Formulation of the three-body ionisation problem

Two types of kinematic range have been most-commonly observed in kinematically-complete experiments. In coplanar asymmetric kinematics 0 = 0, / Eg, Of is fixed at a value less than about 30° and Og is varied. An important subregion is known as the Bethe ridge. Here we are close to the billiard-ball kinematics of a free two-electron collision, for which the recoil momentum p of the ion, given by 

Note that this is a conditon on ko and kf. It is not violated by varying 9g. 

In noncoplanar-symmetric kinematics Ef = Eg, Of = Og = 45° and (f) is varied. For small values of 0 this is again close to billiard-ball kinematics. Both noncoplanar-symmetric and coplanar-asymmetric ranges fix the momentum transfer K, which is conventionally defined by 

The differential cross section for ionisation is given by (6.60). To formulate the T-matrix element we partition the total Hamiltonian H into a channel Hamiltonian K and a short-range potential V and use the distorted-wave representation (6.77). The three-body model is defined as follows. 

The channel Hamiltonian K (10.7) is separable in the electron coordinates. We define the following one-electron states. 

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