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349) How are visible tracks created in CR-39

Ludwik Kowalski
Montclair State University, Montclair, NJ, 07055
April 30, 2008

Being retired often means that one is able to turn a hobby into a full time occupation. Fortunately, I am still healthy enough to do this. So let me describe what I think about the CR-39 track-formation process. Ideas presented here are closely linked with contents of units 346 and 347. They represent an attempt to to develop my own simulation of etching. I will begin with a very simple model and try to make it more realistic, one step at a time. For the time being, each of my models is for latent tracks perpendicular to the surface of the CR-39 chip. What is a latent track? It is cylinder, containing damaged CR-39 material (broken molecules, etc. etc.)

I will use the following notation:

E <--- (in MeV) kinetic energy of the incoming alpha particle
L<---(in microns) the  length of the latent track created by that particle. 
Vt <---(in microns per hour) the etching rate in the damaged CR-39 material. The Vt is caller track etching speed.
Vb <---(in microns per hour) the etching rate in the CR-39 material undamaged by the particle. The Vb is called bulk etching speed. 

Model 1 (not realistic):
Let me make some assumptions: 
a) The damaged material is in a cylinder (centered on the latent track axis) of very small diameter, centered on the latent track axis.
b) The damaged material is uniformly distributed within the latent track cylinder. 
c) Bulk etching speed, Vb, is zero while the track etching speed, Vt, is 4 microns/s.
d) L=16 microns 

Since Vb=0 , the track diameter does not change with the time of etching. The dry track will consist of an air cylinder whose length is y = Vt * t, where t is the etching time smaller than 4 hours. (the time to dissolve all damaged material, t’ = L / Vt = 16 / 4 =4 hours). Nothing changes after 4 hours of etching. 

Model 2 (Vb > Vt or Vb=Vt): 
No track will be seen because the CR-39 material, surrounding the latent track cylinder, will be dissolved faster than the damaged material. The invisible latent track will disappear in 4 hours if L=16 microns and Vb=Vt=4 microns per second.

Model 3 (starts to be realistic):
Assumptions are the same as in Model 1, except that Vb (smaller than Vt) is no longer zero. In order to produce conical tracks I will assume that vertical etching (direction parallel to the latent track) proceeds with the constant speed Vt while horizontal etching proceeds with the constant speed Vb. This crude approximation is probably valid for etching times that are not too long; experimentally determined track profiles (1) were found to be conical, for of not-overetched tracks. A track can be said to be overetched when the etching time is longer than t’= R / Vt, where R is the range of particles in CR-39. For alpha particles of 5.5 MeV, R is about 30 microns. Thus, t’ is 10 hours, when Vt=3 microns per second. Profiles of overetched tracks, according to (1), tend to be come spherical and their radii seem to increas at the Vb rate. This observation is part of Model 3. Fot t > t’, the diameters of tracks are assumed to grow at the same rate as for lower t (see below).

Figure 1 shows the expected evolution of the visible track of a 5.5 MeV alpha particle. Consecutibe profiles refer to etching times of 2, 4, 10, 30 and 40 hours, as indicated. These profiles correspond to Vb=1.2 and Vt=3.6 microns per hour. Note up to t=t’=10 hours, all tracks are conical. The half-angle, A, at the tip of each cone, 18.4 degrees; the tangent of that half-angle is nothing else but the Vb / Vt ratio. The hight of a cone, y, for any given t < t’ (t’=10 hrs) is given by

y = (Vt - Vb) * t. = 2.4 * t

The corresponding diameter, D, at time t, is given by

D = 2 * y * tan(A). = 1.6 *t

= = = = = = = = = = = = = =

Figure 1
Cross sections of track profiles after different etching times. The dashed line shows where the top surface of the CR-39 chip was before etching. Two horizontal segments, at each profile, show where the CR-39 surface is after etching. The bottom of the 10 h cone coincides with the end of the invisible latent track. Note that after 40 hours of etching the latent track layer is totally disolved. The right-side pit, located below that layer, is growing radially at the rate of 1.2 microns per second.
= = = = = = = = = = = = = = = = = = = = = = = = = = = =

* * * * * * * * * * * * * * * * * * * * 
That observation made me think about SPAWAR results described in my Unit 347. According to Figure 3, in that unit, the measured diameters of tracks produced by alpha particles from 241Am became 40 microns after about 22 hours. According to the above formula, D after 22 hours is 35 microns. That is not very different from 40. I consider this to be a good indicator that my model is reasonable. But it is too eraly to start playing with Vt and Vb, in order to get a perfect agreement.
* * * * * * * * * * * * * * * * * * * * * * * * * * * 
Model 4
Damaged material, surrounding a latent track, is probably not uniform. Material closer to the latent track axis is likely to be more damaged than material further away from the axis. Likewise, material closer to the end of a latent track is likely to be more damaged than material closer to the beginning of the track. I am saying this because I know how density of ionization is distributed along the path of an alpha particle in air. That topic has been studied by Bragg, more than one hundred years ago. In the Model 3, the Vt was assumed to be constant; in this model I will assume that it changes along the latent track.

