Quantitation in elemental analysis

Quantitation in This is the first of two parts of an article on Quantitation in Elemental Analysis which grew out of Professor Kaiser's participation as a speaker at the 22nd Annual Summer Symposium on Analytical Chemistry. This meeting was held June 11 to 13, 1969, at the University of Georgia, The second part of the article will appear in the April issue of ANALYTICAL CHEMISTRY

H. Kaiser lnstitut fur Spektrochemie und Angewandte Spektroskopie Dortmund, Germany

I was asked by Prof. G. H. Morrison to give a 40-minute talk about "Quantitation in Elemental Analysis" during the ACS Symposium on Analytical Chemistry at Athens, Ga., in June 1969. Since I did not know, what "quantitation" exactly meant, I started thinking about the general role of numbers in chemical analysis. As we know from Shakespeare's "Julius Caesar," thinking is dangerous and Prof. Morrison may not have been aware what he did to me by choosing such a stimulating topic. The short talk at Athens could give only the main lines and indicate the problems. I t was followed by lively discussions in small groups during the Symposium and thereafter. A first attempt to write a brief article for ANALYTICAL CHEMISTRY failed completely. I realized that the conventional ideas about numbers, statistics, and information, widely held among chemists, are not uniform and are not precise enough to allow a "shorthand" presentation. Basic presuppositions and implications are not generally known. The technical terms have come so much in vogue recently that the omission of a commonly accepted, precise scientific language for our field is overlooked. Hence I was forced t o go through all this once more and to make it clear for myself. This is the result. There are practically no concepts, relations, and thoughts in this article which are new; most of them are at present " i n the air"; I myself do not really know from which person or book I was first introduced to them. For this reason, I had to abstain from giving a haphazard general selection of references from the huge literature. I wish to express my gratitude to all who have contributed to my knowledge. The only responsibility remaining with me refers to the selection and composition of the following mosaic. 24A

ANALYTICAL CHEMISTRY, VOL. 42, NO. 2, FEBRUARY 1970

FORlong

A time chemists have been using numbers in a rather innocent way to describe the results of their analysis. Recently, they have lost their innocence and eaten from the apple in Paradise. The Paradise is lost and lost also is-at least partly-that magic feeling of the born chemist for the transformation of matter, which may be the remainder of the mysterious practice of alchemy. What have we won? Recognition of the good and the evil? The good may be the power of abstract thinking and its success with all its delight to the human intellect. The evil on the other side may be the pitfalls of deception, the constant temptation to take abstract structures as a full picture of nature and, even worse, of human life. I n chemical analysis this dangerous situation is prevalent, due to the advent of the computer, the new possibilities of data processing, and by the progressing adaptation of mathematical statistics. The blind belief in the perfection of measuring instruments, electronics, and formal mathematical relations is a real danger. All this began-slowly -33 years ago in spectrochemical analysis. The main topic of this paper will not be mass production of analysis data, computer programs, methods of numerical mathematics, or some such-all of these may be found elsewhere. I intend to consider the intellectual situation, to point out which decisions are a t stake and which questions about nomenclature and presentation of results must be settled. Many people are aware of these problems. Proposals are made and often repeated independently in slightly different form. However, it is a pity to observe that a good many of these efforts are useless because the newcomers to this realm just do not know enough about the many implications and

Elemental Analysis the long history of scientific thinking in this field. This is the cause of much confusion. We are in the whirlpools between Scylla and Charybdis of overbelief in formal or technical procedures and repulsion caused by misplaced erudition from the side of the missionaries. However, as soon as we have passed these uncomfortable straits, a quiet sea may be before us. The ideas are simple, the necessary formulas are simple also and should be considered as ready-made mental tools. For their correct use, it is not necessary to study in detail the work of the toolmakers-in this case of generations of mathematicians. Description of Scientific Facts by Numbers

Many thousands of years ago, man made a great intellectual discovery when he recognized t h a t five men, five trees, five stars, five words, and five thoughts had something in common: the “fivehood,” if such a word is allowed. Why is it possible not only to count things by numbers but to describe structures by using numbers? We have to consider what happens in the chain of communication between two persons or between an apparatus and its observer. There are signals transferring the information between them. Signals are configurations in space and time-e.g., sound waves, light, letters, punch tapes. All these signals are finite in space and in time. This fact has a remarkable consequence which will be explained by a simple formal example: Let us suppose t h a t we have a signal which occurs during some time. It may be described by a mathematical model, a time-dependent function: F ( t ). With some general presuppositions as t o the mathematical character of such functions (which mainly are valid), we may represent this function by its Fourier

transform. Because the signal is finite i t will have an effective duration time A t = t2 - tl and its Fourier transform will therefore have an effective bandwidth of frequencies. Now let us ask what precision of frequency determination is possible during the lifetime A t of the signal. Let S0v be the smallest difference in frequency which can be distinguished. Then the “resolving power” Ro for frequencies would be given by

