Anal. Chem. 1996, 68, 4326-4334
General Activation and Decay Formulas and Their Application in Neutron Activation Analysis with k0 Standardization Stefaan G. Pomme´,*,† Frank E. M. C. Hardeman,† Piotr B. Robouch,‡ Nestor Etxebarria,‡ Frans A. De Corte,§ Antoine H. M. J. De Wispelaere,§ Robbert van Sluijs,| and Andras P. Simonits⊥
Studiecentrum voor Kernenergie, SCK‚CEN, Boeretang 200, B-2400 Mol, Belgium, Joint Research Centre, European Commission, Institute for Reference Materials and Measurements, Retieseweg, B-2440 Geel, Belgium, Laboratory of Analytical Chemistry, Institute for Nuclear Sciences, University of Gent, Proeftuinstraat 86, B-9000 Gent, Belgium, Radioisotope Applications & Support, DSM Research B.V., P.O. Box 18, 6160 MD Geleen, The Netherlands, and KFKI-Atomic Energy Research Institute of the Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
A general formula for nuclear activation and decay is presented. Besides decay processes, it deals with burnup and successive activation. The representation is compact and has a structure comparable to the classical NAA formulas, used in absence of burnup and activation of daughter products. A solution is presented to the mathematical singularities appearing when different products have equal time constants. Explicit expressions are given for typical situations encountered in neutron activation analysis. Its incorporation into the k0 standardization is demonstrated for several decay types. The processes occurring in neutron activation analysis (NAA) are governed by sets of linear first-order differential equations. They belong to a class of mathematical problems with important applications in a variety of fields, such as physics, chemistry, biology, and health physics. As a consequence, solutions for these systems have already been developed. In the course of time, refinements and alternative representations have become available. The pioneering work of Bateman1 and Rubinson2 is fully recognized in the field. General equations have been reported also by others.3-9 The general solution was translated into recursive formulas by Hamawi,3 later extended by Scherpelz7 to deal with the problem of identical time constants in a single chain and by Miles8 for use in complex chains with branching. Blaauw9 tackled also the problem of backward branching. The general formula presented here is comparable with the Rubinson solution2 but is more compact and eases the link with †
SCK‚CEN. European Commission. § University of Gent. | DSM Research B.V. ⊥ KFKI. (1) Bateman, H. Proc. Cambridge Philos. Soc. 1910, 15, 423. (2) Rubinson, W. J. Chem. Phys. 1949, 17, 542. (3) Hamawi, J. N. Nucl. Technol. 1971, 11, 84. (4) Clarcke, R. H. Health Phys. 1972, 23, 565. (5) Skrable, K.; French, C.; Chabot, G.; Major, A. Health Phys. 1974, 27, 155. (6) Skrable, K.; French, C.; Chabot, G.; Major, A.; Ward, K. Nucl. Saf. 1975, 16, 337. (7) Scherpelz, R. I.; Desrosiers, A. E. Health Phys. 1981, 40, 905. (8) Miles, R. E. Nucl. Sci. Eng. 1981, 79, 239. (9) Blaauw, M. The Holistic Analysis of Gamma-ray spectra in Instrumental Neutron Activation Analysis. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands. ‡
4326 Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
Figure 1. Sequential activation and decay series.
the more classical expressions used in NAA. Also, alternative expressions are presented for cases in which different products have equal disappearance and/or decay rates. Besides demonstrating the general use of the formula, we also focus on its place in the definitions concerning the k0 standardization in NAA.10-13 The concept of k0 standardization of (n, γ) reactor NAA was launched in 197510 to make better use of the potential power of this analytical tool (see, e.g., ref 13 and references therein). The technique enjoys growing popularity, for it adequately exploits the accuracy, traceability, and multielement capability of NAA.
