**2.5. Sources errors**

troduced for application of a more sophisticated formalism [14], the Westcott formalism. These two formalisms should be taken into account in order to preserve the accuracy of

162 Imaging and Radioanalytical Techniques in Interdisciplinary Research - Fundamentals and Cutting Edge Applications

The *k*0-NAA method is at present capable of tackling a large variety of analytical problems when it comes to the multi-element determination in many practical samples. In this part, we have published a paper [15] for which the determination of the Westcott and Høgdahl parameters have been carried out to assess the applicability of the *k*0-NAA method using the

During the three last decades Frans de Corte and his co-workers focused their investigations to develop a method based on co-irradiation of a sample and a neutron flux monitor, such as gold and the use of a composite nuclear constant called k0-factor [3, 16]. In addition, this method allows to analyze the sample without use the reference standard like INAA method. The k-factors have been defined as independent of neutron fluence rate parameters as well as of spectrometer characteristics. In this approach, the irradiation parameter (1+Q0(α)/f) (Eq. (11)) and the detection efficiency ε are separated in the expression (19) of the k-factor, which

0,comp comp comp 0,

1 Q () M

 qs

<sup>1</sup> 1 Q () <sup>M</sup> .. .

 e

 e

*f*

 e

<sup>+</sup> - è ø <sup>=</sup> <sup>+</sup> æ ö

The applicability of HØGDAHL convention is restricted to (n,γ) reactions for which WEST‐ COTT's g-factor is equal to unity (independent of neutron temperature), the cases for which WESTCOTT's g = 1 [3, 4, 17], such as the reactions 151Eu(n, γ) and 176Lu(n, γ) are excluded from being dealt with. Compared with relative method k0-NAA is experimentally simpler (it eliminates the need for multi-element standards [3, 18], but requires more complicated cal‐ culations [19]. In our research reactor, the k0-method was successfully developed using the HØGDAHL convention and WESTCOTT formalism [11, 15]. The k0-method requires tedious characterizations of the irradiation and measurement conditions and results, like the single comparator method, in relatively high uncertainties of the measured values of the masses. Moreover, metrological traceability of the currently existing k<sup>0</sup> values and associated param‐ eters to the S.I. is not yet transparent and most probably not possible. Summarizing, relative calibration by the direct comparator method renders the lowest uncertainties of the meas‐ ured values whereas metrological traceability of these values to the S.I. can easily be demon‐

 e

a

a

0,cal comp 0, cal

*comp comp comp*

i

*f e em*


l

1 Q () (1-e ). (1 ). <sup>1</sup> . . 1 Q () N /t <sup>k</sup>

i

l

0,cal comp p m 0 - t

<sup>+</sup> = = <sup>+</sup> (21)

 g

p m


*d m*

 l

 l

*t t*

*d m*

*e em*

*t t*


N /t

æ ö ç ÷

l

(1-e ). (1 ).

ç ÷ - è ø

l

*cal cal cal*

 g

*comp*

(22)

qs

experimental system and irradiation channels at Es-Salam research reactor.

resulted at the definition of the k0-factor.

*<sup>f</sup> <sup>k</sup> k f*

a

a

0

() ( )

*x unk x comp*

*m m*

*k*0-method.

Many publications reported in literature [20-25] treat the concept of evaluation of uncertain‐ ties in large range of analytical techniques.

We can give in this part of chapter, the evaluation of uncertainties for neutron activation analysis measurements. Among the techniques of standardization the comparator method for which the individual uncertainty components associated with measurements made with neutron activation analysis (NAA) using the comparator method of standardization (calibra‐ tion), as well as methods to evaluate each one of these uncertainty components [1].

This description assumes basic knowledge of the NAA method, and that experimental pa‐ rameters including sample and standard masses, as well as activation, decay, and counting times have been optimized for each measurement. It also assumes that the neutron irradia‐ tion facilities and gamma-ray spectrometry systems have been characterized and optimized appropriately, and that the choice of irradiation facility and detection system is appropriate for the measurement performed. Careful and thoughtful experimental design is often the best means of reducing uncertainties. The comparator method involves irradiating and counting a known amount of each element under investigation using the same or very simi‐ lar conditions as used for the unknown samples. Summarizing, relative calibration by the direct comparator method renders the lowest uncertainties of the measured values whereas metrological traceability of these values to the S.I. can easily be demonstrated. As such, this approach is often preferred from a metrological viewpoint. The measurement equation can be further simplified, by substituting:

$$\mathcal{A}\_0 = \frac{\mathcal{A}N\_p e^{\lambda t\_d}}{(1 - e^{-\lambda t\_n}) \cdot (1 - e^{-\lambda t\_i})} \text{.f.}\_{\text{loc}} \text{.f.}\_{\text{loc}} \tag{24}$$

$$m\_{\rm unk} = m\_{\rm cal} \frac{A\_{0(\rm unk)}}{A\_{0(\rm al.)}} R\_{\theta} R\_{\Phi} R\_{\sigma} R\_{\pi} \text{-blank} \tag{25}$$

