M Dufour

Ires Bat27

P Descouvemont

The 18 F(p,α) 15 O low-energy S-factor: a microscopic approach

The 18 F(p,α) 15 O S-factor is determined in a microscopic cluster model. The wave functions are defined in the Generator Coordinate Method, involving various 18 F+p, 15 O+α, and 18 Ne+n configurations. We analyze the 19 Ne level scheme, and test the model with the 19 F spectrum, better known experimentally. We focus on 1/2 + and 3/2 + states, corresponding to s waves, and expected to represent the main contribution in the 18 F(p,α) 15 O cross section at stellar energies. The 1/2 + partial wave is shown to play an important role in a wide energy range. This property reduces the uncertainties associated with interference patterns in the 3/2 + contribution. The relevant 1/2 + states have been observed in 19 F, but have not been searched for in 19 Ne. An elastic-scattering experiment above 1 MeV is suggested.

Introduction

Information about the nucleosynthesis and the ejection mechanism in novae can be essentially deduced from gamma-ray emission originating from electronpositron annihilations at 511 keV following the β + decay of 18 F. A precise interpretation of the measurements performed by γ-astronomy detectors relies on an accurate knowledge of the destruction modes of 18 F in novae. In this context, 18 F(p,α) 15 O is believed to be the most relevant reaction [1,2].

The typical temperature range in novae is of the order of (1 -4) × 10 8 K. Consequently, 19 Ne resonances relevant in the burning process are located in an interval of about 1 MeV above the 18 F+p threshold (Q = 6.41 MeV) [1,3]. Many uncertainties remain on the determination of the 18 F(p,α) 15 O reaction rate at these energies. Until now, the astrophysical S-factor has been essentially calculated as a sum of contributions of individual states (see, however, Ref. [4], where interference effects are discussed). Among those resonances, only two of them seem to be reasonably well understood. A 3/2 + state at E x = 7.07 MeV (E cm = 0.66 MeV) is an s wave, and is known to be a single particle state, with a large spectroscopic factor [5,6]. It has been widely investigated in the literature, through transfer [7][8][9] or elastic [10,11] reactions, or with indirect methods [12][13][14][15][16][17][18]. On the other hand, the existence of a 3/2 - state at E cm = 0.33 MeV [14] is now well established but, owing to its p-wave nature, plays a role in a limited temperature range only. Some experiments focus on the 19 F mirror nucleus, and derive 19 Ne properties by assuming charge symmetry on the reduced widths and spectroscopic factors. In this case, in addition to the experimental difficulties inherent to each technique, a theoretical uncertainty may arise from the use of charge symmetry. The Q values in both nuclei are quite different and Coulomb effects are expected to be non-negligible. This paper is devoted to a microscopic study of the 18 F(p,α) 15 O reaction at stellar energies. The theoretical framework combines a large-scale variational basis (Extended Two Cluster Model -ETCM) and a new interaction (Extended Volkov Interaction -EVI) (see Refs. [19,20] and references therein). Microscopic cluster models [START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF] are known to be well adapted to the description of reactions relevant in astrophysics, which involve low-energy cross sections and radioactive nuclei. Antisymmetrization between all nucleons, center-ofmass motion, good quantum numbers and boundary conditions are exactly treated. Bound, resonant and scattering states of the system are described in a unified way through the microscopic R-Matrix method [START_REF] Baye | [END_REF] allowing a direct calculation of radiative-capture and transfer cross sections. Furthermore the small number of parameters involved gives them a significant predictive power.

Previously, the same model has been successfully applied to the study of the 13 C(α,n) 16 O transfer reaction [20]. The ETCM provides a large variational space and the EVI improves the hamiltonian by introducing an additional degree of freedom. In the case of transfer reactions, the threshold between the entrance and exit channels and another important property such as the location of a state can be then reproduced simultaneously.

In order to describe the 18 F(p,α) 15 O reaction, the ETCM wave functions of 19 Ne are defined as a sum of 18 F+p, 15 O+α and 18 Ne+n two-cluster basis func-tions. The 18 F, 18 Ne, and 15 O nuclei are described by clusters involving major s, p and sd harmonic oscillator shells. Consequently, the GCM wave function is expanded over many channels. The large variational basis obtained in this way is expected to provide a good description of the asymptotic conditions of the reaction and of some resonances relevant in the 18 F(p,α) 15 O cross section.

Of course the applicability of microscopic models is essentially limited by the level density. In the present case, owing to the large Q value, and to the low isospin, the level density near threshold is significant (typically a few levels per MeV). Consequently a microscopic approach cannot be expected to reproduce all relevant resonances. The present work is more an exploring study of the reaction than an exhaustive review of resonance properties. We essentially focus on the 1/2 + and 3/2 + states, which correspond to the s waves in the 18 F+p motion.

In Section 2, we briefly present the model, and emphasize the definition of the spectroscopic factors. Charge symmetry effects are analyzed in Section 3, before the analysis of the 18 F(p,α) 15 O S factor. Concluding remarks are given in Section 4.

The microscopic model

Wave functions

The wave functions are defined in the Generator Coordinate Method (GCM) which assumes a cluster structure for the A-nucleon system [START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF]. In a given partial wave JM π, the wave function of the system is given by a superposition of 18 F+p, 15 O+α and 18 Ne+n cluster wave functions, each channel involving various excited states (see Ref. [19] for details). More precisely the wave function is given by

Ψ JM π = i Ψ JM π 18 F (i)+p + j Ψ JM π 15 O(j)+α + k Ψ JM π 18 N e(k)+n , (1)
where the sums run over all internal states. The 18 F and 18 Ne wave functions are defined in the sd shell model with all configurations allowed by the Pauli principle. For 15 O, we use a closed p shell with one hole. This procedure provides various states which are presented in Table 1.

In a two-cluster theory, each component of Eq. ( 1) is written as

Ψ JM π α = I A Y (Ω ρ ) ⊗ [φ I 1 π 1 1 ⊗ φ I 2 π 2 2 ] I JM g Jπ α I (ρ), (2)
where (I 1 , I 2 ) are the spin of the colliding nuclei in channel α (α refers to the partition and to the excitation level), I is the channel spin, and the relative angular momentum. In Eq. ( 2), g Jπ α I (ρ) is the radial function depending on the relative coordinate ρ, and A is the antisymmetrization operator. In the GCM, these radial wave functions are expanded over displaced gaussian functions, which makes Eq. ( 2) equivalent to a combination of projected Slater determinants, well adapted to numerical calculations. The gaussian basis functions are centered at different locations called the generator coordinates (see Ref. [START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF] for more information). In the GCM, the asymptotic behaviour of the wave function ( 2) is of course gaussian. This is corrected by the microscopic R-matrix method [START_REF] Baye | [END_REF] which restores the coulombian dependence of the wave function.

A key issue in the treatment of transfer reactions is the threshold problem. When the internal wave functions and the nucleon-nucleon interaction are chosen, the model is parameter free, and the thresholds are an output of the theory. We have recently developed a new nucleon-nucleon interaction [19] based on the Volkov force, but with more flexibility. This force allows to constrain a threshold, and to keep one parameter to reproduce some important property of the system, such as the energy of a resonance. The choice of the interaction will be specified in Section 3. The 18 F and 18 Ne/ 18 O energy spectra have been already discussed in Ref. [19]. Although the interaction adopted here is slightly different, the present spectra are very similar to those of Ref. [19].

