The deflection function expresses the deflection angle of a scattered particle as a function of the collision parameter. In the study of differential cross sections, the deflection function can be used for three purposes: The value of the classical differential cross section is proportional to the derivative of the deflection curve at the relevant angle, while the semiclassical interference structure needs the calculation of phase shifts by integrating over the deflection cube. Moreover, the deflection curve is a handy visual link at the adjustment of the potential parameters to fit the measured and calculated differential cross sections.
Consider two colliding particles interacting via a
spherically symmetric potential function V(r) and suppose that the
relative motion of the particles can be described by classical
mechanics. Fig. MESO-m3c-defl-F1 shows the scattering trajectory in the centre of mass system near the scattering centre M in the case of a
central-force field, consisting of attractive forces at large
distances and short-range repulsive forces. The particles approach
each other with an impact parameter b and a relative kinetic energy
at infinity E.
|Figure MESO-m3c-defl-F1: Scattering of a particle by a symmetric central-force field with centre at M.|
This leads to the classical deflection-angle formula:
We consider the deflection functions for ion-pair formation (in a centre of mass system), which can be described via the crossing of ionic and covalent potential surfaces ).
collision process needs a summation over the different contributions
to the total scattering angle. Incoming on the covalent curve and
outgoing on the ionic one, the scattering angle is given by:
Experimentally, only the absolute value of the classical deflection function is meaningful, because the deflection angles and resulting from repulsive resp. attractive scattering on different sides of the target cannot be distinguished.
Fig. MESO-m3c-defl-F2 shows the typical shape of the deflection function for ion-pair formation.
|Figure MESO-m3c-defl-F2: Typical deflection curves for chemi-ionization scattering (CM system). The two curves due to ionic and covalent scattering are connected . Because of the several interference features, the ionic curve is split up into b, c and e branches, the covalent curve into a and d branches.|
The deflection function shows several peculiar features. Because there are two trajectories leading to ionization, the deflection functions consist of two parts connected at , resulting in a closed deflection function. Of course there is no large-b deflection curve because the classical turning point has to be smaller than or equal to Rc to have the possibility of ionization. In the case in which the classical turning point is equal to Rc, both trajectories merge leading to the connection point of the two branches of the deflection function.
Due to this composite character of the deflection function there are
up to four impact parameters leading to scattering over the same
angle for small-angle scattering. Even at large-angle
scattering there are always two impact parameters for one angle. By
simple addition of the contributions of both trajectories, the classical
chemi-ionization differential cross section is given by