Thomas-Ehrman Shift Research Articles

Double-magicity of proton drip-line nucleus 22Si with ab initio calculation

New magic numbers have been discovered in the neutron-rich region of the nuclear chart. However, there has been a lack of research on proton-rich nuclei. 22O, the mirror nucleus of 22Si, is a double-magic nucleus bearing a high E(21+). Whether 22Si exhibits double-magic characters is an intriguing topic. To investigate this matter, we utilized ab initio valence space in-medium similarity renormalization group for 22Si/22O, and their nearby nuclei. Our ab initio calculations provide good descriptions for the double magicity of 22O, as well as the shell evolution of N=14 and Z=14 through E(21+). The computed E(21+) indicate that the closure of Z=14 sub-shell in proton-rich nuclei is weaker than the N=14 sub-shell closure in their mirror nuclei. Particularly, the calculated E(21+) of 22Si is 800 keV lower than the one of 22O. To further explore the magicity of 22Si, the mirror energy difference (MED) of 26Si/26Mg, 24Si/24Ne, as well as 22Si/22O are calculated. The results demonstrate that the calculated MEDs agree well with available experimental data, and the E(21+) values of 22,24,26Si are all lower than their respective mirror nuclei due to the Thomas-Ehrman shift with large s1/2 occupation. Moreover, our calculation provides that the many-body configurations of the low-lying state of 22Si/22O are nearly identical despite the fact that the states bearing large MED. In conclusion, our ab initio results suggest that 22Si is a double magic nucleus, similar to its mirror nucleus 22O, albeit with a lower E(21+).

<italic>Ab initio</italic> Gamow shell model with chiral three-nucleon force for<sup>14</sup>O isotones

<p indent="0mm">Proton-rich nuclei exhibit many novel phenomena, such as proton emission, proton halo, and Thomas-Ehrman shift. On one hand, due to the Coulomb force, outer protons have a lower nuclear separation threshold and are strongly coupled to the continuum. Such weakly bound nuclei are called the open quantum system. On the other hand, the Coulomb barrier restricts the spatial spread of the weakly-bound outer protons, resulting in that unbound resonance nuclei beyond the proton drip line may have certain lifetimes, e.g., a millisecond lifetime. The interplay of the continuum coupling and Coulomb forces produces a complex pattern in nuclei near the proton-rich drip line, which contains rich and unique structures and dynamic information. These properties lead to new insights into nuclear forces, symmetry breaking between mirror nuclei and many-body correlations. Recent experiment observed the ground state of ¹⁸Mg via its decay into 4p<italic>+</italic>¹⁴O. The decay of the ground state of this nucleus is consistent with two sequential steps of 2p decay passing through the ground state of ¹⁶Ne. And another excited state at <sc>1.84 MeV</sc> was also observed, which may be the first 2⁺ excited state. The excitation energy of this 2⁺ state is significantly higher than that in ¹⁸C, which shows a typical Thomas-Ehrman shift. In addition, this 2⁺ excitation energy of ¹⁸Mg exceeds that of ²⁰Mg. It is meaningful to study these novel phenomena through <italic>ab initio</italic> calculations. However, continuum coupling is important in these proton-rich nuclei, so the continuum degrees of freedom should be treated properly. Using the Berggren basis, the Gamow shell model provides a good description of the interplay between bound state, resonance, and continuum in a self-consistent many-body framework. In the present paper, we employ two- and three-nucleon forces from chiral effective theory, perform the Gamow shell model calculations of ^15–18C and ¹⁵F–¹⁸Mg with continuum coupling included. The Gamow Hartree-Fock method with three-body force included was used to generate the Berggren basis, it can give a good description of the bound, resonant, and scattering states. The Hamiltonian is then normal ordered with respect to the Gamow Hartree-Fock reference state, and the normal ordered two-body approximation is adopted. We use the many-body perturbation theory to construct the effective Hamiltonian for the psd model space with s_1/2 and d_5/2 partial waves treated in Berggren basis. We calculate the <inline-formula content-type="pic" id="INLINE22"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mover accent="true" other="0"><mml:mi other="0">S</mml:mi><mml:mo other="1">^</mml:mo></mml:mover></mml:math></inline-formula>-box up to the third order and <inline-formula content-type="pic" id="INLINE23"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mover accent="true" other="0"><mml:mi other="0">Q</mml:mi><mml:mo other="1">^</mml:mo></mml:mover></mml:math></inline-formula>-box up to the second order. We use the Jacobi-Davidson method to diagonalize the complex-symmetric model-space Hamiltonian, and allow at most two valence particles in the continuum. The energies of the ground and first 2⁺ excited states are calculated for ^15–18C and ¹⁵F–¹⁸Mg. The calculated energies agree well with experimental data or evaluations. By calculating occupation probabilities and effective single-particle energies of the <inline-formula id="INLINE24"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mtext>π</mml:mtext><mml:mn>1</mml:mn><mml:msub other="0"><mml:mtext other="0">s</mml:mtext><mml:mrow other="1"><mml:mn other="1">1</mml:mn><mml:mo other="1">/</mml:mo><mml:mn other="1">2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula id="INLINE25"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mtext>π</mml:mtext><mml:mn>0</mml:mn><mml:msub other="0"><mml:mtext other="0">d</mml:mtext><mml:mrow other="1"><mml:mn other="1">5</mml:mn><mml:mo other="1">/</mml:mo><mml:mn other="1">2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> orbitals, we analyze the continuum-coupling effect in the ground state of ¹⁸Mg. The calculation of the mirror energy difference between the 2⁺ states of ¹⁸C and ¹⁸Mg is improved with continuum coupling included. We have systematically calculated the first 2⁺ states in <italic>Z</italic>=12 isotopes, supporting the experimental observation that the 2⁺ excitation energy in ¹⁸Mg is slightly higher than that in ²⁰Mg. There have been no experimental data for ²⁰Si and ²²Si. We predict that the 2⁺ excitation energy in ²⁰Si is lower than that in ²²Si with the real-energy shell-model calculation.

