Analytical ChemistryTheoretical and Metrological Fundamentals

K.Danzer

Analytical ChemistryTheoretical and Metrological Fundamentals

With 138 Figures and 31 Tables

K. DanzerAm Friedensberg 407745 JenaGermanye-mail: claus.danzer@jetzweb.de

Library of Congress Control Number 2006932265

ISBN-10 3-540-35988-5 Springer Berlin Heidelberg New YorkISBN-13 978-3-540-35988-3 Springer Berlin Heidelberg New YorkDOI 10.1007/b103950

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Preface

Analytical chemistry can look back on a long history. In the course of its de-velopment, analytical chemistry has contributed essentially to the progressof diverse ˇelds of science and technology. Signiˇcant progress in natu-ral philosophy, chemistry and other disciplines has been founded on re-sults and discoveries in analytical chemistry. Well known examples includeArchimedes' principle, iatrochemistry, emergence and overcoming of phlo-giston theory, basic chemical laws on stoichiometry and mass proportionsas well as the discoveries of elements and of nuclear ˇssion.

As chemistry split into inorganic, organic and physical chemistry in themiddle of the nineteenth century, the question occasionally arose as towhether analytical chemistry was likewise an autonomous ˇeld of chemistry.At the beginning of the twentieth century, analytical chemistry was estab-lished by Wilhelm Ostwald and other protagonists on the basis of chemicaland physicochemical laws and principles. In addition, analytical proceduresand instruments spread into other chemical ˇelds. However, in the 1950sand 1960s a new generation of analysts opened their mind to ideas comingfrom other scientiˇc ˇelds dealing with measurements from other points ofview such as information science, metrology, technometrics, etc.

As a result, efforts made towards the acceptance of analytical chem-istry as an autonomous ˇeld of chemistry increased. Analysts in USA andEurope founded organisations and forums that focussed ideas and devel-opments in progressive directions. In this way, not only were national andinternational societies formed but also regular conferences like Pittcon andEuroanalysis. In this connection the so-called Lindau circle should be men-tioned where analysts from Germany, Austria and Switzerland made publicrelevant fundamentals of analytical chemistry derived from statistics, infor-mation theory, system theory, signal theory, game theory, decision theory,metrology, etc. In parallel with this, such modern principles have been suc-cessfully applied in selected case studies, mainly by scientists from USA,UK, and Canada. From this development, real theoretical and metrolog-ical fundamentals of analytical chemistry have crystallized. However, theacceptance of analytical chemistry as an automomous branch of chemistryhas not been realised. Gradually, however, progress in analytical chemistryculminated in the 1990s in the proclamation of a new scientiˇc discipline,analytical science. Protagonists of these activities included Michael Widmerand Klaus Doerffel.

VI Preface

Today, analytical chemistry is still a discipline within chemistry. Al-though characterized as an auxiliary science, analytical chemistry continuesto develop and grow just as before. It is no detraction being characterizedas an auxiliary discipline as mathematics shows. However, it is likely thatefforts to make analytical chemistry a more independent science will berepeated in the future from time to time.

Analytical chemistry possesses today a sound basis of chemical, physical,methodical, metrological, and theoretical fundamentals. The ˇrst of theseare usually taken as the basis of classical textbooks on analytical chemistry.The others are found in diverse publications in the ˇeld of analytical chem-istry and chemometrics. It is essential to state that chemometrics is not thetheoretical basis of analytical chemistry but it contributes signiˇcantly toit. Frequently, analytical chemistry is considered to be a measuring sciencein chemistry. Therefore, its object is the generation, evaluation, interpreta-tion, and validation of measuring signals as well as the characterization oftheir uncertainty. With this aim, the analyst needs knowledge of the generalanalytical process, statistics, optimization, calibration, chemometrical dataanalysis, and performance characteristics.

In this book the attempt is made to summarize all the components thatcan be considered as building blocks of a theory of analytical chemistry.The \building" constructed in this way is a provisional one. It is incompleteand, therefore, extension, reconstruction and rebuilding have to be expectedin the future.

A large number of mathematical formulas will be found in the book. Thismay be regarded as a disadvantage particularly because some of them arenot readily to apply in daily analytical practice. However, great scientists,explicitly Emanuel Kant, said that a scientiˇc branch contains only so muchof science as it applies mathematics. Consequently, all the relationshipswhich can be described mathematically should be so described. It is truedespite Werner Heisenbergs statement: Although natural processes can bedescribed by means of simple laws which can be precisely formulated, theselaws, on the other hand, cannot directly be applied to actions in practice.

Wherever possible, ofˇcial deˇnitions of IUPAC, ISO and other inter-national organizations have been used, in particular in the \Glossary ofAnalytical Terms" compiled at the end of the book. However, uniformitycould not be achieved in every case. In a few instances, special commentsand proposals (characterized as such) have been added. Although progressin the ˇeld of harmonization of nomenclature and deˇnitions has been con-siderable, some things still remain to be done.

