The accuracy and precision of the experimental α-determination in the 1/E 1+α epithermal...

Journal o f Radioanalytical Chemistry, VoL 62, No. 1 -2 (1981) 209-255

THE ACCURACY AND PRECISION OF THE EXPERIMENTAL oL-DETERMINATION

IN THE 1 / E 1 + a EPITHERMAL REACTOR-NEUTRON SPECTRUM

F. DE CORTE, *+ K. SORDO-EL HAMMAMI, *++ L. MOENS,* A. SIMONITS,** A. DE WISPELAERE,* J. HOSTE*

*Institute for Nuclear Sciences, Rijksuniversiteit Gent, Proeftuinstraat 86, B-9000 Gent (Belgium)

**Central Research Institute for Physics, H-1525 Budapest 114, P.O. Box 49 (Hungary)

(Received October 20, 1980)

Some methods for the experimental a-determination in the 1rE l+c~ epithermal reactor- neutron spetrum are critically compared with respect to their accuracy and precision. The analysis is based on the error propagation theory. Besides the general formulae numerical examples are elaborated for specific conditions in the Thetis reactor (Gent) and the WWR-M reactor (Budapest).

Introduction

It has been outlined in earlier publications 1-3 that the knowledge of the factor

a is essential for the correction of resonance integrals, i.e. for the conversion of

Io-values, only valid in an ideal 1/E-epithermal reactor neutron spectrum, to Io(u)-

values, for use in actual 1/El+a-spectra, and vice versa.

The fundamentals can be summarized briefly as follows.

In an ideal epithermal neutron spectrum, where ~ e ( E ) ~ 1/E [~e(E) = epither-

mal flux per unit of neutron energy interval], the resonance integrals can be defined as:

Io T o(E) . . . . dE (1) ECd E

with ECd -- effective Cd cut-off energy;

- 0.55 eV for a detector, having a o ( v ) ~ 1/v activation cross-section

for (n, 7) reaction up to 1 - 2 eV, irradiated in an isotropic neutron

flux as a small sample in a cylindrical Cd-box (height/dia. = 2) with

1 mm wall thickness. 4

+Research Associate of the "Nationaal Fonds voor Wetenschappelijk Onderzoek". + +Present address: University of Rabat, Faculty of Sciences, Maroc.

J. Radioanal. Chem. 62 (1981) 209 14

F. DE CORTE et al.: THE ACCURACY AND PRECISION

In actual, non-ideal epithermal neutron spectra, which in most cases can be approximated by D e ( E ) ~ l /E Z +a (a ~ 0), Eq. (1) should be replaced by:

Io(6) (1 eV) a T o(E) = ~ dE ECd El+a

(2}

For simplicity, the term (1 eV) a = 1 will be omitted in all following expressions.

It can be shown that the relationship between Io and Io(6) is given by:

where

Io(6) = (Io - 0.429) ( E r ) -a -4- ooC a (3)

0.429 C a = - (4)

(26 + 1) (0.55) a

and Oo - 2200 m - s -t (n,~,) cross-section;

Er - effective resonance energy (eV), as defined and tabulated for 96 iso-

topes in Ref. ~

In Eq. (3), the term (Io - 0 . 4 2 9 Oo) represents the reduced resonance integral, i.e. with the 1/v-tail subtracted.

Analogously, the conversion formula for the resonance integral to thermal cross- section ratio (denoting Qo = Io/oo and Q 0 ( 6 ) = Io(6)/Oo) can be written as:

Qo(6) = (Qo - 0.429) (F,r) -a + C a ( 5 )

When putting

with

% ( 6 ) = qo(~r) "~ (6)

qo = Qo - 0.429 (7)

(corresponding to the reduced resonance integral), Eq. (5) becomes:

Qo(a) = % ( 6 ) + C a (8)

210 d. Radioanal. Chem. 62 (1981)


Note that the above conversion formulae are only valid for ECd = 0.55 eV. Indeed, in its most general fbrm Eq. (4) has to be written as:

2(Eo) 1/2 C a = (9)

(2c~ + 1) (Ecd) a+l/2

with Eo = 0.0253 eV. It is important to remark that the factors (Io - 0.429 Oo) - [Eq. (3)] - and

(Qo - 0.429) - [Eqs (5) and (7)] - will not change when ECd :/: 0.55 eV, at least when the isotopes have a a ( v ) ~ l/v dependence up to 1-2 eV. This is due to the fact that these expressions refer to the pure resonance part (1/v-tail subtracted), and thus are insensitive to the Cd cut-off of the Cd-boxes used. Evidently, this will hold only when Ecd is not unreasonably high.

It should be mentioned that, when Io >> oo, i.e. Qo high, Eqs (3) and (5) can be approximated by, respectively:

Io(o0 - Io ( s ~ (10)

and (Io >> Oo)

Qo(ot) ~ Qo(Er) ~ = qo(o 0 (11)

On the other hand, for pure 1/v-detectors, Qo = 0.429 [and qo(a) = qo = 0], so that Eqs (3) and (5) reduce to, respectively:

and

Io(ot)[l/v ! = ooCa (12)

Qo(tX)D/vl = C a (13)

Previously 1'2 various methods have been described for the experimental determination of a, thus enabling the conversion of Io- and Qo'values according to Eqs (3) and (5), respectively. However, when applying these methods in the Thetis reactor (Gent) and WWR-M reactor (Budapest), it was experienced that rather large, and sometimes intolerable uncertainties on a were obtained, the choice of c~-monitor isotopes appearing to be very critical.

The present paper deals with a comparative study of the accuracy and the precision which can be expected with some t~-determination techniques. The ma-

t!:ematical treatment is based on the error propagation theory, with an attempt

J. RadioanaL Chem. 62 (1981) 211 1 4 "


to group the uncertainties into systematic (accuracy) and random ones (precision).

The final goal of this study is to be helpful in a judicious choice of methods

and of activation detectors (denoted hereafter as a-monitors), so as to perform the most accurate and precise a-determination in the given circumstances.

Overall uncertainty on a; calculation of error propagation-factors

In all the following methods for a-determination, the factor a has to be cal-

culated usuaUy from an implicit function of the form:

F ( a , x l , x2, x3 . . . ) = 0 (14)

where x l , x2, Xa . . . are either statistical variables (e.g. measured count rates) or fLxed parameters with an associated uncertainty (e.g. nuclear data).

According to the classical law of propagation of errors, the overall (total) rela-

tive uncertainty on a, as a function of the relative uncertainties on the x's, is given by:

= { (15)

where the error propagation factor Z~(xj) is defined as the multiplier of the relative error on xj to obtain the associated relative error on a.

When writing the relative uncertainties in terms of differentials, one obtains according to the above definition:

a xj (16)

where it should be kept in mind that Za(xi) denotes the "partial" error propagation factor for a, caused by the relative error on xj.

Since partial derivation of the F-function [Eq. (14)] yields:

a F OF d a + - - dxj = 0 (17)

a a ~xj

Eq. (16) becomes finally:

Z~(xi) = I - - a 0xj

212 J. RadioanaL Chem. 62 (1981)


From Eq. (18) it is possible to calculate error propagation factors for the individual variables and parameters in Eq. (14). To obtain the overall uncertainty on a, these Za(xi)-factors have to be introduced in Eq. (15), together with the measured or estimated uncertainties on the xi-parameters under consideration.

It should be noted that the application of Eq. (15), with the introduction of Z~(xj)-factors according to Eq. (18), in principle will only be realistic for moderate s(xi)-values or a = F (xi) relations which do not deviate dramatically from linearity. Nevertheless, in most practical cases the above expressions can be considered as an acceptable approximation. This has been checked numerically for the examples given in this work.

Prec i s ion , fLxed a c c u r a c y and e x p e r i m e n t a l a c c u r a c y o f t h e a ~ l e t e r m i n a t i o n

The xj- and s(xj)-factors of Eq. (15) can be classified into the following cate- gories:

- xj parameters with a random error (index R), which can be described by the laws of probability [XR; S(XR) ]. These parameters influence the precision of the a-determination.

- x i parameters with a systematic error (index S) [xs; S(Xs)]. These parameters influence the fixed accuracy of the a-determination.

- x i parameters with a gross error (index G), which normally should be avoided or corrected for [xG; s(x6)]. These parameters influence the experimental accuracy of the a-determination.

Eq. (15) can then be rewritten as:

with

___ 2 S~,sq_~2 ~112 S~,T (S~,R+ Oa,GJ

s~,~ = { { [Z~(x.) s(xR)] ~ } ,/2

(19)

(20)

and

s~. s = { ~ [z~(xs) s(x~)] = } ,/2 (21)

s~, 6 = {G ~ [Z~(x G) S(XG)12} '/2 (22)

where Y., Y~ and Z denote a summation over all parameters which are causing R S G

random, systematic or gross errors, respectively.

J. RadioanaL Chem. 62 (1981) 213


To perform such a classification, let us consider the practical cases. Most generally, any of the hereafter described u-determination methods proceeds

as follows. A given set of N different a-monitors (thin foils or wires, dilute standardized alloys or small pellets, etc.) is irradiated (with or without Cd-cover, or both in subsequent irradiations, depending on the technique used) in the reactor neutron spectrum under consideration. The induced gamma activities are measured on a Ge(Li) detector, for which the full-energy peak detection efficiency should be known (except in the technique using with and without Cd-cover irradiation). This experiment is carried out in m-fold, by irradiating simultaneously m monitor sets in one container, or by performing m subsequent irradiations of analogous monitor sets.

The t~-factor can then be computed from the specific count rates of the irradiated monitors, from the nuclear data involved, and, if occurring, from the appropriate full-energy peak detection efficiencies (ep).

Thus Eq. (14) can be written explicitly as:

F [t~, Asp, i , (Nuclear Data) i, ~p , i ] = 0 (23)

where the index i refers to all isotopes used as or-monitors.

Parameters influencing the precision

The specific count rate mentioned in Eq. (23) is defined as:

Np/tm (24) Asp - SDC w

with Np

S

D

C

W

- measured number of counts under the full-energy peak, corrected for pulse losses (dead time, true coincidence, pulse pile-up, etc.);

= 1 - e -xtirr, saturation factor with ;k = In 2/T = decay constant and tit r = irradiation period;

= e -xtd, decay factor with t d = decay period;

- (1 -e-Xtm)/Xtm, measuring factor correcting for decay during the measuring period tin;

- weight of the irradiated element.

