J. D. Winefordner, W. J. McCarlhy, and P. A. St. John University of Florida Gainesville 32601
1
I
The Selection of Optimum Conditions for ~~ectrochemical Methods I. Use of signal-to-noise ratio theory
Well-lalow11 expressions and techniques arc i:urrently available for aiding the analyst in the selection of optimum cxperimental conditions for analysis in areas such as elertrochemistry, chromatographil: separations, etc. The romplex nature of tllc instrumentation and experimental t,eoliniques in spectroscopir analysis has t,hus far prevented similar approaches to optimizat,ion of experimental conditions. Unfortunately, a trial and error approach has gcrierally been used in the past to obtain optimum conditions. Such an approach is slow, tedious, and often inaccurate because of the inability of the analyst to study all possible combinations of variables. However, a liriowledge of the signal-to-noise ratio expression for a given experimental system allows the iuvestigator to predict the influencc of varying auy experimental paramcter on his results. Such expressions have been used for many years with great success in the design of electronic circuitry. I t is wcll-known that t,he sensitivity of an instrument cannot be increased ad inlinitunz simply by iucreasiug thc amplifioat~ioubecause superimposed on any signal is a ~:omponcntknown as noise. I t is the purpose of this new approach in spectroscopy to dcscribe quantitatively both the prrduction of the, signal arid the production of noise. This approach is simple to use and is an easily iuidcrstood method for t,hr selection of optimum cxperimental conditions for spectrochemical analyses and thcreforc is a useful tool for the teaching of qumtitative spectroscopy to advanced undergraduates and graduate students. I t is also a useful tool for the investigator to use in designing an approach to analyt,ical speotrochrmi~dmvixsurements. Types of Analytical Spectroscopy
TI] facilitate the discussiou to follow, the thrce nnjor types of spectrochemical methods of analysis will bc defined: enzission spedrome/q which shall be understood t,o include those methods whcreby emitted radiation is produced from species (molccules, atoms, ions, etc.) without a prior radiational activation process, e.g., flame emission spectrometry, chemiluminrscencc
The second part of this paper will appear i u tho AIarah issue of THIS JOUIINAL.I t will cunt,rihute further to the theme by disenssing the quantum efficiency and decay time of luminescent molecules. T h e third pert of this paper will appear in the A p d issue of THIS JOURNAL. It will disews the sertsitivity of atomic fluorescence, absorption, m d emission flame spectrometry.
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Journal o f Chemical Education
spectmnietry, and arc and spark spectromctry; absorption specfrovietry which includes all met,hods in which absorption of radiation is n~easurcd,e.g., atomir absorption spectronictry and n~olecular absorption speotromet,ry; lu~i~inesem.ce spect~mnelrywhich includes those methods whereby cmitt,cd radiation is produced from a sperics cxcit,ed by prior radiational means, e.g., atomic fluoresrenrc flamc spectrometry, and molecular fluorescence, phorph~~reslwux, and other types of luminescence spertrometry. The detection device is a photocmissive cell, e.g., single stage phototuhcs or multiplier phototubes (,marc commonly rttllcd ph~~tomultiplier tubes). Since thermocouples are norn~allyused in infrared spectroscopy and proportional counters and Geiger-lliiller tubes are normally used for high-c~mrgyradiation, the discussion here will bc limited to ultraviolet-visible spectroscopy. However, if sufficient information rcgarding the nonnltraviolct-visible det,eotors is available, this treatment ran be readily cxtended to other spectral regions. It. should bc noted that the treatment to be give11is useful whatever type of eutrance optics, monochromator, amplifier, and readout, system is used to collcct t h ~ radiation and measure tlir result,ing signal. The Measured Intensity
In cmission spectrometry, the iutensity of enlission, Iz, is measured. I E can bc evaluated from spectral parameters, such as, the transition probability, stat,istical weights of the states involved in the transition, thl. wavelength of the emission peak aud from excitatiou parameters, such as temperature of t,he source of excitation and factors affec1,ing production of the measured species. An example of the evaluation of I , for atomir cmission flame spectnimet,ry has been given by Wincfordner, et al. ( 1 ) . In absorption speotn~nietry,the intensity ahsorbed, InbJ,i.e., the difference in the incident intensity transmitted through blank, In,and the intensity transmitted through thc sample absorbers, I,, is not generally measured but rather thc fraction, a, absorbed which is defined as (I0 - I,)/In. At. large values of (I0 - / , ) / I n , i.e., for dih1t.e ahsorbcrs, thc absorbance A is given by
where k: is the average absorptivity coefficient which
ill-
This research was carried out as a part of a. study on the phosphorimetric arralysir of drugs in blood and wine, supported by a grant, frrm t.he IT. S. Puhlir Health Service (GM ll:ii343).
cludes the concentration of the absorber and L is the path length over which absorption occurs. Expressions fork have been derived for atomic species by de Galan, McGee, and Winefordner (t)in terms of spectral and flame compositional parameters for atomic absorption flame spectrometry. I n luminescence spectrometry, the intensity emitted in all directions, I&,is evaluated from Ir.
