2.1 Spherical Gaussian Source

We first calculated the p-p momentum correlation function C_pp(q) by assuming a spherical Gaussian source. In Fig. 1, the results of C_pp(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 C_pp(q) decreases as τ increases for fixed r₀. For larger r₀, the difference of C_pp(q) between different τ becomes larger.

Fig. 1.Full-screen View

(Color online) The p-p momentum correlation function (C_pp(q)) for different τ at source size r₀=0.5 fm (a) and r₀=2.5 fm (b).

In Fig. 2, we can see that C_pp(q) decreases as r₀ increases for a fixed value of τ. For larger τ, the difference of C_pp(q) between different r₀ becomes smaller. In these figures, C_pp(q) has the maximum value at around q_pp = 20 MeV/c.

Fig. 2.Full-screen View

(Color online) The C_pp(q) for different r₀ at emission time τ=0 (a) and τ=400 fm/c (b).

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

Fig. 3.Full-screen View

(Color online) The r₀ dependence of C_max(q) (a) and the FWHM (b) for the p-p momentum correlation function at different τ.

Similarly, the τ dependence of C_max(q) and the FWHM for the p-p momentum correlation function at different r₀ are shown in Fig. 4. The dependence of C_max(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.

Fig. 4.Full-screen View

(Color online) The τ dependence of C_max(q) (a) and the FWHM (b) for the p-p momentum correlation function at different r₀.

Since both r₀ and τ affect the correlation function, it is interesting to see whether or not different r₀ and τ combinations result in the same proton-proton momentum correlation function C_pp(q). To find the answer, contour plots of C_max(q) and the FWHM extracted from C_pp(q) at different r₀ and τ values are given in Fig. 5, in which C_max(q) and the FWHM are the Z axis. From this figure, we can see clearly that each isoline of C_max(q) or the FWHM has only one intersection point with each other. This indicates that a set of C_max(q) and the FWHM values or one proton-proton momentum correlation function has only a uniquely determined r₀ and τ combination. This is due mainly to the monotonic dependence of C_max(q) or the FWHM on r₀ and τ, respectively. Based on the above results, it is shown that the source size r₀ and proton emission time difference τ could be determined uniquely by fitting the experimental C_pp(q) with the CRAB calculation, as demonstrated in [14].

Fig. 5.Full-screen View

(Color online) Contour plot of C_max(q) and the FWHM extracted from the p-p momentum correlation function.

2.2 Ellipsoidal Gaussian Source

Considering the possible deformation of the nucleus, we calculated the proton-proton correlation function C_pp(q) for a deformed nucleus through the CRAB code. The deformation is described by the parameter ϵ=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 C_pp(q) results are shown in Fig. 6. The C_max(q) and the FWHM of the ellipsoidal Gaussian source and the spherical Gaussian source are almost same in the range of ϵ from -0.10 to 0.10. In fact, the C_pp(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.

Fig. 6.Full-screen View

(Color online) The deformation parameter, ϵ, dependence on C_max(q) for different τ at r₀=2.5 fm (a) and for different r₀ at τ=400 fm/c (b); the same dependence of the FWHM for different τ at r₀=2.5 fm (c) and for different r₀ at τ=400 fm/c (d).

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 ΔC_max(q) and ΔFWHM as the difference of C_max(q) and FWHM between the spherical Gaussian source and the double Gaussian source with the same effective source size r₀. The results of ΔC_max(q) and ΔFWHM are presented in Fig. 7. In Fig. 7(a), ΔC_max(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. ΔC_max(q) has the largest value when the source size is near 1.5 fm. As shown in Fig. 7(b), ΔC_max(q) decreases as τ increases. For different r₀, the ΔC_max(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 of τ, as shown in Fig. 7(c) and Fig. 7(d).

Fig. 7.Full-screen View

(Color online) The differences of C_max(q) (ΔC_max(q)) between a spherical Gaussian source and a double Gaussian source for different τ (a) and r₀ (b); the differences of the FWHM (ΔFWHM) between a spherical Gaussian source and a double Gaussian source for different τ (c) and r₀ (d).

ΔC_max(q) decreases with increasing τ, while ΔFWHM does not change with τ, which indicates that the difference of C_pp(q) between the double Gaussian source and the spherical Gaussian source decreases and nears the same value with increasing τ. ΔC_max(q) decreases with increasing r₀, while ΔFWHM gradually increases with r₀, which indicates that the difference of C_pp(q) between the double Gaussian source and the spherical Gaussian source becomes significantly larger as r₀ increases. 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 r₀ and τ. This may provide a possible method for experimentally observing the cluster structure in proton-rich nuclei. For practical applications, further investigations are necessary.