the origin of strong proton-neutron correlations revealed in break-up reactions

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


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


€

Hvalence = H2−body

can describe most correlations

around the Fermi surface!

empty

valence

frozen core

€

HvalenceΨ = EnΨ

N=Z sd-Nuclei


M. Horoi CMU

Decoupling the singlet pn-pairing


M. Horoi CMU

€

26 Al

€

H = HUSDB

Decoupling the singlet pairing


M. Horoi CMU

€

28 Al

€

H = HUSDB

Decoupling the singlet pairing


M. Horoi CMU

€

30 Al

€

H = HUSDB


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”?


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


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)


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


M. Horoi CMU

Spectroscopic Factors: Independent Particle Model (IPM)


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


Can one measure neutron S7/2 for N=28 isotones?


M. Horoi CMU

€

S7 / 2 0+ ↔7

2

− ⎛

⎝ ⎜

⎞

⎠ ⎟= ?

The Magic of Proton-Neutron Correlations


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


M. Horoi CMU€

S f , j , i =< n j >if −

∑€

2654Fe28

S7/2 for N=28 isotones and f7/2 n-occupation


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


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


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


€

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


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.


S7/2 for N=28 isotones described in the f7/2 shell


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


M. Horoi CMU

pn-pairing vs pp/nn pairing: Why the difference?


M. Horoi CMU

Experimental positive parity levels

sd-Nuclei


M. Horoi CMU

Which pieces of H2 are responsible? The case of S5/2 for N=14 isotones


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


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 ?


M. Horoi CMU

Staggering of charge radii of Sn isotopes, MH, PRC 50, 2834 (1995)

€

H3 ∝ Pn+Pnρ p ← Zawischa, PRL 61, 149(1988)


M. Horoi CMU

W. Dickhoff’s talk

Can one measure S7/2 for N=27 isotones?


M. Horoi CMU

€

2754Co27

€

2552Mn27

€

2350V27

€

2148Sc27

€

S7 / 2 0+ ↔7

2

− ⎛

⎝ ⎜

⎞

⎠ ⎟= ?

sd-Nuclei


M. Horoi CMU

26Al Spectroscopic Factors vs (H01)pn


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


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”?


M. Horoi CMU


[Gade et al, PRL 93, 042501] [Gade et al, PRL 93, 042501] g.s. SFs seem to be more reliable


M. Horoi CMU



(2009)

Spectroscopic Factors: (d,p) consistent with (p,d)


M. Horoi CMU


€

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


M. Horoi CMU

Notice the energy scales!

€

3+

€

3+

€

5+

Enhanced realization probability of JT=01,10 g.s. for Odd-Odd nuclei


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?


M. Horoi CMU

Experimental positive parity levels

Spin-triplet vs singlet pn-pairing in 30P


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


M. Horoi CMU

€

Odd − Odd

€

Even − Even

P. Vogel, nucl-th:9805015

transfer versus knockout


[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]


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, …


Limits: 1010 – 1011 m-scheme basis states

pf-Nuclei


M. Horoi CMU

Spectroscopic Factors: SM vs IPM for N=Z nuclei


M. Horoi CMU

IPM:


- One single j-shell: e.g. f7/2


€

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


26Al Spectroscopic Factors: SM vs IPM


M. Horoi CMU

Model:


- One single j-shell: e.g. d5/2


€

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)


Staggering of Ca Charge Radii: A Shell Model Approach


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


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


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


M. Horoi CMU



(2009)

Proton-Neutron Pairing and Binding Energies


M. Horoi CMU

€

Odd − Odd

€

Even − Even

P. Vogel, nucl-th:9805015

Pair Transfer Amplitudes


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


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


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

the origin of strong proton-neutron correlations revealed in break-up reactions

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