2.2. Irradiation interferences

2.4. Irradiation losses and gains

3. Gamma-ray spectrometry stage

standards

2.2.2. Fission interferences

2.2.1. Fast (high energy) neutron interferences

2.2.3. Multiple neutron capture interferences

2.4.4. Target isotope burn up differences

2.3. Effective cross section differences between samples and standards

2.4.2. Transfer of material through irradiation container 2.4.3. Sample loss during transfer from irradiation container

3.3.1. From end of irradiation to start of measurement

3.3.4. Count-rate effects for each measurement

3.5.2. Corrections for gamma-ray self absorption

4. Radiochemical stage (only if radiochemical separations are employed)

3.7. Potential bias due to perturbed angular correlations (-ray directional effects)

3.3.2. Effects of clock time uncertainty 3.3.3. Effects of live time uncertainty

3.4. Corrections for gamma-ray interferences

3.6. Potential bias due to peak integration method

4.2. Losses before equilibration with carrier or tracer

standardization; line numbers in this table represent subsections.

4.1. Losses during chemical separation

2.4.1. Hot atom transfer (losses and gains by recoil, nanometer movement)

2.5. Irradiation timing and decay corrections during irradiation (effects of half life and timing uncertainties)

Concepts, Instrumentation and Techniques of Neutron Activation Analysis

http://dx.doi.org/10.5772/53686

165

3.1. Measurement replication or counting statistics (depending on number of replicates) for unknown samples

3.2. Measurement replication or counting statistics (depending on number of replicates) for comparator

3.3. Corrections for radioactive decay (effects of half life and timing uncertainties for each measurement)

3.3.4.1. Corrections for losses due to pulse pileup for conventional analyzer systems

3.5. Corrections for counting efficiency differences (if necessary), or uncertainty for potential differences 3.5.1. Effects resulting from physical differences in size and shape of samples versus standards

**Table 4.** Complete list of individual uncertainty components for NAA measurements using the comparator method of

3.3.4.2. Effects due to inadequacy of live-time extension for conventional analyzer systems 3.3.4.3. Uncertainties due to hardware corrections for Loss-Free or Zero Dead Time systems

Where: Rθ is the ratio of isotopic abundances for unknown sample and calibrator, Rϕ is the ratio of neutron fluence rates (including fluence gradient, neutron self shielding, and scattering) for unknown sample and calibrator, Rσ is the ratio of effective cross sections if neutron spectrum shape differs from unknown sample to calibrator, Rε is the ratio of counting efficiencies (differ‐ ences due to geometry and γ-ray self shielding) for unknown sample and calibrator, blank is the mass of element x in the blank, fP is the correction for pulse pileup (correction method de‐ pends upon the actual hardware used) and fltc is the correction for inadequacy of live time ex‐ tension (correction method depends upon the actual hardware used)

Note that the R values are normally very close to unity, and all units are either SI-based or dimensionless ratios. Thus an uncertainty budget can be developed using only SI units and dimensionless ratios for an NAA measurement by evaluating the uncertainties for each of the terms in Eqs. (23) and (24), and for any additional corrections required (e.g., interferenc‐ es, dry mass conversion factors, etc.).

Uncertainties for some of the terms in Eq. (24) have multiple components. If we sub-divide the uncertainty for each term in the above equations into individual components, add terms for potential corrections, and separate into the four stages of the measurement process, in‐ cluding: pre-irradiation (sample preparation); irradiation; post-irradiation (gamma-ray spec‐ trometry), and radiochemistry, we arrive at the complete list of individual uncertainty components for NAA listed below in Table 4. Only uncertainties from the first three stages should be considered for instrumental neutron activation analysis (INAA) measurements, while all four stages should be considered for radiochemical neutron activation analysis (RNAA) measurements. More details are given in chapter 2 of reference [1] for each subsec‐ tion of uncertainty component.