Ro

=

V/&V

(1)

The “resolving power” Ro may as well be given for wavelengths or periods if such quantities are considered:

The (practically attained) resolution R is defined correspondingly: R4 = v/--Sv. Always is R 5 Ro. This is a formal definition which has proved itself t o be useful, and therefore it has been generally accepted. As with all definitions, there is nothing to be understoodit is a convention. This smallest distinguishable frequency difference S0v depends on the duration A t of the signal. The relation is At*&v = 1

(2)

The derivation of this basic equation is so simple t h a t its full comprehension may be hampered by its abstract simplicity: Let us suppose we should decide whether the frequencies v of two signals are equal within a precision of 1/100,000. Hence SV = v/lOO,OOO and the required resolution of the measuring device must be a t least R = 100,000. No additional information about the functions constituting the two signals may be given; the only fact known is t h a t they are periodic functions. For example, one may

be a sequence of short pulses, t h e other a sine wave. I n such a case, the only experimental way is the observation of coincidences (equal or similar constellations-e.g., in an interference pattern) and this obviously means t h a t only whole periods can be counted and no fractions thereof. Therefore, to get in our example the required precision of 1/100,000 in frequency discrimination we have to count 100,000 full periods, as many as the required resolving power Ro indicates. This requires a time of R-T and t o perform the experiment, this time must not be longer khan the effeotive lifetime At of the signals, hence At

2 R.7

1/6v For the smallest distinguishable frequency difference, S0v, we must take the equality sign and this gives us our Equation 2, At 60v = 1. Corresponding equations are valid for other pairs of variablese.g., length and spatial frequency (wave number). Equations of this type give the limits of observations; they are general “uncertainty equations.” I n fact, by multiplying both sides of Equation 2 with Plancks constant h and using the symbol A B for the observable difference of energy h a0v, we get = V / ~ V . T=

(3) which is one of the equations of Heisenberg’s Uncertainty Principle in quantum mechanics. Before we use uncertainty Equation 2 t o discuss the informing power of an analytical method, it may be mentioned t h a t Equation 2 leads immediately to the wellknown formulas for the theoretical resolving power Ro of spectroscopic instruments (prisms, gratings, interferometers). One has only to consider which maximal difference A t in traveling time can be At*AE = h


25A

Report for Analytical Chemists

attained experimentally for the extreme rays of the light beam passing the instrument. (This is a nice problem for a student textbook.) ?Tow we are prepared t o ask how many “details” can be, a t best, transmitted by a signal having an effective duration A t and an effective bandwidth of Av. Let us first ask how many different frequencies can be distinguished within the range from va to V b in the bandwidth Av. As the smallest distinguishable difference is S0v, the wanted number K must be Av/Sov or, by using Equation 2,

K = Av.At (4) For each of these K different frequencies, vi, the amplitudes A (vi) are given, from which we constitute the original signal F ( t ) . Again, we are in the same situation: All amplitudes lie in a certain range AA mostly between A = 0 and a possible maximum A,,, with a smallest &A. distinguishable difference Therefore, we have a maximal number S of steps for the amplitude

s = AA/&A

(5)

The maximal number D of discernible different structural details contained in the signal (finite within the ranges Av, At, A A ) , is

D = Av.At.8 (6) The structural details therefore can be counted; they can be labeled by numbers chosen according to suitable rules, and handled by such numbers-all this regardless of their original significance and value in human life. [For the critical reader: I n general, a Fourier transformation leads to a series of sine functions and a series of cosine functions, thus taking care of the phases also. I n such cases Equation 6 should be written D = 2 At . AV . S, because each frequency v occurs twice (has a complex amplitude) . ] This fact explains the overwhelming power of “quantitation,” or the use of numbers in modern life-also in chemistry; but beyond that, the old philosophical question of quality US. quantity presents a surprising new aspect, Before we continue the main line 26A

of our thoughts, a word about SOA, the smallest distinguishable difference of the amplitude, may be appropriate. For many measurements in physics and chemistry, the amplitudes of the Fourier transform (a “spectrum’, in a generalized meaning) are related to energy. Very often A2 is proportional to power or to the flux of elementary particles (photons, electrons). There is a principal lower limit of energy, below which no information a t all can be transmitted. This limit is given by the thermal energy kT of a single particle a t the respective absolute temperature T. It is : 1.38 . . T joule ( T in Kelvin). If information is transmitted by a low flux of photons, the statistical noise of the photons may set the lower limit presupposing their energy h , v is higher than k T . At, room temperature, the thermal noise ( k T ) prevails for wavenumbers p < 200 cm-l (in the far infrared). Informing Power of Analytical Methods

