The results given in Table I11 indicate that both a reactor and an electron linear accelerator provide excellent detection capabilities for potassium in the absence of interference, the reactor being somewhat better. The importance of Equation 22 for detection limit estimation in the presence of interference, however, becomes quite clear upon examination of the last column of Table 111. Here, the detection limit by bremsstrahlung irradiation has become about one thousand times poorer, and thermal neutron activation has become practically useless.
ACKNOWLEDGMENT
Helpful discussions have taken place with a number of my colleagues, in particular: J. R. DeVoe, J. M. Cameron, H. H. Ku, and K. F. J. Heinrich. RECEIVED for review September 21, 1967. Accepted December 15, 1967. The paper was presented in part at the Eighth Eastern Analytical Symposium held in New York, November 1966.
Graphical Technique for Estimating Activity Levels Produced in Thermal- and Fission-Neutron Irradiation Jorma T. Routti Lawrence Radiation Laboratory, University of California, Berkeley, Calq. We describe a rapid and flexible graphical method for estimating the activity induced in neutron irradiations. This method, applicable to several activation problems, is applied here to thermal- and fission-neutron irradiations, which are widely used in activation analysis with reactors. The calculated saturation activities of the products of most activation reactions are represented on two activation charts. We develop a graphical technique, using a transparent overlay, to obtain corrections for the saturation, decay, and counting factors. The princi al advantage of this method is that the corrections f u e to these factors can be applied to any saturation activity of the chart, which already contains all the nuclear information required for estimation of induced radioactivity. Most simple activity calculations can be performed quickly and accurately; the technique is also useful in estimating the best irradiation and decay times to enhance selected activities in composite samples. We give the numerical values for the construction of charts of saturation activities for thermal- and fission-neutron irradiation and for the overlay. Calculation techniques are explained and clarified with typical examples.
THECALCULATION of induced radioactivity is required in many activation problems. Generally, the same mathematical formalism is applied, with different values of nuclear constants and time parameters. The search for nuclear constants and the solution of equations for composite samples is often timeconsuming. We here develop a graphical method for representing nuclear data and solving activation equations; this method can be successfully applied to several activation problems, flux monitoring, and radioisotope production with reactors and accelerators. (The two activation charts and the transparent overlay are available as a courtesy from General Radioisotope Processing Corp., 3120 Crow Canyon Road, San Ramon, Calif. 94583.) The specific applications to thermal- and fission-neutron activation have been chosen because of their wide and frequent use in activation analysis with reactors. Several nomographs for calculating induced radioactivity have been published (1-3). To use most of these, it is first (1) Philip A. Benson and Chester E. Gleit, Nucleonics, 21, No. 8 148 (1963). ( 2 ) E. Ricci, Nucleonics, 22, No. 8, 105 (1964). (3) Edward C. Freiling, Nucleonics, 14, No. 8,65 (1956).
necessary to find the nuclear parameters in the literature Concise summaries of sensitivities for thermal-neutron activation have also been published (4, 5). These generally apply to selected counting and irradiation conditions. The techniques described here combine, in compact form, the essential nuclear data with a quick and flexible graphical calculation method. The various factors appearing in the activity equation, including the saturation, decay, and counting-time corrections, can easily be determined. Also, qualitative information of optimal irradiation and decay times, which are often desired in activation analysis, can be obtained graphically rather than by using numerical computer techniques (6). GRAPHICAL METHOD FOR COMPUTATION
The Activation Equation. In calculating induced radioactivity, we assume that the irradiated sample is so small that it produces no significant attenuation of the flux of incident particles. The total disintegration rate of a radioisotope produced by irradiation of an element of natural isotopic composition in a constant flux is then given by A
=
100 M
- exp(-ln