The roles of three-nucleon force and continuum coupling in mirror symmetry breaking of oxygen mass region

With both three-nucleon force and continuum coupling included, we have developed a self-consistent ab initio Gamow shell model within the Gamow Hartree-Fock (GHF) basis obtained by the realistic interaction itself. With the chiral two-nucleon N3LO and three-nucleon N2LO interactions, the Gamow shell model has been applied to the mirror systems of Z=8 neutron-rich isotopes and N=8 proton-rich isotones, giving good agreements with data in binding energies, dripline positions and excitation spectra. The GHF calculated that the 0d3/2, 1s1/2 and 1p3/2 orbitals are resonances. The resonance states and their interplay with nonresonant continua play a crucial role in the descriptions of nuclei around driplines. Excitation spectra and Thomas-Ehrman shifts observed can be better described when both three-nucleon force and continuum coupling are considered in calculations. The three-nucleon force and continuum coupling produce a combined effect on the Thomas-Ehrman shift, e.g., for the 1/2+ resonance level of 19Na. The calculations help the understandings of related nuclear astrophysical processes.

Proton decays in Ne16 and Mg18 and isospin-symmetry breaking in carbon isotopes and isotones

Proton-rich nuclei possess unique properties in the nuclear chart. Due to the presence of both continuum coupling and Coulomb interaction, phenomena such as halos, Thomas-Ehrman shift, and proton emissions can occur. Experimental data are difficult to be obtained therein, so that theoretical calculations are needed to understand nuclei at drip-lines and guide experimentalists for that matter. In particular, the $^{16}$Ne and $^{18}$Mg isotopes are supposed to be one-proton and/or two-proton emitting nuclei, but associated experimental data are either incomplete or even unavailable. Consequently, we performed Gamow shell model calculations of carbon isotones bearing $A=15\text{-}18$. Isospin-symmetry breaking occurring in carbon isotones and isotopes is also discussed. It is hereby shown that the mixed effects of continuum coupling and Coulomb interaction at drip-lines generate complex patterns in isospin multiplets. Added to that, it is possible to determine the one-proton and two-proton widths of $^{16}$Ne and $^{18}$Mg. Obtained decay patterns are in agreement with those obtained in previous experimental and theoretical works. Moreover, up to the knowledge of authors, this is the first theoretical calculation of binding energy and partial decay widths of $^{18}$Mg in a configuration interaction picture.

