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One considers the core + valence particle approach, where 16O is the core and the valence space consists of
$ spdf $ proton and neutron partial waves. The Hamiltonian used consists of a potential mimicking the effect of the core and of the FHT interaction in the$ spdf $ valence space:$$ H = \sum\limits_i \left( \frac{{ {p_i}}^2}{2 m_i} + \hat{U}^{(i)}_{\rm{core}} \right) + \sum\limits_{i<j} \hat{V}_{ij} , $$ (1) where
$ m_i $ is the effective mass of the nucleon,$ \hat{U}^{(i)}_{\rm{core}} $ is the WS potential of the core acting on the$ i $ -th nucleon, and$ \hat{V}_{ij} $ is the residual interaction, containing the FHT interaction, the Coulomb interaction, and the recoil effects induced by the finite mass of the core[29].The negative-parity states of 17O and 17F are mainly built from 1p-1h excitations from the
$ 0p_{1/2} $ orbitals to the$ sd $ shell, while positive-parity states besides s.p. states mainly consist of 2p-2h excitations from$ 0p_{1/2} $ orbitals to the$ sd $ shell. Consequently, in order to have a sizable configuration mixing, while still dealing with small model space dimensions, one included configurations with up to 3p-3h excitations from$ 0p_{1/2} $ shells. However, this creates a double-counting of nucleon-nucleon correlations in$ H $ (see Eq. (1)), as$ \hat{U}^{(i)}_{\rm{core}} $ mimics the effect of the 16O core when core shells are fully occupied. This spurious effect can be removed with standard correction formulas[32]. The expressions of Ref. [32] are cumbersome, however, so that one has introduced a simplified removal procedure of double-counting terms. We use the following potential, denoted as$ \hat{U}_{dc} $ , which is subtracted from the one-body part of Eq. (1):$$\begin{split} \left\langle {f|{{\hat U}_{dc}}|i} \right\rangle =& \sum\limits_{s \in {\rm{core}},J} {\sqrt {1 \!+\! {\delta _{is}}} } \sqrt {1 \!+\! {\delta _{fs}}}\times\\& \left( {\frac{{2J \!+\! 1}}{{2{j_s} \!+\! 1}}} \right)\left\langle {sf|\hat V|s{i_J}} \right\rangle ,\end{split}$$ (2) where
$ \left| i \right\rangle $ and$ \left| f \right\rangle $ are basis one-body states and$ s $ runs over the active shells of the core, i.e. on proton and neutron$ 0p_{1/2} $ shells. Eq. (2) removes the spurious energy generated by the nuclear interaction already taken care of by$ \hat{U}^{(i)}_{\rm{core}} $ . In fact,$ \hat{U}_{dc} $ resembles the part of the spherical Hartree-Fock potential generated by$ H $ involving fully occupied proton and neutron$ 0p_{1/2} $ shells (see Eqs. (1,2)). Note that$ {\left\langle {si|\hat V|si} \right\rangle _J} = 0$ by definition if$ s,i \in {\rm{core}} $ , as one considers that this matrix element is already included in$ \hat{U}^{(i)}_{\rm{core}} $ .Continuum coupling enters 17O and 17F wave functions only via the coupled channels of GSM-CC (see Sec. 2.2), while 16O target states are well bound. Hence, 16O target states will be built from HO states in our model. For this, one takes the first 5, 4, 4 and 3 HO one-body states of the
$ spdf $ partial waves, respectively, from which the$ 0s_{1/2} $ and$ 0p_{3/2} $ shells are removed as they are inert shells of the core. The harmonic oscillator length is fixed at b = 2 fm. Moreover, Slater determinants can have only one proton or neutron occupied in HO one-body states verifying$ 2n + \ell \geqslant 3 $ . As the resonance spectrum of 17O and 17F is unbound by nucleon emission only, the presented truncation scheme recaptures the main structure of the considered many-body eigenstates of 17O and 17F. -
In GSM-CC, one decomposes the many-body Hamiltonian eigenstates in a basis of reaction channels:
$$\left| {c,r} \right\rangle = \widehat {\cal A}\left| {\{ \left| {\Psi _T^{{J_T}}} \right\rangle \otimes \left| {r\;\ell \;j} \right\rangle \} _{{M_A}}^{{J_A}}} \right\rangle ,$$ (3) $$\left| \Psi \right\rangle = \sum\limits_c {\int_0^{ + \infty } {\left( {\frac{{{u_c}(r)}}{r}} \right)} } {r^2}\left| {c,r} \right\rangle \;{\rm{d}}r ,$$ (4) where
$ c $ denotes the channel quantum numbers,$ \widehat {\cal A}{\rm{ }}\left| {\{ \left| {\Psi _T^{{J_T}}} \right\rangle \otimes \left| {r\;\ell \;j} \right\rangle \} } \right\rangle $ is an antisymmetrized tensor product of the target state$ \left| {\Psi _T^{{J_T}}} \right\rangle $ and projectile state$ \left| {r\;\ell \;j} \right\rangle $ , where$ { {J_A}} = { {J_T}} + { {j}} $ , and$ u_{ c } (r) $ is the radial wave function to determine.Starting from the Schrödinger equation
