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Experimental state of n-n correlation function for Borromean halo nuclei investigation
Nuclear Physics A 790 (2007) 235c–240c
Experimental state of n-n correlation function for Borromean halo nuclei investigation M. Petrascu,a A. Constantinescu,a ∗ I. Cruceru,a M. Duma,a M. Giurgiu,a† A. Isbasescu,a H. Petrascu,a S. Serban,a V. Stoica,a C. Bordeanu,a I. Tanihata,b W.G. Lynch,c M.A. Famiano,d K. Ieki.e a
Horia Hulubei National Institute for Physics and Nuclear Engineering P.O. Box MG-6, Bucharest, Romania
b c
National Superconducting Cyclotron Laboratory, MSU, MI 49008, USA
d e
TRIUMF, Vancouver B.C. V6t, 2A3 Canada
Western Michigan University, Kalamazoo, MI 49008, USA
Rikkyo University, Tokyo 171, Japan
The present experimental and theoretical state of Cnn correlation function for Borromean halo nuclei investigation is reviewed. Some of the consequences of a recently appeared new theory of Cnn , [1] together with the experimental possibilities to test this theory will be presented in this contribution. 1. INTRODUCTION It was predicted in [2] that, due to the very large radius of 11 Li, and due to the very low binding energy of the halo neutrons, one may expect that in a fusion process on a light target, the halo neutrons may not be absorbed together with the 9 Li core, but may be emitted in the early stage of the reaction. Indeed, the experimental investigation of Si(11 Li, fusion) has shown that, a fair amount of fusion events [3] are preceded by the preemission of one or two halo neutrons. In [3] was also found that in the position distribution of the halo neutrons, a very narrow forward neutron peak is present. Considering that this peak may be due to neutron pairs, it was decided to investigate the neutron pre-emission in condition of much higher statistics by means of an array detector [4]. Indeed within the narrow forward peak (9 msr) a large number of n-n coincidences was found [5,6]. Trying to built the n-n correlation function, a serious problem appeared connected with the low values of the correlation strength [7–9], confirming the previous measurements of [10] and [11]. A possible explanation could be the residual correlation of the halo neutrons [11]. In [11], has been proposed an iterative calculation to compensate for the residual correlation. But in [7,9], was shown that the iterative calculation is considerably increasing the error ∗ †
Permanent address, University of Bucharest, Faculty of Physics Permanent address, Technical University of Bucharest
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.03.150
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M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
so that is no more possible to draw any conclusion concerning the theoretical predictions. In [7] was proposed an experiment for getting the intrinsic correlation function by using 11 Li and 11 Be beams. The halo nucleus 11 Be would be ideal for the denominator of the correlation function construction, because it has only one halo neutron and therefore no residual correlation can be possible. In [7] was also proposed to do the new measurements, by using a C target instead a Si target, because due to the target screening which is lower in the case of C target, the yield of halo neutron pre-emission is expected to be higher in this case. A preliminary estimation quoted in [7], indicated that for C target, the yield of halo neutron pre-emission is about 2 times higher than for a Si target. It will be shown in the section dedicated to the target screening effect, that more accurate calculation is indicating that the yield of halo neutron pre-emission is expected to be near 3.5 larger in the case of C target than in the case of Si target. The present experimental state of Cnn investigation, will be reviewed in section 2. Very recently, a new theory for Cnn correlation function has been proposed [1]. In this theory, the 11 Li halo nucleus is modeled as a three body system consisting of 2 neutrons and a core. It is shown that a minimum is present in Cnn , due to the coherence of the 2 halo neutrons. This theory will be briefly reviewed in section 3. An analysis of the possibility to test experimentally this new theory, will be presented in section 4. In section 5, an experimental indication on the validity of the new theory and in section 6, the conclusions will be presented. 