(1) BASIC ACTIVATION AND DECAY FORMULAS Activation. Aware of the general applicability of the equations to be discussed, we will nevertheless interpret the symbolic information merely in the framework of the nuclear activation and decay processes involved in NAA. First we consider the simple case of a linear series of “compartments”, in which matter flows unidirectionally and sequentially from the “first” to the “last” compartment. Afterward, the solution will be extended to complex networks of directly or indirectly interrelated compartments. The simplified case is applicable in most of the common neutron activation reactions. The NAA technique basically gives information about the abundance of an element (1) through the (10) Simonits, A.; De Corte, F.; Hoste, J. J. Radioanal. Chem. 1975, 24, 31. (11) De Corte, F.; Simonits, A. J. Radioanal. Nucl. Chem., Articles 1989, 133, 43-130. (12) De Corte, F.; Simonits, A.; Bellemans, F.; Freitas, M. C.; Jovanovic, S.; Smodisˇ, B.; Erdtmann, G.; Petri, H.; De Wispelaere, A. J. Radioanal. Nucl. Chem., Articles 1993, 169, 125-158. (13) De Corte, F.; Simonits, A.; Lin Xilei; Freitas, M. In Nuclear Analytical Methods in the Life Sciences 1994; Kucˇera, J., Obrusnı´k, I., Sabbioni, E., Eds.; Humana Press: Totowa, NJ, 1994; pp 19-31. S0003-2700(96)00440-4 CCC: $12.00
© 1996 American Chemical Society
activation of one of its (stable) isotopes and the detection of γ-emission during the disintegration of a product (k) in the decay chain. Figure 1 shows such sequential series of products, each product being a decay or activation product of the previous one in the chain (no feedback or joining branches). The abundances of the relevant decay products during activation is then governed by the following set of linear differential equations:
ak,i )
(∏ ) [∑( ∏ )(∏ k
λ* m
m)i+1
λm - λi
i-j
bi,j )
i-1
r)1 s)i+1-r
λ* s
i-r
Λ*t
λs - λi
t)j
Λt+1 - Λj
-
∏λ t)j
N˙ 1 ) -Λ1N1 N˙ i ) -ΛiNi + Λ* i-1Ni-1
(i g 2)
(1.1)
where Ni is the number of target atoms of nuclide i, Λ ) R + λ is the total disappearance constant of a product in the neutron field, Λ* ) R* + λ* is the specific disappearance constant for the considered branch, R is the reaction rate per nucleus, and λ is the nuclear decay constant. The total reaction rate is the sum of the (n,γ), (n,p), (n,2n), etc., contributions, excluding the (n,n′) reactions (unless these would result in an excited state whose lifetime is not insignificant compared to the lifetime of the ground state nucleus in the neutron field). For the solution, fixed neutron field properties are assumed; in other words, the reaction rates per target atom are constant during the irradiation period (tirr). Considering the effect of one single branch, the (average) abundance of product (k) during the activation process is calculated from (see section 2): k-1
∏Λ* n
k
Nk,irr )
∑ i)2
N1,0 Λ*1
n)2
k
∏(Λ
m
- Λi)
(e-Λ1 tirr - e-Λi tirr) Λi - Λ1
m)2 m*i
(k ) 2, 3, ...) (1.2) Since product 1 is mostly stable (λ(*) 1 ) 0), we can also write (*) Λ(*) ) R . 1 1 Subsequent Decay. After the irradiation period, the evolution of the product abundances is only determined by the radioactive decay processes:
(i g 2)
- λj
t+1
j-1
∏Λ n)2
Λ*n
n
)
- Λj
Sj ) (e-Λ1 tirr - e-Λj tirr)
(extended) saturation factor
Di ) e-λi td
decay or cooling factor
Ci ) (1 - e-λi tm)
counting factor
Fi ) λ* i/λi
branching factor
tirr, td, tm
irradiation, decay (cooling), and measurement times
