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Unformatted text preview: Chapter 1
RADIOACTIVE TRANSFORMATIONS THEORY, THE WEAK
FORCE
© M. Ragheb
2/14/2009 1.1 INTRODUCTION
At any given moment thousands of rays of radiation are crisscrossing the human body.
These rays are invisible, and are just part of nature. It is important that we all become literate
about radiation, to alleviate unsubstantiated fears and to be able to protect our coworkers,
families and ourselves when undesirable situations involving radiation arise, such as the
accumulation of Radon222 and its daughter products in some dwellings.
The quest by the middle age alchemists for the philosopher's stone that has the power to
transmute one element into another had been the obsession of many great thinkers throughout
history including Sir Isaac Newton, Robert Boyle and John Locke. This dream of the ancient
alchemists was achieved in 1919 at Manchester University in the UK.
A student of New Zealand born scientist and Nobel Prize winner Ernest Rutherford
noticed that when radioactive materials such as radium were placed in a sealed container of air,
small amounts of hydrogen, which does not exist in ordinary air, began to mysteriously appear.
Ernest Rutherford explained the occurrence that the powerful radioactive alpha particle from the
decay of radium interacted with nitrogen, which constitutes about ¾ of the air we breathe and
transmuted into two other gases: hydrogen and an isotope of oxygen, through the nuclear
reactions:
88
2 Ra 226 → 86 Rn 222 + 2 He4 He4 + 7 N 14 → 1 H 1 + 8 O17 ______________________________
88 (1) Ra 226 + 7 N 14 → 1 H 1 + 8 O17 + 86 Rn 222 The alpha particles being produced by the radium were acting as catalysts in an overall
reaction where the radium and nitrogen nuclei were turning into radon and oxygen while
knocking out single protons which, by combination with electrons became hydrogen atoms then
hydrogen gas molecules.
Nowadays the transmutation of an element into another through nuclear reactions is a
routine process, with this modern nuclear alchemy playing a major role in the way we produce
and use energy. 1.2 MODES OF RADIOACTIVE DECAY Some nuclei in nature, and some artificially created, attempt at reaching a stable
configuration in the nucleus through emitting the excess particles or electromagnetic radiation.
These nuclear transformations are called radioactive transformation, or just radioactivity.
The main modes of radioactive transformations are:
1. Negative beta decay: These are electron emissions from the nucleus. They occur primarily
in neutron rich nuclei in an attempt at reaching stability by increasing the number of protons
in the nucleus.
2. Positron decay: Here positive electrons, the anti matter of negative electrons are emitted by
proton rich nuclei in an attempt at reaching stability by decreasing the number of protons in
the nucleus.
3. Alpha decay: This is the emission of a whole helium nucleus from the parent nuclide. This
mode of radioactivity primarily occurs among the heavy nuclides such as 92Uranium238.
4. Gamma decay: This is an emission of electromagnetic radiation of very short wave length
from excited nuclei on their way to reaching the ground state. It can occur by itself, but
normally accompanies beta decay.
5. Orbital electron capture: Occurs in proton rich nuclides, where an inner shell electron is
grabbed by the nucleus, with the emission of characteristic x rays from the ensuing electron
transition.
6. Delayed Radiations: Can involve neutrons, protons alphas and gamma emissions. Delayed
neutron emission is notable in fission products and affects the control of fission reactors.
7. Isomeric Transitions: Occurs when gamma rays are emitted for an excited nucleus to reach
its ground state.
8. Internal conversion: Involves the direct transfer of energy from the nucleus to one of the
orbital electrons, and the electron is ejected from the atom.
9. Spontaneous fission: Some heavy nuclei decay in a process where the nucleus breaks up
into two intermediate mass fragments and several neutrons. It occurs in with nuclei with a
mass number A > 230.
10. Double beta decay: A rare radioactive event observed for Mo92 and Mo100.
11. Cluster decay: Has been observed in several heavy nuclides where clusters of C12, C14, O20,
Ne20, Mg28, or Si32 are emitted. 1.3 RADIOACTIVE DECAY LAW
HEURISTIC Approach Consider an initial number of radioactive nuclides at time t = 0 as N0. These nuclei
would be undergoing radioactive transformations and the initial number of nuclei is going to
decrease over time. If we consider the time at which the initial number is decreased by one half
to N0/2, we can designate this time as the half life T1/2. These nuclei would continue decaying to
¼ their initial value after 2 half lives, to 1/8 of their initial value after 3 half lives, and so on. In
general, after n half lives, these nuclei would have decayed to (1/2)n their initial value N0 as
shown in Table 1 below.
Table 1: Number of radioactive nuclei present after n half lives. Number of
halflives
0
1
2
3
n Elapsed
time
0
1 T1/2
2 T1/2
3 T1/2
n T1/2 Number of nuclei
present
N0
N0/2
N0/4
N0/8
N0/2n Using mathematical induction the number of nuclei present after n halflives is given by:
⎛1⎞
N (n ) = N 0 ⎜ ⎟
⎝2⎠ n (2) Since the time elapsed t is equal to the number of halflives:
t = n.T1 2 from which:
n= t
,
T1
2 Equation 2 can thus be written as a function of time by substituting for n as;
t ⎛ 1 ⎞ T1
N (t ) = N 0 ⎜ ⎟ 2
⎝2⎠ (3) This law of radioactive decay has a different form that can be derived based on
differential calculus considerations.
EXPONENTIAL DECAY LAW Let the change during a time period dt in the number of radioactive nuclei present be
dN(t). The change dN(t) is both proportional to the number of nuclei present N(t) and the time
interval dt:
dN (t ) α − N (t )dt (4) The negative sign accounts for the fact that the radioactive nuclei are decreasing in
number as a function of time. The proportionality sign can be replaced by an equality sign if we
add a decay constant λ to Eqn. 4 as: dN (t ) = −λN (t )dt (5) To determine N as a function of time, we separate the variables in Eqn. 5:
dN (t )
= −λdt
N (t ) This equation can be integrated from the initial time t = 0 to any time t using limit integration:
N (t ) ∫ N0 t dN (t )
= −λ ∫ dt
N (t )
0 Integrating yields:
N (t ) ln N (t ) N 0 = − λt t
0 Substituting the upper and lower limits, we get: ln N (t ) − ln N 0 = ln N (t )
= −λt
N0 Taking the exponential of both sides yields:
ln e N (t )
N0 = N (t )
= e −λt
N0 This yields a negative exponential process described by the radioactive decay law with N(t)
being the number of nuclei present after a certain time t: N (t ) = N 0e − λ t
where: (6) N0 is the initial number of nuclei present at time t = 0
λ is the radioactive decay constant. 1.4 HALF LIFE AND MEAN LIFE
DEFINITION OF HALF LIFE The half life T1/2 is the time at which a radioactive isotope’s number of nuclei at time t,
N(t), has decayed to one half its initial number of nuclei N0/2. Expressing this fact in the
radioactive decay law above, one can write: N (T 1 ) =
2 − λT 1
N0
= N0e 2
2 Canceling the N0 term on both sides of the equation we get:
− λT 1
1
=e 2
2 To eliminate the exponential we take the natural logarithm of both sides of the equation,
yielding:
ln 1 – ln 2 = –λ T1/2
Substituting ln 1 = 0, we can express the halflife in terms of the decay constant as: T1 =
2 ln 2 λ = 0.6931 λ (7) Similarly, we can express the decay constant in terms of the halflife as: λ= ln 2 0.6931
=
T1
T1
2 (7)’ 2 This suggests another form of the radioactive decay law:
− N (t ) = N 0e 2 − = N 0e ln 2
t
T1
0.6931
t
T1 (8) 2 EQUIVALENCE OF THE TWO FORMS OF THE RADIOACTIVE DECAY LAW One can express the radioactive decay law in terms of the number n of half lives elapsed:
n= t
,
T1
2 in the form: N (n ) = N 0e − n ln 2 We note that:
e − n ln 2 = 1
e n ln 2 = 1
e ln 2n = 1
,
2n then:
n ⎛1⎞
N (n ) = N 0 ⎜ ⎟ ,
⎝2⎠ which is Eqn. 2, and proves that the two forms of the radioactive decay law are equivalent.
