Impulse approximation

In the ideal case being performed at X-ray energy transfers much higher than the characteristic energies of the scattering system, the impulse approximation [14] is applicable. In this case, the dynamical structure factor is directly connected with the electron momentum density p(p) ... 

The i (r (-function was originally introduced as a mathematical intermediate in order to attain high accuracy in calculating EMD or Compton profile J(qz), which is represented under the impulse approximation as [7, 9]... 

Application of the formalism of the impulse approximation to the double differential cross section in terms of the dielectric response (Equation 12), that is, using free-electron-like final states E = p+q 2/2m in the calculation ofU(p+q, E +7ko)... 

In Bohr s theory, only estimates of maximum and minimum impact parameters are necessary. Better computations are required for determining the transverse distribution of lost energy or the effect of secondary electrons. The minimum impact parameter according to classical mechanics is ze2/mv2 from angular momentum consideration in quantum mechanics, it is h /mv. In practice, the larger of these two is taken. Also, the impulse approximation used by Bohr for the maximum impact parameter is not an absolute rule energy transfer beyond bmax falls off exponentially (Orear et al., 1956 Mozumder, 1974). 

Within the impulse approximation, the gas-phase Compton profile Jo q) is related [185,186] to the isotropic momentum density by... 

In binary (e,2e) or electron momentum spectroscopy, an incoming electron collides with a molecule and two electrons leave the molecule. The measured differential cross section is proportional to the spherically averaged momentum density of the pertinent Dyson orbital within the plane-wave impulse approximation. A Dyson orbital v[/ t is defined by... 

This expression for the scattering amplitude, given in terms of the momentum, or velocity, distribution of the Rydberg electrons, is usually termed the impulse approximation. Examination of Eq. (11.29) shows the similarity of the integral over the momentum to the form factor of Eq. (11.8). 

Khare, V., Kouri, D.J. and Hoffman, D.K. (1981). On -preserving properties in molecular collisions. I. Quantal coupled states and classical impulsive approximation, J. Chem. Phys., 74, 2275-2286. 

The simplest way of including the full interaction of the two final-state electrons is to use the impulse approximation. In its simplest plane-wave form this approximation is obtained from (10.14) by neglecting v and vi in the definition of the collision state T ( (k/,kj)). It retains the two-electron function (/> (k, r). In the spirit of this approximation it replaces x + (ko)) with a plane wave. We expect the plane-wave impulse approximation to describe kinematic regions where the two-electron collision dominates the reaction mechanism such as the higher-energy billiard-ball range. 

The T-matrix element in the plane-wave impulse approximation is... 

Fig. 10.2. Noncoplanar-symmetric ionisation of helium at the indicated total energies E (McCarthy and Weigold, 1976). Curve, plane-wave impulse approximation.

Fig. 10.2 shows that the plane-wave impulse approximation is as good for relative helium differential cross sections at different energies as it is for hydrogen. Here p) is the Hartree—Fock orbital. For helium there is an absolute experiment by van Wingerden et al. (1979) for 0 = 0 in symmetric kinematics at different total energies. Fig. 10.3 shows that the plane-wave impulse approximation using the Ford T-matrix element is consistent with the experiment. 

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).

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. 

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... 

Scattering experiments are usually not very sensitive to structure. On the other hand the differential cross section for ionisation in a kinematic region where the plane-wave impulse approximation is valid gives a direct representation (10.31) of the structure of simple targets in the form of the momentum-space orbital of a target electron. 

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. 

The amplitude (k/kj T qko) may be considered as a probe amplitude for the momentum-space structure amplitude (q/ 0). It is the structure amplitude that is essentially probed by the reaction. Increasingly-refined forms may be used for the probe amplitude. The plane-wave impulse approximation is... 

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... 

For valence structure this range is up to about 50 eV. The momentum distribution is observed for each resolved cross-section peak corresponding to an ion eigenvalue —cf. In order to characterise the observation of the target—ion structure we choose a quantity that is as independent as possible of the probe characteristics such as total energy. In conditions where the plane-wave impulse approximation is valid we consider the reaction as a perfect probe for the energy—momentum spectral function... 

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. 

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... 

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