When selecting well-established monitor isotopes and appropriate working conditions, the usually small systematic errors on T, and the gross uncertainties on

tiff, ta and tin, do not contribute sighificantly to the error on Asp. s [See also

214 J. Radioanal. Chem. 62 (1981)


next paragraph (fixed accuracy) for some additional comments on the accuracy of T.] The standard deviation on Asp is then basically determined by counting statistics, and thus is essentially random. However, other factors may add to the error on Asp, such as time variation in detection equipment stability, or, in the case of m subsequent irradiations of analogous monitor sets, time fluctuations in

the reactor neutron spectrum. Nevertheless, it can be assumed that, under well- controlled conditions (no one-directional shifts), these contributions are random in nature as well. It should be mentioned that, when the experiments are performed in m-fold by irradiating simultaneously m monitor sets in one container, an instantaneous recording is performed of the epithermal neutron flux distribu- tion, whereby time-dependent neutron spectrum variations are not involved. Evi- dently, such an instantaneous technique is not possible when a has to be calculated from irradiation of analogous monitor sets with and without Cd-cover (Cd- ratio method), which should be performed necessarily in at least two subsequent irradiations.

When irradiating thin foils or wires of pure metals, the uncertainty on the

weight in Eq. (24) is likely to be negligible. However, problems may arise when using dilute standardized alloys (e.g. Au-A1 wire), which can show statistical ho-

mogeneity fluctuations in their composition, especially important when rather small amounts are used.

All the above mentioned effects will reflect in the observed overall random error on the measured specific count rates. In fact, this will be the only contribution to the random uncertainty, and thus to the precision of the calculated a-factor. Therefore Eq. (20) becomes:

N Scx, R = {~. [Z~(Asp,i) S(Asp,i)]2} 1/2 (25)

1

where, as in Eq. (23), the index i refers to the isotopes used as a-monitors.

Parameters influencing the f txed accuracy

Apart from the above mentioned statistical homogeneity fluctuations of dilute standardized alloys, which influence the precision, the uncertainty on their composition [s(w)] may play an important role. Obviously, the same holds when using home-made pellets as a-monitors. However, this error is systematic in nature, and therefore will effect merely the fixed accuracy of the a-determination method.

Evidently, this is true also for the nuclear data to be introduced in Eq. (23)~ as well as for the detection efflciencies, if involved.



Although, as mentioned above, the systematic error on Asp, induced by the usually small uncertainties on T, is likely to be negligible in well-chosen experimental conditions, special situations may occur where the effect of the error on the half-life is appreciable. 6 This is the case when measuring the 97Zr activity. So as to obtain comparable count rates of both 97Zr and 95Zr, normally a cool- ing period of 3 - 6 days is advised, 7 so that T(97Zr)/td becomes relatively large,

�9 resulting in an appreciable error propagation. For example, if t irr= 5 min, t d = 100 h and t m = 1000 s, 1% error on the half life of 97Zr (formerly accepted as 16.9 h)

would result in about 3% error on Asp, which is intolerable for a-determination. Therefore, it was felt necessary to redetermine T(97Zr) very accurately, resulting in a value of 16.735 h, with an uncertainty of 0.2% at the 99.7% confidence level. 8

This uncertainty is sufficiently low so as to give only a negligible error contribution to the Asp-value.

Finally, it should be mentioned that, for a given type of Cd-covers, the un-

certainty on Ecd, i.e. the deviation of the actual Ecd from 0.55 eV, may contribute significantly to the fixed accuracy. This is due to the fact that the Ca-terms, occurring in the equations for a-determination and given by Eq. (4) for E c a = 0.55 eV,

should be replaced by Eq. (9) for ECd :/: 0.55 eV. It can be expected that the influence of s(Eca) will be large especially for low-Qo monitors [cfr. Eq. (13)]. Obviously, the contribution from s (Eca) will not be involved in the a-determination methods with bare irradiations.

From the above it follows that Eq. (21), which expresses the fixed accuracy of the a-determination, can be written explicitly as:

N Sa, S = { i ~ [(Z~(wi) s(wi)) 2 -I- d ~ (Z~(di) s(di)) 2 a t- (Zc~(Cp,i) S(ep,i)) 2]

+ (Za(Ecd) s(Ecd))2} ,/2 (26)

with, as in Eq. (25), the index i refering to the isotopes used as a-monitors, and where Z denotes a summation over all nuclear data to be introduced in the cal- d culation of a.

Parameters influencing the expenlmental accuracy

Excluding erratic blunders, the experimental accuracy is considered to be determined by gross errors such as: spatial neutron flux gradients, neutron (self) shielding, bulky counting geometry effects, inaccurate dead-time correction, true coincidence, pulse pile-up, gamma-attenuation, inaccurate peak area determination, etc. Although some of these are rather to be considered as correction factors to

the ep-values, we preferred arbitrarily to group them together as errors which affect



the accuracy of the specific count rates to be introduced in Eq. (23). Since

Asp ~ ep, this formalism evidently is justified.

Thus, Eq. (22), which expresses the experimental accuracy of the a-determination, can be written explicitely as:

N Sc~,G = {X i [Za(Asp,i) SG(Asp,i)]:} I/2 (27)

where SG(Asp,i ) denotes the estimated gross experimental error on the specific count rate. In all following examples it is assumed thaf, in carefully controlled, optimal experimental conditions, the gross errors on the Asp-values can be estimated at about 0.5%. Although this might appear as an underestimation at first

sight, it is justified since only Asp-ratios have to be introduced in the expressions for a-calculation, so that common experimental errors are cancelling.

Critical analysis of some a-determination methods nuclear data

Nuclear data

The recommended nuclear data for the a-monitoring isotopes used or suggested in this work are listed in Table 1. They are critically selected as "best values" according to the principles and compilations of Refs. 6's-13

"CA-covered dual monitor"-method, using (randomly selected) nuclear data ~'14

Principle

Two monitors, denoted hereafter as 1 and 2, are simultaneously irradiated under

Cd-cover. The induced activities are measured on a Ge(Li) detector with a known ep-curve. If the two monitors are sufficiently diluted so as to avoid epithermal neutron shielding, and if both have a o ( v ) ~ 1/v dependence up to 1-2 eV, a can be found from the following equation (which, under the conditions stated above, is equivalent to the original expression in Ref.14):

(Asp,l)cd/Fcd,l M2 O1~'1 Oo,1 qo,l(a) + C a ep, 1 F(a) = ~ = 0 (28)

(Asp,2)cd/Fcd,2 M10272 00,2 qo,2(a) + C a ep, 2

where

J. Radioanal. Chem. 62 (1981)

M - atomic weight of the irradiated element; O - isotopic abundance; 7 - absolute gamma-intensity; Fcd - epithermal neutron transmission factor for Cd-filter (Fcd < 1).

217

F . D E C O R T E e t a l . : T H E A C C U R A C Y A N D P R E C I S I O N

~a

o6

~ +

.o .o

. . A ~ . . ~ A ~ ~

. . . . . . . . . . . . . . . . . * ~ **

O

~

o o

~ , ~ o

�9 . . . . . . . , .

+ * * *

N o ~ -

7 7 V 7 ?

�9 �9 . �9

t 4 ~ o6 r v

#

ti o ~

O

r- ,a ,a

2 1 8 Z Radioanal. Chem. 62 (1981]


The factor a can be found from Eq. (28) either graphically, ~4 or mathematically by finding the root of F(a) = 0. The latter calculation can be performed easily on a programmable desk-type calculator.

Error analysis

Precision. According to Eq. (18):

Zc~(Asp,l ) : Zoo(Asp,2 ) --- Za(Asp,i ) =

1 = 0.434

a qo,2 (a)

Qo,2 (a)

1

1 1 1t j , log Er, 2

(29)

where log = logl o and (Asp) stands for (Asp)C d. Eq. (29) reveals that the error propagation factors will be very high for a-values

close to zero. Furthermore, since the term in C a is usually small, it appears that the method will become more precise for Er,1 and Er,2 showing a large spread. This can be demonstrated more clearly when both Qo-values are sufficiently large, thus rendering the C~-term negligible, so that qo(a) -~ Qo(a). Eq. (29) then sim- plifies to:

1 1 l Z~(Asp,i ) ~ 0.434 - - (high Qo,i) (30) a log Er,2 - log Er,l

However, care should be taken not to use a monitor with a too low E-r-value, i.e. close to the Cd cut-off, since this would violate the above requirement for o(v) to follow a 1/v-shape up to 1-2 eV [e.g. l~ 7)x~ : Er = 1.43 eV; 11 s In(n, 7) 116 m In : E r = 1.51 eV].

So as to have an idea about the Za(Asp)-error propagation factors in practice, Table 2 summarizes some numerical examples for two irradiation channels 3 (a = 0.015) and 15 (a = 0.084) of the Thetis reactor (Gent), when using 197Au

(Er = 5.47 eV; Q0 = 15.7) - 94Zr (Er = 4520 eV; Qo = 5.97), one of the best monitor pairs for this a-determination method. The table also shows the relevant precision, calculated according to Eq. (25), for both 198Au and 9SZr activities

measured with a 1% standard deviation. Note that the example for channel 3 of the Thetis reactor is valid as well for channel MILA of the WWR-M reactor (Buda- pest), where the same a = 0.015 has been found.

J. Radioanal. Chem. 62 (1981) 219


Table 2 "Cd-covered dual monitor"-method, using nuclear data,

with 197Au 94Zt in channels 3 and 15 of the Thetis reactor (nuclear data: see Table 1)

Uncertainty sj Channel 3; a = 0.015 Channel 15; ct = 0.084

Parameter j Z~0) Za(J)

Asp

Wa

oob

ep

FCd Qo c

Er ECd d

Asp

1 9 7 A u 9 4 Z r

1% 1%

RANDOM

1% 0.1% 1%

0.3% 3%

1% 1%

0.2%

0.3% 2.3%

1.4% 13%

15%

SYSTEMATIC

0.5% ] 0.5% Y

GROSS

i 9 7 A u 9 4 Z r

10.8 10.8 Y

PRECISION

sa, R = 15.3%

10.8 10.8

10.8 10.8

10.8 10.8

10.8 10.8

10.8 10.8

10.8 10.7

0.16 0.15

0.29

FIXED ACCURACY

sa, S = 46.6%

10.8 I 10.8 J

Y

EXPERIMENTAL ACCURACY s~, G = 7.6%

OVERALL UNCERTAINTY

~ ,T =49 .6%

~97Au 94Zr

2.04

PRECISION

s~, R = 2.88%

2.04 J

2.04 2,04

2.04 2.04

2.04 2.04

2.04 2.04

2.04 2.04

2.04 1.93

0.17 0.15

0.11

FIXED ACCURACY

s~,s = 9.02%

2.04 [ 2.04 e


OVERALL UNCERTAINTY

s~, T = 9.58%

aZr-foil; Au--AI wire. bo o (94Zr) relative to o 0 (I 97Au ) = 98.8 b; s(o0,Au) = s(gAu ) (see text). eQ0 (94Zt) relative to Q0 (197Au ) = 15.7; s(Q0,Au ) = s(gAu ) (see text). d E c d = 0.55 eV assumed.