A radioactive isotope, according to the radioactive decay law in either of its forms, has
substantially decayed after a few half lives. For instance, after seven half lives only:
7 1
⎛1⎞
⎜ ⎟ =
⎝ 2 ⎠ 128
or less than 1 percent of the original amount remains. One can construct a simple Table 2
showing the fraction of a radioactive isotope remaining after n half lives.
A rule of thumb is that after ten half lives; only 1/1000 of the original nuclei are
remaining. After twenty half lives, only 1 millionth of the original nuclei remain. And after
thirty half lives; only one billionth of the original radioactive nuclei would have decayed. Thus
it does not take a large number of half lives for a radioactive sample to decay.
Table 2: Fraction of a Radioactive Isotope remaining after n half lives
Number of half lives (n)
1
2
3
4
5
6
7
8
9
10 Fraction remaining
½
¼
1/8
1/16
1/32
1/64
1/128
1/256
1/512
1/1024 MEAN LIFETIME The process of radioactive transformation is a random process. The mean lifetime or
average life expectancy is the mathematical expectation of the time that it takes a radio nuclide to
decay over the law of radioactive decay as a probability density function. ∞ τ= ∞ ∫ tdN (t ) ∫ tλ N e − λt 0 = 0
∞ 0
∞ ∫ dN (t ) ∫ λ N e
0 0 ∞ dt
= − λt ∫ te
0
∞ ∫e dt 0 − λt ∞ dt
= − λt dt 0 ∫ te − λt 0 e − λt
−λ dt ∞ ∞ = λ ∫ te − λt dt
0 0 Integrating by parts:
∞ τ = λ ∫ te − λt dt =
0 = te
= − λt 0
∞ 0 ∞ λ
tde − λt = ∫ tde − λt
−λ ∫
0
∞
0 0 − ∫ e dt = − ∫ e − λt dt = −
− λt ∞ ∞ 1 − λt 0
e
∞
−λ (9) 1 λ Thus, the mean life is simply the inverse of the decay constant.
The law of radioactive transformations can be expressed in terms of the mean life in
another form as:
N (t ) = N 0 e − t τ (10) DECAY CURVES The transformation of tritium T3 into the He3 isotope is governed by its radioactive decay
equation with N(t) being the number of nuclei present after a certain time t:
ln 2
−
t
N (t )
− λt
12.33
=e =e
N0 where: N0 is the initial number of nuclei present at time t = 0
ln 2 0.6931
=
λ is the decay constant =
.
T1
T1
2 2 T½ is the half life for tritium = 12.33 years.
An American National Standards Institute (ANSI) Fortran90 (f90) procedure that can
display the decay features of the tritium isotopes is listed here:
!
!
!
!
!
! Decay Curve generation for Tritium
N(t)=No*exp(lambda*t)
lambda=decay constant= ln 2 / T
T=halflife
Program written in ANSI Fortran90
Digital Visual Fortran Compiler !
!
! ! ! !
! !
!
! Procedure saves output to file:output1
This output file can be exported to a plotting routine
M. Ragheb, University of Illinois
program decay
real x, lambda
This half life is for the tritium 1T3 nucleus
real :: T = 12.33
integer :: steps=100
real ratio(101), xtime(101)
Calculate decay constant
x = log(2.0)
lambda = x/T
write(*,*) x, lambda
Open output file
open(10,file=’output1’)
Calculate ratio N(t)/No
steps = steps + 1
do i = 1, steps
xtime(i) = i  1
ratio(i) = exp ( lambda*xtime(i))
Write results on output file
write(10,*) xtime(i), ratio(i)
Display results on screen
write(*,*) xtime(i), ratio(i)
pause
end do
end The output file was plotted in Fig. 1 using the Excel Plotting Package where the ratio
N(t)/N0 is shown, and shows the rapid decay of tritium as a function of time. N(t)/N0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.00 20.00 40.00 60.00 80.00 100.00 Time (years) Fig.1 Decay Curve for Tritium 1.5 ACTIVITY
The measurement of the intensity of radioactive transformations is possible if we use the
rate of radioactive transformations from Eqn. 6 as: dN (t ) d
= ( N 0e − λ t ) = −λ N (t )
dt
dt (11) instead of just the number of nuclei present. This is achieved in terms of the radiological
quantity designated as “Activity.” It is defined as the positive value or magnitude of the product
of the radioactive decay constant and the number of radioactive nuclei present at any time t: A(t ) = dN (t )
= − λ N (t ) = λ N (t )
dt (12) This can also be written in the form: A(t ) = λ N 0e − λt = A0e − λt (13) Usually it is estimated at time t = 0 as:
A0 = λ N 0 (13)’ In terms of the mean lifetime, it can be expressed as:
A(t ) = A0e − t τ (14) or: A(t ) =
A0 = N (t ) τ N0 τ (15) (15)’ It is equal to the number of transformations per unit time. The unit used in the Système
International (SI) system of units is:
1 Becquerel = 1 Bq = 1 [Transformation/sec].
In the conventional system of units, the unit used is the Curie, where:
1 Curie = 1 Ci = 3.7 1010 Bq,
which is the amount of activity of 1 gm of the 88Radium226 isotope. Smaller units are used:
1 milliCurie = 1 mCi = 103 Ci,
1 microCurie = 1 μCi = 106 Ci.
Half lives vary widely from one radioactive isotope to another. Excited states of some
nuclei decay with half lives in a range from a thousandth to a trillionth of a second. On the other
hand, the most common naturally occurring isotope of Uranium, 92U238, has a halflife of 4.468
billion years. Similarly, the naturally occurring potassium isotope 19K40 exists in the human
body at an abundance of 0.01 percent and has a half life of 1.27 billion years.
Tritium, 1T3, a man made isotope of hydrogen, does not exist in nature, except for trace
amounts from cosmic ray interaction with hydrogen in the atmosphere, and has a half life of
12.33 years. For tritium to decrease to 1/1000 of its initial amount, would require 10 half lives or
about 123 years. On the other hand, only a few years are needed for a significant decrease in the
amount of tritium initially present. As a consequence, the tritium used in boosted nuclear fission
devices has to be replenished every few years for a viable nuclear weapons arsenal. Interestingly, for 92U238 to decrease to 1/1000 of its initial amount would require 44.68
billion years. This is about three times the age of the universe at about 15 billion years. It will
be there for a long time. On the other hand 94Pu239, the man made fissile isotope, has a half life
of 24,110 years, and decays into the naturally occurring 92U235 isotope through alpha particle
emission and eventually into a stable lead isotope. For this reason, primordial 94Pu239 is not
found in nature any more. Trace amounts of it can be found in uranium ores as a result of
spontaneous fission neutrons capture in U238.
