Journal of Nuclear and Radiochemical Sciences, Vol. 2, No.1-2, pp. R1-R4, 2001

Reviews
Accuracy and Uncertainty in Radiochemical Measurements: Learning from Errors
in Nuclear Analytical Chemistry
Richard M. Lindstrom* and Robert R. Greenberg
Analytical Chemistry Division, National Institute of Standards and Technology, 100 Bureau Drive, Stop 8395
Gaithersburg, Maryland 20899-8395, USA
Received: November 2, 2001; In Final Form: November 16, 2001
A characteristic that sets radioactivity measurements apart from most spectrometries is that the precision of asingle determination can be estimated from Poisson statistics. This easily calculated counting uncertainty permits
the detection of other sources of uncertainty by comparing observed with a priori precision. A good way to test
the many underlying assumptions in radiochemical measurements is to strive for high accuracy. For example, ameasurement by instrumental neutron activation analysis (INAA) of gold film thickness in our laboratory revealed
the need for pulse pileup correction even at modest dead times.
Recently, the International Organization for Standardization (ISO) and other international bodies have formalized
the quantitative determination and statement of uncertainty so that the weaknesses of each measurement are
exposed for improvement. In the INAA certification measurement of ion-implanted arsenic in silicon (Standard
Reference Material 2134), we recently achieved an expanded (95 % confidence) relative uncertainty of 0.38 % for
90 ng of arsenic per sample. A complete quantitative error analysis was performed. This measurement meets the
CCQM definition of a primary ratio method.

Knowledge increases...not by the direct perception of truth but
by a relentless bias toward the perception of error.
- K. E. Boulding, Science 207, 831 (1980).
Introduction
Information is essentially surprise, something that was notpreviously known. Occasionally an analytical chemist can tell
the person who submitted the samples something that he did
not expect, or even does not believe. For instance, in our laboratory a sample of superconducting graphite intercalated withrubidium was found to contain potassium instead. When acustomer is presented with an unforeseen result, it is sometimes comforting for him to decide that the analyst has made ablunder such as mixing up the samples. It is the analyst’s
responsibility to design her procedure and write her report soclearly that the customer must be convinced that the analysis
was done without mistakes (which is not, of course, without
error or uncertainty). It is this integrity of the chemicalmeasurement process that quality systems, audits, and interlaboratory comparisons attempt to insure. Beyond this basic
assurance of reproducibility and reliability, accuracy isimproved and we become more confident of the results if ourprocedures are tested by stress.
Error Detection and Minimization
Quality management systems require a complete anddetailed written procedure, which ensures that the measurement will be done reproducibly. However, aside from theimpossibility of repeating a measurement every time underexactly the same conditions it is not always desirable to try to
do so. For example, because one activation analyst always
carefully duplicated his irradiation and decay parameters, a
decaying interference could not be found until another person
repeated the measurement with less rigorous reproducibility.
*Corresponding author. E-mail: richard.lindstrom@nist.gov.
FAX: +1-301-208-9279.

For robustness, it is desirable to vary parameters to test thesensitivity of the result to the variables, and build that requirement for variation into the protocol. The way to confidence isthrough demonstrated accuracy.
You never know what is enough until you know what is morethan enough - William Blake
These points are illustrated by several attempts in our laboratory to perform neutron activation analysis (NAA) with
better than usual accuracy.
1. A set of gold films of different thicknesses were evaporated on a silicon substrate to provide working standards for
Rutherford backscattering. Instrumental neutron activation
analysis (INAA) was used to measure the quantity of gold on
the wafer.1 In a blind comparison, INAA agreed with independent gravimetric measurements to within 0.3 % on average. In
order to obtain such agreement, several potential sources ofbias were tested. To examine the effect of resonance neutron
self-shielding, a sandwich was made of a 5-mm diameter gold
foil between two 10-mm foils. The package was irradiated andthe foils counted individually. Surprisingly, the apparent
specific activity of the inner foil was larger than the outer foils.
Because the outer foil had four times the mass of the inner, thecounting rate was four times higher, and the random coincidence (pulse pileup) correction2 correspondingly more important. This experience led to the routine incorporation of pileup
correction for all NAA counting at NIST.
Neglect of pileup at low count rates was not confined to ourlaboratory. At about the same time, the International Atomic
Energy Agency (IAEA) organized the G2 intercomparison
exercise to test the adequacy of dead time and pileup corrections. 3 The task of the participants was the simplest
metrology: to determine the activity, relative to a reference
source counted at 1000 counts/s, of four radioactive sources up
to 15 times the activity of the reference. Ninety-eight sets ofresults were submitted by laboratories in 24 countries. The
results were sobering: for the most active source the median© 2001 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 12/28/2001