I will simulate evolution of a track due an alpha particle of E=5.5 MeV (R=32 microns). Simulation steps will be 15 minutes each. The Vb will remain constant but Vt will be assigned a predetermined value for each step. In my simple computer program, the Vb will always be 1.2 microns per second. But my set of Vt values, for consecutive steps, will be treated as a set of adjustabl parameters. This should provide great flexibility. I would be able to see the cross section of a track at any desired etching time. To test the program (to discover and correct hidden errors) I will compare its output with Model 3. The results should be nearly identical, provided Vt=constant=3.6 microns/hour (as above), and time steps are very short, for example, one minute each. That is my plan. The results will be shown later.

Model 5
This model will be similar to Model 4 but it will have an additional assumption. The value of Vt will also have a radial dependence (larger values near the latent track axis and smaller values at larger distances from the axis.

Another thing worth doing is to perform experiments in which CR-39 chips are bombarded with alpa particles of several energies. The chips would then be etched for different times, for example, 2 hrs, 4 hrs, 8 hrs, 16 hrs and 32 hours. Plotting diameters of tracks versus etching time will produce curves to be compared with simulations. Yes, this is a big and ambiguous project. If all goes well, I might be able to publish the results. The project has nothing to do with CMNS; but results might be useful to CMNS researchers.

Appended on 5/1/08
Several things became intuitively clear to me, after I tried to compose an algorithm for Model 4. Let me enumerate them.

a) Strongly overetched tracks will have semi-spherical profiles, as in Model 3.
b) Suppose t’ again stands for the etching time needed to dissolve the CR-39 layer in which the latent track was located. Profiles of tracks etched during times t shorter than t’ will not be exactly triangular; they will resemble profiles of inverted Eiffel towers. But that is not a very big difference.
c) Time t’ in Model 4 will be shorter than what it would be if the value of Vt were not increasing with the depth of the pit. The difference between t’ of Model 4 and t’ of Model 3 increases when the rate at which the Vt changes with the depth became larger.

I do not think that the overall results of Model-3 simulations will be very different from the results of Model-3 simulations. The diameters of strongly overetched pits will keep growing at the same rate close to 2 * Vb=2.4 microns per hour. The pits produced during the etching times shorter that t’ will no longer be proportional to t. But consequences of this will be hard to measure. One thing is clear, Model 4 will not allow me to explain why diameters of tracks created during codeposition stop growing when the etching time is increased.

The same is likely to be true for the Model 5. Here too the diameters of profiles, of strongly overetched pits, should be increasing linearly at the rate close 2.4 microns per hour. Some nuances will probably be different at t < t’ but they will be difficult to notice. Intuition tells me that Model 5 will also not going to explain the experimental fact reported by SPAWAR team -- diameters of tracks, created during codeposition, stop growing when the etching time is increased.

Considering all this I decided not to commit myself to this project. What I learned so far gives me enough confidence to make the following observations.

a) Diameters of tracks due to alpha particles from 241Am increase with the etching time as expected. I know that I can simulate the reported curve; it is shown in Figure 3 at:

b) The diameters of tracks created during codeposition, shown in the same Figure 3, stop growing when the etching time exceeds 20 hours. This experimental fact is not likely to be explained by my model. In fact, I suspect that the model of Nikezic and Yu, used by Pamela, also does not explain this experimental fact. But this is only a suspicion. Perhaps I am not aware of some important effects. Perhaps latent tracks created by alpha particles of ~ 1 MeV are very different from latent track due to common alpha particles. Something is not right somewhere. What is it?

1) F.M.F. Ng, K.Y. Luk, D. Nikezic and K.N. Yu; “Determination of alpha-particle track depths in CR-39 detector from their cross-section and replica heights; Nuclear Instruments and methods in physics research B 263 (2007) 266-270.

2) A.H. Khayrat and S.D. Durrani; “Variation of alpha-particle track diameters in CR-39 as a function of residual energy and etching conditions." Radiation Measurements, vol. 30, Issue 1, (1999) pages 15-18).

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