%% File for SLAC-PUB-5934, published in Phys. Rev. C (April, 1993) \input phyzzx \PHYSREV \overfullrule=0pt \tolerance=2000 %% hyphenation (5000 for narrow column) \linepenalty=100 %% to shorten paragraphs as much as possible \hbadness=10000 %% to avoid reporting underfull boxes!! \clubpenalty=10000 \widowpenalty=10000 \displaywidowpenalty=5000 \let\txt=\textstyle \hsize = 5.8 true in \vsize = 8.5 true in \voffset=.4in \hoffset=.4in \doublespace \parskip=.2\baselineskip plus .001\baselineskip minus .1\baselineskip \lineskip=0pt plus 2pt \parindent=2em \def\str{\penalty-10000\hfilneg\ } %% line break with right adjust \def\nostr{\hfill\penalty-10000\ } %% line break with ragged right \def\up#1{\hbox{$^{#1}$}} %% usage: word.\up{2} \def\down#1{\hbox{$_{#1}$}} \let\dn=\down %% usage: CO\dn{4} \def\pz{{\phantom{0}}} \def\usy#1{\null\undertext{#1\vphantom{$_y$}}} \def\coeff#1#2{{ {\textstyle #1\vphantom{0_y}} \over{\textstyle #2\vphantom{0^s}}}} \def\conf#1#2{{\centerline {\it #1} \baselineskip 12pt \centerline {\it #2}}} \font\twelvebsl=cmbxsl10 scaled\magstep 1 \let\bsl=\twelvebsl \def\mbf#1{\hbox{${\twelvebsl #1}$}} \font\fourteenbsl=cmbxsl10 scaled\magstep 2 \let\bslf=\fourteenbsl \def\mbff#1{\hbox{${\fourteenbsl #1}$}} \def\normalrefmark#1{\relax ~[{#1}]} \def\refmark#1{\relax ~[{#1}]} \def\doeackel{\font\elevenrm=cmr9 scaled\magstep1 \elevenrm \baselineskip 15pt plus 1pt minus 1pt \leftline {\us {\hskip 1.5in}}\hangindent 13.5pt \hangafter 1 \noindent $~^\star$Work supported in part by Department of Energy contracts DOE--AC03--76SF00515 (SLAC), W--7405--ENG--48 (LLNL), DE--AC02-76ER--02853 (U. Mass.); National Science Foundation Grant PHY85--10549 (A.U.); the U.S.--Israel Binational Science Foundation (Tel--Aviv); and the Monbusho International Research Program.} \def\etal{{\it et al.}} \def\etals{{\it et al. }} \def\k{^} \def\t{~} \def\simo{{1}} \def\marc{{2}} \def\hock{{3}} \def\aren{{4}} \def\truh{{5}} \def\gros{{6}} \def\pari{{7}} \def\bonn{{8}} \def\lung{{9}} \def\kiss{{10}} \def\gloz{{11}} \def\yama{{12}} \def\blom{{13}} \def\quas{{14}} \def\pebe{{15}} \def\spec{{16}} \def\make{{17}} \def\elas{{18}} \def\koon{{19}} \def\katt{{20}} \def\dday{{21}} \def\land{{22}} \def\tsai{{23}} \def\gari{{24}} \def\rory{{25}} \def\arrn{{26}} \def\schm{{27}} \def\shut{{28}} \def\tito{{29}} \def\lage{{30}} \PHYSREV %FRONT PAGE IN SLAC-PUB FORMAT \vbox to 8.5in{\null\vskip -.4in \Frontpage {\baselineskip 11pt \rightline {\tenrm SLAC--PUB--5934} \rightline {\tenrm October 1992} \rightline {\tenrm (T/E)} } \singlespace\baselineskip 15.5pt plus 1pt minus 1pt \medskip \centerline {\bf MEASUREMENTS OF TRANSVERSE ELECTRON SCATTERING} \centerline{\bf FROM THE DEUTERON IN THE THRESHOLD REGION} \centerline{\bf AT HIGH MOMENTUM TRANSFERS$^\star$} \bigskip \centerline {\caps M. Frodyma,$\k{(2,a)}$ R. G. Arnold,$\k{(1)}$ D. Benton,$\k{(1,b)}$ P. E. Bosted,$\k{(1)}$ L. } \centerline {\caps Clogher,$\k{(1)}$ G. Dechambrier,$\k{(1)}$ A. T. Katramatou,$\k{(1)}$ J. Lambert,$\k{(1,c)}$ } \centerline {\caps A. Lung,$\k{(1)}$, G. G. Petratos,$\k{(1,d)}$ A. Rahbar,$\k{(1)}$ S. E. Rock,$\k{(1)}$ } \centerline {\caps Z. M. Szalata,$\k{(1)}$ B. Debebe,$\k{(2)}$ R. S. Hicks,$\k{(2)}$ A. Hotta,$\k{(2,e)}$ G. A. Peterson,$\k{(2)}$} \centerline {\caps R. A. Gearhart,$\k{(3)}$ J. Alster,$\k{(4)}$ J. Lichtenstadt,$\k{(4)}$} \centerline {\caps F. Dietrich,$\k{(5)}$ and K. van Bibber$\k{(5)}$} \vskip 3pt\centerline {\it $\k{(1)}$ The American University, Washington, D.C. 20016} \centerline{\it $\k{(2)}$ University of Massachusetts, Amherst, Massachusetts 01003 } \centerline {\it $\k{(3)}$ Stanford Linear Accelerator Center, Stanford, California 94309 } \centerline {\it $\k{(4)}$ Tel Aviv University, Tel Aviv, Israel} \centerline {\it $\k{(5)}$ Lawrence Livermore National Laboratory, Livermore, CA 94550 } \bigskip \centerline{\it Submitted to Physical Review C} \bigskip \vfill \doeackel \vskip .01in} \eject %% BEGIN JOURNAL FIRST PAGE \pagenumber=1 \vbox{\singlespace \vskip .2in \centerline {\bf MEASUREMENTS OF TRANSVERSE ELECTRON SCATTERING} \centerline{\bf FROM THE DEUTERON IN THE THRESHOLD REGION} \centerline{\bf AT HIGH MOMENTUM TRANSFERS} \medskip \centerline {\caps M. Frodyma,$\k{(2,a)}$ R. G. Arnold,$\k{(1)}$ D. Benton,$\k{(1,b)}$ P. E. Bosted,$\k{(1)}$ L. } \centerline {\caps Clogher,$\k{(1)}$ G. Dechambrier,$\k{(1)}$ A. T. Katramatou,$\k{(1)}$ J. Lambert,$\k{(1,c)}$ } \centerline {\caps A. Lung,$\k{(1)}$, G. G. Petratos,$\k{(1,d)}$ A. Rahbar,$\k{(1)}$ S. E. Rock,$\k{(1)}$ } \centerline {\caps Z. M. Szalata,$\k{(1)}$ B. Debebe,$\k{(2)}$ R. S. Hicks,$\k{(2)}$ A. Hotta,$\k{(2,e)}$ G. A. Peterson,$\k{(2)}$} \centerline {\caps R. A. Gearhart,$\k{(3)}$ J. Alster,$\k{(4)}$ J. Lichtenstadt,$\k{(4)}$} \centerline {\caps F. Dietrich,$\k{(5)}$ and K. van Bibber$\k{(5)}$} \vskip 3pt\centerline {\it $\k{(1)}$ The American University, Washington, D.C. 20016} \centerline{\it $\k{(2)}$ University of Massachusetts, Amherst, Massachusetts 01003 } \centerline {\it $\k{(3)}$ Stanford Linear Accelerator Center, Stanford, California 94309 } \centerline {\it $\k{(4)}$ Tel Aviv University, Tel Aviv, Israel} \centerline {\it $\k{(5)}$ Lawrence Livermore National Laboratory, Livermore, CA 94550 }} \vskip .6in \endpage \par \centerline{ABSTRACT} Deuteron electrodisintegration cross sections near $180\k{\circ}$ have been measured near break-up threshold for the four-momentum transfer squared $Q\k2$ range 1.21 to 2.76 (GeV/c)$\k2$. Evidence for a change of slope in the cross section near $Q\k2 = 1$ (GeV/c)$\k2$ has been obtained for the first time. The data are compared to non-relativistic calculations, which predict a strong influence of meson exchange currents. The data are also compared to a hybrid quark-hadron model. None of these calculations agrees with the data over the entire measured range of $Q\k2$. \par The ratio of inelastic structure functions $W_{1}(Q\k2,E_{np})/W_{2}(Q\k2,E_{np})$ is extracted from the present results and previous forward angle data. No prediction is in good agreement with the deduced ratios at small relative energy $E_{np}$. \endpage \twelvepoint \centerline {\bf I. INTRODUCTION} The electrodisintegration of the deuteron near breakup threshold provides one of the most compelling tests of our understanding of the role of meson exchange currents in nuclei. Close to threshold, the dominant mechanism for electrodisintegration is by a spin-flip magnetic dipole transition from the $^3S_1+^3D_1$ ground state to an unbound $^1S_0$ state, a transition that can be most selectively studied by electron scattering at extreme backward angles. This paper presents the results of measurements of the threshold electrodisintegration cross section at 180$^\circ$, in the region where the relative kinetic energy $E_{np}$ of the outgoing nucleons in the center-of-mass system is less than 20 MeV. Previous measurements [\simo] of this cross section extended to a squared four-momentum transfer $Q^2=1.1$ (GeV/c)$^2$. Our data span the range from $Q^2=1.21$ to 2.76 (GeV/c)$^2$, a region where the meson-exchange representation of the nucleon-nucleon force is expected to have diminishing applicability. The results presented here have been previous published [\marc]. This paper describes the experiment in more complete detail, particularly with regard to the procedures employed for extracting the average threshold cross sections. Additional information is provided on a comparison of $W_1 (Q^2,E_{np})$, measured in the present experiment, with values of $W_2(Q^2,E_{np})$ from other experiments. \par The one-photon exchange impulse approximation (IA) diagram is shown in Fig.\t 1 with and without final state interactions (FSI) between the two nucleons. Calculations in the IA predict a diffraction minimum at four momentum transfer squared $Q\k2 \approx 0.5$ (GeV/c)$\k2$, in strong disagreement with existing electrodisintegration data [\simo]. \par Significant improvement is found when meson-exchange currents (MEC) are included. Three important MEC interactions involving pions are shown in Fig.