Tensor Interaction Effects on Stellar Electron Capture and Beta-Decay Rates

Abstract

The Gamow–Teller strength functions as well as electron capture and $\beta$ decay rates for ${}^{56,78}$Ni in hot stellar environment are calculated within a thermodynamically consistent approach based on the thermal quasiparticle RPA. We present a thorough analysis of the tensor interaction effects on the GT strength functions at zero and finite temperatures. To this aim we use self-consistent calculations with the Skyrme interaction. Several Skyrme interactions are used in order to verify the sensitivity of the obtained results to the Skyrme force parameters. It is pointed out that finite temperature amplifies the effect of tensor correlations on the GT strength functions. Our calculations of electron capture and $\beta$ decay rates demonstrate the importance of the tensor interaction for simulation of supernova weak-interaction processes.

Appendix

Here we give explicit form of the TQRPA equations for $J^{\pi}=1^{+}$ charge-changing transitions in hot nuclei taking into account the tensor part of the residual particle–hole interaction (16). More technical details can be found in [11, 13, 15].

Within the TQRPA the thermal Hamiltonian is diagonalized in terms of thermal multipole phonons:

$$\mathcal{H}\approx\sum_{JMi}\omega_{Ji}(T)(Q^{\dagger}_{JMi}Q_{JMi}-\widetilde{Q}^{\dagger}_{JMi}\widetilde{Q}_{JMi}).$$

(A.1)

For charge-changing multipole transitions the thermal phonon operators are defined as a linear superposition of creation and annihilation operators of proton–neutron thermal quasiparticle pairs

$$Q^{\dagger}_{JMi}=\sum_{j_{p}j_{n}}\Bigl{\{}\psi^{Ji}_{j_{p}j_{n}}[\beta^{\dagger}_{j_{p}}\beta^{\dagger}_{j_{n}}]^{J}_{M}+\widetilde{\psi}^{Ji}_{j_{p}j_{p}}[\widetilde{\beta}^{\dagger}_{\overline{\jmath_{p}}}\widetilde{\beta}^{\dagger}_{\overline{\jmath_{n}}}]^{J}_{M}$$

$${}+i\eta^{Ji}_{j_{p}j_{n}}[\beta^{\dagger}_{j_{p}}\widetilde{\beta}^{\dagger}_{\overline{\jmath_{n}}}]^{J}_{M}+i\widetilde{\eta}^{Ji}_{j_{p}j_{n}}[\widetilde{\beta}^{\dagger}_{\overline{\jmath_{p}}}\beta^{\dagger}_{j_{n}}]^{J}_{M}$$

$${}+\phi^{Ji}_{j_{p}j_{n}}[\beta_{\overline{\jmath_{p}}}\beta_{\overline{\jmath_{n}}}]^{J}_{M}+\widetilde{\phi}^{Ji}_{j_{p}j_{n}}[\widetilde{\beta}_{j_{p}}\widetilde{\beta}_{j_{n}}]^{J}_{M}$$

$${}+i\xi^{Ji}_{j_{p}j_{n}}[\beta_{\overline{\jmath_{p}}}\widetilde{\beta}_{j_{n}}]^{J}_{M}+i\widetilde{\xi}^{Ji}_{j_{p}j_{n}}[\widetilde{\beta}_{j_{p}}\beta_{\overline{\jmath_{n}}}]^{J}_{M}\Bigr{\}}.$$

(A.2)

Here, $[]^{J}_{M}$ denotes the coupling of two single-particle angular momenta $j_{p},j_{n}$ to the total angular momentum $J$. In what follows we will drop the index $J$ and assume that $J^{\pi}=1^{+}$. In turn, thermal quasiparticles are normal modes of the mean field and pairing parts of the thermal Hamiltonian

$$\mathcal{H}_{\textrm{mf}}+\mathcal{H}_{\textrm{pair}}$$

$${}\simeq\sum_{\tau=n,p}{\sum_{jm}}^{\tau}\varepsilon_{j}(T)(\beta^{\dagger}_{jm}\beta_{jm}-\widetilde{\beta}^{\dagger}_{jm}\widetilde{\beta}_{jm}),$$

(A.3)

and their vacuum is the thermal vacuum in the BCS approximation. The physical meaning of different terms in (A.2) is explained in [13, 41]. Here we just mention that the creation of a negative-energy tilde thermal quasiparticle corresponds to the annihilation of a thermally excited Bogoliubov quasiparticle. Therefore, the spectrum of thermal phonons contains negative-energy and low-energy states which do not exist at zero temperature. These ‘‘new’’ phonon states are interpreted as thermally unblocked transitions between nuclear excited states.

