Chapter III Theoretical Aspects 26-42
3.5 Fundamental Equation for NAA Method 30
3.5.1 Absolute NAA Method 30
The elements to be determined in a sample are made radioactive by irradiating the sample with neutrons, and the radionuclide formed (characterized by their half-lives) give off their characteristic radiation such as gamma rays, which are then identified and measured. The activity or number of detected gamma rays of a particular energy is directly proportional to the disintegration rate of the radionuclide, which is directly proportional to the amount of its parent isotope in the sample [8]. Measurement of the radionuclide gamma rays provides a measure of the total concentration of the parent element. In a neutron induced reaction, the growth of the product is dependent on the size of the neutron flux. The larger the neutron flux, the greater the rate at which interactions occurs:
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Activation rate neutron flux (φ) (3.1) The activation rate is also directly proportional to the number of target nuclei present,
Activation rate number of nuclei presents (N) (3.2) Here Avogadro’s number (N) represents the total number of atoms in the atomic weight (A) of any element. Therefore, the total number of atoms/g is,
N = NA/A (3.3) And for a mass w, of the element, the total number of target nuclei will be,
N = w NA/A (3.4) However, there may be more than one isotope of an element. In such cases the number of target nuclei must be corrected for the isotopic abundance (θ),
N= w NA θ/A (3.5) The number of target nuclei is therefore proportional to the mass of element present;
therefore the activation rate is proportional to the mass of the element,
Activation rate mass of element (w) (3.6) The relationship between activation rate, the number of target nuclei and the neutron flux is expressed by the term cross section (σ).
Activation rate = σ ϕ N (3.7) where,
N is the number of target nuclei,
ϕ is the neutron flux, in neutrons m-2 s-1, and σ is the cross section of the reaction in m2
Substituting, N = w NA θ/A in Eq. 3.7 the expression for activation rate becomes,
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Activation rate = σ ϕ w NA θ/A (3.8) Cross sections are usually expressed in barns which are 10-28 m2
The activation rate can also be given through:
Activation rate = σγ ϕth N + Iγ ϕepi N (3.9) where,
Φth is the thermal neutron flux, Φepi is the epithermal neutron flux,
Iγ is the radioactive capture resonance integral, and σγ is the neutron radioactive capture cross section.
We know that the number of nuclei is σ ϕ N t.
If there are N* radioactive nuclei, and the rate of decay of the nuclei is given by:
dN*/dt= - λ N*
Here λ is the decay constant, which has a characteristic value for each radionuclide. This equation leads to:
N* = N0* exp (-λt) (3.10) N0*/2 = N0* exp (- λ T1/2)
T1/2 = ln2 0.693
Here the terms have their usual meaning.
If the activation product is radioactive, it decays with its characteristic half-life, consequently the radionuclide will be produced at the rate described by the activation equation. Thus the growth of the activity is governed by the difference between them.
Production rate = activation rate - decay rate dN*/dt = σϕ N- λ N*
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N* = σϕ N (1 - exp(-λt))/λ (3.11) The factor (1- exp (-λt)) is called the saturation factor. The activity or disintegration rate (Ao), at the end of the irradiation time t, is then,
Ao = λN* = σϕ N(1- exp(-λt)) (3.12)
When the irradiation time is very long the expression for activity becomes close to the maximum possible activity for a particular neutron flux called the saturation activity (As), As = σ ϕ N
It is possible to calculate the induced specific activity for a particular length of
irradiation, knowing the nuclear constants for the nuclide of interest and the neutron flux, A0 = σϕ w NAθ (1- exp (-λt))/A (3.13) The weight of the element in the sample is given by
w = Ao A/ σϕ w NAθ (1- exp(-λt)) (3.14) Corrections must also be made for the decay period td before counting; in which case we get
w = AoA/ σϕ w NAθ (1- exp(-λt)) exp(-λtd (3.15) All the factors on the right hand side of the above equation are, in principle, known or can be measured. Thus, it is possible to calculate the mass of the element. The difficulty of accurate measurement of σ leads to the measurement of neutron flux density, φ. The value of φ changes depending on time and the location of the neutron sources like nuclear reactors. Sample and its container cause perturbation of neutron flux density (flux depletion and self-shielding of neutrons), which is very difficult to evaluate precisely.
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The calculation of the weight of the element in the sample can be made from the simple relation:
Wx = Ws
s
Rx
R (3.16) where,
Wx = The weight of the element concerned in the sample to be analyzed Ws = The weight of the element in the standard sample
= The corrected counting rate of the full-energy peak of this gamma rays employed for the identification of the nuclide produced in the sample
= The corrected counting rate of the full-energy peak of the same energy obtained with the standard sample.
Both the counting rates must be measured under the same geometrical conditions, and both have to be normalized at fixed geometrical conditions. When the sensitivity of activation analysis is defined as the activity per weight of an element in the sample, A/W, the following equation can be obtained:
A/w = {σϕ w NAθ (1- exp(-λt)) exp(-λ td)} / Ao (3.17) Therefore, the sensitivity will be greater for higher cross-section, higher isotopic abundance, irradiation time and neutron flux density [9]. The absolute standardization requires accurate determination of several irradiation and counting parameters and also reliable nuclear data that might proliferate uncertainty.