The isotope of strontium 38Sr90, a fission product, has a half life of 29 years, so it would
take a life span for it to decay significantly. Thus if ingested in the human body, where it mimics
calcium, it seeks the bone system, and remains there for practically a lifetime.
An isotope used in medical studies, 8O15, has only a 2 minutes half life. A patient who is
injected with this isotope will have only 1/1000 of the original radioactive dose of the isotope
present after 20 minutes. After one hour, amounting to thirty half lives, only one billionth the
original amount of the radioactivity remains. This is at the core of the beneficial uses of
radioisotopes in nuclear medicine applications. 1.6 SPECIFIC ACTIVITY, ACTIVITY DENSITY
When the activity is estimated per unit mass of a solid material, it is designated as
specific activity. The most commonly used units in this case are:
1 Curie/gram = 1 Ci /gm,
1 Becquerel/gram = 1 Bq / gm
When a liquid is under consideration, the activity density rather than specific activity is
used, such as:
1 Curie/liter = 1 Ci /l,
1 Becquerel/cubic centimeter = 1 Bq/cm3. 1.7 DETERMINATION OF HALF LIFE
The half life, and the decay constant can be determined experimentally. Taking the
natural logarithm of the radioactive transformation law, expressed in terms of the activity of a
sample: ln A(t )
= ln e − λt = −λ t
A0 This appears to be an equation of a straight line with a negative slope of m = λ.
ln A(t ) = ln A0 − λ t
y (t ) = y0 − mt (16) If the logarithm of the measured activity is plotted against the time t, a straight line
should result with a slope of –λ , which itself is equal to ln 2 / T1/2, allowing for the experimental
determination of the half life. 1.8 PRODUCTION OF RADIO NUCLIDES
Radioactive isotopes can be produced by bombardment with charged particles such as
protons or helium ions in particle accelerators. However, the most efficient way is to produce
them with the bombardment with neutrons, since they do not have to overcome the Coulomb
barrier of the nucleus like charged particles have to do. Nuclear reactors being a copious source
of neutrons have been used for the production of radioactive isotopes through the neutron
irradiation of otherwise stable nuclides.
Assuming that the neutron bombardment hardly affects the original material; a good
assumption in high flux reactors, the net rate of change of the number of radioisotopes present in
a reactor will be equal to the production rate (Q) minus the decay rate of the isotope or: dN (t )
= Q  λ N (t )
dt (17) Rearranging: dN (t )
+ λ N (t ) = Q
dt
Multiplying both sides by an integrating factor eλt, converts the left hand side into a total
differential e λt dN (t )
+ e λt λ N (t ) = Qe λt
dt d [ N (t ) eλt ]
= Qe λt
dt
d [ N (t ) eλt ] = Qe λt dt
Integrating both sides yields:
N (t) ∫ N d [ N (t)e λt ] = ∫Qe
0 0 N ( t )eλ t − N 0 = t Q λ ( eλt − 1) λt dt Multiplying both sides by e λt results in: N (t ) = N 0e − λt + Q λ (1 − e − λt ) (18) If the initial number of isotopes is zero, N0 = 0: N (t ) = Q λ (1 − e − λt ) (19) Written in terms of the activity generated, we get:
A(t ) = λ N (t ) = Q (1 − e − λt ) (20) This equation describes a process by which the isotope builds up to a saturation value at t = ∞ of:
A∞ = Q (21) It is worthwhile to bombard only for a period of 2 to 3 half lives, since 3/4 to 7/8 of the
maximum number of nuclei (Q / λ) is then produced. Irradiating the isotope for a longer time
becomes uneconomical since the cost of the irradiation could become prohibitive.
Upon stopping the irradiation, the radioactive isotope decays according to its own halflife as shown in Fig. 2:
A(t ) = Q (1 − e − λts ) e − λ ( t −ts ) (22) where ts is the time at which irradiation has stopped.
The last equation can be rewritten as:
A(t ) = Q ( eλts − 1)e − λt (22)’ EXAMPLE As an example, let us consider the production of the isotope 25Mn56 from 25Mn55. The
latter has a natural abundance of 100 percent and is the only naturally occurring isotope of
manganese. It can be placed in a nuclear reactor where it can undergo 1010 (n, γ) reactions per
second. The reaction is:
55
25Mn + 0n1 → 25Mn56 + γ The formed isotope is radioactive and decays with a half life of 2.58 hours into a stable iron
isotope, with the emission of a negative electron and an antineutrino: 56
25Mn → 26Fe56 + −1e0 + ν∗ The decay constant can be calculated as: λ= ln 2 0.6931
=
= 0.269[hr ]−1
T1
2.58
2 One can calculate the activity reached after 5 hours as:
A(5 hrs) = 0 + 1010 ( 1  e0.269 x 5)
= 7.39 x 109 [Bq]
= 7.39 x 109 / 3.71 x 1010 = 0.2 [Ci]
Here we considered that there was a zero value of the generated isotope at the time of initial
irradiation. 1.9 PROCEDURE FOR THE ESTIMATION OF THE PRODUCTION OF
AN ISOTOPE
The following procedure can be used to estimate the growth of the activity for the
production of an isotope in a nuclear reactor, and its subsequent decay according to Eqns. 20 and
22.
!
!
!
!
!
!
!
!
! !
!
!
!
!
! ! Activity buildup curve for the production of an isotope:
A(t)=Q*(1exp(lambda*t))
Followed by decay after end of irradiation
A(t)=Q*(1exp(lambda*tira))*exp(lambda*t)
tira = irradiation time
lambda = decay constant = ln 2 / T
T=halflife
Program saves output to file:output1
This output file can be exported to a plotting routiine
program isotope_production
real x, lambda
This half life is for the 25Mn56 nucleus in hours
It is formed through neutron irradiation from 25Mn55
It decays to 26Fe56 through negative beta emission
real :: T = 2.54
Q is production rate
real :: Q = 1.0E+10
Conversion ratio from Becquerels to Curies
real :: C =3.71e+10
Irradiation time: 10 hours, decay time: 10 hours
integer :: steps=20
real activity(51), xtime(51)
Calculate decay constant
x = log(2.0)
lambda = x/T
write(*,*) x, lambda !
!
! !
!
! ! !
! 55
25Mn Open output file for plotting in Excel
open(10,file='output1.xls')
Calculate ratio activity in Curies
steps = steps/2.0
Irradiation time
do i = 1, steps
xtime(i) = i  1
activity(i) = Q*(1exp ( lambda*xtime(i)))/C
Write results on output file
write(10,*) xtime(i), activity(i)
Display results on screen
write(*,*) xtime(i), activity(i)
Store irradiation activity in Curies
QQ=activity(i)
end do
jj=steps
Decay time
do i = 1, steps+1
xtime(i) = jj+i1
ti=i
activity(i) = QQ*exp ( lambda*ti)
Write results on output file
write(10,*) xtime(i), activity(i)
Display results on screen
write(*,*) xtime(i), activity(i)
end do
end Figure 2 shows the growth of the activity for the irradiation of the manganese isotope
isotope for 10 hours to produce 25Mn56 followed by 10 hours of its beta decay into 26Fe56. .
Activity [Ci] 0.30 0.25 0.20 0.15 0.10 0.05 25.00 20.00 15.00 10.00 5.00 0.00 0.00 Time [hour] Fig 2: Growth an irradiated isotope in a nuclear reactor, and its decay after the stoppage of
irradiation. 1.10 RADIOISOTOPIC APPLICATIONS
Radioisotopes have wide usage in consumer products, industry, medicine, biology and
scientific research. Most people are unaware that they are safely used in a wide variety of
applications. Some of these peaceful as well as military applications can be listed in Table 3.