Spectroscopic factors

In a microscopic cluster model, the radial wave functions g Jπ α I (ρ) can be transformed as

gJπ α I (ρ) = α I N Jπ α I,α I (ρ, ρ )g Jπ α I (ρ )dρ , (3)
where N Jπ (ρ, ρ ) is the overlap kernel [23,24], which is computed numerically according to the method proposed by Varga and Lovas [25]. From these functions, we derive the spectroscopic factors in a given (α I) channel by

S Jπ α I = ∞ 0 gJπ α I (ρ) 2 ρ 2 dρ. (4)
The spectroscopic factor measures the contribution of a given channel in a many-body wave function. A large value indicates a dominant cluster structure in this channel. Notice that, in most experimental works, the spectroscopic factor is based on simple potential-model wave functions. Physical solutions (in DWBA expressions, for example) are approximated by

gJπ α I (ρ) ≈ S pot α I g Jπ pot (ρ), (5)
where g Jπ pot (ρ) is derived from some potential. In addition to uncertainties related to the choice of the potential, the definition itself of the spectroscopic factor is different. Although both approaches should provide qualitatively consistent results, a precise comparison is often difficult (see, for example, Ref. [26]).

3 Application to 18 F(p,α) 15 O and to the 19 Ne spectroscopy

Conditions of the calculation

As mentioned previously, the GCM basis functions involve 18 F+p, 18 Ne+n and 15 O+α (or mirror) configurations with, for each heavy nucleus, several excited states. In the treatment of transfer reactions, reproducing the Q value is a key issue, and we use the EVI [19] complemented by a zero-range spinorbit interaction [27,19] with the amplitude S 0 as free parameter. The three parameters of the interaction are determined as follows: S 0 is fitted on the 15 O(3/2 -)-15 O(1/2 -) experimental energy (S 0 = 28 MeV.fm 5 ); the remaining parameters are fitted on the 18 F(p,α) 15 O experimental threshold, and on the energy of the 3/2 + resonance in 19 Ne (E x = 7.07 MeV, E cm = 0.66 MeV) which is known to be a single-particle state [5], accurately described by a 18 F+p model. This provides W = 0.7661, M = 0.2339, B = -0.52, H = 0.52 as EVI parameters.

The oscillator parameter, common to all clusters, is chosen as b = 1.6 fm, which represents a compromise between their r.m.s. radii and their binding energies. The generator coordinates are taken from R = 2.4 fm to R = 10.8 fm with a constant step of 1.4 fm. In the R-matrix calculations, we use a channel radius a = 10.1 fm. Large values are necessary to make sure that antisymmetrization effects and nuclear contributions between the clusters are negligible, as required by the R-matrix method. As soon as these conditions are satisfied, the physical results do not depend on the channel radius.

Discussion of Coulomb effects

The use of charge symmetry is widely used in the literature to deduce spectroscopic properties of 19 Ne states from the mirror nucleus 19 F. Here we want to address two important issues: (i) a test of the ability of the GCM to account for Coulomb shifts; this is done on low-lying states where the level scheme of both nuclei is firmly established. (ii) To compute reduced widths and spectroscopic factors of 19 Ne and 19 F; as the model includes the Coulomb force exactly, this will test the accuracy of the charge-symmetry assumption on these quantities. This assumption is used in indirect methods, where the measurements deal with spectroscopic information on the 19 F nucleus.

Concerning item (i), we have readjusted the nucleon-nucleon interaction individually for well known 19 F positive-parity low-lying states of 19 F. Then the theoretical 19 Ne energy is determined with the same nuclear interaction. This allows to quantify the precision of the method to deduce 19 Ne energies. The results are given in Table 2, which shows that the difference between theory and experiment is at most 0.14 MeV. It is reasonable to assume that a similar accuracy can be expected for more excited states, near the proton threshold. Let us now discuss spectroscopic factors and reduced widths. We consider here 1/2 + and 3/2 + states, as mentioned previously. In each case, we have a strongly bound state, a resonance, and a state located near threshold. In Table 3, we present the spectroscopic factors in various channels. The lowlying 1/2 + and 3/2 + states have similar spectroscopic factors in the three partitions. This is not surprising in an antisymmetric model: for compact states, different configurations may describe equally well the wave function.

For 19 F, the spectroscopic factors in the proton channel are in qualitative agreement with the values of Iliadis and Wiescher [26] obtained in various models. The second GCM levels correspond to α cluster states, well known in cluster theories [28,29], and characterized by large α spectroscopic factors.

As expected [5], the 7.07 MeV state in 19 Ne presents a large spectroscopic factor in the proton channel. Note however that the theoretical definition of the spectroscopic factor (4) might give slightly different results from the method based on the potential model. This was already pointed out by Iliadis and Wiescher [26] who compare spectroscopic factors deduced from the shell model, from DWBA analysis, and from the potential model. Using a Woods-Saxon potential, Fortune and Sherr [5,6] recommend slightly lower values (S p (0) = 0.49 ± 0.03), but the single-particle character of this resonance is confirmed by the GCM. A 1/2 + state presents the same properties, and is expected to have some influence on the 18 F(p,α) 15 O S factor.

Concerning charge symmetry, the GCM shows that the spectroscopic factors in 19 F and 19 Ne are very close to each other, even in less important configurations. This supports the assumption of charge symmetry in DWBA analyses (see for example [15]). Reduced widths are presented in Table 4. They are calculated at the R-matrix channel a = 10.1 fm. Reduced widths represent a useful complement to spectroscopic factors: up to a simple factor, they provide the wave function evaluated at ρ = a. Conversely, the spectroscopic factors are obtained from an integral of the wave functions over the whole space. We therefore expect that charge symmetry is less good in reduced widths as, by definition of the R matrix channel, the interaction is dominated by the Coulomb force. This property is confirmed by the results displayed in Table 4. Charge symmetry is a rather poor approximation for reduced widths. This result arises from two reasons: of course the nuclear charges are different, but the Q values in 19 F and 19 Ne are also rather different (Q n = 10.42 MeV in 19 F, and Q p = 6.41 MeV in 19 Ne). At large distances, where the reduced widths are evaluated, the wave functions behave quite differently. The differences should decrease at larger channel radii, but the R-matrix requirements would not be satisfied. Our work therefore suggests that the charge-symmetry approximation should not be used for reduced widths.

F (J = 3/2 + ) 19 Ne (J = 3/2 + ) State E x S n (0) S n (2) S p (2) S α (1) E x S p (0) S p (2) S n (2) S α (1) 1
Table 4 Proton, neutron and α reduced widths (at 10.1 fm) for the first three GCM states in 19 F and 19 Ne. Superscripts represent to the power of 10. All energies are expressed in MeV.

