Experiment 9.001 (30/6/2016) : 20 electrons,
Bfield= 10 gauss, initial kinetic energy of each electron (energy
range) = 100 eV, beam current 0,001 nA, 3 H+ generated with initial
speed=0 (in positions: (0.5, 0.5, 0.25) (0.50125, 0.50125, 0250625),
(0.501246, 0.501246, 0.25125), 3 B+ generated with initial speed=0 (in
positions: (0.5, 0.5, 0.75) (0.50125, 0.50125, 075188), (0.501246,
0.501246, 0.75375) In this experiment the kinetic
energy of all particles is NOT kept constant (from experiment 3 it was
kept constant).
Screenshot after 1,53E5 sec (53 min
computer time)
Screenshot after 6,22E5 sec (3h13 min
computer time)
Just when the 20 e, 3 H+ and 3 B+ were generated
their total potential+kinetic energy = 3,47 E16 J. In the first
screenshot it is 6,23 E16 J. Almost doubled! In the second
screenshot 6,52 E15 J.
Their speeds increase, probably due to the fact that the program is an
aproximation of reality.

Experiment 9.002:
The same experiment again, but now reading the potential and the kinetic
energy apart.
Initial total kinetic energy of all particles = 3,47 E16 J ,
initial potential energy of all particles = 1,21 E24 J
Screenshot after 0,00011 s
Total kinetic energy = 6,2 E14 J, total potential energy = 5,0 E26
The kinetic energy of the particles is 178 times
bigger! Because the distance between the electrons has grown, itīs
understandable that the potential energy has decreased. The
potential energy is, by the way, negligible compared to the kinetic
energy.

Experiment 9.1:
50 electrons, Bfield= 10 gauss, initial kinetic energy of each electron
(energy range) = 100 eV, beam current 0,1 nA, 3 H+ generated with
initial speed=0 (in positions: (0.5, 0.5, 0.25) (0.50125, 0.50125,
0250625), (0.501246, 0.501246, 0.25125), 3 B+ generated with initial
speed=0 (in positions: (0.5, 0.5, 0.75) (0.50125, 0.50125, 075188),
(0.501246, 0.501246, 0.75375).
The
kinetic energy of all electrons is kept constant.
Screenshot 49 electrons generated
(total potential energy of the 49 electrons =
6,9E24 J)
Screenshot just 50 electrons generated
(total potential energy of the 50 electrons
= 17 E24 J)
Video after 7,46E5 s
When the electrons are regrouped, their mean distances become a little
bit smaller, so itīs understandable that their potential energy
increases. As said before, correct should be "slowly" change the
Bfield (during 1 s perhaps?) and simulate the movement of the electrons
during this period. But 1 s real time takes about 80 year computer time
(at the moment, in my own computer), so itīs not possible (at the
moment..).

Experiment 9.22:
Screenshot 50 electrons with some
vertical speed, Bfield not changed (stays 10 G)
The electrons spread out in vertical direction. The
electrons were given a random vertical speed (0.01*sp) < vz < (0.01*sp)
(sp = √(v_{x}^{2} + v_{y}^{2})
sp= speed in the horizontal plan).
The
kinetic energy of all electrons is kept constant.

Experiment 9.23:
Screenshot 50 electrons with some
vertical speed, Bfield changed (from 10 G to 100 G)
Video after 3,11E5 s
It seems to be that the electrons do NOT spread out in
vertical direction. The electrons were given a random vertical speed
(0.01*sp) < vz < (0.01*sp) (sp =
√(v_{x}^{2} + v_{y}^{2})
sp= speed in the horizontal plan)
B+ & H+ are not generated.
The
kinetic energy of all particles is kept constant.
Remark: the kinetic energy is kept constant by mulitplying the
components of the speed vx, vy and vz with a factor 0.999 or
1.001. It could be possible that the component vz was made zero by
multiplying all the time with a factor 0.999.

