Experimental study of intruder components in light neutron-rich nuclei via single-nucleon transfer reaction

Experimental study of intruder components in light neutron-rich nuclei via single-nucleon transfer reaction 1 first-author https://orcid.org/0000-0001-6459-7377 1 corresp Corresponding author, jllou@pku.edu.cn Corresponding author, jllou@pku.edu.cn jllou@pku.edu.cn https://orcid.org/0000-0002-8246-4807 1 https://orcid.org/0000-0001-8938-9152 2 https://orcid.org/0000-0002-2180-0758 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China School of Physics, Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing 100191, China School of Physics, Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing 100191, China With the development of radioactive beam facilities, studies concerning the shell evolution of unstable nuclei has recently gained prominence. Intruder components, particularly s -wave intrusion, in the low-lying states of light neutron-rich nuclei near N = 8 are of importance in the study of shell evolution. The use of single-nucleon transfer reactions in inverse kinematics has been a sensitive tool that can be used to quantitatively investigate the single-particle orbital component of selectively populated states. The spin-parity, spectroscopic factor (or single-particle strength), and effective single-particle energy can all be extracted from such reactions. These observables are often useful to explain the nature of shell evolution, and to constrain, check, and test the parameters used in nuclear structure models. In this article, the experimental studies of the intruder components in low-lying states of neutron-rich nuclei of He, Li, Be, B, and C isotopes using various single-nucleon transfer reactions are reviewed. The focus is laid on the precise determination of the intruder s -wave strength in low-lying states. Single-nucleon transfer reaction Intruder component Light neutron-rich nuclei 1 Introduction 1 Electrons that are confined within atoms by the Coulomb potential have a well-known shell structure. Shell-like phenomena also appear in finite nuclear systems. Almost 70 years ago, Mayer and Jensen developed a theoretical model to describe the nuclear shell structure [ 1 1 , 2 2 ], which was formed by placing the single nucleon (proton or neutron) in a mean field created by all other nucleons. According to the well-established mean field theory, nucleons fill in the single-particle orbitals in shells that are characterized by the conventional magic numbers: 2, 8, 20, 50, 82, and 126. However, for nuclei far from the β -stability line, especially those in the light mass regions where the concept of a mean field is less robust, exotic rearrangements of the single-particle configuration often occur and may result in the inversion of some orbitals or the emergence of various intruder components [ 3 3 ], leading to the disappearance of the traditional magic numbers and the appearance of new magic numbers [ 4 4 ]. This rearrangement may also result in shifts in the single-particle energies (or orbitals), essentially affec ting all the features of the nuclear structure and leading to processes such as deformation [ 4 4 , 5 5 ]. The strongly attractive interaction between neutrons and protons has been considered as the main cause of this rearrangement of orbitals in exotic nuclei [ 4 4 ]. The evolution of the shell structure in unstable nuclei, including the rearrangement of orbitals, intrusion, and shifts in the single-particle energies, can now be partly described in terms of a new mean field model wherein the monopole effect of the tensor force [ 5 5 - 7 7 5-7 ] and three-body forces [ 8 8 ] are implemented. It has been found that the energy gap between the 1 d 5/2 and 2 s 1/2 shells changes significantly for light neutron-rich nuclei near N = 8, leading to the appearance of some s -wave components in low-lying states. These two orbitals can even be inverted, meaning that the 2 s 1/2 orbital can intrude into 1 d 5/2 , and occasionally even into 1 p 1/2 [ 9 9 , 10 10 ]. One widely noted example is the ground state of the one-neutron halo nucleus of 11 Be, which possesses an unusual spin-parity of 1/2 + that is dominated ( ∼ 71%) by an intruder 2 s 1/2 neutron coupled to the inert core of 10 Be(0 + ) [ 11 11 , 12 12 ]. The intrusion of the s -wave into the ground state of 11 Be is clearly responsible for the formation of its novel halo structure. Intruder s -wave components have also been observed in the low-lying states of the nuclei such as 9 He, 10,11 Li, 12,13,14 Be, 13,14 B, and 15,16 C, and have been widely studied in various experiments using different methods in order to understand their exotic structure [ 13 13 ]. The transfer reaction is one of the most common experimental methods used to study the intruder components in the wave functions of exotic nuclei. The transfer reaction, especially the single-nucleon transfer reaction, is a sensitive experimental tool used to selectively populate certain interesting states in nuclei. These populated states can be described using either the original or a residual nucleus as a core with the transferred nucleon in an orbital around it. The spin-parity of the selectively populated state can be assigned because the oscillation behavior of differential cross sections (DCSs) depends on the transferred angular momentum l [ 14 14 , 15 15 ]. The contribution of the transferred nucleon to each populated state is usually described by the spectroscopic factor ( SF ), which is often extracted by comparing the experimental with the theoretical DCSs. It has been disputed for several years whether SF is useful for experimental observation. Can SF be directly connected to nuclear structure? Several experiments have been performed in order to address these questions. The conclusion that has been reached is that the relative or normalized SF is related to the occupancy or vacancy number of a certain orbital [ 11 11 , 16 16 ], rather than the SF that is directly extracted (reaction model dependent). If both the SF and the excitation energy for each populated state with the same orbital j are known, the effective single-particle energy of the orbital j can also be determined [ 14 14 , 15 15 ]. Therefore, transfer reaction is a quantitative tool that can be used to probe the intensity of the single-particle orbital component (or occupancy/vacancy number), the spin-parity of the populated state, and the effective single-particle energy of a nuclei that is at a distance from the β -stability line. Unlike stable nuclei, the half-lives of radioactive beams are relatively short; hence, single-nucleon transfer reactions have to be performed in inverse kinematics. The missing mass method, in which only the energies and angles of the recoil light particles are measured, is a commonly used method for investigating transfer reactions in inverse kinematics. This experimental method has some advantages over normal kinematic measurements. However, in this regard, a number of experimental challenges are generated, such as the precise detection of charged-particles with low energies. Several new experimental techniques have therefore been developed to overcome these issues, and several new detection arrays have been constructed in various laboratories. In this paper, the basic modules and the advantages and disadvantages of the typical experimental setups are introduced in detail. The studies concerning the single-particle orbital intruder components in light neutron-rich nuclei are also reviewed. This paper is organized as follow: the basic concepts of transfer reaction are outlined in Sect. 2; the advantages and disadvantages of normal and inverse kinematics are analyzed, and the missing mass method is introduced in Sect. 3; some typical experimental setups for the single-nucleon transfer reactions used worldwide are summarized in Sect. 4; studies that have been conducted using single-nucleon transfer reactions to investigate the exotic nuclear structure of neutron-rich He, Li, Be, B, and C isotopes are reviewed in Sect. 5; and a brief summary is given in the final section. 2 What can we learn from the transfer reaction? 1 2.1 What is a transfer reaction? 2 Generally, a transfer reaction can be written as <math display="block" id="M1"><mrow><mtext>A</mtext><mo>+</mo><mi>a</mi><mo>→</mo><mtext>B</mtext><mo>+</mo><mi>b</mi><mtext> </mtext><mtext>or</mtext><mtext> </mtext><mtext>A</mtext><mo stretchy="false">(</mo><mi>a</mi><mn>,</mn><mi>b</mi><mo stretchy="false">)</mo><mtext>B</mtext><mn>,</mn></mrow></math> where A and a are the target and the projectile nuclei, and B and b represent the residual and the outgoing particles, respectively. This means that when projectile a collides with target A, a nucleon (proton or neutron) or a cluster (such as 4 He) is transferred to form a new final system that comprises b and B. A stripping reaction describes the situation in which the nucleon or cluster is transferred from projectile a to target A, and a pick-up reaction describes the situation when the nucleon or cluster is transferred to projectile a . If only one nucleon is transferred, the result is known as a single-nucleon transfer reaction. Fig. 1 Fig. 1 shows the different kinds of single-nucleon transfer reactions when 11 Be collides with a proton or deuteron target. When a deuteron beam impinges on a 11 Be target, the one-neutron and one-proton transfer reactions are written as 11 Be( d , p ) 12 Be (or 11 Be( d , t ) 10 Be) and 11 Be( d , 3 He) 10 Li, respectively. If a 11 Be incidence occurs, the experiment has to be performed in inverse kinematics, and the corresponding expressions are changed to d ( 11 Be, p ) 12 Be (or d ( 11 Be, t ) 10 Be) and d ( 11 Be, 3 He) 10 Li. The recoil charged-particle proton (or triton) and 3 He are measured in both cases. Both the d ( 11 Be, t ) 10 Be and d ( 11 Be, 3 He) 10 Li reactions are typical stripping or nucleon-removing reactions in inverse kinematics, while the d ( 11 Be, p ) 12 Be reaction is a pick-up or nucleon-adding reaction. The single-particle knock-out reaction is another typical nucleon-removing reaction, but will not be reviewed in this paper. 2.2 Angular distributions and spin-parity 2 The DCS of A( a , b )B to a given final state in B oscillates as a function of the scattering angle. The structure of the oscillation (that is, the positions of the maxima and minima) depends on the transferred angular momentum l . This behavior can be understood via the simple momentum diagram shown in Fig. 2 Fig. 2 . If we assume that the incident projectile has a momentum <math id="M2"><mover accent="true"><mi>p</mi><mo>→</mo></mover></math> and the momentum that is transferred to the target nucleus is <math id="M3"><mrow><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mtext>t</mtext></msub></mrow></math> , only a small reduction is observed in the magnitude of the momentum of a beam particle for a small scattering angle θ , as seen in the vector diagram ( Fig. 2 Fig. 2 ), which is built according to the conservation of momentum. From the cosine rule, we have: <math display="block" id="M4"><mrow><mi>cos</mi><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msubsup><mi>p</mi><mtext>t</mtext><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mi>p</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> If we make use of the second order expansion, which is referred to as <math display="block" id="M5"><mrow><mi>cos</mi><mi>θ</mi><mo>≈</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msup><mi>θ</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mn>,</mn></mrow></math> then we have, <math display="block" id="M6"><mrow><msup><mi>θ</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>p</mi><mtext>t</mtext></msub></mrow><mi>p</mi></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mi>δ</mi><mi>p</mi></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mi>δ</mi><mi>p</mi></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mn>.</mn></mrow></math> The reduction δ in the length of the vector p is small in comparison to the length of the actual transferred momentum p t , meaning that the term δ/p can be dropped [ 17 17 ]. The expression θ 2 can therefore be simplified to <math display="block" id="M7"><mrow><msup><mi>θ</mi><mn>2</mn></msup><mo>≈</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>p</mi><mtext>t</mtext></msub></mrow><mi>p</mi></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mn>.</mn></mrow></math> In the classical picture of a transfer reaction, the nucleon is usually transferred at the surface of the target nucleus, and the angular momentum <math id="M8"><mover accent="true"><mi>L</mi><mo>→</mo></mover></math> is given by [ 17 17 ] <math display="block" id="M9"><mrow><mover accent="true"><mi>L</mi><mo>→</mo></mover><mo>=</mo><msub><mover accent="true"><mi>p</mi><mo>→</mo></mover><mtext>t</mtext></msub><mo>×</mo><mover accent="true"><mi>R</mi><mo>→</mo></mover><mn>,</mn></mrow></math> where <math id="M10"><mover accent="true"><mi>R</mi><mo>→</mo></mover></math> is the vector radius of the target nucleus. From quantum mechanics [ 15 15 ], we have <math display="block" id="M11"><mrow><msup><mi>L</mi><mn>2</mn></msup><mo>|</mo><mi>ϕ</mi><mo>〉</mo><mo>=</mo><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>ℏ</mi><mn>2</mn></msup><mo>|</mo><mi>ϕ</mi><mo>〉</mo><mn>.</mn></mrow></math> It is therefore easy to deduce that: <math display="block" id="M12"><mrow><msub><mi>θ</mi><mn>0</mn></msub><mo>≈</mo><mfrac><mrow><msub><mi>p</mi><mtext>t</mtext></msub></mrow><mi>p</mi></mfrac><mo>=</mo><mfrac><mrow><msqrt><mrow><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msqrt><mi>ℏ</mi></mrow><mrow><mi>p</mi><mi>R</mi></mrow></mfrac><mn>,</mn></mrow></math> where θ 0 is the angle of the first maxima in the cross section. This indicates that θ 0 is different for each different transferred momentum l , and that θ 0 increases with l . This trend can be seen in Fig. 2 Fig. 2 . The transferred orbital angular momentum l , and therefore the parity of the populated states, can be assigned in conjunction with other experimental information or shell model predictions. Transfer reactions can also be used to assign the total angular momentum j of the selectively populated states [ 14 14 ]. Using the reaction 14 B( d , p ) as an example, Fig. 3 Fig. 3 depicts the angular distributions for this reaction in the 3/2 - ground state (solid curve), the 5/2 - (dashed curve), and the 7/2 - (dotted curve) excited states of 15 B. The DCSs in Fig. 3 Fig. 3 were calculated using FRESCO [ 18 18 ] with global optical potentials obtained from Daehnick [ 19 19 ], and Koning and Delaroche [ 20 20 ] for the entrance and exit channels, respectively. It is evident that the oscillation behavior of the angular distributions depends largely on the transferred angular momentum l . 