2N KNOCKOUT AT THE EXTREMES OF THE PAIR MOMENTUM DISTRIBUTION

D. P. Watts

Department of Physics & and Astronomy, Glasgow University, Scotland, UK

FOR THE PiP/TOF GROUP OF THE MAINZ A2 COLLABORATION



abstract:

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1. Introduction

The study of the photo-emission of nucleon pairs from the nucleus was originally thought of as a good tool for studying short range correlations. It was later realised that several mechanisms will play a role in these reactions and may make it difficult to extract information on correlations. In practice our understanding of the different processes involved has only progressed significantly in quite recent years, which have seen major improvements in both the quality of the data and in the theoretical description of the photo-emission process.

When a photon is absorbed by a nucleus the different absorption and final state interaction (FSI) processes pose a complicated theoretical problem. In particular FSI produce 2-step reaction processes which make information about the initial state difficult to extract. However recent calculations using the Valencia model [1] (VM) indicate that for ($ \gamma$,2N) measurements restricted to low excitation energy in the residual (A-2) system the direct 2N knockout process gives the dominant contribution, with small contributions from FSI or processes other than 2N-knockout. These predictions have been qualitatively confirmed by recent experimental data[2].

Although the Valencia model represents a significant theoretical advance, allowing the treatment of the full complexity of the photonuclear reaction mechanism, this is achieved at significant cost. The model is based on a nuclear matter approach which, although related to real nuclei by a local density approximation, neglects binding effects and averages over nuclear shell structure. Other theories have aimed to understand only the 2N knockout part of the cross section leaving the residual nucleus in a low lying, bound state and this more limited aim allows the models to be based on a more realistic description of nuclear structure. The early theoretical treatment of 2N knockout by Gottfried[3], which has been the basis of many subsequent experimental analyses, uses a `zero range approximation' for the interaction and restricts the photoabsorption to pairs in relative S angular momentum states with the additional assumption of outgoing plane waves. The cross section can then be written as the product of F(P), the probability of finding a pair in the nucleus at zero separation with a CM momentum of P, and a second term Sfi which depends principally on the relative wavefunction of the pair. The F(P) distribution is obtained simply from folding two individual nucleon momentum distributions. Due to parity conservation the relative S state is only allowed with pair angular momenta, L, of 0 or 2 for two (1p) shell nucleons and these two possibilities make up the (1p)2 F(P) distribution.

In recent years far more detailed microscopic models of 2N knockout have become available[4,5]. These models include a distorted wave treatment of the emitted nucleons and a detailed description of the contributing photo-absorption mechanisms, meson exchange currents (MEC), $ \Delta$ contributions and short range correlations (SRC) and are leading to a much deeper understanding of 2N knockout. In these models the formal factorisation, F(P) x Sfi, of the Gottfried treatment is lost, although the pair momentum distribution is still a determining factor in the angular correlation of the outgoing nucleons.

As these models employ the single pair approximation in the calculation of the knockout process the introduction of correlations between the two knocked out nucleons does not affect the CM momentum distribution of the pair. A recent calculation[6] which includes correlation effects between all possible pairs in the nucleus has indicated that at large P, the CM momentum distribution is sensitive to short range correlations with the other (A-2) nucleons. The probability of such large P is small so the prediction does not imply a significant modification of the dominant cross section at lower P. However the study does suggest an interesting further way to access correlation effects in 2N knockout measurements.

The ($ \gamma$,2N) experimental results presented in this contribution are all taken from a low excitation energy region corresponding to that expected to be populated by the knockout of (1p)2 pairs and over a phase space region including regions away from the back-to-back kinematics of previous measurements. The very wide detector acceptance allows study of the 2N knockout process to be extended to kinematic regions where the initial momentum of the nucleon pair is necessarily large. This range of pair momenta and the similarly wide photon energy range of the present data will test the limitations of the previous analyses[2], in which it was assumed that the variation of the ($ \gamma$,NN) cross section is dominated by its proportionality to F(P), which was calculated using harmonic oscillator (HO) wavefunctions and the Gottfried prescription. The pair momenta also reach the region in which sensitivity to SRC is predicted[6].

2. Experimental Setup

Figure 1: Layout of the detectors in the Mainz A2 hall.
\begin{figure} \begin{center} \epsfig{file=tof_ps.ps,height=0.5\textheight} \end{center}\end{figure}

The experiment was carried out at the 855 MeV Mainz microtron (MAMI-B)[7] using the Glasgow tagged-photon spectrometer[8] together with two plastic scintillator arrays to detect the emitted nucleons. The positioning of the detectors for the experiment is shown in Fig. 1.