Experiment 9.24:
Screenshot 50 electrons with some
vertical speed, Bfield changed (from 10 G to 100 G)
The
kinetic energy Ek of all particles is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001. However, v_{z} is
NOT mulitplied by any factor.
The electrons do now YES spread out in
vertical direction. The electrons were given a random vertical speed
(0.01*sp) < v_{z} < (0.01*sp) (sp =
√(v_{x}^{2} + v_{y}^{2})
sp= speed in the horizontal plan)
B+ & H+ are not generated. Itīs a pity, but now they are not stopped by
the magnetic field produced by themselves.
From the difference between experiment
9.22 and 9.23 we can conclude that increasing the magnetic field and let
so the electrons move in small circles, seems to avoid somewhat
the spreading out in the vertical direction.
Itīs a pity that in experiment 9.24 they
yes spreaded out in the vertical directions. We have to find out more
about how much a approximation is the program (because we keep the
kinetic energy of the electrons constant by multiplying the speed of the
electrons with a factor..).
I checked the speed
v_{z} of the electrons and the force Fz on the
electrons in vertical directions, and they never seemed to be oposed..
So the electrons are not stopped spreading out in a vertical directions
by the magnetic field produced by themselves... Itīs a pity.. No,
in the experiment now running I just saw that sometimes they are
opposed... Itīs not clear yet ??
The same experiment again, but with B field 100  1000 G, 100 eV,
0.0000002*sp<vz <0.0000002*sp, 0,5 m 0.001 < z < 0,5m +
0.001 , result: electrons spreading out in vertical direction.
Remark: the speed of the electrons with 100 eV = 5,9E6 ms,
0.0000002*sp = 1,2 m/s vertical speed. I suppose this is quite a lot..
Experiment 9.25:
Pobably wrong experiment, in the program B
was probably not increased!
Ek
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor.
Generated 100 electrons with 100 eV. Then changed Bfield from 10 G to
1000 G and the electrons distributed randomly inside the first orbiting
circle. (beam current = 1 nA)
electron[i].vz:=(0.5random)*sp*0.000000002 (2E9)
> 5,9E3 < vz < 5,9E3 m/s (the initial
vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.001/2+random*0.001);
(s=400, side cube = 1 mtr) > 0,4995 < z < 0,5005 (the
initial vertical position varies +/ 0,5 mm )
Screenshots experiment 9.25
(after 9,76E6 s)
Screenshots experiment 9.25
(after 15,6 E6 s)
At the start of the experiment the total potential energy of the
electrons: 3,1E23 J < Ep < 6,8E23 J
After 15,6E6 s: 2,89E23< Ep < 6,04 E 23 J
The potential energy decreases, so the electrons spread out.
After 1,86E5 s Bfield increased to 10000 G. > 4,656 E23< Ep <
4,653 E23 J slowly decreasing 4,6456 E23< Ep < 4,6409 E23 after
1,88E5 s

