Dynamical Correlations in the Ground State: Transitions between One-Phonon Nuclear States

The probabilities of the E1 transition between the first ${{2}^{ + }}$ and ${{3}^{ - }}$ excited levels in nuclei with pairing have been calculated within the self-consistent many-body nuclear theory and Green’s function method. Calculations for a long chain of even–even tin isotopes have been performed for the first time. The known Fayans energy density functional has been used to calculate the characteristics of phonons and E1 transitions between excited states. A good description has been achieved for existing experimental data for the reduced probabilities of E1 transitions between the first one-phonon states for the ^116–124Sn isotopes but not for the ¹¹²Sn and ¹¹⁴Sn isotopes. Possible reasons for this discrepancy have been discussed; the most probable reason is the deformation in the ground or excited states. It has been shown that new dynamical three-quasiparticle correlations in the ground state should be taken into account to explain the experimental data for ^116–124Sn.

1 INTRODUCTION

Correlations in the ground state (GSCs) or “backward diagrams” have been discussed for a long time in works on nuclear theory. Correlations in the ground state are well known in the conventional random phase approximation (RPA) and quasiparticle RPA (QRPA). Correlations in the ground state in the RPA and QRPA are included in the kernel of the corresponding integral equation whose solution provides multiparticle GSCs owing to the two-particle interaction between nucleons of the nucleus. They are automatically taken into account in the solution of the integral equation. Their quantitative contribution is usually small [1, 2] in spherical and deformed nuclei.

This problem becomes more relevant in the context of GSCs including nuclear phonons (phonon-GSCs), i.e., in the presence of the quasiparticle–phonon interaction [3–8].¹ Here, phonon-GSCs appear in the solution of the corresponding integral equations including phonons and new complex additional terms compared to RPA or QRPA equations. The numerical contribution from these GSCs is noticeable and requires a special analysis [3, 5, 6].

So-called three- [6] and four-quasiparticle [9] correlations in the ground state (GSC3s and GSC4s, respectively), which appear after the integration of three and four single-particle Green’s functions and do not contain phonons, were quite recently considered self-consistently (GSC3s for magic nuclei without self-consistency were considered in [10]). Unlike GSCs in the RPA and phonon-GSCs, they do not originate from the kernel of the integral equation and they depend on the transition energy $\omega$. These “pure” (without phonons and beyond the integral equation) GSC3s appear in the problem of static ($\omega$ = 0) quadrupole moments in the first one-phonon ${{2}^{ + }}$ [6] and ${{3}^{ - }}$ states [11] of tin isotopes. It was shown that the observed effect is the sum of two approximately identical contributions from pure GSC3s and the nuclear polarizability.

Similar results were obtained for physically close E1 transitions between the first one-phonon ${{2}^{ + }}$ and ${{3}^{ - }}$ states in ²⁰⁸Pb and ¹³²Sn magic nuclei [12] and some tin isotopes [13], but the observed effect was determined by the difference between the aforementioned two effects. We refer to such GSCs as dynamic GSC3s (the transferred energy $\omega \ne 0$ in contrast to the static case $\omega = 0$). We emphasize the specificity and importance of this class of problems: the result is proportional to ${{g}^{2}}$, where g is the phonon production amplitude. A reasonable agreement with existing experimental data was obtained in all mentioned cases.

To the best of our knowledge, the $(3_{1}^{ - } \to 2_{1}^{ + })$ E1 transition in the ¹²⁰Sn nucleus [14] and in the ^120,122,124Sn nuclei in [15] was considered only within the quasiparticle phonon model and a good agreement with experiment was obtained. The authors of [14, 15] included the nuclear polarizability by the introduction of coupling to the giant dipole resonance. This technique halved the B(E1) value. In this case, the characteristics of QRPA phonons were fitted to experimental data by varying the parameters of the separable interaction. As will be shown, dynamical GSC3s were disregarded in [14, 15].

In this work, E1 transitions between the first one-phonon ${{2}^{ + }}$ and ${{3}^{ - }}$ states in a long chain of tin isotopes including dynamical GSC3s are considered for the first time. Unlike [14, 15], we use a common self-consistent approach for both phonons and probabilities of the E1 transition between one-phonon states. We cannot explain existing experimental data for the ¹¹²Sn and ¹¹⁴Sn neutron-deficient isotopes.

