&DRIFT: methods: diffusion

longitudinal

Integration of the longitudinal coefficient alone is normally only performed when the drift-line has been computed with the Runge_Kutta_Fehlberg integration method and when in addition the transverse diffusion coefficient is not known.

When the transport properties of a gas have been computed with MAGBOLTZ, then transverse diffusion coefficients are stored - unless they have subsequently been deleted with the RESET command of the gas section.

Such integration can explicitly be requested by calling procedures like DRIFT_ELECTRON after having deleted the transverse diffusion data, if needed. Since the DRIFT command by default uses Runge-Kutta-Fehlberg integration, the diffusion graph made by the DRIFT TRACK command, is computed with this technique if the transverse diffusion is absent.

The computed quantity, the spread in arrival time, is estimated by integrating quadratically the ratio of longitudinal diffusion coefficient and drift velocity over the drift path:

spread\&sup2; = \&int; (\&sigma;/speed)\&sup2; dz

The integration is performed using the Newton-Raphson technique over each step of the drift-line, repeatedly bisecting a step until either the maximum stack depth has been reached, or until the difference between the trapezoidal and the quadratic estimate over a section becomes less than a fraction \ε of the trapezoidal estimate without bisection of the integral over the entire path.

cloud

If a drift-line has been computed with the Runge_Kutta_Fehlberg integration method, and provided both the longitudinal and transverse diffusion coefficient are known, then the diffusion spread is computed by propagating a probability distribution along the drift path.

Starting from a \δ-distribution at the starting point, the cloud is adjusted at each step according to the following phenomena:

additional transverse and longitudinal diffusion at each step, according to the local diffusion coefficients;
longitudinal stretch due to acceleration and deceleration of the electrons along the central drift-line;
transverse compression due to convergence of drift-lines neighbouring to the central drift-line;
alignment of one main axis with the central direction of motion.

The cloud is considered Gaussian at every stage.

When the centre of the cloud comes closer than ncloud wire radii to a wire, the cloud is projected onto the wire using one of several methods.

Monte_Carlo

Monte_Carlo integration includes a simulation of the diffusion process.

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Formatted on 21/01/18 at 16:55.