Fully differential study of interference effects in the ionization of ${\mathrm{H}}_2$ by proton impact

We have measured fully differential cross sections for ionization of ${\mathrm{H}}_2$ by 75-keV proton impact. The coherence length of the projectile beam was varied by changing the distance between a collimating slit and the target. By comparing the cross sections measured for large and small coherence lengths pronounced interference effects could be identified in the data. A surprising result is that the phase angle in the interference term is primarily determined by the momentum transfer and only to a lesser extent by the recoil-ion momentum.

Figure 1

Schematic diagram of the experimental setup. The recoil ions, momentum analyzed by the recoil-ion momentum spectrometer, were measured in coincidence with the scattered projectiles, which were momentum analyzed by the parallel-plate energy analyzer. The vertical collimation slit ($x$ slit) was placed at a variable distance to the target in order to vary the transverse coherence length.

Figure 2

Fully differential, three-dimensional angular distribution of ejected electrons with an energy of 14.6 eV taken for the large (left panel) and small (right panel) slit distance and for a momentum transfer of 0.9 a.u. $p_o$ indicates the initial projectile momentum, which defines the positive $z$ axis. $q_x$ is the transverse component of the momentum transfer, which defines the positive $x$ axis.

Figure 3

(a) Fully differential cross sections for electrons ejected into the scattering plane as a function of the polar emission angle. The $x$ component of the recoil-ion momentum was fixed at −0.2 a.u. (top panel) and +0.2 a.u. (bottom panel). The closed (open) symbols represent the data taken for the large (small) slit distance. The solid lines represent the product of the incoherent data with an interference term of the form 1 + αcos($q_{x}D$) (see text for details). (b) Illustration of the electron ejection geometry for the FDCS plotted in panel (a). Only electrons ejected into the scattering plane ($xz$ plane), defined by $p_o$ and $q_x$, are analyzed, i.e., the azimuthal angle ${\phi }_{\mathrm{el}}$ is fixed at 90° and the polar angle ${\theta }_{\mathrm{el}}$, measured relative to the $z$ axis, is varied.

Figure 4

Fully differential cross sections as a function of the azimuthal electron ejection angle for fixed polar angles of 15° (top panels), 35° (center panels), and 55° (bottom panels). The $x$ component of the recoil-ion momentum was fixed at −0.2 a.u. (left panels) and +0.2 a.u. (right panels). Symbols as in Fig. 3. For a sketch of the ejection geometry see Fig. 5.

Figure 5

Illustration of the electron ejection geometry for the FDCS plotted in Fig. 4. The polar angle ${\theta }_{\mathrm{el}}$ is fixed at 15° (top panels), 35° (center panels), and 55° (bottom panels). Therefore, for all electrons analyzed in Fig. 4 the momentum vectors lie on the surface of a cone with an opening angle of $2{\theta }_{\mathrm{el}}$ centered on the $z$ axis. The FDCS are plotted as a function of the azimuthal angle ${\phi }_{\mathrm{el}}$, which is the angle between the projection of the electron momentum onto the $xy$ plane and the positive $y$ axis.

Figure 6

Fully differential cross-section ratios between the large and small slit distance data of Fig. 4. The solid and dashed lines were obtained from 1 + αcos($q_{x}D$) and Eq. (1) with $D$ = 2 and 1.4 a.u., respectively, and the dotted curve from 1 + αsin($q_{x}D$)/($q_{x}D$) and $D=1.4$ a.u. In each case α = 0.5.