2 CORRELATIONS IN THE GROUND STATE IN THE GREEN’S FUNCTION METHOD

Correlations in the ground state are likely most clearly and completely manifested within the Green’s function method and, correspondingly, the Feynman diagram technique, where two terms initially exist. In the single-particle and one-phonon approximation, the single-particle and one-phonon Green’s functions $G$, $F$, and $D$ have the form

$${{G}_{\lambda }}(\varepsilon ) = \frac{{u_{\lambda }^{2}}}{{\varepsilon - {{E}_{\lambda }} + i\gamma }} + \frac{{v_{\lambda }^{2}}}{{\varepsilon + {{E}_{\lambda }} - i\gamma }} = {{G}^{ + }} + {{G}^{ - }},$$

(1)

$$\begin{gathered} F_{1}^{{(1,2)}}(\varepsilon ) \\ = - \frac{{{{\Delta }_{1}}}}{{2{{E}_{1}}}}\left[ {\frac{1}{{\varepsilon - {{E}_{1}} + i\delta }} + \frac{1}{{\varepsilon + {{E}_{1}} - i\delta }}} \right] = {{F}^{ + }} + {{F}^{ - }}, \\ \end{gathered} $$

(2)

$${{D}_{s}}(\omega ) = \frac{1}{{\omega - {{\omega }_{s}} + i\gamma }} - \frac{1}{{\omega + {{\omega }_{s}} - i\gamma }} = {{D}^{ + }} + {{D}^{ - }}.$$

(3)

Here,

$$\begin{gathered} {{E}_{\lambda }} = \sqrt {{{{({{\varepsilon }_{\lambda }} - \mu )}}^{2}} + \Delta _{\lambda }^{2}} , \\ u_{\lambda }^{2} = 1 - v_{\lambda }^{2} = ({{E}_{\lambda }} + {{\varepsilon }_{\lambda }} - \mu ){\text{/}}2{{E}_{\lambda }}, \\ \end{gathered} $$

(4)

where μ is the chemical potential; $\lambda \equiv ({{n}_{1}},{{l}_{1}},{{j}_{1}},{{m}_{1}})$ is a set of quantum numbers; G^– in Eq. (1) at Δ = 0 describes the hole part of the Green’s function ${{G}^{h}}$ and corresponds to backward diagrams in the time representation; and the second terms in Eqs. (1)–(3) ensure the appearance of GSCs.

In this context, three types of GSCs mentioned in the Introduction can be illustrated by Feynman diagrams for magic nuclei shown in Fig. 1 and corresponding to the propagators

$${{A}_{{12}}}(\omega ) = \int (G_{1}^{ + }G_{2}^{ - } + G_{1}^{ - }G_{2}^{ + })\frac{{d\varepsilon }}{{2\pi i}},$$

$${{A}_{{12s34}}}(\omega ) = \int {{G}_{1}}{{G}_{2}}{{D}_{s}}(\omega ){{G}_{3}}{{G}_{4}}\frac{{d\varepsilon d{{\omega }_{1}}}}{{{{{(2\pi i)}}^{2}}}},$$

(5)

$${{A}_{{123}}}(\omega ) = \int ({{G}_{1}}{{G}_{2}}{{G}_{3}})\frac{{d\varepsilon }}{{2\pi i}}.$$

Fig. 1.

(Color online) Illustration of correlations in the ground state (GSC) mentioned in the Introduction. Integrands for three propagators ${{A}_{{12}}}$, ${{A}_{{12s34}}}$ [7], and ${{A}_{{123}}}$ [11] given by Eqs. (5) are symbolically shown to the left of the double arrow. The double arrow indicates the result of the substitution of Eq. (1) ($G = {{G}^{ + }} + {{G}^{ - }}$) into Eqs. (5) for magic nuclei, in particular, backward diagrams, i.e., correlations in the ground state. The first line includes correlations in the ground state in the random phase approximation, the second and third lines include phonon-GSCs in the ${{g}^{2}}$ approximation (only diagrams with inserts including phonon-GSCs are shown), and the fourth line includes four conventional terms and eight terms with GSC3s (for details see [11]).