2
x tJTl~z)Jexp(-ln2 x td/Tllz>= A,,,SD = N* In 2/T1/2 (1)
where rn is the mass of the irradiated element,fis the isotopic abundance of the isotope that undergoes the reaction, N o is Avogadro's number, q5 is neutron flux, u is the cross section, M is the atomic weight of the irradiated element, is the half-life of the induced activity, ti is the irradiation time, td is the decay time (between the end of irradiation and
(z)
the beginning of counting), A,,, is ,* ' the saturation 100 M activity, S is [(l - exp (-ln 2 X ti/Tlis)],the saturation factor, D is exp (-In 2 X td/Ti,P),the decay factor, and N* is the number of decaying nuclei. (4) W. Wayne Meinke, Sensitivity Charts for Neutron Activation Analysis, ANAL.CHEM., 792 (1959). ( 5 ) Russell B. Mesler, Nucleonics, 18, No. 1, 73 (1960). (6) Thomas L. Isenhour and George H. Morrison, ANAL.CHEM., 36, 1089 (1964). VOL 40, NO. 3, MARCH 1968
593
l6Qm
log[ to8
p
rnc
- I0lu
Q) v)
\
-I t
.-c ." -0
55p
106
1
>r
.t
104
Figure 1. Activity chart for saturation activities induced by thermal neutron flux of 10l2 n/cmz second in a sample of 1 mg of an element of natural composition We obtain measured disintegration rate by multiplying A by the fraction of counted decays, the detector efficiency for measured radiation, the geometry and dead-time factors, the self-absorption factor, and other factors that may affect the counting. We combine all these effects by defining the total efficiency factor E as the ratio of the measured disintegrationrate to the total disintegration rate. This efficiency factor must be estimated separately for each specific set of counting conditions. The measured-count rate is affected by one more term that becomes significant only when the counting time is not small in comparison with the half-life of the radioisotope. The counting factor C, due to the decay of a sample during the counting interval, is given by
C
=
11 - exp(-ln 2 X tc/Tl;lz)l/(ln2 X tc/Tld
(2)
where te is the counting time. The values given later for C for different ratios'fC/TLIP will indicate when it is necessary to apply this correction factor. The measured-count rate A , is now given by
A,
= A,a,SDCE
(3)
For a given sample, flux, and counting system, Asaland E are constants. The factors S , D , and C depend only on the ratio t/TIIz,where t refers to the times of irradiation, decay, and counting, respectively. 594
0
ANALYTICAL CHEMISTRY
Activity Charts. To develop our graphical method we next take logarithms of Equation 3. The addition of the various terms in the equation log A ,
=
log Am,
+ log S + log D + log C + log E
(4)
can be performed graphically if we choose an appropriate coordinate system. This will be a rectangular logarithmic scale, which we will call the activity chart. Its ordinate is called the activity axis; its abscissa is the time axis. If we use fixed values for flux and sample weight, there will be a unique point on the chart corresponding to the saturation activity of each reaction product studied. The location of this point is determined by the saturation activity on the vertical axis and the half-life of the radioisotope on the horizontal axis. Furthermore, the multiplication by S, D , and C can be performed on this scale through the techniques of logarithmic addition described below. We have calculated the saturation activities of most radioisotopes of practical interest produced in the thermal-neutron activation of 1 mg of each of the natural elements in a flux of 10%/cmZ second. The nuclear data required for these computations (a,f, M , T1,z) (3, as well as the calculated (7) David T. Goldman, Chart of the Nuclides, Knolls Atomic Power Laboratory, 9th Edition, July 1966, General Electric Co., Schenectady, N. Y.
Table I. Saturation Activities Induced by Thermal Neutron Flux of 10%/cm2 Second in a Sample of 1.0 mg of an Element of Natural Composition
Induced isotope
F NA IYG AL
20
s
37
24 27 28 SI 3 1 P 32 s 35
CL 3 8 A Q 137 A? K CA CA SC
41 42
v
52
G5
49 46 TI 5 1
C? 5 1
c'i 5 5
56 FE 5 5 FE 5 9 C 9 60
:bird
NI 6 5 CU 54 C ' J 56 ZN 6 5
zq
GA 5.4 GE ,4S SE 89 3? R3
59Y 70 72 75
76 75 813M 82 86
23 38 SR 8 5 SR 0 9 ZR 9 5 ZR 9 7 NR 9 4 MO 9 9 Mol0 1 RU 97 RU193 RU105 ii H 104 M
VOL. 40, NO. 3, MARCH 1968
0
595
~~
~
Table I. (Continued; Induced isotope
Cross section in barns
Rd104 P31:33
?Dl09 .4G108
AGllOM
AGllO CC115M
IN114 I N 116 M s b! L 2 1 SY123 s Id 12 5 53122 59124 TE129 TE131 I 123 CS134;Y CS134 9A139 LA 148'3
CE141 CE143 PR142 ,\I 3 14 7 l\lD14?
94153 EL1152 ElJ152 E!J154 69159
69161 T3163 DY 165M H3166 59.1.53 ER171 Ti4 1 7 3
L'J177 YF175 HFl!3l T.4152 'bi 1s 5 \A!
