Please send comments or questions to John LeRose: lerose@jlab.org

Update: December 13, 2004:

Self-consistent forward and reverse functions are now available for the MAD at 12° with quadrupoles OFF. Click here→12° No Quads Transfer Functions

UPDATE: December 7, 2004:  

Functions, forward and reverse are now available for MAD at 12°, with the quads ON. Click here →12° Transfer Functions.  Forward and reverse functions and instructions on their use are all in the same file. Endplanes are similar to, but not exactly the same as those described below. The functions provided are of the self-consistent type as described below, i.e. 4th order forward with reverse functions based on trajectory parameters described by the 4th order forward functions.

 

Transfer functions for the MAD spectrometer in the 35° minimum central angle configuration

Mad35b configuration (~35° bend angle and 35° minimum central angle)

August 5, 2002

Due to the very large acceptance of the MAD, polynomials characterizing the optical properties as an expansion around a single "central ray" must go to very high orders to accurately reproduce what actually happens with real trajectories. Using very high orders for the forward going transfer functions leads to a need for even higher order reverse transfer functions if the reverse functions are required to be consistent with the forward ones. With limited computing capability this becomes problematic when trying to build a working "model" of the spectrometer that provides both forward and reverse functions that are consistent with each other at a level of precision better than the anticipated level of the performance for the spectrometer itself. In reality this is an artificial problem. By using multiple reference trajectories (more than one "central ray") the need for very high order transfer functions can be eliminated. This is what will almost certainly be done with the physically realized spectrometer. However, doing that at this time would be cumbersome and time consuming. Imagine modeling five spectrometers at the same time! For the present purposes, exploring in a broad general sense, spectrometer performance as it is driven by detector performance, effects of windows, and general overall acceptance, a simpler solution presents itself. Using the single "central ray" approach, and limiting the forward going transfer functions to 4th order, then determining the reverse functions from trajectories calculated, not by tracing rays through the magnetic fields in the spectrometer, but rather from trajectories calculated using those 4th order transfer functions, eliminates the need for very high orders in the reverse functions. Since the effects of finite detector resolution and multiple scattering are generally dominated by lower order terms this seems a reasonable approach. The alternative at this point is to have the uncertainties resulting from inconsistencies between the forward and reverse functions overshadowing the real effects that one wants to study. For completeness sake transfer functions of 2 types are provided. First, for the purists, simple brute force transfer functions using the highest order polynomials possible in both forward and reverse directions with polynomials determined by fitting to a random set of trajectories spanning the acceptance are provided. Second, for those of a more practical orientation, substitute functions for the forward and reverse functions between the target and the 1st VDC using the above strategy are also provided. These second ones will give better consistency between forward and reverse with some, arguably inconsequential, fine details of the transfer properties glossed over.

For going from endplane 0 (the target) to endplane 11 (the 1st VDC) (endplanes are explained below) there are two sets of forward going functions. The 1st set, found in: http://hallaweb.jlab.org/news/minutes/mad/mad35b_funcs.f

are the result of fitting polynomials to traced trajectories up to 9th order in the trajectory parameters at the target. This gives the best possible fit to the traced trajectories. The second set, found at http://hallaweb.jlab.org/news/minutes/mad/madf_4th_fwd.f, are limited to only 4th order in the trajectory parameters. These are simpler but do almost as well as the 1st ones.

There are also two sets of reverse functions which take you from endplane 11 back to endplane 0. They can be found at: http://hallaweb.jlab.org/news/minutes/mad/mad35b_rev_funcs.f and http://hallaweb.jlab.org/news/minutes/mad/madf_4th_rev.f. The 1st set, like the 1st set of forward functions, are the result of fitting up to 9th order polynomials directly to traced trajectories. The 2nd set were determined by fitting to trajectories that were generated using the 2nd set (4th order) of forward functions. That is the forward functions were taken to be correct.

Instructions on the use of these functions can be found in the comments at the top of the file. The spectrometer configuration and the endplane specifications are illustrated in the figure below:

Endplanes (ep's) referred to in the code instructions appear as blue lines in the figure and are labeled. The coordinate system in each of the magnetic regions (boxes in the language of SNAKE) is shown in red (At least I think it's red!). x points generally to the right and SNAKE-y, everybody else's z, points upward. There are 4 regions [numbers in parenthesis are the location and orientation of the origin of the coordinate systems (x mm, snake-y mm, rotation around snake-z degrees)]:

  1. The initial drift: coordinate system centered at the target (0., 0., 0.)
  2. The 1st magnet: coordinate system centered at the center of magnet 1 (-75.00, 3522.39, 5.)
  3. The 2nd magnet: coordinate system centered at the center of magnet 2 (-1050.0, 7707.22, 21.)
  4. Exit drift: coordinate system centered near the center of magnet 2 and rotated (-1550.0, 7707.22, 33.5)

The black line going down the center is actually a bundle of trajectories closely spaced around the central trajectory.

 

 

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