description of measurementapplication-notes.digchip.com/166/166-47949.pdf · x-ray fluorescence...

Estimation of Uncertainty in PlateThickness Measurement By X-rayFluorescence Spectrometry

Glenn E. StaudtAMP Incorporated

ABSTRACTA foundation is developed for assessment of the uncer-tainty of coating layer thickness analysis using commercialx-ray fluorescence instrumentation. Proper evaluation ofthe measurement uncertainty is requisite for effectual ap-plication to plating process control and product disposition,but traditional methods of assessment (by repetitive mea-surement) are inadequate due to exclusion of majorsources of experimental variability. The major sources ofmeasurement uncertainty are identified and their contribu-tions to the overall uncertainty of the analytical results areestimated. The effect of calibration uncertainty due tovariation in calibration standards is found to be dominantunder many practical circumstances. Failure to account forthese important sources of variation properly will yieldsubstantial underestimates of the overall measurementuncertainty that will seriously compromise efforts to im-prove plating process control and plated product quality.

INTRODUCTIONThe thickness of metallic coatings is routinely determinedby x-ray fluorescence spectrometry.l This provides a basisfor assessing the quality of coated materials and thereforeinfluences decisions regarding disposition (acceptability) ofcoated product as well as control of coating processes.Knowledge of the variability associated with these mea-surements of coating thickness is necessary for properinterpretation of test results. This variability is often esti-mated from repetitive measurements either on the samesample (yielding a good estimate of the inherent precisionof the measurement technique) or on similar samples (inwhich case variation between samples is included in theuncertainty estimate). However, such an evaluation yieldssevere underestimation of the measurement uncertaintydue to exclusion of major sources of measurement uncer-

tainty such as substrate and saturation uncertainties andcalibration standard thicknesses. These sources of mea-surement uncertainty can be estimated by other means.

OBJECTIVEThe objective of this work is to provide a basis for accurateevaluation of the measurement uncertainty associated withthe determination of coating thickness using commercialx-ray fluorescence apparatus. This will be accomplishedthrough the identification of the major sources of measure-ment uncertainty and the development of reasonableestimates of the magnitude of their contributions to theoverall uncertainty of the analytical results.

In this context, the term “uncertainty” is used to denotethe overall extent to which the measurement result remainsindeterminate (i.e., the amount to which the measuredvalue may be in error with respect to the actual value of themeasured quantity). This term is used in preference to thestatistical terms “accuracy” and “precision” since it is theoverall effect relative to the “true” value (i.e., the magni-tude of possible error) that is of interest regardless of thenature of its source. Such an approach is appropriate forthe determination of the results of measurement (if devia-tion from an actual value rather than reproducibility of aspecific experiment is desired) since an observed quantitycan be equally uncertain whether its lack of exactitudederives from random or systematic sources. Adoption ofthis perspective avoids the potentially contentious consid-eration of whether individual sources of uncertainty areattributable to random or systematic factors. This perspec-tive reflects a “randomatic” 3

2

interpretation of atraditionally systematic effect (e.g., calibration bias)wherein the bias is considered as a random, precursorysampling of a parent population, which is assumed to be

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AMP Journal of Technology Vol. 3 November, 1993 G.E. Staudt 85

unbiased overall.4

7

The adoption of this randomized con-cept of traditionally systematic errors is justifiable on thebasis of the evident existence of underlying sampling distri-butions (such as from the population of all calibrationstandards), which are expected to be accurate (or verynearly so) on the average. This representation provides aconsistent methodology for the treatment of uncertaintiesregardless of origin (e.g., random or systematic in a tradi-tional viewpoint).5,6 Consequently, the uncertainty of eachquantity is expressed as an estimated standard deviation,and the uncertainty components are combined accord-ingly.

DESCRIPTION OF MEASUREMENTPROCESSX-ray fluorescence does not provide a direct measure ofcoating layer thickness. Rather, the technique operates bycomparison of the level of fluorescent x-radiation detectedfrom an unknown sample with other levels associated withknown thicknesses. Therefore, the initial activity in theanalytical scheme is calibration of the x-ray fluorescenceapparatus with coating layer thickness standards.8 Thebasic instrument response that is measured is the x-rayintensity or counting rate, which is the number of indi-vidual x-rays (photons) collected per unit time. Since theabsolute detected counting rate is highly dependent uponmany factors, the thickness is traditionally related to thenormalized intensity, which expresses the observed inten-sity as a fraction of the range of the counting rate betweenthat from the substrate and that from a sample of satura-tion thickness.9 Consequently, remeasurement of thesubstrate and saturation intensities provides a means forcorrecting the results for instrumental drift. However, tofacilitate compensation of total intensity variations, com-mercial x-ray thickness measurement units generallyfurnish the ability to correct many independent calibrationsby measurement of a single reference standard. This signalcorrection procedure operates by scaling the actual mea-sured counting rates in accordance with the observed ratioof reference intensities at the times of calibration and mea-surement. This presumes that all measured counting ratesare affected equally and change proportionally, which arereasonable assumptions.