R2

Lindstrom

J. Nucl. Radiochem. Sci., Vol. 2, No. 1-2, 2001

error was 6 % and the maximum error was 36 %.
2. Perhaps the most precise and accurate gamma-ray spectrometry ever published was performed in the calibration ofreference materials for nuclear safeguards, certified for 235U
enrichment. 4 The sample-to-sample variations in gamma
assays within each of five sets of 25 samples was 0.06 % - 0.13
% relative standard deviation (rsd), and the uncertainty of relative measurements was comparable with the best mass spectrometry. A significant source of uncertainty was the variationin the peak area as detected by the peak search program. In
order to obtain this precision, a fixed-boundary summation 5
was used in preference to a Gaussian fit to the spectral peaks.
In this work the pileup correction algorithm was shown to beadequate at this degree of precision over a wide range ofcounting rate, from 4 % to 22 % dead time.
Spectrum analysis programs are an often-overlooked sourceof uncertainty in gamma spectrometry.6 Intercomparisons ofdifferent algorithms have always shown differences in peakdetection capability and net area determination, and reveal theimportance of user-selected parameters. 7, 8−10 Overlapping
peaks present a particularly rich field for errors and choices ofalgorithm.11, 12
3. The publication of a high-precision value for the half-lifeof 28Al (Ref. 13) permitted a high-accuracy INAA determination of aluminum as a major element in coal fly ash. Because
the half-life is only 2.2 minutes, precise timing and correct
calculation of the combined effects of pileup in the amplifier,dead time in the multichannel analyzer, and radioactive decay
in the source were crucial. 14, 15 The method was tested byanalyzing single-crystal sapphire (Al 2 O 3 ); the measured
aluminum concentration was within 0.2 % of the theoreticalstoichiometry.
Later examination revealed a pattern in the results. Whenthe sapphire sample was irradiated between two standard Al
foils the apparent concentration was slightly higher than if theinverse sandwich was irradiated. Experiments designed to test
the idea that neutron scattering was important in this well-thermalized, near-isotropic neutron irradiation, as it certainly is forneutron beam experiments, 16 have made this explanation
unlikely.17
4. INAA was used for certification of NIST Standard
Reference Material (SRM) 2134, consisting of 90 ng of arsenic
implanted in 1-cm2 silicon samples. For this measurement acomplete quantitative analysis of all components of uncertainty
was done.18 The expanded relative uncertainty was U = 0.38%, calculated using a coverage factor of 2, which gives a levelof confidence of approximately 95 %. The largest single
contributor to the overall uncertainty was the uncertainty of theconcentration of the reference standards. As discussed below,
this measurement can serve as a prototype for INAA as a
primary method of analysis. The potential of NAA as a reference technique for As in Si has also been suggested byothers.19
In the course of these arsenic measurements, an apparent
systematic error was found in the most recently publishedvalue of the 76As half-life, so an earlier consensus value wasused for the computations in that work. In a subsequent redetermination of this half-life,20 it was inferred that the author of
the recent value may have used a biased procedure for fitting
decay curves. From this we learned that even the mostrecently published and professionally evaluated nuclear data
may still not be sufficiently reliable for the most accurate
work.
Uncertainty Evaluation and Primary Methods
A growing influence on analytical chemistry is the Guide to
the Expression of Uncertainties in Measurement, published by
the International Organization for Standardization: the ISO