Gamow-shell-model description of Li isotopes and their mirror partners

Background: Weakly bound and unbound nuclei close to particle drip lines are laboratories of new nuclear structure physics at the extremes of neutron/proton excess. The comprehensive description of these systems requires an open quantum system framework that is capable of treating resonant and nonresonant many-body states on equal footing. Purpose: In this work, we construct the minimal complex-energy configuration interaction approach to describe binding energies and spectra of selected 5 $\leq$ A $\leq$ 11 nuclei. Method: We employ the complex-energy Gamow shell model (GSM) assuming a rigid $^4$He core. The effective Hamiltonian, consisting of a core-nucleon Woods-Saxon potential and a simplified version of the Furutani-Horiuchi-Tamagaki interaction with the mass-dependent scaling, is optimized in the sp space. To diagonalize the Hamiltonian matrix, we employ the Davidson method and the Density Matrix Renormalization Group technique. Results: Our optimized GSM Hamiltonian offers a good reproduction of binding energies and spectra with the root-mean-square (rms) deviation from experiment of 160 keV. Since the model performs well when used to predict known excitations that have not been included in the fit, it can serve as a reliable tool to describe poorly known states. A case in point is our prediction for the pair of unbound mirror nuclei $^{10}$Li-$^{10}$N in which a huge Thomas-Ehrman shift dramatically alters the pattern of low-energy excitations. Conclusion: The new model will enable comprehensive studies of structure and reactions aspects of light drip-line nuclei.

New isospin-breaking “USD” Hamiltonians for the sd shell

Two new USD-type Hamiltonians, USDC and USDI, have been developed that directly incorporate Coulomb and other isospin-breaking interactions. Starting from ab initio interactions, linear combinations of two-body matrix elements were constrained by experimental energy levels in sd-shell nuclei. With this method, binding energies and excitation energies of proton-rich nuclei in the shell can be added to the data set used in the fit. USDC is based on the same renormalized $G$ matrix used in the derivation of previous USD-type Hamiltonians, while USDI is derived from in-medium similarity renormalization group (IMSRG) interactions. Both contain an analytic Coulomb interaction with Miller-Spencer short-range correlations and an effective isotensor interaction. Also presented are modifications to these interactions, USDCm and USDIm, that have had the Coulomb interaction constrained to better reproduce the experimental $b$ coefficients of the isobaric mass multiplet equation. These Hamiltonians are used to provide new predictions for the proton-dripline and to examine isospin-level mixing and other properties of $sd$-shell nuclei. An empirical expression for the Thomas-Ehrman shift in loosely bound and unbound proton-rich states is presented, and several such states are examined.

Two-neutron knockout as a probe of the composition of states in Mg22,Al23 , and Si24

Simpson and Tostevin proposed that the width and shape of exclusive parallel momentum distributions of the A-2 residue in direct two-nucleon knockout reactions carry a measurable sensitivity to the nucleon single-particle configurations and their couplings within the wave functions of exotic nuclei. We report here on the first benchmarks and use of this new spectroscopic tool. Exclusive parallel momentum distributions for states in the neutron-deficient nuclei $^{22}$Mg, $^{23}$Al, and $^{24}$Si populated in such direct two-neutron removal reactions were extracted and compared to predictions combining eikonal reaction theory and shell-model calculations. For the well-known $^{22}$Mg and $^{23}$Al nuclei, measurements and calculations were found to agree, supporting the dependence of the parallel momentum distribution width on the angular momentum composition of the shell-model two-neutron amplitudes. In $^{24}$Si, a level at 3439(9) keV, of relevance for the important $^{23}$Al(p,$\gamma$)$^{24}$Si astrophysical reaction rate, was confirmed to be the $2^+_2$ state, while the $4^+_1$ state, expected to be strongly populated in two-neutron knockout, was not observed. This puzzle is resolved by theoretical considerations of the Thomas-Ehrman shift, which also suggest that a previously reported 3471-keV state in $^{24}$Si is in fact the ($0^+_2$) level with one of the largest experimental mirror-energy shifts ever observed.

