Until now, the measurements for the kaon-nucleus elastic scattering are very scarce compared to the ones for pion-nucleus scattering. Elastic differential cross sections of 800 MeV/c charged kaon on ¹²C and ⁴⁰Ca targets were measured [1], and good qualitative agreements with the experimental data were obtained from the firstorder optical model calculation. The elastic scattering data of K⁺ + ¹²C system at 635 and 715 MeV/c were reported [2], and analyzed using distorted wave impulse approximation. Several theoretical attempts [3–8] has been made to describe the kaon-nucleus elastic scattering. The experimental data of kaon elastic scattering at 800 MeV/c on ¹²C and ⁴⁰Ca were well reproduced [3] by the phenomenological model. Local optical model calculations have been carried out [5] for elastic and reaction cross sections of K⁺ from ⁶Li, ¹²C and ⁴⁰Ca at kaon momenta ranging from 635 to 800 MeV/c.

On the other hand, the eikonal model [9,10] has been found to be quite successful in describing the high energy hadron scattering from nucleus. The only input of eikonal model for the analysis of elastic scattering data is the optical potential. The eikonal phase shifts are derived from the integral equation by further approximating the Wentzel-Kramers-Brillouin (WKB) results. The basic assumption is that its classical trajectory is little deflected from a straight line because the incident energy is sufficiently high. The eikonal phase shift has been modified to account for the deviation of Coulomb trajectory from straight line trajectory due to the Coulomb field. Higher order corrections to the eikonal phase shift based on Coulomb trajectory have been developed [11– 13] to extend the study of heavy-ion elastic scattering at relatively low energies.

In our earlier work [14], first-order eikonal model, taking into account a first-order correction to the zerothorder eikonal phase shift, was employed to describe the elastic angular distributions of π^± + ¹²C and π^± + ⁴⁰Ca systems at 800 MeV/c. The calculated results are found to be in a good agreement with the experimental data. Success of this model for the description of pion elastic scattering from ¹²C and ⁴⁰Ca motivated us to consider a similar method to the analysis of kaon elastic scattering from same nuclei because of the great similarity between pion and kaon. In this paper, we use the eikonal model formalism to analyze the elastic differential cross sections of K⁺ + ¹²C system at 635, 715, and 800 MeV/c, and of K⁺ + ⁴⁰Ca system at 800 MeV/c, and its results are compared to the ones calculated from strong absorption model formalism suggested by Choudhury et al. [15]. Rutherford and nuclear parts of the elastic scattering amplitude are used to qualitatively understand the structure of the angular distributions. Further, we investigate the critical angular momentum, strong absorption radius and reaction cross sections. The theory employed here is briefly described in Sec. II. In Sec. III, we present our results and discussion for differential and reaction cross sections of K⁺ elastic scattering on ¹²C and ⁴⁰Ca targets. Concluding remarks are given in Sec. IV.

If there is a single turning point in the radial Schrödinger equation, the WKB formula for the nuclear phase shift δ_L, taking into account the deflection effect due to the Coulomb field, is [11,16]

In this formula, k_L(r) and k_c(r) are given by

where U(r) is the nuclear potential and η the Sommerfeld parameter. The turning points r_t and r_c are defined by k_L(r) = 0 and k_c(r) = 0, respectively. If we consider the nuclear potential as a perturbation, the nuclear phase shift can be written as [17]

where r=rc2+z2 and r_c is given by

The form of Eq. (4) is the phase shift of eikonal model. The nuclear potential U(r) entering in Eq. (4) is usually taken as a complex Woods-Saxon shape given by

where V₀, R_v = rvA1/3 T and a_v are respectively, the depth, radius and diffuseness of the real part of the potential, and W0, Rw = rwA1/3 T and aw are the corresponding quantities of the imaginary part. We can use the eikonal phase shift Eq. (4) in the general expression for the scattering amplitude that will be mentioned in below.