21

GUM. Written to facilitate legal metrology and trade, thisstandard requires that all sources of uncertainty be examinedand quantitatively specified, as a consequence of which thegreatest weaknesses of the procedure are exposed for improvement. Supplementary guides to the implementation have been
written by Eurachem/CITAC22 and NIST.23 A guide specifically directed toward nuclear methods is in preparation by the
IAEA. The application of these principles to NAA24 has beenaddressed, and to the k0 standardization method in particular.25
These guides give numerical examples of the application of the
GUM principles to specific cases.
The algorithm for uncertainty assessment is easily describedmathematically: simply determine the uncertainty in each parameter in the measurement equation and compute the degree towhich that uncertainty affects the final result. In practice, ofcourse, for most analytical methods it is tedious to compute thenecessary partial derivatives analytically. However, a spreadsheet approach has been proposed25, 26 that performs differentiation numerically, and displays the influence of each factor in
an instructive way.
For the arsenic in silicon SRM described above,18 twentynine sources of uncertainty were quantitatively evaluated, as
shown in Table 1. Of these, six (in bold) accounted for 95 %of the relative combined uncertainty and only two, the reproducibility of replicate samples and the concentration of thestandards, accounted for 70 %.
The Consultative Committee on the Quantity of Material(CCQM) of the Bureau International des Poids et Mesures(BIPM) has described a primary method of measurement as "...
TABLE 1: Enumeration of Sources of Uncertainty for
INAA Measurement of Arsenic in Silicon
Uncertainty Source

Relative standard
uncertaintyuConcentration of standard solutions
Measurement precision for samples (ν = 9)Counting statistics for standards (ν = 105)
Pulse pileup
Effects of half life
Amount of standard solutions deposited
Neutron fluence exposure - irradiation geometry
Neutron fluence exposure - self shielding
Peak integration
Neutron fluence exposure - scattering
Blank (ν = 100)
Sample area - micrometer accuracyCounting efficiency - geometry effects
Sample Area - Measurement Precision (ν = 26)
Counting efficiency - gamma ray self absorption
Inadequacy of live-time extension
Fast neutron interferences
Decay until start of count - accuracyCount intervals - live time
Effective cross sections
Gamma-ray interferences
Decay until start of count - precision
Irradiation timing
Count intervals - clock time
Fission interferences
Irradiation losses
Target isotope burn-up
Perturbed angular correlations
Isotopic abundances (unknown/standard)

0.106
%
0.081
%
0.073
%
0.066
%
0.056
%
0.041
%
0.039
%
0.024
%
0.019
%
0.015
%
0.013
%
0.012
%
0.009
%
0.009
%
0.004
%
0.001
%
0.0003 %0.00007 %0.00003 %0.000005 %0.000004 %0.000002 %
0
%
0
%
0
%
0
%
0
%
0
%
0
%

Combined uncertainty u

0.188

%

Note: The six sources in bold type together accounted for 95 % of
the relative combined uncertainty. ν denotes degrees of freedom

The Nature of Bonding of Hyperlithiated Molecules Beyohd

J. Nucl. Radiochem. Sci., Vol. 2, No. 1-2, 2001

R3

a method having the highest metrological properties, whose
operation can be completely described and understood, forwhich a complete uncertainty statement can be written down interms of SI units. A primary direct method: measures thevalue of an unknown without reference to a standard of thesame quantity. A primary ratio method: measures the value ofa ratio of an unknown to a standard of the same quantity; its
operation must be completely described by a measurement
equation." 27 If care is taken to document uncertainties, inmany cases INAA can comply fully with the definition of a
primary ratio method.28 In particular, the NIST determinationof arsenic in silicon meets all these criteria. The measurement
and its corresponding relative expanded uncertainty of 0.38 %is of the highest metrological properties, since no other analytical method is able to provide a smaller uncertainty for tracelevels of this element. In addition, equations describing themeasurement process and its uncertainty can be written interms of SI units, the well-understood equations of neutron
activation and radioactive decay.