Experiment 9.252:
The same as exp. 9.25, only the Bfield = 1000 G constant from the
beginning.
Screenshot experiment 9.252
(after 1,9E6 s)
The electrons "quite rapidly" spread out in vertical direction.
This could be explained because the electrons are from the beginning
already more close together than in exp. 9.25, so the Coulomb repelling
forces will be bigger.
Experiment 9.253:
The same as exp. 9.25, only the Bfield = 10 G constant from the
beginning.
Screenshot experiment 9.253
(after 6,64E6 s)
Screenshot experiment 9.253
(after 7,63E6 s)
The electrons spread out in vertical direction.
Experiment 9.254:
Probably wrong experiment, in
the program B was probably increased!
The same as exp. 9.25, but now the Bfield = 10 G > 10000 G
(wrong experiment, in the program B was not realy increased!)
Screenshot experiment 9.254
(after 2,33E6 s)
Screenshot experiment 9.254
(after 1,64E5 s)
Video exp 9.254
t=1,58E7 > 2,08E23 < Ep < 8,05 E23
t=1,64E5 > 2,051E23 < Ep < 7,26E23 
Experiment 9.3:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor.
Generated 100 electrons with 100 eV. Then changed Bfield from 10 G to
1000 G and the electrons distributed randomly inside the first orbiting
circle. (beam current = 0,1 nA)
electron[i].vz:=(0.5random)*sp*0.000000002 (2E9)
> 5,9E3 < vz < 5,9E3 m/s (the initial
vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.001/2+random*0.001);
(s=400, side cube = 1 mtr) > 0,4995 < z < 0,5005 (the
initial vertical position varies +/ 0,5 mm )
At the start of the experiment the total potential energy of the
electrons: 8,67 E23 J < Ep < 8,859 E23 J
Screenshot experiment 9.3 just before
regrouping electrons
8,7 E23 < Ep < 8,86 E23
Screenshot experiment 9.3 after 1,38E6 s
Screenshot experiment 9.3 after 1,59E5 s
5,4 < Ep < 5,5 E23
The potential energy of the electrons decreases. It is clearly to
see in the last screenshot that the electrons have spread out. The
vertical speeds of the electrons have also increased (see
screenshot, from a couple of m/s till a couple of 100 m/s to even
more than 1000 m/s.
 Experiment 9.31: The same as experiment 9.3, but
now the magnetic field changes from B=100 G to B= 1 T.
Screenshot experiment 9.31 after 3,78E6 s
At the start of the experiment Ep= 2,76 E21 J, after 3,78E6 s Ep=
4,8E22 J. The potential energy of the electrons has decreased. The
electrons have spread out.  Experiment 9.33:
electron[i].vz:=(0.5random)*sp*0.0000000002; (2E10)
electron[i].z:=(0.5*s/4000.0001/2+random*0.0001);
(the initial vertical position varies +/ 0,05 mm )
200 electrons, Bfield 10 G to 1000 G, electrons distributed
randomly inside the first orbiting circle.
Screenshot experiment 9.33 after 2,76E6 s
Screenshot experiment 9.33 after 5,14E6 s
At the start of the experiment Ep= about 2,87 E22 J, after 2,76 E6
s Ep= about 2.68 E22 J. The potential energy of the electrons has
decreased. The electrons have spread out.
 Experiment 9.34:
electron[i].vz:=(0.5random)*sp*0.0000000002; (2E10)
electron[i].z:=(0.5*s/4000.0001/2+random*0.0001);
(the initial vertical position varies +/ 0,05 mm )
200 electrons, Bfield 10 G constant
Screenshot experiment 9.34 after 2,81 E6 s
Screenshot experiment 9.34 after 5,24 E6 s
At the start of the experiment Ep= about 2,5448 E22 J, after 2,81
E6 s Ep= about 2.543 E22 J. The potential energy of the electrons
has decreased. The electrons have spread out. 
Conclusion so far: We want that the electrons
stay in the centre of the vacuum chamber (the cube in the simulation
program). We try this by applying the simple trick "increasing the
magnetic field so that the electrons start to move in small circles and
are glued together by the magnetic fields they generate themeselves".
Intuitive explanation of this in
experiment6 .
In experiment 9.3, 9.31 & 9.33 the electrons have spread out. The
"trick" did not work... It's a pity..
 Experiment 9.35:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor.
Generated 100 electrons with 100 eV. Then changed Bfield from 10 G to
1000 G and the electrons distributed randomly inside the first orbiting
circle. (beam current = 0,1 nA)
electron[i].vz:=(0.5random)*sp*0.00000000002; 2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.00001/2+random*0.00001);
(the initial vertical position varies +/ 0,005 mm )
Screenshot exp. 9.35 after 1,01 E5 s
At the start of the experiment Ep= about 7,2 E23 J, after 1,01
E5 s Ep= about 6,3 E23 J. The potential energy of the electrons
has decreased. The electrons have spread out.
 Experiment 9.36:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor.
Generated 100 electrons with 100 eV. Bfield 10 G
constant. (beam current = 0,1 nA)
electron[i].vz:=(0.5random)*sp*0.00000000002; 2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.00001/2+random*0.00001);
(the initial vertical position varies +/ 0,005 mm )
At the start of the experiment Ep= about 6,6 E23 J, after 1,1
E5 s Ep= about 4,07 E23 J. The potential energy of the electrons
has decreased. The electrons have spread out.
Screenshot exp. 9.36 100 e 10 G
 In exp. 9.36 the electrons have
spread out a little bit more than in experiment 9.35. Probably because
they are closer together.  Experiment 9.37:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor.
Generated 100 electrons with 100 eV. Then changed Bfield from 10 G to
1000 G and the electrons distributed randomly inside the first orbiting
circle. (beam current = 0,1 nA)
electron[i].vz:=(0.5random)*sp*0.00000000002; 2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.00001/2+random*0.00001);
(the initial vertical position varies +/ 0,005 mm )
Screenshots experiment 9.37 after 3 E5
Video experiment 9.37 after 3 E5 s
At the start of the experiment Ep= about 8,4 E23 J, after 3
E5 s Ep= about 4,2 E23 J. The potential energy of the electrons
has decreased. The electrons have spread out.
Curious that one electrons up and one down have started to move in
larger circles (probable higher speeds).  Experiment 9.38:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor.
Generated 100 electrons with 100 eV. Bfield constant 10 G . (beam current = 0,1 nA)
electron[i].vz:=(0.5random)*sp*0.00000000002; 2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.00001/2+random*0.00001);
(the initial vertical position varies +/ 0,005 mm )
Screenshot experiment 9.38 after 3 E5
At the start of the experiment Ep= about 6,2 E23 J, after 3
E5 s Ep= about 2,7 E23 J. The potential energy of the electrons
has decreased. The electrons have spread out.
Experiment 9.39:
(error in program, vx:=vx+((Fx+Fxi)/(2*m))*dt;
Fxi=0, should be vx:=vx+((Fx)/(m))*dt; (that is probably why not
in the middle..)
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor. (0,1 nA)
Generated 100 electrons with 100 eV. Then slowly changed
Bfield from 10 G to 1000 G in small steps.
if (amountE=100) and not (Already) then begin
if B<1000E4 then B:=B+0.0000001 else Already:=true;
electron[i].vz:=(0.5random)*sp*0.0000000002;
2E10 m/s
> 5,9E4 < vz < 5,9E4 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.0001/2+random*0.0001);
(the initial vertical position varies +/ 0,05 mm
)
Screenshot experiment 9.39 after
1,86 E6 (Bfield
just became 1000 G)
Screenshot experiment 9.39 after
6,19 E6

