Christopher M Harrington
System Reliability Theory
SMRE Term Project
07 March 2008
Prof. R Ernesto
Introduction
Radioactive material has been found to provide power and forces upwards of 1000 times
the amount of conventional chemical explosion can. This provides a high potential for
energy consolidated into very small packages that would otherwise take up significantly
more size and weight. This application into aerospace structures and systems such as the
Orion Project and Oppenheimer’s Manhattan Project has illustrated our ability to harness
such levels of energy.
Reliability of such systems is paramount due to the fall-out nature of the radioactive
material. Fall-out is the radioactive impact to human life when a substance is forced out
of its natural state and emits alpha, beta or gamma particles into the surrounding
atmosphere. An important design consideration for each respective fuel substance (i.e.
radioactive isotope) is the particle emission timetables. Utilizing data from a study of
time between α-Particle emissions of Americium-241 we can imagine the required
reliability for aerospace structures such to protect precious cargo.
The data table provided in the text will analyze the reliability in understanding the
particle emissions of this particular isotope and how this impacts design. This data is
available from Statistical Methods for Reliability Data, Meeker & Escobar 1998:
Berkson (1966) investigates the randomness of α-Particle emissions of americium-241
which has a half-life of about 458 years. Physical theory suggests that over a short period
of time the inter-arrival times of observed particles is independent and comes from an
exponential distribution. Utilizing this data set we will explore the relationship between
radioactive materials and the reliability of such systems utilizing these isotopes as a
source of power.
The goal of this paper is to begin research into the complex alignment of aerospace
vehicle design requirements and radioactive isotope usage as sustainable fuel.
Understanding of the isotope decay, radioactive fallout and required structural isolation,
and beginning the derivation of proper material thickness and weight iterations. Use of
reliability tools such as MINITAB™ and MAPLE® software as well as basic science and
physics will culminate in a strong foundation and introductory basis to aerospace use of
radioactive isotopes as sustainable fuel. Strong consideration on the public view will be
incorporated in the analysis since it will be the limiting constraint of use of this type of
fuel source.
Reliability of radioactive decay of persistent α-particles and the distribution thereof is an
important factor in understanding the control of such sustainable fuel sources. Continued
research in pulse-rockets requires a significant amount of fuel as well as a highly reliable
system to isolate any radioactive material. Design of such vehicles requires a strong
knowledge of the particle emissions and required materials to isolate such decay.
Methodology & Basic Theory
Aerospace vehicles are constantly seeking new renewable and sustainable forms of
energy: fuel cells, hydrogen fuel and radioactive isotopes. Two basic steps need to be
assessed including the reliability or confidence in the basic isotope decay data, and then
the layering requirements for the isotope.
The radioactive Americium-243 decays into the radioactive isotope Americium-241
through alpha decay resulting in a positively charged ion and Neptunium-237. As an
isotope continues to decay it finds more stability in new isotopes. The half-life is the
time it takes for the isotope to decay to one half of its reactant product.
There are three layers of decay including alpha, beta and gamma particle emissions. The
three different types of decay require varying degrees of protection from the particle
based on the wavelength, energy, and molecular weight of the particle. Basic protective
layers are illustrated below.
Through analysis the radioactive decay of Americium-241 is known to be approximately
85 percent alpha particle decay with the remainder being gamma decay. Different
radioactive decay schemes will cause differing percentages of alpha and gamma particle
emissions. Radiation data for the element is seen below for different reaction yields.
Radiation Data
Percenta
Type
Energy
ge
α
5485 keV
84.5
α
5443 keV
13
γ
59.5 keV
35.9
γ
26.3 keV
2.4
γ
13.9 keV
42
The correlation of radioactive decay data is analyzed in this project. The sample sizes are
varied to illustrate an increasing level of confidence in the range of particle inter-arrival
times. Data is pulled from various sources to utilize the tools for reliability engineering.
Further analysis will begin to derive the requirements for an aerospace vehicle design
utilizing a radioactive isotope.
Further analysis will be required to determine which angles are the worst-case scenarios
for the decay particles. This will determine the final amount of protection required for
the aerospace vehicle. Particle emissions for alpha particles typically follow the
following trajectory.