(In the expressions for ak,i and bi,j, products having an upper counter that is inferior to the lower counter are set equal to 1; e.g., for i ) k, one takes ak,k ) 1 since m ) i + 1 > k.) If the product (k) is created by different branches, their contributions can be summed independently. Even if some branches join again, they can still be treated as separate chains. Beyond the branching point, a factor Fi is associated with each path, corresponding to its fractional weight and hence also determining its importance to the production of a common member. Finally, the solution can be applied to complex networks of products, communicating through nuclear reactions and/or decay. Low Reaction Rate, No Burnup. When moderate neutron field intensities and irradiation times are used, in most cases no burnup effects (R1tirr , 1) or higher-order interference reactions (Ri , λi, i > 1) have to be considered. Then formula 1.4 converges to the more classical expression k
(
k
∑ ∏λ
∆Nk ) N1,0 R*1
i)2 n)2 n n*i
N˙ 1 ) -λ1N1 N˙ i ) -λiNi + λ* i-1Ni-1
)](
λ*t
i-r
λ*n
)
- λi
Fi
Si λi
DiCi
(1.5)
(1.3) with
Making use of the characteristic γ-emission, the analyst will monitor the decay of a number ∆Nk of product k nuclei during a time interval [td, td + tm] after irradiation. As will be shown in section 2, the general formula for the number of monitored k nuclei, after production in a constant neutron field through the activation of a stable product 1 and following one specific branch in the decay processes, can be written as (see also ref 14) k
∑
∆Nk ) N1,0 R*1
i)2
with
i
(
∑b
ak,i Fi Di Ci
j)2
Sj i,j
)
Λj - R1
(1.4)
Si ) (1 - e-λi tirr)
In the absence of burnup and higher-order reactions, the contributions of consecutive irradiations are basically independent and can be summed directly. In Situ Measurements. The activity of the decay products can in principle also be monitored during the activation process; as is, for example, the case with “in-beam” experiments, prompt γ-neutron activation analysis, etc. With t0 ) 0 corresponding to the beginning of the activation, a measurement during a time interval [t1, t2] yields the following integrated activity: Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
4327
This expression can be rearranged to get the typical saturation factors (1 - e-Λi tirr). Indeed, assuming that Nk,0 ) 0 (k > 1), it follows directly from the formulas 2.1 and 2.3 that
k-1
∏
k
∆Nk )
∑ i)2
Λ*n
{(
N1,0 Λ*1
n)2
k
∏(Λ
- Λi)
m
Λi - Λ1
)
e-Λ1 t1 - e-Λ1 t2
λ*k
Λ1
k
m)2 m*i
(
-Λi t1
e
)}
Nk,irr ) Nk,irr - Nk,0 )
Λi
∑A
i)1
-Λi t2
-e
∑
k
Ak,ie-Λi tirr -
(1.6)
k,i
i)1
k
)
∑-A
k,i(1
- e-Λi tirr)
(2.6)
i)1
Again, starting from a stable product, one can replace Λ(*) 1 with R(*) . In the limit of negligible burnup one gets 1 k
(
k
∑ ∏λ
∆Nk ) N1,0 R*1
){
λ*n
- λi
i)2 n)2 n n*i
Evidently, the saturation factor has no relevant meaning for the starting product 1, which does not build up but on the contrary burns up in the neutron field. A more appealing description of the activation is then
Fi (t2 - t1) -
(
)}
e-λi t1 - e-λi t2 λi
(1.7)
(2) DERIVATION OF THE GENERAL EXPRESSIONS Activation. When solving the set of eqs 1.1 we assume constant reaction rates and as initial conditions (tirr ) 0) that N2,0 ) ... ) Nk,0 ) 0 and N1,0 ) N1(tirr ) 0) * 0. Following Rubinson,2 the abundances can be written in following shape: k
∑A
Nk,irr ) Nk(tirr) )
k,ie
-Λi tirr
(2.1)
i)1
Indeed, the validity is readily checked for k ) 1 and k ) 2. For the general case (k) one finds
Nk,irr ) e-Λk t[C +
∫e
Λk t
(Λ*k-1Nk-1) dt] )
k-1
(Nk,0 -
∑A
k-1
k,i)e
-Λk tirr
+
i)1
(
∑A i)1
Λ*k-1 k-1,i
)
Λk - Λi
k
Nk,irr )
∑
k
∑A
- Ak,i(e-Λ1 tirr - e-Λi tirr) + [
i)2
k,i]e
-Λ1 tirr
i)1
(k > 1) (2.7)
in which the last term is zero (see formula 2.3) and the extended saturation factors Si ) (e-Λ1 tirr - e-Λi tirr) converge to the classical ones with decreasing burnup. (Negative values are avoided, since each Si is accompanied by a (Λi - Λ1)-1 factor.) Formula 2.7 is in fact the general activation formula previously presented in formula 1.2. Subsequent Decay. After irradiation, nuclear decay proceeds following the eqs 1.3. At any time t after irradiation, the abundances Nk(t) are described by an expression of the type
e-Λi tirr (2.2)
k
Nk(t) )
∑B
k,ie
-λi t
(2.8)
i)1
This provides a recursive definition of the factors Ak,i:
(