Protons were detected in a charged particle hodoscope (PiP)[9] placed in a backward position covering the polar angular range 78o-158o. The reaction timing was obtained from a segmented half-ring of scintillators ($ \Delta$E PiP) centred on the target and positioned on the PiP side of the photon beam at a radius of $ \sim$11 cm.

Coincident protons and neutrons were detected in an array of time of flight plastic scintillators (TOF)[10] which were positioned to cover a wide angular range (36.7o-142.0o) opposite PiP and 16o-31o on the PiP side of the beam. The TOF flight paths were in the range 4.0-6.2 m, giving a total solid angle $ \Delta$$ \Omega$=0.91 sr. Separation of charged and uncharged particles in the TOF array was carried out using information from two segmented half-rings of scintillator ($ \Delta$E1, $ \Delta$E2) each 2mm thick centred on the target at radii of $ \sim$ 11cm and $ \sim$ 30 cm. Detector calibrations were carried out using a 216.0 mg/cm2 deuterated polythene target (CD2). The combined missing energy resolution for the experimental setup was found to be $ \sim$8 MeV.

3. Results

In the analysis the experimental data have been separated into three kinematic regions according to the angle of the particle detected in TOF, as indicated on Fig. 1. Region I samples the back-to-back kinematics whilst regions II and III sample more extreme kinematics away from this region. The experimental data is studied as a function of two useful variables. The momentum of the recoiling system, $ \bf P_{r}^{}$ = $ \bf P_{\gamma}^{}$ - $ \bf P_{N1}^{}$ - $ \bf P_{N2}^{}$, is obtained from the measured momenta of the incident photon and the emitted nucleons. If FSI and the effect of the nuclear potential on the outgoing nucleon momenta are neglected, then for quasielastic absorption by two nucleons the initial pair momentum, $ \bf P$ is given by $ \bf P$ + $ \bf P_{r}^{}$ = 0. The excitation energy is defined as EX = E$\scriptstyle \gamma$ - TN1 - TN2 - Tr - Q where Q is the Q value for the reaction leading to the ground state, E$\scriptstyle \gamma$ is the incident photon energy, TN1 and TN2 are the energies of the outgoing nucleons and Tr is the (typically small) energy of the recoiling (A-2) system which is calculated from its momentum $ \bf P_{r}^{}$. The excitation region expected to be populated following (1p)2 knockout was estimated from the folding of two single nucleon excitation distributions, obtained from electron scattering (e,e'p) data, and found to be below $ \sim$13 MeV[2].

Figure 2: Recoil momentum distributions for ($ \gamma$,np) in the (1p)2 knockout region (EX $ \leq$12.6 MeV). The lines on the figure show the predictions of the 2N knockout MC (green solid), phase space model (black dotted) and the predicted total (blue dot-dash), direct 2N knockout (red dash) and 2N+FSI (thin black solid) cross sections from the Valencia model. The Valencia model predictions have been multiplied by a factor of 0.5.
\begin{figure} \begin{center} {\epsfig{file=grabs_p4.ps,height=0.63\textheight}} \end{center}\end{figure}

For comparison with the low excitation energy data a Monte Carlo model of the 2N knockout process (MC) is used, the details of which have been described previously[2]. The MC simulation allows the effect of the detector acceptances to be included in the model predictions. The F(P) used in the model is obtained from folding two individual HO nucleon momentum distributions, assuming the resulting nucleon pair is at zero separation in a relative S state of angular momentum and taken from a closed sub-shell. These restrictions lead to a CM pair momentum distribution for an np pair having L components in the ratio $ {\frac{1}{3}}$:0: $ {\frac{2}{3}}$ for L=0, 1, 2 respectively.

The recoil momentum distributions from the 12C($ \gamma$,np) measurement in the (1p)2 knockout region for EX $ \leq$12.6 MeV are shown in Fig. 2. The shape of the region I cross section is well described by the 2N knockout MC (green) for photon energies below and through the $ \Delta$ resonance region. For E$\scriptstyle \gamma$ above 500 MeV the experimental data seem to indicate different ratios of CM angular momentum components than in the simple prescription used in the MC. There is some indication that the reaction mechanism at high E$\scriptstyle \gamma$ is tending to prefer L=0 pairs. Aside from the discrepancies at high photon energies the MC generally describes the data in the back-to-back kinematics well. The MC, normalised to the data in region I, can be seen to reproduce the shape and magnitude of the Pr 1.0 35$ \sim$ 90$ \vee$ 400 MeV/c cross section in the more extreme kinematics. The sensitivity of the cross sections in these kinematic regions to high pair momenta shows the existence of some additional strength in the experimental data for Pr 1.0 35$ \sim$ 90$ \vee$ 400 MeV/c which is not described by the MC. As the excess yield is small compared to the main strength sampled in region I the results indicate the 2N knockout process dominates the cross section at low excitation, even when including regions well away from the usual back-to-back kinematics. Some possible explanations for the small strength at high recoil momenta are discussed in the next section.