The author was puzzled by the still ambiguous nomenclature in this field. The word “information” is used in a t least four different meanings, often in the same text. T o avoid this intellectual traffic jam in this paper written for chemists, the following open language nomenclature will be used:

In general sense : Information = process of instruction Information(s) = (new) knowledge with respect to content (preferably plural)

In the sense of metric quantities : Informing power, Pinf(of a method) Informative volume, Vinf (of a signal) Amount of information, Mini (required technically t o communicate some definite desired knowledge) Information capacity (capability of a technical system to transmit or store an amount of information) Concerning Equation 6: There is a better way t o present its meaning if we use some simple concepts from the mathematical “Theory of Information.” T o get a convenient


measure for the “Informative Volume” Vinf o f the signal, we do not take the number D of structural details, but we state how many binary digits (“bits”) are wanted to label e a c r o f these details by its special (binary) number. For the S steps of the amplitude we need (log, S) bit, where the parens indicate that the next whole number greater than log, S must be taken (log, x I pd x =: 10/3 loglo x). Since we must have a special set of binary numbers for each of the K frequencies the maximal informative volume of the signal is Vin* = [Av.At.(logzS ) ]bit

(6a)

[Here “bit” is regarded as a unit (of dimension 1) to measure the informative volume. Therefore it is -as all units-to be written in the singular form. If in a context bhis term is used as an abbreviation of “binary digit” in this concrete sense, the plural should be used.] No bearing on chemical analysis? Wait a moment; you will be rewarded for your long patience! To come down to practical things, let’s take any method of spectrochemical analysis (optical emission or absorption, ir, uv, Raman, nmr, Mossbauer)-normally, the resolving power R = V / S V will not be constant, but a function of V ; the same is true for S, the number of discernible steps for the “intensity” scale (called “amplitude” in 11). Then the right side of Equation 6a which essentially is a sum, will turn into an integral. We only have to replace A t = 1/Sv by V/SV = R ( v ) / v and the phoenix appears:

(7) the formula for the “informing power” Pinfof a spectrochemical method of analysis, working in the spectral range from va to V b . But not only that, this formula can be applied to all measurements, where the measured quantity 2, follows the same formal rules as the frequency v in our example (which was chosen to avoid an entirely abstract introduction). (The formerly used subscript of 0 in S O V and Ro can now

Report

be dropped, since the quantities vary with v and the relations are valid for the practically attained resolution R as for the theoretical limit, the resolving power Eo.) We now have to spread out what Equation 7 contains. The informing power Pinfis a figure of merit for an analytical method. So far it has not been used much, but in the future it may lead to quantitative definitions of such important, yet only qualitative concepts as selectivity, specifity, universality, etc., of analytical methods. Let us assume for a first discussion t h a t R and S are practically constant, so t h a t we may take their average values. Then we get:

It can immediately be seen t h a t the resolving power R is the decisive factor t o drive the informing power t o very high values; the other factors are logarithms and do not add much : High spectral resolving power is more important than a wide spectral range or high precision in measuring intensities. (All these terms should be taken in a generalized way, not confined t o spectroscopy in a narrow sense.) The second factor log2 S would be 7 for about 100 discernible steps a t the intensity scale; even one million steps (very seldom attainable!) would make for not more than 20. I n contrast to that, the resolving power R may lie between a few units and lo6 or more as in high resolution spectroscopy. For a nondispersive, a monochromatic method, Equation 7a is reduced to [log2 S ] bit (7b) If the intensity is measured not throughout the spectrum but only for several selected frequencies V I , v 2 , . . . v,, (polychromatic method) with practically the same precision, then we have Pinf

=

Pinf= n[logz SI bit (7c) or if S is different for the different frequencies n Pinf

= z=1

[log2 Si]bit

(7d)

