Superoperator Approach to the Theory of Hot Nuclei and Astrophysical Applications: I—Spectral Properties of Hot Nuclei

Abstract

The method of superoperators in Liouville space was applied to study spectral properties of hot nuclei. It is shown that properly defined fermionic superoperators allow us to generalize the equation-of-motion method to hot nuclei. Within the superoperator approach, for the nuclear model with separable particle–hole residual interaction of Landau–Migdal type, we derived the equations of thermal quasiparticle random phase approximation, which allow the spectral densities and strength functions of charge-exchange and charge-neutral excitations of hot nuclei to be calculated in a thermodynamically consistent way, i.e., without violating the principle of detailed balance. For the quasiparticle-phonon nuclear model, a thermodynamically consistent way is proposed for going beyond the random phase approximation by considering the interaction of thermal phonons. Using the Donnelly–Walecka method and the superoperator approach, expressions for cross sections of semileptonic weak reactions with hot nuclei are obtained.

Appendices

APPENDICES

1.1 APPENDIX A: Finding functions $\alpha (m,n)$ and $\beta (m,n)$

For the right superoperators defined according to (47), the conditions ${{[{\kern 1pt} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }_{i}},{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }_{j}}]}_{\sigma }} = 0$, ${{[{\kern 1pt} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{i}^{\dag },\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag }]}_{\sigma }} = 0$ are satisfied regardless of the choice of the functions $\alpha (m,n)$ and $\beta (m,n)$. From the condition ${{[{\kern 1pt} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }_{i}},\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag }]}_{\sigma }} = 0$ for $i \ne j$ and relations

$$\begin{gathered} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }}_{i}}{\kern 1pt} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag }\left| {\left| {mn} \right\rangle } \right\rangle = \alpha (m,n + 1)\beta (m,n)\left| {\left| {\left| m \right\rangle \left\langle n \right|{{a}_{j}}a_{i}^{\dag }} \right\rangle } \right\rangle , \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag }{\kern 1pt} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }}_{i}}\left| {\left| {mn} \right\rangle } \right\rangle = \beta (m,n - 1)\alpha (m,n)\left| {\left| {\left| m \right\rangle \left\langle n \right|a_{i}^{\dag }{{a}_{j}}} \right\rangle } \right\rangle \\ \end{gathered} $$

(310)

we obtain

$$\alpha (m,n)\beta (m,n - 1) = \alpha (m,n + 1)\beta (m,n).$$

(311)

With allowance for this relation, from ${{[{\kern 1pt} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }_{j}},\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag }]}_{\sigma }} = 1$ it follows that

$$\alpha (m,n){\kern 1pt} \beta (m,n - 1) = 1.$$

(312)

Consequently, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{i}^{\dag }{\kern 1pt} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }_{i}}$ satisfies condition (45), i.e., is the superoperator of the number of particles.

Since

$$\begin{gathered} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }}_{j}}\vec {a}{\kern 1pt} _{{i}}^{\dag }\left| {\left| {mn} \right\rangle } \right\rangle = \alpha (m + 1,n)\left| {\left| {a_{i}^{\dag }{\kern 1pt} \left| m \right\rangle \left\langle n \right|{\kern 1pt} a_{j}^{\dag }} \right\rangle } \right\rangle , \\ \vec {a}{\kern 1pt} _{{i}}^{\dag }{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }}_{j}}\left| {\left| {mn} \right\rangle } \right\rangle = \alpha (m,n)\left| {\left| {a_{i}^{\dag }{\kern 1pt} \left| m \right\rangle \left\langle n \right|{\kern 1pt} a_{j}^{\dag }} \right\rangle } \right\rangle . \\ \end{gathered} $$

(313)

Then the requirement ${{[{\kern 1pt} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }_{j}},\vec {a}{\kern 1pt} _{{i}}^{\dag }]}_{\sigma }} = 0$ leads to

$$\alpha (m + 1,n) = - \sigma \alpha (m,n),$$

(314)

whence it follows that

$$\alpha (m,n) = {{( - \sigma )}^{m}}\alpha (0,n).$$

(315)

The same relation follows from ${{[{\kern 1pt} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }_{j}},{{\vec {a}}_{i}}]}_{\sigma }} = 0$. Proceeding similarly, from the conditions ${{[{\kern 1pt} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag },\vec {a}{\kern 1pt} _{{i}}^{\dag }]}_{\sigma }} = 0$ and ${{[{\kern 1pt} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag },{{\vec {a}}_{i}}]}_{\sigma }} = 0$, we get

$$\beta (m + 1,n) = - \sigma \beta (m,n)$$

(316)

or

$$\beta (m,n) = {{( - \sigma )}^{m}}\beta (0,n).$$

(317)