4. Recoil momentum distribution at high Pr - Possible short range effects

The ratio of the (1p)2 pn knockout data to the MC model

Figure 3: The ratio of the experimental data to the 2N knockout MC predictions as a function of recoil momentum. The plot shows data from the kinematic regions I (red circles), II (blue squares) and III (pink triangles) described in the text. The blue solid line shows the predicted excess at high recoil momentum due to SRC. The pink dotted (red dashed) lines show the predicted deviations when Hartree-Fock (Saxon-Woods) wavefunctions are used to calculate the recoil momentum distribution.
\begin{figure} \begin{center} \epsfig{file=2n_data_model_ratio_granada.ps,height=0.55\textheight} \end{center}\end{figure}

prediction is presented in Fig. 3 to highlight the excess at high recoil momentum. Although there is some scatter in the points Fig. 3 seems to indicate an excess over the HO prediction is present in the data at high Pr with the Pr dependence of the excess similar for all measured photon energy regions and kinematics. The calculation of Orlandini and Sarra[6] (blue line) is in reasonable agreement with the data. It shows the ratio of the pair momentum distribution which they obtain for 16O including SRC divided by the distribution obtained without SRC. The agreement with the data should be treated with caution since their calculation includes all excitations in the residual nucleus, whereas the present results include only low lying states. Equivalent calculations for one-nucleon knockout find that the additional high momentum strength produced by SRC is predominantly associated with large excitations in the residual nucleus.

Two other possible sources of the excess yield have also been examined. The red dashed (pink dotted) lines show the ratio of the pair momentum distribution obtained with Saxon-Woods (Hartree-Fock) wavefunctions[11] to the HO result. These indicate that the excess is not a result of the inadequacy of the HO wavefunctions. Second the predictions of the Valencia model [1] were compared to the data in Fig. 2 to see if processes other than direct 2N knockout could be responsible for the measured excess. The result is inconclusive, other processes do produce an excess above Pr $ \sim$450 MeV/c but with only $ \sim$50% of the observed strength. It should be remarked that the Valencia model can only be expected to give rough estimates of the relative contributions of the different processes at low excitation energy due to the neglect of shell structure in the model.

In summary, although Fig. 3 suggests a measurable influence of SRC on the pair momentum distribution, more detailed analysis of other contributing mechanisms and a knowledge of the distribution of the SRC strength over different excitation energy regions is needed.

5. The angular momentum components of F(P)

Figure 4: 12C($ \gamma$,np) (4 left panels) and 12C($ \gamma$,pp) (4 right panels) recoil momentum distributions for two EX cuts within the (1p)2 knockout region. The data have been corrected for phase space and detector effects. The curves show the result of fitting the data with HO pair momentum wavefunctions for L=0 (red dashed), L=1 (green dotted), L=2 (blue dot-dash) and total (black solid).
\begin{figure} \begin{center} \epsfig{file=spectral_pm_pn.ps,height=0.55\textheight} \epsfig{file=spectral_pm.ps,height=0.55\textheight} \end{center}\end{figure}

The recoil momentum spectra can be used to explore the expected sensitivity of 2N knockout to the nature of the residual state and hence to the reaction mechanism. The recoil momentum data are fitted with a combination $ \sum_{L}^{}$aLFL(P) where the CM pair momentum distributions, FL(P), for each angular momentum value are calculated from HO wavefunctions. To conserve spin and parity in the knockout of (1p)2 pairs even (odd) valued quantum states of relative angular momentum must couple to even (odd) valued states of pair angular momentum, L[4,5]. Before being fitted the recoil momentum distributions have been corrected for the effect of detector acceptance using a correction function determined at each photon energy from the 2N knockout MC simulation. To improve the presentation of the fits the P2 phase space factor in the momentum distributions has been divided out.

Fits to the corrected ($ \gamma$,np) and ($ \gamma$,pp) recoil momentum distributions are presented in Fig. 4 for two photon energy bins, chosen to separate knockout from below and also through the peak of the $ \Delta$ resonance. The (1p)2 knockout data is divided into two regions of excitation energy, EX, in the residual (A-2) nucleus, -3 $ \leq$ EX $ \leq$ 3 and 3 $ \leq$ EX $ \leq$ 9. The width of the cuts are comparable to the experimental resolution. For pn knockout to the low lying states both E$\scriptstyle \gamma$ regions can be described dominantly with L=0, 2 wavefunctions with ratios similar to that expected from the zero range prediction. Knockout to the higher lying states does however show an additional contribution from L=1 which decreases with E$\scriptstyle \gamma$. This may indicate a small contribution to the cross section from pairs in relative P states.