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Therefore, a relatively coarse measurement a t a second or third frequency m a y provide m u c h more additional informing power than a considerable and perhaps expensive increase of the precision (characterized by 8 ) . From these formulas, the logical -not the practical-reason will be realized, why the classical procedures of chemical analysis had to be replaced by physical measuring procedures-in particular spectroscopic ones-for the solution of complex, universal analytical problems (as the analysis of lunar minerals). hlost classical measuring procedures, such a s gravimetry, titrimetry, volumetry, coulometry, simple absorptiometry, or refractometry, are in principle nondispersive, monochromatic, or unidimensional whichever one of these terms may be the most appropriate. They all have only one variable, one scale, and no spectral resolution; consequently their informing power is very low, in the order of 10 bit. As soon as a second variable appears, in the role of a parameter which changes the conditions for the measurement of the first, indicating variable (e.g., intensity, absorbance, weight, volume, density, current, or refractive index), the situation can be drastically changed, because the number of discernible steps of the parameter enters as a factor in Equation 7, and its simpler forms (Equations 7a, 7c, 7d). Such parameters may be: frequency, wavelengtlh, time, distance, temperature, voltage, etc. ; they all produce some kind of (spectral) resolution for “indicating variable.” Let us consider an example: .4 gnating spectrograph may have a resolving power Ro = 200,000, a spectral range from 2000 A t o 8000 A and 100 steps in the intensity scale a t each wavelength. The informing power offered by such an instrument would be, according t o Equation 7a, =: 2,000,000 bit. For a high resolving mass spectrometer we get the same order of magnitude. This striking result immediately poses two questions: ( a ) Can such a high informing power, which is offered by the core instrument of an

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analytical procedure, actually be used? (b) Is it needed to solve the analytical problem in question? Obviously the general answer t o both questions must be: “No, certainly not in all cases.” This is so far rather meaningless ; however, a thorough examination of the problem is most enlightening. It would be a grave mistake to consider an instrument or an experimental operation in isolation. It must be emphasized, that these are not our objects, but only “complete analytical procedures.” Figures of merit (e.g., informing power, precision, accuracy, or limits of detection) can only be given for a “complete analytical procedure’’ which is characterized b y a distinct analytical task and a specified working program with all details as to instruments, analytical operations, external conditions, evaluation, and calibration. As soon as anything is altered, the analytical procedure becomes a different one. Difficulties in understanding will arise if this basic concept of a “complete analytical procedure” is not sufficiently grasped and not rigidly enough kept in mind. This is not to say t h a t the potential informing power, offered by an instrument or a n operation, should not be considered; on the contrary, this may be necessary to recognize the bottleneck of the procedure. Now back to our questions! Why is a n offered high potential informing power not always used?

Interpretation of Concept of “Informing Power” and Its Application

For a reasonable answer we may discard accidental restrictions in instrumentation or operation, which can be removed by a better matching of parts or steps. However, the usable fraction of the potential informing power may be determined by the character of the measured quantity-i.e., the analytical signal-by practical considerations or because the relative simplicity of the analytical task does not require the use of the full potential inform-


ing power. This is explained by some examples : 1. Instruments. I n atomic emission spectrochemical analysis, the use of a resolving power Ro z 200,000 is justified by the sharpness of the spectrum lines in order to separate the analytical lines from interfering lines or bands. However, only a few lines are measured; the rest of the spectrum is without interest or empty. Therefore, t o get the “usable informing power,” the integration in Equation 7 must be restricted to those wavelength regions which are of analytical importiance and where the high resolving power is actually needed. With direct reading spectrometers, as they are used for routine production control mainly in the metal industry, the situation is more complicated. These instruments use relatively wide exit slits to keep the narrow spectrum lines safely within the slits. Hence their practical resolution R is lower than the resolving power Ro provided by the grating or prism (the factor may lie between 2 and 5 ) . The high resolving power is not used as such, but as a means t o gain dispersion to have the space for the mounting of many slits. The informing power must be calculated by using R and not Ro in Equation 7. However, this informing powerlowered by the wide exit slits, but calculated for the entire spectral range of the instrument-is further and drastically reduced, when the slits are adjusted in fixed positions, so as to pick out only a selection of analysis lines. I n this case the spectrometer is blind between the slits, and the actual informing power must be calculated according t o Equation 7d where n is the number of “channels.” A direct reading spectrometer has a much lower informing power than a spectrograph of the same optical dimensions. It answers only those questions for which it has been prepared and therefore does not show the unexpected. A spectrograph is therefore indispensable for universal qualitative and quantitative analyses, which demand a high informing power. The high potential informing power offered by its optical parts oan actually be used because

