Shell-Model Approaches to Two-Proton Decay

BABFred Barker (ANU)

General commentsApplication to 2p decay of 45Fe and 48NiApplication to 2p decay widths of light nucleiPredictions for 2p lifetimes of heavier nuclei

45Fe: Two-Proton Radioactivity Observed

GSI GANIL

Q(exp) = 1.14 (5) MeV, T(exp) = 3-10 ms

“…should lead to lifetimes longer than 10-12 sec, a possible lower limit for the process to be called radioactivity.”

“…should lead to lifetimes longer than 10-12 sec, a possible lower limit for the process to be called radioactivity.”

Joseph Cerny and J. C. Hardy, Annu. Rev. Nucl. Part. Sci. 27 (1977) 333

Conditions for Two-Proton Radioactivity

Special cases where two-proton decay is allowed, but one-proton decay is “forbidden”.

Large Coulomb barrier in heavy nuclei makes the states narrow (eV or less), in particular the phase-space “tails” for one-proton decay are much reduced.

Q value for the two-proton decay is just right so that the lifetime is competitive with beta decay, but not too short for experimental observation - about 0.1 to 10 ms.

Very few cases.

Quiz about New York Times article

What is the year?

Who was the group?

What was the decay?

What are the first four?

Definitions

Di-Proton Decay Sequential Decay

12O

11N+p

10C+p+p

12O

10C+2p

10C+p+p

12O 10C+2p11N+p 12O 10C+2p11N+p

Simultaneous Decay

12O

10C+p+p

12O 10C+2p11N+p

Three-Body Models (Grigorenko et al.)L. V. Grigorenko et al., Phys. Rev. Lett. 85 (2000) 22

I will assume the middle picture with the two-protons correlatedin their L=0 (T=1) resonance state.

Two problems for the decay width

• Can we calculate the Q value? – Yes, but not better than to within about 100 keV

• If the experimental Q value is known to a better accuracy we should constrain the calculation to give this value in order to discuss the structure (overlap) questions.

• Can we calculate the overlap?

Basic R-Matrix Formulation

Decay widthP(a) = penetration factor

Reduced widthS = spectroscopic factor

Dimensionless single-particlereduced widthu = cluster wave function from V(r).

Channel radius

Basic R-Matrix Formulation

In many cases the same potential V can be used to calculate the scattering cross section as a function of energy from the phase shift. The widths of resonances obtained this way are the same as the R-matrix model typically within about 20%.

But the R-matrix formalism is useful when the resonance is very narrow and we need to include intermediate states.

Two-Proton Resonance

is replaced by

Where c1 comes from norm

Effective-range resonance parameters; c, A and B.

Many-Body Physics (e.g. for 45Fe)

Cluster-decay model (Brink, Arima etc)(45/43) is a center of mass correctionLambda = 2N+L = 6 (two protons in pf shell)L = 0N = 3 = number of nodes in the cluster wfG = projection of di-proton cluster onto the pf shellA = projection of di-proton cluster onto the specific

two-particle overlap <45Fe|a+a+|43Cr>

Cluster wave function u(r)

Potential depth used to obtainThe experimental Q value.

Potential shape taken fromdeuteron scattering analysis.

N=3 for number of nodes (twoparticles in the pf shell).

45Fe: Di-Proton or Sequential?G. Audi and A.H. Wapstra, Nucl. Phys. A595 (1995) 409

But coulomb energysystematics indicate that the one-proton decay is forbidden and the state should be very narrow.

Q-value results for 45Fe

Q(exp) = 1.14 (5) MeV

Q(th) = 1.15 (9) Brown 1991

1.28 (18) Ormand 1996

1.22 (5) Cole 19

Structure results for 45Fe

• The 45Fe wf has the form [ (proton pf)6 (neutron sd)-1 ]

• But the neutron hole is a spectator and the spectroscopic results essentially comes from the proton (pf)6 -> (pf)4 part.

• From the pf wave functions S=0.195.

• Although the wf are 90% f7/2, the f7/2 part alone would give S=0.06. The small admixtures of the p 3/2 has a large effect.

• In two-neutron transfer the p3/2 was called a “hot” orbit.

Structure results for 45Fe

• Q(exp) = 1.14 (5) MeV, T(exp) = 3-10 ms.

• With S=0.195.

• From the R-matrix (T_o is without the pp interaction)

Q (MeV) T_d (ms) T_o (ms)

1.09 300 0.0601.14 45 0.0131.19 15 0.004

Structure results for 45Fe

• T(exp) = 3-10 ms

• Theoretical rate need may need to be increased by a factor of 2-4. This would imply S=0.6.