Cluster Thomas-Ehrman effect in mirror nuclei

The Coulomb shift is analyzed for the mirror cluster systems of $^{18}\mathrm{O}=\ensuremath{\alpha}+^{14}\mathrm{C}$ and $^{18}\mathrm{Ne}=\ensuremath{\alpha}+^{14}\mathrm{O}$ by applying the orthogonality condition model (OCM). The OCM calculation clearly predicts the suppressed excitation energy of the higher ${0}^{+}$ states in the proton-rich system of $^{18}\mathrm{Ne}$. This results can be interpreted in terms of the extension of the Thomas-Ehrman shift (TES), which is discussed in $^{17}\mathrm{O}=^{16}\mathrm{O}+N$ and $^{17}\mathrm{F}=^{16}\mathrm{O}+P$, to the cluster degrees of freedom. A combination of the cluster TES and the monopole transition is proposed as an experimental probe to identify the cluster structure in mirror systems.

Isospin Non-Conservation in Shell Model Approach and Applications*

Up to now, empirical shell-model Hamiltonians for valence space calculations provide the most accurate description of the low-energy spectra and individual transitions of sd- and pf-shell nuclei. These features made them of particular importance for the description of the isospin-symmetry-breaking phenomena, such as energy splitting of the isobaric multiplets or isospin-forbidden transition rates. In this contribution, we demonstrate the applications of a recently constructed isospin non-conserving (INC) Hamiltonian in sd shell [Lam et al. Phys. Rev. C 87, 054304 (2013)]. First, we explore the partial decay scheme of 24Si and discuss the states affected by the Thomas-Ehrman shift. Second, we show the accuracy of the INC Hamiltonian for the description of the mirror energy differences.

Cluster Thomas-Ehrman Shift in10Be–10C

Coulomb shift of the 10 Be and 10 C nuclei, which can be nicely described by the α + α + N + N four-body degrees of freedom, is investigated by the microscopic cluster model. The Coulomb shift is prominent for the non S-wave configurations, while the shift is suppressed for the S-wave dominant configurations, which have the welldeveloped α cluster structure. This result can be interpreted in terms of the so-called Thomas-Ehrman shift, which is originally discussed in the Coulomb shift of the 17 O–17 F systems.

Light nuclei in the vicinity of the dripline and beyond

After a brief historical overview of the field of physics with radioactive beams, we give an update of the most recent experimental achievements for nuclei at or beyond the nuclear driplines. Long-lived exotic nuclear states are discussed including multi-nucleon radioactivity and exotic isomers. Studies of correlations between decay products in three-body decays and analysis in a Jacobi-coordinate framework are discussed with special emphasis on the difficulty in the interpretations of data obtained in different reactions. We give examples of systematic studies that the vast amount of now existing data allows, such as shell closures, competition between single-particle states in isotopes, isotones and mirror nuclei. The Thomas-Ehrman shift, Garvey-Kelson-type mass relations and IMME analysis of isobaric multiplets with isospin \( T = 3/2\) and 2 are also discussed as well as alternative interpretations in certain cases.

Thomas-Ehrman effect in a three-body model: TheNe16case

The dynamic mechanism of the Thomas-Ehrman shift is studied in three-cluster systems by example of $^{16}$Ne and $^{16}$C isobaric mirror partners. We predict configuration mixings for $0^+$ and $2^+$ states in $^{16}$Ne and $^{16}$C. Large isospin symmetry breaking on the level of wave function component weights is demonstrated for these states and discussed as three-body mechanism of Thomas-Ehrman shift. It is shown that the description of the Coulomb displacement energies requires a consistency among three parameters: the $^{16}$Ne decay energy $E_T$, the $^{15}$F ground state energy $E_r$, and the configuration mixing parameters for the $^{16}$Ne/$^{16}$C $0^+$ and $2^+$ states. Basing on this analysis we infer the $^{15}$F $1/2^+$ ground state energy to be $E_r=1.39-1.42$ MeV.

Studies of continuum states in 16 Ne using three-body correlation techniques

Two-proton decay of the unbound nucleus Ne-16 , produced in one-neutron knockout from a 500 MeV/u Ne-17 beam, has been studied at GSI. The ground state, at a resonance energy 1.388(15) MeV, ( MeV) above the O-14 +p+p threshold, and two narrow resonances at MeV and 7.57(6) MeV have been investigated. A comparison of the energy difference between the first excited 2(+) state and the 0(+) ground state in Ne-16 with its mirror nucleus C-16 reveals a small Thomas-Ehrman shift (TES) of keV. A trend of the TES for the T = 2 quintet is obtained by completing the known data with a prediction for F-16 obtained from an IMME analysis. The decay mechanisms of the observed three resonances were revealed from an analysis of the energy and angular correlations of the O-14 +p+p decay products. The ground state decay can be considered as a genuine three-body (democratic) mode and the excited states decay sequentially via states in the intermediate nucleus F-15 , the 3.22 MeV state predominantly via the F-15 ground-state resonance, while the 7.57 MeV state decays via the 5/2(+) resonance in F-15 at 2.8 MeV above the O-14 +p+p threshold. Further, from an analysis of angular correlations, the spin-parity of the 7.57 MeV state has been determined as and assigned as the third 2(+) state in Ne-16 based on a comparison with C-16.