\t 2. Non-relativistic predictions including only single pion MEC account [\hock] for the discrepancy at $Q\k2\approx 0.5$, but are inadequate at higher $Q\k2$, where short-range effects exert a large influence. Above $Q\k2\approx 1$ (GeV/c)$\k{2}$, non-relativistic predictions have a large model dependence, yielding order-of-magnitude variations in the calculated cross sections. The electromagnetic form factors used in the meson-nucleon coupling of the MEC contribute strongly to this model dependence. Whether calculations should use the Sachs $ G_E(Q\k2) $ or the Dirac $ F_1(Q\k2) $ form factor has been an issue of some debate [\aren,\truh,\gros]. Because previous data [\simo] were better described by models using $F_1$, theoretical arguments were advanced [\truh] in favor of $F_1$. Subsequently, it was shown [\aren,\gros] that these arguments depend on strong, unproven assumptions and in some cases have inconsistencies. \par Other sources of uncertainty are the nucleon-nucleon ($nn$) potential [\pari,\bonn], the $\pi NN$ vertex form factors, and the nucleon electromagnetic form factors. More accurate measurements [\lung] of the neutron electric form factor $G_{En}(Q\k2)$ have recently become available, substantially reducing this last source of uncertainty. \par The strong model dependence at high $Q\k2$ has led to an unsatisfactory situation. There appear to be several plausible combinations of theoretical inputs [\aren], but none of these is in good agreement with all electrodisintegration data for $Q\k{2}\le 2.76$ (GeV/c)$\k{2}$. Such observations underscore the need for a completely relativistic theory in which the number of {\it ad hoc} choices is minimized. \par Another class of predictions for deuteron electrodisintegration are exploratory investigations [\kiss,\gloz,\yama] known as hybrid quark-hadron models. In these models the deuteron is treated as a six-quark cluster when the $NN$ separation is less than a cut-off radius. Unfortunately, the models are quite sensitive to the value of the radius, which is not strongly constrained. These models also yield order-of-magnitude variations in the predicted cross sections at high $Q\k2$. \par This paper is organized as follows. Relevant kinematic and cross section formulae are given in Section II. Since the experimental apparatus has been discussed elsewhere, only a brief overview will be given in Section III. The main steps of the data analysis are discussed in Section IV. A comparison of the electrodisintegration data with several non-relativistic predictions is given in Section V, and concluding remarks are given in Section VI. \bigskip \centerline{\bf II. KINEMATICS and CROSS SECTIONS} \par In the formulas of this section the electron rest mass is neglected. The four-momentum transfer squared $Q\k2$ is given by $$Q\k2 = 4 E E\k{\prime}\sin\k2 (\theta / 2)\ ,\eqn\ac$$ where $E$ and $E\k{\prime}$ are the incident and scattered electron energies, and $\theta$ is the electron scattering angle in the laboratory system. The invariant mass squared $W\k2$ of the two-nucleon recoil system in Fig.\t 1 can be written as $$W\k2 = M_{D}\k2 + 2M_{D}\nu - Q\k2\ ,\eqn\ad$$ where $M_{D}$ is the deuteron mass, and $\nu = E - E\k{\prime}$. \par For elastic scattering, $W\k2 = M_{D}\k2$ and $Q\k2 = 2 M_{D} \nu$. The scaling variable $x_D$ is given by $$x_D = {Q\k2\over 2 M_{D} \nu}\ ,\eqn\ae$$ which is near unity for threshold-inelastic data. A related scaling variable [\blom] $\omega\k{\prime}$ can be written as $$\omega\k{\prime} = 1 + {W_{N}\k2\over Q\k2}\ ,\eqn\af$$ where $W_{N}\k2$ is obtained by substituting the nucleon mass $M_N$ in Eq.\t\ad\ for the deuteron mass. Both $x_D$ and $\omega\k{\prime}$ are used in the data analysis discussed below. \par In the threshold inelastic region, the excitation energy $\omega$ is small compared to the deuteron mass and is given by $\omega = W - M_{D}$. The scattered electron energy is given to first order in ${\omega / M_{D}}$ by $$E\k{\prime} = {E-\omega\over R_{E}}, \eqn\epr$$ where $$R_{E} = 1 + {2 E \sin\k2(\theta / 2)\over M_{D}} \eqn\rec$$ is the recoil factor. \par The electron spectrometer central momentum was set at the deuteron elastic peak for the threshold inelastic data taking. It is useful to express $E\k{\prime}$ in terms of the momentum shift $\delta$ relative to the deuteron elastic peak as $$E\k{\prime} = {E\over R_{E}}\left(1 + \delta \right)\ . \eqn\ede$$ \par The kinetic energy $E_{np}$ of an outgoing nucleon in the neutron-proton rest frame is given to first order in ${\omega\over M_{D}}$ by $$E_{np} = \omega - \omega_o\ , \eqn\aa\ $$ or in terms of $E\k{\prime}$ as $$E_{np} = E - R_{E} E\k{\prime} - \omega_o\ , \eqn\ab$$ where $\omega_o = 2.23$ MeV is the deuteron binding energy. \par The inelastic cross section is written as $${d\sigma\over d\Omega dE\k{\prime}} = {\alpha\k2\over 4 E\k2}\sin\k4 {\left(\theta\over 2\right)} \left[ W_2(\nu,Q\k2)\cos\k2 {\left(\theta\over 2\right)} + 2W_1(\nu,Q\k2)\sin\k2 {\left(\theta\over 2\right)}\right]\ ,\eqn\w$$ where $W_1(\nu,Q\k2)$ and $W_2(\nu,Q\k2)$ are the inelastic structure functions. The inelastic data [\marc,\quas] from the present experiment provide new measurements of $W_1(\nu,Q\k2)$, since all data were taken near 180$\k{\circ}$. Note that $W_1$ and $W_2$ may equivalently be written as functions of any pair of variables such as $E_{np}$ and $x_D$, which depend only on $Q\k2$ and $\nu$. \bigskip \centerline{\bf III. OVERVIEW of the EXPERIMENT} \par The experimental apparatus has been discussed in detail elsewhere [\pebe,\spec], so only a brief overview will be given here. The new threshold inelastic data were obtained during a $180^\circ$ electron scattering experiment in which measurements were also made of quasielastic scattering [\quas,\make], as well as elastic electron-deuteron [\pebe,\elas] and electron-proton [\pebe,\make] scattering. The threshold inelastic data, which only used the $180^\circ$ spectrometer, were taken simultaneously with the elastic $ed$ measurements, in which deuterons recoiling near 0$\k{\circ}$ were detected in coincidence with scattered electrons using a separate spectrometer. \par Experimental conditions such as the spectrometer design could not be simultaneously optimized for the elastic, quasielastic, and threshold inelastic data taking. The elastic data were given priority in order to measure the magnetic form factor of the deuteron. Since elastic events were tagged by detecting recoil deuterons, high energy resolution for the electron spectrometer was not required. Inelastic events could not be tagged by detecting recoil protons in coincidence with scattered electrons since there was a large background of protons from other processes. Also, most of the protons fell outside the recoil spectrometer acceptance. Due to the small elastic cross section, long liquid deuterium targets and spectrometers having a large angular acceptance were needed. These properties compromised the resolution in $E^{\,\prime}$ to the extent that the corresponding resolution in $E_{np}$ was as large as 20 MeV (see Eq.\ab) for the 20 cm liquid targets. Because of this, the data were analyzed using a resolution unfolding procedure in order to make comparisons with theoretical predictions, which are generally constrained to a small $E_{np}$ range near threshold. \par The experiment, identified as NE4, was carried out at the Stanford Linear Accelerator Center (SLAC) in two separate running periods. These occurred during the summer of 1985 (NE4-I) and spring of 1986 (NE4-II) respectively. Data were taken with electron beams of energy $E$ = 0.734, 0.843, 0.885, 0.934, 1.020, 1.102, 1.201, and 1.279 GeV, produced by the Nuclear Physics Injector [\koon] with a maximum intensity of $5 \times 10\k{11}$ electrons per 1.6 $\mu$sec pulse at a repetition rate of 150 Hz. These beam energies correspond to $Q^2$ values at threshold of 1.21, 1.49, 1.61, 1.74, 1.99, 2.23, 2.53, and 2.76 (GeV)/c$^2$ respectively. Energy-defining slits limited the uncertainty in $E$ to $\pm0.35\%$. \par The electron beams were transported to a 180$\k{\circ}$ spectrometer system [\spec] in End Station A. The entire spectrometer system is shown in Fig.