2.3 Spectroscopic factor 2 The single-nucleon transfer reaction is a powerful experimental tool that can be used to study the structure of certain selectively populated states. The contribution of the transferred nucleon to each populated state is usually described by SF , which is often determined by comparing the DCSs that are deduced from experiment with those obtained from theoretical calculation. The functional expression is as follows: <math display="block" id="M13"><mrow><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mtext>exp</mtext></mrow></msub><mo>=</mo><msup><mi>C</mi><mn>2</mn></msup><mi>S</mi><msub><mi>F</mi><mrow><mtext>exp</mtext></mrow></msub><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mtext>theo</mtext></mrow></msub><mn>,</mn></mrow></math> where C 2 is the isospin Clebsch–Gordan coefficient, which is sometimes omitted. The value for this amounts to 1.0 for the ( d , p ) transfer reaction [ 14 14 ]. <math id="M14"><mrow><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi>e</mi><mi>x</mi><mi>p</mi></mrow></msub></mrow></math> and <math id="M15"><mrow><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi>t</mi><mi>h</mi><mi>e</mi><mi>o</mi></mrow></msub></mrow></math> are the DCSs extracted from experiments and various reaction models, respectively. It should be noted that the expressions ( <math id="M16"><mrow><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><msub><mo stretchy="false">)</mo><mrow><mi>t</mi><mi>h</mi><mi>e</mi><mi>o</mi></mrow></msub></mrow></math> are often different when sourced from different reaction codes. For example, the code FRESCO [ 18 18 ] and DWUCK [ 21 21 ], gives <math display="block" id="M17"><mrow><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mtext>theo</mtext></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><msub><mo stretchy="false">)</mo><mrow><mtext>FRESCO</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mn>2</mn><msub><mi>J</mi><mtext>f</mtext></msub><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msub><mi>J</mi><mtext>i</mtext></msub><mo>+</mo><mn>1</mn></mrow></mfrac><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mtext>DWUCK</mtext></mrow></msub><mn>,</mn></mrow></math> where J i and J f describe the spin of the initial and final nuclei. The distorted wave born approximation (DWBA) is the most common approximation theory that is used to calculate the transfer reaction DCSs ( <math id="M18"><mrow><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mtext>theo</mtext></mrow></msub></mrow></math> ). For a given reaction, the theoretical DCS is given by <math display="block" id="M19"><mrow><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><mtext>d</mtext><msub><mi>σ</mi><mrow><mi>β</mi><mi>α</mi></mrow></msub></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mtext>theo</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mi>α</mi></msub><msub><mi>μ</mi><mi>β</mi></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mi>ℏ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>κ</mi><mi>β</mi></msub></mrow><mrow><msub><mi>κ</mi><mi>α</mi></msub></mrow></mfrac><mo>|</mo><msub><mi>T</mi><mrow><mi>β</mi><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>κ</mi><mi>β</mi></msub><mn>,</mn><msub><mi>κ</mi><mi>α</mi></msub><msup><mrow><mo stretchy="false">)</mo><mo>|</mo></mrow><mn>2</mn></msup><mn>,</mn></mrow></math> where μ α ( κ α ) and μ β ( κ β ) are the reduced masses (wave numbers) in the entrance and exit channels, respectively, and T βα is the transition amplitude. This equation takes the distortion of the incoming and outgoing waves caused by the nuclear potential U between the projectile and the target at the entrance ( a + A, α ) and exit partitions ( b + B, β ) into account. Assuming that the transfer reaction occurs in one step (first-order DWA or DWBA), the transition amplitude T βα can be written [ 15 15 ] <math display="block" id="M20"><mrow><msub><mi>T</mi><mrow><mi>β</mi><mi>α</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>χ</mi><mrow><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><msub><mi>k</mi><mi>β</mi></msub><mn>,</mn><mi>r</mi><mo stretchy="false">)</mo><mo>〈</mo><msub><mi>Φ</mi><mi>β</mi></msub><mo>|</mo><mi>Δ</mi><mi>U</mi><mo>|</mo><msub><mi>Φ</mi><mi>α</mi></msub><mo>〉</mo><msup><mi>χ</mi><mrow><mo stretchy="false">(</mo><mo>+</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><msub><mi>k</mi><mi>α</mi></msub><mn>,</mn><mi>r</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>r</mi><mn>,</mn></mrow></math> where r is the relative distance between the projectile and the target ( r α ( β ) in the entrance (exit) channel), and χ (+) ( k α , r ) is the "distorted" wave composed of an incoming plane wave in the α state and the outgoing scattered waves. Similarly, χ (-) ( k β , r ) is the outgoing distorted wave in the channel β . The waves χ (-) ( k β , r ) and χ (+) ( k α , r ) are obtained by solving the Schrodinger equation using an assumed potential for the outgoing and incoming channels, respectively. The potentials are usually extracted from the elastic scattering DCSs using the optical model (OM). The extracted potentials are known as optical model potentials (OPs). In the case of reactions on unstable nuclei, the elastic scattering data is not always available, resulting in larger uncertainties in the DWBA calculations. Moreover, as deuteron is relatively loosely bound (only 2.22 MeV), it is easy to break up in the presence of a target nucleus. This breakup channel can couple with the transfer channel, affecting the SF exp , of which the accurate extraction is important. To account for this mechanism, Johnson and Soper [ 22 22 ] devised the adiabatic wave approximation (ADWA), which uses nucleonic potentials and explicitly includes deuteron breakup. An extension of this method was developed by Johnson and Tandy [ 23 23 ] to include the finite range effects (FR). As stated in [ 11 11 ], the SF s extracted using the adiabatic model (FR-ADWA) are "stable" across measurements at four energies for the ( d , p ) reactions, and are insensitive to the applied OPs, demonstrating the advantages of using this method as compared with extraction using the normal DWBA approach. This is why the FR-ADWA model is generally adopted for analysis of the transfer reaction. It is worth noting that these four measurements are performed under almost identical experimental conditions using the same set of OP parameters [ 11 11 ]. The SF that is extracted experimentally is sensitive to the choice of OP applied and also to several other practical experimental conditions [ 11 11 , 16 16 ]. It has therefore been widely disputed whether it is a good experimental observable. Conceptually, SF is used to describe the occupancy of a valence nucleon at a single-particle orbit in a mean field created by other nucleons. For example, for an orbit with spin j , the total degeneracy number produced by the independent-particle model (IPM) should be 2 j + 1. In the case of configuration mixing based on the shell model approach, this number may be split into several states which are composed of the same j -wave with specific intensities. Shell model calculations that use appropriate effective interactions and model spaces could, in principle, predict the SF of a particular wave (single-particle orbit) in an energy-eigenstate. However, the experimentally observed SF s are often smaller than those predicted by shell-models, an effect that is exhibited as a reduction or quenching factor. This quenching phenomenon was firmly established via (e, e’ p ) knockout reactions [ 24 24 , 25 25 ]. This quenching effect is also generally confirmed in nuclear reactions such as single-particle knockout or transfer reactions. Using these reactions, the SF exp can be extracted by comparing the experimentally measured cross section with that calculated assuming a pure single-particle state [ 26 26 , 27 27 ] (Eq.(9)). As an individual SF exp can be sensitive to the choice of OPs and the practical experimental conditions (as aforementioned), the sum rule method was developed to define the relative SF and the general quenching factor [ 28 28 ]. The quenching factor in the nucleon-transfer reaction is defined as <math display="block" id="M21"><mrow><msub><mi>F</mi><mi>q</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>j</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo stretchy="false">[</mo><mstyle displaystyle="true"><mo>∑</mo></mstyle><msubsup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>σ</mi><mrow><mtext>exp</mtext></mrow></msub></mrow><mrow><msub><mi>σ</mi><mrow><mtext>theo</mtext></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mi>j</mi><mrow><mtext>rem</mtext></mrow></msubsup><mo>+</mo><mstyle displaystyle="true"><mo>∑</mo></mstyle><msubsup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>σ</mi><mrow><mtext>exp</mtext></mrow></msub></mrow><mrow><msub><mi>σ</mi><mrow><mtext>theo</mtext></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mi>j</mi><mrow><mtext>add</mtext></mrow></msubsup><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where the sum of adding and removing the relative cross sections for a given l , j represents the total degeneracy (sum-rule) of that orbit [ 16 16 ]. It is challenging to measure the nucleon-removing and nucleon-adding reactions that take place in radioactive beams using the same experiment because of the limited beam intensity. If the nucleon-adding or nucleon-removing data were available for a given nucleus, the function for the quenching factor requires that the total strength adds up to the number of vacancies in the closed shell, or the number of particles outside it [ 16 16 ], and the quenching factor is therefore modified to <math display="block" id="M22"><mrow><msub><mi>F</mi><mi>q</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>j</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo stretchy="false">[</mo><mstyle displaystyle="true"><mo>∑</mo></mstyle><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>σ</mi><mrow><mtext>exp</mtext></mrow></msub></mrow><mrow><msub><mi>σ</mi><mrow><mtext>theo</mtext></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mi>j</mi></msub><mo stretchy="false">]</mo><mn>.</mn></mrow></math> For example, a consistent quenching factor of approximately 0.55 for a large number of nuclei, with a root-of-mean-square spread of 0.10 was reported in [ 16 16 ]. Once the sum rule is established, the individual SF exp can be normalized through the sum rule to give the intensity (percentage) of the wave component. This normalized SF is much less sensitive to the OP, or other parameters that are used and can therefore be reasonably utilized for comparison with the theoretical predictions. In other words, even for the same reaction, it is hard to directly compare the experimental SF s extracted from different measurements and analyzed using various sets of OP options without any normalization procedure [ 16 16 ]. Only the intensities (percentages) or the normalized SF (also called relative SF in some studies) can be directly compared, unlike the directly extracted experimental SF s of the wave components. It is worth noting that the SF normalization procedure does not change the ratio between the SF exp of different populated states in the final nucleus with the same spin-parity. Hence the ratio, which is equivalent to the SF normalization, is often used in experiments [ 3 3 , 29 29 ]. 2.4 Effective single-particle energy 2 The IPM assumes that nucleons lie on single-particle energy orbits with no correlation between them. The single-particle energy of a particular orbit/state, which is equal to the energy needed for one nucleon to be excited from the ground state to this orbit/state, can therefore be measured simply. The shell model is based on a mean-field theory, in which the correlations between nucleons (protons and neutrons) are taken into consideration [ 15 15 ]. In this case, the single-particle (uncorrelated) energies are not directly observed, as real nuclei are correlated systems by nature. However, they can be obtained from experimental data using the normalized SF and the excitation energy of each populated state according to the sum rule [ 28 28 ]. Based on the Macfarlane–French sum rule [ 28 28 ], for the nucleon-removing reaction of a given nucleus A, the number of nucleons populated in a shell j (occupancy number) of A, G - ( j ), is <math display="block" id="M23"><mrow><msup><mi>G</mi><mo>−</mo></msup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msub><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>F</mi><mo stretchy="false">)</mo></mrow><mi>k</mi></msub><mn>.