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The magnitude of the drift velocity vector is taken from the gas tables. Mainly for historic reasons, the tables list the velocity for electrons and the velocity divided by the field (usually called mobility) for ions, a difference that is of a merely formal nature.
The magnitude of the drift velocity of electrons and the mobility of ions can be entered with the TABLE and ADD gas section commands. Electron velocities can in addition be computed with MAGBOLTZ and MIX.
In the presence of a magnetic field, both electrons and ions experience a force which is perpendicular to both the E field and the local velocity. In vacuum, particles will follow a spiraling trajectory. In chamber gases however, both electrons and ions undergo frequent scattering as a result of which they move in a straight line.
This motion can be described in 3 ways in Garfield:
Q mu / 2 \ v = ----------- | E + mu ExB + mu B (E.B) | 2 \ / 1 + (mu B)where mu stands for the mobility and Q for the charge. The mobility is computed by dividing the velocity entered via DRIFT-VELOCITY by the electric field E.
Since the Lorentz angle for ions can currently not be entered in the gas tables, the latter formula is always used for ions.
The method iterates over the following steps:
accuracy dt' = dt sqrt ----------- |z_2 - z_3|
The iteration ends when any of the following conditions is satisfied:
The strength of this algorithm is that it takes very long steps in areas where the field is nearly constant and small steps in areas with a varying field. This saves computation time and improves the overall accuracy of the calculation.
The method is well adapted to fields that are smooth, such as analytic potentials. Field maps in contrast are sometimes not even continuous. This algorithm has difficulties coping with such rough maps - the Monte Carlo method is then a better choice.
1/2 m v^2 = t Ethen drawing an exponentially distributed time interval with the collision time as mean and multiplying this with a user specified number of collisions to be skipped.
The method iterates over the following steps:
The iteration ends when any of the following conditions is satisfied:
The most striking difference between Monte Carlo and Runge Kutta integration is that the latter will, for a given starting point, always compute the same path while Monte Carlo integration will lead to different trajectories (provided lateral diffusion is appreciable).
Monte Carlo integration is intrinsically robust and is therefore well suited when integrating non-smooth fields such as those encountered in field maps.
The Monte Carlo method is also to be preferred for very small scale detectors at the scale of the diffusion.
However, care must be taken to adjust the step size. If the steps are too large, the method is inaccurate. If they are too small, the maximum number of steps will be reached before the particle has reached an electrode.
The transverse component of diffusion, usually the largest of the two, will make a particle follow a trajectory that differs from the mean. Longitudinal diffusion will only have an effect on the drift time.
If diffusion does not cause the particles starting at one and the same point to reach different electrodes, the dominant effect of diffusion is a fluctuation in arrival time. One should note that this fluctuation is not only due to the longitudinal part of diffusion - also the transverse part plays a role.
Garfield offers several methods to estimate the effect of diffusion:
Additional information on:
integral alpha dz M = e
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 eps of the trapezoidal estimate without bisection of the integral over the entire path.
The integral of the Townsend coefficient over a drift path computed with the Monte_Carlo technique is larger than the integral over a drift path from the same point computed with the Runge_Kutta_Fehlberg method. The reason for this difference is that the path length for Monte Carlo integration is larger than for RKF integration.
Magboltz computes Townsend coefficients which are suitable for use with an Runge-Kutta path. The PROJECTED-PATH-INTEGRATION integration parameter can in certain contexts be used to reduce the step size dependence of the integral over Monte Carlo paths.
Attachment, even at a rate which is orders of magnitude smaller than the gain, profoundly modifies the avalanche fluctuations. Usually, the effect of combined gain and attachment is a loss of energy resolution. The reason for this effect is that the attachment rate, in many gas mixtures, exceeds the gain at the onset of the avalanche. Thus, there is a non-negligible probability that an electron is lost before it can start an avalanche while attachment after the avalanche has reached a certain size has virtually no effect anymore. Fluctuations of the avalanche size can be estimated by using the AVALANCHE and RND_MULTIPLICATION procedures.
- integral eta dz L = e
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 eps of the trapezoidal estimate without bisection of the integral over the entire path.
The integral of the attachment coefficient over a drift path computed with the Monte_Carlo technique is larger than the integral over a drift path from the same point computed with the Runge_Kutta_Fehlberg method. The reason for this difference is that the path length for Monte Carlo integration is larger than for RKF integration.
Magboltz computes attachment coefficients for use with an RKF path. The PROJECTED-PATH-INTEGRATION integration parameter can in certain contexts be used to reduce the step size dependence of the integral over Monte Carlo paths.
When the integration routines are called through commands, the status is, on request, printed as part of the output. If the routines are called via procedures, then the status is usually one of the output arguments.
The status is usually shown or returned as a string, shown in the second column of the table below. Occasionally however, a numeric status code is returned, shown in the third column.
The DRIFT_INFORMATION procedure returns on request the status of the last drift line that has been computed in either of these 2 formats.
Drift line ended because ... | Status string | Numeric status code |
---|---|---|
Particle left the drift area | Left the drift area |
-1 |
More than MXLIST steps | Too many steps |
-2 |
Various faults | Calculations abandoned |
-3 |
Hit an equipotential plane | Hit a plane |
-4 |
Entered a dielectricum | Left drift medium |
-5 |
Left the finite element mesh | Left the mesh |
-6 |
Hit the plane on the left | Hit the minimum x plane |
-11 |
Hit the plane on the right | Hit the maximum x plane |
-12 |
Hit the plane on the bottom | Hit the minimum y plane |
-13 |
Hit the plane on the top | Hit the maximum y plane |
-14 |
Hit the tube wall | Hit the tube |
-15 |
Hit a wire of type X | Hit X wire N |
1 ... MXWIRE |
Hit a replica of an X-wire | Hit a replica of X wire N |
MXWIRE+1 ... 2*MXWIRE |
Ended in a solid of type X | Hit X solid N |
2*MXWIRE+1 ... 2*MXWIRE+MXSOLI |
Other cases (shouldn't occur) | Unknown |
other |
Notes:
Formatted on 0107-04- at 21:33.