220 .I. Radioanal. Chem. 62 [1981)


A special situation arises when one of the a-monitors [e.g. monitor (1)] is a pure 1/v-detector. In that case Eqs (28) and (29) can be simplified, since q o : ( a ) = 0 and Q o , l ( a ) = Ca. Obviously, Er,1 is not involved, and the condition for low Za(Asp)-factors is that Er,2 should be as high as possible, with Qo,2 also sufficiently high. For instance, when combining a 1/v-detector with 94Zr in the above mentioned channels 3 and 15, Za(Asp,i)-factors of 10.2 and 1.86 "respectively, are obtained, which are more or less comparable to the ones for 197Au_94Zr (cfr. Table 2). However, the use of a 1/v-detector is generally not recommended here, since the deviation of the actual Cd cut-off energy from 0.55 eV may cause intolerable systematic errors on the a-determination (see Fixed accuracy).

Fixed accuracy. From Eq. (28) it is obvious that

Za(wi ) ----- Zt~('yi) : Zct(O0,i) ---- Z a ( e p , i ) = Za(Fcd , i )

= Za(Asp,i ) (31)

Note that the factors M and O are not considered here because they are mostly known with good accuracy, so that they will not contribute significantly to the uncertainty on a. However, in some cases the inaccuracy on O-values may be important, mainly when O is around 1% or less (e.g. for SaFe, 64Ni, a6S, 74Se, etc.).

When applying Eq. (18) to Eq. (28), the Za(Qo) and Za(Er)-factors can be ex- pressed as:

(Er,n) -a Qo,n Za(Qo,n ) = Za(Asp,i ) (32)

Qo,n(a) and

Za(Er, n) = Za(Asp,i ) l a [ qo,n (a)

Qo,n(a)

with n = l or 2. When both Qo-factors are high, these equations reduce to [see FXl. (11)]:

(33)

Za(Qo,i) ~ Z~(Asp,i) (34)

and (high Oo,i)

Za(Er,i) ~- Za(Asp,i ) l a l (35)



When introducing the expression for C a, given in Eq. (9), into Eq. (28), Za(Ecd) can be calculated as:

1 1 I Za(Eca) = Za(Asp,i) C a (a + 1/2) ~ ~.)"o,2 `a" (36)

Qo,I(a)

(Note that in practice Iot I < 0.5, thus a + 1/2 is always positive.) If (1) is a pure l/v-detector [Qo,l(a) = Ca], Eq. (36) becomes at high Qo,2:

Za(Eca ) ~ Za(Asp,i ) ( a + 1/2) [(1) = I/v; Qo,2 high] (37)

As seen, Za(Eca ) amounts to approximately half of the value obtained for Za(Asp,i). For instance, when combining a 1/v-detector with 94Zr in channels 3 and 15 of the Thetis reactor, Za(Eca)-factors of 4.85 and 0.95 are obtained, respectively. This is high compared to the 197Au-94Zr pair (cfr. Table 2). Con- sidering that, in practical circumstances, deviations as high as 10-20% can occur from E c a = 0.55 eV, the use of a 1/v de tec tor - 94Zr combination, and of any other 1 / v - non 1/v isotope pair, may lead to unacceptable systematic errors in the a-determination.

Examining the parameters involved in Eqs (31)-(33), the following remarks can be made.

As outlined earlier, the Za(w)-factor can be ignored when using a thin pure metal foil or wire (e.g. 0.125 mm Zr-foil), but will have to be considered when

using alloys, like in the case of Au, where, so as to exclude self-shielding effects, preferably a 0.1% Au-AI wire should be used. In practice, good quality alloyed wires of this type have an accuracy on their composition of about 1%.

With the use of multi-line gamma point sources (1s2 Eu, 226Ra ) and others, ep versus E~-curves can be determined with an accuracy of 1-2%. However, since only ep-ratios have to be introduced, and thus only relative efficiency curves are needed, the accuracy on the individual ep-values can be estimated to 1% or better. Evidently, the above mentioned ep-ratios are only valid for point sources at relatively large distance from the Ge(Li)-detector (e.g. 15 cm) and have to be corrected eventually for the actual a-monitor geometries and for lower source - detector distances. This can be performed either experimentally, with the use of mono-energetic gamma-sources, or mathematically, if the Ge(Li) detector characteristics are known. ~ s The resulting errors of these correction procedures are included in the experimental accuracy.

Usually, Fcd-factors are not significantly different from one, and should not be considered. In some cases, however, where (partial) overlapping occurs of the



resonances of Cd and of the monitor isotope, the Fcd-factor can be markedly lower than one, and is normally not very accurately known. Consequently, FCd occasionally influences the fixed accuracy on a. This is the case for t s6W (see multi monitor methods). Moreover, for monitors with a giant resonance in the 1-10 eV energy range, Fca can also be lower than one due to the high-energy tailing of the dominant 113Cd resonance (0.178 eV). For example, this is the case for 197Au (main resonance at 4.906 eV), giving F c d = 0.991 for a 1 mm Cd-cover, 16'17 with an estimated error of 0.2%. The same holds afort ior i for 11 Sin (main resonance at 1.457 eV), with a Fcd-factor of 0.9516 [s(Fcd) esti-

mated at 2%]. Undoubtedly, the fixed accuracy of the a-determination is drastically influenced

by the usually rather large uncertainties of the nuclear data involved (Oo, Qo and, to a lesser extent, 7, the latter value generally being known more accurately). For instance, in the above cited example in channel 3, an error of 10% on one of these data for 94Zr would result in more than 100% error on a. Detailed infor-

mation about the literature scattering for Oo, Qo and 7-values, and about a critical selection of "best" values may be found in Refs. 6'8-13 Although it is mostly not easy to assign uncertainties, even to the selected best values, at least estimations can be made, thus allowing the calculation of the fixed accuracy on a.

A special situation arises for the assignment of uncertainties to Qo-values, if these are taken from the above cited compilations of best values (see especially Refs. 9'12 '13 ). Since in Eq. (28) only ratios of Qo-factors are involved and since selection of the "best" values was made relative to Qo (197Au) = 15.7, the un-

certainty on the latter value should not be taken into consideration. Some remarks should be made concerning the introduction of t r o - as such or

in Qo - (as done here with the use of the HOGDAHL convention ~s) instead of g Oo (g = WESTCOTT's g-factor), associated with the WESTCOTT 19 or with the STOUGHTON-HALPERIN convention. 2~ Notwithstanding the comments cited in Ref., 21 about the importance of the g-factor in the case of Au, g has not been included in the present calculations for the following reasons. Since in fact g = g(Tn) is depending on the neutron temperature, and since, to preserve the simplicity of the method, no temperature monitoring is included, the more sophisticated WEST- COTT or STOUGHTON-HALPERIN formalisms can as well be replaced by the simple HOGDAHL-convention. According to GRYNTAKIS et al., 22 gAu ranges from 1.0038 at 20 ~ to 1.0068 at 100 ~ Thus for most reactors ( 2 0 ~ < < 100 ~ the uncertainty on gAu will be ~<0.3%. Since in this work Oo-values used for other isotopes were critically evaluated relative to Au (from the ko -technique 6'9-11),

ignoring the gAu-factor and using OO,Au = 98.8 b, no uncertainty should be assigned to the latter value. Thus, in these circumstances, s (oo,Au) can be considered as being



Table 3 "Cd-covered dual monitor"-method, using k0,Au-factors, with 197 Au-94 Zr in channels 3 and 15

of the Thetis reactor (Nuclear data: see Table 1)

Uncertainty s i Channel 3; c~ ---- 0.015 Channel 15 ; c~ ----- 0.084

Parameter j Z~(i) Za0)

Asp

w a

kO,Au ep

Fed Q0 b

Er

ECd c

Asp

97Au 94 Zr

1% 1% Y

RANDOM

15%

n

0.7%

1%

2.3%

13%

1%

1%

0.2%

0.3%

1.4%

SYSTEMATIC

I 0.5% I 0.5%,

y '" f

GROSS

aZr-foil; Au--A1 wire.

1 9 7 A n t 94Zr

10.8 10.8

PRECISION su, R = 15.3%

10.8

10.8

10.8

10.8

10.8

0 .16

0.29 lg "

10.8

10.8

10.8

10.8

10.7

0.15

FIXED ACCURACY

s~, S = 32.4%

10.8 [ 10.8 lr


OVERALL UNCERTAINTY

sa, T = 36.6%

197Au 94zr

2.04 2.04

PRECISION

sa, R = 2.88%

2.04 2.04

2.04 2.04

2.04 2.04

2.04 2.04

2.04 1.93

0.17 0.15

0.11

FIXED ACCURACY

sa, S = 6.43%

2.04 [ 2.04 y

EXPERIMENTAL ACCURACY sa, G = 1.44%

OVERALL UNCERTAINTY

sa, T =7.19%

bQ 0 (94Zr) relative to QO ( 1 9 7 A u ) = 15.7; s(Q0,Au ) ----s(gAu) (see text). C E c d : 0.55 eV assumed.

caused on ly by the possible variat ion o f g (Tn)Au , and hence amount s to m a x i m u m

0.3%. The same reasoning holds o f course for s (Qo,Au).

Er-values for 96 isotopes are tabulated in Ref. 3 (Note that in Table 3 o f Ref. , 3

1 7 6 H f should be changed in to 179Hf) . Al though no associated errors are men t ioned ,

they can be calculated - applying the error propagat ion theory to Eq. (10) o f


F. DE CORTE et aL. THE ACCURACY AND PRECISION

Ref) - from the uncertainties on the different parameters involved in the expression for Er. This has been done f o r the isotopes used in the present work, taking

the data of BNL-325 and of the NEA Data Bank (France). Obviously, rather large uncertainties can be tolerated on Er, since the error propagation factors will be relatively low, due to the I t~ I-multiplier (,~1) in the expression for Za(F-r)[Eq.(33)].

An attempt to perform an error analysis, leading to the calculation of the fixed accuracy on the t~-determination in the above mentioned channels 3 and 15 of the Thetis reactor, is shown in Table 2 for the 197Au_94Zr pair.

Experimental accuracy. The effect of gross errors on the a-determination in channels 3 and 15 of the Thetis reactor is summarized in Table 2 [see Eq. (27)]. In the calculations an estimated error of 0.5% was introduced for all Asp-values.

Overall uncertainty. According to Eq. (19), the overall uncertainty is obtained by quadratic summation of precision, fixed accuracy and experimental accuracy (e.g., see Table 2 for 197Au-94Zr in channels 3 and 15).

Conclusion

From the foregoing considerations it is obvious that, with randomly selected nuclear data - which can show a wild scattering in the literature - the overall error on the a-determination may be rather high or even intolerable. Although

our critical evaluation work of nuclear data, based on the determination of ko- r

factors 6'8-13 may be helpful in this respect (as shown above for 94Zr), for a num-

ber of isotopes no acceptable oo-values and/or 7-values can be selected from the literature. Thus, care must be taken when applying the above described o~-determination method with an arbitrary monitor isotope pair. Therefore, it is generally advised to use the modified versions of the "Cd-covered dual monitor"-method,

replacing the nuclear data by ko-factors, as described hereafter.