</mn></mrow></math> For the neutron-adding reaction of a nucleus A, the number of holes in a shell j (vacancy number) of A, G + ( j ), is <math display="block" id="M24"><mrow><msup><mi>G</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><mfrac><mrow><msub><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>J</mi><mtext>f</mtext></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>k</mi></msub></mrow><mrow><mn>2</mn><msub><mi>J</mi><mtext>i</mtext></msub><mo>+</mo><mn>1</mn></mrow></mfrac><msub><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>F</mi><mo stretchy="false">)</mo></mrow><mi>k</mi></msub><mn>,</mn></mrow></math> where J i and J f is the spin of the initial and final state ( k ), respectively. It should be noted that Eq. (15) and Eq. (16) are simple equations used for a spin zero target such as deuteron. This sum rule has been tested in neutron transfer reactions carried out by Schiffer et al [ 25 25 ]. The neutron-adding, neutron-removal, and proton-adding transfer reactions of the four stable even Ni isotopes were measured in the study, with particular attention paid to the cross section determinations. It was found that the valence-orbit occupancies extracted from the sum rule are consistent with the changing number of valence neutrons, as are the vacancies for protons, both at the < 5% [ 25 25 ] level. This sum rule has also been used in the reactions of d ( 13 B, p ) [ 30 30 ], d ( 19 O, p ) [ 31 31 ], and d ( 12 B, 3 He) [ 32 32 ], among others. If the excitation energies and spectroscopic strengths, referred to as the normalized or relative SF s, respectively, for all the relevant states with the same transferred angular momentum l are available, the effective single-particle energies (ESPE) can be given by the centroid [ 15 15 ]: <math display="block" id="M25"><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msup><mi>G</mi><mo>+</mo></msup><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>k</mi><mo>+</mo></msubsup><mo>−</mo><msub><mi>E</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mi>G</mi><mo>−</mo></msup><mo stretchy="false">(</mo><msub><mi>E</mi><mn>0</mn></msub><mo>−</mo><msubsup><mi>E</mi><mi>k</mi><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow><mrow><msup><mi>G</mi><mo>+</mo></msup><mo>+</mo><msup><mi>G</mi><mo>−</mo></msup></mrow></mfrac><mn>,</mn></mrow></math> where the sum covers all of the final excited states k , ε ( j ) is the ESPE, and <math id="M26"><mrow><msubsup><mi>E</mi><mi>k</mi><mo>±</mo></msubsup></mrow></math> is the excitation energy for the state k in nucleus A ± 1. E 0 is the energy at the ground state of nucleus A. For reactions in which nucleons are added, such as ( d , p ), we have <math display="block" id="M27"><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><mfrac><mrow><msub><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>J</mi><mtext>f</mtext></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>k</mi></msub></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>J</mi><mtext>i</mtext></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><msub><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>F</mi><mo stretchy="false">)</mo></mrow><mi>k</mi></msub><msub><mi>E</mi><mi>k</mi></msub></mrow><mrow><msup><mi>G</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where SF k is the relative or normalized SF for the state k . This can be simplified to <math display="block" id="M28"><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msub><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>J</mi><mtext>f</mtext></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>k</mi></msub><msub><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>F</mi><mo stretchy="false">)</mo></mrow><mi>k</mi></msub><msub><mi>E</mi><mi>k</mi></msub></mrow><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msub><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>J</mi><mtext>f</mtext></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>k</mi></msub><msub><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>F</mi><mo stretchy="false">)</mo></mrow><mi>k</mi></msub></mrow></mfrac><mn>.</mn></mrow></math> This formulation was presented in [ 30 30 ], and is equivalent to that given in [ 33 33 ]. For example, the calculation of ESPE using Eq. (19) for the reaction d ( 13 B, p ) to the low-lying states in 14 B is as follows. The populated low-lying states in 14 B were constructed via the coupling of one sd -shell neutron to the 3/2 - ground state of 13 B. This coupling leads to a (1,2) - doublet for 1 π (1 p 3/2 )- 1 ν (2 s 1/2 ), and (1,2,3,4) - and (0,1,2,3) - multiplets when the transferred neutron populates the 1 d 5/2 and 1 d 3/2 orbitals, respectively. Configuration mixing between states with the same spin and parity in 14 B is possible, especially for the neutron 2 s 1/2 and 1 d 5/2 orbitals because of the smaller energy gap between these two orbitals. Fig. 4 Fig. 4 shows the angular distributions of the d ( 13 B, p ) reaction to four low-lying states in 14 B. Only the <math id="M29"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> , <math id="M30"><mrow><msubsup><mn>1</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> , <math id="M31"><mrow><msubsup><mn>3</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> , and <math id="M32"><mrow><msubsup><mn>4</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> states were populated, therefore only the SF s for these four states could be extracted from this experiment. In Fig. 4(a) Fig. 4(a) and Fig. 4(b) Fig. 4(b) , it was found that it is difficult to fit the <math id="M33"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> and <math id="M34"><mrow><msubsup><mn>1</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> DCSs using only the s -wave ( l = 0) components (blue thick-dashed curves). However, when a little d -wave component was taken into consideration, the fitting of the angular distributions can be improved, indicating that these two states are made up of a mixture of s - and d -waves. If the effect of the 1 d 3/2 orbital is ignored, the spectroscopic strengths or the normalized SF s for the unobserved <math id="M35"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> and <math id="M36"><mrow><msubsup><mn>1</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> states can be deduced using the assumption that the pairs of 2 - and 1 - levels are formed by orthogonal combinations of the 2 s 1/2 and 1 d 5/2 configurations. According to the orthogonal rule, the wave functions for these two states are written as <math display="block" id="M38"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>|</mo><msubsup><mi>J</mi><mn>1</mn><mo>−</mo></msubsup><mo>〉</mo><mo>=</mo><msub><mi>α</mi><mi>J</mi></msub><mi>ν</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>s</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>β</mi><mi>J</mi></msub><mi>ν</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mi>d</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>|</mo><msubsup><mi>J</mi><mn>2</mn><mo>−</mo></msubsup><mo>〉</mo><mo>=</mo><mo>−</mo><msub><mi>β</mi><mi>J</mi></msub><mi>ν</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>s</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>α</mi><mi>J</mi></msub><mi>ν</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mi>d</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where J = 1 and 2, α J α J = SF( l =0) and β J × β J = SF( l =2) for the <math id="M39"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> (or <math id="M40"><mrow><msubsup><mn>1</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> ) state, and β J β J = SF( l =0) and α J × α J = SF( l =2) for the <math id="M41"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> (or <math id="M42"><mrow><msubsup><mn>1</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> ) state. Based on the results of experimental observation of the <math id="M43"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> and <math id="M44"><mrow><msubsup><mn>1</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> states, the s -wave ( d -wave) SF s are determined as 0.17(5)(4) (0.71(5)(20)) and ≤ 0.06 (0.94(20)(20)) for the unobserved <math id="M45"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> and <math id="M46"><mrow><msubsup><mn>1</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> states, respectively. This simple orthogonal method has been widely used in experiments, such as d ( 15 C, p ) [ 29 29 ],and d ( 11 Be, p ) [ 3 3 ]. The excitation energies ( E x ) and SF s for each of the populated states are listed in Tab. 1 Tab. 1 [ 30 30 ]. Using Eq. (16), the holes are determined to be 1.9 ± 0.2 and 5.9 ± 0.3 for the 2 s 1/2 and 1 d 5/2 orbitals, respectively. These two values are in close proximity to the 2.0 and 6.0 predicted by IPM. The ESPE for the 2 s 1/2 and 1 d 5/2 orbitals are calculated at approximately 0.5 ± 0.1 and 2.0 ± 0.4 MeV, respectively, using Eq.(17). These results demonstrate that the s -orbital is lower than the d -orbital in 14 B, which is different from the orbital arrangement described by the conventional shell model but is similar to other N = 9 isotones 13 Be and 15 C. 1 1 Spin-parity 1 1 E x (MeV) 1 1 SF ( l =0) 1 1 SF ( l =2) 1 1 <math id="M47"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> 1 1 0 1 1 0.71(5)(20) 1 1 0.17(5)(14) 1 1 <math id="M48"><mrow><msubsup><mn>1</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> 1 1 0.654 1 1 0.94(20)(20) 1 1 ≤ 0.06 1 1 <math id="M49"><mrow><msubsup><mn>3</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> 1 1 1.38 1 1   1 1 ≡ 1.0 1 1 <math id="M50"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> 1 1 1.86 1 1 [0.17(5)(4)] 1 1 [0.71(5)(20))] 1 1 <math id="M51"><mrow><msubsup><mn>4</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> 1 1 2.08 1 1   1 1 1.0 1 1 ( <math id="M52"><mrow><msubsup><mn>1</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> ) 1 1 4.5 1 1 ≤ 0.06 1 1 [0.94(20)(20)] 1 1 Eq.(16) 1 1   1 1 1.9 ± 0.2 1 1 5.9 ± 0.3 3 Experimental methods 1 3.1 Normal kinematics 2 One of the most effective ways to measure the ( d , p ), ( d , t )m, and ( d , 3 He) reactions when using a beam of deuteron and a stable target is with a high resolution magnetic spectrometer in order to record the recoil charged particles p , t , and 3 He that are emitted from the reactions. The high precision and low background noise are two typical advantages of normal kinematics. The typical energy resolution of the excited states in a final nucleus reaches approximately several tens of keV. For example, the Q -value spectrum for the 13 C( d , p ) 14 C reaction with a deuteron beam at 17.7 MeV in normal kinematics is shown in Fig. 5 Fig. 5 [ 34 34 ]. Three closely spaced excited states can clearly be identified at E x = 6.73, 6.90, and 7.34 MeV in 14 C, with an average resolution of approximately 60 keV [ 34 34 ]. 3.2 Inverse kinematics 2 It is challenging to use normal kinematics with radioactive beams that have relatively short half-lives and low separation energies, because it is nearly impossible to use them as targets. Inverse kinematics is usually applied in such cases wherein the deuteron or the proton is used as target and the radioactive beam is used as the projectile. The vector diagram for the reaction A( a , b )B in inverse kinematics is shown in Fig. 6 Fig. 6 . The center of mass (CM) vector v cm has the same direction as the projectile, with the length <math display="block" id="M53"><mrow><msub><mi>v</mi><mrow><mtext>cm</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>m</mi><mtext>a</mtext></msub></mrow><mrow><msub><mi>m</mi><mtext>a</mtext></msub><mo>+</mo><msub><mi>m</mi><mtext>A</mtext></msub></mrow></mfrac><mo>×</mo><msubsup><mi>v</mi><mrow><msub><mrow/><mi>a</mi></msub></mrow><mrow><mtext>lab</mtext></mrow></msubsup><mn>,</mn></mrow></math> where, m a ( <math id="M54"><mrow><msubsup><mi>v</mi><mrow><msub><mrow/><mi>a</mi></msub></mrow><mrow><mtext>lab</mtext></mrow></msubsup></mrow></math> ) and m A are the mass (velocity) of the projectile and target, respectively. The vector diagram in Fig. 6(a) Fig. 6(a) illustrates the case of (a) d (A, t )B, d (A, 3 He)B, or p (A, d )B. After the reaction takes place, the heavy particle is moving forward with little change in either velocity or direction. According to the conservation of momentum and energy, it is easy to obtain a rough estimate of the vector length of the light (heavy) particle in the CM frame, labelled <math id="M55"><mrow><msub><mi>v</mi><mrow/></msub><msub><mi>l</mi><mrow/></msub><mi>c</mi><mi>m</mi></mrow></math> ( <math id="M56"><mrow><msub><mi>v</mi><mrow/></msub><msub><mi>h</mi><mrow/></msub><mi>c</mi><mi>m</mi></mrow></math> ) in Fig. 6(a) Fig. 6(a) . For example, the mass of the outgoing light particle d is nearly two times that of the target p in the reaction p (A, d )B; however, the momentum that this particle carries in the CM frame is approximately equal to the whole of the CM momentum. The vector <math id="M57"><mrow><msub><mi>v</mi><mrow/></msub><msub><mi>l</mi><mrow/></msub><mi>c</mi><mi>m</mi></mrow></math> is therefore nearly half the length of v cm . The precise value depends upon the reaction Q -value; however, the basic form of the vector diagram is always the same as that seen in Fig. 6(a) Fig. 6(a) . As a result, the light reaction products are emitted in a forward direction, focusing onto a cone of angles at around 40° relative to the direction of the beam. There are two energy solutions for each angle followed by the outgoing light particles, referred to as the low-energy and high-energy branches, which are shown as the thick solid and thin dashed lines, respectively, in Fig. 6(a) Fig. 6(a) . The low-energy branch (solid curves) of these light particles corresponds to the high-energy branch of the heavy particles and to the smaller CM angles of heavy particles, and hence (typically) to the branch with higher DCSs [ 17 17 ]. However, it should be noted that the energy of the light particles in this branch is extremely low, which leads to numerous difficulties in accurate measurement. In the reaction d (A, p )B (or 3 He (A, d )B), the outgoing light reaction products are lighter than the target, which implies that their CM velocities <math id="M58"><mrow><msub><mi>v</mi><mrow/></msub><msub><mi>l</mi><mrow/></msub><mi>c</mi><mi>m</mi></mrow></math> are greater than v cm and the vector diagram is remarkably different from that in case (a) (see Fig. 6(b) Fig. 6(b) ). It is evident that the light particles can be emitted backwards, reaching a maximum angle of 180° in the lab frame. Similar to case (a), there are two branches. The energy of the light particles, corresponding to the branch with the higher cross section, is still extremely low. Therefore, there are significant challenges associated with detecting these particles. The kinematics of the different reaction channels induced when a radioactive beam of 14 B at 20 MeV/nucleon collides with a deuteron target is shown in Fig. 7 Fig. 7 . The energies of the outgoing light particles are given as a function of their angles in the lab frame, corresponding to the part with the high cross section. The dotted, dash-dotted, dashed, and solid curves describe the reactions of d ( 14 B, 3 He), d ( 14 B, t ), d ( 14 B, d ), and d ( 14 B, p ), respectively. The maximum energy of the light particles for each reaction channel is illustrated using the arrowed line, which corresponds to a CM angle of 20°. It is evident that the energies of 3 He, t , d , and p are less than 3.1, 2.0, 1.9, and 6.5 MeV/nucleon, respectively, which are particularly low. These particles have had to force their way through the target before being detected, and have lost energy during this process. Considering the relatively lower beam intensity and smaller DCSs of the transfer reaction (at one or two magnitudes of order lower than elastic scattering), it is therefore better to use a thick target in order to obtain sufficient statistics concerning the angular distribution. However, because of the energy loss of the light particles within the target and the measurement of light particles, a thin target should be used for the accurate detection of light particles in order to obtain a Q -value spectrum with better resolution. The thickness of the target should therefore be determined especially for each particular experiment in inverse kinematics. When plastic CH 2 or deuterated polyethylene CD 2 is used, a typical thickness between several hundreds of μ g/cm 2 and several mg/cm 2 is required. A more detailed account in this regard is given in [ 14 14 ]. It is worth noting that the angular scope of the light particles produced from different reaction channels is highly different. If the solid angles of the detectors are near 4 π , several reaction channels can be measured in one experiment. For a ( d , p ) reaction in inverse kinematics, the light particles are emitted backwards in the laboratory frame, with the highest cross sections. Protons are almost the only possible products at backward angles; hence, particle identification (PID) is relatively easy. In a ( d , t ) or a ( d , 3 He) reaction, the recoil of t or 3 He moves in a forward direction, where several particles from other reaction channels are found, which indicates that accurate PID is more important as well as difficult. In this case, the measurements of the coincidence between the light particles and the residual nuclei are often required. 