2. THE PRESENT EXPERIMENTAL STATE OF Cnn INVESTIGATION The correlation function Cnn (q) is in principle a powerful tool for the space-time proximity investigation of particles emitted from nuclei. In the case of Borromean halo nuclei, due to the fact that the neutrons are emitted in very fast processes (10−23 s) as Coulomb dissociation or pre-emission in the nuclear field, the space proximity is dominant because is characterized by final state interaction (FSI) much larger than QSS (quantum statistical symmetry). Therefore in the case of halo nuclei one should expect n-n correlation strengths much larger in comparison with the neutron evaporation processes dominated by QSS, in which case the correlation strength is lower than 2. One may discuss the Cnn strength by using the r0 parameter representing the variance rms of the Gaussian distribution of the neutrons separation √ inside the nucleus [16]. The r rms between two neutrons is given by r =r0 6. The first measurement on Cnn for 11 Li has been reported in [10]. In this case r0 =5.5 fm was obtained. This corresponds to rrms =13 fm much larger than the experimentally measured value in [14]. The next measurement was reported in [11]. The obtained value for r0 , 4.2 fm, was also considered by the authors of [11] as being too large. The authors interpreted the large value of r0 as being due to residual correlation. Consequently they have applied an iterative calculation, reducing this way the value of r0 from 4.2 fm to 2.7 fm. The last reported measurement on 11 Li is in [7]. The value r0 =5 fm by using a denominator (A) in Cnn , built by single neutron product technique, has been found. This value is close to the value reported in [10]. By using an event mixing denominator (B), r0 =4.2 fm has been obtained. In [7] and [9], was observed that although the iterative procedure is an elegant one, it
M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
237c
drastically increases the error in r0 , so that is no more possible to draw meaningful physical conclusions. A new experiment aiming the determination of the intrinsic Cnn by using 11 Li and 11 Be beams, to avoid residual correlation, and by using 12 C target, for increasing considerably the statistics of n-n coincidence, has been proposed [7–9,13]. 3. BRIEF REVIEW OF THE NEW THEORY Very recently a new theory for the Borromean nuclei correlation function has been achieved [1]. The Borromean halo nucleus is modeled like a three body n-n-A, that is 2 neutrons and a core denoted with the mass number A. To describe such a system, the authors of [1] are using a three body model in the limit of zero-range approximation, which retains the essential physics of the weakly bound and the large two-neutron halo system. The interaction singularity is tamed in a renormalized zero-range model which is appropriate to study weakly bound three-body systems. The model is parametrized by minimal number of physical inputs, which are directly related to known observables: the two neutron separation energy, S(2n), the neutron-neutron and neutron-core scattering length (or the corresponding virtual and or bound state energies). In this new model, the asymptotic limit Cnn =1 is reached at much higher values of the relative momentum q, than was found in previous data analysis. Due to the coherence of the neutrons in the halo and final state interaction, Cnn goes smoothly to the asymptotic limit only after displaying a minimum, as it is shown in the insert of Fig. 2 from ref. [1]. In Fig. 1 of the present contribution it is shown the behavior of Cnn (q) for 11 Li in the 0-40 MeV/c q range ([12]). The error bars in Fig. 1 were simulated starting from the statistics of ∼250 detected neutron pairs in an experiment performed with a Si target [7] and extrapolating to a statistics of ∼1700 detected neutron pairs in the case of a 12 C target, expected on the basis of an investigation of the target screening effect [13]. This investigation is reviewed in the next section. In [1], besides 11 Li were investigated also 14 Be and 6 He. In this presentation, the emphasis was put on 11 Li because the minimum of calculated Cnn is in this case, at a q value which is about half from the q values corresponding to the other 2 nuclei. This makes the experimental testing much more accessible in the case of 11 Li. 4. THE TARGET SCREENING EFFECT For the first time the target screening effect on the pre-emission of halo neutrons from Li has been quantitatively analyzed in [13]. This work has been performed in the 7.5-15 MeV neutron energy range. In this range the target nuclei are likely to behave as opaque and therefore the sharp cutoff calculations are most appropriate for the target screening determination. It has been observed that the ζ probability used in the sharp cutoff calculations is an observable of the experiment, because it can be directly obtained from measured quantities, which are the number of single detected neutrons and the number of detected neutron pairs (see formula (6) of [13]). The value of ζexp obtained this way in the case of Si target, appears to be close to ζ value calculated for the 11 Li halo radius RH =4.8 fm, independently determined in another experiment [14]. This property allows also investigation of Borromean halo nuclei like 6 He, 14 Be, 17 B, for which RH was not yet measured. According to the sharp cutoff model, only the halo neutron being on a “free 11
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M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
Solid line: C nn, ref.[1], FIG.2, insert s.l.
