360
7 Wave Functions in the Vicinity of Particle-Emission Threshold
[118], an effective interaction based on the use of a 12 C core, can properly
describe the energy spectra and observables of 16,17 O and 17 F [118–120].
B. As one considers spectroscopic factors involving (sd) shells, the considered
eigenstates of 13 C, 13 N, and 14 O will have energies close to particle-emission
threshold. As 0p 1/2 shells are well bound, their effect on spectroscopic factors
at S 1n/1p ∼0 will be minimal, contrary to the 1s 1/2 , 0d 5/2 , and 0d 3/2 shells,
which will be loosely bound or narrow resonances. Thus, 0p 1/2 shells will not
significantly impart spectroscopic factors.
Isospin symmetry is exact in the absence of Coulomb Hamiltonian. In this
case, the pairs 13 C/ 14 C and 13 N/ 14 O behave identically. In the studied example,
the Coulomb interaction is included in the Hamiltonian. Consequently, one can
expect different behaviors of spectroscopic factors as the asymptotes of nuclear
wave functions in the pairs 13 C/ 14 C and 13 N/ 14 O will be built from Hankel and
Coulomb wave functions, respectively.
C. One does not obtain Wigner cusps at S 1n/1p = 0 MeV because the partial wave
present in the model space bears = 0 or = 2. One has seen in Sect. 7.4.2.1
that s-waves cannot give rise to Wigner cusps, as one cannot have unbound
states mainly occupied by s-waves. Moreover, neutron d-waves cannot generate
Wigner cusps on spectroscopic factors, as they become visible only on the
derivative of spectroscopic factors (see Sect. 7.4.2).
Thus, the studied spectroscopic factors involving neutrons, i.e. for the pair
13 C/ 14 C, are similar to the spectroscopic factors
11 Li(3/2
−
1 )|[ 10 Li(2
−
1 ) ⊗ s 1/2 ] 3/2 + and 18 O(0
+
3 )|[ 17 O(3/2
+
1 ) ⊗ d 3/2 ] 0 + of
Sect. 7.4.2 for s and d waves, respectively. Values of neutron d-waves spectroscopic factors form a continuous and differentiable function, but its derivative
presents a non-analyticity at S 1n =0 MeV. Spectroscopic factors involving
protons only, that is for the pair 13 N/ 14 O, cannot generate Wigner cusps due
to the presence the Coulomb barrier, as can be seen in Eqs. (7.47), (7.48).
References
1. P. Kleinw˝ achter, I. Rotter, Phys. Rev. C 32, 1742 (1985)
2. I. Rotter, Rep. Prog. Phys. 54, 635 (1991)
3. V. Sokolov, V. Zelevinsky, Phys. Lett. B 202, 10 (1988)
4. V. Sokolov, V. Zelevinsky, Nucl. Phys. A 504, 562 (1989)
5. V. Sokolov, V. Zelevinsky, Phys. Lett. B 202, 10 (1988)
6. S. Dro˙ z ˙
d˙ z, J. Okołowicz, M. Płoszajczak, I. Rotter, Phys. Rev. C 62, 24313 (2000)
7. R. Dicke, Phys. Rev. 93, 99 (1954)
8. N. Auerbach, V. Zelevinsky, Rep. Prog. Phys. 74, 106301 (2011)
9. A. Baz, Soviet Phys. JETP 6, 709 (1957)
10. R. Newton, Phys. Rev. 114, 1611 (1959)
11. C. Hategan, Ann. Phys. (NY) 116, 77 (1978)
12. N. Michel, W. Nazarewicz, M. Płoszajczak, Phys. Rev. C 75, 031301 (2007)
13. Y. Fyodorov, B. Khoruzhenko, Phys. Rev. Lett. 83, 65 (1999)
14. P. Kohler, F. Be˘ cva˘ r, M. Krti˘ cka, J. Harvey, K. Guber, Phys. Rev. Lett. 105, 072502 (2010)
15. G. Celardo, N. Auerbach, F. Izrailev, G. Berman, Phys. Rev. Lett. 106, 042501 (2011)
7 Wave Functions in the Vicinity of Particle-Emission Threshold
[118], an effective interaction based on the use of a 12 C core, can properly
describe the energy spectra and observables of 16,17 O and 17 F [118–120].
