Shell structure in the vicinity of ⁴²Si

J. Fridmann, I. Wiedenhoever, L.T. Baby, P.D. Cottle, E. Diffenderfer, K.W. Kemper FSU

A. Gade, D. Bazin, B.A. Brown, C.M. Campbell, J.M. Cook, D.-C. Dinca, T. Glasmacher, P.G. Hansen, J.L. Lecouey, W.F. Mueller, J.R. Terry, K. Yoneda, and H. Zwahlen
NSCL/MSU

E. Rodriguez-Vieitez
Lawrence Berkeley Laboratory

J.A. Tostevin
University of Surrey

The quest for information about shell structure in N=28 nuclei near the neutron dripline is of central importance to the field of nuclear structure physics for two reasons.  First, the nuclei in the vicinity of ⁴²Si provide the first arena in which ideas about how changes in the spin-orbit force affect shell structure near the neutron dripline [1,2] can be tested.  Indeed, it has been predicted [3-9] that the N=28 shell closure should be less well developed, or even collapse altogether, in the nuclei near ⁴²Si.  Three recent experimental results, one a measurement of the lifetime of the b-decay of ⁴²Si [10], the second the determination that ⁴³Si is bound [11] and the third a mass measurement of ⁴²Si [12] have been used to argue that the N=28 shell closure has narrowed or collapsed in ⁴²Si, resulting in a well-deformed shape for this nucleus.  On the other hand, it has been argued in Refs. [13,14] that a possible proton subshell closure at Z=14 would have a strong effect on the structure of ⁴²Si, preventing it from being well-deformed.  

The second reason that experiments on nuclei near ⁴²Si are important is that they are providing a rigorous testing regime for experimental techniques that will be used heavily at the next generation of radioactive beam facilities.  Among these techniques are the intermediate-energy knockout reactions in which cross sections provide spectroscopic information similar to that obtained for many years from direct transfer reactions used at low-energy stable beam facilities [15,16].

We have performed a set of measurements of the N=28 isotones ⁴²Si, ⁴³P and ⁴⁴S using one- and two-proton knockout reactions from the radioactive beam nuclei ⁴⁴S and ⁴⁶Ar at the National Superconducting Cyclotron Laboratory.  Accounts of this work have been published in Nature [17] and Physical Review C [18].

Knockout reactions were induced on secondary beams of 98.6 MeV/nucleon ⁴⁴S and 98.1 MeV/nucleon ⁴⁶Ar by a beryllium foil.  The measurement of the residual nucleus ⁴²Si was performed via the two-proton knockout reaction on the ⁴⁴S secondary beam, while the ⁴³P measurement was performed with the one-proton knockout reaction on the same secondary beam.  Finally, ⁴⁴S was measured via the two-proton knockout reaction on the ⁴⁶Ar secondary beam.  The spectra from the NSCL’s S800 spectrograph used to identify residual nuclei are shown in Fig. 1.  The inclusive cross sections for production of the ⁴²Si, ⁴³P and ⁴⁴S residual nuclei were 0.12(2) mb, 7.6(11) mb and 0.23(2) mb, respectively. 

The g-ray spectra in coincidence with the ⁴²Si, ⁴³P and ⁴⁴S residual nuclei are shown in Fig. 2.  These spectra are Doppler-reconstructed so that they appear as in the rest frames of the residual nuclei.  There are no discernable g-ray peaks in the ⁴²Si spectrum.  The ⁴³P spectrum includes a single large peak at 184(3) keV.  No other peaks are clearly discernable, although there are background counts up to 1 MeV.  For ⁴⁴S, the 2₁⁺®0_gs⁺ g-ray previously reported in Refs. [19-21] is seen clearly in our spectrum and we assign an energy of 1.329(10) MeV. 