Some of the contents of various sections have been published previouslyand are, of course quoted verbatim. In this connection the author is gratefulto Klaus Doerffel, Karel Eckschlager, G�unter Ehrlich, Andrzej Parczewski,Karol Florian, Mikulas Matherny, G�unter Marx, Dieter Molch, EberhardThan, Ludwig K�uchler and others for long-standing collaboration resultingin mutually complementary papers and books. A chapter on statistics waspreviously published in another book (Accreditation and Quality Assurance

Preface VII

in Analytical Chemistry edited by Helmut G�unzler at Springer). Gratefully Ihave adapted essential parts of the translation of Gaida Lapitajs here in thisbook.

Stimulations and ideas have arisen from discussions with William Hor-witz, Duncan Thorburn Burns, Alan Townshend, Koos van Staden, and othermembers of diverse IUPAC Commissions and Task Groups as well as ofˇ-cers of the Analytical Chemistry Division. From a 20-year membership ofvarious IUPAC bodies I have gained a variety of feedback and experiencewhich has proved to be useful in the writing of this book. In particular,I owe Lloyd Currie and Mattias Otto a great debt of gratitude. They havepermitted me to use essential parts of common publications on calibrationquoted in Chap. 6. Many of the results stem from work carried out withcolleagues, collaborators, postdocs and graduate students. All of them aredeserving of my thanks but, due to the large numbers involved, this cannotbe done here. Some representative colleagues include J�urgen Einax, Hart-mut Hobert, Werner Schr�on, Manfred Reichenb�acher, Reiner Singer, Diet-rich Wienke, Christoph Fischbacher, Kai-Uwe Jagemann, Michael Wagner,Katrin Venth, Raimund Horn, Gabriela Thiel, Demetrio de la Calle Garc��a,and Balint Berente.

The author gratefully acknowledges the cooperation with and the supportof the editorial staff of Springer. Particularly I have to thank Peter Enders,Birgit Kollmar-Thoni, and { last but not least { John Kirby, the English copyeditor, who has essentially improved the English. Many thanks for that.

No book is free of errors and this one will be no exception. Therefore,the author would be grateful to readers who point out errors and mistakesand suggest any improvement.

Jena, July 2006 Klaus Danzer

Contents

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII

Abbreviations and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXV

1 Object of Analytical Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Deˇnition of Analytical Chemistry . . . . . . . . . . . . . . . . . . . . . . . 11.2 Repertoire of Analytical Chemistry . . . . . . . . . . . . . . . . . . . . . . . 5

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The Analytical Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Principles of Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Principles of Analytical Measurement . . . . . . . . . . . . . . . . . . . . 262.4 Analytical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Signals in Analytical Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Signals and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Analytical Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Types and Properties of Analytical Signals . . . . . . . . . . . . . . . . 473.4 Dimensionality of Analytical Signals and Information . . . . . . 533.5 Mathematical Model of Signal Generation . . . . . . . . . . . . . . . . . 60

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Statistical Evaluation of Analytical Results . . . . . . . . . . . . . . . . . . . 654.1 Reliability of Analytical Observations and Measurements . . . 654.1.1 Systematic Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1.2 Random Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Uncertainty Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.1 Null Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.2 Test for Measurement Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3.3 Comparison of Standard Deviations . . . . . . . . . . . . . . . . . . . . . . 814.3.4 Comparison of Measured Values . . . . . . . . . . . . . . . . . . . . . . . . . 82

X Contents

4.4 Reliability of Qualitative Analytical Tests . . . . . . . . . . . . . . . . . 854.5 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5.1 Quality Criteria for Analytical Results . . . . . . . . . . . . . . . . . . . . 904.5.2 Attribute Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.5.3 Sequential Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.5.4 Statistical Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Studying In�uences and Optimizing Analytical Procedures . . . . . . 1015.1 Testing the Signiˇcance of In�uencing Factors . . . . . . . . . . . . 1015.1.1 Analysis of Variance (ANOVA) . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.1.2 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2 Optimization of Analytical Procedures . . . . . . . . . . . . . . . . . . . . 1125.3 Global Optimization by Natural Design . . . . . . . . . . . . . . . . . . . 116

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Calibration in Analytical Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 1236.1 General Fundamentals of Calibration . . . . . . . . . . . . . . . . . . . . . 1246.1.1 Fundamental and Experimental Calibration . . . . . . . . . . . . . . . 1246.1.2 The General Three-Dimensional Calibration Model . . . . . . . . 1266.1.3 Regression and Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Single Component Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.2.1 Linear Calibration Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.2.2 Errors in Linear Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2.3 Weighted Linear Least Squares Estimation (WLS) . . . . . . . . . . 1376.2.4 Linear Least Squares Fitting in Case of Errors

in Both Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.2.5 Statistical Tests and Validation of Calibration . . . . . . . . . . . . . 1406.2.6 Alternative Calibration Procedures . . . . . . . . . . . . . . . . . . . . . . . 1446.2.7 Nonlinear Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.3 Multisignal Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.4 Multicomponent Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.4.1 Classical Multivariate Calibration . . . . . . . . . . . . . . . . . . . . . . . . 1576.4.2 Inverse Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1596.4.3 Validation of Multivariate Calibration . . . . . . . . . . . . . . . . . . . . 1626.5 Calibration by Artiˇcial Neural Networks . . . . . . . . . . . . . . . . . 165