the photographic plate, as a receiver and a memory, has a n enormous geometrical information capacity Cinf (measured in bit/ cm2). Again another situation is found in the case of trace determinations of a few elements-e.g., by flame emission or atomic fluorescence. The spectrum is practically empty ; but high resolution is needed t o dilute the background radiation and to improve the intensity ratio from line t o background. No wavelengths close together must be distinguished, the method is polychromatic (Equation 7d must be applied), and a higher resolving power increases only 8, because the number S of distinguishable steps a t the lower end of the scale is dependent on the radiant flux which is proportional t o the (theoretical) resolving power R, of the instrument and to ( A x ) * . This lowers the limit of detection which is of considerable practical importance, but the effective informing power is not much increased due to the logarithmic function. Use of Fabry-Perot

If the spectrum to be examined is practically empty, tJhe idea may occur t o use a relatively inexpensive high-resolving Fabry-Perot interferometer for trace determinations. The instrument is extremely simple, basically consisting of two parallel flat mirrors. Two parameters are of importance, the distance d between the mirrors, and the “finesse” N , which is a function of the optical quality of the mirrors (flatness, reflectance, transmittance). Physically, N is the effective number of interfering beams produced by multiple reflections between the two mirrors (20 . . . N . . . 100). The instrument can be used over a wide spectral range from the middle uv to the near ir. Its resolving power a t the wavenumber is simply

Ro

=

2d-N.P.

(8)

Fabry-Perots with low resolving power (small distance between the mirrors) are the well-known interference filters; high-resolution Fabry-Perots are used t o investi-

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gate the hyperfine structure of spectrum lines ( R z lo’). Does Equation 8 indicate, we might get many bit of informing power for no money, just by increasing the distance between the mirrors? This would be unjust, we have to pay for every bit of information: Highresolving power is compensated by a small “free spectral range,” A? = 1/2 d-ie., region without overlapping by other orders. Equation 7 shows that the usable informing power of all FabryPerots-having the same optical quality-is the same: Pinf= N log, S,if no additional instrument for spectral preresolution is employed. This explains why FabryPerots-in the form of interference filters-are successfully used to solve simple analytical tasks in flame emission spectroscopy, in some inexpensive absorptiometers, and in the measurement of colors. They provide not sufficient informing power for complex problems. This is the place to make a marginal note about resolving power: I t seems a paradox that it is possible to measure the wavelength of an isolated spectrum line or a single mass number in mass spectroscopy with a precision better than t h a t corresponding to the resolving power Ro of the instrument. However, spectral resolution means separation of adjacent lines and not precise determination of the position of an isolated line. It must be realized that such \a determination cannot be performed solely with the informations produced by the spectrograph ; it requires some pre-informations to overcome the barrier of too low a resolving power. It must be known in advance that the spectrum line, in fact, is isolated; the profile of the line and the slit function of the spectrograph should be known ( a t least t h a t the profile is symmetric), and the noise level must be low enough, for example. This question of pre-informations will come up later in more general form. (By the way: The informing power required to give-for example -9 significant (decimal) figures in a high precision measurement is low -only 30 bit.) 2. Observed phenomenon. The

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potential informing power, offered by the “core instrument or procedure” of an analytical method, may not be usable, because the nature of the observed phenomenon does not allow this, whereas the analytioal problem cries for it. Again, spectrochemical methods give a variety of illustrations: I n optical emission spectroscopy, the analysis lines may be blurred by bands (CN!) or overlapping lines from other elements (e.g., Fe, W, C r ) . If this occurs within the natural width of the lines, higher resolution will not help. However, another type of light source, another atmosphere or preseparation of the disturbing elements, mostly the main constituents, may clear the way. I n analytical infrared spectroscopy where we find broad absorption bands, overlapping is worse. Preseparation methods are by distillation, extraction, or chromatography. But there is a principal restriction: The infrared absorption bands of liquids and solids are relatively wide and do not permit full use of high resolving power; the practical spectral resolution of the method, as a whole, may often be determined by the natural width of the bands. This may be even worse for ultraviolet absorption spectroscopy. Consequently the informing power of the whole method after Equation 7 is low, and perhaps not sufficient to solve a given analytical problem. I n such cases, a combination of different methods (physical and chemical) must be used which complement each other. This ensemble constitutes the “complete analytical procedure” with the required informing power. Of course, this has been done very often in practice, but such “compound procedures” could be built more systematically, if their capability to produce information of a certain kind and quantity were examined in advance. Amount of Information Required by Analytical Problem