Since the production of x-radiation is a random process,every observation of an x-ray intensity contributes uncer-tainty to the resulting thickness even for a perfectmeasurement instrument. Contribution from the analysis ofthe thickness of an unknown sample includes measurementof both the unknown and the signal correction sample(assuming the use of signal correction rather than remea-surement of substrate and saturation samples). Thecalibration procedure also contributes imprecision due tothe measurement of (1) the calibration standards, (2) thesubstrate, (3) the saturation sample, and (4) the signalcorrection sample.

Effects other than random intensity variation also influencethe uncertainty of the resulting thickness. An exampleis fluctuation of the energy calibration of the detection

system, which induces variation in the x-ray intensity ob-served within an established window (region of interest) ofdetector output voltage. The magnitude of variation pro-duced by such phenomena is highly dependent upon thespecific application (such as material combinations andchosen region of interest) and is difficult to assess. Sincethese effects are expected to be modest in well-designedinstruments operated under sound experimental protocols,they will be excluded from further consideration.

Since the analysis is comparative, the accuracy of the cali-bration standards will also exert a profound influence overthe accuracy of resulting thickness determinations. Thisconstitutes another major contribution to the uncertaintyof thickness measurement by x-ray fluorescence instrumen-tation.

Figure 1 exhibits a block diagram of the analytical processcomprising thickness determination by the technique ofx-ray fluorescence using commercial x-ray systems indicat-ing the previously discussed sources of uncertainty.

Figure 1. Shown is a process diagram for a commercial x-rayfluorescence thickness measurement instrument illustratingthe sources of uncertainty and the sequence of theirinteraction.

X-ray fluorescence does not respond to layer thickness butrather to mass per unit area of the element of interest.10

Therefore, the thickness must be derived on the basis ofthe material density. However, accurate knowledge of thedensity of thin coating layers is elusive. Also, material costs(and perhaps product function) are related to the mass ofcoated material rather than to its thickness. Therefore, it isconventional to specify an assumed density to which thethickness measurement is applicable. Consequently, the

86 G.E. Staudt AMP Journal of Technology Vol. 3 November, 1993

(assumed) density of the coating does not contribute uncer-tainty to the measured thickness.

THEORETICAL MODEL OF UNCERTAINTYPROPAGATIONThe quantitative effect of each source of uncertainty on theoverall uncertainty in the measured thickness can be as-sessed theoretically from the relationship between theinstrumental responses (measured fluorescent x-ray inten-sities) and the derived thickness. Under the usualassumptions of constant background (independent of thick-ness) and of the effective wavelength approximation (thatis, treating the x-ray source as monochromatic), the idealrelationship between layer thickness and instrument re-sponse is obtained as shown in Equation 1.11

where

t = thickness of coating layer= calibration constant

Xn = normalized intensity

The normalized intensity is defined in terms of the mea-sured counting rates (absolute intensities) by equation (2).

where

X =Xo =Xs =

Intensity at Thickness tSubstrate Intensity ( t = 0 )Saturation Intensity ( t )

With substrate and saturation intensities measured at thetime of calibration, the expression for the normalized in-tensity is modified to account for signal correction bymeasurement of a reference standard according to Equa-tion 3.

where

Xsc = measured intensity on signal correction sample(at the time of the sample measurement)

Xref = reference intensity on signal correction sample(at the time of calibration)

The uncertainty in the calibration constant results primar-ily from uncertainty in the assigned values of thecalibration standards. Since this originates from the proce-dure employed in characterization of the thicknessstandards, it is expected to be independent of the uncer-tainty in the measured counting rates (and hence in the

normalized intensity), which are dominated by the randomcounting statistics governing the production of fluorescentx-radiation. Consequently, the covariance between theseeffects is considered to be zero. Under this assumption, theexperimental uncertainty (standard deviation) in themeasured thickness is approximated by the first order sen-sitivities to variation in the measured quantities, as shownin Equations 4 and 5.12,13

where

= standard deviation of coating layer thickness= standard deviation of calibration constant= standard deviation of normalized intensity

where

x = standard deviation of intensity at thickness txo = standard deviation of substrate intensity

= standard deviation of saturation intensity= standard deviation of signal correction intensity= standard deviation of reference intensity

Application of the calculus to Equations 1 and 3 yields thepartial derivatives shown in Equations 6 through 12:


xn

t

xs

xsc

xref

ESTIMATION METHODSThe contributions to uncertainty from the various sourcescan be estimated from readily available information. Thesources of uncertainty and associated estimation methodsare represented diagrammatically in Figure 2.

in the intensity measurement. However, it is usually as-sessed by direct experimental observation of the variabilityin the thickness determination from an uninterrupted se-quence of repetitive measurements in the absence ofoperator intervention.