Table 3: Origin and Civilian and Military uses of Radioactive Isotopes
Radioactive Isotope
241
95Americium Half Life T1/2
432 y 109
48Cadmium
47
20Calcium 453 d
4.54 d Usage
Smoke detectors for homes and businesses
Measuring levels of toxic lead in dried paint samples
Online thickness gauges to ensure uniform thickness in rolling processes like paper, steel,
Aluminum, paper, and plastic production
Oil wells logging
Analysis of metal alloys for checking stock, sorting scrap
Biomedical research for the study of cell function and bone
Formation in mammals 98Californium 252 5730 y 14
6Carbon 55Cesium 30.17 y 137 24Chromium
57
27Cobalt 27.71 d
271 d 51 5.27 y 60
27Cobalt 29Copper 61.7 h 67 18.11 y
13.2 h
1.59x107 y 244
96Curium
123
53Iodine
129
53Iodine
53Iodine 8.041 d 131 77Iridium 74.2 d 192 2.7 y
10.72 d 55
26Iron 85
36Krypton 28Nickel 100 y 63 32
15Phosphorus
Plutonium238
94 94Plutonium
84Polonium 239 226 14.28 d
87.74 y 2.411x104 y
138.38 d 210 61Promethium 88Radium 2.64 y 147 13.6 m
1600 y Its neutron emission used in the in airports for the inspection of airline luggage for hidden
explosives
Gauging the moisture content of soils in road construction and
building industries
Measuring the moisture content of materials stored in silos
Formed by cosmic rays neutrons bombardment of 7N14 in the upper atmosphere
Archaeological dating
Research ensuring that new drugs are metabolized without forming harmful byproducts
Its gamma rays emission is used in the treatment of cancers
Sterilization of medical products and food products against harmful
pathogens such as E. Coli 0157:H7, Listeria, Salmonella, and
Campylobacter
Measuring correct patient dosages of radioactive pharmaceuticals
Measuring and controlling liquid flow in oil pipelines
Testing oil wells for sand blockages
Height gauges for fill level for packages and containers of food,
drugs and other products
Research in red blood cells survival studies
In Nuclear Medicine for the interpretation of diagnosis scans
of patients’ organs, and for the diagnosis of pernicious anemia
Its gamma rays emissions used in the sterilization of surgical instruments and medical
products
Treatment of cancers
Meat, poultry, fruits and spice products sterilization against
harmful organisms
Improving the safety and reliability of industrial fuel oil burners
Attached to monoclonal antibodies which seek cancer cells in
the body to destroy them through radioactive emissions
Analysis of materials excavated from pits slurries from drilling operations in mining
Diagnosis of thyroid gland disorders such as Grave’s syndrome
Check some radioactivity counters at in vitro diagnostic testing
laboratories
Formed as a fission product in the fission process
Released from nuclear explosions and postulated reactor accidents
Diagnosis and treatment of thyroid cancer nodules in Grave’s
syndrome
Nondestructive testing the integrity of pipelines welds,
boilers and aircraft parts
Analyzing electroplating solutions
Used in indicator lights in electrical appliances such as cloth
washers and dryers, stereos and coffee makers
Measurement of dust and pollutant levels
In thickness gauges in the manufacturing of thin plastic and sheet
metal, rubber, textiles and paper
Detection of explosive materials
Voltage regulators and current surge protectors in electronic devices
Used in molecular biology and genetics research
Its alpha emissions used as a heat source and through thermionic
conversion as an electrical source in deep space probes and crafts
Thermionic electrical source in imbedded heart pacers
Fuel for future breeder fission reactors
Fission nuclear weapons devices
Through its alpha particles emissions, elimination of static
charges in the manufacturing of photographic film and records
Used in electrical blankets thermostats
Gauging the thickness of thin plastics,thin sheet metal, rubber,
textiles and paper
Daughter nuclide in the decay chain of 92U238
Discovered by Marie and Pierre Curie
Enhances the effectiveness of lightning rods 222
86Radon 3.82 d 75
34Selenium
24
11Sodium 120 d
15.02 h 85
38Strontium 65.2 d 43Technetium
204
81Thallium 90Thorium
90Thorium 1Tritium 229
230 99m 6.02 h
3.77 y
7340 y
7.7x104 y 12.33 y 3 92Uranium 234 2.44x105 y 92Uranium 235 7.04x108 y 92Uranium 238 4.468x109 133
54Xenon 5.25 d Daughter nuclide in the decay chain of 92U238
Formed from the decay of 88Ra226
Concentrates in overly insulated homes.
Health hazard in home construction, uranium mining, and cigarette smoking.
Used in protein studies in life sciences research
Location of leaks in industrial pipelines
Oil well logging studies
Fission product under calcium in the periodic table of the elements
Constituent of fallout from nuclear weapons testing
Studies of bone formation and metabolism
Diagnostic studies in nuclear medicine including brain, bone, liver,
spleen and kidney imaging and for blood flow studies
Measurement of dust and pollutant levels on filter paper
Thickness gauges in plastics, sheet metal, rubber, textiles and paper
manufacturing
Used in making fluorescent lights last longer
Used to breed 92U233 as afissile fuel in thermal fission breeder reactors
As thoriated tungsten, used in electric arc welding rods in the
construction, aircraft, petrochemical, and food processing
equipment industries to produce easier starting, enhanced arc
stability and reduced metal contamination
Life Science and drug metabolism in new drugs studies
Self luminous aircraft and commercial exit signs
Luminous dials, gauges, Liquid Crystal Displays (LCDs), and wrist watches
Production of luminous paint
Fuel for future fusion reactors
In boosted fission, thermonuclear, enhanced neutron, and directed energy weapon devices
Short half life implies the need to regularly remanufacture nuclear weapons
In dental fixtures like crowns and dentures to provide a natural
color and brightness
Fuel for nuclear power plants and naval propulsion systems
Early fission weapons devices
Manufacture of fluorescent glassware
Colored glazing for ceramics and wall tiles
Predominant uranium isotope occurring with a 99.3 percent natural abundance
Cannot be used to create a self sustained chain reaction
Breeder material for breeding Pu239 in fission breeder reactors
Shielding material against xrays, neutron and gamma radiation
Shielding armor against projectiles
Kinetic energy projectiles in anti tank weapons
Energy amplification in boosted fission and thermonuclear weapon devices
In lung ventilation and blood flow studies in Nuclear Medicine 1.11 RADIOACTIVITY IN FOOD
The accumulation in food of the isotopes of Ra226, Th232, K40, C14 and T3 causes a
radiation equivalent dose to the human body averaging 20 mrem/year. The average banana fruit
contains about 400 mg of potassium, leading to a specific activity of 3 pCi/gm from its K40
content. Brazil nuts are notorious for their radium content that causes a specific activity of 14
pCi/gm. Table 4 shows the specific activities in some food items.
Table 4: Specific activities and activity densities of some food items.