MIRROR ASYMMETRY INVESTIGATED BY β DECAY OF PROTON-RICH 24Si

The β-decay spectroscopy on 24 Si was carried out in order to investigate characteristic properties of proton-rich nuclei from a perspective of mirror asymmetry. We observed two β transitions to low-lying bound states in 24 Al for the first time. The B( GT ) of 24 Si were deduced to be 0.13(2) and 0.14(2), which were smaller than those of the mirror transition by 22(11)% and 10(8)%, respectively. Through the comparison with theoretical calculations taking into account the Coulomb force and the Thomas-Ehrman shift, the mirror asymmetry of B( GT ) is attributed to the lowering of the proton 1s1/2 orbital.

βdecay of the proton-rich nucleusSi24and its mirror asymmetry

$\ensuremath{\beta}$-decay spectroscopy of the proton-rich nucleus $^{24}\mathrm{Si}$ was performed. The decay scheme was reconstructed from results of delayed $\ensuremath{\gamma}$-ray and proton measurements. We observed two $\ensuremath{\beta}$ branches to bound states in $^{24}\mathrm{Al}$ for the first time. The branching ratios were determined to be $31(4)%$ and $23.9(15)%$ for the ${1}_{1}^{+}$ state at $0.426$ MeV and the state at $1.090$ MeV, respectively. The observation of an allowed transition to the $1.090$-MeV state enabled us to firmly determine its spin-parity as ${1}^{+}$. In the proton measurements performed with the $\ensuremath{\Delta}E\text{\ensuremath{-}}E$ method, we observed a new unbound level at $6.735$ MeV. The branching ratios to three unbound states, including the new level, were also determined for the first time. Based on the decay scheme, the $B(\mathrm{GT})$ values of $^{24}\mathrm{Si}$ were deduced. The $B(\mathrm{GT})$ values were smaller than those of the mirror nucleus $^{24}\mathrm{Ne}$ by $22%$ and $10%$ for the ${1}_{1}^{+}$ and ${1}_{2}^{+}$ states, respectively. The mirror asymmetries of $B(\mathrm{GT})$, observed in both the ${1}_{1}^{+}$ and the ${1}_{2}^{+}$ states, indicate changes in configuration in the wave function associated with the Thomas-Ehrman shift. To clarify the mechanism of this asymmetry, a comparison with shell-model calculations is also discussed. The calculations attribute the changes in configuration to the lowering of the $1{s}_{1/2}$ orbital.

Beta-decay study of Tz = - 2 proton-rich nucleus 24Si

$ \beta$ -decay spectroscopy on a T z = - 2 proton-rich nucleus 24Si was performed. The decay scheme of 24Si was reconstructed by the $ \beta$ -delayed $ \gamma$ -ray and proton measurements. Two $ \beta$ branches to the bound 11 + and 12 + states in 24Al were observed for the first time. The observation of the allowed transition firmly established the spin-parity assignment for the 12 + states. The branching ratios to the 11 + and 12 + states were determined to be 31(4)% and 23.9(15)%, respectively. The branching ratios to three unbound states in 24Al including a new level at 6.735MeV were also determined for the first time. The level structure of 24Al was compared with its mirror nucleus 24Na . The Thomas-Ehrman shift on the 12 + state indicates s -wave dominance in the state as well as a characteristic behavior of the weakly bound s -wave proton in 24Al .

Proton threshold states in theNa22(p,γ)Mg23reaction and astrophysical implications

Proton threshold states in {sup 23}Mg are important for the astrophysically relevant proton capture reaction {sup 22}Na(p,{gamma}){sup 23}Mg. In the indirect determination of the resonance strength of the lowest states, which were not accessible by direct methods, some of the spin-parity-assignments remained experimentally uncertain. We have investigated these states with shell model, Coulomb displacement, and Thomas-Ehrman shift calculations. From the comparison of calculated and observed properties, we relate the lowest relevant resonance state at E{sub x}=7643 keV to an excited 3/2{sup +} state in accordance with a recent experimental determination by Jenkins et al. From this we deduce significantly improved values for the {sup 22}Na(p,{gamma}){sup 23}Mg reaction rate at stellar temperatures below T{sub 9}=0.1 K.

