Peak Arrival Time

Consider now an experiment such as that of England and Elford [1987], where the time at which the peak collector current occurs is taken as the drift time. A brief summary of the results to be presented in the rest of this chapter has been reported in Standish and Kumar [1987]. We will assume that the swarm is described by the   free space solution at large times ():

In the following analysis, we will assume a delta function initial pulse , and that the velocity   distribution of the source has cylindrical symmetry about the axis defined by the field. Choose cylindrical coordinates , and . All quantities in this system must satisfy cylindrical symmetry, i.e. depend only on and , so that

and

Note that when the swarm is conserved. Then equation () may be written [Skullerud [1974]] :

where  

and

If the total number of charged particles is conserved, the continuity equation for the current density is:  

  Substituting the time dependent transport equation (), we get:

The total current passing through an infinite plane electrode located at d is:

where use has been made of the divergence theorem to substitute equation ().

To calculate the peak arrival time, we differentiate the above expression for the collector current and solve. Since we are interested in drift times much larger than the non-hydrodynamic   relaxation time , the transport coefficients can be replaced by their constant values, so that the required condition is

Substitute for using the transport equation, and the integral

and we find:

Expanding the exponential differential operator, collecting terms in powers of , and using Rodrigues' formula [Abramowitz and Stegun [1965], 22.11.7]

to express    it in terms of Hermite polynomials, we find:

where denotes the greatest integer less than or equal to n. If we assume that the sum over l is appropriately convergent, then is a power series in the inverse drift length. In particular, we expect that the measured drift velocity found by   dividing the drift length by the drift time is given by the true drift velocity plus corrections that decay as a power series in . So we assume that

This power series can be inverted to obtain a power series for [Knopp [1951], 4.3(20)] :

Substituting this into , one obtains:

The leading power of the Hermite polynomials at large drift   distances will be given by the constant or linear term of the polynomial, for even and odd polynomials respectively. So

Substituting this into equation (), we get

The leading power of d will be the one that maximizes . As k increases proportionally to , this will happen for low values of l. In particular, the leading term will have contributions from l=1,2 and 4, and . Setting the leading term to zero yields a linear equation for the unknown , to which the solution is:

The three terms in this expression can be identified. The first is just the non-hydrodynamic term discussed in the previous   section. The second is the effect of diffusion over a plane    electrode [Huxley and Crompton [1974], §(5.8)]

and the third is the contribution from higher order transport effects.

Similarly, the next term in the series () can be calculated. This was done using a computer algebra package (see Appendix ), the result being