3.6 Reaction Cross-Section
39
3.6 Reaction Cross-Section
It is essential to calculate the time required for irradiation and decide the type of
the target material suitable for getting a particular type of product in any of the
nuclear reactions expressed earlier. In the foregoing discussion, we shall concentrate
to understand some of the important parameters on which these nuclear reactions
depend.
One of the important parameters is the probability of occurrence of particular
reaction with the concerned target material. The probability of occurrence of a nuclear
reaction is expressed either for the number of particles emitted, or nuclei undergone
transformation, for a specific number of incident particles. A general method, which
has been widely adopted to express relative efficiency for either of the processes by
means of a quantity, is called nuclear cross-section. This represents a probability
that a given nucleus will undergo a specific nuclear reaction. This probability term
mathematically comes in a unit of cm
2 and hence it is called cross-sectional area for
the reaction. The unit for cross-section is barn, and one barn is equal to 10
−24 cm
2 .
A knowledge of the magnitude of nuclear cross-section helps to calculate the
amount of product formed by irradiating a target material with a nucleon. If for a
particular nuclear reaction with an atom A, the nuclear cross-section is represented
as “(σ )” to undergo a nuclear reaction to produce an atom B and if N is the number
of atoms of A (irradiated), and “ f ” the number of incident particles per cm
2 per
second, then the amount of product formed per second by the nuclear reaction would
be σ N f , where f is assumed to be constant throughout the period of irradiation,
provided the product is not a radioactive material. That is
Rate of production of atom, B = σ N f
(3.1)
But, if the product “B” is radioactive, it will decay with time as well, hence the
net rate of production of “B” from “A” is given by
d N B
dt
= σ N f − λ B N B
(3.2)
in which N B is the number of atoms of B present at time t, and λ B the decay constant
of B. On integration and simplification, Eq. (3.2), becomes
N B =
σ N f
(1 − e −λ B t )
(3.3)
In this calculation, it is assumed that the atom “A” is not a radioactive material.
However, if atom “A” is also a radioactive material which decays by a decay constant
of A, then Eq. (3.3) will need to be modified. This equation thus can be used to
calculate the quantity of radioactive material “B” formed after time “t” of irradiating
material “A”.
39
3.6 Reaction Cross-Section
It is essential to calculate the time required for irradiation and decide the type of
the target material suitable for getting a particular type of product in any of the
nuclear reactions expressed earlier. In the foregoing discussion, we shall concentrate
to understand some of the important parameters on which these nuclear reactions
depend.
One of the important parameters is the probability of occurrence of particular
reaction with the concerned target material. The probability of occurrence of a nuclear
reaction is expressed either for the number of particles emitted, or nuclei undergone
transformation, for a specific number of incident particles. A general method, which
has been widely adopted to express relative efficiency for either of the processes by
means of a quantity, is called nuclear cross-section. This represents a probability
that a given nucleus will undergo a specific nuclear reaction. This probability term
mathematically comes in a unit of cm
2 and hence it is called cross-sectional area for
the reaction. The unit for cross-section is barn, and one barn is equal to 10
−24 cm
2 .
A knowledge of the magnitude of nuclear cross-section helps to calculate the
amount of product formed by irradiating a target material with a nucleon. If for a
particular nuclear reaction with an atom A, the nuclear cross-section is represented
as “(σ )” to undergo a nuclear reaction to produce an atom B and if N is the number
of atoms of A (irradiated), and “ f ” the number of incident particles per cm
2 per
second, then the amount of product formed per second by the nuclear reaction would
be σ N f , where f is assumed to be constant throughout the period of irradiation,
provided the product is not a radioactive material. That is
Rate of production of atom, B = σ N f
(3.1)
But, if the product “B” is radioactive, it will decay with time as well, hence the
net rate of production of “B” from “A” is given by
d N B
dt
= σ N f − λ B N B
(3.2)
in which N B is the number of atoms of B present at time t, and λ B the decay constant
of B. On integration and simplification, Eq. (3.2), becomes
N B =
σ N f
(1 − e −λ B t )
(3.3)
In this calculation, it is assumed that the atom “A” is not a radioactive material.
However, if atom “A” is also a radioactive material which decays by a decay constant
of A, then Eq. (3.3) will need to be modified. This equation thus can be used to
calculate the quantity of radioactive material “B” formed after time “t” of irradiating
material “A”.