Ak,i )
)
Λ*r-1
k
∏Λ
r)i+1
r - Λi
k-1
Ak,k ) Nk,0 -
∑A
The factors Bk,j are obtained in an analogous way as Ak,i (see the formulas 2.2 and 2.3):
Ai,i (2.3)
k,i
i)1
Bk,i )
One can verify that the diagonal elements Aii (i ) 1, 2, ..., k) can be written as
(
Ai,i ) N1,0
Λ*n
i-1
∏Λ n)1
n
)
- λi
Bk,k ) Nk,irr -
∑B
i)1
e
k
∏(Λ
m
-Λi tirr
(2.9)
k,i
The initial conditions (t ) 0) correspond with the situation at the end of the irradiation. This gives the relation between the Bk,i factors and the Ak,i factors describing the activation:
- Λi)
Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
∑ i)1
(2.5)
m)1 m*i
4328
Bi,i
i)1
Nk,irr )
n
∑
k-1
k
∏Λ* n)1
)
λ*n-1
n)i+1 n
k-1
Nk,irr ) N1,0
∏λ
(2.4)
- Λi
This finally leads to the general activation formula as also presented by Rubinson:2
k
(
k
and consequently
k
Bk,i )
∑A i)1
k,ie
-Λi tirr
(2.10)
[( ) ) ] ∑ [∑ ( ∏
∑ ∏Λ
(
j)1
i-1
∏λ r)j
λ*r-1
r)j
Aj,je-Λj tirr -
- Λj
r
i
λs - λi
j)1 r)1 s)i+1-r
i-r
Λ*t
t)j
Λt+1 - Λj
)]
i-r
λ* t
t)j
t+1 - λj
∏λ
-
∆Nk )
)
λ*s
i-1
i-j
Bj,j )
- λj
(∏ r
Λ* r-1
i-1
i-1
Bi,i ) Ai,ie-Λi tirr +
∫
td+tm
td
λ* k Bk,i e-λi td(1 - e-λi tm) λi i)1 (2.16) k
λ* kNk(t) dt )
∑
× In the same notation as used above, this is (k ) 1, 2, ...):
Aj,je-Λj tirr (2.11)
{
∆Nk ) N1,0 ak,1e-Λ1 tirrF1D1C1 + k
∑
Λ* 1
The postirradiation abundances can then be represented by k
Nk(td) ) N1,0
(∏
)
λ* m-1
i-r
Λ* t -
Λt+1 - Λj
t)j
λ* s
)](∏ ) j-1
- λj
t+1
i-1
i-j
∑∑ ∏
j)1 r)1 s)i+1-r
λ* t
∏λ
[(
i
e-λi td
- λi
i)1 m)i+1 m
i-r
t)j
(
k
∑ ∏λ
Λ*n
n)1 Λn
λs - λi
e-Λj tir
- Λj
)
×
(2.12)
Again, to achieve the introduction of the extended saturation factors, one can deduce from the previous formulas 2.8 and 2.9 and the general appearance of the Bi,i factors in formula 2.11 that i
Bk,i )
∑X
i)2
)
j)1
∑-X
-Λ1 tirr i,j[e
j)2
N˙ 1 ≈ -R1N1,0 N˙ 2 ≈ -λ2N2 + R* 1N1,0 N˙ i ≈ -λiNi + λ* i-1Ni-1
- e-Λj tirr] +
i
∑X
i,j]e
The solution is of the type
-Λ1 tirr
(
(i > 1) (2.13) k
j)1
∑
Nk,irr ) in which the last term is zero, because it contains following general structure: i
∑X
i
i,j
with
n
∑(Y )( ∑ A n
)
j)1
n)2
n,m)
)0
(2.14)
m)1
Ck,i )
(
k
λ* n-1
∏ λ -λ
By combination of the expressions for Bk,i, Bi,i, and Aj,j the alternative expression for the post-irradiation abundances is reached:
Nk(td) ) N1,0
(
{(
k
∏λ
m)2 m
(
k
∑ ∏λ
t)j
i-r
- Λj
)
- λi
Λ* t t+1
- λ1
λ* m-1
i)2 m)i+1 m
∏Λ
λ*m-1
i
e-λi td
) [( )](∏ )( )} i-j
λ*t
i-1
∑∑ ∏
t+1
- λj
j-1
Λ*n
n)2 Λn
)
Ci,i
i k-1
∏λ
λ* n
n)2 n
λ* s
λs - λi
×
Sj
(2.17)
(2.18)
)
(k g 2)
- λk
(2.19)
(2.20)
For the subsequent decay, expression 2.8 is still valid; since Nk(0) ) Nk,irr for k ) 1, 2, ..., one finds immediately that
(
)
1 - e-λi tirr λi
(2.21)
and automatically the expression for the postirradiation abundances is found: k-1
∏λ* n
k
∑ ∏(λ
Nk(td) ) N1,0R*1
i)2
In the typical case that the starting product 1 is stable, the (*) first term disappears (Bk,1 ∼ λ* 1 ) 0) and Λ1 can be replaced by (*) R1 . The number of disintegrations during a measurement period follows from an integration of the activity:
(k g 2)
(k g 3, i ) 2, ..., k - 1)
- Λj Λj - Λ1
(2.15)
)
λi
Bk,i ) Ck,i
j)2 r)1 s)i+1-r
∏λ t)j
)
e-Λ1 tirre-λ1 td +
(
n)i+1 n
Ck,k ) N1,0R*1
i-r
)}
Λj - Λ1
(i g 3)
1 - e-λi tirr
Ck,i
i)2
k
(
Sj
j)2
[
Λ* 1
∑
bi,j
which, in the common case of a stable starting product 1, reduces to formula 1.4. Low Reaction Rate, No Burnup. In the case of rather low or moderate neutron fluxes, the burnup of the (stable) analyte or its decay products is negligible (R1tirr , 1 and Ri , λi (i > 1)). Often in these situations higher-order activation can be neglected. Then the specific disappearance constants can be replaced by the specific decay constants
i
-Λj tirr i,je
i
ak,i Fi Di Ci
n)2
k
n
- λi)
Si λi
e-λi td
(2.22)
n)2 n*i
The solution presented in formula 1.5 is obtained from an integration of the activity (formula 2.16). Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
4329
