**b.** The k0-comparator method

*2.4.2. Relative calibration*

**a.** Direct comparator method

calibrator, respectively in g.

cy is required.

the element of interest can be calculated:

( ) ()

*x unk x cal*

*m m*

sample weights defined the k-factor for each element i as:

*i*

*k*

( )

g

geq

*M*

*M*

The unknown sample is irradiated together with a calibrator containing a known amount of the element(s) of interest. The calibrator is measured under the same conditions as the sam‐ ple (sample-to-detector distance, equivalent sample size and if possible equivalent in com‐ position). From comparison of the net peak areas in the two measured spectra the mass of

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

. .(1 ) .

l


l

in which mx(unk), mx(cal) mass of the element of interest, in the unknown sample and the

In this procedure many of the experimental parameters - such as neutron fluence rate, cross section and photopeak efficiency cancel out at the calculation of the mass and the remaining parameters are all known. This calibration procedure is used if the highest degree of accura‐

The relative calibration on basis of element calibrators is not immediately suitable for labo‐ ratories aiming at the full multi-element powers of INAA. It takes considerable effort to pre‐ pare multi-element calibrators for all 70 elements measurable via NAA with adequate degree of accuracy in a volume closely matching the size and the shape of the samples. Sin‐ gle comparator method Multi-element INAA on basis of the relative calibration method is feasible when performed according to the principles of the single comparator method. As‐ suming stability in time of all relevant experimental conditions, calibrators for all elements are co-irradiated each in turn with the chosen single comparator element. Once the sensitivi‐ ty for all elements relative to the comparator element has been determined (expressed as the so-called k-factor, see below), only the comparator element has to be used in routine meas‐ urements instead of individual calibrators for each element. The single comparator method for multi-element INAA was based on the ratio of proportionality factors of the element of interest and of the comparator element after correction for saturation, decay, counting and

> , , ,,,, ,

*i cal i cal i cal i cal eff i cal*

*a i cal comp comp comp eff comp*

eqs

( )

<sup>=</sup> (19)

 s

*N te e*


ç ÷ - è ø

*N te e*

æ ö ç ÷


. .(1 )

*p t t m cal*

*d m*

*d m*

 l

(18)

 l

*p t t m unk*

> The *k*0-based neutron activation analysis (*k*0-NAA) technique, developed in 1970s, is being increasingly used for multielement analysis in a variety of matrices using reactor neu‐ trons [4-10]. In our research reactor, the *k*0-method was successfully developed using the Høgdahl formalism [11]. In the *k*0-based neutron activation analysis the evaluation of the analytical result is based on the so-called *k*0- factors that are associated with each gammaline in the gamma-spectrum of the activated sample. These factors replace nuclear con‐ stants, such as cross sections and gamma-emission probabilities, and are determined in specialized NAA laboratories. This technique has been reported to be flexible with re‐ spect to changes in irradiation and measuring conditions, to be simpler than the relative comparator technique in terms of experiments but involves more complex formulae and calculations, and to eliminate the need for using multielement standards. The *k*0-NAA technique, in general, uses input parameters such as (1) the epithermal neutron flux shape factor (α), (2) subcadmium-to-epithermal neutron flux ratio (*f*), (3) modified spectral index *r*(*α*) *Tn* / *T*0, (4) Westcott's *g*(*Tn*)-factor, (5) the full energy peak detection efficiency (*εp*), and (6) nuclear data on *Q*0 (ratio of resonance integral (*I*0) to thermal neutron cross sec‐ tion (σ0) and *k*0. The parameters from (1) to (4) are dependent on each irradiation facility and the parameter (5) is dependent on each counting facility. The neutron field in a nucle‐ ar reactor contains an epithermal component that contributes to the sample neutron acti‐ vation [12]. Furthermore, for nuclides with the Westcott's *g*(*Tn*)-factor different from unity, the Høgdahl convention should not be applied and the neutron temperature should be in‐

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 *k*0-method.

strated. As such, this approach is often preferred from a metrological viewpoint. The