Consider now the following chains of equalites:

$$\begin{gathered} \left\langle {\left\langle {{{m}_{1}}{{n}_{1}}\left\| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{i}^{\dag }} \right\|{{m}_{2}}{{n}_{2}}} \right\rangle } \right\rangle = \beta ({{m}_{2}},{{n}_{2}}){{\delta }_{{{{m}_{1}}{{m}_{2}}}}}\langle {{n}_{2}}\left| {{{a}_{i}}} \right|{{n}_{1}}\rangle \\ = \beta ({{m}_{2}},{{n}_{2}}){{\delta }_{{{{m}_{1}}{{m}_{2}}}}}{{\delta }_{{{{n}_{1}},{{n}_{2}} - 1}}}\langle {{n}_{2}}\left| {{{a}_{i}}} \right|{{n}_{1}}\rangle \\ \end{gathered} $$

(318)

and

$$\begin{gathered} \left\langle {\left\langle {{{m}_{2}}{{n}_{2}}\left\| {{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }}_{i}}} \right\|{{m}_{1}}{{n}_{1}}} \right\rangle } \right\rangle = \alpha ({{m}_{1}},{{n}_{1}}){{\delta }_{{{{m}_{1}}{{m}_{2}}}}}\left\langle {{{n}_{1}}\left| {a_{i}^{\dag }} \right|{{n}_{2}}} \right\rangle \\ = \alpha ({{m}_{1}},{{n}_{1}}){{\delta }_{{{{m}_{1}}{{m}_{2}}}}}{{\delta }_{{{{n}_{1}},{{n}_{2}} - 1}}}\left\langle {{{n}_{1}}\left| {a_{i}^{\dag }} \right|{{n}_{2}}} \right\rangle \\ = \alpha ({{m}_{2}},{{n}_{2}} - 1){{\delta }_{{{{m}_{1}}{{m}_{2}}}}}{{\delta }_{{{{n}_{1}},{{n}_{2}} - 1}}}\left\langle {{{n}_{1}}\left| {a_{i}^{\dag }} \right|{{n}_{2}}} \right\rangle . \\ \end{gathered} $$

(319)

Then from the condition $\left\langle {\left\langle {{{m}_{1}}{{n}_{1}}\left\| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{i}^{\dag }} \right\|{{m}_{2}}{{n}_{2}}} \right\rangle } \right\rangle = \left\langle {\left\langle {{{m}_{2}}{{n}_{2}}\left\| {{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} }}_{i}}} \right\|{{m}_{1}}{{n}_{1}}} \right\rangle } \right\rangle {\text{*}}$, it follows that

$$\beta (m,n) = \alpha {\text{*}}(m,n - 1).$$

(320)

Consider what the last of the conditions listed on page 892 leads to. Since

$$\left\langle {\left\langle {mn\left\| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag }} \right\|I} \right\rangle } \right\rangle = \beta (m,m)\langle m\left| {{{a}_{j}}} \right|n\rangle $$

(321)

and $\left\langle {\left\langle {\left. {mn} \right|{{a}_{j}}} \right\rangle } \right\rangle = \langle m\left| {{{a}_{j}}} \right|n\rangle$, then the condition $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{a} _{j}^{\dag }\left| {\left| {{\kern 1pt} I} \right\rangle } \right\rangle = c\left| {\left| {{{a}_{j}}} \right\rangle } \right\rangle$ means that

$$\beta (m,m) = c.$$

(322)

Comparing (322) and (317), we obtain

$$\beta (m,n) = c{{( - \sigma )}^{{m + n}}}.$$

(323)

Using this equality on the right-hand side of (320), we get

$$\alpha (m,n) = c{\text{*}}{{( - \sigma )}^{{m + n + 1}}},$$

(324)

which is consistant with (320). Substituting the expressions obtained for $\alpha (m,n)$ and $\beta (m,n)$ into (312), we arrive at the condition $cc^{*} = 1$.

APPENDIX B:

1.1 Secular Equation for Charge-Neutral Thermal Phonons

In addition to condition (213), the amplitudes of charge-neutral thermal phonons satisfy the following relations:

• from $[{{Q}_{{JMi}}},{{Q}_{{JMi{\kern 1pt} '}}}] = 0$, it follows that

$$\begin{gathered} \sum\limits_{{{j}_{1}}{{j}_{2}}} \{ (\psi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\phi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \phi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\psi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\psi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \tilde {\phi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\psi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\eta _{{{{j}_{1}}{{j}_{2}}}}^{{\lambda i}}\xi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \xi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\eta _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\eta }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\xi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \tilde {\xi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\eta }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}})\} = 0; \\ \end{gathered} $$

(325)