Table 1. Explicit Shape of Expressions 1.4 (Left) and 1.5 (Right) for k ) 2, 3, and 4 (See Text) k ) 2:
∆N2 ) [N∞2 S2]F2D2C2
k ) 3:
∆N3 ) [N∞2 S2]
(
)
∆N2 ) [N∞2 S2]F2D2C2
λ* 3 FDC + λ3 - λ2 2 2 2
( [(
∆N3 ) [N∞2 S2]
) ) (
Λ*2 FDC + Λ2 - Λ 3 3 3 3 Λ*2 λ*2 [N∞2 S2] F3D3C3 Λ3 - Λ2 λ3 - λ2
[N∞3 S3]
k ) 4:
∆N4 ) [N∞2 S2]
(
[N∞3 S3] [N∞4 S4] [N∞2 S2] [N∞3 S3]
)]
)
)( )( ) ( ) ( )( ( ) [(
)
Λ*2 λ*4 FDC + Λ 2 - Λ 3 λ4 - λ 3 3 3 3 Λ*2 Λ*3 FDC + Λ2 - Λ4 Λ3 - Λ4 4 4 4 Λ*2 λ*2 λ*4 FDC + Λ3 - Λ2 λ3 - λ2 λ4 - λ 3 3 3 3 Λ*3 λ*3 Λ* 2 FDC + Λ4 - Λ3 λ4 - λ3 Λ2 - Λ3 4 4 4 Λ*2 Λ* λ* λ*3 3 2 + Λ3 - Λ2 Λ4 - Λ2 λ3 - λ2 λ4 - λ2 λ* Λ*2 λ* 3 2 F4D4C4 λ3 - λ 4 Λ3 - Λ2 λ3 - λ2
)
)] ( ) )]( ) ) ( )( ) ) ( )]]
In Situ Measurements. The average number of disintegrations during the activation process, as given by formulas 1.6 and 1.7, is readily obtained from a simple integration of the activity, with Nk(t) represented by expressions 1.2 and 2.22, respectively. (3) EXPLICIT EXPRESSIONS To get a better idea of the composition of the general expressions, the explicit versions of the formulas 1.4 and 1.5 for k ) 2, 3, and 4 are presented in Table 1. It is easily verified that formula 1.4 leads directly to 1.5 in the limit of no burnup and no successive neutron activation reactions, considering
f λ(*) (i > 1, Ri , λi) i -e ) f (1 - e-λi tirr) (R1 , 1/tirr) Si ) (e N N Λ* R* 1,0 1 1,0 1 N∞i ) f Λi - Λ1 λi -Λ1 tirr
Λ(*) i -Λi tirr
(3.1) One can also verify the compactness of expression 1.4 by comparing with the corresponding Rubinson equation.2 (The latter contains 4, 10, and 20 terms for k ) 2, 3, and 4, respectively.) For the most simple case of activation and decay (k ) 2), the interpretation of the applied symbols is straightforward. The achievable abundance of product 2 nuclei through neutron activation is limited by the competition of the natural decay with the (n,γ) reaction, expressed by the ratio N∞2 ) N1,0R* 1/λ2. A high number of nuclei can be built up in the case of a low natural decay probability of the unstable product and a high neutron flux and (n,γ) cross section. The saturation factor S2 demonstrates that the maximum abundance of product 2 is nearly reached after an irradiation time of a few times its typical half-life T1/2. The 4330
∆N4 ) [N∞2 S2]
Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
)
λ*3 FDC + λ3 - λ2 2 2 2
[N∞3 S3]
λ* λ* 3 4 FDC + λ3 - λ2 λ4 - λ2 2 2 2
( ( [( [( [(
[N∞2 S2]
)(
(
(
)
λ*2 FDC λ2 - λ3 3 3 3
(
)(
)
λ*3 λ*4 FDC + λ3 - λ2 λ4 - λ2 2 2 2
( (
)( )(
) )
λ*2 λ* 4 FDC + λ2 - λ3 λ4 - λ3 3 3 3 λ*2 λ* 3 [N∞4 S4] FDC λ2 - λ4 λ3 - λ4 4 4 4
[N∞3 S3]
extended saturation factor also includes burnup, which is a second limiting factor. The decay factor D2 describes the typical exponential decay law. The counting factor C2 is the fraction of the remaining nuclei that will be monitored during the measurement. The branching factor F2 is usually omitted, since the final branching is normally incorporated in the γ-intensity () number of specific γ-emissions per disintegration). (4) AVOIDING SINGULARITIES The activation and decay formulas have to be adapted in cases where some products have equal disappearance and/or decay rates, since the computation leads to singularities. Suppose we have a cluster of (n + 1) products with equal disappearance rates (Λp ≈ Λq ≈ ... ≈ Λz), and m ) k - (n + 1) products with different Λ’s. The activation formula 2.5 is now calculated by replacing ∑j)p,q,...,z Ak,j e-Λj tirr by
[∑ ] n
i tirr
e-Λp tirr i! i)0 n,n-1 (with Hn,n ) -∑(Λj - Λp)-1 k,p ) 1; Hk,p m ) 1: Hk-2,i ) (Λj - Λp)-(k-2-i)(-1)k-2-i k,p m ) 0: Hk-1,k-1 ) 1; k,p k-1,0 k-1,k-2 ) 0) Hk,p ) ... ) Hk,p A′k,p
Hn,i k,p
(4.1)
k-1
∏Λ* i
A′k,p ) N1,0
i)1
∏
i)1 i*p,q,...,z
(Λi - Λp)
(4.2)
Hn,i k,p )
[u]-f
(
n-i n-i-n1
∑ ∑
n-i
(-1)
n-i-n1-...-nm-2
∑
...
n1)0 n2)0
nm-1)0
k
∏
(Λj - Λp)
)
[u],f hk,i,z )1+
j)1 j*p,q,...,z
(
t-n1-...-n(k-i-[u])-2
t-n1
∑ ∑∑ t)1
-n[j]
t
∑
...