The present measurement of ⁴²Si provided two experimental conclusions: first, that the inclusive cross section for producing this nucleus in the two-proton knockout reaction is 0.12(2) mb, and second, that there are no discernable g-rays in the spectrum.  The inclusive cross section is small, smaller than any previously observed for the two-proton knockout reaction [16].  Bazin et al. [16] pointed out that the cross section for the two-proton knockout reaction depends on the number of valence protons, so the small cross section observed here suggests that a shell closure occurs at Z=14 – where the d_5/2 proton orbit fills.  Indeed, in the N=28 isotone ⁴⁸Ca, the (d,³He) reaction [22,23] revealed a large gap between the d_5/2 proton orbit and the d_3/2 and s_1/2 proton orbits (which are nearly degenerate in ⁴⁸Ca).  The small two-proton knockout cross section provides strong evidence that the Z=14 shell gap is substantial in ⁴²Si as well. 

We performed shell model calculations and reaction calculations to examine how neutron and proton shell structure affect the spectroscopy of ⁴²Si and ⁴⁴S.  The shell model  calculations use the interaction of Ref. [24] with the neutrons occupying a model space including the 0f_7/2 and 1p_3/2 orbits and the protons occupying the full sd space.  The two-nucleon amplitudes that resulted were used to calculate two-proton knockout cross sections with a model that extends that described in by including diffractive effects [18]. 

The calculated inclusive two-proton knockout cross section for ⁴²Si, using wavefunctions generated with the shell-model effective interaction of Ref. [24], is 0.32 mb, about a factor of three larger than the experimental value of 0.12(2) mb.  However, taking into account the expected reduction factor R_s(2N) of 0.5, there is a closer agreement with the measured value.  It is also worth noting that 92% of the cross section calculated for ⁴²Si using these parameters is located in the ground state; therefore, the contribution of the excited states to the inclusive cross section measurement is likely to be small.

As mentioned above, the reason that the cross section is small is because of the Z=14 shell gap; that is, the energy gap between the the and the d_3/2 - s_1/2 pair and the d_5/2 orbit is large.  We examined how the theoretical cross section depends on the size of the Z=14 gap by performing several shell model calculations using the parameters of [24] but in which the d_3/2 – d_5/2 proton gap is set to three values – the 5.9 MeV gap prescribed by [24], a 4.9 MeV gap, and a 2.9 MeV gap.  It is worth noting that there is an experimental reason to believe that the prescribed gap may be too large - an analysis of the centroids of proton hole strength observed in ⁴⁷K gives a value for this gap of 4.8 MeV [23].  In addition, the size of the N=28 neutron gap was also reduced by 1 MeV (from its prescribed value of 3.6 MeV for ⁴²Si [24]) to examine how this affects the cross section.  In all, four calculations were performed - the first with both the proton and neutron gaps at the values from [24]; the second with the neutron gap reduced by 1 MeV and the proton gap left at the value from [24]; the third with both the proton and neutron gaps reduced by 1 MeV from the [24] values; and the fourth with the neutron gap reduced by 1 MeV from the [24] value and the proton gap reduced by 3 MeV from the [24] value.

The cross section results from the four calculations, including the R_s(2N)=0.5 suppression, are shown in the top panel of Fig. 3.  The reduction of the neutron gap does not affect the two-proton knockout cross section.  Furthermore, the reduction of the proton gap by 1 MeV does not significantly affect the cross section, either.  This is not surprising since even with this reduction the gap is 4.9 MeV.  However, a 3 MeV reduction in the proton gap does result in a significant increase in the cross section.

The bottom two panels of Fig. 3 show two other sets of spectroscopic results for ⁴²Si from the four shell model calculations - the energy of the lowest 2₁⁺ state, and  B(E2;0_gs⁺→2₁⁺).  Of these two observables, the reduced matrix element is the more reliable indicator of quadrupole collectivity.  Reducing the neutron gap by 1 MeV has a strong effect on both these observables.  Adding the 1 MeV reduction of the proton gap has little additional effect on E(2₁⁺), but causes a significant additional increase in B(E2;0_gs⁺→2₁⁺).  The large (3 MeV) reduction in the proton gap from its original value of 5.9 MeV causes a near-doubling in B(E2;0_gs⁺→2₁⁺) from its value with a 1 MeV proton gap reduction.  At this point, proton excitations are playing a large role in driving deformation.