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7 Analytical Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . 1777.1 Reliability of Analytical Measurements . . . . . . . . . . . . . . . . . . . 1787.1.1 Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.1.2 Precision of Trace Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.1.3 Accuracy and Trueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Contents XI

7.1.4 Remark on the Quantiˇcation of Precision, Accuracyand Trueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.2 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.3 Selectivity and Speciˇcity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.4 Robustness and Ruggedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1957.5 Limit Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.6 Resolving Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

8 Presentation, Interpretation and Validation of Analytical Results 2178.1 Presentation of Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . 2178.2 Factual Interpretation of Analytical Results . . . . . . . . . . . . . . . 2198.2.1 Presentation of Results Near the Limit of Detection . . . . . . . . 2198.2.2 Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218.2.3 Analytical Results in Relation to Fixed Values . . . . . . . . . . . . . 2248.2.4 Interlaboratory Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278.3 Chemometrical Interpretation of Analytical Data . . . . . . . . . . 2288.3.1 Principles of Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2298.3.2 Cluster Analysis: Recognition of Inherent Data Structures . . 2318.3.3 Classiˇcation: Modelling of Data Structures . . . . . . . . . . . . . . . 2358.3.4 Factor Analysis: Causes of Data Structures . . . . . . . . . . . . . . . . 2398.3.5 Exploratory Data Analysis and Display Methods:

Visualization of Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . 2438.3.6 Methods of Artiˇcial Intelligence . . . . . . . . . . . . . . . . . . . . . . . . 2468.4 Analytical Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

9 Assessment of Analytical Information . . . . . . . . . . . . . . . . . . . . . . . 2659.1 Quantiˇcation of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 2659.2 Information Content of Quantitative Analysis . . . . . . . . . . . . . 2689.3 Multicomponent Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2739.4 Process and Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Glossary of Analytical Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Symbols

A Matrix of sensitivity factors inmulticomponent analysis (e.g., ab-sorbance factors)

(6.66)

a Regression coefˇcient (calibrationcoefˇcient): intercept (general andregression in case of errors in bothvariables)

ax Regression coefˇcient a for the esti-mation of y from x

(6.9)

axw Regression coefˇcient a in case ofweighted calibration

(6.37)

ay Regression coefˇcient a for the esti-mation of x from y

(6.10)

acc(x), acc(x) Accuracy of an analytical result xand of a mean x, respectively

(7.10)(7.11)

antilg x = 10x , antilogarithm of x

b Regression coefˇcient (calibrationcoefˇcient): slope (general and incase of regression having errors inboth variables)

bx Regression coefˇcient b for the esti-mation of y from x

(6.8)

bxw Regression coefˇcient b in case ofweighted calibration

(6.36)

by Regression coefˇcient b for the esti-mation of x from y

(6.10)

bias(x), bias(y ) Bias (systematic deviation) of an in-dividual test value x (or y ) from the(conventional) true value xtrue (orytrue)

(4.1)

XIV Symbols

bias(x), bias(y ) Bias of an average test value x (or y )from the (conventional) true valuextrue (or ytrue)

cnf(x), cnf(y ) Total (two-sided) conˇdence inter-vall of a mean

(4.17)

cond(A) Condition number of the matrix A (6.80)

cov(pi; pj) Covariance of the quantities piand pj

CB Conˇdence band (of a calibrationline)

(6.26a)

CR Concordance rate (proportion ofcorrect test results in screening)

(4.53a)

dcrit Critical diameter (in microprobeanalysis)

dij Distance (in multivariate data) (8.12)

df Discriminant function (8.16)

dv Discriminant variable

E Information efˇciency

Ein Input energy

Eout Output energy

E Residual matrix (in multivariate dataanalysis)

e, ex , ey Error term (general, in x, and in y ,respectively), in concrete terms, eymay be the deviation of an individualvalue from the mean or the residualof a mathematical model, see nextline (ex analogous)

ey = yi � y , deviation of an individualmeasured value from the mean (alsody )

ey = yi � y , deviation of an individualmeasured value from the estimateof the corresponding mathematicalmodel

F Test statistic (estimate) of the F test (4.37)

Symbols XV

F1�˛;�1;�2 Quantile of the F distribution at thelevel of signiˇcance 1�˛ and for thedegrees of freedom �1 and �2

Fmax Test statistic (estimate) of Hartley'stest

(4.370)

FNR False negative rate (of screeningtests)

(4.51)

FPR False positive rate (of screeningtests)

(4.50)

fdet(: : :) Deterministic function of . . .

femp(: : :) Empirical function of . . .

