Programmable and tunable cylindrical deflector analyzers

A method of determining the electrostatic field distribution in an electrostatic analyzer, comprising:

• providing an electrostatic analyzer comprising:
• a cylindrical body comprising an inner cylinder and an outer cylinder that are coaxial with one another along a longitudinal axis of the cylindrical body;
• an inner cylindrical electrode positioned on an exterior face of the inner cylinder;
• an outer cylindrical electrode positioned on an interior face of the outer cylinder;
• a first azimuthal electrode positioned on a face of a first azimuthal plane that passes through the longitudinal axis;
• a second azimuthal electrode positioned on a face of a second azimuthal plane that passes through the longitudinal axis;
• a first end electrode positioned on a first end face of the cylindrical body;
• and
• a second end electrode positioned on a second end face of the cylindrical body;
• grouping the boundary conditions of each of the electrodes into six simplified problems;
• defining the electric potential in the electrostatic analyzer by applying the Dirichlet's problem to the six simplified problems;
• expressing the electric potential as a product of three potential functions in terms of Laplace's equation for cylindrical coordinates;
• transferring a function to another side of the equation and separating the variables;
• using the Bessel equation to build a system of equations;
• and
• determining a general solution to the system of equations.

An electrostatic analyzing system comprising:

• an electrostatic analyzer comprising:
• a cylindrical body comprising an inner cylinder and an outer cylinder that are coaxial with one another along a longitudinal axis of the cylindrical body, the inner cylinder and the outer cylinder are spaced apart to form a gap therebetween;
• an entry slit positioned at a first end of the gap;
• an exit slit positioned at a second end of the gap;
• a first azimuthal electrode positioned on a face of a first azimuthal plane that passes through the longitudinal axis;
• a second azimuthal electrode positioned on a face of a second azimuthal plane that passes through the longitudinal axis;
• a first end electrode positioned on a first end face of the cylindrical body;
• and
• a second end electrode positioned on a second end face of the cylindrical body;
• wherein the gap is configured such that a particle beam can enter the gap via the entry slit and exit the gap via the exit slit;
• and
• a detector positioned adjacent the exit slit and configured to receive the particle beam.