Figure 2: Coordinate system for with orientation of polarization axis shown.
Following Donnelly and Raskin[3, 4] we can express the inclusive e-N cross section as a sum of an unpolarized part ( ), that corresponds to the elastic cross section , and a polarized part ( ), that is different from zero only when the beam is longitudinally polarized (helicity h):
The asymmetry is then
As stated above, is just the elastic unpolarized free e-N cross section, and specifically for neutrons it reads
where is the electron's initial (final) energy, , is the neutron mass, is the square of the four- momentum transfer and are the neutron Coulomb and magnetic form factors. The polarized part contains two terms, associated with the possible directions of the target polarization. The full expression is given below, with the kinematic factors and the nucleon form factors both evaluated in the laboratory frame (the elastic recoil factor reduces to in the extreme relativistic limit):
where and are the laboratory angles of the target polarization vector with along the direction and normal to the electron scattering plane. It is clear that to extract the target has to be polarized longitudinally (i.e. ) and perpendicular to ( ). For this special condition, the asymmetry simplifies to
This result was also obtained by Arnold et al. who considered the measurement of the polarization of the recoil neutron, instead of using a polarized target.
The foregoing analysis is valid for free nucleons, and it has been reinterpreted in the case of neutrons in polarized nuclei. For the specific case of polarized deuterium nuclei, the exclusive process involving the detection of the neutron after the electrodisintegration can be similarly described in an expression where the interference between and is contained in the polarized part.
The neutron asymmetry is related to the deuteron asymmetry , as where is a correction factor (0.92) for the D-state of the deuteron.
There are different ways to exploit polarization observables for a determination of . One can either use a polarized beam and target as discussed above, or one can use a polarized beam and measure the polarization of the recoiling nucleon. In practice, the measurement using a polarized beam and target involves determining the experimental asymmetry
which depends on the normalized numbers of counts for two opposite helicities, and . The same expression occurs in the recoil polarimetry method, with the obvious reinterpretation of as the analyzing power of the polarimeter, ; is then the polarization of the recoiling nucleon, and are the numbers of counts in the up(down) segments of the polarimeter.
Our studies of these alternatives have led us to choose the polarized target technique. We have found that it allows us to measure over a larger range of than the alternative, and it avoids the difficult problem of a new calibration of the recoil polarimeter for every neutron energy (for every ). In addition, the same setup (target and detectors) can be used to check the experimental technique and the reaction mechanism, assumed to be quasi-free knockout , by measuring which is known over the range we wish to study.
There are two different polarized targets which provide in effect polarized neutrons, polarized deuteron and polarized He. We have chosen polarized deuteron, as the theoretical description of the (e, e'n) process is on a much firmer footing. For the 2N-system the final state interaction can be treated exactly, while this is questionable for A = 3. The role of the D-state in the ground state wave function and the contributions of MEC, are under better control. Accurate calculations are already available, while for A = 3 we are still speculating on the size of the effects. At the same time, a deuteron target allows the experimental check on procedures and reaction mechanism through the comparison of the d(e,e'n) and d(e,e'p) reactions. Arenhövel et al. have shown that, for the case of the deuteron, the uncertainties introduced by the deuteron structure are very small if one concentrates on the strength corresponding to quasielastic e-n scattering with neutrons of small initial momentum. For such kinematic conditions and for the special case of the two-nucleon system, FSI can be accurately computed, and does not contribute significantly to the systematic errors. The effects of MEC, which for A = 2, also can be calculated with reasonable confidence, are small as well. Effects of both FSI and MEC are much smaller than the statistical and systematical errors of the experiment we propose.
To determine the region of where the proposed technique may be most effective, the evaluation of a figure of merit (FOM) has become customary. In the present case, the figure of merit is related to the time required to accumulate the number of counts needed to determine the asymmetry to a given precision. This number is proportional to the product of the square of the asymmetry times the cross section (averaged over the acceptances of our detectors), so the FOM is defined as
Obviously, this quantity depends on the choice of a model for .
Several models have been tried to describe the existing data, which extend from the photon point to . Among those deserving special attention are the so-called ``dipole'' model which uses the form , with , in fact setting the Dirac form factor to zero, in the full expression for the Sachs form factor ; the phenomenological parameterization of Galster et al., ; and the models that seek a connection between the value of of the form factors at low momentum transfer and the asymptotic values of the Dirac and Pauli form factors and predicted by perturbative QCD, in particular the one proposed by Gari and Krümpelmann.
In Figure 3 we present the dependence of in those three instances. It can be seen that the dipole model is higher than the two others, and in fact it is an upper bound to the experimental data. On the other hand, the Galster parameterization (with the Feshbach-Lomon potential) gives a good fit for p = 5.6. We used these two models, which cover a broad range of possible values for the dependence of , to compute the FOM's. These studies show that the
Figure 3: dependence of for three different models.
scattered electron angle has to be as forward as possible, even though the change of FOM with angle is not large.
For increasing momentum transfer, the FOM drops by a factor of (depending on the model) from its maximum value to the largest momentum transfer considered here, . This places a practical limit on the upper value of the attainable momentum transfer, independent of other technical complications that arise from the high kinetic energy of the recoil neutrons, and the opening of inelastic and channels. Therefore, in the present experiment, we will attempt to extract at four values of , starting at about 0.5 (GeV/c) , up to 2 (GeV/c) .
To obtain these values of the four-momentum transfer, a combination of beam energies and scattering angles are chosen such as to maximize the FOM, within the laboratory capabilities and facilities. The kinematical settings we have chosen are displayed if table 1.
The theoretical studies performed indicate clearly that will provide a clean determination of with small systematic errors. This is an important criterion given the fact that past attempts to measure were all limited by systematic errors in both experiment and, even more so, in the theoretical input necessary to infer .