"Cd-covered dual monitor"-method, using ko-factors Principle

The same principles and conditions hold as mentioned for the above described original "Cd-covered dual monitor"-method. With the introduction of the relevant ko-factors, 6 J ~ Eq. (28) becomes:

(Asp,1)cd/Fcd,1 ko,Au(1) qo j (a ) + C a ep,1 - - ' ~ - 0

(Asp,2)cd/Fcd,2 ko,Au (2) qo,2(a) + C a ep, 2

J. Radioanal. Chem. 62 (1981} 15

(38)

225

F. DE CORTE et ai.: THE ACCURACY AND PRECISION

where ko,Au-factors are theoretically defined as:

ko,Au(i ) = MAu Oi "/i O0,i

Mi OAu "YAu O'O,Au (39)

Experimentally determined ko,Au-factors, with an accuracy <.<2% (usually "~ 1%) are compiled in Refs. 6'11

Error analysis

Precision. Evidently, the expressions and conclusions, mentioned for the original "Cd-covered dual monitor"-method [Eqs (29) and (30)], remain unchanged (see Table 3).

Fixed accuracy. Since the nuclear data of Eq. (28) have been grouped into ko,Au-factors, Eq. (31) should be replaced by:

Za(wi) : Za(k0,Au(i)) = Za(ep,i) = Za(Fcd,i ) = Za(Asp,i ) (40)

whereas Eqs (32)-(37), together with the corresponding comments, remain unchanged.

Error calculation, according to Eq. (26), when using the above cited t97Au-94Zr a-monitor pair in channels 3 and 15, can be found in Table 3. Obviously, ko,Au(Au ) = 1, with no error involved.

Experimental accuracy. Same conclusions as in the original "Cd-covered dual monitor"-method (see Table 3).

Overall uncertainty. Quadratic summation of precision, fixed accuracy and experimental accuracy leads to the results of Table 3.

Conclusion

Undoubtedly, the fixed accuracy and the overall uncertainty of the dual a- monitor method generally will improve seriously when replacing the nuclear data of Eq. (28) - with their associated large uncertainties on Oo and -/ - by the corresponding ko,Au-factors [Eq. (38)]. This holds even after careful selection of nuclear data, as shown above for the 197Au-94Zr example (cfr. Tables 2 and 3).

As a result of a close cooperation between the Institute for Nuclear Sciences (Gent) and the Central Research Institute for Physics (Budapest), experimentally determined ko,Au-factors for most of the analytically tmportant isotopes are nowa- days available with a good accuracy (usually ~1%). Thus, the "Cd-covered dual

226 J. RadioanaL Chem. 62 (1981)


a-monitor" method can in principle be extended to every monitor pai: with otherwise suited characteristics. Still, care should be taken about the accuracy of the Q0-factors to be introduced in Eq. (38), literature data for this parameter showing a large scattering. However, the above mentioned Gent-Budapest group is dealing with this problem as well, with resulting compilations of critically evaluated and recently determined Qo-factors (see Refs 6'1~ and especially Refs9'12'13).

A list of practically suited a-monitor pairs with acceptable error propagation factors is given in Table 13.

"Cd-covered multi monitor"-method, using ko-faetors 1'23 Principle

A set of N monitors is irradiated simultaneously under Cd-cover and subse- quently counted on a Ge(Li) detector with a known detection efficiency curve. When the same conditions hold as mentioned in the first chapter, a can be found as the slope ( - a ) of the straight line when plotting:

(Er,i) "~ Re,i/Oo, i log versus log ]~r,i (41)

qo,i (a) + C a

where i denotes isotope 1, 2 . . . . . N, and with Re

where N A

-- epicadmium reaction rate per nucleus

(Asp)ca - M R e = ; (42)

O N A 3' ep FCd

- Avogadro's number.

The left hand term of Eq. (41) is itself a function of a, and thus an iterative procedure should be applied, as outlined in Ref. ~ [e.g. starting by plotting Eq.(41)

for a = 0, which gives a first approximation of a = a~, and so on]. Since it has been proved in the "Cd-covered dual monitor" method that the

introduction of ko,Au-factors , instead of the nuclear data of Eqs (41) and (42), markedly improves the fixed accuracy of the a-determination, it is advised to

modify Eq. (41) and to plot:

log (Er,i) -a (A sp,i )Cd versus log Er, i (43)

ko,Au(i) "ep,i - Fcd,i [qo,i(a) + Ca]

Theoretically, this procedure will not alter the slope of the obtained straight line.

J. RadioanaL Chem. 62 (1981) 227 15"


Mathematically, the final a-result of this iteration procedure is identical with solving a from the equation:

a +

N N

~ { [ l o g Er i ~i l~ Er ' i ] [ logTi i~ l~ Ti ] } i ' N N

N N[ lOg r,ij Z log Er,i i N

= 0 (44)

with

T i = (Er,i) -a (Asp,i)Cd

k0,Au(i) "ep,i " Fcd,i [qo,i (a) + Cc~ ]

It is interesting to mention that Eq. (44) for N = 2 reduces to the one obtained in the "Cd-covered dual monitor"-method [Eq. (38)]. Thus, the above described multi monitor method can be regarded as an extension of the "Cd-covet~ed dual monitor"-method,

Error analysis [from Eq. (44)]

Precision. For the n-th monitor one obtains:

Za(Asp,n ) = 0.434

N [log r. logEri] 1 ' N

a U i (45)

where

U i = log Er,i

N N i ~ l ~

(46)

with

v i = qo,i (a)

Qo,i(a) log Er,i +

0"26Ca ( 1"67 )1

Q o,i (or) a + 1/2

228 d. Radioanal. Chert 62 (1981)


When all monitors have high Q0-values, qo,i(o)-~Qo,i(t~), and the terms in ca can be neglected, so that:

Za(Asp,n ) - 0.434

N

log Er,n i ~ Iog E r i

O~ N

i logEriN ] (Qo,i high)

2

(47)

Analysis of the above equations shows that the individual error propagation factor for the n-th monitor will decrease (1) for E r,n close to the average effective resonance energy of all monitors; (2) for all Er,i's largely different from the average; and (3) for a large total number of monitors (N large). When considering the monitor set as a whole, the first two conditions are contradictory, but since the denominator of Eq. (47) is quadratic, the condition of all Er,i-values being largely different from the average is mathematically the most important. However,

in practice, it seems better to choose a number of monitors with uniformly distributed Er-vatues, ranging from low to high. This offers the possibility of checking the linearity of the curve [cfr. Eq. (43)], thus proving that a is constant over the whole epithermal neutron energy region in the reactor spectrum under consideration (see RefJ) .

Application of Eq. (45) to the monitor set 197Au _ 1 s 2 S m - i s 6 W - 9 s M o -

96Zr-94Zr yields in the results of Tables 4 and 5 for channels 3 and 15 of the

Thetis reactor, respectively. It seems in principle better to the authors not to include 11 Sin , as was done

in Ref.~ since for this isotope Er = 1.51 eV, i.e. the condition of a ( v ) ~ 1/v up to 1-2 eV does" not hold strictly (cfr. Westcott g-factor = 1.0175 at 20 ~ Note

also that F c d = 0.93 which might introduce an additional error. However, the re- suits of Ref. 1 indicate that in practice no serious systematic error is introduced,

since the ~ i s In-point fits to the straight line.

Fixed accuracy. From Eq. (44) it is obvious that:

Zc~(Wn) = Za[ko,Au(n)] = Za(ep,n) = Za(FCd,n ) = Za(Asp,n ) (48)


F. DE CORTE et al.. THE ACCURACY A N D PRECISION

o

t~

r162

O

~ o

O

o

N

,e

Z

a~

O

e" ~o

C/3 e*

< e- e~ N

O

~o t'-4

E el

< t~

o.

~4

<

r

II

o 0 3

0

Z <

o o o o o o o.

o o _o. ~ �9

,:5 "<

- - - o o ~ u o "~ oq oq c5 < oq

c5 ,-, r i m 0

i o

~ ' ~ : ~ ~ .o

d

c~

I

.<

r

oq II o~

"< b- z

. z

z

> o

y, o ~

I1

< O

~c~ ~ ~g

o~ t . ~ tt~

e~

2 3 0 J. Radioanal. Chem. 62 (1981)

F . D E C O R T E e t a l . . T H E A C C U R A C Y A N D P R E C I S I O N

.g

o 'r. O

O

O

.o

"O ?

N

O

O

O9

II

i

�9

Z

I

O O O O O O

00 0o 0o ~0

r o o r o '

o o o o o ~

o o o o o o

o o o o o o

m

o

o~

II

P~

r

O

O

b., z

Z

�9

II

v E

" ~ II

0 0 r~

J. Radioanal. Chem. 62 (198t) 231


Furthermore one can calculate that:

Qo,n(Er,n) -a Za(Qo,n ) = Qo,n(a ) " Za(Asp,n )

and

Z~(F,r,n) = qo,n(a)

Qo,n(a) �9 l a l �9 Zcz(Asp,n)

Za(Ecd ) = 0.434 C~(a + 1/2).

1

N N - ~ m

1 . . . . ,

i ' N Qo, i (a)

1

~176 N

a U i

with Ui given in Eq. (46). For Qo.n high, Eqs (49) and (50) become:

Z~(Qo, n) ~ Z a ( A s p , n )

and (Qo,n high)

Z~(E~,n) ~ I~1 �9 Za(Asp,n)

(49)

(50)

(51)

(52)

(53)

Error analysis for the above mentioned monitor set is shown in Tables 4 and 5 for channels 3 and 15 of the Thetis reactor, respectively, whereby the following comments should be made.

Since Au - AI, Sm - A1 and W - A1 alloyed wires were used (together with Mo- and Zr-foil on which s(w) can be neglected), s (w)= 1% should be introduced for these monitors.

As mentioned before the Fcd-factor for 1 s 6 w should be taken into account. Indeed, partial overlapping occurs of the 18.40 eV resonance of 113Cd and the

18.84 eV 1 s 6W.resonance, resulting in a F cd-factor of 0.908, with an uncertainty

of 2.3%. 24 It is interesting to note that largely different FCd (186W)-factors can be found in literature. 2s'26

As to the errors on the Qo-values, it can be shown that, when working out the summation terms of Eq. (44) for N detectors, Qo(a)-ratios appear (cfr. dual moni-



tor method). Thus, since selection of the "best" Qo-values 6'8-13 was made relative

to Qo (197Au) the uncertainty on the latter value should not be considered (ex-

cept for the s(gAu)-factor; see earlier).

