Nuclear Engineering

Basic Formulation of Radioactivity

Radioactivity is a process by means of which a nucleus emits various particles such as alpha, beta and gamma without any external perturbation. It is a spontaneous process inside the nucleus and unaffected by changes like temperature, pressure etc. To which the atom may be subjected. Probably, it would not be an exaggeration if we say that the modern nuclear physics begins with the discovery by Becquerel and the Curies.


Discovery of Radioactivity


Way back in 1896, the French Physicists Henry Becquerel observed that a photographic plate wrapped in black paper was affected by some uranium salt kept outside it. In fact he was studying fluorescence from uranium. He was also amazed to notice that it does not matter whether the photographic plate is kept at dark or sunlight. It was he along with his students Marie and Pierre Curie realized that the source of the radiation which affects the photographic plate in all situations is the nucleus and the mechanism was named as radioactivity. All three received Nobel Prize in physics in 1903 for this pioneering work. Eight years later Mme Curie again got the Noble Prize in chemistry for isolating the radioactive element radium.

Let us see the basic equations of radioactivity.



Law of radioactive decay


When a nucleus disintegrates by emitting charged particles (α, β) or a γ– ray photon, or by capturing electron from the atomic shell, the process is called radioactive decay. This decay is spontaneous. The rate of change of nuclei in a radioactive sample is given by law of radioactive decay. Let us take a radioactive sample containing N0 nuclei at time t = 0, i.e., at the beginning. We wish to calculate the number N of these nuclei left after time t. The number of nuclei of a radioactive sample disintegrating per second is called the activity of that sample.
= Rate of decrease of nuclei with time
= Activity at time t.
Experimentally, it has been observed that the activity at any instant of time t is directly proportional to the number N of parent type nuclei present at that time.

or
where λ>0, is the proportionality constant known as decay constant. The negative sign indicates that N decreases as t increases. By integrating Eqn. m4.1 we get,

or
Thus we see that the law of radioactive decay is exponential in nature. Number of radioactive nuclei versus time is plotted in Fig. m4.1.

FIGURE 6.1 Number of radioactive nuclei with time showing the exponential nature of radioactive decay.
On the basis of the radioactive decay law the following physical quantities are defined:

Half Life (T_1/2):

It is convenient to define a time interval during which half of a given sample of radioactive substance decays. This interval is called the half-life or half-value period of that substance, denoted by T_1/2,


or


Mean Life ():
Individual radioactive atoms may have life spans between zero and infinity. Therefore, the average or mean life is defined as,
= Total life time of all nuclei in a given sample/ total number of nuclei in that sample Mathematically, this can be represented as

Where dN₁ atoms have a lifetime t₁, dN₂ atoms a lifetime t₂ and so on.
In the notation of integral calculus

From Eqn. (6.2) we can write,





The mean or average life of a radioactive element is thus not the same as its half-life. The mean life is the reciprocal of the decay constant.

Activity or strength
Differentiating Eqn. (6.2) w.r.t. time we get,

at
Hence from (6.7) we get

or,
Where,
A_t is called the activity or the strength of the sample and is proportional to the rate of disintegration. The activity or the strength A_t of a radioactive sample at any instant of time t is thus defined as, the number of disintegrations occurring in the sample in unit time t, that is,

The activity per unit mass of a sample is called its specific activity. is the initial activity of the sample. The half-life may be defined in terms of activity as well. It is the time in which the activity drops to one-half of the initial activity.

Units of Activity
The customary unit of activity of any radioactive material is curie (Ci), which is defined as the activity of any radioactive substance that disintegrates at the rate of 3.7 x 10¹⁰ disintegrations per second. A thousandth part of a curie is called a millicurie (mCi). A still smalled unit is the microcurie ( μCi). So, by definition
1 Ci = 1 curie = 3.7 x 10¹⁰ disint./sec 1 mCi = 10^-3 Ci; 1 μCi = 10^-6 Ci
Another unit of activity is the rutherford, which is defined as a disintegration rate of 10⁶disintegrations per second.

Example m4.1: Calculate the activity of (i) One gram of radium , whose half- life is 1622 years and (ii) 3 x 10^-9 kg of active gold, whose half- life is 48 minutes.
Solution: (i) Here,
= 2.66 x 1021 atoms.

Activity =
(ii) In the same way number of Au, N = 9.04 x 10¹⁵ atoms λ = 2.406 x 10^-4 sec^-1 and Activity = λN = 58.9 Ci

Let us now discuss how to determine the radioactive constants:
In radioactivity, it is important to determine the values of disintegration constant, half life, average life etc. Considering the activity of any radioactive nuclide we can write,

where, x = ln A_t and C = ln A₀

Therefore, disintegration constant λ can be determined from the slope of the straight-line graph of ln A_t versus t, provided the activity of the nuclide under study is known. The disintegration constant of the radio nuclide having extremely high λ values can be determined from Geiger-Nuttal law, which connects the range of α particles with the disintegration constant by a simple relationship

where, A, B are constants and have different values for different radioactive series. Therefore, if the value of the range R is known (discussed in chapter 2), λ can be calculated from the above equation. Disintegration constant of the radioactive nuclide having short λ values can be determined from the condition of secular equilibrium also. The condition for such equilibrium is N₁λ₁ = N₂λ₂ . Hence, if N₁, λ₁ and N₂ are known, λ₂ can be evaluated. The half-life of a radioactive nuclide can be found by plotting the activity of a sample with time. The time corresponding to half of the activity is the half-life of that particular radioactive nuclide.

Nuclear Engineering

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Basic Formulation of Radioactivity

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