**2.3. Derivation of the measurement equation**

The reaction rate R per nucleus capturing a neutron is given by:

$$R = \bigcap\_{\alpha}^{\circ} \sigma(\nu)\phi'(\nu)d\nu = \bigcup\_{\alpha}^{\circ} \sigma(E)\phi'(E)dE = \bigcap\_{\alpha}^{\circ} n^{\circ}(\nu)\nu\sigma(\nu)d\nu,\tag{1}$$

The (n,γ) cross section function, σ(v) versus v can be interpreted as a σ(v) ~ 1/v dependence, or σ (E) ~ 1/E1/2 dependence [log σ (E) versus log E is linear with slope -1/2], on which (above some eV) several resonances are superposed see Figure 10 taken from http://thor‐

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155

**Figure 9.** A typical reactor neutron energy spectrum showing the various components used to describe the neutron

**Figure 10.** Relation between neutron cross section and neutron energy for major actinides (n, capture).

energy regions.

ea.wikia.com/wiki/Thermal,\_Epithermal\_and\_Fast\_Neutron\_Spectra web page.

where:

σ (v) is the (n,γ) cross section (in cm<sup>2</sup> ; 1 barn (b) = 10-24 cm<sup>2</sup> ) at neutron velocity v (in cm s-1);

σ (E) is the (n,γ) cross section (in cm<sup>2</sup> ) at neutron energy E (in eV);

Φ'(v) is the neutron flux per unit of velocity interval (in cm-3) at neutron velocity v;

n'(v) is the neutron density per unit of velocity interval (in cm-4 s) at neutron velocity v;

Φ'(E) is the neutron flux per unit of energy interval (in cm-2 s-1 eV-1) at neutron energy E.

In Eq.(1), σ (v) = σ (E) with E (in erg = 6.2415.10<sup>11</sup> eV) = ½ mn v2 [m<sup>n</sup> rest mass of the neutron = 1.6749 10-24 g]. Furthermore, per definition, φ'(v) dv = φ'(E)dE (both in cm-2 s-1).

In Eq.1, the functions σ(v) [= σ (E)] and φ'(v) [ φ'(E)] are complex and are respectively de‐ pending on the (n,γ) reaction and on the irradiation site.

In 1987, F De Corte describes in his Aggregate thesis "Chapter 1: fundamentals [3] that the introduction of some generally valid characteristics yields the possibility of avoiding the ac‐ tual integration and describing accurately the reaction rate in a relatively simple way by means of so-called formalisms or conventions. In short, these characteristics are:

In nuclear research reactors – which are intense sources of neutrons – three types of neu‐ trons can be distinguished. The neutron flux distribution can be divided into three compo‐ nents (see Figure 9):


The (n,γ) cross section function, σ(v) versus v can be interpreted as a σ(v) ~ 1/v dependence, or σ (E) ~ 1/E1/2 dependence [log σ (E) versus log E is linear with slope -1/2], on which (above some eV) several resonances are superposed see Figure 10 taken from http://thor‐ ea.wikia.com/wiki/Thermal,\_Epithermal\_and\_Fast\_Neutron\_Spectra web page.

**2.3. Derivation of the measurement equation**

s u f u u s f

*R*

σ (v) is the (n,γ) cross section (in cm<sup>2</sup>

σ (E) is the (n,γ) cross section (in cm<sup>2</sup>

nents (see Figure 9):

where:

The reaction rate R per nucleus capturing a neutron is given by:

In Eq.(1), σ (v) = σ (E) with E (in erg = 6.2415.10<sup>11</sup> eV) = ½ mn v2

pending on the (n,γ) reaction and on the irradiation site.

causing a fission chain reaction in the 235U.

ithermal neutrons induce (n,γ) reactions on target nuclei.

00 0

Φ'(v) is the neutron flux per unit of velocity interval (in cm-3) at neutron velocity v;

= 1.6749 10-24 g]. Furthermore, per definition, φ'(v) dv = φ'(E)dE (both in cm-2 s-1).

means of so-called formalisms or conventions. In short, these characteristics are:

n'(v) is the neutron density per unit of velocity interval (in cm-4 s) at neutron velocity v;

Φ'(E) is the neutron flux per unit of energy interval (in cm-2 s-1 eV-1) at neutron energy E.