Experimental accuracy. Evidently, the experimental accuracy is now determined

by the Asp-Values of N monitors. Results for channels 3 and 15 are shown in Tables

4 and 5, respectively. Overall uncertainty. Results for the overall uncertainty, as obtained for channels

3 and 15 of the Thetis reactor, with the above mentioned monitor set, are presented

in Tables 4 and 5.

Conclusion

Tables 4 and 5 show that, when selecting appropriate activation detectors, the

"Cd-covered multi a-monitor"-method will give somewhat better results for a, with

respect to precision and accuracy, than the "Cd-covered dual monitor" method (cfr. Table 3). Obviously, even the introduction of a monitor like 186 W, with a

large uncertainty on its Fcd-factor, does not influence significantly the fLxed accuracy of the method. As outlined earlier, 1 the use of only three monitors, 197Au- 96Zr-94Zr, with a suitable spread on their Er-values (5.47 eV, 340 eV and

4520 eV) and otherwise interesting characteristics, will give fairly precise and accurate results for a-monitoring. This is illustrated in Table 6. Comparison with

Table 3 reveals that the results are comparable to those of the dual monitor

method, with the additional advantage of enabling a quick control of the con-

stancy of a with neutron energy.

"Cd-Ratio for dual monitor"-method 1 Principle

A set of two monitors (1 and 2) is irradiated with and without Cd-cover. The

induced activities are measured on a Ge(Li) detector. With the same conditions as

mentioned in the first chapter, a can be found by solving the following equation

(graphically, or mathematically e.g. on a programmable desk-type calculator):

where

RCd,2 -- 1 qo,l(a) + C a

RCd,1 -- 1 qo,2(a) + C a = 0 (54)

(Asp,i)bare R cd,i = (5.5)

(Asp,i)ed/Fcd,i



Table 6 "Cd-covered multi(triple) monitor"-method, using k0,Au-factors, with s 9 VAu 9 6 Zr 94 Zr

in channels 3 and 15 of the Thetis reactor (nuclear data: see Table 1)

Para- meter j

Asp

W a

kO,Au ep

Fcd QO b

Er

ECd c

Asp

Uncertainty sj

1 9 7 A u 9 6 Z r 9 4 Z r

Channel 3; c~=:O.O 15 Channel 15 ; t~=0.084

z a o ) z a o )

t 9 7 A u 9 6 Z r 9 4 Z r 1 9 7 A u 9 6 Z r 9 4 Z r

1% 1% 1% r

RANDOM

1%

1%

0.2%

0.3% 1.4%

2.1% 0.7% 1% 1%

2.1% 2.3% 2.3% 13%

15%

SYSTEMATIC

0.5% 1 0 .5 % ] 0 . 5 %

GROSS

11.3 1.61 9.72

PRECISION sa, R = 15.0%

11.3 1.61 9.72 11.3 1.61 9.72 11.3 1.61 9.72 11.3 1.61 9.72

11.3 1.61 9.65 0.17 0.024 0.13

0.24 y

FIXED ACCURACY s~,s = 30.8%

113 II61 19.72 Y

EXPERIMENTAL ACCURACY sa, G = 7.5%

OVERALL UNCERTAINTY

sc~,T = 35.1%

azr-fofl; Au-AI wire. bQ0's relative to Q0 (197Au ) :: 15.7; s(Q0,Au ) = S(gAu ) '(see text). CEcd= 0.55 eV assumed.

2.13 0.30 1.83

PRECISION s~, R = 2.82%

2.13 0.30 2.13 0.30 2.13 0.30 2.13 0.30

2.13 0.30 0.17 0.025

0.097 �9 t

Y

1.83

1.83 1.83

1.83 1.72

0.13

FIXED ACCURACY

su, S = 6.02%

2.13 ] 0 . 3 0 I 1 . 8 3 �9 4

Y

EXPERIMENTAL ACCURACY s~, G =1.41%

OVERALL UNCERTAINTY

sc~,T = 6.80%

Obviously, this me thod has the advantage of avoiding the in t roduc t ion of absolute

(M, | % ao) or c o m p o u n d nuclear data (ko,Au) and enables simple measurements

on a Ge(Li) detector wi thou t the need of the full-energy peak de tec t ion efficiency

curve. On the o ther hand, due to the two types of i rradiat ions required, no in-

s tan taneous epi thermal n e u t ro n flux d is t r ibut ion moni to r ing is possible.

234 .I. Radioanal. Chem. 62 (1981)


Error analysis

Precision. Obviously:

Za(Asp,1)bar e = Za(Asp,l)C d and Za(Asp,2)bar e = Za(Asp,2)c d (56)

1

According to Eq. (18)

Za(Asp,n ) = 0.434 �9 f + Qo,n(Or

.f (57)

_ 1.67 ) [ 1 log Er, 2 q~ logE-r, 1 + 0.26 C a 1

Qo, (o) Qo,7( a

qo,2(c0

Qo,2(a)

where f - thermal (subcadmium) to epithermal flux ratio (see e.g. Ref. 9) n - monitor 1 or 2.

Note that the "absolute value"-term of Eq. (57) is identical to the one for the "Cd-covered dual monitor"-method [Eq. (29)]. However, when using the same a- monitor pair, the error propagation factors become larger with a multiplier [f + Qo(a)]/f, which is especially important for low f and/or large Qo-values.

Analogously to Eqs (29) and (30), if Er,2 >>Er,1 and if Qo,2 is not too small, Eq. (57) can be approximated by:

Za(Asp,n ) - 0.434 f + Qo,n(0t) 1

qo,2 (a) log F,r, 2

Qo,2 (~)

1

o~ (if E'r,2 ~ E'r,,)

(58)

Eq. (58) applied to monitor 1 (n = 1) requires that Qo,1 be not too large compared to the flux ratio. As for monitor 2 (n = 2), mathematical analysis of Eq. (58) shows that the expression [ f + Q0,2(o0]. Qo,2(a)/qo,2(a ) is minimal at the following optimal Q 0,2" value:

[Qo,2(t~)]opt. ~ [(f q_ Ca ) Ca ] 1/2 q_ Ca (59)


F. DE CORTE et al.. THE ACCURACY AND PRECISION

Since Ca shows no great variation with a and amounts only to 0.429 for a = 0, Eq. (59) becomes finally [Qo,2 ( a ) = Qo,2 when a = 0]

[Qo,2]opt. ~ (0.429 f)1/2 (60)

E.g. for flux ratios varying from f = 20 to f = 160, tiffs gives Qo,2-values from 3.34 to 8.68, respectively. Thus, it appears that 94 Zr, with a very high Er-value of 4520 eV and Qo = 5.97, will in general be an excellent a-monitor when using the Cd-ratio method, combined with, for instance, 197Au ' although Q o,Au is relatively large.

A serious drawback of the Cd-ratio method is, of course, that four different specific activities have to be introduced, each of them contributing to the precision on a. This is illustrated in Table 7, giving some numerical examples for channels 3 and 15 of the Thetis reactor, when using the 197Au-94Zr monitor pair. If all activities are counted with a 1% precision, the precision on a can be seen to be nearly the double of the values obtained with the "Cd-covered dual monitor"-methods (cfr. Tables 2 and 3).

Finally, it should be mentioned that the comments and warnings concerning the use of monitors with low Er" or Qo-values (e.g. 1/v-detectors), as mentioned earlier, remain valid for the Cd-ratio method.

Note also that the Cd-ratio measurements are obviously more sensitive to the time stability of the reactor neutron spectrum, due to the double irradiation required.

Fixed accuracy. From Eq. (55) it follows that Za(wn) = Za(Fcd,n ) = Za(Asp,n). However, the uncertainties on w are not involved even when using alloys, since only Asp-ratios have to be introduced in Eq. (54).

Furthermore:

Za(Qo,n) = Za(Asp,i) " (61) f + Qo,i (a) Qo,n (a)

Za(E'-r,n) = [ Za(Asp,i) " f j

f + Q o,i (a)

qo,n(a) a I" (62)

Q0,n (~) and

Za(Ecd ) = Za(Asv,i). f + Q0,i (or)

�9 C,~ �9 (a + 1 /2 ) 1 1

Q o,z(O0 Qo,l( ct )

(63)

236 J. RadioanaL Chem. 62 {1981)

F. DE CORTE et al.. THE ACCURACY AND PRECISION

Table 7 "Cd-ratio for dual monitor"-method, with ~ 9 7 Atl 94 Zr in channels 3 and 15

of the Thetis reactor (nuclear data: see Table 1)

Parameter j

(Asp)Cd

(Asp)bare

Uncertainty sj

97Au 94 Zr

1% 1%

1% 1%

v

RANDOM

Fed 0.2%

Qo a 0.3%

Er 1.4%

ECd b

2.3%

13%

15%

Y

SYSTEMATIC

(Asp)Cd c

(Asp) bare c

0.3% 0.3%

o13% o.3% Y

GROSS

Channel 3; ot=0.015 ; f=25 Channel 15; oc:0.084; f=72

zoto) zoto)

1 9 ~ A u 9 4 Z r ~97Au 9*Zr

17.4 13.1

17.4 13.1

Y

PRECISION

sot,R = 30.8%

17.4 13.1

10.8 10.7

0.16 0.15

0.29 ~r

FIXED ACCURACY

sot,S = 25.5%

17.4 13.1

17.4 13.1

Y

EXPERIMENTAL ACCURACY

Sa,G = 9.2%

OVERALL UNCERTAINTY

sot,T = 41.0%

2.43

2.43 ~r

PRECISION

sot,R = 4.57%

2.13

2.13

2.43 2.13

2.04 1.93

0.17 0.15

0.11

FIXED ACCURACY

sot~S = 5.19%

2.43 2.13

2.43 2.13

EXPERIMENTAL

ACCURACY

sa, G = 1.37%

OVERALL

UNCERTAINTY

sot,T = 7.05%

aQ0 (94Zr) relative to Q0 (197Au ) = 15.7; s(Qo,Au) = s(gAu) (see text).

bEcd = 0.55 eV assumed. epartial cancelling of gross errors (see text).

where Za(Asp , i ) " f / [ f + Qo.i(t~)] is independent of the isotope i, and is in fact identical to the error propagation factor on the specific activity in the "Cd-covered dual monitor"-method [Eq. (29)]. Thus, Eqs (61), (62) and (63) will give identical results to those obtained from Eqs (32), (33) and (36), respectively, and the same comments as in the "Cd-covered dual monitor"-method remain valid.

J. Radioanal. Chem. 62 (198l) 237


Since no other nuclear data, or the detection efficiencies, with their associated uncertainties, have to be considered, the f~xed accuracy of the "Cd-ratio for dual monitor"-method certainly will be better than that of the foregoing dual monitor methods. This is illustrated in Table 7, where examples are given for channels 3 and 15 of the Thetis reactor, using the 197Au-94Zr monitor pair (cfr. Table 3).