2.2. Irradiation interferences

0( ) 0( )

164 Imaging and Radioanalytical Techniques in Interdisciplinary Research - Fundamentals and Cutting Edge Applications

*A m m RR RR A*

*cal*

*unk cal*

tension (correction method depends upon the actual hardware used)

es, dry mass conversion factors, etc.).

tion of uncertainty component.

1.4. Sample and standard blanks

2. Irradiation stage

1. Pre-irradiation (sample and standard preparation) stage 1.1. Elemental content of standards (comparators)

1.2. Target isotope abundance ratio — unknown samples/standards

2.2. Temporal effects (fluence variations with time)

2.1. Physical effects (fluence gradients within a single irradiation)

2.4. Neutron shielding effects from neighbouring samples or standards

1.3. Basis mass (or other sample basis) — including drying


<sup>F</sup> = (25)

q se

Where: Rθ is the ratio of isotopic abundances for unknown sample and calibrator, Rϕ is the ratio of neutron fluence rates (including fluence gradient, neutron self shielding, and scattering) for unknown sample and calibrator, Rσ is the ratio of effective cross sections if neutron spectrum shape differs from unknown sample to calibrator, Rε is the ratio of counting efficiencies (differ‐ ences due to geometry and γ-ray self shielding) for unknown sample and calibrator, blank is the mass of element x in the blank, fP is the correction for pulse pileup (correction method de‐ pends upon the actual hardware used) and fltc is the correction for inadequacy of live time ex‐

Note that the R values are normally very close to unity, and all units are either SI-based or dimensionless ratios. Thus an uncertainty budget can be developed using only SI units and dimensionless ratios for an NAA measurement by evaluating the uncertainties for each of the terms in Eqs. (23) and (24), and for any additional corrections required (e.g., interferenc‐

Uncertainties for some of the terms in Eq. (24) have multiple components. If we sub-divide the uncertainty for each term in the above equations into individual components, add terms for potential corrections, and separate into the four stages of the measurement process, in‐ cluding: pre-irradiation (sample preparation); irradiation; post-irradiation (gamma-ray spec‐ trometry), and radiochemistry, we arrive at the complete list of individual uncertainty components for NAA listed below in Table 4. Only uncertainties from the first three stages should be considered for instrumental neutron activation analysis (INAA) measurements, while all four stages should be considered for radiochemical neutron activation analysis (RNAA) measurements. More details are given in chapter 2 of reference [1] for each subsec‐

2.1. Neutron fluence exposure differences (ratios) for unknown samples compared to standards (comparators)

2.3. Neutron self shielding (absorption and scattering) effects within a single sample or standard

2.2.1. Fast (high energy) neutron interferences

2.2.2. Fission interferences

2.2.3. Multiple neutron capture interferences

2.3. Effective cross section differences between samples and standards

2.4. Irradiation losses and gains

2.4.1. Hot atom transfer (losses and gains by recoil, nanometer movement)

2.4.2. Transfer of material through irradiation container

2.4.3. Sample loss during transfer from irradiation container

2.4.4. Target isotope burn up differences

2.5. Irradiation timing and decay corrections during irradiation (effects of half life and timing uncertainties)

3. Gamma-ray spectrometry stage

3.1. Measurement replication or counting statistics (depending on number of replicates) for unknown samples

3.2. Measurement replication or counting statistics (depending on number of replicates) for comparator standards

3.3. Corrections for radioactive decay (effects of half life and timing uncertainties for each measurement)

3.3.1. From end of irradiation to start of measurement

3.3.2. Effects of clock time uncertainty

3.3.3. Effects of live time uncertainty

3.3.4. Count-rate effects for each measurement

3.3.4.1. Corrections for losses due to pulse pileup for conventional analyzer systems

3.3.4.2. Effects due to inadequacy of live-time extension for conventional analyzer systems

3.3.4.3. Uncertainties due to hardware corrections for Loss-Free or Zero Dead Time systems

3.4. Corrections for gamma-ray interferences

3.5. Corrections for counting efficiency differences (if necessary), or uncertainty for potential differences

3.5.1. Effects resulting from physical differences in size and shape of samples versus standards

3.5.2. Corrections for gamma-ray self absorption

3.6. Potential bias due to peak integration method

3.7. Potential bias due to perturbed angular correlations (-ray directional effects)

4. Radiochemical stage (only if radiochemical separations are employed)

4.1. Losses during chemical separation

4.2. Losses before equilibration with carrier or tracer

**Table 4.** Complete list of individual uncertainty components for NAA measurements using the comparator method of standardization; line numbers in this table represent subsections.
