#### Away from Cathode

Away from the cathode, the following beam parameters were used for the simulation:

spot size: 10 μm (100μm)

pulse length: 20.16 μm

Beam energy: 225 keV

Initial Emittance: 0

Number of Particles: 1000 (10^5)

Barnes Hut(BH) Parameters:

Theta = .3

Plummer Radius = 1 fm

##### Field Deviation

The beam is propogated forward .1 ns and every .01 ns the fields at the position of each particle were recorded. The electric fields in the x,y, and z direction were compared between an exact point to point calculation and the BH Tree Approximation. The deviation (exact field - BH Field)/ exact field was calculated at each particles position at each time step. The deviations are binned and shown in the histogram below, the heights were normalzed so the sum of all bins is 1.

##### Macroscopic Varation:

The emittance (both transverse and longitudional) was tracked over 5 plasma oscillations (15 ns). The results are compared between the direct field calculation, the BH approximation, and a smooth space charge algorithm. The results are below.

#### Close to Cathode

Close to the cathode, the following beam parameters were used for the simulation:

spot size: 1 μm (3μm)

pulse length: 4 mm

Beam energy: 50 keV

Initial Emittance: 0

Number of Particles: 1000 (10^5)

Barnes Hut(BH) Parameters:

Theta = .3

Plummer Radius = 1 fm

##### Field Deviation

The beam is propogated forward .1 ns and every .01 ns the fields at the position of each particle were recorded. The electric fields in the x,y, and z direction were compared between an exact point to point calculation and the BH Tree Approximation. The deviation (exact field - BH Field)/ exact field was calculated at each particles position at each time step. The deviations are binned and shown in the histogram below, the heights were normalzed so the sum of all bins is 1.

##### Macroscopic Varation:

The emittance (both transverse and longitudional) was tracked over 5 plasma oscillations (5 ns). The results are compared between the direct field calculation, the BH approximation, and a smooth space charge algorithm. The results are below.