19 19 F (J = 3/2 + ) 19 Ne (J = 3/2 + )

F (J = 1/2 + ) 19 Ne (J = 1/2 + ) E x γ 2 n (0) γ 2 n (2) γ 2 p (0) γ 2 α (1) E x γ 2 p (0) γ 2 p (2) γ 2 n (2) γ 2 α (
E x γ 2 n (0) γ 2 n (2) γ 2 p (0) γ 2 α (1) E x γ 2 p (0) γ 2 p (2) γ 2 n (2) γ 2 α (1) 1 1
.55 2.04 -5 5.13 -5 2.15 -5 1.06 -5 1.54 7.37 -5 1.61 -4 9.22 -6 1.10 Our goal in this work is essentially to discuss the 18 F(p,α) 15 O cross section, in the framework of a microscopic approach. We do not intend to discuss the level schemes in detail. Here we focus on two specific partial waves, J = 1/2 + and J = 3/2 + , which correspond to s waves and which, consequently, are expected to dominate the cross section at low energies. The level schemes are given in Figures 1 and2, for J = 1/2 + and J = 3/2 + , respectively.

Let us start with a discussion of Fig. 1, where we compare 1/2 + states derived from the GCM, and observed experimentally [3,12,16,18]. Energies are given from the 18 F+n and 18 F+p thresholds for 19 F and 19 Ne, respectively. Let us remind that the 18 F(p,α) 15 O Q value and the 3/2 + (7.07 MeV) level in 19 Ne are fitted by the interaction, but that all other energies (thresholds, level energies) are determined without any fitting procedure. The 18 O+p threshold in 19 F is underestimated by the theory, but the ordering is correct. In 19 Ne, the 18 Ne+n threshold is located above 5 MeV, and is therefore not shown.

E cm ( 19 Ne) p+ 18 O p+ 18 O Fig. 1. 19 F and 19 Ne spectra for J = 1/2 + . All energies are in MeV. The thresholds are displayed as dotted lines. Experimental energies E x are given.

The GCM energy of the ground states are in fair agreement with experiment (within 0.2 MeV). The spectroscopic factors in the p and α channels are similar (see Table 3). The second GCM 1/2 + state presents a dominant component in the 4+15 channel, as indicated by the large spectroscopic factors. Experimentally, the 5.94 MeV state in 19 F is known to be an α+ 15 N state, and has been discussed in previous single-channel α+ 15 N cluster models [28,29].

The GCM gives rise to a third 1/2 + resonance, at E x = 8.3 MeV. Its width (Γ GCM = 180 keV) suggests that the experimental counterpart is the 8.65 MeV state (Γ exp ≈ 300 keV). The predicted energy is quite close to the experimental value.

The study of 1/2 + levels provides two conclusions: (i) In 19 Ne, the third GCM level, analog of the 8.65 MeV state in 19 F, is located at lower energy, near 1 MeV above the 18 F+p threshold. Therefore it might play a role in the 18 F(p,α) 15 O reaction. The large Coulomb shift is supported by the important spectroscopic factor (see Table 3). From theoretical arguments, this state can be considered as a single-particle state, similar to the 3/2 + (7.07 MeV) level in 19 Ne. Although it should play a role in the cross section, as s-wave resonance, it has not been identified yet in 19 Ne.

(ii) Comparing the theoretical and experimental 19 F spectra indicates that three levels are missing in the GCM. Although they are not observed in 19 Ne they might also contribute to the 18 F(p,α) 15 O S-factor. As they are not described by the present two-cluster model, which includes many configurations, these states should have a more complicated structure, such as a three-body structure. It is consequently reasonable to assume that their spectroscopic factors in the p and α channels should be small, and that their role in the cross section should be of minor importance. p+ 18 O p+ 18 O Fig. 2. 19 F and 19 Ne spectra for J = 3/2 + (see caption to Fig. 1).

Let us now discuss 3/2 + states, displayed in Fig. 2. As for J = 1/2 + , they correspond to s waves. The nucleon-nucleon interaction has been fitted on the energy of the 7.07 MeV resonance in 19 Ne, well known to be a single particle state, and hence well adapted to a 18 F+p model. Its proton and α widths are calculated as Γ p = 26 keV, and Γ α = 9 keV. This provides a ratio Γ p /Γ = 0.74 somewhat higher than experimentally, but the resonance strength ωγ = 4Γ p Γ α /6Γ = 4.5 keV is consistent with the data (see Ref. [9] for a review). As expected we find low-lying 3/2 + states, members of the K = 1/2 + ground-state band in both nuclei. Without any fitting procedure the GCM energies are quite close to the experimental values. Table 3 shows that the 18 F+n ( 18 F+p) configuration is dominant, but the α channel also plays a role.

The second 3/2 + GCM state corresponds to an α-cluster molecular resonance, well known for many years [28,29]. As explained previously, the third GCM resonance is assigned to the 7.07 MeV state in 19 Ne. This state determines the 18 F(p,α) 15 O S factor in a wide energy range. As pointed out by Fortune and Sherr [5], the mirror level in 19 F has not been observed. Fortune and Sherr suggest that the Coulomb shift should be quite large, owing to the large spectroscopic factor. This is confirmed by our microscopic approach, which suggests its energy near E x = 7.9 MeV (Fortune and Sherr propose E x = 7.4 ± 0.1 MeV). Such a large Coulomb shift is typical of s waves with a large reduced width [30].

From an experimental point of view, many 3/2 + states are known in 19 F up to 8 MeV. Although the cluster theory does reproduce three of them, some other states are missing. An obvious drawback of the model is that the 3/2 + doublets near 6.5 MeV cannot be described, although their states are very near the 18 F+p threshold in 19 Ne. This problem is well known in cluster models: it arises from the limited number of configurations. When a resonance is missing in the GCM, it should correspond to other cluster configurations, such as threebody structures or other two-body arrangements. This means that the 18 F+p and α+ 15 O configurations, included in the model, should play a minor role, and that the spectroscopic factors in these channels should be significantly lower than the values shown in Table 3. This argument is supported experimentally by the very small Coulomb shifts.

Concerning missing states, it is worth mentioning the experimental 3/2 + 2 states at 3.91 MeV in 19 F and at 4.03 MeV in 19 Ne. These states are expected to determine the 15 N(α, γ) 19 F and 15 O(α, γ) 19 Ne reaction rates, as they are located near the α threshold. Previous α+ 15 N (α+ 15 O) cluster studies concluded that these states do not present an α structure and that, consequently, the reduced α widths should be quite small [29]. They are still missing here, although many additional channels are included. This result seems to support the suggestion of Ref. [29], i.e. that the missing 3/2 + states might belong to a band with a more deformed structure, such as 12 C+ 7 Li ( 12 C+ 7 Be).

3.4 The 18 F(p,α) 15 O reaction A microscopic cluster model provides a unified description of bound and scattering states. The determination of the scattering wave function and of the collision matrix is performed within the R-matrix theory, which exactly accounts for boundary conditions. The transfer cross section from the proton to the α channel is defined as

σ t (E) = π k 2 Jπ 2J + 1 (2I 1 + 1)(2I 2 + 1) II |U Jπ p I,α I (E)| 2 , (6)
where (I 1 = 1, I 2 = 1/2) are the spins of the colliding nuclei, Jπ are the total spin and parity, and k is the wave number in the entrance channel. This definition is valid for resonant as well as for non-resonant processes, and only depends on the collision matrix U Jπ . The transfer cross section is deduced from non-diagonal elements of the collision matrix: ( I) are the angular momentum and channel spin in the entrance channel, whereas ( I ) are the corresponding values in the exit channel.

