** American Journal of Analytical Chemistry ** Vol. 3 No. 3 (2012) , Article ID: 18271 , 7 pages DOI:10.4236/ajac.2012.33033

Influence of the Neutron Flux Characteristic Parameters in the Irradiation Channels of Reactor on NAA Results Using k_{0}-Standardization Method

Research and Development Center for Radiation Technology, Hochiminh City, Vietnam

Email: tranhungkeiko@yahoo.com

Received November 28, 2011; revised January 5, 2012; accepted January 16, 2012

**Keywords:** k_{0}-Standardization Technique; Error Propagation Function; Neutron Flux Characteristics; Dalat Reactor

ABSTRACT

An approximation method using to estimate the influence of the uncertainties of the neutron flux characteristic parameters in the irradiation positions on the NAA results using k_{0}-standardization technique was presented. Those are the epithermal reactor neutron spectrum shape-factor a, the effective resonance energy for a given nuclide and the thermal to epithermal neutron flux ratio f. The method is applied to estimate the effect of the uncertainties in the determination of a, and f on final NAA results for some irradiation channels of the Dalat reactor. It also shows that presented method is suitable in practical use for the estimation of the errors due to the uncertainty of the neutron flux characteristic parameters at the irradiation position.

1. Introduction

Since the k_{0}-standardization method was introduced in NAA [1], it has been broadly applied in the reactor in the world. The fundamental concept of k_{0}-method was being elaborated previously in great detail [1-3]. The concentration of an element in the k_{0}-method is calculated by:

(1)

with k_{0} in Equation (1) defined as:

(2)

In Equations (1) and (2):

M—atomic mass;

q —isotopic abundance;

s_{0}—2200 m∙s^{–1} (n, g) cross-section;

g —absolute gamma-intensity;

N_{p}—peak area corrected for pulse losses;

W—sample weight in gram;

w^{*}—comparator weight in microgram;

S = 1 – exp(–lt_{irr}); t_{irr}—irradiation time; l—decay constant;

D = exp (–lt_{d}); t_{d}—decay time;

C = [1 – exp (–lt_{m})]/lt_{m}; t_{m}—measuring time;

f—thermal to epithermal neutron flux ratio;

Q_{0}(a) = I_{0}(a)/s_{0}; I_{0}(a)—resonance integral corrected for a non-ideal epithermal neutron flux distribution (assumed 1/E^{1+}^{a});

ε_{p}—detector’s efficiency;

When the epithermal neutron flux distribution deviates from ideality, i.e. it does not follow the 1/E-law, Q_{0}(a) of nuclide i can be written by:

(3)

with a—neutron spectrum shape factor deviating from the 1/E-law, independent of neutron energy and.

—effective resonance energy of nuclide i.

The asterisks in Equations (1) and (2) refers to the comparator, which is suitable for coirradiation with the sample; in most case, Au is used as a comparator. The k_{0}-factors to Au for interested isotopes in NAA were experimentally determined and tabulated in report [4] with an accuracy which better than 2% (average ~1%). The relevant nuclear data as Q_{0i} and can be found in a tabulated form or in a computer library. a, f and e_{p} must be experimentally determined and they depend on specific irradiation channel and detector, which are used in practice. The detector’s efficiency (e_{p}) can be determined with an uncertainty about 2%; but the uncertainty of a can be more than 10%, even bigger, depend on the irradiation channels in reactor. Since the term [f + (a)]/ [f + Q_{0}_{ }(a)] in Equation (1), it is clear that an additional parameter, , should be considered, because the uncetainties of of some nuclides are about 20% [4,5].

The accuracy and the applicability of the k_{0}-standardization method were detailly presented in paper [5] by F. De CORTE et al. In paper [6], J. OP De BEEK evaluated the effect of errors of a and on the results in terms of concentration, based on the ^{197}Au comparator; in that Q_{0i}(a) was approximated by :

(4)

However, with this approximation, it led that some results in paper [6] have to be put to discussion (see below).

In this work, we carry out an approximation method to evaluate the effect of errors of a and on the NAA results in the k_{0}-standardization method. The obtained results showed that the approximate method in this work is acceptable with confident accuracy.

2. Base of Approximation

As we know, a value is smaller than unity in absolutte value. In practice, in irradiation channels of reactor, absolute value of a is less than 0.2 (in most cases, < 0.1 and this condition is satisfactory in reactor core). In the approximation of J. OP De BEEK, it is good for the nuclides having Q_{0i} >1, but is not for the nuclides with Q_{0i} <1. Due to, in paper [7,8], we suggest substituting from Equation (3) by the following approximated formula:

(5)

where a_{i} is constant for each nuclide and determined by fitting the values of Q_{0i}(a), which are calculated from Equation (3) in range, then fitting according to function (5) (see reference [7,8]). Note that, a_{i} of each nuclide depends on the sign of a. The values of a_{i} for the interested nuclides in NAA are given in Table 1. Seeing the Equation (5), it differs to Equation (4) of J. OP De BEEK by a correctional coefficient a_{i}. However, it can be used good for all nuclides with uncertainties of the calculated less than about 5% for the nuclides having Q_{0i} < 1 and less than about 2% for Q_{0i} > 1 with. Indeed, we carried out a survey of the ratios of Q_{0i}(a) calculated from Equation (5) (in this work) and Equation (4) (of J. OP De BEEK) to Equation (3) (accurate expression) for Q_{0i} from 0.44 (^{46}Sc) to 248 (^{97}Zr) with a = –0.1. The results are presented in Figure 1 and some results are presented in Table 2. Clearly, the approximated expression in this work is better than one of J. OP De BEEK. Moreover, the calculated Q_{0i}(a) from three expression Equation (3), Equation (4) and Equation (5) for ^{45}Sc(n, g)^{46}Sc presented in Table 3. The another nuclides presented in papers [7,8] also confirm the above conclusion.

Figure 1. Survey of the ratios of Q_{0i}(α) calculated from Equation (5) (in this work) and Equation (4) (of J. OP De BEEK) to Equation (3) (accurate expression) for different Q_{0i} with α = –0.1.

Table 1. The values of a_{i} for the interesting nuclides in NAA.

Table 2. Ratio of Q_{0}(a) calculated by Equations (4) and (5) to Equation (3) of some nuclide in reaction (n, g) using in NAA.

Table 3. The results calculated Q_{0i}(a) from three expression Equations (3)-(5) with a in interval [–0.2, 0.2] for ^{45}Sc(n, g)^{46}Sc.

From Table 1, it shows that coefficients a_{i} of nuclides having Q_{0i} > 1 are close to unity, but a_{i} of the nuclides having Q_{0i} < 1 differs more than unity. Therefore, the approximation of Equation (4) in paper [6] is only acceptable for the nuclides having Q_{0i} > 1, but for the nuclides having Q_{0i} < 1, it is not reliable.

In this work, we use the approximation expression; Equation (5), to evaluate influence of the uncertainties of a, f and on the final element concentration in k_{0}-method in the channels; 7 - 1, neutron trap of Dalat reactor (Vietnam) and channel 17 of THETIS reactor (Belgium) for the nuclides; ^{45}Sc, ^{59}Co, ^{94}Zr, ^{186}W, ^{197}Au, ^{98}Mo, ^{96}Zr. We choose these nuclides, because they differ considerably in Q_{0i} and values. The numerical data of concerning isotopes and irradiation channels used in this work are summarized in Tables 4 and 5.

3. Results and Discussion

The absolute uncertainty in r can be calculated from the uncertainties of the variables (denoted x_{j}) which determine r in Equation (1):

(6)

where ∂a/∂x_{j} are the corresponding partial derivatives.

According to the customary error propagation theory,

Table 4. Characteristics of isotopes used in the calculations of this work.

Table 5. Characteristics of irradiation channels considered: channles 17 of Thetis reactor, Belgium [9], 7 - 1 channel and neutron trap of Dalat reactor, Vietnam.

the error propagation functions can be written as:

(7)

and relative error is:

(8)

3.1. Influence of Uncertainty of on NAA Results

From Equation (8), the uncertainty of the concentration (r) in k_{0}-method due to the uncertainties of the effective resonace energies can be written by:

(9)

Using Equation (7) for the effective resonance energy of the nuclide i, we obtain:

(10)

The values of calculated for chosen nuclides are presented in Table 6. The effect of the effective resonance energy on NAA result include the uncertainties of the effective resonance energies of analytical and comparator nuclides. In this case, Au used as comparator with of 5.65 eV and uncertainty of 7.1% from paper [5], the contribution of the uncertainty of to the error of NAA result in channels 7 - 1, neutron trap of Dalat reactor and channel 17 of THETIS reactor is 0.17%, 0.077% and 0.13%, respectively. Clearly, the effect of the uncertainty of the effective resonance energy of Au is negligible and can be overlooked in the evaluation.

The analysis for 94 nuclides used in NAA showed that the uncertainties of their effective resonace energy are from 0 to 20%, except ^{75}As (34%) [4]. In this measure, we are able to realize that the effect of them on NAA result is also negligible. For example, ^{45}Sc (= 5130 eV, = 17%) and ^{95}Zr (= 338 eV, = 2.1%), the contribution of the uncertainty of the effective resonance energy to the error of NAA result in three above channels is less than 0.01% for ^{45}Sc and 0.1% for ^{95}Zr.

In epicadmium neutron activation analysis (ENAA), the f-term in Equation (10) should be omitted. The error propagation function of can be written:

(11)

The calculated results of for the nuclides;

^{45}Sc, ^{59}Co, ^{94}Zr, ^{186}W, ^{197}Au, ^{98}Mo, ^{96}Zr in ENAA are carried in Table 7. In this case, the error propagation function is higher than in the one of irradiation without cadmium. Generaly speaking, a_{i} < 1 and if a 1, the contribution of to the error of NAA result for almost analytical nuclides is less than 1% and can be omitted in the calculation.