In Eq.1, the functions σ(v) [= σ (E)] and φ'(v) [ φ'(E)] are complex and are respectively de‐

In 1987, F De Corte describes in his Aggregate thesis "Chapter 1: fundamentals [3] that the introduction of some generally valid characteristics yields the possibility of avoiding the ac‐ tual integration and describing accurately the reaction rate in a relatively simple way by

In nuclear research reactors – which are intense sources of neutrons – three types of neu‐ trons can be distinguished. The neutron flux distribution can be divided into three compo‐

**1.** Fission or fast neutrons released in the fission of 235U. Their energy distribution ranges from 100 keV to 25 MeV with a maximum fraction at 2 MeV. These neutrons are slowed down by interaction with a moderator, e.g. H2O, to enhance the probability of them

**2.** The epithermal neutron component consists of neutrons (energies from 0.5 eV to about 100 keV). A cadmium foil 1 mm thick absorbs all thermal neutrons but will allow epi‐ thermal and fast neutrons above 0.5 eV in energy to pass through. Both thermal and ep‐

**3.** The thermal neutron component consists of low-energy neutrons (energies below 0.5 eV) in thermal equilibrium with atoms in the reactor's moderator. At room temperature, the energy spectrum of thermal neutrons is best described by a Maxwell-Boltzmann dis‐ tribution with a mean energy of 0.025 eV and a most probable velocity of 2200 m/s. In general, a 1 MW reactor has a peak thermal neutron flux of approximately 10<sup>13</sup> n/cm<sup>2</sup>

¥¥ ¥

() () () () () () , *d E E dE n d*

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

; 1 barn (b) = 10-24 cm<sup>2</sup>

) at neutron energy E (in eV);

'

 u us u u

=== ¢ ¢ òò ò (1)

) at neutron velocity v (in cm s-1);

[m<sup>n</sup> rest mass of the neutron

.

**Figure 9.** A typical reactor neutron energy spectrum showing the various components used to describe the neutron energy regions.

**Figure 10.** Relation between neutron cross section and neutron energy for major actinides (n, capture).

An NAA technique that employs only epithermal neutrons to induce (n,γ) reactions by irra‐ diating the samples being analyzed inside either cadmium or boron a shield is called epi‐ thermal neutron activation analysis (ENAA).

The production of radioactive nuclei is described by:

$$\frac{dN}{dt} = \mathbf{R}N\_0 - \mathcal{L}N\tag{2}$$

is called the thermal neutron density, with Φth=nv0,

**•** σ0 is the thermal neutron activation cross section, m<sup>2</sup>

*Cd*

good approximation by most of the (n,γ) reactions.

¥

**•** v0 is the most probable neutron velocity at 20 °C: 2200 m s−1.

off energy of 0.55 eV;

is introduced:

with:

**•** Φth is the conventional thermal neutron fluence rate, m−2 s−1, for energies up to the Cd cut-

The second term is re-formulated in terms of neutron energy rather than neutron velocity and the infinite dilution resonance integral I0 – which effectively is also a cross section (m<sup>2</sup>

> n epi 0

*E*

*n*

*epi*

= = ò ò (7)

<sup>=</sup> ò (8)

 j

max

*dE n v vdv <sup>I</sup>*

s

n

*E*

Here, Φepi the conventional epithermal neutron fluence rate per unit energy interval, at 1 eV.

From this definition of I0 it can be seen that it assumes that the energy dependency of the epithermal neutron fluence rate is proportional to 1/En. This requirement is fulfilled to a

In the practice of nuclear reactor facilities the epithermal neutron fluence rate Φepi is not precisely following the inverse proportionality to the neutron energy; the small deviation can be accounted for by introducing an epithermal fluence rate distribution parameter α:

max

n

a

<sup>+</sup> <sup>=</sup> ò (9)

(10)

*n*

*E*

 a

s

E

0 (1 ) E (E ) ( ) (1 eV) *<sup>n</sup>*

a

The expression for the reaction rate can thus be re-written as:

tion as Q0(α)= I0(α)/σ0, an effective cross section can be defined:

*dE <sup>I</sup>*

a

0 0( ) *R I th epi* = + js

 j

Expressing the ratio of the thermal neutron fluence rate and the epithermal neutron fluence rate as f=Φth/Φepi and the ratio of the resonance integral and the thermal activation cross sec‐

Cd

(E ) *<sup>n</sup> n*

E

Cd

E (E ) ( )

max

*dE <sup>I</sup>*

s

E

0 E

Cd

*v n*

j

, at 0.025 eV;

Concepts, Instrumentation and Techniques of Neutron Activation Analysis

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) –

157

In which N0 number of target nuclei, N is the number of radioactive nuclei, λ is the decay constant in s−1. The disintegration rate of the produced radionuclide at the end of the irradia‐ tion time ti follows from:

$$D(t\_i) = N(t\_i)\mathcal{L} = N\_0 R(1 - e^{-\lambda t\_i})\tag{3}$$

where:

D is the disintegration rate in Bq of the produced radionuclide, assuming that N=0 at t=0 and N0=constant.

The dependence of the activation cross section and neutron fluence rate to the neutron ener‐ gy can be taken into account in Eq. (1) by dividing the neutron spectrum into a thermal and an epithermal region; the division is made at En=0.55 eV (the so-called cadmium cut-off en‐ ergy). This approach is commonly known as the Høgdahl convention [4].

The integral in Eq. (1) can then be rewritten as:

$$R = \int\_0^{v\_{\triangle}} n(v) \mathbf{v} \, \sigma(v) dv + \int\_{v\_{\triangle}}^0 n(v) v \sigma(v) dv \tag{4}$$

The first term can be integrated straightforward:

$$\int\_{0}^{v\_{\rm cf}} n(v)vdv = v\_0 \sigma\_0 \int\_{0}^{v\_{\rm cf}} n(v)dv = \text{nv}\_0 \sigma\_0 \tag{5}$$

in which,

$$m = \int\_0^{v\_{\zeta d}} n(v)dv\tag{6}$$

is called the thermal neutron density, with Φth=nv0,


The second term is re-formulated in terms of neutron energy rather than neutron velocity and the infinite dilution resonance integral I0 – which effectively is also a cross section (m<sup>2</sup> ) – is introduced:

$$\int\_{v\_{\rm cl}}^{v} n(v)vdv = \rho\_{\rm epi} \int\_{E\_{\rm cl}}^{E\_{\rm max}} \frac{\sigma(E\_n)dE\_n}{E\_n} = \rho\_{\rm epi}I\_0 \tag{7}$$

with:

An NAA technique that employs only epithermal neutrons to induce (n,γ) reactions by irra‐ diating the samples being analyzed inside either cadmium or boron a shield is called epi‐

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

0 *dN RN N dt* = -

<sup>0</sup> ( ) ( ) (1 )*it Dt Nt NR e i i*

l

ergy). This approach is commonly known as the Høgdahl convention [4].

0 0

*Cd Cd v v*

The integral in Eq. (1) can then be rewritten as:

The first term can be integrated straightforward:

0

*Cd*

*v*

In which N0 number of target nuclei, N is the number of radioactive nuclei, λ is the decay constant in s−1. The disintegration rate of the produced radionuclide at the end of the irradia‐

D is the disintegration rate in Bq of the produced radionuclide, assuming that N=0 at t=0 and

The dependence of the activation cross section and neutron fluence rate to the neutron ener‐ gy can be taken into account in Eq. (1) by dividing the neutron spectrum into a thermal and an epithermal region; the division is made at En=0.55 eV (the so-called cadmium cut-off en‐

( )v ( ) ( ) ( )

*R n v v dv n v v v dv* s

( ) ( ) nv

*n v vdv v n v dv* = = s

0

*Cd v*

( )

*Cd*

0 0 0 0

¥

 s

= + ò ò (4)

 sò ò (5)

*n n v dv* <sup>=</sup> ò (6)

*v*

l


l

(2)

thermal neutron activation analysis (ENAA).

tion time ti follows from:

where:

N0=constant.

in which,

The production of radioactive nuclei is described by:

$$I\_0 = \int\_{E\_{\rm Cd}}^{E\_{\rm max}} \frac{\sigma(E\_n) dE\_n}{E\_n} \tag{8}$$

Here, Φepi the conventional epithermal neutron fluence rate per unit energy interval, at 1 eV.

