Recently measured reaction cross sections with low energy fp-shell ...

PHYSICAL REVIEW C 73, 054601 (2006)

Recently measured reaction cross sections with low energy fp-shell nuclei as projectiles: Microscopic description A. Bhagwat1,2 and Y. K. Gambhir1,∗ 1

Department of Physics, I.I.T. Powai, Mumbai 400076, India 2 KTH (Royal Institute of Technology), Alba Nova University Center, Department of Nuclear Physics, S-106 91 Stockholm, Sweden (Received 9 January 2006; published 3 May 2006) The finite range Glauber model along with the Coulomb modification is used to analyze recently measured reaction cross sections with neutron-deficient Ga, Ge, As, Se, and Br isotopes as low-energy projectiles incident on 28 Si target. The required input, namely the neutron and proton density distributions of the relevant projectiles and the target, are calculated in the relativistic mean-field framework. Though the calculations qualitatively agree with the experiment, on the average, slightly overestimate the cross sections. A phenomenological expression with a single parameter is proposed that consistently improves the agreement with the experiment. DOI: 10.1103/PhysRevC.73.054601

PACS number(s): 21.10.Gv, 21.60.−n, 25.60.Dz, 25.60.Gc

I. INTRODUCTION

The initial experiments [1] with the radioactive ion beams measured the total reaction cross sections σR with high energy (∼800A MeV) projectiles. The root-mean-square rms neutron/matter radii of the projectiles have been subsequently extracted [1,2] from the corresponding measured reaction cross sections, within the Glauber model [3]. The zero range Glauber model in the optical limit (GM-Z) used [1,2] for this purpose was adequate at such high energies. The lowenergy σR measurements, now gradually being reported [4–7], compliment the corresponding available high-energy σR data. Because of the increase in the nucleon-nucleon interaction cross section at low energies, these measured σR are expected to be more sensitive to the nuclear matter distribution in the tail region of the nucleus. However, the GM-Z needs to be modified to include the finite range and Coulomb effects, which now become important at low energies. It is to be noted that such an extraction of the rms radii from the measured reaction cross sections, unfortunately, is model dependent. We stress that it is more appropriate to compute and compare directly the cross sections σR that are the measured quantity rather than emphasizing the comparison of the extracted quantities like rms radii. To supplement the GM-Z analysis at high energies, a finiterange Glauber model (GM-F) analysis has been carried out for recently measured reaction cross sections at low energies for the neutron-deficient isotopes of Ga, Ge, As, Se, and Br of the pf shell as projectiles incident on 28 Si target. The present analysis proceeds in two steps. The ground-state properties like binding energies, deformations, sizes (radii), densities are calculated first in the well tested and most reliable relativistic mean-field (RMF) framework [8–13]. The RMF neutron and proton density distributions of the relevant projectiles and the target required as input are then used in GM-F to compute σR . II. RMF CALCULATION: RESULTS AND DISCUSSION OF GROUND STATE PROPERTIES

The RMF equations are solved [11] in the axially symmetric deformed oscillator basis, using the constant gap approxima∗

Electronic address: [email protected]

0556-2813/2006/73(5)/054601(6)

054601-1

tion with a cutoff (2¯hω). We employ the most widely used Lagrangian parameter set, NL3 [14]. The neutron and the proton gaps are adjusted so as to reproduce separately the neutron and proton pairing energies obtained while solving the relativistic Hartree-Bogoliubov (RHB) equations [12,13] in the isotropic oscillator basis using the Gogny D1S interaction [15,16] that is known to have right content of pairing. For odd-A nuclei, the last odd nucleon does not have a partner to occupy its time reversed state. As a result, the meanfield ground-state wave function does not have time reversal symmetry. For this purpose, we follow the tagged Hartree-Fock procedure, successfully used in the nonrelativistic calculations. Similar procedure has been extended for odd-odd nuclei. The RMF calculations reproduce the experiment rather well as expected. The experimental binding energies [17] are reproduced within ∼0.25%. The deformation parameters β agree with the experiment (where available) and in general closely match with the corresponding Möller and Nix (MN) values [18]. At places the RMF β has opposite sign than that of MN. However, at these very places, there appears another solution at a small (0.5 MeV) excitation with β having the same sign as and value close to that of MN. The calculated charge radii differ from their corresponding experimental values (where available) [19–23] only at second decimal place of a Fermi. These observations are now standard and well established (see, for example, Refs. [24,25]). A systematic study of the densities and nuclear sizes for the neutron-deficient nuclei in the fp shell involved in the rp process of explosive nucleosynthesis may reveal interesting structure information. The present calculation does not reveal any striking abrupt changes in either the density distributions or radii. The calculated proton (neutron) density distributions for a given Z(N ) hardly differ. Further, the proton (neutron) density distributions for a given N (Z) only slightly differ mostly at the surface to satisfy the proper norm condition. The charge radii (rc ) are extracted from the calculated point proton rms radii (rp ), corrected for the finite proton size (0.8 fm) through: rc = rp2 + 0.64. (1) ©2006 The American Physical Society

