Form factors of the hydrogen atom in excited states

An exact closed expression for the form factor (and hence the generalized oscillator strength) for bound-free transitions of the hydrogen atom in highly excited states is derived by use of the Coulomb Green's function. The calculated densities of the generalized oscillator strength of the excited hydrogen atom with an initial principal quantum number $n_0=10\mathrm{}$ are 10.9(-5.0), 13.6(-4.0), 22.7(-3.0), 55.3(-2.0), 96.2(-1.0), 42.1(0.0), and 4.63(1.0) in ${\mathrm{Ry}}^{-1\mathrm{}}$ near the ionization threshold [for $\frac{E}{I\left(n_0\right)}$ = 1.0001]. Here a figure in parentheses denotes the value of $\ln \left(n_0^2{q}^2\right)$, where $q$ is the magnitude of the momentum transfer, $E$ the excitation energy measured from the state $n_0$, and $I\left(n_0\right)$ the ionization potential. Using these results, we draw the Bethe surface, i.e., a three-dimensional plot of the density of the generalized oscillator strength as a function of $E$ and $\ln q^2$. This surface gives a quantitative understanding for the entirety of the inelastic process of this excited atom by charged particles. The validity of the binary-encounter theory is quantitatively discussed. The Born sections for excitations $n_0\to n_0+1\mathrm{}$, $n_0\to n_0+2\mathrm{}$, $n_0\to 2n_0$ transitions) and ionization are evaluated for the case of $n_0=10\mathrm{} \mathrm{and} 20$. Finally, an application of this form factor to the collisional of the highly excited atom with a molecule is briefly mentioned.