Inclusive quasielastic electron-nucleus scattering

This paper presents a review on the field of inclusive quasielastic electron-nucleus scattering. It discusses the approach used to measure the data and includes a compilation of data available in numerical form. The theoretical approaches used to interpret the data are presented. A number of results obtained from the comparison between experiment and calculation are then reviewed. The analogies to, and differences from, other fields of physics exploiting quasielastic scattering from composite systems are pointed out.

Figure 1

Schematic representation of inclusive cross section as a function of energy loss.

Figure 2

Schematic representation of the IA regime, in which the nuclear cross section is replaced by the incoherent sum of cross sections describing scattering off individual nucleons, the recoiling $(A-1)$-nucleon system acting as a spectator.

Figure 3

Nuclear matter spectral function calculated using correlated basis function perturbation theory. From 34.

Figure 4

Calculated momentum distribution per nucleon in ${}^2\mathrm{H}$, ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and uniform nuclear matter (226; 39).

Figure 5

Inclusive electron scattering cross section at $E_e=3.595\mathrm{GeV}$ and $\theta =30°$. The data points represent the extrapolated nuclear matter cross section (86) while the solid and dashed lines show the results of IA calculations carried out with and without inclusion of the inelastic contributions, respectively. From 44.

Figure 6

Inclusive electron scattering cross section at $E_e=7.26\mathrm{GeV}$ and $\theta =8°$ for ${}^3\mathrm{He}$. The data points are from 84; the solid line shows the IA calculation based on the ${}^3\mathrm{He}$ spectral function (181). Approximate values for the scaling variable $x$ are indicated on top.

Figure 7

Response of uniform nuclear matter at equilibrium density and $\mid \mathbf{q}\mid =5{\mathrm{fm}}^{-1}$. The solid and dashed lines have been obtained using the full spectral function (34) and the approximation of Eq. (37), respectively. The diamonds represent the results obtained when shifting the dashed line by $\overline{E}=62\mathrm{MeV}$ (35).

Figure 8

Averaged distribution functions, defined by Eq. (47), of NN (solid lines) and $pp$ (dashed lines) pairs in ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and uniform nuclear matter at equilibrium density (NM). The corresponding quantities obtained neglecting dynamical correlation effects are shown by the dot-dashed and dotted lines, respectively. From 39.

Figure 9

Folding functions $F_{\mathbf{q}}$ of Eq. (49), calculated in uniform nuclear matter at equilibrium density. The solid line and diamonds correspond to momentum transfer $\mid \mathbf{q}\mid =2.2$ and $3.4\mathrm{GeV}∕c$, respectively (32).

Figure 10

Inclusive electron-scattering cross section at $E_e=3.595\mathrm{GeV}$ and $\theta =30°$. The data points represent the extrapolated nuclear matter cross section (86), while the solid and dashed lines show the results obtained in CGA including FSI effects, with and without taking into account correlation effects. For comparison, the IA cross section is also shown by the dot-dashed line (44).

Figure 11

Ratios of inclusive cross sections of iron and deuterium at $E_e=3.595\mathrm{GeV}$ and $\theta =25°$. Solid line, full calculation; dashed line, IA calculation, neglecting FSI in both iron and deuteron; dot-dashed line, calculation carried out using the approximate spectral function of Eq. (37) (47).

Figure 12

(Color online) HMS spectrometer in Hall C at Jefferson Lab with four magnets arranged as QQQD and the heavily shielded detector hut.

Figure 13

HMS detector package, consisting of multiple planes of multiwire drift chambers, fast scintillators, a subatmospheric gas Čerenkov counter, and a lead glass total absorption counter.

Figure 14

Kinematic range covered by the $(e,e^{\prime })$ data for ${}^{12}\mathrm{C}$, plotted as a function of $\mid \mathbf{q}\mid$ and $y$ (for definition, see Sec. 7). The symbols encode the range of $ϵ$ covered by the data (bins of 0.2 in $ϵ$).

Figure 15

A calculation (12, 13, 14) of the electron-deuteron vector asymmetry $A_{ed}^V$ at $Q^2=0.5$ $(\mathrm{GeV}∕c{)}^2$, which demonstrates the insensitivity of $A_{ed}^V$ to the reaction model and subnuclear degrees of freedom at angles near 180° corresponding to neutron along the momentum transfer.

Figure 16

Result from the global analysis of the $L\text{−}T$ (Rosenbluth) data (18). Open (full) points are obtained before (after) two-photon exchange corrections. The dashed line is a fit to the polarization-transfer data.

