Preliminary analysis aid. Check the result against your own analysis and, where needed, a test on the intended part. Full disclaimer.
1. What the tool calculates
Proton test planning arithmetic: beam current to flux (or flux to current) for a given beam area, the ionizing dose rate in silicon at a given proton energy in rad(Si)/s, krad(Si)/min and Gy(Si)/s, the time to reach a target dose, and the dose delivered by a stated fluence. Stopping power comes from Bethe-Bloch with the Sternheimer density correction, or from a stopping power table you paste in. A reference table gives the flux and current for 1 krad/min at ten standard energies.
- Mode. Current to flux: beam current (A to pA) and beam area. Flux to current: target flux in p/cm2/s and beam area.
- Beam area. Circular (diameter, cm or inches), rectangular (width and height), or the area directly (cm2 or in2). Facility presets set the geometry; the UC Davis CNL preset, for example, is a circular 8 cm beam (50.27 cm2). "Custom" restores manual entry.
- Proton energy, MeV.
- Stopping power source. Built in Bethe-Bloch, or a pasted two column table of energy (MeV) and mass stopping power (MeV cm2/g); tab, comma, semicolon or space separated, lines starting with
# ignored, at least two points.
- Target dose (rad, krad, Mrad or Gy) for the time panel; fluence (p/cm2) for the dose panel.
3. Calculation
Flux and current
φ [p/cm2/s] = (I / e) / A I [A] = φ · A · e e = 1.602e-19 C
A = π(d/2)2 circular, w · h rectangular, or as entered
Stopping power
Mass electronic stopping power of protons in silicon, PDG form of Bethe-Bloch with the Sternheimer density effect [1], [3]:
−dE/dx = K · (Z/A) · (1/β2) · L(βγ)
L = ½ ln(2 me β2γ2 Tmax / I2) − β2 − δ(βγ)/2
Tmax = 2 me β2γ2 / (1 + 2γ me/Mp + (me/Mp)2)
K = 0.307075 MeV g-1 cm2 mol, Z/A = 14/28.0855, me = 0.511 MeV, Mp = 938.272 MeV, I = 173 eV
Density correction with x = log10(βγ) and the silicon constants x0 = 0.2014, x1 = 2.8715, a = 0.1492, m = 3.2546, C̄ = 4.4355, δ0 = 0.14:
x ≥ x1: δ = 2 ln(10) x − C̄
x0 ≤ x < x1: δ = 2 ln(10) x − C̄ + a (x1 − x)m
x < x0: δ = δ0 · 102(x − x0)
This is electronic stopping only: the quantity that deposits ionizing dose. Nuclear stopping, the displacement damage quantity, is below 0.1 percent of it above a few hundred keV and is not included.
Dose rate, time and fluence
Ḋ [rad/s] = φ · LET · 1.602e-8 (1 rad = 100 erg/g = 6.242e7 MeV/g)
krad/min = Ḋ · 60 / 1000 Gy/s = Ḋ / 100
t = Dtarget / Ḋ Dtotal [rad] = Φ · LET · 1.602e-8
LETlinear [keV/μm] = LETmass · ρ · 0.1 = 0.233 · LETmass for silicon
The reference table inverts the dose rate for 1 krad/min (16.667 rad/s) at 1, 2, 5, 10, 20, 50, 100, 200, 500 and 1000 MeV: φreq = Ḋtarget / (LET · 1.602e-8), and the current column scales with the active beam area.
Custom table interpolation
log LET = log LET1 + (log LET2 − log LET1) · (log E − log E1) / (log E2 − log E1)
Log log between points, log log extrapolation on the end slopes outside the table. An active table replaces Bethe-Bloch entirely.
4. Assumptions
- The beam is uniform over the stated area. The result is the area averaged flux and dose rate; the center of a Gaussian beam is higher and the edge lower. Read the number against the facility's uniformity specification, typically plus or minus 10 percent over the irradiation area.
- The target is silicon and thin: mean energy loss, no slowing down through the sample, no straggling.
- The dose is from primary proton ionization. Secondaries from nuclear reactions (neutrons, gammas, recoils) add a small local dose that is not modeled.
- The beam stays at the configured current for the whole exposure when a time is quoted.
5. Implementation details
- Bethe-Bloch with the Sternheimer correction reproduces PSTAR and ICRU 49 silicon stopping powers to 1 to 2 percent above about 20 MeV.
- The tool shows a warning when Bethe-Bloch is selected below 20 MeV.
- Results appear in rad/s, krad/min and Gy/s, and times in the most convenient unit from microseconds to days.
- The reference table recomputes when the beam area or the stopping power source changes.
6. Limitations
- Below 20 MeV, do not use the built in Bethe-Bloch. Without shell, Barkas and Bloch corrections it underestimates the stopping power, and the error grows as energy falls. Paste a PSTAR table [2] instead: select silicon, generate the electronic stopping power against kinetic energy, paste the two columns (the total column differs from it by under 0.1 percent above 1 MeV in silicon, so either will do). This is the normal case for medical cyclotrons at 2 to 15 MeV and for degraded beams.
- Straggling. Energy loss fluctuates proton to proton (Landau or Vavilov for thin absorbers); the tool reports the mean only.
- Thick targets. If the proton loses a significant fraction of its energy in the sample, the stopping power changes through the depth and the single value here is wrong; use a transport code.
- Dose rate window. MIL-STD-883 TM1019 high rate testing is 50 to 300 rad(Si)/s. Facilities quote krad/min; the readout is there to compare against it, not to certify compliance.
- Precision. The formula is good to a percent or two above 20 MeV, which is more precision than the beam current reading and the uniformity usually justify. The uncertainty in a proton dose rate is in the dosimetry, not in the stopping power.
7. References
[1] Particle Data Group, "Passage of Particles Through Matter," in Review of Particle Physics, Phys. Rev. D.
[2] M. J. Berger et al., "Stopping-Power and Range Tables for Electrons, Protons, and Helium Ions," NIST PSTAR database, physics.nist.gov/PSTAR.
[3] R. M. Sternheimer, M. J. Berger and S. M. Seltzer, "Density Effect for the Ionization Loss of Charged Particles in Various Substances," Atomic Data and Nuclear Data Tables, vol. 30, pp. 261 to 271, 1984.
[4] ICRU Report 49, "Stopping Powers and Ranges for Protons and Alpha Particles," International Commission on Radiation Units and Measurements, 1993.
[5] J. F. Ziegler, "Stopping of Energetic Light Ions in Elemental Matter," J. Appl. Phys., vol. 85, no. 3, pp. 1249 to 1272, 1999.
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