187
RE185 RE183 3 S 1 ? 1'4 OS191 (Continued)
596
ANALYTICAL CHEMISTRY
Half life
Induced isotope
IR192 IR194 PT197 ?TI99 A!?198
HS197 HG2103 FIG235 TL2r34 TL2.36 P.9 2 (39
74.0 19.0 23#'3 31.C 64.8 55.C 46.6 5.2 3.75
E31219
4a3 3.3 509
TH233 PA232 u 239
2241 1.32 23,s
DAYS HOURS HOURS MINUTES H0Uli.S HC)!JqS DAYS 7.1 I Ntr TES YEARS ?,I I NUTES 3OURS DAYS MINUTES DAYS YINUTES
Table I. (Continued) Half life in minutes
Saturation actitity
1&'>665 +05 1s143E+93 1&2C3E+C3
I@ 1SS;f+39 2rl615+38 7 I 0 3c E t 5 5 8.903E+r)5
3.888E+33 3.903E+O3 Sr71@E+C)4
5@293E+'23 l a 97 If -1-36 4*39'3E+O3 119 3 ? E +r32 702C!OE+33 2. 2 1 0 4 + 0 1 I, I 90 1 E +03 2 r 3 5 ? E +03,
Time
3 r 021E+39 3 I 9 6 7E+9 6 3 I 5 7?'E+C>6 5122?E+r)4
.
9r563E+.t)6 2 285E+:! 5
715C2E+!32 4r323E+34 1r9215+37 5 6 2 1 5E+O8 6 a 7 8 2E+O5
(min)
Figure 2. Activity chart for saturation activities induced by fission neutron spectrum of lo1*n / m 2 second in a sample of 1 mg of an element of natural composition VOL 40, NO. 3, MARCH 1968
0
597
Half life
14e3 2e52 14.3 96.7 5e1,
12r4 83e8 ?e4 153e 44e 4e5
440 3120
Table 11. Saturation Activities Induced by Fission Neutron Spectrum of Threshold in meV
DAYS MINUTES DAYS DAYS '4IVJTES HOU:?S 3AYS DAYS DAYS ';-OURS DAYS !dOURS DAYS
3e95 lr24
-:e92 -(a062 3eG 9.61 1.61 -9 0 3 9 6 2r02 3028 3058 2814 10033
saturation activities and half-lives in minutes for each reaction product, are listed in Table I. The saturation activity and the half-life in minutes for each isotope are also given by the location of the corresponding point on the activation chart in Figure 1. Thus the location of an isotope coordinate on the chart already contains all the nuclear information required for calculation of induced activity. Another chart useful in estimating the effect of interfering reactions has been made for common (n, p), (n, 2n), and (n, CY) threshold reactions induced by fission neutrons. The effective cross sections defined for total flux-density of fission neutrons of all energies and compiled in Reference (8) were used for these computations. The saturation activities of the products of these reactions are given in Table I1 with corresponding nuclear data; the activity chart is shown in Figure 2. Overlay for the Correction Factors S, D,and C. In the activity charts' coordinate system, the saturation, decay, and counting factors can be represented by curves that can be used conveniently with the charts. It is apparent from the defining equations that the factors S, 0,and C depend only upon the ratio t/Tl,z; therefore, we can construct curves for (8) Robert S.Rochlin, Nucleonics, 17, No. 1, 55 (1959).
598
ANALYTICAL CHEMISTRY
lo1*
I 3Pn - 0 I II m I - ' ' I -2 -
-3
-
I I I I I I 1 I I
1 I
I I
I
I
I
I
I
I I I I
I I I I
I
these functions with the value of the function as ordinate and the ratio t/Tl,z as abscissa. Furthermore, since the time axis is logarithmic, the ratio t/T1lz corresponds to the distance log t - log 7 ' 1 , ~ only, and the same curves become universally applicable to any point on the activity chart. Let us consider, for example, the saturation factor S. We plot the logarithm of S as
n/cm2 Second in a Sample of 1.0 mg of an Element of Natural Composition
for a fixed half-life, say T l / ~as' a function of log tt. The resulting curve a in Figure 3 has the same form for any other fixed half-life, say T1/2"; the curve will only be shifted along the time axis to a new position b. A curve of the same shape can thus represent log S as a function of log t i for any fixed Tllz. Now consider log S as a function of the half-life for a fixed irradiation time, say t i ' , represented by curve c in Figure 3. It is again evident that the same curve, when shifted along the time axis, can represent log S as a function of log T1/zfor any other fixed it. Assume that the fixed irradiation-time, say ti!', Then it can be seen that the curve is set equal to T ~ / zabove. ' d, representing log S = f(1og T1lz),is a reflection of curve a ; that is, log S = f(1og t,). The factor S clearly has the same value if we write log tr - log T112' = log tr" - log Tl/Z = -(log
Tl/2
- log t i " ) (6)
which implies a reflection with respect to the vertical line
through log t i v = log TI,^'. Thus log S can be represented by a curve of the same shape in all the cases considered above, In the coordinate system of the activation charts, the saturation factor can thus be represented by a unique curve which we have drawn on an overlay. This curve can be used to deterasZa function of f i by moving it in the mine S for any fixed T I ~ direction of the time axis, and for any fixed t r as a function of TI/Z by turning it over. Similarly, there are unique curves for the decay and counting factors that, when drawn on the same overlay, can be used in the same way as the saturation curve. We have calculated the values of S , D,and C for different values of t/Tl/2and have listed these in Table 111. The curves corresponding to factors are drawn on a transparent overlay as indicated in Figure 4. A common marking line, called the centerline, corresponding to the ratio t/Tllz= 1, for all three curves has been drawn on the overlay; this makes it compatible with the time scale of the charts. The asymptotic line of the curves on the overlay represents the value 1 of S , D,and C. If the overlay is placed on the chart so that the S-curve has a positive slope, the asymptotic line coincides with the ordinate of a given saturation activity, and the centerline with the halflife of an isotope in question, then the activity corresponding VOL 40, NO. 3, MARCH 1968
599
Table 111. Saturation, Decay, and Counting Factors for Induced Radioactivity and Decay Time/half life Saturation factor Decay factor Counting factor
.
