Effect of source size and emission time on the p-p momentum correlation function in the two-proton emission process

Long Zhou · De-Qing Fang

Abstract The effect of source size and emission time on the proton-proton (p-p) momentum correlation function(Cpp（q）) has been studied systematically. Assuming a spherical Gaussian source with space and time profile according to the function S（r，t）～exp（-r2/2r20 -t/τ） in the correlation function calculation code (CRAB), the results indicate that one Cpp（q） distribution corresponds to a unique combination of source size r0 and emission time τ.Considering the possible nuclear deformation from a spherical nucleus, an ellipsoidal Gaussian source characterized by the deformation parameter ∊=ΔR/R has been simulated.There is almost no difference of Cpp（q）between the results of spherically and ellipsoidally shaped sources with small deformation. These results indicate that a unique source size r0 and emission time could be extracted from the p-p momentum correlation function, which is especially important for identifying the mechanism of twoproton emission from proton-rich nuclei. Furthermore,considering the possible existence of cluster structures within a nucleus, the double Gaussian source is assumed.The results show that the p-p momentum correlation function for a source with or without cluster structures has large systematical differences with the variance of r0 and τ.This may provide a possible method for experimentally observing the cluster structures in proton-rich nuclei.

Keywords Two-proton emission · p-p momentum correlation function · Source size · Emission time

1 Introduction

Besides the well-known α,β,and γ radioactivity decays,exotic radioactivity modes also exist in very proton-rich nuclei [1-4]. Two-proton emission is one of the most interesting phenomena in nuclei beyond or close to the proton drip line [5, 6]. Generally speaking, there are three different mechanisms for proton-rich nuclei to emit two protons: (1) two-body sequential emission, (2) three-body simultaneous emission, and (3) diproton emission (also called2He cluster emission).The third mode is an extreme case with the emission of two strongly correlated protons.The2He cluster can only exist for a short while and then separates after penetrating the Coulomb barrier.

The two-particle momentum correlation is influenced by the nuclear force between two particles [7]; consequently,the proton-proton momentum correlation plays an important role in the emission mechanism and causes the twoproton relative momentum（qpp）and opening angle（θpp）to be quite different compared with other emission mechanisms [8, 9]. Generally, the two-proton correlation of diproton emission is much stronger than that of the other two mechanisms. Additionally, the emission time difference between two protons for sequential emission is long,compared to the other two modes. In Ref. [10], the threebody decay of two excited proton-rich nuclei, namely23Al →p+p+21Na and22Mg →p+p+20Ne, has been measured at the RIKEN RI Beam Factory. It has been noted that the emission mechanism of the isospin analogue state (IAS) of22Mg has strong diproton emission probability,based on the analysis of（qpp）and（θpp）.However,it is difficult to determine the emission mechanism of the excited23Al. For three-body simultaneous emission, the two protons are emitted almost at the same time,while the emission time of two-body sequential emission is quite different. Comparing the experimental data with the theoretical simulations, the source size and proton emission time can be extracted [11-13]. In Ref. [14], the p-p momentum correlation function (Cpp（q）) was studied for these two decay channels,and the source size and emission time information were extracted, as well as for the emission mechanisms of two protons from23Al and22Mg.

There are two main factors that affect the p-p momentum correlation function.One is the source size,the other is the emission time difference between the two protons. An increase in the source size will decrease the strength of Cpp（q）, and the emission time difference will also have a similar effect on Cpp（q）. However, the source size and emission time determining Cpp（q）are unique,or a different combination of source size and emission time could give the same Cpp（q）. In this study, the p-p momentum correlation function was investigated systematically by using the code Correlation After Burner (CRAB), which is a widely used method for calculating the momentum correlation function in nuclear physics [15].Assuming the first proton being emitted at time t=0 and the second proton being emitted at time t, a Gaussian source with the form S（r，t）～exp（-r2/2r20-t/τ） is used in CRAB. Here, r0refers to the source size and τ refers to the lifetime for the emission time of the second proton [14]. The effect of source size and emission time on Cpp（q） was studied systematically. The effect of deformation and different configuration of nuclei was also considered [16, 17].

2 Calculation results

2.1 Spherical Gaussian source

We first calculated the p-p momentum correlation function Cpp（q） by assuming a spherical Gaussian source.In Fig. 1,the results of Cpp（q）with source size of r0=0.5 fm and r0=2.5 fm at different values of τ are presented.The figure shows that Cpp（q） decreases as τ increases for fixed r0. For larger r0, the difference of Cpp（q） between different τ becomes larger.

In Fig. 2, we can see that Cpp（q） decreases as r0increases for a fixed value of τ.For larger τ,the difference of Cpp（q） between different r0becomes smaller. In these figures, Cpp（q） has the maximum value at around qpp= 20 MeV/c.

The proton-proton momentum correlation function Cpp（q） increases with qppand saturates at around 1. Two parameters were used to describe Cpp（q） for studying the effect of source size and emission time systematically.One is the maximum value of Cpp（q）at approximately qpp=20 MeV/c (Cmax（q）), and the other is the full width at half maximum(FWHM)of Cpp（q）determined by the difference of the two qppvalues of Cpp（q）=［Cmax（q）-1］/2 located at the left and right side of the Cpp（q） maximum. The r0dependence of Cmax（q）and FWHM for the p-p momentum correlation function with different τ are given in Fig. 3.As shown in Fig. 3a, Cmax（q） decreases gradually with increasing r0. For a large τ, the change of Cmax（q） is very small. As shown in Fig. 3b, the FWHM is inversely proportional to r0.For different τ,the behavior of the FWHM with r0is very similar.