The differential cross section for K⁺-nucleus elastic scattering is calculated using

The scattering amplitude f(θ) in above equation is given by the partial wave formula

where the first-term f_R(θ) is the Rutherford scattering amplitude given by

and the second term is the nuclear scattering amplitude f_N(θ). In Eq. (8), σ_L is the Coulomb phase shift, and k is the kaon wave number in the center of mass system expressed as

where m_K and m_N are the kaon and nuclear rest mass, respectively, and E_K is the total kaon energy in the laboratory system. The nuclear S-matrix element SLN is given by SLN = exp[2iδ_L(r_c)].

Meanwhile, based on a strong absorption model of Frahn and Venter parametrization, Choudhury et al. [15] reported an analytic expression for the elastic scattering amplitude given by

where R₀ is the effective radius and Δ is related with the effective surface thickness a by Δ = ak. We name this model simply a “strong absorption model (SAM)” throughout this paper. This SAM calculations reproduced [15] the most of features of the elastic scattering data of 800 MeV/c charged pions on ¹²C and ⁴⁰Ca nuclei. In this paper, we also use this SAM to calculate the elastic angular distributions of K⁺ + ¹²C and K⁺ + ⁴⁰Ca systems, and its results are compared to the eikonal model results.

Using the eikonal and the strong absorption models mentioned in previous section, we calculated the elastic differential cross sections of K⁺ + ¹²C system at 635, 715, and 800 MeV/c, and of K⁺ + ⁴⁰Ca system at 800 MeV/c. In order to search the best-fit input parameters, we used χ²/N–method (N is the number of data points) which show the quality of fit between the experimental data and our theoretical results. Input parameters for each target nucleus at each incident momentum p are listed in Table 1 (for eikonal model) and Table 2 (for SAM), respectively. The calculated results obtained from two different models are presented in Figs. 1 and 2 along with the experimental data. The solid and dotted curves of these figures denote the elastic differential cross sections calculated from the eikonal model and SAM, respectively. The experimental data are taken from the works of Chrien et al. [2] (K⁺ + ¹²C at 635 and 715 MeV/c) and Marlow et al. [1] ( K⁺ + ¹²C and K⁺ + ⁴⁰Ca at 800 MeV/c).

Table 1 Woods-Saxon optical potential parameters entering in Eq. (6) and χ²/N values in the eikonal model analysis for K⁺ elastic scattering on ¹²C and ⁴⁰Ca.

Target	p (MeV/c)	V0 (MeV)	rv (fm)	av (fm)	W0 (MeV)	rw (fm)	aw (fm)	χ2/N
12C	635	44.5	0.903	0.677	67.5	0.784	0.441	10.64
	715	49.2	0.766	0.526	77.5	0.757	0.515	3.490
	800	35.8	0.961	0.739	91.5	0.801	0.481	0.592
40Ca	800	49.4	0.967	1.135	100.9	0.884	0.628	0.544

Table 2 Input parameters, derived parameters, and χ²/N values in the strong absorption model analysis for K⁺ elastic scattering on ¹²C and ⁴⁰Ca.

Target	p (MeV/c)	R0 (fm)	Δ	s	a (fm)	R0/AT1/3 (fm)	χ2/N
12C	635	1.811	1.813	1.469	0.603	0.791	16.74
	715	1.854	2.162	0.524	0.642	0.810	14.50
	800	1.886	2.406	1.358	0.642	0.824	1.35
40Ca	800	3.396	2.729	1.758	0.690	0.993	1.012

Figure 1. (Color online) Elastic scattering angular distributions at p = 635 and 715 MeV/c for K⁺ + ¹²C system fitted by eikonal (solid curves) and strong absorption (dotted curves) models, respectively. The solid circles with error bars denote the experimental data taken from Chrien et al. [2].

Figure 2. (Color online) Elastic scattering angular distributions at p = 800 MeV/c for K⁺ + ¹²C and K⁺ + ⁴⁰Ca systems fitted by eikonal (solid curves) and strong absorption (dotted curves) models, respectively. The solid circles with error bars denote the experimental data taken from Marlow et al. [1].