• from $[{{Q}_{{JMi}}},\widetilde Q_{{JMi{\kern 1pt} '}}^{\dag }] = 0$, it follows that

$$\begin{gathered} \sum\limits_{{{j}_{1}}{{j}_{2}}} \{ (\psi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\psi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji'}} - \phi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\psi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\psi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \tilde {\phi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\phi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\eta _{{{{j}_{1}}{{j}_{2}}}}^{{\lambda i}}\tilde {\eta }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \xi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\xi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\eta }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\eta _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \tilde {\xi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\xi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}})\} = 0; \\ \end{gathered} $$

(326)

• from $[{{Q}_{{JMi}}},{{\widetilde Q}_{{JMi{\kern 1pt} '}}}] = 0$, it follows that

$$\begin{gathered} \sum\limits_{{{j}_{1}}{{j}_{2}}} \{ (\psi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \psi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\psi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\phi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \tilde {\phi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\phi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\eta _{{{{j}_{s}}{{j}_{2}}}}^{{\lambda i}}\tilde {\xi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \xi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\tilde {\eta }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\eta }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\xi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \tilde {\xi }_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\eta _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}})\} = 0. \\ \end{gathered} $$

(327)

With the help of effective amplitudes (220) four orthonormalization conditions (Eqs. (213), (325)–(337)) can be written as two relations: (223) and

$$\begin{gathered} \sum\limits_\tau \sum\limits_{{{j}_{1}}{{j}_{2}}}^\tau {\{ (\Psi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\Phi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \Phi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\Psi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}})} \\ \times \,\,(1 - y_{{{{j}_{1}}}}^{2} - y_{{{{j}_{2}}}}^{2}) + \\ + \,\,(H_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}\Xi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}} - \Xi _{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}H_{{{{j}_{1}}{{j}_{2}}}}^{{Ji{\kern 1pt} '}})(y_{{{{j}_{2}}}}^{2} - y_{{{{j}_{1}}}}^{2})\} = 0. \\ \end{gathered} $$

(328)

The separable form of the residual interaction allows the TQRPA equations (224) and (225) to be reduced to a system of $4N$ linear homogeneous equations. Indeed, for charge-neutral phonons of normal parity, the formal solution to the problem (224) can be represented in the following form

$$\begin{gathered} G_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}(\tau ) = \frac{{{{{\hat {J}}}^{{ - 2}}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}} - \omega _{{Ji}}^{2}}} \\ \times \,\,\left\{ {\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}u_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}\sum\limits_k f_{{{{j}_{1}}{{j}_{2}}}}^{{(J;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(m;k)}}D_{{Ji}}^{{(k)}}(\rho \tau )} \right)} \right. \\ + \left. {\,\,{{\omega }_{{Ji}}}u_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}\sum\limits_k f_{{{{j}_{1}}{{j}_{2}}}}^{{(JJ;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(s;k)}}D_{{JJi}}^{{(k)}}(\rho \tau )} \right)} \right\}, \\ \end{gathered} $$

$$\begin{gathered} W_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}(\tau ) = \frac{{{{{\hat {J}}}^{{ - 2}}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}} - \omega _{{Ji}}^{2}}} \\ \times \,\,\left\{ {{{\omega }_{{Ji}}}u_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}\sum\limits_k f_{{{{j}_{1}}{{j}_{2}}}}^{{(J;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(m;k)}}D_{{Ji}}^{{(k)}}(\rho \tau )} \right)} \right. \\ \left. { + \,\,\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}u_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}\sum\limits_k f_{{{{j}_{1}}{{j}_{2}}}}^{{(JJ;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(s;k)}}D_{{JJi}}^{{(k)}}(\rho \tau )} \right)} \right\}, \\ T_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}(\tau ) = \frac{{{{{\hat {J}}}^{{ - 2}}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}} - \omega _{{Ji}}^{2}}} \\ \times \,\,\left\{ {\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}v_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}\sum\limits_k f_{{{{j}_{1}}{{j}_{2}}}}^{{(J;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(m;k)}}D_{{Ji}}^{{(k)}}(\rho \tau )} \right)} \right. \\ \left. { + \,\,{{\omega }_{{Ji}}}v_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}\sum\limits_k f_{{{{j}_{1}}{{j}_{2}}}}^{{(JJ;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(s;k)}}D_{{JJi}}^{{(k)}}(\rho \tau )} \right)} \right\}, \\ \end{gathered} $$

(329)

$$\begin{gathered} S_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}(\tau ) = \frac{{{{{\hat {J}}}^{{ - 2}}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}} - \omega _{{Ji}}^{2}}} \\ \times \,\,\left\{ {{{\omega }_{{Ji}}}v_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}\sum\limits_k f_{{{{j}_{1}}{{j}_{2}}}}^{{(J;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(m;k)}}D_{{Ji}}^{{(k)}}(\rho \tau )} \right)} \right. \\ \left. { + \,\,\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}v_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}\sum\limits_k f_{{{{j}_{1}}{{j}_{2}}}}^{{(JJ;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(s;k)}}D_{{JJi}}^{{(k)}}(\rho \tau )} \right)} \right\}. \\ \end{gathered} $$