B. As one considers spectroscopic factors involving (sd) shells, the considered
eigenstates of 13 C, 13 N, and 14 O will have energies close to particle-emission
threshold. As 0p 1/2 shells are well bound, their effect on spectroscopic factors
at S 1n/1p ∼0 will be minimal, contrary to the 1s 1/2 , 0d 5/2 , and 0d 3/2 shells,
which will be loosely bound or narrow resonances. Thus, 0p 1/2 shells will not
significantly impart spectroscopic factors.
Isospin symmetry is exact in the absence of Coulomb Hamiltonian. In this
case, the pairs 13 C/ 14 C and 13 N/ 14 O behave identically. In the studied example,
the Coulomb interaction is included in the Hamiltonian. Consequently, one can
expect different behaviors of spectroscopic factors as the asymptotes of nuclear
wave functions in the pairs 13 C/ 14 C and 13 N/ 14 O will be built from Hankel and
Coulomb wave functions, respectively.
C. One does not obtain Wigner cusps at S 1n/1p = 0 MeV because the partial wave
present in the model space bears = 0 or = 2. One has seen in Sect. 7.4.2.1
that s-waves cannot give rise to Wigner cusps, as one cannot have unbound
states mainly occupied by s-waves. Moreover, neutron d-waves cannot generate
Wigner cusps on spectroscopic factors, as they become visible only on the
derivative of spectroscopic factors (see Sect. 7.4.2).
Thus, the studied spectroscopic factors involving neutrons, i.e. for the pair
13 C/ 14 C, are similar to the spectroscopic factors
11 Li(3/2
−
1 )|[ 10 Li(2
−
1 ) ⊗ s 1/2 ] 3/2 + and 18 O(0
+
3 )|[ 17 O(3/2
+
1 ) ⊗ d 3/2 ] 0 + of
Sect. 7.4.2 for s and d waves, respectively. Values of neutron d-waves spectroscopic factors form a continuous and differentiable function, but its derivative
presents a non-analyticity at S 1n =0 MeV. Spectroscopic factors involving
protons only, that is for the pair 13 N/ 14 O, cannot generate Wigner cusps due
to the presence the Coulomb barrier, as can be seen in Eqs. (7.47), (7.48).
References
1. P. Kleinw˝ achter, I. Rotter, Phys. Rev. C 32, 1742 (1985)
2. I. Rotter, Rep. Prog. Phys. 54, 635 (1991)
3. V. Sokolov, V. Zelevinsky, Phys. Lett. B 202, 10 (1988)
4. V. Sokolov, V. Zelevinsky, Nucl. Phys. A 504, 562 (1989)
5. V. Sokolov, V. Zelevinsky, Phys. Lett. B 202, 10 (1988)
6. S. Dro˙ z ˙
d˙ z, J. Okołowicz, M. Płoszajczak, I. Rotter, Phys. Rev. C 62, 24313 (2000)
7. R. Dicke, Phys. Rev. 93, 99 (1954)
8. N. Auerbach, V. Zelevinsky, Rep. Prog. Phys. 74, 106301 (2011)
9. A. Baz, Soviet Phys. JETP 6, 709 (1957)
10. R. Newton, Phys. Rev. 114, 1611 (1959)
11. C. Hategan, Ann. Phys. (NY) 116, 77 (1978)
12. N. Michel, W. Nazarewicz, M. Płoszajczak, Phys. Rev. C 75, 031301 (2007)
13. Y. Fyodorov, B. Khoruzhenko, Phys. Rev. Lett. 83, 65 (1999)
14. P. Kohler, F. Be˘ cva˘ r, M. Krti˘ cka, J. Harvey, K. Guber, Phys. Rev. Lett. 105, 072502 (2010)
15. G. Celardo, N. Auerbach, F. Izrailev, G. Berman, Phys. Rev. Lett. 106, 042501 (2011)