n1)0 n2)0
n(k-i-[u])-1)0
nm ) n - i - (
∑n )
)1+
s)1
[u]-f
k
λz - λs
( ) [( ) ( )] t+1
t)1
r)i r∉[a,...,z]
λz - λr
∑ ∑
k
∏
λz
λz
-
λz - λr
s)i s∉[a,...,z],s*r
For n[j], the index [j] corresponds with the position of the product j in the list of m products for which Λj * Λp. An alternative for expression 4.3 is
( )) n[s]
λz
(4.8)
[u],f hk,i,z
s
∏
s)i s∉[a,...,z]
λz
(4.3) m-1
k
λz - λs
(4.9)
(k-i-[u])-1
nk-i-[u] ) t -
∑
ng
g)1
∑
n-i
) (-1)
[s] ) the position of product s in the series
(Λj-Λp)-(n-i+m-1)
k
Hn,i k,p
k
∏
j)1 j*p,q,...,z
[(Λj - Λp)-1 - (Λl - Λp)-1]
l)1 l*j;p,q,...,z
(4.4)
n,i is quite compact for high values The latter expression for Hk,p of (n - i) but needs further reduction in the case that additional clusters appear among the m products, in order to avoid singularities. Expression 4.3 can get rather lengthy but always remains n,i directly valid for the computation of Hk,p . Introducing again the extended saturation factors, one finally gets the following activation formula for an arbitrary amount of clusters:
k
∑
Nk,irr )
- Ak,iSi +
i)2 i*clusters
[
∑
n
-Hn,0 k,p Sp
+
clusters
∑
Hn,i k,p
i)1
i tirr
i!
[u] ) 0 for i ∈ [z,k] [u] ) 1 for i ∈ ]y,z], ... [u] ) u for i ∈ [2,a] [u],f n,i The structure of hk,i,a is comparable with that of Hk,p (see expressions 4.3 and 4.4). Ni,irr is calculated from expression 4.5.
(5) GENERAL APPLICATION Branching. Starting from one stable element, the activation can follow different branches, which can join again in a later stage of the decay. We consider a typical situation in which an isomer Bm is populated, which decays back to the monitoring product B. This example illustrates the role played by branching factors Fi. (Burnup effects are omitted here.) The contribution of both branches are summed.
]
e-Λp tirr A′k,p (4.5)
In the subsequent decay, clusters of decay rates are also theoretically possible. Suppose there are clusters of u + 1 (nearly) identical decay rates (λa ≈ ... ≈ λy ≈ λz). The following expression, which is an alternative for formula 1.4, takes care of all clusters in activation and/or decay: k
∆Nk )
∑ i)2
[
[u]
∑∑
clusters f)0
k
∑
Ni,irr
[u],f Mk,i,z
∆NB ) ∆N2(A,B) + ∆N3(A,Bm,B)
0,0 Mk,i,j DjCj +
j)i j*clusters
(
)]
(λz(td + tm))f (λztd)f -λz td e e-λz(td+tm) f! f!
(4.6)
with
( ) ( ∏ )( ∏ ) λ*s
k
0,0 Mk,i,j )
∏λ s)i s*j
s
- λj
k
[u],f Mk,i,z )
s)i λs s∉[a,...,z]
- λz
)
(
)
]
F2λBm SB D C (5.1) λBm - λB λB B B
In case the isomer has a short half-life (λBm . λB), this formula soon (DBm ≈ 0) reduces to
SB D C λB B B
Fj
λ*s
[(
SB λB SBm DBCB + [NARABm] F2 DBmCBm + λB λB - λBm λBm
)[NARAB]
∆NB ≈ NA(RAB + F2RABm)
(5.2)
k
[u],f Fs hk,i,z
s)i s∈[a,...,z]
(4.7)
Backward Branching or Looping. A special case of branching has been left out of consideration up to now, the case in which Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
4331
the abundance of the initial product is fed by a reaction on one of its daughter products. So, as matter “flows” from one “compartment” to another, there is also a feedback mechanism populating the first. This backward branching can in principle still be treated in forward direction, by infinite looping. In practice, summations will be limited according to the requested mathematical precision.
a double activation. The possible natural decay of the intermediate product (λB > 0) is also taken into account (“decay-away”).
∆NC ) ∆N3(A,B,C) ) NA,0RARB ×
[
SB
(RB + λB - RA)(λC - RB - λB) SC
∆Nk ) ∆Nk(A,...,k) + ∆Nk+i(A,...,B,...,A,...,k) +
+
]
(λC - RA)(RB + λB - λC)
DCCC (5.5)
∆Nk+2i(A,...,B,...,A,...,B,...,A,...,k) + ... ) ∞
∑∆
k+ni(A,[...,B,...,A]i,...,k)
(5.3)
When saturation is reached (no burnup of A), one gets
i)0
∆NC ) Whereas such situations are commonly found in the description of biological or chemical systems, they rarely turn up in some chain of nuclear activation reactions. An example can be found in the activation of gold:
Burnup of Daughter Product. The impact by burnup of any of the products on the abundance of the monitoring product follows directly from the general formula 1.4. As an example, we consider the burnup of the daughter products in a motherdaughter decay. The burnup and transformation of other products is considered as being negligible.
