Physics:Trajectory of a charged particle

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The trajectory of a charged particle can be calculated from the equations of motion. Neglecting radiative corrections (see Jackson75) and in the absence of an electric field, the equation has the simple geometrical form:

Hepb img613.gif

For details and units Hepb img54.gif Equations of Motion.

The integral of this second-order differential equation depends on six initial values. Assuming an s0 on a reference surface (e.g. a plane z = const.), the trajectory is determined by five parameters, e.g. x, y, Hepb img614.gif , Hepb img615.gif and Hepb img616.gif in the reference plane.

For constant Hepb img270.gif the solution of the equations is a helix. Choosing Hepb img617.gif , one obtains

Hepb img618.gif

Hepb img619.gif , Hepb img620.gif define the position of the axis of the helix (its ``centre), s is the projected path length, r is the radius of the projection of the helix. We have

Hepb img621.gif

In the ``bubble chamber convention the dip angle Hepb img157.gif is defined by Hepb img622.gif , hence

Hepb img623.gif

Other approximate explicit solutions for the trajectories of particles can be obtained using field symmetries allowing a simple expansion of the magnetic field, e.g. in accelerator theory. In other cases, an approximate expansion of the deviations of the field from an average value can give sufficiently precise correction formulae (e.g. in large detectors with near-homegeneous field, or in polarized targets, see Bradamante77). Trajectories in quadrupole fields allow a particularly elegant explicit solution ( Hepb img54.gif Quadrupole Magnet).

For numerical solutions to the equations of the trajectory in a non-homogeneous field see Bock98 on numerical integration, Runge-Kutta methods, predictor-corrector methods, Numerov's method.

In many cases it is sufficient to know the intersection point of a particle trajectory with only few detector planes, without reference to the track behaviour elsewhere. The intersection coordinates can be expressed in terms of the initial track parameters

Hepb img624.gif

and one can try to parameterize the function Hepb img625.gif . For more details and references see Eichinger81.