Following [15], we first introduce the linear combinations of phonon amplitudes:

$$\binom{g}{w}^{i}_{j_{p}j_{n}}=\psi^{i}_{j_{p}j_{n}}\pm\phi^{i}_{j_{p}j_{n}},$$

$$\binom{\widetilde{g}}{\widetilde{w}}^{i}_{j_{p}j_{n}}=\widetilde{\psi}^{i}_{j_{p}j_{n}}\pm\widetilde{\phi}^{i}_{j_{p}j_{n}},$$

$$\binom{t}{s}^{i}_{j_{p}j_{n}}=\eta^{i}_{j_{p}j_{n}}\pm\xi^{i}_{j_{p}j_{n}},$$

$$\binom{\widetilde{t}}{\widetilde{s}}^{i}_{j_{p}j_{n}}=\widetilde{\eta}^{i}_{j_{p}j_{n}}\pm\widetilde{\xi}^{i}_{j_{p}j_{n}}.$$

(A.4)

Then, applying the thermal state condition (10) for the vacuum of thermal phonons it can be shown that the following relations are valid

$$\binom{g}{w}^{i}_{j_{p}j_{n}}=(x_{j_{p}}x_{j_{n}}-{e}^{-\omega_{i}/2T}y_{j_{p}}y_{j_{n}})\binom{G}{W}^{i}_{j_{p}j_{n}}$$

$$\binom{\widetilde{g}}{\widetilde{w}}^{i}_{j_{p}j_{n}}=\mp(y_{j_{p}}y_{j_{n}}-{e}^{-\omega_{i}/2T}x_{j_{p}}x_{j_{n}})\binom{G}{W}^{i}_{j_{p}j_{n}}$$

$$\binom{t}{s}^{i}_{j_{p}j_{n}}=(x_{j_{p}}y_{j_{n}}-{e}^{-\omega_{i}/2T}y_{j_{p}}x_{j_{n}})\binom{T}{S}^{i}_{j_{p}j_{n}}$$

$$\binom{\widetilde{t}}{\widetilde{s}}^{i}_{j_{p}j_{n}}=\mp(y_{j_{p}}x_{j_{n}}$$

$${}-{e}^{-\omega_{i}/2T}x_{j_{p}}y_{j_{n}})\binom{T}{S}^{i}_{j_{p}j_{n}}.$$

(A.5)

Here, $x_{j}$ and $y_{j}$ ($x^{2}_{j}+y^{2}_{j}=1$) are the coefficients of the so-called thermal transformation which establishes a connection between Bogoliubov and thermal quasiparticles. Note that $y_{j}$ are given by the nucleon Fermi–Dirac function and they define a number of thermally excited Bogoliubov quasiparticles in the thermal vacuum (see [13] for more details). The variables $G$, $W$, $T$, and $S$ are normalized according to

$$\sum_{j_{n}j_{p}}\Bigl{(}G^{i}_{j_{p}j_{n}}W^{i^{\prime}}_{j_{p}j_{n}}(1-y^{2}_{j_{p}}-y^{2}_{j_{n}})$$

$${}-T^{i}_{j_{p}j_{n}}S^{i^{\prime}}_{j_{p}j_{n}}(y^{2}_{j_{p}}-y^{2}_{j_{n}})\Bigr{)}$$

$${}=\delta_{ii^{\prime}}/(1-{e}^{-\omega_{i}/T}).$$

(A.6)

