General Overview of Theory:

The
electrons, after being emitted and colliding with the atoms they came from,
will travel to the detector, where their radial position from the center of the
detector will be found. This position, in the presence of a strong electric
field, will only depend on the vertical component of velocity, or momentum,
since that field will provide such a strong acceleration. We want to be able to
say that if a particle landed at “R” distance from the center of the screen, we
know for certain it had “p” initial vertical momentum. Hence the need for
optimized voltages that will force electrons that start
at slightly different positions to land at relatively the same location. This
will ease detection and will allow us to easier eliminate lower energy ones. The overall goal becomes getting a probability
function of locating electrons with respect to momentum. This can be found
using the Inverse Able Transform of the number of particles on the detector
within a radius, and use the linear relationship between that radius and
momentum to find it.

That’s how I understand it anyway!

Week 1 (5/27-5/30):

Not a lot went on this week. We got to hear about projects and eventually
decide on which we wanted. The REU program was also explained as well as what
is expected of us. I also got introduced to the JRM lab, which is where any
experimentation I do is going to be, as well as the physics machine shop they
have here on campus, just in case I ever need to forge something. The program
SIMION was also introduced to me, since it would help simulate charges moving
through potentials. Also, I read a chapter of a book on velocity map imaging to
get me used to the concept.

Week 2 (6/2-6/6):

I finally meant my mentor along with others working on the same project. I
eventually was given a basic design in SIMION of the setup we would have and I
analyzed, theoretically, what would happen if electrons were sent through at
different energies, such as their final position on our screen and how far
apart they were (resolution). At the end of the week, the design was updated
and I ran the same simulations to see what differences would be created.

Week 3 (6/9-6/13):

I was given the schematics of the imager that was to be made and I looked
through it to try and familiarize myself with how it would work and be
constructed. My mentor also told me to try and simulate real momentum
distributions of ions and to simulate a normal distribution of electrons with
uniform angular distribution and see what if I could use the image on our
screen to get the distribution back out of it using an Abel Transform. I read
through part of the SIMION user manual to become familiar with their
programming language, since this distribution cannot be done by hand. I also
spent some time trying to figure out exactly what an Abel Transform is, as well
as what it actually does physically. I was able to find an equation online that
would simulate random Gaussian numbers, and checked to see if it actually
worked using mathematics software. No work was done on Thursday because of the
tornado damage, and little was done on Friday because of a power outage.

Week 4 (6/16-6/20)

The equation I was given was fine tuned in order to get standard deviation as a
factor. The new equation was put in a premade SIMION program that randomized
both sendoff angle and energy, but would now Gaussian randomize energy and have
a random angle. The code was tested and it appeared to do what it is supposed
to. Most of the time this week was figuring out the programming language SIMION
uses as it is much different than any code I am used to.

Week 5 (6/23-6/27)

Progress
was somewhat halted this week because of asbestos contamination in the air. On
the days I could work, I began working on programs to calculate the density of
electrons on the detector screen simulated in SIMION. The derivative of the
number of electrons as a function of their radial distance from the center of
the detector was key to the Abel Transformation
needed, so the program was used to calculate that, as well as numerically find
N(R), the integral of the derivative data, if it were necessary.

Week 6 (6/30-7/3)

After
fine tuning my MATLAB programs to produce the functions I needed, the difficult
task of actually performing the integral in the transformation came up. The
integral is improper due to each of the bounds, so I am unsure how to go
performing it with strictly numerical data. One idea was to find an expression
for N(R) in Origin, differentiate it, then stick that
function into the transform. I was successful in doing so, but found that N(R)
was an increasing function, and would always be an increasing function due to
the way I defined the derivative. This is not correct, since the number of
electrons should definitely decrease as I move further from the center of the
screen. This led me to believe my thoughts on the derivative were wrong, and
allowed me to recalculate it in MATLAB. After the programs were edited, the
graphs of N(R) and dN/dR
seemed to make more logical sense, as N(R) decreased after a certain point.
However, these graphs verified that trying to find a best fit curve would be
next to impossible due to the noisy nature. I was also given a new possible
design for the VMI that needed to have optimized voltages for minimum
resolution. This proved to be very difficult, however, and no solution came out
that gave results as good as the last design due to an extra piece of metal.

Week 7 (7/7-7/11)

The
beginning of the week entailed some “house keeping”
tasks to make sure no other errors were being made with my programs. First,
since momentum is what we actually care about with these transformations, I had
to see what the relationship was between the final radius
the electrons have from the center of our detector with their initial momentum.
To do this, I went back to the radius and initial kinetic energy data,
converted KE into momentum, and plotted it. The relationship was linear,
meaning N(R) is proportional to N(p). Then, I checked
to make sure that the program I altered in MATLAB to now numerically find dN/dR was correct. Taking the
N(R) data now being estimated from the SIMION data points, Origin was used to
find its derivative, which matched what my program found. The next step is to
now numerically take the Abel Transform, which has proven to be more difficult
than once thought. The bounds making it improper still pose a problem for
numerical integration. However, after a simple change of variables, I found the
integral was transformed into a proper one, and I could move forward in finding
a way to transform by data.

Week 8 (7/14-7/18)

I
started out the week by talking to my mentor, and trying to iron out the parts I
didn’t quite understand with my project.
After speaking with him, a lot more of it made sense. I had to re-edit my MATLAB program again
because of the confusion with N(R). Initially, I had to correct idea by finding
the total number of particles within a radius, but thinking I was wrong,
changed it to a number density, n(R). Now, the MATLAB program used to calculate
this is find N(R) as well as the derivative is doing so correctly. Then, using
the change of variables, I was able to make a program that would calculate the
Inverse transform for me. Now I can analyze data taken on SIMION.