3.3 Missing mass method 2 For the transfer reaction A( a , b )B in inverse kinematics, the whole CM angular range of the residual particles B can be covered in a small range in laboratory frame. A 4 π coverage of a reaction is therefore easily achieved by placing detectors that cover only a small range around the direction of the beam. However, because of the inadequate range covered by laboratory systems, the energy and angle resolution required for this kind of measurement are too high to be achieved in most cases. An alternative solution is to simply identify the residual b particles, without measuring the residual B nuclei in the forward angle. The energies and angles of the light b particles are therefore usually measured, and the excitation energy (or Q -value) spectrum of the heavy B particles is then reconstructed using the following function [ 15 15 ]. <math display="block" id="M59"><mrow><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>m</mi><mtext>a</mtext></msub></mrow><mrow><msub><mi>m</mi><mtext>B</mtext></msub></mrow></mfrac><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><msub><mi>E</mi><mtext>a</mtext></msub><mo>+</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>m</mi><mtext>b</mtext></msub></mrow><mrow><msub><mi>m</mi><mtext>B</mtext></msub></mrow></mfrac><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><msub><mi>E</mi><mi>b</mi></msub><mo>−</mo><mfrac><mrow><mn>2</mn><mo stretchy="false">(</mo><msub><mi>m</mi><mi>a</mi></msub><msub><mi>m</mi><mi>b</mi></msub><msub><mi>E</mi><mi>a</mi></msub><msub><mi>E</mi><mi>b</mi></msub><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mtext>cos</mtext><mi>θ</mi></mrow><mrow><msub><mi>m</mi><mtext>B</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where m a ( E a ), m b ( E b ), and m B are the mass (energy) of projectile a , outgoing light particle b , and heavy particle B, respectively. θ is the outgoing angle of b relative to the direction of the beam in the lab frame. Although the nucleus of the B under investigation is not measured, all bound and unbound states can be derived using Eq.(22). The technique of reconstructing the Q -value spectrum or the excitation energy spectrum ( E x )of one of two ejectiles without actual measurement is called the missing mass (MM) method. One of the few techniques that can be used for the spectroscopic study of unbound states [ 15 15 ], the MM technique is also the most commonly used to investigate the single-nucleon transfer reaction using a radioactive beam in inverse kinematics. The precise measurement of the light particles, including their energies and angles, is the most important aim when the MM method is adopted for use in experimentation. Typical Q -value spectra that have been reconstructed from the energies and angles of the recoil light particles using the MM method are shown in Fig. 8 Fig. 8 . The spectra were simulated using the GEANT4 package [ 35 35 ], taking into consideration the beam profile (double gaussian distributions in a circle with a radius of 10 mm), the dispersion of the radioactive beam (2%), the thickness of the target, the energy threshold (1 MeV), the energy (1%), the angular resolution of the light-particle detectors, and the energy loss, both in the target and the dead layer of the light-particle detectors. The simulation describes the reaction of d ( 14 B, p ) to the ground state, the first ( E x = 1.33 MeV), and the second ( E x = 2.73 MeV) excited states in 15 B with a radioactive beam of 14 B at 20 MeV/nucleon. The typical resolution is less than 1 MeV (FWHM) when the target thickness and angular resolution are 3.0 mg/cm 2 and 1.8°, respectively. The result is shown in the upper picture of Fig. 8 Fig. 8 , where three bound excited states can clearly be discriminated in 15 B. This means that if the energies and the angles of the recoil protons are measured, the resolution of the Q -value spectrum is adequate for the identification of different low-lying excited states in 15 B. If the target thickness is increased to 3.6 mg/cm 2 and the angular resolution is changed to 1.5°, the corresponding Q -value spectrum is as seen in the lower image of Fig. 8 Fig. 8 . Although the resolution is worse than that of the upper image, the three peaks can still be clearly identified. This means that if the target thickness is increased, the Q -value resolution worsens, which can be compensated for by improving the angular resolution. In more complicated cases where the excitation energies of several states in an unmeasured nucleus are closer to each other, it is hard to discriminate these states by merely using the MM method. Several methods are often applied to solve this problem, the simplest of which is to decrease the target thickness to several hundred or tens of μ g/cm 2 . However, new problems concerning statistics are generally created if this simple method is used because of the limitations to the intensity of the radioactive beam and the relatively small cross sections of the transfer reactions. Another method is to maintain (or increase) the target thickness while adding new high-resolution detectors such as γ -ray detectors. It is worth noting that the efficiency of γ -rays is usually extremely low, and the coincidence efficiency of the γ -rays and recoil light particles should therefore be carefully considered. The most advanced method used is the active-target time project chamber (AT-TPC) detector. These is discussed in detail in Sect. 4. 4 Typical experimental setup 1 As aforementioned, it is most important that the recoil light particles are precisely measured if a high-resolution excitation-energy (or Q -value) spectrum for the unmeasured nucleus B is to be obtained. Several different detection arrays have been constructed for this purpose. In this section, the detector setups that are frequently used in different radioactive beam facilities around world for single-nucleon transfer reactions in reverse kinematics are introduced. 4.1 Silicon detector arrays and γ detector arrays 2 The energy and angle of the emitting light particles are often measured using silicon detector arrays. The first layer of a detector array that is constructed using highly segmented silicon detectors is generally known as a Δ E detector, through which light particles can force their way while losing a specific amount of energy. The second or the third layer that is used to stop the light particles is called the E detector. Large area silicon detectors or CsI(Tl) crystals are usually used. Particle identification (PID) is achieved from the amount of energy lost in the Δ E and E detector, while positional or angular information is provided by the segmented silicon detector. The energy of the emitted particles is often too low to penetrate the Δ E detector. In this case, the time-of-flight (TOF) and Δ E method is applied to identify the recoil light particles, as in [ 3 3 ]. Several silicon detector arrays have been constructed over the past three decades, such as MUST2 [ 36 36 ] and TIARA [ 37 37 ] at GANIL. MUST2 is a telescope array that was designed for the detection of light charged particles that are produced via direct reactions using exotic ion beams, with an active area of 10 cm by 10 cm in each module. A typical module is composed of the first stage of a double sided silicon-strip detector with 128 strips on each side, followed by a second stage of 4.5 mm-thick Si (Li) segmented with 2 pads of 2 × 4, and finally a 3 cm-thick CsI segmented into 4 × 4 pads. The structure of one module is presented in Fig. 9 Fig. 9 . The MUST2 array has undergone significant extension over the past few years and now comprises 10 modules, providing a large angular coverage with an efficiency of approximately 70% at angles of up to 45° [ 36 36 ] by reasonable placement. The combination of the hundreds of μm-thick silicon detectors and the several cm-thick CsI allows measurement over a large energy range, while simultaneously measuring both time and position. The reconstruction of the TOF, momentum, total kinetic energy, and trajectory is therefore possible, and the construction of the MM spectrum is achievable using this array. Several experiments have been performed using the MUST2 array, exploring dozens of radioactive isotopes, such as 9 He [ 39 39 ], 10 He [ 40 40 ], 13 O [ 41 41 ], 21 O [ 42 42 ], and 61 Fe [ 43 43 ]. In order to cover a large rangeof angles, a well-organized, quasi-4 π position sensitive silicon array, TIARA, was developed at GANIL. The ultimate goal of TIARA is to perform direct nuclear reaction studies in inverse kinematics with radioactive ion beams. This array is comprised of a set of single-layer silicon detectors. The main part consists of an octagonal barrel that is formed from 8 resistive charge division detectors with a pair of large annular double-sided silicon strip detectors (SiHyBall) covering each end of the barrel ( Fig. 10 Fig. 10 ). Because the target is almost totally covered with silicon detectors, unique target changing mechanisms and electronics and data acquisition systems had to be developed. Detailed information can be obtained in [ 37 37 ]. Despite the good detection performances of the silicon detectors, it is still sometimes necessary to detect coincidental γ -rays for a better determination of energy levels or distinguishing long-lived isomeric states. For example, in a d ( 34 Si, p ) 35 Si experiment [ 44 44 ] performed at GANIL, the energies and angles of the recoil protons were measured with four modules of the MUST2 array, while a more accurate energy determination of the bound excitation levels populating the 35 Si was achieved by analyzing the γ -ray energy spectrum measured by four segmented high purity Germanium (HpGe)detectors in the EXOGAM array. An isomer-tagging technique was used for distinguishing the long-lived isomeric states by directly measuring the cross section of the <math id="M60"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> state of 12 Be that is populated by the reaction d ( 11 Be, p ) [ 3 3 ]; the delayed-correlation technique was also employed in the study of μs-isomers of 67 Ni [ 45 45 ]. The detection of γ -ray plays a key role in numerous single-nucleon transfer reactions [ 3 3 , 46 46 , 45 45 ]. A combined view of the silicon and high-purity Germanium detection array at GANIL is shown in Fig. 11 Fig. 11 . This setup was used for the d ( 16 C, p ) 17 C experiment [ 47 47 ]. Similar arrays have also been constructed at other facilities. Several ( d , p ) transfer reactions in inverse kinematics have been performed at REX-ISOLDE, CERN using a combination of silicon detectors and γ -ray detectors, for the purpose of studying the single-particle properties at the border of the island of inversion [ 48 48 ] or around traditional magic numbers. For example, in the 79 Zn [ 49 49 ] and 67 Ni [ 45 45 ] experiments, the setup (see Fig. 12 Fig. 12 ), which was composed of the T-REX array [ 50 50 ] and Miniball [ 51 51 ], allowed the combined detection of protons recoiling from the ( d , p ) reaction and γ -rays emitted from the residual nuclei. The high-resolution Miniball, which consists of 24 six-fold segmented HpGe crystals, has been used at REX-ISOLDE for over ten years. An overview of the technical details of the full Miniball setup is given in [ 52 52 ]. The silicon detector array T-REX was designed to be used in combination with Miniball, providing the positions (or angles) and the Δ E - E PID of light particles. The silicon detector arrays that were developed at HRIBF at Oak Ridge National Laboratory are named ORRUBA [ 53 53 ], and SuperORRUBA [ 54 54 ]. The latter is an upgrade of the former. These arrays are geometrically similar; however, the former uses position sensitive resistive silicon strip detectors, like those utilized in TIARA and T-REX, while double-sided silicon strip detectors are applied in the upgraded array, which have better energy (and positional) resolution because of the increase in the number of electronics channels. A schematic view of ORRUBA is shown in Fig. 13 Fig. 13 . Focusing on the evolution of nuclear structure away from the stability line and the astrophysical r -processes that occur during supernova, the capability of measuring the transfer reactions in inverse kinematics on unstable nuclei has been tested in several experiments [ 11 11 , 55 55 , 56 56 ]. Both of these arrays can be used in combination with GAMMASPHERE [ 57 57 ] or GRETINA [ 58 58 ] in order to obtain high-resolution excitation energy spectra. The High Resolution Array (HiRA)[ 59 59 ] was developed at NSCL at a similar time to the construction of ORRUBA. Similar to LASSA [ 60 60 ], the HiRA array uses a combination of silicon detectors and CsI(Tl) crystals for PID. Like MUST2, HiRA is not designed specifically for transfer reactions