Substitution of obtained expressions in (226) leads to a system of $4N$ homogeneous equations for the functions $D_{{Ji}}^{{(k)}}(\tau )$ and $D_{{JJi}}^{{(k)}}(\tau )$ ($1 \leqslant k \leqslant N$, $\tau = p,n$). In matrix notation, the resulting system has the form:

$$\left( {\begin{array}{*{20}{l}} {{{\mathbb{M}}_{{mm}}} - 1}&{{{\mathbb{M}}_{{ms}}}} \\ {{{\mathbb{M}}_{{sm}}}}&{{{\mathbb{M}}_{{ss}}} - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\mathbb{D}}_{J}}} \\ {{{\mathbb{D}}_{{JJ}}}} \end{array}} \right) = 0.$$

(330)

The vectors ${{\mathbb{D}}_{J}}$ and ${{\mathbb{D}}_{{JJ}}}$ of dimension $2N$ consist of the elements

$$\begin{gathered} \mathbb{D}_{J}^{{(k)}} = \left( {\begin{array}{*{20}{c}} {D_{J}^{{(k)}}(p)} \\ {D_{J}^{{(k)}}(n)} \end{array}} \right), \\ \mathbb{D}_{{JJ}}^{{(k)}} = \left( {\begin{array}{*{20}{c}} {D_{{JJ}}^{{(k)}}(p)} \\ {D_{{JJ}}^{{(k)}}(n)} \end{array}} \right) (1 \leqslant k \leqslant N), \\ \end{gathered} $$

while the matrices ${{\mathbb{M}}_{{\alpha \beta }}}$ $(\alpha ,\beta = m,s)$ are the $2N \times 2N$ matrices, composed of $2 \times 2$ units

$$\begin{gathered} \mathbb{M}_{{\alpha \beta }}^{{kk{\kern 1pt} '}} = \left( {\begin{array}{*{20}{c}} {\chi _{{ + 1}}^{{(\beta ;k{\kern 1pt} ')}}\mathcal{X}_{{\alpha \beta ;p}}^{{(J;kk{\kern 1pt} ')}}(\omega )}&{\chi _{{ - 1}}^{{(\beta ;k{\kern 1pt} ')}}\mathcal{X}_{{\alpha \beta ;p}}^{{(J;kk{\kern 1pt} ')}}(\omega )} \\ {\chi _{{ - 1}}^{{(\beta ;k{\kern 1pt} ')}}\mathcal{X}_{{\alpha \beta ;n}}^{{(J;kk{\kern 1pt} ')}}(\omega )}&{\chi _{{ + 1}}^{{(\beta ;k{\kern 1pt} ')}}\mathcal{X}_{{\alpha \beta ;n}}^{{(J;kk{\kern 1pt} ')}}(\omega )} \end{array}} \right) \\ (1 \leqslant k,k{\kern 1pt} ' \leqslant N). \\ \end{gathered} $$

Here the following notation for functions of $\omega$ are introduced:

$$\begin{gathered} \mathcal{X}_{{mm;\tau }}^{{(J;kk{\kern 1pt} ')}}(\omega ) = {{{\hat {J}}}^{{ - 2}}}{{\sum\limits_{{{j}_{1}}{{j}_{2}}} }^{\tau }}f_{{{{j}_{1}}{{j}_{2}}}}^{{(J;k)}}f_{{{{j}_{1}}{{j}_{2}}}}^{{(J;k{\kern 1pt} ')}} \\ \times \,\,\left\{ {\frac{{\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}{{{(u_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}} - {{\omega }^{2}}}}} \right.(1 - y_{{{{j}_{1}}}}^{2} - y_{{{{j}_{2}}}}^{2}) \\ + \,\,\left. {\frac{{\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}{{{(v_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}} - {{\omega }^{2}}}}(y_{{{{j}_{2}}}}^{2} - y_{{{{j}_{1}}}}^{2})} \right\}, \\ \end{gathered} $$

$$\begin{gathered} \mathcal{X}_{{ss;\tau }}^{{(J;kk{\kern 1pt} ')}}(\omega ) = {{{\hat {J}}}^{{ - 2}}}{{\sum\limits_{{{j}_{1}}{{j}_{2}}} }^{\tau }}f_{{{{j}_{1}}{{j}_{2}}}}^{{(JJ;k)}}f_{{{{j}_{1}}{{j}_{2}}}}^{{(JJ;k{\kern 1pt} ')}} \\ \times \,\,\left\{ {\frac{{\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}{{{(u_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}} - {{\omega }^{2}}}}(1 - y_{{{{j}_{1}}}}^{2} - y_{{{{j}_{2}}}}^{2})} \right. \\ \left. { + \,\,\frac{{\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}{{{(v_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}} - {{\omega }^{2}}}}(y_{{{{j}_{2}}}}^{2} - y_{{{{j}_{1}}}}^{2})} \right\}, \\ \end{gathered} $$