Additional selection rules apply in the knockout of pp pairs as they only occur in an isospin triplet (T=1) state. The $ \Delta$N $ \rightarrow$ NN knockout mechanism is suppressed for magnetic dipole (M1) photoabsorption on 1S0 proton-proton pairs because of total angular momentum and parity conservation requirements in the decay of the intermediate N-$ \Delta$ state[4,5]. The $ \Delta$ can only contribute to the pp knockout mechanism through photon absorption on pairs in higher relative waves than the S state or from transitions involving higher multipolarity components. In pp knockout the residual 10Be nucleus has a larger spacing of the low lying residual states ( 0+, 2+(3.37 MeV) ) than the 10B states fed in pn knockout. The -3 $ \leq$ EX $ \leq$ 3 cut will emphasize the cross section to the 0+ ground state of 10Be, although the experimental resolution of the present data does not permit a clean separation.

The ($ \gamma$,pp) data for -3 $ \leq$ EX $ \leq$ 3, shown in Fig. 4, indicate the knockout of L=0 pp pairs occurs in both E$\scriptstyle \gamma$ regions. Because of parity restrictions the knockout of L=0 pairs restricts the possible initial relative momenta of the pair to either a 1S0 or 1D2 state. However, only the 1S0 component can leave the residual nucleus in the 0+ state emphasized in this EX region and pairs in relative 1D2 states are generally predicted to give small contributions to the 2N knockout cross section[4]. 1S0 absorption is also implied by the knockout of pairs with L=2, the relative cross section for which increases with E$\scriptstyle \gamma$. An increase in 1S0 knockout with photon energy could be due to either larger SRC contributions or a significant contribution from higher multipolarity photons allowing the contribution of the $ \Delta$ in the knockout of 1S0 pairs. The relative proportion of L=1 knockout is larger than that for the corresponding region in ($ \gamma$,pn).

The data for 3 $ \leq$ EX $ \leq$ 9 show different shapes to those observed at lower excitation. For E$\scriptstyle \gamma$=150-200 MeV the probability of L=0 knockout is significantly reduced and the shape of the cross section is dominated by the knockout of L=1 pairs. This indicates a large cross section for the pair to initially be in a 3P state, the knockout of which is predicted to leave the residual nucleus in the 1+ and 2+ states emphasized in this region[4,5]. At higher E$\scriptstyle \gamma$ a significant 3P component is still visible in the data although the strength at low and high Pr indicates that the relative contribution of 1S0 knockout has increased.

The results indicate that (1p)2 pp knockout cannot be described simply by absorption on 1S0 pairs. The magnitude of the asymmetry from recent 16O ($ \vec{\gamma}\,$, pp)[12] and 12C ($ \vec{\gamma}\,$, pp)[13,14] measurements also support this conclusion. The results presented in this contribution do however indicate that 1S0 knockout does give some contribution to the cross section and it may be possible to emphasize this process using cuts on the momentum and excitation energy of the residual nucleus. 1S0 knockout at low residual excitation has also been inferred from recent 16O(e, e'pp) data[15] where the cross section for 1S0 knockout to the ground and low lying states was predicted to be dominantly due to mechanisms involving SRC. Real photon experiments may be expected to show less sensitivity to SRC due to the lack of a longitudinal component and recent angular distribution measurements for 12C($ \gamma$,pp)[13,16] have indicated that at E$\scriptstyle \gamma$=250-300 MeV the reaction shows features consistent with the contribution of higher photon multipoles. The sensitivity of this ( $ \gamma$, pp) data to SRC is presently being investigated by comparison of the data with a detailed unfactorised model[5].

6. Summary

The study of the 2N knockout process has been extended in both phase space coverage and photon energy range. Further investigations of SRC contributions in the present data are underway. Future experiments to improve the study of the 2N knockout process at high E$\scriptstyle \gamma$ and high recoil momentum[13,14] and with sufficient excitation energy resolution to resolve residual states[17] have been awarded beam time at the Mainz PAC.

Acknowledgments

This work has been supported by grants from the UK EPSERC, the British Council, DFG(Mu750/3), BMFT(06T/u 656), DAAD (313-ARC-XII-98/32) and the EC(SCI.0910.C(JR))

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