NA,0RARB
(5.6)
DCCC
(RB + λB - RA)(λC - RA)
Since the abundances are proportional to RARB, or Φ2, the interference of these second-order reactions to a competing firstorder activation will increase proportionally with the neutron flux Φ. Generally, an nth-order activation reaction is proportional to Φn. Evidently, such transformations by successive neutron capture are of relevance to the theories regarding stellar nucleosynthesis. Consecutive Irradiations. We consider the case of two consecutive irradiations of the same sample. In the absence of burnup, both contributions are independent and can be summed directly. In case the burnup of product 1 has to be considered, the initial abundance N1,0 of the stable isotope is reduced at the start of the second irradiation cycle, with a factor exp( - R1t(1) irr ). Summing (in absence of the burnup of daughter nuclei) is then performed like this
R*1(1) (1) (1) (1) (1) ∆N2 ) N(1) [e-R1 tirr - e-λ2tirr ]e-λ2td C2 + 1,0 λ2 - R(1) 1
[(
)]
(1)t
-R1 N(1) 1,0 e
SB λC DBCB + ∆NC ) NA,0RA λB λC - λB NA,0RA
[
SBΛC - SCλB ΛC(ΛC - λB)
-
]
SB D C (5.4) λC - λB C C
Successive Neutron Capture. Besides the stable elements, the activation products can also undergo nuclear reactions. Sometimes two consecutive neutron captures are the easiest way to produce some product C by neutron activation. We consider 4332
Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
(1)
irr
R*1(2)
λ2 -
(2)t
[e-R1
R(2) 1
(2)
irr
(2)
(2)
- e-λ2tirr ]e-λ2td C2 (5.7)
In the common case of equal neutron flux in both irradiation (2) periods, the reaction rates will also be equal, R(1) 1 ) R1 ) R1. The situation becomes more complicated when higher-order reactions and burnup of daughter products have to be accounted for. Indeed, the contribution from a particular irradiation is no longer unaffected by the next irradiation(s). One could use following formalism to ensure a principally correct evaluation after
Table 2. Activation-Decay Types and Relevant Expressions for the Parameters Involved in the k0 Method type
ka
Q0
∆k
I II/a-c II/d III/a III/b III/c IV/a
NAθσ0γ2/M NAθσ0F2γ3/M NAθσ0F2γ3/M NAθσ0F2F3γ4/M NAθσ0F2F3γ4/M NAθσ0γ4/M NAθσg0γ3/M
I0/σ0 I0/σ0 I0/σ0 I0/σ0 I0/σ0 I0/σ0 Ig0/σg0
∆2(1,2) (1/F2)∆3(1,2,3) (1/F2)[∆3(1,2,3) + (γ2/γ3)∆2(1,2)] (1/F2F3)[∆3(1,2,4) + ∆4(1,2,3,4)] (1/F2F3)∆4(1,2,3,4) ∆3(1,2,4) + ∆4(1,2,3,4) g ∆2(1,3) + (Rm 1 /R1)∆3(1,2,3)
IV/b
g [NAθ(F2σm 0 + σ0)γ3]/M
g F2Im 0 + I0
Rg1∆2(1,3) + Rm 1 ∆3(1,2,3)
g F2σm 0 + σ0
Rg1 + F2Rm 1
IV/c
NAθσm 0 F2γ3/M
m Im 0 /σ0
IV/d
NAθσg0γ3/M
Ig0/σg0
V/a
V/b,c
NAθσg0F3γ4/M
g NAθ(F2σm 0 + σ0)F3γ3/M
VII/a
VII/b
VIII
m2 1 NAθ[F3(σm 0 + F2σ0 ) + g σ0]γ4/M
NAθσg0F3γ4/M
NAθσg0F3γ4/M
NAθF3F4σg0γ5/M
∆2(1,3) +
[
Ig0/σg0
[
Rm 1 Rg1
]
∆3(1,2,3) +
γ2 ∆ (1,2) γ3 2
m
R1 1 ∆ (1,3,4) + g ∆4(1,2,3,4) F3 3 R
g F2Im 0 + I0
F2σm 0 VI
[
g I R1 ∆ (1,3) + ∆3(1,2,3) m F2 R 2 1
+
1
]
]
Rg1∆3(1,3,4) + Rm 1 ∆4(1,2,3,4)
σg0
F3(Rg1 + F2Rm 1)
m2 g 1 F3(Im 0 + F2I0 ) + I0
m2 1 Rg1∆2(1,4) + Rm 1 ∆3(1,3,4) + R1 ∆4(1,2,3,4)
m2 g 1 F3(σm 0 + F2σ0 ) + σ0
m2 1 Rg1 + F3(Rm 1 + F2R1 )
Ig0/σg0
Ig0/σg0
Ig0/σg0
[ [
m
R1 1 ∆ (1,3,4) + g [∆3(1,2,4) + ∆4(1,2,3,4)] F3 3 R 1
m
R1 1 ∆ (1,3,4) + g ∆3(1,2,4) F3 3 R 1
]
] m
1 1 R1 [∆ (1,3,5) + ∆4(1,3,4,5)] + [∆ (1,2,5) + ∆4(1,2,4,5)] F3F4 3 F3F4 Rg 3 1
a
The type definitions are taken from ref 11.