4.0
14000
A
3.5 12000 3.0
N / (2MeV/c)
10000
C(q)
2.5
2.0
1.5
1.0
8000
B
6000
4000
0.5 2000
Err. bars: Simul. for 1700 n-n coinc. 0.0 0
5
10
15
20
25
30
35
40
q (MeV/c)
Figure 1. The solid line represents the Cnn function shown in the insert of Fig. 2 of ref. [1]. The error bars were simulated for a statistics of 1700 detected neutron pairs, calculated by taking into account the target screening effect [13].
2
4
6
8
10
12
14
q (MeV/c)
Figure 2. This figure has been built from the data of[7].By A is denoted the denominator of Cnn obtained by using the single neutron products.The denominator B was built by the event mixing technique.
area” that is an area not hindered by the target, could escape. The escape probability can be defined as: ζ(l) =
Sf (l) Stot
(1)
in which Sf (l) for (l=0) represents, (see Fig.1 of [13]), the free area not hindered by the Si target. The ζ probability for Si target turns out to be ζ Si(l=0)=0.27. For the case of C target (see Fig.2 of [13]), this probability turns out to be ζ C (l=0) = 0.77, that is almost three times larger than for Si. One has to underline that ζ represents the neutron pre-emission probability, if it were only one neutron in the halo. For the calculation of the probability ζ, the sharp cutoff formula for fusion reaction is used. Therefore ζ is
M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
239c
calculated according to the expression: ζ=
l=lcr
(2l + 1)ζ(l) l=0 (2l + 1)
l=0
l=lcr
(2)
The total probability P [1 ] for one halo neutron pre-emission will be given by the following formula: P [1] = 2ζ(1 − ζ)
(3)
The probability P [2 ] for 2 neutron pre-emission will be given by: P [2] = ζ 2
(4)
For completeness, the probability for absorption of both halo neutrons is given by: Pa[2] = (1 − ζ)2
(5)
The calculated within the present approach of the 2-neutron pre-emission yield appears to be 3.5 times larger in the case of 12 C than in the case of Si target (see Table I of [13]). Taking into account that in a C target the number of the nuclei is 2 times larger than in a Si target, the number of detected neutron pairs will be about 7 times larger for the C target. 5. POSSIBLE EXPERIMENTAL INDICATION ON THE VALIDITY OF THE NEW THEORY This indication is relating to Fig.2. This figure was built from data of [7]. By solid squares is denoted the denominator of the correlation function built from single detected neutrons, denoted by (A) in [7]. By open circles is denoted the denominator built by event mixing. In [7] this denominator was denoted (B). The using of denominator of the type B is criticized in published work [15] as in the following: “In the event mixing the denominator is generated by randomly mixing the neutrons from the coincidence sample. This method has the advantage that the uncorrelated distribution corresponds to the same class of collisions as in the case of the numerator, but has the disadvantage that it may distort the correlation one wants to measure because it may not succeed to decorrelate completely the events. In the single product technique the denominator is constructed by the product of single neutron distribution.” For us was a puzzle why in our case the denominator type (B) appears to be lower than denominator type (A). The denominator type (B) should be higher than denominator (A) because denominator (B) is not able to decorrelate completely the events. In a single case denominator type (B) could be lower than denominator type (A). This could happen only in the case there is a sink in the data, so as the minimum predicted by the new theory. 6. CONCLUSIONS The measured Cnn correlation strength for 11 Li appears to be rather low. In [11] has been assumed that the low value of the correlation strength is due to residual
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M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