In practice, the positioning of the sample for measurementmay also contribute to the apparent measurement uncer-tainty this will transpire when the actual plate thicknessvaries with position on the sample over distances on theorder of the positioning capability of the measuring instru-ment (typically about 0.05 mm) or when the samplegeometry is critical. This is not truly part of the measure-ment uncertainty since it represents actual changes in thethickness of the plate. Nevertheless, it contributes to theeffective uncertainty of the measurement result. This effectcan be explicitly included in the precision estimate by com-pletely repositioning the sample between repetitivemeasurements.

The contributions to uncertainty in the thickness due to thecounting statistics of measurements other than that of thesample (i.e., substrate, saturation, and signal correctionintensities) are also determined by propagation from theprecision of the corresponding measured intensities. Al-though these uncertainties in counting rates are alsodirectly accessible empirically, they are usually estimatedfrom the Poisson distribution, as shown in Equations 14through 17:

Figure 2. This modification of the x-ray fluorescencethickness measurement diagram depicts the methods ofestimation of uncertainty components and the propagation ofthe resulting uncertainty estimates.

where

To = measurement time for substrateFluorescent emission of x-radiation is a discrete randomprocess and consequently is Poisson distributed in time.14

This allows estimation of the contributions to measurementuncertainty introduced by random intensity variations sincethe standard deviation associated with an observed numberof Poisson distributed entities (in this case x-ray photons) isthe square root of the number observed. This variationconstitutes the limit of inherent precision for x-ray fluores-cence analysis. Since more x-rays are detected in longertime intervals, this inherent precision limit can be con-trolled directly by manipulation of the measurement time.The inherent precision of the counting rate can be esti-mated from the Poisson distribution of the x-radiation asindicated in Equation 13:

where

Ts = measurement time for saturation sample

where

Tsc = measurement time for signal correction

where where

Tm = measurement time for sample. Tref = measurement time for reference measurement

The inherent variability in the thickness measurement can The reference and signal correction intensities are ascer-consequently be estimated by propagation of the precision tained by identical measurements (one at the time of


calibration and the other at the time of sample analysis).Consequently, the measurement times are the same forthese two determinations (i.e., Tsc = Tref ). Sufficient mea-surement time is allocated to signal correction to restrict itscontribution to the overall uncertainty in the thicknessresult to a low level. In addition, the signal correction in-tensity should not be (and generally does not) allowed todiffer substantially from the reference intensity. That is,signal correction is only permitted to correct instrumentaldrifts of small magnitude; larger changes in intensity re-quire full recalibration. Consequently, the signal correctionintensity can be equated to the reference intensity in thecalculation of the uncertainty component estimates yield-ing the approximation shown in Equation 18.

INDIVIDUAL CONTRIBUTIONS TOUNCERTAINTYThe expression for the total uncertainty in the measuredthickness, Equation 4, can be interpreted as the combina-tion (in a root mean square sense) of variation in thethickness from two sources: calibration and measured nor-malized intensity. The latter can be further expanded intocontributions from four sources (sample, substrate, satura-tion, and reference intensities) by Equation 19. This yieldsa description of the total uncertainty in the measured thick-ness (not normalized intensity) as a combination (again ina root mean square sense) of individual component uncer-tainties as shown in Equation 20; each of these componentsrepresents a thickness uncertainty that includes both thevariation in the source (e.g., a measured intensity) and thesensitivity of the measured thickness to that factor.

This results in simplification of the expression relating theuncertainty in the normalized intensity to variation in mea- wheresured intensities, as indicated in Equation 19. =

=

=

=

=Although the substrate and saturation intensities also varyslightly with time, the impact on the overall uncertainty is

standard deviation of measured thickness due tovariation in calibrationstandard deviation of measured thickness due tosample intensity variationstandard deviation of measured thickness due tosubstrate intensity variationstandard deviation of measured thickness due tosaturation intensity variationstandard deviation of measured thickness due toreference intensity variation

similarly negligible. Consequently, the substrate andsaturation intensities need not be adjusted from the corre- The theoretical estimates of these uncertainty components

spending calibration values. This enables estimation of the are given in Equations 21 through 25; since the sample

uncertainty from direct thickness measurement results and counting rate is generally not reported, these estimates are

parameters of the calibration (without additional experi-expressed only in terms of the measured thickness and

mental work at the time of measurement).calibration parameters (the calibration constant and sub-strate, saturation, and reference counting rates).

