RADIOACTIVE DECAY
Principles of Nuclear Physics
NPE-503
Lecture by: Zahra Ali
Decay Equations
Decay is proportional to the # of atoms present (first order)
dN/dt = - N = AN
where
N=
the number of atoms of the radioactive substance present at time t
=
the first order decay constant (time-1)
The number of parent atoms at any time t can be calculated as follows.
The decay equation can be rearranged and integrated over a time interval.
N Noe
or
A Ae t
where No is the number of parent atoms present at time zero. Integration leads to
Parent-Daughter Relationships
Radioactive Parent (A)
Stable Daughter (B)
A B e.g. 14C 15N (stable)
Production of Daughter = Decay of Parent
dN B
At
A N A A N A,o e
dt
Example: 14C 15N (stable)
t1/2 = 5730 years
2-box model
A
Radioactive Parent (A)
Radioactive Daughter (B)
2-box model
A B
A
source
sink
dN B
A N A B N B
dt
mass balance for B
solution:
B ( N A,0 ) t t
NB
e e
B A
A
B ( AA,0 ) t t
AB
e
e
B A
A
solution after assuming NB = 0 at t = 0
Secular equilibrium
T1/2 daughter = 0.8 hr,
T1/2 parent =
Total Activity
parent
Activity of parent
and daughter at
secular equilibrium
daughter
Activity
(log scale)
t1/2
time (hr)
Example:
Grow in of 222Rn
from 226Ra
After 5 half lives
activity of daughter =
95% of activity of parent
Example: Rate of grow in
Assume we have a really big wind storm over the ocean so that all the inert gas
222Rn is stripped out of the surface ocean by gas exchange. The activity of the parent
of 222Rn, 226Ra, is not affected by the wind.
Then the wind stops and 222Rn starts to increase (grows in) due to decay.
How many half lives will it take for the activity of 222Rn to equal 50% (and then 95%)
of the 226Ra present?
Answer: Use the following equation:
AB AA,0 1 e0.693t / t1/ 2
Radioactive Dating
One application of radioactivity is the dating on
archeological and geological specimens by
measuring the concentration of radioactive isotopes.
Carbon dating: the unstable C-14 isotope, produced
during nuclear reactions in the atmosphere that
result from cosmic-ray bombardment, give a small
proportion of C-14 in the CO2 in the atmosphere.
Plants that obtain their carbon from this source
contain the same proportion of C-14 as the
atmosphere.
Radioactive Dating
When a plant dies, it stops taking in carbon and its C14 undergoes - decay to N-14 with a half-life of 5730
years.
By measuring the proportion of C-14 in the remains,
you can determine how long ago the organism died.
Similar radioactive techniques are used with other
isotopes for dating geological specimens.
Some rocks contain the unstable K-40 isotope, a beta emitter
that decays to the stable Ar-40 nuclide with a half-life of 2.4 x
108 years.
The age of the rock can be determined by comparing the
concentrations of K-40 and Ar-40.
Radioactive Dating
Radioactive Dating
Example: Before 1900 the activity per mass of
atmospheric carbon due to the presence of C-14
averaged about 0.255 Bq per gram of carbon.
a. What number of carbon atoms were C-14?In
analyzing an archeological specimen containing
500 mg of carbon, you observe 174 decays in
one hour.
b. What is the age of the specimen, assuming that
its activity per mass of carbon when it died was
that average value of the air?
Radioactive Dating
a.
365 da 24 hr 3600 s
T1 5730 yr
yr
da
hr
2
T1 1.81 x 1011 s
2
ln 2
0.693
T1
1.81 x 1011 s
2
3.8359 x 10 12 s
A N
A
0.255 Bq
N
3.8359 x 10 12 s
N 6.6478 x 1010 nuclei
Radioactive Dating
b.
174 de cay
hr
0.04833 Bq
hr
3600 s
0.04833 Bq 1000 mg
Bq
0.09666
500 mg
1g
g
A A o e t
ln
ln
A
e t
Ao
A
t ln e
Ao
A
t
Ao
ln
A
ln e t
Ao
ln e 1
A
Ao
ln
t
0.09666 Bq g
ln
0.255 Bq g
t
3.83587 x 10 12 / s
t 2.5289 x 1011 s 8019 .18 yr
Biological Effects of Radiation
As alpha particles, beta particles, neutrons, and EM radiation
such as gamma rays and x-rays, pass through matter, they lose
energy, break molecular bonds, and create ions (which is why
they are called ionizing radiation).
Excessive exposure to radiation, including sunlight, x-rays, and
all the nuclear radiations can destroy tissues.
Mild cases result in a burn, like a sunburn.
