Momentum profile

If the experiment requires the distorted-wave formulation (11.11), the observed momentum profile is distorted. It is still possible to extract ip (q) by a statistical fitting procedure. 

For a given total energy E we can identify a manifold of ion states I/) that all have a momentum profile of the same shape, given by (11.11,11.12,11.14). The shape is characteristic of an orbital a) of the target. The manifold is characterised not only by the symmetry, but by the set of quantum numbers a that includes a principal quantum number. We call it the orbital manifold a. 

The existence of a common momentum profile for the manifold a confirms the weak-coupling binary-encounter approximation. Within these approximations we must make further approximations to calculate differential cross sections. For the probe amplitude of (11.1) we may make, for example, the distorted-wave impulse approximation (11.3). This enables us to identify a normalised experimental orbital for the manifold. If normalised experimental orbitals are used to calculate the differential cross sections for two different manifolds within experimental error this confirms the whole approximation to this stage. An orbital approximation for the target structure (such as Hartree—Fock or Dirac—Fock) is confirmed if the experimental orbital energy agrees with the calculated orbital energy and if it correctly predicts differential cross sections. 

Fig. 11.3 illustrates the relative momentum profile of the 15.76 eV state in a later experiment at =1200 eV, compared with the plane-wave impulse approximation with orbitals calculated by three different methods. The sensitivity of the reaction to the structure calculations is graphically illustrated. A single Slater-type orbital (4.38) with a variationally-determined exponent provides the worst agreement with experiment. The Hartree-Fock—Slater approximation (Herman and Skillman, 1963), in which exchange is represented by an equivalent-local potential, also disagrees. The Hartree—Fock orbital agrees within experimental error. 

Fig. 11.3. The 1200 eV noncoplanar-symmetric momentum profile for the 15.76 eV state of Ar" " (McCarthy and Weigold, 1988). Plane-wave impulse approximation curves are calculated with 3p orbitals. Full curve, Hartree—Fock (Clementi and Roetti, 1984) long-dashed curve, Hartree—Fock—Slater (Herman and Skillman, 1963) short-dashed curve, minimal variational basis.

Fig. 11.4 illustrates the momentum profiles of the other ion states observed in a later experiment with better energy resolution than that of fig. 11.2. All these states have momentum profiles of essentially the same shape. They are thus identified as states of the same orbital manifold, for which the experiment obeys the criterion for the validity of the weak-coupling binary-encounter approximation. Details of electron momentum spectroscopy depend on the approximation adopted for the probe amplitude of (11.1). The 3s Hartree—Fock momentum profiles in the plane-wave impulse approximation identify the 3s manifold. However, the approximation underestimates the high-momentum profile. 

The distorted-wave impulse approximation using Hartree—Fock orbitals is confirmed in every detail by fig. 11.5, which shows momentum profiles for argon at =1500 eV. The whole experiment is normalised to the distorted-wave impulse approximation at the 3p peak. It represents the remainder of the confirmation in this case of the whole procedure of electron momentum spectroscopy. The Hartree—Fock orbitals give complete agreement with experiment for two manifolds, 3p and 3s. The spectroscopic factor Si5.76(3p) is measured as 1, since no further states of the 3p manifold are identified. Later experiments give 0.95 and this is the value used for normalisation. The approximation describes the momentum-profile shape for the first member of the 3s manifold at 29.3 eV within experimental error. The shape for the manifold sum of cross sections agrees and its... 

Fig. 11.4. Noncoplanar-symmetric momentum profiles at the indicated energies for the ionisation of argon to some more-strongly excited ion states above the ion ground state (Weigold and McCarthy, 1978). Full curve, plane-wave impulse approximation for the Hartree—Fock 3s orbital.

Fig. 11.5. The 1500 eV noncoplanar-symmetric momentum profiles for the argon ground-state transition (15.76 eV), first excited state (29.3 eV) and the total 3s manifold (McCarthy et ai, 1989). Hartree—Fock curves are indicated DWIA, distorted-wave impulse approximation PWIA, plane-wave impulse approximation. Experimental data are normalised to the 3p distorted-wave curve with a spectroscopic factor Si5.76(3p) = 0.95. The experimental angular resolution has been folded into the calculations.

Fig. 11.6 shows the noncoplanar-symmetric differential cross sections at 1200 eV for the Is state and the unresolved n=2 states, normalised to theory for the low-momentum Is points. Here the structure amplitude is calculated from the overlap of a converged configuration-interaction representation of helium (McCarthy and Mitroy, 1986) with the observed helium ion state. The distorted-wave impulse approximation describes the Is momentum profile accurately. The summed n=2 profile does not have the shape expected on the basis of the weak-coupling approximation (long-dashed curve). Its shape and magnitude are given quite well by... 

