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Stopping range tables

To correct for this problem, a different approach was introduced in GEANT version 3.14. The stopping range of a particle is defined as the distance that the particle will travel before stopping. By definition the stopping range for a particle of energy E0 is given by:

R = ∫E00{dxdE}&sp;dE=∫0E0-{dxdE}&sp;dE

Note that in the tables the positive quantity -dE/dx is stored. The method used was to build a table of stopping ranges based on ELOW by integrating the inverse of the dE/dx tables in GRANGI. At tracking time the algorithm was the following:

  1. Evaluate the stopping range for the threshold energy ( STOPC). This was done only once at the beginning of each new track.
  2. From the energy of the particle derive the stopping range by a linear interpolation of the range table:

    R0= GEKRT1xRi+ GEKRATxRi+1

    where Ei< E0≤Ei+1 .

  3. Evaluate the stopping range for the particle after the step: R'0= R0- step . If this is less than the stopping range of a particle with threshold energy, the particle is terminated as a stopping particle below the energy cut. Otherwise the following quantities are evaluated:

    Rj< R'0≤Rj+1&sp;GEK= {R'0-RjRj+1-Rj}&sp;GEK1= 1 - GEK

    and the final energy is computed as:

    E'0= GEK1xELOW(j)+ GEKxELOW(j+1)

  4. the energy loss is computed as:

    ΔE = E0-E'0

    This value is then corrected to take into account the energy loss fluctuations (see [PHYS332]).

This method has two main disadvantages. The first is due to the finite precision of computers. As the energy loss in a step is calculated as the difference of two numbers, it is subject to large relative errors. The effect can be particularly serious in the case of light materials, particles near the minimum ionisation point or with very short steps, where ΔE = DESTEP can even result in a negative quantity. As the relative precision of a 32-bit computer is around 10-6 , the error on the energy loss of a 100 GeV track can be as large as 100 keV.

The second problem connected with this method can be easily shown if we compute dE/dx as:

{dEdx} = {ddx}(E0-E'0) &sp;= &sp;{ddx}(E0- GEK1xELOW(j)-GEKxELOW(j+1))

= {ddx}(E0- Ej- {R'0-RjRj+1-Rj}(Ej+1-Ej) ) &sp;= &sp;-{ddx}{ΔEjΔRj}(R'0-Rj)

= -{ΔEjΔRj}{dR'0dx}&sp;= &sp;{ΔEjΔRj}{stepdx}&sp;= &sp;{ΔEjΔRj}

As we can see, the reconstructed dE/dx curve due to continuous energy loss is a step function and constant in each energy bin. Thus, although the results obtained with GEANT 3.14 were very satisfactory, this was felt to be an undesirable feature.


next up previous index
Next: Energy loss in Up: Energy loss Previous: Energy loss tables

Janne Saarela
Mon Apr 3 12:46:29 METDST 1995