Download Lecture 3

Lecture 3
Radioactivity
Objectives
In this lecture you will learn the following
We shall begin with a general discussion on the nucleus.
Learn about some characteristics of nucleons.
Understand some concepts on stability of a nucleus.
In particular, we shall discuss Neutron/Proton ratio for a nucleus to be stable.
Move on to quantify radioactivity and associated concepts.
Understand the radioactive law and learn the concept of half life.
Finally digest the analysis of radioactive chains and observe the general behaviour.
Analyse a typical three element chain and identify its characteristics.
1
2
3
4
5
6
7
8
9
/9
GO
Lecture 3
Radioactivity
Review of Nuclear Physics
Most of the concepts in nuclear engineering can be understood by considering only protons,
neutrons and electrons as described by Rutherford's model.
Just as chemical energy is released by the rearrangement of electrons, nuclear reactions can be
understood by considering rearrangement of protons and neutrons.
Chemical Energy
Nuclear Energy
Rearrangement of Electrons
Rearrangement of Protons and Nuetrons
Mass of Proton and Neutron are approximately equal to 1/N Avogadro in grams.
Every element can be represented by
In the above
Z - No of Protons or Charge Number.
A - No of Protons + No. of Neutrons or Mass Number.
Isotope have Same Z, but different A e,g.
,
,
.
Typically the Radius of Nucleus = 10-15m.
Radius of Atom = 10-10m.
Volume ratio = 1015.
From above we understand that there is enormous hollow space.
This implies that we need a large number of neutrons for collisions to occur.
1
2
3
4
5
6
7
8
9
/9
GO
Lecture 3
Radioactivity
Stability Of Nucleus
In nature all elements are not stable.
For example, Oxygen is stable, while Uranium is not.
Interestingly, Neutron/Proton ratio influences stability.
There is an optimum ratio needed for a nucleus to be stable and this ratio changes with the
mass of the nucleus.
The figure shown below indicates the zone where stable and unstable isotopes lie.
The ratio of N/P varies from 1-1.5.
The stable band is shown by a thick line, while the unstable one is shaded.
The unstable ones emit radiation, transform themselves and move towards stability.
The process of transformation of unstable nucleus by spontaneous emission of radiation is
called radioactivity.
1
2
3
4
5
6
7
8
9
/9
GO
Lecture 3
Radioactivity
Radioactivity
The fundamental radioactive law is that the probability of nuclear disintegration rate is constant. Consider the following:
Population
------>
N
Time
------>
dt
Disintegration
------>
dN
= Constant =
(say)
The above equation is a first order Ordinary Differential Equation (ODE).
If the initial value is
.
Separating the variables and integration leads to
This implies that the population decays exponentially as
shown in the figure.
Thus, technically speaking a radioactive isotope has
infinite life.
1
2
3
4
5
6
7
8
9
/9
GO
Lecture 3
Radioactivity
Activity
Every disintergration results in emission of radiation
dN/dt = Rate of emission of Radiation.
This is termed as Activity and is denoted by α.
α is numerically equal to λN (from radioactive law).
SI unit of α is Becquerel denoted by Bq.
1 Bq = 1 disintegration per second (dps) or s-1.
An older unit is often used and is called Curie.
1 – Curie = 3.7 Χ1010 Bq.
It represents activity of 1g of Radium.
1
2
3
4
5
6
7
8
9
/9
GO
Lecture 3
Radioactivity
Half Life
As pointed out earlier, the life of a radioactive isotope is technically infinite. Hence a half life
is defined for every radioactive isotope
We can now rewrite the radioactive decay
law as
In 5 halflives,
In 10 halflives
Thus in about 10 half lifes the population is reduced by about 1000 times.
1
2
3
4
5
6
7
8
9
/9
GO
Lecture 3
Radioactivity
Analysis of Decay Chains-I
Fission Products are Radioactive A→ B → C → D……..
Practical chains are long.
A three isotope chain has all the characteristics of any long chain.
Consider A → B → C (Stable)
The Population Balance Equation can be written as
Rate of change = Production Rate – Destruction Rate.
Let NA , NB and Ne be the population of A, B and C respectively.
The conservation of species A, B & C can be written as
,
,
The solution of first equation with the initial condition leads to the solution
With NA = NA0 at t = 0
Similarly the second equation can be written as
Using
as Integral factor, we can write,
Integration of the above with the initial condition, NB = NB0 leads to
Since
The following figure shows the variation of population of A, B and C for the data shown
inside the figure.
1
2
3
4
5
6
7
8
9
/9
GO
Lecture 3
Radioactivity
Special Cases-I
For λ B > λ A
After some time, the first term goes to zero and the term 2 in the parenthesis of second term
also becomes negligible. This leads to
This is called transient equilibrium.
Here though both NA and NB varies with time, the ratio of them remains a constant.
It may be observed in the last slide that for t > 3, NB /N A = 1.
1
2
3
4
5
6
7
8
9
/9
GO
Lecture 3
Radioactivity
Special Cases-II
For λ B >> λ A
The above condition is called secular equilibrium.
It implies that irrespective of their initial activities they quickly attain the same activity.
This can be appreciated by the figure shown below, where for t > 0.3, the ratio of the activity
reaches the value 1.05.
1
2
3
4
5
6
7
8
9
/9
GO