Figure 1 shows the calculated results for K⁺ + ¹²C elastic scattering at 635 and 715 MeV/c. For the p = 635 MeV/c, we found the eikonal model result to be in excellent agreement with the observed data. On the other hand, the SAM results reproduced the good description with the elastic data only up to about θ_c.m. <, 30°. In the case of p = 715 MeV/c, eikonal model provided fairly good descriptions of elastic cross section over the whole angular ranges, while SAM results differed from the elastic data at the angle regions larger than about θ_c.m. = 30°. This poor fit is also reflected by the large χ²/N−value shown in Table 2. From Fig. 1, we can see that the agreements of eikonal model calculation with the experimental data are satisfactorily good compared to the results obtained from SAM. Figure 2 is the same as Fig. 1 but for 800 MeV/c positive kaon incident on ¹²C and ⁴⁰Ca targets. The elastic scattering data pattern of K⁺ + ¹²C system is characterized by one minimum and one maximum. The eikonal model calculations provided a reasonable reproduction of this characteristics appearing on the angular distributions. Meanwhile the SAM calculation yields the decreasing pattern as a whole and did not generated this minimum and maximum features. For the K⁺ + ⁴⁰Ca system, the experimental data show somewhat oscillatory structure and are reasonably well reproduced by the eikonal model calculation. But the SAM calculation yields smoothly decreasing shape at the angle regions larger than about θ_c.m. = 20°. From Fig. 2, Tables 1 and 2, we can see that the eikonal model calculations not only give good account of the experimental data but also reproduce reasonably the minima and maxima appearing on the angular distributions, in comparison with the SAM calculations.

As shown in Eq. (8), the elastic scattering amplitude can be separated into the Rutherford and nuclear parts. Two separated amplitudes can be used to understand the qualitative features of the elastic angular distributions for K⁺ on ¹²C and ⁴⁰Ca. The contributions of the nuclear and Rutherford parts to the elastic cross section calculated from eikonal model are shown in Figs. 3 and 4. In these figures, the dotted and the dashed curves are the Rutherford (|fR(θ)|2) and the nuclear (|fN(θ)|2) scattering cross sections, respectively, while the solid curves denote the elastic cross section (solid curves in Figs. 1 and 2) given by |fR(θ) + f_N(θ)|2. For all systems, the Rutherford part shows monotonously decreasing shape as expected. However, the nuclear cross sections of K⁺ + ¹²C system display some structure having one local minimum point, and plays an important contribution to the elastic cross sections as shown in Fig. 3 and Fig. 4(top panel). The dashed curves are closer to the solid ones except for local minimum region appearing in the nuclear cross section. The discrepancies around this local minimum region between solid and dashed curves seem to be arose from the considerable (non-negligible) magnitudes of Rutherford part. In the case of K⁺ + ⁴⁰Ca system (bottom panel of Fig. 4), nuclear cross section generates two local minima. Especially second minimum occurs around θ_c.m. ≈ 38o larger than θ_c.m. ≈ 30° appearing in solid curve. By the way, the Rutherford cross section plays an meaningful role to the K⁺ + ⁴⁰Ca cross section for the angle regions larger than θ_c.m. ≈ 15° in spite of smooth decreasing behavior. Thus the somewhat oscillatory structure appearing in the angular distribution of K⁺ + ⁴⁰Ca system is thought to be due to the interference effect between Rutherford and nuclear scattering amplitudes.

Figure 3. (Color online) Elastic differential cross sections (solid curves) at p = 635 and 715 MeV/c for K⁺ + ¹²C system, their Rutherford (dotted curves) and the nuclear (dashed curves) components calculated from the eikonal model.

Figure 4. (Color online) Elastic differential cross sections (solid curves) at p = 800 MeV/c for K⁺ + ¹²C and K⁺ + ⁴⁰Ca systems, their Rutherford (dotted curves) and the nuclear (dashed curves) components calculated from the eikonal model.