G Test statistic (estimate) of Grubb'soutlier test

(4.36)

Gmax Test statistic (estimate) of Coch-ran's test

(4.38)

G˛;n Signiˇcance limits of Grubb's out-lier test for a risk of error ˛ and nindividual measurements

gaccept(n) Acceptance function in attributetesting

greject(n) Rejection function in attribute test-ing

gaccept(n; s; ˛) Acceptance function in variable test-ing

gaccept(n; s; ˇ) Rejection function in variable testing

H 0 a posteriori information entropy

H a priori information entropy

H0 Null hypothesis

HA Alternative hypothesis

hom(A) Quantity characterizing the homo-geneity of an analyte A in a sample

(2.9)

I Information content (9.1)

IAj Speciˇc strength of an in�uence fac-tor j on the signal of analyte A

I(p ; p0) Divergence measure of information (9.9)

J Information �ow (9.40)

XVI Symbols

L Loading matrix (of principal compo-nent analysis and factor analysis)

(6.44)(8.19)

LCL Lower control limit (in quality con-trol)

LWL Lower warning limit (in quality con-trol)

lx , ly , lz Dimensions (lengths) in x-, y - andz-direction, respectively

lb a Binary logarithm of a

M Information amount

M(n) Information amount of multicompo-nent analysis (of n components)

(9.21)

madfyig Median absolute deviation (aroundthe median), i.e., the median ofthe differences between an individ-ual measured value and the medianof the series: madfyig = medfjyi �medfyigjg

medfyig Median of a set of measured values (4.22)

N Number of repeated measurementsfor the estimation of x by means ofa calibration function

Npp Noise amplitude, peak-to-peak noise

N (�; � 2) Normal distribution with mean �and variance � 2

N (t) Time average of noise

NPR Negative prediction rate (in screen-ing tests)

(4.55)

n Number of individual measurements(also in establishing a calibrationmodel), number of objects (in a dataset)

n(t) Noise component of a signal func-tion in the time domain

neti(t), net(i) Net function (propagation function)in a neural network (layer i)

(6.117)

Oi(t), O (i) Output of a layer i of a neural net-work

(6.119)

Symbols XVII

P Two-sided level of signiˇcance of sta-tistical tests (P = 1� ˛)

P One-sided level of signiˇcance of sta-tistical tests (P = 1� ˛)

P (A) Probability of an event A (ratio ofthe number of times the event occursto the total number of trials)

P (A) Probability that the analyte A ispresent in the test sample

P (A) Probability of the opposite of theevent A (ratio of the number of timesthe event does not occur to the totalnumber, P (A) + P (A) = 1)

P (A) Probability that the analyte A ispresent in the test sample

P (AjB) Conditional probability: probabilityof an event B on the condition thatanother event A occurs

P (T +jA) Probability that the analyte A ispresent in the test sample if a testresult T is positive

P Score matrix (of principal compo-nent analysis)

(6.44)

PB Prediction band (of a calibrationline)

(6.26b)

PPR Positive prediction rate (in screeningtests)

(4.54)

PV Prevalency of screening tests: proba-bility that the analyte A is present ina given number of samples

(4.52)

p Number of primary samples (in sam-pling);Number of calibration points

pi, pj , pij In�uence parameters (4.25)

prec(x) Precision of an analytical result x (7.8)

prd(x), prd(a) Prediction interval of a quantity x(and a, respectively)

q Number of subsamples (in sampling) (2.5)

qR Test statistic (estimate) of the Davidtest

(4.33)

XVIII Symbols

Q Categorial variable characterizingchemical entities (species) being in-vestigated, e.g., elements, isotopes,ions, compounds

Q Test statistic (estimate) of Dixon'soutlier test

(4.35)

Q(z) Evaluation relationship (of quanti-ties characterizing qualitative prop-erties) in form of a function, atlas,or table

Q = f �1(z) Evaluation function of quantitiescharacterizing qualitative properties,viz type of species, Q, in dependenceof signal position, z

R Range (R = ymax � ymin)

R Resolution power

R Correlation matrix (6.4)

RC Risk for customers (in quality assur-ance)

(4.56b)

RM Redundancy (9.31a)

RM Risk for manufacturers (in qualityassurance)

(4.56a)

Rt Temporal resolution power (7.57)

Rz Analytical resolution power (7.53)

RMSP Root mean standard error of predic-tion

(6.128)

RR Recovery rate

Rfqg Range of a quantity q

r Number of measurements at a sub-sample

(2.4)

rM Relative redundancy (9.31b)

rxy Correlation coefˇcient between the(random) variables x and y

(6.3)

rob(A=B : : : ; f1 : : :) Robustness of a procedure to deter-mine an analyte A with regard to dis-turbing components (B; : : :) and fac-tors (f1; : : :)