\t3. A series of three bending magnets $B_1 - B_3$ transported incident electrons toward the target. Dipole $B_2$ was symmetrically located between $B_1$ and $B_3$ and was remotely movable along a line perpendicular to the electron beam. This construction accommodated the different bending angles required for each beam energy. The incident beam then passed through the quadrupole triplet $Q_1 - Q_3$ into 10 or 20 cm long liquid deuterium cells. \par The liquid deuterium and hydrogen target cells were machined out of an aluminum casting, and each 20 cm long cell included two aluminum endcaps of thickness $3.44\times 10\k{-2}$ g/cm$\k{2}$ through which the incident beam passed. Electrons scattered from the target endcaps represented the largest expected source of background, hence the endcaps were made as thin as possible while safely supporting two atmospheres of pressure from the liquid deuterium within. Two aluminum hymens, $6.86\times 10\k{-3}$ g/cm$\k{2}$ thick, isolated the target vacuum chamber and a wire array of average thickness $1.4\times 10\k{-2}$ g/cm$\k{2}$ was used to measure the beam position. The deuteron spectrum at $Q\k2 = 1.21$ (GeV/c)$\k2$ used a 10 cm target cell with $1.92\times 10\k{-2}$ g/cm$\k{-2}$ thick endcaps, while all other threshold data were taken with the 20 cm cell. \par Electrons scattered near 180$\k{\circ}$ returned through $Q_1 - Q_3$ and were momentum-dispersed by spectrometer dipoles $B_3$ and $B_4$. Quadrupoles $Q_1 - Q_3$ provided the focussing strength needed to obtain a large solid angle for the electron spectrometer without unduly disturbing the incident beam. This solid angle $\Delta\Omega$, averaged over $\pm0.5\%$ in relative momentum $\delta$, was $22.4$ msr for the 10 cm target, and $21.5$ msr for the 20 cm target. Corrections for the non-uniformity in the electron spectrometer acceptance [\pebe] were generally small since threshold inelastic data were analysed only in the range $-3.5\%\leq \delta \leq +3.5\%$, where the acceptance was fairly constant. \par Electrons transmitted through the target passed through the quadrupole triplet $Q_4 - Q_6$ and were deflected by $B_5$ into a remotely movable, water-cooled beam dump. The focussing strengths of $Q_4 - Q_6$ were chosen to maximize transmission of deuterons into the recoil spectrometer for the elastic data while maintaining an acceptable beam spot size on the dump. The positively charged nuclei recoiling near 0$\k{\circ}$ were deflected by $B_5$ toward the recoil spectrometer, which was used only in the elastic measurements. The dipole magnets $B_6 - B_8$ of this spectrometer separated recoil deuterons from a large background of lower momentum particles generated in the target. \par For track reconstruction, the electron spectrometer contained six multi-wire proportional chambers (MWPC) spaced 20 cm apart. Two planes of plastic scintillation counters were used for triggering and fast timing. A large background of pions was rejected by a threshold gas \v Cerenkov counter and by measuring the energy deposited in a 40-segment array of lead-glass blocks. \par The various voltage pulses from the detectors were carried by fast Heliax cables to CAMAC electronic modules in the counting house above End Station A. The quantities to be recorded for each scattering event were read from the CAMAC modules by a PDP-11 microcomputer and transferred to a VAX 11/780 computer for logging onto magnetic tape. The same VAX 11/780 computer was used both for analysing data on-line and for most of the subsequent off-line analysis. \bigskip \centerline {\bf IV. DATA ANALYSIS} \par The measured differential cross section per nucleon is given by $${d\k2\sigma(E,E\k{\prime})\over d\Omega dE\k{\prime}} = \left[{1\over S_{f}D\epsilon(\Delta\Omega(\delta))}\right] \left[{R_C(E,E\k{\prime} \over N_e\rho LN_A}\right] {N(E,E\k{\prime})\over \Delta E}\ ,\eqn\bb$$ where $N(E,E\k{\prime})$ is the number of counts in an energy bin of width $\Delta E$ centered on $E\k{\prime}$, corrected for the expected number of counts from $ed$ elastic scattering and for inelastic scattering from the hymens, wire array, and target endcaps. These corrections, as well as the radiative corrections factors $R_C(E,E^\prime)$, are discussed in more detail below. The factor $S_{f}$ ranged from 0.9 to unity, and is a correction for multiple events within a beam pulse, since only the first event in each pulse was analysed. The electronic dead time correction factor $D$ was always within $1\%$ of unity while $\epsilon$, the product of the detector efficiencies, ranged from 94 to 96\%. The factor $N_A$ is Avogadro's number, L is the target length, $\rho$ is the target density, and $N_e$ is the number of incident electrons. \par A correction of $<4\%$ was made for pions misidentified as electrons. Electrons were identified by the large pulse heights they produced in both the \v Cerenkov counter and the shower counter. Misidentification of pions as electrons could only occur when pions produced a large hadronic shower (for example, by charge exchange to $\pi\k{\circ}$), and at the same time either a random hit or a pion-produced knock-on electron ($\approx 1\%$ probability) generated a large pulse height in the \v Cerenkov counter. No correction for electrons from the processes such as: $\gamma d\rightarrow \pi\k{\circ} d$, $\pi\k{\circ}\rightarrow \gamma\gamma$, $\gamma\rightarrow e\k{+}e\k{-}$ were made since estimates for this correction showed it to be $<3\%$. \par As is customary for threshold electron scattering, the cross sections per deuteron were expressed as a differential in $E_{np}$, using $${d^2\sigma \over d\Omega dE_{np}}={1\over 2} {d^2\sigma \over d\Omega dE^\prime} {dE^\prime \over dE_{np}} ,\eqn\cdwd$$ where the factor of two is to convert from cross sections per nucleon to cross section per deuteron. \endpage \centerline{\bf A. Subtraction of Events Originating Outside the Target} \par The measured spectra include a background of electrons scattered from the hymens, wire array, and target endcaps. It was necessary to evaluate this background carefully since its contribution grows to $100\%$ at large negative $E_{np}$, where scattering from the deuteron is kinematically forbidden. Also, the resolution unfolded results discussed below were sensitive to the presence of any residual signal in the electron spectra at large negative $E_{np}$. \par The total background counting rates were measured in separate data runs using empty targets which were replicas of the full ones, except with endcaps thicker by a factor of 8.55 for the 20 cm and 8.20 for the 10 cm cells. The thicker endcaps on the empty target cells provided both a faster counting rate and approximately the same total radiation length as the full targets. This last condition made for similar radiative correction factors for the full and empty target endcaps. \par Evaluation of the background contribution was complicated by the fact that the spectrometer solid angle for the aluminum hymens, wire array, and the two endcaps of the target were all substantially different. Also, if the scattering at $180\k{\circ}$ occurred in the downstream endcap or hymen, both the incident and scattered electrons must traverse the target. Thus, electrons interacting downstream of the target undergo energy losses for the full targets which are not present for the empty cells. These complications are discussed below. \par The spectrometer solid angle $\Delta\Omega$ depends on the location $z$ of the scattering vertex in addition to the relative momentum $\delta$. A Monte-Carlo program [\katt] was used to generate distributions of events in $\delta$ with the scattering vertex held at fixed $z$ positions. An example of such a distribution is shown in Fig.\t 4a, where the scattering vertex was held at the location of the upstream endcap. The distributions for other values of $z$ are similar in shape, but vary considerably in overall magnitude, with the downstream hymen having the smallest solid angle. Each distribution was fit with a sixth order polynomial curve, and the ratios of the fits were used to evaluate the relative contribution of each background source. The ratio of distributions for the downstream to upstream endcaps is shown in Fig.