A calculation of the inclusive two-proton knockout cross section for ⁴⁴S using shell model wavefunctions determined with the parameters of [24] yields a result of 0.66 mb, which is (as in the case of ⁴²Si) much larger than the experimental value of 0.23(2) mb.  Once again, the R_s(2N) systematics lead to an expected theoretical value of around 0.33 mb, in closer agreement with the measured value.  The calculation qualitatively reproduces the increase in cross section from ⁴²Si to ⁴⁴S with the addition of valence protons.

It is reasonable to conclude from comparing the cross section data to these calculations that there is a large gap at Z=14, although these data cannot provide a quantitative measure of the size of this gap.  With respect to the urgent question of whether the N=28 gap has narrowed from its size in ⁴⁸Ca or even disappeared altogether, the present data cannot provide any insights.  Instead, a measurement of B(E2;0_gs⁺→2₁⁺) in ⁴²Si would provide much more information on the size of the neutron gap.

The one-proton knockout reaction preferentially populates states that have the structure of a proton hole in the beam nucleus [15].  Therefore, the states in ⁴³P populated with the largest cross sections in the one-proton knockout reaction on ⁴⁴S are expected to be those that are single protons in the d_3/2 or s_1/2 orbits, or a single d_5/2 proton hole.  In ⁴⁷K, (a proton hole coupled to the doubly-magic nucleus ⁴⁸Ca), the centroids of the strength from the d_3/2 or s_1/2 proton orbits are seen to be separated by only 300 keV [22,23].  In the present measurement of ⁴³P, the 184 keV g-ray suggests that the strength of one of these two single-proton orbits is concentrated in the ground state with the strength of the other orbit concentrated in an excited state at 184 keV.  The d_5/2 strength is expected at higher excitation energies in ⁴³P.  The cross sections for the ground state and 184 keV state are large and therefore support this picture.  The inclusive cross section – which includes both the ground state and the 184 keV state - is 7.6(11) mb.  An examination of the residue-g-ray coincidences shows that the 184 keV state accounts for 75±15% of the cross section.  The combination of this observation regarding the relative cross sections of the two states and calculations based on the prescription given in [25,26] provide a strong argument that the s_1/2 proton strength is concentrated in the ground state, while the d_3/2 strength is concentrated in the 184 keV state.

The spin-orbit splitting of d_3/2 and d_5/2 proton orbits is an important parameter for interpreting the results of the present two-proton knockout  study of ⁴²Si, other measurements of this nucleus and data on nearby nuclei.  While no evidence for the d_5/2 strength was observed in the present data set, the one-proton knockout reaction on ⁴⁴S  provides a means for determining the d_3/2-d_5/2 proton spin-orbit splitting.  A calculation of the distribution of the d_5/2 proton hole strength in ⁴³P using the Ref. [24] parameters, including the 5.9 MeV d_3/2-d_5/2 splitting, and the resulting cross sections for population of these states in the one proton knockout reaction (with the cross section calculation prescription as described in Refs. [25,26]) is shown in the top panel of Fig. 4.  It provides a prediction of a concentration of d_5/2 proton hole strength at 2.24 MeV with a somewhat smaller concentration at 1.54 MeV.  This yields a centroid of 2.1 MeV.  If instead the d_3/2-d_5/2 splitting is set to 4.8 MeV, as shown in the bottom panel of Fig. 4, the centroid of the d_5/2 strength is 1.5 MeV.  We conclude that the location of the d_5/2 proton hole strength in the one-proton knockout reaction provides a sensitive scale for determining the d_3/2-d_5/2 proton spin-orbit splitting, and that this provides a motivation for another ⁴⁴S one-proton knockout experiment with the sensitivity (increased by means of greater statistics than available in the experiment reported here) required to detect the fragments of the d_5/2 strength.