(7.31)

Symbols XIX

rug(A=B : : : ; f1 : : : ; u1 : : :) Ruggedness of a procedure to deter-mine an analyte A with regard to dis-turbing components (B; : : :), factors(f1; : : :), and unknowns (u1; : : :)

(7.33)

S Sensitivity, derivative of the mea-sured quantity (response) y with re-spect to the analytical quantity x

Stotal Total multicomponent sensitivity (7.18)

S Covariance matrixSensitivity matrix

(6.5)(7.17)

SAA Sensitivity of the response of the an-alyte A with respect to the amountxA of the analyte A (also SA)

(7.12)

SAB Cross sensitivity (partial sensitivity)of the response of the analyte A withrespect to the amount xB of the com-ponent B

SAi Cross sensitivity (partial sensitivity)of the response of the analyte A withrespect to the amount xi of species i(i = B; C; : : : N )

(3.11)

Sxx , Sy y Sum of squared deviations (of x andy , respectively)

(6.30)

Sxy Sum of crossed deviations (of x andy )

(6.30)

S�, S�i Sum over a variable index (e.g., atconstant i); the dot replaces the in-dex over that the summation is car-ried out

S(x) Sensitivity function (6.64)

s Estimate of the standard deviation(SD of the sample)

(4.12)

s2 Estimate of the variance (variance ofthe sample)

(4.10)

sN Standard deviation of noise (7.5)spipj , s(pi; pj) Covariance of the quantities pi

and pj(4.27)

sy :x Residual standard deviation (of thecalibration); estimate of the error ofthe calibration model

(6.19)

XX Symbols

sel(A; B; : : : N ) Selectivity of a multicomponentanalysis with regard to the analytesA; B; : : : N

(7.24)

snr(y ) Signal-to-noise ratio of an averagesignal intensity y (related to Npp )

(7.6)

spec(A=B; : : : ; N ) Speciˇcity of the determination of ananalyte A with regard to the accom-panying components B; : : : N

(7.26)

S=N Signal-to-noise ratio of an averagesignal intensity y or a net intensityy net (related to sy )

(7.1)

(S=N )c Critical signal-to-noise ratio (7.51)

SS Sum of squares

ssr(y ) Sum of squares of residuals (of y ) (6.12)

tW Test statistic (estimate) of the gener-alized t test (Welch test)

(4.43)

TE Test efˇciency of screening tests (4.53b)

TNR True negative rate (of screeningtests)

(4.49)

TPR True positive rate (of screening tests) (4.48)

t Test statistic (estimate) of Student'st test

(4.41)

t1�˛;� Quantile of the t-distribution at thelevel of signiˇcance 1� ˛ and for �degrees of freedom

UCL Upper control limit (in quality con-trol)

UWL Upper warning limit (in quality con-trol)

u(x) Combined uncertainty of an analyti-cal value x

(4.31)

u(x(p1; p2; : : :)) Uncertainty of an analytical value x,combined from the uncertainties ofthe parameters p1; p2; : : :

u(y ) Combined uncertainty of a measuredvalue y

u(y (p1; p2; : : :)) Uncertainty of a measured value y ,combined from the uncertainties ofthe parameters p1; p2; : : :

(4.25)

Symbols XXI

U (x) Extended combined uncertainty(limit) of an analytical value x

U (y ) Extended combined uncertainty(limit) of a measured value y

(4.29)

unc(x) Uncertainty interval of a mean x (4.32)

unc(y ) Uncertainty interval of a mean y (4.30)

wyi , wi Weight coefˇcient (in weighted cali-bration)

(6.34)

x Analytical value: analyte amount,e.g., content, concentration

xLD Limit of detection (7.44)

xLQ Limit of quantitation (7.48)

xQ Analytical value (amount) of aspecies Q

xtest Analytical value of a test sample

xtrue (Conventional) true value of a (cer-tiˇed) reference sample

x Arithmetic mean of x (of the sample)

x Estimate of x

x(Q) Sample composition (function)

x = f (Q) Analytical function in the sample do-main, sample composition function

Y (!) Signal function in the frequency do-main, Fourier transform of y (t)

y Measured value, response (e.g., in-tensity of a signal)

yc Critical value (7.41)yzi Measured value (e.g. intensity) of a

signal at position ziy Arithmetic mean of y (of the sam-

ple)

y Total mean of several measured se-ries

y BL Blank

y geom Geometric mean of y (of the sample)

y Estimate of y

XXII Symbols

y � Outlier-suspected value among themeasured values

y (t) Signal function in the time domain

ytrue(t) Component of the signal function inthe time domain that is consideredbeing true (in�uenced by noise)

y (z) Signal record (spectrum, chromato-gram, etc.)