A. BHAGWAT AND Y. K. GAMBHIR


5.0

5.0

4.8

4.8

Gallium

4.6

4.4 RMF Extr.

4.2 4.0

rm (fm)

rm (fm)

4.4

3.8

4.0 3.8 3.6

3.4

3.4

3.2

3.2 63

64

65

66

67

68

3.0 64

69

RMF Extr.

4.2

3.6

3.0 62

Germanium

4.6

65

Mass Number

68

69

70

71

5.0

Arsenic

4.8

4.8

4.6

4.6

4.4

4.4

4.2

4.2

rm (fm)

rm (fm)

67

Mass Number

5.0

4.0 3.8 3.6

Selenium

FIG. 1. The calculated rms matter radii (RMF) for the neutron-deficient Ga, Ge, As, Se, and Br isotopic chains. The corresponding extracted values [7] (“Extr.”) are also shown for comparison.

4.0 3.8 3.6

3.4

67

68

69

RMF Extr.

3.4

RMF Extr.

3.2 3.0 66

66

3.2 70

71

72

3.0 68

73

69

70

71

72

73

74

75

Mass Number

Mass Number 5.0

Bromine

4.8 4.6

rm (fm)

4.4 4.2 4.0 3.8 3.6

RMF Extr.

3.4 3.2 3.0 70

71

72

73

74

75

76

77

Mass Number

The calculated rc for a given Z (set of isotopes) hardly vary with neutron number, only 1% increase for all sets of isotopes except for Ga isotopes where the increase is about 2%. The rms matter radii (rm ) for these neutron-deficient nuclei extracted from the measured cross sections (σR ) in the Glauber model, reported recently [7], exhibit tiny anomalous behavior (deviation from the conventional A1/3 rule). As stated, these so-called experimental radii extracted using such a procedure are model dependent. Further, these measured σR have large error bars. It is known [24,25] that the RMF successfully reproduces such an anomalous behavior observed in the isotopic shift measurements at various places of the periodic table. These isotopic shift measurements are precise and accurate (up to the second decimal place of Fermi). Unfortunately, no such measurements, to our knowledge, exist for these neutron-deficient nuclei. Therefore, it would be interesting to compare our matter radii (rm ) with those

of Ref. [7]. The rms matter radii (rm ) are deduced from the calculated rms proton (rp ) and the neutron (rn ) radii through: N rn2 + Zrp2 . (2) rm = N +Z The calculated rms matter radii rm labeled as RMF for the neutron deficient Ga, Ge, As, Se, and Br isotopic chains are displayed in Fig. 1. The corresponding extracted values [7] indicated by “Extr.” are also shown for comparison. Clearly, the RMF and “Extr.” values, though they agree qualitatively, do show minor deviations. The RMF values are found to increase very slowly and monotonically with the mass or neutron number for all the isotopic chains, whereas the “Extr.” values exhibit small variations. If the calculated rm values are decreased slightly, say ∼0.2 fm, then RMF results match the “Extr.” on the average. We are not sure if this structural behavior

054601-2

RECENTLY MEASURED REACTION CROSS SECTIONS . . .


trate on the calculation and discussion of σR and its comparison with the experiment.

1600 12

1400

12

C on C

III. ESSENTIALS OF GLAUBER MODEL: REACTION CROSS SECTION

σR (mb)

1200

Within the finite range approximation, the transparency function is written as:     T (b) = exp − ij (|b − s + t|)ρ¯iP (t)ρ¯jT (s) ds d t .  

1000 800 600

GM-F GM-Z Fit Expt.

ij

(3)

400 10

100

1000

Here, the summation indices i and j run over neutrons and protons of the target and the projectile. The superscript T (P ) refers to target (projectile) and ρ(s) ¯ is the z direction integrated nucleon density distribution expressed as: +∞ ρ(s) ¯ = dz ρ ( s 2 + z2 ), (4)

Energy (MeV/nucleon) 2000 12

1800

C - Isotopes on C 39.0

σR (mb)

1600

−∞

27.4

1400

with s = (x + y ). The profile function is given by [27,28]: 2 beff 1 ij (beff ) = σij exp − 2 2πβ 2 2β 2

33.4 40.7

20.7

1200

GM-F GM-Z Fit Expt.