in inverse kinematics. However, the HiRA array is useful for measuring transfer reactions because of its energy, angular resolution, and large acceptance. For example, in the study of 33 Ar [ 61 61 ] and 55 Ni [ 62 62 ], the excitation energy spectrum was successfully reconstructed from the recoil light particles measured with the HiRA array by coincidentally detecting the residual with the S800 spectrometer. Fig. 14 Fig. 14 shows HiRA under use in a transfer reaction. Similar to TIARA, ORRUBA, and T-REX, SHARC [ 63 63 ] is a silicon detector array used at TRIUMF for the transfer reaction. SHARC can be used in conjunction with the γ -ray detector array TIGRESS [ 64 64 ], as seen in Fig. 15 Fig. 15 . With highly segmented silicon detectors, SHARC provides a better angular resolution than the other three arrays. The charged particle reaction spectroscopy station IRIS [ 65 65 ] was also developed at TRIUMF in order to perform transfer reactions and the inelastic scattering of rare isotopes for use in inverse kinematics. Besides the regular silicon detectors a nd CsI crystals, IRIS also provides a thin solid hydrogen/deuteron target that is formed by freezing the hydrogen/deuteron gas onto an Ag foil that has been cooled to 4 K. The detection of the aforementioned γ -rays was achieved with the use of HpGe detectors. HpGe detectors have good energy resolution but low intrinsic detection efficiency, which leads to low statistics in the coincidence measurement of the recoil particles and the γ -rays. Two methods that do not involve increasing the beam intensity are usually used to solve this problem. One is to increase the γ -ray detection efficiency by using other detectors with higher efficiency, such as NaI. This method was employed by RIKEN for the study of drip-line nuclei. Although RIKEN can produce radioactive beams with some of the highest beam intensities in the world, the production of adequate statistics remains problematic. For example, in a study of the near drip line nucleus 23 O with d ( 22 O, p ) reaction [ 67 67 ] and the d ( 22 O, dγ ) reaction [ 68 68 ], the total intensity of the secondary beam was only approximately 1500 counts per second (cps), with an average 22 O intensity of 600 cps. Adequate statistics are difficult to produce using the HpGe detector, so a γ -ray detection array DALI2 comprised of NaI crystals was employed. The experimental setup is shown in Fig. 16 Fig. 16 . The residuals were analyzed using RIPS, while the recoil light particles were measured using 156 CsI(Tl) scintillation crystals. DALI2 [ 69 69 ], with 20 % a full-energy photon peak efficiency for 1 MeV γ -rays, was placed around the target in order to detect the γ -rays from the excited states of 22 O that were produced via inelastic scattering [ 68 68 ]. The MUST2 array was also used at RIKEN in order to detect and identify recoil particles; for instance, the d ( 11 Li, 3 He) experiment presented in [ 40 40 ]. No detection arrays are currently available at the EN-course terminal at the Research Center of Nuclear Physics (RCNP) in Osaka university [ 70 70 ]. Therefore, it is an ideal place for users to build their own detection systems for different physical goals using silicon detectors and γ -ray detectors. Fig. 17 Fig. 17 shows the experimental setup inside the large scattering chamber for the d ( 11 Be, p ) 12 Be experiment which was carried out at 26.9 MeV/nucleon [ 3 3 ]. The telescopes TELE0 and TELE1, which are comprised of silicon detectors and CsI crystals, were used for the detection of residual nuclei and scattered light particles, respectively. The annular double-sided silicon strip detector (ADSSD) is responsible for the measurement of recoil protons produced in transfer reactions, and the PID is achieved using the TOF-Δ E method. The Scintillation Counters, composed of NaI and BgO scintillators, were used to discriminate the isomeric state of 12 Be from other bound excited states. A similar setup was also applied for the d ( 14 B, p ) 15 B experiment, which investigated the s -wave intruder components in the low-lying states of 15 B. This experimental setup was also employed for the d ( 16 C, 3 He) 15 B, and the d ( 15 C, 3 He) 14 B experiments, which were performed at the radioactive beam line at Lanzhou (RIBLL1) in 2018 and 2019, respectively. 4.2 Active target time projection chamber 2 With the improvement in the detection techniques used to measure recoil particles, one major contribution to the uncertainties in the excitation energy spectrum comes from the energy loss inside the target. For exotic nuclei that are far from the β stability line, the low intensity and low energy features of the secondary beam make it difficult to conduct transfer reaction experiments with the traditional plastic (CH 2 ) or deuterated polythene (CD 2 ) targets. The combination of a time projection chamber with a gaseous active target provides an alternative solution for studying nuclear systems that are weakly bound [ 71 71 ]. The MAYA detector [ 72 72 ], which is based on the concept of an active target, was developed at GANIL more than ten years ago, allowing the use of a relatively thick gaseous target without any loss in resolution by using the detection gas as the target material. The charged particles inside the detector ionize the filling gas along their trajectories and the released electrons drift toward the amplification area under a high electric field. The projection of the trajectory on one plane is obtained from the segmentation of the readout device, while that of the third dimension is derived from the measured drift time. The reconstruction of the three-dimensional trajectories becomes possible by the analysis of the pad signals and the drift time. The point of reaction can thus be derived. Fig. 18 Fig. 18 shows the ionization process and detection principle of MAYA. This kind of active target time projection chamber (AT-TPC) is also a powerful tool that can be used to study elastic and inelastic scattering [ 73 73 - 76 76 73-76 ], without worrying about the low energy scattered particles being stopped inside the target. A new-generation active target device, ACTAR TPC [ 77 77 , 78 78 ], is currently under construction at GANIL for the purpose of expanding the dynamical range, lowering the thresholds, and increasing the detection efficiency. Other facilities are also making progress in the development of state-of-the-art detectors. The larger AT-TPC that was constructed at NSCL [ 79 79 - 81 81 79-81 ] can be used to measure longer trajectories of recoil particles. The test experiment with a 46 Ar beam shows the potential of this detector for investigating the single-particle states of nuclei that are far from the stability line. Fig. 19 Fig. 19 shows the schematic view of the AT-TPC at NSCL. A compact AT-TPC, MAIKo, has recently been developed at RCNP, together with Kyoto University [ 82 82 ]. The elastic and inelastic scattering experiments of a radioactive beam of 10 C [ 83 83 ] on 4 He have been performed to test the performance of MAIKo [ 82 82 ]. This kind of advanced detector AT-TPC is also under development by the experimental group at Peking University [ 84 84 ], the Institute of Applied Physics in Shanghai [ 85 85 ], and the Institute of Modern physics in Lanzhou, China. 4.3 A new approach with a magnetic spectrometer 2 Unlike the traditional magnetic spectrometers that are used for measuring reactions in normal kinematics, a new type of helical orbit spectrometer, HELIOS [ 87 87 , 86 86 ], has been developed at the Argonne National Laboratory. HELIOS, which was designed specifically to measure recoil light particles with high energy and position resolutions, has been used in several transfer reaction experiments in inverse kinematics [ 29 29 , 88 88 - 90 90 88-90 ]. As the target and detectors are placed inside a uniform magnetic field, the charged particles emitted from the reaction travel on cyclotron orbits in the magnetic field, eventually reaching the beam axis again after one cyclotron period, where they are detected by silicon detectors, as seen in Fig. 20 Fig. 20 . By measuring the arrival time at the silicon array along the beam axis, which is independent of the energy and the scattering angle, the mass to charge ratio A/q can be obtained and the PID attained. For a fixed reaction Q -value, the energy of the emitting particle is proportional to the z position measured by the silicon arrays along the direction of the beam. Because of the lesser influence from the target thickness and the beam spot size, a better resolution can be achieved in terms of excitation energy, as discussed in [ 86 86 ]. In the d ( 12 B, p ) 13 B [ 90 90 ] experiment, the resolution of HELIOS was sufficient to separate two closely spaced excited states at E x = 3.48 MeV and 3.68 MeV in 13 B. Although this method encounters problems when measuring particles with the same A/q value, such as deuterons and α particles, it is adequate for most transfer reactions. Because of the introduction of the magnetic field, the acceptance of the spectrometer is limited by the radius of the solenoid and the homogeneity of the magnetic field strength. Combination with γ -ray detectors remains a challenge because the light particles travel inside the volume where the γ -ray detectors are supposed to be placed. 5 Experimental results of single-nucleon transfer reactions 1 In most light neutron-rich nuclei around N = 8, the ordering of the 2 s 1/2 , 1 p 1/2 , and 1 d 5/2 orbits is usually different from that of stable nuclei, resulting in the emergence of large amounts of s -wave intruder components and the formation of halos [ 13 13 , 91 91 ], such as those seen in 11 Be, 11 Li, and 15 C. The breakdown of the N = 8 shell closure that is due to the 2 s 1/2 orbital intruding into the p -shell is closely related to the appearance of the neutron halo in light neutron-rich isotopes. It is therefore important that the influence of the 2 s 1/2 orbital around N = 8 is studied systematically. Table 2 Table 2 summarizes the intruder s -wave intensities in the ground-state wave functions of loosely bound nuclei around N = 8 from various experiments, together with their single-neutron separation energies ( S n ). It is worth noting that the ground states of most nuclei with smaller S n ( S n 1.22 MeV), such as 11 Be and 15 C, have predominant s -wave components; however, nuclei that contain at least on extra neutron S n ( S n 3.17 MeV), 12 Be and 16 C have few such components. In addition to 11 Be and 15 C, 14 B is another nucleus with smaller S n and a dominant s -wave component [ 30 30 , 92 92 - 95 95 92-95 ], which indicates that the 2 s 1/2 orbit intrudes into the 1 d 5/2 orbital in the ground state of 14 B. It is therefore worth questioning how much of the s -wave component is seen in the ground state of its adjacent isotope 15 B. Intruding components have also been observed in other N = 10 isotones during several experiments, such as the 2 n removal reaction in 14 Be [ 96 96 ] and the 1 n transfer reaction in 16 C [ 29 29 ]. 1 1 Isotope 1 1 s -wave strength 1 1 S n (MeV) 1 1 11 Li 1 1 47 % [ 97 97 - 99 99 97-99 ] 1 1 0.396 1 1 11 Be 1 1 71 % [ 11 11 ] 1 1 0.504 1 1 12 Be 1 1 19 % [ 3 3 ] 1 1 3.17 1 1 14 Be 1 1 87 % [ 96 96 ] 1 1 1.78 1 1 13 B 1 1 little [ 90 90 , 100 100 ] 1 1 4.88 1 1 14 B 1 1 71 % to 89 % [ 30 30 , 92 92 - 95 95 92-95 ] 1 1 0.97 1 1 15 B 1 1 no data 1 1 2.78 1 1 15 C 1 1 88 % [ 101 101 ] 1 1 1.22 1 1 16 C 1 1 30 % [ 29 29 ] 1 1 4.25 Transfer reactions, especially the single-nucleon transfer reaction, can provide useful spectroscopic information for understanding the evolution of the nuclear shell structure by precisely detecting unusual components such as the intruder s -wave. Various single-nucleon transfer reaction experiments carried out using light exotic nuclei in inverse kinematics is reviewed in the next section, focusing on the spectroscopic study of neutron-rich He, Li, Be, B, and C isotopes (including bound and unbound nuclei) around N = 8. The results obtained from other kinds of reaction, such as breakup reactions, charge exchange reactions, and knock-out reactions are not presented in this paper. 5.1 Helium isotopes 2 8 He The "double-borromean" nucleus 8 He [ 102 102 ] is an interesting subject with the largest neutron-to-proton ratio among all the known particle-stable nuclei, exhibiting a neutron halo or thick neutron skin. 8 He is therefore an excellent candidate for testing different models describing nuclear structure [ 103 103 ]. Its neighbors, 7 He and 9 He, are particle unbound, thus it offers an opportunity to study the shell evolution of nuclear structures as a function of the increasing numbers of neutrons. The charge radius of 8 He is smaller than that of 6 He because of the isotropic distribution of the four valence neutrons [ 104 104 ], which is different from the traditional concept. With four loosely bound valence neutrons, 8 He is an unique system that can be used to investigate the role of correlations that occur between neutrons such as pairing [ 105 105 , 106 106 ]. The ground state of 8 He is still an ambiguous topic, attracting attention experimentally as well as theoretically. The cluster orbital shell model approximation (COSMA) assumes that 8 He is comprised of a 4 He core with four valence neutrons filling the 1 p 3/2 sub-shell [ 107 107 ]. This assumption is equal to that of pure jj coupling. The analysis of the p ( 8 He, t ) reaction at incident energies of 15.7 and 61.3 MeV/nucleon [ 103 103 ] consistently indicates that the ground state wave function of 8 He deviates from the pure (1 p 3/2 ) 4 structure. This result is in agreement with the theoretical calculation produced by Hagino’s group [ 108 108 ], which concludes that the probability of the (1 p 3/2 ) 4 and (1 p 3/2 ) 2 (1 p 1/2 ) 2 configurations in the ground state wave function of the 8 He nucleus are 34.9% and 23.7%, respectively. The asymmetry molecule dynamics (AMD) calculation also suggests that the ground state of 8 He has both the jj coupling feature ( 4 He + 4 n ) and the LS coupling feature ( 4 He + 2 n + 2 n ) [ 109 109 ]. The " t + t + valence neutrons" structure [ 110 110 ] and "di-neutron" structure [ 111 111 ] are also predicted by the AMD model for the ground state of neutron-rich He isotopes. The contributions from the first 2 + excited state of 6 He and the cluster of 5 H to the ground state configuration of 8 He have also been investigated using a p ( 8 