Conclusions

Why should we care about this definition? “Strictly, traceability to the SI in measurements of amount of substance – orof any other quantity – requires that the measurements bemade using a primary method of measurement, which iscorrectly applied and stated with an evaluated uncertainty.” 29
Nuclear methods in analytical chemistry are perceived,
correctly, as being more tedious than methods that merely
require that the sample be dissolved and aspirated into a
plasma. Yet the dissolution step can lead to large errors from
insoluble phases, reprecipitation, or volatilization even forsimple biological materials.30 Even if all the sample is in solution, differing response of the atomic and molecular analytical
instruments to different analytes, and still different effects ofthe matrix on calibrations, make it clear that easy and correct
do not necessarily go together.
Because most nuclear techniques are based on simple physical principles, many merely chemical obstacles to accurate
analysis become more tractable. The fact that NAA has effectively zero blank has made it possible to measure ultratracelevels of essential and toxic trace elements in biological tissues
with little concern for sample contamination during the analytical process. For example, using radiochemical NAA thenatural levels of chromium in whole blood were shown to be
more than an order of magnitude lower than previously
believed.31 Values for other difficult elements (Mn, Mo, andV) in tissue have been established by NAA,32 and over half ofthe reference values in blood or serum selected in a criticalreview33 were established by NAA. Results such as these havestimulated the development of ultrapure reagents, clean-roomchemistry, and non-contaminating containers necessary for theapplication of non-nuclear techniques to reliable, routine trace
analysis.
Other analytical problems are also best solved by nuclear
methods. In particular, the use of radioactive tracers is underused, perhaps because the technique is “fictitiously difficult.”34
Because of the complications of chemical trace-element
analyses of high-purity solids, "all reliable values for segregation coefficients for impurities in semiconductors have been
determined by radiotracer methods." 35 Radiotracers can giveexquisite sensitivity in studies of biochemical speciation. 36, 37
and environmental transport processes.38, 39 In the analyticalchemistry laboratory, radioactive tracers can often efficiently
shed light on issues of homogeneity, contamination and losses,
and the yield and purity of separation processes.40 Bowen has
written a thorough introduction to the virtues, applications, andpracticalities of radiotracers.41

Contributions of the National Institute of Standards andTechnology are not subject to United States copyright.

For analytical results to be demonstrably reliable, somebelieve that it is necessary to write and adhere to a detailed
quality manual which sets out every step to be followed inevery unit operation in every analytical procedure. Suchrigidity can only guarantee reproducibility, not accuracy. It
also guarantees scientific boredom. However, the ISO GUM
can also be interpreted as a spur to improved analysis byencouraging the systematic examination of every significant
source of error and uncertainty, and thus lead to improved
accuracy through more complete understanding of the chemical measurement process. Nuclear methods, with their long
history of attention to counting statistics and other uncertainties, can lead the way.
To the ignorant the great results alone are admirable; to theknowing, rather the infinite device and sleight of hand thatmade them possible. - Robert Louis Stevenson

References(1) R. M. Lindstrom, S. H. Harrison, and J. M. Harris, J.
Appl. Phys. 49, 5903 (1978).
(2) A. Wyttenbach, J. Radioanal. Chem. 8, 335 (1971).
(3) H. Houtermans, K. Schärf, F. Reichel, and K. Debertin,
Int. J. Appl. Radiat. Isot. 34, 487 (1983).
(4) R. R. Greenberg and B. S. Carpenter, J. Radioanal. Nucl.
Chem. 111, 177 (1987).
(5) R. M. Lindstrom, Biol. Trace Elem. Res. 43-45, 597(1994).
(6) H. P. Yule, Radioact. Radiochem. 6 (4), 4 (1995).
(7) R. M. Parr, H. Houtermans, and K. Schaerf, in Computersin Activation Analysis and Gamma-Ray Spectroscopy(CONF-780421); B. S. Carpenter, M. D. D'Agostino, and
H. P. Yule, Eds.; U. S. DOE: Springfield, Va. 544 (1979).
(8) C. G. Sanderson, Environ. Internat. 14, 379 (1988).
(9) K. M. Decker and C. G. Sanderson, Appl. Radiat. Isot. 43,323 (1992).
(10) M. Blaauw, V. Osorio Fernandez, P. Van Espen, G.
Bernasconi, R. Capote Noy, H. Manh Dung, and N. I.
Molla, Nucl. Instrum. Methods A387, 416 (1997).
(11) G. L. Ritter and L. A. Currie, in Computers in Activation
Analysis and Gamma-Ray Spectroscopy (CONF-780421);
B. S. Carpenter, M. D. D'Agostino, and H. P. Yule, Eds.;
US Dept. of Energy: Springfield, VA. 39 (1979).
(12) M. Blaauw, Nucl. Instrum. Methods A333, 548 (1993).
(13) R. Van Schandevijl, R. Van Grieken, and J. Hoste, J.
Radioanal. Chem. 9, 55 (1971).
(14) R. F. Fleming and R. M. Lindstrom, J. Radioanal. Nucl.
Chem. 113, 35 (1987).
(15) R. M. Lindstrom and R. F. Fleming, Radioact. Radiochem.
6 (2), 20 (1995).
(16) E. A. Mackey and J. R. D. Copley, J. Radioanal. Nucl.
Chem. 167, 127 (1993).
(17) R. M. Lindstrom, R. R. Greenberg, D. F. R. Mildner, E. A.
Mackey, and R. L. Paul, in Second International k0 Users
Workshop, Jozef Stefan Institute: Ljubljana, Slovenia, 50(1997).
(18) R. R. Greenberg, R. M. Lindstrom, and D. S. Simons, J.
Radioanal. Nucl. Chem. 245, 57 (2000).
(19) M. Grasserbauer, Pure & Appl. Chem. 60, 438 (1988).
(20) R. M. Lindstrom, M. Blaauw, and R. F. Fleming, J.
Radioanal. Nucl. Chem. 2001. (submitted).
(21) ISO, Guide to the Expression of Uncertainties in
Measurement, International Standards Organization,