y = f (x) Calibration function (in the strictersense, i.e. of quantities characteriz-ing quantitative properties, e.g., sig-nal intensity vs analyte amount)

y = f (z) Signal function (measurement func-tion, analytical function in the signaldomain)

z Signal position: measuring quantitythat depends on a qualitative prop-erty of the measurand. Therefore,analytes may be identiˇed by charac-teristic signal positions. The z-scalemay be directly or reciprocally pro-portional to an energy quantity ortime

z z-scores, standardized analytical val-ues

(8.9)

z(Q) Signal assignment function (table,atlas etc.)

z = f (Q) Calibration function of quantitiescharacterizing qualitative properties,viz signal position as a function ofthe type of species

˛ Risk of the error of the ˇrst kind(two-sided)

˛ Regression coefˇcient (intercept) (4.2)

˛i Coefˇcient characterizing in�uences (5.3)

˛ Risk of the error of the ˇrst kind(one-sided)

ˇ Risk of the error of the second kind(two-sided)

Symbols XXIII

ˇ Regression coefˇcient (slope) (4.3)

ˇj Coefˇcient characterizing in�uences (5.4)

ˇ Risk of the error of the second kind(one-sided)

�2 Test statistic (estimate) of the �2 test

�21�˛;� Quantile of the �2 distribution at the

signiˇcance level 1�˛ and for � de-grees of freedom

�xcnf Conˇdence limit of a mean x

�xprd Prediction limit of a mean x

�y cnf Conˇdence limit of a mean y

�2 Test statistic (estimate) for von Neu-mann's trend test

(4.34)

Ffx(t)g Fourier transform of the time func-tion x(t) into the frequency functionX(!)

F�1fX(!)g Fourier backtransform of the fre-quency function X(!) into the timefunction x(t)

� Exponent characterizing nonlinearerrors

(4.5)

� Eigenvalue (characteristic root) of amatrix

M Multiplet splitting according to rele-vant rules

�x Mean of x of the population

�y Mean of y of the population

� Number of the statistical degrees offreedom

� Dispersion factor (4.20)

� Standard deviation (of the popula-tion)

� 2 Variance (of the population) (4.8)

� Kaiser's selectivity (7.21)

XXIV Symbols

A Kaiser's speciˇcity with regard toan analyte A

(7.22)

xx() Autocorrelation function (ACF) of afunction x(t) with time lag

(2.11)

(˛ˇ)ij Coefˇcient characterizing interac-tions of in�uences ˛ and ˇ

(5.4)

(˛ˇ� )ijk Coefˇcient characterizing interac-tions of in�uences ˛, ˇ, and �

(5.5)

fxig Set (series, sequence) of ana-lytical results whose terms arex1; x2; : : : ; xi; : : :

fyig Set of measured values (observationsof a measurement series) whoseterms are y1; y2; : : : ; yi; : : :

a; b ; : : : Estimates of the quantities a; b; : : :

In general, variables are expressed by italics, vectors by bold small letters,and matrices by bold capital letters.

Abbreviations and Acronyms

This list of abbreviations contains both acronyms which are generally usedin analytical chemistry and such applied in the book. In addition to termsfrom analytical methods, essential statistical and chemometrical terms aswell as acronyms of institutions and organizations are included. Terms ofvery particular interest are explained on that spot.

2D Two-Dimensional (e.g., 2D-NMR)3D Three-DimensionalAA Activation AnalysisAAS Atomic Absorption SpectrometryACF Autocorrelation FunctionACV Analytical value at critical (measuring) valueAEM Analytical Electron MicroscopyAES Auger Electron SpectroscopyAES (Atomic Emission Spectrometry)! OAES, OESAFM Atomic Force MicroscopyAFS Atomic Fluorescence SpectrometryANN Artiˇcial Neural NetworksANOVA Analysis Of VarianceAOAC Association of Ofˇcial Analytical ChemistsARM Atomic Resolution MicroscopyARUPS Angle Resolved Ultraviolet Photoelectron SpectrometryASV Anodic Stripping VoltammetryATR Attenuated Total Re�ectanceBASIC Programming language: Beginners All-purpose SymbolicBCA Beckman Glucose AnalyzerBIPM Bureau International des Ponds et MesuresCA Chemical AnalysisCARS Coherent Anti-Stokes Raman SpectrometryCCC Counter Current ChromatographyCCD Charge-Coupled DeviceCE Capillary Electrophoresis