He, t ) transfer reaction with an incoming energy of 25 and 61.3 MeV/nucleon [ 112 112 , 113 113 ]. Early in 2001, a heavy hydrogen 5 H was observed via the p ( 6 He, 2 p ) 5 H reaction at 32 MeV/nucleon. By using triple 2 p - t coincidences, the 5 H ground state was identified as a resonance state at 1.7 ± 0.3 MeV above the n + n + t threshold, with a width of 1.9 ± 0.4 MeV [ 114 114 ]. The transfer reaction is a sensitive and powerful probe that can be used to determine the properties of atomic nuclei [ 115 115 ]. The neutron SF extracted from the p ( 8 He, d ) 7 He reaction includes the structural information of 8 He and 7 He. The ground state of 7 He is a resonant state which often immediately decays into 6 He + n , while the excited state of 7 He at E x = 2.9 ± 0.3 MeV usually decays into 3 n + 4 He [ 116 116 ]. Deuterons that are coincidental with 6 He and 4 He in the reaction channel p ( 8 He, d ) 7 He g.s. and p ( 8 He, d ) 7 He <math id="M61"><mrow><msub><mrow/><mrow><msub><mi>E</mi><mi>x</mi></msub><mo>=</mo><mn>2.9</mn><mi>�</mi><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mo>;</mo><mi>M</mi><mi>e</mi><mi>V</mi></mrow></msub></mrow></math> , respectively, are therefore often measured. The ratio of the 2 n to 1 n transfer cross section for 8 He is expected to be sensitive to the correlations that occur among the valence neutrons [ 108 108 , 117 117 ]. Calculations carried out using the translational invariant shell model (TISM) suggest that the SF s ratio of p ( 8 He, t ) 6 He g.s. to p ( 8 He, t ) 6 He 2+ depends strongly on the assumed structure of the 8 He ground state [ 112 112 , 113 113 ]. It is worth mentioning that this ratio is quite close to 1.0 under the pure jj coupling assumption but deviates from 1.0 for mixed jj and LS coupling. The 6 He ground state with a half-time of 806.70 ms can be directly measured; however, its 2 + excited state immediately decays to 4 He + 2 n . Experimentally, the coincidence of t + 4 He is usually observed in the p ( 8 He, t ) 6 He 2+ reaction. The DCSs of the reactions p ( 8 He, d ) 7 He g.s and p ( 8 He, d ) 6 He 2+ were measured at the relatively high incident energy of 82.3 MeV/nucleon in an experiment performed at RIKEN by the nuclear physics experimental group associated with Peking University. The results of p ( 8 He, d ) 7 He g.s is shown in Fig. 21 Fig. 21 . These results were analyzed using FR-ADWA [ 11 11 ] with FRESCO code [ 18 18 ]. Although little difference was found between the SF s extracted because of the choice of different OP parameters, the values are all significantly smaller than 4.0. This may indicate that the ground state wave function of 8 He is not a pure jj coupling or (1 p 3/2 ) 4 configuration, and there is a possibility that other configurations such as (1 p 3/2 ) 2 (1 p 1/2 ) 2 may occur. The p ( 8 He, t ) 6 He 2+ channel is prior to the p ( 8 He, t ) 6 He g.s. channel at 82.3 MeV/nucleon, which is also observed at incident energies of 25 [ 112 112 ] and 61.3 [ 113 113 ] MeV/nucleon. Analysis of all the existing ( p , t ) reaction data with large angular ranges in the CM frame consistently indicates that the 5 H cluster transfer is necessary for reproducing the angular distributions at angles larger than 90°. Thus, the structure of the cluster 8 He = 5 H + 3 H should not be neglected when investigating the ground state of 8 He. All the ratios of p ( 8 He, t ) 6 He g.s. to p ( 8 He, t ) 6 He 2+ that are extracted from the existing ( p , t ) reaction data deviate from 1.0, which is inconsistent with the hypothesis of four valences filling a closed 1 p 3/2 sub-shell. 9 He As aforementioned, besides the model where four valence neutrons fill the 1 p 3/2 shell in the ground state of 8 He, other configurations such as (1 p 3/2 ) 2 ( p 1/2 ) 2 may occur. For example, the ground state of 9 He, with one more valence neutron than 8 He is possible. Some theoretical calculations predict that, with the same neutron number, the ground state of 9 He is particle unbound and may show the same parity inversion observed in the neighboring 11 Be and 10 Li (see the subsection below). However, in a no core shell model calculation [ 119 119 ], it is found that the 9 He ground-state resonance has a negative parity and thus breaks the parity-inversion mechanism observed in the 11 Be and 10 Li nuclei. The unbound nuclear system 9 He was investigated through the d ( 8 He, p ) transfer reaction by M. S. Golovkov et al using a 8 He beam [ 120 120 ] at a laboratory energy of 25 MeV/nucleon. The lowest resonant state of 9 He was found at 2.0 MeV with a width of approximately 2.0 MeV, which came mainly from the experimental energy resolution of approximately 0.8 MeV. The angular correlation pattern observed can be uniquely explained by the interference of the (1/2) - resonance with a virtual (1/2) + state and with a (5/2) + resonance at an energy of 4.2 MeV. However, the experimental energy resolution of approximately 800 keV prevents this conclusion because of the strong disagreement with a previous two-proton knockout reaction experiment [ 121 121 ], which suggested that a ground-state of 9 He has J π =1/2 + . The same reaction was also performed at GANIL with the 8 He beam [ 39 39 ] at a laboratory energy of 15.4 MeV/nucleon. The MM spectrum was deduced from the kinetic energies and the emission angles of protons detected by four MUST2 telescopes. This MM spectrum displays a structure that lies just above the threshold for neutron emission, which is identified as the ground state of 9 He. Despite the limited statistics describing the angular distribution of the state that was observed close to the 8 He + n threshold (180 ± 85 keV above), the result supports the conclusion that the ground state spin-parity of 9 He is (1/2) + . This result confirms both the parity inversion and the s -wave intrusion. The first excited state lies approximately 1.3 MeV above the neutron threshold and is compatible with J π =(1/2) - , while simultaneously exhibiting a strongly mixed nature. 5.2 Lithium isotopes 2 10 Li The study of the unbound system 10 Li is of great interest as knowledge of this system is necessary for a theoretical description of the Borromean nucleus 11 Li [ 122 122 ]. The existence of a low-lying virtual intruder s -state has also been predicted by some theoretical models. However, because of the unbound nature of 10 Li, the experimental study of this nuclei encounters numerous challenges, such as the coincidence detection of the neutron and 9 Li. Two different d ( 9 Li, p ) 10 Li transfer reactions in inverse kinematics were performed separately at REX-ISOLDE [ 123 123 ] and NSCL [ 124 124 ]. The results of the former experiment [ 123 123 ] supported the existence of a low-lying ( s ) virtual state, with a (negative) scattering length and a p 1/2 resonance with an energy of E r 0.38 MeV. However, the results of the latter experiment [ 124 124 ] were inconclusive as to the possible presence of a low-lying virtual state in 10 Li because of the poor statistics produced. The low amount of statistics and the poor energy resolution have always prevented definitive conclusions concerning the structures of 10 Li. The results from these two experiments were therefore re-examined [ 125 125 ]. From this analysis, it was concluded that both measurements can be described consistently using the same model for the n - 9 Li interaction and the seemingly different features can be understood if the different incident energy and angular range covered by the two experiments is taken into consideration. The s -wave virtual state in 10 Li was found to play a key role in the experiment using the lower beam energy; however, the excitation energy below E x < 1 MeV was found to be dominated by the p 1/2 resonance in the higher energy experiment. Most recently, a new d ( 9 Li, p ) 10 Li transfer reaction in inverse kinematics was performed at TRIUMF with a much higher beam intensity (10 6 pps), which guaranteed the production of more statistics [ 126 126 ]. The 10 Li energy spectrum, which was analyzed using three Fano functions, is shown in Fig. 22 Fig. 22 . The comparison between the experimental data and the theoretical predictions, including the pairing correlation, clearly indicates the presence of a p 1/2 resonance (purple dashed line in Fig. 22 Fig. 22 ) at the excitation energy of 0.45 ± 0.03 MeV. However, there is no obvious evidence for a significant s -wave contribution (blue dotted line) close to the threshold energy. Two high-lying resonances at 1.5 and 2.9 MeV were also observed. The corresponding angular distributions indicate a significant s 1/2 partial wave contribution (blue dotted line) for the 1.5 MeV resonance and a mixing of configurations at E x =2.9 MeV for the first time, with the d 5/2 partial-wave (green dot-dashed line) contributing significantly to the cross section. Several other kinds of experiments have been carried out. Although the excitation energy, parity, and spin assignments are controversial, it can be concluded that the ground state of 10 Li contains a valence neutron in a 2 s 1/2 state at approximately 50 keV or below [ 13 13 ]. This conclusion is different from that obtained from the aforementioned transfer reactions, which might be attributed to the difficulty of a coincidence meas urement between 9 Li and a proton for such a low-energy resonance using the transfer reaction. 11 Li In order to study the configurations of the ground state of 11 Li, a p ( 11 Li, d ) 10 Li reaction was performed at TRIUMF using a solid hydrogen target and a 5.7 MeV/nucleon 11 Li beam [ 127 127 ]. Only one resonance was observed, at E r = 0.62 ± 0.04 MeVwith a width of 0.33 ± 0.07 MeV. The elastic scattering was measured along with the transfer reaction in order to obtain the OP parameters for the entrance channel. The angular distribution of p ( 11 Li, d ) 10 Li to E r = 0.62 MeV is best reproduced by assuming that the neutron is removed from the 1 p 1/2 orbital, as shown by the red solid curve in Fig. 23 Fig. 23 . A SF of 0.67 ± 0.12 is determined for the (1 p 1/2 ) 2 component. This result is much less than that predicted by the conventional shell model ( ≃ 2.0), confirming a relatively small ( p 1/2 ) 2 component in the ground state of 11 Li. Assuming the remaining probability fraction to be s - and d -waves, a large (2 s 1/2 ) 2 probability fraction 44% is deduced for the ground state of 11 Li. This data is also theoretically analyzed in [ 128 128 ], in which the similar p 1/2 -wave component of 31 % is given. Recently, a d ( 11 Li, 3 He) reaction was performed at RIKEN in inverse kinematics with a radioactive beam of 11 Li at 50 MeV/nucleon [ 40 40 ]. This reaction was used to study the configurations of the ground state of 11 Li.Two peaks were observed at 1.4(3) and 6.3(7) MeV when the MM spectrum was gated onto the 8 He residuals. The angular distributions of the 3 He nuclei in coincidence with 8 He residues shown in Fig. 24 Fig. 24 were analyzed using the DWBA calculations. The OPs for the entrance channel of 11 Li + d were obtained from the elastic scattering data that were collected from the same experiment. Use of the DWBA might not be appropriate for dealing with the unbound 10 He. It is therefore argued that the removal of a deeply bound proton in the reaction justifies the simplification of the reaction model. The 11 Li| 10 He overlap function used in the DWBA calculations was calculated from the standard potential model (SPM), source term approach (STA), and the STA, corrected with the geometrical mismatch factor (GMF). The shape of the angular distribution is reproduced sufficiently; however, the magnitude of the cross section for the ground state of 10 He is overestimated in all three cases. The cross sections for the excited states are greater than those predicted by the DWBA calculations with the assumed SF s from the shell model calculations for the <math id="M63"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M64"><mrow><msubsup><mn>1</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> , <math id="M65"><mrow><msubsup><mn>0</mn><mn>0</mn><mo>+</mo></msubsup></mrow></math> states in 10 He. The experimental results may be attributed to the important contribution of a 10 He core excitation in the ground-state wave function of 11 Li. The ground state of 11 Li was also studied using a 2 n transfer reaction, p ( 11 Li, t ) 9 Li [ 99 99 ], with a radioactive beam of 11 Li at an extremely low incidental energy of 3 MeV/nucleon. The experiment was performed [ 99 99 ] at TRIUMF using the active target detector MAYA. Multi-step transfer calculations were applied with different wave functions of 11 Li. Only the wave functions with a strong mixture of p and s neutrons and three-body correlations were found to provide the best fit to the magnitude of the DCSs. 5.3 Beryllium isotopes 2 11 Be The beta decay experiment of 11 Be [ 129 129 ] indicates that the spin-parity of the ground state of 11 Be is 1/2 + [ 130 130 ] instead of 1/2 - , which means that the last neutron prefers to take the 2 s 1/2 orbital instead of the 1 p 1/2 orbital. The disappearance of the N = 8 shell closure has been proposed as an explanation for this phenomenon in neutron-rich light nuclei. A spectroscopic study of a d ( 10 Be, p ) reaction that was carried out at Oak Ridge National Laboratory with a radioactive beam of 10 Be at four different incident energies indicated that the average SF for a neutron in an nlj = 2 s 1/2 state coupled to the ground state of 10 Be is 0.71 ± 0.05 [ 11 11 ]. This large SF indicates a significant s -wave strength in the ground-state of 11 Be. In addition to the naive <math id="M66"><mrow><mi>n</mi><msup><mo>⊗</mo><mrow><mtext>10</mtext></mrow></msup><mi>B</mi><mi>e</mi><msubsup><mrow><mo stretchy="false">(</mo><mn>0</mn></mrow><mrow><mi>g</mi><mi>s</mi></mrow><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> component, the ground state wave function of 11 Be was found to have a considerable overlap with that of a valence neutron coupled to an excited 10 Be (2 + ) core via a ( p , d ) transfer reaction with a 11 Be beam [ 101 101 , 131 131 , 132 132 ]. In the former configuration, the valence neutron surrounding the inert core 10 Be populates the intruder s -orbital rather than the normal p -orbital. However, in the latter, the 10 Be is excited to the 2 + state at 3.37 and the valence neutron fills the intruder d -orbital. The s - and d -wave components in the ground state of 11 Be were calculated as 84 % and 16 %, respectively, from the ( p , d ) transfer reaction with the beam energy at 35.3 MeV/nucleon [ 101 101 , 131 131 , 132 132 ]. The results were confirmed by the same reaction with a radioactive 11 Be beam at 26.9 MeV/nucleon [ 133 133 ]. 