Alpha particles tend to travel away from positively charged surfaces due to the positive
charge on the particle. Beta and gamma particles also hold charges that will direct them
in different trajectories. More detailed analysis will be required to determine the full
layering requirement for the vehicle. Basic layering of materials and material make-up
are selected at the conclusion of this project with further analysis needed for a detailed
design. The methodology will be set forth and be completed in an iterative manner
beyond this project.
Americium-241 is a byproduct of the decay of Americium-243 on its journey to a more
stable elemental state through α-Particle decay. Americium-243 is a byproduct of the
Uranium families that fuel our nuclear power plants. The decay trend (half-life) chart is
derived through experimentation (measurements) as well as through theoretical
chemistry. Basic stoichiometry is utilized to compare between the gathered data and the
chemistry of the decay. As the population of the data increases the accuracy of the
prediction more closely matches the chemistry.
The purpose of this project is to determine the reliability of the predictive variables and
design the most efficient container for aerospace vehicular use of nuclear fuels for
propulsion. As the decay predictive variables increase the design of the container
becomes more efficient (i.e. not oversized) to compensate for the unknown decay
particles that could potentially come out. Protection against alpha and gamma particles is
required, so in the spectrum of protective materials utilizing tissue is a useless addition.
The thickness of the lead or steel container is the driving factor and therefore the focus of
the layering matrix.
Results: Analysis & Discussion
Several basic reliability functions are renamed in order to fit the logical decay and
progression of this project as seen below:
MTTA: is defined as Mean Time to Particle Arrival
MTTA:=G/λ where λ is the arrival rate of individual α-particles
MTTA follows an exponential distribution with arrival rates normally distributed
The project will simulate layers of protection from particle decay emissions. Taking the
data from the theory and the book, reducing the data into a basic formula and correlating
between the two sets yields the decay equation seen below.
t
F (t ,  )  1  exp(  )

Several data points are validated utilizing the basic data provided in a binned α-particle
test for the Americium-241 element. This data is provided below.
Binned α -Particle Interarrival Time Data in 1/5000 Second
Time
Interval Endpoint
Lower
Upper
0
100
100
300
300
500
500
700
700
1000
1000
2000
2000
4000
4000
∞
Interarrival Times Frequency of O ccurance
All Times
Random Samples of Times
n = 10220
n = 2000
n = 200
n = 20
1609
292
41
2424
494
44
1770
332
24
1306
236
32
1213
261
29
1528
308
21
354
73
9
16
4
0
3
7
4
1
3
2
0
0
The data is then normalized to match the derived decay equation from above and be able
to interpret the results. The normalized data is illustrated below.
Normalized Data
1
0
100
300
500
700
1000
2000
4000
100
300
500
700
1000
2000
4000
∞
0.0066378
0.005
0.003651
0.0026939
0.001668
0.0006304
7.302E-05
6.6E-160
1
0.0059109
0.005
0.0033603
0.0023887
0.0017611
0.0006235
7.389E-05
8.1E-160
1
0.0093182
0.005
0.0027273
0.0036364
0.002197
0.0004773
0.0001023
0
1
0.0042857
0.005
0.0028571
0.0007143
0.0014286
0.0002857
0
0
The overall function derived through various MAPLE™ equations and distribution
curves taken from MINITAB® yield the following radioactive decay build-up with a
432.2 year half-life.
The data points are derived from normally distributed functions at each set as illustrated
in the modified decay curve below.
Each data point or set of data points deriving the overall exponentially decreasing
function describing the particle emissions is represented by an individual normally
distributed set of data points. This is where the power of MAPLE™ software comes in
handy to calculate, integrate and derive the overall functions for us. Each data set is
integrated together in a series of normally distributed data, and then derived down to the
final exponentially distributed decay function. This also helps to visualize the required
casing or storage unit used for the Americium-241. An obvious first guess of the
container design point would be to only use partially decayed Americium-241 to save on
the required thickness (i.e. the decay curve has decreased significantly due to the
exponential behavior). Deciding the function between how much fuel power is required
versus the diminishing returns of that weight hit for making a heavier container is an
important trade study to be completed outside of this project.