XXVI Abbreviations and Acronyms

C{E{R Calibration-Evaluation-Recovery (Function)CFA Continuous Flow AnalysisCGC Capillary Gas ChromatographyCMP Chemical Measurement ProcessCPAA Charged Particle Activation AnalysisCRM Certiˇed Reference MaterialCV Critical valueCV-AAS Cold Vapour Atomic Absorption SpectrometryDCM Dielectric Constant Measurement (Dielcometry)Dendral Expert system: Dendritic AlgorithmDIN Deutsches Institut f�ur NormungDSC Differential Scanning CalorimetryDTA Differential Thermal AnalysisDTG Differential ThermogravimetryEBV Errors in Both Variables (Model, Procedure)ECA Electrochemical AnalysisECD Electron Capture DetectorED Electron DiffractionEDX Energy-Dispersive X-Ray (Spectrometry)ED-XFA Energy-Dispersive X-Ray Fluorescence AnalysisEDL Electrodeless Discharge LampEELS Electron Energy Loss SpectrometryEM Electron MicroscopyEMP Electron MicroprobeEN European NormEPA Environmental Protection Agency (USA)EPH ElectrophoresisEPMA Electron Probe MicroanalysisESAC Expert Systems in Analytical ChemistryESCA Electron Spectroscopy for Chemical AnalysisESD Estimated Standard DeviationESR Electron Spin Resonance (Spectroscopy)ETA Electrothermal AtomizerETA-AAS Electrothermal Atomizing Atomic Absorption SpectrometryEXAFS Extended X-Ray Absorption Fine Structure (Spectrometry)FAB Fast Atom BombardmentFANES Furnace Atomization Non-Thermal Emission SpectrometryFD Field DesorptionFEM Field Electron Microscopy

Abbreviations and Acronyms XXVII

FFT Fast Fourier TransformFIA Flow Injection AnalysisFI-AP Field Ion Atom ProbeFID Flame-Ionization DetectorFIM Field Ion Microscopyfn false negative (decisions in screening tests)fp false positive (decisions in screening tests)FNAA Fast Neutron Activation AnalysisFortran Programming language: Formula TranslationFT Fourier TransformFT-ICR-MS Fourier Transform Ion Cyclotron Resonance Mass Spectrom-

etryFT-IR, FTIR Fourier Transform Infrared (Spectrometry)FT-MS Fourier Transform Mass SpectrometryFT-NMR Fourier Transform Nuclear Magnetic Resonance Spectrome-

tryFWHM Full Width at Half MaximumGA Genetic AlgorithmGC Gas ChromatographyGC-IR Gas Chromatography Infrared Spectrometry CouplingGC-MS Gas Chromatography Mass Spectrometry CouplingGDL Glow Discharge LampGDMS Glow Discharge Mass SpectrometryGDOS Glow Discharge Optical SpectrometryGF-AAS Graphite Furnace Atomic Absorption SpectrometryGLC Gas Liquid ChromatographyGLS Gaussian least squares (regression)GLP Good Laboratory PracticeGMP Good Manufacturing PracticeHCL Hollow Cathode LampHEED High Energy Electron DiffractionHEIS High Energy Ion ScatteringHG-AAS Hydride Generation Atomic Absorption SpectrometryHPLC High Performance Liquid ChromatographyHPTLC High Performance Thin-Layer ChromatographyHR High ResolutionHRMS High Resolution Mass SpectrometryIC Ion ChromatographyICP Inductively Coupled Plasma (Spectrometry)

XXVIII Abbreviations and Acronyms

ICP-MS Inductively Coupled Plasma Mass SpectrometryICP-OES Inductively Coupled Plasma Optical Emission SpectrometryICR Ion Cyclotron ResonanceID Ion DiffractionIDMS Isotope Dilution Mass SpectrometryIEC Ion Exchange ChromatographyIEC International Electrotechnical CommissionIFCC International Federation of Clinical ChemistryIM Ion MicroscopyIMS Ion Mobility SpectrometryINAA Instrumental Neutron Activation AnalysisINDOR Internuclear Double ResonanceINS Inelastic Neutron ScatteringIR, IRS Infrared SpectroscopyIRM Infrared MicroscopyISE Ion-Selective ElectrodesIRS Internal Re�ectance SpectroscopyISO International Organisation for StandardizationISS Ion Surface Scattering, Ion Scattering SpectrometryIU Insulin unitIUPAC International Union of Pure and Applied ChemistryIUPAP International Union of Pure and Applied PhysicsLAMMS Laser (Ablation) Micro Mass SpectrometryLASER Light Ampliˇcation by Stimulated Emission of RadiationLC Liquid ChromatographyLC-MS Liquid Chromatography Mass Spectrometry CouplingLD Limit of DetectionLDMS Laser Desorption Mass SpectrometryLEED Low Energy Electron DiffractionLEIS Low Energy Ion ScatteringLEMS Laser Excited Mass SpectrometryLIDAR Light Detection And Ranging (analogous to RADAR)LIMS Laboratory Information Management SystemsLISP List ProcessingLLC Liquid Liquid ChromatographyLM Light MicroscopyLMA Laser Microspectral AnalysisLM-OES Laser Micro Optical Emission SpectroscopyLOD Limit Of Detection