correlation. But this assumption was made within the frame of the previous theory of Cnn based on the independent particle model of [16]. This model is not appropriate for Borromean halo nuclei. In fact the new theory of Cnn which is a three body theory, predicts low value of the correlation strength, as can be seen in Fig.1. However, the concept of residual correlation can be tested for example, in an experiment with 11 Li and 11 Be as was proposed in [7]. The experiments performed up to now, were using either coulomb dissociation or preemission of neutrons on a Si target. In both cases the statistics of events are rather poor. A considerable increase of neutron pair statistics can be obtained by using the target screening effect [13]. In the case one is using a 12 C target it is expected an increase by a factor 7, of the statistics of detected neutron pairs. The authors of the new theory have determined the rms separation between the halo neutrons inside 11 Li, within their model [17] and have found 8.5 fm for a confirmed value of the nA virtual state. This value is very close to COSMA1 rms separation, 8.31 fm [18]. The other rms separation 6.71 fm, given by COSMA2 , is based on an unconfirmed very large virtual state value [19]. This means that a confirmation of the new Cnn theory will be implicitly a confirmation of the COSMA1 model. From the upper mentioned reasons it is hard to overestimate the importance of the experimental testing of the new theory. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
M.T. Yamashita, T. Frederico, and L. Tomio, Phys. Rev. C 72 (2005) 011601. M. Petrascu et al., Balkan Phys. Lett. 3(4) (1995) 214. M. Petrascu et al., Phys. Lett. B 405 (1997) 224. M. Petrascu et al., Rom. J. Phys. 44 (1-2 Suppl.) (1999) 115. M. Petrascu et al., Preprint RIKEN-AF-NP-395 (2001). M. Petrascu, Phys. At. Nucl. 66 (2003) 1528. M. Petrascu et al., Phys. Rev. C 69 (2004) 011602. M. Petrascu et al., Nucl. Phys. A 734 (2004) 237. M. Petrascu et al., Nucl. Phys. A 738 (2004) 503. K. Ieki et al., Phys. Rev. Lett. 70 (1993) 730. F.M. Marques et al., Phys. Lett. B 476 (2000) 219. M.P. thanks Lauro Tomio for sending him Cnn numerical results prior to publication. M. Petrascu et al., Phys. Rev. C 73 (2006) 057601. I. Tanihata et al., Phys. Lett. B 287 (1992) 307. R. Ghetti et al., Nucl. Phys. A 660 20 (1999) 20. R. Lednicky and V.L. Lyuboshits, Sov. J. Nucl. Phys. 35 (1982) 770. M. Yamashita, L. Tomio and T. Frederico, Nucl. Phys. A 735 (2004) 40. M.V. Zhukov et al., Phys. Rep. 231 (1993) 151. Lauro Tomio (private communication).
Experimental state of n-n correlation function for Borromean halo nuclei investigation M. Petrascu,a A. Constantinescu,a ∗ I. Cruceru,a M. Duma,a M. Giurgiu,a† A. Isbasescu,a H. Petrascu,a S. Serban,a V. Stoica,a C. Bordeanu,a I. Tanihata,b W.G. Lynch,c M.A. Famiano,d K. Ieki.e a
Horia Hulubei National Institute for Physics and Nuclear Engineering P.O. Box MG-6, Bucharest, Romania
b c
National Superconducting Cyclotron Laboratory, MSU, MI 49008, USA
d e
TRIUMF, Vancouver B.C. V6t, 2A3 Canada
Western Michigan University, Kalamazoo, MI 49008, USA
Rikkyo University, Tokyo 171, Japan
The present experimental and theoretical state of Cnn correlation function for Borromean halo nuclei investigation is reviewed. Some of the consequences of a recently appeared new theory of Cnn , [1] together with the experimental possibilities to test this theory will be presented in this contribution. 1. INTRODUCTION It was predicted in [2] that, due to the very large radius of 11 Li, and due to the very low binding energy of the halo neutrons, one may expect that in a fusion process on a light target, the halo neutrons may not be absorbed together with the 9 Li core, but may be emitted in the early stage of the reaction. Indeed, the experimental investigation of Si(11 Li, fusion) has shown that, a fair amount of fusion events [3] are preceded by the preemission of one or two halo neutrons. In [3] was also found that in the position distribution of the halo neutrons, a very narrow forward neutron peak is present. Considering that this peak may be due to neutron pairs, it was decided to investigate the neutron pre-emission in condition of much higher statistics by means of an array detector [4]. Indeed within the narrow forward peak (9 msr) a large number of n-n coincidences was found [5,6]. Trying to built the n-n correlation function, a serious problem appeared connected with the low values of the correlation strength [7–9], confirming the previous measurements of [10] and [11]. A possible explanation could be the residual correlation of the halo neutrons [11]. In [11], has been proposed an iterative calculation to compensate for the residual correlation. But in [7,9], was shown that the iterative calculation is considerably increasing the error ∗ †