12 Be With one more neutron, the valence-nucleon configuration of 12 Be has also attracted much attention. Two valence neutrons may populate the d -orbital as well as the s - and p -orbitals, which is different from the situation seen in 11 Li. However, which component is dominant? This question has been investigated through several d ( 11 Be, p ) reactions. In the d ( 11 Be, p ) reaction performed at TRIUMF [ 27 27 ], the s -wave neutron fraction of the first two 0 + levels in 12 Be was investigated for the first time. In this experiment, three peaks were observed in the excitation energy spectrum, including the ground state and a 1 - state located at 2.71 MeV, as well as an unresolved doublet that was constructed from the <math id="M67"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> and 2 + states. Angular distributions for each peak in the excitation energy spectrum were analyzed using the DWBA method using global optical model potentials as well as parameters from the neighboring nuclei. The ground state s -wave SF was determined to be <math id="M68"><mrow><msubsup><mrow><mn>0.28</mn></mrow><mrow><mo>−</mo><mn>0.07</mn></mrow><mrow><mo>+</mo><mn>0.03</mn></mrow></msubsup></mrow></math> while that of the long-lived <math id="M69"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> excited state was <math id="M70"><mrow><msubsup><mrow><mn>0.73</mn></mrow><mrow><mo>−</mo><mn>0.40</mn></mrow><mrow><mo>+</mo><mn>0.27</mn></mrow></msubsup></mrow></math> . The value for the <math id="M71"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> state has a large uncertainty, because this state was not clearly distinguished from the 2 + state. The result has been questioned by H. T. Fortune et al [ 134 134 ], as it is inconsistent with the results from knock-out experiments and theoretical calculations. The same reaction was studied with a lower beam energy (2.8 MeV/nucleon) at the REX-ISOLDE facility [ 135 135 ]. Besides the outgoing proton measured by the T-REX silicon detector array, the γ -rays emitted from the excited states of 12 Be were also detected using the MINIBALL germanium array. The γ -ray detection enabled a clear identification of the four known bound states in 12 Be, and the angular distribution of each state has been studied individually. In this case the SF for the ground state is also smaller than that of the excited <math id="M72"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> state. This experiment suffered from an extremely low beam energy, leading to ineffective detection in the most sensitive angular range, especially for the <math id="M73"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> state. It would be difficult to compare the SF results of the two aforementioned transfer reactions with other results or with each other [ 16 16 ] because of the lack of proper normalization procedures. With the purpose of further studying the intruder configuration in 12 Be, a new measurement of the d ( 11 Be, p ) reaction was performed at the EN-course beam line, RCNP, with special measures taken to deal with the questioned experimental uncertainties of the two previous experiments [ 3 3 ]. Fig. 17 Fig. 17 is a schematic view of the experimental setup. A special isomer-tagging method was used to discriminate the <math id="M74"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> state from the broad excitation-energy peak. Fig. 25 (a) Fig. 25 (a) shows the measured proton energies versus the laboratory angles, gated onto the 12 Be in the zero-degree silicon detector. Fig. 25 (b) Fig. 25 (b) gives the excitation energy spectrum of 12 Be, deduced from the energies and angles of the recoil protons. The elastic scattering of the 11 Be + p was measured simultaneously in order to estimate the hydrogen contamination in the (CD 2 ) n target and to obtain a reliable OP to be used in the analysis of the transfer reaction. The FR-ADWA calculations were employed to extract the SF s of the low-lying states in 12 Be. Fig. 26 Fig. 26 compares the experimental and theoretical cross sections. The extracted s -wave SF s are <math id="M75"><mrow><msubsup><mrow><mn>0.20</mn></mrow><mrow><mo>−</mo><mn>0.04</mn></mrow><mrow><mo>+</mo><mn>0.03</mn></mrow></msubsup></mrow></math> and <math id="M76"><mrow><msubsup><mrow><mn>0.41</mn></mrow><mrow><mo>−</mo><mn>0.11</mn></mrow><mrow><mo>+</mo><mn>0.11</mn></mrow></msubsup></mrow></math> for the <math id="M77"><mrow><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> and <math id="M78"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> states, respectively. The ratio between the SF s of the first two low-lying 0 + states, together with the previously reported results for the p -wave components, were used to deduce the single-particle intensities in these two bound 0 + states of 12 Be, which can be compared directly with the shell model predictions. The s -, d -, p -wave intensity for the <math id="M79"><mrow><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> and <math id="M80"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> states are calculated as 0.19 ± 0.07, 0.57 ± 0.07, 0.24 ± 0.05, and 0.39 ± 0.02, 0.02 ± 0.02, 0.59 ± 0.05, respectively. The error bars are deduced from the statistic uncertainties of the SF s. The results show a small s -wave (0.19 ± 0.07) with a dominant d -wave(0.57 ± 0.07) intruding into the ground state of 12 Be, which is dramatically different from the ground state of the 11 Be when dominated by an intruder s -wave. The experimental results are compatible with those obtained from previous measurements of the transfer reaction, considering the reported uncertainties. In principle, there should be three 0 + states observed within the 12 Be in this p-sd model space. Until now, only the lowest two have been found in the bound region, and the third 0 + has not been identified experimentally. More studies, such as 13 B( d , 3 He) 12 Be or 14 B( d , 4 He) 12 Be, are encouraged to further investigate the excited states of 12 Be. 5.4 Boron isotopes 2 For the N = 8 system, as aforementioned, the halo nucleus 11 Li has a large s -wave component in its ground-state wave function. In contrast, a larger s -wave component was observed in the <math id="M83"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> isomeric state rather than the <math id="M84"><mrow><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> ground state in 12 Be. However, the shell structure of the 13 B nucleus has not been discussed. Its ground state has J π = 3/2 - , and the properties of its low-lying positive parity states can reveal significant amounts of information about the 2 s 1/2 and 1 d 5 /2 single-particle energies and the residual interaction. For the N = 9 isotones, 12 Li and 13 Be are particle unbound; however, 14 B with S n = 0.97 MeV is loosely bound. Therefore, the lightest particle-bound N = 9 isotone, 14 B, provides a unique opportunity to study the evolution of the properties of single-neutron states, especially the gap between the 2 s 1/2 and 1 d 5/2 orbitals. The N = 10 isotones are ideal for studying two-neutron interactions in the sd shell. From the knock-out experiment, it was found that the ground state of 14 Be is dominated by a s -wave (87%) [ 96 96 ]. From the 15 C( d , p ) transfer reaction, the <math id="M85"><mrow><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> ground state of 16 C was found to be dominated by a d -wave [ 29 29 ]. Since 15 B is between 14 Be and 16 C, the questions arise whether the percentage of the s -wave component in the ground state wave function of 13,14,15 B could be ascertained, and whether s -waves dominate in their low-lying states. Several transfer reaction experiments have been carried out to answer these questions. 12,13 B The experiment investigating the d ( 11,12 B, p ) transfer reaction to the low-lying states of 12,13 B was performed at Argonne using a silicon detector [ 100 100 ]. The excitation energy spectra for 12 B (upper) and 13 B (lower), which were constructed using the recoil protons in coincidence with 12 B and 13 B, are shown in Fig. 27 Fig. 27 . The overall Q -value resolution is approximately 250 keV (FWHM), which is not sufficient for the discrimination of the doublet state at E x =3.482 and 3.681 MeV in 13 B. This doublet is expected to contain both s - and d -wave components. A new experiment concerning the d ( 11,12 B, p ) transfer reaction to the low-lying states of 12,13 B was performed at Argonne with the HELIOS setup [ 90 90 ], in which the properties of some of the low-lying states in the neutron-rich N = 8 nucleus 13 B were studied [ 90 90 ]. Two closely spaced states at E x =3.482 and 3.681 MeV were observed in 13 B that could be clearly identified from each other, benefiting from the better energy resolution of HELIOS. The angular distributions are shown in Fig. 28 Fig. 28 . The state at 3.48 MeV shows l = 0 angular distribution and the spin-parity is assigned as (1/2) + . The angular distribution of the state located at a 3.68 MeV excitation energy exhibits a l = 2 shape. J π = (5/2) + can be tentatively assigned, based on comparison with the shell model calculations. The absolute normalization of the cross sections was not possible because the absolute beam intensity was not measured in this experiment. Only the relative SF s are meaningful. As shown in Fig. 29 Fig. 29 , the relative SF of the suggested (5/2) + state, when compared with that of the (1/2) + state, is smaller than the value predicted by the shell model. More d -wave strength is expected at excited states above 4 MeV in 13 B. The absolute excitation energies, as well as the ordering of the excited states, are also not in good agreement with the shell-model predictions as can be seen in Fig. 29 Fig. 29 . The (3/2) + state that was predicted is not observed below 4.8 MeV, which may correspond to a strong transition at 5.1 MeV that was observed in a prior measurement (the lower picture in Fig. 27 Fig. 27 ) [ 100 100 ]. More positive parity states are expected to exist at energies above that of neutron separation; therefore, further experimental studies are required in this area. A state at E x = 4.83 MeV is strongly populated in the 4 He( 12 Be, 13 B γ ) reaction with a secondary beam of 12 Be at 50 MeV/nucleon [ 137 137 ]. The spin and parity of this state were assigned to 1/2 + by comparing the DCSs data with the DWBA calculations, see Fig. 30 Fig. 30 . The DWBA predictions with Δ l = 0, 1, and 2 are shown as the red solid, green dashed, and blue dotted curves in Fig. 30 Fig. 30 . The angular distributions, especially the peak observed at the forward angles, are well described by the Δ l = 0 DWBA calculations. This is therefore interpreted as a proton intruder state. Another 2 p -2 h state at E x = 3.53 MeV in 13 B, with a tentatively assigned spin-parity of 3/2 - , is also suggested to be a neutron intruder state [ 138 138 ]. 14 B The experimental data of the d ( 13 B, p ) 14 B reaction using HELIOS confirm that the ground and first-excited states are predominantly populated by s -waves and are single-neutron halo states [ 30 30 ]. The normalized SF s for the low-lying states are shown in Tab. 1, suggesting that the 2 - ground state has a moderate 2 s 1/2 - 1 d 5/2 configuration mixing and the 1 - first excited state is nearly pure 2 s 1/2 . The effective single-particle energies of the 2 s 1/2 and 1 d 5/2 neutron orbitals in 14 B show the inversion of these two orbitals compared to the normal ordering in the valley of stability [ 30 30 ]. Fig. 31 Fig. 31 shows the excitation-energy spectrum of 14 B obtained in this experiment. The <math id="M86"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> and <math id="M87"><mrow><msubsup><mn>1</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> states, which are expected to be comprised of one 1 p 3/2 valence proton (hole) coupled with one 1 d 5/2 valence neutron, were not observed. However, unbound <math id="M88"><mrow><msubsup><mn>3</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> and <math id="M89"><mrow><msubsup><mn>4</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> states were measured that were composed of the same coupling of π (1 p 3/2 ) -1 ν (1 d 5/2 ) 1 . ( d , 3 He) experiments were also carried out with HELIOS, using 14 C and 15 C secondary beams to provide spectroscopic information about the final states in 13 B and 14 B [ 95 95 ]. Several transitions were observed in the 14 C data, which were reproduced sufficiently using the transferred angular momentum of l = 1. This result indicates that the ground state of 14 C likely possesses a 1 p 1/2 proton character. A weak l = 0 transition was also measured, which may be associated with a possible 1/2 + proton intruder state in 13 B [ 137 137 ]. It is worth noting that such an excitation is not well described by shell-model calculations. The ground state ( <math id="M90"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> ) and the first excited state ( <math id="M91"><mrow><msubsup><mn>1</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> ) at E x = 0.654 MeV in 14 B could not be discriminated from each other in the 15 C data, because of the inferior energy resolution for 3 He produced by HELIOS. In addition to the bound states, a broad unbound excited <math id="M92"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> state was observed clearly by the d ( 15 C, 3 He) reaction at approximately E x = 1.8 MeV in 14 B, which was not seen but was expected in the previous d ( 13 B, p ) 14 B experiment [ 30 30 ]. Fig. 32 Fig. 32 shows the angular distributions of 15 C( d , 3 He) to the bound <math id="M93"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>–</mo></msubsup><msubsup><mn>1</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> doublet and the unbound <math id="M94"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>−</mo></msubsup></mrow></math> state in 14 B. All the DCSs are reasonably well described by a l = 1 proton removal from 15 C. The relative SF s extracted from this experiment are consistent with both the shell-model calculations and expectations based on prior measurements of the d ( 13 B, p ) 14 B reaction. 