Further analysis is completed to determine which of the existing data sets is more
accurate to the normal distribution and therefore having fewer anomalies. Supporting the
thesis of more data is better (i.e. more accurate) are the following probability plots. The
first set of data includes a population of 20 α-particles with a p-value of 0.505.
Probability Plot of n = 20_1
Normal - 95% CI
99
Mean
StDev
N
AD
P-Value
95
90
Percent
80
0.3571
0.3328
8
0.298
0.505
p=0.505
70
60
50
40
30
20
10
5
1
-1.0
-0.5
0.0
0.5
n = 20_1
1.0
1.5
The second set of data includes a population of 200 α-particles (an increase by one factor)
with a p-value of 0.858, a significant increase in accuracy.
Probability Plot of n = 200_1
Normal - 95% CI
99
Mean
StDev
N
AD
P-Value
95
90
p=0.858
Percent
80
70
60
50
40
30
20
10
5
1
0.5682
0.3414
8
0.187
0.858
-1.0
-0.5
0.0
0.5
n = 200_1
1.0
1.5
2.0
Increased distribution confidence on a factor increase of sample by only a factor is an
important reliability function accuracy driver. Since the overall function includes these
distributions, the calculation of the exponential curve will become more and more
accurate as the data population increases. This also will reduce the weight needed to
design the most reliable container to house the fuel when in use on an aerospace vehicle.
Alpha radiation consists of helium-4 nuclei and is stopped by a sheet of paper. Beta
radiation, consisting of electrons, is halted by an aluminum plate. Gamma radiation,
consisting of energetic photons, is eventually absorbed as it penetrates a dense material.
Each of these particle emissions are stopped by different materials of different thickness.
A final matrix will be required in the end of the design for the vehicle to determine the
most efficient and reliable skin. Remember, radioactive material is a public concern and
therefore the reliability for the container will have to exceed normal aerospace
requirements (beyond the rule of nine-zero’s). Our reliability function will need to
exceed 100% to ensure the public that the container is safe for use, especially in a pulse
rocket convention. 241-Am has Decay Energy of 5.638 MeV and requires 0.01 cm of
lead (Pb) for containment of this energy. This is based off of the current data in chemical
journals and databases, which accounts for approximately 150% reliability. This is
illustrated by the red dot in the plot below for absorption coefficients.
Pb
Future Work includes correlating absorption coefficients and angles of particle emissions
with radioactive isotope for aluminum and lead to lead to a more efficient design. The
oversize of the container as it is today is a huge concern for aerospace industries since
weight is such a driving factor in the design. Aluminum is not required for α & λ
particles since the more stringent lead sheeting will take care of those particles and is
needed for higher energy particle emissions.
Conclusion
Increased data sources increase reliability of measurements in the end will help to create
a more efficient design. The limitation of the outliers or anomalies will increase the
overall reliability function since over sizing is a result of less accurate predictions. The
overall standardized errors of the sample populations are included below illustrating the
significant increase in predictive accuracy.
n=20: Std Error 52
n=200: Std Error 13
n=2000: Std Error 3.8
The scale (with respect to Inter Quartile Ranges) increases in accuracy from 3.1 to 2.2
with a correlation factor of 0.966. This supports the need for more data to determine the
final matrix and diminishing power returns needed for the vehicle. All of these factors
will be included in the final matrix to determine the final makeup of the containing
vessel.
For highest reliability a population of n=10220 is utilized. This reduces the standard
error in the distribution to 1.7; however for public concerns due to radioactive fall-out
impacts there is still a need to be higher for public to be comfortable with isotopes. Each
distribution used (normally distributed samples) is illustrated below.
Increased decay analysis can be completed through stoichiometry and Monte Carlo
Process Simulations. Simulations increase in complexity since as each particle decays
(release of an alpha, beta or gamma particle) a new element or isotope emerges. This
continues through the half-life of the parent element until it reaches a stable elemental
state. Further analysis of the half-life data and elemental decay as illustrated in the figure
below is required to fully determine weather or not the Americium-241 has the worstcase decay particles or not. Some elements may have higher energy decay particles that
will size the walls of the container larger than initially anticipated. Below is the decay of
Americium-241 to its stable state, as can be seen are various elemental paths and some
options as to which isotope the decay goes to. When options are included in the
simulation probabilities and different particle energies are listed.
