1124Na13(0+)→11
23 Na12(5 /2+) + n : S5 / 2(USDB) = 0.1
€
1326Al13(0+)→13
25 Al12(5 /2+) + n : S5 / 2(USDB) =1.24
26Al Spectroscopic Factors Experimental Data
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
26 Al : S5 / 2(USDB) =1.24 ≈ 4 × S5 / 2(IPM)
€
(3 He, d) reactions
J. Vernotte et al. NPA 571, 1 (1994)
Are SFs (reliable) “observables”?
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
[Jenny Lee et al, PRL 2010][Jenny Lee et al, PRL 2010]
[Gade et al, PRL 93, 042501] [Gade et al, PRL 93, 042501] g.s. SFs seem to be more reliable
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Spectroscopic factors in the pf-shell
J. Lee et al. Phys. Rev. C 79, 054601 (2009)B. Tsang et al. Phys. Rev. Lett. 102, 062501
(2009)
Spectroscopic Factors: (d,p) consistent with (p,d)
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
B. Tsang, J. Lee, W. Lynch Phys. Rev. Lett. 95, 222501 (2005)
€
σ j− ≈ S j σ j( )
sp
€
σ j+ ≈
2 J f +1
2 Ji +1S j σ j( )
sp
Evolution of lowest 0+ and 1+ states in N=Z nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Notice the energy scales!
€
3+
€
3+
€
5+
Enhanced realization probability of JT=01,10 g.s. for Odd-Odd nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
H = HTBRE
€
PJT =01 ≡# states(JT = 01)
# states(M = 0)= 2.5%
€
PJT =10 ≡# states(JT =10)
# states(M = 0)=1.6%
€
Percentage 1530P15 g.s. realizations
pn-pairing vs pp/nn pairing: Why the difference?
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Experimental positive parity levels
Spin-triplet vs singlet pn-pairing in 30P
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
H = H(USDB) +ε H10(USDB)[ ]
€
ja jb JT | P10 | jc jd JT =
2(−1) ja − jb (2 ja +1)(2 jb +1)(2 jc +1)(2 jd +1)
(1+δ ab )(1+δ cd )
×1/2 ja la
jb 1/2 1
⎧ ⎨ ⎩
⎫ ⎬ ⎭
1/2 jc lc
jd 1/2 1
⎧ ⎨ ⎩
⎫ ⎬ ⎭δ la lb
δ lc ldδ J 1δT 0
€
ja jb JT |V | jc jd JT ∝δ la lbδ lc ld
δ J 1δT 0
PLB 430, 203 (1988)
€
0
Preliminary
Proton-Neutron Pairing and Binding Energies
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
Odd − Odd
€
Even − Even
P. Vogel, nucl-th:9805015
transfer versus knockout
[Jenny Lee et al, PRL 2009][Jenny Lee et al, PRL 2009]
[Gade et al, Phys. Rev. Lett. 93, 042501] [Gade et al, Phys. Rev. Lett. 93, 042501]
[FN, Deltuva, Hong, PRC submitted 2011][FN, Deltuva, Hong, PRC submitted 2011]
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
M. Horoi CMU
Nuclear Configuration Interaction
€
0hω
1hω
(0 + 2)hω
(1+ 3)hω
d = 2 (2 j + 1)
€
H = Hone −body + H two−body + H3b +L
€
|α >= Ciα
i
∑ | i(JT) >
|α >= Ciα
i
∑ | i(JTz) >
|α >= Ciα
i
∑ | i(MTz) >
€
H '= H + β (HCoM − 3/2hω)
€
Ψ(J ) = [ΦCoM (NL)Φ int(J ' )](J ) → ΦCoM (00)Φ int
(J )
JT-scheme: OXBASH, NuShell
Center-of-mass spurious states
Nmax
sd
Lanczos algorithm: provides few lower energies, especially the M-scheme codes.