Food item Salad oil Specific Activity
4,900 pCi/l Milk
Whiskey
Beer
Tap water
Brazil nuts
Bananas
Tea
Flour
Peanuts and peanut butter 1,400 pCi/l
1,200 pCi/l
390 pCi/l
20 pCi/l
14.00 pCi/gm
3.00 pCi/gm
0.40 pCi/gm
0.14 pCi/gm
0.12 pCi/gm 1.12 SUCCESSIVE RADIOACTIVE TRANSFORMATIONS
It can be observed that rarely does a radionuclide decay into other stable isotopes in a
single step. Normally a chain of steps is encountered until a stable nuclide is reached. Consider
the case of a radioactive isotope 1 decaying into another isotope 2, which in turn decays into a
stable isotope 3. The rate equations for such a system are:
dN1 (t )
= − λ1 N1 (t )
dt
dN 2 (t )
= + λ1 N1 (t ) − λ2 N 2 (t )
dt
dN 3 (t )
= + λ2 N 2 (t )
dt (17) This is a coupled set of first order ordinary differential equations. The first equation has a simple
solution obtained by separation of variables:
N1 ( t ) = N10 e λ1 t (18) Inserting this equation into the second rate equation yields: dN 2 (t )
= + λ1 N10 e
dt − λ1 t − λ2 N 2 (t ) or: dN 2 (t )
+ λ2 N 2 (t ) = λ1 N10 e
dt
Multiplying by an integrating factor eλ2 t , we get: − λ1 t dN 2 (t )
+ λ2 N 2 (t )eλ2t = λ1 N10 e
dt
d [ N 2 (t )eλ2t ]
= λ1 N10 e( λ2 −λ1 ) t
dt eλ2t − λ1 t λ2t e Separating the variables and integrating, we get:
N2 (t ) ∫ t d [eλ2t N 2 (t )] = +λ1 N10 ∫ e( λ2 −λ1 ) t dt
0 N 20 λ1 eλ2 t N 2 ( t ) − N 20 = + (λ2 − λ1 ) N10 [e ( λ2 −λ1 ) t − 1] Multiplying both sides by e − λ2t , N 2 ( t ) = N 20e − λ2t + λ1 (λ2 − λ1 )  λ1 t N10 (e − e  λ2 t ) (19) If the initial number of nuclei N20 is zero, the equation reduces to: N2 (t ) = λ1 (λ2 − λ1 ) N10 (e  λ1 t − e  λ2 t ) (20) Upon substitution in the third rate equation, and assuming the initial number of nuclei N30 as
zero, that the third member of the chain is stable, and integrating, we get: N 3 ( t ) = N10 [1 + λ1 (λ2 − λ1 ) e  λ2 t − λ2 (λ2 − λ1 ) e  λ1 t )] (21) Assuming the half life of the first isotope is less than the half life of the second isotope,
the overall result is that the number of nuclei of the isotope 1 will decrease exponentially
according to its own half life. The second isotope number, which is initially zero, increases to a
maximum and then decreases gradually. The third isotope as an end product will increase
steadily with time and approaches N10, since all the nuclei of the initial isotope will eventually
decay to the stable end product. 1.13 SUCCESSIVE DECAYS GENERAL SOLUTION
As a generalization of the previous analysis, consider a radioactive decay chain
containing n members. dN1 ( t )
= − λ1 N1 ( t )
dt
dN 2 ( t )
= + λ1 N1 ( t ) − λ2 N 2 ( t )
dt
dN 3 ( t )
= + λ2 N 2 ( t ) − λ3 N 3 ( t )
dt
........................................................ (22) dN n ( t )
= + λn −1 N n −1 ( t ) − λn N n ( t )
dt
Further assume that initially at time t=0:
N10 ≠ 0
N 20 = N 30 = … = N n 0 = 0
This expresses a situation with a pure sample where only the parent substance (N10 ≠ 0) is
initially present.
We can write the product form of a general solution for this case as:
n N n (t ) = ∑ Am e − λm t (23) m =1 where:
n −1 A1 = ∏λ
i =1 n ∏ (λ i i − λ1 ) .N 10 i=2 n −1 A2 = ∏λ
i =1 n ∏ (λ i i − λ2 ) .N10 i =1
i≠2 ………………………. n −1 Am = ∏λ
i =1 i . N10 n ∏ (λ − λ
i =1
i ≠m i m ) This form is a modified form of equations first derived by Bateman. The equations
derived in the previous section can be directly derived from Eqn. 23, which lend themselves
readily for numerical computations.
To obtain a more general solution for the case where:
N 20 , N 30 ...N n 0 ≠ 0,
the overall solution is obtained by adding to the solution above for Nn(t) in an nmember chain, a
solution for Nn(t) in an (n1)member chain with now isotope 2 as the parent nuclide. Therefore
N2(t) = N20 at t=0, and a general solution for Nn(t) in an (n2)member chain, and the process is
repeated.
When branching occurs in the radioactive decay chain, the decay constants in the
numerators above should be replaced by the partial decay constants. Once the branches rejoin,
the two branches are treated as separate chains. The two branches are followed, and then the
contributions of the two paths are added out for any common member. 1.14 RADIOACTIVE EQUILIBRIA
SECULAR EQUILIBRIUM Equilibrium is normally reached when the time derivatives in the rate equations are equal
to zero, resulting in:
dN1 ( t )
= − λ1 N1 ( t ) = 0
dt
λ1 N1 ( t ) = λ2 N 2 ( t ) λ2 N 2 ( t ) = λ3 N 3 ( t ) (24) ........................................................ λn −1 N n −1 ( t ) = λn N n ( t )
This cannot be strictly achieved since this would lead to contradiction in the first equation
when the decay constant would have to be equal to zero. However, a case close to equilibrium
can be reached if the half life of the first parent is must longer than the half life of the daughter
nuclide. This situation occurs in the naturally occurring decay chains. For instance, Uranium238
has a half life of 4.5 billion years. In this case we can take the number of atoms of the initial parent N1 as a constant, and the value of the decay constant is much smaller than other decay
constants in the chain. This type of equilibrium is called "secular equilibrium," where: λ1 N1 ( t ) = λ2 N 2 ( t ) = λ3 N 3 ( t ) = … = λn −1 N n −1 ( t ) = λn N n ( t ) (25) This means that the activities of the chain members are equal:
A1 ( t ) = A2 ( t ) = A3 ( t ) = … = An −1 ( t ) = An ( t ) (26) This situation applies whenever a succession of short lived isotopes arises from the decay
of a relatively long lived parent.
This kind of equilibrium can also be attained when a radioactive substance is produced at
a steady state from an artificial method, such as in a nuclear reactor or a particle accelerator.
The last expression can be used to determine the half life of an isotope having a long half
life in terms of the half life of a short half life isotope, if we rewrite in the form:
N1 ( t )
N2 (t )
N3 (t )
N n −1 ( t )
Nn (t )
=
=
= … =
=
T1
T2
T3
Tn −1
Tn (27) If the ratio of two isotopes in secular is known, and the half life of the daughter is known, then
the half life of the parent can be calculated from:
⎛N ⎞
T1 = ⎜ 1 ⎟ T2
⎝ N2 ⎠ (28) EXAMPLE The relative atomic abundance ratio for Uranium238 to Radium226 is 2.7925 million, and
the half life for Radium226 is about 1,600 years, leading to:
NU 238
N Ra 226 = 2.7925 x106 TRa 226 = 1,600 [ years ]
⎛ N 238 ⎞
TU 238 = ⎜ U ⎟ TRa 226
⎜ N 226 ⎟
⎝ Ra ⎠
= 2.7925 x106 x1,600 [ years ]
= 4.468 x109 [ years ]
We would like to study the approach to secular equilibrium. Consider a parent nuclide
and its daughter, with the parent with a long half life such that: T1 >> T2 or: λ1 << λ2
Using the equation for the parent and daughter under this condition,
N1 ≅ N10
⎛λ ⎞
N 2 ( t ) ≅ ⎜ 1 ⎟ N10 1 − e
⎝ λ2 ⎠ ( λ2 t ) This implies that the activity of the daughter reaches a saturation level according to the
relationship:
A 2 (t) = λ2 N 2 ( t ) ≅ λ1 N10 (1 − e  λ2 t ) (29) The last equation shows as the daughter's activity grows and eventually reaches the activity of
the parent, as implied by secular equilibrium.