Similarly, the τ dependence of Cmax（q） and the FWHM for the p-p momentum correlation function at different r0are shown in Fig. 4. The dependence of Cmax（q） on τ is quite similar with r0, except for the FWHM results. The difference of the FWHM values is much larger for different source sizes.

Since both r0and τ affect the correlation function, it is interesting to see whether or not different r0and τ combinations result in the same proton-proton momentum correlation function Cpp（q）. To find the answer, contour plots of Cmax（q） and the FWHM extracted from Cpp（q） at different r0and τ values are given in Fig. 5, in which Cmax（q）and the FWHM are the Z axis.From this figure,we can see clearly that each isoline of Cmax（q） or the FWHM has only one intersection point with each other. This indicates that a set of Cmax（q） and the FWHM values or one proton-proton momentum correlation function has only a uniquely determined r0and τ combination. This is due mainly to the monotonic dependence of Cmax（q）or the FWHM on r0and τ, respectively. Based on the above results, it is shown that the source size r0and proton emission time difference τ could be determined uniquely by fitting the experimental Cpp（q） with the CRAB calculation, as demonstrated in [14].

2.2 Ellipsoidal Gaussian source

Considering the possible deformation of the nucleus,we calculated the proton-proton correlation function Cpp（q）for a deformed nucleus through the CRAB code. The deformation is described by the parameter ∊=ΔR/R,where R is the nuclear radius with no deformation and ΔR is the difference in radius before and after the deformation.

After considering the deformation of the source using CRAB, the calculated Cpp（q） results are shown in Fig. 6.The Cmax（q） and the FWHM of the ellipsoidal Gaussian source and the spherical Gaussian source are almost same in the range of ∊from-0.10 to 0.10.In fact,the Cpp（q）of the ellipsoidal Gaussian source and the spherical Gaussian source are almost identical, with the same effective source radius r0. This indicates that the deformation (not very large)of the nucleus has little effect on the p-p momentum correlation function.

2.3 Double Gaussian source

The α cluster structure is one of the most common aspects of a nucleus [18]. If two protons are emitted from this kind of nucleus, the protons may come from the same or different α cluster within it.It would be interesting to see the effect of the α cluster on the p-p momentum correlation function. To study the cluster structure in the nucleus, a double Gaussian source was used in CRAB to simulate two clusters inside the nucleus.

We assume that the source of two protons is not distinguished. Thus, the two emitted protons may come from the same cluster or from two different clusters. Define ΔCmax（q） and ΔFWHM as the difference of Cmax（q） and FWHM between the spherical Gaussian source and the double Gaussian source with the same effective source size r0. The results of ΔCmax（q） and ΔFWHM are presented in Fig. 7. In Fig. 7a, ΔCmax（q） first increases and then decreases with r0, and the maximum value appears near r0=1.5 fm, i.e., the p-p momentum correlation function has the largest difference for two protons emitted from ordinary nuclei and from nuclei with clusters. ΔCmax（q）has the largest value when the source size is near 1.5 fm.As shown in Fig. 7b, ΔCmax（q） decreases as τ increases.For different r0,the ΔCmax（q）values are very large when τ is small, but they are quite close when τ is large enough.We can also see that ΔFWHM gradually increases with r0,but there is almost no change with the increase in τ, as shown in Fig. 7c, d.

ΔCmax（q） decreases with increasing τ, while ΔFWHM does not change with τ,which indicates that the difference of Cpp（q） between the double Gaussian source and the spherical Gaussian source decreases and nears the same value with increasing τ. ΔCmax（q） decreases with increasing r0, while ΔFWHM gradually increases with r0, which indicates that the difference of Cpp（q） between the double Gaussian source and the spherical Gaussian source becomes significantly larger as r0increases. These results indicate that the p-p momentum correlation function for a source with or without cluster structure will have large systematical differences with the variance of r0and τ.This may provide a possible method for experimentallyobserving the cluster structure in proton-rich nuclei. For practical applications, further investigations are necessary.

3 Summary and outlook

In summary, the proton-proton momentum correlation functions (Cpp（q）) for a sphere, ellipsoid, and double Gaussian source were investigated using the CRAB code.From systematical studies of the varied effects of the source size r0and emission time τ on Cpp（q）,it was shown that one Cpp（q） distribution corresponds to unique values of r0and τ. There is almost no difference in Cpp（q）between a spherical Gaussian source and an ellipsoidal Gaussian source,i.e.,a small nuclear deformation has very little effect on Cpp（q）. The proton-proton momentum correlation function has the largest difference between ordinary nuclei and clustered nuclei when the source size is near 1.5 fm; this may give a possible method for experimentally observing the cluster structure of proton-rich nuclei. Recently, artificial neural networks have been widely used in the research of many practical problems that are difficult for modern computers to solve [19-22].Extracting the source size and emission time of two particles from experimental data is relatively difficult. It may be interesting to study systematically the p-p momentum correlation functions by using artificial neural networks in future studies.