Analysis of Scattering and Excitation Amplitudes in Polarized-Electron-Atom Collisions. I. Elastic Scattering on One-Electron Atoms and the Excitation Process ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{\frac{1}{2},\frac{3}{2}}^{}{}_{}{}^{}$

Scattering amplitudes are analyzed for the scattering of polarized or unpolarized electrons on polarized or unpolarized atoms (one-electron atoms). The elastic scattering of unpolarized electrons on partially polarized atoms provides the direct and exchange cross sections when the polarization of the scattered electrons and atoms are observed, respectively. The same quantities can also be obtained from the scattering of partially polarized electrons on unpolarized atoms, while the difference between the phases of the direct and the exchange amplitude can be derived from the spin analysis with polarized electrons and polarized atoms. When the degree of polarization of the colliding particles is known, the intensity of the scattered collision particles determines the interference term of the elastic cross section. The excitation process ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}P_{\frac{1}{2},\frac{3}{2}}^{}{}_{}{}^{}$, carried out with any combination of polarized and unpolarized electrons and atoms, has been analyzed in terms of differential and integral excitation amplitudes distinguished in terms of direct, exchange, and interference processes and also in terms related to the excitation of the magnetic substates $m_l=0\mathrm{} or \pm 1$. The analysis of the polarization of the inelastically scattered electrons gives differential excitation amplitudes for the direct, exchange, and interference terms of the excitation cross section without distinguishing between the magnetic substates. Based upon the observations of the intensity and the circular polarization of the emitted line radiation, the analysis of the ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2},\frac{3}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ deexcitation process provides integral excitation amplitudes. Observation of the total intensity of the ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2},\frac{3}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ transition in an excitation process with polarized electrons and polarized atoms enables one to determine interference terms of the excitation cross section. Equal polarization for the exciting electrons and target atoms allows us to separate the interference amplitudes for the magnetic substates ($|F_1-G_1|$ and $|F_0-G_0|$). Analysis of the circular polarization of the ${}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ transition excited with unpolarized electrons and partially polarized atoms results in partial cross sections describing excitation processes in which the quantum number $m_j$ of the excited state differs from that of the ground state at most in sign. Excitation with polarized electrons and unpolarized atoms should also result in the emission of circular polarized light of the $P\to S$ transitions. In this context Farago's proposal for measuring electron-spin polarization by circularly polarized line excitation is discussed as it applies to alkali atoms. Finally, it is pointed out how sets of integral excitation amplitudes determined by collisions, as discussed above, can be used for comparing single-excitation amplitudes with theory.