Electron momentum spectroscopy

Some atomic bound states have simple structure in the sense that a straightforward calculation obtains correct energy levels. In some cases optical oscillator strengths probe further detail. Collision theory has reached the stage where experimental observables for electron collisions involving such states can be calculated within experimental error. Observables whose calculation is sensitive to structure details constitute a probe for structure which verifies the details in more-difficult cases. 

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 reaction depends as much on the observed state /) of the residual ion as on the ground state 0) of the target. Not only the single-particle structure but electron correlations in each state are sensitively probed in different circumstances. 

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The measurement of spectral momentum densities of solids by electron momentum spectroscopy... 

Several experimental techniques such as Compton scattering, positron annihilation, angular correlation, etc., are used for measuring momentum densities. One of the most popular techniques involved in measuring momentum densities is termed as electron momentum spectroscopy (EMS) [29]. This involves directing an electron beam at the surface of the metal under study. Hence EMS techniques fall under what is classified as coincidence spectroscopy. 

Section III. Methods for obtaining momentum densities, both experimental and computational, are reviewed in Section IV. Only a sample of representative work on the electron momentum densities of atoms and molecules is summarized in Sections V and VI because the topic is now too vast for comprehensive coverage. Electron momentum densities in solids and other condensed phases are not considered at all. The literature on electron momentum spectroscopy and Dyson orbital momentum densities is not surveyed, either. Hartree atomic units are used throughout. 

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... 

C. E. Brion, Chemical applications of the (e,2e) reaction in electron momentum spectroscopy, in Correlations and Polarization in Electronic and Atomic Collisions and (e,2e) Reactions, P J. O. Turner and E. Weigold, eds. (Institute of Physics, Bristol, 1992), Vol. 122 of Inst. Phys. Conf. Series, pp. 171-179. 

E. Weigold and I. E. McCarthy, Electron Momentum Spectroscopy (Kluwer Academic, New York, 1999). 

J. Ullrich, R. Moshammer, A. Dorn, R. Domer, L.Ph.H. Schmidt, H. Schmidt-Bocking, Recoil-ion and electron momentum spectroscopy Reaction-microscopes, Rep. Prog. Phys. 66 (2003) 1463. 

S. Wolfe, Z. Shi, Isr. J. Chem. 40, 343 (2000). The SCO (O-C-S) Edward-Lemieux Effect is Controlled by p-Orbital on Oxygen. Evidence from Electron Momentum Spectroscopy, Photoelectron Spectroscopy, X-Ray Crystallography, and Density Functional Theory. 

Additional information on orbital type and composition is available from (e,2e) or electron momentum spectroscopy (Moore et al., 1982 see Appendix B) performed on Sip4 by Fantoni et al. (1986). Electron momentum distributions measured at various binding energies have been compared with those from ah initio Hartree-Fock-Roothaan SCF calculations using a double- wave function with a single Si 3of polarization... 

Fig. 4. 3. Measured momentum distributions (dots) of the outer valence band of SiF4 obtained from electron momentum spectroscopy and compared with values obtained using ab intitio Hartree-Fock-Roothaan (SCF) calculations (solid line) (after Fantoni et al., 1986 reproduced with the publisher s permission).

Bagawan, A. O., R. Muller-Fiedler, C. E. Brion, E. R. Davidson, and C. Boyle (1988). The valence orbitals of NH3 by electron momentum spectroscopy quantitative comparisons using Hartree-Fock limit and correlated wave-functions. Chem. Phys. 120, 335-57. 

While the multiconfiguration methods lead to large and accurate descriptions of atomic states, formal insight that can lead to a productive understanding of structure-related reaction problems can be obtained from first-order perturbation theory. We consider the atomic states as perturbed frozen-orbital Hartree—Fock states. It is shown in chapter 11 on electron momentum spectroscopy that the perturbation is quite small, so it is sensible to consider the first order. Here the term Hartree—Fock is used to describe the procedure for obtaining the unperturbed determi-nantal configurations pk). The orbitals may be those obtained from a Hartree—Fock calculation of the ground state. A refinement would be to use natural orbitals. 