(331)

$$\begin{gathered} \mathcal{X}_{{ms;\tau }}^{{(J;kk{\kern 1pt} ')}}(\omega ) = {{{\hat {J}}}^{{ - 2}}}\omega {{\sum\limits_{{{j}_{1}}{{j}_{2}}} }^{\tau }}f_{{{{j}_{1}}{{j}_{2}}}}^{{(J;k)}}f_{{{{j}_{1}}{{j}_{2}}}}^{{(JJ;k{\kern 1pt} ')}} \\ \times \,\,\left\{ {\frac{{u_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}u_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}} - {{\omega }^{2}}}}} \right.(1 - y_{{{{j}_{1}}}}^{2} - y_{{{{j}_{2}}}}^{2}) \\ + \,\,\left. {\frac{{v_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}v_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}} - {{\omega }^{2}}}}(y_{{{{j}_{2}}}}^{2} - y_{{{{j}_{1}}}}^{2})} \right\} \\ \end{gathered} $$

and $\mathcal{X}_{{sm;\tau }}^{{(J;kk{\kern 1pt} ')}}(\omega ) = \mathcal{X}_{{ms;\tau }}^{{(j;k{\kern 1pt} 'k)}}(\omega )$. The solvability condition for the system of homogeneous equations (330) leads to a secular equation for finding the energy ω_Ji of thermal phonons

$$\det \left( {\begin{array}{*{20}{l}} {{{\mathbb{M}}_{{mm}}} - 1}&{{{\mathbb{M}}_{{ms}}}} \\ {{{\mathbb{M}}_{{sm}}}}&{{{\mathbb{M}}_{{ss}}} - 1} \end{array}} \right) = 0.$$

(332)

For thermal phonons of anomalous parity, the formal solution of the system of TQRPA equations (225) has the form

$$\begin{gathered} W_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}(\tau ) = \frac{{{{{\hat {J}}}^{{ - 2}}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}} - \omega _{{Ji}}^{2}}}\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}u_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}} \\ \times \,\,\sum\limits_k \sum\limits_{L = J \pm 1} f_{{{{j}_{1}}{{j}_{2}}}}^{{(LJ;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(s;k)}}D_{{LJi}}^{{(k)}}(\rho \tau )} \right), \\ S_{{{{j}_{1}}{{j}_{2}}}}^{{Ji}}(\tau ) = \frac{{{{{\hat {J}}}^{{ - 2}}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}} - \omega _{{Ji}}^{2}}}\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}v_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}} \\ \times \,\,\sum\limits_k \sum\limits_{L = J \pm 1} f_{{{{j}_{1}}{{j}_{2}}}}^{{(LJ;k)}}\left( {\sum\limits_{\rho = \pm 1} \chi _{\rho }^{{(s;k)}}D_{{LJi}}^{{(k)}}(\rho \tau )} \right). \\ \end{gathered} $$

(333)

As in the case of the normal parity phonons, a substitution of the obtained expressions in (226) leads to a system of $4N$ homogeneous equations for the functions $D_{{J \pm 1J}}^{{(k)}}(\tau )$ ($1 \leqslant k \leqslant N$, $\tau = p,{\kern 1pt} {\kern 1pt} n$):

$$\left( {\begin{array}{*{20}{l}} {{{\mathbb{M}}_{{J - 1J - 1}}} - 1}&{{{\mathbb{M}}_{{J - 1J + 1}}}} \\ {{{\mathbb{M}}_{{J + 1J - 1}}}}&{{{\mathbb{M}}_{{J + 1J + 1}}} - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\mathbb{D}}_{{J - 1J}}}} \\ {{{\mathbb{D}}_{{J + 1J}}}} \end{array}} \right) = 0.$$

(334)

Here ${{\mathbb{M}}_{{LL{\kern 1pt} '}}}$ are the $2N \times 2N$ matrices, composed of $2 \times 2$ units

$$\begin{gathered} \mathbb{M}_{{LL{\kern 1pt} '}}^{{(kk{\kern 1pt} ')}} = \left( {\begin{array}{*{20}{c}} {\chi _{{ + 1}}^{{(s;k{\kern 1pt} ')}}\mathcal{X}_{{LL{\kern 1pt} ';p}}^{{(J;kk{\kern 1pt} ')}}(\omega )}&{\chi _{{ - 1}}^{{(s;k)}}\mathcal{X}_{{LL{\kern 1pt} ';p}}^{{(J;kk{\kern 1pt} ')}}(\omega )} \\ {\chi _{{ - 1}}^{{(s;k{\kern 1pt} ')}}\mathcal{X}_{{LL';n}}^{{(J;kk{\kern 1pt} ')}}(\omega )}&{\chi _{{ + 1}}^{{(s;k)}}\mathcal{X}_{{LL{\kern 1pt} ';n}}^{{(J;kk{\kern 1pt} ')}}(\omega )} \end{array}} \right) \\ (1 \leqslant k,k{\kern 1pt} ' \leqslant N), \\ \end{gathered} $$