The indicated GSCs in the RPA and phonon-GSCs appear in the corresponding integral equations whose kernels include not only propagators but usually also two-particle interaction between nucleons. For this reason, the iteration of the RPA equation results in the appearance of multiparticle GSCs. As known, if the second term in Fig. 1 for the RPA is omitted in the RPA or QRPA equation, the Tamm–Dancoff method free of GSCs is obtained. The iteration of the integral equation including phonons results in the appearance of phonon-GSCs. As mentioned in the Introduction, GSC3s are not included in the integral equation and do not contain phonons. The absence of the small ${{g}^{2}}$ parameter ensures a large quantitative contribution from our dynamical GSC3s; see the next section and Fig. 2.

Fig. 2.

(Color online) Reduced probabilities $B(E1)(3_{1}^{ - } \to 2_{1}^{ + })$ of the transition between one-phonon states according to the calculations with and without GSC3s and experimental data from [18, 19].

3 CALCULATION FORMULAS. COMPARISON WITH OTHER APPROACHES

A detailed derivation of an expression for the transition amplitude ${{M}_{{ss'}}}$ between one-phonon states s and s' in nuclei with pairing is described in [13] (phonons are described in the RPA or QRPA). The amplitude ${{M}_{{ss'}}}$ contains eight terms shown in the diagrammatic representation in [13, Fig. 2]. After the separation of angular variables and summation over magnetic quantum numbers, we obtain the reduced probability of the ${{I}_{s}} \to {{I}_{{s'}}}$ transition with the energy $\omega = {{\omega }_{{s'}}} - {{\omega }_{s}}$ in the form

$$B({{E}_{L}}) = \frac{1}{{2{{I}_{s}} + 1}}{\text{|}}\langle {{I}_{s}}{\kern 1pt} ||{\kern 1pt} {{M}_{L}}{\kern 1pt} ||{\kern 1pt} {{I}_{{s'}}}\rangle {{{\text{|}}}^{2}},$$

(6)

where the reduced matrix element is given by the expression

$$\begin{gathered} \langle {{I}_{s}}{\kern 1pt} ||{\kern 1pt} {{M}_{L}}{\kern 1pt} ||{\kern 1pt} {{I}_{{s'}}}\rangle = \sum\limits_{123} \left\{ {\begin{array}{*{20}{c}} {{{I}_{s}}}&{{{I}_{{s'}}}}&L \\ {{{j}_{2}}}&{{{j}_{1}}}&{{{j}_{3}}} \end{array}} \right\}{{V}_{{12}}}g_{{31}}^{s}g_{{23}}^{{s'}} \\ \times \;\left[ {A_{{123}}^{{(12)}} + A_{{123}}^{{(34)}} + {{{( - 1)}}^{{L + {{I}_{s}}}}}A_{{123}}^{{(76)}} + {{{( - 1)}}^{{L + {{I}_{{s'}}}}}}A_{{123}}^{{(58)}}} \right]. \\ \end{gathered} $$

(7)

Here, the reduced matrix elements of the vertex and phonons are denoted as $\langle 1{\kern 1pt} ||{\kern 1pt} V{\kern 1pt} ||{\kern 1pt} 2\rangle = {{V}_{{12}}}$, $\langle 3{\kern 1pt} ||{\kern 1pt} {{g}^{s}}{\kern 1pt} ||{\kern 1pt} 1\rangle = g_{{31}}^{s}$, and $\langle 2{\kern 1pt} ||{\kern 1pt} {{g}^{{s'}}}{\kern 1pt} ||{\kern 1pt} 3\rangle = g_{{23}}^{{s'}}$, and ${{A}^{{(ik)}}} = {{A}^{{(i)}}} + {{A}^{{(k)}}}$ are the sums of two propagators, which are integrals of three Green’s functions corresponding to one of eight diagrams in [13, Fig. 2]. The sum of the propagators in Eq. (7) can be represented in the form

$$\left[ {A_{{123}}^{{(12)}} + A_{{123}}^{{(34)}} + {{{( - 1)}}^{{L + {{I}_{s}}}}}A_{{123}}^{{(76)}} + {{{( - 1)}}^{{L + {{I}_{{s'}}}}}}A_{{123}}^{{(58)}}} \right]$$

$$ = \left[ {\frac{1}{{({{E}_{{13}}} - {{\omega }_{s}})({{E}_{{23}}} - {{\omega }_{{s'}}})}} + \frac{1}{{({{E}_{{13}}} + {{\omega }_{s}})({{E}_{{23}}} + {{\omega }_{{s'}}})}}} \right]$$