Experimental accuracy. If similar monitor sets are irradiated with and without

Cd-cover, the cadmium ratio method has the important advantage that gross errors

due to 7-attentuation, different counting geometries for both monitors, tree coincidence effects and inaccurate peak area determination (if systematic), are can- celled in the Cd-ratio measurements.

" On the other hand, since four Asp-Values have to be introduced in Eq. (54), the contribution of the remaining gross errors (see earlier) on the experimental accuracy still can be relatively important. This can be seen in Table 7, where the

gross errors on the Asp-Values are estimated at 0.3% (cfr. Table 3). Overall uncertainty. The overall uncertainty is obtained by quadratic summation

of the precision, fixed accuracy and experimental accuracy (see Table 7).

Conclusion

Although the outlined "Cd-ratio for dual monitor"-method is undoubtedly su- perior to the "Cd-covered dual monitor"-method witt~ respect to the fixed accuracy, the overall uncertainty may become worse due to the contribution of counting statistics. This holds especially for isotopes with high Qo-values, which should not be used. The only possibility to reduce this contribution is trying to obtain good precision on the specific count rates, e.g. by performing a large number of experi-

ments, thus reducing the standard deviations on the means of the Asp-Values. For instance, if all 198Au and 95Zr specific count rates are determined with 0.3% precision, instead of 1% as assumed in Table 7, for channel 3 a precision of 9.2%,

and an overall uncertainty of 28.6% is obtained (cfr. the results of Table 7).

A list of suited monitor pairs with acceptable error propagation factors is given

in Table 13.

"Cd-Ratio for multi monitor"-method

Principle

The above cited "Cd-ratio for dual monitor"-method can be easily extended to a multi-monitor method. A set of N monitors is irradiated with and without Cd- cover, and the induced activities are measured on a Ge(Li) detector. It can be shown that when the same conditions hold as mentioned, a can be found as the

slope ( - a ) of the straight line when plotting:

238 J. Radioanal. Che~ 62 (1981)


(Er,i)-a log versus log Er,i (64)

(Rcd,i -- 1) Qo,i(ot)

with Rcd,i given in Eq. (55). Analogously to the "Cd-covered multi monitor"-method, a should be solved

from the equation:

N N log

(Er,i) i �9 , l o g / - l o g E r i N

(Rcd,i -- 1) Qo,i(a) N

(Er,i)

(Rcd,i-- 1) Qo,i(a) ]} = 0 N

log Er,i 2

' N (65)

For N = 2, Eq. (65) reduces to Eq. (54), obtained for the "Cd-ratio for dual monitor"-method. The same comments as mentioned for the latter remain valid.

Error analysis

Precision. For the n-th monitor one obtains

Z,.,(Asp,n)bar e = Za(Asp,n)C d =

= Za(Asp,n ) = 0.434 " f + Qo,n (~ 1

N [lOg r,n lOg r,i]N a U i

(66)

with U i given in Eq. (46).

Equation (66) is equal to Eq. (45), after multiplication of the latter with the

factor [ f + Qo,n(a)]/f. Thus, as usual with Cd-ratio measurements, high Q0,i-values will lead to important error propagation factors, especially in low flux-ratio irradiation sites.

Evidently, concerning the spread on the E'r-values the same comments can be made as in the "Cd-covered multi monitor"-method. As an example, the results of Eq. (66) applied to the monitor set 197Au_1S2Sm 186W 19apt l OOMo 94Zr

s Radioanal. Chem. 62 (1981] 239


are shown in Tables 8 and 9, for channels 3 and 15 of the Thetis reactor, respectively.

Fixed accuracy. The only factors to be considered are Za(FCd,n), Za(Qo,n), Za(Er,n) and Za(Ecd).

Obviously Za(Fca,n) = Z~(Asp,n), whereas the expressions for Za(Qo,n) and Za(Er,n) can be written as:

f ]. Za(Qo,n) = [ Za (Asp,n) �9 f + Qo,n(a)

(Er,n) -a Qo,n

Qo,n(a)

----- Zc~(Asp,n ) high] f + Qo,n (a)

(67)

Za(E'r,n) = [Zc~(Asp, n)" q0,n(a)

�9 l a [ �9 ( 6 8 )

f + Qo,n(a) Qo,n(a)

- Z~(Asp,n ) �9 f + Qo,n (a)

a[ [Qo,n high]

Thus, Eqs (67) and (68) are identical to Eqs (49) and (50), respectively. Finally, the expression for calculating Z a(Ecd ) is also identical to the one obtained in the "Cd-covered multi monitor"-method [Eq. (51)]. Thus, the comments under "Fixed accuracy" remain unchanged.

Error analysis for the above mentioned monitor set is shown in Tables 8 and 9, for channels 3 and 15 of the Thetis reactor, respectively.

Experimental accuracy. The same comments can be made as mentioned in the "Cd-ratio for dual monitor"-method. Obviously, the experimental accuracy is now determined by the Asp-values of N monitors. Results for channel 3 and 15, for s(Asp ) estimated at 0.3%, can be found in Tables 8 and 9, respectively.

Overall uncertainty. As usual, the overall uncertainty is obtained by quadratic summation of the precision, fixed accuracy and experimental accuracy (see Tables 8 and 9).

Conclusion

The "Cd-ratio for multi monitor"-method is, with respect to its accuracy, supe- rior to the "Cd-covered multi monitor"-method (cfr. Tables 4, 5 and Tables 8, 9), on condition that a set of suited a-monitors is used. Moreover, if the specific-

240 J. RadioanaL Chem. 62 (1981J

F. DE CORTE et al.: THE ACCURACY A N D PRECISION

t~ [-.,

0

0

0

0

0

m

e~

e-

0

o

~ tz

II

i

0

z

~ 5 o 6 ~

,6 .4 r

r r ,::5

r t ~

.n- oi r

t t ) t ~ e q

o'~ ',,tl"

oo

,:5

~o

e4 en

t t )

e4 r

r

("4 f ~

~ d

r ~

, d

II [..,

z

o~

tl

�9 < ..<

o ,

ii ~ >o "~ . . ~

"~ t t - O ~

Z Radioanal. Chem. 62 (1981) 2 4 1 1 6

F. DE C O R T E et al.: T H E A C C U R A C Y A N D P R E C I S I O N

0

0 . ~

~" 2 2 . e

t - q t - q

t ~ l f q

r

o .....4 o M

r,

e~

e ~

e~

,%,-~

Z

I1

II m

r..)

I

t t ~

6 ~

�9

,t

6 ( : 5

6 ~

II

z

I

r ~

,d II

Z

r,.)

o

% .r

tl %

o,

i, i >~ < . ~

~ , ~, ,,4

2 4 2 J. Radioanal. Chem. 62 {1981)


count rates are determined with high accuracy (e.g. counting statistics 0.3% instead of 1%, as assumed in Tables 8, 9), also the precision and thus the overall uncertainty become better.

Note that wmonitor isotopes with high Qo-values (cfr. Tables 4, 5: 98Mo and 96Zr) are replaced in Tables 8, 9 by others with moderate Qo's (19apt and l~176 but with comparable ~-values. However, it should be remarked that, when performing such replacements, the introduction of itosopes with too low Qo'values should be avoided. Indeed, as to be expected, this operation would markedly increase Za(Eca ). For instance, when l~176 (/~r = 513 eV; Qo = 19.4) in the monitor set of Table 8 is replaced by 6 S Zn with comparable Er (515 eV) but lower Qo (3.51), Za(Ecd ) approximately doubles from 0.206 to 0.434.

"Bare triple monitor"-method, using ko-factors 2 Principle

A set of three detector (1, 2 and 3) is irradiated without Cd-cover. The induced activities are measured on a Ge(Li) detector with a known efficiency curve. Under the same conditions as mentioned, a can be found from the following equation (graphically, or mathematically e.g. on a programmable desk-type calculator):

with

and

( a - b)qo, l (a) - a qo,2(a) + b qo,a(a) = 0

1 a =

Asp,2 ko,Au(1) ep,1

Asp,1 k0,Au(2) ep,2

(69)

1 b = (70)

Asp,3 k0,Au(l) ep,l

Asp,1 ko,Au(3) 6p,3

When one of the monitors is a pure 1/v-detector, e.g. (1) = I/v, Eq. (69) can be transformed into an explicit function of a [qo,l(a) = 0]:

a qo,2 l o g ~

b qo,3 a = [(1) = I/v] (71)

Er,2 log

Er,3 Z RadioanaL Chem, 62 {1981} 243

16"


As far as known by the authors, the bare monitor method is the only technique enabling wdetermination without Cd-covered irradiation. It is therefore especially important in (n, 3,) activation analysis, using the k0-standardization method 2 and in all nuclear experiments (cross-section measurements, etc.) where a cadmium cover is not allowed to be irradiated.

Error analysis

Precision. Application of Eq. (18) gives:

Za(Asp,l ) = 0.434 [f + Qo,l (a)] 1 qo,2(a) - qo,3 (or)

ct W (72)

Za(Asp,2) = 0.434 [f + Qo,2(ot)] 1 qo,3 (~) - qo,l(a) a W

(73)

Za(Asp,3 ) = 0.434 [ f+ Qo,3(ot)] I 1 m �9

ot

q0,1 (00 -- q0,2(0t)

W (74)

with

W = qo,l(a) qo,2 (t~) [log Er,2 - log Er,t] + qo,2(r qo,3(Or) [log Er,3 - log Er,2]

-I- qo,3 (Or) qo,l(a) [log Er , l - log l~r,a I (75)

Mathematical analysis of the above Za-functlons shows that, theoretically, the best condition would be a combination of one pure I/v-detector with two monitors showing a large spread on their Er-values and with almost equal, high Q0" factors (note that, since no Cd-cover is used, the uncertainty on ECd is not involved in this case, thus allowing the use of a 1/v-detector). In a hypothetical case, for example in channel 3 ( f = 25; a = 0.015) one obtains

Qo,l = 0.429; Er,l not relevant Z~(Asp,l ) = 0.110

Qo,2 = 250; Er,2 = 5000 eV Za(Asv,2 ) : 10.8

Qo,3 = 250; Er,3 = 5 eV Za(Asp,3 ) = 10.7

However, in practice it seems impossible to find such a pair of high Qo isotopes.

244 J. Radioanal. Chem. 62 (1981}


Another, second best choice appears to be a set of three monitors with a large

spread in E r- and in Qo-Values, and if Q0,t < Qo,2 < Qo,3, preferably Er, 2 <( ErA <(Er, 3. Since this condition is also difficult to fulfdl in practice, other sequences of Er have to be accepted, whereby it should be absolutely avoided that the middle Er-value corresponds to the middle Qo-value. The latter situation would lead to unacceptably high error propagation factors.