with the matrix elements

$$\begin{gathered} \mathcal{X}_{{LL{\kern 1pt} ';\tau }}^{{(J;kk{\kern 1pt} ')}}(\omega ) = {{{\hat {J}}}^{{ - 2}}}{{\sum\limits_{{{j}_{1}}{{j}_{2}}} }^{\tau }}f_{{{{j}_{1}}{{j}_{2}}}}^{{(LJ;k)}}f_{{{{j}_{1}}{{j}_{2}};}}^{{(L{\kern 1pt} 'J;k{\kern 1pt} ')}} \\ \times \,\,\left\{ {\frac{{\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}}{{{(u_{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}} - {{\omega }^{2}}}}} \right.(1 - y_{{{{j}_{1}}}}^{2} - y_{{{{j}_{2}}}}^{2}) \\ + \,\,\left. {\frac{{\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}}{{{(v_{{{{j}_{1}}{{j}_{2}}}}^{{( + )}})}}^{2}}}}{{{{{(\varepsilon _{{{{j}_{1}}{{j}_{2}}}}^{{( - )}})}}^{2}} - {{\omega }^{2}}}}(y_{{{{j}_{2}}}}^{2} - y_{{{{j}_{1}}}}^{2})} \right\}, \\ \end{gathered} $$

while the vector ${{\mathbb{D}}_{{LJ}}}$ of $2N$ dimension has the following components:

$$\mathbb{D}_{{LJ}}^{{(k)}} = \left( {\begin{array}{*{20}{c}} {D_{{LJ}}^{{(k)}}(p)} \\ {D_{{LJ}}^{{(k)}}(n)} \end{array}} \right),\,\,\,\,1 \leqslant k \leqslant N.$$

The thermal phonon energy is found from the condition for the existence of a nontrivial solution for the system (334), therefore, it is a solution to the secular equation

$${\text{det}}\left( {\begin{array}{*{20}{c}} {{{\mathbb{M}}_{{J - 1,J - 1}}} - 1}&{{{\mathbb{M}}_{{J - 1,J + 1}}}} \\ {{{\mathbb{M}}_{{J + 1,J - 1}}}}&{{{\mathbb{M}}_{{J + 1,J + 1}}} - 1} \end{array}} \right) = 0.$$

(335)

APPENDIX C:

1.1 Secular Equation for Charge-Exchange Thermal Phonons

The requirement to preserve the bosonic commutation relations for charge-exchange thermal phonons leads to four orthonormalization conditions for the amplitudes:

• from $[{{\Omega }_{{JMi}}},\Omega _{{J{\kern 1pt} 'M{\kern 1pt} 'i{\kern 1pt} '}}^{\dag }] = {{\delta }_{{JJ{\kern 1pt} '}}}{{\delta }_{{MM{\kern 1pt} '}}}{{\delta }_{{ii{\kern 1pt} '}}}$, it follows that

$$\begin{gathered} \sum\limits_{{{j}_{p}}{{j}_{n}}} \{ (\psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\psi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\psi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\eta _{{{{j}_{p}}{{j}_{n}}}}^{{\lambda i}}\eta _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\eta }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\eta }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \tilde {\xi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\xi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}})\} = {{\delta }_{{ii{\kern 1pt} '}}}; \\ \end{gathered} $$

(336)

• from $[{{\Omega }_{{JMi}}},{{\Omega }_{{J{\kern 1pt} 'M{\kern 1pt} 'i{\kern 1pt} '}}}] = 0$, it follows that

$$\begin{gathered} \sum\limits_{{{j}_{p}}{{j}_{n}}} \{ (\psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\psi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\psi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\eta _{{{{j}_{p}}{{j}_{n}}}}^{{\lambda i}}\xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\eta _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\eta }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\xi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \tilde {\xi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\eta }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}})\} = 0; \\ \end{gathered} $$

(337)

• from $[{{\Omega }_{{JMi}}},\widetilde \Omega _{{J{\kern 1pt} 'M{\kern 1pt} 'i{\kern 1pt} '}}^{\dag }] = 0$, it follows that

$$\begin{gathered} \sum\limits_{{{j}_{p}}{{j}_{n}}} \{ (\psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\psi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\psi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\eta _{{{{j}_{p}}{{j}_{n}}}}^{{\lambda i}}\widetilde \eta _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\widetilde \xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\eta }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\eta _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \tilde {\xi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}})\} = 0; \\ \end{gathered} $$

(338)