pf
€
< i | H | j > C jα
j
∑ = Eα Ciα
J-scheme: NATHAN, NuShellX
M-scheme: Oslo-code, Antoine, MFDN, MSHELL, CMichSM, …
NN2012 San Atonio, May 31, 2012
Limits: 1010 – 1011 m-scheme basis states
pf-Nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Spectroscopic Factors: SM vs IPM for N=Z nuclei
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
IPM:
- n identical nucleons in
- One single j-shell: e.g. f7/2
- Only pairing interaction
€
S j = n n − even
S j =(2 j +1) − (n −1)
(2 j +1)n −odd
54Co2750Mn25
€
2754Co27 : S7 / 2(GXPF1A) =1.9 ≈ 8 × S7 / 2(IPM)
€
2754Co27 : S7 / 2(IPM) =
8 − 6
8= 0.25
B. Tsang, J. Lee, W. Lynch Phys. Rev. Lett. 95, 222501 (2005)
26Al Spectroscopic Factors: SM vs IPM
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Model:
- n identical nucleons in
- One single j-shell: e.g. d5/2
- Only pairing interaction
€
S j = n n − even
S j =(2 j +1) − (n −1)
(2 j +1)n −odd
€
26 Al : n = 5 ⇒ S5 / 2(IPM) =6 − 4
6= 0.33
€
26 Al : S5 / 2(USDB) =1.24 ≈ 4 × S5 / 2(IPM)
B. Tsang, J. Lee, W. Lynch Phys. Rev. Lett. 95, 222501 (2005)
Staggering of Ca Charge Radii: A Shell Model Approach
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Valence space: d3/2 , s1/2 , f7/2 , p3/2
€
Hvalence ≡ H2−body
€
r2
ch= r2
sdnsd
p + r2
pfnpf
p
E. Caurier et al, PLB 522, 240 (2001)
I. Talmi NPA 423, 189 (1984)
sd-nuclei: N=13 odd-odd isotones
NN2012 San Atonio, May 31, 2012
M. Horoi CMU€
1326Al13(0+)→13
25 Al12(5 /2+) + n : S5 / 2(USDB) =1.24
€
1124Na13(0+)→11
23 Na12(5 /2+) + n : S5 / 2(USDB) = 0.1
sd-nuclei: N=13 odd-odd isotones
NN2012 San Atonio, May 31, 2012
M. Horoi CMU€
1326Al13(0+)→13
26 Al12(5 /2+) + n : S5 / 2(USDB) =1.24
€
26 Al
€
24 Na
€
1124Na13(0+)→11
23 Na12(5 /2+) + n : S5 / 2(USDB) = 0.1
€
22 Na
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Spectroscopic factors in the pf-shell
J. Lee et al. Phys. Rev. C 79, 054601 (2009)B. Tsang et al. Phys. Rev. Lett. 102, 062501
(2009)
Proton-Neutron Pairing and Binding Energies
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
Odd − Odd
€
Even − Even
P. Vogel, nucl-th:9805015
Pair Transfer Amplitudes
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
26Al -> 24Mg <(0T)_f ||| a_j a_j ||| (0T)_i>
01 -> 00
j=5/2 0.198
j=3/2 0.185
j=1/2 0.193
01 -> 01
j=5/2 0.045
j=3/2 0.244
j=1/2 0.155
Si isotopes: Staggering of Occupation Probabilities
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
Overview• The status of the effective Hamiltonians
• Shell model description of the low-lying states
• Shell model spectroscopic factors
• Nuclear structure analysis of charge-exchange reactions
• Shell model analysis of double-beta decay
• Novel developments with shell model nuclear level densities and reaction rates
M. Horoi CMUNN2012 San Atonio, May 31, 2012
Proton Breakup of 23Al
NN2012 San Atonio, May 31, 2012
M. Horoi CMU
€
4.3 =< nd 5 / 2 >5 / 2+
USDB
€
12C(23 Al, 22Mg) E = 50 MeV / A€
S f , k, i =< nk >if −
∑
€
Jπ 23 Al( ) =5
2
⎛
⎝ ⎜
⎞
⎠ ⎟+
€
Sp23 Al( ) =141 keV
€
S f , d 5 / 2, 5 / 2+ = 3f −
∑ 70%( )
€
22Mg(p,γ)23 Al