Greater exposures can cause severe illness or death by a variety of
mechanisms, including massive destruction of tissue cells, alterations
of genetic material, and destruction of the components in bone marrow
that produce red blood cells.
Calculating Radiation Doses
Radiation dosimetry is the quantitative
description of the effect of radiation on living
tissue.
Absorbed dose (AD) of radiation is defined as the
energy delivered to the tissue per unit mass.
SI unit of absorbed dose, the J/kg, is called the Gray
(Gy); 1 Gray = 1 J/kg.
The unit in more common use is the rad (radiation
absorbed dose) , defined as 0.01 J/kg; 1 rad = 0.01
J/kg = 0.01 Gy.
Calculating Radiation Doses
Absorbed dose by itself is not an adequate
measure of biological effect because equal
energies of different kinds of radiation cause
different extents of biological effect.
The variation in biological effect is described by a
numerical factor called the relative biological effect
(RBE), also called the quality factor (QF), of each
specific radiation.
The values for RBE depend somewhat on the kind
of tissue in which the radiation is absorbed and on
the energy of the radiation.
Calculating Radiation Doses
X-rays with 200 keV of energy are defined to have
an RBE of 1.
The biological effect is described by the product of
the absorbed dose and the RBE of the radiation,
this is called the biological equivalent dose (or the
equivalent dose, ED).
SI unit of equivalent dose is the Sievart (Sv).
1 Sv = 100 rem.
RBE units: Sv/Gy or rem/rad
1 rad = 1 rem (Rngen equivalent for man) = 0.01
J/kg.
RBE for Several Types of Radiation
Radiation
RBE (Sv/Gy or
rem/rad)
X-rays and rays
Electrons
1 1.5
Slow neutrons
35
Protons
10
Particles
20
Heavy ions
20
Equations and Example
Equations:
Bio log ical effect of radiation
RBE
effect of equal dose (in rad ) of 200 keV x rays
E D ( Sv) RBE AD (Gy )
E D (rem) RBE AD (rad )
Example: During a diagnostic x-ray examination a
1.2 kg portion of a broken leg receives an equivalent
dose of 0.4 mSv.
a. What is the equivalent dose in mrem?
b. What is the absorbed dose in J/kg?
c. If the x-ray energy is 50 keV, how many x-ray photons
are absorbed?
Example
a.
ED 0.4 mSv
b. A D
Sv
1 rem 1000 mrem
40 mrem
1000 mSv 0.01 Sv
1 rem
J
0
.
01
1 rem
1 rad
kg
40 mrem
1000 mrem 1 rem
1 rad
A D 0.0004 J
c.
kg
J
E 0.0004
1.2 kg 0.00048 J
kg
1 eV
15
E 0.00048 J
2
.
996
x
10
eV
19
1.602 x 10
J
15
photons 2.996 x 10
photon
eV
5.9925 x 1010 photons
50000 eV
Radiation Hazards
An ordinary chest x-ray delivers about 0.2 to 0.4 mSv to
about 5 kg of tissue.
Radiation exposure from cosmic rays and natural
radioactivity in soils, etc, is about 1 mSv (0.001 J/kg) per
year at sea level and twice that at an elevation of 5000 ft.
A whole-body dose of up to about 0.2 Sv (0.2 J/kg)
causes no immediate detectable effect.
A short-term whole-body dose of 5 Sv (5 J/kg) or more
usually causes death within a few days or weeks.
A localized dose of 100 Sv (100 J/kg) causes complete
destruction of the exposed tissues.
Radiation Hazards
Long term exposure to radiation can cause various
cancers and genetic defects.
U.S. government regulations are based on maximum
yearly exposure, from all except natural resources, of 2 to
5 mSv.
Workers with occupational exposure to radiation are
permitted 50 mSv per year.
Radiation levels from nuclear power plants is not
negligible, but the health hazards from coal smoke are
serious and the natural radioactivity in the smoke from a
coal-fired power plant is believed to be 100 times as great
as that from a properly operating nuclear power plant.
Radiation Units
Radiation Unit
Basis
Roentgen (R)
1 R the quantity of x-rays or gamma rays
that produces an ionization charge of
0.000258 C/kg in air.
rad (radiation
absorbed dose)
1 rad an absorbed dose of radiation of
0.01 J/kg
Gray (Gy)
SI absorbed dose unit; 1 Gy = 1 J/kg = 100
rad
rem (rad
equivalent man)
Effective dose. Relative effectiveness
depends on type of radiation and is
characterized by RBE.
Sievert (Sv)
SI unit of effective dose; 1 Sv = 100 rem
Effective dose (in Sv) = dose (in Gy) x RBE