In the present calculation, only s-waves are included, which means that the summation over Jπ in Eq. ( 6) is limited to Jπ = 1/2 + and 3/2 + . The first step is to determine the transfer cross section from the GCM wave functions. Then, as some states are missing or slightly shifted with respect to experiment, we will use the ability of the R-matrix theory to combine GCM results with experimental information. The GCM S-factor is shown in Fig. 3. Near the 3/2 + resonance, the agreement with the data is excellent, as only the energy has been fitted by the N N interaction. Below 0.3 MeV, the 3/2 + contribution presents an interference pattern, due to other 3/2 + resonances. The 1/2 + contribution is essentially determined by the properties of a broad resonance near 1 MeV, and of subthreshold state (see Fig. 1). One should of course keep in mind that some 3/2 + states are missing in the region 0-0.5 MeV, but our calculation suggests that the 1/2 + partial wave plays an important role.

This S-factor can be considered as a fair estimate, but some resonances are missing, and others are not located at the experimental energy. To improve the accuracy on the theoretical S-factor, we have combined GCM information with experimental data. This procedure has been successfully used in other reactions, such as 12 C(α, γ) 16 O for example [31]. Let us start with the 1/2 + contribution. As shown in Fig. 3, two states determine the S-factor. The microscopic calculation suggests that interference effects are negligible, and that the 1/2 + S-factor can be approximated by the sum of two isolated states. These states are known experimentally in 19 F, but are not observed in 19 Ne.

It is reasonable to assume that the differences between the GCM and experimental energies are identical for the mirror nuclei. The 19 Ne energies have been therefore corrected by this difference, and the partial widths, unknown in 19 F, are modified according to the energy shift. The 19 Ne properties of these states are given in Table 5, and the two-level S-factor in Fig. 4. According to the previous discussion, interference effects have been neglected. Comparing with Fig. 3 provides a reduction of the 1/2 + contribution since the resonance is shifted upwards and the subthreshold state downwards. For the 3/2 + partial wave, a similar procedure has been used. We have complemented the microscopic R-matrix calculation with the two states of the doublet (E cm = 8 keV and 38 keV). Partial widths have been taken from the literature [12]. As the interference signs are not known, we have considered the four possibilities. This procedure provides the different curves in Fig. 4. They correspond to the total S factor. In Ref. [4], the authors investigate 3/2 + interference effects in an R-matrix analysis. Considering three 3/2 + levels, Chae et al. find that each of the 8 sign combinations gives a different χ 2 . This conclusion is surprising since a common change of all signs should not modify the S-factor. Within the resonance, Fig. 4 shows that the theory slightly underestimates the data. This could be compensated by a slight readjustment of the partial widths, but our goal here is a qualitative discussion, rather than an accurate fit of the data. The main conclusion from Fig. 4 is that, in the temperature range typical of novae (corresponding to E cm ≈ 0.2 -0.3 MeV) the uncertainties of the S-factor due to the interference patterns is much smaller than expected [32]. The reason is that the main contribution at nova energies comes from the 1/2 + partial wave.

3.5 Discussion of J = 1/2 + states Currently, the relevant 1/2 + states are observed in 19 F, but have not been experimentally studied in 19 Ne. A spectrum of expected 1/2 + states in 19 Ne is displayed in Fig. 5. The known states are shown as solid thin lines, and the predicted levels (from Table 5) are shown as thick lines. The theoretical widths are given in Table 5. Experimental information about these resonances seems to be crucial for a good knowledge of the 18 F(p,α) 15 O cross section at novae temperatures. The main novelty of this work deals with new 1/2 + states near the proton threshold in 19 Ne. Experimentally, the second state could be observed by elastic scattering. It is interesting to point out that, in a 18 F+p elastic-scattering experiment, Bardayan et al. [11] mention the need for a broad s state to fit their cross section. This experiment was however limited to energies E cm < 1.01 MeV, and a clear identification could not be performed.

We present in Fig. 6 the theoretical scattering cross section at θ cm = 180 • (corresponding to θ lab = 0 • in reverse kinematics). This cross section should be considered as qualitative only, as only the 1/2 + and 3/2 + partial waves have been included (for other partial waves, up to = 2, we use the hard-sphere approximation of the phase shift). The subtreshold 1/2 + state does not play any role near 1.5 MeV. Its contribution is limited to lower energies, where the Coulomb term dominates. Accordingly, the elastic cross section is not sensitive to its properties. The main emphasis should be put on the difference between the cross section with and without the broad 1/2 + state. Its presence provides a clear signature in the cross section, and should be observable in future experiments. 3/2 + Fig. 6. 18 F+p elastic cross section at θ cm = 180 • , with the 3/2 + partial wave only, and with the addition of J = 1/2 + .

Conclusion

This work is aimed at providing an exploring study of the 18 F(p,α) 15 O reaction from a microscopic viewpoint. This approach cannot be expected to fit the experimental data but, instead, to give qualitative arguments about the energy spectra, cross sections, and other physical quantities. Consequently, it represents an ideal complement to fitting procedures, such as the phenomenological R-matrix method, which provides more accurate extrapolations, but necessarily requires cross-section or/and spectroscopic data.

The high number of channels included in the present GCM basis represents a challenge. As mentioned in Section 2, the basis involves 28 18 F+p channels, 2 15 O+α channels and 14 18 Ne+n channels. This property allows us to span the sd shell. The main advantage of the GCM is to provide, in addition to spectroscopy, scattering wave functions of the system. This is crucial in lowenergy reactions where the Coulomb interaction is dominant.

We have essentially focused on 1/2 + and 3/2 + states, i.e. s-waves in the 18 F+p system. Our nucleon-nucleon interaction is adjusted on the energy of the 7.07 MeV single-particle state in 19 Ne. All other quantities (resonance energies and widths, cross sections, etc) are obtained without any fitting. As expected for relatively high level densities, the GCM does not reproduce all 19 Ne experimental states. Missing states should have more complicated structures, such as three-body structures and, consequently, should be characterized by small 18 F+p spectroscopic factors. In particular the experimental 3/2 + doublets in 19 F and 19 Ne are not described by the model. In addition it is known (see, for example, Ref. [14]) that p waves should play a role.

Our main conclusion is that the 1/2 + partial wave plays an important role in the 18 F(p,α) 15 O cross section. This is not surprising as it represents an s wave, but has been dismissed until now. Our model suggests that the 1/2 + contribution should essentially arise from a subthreshold state, and from a broad resonance near 1.4 MeV. In these conditions, the S-factor at typical novae temperatures (E cm 0.2 -0.3 MeV) is dominated by the 1/2 + term, and the interference problem between 3/2 + resonances is reduced. Accordingly, if the properties of these 1/2 + states are confirmed, the uncertainties on the 18 F(p,α) 15 O reaction rate should be lower than expected. From the energy and width, the 8.65 MeV level in 19 F is a very good candidate for the broad 1/2 + resonance. It is expected near 7.7 MeV in 19 Ne (E cm ≈ 1.5 MeV). The existence of a new s state at E cm > 1.01 MeV is suggested by Bardayan et al. [11], and might correspond to the GCM state. From the theoretical point of view, this level can be considered as a single-particle state, with a structure very similar to the 3/2 + resonance at 7.07 MeV. An experimental confirmation of this 1/2 + state, with an elastic-scattering experiment, should provide an important step in a better understanding of the 18 F(p,α) 15 O process.