r irradiation periods:
assuming pulse loss correction, can be calculated from
Np ) N1,0 R*1 ∆k γp
(r-1) ∆N(r) ∆′k(1,...,k) + N(r-1) ∆′k-1(2,...,k) + ... + k ) N1 2
(6.1)
∆′1(k) (5.8) N(r-1) k where is the abundance of product i at the beginning of where N(r-1) i irradiation (r); this is after (r - 1) irradiations and possibly as much cooling periods (see formula 2.12 or 2.15). ∆′i is the general activation and decay formula of order iswithout the condition of stability of the initial nucleisexcluding the factor N1,0 (i.e., the part within braces of formula 2.17). (6) SPECIFIC EXPRESSIONS FOR THE K0 METHOD IN NAA The concentration F of an analyte (1) in the matrix can be derived from γ-spectrometry, by determining the area of the fullenergy peak Np corresponding to a characteristic γ from a decay product (k). The average number of counts in the photopeak,
N1,0 )
NAWFθ M
(6.2)
is the initial number of nuclei (from one isotope of the analytical element of interest, leading to the observed activation reaction), with NA being the Avogadro number, W the sample mass, F the element concentration (F ) wa/W), θ the isotopic fraction, and M the molar mass of the analyte,
R* 1 ) Φeσ0[Gth f + GeQ0(R)]
(6.3)
is the reaction rate per nucleus (in the Høgdahl convention), Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
4333
where Φe is the epithermal neutron flux, σ0 is the (n,γ) cross section at v0 ) 2200 m/s, Gth and Ge are the thermal and epithermal neutron self-shielding factors, respectively, f ) Φs/Φe is the “thermal” (subcadmium) to epithermal flux ratio, Q0(R) ) I0(R)/σ0 is the resonance integral to thermal cross section, and R is the shape parameter of the 1/E1+R epithermal neutron spectrum.
∆k )
∆Nk N1,0R*1
(6.4)
is the “probability” per product (1) nucleus and at unity reaction rate for the decay of a product (k) nucleus during a measurement period tm, started at a time td after an irradiation at constant neutron flux during tirr (see also the formula 1.4). γ is the absolute gamma intensity, and p is the full-energy peak detection efficiency for the sample-detector geometry (including attenuation and true coincidence effects). The concentration (g/g) of the analyte in the sample is then
[
F ) Np ∆k
NAWθγ pΦeσ0(Gth f + GeQ0(R)) M
]
-1
removed from the decay formulas and included into the k0 factor. This makes the definition of the k0 factor, and consequently also of ∆k, less transparent. Reference has to be made to a table of explicit definitions for different types of activation and decay. In the existing version,11 burnup effects are not specifically included in the formulas. As a final application of the general formulas presented in this work, the relevant definitions are upgraded in Table 2 (see also ref 14). (Note that, in comparison with ref 11, the factor (λtm)-1 has been removed from the definition of the counting factor.) In low-flux situations often simplified decay formulas are used when, for example, one of the radionuclides involved has a relatively short half-life. These approximations may become invalid in high-flux regimes, since some extra conditions for their use can come into play. Therefore, a systematic use of the general expressions is recommended. In some situations, activation takes place through two or three isomers. Unavoidably, the ratio of their respective reaction rates ends up in the decay formula (see types IV-VIII). It is calculated from (Høgdahl convention)
(6.5)
Rm 1 Rg1
The concentration formula contains a compound nuclear constant, which is specific to one γ transition and its corresponding analyte:
ka )
NAθσ0γ M
(6.6)
It was recognized that assembling this constant with nuclear data from literature causes unnecessary buildup of uncertainties. In the k0 standardization it is considered as a whole and obtained relatively from coirradiation of a standard with a comparator, leading to the well-known k0 factor:10
k0,c(a) )
Mc θ γ σ0 ka ) M θc γc σ0,c kc
(6.7)
with the 411.8 keV γ line in the decay of 198Au, from the activation reaction 197Au(n,γ)198Au, as the ultimate comparator (kAu ) 0.2883). A nuclear data library has been published for the most commonly used reactions.11,12 In more complex decay schemes, the explicit expression for the k0 factor changes slightly, however not fundamentally.11 The thermal activation cross section is in some cases replaced by the sum of two or three contributing cross sections. For accuracy reasons, (some of) the branching factors have been (partly) (14) Pomme´, S.; Hardeman, F.; Robouch, P.; Etxebarria, N. Neutron Activation Analysis with k0-standardisation: General Formalism and Procedure. Internal report SCK‚CEN, BLG 700, unpublished.
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Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
m σm 0 Gth f + GeQ0 (R)
)
σg0 Gth f + GeQg0(R)
m km 0 Gth f + GeQ0 (R)
)
kg0 Gth f + GeQg0(R)
(6.8)
involving thermal cross-section data or, alternatively, k0 factors (see ref 11). (7) CONCLUSIONS A general solution is presented for a set of linear first-order differential equations describing the different activation and decay processes involved in neutron activation analysis. It is also applicable in other areas of research where comparable problems arise (e.g., biology, chemistry, health physics, and astrophysics). The use of the formulas has been demonstrated with basic examples. As a major application, the activation and decay formulas applied in k0 NAA have been upgraded to include burnup effects. An effort has been made to preserve the structure of the classical formulas and to reduce the number of terms to a minimum. Also a solution is provided to the occurrence of singularities due to equal time constants. ACKNOWLEDGMENT Grateful acknowledgement is made to the Belgian National Fund for Scientific Research (FDC, ADW) and to the Hungarian National Foundation of Scientific Research, Contract OTKA-1838 (AS), for financial support. Received for review May 3, 1996. Accepted August 30, 1996.X AC9604402 X
Abstract published in Advance ACS Abstracts, November 1, 1996.