Since the uncertainty in the calibration constant is contin-gent upon the accuracy of the calibration standards, directobservation of this uncertainty requires repetitive calibra-tion with sets of standards selected randomly from a vastcollection of independent standards. This is impractical (ifnot impossible) due to the paucity of sources of accuratestandards, the large expense of such standards, and therelatively long duration of the calibration procedure. Con-sequently, the uncertainty in the calibration is taken to bethe uncertainty within which the standards are certified.Certification of optimal commercial standards is typicallyspecified as ±5 percent of the assigned coating thickness.This is considered to be indicative of a 95-percent confi-dence interval for a normal distribution (1.96 standarddeviations), although the exact statistical interpretation ofthe certified tolerance is frequently unstated. It should benoted that this assumption of a confidence level yields asmaller estimate of the associated standard deviation thanwould be derived from the assumption of a uniform distri-bution across the stated interval.


xo

txref

txs

tx

t

EXPERIMENTAL WORKEmpirical estimates of inherent measurement precisionwere obtained by repetitive analysis at a single location oneach of four gold plate thickness standards using a com-mercial x-ray fluorescence thickness measurementinstrument (FISCHERSCOPE X-RAY 1510Z fromFischer Technology, Inc., 750 Marshall Phelps Rd., Wind-sor, Connecticut 06095) with a 0.015 by 0.3 mm rectangularcollimator. The distribution of the fundamental instrumentresponse (measured intensity) was verified by repeatedmeasurement of the counting rate on the four gold thick-ness standards as well as on a bare nickel substrate and asaturation (effectively infinite thickness) gold sample. Theeffect of the intensity variation on the measured thicknesswas assessed by repetitive thickness measurements on thefour plating thickness standards. Both of these sets of ex-periments were performed at each of two measurementtimes to demonstrate the effect of measurement time onprecision.

Direct empirical assessment of the effects of variation insubstrate, saturation, and reference intensities is not ex-perimentally accessible because measurements cannot bemade in the absence of variation in the counting rate of themeasured sample. Therefore, the following procedure wasemployed as illustrated for the case of variation in the sub-strate. The substrate intensity was determined (by partialapplication of the function called “Normalization” on theFischer x-ray unit ) using a short measurement time (1second). The thickness of each of the four plating thicknessstandards was then measured with a long measurementtime (100 seconds) using the newly measured substrateintensity. This series of two measurements was repeatedpositioning to the same points on both the substrate andthe plating thickness standards in each iteration. A longacquisition time for the thickness measurement but a shorttime for the determination of the substrate counting ratewere employed in an attempt to obtain relatively goodprecision in the former; this approach produced poor pre-cision in the latter, thereby causing the substrate effect todominate the observed variation. Since the variability ofthe obtained measurements derives from two sources(sample and substrate intensities), the corresponding theo-retical estimate is obtained by combining the individualcontributions (in a root mean square sense).

seconds. Accordingly, the thickness measurement time islengthened (to 300 seconds) in an attempt to retain domi-nance of the effect of interest in the observed variation.Since the reference measurement is a composite of severalfunctions (energy calibration of the detection system andintensity correction), the measurement time associatedwith the intensity portion of the signal correction proce-dure is not directly observable. Consequently, the effectivemeasurement time for determination of the reference in-tensity (Xref) is estimated empirically from repetitive signalcorrections.

Calibration of the Fischer x-ray instrument for thicknessmeasurement was performed by fitting the ideal calibrationrelationship to intensity data from 30 measurements oneach of 39 gold plate thickness standards on three differentsubstrates (nickel, copper, and nickel-plated brass) fromthree different sources including the National Institute ofStandards and Technology (NIST), formerly the-NationalBureau of Standards (NBS). The fitting procedure em-ployed was a weighted linear least squares regression withonly one adjustable parameter (a), in accordance with thesimplified model of Equation 1. As is usual, the weighingswere taken to be the reciprocals of the estimated variancesin the thicknesses.15 Since the assigned thicknesses of cali-bration standards are certified to a relative tolerance (±5percent), the variances were taken to be proportional tothe squares of the thicknesses. This assumption of a con-stant coefficient of variation is also appropriate to theinherent precision of coating thicknesses measured by x-rayfluorescence in the middle of the measurable range.16 Cor-respondingly, the uncertainties in the thicknessmeasurement are also expressed in relative terms as thecoefficient of variation (CV), which is also known as therelative standard deviation (standard deviation divided bythe mean).