Abbreviations and Acronyms XXIX

LOS, LoS Level of signiˇcanceLQ Limit of quantitationLRMA Laser Raman Micro AnalysisMALDI Matrix-Assisted Laser Desorption and IonizationMALDI-TOF MALDI Time-Of-Flight (Mass Spectrometry)MEIS Medium Energy Ion ScatteringMES M�ossbauer Effect SpectroscopyMIP Microwave-Induced PlasmaMLD Measured value at Limit of DetectionMLQ Measured value at Limit of QuantitationMORD Magneto Optical Rotary DispersionM �OS M�ossbauer SpectrometryMS Mass SpectrometryMS-MS Tandem Mass SpectrometryMSn n-fold Tandem Mass SpectrometryM-OES Micro(spark) Optical Emission SpectroscopyMWS Microwave SpectroscopyNAA Neutron Activation AnalysisNBS National Bureau of Standards, USA (today: NIST)ND Neutron DiffractionNEXAFS Near-Edge X-Ray Absorption Fine Structure (Spectroscopy)NIR, NIRS Near Infrared (Spectrometry)NIST National Institute of Standards and Technology, USANMR Nuclear Magnetic Resonance (Spectroscopy)NOE Nuclear Overhauser EffectNQR Nuclear Quadrupole ResonanceOAES Optical Atomic Emission SpectroscopyOCS Out-of-control situations (in quality control)OES Optical Emission SpectroscopyOIML Organisation Internationale de M�etrologie L�egaleOLS Ordinary least squares (regression)ORD Optical Rotary DispersionPARC Pattern RecognitionPAS Photo Acoustic SpectroscopyPASCAL Programming language called by Blaise Pascal (1623{1662)PC Paper ChromatographyPCR Principal component regressionPES Photoelectron SpectrometryPIXE Particle Induced X-Ray Emission

XXX Abbreviations and Acronyms

PLS Partial least squares (regression)PMT Photomultiplier Tubeppm part per million (10�6 corresponding to 10�4 per cent)ppb part per billion (10�9 corresponding to 10�7 per cent)ppt part per trillion (10�12 corresponding to 10�10 per cent)ppq part per quatrillion (10�15 corresponding to 10�13 per cent)PROLOG Programming language: Programming in Logic

QCC Quality control chartsQMS Quadropol Mass SpectrometerR&D Research and DevelopmentRADAR Radiowave Detection And RangingRBS Rutherford Backscattering SpectrometryREELS Re�ection Electron Energy Loss SpectrometryREM Re�ection Electron MicroscopyRHEED Re�ection High Energy Electron DiffractionRF RefractometryRIMS Resonance Ionization Mass SpectrometryRM Reference MaterialRMSP Root mean standard error of predictionRPC Reversed Phase ChromatographyRPLC Reversed Phase Liquid ChromatographyRRS Resonance Raman ScatteringRS Raman SpectroscopySAM Scanning Auger MicroscopySAM Standard Addition MethodSEC Size Exclusion ChromatographySEM Scanning Electron MicroscopySERS Surface Enhanced Raman ScatteringSFC Supercritical Fluid ChromatographySI Syst�eme International (d'Unit�es)SIMS Secondary Ion Mass SpectrometrySNMS Sputtered Neutrals Mass SpectrometrySNR Signal-to-Noise RatioSOP Standard Operating ProcedureSPE Solid Phase ExtractionSPME Solid Phase Micro ExtractionSRM Standard Reference MaterialSSMS Spark Source Mass SpectrometrySTEM Scanning Transmission Electron Microscopy

Abbreviations and Acronyms XXXI

STM Scanning Tunneling MicroscopySV Standard valueTA Thermal AnalysisTCD Thermal Conductivity DetectorTEELS Transmission Electron Energy Loss SpectrometryTEM Transmission Electron MicroscopyTG ThermogravimetryTGA Thermogravimetric AnalysisTHEED Transmission High Energy Electron DiffractionTIC Total Ion ChromatogramTID Thermoionic DetectorTLC Thin-Layer ChromatographyTMA Thermomechanical Analysistn true negative (decisions in screening tests)TOF-(MS) Time-Of-Flight (Mass Spectrometry)tp true positive (decisions in screening tests)TXRF Total Re�ection X-ray Fluorescence (Spectrometry)UPS Ultraviolet Photoelectron SpectrometryUV Ultraviolet (radiation)UV-VIS Ultraviolet-Visible (Spectrometry)VIS Visible (radiation)WD-XFA Wavelength-Dispersive X-Ray Fluorescence (Spectrometry)XAS X-Ray Absorption SpectrometryXD X-Ray DiffractionXFA X-Ray Fluorescence AnalysisXFS X-Ray Fluorescence SpectrometryXPS X-Ray Photoelectron SpectrometryXRD X-Ray DiffractionXRF X-Ray Fluorescence (Spectrometry)ZAF Z (stands for atomic number) Absorption Fluorescence (Cor-

rection) (XFA)ZAAS Zeeman Atomic Absorption Spectrometry