Permanent address, University of Bucharest, Faculty of Physics Permanent address, Technical University of Bucharest
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.03.150
236c
M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
so that is no more possible to draw any conclusion concerning the theoretical predictions. In [7] was proposed an experiment for getting the intrinsic correlation function by using 11 Li and 11 Be beams. The halo nucleus 11 Be would be ideal for the denominator of the correlation function construction, because it has only one halo neutron and therefore no residual correlation can be possible. In [7] was also proposed to do the new measurements, by using a C target instead a Si target, because due to the target screening which is lower in the case of C target, the yield of halo neutron pre-emission is expected to be higher in this case. A preliminary estimation quoted in [7], indicated that for C target, the yield of halo neutron pre-emission is about 2 times higher than for a Si target. It will be shown in the section dedicated to the target screening effect, that more accurate calculation is indicating that the yield of halo neutron pre-emission is expected to be near 3.5 larger in the case of C target than in the case of Si target. The present experimental state of Cnn investigation, will be reviewed in section 2. Very recently, a new theory for Cnn correlation function has been proposed [1]. In this theory, the 11 Li halo nucleus is modeled as a three body system consisting of 2 neutrons and a core. It is shown that a minimum is present in Cnn , due to the coherence of the 2 halo neutrons. This theory will be briefly reviewed in section 3. An analysis of the possibility to test experimentally this new theory, will be presented in section 4. In section 5, an experimental indication on the validity of the new theory and in section 6, the conclusions will be presented. 2. THE PRESENT EXPERIMENTAL STATE OF Cnn INVESTIGATION The correlation function Cnn (q) is in principle a powerful tool for the space-time proximity investigation of particles emitted from nuclei. In the case of Borromean halo nuclei, due to the fact that the neutrons are emitted in very fast processes (10−23 s) as Coulomb dissociation or pre-emission in the nuclear field, the space proximity is dominant because is characterized by final state interaction (FSI) much larger than QSS (quantum statistical symmetry). Therefore in the case of halo nuclei one should expect n-n correlation strengths much larger in comparison with the neutron evaporation processes dominated by QSS, in which case the correlation strength is lower than 2. One may discuss the Cnn strength by using the r0 parameter representing the variance rms of the Gaussian distribution of the neutrons separation √ inside the nucleus [16]. The r rms between two neutrons is given by r =r0 6. The first measurement on Cnn for 11 Li has been reported in [10]. In this case r0 =5.5 fm was obtained. This corresponds to rrms =13 fm much larger than the experimentally measured value in [14]. The next measurement was reported in [11]. The obtained value for r0 , 4.2 fm, was also considered by the authors of [11] as being too large. The authors interpreted the large value of r0 as being due to residual correlation. Consequently they have applied an iterative calculation, reducing this way the value of r0 from 4.2 fm to 2.7 fm. The last reported measurement on 11 Li is in [7]. The value r0 =5 fm by using a denominator (A) in Cnn , built by single neutron product technique, has been found. This value is close to the value reported in [10]. By using an event mixing denominator (B), r0 =4.2 fm has been obtained. In [7] and [9], was observed that although the iterative procedure is an elegant one, it