Thus, among others (see Table 14), the combination of Z97Au (Q0 = 15.7; Er = 5.47 eV), 94Zr (Qo = 5.97; F-r = 4520 eV) and 96Zr (Qo = 280; Er = 340 eV)

may be expected to give relatively precise results. This is shown in Table I0 for channels 3 and 15 of the Thetis reactor.

It should be emphasized that the use of a suited isotope, like 197Au ' in combination with the two zirconium isotopes is a favourable case, since Zr additionally can serve as a flux ratio monitor. 7 Another advantage of the 197Au-94Zr-96Zr

set is that none of the measured isotopes emits gamma-rays in cascade, so that no true coincidence effect should be corrected for at small source-detector distances. Thus, simple coirradiation of a standardized Au-A1 wire and a thin Zr-foil enables the simultaneous determination of o~ and f. It may be interesting to note that epi-

thermal neutron self-shielding factors Gep i (to be multiplied with I0 or Qo) for dilute A u - A1 wires of different composition can be found in Ref. 27 Recently, G epi-factors for thin Zr-foils have been determined experimentally. 8 E.g., for com- monly used 0.125 mm thick Zr-foil, Gep i = 0.98 for 94Zr and Gep i = 0.97 for 96 Zr has been found.

It might be tempting to replace ~ 97Au, for which the Qo-factor is rather close to the one of 94Zr, by 23SU (Qo = 102.3; Er = 15.8 eV), thus obtaining a better

spread in Qo's. Although indeed lower error propagation factors will be obtained, shielding of the 238 U-resonances by the reactor fuel may cause a considerable re- duction of the 239u/z39Np specific count rate, especially when irradiating near

to the core. This will result in an appreciable systematic error on a.

Fixed accuracy. Obviously:

Z~(Wn) = Z~[ko,Au(n)] = Z~t(ep, n) -- Ztx(Asp,n) (76)

where n = 1, 2 or 3. Furthermore one obtains:

Z,,,(Qo,n ) = Zoc(Asp,n ) " Qo,n (Er,n) ~

f + Qo,n(a)

(77)

s Radioanal. Chem. 62 (1981) 245


and

qo,n(a) Za(Er, n) = Z,,(Asp,n) " [ a l (78)

f + Qo,n (a)

Some comments can be made concerning the use of a 1/v-detector. Although as mentioned earlier, the uncertainty on Eca is not involved in the triple com- parator method, one should consider the fact that Q0-values for "so called" 1/v- detectors are generally known with a much worse accuracy than for high Q0" detectors. Fortunately, when using a good combination of a 1/v-detector with two other suited monitors, Za[Qo(1/v)] will be rather low, since Qo,n in Eq. (77) should be replaced by Qo(1/v) = 0.429. However, the set 1 / v - 9 4 Z r - 9 6 Z r turns out to give unacceptable high Za(Asp)-error propagation factors, so that a combination of a 1/v-detector with Zr cannot be recommended.

The error analysis tbr a-determination with the "bare triple monitor"-method in channels 3 and 15 of the Thetis reactor, using 197Au-94Zr--96Zr, iS shown in Table 10.

Experimental accuracy. The experimental accuracy, which is determined by three Asp-ValUes is summarized in Table 10.

Overall uncertainty. As usual, quadratic summation of precision, ftxed accuracy and experimental accuracy yields the overall uncertainty of the triple a-monitor method (see Table 10).

Conclusion

Although the "bare triple a-monitor"-method suffers from rather high error propagation factors, and thus a considerable uncertainty on the determined a has to be expected (especially for low a), it should be accepted as such, since no other simple alternative technique without Cd-covered irradiations is available (except the hereafter described "bare multi monitor"-method, which is, of course, experimentally more complicated). When using this technique, three ways of reducing the uncertainties should be considered:

- judicious choice of monitors, having accurately known ko- and Qo-values, and giving rise to relatively low error propagation factors;

- utmost experimental care, so as to minimize gross errors. In this respect, the importance of small corrections for neutron self-shielding, gamma-attenuation and differences in counting geometry, depending on the shape, thickness and composition of the foils and wires, should not be underestimated. So as to eliminate some of the gross errors, it would be interesting to make use of alloyed foils or wires, containing three monitors with an accurately known concentration. At

246 J. Radioanal. Chert 62 H981)


Table I0 "Bare triple rnonitor"-method, using k0,Au-factors, t 9 ~Au-94 Zr 9 ~ Zr

in channels 3 and 15 of the Thetis reactor (nuclear data: see Table 1)

Para- meter j

Asp

Uncerta inty sj

1 9 7 A u 94Zr 9 6 Z r

Channel 3; r f--2S Channel 15; cx---0.084; f=72

z~O) z~o)

t 9"/Au 9 4 Z r 96Zr t g ~ A u 94Zr

1% 1% 1% 36.5 16.6

RANDOM

w a 1% - - 36.5

ko,A u - 0.7% 2.1% 36.5

ep 1% 1% 1% 36.5

Qo h 0.3% 2.3% 2.1% 13.9

Er 1.4% 13% 2.3% 0.202

9 6 Z r

Asp

26.3 10.1 i Y

PRECISION

sc~,R = 46.1%

26.3 10.1

26.3 10.1

26.3 10.1

4.57 9.24

0.0637 0.138 qr �9

SYSTEMATIC

0.5% 105 10.5% Y

GROSS

FIXED ACCURACY

s,~,S = 68.9%

36.5 1 2 6 . 3 [ 1 0 . 1

T

EXPERIMENTAL ACCURACY

sa, G = 23.1%

OVERALL UNCERTAINTY

sa, T = 86.1%

13.7 2.95 Y

PRECISION

Sa,R = 21.7%

16.6 13.7 2.95

16.6 13.7 2.95

16.6 13.7 2.95

2.64 0.536 2.08

0.216 0.041[ 0.174 1r

FIXED ACCURACY

sa,s = 30.0%

16.6 113.7 12.95 J

"If

EXPERIMENTAL ACCURACY Sa,G= 10.9%

OVERALL UNCERTAINTY

s~t,T = 38.6%

azr-foil; Au-AI wire.

bQ0's relative to QO (197Au ) = 15.7; s(Q0,Au ) = s(gAu ) (see text).

present, the authors are considering this possibility, to start with a low-content Au (~100 ppm) - Zr alloy of accurate and homogeneous composition.

- improvement of counting statistics, e.g. by coirradiating several sets o f mon-

itors, so as to obtain better standard deviations on the means of the Asp-Values. For instance, if all activities are determined with 0.3% precision, instead of 1% as assumed in Table 10, the following precision and overall uncertainty will be ob-

tained for channel 3 (with Au-Zr): sa, R = 13.8%; Sa,T = 74.0%.



In Table 14, a list of practically suited a-monitor sets with acceptable error propagation factors is presented. Note that the use of Lu (e.g. in the triplet 1 7 S L u - 9 4 Z r - 9 6 Z r ) might be an interesting alternative, since additional meas- urement of 177 Lu enables a neutron temperature monitoring. 2 s

"Bare multi monitor"-method, using k0-factors

Principle

Keeping in mind the rather high Z~-factors of the "bare triple monitor"-method, it might be interesting to perform an extension into a "bare multi monitor"-method.

A set of N monitors, together with a "reference" monitor isotope, is irradiated without Cd-cover, whereafter the induced activities are measured on a Ge(Li) de-

tector with known ep versus E~ curve. If the conditions mentioned are met, a can be found from the slope ( - a ) of the straight line when plotting:

log [(Er~) -a Ai] versus log Er,i (79)

with

Ai = Asp'i/k~ eP'i - Asp'ref/k~ ep~ref (80)

q0,i(a) - qo~ef(a)

and the index "ref" denotes the reference monitor isotope. Thus, a should be solved from the equation:

a +

N

log [(Er,i) "a "Ai] - N

N

N = 0 N

logEr i I~ log F"r,i i N

(81)

("ref" not included in the i-series).

Note that Eq. (81) for N = 2 (plus one reference monitor, e.g. " ref" = 1) reduces to Eq. (69). This is the minimum number of detectors which can be used in the bare a-monitor method.



Error analysis

Precision. Application of Eq. (18) to (81) gives for the n-th monitor (n ~: re0:

1

Za(Asp,n ) = 0.434 [ f + Qo,n(a)].

N

ilog rn log ri]N a

[qo,n (a) - qo,ref(a)] ~ l II~ Er,i

N Z log Er,i

N

with

B i --- q0,i(a) l~ Er , i - qo,rer(a) l~

qo,i(a) - qo,ref(a)

For the reference monitor one obtains analogously:

Za(Asp,ref ) = 0.434 [f + Qo,ref(a)] -

N (82) Y" Bi I i

N

(83)

N I N N

1 ~ { [ l~ iE l~ l [ 1 i q o , i ( a ) - q o , ~ e r ( a ) ] }

�9 ' N j r . qo, i (a)-qo,ref(a) - N N N Z log Er,i Z B i

1[ '1/ I~ log F"r,i i Bi i N N

(84)

[in Eqs (82) and (84), "ref" is not included in the i-series)�9 For N = 2, plus one reference monitor (e.g. "ref" = 1), Eqs (82) and (84) can

be easily converted to Eqs (72)--(74) of the "bare triple monitor" method. In the latter, the three a-monitor isotopes are permutational without altering the results for the error propagation factors, i.r any of the three isotopes can be chosen as the reference monitor, which does not play a special role. This is not the case in the "bare multi a-monitor" method. Indeed, for a given monitor set, the error propagation factors calculated according to Eqs (82) and (84) will depend on the


F. DE CORTE et al.: THE ACCURACY ~ D PRECISION

reference isotope selected from this set (but evidently, for a given Asp-Series, the resulting a-value [Eq. (81)] is independent of such a selection). This means that, with respect to the error on a, the reference isotope has an exceptional function and should be chosen critically.

Analysis of the above Za(Asp)-functions [Eqs (82) and (84)] reveals that in practice, one should select a number of monitors with uniformly distributed E r- values, ranging from low to high. Furthermore, the reference monitor should preferably have a relatively low Qo-value, with all other monitors of the i.series showing high Q0-values. This conclusion is analogous to the one of the "bare triple a-monitor"-method.

As an example, an analysis of the precision for the monitor set 94Zr (reference) 181Ta-121Sb-12sSb-l12Sn-96Zr is shown in Tables 11 and 12 for channels 3 and 15 of the Thetis reactor, respectively. Note that comparable results are obtained when replacing the reference 94Zr by 6SZn (Qo-values resp. 5.97 and 3.51).

Fixed accuracy. As well for "ref" as for "n" one obtains

and

z~(w) = Z~(ko,Au) = %(%) = Z~(Asp) (85)

Qo(Er) -a �9 ( 8 6 ) za(Oo ) = Za(Asp ) f + Qo(a)

qo(a) Za(Er) = Z~(Asp) l a l (87)

f + Qo(a)

Analogous comments as in the discussion of the "bare triple monitor"-method can be made.