The 1200 eV experiment of Cook et al (1984) showed that the 5p2/2 and 5pi/2 momentum profiles differed significantly. They are not consistent with nonrelativistic Hartree—Fock orbitals but can be described within experimental error by the distorted-wave impulse approximation using Dirac—Fock orbitals. The 5p2/2 Pi/i branching ratio is shown in fig. 11.8, where it is compared with the distorted-wave impulse approximation using relativistic and nonrelativistic orbitals. The 5p3/2 orbital... 

Fig. 11.11. The 1000 eV noncoplanar-symmetric momentum profiles for the summed (a) 5p and (b) 5s manifolds of xenon (McCarthy and Weigold, 1991). Distorted- and plane-wave impulse approximations are indicated respectively by DW and PW. Dirac—Fock and Hartree—Fock orbitals are indicated respectively by DF and HF. The experimental angular resolution has been folded into the calculation. The experimental data are normalised at the peak of the 5p profile.

Fig. 11.13. The 1000 eV noncoplanar-symmetric momentum profiles for lead (Frost et al., 1986). Curves show the plane-wave impulse approximation. The experiment is normalised at the peak of the 6p-manifold profile (a). The 14.6 eV and 18.4 eV states of the 6s manifold are indicated by (b) and (c). Spectroscopic factors are given in table 11.2. For (a), (b) and (c) respectively the Hartree—Fock calculation (broken curve) is normalised to multiconfiguration Dirac—Fock (solid curve) by factors 0.82, 0.70 and 0.64.

Fig. 11.13(a) shows the summed momentum profiles for the states of the 6p manifold at 7.4 eV and 9.2 eV. Figs. 11.13(b) and (c) describe states that are identified by the plane-wave impulse approximation with the Dirac—Fock orbital as belonging to the 6s manifold. Since the valence states of lead are diffuse in coordinate space most of the momentum profile is within the 1 a.u. limit of validity of the plane-wave impulse approximation for the profile shape. The experiment agrees with the Dirac—Fock profile but rules out the nonrelativistic Hartree—Fock method. 

The momentum distributions for the 3s ground state and the 3pi state are shown in fig. 11.15. They are compared with the momentum distributions calculated using Hartree—Fock orbitals and folding in the experimental momentum resolution function. Because the 3s and 3p orbitals are very diffuse in coordinate space the momentum profile is well within the p=l limit of validity of the plane-wave impulse approximation. 

Fig. 11.15. The 800 eV noncoplanar-symmetric momentum profiles for the laser-assisted ionisation of sodium (Zheng et al, 1990). Hartree—Fock curves for the indicated states are calculated in the plane-wave impulse approximation. From McCarthy and Weigold (1991).

Proper description of hydrodynamic effects and the momentum balance is often neglected in reactor modeling today. Assumptions of plug flow or perfectly mixed are common and simplify the calculations tremendously. A trend towards full calculation of flow and momentum profiles is starting to take shape in the literature, but it is still hampered by excessive computing times. 

Electron Momentum Spectroscopy [1—4] is a powerful orbital imaging technique, which enables straightforward reconstructions of electron momentum distributions associated with specific ionization channels (i.e., of orbital momentum profiles in a one-electron picture of ionization), according to an angular analysis of intensities in electron impact (e, 2e) ionization experiments [M -F e Eq +... 

However, the more recent EMS experiments by K. Liu et al. [45] on W(CO)6 at electron impact energies of 1.2 and 2.4 keV gave at first glance quite similar momentum profiles for the HOMO and led thus on the contrary to the conclusion that distorted wave and post-collision effects are too weak to explain the experimentally observed turnups at low electron momenta. Since the target compounds contain relatively heavy metal atoms, this discrepancy between theory and experiment was then also thought to be the outcome of the limitations inherent to a non-relativistic depiction. Further investigations of scalar relativistic and spin-orbit coupling effects indicated, however, a very... 

The first purpose of the present work is thus to evaluate whether the structural distortions which were proposed by Liu et al. [45] are compatible with thermal fluctuations at or near standard room temperature, that is, at 298 and 310 K, according to Maxwell-Boltzmann (MB) statistics [53, 54] on vibrational energy levels. The analysis is supplemented by Born-Oppenheimer Molecular Dynamical (BOMD) simulations [55-57] at the same temperatures of momentum profiles inferred from vertical (e, 2e) ionization cross-sections. A main advantage of this approach is that, by virtue of ergodicity [58], it enables a complete exploration of phase space which is equivalent to an ensemble... 

Fig. 3 Spherically averaged theoretical momentum profiles (without resolution folding) of the HOMO of W(CO)s...

Similar remarks can be made regarding the influence of nuclear dynamics in the initial ground state upon the momentum profile characterizing the HOMO of Cr(CO>6 and Mo(CO)g. As our results indicate (Figs. 6, 7), compared with the available measurements, nuclear dynamics in the initial state is found in both cases to yield a... 

Fig. 4 Study of the consequences of structural distortions along the three first vibrational eigenmodes of W(CO>6 upon the momentum profile characterizing the HOMO (without resolution folding). See text and Table 4 for a definition of the scaling factor a (results of single point calculation results at the B3LYP/aug-cc-pVTZ-PP level)...

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