Experiment 9.392:
(error in program, vx:=vx+((Fx+Fxi)/(2*m))*dt;
Fxi=0, should be vx:=vx+((Fx)/(m))*dt; (that is probably why not
in the middle..)
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor. (0,1 nA)
Generated 200 electrons with 1000 eV. Then slowly changed
Bfield from 10 G to 1000 G in small steps.
if (amountE=200) and not (Already) then begin
if B<1000E4 then B:=B+0.0000001 else Already:=true;
electron[i].vz:=(0.5random)*sp*0.00000000002;
2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.00001/2+random*0.00001);
(the initial vertical position varies +/ 0,005
mm
Screenshot experiment 9.392
after 4,58 E6 s 
Experiment 9.393:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor. (0,1 nA)
Generated 100 electrons with 100 eV. Then slowly changed
Bfield from 10 G to 1000 G in small steps.
if (amountE=100) and not (Already) then begin
if B<1000E4 then B:=B+0.0000001 else Already:=true;
{Bh:=B + 2r1 + 10r2 (r1= radius in xyplane, r2=vertical distance
from center cube, at side r1=0,5 & r2=0,5 m)
electron[i].vz:=(0.5random)*sp*0.00000000002;
2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.0001/2+random*0.00001);
(the initial vertical position varies +/ 0,05 mm
)
Screenshot experiment 9.393 after
1,7 E6 (Bfield 880 G,
not yet 1000 G)
The electrons did spread out already quite a lot.
 Experiment 9.394:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor. (0,1 nA)
Generated 200 electrons with 100 eV. Then slowly changed
Bfield from 10 G to 1000 G in small steps.
if (amountE=200) and not (Already) then begin
if B<1000E4 then B:=B+0.0000001 else Already:=true;
{Bh:=B + 2r1 + 10r2 (r1= radius in xyplane, r2=vertical distance
from center cube, at side r1=0,5 & r2=0,5 m)
B at the sides stronger. B up and down also stronger.
But after generated 200 e Bfield constant?
electron[i].vz:=(0.5random)*sp*0.00000000002;
2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.0001/2+random*0.00001);
(the initial vertical position varies +/ 0,05 mm
)
Screenshot experiment 9.394 after
3,03 E6
Screenshots experiment 9.394 after 6.05
E6
The electrons did spread out quite a lot.