Fig. 3 .

 3 Fig.3.18 F(p,α)15 O S-factor for J = 1/2 + and J = 3/2 + . The experimental data are taken from Ref.[9] (full circles) and Ref.[14] (open circles).

Fig. 4 .

 4 Fig. 4. Corrected 18 F(p,α) 15 O S-factor, with the individual contribution of J = 1/2 + (see text). The signs correspond to different interference signs in the 3/2 + partial wave. The data are as in Fig 3.

Fig. 5 .

 5 Fig.5. Theoretical19 Ne level scheme for J = 1/2 + . Thin levels are experimentally known, and the thick lines correspond to GCM energies corrected by the energy difference in19 F.

Table 1

 1 Shell-model states included in the two-cluster GCM basis. The bracketed values correspond to the number of states for a given spin.

	Nucleus	States
	18 F	0 + (3), 1 + (7), 2 + (8), 3 + (6), 4 + (3), 5 + (1)
	18 Ne / 18 O 0 + (3), 1 + (2), 2 + (5), 3 + (2), 4 + (2)
	15 O / 15 N	1/2 -(1), 3/2 -(1)

Table 2 19

 2 Ne energies (in MeV) after tuning of the interaction on19 F levels (see text).

	J π	19 F (exp) 19 Ne (exp) 19 Ne (GCM)
	1/2 +	-10.42	-6.41	-6.27
	5/2 +	-10.24	-6.17	-6.13
	3/2 +	-8.88	-4.87	-4.73
	7/2 +	-6.05	-2.03	-2.07

Table 3 Proton

 3

	1	0	0.156 0.009 0.198 0.165	0	0.173 0.010 0.183 0.159
	2	5.94 0.045 0.003 0.008 0.806		0.082 0.003 0.003 0.768
	3	8.65 0.456 0.002 0.195 0.084		0.707 0.000 0.103 0.094
			19

, neutron and α spectroscopic factors for the first three GCM states in

19

F and

19

Ne. The values are given in brackets. Energies E x (in MeV) correspond to the experimental values.

19 F (J = 1/2 + )

19

Ne (J = 1/2 + ) State E x S n (0) S n (2) S p (0) S α (1) E x S p (0) S p (2) S n (0) S α (1)

  -5 1.61 -6 5.88-5 2.06 -6 0 1.91 -4 4.14 -6 2.46 -5 2.15 -6 2 5.94 1.48 -4 2.38 -6 2.05 -6 3.95 -2 1.95 -3 1.21 -5 4.44 -7 4.06 -2 3 8.65 8.26 -3 1.25 -5 1.77 -3 1.55 -2 9.20 -2 2.71 -6 1.46 -4 1.38 -2

			1)
	1	0	6.31

  -5 2 5.50 5.19 -5 2.56 -4 2.60 -6 1.62 -2 4.73 -4 1.21 -3 1.88 -6 1.39 -2 3 1.19 -2 2.32 -4 2.09 -5 3.51 -3 7.07 7.21 -2 1.05 -3 3.40 -7 1.06 -3 3.3 Spectroscopic properties of 19 F and 19 Ne

Table 5

 5 Spectroscopic properties (in MeV) of 1/2 + states in19 Ne near the proton threshold.

	E GCM E mod	Γ p	Γ α
	-0.15 -0.41 1.95 × 10 -3 a) 0.231
	1.11	1.49	0.157	0.139
	a) Reduced width at 10.1 fm.

Acknowledgements

We are grateful to Carmen Angulo and Nicolas de Séréville for helpful discussions. This text presents research results of the Belgian program P5/07 on interuniversity attraction poles initiated by the Belgian-state Federal Services for Scientific, Technical and Cultural Affairs.


Danillo Sprovieri
email: danillo.sprovieri@malix.univ-paris1.fr

Dynamic Re-Configuration of Software Product Lines Towards an Exploratory Study on DSPLs

Keywords: Run-Time Adaptation, SPLs, DSPLs

Context: Adaptations need to be considered at design-time (adapting complex systems to new technologies, reengineering due to new regulations etc.), but also during runtime (e.g. new emerging functional and non-functional requirement, context-specific decisions). Objective: I use SPLs as a strategy for coping with uncertainty and adapting to change, where conventionally change occurs in the requirements of the software product lines' market. Our idea is to design a variability mechanism in the domain of dynamic software product lines engineering in order to enable continuous evolution and adaptation of the software product lines at run-time. Method: I investigate dynamic change propagation of SPLs at run-time through an explorative study. A literature review and semistructured personal interviews with relevant actors in the domain of SPLs are the fundament of our research. This analysis enables us to understand how SPLs are dynamically adapted and evolved in practice. Conclusion: This study will give us an overview of the domain of DSPLs and allows us to identify the research gap regarding run-time adaptation and evolution of SPLs.

I. INTRODUCTION

Due to market dynamics, changing business conditions and new regulations [START_REF] Goodhue | Addressing business agility challenges with enterprise systems[END_REF], the success of an enterprise depends increasingly on its ability to be flexible and react to environmental changes in a quick and cost-effective way [START_REF] Li | Mining business process variants: Challenges, scenarios, algorithms[END_REF]- [START_REF] Dadam | Prozessmanagementsysteme[END_REF]. Under these conditions, adaptations are unavoidable and crucial in most application domains [START_REF] Lenz | IT support for healthcare processespremises, challenges, perspectives[END_REF], [START_REF] Dadam | Prozessmanagementsysteme[END_REF]- [START_REF] Reijers | BPM in practice: Who is doing what?[END_REF]. These adaptations need to be considered at design time (adapting complex systems to new technologies, reengineering due to new regulations, etc.) [START_REF] Hallerbach | Managing process variants in the process life cycle[END_REF], [START_REF] Rosemann | A Configurable Reference Modeling Language[END_REF], but also during run-time (e.g. context-specific decisions) [START_REF] Weber | Change patterns and change support features -Enhancing flexibility in process-aware information systems[END_REF], [START_REF] Reichert | ADEPTflex -supporting dynamic changes of workflows without losing control[END_REF]. For several decades, all software was custom software. Every application was tailored specifically to one customer's requirements. Software reuse has gained more and more attraction from the research community to find a solution for developing complex systems. Reuse software parts leads to improvements such as productivity, quality and cost reduction [START_REF] Pohl | Software Product Line Engineering: Foundations, Principles and Techniques[END_REF]. The idea of software reuse was already introduced by McIlroy [START_REF] Mcilroy | Mass Produced Software Components[END_REF], who proposed at that time mass production of software components as the base for software development. Later on, Parnas [START_REF] Parnas | On the criteria to be used in decomposing systems into modules[END_REF]- [START_REF] Parnas | Designing Software for Ease of Extension and Contraction[END_REF] created the artefact of program family, which is the foundation until today in engineering for reusable component and reuse-based application development [START_REF] Kang | Applied Software Product Line Engineering[END_REF]. There are several advantages to the product lines production strategy. According to the study realized by Clements and Northrop [START_REF] Clements | Software Product Lines: Practices and Patterns[END_REF] the product lines production approach decreases not only the cost per product, but also the time to market, the labor need and improves the productivity, the quality of each derived product and increases the portfolio size and therefore the possibility to gain new markets. One of the most effective means to manage software reuse in an industrial context is software product lines engineering. In particular, product lines engineering allows managing a family of applications that share a common architecture. These applications are configured as a set of components that provide core functions, but will exhibit variability in the features they provide to users because each family member will use a unique set of non-core components bound to the architecture at variation points.