=
I*.
rn&
(2)
where @L is the power efficiency, i.e., the ratio of the radiant power emitted by luminescence to the radiant power absorbed. The radiant power absorbed, Iaas, is given by I*.
=
I" - I t
=
1°(1
- ckL)
(3
I n eqn. (3), Inrepresents the intensity of radiation incident on the absorbing sample. The intensity IL from a unit surface area and from a unit solid angle and corrected for self-absorption by similar species and for absorption by foreign species has been evaluated by St. John, McCarthy, and Winefordner (3). Exact expressions for specific systems for I g , A or or, and I L will not be given here hut rather the purpose of this article is to indicate how these intensities affect the signal and ultimately the signal-to-noise ratio.
aT,
a=-
df+
+a
...
When using a phototube detector at very low light levels, two noises, namely the shot and Johnson noises, will generally become significant. The shot noise, is the random fluctuation of the electron current from any electron emitting surface of a phototube. This results in a variational (ac) noise current superimposed on the phototube current. The r m shot noise photoanodic current, in amperes, is given by A ~ N=
The measured photoanodic current from the photodetector, i, in amperes, for an emission intensity Ie or for a luminescence intensity, I', is given by i = rko Ix or i = y ko Ir. (4) where y is the ratio of the number of amperes produced at the photoanode of the detector to the number of watts of power incident on the photocathode, ko is the monochromator-entrance optics factor in units of cm2 steradian, and Ix or I L are the number of watts emitted per cm2 of source area per steradian over the spectral bandwidth, s, of the monochromator. If the emitted radiation has a half-intensity bandwidth considerably less than s, then the entire line or band of radiation is measured and the intensity, Is or IL, has units of watts emitted per cm2 per steradian. If the emitted radiation has a half-intensity bandwidth greater than s, then only a portion of the emitted radiation is measured and that portion is given by IESor ILS which has units of watts emitted per cm2per steradian. The measured readout voltage, E, for an emission or luminescence intensity is given by E
=
GR'i
(5)
where R5 is the lead resistor, in ohms, connected to the photoanode of the photodetector and G is the overall voltage gain (no units) of the amplifier-readout system. The measured photoanodic signal in absorption is the same as or, the fraction of intensity absorbed, i.e., a=-
I0
- I* = zU--' I0
21
iO
(6)
where the currents io and i, are defined as and
i t = y ko It
(7
The Noise
I n any spectrochemical method of analysis a number of noises are present. When the noises from two or
(8)
where Af, the frequency response bandwidth of the amplifier-readout system is assumed to be the same for d l noise components and the noise is assumed to have equal amplitudes at all frequencies, i.e., the noise is white. Therefore, when a particular random noise signal is passed through an amplifier with a variablehandpass, the root-mean-square (rrns) noise output, AET, is proportional to the square root of frequency response bandwidth, Af. Noises that are dependent -add -linearly = AEldAj rather than quadratically, e.g.,
-
The Signal
i" = 7 ko I0
more inhpendent sources are combined, the quadratic content of the resultant noise signal, is the sum of the square of the individual quadratic components, namely A=A~, ~ e i ? A f , etc., for example,
+
d ~ \ / i i d ) Af BM
(9
where i is the photoauodic current flowing through the phototube due to light striking the photocathode, id is the photoanodic dark current due primarily to thermionic emission from the photocathode, BIM is the overall gain of the phototube, and e, is the charge of the electron (1.6 X 10-lg coulombs). The is given by shot noise readout voltage, EN,
a
=
d2\/2e.(i
+ id)Af BM G2 RL*
(10)
The rrns Johnson noise voltage, AX,is a randomly varying potential difference arising between the terminals of a resistor caused by thermal agitation of electrons withim the resistor and is given by = ~ ~ ~ T L R L G " ~
(11)
where TLis the temperature of the load resistor and k is the Boltzmann constant. The shot and Johnson noise terms add quadratically, but because the Johnson noise current is generally negligible at room temperatures compared to the shot noise current, Johnson noise will he neglected in all further discussions of noise. All other noises in spectrometric measurements result from amplifier-readout noise and from fluctuations in the sources of excitation or emission. For example, in the -case of the latter, the rrns noise fluctuation voltage, AE,,in volts, is due to the flickering of a light source of intensity, I+,where the subscript i could refer to E or L or any other intensity, is given by %A
=
tiid$
RL G
(12)
where i, is the photoanodic current, in amperes, due to intensity I t striking the photocathode and E is the rrns fluctuation in intensity, per unit of intensity, I , per unit frequency response hand width, i.e., E = A I t / I , for a Af = =Hz. The amplifier-readout rrns noise voltage, AE., is given
z,
Volume 44, Number 2, February
1967 / 81
by the sum of two terms. The first noise term is proportional to the measured signal, ((i &)dZ where is the rms fluctuation factor, i.e., the noise current due to the amplifier-readout system per measured current (i id)for a Af of lHz, and the secondnoise term is a constant, C,, which is inherent in any electronic system. The total amplifier-readout noise voltage is given by
+
+
A~
=
d[r(\/[r(i f \/[r(i~)dv]z + CaP
(13)
The specific types of noises present in any given spectrochemical analysis depend entirely on the specific method. In Table 1 some of the major noises present Some of the Major Noises in Several Spectro-
Table 1.