TRANSIENT EQUILIBRIUM If the parent is long lived, but the halflife of the parent is not very long, the following
condition is satisfied:
or: T1 > T2
λ1 < λ 2 Consequently:
⎡ λ1
⎤
N2 (t ) ≅ ⎢
N e
( λ2 − λ1 ) ⎥ 10
⎣
⎦  λ1t ⎡ λ1
⎤
N2 (t ) ≅ ⎢
N (t )
( λ2 − λ1 ) ⎥ 1
⎣
⎦ ⎡ λ1
⎤
N2 (t )
≅⎢
⎥
N1 ( t )
⎣ ( λ2 − λ1 ) ⎦
In terms of the activities: (31) A2 ( t )
λ N (t )
λ2
≅ 2 2 ≅
λ1 N1 (t ) ( λ2 − λ1 )
A1 (t ) (32) This implies that the daughter's activity is less than the parent's activity by the factor: ( λ2 λ2 − λ1 ) . CASE OF NO EQUILIBRIUM If the parent is shorter lived than the daughter, the following condition is satisfied:
T1 < T2 or: λ1 > λ 2
and no state of equilibrium is attained. The daughter nuclide will increase, pass through a
maximum, and decay with the half life of the daughter, as the parent decays with its own half
life. 1.15 RADIOACTIVE DECAY CHAINS
Elements found in nature with an atomic number above Bismuth (Z=83), are radioactive.
They belong to chains of successive transformations, starting from a radioactive element, and
each ending with a stable isotope. There exist three series containing all the natural activities in
this region of the Chart of the Nuclides. There is also an artificially created chain.
These chains are closely similar, and contain an interesting feature of branching
transformations.
In each of these chains, we notice the occurrence of an isotope of radon. It is released in
the form of a gas at room temperature. It constitutes a health hazard in uranium mines, in water
wells using uranium containing aquifers, in natural gas, in homes built on rocks containing
uranium, and in cigarette smoking.
THE URANIUM238 CHAIN The parent substance in this case is U238. It contains 14 transformations, 8 through alpha
decay and 6 through beta emission. The end of the chain is the stable lead isotope Pb206. This
chain contains radium and its decay products. The atomic mass number is here modified in units
of four in each alpha transformation. The beta decays do not affect the mass number. A general
formula for the mass number can be mathematically induced as:
(4n +2) where n is an integer. Radon222, which is a health hazard in uranium mines and some human
dwellings through its Po210 daughter, with a half life of 3.825 days occurs in this chain. This
chain is shown in Fig. 3.
THE THORIUM232 CHAIN The parent nuclide here is Th232, and the stable end product is Pb208. A general formula
for the mass number can be mathematically induced as:
4n
where n is an integer.
Another radon isotope Rn220 with a halflife of 54.5 seconds appears in this chain. This
chain is shown in Fig. 4.
THE URANIUM235, ACTINIUM CHAIN This chain has Pb207 as the stable end product and has U235 as its parent nuclide. A
general formula for the mass number can be mathematically induced as:
(4n +3)
where n is an integer. The radon isotope existing in this chain is Rn219 with a halflife of 3.92
seconds. This chain is shown in Fig. 5.
THE NEPTUNIUM237, ARTIFICIAL CHAIN This chain is artificially created and does not exist in nature. It starts with Np237 and ends
with 83Bi209 as a stable element. Its general formula is:
4n + 1.
This chain is shown in Fig. 6. 1.16 OTHER NATURALLY OCCURRING ISOTOPES
Other than the members of these chains, many radioactive isotopes have been discovered
with long halflives and small abundances, as shown in Table 4. Notably is K40, with a half life
of 1.27 x 109 years, in the same range as that for U238 at 4.52 x 109 years. This isotope of
potassium exists in living organic matter and cannot be separated from its other isotopes by
chemical means (Table 5). The detection of these isotopes is rather difficult because of the
existing radiation background in laboratories. This radiation background is caused by traces of
uranium, thorium, potassium, and in a larger part due to cosmic radiation. Table 5: Other naturally occurring or otherwise available radioisotopes
Radioisotope
T3
C14
K40
V50
Rb87
Cd113
In115
Te123
La138
Ce142
Nd144
Nd145
Sm147
Sm148
Sm149
Gd152
Dy156
Lu176
Hf174
Ta180m
Re187
Os186
Pt190
Pb204
Bi209
Th232
U234
U235
U238
ε = electron capture Isotopic
Abundance
(a%)
0.0117
0.25
27.83
12.22
95.71
0.89
0.09
11.08
23.8
8.3
14.99
11.24
13.9
0.20
0.057
2.59
0.16
0.012
62.6
1.59
0.014
1.42
100
100
0.0054
0.72
99.2745 Halflife
(years) Transformation
type 12.33
5370
1.27 x 109
1.40 x 1017
4.88 x 1010
7.7x1015
4.4 x 1014
6.0 x 1014
1.05 x 1011
>5.0 x 1016
2.38 x 1015
>1.0 x 1017
1.06 x 1011
7.00 x 1015
>1.00 x 1016
1.1 x 1014
>1.00 x 1018
3.75x1010
2.0x1015
>1.2x1015
4.12x1010
2.0x1015
6.5x1011
1.4x1017
>2.0x1018
1.40x1010
2.44x105
7.04x108
4.468x109 betabetabeta,beta+, ε
beta, ε
betabetabetaε
ε, betaalpha
alpha
alpha
alpha
alpha
alpha
alpha
betaalpha
ε, beta+
betaalpha
alpha
alpha
alpha
alpha
alpha
alpha
alpha Stable
Products He3
N14
Ca40,Ar40
Cr50,Ti50
Sr87
In113
Sn115
Sb123
Ba138,Ce138
Ba138
Ce140
Pr141
Nd143
Nd144
Nd145
Sm148
Hf176
Yb170
Hf180,W180
Os187
W182
Os186
Hg200
Tl205
Pb208
Pb206
Pb207
Pb206 1.17 SPONTANEOUS FISSION
Some heavy nuclei decay in a process where the nucleus breaks up into two intermediate
mass fragments and several neutrons. It occurs in with nuclei with mass number A > 230.
Since the maximum binding energy per nucleon occurs at A = 60, nuclides above A >
100 are unstable with respect to spontaneous fission, sine a condition for spontaneous fission is: m( A, Z ) > m( A ', Z ') + m( A − A ', Z − Z ') (33) in the spontaneous fission reaction:
Z XA→ Z' X A ' + Z − Z ' X A− A ' (34) Because of the high Coulomb barrier for the emission of the fission fragments,
spontaneous fission is only observed in the heaviest nuclei. 1.18 NEGATIVE BETA DECAY
Below the atomic number Z = 83, radioactive nuclides seek stability by either increasing
decreasing their nuclear charge through either negative beta positron decay, or electron capture.