The effective calibration precision and the uncertainty ofcalibration imposed by the calibration standards were as-sessed using the previously cited calibration data for goldplate thickness measurement from the Fischer x-ray sys-tem. Analogous calibration data from 17 gold flash (goldplate less than approximately 0.25 µm) thickness standards(encompassing both foils and electrodeposits on nickelfrom four different sources, including the NIST) were alsoconsidered as an additional example. The assigned thick-

The effect of the saturation intensity is assessed in a man- nesses for all gold standards assume wrought gold density

ner exactly analogous to the assessment of the effect of the (19.3 g/cm3).

substrate. For the effect of variation in the reference inten-sity, the methodology was also analogous but with severalslight differences. The reference intensity is only assignableby performing the instrument’s signal correction proce-dure. Therefore, the Fischer x-ray “Reference”measurement function was invoked between each set ofrepetitive thickness measurements. Since this function alsoaccomplishes energy calibration of the detection system,the resulting uncertainty estimate may by contaminated bythe effect of variation in energy calibration (which is beingexcluded from consideration). The measurement time forthe intensity portion of the signal correction is fixed at 100

EXPERIMENTAL RESULTSThe gold plate thickness calibration data are presented inTable 1 and displayed graphically in Figure 3. The resultingcalibration parameters are shown in Table 2. The calibra-tion data for gold flash are presented in Table 3.

Since portions of the experimental procedure encompassedlarge numbers of long measurements, the total time periodspanned by a single set of measurements was quite exten-sive in some cases. Consequently, the overall variation


Table 1. Gold plate thickness calibration data.ments did not produce reported thickness values due tomeasured counting rates approaching or exceeding thesaturation intensity (whereupon the measured thicknessbecomes unfoundedly large). This is a natural conse-quence of high variability imposed by the shortmeasurement time.

Table 2. Gold plate thickness calibration parameters.

Table 3. Gold flash thickness calibration data.

observed in these measurements may include effects ofinstrumental drift (which could be compensated by morefrequent application of the signal correction procedure).To eliminate this potential corruption of the experimentaldata, the standard deviation was computed between succes-sive pairs of data, and these values were pooled (bycalculating the root mean square) to obtain the final em-pirical estimate of the variability.

Although 1000 measurements were taken in each series,only 997 results were obtained for the saturation samplewith the short measurement time. Three of the measure-

COMPARISON BETWEEN EMPIRICAL ANDTHEORETICAL ESTIMATESThe theoretical estimates of uncertainty corresponding tothe measured variations in counting rate and thickness areobtained from Equation 13 and Equations 22 through 25using the calibration parameters indicated in Table 2.These theoretical and empirical values are compared inTables 4 through 6. The statistical significance of the dis-parities between these two estimates is assessed by meansof the ratio (experimental to theoretical) of the variances(uncertainty values squared), which is expected to follow a

2/df distribution. Accordingly, the pertinent variance


ratios are plotted against thickness in Figures 4 through 6,along with the critical values of 2/df at the 95-percent Table 5. Empirical and theoretical thickness precision

confidence level. estimates 1.127).

Table 6. Empirical and theoretical uncertainty estimates forsubstrate, saturation, and reference components

(50) = 0.638, (50) = 1.418).( 2/d f0.025 0.975

Figure 3. X-ray fluorescence calibration data is displayed inthis plot of the measured x-ray intensity as a function of theassigned thickness of gold calibration standards. The linerepresenting the best fit of this data to the ideal calibrationrelationship is also shown.

Table 4. Empirical and theoretical intensity precision(500) = 0.879,estimates ( 2/df0.025

2/df0.975(500) =1.127).

producing superior precision on one given sample isenigmatic. Consequently, the origin of the observeddiscrepancies is deemed to be statistical rather thansystematic.

Two out of the twelve experimental determinations of theinherent precision of the measured counting rates arelower than the theoretical estimates by an amount that isstatistically significant. Since the production of x-radiationis known to be Poisson distributed in time, the theoreticalresults constitute limits to observed precision upon whichimprovement is implausible. Further, both of the inconsis-tencies occur when measuring the lowest gold platethickness standard (0.035 µm); the nature of a mechanism

For the sample intensity component of the thickness mea-surement uncertainty, the 95-percent confidence limits areexceeded in none of the eight cases.