3.2. Influence of Uncertainty of ( on NAA Results

Also from Equation (8), the uncertainty of r due to the uncertainty of a can be written:

(12)

and error propagation function of a:

(13)

The values of the error propagation function of a in the channels; 7 - 1 and neutron trap of Dalat reactor (Vietnam) and channel 17 of THETIS reactor (Belgium) for the nuclides; ^{45}Sc, ^{59}Co, ^{94}Zr, ^{186}W, ^{197}Au, ^{98}Mo, ^{96}Zr were shown in Table 8. From Table 8, for the nuclides having Q_{0} < Q_{0Au} in three these channels, the contribution of the uncertainty of a to the error of NAA result is not significant, about less than 1%. But for nuclides having Q_{0} Q_{0Au}, this effect is noticeable. For instance, in channel 7 - 1 of Dalat reactor (a = –0.044, Da = 12%

Table 6. Calculation results of for chosen nuclides.

Table 7. Calculated results of for the nuclides in ENAA.

Table 8. Calculation results of Z_{r}(a) for chosen nuclides.

[7,8]), the contribution of the uncertainty of a on the error of result of ^{45}Sc (Q_{0} = 0.44) is 0.42%, but for ^{99}Mo and ^{96}Zr is 1.36% and 2.4%, respectively. As a comment, for RNAA using ^{197}Au comparator, the systematic effect for a value up to 0.1 is practically negligible for all nuclides with a low enough Q_{0} value (e.g. ^{45}Sc, ^{59}Co, ^{58}Fe, ect.). On the other hand, for nuclides with a relatively large Q_{0} value, a correction for the a effect becomes really necessary. To reduce the a effect, it is either to develop more accurate and precise techniques for a determination or to choose the irradiation channels with the a value low enough.

In the case of the epicadmium neutron activation, Equation (13) can be changed into:

(14)

The values of the error propagation of a in this case were carried in Table 9. In this case, it clearly shows the inaccuracy of the approximation expression in [6] (Equation (4) in this report). Really, according to Equation (4), the error propagation function of a in the irradiation with cadmium can be written:

(15)

Equation (15) is different to Equation (14) by the correctional coefficients a_{i}. However, the value of the error propagation function in channel 7 - 1 of Dalat reactor, for ^{45}Sc is 0.0083 from Equation (14) and 0.2997 from Equation (15). If the uncertainty of a in experiment is 100%, the contribution of uncertainty of a on NAA result is 0.83% and 29.97%, respectively. It differs by a factor of 30 (!). Similarly, in channel 17 of Thetis reactor, the error propagation function for ^{45}Sc is 0.0053 and 0.1907. The difference is huge. This comment is also correct for nuclides having Q_{0} < 1. It once more confirms that the approximation expression in paper [6] is not good for nuclides having Q_{0} < 1.

From Equation (13) or Equation (14), we easily estimate the influence of a on NAA results, if we know uncertainty of a in the irradiation channel. However, for ENAA (epicadmium neutron activation analysis) the situation is much more dramatic, especially for nuclides with low Q_{0} value.

3.3. Influence of Uncertainty of f on NAA Results

The error propagation function Z_{r}(f) can be written:

(16)

The values of the error propagation function of f in the channels; 7 - 1 and neutron trap of Dalat reactor and channel 17 of THETIS reactor for the nuclides; ^{45}Sc, ^{59}Co, ^{94}Zr, ^{186}W, ^{197}Au, ^{98}Mo, ^{96}Zr were carried in Table 9. The uncertainty of f contributes on the error of NAA results is:

(17)

Generally seeing, the uncertainty of f in experiment is about less than 4%, therefore, from Table 10, the contribution of the uncertainty of f on the error of NAA result

Table 9. Calculation results of Z_{r}(a) for the nuclides in ENAA.

Table 10. Calculation results of Z_{r}(f) for chosen nuclides.

is about less than 2%.

3.4. Collective Influence of Uncertainties of (, and f on NAA Results

In view of the above, we can estimate the influence of the uncertainties of a, and f on final NAA results. The contribution of these parameters on the errors of the analysis results is written as:

(18)

However, as discussion above, the effect is negligible and can be omitted in Equation (18). Thus, the contribution on error of NAA results in this case is primarily due to the uncertainties of a and f. Finally, as well as estimation above, this overall contribution of a and f is about 2% on the error of NAA results. It was also confirmed by actual analysis.

4. Conclusion

For a in the irradiation position relatively small (), Equation (5) is a good approximation to estimate influence of the neutron flux characteristics on NAA result using the k_{0} standardization method. From this approximative expression, the error propagation functions of the parameters were presented. They can be used for the estimation of the errors on NAA due to the uncertainty of the neutron flux characteristic parameters at the irradiation position. From the results of this report, it was also confirmed that the approximation in paper [6] is only acceptable for the nuclides having Q_{0i} > 1, but not for the nuclides having Q_{0i} < 1.

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