 Quite recently, a group from GANIL published results of a new study of ⁴²Si and ^41,43P, with ⁴²Si and ⁴³P being populated via the same reactions used for our experiments, but with greater statistics [27].  They identified a 770 keV g-ray in ⁴²Si and assigned it to be the 2₁⁺®0_gs⁺ transition.  On the basis of this, they concluded that ⁴²Si is deformed.  In addition, the GANIL group placed two more g-rays in ⁴³P.   

 

[1] W. Nazarewicz and R.F. Casten, Nucl. Phys. A 682, 295c (2001).

[2] D. Warner, Nature 430, 517 (2004).

[3] T.R. Werner et al., Phys. Lett. B 333, 303 (1994).

[4] T.R. Werner et al., Nucl. Phys. A 597, 327 (1996).

[5] J. Terasaki, H. Flocard, P.-H. Heenan, and P. Bonche, Nucl. Phys. A 621, 706 (1997).

[6] G.A. Lalazissis, A.R. Farhan, and M.M. Sharma, Nucl. Phys. A 628, 221 (1998).

[7] G.A. Lalazissis, D. Vretenar, P. Ring, M. Stoitsov, and L.M. Robledo, Phys. Rev. C 60, 014310 (1999).

[8] S. Peru, M. Girod, and J.F. Berger, Eur. Phys. J. A 9, 35 (2000).

[9] R. Rodriguez-Guzman, J.L. Egido, and L.M. Robledo, Phys. Rev. C 65, 024304 (2002).

[10] S. Grevy et al., Phys. Lett. B 594, 252 (2004).

[11] M. Notani et al., Phys. Lett. B 542, 49 (2002).

[12] H. Savajols et al., Eur. Phys. J. A {\bf 25}, s01, 23 (2005).

[13] P.D. Cottle and K.W. Kemper, Phys. Rev. C 58, 3761 (1998).

[14] E. Caurier, F. Nowacki, and A. Poves, Nucl. Phys. A 742, 14 (2004).

[15] P.G. Hansen and J.A. Tostevin, Annu. Rev. Nucl. Part. Sci. 53, 219 (2003).

[16] D. Bazin et al., Phys. Rev. Lett. 91, 012501 (2003).

[17] J. Fridmann et al., Nature 435, 922 (2005).

[18] J. Fridmann et al., Phys. Rev. C 74, 034313 (2006).

[19] T. Glasmacher et al., Phys. Lett. B 395, 163 (1997).

[20] D. Sohler et al., Phys. Rev. C 66, 054302 (2002).

[21] S. Grevy et al., Eur. Phys. J. A 25, 111 (2005).

[22] P. Doll et al., Nucl. Phys. A 263, 210 (1976).

[23] S.M. Banks et al., Nucl. Phys. A 437, 381 (1985).

[24] S. Nummela et al., Phys. Rev. C 63, 044316 (2001).

[25] J.A. Tostevin, Nucl. Phys. A 682, 320c (2001).

[26] A. Gade et al., Phys. Rev. C 69, 034311 (2004).

[27] B. Bastin et al., Phys. Rev. Lett. 99, 022503 (2007).

Figure 1

Particle spectra used to identify ⁴²Si from the two-proton knockout reaction from ⁴⁴S (top), ⁴³P from the one-proton knockout reaction from ⁴⁴S (middle), and ⁴⁴S from the two-proton knockout reaction from ⁴⁶Ar (bottom).  The energy loss in the ion chamber at the focal plane of the S800 spectrograph is plotted on the vertical axis, and the horizontal axis plots the path-corrected time of flight between the object point of the spectrograph and the focal plane, with shorter flight times to the right.

Figure 2

Spectra of g-rays detected in coincidence with the residual nuclei shown.

Figure 3

Spectroscopic observables in ⁴²Si calculated with the shell model and four sets of parameters for the Z=14 and N=28 (sub)shell closures, as described in the text. 

Figure 4

Distribution of the proton strength in ⁴³P from the one-proton knockout reaction on ⁴⁴S calculated using two different values for the d_3/2-d_5/2 proton spacing and (otherwise) the parameters from Ref.[24].  The top panel uses the spacing from Ref. [24], while the bottom panel uses a spacing 1 MeV smaller.