STUDY OF NUCLEAR MEDIUM EFFECTS USING THE ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B AND ⁴He ($\gamma$,$\pi^{+}_{}$n) REACTIONS

D. Branford⁽¹⁾,
for the Edinburgh, Glasgow, Tübingen, Mainz PiP/TOF Collaboration, which is part of the A2 Collaboration at Mainz.

⁽¹⁾ Department of Physics and Astronomy,
University of Edinburgh,Edinburgh, EH9 3JZ, Scotland.

printings:

abstract:

$\parbox{15truecm}{\footnotesize \parindent=0pt Measurements of the... ...improved calculations are required to quantify $\Delta$\ medium effects.\par }$

1. Introduction

It is well known that excitation of the nucleon to the $\Delta$(1232) resonance plays an important role in intermediate energy photonuclear reactions. However, although modifications to the excitation, propagation and decay of the $\Delta$, brought about by the surrounding nuclear medium, have been predicted (e.g. [1]), few measurements against which to test these predictions have been made. To provide a comprehensive survey, we have made measurements of the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B and ⁴He ($\gamma$,$\pi^{+}_{}$n) reactions using tagged photons up to E_$\gamma$ $\sim$ 400 MeV. Large position sensitive detectors were employed so that broad range angular distributions could be measured. The reasoning behind the measurements was that changes to the $\Delta$ mass and width will tend to redistribute the strength and this can only be seen when the reaction is studied over a wide range of energies and angles.

2. Experimental Arrangement

Fig. 1 shows the experimental arrangement. A bremsstrahlung photon beam was produced by a $\sim$ 15 nA beam of electrons from the Mainz microtron (MAMI-B) incident on a 4 $\mu$m Ni radiator. Photon energies were analysed using the Glasgow tagging spectrometer [2,3], giving typical tagged photon resolutions and counting rates of $\Delta$E_$\gamma$ = 2 MeV and 5 x 10⁷ s^-1, respectively.

Figure 1: Experimental arrangement used for the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B measurement.

The target for the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B reaction study was either 0.839 gcm^-2 Carbon or 0.915 gcm^-2 CH₂ inclined at an angle of 20^o. An 8 cm long Kapton walled cell filled with liquid ⁴He was used for the ⁴He ($\gamma$,$\pi^{+}_{}$n) measurement. On one side of the beam, a large plastic scintillator hodoscope (PiP) [4] was used to detect charged particles in the angular range $\theta$ = 50-130^o and $\phi$ = (-24)-24^o. A time of flight detector array (TOF) was placed on the opposite side of the beam. This detector [5] consisted of 6 stands each containing 16 vertically mounted scintillators of dimension 3000x200x50 mm³ in two ranks of 8 (see Fig. 1) and covered the range $\theta$ = 10-150^o. The range of azimuthal angles $\phi$ covered by each element depended on the distance from the target which was between 6 and 14 m. Surrounding the target at a radius of 30 cm was a ring of 15 thin $\Delta$E scintillator detectors. Used in coincidence with PiP, they produced a trigger pulse for each detected charged particle. On the TOF side, the presence or absence of a signal in the appropriate element indicated a charged or neutral particle, respectively.

Figure 2: Schematic diagram of the PiP detector showing the orientations of the thin $\Delta$E-detectors (vertical) and the thick E-detector bars (horizontal), assembled with the light guides.

Fig. 2 shows a schematic of the PiP detector, which consisted of a 2 mm thick $\Delta$E and four E layers of thickness 110, 175, 175 and 175 cm, all constructed from NE110 plastic scintillator material[4]. This gave a total stopping power of 180 MeV for pions. Events due to detecting $\pi$ particles were separated from those due to protons and electrons by selecting a region of the $\Delta$E-E distribution. Positive pion events were selected by demanding that an afterpulse occurred within the time interval 0.2-6.2 $\mu$s after the initial interaction.

An undesirable property of this type of detector, is the degradation in resolution that occurs due to inelastic nuclear reactions in the material. However, the multi-layer structure of the detector provides the means to reject most events of this type. Given that the problem is most serious for the higher energy pions, which traverse two, three or four E-layers, it is possible to demand in such cases that the energy deposited in each layer is consistent with purely electronic stopping. An algorithm to implement this condition was tested using a sample of pions from the p ($\gamma$,$\pi^{+}_{}$)n reaction obtained with the CH₂ target.