In parallel with advances in software engineering for SPLs, a new generation of middleware and service-oriented architectures has emerged that is capable to adapt its behaviour according to changes in operation or environment conditions, and/or user requirements. This may require a system to adapt dynamically to environmental change in a way that is far more radical than is possible using mere modes of operation supported by parametric adaptation [START_REF] Mckinley | Composing adaptive software[END_REF]. The emergence of such re-configurable software machinery has profound implications for software development. In particular, it permits: Software to be developed that can tolerate a wide range of environmental contexts without forcing the developer to enumerate how the system will behave in each; Software to be developed in the presence of uncertainty [START_REF] Welsh | Understanding the scope of uncertainty in dynamically adaptive systems[END_REF] about the environmental contexts that may be encountered at run time. Effectively, a re-configurable capability permits design decisions to be deferred from design time to run time. This is only possible if the system is able to monitor its environment at run time, using the monitoring data and the system's own internal state to trigger adaptations as necessary. However, such adaptations must be driven by the need to satisfy the system's requirements. However, there is currently a wide conceptual gap between the capabilities of re-configurable machinery, expressed in terms of components and services, and requirements, expressed in terms of user or customer needs, with only an emerging understanding of what changing environmental context means. The latter characterizes the extent to which a system can tolerate change in the environment; from optimizing the way a fixed set of requirements are satisfied, to adapting to satisfy new requirements that emerge dynamically.

In this paper I intend to study this aspect in the context of adaptive middleware and service-oriented architectures. The possible solutions will be empirically evaluated to improve a selected solution that will be implemented into a tool be used by software practitioners and other research laboratories around the world.

A. Software Product Lines

In an SPL, "a set of software-intensive systems …. share a common, managed set of features satisfying the specific needs of a particular market segment or mission and that are developed from a common set of core assets in a pre-scribed way" 1 . Thus, a family of applications will share a common architecture, configured as a set of components that provide core functions, but will exhibit variability in the features they provide to users because each family member will use a unique set of non-core components bound to the architecture at variation points.

II. CHALLENGES AND PROBLEMATIC SITUATION

It is not possible to anticipate all possible execution contexts already at design-time (e.g. if the change is a user interface that must be adapted to an as yet unknown UI technology). At design-time, the software product lines must be specified in a way that is able to sense and react to context changes at run-time. So, the system is able to cope to a certain degree with new emerging requirements. Andersson et al. [START_REF] Andersson | Modeling Dimensions of Self-Adaptive Software Systems[END_REF] characterize the scope of change as follows: (i) taken care of, (ii) planned for or (iii) not planned for. Adapting dynamically to satisfy new requirements is unplanned change and for fully autonomous systems is currently possible only in the realm of science fiction. Adaptive middleware and service-oriented architectures are now able to support change, but dealing with change that is planned for but not taken care of at design-time is an open challenge that, when solved, will significantly extend the types of problems for which resilient software systems can be developed. However, even where adaptation involves optimizing for a fixed set of requirements (as is the case for (i) and (ii)), established techniques for reasoning about requirements are insufficient, since not only must the system's behaviour be specified when the environment is in a steady state, but the adaptive behaviour (i.e. under what circumstances must the system adapt?) have to be specified also. Worse, if, as is frequently the case, knowledge about the environment is incomplete at design time, the system's behaviour must be specified not only for the environmental contexts that the analyst believes the system will encounter, but also for those the system will encounter if the analyst's belief evidences to be mistaken. Risks of evolution are mainly related on the one hand to loss of consistency and accuracy in the components and on the other hand the loss of the benefits associated with the product lines technique. These risks are increasing directly proportional to the number of components and the complexity of the relations between them. Given the importance and the need for solutions that avoid these risks [START_REF] Mcgregor | The Evolution of Product Line Assets[END_REF], L. Yu and S. Ramaswamy [START_REF] Yu | A Configuration Management Model for Software Product Line[END_REF], Dhungana et al. [START_REF] Dhungana | Supporting the Evolution of Product Line Architectures with Variability Model Fragments[END_REF], Borba [START_REF] Borba | A theory of software product line refinement[END_REF], Neves [START_REF] Neves | Investigating the Safe Evolution of Software Product Lines[END_REF], have made several proposals to address these challenges.

1 www.sei.cmu.edu/productlines However, the arguments and empirical evidences presented by Dhungana et al. [START_REF] Dhungana | Supporting the Evolution of Product Line Architectures with Variability Model Fragments[END_REF] and Neves [START_REF] Neves | Investigating the Safe Evolution of Software Product Lines[END_REF] suggest that the few existing proposals are far from representing a satisfactory solution to the problems and challenges associated to the evolution of PLs systems. The use of SPLs can be seen as a strategy for coping with uncertainty and adapting to change, where conventionally, change occurs in the requirements of the software product lines' market. In contrast to autonomous reconfigurable systems, the derivation of new SPLs variants typically involves satisfying new goals rather than the ways in which a fixed set of goals are realized. Moreover, these goals could not be included when the SPLs was conceived. Thus, SPLs deal with unplanned change. This is possible because adaptation is done offline by human developers rather than automatically at run-time. Nevertheless, the techniques used in SPLs are exploitable. Thus, a re-configurable system can be conceptualized as a dynamic SPLs (DSPLs) [START_REF] Hallsteinsen | Dynamic Software Product Lines[END_REF], [START_REF] Sawyer | Constraint Programming as a Means to Manage Configurations in Self-Adaptive Systems[END_REF]; a set of variants within a product family, with each variant optimized to a particular environmental context. The key difference between an SPLs and a DSPLs is the time at which variants are bound; at design or deployment time for SPLs or at run-time for DPSLs.