• Similar situation to the pf-shell (p,t) reaction cross sections in the 1970’s.

• One requires an “enhancement factor” which is understood in terms of an enhanced pairing from small mixing with orbits outside the pf shell.

Q (MeV) T_d (ms)

1.09 300 1.14 45 1.19 15

Results for 48Ni

• Q (predicted) = 1.36 (13) MeV (Brown 1991)

• S=0.14

Q (MeV) T_d (ms) T_d (ms) (with enhancement)

1.23 260 651.36 8 21.49 0.4 0.1

48Ni belongs to a class of “new magic nuclei” (ask at the end)

Two-Proton Decay of 6Be

g.s.

1.67

0.59

-1.374He+2p

5Li+p6Be

(0+)(3/2-)

(2+)

D.F. Geesaman et al., Phys. Rev. C15, 1835 (1977).

Structure results for 6Be (Barker paper)

• Q(exp) = 1.37 MeV, Gamma(exp) = 92(6) keV.

• S=1.25 from the p-shell wf.

• Di-proton (th) = 85 keV (2 MeV without pp correlations)

• Inclusion of sequential would give 64-80 keV (a reduction!)

• Perhaps another sign of enhanced pairing correlations.

Two-Proton Decay of 12O

Old

New

??2.0 MeV1.8 MeV

1.8 MeV ~1.5 MeV

Di-Proton

Sequential

R.A. Kryger et al., Phys. Rev. Lett. 74, 860 (1995)

12O 11N+p 10C+2p

Structure results for 12O (Barker paper)

• Q(exp) = 1.77 MeV, Gamma(exp) = 400(250) keV.

• S=1.0 estimate from the p-sd shell wf (dominated by large sd-shell admixture)

• Di-proton (th) = 5 keV

• Inclusion of sequential (dominated by s-wave) would give about 100 keV.

• Dominated by sequential.

Structure results for 16Ne

• Q(exp) = 1.41 MeV, Gamma(exp) = 122 keV.

• S=1.0 (estimate)

• Di-proton (th) = 0.1 keV

• Dominated by s-wave sequential.

Two-Proton Decay of 18Ne?J. Gomez del Campo et al., Phys. Rev. Lett. 86 (2001) 43

• One-proton decay width (keV)

§ Sequential two-proton decay though the ½+ ghost (eV)

§ Di-proton decay width (eV)6 3 10

§18Ne total two-proton decay width (eV)

16 23 19

Exp

50(5)

20-60

• Two-proton decay width (eV)

10 20 9

Watch out for the ghosts!

N=Z line

Predicted separation energiesfor proton-rich nuclei used for the rp-processA=53-76

Results for Heavier Nuclei

Nucleus Q (MeV) T_d (ms)

54Zn 1.33 50763Se 1.51 600067Kr 1.76 26271Sr 2.06 12

Assume S=1

Two-Proton Decay - A New Type of Radioactivity

• The new mode of two-proton radioactivity has recently been observed for 45Fe [1-2].

• The Q value is in excellent agreement with a prediction [3].

[1] J. Giovinazzo, et. al., Phys. Rev. Lett. 89, 102501 (2002).[2] M. Pfutzner, et. al., Eur. Phys. J. A 14, 279 (2002).[3] B. A. Brown, Phys. Rev. C43, 1513 (1991).[4] B. A. Brown and F. C. Barker, submitted to Phys. Rev. C.

• We have applied a new R-matrix model which includes the two-proton resonance as an intermediate state to calculated the di-proton decay lifetime [4].

• The spectroscopic factor is calculated microscopically from the full pf-shell wave functions and is sensitive to the pairing correlations.

• The results are near the upper limit of the experiment with agreement being obtained with an enhanced pairing. 48Ni is another good candidate.

45Fe

43Cr

Answers to Quiz

1983

Cerny et al.

Beta delayed 2p (22Al)

Alpha, beta, fissionone-proton

New Magic Nuclei

• New data for 22O, 24O and N=34 taken together with previous observations, one finds a new “rule” for the observed magic numbers:

• If there is an oscillator magic number (2, 8, 20 or 40) for one kind of nucleon, then the other kind of nucleon has a magic number for the filling of every possible (n,l,j) value.

• This rule accounts for 16 doubly-magic nuclei shown by the circles.

• There are no exceptions to the rule.• New magic nuclei are predicted

(open circles).• These results were predicted but are

not yet fully understood.