The above conditions determine the efficiency of PiP as a $\pi^{+}_{}$ detector. In order to measure this efficiency, $\pi^{+}_{}$ particles produced by the p( $\gamma$,$\pi^{+}_{}$n) reaction in a CH₂ target, tagged by detecting the correlated neutron in TOF, were employed. For each event, it was possible, using the kinematics of the reaction, to deduce the direction and energy of the corresponding pion. The efficiency ( $\epsilon_{\pi}^{}$) was obtained by comparing the number of tagged pions incident on the detector at a given energy and angle, to the number that actually survived to give an afterpulse and an energy consistent with the p( $\gamma$,$\pi^{+}_{}$n) reaction. The neutron TOF technique is fairly standard and has been described elsewhere [6].

In order to separate events corresponding to the removal of protons from the 1p and 1s shells of ¹²C, it was necessary to obtain an overall energy resolution of better than 10 MeV. Hence, a good calibration was crucial. For PiP this was achieved using cosmic rays. Using these calibrations, the signals from the individual blocks were combined and the overall calibration obtained was checked later using the two body p ($\gamma$,$\pi^{+}_{}$n) reaction. Fig. 3

Figure 3: Missing energy spectra for the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B reaction determined using prompt events (solid line), and after subtracting random backgrounds (dashed line).

shows the missing energy spectrum for the reaction ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B . The missing energy E_m is defined by the equation,

$$\gamma \pi$$ (1)

where E_$\gamma$ is the energy of the tagged photon, E_$\pi$ the kinetic energy of the pion, E_n the kinetic energy of the neutron and E_recoil the kinetic energy of the recoiling A = 11 system determined using momentum conservation. E_x and Q are the excitation energy associated with the A = 11 system and the Q-value for the reaction leading to the ground state, respectively.

As the aim of the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B experiment was to measure exclusive pion production on p-shell nucleons, this channel was isolated by applying a cut from E_m = 150-165 MeV. No cut was made for the ⁴He ($\gamma$,$\pi^{+}_{}$n) reaction since the missing energy spectrum shows a single peak due to the removal of 1s shell protons. In order to obtain cross sections with a reasonable statistical accuracy, it was necessary to combine the data into fairly large energy and angle bins. The bins are noted in table 1.

Table 1: The binning regions used in extracting the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B cross sections.

Quantity	Range	Bin Size	No. of Bins
E$\gamma$	240-400 MeV	40 MeV	4
E$\pi$	20-180 MeV	10 MeV	16
$\theta_{\pi }^{}$	60-120o	15o	4
$\phi_{\pi }^{}$	(-15)-15o	30o	1
$\theta_{n}^{}$	10-150o	5o	28
$\phi_{n}^{}$ - $\phi_{\pi }^{}$	170-190o	20o	1

Results for the p( $\gamma$,$\pi^{+}_{}$n) reaction were obtained in a similar way using data taken with the CH₂ target. The $\gamma$p $\rightarrow$ $\pi^{+}_{}$n events were separated on the basis of their missing energy and the cross section was obtained by integrating over allowed E_$\pi$ and n angles. These cross sections were compared to calculations using the expressions of Blomqvist and Laget [7], which reproduce the previously measured cross sections [8]. An overall normalisation factor of 1.20 was found to be required to bring the present p( $\gamma$,$\pi^{+}_{}$n) results into agreement with the calculations. Since this factor is consistent with the total estimated systematic error of 20%, it was decided to renormalise the ¹²C and ⁴He data by the same factor of 1.2. A systematic error of $\pm$ 10% for the renormalised data was obtained by combining the statistical errors of the present p( $\gamma$,$\pi^{+}_{}$n) data with the systematic errors of about $\pm$ 4% for the previous data [8].

Figure 4: Differential cross sections for the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B reaction versus n laboratory angle obtained at average $\pi$ detection angles of $\theta_{\pi }^{}$ = 67^o and 82^o, and average photon energies E_$\gamma$ = 260, 300, 340, and 380 MeV. The E_$\gamma$, $\theta_{\pi }^{}$, $\phi_{\pi }^{}$ and $\phi_{n}^{}$ bins are as shown in table 1. The results are integrated over E_$\pi$ and E_n. The theory curves are LWPW (dot dashed), LWDW (solid), VPW (long dashed) and VDW (dashed) calculations.