1 9 3 12
600
0
-iv NoGT.4
ANALYTICAL CHEMISTRY
6.93E-( N+1) l r O i J G E + ( J O
Io
-~
I d4
I
f3
16'
I.o
10.0
Io2
I o3
L.
0 t
0 0
IL
TI/, 1 t Figure 4. Overlay showing saturation factor S, decay factor D, counting factor C, centerline CL, and lines L and L X 10relating activity to the number of decaying nuclei to irradiation-time t can be directly read at the ordinate of the S-curve at time t . Decay or counting-factor corrections can be determined in a similar manner. If the overlay is placed on the chart so that the S curve has a negative slope and the asymptotic line coincides with a decade line of the activity axis, then the centerline marks a fixed irradiation, decay, or counting time and the factors S, D,and C for different halflives can be read at the ordinate of the respective curve at time T. The decade tickmarks on the centerline may be used to overcome difficulties in scaling when reading small activities. For simple multiplications which we included in the total efficiency factor E, a logarithmic scale compatible with the charts' activity-scale has been added to the side edge of the overlay. This can be used as an ordinary slide rule and it is useful in making corrections when the values of the flux and the sample weight differ from those used in calculating the saturation activities on the activation charts. The number of decaying nuclei N* is related to the activity through the expression
N* = ATi,2/ln 2
(7)
In logarithmic terms this yields log N* = log A
+ log Til2 + log Ci
(8)
which is represented by the straight line L on the overlay. The constant Ci is determined so that the L line is compatible with the centerline of the overlay and the left activity scale of the charts; the L line has a slope - 1 and it crosses centerline at 0.866. If we now place the overlay on the appropriate chart so that the top of the centerline coincides with the activity point in question, then the number of decaying nuclei can be read at the intersection of the left activity scale and the L line or its extension. Another line drawn parallel to L six decades lower may be more convenient to read N* 10-6. Resolution of the points on the charts and of the curves on the overlay permits the estimation of activities within 5 to 15 %. This accuracy is generally adequate when compared to the relatively large error-limits which must be assigned to the values of the cross sections and the flux. The accuracy of the determination of induced radioactivity should be estimated for each specific case, using the uncertainties in the values of the cross section and the flux, which often are even orders of magnitude. Quantitative analysis is almost always based on the comparison of unknown and standard samples, by which these uncertainties can be largely avoided, and a more accurate calculation for saturation, decay, and counting corrections may be necessary. Even then, our method is useful in estimating activity levels and determining irradiation conditions. VOL 40, NO. 3, MARCH 1968
601
COMPUTATION PROCEDURES AND EXAMPLES
To determine the activity induced for a given set of irradiation and decay conditions, set the overlay on the appropriate chart so that the S-curve has a positive slope and the top of the centerline coincides with the ordinate of the isotope considered. Read the activity after the chosen irradiation time at the ordinate of the S-curve corresponding to the time t,. Next, move the overlay down vertically until the top of CL is at this activity level. Read the activity after the decay time Finally, t d at the ordinate of the D-curve corresponding to t d . correct for efficiency, sample weight, and flux, using the logarithmic scale on the edge of the overlay as a slide rule. This now yields the count rate of the sample after the chosen irradiation and decay times. Multiplication by the counting factor, if required, is performed by using the C-curve and the time-line t,. The induced or measured activity in a given sample for any set of experimental irradiation conditions can thus be easily calculated, using only the chart and the overlay. The amount of an element that gives rise to a measured activity, or any other unknown in the activity equation, can also be determined. As mentioned before, the saturation and decay curves are also useful in determining optimal irradiation and decay times in cases where additional or competing reactions may