1000

800 10 11 12 13 14 15 16 17 18

Mass Number FIG. 2. The calculated reaction cross sections, along with the corresponding experimental values (Expt.) [5,28,32,33].

of “Extr.” is genuine or is just an outcome of the uncertainties inherent in the procedure adopted. The measured cross sections σR have large error bars. In addition the Glauber model requires the neutron and the proton densities of both the projectile and the target. The densities used by Ref. [7] are extrapolated from the corresponding charge densities of the stable isotopes with additional assumptions for the Woods-Saxon shape of neutron densities. The number of parameters involved exceeds the number of the experimental data (σR ) to be fitted. Further, it is expected that the extracted radii “Extr.” should show the trend as observed in the measured cross sections. Clearly, it is not so; the trend of “Extr.” data does not match (see Fig. 3) the trend observed in σR . However, the RMF has been shown to provide exceptionally accurate description of such anomalous behavior. For example, the very rich structure in the variation of charge radius (rc ) of Ne isotopes revealed in the isotopic shift measurements accurately emerges from the RMF studies [24]. Therefore, we suggest that before the “Extr.” values of Ref. [7] are to be taken seriously, more precise and reliable analysis for the extraction of the radii is required. It has been verified that the use of the other Lagrangian parameter sets like NLSH [26] or NL1 [9] do not alter the systematics and the conclusions drawn here. As remarked before, it is more opportune to compute and compare directly the cross section σR that is the measured quantity. Therefore, with this view in mind, we now concen-

2

2

(5)

with the requirement that its norm should be σij . In this expression, beff = |b − s + t|. (6) b is the impact parameter and s and t are just the dummy variables for integration over the z-projected target and projectile densities. The range parameter β is [28],

E β = βNN = 0.996 × exp − + 0.089, (7) 106.679 where, E is the projectile energy. This range parameter is obtained by fitting 12 C on 12 C cross sections from 30A MeV to 1A GeV energies. The nucleon-nucleon (NN) cross section σij is as usual taken from the experiment or taken from some empirical fit to the experimental nucleon-nucleon cross sections (see, for example, Ref. [29]). Especially for the lower energies, apart from the finite range effect, another important aspect is required to be taken into account: the Coulomb effects. The straight-line trajectories assumed in the Glauber-model get distorted because the Coulomb force becomes significant at lower energies. This effect can be incorporated in the Glauber model through the classical perihelion. Under this assumption, the Coulomb modified impact parameter (b ) can be written as [30]: 1 (8) [η + (η2 + k 2 b2 )1/2 ], k where, η is the usual Sommerfeld parameter and k is the wave number of projectile. With this correction, the total reaction cross section is expressed as: ∞ η σR = 2π b 1 − [1 − T (b )]db . (9) kb 2η/k b =

Replacing the nucleon profile function in Eq. (3) by δ function times the experimental NN cross section σij (zero

054601-3



range limit), the transparency function T (b) reduces to: σij ρ¯iP (s)ρ¯jT (|b − s|) ds . (10) T (b) = exp −

reaction cross section at low projectile energies, we present the calculated σR for 12 C and C isotopes projectiles incident on 12 C target in Fig. 2. The density of the 12 C target has been taken from an earlier work [31]. The σR obtained by using the finite (zero) range Glauber model with Coulomb corrections, in the optical limit, are labeled by GM-F (GM-Z), whereas the results obtained by using the phenomenological single parameter expression [Eq. (11)] are denoted by “FIT.” The corresponding experimental values (Expt.) [5,28,32,33] are also included for comparison. It turns out that the GM-F results consistently are larger than the corresponding GM-Z values by about ∼15% at very low energies. The difference between GM-F and GM-Z values decreases with the increase in the projectile energy and eventually at higher energies (above 300A MeV) both GM-F and GM-Z yield almost identical results close to the corresponding experimental values, as expected. At lower energies GM-F is clearly superior and agrees relatively better with the experiment. We now present and discuss the most recently reported [7] σR for low energy (50A–60A MeV) neutron-deficient

i,j

Calculation, results and discussion

The calculation of the reaction cross section in the Glauber model requires the NN cross section and the density distributions of both the target and the projectile. The former is taken from experiments and is then multiplied by a phenomenological factor (usually taken to be 0.8 [2]) to partly incorporate the effects because of the nuclear medium. All the projectile and the target (28 Si) densities are obtained in the RMF framework. In the case of the deformed densities, the L = 0 component (spherical) is projected out and then renormalized separately for the protons and the neutrons. The resulting nucleon density distributions closely agree with the experiment (where available). To reaffirm the necessity to use finite range Glauber model with Coulomb corrections for the calculation of the 3.6

3.5 28

28

Ge - Isotopes on Si

3.2

3.1

2.8

2.7

σR (b)

σR (b)

Ga - Isotopes on Si

2.4 2.0 1.6 62

GM-F GM-Z FIT Expt.