In summary, structure calculations can obtain 1 or 2% agreement with accurate optical data. A broader perspective is given in chapter 11 by electron momentum spectroscopy. Hartree—Fock calculations agree with one-electron momentum densities within experimental error, but configuration-interaction calculations agree only qualitatively with detailed data on correlations. 

Up to now there has been no calculation of differential cross sections by a method that is generally valid. We use a formulation due to Konovalov (1993). Understanding of ionisation has advanced by an iterative process involving experiments and calculations that emphasise different aspects of the reaction. Kinematic regions have been found that are completely understood in the sense that absolute differential cross sections in detailed agreement with experiment can be calculated. These form the basis of a structure probe, electron momentum spectroscopy, that is extremely sensitive to one-electron and electron-correlation properties of the target ground state and observed states of the residual ion. It forms a test of unprecedented scope and sensitivity for structure calculations that is described in chapter 11. 

Electron momentum spectroscopy can therefore be considered in terms of (q/ 0). For the one-electron model of the target... 

The valence structure of argon provides a complete illustration of the application of electron momentum spectroscopy to correlations in the ion. The Hartree—Fock single-electron level diagram of fig. 11.1 illustrates the values of the separation energy e to be expected on the basis of the independent-electron model. The experimental situation is illustrated in fig. 11.2 by the first experiment in the field (Weigold, Hood and Teubner, 1973). The noncoplanar-symmetric differential cross section at 10° is plotted against Eq for =400 eV. There is an ion state at 15.76 eV, as predicted by Hartree—Fock, but there are at least two further states rather than the predicted one. 

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... 

The principles derived and illustrated in section 11.1 show that electron momentum spectroscopy gives information about orbitals, about orbital manifolds that are split by electron correlations in the ion, and about correlations in the target ground state. We give examples of the kind of information that is obtained. 

The dominant features of the argon ion spectrum observed in an ionisation reaction have illustrated the electron momentum spectroscopy of the 3p... 

R(q) is momentum-dependent. Both the extended-average-level and the optimal-level multiconfiguration Dirac—Fock methods were tried. The results are illustrated in fig. 11.12. The sensitivity of electron momentum spectroscopy is shown by the fact that the former calculation is completely... 

Excited target atoms can be prepared in well-defined states by optical pumping with a tunable laser. Specific magnetic substates are excited by polarised light. Momentum distributions are observed for these states by electron momentum spectroscopy. 

It has been shown that, under appropriate conditions, the momentum distribution uj py for an individtial electronic state is directly meastired by electron-momentum spectroscopy (ref. 29). Figure 3.8 compares the experimental momentum distribution for the hydrogen atom ground state with the function calculated by the Fourier transform of the hydrogen Is orbital. In general, the electron-momentum spectroscopy results serve to evaluate wavefunctions at various levels of theory for a variety of atomic (and molecular) systems. 

F. RIOUX, Electron-momentum spectroscopy and the measurement of orbitals interesting results for chemists from the American Journal of Physics. J. Chem. Educ., 76, 156 (1999) and references therein. 

Accurate investigation of the valence ionization spectra is important subject to elucidate the electronic structure of molecules. Ionization spectra of five-membered aromatic compounds have also been intensively studied. The high-resolution synchrotron radiation photoelectron spectra (SRPES) of furan and thiophene were measured and analyzed with asymmetry parameter up to about 40 eV [63,64]. The electron momentum spectroscopy (EMS) was also applied to furan up to 30-40 eV [65]. The ionization spectra of these molecules were also studied by several theoretical methods. However, there were some controversial assignments even for the outer-valence region, in particular for the peak position of Ibi(TTi) state and the inner-valence spectra have not been theoretically reproduced. 

I.E. McCarthy and E. Weigold. Electron Momentum Spectroscopy for Atoms and Molecules. Rep. Prog. Plus.. 82 (1991). 827-840. 

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