\t 4b. \par A further complication is the difference between the cross sections per nucleon for the copper wire array and aluminum target endcaps and hymens, due to the larger Fermi momentum for copper compared to aluminum. The ratio of these cross sections was obtained from a $y$-scaling analysis of existing data (see Fig.\t 4 of Ref. [\dday]) and yielded a correction factor of 1.1 for the wire array contribution. \par The experimentally determined quantities were $C_f$ and $C_e$, the total counts per unit incident electron for full and empty targets, given by $$ C_{i}(E,E\k{\prime}) = {1\over S_{f}\epsilon N_{e} D } N_{R}(E,E\k{\prime})\ ,\eqn\cf$$ where $N_{R}(E,E\k{\prime})$ is the raw number of counts corrected for spectrometer acceptance only. For example, $C_f$ is given by $$C_f=C_{h}+C_w+rC_E+C_D+rC_E\k{\prime}+C_{h}\k{\prime}\ ,\eqn\ek$$ where $C_D$ is the desired contribution from liquid deuterium alone, $C_h$, $C_w$, and $C_E$ are the contributions from the hymens, wire array, and target endcaps, and $r$ is the ratio of full/empty target endcap thicknesses. The primes on $C_{E}\k{\prime}$ and $C_{h}\k{\prime}$ indicate that these quantities have been corrected for ionization losses in the full targets. To correct for these ionization losses, $C_E\k{\prime}$ and $C_{h}\k{\prime}$ were evaluated at $(E-\Delta E,E\k{\prime}+\Delta E)$ instead of $(E, E\k{\prime})$, where $\Delta E$ is the most probable energy loss [\land], approximately 5.8 MeV for 20 cm of liquid deuterium. Corresponding losses within the endcaps, hymens, and wire array were found to be negligible. \par The total measured empty target contribution $C_e$ is given by a similar expression. The ionization losses were neglected in this case as they were not significant. Since these data had poor statistics compared to the full target data, a smooth fit to the empty target data was used. \par It was found best to fit the data using the quantity $E\k2 C_e(E,E\k{\prime})$, which is proportional to the inelastic structure function $W_1(Q\k2,\nu)$. Fig.\t 5a shows this quantity for all incident energies $E$ as a function of the scaling variable $\omega\k{\prime}$. The data define a relatively smooth curve except for the spectrum at the highest $\omega\k{\prime}$, corresponding to $Q\k{2} = 1.21$ (GeV/c)$\k2$. A three parameter fit to the empty target data was obtained using the form $$\ln(E\k2C_e(E,E\k{\prime}))=a_1+a_2E+a_3E\omega\k{\prime}\ .\eqn\ec$$ This fit yielded a $\chi\k2$ value of 1.06 per degree of freedom. The result, shown in Fig.\t 5b and c, was used in the endcap subtraction for all of the threshold inelastic data. The resulting errors in $C_e(E,E\k{\prime})$ ranged typically from $5\%$ to $30\%$. Using the ratios of solid angles and thicknesses of each background source and the fits to the empty target data, the desired contribution from deuterium could be extracted. \par In order to determine the sensitivity to the choice of fit to the empty target data, several fits with up to nine free parameters were obtained. The variation in the final cross sections due to the choice of fit is discussed in Sec. IV C, and was only significant for the $Q\k2 = 1.21$ GeV/c$\k2$ data. \par The counts per unit charge before and after background subtraction are shown in Fig.\t 6 for the lowest and highest values of $Q\k2$: 1.21 and 2.76 (GeV/c)$\k2$. This correction is relatively small for momenta $\delta\leq -2\%$, where the deuterium cross section is large. However, the size of the correction is essentially 100$\%$ for $\delta\geq 1\%$, as expected. After subtracting the non-deuterium contributions, all spectra were consistent with zero for large negative $E_{np}$. \medskip \centerline{\bf B. Radiative Corrections} \par Radiative corrections were performed to correct for bremsstrahlung and straggling of the incident and scattered electrons in the target medium. Bremsstrahlung occurs both as external radiation in the fields of nuclei distinct from the scattering nucleus, and as internal radiation at the scattering vertex. The radiative corrections were carried out using the equivalent radiator procedure of Mo and Tsai [\tsai]. In this approach, the internal bremsstrahlung is modelled by two external radiators, placed before and after the scattering vertex. Since both $E$ and $E\k{\prime}$ depend on the radiated photon energy, the procedure involves integrations over a model for the unradiated cross section $\sigma (E,E\k{\prime})$. The ``radiated'' cross sections $\sigma_{R}(E,E\k{\prime})$ are obtained by convoluting $\sigma (E,E\k{\prime})$ with a normalized bremsstrahlung function. In order to perform the required integrations, it was necessary to interpolate the models of $\sigma (E,E\k{\prime})$ in both $E$ and $E\k{\prime}$. For a given incident energy $E$, the theoretical models [\aren] used for $\sigma (E,E\k{\prime})$ were calculated at discrete values of $E\k{\prime}$. Cross sections at intermediate values of $E\k{\prime}$ were obtained by linear interpolation. For the interpolation in incident energy $E$ a simple power law fit was used. The $E$-dependence of a typical cross section model is shown for $E_{np}=1$ and 12 MeV in Fig.\t7. Since only the threshold region was investigated, the required range in $E$ and $E^{\,\prime}$ was only a few percent. \par The large range of material in the target before and after scattering caused substantial differences in the radiative correction factors as a function of target length. This was taken into account by calculating the corrections at each of 40 positions equally distributed along the target length. The most probable energy loss corresponding to the thickness of each layer was used to correct $E$ and $E\k{\prime}$. Radiative correction factors $R_{C}(E, E\k{\prime})=\sigma(E,E\k{\prime})/ \sigma_{R}(E,E\k{\prime})$ were calculated separately for each target section with $E_{np} > 0$. The correction factors increased approximately linearly with increasing depth into the target, as expected. \par Shown in Fig.\t 8a,b are the separate contributions to the radiated cross section $\sigma_{R}(E,E\k{\prime})$ from Landau straggling and bremsstrahlung, for ($E,E\k{\prime})$ = (0.734, 0.3958) GeV, as a function of $\Delta$, a convergence parameter [\tsai] for the improper integrations over $E$ and $E\k{\prime}$. In the present case, $\Delta$ is constrained to a few MeV, and the Landau contribution is small relative to the bremsstrahlung effect. \par Unfortunately, as shown in Fig.\t 8c, the calculated radiative correction factors displayed a sizable dependence on $\Delta$. This occurred because the straggling energy loss was comparable to the relative energy $E_{np}$, and the Mo-Tsai approximations break down under these conditions. Because the Landau terms were small, the radiative correction factors $R_{C}(E, E\k{\prime})$ were calculated using the bremsstrahlung terms only. This removed the lower constraint on $\Delta$, which could then be made arbitrarily small, though still nonzero. The final correction factors $R_{C}(E, E\k{\prime})$, using bremsstrahlung only and averaged over target segments, had negligible dependence on $\Delta$ for any value below $\Delta = 1$ eV. \par The radiative correction factors averaged over target segment are shown in Fig.