Dear Users:
1. The next order term of the transformation at the linear fringe
of a bend is added. The new part is shown by ^^^ below. This may affect
the dynamic aperture with short bends or wigglers. Strangely, this term
is inversely proportional to the length of the fringe, F1.
Length of the slope of the field at the edge as:
By(s) | *******
| *
| *
|*
*
*|
* |
* |
----*******---+--------- s
| |
|<----->|
| F1 |
Only the effects up to y^4 in Hamiltonian are taken into account.
^^^^^^^^^^^^^^^^^^^^
More rigorous definition is
F1 = 6 Integrate[By(s)/B0 - (By(s)/B0)^2, {s, -Inf, Inf}] ,
where integration is done over one fringe.
The transformation of the linear fringe of the entrance of a bend is
exp(:V:), V = -f^2/rhob px/p/24 - f/rhob^2 y^2/p/12
+ 1/rhob^2/f y^4/p/6 ,
^^^^^^^^^^^^^^^^^^^^^
where where f is the length of fringe given by F1, and rhob bending radius
at the design momentum. At the exit, the sign of rhob is changed.
This linear fringe also changes the path length in the body of the bend as
l'=l-(phi0 f)^2/l/24 Sin[phi0(1-E1-E2)/2]/Sin[phi0/2]
to maintain the geometric position of the design orbit, i.e., you have to
increase the bend field a little bit to keep the orbit unchange. Unlike a
quadrupole, the effect of linear fringe is always applied at both the
entrance and the exit, otherwise you cannot obtain a circular design orbit.
2. The Fit function now tries to use an analytic expression for the derivatives
of the fitting function by the fit parameters, if possible. Numerical differentiation was used before. Though the speed of fitting did not
improved by the analytic expressions, the worry about the size of the step for the
differentiation is gone.