obscure the activity of interest. For this purpose, place the overlay so that the saturation curve has a negative slope and compare the saturation and decay factors for competing reactions for different irradiation and decay times by setting the top of the centerline at the corresponding points on the chart. An iterative procedure will generally yield a quick solution that meets the requirements for absolute and relative activities. The following examples illustrate the use of the charts and the overlay. Example 1. Calculate the activities of copper-64 (T,/z = 12.9 hours) and copper-66 (TlI2 = 5.1 minutes) induced in a sample of 0.5 mg of natural copper, irradiated in a flux of 1012 n/cm2 second for 3 minutes followed by a decay time of 10 minutes. What is the additional decay time required to reduce the copper-66 activity to 1 % of the copper-64 activity? First find the copper-66 point on the thermal-neutron activation chart, looking at the time line of 5.1 minutes. The saturated activity is 6.7 X 106. Use the overlay so that the slope of the saturation curve is positive. Set the overlay on the chart so that the top of the centerline is at this point. Read the induced activity at the intersection of the saturation curve and the time line of 3 minutes. It is 2.3 X 106. Next, set the top of the centerline on this level, keeping the center1 ne at the 5.1-minute time line. Read the activity after the decay time at the intersection of the decay curve and the ti.ne line of 10 minutes. It is 5.8 X lo5. Set the top of the vertical scale of the overlay at this level and multiply the activity of the sample by 0.5. The copper-66 activity in disintegrations per second is thus 2.9 X lo5. The copper-64 activity is determined in the same way after locating copper-64 at the half-life line of 12.9 hours = 774 minutes. It is 4.0 X 104 disintegrations per second. To find the additional decay time required to decrease the Asscu to ljl00 AB'^^, first assume that the latter remains constant. Set the overlay so that the centerline coincides
602
ANALYTICAL CHEMISTRY
with the time line of 5.1 minutes and the second highest tickmark is at AWC,= 2.9 X lo6. Read the decay time on the decay curve at the activity level Asrcu = 4.0 X lo4. It is 50 minutes for which the assumption that Asdcu remains constant is justified and no correction is required. Example 2. Compare for different half-lives the activities induced by a reactor pulse of 250 MW X 13 msecond with the saturation activities reached at a constant power level of 100 kW. What are the counting factors corresponding to 3seconds counting time for the same half-lives ? Position the overlay so that the slope of the saturation curve is negative. For convenience, set the asymptotic line of the overlay at the activity level of 2.5 X 10s (corresponding to 2.5 X 10s W = 250 MW) and the centerline at 13, assuming that the time scale is in mseconds. Read the relative activities compared with the activity level of lo5 (corresponding to 105 W = 100 kW) at the intersections of different time lines and the saturation curve. To get the counting factors, set the centerline at 3 X l o 3(corresponding to 3 seconds) and the top of the curve at a decade line. Read the counting factors at the intersections of different time lines and the counting curve. The results can be directly read:
Tli2(seconds)
0.01
0.1
1 . 0 10
22
60
220
Apu1a6(250MW)1500 210 22 2 . 2 1 . 0 0.38 0.10 A d 1 0 0 kW) Counting factor 0.0048 0.048 0.42 0.90 0.95 0.98 1 .O ACKNOWLEDGMENTS
The cooperation and support of S. G . Prussin and R. W. Wallace of Lawrence Radiation Laboratory as well as encouragement by Pekka Jauho of Technical University of Helsinki are gratefully acknowledged. RECEIVED for review August 2, 1967. Accepted December 13, 1967. This work was done under the auspices of the U. S. Atomic Energy Commission.
Correction Quantitative Infrared Multicomponent Determination of Minerals Occurring in Coal In this article by Patricia A. Estep, John J. Kovach, and Clarence Karr, Jr. [ANAL.CHEM.,40, 358 (1968)l the frequency of the analytical absorption band for pyrite should read 41 1 cm-l, instead of 406 cm-', in Tables I and 11, and Figure 2. The correct value of 41 1 cm-l was given in reference (2) by Karr, Estep, and Kovach [Chern. Znd. (London), 9, 356 (196711. Also, the value for marcasite iq Table I should read 412 cm-' instead of 407 cm-'.