Figure 5 shows the transparency function, T_L = |SLN |2, versus the angular momentum L. As this figure shows, the T_L function converges smoothly to unity as L increase. For the K⁺ + ¹²C system, the T_L functions are found to be shifted toward the right as the p is increased from 635 MeV/c to 800 MeV/c. At the same incident momentum of 800 MeV/c, the T_L function of K⁺ + ⁴⁰Ca system is moved to the higher angular momentum compared to the one of K⁺ + ¹²C system. The transparency function T_L provides a critical angular momentum L_1/2 corresponding to T_L = 0.5, and finally estimates a strong absorption radius Rs given by the relation : R_s = (η + η2+L1/2L1/2+1)/k. The reaction cross sections σ_R are calculated from the well known formula σ_R = (π/k2) ∑ L=0∞2L+11−TL. The values of L_1/2, Rs and σ_R obtained from eikonal model analysis are listed in Table 3. We notice the followings from the analysis results : (1) For K⁺ + ¹²C system, the L_1/2 value is found to increase as the incident momentum increases with a slight increase in Rs values. (2) The σ_R values are in reasonable comparison to those determined from local optical model [5] except for K⁺ + ⁴⁰Ca at 800 MeV/c. The σ_R values for K⁺ + ¹²C system are found to show a gradual increase as the p increases from 635 MeV/c to 800 MeV/c. It is also noticed that σ_R values increase as the mass number of target nucleus increases.

Table 3 Derived quantities obtained from the eikonal model analysis for K⁺ elastic scattering on ¹²C and ⁴⁰Ca. All quantities are defined in the text.

Target	p (MeV/c)	L1/2	Rs (fm)	σR (mb)
12C	635	5.258	1.920	134.2
	715	6.124	1.970	149.2
	800	7.295	2.083	158.8
40Ca	800	14.171	3.730	474.0

Figure 5. (Color online) Transparency functions obtained from eikonal model calculation for K⁺ + ¹²C system at p = 635, 715, and 800 MeV/c, and for K⁺ + ⁴⁰Ca system at p = 800 MeV/c.

If there is a single turning point in the radial Schrödinger equation, the WKB formula for the nuclear phase shift δ_L, taking into account the deflection effect due to the Coulomb field, is [11,16]

In this formula, k_L(r) and k_c(r) are given by

where U(r) is the nuclear potential and η the Sommerfeld parameter. The turning points r_t and r_c are defined by k_L(r) = 0 and k_c(r) = 0, respectively. If we consider the nuclear potential as a perturbation, the nuclear phase shift can be written as [17]

where r=rc2+z2 and r_c is given by

The form of Eq. (4) is the phase shift of eikonal model. The nuclear potential U(r) entering in Eq. (4) is usually taken as a complex Woods-Saxon shape given by

where V₀, R_v = rvA1/3 T and a_v are respectively, the depth, radius and diffuseness of the real part of the potential, and W0, Rw = rwA1/3 T and aw are the corresponding quantities of the imaginary part. We can use the eikonal phase shift Eq. (4) in the general expression for the scattering amplitude that will be mentioned in below.

The differential cross section for K⁺-nucleus elastic scattering is calculated using

The scattering amplitude f(θ) in above equation is given by the partial wave formula

where the first-term f_R(θ) is the Rutherford scattering amplitude given by

and the second term is the nuclear scattering amplitude f_N(θ). In Eq. (8), σ_L is the Coulomb phase shift, and k is the kaon wave number in the center of mass system expressed as

where m_K and m_N are the kaon and nuclear rest mass, respectively, and E_K is the total kaon energy in the laboratory system. The nuclear S-matrix element SLN is given by SLN = exp[2iδ_L(r_c)].

Meanwhile, based on a strong absorption model of Frahn and Venter parametrization, Choudhury et al. [15] reported an analytic expression for the elastic scattering amplitude given by

where R₀ is the effective radius and Δ is related with the effective surface thickness a by Δ = ak. We name this model simply a “strong absorption model (SAM)” throughout this paper. This SAM calculations reproduced [15] the most of features of the elastic scattering data of 800 MeV/c charged pions on ¹²C and ⁴⁰Ca nuclei. In this paper, we also use this SAM to calculate the elastic angular distributions of K⁺ + ¹²C and K⁺ + ⁴⁰Ca systems, and its results are compared to the eikonal model results.

This research was supported by the 2021 scientific promotion program funded by Jeju National University.