Applying the equation of motion method, we get the system of TQRPA equations for unknown variables $G$, $W$, $T$, and $S$ and phonon energies $\omega_{i}$

$$G^{i}_{j_{p}j_{n}}\pm W^{i}_{j_{p}j_{n}}=\frac{2}{3}\frac{1}{\varepsilon^{(+)}_{j_{p}j_{n}}\mp\omega_{i}}$$

$${}\times\sum_{n=1}^{2N+2}f^{(n)}_{j_{p}j_{n}}\bigl{(}u^{(+)}_{j_{p}j_{n}}D^{in}_{+}\pm u^{(-)}_{j_{p}j_{n}}D^{in}_{-}\bigr{)},$$

$$T^{i}_{j_{p}j_{n}}\pm S^{i}_{j_{p}j_{n}}=\frac{2}{3}\frac{1}{\varepsilon^{(-)}_{j_{p}j_{n}}\mp\omega_{i}}$$

$${}\times\sum_{n=1}^{2N+2}f^{(n)}_{j_{p}j_{n}}\bigl{(}v^{(-)}_{j_{p}j_{n}}D^{in}_{+}\pm v^{(+)}_{j_{p}j_{n}}D^{in}_{-}\bigr{)}.$$

(A.7)

In the above equations we have introduced the following combinations of the thermal quasiparticle energies and the Bogoliubov $(u,v)$ coefficients: $\varepsilon^{(\pm)}_{j_{p}j_{n}}=\varepsilon_{j_{p}}\pm\varepsilon_{j_{n}}$, $u^{(\pm)}_{j_{p}j_{n}}=u_{j_{p}}v_{j_{n}}\pm v_{j_{p}}u_{j_{n}}$, and $v^{(\pm)}_{j_{p}j_{n}}=u_{j_{p}}u_{j_{n}}\pm v_{j_{p}}v_{j_{n}}$. The factors $f^{(n)}_{j_{p}j_{n}}$ are given by

$$f^{(n)}_{j_{p}j_{n}}$$

$${}=\begin{cases}\kappa_{1}^{(n)}g^{(01k)}_{j_{p}j_{n}},\quad\textrm{if }n=1,k,\\ \kappa_{1}^{(n)}g^{(21k)}_{j_{p}j_{n}},\quad\textrm{if }n=N+k,\\ \lambda_{1}t^{(21)}_{j_{p}j_{n}},\quad\textrm{if }n=2N+1,\\ \lambda_{1}t^{(01)}_{j_{p}j_{n}}-\lambda_{2}t^{(21)}_{j_{p}j_{n}},\quad\textrm{if }n=2N+2.\end{cases}$$

(A.8)

Here, the isovector strength parameters $\kappa_{1}^{(n)}$ and the reduced matrix elements $g^{(LJk)}_{j_{p}j_{n}}$ stem from the central part of the residual interaction and they are defined in [24], while $t^{(LJ)}_{j_{p}j_{n}}$ denotes the reduced single-particle matrix elements due to tensor interaction

$$t^{(LJ)}_{j_{p}j_{n}}=\langle j_{p}||i^{L}r^{L}T_{LJ}||j_{n}\rangle.$$

(A.9)

And finally, $D^{in}_{\pm}$ are defined as

$$D^{in}_{+}=\sum_{j_{p}j_{n}}f^{(n)}_{j_{p}j_{n}}\Bigl{\{}u^{(+)}_{j_{p}j_{n}}(1-y^{2}_{j_{p}}-y^{2}_{j_{n}})G^{i}_{j_{p}j_{n}}$$

$${}-v^{(-)}_{j_{p}j_{n}}(y^{2}_{j_{p}}-y^{2}_{j_{n}})T^{i}_{j_{p}j_{n}}\Bigr{\}},$$

$$D^{in}_{-}=\sum_{j_{p}j_{n}}f^{(n)}_{j_{p}j_{n}}\Bigl{\{}u^{(-)}_{j_{p}j_{n}}(1-y^{2}_{j_{p}}-y^{2}_{j_{n}})W^{i}_{j_{p}j_{n}}$$

$${}-v^{(+)}_{j_{p}j_{n}}(y^{2}_{j_{p}}-y^{2}_{j_{n}})S^{i}_{j_{p}j_{n}}\Bigr{\}},$$