Of the twelve cases representing the substrate, saturation,and reference intensity components of the thickness mea-surement uncertainty, the upper 95-percent confidencelimit is exceeded in one instance (of the reference intensitycomponent) indicating an experimental observation signifi-cantly higher than the theoretical estimate. This may resultfrom variation due to energy calibration of the detectionsystem, which is included in the signal correction procedure(reference measurement) but is not considered in thetheoretical estimate.


(500) = 0.879,( 2/d f0.025 (500) =( 2/d f0.975

2/d f

Figure 4. The ratios of the observed variances from repetitiveintensity measurements to theoretical estimates are plottedversus thickness to verify the applicability of Poisson count-ing statistics. The 95% confidence limits of the 2 /dfdistribution with 500 degrees of freedom are shown forcomparison.

Figure 5. The ratios of the observed variances from repetitivethickness measurements to theoretical estimates are shownas a function of thickness to validate the estimation methodfor the component of uncertainty in measured thickness dueto variation in the sample intensity. The 95% confidencelimits of the 2/df distribution with 500 degrees of freedomare shown for comparison.

In total, three inconsistencies are observed in 32 cases; thisis only slightly in excess of expectation for the 95-percentconfidence level. Consequently, the proposed theoreticalmodel is determined to be suitable for estimation of theuncertainty of thickness measurement by x-rayfluorescence.

Figure 6. The ratios of the observed thickness variances totheoretical estimates are shown as a function of thickness.The observed values were obtained from series of thicknessmeasurements with intervening redetermination of the sub-strate, saturation, or reference counting rates. This providesan assessment of the validity of the estimation methods forthe contributions to thickness measurement uncertaintyfrom the substrate, saturation, and reference intensities. The95-percent confidence limits of the /df distribution with50 degrees of freedom are shown for comparison.

COMPARISON OF MAGNITUDES OFUNCERTAINTY COMPONENTSThe relative importance of the uncertainty components canbe assessed by comparison of the theoretical estimatesunder typical conditions. As before, the theoretical esti-mates are obtained from Equations 22 through 25 using thecalibration parameters from Table 2. The measurementtime for determination of the substrate and saturationcounting rates was assumed to be ten times that of themeasurement time for samples. As previously cited, thecomponent of uncertainty due to calibration is estimated as2.55 percent from the certified uncertainty of the calibra-tion standards (±5 percent at a 95-percent confidencelevel). These estimates are plotted in relative terms (coeffi-cient of variation) as a function of thickness in Figure 7 fora measurement time of 10 seconds.

Several features of interest emerge from this plot. Thesubstrate component becomes important at very low thick-ness but is otherwise negligible. The saturation componentis insignificant at low thickness but increases with thick-ness; it will become a dominant contribution at very highthickness although the plot does not extend to sufficientlyhigh thickness to show this. As is desired, the referencecomponent is inconsequential for all thicknesses. The com-ponent of uncertainty due to the sample intensitydominates at the thickness extremes. However, throughoutthe majority of the pertinent thickness range, the calibra-


2

tion component is predominant. Numerical values of theseuncertainty components are presented in Table 7 for atypical application of a 0.7-µm thick gold coating. Theseresults show the dominance of the calibration uncertaintyand the insignificance of the substrate, saturation, andreference components in a typical measurement situation.

Figure 8. The effect of measurement time on the overalluncertainty of gold plate thickness as measured by x-rayfluorescence is illustrated in this plot of theoretical uncer-tainty estimates versus thickness for several measurementtimes. Ten measurements of the substrate and saturationintensities and standards certified to ±5% at the 95%confidence level are assumed.

Figure 7. The magnitudes of the various components ofuncertainty in thickness as measured by x-ray fluorescenceare compared by plotting the theoretical estimates againstthickness. These are typical values for a measurement timeof 10 seconds assuming ten measurements of the substrateand saturation intensities and standards certified to ± 5% atthe 95% confidence level.

Table 7. Typical components of uncertainty in measuredthickness for 10 second measurement of 0.7 micrometergold.

CALIBRATION UNCERTAINTYUnder most situations of interest, the total measurementuncertainty is dominated by the calibration uncertainty.Consequently, it would be most desirable to reduce thisuncertainty component. Unfortunately, the contribution ofcalibration to measurement uncertainty is among the leastamenable to control since it originates from the certifiedtolerance of the calibration standards. Plating thicknessstandards certified to better than ±5 percent are generallynot commercially available. Therefore, prospects are scantfor substantial reduction in the calibration uncertainty byutilization of superior standards.