$ H \left| \Psi \right\rangle = E\left| \Psi \right\rangle $ , one derives the coupled-channel equations to solve in GSM-CC:$$ \!\sum\limits_{c}\int_{0}^{\infty} \!\!\! r^{ 2 } \left( H_{ cc' } (r , r') - E N_{ cc' } (r , r') \right) \left( \frac{ { u }_{ c } (r) }{ r } \right) \; {\rm d}r = 0\,, $$ (5) where
$ H_{ cc' } (r,r') $ and$ N_{ cc' } (r,r') $ arise from the integration of the Schrödinger equation over all degrees of freedom except$ r $ :$$ \begin{split} H_{ cc' } (r , r') =& \frac{ { \hbar }^{ 2 }}{ 2 m_p} \left( -\frac{ { d }^{ 2 }}{ {\rm d} r^{ 2 }} + \frac{\ell(\ell + 1)}{r^2} + E_{ T } +\right.\\& { U_{\rm{basis}}} (r) \left) \frac{ \delta (r - r') }{ r r' } { \delta }_{ cc' } \right. + { \tilde{ V }}_{ cc' } (r , r') \,, \\ N_{ cc' } (r , r') = & \frac{ \delta (r - r') }{ r r' } { \delta }_{ cc' } + \Delta N_{ cc' } (r , r') \,, \end{split} $$ (6) where
$ U_{\rm{basis}} (r) $ is the basis-generating potential, of WS type, and$ \tilde{ V }_{ cc' } $ ,$ \Delta N_{ cc' } (r , r') $ consist of the residual short-range terms of the Hamiltonian and norm kernels, respectively. As target states are expressed with a basis of Slater determinants, matrix elements involving target states are straightforward to compute using HO shell model (HO-SM) formulas.Eq. (5) is, in fact, a generalized eigenvalue problem, due to the presence of the norm kernel
$ N_{ cc' } (r , r') $ . The antisymmetry requirement between projectile and target indeed generates non-orthogonality between compound target-projectile basis states. A convenient method to solve generalized eigenvalue problems consists to apply a unitary transformation to the Hamiltonian so that one obtains a standard eigenvalue problem. For this, one introduces the overlap operator$ \hat{O} $ . The demanded standard eigenvalue problem equivalent to Eq. (5) then reads:$$ \sum\limits_c {\int_0^\infty {\rm{d}} } r\;{r^2}\left\langle {c',r'|{{\hat H}_m} - E|c,r} \right\rangle \left( {\frac{{{w_c}(r)}}{r}} \right) = 0,\;\;\; $$ (7) where
$ \hat{H}_{m} = \hat{O}^{ - \frac{1}{2} } \hat{H} \hat{O}^{ - \frac{1}{2}} $ is the modified Hamiltonian and where:$${w_c}(r) = \frac{{\left\langle {c,r|{{\hat O}^{\frac{1}{2}}}|\Psi } \right\rangle }}{r}{\mkern 1mu} .$$ (8) Thus, the
$ {w}_{c} (r) $ radial wave functions are firstly calculated from the standard eigenvalue problem of Eq. (7). The$ {u}_{c} (r) $ radial wave functions of Eq. (5) are then straightforward to recover from$ {w}_{c} (r) $ and the overlap operator$ \hat{O} $ . Once the$ {u}_{c} (r) $ radial wave functions are obtained, cross sections can be calculated from standard reaction theory. Indeed, besides kinetic and angular-momentum recoupling terms, cross sections are functions of the phase shifts of the$ {u}_{c} (r) $ radial wave functions, which are straightforward to obtain from the asymptotic form of$ {u}_{c} (r) $ functions.The coupled-channel representation of the Schrödinger equation of Eq. (7) will be solved using the Berggren basis[33]. It consists of resonant and scattering one-body states generated by a finite-depth potential[33]:
$$ \sum\limits_{n} u_{n}(r) \; u_{n}(r') + \int_{L^{+}} u(k,r) \; u(k,r') \; {\rm d}k = \delta(r - r')\,, $$ (9) where
$ u_n(r) $ and$ u(k,r) $ are resonant and scattering states, respectively, and$ L^+ $ is the contour of scattering states in the complex plane, which encompasses unbound$ u_n(r) $ states. All one-body states have the same angular quantum numbers, so that the three-dimensional completeness relation is recovered by summing over all partial waves. In numerical calculations, the scattering part of Eq. (9) is discretized using the Gauss-Legendre quadrature[34]:$$ \sum\limits_{n = 1}^{N} u^{(n)}(r) \; u^{(n)}(r') \simeq \delta(r - r')\,, $$ (10) where
$ N $ is the number of resonant and discretized scattering states, and$ u^{(n)}(r) $ is either a resonant$ u_n(r) $ wave function or the renormalized scattering$ \sqrt{w_n} \; u(k_n,r) $ function, with$ k_n $ and$ w_n $ the Gauss-Legendre abscissa and weight on the$ L^+ $ discretized contour, respectively. The latter renormalization is necessary to replace the initial Dirac delta normalization of$ u(k,r) $ functions in Eq. (9) by a Kronecker delta normalization.Let us firstly consider the calculation of bound and resonance many-body states.