chemical Methods
Flt~meEmission Spectrometry Flame Absorption S~eetrometrv Molecular Luminescence In Condensed Phase -~ ~-~~ Absorptimetry (Molecules in Condensed Phme)
Shot Noiseo
Flicker Koiseb
Amplifier Noisee
Yes
Flame Background
Yes
Yes
FlemeBackgroundSource of Excitatmn Source oi Excitation
Yes
Yes
Yes
~
Yes
Source of Excitation
Yes
At low signals, shot noise is generally the most significant 110188.
At high signals, flicker noise is generally the most significant noise. With a good electronic system, amplifier noise is generally insignificant compared to other noises.
for several specific spectrochemical methods are given. I t should not be inferred that these are the only noises of importance for t,he designated methods. If high-speed computers are available, then it is just as simple to consider all noises in computations. If however, the investigator is experimentally attempting to reduce the total noise in a system, then the most logical approach would be to reduce the major source of noise. For instance, if shot noise is significant, then it could he reduced by cooling of the photocathode. The Signal-to-Noise Ratio, S/N
The signal-to-noise ratio can be calculated by simply taking the ratio of the signal to the total noise. Once an expression for S I N is known, then optimum values of each experimental parameter can be calculated by the standard maximization technique of taking the partial derivative of S I N with respect to the parameter of interest, X, setting the partial derivative equal to zero and solving for the optimum value of X (X,,,). I n other words, if all instrumental parameters are independent of each other (which is generally a good approximation), then
(%I)
= Y ,7 . 1
0 (then solve for X.,,)
/
W, the frequency response bandwidth, Af, and the photoanodic sensitivity factor, y, of the phototube (y can be varied by varying the voltage applied to the photocathode). If the background is due to band emission, and the signal, on the other hand, is due to line emission (such as in atomic emission flame spectmmetry) then there exists an optimum value of W (Fig. 1, curve A ) . If the background and signal are both due to band emission (such as in molecular emission flame spectrometry) then there is no optimum value of W but rather a plateau of optimum values (Fig. 1,curve B). There is, however, no optimum value of AJ. The plot of S I N versus l / A f results in a curve like curve C, Figure 1 , and therefore a value of Af should be chosen which is consistent with the time available for the quantitative analysis. I n other words, Af can be made as small as desired by making more measurements of S I N or by using integration or time-averaging techniques ~vbich require long periods of time. Theoretically the noise can be made infinitely small, if one is content to take measurements for an infinite period of time. I t should be noted that although y various rapidly with photocathodic voltage, S I N changes only slightly (actually decreases slowly due to phototube fatigue) if photocathodic voltage is increased (Fig. 1, curve D). If variables X, Y, and Z are not independent parameters, then X is optimized first with Y and Z held constant at values midway within the range of variation of Y and Z. Then Y is optimized holding X at X,,, and Z at a value near the midpoint of the range, and finally Z is optimized holding X and Y a t X,,, and Yo,,, respectively. Then holding Y and Z at Ye,, and Z ,, respectively, a new value of X,,, is found. This process
(14)
Cte.
where Y, Z, etc., are other experimental variables which are held constant during the optimization process. If all experimental parameters are known or can be approximated, this method is fast, accurate, and simpIe to use. If optimum values for various parameters exist, 82
they can be found by this procedure (Fig. 1, curve A ) . For some parameters an optimum value may not exist. For example, a plot of S I N versus X results in a curve with a plateau a t large X values (Fig. 1, curve B) or results in a continually increasing S I N as X increases (Fig. 1, curve C). If an optimum value of X exists then X should be adjusted to XoDL to obtain the greatest precision and sensitivity of measurement, i.e., the greatest S I N . If an optimum value of X does not exist, then X should be adjusted to thevalue giving the largest S I N with the existing experimental system and with the time available for the analysis. The only instrumental
Journd o f Chemical Education
X Figure 1. Types of Signal-to-Noise Rotios(S/N) Versus Values of Experimental Voriobler(Xl. (See text for dixurrion of curves.)