The effect of measurement time on the overall uncertaintyis displayed in Figure 8. Due to the dominance of thecalibration uncertainty (which is not a function of mea-surement time), extensive measurement times (beyondapproximately 10 seconds for gold) do not yield substantialimprovements in the overall measurement uncertaintydespite an apparent improvement in the observed precision(from repetitive sample measurements).

Improvements in the uncertainty maybe attainablethrough the application of statistical estimation of the con-tribution due to calibration. Since the calibrationrelationship incorporates a single adjustable parameter ( ),this calibration constant is obtainable from a single thick-ness standard (presuming measurement of substrate andsaturation samples). Therefore, if multiple calibrationstandards are available, the precision of the calibrationconstant can be estimated also. For calibration with mul-tiple standards, the overall calibration constantcorresponds conceptually to the average (properlyweighted) of those from individual standards, and the pre-cision estimate corresponds conceptually to the associatedstandard deviation among them. In practice, these valuesare obtained from a weighted linear least squares fit of thecalibration relationship; the precision is taken to be thestandard error of the regression parameter, which derivesfrom the residuals from the line of best fit.17 It must beemphasized that a precision calculated in this manner is areasonable estimate of the calibration uncertainty only if


the complement of standards employed in the calibrationcomprises an unbiased set of independent standards.

Such a statistical interpretation of the calibration constantobtained from multiple standards implies that the precisionof calibration can be enhanced by inclusion of a large num-ber of calibration standards. In general, the standard errorof a mean value is expected to decrease as the square rootof the number of individuals included in the randomsample in accordance with Equation 26.18-20

where

= standard deviation of distribution of individuals

n = standard deviation of the mean of n samplesn = number of samples

To investigate this concept, the calibration relationship wasfit using a variable number of standards by bootstrap sam-plinglg19,20 of the observed standards (see Tables 1 and 3).Two estimates of the precision for a fixed sample size canbe obtained from the regression results: the standard de-viation among the calibration constants and the root meansquare of the standard errors of the calibration constants(for all cases except a single standard). The results of re-gression with the bootstrap samplings are listed in Table 8and displayed graphically in Figure 9. The precision esti-mates are given in relative terms as the coefficient ofvariation among calibrations and the root mean square ofthe relative standard errors.

As expected, the calibration precision estimates are verynearly proportional to the reciprocal of the square root ofthe number of standards (n). The only observable devia-tions occurs in the coefficient of variation among gold flashcalibrations with a single standard; this is probably a resultof one highly aberrant standard (0.071 µm). Fitting thesecoefficients of variation to a relationship proportional tothe reciprocal of the square root of the number ofstandards yields the results shown in Table 9.

The coefficient of determination is very close to 1,indicating an excellent fit in accord with expectation. Inter-pretation of the certified tolerance of ±5 percent as a 95percent confidence interval yields an expected coefficientof variation between unbiased standards of 2.55 percent.Comparison of this value with the slopes (CV/[1/n1/2]),which correspond to the projections of the calibration pre-cision to a single standard, indicates that the thicknessesassigned to the gold standards are slightly less accuratethan certified (±6.4 percent at the 95-percent confidencelevel) whereas the gold flash standards are inferior (±21percent at the 95-percent confidence level). Consequently,reduction of calibration uncertainty through utilization ofincreased numbers of standards was observed with two setsof standards of vastly diverse quality.

AMP Journal of Technology Vol. 3 November, 1993

Table 8. Effect of number of standards on calibrationuncertainty by bootstrap estimation.

This simulation of the effect of the number of standards oncalibration variability implies that implementation of cali-bration procedures exploiting multiple standards may beeffective in alleviating the deleterious effects of calibrationuncertainty. In practice, statistical determination of theuncertainty of calibration (from lack of fit in the calibrationdata) will encompass all sources of uncertainty associatedwith the calibration process including the counting statis-tics of measured quantities as well as the assignedthicknesses of calibration standards. However, this strategyis only applicable if an unbiased complement ofindependent standards is available.

For a biased set of standards, a precision assessment basedon comparison between standards (such as lack of fit) will

G.E. Staudt 95

Figure 9. The effect of the number of calibration standardson the statistically determined calibration uncertainty isdepicted. Observed variations in the calibration constants areplotted as a function of the reciprocal of the square root ofthe number of standards (to which it is expected to be pro-portional). These results were derived from variously sizedbootstrap samplings of the full set of calibration standardswith three replicate runs often thousand trials at eachsample size. For each of two calibrations (gold and goldflash), two estimates of calibration uncertainty are repre-sented: 1) variation among calibrations with a fixed samplesize as given by the coefficient of variation of the calibrationconstants (between Cal. CV); and 2) the root mean square(over all calibrations with a fixed sample size) of the relativestandard error of the calibration constant as determined byregression (rms within Cal. CV).