15 B Previous experiments have shown a large s -wave component in the ground state of 14 B. With an extra neutron, this poses an interesting question about how much s -wave component exists in the ground state and low-lying excited states of its adjacent isotope 15 B; however, there is no experimental data available that can be used to address this question. Recently, ( d , 3 He) and ( d , p ) transfer reactions in inverse kinematics were performed in order to study the spectroscopic information of 15 B. In principle, there should be two J π =(3/2) - states, including the ground state, in 15 B because of the mixture of s - and d -wave components. However, the second (3/2) - has not yet been found experimentally. The s -wave SF s of the known states, including the ground state of 15 B are unknown. In theory, the shell model predicted a small s -wave SF of 0.48 for the ground state [ 13 13 ],which was significantly smaller than the expected value of 2.0, which implies that there is a larger s -wave SF present in the excited (3/2) - state, as seen in the situation surrounding 12 Be and 16 C. A d ( 16 C, 3 He) 15 B experiment was recently conducted at RIBLL1 in Lanzhou, China. The preliminary result of the light particle PID is shown in Fig. 33 Fig. 33 . It was found that some 3 He particles clearly recoil from the transfer reaction under coincidence with 15 B. Further analysis of the data collected from this experiment is in progress. Shortly after the ( d , 3 He) experiment, a d ( 14 B, p ) 15 B experiment was performed at RCNP, which used a similar setup to that shown in Fig. 17 Fig. 17 . The elastic scattering of 14 B on the proton and deuteron targets were measured using data from the same experiment in order to extract OPs for the theoretical calculations of the transfer reaction. The energy versus angle for protons and deuterons emitted from the elastic scattering channels are shown in Fig. 34 Fig. 34 and Fig. 35 Fig. 35 , respectively. The protons from the transfer reaction were measured with a single-layer annular silicon detector at the backward angle, and the PID was achieved using the TOF-Δ E method. As shown in Fig. 36 Fig. 36 , the proton from the transfer reaction is clearly distinguished from other particles emerging from the fusion-evaporation reaction between the 14 B and the silicon detectors in the zero-degree telescope. 5.5 Carbon isotopes 2 The intruder components have been studied for the neutron-rich Li, Be, and B isotopes via transfer reactions. The experimental studies on carbon isotopes are discussed next. With 8 neutrons filling the p shell and the high excitation energy of (6.09 MeV) for the first excited (2 + ) state, 14 C shows a good magic character for N = 8. However, the spin-parity of the 15 C ground state is (1/2) + , which clearly indicates that the 2 s 1/2 orbital is below the 1 d 5/2 orbital. Studies of the 14 C( d , p ) 15 C reaction [ 139 139 ] show that the ground state of 15 C is well described as a valence neutron in a single-particle state around a 14 C core, with an s -wave SF as large as 0.88. It is also indicated that the ground state of 15 C has only a small fraction of the 14 C(2 + )× d configuration [ 140 140 ]. This component is similar to the core excitation component in the ground state of 11 Be. However, the study of the d ( 15 C, p ) 16 C reaction [ 29 29 ] shows a different picture, suggesting that 16 C may not need to be described with exotic phenomena. The d ( 15 C, p ) 16 C reaction was carried out at Argonne National Laboratory. A highly thin deuterated polyethylene ((C 2 D 4 ) n ) was bombarded by the 15 C beam with a high beam intensity of 1 ∼ 2 × 10 6 pps. The recoil protons were detected with the HELIOS spectrometer. The <math id="M97"><mrow><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> , <math id="M98"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> , and <math id="M99"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> states and a <math id="M100"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>3</mn></mrow><mn>1</mn><mo>−</mo></msubsup></mrow></math> doublet in 16 C were observed in the excitation-energy spectrum reconstructed with the recoil protons. Although a resolution of approximately 140 keV (FWHM) was used, it was still insufficient for resolving the closely spaced <math id="M101"><mrow><msubsup><mn>2</mn><mn>2</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>3</mn></mrow><mn>1</mn><mo>−</mo></msubsup></mrow></math> doublet near E x = 4 MeV. The angular distributions for these four populated states are shown in Fig. 37 Fig. 37 , compared with the DWBA calculations using four sets of optical-model parameters. Absolute SF s were obtained by comparing the experimental cross sections with the DWBA calculations. The absolute SF s were normalized by requiring that the SF s of the first two 0 + states were summed to 2.0. The normalized s -wave SF s calculated for the first two 0 + states were 0.60 ± 0.13 and 1.40 ± 0.31. This result indicates that, similar to 12 Be, more s -wave components appear in the excited <math id="M102"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> state (70%) than in the <math id="M103"><mrow><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> ground state (30%). The s -wave strength also indicates that each state has a large number of (1 s 1/2 ) 2 components, revealing a strong mixture of the 2 s 1/2 and the 1 d 5/2 orbital configurations in the low-lying states of 16 C. Note that the 30 % s -wave component in the ground-state wave function of 16 C is much smaller than that that of 15 C but much larger than that of 17 C, which is dominated by a d -wave. Further experimental studies, such as d ( 16 C, 3 He) 15 B, d ( 17 N, 3 He) 16 C, and d ( 16 C, p ) 17 C [ 47 47 ] are required in order to explain the differences between these neutron-rich carbon isotopes. 5.6 Summary of experimental results 2 A brief summary of the experimental studies that have been carried out on the intruder components found within neutron-excess He, Li, Be, B, and C isotopes is provided in this subsection. The variation in the energy of the 2 s 1/2 orbital in the N = 7 and N = 9 systems, relative to the 1 d 5/2 orbital, is presented in Fig. 38 Fig. 38 and Fig. 39 Fig. 39 , respectively. The order of the s and d orbitals is reversed in the N = 7 isotones, and the energy gap between these two orbitals decreases as Z increases. The s orbital also moves rapidly with respect to the p orbital for the N = 7 nuclei. The sequences of these two orbitals are inverted in 9 He and 11 Be but are reverted to the normal order in 12 B and 13 C. For N = 9 isotones, the effective single-particle energies for 2 s 1 =2 and 1 d 5 =2 orbitals relative to the one-neutron separation energy, clearly shows the inversion of s and d . Two valence neutrons are present in N = 6, N = 8, and N = 9 nuclei, meaning that the experimental studies of these isotopes are more complicated and the results are not generally conclusive, except for 8 He, 11 Li, 12 Be, and 16 C. Conclusions from these studies include the following. The intruder (1 p 1/2 ) 2 component cannot be neglected in the ground state of 8 He. In the ground state of 11 Li, the intruder (2 s 1/2 ) 2 configuration is as important as the normal (1 p 1/2 ) 2 component. For 12 Be and 16 C, the (2 s 1/2 ) 2 intrusion appears more in the excited <math id="M108"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> state than that in the <math id="M109"><mrow><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> ground state. The core 6 He ( 10 He) was found to be partially excited to the 2 + state in the ground state of 8 He ( 11 Li). Some intruder configurations have also been found in the low-lying states of light neutron-rich nuclei from these transfer reactions. An exotic s -wave proton intruder state was observed at E x = 4.83 MeV in 13 B using a proton transfer reaction of 12 Be, which was re-examined using a new 14 C( d , 3 He) experiment. The positive-parity low-lying states in 13 B, which are expected to be mixed with the 2 s 1/2 and 1 d 5/2 orbitals, have been studied in several d ( 12 B, p ) experiments. Some of the strengths of the s - and d -waves that were found via the shell model predictions were not observed experimentally. The first excited state at E x = 0.74 MeV in 14 B was most likely a s -wave halo state. The second 3/2 - state in 15 B, which was similar to the <math id="M110"><mrow><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> state in 12 Be and 16 C, was predicted to be an s -wave dominant state but was not found in experiment. Two single-nucleon transfer reaction experiments, d ( 14 B, p ) and d ( 16 C, 3 He), have been performed by the nuclear physics experimental group at Peking University in order to search for such an exotic state and study the intruder component in low-lying states of 15 B; the data are currently under analysis, and there has been no conclusion as yet. These results clearly show that the experimental studies on the intruder component in light neutron-rich nuclei offers an opportunity to understand shell evolution in exotic nuclei. The transfer reaction in inverse kinematics has also been used to study the shell evolution of nuclei around the conventional magic numbers N = 20, 28, 50, and 82 as well as the new magic numbers such as N = 14, 16, and 32. Table. 3 Table. 3 summarizes the recently reported transfer reaction experiments focused on investigating heavier nuclei. The experimental results have been partly reviewed in [ 14 14 ]. In addition to shell evolution, these experiments were also used to study proton–neutron asymmetry, which is related to short-range correlations [ 159 159 ], and to provide the important reaction ratio for nuclear astrophysics; a summary in this regard can be found in [ 160 160 ]. 1 1 Experiment 1 1 Facility 1 1 Year 1 1 Reference 1 1 d ( 15 N, p ) 16 N 1 1 ORNL 1 1 2008 1 1 [ 142 142 ] 1 1 d ( 14 O, t ) 13 O 1 1 GANIL 1 1 2013 1 1 [ 41 41 ] 1 1 d ( 14 O, 3 He) 13 N 1 1 GANIL 1 1 2013 1 1 [ 41 41 ] 1 1 d ( 16 O, p ) 17 O 1 1 CIAE 1 1 2019 1 1 [ 143 143 ] 1 1 d ( 19 O, p ) 20 O 1 1 ANL 1 1 2012 1 1 [ 31 31 ] 1 1 d ( 20 O, p ) 21 O 1 1 GANIL 1 1 2011 1 1 [ 42 42 ] 1 1 d ( 22 O, p ) 23 O * 1 1 GANIL 1 1 2007 1 1 [ 67 67 ] 1 1 d ( 18 F, p ) 19 F 1 1 ANL 1 1 2018 1 1 [ 88 88 ] 1 1 d ( 21 F, p ) 22 F 1 1 ANL 1 1 2018 1 1 [ 144 144 ] 1 1 d ( 19 Ne, n ) 20 Na 1 1 FSU 1 1 2016 1 1 [ 55 55 ] 1 1 d ( 24 Ne, p ) 25 Ne 1 1 GANIL 1 1 2010 1 1 [ 46 46 ] 1 1 d ( 26 Ne, p ) 27 Ne 1 1 GANIL 1 1 2012 1 1 [ 145 145 ] 1 1 d ( 25 Na, p ) 26 Na 1 1 TRIUMF 1 1 2016 1 1 [ 146 146 ] 1 1 d ( 28 Mg, pγ ) 29 Mg 1 1 TRIUMF 1 1 2019 1 1 [ 147 147 ] 1 1 d ( 26 Al m , p ) 27 Al 1 1 ANL 1 1 2017 1 1 [ 89 89 ] 1 1 d ( 34 Si, p ) 35 Si 1 1 GANIL 1 1 2014 1 1 [ 44 44 ] 1 1 p ( 34,46 Ar, d ) 1 1 NSCL 1 1 2010 1 1 [ 61 61 ] 1 1 d ( 44 Ar, p ) 45 Ar 1 1 GANIL 1 1 2008 1 1 [ 148 148 ] 1 1 d ( 46 Ar, p ) 47 Ar 1 1 GANIL 1 1 2006 1 1 [ 149 149 ] 1 1 d ( 60 Fe, p ) 61 Fe * 1 1 GNAIL 1 1 2017 1 1 [ 43 43 ] 1 1 p ( 56 Ni, d ) 55 Ni 1 1 NSCL 1 1 2014 1 1 [ 62 62 ] 1 1 d ( 66 Ni, p ) 67 Ni 1 1 CERN 1 1 2014 1 1 [ 45 45 ] 1 1 d ( 66 Ni, p ) 67 Ni 1 1 CERN 1 1 2015 1 1 [ 150 150 ] 1 1 d ( 70 Zn, 3 He) 69 Cu 1 1 Alto 1 1 2016 1 1 [ 151 151 ] 1 1 d ( 72 Zn, 3 He) 71 Cu 1 1 GANIL 1 1 2015 1 1 [ 152 152 ] 1 1 d ( 78 Zn, p ) 79 Zn 1 1 CERN 1 1 2015 1 1 [ 49 49 ] 1 1 d ( 82 Ge, p ) 83 Ge 1 1 ORNL 1 1 2005 1 1 [ 153 153 ] 1 1 d ( 84 Se, p ) 85 Se 1 1 ORNL 1 1 2005 1 1 [ 153 153 ] 1 1 d ( 95 Sr, p ) 96 Sr 1 1 TRIUMF 1 1 2018 1 1 [ 154 154 ] 1 1 d ( 94,95,96 Sr, p ) 1 1 TRIUMF 1 1 2019 1 1 [ 155 155 ] 1 1 d ( 132 Sn, p ) 133 Sn 1 1 ORNL 1 1 2010 1 1 [ 156 156 ] 1 1 d ( 132 Sn, t ) 131 Sn 1 1 ORNL 1 1 2018 1 1 [ 157 157 ] 1 1 p ( 154,159 Gd, d ) 1 1 LBNL 1 1 2014 1 1 [ 158 158 ] 6 Summary 1 The single-nucleon transfer reaction in inverse kinematics is a sensitive tool to investigate the exotic structure in unstable neutron-rich nuclei. The DCSs (or angular distributions) of the selectively populated final states contain essential structural information that can help understand the nature of the nuclear force and the interactions in unstable nuclei. The transferred momentum l (or the spin-parity of the low-lying states in the final nucleus), the SF s, and the effective single-particle energy can be deduced from the single-particle transfer reaction. These observations are useful for interpreting the shell evolution of neutron-excess nuclei. Inverse kinematics is preferred to normal kinematics in transfer reactions with radioactive beams. Several experimental setups including silicon detection arrays (with or without a γ -ray array), the special spectrometer HELIOS, and the newly developed AT-TPC, which are appropriate for use in single-nucleon transfer reactions in inverse kinematics, have been developed at various laboratories around the world. The basic modules, advantages and disadvantages of each array are illustrated in this paper. A large number of single-nucleon transfer reaction experiments have been carried out using radioactive beams in order to quantitatively study the intruder components in the low-lying states of neutron-rich He, Li, Be, B, and C isotopes. The experimental results are conclusive for the ground states of the N = 7 and N = 9 isotones. For example, the intruder s -wave component dominates in 9 He, 10 Li (from other reactions), 11 Be, 14 B, and 15 C. However, for the ground state of the N =6, N =8, and N =10 isotones, the existence of two valence neutrons leads to more complications. Thus, although some experimental results are disputed, some of the intruder information on the ground state of 8 He, 11 Li, 12 Be and 16 C can be conclusive. First, the intruder 1 p 1/2 ) 2 component in 8 He cannot be ignored. 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Phys . 43 : 043001 2016 . https://iopscience.iop.org/article/10.1088/0954-3899/43/4/043001 https://iopscience.iop.org/article/10.1088/0954-3899/43/4/043001 Transfer reaction in nuclear astrophysics 10.1007/s41365-020-0731-y This work was supported by the National Key R&D program of China (No. 2018YFA0404403), and National Natural Science Foundation of China (Nos. 11775004, U1867214, and 11535004). 2019-12-15 2020-01-09 2020-01-11 2020-02-06 2020-02-01 2021-04-10 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2020 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-020-0731-y http://dx.doi.org/10.1007/s41365-020-0731-y . 2020 Nuclear Science and Techniques 2 31