Next: C Progam for calculating Up: Mathieu's Equation, solution, and Previous: Hill's Method solution

## Sträng's recursion formula for

First we note that by the symmetry of , . Following Sträng, we define

 (2.10)

We have We can decompose in terms of ,

 (2.11)

A Laplace decomposition yields

 (2.12)

Here represents with its left most column chopped off. Again, following Sträng we define as the matrix A with its rightmost column removed, the matrix A with its lowest row removed, the matrix A with its upper most row removed. Ultimately, , and given the symmetry involved, detdet . Following this procedure we find

 det (2.13)

We also note, similarly using Lapalcian decomposition,

detdet

so that

det

and

Plugging this into Equation 2.13,

Define and and find,

 (2.14)

We can recursively solve for to as much accuracy as nessessary, though the program presented below found convergence to a fair tolerance quite quickly. We first must seed'' the recurssion with the first three . This can be done by hand, though we have deferred to the kindess of our computer algebraic program Maple instead.

Maple finds,

with(linalg):
C:=matrix([[1,e6,0,0,0,0,0],[e4,1,e4,0,0,0,0],[0,e2,1,e2,0,0,0],
[0,0,e0,1,e0,0,0],[0,0,0,e2,1,e2,0],[0,0,0,0,e4,1,e4],[0,0,0,0,0,e6,1]]):
dc:=det(C);
dc := -2*e2^2*e0*e4^2*e6+e2^2*e4^2-2*e4^2*e2*e0*e6^2+2*e2*e4^2*e6
+e4^2*e6^2+2*e2^2*e0*e4+4*e2*e0*e6*e4-2*e2*e4-2*e6*e4-2*e2*e0+1

A:= matrix([[1,e4,0,0,0],[e2,1,e2,0,0],[0,e0,1,e0,0],
[0,0,e2,1,e2],[0,0,0,e4,1]]):
da:=det(A);
da := 1-2*e2*e4-2*e2*e0+2*e2^2*e0*e4+e2^2*e4^2

B:=matrix([[1,e2,0],[e0,1,e0],[0,e2,1]]):
db:=det(B);
db := 1-2*e2*e0


Our program seeks to find all stable values of , i.e. those that satisfy Equation 2.9 as real values (i.e. all iso- for which is exclusively imaginary.

Our code finds all such iso- by looping through the a and q axis. If our formula returns nan'' which is the C language's way of saying not a real number, then we set the value of to zero, though of course it is only the imaginary part of which is actually zero. We perform a contour plot on our data output and find the elegant avian like image of the stability region of Mathieu's equation (Figure 2.1).

For the quadropole field, the rf linear Paul trap, we have the following stability regime (Figure 2.2). The original stability diagram is simply reflected about the x-axis as between the two.

For the chamber rf Paul trap, we recall that for the z direction we must allow for (Equation 1.3), and so the lowest region of stability (and the largest region at that) has a slightly different look (Figure 2.4).

Next: C Progam for calculating Up: Mathieu's Equation, solution, and Previous: Hill's Method solution
tim jones 2008-07-07