What is the meaning of “amount of information”? We have touched on this idea several times before, but now it must be formally introduced.


of a whole method, we can decide What is the meaning when we in advance whether a given probstate t h a t a big dictionary contains lem can be solved with given means “more’ information than a small or if this task is hopeless. one, that a course conveys “more” We may realize t h a t we want knowledge than a single lecture, more “money” in the form of bits t h a t a picture on a movie screen is “better” than t h a t shown in TV? ( M i n f j , and we can start to think Obviously, these are purely formal how and where to get it-or we may ask how excess money (“redunstatements, with no regard t o condancy”) could reasonably be used. tent, kind of information, style, importance, morals, and so forth. The This consideration will now be abstract metric quantity which can applied to chemical analysis. It be distilled from statements of this opens new ways for a strategy t o type plays the same formal role as develop complete analytical prothe monetary value, the price, for cedures in a more systematic way the exchange of goods and services than heretofore. The simple conin daily life. What is the common cepts taken over from the modern denominator in the field of informa“Theory of Information” are mental tion? W e have stated previously tools by which a better understandthat all “signals” which are used t o ing can be achieved-how prelimiproduce, transfer, or store informanary knowledge, instruments, and tions (in the sense of knowledge) operations work together to create are finite in space and time. Therenew knowledge. fore, they have only a limited numThis shall be demonstrated first ber, D , of structural details (from by a simple example-it is a famous Equation 6) to which some meaning puzzle, but also an analytical task: can be attributed by convention in Twelve silver coins are given; order to carry informations. one of them is a counterfeit. These details cannot be subdiThe only analytical instrument vided (atomos!j ; they can be either admitted is a lever balance present or absent; their interpretawithout weights. The task is tion can give not more than a t o find out the counterfeit and simple yes-no statement. For conwhether it is too heavy or too venience, the number of structural light. details is represented by the number Questions: Vinf of binary digitals (bits), which ( a ) How many weighing operaare needed t o label all details tions a t least are needed, (Equation 6a). This informative t o find this out? volume in bit describes the capa(b) How must one proceed to bility of a signal to carry informasolve the problem with tion regardless of any meaning; it t h a t minimal number of is its “value” in the currency of weighings? “Information Land”! Which price, Solution of (a). We have t o find in this currency, must be paid to get out for the 12 coins whether one of some distinct information (here in them is too light. This requires 12 the sense of scientific knowledge) ? yes-no decisions. We also must deW e get this price which is called cide for tall 12, whether one is too “amount of information” ( M i n f ) by heavy or not. Therefore a total of a logical analysis. How many yes24 alternative decisions are t o be no decisions are to be made t o commade-23 answers will be “no”; 1 pletely build the desired informaanswer will be “yes” indicating the tion (knowledge, answer to a probcounkerfeit and whether it is too lem, message, s t a t e m e n t a s the heavy or too light. case may be) right from the botT o label all the possible answers tom? The answer to this question to these 24 alternatives with binary is the number of bits which are renumbers, 5 binary digits-ie., bits quired. If we compare this numare required [log, 241 = 5. Howber, the “price” (in bit) of the ever 5 digits provide Z5 = 32 difdesired information with the ferent alternatives (corresponding “money” we have--i.e., the informat o 64 answers) ; 8 of them are not tive volume, Vinf, of the available used for our problem. This fact signals or the informing power Pinf t h a t the five bits are - necessary, but

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not filled up, can be expressed in a n odd, but convenient, way by saying that the problem requires log, 24 = 4.58 bit to be solved. Now for the analytical procedure: Each comparison on the balance gives 3 possible decisions-equal and overweight left or right. This corresponds t o log, 3 = 1.58 bit. The answer to (a) is 3 weighings are sufficient, for they have an informing power of 3:1.58 = 4.74 bit. Or expressed in open language: Each weighing opens 3 different ways for the next one. Therefore 3 weighings give 3 . 3 3 = 27 alternatives, 3 more than required by the problem, log, 27 = 4.74. Solution of (b). The procedure itself will not be given here in order t o leave the pleasure for the reader. However there is a “strategic” consideration, which may lead the way: The informing power offered by the three-step procedure is only a little greater thfanthe required amount of information; nothing must be wasted. I n consequence, the three weighings must contribute as much as possible and nearly equal amounts of information. This means they must be chosen in such a way t h a t the three possible results of each of them may occur with equal probability. [For further discussion, see I. D. Fast a . F.L.H.M. Stumpers, Philips Tech. Rundsch., 18 164-176 (1956/57) 1. Now some analytical problems shall be investigated with regard t o the amount of information which they require. 1. A sample of unknown composition shall be analyzed for all elements. The concentrations shall be given in a scale of 1000 steps. This is reasonable: If the concentrations we go from 100% down t o have 100 different values within each order of magnitude. On an equally subdivided logarithmic scale, the concentration would grow from step t o step by a factor 1.023. Of course, nothing forbids choosing another subdivision with coarse steps in the lower range and finer ones in the upper range. According t o Equation 7c, the amount of information Mini required for about 100 elements is 100 . log, 1000 = 1000 bit. There are three universal analyti-