The T=2 mirrors 36Ca and 36S: A test for isospin symmetry of shell gaps at the driplines

Abstract The first excited 2 + state of 36Ca has been identified by its γ-decay, exploiting the two-step fragmentation technique at the FRS-RISING setup at GSI. This is the heaviest T z = − 2 nucleus in the Segre chart in which a γ-decay of an excited state has been observed. A stable beam of 40Ca at 420 A MeV impinged on a primary 9Be target. Out of the secondary beam of fragmentation products, 37Ca was separated by the FRS and struck on a second 9Be target at the final focus of the FRS. The energy for the 2 1 + decay of 36Ca was determined to be 3015(16) keV, which is 276 keV lower than in its T = 2 mirror 36S. This mirror energy difference (MED) is discussed in the framework of shell model calculations using a 16O core, the sd shell isospin symmetric interaction USD and experimental single-particle energies from 17O and 17F. The results show that the MED within the sd shell provide a sensitive test for the evolution of the N, Z = 14 , 16 subshell gaps towards the driplines. Especially the N, Z = 16 gap is determined by Thomas–Ehrman shift in the A = 17 , T = 1 / 2 isospin doublet, while Coulomb effects are found to have marginal influence.

Charge-dependent calculations of single-particle energies in nuclei aroundO16with modern nucleon-nucleon interactions

The binding energies of the ground states and several excited states related to single-particle and -hole states in nuclei around ^{16}O are calculated taking charge dependence into account. Effective interactions on the particle basis are constructed from modern charge-dependent nucleon-nucleon interactions and the Coulomb force within the framework of the unitary-model-operator approach. Single-particle (-hole) energies are obtained from the energy differences of the binding energies between a particle (hole) state in ^{17}O or ^{17}F (^{15}N or ^{15}O) and the ground state of ^{16}O. The resultant spin-orbit splittings are small for the hole state and large for the particle state in comparison with the experimental values though the differences between the experimental and calculated values are not very large. The charge dependence of the calculated single-particle energies for the ground states are in good agreement with the experimental values. Furthermore, the Thomas-Ehrman shift due to the Coulomb force for the 1s_{1/2} states in ^{17}O and ^{17}F can be observed.

Resonance structure of9Beand9Bin a microscopic cluster model

Resonance structures of excited states in ${}^{9}\mathrm{Be}$ and ${}^{9}\mathrm{B}$ are studied by means of a microscopic multicluster model. The two-body scattering of ${{}^{8}\mathrm{Be}{(0}^{+}{,2}^{+}{,4}^{+})+N;$ ${}^{5}\mathrm{He}$ or ${}^{5}\mathrm{Li}{(3/2}^{\ensuremath{-}}{,1/2}^{\ensuremath{-}})+\ensuremath{\alpha}}$ is solved by the microscopic R-matrix method where two-cluster wave functions of ${}^{8}\mathrm{Be}$ and ${}^{5}\mathrm{He}$ are employed. These results are compared with the $\ensuremath{\alpha}+\ensuremath{\alpha}+N$ three-body complex scaling method to check the validity of neglecting three-body channels in the microscopic R-matrix method. The ${1/2}^{+}$ first excited state of ${}^{9}\mathrm{Be},$ which cannot be identified in the complex scaling method, is given as a ${}^{8}\mathrm{Be}{(0}^{+})+n$ virtual state with a purely imaginary complex momentum, and this resonance lies on the second Riemann sheet only for the ${}^{8}\mathrm{Be}{(0}^{+})+n$ channel. The ${1/2}^{+}$ resonance of the mirror nucleus ${}^{9}\mathrm{B}$ has a very large width, and its excitation energy as obtained from the analytic continuation of the S matrix to the complex energies shows a normal Thomas-Ehrman shift. The dimensionless reduced widths and the reduced width amplitudes are calculated for low-lying resonances, and the cluster structure of these states in ${}^{9}\mathrm{Be}$ is discussed.