Research Report of Laboratory of Nuclear Science, Tohoku University Vol. 38 (2005)

18(LNS Experiment : #2507)

Comparison of the 40Ca(e, e' p) Cross Section at Low
Momentum Transfer Region with Relativistic Calculations
R. Hashimoto1, T. Tamae1, T. Fujibayashi2, O. Hashimoto2, K. Hirose2,T. Ishikawa1, H. Kanda2, O. Konno3, K. Maeda2, H. Miyase2, S.N. Nakamura2,
M. Nanao1, I. Nishikawa1, T. Otsuki1, T. Saito4, Y. Sato1, K. Takahashi2,H. Tamura2, H. Tsubota2, M. Wakamatsu2, H. Yamazaki1, and H. Yuki1
13

Laboratory of Nuclear Science, Tohoku University, Sendai 982-0826
2
Department of Physics, Tohoku University, Sendai 980-8578

Department of Electrical Engineering, Ichinoseki National College of Technology, Ichinoseki 021-8511
4

Faculty of Engineering, Tohoku Gakuin University, Tagajo, 985-8537

The (e, e' p0) cross section of 40Ca measured using a 199.53 MeV continuous electron beam iscompared with theoretical calculations based on the relativistic distorted-wave impulse approximation(RDWIA). The theoretical values overestimate the experimental ones by a factor of 2 when thespectroscopic factor is obtained from the (e, e' p) experiment in parallel kinematics, or by a factor of 1.4
when spectroscopic factor is obtained in constant (ω, q) kinematics.

§1. Introduction
Recently, the (e, e' p) reaction in the quasi-elastic region and the (γ, p) reaction above the giant
resonance region have been investigated in the relativistic framework [1-3]. The contributions of mesonexchange currents (MEC) and the choice of the one-body current operator were discussed in Refs. [1, 2].
In previous reports [4-6] we compared the reduced cross section obtained from 12C(e, e' p0) and thedifferential cross section obtained from 16O(e, e' p0) with theoretical calculations based on the relativistic
distorted wave impulse approximation (RDWIA). Both experiments were performed at a low momentum→|= 105.2 MeV/c. The result oftransfer region: energy transfer ω = 60 MeV and momentum transfer |qthese reports was that RDWIA overestimates the experimental data by a factor of 2. A large
contribution of the two-body seagull term was observed at high missing momentum region.
To investigate this reaction mechanism further, we choose 40Ca as a target and performed theexperiment. The 40Ca(e, e' p0) cross section obtained from this experiment is compared with calculations
based on RDWIA.

§2. Experiment
The experiment was performed at Laboratory of Nuclear Science, Tohoku University (LNS) using a199.53 MeV continuous electron beam from the stretcher-booster (STB) ring. Incident electrons werescattered with a natural calcium target of 19.0 mg/cm2 thickness. Scattered electrons were analyzed

19
their momentum at θe = 30° by a magnetic spectrometer (LDM) and detected with a vertical drift
chamber (VDC) placed on the focal plane of the spectrometer and with a backup counter, layered threeplastic scintillators behind the VDC. In this setup, ω is 60.0 ± 4.2 MeV and |q→| is 105.2 MeV/c.
Knocked-out protons were detected by SSD telescopes, which consisted of three layers of 1mm-thick
SSD' s. In order to degrade the energy of protons, a 6mm-thick aluminum disk was put in front of eachtelescope.
The normalization of our measured (e, e' p) cross section was performed by comparing the elastic
scattering cross section with that of Ref. [7]. We measured the elastic scattering cross section at θe =58.5°. The form factor of our experiment corresponds to that at θe = 46.0° with a beam energy 249.3

The measured cross sections are shown in
Table 1 and Fig.1. The 40Ca(e, e' p0) cross section
obtained from our experiment is compared with2§3. Result and Discussion[nb/sr /MeV]MeV of Ref. [7].

10

theoretical calculations based on RDWIA [8].
The RDWIA treatment is same as in Refs. [1, 2].
The bound state wave function is a solution of1

a relativistic Hartree-Bogoliubov equation
using parameters NLSH [9]. The EDAD1
optical potential parameters [10] are used for
calculations of the scattering wave function. As-1

10

the choice of the electromagnetic operator is0

arbitrary, three current conservation operators,
cc1, cc2 and cc3, are used in the calculations.

102030405060θ p [deg]

Fig.1. The 40Ca(e, e' p0) cross section.