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cal principles (not methods !) which are used to derive analytical procedures having the required informing power and providing information of the right type: optical emission, X-ray emission, and mass spectroscopy. They all use properties of the atoms which are common to all elements; the first two use characteristic radiation coming from the atomic shell, the third uses the mass of the nuclides, properties which are correlated to the chemical behavior of the elements. They do not give “circumstantial evidence,” as most other analytical procedures do; they present the “calling cards” of the elements. This explains their universality and potency. What are the figures? A medium quartz spectrograph has an average resolving power of 10,OOO. The intensity scale may give 30 steps for each spectrum line on a photographic plate, corresponding to 5 bit. The informing power is then in the order of 50,000, much more than required. The excess is used to overcome difficulties, )as overlapping of lines and bands, to apply several analysis lines for the element, to increase the precision, and so forth. Together with a set of different light sources (arc, spark, glow-discharge) , the problem in general can be solved. This was the state of the art for several decades. The new grating instruments with high resolving power are used for convenience of observation to arrive a t lower limits of detection and to disentangle spectra which are extremely rich in lines. The K and L lines in X-ray spectra behave much more orderly according to hloseleys l a w i t h e r e fore a resolving power of 300 and the possibility to count lo6 pulses with a relative standard deviation of 0.001, together with the wide wavelength range from BK, (68 A) to CeK, (0.34 A) (In y b / y a M 5 ) should formally give an informing power of 10,000 to 15,000 bit. [For the elements from Ce to U the L-spectra are used, UL, ( 1 A)]. However, most commercial instruments are built for production control; they use electronic detectors adjusted to determine only those elements which are of special inter-


est; only a small fraction of the potential informing power is used; the X-ray spectrometers are rarely employed to detect and determine all elements simultaneously in totally unknown samples. Their primary field of application is the very precise determination of the main constiituents (0.1% . . . 100 % ) in technical materials, an important analytical task which justifies the costs of these instruments. The a m o u n t of information in bit is not very high, but the t y p e of information, which they provide, is of high economic value. It is most instructive to consider another type of X-ray spectrometer -the electron microprobe. When such an instrument is used t o determine 5 elements and gives 10 steps for the concentration of each of them, there are about 15 bit for the analysis of a single spot-not very much. However, if the focus of the electron beam has a diameter of 1 pm, such an analysis can be gained for 1 million different places within a scanned area of 1 mm2, and can be shown on a plake or screen. The analytical task-to give a picture of the distribution of the elements over the surface of the sample-requires a very high geometrical “information capacity”; it is in the order of lo7 bit/ mm2. The technical means t o get this, are expensive. Relation between Analytical Problem and Analytical Procedure

Commercial spark source mass spectrometers have a resolving power Ro in the order of 5000, a wide concentration range of 108, which may be subdivided in 200 steps, and a quotient mb/ma 35 for one fixed adjustment of the spectrometer. ( I n mass spectrometry, the “spectroscopic variable” commonly used is not the de Broglie wavelength of the ion beam, but the mass number, m, going from 1 for hydrogen, over 238 for uranium, t o several thousand for big molecules. Therefore in Equations 7 and 7a, the letter v must now be replaced by m. This is only a formality.) Therefore the informing power is in the order of 150,000 bit and our universal analytical problem can be solved, presupposing the