• from $[{{\Omega }_{{JMi}}},{{\widetilde \Omega }_{{J{\kern 1pt} 'M{\kern 1pt} 'i{\kern 1pt} '}}}] = 0$, it follows that

$$\begin{gathered} \sum\limits_{{{j}_{p}}{{j}_{n}}} \{ (\psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\tilde {\psi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \tilde {\phi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ + \,\,(\eta _{{{{j}_{p}}{{j}_{n}}}}^{{\lambda i}}\tilde {\xi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\tilde {\eta }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}}) \\ \,\, + (\tilde {\eta }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \tilde {\xi }_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\eta _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}})\} = 0. \\ \end{gathered} $$

(339)

For effective amplitudes, the above conditions take the form of two relations: (265) and

$$\begin{gathered} \sum\limits_{{{j}_{p}}{{j}_{n}}} \{ (\Psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\Phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \Phi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\Psi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}})(1 - y_{{{{j}_{1}}}}^{2} - y_{{{{j}_{2}}}}^{2}) \hfill \\ + \,\,(H_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}\Xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}} - \Xi _{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}H_{{{{j}_{p}}{{j}_{n}}}}^{{Ji{\kern 1pt} '}})(y_{{{{j}_{n}}}}^{2} - y_{{{{j}_{p}}}}^{2})\} = 0. \hfill \\ \end{gathered} $$

(340)

In proving the fulfillment of the Ikeda sum rule in TQRPA approximation (277), we used the completeness properties of effective amplitudes

$$\begin{gathered} \sum\limits_i G_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}W_{{{{j}_{{p{\kern 1pt} '}}}{{j}_{{n{\kern 1pt} '}}}}}^{{Ji}} = \frac{{{{\delta }_{{{{j}_{p}}{{j}_{{p{\kern 1pt} '}}}}}}{{\delta }_{{{{j}_{n}}{{j}_{{n{\kern 1pt} '}}}}}}}}{{1 - y_{{{{j}_{p}}}}^{2} - y_{n}^{2}}}, \\ \sum\limits_i T_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}S_{{{{j}_{{p{\kern 1pt} '}}}{{j}_{{n{\kern 1pt} '}}}}}^{{Ji}} = \frac{{{{\delta }_{{{{j}_{p}}{{j}_{{p{\kern 1pt} '}}}}}}{{\delta }_{{{{j}_{n}}{{j}_{{n'}}}}}}}}{{y_{{{{j}_{n}}}}^{2} - y_{p}^{2}}}, \\ \sum\limits_i G_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}S_{{{{j}_{{p{\kern 1pt} '}}}{{j}_{{n{\kern 1pt} '}}}}}^{{Ji}} = \sum\limits_i T_{{{{j}_{p}}{{j}_{n}}}}^{{Ji}}W_{{{{j}_{{p{\kern 1pt} '}}}{{j}_{{n{\kern 1pt} '}}}}}^{{Ji}} = 0. \\ \end{gathered} $$

(341)

These properties can be easily obtained by writing down the completeness conditions for phonon amplitudes and then expressing $\phi , \psi$, etc. through effective amplitudes.

As in the case of charge-neutral phonons, the separable form of the residual interaction makes it possible to reduce the system of TQRPA equations (266) to $4N$ homogeneous equations

$$\left( {\begin{array}{*{20}{c}} {{{\mathbb{M}}_{ + }} - 1}&{{{\mathbb{M}}_{{ + - }}}} \\ {{{\mathbb{M}}_{{ + - }}}}&{{{\mathbb{M}}_{ - }} - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\mathbb{D}}_{ + }}} \\ {{{\mathbb{D}}_{ - }}} \end{array}} \right) = 0.$$

(342)

Here ${{\mathbb{M}}_{\sigma }}$ ($\sigma = + , - , + -$) are the $2N \times 2N$ matrices, composed of $2 \times 2$ units

$$\begin{gathered} \mathbb{M}_{\sigma }^{{kk{\kern 1pt} '}} = \left( {\begin{array}{*{20}{c}} {\chi _{1}^{{(a;k{\kern 1pt} ')}}\mathcal{X}_{\sigma }^{{(aa;kk{\kern 1pt} ')}}(\omega )}&{\chi _{1}^{{(b;k{\kern 1pt} ')}}\mathcal{X}_{\sigma }^{{(ab;kk{\kern 1pt} ')}}(\omega )} \\ {\chi _{1}^{{(a;k{\kern 1pt} ')}}\mathcal{X}_{\sigma }^{{(ba;kk{\kern 1pt} ')}}(\omega )}&{\chi _{1}^{{(b;k{\kern 1pt} ')}}\mathcal{X}_{\sigma }^{{(bb;kk{\kern 1pt} ')}}(\omega )} \end{array}} \right), \\ 1 \leqslant k,\,\,\,k{\kern 1pt} ' \leqslant N. \\ \end{gathered} $$