$$ \times \;(u_{1}^{2}u_{2}^{2}v_{3}^{2} - v_{1}^{2}v_{2}^{2}u_{3}^{2} + {{C}_{{12}}}(u_{3}^{2} - v_{3}^{2}) + {{( - 1)}^{{L + {{I}_{s}}}}}{{C}_{{13}}}$$

$$(u_{2}^{2} - v_{2}^{2}) + {{( - 1)}^{{L + {{I}_{{s'}}}}}}{{C}_{{23}}}(u_{1}^{2} - v_{1}^{2})) + \frac{1}{{(E_{{12}}^{2} - {{\omega }^{2}})}}$$

$$\begin{gathered} \times \;\left[ {\frac{{2({{E}_{{32}}}{{E}_{{12}}} + \omega {{\omega }_{{s'}}})}}{{(E_{{32}}^{2} - \omega _{{s'}}^{2})}}} \right.(u_{1}^{2}v_{2}^{2}u_{3}^{2} - v_{1}^{2}u_{2}^{2}v_{3}^{2} + {{C}_{{12}}}(u_{3}^{2} - v_{3}^{2})) \\ + \;\frac{{2({{E}_{{32}}}{{E}_{{12}}} - \omega {{\omega }_{{s'}}})}}{{(E_{{32}}^{2} - \omega _{{s'}}^{2})}}(( - {{1)}^{{L + {{I}_{s}}}}}{{C}_{{13}}}(u_{2}^{2} - v_{2}^{2}) \\ \end{gathered} $$

(8)

$$ + \;{{( - 1)}^{{L + {{I}_{{s'}}}}}}{{C}_{{23}}}(u_{1}^{2} - v_{1}^{2}))$$

$$ + \;\frac{{2({{E}_{{31}}}{{E}_{{21}}} - \omega {{\omega }_{s}})}}{{(E_{{31}}^{2} - \omega _{s}^{2})}}(v_{1}^{2}u_{2}^{2}u_{3}^{2} - u_{1}^{2}v_{2}^{2}v_{3}^{2} + {{C}_{{12}}}(u_{3}^{2} - v_{3}^{2}))$$

$$ + \;\frac{{2({{E}_{{31}}}{{E}_{{21}}} + \omega {{\omega }_{s}})}}{{(E_{{31}}^{2} - \omega _{s}^{2})}}(( - {{1)}^{{L + {{I}_{s}}}}}{{C}_{{13}}}(u_{2}^{2} - v_{2}^{2})$$

$$ + \;{{( - 1)}^{{L + {{I}_{{s'}}}}}}{{C}_{{23}}}(u_{1}^{2} - v_{1}^{2}))].$$

The first part of the right-hand side of Eq. (8) (before the term with the factor ${{(E_{{21}}^{2} - {{\omega }^{2}})}^{{ - 1}}}$) corresponds (except for the phase) to the result obtained within the quasiparticle–phonon model [14, Eq. (9)] for transitions between the first one-phonon states. Indeed, it is easily seen that, in terms of the quasiparticle–phonon model,

$$\begin{gathered} v_{{12}}^{ - }u_{{23}}^{ + }u_{{31}}^{ + } = u_{1}^{2}u_{2}^{2}v_{3}^{2} - v_{1}^{2}v_{2}^{2}u_{3}^{2} \\ + \;{{C}_{{12}}}(u_{3}^{2} - v_{3}^{2}) + {{C}_{{13}}}(u_{2}^{2} - v_{2}^{2}) + {{C}_{{23}}}(u_{1}^{2} - v_{1}^{2}), \\ \end{gathered} $$

(9)

where

$${{C}_{{12}}} = \frac{{{{\Delta }_{1}}{{\Delta }_{2}}}}{{4{{E}_{1}}{{E}_{2}}}}.$$

(10)