Table 9. Effect of number of standards on calibration preci-sion by proportional fit of coefficient of variation to 1 /n .1/2

underestimate the calibration uncertainty. Similarly, if thestandards are not independent (for example, if there aresystematic effects with thickness level), errors. in the uncer-tainty estimation (possibly varying with thickness) mayensue. Unfortunately, these stipulations are difficult tosatisfy due to the dearth of availability of high-qualitythickness standards. Generally, sets of standards obtainedfrom a single source (which is common) maybe expected tobe biased and are not anticipated to be independent. Fur-

thermore, the experimental burden associated withimprovement of calibration precision by proliferation ofstandards is onerous because the uncertainty decreasesonly as the square root of the number of standards. Conse-quently, abatement of the calibration uncertainty will entailconsiderable expense and experimental effort and shouldonly be attempted with great prudence. Nevertheless, if alarge number of independent standards is available, thecalibration uncertainty can be estimated directly from thestandard error of the calibration constant as determined byregression.

CONCLUSIONSInclusion of all major sources of measurement uncertaintyin empirical estimates minimally requires each measure-ment in the repetitive set to be made with a differentcalibration derived from an independent set of calibrationstandards. Since this is not practical, contributions to mea-surement uncertainty (other than the inherent precisiondue to the counting statistics) must be estimated by othermeans. Since these other sources of variability are substan-tial (relative to the inherent precision), their effects mustbe included in order to obtain meaningful overall estimatesof the measurement uncertainty. In particular, the effect ofcalibration bias (due to limitations of certification of cali-bration standards) predominates in many situations.Failure to account properly for these sources of variationwill yield substantial underestimates of the overall mea-surement uncertainty and may consequently engendererroneous decisions regarding the disposition of product.

ACKNOWLEDGMENTThe author is indebted to Marsha K. Lower for her assis-tance in the experimental work.

DEFINITION OF SYMBOLSn = number of samplest = thickness of coating layer

Tm = measurement time for sampleTo = measurement time for substrateTref = measurement time for reference measurementTs = measurement time for saturation sample

Tsc = measurement time for signal correctionX = intensity at thickness t

Xn = normalized intensityXo = substrate intensity (t = 0)

Xref = reference intensity on signal correction sampleXs = saturation intensity (t )

Xsc = measured intensity on signal correction sample= calibration constant

standard deviation of distribution of individualsstandard deviation of the mean of n samplesstandard deviation of coating layer thicknessstandard deviation of measured thickness due tosample intensity variationstandard deviation of measured thickness due tosubstrate intensity variation


tXref = standard deviation of measured thickness due toreference intensity variation

tXs = standard deviation of measured thickness due tosaturation intensity variation

t = standard deviation of measured thickness due tovariation in calibration

X = standard deviation of intensity at thickness tXn = standard deviation of normalized intensityXo = standard deviation of substrate intensity

Xref = standard deviation of reference intensity= standard deviation of saturation intensity= standard deviation of signal correction intensity= standard deviation of calibration constant

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ASTM Standard B568-90, “Standard test method formeasurement of coating thickness by x-rayspectrometry” (American Society for Testing and Ma-terials, Philadelphia, PA 1990).

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J.W. Müller, in Precision Measurement and Fundamen-tal Constants II (Special Publication 617), edited byB. N. Taylor and W.D. Phillips (National Bureau ofStandards, Gaithersburg, MD, 1984), p. 375.

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G.E.P. Box, W.G. Hunter, and J.S. Hunter, Statisticsfor Experimenters (John Wiley & Sons, New York,1978), p. 505.

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Glenn E. Staudt is a Member of Technical Staff in theTechnology Group at AMP Incorporated in Harrisburg,Pennsylvania.

Mr. Staudt received a Bachelor of Science degree in chem-istry from Bucknell University, Lewisburg, Pennsylvania in1974 and a Master of Arts degree in chemistry from Prince-ton, New Jersey in 1976. From 1977 to 1981, he wasemployed as a chemist at Hamilton Technology, Inc. inLancaster, Pennsylvania. Since joining the TechnologyGroup of AMP Incorporated in 1981, his activities haveincluded technological support and research in the areas ofelectrodeposition and electrodeposited materials. Currentresearch interests focus on the characterization of metalliccoating systems.


Xs

Xsc

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