From this definition of I0 it can be seen that it assumes that the energy dependency of the epithermal neutron fluence rate is proportional to 1/En. This requirement is fulfilled to a good approximation by most of the (n,γ) reactions.

In the practice of nuclear reactor facilities the epithermal neutron fluence rate Φepi is not precisely following the inverse proportionality to the neutron energy; the small deviation can be accounted for by introducing an epithermal fluence rate distribution parameter α:

$$I\_0(\alpha) = \begin{pmatrix} 1 & \text{eV} \end{pmatrix}^{\alpha} \int\_{E\_{\text{Cd}}}^{E\_{\text{max}}} \frac{\sigma(\mathcal{E}\_n) dE\_n}{E\_n^{(1+\alpha)}} \tag{9}$$

The expression for the reaction rate can thus be re-written as:

$$R = \varphi\_{lh}\sigma\_0 + \varphi\_{epi}I\_0(\alpha) \tag{10}$$

Expressing the ratio of the thermal neutron fluence rate and the epithermal neutron fluence rate as f=Φth/Φepi and the ratio of the resonance integral and the thermal activation cross sec‐ tion as Q0(α)= I0(α)/σ0, an effective cross section can be defined:

$$
\sigma\_{\rm eff} = \sigma\_0 (1 + \frac{Q\_0(a)}{f}) \tag{11}
$$

**•** mx is the mass of the irradiated element in g;

photon of Eγ (photons disintegration−1);

resolution of the detector; this effect is called summation.

*a*

q

*<sup>M</sup> m N*

sample under given experimental conditions:

clides. The other parameters Np, mx, Φ, ε, t<sup>i</sup>

the given circumstances and uncertainties can be established.

Both absolute and relative methods of calibration exist.

**•** I is the gamma-ray abundance, i.e. the probability of the disintegrating nucleus emitting a

**•** ε is the full energy photopeak efficiency of the detector, i.e. the probability that an emitted photon of given energy will be detected and contribute to the photopeak at energy E<sup>γ</sup> in

Although the photons emitted have energies ranging from tens of keVs to MeVs and have high penetrating powers, they still can be absorbed or scattered in the sample itself depend‐ ing on the sample size, composition and photon energy. This effect is called gamma-ray selfattenuation. Also, two or more photons may be detected simultaneously within the time

Eq. (15) can be simply rewritten towards the measurement equation of NAA, which shows

<sup>m</sup> - t (1 ) (1-e ) *i d*


 l l

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159

*M*<sup>=</sup> (17)

, td, tm are determined, calculated or measured for

l

l

how the mass of an element measured can be derived from the net peak area C:

*x p t t av th eff*

js

*N e e*

 e

*Np <sup>F</sup>*

Standardization is based on the determination of the proportionality factors F that relate the net peak areas in the gamma-ray spectrum to the amounts of the elements present in the

The values of the physical parameters determining the proportionality factor θ, NAv, M, σeff I, λ, are taken from literature. The parameters σeff respectively I, λ are not precisely known for many (n,γ) reactions and radionuclides, and in some cases θ is also not accurately known. Since the various parameters were often achieved via independent methods, their individual uncertainties will add up in the combined uncertainty of measurement of the ele‐ mental amounts, leading to a relatively large combined standard uncertainty. Moreover, the metrological traceability of the values of the physical constants is not known for all radionu‐

**•** Ma is the atomic mass in g mol−1;

the spectrum.

**2.4. Standardization**

*2.4.1.Absolute calibration*

It simplifies the Eq. (10) for the reaction rate to:

$$R = \phi\_{\rm th} \sigma\_{\rm eff} \tag{12}$$

This reaction rate applies to infinite thin objects. In objects of defined dimensions, the inside part will experience a lower neutron fluence rate than the outside part because neutrons are removed by absorption.