The expression in parentheses with two fractions near the factor $v_{{12}}^{ - }u_{{23}}^{ + }u_{{31}}^{ + }$ corresponds in terms of the quasiparticle–phonon model to $\psi _{{{{j}_{2}}{{j}_{3}}}}^{{{{\lambda }_{1}}{{i}_{1}}}}\psi _{{{{j}_{3}}{{j}_{1}}}}^{{{{\lambda }_{2}}{{i}_{2}}}} - \phi _{{{{j}_{2}}{{j}_{3}}}}^{{{{\lambda }_{1}}{{i}_{1}}}}\phi _{{{{j}_{3}}{{j}_{1}}}}^{{{{\lambda }_{2}}{{i}_{2}}}}$ [14, Eq. (9)] because $\psi _{{{{j}_{2}}{{j}_{1}}}}^{{{{\lambda }_{1}}{{i}_{1}}}} \sim \frac{{u_{{{{j}_{2}}{{j}_{3}}}}^{ + }}}{{{{\varepsilon }_{{{{j}_{2}}{{j}_{3}}}}} - {{\omega }_{{{{\lambda }_{1}}{{i}_{1}}}}}}}$ and $\varphi _{{{{j}_{2}}{{j}_{1}}}}^{{{{\lambda }_{1}}{{i}_{1}}}} \sim \frac{{u_{{{{j}_{2}}{{j}_{3}}}}^{ + }}}{{{{\varepsilon }_{{{{j}_{2}}{{j}_{3}}}}} + {{\omega }_{{{{\lambda }_{1}}{{i}_{1}}}}}}}$ and the coefficients in these expressions correspond to ${{V}_{{12}}}g_{{31}}^{s}g_{{23}}^{{s'}}$. Terms corresponding to all six diagrams containing anomalous (“pairing”) Green’s functions ${{F}^{{(1)}}}$ and ${{F}^{{(2)}}}$ given by Eq. (2) are partially included on the right-hand side of Eq. (8).

The second part of the right-hand side of Eq. (8), which includes the factor ${{(E_{{21}}^{2} - {{\omega }^{2}})}^{{ - 1}}}$) and depends on $\omega$, describes dynamical GSC3s that do not appear in the quasiparticle–phonon model. As shown below, they make a significant contribution to the probabi-lity $B(E1)$.

The limiting cases of Eqs. (6)–(8) with $\Delta = 0$ and $\omega = 0$ were discussed in [12] and [6, 11], respectively.

4 CALCULATIONS FOR TIN ISOTOPES

In all calculations, well-known parameters of the DF3-a Fayans energy density functional [16, 17] were used for the effective field, effective interaction, and phonon creation amplitudes.

Equations for the effective field V and the phonon creation amplitude g were solved in the coordinate representation using the self-consistent basis of the DF3-a Fayans functional. Since the sum in Eq. (7) is incoherent, we determined the maximum energy to which summation in Eq. (7) should be performed for convergence. As shown in [6, 11], summation up to 100 MeV provides sufficient accuracy. This summation limit ensures the inclusion of the continuous single-particle spectrum, which is a good analog of the coordinate representation. The same basis was used in the calculations in [12, 13].

The characteristics of the first ${{2}^{ + }}$ and ${{3}^{ - }}$ levels were calculated with the same calculation scheme and the same parameters of the DF3-a Fayans functional. The results are summarized in Tables 1 and 2. Agreement with experimental probabilities for phonons is satisfactory taking into account experimental errors. This is more significant for the final result than discrepancy in energies because the effect is determined by the product $g_{{31}}^{s}g_{{23}}^{{s'}}$ according to Eq. (7). The resulting discrepancies are not surprising because we used the universal self-consistent approach, where both the mean field and the effective interaction are described by the same parameters of the Fayans functional, which also reproduce many other nuclear characteristics (see review [17]). Tables 1 and 2 do not demonstrate a noticeable specificity of ¹¹²Sn and ¹¹⁴Sn.

Table 1. Characteristics of low-lying $2_{1}^{ + }$ phonons for tin isotopes ${{\omega }_{2}}$ and $B(E2) \uparrow$

Table 2. Characteristics of low-lying $3_{1}^{ - }$ phonons for tin isotopes

The final results of our calculations are presented in Table 3 and Fig. 1. Good agreement for the probabilities $B(E1)(3_{1}^{ - } \to 2_{1}^{ + })$ is seen for all isotopes except for ¹¹²Sn and ¹¹⁴Sn. To estimate the contributions from different effects to the calculated probability, the results obtained disregarding the nuclear polarizability and GSCs are also given in Table 3. As seen in the table, as in [12], the inclusion of the polarizability reduces B(E1) for magic nuclei (columns 2 and 3), but this decrease (by about a factor of 3) is less than that in [12] (by an order of magnitude). The inclusion of GSCs increases B(E1) by more than an order of magnitude (columns 2 and 4), whereas the inclusion of the polarizability reduces B(E1) by almost an order of magnitude (columns 4 and 5) and leads to good agreement with experimental data. Thus, good agreement with experimental data was due to the difference of two large effects, which emphasizes the importance of the used self-consistent scheme. This conclusion is illustrated in Fig. 1, where it is clearly seen that the inclusion of the contribution from GSCs is obligatory to reproduce experimental data for the ^116–124Sn nuclei.