III. RELATED WORK

Lapouchnian [START_REF] Lapouchnian | Exploiting Requirements Variability for Software Customization and Adaptation[END_REF] defines Awareness Requirements (or AwReqs) as requirements that talk about the success or failure of other requirements. AwReqs can refer to goals, tasks, quality constrains and domain assumptions. However, while I plan to use in-memory representation and a limited form of manipulation of goal models, Lapouchnian does not deal with the runtime representation of the requirements, i.e. not explicit runtime representation of AwReq is provided. Instead, they focus on the mapping from requirement models to feedback loops using the requirements monitoring framework proposed by Robinson [START_REF] Robinson | A Requirements Monitoring Framework for Enterprise Systems[END_REF] to monitor AwReq. Lapouchnian [START_REF] Lapouchnian | Exploiting Requirements Variability for Software Customization and Adaptation[END_REF] does reasoning about partial satisfaction of requirements. They offer high-level monitoring capabilities that can be used to determine satisfaction levels for AwReqs. In our case, the DSPL itself verifies if an assumption is true. If the assumption is falsified, the system automatically re-evaluates the trade-off among the softgoals, triggering an adaptation to rebalancing the softgoals if necessary. Kirsch-Pinheiro et al. [START_REF] Kirsch-Pinheiro | Requirements Analysis for Context-oriented Systems[END_REF] explored the readaptation of context-oriented systems proposing a roadmap to context management and a requirements elicitation process. DeLoach & Miller [START_REF] Deloach | A goal model for adaptive complex systems[END_REF] explore how to maintain a runtime representation of goals. However, they do not deal with the runtime representation of softgoals or goal realization strategies. As far as I am able to understand, in their research the running system interacts with the runtime goal model to trigger an update of the runtime goal model (a goal can be triggered to go from active to achieved, failed, obviated, or removed). Its main utility has been for understanding what the system is doing in terms of goals. No reasoning about partial satisfaction is done. This contrasts with Letier & Van Lamsweerde [START_REF] Letier | Reasoning about partial goal satisfaction for requirements and design engineering[END_REF], which formalizes a mean for representing partial goal satisfaction based on KAOS. A contrasting approach to partial goal satisfaction is covered by RELAX proposed by Whittle et al. [START_REF] Whittle | RELAX: a language to address uncertainty in self-adaptive systems requirement[END_REF]. Although RELAX is not goalbased per se, Cheng et al. [START_REF] Cheng | A goal-based modeling approach to develop requirements of an adaptive system with environmental uncertainty[END_REF] illustrates the use of RELAX, with KAOS goal models, using obstacle analysis to identify when to RELAX a goal. Baresi & Pasquale [39] propose adaptive goals that are aware of their own degree of satisfaction during runtime and a means to trigger adaptation.

Raúl Mazo et al. [START_REF] Muñoz-Fernández | VariaMos : an Extensible Tool for Engineering ( dynamic ) Product Lines VariaMos : an extensible tool for engineering ( dynamic ) product lines[END_REF] presented a Java-based tool (VariaMos) for defining variability modeling languages, (dynamic) product lines and re-configurable systems. In addition to that, the tool enables automated verification, analysis, configuration and simulation. Models designed with VariaMos refer to the specification of variability of dynamic product lines. So, it enables at a certain degree to modify the modeling languages at run-time. In the same place, at the same time, Muñoz-Fernández et al. [START_REF] Muñoz-Fernández | Towards a Requirements Specification Multi-View Framework for Self-Adaptive Systems[END_REF] investigated in more detail the requirements specification for re-configurable systems. They presented a multi-view framework. This framework fosters to manage uncertainty and is sufficiently expressive for re-configurable systems.

IV. RESEARCH DESIGN

In this Section, I outline how I conduct research and the reasoning about the applied concepts.

A. Research Goal

I investigate, how the idea of dynamic software product lines could help to deal with the challenges of developing efficient re-configurable software. I also offer insight into the different approaches that use dynamic software product lines engineering for developing re-configurable systems focusing on practical approaches

B. Research Questions

The research goal, combined with the problems and approaches mentioned in Section 1 lead to the following research questions:

RQ1. What elements are common in evolutionary and reconfigurable SPLs? Why is it not possible to anticipate at design-time all re-configurations and what are the requirements to adapt and evolve at run-time? The answer should help to establish a conceptual architecture to facilitate reasoning about conceptual adaption. Literature review and analysis of real interviews are fundamental to identify relevant information about how to re-configure and evolve SPLs at run-time.

RQ2.

How can I support run-time re-configuration and evolution of SPLs? Is the context required to evolve from dynamic adaptation to dynamic re-configuration? The answers to these question refers to the main goal of this project. The information is required to suggest and develop a solution that can be applied to derive a solution of which configurations are appropriate to a given context at run-time. Dynamic software product lines engineering is a mean to flexibly handle context changes at run-time. Techniques in this area are used to develop an architecture that support run-time re-configuration of re-configurable systems.

RQ3. How can the conceptual architecture be validated?

This should establish a validation framework.

C. Research Strategy

Based on the problem description, I chose Design Science Research (DSR), proposed by Hevner et al. [START_REF] Hevner | Design Science in Information Systems Research[END_REF] and V. Vaishnavi and B. Kuechler [START_REF] Vaishnavi | Design Science Research in Information Systems[END_REF], as a research strategy, as my goal is to derive an artefact, which is able to provide empirical evidence for filling the research gap. Also, because DSR determines iteratively the reality and literature foundation that emerge from research efforts. DSR as research strategy provides both the methodology of design science research by Vaishnavi et al. [START_REF] Vaishnavi | Design Science Research in Information Systems[END_REF], [START_REF] Vaishnavi | Design Science Research Methods and Patterns: Innovating Information and Communication Technology[END_REF] and the information system research framework by Hevner and Chatterjee [START_REF] Hevner | Design Research in Information Systems: Theory and Practice[END_REF]. The methodology describes the process steps and the circulation of knowledge within these steps. The framework considers previous scientific work (rigor cycle) and a real environment (relevance cycle), during the artefact development and evaluation in the field of information systems.

To apply the research strategy, the approach is based on the methodology of design science research by Vaishnavi et al. [START_REF] Vaishnavi | Design Science Research in Information Systems[END_REF] and further developed the framework of information system research by Hevner and Chatterjee [START_REF] Hevner | Design Research in Information Systems: Theory and Practice[END_REF]. Interviews with domain experts serve as our environment and the literature provide evidence for flexible adaption of reconfigurable systems at run-time is a real issue to solve and therefore has the need for a framework to support reconfigurable adaptions. The research methodology is illustrated and structured in five phases. In the subsequent Section, I elaborate in more detail how these phases are executed. The knowledge base in the right column refers to the literature. I take existing re-configurations methods into account. The framework considers both applicable knowledge (rigor cycle) and business needs (relevance cycle) during the artefact development and evaluation in the field of information systems. The rigor cycle refers primarily to the literature review in order to define the research problem. The literature is also referred in the suggestion and development of the artefact. I choose appropriate techniques and improve or adapt what is already available. The relevance cycle serves to integrate the problem domain into the research by deriving the needs from the interview sections and applying them to the artefact. It thus supports the alignment of the solution with the environment. The rigor and relevance cycles are applied iteratively during the execution of the five phases.

In the Section research methodology, I describe how the methodology of Vaishnavi and Kuechler [START_REF] Vaishnavi | Design Science Research in Information Systems[END_REF] is applied on this research. Specifically, I outline how the phases of the methodology are related to the research questions and thus the research goal. Next, I explain how the problem domain affects the research approach. An inductive approach is conducted in order to develop a new artefact for supporting run-time adaptation and evolution of SPLs.

D. Research Methodology

In this Section, I outline the applied research methodology of Vaishnavi and Kuechler [START_REF] Vaishnavi | Design Science Research in Information Systems[END_REF]. The research methodology structures, defines and justifies the research procedure applied, how I conducted research, and how results are achieved. The methodology is divided into five phases: awareness of the problem, suggestion, development, and evaluation. The following section describes the main characteristics of the five phases of the methodology.

