Distorted wave impulse approximation

Fig. 10.3. The differential cross section for electron—helium ionisation at < = 0 in symmetric kinematics, plotted against total energy (van Wingerden et al., 1979). Full curve, distorted-wave impulse approximation broken curve, plane-wave impulse approximation. From McCarthy and Weigold (1988).

Fig. 10.5 compares the plane- and distorted-wave impulse approximations for the 3p orbital of argon in noncoplanar-symmetric kinematics at =1500 eV. Here distortion makes a difference beyond p=1.5 a.u. The experiment is described excellently (within an unmeasured normalisation) by the distorted-wave impulse approximation. Figs. 10.3 and 10.5 support... 

Electron momentum spectroscopy (McCarthy and Weigold, 1991) is based on ionisation experiments at incident energies of the order of 1000 eV, where the plane-wave impulse approximation is roughly valid. The differential cross section is measured for each ion state over a range of ion recoil momentum p from about 0 to 2.5 a.u. Noncoplanar-symmetric kinematics is the usual mode. In such experiments the distorted-wave impulse approximation turns out to be a sufficiently-refined theory. Checks of this based on a generally-valid sum rule will be described. 

In most experiments the target is not oriented. The differential cross section is averaged over the solid angle p. In the distorted-wave impulse approximation it is... 

In practice transfer of momentum to the ion by the mechanism described by distortion renders the probe imperfect. Experience has shown that the distorted-wave impulse approximation is sufficient to describe the distorted spectral function... 

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. 

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. 11.5. 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... 

Fig. 11.7. Noncoplanar-symmetric momentum profiles at the indicated energies for the indicated transitions in argon, compared with calculated profiles (McCarthy et ai, 1989). Experimental data are normalised to the distorted-wave impulse approximation for the summed 3s manifold. Calculations are indicated by the square of a Hartree—Fock orbital multiplied by a spectroscopic factor. Configuration-interaction curves (Cl) are described in the text.

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... 

The factorisation characteristic of the impulse approximation is retained, but the plane waves in (10.34) are replaced by distorted waves. The approximation is calculated by substituting <5(ri — r2)/rj in (10.32) for the multipole of U3 (see equn. (3.102)). The resulting short-range onedimensional radial integrals are much simpler to compute than (10.32). 

The validity of the impulse approximation can be tested by factorising the distorted-wave Born approximation in the same way. The differential cross section in the factorised distorted-wave Born approximation, obtained by replacing the two-electron T-matrix element in (10.42) by the potential matrix element (10.36), is compared with that of the full distorted-wave Born approximation in fig. 10.4 for the 2p orbital of neon in coplanar-asymmetric kinematics for =400 eV, s=50 eV. In this case the Bethe-ridge condition is Of = 20°, and p is less than 2 a.u. for 6s between 0° and 120° with this value of 6f. The impulse approximation is verified in Bethe-ridge kinematics for p less than 2 a.u. 

The complete understanding of the reaction is summarised by fig. 11.11, in which the 1000 eV 5p and 5s manifold sums (normalised at the 5p peak) are compared with the distorted- and plane-wave impulse approximations... 

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.

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