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type of sample allows the use of this instrument. A masg spectrum is very orderly; the lin$s produced by the singly ionized atoms of the elements form a sequence with (nearly) equal distances. Overlapping of analysis lines by lines from doubly ionized atoms can, in general, be overcome by switching t o other natural isotopes. The same can be said about interferences from isobar nuclides. There are also tricks to avoid possible mistakes due t o molecules formed by the most abundant elements. All these helpful operations do not require much additional bits. There is still a large reserve due t o the high resolving power and the wide range of the spectrometer. This otherwise unusable excess is transformed-yet a t a very low rate of exclhange (log function)-into information about the intensity of the spectrum lines: The entrance slit is widened to produce broad lines on the photographic plate. This results in a higher precision for intensity measurements. Before using a wide slit, the operator should be sure no lines from vagabonding hydrocarbons occur close to the analysis lines t o be measured. T o summarize, we have three competing analytical principles which can be used t o develop “complete analytical procedures’’ for the task t o determine the concentrations of all elements in a sample a t one blow. It will be admitted t h a t the concepts taken over from the theory of information have given us a better understanding of some logical relations effective in analytical procedures. However, this is not sufficient to decide which concrete procedure will be the best to solve a distinct universal analytical problem. To clarify this, a n analogy from daily life will be helpful: When a man wishes t o build a house, the first step is t o define his wishes (task), then he estimates the costs (amount of information). The next formal action will be t o procure the necessary money, either his own (informing power) or borrowed money from others (e.g., pre-information). At this stage a very important decision is due: If the man has got enough money, he

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Signal Averaging... Principles and Practices

Number 7 of a Series

NMR Fourier Spectroscopy In Number 3 of this series we discussed how and why Fourier transform techniques are used in infrared spectroscopy and said that the same principles could be extended to Nuclear Magnetic Resonance (NMR) spectroscopy. We recently applied our 1070 System to a Bruker Scientific, Inc. (Elmsford, N.Y.) spec-

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trometer and the above 1 IC interferogram and spectrum are early results of this joint effort. The interferogram is only the first 1024 data points of a 4096 point free induction signal and is the average of 512 sweeps taking a total of approximately 200 seconds to accumulate. The Fourier transformed high resolution spectrum is shown below it. The equivalent sweep width is 5 KHz. The significance of this technique lies in the time savings that can be achieved and the fact that this time savings can be put to good use in improving sensitivity o r signal-to-noise ratio. The above spectrum of naturally low abundance 13C, if recorded by conventional (CW) NMR techniques, would have taken 10 to 12 hours instead of 3 to 4 minutes. While these data demonstrate approximately a 200 :1 time savings, much longer savings are theoretically possible. Ernst and Anderson1 have shown that the theoretical time reduction increases in proportion to the spectral resolution. The relaxation times ( T I ) of the nuclear spin systems have been considered a limiting factor in approaching these theoretical values. However, recent experiments2 indicate that the T, limitation can be overcome by newly developed techniques and that savings in the order of 1 0 , O O O : l may soon be practical. If part of these time savings were used for sensitivity improvement through signal averaging, signal-to-noise ratio improvements of one to two orders of magnitude could be achieved. Interested? Why not call or write to arrange to see a FabriTek 1070 System References : 1. R. R. Ernst and W. A. Anderson, Rev. Sei. Instr., Vol. 37,93 (1966). 2. E. D. Becker, J. A. Ferretti, and T. C. Farrar, J. Am. Chem. SOC. (tentative publishing date Jan. 1970).

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Report

may go on; if not, he must give up or he may reduce his wishes (analytical task) and start anew. This is precisely the point t o which we are carried by discussing the required and available amount of information. But from here on, new problems raise their heads: Is the money of the prospective builder good money, can he buy the proper building materials and services (type of informations with respect to contents) ? It will be clear that these are questions of another level. I n daily life, the builder will study a lot of technical questions (and also the market) t o find the best solution. What is the corresponding procedure in chemical analysis? At present there is no systematic approach t o answer the questionwhich one may be the best analytical procedure t o solve a given analytical problem. We are relying on the knowledge and experience of the analyst, we are scanning the literature, we are dependent on habits and fashions. Isn't t h a t muddling through? I dare to sketch an idea of how a more systematic approach might be achieved : First, a set of independent parameters (coordinates) must be selected t o describe analytical problems by numbers. Such parameters may be correlated t o the following items: (a) Elements or compounds to be detected and determined and respective ranges of concenrtrations (b) Type, composition, homogeneity, variety of the samples to be analyzed. (c) Quantity of sample available (in particular restricted or practically nonrestricted) (d) General requirements (precision, accuracy, limits of detection) (e) Special and practical requirements (local analysis, microanalysis, production control) ( f ) Restriction with respect to costs, laboratory space, time, skill

The scales for the different pa-

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Quantitation in elemental analysis

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