\t 9 for $Q\k2 = 1.21$ and 2.76 (GeV/c)$\k2$. The values of $R_{C}(E, E\k{\prime})$ were calculated separately for each of two widely-divergent input models [\aren]. One model used $F_1(Q\k2)$ coupling for the MEC and went smoothly to zero at the break-up threshold, while the other model had $G_E(Q\k2)$ coupling and a strong enhancement at threshold. Since these two models represent the largest variation in the $E_{np}$ dependence near threshold (other predictions [\aren,\yama] lie in between), the adopted set of radiative correction factors was the average of correction factors obtained from the two input models. Errors were assigned as half the difference between the two sets of correction factors and ranged typically from $\pm3\%$ to $\pm8\%$ of the average correction factor. \medskip \centerline{\bf C. Resolution Unfolding} \par As previously noted, the data have relatively coarse energy resolution due to the intrinsic spectrometer resolution, ionization energy losses, multiple scattering, and the spread in incident beam energy. This total resolution ranged from $\pm 5$ to $\pm 9$ MeV in $E_{np}$. The attempt to unfold resolution effects from the data was motivated by the objective of determining the $Q\k2$ dependence of the electrodisintegration cross section near threshold. Since the true cross section near the deuteron break-up threshold may vary rapidly with $E_{np}$, the resolution unfolding procedure is necessarily model-dependent. \par Resolution effects have been treated using two different methods. In the first method, theoretical models were convoluted with Monte-Carlo determined [\katt] resolution functions and compared with the data. These results will be described below. In the second method, a model-dependent procedure was used to extract resolution unfolded cross sections, {\it i.e.}, cross sections free of resolution smearing effects, given by $$\sigma_{exp}(E, \delta) = \int_{-\infty}\k{\delta_{T}} R(\delta-\delta\k{\prime}) \sigma(E, \delta\k{\prime})d\delta\k{\prime}\left[ \int_{-\infty}\k{+\infty}R(\delta\k{\prime}) d\delta\k{\prime}\right]\k{-1}\ ,\eqn\res$$ where $R(\delta\k{\prime})$ is the Monte-Carlo calculated resolution function, $\sigma_{exp}(E, \delta)$ represents the experimental data, and $\delta_{T}$ is the electron momentum at threshold relative to the deuteron elastic peak. Resolution functions were obtained by Monte-Carlo methods using the known electron spectrometer matrix elements. The spread in beam energy, and energy losses in the and targets and the wire chambers were all taken into account. The true cross section, $\sigma(E, \delta)$, was represented by a polynomial expansion, $$\sigma(E, \delta) = \Sigma_{i=1}\k{N}a_{i}\delta\k{i}\ \ E_{np} > 0 \ ,\eqn\ss$$ $$\sigma(E,\delta) = 0\ \ E_{np} < 0\ ,$$ where $N$ ranged from 2 to 4. These polynomials were inserted into Eq.\ \res , and the coefficients adjusted to give the best fit to the experimental data using a least-squares fitting routine. Such polynomial representations adequately describe available theoretical predictions for the shape of deuteron cross sections near threshold. Choices other than polynomials are feasible, but were not investigated. \par The dominant systematic error arose from an uncertainty [\spec] of $\pm0.25\%$ in the scattered electron energy $E\k{\prime}$. This yielded errors of $\pm10\%$ to $\pm30\%$ in the cross sections and contributed the largest variations in the resolution unfolded results. The size of these variations in the unfolded cross sections was evaluated by shifting the data by $\pm 0.25\%$ in $\delta$ and repeating the least-squares fit in each case. The reduced chi-squares for these fits ranged from 1.0 to 1.9 with an average of 1.3. \par Typical fits to the radiatively corrected data for $Q\k2 = 2.53$ (GeV/c)$\k2$ are shown in Fig.\t 10. The three solid curves correspond to momentum shifts of $\pm0.25\%$ and $0\%$. %the dotted curve is the resolution function derived from %Monte Carlo studies. \par Figs.\t 11, 12, and 13 show, for three values of $Q\k2$, the cross sections $\sigma (E, \delta)$ from 2nd, 3rd, and 4th order polynomial fits to the radiatively corrected data. In each figure panel, the three curves correspond to the three momentum shifts $\delta$ of $\pm 0.25\%$ and $0\%$ for a given choice of polynomial order. Although these cross sections fits are consistent with a non-zero cross section at the break-up threshold, the large dependence on the shifts in $\delta$ makes it impossible to draw any firm conclusions regarding the shape of the true cross section at threshold. \par The resolution unfolded cross sections for all $Q\k2$ were averaged over the relative kinetic energy $E_{np}$ from 0 to a maximum $E_{np}\k{M}$ for comparison with averaged theoretical predictions as well as previous data. The $E_{np}$-averaged results for each value of $Q\k2$ are shown as a function of $E_{np}\k{M}$ in Figs.\t 14 to 16. The curves in each figure panel correspond to the various choices of momentum offset and polynomial order. For a given $E_{np}\k{M}$, the final unfolded result at each $Q\k2$ was chosen as the centroid of the curves. Results from earlier experiments have usually been averaged over $E_{np}$ from 0 to 3 MeV. As seen in Figs.\t 14 to 16, the large systematic spreads in the resolution unfolded results are dramatically reduced by averaging over a larger range of $E_{np}$, 0 to 10 MeV. The 0 to 10 MeV range was chosen since it is comparable to the experimental resolution. The present results ware compared to similarly averaged predictions in Section V. \par The spectrum at $Q\k2=1.21$ (GeV/c)$\k2$ was analysed using both a three and nine-parameter fit to the corresponding empty target data, and the results are shown in Figs. 14a,b. The final cross sections in this case were obtained as the average of the two set of results. \par The systematic errors in the unfolding procedure were estimated from the observed variation among the curves for each $Q\k2$ in Figs.\t 14 to 16. For example, the $E_{np}$-averaged cross sections tend to fall into three groups corresponding to the momentum shifts applied to the data. This variation in the results was the largest systematic uncertainty, ranging from $\pm20\%$ of the centroid for a 0 to 10 MeV range of $E_{np}$ to $\pm70\%$ for a 0 to 5 MeV range. Systematic errors due to the choice of polynomial order for the unfolded results were similarly estimated, and they varied from $\pm5\%$ for a 0 to 10 MeV range to $\pm30\%$ for a 0 to 5 MeV range. An additional error of $<\pm10\%$ was due to the estimated uncertainty in the width of the Monte-Carlo resolution function. All of the systematic errors discussed above were added in quadrature to form the total error. Statistical errors in the resolution unfolded cross sections were negligible in comparison. \bigskip \centerline {\bf V. COMPARISON WITH THEORY} \medskip \centerline{\bf A. Predictions Folded With the Experimental Energy Resolution} \par One of the present experimental goals is to test for the influence of non-nucleonic effects such as MEC and, possibly, quark clusters in the deuteron wavefunction. If, for example, MEC have a strong effect on the predictions up to $E_{np} = 20$ MeV, then the present resolution unfolded results constitute a legitimate test of $E_{np}$-averaged models. \par Theoretical indications for the importance of MEC at large $E_{np}$ are presented in Fig.\t 17, where several meson-nucleon predictions [\aren] and a hybrid quark-hadron prediction [\yama] are shown for the lowest and highest $Q\k2$ values of the present experiment. The calculations including MEC are all considerably lower than the IA calculation, with the differences decreasing slowly with increasing $E_{np}$. At $E_{np}=20$ MeV, the calculation [\aren] using $F_{1}$ coupling for the MEC is about 50\% of the IA calculation at $Q\k2 = 1.21$ (GeV/c)$\k2$, and only 15\% of the IA calculation at $Q\k2 = 2.76$ (GeV/c)$\k2$. Since any deviation from the IA is a measure of the influence of non-nucleonic effects, it is clear that for the models studied, MEC contribute strongly over a relatively large range of $E_{np}$. \par Fig. 17 emphasizes the large differences that exist between calculations with different treatments of the MEC. For $E_{np}\leq 10$ MeV these variations can exceed an order-of-magnitude, and they remain large for $E_{np}$ up to 20 MeV. Also evident in Fig.\t 17 are the substantial differences between the Yamauchi {\it et al.} [\yama] hybrid quark-hadron model calculations and the Arenh\"ovel {\it et al.