Isospin symmetry in mirror decays

1
Isospin symmetry in mirror ?-decays

• N.K. Timofeyuk, R.C. Johnson
• University of Surrey
• P.Descouvemont
• Université Libre de Bruxelles

2
Nuclear reactions can be thought of as sequences
of virtual or real decays
A
B
Direct transfer reactions
?
a
b
C
A
B
Resonance reactions
p
?
3
Virtual decays
The amplitude of a virtual decay A ? (A
?) ?
Gl -il p1/2 (h/µc) Cl Cl is ANC Gl is a vertex
constant (an analogue of a coupling
constant in particle physics)
A
A-?
?
On the other hand, at r ? ? ?A-??? ?A? ? Cl W
-?,l½ (2?r)/r Ylm( )
Cl determines the strength of the overlap
integral ?A-? ? ? ?A? at large distances
Real decays are characterized
by the width ?
4
A ratio of ANCs for mirror nucleon decays (N.K.
Timofeyuk, R.C. Johnson and A.M. Mukhamedzhanov,
PRL 91, 0232501 (2003))

• Assumptions
• The sum of the Coulomb interactions of the A-th
  nucleon with the protons of the core A-1 is a
  constant inside the range of the strong
  interaction
• This constant is equal to the separation energies
  difference ?n- ?p
• The wave functions of mirror nuclear states are
  the same inside the range of strong interaction
• The main contribution to ANC comes from r lt RN

Then
where Fl is the regular Coulomb wave function,
jl is the Bessel function
?p (2µep)1/2, ?n (2µen)1/2
RN is some radius beyond the range of nuclear
force
5

• Isospin symmetry in mirror ? decays
• ANZ ? A-4 (N-2)Z-2 ?
• AZN ? A-4 (Z-2)N-2 ?
• Ratio of ANCs in bound mirror analogs

RN must be taken at a point where the product
V?-core ? ?-core has maximum, RN ?
(1.1-1.3)(41/3 C1/3)
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Potential model Overlap integral S1/2
? potential model wave function where S is a
spectroscopic factor ? C S1/2 b,
where b is a potential model ANC If
1) ?-particle potential wells for mirror
nuclei are equal and 2) spectroscopic
factors for mirror systems are equal Then
equation ? ?PM (b2/b1)2
could become a recipe for finding ? because in
this case ?PM almost potential independent.
7
Microscopic cluster calculations (MCM)
Calculations of ANCs for a range of 0p and sd
shell nuclei ?A ? ( ?A-4 ? g(r)?? ) gl(r) ?
Cl W-l,?1/2 (2?r)/r , r ?
? Microscopic R-matrix approach is used to find
Cl. Nucleus A 4 is described in one- or
two-centre translation-invariant oscillator shell
model Effective NN potentials used Volkov (V2)
(describes binding energies of triton and
4He) Minnesota (MN) (describes the two-nucleon
low-energy scattering data but also the essential
properties of the deuteron, triton and
4He) Majorana parameter m of V2 and the
parameter u of MN are adjusted to fit the
?-particle separation energies
8
MCM calculations of ANCs for mirror virtual
?-decays (N.K. Timofeyuk, P. Descouvemont and
R.C. Johnson, submitted to PRC) Two NN
potentials are used, V2 and MN, two oscillator
radii are used. ANCs change by a factor of 2 20
with different choice of NN potential and
oscillator radius.
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Ratio of mirror ANCs
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Ratio of mirror spectroscopic factors
11
Bound-unbound mirror pairs
alpha decay threshold
G? , ER
bound state
resonance
alpha decay threshold
C?
AZXN
ANXZ
12
__________________________________________________
_________________ Mirror pair J? ?b.s.
ER l? ?

analytical potential model MCM

formula ________________________________________
___________________________ 11B11C 3/23
0.104 0.56 0 1.18?10-9 1.05?10-9
(1.05?0.06)?10-9
2 1.52?10-9
(1.48?0.01)?10-9 (1.47?0.03)10-9 19F-19Ne
3/22 0.106 0.50 1 3.30?10-33
(3.25?0.04)?10-33 (3.42?0.04)?10-33
7/22- 0.015 0.67 4 7.86?10-84
(8.0?0.18)?10-84 (1.32?0.12
)?10-83 __________________________________________
__________________________________________________
_________
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Unbound mirror pairs
Resonance 2
Resonance 1
G? (1), ER (1)
G? (2), ER (2)
alpha decay threshold
alpha decay threshold
AZXN
ANXZ
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Traditional way to link width of two mirror
resonances is based on equality of reduced widths
??
Width
Penetration probability
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Old formula
New formula
Ratio New/Old
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J? l? MCM(2c) MCM(3c)
PM old R new R exp R

7Li 7Be 7/2- 3 1.795?0.005
1.82 1.74 1.79
1.88?0.24
11B 11C
5/22- 2 1610?40 1740 1434
1493 1530 2140?970
4 (1.02?0.04)?104 11400 9964
9982 10000 7/22 3 38.4 ?0.5
40.3 37 38.1 38.3
5 151 ?7 183
152 152.3 152.2
19F 19Ne
7/21 3 (1.28?0.03)?105 1.29?105
1.31?105 1.30?105 5/22- 2 204 ? 7
203 209
207 121 ? 55
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Implications for nuclear astrophysics Direct
radiative capture A-4(Z-2)N-2(?, ?)AZN is
determined by ANC for AZN ? A-4(Z-2)N-2 ?. If
for some reason this ANC is difficult to
measure than the ANC for a mirror decay ANZ ?
A-4(N-2)Z-2 ?, the latter can be determined
and then used together with the formula for the
ratio of mirror ANC to determine the ANC of
interest. The astrophysical S-factors at zero
energy for 3H(?, ?)7Li and 3He(?, ?)7Be reactions
should be related.
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Resonant (?,?) reactions
Resonant capture rate is proportional to
???? /(????)
Gamov window
narrow resonance
?-decay threshold
If ?? ? ?? then knowledge of ?? is important
AZ
Determination of G? is possible using ANC C?
For example, 15O(?, ?)19Ne(4.033 MeV) using ANC
for 19F(3.908 MeV) ? 15N ? A good reaction to
determine this ANC would be 19F(15N,15N)19F(3.908
)
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(?,p) and (?,n) reactions
? decay threshold
neutron decay threshold
Gp , G? , ER
bound state
resonance
Cn , C?
? decay threshold
proton decay threshold
Example, 17F(p,?)14O requires widths for p17F
and ?14O. These can be found from n17O and
?14C.
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Conclusions

• A link exists between the ANCs of mirror ?-decays
  due to the charge symmetry of strong
  interactions.
• A link also exist between the width of a narrow ?
  resonance and the ANC of its mirror stable
  analog.
• A new formula for a ratio of resonances for
  mirrror narrow resonance is available
• The analytical formulae for mirror ?-decays are
  in good agreement with potential model and
  microscopic model calculations. Some deviations
  may be expacted if core excitations are strong
• The link between mirror ?-decays can be used to
  determine astrophysically relevant capture
  reactions.