Momentum orbital impulse

The orbital impulse momentum and the spin momentum are vectors and determine the total impulse momentum of the electron J as ... 

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... 

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. 

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.

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(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. 

When atoms have more than one valence electron, the term schemes become more complex as a coupling between the impulse and orbital momentums of the individual electrons occurs. According to Russell and Saunders (L — S) a coupling applies, where the orbital moments of all electrons have to be coupled to a total orbital momentum, as with the spin momentum. This coupling applies for elements with Z below 20, where it is accepted that the spin-orbital interactions are much lower than the spin-spin and the orbital-orbital interactions. The fact that none of the electrons in an atom can have the same set of quantum numbers is known as the Pauli rule. The total quantum number I is obtained as L = 2,1, S = Es and J = L — S,..., I + S. The term symbol accordingly becomes ... 

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