63

64

2.3 GM-F GM-Z FIT Expt.

1.9

65

66

67

68

1.5 64

69

65

66

Mass Number 3.6

σR (b)

3.0


69

70


71

72

1.8 67

73

68

Mass Number

69

70

71

72

Mass Number

3.8 28

Br - Isotopes on Si 3.4 3.0 2.6 GM-F GM-Z FIT Expt.

2.2 1.8 70

71

FIG. 3. The calculated and the corresponding experimental [7] reaction cross sections for fpshell nuclei as projectiles on 28 Si target.

2.2

σR (b)

σR (b)

2.8

68

70

28

3.4

67

69

Se - Isotopes on Si

3.2

1.6 66

68

3.8

28

As - Isotopes on Si

2.0

67

Mass Number

71

72

73

74

75

76

77

Mass Number

054601-4

73

74

RECENTLY MEASURED REACTION CROSS SECTIONS . . .


Ga, Ge, As, Se, and Br isotopes incident on 28 Si target. It is to be mentioned that no corresponding high-energy data exist. The results are presented in Fig. 3. Overall the calculation are in qualitative agreement with the experiment. It is to be noted that the experimental σR (Expt) have large uncertainties. This aspect hampers the discussion and the quantitative comparison between the calculation and the experiment. The GM-F values are larger than the corresponding GM-Z values by 10–15%. The GM-F and so also GM-Z results for a given chain of isotopes show very little (few percentages) increase with the addition of neutrons. This reflects very little increase in the corresponding radii and is consistent with the systematics of the calculated radii. However, the experimental σR show small variations with the neutron number for a given set of isotopes. However, because of the large uncertainties in the measured σR , the reliability and the genuineness of these small variations may not be certain. Therefore, more precise measurements of σR are required before arriving at definite conclusion(s). The finite range GM-F overestimates the cross sections. This overprediction may be attributed to the use of the value of the range parameter solely determined from the fit to the 12 C-12 C data. This range parameter may have a different value for heavier target (28 Si) and relatively heavy projectiles. As stated before, we do not intend to introduce additional parameter(s). Instead, we propose the following one parameter expression based on the inspection and study of the variation of the calculated σR with energy and different projectile-target combinations: σR = σZR 1 +

α exp(−E 0.2 ) . AT AP

The minimum χ 2 for FIT is amply reflected from its resulting improved agreement with the experiment and is clearly visible from the figures. In addition, it does not destroy the good agreement of GM-F with the experiment already achieved [34] for low-energy lighter projectiles. This indeed is gratifying.

IV. SUMMARY AND CONCLUSION

In summary, finite-range Glauber model with Coulomb modifications (GM-F) analysis has been carried out for recently measured reaction cross sections at low projectile energies for the neutron-deficient isotopes of Ga, Ge, As, Se, and Br incident on 28 Si target. The calculations proceed in two steps. In the first step, the ground-state properties of the relevant nuclei are calculated using the RMF formulation. As expected, the RMF calculations give an excellent account of the ground-state properties (e.g., binding energies, deformations, radii, densities, etc.) of the relevant nuclei. In the second step, the calculated (RMF) point target/projectile densities of both the neutrons and the protons are used in the Glauber model to compute the cross sections. Overall, the GM-F reproduce the experiment well. It does overestimates σR by about 10–15%. A simple expression with a single parameter is proposed for σR based on the systematic inspection and study of the variation of the calculated σR with projectile energy and with the projectile-target combinations. It consistently yields better fit to the experiment.

(11) ACKNOWLEDGMENTS

Here, σZR corresponds to the GM-Z (zero range range Glauber model) value, AT (AP ) refers to the target (projectile) mass number, and E denotes the energy of the projectile. The parameter α is determined through χ 2 fit to all the data (σZR ) [34]. Its value is α = 170.56 ∼ 12(π r0 )2 , with r0 = 1.2 fm.

The authors are thankful to S. H. Patil, P. Ring, and J. Meng for their interest in this work. Partial financial support from the Department of Science and Technology (DST), Government of India (project SR/S2/HEP-13/2004), is gratefully acknowledged. A.B. acknowledges financial support from the Swedish Institute (SI).

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