(343)

For the normal parity phonons, the indices $a,{\kern 1pt} b$ assume the values $a = J$, $b = JJ$, while for the anomalous parity phonons $a = (J - 1)J$, $b = (J + 1)J$. In addition, $\chi _{1}^{{(J;k)}} = \chi _{1}^{{(m;k)}}$ and $\chi _{1}^{{(LJ;k)}} = \chi _{1}^{{(s;k)}}$. The functions $\mathcal{X}_{\sigma }^{{(cd;kk{\kern 1pt} ')}}(\omega )$ ($c = a,b$, $d = a,b$) are defined as

$$\begin{gathered} \mathcal{X}_{ \pm }^{{(cd;kk{\kern 1pt} ')}}(\omega ) = 2{{{\hat {J}}}^{{ - 2}}}\sum\limits_{{{j}_{p}}{{j}_{n}}} f_{{{{j}_{p}}{{j}_{n}}}}^{{(c;k)}}f_{{{{j}_{p}}{{j}_{n}}}}^{{(d;k{\kern 1pt} ')}} \\ \times \,\,\left\{ {\frac{{\varepsilon _{{{{j}_{p}}{{j}_{n}}}}^{{( + )}}{{{(u_{{{{j}_{p}}{{j}_{n}}}}^{{( \pm )}})}}^{2}}}}{{{{{(\varepsilon _{{{{j}_{p}}{{j}_{n}}}}^{{( + )}})}}^{2}} - {{\omega }^{2}}}}} \right.(1 - y_{{{{j}_{p}}}}^{2} - y_{{{{j}_{n}}}}^{2}) \\ + \,\left. {\,\frac{{\varepsilon _{{{{j}_{p}}{{j}_{n}}}}^{{( - )}}{{{(v_{{{{j}_{p}}{{j}_{n}}}}^{{( \mp )}})}}^{2}}}}{{{{{(\varepsilon _{{{{j}_{p}}{{j}_{n}}}}^{{( - )}})}}^{2}} - {{\omega }^{2}}}}(y_{{{{j}_{n}}}}^{2} - y_{{{{j}_{p}}}}^{2})} \right\}, \\ \mathcal{X}_{{ + - }}^{{(cd;kk{\kern 1pt} ')}}(\omega ) = 2\omega {{{\hat {J}}}^{{ - 2}}}\sum\limits_{{{j}_{p}}{{j}_{n}}} f_{{{{j}_{p}}{{j}_{n}}}}^{{(c;k)}}f_{{{{j}_{p}}{{j}_{n}}}}^{{(d;k{\kern 1pt} ')}} \\ \times \,\,\left\{ {\frac{{u_{{{{j}_{p}}{{j}_{n}}}}^{{( + )}}u_{{{{j}_{p}}{{j}_{n}}}}^{{( - )}}}}{{{{{(\varepsilon _{{{{j}_{p}}{{j}_{n}}}}^{{( + )}})}}^{2}} - {{\omega }^{2}}}}} \right.(1 - y_{{{{j}_{p}}}}^{2} - y_{{{{j}_{n}}}}^{2}) \\ + \,\,\left. {\frac{{v_{{{{j}_{p}}{{j}_{n}}}}^{{( + )}}v_{{{{j}_{p}}{{j}_{n}}}}^{{( - )}}}}{{{{{(\varepsilon _{{{{j}_{p}}{{j}_{n}}}}^{{( - )}})}}^{2}} - {{\omega }^{2}}}}(y_{{{{j}_{p}}}}^{2} - y_{{{{j}_{n}}}}^{2})} \right\}. \\ \end{gathered} $$

(344)

The vectors ${{\mathbb{D}}^{{( \pm )}}}$ of the $2N$ dimension consist of the functions $D_{J}^{{( \pm ;k)}}$, $D_{{LJ}}^{{( \pm ;k)}}$ (267)

$$\mathbb{D}_{k}^{{( \pm )}} = \left( {\begin{array}{*{20}{c}} {D_{a}^{{( \pm ;k)}}} \\ {D_{b}^{{( \pm ;k)}}} \end{array}} \right), \,\,\,\,1 \leqslant k \leqslant N.$$

The condition for the existence of a nontrivial solution for system (342) leads to a secular equation for finding the energy of charge-exchange thermal phonons

$${\text{det}}\left( {\begin{array}{*{20}{c}} {{{\mathbb{M}}_{ + }} - 1}&{{{\mathbb{M}}_{{ + - }}}} \\ {{{\mathbb{M}}_{{ + - }}}}&{{{\mathbb{M}}_{ - }} - 1} \end{array}} \right) = 0.$$

(345)

For the QPM Hamiltonian, the secular equation for determining the energy of charge-exchange phonons in hot nuclei is given in our paper [134].