M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
237c
drastically increases the error in r0 , so that is no more possible to draw meaningful physical conclusions. A new experiment aiming the determination of the intrinsic Cnn by using 11 Li and 11 Be beams, to avoid residual correlation, and by using 12 C target, for increasing considerably the statistics of n-n coincidence, has been proposed [7–9,13]. 3. BRIEF REVIEW OF THE NEW THEORY Very recently a new theory for the Borromean nuclei correlation function has been achieved [1]. The Borromean halo nucleus is modeled like a three body n-n-A, that is 2 neutrons and a core denoted with the mass number A. To describe such a system, the authors of [1] are using a three body model in the limit of zero-range approximation, which retains the essential physics of the weakly bound and the large two-neutron halo system. The interaction singularity is tamed in a renormalized zero-range model which is appropriate to study weakly bound three-body systems. The model is parametrized by minimal number of physical inputs, which are directly related to known observables: the two neutron separation energy, S(2n), the neutron-neutron and neutron-core scattering length (or the corresponding virtual and or bound state energies). In this new model, the asymptotic limit Cnn =1 is reached at much higher values of the relative momentum q, than was found in previous data analysis. Due to the coherence of the neutrons in the halo and final state interaction, Cnn goes smoothly to the asymptotic limit only after displaying a minimum, as it is shown in the insert of Fig. 2 from ref. [1]. In Fig. 1 of the present contribution it is shown the behavior of Cnn (q) for 11 Li in the 0-40 MeV/c q range ([12]). The error bars in Fig. 1 were simulated starting from the statistics of ∼250 detected neutron pairs in an experiment performed with a Si target [7] and extrapolating to a statistics of ∼1700 detected neutron pairs in the case of a 12 C target, expected on the basis of an investigation of the target screening effect [13]. This investigation is reviewed in the next section. In [1], besides 11 Li were investigated also 14 Be and 6 He. In this presentation, the emphasis was put on 11 Li because the minimum of calculated Cnn is in this case, at a q value which is about half from the q values corresponding to the other 2 nuclei. This makes the experimental testing much more accessible in the case of 11 Li. 4. THE TARGET SCREENING EFFECT For the first time the target screening effect on the pre-emission of halo neutrons from Li has been quantitatively analyzed in [13]. This work has been performed in the 7.5-15 MeV neutron energy range. In this range the target nuclei are likely to behave as opaque and therefore the sharp cutoff calculations are most appropriate for the target screening determination. It has been observed that the ζ probability used in the sharp cutoff calculations is an observable of the experiment, because it can be directly obtained from measured quantities, which are the number of single detected neutrons and the number of detected neutron pairs (see formula (6) of [13]). The value of ζexp obtained this way in the case of Si target, appears to be close to ζ value calculated for the 11 Li halo radius RH =4.8 fm, independently determined in another experiment [14]. This property allows also investigation of Borromean halo nuclei like 6 He, 14 Be, 17 B, for which RH was not yet measured. According to the sharp cutoff model, only the halo neutron being on a “free 11
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M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
Solid line: C nn, ref.[1], FIG.2, insert s.l.
4.0
14000
A
3.5 12000 3.0
N / (2MeV/c)
10000
C(q)
2.5
2.0
1.5
1.0
8000
B
6000
4000
0.5 2000
Err. bars: Simul. for 1700 n-n coinc. 0.0 0
5
10
15
20
25
30
35
40
q (MeV/c)
Figure 1. The solid line represents the Cnn function shown in the insert of Fig. 2 of ref. [1]. The error bars were simulated for a statistics of 1700 detected neutron pairs, calculated by taking into account the target screening effect [13].