} [\aren] meson-nucleon calculations. Such variations between the theoretical predictions are preserved, even for a resolution in $E_{np}$ as large as 10 MeV. The coarse energy resolution of the present data motivated the use of two methods of comparison with theoretical predictions. The model-dependent resolution unfolding procedure has already been discussed, and the resulting comparisons with theory will be presented below. A less model-dependent procedure is to compare the actual data with predictions folded with Monte-Carlo determined resolution functions. \par The convolution integral with respect to $E_{np}$ can be written as $$\sigma_{s}(E,E_{np}) = \int_{0}\k{\infty}R(E_{np}- E_{np}^{\prime}) \sigma(E,E_{np}^{\prime})dE_{np}^{\prime}/ \int_{-\infty}\k{\infty}R(E_{np}^{\prime})dE_{np}^{\prime}\ ,\eqn\zgx$$ where $\sigma(E,E_{np})$ is the theoretical cross section, $R(E_{np})$ is the resolution function, and $\sigma_{s}(E,E_{np})$ is the resolution-smeared cross section. \par Radiatively corrected data at six values of elastic four-momentum transfer squared $Q\k2$ are shown in Figs.\t 18 and 19. The error bars represent total statistical and systematic uncertainties, added in quadrature. The $\pm 0.25\%$ uncertainty in scattered electron energy $E\k{\prime}$ produced the largest systematic error in the cross sections. \par Also shown in Figs.\t 18 and 19 are several non-relativistic predictions [\aren,\yama] smeared by the experimental resolution function according to Eq. $\zgx$. Within $\approx 3$ MeV of threshold, electroproduction proceeds primarily through an M1 spin-flip transition to an unbound $^1S_0$ $T=1$ scattering state. However, for $E_{np}$ greater than a few MeV, higher order partial waves contribute to the electrodisintegration cross section. The meson-nucleon predictions of Arenh\"ovel \etals [\aren] take account of all electric and magnetic transitions with $L\leq 4$, where $L$ is the orbital angular momentum of the final state. The hybrid quark-hadron calculations of Yamauchi \etals [\yama] take account of 12 different final $np$ states and 28 transitions. Thus, the comparison of these predictions with the present data for $E_{np}$ up to 20 MeV is justifiable. \par The meson-nucleon predictions shown in Figs.\t 18 and 19 all use the Paris potential [\pari] to describe the deuteron wave function. Calculations with both the $G_{E}(Q\k2)$ and $F_{1}(Q\k2)$ electromagnetic form factors for the MEC are represented. The calculations employing $G_{E}(Q\k2)$ use two different choices for the neutron form factor $G_{En}(Q\k2)$: $G_{En}(Q\k2) = 0$ and the model of Gari and Kr\"umpelmann [\gari]. These choices have a sizeable effect on the calculations, although it should be noted that the first choice is strongly favored by recent data [\lung]. The models with Dirac coupling describe the data better up to $Q\k2\approx 2$ (GeV/c)$\k2$, while those with Sachs coupling exhibit comparable agreement at higher $Q\k2$ values. \par The effects of six-quark clusters in the deuteron wave function are generally expected to be small. Exploratory quark-inspired models [\kiss,\gloz,\yama] are plagued by high sensitivity to poorly-constrained parameters. The hybrid quark-hadron model of Ref. $\yama$ is in fair agreement with the higher $Q^2$ data shown in Fig.\t 19, but lies below the lower $Q\k2$ data shown in Fig. \t 18. \par To summarize this section, none of the non-relativistic predictions [\aren,\yama] is in quantitative agreement with the data over the entire $Q\k2$-range of 1.2 to 2.7 (GeV/c)$\k2$, although some calculations can describe the data in a more limited $Q^2$ range. In particular, understanding of the present data relies heavily on resolving the issue of what electromagnetic form factor is appropriate for the MEC. Fully relativistic meson-nucleon calculations and more rigorous quark-hadron models are needed. \medskip \centerline{\bf B. Predictions Compared With Resolution Unfolded Data} \par In Section IV, a model-dependent procedure for extracting resolution unfolded cross sections was described. The results for each $Q\k2$ were averaged over various ranges of $E_{np}$. A range of 0 to 10 MeV in $E_{np}$ was chosen to be compatible with the present energy resolution, and much larger than the $\pm 0.25\%$ uncertainty in $E\k{\prime}$. Also, the model dependence was found to be substantially reduced for larger averaging ranges. \par Averaging over a range of 0 to 10 MeV requires some justification since previous experiments [\simo] at lower $Q^2$ have better resolution than the present high $Q^2$ experiment, and the published results were averaged over $E_{np}=0$ to 3 MeV. For comparison, Fig. 20 shows three different theoretical predictions [\aren,\yama] averaged both over the range of $E_{np}$ from 0 to 3 and over the range from 0 to 10 MeV. For the model of Yamauchi \etals [\yama] and the $G_E$ calculation of Arenh\"ovel \etals [\aren], the 0 to 3 MeV averaging range to gives somewhat larger results than the 0 to 10 MeV range. This is expected since these models predict an enhancement in the cross section close to threshold. However, the differences are small, on the same order as the experimental errors, and the differences between the models is much larger than differences due to the averaging range. The $F_1$ calculation of Arenh\"ovel \etals [\aren] shows a larger difference between the two averaging ranges, especially at low $Q^2$. In this case the 0 to 10 MeV results are higher than the 0 to 3 MeV results, because this model predicts no enhancement at threshold. Nevertheless, the differences due to the choice of MEC coupling ($F_1$ versus $G_E$) are much larger than the differences due to the averaging range. \par In short, at least up to $E_{np} = 20$ MeV, differences between various predictions are much larger than effects from different $E_{np}$-averaging ranges and errors introduced by the resolution unfolding procedures. We therefore feel it is reasonable to compare the present experimental results, averaged over 0 to 10 MeV, with similarly averaged theoretical predictions. \par Resolution unfolded results from the present experiment averaged over 0 to 10 MeV are compared with similarly averaged predictions [\aren,\yama] shown on the right hand side of Fig.\t 21. The error bars include both statistical and systematic uncertainties, and primarily reflect the uncertainty in $E\k{\prime}$. Higher resolution data from a recent experiment [\rory] performed at the Bates Linear Accelerator Center up to $Q^2=1.6$ (GeV/c)$^2$ are in reasonable agreement with the present data. On the left hand side of Fig.\t 21 finer resolution data from previous experiments [\simo,\rory] are compared with the theoretical predictions of Ref. $\yama$ at $E_{np} = 1.5$ MeV, and of Ref. [\aren], averaged over the range 0 to 3 MeV. The differences due to averaging over 0 to 3 {\it versus} 0 to 10 MeV, are indicated by the small discontinuities in the curves at $Q\k2 = 1.1$ (GeV/c)$\k2$. Despite the relatively coarse resolution in $E_{np}$ and systematic errors from resolution unfolding, the present data can discriminate between the available models. The data indicate a change in slope with increasing $Q\k2$ around 1 (GeV/c)$\k2$, which is qualitatively consistent with ``diffraction features'' observed in all of the models. \par Although several models predict the change of slope shown in Fig.\t 21 at roughly the correct $Q\k2$ value, they are not in accord with the data over the entire $Q\k2$ range. While the inclusion of MEC certainly improves the agreement for $Q\k2 < 1$ (GeV/c)$\k2$, severe discrepancies remain at higher $Q\k2$. Comparisons of the present data with other predictions are given elsewhere [\rory,\arrn,\schm]. The dependence on nucleon-nucleon potential, nucleon form factor parametrization, treatment of MEC and isobars, and possible quark clusters are examined in these references. All of these inputs are found to have a substantial influence on non-relativistic predictions. The extensive investigations in [\schm] show that certain choices of nucleon-nucleon potential and neutron form factor parametrization lead to agreement with most of the available data in their non-relativistic model, although the calculations always lie below the data in the region $1.2 50$ MeV, but decreases as $E_{np}\rightarrow 0$, in agreement with earlier results by Titov [\tito] at lower $Q\k2$. This indicates values of $R$ which become relatively large near threshold. The curves in Fig.