3. Results and Discussion

The large peak in the missing energy spectrum for the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B reaction shown in Fig. 3 occurs at 160 MeV and has a FWHM of $\Delta$E_m $\sim$ 12 MeV. There is an indication of a broader peak at E_m $\sim$ 180 MeV with $\Delta$E_m $\sim$ 25 MeV. Very similar spectra have been observed in the ¹²C(e, e'p)¹¹B [9], ¹²C(p,2p)¹¹B [10] and ¹²C(p,d $\pi^{+}_{}$)¹¹B [11] reactions. As in those cases, the two peaks are interpreted as arising from removing a proton from the 1p and 1s shell, respectively.

Figure 5: As for Fig. 4 for $\theta_{\pi }^{}$ = 97^o and 112^o.

Figs. 4-5, which were obtained using the bins shown in table 1, show the $\theta_{n}^{}$ dependence of the cross sections for the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B reaction. Similar results were obtained for the ⁴He ($\gamma$,$\pi^{+}_{}$n) reaction. Fig. 6-7 show the cross sections, integrated over the $\theta_{n}^{}$ angles, as a function of of $\theta_{\pi }^{}$ at 4 average photon energies. All the results show that the cross section is largest for E_$\gamma$ $\sim$ 340 MeV, which is consistent with a reaction proceeding through excitation of the $\Delta$(1232 MeV) resonance. A quantitative assessment of our data was obtained by comparing the results to PWIA and DWIA calculations made by Lee and Wright [12], (referred to as LWPW and LWDW, respectively), and by Vanderhaeghen [13], (VPW and VDW, respectively). Both models have three basic ingredients: (1) single nucleon bound state wavefunctions and associated spectroscopic factors, (2) an elementary pion photoproduction operator, and (3) pion and nucleon optical model potentials.

Figure 6: Differential cross sections for the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B reaction versus pion angle. The results are integrated over E_n, E_$\pi$ and $\theta_{n}^{}$. The E_$\gamma$, $\theta_{\pi }^{}$, $\phi_{\pi }^{}$ and $\phi_{n}^{}$ bins are as shown in table 1. The theory curves are LWPW (dot dashed), LWDW (solid) and VDW (dashed).

Figure 7: Differential cross sections for the ⁴He ($\gamma$,$\pi^{+}_{}$n) reaction versus pion angle. The results are integrated over E_n, E_$\pi$ and $\theta_{n}^{}$. The E_$\gamma$, $\theta_{\pi }^{}$, $\phi_{\pi }^{}$ and $\phi_{n}^{}$ bins are as shown in table 1. The theory curves are LWPW (dot dashed) and LWDW (solid).

The calculations of Lee and Wright (LW) were performed using the model (LWB) of Li, Wright and Bennhold [14]. The proton 1p_3/2 bound-state was represented by a harmonic oscillator wave function. For the pion photoproduction process, the full Blomqvist-Laget production operator [7,15] was used. The use of this phenomenological operator allows possible $\Delta$ nuclear medium effects to be investigated by varying the mass M_$\Delta$ and the decay width $\Gamma_{\Delta}^{}$, as shown in LWB for selected kinematics in the $\Delta$ resonance region.

The Vanderhaeghen results (V) were obtained using Hartree-Fock-Skyrme wavefunctions to describe the 1p_3/2 bound-state [13]. The elementary pion photoproduction operator was derived from a fully relativistic and unitary model developed by Vanderhaeghen [13]. For the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B calculations, $\Delta$ medium effects derived by Oset et al. [16] were included.

A spectroscopic factor S = 2.6, which is derived from ¹²C(e, e'p)¹¹B measurements [14], was used in both sets of calculations. The outgoing pion wavefunctions were calculated using an effective description based on the results of Stricker, McManus and Carr [17], Seki et al. [18] and Nieves et al. [19], and Cottingame et al. [20] and Gmitro et al, [21]. The neutron optical potentials used was the global phenomenological potential of Schwand et al. [22] and of Meyer et al. [23].