( ) <sup>1</sup> ( ) . . . 10

*p m Au x th x epi x x*

*SDCW G fG Q ppm N t k G fG Q*

*x th Au epi Au Au*

Where: the indices x and Au refer to the sample and the monitor, respectively; WAu and Wx represent the mass of the gold monitor and the sample (in g); Np is the measured peak area, corrected for dead time and true coincidence; S, D, C are the saturation, decay and counting factors, respectively; tm is the measuring time; Gth and Ge are the correction factors for ther‐

Many publications reported in literature [20-25] treat the concept of evaluation of uncertain‐

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‐

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

> <sup>0</sup> p ltc .f .f (1 ) (1 ) *d m i t p t t N e*

l

 l


*e e*

l

l

*A*

tion), as well as methods to evaluate each one of these uncertainty components [1].

ê ú <sup>+</sup> ë û = ´ é ù <sup>+</sup> ê ú

, , 0, p,Au 6

( )

a e

a e

Concepts, Instrumentation and Techniques of Neutron Activation Analysis

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163

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

0, ( ) , , 0, p,x

concentration of an element can be determined as:

*x*

r

**2.5. Sources errors**

*p m*

*N t*

é ù ê ú

*SDCW*

mal and epithermal neutron self shielding, respectively.

ties in large range of analytical techniques.

be further simplified, by substituting:

in :

ê ú ë û

*Au*

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

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 resulted at the definition of the k0-factor.

$$k\_0 = \frac{1}{k} \cdot \frac{1 + \mathcal{Q}\_{0,\text{comp}}(a) / f}{1 + \mathcal{Q}\_{0,\text{cal}}(a) / f} \cdot \frac{\varepsilon\_{\text{comp}}}{\varepsilon\_{\text{comp}}} = \frac{\mathcal{M}\_{\text{comp}}}{\theta\_{\text{comp}} \sigma\_{0,\text{comp}} \mathcal{Y}\_{\text{comp}}} \cdot \frac{\theta\_{\text{cal}} \sigma\_{0,\text{cal}} \mathcal{Y}\_{\text{cal}}}{\mathcal{M}\_{\text{cal}}} \tag{21}$$

$$m\_{\rm x\{unk\}} = m\_{\rm x\{comp}} \frac{1 + \mathcal{Q}\_{0, \rm comp}(a) / f}{1 + \mathcal{Q}\_{0, \rm cal}(a) / f} \cdot \frac{\varepsilon\_{\rm comp}}{\varepsilon\_{\rm comp}} \frac{\left(\frac{\mathcal{N}\_{\rm p} / t\_{\rm m}}{(1 - e^{-\lambda t\_{\rm i}}) e^{-\lambda t\_{\rm d}} (1 - e^{-\lambda t\_{\rm m}}) m}{\mathcal{E}\_{\rm p} / t\_{\rm m}}\right)\_{\rm unk}}{\left(\frac{\mathcal{N}\_{\rm p} / t\_{\rm m}}{(1 - e^{-\lambda t\_{\rm i}}) e^{-\lambda t\_{\rm d}} (1 - e^{-\lambda t\_{\rm m}}) m}\right)\_{\rm comp}} \tag{22}$$

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‐ strated. As such, this approach is often preferred from a metrological viewpoint. The concentration of an element can be determined as:

$$\rho\_x(ppm) = \frac{\left[\frac{N\_p \int t\_m}{SDCVW}\right]\_x}{\left[\frac{N\_p \int t\_m}{SDCVW}\right]\_{A\mathbf{u}}} \cdot \frac{1}{k\_{0,Au(x)}} \cdot \frac{G\_{\text{th},Au}f + G\_{\text{qpl},Au}Q\_{0,Au}(\alpha)}{G\_{\text{th},x}f + G\_{\text{qpl},x}Q\_{0,x}(\alpha)} \cdot \frac{\varepsilon\_{\text{p,Au}}}{\varepsilon\_{\text{p,x}}} \times 10^6\tag{23}$$

Where: the indices x and Au refer to the sample and the monitor, respectively; WAu and Wx represent the mass of the gold monitor and the sample (in g); Np is the measured peak area, corrected for dead time and true coincidence; S, D, C are the saturation, decay and counting factors, respectively; tm is the measuring time; Gth and Ge are the correction factors for ther‐ mal and epithermal neutron self shielding, respectively.