(A.10)

with

$$d^{(n)}_{j_{p}j_{n}}=\begin{cases}g^{(01k)}_{j_{p}j_{n}},\quad\textrm{if }n=1,k,\\ g^{(21k)}_{j_{p}j_{n}},\quad\textrm{if }n=N+k,\\ t^{(01)}_{j_{p}j_{n}},\quad\textrm{if }n=2N+1,\\ t^{(21)}_{j_{p}j_{n}},\quad\textrm{if }n=2N+2.\end{cases}$$

(A.11)

Because of the separable form of the residual interaction the TQRPA equations for phonon amplitudes and phonon energies $\omega_{i}$ can be reduced to the set of equations for $4N+4$ unknown variables $D^{in}_{\mp}$

$$\left(\begin{array}{cc}\mathcal{M}_{1}-\frac{1}{2}I&\mathcal{M}_{2}\\ \mathcal{M}_{2}&\mathcal{M}_{3}-\frac{1}{2}I\end{array}\right)\left(\begin{array}{c} D_{+}\\ D_{-} \end{array}\right)=0.$$

(A.12)

The matrix elements of the $(2N+2)\times(2N+2)$ matrices $\mathcal{M}_{\beta}$ are the following:

$$\mathcal{M}^{nn^{\prime}}_{1,3}=\frac{1}{3}\sum_{j_{p}j_{n}}d^{(Jn)}_{j_{p}j_{n}}f^{(Jn^{\prime})}_{j_{p}j_{n}}$$

$${}\times\left\{\frac{\varepsilon_{j_{p}j_{n}}^{(+)}(u^{(\pm)}_{j_{p}j_{n}})^{2}}{(\varepsilon_{j_{p}j_{n}}^{(+)})^{2}-\omega^{2}_{Ji}}(1-y^{2}_{j_{p}}-y^{2}_{j_{n}})\right.$$

$${}-\left.\frac{\varepsilon_{j_{p}j_{n}}^{(-)}(v^{(\mp)}_{j_{p}j_{n}})^{2}}{(\varepsilon_{j_{p}j_{n}}^{(-)})^{2}-\omega^{2}_{Ji}}(y^{2}_{j_{p}}-y^{2}_{j_{n}})\right\},$$

$$\mathcal{M}^{nn^{\prime}}_{2}=\frac{1}{3}\omega_{i}\sum_{j_{n}j_{p}}d^{(Jn)}_{j_{p}j_{n}}f^{(Jn^{\prime})}_{j_{p}j_{n}}$$

$${}\times\left\{\frac{u^{(+)}_{j_{p}j_{n}}u^{(-)}_{j_{p}j_{n}}}{(\varepsilon_{j_{p}j_{n}}^{(+)})^{2}-\omega^{2}_{Ji}}(1-y^{2}_{j_{p}}-y^{2}_{j_{n}})\right.$$

$${}-\left.\frac{v^{(+)}_{j_{p}j_{n}}v^{(-)}_{j_{p}j_{n}}}{(\varepsilon_{j_{p}j_{n}}^{(-)})^{2}-\omega^{2}_{Ji}}(y^{2}_{j_{p}}-y^{2}_{j_{n}})\right\},$$

where $1\leq n$, $n^{\prime}\leq 2N+2$. Thus, the TQRPA eigenvalues $\omega_{Ji}$ are the roots of the secular equation

$$\textrm{det}\left(\begin{array}{cc}\mathcal{M}_{1}-\frac{1}{2}I&\mathcal{M}_{2}\\ \mathcal{M}_{2}&\mathcal{M}_{3}-\frac{1}{2}I\end{array}\right)=0,$$

(A.13)

while the phonon amplitudes corresponding to the TQRPA eigenvalue $\omega_{i}$ are determined by Eqs. (A.4), (A.5), and (A.7), taking into account the normalization condition (A.6).