Figure 17

Integration region of $S(\mathbf{k},E)$ in Eq. (59) (below corresponding curves) contributing to $F\left(y\right)$ at $y=-0.2\mathrm{GeV}∕c$ and given $\mid \mathbf{q}\mid$ (87). The long-dashed line shows the location of the “ridge” of $S(\mathbf{k},E)$ where, at large $\mid \mathbf{k}\mid$, most of the strength is expected to occur.

Figure 18

Cross sections for ${}^3\mathrm{He}(e,e^{\prime })$ as a function of incident electron energy and energy loss, for different incident electron energies.

Figure 19

Scaling function for ${}^3\mathrm{He}(e,e^{\prime })$. The various data sets are labeled by electron energy (MeV) and angle (deg).

Figure 20

Convergence of $F(y,Q^2)$ with increasing $Q^2$, for different values of $y$ (20). The rise of $F(y,Q^2)$ for $y=0$ and increasing $Q^2$ is due to the $\Delta$ contribution.

Figure 21

Per nucleon cross section ratio of nuclear matter and the deuteron taken at $3.6\mathrm{GeV}$ and $30°$, as a function of the scaling variable $y$. The result of the CBF calculation is shown as a solid line (45).

Figure 22

Scaling function for nuclei $A=4–197$ and fixed kinematics $(\mid \mathbf{q}\mid \sim 1\mathrm{GeV}∕c)$ as a function of ${\psi }^{\prime }$, which corresponds to the variable $y$ scaled by a “Fermi momentum.”

Figure 23

Longitudinal superscaling function extracted (100) from the available data on C, Ca, and Fe, for momentum transfers between 300 and $570\mathrm{MeV}∕c$.

Figure 24

Comparison of experimental and theoretical response (112) for Argonne 18 potential (dotted) and Argonne V18+3BF (solid). Data are shown as open circles (104) and squares (177).

Figure 25

Comparison of theoretical (136) and experimental response at $\mid \mathbf{q}\mid =300\mathrm{MeV}∕c$ for (a) ${}^3\mathrm{He}$ and (b) ${}^3\mathrm{H}$. Dotted, two-body breakup; dashed, three-body breakup. Data are shown as open circles (104) and squares (177).

Figure 26

Comparison of theoretical (96) and experimental response at $\mid \mathbf{q}\mid =500\mathrm{MeV}∕c$. Solid, coupled-channel nucleon-$\Delta$; dashed, purely nucleonic case. Data are shown as circles (104) and squares (177).

Figure 27

Comparison of deuteron momentum distribution obtained from $(e,e^{\prime })$ (squares), $(e,e^{\prime }p)$ (triangles), and theory (curves) (74).

Figure 28

Model responses derived from the “normal” response plus various modifications (top). The corresponding Euclidean response in terms of the ratio to the one from the normal one (bottom).

Figure 29

Transverse Euclidean response divided by the proton magnetic form factor for ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$ at $600\mathrm{MeV}∕c$ momentum transfer, scaled by $e^{\tau {\omega }_{qe}}$. Data $(+)$, IA $(○)$, and full calculation $(\times )$.

Figure 30

Longitudinal (lower data set) and transverse responses of ${}^{12}\mathrm{C}$ (122), plotted in terms of the scaling function $F\left(y\right)$.

Figure 31

Rosenbluth separation for ${}^{56}\mathrm{Fe}$ and $\mid \mathbf{q}\mid =570\mathrm{MeV}∕c$ (158). The dashed lines are fits to the Saclay data alone; the solid lines are fits to the world data, which include the forward-angle SLAC data and emphasize the importance of a large range in $ϵ$.

Figure 32

Separated response functions for ${}^{56}\mathrm{Fe}$ and $\mid \mathbf{q}\mid =570\mathrm{MeV}∕c$. The solid curve corresponds to the CBF calculation (120).

Figure 33

Extrapolation of nuclear response per nucleon at fixed $q$ and $\omega$ ($E=3.6\mathrm{GeV}$, $\theta =16°$, and $\omega =180\mathrm{MeV}$) as a function of $A^{-1∕3}$ (top) and as a function of $A$ (bottom) where the extrapolated value of the nuclear matter response is indicated by the arrow.

Figure 34

Nuclear matter response at momentum transfers up to 3.5 $(\mathrm{GeV}∕c{)}^2$. From 86.

Figure 35

Compton profile of Na as a function of electron momentum (in atomic units), with the contribution of core electrons removed and finite-resolution effects unfolded (116).

Figure 36

Profile for neutron scattering from liquid helium at $0.35\mathrm{K}$. The curves, calculated using GFMC and including FSI effects, are shown for different fractions of the Bose condensate, 10% giving the best fit. From 235.

Figure 37

Structure function $F_2$ as a function of the momentum transfer (shifted vertically by factors $2^i$), for different values of the Bjorken scaling variable $x$. From 115.