Experiment 9.395:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor. (0,01 nA)
Generated 50 electrons with 100 eV. Then slowly changed
Bfield from 10 G to 1000 G in small steps.
if (amountE=50) and not (Already) then begin
if B<1000E4 then B:=B+0.0000001 else Already:=true;
{Bh:=B(1 + 2r1 + 10r2) (r1= radius in xyplane, r2=vertical
distance from center cube, at side r1=0,5 & r2=0,5 m)
B at the sides stronger. B up and down also stronger.
electron[i].vz:=(0.5random)*sp*0.00000000002;
2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.0001/2+random*0.00001);
(the initial vertical position varies +/ 0,05 mm
)
Screenshot experiment 9.395
1.7 E5 s
The electrons did spread out quite a lot.

Experiment 9.396:
The total kinetic energy Ek of the electrons
is kept constant by
mulitplying the components of the speed v_{x} and v_{y}
with a factor 0.999 or 1.001; v_{z} is
NOT mulitplied by any factor. (0,1 nA)
Generated 100 electrons with 1000 eV. Then slowly changed
Bfield from 10 G to 1000 G in small steps.
if (amountE=100) and not (Already) then begin
if B<1000E4 then B:=B+0.0000001 else Already:=true;
{Bh:=B(1 + 2r1 + 10r2) (r1= radius in xyplane, r2=vertical
distance from center cube, at side r1=0,5 & r2=0,5 m)
B at the sides stronger. B up and down also stronger.
electron[i].vz:=(0.5random)*sp*0.00000000002;
2E11 m/s
> 5,9E5 < vz < 5,9E5 m/s (the
initial vertical speed of the electrons)
electron[i].z:=(0.5*s/4000.0001/2+random*0.00001);
(the initial vertical position varies +/ 0,05 mm
)
Screenshots experiment 9.396 7.99
E6 s
Screenshot experiment 9.396 1,2 E5
s It seems to be
that the electrons do not spread out.....
(one electron moving very fast, the others seem not to move,
anyway as can be seen in the screen)
The same experiment again,
starting from t=0:
Screenshot experiment 9.396 2.02E5
s
The electrons do spread
out...
The Bfield was reduced to 0,1 G. Only one electron started to
move in a wide orbit. The other ones seemed not to move,
what means they had a very small speed. Probably because the
procedure that kept the total kinetic energy constant by
reducing (or augmenting) the speed of all electrons, did reduce
their speed (in the xy plane) to zero (except for one
electron).

Experiment 10:
To make the simulation more accurate we implemented the Leapfrog method,
see experiment 10
