Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical Techniques

A.R. Filippelli; Chun C. Lin; L.W. Anderson; J.W. McConkey    Thermophysics Division, National Institute of Standards and Technology, Gaithersburg, Maryland
Department of Physics, University of Wisconsin, Madison
Department of Physics, University of Windsor, Ontario

I Introduction

In a collision between an electron and an atom or an electron and a molecule, the internal energy (rotational, vibrational, and electronic) of the atom or molecule as well as the linear momentum of the colliding objects, may be altered. Two routes are open for the experimental study of such electron impact excitation of atoms and molecules. One can measure the flux and angular distribution of those scattered electrons that have lost an amount of energy corresponding to some particular internal excitation of the target atom or molecule. This energy loss method is used to study vibrational excitations of molecules and sometimes electronic state excitation in atoms and molecules. It is discussed more fully in this volume by Trajmar and McConkey. If, however, one is interested primarily in the excitation of the target rather than the angular distribution of the scattered electrons, one uses the other, complementary, approach and measures the density or number of atoms or molecules produced in the excited state of interest by the collision. For those cases in which the excited state can radiatively decay by means of short-lived (dipole-allowed) transitions, the optical method may be used to study the collision process and to determine the corresponding collision cross section. The optical method is so named because it relates the intensity of the decay radiation to the cross section for production of the excited state by the collision process.

II Principles of the Optical Method

In the following development, the target is assumed to be a free atom in its ground state, and the incident particle is assumed to be an electron. However, the method can also be applied to study excitation of electronic states in molecules and, in principle, to situations in which the target atoms or molecules are initially in an excited state (Section VIII of this chapter), or where the incident particles are not electrons, e.g., protons or hydrogen atoms (Allen, Anderson, and Lin, 1988).

A ASSUMPTIONS AND APPROXIMATIONS

To study the production of some excited atomic state j by electron collision with ground-state atoms, one sends a collimated, monoenergetic electron beam through the target gas, and measures the steady-state intensity and polarization of the resulting radiation emitted by the state-j atoms as they decay to states of lower energy. For this method to be valid, three basic conditions must apply.

(1) The electron beam current density J and the target atom number density no must be small enough that collisions of excited atoms with electrons or with other atoms occur at a negligible rate. This condition is verified by demonstrating that the emission intensity is proportional to beam current and to target gas density.

(2) The natural radiative lifetime τk of higher lying states which can decay into state j, as well as the natural radiative lifetime τj of the state-j atoms themselves, is assumed to be short with respect to the time required for an atom to travel radially outward from the electron beam region to the walls of the apparatus. This requirement means that the emitting region coincides with, or is only slightly larger than, the electron beam, and it simplifies the task of determining what fraction of the total emission per unit beam length is measured. Experimentally, it is desirable to have 〈v〉τ less than a few millimeters, where 〈v〉 is the average speed of the target atoms.

(3) For the case in which the radiative decay channels of the excited state j include an allowed (dipole) transition to the ground state g, there may be a nonnegligible probability for absorption of the j → g photon by the target gas, since virtually 100% of the atoms are in the ground state. Repeated absorption and re-emission of the radiation can lead to a diffuse emitting region much larger in extent than the electron beam. This process is called radiation trapping. Also, each time the j → g photon is absorbed, there is some probability that the j-state atom will next branch to some intermediate lower level i. In this way, the j ↔ g absorption/re-emission process can alter the intensity and spatial extent of the j → i emission intensity, as well as that of the j → g radiation. The j ↔ g absorption/re-emission process can also alter the polarization of the emission from the state j. In the development that follows, we assume a gas density low enough that the absorption/re-emission process has a negligible effect on the intensities and their spatial distribution. In some electron excitation experiments, however, this condition is not met and care must be taken to apply the appropriate corrections.

B THE DIRECT CROSS SECTION

Under the assumptions above, the only significant processes that can produce atoms in excited state j are direct electron-impact excitation from the ground state into state j, and spontaneous radiative decay of higherlying states k into state j (cascade). The population of the higher states k are themselves also the result of some combination of direct electron impact excitation and cascade from still higher levels. The only significant loss mechanism for state-j atoms is radiative decay to lower lying states i. After the electron beam is turned on, the spatial density of the state j atoms will build up in a matter of microseconds or less to some steady-state distribution, corresponding to a balance between the production (direct excitation and cascade), and loss (radiative decay) mechanisms. In this steady state, the time rate of change of the volume number density nj(r) of state-j atoms at position r, will be zero; i.e.,

ge∫JrEQjdEdE+∑knkτkΓkj=njτj

where Qjd is the cross section for direct excitation of state j by electron collision with a ground state atom, E is the incident electron energy as seen in the rest frame of the target atom, ng is the steady-state volume number density of ground state atoms, e is the absolute value of the electron charge, J(r, E) is the distribution function over electron energies of the electron beam current density, and k is a label referring to higher lying states that can radiatively decay into state j with optical branching ratio Γkj. The electron beam is assumed to be sufficiently well collimated, and is typically 1–2 mm in diameter, but the magnitude of the current density may vary with position over a plane perpendicular to the direction of the electron beam. Integrating Eq. (1) over a surface S perpendicular to the electron beam, and assuming that ng does not vary with position over this surface, yields

ge∫∫JrEQjdEdEdS+∑kNkτk−1Γkj=Njτj

where Nk and Nj are the steady-state number of state-k and state-j atoms per unit beam length. To further simplify Eq. (2), it is assumed that the position and energy dependence of the electron current density J(r, E) is separable into a product j(r) f(E), where ʃf(E)dE = 1 and the electron beam current I = ʃ j(r)dS. This is justified since the electron energy distribution is determined primarily by the electron source. The electron optics of the experimental apparatus may in fact introduce some position dependence to the electron energy distribution in the observed collision region. However, in the present development, it is assumed that this position dependence is negligible. Then, the first term on the lefthand side of Eq. (2) becomes (ng/e)I〈Qjd〉, where

jd=∫QjdEfEdE

The energy distribution function f(E) for the beam electrons will be significantly different from zero only over an energy range that is typically less than...