Beam Field

    One of the basic assumptions of CAIN is that the most particles in the beams have high energy and are almost either right- or left-going. This assumption leads to the following facts:

• A field due to a particle is almost concentrated in a transverse plane with the same s-coordinate of the particle because of the Lorentz contraction.
• If the electric field is , the magnetic field is given by , where is the unit vector along the s-axis, c the velocity of light, and the upper (lower) sign is for the field created by the right(left)-goin beam.

In contrast to ABEL, CAIN does not assume that all the particles have the above property. Some particles may have low energies and large angles with respect to the s-axis. CAIN will work, unless the sum of their weight becomes a significant fraction of the beam. The equation of motion under the Lorentz force is integrated with possible low energies and large angles taken into account.

The calculation of the beam electric field is done in the following way. First, cut the right(left)-going particles into longitudinal slices (the width is defined by the parameter Smesh).    Within each slice the following Poisson equation  is solved.
 
where m is the electron mass in units of eV/c, the classical electron radius in meters, is the charge (divided by the elementary charge) per unit transverse area, then is given in units of V/m.

For each slice and for each of right- and left going beams, a region is selected, where is the center-of-mass and is the width determined by the input parameters. The field created by the particles outside this region is ignored. Let us name this region [O].

  In the region [O], the Poisson equation  is solved using the FFT. Eq.(42) can formally be solved as
 
Divide this region by grid. Within each cell (i,j) (, ), the the density is approximated by , where , , and is the total charge in the cell:

where () is the cell center coordinate. Then, eq.(43) becomes a sum over the cells.
 
The kernel matrix has to be calculated by taking average over the source cell:

This averaging is important when is far from unity. The convolution in eq.(45) can be done efficiently by using FFT.

However, if we apply FFT for the finite region instead of the infinite region in eq.(43), we would be assuming a periodic charge distribution, i.e., the charge distribution in is infinitely repeated. To avoid this problem, we use the following trick. First double the region to by padding zero in the extended region and carry out FFT. This still means a periodic charge distribution as depicted in Fig.3. However, if we use the kernel matrix with zero padded   in the extended region ( if or ), the field due to the ghost charges will never reach the real charge region because their horizontal(vertical) distance is larger than (). Thus, the potential in the region is calculated correctly although incorrect in the extended region.

  
Figure 3: Doubled region for FFT. The solid frame indicates the doubled region for FFT and its left-bottom quadrant is the charge region . The region hatched by solid lines is the real charge region and that by dotted lines the ghost charge due to the periodicity of Fourier transformation.

The obtained values of the potential are those at cell centers. They are interpolated by 2-dimensional cubic spline and differentiated to get and .

When a charged particle gets out of the mesh region, the field created by it is ignored in CAIN. However, the force by the other beam is taken into account even if the particle is outside the mesh region of the other beam. To this end, CAIN  adopts three methods, namely, [A] direct Coulomb force by the charge distribution in the mesh, [B] harmonic expansion in polar coordinate, and [C] harmonic expansion in elliptic coordinate.

Let be the total width of the mesh region. If it is close to a square, or more precisely, if , the whole region is divided into three regions [O],[A],[B], as depicted in Fig.4a. If the mesh region is far from square, the whole region is divided into four, [O],[A],[B],[C], as in Fig.4b. In the region [O] the mesh is used for calculating the field. In other regions, the methods mentioned above are used.

  
Figure 4: Regions for calculating the beam field

Since the sum is time consuming, this is used only in region [A], where two other methods fail to converge. The method is trivial and given by

However, this formula is not accurate when the bin size ratio is far from unity. It is needed to take average over a bin when the bin is close to the field point (x,y). CAIN makes a table for the Coulomb force by a bin () for faster computation.

  In the region [B] the following formula is used.



Here, is arbitrary (introduced for avoiding overflow/underflow). The formula is valid for , where = is the maximum radius of the mesh region.

  When (otherwise, exchange x and y), the elliptic coordinate (u,v) defined by

is used. Here, f is chosen as

The maximum of the radial-like coordinate u in the mesh region is

which is taken at the four corners.

Then, the expansion of is



Actually, there is a finite relation between and :

The formula converges if , which corresponds to the region [C] (and [B]) in Fig.4b. The truncation of the series is defined by the operand NMOM of the command BBFIELD (common to the two types of expansions for simplicity).