R4

J. Nucl. Radiochem. Sci., Vol. 2, No. 1-2, 2001

Geneva, 1993.
(22) S. L. R. Ellison, M. Rosslein, and A. Williams, Eds.
Quantifying Uncertainty in Analytical Measurements (2nded.; QUAM:2000. P1); Eurachem/CITAC, 2000.
(23) B. N. Taylor and C. E. Kuyatt, Guidelines for Evaluatingand Expressing the Uncertainty of NIST Measurement
Results, NIST Tech. Note 1297 (National Institute ofStandards and Technology, Gaithersburg, 1993).
(24) J. Kucera, P. Bode, and V. Stepanek, J. Radioanal. Nucl.
Chem. 245, 115 (2000).
(25) P. Robouch, G. Arana, M. Eguskiza, S. Pommé, and N.
Etxebarria, J. Radioanal. Nucl. Chem. 245, 195 (2000).
(26) J. Kragten, Analyst 119, 2161 (1994).
(27) Minutes from the Fifth Meeting (February 1998) of theConsultative Committee on the Quantity of Material(CCQM) of the Bureau International des Poids et Mesures(BIPM) (Sevres, France, 1998).
(28) P. Bode, E. A. De Nadai Fernandes, and R. R. Greenberg,
J. Radioanal. Nucl. Chem. 245, 109 (2000).
(29) Report of the CCQM (2nd meeting), 1996.
(30) R. R. Greenberg, H. M. Kingston, R. L. Watters Jr., andK. W. Pratt, Fresen. J. Anal. Chem. 338, 394 (1990).
(31) R. R. Greenberg, R. Zeisler, and S. Mazolowski, in Trace
Element Analytical Chemistry in Medicine and Biology
edited by P. Brätter and P. Schramel, (de Gruyter: Berlin,

Lindstrom
5, 220 (1988)).
(32) J. Versieck, Biol. Trace Elem. Res. 43-45, 407 (1994).
(33) J. Versieck, CRC Crit. Rev. Clin. Lab. Sci. 22, 97 (1985).
(34) E. W. Haas and R. Hofmann, Solid-State Electronics 30,
329 (1987).
(35) P. F. Kane, G. B. Larrabee, Characterization of Semiconductor Materials, (McGraw-Hill: New York, 1970).
(36) D. Behne, C. Weiss-Nowack, M. Kalcklösch, C. Westphal,
H. Gessner, and A. Kyriakopoulos, Biol. Trace Elem. Res.
43-45, 287 (1994).
(37) R. Cornelis, J. Radioanal. Nucl. Chem. 244, 255 (2000).
(38) Y. Muramatsu, S. Uchida, M. Sumiya, Y. Ohmomo, and
H. Obata, Water, Air, and Soil Pollution 45, 151 (1989).
(39) J. P. McKinley, C. J. Zeissler, J. M. Zachara, R. J. Serne,
R. M. Lindstrom, H. T. Schaef, and R. D. Orr, Environ.
Sci. Technol. 35, 3433 (2001).
(40) V. Krivan, in Treatise on Analytical Chemistry, Part I:
Theory and Practice, 2nd ed., v. 14, I. M. Kolthoff, P. J.
Elving, and V. Krivan, Eds. (Wiley: New York, 1986),
pp.339-417.
(41) H. J. M. Bowen, in Treatise on Analytical Chemistry, Part
I: Theory and Practice, 2nd ed., v. 14, I. M. Kolthoff, P. J.
Elving, and V. Krivan, Eds. (Wiley: New York, 1986),
pp.193-240.