2
4
6
8
10
12
14
q (MeV/c)
Figure 2. This figure has been built from the data of[7].By A is denoted the denominator of Cnn obtained by using the single neutron products.The denominator B was built by the event mixing technique.
area” that is an area not hindered by the target, could escape. The escape probability can be defined as: ζ(l) =
Sf (l) Stot
(1)
in which Sf (l) for (l=0) represents, (see Fig.1 of [13]), the free area not hindered by the Si target. The ζ probability for Si target turns out to be ζ Si(l=0)=0.27. For the case of C target (see Fig.2 of [13]), this probability turns out to be ζ C (l=0) = 0.77, that is almost three times larger than for Si. One has to underline that ζ represents the neutron pre-emission probability, if it were only one neutron in the halo. For the calculation of the probability ζ, the sharp cutoff formula for fusion reaction is used. Therefore ζ is
M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
239c
calculated according to the expression: ζ=
l=lcr
(2l + 1)ζ(l) l=0 (2l + 1)
l=0
l=lcr
(2)
The total probability P [1 ] for one halo neutron pre-emission will be given by the following formula: P [1] = 2ζ(1 − ζ)
(3)
The probability P [2 ] for 2 neutron pre-emission will be given by: P [2] = ζ 2
(4)
For completeness, the probability for absorption of both halo neutrons is given by: Pa[2] = (1 − ζ)2
(5)
The calculated within the present approach of the 2-neutron pre-emission yield appears to be 3.5 times larger in the case of 12 C than in the case of Si target (see Table I of [13]). Taking into account that in a C target the number of the nuclei is 2 times larger than in a Si target, the number of detected neutron pairs will be about 7 times larger for the C target. 5. POSSIBLE EXPERIMENTAL INDICATION ON THE VALIDITY OF THE NEW THEORY This indication is relating to Fig.2. This figure was built from data of [7]. By solid squares is denoted the denominator of the correlation function built from single detected neutrons, denoted by (A) in [7]. By open circles is denoted the denominator built by event mixing. In [7] this denominator was denoted (B). The using of denominator of the type B is criticized in published work [15] as in the following: “In the event mixing the denominator is generated by randomly mixing the neutrons from the coincidence sample. This method has the advantage that the uncorrelated distribution corresponds to the same class of collisions as in the case of the numerator, but has the disadvantage that it may distort the correlation one wants to measure because it may not succeed to decorrelate completely the events. In the single product technique the denominator is constructed by the product of single neutron distribution.” For us was a puzzle why in our case the denominator type (B) appears to be lower than denominator type (A). The denominator type (B) should be higher than denominator (A) because denominator (B) is not able to decorrelate completely the events. In a single case denominator type (B) could be lower than denominator type (A). This could happen only in the case there is a sink in the data, so as the minimum predicted by the new theory. 6. CONCLUSIONS The measured Cnn correlation strength for 11 Li appears to be rather low. In [11] has been assumed that the low value of the correlation strength is due to residual
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M. Petrascu et al. / Nuclear Physics A 790 (2007) 235c–240c
correlation. But this assumption was made within the frame of the previous theory of Cnn based on the independent particle model of [16]. This model is not appropriate for Borromean halo nuclei. In fact the new theory of Cnn which is a three body theory, predicts low value of the correlation strength, as can be seen in Fig.1. However, the concept of residual correlation can be tested for example, in an experiment with 11 Li and 11 Be as was proposed in [7]. The experiments performed up to now, were using either coulomb dissociation or preemission of neutrons on a Si target. In both cases the statistics of events are rather poor. A considerable increase of neutron pair statistics can be obtained by using the target screening effect [13]. In the case one is using a 12 C target it is expected an increase by a factor 7, of the statistics of detected neutron pairs. The authors of the new theory have determined the rms separation between the halo neutrons inside 11 Li, within their model [17] and have found 8.5 fm for a confirmed value of the nA virtual state. This value is very close to COSMA1 rms separation, 8.31 fm [18]. The other rms separation 6.71 fm, given by COSMA2 , is based on an unconfirmed very large virtual state value [19]. This means that a confirmation of the new Cnn theory will be implicitly a confirmation of the COSMA1 model. From the upper mentioned reasons it is hard to overestimate the importance of the experimental testing of the new theory. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
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