Table 3. Reduced probabilities B(E1) (in e² fm²) of the transition between the first ${{3}^{ - }}$ and ${{2}^{ + }}$ phonons for tin isotopes according to (columns 2–5) the calculations (2) disregarding the polarizability and GSCs), (3) including the polarizability but disregarding GSCs, (4) disregarding the polarizability but including GSCs, and (5) including the polarizability and GSCs and (6) experiment

5 POSSIBLE REASONS FOR DISCREPANCY WITH EXPERIMENTAL DATA FOR THE ¹¹²Sn AND ¹¹⁴Sn NUCLEI

(i) We compared our results with experimental data obtained by different methods. Data for $^{{116 - 124}}$Sn nuclei were obtained by analyzing γ radiation from the inelastic scattering of fast reactor neutrons on tin [18]. Data for the ¹¹²Sn and ¹¹⁴Sn nuclei were obtained by different, very reliable experimental methods involving the ¹¹²Cd$(\alpha ,2n)$¹¹⁴Sn reaction [19, 20]. The authors of [20] concluded that the $3_{1}^{ - }$ → $2_{1}^{ + }$ transition occurs from the deformed band to the spherical band (see Table 6 in [20]). This is the reason for discrepancy between experimental data and our calculations, where deformation was not taken into account. Low-lying excited states in other nuclei can also have different deformations, as shown, e.g., in [21]. Furthermore, some self-consistent RPA calculations with separable forces predict that the ¹¹²Sn and ¹¹⁴Sn nuclei are deformed in the ground state [22].

(ii) The author of [23] jointly analyzed the data for the stripping and pickup reactions [24] and emphasized that the ¹¹⁴Sn nucleus is specific compared to neighboring nuclei because it has a local magic number OF N = 64. These results were based mainly on the analysis of the corresponding data for odd nuclei from the stripping and pickup reactions from which data on observed single-particle energies and spectroscopic factors in the corresponding even–even nuclei were extracted. To explain experimental results obtained for even–even nuclei in [23] within the consistent microscopic approach, it is necessary to calculate the phonon coupling effects in the pairing gap and in spectroscopic factors within the single-particle scheme obtained with the energy density functional. Such calculations for nuclei with pairing were carried out in [25, 26]. We compared experimental results obtained in [23] with existing non-self-consistent calculations in [25] only for ¹²⁰Sn. Taking into account experimental errors, good agreement was obtained for single-particle energies and populations of the 2d3/2, 1g7/2, 3s1/2, and 1h1/2 levels. However, this does not explain our results for B(E1) in ^112,114Sn summarized in Table 3. More detailed and self-consistent calculations are necessary to explain single-particle characteristics in nuclei with pairing including the quasiparticle–phonon interaction. Such calculations are beyond the scope of this work. This problem is relevant, as confirmed in recent review [27], where the Green’s function method is also used.

6 CONCLUSIONS

The probabilities of the E1 transitions between the first ${{2}^{ + }}$ and ${{3}^{ - }}$ collective one-phonon levels in a long chain of even–even tin isotopes have been calculated self-consistently. The same self-consistent approach with the same parameters of the Fayans functional has been used to calculate the characteristics of the indicated one-phonon levels. This common self-consistent description emphasizes the importance of self-consistency and allows one to hope to obtain possible indications of a new physical nature of the described characteristics. It has been shown for the first time that the inclusion of new correlations in the ground state (dynamical GSC3s) is necessary to explain experimental data for ^116–124Sn. Consequently, the self-consistent theoretical analysis of transitions between excited states is very promising for low-energy physics.

A reasonable agreement with existing experimental data has been achieved for all considered characteristics except for the probabilities B(E1) for the ¹¹²Sn and ¹¹⁴Sn neutron-deficient nuclei. The most probable reason for such inconsistency is the deformation of one of the considered one-phonon states. At the same time, the modern self-consistent theoretical analysis of data on single-particle energies and spectroscopic factors in nuclei with pairing is also necessary.