Nuclides possessing an excess number of neutrons, or neutron rich nuclides tend to
undergo negative beta decay. Internally in this process, a neutron is transformed into a proton
and a negative electron with the emission of an antineutrino for conservation of parity:
0 n1 → 1 H 1 + −1 e0 +ν * (35) The atomic number of the decaying nucleus is increased by one unit in this process:
Z XA→ Y A + −1 e0 +ν * Z +1 (36) For this process to occur, the condition for a negative beta decay to occur is: m( A, Z ) ≥ m( A, Z + 1) (37) The beta decay energy is defined as:
E ( β − ) = m( A, Z ) − m( A, Z + 1)
It should be positive for the process of beta decay to be energetically possible. For instance, for
Po210, E(β)=  3.981 MeV, and in fact it decays through alpha decay into Pb206.
Fission products, being neutron rich nuclei, undergo a succession of negative beta decays
forming decay chains. An example of a beta decay is the decay of the hydrogen isotope tritium: T 3 → 2 He3 + −1 e0 +ν * 1 The beta decay energy for this reaction is 18.591 keV, which makes it energetically possible. 1.19 POSITRON DECAY
Nuclides possessing an excess number of protons, or proton rich nuclides tend to undergo
a positron decay. Internally in this process, a proton is transformed into a neutron and a positron
with the emission of a neutrino for conservation of parity:
1 H 1 → 0 n1 + +1 e0 +ν (38) The atomic number of the decaying nucleus is decreased by one unit in this process:
Z XA→ Y A + +1 e0 +ν Z −1 (39) For this process to occur, the condition for a positron decay to occur is:
m( A, Z ) ≥ m( A, Z + 1) + 2me ,
me is an electron mass = 0.51 MeV . (40) Fission products, being neutron rich nuclei, undergo a succession of negative beta decays
forming decay chains.
An example of a positron decay is the decay of the N13 isotope:
7 N 13 → 6 C13 + +1 e0 +ν Another example is the F18 positron decay used in Positron Emission Tomography or PET
nuclear medicine scanning: F 18 → 8 O18 + +1 e0 + ν 9 1.20 ELECTRON CAPTURE
Similar to positron decay possessing an excess number of protons, or proton rich nuclides
undergo an electron capture process. Internally in this process, a proton combines with an inner
shell electron into a neutron with the emission of a neutrino for conservation of parity:
1 H 1 + −1 e0 → 0 n1 +ν (41) The atomic number of the decaying nucleus is decreased by one unit in this process:
Z X A + −1 e0 → Y A +ν Z −1 (42) For this process to occur, the condition for a positron decay to occur is: m( A, Z ) ≥ m( A, Z − 1) (43) Fission products, being neutron rich nuclei, undergo a succession of negative beta decays
forming decay chains.
An example of an electron capture decay is the decay of the Be7 isotope:
4 Be7 + −1 e0 → 3 Li 7 +ν Positron decay is possible only if the initial and final masses differ by two electron
masses = 2 me = 2 x 0.51 = 1.02 MeV. Electron capture, on the other hand, is possible if the
initial ,ass is just larger than the final mass. If positron decay is possible, so is electron capture,
and nuclei that cannot undergo positron decay can undergo electron capture decay. 1.21 DOUBLE BETA DECAY
This is a rare radioactive event observed for Mo92 and Mo100. Two beta particles and two
antineutrinos are emitted, resulting in the original nucleus gaining two protons and losing two
neutrons.
Nuclides such as Se82, Cd116 and Te130 undergo this type of decay, albeit with half lives
exceeding 1019 years
The decay process with the emission of two beta particles and no antineutrinos is possible
according to theories requiring antineutrinos to have zero mass. Such decays have not been
experimentally verified yet. 1.22 ALPHA DECAY
For nuclides with a large mass number A, alpha decay becomes possible.
Z XA→ Y A− 4 + 2 He4 Z −2 (44 For this process to occur, the condition for an alpha decay to occur is: m( A, Z ) ≥ m( A − 4, Z − 2) + m( 2 He4 )
An example of an alpha decay is the decay of the Pu239 isotope:
94 Pu 239 → 92U 235 + 2 He4 (45) Another example is the decay of Po210 at the end of the U238 decay chain into the Pb206 stable
lead isotope:
84 Po 210 → 82 Pb206 + 2 He4 The binding energy per nucleon for the alpha particle is 7.1 MeV, consequently the total
binding energy is 4 x 7.1 = 28.4 MeV. For some nuclides around A = 140, the biding energy per
nucleon is around 7 MeV, and alpha decay is possible. It becomes a dominant decay mode for
proton rich nuclides with A > 160 and for neutron rich nuclides with A > 180. 1.23 ISOMERIC TRANSITIONS
A metastable or isomeric state of a nuclide will decay to its ground state by an isomeric
transition (IT) gamma ray emission followed by one or more gamma rays in a cascade. It is
possible to have just one IT gamma ray to the ground state, or a more complicated scheme with
more than one IT gamma ray, each with its cascading gamma rays. An example of an isomeric
transition is the Technetium99m decay with a half life of 6.02 hours: Tc99 m → 43Tc99 + γ 43 This is different from the normal Tc99 beta decay with a half life of 0.211x106 years: Tc 99 → 44 Ru 99 + −1 e0 + ν * 43 1.24 INTERNAL CONVERSION
This is a process in which the nucleus interacts with its extranuclear electrons. It
competes with gamma ray emission. The excitation energy of the nucleus is usually transferred
to a K shell orbital electron, and the electron is emitted from the atom instead of a gamma ray.
Neither the atomic number Z nor the mass number A change in this process. The conversion
electrons will have a kinetic energy equal to the difference between the energy of the nuclear
transition involved and the binding energy of the electron in the atom. An example showing the
competition with gamma rays emission occurs to Tc99m as: Tc99 m → 43Tc99 + −1 e0 43 1.25 DELAYED PARTICLE EMISSION
Nitrogen17 with a half life of 4.174 seconds decays through negative beta decay into short
lived states of O17 which in turn emit neutrons. Thus N17 is considered to emit delayed neutrons
with a half life of 4.174 seconds.
Delayed protons emission occurs in Si25 which decays by positron emission to its
daughter Al25 which emits protons.
Delayed neutron emissions occur in some fission products, and greatly influence the
control of fission reactors. 1.26 CLUSTER DECAY
Cluster decay has been observed in several heavy nuclides where clusters of C12, C14,
O20, Ne20, Mg28, or Si32 have been observed.
Table 6: Nuclides undergoing Cluster Decay.