Further analysis of this through non-linear programming will yield a more accurate
predictive model that will come up with more effective normally distributed sample
distributions. These distributions can in turn be recalculated through the decay cycle and
run through MAPLE™ and deriving out new mean times to arrival and sizing codes. The
process iterates on more data in the future state.
References
Argonne National Laboratory, EVS Human Health Fact Sheet August 2005, Americium:
http://www.ead.anl.gov/pub/doc/Americium.pdf, Google 2005.
ChemGlobe Periodic Table of the Elements:
http://pol.spurious.biz/projects/chemglobe/ptoe/, Google: ©2000.
Chemical & Biological Warfare Information, Factsheets for Chemical & Biological
Warfare Agents: http://www.cbwinfo.com/Radiological/radmat/am241.shtml, Nolo Press,
Berkeley CA, 1997.
The Encyclopedia of Alternative Energy and Sustainable Living:
http://www.daviddarling.info/encyclopedia/AEmain.htm, Darling, David.
Hyperphysics Radioactivity: http://hyperphysics.phyastr.gsu.edu/hbase/nuclear/radact.html.
Jefferson Laboratories, It’s Elemental: http://education.jlab.org/itselemental/ele095.html,
12000 Jefferson Avenue, Newport News, VA 23606.
NASA Orion Project Office: http://spaceflightsystems.grc.nasa.gov/Orion/, NASA:
March 7, 2008.
Reliability Engineering and Risk Analysis, M. Modarres et al, CRC, Boca Raton, 1999.
Statistical Methods for Reliability Data, W. Meeker and L. Escobar, Wiley, NY, 1998.
System Reliability Theory, 2nd ed., M. Rausand and A. Hoyland, Wiley, Hoboken, New
Jersey, 2004.
Appendix
//Basic build-up of alpha particle emissions.
//This simplified version is to simulate a reliability
//experiment on a complex system for
//RPI Cohert IPM V.
//This example will illustrate the reliability of a predictive function: the data supports
//an exponential distrubution of a natural event; the functions derived form this data
//illustrate the predictions and classification of the particular elements' decay.
//As radioactive materials become more integrated as sustainable energy
//reliability of the predictive functions must provide sufficient safety data to be
//precieved as a system just as safe as automotive or aerial vehicles and sources of
energy.
Histogram of n = 10220, n = 2000, n = 200, n = 20
Normal
50
Variable
n = 10220
n = 2000
n = 200
n = 20
Frequency
40
Mean
1278
250
25
2.5
30
20
10
0
-500
0
500
1000
1500
Data
2000
2500
3000
StDev
772.6
152.9
15.02
2.330
N
8
8
8
8
Distribution Overview Plot for n = 20, n = 200, n = 2000
LSXY Estimates-Complete Data
Probability Density Function
V ariable
n = 20
n = 200
n = 2000
Normal
99
0.16
Percent
PDF
90
0.08
50
10
0.00
0
200
Data
400
1
600
0
Survival Function
200
Data
400
600
400
600
Hazard Function
100
Rate
Percent
2
50
0
1
0
0
200
Data
Table of S tatistics
M ean S tDev C orr
2.5
2.477 0.958
25.0 16.412 0.985
250.0 163.748 0.966
F
8
8
8
400
600
0
200
Data
C
0
0
0
Main article: Isotopes of americium
NA
half-life
DM
DE (M eV)
SF
241
α
5.638
Am
syn
432.2 y
IT
0.049
α
5.637
SF
242mAm syn
141 y
SF
243
α
5.438
Am
syn
7370 y
iso
Radiation Data
Percenta
Type
Energy
ge
α
5485 keV
84.5
α
5443 keV
13
γ
59.5 keV
35.9
γ
26.3 keV
2.4
γ
13.9 keV
42
paper:
aluminum:
lead:
0.8 g/cm^3
2.7 g/cm^3
11.34 g/cm^3
DP
237Np
238Np
239Np