The nuclear transformations are established by measurement of the number of nuclear de‐ cays. The number of activated nuclei N(ti ,td) present at the start of the measurement is given by:

$$\text{IN}(t\_{i'}, t\_{d'}, t\_m) = \frac{RN\_0}{\mathcal{N}} (1 - e^{-\lambda t\_i}) e^{-\lambda t\_d} \tag{13}$$

and the number of nuclei ΔN disintegrating during the measurement is given by:

$$
\Delta\text{NN}(t\_{i'}t\_{d'}t\_m) = \frac{RN\_0}{\lambda}(1 - e^{-\lambda t\_i})e^{-\lambda t\_d}(1 - e^{-\lambda t\_m})\tag{14}
$$

in which td is the decay or waiting time, i.e. the time between the end of the irradiation and the start of the measurement tm is the duration of the measurement. Additional correction resulting from high counting rates may be necessary depending upon the gamma-ray spec‐ trometer hardware used as illustrated in chapter 2 [1]. Replacing the number of target nuclei N0 by (NAvm)/M and using the Eq. (12) for the reaction rate, the resulting net counts C in a peak in the spectrum corresponding with a given photon energy is approximated by the ac‐ tivation formula:

$$N\_p = \Delta N \jmath \varepsilon = \varrho\_{\rm th} \sigma\_{\rm eff} \frac{N\_{a\rm av} \theta m\_{\rm x}}{M\_a} (1 - e^{-\lambda t\_i}) e^{-\lambda t\_d} (1 - e^{-\lambda t\_m}) \text{I.c} \tag{15}$$

with:


0

= + (11)

(12)

,td) present at the start of the measurement is given


l

l

e


0 ( ) (1 ) *eff Q f* a

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

*R th eff* = j s

This reaction rate applies to infinite thin objects. In objects of defined dimensions, the inside part will experience a lower neutron fluence rate than the outside part because neutrons are

The nuclear transformations are established by measurement of the number of nuclear de‐

l l

<sup>0</sup> ( , , ) (1 )*i d t t*

<sup>m</sup> <sup>0</sup> - t ( , , ) (1 ) (1-e ) *i d t t*

l l

in which td is the decay or waiting time, i.e. the time between the end of the irradiation and the start of the measurement tm is the duration of the measurement. Additional correction resulting from high counting rates may be necessary depending upon the gamma-ray spec‐ trometer hardware used as illustrated in chapter 2 [1]. Replacing the number of target nuclei N0 by (NAvm)/M and using the Eq. (12) for the reaction rate, the resulting net counts C in a peak in the spectrum corresponding with a given photon energy is approximated by the ac‐

<sup>m</sup> - t (1 ) (1-e ) *i d av x t t*


l l

l

*RN Nt t t e e*

and the number of nuclei ΔN disintegrating during the measurement is given by:

*RN Nt t t e e*

l

*a*

*M* q

*N m N N e e*

*i dm*

*idm*

*p th eff*

ge j s

**•** Np is the net counts in the γ-ray peak of Eγ ;

**•** θ is isotopic abundance of the target isotope;

**•** NAv is the Avogadro's number in mol−1;

 s

s

It simplifies the Eq. (10) for the reaction rate to:

cays. The number of activated nuclei N(ti

removed by absorption.

tivation formula:

with:

by:


Although the photons emitted have energies ranging from tens of keVs to MeVs and have high penetrating powers, they still can be absorbed or scattered in the sample itself depend‐ ing on the sample size, composition and photon energy. This effect is called gamma-ray selfattenuation. Also, two or more photons may be detected simultaneously within the time resolution of the detector; this effect is called summation.

Eq. (15) can be simply rewritten towards the measurement equation of NAA, which shows how the mass of an element measured can be derived from the net peak area C:

$$m\_x = N\_p \cdot \frac{M\_a}{N\_{av} \cdot \theta} \cdot \frac{\lambda}{\varphi\_{th} \cdot \sigma\_{eff} \cdot \mathbf{I} \cdot \boldsymbol{\varepsilon} \cdot (1 - e^{-\lambda t\_i}) \cdot e^{-\lambda t\_d} \cdot (1 - e^{-\lambda t\_m})} \tag{16}$$

#### **2.4. Standardization**

Standardization is based on the determination of the proportionality factors F that relate the net peak areas in the gamma-ray spectrum to the amounts of the elements present in the sample under given experimental conditions:

$$F = \frac{N\_p}{M} \tag{17}$$

Both absolute and relative methods of calibration exist.

