68
2 The Discrete Spectrum and the Continuum
state, for example, −0.1 keV, and then slowly vary the potential depth so that
the calculated state becomes a resonance.
B. Notice that both numerical integration and current formula give the same
results when Γ > 10 −6 keV. Check that only the current formula can provide
with a nonzero width when Γ < 10 −6 keV.
In the proton case, u(k, R) becomes bound at zero energy as well, so that one
obtains similarly to the neutron case:
C
+
∼ A 0 C (η) k
∼ A 1 exp (−πη) η
1/2 ,
(2.200)
where A 0 and A 1 are constants depending on Hamiltonian parameters only, and
where Eq. (2.57) has been used. Thus Eq. (2.197) for protons becomes:
Γ ∼ A p exp (−2π (η)) ,
(2.201)
where A p is a constant depending on the proton Hamiltonian only (see Exercise XVII for numerical applications). Note that approximate expressions for the
energy and width of resonance states have been developed in the context of the
two-potential method [51], where only bound and real-energy scattering solutions
of Eq. (2.2) are needed for their evaluation.
Exercise XVII
One will demonstrate in a numerical example that the relations between
energy and width in the resonance case of Eqs. (2.199) and (2.201) hold in
practice.
A. Run the one-particle code of radial wave functions to generate plots of complex
linear momenta, energies, and widths close to the particle emission threshold,
for both proton and neutron case, and different orbital angular momenta
B. Determine the overall factor A in Eqs. (2.199) and (2.201), so that the
asymptotic behavior of width in the vicinity of particle emission threshold is
reproduced.
C. Check that the expressions in Eqs. (2.199), and (2.201) provide with the correct
behavior of Γ in the vicinity of particle emission threshold.
2 The Discrete Spectrum and the Continuum
state, for example, −0.1 keV, and then slowly vary the potential depth so that
the calculated state becomes a resonance.
B. Notice that both numerical integration and current formula give the same
results when Γ > 10 −6 keV. Check that only the current formula can provide
with a nonzero width when Γ < 10 −6 keV.
In the proton case, u(k, R) becomes bound at zero energy as well, so that one
obtains similarly to the neutron case:
C
+
∼ A 0 C (η) k
∼ A 1 exp (−πη) η
1/2 ,
(2.200)
where A 0 and A 1 are constants depending on Hamiltonian parameters only, and
where Eq. (2.57) has been used. Thus Eq. (2.197) for protons becomes:
Γ ∼ A p exp (−2π (η)) ,
(2.201)
where A p is a constant depending on the proton Hamiltonian only (see Exercise XVII for numerical applications). Note that approximate expressions for the
energy and width of resonance states have been developed in the context of the
two-potential method [51], where only bound and real-energy scattering solutions
of Eq. (2.2) are needed for their evaluation.
Exercise XVII
One will demonstrate in a numerical example that the relations between
energy and width in the resonance case of Eqs. (2.199) and (2.201) hold in
practice.
A. Run the one-particle code of radial wave functions to generate plots of complex
linear momenta, energies, and widths close to the particle emission threshold,
for both proton and neutron case, and different orbital angular momenta
B. Determine the overall factor A in Eqs. (2.199) and (2.201), so that the
asymptotic behavior of width in the vicinity of particle emission threshold is
reproduced.
C. Check that the expressions in Eqs. (2.199), and (2.201) provide with the correct
behavior of Γ in the vicinity of particle emission threshold.