Collision state

The nine-fold integration for the gain term is over both pre-collision velocities and over the second (all but the first) post-collision velocity. Both pre-collision states are folded with their corresponding distribution function. 

With these specifications, and with the appropriate neutral particle-plasma collision terms put into the combined set of neutral and plasma equations, internal consistency within the system of equations is achieved. Overall particle, momentum and energy conservation properties in the combined model result from the symmetry properties of the transition probabilities W indices of pre-collision states may be permuted, as well as indices of postcollision states. For elastic collisions even pre- and post collision states may be exchanged in W. 

The collision state Fy) is in a one-to-one correspondence with the channel state the entrance channel for the colUsion. The physical collision state is Fo), but the index j is needed for some formal purposes when we use the spectral representation of H. 

The relationship of the mathematical constructions to the physical situation is given by the interpretation of section 3.2. The amplitude for detecting the channel state i(t)) at time t for the collision state Fo(t)) is... 

We are interested in the rate of transition from the entrance channel o(t)) to the exit channel ,(0) caused by the interaction V. We choose t = 0 to be the time at which the collision occurs and consider the mathematical representation of the dependence of the collision state Fo(t)) on o(T)), its state at time T in the distant past (on an atomic scale) when the system was so well separated that the potential V did not act. Applying the time-development operator (3.45) we have... 

We use (6.22) to obtain from (6.21) the box-normalised wave-packet collision state at t = 0. 

This is an explicit expression for the collision state in terms of the corresponding channel state. The difficulty is in the integral operator. We can turn this into numbers by introducing the spectral representation of H, but a knowledge of this requires a solution of the problem. To obtain a form that leads to a solution we use the operator identity... 

This is substituted into (6.23) to obtain the integral equation for the collision state... 

We first consider the limit L — oo, applied to R q We normalise both the channel and collision states of (6.28) in the same way by considering the part of that describes the relative motion of the projectile and the target. After taking the limit this is an eigenstate of momentum kj). 

In order to evaluate it we need an identity obtained by writing the time-dependent collision state in (6.13) as the time-development of the collision state at t = 0 and using (6.5). 

In this section we first summarise the meaning of the notation for the channel and collision states with box normalisation and in the continuum limit L —> 00. We then define notation for the limit 6 —> 0-1- and write the corresponding integral equations. 

The same meaning is given to the relative motion described by the collision state For this state we must also consider the limit... 

Introduction of the energy width e enabled us to write an integral equation (6.26) for the collision state. The limiting procedure is represented by... 

The time-reversed collision state and Green s function operator are denoted by... 

The scattering problem is formulated in terms of one-electron states, which we call orbitals to distinguish them from the A/ -electron target states and the (AZ -l-l)-electron collision and channel states of scattering theory. The space of collision states is spanned by products of N+l orbitals, which we explicitly antisymmetrise in this section. 

The antisymmetric multichannel expansion (7.3) of the collision state T S,+ (k)) has a serious difficulty. It is not unique. This is seen by considering the orbital-cofactor expansion (7.5) in the elements of the u Vk)) column. If we add A(xr a) to (xr u k)), where the orbital a) is occupied in all the determinants comprising ly), we are adding determinants with two identical columns, which are zero. Therefore u (k)) is ambiguous, at least with respect to the addition of a linear combination of occupied orbitals. 

So far the method has only been fully tested for one-electron atoms. In the case of hydrogen a complete check is available for a very restricted subset of angular-momentum states, namely LS-coupled collision states with / = X = 0. This is the Temkin—Poet problem (Temkin, 1962 Poet, 1978, 1981). The three-body potentials are separable in the radial coordinates. This enables a convergent numerical solution to be obtained. 

The distorted-wave integral equation for the full collision state, corresponding to (6.81), is... 

With the appropriate definition of the coordinate—spin variables Xj, and using the spin wave functions (3.79), the asymptotic form of the coordinate—spin representation of the collision state T ( (ky, kj)) is... 

The weak-coupling approximation for the collision state in (10.13) involves neglecting the possibility of exciting the target, except perhaps by including excitations through an optical potential. The approximation is... 

The simplest way of including the full interaction of the two final-state electrons is to use the impulse approximation. In its simplest plane-wave form this approximation is obtained from (10.14) by neglecting v and vi in the definition of the collision state T ( (k/,kj)). It retains the two-electron function (/> (k, r). In the spirit of this approximation it replaces x + (ko)) with a plane wave. We expect the plane-wave impulse approximation to describe kinematic regions where the two-electron collision dominates the reaction mechanism such as the higher-energy billiard-ball range. 

The stationary collision-state is an eigenstate (P which satisfies the boundary conditions corresponding to the experiment. Let us define these conditions. 

The stationary collision state is a state V )) which is associated with 4> with the same energy, and which is a solution of the equation... 

H. Nakamura, Semiclassical approach to charge-transfer processes in ion-molecule collisions, State-Selected and State-to-State Ion-Molecule Reaction Dynamics. Part 2 Theory, Advances in Chemical Physics LXXXII (M. Baer and C. Y. Ng, eds.), Wiley, New York, 1992, p. 243. 

Ho is assumed to be diagonal in the basis of the collision states. Ea) is the collision (or scattering) state of energy E belonging to the channel a. In the following the indexes a,b,c--- will label the channels. The collision states are normalized in energy... 

Figure 2. Collision state between drill pipe and hole wall. 

Collision state theory is useful for gas-phase reactions of simple atoms and molecules, but it cannot adequately predict reaction rates for more complex molecules or molecules in solution. Another approach, called transition-state theory (or activated-complex theory), was developed by Henry Eyring and others in the 1930s. Because it is applicable to a wide range of reactions, transition-state theory has become the major theoretical tool in the prediction of chemical kinetics. 

After the pulse, the presence of a dynamical hole means that the wavepacket in the E+state differs from the initial stationary collision state, with well defined collision energy it has now significant projection on bound levels (pgy and on continuum... 

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