The spectroscopic factor Z(d 3/2 ) was

obtained from analysis of the quasi-elastic (e, e' p0) reaction [11], which included the results of two
different kinematics, parallel kinematics and constant (ω, q) kinematics. The factor was calculated forthe three current operators on each kinematics. The spectroscopic factors obtained from parallel
kinematics and constant (ω, q) kinematics are listed in Table 2. The present data are compared with
Table 1. Differential cross section of the40Ca(e, e' p0) reaction.
θeCross section [nb/MeV/sr2]

0°
12°
24°
36°
48°
60°

6.901 ± 0.427
5.832 ± 0.392
4.512 ± 0.347
1.662 ± 0.213
0.791 ± 0.150
0.179 ± 0.078

Table 2. Spectroscopic factors for three current
operators in two kinematics.

cc1cc2cc3parallel kinematics

constant (ω , q)

0.627
0.713
0.766

0.446
0.594
0.701

[nb/sr /MeV]

1011-110100102030405060-170 80
θp [deg]

02030

4050

6070 80
θp [deg][nb/sr /MeV][nb/sr /MeV]

21010

11-1

101010The 40Ca(e, e' p0) cross section compared with theoretical calculations, using spectroscopic
factors obtained from the (e, e' p) experiment in the parallel kinematics. Calculations
include the MEC effect in the left panel, and does not in the right panel. Closed circles
show the present data, and solid, dashed, and dot-dashed lines represent theoretical
calculations with cc1, cc2, and cc3 current operators, respectively. θp is the out-going angleof the proton.

Fig.2.

2102

2[nb/sr /MeV]

20010203040506070 80
θp [deg]-10102030

40506070 80
θp [deg]Fig.3. Same as Fig. 2, but comparison with theoretical calculations using the spectroscopic factor
obtained from the (e, e' p) experiment in the constant (ω, q) kinematics.

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theoretical calculations in Figs.2 and 3. The theoretical values are obtained using 3 one-body currentconservation operators and spectroscopic factors listed in Table 2. According to Figs.2 and 3, the MEC
effect is not important in the small θp region, corresponding to the lower momentum transfer region.
Calculations with spectroscopic factors obtained from parallel kinematics overestimate the data, atforward angles, and are different from each other, as shown in Fig.2. In order to ﬁt calculations to thedata, a normalizing factor 0.423 is needed for the cc1 current, and 0.523 and 0.588 for the cc2 and, cc3
currents, respectively. Thus, theoretical values overestimate the experimental ones by a factor about 2.
This result is same as in previous reports [4-6]. As shown in Fig.3, calculations with spectroscopic
factors obtained from constant (ω , q) kinematics also overestimate the data by a factor of about 1.4,
but the calculation is equivalent to each other in the small θp region. The theoretical values are closer
to the experimental one than using a spectroscopic factor obtained in parallel kinematics, but still toolarge.

§4. Summary
Both theoretical calculations using the spectroscopic factor obtained in the parallel and constant (ω,q) kinematics. overestimate the experimental cross section, about factors 2 or 1.4, respectively. The (e,e' p) experiment at the low momentum transfer in 12C, 16O and 40Ca show some important contributions
is missing in the theoretical approach.
The authors would like to thank C. Giusti and A. Meucci for their relativistic calculations. They
also thank the accelerator group, the computer group and Mr. Matsuda for their assistance to theexperiment. This work has been supported by Grant-in-Aid for Scientific Research (KAKENHI)(No.14540239, No.17540229) from Japan Society for the Promotion of Science (JSPS).

References[1]

A. Meucci, C. Giusti, and F. D. Pacati: Phys. Rev. C64 (2001) 064615.

[2]

A. Meucci, C. Giusti, and F. D. Pacati: Phys. Rev. C66 (2002) 034610.

[3]

J. I. Johansson, H. S. Sherif, and G. M. Lotz : Nucl. Phys. A605 (1996) 517.

[4]

Y. Sato et al.: Research Report of LNS, Tohoku Univ. 33 (2000) 7.

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T. Tamae et al.: Research Report of LNS, Tohoku Univ. 36 (2003) 19.

[6]

Y. Hayashi et al.: Research Report of LNS, Tohoku Univ. 37 (2004) 1.

[7]

B. B. P. Sinha et al.: Phys Rev C7 (1973) 1930.

[8]

C. Giusti and A. Meucci: private communication.

[9]M. M. Shaeme, M. A. Nagarajan, and P. Ring: Phys. Lett. B312 (1993) 377.

[10]E. D. Cooper et al.: Phys. Rev. C47 (1993) 297.

[11]

G. J. Krammer: private communication.