1) Awareness of the Problem

In the first phase awareness of the problem, I gain an in depth understanding of the problems that should be solved with the developed artefact. In order to do so, I identify the requirements of run-time evolution and adaptation of SPLs. More precisely, I identify relevant information about why it is not possible to predict all re-configurations at design-time and what are the requirements to evolve at run-time by analyzing the process and arising problems found in the literature. It represents the problem where change can be planned at designtime. Doing so, the needs are properly reflected and are sound to build a solution upon. In order to gain a broader overview of the different facets of the problem, I conduct a literature review of concepts for similar solutions. To strengthen the relevance cycle, I investigate if the process contains the same issues found as during the literature review, which led to the problem description (RQ1).

2) Suggestion

In the second phase, based on the insight of the previous phase, I present a concept for a possible solution. First, I analyze the previously found requirements to find a solution for the investigated scenario. This information is needed to suggest a solution that supports run-time re-configuration and evolution of SPLs. As a next iteration in the rigor cycle, the derived requirements are validated against the problem description. This proofs that the proposed solution is valid according to the problem domain. Furthermore, the suggestion is ground while comparing it to similar solutions found in the literature review. This phase describes how to develop the desired artefact in order to solve the identified problem (RQ2).

3) Development

In the third phase, I elaborate the suggested solutions according to the results of the requirements analysis (relevance cycle iteration). To develop the artefact, I use dynamic product lines engineering techniques and design architecture, which support run-time re-configuration and evolution of SPLs. The run-time adaptation and evolution is demonstrated on the basis of real practices. The overall goal of this phase is to design an architecture, which supports run-time adaptation and evolution of SPLs (RQ2).

4) Evaluation

In the fourth phase (evaluation), I perform a validation of my results. It provides empirical evidence that the needs are reflected properly by using real instances to test the functionality of the architecture. The results of the evaluation provide answer to the question, if it is possible to evolve and re-configure SPLs at run-time (RQ3).

5) Conclusion

Finally, in the fifth phase (conclusion), the results are critically discussed.

In addition, I conduct in each phase of the research methodology personal interviews with relevant actors in the domain of SPLs. This approach strengthens the relevance cycle and increases the quality of research. The research presented has a qualitative approach and the aim of an exploratory nature.

V. ACTION PLAN

The main aim of the action plan is to conduct a study on reconfigurable and evolutionary SPLs involving literature and prominent actors in the software engineering communities and technology industries. All participants are experienced software engineers. The results of the action plan will be reported as an empirical study conducted with software engineers from different domains (e.g. technology industries) and will be useful for further research in the software engineering community. In addition, the study will be empirically validated.

The goal of the action plan is to elicit the requirements of re-configurable and evolutionary SPLs. To achieve the goal, the action plan has been organized into several process steps:

A. Identification and Formalization of the Problem Domain

The first process step is conceived as the very first interaction point of the action plan where I define and elicit the need and the vision for run-time re-configuration and evolution of SPLs. Specifically, I discuss the requirements from research, methodology, technical, architectural and user-perspectives. Indications about constraints and standards will also be defined supporting the final architecture. This phase explicitly represents the rationale about the study and determines the research gap identified during the literature review. So, new novel theory and hypotheses can be generated.

B. Objective and Preparation

To execute the reviewing activities, we use the general process of reviewing protocols proposed by Kitchenham and Charters [START_REF] Kitchenham | Guidelines for performing Systematic Literature Reviews in Software Engineering[END_REF]. We refer to the reviewing process that has been conceived with a particular emphasis on reviews conducted within the domain of software engineering [START_REF] Kitchenham | Guidelines for performing Systematic Literature Reviews in Software Engineering[END_REF]. To systematically retrieve research works in the domain of DSPLs, I perform a search using carefully a planned search query on scientific electronic databases that were accessible online (e.g. IEEExplorer, ACM Digital Library, Citeseerx Library, ScienceDirect, SpringerLink). The query will be designed based on keywords derived from the research questions (RQ1 -RQ3). This systematic literature review intends to provide an objective summary of the existing knowledge in the area of DSPLs. At the same time, I want to highlight open issues in order to suggest the areas that are in need of further contributions. Based on the gained knowledge, I will create together with domain experts a protocol for the interview sections. Important is the fact that there must be no limitation to new ideas occurring during the interview sections. So, for example questions can be added or modified during run-time. Each interview should start with general questions about the participants and their experiences. Next, I ask them which challenges they face during engineering re-configurable and evolvable SPLs. Then, possible research initiatives, which could be helpful in their current organization, are discussed. The objective is refined into three questions, which will be answered through the data collection and analysis:  How do they deal with unplanned changes?  Have they experienced a problem with unplanned changes?

 And if there is a problem: What do they expect from a solution?

An important element of the preparation is the implementation of a pilot test. The first interview will be conducted in our own laboratory and will support to improve the quality of the study. Possible weaknesses can be uncovered and eliminated.

C. Interview Design

I will conduct semi-structured interviews, which allow to consider new arising ideas during the interview sections. In general, open-ended questions are primarily. We specify a protocol that defines at design-time the general topics of interest. This relieves the collection of similar information from all participants. The protocol contains a list of questions and topics that need to be covered during run-time. The protocol can be modified depending on the appropriateness during the conversation. Since the data will be collected from human beings, it is very important to carefully select participants for the interview sections. This fosters a more reliable and valid study.

D. Collection, Analysis, and Formalization of Results

This process step involves collection and analysis of quantitative and qualitative data. Furthermore, it supports the formalization of the results of both research and development. This takes place in strict cooperation with domain experts. I will use multiple sources of data to create the study database. The collection of data will be stored in a semi-structured way for processing, analysis and dissemination. Resulting from this phase, criteria for interpreting the collected data should be identified and it should be also identified, which data element referrer to which research question.

E. Validation

We plan to apply an observational method (e.g. case study) to our architecture. This allows us to monitor and collect data over the whole study with a relatively minimal addition to cost. Valuable information can be obtained characterizing the development of the artefact. The main goal of the validation is to demonstrate a causal relationship between outcome and intervention. Furthermore, it must be analyzed in more detail, if the findings can be further generalized and applied to different domains.

F. Dissemination of Results

The last step deals with supporting activities including dissemination, exploitation, and standardization are undertaken throughout the whole action plan. Specific documentation, presentations, press releases and other materials will be prepared and disseminated to encourage further research and ongoing development of the middleware and its related technologies also after action plan completion.

VI. CONCLUSION

In this paper, I present the scientific rationale of the exploratory study. The results of this study will be useful to software engineering professionals and researchers worldwide by revealing fundamental concepts of re-configurable and evolving SPLs. The study could raise new research directions: (i) what are the challenging experiences, (ii) research initiatives, (iii) best practices.

A. Future Work

I will use the results of the interview sections and combine them with the findings of the literature (e.g. DSPLs techniques) in order to define a middleware that supports dynamic adaptation and evolution of SPLs. The aim is to develop an architecture of the middleware. The architecture will permit to the technical and research work to establish their interconnections. We provide feedback on the architecture of the middleware with respect to the elicited requirements. The assessment will be done to constantly check that research is done correctly with respect to the expectations. In addition, gathering empirical evidence is already included in our working agenda. Nonetheless, some further work is also required to perform run-time reasoning.

ACKNOWLEDGMENT

This Ph.D. study is part of research work funded by the Ministère de l'Enseignement supérieur et de la Recherche (MESR). The research project is supervised by Prof. Carine Souveyet (carine.souveyet@univ-paris1.fr), Université Paris-Sorbonne, France.