Test2d05, 5th 'benchmark test' data file for CPO2D
Relativistic motion in a uniform electric field
The set-up is the same as in test2d03.dat (parabolic motion in a uniform field), except that the energy is 1 MeV and the voltage is 1 MV. At the turning point z should be 0.38312, and is found to be 0.38309 (an error of 0.008%).
The number of segments used in the present example is small enough for the example to be run with the ‘demo’ version of CPO2D. Higher accuracy could of course be obtained with more segments, using the standard or full versions of CPO2D.
The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.
The set-up is identical to that in the 3rd benchmark file test2d03.dat, except that the applied potential is 1000000 V instead of 1 V and the initial energy is 1000000 eV instead of 1 eV, and the test plane is at r = 0.3069.
Solving the relativistic equations of motion for the present conditions, the turning point of the ray should occur at r = rm, z = zm, t = tm, where
rm = r_initial + (U/T)*ln(W/(1+T-U)),
zm = z_initial + (1+T-W)/T,
tm = U/(T*c),
T = initial_kinetic_energy/(m0*c**2),
U = sqrt(T+0.5*T**2),
W = sqrt(1+T+0.5*T**2).
For the energy used in this example,
rm = 0.30692 mm,
zm = 0.38312 mm,
tm = 3.354E-9 ms (that is, 3.354 ps).
The test plane has been put at r = 0.30692, and it can be seen from the output file that zm = .38309 (an error of 0.008%, which is presumably this small by chance because the error increases slightly when the number of segments is increased to 320 and the ray inaccuracy is decreased to 0.00001) and that tm is correct (to within the 3 significant figures given in the output file). It can also be seen from the detailed information on the ray steps, or from the screen plot, that the crossing point is at the turning point of the ray.