is repeated until consistent values of X.,,, Y,,,, and Z,,, axe obtained. Experimentally, optimization of dependent variables can be achieved simply and rapidly only if methods of statistical experimental designs are used. For example, the most thorough design by which several parameters can be simultaneously optimized is called a factorial design. This method of statistical experimental design requires kn observations ( 6 ) ,where n is the number of variables, and k is the number of values which the variable may assume, and thus the factorial design is extremely time consuming in practice. Central composite rotatable experimental designs lend themselves naturally to the problems usually encountered in analytical methods. Essentially all the useful information obtained from a factorial design is produced by a rotatable design ( 6 ) and the latter is considerably less time consuming. The Minimum Detectable Sample Concentration
The minimum detectable sample concentration, Cmt,, is defined as that concentration which results in a signalto-noise ratio which has a value of
where n is the number of combined signal and blank measurements made. By substituting in the expressions for the s'gnal in terms of the intensity expression and for the noises, the limiting detectable intensity (or signal! can be calculated. Using expression (15) and the expression for the intensity, Cmt, can be calculated for optimum or near optimum values of all experimental parameters. Equation (15) is also valid from a statistical standpoint. A signal-to-noise ratio of 3d2/di gives a minimum detectable concentration, C,,,, with about 99.7% confidence, i.e., the value of Cmt, lies within approximately 3 standard deviations. The signal-to-noise ratio, SIN in any given experiment can be calculated by measuring the detector signal, in amperes or in volts, and dividing by one-fifth of the pealc-to-peak noise signal, in amperes or in volts (4, 5). This procedure gives the most valid indication of S I N . The v2 ' in equation (15) occurs because the signal is the difference of 2 measurements (7). It should be noted that eqn. (15) is identical to the limit of detection as defined by Baumans and hlaessen ( 7 ) :i.e., the signal, S, is the diierence between the smallest sample meter reading that can be discerned a t a certain level of
significance and the average blank meter reading; the noise, N , is the standard deviation of the blank or sample which is the sa,me for a number of measurements at the limit of detection. Summary
Expressions for the signal, the noise, and the signalto-noise ratio, S I N , can be derived by consideration of the intensity expressions and the sources of noise for any given spectrochemical method. Such expressions are useful in predicting the shapes of analytical curves (signal versus sample concentration) and optimum conditions for analysis. Optimum instrumental conditions for analysis can be simply obtained by setting ( d [ S / N ] / d X )= 0 and solving for X,,,, where X.,, is the optimum value of parameter X which is independent of other instrumental parameters. If X is not independent of other instrumental parameters, a statistical design experiment must be used to obtain XopL. Generally most instrumental parameters are not variable and so relatively few can be optimized. If instrumental parameters are optimized, the precision of measurement is maximal and the minimum detectable sample concentration, Cma, is minimal, i.e., maximum sensitivity. If expressions for SIN and C,,, are available, then it is also possible to compare instruments with respect to precision and sensitivity of measurement prior to purchase and prior to use. I n addition, the most sensitive and precise method for a given measurement can be chosen prior to analysis, and methods can be directly compared with respect to recision (maximum SIN ratio) and sensitivity (minimum C m J . Literature Cited (1) WINEPORDNER, J. D., MCGEE,W. W., MANSFIELD, J. A[., P ~ t s o ~ M. s , L., AND Z.~CHA, K. E., Anal. Chim. Acla., 36,25 (1966). (2) Dm G.ALAN, L., MCGEE, W. W., AND WINEFORDNER, J. D., Anal. Chim. Ada., in press. (3) ST. JOHN,P. A,, MCCIRTHY,W. J., AND WINEFORDNER, J. D., Anal. Chem., 38,1828, (1966). V. D., PIOC. Inst. of Radio Engineers, 29, 50 (1941). (4) LANDON, (5) PRAGLIN, J., "The Measurement of Nanovolts," Keithley Instruments, Inc., bulletin 102.1, Cleveland, Ohio, 1966. (6) COCHRAN, W. G. AND COX,G. M., "Experimental Designs," John Wiley & Sons, Inc., New York, 1951. (7) BOUM.ANS,P. W. J. M. AND MIESSEN,F. J. 31. J., 2.anal. C h .220,241 (1966).
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