The main difference between the two sets of calculations is the absence of $\Delta$ medium effects in LW. Although these effects could have been simulated by varying M_$\Delta$ and $\Gamma_{\Delta}^{}$, this was not carried out due to the large computer time requirement. For comparison with the data, the calculations are averaged over the detector acceptances given in table 1. When making these comparisons, we note that only the statistical errors in the data need to be taken into account since the experimental results are normalised to the p( $\gamma$,$\pi^{+}_{}$n) data which were fitted to obtain the elementary pion photoproduction operator. The main uncertainties in the calculations are associated with the optical model estimates of the FSI. In considering this, we estimate an overall uncertainty in the DWIA results due to the optical model parameters used of $\pm$ 15%. In the case of the LWDW results, there is an additional uncertainty of $\pm$ 10% due to the use of the local potential approximation.

Reasonable qualitative agreement in shape between the DW calculations and the data is obtained for all the results, which adds further weight to the assumption that in our kinematic regime, the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B and ⁴He ($\gamma$,$\pi^{+}_{}$n) reactions proceed through a quasifree pion production mechanism. It is interesting to note that on average the VDW curves are in closest agreement with the measured cross sections, whereas the LWDW curves tend to lie above the data. Although these results are qualitatively as expected if $\Delta$ medium effects are present, it is not possible at this time to conclude that $\Delta$ medium effects have been observed due to the uncertainties associated with the absolute normalisations of the data and the calculations. Clearly, a quantitative estimate of the magnitude of $\Delta$ medium effects will require higher statistics data and a more detailed theoretical treatment.

To consider the LWDW results further, we note that the calculated curves shown in Fig. 4 lie 10-100% above the data, the largest differences occuring at the largest $\theta_{\pi }^{}$. Calculations shown in LWB for the ¹⁶O ($\gamma$,$\pi^{-}_{}$p)¹⁵N reaction indicate that a reduction in M_$\Delta$ by $\sim$2-3%, reduces the cross section on average by $\sim$ 30%. However, the largest reduction ($\sim$ 40%) occurs at forward angles close to $\theta_{\pi }^{}$ = 67^o with smaller effects ($\sim$ 20%) occuring at large angles such as 112^o. A similar reduction in M_$\Delta$ would therefore improve the agreement between LWDW and the present data on average, but large discrepancies would remain at $\theta_{\pi }^{}$ = 67^o and 112^o. A possible explanation is given below.

A comparison of the theory to the E_$\pi$ dependence of the differential cross section (not shown here) suggests that the optical potentials used in the calculations may need to be modified. Overall, the LWPW and LWDW curves describe the shapes of the spectra quite well. However, there is evidence that the LWDW theory underestimates the data at the higher $\pi$ energies. To investigate this, the weighted mean of the ratio, R = theory/experiment, was obtained and fitted with a straight line. Based on the slope of this line, we conclude that the attenuation of the pion, neutron or both outgoing waves has the wrong energy dependence and may overestimate the flux losses for high values of E_$\pi$, which also correspond to low E_n. In both cases, this is the energy region where the absorption is highest. Since, on average, forward emitted pions have higher energies, the effect of using $\pi$ and n optical model potentials that are too absorptive is to reduce the calculated cross sections more at forward than backward $\theta_{\pi }^{}$. This suggests that a justifiable reduction of M_$\Delta$ in conjunction with less absorptive optical potentials could lead to acceptable fits to the data. Similar arguments apply to the ⁴He ($\gamma$,$\pi^{+}_{}$n) data, although in that case the largest differences between the LWDA calculations and the data occur at $\theta_{\pi }^{}$ = 67^o as shown in Fig. 7.

4. Conclusion

Data of good statistical accuracy have been obtained for the ¹²C ($\gamma$,$\pi^{+}_{}$n)¹¹B and ⁴He ($\gamma$,$\pi^{+}_{}$n) reactions over a wide range of E_$\gamma$, $\theta_{\pi }^{}$ and $\theta_{n}^{}$. The missing energy resolution was sufficiently good to resolve events due to the removal of 1p shell protons in ¹²C from those involving the 1s shell. A comparison of DWIA calculations to the data is consistent with the explanation that modifications to the amplitudes describing $\Delta$ excitation, propagation and decay may be occuring in the nuclear medium. A comparison of the LWDW results with the E_$\pi$ dependence of the differential cross sections suggests that the optical model parameters used in the DWIA calculations may lead to an overestimate of the loss of $\pi$ and n fluxes at high E_$\pi$ and low E_n, respectively. Clearly, the interpretation of these results would benefit from a more careful and detailed theoretical investigation.

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