$ { { u }_{ c }(r) } $ (see Eq. (5)) can be expanded in the basis of coupled-channel Berggren basis states arising from Eq. (10). The obtained matrix problem is formally identical to that of GSM, where a basis of Slater determinants is utilized, so that bound and resonance eigenstates can be calculated using the Jacobi-Davidson and overlap methods devised in GSM[28].Scattering eigenstates can be conveniently implemented by making use of the Hamiltonian Green's function[27]. For this, one uses the basis-generating Hamiltonian, whose eigenvector at the considered scattering energy
$ E $ is denoted as$\left| {{\Psi ^{(b)}}} \right\rangle $ . One can then separate$ \hat{H} $ and$ \left| {\Psi _{{M_A}}^{{J_A}}} \right\rangle $ into two parts, involving the basis-generating Hamiltonian,$ \left| {{\Psi ^{(b)}}} \right\rangle $ , and a rest part:$$ \hat{H} = t + \hat{U}_{\rm{basis}} + \hat{H}_{\rm{rest}}\,, $$ (11) $$\left| {\Psi _{{M_A}}^{{J_A}}} \right\rangle = \left| {{\Psi ^{(b)}}} \right\rangle + \left| {{\Psi _{{\rm{rest}}}}} \right\rangle . $$ (12) Using Eqs. (11, 12), one obtains:
$$ (\hat H - E)\left| {{\Psi _{{\rm{rest}}}}} \right\rangle = - {\hat H_{{\rm{rest}}}}\left| {{\Psi ^{(b)}}} \right\rangle . $$ (13) All values entering Eq. (13) are finite-ranged, so that they can be represented with a Berggren basis. Eq. (13) then becomes a linear system in the Berggren coupled-channel basis representation. Note that Eq. (13) is not invertible on the real energy axis. In order to avoid this problem, the contour of Berggren basis states is defined so that their complex energy always has a non-zero imaginary part[27]. This procedure is, in fact, equivalent to the standard replacement of
$ E $ by$ E + i \epsilon $ , with$ \epsilon \rightarrow 0^+ $ , which imposes an outgoing wave function character for$ \left| {{\Psi _{{\rm{rest}}}}} \right\rangle $ on all outgoing channels[27].As a consequence, the calculation of bound, resonance and scattering states in GSM-CC is handled with standard eigenvalue problems and linear systems, respectively, so that the determination of the solutions of Eqs. (5, 7) in GSM-CC is both fast and numerically stable.
Calculations of the 17O and 17F Spectra and 16O(p,p) Reaction Cross Sections in the Coupled-channel Gamow Shell Model
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摘要: 利用耦合道Gamow壳模型计算了17O和17F的低激发能谱以及16O(p,p)反应的低能弹性散射截面。结果表明,17O和17F中非束缚共振态能级的核子发射宽度的计算需要合理地考虑连续态耦合效应。计算得到的17O和17F的低激发能谱以及16O(p,p)反应的低能弹性散射激发函数都与实验数据吻合较好。这说明基于现实核力的计算可更好地描述16O(p,p)反应的低能弹性散射截面。Abstract: We apply the coupled-channel Gamow shell model to calculate the spectra of 17O and 17F, as well as 16O(p,p) elastic cross sections at low energies. It is shown that continuum coupling is necessary to account for the particle-emission width of the unbound eigenstates of 17O and 17F. The low-lying spectrum of 17O and 17F and 16O(p,p) excitations functions are in fair agreement with experimental data. Nevertheless, it is also shown that the use of a realistic nuclear Hamiltonian is needed to have an optimal reproduction of 16O(p,p) elastic cross sections in the low-energy region.
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Figure 1. Low-lying energy spectrum of 17F calculated with HO-SM and GSM-CC and compared with experimental data (Exp). Positive-parity states only are depicted in this figure. Energies are given with respect to the 16O ground state and are given in units of MeV. The proton-emission width of unbound states is written next to their energy and is given in keV. Experimental data are taken from Ref. [11].
Figure 2. Same as Fig. 1, but for negative-parity states.
Figure 3. Same as Fig. 1, but for 17O positive-parity states, and where neutron-emission widths are depicted.
Figure 4. Same as Fig. 3, but for negative-parity states.
Figure 5. Excitation function of the 16O(p,p) elastic scattering reaction calculated with GSM-CC at
$ \theta = 140.76^\circ $ . Experimental data are taken from Ref. [36]. Energies and angles are given in the center of mass frame.Figure 6. Same as Fig. 5, but with
$ \theta = 166.75^\circ $ . -
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