Tables 11 and 12 show the results of the fixed accuracy obtained with the "bare multi monitor"-method in channels 3 and 15 of the Thetis reactor, using the above mentioned monitor set.

Experimental accuracy. The experimental accuracy is now determined by N + 1 monitor isotopes - including the reference monitor (see Tables 11 and 12).

Overall uncertainty. The overall uncertainty is, as usually, obtained by quadratic summation of precision, fixed accuracy and experimental accuracy (see Tables 11 and 12).

Conclusion

Comparison of Tables 11 and 12 with Table 10 shows that, as expected, the accuracy and the precision on a are markedly improved by increasing the number

250 J. Radioanal. Che~ 62 (1981)

F. D E C O R T E et al.: T H E A C C U R A C Y A N D P R E C I S I O N

iv )

e~

~ g 0

0 " '

e~ "~,

0

E

g

el

e ,

an

r / ) el

e *

*e M

~t

II

e4

Z ,r a~

t ~

O

�9 . ~ . ~ ~ e 4 d

�9 �9

~4

II

,r

q)

>, r /3

O~

v-4

O

tr

~5

c5

O

O

O

II

Z

o ~

II E-

z

t -

O

II

O

II

�9 "-d r~

.~ ~ .4

e~

J. Radioanal. Chem. 62 (1981) 2 5 1

F. D E C O R T E e t al.: T H E A C C U R A C Y A N D P R E C I S I O N

"d

,=..~

Q

I= 0

t ~

~q

I=

, .o

m

q

L .

.<

0

.<

I ~ ~

o~

d

b o

II e,,

z

,.d

0

<

II

, 0

m

II

, = ,...

~ o

2 5 2 J. Radioanal. Chem. 62 (1981)


Table 13 Some possible ~-monitors suggested for use in the "Cd-covered dual monitor"-method

and the "Cd-ratio for dual monitor"-method ~ Nuclear data: see Table 1)

"Cd-covered dual monitor"-method "Cd-ratio for dual monitor"-method Combination ( 1 ) - ( 2 ) f = 25 ; r, = 0 . 0 1 5 f = 2 S ; a = 0 . 0 1 5

Zol(Asp, 1) = Zot(Asp,2) Zct(Asp, 1) Zoo(Asp,2)

19 "t Au_94 Zr

~alTa_94Zr

lOgAg_94Zr

9 1 A u _ t o 9 M o

197Au_6aZn

10.8

11.4

12.7

15.0

17.0

17.4

26.2

21.4 24.2

27.3

13.1

13.8 15.4

25.7

19.2

of monitors in the bare method. However, it should be emphasized that such an

improvement can only be obtained after careful selection of the monitor isotopes,

and especially of the reference monitor. Even then, the uncertainties on a remain

rather high, so that the comments and suggestions, outlined in the conclusion of the "bare triple monitor"-method, remain valid.

General conclusion

The error analysis, together with the corresponding considerations and comments, as outlined in this paper, should allow a judicious choice of methods and

of activation detectors for or-monitoring in a 1/E t+a-epithermal neutron spectrum.

Although numerical examples were mostly worked out for the combination

Au-Zr , which was tested extensively in the Thetis reactor and the WWR-M reactor,

other monitor sets may be used, giving also results with acceptable or even better

precision and accuracy. However, it should be emphasized that, even with otherwise suited characteristics, extreme care should be taken about the choice of ac-

curate Qo-values, which have to be introduced in the expressions for ~-calculation.

As a guide, Tables 13 and 14 give a listing of some possible combinations sug-

gested for use in the "Cd-covered and Cd-ratio for dual monitor"-methods, or in the "bare triple monitor"-method, respectively. The nuclear data of interest can be found in Table 1. In Tables 13 and 14, as an indication only Za(Asp)-values

are tabulated for channel 3 of the Thetis reactor, to facilitate more easy selection of the suitable monitor pairs.

In practice, a-determination is required mainly in two important fields:

1. Calibration of an irradiation channel for nuclear data measurements [I0 (a = 0), Q0(ot = 0), resonance parameters etc.]



Table 14 Some possible a-monitors suggested for use in the "Bare triple monitor"-method

(Nuclear data: see Table 1)

Combination (t)-(2)-(a)

I 9 7 A u _ 9 4 Z r _ 9 6 Zr

t o 9 A g _ 9 4 Z r _ 9 6 Zr

1 a t T a _ 9 4 Z r _ 9 6 Zr

7 S L u _ 9 4 Z r _ 9 ~ Z r

t a l S b _ 9 4 Z r _ 9 6 Zr

t 2 ~ S b _ 9 4 Z r 9 6 Z r

4 , S c _ t ~ S L u _ 9 6 Zr

SOCr _ t 8 t T a _ 9 6 Zr

t s t T a _ 6 8 Z n _ 9 6 Zr

t 9 7 A u t t 2 S n _ 9 4 Z r

"Bare triple monitor"-method; f = 25; a = 0.015

Zct(Asp,l) Za(Asp,2)

36.5 26.3 43.1 29.7 28.8 13.6 29.2 13.4 31.9 15.2 42.5 22.4 12.0 31.8 12.3 31.5 31.2 13.7 45.9 19.9

Z~(Asp,3)

10.1 13.5 15.2 15.8 16.7 20.1 19.8 19.2 17.6 26.0

2. Performing reactor-neutron activation analysis without standards (e.g. using

the ko-standardization method).

The former case requires the most accurate a-determination possible. Therefore

the "Cd-covered multi a-monitor"-method or the "Cd-ratio for multi a-monitor"-

method should be recommended for use. When (n, 7) reactor neutron activation

analysis is concerned, less stringent a-determination is adequate. For most analytical cases the "bare triple monitor"-method using the t 9 7Au 96 Zr 94 Zr set provides

acceptable precision and accuracy. Some a-determination methods are extremely

sensitive to the accuracy of the nuclear data used. Therefore the use o f ko-factors

and of carefully selected Q0, Er, etc. nuclear parameters is strongly recommended.

Note that in some reactors and especially for long irradiation periods, a Cd-cover

is not allowed to be irradiated (melting of Cd, strong neutron flux perturbation,

etc.). In such case the only possibility for a-determination is to apply the "bare

triple or multi monitor"-method.

Grateful acknowledgment is made to the "Nationaal Fonds voor Wetenschappelijk Onder- zoek" and to the "Interuniversitair Instituut voor Kernwetenschappen" for financial support (F.D.C.).

254 .i. Radioanal. Chem. 62 (1981)


References

1. F. DE tORTE, L. MOENS, K. SORDO-EL HAMMAMI, A. SIMONITS, J. HOSTE, J. Radio- anal. Chem., 52 (1979) 305.

2. F. DE tORTE, L. MOENS, A. SIMONITS, A. DE WISPELAERE, J. HOSTE, J. RadioanaL Chem., 52 (1979) 295.

3. L. MOENS, F. DE CORTE, A. SIMONITS, A. DE WISPELAERE, J. HOSTE, J. RadioanaL Chem., 52 (1979) 379.

4. R. W. STOUGHTON, J. HALPERIN, Nucl. ScL Eng., 15 (1963) 314. 5. T. BEREZNAI, G. KE~SMLEY, Radiochem. Radioanal. Lett., 17 (1974) 305. 6. L. MOENS, F. DE CORTE, A. SIMONITS, A. DE WISPELAERE, A. ELEK, J. HOSTE, J.

Radioanal. Chem., to be published. 7. A. SIMONITS, F. DE CORTE, J. HOSTE, J. Radioanal. Chem., 31 (1976) 467. 8. F. DE CORTE, A. SIMONITS, L. MOENS, J. HOSTE, J. RadioanaL chem., to be published. 9. L. MOENS, F. DE CORTE, J. HOSTE, A. SIMONITS, J. RadioanaL Chem., 45 (1978) 221.

10. A. SIMONITS, L. MOENS, F. DE CORTE, J. HOSTE, A. DE WlSPELAERE, A. ELEK, Pre- limin. Rept. 5th Symp. Recent Development in Activation Analysis, Oxford, 17-21 July 1978.

11. A. SIMONITS, L. MOENS, F. DE CORTE, A. DE WlSPELAERE, A. ELEK, J. HOSTE, J. Radioanal. Chem., 60 (1980) 461.

12. L. MOENS, A. SIMONITS, F. DE CORTE, J. HOSTE, J. RadioanaL Chem., 54 (1979) 377. 13. A. SIMONITS, L. MOENS, T. EL NIMR, F. DE CORTE, J. HOSTE, J. Radioanal. Chem.,

to be published. 14. T. B. RYVES, Meteorologia, 5 (1969) 119. 15. L. MOENS, J. DE DONDER, LIN XILEI, F. DE CORTE, A. DE WlSPELAERE, J. HOSTE,

A. SIMONITS, NucL Instr. Meth., in press. 16. G. S. STANFORD, USAEC Rept., ANL-7010 (1965) 203. 17. T. B. RYVES, K. J. ZIEBA, J. Phys. (A), 7 (1974) 2318. 18. O. T. HOGDAHL, Pro~ Symp. Radioehemical Methods of Analysis, Salzburg, October

19-23, 1964, Vienna 1965, p. 23. 19. C. H. WESTCOTT, W. H. WALKER, T. K. ALEXANDER, Proc. Intern. Conf. Peaceful Uses

of Atomic Energy, 16 (1958) 70. 20. R. W. STOUGHTON, J. HALPERIN, Nuel. Sci. Eng., 6 (1959) 100. 21. A. AHMAD, T. D. MAC MAHON, M. MACWANI, M. J. MINSKI, J. G. WILLIAMS, T.

BEREZNAI, "NAA without Multi-element Standards", Proc. Conf. "Nuclear Methods in En- vironmental and Energy Research", Columbia, Missouri, April 1980 (to appear).

22. E. M. GRYNTAKIS, J. I. KIM, Radiochim. Acta, 22 (1975) 128. 23. P. SCHUMANN, D. ALBERT, Kernenergie, 2 (1965) 88. 24. A. SIMONITS, L. MOENS, F. DE CORTE, J. HOSTE, J. Radioanal. Chem., to be published. 25. N. P. BAUMANN, Rept DP-817, 1963. 26. C. R. 'PIERCE, D. F. SHOOK, Nucl. Sci. Eng., 31 (1968) 431. 27. L. M. HOWE, R. E. JERVIS, T. A. EASTWOOD, Nucl. Sci. Eng., 12 (1962) 185. 28. IAEA Technical Reports Series No. 107, "Neutron Fluence Measurements", Vienna 1970.


The accuracy and precision of the experimental α-determination in the 1/E 1+α epithermal...

Documents

Transcript of The accuracy and precision of the experimental α-determination in the 1/E 1+α epithermal...