Deflection of a particle beam by a bending magnet

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Hilda Tucker
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1 4/7/ Lorentz Eqns. of Motion Bending of a particle beam Deflection of a particle beam b a bending magnet The Runge-Kutta method will be used on the Lorentz equations of motion to follow a charged particle beam through a B field between the poles of a bending magnet. The pole faces of the magnet are an equilateral triangle. Bending magnets have been used to separate beams of ions according to their mass in order to identif the isotopes of elements and their relative abundances. Mass spectrographs were designed to operate analogousl to prism spectrographs for light. A good design causes all of the ions entering through a slit to exit through another slit if their masses are the same. Hence the magnet must focus the beam as well as bend it. 3 Deflection of a particle beam (in red) b a bending magnet (outlined in blue). The particle coordinates will be a 4-vector with subscripts and for x and, and subscripts and 3 for v x and v. The vector Zstart will hold the starting coordinates of the particle. In the example at right the particles starts at x = -4, = with velocities v x = and v =. The meanings of the 4 components are listed to the right of Zstart. Zstart 4 x v x v The pole faces of the magnet will be an equilateral triangle. The half-angle between the faces is 3 degrees or = /6. The B field will have a magnitude of. between the pole faces and will be zero outside the pole faces. The region with the field is x < tan. The angle between one face of the triange and the vertical axis: Θ π 6 The magnetic field as a function of position: Recall that Z is x and Z is. If Z is within the magnet, B =, otherwise B =.. BZ ( ) if Z Z tan( Θ) For example: B B The field B is independent of Z and Z 3. The point (,) is outside the magnet and the point (,) is within the magnet.

2 4/7/ Lorentz Eqns. of Motion Bending of a particle beam The Lorentz equations of motion in two dimensions: We will let the charge q = and the mass m = to avoid scientific notation: q m DZ is the derivative of the 4-vector Z. The bottom terms, the accelerations, are the Lorentz force divided b the mass. This definition of the derivatives will be used b the Runge Kutta integrator. DZ( tz) Z Z 3 q m Z BZ ( ) 3 q Z BZ ( ) m dx/dt d/dt dv x /dt dv /dt Find and plot the particle trajector We will choose a trajector so that the particle enters the magnet perpendicular to the face. This means that the initial velocit vector has an angle of. The magnitude of the velocit will be.. Zstart.cos( Θ).sin( Θ) x v x v For accurac, we will want the trajector to be divided so finel that there will be man points within the magnet. If we seek accurac (in bending angle) of order -, then there should be about points. The width of the magnet is about unit and the velocit is about unit, so we will use a time step of. which will place the points in the trajector about. unit apart. These points will be so close together that a point plot will appear to be a continuous line. Let t be the total time interval: t 5.4 This value was determined b trial and error to give a nice plot. This definition for the number of integration points gives the desired accurac: Integrate the equation of motion with Runge Kutta: npoints t. The list of points that is generated b the Runge Kutta integrator is put in an answer matrix M. M rkfixed( Zstart t npointsdz)

3 4/7/ Lorentz Eqns. of Motion Bending of a particle beam 3 The answer matrix M is labelled here with the meaning of the columns. t x v x v M Plot of one trajector: For plotting, it will be helpful to define the left and right boundaries of the bending magnet x and x:..5 x( ) tan( Θ) x( ) tan( Θ) It is also useful to have the horizontal axis defined: xaxis.9 3 The plot below shows that the trajector is bent within the magnet and is straight outside of the magnet..5 M M x ( ) x( ) xaxis

4 4/7/ Lorentz Eqns. of Motion Bending of a particle beam 4 Trajectories for a divergent beam An entrance slit is useful for limiting the spread in the starting positions of the beam particles. A beam has a divergence, which is a spread in the angles of the velocit vectors of the particles. If the magnet is made focusing, then particles from the entrance slit with slightl different angles of incidence will all pass through the exit slit. If the magnet is not focusing the number of particles passing through the exit slit will be smaller and thus harder to detect. Another problem is that an isotope of a different mass can pass through the exit slit if it enters with a velocit vector different of that of the isotope with the desired mass. We will show that the magnet is focusing b finding trajectories for a range of angles of incidence. Define 5 angles of incidence k that are near to the used above: Define Zstart k as the 5 starting vectors: k 4 θ Θ k Zstart k k cos θ k sin θ k k in degrees: θ deg The matrices M k will have the 5 individual trajectories and the matrix MM will have them all. The lines below run the Runge Kutta integrator 5 times for the 5 trajectories, then combine the trajectories into one matrix MM for plotting: M rkfixed Zstart t npointsdz MM stack M M M M M k k 3 4 This plot of the 5 trajectories shows that the converge at a focus that is on the horizontal axis. Analsis of bending and focusing magnets shows that this is a propert of magnets with faces 6 degrees apart if B is adjusted so that the Larmor radius originates at the apex of the triangle..5 MM MM x ( ) x( ) xaxis Tr it: Investigate a magnet with 9 degrees between the faces b changing to /4 and changing the B field to.77.

5 4/7/ Lorentz Eqns. of Motion Bending of a particle beam 5 Tr it: Find the spread in the trajectories at the focal point b finding where the trajectories cross the horizontal axis. Tr it: Suppose that the edge ras deviate from the central ra b a small angle. The position of the ra at the exit slit is independent of to first order, but not to second order. Increase the angular spread in the ras and show that the size of the focal point increases. Reference: A. Septier, Focusing of charged particles (Academic Press, New York, 967), in two volumes.

3/7/2019 N S N S. Magnetism. Magnetism

3/7/2019 N S N S. Magnetism. Magnetism Magnetism Magnetic charges Called poles Two types, North and South Like poles repel each other Opposite poles attract each other Found only in North/South pairs (Dipoles) Magnetism Magnetic poles Found