Nuclide
Ba114
Fr221
Ra221
Ra222 Q value
[MeV]
18.3 – 20.5
31.28
32.39
33.05 Cluster C12
C14
C14
C14 Ra223
Ra224
Ac225
Ra226
Th228
Pa231
Th230
Th232
Pa231
U232
U233
U234
U235
U236
U232
U233
U234
U235
U236
Np237
Pu236
Pu238
Pu238
Pu240
Am241 31.85
30.54
30.48
28.21
44.72
51.84
57.78
55.62, 55.97
60.42
62.31
60.50, 60.75
58.84, 59.47
57.36, 57.83
55.96, 56.75
74.32
74.24
74.13
72.20, 72.61
71.69, 72.51
75.02
79.67
75.93, 77.03
91.21
90.95
93.84 C14
C14
C14
C14
O20
F23
Ne24
Ne24, Ne26
Ne24
Ne24
Ne24, Ne25
Ne24, Ne26
Ne24, Ne25
Ne24, Ne26
Mg28
Mg28
Mg28
Mg28, Mg29
Mg28, Mg30
Mg30
Mg28
Mg28, Mg30
Si32
Si34
Si34 The process starts with the formation of a cluster of nucleons within the nucleus followed
by the cluster tunneling through the Coulomb barrier of the nucleus. Atomic Number
Z
92 Element 91 Pa 90 Th 89 Ac 88 Ra 87 Fr 86 Rn 85 At 84 Po 83 Bi 82 Pb 81 Tl 82 U U238
4.51x109 y Th234
24.1 d Pa234m
1.18 m
Pa234
6.7 h
(0.15%) U234
2.48x105 y
(98.85%) Th230
7.52x104 y
Ra226
1622 y
Rn222
3.825 d Po218
3.05 m At218
1.3 s
(0.02%) Bi214
19.7 m Po214
1.6x104 s
(99.96%) Pb214
3.3 h
(99.98%) Po210
138.4 d
Bi210
5.01 d Pb210
22 y
Tl210
2.2 m
(0.04%) Pb206
Stable
Tl206
4.3 m (5x105%) Hg Hg206
8.5 m (1.8x106%) Fig. 3: The uranium238 (4n + 2) radioactive decay series Atomic Number
Z
90 Element 89 Ac 88 Ra 87 Fr 86 Rn 85 At 84 Po 83 Bi 82 Pb 81 Tl Th Th232
1.39x1010 y Th228
1.9 y
Ac228
6.13 h Ra228
6.7 y Ra224
3.64 d Rn220
54.5 s Po216
0.158 s
Bi212
60.6 m
Pb212
10.6 h Fig. 4: The thorium232 (4n) radioactive decay series Po212
3.0x107 s
(66.3%) Pb208
Stable
Tl208
4.79 m
(33.7%) Atomic Number
Z
92 Element 91 Pa 90 Th 89 Ac 88 Ra 87 Fr 86 Rn 85 At 84 Po 83 Bi 82 Pb 81 Tl U U235
7.13x108 y
Pa 231
3.48x104 y
Th231
25.6 h
Ac227
22 y Th227
18.17 d
(98.8%) Ra223
11.7d
Fr223
22 m
(1.22%) At219
0.9 m Rn219
3.92 s
(3%) At215
104 s Po215
1.83x103 s
Bi215
8m
(97%) Bi211
2.15 m
Pb211
36.1 m Po211
0.52 s
(0.32%) Pb207
Stable
Tl207
4.79 m
(99.68%) Fig. 5: The uranium235 or actinium (4n + 3) radioactive decay series Atomic Number
Z
93 Element 92 U 91 Pa 90 Th 89 Ac 88 Ra 87 Fr 86 Rn 85 At 84 Po 83 Bi 82 Pb 81 Tl Np Np237
2.2x106 y
U233
1.62x105 y
Pa233
27 d
Th229
7340 y
Ac225
10 d
Ra225
14.8 d
Fr221
4.8 m At217
0.018 s Bi213
47 m Po213
4.2x106 s
(98%) Pb209
3.3 h
Tl208
2.2 m
(2%) Fig. 6: The artificial neptunium237 (4n + 1) radioactive decay series. Bi209
Stable EXERCISES
1.
Consider the isotope Ra226. Using Avogadro’s law, calculate its specific activity or the
activity of 1 gram of material, and discuss its relationship to the Curie unit of activity. You can
obtain the half life of the radium226 isotope from the Table of the Nuclides.
2.
The naturally occurring isotope K40 is widely spread in the environment. In fact, the
average concentration of potassium in the crustal rocks is 27 [g/kg] and in the oceans is 380
[mg/liter]. K40 occurs in plants and animals, has a halflife of 1.3 billion years and an abundance
of 0.0119 atomic percent.
Potassium's concentration in humans is 1.7 [g/kg]. In urine, potassium's concentration is 1.5
[g/liter].
a) Calculate the specific activity of K40 in Becquerels per gram and in Curies/gm of K40.
b) Calculate the specific activity of K40 in Becquerels per gram and in Curies per gm of overall
potassium.
c) Calculate the specific activity of K40 in urine in [Bq/liter].
d) A beta activity above 200 transformations (disintegrations) per minute per liter of urine
following accidental exposure to fission products is indicative of an internal deposition in the
body, and requires intervention. How does this "body burden" criterion compare to the activity
caused by the one due to the naturally occurring potassium?
3.
The production of Carbon14 with a half life of 5730 years is an ongoing nuclear
transformation from the neutrons originating from cosmic rays bombarding Nitrogen14 in the
Earth’s atmosphere:
1
14
→ 1 H 1 + 6 C14
0n + 7 N
6 C14 → −1 e0 + 7 N 14 −−−−−−−−−−−−−−
n1 → −1 e0 + 1 H 1
where Nitrogen14 and Carbon14 appear as catalysts in the overall reaction leading to the
disintegration of a neutron into a proton and an electron.
The atmospheric radiocarbon exists as C14O2 and is inhaled by all fauna and flora. Because only
living plants continue to incorporate C14, and stop incorporating it after death, it is possible to
determine the age of organic archaeological artifacts by measuring the activity of the carbon
present.
Two grams of carbon from a piece of wood found in an ancient temple are analyzed and found to
have an activity of 20 disintegrations per minute. Estimate the approximate age of the wood, if it
is assumed that the current equilibrium specific activity of C14 in carbon has been constant at
13.56 disintegrations per minute per gram.
4.
Using the chart of the nuclides, generate the decay chains for U238 and Th232.
a. Identify the two gaseous radon isotopes in the chain and find their decay graphs.
b. Identify the solid products of the radon chain that are of particular health interest, and show
their decay diagrams, decay products, half lives, and decay energies.
5. Upon manufacturing for nuclear fuel, the decay chain of 92U238 is broken during the process of
the chemical reduction of Uranium Hexafluoride UF6 into the uranium metal.
0 a. Calculate the specific activity of a fresh sample of uranium238, which has a half life of
4.51x109 years, in Curies/gm and in Bq/gm.
b. The isotopes 90Th234 and 91Pa234m are daughters of U238 and have as halflives: 24.1 days
and 1.18 minutes respectively. These halflives are short compared with the halflife of
the parent U238. Hence they can be considered to be in “secular equilibrium” with the
parent U238. Calculate the total activity of a freshly manufactured sample of 1 gram of
U238 under secular equilibrium conditions containing its daughters 90Th234 and 91Pa234m in
Curies and in Becquerels. REFERENCES
1. Bjorn Wahlstrom, “Understanding Radiation,” Medical Physics Publishing, Madison
Wisconsin, 1995.
2. Suzanne Jones, Ray Kidder, and Frank von Hippel, "The Question of PureFusion Explosions
under the CTBT," Opinion, Physics Today, September, 1998.
3. H. Bateman, "The Solution of a System of Differential Equations Occurring in the Theory of
RadioActve Transformations," Proc. Cambridge Phil. Soc. 16, 423, 1910.
4. Irving Kaplan, "Nuclear Physics," AddisonWesley Publishing Company, 1962.
5. E. M. Baum, H. D. Knox and T. R. Miller, “Nuclides an Isotopes,” Kno;;s Atomic Power
Laboratory, Lockheed Martin, 2002. ...
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This note was uploaded on 06/16/2010 for the course NPRE 402 taught by Professor Ragheb during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 RAGHEB

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