#### *2.4.1.Absolute calibration*

The values of the physical parameters determining the proportionality factor θ, NAv, M, σeff I, λ, are taken from literature. The parameters σeff respectively I, λ are not precisely known for many (n,γ) reactions and radionuclides, and in some cases θ is also not accurately known. Since the various parameters were often achieved via independent methods, their individual uncertainties will add up in the combined uncertainty of measurement of the ele‐ mental amounts, leading to a relatively large combined standard uncertainty. Moreover, the metrological traceability of the values of the physical constants is not known for all radionu‐ clides. The other parameters Np, mx, Φ, ε, t<sup>i</sup> , td, tm are determined, calculated or measured for the given circumstances and uncertainties can be established.

#### *2.4.2. Relative calibration*

#### **a.** Direct comparator method

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 the element of interest can be calculated:

$$m\_{x\text{(unk)}} = m\_{x\text{(cal)}} \cdot \frac{\left(\frac{N\_p}{t\_m \, e^{-\lambda t\_d} \cdot (1 - e^{-\lambda t\_m})}\right)\_{\text{unk}}}{\left(\frac{N\_p}{t\_m \, e^{-\lambda t\_d} \cdot (1 - e^{-\lambda t\_m})}\right)\_{\text{cal}}} \tag{18}$$

Masses for each element i then can be calculated from these ki

where: mx(comp) is the mass of element x in comparator in g.

*x unk x comp*

*m m*

demonstrated.

**b.** The k0-comparator method

mined via a directly produced radionuclide the mass mx(unk) follows from:

() ( ) i


(1 ). . .(1 ) . .k

lll

*p ttt*


*N e te e*

æ ö ç ÷

*md m*

*md m*

*m unk*

Concepts, Instrumentation and Techniques of Neutron Activation Analysis

*m comp*

(1 ). . .(1 )


These experimentally determined k-factors are often more accurate than when calculated on basis of literature data as in the absolute calibration method. However, the k-factors are only valid for a specific detector, a specific counting geometry and irradiation facility, and remain valid only as long as the neutron fluence rate parameters of the irradiation facility remain stable. The single comparator method requires laborious calibrations in advance, and finally yield relatively (compared to the direct comparator method) higher uncertainties of the measured values. Moreover, it requires experimental determination of the photopeak effi‐ ciencies of the detector. Metrological traceability of the measured values to the S.I. may be

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‐

*N e te e*

*p ttt*


lll

factors; for an element deter‐

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(20)

161

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

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‐ cy is required.

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

$$k\_i = \frac{(M\_a)\_{i,\text{cal}} \gamma\_{\text{comp}} \mathcal{E}\_{\text{comp}} \theta\_{\text{comp}} \sigma\_{\text{eff,comp}}}{M\_{i,\text{cal}} \gamma\_{i,\text{cal}} \varepsilon\_{i,\text{cal}} \theta\_{i,\text{cal}} (\sigma\_{\text{eff}})\_{i,\text{cal}}} \tag{19}$$

Masses for each element i then can be calculated from these ki factors; for an element deter‐ mined via a directly produced radionuclide the mass mx(unk) follows from:

$$m\_{\mathbf{x}\{\mathrm{unk}\}} = m\_{\mathbf{x}\{\mathrm{comp}\}} \cdot \frac{\left(\frac{N\_p}{(1 - e^{-\lambda t\_m}) \, t\_m \, e^{-\lambda t\_d} \, (1 - e^{-\lambda t\_m})}\right)\_{\mathrm{unk}}}{\left(\frac{N\_p}{(1 - e^{-\lambda t\_m}) \, t\_m \, e^{-\lambda t\_d} \, (1 - e^{-\lambda t\_m})}\right)\_{\mathrm{comp}}} \cdot \mathbf{k}\_i \tag{20}$$

where: mx(comp) is the mass of element x in comparator in g.

These experimentally determined k-factors are often more accurate than when calculated on basis of literature data as in the absolute calibration method. However, the k-factors are only valid for a specific detector, a specific counting geometry and irradiation facility, and remain valid only as long as the neutron fluence rate parameters of the irradiation facility remain stable. The single comparator method requires laborious calibrations in advance, and finally yield relatively (compared to the direct comparator method) higher uncertainties of the measured values. Moreover, it requires experimental determination of the photopeak effi‐ ciencies of the detector. Metrological traceability of the measured values to the S.I. may be demonstrated.