\t 23 represent calculations [\aren,\lage] that use wave functions derived from the Paris potential and take into account final state interactions, MEC, and $\Delta$ resonances. All of the predictions of Ref. [\aren] yield $W_{1}/W_{2}\approx 1$ for $E_{np} > 40$ MeV, in agreement with the data. Below $E_{np} = 50$ MeV, the IA calculation and a calculation that includes MEC with Sachs coupling produce essentially constant values of $W_{1}/W_{2}$ over the entire range of $E_{np}$, in marked disagreement with the data. Even though both $W_1$ and $W_2$ are strongly affected by MEC near threshold (see Figs. 18 and 19), with the Sachs coupling $R$ remains small, so that $W_1/W_2$ remains close to unity. The prediction that uses Dirac coupling for the MEC decreases rapidly as $E_{np}\rightarrow 0$, in qualitative agreement with the data. This is due to the absence of any enhancement of $W_1$ near the deuteron break-up threshold (see Fig.\t 17), resulting in large values of $R$. The prediction of Ref. $\lage$ does not extend into the threshold region and lies somewhat below the results of Ref. [\aren] at large $E_{np}$. It will be interesting to see if future calculations will be able to describe the shape and magnitude of both $W_1$ and $W_2$ for $E_{np}<40$ MeV. \par \centerline{\bf VI. CONCLUSIONS} \par Inelastic cross sections for electron scattering from deuterium near threshold have been extended by this experiment to $Q^2=2.7$ (GeV/c)$^2$. The present resolution unfolded results, when compared with earlier data [\simo] at lower $Q\k2$, have provided the first evidence of a change of slope in the cross section near $Q\k2\approx 1$ (GeV/c)$\k2$. This change of slope is consistent with recent experimental results [\rory] with improved energy resolution. Although the present data have relatively poor resolution in $E_{np}$, calculations convoluted with the experimental energy resolution maintain a strong sensitivity to the effects of MEC. Comparisons to several non-relativistic calculations [\aren,\yama,\schm] show that the inclusion of MEC substantially improves the agreement with data, although none of the theoretical curves passes through the error bars of all the available data. \par It is clear that the present experimental results have opened many new questions in a region where the deuteron wave function, non-nucleonic degrees of freedom, and relativistic effects are all important. \endpage \par \centerline{ACKNOWLEDGEMENTS} We would like to acknowledge the support of J. Davis, R. Eisele, C. Hudspeth, J. Mark, J. Nicol, R. Miller, L. Otts, and the rest of the SLAC staff. We also thank H. Arenh\"ovel, J.-M. Laget, and Y. Yamauchi for providing numerical results of their calculations. This work was supported in part by the Department of Energy, contracts DOE--AC03--76SF00515 (SLAC), W--7405--ENG--48 (LLNL), DE--AC02-76ER--02853 (U. Mass.); National Science Foundation Grant PHY85--10549 (A.U.); the U.S.--Israel Binational Science Foundation (Tel--Aviv); and the Monbusho International Research Program (A. Hotta). \endpage \twelvepoint \baselineskip 17pt \noindent Table I. Cross sections per deuteron nucleus for inelastic electron-deuteron scattering near break-up threshold. The beam energy $E$ and relative energy $E_{np}$ are evaluated at the center of the target. The errors include statistical and systematic contributions added in quadrature. \vskip .05in \input tables \def\tstrut{\vrule height 12pt depth 4pt width 0pt} \begintable \multispan{2} \tstrut\hfil $E=$0.734 GeV\hfil | \multispan{2} \tstrut\hfil $E=$0.843 GeV\hfil | \multispan{2} \tstrut\hfil $E=$0.885 GeV\hfil | \multispan{2} \tstrut\hfil $E=$0.934 GeV\hfil \nr $E_{np}$ & $d\sigma/d\Omega dE_{np}$ | $E_{np}$ & $d\sigma/d\Omega dE_{np}$ | $E_{np}$ & $d\sigma/d\Omega dE_{np}$ | $E_{np}$ & $d\sigma/d\Omega dE_{np}$ \nr (MeV) & (fb/sr-MeV) | (MeV) & (fb/sr-MeV) | (MeV) & (fb/sr-MeV) | (MeV) & (fb/sr-MeV) \nr \hfill -22.4 & \hfill $ 0.12\pm 0.29 $ | \hfill -25.4 & \hfill $ 0.01\pm 0.06 $ | \hfill -26.6 & \hfill $-0.02\pm 0.03 $ | \hfill -27.9 & \hfill $-0.02\pm 0.03 $ \nr \hfill -20.6 & \hfill $ 0.03\pm 0.29 $ | \hfill -23.3 & \hfill $ 0.01\pm 0.06 $ | \hfill -24.4 & \hfill $-0.01\pm 0.04 $ | \hfill -25.6 & \hfill $ 0.00\pm 0.03 $ \nr \hfill -18.7 & \hfill $-0.49\pm 0.25 $ | \hfill -21.2 & \hfill $-0.01\pm 0.06 $ | \hfill -22.2 & \hfill $-0.07\pm 0.04 $ | \hfill -23.3 & \hfill $-0.01\pm 0.03 $ \nr \hfill -16.9 & \hfill $-0.81\pm 0.31 $ | \hfill -19.1 & \hfill $-0.05\pm 0.06 $ | \hfill -20.0 & \hfill $-0.03\pm 0.04 $ | \hfill -20.9 & \hfill $ 0.01\pm 0.03 $ \nr \hfill -15.1 & \hfill $ 0.16\pm 0.35 $ | \hfill -17.0 & \hfill $ 0.03\pm 0.07 $ | \hfill -17.7 & \hfill $ 0.03\pm 0.04 $ | \hfill -18.6 & \hfill $ 0.00\pm 0.03 $ \nr \hfill -13.2 & \hfill $-0.32\pm 0.34 $ | \hfill -14.9 & \hfill $ 0.05\pm 0.08 $ | \hfill -15.5 & \hfill $-0.04\pm 0.04 $ | \hfill -16.2 & \hfill $ 0.01\pm 0.03 $ \nr \hfill -11.4 & \hfill $-0.18\pm 0.41 $ | \hfill -12.8 & \hfill $-0.05\pm 0.07 $ | \hfill -13.3 & \hfill $-0.05\pm 0.04 $ | \hfill -13.9 & \hfill $ 0.01\pm 0.03 $ \nr \hfill -9.6 & \hfill $ 0.27\pm 0.49 $ | \hfill -10.7 & \hfill $ 0.01\pm 0.08 $ | \hfill -11.1 & \hfill $-0.08\pm 0.06 $ | \hfill -11.6 & \hfill $ 0.02\pm 0.04 $ \nr \hfill -7.7 & \hfill $ 0.54\pm 0.59 $ | \hfill -8.6 & \hfill $ 0.00\pm 0.10 $ | \hfill -8.9 & \hfill $ 0.07\pm 0.09 $ | \hfill -9.2 & \hfill $ 0.04\pm 0.04 $ \nr \hfill -5.9 & \hfill $ 0.52\pm 0.56 $ | \hfill -6.4 & \hfill $ 0.04\pm 0.13 $ | \hfill -6.7 & \hfill $ 0.09\pm 0.10 $ | \hfill -6.9 & \hfill $ 0.09\pm 0.05 $ \nr \hfill -4.1 & \hfill $ 0.84\pm 0.59 $ | \hfill -4.3 & \hfill $ 0.14\pm 0.14 $ | \hfill -4.4 & \hfill $ 0.32\pm 0.12 $ | \hfill -4.6 & \hfill $ 0.12\pm 0.07 $ \nr \hfill -2.2 & \hfill $ 2.30\pm 0.64 $ | \hfill -2.2 & \hfill $ 0.52\pm 0.17 $ | \hfill -2.2 & \hfill $ 0.29\pm 0.12 $ | \hfill -2.2 & \hfill $ 0.21\pm 0.08 $ \nr \hfill -0.4 & \hfill $ 0.53\pm 0.56 $ | \hfill -0.1 & \hfill $ 0.57\pm 0.18 $ | \hfill 0.0 & \hfill $ 0.55\pm 0.14 $ | \hfill 0.1 & \hfill $ 0.33\pm 0.08 $ \nr \hfill 1.4 & \hfill $ 2.14\pm 0.73 $ | \hfill 2.0 & \hfill $ 0.62\pm 0.19 $ | \hfill 2.2 & \hfill $ 0.77\pm 0.14 $ | \hfill 2.4 & \hfill $ 0.42\pm 0.07 $ \nr \hfill 3.3 & \hfill $ 3.43\pm 0.87 $ | \hfill 4.1 & \hfill $ 1.00\pm 0.19 $ | \hfill 4.4 & \hfill $ 0.93\pm 0.15 $ | \hfill 4.8 & \hfill $ 0.43\pm 0.07 $ \nr \hfill 5.1 & \hfill $ 3.14\pm 1.04 $ | \hfill 6.2 & \hfill $ 1.50\pm 0.21 $ | \hfill 6.6 & \hfill $ 0.97\pm 0.17 $ | \hfill 7.1 & \hfill $ 0.56\pm 0.08 $ \nr \hfill 6.9 & \hfill $ 4.86\pm 1.11 $ | \hfill 8.3 & \hfill $ 1.24\pm 0.20 $ | \hfill 8.8 & \hfill $ 1.02\pm 0.19 $ | \hfill 9.4 & \hfill $ 0.64\pm 0.10 $ \nr \hfill 8.8 & \hfill $ 5.71\pm 1.18 $ | \hfill 10.4 & \hfill $ 1.45\pm 0.26 $ | \hfill 11.1 & \hfill $ 1.44\pm 0.24 $ | \hfill 11.8 & \hfill $ 0.72\pm 0.12 $ \nr \hfill 10.6 & \hfill $ 7.99\pm 1.29 $ | \hfill 12.5 & \hfill $ 1.69\pm 0.31 $ | \hfill 13.3 & \hfill $ 1.74\pm 0.26 $ | \hfill 14.1 & \hfill $ 0.79\pm 0.12 $ \nr \hfill 12.4 & \hfill $ 7.44\pm 1.27 $ | \hfill 14.6 & \hfill $ 1.88\pm 0.37 $ | \hfill 15.5 & \hfill $ 1.85\pm 0.27 $ | \hfill 16.5 & \hfill $ 1.04\pm 0.13 $ \nr \hfill 14.3 & \hfill $ 9.61\pm 1.45 $ | \hfill 16.7 & \hfill $ 3.12\pm 0.40 $ | \hfill 17.7 & \hfill $ 2.25\pm 0.29 $ | \hfill 18.8 & \hfill $ 1.15\pm 0.14 $ \nr \hfill 16.1 & \hfill $ 9.98\pm 1.55 $ | \hfill 18.9 & \hfill $ 2.77\pm 0.35 $ | \hfill 19.9 & \hfill $ 2.53\pm 0.31 $ | \hfill 21.1 & \hfill $ 1.01\pm 0.14 $ \nr \hfill 17.9 & \hfill $12.10\pm 1.76 $ | \hfill 21.0 & \hfill $ 3.21\pm 0.37 $ | \hfill 22.1 & \hfill $ 2.68\pm 0.32 $ | \hfill 23.5 & \hfill $ 1.41\pm 0.17 $ \nr \hfill 19.8 & \hfill $13.80\pm 1.83 $ | \hfill 23.1 & \hfill $ 2.95\pm 0.33 $ | \hfill 24.3 & \hfill $ 2.97\pm 0.33 $ | \hfill 25.8 & \hfill $ 1.47\pm 0.18 $ \nr \hfill 21.6 & \hfill $16.02\pm 1.89 $ | \hfill 25.2 & \hfill $ 3.58\pm 0.36 $ | \hfill 26.6 & \hfill $ 3.53\pm 0.35 $ | \hfill 28.1 & \hfill $ 1.82\pm 0.20 $ \nr \hfill 23.5 & \hfill $16.99\pm 1.91 $ | \hfill 27.3 & \hfill $ 4.44\pm 0.40 $ | \hfill 28.8 & \hfill $ 3.67\pm 0.35 $ | \hfill 30.5 & \hfill $ 1.84\pm 0.20 $ \endtable \vskip -.15in \hskip 4.9in {\it continued} \endpage \noindent Table I. continued. \vskip .15in \begintable \multispan{2} \tstrut\hfil $E=$1.020 GeV\hfil | \multispan{2} \tstrut\hfil $E=$1.102 GeV\hfil | \multispan{2} \tstrut\hfil $E=$1.201 GeV\hfil | \multispan{2} \tstrut\hfil $E=$1.279 GeV\hfil \nr $E_{np}$ & $d\sigma/d\Omega dE_{np}$ | $E_{np}$ & $d\sigma/d\Omega dE_{np}$ | $E_{np}$ & $d\sigma/d\Omega dE_{np}$ | $E_{np}$ & $d\sigma/d\Omega dE_{np}$ \nr (MeV) & (fb/sr-MeV) | (MeV & (fb/sr-MeV) | (MeV) & (fb/sr-MeV) | (MeV & (fb/sr-MeV) \nr \hfill -30.3 & \hfill $ 0.038\pm 0.012 $ | \hfill -32.6 & \hfill $ 0.001\pm 0.007 $ | \hfill -35.3 & \hfill $ 0.000\pm 0.004 $ | \hfill -37.4 & \hfill $-0.002\pm 0.002 $ \nr \hfill -27.7 & \hfill $ 0.001\pm 0.013 $ | \hfill -29.8 & \hfill $ 0.003\pm 0.008 $ | \hfill -32.3 & \hfill $-0.004\pm 0.004 $ | \hfill -34.2 & \hfill $-0.005\pm 0.002 $ \nr \hfill -25.2 & \hfill $ 0.006\pm 0.013 $ | \hfill -27.0 & \hfill $ 0.007\pm 0.010 $ | \hfill -29.3 & \hfill $ 0.007\pm 0.006 $ | \hfill -31.0 & \hfill $ 0.000\pm 0.003 $ \nr \hfill -22.6 & \hfill $-0.006\pm 0.013 $ | \hfill -24.3 & \hfill $-0.001\pm 0.009 $ | \hfill -26.3 & \hfill $-0.002\pm 0.006 $ | \hfill -27.8 & \hfill $-0.003\pm 0.002 $ \nr \hfill -20.1 & \hfill $ 0.014\pm 0.018 $ | \hfill -21.5 & \hfill $ 0.009\pm 0.009 $ | \hfill -23.3 & \hfill $ 0.013\pm 0.007 $ | \hfill -24.6 & \hfill $ 0.000\pm 0.003 $ \nr \hfill -17.5 & \hfill $ 0.004\pm 0.016 $ | \hfill -18.8 & \hfill $ 0.026\pm 0.011 $ | \hfill -20.3 & \hfill $ 0.002\pm 0.005 $ | \hfill -21.4 & \hfill $ 0.002\pm 0.004 $ \nr \hfill -15.0 & \hfill $-0.009\pm 0.016 $ | \hfill -16.0 & \hfill $-0.002\pm 0.008 $ | \hfill -17.3 & \hfill $-0.002\pm 0.005 $ | \hfill -18.2 & \hfill $ 0.003\pm 0.004 $ \nr \hfill -12.4 & \hfill $ 0.027\pm 0.019 $ | \hfill -13.3 & \hfill $ 0.016\pm 0.012 $ | \hfill -14.3 & \hfill $ 0.016\pm 0.008 $ | \hfill -15.0 & \hfill $ 0.005\pm 0.005 $ \nr \hfill -9.9 & \hfill $ 0.023\pm 0.021 $ | \hfill -10.5 & \hfill $ 0.019\pm 0.014 $ | \hfill -11.2 & \hfill $ 0.006\pm 0.008 $ | \hfill -11.8 & \hfill $ 0.010\pm 0.005 $ \nr \hfill -7.3 & \hfill $ 0.055\pm 0.024 $ | \hfill -7.7 & \hfill $ 0.034\pm 0.017 $ | \hfill -8.2 & \hfill $ 0.010\pm 0.009 $ | \hfill -8.6 & \hfill $ 0.004\pm 0.006 $ \nr \hfill -4.8 & \hfill $ 0.069\pm 0.026 $ | \hfill -5.0 & \hfill $ 0.049\pm 0.019 $ | \hfill -5.2 & \hfill $ 0.036\pm 0.010 $ | \hfill -5.4 & \hfill $ 0.025\pm 0.007 $ \nr \hfill -2.2 & \hfill $ 0.083\pm 0.032 $ | \hfill -2.2 & \hfill $ 0.068\pm 0.021 $ | \hfill -2.2 & \hfill $ 0.053\pm 0.012 $ | \hfill -2.2 & \hfill $ 0.028\pm 0.008 $ \nr \hfill 0.3 & \hfill $ 0.160\pm 0.040 $ | \hfill 0.5 & \hfill $ 0.098\pm 0.024 $ | \hfill 0.8 & \hfill $ 0.042\pm 0.011 $ | \hfill 1.0 & \hfill $ 0.047\pm 0.009 $ \nr \hfill 2.9 & \hfill $ 0.207\pm 0.041 $ | \hfill 3.3 & \hfill $ 0.157\pm 0.026 $ | \hfill 3.8 & \hfill $ 0.063\pm 0.013 $ | \hfill 4.2 & \hfill $ 0.042\pm 0.008 $ \nr \hfill 5.4 & \hfill $ 0.275\pm 0.047 $ | \hfill 6.0 & \hfill $ 0.145\pm 0.026 $ | \hfill 6.8 & \hfill $ 0.081\pm 0.014 $ | \hfill 7.4 & \hfill $ 0.067\pm 0.010 $ \nr \hfill 8.0 & \hfill $ 0.358\pm 0.054 $ | \hfill 8.8 & \hfill $ 0.183\pm 0.029 $ | \hfill 9.8 & \hfill $ 0.083\pm 0.016 $ | \hfill 10.6 & \hfill $ 0.041\pm 0.009 $ \nr \hfill 10.5 & \hfill $ 0.263\pm 0.054 $ | \hfill 11.6 & \hfill $ 0.199\pm 0.031 $ | \hfill 12.8 & \hfill $ 0.109\pm 0.018 $ | \hfill 13.8 & \hfill $ 0.069\pm 0.011 $ \nr \hfill 13.1 & \hfill $ 0.410\pm 0.065 $ | \hfill 14.3 & \hfill $ 0.262\pm 0.038 $ | \hfill 15.8 & \hfill $ 0.100\pm 0.017 $ | \hfill 17.0 & \hfill $ 0.078\pm 0.011 $ \nr \hfill 15.6 & \hfill $ 0.438\pm 0.067 $ | \hfill 17.1 & \hfill $ 0.263\pm 0.038 $ | \hfill 18.8 & \hfill $ 0.139\pm 0.020 $ | \hfill 20.2 & \hfill $ 0.071\pm 0.011 $ \nr \hfill 18.2 & \hfill $ 0.492\pm 0.066 $ | \hfill 19.8 & \hfill $ 0.259\pm 0.038 $ | \hfill 21.8 & \hfill $ 0.150\pm 0.020 $ | \hfill 23.4 & \hfill $ 0.062\pm 0.011 $ \nr \hfill 20.7 & \hfill $ 0.608\pm 0.074 $ | \hfill 22.6 & \hfill $ 0.334\pm 0.041 $ | \hfill 24.8 & \hfill $ 0.129\pm 0.022 $ | \hfill 26.5 & \hfill $ 0.091\pm 0.014 $ \nr \hfill 23.3 & \hfill $ 0.679\pm 0.074 $ | \hfill 25.4 & \hfill $ 0.365\pm 0.047 $ | \hfill 27.8 & \hfill $ 0.162\pm 0.024 $ | \hfill 29.8 & \hfill $ 0.100\pm 0.015 $ \nr \hfill 25.8 & \hfill $ 0.639\pm 0.074 $ | \hfill 28.1 & \hfill $ 0.352\pm 0.051 $ | \hfill 30.8 & \hfill $ 0.208\pm 0.027 $ | \hfill 33.0 & \hfill $ 0.103\pm 0.015 $ \nr \hfill 28.4 & \hfill $ 0.735\pm 0.076 $ | \hfill 30.9 & \hfill $ 0.422\pm 0.050 $ | \hfill 33.8 & \hfill $ 0.234\pm 0.028 $ | \hfill 36.2 & \hfill $ 0.144\pm 0.017 $ \nr \hfill 30.9 & \hfill $ 0.756\pm 0.076 $ | \hfill 33.6 & \hfill $ 0.502\pm 0.056 $ | \hfill 36.8 & \hfill $ 0.209\pm 0.027 $ | \hfill 39.3 & \hfill $ 0.134\pm 0.016 $ \nr \hfill 33.5 & \hfill $ 0.870\pm 0.077 $ | \hfill 36.4 & \hfill $ 0.556\pm 0.059 $ | \hfill 39.8 & \hfill $ 0.246\pm 0.029 $ | \hfill 42.5 & \hfill $ 0.114\pm 0.015 $ \endtable \endpage \noindent Table II. Ratio of the inelastic structure functions $W_1/W_2 $ for inelastic electron-deuteron scattering. The relative energy $E_{np}$ in units of MeV is evaluated at the target center, and the errors include both statistical and systematic contributions. \vskip .1in \begintable \multispan{2} \tstrut\hfil $=$1.36 (GeV/c)$^2$ \hfil | \multispan{2} \tstrut\hfil $=$1.84 (GeV/c)$^2$ \hfil | \multispan{2} \tstrut\hfil $=$2.33 (GeV/c)$^2$ \hfil \nr $E_{np} $ & $W_1/W_2$ | $E_{np} $ & $W_1/W_2$ | $E_{np} $ & $W_1/W_2$ \nr \hfill 9.4 & $0.178\pm 0.024 $ | \hfill 11.8 & $0.237\pm 0.035 $ | \hfill 14.3 & $0.386\pm 0.071 $ \nr \hfill 13.6 & $0.211\pm 0.027 $ | \hfill 16.9 & $0.317\pm 0.047 $ | \hfill 20.3 & $0.534\pm 0.094 $ \nr \hfill 17.8 & $0.320\pm 0.060 $ | \hfill 22.0 & $0.435\pm 0.068 $ | \hfill 26.3 & $0.525\pm 0.103 $ \nr \hfill 22.0 & $0.310\pm 0.061 $ | \hfill 27.1 & $0.441\pm 0.071 $ | \hfill 32.3 & $0.774\pm 0.147 $ \nr \hfill 26.2 & $0.368\pm 0.075 $ | \hfill 32.2 & $0.489\pm 0.077 $ | \hfill 38.3 & $0.730\pm 0.132 $ \nr \hfill 32.4 & $0.409\pm 0.065 $ | \hfill 38.5 & $0.428\pm 0.082 $ | \hfill 49.9 & $0.525\pm 0.129 $ \nr \hfill 40.3 & $0.521\pm 0.071 $ | \hfill 48.1 & $0.795\pm 0.113 $ | \hfill 61.4 & $0.734\pm 0.145 $ \nr \hfill 48.2 & $0.746\pm 0.089 $ | \hfill 57.6 & $0.541\pm 0.080 $ | \hfill 72.8 & $0.847\pm 0.147 $ \nr \hfill 56.1 & $0.812\pm 0.092 $ | \hfill 67.2 & $0.923\pm 0.115 $ | \hfill 85.6 & $0.711\pm 0.118 $ \nr \hfill 64.0 & $0.645\pm 0.073 $ | \hfill 76.8 & $0.743\pm 0.093 $ | \hfill 96.4 & $0.702\pm 0.106 $ \nr \hfill 71.9 & $0.796\pm 0.085 $ | \hfill 86.4 & $0.820\pm 0.097 $ | \hfill 107.1 & $0.857\pm 0.117 $ \nr \hfill 80.8 & $0.876\pm 0.114 $ | \hfill 96.0 & $0.827\pm 0.108 $ | \hfill 117.9 & $1.010\pm 0.130 $ \nr \hfill 88.3 & $0.931\pm 0.109 $ | \hfill 105.0 & $0.927\pm 0.108 $ | \hfill 128.7 & $0.764\pm 0.098 $ \nr \hfill 95.7 & $0.811\pm 0.095 $ | \hfill 114.0 & $0.931\pm 0.105 $ | \hfill 139.5 & $1.000\pm 0.120 $ \nr \hfill 103.1 & $0.892\pm 0.101 $ | \hfill 123.0 & $0.843\pm 0.094 $ | \hfill 151.5 & $0.877\pm 0.150 $ \nr \hfill 110.5 & $0.825\pm 0.094 $ | \hfill 132.0 & $0.856\pm 0.093 $ | \hfill 161.6 & $0.974\pm 0.140 $ \nr \hfill 118.0 & $0.924\pm 0.102 $ | \hfill 141.0 & $0.849\pm 0.090 $ | \hfill 171.7 & $1.050\pm 0.150 $ \nr \hfill 126.3 & $0.979\pm 0.122 $ | \hfill 151.3 & $1.010\pm 0.130 $ | \hfill 181.9 & $0.947\pm 0.130 $ \nr & | \hfill 159.8 & $1.110\pm 0.130 $ | \hfill 192.0 & $1.040\pm 0.130 $ \endtable \endpage \baselineskip 23 pt \centerline{REFERENCES} \noindent$\k{(a)}$ Present address: Saddleback College, Mission Viejo, CA 92691 \hfill \break \noindent$\k{(b)}$ Present address: University of Pennsylvania, Philadelphia, PA 19104 \hfill \break \noindent$\k{(c)}$ Permanent address: Georgetown University, Washington, D.C. 20057 \hfill \break \noindent$\k{(d)}$ Present address: Stanford Linear Accelerator Center, Stanford, CA 94309 \hf